Combustion waves and fronts in flows : flames, shocks, detonations, ablation fronts and explosion of stars 978-1-107-09868-8

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C O M B U S T I O N WAV E S A N D F RO N T S I N F L OW S

Combustion is a fascinating phenomenon coupling complex chemistry to transport mechanisms and nonlinear fluid dynamics. This book provides an up-to-date and comprehensive presentation of the nonlinear dynamics of combustion waves and other non-equilibrium energetic systems. The major advances in this field have resulted from analytical studies of simplified models performed in close relation with carefully controlled laboratory experiments. The key to understanding the complex phenomena is a systematic reduction of the complexity of the basic equations. Focusing on this fundamental approach, the book is split into three parts. Part I provides physical insights for physics-oriented readers, Part II presents detailed technical analysis using perturbation methods for theoreticians and Part III recalls the necessary background knowledge in physics, chemistry and fluid dynamics. This structure makes the content accessible to newcomers to the physics of unstable fronts in flows, whilst also offering advanced material for scientists who wish to improve their knowledge. pau l c l av i n is Professor Emeritus at Aix-Marseille Universit´e and is an honorary member of the Institut Universitaire de France (Chair of M´ecanique Physique 1993– 2004). In 1995 he founded a renowned research institute, the Institut de Recherche sur ´ les Ph´enom`enes Hors Equilibre (IRPHE), and has received major awards from the Soci´et´e Franc¸aise de Physique (Plumey 1988), French Academy of Sciences (Grand Prix 1995) and the Combustion Institute (Zeldovich Gold Medal, San Francisco, August 2014). g e o f f s e a r b y is retired Director of Research at the Institut de Recherche sur les ´ Ph´enom`enes Hors Equilibre (IRPHE). He is a renowned specialist of the physics of thermoacoustic instabilities in combustion chambers and rocket motors, and his experiments have made major contributions to the understanding of the dynamics of flame fronts. In 2004 he obtained a major award from the French Academy of Sciences.

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C O M BU S T I O N WAV E S A N D F RO N T S IN FLOWS Flames, shocks, detonations, ablation fronts and explosion of stars PAU L C L AV I N Aix Marseille Universit´e, France

GEOFF SEARBY Formerly of the Centre National de la Recherche Scientifique (CNRS), Marseille, France

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University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107098688 © Cambridge University Press 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication data Names: Clavin, Paul, author. | Searby, Geoff, 1945– author. Title: Combustion waves and fronts in flows : flames, shocks, detonations, ablation fronts and explosion of stars / Paul Clavin (Aix Marseille Universit´e, France), Geoff Searby (Formerly Centre National de la Recherche Scientifique (CNRS), Marseille, France). Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2016. | ©2016 | Includes bibliographical references and index. Identifiers: LCCN 2016011752| ISBN 9781107098688 (hardback ; alk. paper) | ISBN 1107098688 (hardback ; alk. paper) Subjects: LCSH: Combustion. | Waves. | Chemical reactions. | Dynamics. | Nonlinear theories. | Fluid mechanics. Classification: LCC QD516 .C57 2016 | DDC 541/.361–dc23 LC record available at http://lccn.loc.gov/2016011752 ISBN 978-1-107-09868-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Contents

Preface

page ix

Introduction Brief Historical Introduction Energy and Modern Society Combustion and Related Phenomena Scope of the Book Part One

1 1 6 8 8

Physical Insights

11

1 General Considerations 1.1 Introductory Remarks 1.2 Combustion Waves on Earth 1.3 Fronts and Thermal Waves in Extreme Conditions 1.4 Appendix: Physical Constants and Conversion of Units

13 14 15 30 43

2 Laminar Premixed Flames 2.1 Main Characteristics 2.2 Hydrodynamic Instability 2.3 Flame Stretch and Markstein Numbers 2.4 Thermo-Diffusive Phenomena 2.5 Thermo-Acoustic Instabilities 2.6 Curved Fronts 2.7 Nonlinear Dynamics of Unstable Flame Fronts 2.8 Additional Laboratory Experiments 2.9 Appendix

45 48 56 69 77 101 121 135 146 156

3 Turbulent Premixed Flames 3.1 Basic Considerations 3.2 Turbulent Wrinkled Flames 3.3 Turbulent Combustion Noise 3.4 Appendix

174 176 188 197 203

v

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vi

Contents

4 Gaseous Shocks and Detonations 4.1 Introductory Remarks 4.2 Planar Supersonic Waves 4.3 Initiation of Detonation 4.4 Dynamics of Shock Fronts 4.5 Instabilities of Detonation Fronts 4.6 Appendix

210 213 214 231 261 279 300

5 Chemical Kinetics of Combustion 5.1 Introduction 5.2 Basics Concepts in Chemical Kinetics 5.3 Combustion of Hydrogen 5.4 Combustion of Lean Methane Mixtures

305 306 307 311 317

6 Laser-Driven Ablation Front in ICF 6.1 Approximations and Constitutive Equations 6.2 Dynamics of the Ablation Front

323 324 329

7 Explosion of Massive Stars 7.1 Constitutive Equations of Stars 7.2 Stellar Equilibrium 7.3 Instability and Gravitational Collapse 7.4 Appendix

339 341 346 352 370

Part Two

Detailed Analytical Studies

375

8 Planar Flames 8.1 General Formulation 8.2 Thermal Propagation 8.3 Reaction–Diffusion Waves 8.4 Chain Branching and Flame Propagation 8.5 Flame Quenching

377 379 381 393 403 411

9 Flame Kernels and Flame Balls 9.1 Flame Kernels Near the Flammability Limits 9.2 Stability Analysis of Spherical Flame Kernels 9.3 Flame Expansion at Lewis Number Smaller Than Unity

429 431 440 447

10 Wrinkled Flames 10.1 Hydrodynamics 10.2 Thermo-Diffusive Instabilities of Planar Flames 10.3 Hydrodynamics and Diffusion Coupling 10.4 Appendix

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460 463 473 481 500

Contents

vii

11 Ablative Rayleigh–Taylor Instability 11.1 Linear Analyses of Simplified Models 11.2 Asymptotic Analysis of the Quasi-Isobaric Model 11.3 Nonlinear Dynamics in the Limit of a Large Power Index

504 506 511 522

12 Shock Waves and Detonations 12.1 Linear Dynamics of Wrinkled Shocks 12.2 Dynamics of Detonation Fronts

527 529 543

Part Three

565

Complements

13 Statistical Physics 13.1 Statistical Thermodynamics 13.2 Ideal Gases 13.3 Physical Kinetics

567 569 580 591

14 Chemistry 14.1 Elementary Combustion Chemistry 14.2 Chemical Equilibrium 14.3 Elements of Thermonuclear Fusion

606 607 620 633

15 Flows 15.1 Macroscopic Conservation Equations 15.2 Approximations 15.3 One-Dimensional Compressible Flows

637 640 654 663

References Index

685 704

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Preface

Combustion is a fascinating phenomenon coupling complex chemistry to transport mechanisms and nonlinear fluid dynamics. The combustion of reactive mixtures, frozen far from chemical equilibrium, is an irreversible process in which the approach to equilibrium proceeds through the propagation of nonlinear waves in the form of sharp fronts exhibiting complex geometrical forms. These waves were discovered in the nineteenth century, but the understanding of their structure and dynamics is quite recent. In gaseous mixtures, the rate of chemical heat release is small compared with the rate of elastic collisions, so that combustion is described by the macroscopic equations for the conservation of mass, species, momentum and energy, assuming local equilibrium (except for the inner structure of shock waves). The full system of equations is complicated and is not useful to describe each of the elementary phenomena, separate from the others. Even the coupling of two phenomena, as for example a quasi-isobaric flame and acoustic waves, is represented by simplified equations. The major advances have resulted from analytical studies of simplified models performed in close relation to carefully controlled laboratory experiments. A systematic reduction of the complexity of the basic equations, validated by the confrontation with experiments, is the key to understanding. It is also the most difficult step. The analytical and numerical solutions of simplified equations of relevant models can be completed in a second step by direct numerical simulations of a more detailed system of equations. The book is written along this line and attention is focused on fundamental aspects. It is meant to be a survey of the nonlinear dynamics of combustion waves, which now constitute a mature scientific field. A similar approach is used to improve the understanding of other types of waves such as ablation fonts in inertial confinement fusion. The approach is also tentatively extended to the explosion of stars at the end of their lifetime, the famous supernovae. A large variety of phenomena is presented. The purpose is to provide a wide view of the physical problems involved in different domains that can benefit from crossfertilisation. The most important scientific results are reported, ranging from the pioneering works of the last century to the advanced research of the last decade. The book is aimed at both newcomers to the physics of unstable fronts in flows and scientists who wish to improve their knowledge. It is self-contained and can also be used as a textbook by students. The background in physics, chemistry and fluid mechanics is given in the last part as complements. Physical insights into the phenomena occupy half ix

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x

Preface

of the book and are presented first. Detailed analytical studies are developed in the second part and no prerequisites in applied mathematics are required concerning the perturbation methods that are used (multiple-scale and asymptotic analyses). This book would never have been written without 40 years of activity in our research group, interacting closely with our friends and colleagues, Louis Boyer, Bruno Denet, Alain Pocheau, Joel Quinard and Emmanuel Villermaux. We are also grateful to our close friends and outstanding theoretical physicists for fruitful collaboration and enlightening discussions of great help to improve our understanding of nonlinear problems related to combustion, particularly Guy Joulin, Amable Li˜nan, Yves Pomeau, Grisha Sivashinsky and Forman A. Williams.

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Introduction

Brief Historical Introduction Fire, Humanity and Civilisation Combustion has played a decisive role in the development of mankind and civilisation. At the present time it is the principal source of energy and the main support of economic development. Mastering the use of fire was the first major step in the development of humanity and marked a decisive step in its evolution. The oldest traces of the use of fire, limited to the occasional use of naturally occurring embers, date back to more than one million years, after the emergence of Homo erectus. However, the first fireplaces in dwellings appeared much later in Eurasia, about 450000 years ago. Neanderthal man had well mastered the art of fire, greatly improving living conditions through the ability to create light after sunset, to fight against cold, and to cook food. The first technical uses of fire were limited to hardening the points of wooden arrows and spears. Small lamps made of stone, bone or shells burning animal fat or vegetable oil, and dating back to the upper Palaeolithic 35000 years ago, can be directly associated with the development of parietal arts. The development of other technologies using fire appeared with the advent of farming and a more sedentary existence in the Neolithic era and the early Bronze Age, some 10 000 years ago. At this time, fire was used to transform clay into pottery, and ore into metals. Because of its capacity to transform and purify(?) or destroy matter and its dancing flames, source of light and heat, fire quickly acquired a mystical dimension as attested by the central place of fire in the Vedic ritual and in the legends of antique Greece. Let us cite, for example, the legend of Prometheus, a Titan who stole fire from Zeus and gave it to the mortals, as did his Hindu counterpart, Pramatha, the hero who brought civilising fire to earth. The holders of the secrets of fire were at the same time priests and scientists, as for example the ‘Guild of Chinese Blacksmiths’ (eighth century BC), which was at the origins of Taoism at the time of Laozi (sixth century BC). From this stems the Taoist alchemy, more than one and a half millenniums before the alchemy of the Middle Ages, whose esoteric gnosis reserved a major role for the sacred fire.

1

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2

Introduction

Fire, War and Industry The mastery of bronze, iron and steel is a marker in the history of a civilisation that evolved with the progress of metallurgy in the hands of the master blacksmiths, controlling the quality of tools and weapons. In the fifteenth century, the arrival of gunpowder changed the tactics of combats. The earliest known formula is mentioned in a Chinese military text, the Wujing Zongyao, and the transfer of knowledge to Western Europe probably occurred via Arabian literature.[1] Its composition – approximately 70% saltpetre, 15% sulphur and 15% charcoal – was known in western Europe as early as the thirteenth century, and more precisely in 1267 by Roger Bacon, the ‘Doctor Mirabilis’, in his description of firecrackers. However the effective use of firearms commenced only towards the end of the fourteenth century, first during sieges and a little later on the battlefield, changing the course of history. During the Middle Ages they were a decisive help to the armies of Charles VII in bringing the Hundred Years’ war to an end (the battles of Formigny and Castillon).[1] In Russia, as early as 1480, it was the use of cannons and muskets that permitted the Muscovite princes to overcome the Tatar domination (Mongol invaders, called ‘Tatars’ by the Russians), the Muscovite Tsar then replacing the Khan of the Golden Horde. As noted by Fernand Braudel:[2] ‘The Asian invader had penetrated Occident with its horses, but was finally halted by gunpowder.’ Despite this decisive technical advantage, the fight was long and difficult, and it was not until a century later that Ivan the Terrible managed to take control of Kazan (1551) and Astrakhan (1556). In France towards 1580, during the wars of religion, the invention of cannons made town walls an illusory defence.[3] A century later, during the reign of Louis XIV, a new generation of fortresses, adapted to artillery, were built by Vauban on the borders of the country. Coal was the primary ingredient of the Industrial Revolution that took place at the end of the eighteenth and during the nineteenth centuries. It was marked by the development of thermal machines and heavy industry, with its blast furnaces for the production of steel and coke. The internal combustion engine, invented at the end of the nineteenth century, revolutionised terrestrial and maritime transportation before giving birth to aeronautics. This latter had its own revolution in the 1950s with the development of the turbo-reactor. Rocket engines, propelled by liquid fuels or solid propellants, were developed at the same time, giving rise to a new era in communications with the advent of artificial satellites.

Science and Combustion The scientific comprehension of combustion progressed tardily, perhaps because of the strong symbolism attached to flames. With few exceptions, the pre-scientific mind was ‘very symptomatic of the dialectics of ignorance that goes from darkness to blindness’, as [1] [2] [3]

ˆ Contamine P., 1999, La guerre au Moyen Age. PUF, 5th ed. Braudel F., 1987, Grammaire des civilisations. Arthaud. Ferro M., 2001, Histoire de France. Odile Jacob.

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Introduction

3

Gaston Bachelard said so beautifully in the middle of twentieth century, adding that ‘fire, unlike electricity, has not found its science’.[4] Nevertheless, the first scientific observations of a candle flame date back to Francis Bacon in 1600, followed half a century later by Boyle and Hooke. But understanding was obscured by European alchemy, reviewed by the Abbot Lenglet-Dufresnoy,[5] one of the first historians of alchemy in the eighteenth century. The alchemists, who were qualified as the most illustrious dreamers that humanity has produced, considered flames and combustion to be the release of an imponderable agent, or phlogiston, contained within each combustible body. This view was still shared by most of the scientists up to the end of the eighteenth century. The first decisive progress was made in the eighteenth century. An original reflexion is presented in the 1738 dissertation of Leonhard Euler, Dissertatio de igne in qua ejus natura et proprietates explicantur[6] (Dissertation on fire in which its nature and properties are explained). How can one explain that a small local cause can have a significant effect on a large scale, such as a spark that lights a huge fire, or a charge of explosive that explodes from the combustion of one of its grains? To answer this question, Euler proposed that a combustible material be assimilated to a network of fragile hollow spheres containing a compressed ‘igneous’ fluid. These spheres can break as a result of a perturbation caused by the destruction of their neighbours. They thus release their ‘igneous’ material (subtle matter) which, ‘expanding’ into the surrounding atmosphere in the form of flames, breaks other spheres. In more technical modern terms, Euler proposes an analogy between a fire and a nonlinear wave: a disturbance of finite amplitude propagates over a network of metastable elements whose state is changed by the passage of the wave, the initial shock acting as the initiator. Assimilating heat to the agitation of particulate matter, he introduced the concept of thermal runaway: combustion is accompanied by heat release which, in turn, can ‘give birth to and spread’ fire. Indeed, the spheres break more easily and more frequently when they move and collide more violently. What a great insight! The ‘illustrious’ Royal Academy of Sciences of Paris, in the words of Euler, therefore showed great discretion in awarding him the prize it had put to contest on the theme of fire, a competition in which Voltaire and the Marquise of Chˇatelet had also participated. Far ahead of his time, the ideas of Euler were not followed and did not give rise to any experimental study, the first experiments being performed at the end of the eighteenth and early nineteenth century. It was not until the middle of the twentieth century that the speed and structure of flames were described correctly through the analysis of a mathematical model actually quite close to that of Euler. The ‘igneous’ matter of Euler’s model, enclosed in spheres that break open in cascade, is reminiscent of the phlogiston of alchemists. However, it anticipates with great discernment the release of energy by the formation of strong chemical bonds and the subsequent thermal runaway in combustion. Two key concepts were unknown to Euler; one is chemical, the [4] [5] [6]

Bachelard G., 1928, La psychanalise du feu. Gallimard. Lenglet-Dufresnoy N., 1742, Histoire de la philosophie herm´etique. Coustelier, Quai des Augustins. Euler L., 1944, Cinq m´emoires sur la nature et la propagation du feu. Association pour la sauvegarde du patrimoine m´etallurgique du Haut-Marnais.

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4

Introduction

notion of species (fuel, oxidant and combustion products), the other physical, conservation of mass and total energy (kinetic plus chemical). Based on experiments more detailed than those of Boyle, Lavoisier, the founder of modern chemistry, refuted the concept of phlogiston in his 1777 book Reflections on Phlogiston, more than a century after its introduction. Combustion was described for the first time as a chemical reaction between two bodies, fuel and an oxidant, primarily oxygen, that are transformed into products, the total mass remaining unchanged. The analogy with breathing was also highlighted. In the same period, soon after the discovery of hydrogen, the introduction of a burner into a tube produced the ‘singing flame’[1] that became an attraction in nineteenth-century parlours lit by gas chandeliers. However, the analyses of Lavoisier lacked the concept of energy conservation. Despite ideas dating back to the seventeenth century on the nature of heat being linked to the agitation of matter (Francis Bacon, Robert Boyle, etc.), and despite the first quantitative experiments of Benjamin Thompson (Count Rumford) in the late eighteenth century on the transformation of work into heat, a misconception of heat, called ‘caloric’, considered to be a self-repellent indestructible fluid, persisted until the middle of nineteenth century. This misconception is found in 1824 in the remarkable work of Sadi Carnot,[2] selfpublished by the author at the age of 28. He postulates that the production of work requires the existence of at least two sources of heat, a hot source from which heat is extracted and a cold source into which it is ejected. He then considers an alternating succession of irreversible cycles and reversible inverse cycles. Based on the inability to produce work without extracting heat from the outside environment, he concludes that the maximum efficiency of a heat engine is obtained with reversible cycles. He also shows that the performance depends only on the temperature of the sources, independent of the nature of the fluid used to perform the cycles. Despite the misconception of heat, based on an analogy with the flow of water from a mill (the amount of heat extracted from the hot source is equal to that discharged into the cold source), the reasoning leads Carnot to a brilliant and exact conclusion concerning the maximum efficiency of a heat engine. Made public in 1834 by Clapeyron, this work served as the foundation for the introduction of entropy a few years later by Clausius. This new concept, unknown to Newtonian mechanics, opened opportunities hitherto unsuspected in thermodynamics and physics. The next milestones were the works of Maxwell in 1860 and Boltzmann in 1872. Meanwhile, it was not until the first work of Joule, published in 1847, that the equivalence between heat and work was established experimentally[3] and fully recognised, half a century after the experiments of Benjamin Thompson.[4] In notes written after the publication of his book, but published only later after his untimely death in 1832, Carnot imagined experiments to demonstrate the equivalence of heat and work.[5] This would surely have led him to correct the initial error in his work if he had had the time. [1] [2] [3] [4] [5]

Rayleigh J., 1945, The theory of sound, vols. 1 and 2. New York: Dover. Carnot S., 1824, R´eflexions sur la puissance motrice du feu et sur les machines propres a` d´evelopper cette puissance. Bachelier. Prigogine I., Kondepudi D., 1999, Thermodynamique: Des moteurs thermiques aux structures dissipatives. Odile Jacob. Longair M., 2003, Theoretical concepts in physics. Cambridge University Press. Mendoza E., ed., 1977, Reflections on the motive power of fire by Sadi Carnot and other papers. Gloucester, Mass: Peter Smith.

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Introduction

5

Mention should also be made of a result obtained during the same period concerning the propagation of sound waves. In 1823, realising that the compressions and expansions of the gas in an acoustic wave are adiabatic, and not isothermal (as supposed by Newton), the marquis Pierre-Simon Laplace corrected the speed of sound predicted by Newton a century earlier but with a value that was too low. This problem had also been discussed by Euler in his 1738 dissertation, but without providing a relevant insight. The systematic study of flame propagation was motivated by accidents in mines caused by the explosion of firedamp. The earliest work, dating back to 1830, is due to Sir Humphry Davy, who gave his name to the miners’ safety lamp. Some time later, his former student Michael Faraday gave the Royal Society a famous series of conferences entitled ‘The chemical history of a candle’.[6] Detonations, or supersonic combustion waves, were discovered in a condensed explosive, gun-cotton, by Sir Frederick Abel. He measured the propagation velocity of these waves, of the order of 6000 m/s,[7] a remarkable feat for the time. In the October 1873 issue of Nature[8] it was remarked that ‘Indeed with the exception of light and electricity the detonation of gun-cotton travels faster than anything else we are cognizant of’. Detonations were discovered in gases later, in 1882, by Berthelot and Vieille. At about the same time, Mallard and Le Chatelier,[9] two engineers of the French Corps des Mines, charged by La Commission du Grisou to study the propagation of flames in various mixtures of reactive gases, submitted their report presenting a detailed series of measurements on the propagation of subsonic combustion waves. During the same period, Lord Rayleigh[1] established his famous criterion for the stability of combustion confined in an enclosure (combustion chamber): an instability of the acoustic modes can develop if the energy release rate fluctuates in phase with pressure. This thermo-acoustic instability is responsible both for the singing flame discovered a century earlier and also much later, from the middle of the twentieth century, for the violent vibrations observed in rocket engines that became the nightmare of engineers in charge of propulsion in astronautics. At the beginning of twenty-first century, these issues remain relevant in the engines of the Space Shuttle and the Ariane rockets. The internal structure of the two types of combustion waves, flames and detonations was not fully understood and described theoretically until 1938, thanks, in particular, to the work of Zeldovich.[10] The understanding of the instabilities, the dynamics and the spatiotemporal structure of these fronts did not really progress until 1975 for flames, following the pioneering work of Landau and Darrieus in 1940, and only after 1990 for detonation fronts, although their cellular structure had been observed experimentally in the early 1950s[11] . Optical diagnostics for nonintrusive measurements of concentration and temperature, as well as visualisations and fast scans, were developed in research laboratories in the years [6] [7] [8] [9] [10] [11]

Biblioth`eque des succ`es scolaires, ed., 1868, Histoire d’une chandelle. J. Hetzel et Cie. Abel F., 1874, Philos. Trans. R. Soc. London, 164, 337–395. Editorial, 1873, Nature, 8(208), 534. Mallard E., Le Chatelier H., 1883, Annales des Mines, Paris, Series 8(4), 296–378. Ostriker J., ed., 1992, Selected works of Ya.B. Zeldovich, vol. I, p. 193. Princeton University Press. Markstein G., 1953, Proc. Comb. Inst., 4, 44–59.

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6

Introduction

1970–80 thanks to lasers and the development of electronics and information technology. These methods have opened a new era in experimental studies of combustion. They are now currently used in industrial combustion chambers (engines, turbo-machines, furnaces, etc.).

Energy and Modern Society The production and use of energy are essential elements of economic activity. In the second half of the twentieth century, the global annual production of energy doubled in less than half a century, reaching about 13 gigatonnes (Gt) (1 Gt is equivalent to one billion metric tons) of oil equivalent per year, composed of 31.5% oil, 21% natural gas and 29% coal, according to the figures given in 2014 by the International Energy Agency (IEA) of the Organization for Economic Cooperation and Development (OECD). Renewable energies, hydropower, geothermal, wind, solar, biofuels, biomass, and others, along with waste combustion together represent 13.5% of the overall production; nuclear power contributes less than 5%. Fossil fuels currently provide more than 80% of the energy used by man. In France the share of nuclear power is much higher than the rest of the world; it represents approximately 76% of electricity production. In 2014 the average annual energy consumption per capita in industrialised countries was around 4.2 tons of oil equivalent, to be compared with the total consumption of 13 Gt for 7 billion people, or 1.9 tonnes on average per inhabitant of the planet (1 ton of oil is roughly equivalent to 11.6 MWh). Based on current consumption in Western countries and the rapid development of the economy of countries like China, India and Africa, representing more than half the population on earth, energy consumption could continue to increase at a pace not only harmful to the health of our planet, but also worrying with regard to the reserves of fossil fuels accessible by current technologies. At its present trend, the total energy consumption of the planet will more than double over the next half-century to come. The assessment of usable fossil fuel reserves is subject to discussion; it depends on the extraction efficiency which is currently 30% for oil, but could fairly quickly reach 40%. The figures announced by companies and oil exporting countries are unreliable and the IEA calls for more transparency and rigour in the assessment. At the end of 2011, recoverable reserves were estimated to be about 1030 Gt (1012 tonnes) of oil equivalent, consisting of 22% oil, 17% gas and 61% coal (World Energy Council 2013). The reserves of shale oil are estimated to be around 655 Gt; however, the economic and ecological viabilities of this resource remain uncertain. Other major unconventional resources such as methane hydrates under the oceans are also known to exist[1] , but accurate data is not available. The production peak of oil will mark the beginning of the depletion of resources, when supply can no longer meet demand, creating a very high tension on the oil market that will disturb the world economy. There is no general consensus when this will occur but it could be in the second half of the century.

[1]

Dautray R., 2004, Quelle e´ nergie pour demain. Odile Jacob.

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Introduction

7

To address both the limitation of the release of CO2 into the atmosphere and the inevitable decline of fossil fuel reserves, major changes in energy policy will be needed in the next decade. Technologies for renewable energies, currently under-utilised, as well as energy storage, will grow at a much faster rate than today. These technologies will soon become competitive. However, even associated with a drastic reduction of wastage, especially in housing and transport, renewable energy alone will not meet the needs. Nuclear fusion will never replace all other energy sources, even supposing that it will be available in the foreseeable future. The scientific and technical problems to be solved are so gigantic that there can be no serious prognosis for the date of its industrial use. The feasibility of the process is far from proven. Also there are serious problems for the preservation of materials in the neutron flux, with no prediction for when they will be resolved, not to mention possible risks of accidental pollution by tritium. It will be necessary to use other technologies whose medium-term prospects are more realistic. Nuclear fission is a more serious asset, at least for several decades, even if the technical solutions and the actual costs of reprocessing waste and the demolition of obsolete plants are always controversial, to say nothing of the risk of proliferation of nuclear weapons. These difficulties can be overcome within a reasonable time provided that the will and the effort are coordinated across the planet. The production of biofuels and the use of solar technologies and wind power will continue to increase, but will ultimately represent only a fraction of the total energy production. The combustion of hydrogen and/or its use in fuel cells are viable solutions. However it will still be necessary to produce it in sufficient quantities – by electrolysis using nuclear electricity or from renewable energy sources? – and store it at a low cost without risk of leakage. Consumption of coal has more than doubled since 1973 with the industrialisation of China and other parts of Asia. Its substantial reduction is not an easy task, and cannot be expected in a near future in underdeveloped countries that have an increasing growth of population. Clean combustion of fossil fuels remains a priority in research and technology. Burning coal and oil shale or their derivatives is expected to continue for one more century in thermal power plants from which the emissions of carbon dioxide and pollutants like sulphur dioxide and nitrous oxides will be captured. The necessity to develop scientific knowledge and technological expertise of combustion phenomena remains more important than ever. Many scientific problems whose applications are important, especially for the production and economy of energy, as well as for safety, require further fundamental analyses for a better control of the technologies. In addition to issues of thermal and chemical pollution, let us cite micro-combustion; the deflagration-to-detonation transition, a phenomenon of importance for safety, including major incidents in nuclear power plants; the critical conditions for the spontaneous initiation and dynamic extinction of detonations; cellular detonations; turbulent and even the self-turbulent flame propagation; combustion chamber instabilities, a serious problem in rocket engines; super knock in gasoline engines working at high compression ratios; lean combustion of hydrocarbon fuels, more economical and less polluting, but difficult to ignite and to stabilise with the risk of flash-back in the injectors of turbo-reactors; and so on.

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8

Introduction

Combustion and Related Phenomena The study of flames, detonations and explosions on earth shows they generally have aspects in common with other phenomena of great importance. This is the case with inertial confinement fusion, the principle of the H-bomb, currently being studied for both military and civilian purposes. The principle of this method is to generate energy by micro-explosions in which thermonuclear ignition is obtained by the implosion of a shell of combustible material.[1] Such small-scale experiments of nuclear explosions can be used to test the numerical codes that are used in the development of nuclear weapons. Civilian production of energy is also envisaged in this way as a competitor for magnetic confinement fusion, but with not much chance of success, to say the least. The most advanced programs were launched in the 1970s and 1980s in the United States and in France. In 2007 the European Union funded a civil scientific research program on the subject. Similar phenomena occur at a very different scale in the gravitational collapse and subsequent explosion of dying stars and white dwarfs. These supernovae merit interest; for the astrophysicists they are the ‘candles of the universe’. They are also responsible for the production and dissemination into the cosmos of the heavier elements forming the planets and are necessary for the emergence of life forms. Despite more than 40 years of intense research in nuclear physics and impressive numerical simulations, the basic mechanism of the explosion of supernovae, responsible for the liberation of huge quantities of energy, partly of gravitational origin and partly from nuclear reactions, is still not understood Scope of the Book This book is mainly concerned with the properties, structure and dynamics of wave propagation in fluids. Subsonic waves, flames, deflagrations (fast flames) or ablation fronts and supersonic waves, shocks and detonations share similar properties. First, the front and the flow are strongly coupled. This is a consequence of the density variation across the wave, produced by the quasi-isobaric expansion associated with heat release in subsonic propagation and/or by compression in supersonic propagation. Moreover the upstream flow of subsonic waves is modified by wrinkling of the front. Second, their structure is controlled by coupled nonlinear mechanisms of different origins (fluid mechanics, chemical kinetics or dissipative transports). Consequently, different time and length scales (having many orders of magnitude difference) are involved in the dynamics and the geometry of the front, so that multiple-scale methods are required in the analyses. Generally speaking, the understanding of combustion waves and/or explosions is a difficult task and the research in the field is still very active. Flames, shocks, detonations and explosions have been investigated for a long time. Their understanding has greatly improved during the three last decades, but many aspects, in particular those concerning the strongly nonlinear regimes and their transition, require further investigation. [1]

Atzeni S., Meyer-Ter-Vehn J., 2004, The physics of inertial fusion. Clarendon Press–Oxford Science Publications, 1st ed.

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Introduction

9

The success of the approach of Ya. B. Zeldovich for the study of combustion waves has served as a guide in the analyses of the last 50 years. Understanding is greatly improved by the analytical studies of simplified models, obtained in a systematic way through deep physical insight. A relevant reduction of the complexity is the main difficulty. Efficient perturbation analyses have then to be carried out to solve the simplified models. Usually, the purely mathematical contents of the technical part are not the most useful points, and many of the perturbation methods have been set up by a physical approach. The essential ingredient for a successful input is the development in parallel of small-scale, carefully controlled laboratory experiments. The lack of such controlled experimental input is a major obstacle to progress in the understanding of astrophysical phenomena such as the explosion of stars. The main body of the book concerns combustion waves on earth. It differs in spirit and in content from other books on combustion. It is a successor to three pioneering works in the theory of combustion[2,3,4] and is complementary to more recent books.[5,6,7] Two other topics that may benefit from the advances in the theory of flames and detonations are also discussed: ablation fronts in inertial confinement fusion and the explosion of massive stars. According to the current views, the formation and the propagation of a shock wave in the flow of the imploding stellar core is the key mechanism for turning the gravitational collapse into a devastating explosion of the star. Mechanisms that have been identified in combustion on earth could be useful to help decipher the explosion of stars (supernovae), a subject that is still not well understood. The book is split into three parts. Part I is written for physics-oriented readers. Physical insights are provided by solutions to well-posed problems. Here, the physical analyses are almost free from the technical difficulties inherent in perturbation methods, used in the second part. The small-scale experiments on flames, which have proved to be essential to develop a relevant flame theory, are also reported in the first part. A chapter is also devoted to the systematic reduction of chemical kinetic schemes in combustion, illustrating the power of multiple-time-scale considerations. It also points out the limitations of the simplest models. This can serve as an interesting example for nuclear reactions. The prerequisites in applied mathematics and physics are those taught during the first year at university, some useful complements being given in appendices. The necessary background in physics (thermodynamics and statistical physics), chemistry and fluid mechanics (subsonic and supersonic flows) is recalled in Part III. Part II is for theoreticians. Detailed analytical studies are presented here on the basis of the physical insights presented in the first part. Conversely, these calculations provide the solid foundations for the physical analyses of Part I.

[2] [3] [4] [5] [6] [7]

Markstein G., 1964, Nonsteady flame propagation. New York: Pergamon. Williams F., 1985, Combustion theory. Menlo Park, Calif.: Benjamin/Cummings, 2nd ed. Zeldovich Y., et al., 1985, The mathematical theory of combustion and explosions. New York: Plenum. ´ Borghi R., Champion M., 2000, Mod´elisation et th´eorie des flammes. Edition Technip. Kuo K., 2005, Principles of Combustion. Hoboken, N.J.: John Wiley and Sons, 2nd ed. Law C., 2006, Combustion physics. Cambridge University Press.

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Part One Physical Insights

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17:00:04

1 General Considerations

Nomenclature Dimensional Quantities a cp cv dL D D e E EG kB  L m M n p qm R s S t T T∗ UL Ub v

Description Mean molecular velocity. Sound speed Specific heat at constant pressure Specific heat at constant volume Scale of laminar flame thickness Normal propagation velocity Molecular diffusivity Energy Activation energy Gamow energy Boltzmann’s constant Molecular mean free path Length of tube, burner or space scale Mass flux Mass of the sun Number density Pressure Heat of combustion per unit mass Radius Entropy Surface area Time Temperature Cutoff temperature Laminar flame speed Laminar flame speed w.r.t. burnt gas Impact velocity

S.I. Units m s−1 J K−1 kg−1 J K−1 kg−1 m m s−1 m2 s−1 J ≡ kg (m/s)2 J mole−1 J J K−1 m m kg m−2 s−1 kg m−3 Pa J kg−1 ≡ (m/s)2 m J K−1 kg−1 m2 s K K m s−1 m s−1 m s−1 13

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14

ρ σo σnr τ

General Considerations

Density Collision cross section Collision cross section for nuclear reaction A characteristic time

kg m−3 m2 m2 s

Nondimensional Quantities and Abbreviations cst. M Z γ ϑ φ CJ ICF DDT D-T SNI SNII ZND

Constant Mach number Mixture fraction Ratio of specific heats Stoichiometric coefficient Equivalence ratio Chapman–Jouget Inertial confinement fusion Deflagration-to-detonation transition Deuterium-tritium Supernovae type I Supernovae type II Zeldovich–von Neumann–D¨oring

U/a cp /cv

Superscripts, Subscripts and Math Accents a∗ ab acoll aCJ adiff ae anr aN ar au

Critical value Burnt gas Collisions Chapman–Jouget conditions Diffusion Elecron Nuclear reaction Neumann state (behind a shock) Reaction Unburnt gas 1.1 Introductory Remarks

In this introductory chapter we briefly introduce the physical background and the context of the phenomena that are studied in this book. The discussion is limited to orders of magnitude and dimensional analysis. As first demonstrated by the experiments of Lavoisier at the end of the eighteenth century, combustion is an exothermic chemical reaction between a fuel, such as hydrogen or a hydrocarbon, and an oxidant, generally the oxygen of ambient air. The reaction rate is a strongly increasing function of temperature, leading to thermal self-acceleration. Due to the

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1.2 Combustion Waves on Earth

15

thermal sensitivity, combustion occurs almost exclusively in thermal fronts. The concept of a well-stirred (homogeneous) reactor, widely used by chemists, is of only limited interest in practical combustion. Self-propagating exothermic reaction waves, flames (subsonic waves called also deflagrations) and detonations (supersonic waves) were identified in the nineteenth century. Self-consistent theoretical analyses of the dynamics of such waves and of the instabilities of their fronts started in the mid-twentieth century and are still in progress. The kinetics of heat release in combustion is complex; it involves hundreds of elementary reactions and tens of species. Examples are given in Chapter 5, while the basic elements of chemistry are recalled in Chapter 14. Hopefully most of the phenomena can be described using ultra-simplified chemistry schemes. The dynamics of combustion waves is a key issue in different areas of physics. For example the deflagration-to-detonation transition, which is not yet fully understood, concerns not only the safety of power plants but also the explosion of stars in astrophysics. With the development of nuclear physics during the twentieth century, astrophysicists explained how stars can maintain a quasi-stationary state of high temperature thanks to the energy released by nuclear fusion reactions, compensating the losses by radiation. These reactive phenomena have some analogies with ordinary combustion, but have also differences, arising mainly from the extreme conditions of temperature and density at which thermonuclear reactions occur. A short introduction to thermonuclear fusion is given in Section 1.3.1. In contrast to combustion on earth, the dynamics of reactive fronts in such extreme conditions is not yet a mature field, and a detailed presentation is outside the scope of this book. The explosion of stars during the gravitational collapse at the end of their lifetime, the famous supernovae that are typically 109 –1010 times brighter than the sun, are fascinating phenomena that are not fully explained. Shock formation and/or the deflagration-to-detonation transition are expected to play essential roles in the outbursts of supernovae. In this book, which is mainly concerned with combustion on earth, these phenomena will be discussed in simple physical terms with the objective of extracting the basic mechanisms without entering into the details of nuclear physics. Ablation fronts, which arise in inertial confinement fusion (ICF), will be discussed in the same spirit. The concept of this method of energy production, whose feasibility is not yet proved, is to burn a few milligrammes of nuclear fuel by imploding a spherical shell using high-power laser radiation. The ignition of a nuclear reaction at the central hot spot, compressed to more than 1000 times the density of liquids, depends critically on the control of hydrodynamic instabilities of the ablation front during the implosion. The similarities and differences with gaseous flames open new perspectives in the study of the dynamics of such thermal fronts. 1.2 Combustion Waves on Earth 1.2.1 Combustion Modes Two different modes of combustion are identified according to whether the fuel and the oxidant are initially mixed at the molecular level or spatially separated. The former is called premixed combustion and the latter non-premixed.

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16

General Considerations

Figure 1.1 Planar flame propagating from the open to the closed end of a tube. The unburnt gas is at rest, and the flame propagates at a speed UL with respect to the tube. The hot burnt gas flows out of the tube with velocity UL (Tb − Tu )/Tu ; see (2.1.2).

Propagating Waves In premixed combustion (gaseous, liquid or solid), the reaction generally propagates through the initial mixture as a reaction wave, transforming reactants into combustion products. There are two very distinct types of propagating combustion waves, depending on the conditions of ignition: • Premixed flames are very subsonic and quasi-isobaric since their propagation is governed by thermal conduction, and the term deflagration referred usually to fast flames essentially in turbulent flows. • Detonations are supersonic waves composed of a shock wave with a strong pressure pulse, followed by a reaction zone. These waves are thin, of a few tenths of a millimetre thick for flames and a few millimetres to centimetres for gaseous detonations in ordinary conditions. They are thus often assimilated to infinitely thin fronts whose instantaneous position is well defined. They transform fresh gas at temperature Tu ≈ 300 K into burnt gas at temperature Tb ≈ 1200–3000 K. For the case of a curved flame front propagating in a nonhomogeneous and/or nonstationary flow, the propagation speed is defined as the speed of the local normal to the front with respect to the initial reactive mixture just ahead of the front. This notion is well defined if the spatial and temporal scale of velocity variations in the upstream flow are much larger than the thickness of the flame and the transit time of a fluid particle across the flame, respectively. This is often the case. The propagation speed of a planar deflagration front in a quiescent reactive mixture, called the laminar flame speed and designated by UL in the rest of this book, is defined without ambiguity; see Fig. 1.1. In automobile petrol engines, a flame is ignited by a spark plug and propagates as a wrinkled premixed flame through the turbulent gas mixture confined in the cylinder. In a Bunsen flame such as that of Fig. 1.2, the conical premixed flame is stationary in the frame

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1.2 Combustion Waves on Earth

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(a)

(b)

Figure 1.2 Bunsen flame. (a) Tomographic cut through a Bunsen flame. The unburnt gas is seeded with microscopic refractory solid particles, which diffuse the laser light sheet used to visualise the flow (courtesy of J.Quinard, IRPHE Marseilles). (b) The flow lines are strongly deflected through the flame front; see also Fig. 2.6.

of the burner. The unburnt reactive mixture flows out of the orifice in a laminar flow with a velocity greater than flame speed, UL , and the inclination of the flame spontaneously adjusts itself so that the component of the gas flow normal to the flame front is equal to UL . Increasing the flow rate will decrease the angle of the summit and increase the height of the cone. For a planar flame, the product of the initial density of the unburnt gas and the laminar flame speed, ρu UL , defines the mass of reactive mixture transformed into burnt products per unit time and per unit surface of a planar flame, m = ρu UL . When the internal structure of the flame is not significantly modified by wrinkling of the front, the mass flow rate remains close to ρu UL and the total mass of gas burnt per unit time is given by ρu UL S, where S is the total flame surface area. This is true when the wavelength of wrinkling is much greater than the thickness of the flame. However, at the tip of a Bunsen cone the curvature is very high and a particular study is required. Non-Premixed Combustion In non-premixed combustion, the fuel and the oxidant are separated by the reaction front, fed by diffusive fluxes, and the flame is called a diffusion flame; see Fig. 1.3. This mode of combustion does not propagate in the preceding sense, but propagation along the wall of a solid combustible is possible. It is the type of flame found in a candle or a cigarette lighter. In industrial gas-fired burners (furnace, boilers, ovens, etc.) the chemical reaction develops in the mixing layer between the jet of fuel gas and the ambient air (oxidant); see Fig. 1.4. In this book attention is focused on premixed systems, and diffusion flames will not be covered. For more detail, readers are referred to classical books.[1,2,3] A brief outline [1] [2] [3]

Williams F., 1985, Combustion theory. Menlo Park, Calif.: Benjamin/Cummings, 2nd ed. Turns S., 2000, An introduction to combustion. McGraw-Hill, 2nd ed. Law C., 2006, Combustion physics. Cambridge University Press.

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General Considerations (a)

(b)

Figure 1.3 Photo of a candle flame in micro-gravity. In the absence of gravity-induced convection, the reaction zone is blue and the flame is hemispheric. (b) Diffusive fluxes in the region of the flame. The heat is evacuated in the same direction as the combustion products. Photo from nasa.gov website with thanks.

Figure 1.4 Diffusion flame in a sheared mixing layer. The black arrows represent the flow of fuel and oxidant separated by a wall, at the end of which the diffusion flame is anchored. Between pairs of roll-ups, stagnation points appear similar to that shown in Fig. 1.7, where the flame is locally stretched.

is presented in Section 1.2.4. The two types of flame (premixed and diffusion) can coexist in highly turbulent combustion chambers, when fuel and oxidant are injected separately.

1.2.2 Premixed Combustion In this section we recall the basic concepts in the theory of premixed combustion.[1] Combustion Temperature and Equivalence Ratio The energy liberated by ordinary combustion comes from the change in binding energy between the atoms forming the molecules. The binding is due to interactions of the electrons in the external layers of the atomic structure. The order of magnitude of this energy is a few electron-volts (eV) per molecule, leading to a temperature increase of few thousand Kelvin; see Section 14.1.1. The temperature of combustion of premixed gases results from a chemical equilibrium; see Section 14.2. This temperature varies with the composition of the mixture. It reaches a maximum when the proportion of fuel and oxidant is near stoichiometry, defined as the [1]

Clavin P., 1994, Ann. Rev. Fluid Mech., 26, 321–352.

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Figure 1.5 Calculated burnt gas temperature of methane–air flames as a function of equivalence ratio.

composition for which both are totally consumed, for example two moles of hydrogen mixed with one mole of oxygen, 2H2 +O2 → 2H2 O. More generally, if ϑF+ moles of fuel F fully react with ϑO+ moles of oxidant O to form combustion products P: ϑF+ F + ϑo+ O  P, then a fresh mixture containing NF moles of fuel and NO moles of oxidant is said to have a stoichiometric composition if NF /NO = ϑF+ /ϑO+ . The equivalence ratio φ of the mixture is defined by φ=

NF /NO . ϑF+ /ϑO+

(1.2.1)

A stoichiometric mixture is characterised by φ = 1. In a rich mixture, φ > 1, the fuel is in excess and the combustion is limited by the oxidant. For lean mixtures, φ < 1, the oxidant is in excess. An example of the evolution of the combustion temperature of methane–air flames with equivalence ratio is shown in Fig. 1.5. Chemical Kinetics The chemical kinetics controlling the reaction rate of the combustion of hydrogen and common hydrocarbon fuels is complex. Nevertheless the system has overall global properties that can be reasonably well approximated by simple kinetic models, described in Chapter 5. The overall chemical heat release rate increases very strongly with temperature, and the combustion rate undergoes a thermal runaway, which stops only when the minority reactant has been completely consumed. The energy release rate is significant only for temperatures typically above 1000 K. At room temperature a reactive mixture can be conserved indefinitely: the composition of the mixture is totally frozen far from thermochemical equilibrium. However, at high temperature, the characteristic reaction time (time to reach thermo-chemical equilibrium) is short, of the order of a few microseconds at the temperature produced by the exothermic reaction, typically 2000 K. The thermal runaway in combustible mixtures can be described in macroscopic terms, as a function of the temperature and pressure. It is not necessary to invoke Boltzmann’s equation. This is because the frequency of inelastic collisions, which liberate chemical

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20

General Considerations

energy, is much lower than the frequency of the elastic collisions, which maintain the (Maxwell–Boltzmann) equilibrium distribution of velocity. In other words, a reactive molecule undergoes many elastic collisions before being transformed. The liberation of energy thus proceeds on a time scale that is long compared with the time needed to reach local equilibrium, and the temperature remains locally and globally a well-defined quantity during the chemical reaction. Thermal Sensitivity and the Arrhenius Law The characteristic reaction time, τr , of a first-order elementary reaction between two reactive species, R1 + R2 → P1 + P2 , is defined in a homogeneous medium by the evolution equation dn1 /dt = −n1 /τr , where n1 = N1 /V is the number density of the limiting reactive species R1 . Usually, the reaction time τr is a function of the temperature, T, and proportional to the density n2 of the species R2 . To simplify the presentation, we will limit the discussion in this section to the situation where the species R2 is in great excess, such that the relative quantity consumed by the reaction is very small. The change of concentration is negligible in the expression for τr , which is then only a function of the temperature, T. For most gaseous combustible mixtures, far from the flammability limits, and over a wide range of temperature, the reaction rate is described by the Arrhenius law, 1/τr ≈ e−E/kB T /τcoll

with E/kB T  1,

(1.2.2)

where τcoll is the mean time between elastic collisions of species R1 with R2 , and E is an activation energy that is systematically greater than the mean energy of thermal agitation, kB T, typically by an order of magnitude. The activation energy E can be interpreted as the energy that must be brought by the molecules into a collision in order to open bonds and initiate the chemical reaction. The exponential factor in the Arrhenius law comes from the Maxwell–Boltzmann distribution of velocities. It represents the fraction of the number of collisions (per unit time) in which the relative energy carried by the two molecules is greater than E. According to (1.2.2), the thermal sensitivity of the reaction rate increases as the ratio of activation energy to mean thermal energy, E/kB T, increases: δτr /τr ≈ δτcoll /τcoll − (E/kB T)δT/T, where, according to (1.2.3), the sensitivity of the collision rate, 1/τcoll , to temperature at constant pressure is only T 1/2 , δτcoll /τcoll ≈ −(1/2)δT/T, and can be neglected, δτr /τr ≈ −(E/kB T)δT/T, since, typically, E/kB Tb ∼ 8–12, where Tb is the maximum temperature in the flame, Tb ∼ 1200–3000 K. The collision frequency can easily be evaluated by the elementary kinetic theory of gases as follows. The distance travelled by a molecule during a lapse of √ time δt is aδt where a ∝ kB T/mR is the mean velocity of the molecules (of the order of the speed of sound), a few hundred metres per second for normal conditions, and mR is the mass of a molecule. The volume swept, aσo δt, is expressed in terms of the collision cross section σo = 4π ro2 where ro is the range of action of inter-molecular forces (a few angstroms, 10−10 m). Introducing n, the number density of particles, the number of

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1.2 Combustion Waves on Earth

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collisions experienced by the test particle is naσo δt, and the number of collisions per unit time is naσo . The mean free path , namely the mean distance travelled by a molecule between two collisions, is obtained by the ratio of the distance travelled to the number of collisions,  = 1/(nσo ),

1/τcoll = naσo ;

(1.2.3)

see Section 13.3. This leads to a temperature dependence of τcoll as T −1/2 at constant pressure, p = nkB T. The Arrhenius law (1.2.2) then describes a strong variation of the reaction rate with temperature. For example, taking a typical value for E/kB Tb = 8, the Arrhenius factor at the temperature of the burnt gas is e−E/kB Tb ≈ 3 × 10−4 . So, for a typical mean time between elastic collisions of one nanosecond, τcoll ≈ 10−9 s, the characteristic reaction time in the burnt gas is of the order of 1 μs, τr (Tb ) ≈ τcoll eE/kB Tb ≈ 3×10−6 s. However, at room temperature Tu , Tb /Tu ≈ 8, E/kB Tu = 64, the characteristic reaction time τr (Tu ) ≈ τcoll eE/kB Tu is of the order of 6 × 1018 s, which is longer than the estimated age of the universe ≈ 4 × 1017 s, or 13 × 109 years! The reaction rate given by the Arrhenius law is thus completely negligible at room temperature. The high sensitivity of the reaction rate to temperature makes premixed flames very different from the reaction–diffusion waves encountered in other fields, as for example in biophysics; see Section 8.3. Cut-Off Temperature, Flammability Limits The Arrhenius law (1.2.2) is a drastic simplification that is nevertheless useful to analyse the thermal propagation of flames and detonations. However, certain details of chemical kinetics also play an essential role, as explained in Chapter 5. This is the case, for example, when the limits of flammability or detonability are approached. These limits correspond to brutal transitions in the energy release rate. The behaviour of flames at these limits can be qualitatively represented in a simple but discontinuous model by introducing a cut-off temperature T ∗ ≈ 1000 K; see (9.1.1) and (9.1.2). The reaction rate is given by (1.2.2) for temperatures above cut-off, T > T ∗ , but drops to zero below T ∗ , corresponding to the crossover of chain-branching and chain-breaking reactions; see Section 5.2.2. Cool Flames The chemical reaction never stops suddenly below T ∗ , but the nature of the oxidation reaction changes completely. The combustion is incomplete; the initial reactants are not transformed into stable species such as water vapour and carbon dioxide, which are the major species produced at thermo-chemical equilibrium. Instead, the initial reactive species are degraded to form relatively stable intermediate species, or more exactly metastable species, such as peroxides, or aldehydes, such as formaldehyde. Only a small part of the available chemical energy is released, and the characteristic reaction time increases rapidly as the combustion temperature drops below T ∗ . For common hydrocarbon molecules, the cut-off temperature is in the range 900–1300 K. The chemical reactions that take place at low temperature, T < T ∗ , play no significant role in the propagation of normal flames.

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General Considerations

Nevertheless, chemical waves, and even successions of waves (called ‘cool flames’ since they release little energy) can propagate below T ∗ . They can be produced in special laboratory conditions.[1] The analytical study of such reaction–diffusion waves[2] show that they have nothing to do with ordinary flames, and they will not be treated in this book. Dimensional Analysis Several mechanisms participate in the propagation of laminar combustion fronts: the amount and rate of heat release, the transport of heat and mass, and also compressible phenomena characterised by the speed of sound. In usual gaseous flames and detonations, the radiative transfer of heat is weak and can be neglected compared with diffusive transport. The physical mechanisms can then be characterised by the following quantities: • The amount of chemical energy, qm , released per unit mass of the mixture. This energy is of the order of the square of the sound speed in the burnt gas, a2b , which is typically four to ten times larger than in the fresh mixture, (ab /au )2 ≈ Tb /Tu . • The exothermic reaction rate, 1/τr , which is a strongly increasing function of the temperature of the gas. • The molecular and thermal diffusion coefficients, D, which all have the same order of magnitude in a gas. Using dimensional analysis, it is possible to construct different velocities from these parameters. It will be seen that the orders of magnitude for the propagation velocity of laminar √ √ flames are D/τr (Tb ), and qm for detonations.

1.2.3 Flames in Premixed Gas In ordinary premixed combustion, the initial thermodynamic state of the reactants can be gaseous, liquid (such as nitromethane or nitroglycerin1 ) or solid, such as the propellants used in missiles and rocket engines. The temperature of the combustion products Tb is determined by the heat of reaction (the chemical energy that is released) and is generally in the range 1200–3000 K for air-based combustion. Since the exothermic reaction can proceed only if the temperature is sufficiently high, typically above 1000 K, the reaction zone is almost systematically located in the gas phase.2 In the rest of this book we will consider gaseous mixtures. 1 To be exact, nitromethane and nitroglycerin are not mixtures of reactants, but monopropellants, i.e. metastable molecules, which

decompose under the action of heat to release heat and combustion products. 2 One notable exception is the oxidation of solid carbon (charcoal).

[1] [2]

Lewis B., von Elbe G., 1961, Combustion flames and explosions of gases. Academic Press. Nicoli C., et al., 1990, In P. Gray, G. Nicolis, F. Barras, P. Borkmans, S. Scott, eds., Spatial inhomogeneities and transient behavior in chemical kinetics, 317–334, Manchester University Press.

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Laminar Flame Speed, Thickness and Transit Time The propagation speed of a planar laminar flame in a premixed reactive medium at rest is the result of a direct coupling between the reaction rate (1.2.2) and the transport of heat and species. Roughly speaking, the chemical heat, which is released on the hot side of the reacting zone, is transferred by conduction to warm up the adjacent cold mixture to a temperature at which the exothermic reaction starts, and so on. According to the elementary kinetic theory of gases, the molecular transport of heat and species is controlled by Fourier’s law (15.1.28), Jq = −λ∇T, and Fick’s law (15.1.10), respectively, with coefficients for thermal diffusion, DT ≡ λ/ρcp , and molecular diffusion Di that have the same order of magnitude, D ≈ a2 τcoll ≈ a ≈ 2 /τcoll

(1.2.4)

(see (13.3.29)) where the thermal velocity of molecules v is of the order of the speed of sound a. The collision frequency 1/τcoll , and the mean free path , have been evaluated above; see (1.2.3). Using√the perfect gas law at constant pressure, nT ≈ cst., and remembering that a varies as T, the diffusion coefficients increase with temperature as T 3/2 and nD as T 1/2 . However, to simplify the presentation, this weak temperature dependency is often neglected in flames compared with the exponential factor in the Arrhenius law (1.2.2). This is not the case for the ablation front in ICF, presented in Section 1.3. The large value of the activation energy in the expression for the reaction rate (1.2.2) is an essential characteristic of combustion. An important consequence is that the propagation speed of deflagrations is very subsonic. Using dimensional analysis to construct a flame speed, UL , from the reaction rate, τr , and the diffusion coefficient, D, the flame speed takes √ the form UL ∝ D/τr . The order of magnitude of the Mach number of the flame speed, M ≡ UL /a, can then be found using (1.2.2) and (1.2.4):   UL ∝ D/τr ⇒ M ≡ UL /a ∝ e−E/kB T 1. (1.2.5) No matter which temperature (between the fresh gas temperature, Tu , and the burnt gas temperature, Tb ) is used in (1.2.5), the Mach number is very small since E/kB Tb ≈ 8 in usual mixtures. Then, as explained in Section 2.1.1 the relative pressure variation across the flame is negligible in front of the thermal energy. Therefore, according to (15.2.3), the temperatures of the fresh and burnt gases are related by a simple thermal balance cp (Tb − Tu ) = qm ,

(1.2.6)

where cp is the specific heat per unit of mass at constant pressure. Since the reaction proceeds at high temperature, the relevant temperature in (1.2.5) is Tb and the Mach number of flame propagation is found to be of order 10−2 , yielding a flame velocity of the order of a few metres per second. This value is somewhat greater than those measured experimentally, between a few tens of centimetres per second and 1 m/s for typical flame speeds in air (it reaches 10 m/s only in very energetic mixtures such as a stoichiometric mixture of hydrogen or acetylene in pure oxygen). This difference, discussed in Section 2.1.2, will be explained

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General Considerations

by the analytical study of the thermal propagation of flames in Section 8.2. This points out the limitations of dimensional analysis when large nondimensional parameters such as E/kB Tb are involved in the problem. The laminar flame thickness, dL , is the characteristic length over which the gas temperature rises from Tu to Tb ; see Fig. 1.1. Dimensional analysis using (1.2.2) and (1.2.4) shows that dL is proportional to the inverse of the pressure and is much greater than the mean free path,  = D/a, when the activation energy is large:   (1.2.7) dL ∝ UL τr ≈ Dτr ≈  eE/kB T  . For a typical stoichiometric methane–air flame this estimation yields a flame thickness of the order of 10−5 m. For the same reason as above, this value is an order of magnitude too small compared with experimentally measured values. The flame transit time, τL , the characteristic time taken by the gas to traverse the flame structure, of the order of the ratio of flame thickness to flame speed, is, according to (1.2.2), much larger than the elastic collision, τL ≡ dL /UL ≈ τr  τcoll .

(1.2.8)

These results confirm that, thanks to the large activation energy, the structure of flames can be described in terms of macroscopic variables, without resorting to the kinetic Boltzmann equation.

1.2.4 Diffusion Flames The discussion is limited here to a general presentation. Internal Structure In diffusion flames the reactants are initially separated in space. They may be injected with different phases, as for instance a drop of liquid fuel burning in air. However, when one of the reactants is initially in a condensed phase, it is vaporised before reaching the reaction zone, except in a few exceptional cases, such as the combustion of carbon (charcoal). The high temperature reaction zone is thus generally situated in the gaseous phase. It is fed by molecular diffusion of the fuel from one side and oxidant from the other side. The reaction products and the heat of combustion are evacuated by diffusion towards both sides, as indicated in Fig. 1.3. Mixing of fuel and oxidant must take place on the molecular scale before the reaction can proceed. When the characteristic time of the exothermic reaction is sufficiently short, in a sense to be defined later, the reaction zone is thin and the concentration of reactants is very small in this zone, where they coexist but are rapidly consumed. However, the diffusive fluxes of reactants, and thus the reaction rate, can be high. The fluxes of fuel and oxidant have quasi-stoichiometric proportions and both reactants are completely consumed. The combustion temperature is then close to that of a stoichiometric premixed flame. Since the temperature around the reaction zone is high, fuel and oxidant cannot coexist outside the

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25

Figure 1.6 Sketch of profiles of temperature and concentration through a diffusion flame.

Figure 1.7 Flow field around a stagnation point.

reaction zone. The position of the reaction zone must then adjust itself so that the reaction occurs with a stoichiometric composition; see Fig. 1.6. The diffusion time, τdiff = L2 /D, is usually controlled by a geometrical length L, and D is the diffusion coefficient of the reactants. For example in fuel-droplet combustion, L is the radius of the flame, of the same order of magnitude as the radius of the droplet. This time is usually much longer than the reaction time, τdiff  τr (Tb ), τr (Tb ) ≈ 10−5 –10−6 s. The reaction zone is thus much thinner than the external diffusion zone, as anticipated. Moreover, the reaction rate, being limited by diffusion, is of the order of diffusion rate, 1/τdiff . For example, the combustion time of a stationary drop of liquid hydrocarbon in a quiescent atmosphere is given (to a numerical factor of order unity) by τd ∝ d2 /D, where d is the initial diameter of the droplet and D is the molecular diffusion coefficient of oxygen in the gaseous mixture containing the combustion products. For a diffusion flame stabilised in a stagnation flow between two opposed jets of fuel and oxidant as in Fig. 1.7, the two parameters characterising the external diffusion zones are the stretch time, τs , equal to the inverse of the velocity gradient, and the diffusion coefficient, D, of the reactants. The consumption of reactants per unit surface and per unit time is given

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26

General Considerations

Figure 1.8 Sketch of the configuration studied by Burke and Schumann to represent a diffusion flame at an orifice.

√ by the mass flux of reactants, whose order of magnitude is ρYR D/τs where YR is the initial mass fraction of reactant in the jet of density ρ. In the more general case, such as a jet of fuel entering a flow of oxidant, the problem is complicated by the fact that the shape of the flame front is no longer determined by simple geometrical constraints. Nevertheless, the problem can be simplified by supposing that the diffusion coefficients of fuel and oxidant are equal and that the reaction is a single-step irreversible reaction. With this simplification, the term containing the chemical consumption (reaction rate) can be eliminated from the conservation equations by a introducing a linear combination, Z, of the concentrations of fuel and oxidant. This combination Z, called the mixture fraction , is a conserved scalar governed by a convection–diffusion equation. It can be solved using the boundary conditions of the configuration. The first analysis of this type was performed by Burke and Schumann[1] for the geometry of a uniform coaxial jet sketched in Fig. 1.8. In the limit of infinitely fast chemistry, the reaction zone is infinitely thin and on this surface the concentrations of fuel and oxidant are zero. The location of the reaction zone is given by the location of the surface Z = 0. The effect of finite rate chemistry on a diffusion flame stabilised between two opposed jets (Fig. 1.7) or in a mixing layer (Fig. 1.4) can lead to local extinction through stretch of the reaction front. The first analytic studies of stretched diffusion flames were performed in 1974 by Li˜nan.[2] Diffusion flames and premixed flames can coexist in the same flow. This is the case, for example, in the region where a diffusion flame is anchored in the mixing layer of Fig. 1.4. Molecular mixing takes place in the near wake of the plate separating the flows of fuel and oxidant. A zone of premixed gases is thus present in the wake. The flame is then anchored by a triple flame formed by a premixed flame, with lean and rich branches stabilised in this nonuniform premixed flow, followed by a diffusion flame, which develops downstream in the mixing layer.[3] These triple flames can also appear on the edges of holes produced in a diffusion flame by local extinction due to stretch. Another type of diffusion flame anchoring, purely thermal in nature, can exist where the flame is directly attached to the trailing edge of a wall that is at a high temperature.[3] [1] [2] [3]

Burke S., Schumann T., 1928, Ind. Eng. Chem., 20(10), 998–1004. Li˜nan A., 1974, Acta Astronaut., 1(7-8), 1007–1039. Fern´andez-Tarrazo E., et al., 2006, Combust. Flame, 144(1-2), 261–276.

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1.2.5 Gaseous Detonations In premixed combustion, there is also a supersonic mode of propagation, called detonation. The history of the discovery of this phenomenon in the nineteenth century was briefly recalled in the introduction. ZND Structure The structure of a detonation, called ZND in honour of the works of Zeldovich, von Neumann and D¨oring in the middle of the twentieth century, is composed of an inert shock wave followed by and coupled to an exothermic reaction zone. This structure was initially imagined half a century earlier by the chemist Paul Vieille.[4] As explained in the complements, recalled in Part III, the thickness of shock waves is microscopic (see Section 15.1.7), so that these waves can be considered as hydrodynamical discontinuities of the flow field. Their formation is recalled in Section 15.3. Since the propagation speed of the wave, D, is supersonic, of the order of 1800 m/s for detonations of hydrocarbon– air mixtures at room temperature and pressure, the compression in the leading shock wave raises the pressure, density and temperature to values sufficiently high to initiate the chemical reaction. The values of these quantities, pN , ρN , TN , just downstream of the shock, called the Neumann conditions, are expressed in terms of the propagation Mach number M ≡ D/au by the Rankine–Hugoniot relations (4.2.14)–(4.2.17). For M of the order of five, the Neumann conditions are typically pN /pu ≈ 30, ρN /ρu ≈ 5 and TN ≈ 1800 K. The high pressure in the burnt gas behind the front causes detonations to be very destructive. The study of the structure of planar detonation waves is presented in Section 4.2. The ZND structure of a detonation can be understood from Fig. 1.9, which represents schematically a (very) long tube filled with a reactive gaseous mixture. One extremity of the tube is closed by a piston advancing at a constant speed. Viscous effects and the existence of boundary layers at the walls are neglected (slip conditions at the walls). If the piston is started initially from rest, the upstream gas is put in motion by a compressible wave, which soon becomes a shock wave; see Section 15.3.4. If the shock is strong enough, the chemical reaction is initiated in the gas at the Neumann condition just behind the shock, and, under certain conditions, the reaction zone remains attached to the shock. The complex shock– reaction structure then propagates at a constant supersonic speed, D, into the fresh gas. The conservation equations for mass, momentum and energy show that the existence of a stationary solution for a planar wave driven by a piston at constant speed, called an ‘over-driven detonation’, is possible only if the piston speed exceeds a minimum value; see Section 4.2.3. Chapman–Jouguet Regime The Mach number of the downstream flow relative to the leading shock increases, as chemical energy is released, from its initial value MN < 1 of the Neumann state to a value Mb ≤ 1 at the downstream edge of the reaction zone in the burnt gas. The latter [4]

Vieille P., 1900, C. R. Acad. Sci. Paris, 131, 413.

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General Considerations

Figure 1.9 In the reference frame of the laboratory (top), or in other words when the fresh gas is at rest, the burnt gas behind a detonation moves in the same direction as the front, with a speed equal to that of the piston. In the reference frame of the detonation (bottom), the fresh gas enters the front with a supersonic velocity, D, greater than the subsonic speed UN at which the compressed burnt gas leaves the front. Because of compressible effects, the temperature of the reacting gas reaches a maximum before the exothermic reaction goes to completion.

value, Mb , increases towards unity as the speed of the driving piston is reduced, whilst the detonation velocity, D, decreases towards a lower bound DCJ . The limiting value Mb = 1, D = DCJ , corresponds to a critical nonzero value of piston velocity below which there is no steady solution to the conservation equations across the detonation front, as explained in Section 4.2.3. This marginal condition with unit Mach number (relative to the front) of the burnt gas leaving the reaction zone is called the Chapman–Jouguet (CJ) detonation. It will be seen in Section 4.2.3 that this solution can also propagate as an autonomous detonation, that is, without the support of a piston. The propagation speed, DCJ , can be derived directly from the conservation equations for mass, momentum and energy (15.1.45)–(15.1.48). These conservation equations were established at the end of the nineteenth century, and the expression for DCJ was already known at that time; see, for example, the reference to Michelson (1893) in the books of Shchelkin and Troshin[1] and of Zeldovich and Kompaneets.[2] The expression for DCJ simplifies when the chemical energy released in the reaction is large compared with the initial thermal energy of the flow. This is the case for almost all usual reactive mixtures where qm /(cp Tu ) is of the order of six to ten in detonations. The propagation Mach number of a CJ detonation can then be written [1] [2]

Shchelkin K., Troshin Y., 1965, Gasdynamics of combustion. Baltimore, Md.: Mono Book Corp. Zeldovich Y., Kompaneets A., 1960, Theory of detonation. Academic Press.

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qm /cp Tu  1:

MCJ

29

   cp qm ≈ 2 +1 , cv cp Tu

(1.2.9)

where cp and cv are the specific heats per unit mass, supposed constant to simplify the presentation; see Equation (4.2.44). Typically the planar detonation wave is unstable, leading to galloping detonations and a cellular structure on the detonation front. The analytical studies of these phenomena are recent and are presented in Chapter 4. Detonation Thickness The thickness of a CJ detonation is greater than that of a subsonic flame for the same thermodynamic conditions. Its order of magnitude is given by the product of gas velocity UN and the reaction time, τr (TN ), at the Neumann state, τr (TN ) = τcoll eE/kB TN , according to the Arrhenius law (1.2.2). In the reference frame of the detonation, the velocity, UN , of the gas leaving the shock and entering the reaction zone is a fraction α of the local speed of sound, UN = aN /α, with typically α ≈ 5 for a CJ detonation. A dimensional analysis, similar to that used in (1.2.5), then shows that diffusive transport is negligible (see (4.2.45) for more details), and the estimated detonation thickness dN is dN = UN τr (TN ) ≈ (aN /α)τcoll eE/kB TN .

(1.2.10)

This is a rough evaluation of the detonation thickness, sufficient for our purpose here. A more accurate evaluation is given in Chapter 4; see (4.3.14). According to (1.2.7), and more precisely to the result from asymptotic analysis (2.1.9)–(2.1.11), the thickness of a flame can be rewritten dL ≈ ab (E/kB Tb )τcoll eE/(2kB Tb ) ,

(1.2.11)

where ab is the speed of sound in the burnt gas at temperature Tb . Expressions (1.2.10) and (1.2.11) then yield dL E ab − k ET ≈α e Bb dN kB Tb aN



Tb TN

− 12



.

(1.2.12)

The burnt gas temperature of a deflagration is higher than the Neumann temperature of a CJ detonation, Tb /TN > 1. The exponential term in (1.2.12) is thus much smaller than the prefactor, which is typically 30. For instance, for a CJ detonation in a stoichiometric methane–air mixture with ambient conditions, TN ≈ 1525 K, aN ≈ 750 m/s, α ≈ 5, τcoll ≈ 10−9 s, E/kB TN ≈ 11, and the value of the exponential is ≈ 5 × 10−4 , leading to a detonation thickness of the order of a centimetre. This is two orders of magnitude greater than the thickness of a flame of the same mixture. According to (1.2.10), the thickness of overdriven detonations is smaller since the Neumann temperature increases with the speed of the shock.

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General Considerations

1.3 Fronts and Thermal Waves in Extreme Conditions Explosions in astrophysical and terrestrial contexts have some aspects in common[1] but also many differences. In principle, flames, shocks and detonations can also be produced in the extreme conditions of thermonuclear fusion, namely at high temperature ≈107 –1010 K and high density, 106 kg/m3 in ICF and much higher densities in massive stars: 1010 kg/m3 in the burning silicon shell surrounding the iron core and ≈1017 kg/m3 at the end of core collapse.[2,3]

1.3.1 Thermonuclear Fusion In 1926 Eddington[4] postulated nuclear reactions as the mechanism powering the stars. Nuclear fusion reactions have some analogies with ordinary combustion. Nucleons (protons, neutrons) are held together by a nuclear binding energy to form an atomic nucleus. This is analogous to the way in which atoms are held together to form a molecule. However, nuclear forces are quite different, in strength and nature, from interatomic forces. Nevertheless, in the same way that energy is released in a chemical reaction when atoms are rearranged to form different molecules having a higher binding energy, energy is released if nucleons are rearranged to form more stable atomic nuclei. Also, a barrier of energy has to be crossed in both cases before the energy can be released. The differences arise from the binding energy of nucleons, several million electronvolts (MeV), six orders of magnitude higher than the electronic binding energy of atoms in molecules, typically a few electron-volts (eV). To fix ideas, we recall that an energy of 1 eV per particle (1 eV ≈1.6×10−19 joule) corresponds to a temperature of the order of 11 600 K. The energy released in nuclear reactions is thus typically six orders of magnitude (106 ) greater than in chemical reactions. In order to make two nuclei interact, it is necessary to overcome the potential barrier created by the electrostatic repulsion of their positive charges; see Section 14.3.1. This barrier, called Gamow energy, EG , is also typically of the order of a million electron-volts, implying that the temperatures needed to initiate nuclear fusion are typically larger than 107 K. The conditions for nuclear fusion thus involve completely ionised plasmas, and nuclear reactions occur at extreme conditions that are far from that of ordinary combustion. Another difference is that the distinction between fuel and oxidant can be irrelevant in nuclear fusion: two identical nuclei can combine and release energy, as in the sun, for example, where protons (hydrogen nuclei) combine in a succession of reactions to form a helium nucleus. Fig. 1.10 shows the mean binding energy per nucleon as a function of the number of nucleons in a nucleus. A flat maximum is reached for 56 nucleons, corresponding to atomic iron 56 Fe (26 protons and 30 neutrons). The number in front of the symbol denotes the [1] [2] [3] [4]

Wheeler J., 2012, Philos. Trans. R. Soc. London Ser. A, 370, 774–799. Janka H.T., et al., 2007, Phys. Rep., 442, 38–74. Janka H., 2012, Annu. Rev. Nucl. Part. Sci., 62, 407–451. Eddington A., 1926, The internal constitution of stars. Cambridge University Press.

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Average binding energy (MeV)

1.3 Fronts and Thermal Waves in Extreme Conditions 9 8

12

C

7

4

6

7 Li 6

5

16

He

56

O

235

Fe 238

31

U

U

Li

4 3 2

3 H 3

He

2 1 H 1 0 H 0 30

60

90

120 150 180 210 240 270

Number of nucleons in nucleus

Figure 1.10 Binding energy per nucleon as a function of the number of nucleons in a nucleus. The nucleus of iron (56 Fe) has the highest binding energy.

atomic mass (number of nucleons). The proton, denoted p, is simply 1 H and the nucleus of the stable helium molecule, following hydrogen in the Mendeleev’s classification (see Figs. 14.1 and 14.2), is denoted 4 He (2 protons and 2 neutrons). Energy can thus be released either by the fission of heavy nuclei such as uranium or by the fusion of light nuclei, such as 1 H, 3 He (2 protons and 1 neutron), lithium 6 and 7 (6 Li, 7 Li), carbon 12 C, or oxygen 16 O. It is fairly clear from Fig. 1.10 that potentially much more energy is released by the successive fusion of light nuclei than by fission of heavy nuclei. Both processes stop with iron. Due to the large energy barrier EG the rate of fusion reaction is very sensitive to temperature in a way somehow similar to the combustion rate on earth but for different time and energy scales; see (1.2.2) and (14.3.1)–(14.3.3). For a temperature about 2 × 108 K, appreciably higher than the minimum (≈107 K) required for hydrogen burning into helium recalled in Section 14.3.2, the gradual fusion of several 4 He nuclei into 12 C and 16 O develops. Nuclear burning of mixtures of carbon 12 C and oxygen 16 O will start at higher ignition temperatures, about 6 × 108 K and 1.5 × 109 K, respectively, to form essentially 28 Si and 24 Mg. Finally the fusion of two 28 Si nuclei into 56 Fe occurs at an even higher temperature, about 4 × 109 K. Most of these thermonuclear reactions proceed through a complex network of chain reactions that are not necessary to describe here.

1.3.2 Stellar Evolution and Supernovae Stars are born from clouds of gas (essentially hydrogen and helium) collapsing under gravitational attraction. They undergo a sequence of changes during their lifetime. The knowledge of how stars evolve is developed by observing different stars at various states of their lifetime. Depending on their initial mass, M, the lifetime of stars ranges from few million years for the most massive stars to a time much longer than the age of the universe for the least massive ones. This is because the core temperature increases with the size of the star and because of the high sensitivity of thermonuclear burning to temperature, so that massive stars burn all their fuel much faster than small stars.

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General Considerations

Stellar Structure Most of the energy transfer inside stars is due to radiation and can be roughly modelled by a Fourier law (15.1.28) with a diffusion coefficient of the same form as (1.2.4), where a is the speed of light and  the mean free path of photons. In its quasi-steady state, a star is in hydrostatic and thermal equilibrium:[1,2] the gravitational force is balanced by pressure, while the radiative losses are balanced by the energy released in nuclear reactions. The inner core temperature (roughly evaluated with an adiabatic hydrostatic equilibrium) increases with the mass of the star. For a mass above 8% of the mass of the sun (M > 0.08 M , M ≈ 2 × 1030 kg is the solar mass), the core temperature of a star becomes sufficiently high to ignite nuclear burning of hydrogen into helium. In a star, the nuclear combustion inside the hot plasma cannot take the form of a laminar flame because of the strong Rayleigh–Taylor instability associated with the gravitational force; see (2.2.15). The resulting unsteadiness could lead to a combustion regime similar to that of a wellstirred reactor or, more likely, of a strongly turbulent flame in the well-stirred regime; see Section 3.1.3. This question is open and one cannot do better than to use empirical models of nuclear combustion in stars. In the sun the radius of the central region of nuclear burning is evaluated at about 25% of the total radius. Unicity and stability of the equilibrium state solution are subtle questions[2,3] that are briefly discussed in Chapter 7. In a stable configuration, contraction (expansion) of the star increases (decreases) the pressure and the temperature of the core. The regulation mechanism is roughly explained as follows (see the discussion following Equation (7.2.10)): • If the fusion reaction runs too fast, the core heats up, producing a higher pressure and making the core expand against gravity. Expansion cools core, slowing the rate of fusion. • If the reaction runs too slow, the core cools, leading to a lower pressure and making the core contract. Contraction heats the core, increasing the reaction rate. When the hydrogen has been consumed and its nuclear burning rate is no longer sufficient to balance energy loss by radiation, the star contracts, the core temperature increases and the burning of helium into carbon and oxygen can be ignited. For stars of intermediate mass (less than about eight times the mass of the sun), when the helium is exhausted and after sufficient radiative cooling, the core of the star can take the form of an electrondegenerate white dwarf composed of carbon and oxygen[4] in which degenerate electrons offer resistance to further compression. For more massive stars, M > 8 M , the nuclear burning of carbon and oxygen is ignited, producing heavy-element cores. For a mass greater than ≈15 M the inner temperature increases sufficiently to ignite further fusion reactions, and the final core is composed essentially of 56 Fe (the maximum nuclear binding energy).

[1] [2] [3] [4]

Chandrasekhar S., 1967, An introduction to the study of stellar structure. Dover. Zeldovich Y., Novikov I., 1971, Stars and relativity. Dover. Kippenhahn R., Weigert A., 1994, Stellar structure and evolution. Springer-Verlag, 3rd ed. Wheeler J., 2012, Philos. Trans. R. Soc. London Ser. A, 370, 774–799.

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Supernovae It was suggested as early as 1934 that the origin of cosmic rays is the explosion of massive stars at the end of their life cycle. More quantitative considerations were developed later.[5] Stellar explosions, the sudden disruption of an entire star at the end of its lifetime, are called supernovae. They are relatively rare events, occurring on average one to three times per century in a galaxy of the size of the Milky Way. They are observed with powerful telescopes and the collected data are essentially the intensity and the spectrum of the light versus time. The luminosity is typically 1 to 10 × 109 times brighter than the sun, often as bright as the host galaxy of the exploding star. More precisely, the rate of radiated energy is approximately 1034 –1037 J/s. These catastrophic events produce typically 1043 J of total radiant energy, and, according to recent evaluations, some 100 times more energy is carried away in the flux of neutrinos, most of of which are liberated in a violent short burst (a few tenths of a second) at the beginning of the explosion of massive stars (neutrinos are light and very energetic particles that are not easy to detect because they interact weakly with matter). In general the luminous emission reaches a peak in a few weeks, and then the luminosity fades over the course of months or years, and remnants can be observed during hundreds and thousands years after the explosion of the star. Remnants are materials that are ejected by the explosion at very high velocity (as much as few tenths of the speed of light) into the interstellar medium. This medium is a very low density hydrogen gas with temperature ranging from 10 K to 104 K and number density ranging from 106 molecules per cm3 to less than one per cm3 , to be compared with 1019 per cm3 for a gas at ordinary conditions on earth. According to (1.2.3), the order of magnitude of the mean free path in the interstellar medium ranges from 103 to 109 km, still relatively small on the length scale of one light-year, ≈ 1013 km, but not on the scale of the size of a star, typically few 103 km for a white dwarf and 107 km for the external envelope of a highmass star. The freely expanding stellar material acts as a spherical piston, and, far away from the initial radius of the star, it takes the form of a strong blast wave (100 times faster than the speed of sound) that compresses and heats the interstellar medium up to 107 –108 K. The radius of remnants reaches typically 10 light-years. The theoretical study of the structure of these astrophysical blast waves has a long history, reported in the literature.[6] They are more complicated than the blast waves observed on earth and will not be discussed in this book. Supernovae play an important role in the universe by enriching interstellar structures with higher mass elements. They are considered as standard candles by astronomers, useful for measuring distance (using the luminosity) and expansion velocity (using the Doppler shift). Supernovae are classified into two groups based on the presence or not of Balmer series hydrogen lines in their spectra at maximum brightness. Those without hydrogen lines are classified as type I (SNI), and those with hydrogen lines are classified as type II (SNII). Type I supernovae (SNI) must have burned out, or ejected, their external shell of hydrogen [5] [6]

Colgate S., Johnson M., 1960, Phys. Rev. Lett., 5, 235–238. Truelove J., McKee C., 1999, Astrophys. J., 120, 299–326.

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General Considerations

before exploding. Type I supernovae (SNI) are brighter than type II (SNII) and decay faster. There are also different subgroups of SNI and SNII that are not worth discussing here. Astrophysics is an observational science. The current understanding of supernovae is poor. More data can now be obtained from modern astronomical instruments, especially concerning the neutrino outburst at the very beginning of the explosion (just before the emission of visible light). These data are needed to improve our understanding. This could be the case during this century if, by chance, a sufficient number of supernovae are observed in nearby galaxies, as was the case in 1987 for the famous SN 1987A in the Large Magellanic Cloud, visible to naked eye (approximately 166 000 light-years from earth; by comparison, the diameter of the Milky Way is 100 000 light-years). Few supernovae have exploded nearby; the last one in our galaxy was observed by Kepler in 1604. Most of the hundreds of the present-day observations concern old supernovae far beyond the Milky Way. Without the possibility to perform experiments on earth, the current understanding of star explosion is based on sophisticated numerical simulations. Due to a huge range of scales, direct numerical simulations cannot be performed and empirical subgrid scale models have to be introduced (turbulent nuclear combustion, transport phenomena). Because of their sensitivity to the models that control the star progenitor (initial condition), the numerical results are questionable. Despite the ever-increasing complexity of the advanced models in nuclear physics[1,2,3] (and an ever-increasing rate of publication), the key mechanisms of star explosion have not yet been identified. The understanding has not progressed much during the last 50 years. Two types of star explosions are currently retained: a sudden reignition of nuclear fusion in the form of a thermonuclear explosion, and a gravitational collapse-induced explosion. The first is retained for less massive progenitors, such as white dwarf stars (SNI), while the second concerns the iron core of massive stars (SNII). Both mechanisms involve waves of the same nature as those presented in this book for earth conditions (flames and/or detonations for SNI, shock waves for SNII) but with scales of energy, length and time that differ by many orders of magnitude. Thermonuclear Explosion (SNI) It has been known since 1931 that the equilibrium between the pressure of a relativistic electron-degenerate gas and the gravitational force is no longer stable when the mass increases above the Chandrasekhar mass, ≈ 1.4 M , and a gravitational collapse occurs.[4] This concerns only matter sufficiently dense that the highly degenerated electron gas is ultrarelativistic[5] with an adiabatic index γ = 4/3; see (13.2.45). The hydrostatic equilibrium is stable for γ > 4/3; see Sections 7.2 and 7.3.

[1] [2] [3] [4] [5]

Bethe H., 1990, Rev. Mod. Phys., 62(4), 801–866. Janka H.T., et al., 2007, Phys. Rep., 442, 38–74. Burrows A., 2013, Rev. Mod. Phys., 85, 245–261. Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed. Landau L., Lifchitz E., 1982, Statistical physics. Part I. Oxford: Pergamon Press, 3rd ed.

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Carbon-oxygen white dwarfs are small dense stars, ρ ≈ 109 –1011 kg/m3 . Their radius is few thousands kilometres and their mass cannot be larger than the Chandrasekhar mass since the degenerate electron gas is ultrarelativistic at high electron density ne > 1036 electrons/m3 ; see Section 13.2.4. White dwarfs can accumulate material from a stellar companion (binary star system) and acquire a mass near to the critical Chandrasekhar mass. They then begin to contract and quickly explode. The nature of the explosion is subject to debate. A first scenario is the following. As a result of contraction-induced heating, carbon fusion is ignited at the centre and stops the contraction. The nuclear combustion then takes the form of a turbulent flame that is assumed to transit quickly to detonation. Due to the Rayleigh–Taylor instability, a flame that is ignited at the centre and propagates outwards is expected to be strongly unstable. Unfortunately the deflagration-to-detonation transition (DDT) in such conditions is an open question, as it is also the case on earth for flames expanding freely in open space; see Section 4.3.5. Another scenario, more popular today, is the spontaneous initiation of a detonation similar to the one described in Section 4.3.4 for ordinary combustion. None of these unsteady combustion processes of ignition and/or transition to detonation evoked to explain SNI explosions is clear. There is not even a consistent evaluation of the corresponding orders of magnitude of time and length scales. However, detonations seem to reproduce the characteristics of the light emitted by ejecta more correctly than flames. The structure of unsteady planar detonation waves, sustained by the nuclear burning of homogeneous mixtures of carbon–oxygen at the temperature and pressure encountered in SNI, has been investigated numerically.[6] The results yield typical detonation velocities ≈ 104 km/s in agreement with (1.2.9). According to (1.2.10), the detonation thickness varies strongly with the Neumann temperature because of the thermal sensitivity of the reaction rate. For nuclear fusion–sustained detonations the effect is spectacular: the thickness of the detonation varies by many orders of magnitude with the location inside the star,[7] from few centimetres at high density, ≈ 1012 kg/m3 , to 104 km at density below 1010 kg/m3 , the latter being larger than the typical size of a white dwarf. As developed throughout this book, the initiation, quenching and the various mechanisms of instability of flames and detonations, as well as the transition from flame to detonation, are relatively well understood in terrestrial conditions, even though they still require further investigation, especially for the latter topic. Combustion knowledge results from long-standing confrontations of analytical studies of simplified models and well-controlled experiments. The situation is different in astrophysics. The mechanisms of SNI thermonuclear explosion have not yet been deciphered. They will not be discussed in the rest of this book. Core-Collapse Scenario for Supernovae (SNII) The current status of the core-collapse scenario for high-mass stars is the result of more than half a century of numerical investigations. It can be summarised as follows.[3,8] [6] [7] [8]

Dominguez I., Khokhlov A., 2011, Astrophys. J., 730, 87–102. Khokhlov A., 1993, Astrophys. J., 419, 200–206. Woosley S., Janka H., 2005, https://arxiv.org/abs/astro-ph/0601261, 1–11.

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36

General Considerations

High-mass stars pass through successive stages of fusion, starting with the lightest elements, hydrogen into helium, and ending with the the heaviest ones involved in nuclear fusion, namely silicon into iron. The reason for the successive character of the fusion steps, H→He, He→ C & O, C→ Ne & Mg, ..., O & Mg→Si, Si→Fe, is that each successive step requires a higher temperature and density than the previous one: the first one, H→He, becomes significant for temperature of a few 107 K and density of a few thousand kg/m3 , while O & Mg→Si requires T ≈ 1.9 × 109 K, ρ ≈ 8.8 × 109 kg/m3 , and the last one, Si→Fe, T ≈ 3.3 × 109 K, ρ ≈ 4.8 × 1010 kg/m3 . The numerical results[1,2,3] highlight an important mechanism of nuclear physics in the conditions of high-mass stars: the neutronisation of nuclear matter by electron capture by free protons or nuclei along with the emission of a neutrino. This production of neutrinos is an endothermic process that absorbs ≈ 0.78 MeV per neutron (essentially the difference between the rest mass of a neutron and the couple proton plus electron), accompanied by the emission of a neutrino. There are other mechanisms leading to neutrino production and emission. At temperatures above 109 K, the production of electron–positron pairs by gamma rays increases rapidly (a positron is the antimatter version of an electron, predicted by Dirac in 1928 and observed in experiments shortly after). At temperatures above 1011 K, an electron–positron pair can annihilate with the production of a pair of neutrinos instead of photons.[2] Neutrino production increases strongly with increasing density and temperature. However, for densities below 1014 kg/m3 the neutrinos, unlike photons, escape easily from the star and produce an important cooling that significantly reduces the heating by gravitation-induced compression. For densities above 1015 kg/m3 , neutrinos are trapped and participate in the local thermal equilibrium.[2] These mechanisms are difficult to evaluate quantitatively but play an important role in the energy balance and in the equation of state of nuclear matter. When the fuel of a given fusion step runs out, the release rate of nuclear energy becomes insufficient to balance energy loss by radiation and/or neutrinos. The star then begins to contract; temperature and density both increase so that the ashes can be eventually ignited. The next fusion step then produces energy release at a sufficient rate to balance the losses and stop the contraction. These successive steps proceed up to the last one, namely the production of iron, if the mass of the star is sufficient to generate the required temperature and density at its centre. Otherwise the chain of fusion steps is stopped to form, for example, a carbon–oxygen white dwarf. Also, the reaction rate of each step is very different. For example, in a 15-solar-mass star, the duration of the first one, H→He, is typically 11 million years and decreases regularly in the next steps; it is only 2 million years for the second one, 2000 years for the third and only 2 weeks for the last one.[4]

[1] [2] [3] [4]

Burrows A., 2013, Rev. Mod. Phys., 85, 245–261. Bethe H., 1990, Rev. Mod. Phys., 62(4), 801–866. Janka H.T., et al., 2007, Phys. Rep., 442, 38–74. Woosley S., Janka H., 2005, https://arxiv.org/abs/astro-ph/0601261, 1–11.

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The current view is as follows.[1,2,3,4,5] Before the explosion of a high-mass star, the progenitor model has an onion-skin structure, delimited by an external shell of residual hydrogen surrounding successive layers of increasingly heavier elements, at increasingly higher temperature and the pressure. The main characteristic of a high-mass star at the end of its lifetime is the existence of a dense central core (ρ  1011 kg/m3 ) of iron and neighbouring elements. There is no further fusion step to counteract contraction. The mass of the iron core increases because it is fed by the surrounding shell burning silicon into iron. The electrons of the core are strongly degenerated and ultrarelativistic (γ = 4/3). This is not the case in the outer layers (γ > 4/3) that are less dense and not subject to an eventual gravitational collapse; see Section 7.3. For a constant adiabatic index γ = 4/3 the mass of a spherical structure in stable equilibrium must be smaller than an upper bound, called the Chandrasekhar mass; see Sections 7.2 and 7.3. A catastrophic event occurs when the mass of the iron core reaches the Chandrasekhar mass. The core collapses suddenly with a maximum infall velocity as high as few tenths of the speed of light. Its size decreases from a few thousand kilometres to a few tens of kilometres in a tenth of a second, or even less. The response time of the stable outer shells being much larger (hours), the collapsing core is decoupled from the outer shells.[1] Before this event, the mass of the γ = 4/3 core increases slowly and the onion-skin structure of the progenitor star is in quasi-steady state. The response of the whole star to the sudden core collapse deserves further study. To the best of our knowledge no stability analysis of the whole star has been performed; only semi-phenomenological analyses are available.[6] Nevertheless, the case with a uniform value of the polytropic index γ is more or less well understood; see Sections 7.2 and 7.3. Analytical studies have been performed[7] for the marginal case γ = 4/3 showing the existence and the stability of a homologous solution for the collapsing core presented in Section 7.3.2. During the collapse, the density of the iron core increases abruptly and reaches ρ ≈ 4– 5 × 1017 kg/m3 in a few tens of seconds, typically twice the density of atomic nuclei. This dense and neutron-rich sphere emits a huge flux of neutrinos during a short time, releasing an energy equivalent to about 10% of the rest mass (a few 1046 J) in less than a second. If it does not collapse into a black hole, it eventually settles down as a 10-km-radius neutron star. A strong neutrino burst is the signature of a catastrophic core collapse. A first difficulty in the theoretical analyses is the inhomogeneity of the steady state. Due to gravitation and to entropy increase through reactions and dissipative transfers of energy, the thermodynamic conditions (temperature, density) and the properties of the matter (polytropic index, γ , and equation of state) vary with the radius. Unfortunately, this is also the case across the collapsing core, essentially because new phenomena of nuclear physics appear as the density increases. The greatest difficulty is to explain how the SNII explosion involving typically 1044 J of kinetic energy can be produced and how it can eject such an important fraction of the mass [5] [6] [7]

Cooperstein J., Baron E., 1990, In A. Petschek, ed., Supernovae, chap. 9, 213–266, Springer-Verlag. Zeldovich Y., Novikov I., 1971, Stars and relativity. Dover. Goldreich P., Weber S., 1980, Astrophys. J., 238, 991–997.

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38

General Considerations

of the star, with such a high velocity, into the interstellar medium. The situation is currently not clear and the SNII explosion is not understood. According to recent analyses, nuclear burning of the external layers might contribute at most to ≈ 10% of the blast energy, so that core-collapse supernovae must be gravitationally powered.[1] In the current view, neutrinos are thought to play an essential role through energy transfers, as is discussed a few lines below. It is worth first recalling the situation before advanced nuclear physics was introduced into the model. In the 1950s and 1960s it was noticed,[2] in the simplest hydrodynamical calculations of gravitational core collapse in spherical geometry, that a strong outwardspropagating shock wave is quickly formed. The shock appears suddenly near to the sonic point, enclosing about half the mass of the collapsing core, not far from the radius where the infall velocity is maximum, delimiting roughly the inner region where the velocity profile is homologous (infall velocity proportional to the radius); see Section 7.3. As a result, behind the shock wave, the flow velocity of the shocked material is reversed and flows outwards. The rest of the inner part of the core is suddenly halted and set almost at rest. The material outside the shocked sphere still flows inwards at a supersonic velocity and is both compressed and reversed when crossing the shock. This abrupt transition is called a rebound. In some conditions, the numerical calculations showed that the mass flow rate across the shock is sufficiently large for the shock to escape from the collapsing core. It can then propagate through the external shells and reach the exterior of the star. This scenario is called the prompt shock. As the external surface of the star is approached, the density decreases to zero as a power law due to radiative losses[3] (the external shell is transparent to radiation). According to an analytical solution[2] (self-similar solution of the second kind), the temperature and the velocity of the shocked gas diverges when the shock wave emerges at the surface of the star. This phenomenon is due to a divergence of the energy per unit mass, and can explain the acceleration to a tremendously high speed of the particles ejected into the interstellar medium. There is no analytical solution of the rebound. In contrast to what is generally said in the literature, the rebound is not necessarily related to incompressibility of matter at high density. It is also produced when the compressibility does not change much with the density; see Section 7.3. In contrast to simple waves in quasi-planar compressible flows recalled in Sections 15.3.1–15.3.4, the quasi-spontaneous formation of the shock during gravitational collapse in spherical geometry is poorly understood. Simple models with a ‘pressure versus density’ law introduced into Euler’s equations deserve further study. When the most advanced models of nuclear physics at high density, ρ > 1010 kg/m3 , and temperature T > 109 K are introduced into the energy equation (insofar as the physics is known in such extreme conditions), the numerical simulations do not reproduce the explosion.[1,4,5] The rebound is still produced but the prompt shock scenario does not work [1] [2] [3] [4] [5]

Burrows A., 2013, Rev. Mod. Phys., 85, 245–261. Zeldovich Y., Raizer Y., 1967, Physics of shock waves and high-temperature hydrodynamic phenomena II. Academic Press. Zeldovich Y., Raizer Y., 1966, Physics of shock waves and high-temperature hydrodynamic phenomena I. Academic Press. Bethe H., 1990, Rev. Mod. Phys., 62(4), 801–866. Janka H.T., et al., 2007, Phys. Rep., 442, 38–74.

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39

because the shock cannot escape from the collapsing core.[4] The strength of the shock is substantially weakened by the copious energy loss due to both escaping neutrinos and the dissociation of heavy nuclei[4] (mainly Fe) into nucleons (≈ 8.8 Mev per nucleon). A few milliseconds after its formation the shock takes the form of an accretion surface of discontinuity, delimiting a small sphere of very hot and dense material quasi at rest (a hot neutron-rich core called a proto-neutron star) that is accreting mass at a few tenths of solar mass per second. This inner dense part of the core is further compressed by gravitation, emitting neutrinos. In the numerical simulations[1,6] the shock appears typically at about 10 km from the centre and its radius does not grow beyond 100–200 km. The objective of current investigations is to find a mechanism that uses a small part of the tremendous energy emitted in the form of neutrino flux, sufficient to turn the core collapse and imploding matter into an explosion of the star. The main part of this neutrino flux is emitted from the so-called neutrino sphere, located at density about 1014 kg/m3 , at a radius smaller than the stalled accretion shock. The most popular mechanism for a neutrinodriven explosion is absorption of neutrinos, heating the material. This requires a good knowledge of the transport, emission, absorption and re-emission of neutrinos. Because of the lack of experimental data to validate the theoretical predictions in nuclear physics, the corresponding laws are not well known. In spherical geometry, two ways to take advantage of a hypothetical neutrino-heating mechanism have been envisaged, either through a local revival of the shock wave[5] or through a parametric analysis of the nonexistence and/or of the global stability of the quasi-steady state solution involving the stalled accretion shock and the flow modified by the flux of neutrinos.[1,7] Other ideas investigated concern multidimensional instabilities. Modest successes have been obtained so far. Back to Gravitational Collapse After 40 years of intensive investigation, the current status of the theory of core-collapse explosion of supernovae is somewhat confusing. Knowledge could be improved by following an approach similar to that successful in the theory of combustion. In the same way as for the transient regimes of ordinary flames and detonations studied in this book, a drastic simplification of the complexity might be useful (necessary?) to decipher the explosion of stars. In that respect, parametric studies of simplified models, identifying the critical parameters for the transition from ‘prompt shock to stalled shock’ would be welcome. An eventual transition to explosion in the onion-skin structure of the external shells, subsequent to a sudden core collapse, could also be investigated without invoking a prompt shock scenario. Attention could also be paid to an eventual transition in the combustion regime of silicon into iron (DDT?). Even though the nuclear energy available is too small (smaller than the kinetic energy of the ejecta) to postulate such phenomena as the only mechanisms exploding the star, they could play an important role in the whole process. The difficult problems in nuclear physics involving neutrinos will not be addressed in this book, in which only simple models are considered. The classical hydrodynamical [6] [7]

Woosley S., Janka H., 2005, https://arxiv.org/abs/astro-ph/0601261, 1–11. Keshet U., Balberg S., 2012, Phys. Rev. Lett., 108, 251101.

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40

General Considerations

results, at the root of gravitational collapse, which have been well established for a long time, are presented in Sections 7.1 and 7.2. More recent results concerning the dynamics are presented in Section 7.3. The physics of ultrarelativistic degenerated electrons, responsible for the critical value 4/3 of the adiabatic index γ , is recalled in Section 13.2.4. An illustration of the sensitivity of the evolution of the shock wave to the equation of state[1] is given in Section 7.3.3 where a nonintuitive result is presented with an ultrasimplified model: the so-called rebound and the prompt shock scenario are easily produced by a soft equation of state having only a small variation of the compressibility with density. In contrast, the prompt shock scenario is not observed with the rebound produced by a stiff equation of state in which the compressibility of the dense material (the nuclear matter at the centre of the core) is negligible compared with that of the less dense material at the periphery. Moreover a mechanism of neutrino-driven explosion, based on a thermoacoustic instability, is suggested in Section 7.3.3.

1.3.3 Inertial Confinement Fusion Another example of thermal waves in extreme conditions is the ablation front in ICF.[2] The notion of ablation front relies on the steepness of the temperature gradient created by the nonlinear temperature dependence of electron heat conductivity in fully ionised plasma.[3] The control of the hydrodynamic instabilities of the ablation front is a major challenge facing ICF.[2] Context In January 1951 Ulam and Teller developed the principle of a radiation-driven nuclear explosion. On 31 October 1952 a successful test of a nuclear weapon (H-bomb) with a yield of 10.4 megatons (of high explosive) was carried out on Enewetak Atoll. With the development of pulsed high-power lasers after the 1960s (10 kJ with a pulse duration of few nanoseconds at 1 μm wavelength in 1980), the idea of laser-driven implosion of small cryogenic fuel spheres filled with a mixture of deuterium and tritium (D-T) was developed with the objective of achieving the necessary 103 - to 104 -fold compression and the temperature of a few 107 K required to ignite a nuclear fusion reaction. A brief summary of the D-T reaction is recalled in Section 14.3.3. In this process, called inertial confinement fusion (ICF), the mechanism to confine the matter is mass inertia. The initial objective was the production of (nonweapon) energy through micro-explosions. However, the more recent programs in France and the United States are more weapons oriented. In 2014 the focus of ICF experiments is on understanding thermonuclear ignition under the conditions found within the nuclear weapons, rather than production of energy. The objective is mainly to provide experimental data for the validation of numerical simulation codes that are used in the development of newer generations of nuclear weapons. The National Ignition [1] [2] [3]

Swesty F., et al., 1994, Astrophys. J., 425, 195–204. Atzeni S., Meyer-Ter-Vehn J., 2004, The physics of inertial fusion. Clarendon Press–Oxford Science Publications, 1st ed. Spitzer L.J., 1962, Physics of fully ionized plasmas. New York: Wiley Interscience, 2nd ed.

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41

Facility (NIF), completed in 2009 in Livermore, California, aims to deliver a peak flash of ultraviolet light of more than 106 J and 1014 W of energy and power into the target chamber. The total amount of energy deposited in the fuel is a few 105 J and the laser is driven by more than 108 J of electricity! In France an ICF facility, the Laser Megajoule (LMJ), built near Bordeaux and equivalent to the NIF, has been officially operational since 2014. Both the NIF and the LMJ use indirect drive. In this approach the target is surrounded by a metal cylinder which is irradiated by the laser beams that are focused inside the cylinder. The inner side of the cylinder is heated to a hot plasma, which radiates X-rays that are then absorbed by the target surface. This is more efficient than direct absorption of laser light by the small fuel sphere. At the end of 2013, experiments at the NIF are reported showing a production of typically 5 × 1015 neutrons generated per shot. The nuclear energy released is slightly more than the laser energy absorbed by the target. However, this is still a long way from ignition. Conditions for Ignition For obvious reasons, the nuclear energy released from a single micro-explosion in a laboratory experiment has to be limited to typically a few 108 J, representing the complete burn of no more than 10−3 g of a D-T mixture. This corresponds to the energy released by approximately 80 kg of high explosive, but the blast of such a nuclear explosion is considerably weaker because the neutrons and photons carry relatively little momentum per unit energy. An extreme level of compression has to be achieved in order to ignite such a small D-T mass. As a minimum, the mean free path, defined in (1.2.3), must be much smaller than the size of the fuel sample R. In reality the criterion is more elaborate because the confinement time in ICF is very short, typically a few 10−10 s, and the reaction, which is expected to be ignited at the centre, must have sufficient time to propagate throughout the compressed fuel. A rough estimate is obtained by comparing the reaction rate (14.3.2), n σnr v, σnr v ≈ 5 × 10−16 cm3 /s at T = 5 × 108 K, and the strong expansion rate that develops immediately after the maximum of compression. The later can be evaluated as a/R, where R is the size of the compressed fuel sample and a the speed of sound. Requiring that the reaction rate be larger than the expansion rate yields a lower bound for the quantity nR, where n is the number density. For the maximum cross section of the D-T nuclear reaction shown in Fig. 14.5, namely for T ≈ 108 K, this imposes that ρR be greater than few g/cm2 , where ρ is the fuel density. For 1 mg of D-T this corresponds to a density of a few 102 g/cm3 , which is typically a density ratio of 103 relative to the initial density of the cryogenic D-T sample in solid state (ρ ≈ 0.2 g/cm3 ). Such a high compression is achieved by ICF; see below. It is worth stressing that, as illustrated by quasi-isobaric flame ignition in Sections 2.4.2–2.4.5 and by the initiation mechanisms of detonations in Section 4.3, the full determination of the conditions of ignition in ICF is a difficult problem. It is not addressed in this book. Attention will be limited to the hydrodynamical instability of the ablation front during the implosion of the irradiated capsule. Various other scientific problems involved in ICF are discussed in the specialised literature.[2,4] [4]

Lindl J., 1998, Inertial confinement fusion. Springer.

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42

General Considerations

Figure 1.11 The principles of inertial confinement fusion.

Orders of Magnitude in Direct Drive The order of magnitude of parameters and performances that are expected for a directdrive laser target[1,2] are the following. The target consists of a hollow shell capsule of 1.7×10−3 g cryogenic D-T mixture initially in solid state (T ≈ 17.9 K, ρ ≈ 0.22 g/cm3 at atmospheric pressure). The radius is slightly less than 2 mm and the thickness is typically 0.3 mm. The capsule is filled with D-T vapour (ρ ≈ 0.3×10−3 g/cm3 at atmospheric pressure) and is coated with an outer plastic layer of thickness slightly less than 0.4 mm; see Fig. 1.11. Irradiation leads to surface ablation; the plastic layer is vaporised, ionised and expelled outwards. Due to momentum conservation, the capsule moves inwards at high velocity (≈ 4×105 m/s) under the ablation pressure (≈ 108 bar). The fuel of the imploding shell comes suddenly to rest near the centre. The pressure and density suddenly increase to a few 1011 bar and ρ ≈ 400 g/cm3 , while the temperature of the compressed D-T mixture inside the shell reaches about 10 keV (≈ 108 K). Ignition occurs in the central hot spot and propagates outwards into the compressed fuel of the shell in the form of a very fast thermal wave (5×106 m/s), driven by electron heat conduction and the escaping αparticles. The fusion energy, typically few 108 J, is liberated in less than 10−10 s, before the subsequent expansion of the hot matter quenches the nuclear reaction. Inertial confinement is the best method on earth to reach the high density by compression (density ratio) that is required for thermonuclear ignition. By comparison, the compression across a planar shock front in a gas is limited to a factor typically between 4 and 6 in an ideal gas, ρN /ρu < (γ + 1)/(γ − 1); see (4.2.14). Moreover, according to the first law of thermodynamics (13.1.3), de = Tds − pd(1/ρ); isentropic compression (ds = 0) minimises the mechanical energy −pd(1/ρ) > 0 needed to produce a given increase of internal energy de. It follows that the formation of strong shocks should be avoided and temporal shaping of the driving pulse helps to optimise a quasi-isentropic compression. The spherical shape of the capsule is optimal to reach an implosion velocity sufficiently high to achieve the high density [1] [2]

Atzeni S., Meyer-Ter-Vehn J., 2004, The physics of inertial fusion. Clarendon Press–Oxford Science Publications, 1st ed. Bychkov V., et al., 2015, Prog. Energy Combust. Sci., 47, 32–59.

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43

required for ignition. The high temperature is attained through compression of the D-T mixture inside the imploding shell. Hydrodynamic Instability At the peak irradiation density (typically 1015 W/cm2 ), the ablation pressure and the implosion velocity reach 108 bar and a few 105 m/s in few tens of nanoseconds. Such a large acceleration produces a very strong Rayleigh–Taylor instability (2.2.14) and (2.2.15) of the ablation front since the density of the vaporised matter is much smaller than that of the plastic layer. This tends to deform the imploding spherical shell and hinders the formation of the central hot spot by breaking the spherical symmetry. One of the major challenges facing ICF is to control this instability. Fortunately, heat transfer by electron heat conduction, involved in the ablation mechanism, stabilises the small wavelength disturbances reducing the linear growth substantially. This scientific problem has a long history, starting in the 1970s.[3,4,5,6] A better understanding and appropriate description of the hydrodynamic instability of the ablation front in ICF were developed later by comparison with flame theory,[2,7,8,9] ending up with a simple model equation to describe the nonlinear dynamics of the unstable ablation front.[10] This topic is presented in Chapter 6, and the details of the analyses are given in Chapter 11.

1.4 Appendix: Physical Constants and Conversion of Units G = 6.674 08 × 10−11 N m2 /kg2

Gravitational constant

c = 2.997 92 × 108 m/sec

Velocity of light in a vacuum

−34

J sec

−23

J/K

Boltzmann’s constant

N = 6.022 14 × 10 /mole

Avogadro’s number

h¯ = h/2π = 1.054 57 × 10 kB = 1.380 65 × 10

Planck’s constant

23

qe = 1.602 18 × 10−19 Coulomb kg

Electron rest mass

−27

kg

Neutron rest mass

me = 9.109 38 × 10 mn = 1.674 93 × 10

[3] [4] [5] [6] [7] [8] [9] [10]

Electronic charge

−31

Bodner S., 1974, Phys. Rev. Lett., 33, 761–764. Takabe H., et al., 1983, Phys. Fluids, 26(2299-2307). Sanz J., 1994, Phys. Rev. Lett., 73, 2700–2703. Bychkov V., et al., 1994, Phys. Plasmas, 1, 2976–2986. Clavin P., Masse L., 2004, Phys. Plasmas, 11, 690–705. Clavin P., Almarcha C., 2005, C. R. M´ecanique, 333(379-388). Sanz J., et al., 2006, Phys. Plasmas, 13, 102702. Almarcha C., et al., 2007, J. Fluid Mech., 579, 481–492.

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General Considerations

mp = 1.672 62 × 10−27 kg −11

aB = 5.291 77 × 10

Proton rest mass Bohr radius (h¯ 2 /me qe )

m

1 bar = 105 Pa = 106 erg/cm3 1 J = 107 erg 1 eV = 1.602 18 × 10−19 J 1 eV/kB = 11 604.5 K

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2 Laminar Premixed Flames

Nomenclature Dimensional Quantities a A cp cv d da dL D DT D e E E˙ t g g k km kB l ls L m p qm q˙ v q˙ γ

Description Mean molecular velocity. Sound speed Amplitude of perturbation Specific heat at constant pressure Specific heat at constant volume Thickness Acoustic displacement Scale of laminar flame thickness Molecular diffusivity Thermal diffusivity Normal propagation velocity Energy Activation energy Rate of energy transfer per unit surface Acceleration of gravity Periodic acoustic acceleration Wavenumber Marginal wavenumber Boltzmann’s constant Spatial scale. Curvilinear coordinate Length proportional to Q˙ s Length of tube, burner or space scale Mass flux Pressure Heat of combustion per unit mass Heat release rate per unit volume (γ − 1)˙qv

S.I. Units m s−1 m J K−1 kg−1 J K−1 kg−1 m m m m2 s−1 m2 s−1 m s−1 J ≡ kg (m/s)2 J mole−1 J s−1 m−2 m s−2 m s−2 m−1 m−1 J K−1 m Q˙ s /(4πρDcp (Tb − Tu )) m kg m−2 s−1 Pa J kg−1 ≡ (m/s)2 J m−3 s−1 J m−3 s−1 45

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46

Q˙ s r r R Rf s S t T u ua ut u Ub UL Un v w wθ x, y x α δ δs2  ρ σ τ τL τh ω

Laminar Premixed Flames

Power of point heat source Radius. Radial coordinate Vector of coordinates Radius, radial coordinate, radius of curvature Radius of reaction zone Arclength Surface area Time Temperature Mean velocity. Longitudinal velocity Acoustic displacement velocity Transverse velocity Velocity vector. Flow field Laminar flame speed w.r.t. burnt gas Laminar flame speed Local (stretched) flame speed Velocity fluctuation. Transverse velocity Transverse velocity Tangential velocity Coordinates Radial coordinate attached to spherical front Position of front A characteristic thickness Element of flame area Wavelength Density (Complex) growth rate of perturbation A characteristic time Transit time through laminar flame Hydrodynamic time scale, = (UL k)−1 Angular frequency

J s−1 m m m m m m2 s K m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m m m m m2 m kg m−3 s−1 s s s s−1

Nondimensional Quantities and Abbreviations A A B B cst. C D

σ/(UL k) (ρu − ρb )/(ρu + ρb )

Dimensionless coefficient Atwood number Dimensionless coefficient, see (2.2.7) Dimensionless coefficient, see (2.5.22) Constant Dimensionless coefficient, see (2.5.22) Dimensionless coefficient, see (2.5.22)

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Laminar Premixed Flames

F G Go h H (.) l Le Fr M M M2 Mc Mu n N O(.) t Tr ua w X Y Z Z β γ   η θ ϑ κ κm λ  τ τ υb φ φ φ

Geometrical gain factor Reduced periodic acoustic acceleration, see (2.5.13) Scaled inverse Froude number squared (ρb /ρu )(1/Fr2 ) Amplitude of forcing in Mathieu’s equation Darrieus–Landau operator ≡ Hilbert transform of space derivative Lewis dependence in Markstein number β(Le − 1) Lewis number DT /D √ Froude number UL / gdL Mach number u/a First Markstein number Second Markstein number Mc − Mu Markstein number for curvature of front Markstein number for stretch of front Local unit vector normal to front Dimensionless coefficient, see (2.5.14) Of the order of Local unit vector tangential to front Acoustic transfer function for velocity, see (2.5.12) Reduced acoustic displacement velocity ua /UL = ωda /UL Reduced reaction rate, see (2.1.5) (1 − θ )e−β(1−θ) Nondimensional flame kernel parameter R˙ f /UL )(Rf /dL ) Mass fraction of reactant Mixture fraction Admittance function, see (2.5.9) Zeldovich number of flames E(Tb − Tu )/kB Tb2 Ratio of specific heats cp /cv A small parameter, generally dL /λ (In Section 2.7) Small expansion ratio ρu /ρb − 1 (In Section 2.7) Reduced transverse coordinate km y Reduced temperature (T − Tu )/(Tb − Tu ) Stoichiometric coefficient. Order of reaction Dimensionless wavenumber kdL Dimensionless wavenumber at stability limit Ratio of thermal conductivity to value in fresh mixture Reduced acoustic frequency ωτh = (ωτL )/(kdL ) (In Section 2.7) Reduced time, see (2.7.7) (ρu /ρb − 1)km UL t/2 Reduced acoustic time t/τh = t UL k Gas expansion ratio ρu /ρb (= Tb /Tu ) Equivalence ratio (In Section 2.6) Potential of fluid flow (In Section 2.7) Reduced amplitude, see (2.7.7) 2km α/(ρu /ρb − 1)

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47

48

ψ ψ   DL RT ZFK

Laminar Premixed Flames

Reduced mass fraction of limiting species (In Section 2.6) Stream function of fluid flow Frequency in Mathieu’s equation (In Section 2.6) Vorticity Darrieus–Landau Rayleigh–Taylor Zeldovich and Frank-Kamenetskii

Y/Yu

Superscripts, Subscripts and Math Accents a∗ a− a+ ax aa ab ac adiff a(e) af ains aL am an ao ar as at aT au ax aθ a a˜ aˆ

Critical value Value on upstream side of front Value on downstreamside of front Derivative w.r.t. x Acoustic Burnt gas Critical value. Curvature. Cutoff value Diffusion Excitation or perturbation flow (vortex, turbulence, . . . ) At flame front Instability Laminar flame Marginal value Normal component Unperturbed value Reaction rate Surface. Stretch Transverse component Thermal Unburnt gas At position x Tangential component Average value or unperturbed value Fourier component of a, a(y, t) = a˜ (t)eiky Amplitude of linear harmonic perturbation, a(y, t) = aˆ eiky+σ t

2.1 Main Characteristics The theory of flames is performed in the framework of the quasi-isobaric approximation, a generalisation of the incompressible approximation in hydrodynamics.

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2.1 Main Characteristics

49

2.1.1 The Quasi-Isobaric Approximation The speed of sound in a perfect gas is, according to Laplace, a = (γ p/ρ)1/2 , where ρ and γ are, respectively, the density and the ratio of specific heats, γ ≡ cp /cv , of the gas. This expression shows that p/ρ is of the order of a2 and thus of the order of the internal thermal energy per unit mass eT . The order of magnitude of the pressure is thus p ≈ ρa2 . The above remark can be used to show that the full equations of reactive fluid mechanics (15.1.33)–(15.1.35) can be greatly simplified under the following two conditions: 1. The flow is very subsonic (the Mach number of the flow is much smaller than unity, M ≡ u/a 1). 2. The evolution of the flow is slow on the time scale of acoustics. The first condition implies that the kinetic energy of the fluid can be neglected compared with the internal energy, u2 /2 eT ≈ a2 . The second condition is satisfied when the mechanisms controlling the dynamics of the flow (such as the motion of flame fronts) change slowly on the time scale needed by an acoustic wave to cross the domain. It follows that ∂/∂t ≈ u.∇ a |∇| . Euler’s equation (15.1.18) then shows that the relative changes in pressure are only of order M 2 when the relative changes in velocity are of order unity, δu/u = O(1), ρ(u.∇)u ≈ −∇p ⇒ p ≈ ρa

2

δp ≈ ρuδu,

⇒ δp/p ≈ M 2 .

(2.1.1)

With these conditions, the variations of pressure and kinetic energy, in a flow with combustion, are negligible compared with the heat release, and the equation for energy conservation reduces to a purely thermal equation; see (15.2.3). For a planar flame, the fresh and burnt gas temperatures are then related by the simple thermal balance of (1.2.6), cp (Tb − Tu ) = qm , Moreover, the relative fluctuations of pressure, temperature and density are related by the equation of state, which for a perfect gas can be written δp/p = δρ/ρ + δT/T. In the region of the flame front, the relative changes in temperature are at least of order unity. Since the changes in pressure are only of order M 2 , they can be neglected and the density varies as the inverse of the temperature (ρT is constant). Away from the flame front, the relative changes in density can be neglected in the conservation equations for the mass and momentum, as in hydrodynamics for incompressible flows. These approximations lead to the system of quasi-isobaric conservation equations (15.2.2)– (15.2.5). The terminology is somewhat misleading since flame wrinkling produces largescale flows with associated pressure gradients that must be retained in the equations for the conservation of momentum; see Section 2.2. However these gradients are sufficiently small that they can be neglected in the relation between density and temperature in the flame front, leading to a simple expression for the density ratio between the fresh and burnt gas ρu /ρb ≈ Tb /Tu , ρb < ρu .

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Figure 2.1 Gas velocities in reference frame of flame.

Figure 2.2 Profiles of temperature and mole fraction of the main species in a lean methane–air flame (equivalence ratio 0.65). The molar fractions of reactants and stable products are shown by solid lines, the main intermediate species are shown by dotted lines.

2.1.2 The Structure of Planar Flames By definition, the laminar flame speed UL is the speed at which a planar flame propagates into fresh gas at rest. In the reference frame of the flame – see Fig. 2.1 – UL is the speed at which gas enters the stationary front. By mass conservation the burnt gas leaves the flame front at a speed Ub greater than UL , ρu UL = ρb Ub ,

Ub /UL ≈ Tb /Tu , ≈ 4−9.

(2.1.2)

As already mentioned in Section 1.2.3, the flame thickness and transit time are greater than those predicted by dimensional analysis; see (2.1.9). Their orders of magnitude under standard conditions are typically a few tenths of a millimetre and a few tenths of a millisecond, respectively. The equations governing the flame structure will be derived in Section 8.1; see equations (8.1.1)–(8.1.2). They form a system of second-order nonlinear ordinary differential equations, completed by boundary conditions (8.1.3). The mass flux, m ≡ ρu UL , in these equations is an eigenvalue of the problem. The large number of these equations reflects the complexity of the chemical kinetics that govern the structure of real flames; see Chapter 5. Numerical solutions to these equations can now be obtained with good accuracy using computer codes. A typical example of the structure of a lean methane-air flame is shown in Fig. 2.2. Fig. 2.3 shows the evolution of methane–air flame speed with

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Figure 2.3 Methane flame speeds from experimental measurements and from numerical simulations using two different detailed chemical schemes. The values 0.6 and 1.6 of the equivalence ratio correspond to the lean and rich flammability limits, respectively.

equivalence ratio (see Section 1.2.2) at ambient temperature and pressure. The symbols are experimental data[1,2,3] and the full lines are numerically calculated flame speeds using the detailed chemical kinetics of the widely used GRIMech-3 scheme[4] (53 species and 325 reactions) and also a more detailed scheme proposed by A.A Konnov[5] with 127 species and 1207 individual reactions. Flammability Limits The main features can be described by reduced kinetic models obtained via a more or less systematic reduction of the complete system. Some examples of such reductions will be presented in Sections 5.3 and 5.4. Two mechanisms lead to self-amplification of the reaction rate and control the propagation of flames. The first is the temperature sensitivity, and the second concerns autocatalytic reactions (chain branching) that are in competition with recombination reactions (chain breaking); see Section 5.2. The latter are essential to describe ignition and also flammability limits beyond which planar flames cannot propagate. As already mentioned in Section 1.2.2, the chain-branching and chain-breaking competition leads to a crossover temperature T ∗ , a purely chemical kinetic property, defined in Section 5.2.2; see (5.2.7). The production of intermediate radicals, which are indispensable for the complete release of energy, is not possible below T ∗ , as explained in more detail in Chapter 5. The flammability limits are then defined by the compositions for which the flame temperature (given by the energy balance between initial and final states; see Section 14.2.3) is equal to the cut-off temperature, Tb = T ∗ . An ordinary planar flame cannot propagate when Tb < T ∗ . For flame ignition, or flame propagation close to the flammability [1] [2] [3] [4] [5]

Van Maaren A., et al., 1994, Combust. Sci. Technol., 96(4-6), 327–344. Bosschaart K., De Goey L., 2004, Combust. Flame, 136, 264–269. Vagelopoulos C., Egolfopoulos F., 1998, Proc. Comb. Inst., 27, 513–519. Smith G., et al., 2000, www.me.berkeley.edu/gri mech/. Konnov A., 2009, Combust. Flame, 156, 2093–2105.

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limits, the effect of chain branching may be qualitatively well represented by two-step chain-branching models (5.1.1) or (5.2.7), derived in Chapter 5. The interest in simplified models is that they can be solved analytically; see Section 8.5.5. Flammability limits have been investigated experimentally for many years, but there is little agreement among investigators.[1] Because of natural convection, the results obtained on earth are very sensitive to the experimental conditions. Well-defined flammability limits are more easily studied under micro-gravity conditions.[2,3] We discuss the flammability limits and ignition problems in Section 2.4.2. The Zeldovich and Frank-Kamenetskii Analysis Far from the flammability limits, the propagation is mainly controlled by the coupling between heat diffusion and the rate at which heat is released. The thermal propagation of flames is then well described by a single irreversible exothermic reaction of order unity, controlled by an Arrhenius law (1.2.2) with a large energy of activation, R → P + Q,

(2.1.3)

where R represents the reactive species (fuel or oxidant) that limits the reaction rate, P represents the combustion products and Q the net energy release per mole of R consumed. It is supposed that the other reactive species (fuel or oxidant) is sufficiently in excess that its concentration varies little and its consumption does not affect the reaction rate. The first analytical solution for the thermal propagation of premixed flames was obtained in 1938 by Zeldovich and Frank-Kamenetskii[4] using the simple one-step mode l (2.1.3) in the limit of a large activation energy, E/(kb T) → ∞; see (1.2.2). Introducing the reduced temperature, θ (x) = (T(x) − Tu )/(Tb − Tu ), the reduced mass fraction ψ of species R (fresh mixture at x = −∞: θ = 0, ψ = 1, burnt gas at x = +∞: θ = 1, ψ = 0), the unknown mass flux m ≡ ρu UL , the reaction time at the temperature of the burnt gas τrb = τr (Tb ), the reduced activation energy (Zeldovich number), β ≡ E(Tb − Tu )/kB Tb2 , the thermal diffusivity of the mixture DT and the molecular diffusivity D of species R, the flame structure is, according to (8.1.1)–(8.1.3), controlled by two nonlinear equations, m

d2 θ dθ ρb −β(1−θ) − ρDT 2 = ψe , dx τrb dx

m

d2 ψ dψ ρb − ρD 2 = − ψe−β(1−θ) , dx τrb dx

(2.1.4)

where, for simplicity, ρDT and ρD are supposed constant and where only the dominant term has been retained in the expression of the reaction rate; see (8.2.13) and (8.2.14). The reaction rate (the right-hand side) contains the reduced mass fraction ψ and denotes a first-order reaction rate (ϑ = 1) that decreases to zero when the reactants is consumed (ψ = 0). The asymptotic solution of this model, called the ZFK solution in the following, is given in Section 8.2. Here we give just a sketch of the asymptotic method that is so [1] [2] [3] [4]

Ronney P.D., Wachman H., 1985, Combust. Flame, 62, 107–119. Ronney P., 1985, Combust. Flame, 62, 121–133. Ronney P., 1990, Combust. Flame, 82, 1–14. Zeldovich Y., Frank-Kamenetskii D., 1938, Acta Phys. Chim., IX, 341–350.

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useful in combustion theory. In the simplest case, for DT = D (Lewis number Le ≡ DT /D unity), ψ = (1 − θ ), the problem is reduced to solving a single equation for the reduced temperature θ (x) that describes the balance between heat release and thermal transfer by advection and diffusion, m

d2 θ ρb  dθ − ρDT 2 = w (θ ), dx τrb dx

w ≈ (1 − θ )e−β(1−θ) ,

(2.1.5)

with the two boundary conditions at x = ±∞ x = −∞: θ = 0,

x = +∞: θ = 1.

(2.1.6)

The unknown mass flux m ≡ ρu UL is an eigenvalue of the problem; all the other parameters are known. In the limit β → ∞, the nonlinear term, w (θ ), is concentrated in a thin reaction zone on the hot side of the flame, where θ − 1 is small; see Fig. 2.4. The asymptotic analyses in Section 8.2.2 for Le = 1 and in Section 8.2.3 for Le = 1 can be summarised as follows. First, the variation of temperature occurs essentially in an inert preheated zone (outer zone w = 0) where the diffusive flux of energy by conduction, namely the second term in the left-hand side of (2.1.5), is balanced by the advection represented by the first term in the left-hand side of (2.1.5), mdθ /dx − ρDT d2 θ /dx2 ≈ 0, θ = emx/ρDT ; the origin x = 0 is the location of the thin reaction zone. The heat flux at the hot side of the preheated zone is obtained from dθ/dx|x=0 = m/ρDT . Second, advection is negligible in the thin reaction layer (inner zone) where (1 − θ ) = /β is small ( is of order unity). This is because, using a reduced longitudinal coordinate, the second derivative with respect to space is larger than the first derivative, so that the diffusive heat flux is balanced by the reaction rate, −ρb DTb d2 θ /dx2 ≈ (ρb /τrb )w (θ  ), where the notation DTb ≡ DT (Tb ) has been introduced. Expressed in terms of , the downstream and upstream boundaries of the Reaction zone

Reduced temperature

Preheat zone

Unburnt gas

Burnt gas

Figure 2.4 Structure of a premixed flame.

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thin reaction zone are, respectively,  = 0 and  → ∞, the latter being valid in the limit β → ∞. The solution in the reaction zone     d2  1 1 d 2 1 − e , ≈ Xe−X dX, (2.1.7) β → ∞: DTb 2 ≈ τrb 2 dx DTb τrb 0 dx leads to the expression for the heat flux leaving the reaction zone to warm up the upstream gas,  dθ DTb (2.1.8) β  1: lim DTb = 2 2 . →∞ dx β τrb Matching this inner heat flux and the outer flux at the boundary separating the two zones, DTb dθ/dx|x=0 = Ub , yields the eigenvalue m ≡ ρu UL = ρb Ub ,  τb ≡ β 2 τrb /2, (2.1.9) β  1: Ub = DTb /τb , where the relation ρDT = ρb DTb has been used. This gives the flame velocity (ρu UL = ρb Ub ) when the reduced activation energy is large. In mathematical terms the result is the leading order of an asymptotic expansion in the limit β → ∞. The laminar flame speed UL and thickness dL of the laminar flame have the same form as that given by dimensional analysis of Section 1.2.3, but the relevant characteristic time, τb , is greater than the reaction time at the burnt gas temperature, τrb , by a large numerical factor β 2 /2 ≈ 50. More general expressions (8.2.27) and (8.2.39) are obtained in the same way for an order of reaction and a Lewis number different from unity, ϑ = 1 and Le = 1. The corresponding values, UL and Ub , are in agreement with experimental data. The characteristic time τb is a measure of the transit time of a particle of gas through the flame structure. However as in (1.2.8) – see also (8.2.27) and (8.2.28) – it is often convenient to introduce a slightly different transit time, τL , τL ≡

dL ρu = τb , UL ρb

(2.1.10)

and the flame thickness, dL , is given by   dL ≡ DTb τb = DTu τL = DTu /UL .

(2.1.11)

Since the gas velocity increases from UL to Ub through the flame thickness, the real transit time of a gas particle lies between τb and τL . Asymptotic analysis also reveals a structural difference between flame fronts and the reaction–diffusion waves encountered in biophysical systems in which the reaction term w (θ ) is not a stiff function of θ ; see Section 8.3. The analysis also shows that weak thermal losses or flame stretching by a velocity gradient can lead to a brutal extinction of flame fronts at a finite propagation speed; see Sections 8.5.1 and 8.5.2. Flames become very sensitive to thermal losses near the flammability limits where the flame quenching occurs for very small thermal losses that have no noticeable effects far from these limits. Extinction and ignition will be presented in Chapter 9. All these

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phenomena are direct consequences of the strong sensitivity of the exothermic reaction rate to temperature. In the 1990s, the technique of asymptotic analysis was generalised to more complicated chemical kinetic schemes, such as reduced schemes containing three or four steps modelling the combustion of hydrogen and hydrocarbons.[1,2] The asymptotic analyses in Sections 8.4 and 8.5 of the two-step chain-branching model (5.1.1) are sufficient to represent these phenomena, at least qualitatively.

2.1.3 Instabilities of Flame Fronts Freely propagating planar flames are generally unstable with respect to geometrical deformations and are observed only in special cases; see Section 2.2.4. Nevertheless, the structure of a planar front is a useful reference case to study the structure of weakly wrinkled flames. In this chapter we will restrict the presentation to physical insights into the instabilities. The technical details of the calculations will be given later in Chapter 10. The instabilities of flame fronts can be grouped into three families: • Hydrodynamic instabilities • Thermo-diffusive instabilities • Thermo-acoustic instabilities. The first two are intrinsic instabilities of flame fronts, eventually in the presence of gravity. The third type results from a coupling between a flame and the acoustic waves generated by an unsteady flame propagating in a cavity. The hydrodynamic instability and some mechanisms of thermo-acoustic instability have their origin in the change of density through the flame front. They concern flame wrinkling on a wavelength greater than the flame thickness. The hydrodynamic instability was described independently by G. Darrieus[3] and by L. Landau[4] early in the 1940s using a model in which the flame front is treated as a hydrodynamic discontinuity of zero thickness. The effects of finite flame thickness, discussed in Section 2.2.3, were studied much later. They play an essential role controlling the form and dynamics of flame fronts. Because of the lack of noninvasive tools to observe real flames, these phenomena were not the subject of quantitative experimental studies until the 1980s.[5,6,7] The difference of density between the cold (heavy) fresh mixture and the hot (light) burnt gas makes the flame front sensitive to the action of gravity. When a flame front propagates upwards in a quiescent gas, the flame takes a shape similar to that of an air bubble in a liquid, as shown in Fig. 2.11. For the case of a flame propagating downwards in a wide

[1] [2] [3] [4] [5] [6] [7]

Peters N., Williams F., 1987, Combust. Flame, 68(2), 185–207. Peters N., 1997, Prog. Astronaut. Aeronaut., 173, 73–91. Darrieus G., 1938, in La technique moderne. Landau L., 1944, Acta Phys. Chim., 19, 77–85. Searby G., et al., 1983, Phys. Rev. Lett., 51(16), 1450–1453. Searby G., Quinard J., 1990, Combust. Flame, 82(3-4), 298–311. Clanet C., Searby G., 1998, Phys. Rev. Lett., 80(17), 3867–3870.

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tube with a propagation speed sufficient to make gravity play only a minor role (large √ Froude number, Fr = UL / gdL ), a number of cellular bulges appear on the flame front, separated by ridges, as seen in left-hand photo of fig. 2.19. This is a manifestation of the hydrodynamic instability. In rich mixtures of heavy hydrocarbon fuels, the cellular aspect of the front is more pronounced, with smaller structures, as seen in Fig. 2.19b. These structures can sometimes have a chaotic aspect, referred to as self-turbulence. This behaviour is the signature of another type of instability, whose origin lies in the diffusion of heat and species and which adds itself into the hydrodynamic instability. The effects of diffusion inside the flame structure are systematically stabilising for perturbations with a wavelength smaller than the flame thickness. However, for intermediate wavelengths, the competition between the diffusive transport of heat and the diffusive transport of molecular species can create an instability known as the ‘thermo-diffusive’ instability (preferential diffusion). A physical insight into this instability mechanism is given in Section 2.4.1. A detailed analysis is performed in Section 10.2 in the framework of the thermo-diffusive model that neglects hydrodynamic effects generated by the change in density. The difference in density between the fresh and burnt gas not only is responsible for the Darrieus–Landau (DL) instability, it also makes the flame respond to unsteady accelerations, such as that produced by an acoustic wave. In the presence of the latter, the behaviour of the flame changes according to the frequency and amplitude of the wave. The response can be extremely different, leading either to a stabilisation of the planar front or to a violent thermo-acoustic instability such as that shown in Fig. 2.27d. This phenomenon was first observed in the nineteenth century by Mallard and Le Chatelier[1] for flames propagating in tubes. The mechanism of instability was understood much later.[2,3] It is presented in Section 2.5. Most of the topics concerning the dynamics of freely propagating flames in premixed gases are presented in this book. Except for the combustion of a vortex tube presented in Section 3.1.5, the dynamics of swirling flames is not considered. Much work remains to be carried out on this topic for its important practical value[4] (swirling burners). 2.2 Hydrodynamic Instability Hydrodynamic instabilities can be explained by simple mechanisms involving the difference in density between the fresh and burnt gas. 2.2.1 Gas Expansion We will first examine the flow associated with a planar flame.

[1] [2] [3] [4]

Mallard E., Le Chatelier H., 1883, Annales des Mines, Paris, Series 8(4), 296–378. Markstein G., 1964, Nonsteady flame propagation. New York: Pergamon. Searby G., Rochwerger D., 1991, J. Fluid Mech., 231, 529–543. Candel S., et al., 2012, C. R. M´ecanique, 340(758-768).

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Piston Effect In premixed flame propagation, the density of the burnt gas is smaller than that of the initial fresh gas, ρb /ρu ≈ Tu /Tb 1. This difference of density is responsible for the socalled piston effect in which a flame propagating away from the closed end of a tube pushes the fresh gas ahead of the flame. In a coordinate system attached to the flame front – see Fig. 2.1 – the fresh gas enters the flame with a normal velocity UL . Mass conservation equation (2.1.2) imposes that ρu UL = ρb Ub , so the burnt gas leaves the flame with a velocity Ub greater than UL and equal to (ρu /ρb )UL . Since ρT is almost constant in subsonic premixed flames, the velocity ratio is also approximately equal to the temperature ratio; see (2.1.2). This ratio is typically in the range of 5 to 9. Ub and UL are the normal components of the gas velocities relative to the flame front. For the case of a flame propagating from the open end towards the closed end of a tube, initially filled with fresh reactive mixture, the fresh gas is at rest and the flame propagates at speed UL in the reference frame of the tube. The burnt gas escapes in the opposite direction with speed Ub − UL ; see Fig. 1.1. However, if the flame propagates from the closed end of the tube towards the open end, as shown in Fig. 2.5, the burnt gas is at rest. The flame now moves down the tube with speed Ub and pushes the fresh gas forwards at speed Ub − UL , so that the relative speed of the flame with respect to the fresh gas is UL . The flame thus behaves as a semi-permeable piston. This effect plays an important role in the transition from deflagration to detonation and in explosive accidents. Contrary to a description in which compressible phenomena are ignored, the transition from fresh gas at rest to fresh gas in motion cannot be global and instantaneous. It occurs progressively through the propagation of acoustic waves (a compressible phenomenon), which lead, in a finite time, to the formation of a supersonic shock wave that moves quickly away from the flame. The theory of formation and propagation of shock waves is recalled in Section 15.3. If the tube is closed at both ends, the mean pressure in the vessel increases, and flame propagation is necessarily accompanied by an overall increase in pressure and density. These schematic descriptions of flame propagation are oversimplified because planar flame fronts are unstable. In practice, flame fronts are curved and/or turbulent. However, the overall phenomenology is not substantially modified when observed at scales much greater than that of the wrinkling, provided that the laminar flame velocity, UL , is replaced by the global velocity of the wrinkled flame; see (3.1.11) and Fig. 3.1.

Figure 2.5 In the reference frame of a tube closed at the burnt gas side, the flame moves with speed Ub and the fresh gas is pushed ahead with speed Ub − UL . The flame behaves as a semi-permeable piston.

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Weakly Wrinkled Flames The study of hydrodynamic instabilities is greatly simplified when the characteristic length of wrinkling of the flame and that of flow of fresh gas are much greater than the flame thickness, /dL  1. The flame can then be treated as a surface of hydrodynamic discontinuity at which the fresh gas is transformed into hot, less dense, burnt gas. The internal structure of the flame is only weakly modified compared with that of the planar flame. The same is true for the values of the local parameters. As a first approximation these small differences can be neglected and we can use the mass consumption, density jump, temperature jump and pressure jump of a planar flame. This approximation, valid in the limit dL / → 0, highlights hydrodynamic instabilities. However, it is necessary to include the effects of curvature (of the front and of the flow) to describe the form and the dynamics of unstable flame fronts. In the approximation of a weakly wrinkled flame these weak effects can be treated by a perturbation analysis of the planar front, presented in Section 10.3. Deflection of the Streamlines We first consider a steady planar front that is inclined with respect to a uniform flow, represented schematically in Fig. 2.6. This is the case, for instance, of the inclined front of the Bunsen flame in Fig. 1.2. In the reference frame of the flame, the incident gas flow can be decomposed into the normal component, equal to UL and a tangential component ut . By mass conservation, the normal component in the burnt gas is equal to Ub ; see (2.1.2). By conservation of momentum, the velocity of the tangential component of the flow, ut , is the same on both sides of the flame; see Section 15.1.6. As a consequence, the streamlines are deviated towards the normal to the flame. This mechanism is similar to the refraction of a light beam at the interface between two media in which the speed of light is different. 2.2.2 The Darrieus–Landau (DL) Instability If the inclined flame front is planar, the situation shown in Fig. 2.6 has translational invariance. However, if the flame front is wrinkled the situation is more complicated. Inclined flame front

Burnt gas

Fresh mixture

Figure 2.6 Deflection of streamlines through a flame front inclined with respect to the velocity of the fresh gas. The normal components of the fresh gas and burnt gas velocities are UL , and Ub > UL , respectively. The tangential component of the gas velocity, ut , is unchanged.

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Figure 2.7 Deviation of the streamlines through a wrinkled flame, in the reference frame of the stationary front.

The deviation of the streamlines varies along the flame front and induces a nonlocal hydrodynamic effect that modifies the flow field, both upstream and downstream, over a distance of the order of the wavelength of wrinkling. Instability Mechanism The nonlocality of perturbations to the flow is a characteristic of incompressible hydrodynamics. Through momentum conservation, a velocity gradient is always associated with a pressure gradient. In the limit of slow subsonic flows, the speed of propagation of pressure perturbations (speed of sound) can be treated as infinite; see the discussion in Section 2.1.1. In this limit and in the linear approximation, pressure perturbations are described by a Laplace equation – see (10.1.14) – and extend throughout the flow. Any local perturbation to the flow thus has an instantaneous effect on the whole flow field. The deviation of the streamlines at the wrinkled flame is shown in Fig. 2.7. Anticipating that, in the linear approximation, the velocity perturbations vanish at infinity – see equation (10.1.21) – the burnt gas velocity at points ‘A’ and ‘B’ must be respectively smaller and greater than Ub . In order to maintain the same velocity Ub relative to the flow, the flame front must move and will advance with respect the the flow at point ‘A’, and recede at point ‘B’. The flow perturbation induced by the presence of flame wrinkling thus tends to increase the amplitude of wrinkling and amplifies the initial deformation. Planar flames are thus unstable with respect to transverse amplitude perturbations. Linear Stability Analysis The detailed analytical study of this phenomenon is presented in Section 10.1. Generally speaking, the objective of a linear stability analysis is to describe the short-term evolution of a small initial perturbation of the front. If the amplitude increases, the front is unstable, and if the amplitude decreases, the front is stable. The analysis is performed in the approximation of vanishingly small amplitudes (linear approximation) and is limited to short times. It cannot provide information about the long-term behaviour of unstable fronts, for which it is necessary to perform a nonlinear study; see section 2.7.

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Three-dimensional effects do not introduce significant changes in the analysis of flame stability, which is generally performed in a two-dimensional geometry (in which the front is represented by a line) to simplify the presentation. It is convenient to take the spatiotemporal Fourier transform of the perturbed front, described by an equation of the form x = α(y, t), where α is the perturbation of the position of the front. In the linear approximation there is no coupling between modes, and it is sufficient to consider a harmonic perturbation ik.y + c.c. , α( y, t) = α(t)e ˜

(2.2.1)

where k is the given transverse wave vector (a real number) and c.c. stands for complex conjugate. In order to simplify the notation, in the following k will be used to denote the modulus of the wave vector, k = 2π/, where  is the wavelength of wrinkling and we will omit the complex conjugate. We look for a solution of the form α(t) ˜ = αe ˆ σ t,

(2.2.2)

where αˆ is the initial amplitude of wrinkling, αˆ ≡ α(t ˜ = 0), and the growth rate σ is, in general, a complex function of k. The problem is then to calculate σ (k). If Re(σ ) is positive (negative), the front is unstable (stable) and the amplitude of wrinkling grows (decreases) exponentially with time. If the imaginary part of σ is nonzero, then the perturbation oscillates with an amplitude that increases or decreases according to the sign of the real part. When Re(σ ) = 0 the stability has to be studied in a different way; the growth or the damping may involve a power law in time. This is the case, for example, in gaseous shock waves; see Section 12.1. Growth Rate of the DL Instability The essential results of flame stability analysis can be discussed on physical grounds without needing the detailed analysis of Section 10.1. In the absence of a characteristic length other than , dimensional analysis tells us that the linear growth rate of the instability, σDL , must be proportional to the product of the flame speed, UL , and the modulus of the wavenumber of wrinkling, k ≡ 2π/, σDL ≡ 1/τDL = AUL k.

(2.2.3)

The coefficient of proportionality, A, is a positive dimensionless number that is a function of the density ratio between the initial mixture and final products (fresh and burnt gases for flames); see below. Because of the inertia of the fluids, the amplitude of the perturbation is governed by a second-order differential equation     ρu dα˜ ρb d2 α˜ − + 2(UL k) − 1 (UL k)2 α˜ = 0 (2.2.4) 1+ ρu dt2 dt ρb (see (10.1.30)), with g(t) = 0. The first term in (2.2.4) arises from the effect of inertia, which appears also in the Rayleigh–Taylor instability (2.2.14), and the two other terms from

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the propagation of the front, UL = 0. Substituting α(t) ˜ = αe ˆ σ t gives the corresponding dispersion relation for σDL     ρb ρu 2 1+ σDL + 2(UL k)σDL − − 1 (UL k)2 = 0. (2.2.5) ρu ρb The initial mixture being more dense than the reaction products, ρu /ρb > 1, one of the roots is positive, so that flame fronts are systematically unstable. The other root is negative, describing a contribution that disappears quickly after the initial instant, t > 1/(UL k). To fix ideas, when the gas expansion ratio is very large, ρu /ρb  1, the coefficient A tends √ to A ≈ ρu /ρb ; see (10.1.32). In the opposite limit of weak gas expansion inertial effects are negligible, and Equation (2.2.4) reduces to a first-order differential equation,     ρu ρu dα˜ − − 1 1: 2 − 1 UL kα˜ ≈ 0, (2.2.6) ρb dt ρb corresponding to (2.2.3) with A ≈ (ρu − ρb )/(2ρb ). This describes an instability that is increasingly violent for small wavelengths and led Landau[1] to the erroneous conclusion that laminar flames could not exist and that propagating flames are necessarily turbulent with the size of the turbulent brush equal to the diameter of the tube. Nevertheless, he remarked that for a liquid fresh mixture, the pressure jump induced by surface tension will stabilise the wrinkling at sufficiently small wavelengths (large k). It turns out that a similar mechanism exists for gaseous flames;[2] see Sections 10.1.3 and 10.3.3.

2.2.3 Stabilisation at Small Wavelengths The analysis of DL ceases to be valid at small wavelengths when the scale of wrinkling becomes comparable to the flame thickness, dL . The curvature of the front produces transverse fluxes of energy and species. There are two types of induced fluxes: convective and diffusive. The former arise from the tangential component of the flow within the flame thickness, and the latter from the shift of the profiles of temperature and species concentration; see Fig. 2.8. The gradients of these fluxes modify the local internal structure of the wrinkled flame and the reaction rate, leading to a modification to the local flame speed. A simple representation of this phenomenon is given by the ZFK model. According to (2.1.4), the temperature and the mass fraction of the limiting species vary in opposite direction inside the preheated zone, as sketched in Fig. 2.8. When the flame front is locally convex towards the fresh gas, as in the lower part of Fig. 2.8, the transverse diffusive flux of heat (dotted dark grey arrows) removes heat and tends to cool the reaction zone and decrease the local flame speed. However, the transverse flux of fresh species (dotted light grey arrows) brings extra reactants into the reaction zone, with the opposite effect. The net result depends on the ratio of heat and species diffusivities (Lewis number, [1] [2]

Landau L., 1944, Acta Phys. Chim., 19, 77–85. Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237.

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Figure 2.8 Diffusive fluxes in a wrinkled flame. The solid dark grey and light grey arrows represent, respectively, the diffusive fluxes of heat and species in the direction of the local gradient. The dotted arrows show the transverse fluxes induced by the curvature.

Le ≡ DT /D), the chemical kinetics and the transverse convective fluxes. Since the direction of the transverse gradients depends on the sign of the curvature, the change in flame velocity also depends on the sign of the curvature. It can be expressed as a function of the local geometry of the flame front and gradients of the incident flow field; see Section 2.3 and Chapter 10 for a detailed analytical study. In the linear approximation, the amplitude of the modifications is proportional to (kdL )2 . When the wrinkling of the front occurs on a scale that is large compared with the flame thickness, these local modifications are small compared with hydrodynamic effects. However, when the wavelength of wrinkling is comparable to the flame thickness, the diffusive effects can dominate. The resulting change in growth rate can be calculated by a perturbation technique in which the solution for the weakly perturbed flame is developed to first order around the planar solution using kdL as the small parameter; see Sections 10.1 and 10.3. We begin with simple considerations. Thermal Relaxation Rate As first imagined by Markstein,[1] diffusive effects introduce a relaxation or growth rate proportional to the square of the wavenumber σdiff ≡ 1/τdiff = −BDT k2 ,

(2.2.7)

where DT is the thermal diffusion coefficient and B a dimensionless constant. This corresponds to a diffusion equation for the position of the flame front, x = α(y, t), ∂ 2α ∂α = BDT 2 . ∂t ∂y [1]

(2.2.8)

Markstein G., 1964, Nonsteady flame propagation. New York: Pergamon.

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Figure 2.9 Growth rate as a function of wavenumber for B > 0.

This equation effectively describes the linear evolution of a flame front in the limit of no change in density, ρu = ρb , A = 0 (no hydrodynamic effects); see (10.2.25). With this approximation, the constant of proportionality, B, is a function of the Lewis number, Le, and the chemical kinetics. If B is positive, Equation (2.2.7) describes a relaxation. If B is negative, a new instability, called the thermo-diffusive instability, appears at small wavelengths superposed on the hydrodynamic instability. We will come back to thermodiffusive instabilities in Sections 2.4 and 10.2. Assume for the moment B > 0. Marginal Wavenumber, a Heuristic Approach In the presence of the hydrodynamic instability, diffusive effects introduce a corrective term in the expression for the DL growth rate, which can obtained by a perturbation calculation for small kdL . This finite thickness effect can be found heuristically by combining Equations (2.2.3) and (2.2.7) and using the relation DT = UL dL . The linear growth rate, σ , of a wrinkle of wavelength 2π/k has the form kdL < 1:

σ = AUL k [1 − (B/A)kdL + · · · ] .



 σ = AUL k 1 − k/km + ··· ,

 dL km ≡ A/B.

(2.2.9) (2.2.10)

 has a simple meaning where A > 0 is given in (10.1.32). For B > 0, the quantity km  if (2.2.10) is extrapolated to k/km of order unity. It represents the upper bound for the wavenumber of unstable wrinkles. However, the perturbation analysis for kdL 1 does  . not guarantee the accuracy of the marginal wavelength given by 2π/km

Cells and Cusps The formation of cellular structures may be decomposed into two stages. The initial disturbance grows exponentially in time with the linear growth rate (2.2.10) on a time scale of order (kUL )−1 . The propagation of a front travelling with constant normal velocity produces a nonlinear geometrical deformation on the time scale (|α|k2 UL )−1 , which is longer than that of the linear growth for small amplitudes, |α|k 1. It is represented by

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Laminar Premixed Flames (b)

(a)

Figure 2.10 (a) Huygens’ construction for a wrinkled front propagating at constant normal velocity showing the formation of cusps. (b) Photograph of a 140 mm diameter lean methane flame, curved and wrinkled by the DL instability showing cusps. Notice also that the smallest cells have a size much greater than the flame thickness.

Huygens’ construction (see Fig. 2.10): the radius of curvature increases in regions where the front is convex towards the fresh gas and decreases for the contrary, leading to convex cells separated by abrupt cusps, shown in Fig. 2.10b. Similarly to the tip of the Bunsen burner, the sharpness of the cusps is controlled by the stabilising diffusive mechanisms. However, the mean size of the cells on these curved fronts is substantially greater than the most unstable wavelength, k∗ . This ‘abnormal’ stability of curved fronts will be discussed in Section 2.6. Linear Evolution Equation of the Flame Front A model equation combining the linear instability and the nonlinear geometrical effect was obtained for a small density contrast[1] where inertia of the gas is negligible; see Section 2.7.1. Inertia is important in the presence of external accelerations, such as gravity, or that induced by acoustic waves; see Section 2.5.5. The evolution equation for the flame front including both inertia and diffusive stabilisation at small length scales is obtained in the linear approximation by a perturbation analysis, limited to first order in small kdL . The results are summarised below, and the detailed analysis is presented in the second part of the book; see Section 10.3. The analysis leads to a linear equation of evolution, which is of second order in time, similar to (2.2.4), but containing additional corrective contributions of order kdL ; see Section 2.9.5 for a detailed equation. The simplest form representing qualitatively the phenomena may be written as an extension of (2.2.4),       ρu k ρb d2 α˜ dα˜ 2 − 1 − α˜ = 0, (2.2.11) 1+ + 2(U k) − 1 (U k) L L ρu dt2 dt ρb km [1]

Sivashinsky G., 1977, Acta Astronaut., 4, 1177–1206.

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where the expression for km in terms of the flame parameters is given in (2.9.54). The extra ˜ by αe ˆ σ t gives a term k/km stabilises the wrinkles with small wavelengths. Replacing α(t) quadratic equation for the growth rate σ :       k ρb ρu σ 2 + 2(UL k)σ − = 0. (2.2.12) 1+ − 1 (UL k)2 1 − ρu ρb km The first-order correction to the positive root for small kdL yields (2.2.10) with a coefficient   = 1/k . They are equal only for small gas expansion, for which the inertial (first) 1/km m term is negligible:       ρu dα˜ k ρu α˜ ≈ 0. (2.2.13) − − 1 1: 2 − 1 UL k 1 − ρb dt ρb km 2.2.4 Effect of Gravity A flame front shows new instabilities when accelerated. Rayleigh–Taylor Instability If a flame front propagates upwards in a vertical tube, the light burnt gas is underneath the heavier fresh gas and is subject to the Raleigh–Taylor (RT) instability, well known for an inert interface between a heavy fluid placed above a lighter fluid; see Sections 2.6.1 and 10.1.2. Let g be the acceleration of gravity. Assuming inviscid incompressible fluids, and neglecting surface tension effects, the linear evolution equation for the amplitude of a harmonic perturbation on an inert interface[2] is     ρb d2 α˜ ρb gkα˜ = 0; (2.2.14) 1+ − 1− ρu dt2 ρu see (10.1.31). The corresponding linear growth rate, 1/τRT , is  1 ρu − ρb = A gk, A ≡ , τRT ρu + ρb

(2.2.15)

where A > 0 and the Atwood number. It tends to unity in the limit of a high-density contrast, and to zero for no density contrast. The first term on the left-hand side of Equation (2.2.14) is the same inertial term as that in Equation (2.2.4). The second term is the buoyancy term of Archimedes. For the case of a flame, the linear equation of evolution for superposed DL and RT instabilities can be obtained in the limit dL / → 0 (no diffusive relaxation) by combining Equations (2.2.4) and (2.2.14):      ρu dα˜ ρb ρb d2 α˜ 2 − + 2(U k) − 1 k g + U k α˜ = 0. (2.2.16) 1+ L L ρu dt2 dt ρb ρu This equation is derived in the second part of the book; see (10.1.30). For flames propagating upwards, the term in square brackets is positive (g > 0). The growth rate of this [2]

Taylor G., 1950, Proc. R. Soc. London, A 201, 192–196.

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Figure 2.11 Photograph of a propane flame propagating upwards in a vertical Pyrex tube, 5 cm internal diameter.

instability increases with the magnitude of this term. The RT contribution to the instability (first term in the square bracket) dominates for small wavenumbers (large wavelengths). This explains why the shape of an ascending flame in a tube closely resembles that of an air bubble rising in a tube filled with water; see Fig. 2.11. As above, the effect of diffusive fluxes can be obtained by a perturbation calculation for kdL 1. This introduces corrective terms of order kdL in the coefficients of (2.2.16). The phenomenology can be understood by combining (2.2.11) and (2.2.16) to give an equation similar to (2.2.16) with the term in square brackets replaced by    k ρb , (2.2.17) g + UL2 k 1 − ρu km showing that small wavelengths are stabilised. The perturbation analysis performed in Section 10.3.4 and the calculation of km are tedious; see the expression for km in (2.9.54). Threshold of Instability for Downwards-Propagating Flames For a flame propagating downwards, the lighter fluid is above the heavy fluid. The effect of gravity on a passive interface (UL = 0) is to give rise to propagating waves, described by Equation (2.2.14) with g negative, (ρu − ρb )g < 0. For flames (UL = 0), gravity produces a stabilising effect at long wavelengths,   ρb d2 α˜ dα˜ 1+ (2.2.18) + 2(UL k) ρu dt2 dt      ρu ρb k α˜ = 0; − − 1 k − |g| + UL2 k 1 − ρb ρu km see (10.1.42). This equation is the simplest one that describes flame dynamics qualitatively well. However a quantitative comparison with experiments requires a more detailed equation, as for example, that presented in Sections 2.9.5 and 10.3.4. When the term in square brackets is positive, one of the roots of the equation for σ is positive and the flame front is unstable. When it is negative, both roots are negative, or complex with a negative real part,

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so the flame front is stable but can have damped travelling waves. The sign of this term thus controls the stability of the flame front. The parameter that measures the combined effect of gravity and flame propagation is the square of the Froude number Fr2 ≡

UL2 UL3 = . |g|dL |g|DTu

(2.2.19)

The range of unstable wavenumbers is delimited by the two roots κ− and κ+ (with κ ≡ kdL ) of the quadratic equation obtained by setting the square bracket in (2.2.18), denoted N, equal to zero:      κ ρb = 0, Go ≡ Fr−2 . (2.2.20) N ≡ −Go + κ 1 − κm ρu There is a critical value of the parameter Go for which the discriminant of (2.2.20) is zero and the two roots κ− and κ+ coincide: Go = Goc ≡ κm /4,

κ− = κ+ ≡ κc = κm /2.

(2.2.21)

This provides a relation between the critical flame speed ULc and the wavelength 2π/kc of the cellular structure appearing at the stability limit of flames propagating downwards (κc = kc dL ),  kc dL km ρb |g| Goc = . (2.2.22) , kc = , ULc = 2 2 2 ρu kc For slower flames, the value of Go is greater than the critical value, UL < ULc : Go > Goc ; the roots for κ in (2.2.20) are complex and the term in square brackets in (2.2.18) is negative for all values of the wavenumber k, so the flame is stable at all wavelengths. For faster flames, UL > ULc , (2.2.20) has two positive real roots, κ− and κ+ , and the planar flame front is unstable for wavelengths in the finite range (2π/κ+ )dL <  < (2π/κ− )dL . The range of unstable wavenumbers increases as the flame speed increases (Go decreases). This behaviour is shown schematically in Fig. 2.12. Theory and Experiments Experiments have been carried out with hydrocarbon–air and hydrogen–air flames sufficiently diluted in an inert gas (e.g. N2 ) to achieve the necessary low flame velocity.[1] The experimental results for the stability limits were compared with the theoretical results[2,3] corresponding to an extension of (2.2.18), obtained by a perturbation analysis of the ZFK model developed in Section 10.3; see also Section 2.9.5. In this equation the nondimensional marginal wavenumber, κm ≡ km dL , given by (2.9.54), involves the expansion coefficient (υb ≡ ρu /ρb = Tb /Tu ) and a dimensionless parameter, M, called the first Markstein number, that measures the strength of the stabilising diffusive mechanisms and whose [1] [2] [3]

Searby G., Quinard J., 1990, Combust. Flame, 82(3-4), 298–311. Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237. Clavin P., Garcia P., 1983, J. M´ec. Th´eor. Appl., 2(2), 245–263.

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d c b

Stable

a

Figure 2.12 Real part of the growth rate of perturbations on a downwards-propagating flame for different values of the Froude number. Curve ‘a’ is unconditionally stable. Curve ‘b’ is marginally stable for κ = κc . Curve ‘d’ is for zero gravity. Along the dotted line the growth rate is complex (damped propagating waves). (b)

(a)

Figure 2.13 (a) Slowly downwards-propagating flame, 10 cm diameter, with a Froude number sufficiently small for the flame to be stable at all wavelengths. (b) Downwards-propagating flame with a Froude number just above threshold. The flame is cellular, but flat on average. Courtesy of J. Quinard, IRPHE Marseilles.

physical interpretation is discussed below in Section 2.3.1. With this expression for km , the relation (2.2.22) between the critical wavelength and flame speed is approximately verified in the experiments. The experimental threshold of instability for downwards-propagating flames occurs for laminar flame speed in the range 7–11.5 cm/s, yielding, by comparison with the theoretical analysis, 2 < M < 4.5. Higher values of the critical flame velocity ≈ 15 cm/s compatible with M ≈ 5.7 are found for a rich mixture of hydrogen–air diluted with an inert gas. Fig. 2.13a shows a 10 cm diameter lean propane–air flame diluted with nitrogen and freely propagating downwards with a velocity of 10 cm/s. The front is planar, except in the boundary layers of the tube. Fig. 2.13b shows weak cellular structures developing on a slightly faster flame just above the instability threshold. The size of the cellar structures, 2π/km , is close to 1 cm. The experiment is not trivial since the critical flame speed is close to the extinction limit (see Section 5.2.1) and the thermal boundary layer must be well controlled both upstream and downstream of the front. Such freely propagating planar

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flames were stabilised in the laboratory frame only in the 1980s. They were of great help to study the dynamics of premixed flames.[1]

2.3 Flame Stretch and Markstein Numbers To summarise, the dynamics of a flame front are controlled not only by the pre-existing flow field, but also by two other distinct mechanisms: • The large-scale flow induced upstream and downstream of the front by the change in density of the gas traversing the flame • The local modification to the internal structure of the wrinkled flame, induced by diffusive and convective transverse transport of heat and species within the flame thickness. The latter is the topic of this section.

2.3.1 The First Markstein Number The study of the second mechanism can be generalised to structures of finite amplitude, provided that the radius of curvature of the front, and the spatial and temporal scales of the outer flow, remain much greater than the scales characterising the internal structure of the flame. Thanks to scale separation, the flame can then be treated as a hydrodynamic discontinuity with a weakly variable propagation speed. The mass flux, m, of fresh gas transformed into burnt gas (per unit time and unit surface) varies from one point to another on the flame front, according to the changes in the internal structure of the flame. When the flame is considered as a surface, the local flame speed (normal burning velocity) Un− − can be associated with the local mass flux of fresh mixture at the flame, m− f = ρu Un . It generalises the laminar flame speed of the planar flame. It is the normal component of the cold gas flow relative to the front Un− = u− n − Df ,

− with u− n ≡ nf .uf ,

(2.3.1)

where nf is the unit normal to the front, directed towards the burnt gas, at a point rf on − − − the flame front, u− f is the velocity of the fresh mixture u (r) at that point, uf = u (rf ), and Df is the normal velocity of the flame front at rf , seen in the laboratory frame; see Fig. 2.14. Since the normal vector nf is oriented towards the burnt gas, then Df < 0 when the front moves towards the fresh mixture as, for example, when the fresh mixture is at rest u− = 0, and, by definition, Un− > 0. Since the local change in the internal structure of the flame is weak, the change in flame speed is small. A perturbation analysis may be carried out by using  ≡ dL / as a small parameter, where  is the wavelength of wrinkling. The first result for a large

[1]

Searby G., et al., 1983, Phys. Rev. Lett., 51(16), 1450–1453.

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Figure 2.14 Schematic cut through a wrinkled flame front.

amplitude of wrinkling was obtained with the ZFK model.[1,2] The detailed analysis is presented in Section 10.3.2. At the leading order, the perturbation analysis provides an expression for the local flame speed as a function of the geometry of the front and the gradients of the flow field. The modification to the local flame speed is found to be proportional to the rate of stretch of an element of flame surface 1/τs whose expression is given in (2.3.8), (Un− − UL )/UL = −M(τL /τs ),

(2.3.2)

where τL ≡ dL /UL is the transit time (1.2.8), and τL /τs is of order . The constant of proportionality, M, is a dimensionless number of order unity, called the first Markstein number in tribute to Markstein’s work in the 1960s.[3] Its analytic expression was first obtained for small amplitudes of wrinkling;[4] see (10.3.36). The simplicity of this expression arises from the simplicity of the one-step ZFK model; see Section 2.3.3 for a more general expression. Before discussing further this topic, it is worth recalling the stretch rate of a surface. 2.3.2 Stretch Rate, Strain and Curvature of a Flame The stretch rate of a surface, 1/τs , is a local quantity, defined positive if an element of surface area, δ 2 s, increases in time and negative on the contrary, 1 1 dδ 2 s . = 2 τs δ s dt Different ways of obtaining an expression for the stretch rate of a flame can be found in the literature;[5] see also Section 2.9.1. Here we give a simple geometrical derivation.

[1] [2] [3] [4] [5]

Matalon M., Matkowsky B., 1982, J. Fluid Mech., 124, 239–259. Clavin P., Joulin G., 1983, J. Phys. Lett., 44, L–1– L–12. Markstein G., 1964, Nonsteady flame propagation. New York: Pergamon. Clavin P., Williams F., 1982, J. Fluid Mech., 116, 251–282. Buckmaster J., 1979, Acta Astronaut., 6, 741–769.

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Passive Surface We start from the stretch of a passive surface in a nonuniform flow, as presented in textbooks on fluid mechanics.[6] The velocity u(e) (rf ) of a point rf of the surface is, by definition, that of the flow, drf /dt = u(e) (rf ), where the subscript f means that the value is taken on the front at rf . The rate of stretch is related to the gradients of the flow field. As will be shown below, it can be written quite generally in terms of the normal to the surface, nf , and the gradients of the flow field u(e) (r) at the point rf , 1 1 d 2 δ s = ∇.u(e) |f − nf .∇u(e) |f .nf . ≡ 2 τs δ s dt

(2.3.3)

Only the symmetric part of the tensor ∇u(e) (deformation rate) contributes in (2.3.3), since the antisymmetric part (rotation) yields zero. This expression is easily obtained by first noting that the rate of change of volume of an element δ 3 r centred on the point rf is, by definition, the divergence of the flow field u(e) (r) (see equation (15.1.5)), 1 d 3 δ r = ∇.u(e) |f . δ 3 r dt

(2.3.4)

The volume element centred on a point rf of the surface is δ 3 r = δ 2 s δζ , where ζ is the curvilinear coordinate normal to the front. It follows that 1 d 3 1 d 2 1 d δ r= 2 δ s+ δζ , 3 δζ dt δ r dt δ s dt where the rate of change along the normal is

 dδζ /dt = nf . u(e) (rf + δζ nf ) − u(e) (rf ) .

(2.3.5)

A Taylor expansion of u(e) (rf + δζ nf ):



u(e) rf + δζ nf ≈ u(e) rf + δζ nf .∇u(e) ,

(2.3.6)

then yields (2.3.3). A more detailed (and pedestrian) demonstration of (2.3.3) using fixed Cartesian coordinates is given in appendix; see Section 2.9.1. Propagating Fronts Flame fronts are not a passive interfaces. By definition, the normal to the front moves through the fluid with a local velocity −Un− nf with respect to the flow of unburnt gas, whose velocity is u− ; see Equation (2.3.1). This motion gives rise to an additional contribution to the stretch rate. In the spirit of a perturbation analysis we neglect the small difference between curved and planar flame velocities in the expression for the stretch rate, Un− ≈ UL . If we suppose that a point on the flame front moves in the tangential direction [6]

Batchelor G., 1967, An introduction to fluid dynamics. Cambridge University Press.

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with the tangential velocity of the flow, the velocity of a point on the flame front, seen in the laboratory frame of reference, is drf /dt = u(e) (rf ),

with u(e) (rf ) = u− f − UL nf ,

(2.3.7)

since nf .u(e) (rf ) = Df ; see (2.3.1) for Un− ≈ UL and t.u(e) (rf ) = t.u− f (t.nf = 0), where t is any unit tangent to the front. The stretch rate of a flame front is found by introducing expression (2.3.7) in (2.3.3), 1/τs = −UL ∇.nf + ∇.u− |f − nf .∇u− |f .nf ,

(2.3.8)

where we have used the fact that nf is a unit vector, nf .nf = 1, so nf .∇n|f .nf = 0. According to (2.9.2), the quantity −∇.nf is the mean radius of curvature of the front, −∇.nf = 1/R ≡ (1/R1 + 1/R2 ), where the principal radii of curvature R1 and R2 are defined as positive when the burnt gas forms a locally convex volume; see Fig. 2.14. Noting also that ∇.u− |f = 0 for an incompressible flow (unburnt gas), the stretch rate of the flame surface is 1/τs = UL /R − nf .∇u− |f .nf .

(2.3.9)

Expression (2.3.9) is the dominant order of the stretch rate of a flame surface, to be used in (2.3.2). The stretch rate in (2.3.9) has two contributions: the first term on the right-hand side is the stretch rate arising from curvature of the propagating front; it disappears if the front does not propagate or if it is planar. The second term represents flame stretch induced by gradients of the upstream flow; it is generally referred to as strain rate. Pure Strain Strain is the only stretch acting on a planar flame stabilised in a stagnation point flow against a wall, as shown in Fig. 2.15. The corresponding change in laminar flame speed, given by (2.3.2), arises only from gradients in the flow field. The strain rate nf .∇u|f .nf is the gradient, along the normal to the flame, of the component of flow velocity along the same normal. In incompressible two-dimensional flow it is also equal to the tangential gradient of the tangential velocity. If the normal to the flame front is directed along one of the Cartesian axes in the laboratory frame, for instance along the x axis as in Fig. 2.15, then the flame

Figure 2.15 Schematic representation of a flame stabilised in a stagnation point flow.

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Figure 2.16 Schematic representation of a spherical flame propagating inwards.

stretch reduces simply to 1/τs = −du− /dx, where u− is the x-component of the velocity of the incompressible cold flow, du− /dx = −dv− /dy, where du− /dx  < 0 for the− stagnation  − flame of Fig. 2.15. From (2.3.2) the flame velocity is Un /UL = 1 + M τL (du /dx) . Pure Curvature The first term on the right-hand side of (2.3.9) represents the stretch in the absence of unburnt gas flow. It is a pure effect of flame curvature. It is the only term present for a spherical flame propagating inwards into quiescent fresh gas, u− = 0, dR/dt = −Un− ; see Fig. 2.16. The mean curvature is −2/Rf and from (2.3.2) the corresponding flame velocity is Un− /UL = 1 + 2M(dL /Rf ) . 2.3.3 The Second Markstein Number These two types of stretch, strain and curvature, are different in nature. It is then quite surprising to find that, according to (2.3.2), the two types of stretch affect the local flame speed in exactly the same way. One would have expected two different Markstein numbers[1] for curvature and strain, Mc = Mu : (Un− − UL ) dL = −Mc + Mu τL nf .∇u− |f .nf . UL R

(2.3.10)

The equality Mc = Mu is not a general result, but only a consequence of the simplicity of the ZFK model. A numerical simulation in spherical geometry, using detailed chemical kinetics representative of methane combustion – see Section 5.4 – shows that these two Markstein numbers are different.[2] A recent analytic study[3] of flames sustained by a multiple-step chemistry has confirmed that Mc = Mu . The analysis is presented in Section 10.3.6. The erroneous assumption that the total flame stretch is the only geometrical scalar controlling the normal burning velocity of flames dates back to Karlovitz et al.[4] For the [1] [2] [3] [4]

Clavin P., Joulin G., 1989, In R. Borghi, S. Murthy, eds., Turbulent Reactive Flows, Lecture Notes in Engineering, 213–240, New York: Springer. Bradley D., et al., 1996, Combust. Flame, 104, 176–198. Clavin P., Gra˜na-Otero J., 2011, J. Fluid Mech., 686, 187–217. Karlovitz B., et al., 1953, Proc. Comb. Inst., 4, 613.

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purpose of comparison with (2.3.2), it is useful to write (2.3.10) in the form (Un− − UL ) dL τL = −M − M2 , UL τs R

(2.3.11)

where M2 = Mc − Mu is called the second Markstein number, the first one being M = Mu . Generally M2 = 0 in real flames. Effect of Finite Thickness A basic difficulty with Equation (2.3.2) comes from the finite thickness of flames, across which the flow velocity changes with density. When Equation (2.3.2) is used to determine the Markstein number, the quantity (Un− − UL )/UL must be known with a precision better than . The evaluation of Un− , given by (2.3.1), necessitates a precise location of the − flame front rf at which the normal velocity of the cold mixture, u− n ≡ nf .u (rf ), is − defined. However, un varies with a relative amplitude of order  = dL / when the normal coordinate xf , used to define the location of flame front, rf = xf nf , is allowed to move within the flame thickness. On the other hand, up to order , the normal flame velocity Df and the small nondimensional stretch rate, τL /τs = O(), do not depend on the choice of xf . It follows that the calculated first Markstein number varies with a magnitude of order unity when different locations are chosen for the position of the flame surface within the flame structure. This may be an explanation for the scatter of the experimental data.[1] Moreover, the form of the law in (2.3.2) is not conserved when xf is varied. The change in the left-hand side, produced by a shift of xf along the normal nf within the flame structure, δrf = nf δxf , is τL nf .∇u− |f .nf (δxf /dL ). For curved front, this term cannot be balanced by a change δM since this would introduce simultaneously a curvature term in the righthand side without counterpart in the left-hand side of (2.3.2). This shows that two different Markstein numbers, Mc and Mu , must be introduced, as in (2.3.10), where Mc must be invariant to a change in xf , and Mu must vary with xf to compensate for the shift coming from u− n , δM c = 0, δM u = δxf /dL . Markstein Numbers Measured in the Burnt Gas The reasoning presented above can also be applied considering the motion of the flame with respect to the burnt gas. It leads to a result for the change in flame velocity having the same structure as Equation (2.3.10), with a normal velocity defined in the same way as (2.3.1), but with respect to the burnt gas u+ : Un+ = u+ n − Df ,

+ with u+ n ≡ nf .u (rf ),

(Un+ − Ub ) dL = −Mc+ + Mu+ τL nf .∇u+ |f .nf . Ub R

[1]

Davis S., et al., 2002, Combust. Flame, 130, 123–136.

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Figure 2.17 Schematic representation of a stationary spherical flame stabilised around a point source.

The mass fluxes in the fresh mixture ρu Un− and in the burnt gas ρb Un+ are different[2,3] ρu Un− = ρb Un+ – see (10.3.99) – and they do not represent exactly the burnt mass rate; see Section 10.3.5. However, the total stretch rate of the flame is the same on both sides of the front:[4] 1 UL Ub = (2.3.12) − nf .∇u− |f .nf = − nf .∇u+ |f .nf ; τs R R see (10.3.97). The two Markstein numbers defined with respect to the burnt gas, Mc+ and Mu+ , are different from those defined with respect to the fresh gas, Mc+ = Mc , Mu+ = Mu . This has the nonintuitive consequence that Markstein number, Mc+ , deduced from the propagation speed of spherically expanding flames is not the Markstein number obtained from velocity measurements in the fresh gas;[5] see (2.3.13). The geometry of spherical flames is interesting because the two contributions from curvature and strain in Equation (2.3.10) have particularly simple expressions, and this geometry can be used to investigate the existence of two different Markstein numbers, Mc = Mu . Imploding Spherical Flame For the case of the inwards-propagating flame shown in Fig. 2.16 the fresh gas is at rest, so the hydrodynamic strain rate is zero. The curvature is −2/Rf and   from (2.3.10) the normal burning velocity can be written Un− = UL 1 + 2Mc (dL /Rf ) . This is the velocity of the flame front observed in the laboratory frame, Un− = −dRf /dt. Stabilised Spherical Flame If the spherical flame is stabilised in the expanding spherical flow provided by a point source, as shown in Fig. 2.17, the curvature is −2/Rf , the strain rate is du/dr = −2Un− /Rf ,  and the flame velocity is Un− /UL = 1 + 2(Mc − Mu )(dL /Rf ) . This configuration is appropriate to show that the second Markstein number is nonzero. [2] [3] [4] [5]

Frankel M., Sivashinsky G., 1983, Combust. Sci. Technol., 31, 131–138. Clavin P., 1985, Prog. Energy Combust. Sci., 11, 1–59. Clavin P., Gra˜na-Otero J., 2011, J. Fluid Mech., 686, 187–217. Davis S., et al., 2002, Combust. Flame, 130, 112–122.

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Figure 2.18 Schematic representation of a spherical flame propagating outwards.

Expanding Spherical Flame Finally, for the case of a spherical flame propagating outwards from an ignition point (see Fig. 2.18), the burnt gas is at rest and the flame front is pushed outwards with the velocity (ρu /ρb )Un− . The unburnt gas ahead of the flame is pushed outwards at velocity (ρu /ρb − 1)Un− by the ‘piston effect’; see Section 2.2.1. The curvature is +2/Rf , the strain rate is −du− /dr = +(2Un− /Rf )(ρu /ρb − 1) and the normal burning velocity is  dL , Un− = UL 1 − 2[Mc + Mu (ρu /ρb − 1)] Rf  (2.3.13) dL ρu Un+ = UL 1 − 2Mc+ . ρb Rf These three spherical configurations (expanding, stationary, imploding) can be used to investigate the two Markstein numbers Mc and Mu in numerical simulations with detailed chemistry. In laboratory experiments, the configuration most easily implemented is the expanding flame. High-Frequency Response The high-frequency response of premixed flames to weak stretch and curvature has been analysed using the one-step ZFK model.[1,2] The frequency-dependent responses to stretch and curvature are different, even for the one step model. Moreover, in the high-frequency limit when the characteristic time of the external flow is smaller than the transit time across the flame, the response is independent of the Lewis number. 2.3.4 Strong Stretch Tip of Bunsen Flame At the tip of a conical Bunsen flame, (see Fig. 1.2), the curvature of the flame front is negative and the flow in the fresh mixture is quasi-uniform, ∇.u− |f = 0. If the Markstein number, Mc , is positive, Equation (2.3.10) shows that the the normal flame speed, Un− , is [1] [2]

Joulin G., 1994, Combust. Sci. Technol., 97, 219–229. Clavin P., Joulin G., 1997, Combust. Theor. Model., 1, 429–446.

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greater than the laminar flame speed, UL , allowing the existence of a steady conical flame front in a uniform flow faster than UL , u− = (0, 0, w− ), w− > UL . When w−  UL this effect is very strong at the tip of a Bunsen flame, and the radius of curvature of the front is very small and may become of the order of the flame thickness. In this case Equation (2.3.10) obtained by a perturbation calculation is no longer strictly valid; however, it remains qualitatively correct. If Mc is negative, the effect of curvature cannot make the flame speed match the flow velocity, and the flame tip opens as observed in experiments.[3,4] A detailed theoretical study of the Bunsen flame can be found in the literature.[5,6] We will come back to the tip of the Bunsen burner in Section 2.4.5. Extinction by Strong Stretch Sufficiently high stretch rates, τL /τs = O(1), may also lead to local extinction. This nonlinear phenomenon cannot be analysed by considering the flame stretch as a linear perturbation to the flame structure. The analyses[7,8,9,10] have been carried out with the onestep ZFK model for planar stretched flames in a stagnation point flow, (see Fig. 2.15), or in a counterflow configuration. Sudden quenching is a consequence of the high sensitivity of the reaction rate to temperature. It occurs when the flame temperature is decreased by the strain rate of the flow. As more easily shown in the thermo-diffusive approximation[7,8] (zero gas expansion ρb = ρu ), this is the case when heat conduction is sufficiently stronger than molecular diffusion of the limiting species (Le > 1); see Section 8.5.2. However, when the hydrodynamic effects that are induced by a gas expansion of practical interest are taken into account[10] (ρu /ρb = 5−9), abrupt extinction requires an unrealistic large deviation of the Lewis number from unity for such transitions to occur without additional phenomena such as thermal loss or complex chemical kinetic effects. The analytical studies of these phenomena are presented in Section 8.5.4. 2.4 Thermo-Diffusive Phenomena Other types of flame instability may occur independently of hydrodynamic effects. Called thermo-diffusive instabilities, they result from the competition between the molecular diffusion of species and the conduction of heat. They take the form of two different phenomena, cellular flames and pulsating flames. Diffusive competition plays also an important role in flame ignition, flammability limits and the spectacular phenomenon of flame balls observed in microgravity conditions.[11] These phenomena are discussed in this section, and detailed analytical studies are presented in Chapters 9 and 10. [3] [4] [5] [6] [7] [8] [9] [10] [11]

Lewis B., von Elbe G., 1961, Combustion flames and explosions of gases. Academic Press. Law C., et al., 1982, Combust. Sci. Technol., 28, 89–96. Higuera F., 2009, Combust. Flame, 156, 1063–1067. Higuera F., 2010, Combust. Flame, 157(8), 1586–1593. Zeldovich Y., et al., 1985, The mathematical theory of combustion and explosions. New York: Plenum. Buckmaster J., Mikolaitis D., 1982, Combust. Flame, 47, 191–204. Libby P., Williams F., 1982, Combust. Flame, 44(1-3), 287–303. Libby P., Williams F., 1987, Combust. Sci. Technol., 54(1-6), 237–273. Ronney P., 1990, Combust. Flame, 82, 1–14.

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(b)

Figure 2.19 Propane flames with a large Froude number in a 140 mm diameter tube. The two flames have approximately the same propagation speed. (a) Lean flame, equivalence ratio = 0.60. (b) Rich flame, equivalence ratio = 1.53.

2.4.1 Cellular and Pulsating Flames Cellular flames are observed in rich mixtures of heavy hydrocarbon fuels and in lean mixtures of light fuels such as hydrogen or methane. The molecular diffusion coefficient of the limiting species, oxygen in the first case and hydrogen or methane in the second case, is greater than the thermal diffusion coefficient of the mixture, Le < 1. The flame front becomes very cellular and may have a chaotic aspect. A comparison of lean and rich propane flames having the same laminar flame speed (≈ 22.5 cm/s) is shown in Fig. 2.19. Both flames are curved because of the hydrodynamic instability. The rich flame on the right is also affected by the thermo-diffusive instability. There are many cells of smaller size; the small structures have a deeper aspect ratio; and the flame has an overall chaotic aspect. Moreover, lean and rich hydrocarbon flames do not have the same colour: lean hydrocarbon flames have deep blue colour that comes from de-excitation of a C-H bond emitting a photon at ≈ 430 nm, whereas rich flames have a greenish tint that comes from the de-excitation of a C=C bond at ≈ 510 nm.

Mechanism of the Cellular Instability The underlying mechanism of multidimensional thermo-diffusive instabilities, responsible for cellular flames, is sketched in Fig. 2.8 and occurs if molecular diffusion of the limiting species is sufficiently stronger than heat conduction so that the Lewis number Le ≡ DT /D, the ratio of the thermal diffusivity to the molecular diffusivity of the species limiting the reaction rate, is smaller than unity, Le < 1. As mentioned in Section 2.2.3, the multidimensional instability occurs when the coefficient B in Equations (2.2.7)–(2.2.9) is negative, B < 0. This diffusive instability is similar in nature to that presented in 1952

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by Turing to represent morphogenesis.[1] For the case of flames, it was imagined and described in simple terms by Zeldovich[2] in 1944 neglecting hydrodynamic effects, so that Equation (2.2.8) is the linear approximation to (2.3.10) in which u− = 0, and B < 0 corresponds to Mc < 0. In the ZFK model, this is the case if the Lewis number is smaller than a critical value close to unity. As explained in Section 2.2.3, wrinkling of the front produces transverse gradients of temperature and of the mass fraction of the species limiting the reaction. The corresponding transverse fluxes of heat and species compete to determine the flame temperature. If molecular diffusion is sufficiently stronger than thermal diffusion (Lewis number below the critical value), the combustion temperature increases (decreases) when the flame is convex (concave) towards the fresh gas; see the lower (upper) part of Fig. 2.8. This is due to an unbalance of the diffuse fluxes of energy in the transverse direction. The chemical energy brought in (taken out) by diffusion of the limiting species is larger (smaller) than the thermal energy lost (brought in) by conduction. As a consequence, due to the high sensitivity of the exothermic reaction rate to temperature, the flame speed is increased (decreased) in the lower (upper) part of Fig. 2.8, so that the instability develops (amplification of the initial wrinkling). Cellular Instability in the Thermo-diffusive Model An expression for the critical value of the Lewis number was first obtained using an extension of the flame model (2.1.4) to unsteady and multidimensional conditions, when the hydrodynamic effects are neglected. This model, called thermo-diffusive model, consists in neglecting the flow velocity and the density variation (ρu = ρb ) in the equations for conservation of energy and species. The constitutive equations thus reduce to two coupled diffusion–reaction equations, ψ −β(1−θ) ∂θ − DT θ = e , ∂t τrb

ψ ∂ψ − Dψ = − e−β(1−θ) , ∂t τrb

(2.4.1)

with boundary conditions at x = ±∞ x = −∞: θ = 0, ψ = 1,

x = +∞: θ = 1, ψ = 0.

(2.4.2)

The problem is ill-posed with the reaction rate in (2.4.1) because the reaction rate does not go to zero in the fresh mixture at x = −∞. In principle, a cut-off temperature 0 < θc < 1 must be introduced, below which the reaction rate is switched off. As for planar flames, this is not useful if the solutions are investigated in the limit of an infinitely large activation energy, that is, in the limit β → ∞. Such an analysis was first performed[3] in the 1960s. Difficulties in the pioneering work come from the fact that the planar wave solution to these equations is unstable not only for a Lewis number Le ≡ DT /D smaller than unity, Le < 1 (cellular flame), but also for Le > 1 (pulsating flames) and both critical values of Le very close to unity, so that the stability domain decreases to zero in the limit β → ∞. [1] [2] [3]

Turing A., 1952, Philos. Trans. R. Soc. London, B 237, 37–72. Ostriker J., ed., 1992, Selected works of Ya.B. Zeldovich, vol. I, p. 193. Princeton University Press. Zeldovich Y., et al., 1985, The mathematical theory of combustion and explosions. New York: Plenum.

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The study was further developed in the late 1970s, still in the limit β → ∞, anticipating that the bifurcation parameter of order unity is β(Le − 1), and the cellular instability was investigated using a power expansion in small κ ≡ kdL .[1,2] Near the instability threshold, the small linear growth rate takes the form (2.2.9) where the coefficients are given in (10.2.27) for the model (2.4.1)–(2.4.2) in the limit β → ∞. The result shows how disturbances with long wavelengths (small kdL ) become unstable, and how disturbances with sufficiently short wavelengths are systematically stabilised by heat transfer. Equation (10.2.27) is equivalent to the following partial differential equation for the position of the flame front x = α(y, t), τL

[β(Le − 1) + 2] 2 ∂ 2 α ∂α ∂ 4α = dL 2 − 8dL4 4 . ∂t 2 ∂y ∂y

(2.4.3)

For a weakly unstable solution, the parameter β(Le − 1) + 2 is a small negative number. The instability threshold and the unstable domain correspond to β(Le − 1) = −2 and β(Le − 1) < −2, respectively. For unstable solutions, the marginal wavenumber km separating the unstable and stable domains of wavelength is given by (km dL )2 = −[β(Le − 1) + 2]/16 > 0. When the flame is close to the instability threshold, the ratio of marginal wavelength to laminar flame thickness is large, km dL 1; see Fig. 10.3. For strongly unstable flames the marginal wavelength becomes small and can be of the order of the flame thickness, km dL = O(1), a case that is outside the limit of validity of the perturbation analysis for small wavenumber km dL 1. Extension of the analysis to an arbitrary density contrast, coupling transverse diffusion and convection, was carried out in 1982[3,4] and provided more realistic expressions for the critical Markstein and Lewis numbers; see (2.9.44) and (10.3.36). Kuramoto–Sivashinsky Equation The thermo-diffusive cellular instability is superimposed on the DL hydrodynamic instability. In order to shed light on the purely diffusive effects, a nonlinear analysis was first performed with the thermo-diffusive model (2.4.1)–(2.4.2). Equation (2.4.3) is written in the reference frame attached to the mean planar solution propagating with constant velocity (equal to UL in the linear approximation). This equation corresponds to the linear approximation of the normal burning velocity (UL − ∂α/∂t)/ 1 + (∂α/∂y)2 ; see (10.1.6). Near the threshold, the instability is weak and the slope of the flame front small, |∂α/∂y| 1. So the first nonlinear correction to (2.4.3) is UL (∂α/∂y)2 /2. When the length scale is reduced by the marginal wavelength, and the time scale conveniently reduced (see (2.7.7)), the dimensionless form of the nonlinear equation becomes free from parameters

[1] [2] [3] [4]

Sivashinsky G., 1977, Combust. Sci. Technol., 15, 137–146. Joulin G., Clavin P., 1979, Combust. Flame, 35, 139–153. Clavin P., Williams F., 1982, J. Fluid Mech., 116, 251–282. Clavin P., Garcia P., 1983, J. M´ec. Th´eor. Appl., 2(2), 245–263.

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and takes the form of the so-called Kuramoto–Sivashinsky equation in the unstable case, β(Le − 1) < −2:   ∂ 2φ ∂φ ∂ 4φ 1 ∂φ 2 + 2 + 4 + = 0. (2.4.4) ∂τ 2 ∂η ∂η ∂η The increase of propagation velocity is the time average of 1/2 (∂φ/∂η)2 . The only remaining parameter concerns the boundary conditions. For periodic solutions it is the reduced length of the box: as the ratio of this length to the marginal wavelength is increased, so does the number of linearly unstable modes involved in the solution. Equation (2.4.4) was derived by Kuramoto[5,6] to describe propagating patterns in chemical reaction–diffusion systems and by Sivashinsky[1] in the context of cellular flames. For a sufficiently large box, the patterns that are generated present an intrinsic stochasticity, despite the existence of steady state solutions. For this reason Equation (2.4.4) has become very popular as the simplest model of phase turbulence. It is of limited use for wrinkled flames since their dynamics are dominated by the hydrodynamical phenomena described in Section 2.2. A better model for cellular flames is obtained by taking into account small gas expansion (weak hydrodynamical instability);[7] see (2.7.10). Oscillatory Instability A different type of diffusive instability is predicted theoretically for large values of β(Le − 1) close to 10; see Fig. 10.5. It is a longitudinal instability that gives rise to a pulsating planar mode of propagation, first observed in one-dimensional numerical analyses of planar flames in the early 1970s in the Russian literature.[8,9] The Lewis number has to be so large that this pulsating mode has been observed only in solid combustion where molecular diffusion is quasi-blocked and also in the self-propagating high-temperature synthesis of materials.[9] However, theoretical studies show that it may also be possible to observe this phenomenon in some gaseous flames close to the extinction limit;[2] see Section 10.2 and Fig. 10.5. Relaxation oscillations that develop not far from the stability limit have been analysed in detail.[10]

2.4.2 Flame Kernels and Quasi-isobaric Ignition The competition between molecular and thermal diffusion also manifests itself in flame ignition and extinction. Analytical studies of these problems are presented in Chapter 9. The discussion is limited here to physical insights and also to simplified analyses, sufficient

[5] [6] [7] [8] [9] [10]

Kuramoto Y., Tsuzuki T., 1976, Prog. Theor. Phys., 55(2), 356–369. Kuramoto Y., 1978, Prog. Theor. Phys. Supp., 64, 346–367. Sivashinsky G., 1977, Acta Astronaut., 4, 1177–1206. Zeldovich Y., et al., 1985, The mathematical theory of combustion and explosions. New York: Plenum. Merzhanov A., Khaikin B., 1988, Prog. Energy Combust. Sci., 14(1), 1–98. Gra˜na-Otero J., 2009, Ph.D. thesis, Universidad Polit´ecnica de Madrid, ETSIA.

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to understand the essential features of the experiments of Ronney[1,2] performed in microgravity conditions in order to eliminate natural convection; see Section 2.4.3. Ignition is a key problem in many technological applications, as for example in automobile petrol engines or in turbojets working with lean hydrocarbon–air mixtures. It is also the case in cryogenic hydrogen–oxygen rocket engines where liquid hydrogen is used to cool the wall of the combustion chamber. The ignition of rich hydrogen or lean heavy hydrocarbon mixtures is difficult, especially close to the flammability limits. Quasi-isobaric ignition should not be confused with the question of the flammability limits of a mixture. As already mentioned, the later are defined by the critical compositions (very lean or very rich; see Fig. 2.3) beyond which a planar flame can no longer propagate because the flame temperature is below the crossover temperature T ∗ , a purely chemical kinetic parameter independent of the initial composition; see the preliminary discussion in Section 2.1.2 and the detailed analyses in Section 8.5.5 and Chapter 9. Flammable reactive mixtures can be more or less easy to ignite, particularly when approaching the flammability limits. Moreover, in some nonflammable mixtures with a small Lewis number, combustion can still proceed in the form of small spherical flames that have been observed in microgravity conditions.[1,2] Two successive steps are involved in flame initiation by a local input of energy. During the first stage, the hot spot created in the reactive mixture must undergo a thermal runaway. This requires that the rate of heat release by the ignition device be larger than the cooling rate by conduction towards the fresh mixture, so that the temperature is locally raised enough to initiate exothermic reactions. In such conditions the reactants are consumed and a flame builds up. However, because of the competition between diffusion of reactants and heat conduction in the radial direction, this condition is not sufficient to ensure that the flame will grow indefinitely. An energy input sufficient for the first step may be too weak to initiate an ever-expanding flame. Here we focus attention on the second step. Zeldovich Critical Radius (Spherical Flame Kernel for Le = 1) The criterion for quasi-isobaric flame ignition was formulated by Zeldovich[3] as a nucleation problem. The critical radius of a spherical hot spot required to initiate the propagation of an ever-expanding spherical flame in a cold mixture initially at rest is related to the solution for a steady spherical flame (called flame kernel in the following). This solution is unstable, in the sense that flames with a smaller radius collapse, while those with a larger radius grow indefinitely in size. The larger the critical radius, the more difficult is ignition. Preferential diffusion plays an essential role in this problem. In lean hydrocarbon–air or rich hydrogen–air mixtures, corresponding to a large Lewis number, the critical radius diverges before the flammability limit,[4] so that close to the flammability limits, there exist [1] [2] [3] [4]

Ronney P., 1985, Combust. Flame, 62, 121–133. Ronney P., 1990, Combust. Flame, 82, 1–14. Zeldovich Y., et al., 1985, The mathematical theory of combustion and explosions. New York: Plenum. He L., Clavin P., 1993, Combust. Flame, 93, 408–420.

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as

at

s

Concentration

83

He

Temperature Reaction rate

Figure 2.20 Sketch of a steady spherical solution obtained with the ZFK model. The heat flux and temperature profile are shown in dark grey. The mass flux and species concentration are shown in light grey.

flammable mixtures that cannot be ignited by a spherical hot spot; see Section 9.1.2 and Fig. 9.1. In the opposite conditions, for rich hydrocarbon–air or lean hydrogen–air mixtures (small Lewis number), spherical flame balls may exist for compositions that are beyond the flammability limit, that is, in conditions for which planar flames cannot propagate. This point is discussed in more detail after the Zeldovich analysis[3] presented next. Consider first the simplest case, namely adiabatic flames far from the flammability limits and well characterised by the one-step ZFK model (2.1.3). A steady spherical solution in a premixed gas is sketched in Fig. 2.20. In the limit of a large activation energy, the reaction zone is much thinner than the radius of the flame and may be considered as a thin sheet, as in the planar flame sketched in Fig. 2.4. In spherical geometry, both mass and heat diffusion fluxes can have a zero divergence (∇. j = 0) in the preheated zone of a steady spherical flame. In contrast to the planar case, no motion of the flame front relative to the fresh mixture is required to sustain a steady flame structure. There is no flow; the gas is at rest so that the steady spherical flame is a solution to a pure reaction–diffusion problem. Denoting R the radial coordinate and Rf the unknown radius of the thin reaction sheet (which is well defined in the limit β → ∞), and using the simplest one-step ZFK model, Equations (2.1.4) are replaced by     1 d ψ −β(1−θ) 1 d dθ dψ R2 =D 2 R2 = e , (2.4.5) − DT 2 dR dR τrb R dR R dR with the boundary conditions R  Rf :

θ = θf ,

ψ = 0;

R → ∞:

θ = 0,

ψ = 1.

(2.4.6)

For simplicity, both DT and D are supposed constant. The state of the burnt gas is uniform in the hot kernel; see Fig. 2.20. The temperature of the burnt gas is given simply by conservation of energy between the fresh mixture at Tu and the burnt gas. For a large activation energy, (β  1) this burnt gas temperature is also the flame temperature Tf at which the exothermic reaction proceeds. Denoting θf the nondimensional flame temperature, reduced by the flame temperature Tb of the planar adiabatic flame, θf ≡ (Tf − Tu )/(Tb − Tu ), and

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integrating Equation (2.4.5) with the boundary condition (2.4.6), yields DT θ = D(1 − ψ)

⇒

θf = 1/Le;

(2.4.7)

see (9.1.19) for more detail. This shows that Tf is independent of the flame radius Rf but is smaller or larger than Tb depending on the sign of (Le − 1), Le > 1 ⇒ Tf < Tb , (Le < 1 ⇒ Tf > Tb ). The heat and mass fluxes compete here in the radial direction to determine Tf : heat conduction cools the hot pocket, while molecular diffusion of the species limiting the reaction brings the chemical energy that is liberated by the exothermic reaction in the reaction zone; see Fig. 2.20. Assuming that Rf is much larger than the thickness of the reaction layer, the solution in this zone is obtained in the same way as for the planar flame, (see (2.1.7)), but where ψe−β(1−θ) = eβ(θf −1) (Le/β)e− ,  ≡ β(θf − θ ) and ψ ≡ Le(θf − θ ), so that the heat flux leaving the reaction zone has the form  dθ DT = eβ(θf −1)/2 2Le 2 , β → ∞: − lim DT (2.4.8) →∞ dR β τrb where θf is given by (2.4.7). The difference with the planar flame (2.1.8) comes from the flame temperature that is different from Tb , θf = 1 when Le = 1. In the preheated zone where the reaction rate in (2.4.5) is negligible,   d 2 dθ R = 0, (2.4.9) dR dR and the temperature profile satisfying the boundary conditions (2.4.6) is R  Rf :

θ=

1 Rf , Le R

R = Rf :

DT

1 DT dθ =− , dR Le Rf

(2.4.10)

where θf = 1/Le has been used. The flame radius Rf is obtained by matching the inner and outer heat fluxes at the boundary of the two zones, yielding     β 1 β 1 Rf 1 DT DT −1 −3/2 2 1− Le 2 Le β → ∞: =e Le ⇔ = Le e , (2.4.11) Le Rf τb dL √ where here τb ≡ β 2 τrb /2, and dL = DT τb is the planar flame thickness for Le = 1; see (2.1.9) and (2.1.11). The result (2.4.11) is written for a first-order reaction rate (ϑ = 1). Due to the large activation energy, β  1, the kernel radius is much larger than the flame thickness when Le > 1, explaining the difficulty in igniting lean mixtures of a heavy hydrocarbon in air or rich hydrogen mixtures. The difficulty of ignition is reinforced as the flammability limits are approached. We will come back to this point later. The instability of the spherical solution in adiabatic conditions is proved by the stability analysis in Section 9.2.1. It may be understood from simple considerations: for a flame radius larger (smaller) than that obtained in the steady solution, the diffusion fluxes of heat and species at the reaction sheet, which are proportional to 1/R2f if they were kept in steady state, would decrease (increase), but not the reaction rate since Tf in (2.4.7) is independent of Rf . Therefore, in order to satisfy energy and mass conservation at the reaction sheet,

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a convective motion with dRf /dt > 0 (dRf /dt < 0) must exist, amplifying the initial perturbation. This simple explanation works well for adiabatic flames but must be revised when radiative heat losses are taken into account;[1] see Section 9.2.2. To conclude, due to differential diffusion, mixtures in which heat diffusivity is larger than the molecular diffusion coefficient of the limiting species, Le > 1 (lean heavy hydrocarbon–air mixtures or rich hydrogen mixtures), are difficult to ignite in a quiescent medium since the critical Zeldovich radius can be large. Facilitated Ignition in Turbulence experiments[2]

have shown that such mixtures (Le > 1) are more easily ignited Recent in turbulent flows, although turbulence makes ignition more difficult for mixtures that are easily ignited in laminar regimes, Le < 1. A simple explanation could be that, according to Section 3.1.2, the turbulent diffusion coefficients are all the same, so that the effective Lewis number becomes unity in turbulent flows having small length and time scales. Heat Losses and Flame Balls Small heat losses may quench flames. This problem has been studied first in planar geometry in the framework of the ZFK model; see Section 8.5.1. The larger the thermal sensitivity of the reaction rate, the smaller is the heat loss that produces flame quenching. For more realistic flame models in which a crossover temperature appears, it turns out that the thermal sensitivity increases and diverges as the flammability limits are approached; see (8.5.81). Even tiny radiative losses can thus never be neglected when approaching the flammability limits. In spherical geometry, thermal quenching of the steady state solution systematically occurs before the divergence of the critical radius, no matter how small be the radiative loss; see the detailed analysis in Section 9.1.3. In a way similar to the planar case, sketched in Fig. 8.16, the quenching mechanism corresponds to a turning point where two branches of solution merge; see Fig. 9.2. Above the quenching value of radiative loss, there is no steady state solution, while there are two branches of solutions for smaller intensities of heat loss. One branch is the prolongation of the adiabatic solution. The other, called the second branch in the following, is new and does not exist in the adiabatic case. In planar geometry, sketched in Fig. 8.16, the latter is unstable. For the steady state in spherical geometry, the situation is different[1,3] when Le < 1. As in the adiabatic case, the spherical solutions of the branch that is the prolongation of the adiabatic solution are systematically unstable – see Section 9.2.2 – and this is also the case for the solutions of the second branch if Le > 1. But for Le < 1, the stability analysis in Section 9.2.2 shows that part of the second branch corresponds to a stable spherical solution. In other words, small heat losses can stabilise stationary spherical flame balls when Le < 1. [1] [2] [3]

Buckmaster J., et al., 1990, Combust. Flame, 79, 381–392. Wu F., et al., 2014, Phys. Rev. Lett., 113, 024503. Buckmaster J., Weeratunga S., 1984, Combust. Sci. Technol., 35, 287–296.

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Flame Kernel in the Presence of a Point Source of Constant Power The Zeldovich analysis of a spherical flame kernel has been extended[1] to take into account the presence of a central heat source of constant power Q˙ s . The formulation of the problem is similar to (2.4.5) but the boundary conditions at the centre differs from (2.4.6): R → 0: R2

dθ = ls ; dR

R < Rf : ψ = 0;

R → ∞: θ = 0, ψ = 1,

(2.4.12)

where ls ≡ Q˙ s /(4πρDcp (Tb − Tu )) is a length proportional to the power of the heat source. The solution in the preheated zone is the same as in the Zeldovich solution, R  Rf :

θ = θf

Rf , R

(1 − ψ) =

Rf . R

Due to the energy source, the temperature is not uniform in the burnt gas:   1 1 R  Rf : , ψ = 0. − θ = θf + ls R Rf

(2.4.13)

(2.4.14)

The temperature gradient in the burnt gas influences the inner structure of the thin reaction layer. The problem simplifies in the limit β → ∞ when the gradient in the burnt gas is smaller than in the preheated zone by a factor 1/β, namely when ls /Rf is of order 1/β, qs ≡ βls /Rf = O(1). Equation (2.4.8) is still valid to leading order, but the energy balance and the flame temperature θf are slightly modified. The flame temperature θf is given by conservation of energy. Equality of the total energy fluxes on both sides of the thin reaction layer yields  1 dψ Rf + dθ + ≈ 0. (2.4.15) β → ∞: dR Le dR Rf − This relation is easily obtained from (2.4.5) when the effect of thickness of the reaction zone is neglected so that the first derivative with respect to R is negligible in front of the second derivative. For a small departure from unity of the Lewis number,[2] β(Le−1) = O(1), and for a small gradient in the burnt gas, βls /Rf = O(1), the jump relation in (2.4.15) is valid up to the order 1/β. More precisely the left-hand side of (2.4.15) times Rf is a small number of order 1/β 2 . More details can found in Section 8.2.4 where matching of the external and the inner solutions is performed for a temperature gradient in the burnt gas smaller than in the preheated zone by an order 1/β. Equation (2.4.15) leads to the small modification to θf in the presence of the energy source,   ls 1 1 + qs ⇒ β(θf − 1) = β −1 +β , (2.4.16) β → ∞: θf = Le Le Rf

[1] [2]

Deshaies B., Joulin G., 1984, Combust. Sci. Technol., 37, 99–116. Joulin G., Clavin P., 1979, Combust. Flame, 35, 139–153.

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Reduced kernel radius,

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87

1

0 Reduced strength of heat source,

Figure 2.21 Solutions for radius of stationary spherical flame kernel in the presence of a constantpower heat source.

instead of (2.4.7). Because of the term β(θf − 1) in (2.4.8) a small variation of order 1/β of the flame temperature θf changes the result by an order unity. Equations (2.4.8) and (2.4.16) then lead to a nonlinear equation for Rf ,       β 1 RfZ RfZ 1 DT DT ls 2 Le −1 ⇔ = 1, =e Le exp β exp −K Le Rf τb Rf Rf Rf where RfZ denotes the Zeldovich radius (2.4.11) and K ≡ constant power of the heat source. Written in the form β → ∞:

β

β ls 2 2 dL e

(Rf /RfZ ) ln(Rf /RfZ ) = −K,



1 Le −1



is a measure of the

K = O(1),

(2.4.17)

the nonlinear equation shows that there is no solution for a sufficiently large power of the heat source, K > Kc ≡ 1/e, and there are two branches of solutions Rf + > Rf − for 0 < K < Kc . The two solutions merge for K = Kc , Rf + = Rf − = RfZ /e. The branch of solution Rf + is an extension of the Zeldovich solution (K = 0: X = 1 ⇔ Rf + = RfZ ) and the other branch starts from zero (K = 0: Rf − = 0); see Fig. 2.21. If the power of the constant heat source, Q˙ s , is sufficient large, namely if K > Kc , there is no steady spherical solution. One can then expect that an expanding flame will propagate indefinitely, indicating successful ignition. By noticing value of  the critical  that  β 1 ˙ Qs corresponding to Kc varies with the Lewis number as exp − 2 Le − 1 , ignition of mixtures with Le > 1 requires a much larger intensity of the constant source of energy than mixtures with a small Lewis number Le < 1. The conclusion is qualitatively the same as that obtained from the Zeldovich analysis. The one-dimensional stability analysis[1] shows that the Rf + solutions are systematically unstable, as is the case for the Zeldovich solution. The  stability  of the Rf − solutions depends on the Lewis number. For a small Lewis number β 1 −

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< 4, namely for Le < 1+4/β,

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  1 > 4, β  1, the spherical Rf − solutions are stable. For a large Lewis number, β 1 − Le the Rf − solutions are stable below a critical value of K, just below Kc . These studies have been also extended to spherical flames stabilised in the flow generated by a sink or a source of mass at the origin.[1] Spherical Flame Kernel Near the Flammability Limits (Adiabatic Case) The adiabatic flame temperature Tb of planar flames is determined by the initial composition of the reactive mixture; see (8.2.3) for the ZFK model. In contrast, in this model the crossover temperature T ∗ (below which the reaction rates decreases abruptly and becomes negligible) does not depend on the composition; see Section 5.2.2. In real mixtures the dependence is weak. The propagation of a planar flame is not possible for compositions in which the chemical energy liberated by the reaction is not sufficient to raise the temperature above T ∗ (Tb < T ∗ ). Planar flames exist only if Tb > T ∗ , explaining the two flammability limits of rich and lean mixtures. Near to the flammability limits, Tb approaches the crossover temperature T ∗ from above, and the reaction rate decreases to zero. The laminar flame speed of such planar flames is studied in Section 8.5.5 with a simple one-step model (8.5.71)–(8.5.72). In spherical geometry, according to (2.4.7), for a given composition of the initial mixture the flame temperature Tf differs from Tb when Le = 1, (Tf −Tu ) = (Tb −Tu )/Le. The structure of the flame kernel (steady spherical solution) is studied near the flammability limit in Section 9.1.2 for adiabatic flames and in Section 9.1.3 when radiative heat losses are taken into account. In the vicinity of the flammability limit flame kernels have a radius much larger than far from the limit, where Zeldovich’s solution (2.4.11) is valid; see (9.1.26). For all values of the Lewis number, the radius of flame kernel diverges systematically at the critical composition, Tf = T ∗ . This explains the difficulty of ignition in such mixtures and more specially for Le > 1 where the Zeldovich radius can be itself much larger than the flame thickness. The explanation is simple. Since the reaction rate decreases to zero when Tf approaches T ∗ from above, the diffusive fluxes also decrease to zero at the reaction sheet, so that, according to (2.4.10), the critical radius increases and diverges as Tf → T ∗ . The relative values of Tf and T ∗ control the existence of a steady spherical solution. Since Tf < Tb when Le > 1, the divergence of the critical radius (Tf = T ∗ ) occurs for a composition for which the propagation of a planar flame is still possible, Tb > T ∗ . In this case there exist flammable mixtures that cannot be ignited by a quasi-isobaric process;[2] see Fig. 9.1. Conversely, for Le < 1, Tf > Tb , spherical solutions do exist for mixtures that are nonflammable, T ∗ > Tb . Planar flames cannot propagate but the combustion can proceed in the form of spherical flame balls; see Fig. 2.22 and the detailed explanation in Section 9.1.2.

[1] [2]

Daou J., et al., 2009, Combust. Theor. Model., 13(2), 1–26. He L., Clavin P., 1993, Combust. Flame, 93, 408–420.

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89

(b)

Figure 2.22 Radius of the spherical solution Rf and thickness of the planar flame dε versus the mass fraction of the limiting component YRu . The thickness of the planar flame far from the flammability ∗ and the divergence of the radius limit (ZFK model) is dL . Flammability limits are represented by YRu s∗ . The shaded region represents nonflammable mixtures. (a) Le > 1, of the flame kernel occurs at YRu Tf < Tb . (b) Le < 1, Tf > Tb .

The stability of flame kernels near the flammability limits is studied in Section 9.2 for both adiabatic and nonadiabatic flames. As already mentioned, flame kernels are unstable in adiabatic conditions. Consider a steady spherical solution in a nonflammable mixture with Le < 1, Tb < T ∗ < Tf . For an initial flame radius slightly larger than the radius of the steady state solution, the radius of the flame front is expected to increase with time. Since planar flames cannot propagate in such a mixture, expanding flames cannot tend towards the planar flame solution in the long time limit. Their evolution is discussed now.

2.4.3 Flame Balls and Self-Extinguishing Flames Unsteady expanding flames with a radius growing approximately as the square root of time have been reported[3] in micro-gravity spark ignition experiments that are free from the buoyancy-induced disturbances. These experiments concern mixtures beyond the flammability limits and characterised by Lewis numbers smaller than unity (Le < 1). This is a new regime of propagation, different from that of the planar flame. However, these expanding spherical flames suddenly extinguish at a radius of a few centimetres. This regime was called ‘self-extinguishing flames’ by Ronney.[3] The quenching mechanism arises from the coupling between unsteady effects and tiny radiative heat losses, as explained in Sections 9.2 and 9.3.

[3]

Ronney P., 1985, Combust. Flame, 62, 121–133.

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Self-Extinguishing Flames in Lean Methane–Air Mixtures Self-extinguishing flames were first observed in experiments performed in a drop tower,[1] with very lean methane–air mixtures (Le  1), beyond the flammability limits. The mixture was ignited by a spark with an ignition energy ranging from a millijoule to a few joules. The minimum ignition energy plotted versus fuel concentration is qualitatively similar for different pressures ranging from 0.07 to 2 atmospheres. At 1 atmosphere the results may be summarised as follows. Starting from a stoichiometric mixture (≈ 9.5 mole percent methane in air) the minimum ignition energy increases regularly from 0.3 mJ to 5 mJ as the methane content decreases from stoichiometry to 5.1 mole percent methane in air. In all these mixtures, an ever-expanding flame (tending towards a steadily propagating flame at constant velocity) is observed above the minimum ignition energy. The situation changes in leaner mixtures. Below 5.1 mole percent methane in air everexpanding flames are no longer observed. This limiting composition should correspond to the flammability limit of planar flames for which Tb = T ∗ . Between 5.1 and 4.7 mole percent methane in air self-extinguishing flames are observed. For a given composition in this range the flames extinguish at a radius which first increases with the spark energy, but, above a spark energy of typically 100 mJ, the flame radius never exceeds a maximum value however high be the spark energy. The maximum radius at which the flame is quenched decreases with decreasing methane content, from 5 cm just below 5.1 mole percent methane in air to 2 cm just above 4.7 mole percent methane. No spherical flames are observed for mixtures below approximately 4.7 mole percent methane in air, no matter how high the spark energy. This limiting composition could correspond to a spherical kernel for which Tf = T ∗ , sketched in Fig. 9.1b. In a small transition region around 5.8 mole percent methane, corresponding to Tb ≈ T ∗ , self-extinguishing flames are observed for spark ignition below 72 mJ and steady propagation of normal flames occurs at a higher spark energy. A theoretical description of self-extinguishing flames is presented in Section 9.3. Flame Balls in Lean Hydrogen Mixtures Different phenomena were observed in micro-gravity experiments performed with lean hydrogen–air and lean H2 –O2 –N2 mixtures[2] (equivalence ratio φ less than about 0.07) for which the Lewis number is much smaller than in lean methane–air flames, typically Le  0.3. Self-extinguishing flames were not observed in these mixtures. The propagation and structure of lean hydrogen flames were found to be dominated by the initial cellular structure resulting from the thermo-diffusive instability described in Section 2.4.1. For sufficiently reactive (but still lean) mixtures, the cellular front propagates quasispherically outwards, and the individual cells split and do not increase in size. For leaner (less reactive) H2 mixtures, individual cells propagate away from the ignition source and separate. When the cells have moved far enough apart, they become spherical with a [1] [2]

Ronney P., 1985, Combust. Flame, 62, 121–133. Ronney P., 1990, Combust. Flame, 82, 1–14.

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(b)

Figure 2.23 Stable flame balls in lean hydrogen mixtures. The field view is 300 × 225 mm. (a) 4.0% H2 , 20.2% O2 , 75.8% N2 , flame ball diameters ≈ 9–16 mm. (b) 4.9% H2 , 9.8% O2 , 85.3% CO2 , flame ball diameters ≈ 2.5–4 mm. Both mixtures are below the flammability limits for planar flames. Photos taken in the space shuttle. Courtesy of P. Ronney, USC, Los Angeles.

stationary radius. In other words the cellular front degenerates into many spherical and quasi-steady spherical flames of smaller radius, called ‘flame balls’. These phenomena have been observed during the space shuttle SOFBALL experiments.[3,4] Two examples of stable flame balls in very lean hydrogen mixtures below the flammability limit are shown in Fig. 2.23. These hydrogen flames have a very weak luminosity and were photographed using intensified video cameras. Typical burning times were between 5 and 80 minutes, the earlier experiments being limited by experimental timeout. The weaker flame balls (3.2% H2 in air) produced about 0.5 W of thermal power (by comparison, a birthday candle produces about 50 W) and they were generated by a small spark ignition energy, typically 10 mJ. The interpretation of flame balls is that they correspond to ‘flame kernels’ stabilised by radiative heat loss.[5,6] The stability analysis of nonadiabatic flame kernels near to the flammability limits is performed in Section 9.2.2. Typically, on earth, the limiting concentration for planar hydrogen–air flames propagating downwards (lean flammability limit) is about 10% H2 (by comparison, stoichiometry is 29.6% H2 in air). For upwards propagation, the limit is 4.5% H2 . In this limit, small disconnected semi-spherical flame caps are observed. They are analogous to micro-gravity flame balls but are stabilised by the strain rate of the stagnation point flow at the tip of the cap resulting from the buoyancy-driven upwards motion. The stabilisation is described in Section 2.6.4.

[3] [4] [5] [6]

Ronney P., et al., 1998, AIAA J., 36, 1361–1368. Kwon O., et al., 2004, In 42nd AIAA Aerospace Sciences Meeting, Reno, Paper No. 2004–0289. Buckmaster J., Weeratunga S., 1984, Combust. Sci. Technol., 35, 287–296. Buckmaster J., et al., 1990, Combust. Flame, 79, 381–392.

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2.4.4 Quasi-Steady State Approximation The effects of the density change are not essential to describe flame dynamics near the critical conditions of quasi-isobaric ignition. The thermo-diffusive model (2.4.1) is sufficient in this context, correctly describing the competition between heat conduction and molecular diffusion of species. Simplified analyses of converging[1] and expanding flames,[2] including a constant power source and radiative heat losses,[3] have been performed with the drastic simplification of the quasi-steady state approximation. As discussed in this section, the validity of the quasi-steady state approximation is questionable for determining the critical conditions of flame ignition. The analyses presented in Section 9.3 for a Lewis number smaller than unity lead to quite different results, explaining the micro-gravity experiments reported in Section 2.4.2. The quasi-steady state approximation is not valid, neither near to the flame kernel (steady spherical solution) nor in the dynamics of self-extinguished flames. However, this approximation is useful, in particular, to describe open-tipped Bunsen flames, observed in rich heavy hydrocarbon mixtures, as explained in Section 2.4.5. General Formulation In spherical flames, the boundary conditions depend on the direction of propagation: • For converging flames they are dθ dψ = 0, R2 = 0; R → ∞: θ = 1, ψ = 0. (2.4.18) dR dR • Ignition concerns expanding flames (R˙ f  0) in a reactive mixture at rest. After the heat source is switched off, the boundary conditions are R → 0: R2

dθ = 0, ψ = 0; R → ∞: θ = 0, ψ = 1. (2.4.19) dR Using the thermo-diffusive model, the dynamics of flame is reduced to solving (2.4.1) with (2.4.19) for a given initial condition, for example a hot pocket of burnt gas. An even simpler initial condition is a concentrated instantaneous heat source in an infinite domain of fresh mixture at initial temperature Tu . In a nonreactive mixture the temperature is obtained by the self-similar solution (Green’s function of the diffusion equation) R → 0: R2

T(R, t) − Tu =

E/(ρcp ) exp (−R2 /4DT t). (4π DT t)3/2

(2.4.20)

For the ignition problem, Equation (2.4.20) can be also used as an initial condition because it holds during the short time when heat release by the reactions is negligible in front of the deposited energy E. The problem is then to determine the critical value of E needed to initiate an ever-expanding flame.

[1] [2] [3]

Frankel M., Sivashinsky G., 1984, Combust. Sci. Technol., 40, 257. He L., 2000, Combust. Theor. Model., 4, 159–172. Chen Z., Ju Y., 2007, Combust. Theor. Model., 11(3), 427–453.

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No exact solution to this unsteady problem is available, even in the limit of a large activation energy β → ∞ where the reaction is concentrated in a thin reaction layer of radius R = Rf (t), called the flame radius, the objective being to calculate the radius as a function of time. As soon as a spherical reaction layer is formed, two characteristic times are involved: the reaction time at the flame temperature τr (Tf ) and the diffusion time tdiff ≡ d2 /(4DT ), where d is the flame thickness, which is the characteristic length of variation for the temperature and the species concentration. Depending on the flame regime, this length d may be as small as the laminar flame thickness dL or as large as the flame radius Rf in spherical geometry. Quasi-Steady State Equations When the characteristic time of evolution tev ≡ Rf /|R˙ f | is sufficiently long, it is tempting to look for approximate solutions using a quasi-steady state approximation by neglecting the derivatives of temperature and species concentration with respect to time in the moving referential frame of the spherical reaction layer. Equations (2.4.1) are then transformed into ordinary differential equations. Introducing the Laplace operator in spherical geometry, =

  2 d d2 1 d 2 d R = + 2, 2 dR R dR dR R dR

the space coordinate x ≡ R − Rf (t) and the notation R˙ f ≡ dRf /dt, ∂/∂t → ∂/∂t − R˙ f ∂/∂x , ∂/∂R → ∂/∂f , these equations are obtained by writing ∂/∂t ≈ −R˙ f ∂/∂x , 

− R˙ f + 2

DT R



dθ d2 θ ψ −β(1−θ) − DT 2 = e , dR τrb dR

 D dψ ψ d2 ψ − R˙ f + 2 − D 2 = − e−β(1−θ) , R dR τrb dR

(2.4.21) (2.4.22)

where the variable R ∈ [0, ∞] has been used instead of x ∈ [−Rf , ∞]. Attention is limited here to adiabatic conditions after the deposition of energy is switched off. When |R˙ f |Rf < 2DT the unsteady term in the square brackets of (2.4.21)–(2.4.22) is small in front of the geometrical curvature term, and the solution is close to the Zeldovich kernel (2.4.11). In the opposite case, |R˙ f |Rf > 2DT , the curvature term is negligible and Equations (2.4.21)–(2.4.22) reduce to (2.1.4) for a planar flame, ρ R˙ f → m > 0, R − Rf (t) → −x, and the solution is close to a propagation at constant velocity, R˙ f ≈ UL ; see (2.4.36) (there is no difference between UL and Ub in the thermo-diffusive model). Both limits are included in (2.4.21)–(2.4.22) so that the quasi-steady state approximation bridges the gap between the spherical flame kernel of Zeldovich and the planar flame. Unfortunately the quasi-steady state approximation is not valid neither near to the Zeldovich flame kernel nor during the flame expansion in general. Its validity for flame ignition is discussed next.

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Limitations of the Quasi-Steady State Approximation. Far Field Dynamics In the reference frame attached to the spherical reaction zone, R = Rf (t), x ≡ R − Rf (t), Equation (2.4.1) for θ in the preheated zone is  ∂ 2θ ∂θ DT ∂θ − R˙ f + 2 − D = 0, (2.4.23) T ∂t R ∂x ∂x 2 and the quasi-steady state approximation is recovered when the first term is neglected. Consider first the case |R˙ f |Rf  DT (≈ UL dL ). This includes large flame radius Rf  dL when the flame structure is close to that of the planar case, R˙ f ≈ UL , and also small flame radius Rf (however, not smaller than ≈ dL ) when the flame velocity is large, |R˙ f |  UL . For |R˙ f |Rf  DT , the curvature term, namely the third term in (2.4.23), is small everywhere in the preheated zone, DT |R˙ f |R. Disregarding the unsteady term, ∂θ/∂t, the flame thickness is obtained from the balance |R˙ f |/d ≈ DT /d2 , yielding d ≈ DT /|R˙ f |. The unsteady term ∂θ/∂t is then found to be effectively negligible since ∂/∂t ≈ |R˙ f |/Rf is smaller than |R˙ f |/d ≈ |R˙ f |2 /DT . However it is not guaranteed that the curvature term can be retained in a consistent way when the small unsteady term ∂θ/∂t is neglected, since these two terms may be of same order. The quasi-steady state approximation (2.4.21)–(2.4.22) is relevant only when the flame structure is close to that of the planar case (R˙ f ≈ UL , |R˙ f |  UL ). In this case θ (x , t) is close to the steady planar solution θ(x ), θ = θ(x ) + δθ (x t), where θ is of order unity and δθ is small of order τL |R˙ f |/Rf ≈ dL /Rf , so that ∂θ/∂t = ∂δθ/∂t is of order (dL /Rf )2 , smaller than the curvature term DT /Rf dL by a factor dL /Rf 1. But it is doubtful that the slow motion, R˙ f UL , observed near critical conditions of ignition, can be well described by (2.4.21)–(2.4.22). Consider now the case of slow spherical flames, |R˙ f |Rf DT . For a radius R in the preheated zone of the same order as the flame radius Rf , the term R˙ f in the square brackets of (2.4.23) is negligible and the quasi-steady state approximation reduces to the Zeldovich kernel. The characteristic length of variation of species concentration and temperature is then the radius R; see (2.4.10). The quasi-steady state approximation is effectively valid in the region R ≈ Rf since ∂/∂t ≈ |R˙ f |/Rf is smaller than DT /R2 ≈ DT /R2f , but this approximation is not uniformly valid in the preheated zone. It fails in the far field as soon as the diffusion term DT /R2 becomes as small as the unsteady term |R˙ f |/Rf , that is, for R of order (DT Rf /|R˙ f |)1/2 , R/Rf ≈ DT 1/2 /(Rf |R˙ f |)1/2 . This can also be seen from the ˙ self-similar solution (2.4.20): for a point source with a varying rate of heat release Q(t) = dE/dt, switched on at time t = 0 in an infinite medium initially at uniform temperature,[1,2] t  0: T = Tu , one gets  t ˙ Q(t − τ ) exp(−R2 /4DT τ ) t > 0: T(R, t) − Tu = dτ , (2.4.24) ρcp (4π DT τ )3/2 0

[1] [2]

Carslaw H., Jaeger J., 1959, Conduction of heat in solids. Clarendon Press–Oxford Science Publications. Crank J., 1986, The mathematics of diffusion. Clarendon Press–Oxford Science Publications, 2nd ed.

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Le=1.8,

U

1 Qs = 0 2 Qs = 1.5 3 Qs = 2 4 Qs = 3 5 Qs = 4 3

4

5

(Rc, Uc) 2 1

2

R

Figure 2.24 Numerical simulation of spherical flame initiation with a constant central heat source. The reduced radial velocity of the reaction zone (U ≡ R˙ f /UL ) is plotted against the reduced radius (R ≡ Rf /dL ) for five values of the reduced constant heat source (Qs ≡ dE/dt/(4πρcp νTb dL )). Reproduced from He L., 2000, Combustion Theory and Modelling, 4, 159–172, with permission from Taylor and Francis Ltd. www.informaworld.com.

˙ = 0; t > 0: Q ˙ =cst., yielding for a constant rate of heat release, t  0: Q  ∞ ˙ 1 2 Q 1 2 dX  e−X , (2.4.25) t > 0: θ = √ 4π DT ρcp (Tb − Tu ) R π R/√4DT t √ ∞ 2 where (2/ π ) 0 dX  e−X = 1. The relaxation time of the temperature profile towards the steady state, θ ∝ 1/R, increases with the radius as R2 /DT and the quasi-steady state approximation is not valid at large distance. In the intermediate regime, |R˙ f |Rf ≈ DT , all the terms in (2.4.23) are of the same order of magnitude and the quasi-steady state approximation is not valid. These order-of-magnitude estimates cast doubt on the relevance of the quasi-steady state approximation to study the critical conditions of flame ignition. This is because, near the critical condition, the flame position reaches a plateau value where the flame velocity is very small, R˙ f UL , so that the flame structure becomes closer to that of the Zeldovich kernel than that of the planar flame. Nevertheless, experiments[3] of flame initiation in mixtures with a Lewis number larger than unity, Le > 1, yield trajectories R˙ f -Rf in qualitative agreement with those obtained by the semi-phenomenological analysis using the quasi-steady state approximation;[4] see Fig. 2.24. For Le < 1 the slow motion of flame dynamics[5] is controlled by the unsteady effects in the far field (and not by the quasi-steady state approximation), leading to a new diffusioncontrolled regime of propagation, different from planar (or quasi-planar) propagation. This is especially the case for mixtures near to the flammability limits where the flame radius

[3] [4] [5]

Kelley A., et al., 2009, Combust. Flame, 156, 1006–1013. He L., 2000, Combust. Theor. Model., 4, 159–172. Joulin G., 1985, Combust. Sci. Technol., 43, 99–113.

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is larger than the laminar flame thickness dL . The corresponding study is performed in Section 9.3. Solutions to (2.4.21)–(2.4.22) are relevant for quasi-steady and quasi-planar flames in spherical or cylindrical geometry, for converging flames stabilised in the flow of a point source or at the tip of a Bunsen burner flame where curvature-induced quenching is observed in mixtures with Le < 1; see Section 2.4.5. In such case R˙ f is not the velocity of the flame radius in the laboratory frame but the flame speed Uf , which is close to UL . Quasi-Steady State Solutions The method of solution of Equations (2.4.21)–(2.4.22) in the limit β → ∞ is relatively simple and follows the same lines as for the Zeldovich kernel (2.4.11). Considering a flame radius Rf (t) larger than the thickness of the reaction zone, dL /β, the structure of the reaction layer is the same as in the planar flame. Equation (2.4.8) is still valid to leading order in the limit β → ∞. This gives the boundary condition at the reaction sheet in the preheated zone  dθ DT τb (2.4.26) = ±eβ(θf −1)/2 , β → ∞, R = Rf : θ = θf , dR where here τb ≡ β 2 τrb /(2Le). The minus (plus) sign is for expanding (converging) flames. According to (2.4.21)–(2.4.22), the equations in the external zones can be written         DT d D d 2 dθ 2 dθ 2 dψ 2 dψ + R = 0, R + R = 0. (2.4.27) R dR dR dR dR R˙ f dR R˙ f dR For expanding flames, R˙ f  0, the solution in the preheated zone takes the form R > Rf :

˙ θf e−(Rf R/DT ) dθ = − 2 ∞ ,  ˙  dR R e−(Rf R /DT ) dR Rf

˙

1 e−(Rf R/D) dψ = 2 ∞ ,  ˙  dR R e−(Rf R /D) dR Rf

R2

(2.4.28)

R2

where both the boundary condition at the front R = Rf :

θ = θf ,

ψ = 0,

(2.4.29)

and (2.4.19) at R → ∞ have been used. In the burnt gas, the solution to (2.4.27) satisfying the boundary conditions (2.4.19) at R = 0 is uniform, R  Rf :

θ = θf ,

ψ = 0,

and the energy balance across the reaction zone (2.4.15) reduces to  1 dψ dθ + = 0. dR Le dR R=Rf +

(2.4.30)

(2.4.31)

The expression for the flame temperature θf in terms of Rf and R˙ f is obtained by introducing (2.4.28) into (2.4.31). A nonlinear relation between R˙ f and Rf is then obtained when this expression for θf is introduced into (2.4.26), where dθ/dR at R = Rf is computed from the first equation (2.4.28). This ordinary differential equation for Rf (t) has been investigated

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numerically;[1] see Fig. 2.24. Analytical expressions can be obtained by noticing that the transcendentally small or large terms in (2.4.26) are eliminated in the limit β → ∞ when (θ − 1) is small, of order 1/β. This is the case if the Lewis number is close to unity,[2] β(Le − 1) = O(1). It can also be the case for (Le − 1) = O(1) when the radius of the flame is sufficiently large Rf /dL = O(β), as we shall see later in (2.4.38). Let’s consider the leading order in the limit β → ∞, β(Le − 1) = O(1). For small values of (Le − 1), the expression for (θf − 1), obtained from (2.4.31), is a function of X ≡ R˙ f Rf /DT  ∞ −X      e dX /X 1 − 1 [1 + X − J(X)] , J(X) ≡  X∞ −X    2 , θf − 1 = (2.4.32) Le dX /X X e proportional to (Le−1 − 1). To leading order, θf = 1, the first equation (2.4.28) at Rf yields R = Rf :

1 dθ = − I(X), dR Rf

I(X) ≡  ∞ X

e−X /X   . e−X dX  /X 2

(2.4.33)

Introducing the velocity UL and the thickness dL of the planar front of the ZFK model √ (8.2.40), dL ≡ DT τb , DT = UL dL , the variable X takes the form X ≡ (R˙ f /UL )(Rf /dL ). A differential equation is obtained by introducing (2.4.32) and (2.4.33) into (2.4.26). This relation between Rf /dL and R˙ f /UL depends on the activation energy and on the Lewis number through a single parameter of order unity, l ≡ β(Le − 1). The Zeldovich flame kernel (2.4.11) is recovered in the limit 0 < X 1: J(X) ≈ −X ln X → 0, θf → 1/Le and I(X) ≈ 1 − X ln X → 1, dL dθ /dR|R=Rf → dL /Rf . However, because of the logarithmic term, the relation between R˙ f and Rf is not analytic in the limit X → 0. This is in contradiction with the stability analysis of the Zeldovich kernel presented in Section 9.2.1, where perturbations to the radius are shown to grow exponentially with time involving a linear growth rate, usually of the order of the inverse of the transit time. This indicates that, as already said, the quasi-steady state approximation (2.4.21)–(2.4.22) is not valid for the dynamics near to the steady spherical solution (flame kernel). Weakly Curved Flame The case of weakly curved flames is obtained from (2.4.32)–(2.4.33) in the limit of large X ≡ (R˙ f /UL )(Rf /dL ) using partial integration to evaluate I(X) and J(X) for X  1, limX→∞ J(X) = X + 1 − 2/X + O(1/X 2 ), limX→∞ I(X) = X[1 + 2/X + O(1/X 2 )]. This yields the following expressions for the flame temperature and the flame velocity,     R˙ f (θf − 1) 2 1 1 dθ  1 + =β −1 + · · · , = X  1: β + · · · , −d (1 ) L 2 Le X dR R+ UL X f     R˙ f 2 1 1 1 + + · · · = exp β (2.4.34) −1 (1 + · · · ) , UL X Le X

[1] [2]

He L., 2000, Combust. Theor. Model., 4, 159–172. Joulin G., Clavin P., 1979, Combust. Flame, 35, 139–153.

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where the last equation expressing R˙ f in terms of X  1 is obtained from (2.4.26) and (2.4.32)–(2.4.33) when relations DT = UL dL and (dL /Rf )X = (R˙ f /UL ) are used. For a  the 

1 parameter l ≡ −β Le − 1 ≈ β(Le − 1) of order unity in the limit β → ∞, l = O(1), the exponential in (2.4.34) can be expanded in powers of 1/X. To leading order in the limit X → ∞, the planar flame is recovered, β(θf − 1) → 0, R˙ f ≈ UL . A weak curvature correction to the propagation velocity is obtained at the following order   R˙ f 2 l 1 + + · · · = e−(l/X+··· ) ≈ 1 − + · · · . (2.4.35) X  1, l = O(1): UL X X

Equation (2.4.35) is consistent with the limit X  1 if the flame radius is much larger than the thickness of the planar flame, dL /Rf R˙ f /UL ≈ 1. Using the relation 1/X = dL /Rf + · · · one gets the front velocity of a spherical flame in the form of an expansion in powers of the curvature dL /Rf , β → ∞, l = O(1), dL /Rf 1:

R˙ f /UL ≈ 1 − (l + 2)(dL /Rf ) + · · · .

(2.4.36)

The expansion velocity R˙ f of the spherical flame is smaller or larger than the propagation velocity UL of the planar flame, depending on the sign of [Le − (1 − 2/β)]; l > −2 ⇒ R˙ f < UL , and l < −2 ⇒ R˙ f > UL . This is in agreement with the stability analysis of the planar flame in the thermo-diffusive model – see (2.4.3) and (10.2.26) – the quantity (l + 2)/2 being the Markstein number (10.3.36) in the limit ρb → ρu (υb → 1). The dynamics of ever-expanding spherical flames with a structure close to that of a planar flame is given more generally, including the gas expansion, by (2.3.13) for the modification to the normal burning velocity with a Markstein number computed in (10.3.36) for the one-step ZFK model or in (10.3.102) for a more sophisticated model of the reaction rate.

2.4.5 Curvature-Induced Quenching. Open-Tip Bunsen Flames In this section, we describe nonlinear phenomena in spherical or cylindrical flames produced by small curvature. The effect is amplified by the large reduced activation energy β. Attention is focused on flames of radius larger than dL by a factor β, producing modifications to the front velocity δ R˙ f of order of the laminar flame speed UL . Nonlinear solutions to (2.4.21)–(2.4.22) are sought in the following conditions: β → ∞:

Rf /dL = O(β),

δ R˙ f /UL = O(1).

(2.4.37)

Quenched Curved Flame for Le > 1 For expanding flames a nonlinear relation between R˙ f /UL and dL /Rf is obtained from (2.4.34) if the exponential in the right-hand side is of order unity in the limit β → ∞      R˙ f 1 UL dL 1 1 . (2.4.38) −1 = O(1), −1 ≈ exp β β Le X UL Le R˙ f Rf

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99

Curved flame velocity,

1

0

5 Scaled flame radius,

10

Figure 2.25 Reduced velocity of a quasi-steady curved flame as a function of scaled radius.

This equation is compatible with the expansion for large X, used in (2.4.34), if the quantity (1/Le − 1) is of order unity and X is of order β, namely when the conditions (2.4.37) are fulfilled. Equation (2.4.38) can be also be obtained directly from (2.4.21)–(2.4.22) when the curvature term DT /R is replaced by a small constant term, DT /Rf = O(UL /β). Written in the form    ˙  ˙   Rf Rf 1 dL 1 dL ln = : −1 β , =O (2.4.39) (Le − 1) = O(1), Rf β UL UL Le Rf Equation (2.4.38) shows the existence of a turning point for Le > 1. Expanding flames 1 with a radius smaller than a minimum value Rfc ≡ 1 − Le (βdL e) cannot propagate in the quasi-steady state approximation. The critical flame velocity at quenching is smaller than the laminar flame speed, but the relative modification is of order unity, R˙ f /UL = 1/e (see Fig. 2.25), so that the flame thickness is still of order dL . For Le > 1 in the limit β → ∞, the critical radius at quenching Rfc is smaller than the radius of the Zeldovich 

kernel (2.4.11), Rfc < RfZ ≈ dL exp

β 1 2 (1 − Le )

.

It is tempting to define a new criterion for ignition[1] based on Rfc (and not on Rf Z ) by assuming that the branch R˙ f + should attract the trajectory in the phase space R˙ f -Rf . This is not clear for the following reasons. Equation (2.4.39) tells us that the quasi-steady state approximation cannot be verified for a flame radius Rf smaller than Rfc . Therefore the fully unsteady solution of (2.4.1) cannot satisfy the quasi-steady approximation when the flame radius Rf (t) approaches Rfc . Moreover, near the critical conditions, the numerical study of the trajectories R˙ f -Rf using the steady-state approximation from the beginning of a point ignition process[1] shows a drastic slow-down, R˙ f UL (and an increase in flame thickness), followed by a sudden strong acceleration, just before catching the R˙ f + branch (see Fig. 2.24), so that the quasi-steady state approximation is not valid. If the slow motion lasts a sufficiently long time, of order R2f /DT ≈ (Rf /dL )2 τL , sufficient for the Zeldovich solution to develop, the critical radius Rf Z could still be the appropriate [1]

He L., 2000, Combust. Theor. Model., 4, 159–172.

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minimum radius to be reached for a successful ignition. However, numerical simulations[1] of the fully unsteady equations for Le = 1.8 and β = 15 show that the transition towards a self-sustaining spherical flame occurs at a radius significantly smaller than RfZ . Much work remains to be done to clarify the problem of flame initiation. Open-Tipped Bunsen Flames for Le < 1 Interesting results are obtained for converging cylindrical flames:[2]   d 1 d 1 d d2 R = = + 2 , Uf ≡ −R˙ f > 0. R dR dR R dR dR This configuration is useful to explain the opening of Bunsen flame tips, and of highly curved flame tips in general. As already mentioned in Section 2.3.4, this phenomenon was observed long ago in lean hydrogen–air flames and rich heavy hydrocarbon–air flames (Le < 1). Considering large flame radius, DT /(UL Rf ) = O(1/β), Uf /UL = O(1), the curvature terms in the left-hand side of (2.4.21)–(2.4.22) are small, of relative order 1/β. Up to first order, the radial coordinate R in DT /R can be replaced by the flame radius Rf , (DT /R)dθ/dR → (DT /Rf )dθ/dR. The solution in the preheated zone takes the form  

Uf 1 R − Rf , R  Rf : (2.4.40) θ = θf exp − DT Rf  

Uf 1 R − Rf , 1 − ψ = exp Le (2.4.41) − DT Rf where the boundary conditions at the flame front (2.4.29) have been used. These expressions, in which the curvature 1/Rf introduces a correction of order 1/β, are valid up to the order 1/β. The temperature and the mass fraction decreasing exponentially with a length scale DT /Uf , the boundary condition at R = 0 in (2.4.19) is satisfied by (2.4.40)– (2.4.41) for large radius Rf Uf /DT = O(β) when transcendentally small terms in the limit β → ∞ are neglected. The flame temperature is obtained by introducing (2.4.40)–(2.4.41) into (2.4.31):   DT 1 −1 [1 + O(1/β)]. (2.4.42) (θf − 1) = − Le Rf Uf Thanks to the small departure of the flame temperature Tf from the adiabatic flame temperature Tb (planar flame) at R = ∞, (θf − 1) = O(1/β), the gradient of temperature in the burnt gas is expected to be of an order of magnitude smaller than 1/β, as it should be for the validity of (2.4.31). Introducing (2.4.42) into (2.4.26) with the leading order of the temperature gradient in the preheated zone dθ/dR|R=Rf ≈ Uf /DT , obtained from (2.4.40), yields a nonlinear equation for Uf of the same form as (2.4.39),         Uf Uf 1 dL dL 1 ln =− : − 1 β . (2.4.43) (Le − 1) = O(1), =O Rf β UL UL Le Rf [1] [2]

He L., 2000, Combust. Theor. Model., 4, 159–172. Frankel M., Sivashinsky G., 1984, Combust. Sci. Technol., 40, 257.

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Figure 2.26 Extinction by curvature. This photo shows extinction at the cusps of a rich (φ = 1.4) cellular two-dimensional propane–air flame propagating in a Hele-Shaw cell. Courtesy of C. Almarcha and J. Quinard, IRPHE Marseilles.

  1 For Le < 1 this expression has a turning point at a critical flame radius Rfc = Le −1 (βdL e). There is no quasi-steady cylindrical solution for a converging flame with a radius smaller than Rfc , Rf < Rfc . The quenching occurs at a nonzero flame velocity Uf = UL /e. This provides a simple explanation of the opening of Bunsen flame tips in which the planar cross sections, perpendicular to the flow, of the conic flame can be approximated by a converging cylindrical flame in equilibrium with the normal component of the vertical flow. Comparison with experiments is not so easy because the flame can be cellular (Le < 1) and the cross section can differ from a circle. Open cusp tips are also observed in the quasitwo-dimensional flame front of rich hydrocarbon flames propagating in a Hele-Shaw cell, shown in Fig. 2.26. 2.5 Thermo-Acoustic Instabilities The dynamics of flame fronts may couple to acoustic waves in a combustion chamber or in a burner holding the flame. This coupling can lead to thermo-acoustic instabilities. All systems using confined combustion (boilers, gas turbines, rocket engines, etc.) are prone to thermo-acoustic instabilities. These instabilities may be dangerous, especially in rocket engines where the energy density is extremely high, so a small amount of the chemical energy of combustion when transformed into mechanical energy is sufficient to produce destructive damage. The industrial difficulty lies in the identification of the exact mechanism(s) of coupling and the quantification of acoustic losses. There are many possible coupling mechanisms, particularly when the fuel is injected in the liquid phase. There are also different damping mechanisms, such as viscosity, thermal transfer across the boundary layers at the walls, acoustic radiation through exhaust holes, inhomogeneities of the basic flow, phase lags etc and so on. The amplification rates are often of the same order of magnitude as the damping rates, making the predictive design of stable high power combustion chambers a difficult task. Moreover, the turbulence-induced noise can influence high-frequency thermo-acoustic instabilities.[3,4]

[3] [4]

Clavin P., et al., 1994, Combust. Sci. Technol., 96(61-84). Noiray N., Schuermans B., 2013, Int. J. NonLin. Mech., 50, 152–163.

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Thermo-acoustic instabilities were first observed in simple laboratory experiments a long time ago. At the end of the eighteenth century Byron Higgins discovered the ‘singing flames’ produced when a hydrogen diffusion flame stabilised at the mouth of a burner is introduced into a tube.[1,2] Another classical example is the Rijke tube: when an electrically heated gauze is placed in certain positions in an open vertical tube, a continuous loud tone is spontaneously produced, as first observed by Rijke in 1859.[3] During the nineteenth century, people were fascinated by various gauze tone phenomena, in particular in gas lamps. A century later thermo-acoustic instabilities became the nightmares of the engineers in charge of building rocket engines, and even now, the design of stable rocket engines remains an industrial challenge.[4] High-frequency instabilities coupled by transverse acoustic modes are the most dangerous ones in liquid rocket engines. The understanding of this phenomenon has much improved recently.[5] The same is true for annular gas turbine combustion chambers.[6,7] We will not consider technical aspects here, but focus our attention on scientific problems. The topic is old and the state of the art before 1980 is well documented in the literature.[2,8] In the absence of a systematic analysis of the structure of wrinkled flames, the pioneering theoretical developments were based on semi-phenomenological considerations. Analytical studies of the structure of wrinkled flames, coupling the diffusive transport mechanisms with the gas expansion, were achieved only after 1980.[9] Moreover, diagnostic techniques have spectacularly improved since 1980. In this section, attention is focused on some laboratory experiments carried out after 1990 in continuity with the earlier works. A simple experiment demonstrates the rich variety of thermo-acoustic instabilities of flames. It is provided by a premixed flame propagating from the top to the bottom of a vertical tube. For a sufficiently high laminar flame velocity, strong acoustic oscillations are produced, leading to a strong oscillatory acceleration of the flame, ending by a violent disruption of the initially regular front as shown in Fig. 2.27d, where a instantaneous cut through such a flame front has been obtained using the tomographic technique.[10] A phenomenon of this type was first documented in the nineteenth-century experiments of Mallard and Le Chatelier,[11] but the first explanation was not given until much later.[8] Before going into the theoretical analysis it is worth recalling the results of experiments carried out with modern diagnostics.[12]

[1] [2] [3] [4]

Higgins B., 1802, A Journal of Natural Philosophy, Chemistry and the Arts, 1, 129–131. Strehlow R., 1979, Fundamentals of combustion. New York: Kreiger. Rijke P., 1859, Phil. Mag., 17, 419–422. Yang V., Anderson W., 1995, Liquid rocket engine combustion instability, Progress in Astronautics and Aeronautics, vol. 169. Washington, D.C.: AIAA. [5] Mery Y., et al., 2013, C. R. M´ecanique, 341, 100–109. [6] Noiray N., Schuermans B., 2013, Int. J. NonLin. Mech., 50, 152–163. [7] Noiray N., Schuermans B., 2013, Proc. R. Soc. London Ser. A, 469, 20120535. [8] Markstein G., 1964, Nonsteady flame propagation. New York: Pergamon. [9] Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237. [10] Boyer L., 1980, Combust. Flame, 39, 321–323. [11] Mallard E., Le Chatelier H., 1883, Annales des Mines, Paris, Series 8(4), 296–378. [12] Searby G., 1992, Combust. Sci. Technol., 81, 221–231.

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2.5.1 Vibratory Instability of Flames in Tubes In a simple laboratory experiment, a flame propagates downwards from the open end towards the closed end of a vertical tube containing premixed propane–air initially at rest. In the experiments described here the tube is 120 cm long, 10 cm in diameter and contains a lean to stoichiometric mixture of propane and air. Depending on the burning rate, four different types of behaviour of the flame can be observed:[12] (i) For sufficiently lean mixtures, below a laminar flame speed of ≈ 16 cm/s, the flame propagates to the bottom of the tube producing no sound. (ii) For lean laminar flame speeds in the range ≈ 16–25 cm/s, a primary acoustic instability occurs when the flame is in the lower part of the tube. (iii) For lean laminar flame speeds in the range ≈ 25–30 cm/s, a violent secondary instability occurs after the primary instability. (iv) For even faster flames, the secondary instability reaches an extremely high acoustic intensity (up to 180 dB) and the cellular nature of the flame front eventually degenerates into incoherent turbulent motion before reaching the base of the tube. A similar behaviour, but with different thresholds on the laminar flame speeds, was observed in tubes of various lengths ranging from 50 to 180 cm, and with internal diameters between 6 and 10 cm. Other authors have observed equivalent behaviours in lean methane– air[13] and not too-rich hydrogen–air mixtures.[14] For all cases, in the first half of the tube the flame has a curved shape generated by the DL instability (see Fig. 2.27a), and no sound is generated. In these conditions the combustion region propagates at roughly twice the laminar flame speed, as would be expected from the ratio of the surface area of the flame to the cross section of the tube. In cases (ii) and (iii) a thermo-acoustic instability develops when the flame is near to the midpoint of the tube. The growth rate of the acoustic pressure is of the order of 10 s−1 . After few hundred milliseconds the flame becomes quasi-planar, as shown in Fig. 2.27b, and the acoustic pressure saturates around 500 Pa (≈ 145 dB). This phenomenon was called a ‘vibrating flat flame’.[15] The pressure continues to increase but at rate (typically 0.3 s−1 ) much lower than during the first stage of the acoustic instability when the flame front was curved. In case (ii) the vibrating flat flame propagates down to the bottom of the tube. In case (iii), a strong secondary instability develops during the propagation in the lower half of the tube. Small cells appear on the vibrating flat flame pulsating with a period (≈ 14 ms) exactly twice the acoustic period (≈ 7 ms) and increase rapidly in amplitude; see Fig. 2.27c and d.. This is the signature of a parametric instability, similar to the Faraday instability[16] observed for free liquid surfaces subjected to a periodic vertical acceleration. During the first stage of the secondary instability, which lasts approximately 0.1 s, the acoustic pressure grows at a fast rate, typically 30 s−1 . The acoustic amplitude reaches a maximum [13] [14] [15] [16]

Aldredge R., Killingsworth N., 2004, Combust. Flame, 137, 178–197. Yanez J., et al., 2015, Combust. Flame, 162, 2830–2839. Kaskan W., 1953, Proc. Comb. Inst., 4, 575–591. Faraday M., 1831, Philos. Trans. R. Soc. London, 121, 299–338.

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(b)

(c)

(d)

Figure 2.27 Tomographic cuts showing successive stages of a thermo-acoustic instability of a propane flame in a 10 cm diameter tube. The fresh gas is lit up by a short (10−9 s) vertical laser sheet. The self-excited acoustic wave leads to a violent oscillating disruption of the initially regular front, as shown in (d). Photos courtesy of Geoff Searby.

of ≈ 5000 Pa and then decreases. The flame reaches the bottom of the tube after a few more tenths of a second. The average propagation speed between the start of the parametric instability and the end of propagation is typically 1.7 m/s, much higher than the laminar flame speed (UL = 28 cm/s). Typical pressure recordings are shown in Fig. 2.28. In case (iv) the secondary instability is even more violent. For flames close to stoichiometry (UL = 42 cm/s), the onset of the secondary instability appears during the growth of the primary instability. At early stages of sound production, the front flattens for three or four acoustic cycles, and the growth of the parametric cellular structures lasts another 10–12 cycles. During this short period of time the growth rates of the acoustic pressure and of the cell amplitude are 31 s−1 and 60 s−1 , respectively. The acoustic pressure reaches 25 kPa, after which the coherent pulsating cells destabilise quickly into an incoherent highly turbulent motion, while the velocity of the combustion region reaches 7.5 m/s. The breakdown of the cellular structure is accompanied by a rapid decay of the acoustic pressure with a decay rate (> 60 s−1 ) much higher than the natural decay rate of the acoustic in the tube ≈ 5 s−1 . This fast decay is not yet well explained. It would seem that the energy of the acoustic waves is transferred rapidly into the turbulence of the fluid. The pressure waves generated by the acceleration of the flame may play an important role in the final stage. After being reflected at the bottom of the tube they may break down the cellular structure. Except for

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Figure 2.28 Recordings of acoustic pressure and flame position in a tube during thermo-acoustic instabilities for cases (ii), (iii) and (iv) (propane flame speeds 22, 27 and 42 cm/s, respectively). Note the changes in pressure scales.

the final fast decay, the other behaviours are well understood from a linear equation similar to (2.2.18), including the acoustic acceleration; see (2.5.13). To begin with, let’s recall some general considerations on thermo-acoustic instabilities.

2.5.2 Rayleigh Criterion, Admittance Function Generally speaking, quasi-isobaric combustion increases the volume of the reactive mixture when it is transformed into burnt gas. Therefore unsteady combustion acts as an unsteady volume source, generating fluctuations in the flow velocity. This excites the acoustic modes

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of the combustion chamber. The fluctuations of temperature and/or flow velocity associated with the resulting acoustic wave can in turn modulate the local combustion rate (feedback). Depending on the phase and gain of the feedback, this loop may lead to instability. Rayleigh Criterion for a Distributed Heat Release The general principle of thermo-acoustic instabilities can be understood in simple terms as follows. Consider the equation for the pressure of acoustic waves (d’Alembert’s equation) in the presence of heat release when the Doppler effect and the gradients in the mean flow are neglected, ∂ 2 p/∂t2 − a2 p = ∂ q˙ γ /∂t,

(2.5.1)

where a is the sound speed, q˙ γ ≡ (γ − 1)˙qv , γ ≡ cp /cv is the ratio of specific heats and q˙ v is the rate of heat release per unit of volume; see Section 15.2.4. As first remarked by Rayleigh,[1] this system is unstable if there is a coupling that produces oscillations of heat release rate in phase with the oscillations of pressure (positive feedback). This is easily shown from Equation (2.5.1) in one-dimensional geometry. Consider the nonrealistic case of a heat release rate homogeneously distributed in the tube. Assume for simplicity that the fluctuations of heat release rate are directly proportional to that of pressure with a positive coefficient, 1/τins > 0, δ q˙ γ = δp/τins .

(2.5.2)

Using a spatial Fourier representation δp(x, t) =

∞ 

p˜ k (t)eikx ,

(2.5.3)

k=−∞

where k is the longitudinal wavenumber of an acoustic mode, Equation (2.5.1) shows that each pressure mode is a solution to an equation describing an unstable oscillator, 1 d˜pk d2 p˜ k + a2 k2 p˜ k = 0. − τins dt dt2

(2.5.4)

In the case where the fluctuations of heat release rate and pressure are in phase opposition (1/τins < 0), combustion is a damping mechanism for acoustic waves. This simple analysis is easily extended to the case of a linear relation nonlocal in time and space between δ q˙ γ (r, t) and δp(r, t); see Section. Nonlinear terms added to (2.5.2) may lead to limit cycles and oscillation relaxations. However, terms that are neglected in (2.5.1) introduce damping, which counteracts the instability. Localised Heat Release: Admittance Function The case of a heat release sharply localised in space is treated in a similar way. Consider a planar flame propagating downwards in a tube; the thickness of the flame is negligible [1]

Rayleigh J., 1945, The theory of sound, vols. 1 and 2. New York: Dover.

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Figure 2.29 Definition of the positive direction for oscillating acoustic velocity fluctuations, δuu and δub . In the absence of interaction δuu = δub and the flame propagates at velocity UL with respect to the fresh gas.

compared with the acoustic wavelength. The region of heat release may thus be considered as a discontinuity of the flow velocity. This is also the case for a cellular and/or turbulent flame when the thickness of the flame brush (usually not much larger than the tube diameter) is much smaller than the length of the tube. Because of the density change, there is a jump of flow velocity across the flame, ub − uu = 0. Neglecting relative modifications to the pressure of the order of the square of the Mach number, the pressure pf is continuous across the flame. Focusing our attention on the effect of combustion, the mean energy per unit flame surface transferred per unit time (energy flux) to the acoustic modes, E˙ t , is equal to the product of the pressure fluctuation and the velocity jump fluctuation δu across the thin flame, E˙ t = (δub − δuu )δpf ,

(2.5.5)

where the overbar means ‘time average during a period of oscillation’, δpf (t) is the pressure fluctuation of the acoustic wave at the flame and δub (t) and δuu (t) are the velocity fluctuations at the burnt gas and fresh mixture (unburnt) sides of the discontinuity respectively; see Fig. 2.29. Note that the pressure fluctuation in acoustic waves, δp, is of order ρa δu. The pressure jump across a laminar flame is of order ρu UL2 . Thus, for δu of order UL , it is effectively negligible compared with the pressure fluctuations at the front by a factor of the order of the Mach number M ≡ UL /a. The velocity difference (δub − δuu ) results from gas expansion with quasi-isobaric heat release. According to the mass and energy conservation equations in the low Mach number approximation, the divergence of the flow velocity is proportional to the heat release rate per unit volume, q˙ v , ∇.u =

q˙ γ 1 DT q˙ v = = ; T Dt ρcp T ρa2

(2.5.6)

see Equations (15.2.2)–(15.2.5) along with (15.1.29). In one-dimensional geometry, spatial integration across the heat release zone yields  δ q˙ γ (δub − δuu ) = dx. (2.5.7) ρa2 If the reaction zone is not homogeneous in the transverse direction, δ q˙ γ should be simply replaced by its average in the transverse direction. The difference of velocity (δub − δuu )

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thus fluctuates with the heat release rate, whose perturbations are generated by the local properties of the acoustic wave at the flame. The difficult problem left in (2.5.5) is to determine the relation linking (δub − δuu ) to δpf . In the linear approximation, using δu(t) = (1/2)ˆueiωt ,

δpf (t) = (1/2)ˆpf eiωt ,

(2.5.8)

where |ˆu| and |ˆpf | are the amplitudes of the velocity and pressure oscillations at the flame front, we introduce a nondimensional admittance function, Z, relating (ˆub − uˆ u ) to pˆ f : (ˆub − uˆ u ) = Z pˆ f /ρb ab .

(2.5.9)

The admittance is a complex function of the acoustic frequency ω. It is convenient to introduce the nondimensional frequency reduced by the transit time across the planar flame τL ≡ dL /UL . In the experiments reported in Section 2.5.1, the reduced frequency ωτL is typically of order unity. The admittance function Z(ωτL ) can be obtained from an unsteady analysis of the inner structure of the flame (or flame brush). Using (2.5.8) and (2.5.9) in (2.5.5), the energy flux into a given acoustic mode is  1 |ˆpf |2 1 Z pˆ f pˆ ∗f + Z ∗ pˆ ∗f pˆ f = Re(Z(ωτL )) , (2.5.10) E˙ t = 4ρb ab 2 ρb ab where ∗ denotes the complex conjugate. All acoustic modes are thus potentially unstable when Re(Z(ωτL ) > 0. By definition, the instability growth rate is 1/τins = E˙ t /E, where E ≈ 0.5|ˆp|2 L/(ρa2 ) is the total energy of the acoustic wave per unit area of the tube, L is the length of the tube and |ˆp| is the amplitude of the acoustic mode in the tube (at an antinode). The order of magnitude of the linear growth rate obtained from (2.5.10) is E˙ t /E ≈ Re(Z(ωτL ))(a/L), τa /τins = O (Re(Z)) , where τa = L/a ≈ 1/ω is the characteristic acoustic time. The ratio |ˆpf |2 /|ˆp|2 = O(1), namely the relation between the acoustic pressure at the flame, |ˆpf |, and the acoustic energy per unit area in the tube, E, is a function of the position of the flame in the tube. The growth rate τa /τins thus also depends on a geometrical factor 0 < F (r) < 1, where r is the relative position of the flame in the tube.[1,2] Typical curves for the geometrical factors of the fundamental mode of a tube, open at one end and closed at the other, are shown in Fig. 2.30. They are plotted for two different types of coupling studied below: pressure coupling and velocity/acceleration coupling. The geometrical factors go to zero at the open end of the tube, r = 1, where the pressure fluctuations disappear. For velocity/acceleration coupling the geometrical factor also goes to zero at the closed end where the velocity fluctuations disappear. As already mentioned, damping mechanisms such as heat transfer and viscous friction at the walls, acoustic radiation at the open end, inhomogeneities of the unperturbed flow (termed ‘flow turning’ in rocket engine contexts[3] ) and so on counteract the gain of combustion. The stability limits are obtained by balancing the gain and the loss. The detailed [1] [2] [3]

Clavin P., et al., 1990, J. Fluid Mech., 216, 299–322. Clanet C., et al., 1999, J. Fluid Mech., 385, 157–197. Culick F., 1975, Combust. Sci. Technol., 10, 109–124.

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Gain factor

2.5 Thermo-Acoustic Instabilities

Relative position of flame in tube,

Figure 2.30 Geometrical dependence of thermo-acoustic gain for pressure and velocity/acceleration couplings as a function of the relative position, r, of the flame in a half-open tube. Calculated for the fundamental frequency. The open end corresponds to r = 1.

analysis of the thermo-acoustic growth rate requires not only the identification of the coupling mechanism and the computation of the different transfer functions for gain and loss, but also the solution of the full acoustic problem taking into account the difference of density and sound speed in the burnt gas and the fresh mixture.

2.5.3 Pressure and Velocity Coupling There are many possible coupling mechanisms, each of which is described by an admittance function. The simplest mechanism is the sensitivity of the heat release rate to temperature: adiabatic compression changes the temperature of the gas, which in turn changes the heat release rate. This is called a pressure coupling. Pressure Coupling The calculation of Z(ωτL ) for pressure coupling has been done for planar gaseous flames in the framework of the ZFK model[1] and also with two-step chain-branching models;[4] see Fig. 2.31. The detailed thermo-acoustic stability of planar flames propagating in tubes, including geometrical factors, has also been carried out.[1] The function Re(Z(ωτL )) shown in Fig. 2.31 has a wide maximum for ωτL of order unity, which is typically the case for longitudinal acoustic modes in tubes with a length L ≈ 1 m. The maximum is more pronounced for high Lewis numbers (heavy fuels) and the high-frequency response is different in the singlestep and two-step models. The admittance function is everywhere positive. The peak value corresponds to a small reduced linear growth rate τa /τins whose order of magnitude is   τa E , (2.5.11) = O (γ − 1)Mb τins kB Tb [4]

Clavin P., Searby G., 2008, Combust. Theor. Model., 12(3), 545–567.

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Figure 2.31 Real part of admittance function for the ZFK model (solid line) and for a flame with twostep chemistry (dotted line) with radical recombination allowed everywhere; see Equation (5.1.3). Note that the admittance function is divided here by the Mach number of the burnt gas, so the order of magnitude of Re(Z) is 10−3 . The parameters are representative of a stoichiometric flame for fuels with Lewis numbers of 0.8 and 1.4.

where Mb ≡ Ub /ab is the Mach number of the flow in the burnt gas. The presence of the Mach number Mb comes from the acoustic pressure δp/p ≈ Mb (δub /Ub ), and the reduced activation energy E/kB Tb results from the Arrhenius law (1.2.2). In comparison, the effect of pressure on the combustion rate through the density is usually a power law with an exponent n smaller than E/kB Tb . The effect of pressure on density is thus usually smaller than the pressure coupling through temperature. In the ordinary conditions of laboratory experiments, the pressure coupling leads typically to τa /τins  10−3 . This is too small by a factor 102 to explain the initial growth of the primary instability of cellular gaseous flames propagating in tubes, case (ii) in Section 2.5.1. The typical growth rate obtained from the experiments, when the damping rate by the losses is subtracted, is about 15 s−1 (τa /τb ≈ 0.05).[1] However, pressure coupling may explain the small acoustic growth rate (≈ 0.1 s−1 ) observed during the vibrating flat flame in Fig. 2.27b; see also Fig. 2.28b and c. It was also argued that the unsteady effects in the edge of the flame front at the wall may play a role;[2] however, because of quenching at the wall, it is difficult to evaluate this last effect. The admittance function of a homogeneous solid propellant has also been determined by theoretical analyses. The heat is released in the gas phase near the wall of solid propellant and the admittance function relates the velocity fluctuations of the burnt gas to the pressure at the wall, yielding a boundary condition for the acoustic waves. The first calculation of the fluctuation of the combustion rate was obtained by Zeldovich in 1942 by assuming that the gas phase is in a quasi-steady-state, ωτL 1. More recently, nonsteady effects in the gas phase have been taken into account in the calculation of the admittance function for investigating higher frequency acoustic modes, ωτL  1;[3] see Fig. 2.32. A detailed [1] [2] [3]

Clanet C., et al., 1999, J. Fluid Mech., 385, 157–197. Kaskan W., 1953, Proc. Comb. Inst., 4, 575–591. Clavin P., Lazimi D., 1992, Combust. Sci. Technol., 83, 1–32.

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111

Solid propellant Reaction zone

Figure 2.32 Typical quasi-steady state and unsteady calculations of admittance function for solid propellants. The parameters used here are representative of high-pressure rocket combustion.

stability analysis for longitudinal acoustic modes in slender solid propellant rockets[4] predicts that this pressure coupling may lead to an overall instability if the nonsteady effects of the gas phase are taken into account in the admittance function. Rijke Tube, Velocity Coupling and Transfer Function The heat release may also be directly modified by the flow velocity. This simplest case of this coupling, called ‘velocity coupling’, occurs in the Rijke tube, where the heat transfer from the gauze to the gas is proportional to the flow velocity. A simplified model has been used to compute the transfer function and the stability limits of the Rijke tube.[5] Because of buoyancy effects, the hot gauze gives rise to a permanent mean flow in the tube, and for reasonable levels of acoustic excitation, the mean flow velocity is generally greater than the maximum acoustic velocity, U > δuu , so that the unsteady heat transfer to the gas is directly proportional to the acoustic velocity. In such a case, it is convenient to introduce the transfer function Tr (ωτL ) (ˆub − uˆ u ) = Tr (ωτL )ˆuu ,

(2.5.12)

where the subscripts b and u denote here the hot and cold side of the gauze, respectively, and the frequency is reduced by the time taken for the mean flow to cross the region of thermal diffusion. According to (2.5.5), with (2.5.8) and (2.5.12), E˙ t = (1/4)(Tr uˆ u pˆ ∗f + Tr ∗ uˆ ∗u pˆ f ). For an acoustic mode of a tube, the acoustic pressure and velocity are in phase quadrature, so uˆ pˆ ∗f = −ˆu∗ pˆ f and the gain is proportional to the imaginary part of Tr , E˙ t = Im(Tr (ωτL ))(iˆuu pˆ ∗f )/2, where iˆuu pˆ ∗f is a real quantity prescribed by the one-dimensional acoustic mode. Its sign depends on the position in the tube. For the fundamental frequency, it is negative in the [4] [5]

Garcia-Sch¨afer J., Li˜nan A., 2001, J. Fluid Mech., 437, 229–254. Nicoli C., Pelc´e P., 1989, J. Fluid Mech., 202, 83–96.

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Hot xxxxxxxx gauze

Figure 2.33 Real and imaginary parts of the transfer function for the Rijke tube, in the limit of small gas expansion ratio.

lower half of the Rijke tube and positive in the upper half. Fig. 2.33 shows Tr calculated in the limit of small gas expansion ratio Tb /Tu 1. The imaginary part of Tr is negative at all frequencies so the instability develops when the grid is in the lower half of the tube. The gain is maximum for reduced frequencies of order unity, which is the typical order of magnitude in laboratory experiments. For higher harmonics iˆupˆ ∗ changes sign several times with position of the gauze in the tube.

2.5.4 Acceleration Coupling and Primary Instability It was suggested in the 1960s[1] that the coupling mechanism responsible for the primary instability of flames propagating in a tube, case (ii) in Section 2.5.1, is due to the modulation of the surface area of the curved flame front by the periodic acceleration of the acoustic wave leading to a modulation of the global heat release rate. The analytical calculation of the corresponding transfer function for curved flames with an amplitude of the same order as the tube radius is a difficult problem, which has not yet been solved. The transfer function has been computed[2] for a weakly cellular flame near the instability threshold of a planar flame propagating downwards described by (2.2.22). In the absence of acoustics, the unperturbed cellular structure is stationary with a fixed ˜ c 1; see (2.2.22) and Fig. 2.13. The wavenumber kc ≈ km /2 and a small amplitude |α|k calculation is based on the linear evolution equation of the flame front (2.2.18) where |g| is replaced by g(t) ≡ |g| + g  (t) and g  (t) = duu /dt is the acceleration of the front due to the acoustic wave,      ρu kdL dα˜ kdL  ρb d2 α˜ − −G  (t) + N α˜ = 0, + 2 − 1 (2.5.13) 1+ 2 2 ρu dt τL dt ρb τL

[1] [2]

Rauschenbakh B., 1961, Vibrational combustion. Fizmatgiz, Moscow: Mir. Pelc´e P., Rochwerger D., 1992, J. Fluid Mech., 239, 293–307.

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G  (t) ≡



ρb ρu



g  (t)dL , UL2

  k , N(kdL ) ≡ −Go + kdL 1 − km

113

(2.5.14)

where Go is defined in (2.2.19)–(2.2.20). Consider now a weakly cellular flame α(y, t) = α˜ cos(kc y) with α˜ = α˜ o + αˆ 1 eiωt + c.c., where α˜ o is the small amplitude of the steady cellular front, αˆ 1 (kc , ω) the complex amplitude of the perturbation (|αˆ 1 | |α˜ o |) created by the acoustic acceleration, g  (t) = iωuˆ u eiωt + c.c. and c.c. stands for complex conjugate. For small amplitudes, kc |α˜ o | 1, the fluctuation of flame surface area per unit cross-sectional area, computed in two-dimensional geometry, is δS/So = (kc2 /2)α˜ o αˆ 1 eiωt + c.c.

(2.5.15)

Since the fluctuation of the rate of heat release per unit surface area is  δ q˙ v dx = ρu UL cp (Tb − Tu ) δS/So , the jump of velocity is, according to (2.5.7) together with ρa2 = (γ − 1)cp ρT, δub − δuu = (Tb /Tu − 1)UL δS/So .

(2.5.16)

Introducing a linear approximation in the third term of (2.5.13), G  (t)α˜ ≈ G  (t)α˜ o , and using the relation g  (t) = iωuˆ u eiωt + c.c., the fluctuation of the wrinkles, αˆ 1 , can be expressed in terms of α˜ o from (2.5.13), αˆ 1 = H(ωτL )(ˆuu /UL )α˜ o ,   −i(ωτL )(kc dL ) ρb   , (2.5.17) H(ωτL ) = 1 − ρu −(ωτ )2 1 + ρb + 2i(ωτ )(k d ) L L c L ρu where we have used N = 0 at the threshold of instability, k = kc . According to (2.5.15)– (2.5.17) the transfer function, Tr (ωτL ) = (ˆub − uˆ u )/ˆuu , is then given by   Tb (kc α˜ o )2 H(ωτL ). −1 (2.5.18) Tr (ωτL ) = Tu 2 This shape of this transfer function is plotted in Fig. 2.34. According to (2.5.5), the energy flux is given by  1 1 Tr uˆ u pˆ ∗f + Tr ∗ uˆ ∗u pˆ f = Im(Tr (ωτL ))(iˆuu pˆ ∗f ), E˙ t = 4 2 so for a standing wave, the gain is again proportional to Im(Tr ). The reduced growth rate of the instability τa /τins is proportional to the relative increase of area of the steady cellular flame (kc α˜ o )2 /2. The imaginary part of the transfer function is positive for all acoustic frequencies, and detailed analysis[2] shows that the geometrical factor (iˆuu pˆ ∗f ) is everywhere positive for the fundamental mode of a tube closed at one end, with a maximum in the lower half of the tube and going to zero at each extremity, see Fig. 2.30. The amplitude of the cells in the vicinity of the stability threshold of a planar flame propagating downwards is very small, (kc αo )2 1. Therefore the growth rate of the thermo-acoustic instability is systematically smaller than the damping rate, so that, according to this analysis, the planar

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Figure 2.34 Real and imaginary parts of the transfer function for acceleration coupling to cellular flames, plotted for UL = 0.1 m/s, Tb /Tu = 7 and kc dL = 0.06.

flame close to the threshold of cellular instability should be thermo-acoustically stable. This is in agreement with the experimental observations since the critical velocity for a planar flame propagating downwards (UL ≈ 10 cm/s) is smaller than the minimum velocity of cellular flames for which sound is generated in experiments[1] (UL  16 cm/s). The above theoretical result, obtained for a small amplitude cellular structures,[2] has been extended in a nonrigorous manner to the experimental case of flames with large amplitude cells far from the stability threshold k2 |α˜ o |2 /2 = O(1)[3] . This was done by arbitrarily modifying the transfer function in three ways: 2 • The increased area of the small amplitude sinusoidal flame (kc α˜ o ) was replaced by the increased area measured experimentally on the cusped flame. • The coefficient N(k) = 0 was retained in the denominator of (2.5.17) for the function H.

When this is done, the agreement between experiments and the calculated growth rate is quite reasonable, as shown in Fig. 2.35. This provides good evidence in favour of the explanation of the acceleration coupling of curved flames being responsible for the primary thermo-acoustic instability of gaseous flames propagating in tubes. A numerical analysis, reproducing qualitatively well the primary instability and the threshold of the parametric instability, has been performed recently,[4] based on a semiphenomenological equation in which quadratic terms for a weakly curved flame have been added to (2.5.13). Two-Phase Flames Another type of acceleration coupling has been observed experimentally for a flame propagating in a spray of decane droplets of a few microns in diameter in air.[3] The experiments [1] [2] [3] [4]

Searby G., 1992, Combust. Sci. Technol., 81, 221–231. Pelc´e P., Rochwerger D., 1992, J. Fluid Mech., 239, 293–307. Clanet C., et al., 1999, J. Fluid Mech., 385, 157–197. Assier R., Wu X., 2014, J. Fluid Mech., 758, 180–220.

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115

Figure 2.35 Comparison of calculated and measured growth rates of acceleration coupled instability. The empty circles are measurements on gaseous propane–air flames and the solid diamonds are measurements on liquid decane spray flames. From Clanet C., et al., 1999, J. Fluid Mech., 385, 157–197, reproduced with permission.

show that the linear growth rate of the instability is at least an order of magnitude higher than for the corresponding case of gaseous cellular flames; see Fig. 2.35. For lean flames, this may be explained as follows.[5] In the presence of a periodic acceleration, the inertia and Stokes’ drag of the droplets produce a phase shift between the velocities of the gas and the liquid droplets. This velocity difference in turn produces oscillations in the flux of fuel into the reaction zone and leads directly to a modulation of the heat release by the acoustic wave. Introducing the viscous relaxation time of the droplets τvis = (2/9)(ρl /ρu )rl2 /ν, where ρl and rl are the density and the radius of the liquid droplets, ρu and ν the density and the kinematic viscosity of the cold air, the order of magnitude of the transfer function, computed with a simple model in which the droplets vaporise before reaching the reaction zone,[5] is   τa E τvis . (2.5.19) ≈ Im(Tr ) = O τins kB Tb τL The assumption of total vaporisation in the preheat zone is valid for droplet diameters less than ≈ 6 μm. Moreover, for droplets larger than 1 μm, we have τvis /τL  (γ − 1)Mb . The comparison of (2.5.11) and (2.5.19) then explains why the acceleration coupling in sprays leads to a much stronger primary thermo-acoustic instability than the pressure coupling. The experiments show also that, in contrast to premixed gaseous flames, there is a cut-off frequency for the instability. To see why this is so, it is necessary to examine the details of the mechanism and the resulting frequency dependence of the transfer function, Tr . The small liquid droplets vaporise in the preheat zone at an isotherm close to their boiling temperature. The flux of gaseous fuel is given by the flux of droplets into this isotherm. However, the gaseous fuel must first be transferred, by convection and diffusion, into the reaction zone before the corresponding heat is released. This introduces a finite delay [5]

Clavin P., Sun J., 1991, Combust. Sci. Technol., 78, 265–288.

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between the action of the acoustic wave and the heat release rate. The transfer function is thus frequency dependent and changes sign when the delay time is equal to half the acoustic period. The details of the calculation are tedious[1] and in Fig. 2.36a we simply plot the results for the imaginary part of the transfer function. The transfer function is negative at low frequency, but changes sign at ωτL ≈ 0.4 and remains positive for all higher frequencies. The geometrical gain factor for flames propagating from the open to the closed ends of a tube is positive (see Fig. 2.30), so the spray system is predicted to be unstable only for reduced frequencies greater than ≈ 0.4. Fig. 2.36b shows experimental measurements of the reduced frequency of instability for spray flames of 3.8 μm decane droplets,[2] plotted as a function of the ratio of flame transit time to the acoustic time τa = L/au . For the slowest flames, on the right-hand side of the figure, the instability occurs at the fundamental frequency of the tube, but as the transit time of the flame is reduced by increasing the flame speed, the frequency of instability suddenly jumps to the next harmonic, increasing the reduced frequency. As the flame speed is increased further, the reduced frequency decreases until it reaches the same limit and again jumps to the next harmonic of the tube. The minimum value of reduced frequency for which instability can occur is ωτL ≈ 0.3. This is in good agreement with the result of the simple analytical model[1] in Fig. 2.36a, ω∗ τL ≈ 0.4. 2.5.5 Acoustic Restabilisation and Parametric Instability This section is devoted to a theoretical analysis of the vibrating flat flame and of the secondary instability described in Section 2.5.1. The analysis[3] is based here on (2.5.13). The results obtained with a more detailed model are given in appendix; see Section 2.9.5. The acoustic acceleration may be written g  (t) = ωua UL cos(ωt), where ua is the nondimensional acoustic displacement velocity scaled by the laminar flame velocity. Introducing the nondimensional quantities for the acoustic time and acoustic frequency, τ  and  , τ  ≡ t/τh ,

τh ≡ 1/(UL k),

 ≡ ωτh , = (ωτL )/(kdL ),

where τh is the hydrodynamic time scale, Equation (2.5.13) takes the form

 dα˜ d2 α˜ 2  + 2B + −D +  C cos( τ ) α˜ = 0, dτ dτ 2    υb − 1 N υb − 1 ua υb , C≡ , D ≡ υb , B≡ υb + 1 υb + 1  υb + 1 κ

(2.5.20)

(2.5.21) (2.5.22)

where υb ≡ ρu /ρb > 1 and κ = kdL is the nondimensional wavenumber. According to (2.5.14), the nondimensional coefficient N(κ) ≡ −Go + κ − κ 2 /κm [1] [2] [3]

Clavin P., Sun J., 1991, Combust. Sci. Technol., 78, 265–288. Clanet C., et al., 1999, J. Fluid Mech., 385, 157–197. Markstein G., 1964, Nonsteady flame propagation. New York: Pergamon.

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(2.5.23)

2.5 Thermo-Acoustic Instabilities (a)

117

(b)

1st harmonic 2nd harmonic

Figure 2.36 (a) Transfer function of spray flame,[3] calculated for 3.8 μm diameter decane drops with β(Le−1) = 6. (b) Experimental reduced frequency of instability in decane spray flames as a function of the ratio of flame transit time to acoustic time. For comparison, the behaviour of gaseous propane flames is also plotted. The arrows indicate the direction of decreasing transit time (increasing flame speed). From Clanet C., et al., 1999, J. Fluid Mech., 385, 157–197, reproduced with permission.

is a function of κ, involving two parameters: the reduced marginal wavenumber κm and the Froude number through Go ≡ υb−1 |g|dL /UL2 . As already explained in Section 2.2.4, for fast flames, when Go < Goc ≡ κm /4, there is a range of wavenumbers, κ− < κ < κ+ , for which N > 0, D > 0, and the planar flame is unstable (DL instability) in the absence of acoustics (ua = 0 ⇒ C = 0); see Fig. 2.12. When Go > Goc (slow flames), N < 0, D < 0 ∀k, the planar flame √ is stable and Equation (2.5.21) describes a damped oscillator whose natural frequency −D is modulated at a frequency  . Floquet’s theory predicts that solutions to such linear ordinary differential equations with periodic coefficients may be unstable;[4,5] see Section 2.9.4. These parametric instabilities are of the same type as the Faraday instability discovered in 1831.[6] Moreover, as shown by Kapitza’s pendulum,[7] unstable positions of a pendulum (D > 0) may be restabilised when its point of suspension vibrates (periodic restoring force C = 0); see Section 2.9.3. The vibrating flat flame and the secondary (parametric) instability of gaseous flames presented in Section 2.5.1 were explained by Markstein[3] from (2.5.21) transformed into  ˜ Mathieu’s equation, by using the change of variables, t ≡  τ  , Y(t) ≡ eBτ α, d2 Y + { + h cos(t)} Y = 0, dt2

h = C,

=−

(D + B2 ) . 2

(2.5.24)

The general properties of the solutions to Mathieu’s equation are recalled in Section 2.9.2.

[4] [5] [6] [7]

Arnold V., 1973, Ordinary differential equations. MIT Editions. Bender M., Orszag S., 1984, Advanced mathematical methods for scientists and engineers. McGraw-Hill. Faraday M., 1831, Philos. Trans. R. Soc. London, 121, 299–338. Kapitza P., 1951, Sov. Phys.–JETP, 21 (in Russian).

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Downwards-Propagating Flames in Tubes In the notations (2.5.24), the limit of stability √ of the flame model (2.5.21) and (2.5.22) is given by Re(σ  ) = B. The resonant frequency  and the driving force h are functions of the wavenumber κ = kdL through C and D; see (2.5.22) and (2.5.24). The phenomenological equations (2.5.13) and (2.5.21) are not fully coherent with a perturbation analysis for small κ. The analytical studies of the ZFK flame model presented in Chapter 10 lead to similar equations but include additional κ correction terms that depend on the (first) Markstein number M; see (10.3.68)–(10.3.74). These terms can be useful from a quantitative point of view, but they do not introduce new phenomena. Additional effects such as variation of the (molecular, thermal, and viscous) diffusion coefficients with the temperature have also been included.[1,2] This model, called the ‘detailed model’ in the following, is briefly presented in appendix; see Section 2.9.5. Fig. 2.37 shows typical stability diagrams for the detailed model with parameters representative of the propane flames in the experimental study[3] of Section 2.5.1. The downwards-propagating flames were systematically above the threshold for stabilisation by gravity, Go < Goc . In the range κ− < κ < κ+ the diagrams correspond to the solutions to the left of the origin in Fig. 2.58,  < 0. The stability limits are plotted in the parameter space of nondimensional acoustic velocity and reduced wavenumber (ua ≡ ua /UL , κ ≡ kdL ). There are two distinct unstable regions, labelled I and II: • The lower unstable region, labelled I, extends down to zero amplitude of acoustic excitation where it corresponds to the DL hydrodynamic instability of planar flames in the wavenumber range [κ− , κ+ ]. It also corresponds to the domain labelled I in Fig. 2.58 for Mathieu’s equation. In this region the amplitude of the cells oscillates at the acoustic frequency. The wavenumber range decreases as the amplitude of acoustic excitation ua is increased and it shrinks to zero at a finite acoustic amplitude ua = u∗aI . The hydrodynamic instability is thus suppressed by an acoustic wave above a finite amplitude, ua > u∗aI . In Fig. 2.58 it corresponds to the narrow tongue of stability in the region where  < 0, h > 0. • The upper unstable domain, labelled II, concerns the secondary instability, which develops for a sufficiently large acoustic excitation, ua > u∗aII , and has a well-defined wavenumber at the threshold, κ = κII∗ . In Fig. 2.58 it belongs to the unstable domain, also labelled II, which has its minimum close to  = +1/4 (n = 1). Here, the cellular structure oscillates at one-half the acoustic frequency (parametric instability). When u∗aII > u∗aI , the planar flame is stable at all wavenumbers for intermediate acoustic amplitudes u∗aI < ua < u∗aII ; see Fig. 2.37a. This explains the vibrating flat flame observed in experiments; see Section 2.5.1. This restabilisation window between the primary and secondary instability disappears when the Markstein number M and/or the reduced frequency ωτL are decreased; see Fig. 2.37b–d. In Fig. 2.37a, c and d, the secondary parametric [1] [2] [3]

Clavin P., Garcia P., 1983, J. M´ec. Th´eor. Appl., 2(2), 245–263. Garcia P., et al., 1984, Combust. Sci. Technol., 42, 87–109. Searby G., Rochwerger D., 1991, J. Fluid Mech., 231, 529–543.

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Reduced acoustic velocity

(a)

(c)

119

(b)

(d)

Figure 2.37 Stability diagrams for acoustically driven flames. The unstable regions are shaded. The four panels demonstrate the effect of changing the frequency, the Markstein number and the flame speed. The figures are calculated from numerical resolution of the detailed model presented in Section 2.9.5. The parameters, representative of propane with a flame speed of 0.13 m/s and an excitation frequency of 128 Hz, are those given in the Table 2.1 in Section 2.9.5, except for panel b, where the frequency is reduced to 6.4 Hz, and for panel d, where the flame speed is increased to 0.29 m/s.

instability appears at a wavelength smaller than the cell size of the primary instability, as in the experiments; see Fig. 2.27. Experimental Data. Quantitative Comparison with the Analysis The critical cell size 2π/kII∗ and the threshold u∗aII may both be measured with accuracy in the experiments since the transition of the parametric instability is sharp. According to the perturbation analysis of the ‘detailed ZFK model’, presented in appendix; see Section 2.9.5, they are functions of the reduced frequency ωτL and depend on the physico-chemical parameters of the flames, all of which are known except the first Markstein number M defined in (2.3.2). A simultaneous good fit of the two functions kII∗ (ωτL ) and u∗aII (ωτL ) to the experimental data for propane flames has been obtained[3] from numerical resolution of this model; see the solid lines in Fig. 2.38. The values of the parameters are given in Table 2.1 in the Section 2.9.5. This fit gives an evaluation of the only unknown parameter, M ≈ 4.5. This agreement is surprisingly good considering not only the simplicity of the

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(a)

Parametric threshold

Critical wavenumber

(b)

Reduced frequency

Reduced frequency

Figure 2.38 Comparison of experimental and theoretical values of threshold and cell size of parametric instability. The measurements are made on propane flames over a range of flame speeds, 7.3 ≤ UL ≤ 21 cm/s, and acoustic frequencies, 57 ≤ f ≤ 265 Hz. The solid lines are obtained by numerical resolution of the detailed model supposing a Markstein number M = 4.5 and parameter values given in Table 2.1 in Section 2.9.5. Dotted lines are from analytical resolution of the model, (2.9.57)–(2.9.58).

ZFK model, but also the frequency range, which extends beyond the limit of validity of the theory, restricted to ωτL 1. An analytical resolution of the detailed model is also given in Section 2.9.5 using approximations that are not well respected by the parameters of usual flames. The analytical results are nevertheless plotted as dotted lines. The agreement with the acoustic amplitude at threshold is good, but the cell size overestimated. It is not possible to fit the experimental data with the simpler model (2.5.22), which is sufficient only for a qualitative understanding. Flattening of Conical Flames in an Acoustic Field A similar phenomenon has been observed for conical flames anchored to the exit from a pipe (often called ‘Bunsen’ flames although the name refers in fact to the type of burner that creates premixed air–gas by means of a Venturi inlet at the base). Since the gas velocity is greater than the laminar flame speed, the flame front is usually inclined to form a conical premixed flame. However, as first observed in 1943 by von Hahnemann and Ehret[1] and as shown in Fig. 2.39, in the presence of an intense axial acoustic field the conical front deforms to a shape that resembles a flattened hemisphere. These authors remarked that the new shape of the front strongly resembles an equipotential surface of the acoustic field. This phenomenon has been investigated experimentally more recently using laser diagnostics.[2,3] These studies confirm that the total flame surface area and the local normal propagation speed of the front remain unchanged and that the upstream gas flow is strongly [1] [2] [3]

von Hahnemann H., Ehret L., 1943, Zeitschrift f¨ur Technische Physik, 24, 228–242. Durox D., et al., 1997, J. Fluid Mech., 350, 295–310. Baillot F., et al., 1999, Combust. Sci. Technol., 142, 91–109.

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121 (c)

(b)

Figure 2.39 Conical flame in an axial acoustic field. (a) Conical rich methane flame in the absence of acoustics, equivalence ratio = 1.5, duct diameter = 34 mm. (b) Same flame in the presence of an axial acoustic field, frequency = 140 Hz. (c) Calculated three-dimensional acoustic equipotential surfaces at exit from a cylindrical duct of same dimensions, but in the absence of flame.

deformed to conform to the modified shape of the flame front. They also find that the acoustic level at the flame front producing the collapse of the conical flame front into the flattened hemisphere is close to that needed for parametric stabilisation of the wrinkled flame in a tube, u∗aI , and moreover oscillating parametric cells appear on the flattened hemisphere when the acoustic level is further increased to a value close to the threshold of parametric instability, u∗aII . This evidence strongly suggests that the deformation of the conical flame by a radiating acoustic field is driven by the same mechanism as that of parametric stabilisation of a planar flame by a standing acoustic wave (with planar equipotential surfaces); however, the analytical resolution of the problem introduces technical problems that have not yet been resolved.

2.6 Curved Fronts For ordinary gas expansion there is no exact solution, neither for the shape nor for the propagation velocity of curved flame fronts. In the framework of Euler’s equations, the flow is potential in the fresh mixture, but vorticity is generated by the jump conditions across the flame – see Section 2.6.2 – so that the flow of burnt gas is rotational. This is not the case for a gas bubble rising in a channel filled with liquid: the interface is a vorticity sheet separating two potential flows. When the gas density is negligible, the bubble problem may be reduced to that of an incompressible fluid above a vacuum with a tensionless interface at which the pressure is zero. The problem can be further simplified by considering twodimensional periodic solutions whose wavelength is the width of the channel. Even in this simple case, there is no exact analytical solution for the evolution of the entire interface. However, in contrast to the propagation velocity of a curved flame, the rising velocity of the bubble vertex may be obtained by a local analysis. The rising gas bubble is much simpler than the curved flame and is presented first.

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Figure 2.40 Sketch showing the evolution of a rising Rayleigh–Taylor bubble.

2.6.1 Rayleigh–Taylor Bubble Consider a bubble rising in a fluid above a vacuum in the presence of gravity, as sketched in Fig. 2.40. We assume that the flow can be described by Euler’s equations. For simplicity, surface tension is neglected and attention is limited to two-dimensional x–y slab geometry with periodic boundary conditions, of period 2R, in the y direction. The reference system is attached to the vertex of the bubble and moves upwards. It has a height h(t) relative to the position of the unperturbed planar interface. The bubble surface is described by x = α(y, t), the fluid fills the region x < α and the vacuum concerns x > α. The interface is fully unsteady. The upwards-rising bubble vertex (y = 0) is accompanied by spikes at y = ±R that propagate downwards with an unsteady velocity approaching that of free fall in the long time limit. Formulation The fluid flow is uniform at infinity, x → −∞, so the flow u = (u, w) is potential, u = ∇φ(r, t), and satisfies Bernoulli’s equation, ∂φ/∂t + p/ρ + |u|2 /2 − (g + h¨ tt )x = C(t)

φ = 0,

(2.6.1)

(see Section 15.2.2), where h¨ tt ≡ d2 h/dt2 is the acceleration of the reference system attached to the bubble vertex, and g > 0 is the acceleration of gravity. Using Fourier series, the solution of Laplace’s equation may be written in the form φ = h˙ t x +

∞ 

Bn (t)ekn x cos(kn y),

n=1

∞ 

kn Bn = −h˙ t ,

(2.6.2)

n=1

where kn ≡ (π/R)n, and h˙ t ≡ dh/dt denotes the rising velocity of the bubble vertex. The time-dependent coefficients h(t) and Bn (t) in (2.6.2) are such that the flow velocity satisfies the boundary conditions at the bubble vertex (x = 0: u = ∂φ/∂x = 0, w = ∂φ/∂y = 0) on the walls (y = ±R: w = 0) and at x = −∞, where the flow is at rest in the laboratory ˙ w = 0. Two boundary conditions are provided at the interface by frame, x → −∞: u = h, a kinematic condition and by Bernoulli’s equation. The surface tension being neglected, the interface is convected by the flow; the normal velocity of the front D, defined in (10.1.4), is equal to the component of flow velocity normal to the interface, un ≡ n.u|x=α , defined in (10.1.5),

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α˙ t =

123

u|x=α − αy w|x=α ,

(2.6.3)

where α˙ t ≡ ∂α/∂t, and αy ≡ ∂α/∂y. Assuming that the pressure is zero at the interface, Bernoulli’s equation (2.6.1) yields ∂φ/∂t|x=α + |u|2x=α /2 − (g + h¨ tt )α =

∞ 

B˙ nt ,

(2.6.4)

n=1

 where B˙ nt ≡ dBn /dt. The gauge function C(t) = B˙ nt has been determined at the bubble vertex, x = y = 0: α = 0, u = 0. In principle, the problem may be solved by using a Fourier series for α(y, t); however, the method is not accurate away from the bubble vertex since the curvature of the spike at y = ±R increases with time as a power law[1,2] in the long time limit. Local Solution at the Bubble Vertex An approximation, initially suggested by Taylor[3] to determine the constant (long time limit) rising velocity of the bubble vertex, consists in retaining only the first mode in the series (2.6.2). This approximation is accurate for the flow velocity in the vicinity of the vertex where the interface reaches a constant radius of curvature (not small compared with R) and where the flow takes the form of a stagnation point flow. The calculation is then performed by a perturbation analysis using an expansion for small y. The same method was also used to obtain the time evolution of the shape of the interface near the vertex.[4] For a single mode, equation (2.6.2) yields φ = h˙ t (t)x − h˙ t (t)ek1 x cos(k1 y),

k1 ≡ π/R.

(2.6.5)

The power expansion around y = 0 is limited to the first term, α = κ(t)y2 /2 + · · · , where κ(t) is the curvature (inverse of the radius of curvature) of the interface at the vertex. The coefficients of y2 in (2.6.3)–(2.6.5) lead to a system of two ordinary differential equations for h(t) and κ(t), κ˙ t = −h˙ t k1 (3κ − k1 ),

h¨ tt (k1 − κ) + h˙ 2t k12 − gκ = 0,

(2.6.6)

where κ˙ t ≡ dκ/dt, h˙ t ≡ dh/dt and h¨ tt ≡ d2 h/dt2 . The steady state solutions (κ˙ t = 0, h¨ tt = 0) of (2.6.6) for the bubble curvature, κ = k1 /3, √ and the rising velocity, h˙ t = g/(3k1 ), are the same as Taylor’s solution.[3] A similar √ analysis[4] shows that the asymptotic velocity in cylindrical geometry is h˙ t = 0.361 g, where  is the diameter of the cylinder. The linear growth rate (10.1.31) of the Rayleigh– Taylor instability is recovered from the linearised versions of (2.6.6), κ ≈ −k12 h, h¨ tt ≈ gκ/k1 ≈ −gk1 h. The full history of the dynamics of the bubble vertex can be found by integrating the system of equations in (2.6.6) from small initial conditions t = 0: h = h0 , [1] [2] [3] [4]

Clavin P., Williams F., 2005, J. Fluid Mech., 525, 105–113. Duchemin L., et al., 2005, Phys. Rev. Lett., 94, 224501. Taylor G., 1950, Proc. R. Soc. London, A 201, 192–196. Layzer D., 1955, Astrophys. J., 122, 1–12.

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h˙ t = h˙ t0 . The result shows that the rising velocity of Taylor’s solution is reached in the long time limit. Local Solution at the Spikes The asymptotic dynamics of the interface near the spikes, in the long time limit, is obtained by a quite different approach.[1,2] Assuming that the asymptotic flow field near the spike, √ y = R, x = xs (t), is close to that imposed by steady parallel free fall, u ≈ 2gx, xst ≈ gt2 /2, the curvature of the spike is found to increase as t3 in planar geometry and as t2 in cylindrical geometry. Surprisingly, the acceleration of the spike approaches g from above, as 1/t5 and 1/t4 in planar and cylindrical geometry, respectively. These results have been confirmed by accurate numerical simulations[2] using a boundary integral method that is free from numerical dissipation. These numerical solutions were obtained starting from small initial disturbances to the planar interface, which is unstable at all wavelengths. No singularity in finite time was observed, despite a linear growth rate (10.1.31) that increases as the square root of the wavenumber and the absence of viscous damping. However, the proof of the existence of such regular solutions in the long time limit is still an open question. Multimode Dynamics Many analyses of the nonlinear multimode dynamics of Rayleigh–Taylor unstable interfaces have been carried out.[3,4] Despite impressive simulations[5] using high-resolution three-dimensional numerical codes, the problem is still open. Bubble competition has been reproduced numerically, showing that bubbles are mutually attracted and merge to form larger and faster bubbles. For two different fluids, turbulent mixing is also observed both in experiments and in numerical simulations.

2.6.2 Vorticity Production across a Curved Flame Front The case of a flame front is different from that of a passive front. There is a flux of fluid through the front, and if the front is nonplanar, the associated gas expansion leads to production of vorticity. The burnt gas flow behind a curved front is always rotational, as shown next. Consider a steady curved flame front whose radius of curvature is very large compared with the laminar flame thickness. In this approximation the modifications to the internal flame structure are negligible and the curved front can be treated as a hydrodynamical discontinuity. For simplicity we consider two-dimensional planar geometry. The normal and tangential components of the flow velocity at the flame are denoted un and

[1] [2] [3] [4] [5]

Clavin P., Williams F., 2005, J. Fluid Mech., 525, 105–113. Duchemin L., et al., 2005, Phys. Rev. Lett., 94, 224501. Kull H., 1991, Phys. Rep., 206(5), 197–325. Atzeni S., Meyer-Ter-Vehn J., 2004, The physics of inertial fusion. Clarendon Press–Oxford Science Publications, 1st ed. Dimont G., al., 2004, Phys. Fluids, 16(5), 1668–1693.

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wθ , respectively, un ≡ n.u|f , wθ ≡ t.u|f , where n and t are the unit vectors, normal and tangential to the flame front. Neglecting curvature effects, the normal burning velocity is the laminar flame velocity, u− n = UL ,

u+ n = Ub ,

ρb Ub = ρu UL ,

(2.6.7)

where the superscripts − and + refer to the unburnt and burnt gas sides, respectively. Normal and tangential momentum conservation (15.1.46)–(15.1.47) yield  1 + − 2 1 , (2.6.8) − pf − pf = −(ρu UL ) ρb ρu − w+ θ = wθ ≡ wθ ,

(2.6.9)

where the subscript f refers to the value at the flame front. Equation (2.6.8) shows that the pressure jump across the flame front is constant. Introducing the stream function ψ (see (15.2.10)) and the arclength s of the flame front, the tangential derivative, t.∇ = ∂/∂s, of ψ along the flame front is dψf− /ds = −UL ,

t.∇ψ|f = −un ,

dψf+ /ds = −Ub .

(2.6.10)

Assuming that the flow of fresh mixture is uniform at infinity, the upstream flow is potential everywhere, − = 0; see (15.2.8). Euler’s equations written in the form (15.2.11) and applied to both sides of a steady flame front yield, according to (2.6.7)–(2.6.10), dp+ f

dwθ , ds ds dp− dwθ f 0= + ρu wθ , ds ds

−ρb Ub + f

=

+ ρb wθ

(2.6.11) (2.6.12)

where the relation du± n /ds = 0 has been used; see (2.6.7). Since the pressure jump (2.6.8) is constant along the flame front, the difference (2.6.11) minus (2.6.12) yields the production of vorticity across the flame in terms of the tangential derivative of the tangential component of the flow velocity at the front,   ρb wθ dwθ + . (2.6.13) f = 1 − ρu UL ds According to (2.6.12), the tangential gradient of the pressure at the front is dp+ f ds

=

dp− f ds

=−

ρu dw2θ . 2 ds

(2.6.14)

2.6.3 Curved Flames Propagating in a Channel Consider a steady curved flame front propagating in a two-dimensional channel of thickness 2R (x–y slab geometry) schematised by a periodic solution of period 2R; see Fig. 2.41. In the limit dL /R → 0, the flame is considered as a hydrodynamic discontinuity of equation

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Figure 2.41 Sketch of a curved flame propagating in a channel. The flame propagates in the direction x < 0. The stagnation regions are in grey.

x = α(y) in the reference system attached to the flame vertex. The effect of gravity is neglected for simplicity. The flow is steady in this reference flame, u = u(r)ex + w(r)ey , where ex , ey are the unit vectors, and r = xex + yey . The fresh mixture and the burnt gas are in the regions x < α and x > α, respectively. We assume that the flow is described by the Euler equations (viscous effects are neglected), and a slip condition holds at the walls, y = ±R: w = 0. As in Section 2.6.1, the flow is potential in the region x < α, − = 0, u− = ∇φ − (r, t), φ − = 0, limx→−∞ w = 0, limx→−∞ u = U, where U is the propagation velocity of the curved front, and φ is the flow potential. According to Section 2.6.2, the flow of burnt gas, x > α, is rotational, + = 0. Due to deflection of the stream lines, sketched in Fig. 2.6, stagnation regions appear at the walls in the burnt gas whenever the angle between the front and the wall differs from π/2. These regions, where the gas is at rest in the reference frame attached to the flame, are coloured in grey in Fig. 2.41. The pressure in these stagnation zones is uniform and equal to the pressure p+ ∞ at x = +∞, + /∂x = 0, + = −∂u+ /∂y  = 0, = 0, ∂u where the flow is a pure shear flow, w+ ∞ ∞ ∞ ∞ ∇p+ ∞ = 0. The unknowns in the problem are the propagation velocity, U, and flame shape, α(y). The parameters are the laminar flame speed UL and the gas expansion ratio υb ≡ ρu /ρb = Ub /UL > 1. In the absence of time and length scales other than those associated with the period 2R and the laminar flame velocity UL , it can be anticipated that the solution U/UL will not depend on R and that α/R will be a function of y/R. Such a simplified model is far from being a good approximation to a real curved flame propagating in a tube. The first reason is that the streamlines on the free boundaries of the stagnation zones are not stable (Kelvin–Helmholtz instability). Another more serious reason is that the viscous and heat transfers at the walls have been neglected. Moreover, for periodic solutions, the curvature effects described in Section 2.3 regularise the extremities of the flame front at y = ±R. It is not obvious that curvature effects can be systematically

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127

neglected even for very wide boxes, R  dL . For small gas expansion, considered in Section 2.7, it will be shown that they cannot be neglected for all periodic solutions, whose the number increases with the size of the box, km R  1. It is possible only for the most stable solution, corresponding to a single cell in the box; see the comments below (2.7.18). Nevertheless, the simplified model in which all dissipative effects are neglected highlights the hydrodynamic effects of gas expansion. There are two types of free boundaries in this problem, one at the flame front and another at the boundary of the stagnation zones. This is a very tough problem, so difficult that no analytical solution has yet been found, except in the limit of small heat release; see Section 2.7.2. For realistic gas expansion, in the absence of a systematic analysis, an approximate solution based on an exact integral equation (2.6.25) has been obtained.[1] Even though the approximation used by Zeldovich et al.[1] to solve this equation is not justified by an asymptotic analysis, the result is quite interesting. Overall Conservation Relations According to (2.6.7), the normal flow velocity at the front is equal to UL ,    − ⇒ u− − α w = U 1 + αy2 , u− L n = UL y f f

(2.6.15)

− − − where u− f (y) and wf (y) are the values of u (x, y) and w (x, y) at the front x = α(y); see (10.1.5). Equation (2.6.15) gives a quadratic equation for αy , whose solution is an ordinary differential equation of first order for the flame front α(y). Overall mass conservation through the flame front and in the far downstream flow yields  R  R∗  1 + αy2 dy, ρu RU = ρb u+ (2.6.16) RU = UL ∞ (y)dy, 0

0

where the left-hand side represents the mass  S/2 flux in the upstream flow. The integral in the first equation is half the flame length S, 0 ds, and R∗ in the second equation is the halfwidth of the funnel through which the parallel flow of burnt gas escapes at infinity with the velocity u+ ∞ (y); see Fig. 2.41. The first equation in (2.6.16) is automatically satisfied if the upstream flow field verifies the kinematic condition (2.6.15). The second equation is a constraint on the downstream flow, imposed by ρu UL = ρb Ub . Another overall equation concerning the conservation of momentum is obtained by a spatial integration over the tube volume of the Euler equations, written in the conservative form, ∇.(pI + ρuu) = 0; see (15.1.44). The effect of pressure at the side walls cancels by symmetry and the y-integrals at x = ±∞ yield  R∗ 2 R + ρ (u+ (2.6.17) (pu + ρu U 2 )R = p+ b ∞ ∞ ) dy, 0

where pu is the initial pressure of the fresh gas. This shows that the effect of pressure at ∗ the free boundaries of the stagnation zones, acting on the downstream flow, is p+ ∞ (R − R ). [1]

Zeldovich Y., et al., 1980, Combust. Sci. Technol., 24, 1–13.

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According to (15.2.10) the stream function ψ is constant along streamlines, so that u+ ∞ = −dψ + /dy|x=+∞ . Introducing the value ψ∗+ of the stream function on the free boundary of the stagnation region located at y > 0, the variable y may be eliminated from (2.6.16) and (2.6.17) in favour of ψ + , to give ρu RU = −ρb ψ∗+ and  ρb −(ρu /ρb )RU + + 2 + ρu U = (p∞ − pu ) − u∞ (ψ )dψ + . (2.6.18) R 0 We will show now that the integral in (2.6.18) may be expressed as an integral on the flame front of a function of the component of the flow velocity tangent to the front, wθ (y), which, according to (2.6.9), is continuous across the front. Bernoulli Equations In the upstream potential flow, the quantity p− + ρu |u− |2 /2 is constant; see (15.2.9). Therefore it is constant on the flame and equal to its value at infinity in the fresh gas,

 2 2 2 U + ρ + w (2.6.19) p− u L θ /2 = pu + ρu U /2, f where the pressure p− f (y) and the tangential component of the flow velocity wθ (y) vary along the flame front. In the downstream flow, according to (15.2.12), the quantity p+ + ρb |u+ |2 /2 is constant along the streamlines, but, due to the vorticity, it differs from streamline to streamline and varies with ψ + (see Section 2.6.2),

 2 2 + + 2 (2.6.20) p+ f + ρb Ub + wθ /2 = p∞ + ρb (u∞ ) /2, where both the pressure at the front, p+ f , and the tangential component of the flow velocity, + is to be written as u+ wθ , vary with ψ + and where u+ ∞ ∞ (ψ ). Subtracting (2.6.19) from + 2 (2.6.20), and using the jump conditions (2.6.8)–(2.6.9), the quantity p+ ∞ + ρb (u∞ ) /2 can be expressed in terms of the component of the flow velocity tangent to the flame front on the same streamline, wθ (ψ + ),  (ρu − ρb ) 2 ρb + 2 ρu 2 (ρu UL )2 1 1 + − wθ . p∞ + (u∞ ) = pu + U − − (2.6.21) 2 2 2 ρb ρu 2 This equation is valid on any streamline and, in particular, on the boundary of the stagnation region at the wall, which will be denoted by the subscript ∗. Subtracting this last equation from (2.6.21) yields    ρ

u 2 + 2 2 2 , (2.6.22) ) − (u ) ) = − 1 (w ) − (w ) (u+ θ∗ θ ∞ ∞∗ ρb where wθ∗ is the value of wθ at the point where the flame touches the wall, y = R. The same equation would be obtained by writing that, according to (15.2.8), the vorticity is constant on each streamline. The pressure is uniform in the stagnation regions and equal to p+ ∞ . Therefore, according to Bernoulli’s equation (15.2.12), the modulus of the flow

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+ 2 + 2 velocity |u+ ∗ | is constant along the boundary of the stagnation regions |u∗ | = (u∞∗ ) = 2 2 Ub + (wθ∗ ) . Equation (2.6.22) yields    + 2 2 2 2 u+ (2.6.23) ∞ (ψ ) = Ub + (wθ∗ ) + [(ρu /ρb ) − 1] (wθ∗ ) − (wθ ) ,

where the only quantity that varies with ψ + in the right-hand side is wθ . The constant pressure difference p+ ∞ − pu in (2.6.18) and (2.6.21) is obtained from the Bernoulli equation along the wall and the jump conditions across the flame, using (2.6.22) to eliminate + simultaneously both u+ ∞ (y) and wθ (ψ ), ρu 2 p+ (2.6.24) (U + UL2 − w2θ∗ ) − ρb Ub2 . ∞ − pu = 2 The integral in (2.6.18) may be transformed into an integral over the front by using (2.6.10),  where ds = (1 + αy2 )1/2 dy is the element of arc length of the flame front. The righthand side of Equation (2.6.18) is a scalar, functional of the position of the curved front α(y) and of the tangential component of flow velocity wθ (y). Introducing the density ratio υb ≡ ρu /ρb = Ub /UL > 1, Equation (2.6.18) may be written in the dimensionless form  1    u2 1 − v2∗  = − υb + (2.6.25) dη (1 + aη2 ) (1 − υb )v2 + υb v2∗ + υb2 , 2 2 0 u ≡ U/UL ,

η ≡ y/R,

a(η) ≡ α(y)/R,

v(η) ≡ wθ (y)/UL ,

v∗ ≡ v(η = 1),

where v(η) and aη (η) ≡ da/dη are two unknown functions of η that have to be determined by solving the upstream flow. Notice that the planar solution, u = 1, v = 0, aη = 0, is a solution for all values of the gas expansion ratio υb . Approximate Solution The upstream flow velocity being potential, u− = ∇φ − , φ − = 0 (see (15.2.9)), it may be written in a general manner, as in (2.6.2), by using a Fourier series, ∞ ∞   kn An ekn x cos(kn y), kn An = −(U − UL ), (2.6.26) u− = U + n=1

w− = −

∞ 

n=1

kn An ekn x sin(kn y),

(2.6.27)

n=1

where kn = nπ/R, and where four boundary conditions at the wall (y = ±R: w = 0) at the flame vertex (x = 0, y = 0: u− = UL ) and at infinity (x = −∞: u− = U, w− = 0) have been used. As in the Taylor analysis of the bubble vertex in Section 2.6.1, the approximation used by Zeldovich and co-workers[1] consists in limiting the Fourier series to the first nontrivial term. The upstream flow velocity u− (x, y)/UL is thus expressed in terms of the unknown scalar u,   u− /UL ≈ 1 + (u − 1) 1 − eπ x/R cos(π y/R) , (2.6.28) w− /UL ≈ (u − 1)eπ x/R sin(π y/R). [1]

Zeldovich Y., et al., 1980, Combust. Sci. Technol., 24, 1–13.

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Figure 2.42 Solution of the integral equation (2.6.25) for the propagation velocity of a twodimensional curved flame in a channel as a function of inverse gas expansion ratio.

As already mentioned, the kinematic equation (2.6.15) is a first-order differential equation for a(η). Its solution, obtained with the condition y = 0: dα/dy = 0, provides the front position, a(η). Using the flow field (2.6.28) in the expression in (10.1.5) for the component of the velocity tangential to the front, v(η), yields

 (u − 1)eπ a(η) sin(π η) − aη (η) cos(π η) + uaη (η) , (2.6.29) v(η) =   1 + aη2 (η) where the two functions v(η) and a(η) depend on a single parameter, the unknown eigenvalue u. Introducing (2.6.29) into (2.6.25) leads to an integral equation whose solution yields the eigenvalue u (the propagation velocity of the curved flame) in terms of a single parameter, the gas expansion ratio υb . An analytical expression can be obtained by a perturbation analysis for small υb − 1, (υb − 1)2 . (2.6.30) 2 In the general case, the integral equation has to be solved numerically. Fig. 2.42 shows the result for the two-dimensional velocity field (2.6.28). u≈1+

Extension and Concluding Remarks To conclude, we emphasise again that, in contrast to the rising velocity of a bubble above a vacuum, obtained by a local analysis in Section 2.6.1, the propagation velocity of a curved flame is an eigenvalue of a nonlocal problem. Once the jump conditions at the flame front are known,[1,2] an equation for the front can be obtained if the flow velocity on both the upstream and downstream sides of the front can be expressed in terms of the front position. According to (2.6.26)–(2.6.27) this is not difficult for the upstream flow, which is a potential flow when it is uniform far upstream from the flame front. The difficulty comes [1] [2]

Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237. Matalon M., Matkowsky B., 1982, J. Fluid Mech., 124, 239–259.

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from the rotational part of the burnt gas flow. According to Section 2.6.2, vorticity + f is produced across the curved front by the tangential gradient of tangential velocity (see (2.6.13)) and is propagated downstream according to the Thomson circulation theorem in the form (15.2.8), u+ .∇+ = 0. This was achieved by Kazakov[3] for a periodic smooth flame front (with no stagnation zone) in two-dimensional geometry. The result is a very complicated system of equations including a complex integro-differential equation that can be analysed numerically in limiting cases.[4,5] The preceding analysis leading to (2.6.25) and also to the rising velocity could be extended to include the effect of gravity, which to the best of our knowledge has not yet been done in the general case. The problem was recently considered in the limit of a strong gravity field, for an elongated flame front propagating upwards[6] on the basis of the integro-differential equation mentioned above. For a flame propagating upwards, as shown √ in Fig. 2.11, one may expect that in the limit of a small Froude number, UL / gR → 0, the laminar flame velocity becomes negligible near the flame vertex. However, this cannot be true in the thin regions of fresh mixture descending near the walls where any nonzero flame velocity is sufficient to prevent unlimited growth of the spikes. Moreover, flame quenching at the wall complicates the problem. However, in the small Froude limit, the propagation velocity of the curved flame should be close to that of the Taylor bubble, at least in the limit of a strong density contrast, υb ≡ ρu /ρb → ∞, since, according to (2.6.8), the pressure − − on the downstream side of the front becomes negligible (p+ f − pf )/pf ≈ 1/υb , and the problem near the vertex of a curved flame front approaches that of the Rayleigh–Taylor bubble presented in Section 2.6.1. In the next sections it will be seen that curved flame fronts are more stable than planar flames. 2.6.4 Stability of Curved Flame Fronts The same as for planar flames stabilised in a stagnation point flow, the stability of curved flame fronts is different from that of a planar flame propagating in a uniform flow. Disturbances on a Curved Front When an unstable planar flame front propagates in a quiescent medium, a small perturbation to the front grows at the same location. In the presence of a tangential flow, the perturbation is also convected along the front with the tangential component of the flow velocity. This also happens on curved fronts, such as that of Fig. 2.11, since the curvature induces an upstream flow field with tangential velocity components. The upstream flow in the region where the front is convex towards the fresh gas has a form that is similar to that of the stagnation flow sketched in Fig. 2.15. This flow sweeps a local perturbation [3] [4] [5] [6]

Kazakov K., 2005, Phys. Rev. Lett., 17, 032107. Kazakov K., 2012, Phys. Fluids, 24, 022108. Kazakov K., 2013, Phys. Fluids, 25, 082107. Kazakov K., 2015, Phys. Rev. Lett., 115, 264051.

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Figure 2.43 Sketch of a perturbation on a curved front. The thick curved arrows show the streamlines in the fresh gas.

away from the centre of the bulge, as sketched in Fig. 2.43. The growth of the perturbation must be calculated taking into account not only the displacement of the perturbation along the flame front, but also the strain rate created by the gradient of tangential velocity. Here we present the simplified semi-phenomenological analysis of Zeldovich and coworkers[1] showing clearly the mechanism at work, but without going into a complicated formalism. Consider a small perturbation initially localised close to the summit of a curved front; see Fig. 2.43. Let A(0) and (0) be, respectively, the initial amplitude and wavelength of the perturbation. We will suppose that the wavelength  remains smaller than the radius of curvature of the unperturbed front during the whole period of growth,  < R. Let wθ (l) be the tangential component of flow at the front, at a distance l from the summit, wθ (l) ≈ lwθl , where wθl = dwθ /dl ≈ UL /R is the gradient of tangential velocity, which is quasi-uniform near to the summit, dwθ /dl ≈ cst. This approximation is sufficient to calculate the final amplitude of the perturbation since, as will be shown, the growth in the region of a summit is much greater than elsewhere on the front. There is no ambiguity in the definition of the tangential velocity, wθ (l) since it is conserved through the flame front. Now, the perturbation is not only swept along the curved front; it is also elongated by the gradient of the tangential velocity dwθ /dl, in a manner similar to that of the wavelength of electromagnetic radiation in an expanding universe, as noted by Zeldovich,[1] d = wθ (l + /2) − wθ (l − /2) ≈ wθl , (t) ≈ (0) exp(wθl t). dt So that, using wθl /wθ = 1/l, dl = wθ dt and d/dt = wθ d/dl,

(2.6.31)

dl d ≈ ,  ∝ l. (2.6.32)  l Since the size of the perturbation is supposed small compared with the radius of curvature, the flame is locally planar. The instantaneous growth rate of the perturbation (in the absence of gravity) is given by (2.2.10). However, the wavelength changes in time, so the growth [1]

Zeldovich Y., et al., 1980, Combust. Sci. Technol., 24, 1–13.

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rate is also a function of time σ (t) = AUL k(t)[1 − k(t)/km ] and the evolution of amplitude is no longer exponential as in (2.2.2). When the growth rate is fast compared with the rate of change of the growth rate, the amplitude is given to a good approximation by t the solution of dA/dt = σ (t)A, A(t) ≈ A(0) exp 0 σ (t )dt . Changing the variable

of integration dt = dl/wθ ≈ (1/wθl )(dl/l) ≈ (1/wθl )(d/) then leads to A(t) ≈ 

 (t) A(0) exp (1/wθl ) (0) σ ()d/ , where, according to Equation (2.2.10),  1 2π . (2.6.33) − σ () = 2π AUL  km 2

After sufficient stretching, (t)  (0), l  l(0), or according to (2.6.31) and (2.6.32), for wθl t  1, Equation (2.6.33) gives an amplitude that saturates in time, A(t) → Af , with Af = A(0) exp , where      dL dL 2 π UL −  = 2π A  , (2.6.34) wθl dL (0) km dL (0) and where A is given in (10.1.32), A ≈ 1.62 for υb ≈ 7. Thus, as anticipated above, if the initial perturbation occurs sufficiently close to the summit, l(0) R, the saturation in amplitude also occurs in the region of the summit, l < R. Equation (2.6.34) shows that the maximum amplitude results from a perturbation whose initial wavelength, (0), is equal to the marginal wavelength, m ≡ 2π/km , σ (m ) = 0. The most amplified wavelength is m because, during the stretching process, it is the one that spends the longest time in the region of positive growth rate of Fig. 2.9. For a smooth spectrum of noise in the incoming flow, the amplitude of the perturbation reaching the edge of the flame is then given approximately by Af = Am em ,

m = π AUL /(wθl m ) ≈ π AR/m ,

(2.6.35)

where the relation wθl R ≈ UL has been used, and where Am is the amplitude of perturbations to the flame front, induced by the fluctuations of flow velocity having a wavelength equal to the marginal wavelength. The amplification Af /Am increases exponentially with the radius of curvature of the front and reaches very high values when the size of the flame is much greater than the marginal wavelength, R/m > 1. This semi-phenomenological analysis has a number of shortcomings, such as the use of the growth rate (2.6.33) for a localised perturbation at a nonzero distance from the summit, l(0). A more systematic stability analysis can be performed for a planar front stabilised in a stagnation point flow. This configuration contains the dominant feature of the stability of curved fronts, namely the stretching of the wavelength; its study will give more weight to the preceding rough analysis. Stability of Planar Flames in a Stagnation Point Flow Consider the stability of a planar front stabilised at x = 0 in a stagnation point flow sketched in Fig. 2.15. The main characteristic of this configuration is the presence of a tangential flow

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that increases linearly with the distance from the stagnation point at y = 0, w ≈ y/τs , where 1/τs is the strain rate of the incompressible upstream flow ∂u/∂x = −∂w/∂y. The linear equation for evolution of small disturbances of the front, x = α(y, t), is that for a freely propagating flame front plus a stretching term −(y/τs )∂α/∂y, ∂α/∂t = L(α) − (y/τs )∂α/∂y,

(2.6.36)

where L(.) is a linear differential operator acting on functions of y. In agreement with (2.2.10), it is defined in Fourier space as multiplication by AUL k(1 − k/km ), where k is the modulus of the wavevector. Guided by the preceding analysis, consider a solution to (2.6.36) in the form of a harmonic function whose wavenumber is a function of time, ik(t).y + c.c. α(y, t) = α(t)e ˜

(2.6.37)

Two types of terms appear when (2.6.37) is introduced into (2.6.36): terms proportional to yeik(t).y and terms proportional to eik(t).y . They must vanish separately, leading to two equations: dk(t)/dt = −k(t)/τs ,

dα(t)/dt ˜ = AUL k(t)[1 − k(t)/km ]α(t). ˜

These equations are easily integrated to give k(t) = ko e−t/τs ,

lim α(t) ˜ = e AUL τs [ko −ko /(2km )] α˜ o , 2

t→∞

which shows that, as above, the amplitude of disturbances saturates at long time, with a maximum amplification for an initial wavelength equal to the marginal wavelength, ko = ˜ → e AUL τs km /2 α˜o . km , α(t) Semi-phenomenological Stability Criterion Equation (2.6.35) gives a relation between the final amplitude of perturbations and the amplitude of external noise. This provides a criterion for the stability of curved fronts. A perturbation is negligible if its final amplitude is much smaller than the size of the flame, Af R. However, if the final amplitude is of the same order as the size of the flame, or of a cellular structure, then we can consider that the initial structure has been destroyed. The relation Af ≈ R, Am eπ AR/m ≈ R,

(2.6.38)

is an equation for the maximum size R of a curved flame in the presence of external disturbances, since larger flames will split into smaller units. This maximum size is a function of the amplitude Am of the initial disturbance, whose wavelength is m and which is produced at the tip by an external noise. Equation (2.6.38) thus defines the stability limit of a curved front submitted to external noise. The effect of weak turbulence on the morphology of premixed flames has been observed experimentally.[1] The experiments on spherical flames show a cellular front with a characteristic cell size that decreases as the turbulence intensity is increased. We will come back [1]

Strehlow R., 1979, Fundamentals of combustion. New York: Kreiger.

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to this question in Sections 2.7.3 and 3.1.3. The criterion (2.6.38) may be used to evaluate the characteristic size of cells on a weakly turbulent flame front.[2] Equation (2.6.38) for R presents a turning point: there is a critical amplitude A∗m ≡ m /(eπ A) and a minimal cell size m /(π A), since for Am > A∗m , Equation (2.6.38) has no solution for R. Therefore Equation (2.6.38) predicts that there are no long living cells with a size smaller than m /(π A). The maximum amplitude A∗m defines the limit of the regime of weakly turbulent unstable flames, called cusped flame regime.[2] In a perfectly controlled uniform laminar flow, the effect of thermodynamic fluctuations, whose amplitudes are microscopic, cannot be neglected. The amplification factor is so large that even the thermodynamic fluctuations can induce a macroscopic response, visible on the laboratory scale.[3] The analysis has been generalised[3,4] to other types of fronts such as Saffman–Taylor fingers or crystal dendrites.

2.7 Nonlinear Dynamics of Unstable Flame Fronts In the absence of stabilising effects at large wavelengths, such as gravity, the description of freely propagating cellular flames requires a nonlinear analysis. In many aspects cellular flame fronts are different from most other nonlinear patterns in physics. This is exemplified by the nonlinear differential equation obtained by G.I. Sivashinsky[5,6] in the limit of small density changes.

2.7.1 Sivashinsky’s Equation for Small Heat Release In this section we consider the hydrodynamic instability of freely propagating planar flames in the absence of thermo-diffusive instability, B > 0, and acceleration of gravity, g = 0. Linear Equation For a small density contrast  ≡ (ρu /ρb − 1), according to (2.2.6), the linear evolution equations (2.2.9)–(2.2.10) for a flame propagating in a uniform flow UL may be written in the equivalent form of a partial differential equation for the amplitude of a perturbation α(y, t),  1 ∂ 2α 1 ∂α ≈ 0. (2.7.1) − UL H (α) +  1: ∂t 2 km ∂y2

[2] [3] [4] [5] [6]

Clavin P., 1988, In E. Guyon, J. Nadal, Y. Pomeau, eds., NATO ASI Series E. Disorder and Mixing, vol. 152, 293–315, Kluwer Academic Publishers. Pelc´e P., 2004, New visions on form and growth. Oxford University Press. Pelc´e P., Clavin P., 1987, Europhys. Lett., 3, 907–913. Sivashinsky G., 1977, Acta Astronaut., 4, 1177–1206. Michelson D., Sivashinsky G., 1977, Acta Astronaut., 4, 1207–1221.

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Figure 2.44 System of axes and local flow components at the flame front. The quantities with the superscript − refer to the flow in the unburnt gas, and the subscript f refers to the value at the flame front.

The linear operator H (.) representing the DL instability is an integral operator that operates on functions of the space coordinate y. It is defined in Fourier space as multiplication by (α) ≡ |k|α, the modulus of the wavenumber, H ˜ H (α) ≡

1 2π



∞



∞

−∞ −∞



|k|eik(y−y ) α(y , t)dk dy .

(2.7.2)

It is also the Hilbert transform of the space derivative, namely the Cauchy principal value of (y−y )−1 ∂α/∂y . According to (2.2.6) and (2.2.10), the marginal wavenumber km dL > 0 is small in the limit of a small density contrast, and all the unstable wavenumbers are small kdL = O(). The last two terms in Equation (2.7.1) are thus of order  2 (α/dL )UL . The maximum linear growth rate is then of order  2 /τL , where τL is the flame transit time τL ≡ dL /UL . This scale separation in both space and time makes it possible to obtain a nonlinear equation for the dynamics of the front. Equation (2.7.1) is obtained as follows. In the limit  1, the effect of strain is negligible compared with that of curvature and (2.3.10) reduces to (Un− − UL )/UL ≈ −Mc dL /R,

where

Mc = O(1),

(2.7.3)

and where 1/R ≈ ∂ 2 α/∂y2 in the linear approximation, and Mc > 0 (no thermodiffusive instability). Equation (2.7.1) is the linear approximation to Equation (2.7.3), δUn− ≈ −Mc dL ∂ 2 α/∂y2 , as shown now. According to (10.1.4)–(10.1.6), the local propagation speed of the front, Df , and the local flame speed with respect to the fresh gas, Un− , defined by (2.3.1), written in the linear approximation δDf = ∂α/∂t, δUn− = δu− f − ∂α/∂t, yields 2 2 ∂α/∂t = δu− f + M c dL ∂ α/∂y ,

(2.7.4)

where δu− f is the perturbation of the longitudinal component of the fresh gas velocity induced at the flame front by the front wrinkling; see Figs. 2.44 and 2.7. It may be obtained

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in the linear approximation from the calculation in Section 10.3.4 for the general case of an arbitrary density contrast. For a small density contrast, the linear results are δu− f = UL H (α)/2,

km dL = /(2Mc ),

σ = O(UL k),

(2.7.5)

showing that (2.7.4) reduces to (2.7.1). Nonlinear Equation There are two types of nonlinear terms, depending on whether they come from the flow or from the geometry of the front. The latter, UL (∂α/∂y)2 /2, appears in the geometrical − definition of the normal burning velocity (10.1.6) when the decomposition u− f = UL + δuf is used, and the square root in the denominator is expanded for small ∂α/∂y. This nonlinear term represents Huygens’ construction; see Fig. 2.10. The order of magnitude of the nonlinear terms in Euler’s equations (Reynolds tensor) is km (δu− )2f ≈  5 (UL /τL )(α/dL )2 , 4 while the unsteady term is of order σ δu− f ≈  (UL /τL )(α/dL ). The nonlinear Reynolds terms in Euler’s equations are thus negligible provided that α/dL is not large and the linear approximation can be used for the flow field. According to continuity, the transverse − 2 component δw− f has the same order of magnitude as δuf ,  (α/dL )UL , given by (2.7.5). The order of magnitude of the nonlinear terms in (2.7.3) can be evaluated from (10.1.4)– (10.1.7) using the linear results. Therefore, the nonlinear term δw− f (∂α/∂y) in (2.7.3) com− ing from Un (see (10.1.6)), is smaller than the geometrical term UL (∂α/∂y)2 by a factor . The geometrical term is thus the dominant nonlinear term in (2.7.3). Comparing this geometrical term with ∂α/∂t gives the order of magnitude of the amplitude of wrinkling, α/dL = O(1). The nonlinear equation for the evolution of the front is then obtained by adding the geometrical term to the left-hand side of (2.7.1),   2 1 ∂ 2α 1 ∂α ∂α  + UL = 0. (2.7.6) − UL H (α) + 2 ∂t 2 km ∂y 2 ∂y This equation can be put into a nondimensional form by introducing the dimensionless variables, η (space), τ (time) and φ (amplitude), constructed using the marginal wavelength and the inverse of the linear growth rate as units of length and time, and taking account of the small slope of the front, αkm = O(), η ≡ km y,

τ ≡ km UL t/2 = ( 2 /4Mc )(t/τL ),

φ ≡ 2km α/.

(2.7.7)

Dividing the resulting equation by  2 UL /4 gives the following nondimensional equation in two-dimensional geometry,   ∂ 2φ 1 ∂φ 2 ∂φ − H (φ) − 2 + = 0. (2.7.8) ∂τ 2 ∂η ∂η The generalisation to three dimensions is immediate: 1 ∂φ − H (φ) − φ + |∇φ|2 = 0. ∂τ 2

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(2.7.9)

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This equation obtained by Sivashinsky[1] was shown to be valid[2] for flames fronts up to second order in an expansion in powers of  ≡ (ρu /ρb − 1). It belongs to a class of equations derived in plasma physics for which exact solutions are available. For more details the reader is referred to the original papers[3,4] and to the 1998 review of Joulin and Vidal.[5] These solutions are discussed in Section 2.7.2 and compared with experiments in Section 2.8.2. Extensions One of the simplifications associated with the leading orders in the limit of a small density contrast,  1, is that vorticity generation through the flame is of higher order and thus negligible. To leading order, the burnt gas flow is irrotational. Some artificial extensions to a larger density contrast have been proposed.[6,7,8] An integral equation describing large distortions of the flame front has been obtained without restriction on the thermal expansion by imposing the flows to be potential in a brute-force manner.[6] As explained at the end of Section 10.1.1 this requires relaxing the constraint in (10.1.11) for conservation of tangential momentum across the flame front. Another approximate extension[7] to large thermal expansion was obtained by changing the coefficient in (2.7.6). This introduces an additional term, depending only on time, which ensures that the increase in propagation velocity of the flame brush is proportional to the increase in surface area as in (3.1.11); see (2.8.1). A fairly good agreement for the shape of the wrinkled flame front in a laboratory experiment is shown in Section 2.8.2. Another nonlinear equation, valid also only for a small density contrast, was also obtained by Sivashinsky[1] when a thermo-diffusive instability is superimposed on the hydrodynamic instability. It is similar to (2.7.8) but with a positive sign in front of the second derivative term and an additional fourth derivative term that stabilises small wavelengths by heat conduction as in (2.4.4),   ∂ 2φ ∂ 4φ 1 ∂φ 2 ∂φ − H (φ) + 2 − b 4 + = 0, (2.7.10) ∂τ 2 ∂η ∂η ∂η where b is a positive dimensionless coefficient of order unity; see (10.2.27). A more systematic but less transparent (to say the least) approach to flame front dynamics, with no restriction on the thermal expansion, was considered by extending the integrodifferential equation mentioned at the end of Section 2.6.3 to unsteady cases.[9,10] [1] [2] [3] [4] [5]

Sivashinsky G., 1977, Acta Astronaut., 4, 1177–1206. Sivashinsky G., Clavin P., 1987, J. Phys., 48, 193–198. Lee Y., Chen H., 1982, Phys. Scripta, T2, 41–47. Thual O., et al., 1985, J. Phys., 46(9), 1485–1494. Joulin G., Vidal P., 1998, In G. Godr`eche, P. Manneville, eds., Hydrodynamics and nonlinear instabilities, 493–675, Cambridge University Press. [6] Frankel M., 1990, Phys. Fluids, A 2(10), 1897–1883. [7] Joulin G., Cambray P., 1992, Combust. Sci. Technol., 81, 243–256. [8] Boury G., 2003, Thesis, Universit´e de Poitiers. [9] Joulin G., et al., 2008, J. Fluid Mech., 608, 217–242. [10] El-Rabii H., et al., 2010, Phys. Rev. E, 81, 066312.

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2.7.2 Nonlinear Solutions for Wrinkled Flame Fronts Nonperiodic Solutions Equation (2.7.8) possesses a pole decomposition; this property allows the construction of an infinite number of exact solutions in the infinite domain η ∈ (−∞, +∞), in the form[4] α=2n  ∂φ 1 = −2 , ∂η η − zα(τ )

zα(τ ) = xα(τ ) + iyα(τ ) ,

(2.7.11)

α=1

where the zα are poles in the complex plane. These poles exist in complex conjugate pairs and move according to the law  1 − i sign(yα ), (2.7.12) α = 1, 2, . . . , 2n: z˙α = −2 zα − zβ β=α

where z˙ denotes time derivative of z, z ≡ dz/dτ . The proof of (2.7.11)–(2.7.12) is presented in Section 2.9.6. The polar representation was introduced in the 1970s to study Burgers’ equation and, more generally, solitons. Details may be found in the 1981 paper of Frisch and Morf[11] and the 1994 paper of Joulin.[12] Consider first the simplest case of two complex conjugate poles, z1 = x1 + iy1 , z2 = z∗1 = x1 − iy1 , y1 > 0. Equation (2.7.12) yields x˙ 1 = 0,

y˙ 1 = y−1 1 − 1.

(2.7.13)

As t → ∞, the solution tends to a steady state, x1 , y1 = 1, corresponding to a localised structure with a single maximum at η = x1 , ∂φ (η − x1 ) , = −4 ∂η (η − x1 )2 + 1

φ = −2 log[(η − x1 )2 + 1] + cst.

(2.7.14)

When many pairs of poles are involved, the integration of (2.7.12) must be carried out numerically. However, roughly speaking, the poles tend to attract each other in the direction parallel to the real axis, and to repel each other in the direction parallel to the imaginary axis. Therefore the poles form vertical alignments, eventually coalescing into a single line.[4] This is illustrated by the simple case of two poles sufficiently close to each other in the upper half of the complex plane so that the influence of the other poles may be neglected. According to (2.7.12), z˙1 ≈ −

2 − i, z1 − z2

z˙2 ≈ −

2 − i; z2 − z1

(2.7.15)

the difference ζ ≡ z1 − z2 , ζ˙ = −4/ζ , gives ζ 2 ≈ −8τ + cst.,

(2.7.16)

showing that the product of the real and imaginary parts of ζ remains constant, while the imaginary part increases linearly with time. This tendency to form a single alignment of [11] Frisch U., Morf R., 1981, Phys. Rev. A, 23(5), 2673–2705. [12] Joulin G., 1994, Phys. Rev. E, 50(3), 2030–2047.

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poles corresponds to the ultimate formation of a cellular structure with a wavelength much greater than that of the disturbance most amplified by the linear instability. This agrees with the Zeldovich mechanism for the stability of curved flame fronts described in Section 2.6. Periodic Solutions When the flame front in (2.7.6) is confined in a box of size L, y ∈ [0, L], one may look for periodic solutions, α(y + L) = α(y), that is, φ(η + km L) = φ(η). When periodicity is assumed along the η-axis, introducing the notation r ≡ 2π/(km L) and using the same notation as in (2.7.11)–(2.7.12), the pole decomposition takes the form α=2n  1 + e−ir[η−zα(t) ] ∂φ = −i r , ∂η 1 − e−ir[η−zα(t) ] α=1

z˙α = i r

 e−ir(zα −zβ ) + 1 − i sign(yα ), e−ir(zα −zβ ) − 1 β=α

(2.7.17) (2.7.18)

where, roughly speaking, 1/r is the number of unstable normal modes in the box of size L; see Section 2.9.6. The pole dynamics are similar to the previous case of nonperiodic solutions, and vertical alignment of poles still occurs, essentially because the short distance interaction is unaffected. Typically, condensation to a single steady vertical line of poles (all the poles have the same real part), or in other words a single wrinkle (single-peak structure), is obtained in the long time limit of (2.7.18), even though steady solutions with more than one vertical alignment do exist but are not stable.[1] Therefore, in the limit of a large number of linearly unstable modes (large wavelength limit), the stable steady solution has a single fold per wavelength, as in the hydrodynamical solution for curved flames propagating in a channel, presented in Section 2.6.3. Considering a steady solution of (2.7.18) in the form of a single pole condensation, xα = xβ ∀(α, β); the number of condensed poles at finite distance from the real axis, |yβ | < ∞ ∀β, is constrained,[2] 2n−1  1/r: the number of poles cannot be larger than the number of unstable linear modes in the box of size L. This is a consequence of the condition that all poles are stationary. Consider the pole α that has the largest imaginary part, yα > yβ ∀β = α and notice that each term of the sum in the right-hand side of (2.7.18) is larger than unity, e[r(yα −yβ )] + 1 > e[r(yα −yβ )] − 1; it follows that 2n − 1  1/r. If more poles (than in the steady state solution corresponding to a single vertical alignment) are present in the initial conditions, the extra poles are ejected to infinity along the imaginary axis.[3] The properties of the pole solutions in the periodic case agree with the numerical solutions of (2.7.8) starting with small arbitrary initial disturbances. These numerical solutions show that a single fold with the maximum admissible wavelength is reached in the long time limit.[4] Moreover, the asymptotically stable steady solution corresponds to the single pole [1] [2] [3] [4]

Vaynblat D., Matalon M., 2000, SIAM J. Appl. Math., 60(2), 703–728. Thual O., et al., 1985, J. Phys., 46(9), 1485–1494. Joulin G., Vidal P., 1998, In G. Godr`eche, P. Manneville, eds., Hydrodynamics and nonlinear instabilities, 493–675, Cambridge University Press. Matalon M., 2007, Ann. Rev. Fluid Mech., 39, 163–191.

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condensation with the maximum number of poles. The situation is different for Neumann boundary conditions (zero gradient) for which more than one stable solution, typically two, are observed.[5] These numerical simulations are difficult to perform because the solutions are extremely sensitive to noise; see Section 2.7.3. The periodic solution corresponding to two complex conjugate poles, z1 = x1 +iy1 , z2 = z∗1 = x1 − iy1 , y1 > 0, has been observed in the inverted ‘V’ flame studied experimentally in Section 2.8.2. According to (2.7.17), this solution may be written  cos [r(η − x1 )] + f (τ ), (2.7.19) φ(η, τ ) = −2 log 1 − cosh[ry1 (τ )] y˙ 1 = r

(e2ry1 + 1) − 1, (e2ry1 − 1)

x˙ 1 = 0;

(2.7.20)

see Section 2.9.6. This solution is valid for r < 1; that is, when one unstable normal mode, at least, is in the box of size L. The term f (τ ) comes from averaging (2.7.8) over the spatial k L coordinate 2∂f /∂τ = −(km L)−1 0 m dη(∂φ/∂η)2 . According to (2.7.20), a steady state solution, e2ry1 = (1 + r)/(1 − r), is reached in the long time limit, and f → −μτ where μ represents the increase of flame speed due to the increase in flame surface by wrinkling; see also (3.1.11). For ry1  1 the wrinkle is sinusoidal with a small amplitude φ ≈ 4e−ry1 cos[r(η − x1 )],

y1 ≈ −(1 − r)τ .

(2.7.21)

This corresponds to the linear growth of an unstable linear mode of wavelength 2π/L, φ ≈ 4er(1−r)τ cos[r(η − x1 )].

2.7.3 Effect of External Noise: The Case of Expanding Flames The solutions of (2.7.8) are highly sensitive to a fluctuating forcing term u(e) (η, τ ) (additive noise)   ∂ 2φ 1 ∂φ 2 ∂φ − H (φ) − 2 + = u(e) (η, τ ). (2.7.22) ∂τ 2 ∂η ∂η As we shall see, external noise, even of small amplitude, is needed to represent the experimental observations. The origin of this noise may be a residual turbulence in the quasiquiescent fresh mixture. In numerical simulations, computational noise qualitatively influences the results. The sensitivity is so high that the effect of thermal noise cannot be excluded. Noise-Induced Cells on Quasi-planar Flame Fronts The drastic effect of very weak turbulence on cellular structures was first observed in nearly spherical expanding flames.[6] If the second and fourth terms in the left-hand side [5] [6]

Denet B., 2006, Phys. Rev. E, 74, 036303–1–9. Palm-Leis A., Strehlow R., 1969, Combust. Flame, 13, 111–129.

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(hydrodynamic and nonlinear geometrical terms) are omitted, Equation (2.7.22) reduces to Langevin’s equation describing the diffusive thickening of the flame brush. The solutions of (2.7.22) are quite different from the solution of Langevin’s equation. Crests that are sharply pointed towards the burnt mixture (local maxima of φ(η, τ )) appear on the flame front and their number fluctuates.[1] For a given flame length (diameter), L, the average number of crests or, in other words, the average number of cells depends on the statistical properties of the external noise and increases with the intensity of the noise. Numerical studies[1] of (2.7.22) show that the semi-phenomenological criterion[2] (2.6.38) based on the Zeldovich criterion, where R represents here the average size of the noise-induced cells, provides a fairly good relation between cell size and noise. A small intensity of noise is sufficient to produce cells of size much smaller than the size of the box, L, or the diameter of the tube in which the flame propagates. Numerical studies of an extension of (2.7.22) to twodimensional geometry show similar behaviour.[3] Morphology of Expanding Spherical Flames Two types of experiments concerning expanding flames in quiescent mixtures are reported. 1. Laboratory experiments of expanding flames have been carried out in constant volume[4,5] or constant pressure[6] vessels whose typical size is 10–30 cm. The combustible mixture is spark-ignited at the centre of the vessel. Flame morphology is observed by Schlieren photography or a high-speed digital motion camera. A similar scenario is observed in all conditions. Initial disturbances associated with the ignition device first grow in proportion to flame radius and evolve to a few large-scale ridges which stay at fixed polar angles. For a short time after ignition, typically a few tens of millisecond for thermo-diffusively stable flames, the flame front remains smooth except for these initial large ‘cracks’. At a later stage, the front becomes suddenly covered with a pattern of small-scale cells a few millimetres in size. This occurs almost instantaneously, even for reactive mixtures that do not produce thermo-diffusive instabilities in flames. The critical radius of the flame at the onset of this cellular structure is typically a few centimetres, and the cell size is few millimetres. As we shall see, this behaviour cannot be fully explained without introducing the substantial role played by noise, a suggestion made first by Joulin.[7,8] 2. Large-scale free spherical flames of several metres in radius have also been studied experimentally.[9,10] A self-similar acceleration is reported corresponding to a thickness [1] [2]

Cambray P., Joulin G., 1994, Combust. Sci. Technol., 97, 405–428. Clavin P., 1988, In E. Guyon, J. Nadal, Y. Pomeau, eds., NATO ASI Series E. Disorder and Mixing, vol. 152, 293–315, Kluwer Academic Publishers. [3] Creta F., et al., 2011, Combust. Theor. Model., 15(2), 267–298. [4] Groff E.G., 1982, Combust. Flame, 48, 51–62. [5] Bradley D., et al., 2000, Combust. Flame, 122(1-2), 195–209. [6] Jomaas G., et al., 2007, J. Fluid Mech., 583, 1–26. [7] Joulin G., 1989, J. Phys-Paris., 50, 1069–1082. [8] Joulin G., 1994, Phys. Rev. E, 50(3), 2030–2047. [9] Gostintsev Y., et al., 1988, Combust. Expl. Shock Waves, 24(5), 563–569. [10] Bradley D., et al., 2001, Combust. Flame, 124, 551–559.

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Figure 2.45 Three-dimensional numerical simulation of the model equation for a spherical expanding flame in the presence of an external white noise. The three figures correspond to different instants of time, increasing from left to right. The radius of the flame, which also increases in time, has been rescaled. The noise-induced cells appear at a sufficiently large radius (central figure). Their number increases by tip splitting as the mean radius of the flame increases (right figure). The crests on the left figure result from the traces left by the initial inhomogeneities of ignition. Courtesy of Yves d’Angelo.

of the flame brush growing with time as t3/2 , and a wrinkled flame velocity increasing as t1/2 . According to the available data, the behaviour is universal, meaning that it does not depend on the nature of the flammable mixture. Furthermore, buoyancy effects seem negligible. The increase of velocity is related to the increase of flame surface area produced by the cellular structure. The acceleration of the flame brush should result from a continuous increase of the corrugated conformation. It was suggested that the self-similar acceleration is linked to development of fractal structures on the flame surface.[9] Despite many theoretical attempts, self-fractalisation of a freely outwardly propagating flame front is still an open question,[3,8,11,12] especially in the absence of thermo-diffusive instability. It is more probable that this phenomenon is related to noise playing an increasingly important role as the flame grows larger.[13,14] Analysis of a Model Equation In expanding flames, expansion-induced time-dependent stretch of the flame surface alters the hydrodynamic instability. A model equation extending (2.7.22) to such cases has been proposed[8,15] in order to study the morphology of expanding wrinkled flames in the simplest possible terms. Numerical studies of the model equation in three-al geometry[13] show the same morphology as observed in experiments;[4,16] see Fig. 2.45. The results have also been confirmed by direct numerical simulations.[17] Two-dimensional cylindrical geometry is sufficient to catch the essential mechanisms. Using polar coordinates, φ(θ , ρ), dη = ρdθ , the equation of evolution for corrugations on the front is obtained from (2.7.22) in the simple form [11] [12] [13] [14] [15] [16] [17]

Rahibe M., et al., 1995, Phys. Rev. E, 52(4), 3675–3686. Ashurst W., 1997, Combust. Theor. Model., 1, 405–428. D’Angelo Y., et al., 2000, Combust. Theor. Model., 4, 317–338. Karlin V., Sivashinsky G., 2006, Combust. Theor. Model., 10(4), 625–637. Filyand L., et al., 1994, Physica D, 72, 110–118. Palm-Leis A., Strehlow R., 1969, Combust. Flame, 13, 111–129. Albin Y., D’Angelo Y., 2012, Combust. Flame, 159, 1932–1948.

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1 ∂φ 1 ∂ 2φ 1 − Hθ (φ) − 2 2 + 2 ∂τ ρ ρ ∂θ 2ρ



∂φ ∂θ

2 = u(e) ,

(2.7.23)

where ρ = τ is the reduced mean radius of the flame. Attention is limited to case 1, presented above, in which the velocity of the expanding flame brush is constant, so that, using conveniently reduced variables, ρ is replaced by τ in (2.7.23). A similar equation could be obtained from (2.7.10) for flames that are thermo-diffusively unstable. In the absence of the forcing term ue = 0, Equation (2.7.23) possesses a pole decomposition.[1,2] However, the corresponding solutions are not very useful for explaining the cellular structure of expanding flames since, for the same reasons as in Section 2.7.2, these particular solutions are characterised by a constant number of cells whose size increases with the flame radius. Linear solutions cannot fully explain the experimental data. However, they give a useful insight.[1] Consider first the stability analysis of (2.7.23) in the absence of external noise, ue = 0. For a single angular mode, φ(θ , τ ) = φ˜ n (τ )einθ , Equation (2.7.23) yields  |n|  1 1  1 dφ˜ n |n| n2 τ n2 − = − 2, φ˜ n (τ ) = φ˜ n (τ0 ) e τ τ0 . (2.7.24) τ τ0 τ φ˜ n dτ Similar results have been obtained from a stability analysis starting with the basic equations of fluid mechanics.[3,4] Notice that, in the long time limit, the amplitude of disturbances with a fixed polar angle grows in time with a power law. Following an argument due to Istratov and Librovich,[3] an expanding spherical front can be considered as unstable if the amplitude of a stretched disturbance grows faster than the mean flame radius. This leads to consideration of the relative amplitude of a mode ψ˜ n (τ ) ≡ φ˜ n /τ ,  |n|−1  1 1  ψ˜ n (τ ) τ n2 − e τ τ0 . (2.7.25) = τ0 ψ˜ n (τ0 ) According to (2.7.25), for |n| > 1, the relative amplitude ψ˜ n (τ ) first decreases with time. This occurs at small times due to the strong stretch effect of an expanding flame with a small radius. However, the relative amplitude increases at larger times when the stretch effect is sufficiently weakened for the hydrodynamical instability to take over. The angular mode characterised by n starts to grow after its relative amplitude ψ˜ n (τ ) has reached its minimum value, dψ˜ n /dτ = 0, which corresponds to the time τ = τn ≡ n2 /(|n| − 1). According to this criterion, the instability initially appears at time τ = τ ∗ ≡ 4 for the mode n = 2, which is the minimum of τn . It is thus natural to consider that the onset of the hydrodynamical instability occurs at time τ = τ ∗ ≡ 4. Notice that the time delay τ ∗ characterising the instability threshold corresponds to the linear growth rate of the most amplified mode of a planar flame. Notice also that the spatial wavenumber associated with the angular mode n is κn = n/τ since the size of the expanding flame increases like τ . Therefore for large n, such [1] [2] [3] [4]

Joulin G., 1994, Phys. Rev. E, 50(3), 2030–2047. Rahibe M., et al., 1995, Phys. Rev. E, 52(4), 3675–3686. Istratov A., V.B. L., 1969, Acta Astronaut., 14, 453–457. Bechtold J., Matalon M., 1987, Combust. Flame, 67, 77–90.

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a mode starts to grow on the expanding flame front at a time τ = τn ≈ n increasing linearly with n, but involving a fixed spatial wavenumber corresponding to that of the marginal mode of the planar case, κ = 1. Ultimately the mode grows as τ n while being stretched. These growth rates and the critical times τ ∗ are too small to explain the sudden onset of the cellular structure appearing at a critical flame radius of few centimetres reported in experiments. However, this linear analysis is consistent with the cracks that are observed on the flame surface just after ignition. Noise-Induced Cells on Expanding Flame Fronts Consider now the linear response to external noise.[1] For a single angular mode, Equation (2.7.23) takes the form of a Langevin equation,  |n| n2 dφ˜ n = − 2 φ˜ n + u˜ n(e) (τ ), dτ τ τ

(2.7.26)

(e)

where u˜ n (τ ) is a fluctuating velocity. Focusing attention on the flame response to noise, the initial conditions are neglected in a first step. The solution to (2.7.26) for a zero initial condition, φ˜ n (τ = 0) = 0, is

  2  τ 2 − |n| ln τ  + nτ  (e)  |n| ln τ + nτ e u˜ n (τ )dτ  . (2.7.27) φ˜ n (τ ) = e 0

The essential feature is obtained by considering (2.7.27) in the limit of large n. This can be roughly seen as follows, without entering into the detailed formalism of stochastic fields. The first point is that for a forcing term resulting from a random spatial field, as for example the residual turbulence of an otherwise quiescent initial mixture, the forcing felt by the front depends on the spatial wavenumber of the external field. Assume for simplicity that this random field is homogeneous with a single length scale, namely a single (e) spatial wavenumber κe . Therefore |n| = κe τ in u˜ n (τ ), so that, at sufficiently long time, the relevant angular n  1. Assume for simplicity that we are dealing with   indices n are large, (e) (e) a white noise, u˜ n (τ  )˜u−n (τ  ) = δ(τ  − τ  )Dtur , where Dtur is the diffusion coefficient of the random walk associated with the rapidly fluctuating velocity u(e) (τ ). Therefore, according to (2.7.27), the amplitude of corrugations on the expanding flame front in the presence of external noise is given by

  2  τ 2   2 |n| ln τ + nτ −2 |n| ln τ  + nτ  |φ˜ n |2 ≈ Dtur e e dτ  , (2.7.28) 0

contributions to valid for |n| = κe τ ; the other angular indices, |n| = κe τ , give negligible        the mean square of the amplitude of the wrinkles at time τ , φ 2 = n |φ˜ n |2 ≈ |φ˜ κe τ |2 . This formula is similar to (2.7.27), except for the coefficient 2 in the exponents. Using the relation |n| = κe τ and the variable of integration υ ≡ τ  /τ , dτ  = τ dυ, Equation (2.7.28) yields

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   2 φ 2 ≈ Dtur τ e2κe τ

1

e−2κe τ [ln υ+κe /υ] dυ.

(2.7.29)

0

In the limit κe τ → ∞, the function e−2κe τ [ln υ+κe /υ] becomes sharply peaked at the minimum υ = κe of the function h(υ) ≡ ln υ + κe /υ. Assume that the spatial wavelength of the external noise is in the unstable range of the planar flame, κe < 1. At large time, κe τ  1, the integral in (2.7.29) may then be evaluated by Laplace’s method,[1] using the expansion h(υ) ≈ (ln κe + 1) + (υ − κe )2 /(2κe2 ) + · · · , and extending the integral domain of υ  ≡ υ − κe to υ  ∈ (−∞, +∞),   √ κe < 1, κe τ  1: φ 2 ≈ Dtur π κe τ e2τ s(κe ) , s(κe ) ≡ κe (κe − ln κe − 1), (2.7.30) √ 2 and where the relation −∞ eaX dX = π/a has been used. When the characteristic length scale of the noise belongs to the hydrodynamical unstable domain of planar flames, Equation (2.7.30) shows that the noise-generated cells have the same size as the noise length scale, and, ultimately, their amplitude grows exponentially with time. Therefore the effect of noise quickly overtakes that of disturbances generated by the initial conditions, which grow as power laws. Moreover, according to (2.7.30), the growth rate is maximum for a noise having a reduced wavenumber κe ≈ 0.2 corresponding to the maximum value of s(κe ) in the range κe < 1 and to a maximum reduced growth rate of 0.16. Coming back to the original dimensional variables, these results are found to be compatible with the observed trends.[2] where

 +∞

2.8 Additional Laboratory Experiments 2.8.1 Measurements of Growth Rate of the DL Instability Validation of Dispersion Relation for Slow Flames Experimental measurement of the growth rate of the DL instability is not easy: the typical growth time of the most unstable wavelengths, τDL ≈ (UL km /2)−1 ≈ 10–50 ms, is shorter than the time needed to establish a free flame front a few centimetres in diameter. The first experimental measurement was performed using acoustic stabilisation, as described in Section 2.5.5, to maintain a planar laminar flame in conditions where the flame is intrinsically cellular in the absence of acoustics. The imposed acoustic field is then removed on a short time scale, ≈ 1 ms, and the unconstrained growth of the DL instability is observed.[3] Lean premixed propane mixtures are fed into the bottom of a Pyrex glass tube, as shown in Fig. 2.46. The plug flow is laminarised by a 50 μm porous plate and the flame is held

[1] [2] [3]

Bender M., Orszag S., 1984, Advanced mathematical methods for scientists and engineers. McGraw-Hill. Joulin G., 1994, Phys. Rev. E, 50(3), 2030–2047. Clanet C., Searby G., 1998, Phys. Rev. Lett., 80(17), 3867–3870.

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Figure 2.46 Schematic diagram of apparatus to study DL growth rate. Reproduced with permission from Clanet C., Searby G., Physical Review Letters, 80(17), 3867–3870. Copyright 1998 by the American Physical Society.

stationary by adjusting the flow rate. An aluminium honeycomb structure, placed a few centimetres upstream from the flame, helps maintain the laminar plug flow. A loudspeaker imposes a standing acoustic wave with a velocity anti-node close to the flame front. Using this technique, lean propane flames can be stabilised with flame speeds up to 0.2 m/s. For faster flames, two regions of instability, DL and parametric, overlap (see Fig. 2.37) and it is not possible to obtain a planar flame. In order to control the wavelength and orientation of the structures that develop when the acoustic stabilisation is removed, the upstream flow is perturbed by an array of parallel wires placed on the honeycomb. The spacing between the wires is close to the most unstable wavelength, 2π/kc ≈ 2 cm. The luminous emission from the flame front is filmed edge-on in a direction parallel to the axis of the wires. Fig. 2.47a shows images taken from a high-speed film after removal of the acoustic field. The apparent thickening of the flame indicates the presence of slight three-dimensionality of the wrinkling. The nonlinearity visible in last images indicates the onset of saturation; see Section 2.7. The peak-to-peak amplitude of the wrinkling is fitted to an exponential function of the form     v σ t v −σ t 1 , a˜ o + e + a˜ o − e a˜ (t) = 2 σ σ which is the general solution of ∂ 2 a˜ /∂t = σ 2 a˜ with the initial conditions a˜ (0) = a˜ 0 and ∂ a˜ (0)/∂t = v. Here, v is the rate of increase of the wrinkling at time t = 0, supposed equal to the measured peak-to-peak velocity modulation produced by the wires in the flow. The experimentally measured growth rates are plotted as a function of laminar flame speed in Fig. 2.47b. The full line shows the theoretical growth rates obtained from the ‘detailed model’ presented in Section 2.9.5. For lean propane flames, a Markstein number M = 4 was found to give best agreement with the experimental data. The experimental points agree with the theoretical curve to within experimental error, except for the measurement at the

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(a)

(b)

Figure 2.47 (a) Images from a high-speed film showing the growth of the DL instability. Framing rate 500 i/s, wavelength 2 cm, propane flame, speed 0.12 m/s. (b) Comparison of measured and theoretical growth rates. Reproduced with permission from Clanet C., Searby G., Physical Review Letters, 80(17), 3867–3870. Copyright 1998 by the American Physical Society.

lowest flame velocity. The value M = 4 is comparable to value 4.5 found by fitting the threshold of parametric instability; see Fig. 2.38. The corresponding marginal wavenumber km varies with laminar flame speed from km = 0.079 for UL = 0.12 m/s to km = 0.111 for UL = 0.20 m/s. Validation of Dispersion Relation for Fast Flames Measurements on faster propane flames have been made on an inverted ‘V’ flame using a laminar slot burner[1] 80 mm long and 8 mm wide; see Fig. 2.48a. On one side of the slot, the flame is anchored on a thin tungsten rod. A sinusoidal high voltage (≈ 2–4 kV) on the tungsten rod produces an electrostatic deflection of the flame front with an amplitude (≈ 50–100 μm) and frequency (≈ 1–4 kHz) that are easily varied. The resulting sinusoidal perturbations on the flame front grow both in space and in time as they are are convected downstream to the tip of the flame by the tangential component of the flow velocity (≈ 5– 8 m/s). The flow is sufficiently laminar that perturbations induced by residual turbulence are totally negligible compared with the electrostatic deflection, as confirmed by the fact that the nonexcited flame front does not show wrinkling before the two flame fronts are close enough to interact through the upstream modification to the flow field. This interaction extends a distance of the order of 1/k = /(2π ), upstream (see Fig. 2.7 and (10.1.20)), [1]

Truffaut J., Searby G., 1999, Combust. Sci. Technol., 149, 35–52.

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149 (b)

10 mm

(a)

Figure 2.48 (a) The slot burner and electrostatic deflection system. (b) Instantaneous photo of the growth of the DL instability on a flame.

limiting the longest wavelengths that can be studied. An intensified camera takes shortexposure (100 μs) images, shown in Fig. 2.48b, which are used to obtain a spatial growth rate, σx . The temporal growth rate, σt , is related to the spatial growth rate by the convection velocity: Uc , σt = σx Uc , provided that the growth is small during the time needed to convect the structures a distance of one wavelength, σ  Uc . Since the frequency, and thus the wavelength, of excitation is continuously variable, it is possible to explore the growth rate as a function of wavenumber for constant flame parameters. Fig. 2.49 shows the reduced temporal growth rate as a function of the reduced wavelength of wrinkling for four propane–air flames with equivalence ratios 1.05, 1.15, 1.25 and 1.33, corresponding to flames speed of 0.43, 0.41, 0.35 and 0.27 m/s, respectively. The lines are calculated using the dispersion relation of the ‘detailed model’ presented in Section 2.9.5, but with g = 0 since the effect of gravity is negligible in these experiments. The only unknown parameter is the (first) Markstein number M. The full lines show the best fits; the dotted lines give an indication of the sensitivity. It is found that M tends to decrease with decreasing flame speed.

2.8.2 Validation of the Nonlinear Dynamics The growth of the DL instability in the nonlinear domain has been investigated using an inverted ‘V’ flame excited by electrostatic deflection.[2] The apparatus is identical to that described in the previous section (see Fig. 2.48), except that the mixture is enriched with oxygen, increasing both the flame speed and the growth rate of the DL

[2]

Searby G., et al., 2001, Phys. Fluids, 13, 3270–3276.

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Reduced growth rate

150

Reduced wavenumber

Figure 2.49 Measured and calculated dispersion relation for four propane flames. The solid line is a best fit. Reproduced from Truffaut J., Searby G., 1999, Combustion Science and Technology, 149, 35–52 with permission from Taylor and Francis Ltd. www.informaworld.com.

instability so that the final amplitude the cellar instability is strongly nonlinear, as shown in Fig. 2.50. The nonlinear evolution equation (2.7.6) of the wrinkles has to be extended to accommodate realistic gas expansion and spatio-temporal behaviour before it can be compared with the flame of Fig. 2.50. Equation (2.7.6) was derived to first order in the limit of small gas expansion; see Section 2.7.1. The analysis has been pushed up to second order in an expansion in powers of (ρu /ρb − 1), and the result[1] shows that the wrinkled flame front is described by        ∂α 2 1 ∂ 2α UL ∂α 2 ∂α − − (a − 1) , (2.8.1) = AUL H (α) + a ∂t km ∂y2 2 ∂y ∂y

[1]

Sivashinsky G., Clavin P., 1987, J. Phys., 48, 193–198.

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Figure 2.50 Growth of DL instability of a propane–air–oxygen flame in the nonlinear domain. Equivalence ratio = 1.33, 28% oxygen, flow velocity 8.56 m/s, excitation frequency 2500 Hz. Reproduced with permission from Searby G., Truffaut J.-M., Joulin G., Physics of Fluids, 13, 3270– 3276. Copyright 2001, AIP Publishing LLC.

where a ≡ 2ρu /(ρu + ρb ), AUL k is the linear growth rate of the DL instability (see (10.1.32)), and < . > denotes the y-average. The last term on the right-hand side, which was omitted in the original paper,[1] was added[2,3] to ensure that, in the reference frame of the unperturbed front (α = 0), the flame brush propagates towards the fresh mixture  at a speed (1/2)UL (∂α/∂y)2 given by the fractional increase in total front length (see (3.1.11)),  1/2   (1 + (∂α/∂y)2 − 1 ≈ (1/2) (∂α/∂y)2 . It turns out that this nonlinear equation describes accurately the cellular flame for ordinary values of the gas expansion in the absence of external forces, although the inertia term is missing. In the presence of a tangential flow velocity, ut , as is the case in the experiment of Fig. 2.50, the wrinkles are convected with the velocity ut . Equation (2.8.1) is still valid[3] in the Lagrangian frame that moves in the tangential direction at velocity ut , and y has to be interpreted as y − ut t. Using the pole decomposition presented in Section 2.7.2, an exact solution of (2.8.1) can be constructed in the form[3] " ! 2A cos[k(y − yo )] , (2.8.2) α(y, t) = −A(t) − log 1 − a km cosh [kB(t)] where the origin yo in the moving frame and the wavenumber k > 0 are arbitrary given constants, and         Ak 2 1 1 Ak dB dA = 4UL , +4 2kB 2kB dt akm akm dt e −1 e +1  dB k = AUL coth (kB) − 1 ; dt km see (2.7.19)–(2.7.21). The function B → ∞ for t → −∞. At fixed position y in the laboratory frame y + ut t = y = cst., the above solution for α(y, t) oscillates in time with [2] [3]

Joulin G., Cambray P., 1992, Combust. Sci. Technol., 81, 243–256. Searby G., et al., 2001, Phys. Fluids, 13, 3270–3276.

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(a)

Figure 2.51 (a) Plot of flame profile from fig. 2.50. (b) Peak-to-peak amplitude of cells and calculated amplitude. ρu /ρb = 8.33, A = 1.82, k/km = 0.38. Reproduced with permission from Searby G., Truffaut J.-M., Joulin G., Physics of Fluids, 13, 3270–3276. Copyright 2001, AIP Publishing LLC.

a pulsation ω = ut k. This yields k if ut and ω are known. The local maxima and minima occur when cos[k(y − yo )] = ±1, yielding a peak-to-peak amplitude of wrinkling, " ! 2A cosh [kB (y/ut )] + 1 αp−p (y) = , (2.8.3) ln a km cosh [kB (y/ut )] − 1 where t has been replaced by y/ut , an approximation valid in the limit ut  UL . There are no new parameters in (2.8.3). The wavenumber k is easily measured, A and a are given by the density ratio, and km can be deduced from measurements of the linear growth rate as a function of wavenumber (see Section 2.8.1) or from knowledge of the first Markstein number M; see (2.9.54) in Section 2.9.5. Fig. 2.51a shows the digitised profile of the flame in fig. 2.50, along with αp−p (y). The plot on the right shows the comparison between the measured peak-to-peak amplitude and that calculated from (2.8.3). The agreement is surprisingly good, both for the amplitude at saturation and for the width of the crossover region between exponential growth and saturation. Comparison of the flame shape is also very good.[1]

2.8.3 Richtmyer–Meshkov Instability and Tulip Flames Interaction with a Weak Shock Markstein[2]

that when a curved flame front is traversed by a shock wave, It was noticed by the curvature of the front inverts a short time after the interaction to form a characteristic shape known as the tulip flame. An example of this type of interaction is shown in Fig. 2.52. In this experiment a rich methane–air flame propagates freely downwards from the open end towards the closed end of a Pyrex tube 450 mm long and 47 mm in internal diameter. The tube is allowed to drop freely until the closed base impacts on a hard surface. This [1] [2]

Searby G., et al., 2001, Phys. Fluids, 13, 3270–3276. Markstein G., 1956, Proc. Comb. Inst., 6, 387–398.

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Figure 2.52 Tulip flame formed by the impulsive acceleration of a methane–air flame towards the burnt gas. Equivalence ratio = 1.35, laminar flame speed = 0.22 m/s, tube internal diameter 46 mm, length = 450 mm. Time after impact. The impulsive velocity jump is V = −3.4 m/s. Courtesy of E. Villermaux and G. Searby, IRPHE, Marseilles.

impact creates a strong acceleration of short duration < 10−4 s that is transmitted through the flame as a weak shock propagating upwards through the gaseous mixture. The sequence of images in Fig. 2.52 shows short-exposure (1/2000 s) images of the flame. The frame of the images is stationary in the reference frame of the tube, and the time scale is the time after impact. A similar experiment was done earlier by Pokrovski with a tube filled with water.[3] The phenomenon is similar to that produced by an impulsive acceleration of an interface separating two fluids of different density,[4] called Richtmyer–Meshkov instability, ˜ of a although the instability was first described by Markstein.[5] The amplitude, α(t), harmonic wrinkle of the flame is described by Equation (2.2.18) in which the acceleration of gravity |g| is replaced by a time-dependent acceleration g(t). This equation can be solved by numerical integration; however, it can be greatly simplified by noticing that the mean position of the front has changed very little during the formation of the tulip flame, meaning that the local propagation speed can be neglected, UL = 0. Therefore, the initial evolution of the front is described to good approximation by (2.2.14) for a passive interface: ˜ 2 − (1 − ρb /ρu ) g(t)kα˜ = 0. If the duration of the impact is much (1 + ρb /ρu ) d2 α/dt shorter than the time scale of evolution of the flame front, as can be checked afterwards (see ∞ (2.8.6)), the acceleration, g(t), can be replaced by a delta function g(t) = δ(t) −∞ g(t ) dt , and the equation can then be integrated once to give  ∞ dα˜ = A kV α˜ 0 , V = g(t )dt , (2.8.4) dt −∞ where A = (ρu − ρb )/(ρu + ρb ) is the Atwood number, α˜ 0 is the amplitude before impact, and g is negative when a shock front traverses the flame from the cold (heavy) side towards [3] [4] [5]

Lavrentiev M., Chabat B., 1980, Effets hydrodynamiques et mod`eles math´ematiques. Editions MIR. Richtmyer R., 1960, Commun. Pure Appl. Math., 13, 297–319. Markstein G., 1957, J. Aero. Sci., 24, 238–239.

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Figure 2.53 Photograph of luminous emission from a flame ignited at the closed end of a long tube, showing the elongated shape of the front. Reproduced from Clanet C., Searby G., Combustion and Flame, 105, 225–238, Copyright 1996 with permission from Elsevier.

the burnt gas, V < 0. Integrating a second time yields α(t) ˜ = α˜ 0 (1 + A kVt).

(2.8.5)

According to (2.8.5) the amplitude of wrinkling, α, ˜ goes through zero after a time tinv given by tinv = −(A kV)−1 ;

(2.8.6)

for times longer than τinv the local curvature of the font is inverted. The flame front in Fig. 2.52 is not harmonic, but if it is supposed that the equivalent wavelength is equal to twice the tube diameter, using ρu /ρb = 7.2 and V = −3.4 m/s, (2.8.6) predicts tinv = 5.7 ms. This value is in reasonable agreement with the time 4.5 ms observed in Fig. 2.52, considering that the curvature is not harmonic and also that reflexion at the extremities has been neglected. Auto Acceleration and Tulip Formation Tulip flame formation occurs when a flame is ignited at the closed end of a long tube.[1,2] If the flame is ignited on the axis of the tube, it initially expands as a hemispherical front pushing the unburnt gas spherically outwards until the flame reaches the vicinity of the cylindrical wall. At this time the wall blocks the radial component of the flow and the increasing volume of burnt gas is accommodated essentially by an extension of the flame along the axis of the tube, as shown in Fig. 2.53, increasing the surface area of the flame and the rate of production of burnt gas. This configuration thus leads to an exponential acceleration of the velocity of the tip of the flame that lasts until the side wall of the elongated flame front touches the wall of the tube, after which time a large part of the side wall of the flame is rapidly extinguished, creating a rapid drop in the rate of production of burnt gas and the velocity of the flame tip. The flame tip is then subject to a rapid deceleration, that is, an acceleration towards the burnt gas, leading to the formation of a tulip flame (see Fig. 2.54), by a mechanism[3] similar to that described above for interaction with a weak shock. Noticing that the side wall of the flame in Fig. 2.53 is almost parallel to the tube wall, the accelerating flame can be modelled to a good approximation as a cylindrical flame with a hemispherical cap; see Fig. 2.55. The volume of burnt gas inside the [1] [2] [3]

Ellis O.d.C., 1928, J. Fuel Sci., 7(11), 502–508. Salamandra G., et al., 1958, Proc. Comb. Inst., 7, 851–855. Clanet C., Searby G., 1996, Combust. Flame, 105, 225–238.

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Figure 2.54 Superposition of three images showing formation of a tulip flame in a half-closed tube. Reproduced from Clanet C., Searby G., Combustion and Flame, 105, 225–238, Copyright 1996 with permission from Elsevier.

Figure 2.55 Geometrical model of accelerating flame in a long tube. Reproduced from Clanet C., Searby G., Combustion and Flame, 105, 225–238, Copyright 1996 with permission from Elsevier.

flame is Vb = π r2 (ztip −r)+2/3π r3 and the surface area of the flame is S ≈ 2π rztip , where ztip (t) is the position of the tip of the flame with respect to the base of the tube. The rate of production of burnt gas is proportional to the flame area, dVb /dt = UL Sρu /ρb , neglecting dr/dt ∼ = UL compared with dztip /dt; and putting r = R leads to a simple evolution equation for the position of the flame tip: dztip /dt = ztip /τa

1/τa = 2(ρu /ρb )UL /R,

(2.8.7)

which can be integrated to give ztip /R = exp (t − to )/τa ,

(2.8.8)

where to is a measure of the time at which the initial hemispherical flame changes to a quasi-cylindrical shape. Despite its simplicity, this model gives a reasonable approximation for the evolution of the flame tip, up to the time tw when the flame skirt first touches the tube wall, as shown in Fig. 2.56. For t0 < t < tw the flame is accelerated towards the dense fresh gas and the curved front is stable, as for the flame propagating upwards in Fig. 2.11. For t > tw the flame surface area decreases rapidly by extinction of the skirt, which is quasi-parallel to the tube wall. The corresponding strong deceleration destabilises the flame tip in a way similar to the interaction with a weak shock of Fig. 2.52. The problem

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Reduced position,

Reduced pressure,

Reduced time,

Figure 2.56 Experimental measurement of position of leading tip of a propane–air flame (equivalence ratio 0.7) ignited at the closed end of a tube 1.5 m long and 0.1 m in diameter. Solid line: experiment. Dotted line: exponential from (2.8.8). Open squares: measured position of flame skirt. The reduced pressure at the closed end is also shown. Reproduced from Clanet C., Searby G., Combustion and Flame, 105, 225–238, Copyright 1996 with permission from Elsevier.

is complicated by the fact that the characteristic deceleration time is not small compared with the evolution time. A semi-phenomenological analysis leads to an inversion time, tinv , in reasonably good agreement with the experiment.[1]

2.9 Appendix 2.9.1 Curvature and Stretch of a Surface in R3 In this section we derive the curvature and the rate of stretch of a surface, using fixed Cartesian coordinates. Consider a surface function of time whose equation takes the form x = α(y, z, t). The coordinates of a point on the surface are r = (α, y, z). At each point two tangential vectors ry = (αy , 1, 0), rz = (αz , 0, 1), and a unit vector normal to the surface  nf = (ry × rz )/ 1 + αy2 + αz2 are defined, ⎛

⎞ 1 1 ⎝ −αy ⎠ , nf =  (2.9.1) 2 2  1 + αy + αz −αz   as well as unit tangential vectors tgy = ry / 1 + αy2 and tgz = rz / 1 + αz2 . The coefficients of the first fundamental quadratic form,[2] useful for measurements of length, area and angle on surfaces, are E ≡ ry .ry = 1 + αy2 , [1] [2]

F ≡ ry .rz = αy αz ,

G ≡ rz .rz = 1 + αz2 .

Clanet C., Searby G., 1996, Combust. Flame, 105, 225–238. Stoker J., 1989, Differential geometry. Wiley-Interscience.

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157

The coefficients of the second fundamental quadratic form, denoting the deviation of the surface from its tangent plane, are[2]   L ≡ ryy .nf = αyy / 1 + αy2 + αz2 ,   M ≡ ryz .nf = αyz / 1 + αy2 + αz2 ,   / 1 + αy2 + αz2 . N ≡ rzz .nf = αzz Mean Curvature The sum of the curvatures in the principal directions takes the form[2] 1 1 EN − 2FM + GL + = R1 R2 EG − F 2  (1 + α 2 ) + α  (1 + α 2 ) − 2α  α  α  αyy z zz y y z yz = (1 + αy2 + αz2 )3/2 and can be written −∇.nf . The element of area is given by   dS = H(y, z, t)dydz, H ≡ EG − F 2 = 1 + αy2 + αz2 .

(2.9.2)

(2.9.3)

Velocity of the Points on the Surface If the points move on the surface δα = αy δy + αz δz + α˙ t δt, δy = y˙ t δt, δz = z˙t δt, where y˙ t (y, z, t) and z˙t (y, z, t) are functions of y, z and t, the velocity of the point is dr = (˙yt αy + z˙t αz + α˙ t , dt

y˙ t ,

z˙t ),

(2.9.4)

with the following normal and tangential components n.

tgy .

α˙ t dr = , dt 1 + αy2 + αz2

(2.9.5)

y˙ t (1 + αy2 ) + z˙t αz αy + α˙ t αy dr = ,  dt 1 + α 2

(2.9.6)

y

tgz .

dr = dt

z˙t (1 + αz2 ) + y˙ t αy αz 

+ α˙ t αz

1 + αz2

.

(2.9.7)

Consider a surface in a flow u(r, t) = (u, v, w) moving with a normal velocity Un relative to the flow, Df = nf .u − Un ,

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and assume that the points on the surface move with the tangential velocity of the flow, u − vαy − wαz dr = − Un , dt 1 + αy2 + αz2

(2.9.8)

tgy .

uαy + v dr , = dt 1 + αy2

(2.9.9)

tgz .

uα  + w dr ; = z dt 1 + αz2

(2.9.10)

n.

comparison between (2.9.5)–(2.9.7) and (2.9.8)–(2.9.10) leads to expressions for α˙ t , y˙ t , z˙t in terms of the flow velocity (u, v, w) and of αy and αz ,  α˙ t = u − vαy − wαz − Un 1 + αy2 + αz2 , y˙ t = v + 

Un αy 1 + αy2 + αz2

z˙t = w + 

(2.9.11)

,

(2.9.12)

,

(2.9.13)

Un αz 1 + αy2 + αz2

where the flow velocity is taken at the front, u(α, y, z), v(α, y, z), w(α, y, z), ∂u = ux αy + uy , ∂y ∂u = ux αz + uz , ∂z

∂v = vx αy + vy , ∂y ∂v = vx αz + vz , ∂z

∂w = wx αy + wy , ∂y ∂w = wx αz + wz . ∂z

(2.9.14) (2.9.15)

Flame Stretch Using dδy/dt = δ˙yt , dδz/dt = δ˙zt , the evolution of a surface element δS = H(y, z, t)δyδz yields the rate of stretch in the form

A≡

δ˙yt δ˙zt + , δy δz

1 d δS = A + B + C, δS dt ∂H/∂y ∂H/∂z B ≡ y˙ t + z˙t , H H

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C≡

∂H/∂t , H

2.9 Appendix

159

with, according to (2.9.3), ∂ y˙ t ∂ z˙t + , ∂y ∂z  + α  α  ) + z˙ (α  α  + α  α  ) y˙ t (αy αyy t z zz z zy y zy , B=  1 + αy2 + αz2

A=

 + α α  αy α˙ yt z ˙ zt . C=  1 + αy2 + αz2

(2.9.16) (2.9.17)

(2.9.18)

The rate of stretch of a surface element is then obtained when the expressions (2.9.11)–  and α ˙ zt , computed from (2.9.14)–(2.9.15), are introduced into (2.9.13) and those for α˙ yt (2.9.16)–(2.9.18), ⎧  (1 + α 2 ) − 2α  α  α  ] U [α  (1 + αz2 ) + αzz ⎪ y y z yz ⎪ n yy ⎪ ⎪ 2 + α 2 )3/2 ⎪ (1 + α ⎪ y z ⎪ ⎪ ⎪ ⎪ ⎪ ⎨       A = + (vy + αy vx ) + (wz + αz wx ) ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ 1 ⎪  ∂Un  ∂Un ⎪ + α + α ,  ⎪ y z ⎪ ⎩ ∂y ∂z 1 + α 2 + α 2 y

B=

and

z

⎧  + α  α  ) + w(α  α  + α  α  ) v(αy αyy ⎪ z yz z zz y yz ⎪ ⎪ ⎪ 2 2 ⎪ (1 + αy + αz ) ⎪ ⎨ ⎪ ⎪  + α 2 α  + 2α  α  α  ] ⎪ Un [αy2 αyy ⎪ z zz y z yz ⎪ ⎪ , + ⎩ (1 + αy2 + αz2 )3/2

⎧ ⎪ ux (αy2 + αz2 ) + uy αy + uz αz − αy αz (wy + vz ) ⎪ ⎪ ⎪ ⎪ ⎪ (1 + αy2 + αz2 ) ⎪ ⎪ ⎪ ⎪ ⎪ (αy vx + αz wx )(αy2 + αz2 ) + (αy2 vy + αz2 wz ) ⎪ ⎪ ⎪ − ⎪ ⎪ ⎪ (1 + αy2 + αz2 ) ⎪ ⎪ ⎪ ⎪  + α  α  ) + w(α  α  + α  α  ) ⎨ v(αy αyy z yz z zz y yz − C= 2 + α 2 ) (1 + α ⎪ y z ⎪ ⎪ ⎪ ⎪ 2 α  + α 2 α  + 2α  α  α  ] ⎪ [α U n ⎪ y yy z zz y z yz ⎪ ⎪ − ⎪ 2 + α 2 )3/2 ⎪ (1 + α ⎪ y z ⎪ ⎪   ⎪ ⎪ 1 ∂U ∂U ⎪ n n ⎪ ⎪ + αz − αy .  ⎪ ⎪ ∂y ∂z ⎩ 1 + αy2 + αz2

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The first line in the expression of A is the mean curvature (2.9.2). The three last lines of C cancel with B and with the third line of A. Combination of the rest yields ⎧   1 1 ⎪ ⎪ Un + ⎪ ⎪ ⎪ R1 R2 ⎪ ⎪ ⎪ ⎪ ⎨ ux (αy2 + αz2 ) + uy αy + uz αz − αy αz (wy + vz ) (2.9.19) A+B+C = + (1 + αy2 + αz2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (vy + wz ) + (vy αz2 + wz αy2 ) + (vx αy + wz αz ) ⎪ ⎪ ⎪+ . ⎩ (1 + αy2 + αz2 ) Expression (2.3.9) is then obtained by comparison with ∇.u − nf .∇u.nf where the components of the tensor ∇u are ⎞ ⎛  ux uy uz ⎟ ⎜   ⎟ (2.9.20) ∇u = ⎜ ⎝ vx vy vz ⎠ .    wx wy wz

2.9.2 Mathieu’s Equation Mathieu’s equation can be written d2 Y + { + h cos(t)} Y = 0. dt2

(2.9.21)

√ When  > 0, it represents a harmonic oscillator whose natural frequency ωo =  is modulated by an oscillatory forcing term, h. When  < 0, the solutions to the equation for √ h = 0 are unstable and grow exponentially as exp(t −). The equation is even with respect to the reflexion t → −t. Thus, if Y(t) is a solu tion, Y(−t) is also. The solutions to (2.9.21) take the general form Y = A1 eσ t φ(t) +  A2 e−σ t φ(−t), where φ(t) is a 2π -periodic function[1,2] . Therefore the stable domain corresponds to Re(σ  ) = 0. The stability limits may be computed numerically (or analytically by a perturbation method[1,2] for small h; see Section 2.9.4) in the parameter space (, h). They are plotted in standard handbooks[2] and a sketch is shown in Fig. 2.57. For a harmonic oscillator,  > 0, the unstable domains meet the -axis at a discrete set of points,  = n2 /4, n = 1, 2, 3, . . . . The corresponding ratios of the frequency of the oscillator to that of the forcing term are ωo = n/2. The unstable domains exist for all values of the driving term h, no matter how small. This instability is called a parametric instability. The first tongue of instability, which touches the -axis at  = 1/4 (n = 1, ωo = 1/2), is the most relevant. This is because the solutions √are stabilised for small h as soon as nonzero damping is added to the basic oscillator for  = 1/2; see Section 2.9.4 ( = 2o in the notations of [1] [2]

Arnold V., 1973, Ordinary differential equations. MIT Editions. Bender M., Orszag S., 1984, Advanced mathematical methods for scientists and engineers. McGraw-Hill.

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161

II

I

Figure 2.57 Sketch of the stability limits of solutions to Mathieu’s equation (2.9.21). The solutions are unstable in the grey regions and stable in the white regions. Note the narrow tongues of stability in the region of negative .

II

I

Figure 2.58 Sketch of the stability limits of solutions to the Mathieu equation when a small damping term is added. The solutions are unstable in the grey regions and stable in the white regions. Note the narrow tongues of stability in the region of negative .

this appendix). The parametric instability appears above thresholds, h  hc (n), and hc (n) increases rapidly with n; see Fig. 2.58. Therefore the threshold for n = 1 is that usually observed in experiments. Another remarkable feature is that narrow tongues of stable solutions exist in the region of negative restoring force,  < 0. In other words, in the presence of forcing, h > 0, the parametric oscillator can be stable in the region where the solutions to the equation without forcing (h = 0) are unstable; see Figs. 2.57 and 2.58. This corresponds to the re-stabilisation of the Kapitza pendulum;[3] see Section 2.9.3. 2.9.3 Parametric Stabilisation The restabilisation of the DL hydrodynamic instability during the ‘vibrating flat flame’, described in Section 2.5.5 (the upper limit of unstable region I of Fig. 2.37), can be [3]

Kapitza P., 1951, Sov. Phys.–JETP, 21 (in Russian).

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analysed[1] using the method initiated by Kapitza (1951) for rapid oscillations of the restoring force of a pendulum.[2,3] Consider the case of a flame that is unstable in the absence √ of acoustic excitation, D > 0 in (2.5.21), κ− < κ < κ+ , N(κ± ) = 0. Assuming that D is of order unity for simplicity, the growth rate of the DL instability is of order of τh−1 = UL k; see (2.5.20). Consider an acoustic frequency, sufficiently high that the acoustic period is small compared with the growth time, τh , of the instability τa τh ,  ≡ ωτh  1. The evolution of the flame front then involves a slow and a fast time scale, τh and τa , respectively, with reduced times, τ  = t/τh and  τ  = t/τa . The method of resolution is that used for the motion of a particle in a rapidly oscillating field.[3] The flame front is decomposed into two parts, one evolving on the slow time scale, α˜ 0 (τ  ), plus a small perturbation, α˜  ( τ  ), evolving on the fast time scale, |α˜  | |α˜ 0 |, α˜ = α˜ 0 (τ  ) + α˜  ( τ  ).

(2.9.22)

When (2.9.22) is introduced into (2.5.21), two different types of terms appear: rapidly oscillating perturbation terms and dominant terms evolving on the slow time scale. They must cancel independently. Collecting the rapidly oscillating terms, and noting that 2B dα˜  /dτ  d2 α˜  /dτ 2 and also α˜   2 α˜ 0 , leads to d2 α˜  +  2 C cos( τ  )α˜ 0 = 0. (2.9.23) dτ 2 Neglecting the evolution of α˜ 0 during one period of fast oscillation, Equation (2.9.23) may be integrated, α˜  = C cos ( τ  )α˜ 0 (τ  ).

(2.9.24)

As a first step, α˜ 0 is considered to be constant. A more rigorous multiple-scale analysis can take into account the fact that α˜  varies also on the slow time scale, α˜  (τ  ,  τ  ). We will come back to this point later. Introducing (2.9.24) into (2.5.21) produces a term of the form  2 [cos ( τ  )]2 C2 α˜ 0 whose time average on the fast period is  2 C2 α˜ 0 /2. Taking the time average on the fast time scale of (2.5.21) then leads to an equation for α˜ 0 (τ  ),  d2 α˜ 0 dα˜ 0 (υb − 1)2 u2a , (2.9.25) + 2B + G α ˜ = 0, with G = −D + 0 dτ  dτ 2 (υb + 1)2 2 The disturbances are stable when G > 0, so the acoustic wave has a stabilising effect. For flames propagating downwards, or in zero gravity, the hydrodynamic instability may be completely suppressed (at all wavenumbers) by a sufficiently strong acoustic intensity, as shown by     k (υb − 1) ua 2 (υb − 1) |g| + − υb 1 − G= , (2.9.26) (υb + 1) UL2 k km (υb + 1) 2 [1] [2] [3]

Bychkov V., 1999, Phys. Fluids, 11(10), 3168–3173. Kapitza P., 1951, Sov. Phys.–JETP, 21 (in Russian). Landau L., Lifshitz E., 1976, Mechanics. Butterworth-Heinemann.

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163

obtained from (2.5.14) and (2.5.22). The critical acoustic intensity and wavenumber, u∗aI and kI∗ , are given by G = dG/dk = 0, u∗2 aI

  ULc (υb + 1) , 1− ≈ 2υb (υb − 1) UL

kI∗ 1 ULc ≈ , km 2 UL

(2.9.27)

where ULc is the critical flame velocity below which the planar flame propagating downwards is stable in the absence of acoustics; see (2.2.22). The critical wavenumber can be rewritten as kI∗ dL = 2(UL /ULc )Go . It is also clear from (2.9.26) that the hydrodynamic instability can never be completely suppressed for a flame propagating upwards |g| → −|g|. These analytical results are in satisfactory agreement with those of the numerical study presented in Fig. 2.37. A multiple-scale analysis overcomes the difficulty raised by Equation (2.9.24). It is convenient to introduce two reduced times, τ0 ≡ τ  and τ1 ≡  τ  , d/dτ  = d/dτ0 +  d/dτ1 . We will assume that the coefficient C in (2.5.21) is small, of order 1/ , C = C0 / , C0 = O(1), meaning that the acoustic displacement is small compared the wavelength of wrinkling. In the limit  → ∞, we look for solutions to (2.5.21) in the form α˜ = α˜ 0 (τ0 ) + +

1 α˜ 1 (τ0 ) cos(τ1 ) 

 1  α˜ 21 (τ0 ) sin(τ1 ) + α˜ 22 (τ0 ) cos(2τ1 ) + · · · . 2 

(2.9.28)

To leading order, O( ), Equation (2.5.21) yields (2.9.24) in the form − α˜ 1 + C0 α˜ 0 = 0.

(2.9.29)

When Equation (2.9.29) is introduced into (2.5.21) with the relation cos2 (τ1 ) = [cos(2τ1 )+ 1]/2, the terms of order unity, O(1), are classified into three categories: terms that are independent of τ1 , terms varying with τ1 as sin(τ1 ) and as cos(2τ1 ). They must vanish separately, leading to three equations for α˜ 0 , α˜ 21 and α˜ 22 : d2 α˜ 0 dα˜ 0 + 2B + [−D + C20 /2]α˜ 0 = 0, 2 dτ dτ0 0 dα˜ 0 − 2BC0 α˜ 0 = 0, −α˜ 21 − 2C0 dτ0 −4α˜ 22 + (C20 /2)α˜ 0 = 0. The first equation is equivalent to (2.9.25) and the two others give α˜ 21 and α˜ 22 in terms of α˜ 0 .

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2.9.4 Parametric Instability The threshold of the secondary instability in Section 2.5.5 may be analysed[1] by the perturbation method for studying the parametric resonance of an oscillator.[2] Consider situations similar to Fig. 2.37a. The dynamics of wrinkles with wavelength such that κ > κ+ is described by (2.5.21) with D < 0, which can be rewritten as   d2 α˜ 1 dα˜ +2 + o2 1 +  cos( τ  ) α˜ = 0, τd dτ  dτ 2  ≡ ( /o )2 C, 1/τd ≡ B, o2 ≡ −D.

(2.9.30)

The perturbation analysis works for a weakly damped oscillator whose natural frequency √ o ≡ −D is weakly modulated.[2] Since B is of order unity, the basic assumptions are  o ≡ −D  1,  ≡ ( /o )2 C 1. (2.9.31) In the perturbation analysis, √ the relative damping rate will be assumed to be of order , o τd = O(1/), that is, C −D = O(1). The most dangerous parametric resonance appears when the forcing frequency is twice the natural frequency of the oscillator  = 2o (see Figs. 2.57 and 2.58), so we introduce the small nondimensional quantity δw1  /o = 2 + δw1 .

(2.9.32)

Introducing the notation w ≡  /o , t ≡ o τ  and td ≡ o τd = O(1), Equation (2.9.30) can be rewritten in a more transparent form for perturbation analysis in the limit  → 0,  dα˜ d2 α˜ +2 + [1 +  cos(wt)] α˜ = 0, 2 td dt dt w ≡  /o = 2 + δw1 , td ≡ o τd = O(1), t ≡ o τ  .

(2.9.33)

Anticipating a solution in the form of an expansion involving slowly varying amplitudes, α˜ = a(t) cos(wt/2) + b(t) sin(wt/2) 1 + [a(t) cos(3wt/2) + b(t) sin(3wt/2)] + O( 2 ), 16 the last term,  α˜ cos(ωt), in (2.9.33) reads   a [cos(wt/2) + cos(3wt/2)] + b [− sin(wt/2) + sin(3wt/2)] + O( 2 ). 2 2

(2.9.34)

(2.9.35)

According to (2.9.32) and (2.9.34), Equation (2.9.33) is verified to leading order in the limit  → 0. At order , the terms involving sin(3wt/2) and cos(3wt) in (2.9.35) are balanced ˜ 2 in (2.9.33) and (2.9.34). This explains the numerical factor by those coming from d2 α/dt 1/16 in (2.9.34). The remaining terms of order  are proportional to either cos(wt/2) or

[1] [2]

Bychkov V., 1999, Phys. Fluids, 11(10), 3168–3173. Landau L., Lifshitz E., 1976, Mechanics. Butterworth-Heinemann.

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2.9 Appendix

165

sin(wt/2), leading to a system of two equations     db da a 1 1 b 1 1 δw1 + b = 0, δw1 − a = 0, + + + − dt td 2 2 dt td 2 2 



where t = t. Looking for a solution in the form a(t ) = Aest and b(t ) = Best yields 2(s + 1/td )A + (δw1 + 1/2)B = 0,

2(s + 1/td )B − (δw1 − 1/2)A = 0.

The nondimensional growth rate s is obtained by the compatibility condition, 4(s+1/td )2 = 

1/4 − δw21 , s = (1/2) 1/4 − δw21 − 1/td . The stability limits, Re(s) = 0, are then given by   4 4 1  − 2o 2 2 2 − δw1 = − 2 , that is, =  0. (2.9.36) 4 td o 4 (o τd )2

The threshold of the parametric instability of the oscillator (2.9.30), δw21 = 0, corresponds effectively to the forcing frequency  = 2o for a forcing amplitude  = 4/o τd . According to (2.9.30), these equations for the threshold can be written   C∗II = 2B∗II , (2.9.37)  = 2 −D∗II , where A∗II denotes the value of any quantity A at the threshold. When the simplified equation in (2.5.13) is used for the flame dynamics, the expressions of the coefficients B, C and D are given in (2.5.22). The frequency disappears from the second equation in (2.9.37) so that the acoustic threshold does not vary with the frequency in this model, u∗aII = 2υb /(υb − 1),

(2.9.38)

and the parametric instability develops for a reduced acoustic velocity above threshold ua  u∗aII . The first equation in (2.9.37) gives an equation for the corresponding nondimensional wavenumber κII∗ ,     κII∗2 υb + 1 1 ∗ ∗ (ωτL )2 . κII Go − κII + (2.9.39) = κm 4υb υb − 1 Only one of the roots is relevant. Notice that, for a large contrast of density, υb  1, and a reduced frequency of order unity, the critical wavenumber κII∗ is close to the roots of N = 0; see (2.5.23). In practical situations (see Fig. 2.37), only the largest root κ+ is of interest, and, typically, κII∗ is larger than κ+ ,   (ωτL )2 1 υb + 1 (ωτL )2 /(υb κ+ ) 1: (κII∗ − κ+ ) ≈ , (2.9.40) (υb κ+ ) 4 υb − 1 (κ+ /κc − 1) where, according to (2.2.21) and (2.2.22), κ+ > κc . In the vicinity of the threshold, the stability limit is given by (2.9.36) in the form  2 C2 − 4B2 = ( − 2o )2 ,

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(2.9.41)

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Laminar Premixed Flames

where, according to (2.5.22),  2 C2 = (υb − 1)2 u2a /(υb + 1)2 ,

B2 = υb2 /(υb + 1)2 ,

(2.9.42)

so that, according to (2.9.38), the left-hand side of (2.9.41) is proportional to ua − u∗a . In the right-hand side, the quantity√ ( − 2o ) is proportional to (κ − κII∗ ), as can be seen by using the expressions 0 (κ) = −D and  (κII∗ ) given by (2.9.39). Therefore the stability limit (2.9.41) has a parabolic shape in the plan ua − κ, as in Fig. 2.37. From a quantitative point of view, the results obtained in this section by using the simplified Equation (2.5.13) do not fit the experimental data well. For example, Equation (2.9.38) predicts a threshold typically 50% smaller than in experiments. However, the stability limits obtained from numerical resolution of the ‘detailed model’ presented in the next section are in reasonably good agreement with experiments.

2.9.5 The Detailed Linear Equation The equations used in this chapter to describe the linear dynamic of a wrinkled flame front are semi-phenomenological; see for example (2.2.18) for the effect of gravity and (2.5.21) or (2.9.30) for a flame under the influence of an oscillatory acceleration of an acoustic wave. They include a small diffusive term involving the flame thickness dL , which kills the DL instability at small wavelengths. This term is a small correction of order κ ≡ kdL 1, but which is essential to describe the phenomena qualitatively. The perturbation analyses presented in the second part of the book show that other correction terms appear in the coefficients of the equation. Although they do not change the qualitative behaviour, they are useful for a quantitative comparison with experiments. In this section we present the results obtained by a perturbation analysis for small κ, based on the one-step flame model (2.1.3) in the limit of a large activation energy (ZFK model β ≡ E(Tb − Tu )/kB Tb2 → ∞) leading to a single Markstein number M. This detailed analysis[1] includes different diffusive effects: preferential diffusion represented by the Lewis number Le ≡ DT /D or, more precisely, by the scalar of order unity l ≡ β(Le − 1); gas viscosity represented by the Prandtl number Pr. The analysis is developed in Section 10.3 for a constant heat conductivity. When the temperature variation of the diffusion coefficients represented by λ(T/Tu ), the ratio of the heat conductivity to its value in the fresh mixture, is taken into account,[1] the linear equation may be written in the same form as (2.5.21) with the notations (2.5.20),

 d2 α˜ dα˜ 2  + 2B + −D +  C cos( τ ) α˜ = 0, dτ  dτ 2

[1]

Clavin P., Garcia P., 1983, J. M´ec. Th´eor. Appl., 2(2), 245–263.

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(2.9.43)

2.9 Appendix

167

but with expressions for the coefficients B, C and D that include corrective terms of order κ, B=

1 + (M + lD/2)κ 1 + υb−1 (1 + lDκ/2)

,

M=

l D υb J+ , υb − 1 2 (υb − 1)

(2.9.44)

where M is the Markstein number and where the coefficients J > 0 and D > 0 are functions of the gas expansion ratio υb ≡ ρu /ρb > 1,  J = 0

1

(υb − 1)λ dθ , 1 + (υb − 1)θ



1

D=− 0

(υb − 1)λ ln θ dθ , 1 + (υb − 1)θ

(2.9.45)

where λ(θ ) is the reduced thermal conductivity and θ ≡ (T − Tu )/(Tb − Tu ). Although the second equation in (2.9.44) gives an analytical expression to calculate the Markstein number, the parameter l ≡ β(Le − 1) is not easily evaluated. In particular the diffusivity D of the species limiting the reaction rate in the ZFK model is not well defined when real multistep chemistry is considered. Moreover in this case there are two different Markstein numbers, as explained in Section 2.3.3. In practice we will use the ZFK model with a single Markstein number, M, considered to be an unknown parameter that must be measured experimentally. For a nondimensional acoustic velocity ua cos(ωt), reduced by the laminar flame speed UL , as in (2.5.20), the coefficients C and D are   (υb − 1) 1 − (υb − 1)−1 lDκ/2 ua , C = υb [1 + υb−1 (1 + lDκ/2)] ˜ N (υb − 1) , D= −1 [1 + υb (1 + lDκ/2)] κ ˜ ≡ −Go + [1 + (υb − 1)−1 lDGo /2]κ − Mκ ˜ 2 N

where

(2.9.46) (2.9.47)

(2.9.48)

and ˜ ≡1+ M



1 0

 (λ − λu )dθ + 2Pr

1

(λb − λ)dθ +

0

2υb 3υb − 1 M− J. υb − 1 υb − 1

(2.9.49)

For a flame propagating downwards, the parameter Go ≡ υb−1 |g|dL /UL2 should be small, of the same order as κ for the validity of the perturbation analysis and the κ expansion of the coefficients B, C and D should be limited to the first correction term of order κ. When expressed in terms of the Markstein number, M, and the expansion parameter, υb ,

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Laminar Premixed Flames

eliminating lD/2 with the help of (2.9.44), this gives . / . / 0 υb2 + 1 υb2 J υb B= 1+ M− κ , υb + 1 υb + 1 υb + 1 0 . / 

 υb − 1 2υb (υb − 1)M − υb J κ ua , C = 1− υb + 1 υb2 − 1   N υb − 1 υb , D= υb + 1 κ   υb J 2υb κ2 N ≡ −Go + κ + M− Go κ − , υb + 1 υb − 1 κm

(2.9.50) (2.9.51) (2.9.52) (2.9.53)

where the marginal wavenumber, κm , is given by 1 ≡1+ κm





1

1

(λ − λu )dθ + 2Pr (λb − λ)dθ 0 .0 / . / 4υb2 3υb + 1 + M− υb J . υb2 − 1 υb2 − 1

(2.9.54)

When the thermal variation of the diffusion coefficients is neglected, λ = λu = λb , the expression for the Markstein number in (2.9.45) reduces (10.3.36) and Equations (2.9.50)– (2.9.54) reduce to (10.3.71)–(10.3.74). The stability limits of a planar flame propagating downwards in the absence of acoustics is defined by N = 0 and dN/dκ = 0. The third term in the right-hand side of (2.9.53) is typically negligible so that the expressions (2.2.22) of the critical quantities Gc and kc in terms of km are still valid. Typical values of the flame parameters in Equations (2.9.44)–(2.9.54), evaluated for a lean propane flame with an equivalence ratio 0.58, are given in Table 2.1. These values were used to plot experimental results in Sections 2.5 and 2.8. The study of the parametric stabilisation follows the same way as in Section 2.9.3 but with a function G(κ) ≡ −D + ( C)2 /2 involving extra correction terms of order κ. The critical acoustic intensity u∗aI and the critical wavenumber κI∗ are still defined by G = 0 Table 2.1 Parameter values for a typical lean propane flame, equivalence ratio = 0.58, M = 4.5. UL Tu Pr λ+ J

= = = = =

0.13 m/s 293 K 0.691 3.176 3.332

υb Tb G 1 o λ(θ)dθ 0 κm

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= = = = =

5.689 1667 K 0.0164 2.215 0.0893

2.9 Appendix

169

and dG/dκ = 0. The resulting expressions for u∗aI and κI∗ are   /0 .   κI∗ υb J Go Go 2υb   M− κm = 1+ 1− ,  ≡ κm κm κm υb + 1 υb − 1   /0 .   υ + 1 G Go 3 b o u∗2 1 −  1 − . 1−2 aI = 2υb υb − 1 κm 2 κm

(2.9.55)

When the correction terms are neglected (  → 0) the above expressions reduce to those √ of the simplified model (2.9.27) for which, according to (2.2.22), 2 Go /κm = ULc /UL . Using the values given in Table 2.1 the correction is effectively small,   ≈ 0.07. The acoustic threshold u∗aII for parametric destabilisation, studied in Section 2.9.4 with the simplified equation, is still given by  C = 2B, [1 + (M + lD/2)κII∗ ] 2υb  , (υb − 1) 1 − (υb − 1)−1 lDκII∗ /2  0  υb2 2υb J κII∗ , = 1 + (υb + 1)M − (υb − 1) (υb − 1)

u∗aII =

(2.9.56) (2.9.57)

where κII∗ is the real solution of the cubic equation −4D =  2 − κN(κ) =

(ωτL )2 (υb + 1) . 4 υb (υb − 1)

(2.9.58)

Equation (2.9.56) yields a critical intensity higher than that predicted by the simplified Equation (2.9.38). The κ correction term in the numerator of (2.9.56) turns out to be of order unity because of the large value of the coefficient in front of κ. This may shed doubt on the relevance of the perturbation analysis leading to the detailed model. However, comparison of numerical resolution of (2.9.43) and experiments does not work so badly; see Fig. 2.38. The differences between the analytical results (2.9.56)–(2.9.58) and the numerical resolution of (2.9.43), also shown in Fig. 2.38, is not so surprising considering the fact that the initial assumptions in (2.9.31) are not satisfied with the values of the parameters for usual flames.

2.9.6 Pole Decomposition This appendix presents the details of the pole decomposition of the solution to (2.7.8). Hilbert Transform of the Derivative Consider the Fourier transform (κ and η real)  +∞  +∞ 1 ˜ τ )dk, ˜ τ) = eiκη φ(κ, e−iκη φ(η, τ )dη, φ(κ, φ(η, τ ) = 2π −∞ −∞ and the function pz (η) ≡ 1/(η − z) in the complex plane z = x + iy. Its Fourier transform +∞ p˜ z (κ) = (1/2π ) −∞ e−iκη pz (η)dη is computed by integration around a closed contour in

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Laminar Premixed Flames

the complex η-plane using the residue theorem. The contour is composed of a segment on the real axis and a semi-circle, both centred on the origin η = 0. The result is obtained in the limit of an infinitely large radius if the contour is closed either from above, Im(η) > 0, if κ < 0, or from below, Im(η) < 0 if κ > 0. The result thus depends on the signs of both y and κ, y > 0:

p˜ z (κ) = ie−iκz if κ < 0,

p˜ z (κ) = 0 if κ > 0,

y < 0:

p˜ z (κ) = 0 if κ < 0,

p˜ z (κ) = −ie−iκz if κ > 0,

so that the Fourier transform of 1/(η − z) (η real, z complex) is  0 1 =i eiκ(η−z) dκ, η−z −∞  ∞ 1 = −i y < 0: eiκ(η−z) dκ, η−z 0 y > 0:

   0 1 ∂ =− κeiκ(η−z) dκ, ∂η η − z −∞    ∞ 1 ∂ = κeiκ(η−z) dκ, ∂η η − z 0

as it can be checked directly. This can also be written using the expression of p˜ z (κ) computed just above    1 1 ∞ ∂ = |κ|eiκη p˜ z (κ)dκ, ∂η η − z i −∞    1 1 ∞ ∂ =− |κ|eiκη p˜ z (κ)dκ, ∂η η − z i −∞

y > 0: y < 0:

or, using the H operator defined in (2.7.2),  H (pz ) ≡

∞ −∞

|κ|eiκη p˜ z (κ)dκ = −sign(y)

i . (η − z)2

(2.9.59)

Proof of (2.7.11)–(2.7.12) Introducing the function v(η, τ ) ≡ ∂φ/∂η, Equation (2.7.8) reads ∂v/∂τ = H (v) + ∂ 2 v/∂η2 − v∂v/∂η,

(2.9.60)

which takes the form of Burgers’ equation excited by the hydrodynamical instability H (v). Looking for a solution in the form (2.7.11),  z˙α ∂v = −2 , ∂τ (η − zα )2 2n

H (v) = 2i

α=1

α=2n  α=1

sign(yα ) , (η − zα )2

(2.9.61)

where the last relation results from (2.9.59). Therefore the property of Equation (2.7.8) to possess a pole decomposition results from the same property of Burgers’ equation.

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2.9 Appendix

171

Separating the diagonal terms, ⎤ ⎡ 2n    ∂v 1 1 1 ⎦ v + = −4 ⎣ 2 ∂η (η − z ) (η − zα )3 (η − z ) α β α = β α=1    2   ∂v 1 ∂ v 1 1 1 v − − = −4 ∂η ∂η2 (zα − zβ ) (η − zα ) (η − zβ ) (η − zβ )2 α

(2.9.62)

= β

  =4 α = β

1 1 , (zα − zβ ) (η − zβ )2

where the first terms in the right-hand side disappear by antisymmetry. Putting together (2.9.60)–(2.9.62) yields (2.7.12). Proof of (2.7.17)–(2.7.18) The proof proceeds in the same way as before. Consider the Fourier series of a periodic function φ(η + km L) = φ(η), n=∞ 

φ(η) =

˜ einrη φ(n),

˜ φ(n) =

n=−∞

r 2π



2π/r

e−inrη φ(η)dη,

0

where r ≡ 2π/(km L). The Fourier coefficient p˜ z (n) of pz (η) ≡ −i

1 + e−ir(η−z) , 1 − e−ir(η−z)

z = x + iy,

(2.9.63)

may be written p˜ z (n) = e−inrz (I1 + I2 )/2π , where 

2π/r

I1 ≡ ir 0



I2 ≡ −ir 0

e−inr(η−z)   dη = − −ir(η−z) e −1

2π/r

5

e−inr(η−z)   dη = − eir(η−z) − 1

5

Z1n−1 dZ1 , Z1 − 1 1 Z2n+1 (Z2

− 1)

dZ2 ,

and where the contour is defined by the definitions of Z1 and Z2 , Z1 ≡ e−y e−ir(η−x) , Z2 ≡ ey eir(η−x) , namely the clockwise (anticlockwise) circle around the singularity, of radius ey (e−y ) for Z2 (Z1 ). The integrals may be computed by the residue theorem. Singularities may or may not appear at Z1,2 = 0 and Z1,2 = 1, depending on n and the sign of y. The final result is y < 0:

p˜ z (n) = 2ie−inrz if n > 0;

p˜ z (n) = 0 if n < 0,

y > 0:

p˜ z (n) = 0 if n > 0;

p˜ z (n) = −2ie−inrz if n < 0.

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Laminar Premixed Flames

so that y < 0:

n=∞  ∂pz (η) = −2r neinr(η−z) , ∂η

(2.9.64)

n=−1  ∂pz (η) neinr(η−z) , = 2r ∂η n=−∞

(2.9.65)

n=1

y > 0:

n=∞ 

H (pz ) ≡ r

|n|einrη p˜ z (n) = i sign(y)

n=−∞

∂pz . ∂η

(2.9.66)

Consider the pole decomposition (2.7.17) v=r

2n  α=1

 ∂pz ∂v α = −r , z˙α ∂τ ∂η 2n

pzα (η),

(2.9.67)

α=1

H (v) = i r sign(y)

2n  ∂pz

α

α=1

∂η

.

(2.9.68)

The property (2.7.18) then results directly from the pole dynamics of the Burgers’ equation, which is obtained by using the two relations ∂ 2 pzα ∂pzα = , ∂η ∂η2     eirzα + eirzβ ∂pz   ∂pzα α =i . pzβ irzβ − eirzα ∂η ∂η e α α β β pzα

=

(2.9.69) (2.9.70)

=

Denoting Zα,β ≡ e−ir(η−zα,β ) , Equation (2.9.70) takes the form   1 + Zβ   Zα + Zβ Zα Zα = , 2 1 − Z Z − Z (1 − Z ) (1 − Zα )2 β α β α α α = β

(2.9.71)

= β

which results from the following relations obtained by cancellation of the antisymmetric terms   1 + Zβ  1 Zα 1 − Z − Z 1 − Z 1 − Z (1 − Zα ) β α β α α = β    Zα 1 + Zβ 1 1 = − Z − Z 1 − Z 1 − Z (1 − Zα ) β α β α α = β    Zα Zβ 1 + Zβ 1 1+ = − Zβ − Zα 1 − Zβ 1 − Zα (1 − Zα ) α = β    Zα 1 + Zβ 1 1− . = Z − Z 1 − Z (1 − Zα ) β α α α = β

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2.9 Appendix

173

Proof of (2.7.19)–(2.7.20) For a pair of conjugated poles, z1 = x1 + iy1 and z∗1 = x1 − iy1 , Equation (2.7.17) yields  ir  ir ir ir ∗ ∗ e 2 (η−z1 ) + e− 2 (η−z1 ) e 2 (η−z1 ) + e− 2 (η−z1 ) ∂φ = −ir ir + ir , ir ir ∗ ∗ ∂η e 2 (η−z1 ) − e− 2 (η−z1 ) e 2 (η−z1 ) − e− 2 (η−z1 ) . ir / . ir / ir ir ∗ ∗ e 2 (η−z1 ) − e− 2 (η−z1 ) e 2 (η−z1 ) − e− 2 (η−z1 ) φ(η) = −2 log + f (τ ), 2i i k L k L where f (τ ) is obtained from 0 m φ(η)dη = −(1/2) 0 m (∂φ/∂η)2 dη resulting from (2.7.8). Equation (2.7.19) is obtained by noticing that the square bracket is equal to cosh(r y1 ) − cos[r(η − x1 )]. Equation (2.7.20) results from (2.7.18).

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3 Turbulent Premixed Flames

Nomenclature Dimensional Quantities a cp dL dtur Dtur DT I k km l lI L ˙ M r r S t T u u U UL Utur v v V

Description Sound speed Specific heat at constant pressure Scale of laminar flame thickness Scale of turbulent flame thickness Turbulent diffusivity Thermal diffusivity Intensity of sound Wavenumber Marginal wavenumber Spatial scale Integral length scale Length of tube, burner or space scale Mass flow rate Radius. Radial coordinate Vector of coordinates Surface area Time Temperature Mean velocity. Longitudinal velocity Velocity vector. Flow field Flame speed Laminar flame speed Turbulent flame speed r.m.s. velocity fluctuation (turbulence intensity) Turbulent velocity field (vector), Volume

174

17:02:00 .005

S.I. Units m s−1 J K−1 kg−1 m m m2 s−1 m2 s−1 J s−1 m−1 m−1 m m m kg s−1 m m m2 s K m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m3

Turbulent Premixed Flames

w x, y, z α  δ   μ ν ρ σ τ τI τL τrb φ ω

Transverse velocity Coordinates Position of front Circulation of a vortex A characteristic thickness Energy flux/unit mass in Kolmogorov cascade Wavelength Shear viscosity Viscous diffusivity μ/ρ Density Growth rate of perturbation A characteristic time Integral time scale Transit time through laminar flame Reaction time at burnt gas temperature Acoustic potential Angular frequency

175

m s−1 m m m2 s−1 m m2 s−3 m Pa s m2 s−1 kg m−3 s−1 s s s s m2 s−1 s−1

Non–dimensional Quantities and Abbreviations A Df n Ni,j O(.) Re t w θ ZFK

Dimensionless coefficient, see (10.1.32) Fractal dimension Local unit vector normal to front Number of boxes intersecting the flame front, see (3.4.10) Of the order of Reynolds number u l/ν Local unit vector tangential to front Reduced reaction rate see, (2.1.5) (1 − θ )e−β(1−θ) Reduced temperature (T − Tu )/(Tb − Tu ) Zeldovich and Frank-Kamenetskii Superscripts, Subscripts and Math Accents

a− a+ a˙ ay ab af aG ai aj

Value in fresh (upstream) flow Value in burnt (downstream) flow Derivative w.r.t. time Derivative w.r.t. y Burnt gas At flame front Gibson scale Scale i Scale j

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al aI aK an ar atur au

Turbulent Premixed Flames

Scale l Integral scale Kolmogorov scale Normal component Reaction rate Turbulent Unburnt gas

When an unstable flame propagates in a medium at rest, the cellular flame front can have a chaotic appearance, such as those shown in Fig. 2.19. However the term ‘turbulent flame’ is generally reserved to describe flames propagating in an initially turbulent flow. This is the case of almost all industrial applications, in which the flows are highly turbulent. Nonreactive turbulent flows is a vast field of investigation, still in progress, and the subject of several books.[1,2,3] The transition to turbulence in wall bounded flows, first studied by Reynolds in 1883, remains an open problem, especially the development of turbulent spots.[4] The coexistence of laminar and turbulent domains has been interpreted by Pomeau[5,6] as the result of a subcritical bifurcation in the flow regimes. There is still no theory based on the equations of fluid mechanics that describes such a bifurcation. Much work remains to be done in that direction. We will not discuss the difficult problem of turbulent bursting and we will limit our attention to fully developed turbulence, neglecting its intermittent character. We will need only the basic foundation of this theory, namely the 1941 Kolmogorov cascade, recalled in appendix; see Section 3.4.1. 3.1 Basic Considerations There is no satisfactory theory for the propagation of highly turbulent flames, even for the simplest models of turbulence and flame structure. The present-day theories incorporate closure hypotheses that are more or less well controlled and validated. Comparison with experiments is limited to a small number of flow configurations, and even then are unsatisfactory, except in limiting cases, such as the one presented in Section 3.2.2. Generally speaking, the experimental data by different authors are quite dispersed and the results seem to be particularly sensitive to the flow configuration. Turbulent combustion is a particularly complicated field and will not be treated in detail in this book. It has been the subject of many investigations and the state of the art has been discussed extensively in specialised books.[7,8] In this chapter we will limit our presentation either to simple flows or to [1] [2] [3] [4] [5] [6] [7] [8]

Monin A., Yaglom A., 1971, Statistical fluid mechanics, vols. 1 and 2. MIT Press. Frisch U., 1995, Turbulence. Cambridge University Press. Pope S., 2000, Turbulent flows. Cambridge University Press. Manneville P., 2014, J. Mech. B/Fluids, 49(SI), 345–362. Pomeau Y., 1986, Physica D, 23, 3–11. Pomeau Y., 2014, C. R. Acad. Sci. A, 343(3), 210–218. Peters N., 2000, Turbulent combustion. Benjamin Cummings. ´ Borghi R., Champion M., 2000, Mod´elisation et th´eorie des flammes. Edition Technip.

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estimations of orders of magnitude, based on dimensional arguments and/or scaling laws[9,10,11] that are useful in limiting cases. The symbol ≈ will be used to mean ‘nearly equal to’ or ‘of the order of’ without specifying the constant of proportionality, which should be of order unity.

3.1.1 Elementary Model of Turbulent Flames Profound insight into the theory of turbulent flames can be gained by working in the absence of flame instability and with a schematic model that illustrates some, but not all, of the basic difficulties. Unstable turbulent flames have been analysed only for weak turbulence and in the limit of small gas expansion. This work, presented in Section 2.7.3, is not considered below. Using the ZFK model for a unity Lewis number in the thermo-diffusive approximation in which the hydrodynamic effects of gas expansion are neglected, the turbulent flame model reduces to a reaction–diffusion wave propagating in a random velocity field v(r, t) of zero mean, v = 0, representative of the fluctuations of velocity in a turbulent flow. Introducing the divergence of the convective flux in a turbulent and incompressible flow, ∇.v = 0, ∇.(vθ ) = v(r, t).∇θ , the reduced temperature field, θ (r, t), is the solution to a stochastic equation that is an extension of (2.1.5), with the same boundary conditions as in (2.1.6), ∂θ/∂t + v(r, t).∇θ − DT θ = w (θ )/τrb .

(3.1.1)

In combustion, the nonlinear production term w (θ ) is a stiff function of the reduced temperature θ , concentrated near the maximum temperature θ = 1. No solution to such a stochastic equation is known, except for small turbulence-induced disturbances when the turbulent flame is close to the planar reaction–diffusion wave. In such a case the problem can be solved by a perturbation analysis. In all the other cases the problem is widely open even for the simplest stochastic flow fields. Key issues are the mean propagation velocity, Utur , called turbulent flame velocity, and the thickness of the flame brush, dtur . However, even the very existence of a steady state planar solution, propagating at a constant mean velocity with a constant mean thickness of the turbulent flame brush, is not yet proved from (3.1.1). Therefore the analyses of turbulent flames are far from being as rigorous as those of laminar flames.

3.1.2 Turbulent Diffusion The two first terms in (3.1.1) correspond to an equation for a scalar, θ (r(t), t), evolving by turbulent diffusion of fluid particles, dr/dt = v(r, t). [9] Peters N., 1986, Proc. Comb. Inst., 21, 1231–1250. [10] Clavin P., 1988, In E. Guyon, J. Nadal, Y. Pomeau, eds., NATO ASI Series E. Disorder and Mixing, vol. 152, 293–315, Kluwer Academic Publishers. [11] Clavin P., Siggia E., 1991, Combust. Sci. Technol., 78, 147–155.

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Taylor’s Diffusion Coefficient (1922). Richardson’s Law (1926) Turbulent diffusion is well documented in textbooks[1,2] and can be understoodusing the t Lagrangian picture, in one dimension for simplicity, dx/dt = v(t), x(t) = 0 v(t )dt . The analysis is based on an analogy between the stochastic motion of a fluid particle in turbulence and the Brownian motion of molecules (random walk of microscopic particles) described by Einstein (1905) and recalled in Section 13.3.3. A statistical treatment is introduced by averaging over many such particle paths. Assuming v = 0, the displacement is positive as often as it is negative, moment is the mean  t the lowest  order statistical  and   t square of the particle position x2 (t) = 0 dt 0 dt v(t )v(t ) , which can be written using t t  t  t the change of variable t = t −τ and symmetry arguments, 0 dt 0 dt = 2 0 dt 0 dτ , as  t  t     x2 (t) = 2 dt dτ v(t )v(t − τ ) . (3.1.2) 0

0

In the Lagrangian frame, assuming that the turbulence is homogeneous in time, the velocity autocorrelation function takes the form   v(t)v(t − τ ) = v2 g(τ ), where g(0) = 1, lim g = 0, (3.1.3) τ →∞

and where the correlation function is symmetric, g(τ ) = g(−τ ), dg/dτ |τ =0 = 0. The ∞ Lagrangian integral time scale τI is defined as τI ≡ 0 g(τ )dτ , τ  τI ⇒ g = 0. Integration by parts with respect to t in (3.1.2) yields     t x2 (t) = 2 v2 (t − τ )g(τ )dτ . (3.1.4) 0

Two limiting cases are of interest. When t is sufficiently small, smaller than the Taylor function can be approximated by unity, g(τ ) ≈ 1, scale, (d2 g/dτ 2 |τ =0 )−1/2  correlation    , the in (3.1.4). This yields x2 (t) ≈ v2 t2 , which is the ballistic approximation, valid at short t time. When t is large, t  τI , the term 0 τ g(τ )dτ is a bounded quantity of order τI2 in the limit t → ∞, negligible compared with τI t, so that Equation (3.1.4) leads to a result similar to classical random walk,   t  τI , 1-D: x2 (t) = 2Dtur t,

  3-D: x2 (t) = 6Dtur t,

  Dtur ≡ v2 τI .

(3.1.5)

Intermediate time scales have been considered by Richardson (1926).[2] On the basis of experimental results on the separation of two particles y(t) in the atmosphere, he obtained   2/3    the empiric laws d y2 (t) /dt ∝ y2 (t) , y2 (t) ∝ t3 . Obukhov derived this result later (1941) by dimensional analysis when y(t) is in the inertial range of the Kolmogorov cascade  2/3   (see Section 3.4.1), d y2 (t) /dt ≈  1/3 y2 (t) . This result illustrates the fact that the

[1] [2]

Hinze J., 1975, Turbulence. McGraw-Hill. McComb W., 1990, The physics of fluid turbulence. Clarendon Press–Oxford Science Publications.

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motions of the two particles are correlated when their distance is smaller than the integral length scale. By analogy with molecular transport processes, Equation (3.1.5) leads to a rough model for turbulent transport vθ  ≈ −Dtur ∇ θ  ,

∇.(vθ ) ≈ −Dtur  θ  ,

(3.1.6)

where Equation (3.1.5) has to be extended to the Lagrangian picture in three dimensions.   To do this it is necessary to introduce the two-point correlation function v(r, t)v(0, 0) = v.v g(r, t). It is not useful to go into these details here. It is sufficient to keep in mind that isotropic homogeneous turbulence is characterised by an integral length scale lI and an integral time scale τI and that the expression for the turbulent diffusion coefficient is similar to (3.1.5), Dtur ≈ lI2 /τI ≈ lI v ≈ v2 τI ,

where

v ≡ v.v1/2 .

(3.1.7)

Limitations In the same way as for the Fick and Fourier laws for molecular transport, which are based on the separation of microscopic and macroscopic scales of length and time, the validity of (3.1.6)–(3.1.7) is limited to random displacements that are statistically independent and small compared with the size of the mean profile (macroscopic length), in such a way that the stochastic process satisfies the central limit theorem. These conditions are not satisfied in many examples of scalar mixing by random stirring of sheets, blobs or strips of dyed fluid. In these cases, the picture is substantially altered by the competition between stretching and molecular diffusion.[3,4] Another counterexample is provided by the model of a flame considered as a fluctuating interface of zero thickness across which the reduced temperature θ and the flow velocity v are discontinuous. The mass-weighted average of the reduced temperature times the component of flow velocity normal to the front is positive and leads to the so-called countergradient diffusion,[5] in contradiction with (3.1.6)–(3.1.7).

3.1.3 Turbulent Flame Regimes Different regimes of turbulent combustion can be identified according to the ratio of the gas velocity fluctuation, v ≡ v.v1/2 , to the laminar flame velocity UL , and to the relative spatio-temporal scales of turbulence and laminar flames. Numerous time and length scales are involved in turbulent flows, whereas essentially two scales, the laminar flame thickness and that of the thinner reaction zone, characterise flames. Refinements of the classification of turbulent combustion regimes have been introduced, reflecting this multiplicity of scales.

[3] [4] [5]

Meunier P., Villermaux E., 2010, J. Fluid Mech., 662, 134–172. Villermaux E., 2012, C. R. M´ecanique, 340, 933–943. Libby P., Bray K., 1981, AIAA J., 19(205-213).

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We will not enter into these details that can be found in the literature,[1,2,3] plotted usually in different versions of the so-called Barr`ere–Borghi diagram.[4,5] Two extreme regimes are easily identified: • The well-stirred flame regime in which the length scales of turbulence are smaller than the thickness of the reaction zone of the laminar flame • The wrinkled flame regime in which the smallest length scales of turbulence are larger than the laminar flame thickness. In Section 3.2 attention is focused on the latter for a well-developed turbulence represented by the Kolmogorov cascade. Well-Stirred Flame Regime of Damk¨ohler (1940) In the extreme limit of turbulent flows having a sufficiently strong agitation at scales smaller than the thickness of the reaction zone, the notion of a laminar front loses its meaning. If, in addition, stirring to make both temperature fluctuations and molecular transport  sufficient   is  negligible, w (θ ) ≈ w (θ ), D Dtur , the mean of (3.1.1), using (3.1.6), takes the same form as the basic model equation for laminar flame, provided that D is replaced by Dtur . This regime is attributed to Damk¨ohler (1940).[2,3] The turbulent flame velocity, Utur , and the thickness of the turbulent flame, dtur , then take the same form as for laminar flames (2.1.9)–(2.1.11),  Dtur Dtur Utur ≈ , dtur ≈ , (3.1.8) lI dL : τb Utur     lI v dtur Utur Utur  1, ≈ ≈  1, (3.1.9) UL dL UL dL UL where (3.1.7) has been used. These results,which are obtained assuming lI dL and Dtur  D, lead to Utur  UL and are valid for a very high turbulence intensity v  UL at very small scales. This well-stirred flame regime is of little practical importance in the industrial situations in which localised laminar flamelets, flapping back and forth, are more or less well defined at any instant of time. Another extreme limit, referred to as a well-stirred reactor in chemistry, in which the time scale of agitation at large length scales (size of the vessel) is shorter than the reaction time, is not useful in combustion on earth. Wrinkled Flame Regimes When the smallest perturbations to the flame front that are induced by the turbulent flow are larger than the laminar flame thickness, dL , and when the characteristic time scales of [1] [2] [3] [4] [5]

Williams F., 1985, Combustion theory. Menlo Park, Calif.: Benjamin/Cummings, 2nd ed. Peters N., 2000, Turbulent combustion. Benjamin Cummings. ´ Borghi R., Champion M., 2000, Mod´elisation et th´eorie des flammes. Edition Technip. Borghi R., 1985, In C. Bruno, C. Casci, eds., Recent advances in aerospace sciences, 117–138, Plenum Pub. Corp. Borghi R., 1988, Prog. Energy Combust. Sci., 14(4), 245–292.

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turbulence are longer than the flame transit time τL , the flame front is wrinkled, but its internal structure remains almost identical to that of the planar flame. This is the case in the extreme regime when the smallest scale of turbulence, the Kolmogorov scale lK defined in appendix; see Section 3.4.1, is greater than the laminar flame thickness, lK  dL

⇒

τK  τL .

(3.1.10)

The large time ratio τK /τL  1 follows from the definition of the laminar flame scales, τL = dL /UL , UL ≈ DT /dL (see (2.1.10)–(2.1.11)) and, assuming a Kolmogorov cascade, that of the Kolmogorov scales τK = lK /vK , vK ≈ ν/lK (see (3.4.1)–(3.4.2)), by noticing that the thermal and viscous diffusion coefficients are of same order of magnitude in gases, ν ≈ DT , leading to τK /τL ≈ (lK /dL )2 . When the inequality (3.1.10) is verified, the normal velocity of the flame with respect to the unburnt gas, Un− , defined in (2.3.1), is everywhere close to the laminar velocity UL . The difference (Un− − UL )/UL can be neglected, except at certain particular locations such as cusps, where, as for the tip of a Bunsen flame in Fig. 1.2, a change in flame velocity is necessary to avoid the presence of a singularity in the curvature. The flame can be treated as a surface of discontinuity propagating with a constant normal velocity UL = 0 between two fluids of different temperature and density. This specificity, compared with the turbulent mixing of passive scalars, introduces two new problems of a different nature: • The propagation with a nonzero normal velocity, UL = 0, has a tendency to decrease the surface area (see the Huygens’ construction in Fig. 2.10), and can produce a nonlinear saturation in the turbulent growth of surface area. • The change in density induces a retroaction of the wrinkled front on the turbulent flow field, illustrated in Fig. 2.7. Both problems are still open and physical insights can be obtained only for limiting cases discussed in Section 3.2. Attention is limited here to basic considerations. The only obvious property is that turbulent flames propagate faster than laminar flames since the velocity ratio is given by the ratio of the total flame area to the normal cross section, as explained now. Consider a turbulent flame which on average is one-dimensional. If S(t) is the total flame area at an instant t, the mass of unburnt gas transformed into combustion products per unit time is ρu UL S(t). If we suppose that an equilibrium can be reached between the increase in flame surface area by turbulence and the consumption of surface by the normal flame velocity, the time-averaged flame area S is a well-defined constant quantity (a reasonable but not yet proved assumption). Mass conservation then gives the following law for the turbulent flame speed, Utur , Utur So = UL S ,

(3.1.11)

where So is the planar cross section of the flame and Utur is defined as the mean flow velocity necessary to main the average flame position stationary in the reference frame of the laboratory; see Fig. 3.1. For example, the increase of flame speed is essential in internal combustion engines. In order for a flame to propagate several centimetres in a few

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Turbulent Premixed Flames y x z

Figure 3.1 Sketch of a flame stabilised in a turbulent flow.

milliseconds, as in a piston engine running at 6000 rev/min, the flame speed has to be increased by more than an order of magnitude. The remaining difficulty is to determine the increase of flame surface S /So − 1 in terms of the characteristics of the turbulent flow. The kinematic equation of a front, propagating in a turbulent flow v(r, t) with a normal velocity relative to the flow, Un , leads to a stochastic equation which looks somehow simpler than (3.1.1). Introducing the equation of flame surface G(r, t) = G0 , ∂G/∂t + (dr/dt).∇G = 0, with dr/dt = v(r, t) − Un n, where n = ∇G/|∇G| is the normal to the front oriented towards the burnt gas, all these quantities being defined at the flame front, G(r, t) = G0 , the geometrical equation of the flame front takes the form, called G equation in combustion literature,[1] ∂G/∂t + v(r, t).∇G = Un |∇G|,

(3.1.12)

where, in the wrinkled flame regime (3.1.10), the normal burning velocity Un is close to the constant laminar flame velocity UL . The small difference in (2.3.10) is useful to prevent the formation of cusps (discontinuities of tg .∇G|G=G0 , where tg is a tangent vector) by Huygens’ construction. For this purpose it is sufficient to retain only the curvature term, with a positive Markstein number. When, in the same geometrical configurations as in Fig. 3.1, the flame surface is monovalued, x = α(y, z, t), G − G0 = x − α(y, z, t), Equation (3.1.12), written in the laboratory frame where the mean position of the front is planar and steady, ∂α/∂t = 0, takes the form  (3.1.13) ∂α/∂t − u(rf , t) + w(rf , t).∇ yz α = Utur − Un 1 + |∇ yz α|2 , where ∇ yz = (∂/∂y, ∂/∂z), u is the longitudinal fluctuating velocity, w = (wy , wz ) stands for the transverse components, and rf (y, z, t) ≡ (α(y, z, t), y, z) is a point on the flame surface. Equation (3.1.11) is obtained from (3.1.13) by assuming that there is no thickening of the flame brush by turbulent diffusion, or more precisely that the Lagrangian derivative of α(y, z, t) is zero, dα/dt = 0. In this case the mean of the left-hand side of (3.1.13) is zero. Neglecting stretch effects, (Un − UL )/UL 1, averaging (3.1.13) yields    S = Utur /UL = (3.1.14) dxdy 1 + |∇ yz α|2 , 1 + |∇ yz α|2 ,

[1]

Peters N., 2000, Turbulent combustion. Benjamin Cummings.

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 where 1 + |∇ yz α|2 ≡ 1 + |αy |2 + |αz |2 is a fluctuating function of y and z, statistically homogeneous and isotropic. The difference between Equation (3.1.12) and the turbulent mixing of a passive scalar comes from the nonlinear term in right-hand side describing the consumption of flame surface area by the Huygens’ mechanism. For a given stochastic field, v(r, t), and a given normal flame velocity, Un , no solution to such stochastic equations is known, except for small turbulence intensity. The latter case is called the weakly wrinkled flame regime, analysed in Section 3.2.1. Corrugated flamelets and Intermediate Regimes In the wrinkled flame regime (3.1.10), the topology of the flame front becomes increasingly tortuous as the intensity of turbulence becomes large compared with the laminar flame speed, v/UL > 1. This is called the corrugated flamelet regime, analysed in Sections 3.2.2 and 3.2.3. Intermediate regimes, between the well-stirred flame regime of Damk¨ohler (1940) and the wrinkled flame regime, have also been identified. These regimes, called thickened flame regimes, correspond to turbulence length scales that are smaller than the thickness of the premixed zone but larger than the thin reaction zone. In such a case the inner structure of the reaction sheet is not modified and Equation (3.1.11) is still valid if S(t) denotes the surface area of the reaction sheet. However, the overall thickness of the turbulent flame dtur should involve the turbulent diffusion coefficient (3.1.5) if Dtur > D. The existence of such regimes in practical situations remains controversial. The best that has been done in the study of highly turbulent flames is either derivation of scaling laws based on dimensional analyses for the strongly corrugated regime, v/UL  1, presented in Section 3.2.2, or phenomenological modelling using closure assumptions that have to be validated by comparison between numerical computations and experimental data. Much effort has been made in this direction over the last 60 years and will continue. It involves various statistical approaches: distribution functions of the progress variable,[2] as θ in (3.1.1), flame surface density models[3] or the so-called level set method[1] based on (3.1.12). This work has been extensively reviewed[1,4,5] and will not be discussed here. Thanks to it, significant progress has been made in modelling turbulent combustion, but the physical understanding of strongly turbulent flames remains poor.

3.1.4 Propagation in a Periodic Parallel Shear Flow Exact solutions can be obtained only in a few examples of particular flows. The following one is worth mentioning since it illustrates the nonunicity of fluctuating solutions. [2] [3] [4] [5]

Bray K., Moss J., 1977, Acta Astronaut., 4(3-4), 291–319. Poinsot T., et al., 1996, Prog. Energy Combust. Sci., 21, 531–576. ´ Borghi R., Champion M., 2000, Mod´elisation et th´eorie des flammes. Edition Technip. Bilger R., et al., 2005, Proc. Comb. Inst., 30(21-41).

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Bunsen tip

Flame front

Shear flow Unburnt gas

Figure 3.2 Flame front in a periodic shear flow

Flame Front in a Steady Parallel Flow Consider a large-scale permanent periodic shear flow having a longitudinal velocity u = v sin(2π y/l) in the laboratory frame, l  dL . A solution corresponding to a steady form of the wrinkled flame front propagating at constant velocity exists and is shown in Fig. 3.2. The increase in surface area and propagation velocity may be computed for this particular solution. The radius of curvature of the front at the point A is large, so the change in local velocity of the front can be neglected, Un ≈ UL , as can also the flame-induced modification to the upstream flow. In the laboratory frame, the velocity of this point is directed along the negative x-axis and has the value UL +v. This velocity is also the propagation velocity of the complete periodic structure of the front. Away from the point A, the flame front becomes inclined, similar to the case of the Bunsen flame (see Figs. 1.2 and 2.6), so that the normal component of the flow relative to the flame front remains equal to the flame velocity UL . At the points B, the situation is the same as that of the tip of a Bunsen flame: the curvature of the font at B increases (smaller radius of curvature) until the form of the front becomes stationary, as explained in Section 2.3.4 for the tip of the Bunsen flame. For this to happen, the local flame speed at the tip must be equal to the local flow velocity seen by the flame Un− = 2v + UL . This is possible for a positive Markstein number, Mc (see (2.3.10)), when v is not too large, typically not larger than three to five times the laminar flame velocity UL ; above this, a steady nonlinear curvature effect cannot compensate for the flow velocity. Fluctuating Front in an Unsteady Shear Flow Assuming a quasi-steady state approximation, the analysis can be easily extended to a pulsed shear flow u = v cos ωt sin(2π y/l), ω < vl. After half a period, the initial geometry of the flame front is recovered, translated in the transverse direction by half a wavelength. A time integration during half a period then gives the mean propagation velocity Utur = 2v/π + UL . A similar solution is also obtained if the velocity fluctuation is a random function of time, u = v(t) sin(2π y/l), v = 0. Even additional small fluctuations of the

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wavelength 2π/l(t) will not destroy this type of solution. For small |v|, |v|/UL 1, this leads to (Utur /UL − 1) ∝ |v|/UL , in contrast to the result obtained below for a turbulent flame in the weakly wrinkled regime; see (3.2.3). The discrepancy comes from the fact that the assumption in (3.2.1) is not verified here, ∂α/∂t = u, since the flame displacement α (and not the derivative with respect to time ∂α/∂t) is linearly related to u through front tilting. The flame is anchored at the unsteady leading edges, like the point A in Fig. 3.2, which propagate at a mean velocity exceeding the laminar flame velocity by a quantity proportional to |v|. Notice that, for fluctuating flows (random or not), another solution also exists, planar in the mean, satisfying ∂α/∂t = u; see (3.2.1)–(3.2.3). Which solution is selected, by which mechanisms, and in which conditions, are open questions.

3.1.5 Flame-Vortex Interaction Turbulence may be viewed as a collection of eddies interacting with each other. The interaction between a propagating front (UL = 0) and a single permanent vortex is analysed below in the spirit of the study of Marble[1] for diffusion flames. Combustion of a Single Laminar Vortex Tube Consider a tubular cylindrical vortex, solution to the Navier–Stokes equations with an initial condition corresponding to the inviscid solution   1 − exp(−r2 /4νt) , (3.1.15) vθ = 2π r where r is the radius, ν is the viscosity, and  is the circulation of the vortex. This vortex √ √ is characterised by a quasi-solid core of radius l, increasing with time as νt, l ≈ νt, √ having a maximum rotational velocity vm , which decreases in time as / νt, followed by a gradual decrease to zero. At large radii, viscous dissipation becomes negligible, the flow is potential, and the tangential velocity decreases as 1/r, vθ = /2π r. The turnover time of the solid core is τm ≡ l/vm , τm ≈ νt/ . Limiting ourselves to the geometrical case of a wrinkled flame, τL τm , dL l, consider the initial condition of a planar flame front in which the axis of the vortex is inserted. This two-dimensional problem can be characterised by a cut perpendicular to the axis of the vortex, shown in Fig. 3.3. If UL  vm the core is burnt out after a time l/UL shorter than the turnover time τm and the flame front is weakly wrinkled after the passage through the vortex core. Consider now the opposite limit UL vm . The roll-up in the core is shown schematically in Fig. 3.3. The combustion time, tcomb , of the solid core can be estimated as follows. After sufficient time, the number of revolutions is of order vm t/l  1, and the length of the rolled-up interface is L = vm t  l. Neglecting the growth of the size of the vortex core, νt/l2 1, the average thickness of the engulfed tongues of fresh mixture, δ, can be evaluated from mass conservation, Lδ ≈ l2 , giving δ ≈ l2 /vm t l. This estimation is valid during the time that [1]

Marble F., 1985, Recent advances in the aerospace sciences, chap. Growth of a diffusion flame in the field of a vortex, 395–413. New York: Plenum Press.

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Figure 3.3 Sketch of a passive surface rolled up by a vortex.

the flame speed remains negligible compared with the speed at which the thickness of the tongues decreases by roll-up, UL l2 /vm t2 , leading to a transition time t∗2 ≈ l2 /(vm UL ),

t∗ ≈ δ ∗ /UL ,

(3.1.16)

where δ ∗ is the characteristic thickness of the tongues at time t∗ , δ ∗ = l2 /vm t∗ . According to the second equation in (3.1.16), t∗ is also the characteristic time to burn these tongues and thus the vortex core, tb ≈ t∗ ,   tb ≈ l/ vm UL , tb /τm ≈ vm /UL  1, (3.1.17) UL vm : so that the assumption νtb /l2 1 is verified. The combustion propagates in the outer potential region of the vortex at a lower rate, as shown by the self-similar solution obtained by Peters and Williams[1] for the propagation of the flame in this region (where v ∝ 1/r). Ignoring the viscous core, the roll-up speed diverges on the axis; the central region burns the fastest; and the radius of the burnt region grows as t2/3 : r(t) = π −2/3  1/3 UL t2/3 . 1/3

(3.1.18)

Putting r = l and  ≈ vm l into (3.1.18) gives t ≈ tcomb defined by (3.1.17), showing that the self-similar solution (3.1.18) is valid for a long time, t > tb . Propagation Through a Stationary Array of Cylindrical Vortices The propagation of a reaction–diffusion wave (no gas expansion) through a regular and stationary array of identical cylindrical vortices, of size l and rotational velocity v, has been the subject of analytic, numerical and experimental studies.[2,3,4,5] Several different regimes have been studied, characterised by the relative orders of magnitude of the three time scales of the problem:[4] [1] [2] [3] [4] [5]

Peters N., Williams F., 1988, Proc. Comb. Inst., 22, 495–503. Audoly B., et al., 2000, C. R. Acad. Sci. Paris, 328(3), 255–262. Kagan L., Sivashinsky G., 2000, Combust. Flame, 120(1-2), 222–232. Vladimirova N., et al., 2003, Combust. Theor. Model., 7(3), 487–508. Pocheau A., Harambat F., 2008, Phys. Rev. E, 77(3), 036304.

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• The transit time through the laminar flame, τL = dL /UL • The turnover time of the vortices, τl = l/v 2 • The diffusion time on the scale of the vortex, τD = l /D. Here we will simply recall the results for the case where v  UL with vortices sufficiently large so that the diffusion time, τD , is much longer than τL and τl . In the geometrical limit of a wrinkled flame, τL τl , the numerical results[4] yield τl  τL , v  UL :

Utur (v/UL ) , ≈ UL ln(v/UL )

in agreement with a previous study.[6] In this regime, the turbulent flame speed is observed to be independent of the size of the vortices. In the opposite limit, τl τL , where the wrinkled flame approximation is not applicable, a numerical study[4] yields  1/4  3/4 l Utur v τL  τl , v  UL : ≈ . UL UL dL This result is in agreement with the predictions of an analytical study,[2] in which it was found that Utur /UL ≈ (v/UL )1/4 when the propagation of the front is controlled by molecular diffusion through the stagnation separators between the vortex cells (the longest time in the propagation sequence). This result is obtained essentially for a mild reaction rate, of the quadratic type discussed in Section 8.3.2. A different result is obtained for a more realistic reaction rate for combustion, shown in (2.1.5). There is a threshold level v = vcr at which the propagation speed reaches a maximum.[3] Propagation in a Chaotic Flow The chaotic nature of the flow, in both space and time, plays an essential role in turbulent flames, leading to results that are different from those for a stationary array of vortices. In the same spirit as for the Kolmogorov scaling laws of turbulence in the inertial range, where molecular processes can be ignored, it has been conjectured for a long time that the overall combustion rate in a strongly turbulent flame is mainly controlled by the turbulent flow and not by the laminar burning. At first sight, this seems to be in contradiction with (3.1.11) for the turbulent wrinkled flame regime. We will see in Sections 3.2.2 and 3.2.3 how this is possible. In the strongly corrugated flamelet regime, the fastest propagation velocity of a turbulent flame, planar in the mean, across a fully developed turbulence is the velocity at the integral scale vI . This corresponds to the highest amplitude of velocity fluctuations in the power spectrum, and can be seen as the turnover velocity of the largest vortex blobs (of size the integral scale lI ). In other words the fastest propagation is by convective contamination, as

[6]

Shy S., et al., 1992, Proc. Comb. Inst., 24, 543–551.

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Figure 3.4 Schematic illustration of convective contamination.

illustrated schematically in Fig. 3.4, where the flame front is transported over the scale lI at the speed vI in a time τI ≡ lI /vI . For this to be true, the combustion time, tb , of a vortex blob of size lI cannot be much longer than the turnover time, τI ≡ lI /vI , otherwise the amount of burnt gas transported by each vortex blob would decrease at each step (as for a passive contaminant) until ignition fails. The combustion time (3.1.17) of a single isolated vortex is too long to explain a fast propagation. Two other mechanisms that characterise the mixing process in a fully developed turbulence must be invoked: the interaction between vortices leading to the well-known vortex stretching, and the multiple scale character of the flow. Concerning the stretching of passive interface, it is generally accepted that the surface area increases exponentially in time (a positive Lyapunov exponent) and not linearly as in an isolated tubular vortex. Therefore, using arguments similar to those leading to (3.1.16), the decay of thickness of the enrolled tongues in a vortex of size l is δ/l ≈ e−t/τl . For UL = 0, the thickness δ(t) of the tongues of unburnt gas enrolled into the vortex should then follow a law of the type v  UL : dδ/dt = −(δ/τl + 2UL ), where the second term on the right-hand side is a small correction describing the effect of laminar combustion of the unburnt gas. If the initial condition is taken as δ(t) = l for t = 0, the solution is δ/l + 2UL /v = [1 + 2UL /v] e−t/τl . If we now define the combustion time tb as the time at which δ(t) = 0, v  UL :

tb /τl ≈ ln (v/2UL ) ,

(3.1.19)

it is then seen that the combustion time is still too large compared with the turnover time, tb  τl , to explain a propagation velocity controlled by contamination Utur ≈ v. As we shall see in Section 3.2.2, the combustion time of the large vortices in a fully developed turbulence is shortened by the multiple-scale nature of turbulence, and becomes of the order of the turnover time at the integral scale, tb ≈ τI .

3.2 Turbulent Wrinkled Flames In this section the weakly wrinkled flame regime is considered first. In the second part, it will be argued that the multiscale character of fully developed turbulence leads to simple universal laws for the structure and the propagation of turbulent wrinkled flames.

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3.2.1 Weakly Wrinkled Flamelet Regimes In the limit v UL of wrinkled flames (3.1.10), called the weakly wrinkled regime, the amplitude of wrinkling remains small. For a stable flame front (propagating downwards), one-dimensional in the mean, this regime is easily solved even when the gas expansion is included.[1,2] The incident turbulence acts only as a small perturbation to the solution for a planar flame front. The complete solution to this academic problem illustrates one of the difficulties already mentioned: not only is the turbulent flow modified by gas expansion as it traverses the flame front, but the incident turbulence is also modified by nonlocal hydrodynamic effects, illustrated in Fig. 2.7. It then follows that the turbulence seen by the flame is different from the turbulence that would have been present in the absence of the flame. This effect has been observed experimentally.[1] Assuming that the equation of the flame front is x = α(y, z, t), the increase in flame surface area can be obtained using scaling laws for turbulence. For a weakly wrinkled flame Equation (3.1.14) yields

   S ≈ S0 1 + αy2 + αz2 /2 . (3.2.1) The turbulent flame velocity is obtained by evaluating the longitudinal fluctuating displacement of the flame front, using the linear approximation of (3.1.13):  t ∂α(y, z, t) ≈ u(α, y, z, t), ∇ yz α ≈ ∇ yz u dt. (3.2.2) ∂t If an extended Taylor hypothesis is introduced, linking time and length scales, Equa   2  2 tion (3.2.2) then leads to αy = αz ≈ v2 /UL2 , where an isotropic assumption has also been used.[3] Expression (3.2.1) then gives a quadratic correction to the flame speed, as suggested by Shchelkin (1943) but with a different coefficient: lK  dL ,

v/UL 1:

Utur /UL = 1 + (v/UL )2 + · · · .

(3.2.3)

This result was originally derived from the equations for temperature and mass fraction of the limiting species in the framework of the ZFK model.[3] It has also been confirmed by numerical simulations.[4] In addition to the separation of lengths, lK  dL , and velocities, v/UL 1, the validity of (3.2.3) is limited by the following assumptions. • Equation (3.2.3) rests on (3.2.2); a simple counterexample has been given in Section 3.1.4 for parallel shear flows. • A constant turbulent flame velocity and a stationary state for the mean flame brush is questionable for a large size in the transverse directions. The one-dimensional mechanism of turbulent diffusion in the longitudinal direction, associated with (3.2.2) and discussed [1] [2] [3] [4]

Searby G., et al., 1983, Phys. Rev. Lett., 51(16), 1450–1453. Searby G., Clavin P., 1986, Combust. Sci. Technol., 46, 167–193. Clavin P., Williams F., 1979, J. Fluid Mech., 90 part 3, 589–604. Matalon M., Creta F., 2012, C. R. M´ecanique, 340(845-858).

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in Section 3.1.2, predicts that the distance between two points on the flame surface increases in time if they are separated in the transverse direction by a distance larger than the integral scale (correlation length of the fluctuating velocities). Equation (3.2.2) thus describes a nonstationary stochastic process when the transverse direction is sufficiently large. The question remains open for the nonlinear stochastic equation in (3.1.12). In any case, the problem does not concern laboratory experiments or practical applications since the size of the vessel (e.g. diameter of the tube) is limited. Combustion of large unconfined clouds or in astrophysics deserves further study. • The planar flame was assumed to be stable. The dynamics and geometry of an unstable flame front in a weakly turbulent flow is different. Cells separated by sharp cusps appear on the flame front. The flame surface area is related to the cell size, which is controlled by the weak turbulence, as described roughly in Section 2.6.4. The problem has been treated more rigorously in the limit of small gas expansion by considering the turbulence as an external noise, leading to the cellular structure of expanding flames;[1,2,3] see Section 2.7.3. Self-similar acceleration has also been reported in experiments.[4] Statistical properties of the flame geometry (one-dimensional on the average) have been obtained in the numerical simulations[5,6] using a hybrid Navier–Stokes/front-capturing methodology. 3.2.2 Strongly Corrugated Flamelet regimes In a real turbulent flow there is a large range of time and space scales. In the wrinkled flame regime, none of these scales is small enough to penetrate into the front, the internal structure of the flame is unchanged and (3.1.11) still applies. The problem is to calculate the average total surface area, S, and the geometrical properties of the flame front as a function of the characteristics of the turbulence. As said before, this problem has no known mathematical solution, even when the retroaction of the flame front on the turbulent flow is neglected. It is nevertheless possible to evaluate orders of magnitude in limiting cases. Let us consider a fully developed turbulence characterised by a Kolmogorov cascade recalled in Section 3.4.1. We limit our attention to intermediate regimes in which the Kolmogorov and integral scales are well separated with turnover velocities respectively much smaller and much larger than the laminar flame velocity: vK UL vI .

(3.2.4)

The first inequality, vK UL ensures that (3.1.10) is verified so that the internal structure of the flame is not modified by the turbulence. This can be seen by remembering that vK lK ≈ ν, from the definition of the viscous scale lK , and according to (2.1.9)–(2.1.11), UL dL ≈ DT . Since all diffusion coefficients have similar values in the gas phase, DT ≈ ν, [1] [2] [3] [4] [5] [6]

Joulin G., 1989, J. Phys-Paris., 50, 1069–1082. D’Angelo Y., et al., 2000, Combust. Theor. Model., 4, 317–338. Karlin V., Sivashinsky G., 2006, Combust. Theor. Model., 10(4), 625–637. Gostintsev Y., et al., 1988, Combust. Expl. Shock Waves, 24(5), 563–569. Matalon M., Creta F., 2012, C. R. M´ecanique, 340(845-858). Creta F., et al., 2011, Combust. Theor. Model., 15(2), 267–298.

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Figure 3.5 Tomographic cut through a turbulent premixed flame. Reproduced with permission from Pocheau A., Queiros-Cond´e D., Physical Review Letters, 76(18), 3352–3355. Copyright 1996 by the American Physical Society.

the following relations can be immediately deduced: vK /UL ≈ dL /lK , τK /τL ≈ (lK /dL )2 , as already mentioned below (3.1.10). The combustion regime (3.2.4) thus belongs to the geometrical regime of wrinkled flames defined in (3.1.10). In the absence of flame propagation, UL = 0, each scale of turbulence would produce smaller and smaller scales of wrinkling. A nonzero propagation velocity, UL = 0, modifies the enfolding scenario. The findings presented below may be summarised as follows. By a mechanism of renormalisation, the interaction time with each vortex structure is limited to a time of the order of the local turnover time, and the interaction between the turbulent flow and the flame front becomes local in the space of scales of turbulence. To be more precise, a fluctuation of velocity on a scale l creates only wrinkling on the same scale l. As a consequence, the wrinkling of the flame front reflects the properties of the scales of turbulence and the front acquires a fractal geometry whose rudiments are recalled in appendix; see Section 3.4.2. An example of such a turbulent flame front is shown in Fig. 3.5. The Gibson Scale The smallest scale of wrinkling of the flame, called the Gibson scale,[7,8] lG , is the scale at which the turnover velocity is equal to the laminar flame velocity, and can be estimated using Kolmogorov’s scaling laws (3.4.4) UG ≈ UL

⇒

lG ≈ UL3 /.

(3.2.5)

These relations define an element of fluid of size lG , which is traversed by a laminar flame at speed UL in a time equal to the characteristic time of turbulence at this scale. Fluid elements much smaller than this scale will be completely burnt before the turbulence has the time to deform the front. Thus the flame front is significantly wrinkled only on scales [7] [8]

Peters N., 1986, Proc. Comb. Inst., 21, 1231–1250. Peters N., 2000, Turbulent combustion. Benjamin Cummings.

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equal to or greater than lG . The amplitudes of wrinkling on scales smaller than the Gibson scale are small and their contribution to the increase in flame surface area is negligible. The inequality UL vI implies that there is a large number of scales involved in the wrinkling of the flame, whereas the inequality vK UL implies that the Gibson scale is large compared with the Kolmogorov (dissipation) scale. Thus the relations (3.2.4) lead to lK lG lI ,

(3.2.6)

and the flame surface is well defined at all scales of wrinkling, including the smallest ones, li  lG , since dL lK . Moreover, there is a large band of scales in the wrinkled flame front since lG lI , and a fractal property of the front geometry can be meaningful in the range lG < li < lI ; see Section 3.4.2. Self-Similarity and Turbulent Flame Velocity Experimental resolution is often not sufficient (or fast enough) to resolve the details of the smallest structures. When the geometry of a flame front is analysed with a scale of resolution li that ignores the details of smaller scale wrinkling, the front is seen as an effective front having a thickness li , as we shall see in (3.2.14), and whose normal propagation speed Ui is, according to (3.1.11), Ui /UL = Si /li2 ,

(3.2.7)

where Si is the total flame surface in a box of size li . The largest vortices, with the integral scale lI , are the most effective in contaminating the unburnt gas. They transport the flame front over a distance lI in a time τI , at a velocity that is the largest turnover velocity vI = lI /τI since, according to Kolmogorov scaling (3.4.4), the turnover velocity, vi , and the time scale, τi = li /vi , decrease as the spatial scale decreases, 1/3 2/3 vi ≈  1/3 li , τi ≈  −1/3 li . As explained in Section 3.1.5, the turbulent velocity of propagation can be controlled by the fastest contamination process if the combustion time is not much longer than the turnover time. This is the case if a self-similar property of combustion holds in a fully developed turbulence, as shown now. By definition of the Gibson scale, lG , vG = UL , the smallest vortices involved in front wrinkling are consumed in a turnover time tbG = τG . Since the time scale decreases with the length scale, small vortices are expected to burn faster than larger ones, so that the combustion time tbi of a blob of fresh mixture of size li is tbi = li /Ui . Assuming a selfsimilar property, the combustion time at scale li should follow then the same law at all scales. The law is given by the smallest scale lG : the combustion time is approximately equal to the turnover time, tbi ≈ τi = li /vi . If an element of fluid of size li burns in a time τi , then the propagation speed of the effective flame front is of the order of vi ≈ li /τi , Ui ≈ vi ,

(3.2.8)

and the amplitude of front wrinkling at the scale li is of order of li . Consequently vortices at the integral scale are completely consumed in their turnover time lI . Thus the quantity of burnt gas brought forward at the integral scale is the same at each time step τI , and the

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193

flame propagates at the contamination speed vI without losing intensity. In conclusion, the propagation velocity of a turbulent flame, Utur , is of the order of the intensity of turbulence, as first suggested by Damk¨ohler[1] in 1940, and the thickness of the turbulent front, dtur is of the order of the integral length scale, Utur ≈ vI ,

dtur ≈ lI .

(3.2.9)

This result can be understood both with and without making use of the notion of coherence time of a vortex. If the small vortices are permanently transported by the larger ones, the conclusion is immediately evident. If the turbulent cascade is considered to be a sequential process in which large vortices of size li destabilise after a coherence time τi to form smaller vortices, the total combustion time at the integral scale is given by the sum of the successive  combustion times τi ≈ τI where the sum concerns li  lI , for example li = lK /bi , b  2. For a Kolmogorov cascade (3.4.4), this sum converges to a value equal to τI times a numerical factor of order unity, so that the combustion time of a blob of fresh mixture of size lI is still of order τI . The Fractal Dimension of Flame Fronts in the Kolmogorov Cascade In the following we will use the notations of Section 3.4.2. The system of counting cubes that intersect the flame yields the relation Ni,G = Ni,j Nj,G , which from (3.4.7)–(3.4.8) can also be written, Si = Sj Si,j /lj2 . Assuming that the fluctuating quantities Ni,j and Nj,G are not correlated, with the help of (3.2.7) one gets   2 Si,j /li = Ui /Uj . (3.2.10) Let j now be the Gibson scale; relation (3.2.8) then yields Si  /li2 ≈ vi /UL ,

and in particular

SI  /lI2 ≈ vI /UL ≈ (lI /lG )1/3 ,

(3.2.11)

where Kolmogorov’s scaling (3.4.4) between the Gibson scale defined in (3.2.5) and the integral scale has been used in the second equation. If the geometry of the flame surface is fractal between the scales lI and lG , with a fractal dimension Df , the total flame area in a volume li3 of intermediate size, lG li lI , can be expressed in term of Df by using (3.4.9) for lj = lG , Si /li2 ≈ (li /lG )Df −2

and

SI /lI2 ≈ (lI /lG )Df −2 .

(3.2.12)

Comparing (3.2.11) and (3.2.12) then yields the fractal dimension of a flame, Df = 2 + 1/3 = 7/3.

(3.2.13)

The result can also be obtained by the following arguments. According to (3.2.12) the turbulent flame velocity Utur /UL = SI /lI2 takes the form Utur = UL (lI /lG )Df −2 , which agrees with (3.2.9) for Df = 7/3. This is the only value of the fractal dimension for which the turbulent flame velocity becomes independent of the characteristics of the laminar flame [1]

Damk¨ohler G., 1940, F. Elecktrochem., 601–652.

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when the condition in (3.2.6) is fulfilled. The philosophy is similar to that of fully developed turbulence in the inertial regime where the viscosity of the fluid does not appear in the scaling laws. Finally we can remark that by considering two successive scales, li and li+1 = li /b, where b = 2 for example, relation (3.4.9) can be used to find the surface area density of the effective flame front at the scale li :     Si,i+1 /li2 = b−1/3 , Si,i+1 /li3 ≈ 1/li . (3.2.14) As mentioned before, for each scale li , the amplitude of wrinkling of the effective front is of the order of the wavelength of wrinkling. Limitations The above arguments do not constitute proofs and must be taken with a fair degree of wariness. First, the experimental verifications are not satisfactory. When the experimental results obtained by different authors are grouped and plotted in the same representation (Utur /UL versus vI /UL ), they are quite dispersed and not always evenly distributed around the bisector.[1] Nevertheless, the orders of magnitude are correct. Moreover, for a given turbulent flow, it is found that the flame position in industrial burners varies notably with the composition and the nature of the reactive mixture. These disagreements arise mainly from the fact that it is very difficult to perform experiments in conditions that fully respect the hypotheses of the theoretical analysis (3.2.4). In particular, the turbulence must not be too intense, so that the internal structure of the flame is not significantly modified, lG  dL , but the integral scale must be sufficiently large so that there are several orders of magnitude between the integral scale and the Gibson scale, lI  lG . However, we should also question the validity of neglecting the hydrodynamic effects introduced by gas expansion through the flame. In particular, under which conditions do Kolmogorov’s scaling laws remain applicable in the presence of a flame front? The answer to this question is twofold.[2] 1. Strong arguments are in favour of the fact that the Darrieus–Landau instability cannot play a dominant role. According to (2.2.3) and (3.2.8), the linear growth time of the instability of the effective flame front, ≈ li /Ui , is of the same order of magnitude as the turnover time, li /vi . The instability does not have time to grow appreciably and thus it cannot change the order of magnitude of the amplitude of wrinkling. 2. For a large density ratio, the kinetic energy injected into the flow by the flame front can modify the spectrum of turbulence on scales larger than the Gibson scale. For the second point we first note that the energy injected by the flame, per unit surface and unit time, is ρu UL (Ub2 − UL2 ) = ρu UL3 [(ρu /ρb )2 − 1]. If the density ratio is not much greater than unity, this quantity is of the order ρu UL3 . The energy injected per unit [1] [2]

Peters N., 2000, Turbulent combustion. Benjamin Cummings. Clavin P., Siggia E., 1991, Combust. Sci. Technol., 78, 147–155.

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time by a structure of volume li3 is obtained by multiplying by the corresponding surface, < Si >≈ li2 vi /UL ; see Equation (3.2.11). Dividing by the volume li3 then gives the energy injected per unit volume. This transfer of energy occurs only during the combustion of the structure, equal to the turnover time, li /vi . The energy per unit time and unit volume transferred into the flow by structures of size li is thus ρu UL2 [(ρu /ρb )2 − 1]. This transfer of energy is independent of the scale of the structure. It can now be compared with the  ρu v2k /2 ≈ ρu v2i /2, energy per unit volume contained in turbulent structures of size li , where the sum concerns smaller scales k  i. For density ratios not much larger than unity, the Kolmogorov cascade is significantly modified only for scales in the vicinity of the Gibson scale lG , where vG = UL . For smaller scales the flames is not wrinkled and for larger scales the injected energy becomes negligible. However, if the density ratio is much larger than unity, the cascade is modified for larger scales up to li ≈ lG (ρu /ρb )3 . This modification concerns primarily the flow of burnt gas, but also locally modifies the upstream flow through the non local effects of hydrodynamics. 3.2.3 Covariant Laws The theoretical arguments presented in the preceding paragraph are based on the multiscale nature of the flame front (3.2.6) and on two hypotheses: 1. A local law for flame propagation (3.2.8) 2. The scaling laws for Kolmogorov’s cascade (3.4.4). A more general approach has been proposed by Pocheau[3] by examining the coherence of general laws of turbulent flame velocity having a form Utur (v, UL ), which should also be valid at each length scale li , where the corresponding effective flame speed is Ui . If v is the  total intensity of turbulence, v2 /2 ≡ v2k /2, the only law that is internally coherent is[3] 2 Utur = UL2 + c v2 ,

(3.2.15)

where c is a constant coefficient of order unity characterising the turbulent flow. This is because the only local law that is covariant (independent of the scale, i.e. valid for all possible pairs of scales) connecting the flame propagation speed at two scales lj and li (j > i), and which also satisfies additivity of the energy of turbulence, is Ui2

=

Uj2

+ c v2i,j ,

where

v2i,j

≡

j−1 

v2k

(3.2.16)

k=i

and where v2i,j denotes the turbulent energy contained in the range of scales between li and lj , and v2k is the turbulent energy in the range of two successive scales lk to lk+1 . Iterating (3.2.16) gives the relation between the propagation velocity Ui of the effective front at scale

[3]

Pocheau A., 1994, Phys. Rev. E, 49, 1109–1122.

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li , the laminar flame speed UL , and the turbulent flow,  Ui2 = UL2 + c v2k .

(3.2.17)

ki

This relation leads to (3.2.15) for the propagation speed at the integral scale. These laws are compatible with (3.2.9) and (3.2.8) for the combustion regime of fully developed turbulence characterised by (3.2.4) and Kolmogorov’s cascade in which v2i,j ≈ v2i . They are also compatible with relation (3.2.3) for a weakly turbulent flow. The geometry of the flame surface is described by the same local laws (3.2.7) and (3.2.10) that relate the local flame speed, Ui , to the mean increase in flame area. When (3.2.7) is inserted into (3.2.17), we obtain a relation between the total flame area contained in a volume li3 and the turbulence of the flow: 2

 Si /li2 = 1 + c v2k /UL2 . (3.2.18) ki

As expected, this expression reduces to (3.2.11) for turbulence characterised by Kol mogorov’s cascade in which ki v2k ≈ v2i  UL2 . These results[1] are based only on the existence of a multiplicity of scales and on the assumption of covariant laws, meaning that they keep the same form at all scales. They are not restricted to the strongly corrugated regime of (3.2.4) nor are they restricted to the inertial domain of Kolmogorov’s cascade, and they do not presuppose the scaling law (3.4.4). They remain valid when the intensity of turbulence is not large compared with the laminar flame velocity UL , (Utur − UL )/UL = O(1). In this case, the cut-off at Gibson’s scale is no longer pertinent; the increase in flame area is only of order unity and so the contribution of all scales must be taken into account, including those for which the amplitude of wrinkling is only moderate. These results have been tested in a series of experiments[2,3] in which the intensity of turbulence was of the order of UL , with a Reynolds number of the integral scale ReI ≈ 880 and a range of scales lI /lK ≈ 180. These conditions are those of Fig. 3.5. The multiplicity of scales of wrinkling is clearly apparent in this figure, showing that it exists even when the intensity of turbulence is not high (v/UL close to unity). Supposing that the ratio of successive scale sizes is not too large, li+1 = li /a with a = 1.2, expression (3.2.18) has been validated experimentally with a surprisingly good degree of accuracy. A synthesis of this work can be found in a review paper.[4]

[1] [2] [3] [4]

Pocheau A., 1994, Phys. Rev. E, 49, 1109–1122. Pocheau A., Queiros-Cond´e D., 1996, Phys. Rev. Lett., 76(18), 3352–3355. Pocheau A., Queiros-Cond´e D., 1996, Europhys. Lett., 35, 439–444. Pocheau A., 2000, In H. Chat´e, J. Chomaz, E. Villermaux, eds., Chaos and turbulence, Series B, vol. 373, 187–204, NATO ASI.

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Figure 3.6 Sketch of an open premixed turbulent burner.

3.3 Turbulent Combustion Noise In this section we will discuss the acoustic noise generated by unconfined turbulent flames. The discussion will be restricted to two limiting cases: that of a strongly wrinkled highly turbulent flame and that in which the turbulence is weak, flame wrinkling being amplified by the intrinsic instability of the flame front, as is the case for blowtorches. The general formalism for sound generation is presented in the third part of the book (see Section 15.2.4), and the monopolar sound emission by a deformable body is recalled in appendix; see Section 3.4.4.

3.3.1 Mechanism of Sound Generation In unsteady situations, changes in flame surface area generate acoustic waves. Consider a burner tube from which a turbulent flow of fresh premixed gases exits into an infinite free volume. Suppose that the flame attached to the tube exit is in the regime of strongly turbulent wrinkled flames, studied in the previous section. Large fluctuations in the surface area of the flame are produced by the direct action of turbulence on the flame front, as sketched in Fig. 3.6. Under the simplifying hypothesis that the mass of burnt gas produced per unit flame area and per unit time is constant, ρb Ub = ρu UL , fluctuations in the flame area produce fluctuations in the mass and volume flux of burnt gas, even when the mass flow of fresh gas from the tube exit is constant. The flow of burnt gas can be decomposed into the sum of the (constant) mean flow and a fluctuating part. A flame generates monopolar sound in the same way as a pulsating membrane or as body whose volume fluctuates in time, such as a pulsating sphere, recalled in Section 3.4.4. This is true if the dimensions of the flame are small compared with the wavelength of the radiated sound so that the flame can be considered as a point source; see (3.4.17)–(3.4.19) in the Appendix to this chapter. The radiated intensity is much greater

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Turbulent Premixed Flames

than that produced by a nonreactive turbulent jet whose flow rate is constant. The analysis of this latter[1,2,3] is more complicated than for turbulent burners presented below.[4] The intensity of sound radiated by a turbulent wrinkled flame, having dimensions that are small compared with the acoustic wavelengths, can be found by replacing the volume ˙ b (t) is the instantaneous mass flow rate of ˙ b /ρb , where M flow rate dV/dt in (3.4.19) by M burnt gas in the laboratory frame. In the wrinkled flame regime this quantity is given by the integral of the mass flow over the flame surface S(t),  ˙ b = ρb (Df + Ub )d2 S, M S

where Df is the propagation velocity of the front in the direction of the normal oriented towards the burnt gas; see (10.1.4). This latter quantity can be eliminated using the equivalent expression for the mass flow rate of fresh gas,  ˙ u = ρu (Df + UL )d2 S, M S

along with the mass conservation across the flame surface, ρb Ub = ρu UL , to give ˙ u /ρu + (Ub − UL )S. ˙ b /ρb = M dV/dt = M

(3.3.1)

˙ u , of fresh gas is constant, the intensity of radiated sound according to If the flow rate, M (3.4.19) is   (3.3.2) I = (ρ/4π a)(Ub − UL )2 (dS/dt)2 , where ρ and a are, respectively, the density and sound speed of the surrounding medium. This expression is in agreement with that given by Strahle[5] in terms of the rate of ˙ release of chemical energy, (t). For the case of a wrinkled flame this latter is given by ˙ (t) = cp (Tb − Tu )ρu UL dS/dt; see Section 15.2.4 and in particular (15.2.23) for a general presentation.

3.3.2 Intensity and Power Spectrum of the Radiated Sound The characteristics of the sound radiated by a flame can be obtained from Equation (3.3.2), provided that the fluctuations of flame surface S(t) are known. The calculation is carried out in Section 3.4.3. Using the the notations of Section 3.4.2 (see (3.4.7) and (3.4.8)), the 2 , and, according to Section 3.4.3, the relative fluctuations in flame flame area is SI = NI,G lG

[1] [2] [3] [4] [5]

Lighthill M., 1952, Proc. R. Soc. London Ser. A, A221, 564–587. Lighthill M., 1954, Proc. R. Soc. London Ser. A, 222, 1–32. Kampe T., 1986, J. Fluid Mech., 173, 643. Clavin P., Siggia E., 1991, Combust. Sci. Technol., 78, 147–155. Strahle W., 1985, In C. Casci, C. Bruno, eds., Recent advances in the aerospace sciences, 103–114. New York: Plenum Press.

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3.3 Turbulent Combustion Noise

surface area may be written[4]   (δSI )2 SI 2

=

n−1  i=1

  1 (δSi,i+1 )2     . NI,i Si,i+1 2

199

(3.3.3)

The intensity of sound radiated by a strongly turbulent flame can now be found from (3.3.3) using Kolmogorov’s cascade. Intensity of Sound Dividing Equation (3.3.3) by

δt2 ,

with δt → 0,

  2 n−1   S˙ i,i+1  1   S˙ I2 = SI 2  , NI,i Si,i+1 2 i=1

where S˙ ≡ dS/dt, and where, according to (3.4.9),   NI,i ≈ (lI /li )Df .

(3.3.4)

(3.3.5)

Supposing that the characteristic time associated with a fluctuation of the effective flame front at scale li is just the turnover time τi ≡ li /vi , we can write    2 2 / Si,i+1 ≈ 1/τi2 = (vi /li )2 . (3.3.6) S˙ i,i+1 According to Kolmogorov’s scaling laws, (3.4.4) and (3.4.5), the turnover time at scale li is given by 1/τi2 ≈ (vI /lI )2 (lI /li )4/3 ,

(3.3.7)

and so Equation (3.3.4) can be written, using (3.3.5), n−1    S˙ I2 ≈ SI 2 (vI /lI )2 (li /lI )Df −4/3 ,

(3.3.8)

i=1

where, according to (3.2.13), Df − 4/3 = 1. Equation (3.3.8) shows that the time scale controlling the dynamics of turbulent flames is the integral time scale, as is also the case for mixing in fully developed turbulence.[6,7,8] Now li /lI = 1/bi , with b  2, so for n > 4 the sum in the right-hand side of (3.3.8) converges rapidly towards 2 when b = 2, and to a value close to unity when b is large. With the help of (3.2.11), SI  ≈ lI2 vI /UL , we then find that   (3.3.9) S˙ I2 ≈ lI2 v4I /UL2 .

[6] [7] [8]

Onsager L., 1949, Nuovo Cimento, VI, 279–287. Corrsin S., 1951, J. Appl. Phys., 22, 469–473. Villermaux E., et al., 2001, Phys. Fluids, 13(1), 284–289.

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Turbulent Premixed Flames

The intensity of sound radiated by the flame surface in an integral volume is given by (3.3.2),  2 Tb ρ − 1 lI2 v4I . (3.3.10) II ≈ 4π a Tu Let V be the total volume of the turbulent flame brush. This brush contains V/lI3 elements having the size of an integral volume, S = (V/lI3 ) SI . These elements are statistically independent,   so the mean square fluctuations of flame surface are additive,  3 2 2 (δS) = (V/lI ) (δSI ) , and so     (3.3.11) S˙ 2 = (V/lI3 ) S˙ I2 , giving the following expression for the total sound intensity from the flame:  2 v4 1 Tb I≈ − 1 (ρV) I , 4π Tu alI

(3.3.12)

where ρ and a are, respectively, the density and sound speed of the medium into which the sound is radiated. This latter is the burnt gas if there no mixing with the ambient air. For a classical situation of a turbulent flame anchored on the outlet of tube, lI and vI are, respectively, the diameter of the tube and the turbulence intensity of the outflowing gas. Power Spectrum Equation (3.3.4) remains valid when the second-order moments are replaced by the correlation functions         2 → S˙ i,i+1 (t)S˙ i,i+1 (0) . S˙ I2 → S˙ I (t)S˙ I (0) and S˙ i,i+1 According to (3.3.2), the power spectrum of the sound, dI˜(ω), is the Fourier transform of   ρ ˙ S(0) ˙ , (Ub − UL )2 S(t) 4π a which can be obtained directly from Equation (3.3.8), with Df = 7/3 and the help of the scaling laws of Kolmogorov’s cascade (3.4.4) to provide a relation between frequency −2/3 , ωi ≡ 2π/τi and the scale of fluctuations: ωi = 2π  1/3 li n−1   v2  −3/2 ω . S˙ I2 ≈ SI 2  1/2 3I lI i=0 i

(3.3.13)

The transformation to a continuum formulation can be obtained by remarking that the frequency  increment ωi ≡ (ωi+1 − ωi ) is proportional to frequency, ωi , or in other words  → dωi /ωi , giving finally a power spectrum of combustion noise that decreases with i frequency[1] as ω−5/2 ,

[1]

Clavin P., Siggia E., 1991, Combust. Sci. Technol., 78, 147–155.

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3.3 Turbulent Combustion Noise (a)

201

(b)

0.65 0.85 1.15 1.40 1.60

Figure 3.7 (a) Outline of an instantaneous tomographic cut through a turbulent propane–air flame superimposed on a longer exposure (1/50 s) of the turbulent flame brush, equivalence ratio 0.75. (b) Noise power spectra of turbulent propane–air flames for a range of equivalence ratios φ. The dotted line has slope ω−5/2 . Courtesy of A. Belliard and G. Searby, IRPHE Marseilles. −5/2

dI˜(ω) ∝ ωi

dωi .

(3.3.14)

This ω−5/2 law for the decrease of the noise power spectrum was first observed in 1978 on highly turbulent Bunsen burners with oxygen-enriched propane-air flames[2] and more recently in 1997 on a propane–air burner;[3] see Fig. 3.7.

3.3.3 Blowtorch Noise The case of weakly turbulent burners is different, as for example the noise radiated by a blowtorch. In this type of burner, the decrease in diameter of the flow at the nose of the blowtorch causes a strong increase in flow velocity with almost no change in the amplitude of the velocity fluctuations. As a consequence, the relative level of turbulence in the flow of unburnt gas is quite low. Moreover, the reactive gas in most blowtorches is a mixture of a hydrocarbon fuel (acetylene, propane, etc.) and pure oxygen, with no nitrogen dilutant. The resulting laminar flame velocity is an order of magnitude higher than for ordinary hydrocarbon–air flames. Consequently, the combustion regime of these flames is that of a weakly turbulent flame, vI /UL < 1. Although the noise level produced by these flames is compatible with the observed fluctuations of flame surface area, it cannot be explained by the above mechanism in which the fluctuations are produced by the direct action of upstream turbulence on the flame front. It is observed experimentally that the combustion noise from these flames [2] [3]

Abugov D., Obrezkov O., 1978, Combust. Expl. Shock Waves, 14, 606–612. Belliard A., 1997, University thesis, Universit´e d’Aix-Marseille-I.

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202

Turbulent Premixed Flames (a)

(b)

Figure 3.8 (a) Photograph of a propane–oxygen blowtorch flame. Exposure time = 1/50 s. (b) Same flame seen with an exposure time of 1/10,000 s. Small turbulence-induced perturbations at the base of the flame are amplified by the Darrieus–Landau instability. The wavelength of the resulting structures is close to that of the maximum calculated growth rate. Courtesy of G. Searby and J.-M. Truffaut, IRPHE Marseilles.

is two orders of magnitude higher than that predicted by Equation (3.3.12).[1] The most important contribution to the fluctuations of flame area comes from the tip of the flame. These fluctuations are generated by small perturbations at the base of the flame that are amplified by the intrinsic instability of the flame front (Darrieus–Landau instability; see Section 2.2) as they are advected towards the flame tip; see Fig. 3.8, and Section 2.8.1. The most visible structures have a wavelength close to that having the maximum growth rate. According to Equation (2.2.10) the growth rate of the perturbations is proportional to the laminar flame speed, σ = AUL km /2. The growth rate of this instability on blowtorch flames is very high because of the high laminar flame velocity, and their dynamics dominate the effect of weak turbulence. The radius of the flame cone decreases at the summit, and the presence of these structures causes rapid variations in the flame area near to the summit, often with the formation of detached pockets of fresh gas. The exact calculation of the time-dependent variations in flame surface area and production of burnt gas is difficult[2] due to geometrical randomness coupled to the changes in local burning speed caused by the very small radii of curvature. Fig. 3.9 shows a sequence of images demonstrating the formation of a pocket of unburnt gas induced by controlled sinusoidal structures advected towards the summit of a two-dimensional Bunsen tip.[1] Notice the sudden increase in luminosity of the flame front (increased combustion rate) at the neck when the pocket separates from the tip of the cone. The acoustic level produced by the formation and extinction of the pocket is quite high, ≈ 94 dB. In this simple case the separation of the neck (sudden disappearance of flame surface) was identified as being the main source of acoustic noise. The plot shows the measured acoustic pressure radiated [1] [2]

Truffaut J., 1998, University thesis, Universit´e d’Aix-Marseille I. Searby G., et al., 2001, Phys. Fluids, 13, 3270–3276.

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3.4 Appendix (a)

203

(b)

Figure 3.9 (a) Sequence of high-speed photographs showing a cycle of formation of a pocket of fresh gas at the tip of a sinusoidally excited two-dimensional Bunsen cone. The time is in milliseconds. (b) Acoustic pressure radiated by the flame. Solid line: measured pressure. Dotted line: pressure calculated from variation of flame area. Courtesy of G. Searby and J.-M. Truffaut, IRPHE Marseilles.

by the flame, along with the acoustic pressure calculated from the flame surface area using expression (3.3.2).

3.4 Appendix 3.4.1 Turbulence and Kolmogorov’s Cascade Turbulence results from instabilities of fluid flow. It is characterised by a random distribution of velocity fluctuations, v(r, t), whose average value is zero, v = 0, around some mean flow velocity, u, which we will consider to be uniform in the following. The standard (r.m.s.) deviation of the velocity, denoted by v, is called the turbulence intensity. We will limit our discussion to the ideal case of homogeneous isotropic turbulence. This ideal turbulence is well represented by grid-generated turbulence. The Kolmogorov Scale The field of random velocity fluctuations can be decomposed into a sum of fluctuations, each characterised by its individual length and time scale, using for instance the Fourier transform of the velocity autocorrelation function. It is then possible to define the turbulence intensity vl associated with a spatial scale l, and the associated coherence time, τl , also called the turnover time, of a vortex of size l. In fully developed turbulence, this turnover time is of the order of l/vl , τl ≈ l/vl .

(3.4.1)

The largest scale present in the flow, lI , is called the integral scale. It is the scale at which energy is injected into the flow by some external mechanism, for example by boundary

17:02:00 .005

204

Turbulent Premixed Flames

Figure 3.10 Schematic representation of vortex cascade in a turbulent flow.

layer separation at the walls. For flow in a tube, lI is of the order of the radius of the tube. For grid generated turbulence, it is the pitch of the grid. The scale of turbulence also has a lower bound, called the Kolmogorov scale, lK . This is the scale at which viscous dissipation transforms mechanical energy into thermal energy. Dimensional analysis equating the inertial term ρv.∇ v to the viscous dissipation term μ∇ 2 v, in the Navier–Stokes equations (15.1.17) suggests that the Kolmogorov scale is characterised by a local Reynolds number, Re ≡ vl l/ν, of order unity, vK lK /ν ≈ 1,

(3.4.2)

where ν = μ/ρ is the viscous diffusion coefficient. For larger scales, l  lK , viscous forces are negligible compared with inertial forces. On the other hand, for scales smaller than the Kolmogorov scale, l lK , the viscous forces dominate, dissipation is strong, and the intensity of turbulence becomes negligible. When the Reynolds number at the integral scale is large, ReI ≡ vI lI /ν  1, the integral and Kolmogorov scales are well separated, lI  lK . The length scales form a continuum, but to simplify, we will consider a discrete series of lengths li = lI /bi , i = 0, 1, 2, .., and b is an integer greater than unity, generally taken equal to 2. We also define the associated velocities, vi , and time scales, τi = li /vi . Thus v2i /2 is the turbulent energy per unit mass for length scales in the elementary range [li , li+1 ], and the intensity of turbulence is given  2 by the sum v2 = i vi . Choosing b = 2 suggests a representation of turbulence as an ensemble of interlocked vortices, one inside the other, as sketched schematically in Fig. 3.10, in which the vi are simply the characteristic velocity of the vortex of diameter li . This schematic representation can be useful, but it is also misleading. It gives only a vague idea of the multiplicity of scales in real turbulent flows and gives no idea of the random behaviour in space and time. Scaling Laws A synthetic historical presentation of the theory of turbulent flows, from the work of pioneers up to recent developments, can be found in the literature.[1,2] Homogeneous isotropic fully developed turbulence is an idealised concept whose theoretical description, called K41, was developed by Obukhov and Kolmogorov in 1941. This theory is valid only in [1] [2]

Monin A., Yaglom A., 1971, Statistical fluid mechanics, vols. 1 and 2. MIT Press. Frisch U., 1995, Turbulence. Cambridge University Press.

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205

the inertial domain, which characterises turbulence at a high Reynolds number, ReI  1. The scaling laws can be found from a dimensional analysis of the Navier–Stokes equations 15.1.17. They presuppose a steady semi-permanent regime, first imagined by Richardson in 1922, in which the energy is injected into the large scales and is transferred successively to smaller scales by flow instabilities, and is finally dissipated at a small scale, lK , at a fixed rate equal to the rate of injection. When the integral and Kolmogorov scales are well separated, lK lI , there is an intermediate domain, called the inertial domain, lK l lI ,

(3.4.3)

in which the details of the mechanisms of energy injection at large scales and energy dissipation at small scales are not ‘felt’, or, in other words, the only relevant length scale in the intermediate domain is the local scale, l. In a given volume, the turbulent energy transferred per unit time from the larger scale to the smaller scale is a constant and is the same at all scales. This cascading transfer of energy is commonly called the Kolmogorov cascade. The rate of energy transfer can be estimated from the nonlinear term in the Navier– Stokes equation (multiplied by v) as ρ(v.∇)v2 /2. A dimensional analysis for the energy flux per unit mass at a given scale l yields v3l /l, giving the following scaling law: v3l /l ≈ cst. It is a measure of the turbulence intensity vi at the scale li or, more exactly, the intensity at scales in the interval [li , li+1 ], 1/3

vi ≈  1/3 li ,

2/3

v2i ≈  2/3 li ,

τi ≈  −1/3 li , 2/3

(3.4.4)

where  is the rate of energy transfer between scales, which is assumed here to be constant and nonfluctuating. It is the rate at which energy is injected per unit mass  ≡ v3I /lI ,

(3.4.5)

and also the rate at which energy is dissipated by viscosity at the Kolmogorov scale, 2 .  = νv2K /lK

The following relations hold between the integral and dissipation scales: 3/4

lI /lK ≈ ReI ,

1/4

vI /vK ≈ ReI ,

1/2

τI /τK ≈ ReI .

(3.4.6)

Steady homogeneous isotropic turbulence is thus completely characterised by just three quantities: the rate at which energy is injected (per unit mass), ; the size of the integral scale, lI ; and the viscous diffusion coefficient, ν. According to the initial hypothesis, from which Kolmogorov’s relations (3.4.4) are obtained, neither lI nor ν plays any role in the inertial domain (3.4.3), other than through . These scaling laws (3.4.4) have been verified experimentally over twoor three decades of the energy spectrum obtained by Fourier ∞ decomposition, v2 /2 = 0 E(k)dk, where E(k) ≈  2/3 k−5/3 . This expression was first obtained by Obukhov in 1941, using (3.4.4) and the approximation dk ≈ k ≈ l−1 . According to (3.4.4), the turbulent velocities, vi , and also the turnover times, τi , decrease with

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206

Turbulent Premixed Flames

li/16 li/8 li/4 li/2 li

Figure 3.11 Paving space with successively smaller cubes.

 decreasing scale such that the series i v2i converges to a finite value. We can then deduce that the total turbulent intensity is given by the intensity at the integral scale, to within a  numerical constant, ji v2j ≈ v2i , and v ≈ vI . 3.4.2 Elements of Fractal Geometry The standard way to study the geometry of a front and measure its fractal dimension is to in pave space successively with cubes of linear size lj , each successive set of cubes being smaller than the previous set, lj < li for j > i, lj = lI /bj , b > 1, and counting the number of cubes that intersect the front at each scale j. Fig. 3.11 shows such a paving for b = 2. Consider a front whose the smallest scale of wrinkling is lG (the Gibson scale for flames). The total surface area in a cube of size li (lG < li < lI ) can be found by counting the number of cubes of size lG , within the volume li3 , that intersect the flame front. This is schematised (in two dimensions) in Fig. 3.11 by the grey squares, supposing lG = li /16. If this number of cubes is Ni,G , then the surface area is 2 . Si ≈ Ni,G lG

(3.4.7)

Since the front is not significantly wrinkled on scales smaller than lG , the use of smaller cubes will give essentially the same surface area. Now let Si,j ≈ Ni,j lj2

(3.4.8)

be the surface area seen at a weaker resolution lj , lG < lj < li , obtained by counting the number of cubes Ni,j of size lj that intersect the surface in the volume li3 (with the

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3.4 Appendix

207

same notation Si = Si,G ). Since the details of small-scale wrinkling are lost as the size of the cubes increases, the flame area at scale j will be smaller than that at the scale lG , Si,j < Si,G = Si . A surface is said to be fractal, with a fractal dimension Df > 2, if the area Si,j increases with a power law Df − 2 > 0 larger than 2 as the scale lj decreases:

D

D −2 Ni,j ≈ li /lj f , Si,j /li2 ≈ li /lj f . (3.4.9) The total area Si of a regular surface included in a box of size li is obtained in the limit lj → 0, Si = limlj →0 Si,j = limlj →0 Ni,j lj2 . For a physical surface having a nonzero thickness dL , the surface is well defined only if the smallest wrinkles have a scale greater than dL . Obviously, a fractal geometry can concern only scales greater than that of the smallest wrinkles, since below this, the surface area becomes finite and independent of the scales. Thus for a turbulent flame in the regime (3.2.6), the fractal property of the flame front concerns scales between the Gibson scale and the integral scale, lG < lj < li < lI .

3.4.3 Fluctuations of Turbulent Flame Surface The objective of this appendix is to calculate the fluctuations of flame surface.[1] Using the same approach as that of Section 3.2.2, let NI,i be the number of boxes of size li that intersect the flame in a volume equal to the integral volume. Each box can be identified by a dummy index i , i ∈[1, NI,i ]. Looking now at smaller boxes of size li+1 , we can write NI,i+1 =

NI,i 

Ni, i+1 (i ),

(3.4.10)

i =1

where, in the notation of (3.4.8), Ni, i+1 (i ) is the number of boxes of size li+1 that intersect an element of flame surface contained in box i (of size li ) with li+1 = li /b, where b  2 and 1  Ni, i+1 (i )  b3 . Equation (3.4.8) can be written as 2 Si,i+1 = Ni, i+1 li+1 .

(3.4.11)

The quantities NI,i and Ni, i+1 (i ) are random variables that we will to be inde suppose  pendent (no correlation) for i >  1. Their mean values NI,i and Ni, i+1  along with their fluctuations δNI,i ≡ NI,i − NI,i and δNi, i+1 (i ) ≡ Ni, i+1 (i ) − Ni, i+1 then satisfy the following relations (see (3.4.10)),      (3.4.12) NI,i+1 = NI,i Ni, i+1 , δNI,i+1 =

NI,i  



δNi, i+1 (i ) +

i =NI,i +1

i =1

[1]

NI,i 

Clavin P., Siggia E., 1991, Combust. Sci. Technol., 78, 147–155.

17:02:00 .005

Ni, i+1 (i ).

(3.4.13)

208

Turbulent Premixed Flames

In the linear approximation (i.e. to first order in the intensity the second  of fluctuations),  term on the right-hand side of (3.4.13) can be rewritten δNI,i Ni,i+1 , and so, after dividing by (3.4.12) we obtain NI,i  δNI,i+1 1  δNi,i+1 (i ) δNI,i  =    +  . NI,i+1 NI,i  Ni,i+1 NI,i

(3.4.14)

i =1

Supposing that the random variables δNI,i+1 and δNi,i+1 (i ) are statistically independent (noncorrelated),   δNI,i+1 δNi,i+1 (i ) = 0,     δNi,i+1 (i )δNi,i+1 (j ) = δi j (δNi,i+1 )2 , or in other words, supposing that the different scales of turbulence fluctuate independently of each other, so that the same is also true for the fluctuations of flame wrinkling, the mean square value of the fluctuations is       (δNI,i )2 (δNI,i+1 )2 1 (δNi,i+1 )2 (3.4.15)  2 =     +  2 . NI,i Ni,i+1 2 NI,i+1 NI,i Let n be the total number of different scales between the integral scale and the Gibson scale. With li+1 = li /b, where i ∈ [0, n], we have l0 = lI , ln = lG = lI /bn , and iterating expression (3.4.15) leads to   n−1    1 (δNi,i+1 )2 (δNI,G )2    (3.4.16)  2 =  , NI,i Ni,i+1 2 NI,G i=1 3 that intersect where, in the notation of (3.4.8), NI,G is the number of boxes of volume lG 3 the flame surface contained in an integral volume lI . The corresponding flame area is SI = 2 ; see (3.4.7). Using (3.4.11) in Equation (3.4.16), we obtain the expression for the NI,G lG relative fluctuations in flames surface area; see (3.3.3).

3.4.4 Monopolar Sound Emission The velocity field u(r, t) of an acoustic wave propagating in a fluid at rest is given by the gradient of an acoustic potential that is a solution of the wave equation (also called d’Alembert’s equation), u = ∇φ(r, t),

∂ 2 φ/∂t2 − a2 φ = 0,

where the  is the Laplacian operator, and a is the speed of sound; see (15.2.22). In the case of a source (such as a flame) if the unperturbed flow is very subsonic it can be neglected (Doppler effects neglected). Consider now a deformable body whose instantaneous volume is V(t), and let ω be the characteristic frequency of changes in V. We restrict the discussion to the case of low

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3.4 Appendix

209

frequencies for which the wavelength of the acoustic wave  ≡ a/ω is much greater than the characteristic dimension of the body,   L. At distances much greater than L, the acoustic potential has spherical symmetry and can be written as[1] r  L:

˙ − r/a), φ = −(4π r)−1 V(t

(3.4.17)

˙ ≡ dV/dt is the time derivative of the volume of the body and is also the volume where V(t) flow rate of the equivalent source. This classical result is easily retrieved. To do this we first note that at intermediate distances from the source, L r , the potential must satisfy Laplace’s equation, φ ≈ 0, since ∂ 2 φ/∂t2 a2 φ. So at these intermediate distances, the acoustic field is an incompressible open flow generated by a point source ˙ ˙ of variable volume flow rate V(t); the resulting potential is φ ≈ −V(t)/(4π r). However, at distances much greater than the acoustic wavelength, r  , the acoustic flow takes the form of spherically diverging waves, whose potential can be written in a general form ˙ as φ = f (t − r/a)/r; see (15.2.23). Matching these two solutions imposes f (t) = V/4π , leading to relation (3.4.17). The fluid velocity in the acoustic wave radiated by the body, v = ∂φ/∂r, can be written as r  L:

¨ − r/a), v = (4π ar)−1 V(t

¨ ≡ d2 V(t)/dt2 . V(t)

(3.4.18)

The derivative of the denominator in (3.4.17) has been neglected since it gives a contribution that is of the following order in an expansion in powers of /r. The acoustic energy  2  per  2 2 unit mass of fluid is v , the time average of v . The radial flux of energy is thus ρa v , and the total acoustic energy radiated per unit time (intensity of sound) is found by integrating this quantity over a sphere of arbitrary radius much greater than L,  2 , (3.4.19) I = (ρ/4π a) d2 V/dt2 which is a classical result in acoustics.[1]

[1]

Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed.

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4 Gaseous Shocks and Detonations

Nomenclature Dimensional Quantities a cp cv d D D DT E E h k kB l  L m p qm qv r R s s S t tN

Description Mean molecular velocity. Sound speed Specific heat at constant pressure Specific heat at constant volume Thickness (of a structure) Normal propagation velocity Molecular diffusivity Thermal diffusivity Activation energy Energy Enthalpy Transverse wavenumber Boltzmann’s constant Longitudinal evolution parameter, see (12.1.11) Molecular mean free path A length scale Mass flux Pressure Heat of combustion per unit mass Heat of combustion per unit volume Radius. Radial coordinate Radius of tube or combustion chamber Entropy Real part of growth rate Surface area Time Reaction time at Neumann state, τr (TN )

S.I. Units m s−1 J K−1 kg−1 J K−1 kg−1 m m s−1 m2 s−1 m2 s−1 J molecule−1 J J kg−1 m−1 J K−1 m−1 m m kg m−2 s−1 Pa J kg−1 ≡ (m/s)2 J m−3 m m J K−1 kg−1 s−1 m2 s s

210

.006

19:26:43

Gaseous Shocks and Detonations

T T∗ u UL Ub v ve vp

v w ˙ W x, y, z x α δ μ ν λ ρ σ τ τr ω 

Temperature Crossover temperature Mean velocity. Longitudinal velocity Laminar flame speed Laminar flame speed relative to burnt gas Velocity in laboratory frame Turnover velocity of a vortex Piston velocity Specific volume 1/ρ Transverse velocity Reaction rate Coordinates Self-similar variable Local position of front A characteristic thickness Shear viscosity Viscous diffusivity μ/ρ Thermal conductivity Density (Complex) growth rate of perturbation A characteristic time Reaction time Angular frequency Heat release rate

K K m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m3 kg−1 m s−1 kg m−3 s−1 m x/t, r/t m m Pa s m2 s−1 J s−1 m−1 K−1 kg m−3 s−1 s s s−1 s−1

Nondimensional Quantities and Abbreviations b cst. K lN m M n O(.) Pr P qN Q r Re

Temperature sensitivity parameter, see (4.5.21) Constant see (4.3.40) and (4.3.47) Length scale of heat release, see (4.5.26) Reduced mass flux through shock, see (4.5.3) Mach number u/a Parameter characterising strength of shock, see (4.4.4) Of the order of Prandtl number μ/(ρDT ) Normalised pressure, see (4.2.11) Reduced heat of reaction of detonation, see (4.5.11) Normalised heat of reaction, see (4.2.42) Parameter characterising the fluid, see (4.4.4) Reynolds number u l/ν

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212

s S Sa, Si t V w ˙ x βN γ  ζ η ε  κ ν ξ π σ φ ψ  CJ DDT ZND

Gaseous Shocks and Detonations

Degree of folding, relative increase of surface S /So Reduced complex growth rate, see (4.4.3) Parts of reduced growth rate, see (4.5.31) Reduced time, see (4.5.1) Normalised specific volume, see (4.2.11) ˙ tN W Reduced reaction rate Reduced mass-weighted coordinate, see (4.5.1) Reduced activation energy of detonation E/kB TN Ratio of specific heats cp /cv A small parameter Reduced transverse coordinate, see (4.4.40) z/L Reduced transverse coordinate, see (4.4.40) y/L A second small parameter Reduced temperature βN (T − T N )/T N Reduced wavenumber, see (4.5.29) Reduced specific volume (v − vu )/vu Reduced length x/, r/(Dt) Reduced pressure (p − pu )/pu (In Section 4.5) reduced growth rate, see (4.5.29) Reduced amplitude of wrinkling, see (4.4.40) α/(L) Reaction progress variable (heat release), see (15.1.42) Distribution of reduced reaction rate Chapman–Jouget Deflagration-to-detonation transition Zeldovich, von Neumann and D¨oring

Superscripts, Subscripts and Math Accents a∗ a1 a(a) a(i) ay a˙ t ab ac acoll aCJ ae af aind

A critical or particular value Value in upstream (unshocked) flow Compressible (acoustic) part of flow Incompressible part of flow ∂a/∂y ∂a/∂t Burnt gas Critical value Collisions Chapman–Jouget conditions Excitation or perturbation flow (vortex, etc.) Flame front Induction (time or length)

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aint aL aN ao ap ar atur au a a˜ aˆ aˆ a˘

213

Interaction (time) Laminar flame Neumann state (behind a shock) Initial, or unperturbed or reference value On the piston Reaction rate Turbulent Fresh gas Average value or unperturbed value Fourier component of a, a(y, t) = a˜ (t)eiky Amplitude of linear harmonic perturbation, a(y, t) = aˆ eiky+σ t (In Section 4.5) Dimensional quantities Laplace transform of a 4.1 Introductory Remarks

Detonations are supersonic waves of combustion. They were briefly presented in Section 1.2.5. The difference of time scales between the rate of elastic collisions and that of inelastic collisions producing the heat release makes the structure of a gaseous detonation decompose into an inert shock wave (lead shock) followed by an exothermic reaction zone.[1] Shock waves are thin regions of strong gradients propagating at supersonic speed in an inert fluid. As recalled in Section 15.1.7, the internal structure of shock waves is controlled by elastic collisions and their thickness is microscopic, a few tens of mean free paths in gases, except for weak shocks (propagating at a velocity close to that of sound) that are thicker; see Section 4.2.2. Ordinary shock waves appear as discontinuities in the solutions to Euler’s equations. Their formation is discussed in Section 15.3. The exothermic reactions are produced in gaseous mixtures by inelastic collisions that are much less frequent than elastic collisions, a phenomenon that can be represented by an Arrhenius law with an activation energy larger than the thermal energy; see Section 1.2.2. The thickness of the reaction zone of gaseous detonations is thus macroscopic, ranging from several millimetres to one centimetre in normal conditions, see Section 1.2.5. Unlike inert shocks, there is a regime of autonomous propagation of detonations, without external support such as a piston or a projectile. This self-propagating regime, with a constant velocity of propagation, is called the Chapman–Jouguet (CJ) regime. Detonations are observed not only in gaseous mixtures, but also in solid and liquid explosives. In contrast to gaseous detonations, detonations in condensed phases are microscopically thin. They will not be studied in this book. The study of gaseous detonations was initiated during the nineteenth century,[2,3,4] but the multidimensional dynamics of the front was not understood until much later and is [1] [2] [3] [4]

Vieille P., 1900, C. R. Acad. Sci. Paris, 131, 413. Rankine W., 1870, Philos. Trans. R. Soc. London, 160, 277–288. ´ Hugoniot P., 1889, Journal de l’Ecole Polytechnique, 58(1), 1–125. Rayleigh J., 1910, Proc. R. Soc. London, 84, 247–284.

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Shocked gas

Figure 4.1 Sketch of flow through a shock wave. The flow velocities are given in the reference frame of the shock.

still under investigation. The basic knowledge of shock waves[1] and detonations[2] is summarised in Section 4.2, while the general theory of one-dimensional compressible flows is recalled in Section 15.3. The results of the two last decades are presented in Sections 4.3– 4.5 where the emphasis is put on simple theoretical analyses based on physical insights. Technical details of the more complicated analyses are presented in Chapter 12. 4.2 Planar Supersonic Waves In this section attention is limited to planar waves propagating at constant velocity in a simple fluid. The multidimensional dynamics resulting from intrinsic instabilities is discussed later. We start with planar shock waves supported by a piston in an inert fluid. 4.2.1 Shock Waves: Rankine–Hugoniot Relations In the steady planar case, a shock wave separates two homogeneous and stationary flows. Its velocity depends on the thermodynamic state of the upstream fluid and the velocity of the driving piston, regardless of the dissipative mechanisms that control its internal structure. For a given velocity of the piston, the propagation speed and the density and pressure of the shocked gas are obtained using the jump relations for the conservation of mass, longitudinal momentum and total energy in (15.1.45)–(15.1.48), where the thermal gradients are neglected in the external regions on both sides of the shock. The propagation velocity D of a planar shock wave supported by a piston moving at constant speed vp in a medium initially at rest (density ρu and and pressure pu ) is obtained as follows. In the reference frame of the shock wave, fluid of density ρu enters the wave with velocity D in the direction normal to the plane of the shock. After crossing the shock, the fluid density and velocity are different; see Fig. 4.1. Let uN be the speed at which the ‘shocked’ fluid leaves the wave, denoted by UN in Section 1.2.5 and in (15.1.45)–(15.1.48). It flows in the normal direction, and the state of shocked gas is called the ‘Neumann state’ with density ρN and pressure pN . According to conservation of mass, the mass flow rate m through the wave is constant, m ≡ ρu D = ρN uN . [1] [2]

(4.2.1)

Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed. Zeldovich Y., Kompaneets A., 1960, Theory of detonation. Academic Press.

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The wave propagates at speed D in the laboratory frame where the upstream gas is at rest, and the shocked gas velocity is equal to the piston velocity vp in this frame, so that D − uN = vp .

(4.2.2)

The state of shocked gas (ρN , pN ) is found as a function of m from conservation of longitudinal momentum (15.1.46) and energy (15.1.51). With the help of (4.2.1) it is then possible to express D and uN as functions of m, which can finally be obtained as a function of vp using (4.2.2). Case of an Arbitrary Fluid Momentum conservation (15.1.46) can be written pu + m2 /ρu = pN + m2 /ρN , or   1 1 2 . (4.2.3) pN − pu = −m − ρN ρu Energy conservation (15.1.51) in the absence of heat release (qm = 0) and external fluxes, hN − hu + (u2N − D 2 )/2 = 0,

(4.2.4)

can be written, using (4.2.1), D 2 = m2 /ρu2 , u2N = m2 /ρN2 , in the form / . m2 1 1 − 2 , hu − hN = 2 ρN2 ρu where hu and hN are the enthalpies in the initial and shocked states. Eliminating m with the help of (4.2.3) gives   1 1 1 (pN − pu ) = 0. + (4.2.5) h(ρu , pu ) − h(ρN , pN ) + 2 ρu ρN When the enthalpy h(ρ, p) is expressed as a function of density and pressure using the thermodynamics of the fluid, the Hugoniot relation,   1 1 1 (p − pu ) = 0, + (4.2.6) h(ρu , pu ) − h(ρ, p) + 2 ρu ρ defines a curve in the (1/ρ, p) plane that goes through the point (1/ρu , pu ). In this same plane, momentum conservation (4.2.3),   1 1 , (4.2.7) − p − pu = −m2 ρ ρu defines a straight line with negative slope that also goes through the point (1/ρu , pu ); see Fig. 4.2. The absolute slope of this line, called the ‘Michelson–Rayleigh’ line, is proportional to the square of the mass flux. Its intersection with the Hugoniot curve defines the point (1/ρN , pN ) and yields the shock speed D as a function of the piston velocity vp .

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Michelson– Rayleigh line Hugoniot curves

0

Figure 4.2 Hugoniot curves and the Michelson–Rayleigh line representative of a perfect gas subject to a strong shock propagating at Mach 6. The solid curve is the Hugoniot curve corresponding to the initial condition U. The dotted curve is the back projection of the Hugoniot curve corresponding to the initial condition N. They are different, although both pass through the two points; see the discussion following (4.2.23).

Case of a Polytropic Gas Consider now the particular case of a polytropic gas, namely an ideal gas with constant specific heats per unit mass cp and cv , γ = cp /cv > 1, p = (cp − cv )ρT,

h = cp T =

γ p . γ −1ρ

(4.2.8)

In this case Equations (4.2.2)–(4.2.4) for mass, momentum and energy conservation can be obtained directly from the kinetic theory of gases; see Section 4.6.1. According to (4.2.8), the Hugoniot relation (4.2.6) can then be written as     1 p pu 1 1 γ − (p − pu ) − = 0. (4.2.9) + γ −1 ρ ρu 2 ρu ρ Multiplying by 2(γ − 1)ρu /pu yields        p ρu ρu p −1 +2 − 1 = 0. −1 − 1 + 2γ (γ + 1) pu ρ pu ρ Introducing the notations, P≡

(γ + 1) 2γ



 p −1 , pu

V≡

(γ + 1) 2



 ρu −1 , ρ

(4.2.10)

(4.2.11)

Equation (4.2.10) multiplied by (γ +1)/4γ reduces to an equilateral hyperbola in the P–V plane with the equation PV + P + V = 0,

(P + 1)(V + 1) = 1.

(4.2.12)

This curve goes through the origin P = 0, V = 0, and its asymptotes are the lines P = −1 and V = −1. Using the speed of sound in the initial gas, along with the conservation equations for mass, (4.2.1), the Michelson–Rayleigh line (4.2.7) yields a linear relation between P and V,

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217

P = −Mu2 V,

(4.2.13)

where Mu ≡ D/au is the Mach number of the shock. The two relations (4.2.12) and (4.2.13) give a quadratic equation for V. One root is zero. The other, VN = −(Mu2 −1)/Mu2 , corresponds to the Neumann state and permits the calculation of the state of the gas just behind the shock as a function of the thermodynamic state of the fresh gas (pu , ρu ) and the Mach number of propagation Mu , yielding uN (γ − 1)Mu2 + 2 ρu = , = D ρN (γ + 1)Mu2

(4.2.14)

pN 2γ Mu2 − (γ − 1) , = pu (γ + 1)    2γ Mu2 − (γ − 1) (γ − 1)Mu2 + 2 TN = , Tu (γ + 1)2 Mu2 MN2 =

(4.2.15) (4.2.16)

(γ − 1)Mu2 + 2 . 2γ Mu2 − (γ − 1)

(4.2.17)

These are known as the Rankine–Hugoniot relations for a polytropic gas. The last expression is a symmetrical relation between Mu2 and MN2 , 2γ Mu2 MN2 − (γ − 1)(Mu2 + MN2 ) − 2 = 0,

Mu2 =

(γ − 1)MN2 + 2 2γ MN2 − (γ − 1)

.

(4.2.18)

General Comments (Arbitrary Fluid) The Hugoniot curve is tangent to the isentropic at the point (1/ρu , pu ). To see this it is convenient to introduce the specific volume v ≡ 1/ρ and use Gibbs’ relation Tds = deT + pdv in the form dh = Tds + v dp, where s is the entropy and eT the internal energy. A Taylor expansion of h(s, p) around (su , vu ) limited to first order in δs = s − su , but extended to third order in δp = p − pu , yields     1 ∂v 1 ∂ 2v 2 h = Tu δs + vu δp + (δp) + (δp)3 + · · · . (4.2.19) 2 ∂p s 6 ∂p2 s This can be compared with an expansion of the enthalpy h along the Hugoniot relation (4.2.6) in powers of δp,       ∂v ∂v 1 ∂ 2v 1 δs + δp + (δp)2 + · · · , h = (2vu + v )δp, where v = 2 ∂s p ∂p s 2 ∂p2 s which can be written to first order in δs and third order in δp:       1 ∂v 1 ∂ 2v 1 ∂v h = vu δp + (δp)2 + (δp)3 + · · · . (∂s∂p) + 2 ∂s p 2 ∂p s 4 ∂p2 s

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(4.2.20)

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Comparison of (4.2.19) and (4.2.20) shows that the entropy change along a Hugoniot curve is of third order,   1 1 ∂ 2v δs = (δp)3 . (4.2.21) 12 Tu ∂p2 s The Hugoniot relation (4.2.6) does not correspond to an isofunction of state since it cannot be put in the form H (1/ρ, p) = H (1/ρu , pu ). If we choose a point 1/ρu = 1/ρu , pu = pu , on the Hugoniot curve (4.2.6),   1 1 1   h(ρu , pu ) − h(ρu , pu ) + +  (pu − pu ) = 0, (4.2.22) 2 ρu ρu and consider the Hugoniot curve using this point as the initial condition,   1 1 1   (p − pu ) = 0, h(ρu , pu ) − h(ρ, p) + + 2 ρu ρ

(4.2.23)

the new curve is not the same as the original (4.2.6). The Hugoniot curve with the initial condition (1/ρN , pN ),   1 1 1 h(ρN , pN ) − h(ρ, p) + + (4.2.24) (p − pN ) = 0, 2 ρN ρ still goes through the point (1/ρu , pu ), because of the symmetry of N and U, but it is different from (4.2.6); see Fig. 4.2. According to the second law of thermodynamics, compressibility is a positive quantity, with pressure. The coef−(∂ v /∂p)s > 0. In addition, compressibility usually decreases

ficient in Equation (4.2.21) is then positive, ∂ 2 v /∂p2 s > 0. So, according to (4.2.21), entropy changes in the same direction as pressure on the Hugoniot curve and its first and second derivatives are zero. The Hugoniot curve (4.2.6) is tangent to the isentropic at the point (1/ρu , pu ) where the slope of the Hugoniot curve is thus related to the sound speed in the unshocked mixture, (δp/δ v )u = −ρu2 (δp/δρ)u = −ρu2 a2u ,

√ where a = (∂p/∂ρ)s . This is valid only at the initial state, and, for example the speed of sound at the point N in Fig. 4.2 is given not by the slope of the solid curve but by that of the dotted curve. The conservation equations do not indicate any preferential direction of propagation. However, there are no steady rarefaction waves that propagate at constant velocity into a compressed medium, represented by N in Fig. 4.2, transforming the matter into a less dense state represented by U. The only steady waves are compression waves, called shock waves, while rarefaction waves are unsteady processes, described in Section 15.3.4; see Fig. 15.11. In gases, this is proved by the steady solution to Boltzmann’s equation showing that, in a steady wave, the entropy of the final state should be larger than in the initial state; see Section 4.6.1.

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Because of the concavity of the Hugoniot curves, the absolute value of the slope of the Michelson–Rayleigh line (4.2.3), m2 , is greater than the tangent to the Hugoniot curve (4.2.6) at the initial point (1/ρu , pu ); see Fig. 4.2 (solid curve). Anticipating that the pressure and density of the shocked fluid is larger than the initial fluid (a ‘rarefaction shock’ cannot exist, as we shall see Section 4.2.2), the propagation speed of the shock wave is thus supersonic, ρu2 D 2 > ρu2 a2u . However, the absolute value of the slope of the Michelson– Rayleigh line is smaller than the slope of the tangent to the Hugoniot curve (4.2.24) at the final state (1/ρN , pN ) (dotted line in Fig. 4.2), ρN2 u2N < ρN2 a2N , so the flow leaving the shock is subsonic. In summary, the propagation speed of the shock wave is supersonic and the flow of shocked fluid is subsonic relative to the shock wave, D > au ,

uN < aN ,

with aN > au since pN > pu and hence TN > Tu . 4.2.2 Inner Structure of Weak Shock The inner structure of shock waves is controlled by dissipative transport. Under ordinary conditions, the thickness of a gaseous shock is of the order of a few mean free paths, d/ = O(1); see Section 15.1.7. Its inner structure is fully out of equilibrium and must be studied using Boltzmann’s equation, recalled in Section 13.3.1. However, the thickness increases and the jumps through the shock decrease as the supersonic shock velocity decreases. In the limit of a propagation Mach number approaching unity, the ratio d/ diverges. The precise form of the divergence is an outcome of the analysis presented below; see (4.2.40). Anticipating that, for a very weak shock propagating at a velocity slightly larger than (but close to) the sound speed, the thickness becomes macroscopic, d/  1, the approximation of local equilibrium becomes valid and the fluid mechanic equations can be used to describe the detonation structure. The problem is reduced to solving a onedimensional steady transonic flow between the initial state and the shocked gas; see Section 13.3.2, The problem was first solved by Rankine[1] when the viscous effects are neglected, keeping only heat conduction. Rankine did not know the conservation of total energy that was introduced later by Hugoniot.[2] The temperature gradient being zero on both sides of the wave, he wrote, ‘The integral amount of heat received must be nothing’, and he used the balance of entropy, called ‘thermodynamic function’, in the form of the first entropy equation in (13.3.30); see also (15.1.61), for μ = 0,  s+∞ Tds = 0, (4.2.25) μ = 0: mTds = d(λdT/dx), x = ±∞: dT/dx = 0 ⇒ s−∞

where the initial mixture is at x = −∞: s−∞ = su and the shocked gas is at x = +∞: s+∞ = sN . These relations are valid only for weak shocks for which, as we will see below in (4.2.37), the total jump of entropy sN − su is small. For a given mass flux m = ρu D, [1] [2]

Rankine W., 1870, Philos. Trans. R. Soc. London, 160, 277–288. ´ Hugoniot P., 1889, Journal de l’Ecole Polytechnique, 58(1), 1–125.

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Rankine then integrated the first equation in (4.2.25) using the thermodynamic function (13.2.7), s(ρ, T) where ρ = nm, the ideal gas law, p/ρ = (cp − cv )T, the conservation of mass ρu = m > 0 and momentum p + ρu2 =cst. He obtained jump relations that do not depend on heat conductivity, the latter controlling only the thickness of the shock, d. This result illustrates that the jump conditions result from conserved quantities, in particular the total energy (internal + kinetic energy of the centre of mass of a ‘macroscopic’ fluid particle), as Hugoniot realised almost 20 years later, ignoring the work of Rankine. The advantage of Hugoniot’s method to compute the jumps through the shock is to be easily extended to any strength of the shock in any material. Formulation Including Viscosity When the viscous dissipation is taken into account,[1] a quadratic term appears in the entropy equation (15.1.61). It is then more convenient to use the conservation of energy, instead of the entropy equation. In the reference frame moving with the planar wave, equations (15.1.49)–(15.1.50) for ρ, p, u and T yield du = cst., ρu = m, p + ρu2 − μ dx   u2 dT du m h+ −λ − μu = cst., 2 dx dx

(4.2.26) (4.2.27)

where, in an ideal gas, T and h are proportional to p/ρ; see (4.2.8). The choice m > 0 (u > 0) corresponds to an axis oriented toward the shocked gas, namely to a wave propagating into the initial medium at x = −∞. Using the same trick as for deriving Hugoniot’s relations (4.2.7)–(4.2.9), Equations (4.2.26)–(4.2.27) yield two coupled Equations for p and v ≡ 1/ρ: (p − pu ) + m2 (v − vu ) = μm

dv , dx

(4.2.28)

λ d(pv ) μm γ 1 γ dv (pv − pu vu ) − (p − pu )(v + vu ) = + (v − vu ) , γ −1 2 γ − 1 mcp dx 2 dx (4.2.29) that are extensions of the Rankine–Hugoniot relations, including the dissipative transports. According to the discussion in Section 13.3.2, these equations are valid for states in local equilibrium. As we shall see at the end of the calculation, this is the case for weak shocks,  ≡ Mu − 1 1,

m = ρu au (1 + ).

(4.2.30)

In the limit  → 0, the variations of v and p across the shock are small; the nondimensional quantities ν ≡ (v − vu )/vu , π ≡ (p − pu )/pu are both of order . Introducing the √ nondimensional length ξ ≡ x/, where  ≡ DTu /au , DTu ≡ λ/(ρu cp ) and au = γ pu /ρu , are the mean free path, the thermal diffusivity and the speed of sound, respectively, the

[1]

Rayleigh J., 1910, Proc. R. Soc. London, 84, 247–284.

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terms dν/dξ and dπ /dξ are expected to be smaller than ν and π if the thickness of the shock wave d is anticipated to be larger than the mean free path  (local equilibrium):  1:

d = O(), ν = O(), π = O(), 

dν = O( 2 ), dξ

dπ = O( 2 ). (4.2.31) dξ

Using an expansion in powers of , limited to the second order, (4.2.28)–(4.2.29) reduce to 1 dν π + (1 + 2)ν = Pr , γ dξ   γ +1 1 dπ dν πν + π + ν = + , 2γ γ dξ dξ

(4.2.32) (4.2.33)

valid up to  2 , where Pr ≡ μ/(ρu DTu ) is the Prandtl number. When the dissipative terms in the right-hand side are neglected, the Rankine–Hugoniot jumps are recovered, π ≈ γ4γ +1 ,

4 ν ≈ − γ +1 , in agreement with (4.2.14)–(4.2.15) in the limit  → 0, Mu2 ≈ 1 + 2. According to (4.2.32), the quantity ν + π/γ is of second order, ν + π/γ = O( 2 ), so that the relations dπ /dξ ≈ −γ dν/dξ and π ν ≈ −γ ν 2 can be used in (4.2.33) valid up to order  2 . When the expression for ν + π/γ , obtained from (4.2.32), is introduced into (4.2.33) a first-order differential equation is obtained for ν,   γ +1 dν = ν + 2 ν, (4.2.34) [(γ − 1) + Pr] dξ 2

where the factor [(γ − 1) + Pr] appears also in the attenuation for acoustic propagation.[2] Thickness of Weak Shock. Irreversibility When the initial and final state, ν = ν−∞ and ν = ν+∞ at ξ = −∞ and ξ = +∞, respectively, are introduced into (4.2.34), the first-order differential equation describing the decrease of the reduced specific volume ν from the initial state to the final state takes the form dν 2 [(γ − 1) + Pr] = (ν − ν−∞ )(ν − ν+∞ )  0 (4.2.35) γ +1 dξ 4 , (4.2.36) ξ = +∞: ν = ν+∞ ≡ − ξ = −∞: ν = ν−∞ ≡ 0, γ +1 where the coefficient [(γ − 1) + Pr] is positive so that dν/dξ  0. The solution of (4.2.35)– (4.2.36) exists only for a supersonic wave (Mu > 1,  > 0, ν+∞ < ν−∞ = 0). A planar rarefaction wave, ν+∞ > ν−∞ = 0, propagating at constant subsonic velocity (Mu − 1 < 1,  < 0) with a steady inner structure, cannot exist because, according to (4.2.35), ν should decrease, dν/dξ  0. Rarefaction waves are necessarily unsteady; see the self-similar solution of

centred waves in Fig. 15.11. According to (4.2.21), using π+∞ > π−∞ and ∂ 2 v /∂p2 s > 0 in an ideal gas, the total increase of entropy through a weak shock is a small positive quantity of order  3 . However, according to (4.2.35), [2]

Morse P., Ingard K., 1986, Theoretical acoustics. Princeton University Press.

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the entropy first increases inside the shock, reaches a maximum of order  2 , and then decreases. This is easily verified using the thermodynamic expression (13.2.8) for the entropy of an ideal gas (s − su )/cv = ln [(1 + π )(1 + ν)γ ]. Expanded up to the second order, (s − su )/cv = π + γ ν − (π 2 + γ ν 2 )/2γ + O( 3 ), using (4.2.31) and, according to (4.2.32) π ≈ −γ ν + O( 2 ), the variation of entropy (s−su )/(γ cv ) is given by the left-hand side of (4.2.33), while the right-hand side yields −(γ − 1)dν/dξ , (s − su ) dν = −(γ − 1) + O( 3 ), cp dξ

(4.2.37)

where ν = O() and, according to (4.2.35), dν/dξ = O( 2 ). The fact that the entropy goes through a maximum is shown by noticing that the function ν(ξ ) has an inflexion point, because ν−∞ and ν+∞ are bounded and dν/dξ  0. According to (4.2.37) the entropy inside the shock cannot be smaller than in the initial state, −(γ − 1)dν/dξ  0, s  s−∞ . However, the total entropy jump is of the following order since dν/dξ = 0 on both sides of 3 shows the shock wave, (sN − su )/cp = O( 3 ). The calculation pushed up to order

that the jump is positive, sN > su , in agreement with (4.2.21) for pN > pu and ∂ 2 v /∂p2 s > 0 in an ideal gas. This is also in agreement with the entropy balance in (15.1.65), obtained under the approximation of local equilibrium, and also, as shown in Section 4.6.1, with Boltzmann’s equation valid also for strong shocks whose structure is fully out of equilibrium. Introducing variables of order unity y ≡ ν/ and x ≡ ξ [(γ + 1)/2] / [(γ − 1) + Pr], where ξ = x/(/), Equation (4.2.34) takes the nondimensional form with no parameters, γ excepted,   4 dy = y+ y, (4.2.38) dx γ +1 where the roots of the right-hand side correspond to the initial gas (ξ = −∞: ν = 0) and 4 ). Choosing the origin to be where y = −2/(γ +1), the shocked gas (ξ = +∞: ν = − γ +1 4

y=−

x

e γ +1 4  4 , (γ + 1) e γ +1 x + 1

the solution shows that the thickness of a weak shock is (Mu − 1 1):

1  d = [(γ − 1) + Pr] , 2 (Mu − 1)

d =O 

(4.2.39)



1 Mu − 1

 (4.2.40)

and is effectively much larger than the mean free path  for Mu → 1. 4.2.3 ZND Structure of Detonations When a shock wave propagates in a reactive flow, the increase in temperature of the shocked gas can be sufficient to initiate an exothermic reaction behind the shock. Under certain

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Reference frame of laboratory Detonation wave Reactive mixture at rest

Piston

Burnt gas at piston speed

Reference frame of shock wave

Figure 4.3 Flow through a planar detonation driven by a piston, presented in the laboratory frame (upper) and in the reference frame of the shock (lower).

conditions, this reaction zone may remain ‘attached’ to the shock wave to form a detonation, with a Zeldovich, von Neumann and D¨oring (ZND) structure, sketched in Fig. 1.9 and computed with a detailed chemical kinetic of combustion in Fig. 4.29a. State of the Burnt Gas. Family of Solutions Overdriven detonations are planar waves driven by a piston; see Fig. 4.3. The jump conditions are obtained in a manner similar to (4.2.5) for shock waves, but including chemical energy in the energy conservation equation; see (15.1.48), in which u is denoted U. We will use the enthalpy of a perfect gas with constant specific heats for simplicity. The thermodynamic conditions at the end of chemical reaction thus satisfy cp (T − Tu ) + (u2 − D 2 )/2 = qm ,     1 p pu γ 1 1 − (p − pu ) − = qm , + (4.2.41) γ −1 ρ ρu 2 ρu ρ where qm is the chemical energy released per unit mass. After multiplication by [(γ 2 − 1)/ 2γ ](ρu /pu ) this equation can be rewritten in the notations of (4.2.11)–(4.2.12), PV + P + V = Q,

where by definition,

Q≡

γ + 1 qm . 2 cp Tu

(4.2.42)

In the P–V plane, (4.2.42) is an equilateral hyperbola, (P + 1)(V + 1) = 1 + Q,

(4.2.43)

that lies above the Hugoniot curve (4.2.12), since Q > 0, but has the same asymptotes. The equation for momentum conservation is identical to the nonreactive case and is represented by the Michelson–Rayleigh line (4.2.13), where Mu = D/au is the propagation Mach number of the detonation. The final state of the burnt gas, (1/ρb , pb ), is given by the intersection of the Michelson–Rayleigh line and the hyperbola (4.2.41), as sketched in Fig. 4.4.

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Hugoniot after heat release Michelson– Rayleigh line CJ Michelson– Rayleigh line

Hugoniot of shock 0

Figure 4.4 Sketch of a detonation in the (v , p) plane (v ≡ 1/ρ). The curve (4.2.41) is shown in bold. The point U is the initial state, B is the state of the burnt gas and N is the Neumann state.

For a given Mach number Mu , the quadratic equation for P, obtained by substituting (4.2.13) in (4.2.43), has two real roots if the discriminant is positive: (Mu2 − 1)2 − 4QMu2 > 0. Real roots exist only for a propagation Mach number greater than a threshold value MuCJ , called the Chapman–Jouguet Mach number in the literature, but which was first mentioned by Michelson in 1893,[1] √ √ Mu  MuCJ ≡ Q + Q + 1, (4.2.44) where the + sign is imposed by the condition MuCJ > 0, MuCJ = DCJ /au . There is no solution for a steady planar detonation propagating at a constant velocity less than DCJ (the Michelson–Rayleigh line does not intersect the Hugoniot after heat release). The corresponding minimum velocity of the driving piston, vpCJ , is given by mass conservation through the detonation, ρb (D − vp ) = ρu D. The tangency of the CJ Michelson–Rayleigh line at the point BCJ indicates that the exhaust gas flow is sonic in the reference frame of the CJ detonation but subsonic for all other regimes Mu > MuCJ (see Fig. 4.5 for more details). As explained below, the minimum speed DCJ is also the propagation speed of an autonomous detonation, that is, a detonation that propagates without the support of a driving piston. In the limit Q  1 the expression for the CJ velocity in (4.2.44) reduces to (1.2.9). ZND Structure (1940) For a constant Mach number of propagation Mu > MuCJ , the state of the burnt gas (1/ρb , pb ) of a steady planar detonation corresponds to the point B in Fig. 4.4, where the point U represents the initial state. The other root, represented by the point B , is not [1]

Shchelkin K., Troshin Y., 1965, Gasdynamics of combustion. Baltimore, Md.: Mono Book Corp.

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physical, as explained few lines below; see Fig. 4.5. The transition from the initial state to the state of the burnt gas occurs in two steps. In the first step, the leading shock wave takes the frozen mixture from point U to point N on the Hugoniot curve plotted in Fig. 4.4, and the exothermic reaction takes the gas from point N to point B in a second step. This results from the difference of time scales between elastic collisions controlling the inner structure of inert shock waves and the inelastic collisions controlling the heat-release rate (large activation energy). Since a gas molecule undergoes only a small number of elastic collisions as it crosses the shock wave, the exothermic reaction cannot develop until later downstream. To describe the internal structure of a detonation, it is useful to introduce the progress variable ψ of the exothermic reaction, ψ = 0 in the initial state and ψ = 1 in burnt gas; see Equation (15.1.42). For a one-step Arrhenius reaction ψ = θ = (1 − ψ), where ψ is the reduced mass fraction of reactant. The reaction is supposed irreversible to simplify the presentation. Extension to more general cases is possible.[2] The speed of shocked gas with respect to the shock is subsonic u < a, except at the CJ point where it is sonic (ubCJ = abCJ ), but it is of the same order of magnitude as the local speed of sound, u/a = O(1), typically u/a = 0.2. The dimensional analysis in (1.2.5) then shows that the flow speed is high compared with diffusion speeds when the activation energy is sufficiently large,   (4.2.45) E/kB T > 1 ⇒ u  D/τr ≈ a e−E/kB T , where 1/τr is the reaction rate and D is a diffusion coefficient, given in (1.2.2) and (1.2.4), respectively. In other words, downstream of the shock, diffusive transport (heat conduction, molecular diffusion, viscosity) is negligible compared with convective transport, D/(uτr )2 u/(uτr ), where uτr is the length of variation associated with the reaction rate, D ≈ a2 τcoll , with τcoll the time between molecular collisions, u ≈ a and τcoll τr . Under these conditions, the conservation equations (15.1.33)–(15.1.34) controlling the internal structure of the detonation, written in the coordinate system of the steady plane wave, are d(ρu) = 0, dx du dp + ρu = 0, dx dx

γ d γ − 1 dx

  p du dψ + u − qm = 0, ρ dx dx

(4.2.46) (4.2.47) (4.2.48)

dψ ρ = , (4.2.49) dx τr where the ideal gas law, cp T = [γ /(γ − 1)]p/ρ, the progress variable ψ and the rate of heat release 1/τr have been introduced. The latter is a chemical kinetic quantity that depends on the composition and on the thermodynamic conditions (ρ, T). The quantity qm ψ(x) is ρu

[2]

Fickett W., Davis W., 1979, Detonation. University of California Press.

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the heat released in the slab of compressed gas delimited by x = 0 (Neumann state, just downstream of the inert shock wave, ψ = 0) and x. The first three equations are written in a conservative form that can be integrated directly from the initial condition ψ = 0 to any intermediate state ψ = 0, ψ  1. This allows us to express the intermediate thermodynamic states as functions of ψ. In the notation of (4.2.42), we obtain the relation PV + P + V = ψQ

(4.2.50)

in which Equation (4.2.13) must be used. This equation is a generalisation of (4.2.42); it describes hyperbolas that lie between those of the Hugoniot and the burnt gases. A relation involving the speed of sound a2 = γ p/ρ, equivalent to (4.2.50), is obtained by integration of Equation (4.2.48) from the initial state: 1 1 1 1 a2 + u2 − qm ψ = a2u + D 2 . γ −1 2 γ −1 2

(4.2.51)

For ψ = 0 this equation for the conservation of total energy across the inert shock wave has two solutions, represented by two points in Fig. 4.4, U corresponding to the initial state (fresh gas), ψ = 0:

u2 = D 2 , a2 = a2u ,

and N corresponding to the Neumann state just behind the inert leading shock, ψ = 0:

u2 = u2N , a2 = a2N .

For a given initial condition (1/ρu , pu ), the internal structure of the wave and the propagation velocity Mu are then fully determined by the function ψ(x), which can be determined by solving the chemical kinetic problem, that is, by integration of Equation (4.2.49). Instructive information can be obtained without considering chemical kinetics by studying the solutions of the system (4.2.46)–(4.2.48) in the phase space u2 –ψ. Using ρu = cst., Equation (4.2.47) and a2 = γ p/ρ, the first term on the left-hand side of (4.2.48) can be written as   p du 1 dp 1 a2 du γ du γ + = − u , γ − 1 ρu dx ρ dx γ − 1 u dx γ − 1 dx and the equations describing the inner structure of a detonation take the form (a2 − u2 )

du dψ = (γ − 1)qm u , dx dx

du2 u2 = 2(γ − 1)qm 2 , dψ (a − u2 )

(4.2.52)

where the expression for a2 , as a function of u2 and ψ for a given value of D, is obtained from (4.2.51). The solutions of Equation (4.2.52) are represented by trajectories in the phase space u2 –ψ that pass through the two points U and N. The phase portraits of these solutions are shown schematically in Fig. 4.5. The solid lines represent trajectories of the physical solutions for D  DCJ . They describe the passage of shocked gas from the Neumann condition (just downstream of the inert shock wave, ψ = 0) to that of the burnt gas (ψ = 1). The flow velocity, u, in the frame of the detonation is an increasing function

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(Initial state)

Supe

rson

ic

c

soni

Sub

(Burnt gas)

(Neumann state)

Figure 4.5 Phase portraits of the solutions to Equation (4.2.52).

of ψ; since the flow is subsonic, u < a. For D = DCJ , the slope is vertical at ψ = 1 (the point BCJ ); the flow velocity in the burnt gas is sonic, u = ubCJ = abCJ . The part of the trajectories shown as dashed lines, connecting the initial state (ahead of the lead shock wave, point U, ψ = 0) to the burnt gas (point B , ψ = 1), describes a supersonic flow (u > a) where the flow velocity decreases as energy is released. This solution cannot describe a self-propagating wave, because the chemical reaction is frozen at the initial conditions (zero reaction rate, τr = ∞), and, in the absence of molecular transport (heat diffusion and molecular diffusion) ψ cannot increase. Moreover, the wave velocity selected by heat diffusion is very subsonic; see (1.2.5). In addition, the structure of a supersonic wave attached to a flame holder at room temperature Tu ≈ 300 K would necessarily have a very large thickness ≈ Dτr (Tu ), much larger than any terrestrial macroscopic length. For example this length would be of the order of 3 × 1021 m (or 3 × 105 light-years!) for D ≈ a ≈ 5 × 102 m/s and τr (Tu ) having the order of magnitude discussed at the end of Section 1.2.2 for an Arrhenius law. There is therefore no physical solution going from U to B in Figs. 4.4 and 4.5. For D < DCJ , there is no stationary solution that goes from the state of the initial fresh gas ψ = 0 to the state of the burnt gas ψ = 1, even though the reaction rate is nonzero. In summary, overdriven planar detonations can be created by a piston whose velocity

vp is greater than vpCJ = 1 − ρu /ρbCJ DCJ , vp > vpCJ . The velocity of the detonation decreases with that of the piston and has a lower bound DCJ given by Equation (4.2.44). For piston speeds less than vpCJ , supersonic combustion waves with a uniform flow between the end of the exothermic reaction and the piston, and propagating at constant speed, do not exist. Detonation waves consist of an inert shock wave followed by a reactive subsonic flow (in the frame of the wave). Under the effect of energy release, the density and pressure decrease from (ρN , pN ) to (ρb , pb ), while remaining always greater than their initial values, ρb > ρu pb > pN ; see Fig. 4.4. In the reference frame of the motionless fresh gas, the

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burnt gas velocity is nonzero and oriented in the direction of propagation, vp = D − ub = D(ρb −ρu )/ρb > 0; see Fig. 4.3. The marginal case, D = DCJ , is characterised by a speed of burnt gas at the end of heat release that is equal to the sound speed, ubCJ = abCJ . Profiles of Temperature, Velocity and Pressure The detonation structure is found by expressing the flow velocity and the thermodynamic variables in terms of the progress variable ψ(x), which increases from ψ = 0 at the Neumann state to ψ = 1 in the burnt gas. Consider a detonation propagating at a given velocity D; Mu ≡ D/au  MuCJ is fixed. Using the notations of (4.2.11), the expression for the specific volume downstream from the shock is given in terms of ψ by the solution of the quadratic equation for V obtained when (4.2.13) is introduced into (4.2.50). The root that corresponds to the Neumann state VN = −(Mu2 − 1)/Mu2 at ψ = 0 is an increasing function of ψ ∈ [0, 1],    (Mu2 − 1) 4Mu2 Qψ , (4.2.53) 1− 1− V − VN = 2Mu2 (Mu2 − 1)2 where the square root is real for Mu  MuCJ , [4Mu2CJ /(Mu2CJ − 1)2 ]Q = 1. The density and the pressure, obtained from (4.2.53) with (4.2.13), decrease continuously from the Neumann state to the burnt gas. The temperature is then obtained using the ideal gas law in the following form, written with the notation   

2 Mu2 4Mu2 (Mu2 −1) (M −1) T−TN , X ≡ (M 2 −1) , B ≡ (γ − 1) (γu+1) + γ1 , Y ≡ (γ γ+1) (M 2 −1) 2 Qψ, A ≡ T γ +1 u u u √ Y + B = AX + B 1 − X, X ∈ [0, Xb  1], (4.2.54) where for ordinary detonations B < 2A, (γ − 1)/γ < [(Mu2 − 1)/(γ + 1)](3 − γ ). Equation (4.2.54) then shows that the temperature first increases when ψ increases from 0, √ reaches a maximum at X = Xm , 1 − Xm = B/(2A) and decreases down to the burnt gas temperature[1] at the end of the exothermic reaction, ψ = 1. In the last part of the reaction zone the temperature decreases because the rate of heat release is smaller than the rate of adiabatic cooling due to gas expansion in the thermal equation (15.1.34)) where Dp/Dt < 0. The burnt gas of a CJ wave corresponds to X = 1 in (4.2.54), VbCJ − VNCJ = (Mu2CJ − 1)/2Mu2CJ , and the total increase of temperature in the shocked gas is Mu2CJ  1:

(TbCJ − TNCJ ) (2 − γ )γ 2 ≈ M , Tu (γ + 1)2 uCJ

(4.2.55)

yielding (TbCJ − TNCJ )/Tu ≈ 0.17 Mu2 for Mu  1 and γ = 1.3. In ordinary detonations, the heat release is larger than the enthalpy of the initial gaseous mixture, about 10 times larger, so according to (4.2.44), the square of the propagation Mach number is also large: 6  qm . (4.2.56) qm  cp Tu ⇒ DCJ ≈ 2(γ 2 − 1)qm , MuCJ ≈ 2(γ + 1) cp Tu [1]

Zeldovich Y., Kompaneets A., 1960, Theory of detonation. Academic Press.

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Speed of discontinuities in laboratory frame

Shock

Static piston

Detonation

Detonation

Moving piston

Rarefaction wave

Gas speed in laboratory frame

Shock

Figure 4.6 Instantaneous arrest of a piston supporting an overdriven detonation, ub < ab ; see Fig. 4.3. The piston, initially at speed vp > vpCJ , is instantaneously brought to rest at a time just after t = to . The plots show the profiles of flow velocity in the laboratory frame (where the fresh gas is at rest) at times to and t > to . The propagation speeds of the discontinuities are shown above the plots. The sound speed close to the piston is given by (15.3.47), ao = ab − 0.5(γ − 1)(D − ub ).

Neglecting the quantity (γ − 1) < 1 in front of 2γ Mu2 , the Neumann conditions (4.2.15)– (4.2.17) yield

 (γ − 1) TN 2γ pN 2γ 1 2 (γ − 1)M γ MN2 ≈ ≈ + 2 , ≈ + 2, M2, u 2 2 Tu pu γ +1 u Mu (γ + 1) (4.2.57) showing that the flow Mach number at the Neumann state is relatively small but the pressure jump across the lead shock is large, scaling as the square of the propagation Mach number. For typical gaseous mixtures, qm /cp Tu = 8, γ = 1.3, the Mach numbers are, according to (4.2.56)–(4.2.57), MuCJ = 6.22 and γ MN2 CJ = 0.176 and the temperature jump across the lead shock is relatively large, TNCJ /Tu ≈ 6.66. According to (4.2.55), the increase of temperature in the shocked gas is not large, TbCJ /TNCJ ≈ 2, so that the most important part of the total temperature jump in a CJ detonation is across the leading shock wave. The increase of the flow velocity across the exothermic reaction zone is of order unity, ubCJ /uNCJ = 2.88, where ubCJ = abCJ and the pressure is divided approximately by two across the reaction zone, pbCJ /pNCJ ≈ 0.5. Dynamics of the Burnt Gas in a Planar Chapman–Jouguet (CJ) Wave The selection mechanism leading to the CJ regime of an autonomous detonation (no supporting piston) can be explained in planar geometry using the simple waves described in Section 15.3.4; see Fig. 15.11. When the piston supporting an overdriven detonation, initially at the speed vp , is suddenly stopped, a simple rarefaction wave leaves the piston, reducing the velocity of the burnt gas flow (measured in the laboratory frame) from vp to zero; see Fig. 4.6. The rarefaction wave propagates and widens linearly in time until it meets and weakens the detonation. The velocity of the leading shock then decreases until it reaches the limit DCJ . Starting at this instant, the burnt gas velocity in the frame of the CJ

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Gaseous Shocks and Detonations Speed of discontinuities in laboratory frame

Rarefaction wave

Static piston

Detonation

Gas speed in laboratory frame

Shock

Figure 4.7 Autonomous CJ detonation. The velocity profile in the detonation is identical to that of Fig. 4.6 just after the rarefaction wave has encountered the detonation.

wave becomes sonic, ubCJ = abCJ ; the rarefaction wave can no longer penetrate the internal structure of the detonation which then continues to propagate autonomously at the velocity DCJ ; see Fig. 4.7. After a sufficiently long time, the initial conditions for initiation of the detonation at the closed end of the tube are forgotten. The whole CJ detonation wave can be considered as a hydrodynamic discontinuity as soon as its thickness becomes negligible compared with the characteristic length of the burnt gas flow. This flow is described by a self-similar solution, obtained in 1941 by G. Taylor[1] but published later. The detonation front propagates at velocity DCJ , and the velocity of the burnt gas (in the laboratory frame) at the detonation front is also constant, vbCJ = DCJ − abCJ ,

(4.2.58)

since, according to the CJ wave, ubCJ = abCJ , where abCJ is the sound speed in the burnt gas at the detonation front. For a constant propagation velocity, the entropy is constant at the detonation front and is thus also constant in the entire flow behind the detonation front. This is the key simplification that is associated with a constant propagation velocity of the detonation front. For reasons similar to those in Sections 15.3.4 and 15.3.5, the flow of burnt gas is sought in the self-similar form of a function of space and time through the ratio x ≡ x/t, ρ(x), v(x), where v ≡ DCJ − u denotes the flow velocity in the laboratory frame (u is the flow velocity relative to the detonation front). The Euler equations take the form (v − x)

dv 1 dρ + = 0, ρ dx dx

a2

dv 1 dρ + (v − x) =0 ρ dx dx

(4.2.59)

(see (15.3.56)), where, with the isentropic condition, the sound speed is a function of the mass density, a(ρ). In an ideal gas with a constant ratio of specific heats (polytropic gas),

γ −1 p/ρ γ = cst., a(ρ)/abCJ = ρ/ρbCJ 2 , (4.2.60) [1]

Taylor G., 1950, Proc. R. Soc. London Ser. A, 200, 235–247.

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the existence of a nontrivial solution to the system (4.2.59) requires that the determinant be zero, a2 = (v − x)2 , so that v + a = x/t as for the simple waves corresponding to the linear characteristics C+ introduced in Section 15.3.4. Integrating the first equation in (4.2.59) yields  ρb

CJ a(ρ) 2 dρ = abCJ − a , (4.2.61) v + a = x/t, vbCJ − v = ρ γ −1 ρ where the last relation is obtained using (4.2.60). This shows that the flow velocity v and the sound speed a are linearly related. Combining the two equations in (4.2.61) leads to an expression for the flow field in the form     γ +1 γ −1 x v + abCJ − vbCJ = , (4.2.62) 2 2 t obtained using the boundary conditions at the detonation front given by the burnt gas state of the CJ wave, v = vbCJ = DCJ − abCJ and a = abCJ . The result in (4.2.62) shows the coherence of the self-similar solution since the point where v = vbCJ , which is the detonation front, propagates at the CJ velocity, x/t = abCJ + vbCJ ≡ DCJ . The solution corresponds to a planar self-similar solution, introduced in general terms in Section 15.3.5, with, in the notations of (15.3.61)–(15.3.63), ξ = x/(DCJ t), A = 1/DCJ , α = 1 and βN = 0. When the rear end of the tube (x = 0 where the detonation was ignited) is closed, the velocity of the flow at that point is zero. According to (4.2.62), the flow velocity decreases to zero, v = 0, at the point x = xo (t) such that xo /t = abCJ − vbCJ (γ − 1)/2. This point propagates at the local sound speed of the gas at rest, x/t = ao , since, according to the second equation in (4.2.61), ao = abCJ − vbCJ (γ − 1)/2. In the limit of a strong CJ detonation, the sound speed ao takes a very simple form ao ≈ DCJ /2, as shown now. In this limit the temperature and pressure of the fresh mixture are negligible in front of those in the burnt gas, TbCJ  Tu and pbCJ  pu . Using the relation a2b = γ pb /ρb and ρb ab = ρu DCJ , Equation (4.2.41) leads to a quadratic equation for X ≡ DCJ /abCJ , X 2 + (γ 2 − 1)X/2 −(γ + 1)2 /γ = 0, yielding qm  cp Tu

⇒

DCJ ρb (γ + 1) , = CJ ≈ abCJ ρu γ

DCJ ≈ (γ + 1). vbCJ

(4.2.63)

The gas is at rest in the region between the end of tube, x = 0, and the weak discontinuity at x = xo (t) = ao t, where, according (4.2.62)–(4.2.63), ao = DCJ /2 in the limit of strong CJ detonation; see Fig. 4.8. When the end of the tube at x = 0 is open, the solution in (4.2.61) extends to negative values of the flow velocity v, corresponding to burnt gas flowing out from the tube.

4.3 Initiation of Detonation As explained in Sections 5.2.1 and 5.2.2, chemical kinetics introduce a crossover temperature T ∗ , in the range 950–1400 K for normal conditions (see Sections 5.3 and 5.4) for

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0 0

1/2

1

Figure 4.8 Planar self-similar solution for the flow of burnt gas in the limit (4.2.63) for a strong CJ detonation in a polytropic gas.

the combustion of hydrogen and methane. The rate of the exothermic reaction changes drastically, by two to three orders of magnitude, in a small temperature range around T ∗ of a few percent T/T ∗ ∼ 10−1 to 10−2 . For example, the induction delay (the time delay before thermal runaway) varies from 2 × 10−4 s at 1000 K to 10−1 s at 900 K for a stoichiometric mixture of hydrogen–air at atmospheric pressure.[1] It is of the order of 102 s and 10−6 s at 750 K and 2000 K, respectively. A simple model of induction delay is presented in Section 5.2.2. 4.3.1 Detonability Limits A planar detonation cannot propagate unless the temperature of the shocked gas just behind the inert lead shock, TN , is sufficient to initiate the exothermic reaction downstream of the shock after a time (induction delay) sufficiently short for the region of heat release to be located not too far from the lead shock. If this is not the case, the slightest heat or radical loss in the transverse directions quenches the exothermic reaction. Roughly speaking, the limiting composition (dilution, equivalence ratio) that defines the detonability limits of a reactive mixture is given by the equality TNCJ = T ∗ , where TNCJ is the Neumann temperature of the CJ detonation. No detonation can propagate autonomously if TNCJ < T ∗ . The temperature TNCJ , like DCJ , is directly associated with the chemical energy released per unit mass qm and is thus fully controlled by the composition of the reactive mixture, while T ∗ depends on the detailed chemical kinetics. The detonability limits TNCJ = T ∗ concern both rich and lean mixtures. 4.3.2 Mechanisms of Detonation Initiation In a reactive mixture whose composition lies inside the detonability limits, the ways to initiate a self-propagating detonation have been classified into three categories: spontaneous initiation, direct initiation and the so-called deflagration to detonation transition (DDT).

[1]

Sanchez A., Williams F., 2014, Prog. Energy Combust. Sci., 41, 1–55.

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The two last mechanisms rely on the formation of an inert shock wave that becomes sufficiently strong to generate a pressure larger than, or at least equal to, pNCJ . Due to the large value of the ratio pNCJ /pu > 30 for ordinary combustible mixtures (see (4.2.57)), this condition makes DDT difficult to be observed in open space. In a channel DDT arises through flame acceleration and/or DDT is coupled to spontaneous ignition in hot spots ahead of the flame. Direct initiation refers to the formation of a detonation in open space in the decay of a strong blast wave produced by a powerful concentrated energy source. If sufficient energy is deposited (quasi-)instantaneously, E  Ec , a CJ wave is formed at a certain distance from the source. The challenge is to predict the critical initiation energy Ec . Spontaneous initiation can occur in a preconditioned medium through a gradient of induction time. Roughly speaking, a detonation is initiated when the velocity of the induction front is decreased to the velocity of pressure wave (sound speed) so that a runaway of the pressure occurs, leading quasi-spontaneously to a CJ wave. This mechanism is involved in some dangerous explosions and in the transition from deflagration to detonation. The intense pressure peak, observed in bad ignition of liquid rocket engines, could be related to the spontaneous initiation of a detonation, followed rapidly by detonation quenching. The fundamental studies of these initiation mechanisms are presented in the next sections. 4.3.3 Direct Initiation of Detonation After a successful initiation in open space, E  Ec , at large distance from the ignition region, much larger than the detonation thickness, a spherical detonation front is locally planar and propagates at the CJ velocity, DCJ . The self-similar solution[2] for a spherical detonation propagating at constant velocity DCJ is first presented below. The solution is obtained in a way similar to, but more complicated than, the planar solution in Fig. 4.8. In a second step we consider the solution for spherical detonations including the modifications to the inner structure due to the front curvature. Finally, an expression for the critical energy for direct initiation, Ec , is derived. Self-Similar Flow of Burnt Gas in a Spherical CJ Wave (1950) When the intrinsic instabilities of a detonation wave are disregarded, curvature of the front and internal unsteady phenomena can be neglected when the detonation radius is much larger than the thickness so that the internal structure of the wave is that of the planar wave. The detonation can then be considered as a spherical discontinuity of radius rf (t) = DCJ t, across which the Rankine–Hugoniot condition (4.2.43) is satisfied with, according to (4.2.13), P = −Mu2CJ V, where MuCJ is a constant quantity given by (4.2.44). The radial flow field is sought in the form of a self-similar form, v(x), ρ(x), where x ≡ r/t, r being the radial coordinate. The Euler equations become (v − x) [2]

dv 2v 1 dρ + + = 0, ρ dx dx x

a2

dv 1 dρ + (v − x) = 0, ρ dx dx

Taylor G., 1950, Proc. R. Soc. London Ser. A, 200, 235–247.

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(4.3.1)

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so that after elimination of (1/ρ)dρ/dx, x dv 2 = − v−x 2  > 0. v dx 1−

(4.3.2)

a

The first equation in (4.3.1) differs from the planar case (4.2.59) by the additional geometrical term 2v/x, and numerical solutions are required. Since the CJ velocity, DCJ , and the thermodynamic conditions in the burnt gas at the detonation front, pbCJ , ρbCJ , are constant and fully determined by the conditions of the initial mixture, pu , ρu , the entropy is constant in the burnt gas, ∂p/∂r = a2 (ρ)∂ρ/∂r, where a2 (ρ) is given in (4.2.60). Following the method in Section 15.3.5 for self-similar solutions of first kind, the solution is sought in the form ξ ≡ r/(DCJ t),

v = vbCJ U(ξ ),

ρ = ρbCJ R(ξ ),

with a/abCJ = R (γ −1)/2 . Using the strong detonation limit (4.2.63) for simplicity, DCJ /vbCJ = γ + 1, a/vbCJ = γ R (γ −1)/2 , Equations 4.3.1 take the form dU 2U 1 dR + + = 0, R dξ dξ ξ dU 1 dR γ 2 R γ −1 + [U − (γ + 1)ξ ] = 0, R dξ dξ [U − (γ + 1)ξ ]

(4.3.3) (4.3.4)

where the only parameter is γ . The solution is obtained by integrating this system of equations numerically, with the boundary conditions ξ = 1:

U = 1,

R = 1;

ξ = 0:

U = 0,

where the last relation comes from the spherical geometry. The numerical calculation starts from the front, ξ = 0, and is stopped at the point ξ = ξo , where U = 0, 0 < ξo < 1. According to (4.3.2), the derivative of the velocity with respect to the radius diverges at the detonation front since, according to (4.2.58), a2bCJ − (vbCJ − DCJ )2 = 0, and, according to (4.3.3)–(4.3.4), limξ →0 (1 − U)2 = [2γ /(γ + 2)] (1 − ξ ). The numerical solutions[1] are shown in Fig. 4.9 for γ = 1.25. The flow of burnt gas is bounded by two spheres, the detonation front rf (t) = DCJ t and an inner sphere of radius ro (t) = ao t, where the velocity is zero and ao denotes the sound speed in the burnt gas at rest inside this sphere. This can be seen as follows. Let xo ≡ ro /t = 0 be the point at which v becomes zero; one has limx→xo d ln v/dx = +∞ since limv→0 ln v = −∞, so that, according to (4.3.2), xo = ao . This point is a weak discontinuity where the velocity and its first derivative with respect to the radius are both zero, but the second derivative is discontinuous.[2]

[1] [2]

Li˜nan A., et al., 2012, C. R. M´ecanique, 340, 829–844. Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed.

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Figure 4.9 Self-similar solutions for the normalised temperature, density, pressure and gas velocity inside a spherical CJ detonation. From Li˜nan A. et al., 2012, C. R. M´ecanique, 340(11–12), 829–844 with permission. Copyright 2012, published by Elsevier Masson SAS. All rights reserved.

Zeldovich Criterion (1956) The direct initiation of gaseous detonation by an energy source, and the determination of the critical energy Ec are old problems that have been extensively studied and reviewed[3,4] since the pioneering work of Zeldovich et al..[5] Much experimental data for Ec is available since the 1980s, but the origin of the critical nature of the phenomenon was not understood and described analytically until a decade later.[6] Let us first summarise the pioneering analyses. Consider an igniter of negligible size that deposits an energy E in a negligibly short time. The 1941 self-similar solution for a point blast explosion,[7] recalled in Section 15.3.5, may be considered as the initial condition, since, at early times, the chemical heat release is negligible compared with E. Since the propagation of a spherical CJ wave is also described by a self-similar solution, the successful direct initiation of detonation may be then considered as the transition between the two self-similar solutions, at least when multidimensional and/or unsteady effects (pulsating or cellular detonations) are disregarded. The first numerical solutions[8] were obtained by considering the detonation as a discontinuity across which the planar jump conditions are satisfied. This approximation, often called ‘infinitely fast chemistry’, does not have a critical energy: the strongly overdriven detonation that is initially generated by the sudden deposition of heat at a point relaxes systematically to a CJ wave no matter how small the value of E, and the onset of the spherical CJ detonation occurs at a well-defined radius of the order of (E/ρu qm )1/3 . A more detailed analysis of this problem has been recently performed.[9] These studies show that the critical [3] [4] [5] [6] [7] [8] [9]

Lee J., 1977, Ann. Rev. Phys. Chem., 28, 75–104. Lee J., Higgins A., 1999, Proc. R. Soc. London Ser. A, 357(3503-3521). Zeldovich Y., et al., 1956, Sov. Phys.–Tech. Phys., 1, 1689–1713. He L., Clavin P., 1994, J. Fluid Mech., 277, 227–248. Taylor G., 1950, Proc. R. Soc. London Ser. A, 201(1065), 159–174. Korobeinikov P., 1971, Ann. Rev. Fluid Mech., 3, 317–346. Li˜nan A., et al., 2012, In Vazquez-Cendon E., et al., eds., Numerical methods of hyperbolic equations, vol. 61-74, Taylor and Francis.

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energy Ec is associated with the finite thickness of the inner structure of the detonation wave. A first criterion was proposed by Zeldovich et al.[1] in the following way. The time taken ∗ ≈ (E/ρ )1/3 (1/D )5/3 (see for the blast wave velocity to decrease to the CJ velocity, τCJ u CJ (15.3.67)), must not be shorter than the induction delay for ignition of the mixture at the Neumann state of the CJ wave. The induction delay is of same order of magnitude as the transit time of a fluid particle across the wave, τCJ ≈ dCJ /uNCJ , where dCJ ∗ ≈ τ is the thickness of the planar CJ detonation. The condition τCJ CJ yields Ec ≈ 5 3 3 2 3 ρu DCJ τCJ ≈ ρu (DCJ /uNCJ ) DCJ dCJ for the critical energy. Equations (4.2.44) and 2 ≈ 2(γ 2 − 1)q , DCJ ≈ (γ +1) , then (4.2.14) in the limit of a strong CJ detonation, DCJ m uN (γ −1) CJ give Ec ≈ 2ρu qm

(γ + 1)4 3 d (γ − 1)2 CJ

(4.3.5)

for the critical energy obtained by Zeldovich et al..[1] The radius at which the CJ wave is formed is of the order of the thickness of the CJ wave, dCJ . The expression for Ec in (4.3.5) is related to the chemical energy in a sphere of fresh mixture of radius of same order as dCJ . This critical value of energy is smaller than the experimental data[2] for Ec by a factor 10−5 to 10−6 . This suggests that the modifications to the planar structure of detonations play an essential role. There are two possibilities: either unsteady effects inside the detonation structure or curvature effects. What is surprising at first sight is that such small effects may have such a large influence. As we shall see, this is due to the high sensitivity of the reaction rate to temperature. A correct order of magnitude for Ec was obtained[3] in the limit of a high sensitivity of the reaction rate to temperature by taking account of nonlinear curvature effects in the inner structure of the detonation wave. This analysis is presented next. Inner Structure of Detonations in Spherical Geometry (1994) Neglecting viscosity and heat conduction, Euler’s equations (conservation of mass, momentum and entropy) in spherical geometry take the form (15.3.58)–(15.3.59). To study the inner structure of the detonation wave it is more convenient to use the reference frame of the lead shock. Denoting by rf (t) and D ≡ drf /dt the radius and the velocity of the lead shock, we introduce the flow velocity in the reference frame of the shock, u = D(t) − v, and the distance from the shock, x = rf (t) − r, where v and r denote the flow velocity and the radius in the laboratory frame, x > 0 and u > 0 in the shocked gas. Using the change of variables, r → x, ∂/∂r → −∂/∂x, ∂/∂t → ∂/∂t + D∂/∂x, Equations (15.3.58)–(15.3.59) become [1] [2] [3]

Zeldovich Y., et al., 1956, Sov. Phys.–Tech. Phys., 1, 1689–1713. Lee J., 1984, Ann. Rev. Fluid Mech., 16, 311–336. He L., Clavin P., 1994, J. Fluid Mech., 277, 227–248.

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∂(ρu) 2 ∂ρ + + ρ(D − u) = 0, ∂t ∂x rf − x   ∂ ∂ 1 ∂p dD +u u=− + , ∂t ∂x ρ ∂x dt

237

(4.3.6)

which can be rewritten ∂(ρu) ∂(ρu2 + p) 2 dD + + ρ(D − u)u = , ∂t ∂x rf − x dt

(4.3.7)

where the last term in the left-hand side is introduced by the flow divergence in spherical geometry (in cylindrical geometry, the 2 is replaced by 1). When heat conduction is neglected, Equation (15.1.34) for conservation of energy, written in the moving frame of the lead shock, D/Dt → ∂/∂t + u∂/∂x, takes the form   ∂ ∂ ∂ 1 ∂ +u (cp T − qm ψ) − +u p = 0, ∂t ∂x ρ ∂t ∂x where the notations below (15.1.42) have been used. Using the expression for (1/ρ)(∂p/∂x) from the second equation in (4.3.6), one gets 

∂ ∂ +u ∂t ∂x

  dD u2 1 ∂p cp T + − qm ψ − −u = 0. 2 ρ ∂t dt

(4.3.8)

Equations (4.3.6)–(4.3.8) are valid both in the burnt gas (ψ = 1), where they reduce to (15.3.58)–(15.3.60), and inside the inner structure of the detonation front, where ψ(x, t) is given by the solution of the chemical equation in (12.2.2). The geometrical term that introduces the curvature effect is the divergence of the flow in the mass conservation in (4.3.6). Without the unsteady and curvature terms, Equations (4.3.6)–(4.3.8) reduce to (4.2.46)– (4.2.48) for the structure of planar waves propagating at constant speed, leading to the jump conditions (15.1.45)–(15.1.48) (where u is denoted U) with no external heat flux. When these jumps are incorporated into Euler’s equations, the numerical results[4] show systematic initiation of a detonation, for any value of the deposited energy, when the selfsimilar solution of point blast explosion is used as an initial condition. The perturbation analysis of the inner structure of detonations, presented next, is performed in the limit of a large activation energy for a weakly curved front and in a quasisteady-state approximation. The latter is valid if the characteristic time for evolution of the radius rf (t) and of the propagation velocity D(t) is much longer than the transit time of a fluid particle across the inner structure of the detonation wave. In this case the equations reduce to the system

[4]

Korobeinikov P., 1971, Ann. Rev. Fluid Mech., 3, 317–346.

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2 ∂(ρu2 + p) 2 ∂(ρu) ≈− ρ(D − u), ≈− ρ(D − u)u, ∂x rf − x ∂x rf − x   2 ∂ γ γ−1 ρp + u2 − qm ψ ≈ 0. ∂x

(4.3.9)

(4.3.10)

The validity of the approximation is limited not only by the conditions in the external flows, outside the inner structure of a detonation, but also by the stability properties of the detonation structure itself since instabilities may lead to pulsating phenomena and cellular fronts, both evolving on a time scale as short as the transit time; see Sections 4.5.1 and 4.5.2. Integrating (4.3.9)–(4.3.10) across the inner structure of the detonation gives the curvature induced modifications to the jump conditions,

where

(ρb u2b + pb ) − (ρu D 2 + pu ) (ρb ub − ρu D) ≈ −I2 , ≈ −I1 , ρu D ρu D 2 . /   u2b γ pb D2 γ pu + qm + + ≈ γ − 1 ρb 2 γ − 1 ρu 2  d  d ρ(D − u) ρu(D − u) 2 2 I1 ≡ dx, I2 ≡ dx, 2 ρu D 0 (rf − x) ρu D 0 (rf − x)

(4.3.11) (4.3.12)

and d is the detonation thickness. The integrals in I1 and I2 are well defined when there is a clear separation of length scales between the inner structure of the detonation and the external flow in the inert burnt gas. This is possible for weakly curved detonations,  ≡ d/rf 1. Since I1 and I2 are of order , the inner structure of a quasi-steady and weakly curved detonation can be obtained by a perturbation analysis in the limit  1. To first order in this limit the perturbation terms in (4.3.11) take the form    d  d dx ρu ρ(x) dx , I2 ≈ 2 , (4.3.13) 1− −1 I1 ≈ 2 ρ d ρ(x) d u 0 0 where ρ(x) denotes the density distribution in the planar detonation at velocity DCJ , ρ(x)uN (x) = ρu DCJ . Consider an expanding spherical detonation and look for the propagation velocity as a function of its radius, D(rf ). Because of the high sensitivity of the induction delay to temperature variations, a nonlinear analysis can be performed in the framework of the weakly curved approximation. The solution for D(rf ) has a turning point that plays a key role for the critical energy; see Fig. 4.10. This is shown as follows. According to the chemical kinetics of gaseous combustion, studied in Chapter 5, the internal structure of a detonation can be decomposed into two parts: an induction zone of thickness dind (TN ) (induction length) where the release of chemical energy is negligible, followed by an exothermic reaction zone of thickness dexo ; see Fig. 4.29. The thickness of these two zones is typically of same order of magnitude in ordinary gaseous detonations, so that the detonation thickness d is effectively of order of the induction length d/dind = O(1).

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The expression for dind in terms of the Neumann temperature TN (t) is easily computed for the one-step model (2.1.3) with a large activation energy βN ≡ E/(kb TN ), using a quasisteady-state approximation and limiting attention to small variations, δTN /TN = O(1/βN ), sufficient to produce a strong variation of the reaction rate. For simplicity, consider first a quasi-isobaric approximation in which the compressible effects are neglected; the extension to the more general case is straightforward. The modifications to the distribution of temperature T and reduced species concentration Y/Yu = 1 − ψ are of order unity. They are solutions to (8.1.1)–(8.1.2) (without diffusion terms) in which the small modifications to the convection terms, of relative order 1/βN , are neglected:    d(cp T/qm ) 1−ψ E 1 1 dψ ≈ uN ≈ exp − − uN dx dx τr (TN ) kB T TN x = 0:

T = TN (t),

ψ = 0.

Here, the varying reaction time at the Neumann state, τr (TN ), has been used as the reference time and the overbar denotes the unperturbed solution for TN = T N . The thermal runaway, which marks the end of the induction zone, occurs when (T − TN )/TN becomes of order 1/βN , so that ψ ≈ 0 throughout the induction zone. Introducing the notation  ≡ βN (T − TN )/TN , the solution takes the form  cp T N uN τr (TN ) d cp T N uN τr (TN ) 1 de− . ≈ exp , dind ≈ qm βN dx qm βN 0 When compressible effects are taken into account, namely when the term udu/dx is retained in (4.2.48), the result takes a similar form[1] . / 2 cp T N 1 − MN 1 δTN dind δdind = βN  1: , = −βN , (4.3.14) 2 uN τr (TN ) βN 1 − γ M qm dind TN N

where the second relation is valid to leading order in the limit βN → ∞ and comes from the fact that the variation of dind with TN is given by the exponential term in the reaction −1 −E/kB TN e . rate τr−1 (TN ) = τcoll The solution for D(rf ) is obtained easily using the so-called square-wave model. This model is the large activation energy limit of the one-step ZFK model (2.1.3). The ratio of the length of the exothermic zone to the induction length vanishes as 1/βN in the limit βN ≡ E/kB TN → ∞, dexo /dind → 0, so that the detonation thickness becomes equal to the induction length, d = dind . The exothermic reaction then proceeds inside an infinitely thin zone, considered as a discontinuity and located at a distance dind from the shock front. The variables, u(x), ρ(x) and p(x), which are discontinuous across the shock front, are also discontinuous across the thin reaction layer and are constant inside the induction layer separating the two discontinuities. The quantities I1 and I2 in (4.3.13) are then simple functions of the Neumann state, [1]

He L., Clavin P., 1994, J. Fluid Mech., 277, 227–248.

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dind I1 ≈ 2 rf



 ρN −1 , ρu

dind I2 ≈ 2 rf

  ρu ρu 1− = I1 . ρN ρN

(4.3.15)

The two quantities I1 and I2 are proportional to the curvature of the front with a coefficient involving the induction length, dind (TN ), which is, according to (4.3.14), a nonlinear function of TN and thus of the detonation velocity D. For a given initial mixture (qm , ρu , pu ), the three quantities defining the burnt state (ub , ρb , pb ) are solutions to three equations, (4.3.11)–(4.3.12), involving rf and D through I1 and I2 . The condition for self-sustained propagation introduces an additional constraint, the sonic condition in the burnt gas, u2b = γ pb /ρb ,

(4.3.16)

which, in principle, determines the function D(rf ). The analysis thus starts by computing I1 and I1 in terms of D and rf . The sensitivity of the reaction rate 1/τr (T) to temperature implies that the quantities I1 and I2 are very sensitive to variations of the detonation velocity, δD/D 1 ⇒ δI1,2 /I1,2 = O(1). According to the Arrhenius law (1.2.2) with a large reduced activation energy, 

 TN − TNCJ E τr (TN ) ≈ exp −βN  1: βN = , (4.3.17) kB TN τr (TNCJ ) TNCJ where TNCJ is the Neumann temperature of the planar CJ wave and TN is a function of D given by (4.2.16). In the limit βN  1, it is sufficient to consider small variations of TN around TNCJ , (TN − TNCJ )/TNCJ = O(1/βN ), and thus also small departures of the detonation velocity from its CJ value: (D − DCJ ) /DCJ = O(1/βN ) with βN ≈ E/kB TNCJ  1. Neglecting small terms of order 1/βN coming from uN and ρN , and using (4.2.14) and (4.2.16) in the strong shock limit for simplicity,   TN (γ − 1) ρu γ −1 TN D 2 , ≈ ≈ 2γ Mu2 , ≈ , (γ − 1)Mu2  1 ⇒ ρN γ +1 Tu TNCJ DCJ (γ + 1)2 (4.3.18) where Mu = D/au ; Equations (4.3.14) and (4.3.17) yield  (D − DCJ ) , (4.3.19) βN  1, dind = dCJ exp −2βN DCJ where dCJ is the thickness of the planar CJ detonation, equal to the induction length in the square-wave model. Therefore, according to (4.3.15),  4 dCJ γ −1 (D − DCJ ) , I2 ≈ I1 . exp −2βN (4.3.20) I1 (D) ≈ γ − 1 rf DCJ γ +1 Considering small curvature, dCJ /rf , and small changes in the detonation velocity, δD/DCJ ≡ (D − DCJ )/DCJ , both of order 1/βN , dCJ /rf = O(1/βN ) and δD/DCJ = O(1/βN ), the two quantities I1 (D) and I2 (D) are nonlinear functions of βN δD/DCJ = O(1). However, the values I1 and I2 are still small, of order 1/βN . According to (4.3.11)–(4.3.12), the same is true for the quantities δub (D)/abCJ , δρb (D)/ρbCJ and

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δpb (D)/pbCJ , where the notations δub ≡ ub − abCJ , δρb ≡ ρb − ρbCJ and δpb ≡ pb − pbCJ have been introduced, and where the subscript CJ refers to the planar CJ wave. These quantities can all be computed by using a linear approximation in the left-hand sides of (4.3.11) and (4.3.12), δub δD δρb + = − I1 , ρbCJ abCJ DCJ    δD 1 δpb δρb δub DCJ 2 + +2 = − I2 , γ pbCJ ρbCJ abCJ abCJ DCJ 2  1 δpb δD 1 δρb δub DCJ − + = , γ − 1 pbCJ γ − 1 ρρCJ abCJ abCJ DCJ

(4.3.21) (4.3.22) (4.3.23)

where the relations γ pbCJ /ρbCJ = a2bCJ and ubCJ = abCJ have been used. The sonic condition (4.3.16) yields the additional relation δρb δub δpb − −2 = 0. pbCJ ρρCJ abCJ

(4.3.24)

The four quantities δpb /pbCJ , δρb /ρbCJ , δub /abCJ and δD/DCJ , solutions to the four linear equations (4.3.21)–(4.3.24), are thus expressed in terms of I1 and I2 that are, according to (4.3.20), nonlinear functions of δD/DCJ . The expression δD(I1 , I2 )/DCJ then yields a nonlinear equation for the detonation velocity, 0     γ +1 γ +1 DCJ DCJ δD γ − 1 DCJ 2 I1 (D) − = I2 (D), −2 1+ γ γ + 1 abCJ abCJ DCJ γ abCJ which takes a simple form in the limit of strong detonation (4.2.63):     −D DCJ − D −2βN DCJ dCJ 16γ 2 DCJ 2βN e βN = 2 . DCJ rf γ −1

(4.3.25)

This equation, of the type Xe−X = h, gives a ‘C’-shaped curve for the detonation velocity D versus the radius rf of the lead shock, with a turning point located at rf = r∗ , D = D ∗ , where   16eγ 2 1 ∗ ∗ DCJ βN dCJ , D = 1− (4.3.26) r = 2 2βN γ −1 (see Fig. 4.10), where the curve was obtained numerically without introducing the approximation of the square-wave model. The latter introduces only a small quantitative difference. This shows that the critical radius is much larger than dCJ when βN is large, typically by a factor greater than 102 ! There are two branches of solutions to (4.3.25) for rf > r∗ with D + (rf ) > D − (rf ). Both are close to DCJ when βN  1 and the upper branch corresponds to a propagation velocity slightly smaller than the CJ velocity, D +  DCJ with limrf →∞ D + = DCJ . The two branches merge at rf = r∗ , D + = D − = D ∗ , and there is no solution to (4.3.25) at a smaller radius, r < r∗ . There is no spherical detonation, with a radius smaller than r∗ , whose the inner structure is quasi-steady and having a velocity of

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(a)

(b)

Figure 4.10 D + and D + branches of the reduced detonation velocity versus reduced radius, obtained numerically for γ = 1.4 and βN = 5.33. (a) The dashed line shows the trajectory of a CJ blast wave with an initial energy just sufficient to reach r∗ with the velocity D ∗ . (b) The dotted curves show the numerical solutions of the full unsteady equations (15.1.33)–(15.1.34), with zero thermal conductivity, for the initiation of spherical detonations sustained by an Arrhenius law in (15.1.40) with 3 )= qm = 12.5. Curves 1–4 correspond to four different values of the energy source, E/(ρu DCJ dCJ 3.3 × 107 , 5.69 × 107 , 1.34 × 108 and 2.64 × 108 , respectively. Reproduced from He L., Clavin P., 1994, J. Fluid Mech., 277, 227–248, with permission

burnt gas at the end of the reaction zone ub equal to the local sound velocity, ub = ab . All such solutions have a burnt gas velocity smaller than the sound velocity, ub < ab , so that, in the presence of an expanding wave in the burnt gas, these detonations are systematically slowed down. For r > r∗ , spherical detonations having a quasi-steady internal structure do exist for D > D + and D < D − but with ub < ab so that the latter solutions are slowed down. No solution exists for an intermediate velocity D − < D < D + and a radius larger than r∗ , rf > r∗ . The lower branch of solutions to (4.3.25) is thus not physical since these spherical detonations slow down and will extinguish as soon as TN decreases below the crossover temperature mentioned at the beginning of this section, Section 4.3. To summarise, only the upper branch of solutions may attract quasi-steady detonations in spherical geometry. Critical Ignition Energy (1994) In view of the above results, it is reasonable to assume that a detonation can be initiated by a blast wave only if the radius of the shock rf is equal to or larger than r∗ in (4.3.26) when the shock velocity reaches DCJ , so that the radius at which the CJ wave is formed cannot be smaller than r∗ , which is typically 102 larger than the detonation thickness, r∗ /dCJ ≈ 102 . The marginal blast wave defined by this criterion is shown in Fig. 4.10a. Neglecting the chemical heat release before ignition, and using the relation between the radius rf , the velocity D and the deposited energy E in the self-similar solution for inert blast waves (15.3.67), E ≈ ρu rf3 D 2 , the critical energy using (4.3.26) is[1] 2 ∗3 Ec ≈ ρu DCJ r ≈ 2ρu qm (γ 2 − 1)r∗3 . [1]

He L., Clavin P., 1994, J. Fluid Mech., 277, 227–248.

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(4.3.27)

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243

The ratio between the two expressions (4.3.27) and (4.3.5) for the critical energy is extremely large, of order 106 for βN = 8 and γ = 1.3. This is a consequence of the large ratio of lengths, r∗ /dCJ ≈ 103 . The orders of magnitude of r∗ and Ec given by (4.3.26) and (4.3.27) are in agreement with the experimental data. This estimation is also confirmed by direct numerical simulations,[1] shown by curves 1–4 in Fig. 4.10b. The energy source of curve 1 has the value given by (4.3.27). The detonation is effectively ignited by curve 3, which has an energy source four times greater. These curves also indicate that onedimensional detonation fronts are easily destabilised to have an oscillating propagation velocity. Three assumptions limit the quantitative applicability of this result, namely that the blast wave is inert and described by the Taylor self-similar solution (15.3.67), an oversimplified chemical kinetic model and a quasi-steady-state structure of the inner detonation wave. The first approximation is valid for a radius smaller than r∗ but larger than the size of the igniter, and for times larger than the energy deposition time. A promising approach to take into account a finite size of the igniter and a finite deposition time is presented in the 2013 analysis of Li˜nan et al..[2] A complex chemical kinetic scheme will not change the results drastically, since the thermal sensitivity of the induction length is the key mechanism of the phenomenon. Concerning the quasi-steady-state approximation, direct numerical calculations[1] in spherical geometry shows that unsteadiness of the inner structure of the detonation may increase the critical energy but does not modify the order of magnitude predicted by (4.3.27); see Fig. 4.10 and the references in a 2012 review paper.[3] The above analysis points out the essential role of nonlinear curvature effects that can be described in an approximation of weak curvature, thanks to the strong thermal sensitivity. However, unsteadiness is unavoidable since detonation waves are unstable and exhibit both one-dimensional oscillations, called galloping detonations, and transverse structures on the detonation front, called cellular detonations. Addressing intrinsic unsteadiness requires considering times as short as the transit time. Such effects therefore cannot be correctly taken into account by assuming slow temporal evolution, and their analytical study has been performed only in some limiting cases; see Section 4.5. 4.3.4 Spontaneous Initiation and Quenching of Detonations Ignition of a detonation in the absence of a strong blast wave is not an easy task since a pressure rise higher than pNCJ is required in order to generate a self-propagating CJ wave. For example, such a high pressure cannot be obtained by the combustion of a pocket of fresh mixture of fixed volume, since, typically, pb /pu = Tb /Tu  10 for adiabatic combustion while pNCJ /pu > 30. However, a detonation can be initiated inside a nonhomogeneous hot spot, without need for the deposition of a high energy density in a tiny kernel. Such a spontaneous initiation was predicted by Zeldovich et al.[4,5] in a preconditioned medium with a [2] [3] [4] [5]

Li˜nan A., et al., 2012, C. R. M´ecanique, 340, 829–844. Clavin P., Williams F., 2012, Philos. Trans. R. Soc. London Ser. A, 370, 597–624. Zeldovich Y., et al., 1970, Acta Astronaut., 15, 313. Zeldovich Y., 1980, Combust. Flame, 39, 211–214.

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gradient of induction time (e.g. produced by a temperature gradient). This mechanism was also identified by Lee et al.[1] The research on this topic was still active more than 20 years later.[2,3] Criterion for Spontaneous Ignition (1970) Suppose that the chemical energy is liberated by combustion after an induction delay τind . Suppose also that a gradient of induction delays, dτind /dx > 0, has been created in a reactive medium during a time scale shorter than the shortest induction time. This can be done in the laboratory by heating the reactive mixture with a high-power laser pulse. Other less intuitive possibilities mentioned in the literature are heating in the boundary layer developing behind a strong shock propagating through a reactive mixture in a tube, and also the fast turbulent mixing of strong hot jets injected into a reactive mixture. A propagating induction front is generated by a gradient of induction delay; the slices of gas having a longer induction delay react after those having a shorter induction delay. Roughly speaking, a detonation is initiated when the velocity of the induction front, namely the inverse of the gradient of induction delay, is equal to the propagation velocity of pressure disturbances (sound speed au ) so that a runaway of the pressure occurs, leading quasi-spontaneously to a CJ wave. For small induction gradients, the induction front propagates faster than acoustic waves and combustion proceeds approximately at constant volume. For very steep induction gradients the combustion proceeds at constant pressure. In both cases the pressure increase remains small, well below pNCJ , and the reaction wave is systematically damped out by the expansion waves developing behind the reaction region. Quantitative insight is provided by a study of the coupling between acoustic waves and heat release. Suppose that after the induction delay, τind (x), the release of chemical energy takes place on a time scale shorter than that of acoustics, and also that the heat conductivity can be neglected on this short time scale. Each slice of gas reacts, one after the other, at constant volume without affecting the neighbouring slices. In planar geometry the gradient of induction delays produces a distribution of energy release rate (per unit volume) q˙ v = qv  (t − τind (x)) , where, by definition, the heat release rate  (t) is positive for t > 0, is nullfor t < 0, has the dimension of the inverse of the reaction time and is normed to unity,  (t)dt = 1. The time scale of the rate of heat release is usually of same order as the induction time τind . To simplify the presentation we assume that it is shorter, τind  (t) > 1. Consider o + x(dτ /dx). The a constant gradient in planar geometry, dτind /dx = cst., τind = τind ind instantaneous distribution of q˙ v results from an induction front propagating at constant a speed (dτind /dx)−1 , giving rise to a source term in d’Alembert’s equation for the acoustic

[1] [2] [3]

Lee J., et al., 1978, Acta Astronaut., 5, 971–982. Bartenev A., Gelfand B., 2000, Prog. Energy Combust. Sci., 26, 29–55. Kapila A., et al., 2002, Combust. Theor. Model., 6, 553–594.

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pressure (2.5.1) or (15.2.21) of the form

o q˙ v = qv  t − τind − x(dτind /dx) ,

245

(4.3.28)

o is a constant. Suppose now that the propagation speed of the induction where the delay τind front is equal to the sound speed, assumed to be constant for simplicity,

(dτind /dx)−1 = a.

(4.3.29)

The simple acoustic wave propagating in the same direction as the induction front, solution to Equation (2.5.1),

∂ ∂ o δp + a δp = (γ − 1)qv  t − τind − x/a , (4.3.30) ∂t ∂x has a secular term whose amplitude increases linearly in time:

o δp = t(γ − 1)qv  t − τind − x/a . (4.3.31) The synchronisation of a simple acoustic wave with the induction front thus leads to a rapid increase of pressure (local runaway), on the time scale of energy release, which is short compared with the acoustic time. A strong shock wave with a pressure peak sufficiently high to ignite a CJ detonation quasi-instantaneously is created at the point where relation (4.3.29) is verified. The increase of pressure is saturated by the damping due to the expansion wave which develops behind the shock to match the conditions at the origin. For the Arrhenius law (1.2.2) and a gradient of initial temperature characterised by a length scale L ≡ Tu /|dTu /dx|, the criterion (4.3.29) yields the following condition for the temperature in the fresh mixture, Tu∗ , at which spontaneous ignition occurs: E τind (Tu∗ )a(Tu∗ ) ≈ 1, kB Tu∗ L(Tu∗ )

(4.3.32)

−1 since τind dτind /dx = (E/kB T)T −1 dT/dx, valid for a large activation energy E/kB Tu∗  1. Therefore the condition in (4.3.32) is consistent with a characteristic time of heat release ( τind ) smaller than the acoustic time L/a as required for spontaneous initiation. Such a separation of time scales is possible at a sufficiently high temperature Tu∗ . For a bellshaped distribution of initial temperature, the acoustic time L/a is too large near the centre where the temperature gradient is small, limx→0 L = ∞. The pressure runaway can occur at a distance from the centre where the temperature gradient is still sufficiently high, more precisely at the location where L(Tu∗ ) becomes sufficiently small for the relation (4.3.32) to be satisfied. However some 20 years after the discovery of spontaneous ignition, an antagonistic phenomenon was identified;[4] the temperature gradient that causes spontaneous ignition of a detonation at high temperature can also lead to spontaneous quenching of the CJ detonation at a lower temperature, showing that the conditions for ignition of a detonation are a delicate compromise.

[4]

He L., Clavin P., 1992, Proc. Comb. Inst., 24, 1861–1867.

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Spontaneous Quenching by Thermal Gradient (1992) The quenching mechanism is explained in simple terms by the rate of increase of the induction length dind as TN decreases. In the quasi-steady-state approximation, when the −1 ≡ rate of variation of dind is negligible compared with the inverse of the transit time, τind uN /dind , the induction length responds quasi-instantaneously to changes in the Neumann temperature, and according to (4.3.14)   1 d 1 d E E (4.3.33)  1: dind ≈ − TN . kB TN dind dt kb TN TN dt In the strong shock limit (γ − 1)Mu2  1, used for simplicity (see the relations (4.3.18)), the variations of TN and D are simply related, (γ − 1) TN ≈ 2γ Mu2 Tu (γ + 1)2

⇒

δTN δD ≈2 , TN D

(4.3.34)

where the relation Mu2 ∝ D 2 /(cv Tu ) has been used. According to (4.2.44), the variation of the CJ velocity DCJ with the temperature of the fresh mixture Tu is small for a fast detonation, 1 δTu 1 1 δTu δDCJ = ≈ .  DCJ Tu 2Mu2CJ 1 + cp Tu 2Mu2CJ Tu (γ +1)qm /2

(4.3.35)

Then, for a large temperature sensitivity, βN ≡ E/kB T N  1, where the overbar denotes a steady reference detonation, and for small variations of the detonation velocity, (D−D)/D of order 1/βN , the induction length is a nonlinear function of the detonation velocity D that takes a simple exponential form   β (D−D) 1 dD −2 βN (D−D) dind 1 d −2 N D D dind ≈ −2βN e ≈e , , (4.3.36) D dt dind dind dt in agreement with (4.3.19). For small departures from the planar CJ wave in a temperature gradient characterised by the length L, using the CJ wave at Tu as the reference and the relation d/dt ≈ DCJ d/dx, Equations (4.3.35)–(4.3.36) yield, . / CJ ) βN DCJ dind −2 βN (D−D τ ind d DCJ dind ≈ , (4.3.37) e 2 MuCJ uNCJ L dind dt where the relation dind = uNCJ τ ind has been used. According to (4.2.14), the quantity (1/Mu2CJ )(DCJ /uNCJ ) is of order unity for (γ − 1)Mu2CJ = O(1). Therefore, for small departures from CJ waves, (D − DCJ )/DCJ of order 1/βN , the quasi-steady-state assumption (τind /dind )(ddind /dt) 1 is satisfied when the temperature gradient is not too large on the scale of the detonation thickness, βN (dind /L) 1,

(4.3.38)

a condition which is typically satisfied in experiments.

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Hugoniot after heat release

Hugoniot of shock

Figure 4.11 Hugoniot diagram for a detonation propagating in a negative temperature gradient. The line NB has a smaller absolute slope than the Rayleigh line NU showing that the mass flux through the reaction zone is smaller than the mass flux through the lead shock. Reproduced from He L., Clavin P., Proc. Comb. Inst., 24, 1861–1867. Copyright 1992, with permission from Elsevier.

The small rate of increase of the induction length (4.3.36), which is a consequence of the changing conditions in the fresh mixture, induces a small difference of mass flux across the lead shock and across the reaction layer, ρNCJ (d/dt)dind = 0. This is illustrated by the numerical results.[1] They show that the inner structure of the detonation that is formed in a temperature gradient is effectively in a quasi-steady-state, close to that of a unperturbed CJ wave, but there are small differences that lead to the drastic quenching effect in (4.3.40). Plotting the pressure p as a function of the specific volume 1/ρ through the reaction zone, a straight line NB is obtained, linking the Neumann state (labelled N) to the burnt gas (labelled B). As in CJ waves, this line is tangent to the equilibrium curve for burnt gas corresponding to the local temperature Tu of the initial state of the fresh gas in the temperature gradient; see Fig. 4.11. This shows that the exothermic reaction zone (where the chemical energy is released) is in a steady state. However, the slope of the line NU, linking the Neumann state to the initial state, pu , 1/ρu (labelled U) across the lead shock wave, is slightly larger than the slope of the line NB, whereas the two lines NB and NU are aligned in a CJ wave; see Fig. 4.4. This shows that the mass flux across the inert lead shock is slightly larger than the mass flux across the exothermic zone, so that the latter separates slowly from the lead shock on a time scale longer than the transit time τind (TN ) = dind (TN )/uN . The fact that the exothermic zone is in full equilibrium while the induction length is slowly increasing is due to the fact that the transit time across the exothermic zone is substantially shorter than across the induction zone, essentially because the flow velocity increases with the temperature. The difference of mass flux corresponding to the difference of slopes of the two lines NI and NU may be computed by a geometrical construction around the CJ wave as a linear function of the departure of the shock velocity D from the CJ velocity DCJ , so that the result is proportional to ρu (DCJ − D). The presentation simplifies in the limit of a strong [1]

He L., Clavin P., 1992, Proc. Comb. Inst., 24, 1861–1867.

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shock, (γ − 1)Mu2  1: the coefficient of proportionality goes to unity in this limit.[1] Thanks to this geometrical construction, which is a consequence of the particular quasisteady state of the inner structure of the detonation, the time derivative (d/dt)dind can be computed in a way totally different from (4.3.36), without the need to consider the chemical kinetics, (γ − 1)Mu2  1:

(DCJ − D) d . dind ≈ uNCJ dt DCJ

(4.3.39)

Combining the two expressions (4.3.36) and (4.3.39) gives a nonlinear equation for the local velocity D of the lead shock of the detonation propagating in the temperature gradient Tu−1 |dTu /dx| ≡ L−1 (Tu ). According to (4.3.36) in which dind = τind (TNCJ )uNCJ , D = DCJ , and using again the relation dD/dt ≈ DCJ dDCJ /dx, since (D − DCJ )/DCJ is small, of order 1/βN , the equation for X ≡ 2βN (DCJ − D)/DCJ takes the form       −D DCJ − D −2βN DCJ dDCJ 2 DCJ 2βN = K with K ≡ 4βN τind (TNCJ ) − e , DCJ dx (4.3.40) where βN ≡ E/kB T N  1. This equation has a turning point for a critical value K = 1/e corresponding to detonation quenching; there is no solution for D when K > 1/e. There are two branches of solution for K < 1/e, but only the one going to DCJ in the limit K → 0 is physical. The detonation velocity at quenching is close to that of the CJ wave in the same conditions, (DCJ − D)/DCJ = 1/(2βN ). According to (4.3.35), the velocity gradient and thus the coefficient K in (4.3.40) may be expressed in terms of the temperature gradient, 1 au dDCJ =− , dx 2MuCJ L(Tu )

⇒

K=2

βN2 au τind (TNCJ ) . MuCJ L(Tu )

(4.3.41)

A comparison with numerical simulations[2,3] (see Fig. 4.12) shows a good agreement with the spontaneous quenching predicted by (4.3.40)–(4.3.41). The coefficient K may be expressed in terms of the conditions for spontaneous ignition (labelled by ∗ ) by introducing (4.3.32) into (4.3.41), K=

βN2 au τind (TNCJ ) L(Tu∗ ) 2 MuCJ (E/kB Tu∗ ) a∗u τind (Tu∗ ) L(Tu )

(4.3.42) E Tu τind (TNCJ ) L(Tu∗ ) 2 , ≈ ∗ MuCJ kB TNCJ TNCJ τind (Tu∗ ) L(Tu )   where the relations MuCJ /Mu∗CJ ≈ Tu∗ /Tu and au /a∗u = Tu /Tu∗ have been used and where Mu∗CJ is the Mach number of propagation of the CJ wave for a temperature of fresh mixture equal to that at spontaneous initiation, Tu∗ . [1] [2] [3]

Clavin P., Williams F., 2012, Philos. Trans. R. Soc. London Ser. A, 370, 597–624. He L., Clavin P., 1992, Proc. Comb. Inst., 24, 1861–1867. He L., Clavin P., 1994, Proc. Comb. Inst., 25, 45–51.

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249

CJ Numerical simulation

Theory

Initiation

Propagation

Quenching (cm)

Figure 4.12 Maximum overpressure as a function of distance in a detonation initiated in a negative temperature gradient of 600 K/cm satisfying conditions (4.3.29)–(4.3.32) for spontaneous initiation of a detonation in a gradient. The dashed grey curve is the local CJ value. The black curve is the prediction of (4.3.40)–(4.3.41) and shows the turning point. The solid grey curve is the result of direct numerical simulation, showing initiation followed by quenching. Reproduced from He L., Clavin P., Proc. Comb. Inst., 24, 1861–1867. Copyright 1992, with permission from Elsevier.

Conclusion A detonation which is initially spontaneously initiated by a hot pocket of fresh mixture can be transmitted to the surrounding uniform medium only if the temperature profile of the pocket is such that K is nowhere larger than 1/e. If this is not the case, the detonation is quenched inside the hot pocket before reaching the uniform medium. For L(Tu ) = cst., the condition for quenching by the same temperature gradient that is responsible for spontaneous ignition at high temperature then takes the form K≈

2 E Tu τind (TNCJ ) = 1/e. ∗ MuCJ kB TNCJ TNCJ τind (Tu∗ )

(4.3.43)

This expression gives a relation between the Neumann temperature of the CJ wave at quenching, TNCJ , and the temperature of fresh gas at spontaneous ignition, Tu∗ ; the temperature gradient is involved only through τind (Tu∗ ) (see (4.3.32)). For a detonation propagating into a region of decreasing temperature Tu near the temperature of spontaneous initiation Tu∗ , defined in (4.3.32), with Tu  Tu∗ , the transit time τind (TNCJ ) across the induction zone of the CJ detonation increases quickly as Tu decreases, since, according to (4.3.14), a small variation of TNCJ , of order 1/βN , produces a variation of order unity of τind and thus also of K. For a constant temperature gradient, a rough estimate of the critical condition for quenching is obtained from (4.3.43) as follows. In contrast with τind , the variation of the coefficient (2/Mu∗CJ )(E/kB TNCJ )(Tu /TNCJ ) is not large and its value is of order unity, so that quenching occurs when the transit time across the induction zone, τind (TNCJ ), increases to reach a value comparable to the induction delay at spontaneous ignition τind (Tu∗ ). This corresponds roughly to TNCJ close to Tu∗ , relating quenching to spontaneous ignition. It is worthwhile stressing that the quenching mechanism described

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here is not associated with the crossover temperature T ∗ studied in Sections 5.3 and 5.4, TNCJ = T ∗ . The quasi-steady-state approximation, which is the basic assumption in the above analysis, is not valid if the CJ wave is unstable. Therefore, the validity of the analysis is ensured only if the thermal sensitivity is smaller than that of the instability threshold of galloping detonations (pulsating instability) studied in Section 4.5.1. This is the case at sufficiently high temperature Tu . Another limitation is the planar geometry. In spherical and cylindrical geometry a critical radius, such as the one in Section 4.3.3, must also appear in the criterion for spontaneous initiation.[1]

4.3.5 Deflagration-to-Detonation Transition (DDT) In contrast to Section 4.3.3, we consider here mild energy discharges that initiate a subsonic flame, also called a deflagration (see Section 2.4.2 for the condition of initiation), and we discuss the deflagration-to-detonation transition (DDT). The basic DDT mechanisms are extremely complex. Despite more than a century of research, identification of these mechanisms is not yet achieved, and, according to various reviews of the last 40 years,[2,3,4,5,6,7] full understanding of DDT remains one of the major challenges of combustion theory; see also the special issue of the Philosophical Transactions of the Royal Society A, (2012) 370, no. 1960, pp. 531–799. However, understanding of DDT has improved recently thanks to theoretical analyses and accurate numerical solutions of simplified models. We limit our presentation to a summary of the state of the art of the knowledge, without entering into the detailed analyses, the simplest ones being the most instructive. Phenomenology of DDT To the best of our knowledge, in the modern literature, DDT is not reported for spherical flames propagating in free space filled with a quiescent and uniform gaseous combustible mixture. Such a transition was reported in early Soviet experiments[8] around 1950, concerning very energetic mixtures such as acetylene–oxygen mixtures (UL ≈ 5– 10 m/s) enclosed in a soap bubble or a rubber sphere. The phenomenon was attributed to flame acceleration by front wrinkling, due either to intrinsic instabilities (self-turbulising flames) or to the interaction between the flame and shock waves reflected from the boundary delimited by the bubble. It is not clear whether or not it was a DDT or a direct initiation, as discussed in Section 4.3.3. However, severe explosion events are reported in large vapour [1] [2] [3] [4] [5] [6] [7] [8]

He L., Law C., 1996, Phys. Fluids, 8(1), 248–257. Oppenheim A., Soloukhin R., 1973, Ann. Rev. Fluid Mech., 5, 31–58. Lee J., 1977, Ann. Rev. Phys. Chem., 28, 75–104. Lee J., Moen I., 1980, Prog. Energy Combust. Sci., 6, 359–389. Lee J., Berman M., 1997, Advances in Heat Transfer, 29, 59–126. Lee J., 2008, The detonation phenomenon. Cambridge University Press. Ciccarelli G., Dorofeev S., 2008, Prog. Energy Combust. Sci., 34, 499–550. Shchelkin K., Troshin Y., 1965, Gasdynamics of combustion. Baltimore, Md.: Mono Book Corp.

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251

clouds.[9] They are not fully explained, even though obstacle-generated turbulence (e.g. the trees) is thought to be the principal ingredient. Since the discovery of detonation waves in gases by Berthelot and Vieille (1881) and Mallard and Le Chatelier (1883), it is known that, for many combustible mixtures ignited at the closed end of a tube, a flame ignited by a weak spark can accelerate to a detonation.[2] For sensitive fuel oxygen mixtures involving acetylene, ethylene or hydrogen in tubes with cross sections of a few tens of square centimetres, the onset of detonation occurs abruptly after a flame travel typically of the order of a metre and acceleration to a velocity above 300 m/s (in the laboratory frame). For most hydrocarbon–air mixtures the transition distance (if it exists in smooth tubes) is much larger, exceeding the length of laboratoryscale experiments.[3] Flame acceleration and preconditioning of the fresh mixture ahead of the flame due to compression waves are key features of DDT in an open-end tube ignited at the closed end. The early experimental detonation studies were performed in smooth tubes. However, as early as 1926, obstacles (orifice plates) were placed in tubes to promote flame acceleration up to few hundred metres per second.[5,7] Different mechanisms of DDT have been proposed since 1940. The mechanism of flame acceleration still gives rise to some controversy. More likely, several mechanisms promoting DDT are involved in real experiments, not a single one. Before discussing the orders of magnitude and the experimental results, it is convenient for pedagogical reasons to present a qualitative overview of the general ideas on DDT that have emerged from experiments, analyses and, more recently, from numerical studies. Turbulence-Induced DDT Between 1940 and 1945 Shchelkin proposed that flame acceleration in tubes is governed by the turbulent fluctuations of the flow in the fresh mixture ahead of the flame, leading to an increase of the flame surface; see Section 3.2. The general idea is as follows. For flame ignition at the closed end of a tube, a flow of fresh mixture is generated upstream from the flame front, due to expansion of the burnt gas; the flame acts as a semi-permeable piston (see Section 2.2.1). For a sufficiently large Reynolds number, the expansion-induced flow becomes turbulent and a coherent feedback mechanism can develop: the turbulenceinduced increase of propagation speed of the wrinkled flame brush (3.1.11) leads to an increase of the flow velocity and then of the turbulent intensity that in turn increases the turbulent flame speed. The nonslip condition at the wall creates transverse gradients of the longitudinal flow velocity. In tubes without obstacles the turbulence develops in the boundary layers at the wall and is strongly affected by wall roughness. Obstacles facilitate the development of turbulence and DDT.

[9]

Bradley D., et al., 2012, Philos. Trans. R. Soc. London Ser. A, 370, 544–566.

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Localised Thermal Explosion (1973) Another feedback mechanism, of a quite different nature, involving gas compressibility, also develops. As recalled in Section 15.3 for the case of a piston, the fresh mixture is put into motion by a quasi-planar compression wave propagating upstream from the flame. After a sufficiently long time in sufficiently long tubes a shock wave formed, somewhere upstream, depending on the details of flame acceleration. Due to this compression wave, the temperature on the cold side of the flame front increases with time and a longitudinal temperature gradient develops to match the initial conditions far upstream. This increase of temperature in the fresh mixture has two effects: it increases the laminar flame speed, thus producing a feedback mechanism for flame acceleration, and it shortens the induction delay. If the temperature increase is sufficient, typically 1000 K, self-ignition can occur ahead the flame. There is now a general consensus that such temperature gradients can lead locally to spontaneous formation of detonation by a gradient of induction delay (Zeldovich’s mechanism, described in Section 4.3.4). This phenomenon was called ‘localised thermal explosion’ or ‘explosion in the explosion’.[1,2] As discussed later in more quantitative terms, such a mechanism, based on compressibility-induced preconditioning of the fresh mixture, is possible only for a very fast flame brush, (deflagration) propagating at a velocity above many hundreds of metres per second. In insulated tubes, heating is reinforced in the boundary layers at the wall of the tube where the transverse velocity gradient produces a temperature gradient by viscous dissipation of mechanical energy. The faster the longitudinal flow, the stronger is such a dissipation mechanism. However, heat loss at the wall is expected to weaken the dissipation-induced DDT mechanism. Friction Effect (2000–2015) In a more recent series of accurate numerical simulations in simple configurations, still in open-end tubes, it has been shown by Sivashinsky and co-workers[3,4] that DDT can be produced by the increase of temperature due to the friction-induced adiabatic compression. To be more specific, in the presence of sufficiently strong friction, the dominant effect is due not necessarily to viscous dissipation, described by the last term in the right-hand side of (15.1.37) or (15.1.61), but to the first term in the right-hand side of (15.1.27) or (15.1.34). This mechanism has been first schematised by a planar model for shockless initiation of detonation, called hydraulic resistance.[3,4] In this model, a volumetric friction force is introduced in the momentum equation. As a result, a shockless temperature gradient develops in the fresh mixture and DDT can be observed even when the dissipative counterpart of friction in the energy equation is artificially turned off.

[1] [2] [3] [4]

Oppenheim A., Soloukhin R., 1973, Ann. Rev. Fluid Mech., 5, 31–58. Ciccarelli G., Dorofeev S., 2008, Prog. Energy Combust. Sci., 34, 499–550. Brailovsky I., Sivashinsky G., 2000, Combust. Flame, 122, 492–499. Brailovsky I., et al., 2012, Philos. Trans. R. Soc. London Ser. A, 370, 625–646.

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Hot gas at rest

253

Cold gas at rest Flame brush

Shock wave

Figure 4.13 Sketch of the one-dimensional runaway configuration showing the folded turbulent flame brush and associated preceding shock wave.

More surprising, when a cut-off temperature T ∗ is introduced in the reaction rate so that the chemical heat cannot be released for T < T ∗ , DDT can be still observed in numerical simulations of simplified flame models even when the increase of temperature in the fresh mixture just ahead of the the flame is smaller than T ∗ , ruling out the Zeldovich mechanism of a localised thermal explosion in the fresh mixture.[5,6] A few years previously, DDT with a chemically frozen flow adjacent to the flame below T ∗ ≈ 550 K was observed experimentally for flames propagating at a high velocity, typically 500 m/s or even more,[7] in highly sensitive stoichiometric hydrogen–oxygen and ethylene–oxygen mixtures at ordinary initial temperature Tu = 293 K and for different initial pressures pu = 0.5–0.75 bar. Even though a Zeldovich mechanism cannot be fully excluded inside the flame structure, a different DDT mechanism seems to be involved. It could be related to the 26-year-old analytical work of Deshaies and Joulin,[8] presented below, showing that there is an upper bound for the propagation velocity of a flame brush, a phenomenon observed later in direct numerical simulations.[9] Runaway Mechanism (1989) A DDT mechanism, different from the localised thermal explosions induced by a thermal gradient, was identified by considering a simplified configuration,[8] namely the onedimensional self-similar solution of a weak shock at constant velocity D > au , generated by a turbulent flame brush propagating at constant velocity, Utur , from the closed end of a tube; see Fig. 4.13. Due to thermal sensitivity, the small increase of flame temperature by the compressible effects, namely here by the weak shock, produces a velocity increase of the flame brush leading to a constructive feedback. The burnt gas being at rest, the velocity of the flame brush is given by (3.1.11) when UL is replaced by Ub = (ρu /ρb )UL , Utur = Ub s, where s ≡ S /So is the degree of folding and Ub is subsonic. As shown below, there is a critical degree of folding s∗ (critical velocity of the flame brush) above which there is no self-similar solution[8] (a shock wave followed by a flame, both at constant velocity; see Fig. 4.14). If s increases for any reason, for example because of flame instabilities or [5] [6] [7] [8] [9]

Kagan L., Sivashinsky G., 2014, In G.S. Roy, S. Frolov, eds., Transient Combustion and Detonation Phenomena, Torus Press, Moscow. Kagan L., et al., 2015, Proc. Comb. Inst., 35, 913–920. Kuznetsov M., et al., 2010, Combust. Sci. Technol., 182(1628-1644). Deshaies B., Joulin G., 1989, Combust. Flame, 77, 201–212. Gamezo V., et al., 2011, In U. Irvine, ed., Proceedings of 23rd ICDERS, 24–29.

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some positive feedback with the upstream turbulent flow, a runaway of the velocity of the flame brush with a sudden increase in the flame temperature is expected to occur when s reaches s∗ . This could correspond to an abrupt transition to detonation. Further analytical studies of the runaway will improve the understanding of DDT, for example by considering a slow increase of s, meaning slow on the time scale of the acoustic waves. Assuming adiabatic conditions (insulated walls) the temperature of the gas just ahead of the flame brush is the temperature TN behind the shock. According to the Rankine– Hugoniot conditions (4.2.14) and (4.2.16) for weak shock (Mu2 −1) 1:

D − UN 2 (M 2 −1), ≈ au γ +1 u

TN 2(γ − 1) 2 (Mu −1), −1 ≈ Tu γ +1

(4.3.44)

the temperature of the gas ahead of the flame, TN , can be expressed in terms of the velocity of the flame brush Utur as follows. The flow velocity ahead of the flame brush, namely downstream of the weak shock, is v = D − UN , where UN denotes the flow velocity relative to the shock at the Neumann state. Neglecting the small difference between ρN and ρu , mass conservation across the flame brush gives also v = (1 − ρb /ρu )Utur . Using the relation D − UN = (1 − ρb /ρu )Utur , Equations (4.3.44) lead to an expression for the increase of temperature in the fresh mixture in terms of the velocity of the flame brush Utur and the speed of sound au :   TN ρb Utur − 1 = (γ − 1) 1 − . Tu ρu au Using (8.2.3), Tb = Tu + qm /cp , δTb /Tb = (ρb /ρu )δTu /Tu , the small increase of flame temperature due to the presence of the shock wave takes the form     ρb TN ρb ρb Utur Tb −1= − 1 = (γ − 1) 1 − , (4.3.45) Tbo ρu Tu ρu ρu au where Tbo is the flame temperature for the initial temperature Tu ahead of the shock. The thermal sensitivity of the laminar flame speed Ub (2.1.9) yields E/2kB To  1:

E

Ub /Ubo ≈ e 2kB Tbo



Tb −Tbo Tbo



,

(4.3.46)

and the relation Utur = sUb then leads to a nonlinear equation for X ≡ K(Ub /Ubo ) when (4.3.45) is introduced into (4.3.46), Xe−X = K,

K ≡ s(γ − 1)

E (Tbo − Tu ) ULo , 2kB Tbo Tbo au

(4.3.47)

where ULo = (ρb /ρu )Ubo is the laminar flame speed for the initial flame temperature Tu and K is a nondimensional parameter proportional to the folding s. The solution Ub /Ubo to (4.3.47), plotted versus K, that is also the velocity of the flame brush Utur versus the folding s, has two branches of solution with a turning point for the critical value K = 1/e. There is no solution for K > 1/e and there are two solutions for K < 1/e; see Fig. 4.14. For ordinary hydrocarbon–air mixtures with a flame velocity UL about 0.35 m/s, and for E(Tb − Tu )/kB Tb2 ≈ 8, γ = 1.4, the critical value for the degree

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Figure 4.14 Plot of the solution to (4.3.47) for the folded flame brush velocity. There is no solution beyond a critical degree of folding, proportional to K.

of folding is about 2 × 102 . For very energetic mixtures such as a stoichiometric ethylene– oxygen mixture, UL ≈ 10 m/s, the critical degree of folding is about 8. This corresponds to a velocity of the flame brush in the laboratory frame, Utur , about 500 m/s. A preliminary dynamical study of the problem was performed by solving the acoustic problem in the burnt gas.[1] The result shows that the lower branch Ub− of solution is stable and the upper branch Ub+ unstable. Flame Acceleration Numerical studies have shown that one of the key ingredients for DDT, namely flame acceleration, can be produced by mechanisms other than flow turbulence: • A first mechanism is the flame front folding by the different instabilities described in Sections 2.2 to 2.5, especially if the initial flame is initiated in a quasi-planar geometry. • Another possibility is the Richtmyer–Meshkov type of instability in Section 2.8.3 if shock waves are reflected by the walls and impinge the flame. • One may also invoke the initial flame elongation just after ignition by a spark at the centre of the closed end of the tube, described in (2.8.8). This leads to an exponential increase of the flame surface area at a rate of the order 2Ub /R, where R is the radius of the tube. This acceleration lasts only during a short lapse of time after ignition, ≈ R/Ub . However, for a very energetic reactive mixture, such as a stoichiometric mixture of hydrogen–oxygen, UL ≈ 10 m/s, Ub ≈ 85 m/s, (Ub − UL ) ≈ 75 m/s, the flow velocity can reach a velocity greater than 550 m/s (e2 ≈ 7.39) at the end of this short delay. • A subsequent flame acceleration has also been considered. It is associated with the twodimensional character of a parallel flow that is generated ahead of the flame by expansion of the burnt gas. Due to viscous effects (nonslip boundary condition at the wall), this parallel flow is nonuniform in the transverse direction, zero at the wall and maximum at the centre of the tube. The flame then takes an elongated form with a surface increasing with time. Assuming a quasi-isobaric approximation and a laminar flow, an approximate analysis[2] predicts an exponential increase in time of the propagation velocity of the

[1] [2]

Deshaies B., Joulin G., 1989, Combust. Flame, 77, 201–212. Bychkov V., et al., 2005, Phys. Rev. E, 72(4), 046307.

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flame tip with a growth rate ≈ UL /R, smaller than (2.8.8) but lasting in principle indefinitely. Further analyses predict that compressible effects in the burnt gas can saturate the growth rate.[1] • An antagonistic effect can be produced if the boundary layer at the wall becomes turbulent and accelerates the flame propagation near the wall, leading to a tulip-shaped flame. • As already mentioned, a constructive feedback can also be produced by an increase of temperature ahead of the flame due to compression waves in the presence of friction without shock formation.[2] In addition the role of viscous dissipation cannot always be neglected, especially in narrow channels; see the end of this section. During the last decade a huge number of numerical studies have investigated these different possibilities.[3,4,5,6,7,8,9,10,11,12] Each of these numerical simulations succeeded in producing a DDT, but it is hard to say to what extent the real experimental phenomena are represented. It is probable that a runaway phenomenon similar to the one described by Deshaies-Joulin[13] is involved in most of these simulations, whatever be the mechanism of flame acceleration. DDT Experiments in Ordinary Tubes review[14]

of the DDT experiments in tubes of typically 5 cm transverse size was preA sented in 2008, prior the 2010 experiment already mentioned[15] ; see also the book of Lee.[16] Much information was already obtained from the pioneering Soviet experiments during the first half of the last century.[17] About 20 years later, the sequence of high-resolution stroboscopic Schlieren photographs by Oppenheim and coworkers[18,19,20] represents a milestone in the study of DDT. These experiments concern mainly a stoichiometric hydrogen–oxygen mixture at atmospheric pressure (UL ≈ 10 m/s) in a rectangular tube [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Valiev D., et al., 2009, Phys. Rev. E, 80, 036317. Kagan L., et al., 2015, Proc. Comb. Inst., 35, 913–920. Kagan L., Sivashinsky G., 2003, Combust. Flame, 134, 389–397. Liberman M., et al., 2006, Int. J. Transp. Phenomena, 8, 253–277. Oran E., Gamezo V.N., 2007, Combust. Flame, 148, 4–47. Kagan L., Sivashinsky G., 2008, Combust. Flame, 154, 186–190. Valiev D., et al., 2008, Phys. Lett. A, 372, 4850–4857. Kessler D., et al., 2010, Combust. Flame, 157, 2063–2077. Akkerman V., et al., 2010, Phys. Fluids, 22, 053606. Ivanov M., et al., 2011, Phys. Rev. E, 83, 056313. Dzieminska E., et al., 2012, Combust. Sci. Technol., 184, 1608–1615. Ivanov M., et al., 2013, J. Hydrogen Energy, 38, 16427–16440. Deshaies B., Joulin G., 1989, Combust. Flame, 77, 201–212. Ciccarelli G., Dorofeev S., 2008, Prog. Energy Combust. Sci., 34, 499–550. Kuznetsov M., et al., 2010, Combust. Sci. Technol., 182(1628-1644). Lee J., 2008, The detonation phenomenon. Cambridge University Press. Shchelkin K., Troshin Y., 1965, Gasdynamics of combustion. Baltimore, Md.: Mono Book Corp. Urtiew P., Oppenheim A., 1966, Proc. R. Soc. London Ser. A, 295(13-28). Meyer J., et al., 1970, Combust. Flame, 14(1), 13–20. Oppenheim A., Soloukhin R., 1973, Ann. Rev. Fluid Mech., 5, 31–58.

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Flame Shock wave Flame

Boundary layer Shock wave

Flame

Figure 4.15 Formation of a tulip flame by acceleration in the flame-induced turbulent boundary layers along the walls. The full curve u(x) represents the velocity profile of the induced flow ahead of the flame, and the dashed line schematises the width of the boundary layer (not to scale).

of cross section 9.7 cm2 with a test section located about 3 m away from the igniter at the closed end. The sequence of events has been established as follows: 1. Initial acceleration of a wrinkled flame generating shock waves 2. A long period of turbulent flame propagation following the formation of a tulip shape 3. Transition to detonation triggered by a local explosion within the shock–flame complex, most of the time either near the wall or at the flame tip. Following the early work of Shchelkin, it has been established experimentally by different authors that flame acceleration is strongly affected by wall roughness: the larger the roughness, the faster the growth of the turbulent boundary layers, leading to flame acceleration close to the walls, as sketched in Fig. 4.15. For a wall roughness of 1 mm in a square channel of 5 cm height, the formation of a tulip-shaped flame, propagating at more than 300 m/s, has been clearly observed before onset of detonation.[21] Propagation is no longer possible below a critical radius of the tube. Just above this radius, spinning detonations are observed. They were first observed in cylindrical tubes as early as 1926.[17] In this marginal regime, burning is concentrated around a triple point rotating around the tube axis, of the same type as that described later in Section 4.4.2. This phenomenon appears when the tube diameter is of the order of the typical size of the cellular structure of the detonation front; see Section 4.5 for a discussion of the cellular detonations. Spinning detonations are more easily observed when approaching the detonation limit, namely when the Neumann temperature approaches the crossover temperature; see the beginning of Section 4.3. This phenomenon can be easily understood by noticing that the length of the induction zone increases near the detonability limit so that transverse propagation of detonation waves is possible inside the induction zone, leading to a spinning detonation front even in conditions for which the planar detonation cannot exist.

[21] Kuznetsov M., et al., 2005, Shock Waves, 14(3), 205–215.

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Gaseous Shocks and Detonations Lead shock Hugoniot of shock

Michelson– Rayleigh lines Hugoniot after heat release

0

Figure 4.16 Sketch in the (v , p) plane (v ≡ 1/ρ) showing the CJ detonation point, BCJ , and the CJ deflagration point, FCJ , characterised by the lower tangency point of a Michelson–Rayleigh line with the Hugoniot of the reacted gas.

Choked Regime of Turbulent Flame In long tubes of large diameter, equipped with regularly spaced orifice rings, turbulent flames propagating at a steady velocity of the order of the speed of sound in the combustion products, 600–1000 m/s, have been observed.[1] This fast propagation regime of turbulent flames, called the ‘choked regime’, is sometimes considered as the so-called CJ deflagration speed, determined by the lower tangency point of the Michelson–Rayleigh line with the Hugoniot curve and characterised by a decrease of pressure in the combustion products; see Fig. 4.16. For a convenient blockage ratio in an obstacle-laden tube, turbulence can sustain flames in the ‘choked regime’ without transition to detonation, while such fast flames normally transit systematically to a detonation in smooth tubes. In the absence of a lead shock wave, the fresh mixture is not warmed up and DDT cannot be produced. Another interpretation of the choked regime is based on the multiplicity of the friction-controlled propagation regimes.[2,3] DDT Experiments in Capillary Tubes (2007) For ordinary fuel–air mixtures (UL ≈ 0.5 m/s), flames cannot propagate in capillarity tubes because they are quenched by thermal loss at the wall; see the detailed study in Section 8.5.1. According to the rough estimates in (8.5.1) and (8.5.4), the nondimensional parameter characterising the critical heat loss for flame quenching is D2T /(βN R2 UL2 ), where βN is the reduced activation energy (8.2.8), so that more energetic mixtures (faster flames) have a smaller quenching radius R. Motivated by gaseous combustion in porous media, by possible applications of microscale combustion to energy production and by numerical and

[1] [2] [3]

Lee J., Berman M., 1997, Advances in Heat Transfer, 29, 59–126. Brailovsky I., Sivashinsky G., 2000, Combust. Flame, 122, 492–499. Sivashinsky G., 2002, Proc. Comb. Inst., 29, 1737–1761.

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theoretical studies of the last decade,[2,4] recent experiments[5] have shown that stoichiometric ethylene–oxygen flames (UL ≈ 10 m/s) can also accelerate to DDT in smoothwalled capillary tubes with diameters as small as 2 mm or even 0.5 mm. The onset of detonation occurs at a distance about 100 times the diameter. This mixture is very energetic and sensitive. The typical size of the cellular structure is about 0.7 mm and the critical diameter is below 0.3 mm. Three typical scenarios were observed: • Abrupt transition to a CJ detonation (DCJ = 2373.5 m/s), typically 100 μs after ignition during turbulent flame propagation at 300 m/s (Reynolds number Re ≈ 4 × 104 ) • CJ detonation initiation followed by extinction • Flame acceleration to a constant speed about 1600 m/s (choked regime?). Other behaviours have been observed[6] for different equivalence ratios, for example an oscillating flame mode with a propagation velocity between 5 and 10 m/s in a 2 mm tube filled with lean mixtures (conditions for which DDT does not occur), galloping detonations (spinning?) travelling at 2000 m/s, and also flame or detonation quenching in rich mixtures.

Orders of Magnitude. Discussion Whatever be the basic mechanism responsible of DDT, runaway or localised explosion, the fact that the onset of detonation in tubes is observed systematically for a propagation velocity (in the laboratory frame) of the flame brush always greater than 300 m/s can be understood as follows. First, as already mentioned, the critical velocity of the flame brush involved in the runaway mechanism (4.3.47) of Deshaies and Joulin[7] is about 500 m/s in the laboratory frame. Second, a localised explosion through Zeldovich’s mechanism also requires such a large flame velocity as shown now. Roughly speaking, according to this scenario, the temperature of the fresh mixture ahead the flame must be sufficiently high for the induction (ignition) time to be sufficiently small. Because of the drastic increase of the induction time below the crossover temperature T ∗ , an ignition time, sufficiently short for the onset of detonation, requires a relatively high temperature and therefore a large flow velocity. For example, the ignition time of a stoichiometric hydrogen–oxygen mixture[8] at 1 atm is 10 s at 800 K, about 2×10−1 s at 900 K, 10−4 s at 1000 K and 10−5 s at 1250 K. Heating the fresh mixture to a temperature above 1000 K by compression through a shock wave and/or an isobaric compression wave is possible only for a sufficiently high propagation velocity of the flame brush. The temperature in an adiabatic compression wave may be evaluated using the relation a/ao = (T/To )1/2 and the Riemann invariant (15.3.47) of a simple wave in planar geometry, (T/To )1/2 = 1 + [(γ − 1)/2] (u/ao ), where the subscript o denotes the initial state of the fresh mixture To ≈ 273 K, p = 1 atm and γ ≈ 1.4. [4] [5] [6] [7] [8]

Bychkov V., et al., 2005, Phys. Rev. E, 72(4), 046307. Wu M., et al., 2007, Proc. Comb. Inst., 31, 2429–2436. Wu M., Wang C., 2011, Proc. Comb. Inst., 33, 2287–2293. Deshaies B., Joulin G., 1989, Combust. Flame, 77, 201–212. Sanchez A., Williams F., 2014, Prog. Energy Combust. Sci., 41, 1–55.

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For a flow velocity of the order of the sound speed, u ≈ ao , this leads to a gas temperature below 500 K, not high enough for ignition. Also, the temperature increase behind the shock wave generated by such a flow is too low. According to the Rankine–Hugoniot relations (4.2.14)–(4.2.16), a shock wave with Mu = 2 in a quiescent gaseous mixture at ordinary conditions requires a gas velocity in the laboratory frame, (D − UN ), larger than the initial sound speed and corresponds to a temperature of the compressed gas also less than 500 K. Therefore, according to the typical laws controlling ignition delay, the compressibilityinduced increase of temperature for a flow velocity of the order of the sound speed is not high enough to produce a detonation. So, for a flow velocity of few hundred metres per second, an additional heating mechanism must be involved. Denoting the velocity of the fresh mixture by u and the local sound speed by a, an evaluation of the heating rate by viscous dissipation using entropy production, recalled in (15.1.60), yields (ν/δ 2 )(u2 /a2 ), where ν = μ/ρ is the viscous diffusion coefficient and δ is the thickness of the laminar boundary layer that develops downstream from the shock. The latter grows with time as δ 2 ≈ νt. Discarding heat loss at the wall, the relative temperature increase of a fluid particle inside the boundary layer after the transit time t from the leading edge is thus predicted to be of order (u/a)2 . Thus, for a flow velocity of the order of the sound speed, the cumulative effect of viscous dissipation and compressibility could create sufficient heating to produce a local explosion and transition to detonation. DDT by a localised explosion through the Zeldovich mechanism cannot occur ahead of the flame for smaller flow velocities. Moreover, for flow velocity of the order of a, the Reynolds number is large, for example already greater than 104 in a 1 mm capillarity tube, and so turbulent boundary layers develop in larger tubes, as observed in experiments.[1] A friction-driven increase in temperature, resulting from the existence of the boundary layer in a compressible flow, could also play a role. The remaining problem to be understood is flame acceleration to a velocity higher than 300 m/s during the period after flame ignition when the induced flow is still laminar and the lead shock too weak to produce a significant increase in temperature. For very energetic mixtures (UL ≈ 10 m/s) this is not difficult to understand since, as already explained, the initial quasi-isobaric acceleration (2.8.8), at a rate 2Ub /R, following a point ignition at the closed end of the tube, is sufficient to rapidly produce an upstream flow velocity exceeding 500 m/s. For hydrocarbon–air mixtures (UL < 0.5 m/s) in smooth tubes, DDT is generally not observed for a tube length less than 30 m. Flame acceleration leading to detonation in such conditions (if any) is much more difficult to decipher. The quasi-isobaric elongation mechanism of the flame shape, investigated by Bychkov et al.[2] leading to a parallel flow velocity increasing exponentially in time with a growth rate of the order UL /R, could be

[1] [2]

Kuznetsov M., et al., 2005, Shock Waves, 14(3), 205–215. Bychkov V., et al., 2005, Phys. Rev. E, 72(4), 046307.

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relevant in capillary tubes (R ≈ 1 mm), as shown in numerical simulations.[3] However, the experiments[4] were not able to confirm the corresponding tubular flame structure. To summarise, DDT in smooth tubes is understood for energetic mixtures such as stoichiometric acetylene–oxygen or hydrogen–oxygen (UL ≈ 10 m/s); however, the situation is not so clear for much less energetic mixtures such as ordinary fuel–air mixtures (UL < 0.5 m/s). To conclude this section on DDT, recent large-scale experiments in methane–air mixtures (UL ≈ 0.38 m/s), performed in the context of explosion in coal mines,[5] are worth mentioning. Detonations in such conditions have been ignited by a very strong explosion at the closed end of a tube filled with very lean mixtures near to the limit where a planar detonation cannot be initiated in ordinary tubes. The wave takes the form of a spinning detonation. The following explanation could be relevant: just downstream of a planar shock near to (or even just below) the detonability limit, the Neumann temperature is too low to sustain an ordinary planar detonation, but a detonation wave can propagate in the transverse direction into the shocked gas, forming triple point structures whose trajectory is a spiral.

4.4 Dynamics of Shock Fronts The multidimensional dynamics of ordinary detonation waves is strongly associated with that of the inert lead shock. It is thus useful to begin with a study of the latter.

4.4.1 Linear Stability of Shock Waves It has been known for a long time that gaseous planar shock waves are stable. Experimental and theoretical pioneering studies[6,7,8] reported power law relaxations of initial disturbances. Typically, the initial disturbances were produced either when the shock wave meets wedges on the wall of the tube or when the shock is reflected normally from a perturbed flat wall. Attention was focused on acoustic disturbances propagating in the shocked gas. Moreover, stroboscopic Schlieren photographs, taken to study the relaxation, also showed the formation of Mach stems (triple points) propagating in the transverse direction on the shock front. Power laws cannot be obtained by a normal-mode analysis; it is necessary to solve the initial-value problem using the Laplace transform method.[9,10] Nevertheless, it is instructive to start with a normal-mode analysis because it illustrates the physical mechanisms in the simplest terms. This topic was first investigated by D’yakov[11] and [3] [4] [5] [6] [7] [8] [9] [10] [11]

Valiev D., et al., 2013, Phys. Fluids, 25, 096101–16. Wu M., et al., 2007, Proc. Comb. Inst., 31, 2429–2436. Oran E., et al., 2015, Combust. Sci. Technol., 187, 324–341. Lapworth K., 1959, J. Fluid Mech., 6, 469–480. Briscoe M., Kovitz A., 1968, J. Fluid Mech., 31(3), 529–546. Van-Mooren K., George A., 1975, J. Fluid Mech., 68(1), 97–108. Erpenbeck J., 1962, Phys. Fluids, 5(10), 1181–1187. Bates J., 2004, Phys. Rev. E, 69, 056313. D’yakov S., 1954, Zh. Eksp. Teor. Fiz., 27, 288.

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Kontorovich[1] for arbitrary media. Earlier references, some of them published in the form of reports, as well as recent works, may be found in the modern literature.[2,3] Here we focus attention on polytropic gases, including both inert and reactive shock waves. The normal-mode analysis is straightforward but its interpretation is laborious. The details of the calculation are presented in the second part of the book (Section 12.1), in a simpler way than in the original works and in textbooks.[4] In this section we limit the presentation to the method of solution and to a discussion of the main results. Formulation and Method. Acoustic Wave and Entropy–Vorticity Wave The shock wave is considered as a hydrodynamic discontinuity of zero thickness. To simplify the notation we consider a two-dimensional geometry. The extension to three dimensions is straightforward. Consider a planar shock wave propagating in the negative x direction at a constant (supersonic) velocity D in a uniform medium. This requires an external device to trigger the shock, for example a piston at constant velocity in the shocked gas. We assume that such a piston is at infinity and does not generate disturbances. This is called an ‘isolated shock’ in the literature. In the reference frame of the unperturbed planar solution, the unperturbed front stands perpendicular to the x-axis at x = 0 and the shocked material flows at a constant (subsonic) velocity uN > 0 in the positive x direction. Let x = α(y, t) represent the perturbed position of the shock front at the transverse position y and at time t. For any physical quantity f we introduce the decomposition f = f +δf , where f represents the unperturbed solution. The wave being supersonic, the upstream medium is unperturbed and is assumed to be uniform. The x and y components of the flow velocity in the shocked gas, written in the reference frame of the unperturbed shock, are denoted by u = uN + δu and w = δw. The flow is considered as inviscid and heat conduction is neglected. The perturbed flow δu, δw, δρ and δp is the solution of the four linearised Euler equations (12.1.1)–(12.1.3) in which the entropy equation (12.1.3) is introduced since the entropy is modified at the Neumann state of the wrinkled shock front. The material can be characterised by an equation of state written in the entropy form. Expressing the entropy in terms of the density and pressure, s(p, ρ), the sound speed is a = ∂p/∂ρ|s=cst. . The boundary conditions are given at x = 0 by the linearised Hugoniot relations, (12.1.25)– (12.1.26), and a boundedness condition is enforced downstream in the shocked material for x → ∞. Generally speaking, the disturbance of the flow, the solution to the linearised Euler equations around a uniform state, is decomposed into two different linear modes: acoustic waves, solutions to d’Alembert’s equation (12.1.5), and an isobaric incompressible rotational flow, called a vorticity wave, which is simply convected downstream at the unperturbed flow velocity, along with the entropy disturbance; see (12.1.7)–(12.1.8). The method of solution is as follows. The acoustic waves in the shocked gas are solved in terms [1] [2] [3] [4]

Kontorovich V., 1957, Zh. Eksp. Teor. Fiz., 33, 1525. Bates J., 2007, Phys. Fluids, 19, 094 102–1–6. Bates J., 2012, J. Fluid Mech., 691, 146–164. Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed.

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of the dynamics of the shock front using the pressure disturbances at x = 0 given by the Rankine–Hugoniot conditions. The usual acoustic relations then yield the flow velocity in the acoustic waves, and in particular the value at the Neumann state, x = 0. The vorticity wave is then computed using its value at x = 0, which is given by the difference between the total flow velocity and the flow velocity in the acoustic wave; see (12.1.16)–(12.1.17). An equation for the evolution of the wrinkles on the front is then obtained by imposing that the vorticity wave is an incompressible flow and that the acoustic flow is bounded at infinity. Dispersion Relation from a Normal-Mode Analysis For a harmonic perturbation of the front position, a normal-mode decomposition is used, as in (2.2.1)–(2.2.2), iky , α(y, t) = α(t)e ˜

δf (x, y, t) = f˜ (x)eiky+σ t ,

α(t) ˜ = αe ˆ σ t,

(4.4.1)

where k is the transverse wave vector (a real quantity), σ = s + iω is a complex whose real part s is the linear growth rate (or damping rate if s < 0) and the imaginary part ω is the frequency of oscillation in time. Power laws α(t) ˜ ∝ tν can be obtained for s = 0 by using the Laplace transform, as discussed later. Following the method outlined above, the solution takes the form of an equation for σ in terms of k, called the dispersion relation. In the absence of other reference length and time scales (the shock being considered as a hydrodynamic discontinuity), the complex linear rate σ must be proportional to the inverse of the wavelength times a velocity, namely the unperturbed shock velocity, σ ∝ D|k|. Considering the shocked material in the reference frame of the unperturbed wave, the natural velocities are the sound speed aN and the flow velocity uN = M N aN , both of which are proportional to D with a coefficient of proportionality involving the Mach number M N ; see for example the Rankine–Hugoniot relations (4.2.14)–(4.2.17) for a polytropic gas. The dispersion relation then takes the form of an equation for σ/(aN |k|), or more precisely for S,  ±2M N S 1 + S2 = (1 + r)S2 + (1 − r)n, (4.4.2) 2

(1 − M N )1/2 S ≡ σ/(aN |k|)

where

(4.4.3)

(see Section 12.1.2), and where r and n are nondimensional parameters characterising the material and the strength of the shock wave, r≡−

(ρu D)2 dpN /dρ −1 N

2

> 0,

n≡

ρN MN  . ρu 1 − M 2 N

(4.4.4)

The parameter r involves the mass flux across the planar wave ρu D and the slope of the Hugoniot curve (4.2.24), at the Neumann state, in the plane (p, 1/ρ). For usual gases 0 < r < 1; see Fig. 4.2. The parameter n > 0 involves the density ratio ρ N /ρu and the Mach number of the flow of shocked gas at the Neumann state of the unperturbed planar wave,

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Neutral

Radiating

Stable

Nonradiating

Figure 4.17 General stability limits for shock waves; σ is the complex linear growth rate. The critical value r∗ is given in (12.1.28).

M N < 1. The square-root term on the left-hand side of (4.4.2) is a pure pressure term, whereas the right-hand side involves the vorticity wave. The sign must be chosen such that the downstream acoustic waves are bounded; see the text below (12.1.12). The solutions to (4.4.2) are also roots of the quadratic equation for S2 , obtained by taking the square of (4.4.2). Only the roots that satisfy both Equation (4.4.2) and the boundedness condition at infinity should be retained to represent the physical solution. Relation with the Initial-Value Problem. Power Laws In parameter space, the regions of instability, Re(σ ) > 0, and of linear stability, Re(σ ) < 0, are separated by a wide region (of finite size) where the normal modes are neutral, Re(σ ) = 0; see Fig. 4.17. This region corresponds to intermediate values of r, −(1+2M N )  r  r∗ , where r∗ is defined in (12.1.28). All the normal modes of gaseous inert shock waves and of reactive gaseous shocks (detonations without modifications to the inner structure), are neutral; see Sections 12.1.3–12.1.5. Neutrality of modes does not necessarily mean that the amplitude of initial disturbances does not evolve in time in the linear approximation. It only excludes exponential growth or damping. Neutral modes may well evolve with power laws in time or with even more complicated laws. This can be better understood by considering the linear solution to the initial-value problem obtained by Laplace transform (with respect to time) α(z), ˘  s+i∞  ∞ 1 e−zt α(t)dt, ˜ α(t) ˜ = ezt α(z)dz, ˘ α(z) ˘ ≡ 2iπ s−i∞ 0 where s is taken to be the largest imaginary part of all the singularities in α(z). ˘ A neutral mode represents a pure oscillation of an initial harmonic disturbance of the shock front (neither damped and nor amplified) only if it corresponds to a simple pole z = iω of α(z) ˘ on the imaginary axis of z. Denoting by F(σ ) = 0 the dispersion relation obtained by the normal-mode analysis, the Laplace transform of the amplitude of the wrinkles takes the form α(σ ˘ )/α(0) ˜ = N(σ )/F(σ ), where N(σ ) is a function of the complex variable that has no other singularity than a discontinuity through a branch cut.[1,2] Due to the √presence of acoustics, according to (4.4.2), the function F(σ ) involves a square-root term, S2 + 1, that introduces a branch cut in the complex plane for σ . Therefore, purely imaginary roots of [1] [2]

Bates J., 2007, Phys. Fluids, 19, 094 102–1–6. Bates J., 2012, J. Fluid Mech., 691, 146–164.

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the dispersion relation (neutral normal modes) do not necessarily correspond to undamped oscillations of harmonic disturbances of the shock front. This may be illustrated by√simple mathematical examples; the inverse Laplace transforms of 1/zν , n! /(z−a)1+n , 1/ z2 + 1 √ and 1/[ z2 + 1 + z] are, respectively, t(ν−1) / (ν), tn eat , J0 (t) and J1 (t)/t, where J0 (t) and J1 (t) are the zero-order and first-order Bessel functions of the first kind.[3] The third example shows that the roots σ = ±i do not yield a simple harmonic oscillation. It turns out that the initial disturbances of fronts in the domain (n−1)/(n+1) < r < r∗ are damped,[1,4] with an asymptotic dependence for t → ∞ as t−3/2 times an oscillatory function. Spontaneous Emission of Sound. Acoustic Wave Reflexion The stability of shock waves is characterised by neutral normal modes; see Section 12.1.3. The stability analysis was discussed initially in a different way by considering the reflexion of planar acoustic waves impinging on the front from the shocked gases.[5,6] This analysis is presented in Section 12.1.4. The front is considered to be unstable if there exist particular incident waves for which the reflexion coefficient diverges. This is the case when the acoustic wave of a neutral normal-mode is ‘radiating’, that is, when it propagates from the shock front towards infinity in the shocked gases. This situation is called ‘spontaneous emission of sound’. More precisely, the divergence of the reflexion coefficient occurs when the reflected acoustic wave matches the radiating normal mode. Such a divergence never occurs if all acoustic modes are nonradiating, that is, when they propagate from infinity towards the shock front. Therefore the instability threshold is defined by the critical condition separating the cases for which all acoustic modes are nonradiating from those for which one, at least, of the acoustic modes is radiating (spontaneous sound emission). The corresponding critical value for r, (n − 1)/(n + 1), in the middle of the range of neutral modes in Fig. 4.17 is considered to be the stability limit in parameter space. The fronts characterised by neutral modes are unstable for r < (n − 1)/(n + 1) and stable in the opposite situation. Inert gaseous shock waves are stable since they have only neutral modes with nonradiating acoustic waves, (n − 1)/(n − 1) < r < r∗ ; see Section 12.1.4. Concluding Remarks The nonradiating acoustic waves of the normal modes of an ‘isolated’ shock could lead one to question the physical relevance of such modes, since no energy in the compressed gas can be sent to the front from infinity (causality condition). The shock–vortex interaction helps to clarify the role of normal modes with nonradiating acoustic waves. If the turnover velocity of a subsonic vortex of size L in the initial gas is small compared with the shock velocity D, the lapse of time, τint = L/D, during which the shock crosses the vortex is shorter than the turnover time. After τint , the shock front is weakly wrinkled and propagates into a quiescent medium. The transmitted vortex emits acoustic waves in every direction [3] [4] [5] [6]

McQuarrie D., 2003, Mathematical methods for scientists and engineers. University Science Books. Majda A., Rosales R., 1983, SIAM J. Appl. Math., 43(6), 1310–1334. Kontorovich V., 1957, Zh. Eksp. Teor. Fiz., 33, 1525. Fowles G., 1981, Phys. Fluids, 24(2), 220–227.

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Gaseous Shocks and Detonations Shock wave

Slip line Secondary shock

Secondary shock Slip line

Shock wave

Figure 4.18 Structure of triple points propagating in the positive and negative y-directions.

(backward and forward) in the compressed gas. This is also the case when a shock front is perturbed by crossing a bump on the wall of the tube. The normal modes with nonradiating acoustic waves may then be useful to describe the evolution of the wrinkles of the shock front for t > τint . After a sufficiently long time, nonlinear effects dominate the linear dynamics of the wrinkled shock front. The linear analysis is a preliminary step before performing a nonlinear study of the multidimensional dynamics of shock and detonation fronts.

4.4.2 Formation of Mach Stems The formation of singularities could be anticipated by Huygens’ construction shown in Fig. 2.10 for flames. However, this is not exactly the mechanism at work in shock waves. The Mach stems (triple points), which are systematically observed on weakly perturbed shock fronts, are formed by a similar but different mechanism. In agreement with the oscillation frequency of the normal modes, these triple points propagate in the transverse direction at a phase velocity close to the speed of sound in the shocked gas. The triple points are constituted by a cusp of the shock front, a slip line and a secondary shock in the shocked gas; see Fig. 4.18. The formation of singularities of the slope of the front is important for understanding the cellular patterns of detonations, which are also characterised by triple points; see Section 4.5.2. Patterns similar to those of cellular detonations are also observed on inert shock waves reflected normally from an undulated wall;[1] see the Schlieren pictures in Fig. 4.19, extracted from a recent experiment.[2] Triple points are also formed during the shock–vortex interaction, reproduced in two-dimensional geometry by direct numerical simulations for about 20 years.[3,4] A recent simulation[5] is presented in Fig. 4.20. From the theoretical point of view the first analyses focused attention on the sound wave generated by shock–vortex interaction.[6]

[1] [2] [3] [4] [5] [6]

Briscoe M., Kovitz A., 1968, J. Fluid Mech., 31(3), 529–546. Biamino L., et al., 2011, Pattern of triple points on a shock wave reflected from an undulated wall, private communication. Guichard L., et al., 1995, AIAA J., 33(10), 1797–1802. Ellzey J., et al., 1995, Phys. Fluids, 7(1), 172–184. Lodato G., Vervisch L., 2014, DNS of shock-vortex interaction using spectral difference high-order methods, private communication. Ribner S., 1985, AIAA J., 23(11), 1708–1715.

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(b)

267 (c)

Figure 4.19 Schlieren images showing a regular pattern of Mach stems propagating in the transverse direction on a shock front after reflexion on a undulated wall. The reflected shock propagates from right to left. Shock Mach number: 1.1. Time between pictures: 10−4 s, extracted from a film at a frame rate of 106 image/s. Courtesy of L. Biamo, G. Jourdan and L. Houas, IUSTI Marseilles. (a)

(b)

Figure 4.20 Direct numerical simulation of shock–vortex interaction in two-dimensional geometry. (a) Pressure field; the initial gas is on the left and the shocked gas on the right, pressure ratio 4.5. (b) Density gradient obtained by a simulated Schlieren method. Shock Mach number: 2. Rotation Mach number of the initial cylindrical vortex: 0.8. Courtesy of G. Lodato and L. Vervisch 2014, CORIA Rouen.

Triple Points The formation of cusps of the shock front is more easily analysed in two-dimensional geometry. The linear flow of the compressed gas helps us to understand both the secondary shock and the slip line of the Mach stem. According to the linear analysis, the flow takes the form (12.1.7)–(12.1.8) δu = δu(i) (y, t − x/uN ) + δu(a) (x, y, t), δw = δw(i) (y, t − x/uN ) + δw(a) (x, y, t), where the superscripts (i) and (a) denote the incompressible vorticity wave and the acoustic wave, respectively; see Fig. 4.21. Using the normal mode decomposition in (4.4.1) and the notations of (12.1.10) and (12.1.37), the flow in the progressive waves of a normal mode may be written δu(i) = u˜ (i) eik[y−(ω/uN k)x+(ω/k)t] ,

δw(i) = w˜ (i) eik[y−(ω/uN k)x+(ω/k)t] ,

δu(a) = u˜ (a) eik[y+(l/k)x+(ω/k)t] ,

δw(a) = u˜ (i) eik[y+(l/k)x+(ω/k)t] ,

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Gaseous Shocks and Detonations Normal mode

Nonradiative acoustic wave

dary Secon o h s ck Slip line

Vorticity wave (shear flow)

Mach stem (triple point)

Figure 4.21 Flow of shocked gas associated with a progressive normal mode of a wrinkled shock front propagating into a quiescent gas (two-diimensional geometry). The picture shows the upward propagation of the progressive wave associated with a harmonic disturbance of the front. The symmetric picture is obtained for the progressive disturbance propagating downwards. The shear flow in the vorticity wave is indicated by the dashed lines; the location of the velocity extrema of the nonradiative acoustic wave is indicated by the dotted lines. The sketch on the right shows Mach stems formed by the nonlinear evolution of the progressive disturbance propagating upwards. To leading order in the limit (4.4.9) the phase velocity, ω/k, of the wrinkles on the front is equal to the sound speed in the shocked gas aN .

where ω and l are positive; see (12.1.36)–(12.1.38). The waves travelling in the opposite y-direction are obtained by changing the sign of y. The vorticity wave being incompressible,

∂δu(i) /∂x + ∂δw(i) /∂y = 0,

∂δu(i) /∂x = −(1/uN )∂δu(i) /∂t,

(4.4.5)

it is a shear flow since the velocity vector is aligned with the iso-velocity line, w˜ (i) /˜u(i) = ω/(uN k); see Fig. 4.21. If, for any reason, a cusp is formed on a wrinkle of the shock, the corresponding part of the shear flow degenerates into a slip line with a slope ω/uN k, and a secondary shock with a slope close to l/k is created on the acoustic wave. The triple point associated with these progressive waves then propagates in the transverse direction on the front with a phase velocity close to ±ω/k; see Fig. 4.18. The difficult problem is to decipher the basic mechanism of cusp formation: is it created by wave breaking of the acoustics? Or, inversely, is the secondary shock a consequence of cusp formation? The intriguing relation between the formation of cusps on the shock front and that of the secondary shock is described analytically below in a limiting case.

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Generally speaking, the pressure and density jumps at the front x = α(y, t) are obtained when the expression for the local Mach number of the propagation in the normal direction   D + δu1f − α˙ t − δw1f αy , α˙ t ≡ ∂α/∂t, αy ≡ ∂α/∂y, (4.4.6) Mu =  1/2 2 au 1 + αy is introduced into (4.2.14)–(4.2.15). Here the subscript f denotes the value at the front, and a disturbance of the upstream flow (δu1 , δw1 ) has been retained in view of the study of the shock–vortex interaction. Conservation of mass (15.1.45) and transverse momentum (15.1.47) yield (see also (12.1.21)),     (4.4.7) ρuf D + δu1f − α˙ t − δw1f αy = ρN uN + δuN − α˙ t − δwN αy

  δw1f + D + δu1f αy = δwN + (uN + δuN ) αy , (4.4.8) where ρu ≡ ρ u + δρ1 , ρN ≡ ρ N + δρ and (δuN , δwN ) are the modifications to the components (in the frame attached to the planar and unperturbed front) of the flow at the Neumann state (shocked gas at the shock front). Equations (4.4.6)–(4.4.8) are general; they are limited not by the amplitude of wrinkling compared with the wavelength, but only by the approximation of a shock wave considered as a hydrodynamic discontinuity separating two inviscid gas flows. Strong Shock in the Newtonian Approximation Except for a semi-phenomenological geometrical approach[1] and a weakly nonlinear analysis for radiating acoustic waves,[2] there was no analysis of the Mach stem formation on wrinkled shock fronts until recently when a nonlinear analysis was performed[3] for a strong shock M u  1 in the Newtonian limit (γ − 1) 1, 2

M u  1,

M u (γ − 1) = O(1).

(4.4.9)

According to the Rankine–Hugoniot relations (4.2.14)–(4.2.17), the flow in the shocked gas is strongly subsonic in this limit, 2

2

M N ≈ (γ − 1)/2 + 1/M u ,

(4.4.10)

and a perturbation analysis can be performed using the Mach number in the shocked gas as the small parameter,  2 ≡ γ MN2 1, −1

 ≈ (aN /au )M u = aN /D,

2

(aN /au )2 ≈ [2 + (γ − 1)M u ]/2 = O(1),

uN /D = ρ u /ρ N ≈  , 2

pN /pu ≈

2 Mu

= O(1/ ).

−2

2

(4.4.11) (4.4.12)

Both quantities M u > 0 and (γ − 1) > 0 are of order  2 , (γ − 1)/(2 2 ) < 1. The perturbation analysis in the limit (4.4.9) provides a qualitatively good physical insight into [1] [2] [3]

Whitham G., 1957, J. Fluid Mech., 2(02), 145–171. Majda A., Rosales R., 1983, SIAM J. Appl. Math., 43(6), 1310–1334. Clavin P., 2013, J. Fluid Mech., 721, 324–339.

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the formation of Mach stems. In the linear approximation, the jump relations (4.2.14)– (4.2.15) with (4.4.6) take the form



 2 δu1f − α˙ t δu1f − α˙ t δρ1f δp1f au δρN δpN − =2 − ≈2 , (4.4.13) , ρN ρu aN pN pu D D 2

(δuN − α˙ t ) =

[(γ − 1)M u − 2] δu , − α ˙ 1f t 2 2M u

δwN ≈ Dαy + δw1f ,

(4.4.14)

where, for simplicity, some unimportant terms of order  2 have been omitted in the first equation (4.4.13) and in the second equation (4.4.14). To leading order in the limit (4.4.9), the dispersion relation (4.4.2), in the form (12.1.29) for a polytropic gas, reduces to S2 ≈ −1, S ≈ ±i, σ 2 + a2N k2 = 0, ω ≈ aN k, corresponding to a wave equation for the wrinkles on the front ∂ 2 α/∂t2 − a2N ∂ 2 α/∂y2 = 0.

ω ≈ aN k,

(4.4.15)

This is because the flow in the acoustic wave is negligible compared with the incompressible shear flow in this limit, as shown now. According to (12.1.15) and (12.1.37), l/|k| = O(), the order of magnitude of the acoustic wave is  

δw(a) = O δpN /ρ N uN . (4.4.16) δu(a) = O  2 δpN /ρ N uN , According to (4.4.13) and (4.4.14), (δuN − α˙ t ) = O( 2 α˙ t ),

δwN = O(α˙ t /),

δpN /pN = O(α˙ t /D).

(4.4.17)

The order of magnitude of the acoustic wave is then obtained by introducing (4.4.17) into (4.4.16) using pN ≈ ρ N uN aN /M N and aN /M N = (aN /au )D/(M u M N ) = O(D): δpN /(ρ N uN ) = O(α˙ t )

⇒

δu(a) = O( 2 α˙ t ),

δw(a) = O( α˙ t ).

The comparison with (4.4.14), δuN ≈ α˙ t , δwN ≈ Dαy = O(α˙ t /), shows that the ratio of the acoustic flow to the flow in the vorticity wave is of order  2 , the latter being a shear flow quasi-parallel to the unperturbed shock front, |δu(a) /δu(i) | = O( 2 ),

|δw(a) /δw(i) | = O( 2 ),

|δu(i) /δw(i) | = O().

Therefore, to leading order, the linear dynamics of a wrinkled shock wave propagating in a quiescent gaseous medium is controlled by the vorticity wave. The vorticity wave is obtained from the leading order of the linearised Rankine–Hugoniot relations, (4.4.13)– (4.4.14),

to give

δu(i) |x=0 = δuN ≈ α˙ t ,

δw(i) |x=0 = δwN ≈ Dαy ,

δu(i) = α˙ t (y, t − x/uN ),

δw(i) = Dαy (y, t − x/uN ).

(4.4.18)

The wave equation (4.4.15) is then obtained directly from the incompressible condition (4.4.5) using (4.4.11)–(4.4.12) in the form DuN ≈ a2N . The transverse propagation, at the

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sound velocity a2N of disturbances on the front, is produced by the tilted shear flow. Compressible effects (acoustic waves) in the shocked gas are not involved in this phenomenon.

Weakly Nonlinear Analysis for the Formation of Folds In the same conditions and for sufficiently small amplitude of the wrinkles, acoustic waves are also negligible in the nonlinear dynamics of a weakly wrinkled shock propagating in a quiescent medium. Denoting the quadratic terms U and W, the Euler equations in the compressed gas can be written in the form 1 ∂p ∂u ∂u + uN =U− , ∂t ∂x ρ ∂x ∂u ∂u −U ≡ δu +w , ∂x ∂y

∂w 1 ∂p ∂w + uN =W− , ∂t ∂x ρ ∂y ∂w ∂w − W ≡ δu +w , ∂x ∂y

where δu ≡ u − uN and U = W = 0 in the unperturbed flow. These equations are completed by the equation for mass conservation. A weakly nonlinear approximation is valid when the quadratic terms U and W are small compared with the unsteady terms. When expressed in terms of the vorticity wave (4.4.18) and using the relation DuN ≈ a2N , the quantities U and W take the form U≈ where

1 ∂H , 2 ∂x

W≈−

1 D ∂H , 2 uN ∂y

(4.4.19)



H ≡ [−α˙ t2 (y, t − x/uN ) + a2N αy2 (y, t − x/uN )].

The relative order of magnitude of these source terms compared with the linear terms is ε ≡ |αy |/ ≈ |α˙ t |/uN . A perturbation analysis using the small parameter ε 1 is thus valid for sufficiently small amplitudes of the wrinkles |αy | . The acoustic waves introduce quadratic terms that are smaller by at least a factor . The weakly nonlinear analysis is thus performed in the limits  1

and

ε ≡ |αy |/ ≈ |α˙ t |/uN 1.

(4.4.20)

The source terms U and W in (4.4.19) satisfy the same equations as the linear vorticity wave, ∂U/∂t +uN ∂U/∂x = 0 and ∂W/∂t +uN ∂W/∂x = 0, so that they would introduce a secular contribution (a term growing linearly in time) to u and w if they were not balanced by a pressure term. However, it is not necessary to carry out this calculation here, since the source terms U and W are zero for simple progressive waves, α˙ t = ±aN αy ⇒ H = 0. This is because the shear flow associated with the progressive waves of normal modes is an exact solution of the incompressible Euler equations. More generally, for an initial disturbance of the front of finite size L, the source terms U and W cannot influence the front geometry, since, after a short finite time, the term H vanishes for t > L/aN , namely when the two progressive waves (propagating up and down) no longer overlap. These terms cannot play a significant role in the ultimate formation of cusps on wrinkles of small amplitude. However, this is not true for initial wrinkles whose amplitude is of the same order as the wavelength,

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which is a case beyond the scope of the present analysis. To leading order of the weakly nonlinear analysis, the Euler equations then reduce to their linear form and the flow takes the same form as in the linear analysis, (12.1.7)–(12.1.8), δu ≡ u − uN = u(i) (y, t − x/uN ) + u(a) (x, y, t), w = w(i) (y, t − x/uN ) + w(a) (x, y, t),

(4.4.21) (4.4.22)

where the pressure fluctuations are fully taken into account by the acoustic flow (u(a) , w(a) ), and the flow (u(i) , w(i) ) is incompressible, ∂u(i) /∂x + ∂w(i) /∂y = 0. In the laboratory frame of the unperturbed front, the quadratic terms that introduce corrections of order ε to the dynamics come from the boundary conditions at the front. The latter are obtained by introducing (4.4.6)–(4.4.8) into (4.2.14)–(4.2.15)

 (δuN − α˙ t ) 1 −

a2u a2N

α˙ t pN 2 = 1 − 2 − αy , pN D 

 2

α˙ t

D

+ αy

2

= wN αy ,

(4.4.23) wN = Dαy − δuN αy ,

(4.4.24)

where the notation δuN ≡ uN − uN has been used and where, for simplicity, some unimportant terms of order uN /D ≈  2 in the linear approximation (4.4.13)–(4.4.14) have been neglected in (4.4.24). They do not change the nonlinear dynamics but introduce corrections to the linear dynamics that are not useful to describe cusp formation in a weakly nonlinear analysis. Anyway they are negligible in the intermediate regime  3 < |αy | < . Limiting attention to nonlinear corrections of order ε, the boundary conditions in (4.4.24) for the flow velocity in the shocked gas reduce to x = α:



w ≡ wN (y, t) ≈ Dαy ,

δu ≡ δuN (y, t) ≈ α˙ t + Dαy2 ,

(4.4.25)

where the terms of order  2 and the quadratic term in the expression (4.4.24) for wN ,  −δuN αy , which is of order |α˙ t |αy = aN αy2 , have been neglected. The latter corresponds to a correction of order |αy | =  2 ε relative to the linear approximation. The only correction   term of order ε ≡ |αy |/ is Dαy2 , D|αy2 /α˙ t | = O(|αy |/) since |α˙ t | = O(aN αy ) and D/aN = O(1/). The shift of the front position, induced by the wrinkling, also introduces quadratic terms into the boundary values at the origin x = 0:

δu ≡ uf (y, t) ≈ δuN − αux ,

w ≡ wf (y, t) ≈ wN − αwx ,

(4.4.26)

where ux (y, t) ≡ ∂u/∂x|x=0 , and wx (y, t) ≡ ∂w/∂x|x=0 . The coupling between the acoustic wave and the vorticity wave is negligible since it introduces nonlinear contributions that are smaller than those in (4.4.25) by a factor . The correction to the pressure in (4.4.23) being even smaller, the flow (u(a) , w(a) ) is negligible in front of (u(i) , w(i) ), to leading order in the weakly nonlinear analysis. Consequently the boundary conditions at x = 0 for (u(i) , w(i) ) are given by (4.4.26). The incompressible condition ∂u(i) /∂x + ∂w(i) /∂y = 0 then leads to a nonlinear equation for the wrinkles in the form −u−1 N ∂uf /∂t+ ∂wf /∂y = 0. As shown

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just below, the nonlinear terms coming from the shift of origin in (4.4.26) do not contribute to this equation, so that the nonlinear equation for the front wrinkles reduce to −

∂wN 1 ∂uN + = 0, uN ∂t ∂y

(4.4.27)

where uN and wN are given in (4.4.25). This is shown by computing −uN ∂(αux )/∂t = −uN α˙ t ∂u(i) /∂x|x=0 + α∂ 2 u(i) /∂x2 |x=0 , ∂(αwx )/∂y = αy ∂w(i) /∂x|x=0 + α∂ 2 w(i) /∂x∂y|x=0 . According to the incompressible condition, the sum of the last terms in the right-hand side  of these two equations is zero. It is also the case for the first terms for H ≡ a2N αy2 − α˙ t2 = 0 when u(i) and w(i) in the quadratic terms are replaced by the linear approximation (4.4.18), −uN α˙ t

∂δu(i) ∂δw(i) D   1 ∂H + αy = α˙ t α¨ t − = 0. α α˙ ≈ ∂x ∂x uN y ty 2 ∂t

According to (4.4.25) and (4.4.27), the weakly nonlinear equation for the evolution of the shock is then[1]   2 ∂ 2α ∂ ∂α 2 2 ∂ α − a + D = 0, (4.4.28) N ∂t ∂y ∂t2 ∂y2 where the relation a2N ≈ DuN , valid to leading order in the limit (4.4.9), has been used.  The ratio of the nonlinear term to the linear terms is effectively of order ε, Dαy2 /|α˙ t | ≈ (D/aN )|αy | ≈ ε since |α˙ t | ≈ aN |αy |. This result can also be obtained when working in the frame attached to the front; see Section 4.6.2. Equation (4.4.28) has two time scales: a short time τs ≡ L/aN , where L is the wavelength of the wrinkles, and a longer time τl ≡ τs /ε for the effect of the nonlinear term. This is more easily seen by introducing the reduced time scale and length scale τ ≡ t/τs , η ≡ y/L, and the reduced amplitude of order unity, A ≡ α/(εL) (see (4.4.20)),   ∂2A ∂ ∂A 2 ∂2A − +ε = 0. (4.4.29) ∂τ ∂η ∂τ 2 ∂η2 This equation can be written in a parameter-free form by introducing εA, but the form (4.4.29) is convenient to point out the two-time-scale nature of the problem. For any smooth initial wrinkle of amplitude of order unity, A = O(1), the slope ∂A/∂η develops a singularity (a corner of the front) after a finite time on the long time scale, τ  = τ/ε. Considering a simple progressive wave η = η ± τ , A = A(η , τ  ), Equation (4.4.29) takes the form of Burger’s equation (without dissipation) for the slope, A ≡ ∂A/∂η , ∂A /∂τ  + A ∂A /∂η = 0,

[1]

Clavin P., 2013, J. Fluid Mech., 721, 324–339.

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(4.4.30)

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known to produce a discontinuity after a finite time; see Section 15.3.1. This describes in simple terms the basic mechanism of Mach stem formation on gaseous shock waves in the limit (4.4.9). Concluding Remarks To summarise, Mach stem formation on a strong shock in the Newtonian approximation is associated with the formation of folds (lines of singularity of slopes) on the shock front. The nonlinear mechanisms responsible for their formation come from the boundary condition at the front (4.4.7) for the vorticity wave. They are different from Huygens’ construction even though the nonlinear term in the equation controlling the weakly nonlinear dynamics of front has the same form. More precisely, the dominant nonlinear effect comes from the term wN αy in the normal component of the flow velocity at the Neumann state. This   introduces a corrective term Dαy2 to α˙ t ; see (4.4.25). By comparison, a term Dαy2 /2 would have been introduced by Huygens’ construction for a front propagating at a constant normal   velocity Un = D in a quiescent medium, Un ≡ (D − α˙ t )/(1 + αy2 )1/2 ≈ D − α˙ t − Dαy2 /2; see (10.1.6). In the limit (4.4.9) the acoustic waves introduce smaller corrections that could be obtained by pushing the weakly nonlinear analysis to the next order. Such an extension of the perturbation analysis is necessary to describe the secondary shock wave. This has not yet been done. Numerical simulations[1] of (4.4.29) or, more precisely, its parameter-free form, using an initial harmonic disturbance of the shape of the front show qualitatively good agreement with the experiments reported in Fig. 4.19. Moreover, an interesting coalescence property is also observed when the simulation is started using a white noise disturbance of the shape of the front: the formation of a large-scale structure delimited by folds is observed on the front.[1] This is similar to the large-scale structure of the universe obtained, following the Zeldovich model[2] with a multidimensional form of Burgers’ equation.[3]

4.4.3 Shock–Vortex and Shock–Turbulence Interaction As already mentioned, the shock–vortex interaction has been extensively studied by numerical simulation in two dimensions. From the theoretical point of view the analyses focused attention on the sound wave generated by shock–vortex interaction.[4] Impressive direct numerical simulations for the shock–turbulence interaction have been performed by the Standford group.[5] In their work the attention is focused on the transmitted turbulence in the shocked gas. On the theoretical side, extensive analytical studies have [1] [2] [3] [4] [5]

Denet B., et al., 2015, Combust. Sci. Technol., 187(1-2), 296–323. Shandarin S., Zeldovich Y., 1989, Rev. Mod. Phys., 61(2), 185–222. Gurbatov S., et al., 2012, Sov. Phys.–Uspeki, 55(3), 223–249. Ribner S., 1985, AIAA J., 23(11), 1708–1715. Larsson J., Lele S., 2009, Phys. Fluids, 21, 126101.

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275

Weak acoustic pulse ity r tic Vo ave w

Initial vortex

Transmitted vortex Strong acoustic burst

Figure 4.22 Sketch of the shock–vortex interaction. Reproduced from Clavin P., 2013, J. Fluid Mech., 721, 324–339, with permission.

been carried out in the linear approximation.[6] A nonlinear analytical study has been recently performed in the limiting case (4.4.9) and for very subsonic velocity fluctuations of the upstream flow[7] , ve /au . The objective of this study, summarised below, is to set up a simple model equation for the dynamics and geometry of the shock front; see (4.4.41). Interaction of a Strong Shock and a Weak Vortex Consider a cylindrical and very subsonic vortex of diameter L and turnover velocity ve /au  (vortex strength:  = π Lve ). The axis of the vortex is parallel to the planar shock front and perpendicular to the plan Oxy. The interaction is sketched in Fig. 4.22. The velocity of the vortex centre, relative to the upstream gas, will be neglected compared with D. The analysis is performed below for a vortex of finite extension, L. It can be extended to take account of the long tail of real vortices.[7] The method of solution for a strong shock in the Newtonian approximation (4.4.9) can be summarised as follows. In the limiting case under investigation, a small distortion of the shock front is generated during the short lapse of time τint = L/D, taken by the vortex to cross the shock (interaction time). The amplitude of the wrinkle is much smaller than its transverse extension L, the 2 order of magnitude of the slope of the wrinkle being of order ve aN /D ; see (4.4.38). After the interaction time, t > τint , the wrinkled shock front propagates in a quiescent medium. According to the previous linear analysis, the wrinkles on the front are then expected to propagate in the transverse direction at a velocity aN . This introduces a time scale of order L/aN that is larger than τint by a factor 1/ in the limit (4.4.9). Because of this difference of time scales, the initial condition for the subsequent nonlinear evolution of the front wrinkling t > τint is, roughly speaking, provided by a linear analysis of the vortex–shock crossover, 0 < t < τint .

[6] [7]

Wouchuk J., et al., 2009, Phys. Rev. E, 79(066315). Clavin P., 2013, J. Fluid Mech., 721, 324–339.

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Solution during the Short Period of Crossover The small short-lived perturbations of the upstream flow, δu1f , δw1f , δp1f , to be introduced into the linear jump conditions (4.4.13)–(4.4.14), are δu1f (y, t) = ue |x=−Dt ,

δw1f (y, t) = we |x=−Dt ,

δp1f (y, t) = pe |x=−Dt ,

(4.4.31)

where ue (r), we (r), pe (r) denote the turnover flow and pressure of the vortex and where the origin of time t = 0 is the beginning of the shock–vortex crossover. Consider first the pressure disturbances that are generated in the compressed gas during the vortex–shock crossover. To leading order in the limit (4.4.9), the linear equations of acoustics in the compressed gas yield  2  ∂ p ∂ 2p 1 ∂p ∂w(a) 1 ∂p ∂ 2p ∂u(a) 2 , (4.4.32) ≈− , ≈− , ≈ a + N ∂t ρ N ∂x ∂t ρ N ∂y ∂t2 ∂x2 ∂y2 where the Doppler shift has been neglected for 0 < t < τint . The time scale, τint , and transverse length scale, L, are imposed by the quasi-impulsive source terms at the front x = 0, x = 0,

0 < t < τint :

∂/∂t = O(D/L),

∂/∂y = O(1/L).

(4.4.33)

According to the last equation in (4.4.32), the pressure burst generated in the compressed gas during this short lapse of time is then quasi-planar, ∂p/∂x ≈ (1/aN )∂p/∂t ≈ (D/aN )δp/L ≈  −1 (∂p/∂y); see (4.4.11). A simplification occurs at low turnover Mach number, ve /D 1. In this limit the disturbances of pressure and density, δp1 /pu and δρ1 /ρ u , are of order (ve /au )2 . The upstream pressure fluctuation δp1f becomes negligible in the jump conditions for the pressure (4.4.13) for a sufficiently small turnover velocity

δpN /pN ≈ 2 δu1f − α˙ t /D, (4.4.34) ve /au : since δp1f /pu is smaller than the first term in the right-hand side of the first equation in (4.4.13), (δp1f /pu )/(ve /D) ≈ (ve /au )2 /(ve /D) ≈ (ve /au )/ 1. In this limit the pressure pulse in the compressed gas is generated by an impulsive source, of lifetime τint and transverse extension L, constituted mainly by a fluctuation of the longitudinal flow. The longitudinal component of flow velocity associated with the quasi-planar pressure pulse radiated in the compressed gas during this short lapse of time, δu(a) ≈ δp/(ρ N aN ), takes the form ve /au ,

0 < t < τint :

δu(a) |x=0 ≡ δu(a) ˙ t ), N ≈ 2(aN /D)(δu1f − α

(4.4.35)

and, according to the first and the second equations in (4.4.32), |δw(a) /δu(a) | = O(). According to the jump conditions (4.4.14), δuN ≈ α˙ t , δwN ≈ Dαy + δw1f , and to the

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277

flow splitting in (4.4.21)–(4.4.22), the flow velocity of the vorticity wave at the front, (i)

(a)

δw(i) |x=0 ≡ δwN = δwN − δwN ,

(i) (y, t) ≈ α˙ t − 2(aN /D)δu1f , δuN

(i) δwN (y, t) ≈ Dαy + δw1f − δw(a) N ,

δu(i) |x=0 ≡ δuN = δuN − δuN ,

(i)

(a)

takes the form (4.4.36)

where a correction of order  2 has been neglected in the first equation. The equation for the disturbances of the front position during the short interaction time 0 < t < τint is obtained by the incompressible condition of the vorticity wave, ∂δu(i) /δx + ∂δw(i) /δy = 0, in the form ve /au ,

0 < t < τint :

α˙ t ≈ 2(aN /D)δu1f ,

(4.4.37)

as shown now. According to the expressions for the vorticity wave, δu(i) (x, y, t) = (i) (i) δuN (y, t − x/uN ), δw(i) (x, y, t) = δwN (y, t − x/uN ), the relation between the derivatives with respect to space and time, ∂/∂x = −(1/uN )∂/∂t, with, according to (4.4.33), ∂/∂t = O(D/L), gives the order of magnitude for the derivative with respect to x of the longitudinal component, ∂δu(i) /∂x = O(|δu(i) |/ 2 L), where the order of magnitude, uN /D ≈  2 in (4.4.12), has been used. Using ∂δw(i) /δy = O(δw(i) /L), it is then found that the vorticity wave must be quasi-transverse, |δu(i) /δw(i) | = O( 2 ). The second term in the right-hand side of the first equation in (4.4.36) leads to a contribution to δu(i) that is of order δu1f . This term is too large, by a factor 1/, to satisfy the order of magnitude imposed by the incompressible condition, |δu(i) | = O( 2 |δw(i) |), as can be seen by anticipating that, to leading order, the second equation in (4.4.36) reduces to (i) δwN ≈ δw1f , where |δw1f /δu1f | = O(1). Therefore the second term, 2(aN /D)δu1f , in (4.4.36) must be balanced by the first term, α˙ t . This leads to (4.4.37) with, according to (4.4.31), δu1f (y, t) = ue (−Dt, y), where ue (x, y) is the longitudinal velocity of the vortex in the upstream gas. Then the amplitude of wrinkling is  aN 0 ue (x, y) ve /au , 0 < t < τint : α(y, t) = 2 dx , (4.4.38) D −Dt D 2

corresponding to a very small slope of the front |αy | of order ve aN /D = O( 2 ve /au ). The symmetry property of the wrinkle α(y, t) = −α(−y, t) results from the limiting case of a weak symmetrical vortex. However, because of the nonlinear character of the engulfment mechanism, the wrinkling is no longer symmetric for a large turnover velocity, ve of the order of au ; see Fig. 4.20. This effect is not taken into account in the present analysis. The shock–vortex interaction during the crossover, 0 < t < L/D, for ve /au  in the limit (4.4.9), can be summarised as follows: (i) A quasi-planar acoustic pulse is generated by the longitudinal component of turnover velocity of the vortex, δpN /pN ≈ 2δu1f /D,

(a)

δuN ≈ 2δu1f ,

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(ii) The transmitted vortex is a vorticity wave quasi-parallel to the front generated by the (i) (y, t − x/uN ), transverse component of turnover velocity of the vortex, δw(i) = δwN (i)

δwN ≈ δw1f ,

|δu(i) | = O( 2 |δw(i) |).

(iii) Wrinkles of very small amplitude are generated on the shock front during the crossover, α˙ t ≈ 2δu1f ,

|αy | = O(ve /D).

Evolution after Crossover After the crossover, t > τint , the shock wave propagates in the quiescent medium. The pressure pulse in the compressed gas becomes multidimensional and takes a quasi-cylindrical shape centred on the transmitted vortex core, as sketched in Fig. 4.22. However, in the limiting case considered here, the sound intensity varies with the azimuthal angle and decreases to negligible values near the shock front. This is because the sound is initially radiated from the shock front at x = 0 by a pulse of longitudinal flow of transverse extension L, while the transmitted vortex is an isobaric vorticity wave quasi-parallel to the front. The conditions for the subsequent evolution of the wrinkled shock front are then the same as in Section 4.4.2; Equation (4.4.28) holds for t > τint and describes the subsequent Mach stem formation. As mentioned at the beginning of Section 4.4.3, the key point for the validity of this simple description lies in the difference of time scales describing the crossover and the transverse propagation of the wrinkles, aN τint /L being a small quantity of order . Model Equation for Shock–Turbulence Interaction (2013) A composite solution for the wrinkled shock front produced by the shock–vortex interaction is given by a nonlinear equation that interpolates the linear equation (4.4.37) for 0 < t < τint and the nonlinear equation (4.4.28) describing the wrinkle dynamics for t > τint and cusp formation in the long time limit, |δu1f | : au

2 ∂ 2α ∂ 2 ∂ α − a +D N 2 2 ∂t ∂t ∂y



∂α ∂y

2 =2

aN ∂δu1f , D ∂t

(4.4.39)

where the forcing term is given by the vortex blob in (4.4.31) and varies on the short time scale of crossover, τint = L/D. The extension to a two-dimensional surface α(y, z, t) is straightforward. The corresponding equation can also be used as a model for shock–turbulence interaction. In this case the forcing term δu1f is the fluctuation of the longitudinal component of the velocity of the turbulent flow at the front. In contrast to flames, due to the supersonic propagation of the shock wave, the upstream turbulence is not modified by the wrinkling when approaching the front. However, the turbulence in the compressed gas is strongly different from that upstream.

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279

Introducing the reduced variables η ≡ y/L,

ζ ≡ z/L,

τ ≡ aN t/L,

φ ≡ α/(L),

(4.4.40)

Equation (4.4.39) multiplied by (L)−1 (L/aN )2 yields the nondimensional equation for the surface of the shock front φ(η, ζ , τ ) in the form   δu1f ∂ψ ∂ 2φ ∂|∇φ|2 (4.4.41) = , where ψ ≡ 2 − φ + aN ∂τ ∂τ ∂τ 2 and where, according to (4.4.11), the relation  ≈ aN /D has been used, aN /au = O(1). For ue /au  the forcing term ∂ψ/∂τ is a small function of order ue /(au ) varying in space with the transverse variables (η, ζ ) and in time with the short reduced time scale τ/. Focusing attention on the geometry and the dynamics of folds on the wrinkled shock front, representative of lines of Mach stems, instructive results are obtained from the numerical study[1] of (4.4.41) in a two-dimensional periodic box. The forcing term ψ(η, ζ , τ/) is extracted from a turbulence generator characterised by a k−5/3 Kolmogorov cascade. The main physical outcome is related to the coalescence property mentioned earlier; in the long time limit, the characteristic size of the cells delimited by the folds is not that of the length scales of the turbulent flow. The average cell size increases continuously with time and soon becomes much larger than the integral scale of the turbulent flow upstream from the shock front; see Fig. 4.23. The saturation mechanism of the cell size and the role of the size of the box (periodicity of η and ζ ) are not yet clearly identified.

4.5 Instabilities of Detonation Fronts It has been known for a long time that gaseous detonations are unstable and exhibit transverse structures. The experimental studies started as early as 1926 (spinning detonations). A detailed history of the topic can be found in the literature.[2,3] The cellular structure of detonation fronts was first observed in the 1960s by the markings left on soot-coated foils on the walls,[2] showing more or less regular diamond-shaped patterns. These markings correspond to the trajectories of triple points (Mach stems) on the lead shock, coupled to a pulsation of the internal structure of the detonation; see Fig. 4.30. More recent experiments[4] have shown more complex patterns in H2 -(NO2 /N2 O4 ) mixtures, such as double cellular structures; see Fig. 4.24. Modern optical methods,[5] including laser diagnostic techniques for imaging of chemical species within the detonation,[6] have improved the observation of cellular structures. The research in this field is still active;[7] see also

[1] [2] [3] [4] [5] [6] [7]

Denet B., et al., 2015, Combust. Sci. Technol., 187(1-2), 296–323. Shchelkin K., Troshin Y., 1965, Gasdynamics of combustion. Baltimore, Md.: Mono Book Corp. Strehlow R., 1979, Fundamentals of combustion. New York: Kreiger. Joubert F., et al., 2008, Combust. Flame, 152, 482–495. Presles H., et al., 1987, Combust. Flame, 70, 207–213. Mevel R., et al., 2014, J. Hydrogen Energy, 39, 6044–6060. Taylor B., et al., 2013, Proc. Comb. Inst., 34, 2009–16.

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(b)

Figure 4.23 Numerical simulation of (4.4.41) modelling the propagation of a shock wave in a turbulent flow. The simulations are performed in a square box of reduced size 5, with a reduced turbulent kinetic energy 0.078 and an integral scale of 0.15, corresponding to an amplitude of the forcing term ψ = 0.396 and a time scale of fluctuations of 0.1. The small integral scale is selected in order to highlight the ‘coalescence property’; the length scale of the structures on the wrinkled front increases with time and soon becomes notably larger than the integral scale of the vorticity in the flow, which is also the length scale of variation of ψ. The initial wrinkled front shown in (a) is characterised by integral scale of the upstream turbulence. The scale of the wrinkles in (b) at reduced time τ = 4 is typically 10 times larger. Courtesy of B. Denet, IRPHE Marseilles.

(a)

(b)

Figure 4.24 (a) Visualisation of a cellular detonation obtained by an optical method recording the deformation of an aluminised Mylar sheet hit by the detonation front. Reproduced from Presles H., et al., Combustion and Flame, 70, 207–213, Copyright 1987 with permission from Elsevier. (b) Markings of double cellular structure left on the wall by a detonation in a H2 -(NO2 /N2 O4 ) mixture. Reproduced from Joubert F., et al., Combustion and Flame, 152, 482–495, Copyright 2008 with permission from Elsevier.

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281

the special issue Philos. Trans. R. Soc. A 370 (2012). The explanation of detonation instabilities has been elusive for a long time, even though the diamond pattern of the cellular structure of an unstable detonation front has been reproduced since the 1990s in two-dimensional geometry by direct numerical simulations,[1,2,3] including more recently double cellular structures.[4,5] The problem is too complicated to be studied analytically in the general case. The purpose of this section is to provide understanding of the basic mechanisms through analytical solutions of simplified models. The nonlinear phenomena that control the spatio-temporal patterns of cellular detonation fronts have been analysed in a systematic way by weakly nonlinear analyses that are valid near to the instability threshold. These analyses are expected to point out the dominant physical mechanisms at work in real detonations. Two opposite limiting cases have been investigated using the Newtonian approximation: strongly overdriven detonations[6] and weakly overdriven detonations,[7] including the CJ wave. Unfortunately, in both cases, the multidimensional stability limits concern a small heat release, in contrast to ordinary detonations that are strongly unstable. However, instructive insights are provided in the first case, the analysis of which is summarised in Section 4.5.2 and developed in Section 12.2.4. The cellular instability of the second case, near the instability threshold of CJ waves, is presented in Section 12.2.3. Such waves are not physically relevant because the flow is transonic throughout the detonation structure, including the lead inert shock, so that the ZND structure is possible only for reaction rates that are quite artificial. However, the analysis is worth performing because it shows that the oscillatory instability is similar to that at large overdrive. The underlying physical mechanisms are better understood by considering first the onedimensional instability, presented now. 4.5.1 One-Dimensional Pulsations. Galloping Detonations One-dimensional oscillations, called galloping detonations, were first observed in the 1960s in direct numerical simulations.[8] They were visualised in the 1970s in the Schlieren photograph of a blunt projectile traversing a combustible gaseous mixture at a velocity close to the CJ detonation velocity.[9] Instability Mechanism: Loop and Phase Shift The origin of the one-dimensional instability can be roughly understood from the ZND structure of a detonation wave, sketched in Fig. 1.9 (see also Fig. 4.29), when the variation [1] [2] [3] [4] [5] [6] [7] [8] [9]

Boris J., Oran E., 1987, Numerical simulation of reactive flow. New York: Elsevier. Bourlioux A., Majda A.J., 1992, Combust. Flame, 90, 211–229. Mevel R., et al., 2014, J. Hydrogen Energy, 39, 6044–6060. Guilly V., et al., 2006, C. R. Acad. Sci. Paris, 334(11), 679–685. Joubert F., et al., 2008, Combust. Flame, 152, 482–495. Clavin P., Denet B., 2002, Phys. Rev. Lett., 88(4), 044502–1–4. Clavin P., Williams F., 2009, J. Fluid Mech., 624, 125–150. Fickett W., Wood W., 1966, Phys. Fluids, 9, 903–916. Lehr H., 1972, Acta Astronaut., 17, 589–597.

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of the induction delay with the temperature is taken into account. A disturbance that reinforces the lead shock increases the temperature at the Neumann state, shortening the induction delay and causing the heat release zone to move upstream relative to the lead shock. The upstream motion of the heat release zone acts as a semi-transparent piston generating an acoustic pulse that runs upstream and, in turn, further strengthens the lead shock, thereby intensifying the initial disturbance. The time delays for the temperature perturbations to be transmitted back and forth between the lead shock and the exothermic zone induce a phase shift that is responsible for the oscillatory nature of the instability. The transmission upstream is due to acoustics but the transmission downstream involves both an acoustic wave and an entropy wave that propagates with the subsonic velocity of the flow relative to the shock. The latter carries the strongest disturbances and imposes the longest delay in the loop. It is thus the dominant mechanism in the one-dimensional oscillatory instability, and also of the cellular instability described in Section 4.5.2, at least for strongly overdriven detonations in the limit (4.4.9). This one-dimensional analysis[1] of galloping detonations is presented now. General Formulation (Mass-Weighted Coordinate) According to the discussion below (4.2.45), molecular transport mechanisms are negligible (λ = 0 and Di = 0) and Equations (12.2.1)–(12.2.2) govern the unsteady structure of ZND detonations. Working in the reference frame of the unperturbed detonation located at x = 0 (with x < 0 in the fresh mixture) and denoting by α(t) the trajectory of the lead shock of the perturbed detonation, it is convenient to introduce the reduced mass-weighted distance from the shock, x, and the time, t, reduced by the unperturbed reaction time, tN , x≡

1 ρu DtN



x

ρ(x , t)dx ,

t≡

α(t)

t , tN

tN ≡ τr (T N ),

(4.5.1)

where the subscript N denotes the Neumann state. According to Section 4.6.3, the particulate derivative D/Dt ≡ ∂/∂t + u∂/∂x takes the reduced form ∂ ∂ D D = tN = + m(t) , Dt Dt ∂t ∂x

 m(t) ≡

ρ(x, t)[u(x, t) − α˙ t ] ρu D

,

(4.5.2)

x=α(t)

where m(t) is here the reduced mass flux across the moving shock, m(t) = D(t)/D,

where

D(t) ≡ D − α˙ t .

(4.5.3)

The velocity of the perturbed detonation relative to the quiescent fresh gas, D − α˙ t , and m(t) are the unknown functions of time characterising the dynamics of the planar detonation.

[1]

Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378.

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Equations (12.2.1)–(12.2.2) then take the form     ∂ u D ρu ⇔ = Dt ρ ∂x D / .   D u ∂ p =− , Dt D ∂x ρu D 2

D Dt



ρN ρ



283

∂ = ∂x



u uN

 ,

(4.5.4)

p = (cp − cv )ρT,

1 DT (γ − 1) 1 Dp qm − = w, ˙ T Dt γ p Dt cp T

Dψ = w, ˙ Dt

(4.5.5)

where we have used a shorthand notation for the species and the chemical kinetic scheme by introducing the progress variable ψ ∈ [0, 1], as in (4.2.49), and also the reduced heat ˙ ρqm w ˙ (j) ; see (15.1.42). For the simplest model of ˙ = tN j Q(j) W release rate, w ˙ = tN W, a one-step irreversible reaction governed by an Arrhenius law (1.2.2), one has w(ψ, ˙ θ) = (tN /τcoll )(1 − ψ) exp(−E/kB T). The boundary conditions at the Neumann state, x = 0, ψ = 0, x = 0:

ρ = ρN (t),

p = pN (t),

T = TN (t),

(4.5.6)

are functions of m(t), since uN , ρN , pN and TN are given by the Rankine–Hugoniot relations (4.2.14)–(4.2.16) in which the propagation Mach number Mu is a function of time Mu (t) = (D − α˙ t )/au = m(t)M u , δMu = −α˙ t /au . The boundary condition for the velocity u in the laboratory frame (relative to the unperturbed front) is given by mass conservation x = 0:

ρN (t)(u − α˙ t ) = ρu (D − α˙ t ).

(4.5.7)

The intrinsic longitudinal dynamics develop on the induction time scale and are not very sensitive to the rear boundary condition (in the burnt gas). In particular few differences are observed between an overdriven detonation and a CJ detonation in the numerical analyses. For an overdriven detonation with the piston at infinity (isolated detonation), the boundary condition is x → ∞:

u = ub .

(4.5.8)

Similar results are obtained using a radiation condition at the end of the reaction zone for the acoustic waves propagating in the burnt gas (see the discussion in the original article[1] ), x → ∞:

(p − pb ) − ρ b ab (u − ub ) = 0.

(4.5.9)

In the induction zone where the heat release is negligible, qm w/c ˙ p TN 1, the flow, the solution to (4.5.2)–(4.5.5) with the boundary conditions (4.5.6), may be decomposed into an isobaric entropy wave and acoustic waves, at least for small fluctuations. However, in general, this splitting is not possible in the exothermic zone. One option to overcome this technical difficulty would be to consider the limiting case of a reaction sheet of negligible thickness compared with that of the induction layer, namely the square-wave model, used in Section 4.3.3 for quasi-steady state regimes. This model is not convenient here because it leads to singular dynamics, as we shall see later. Other approximations must be considered to obtain analytical results.

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Overdriven Detonations in the Newtonian Approximation Generally speaking, the quasi-isobaric approximation is valid for a sufficiently subsonic flow when the evolution is slow on the time scale of acoustics; see Section 2.1.1. This is the case in the shocked gas of a strong shock in the Newtonian approximation (4.4.9)–(4.4.12), 2  2 ≡ γ M N 1, 2

M u = O(1/ 2 ),

(γ − 1) = O( 2 ),

h≡

(γ − 1) = O(1). 2

(4.5.10)

In the same limit, the flow of shocked gas is also quasi-isobaric throughout the detonation structure if the heat release is of same order of magnitude as the thermal enthalpy at the Neumann state, qN ≡ qm /cp T N = O(1),

(T b − T N )/T N ≈ qN .

(4.5.11)

This is the case for strongly overdriven detonation, as shown now. Conservation of momentum in (4.2.47) and mass, ρu = ρN uN , lead to a linear relation between velocity and pressure downstream of the lead shock expressed in terms of the Mach number at the Neumann state, MN ,

p/pN − 1 = − 2 (u/uN − 1) ,

2

 2 ≡ γ MN ,

(4.5.12)

where the relation a2N /γ = (pN /ρN ) has been used. The temperature ratio T b /T N , the velocity ratio ub /uN and the density ratio ρ b /ρ N being of order unity while the relative pressure variation is of order  2 , the flow in the compressed gas satisfies the quasi-isobaric approximation. However, the two conditions (4.5.10) and (4.5.11) are compatible only for strongly overdriven detonations: according to (4.4.12), the ratios aN /au and T N /Tu are of order unity, consequently, according to (4.2.44), the condition in (4.5.11) implies that MuCJ is also of order unity, as is qm /cp Tu , so that M u  MuCJ . The problem of overdriven detonations in the Newtonian limit then reduces to solving the equations for conservation of energy and species in (4.5.5), which form a closed set of equations since the term (γ − 1)Dp/Dt is negligible, ∂T qm ∂T + m(t) = w(ψ, ˙ T), ∂t ∂x cp

∂ψ ∂ψ + m(t) = w(ψ, ˙ T). ∂t ∂x

(4.5.13)

Using the boundary condition (4.5.6), x = 0:

ψ = 0,

T = TN (t),

(4.5.14)

the solution to (4.5.13)–(4.5.14) yields the spatial distributions of temperature and heatrelease rate in terms of TN (t). The boundary condition in the burnt gas yields a nonlinear equation for the dynamics of the front. A useful integral equation is obtained by spatial integration of the equation for energy conservation in (4.5.13) written in the quasi-isobaric

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approximation, ρT = ρ N T N using the boundary conditions (4.5.7),     u ∂ T ∂ ∂ , + m(t) = ∂x uN ∂t ∂x T N    ∞ α˙ t ub ρu D ρu − − 1− = qN w ˙ dx, uN ρN (t)uN ρN (t) uN 0

285

(4.5.15) (4.5.16)

where the reduced heat release qN is a parameter of order unity; see (4.5.11). The remaining problem is to find the instantaneous distribution of rate of heat release w ˙ as a function of x and t. This is more easily done with the additional simplification of a large activation energy. In ordinary gaseous mixtures, the reaction rate is strongly sensitive to temperature, and, more specifically, the induction delay varies with order unity for small fluctuations of the temperature TN at the Neumann state x = 0. Attention is then limited to small fluctuations of TN (t), that is, to small fluctuations of the shock velocity δD/D = O(1/βN ) where βN is a large parameter, βN  1, that characterises the sensitivity of the distribution of heat release rate to the Neumann temperature, N (t) ≡ βN (TN − T N )/T N = O(1),

(uN − uN )/uN = O(1/βN ),

(4.5.17)

and the solution is sought in the limit βN → ∞. We assume that the inner structure of the detonation varies by order unity when the variation of Neumann temperature is small, of order 1/βN , δTN /TN = O(1/βN ). In the limits (4.5.10)–(4.5.11) and for βN  1, the leading order of the left-hand side of (4.5.16) is (ub /uN − 1) − α˙ t /uN , and a simplified form of the nonlinear equation for the front dynamics is obtained:    ∞ ub α˙ t −1 − ≈ qN w ˙ dx. (4.5.18) uN uN 0 Since the variation of D is small, δD/D = O(1/βN ), the linear approximation of the Rankine–Hugoniot relations (4.2.14)–(4.2.16) can be used. To leading order, the variations of the profiles [T(x, t) − TN (t)] /TN (t) and ψ(x, t) are of order unity, whereas the variations of the mass flux at the Neumann state remain small δm(t) ≈ O(1/βN ), and the reduced mass flux m(t) can be replaced by unity in (4.5.13), ∂T qm ∂T + = w(ψ, ˙ T), ∂t ∂x cp

∂ψ ∂ψ + = w(ψ, ˙ T), ∂t ∂x

(4.5.19)

the dominant effect coming from the unsteady boundary condition (4.5.14) for the small temperature fluctuations at the Neumann state, δTN (t)/TN = O(1/βN ). Limitation of the Quasi-isobaric Approximation The CJ regime is excluded by considering qN to be of order unity in the limit (4.5.10). However, the discrepancies are not very important in real CJ detonations. For a typical detonation propagating in an ordinary gaseous mixture such as that considered just below (4.2.57) with qm /cp Tu = 8, γ = 1.3, and MuCJ = 6.22, MN2 CJ = 0.176, T NCJ /Tu = 6.66, the quantity qN ≡ qm /(cp T NCJ ) = 1.2 is of order unity, as is assumed in (4.5.11). The

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increase of temperature in the shocked gas is of order unity, (T bCJ /T NCJ − 1) = 0.93. The difference with the quasi-isobaric approximation, 1.2, is not large, 25% lower due to compressible effects. The temperature change due to compressible effects represents less than 25% of that due to gas expansion in the quasi-isobaric approximation. According to (4.2.53) and (4.5.12), the increase of the flow velocity and the decrease of pressure across the exothermic reaction zone are of order unity, abCJ /uNCJ = 2.88, pbCJ /pNCJ ≈ 0.5, as is well known for CJ waves. A pressure variation of order unity in the reaction zone does not mean that the quasi-isobaric approximation is irrelevant for the dynamics of real detonations near the CJ regime. The large variations of the induction length, due to thermal sensitivity, is the essential mechanism modifying the reaction rate at a fixed distance from the lead shock. The induction zone being very subsonic, M NCJ = 0.135 in the case considered above, the variation of the induction length is accurately described by a quasi-isobaric approximation. The effects of the compressible phenomena in (4.5.5) may be decomposed into two parts, unsteady and steady. The quasi-steady effect of the pressure on the temperature profile is not very important from a quantitative point of view and can easily be taken into account in a modified version of the quasi-isobaric theory presented at the end of this section. Concerning the purely unsteady effects of pressure on ordinary galloping detonations in the self-propagating CJ regime, the acoustic waves introduce a small additional time delay and a small additional phase shift in the loop mentioned at the beginning of this section. The phase shift is small because of the subsonic character of the induction zone, and a perturbation analysis[1] shows that it produces a stabilising effect. However, the pressure fluctuations become important in the dynamics of strongly unstable detonations and also in the transonic CJ regime for small heat release studied in Section 12.2.2. To summarise, in ordinary detonations the quasi-isobaric propagation of the entropy wave across the induction zone controls the position of the exothermic layer relative to the shock. It is the basic mechanism that drives the galloping instability of real detonations. The accuracy of the results in the quasi-isobaric approximation is increasingly better for smaller values of (γ − 1). Integral Equation Using N defined in (4.5.17), let T = T (N , x) and ψ = Y(N , x) denote the steady state ˙ T ), satisfying the boundary condition x = 0: solution to (4.5.19), dT /dx = (qm /cp )w(Y, ψ = 0, T = TN . The solution to (4.5.19) satisfying (4.5.14) is T(x, t) = T (N (t − x), x) ,

ψ(x, t) = Y (N (t − x), x) .

(4.5.20)

In this retarded solution, the quantity (t − x) is the reduced time at which a fluid particle, located at position x at time t, had crossed the lead shock. In other words, x is the reduced time lag taken by a fluid particle to go from the lead shock to the position x, since, according to (4.5.20), the reduced propagation velocity is unity. The general solution to (4.5.13) for [1]

Clavin P., He L., 1996, Phys. Rev. E, 53(5), 4778–4784.

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287

(b) 2

3 2

1.5

1 1 0 0.5

−1 −2

0 0

2

0

4

2

4

Figure 4.25 Profiles of 0 (x) and N (x) for a modified Arrhenius law (4.5.25) with qN = 1 and βT = 1, 2, 4. From Daou R., Clavin P., 2003, J. Fluid Mech., 482, 181–206, with permission.

m(t) = 1 takes the same form as (4.5.20) when the delay x in t − x in the right-hand side is replaced by τ (x, t), the solution to Dτ/Dt = 1 satisfying the condition x = 0: τ = 0. Denoting (N , x) the distribution of the reduced reaction rate of the steady state ∞ solution to (4.5.19) for T|x=0 = TN , 0 (N , x)dx = 1, (Tb − TN ) = qm /cp , the instantaneous distribution of reaction rate w(T, ˙ Y) for a fluctuating Neumann temperature TN (t), that is for (4.5.20), is the retarded function (N (t − x), x). In the limit (4.5.10)– (4.5.11) Equation (4.5.18) then reads  ∞   α˙ t (N (t − x), x) − (N , x) dx. = qN − uN 0 Neglecting (γ − 1) in front of 2γ Mu2 , the linearised Rankine–Hugoniot relation (4.2.16) yields δTN /T u ≈ [4γ (γ − 1)/(γ + 1)2 ]M u δMu ,

δTN /T N ≈ −(γ − 1)α˙ t /uN .

Expressing α˙ t in terms of N then leads to a nonlinear integral equation for N (t),  ∞ (N (t − x), x)dx, b−1 ≡ βN (γ − 1)qN , (4.5.21) 1 + bN (t) = 0

where the parameter b is of order unity for a high temperature sensitivity, (γ − 1)βN = O(1),

that is,

2

βN = O(M u ).

(4.5.22)

In these conditions, the instability threshold for planar disturbances is described by the linear integral equation obtained from (4.5.21)  ∞ N (x)δN (t − x)dx, N (x) ≡ ∂/∂N |N =0 , (4.5.23) δN (t) = b−1 0

∞ = 0 since 0 (N , x)dx = 1; see Fig. 4.25. Looking for the solution where 0 ˆ the reduced complex growth rate σˆ is the solution in the form δN (t) ∝ eσˆ t with σˆ = sˆ +iω, ∞

N (x)dx

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to the integral equation 

∞

b= 0

N (x)e−σˆ x dx.

(4.5.24)

For a given regular function N (x), the detonation becomes unstable when b−1 is −1 > b−1 : increased: above a critical value b−1 c the linear growth rate becomes positive (b c sˆ > 0) and the imaginary part of the growth rate is nonzero, describing an oscillatory instability. This is a Poinca´e[1] –Andronov bifurcation (ˆs = 0, ωˆ = 0), also called a Hopf bifurcation. For a given value of the parameter b−1 , the bifurcation occurs when the function N (x) becomes stiffer. Two simple examples describing two extreme cases are instructive: • Consider first a reaction rate that increases rapidly as the Neumann temperature is increased, but which is independent of changes of temperature downstream from the lead shock. The progress variable then increases as an inverse exponential with the distance −1 −x/lN e , and the length scale lN can from the shock, 1 − ψ = e−x/lN , (N , x) = lN be assumed to vary exponentially with the Neumann temperature, lN = e−N so that N (x) = (1 − x)e−x . Equation (4.5.24) reduces to σˆ 2 + (2 − b−1 )σˆ + 1 = 0 with an instability threshold (ˆs = 0, ωˆ = 1) at b−1 c = 2. • Consider now the opposite situation, described by the square-wave model, for which the reaction time is so sensitive to temperature that the heat release is located in a thin reaction zone located at the reduced distance lN = e−N from the shock, (N , x) = δ(x − lN ), N (x) = δ  (x − 1), where δ(x) denotes the Dirac distribution and δ  (x) is its derivative. Equation (4.5.23) takes the form of an advanced-time difference-differential equation, dN (t − 1)/dt = bN (t), which is known to be quite singular, as shown by (4.5.24), σˆ e−σˆ = b. This transcendental equation has an infinite set of discrete unstable modes with unbounded amplification rates increasing with frequency (ˆs → ∞, ωˆ → ∞), whatever be the value of the parameter b. This square-wave pathology is also observed if compressible effects (acoustic waves) are taken into account.[2] Modified Arrhenius Model A chemical kinetic model, representative of gaseous detonations and free from the spurious square-wave pathology, must represent both an induction delay that is very sensitive to changes of the Neumann temperature and a heat-release rate that remains finite. The second condition is not fulfilled by the one-step Arrhenius model in the limit of a infinitely large activation energy, which leads to the square-wave model. A convenient and accurate model for gaseous detonations is an extension of the ZFK model, involving two different activation energies, EN and ET , describing two thermal sensitivities, one for the induction delay and

[1] [2]

Poincar´e H., 1908, Revue d’´electricit´e, 387. Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378.

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Modified 1-step model

Reduced growth rate,

Arrhenius law

Reduced frequency,

Figure 4.26 Numerical results for the frequency spectrum of a galloping detonation for the modified one-step model (solid lines) with b−1 = 2.03 and βT = 8.44. Time scales are reduced by the unperturbed induction delay. For comparison, the numerical results from the full equations for an Arrhenius law (βN = βT = 8.44) are plotted with dotted lines. Adapted from Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378, with permission.

the other for the heat release.[2] The corresponding reaction rate −1 ˙ W(ψ, T) = Bτcoll (1 − ψ)e−EN /kB TN eβT (T−TN )/TN

(4.5.25)

is investigated in the limit βN ≡ EN /kB TN → ∞, βT = O(1). In the distinguished limit (4.5.10)–(4.5.11) and (4.5.22), this modified one-step model leads to the nonlinear integral ˜ N , x), where equation (4.5.21) with (N , x) = (l ˜ N , x) = l−1 0 (x/lN ), (l N

lN ≡ e−N ,

N (x) = d(x0 )/dx,

(4.5.26)

and where 0 (x) is the profile of the reaction rate for N = 0, that is, TN = T N . For particular examples of functions 0 (x), Equation (4.5.24) becomes polynomial and σˆ can be obtained analytically; see (12.2.51)–(12.2.56). The distribution 0 (x) can also be computed from the steady state solution to (4.5.19) for the boundary condition x = 0: T = T N and in which w(ψ, ˙ T) = (1 − ψ)eβT (T−T N )/T N .

(4.5.27)

In this limit, the one-dimensional nonlinear dynamics are thus fully controlled by two parameters of order unity, b−1 and βT . The first parameter, defined in (4.5.21), characterises both the degree of overdrive through qN and the thermal sensitivity of the induction length through βN . The second characterises the increase of heat-release rate with the increase of temperature inside the exothermic layer. This model has been investigated numerically.[2] A typical result for the spectrum of the linear equation (4.5.23) with the kinetic model (4.5.25)–(4.5.27) is shown in Fig. 4.26 for βT = 8.44 and b−1 = 2.03. The number of unstable modes, the linear growth rate and the frequency of oscillation increase without upper bound when βT is increased, leading to the spurious square-wave pathology in the limit βT → ∞. For the sake of comparison, the spectrum of the linear version of the full −1 ˙ (1−ψ)e−E/kB T is plotted equations (4.5.4)–(4.5.5) with the Arrhenius law W(ψ, T) = Bτcoll

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III Dynamical quenching

I Stable

II Unstable

Figure 4.27 Stability thresholds of galloping detonation in the b−1 , βT plane. Squares and solid line: stability limits obtained from (4.5.23) with the model (4.5.26) where 0 (x) is computed with (4.5.27). For comparison, the numerical stability limits are given by the circles plus dashed line for an Arrhenius law (βN = βT ) and for different values of f and γ , the heat release being given by the expression of b−1 in (4.5.21). Diamonds plus dotted line: limit of dynamical quenching for an Arrhenius law. Adapted from Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378, with permission.

in the same figure for βN = βT = 8.44, γ = 1.2 and a overdrive factor corresponding to the value of qN yielding b−1 = 2.03. Comparison shows a qualitative agreement but also quantitative differences. The stability limit in the plane b−1 –βT is plotted in Fig. 4.27. Near the instability threshold, the time scale of the dynamics of a galloping detonation is of the same order as the transit time of a fluid particle across the inner structure of the detonation.

Nonlinear Dynamics. Dynamic Quenching Beyond the stability limit, the numerical results for the nonlinear behaviour, obtained from (4.5.21), show period doubling and a transition to chaos as b−1 is increased. Intermittent bursts of large-amplitude high-frequency oscillations, separated by long periods of almost constant velocity propagation with a velocity and a Neumann temperature both well below their values in the steady state solution, are also observed as b−1 is further increased; see Fig. 4.28. Such large fluctuations of the Neumann temperature lead to dynamic quenching of planar detonations. This phenomenon has also been observed in direct numerical simulations of planar detonations, sustained by a one-step Arrhenius reaction, if the activation energy has sufficiently high value.[1] In real detonations the dynamical quenching is reinforced by the crossover temperature T ∗ below which the exothermic reaction is quasiquenched; see Section 1.2.2 and Chapter 5 for more detail. This is easily understood: if the dynamically induced fluctuations are sufficiently strong to bring the Neumann temperature TN (t) below T ∗ , the coupling of the nonlinear galloping instability to the nonlinear chemical kinetics leads to quenching of the planar detonation. The multidimensional case is not yet known. [1]

He L., Lee J., 1995, Phys. Fluids, 7, 1151–1158.

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Figure 4.28 Galloping detonation: evolution of N (t) obtained from (4.5.21) with βT = 5 for increasing values of b−1 showing period doubling, chaotic bursts and quasi-quenching. From Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378, with permission.

Concluding Remarks. Further Studies and Extended Models A comparison with direct numerical simulations shows that the dynamics of the lead shock, obtained by the model equation (4.5.21) represents the essential mechanism of the galloping detonation, at least for the unstable domain not too far from the instability threshold. Compressible effects have been taken into account in a perturbation analysis.[2] The result takes the same form as (4.5.21) but with a correction of order  to the coefficient b and to the time lag inside the integral equation. Moreover, acoustic waves are found to be stabilising, so that the instability does not arise from thermo-acoustic instability, contrary to what was often thought before. Extended analyses have been performed[3] using a three-step chemistry model that describes chain-branching processes discussed in Section 5.2.1. They are seen to successfully reproduce many of the characteristics observed in computations employing detailed chemistry for hydrogen–air mixtures.[4] More accurate quantitative results can be obtained with the model equation (4.5.21) by using a distribution 0 (N , x), computed from a numerical simulation of the steady state solution for a complex kinetics scheme (e.g. the one used in Fig. 2.2 for the [2] [3] [4]

Clavin P., He L., 1996, Phys. Rev. E, 53(5), 4778–4784. Sanchez A., et al., 2001, Phys. Fluids, 13(3), 776–792. Yungster S., Radhakrishnan K., 2004, Combust. Theor. Model., 8, 745–770.

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(a) m/s, m/s, m/s,

K K K

Figure 4.29 (a) Spatial distribution of the reaction rate in overdriven detonations propagating in a stoichiometric hydrogen–oxygen mixture for different Neumann temperatures (different degrees of overdrive). The reaction rate and the distance are scaled by those of the detonation propagating at 3100 m/s. The calculation is performed with a detailed chemical kinetic scheme of hydrogen–oxygen combustion; see Section 5.3. (b) Same data rescaled, showing that the extended ZFK model is a fairly good approximation. From Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378, with permission.

structure of a methane flame). For better accuracy, compressible effects may easily be included in the computation of (N , x). For example, the inner structure of hydrogen– oxygen detonations, computed with a detailed kinetic scheme for different propagation velocities (different overdrive factors), is shown in Fig. 4.29a. The scaling law ˜ N , x) = l−1 0 (x/lN ) in (4.5.26) is shown to be a reasonable approximation;[1] (l N see Fig. 4.29b. The instability threshold of galloping detonations in the CJ regime has been investigated[2] for a small heat release. This extreme condition corresponds to the opposite limit of the quasi-isobaric approximation since the dominant mechanism is now acoustics. However, the nature of the galloping instability is not very different from the one in the limit (4.5.10)–(4.5.11). It still results from a phase shift in the loop involving the lead shock and the reaction zone, but the time delay is now due to the upstream running acoustic wave instead of the downstream running entropy wave. The sensitivity of the heat-release rate to the Neumann temperature is still the mechanism responsible for the one-dimensional instability, and the detonation becomes stable against planar disturbances when the thermal sensitivity is turned off. The corresponding analysis[2] is presented in Section 12.2.2. 4.5.2 Cellular Detonations. Diamond Pattern Gaseous detonations are known to exhibit transverse structures that are larger than the detonation thickness. The underlying mechanism of the ‘diamond’ or ‘fish scale’ pattern, [1] [2]

Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378. Clavin P., Williams F., 2002, Combust. Theor. Model., 6, 127–129.

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Figure 4.30 Time evolution of a cellular detonation showing how the diamond pattern is produced by transverse propagation of triple points coupled to the longitudinal pulsation of the internal and quasi-planar structure.

recorded on soot-coated foils (see Fig. 4.24), is sketched in Fig. 4.30. The cellular pattern results from the coupling of the one-dimensional pulsation of the internal structure, studied in Section 4.5.1, and the formation of Mach stems on the wrinkled shock front discussed in Section 4.4.2. Using a pulsation time of the order of the induction delay tN and the transverse velocity of the Mach stem aN , the order of magnitude of the cell size in the transverse and longitudinal directions would be (aN /uN )lN and (D/uN )lN , respectively. These two lengths are larger than the thickness of the induction zone lN ≈ uN tN by the factors aN /uN and D/uN , respectively, in rough agreement with observations. The real problem is more complicated since planar waves can be strongly unstable to disturbances with transverse components, even when the detonations are stable to purely longitudinal disturbances. In this sense the cellular instability is stronger than the galloping detonation. However, the underlying oscillation, even when it is damped in planar geometry, is essential to the cellular structure. As for most of the nonlinear patterns, purely analytical studies cannot be performed in the general case. An analytical description of cellular detonations has been obtained for strongly overdriven regimes by a weakly nonlinear analysis at the threshold of instability.[3, 4] This analysis is summarised below; the details are given in Section 12.2.4.

[3] [4]

Clavin P., 2002, In H. Berestycki, Y. Pomeau, eds., Nonlinear PDE’s in condensed matter and reactive flows, 49–97. Kluwer Academic Publishers. Clavin P., Denet B., 2002, Phys. Rev. Lett., 88(4), 044502–1–4.

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Instability Threshold of Strongly Overdriven Detonations The preliminary step is to carry out a multidimensional linear analysis to find the instability threshold. The pioneering analyses of linear stability[1,2] were performed as initial-value problems, using a Laplace transform and numerical analysis. The existing weakly nonlinear analysis[3,4] is based on linear analyses[5,6,7] performed for overdriven gaseous detonations in the limit (4.5.10)–(4.5.11), using a normal mode decomposition (4.4.1). The dispersion relation, namely the relation linking the complex linear growth rate σ and the wavenumber k, is obtained for a polytropic gas by imposing the Rankine–Hugoniot conditions (12.1.25)– (12.1.26) at x = 0 and a boundedness condition at infinity, x → ∞ on the solutions to the reactive Euler equations (12.2.1)–(12.2.2). In the general case there are different branches of solutions to the dispersion relation σ (k), and the overall picture is rather complicated.[5] In the limit (4.5.10) the instability threshold occurs at small values of qN , for qN /(γ − 1) of order unity,[7] that is, 2

 2 ≡ γ M N 1,

(γ − 1) = O( 2 ),

qN ≡ qm /cp T N = O( 2 ).

(4.5.28)

Real detonations are always unstable because they do not correspond to such a small heat release. A weakly nonlinear analysis near to the multidimensional stability limit is physically meaningful if it retains the dominant nonlinear effects of real detonations, which is usually the case if there is no secondary bifurcation. The analysis simplifies near the stability limit but is still complicated. The presentation is limited here to the discussion of the results, and the details are given in Section 12.2.4. In a way similar to the one-dimensional analysis in Section 4.5.1 the chemical reaction occurs only through two distributions, the steady state distribution (x) ≡ (N x) and N (x) that describes its modification when the detonation speed is modified; see Fig. 4.29a. We assume that they are still meaningful in the limit of small heat release in (4.5.28). According to the discussion at the beginning of this section, we introduce the following scaling of the reduced wavenumber κ and the reduced linear growth rate, σ = s ± iω, s and ω > 0 being real numbers, scaled using the reaction time tN and the flow velocity uN at the Neumann state of the unperturbed solution σ ≡ σˆ tN ,

ˆ N tN /, κ ≡ |k|u

σ (κ) = s(κ) ± iω(κ),

(4.5.29)

where in this section the hat denotes dimensional quantities. The function σ (κ) is obtained by a perturbation analysis in the form of an expansion in powers of  2 1 in the limit (4.5.10) and (4.5.28), σ = σ0 (κ) +  2 σ2 (κ) + O( 4 ). [1] [2] [3] [4] [5] [6] [7]

(4.5.30)

Erpenbeck J., 1962, Phys. Fluids, 5, 604–614. Erpenbeck J., 1966, Phys. Fluids, 9, 1293–1306. Clavin P., 2002, In H. Berestycki, Y. Pomeau, eds., Nonlinear PDE’s in condensed matter and reactive flows, 49–97. Kluwer Academic Publishers. Clavin P., Denet B., 2002, Phys. Rev. Lett., 88(4), 044502–1–4. Clavin P., et al., 1997, Phys. Fluids, 9(12), 3764–3785. Clavin P., He L., 2001, C. R. Acad. Sci. Paris, 329(IIb), 463–471. Daou R., Clavin P., 2003, J. Fluid Mech., 482, 181–206.

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The leading order, σ0 (κ), corresponds to the oscillatory modes of the lead shock (see (4.4.15)), s0 = 0, ω0 = κ. The linear growth rate appears at the following order Re(σ ) =  2 s2 (κ), and, according to (12.2.87)–(12.2.88) takes the form[7] 2

 MN Re(σ )/κ, = −Im S (i) (κ) − √ S (a) (κ), qN qN

(4.5.31)

where by definition (i)

S (i) (κ) ≡ βN (γ − 1)sβN (κ) + sq(i) (κ),  ∞  ∞ N (x)e−iκx dx, sq(i) (κ) ≡ (1 + iκx)(x)e−iκx dx, sβ(i)N (κ) ≡ 0 0      (γ − 1)   (a) (i) S (κ) ≡ 2 Im + S (κ) − 1 .   2qN

(4.5.32) (4.5.33) (4.5.34)

The result σ0 = iκ has been used an  in (4.5.31)–(4.5.33). Equation (4.5.31) describes √ isobaric mechanism, Im S (i) (κ) , and a compressible effect proportional to M N / qN , involving S (a) . In addition to the functions of order unity (x) and N (x) controlled by the chemical kinetics, the reduced growth rate Re(σ )/qN depends on three positive parameters √ of order unity, βN (γ − 1) measuring the thermal sensitivity, (γ − 1)/qN and M N / qN . A small correction to the oscillatory frequency is also obtained at this order ω/κ = 1+ O( 2 ). Using (4.4.10), it is useful to rewrite (4.5.34) in the form      M N (a) (γ − 1)  2 2qN  (i) (4.5.35) S (κ) − 1  . Im 1 + √ S = 1+ 2  qN qN  (γ − 1) (γ − 1)M u

The function S (i) (κ) describes quasi-isobaric mechanisms of instability resulting from the coupling of the vorticity–entropy wave to the rate of heat release. There are two different contributions: (i)

• The function sβN (κ) comes from the sensitivity of the induction delay to the Neumann temperature, responsible for the one-dimensional instability; see (4.5.23)–(4.5.24). In the conditions of (4.5.28), the parameter b−1 = βN (γ −1)qN is small, of order  2 , well below −1 the critical value b−1 c in (4.5.24), since bc is typically of order unity (for a thickness of the exothermic reaction layer of the same order as the induction length). Therefore the detonation is stable against one-dimensional longitudinal disturbances. (i) • The function sq (κ) describes the interaction of the heat release rate with the deflection of the flow velocity across the wrinkled wave. This leads to a strong instability against transverse disturbances that does not involve the sensitivity to temperature. This instability mechanism has some analogy with the Darrieus–Landau hydrodynamic instability in flames. The quasi-isobaric instability is strong in the sense that the overall growth rate remains positive and reaches its maximum in the limit of small wavelengths,

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Gaseous Shocks and Detonations 7 6 5 4 3 2 1 0 −1 0

2

4

6

8

10

Figure 4.31 −κIm[S (i) (κ)] versus κ for an Arrhenius law with βN qN = 0.1 and βN (γ − 1) = 0, 1, 5. From Daou R., Clavin P., 2003, J. Fluid Mech., 482, 181–206, with permission.

limκ→∞ −κIm[S (i) (κ)] = cst. > 0; see Fig. 4.31. This is also true in the absence of (i) thermal sensitivity, βN (γ − 1) = 0, limκ→∞ −Im[sq (κ)] > 0. The positive function S (a) (κ) > 0 describes a stabilisation mechanism that results from the coupling of the acoustic waves to the heat release rate. The origin of the stabilisation is the modification of the longitudinal flow velocity at the exit of the reaction zone due to radiating acoustic waves in the burnt gas, leading to a weakening of the piston effect mentioned at the beginning of Section 4.5.1. This is called a ‘negative velocity coupling’ in the thermo-acoustic literature. is stronger than the isobaric instability For small qN /(γ − 1) the damping mechanism   √ at all wavelengths, (M N / qN )S (a) (κ) > |Im S (i) (κ) | ∀κ, as shown from an expansion 2

of (4.5.35) for small qN /(γ − 1) 1 by noticing that 1 + 2/[(γ − 1)M u ] > 1. The planar front is unconditionally stable in this condition. In the opposite limit, for large qN /(γ − 1), the isobaric instability mechanism dominates the acoustic damping in an intermediate range of wavelengths corresponding to κ = O(1). Compressible effects stabilise the small wavelength disturbances,

 lim κS (a) (κ) = 2κ, lim S (a) (κ) → 2, κ→∞

κ→∞

as shown from (4.5.34) when qN /(γ − 1)  1 using limκ→∞ S (i) (κ) = 0. Thus, as qN /(γ − 1) is increased, a Poincar´e–Andronov (Hopf) bifurcation occurs at a finite wavelength, larger than the detonation thickness by a factor 1/; see (4.5.29). An example of the dispersion relation near the onset of instability, obtained from (4.5.31) for an Arrhenius law, is shown in Fig. 4.32. The expression for the linear growth rate in terms of the wavenumber, σ (κ), is a combination of the two roots of a quadratic equation; see (12.2.87). Only the branch of the solution satisfying the boundedness condition of the acoustic waves in the limit x→ ∞ is retained. The overall result consists in two pieces of two different branches of the solutions. This is responsible for the small jump

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297

(b)

(a)

2.5

0.005

2.0 0.0025

Unstable branch

Nonphysical

1.5 0

1.0 −0.0025

Stable branch

0.5 Unstable branch 0.0

−0.005 0

0.5

1.0

1.5

2.0

0

2.5

0.5

1.0

1.5

2.0

2.5

Figure 4.32 (a) Growth rates from (4.5.31) with (γ − 1) = 0.1, βN = 10 and Mu2 = 50 for an unstable case, qN = 0.04 (solid line) and a stable case, qN = 0.03 (dashed line). The nonphysical parts of the solutions are shown by thin lines. (b) Frequency as a function of κ for the unstable branch of the solution. The thin lines correspond to the nonphysical part. From Daou R., Clavin P., 2003, J. Fluid Mech., 482, 181–206, with permission. 0.002 Numeric 0

Analytic

−0.002

−0.004 0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 4.33 Determination of the threshold of linear instability. The dispersion relations at the threshold, σ (κ), given by (4.5.31) (dashed line) and by numerical analysis of the linear equation (solid line) for βN = 0, γ = 1.05, (γ −1)Mu2 = 1. The numerical result corresponds to 2qN /(γ −1) = 10.4 and the analytical result to 2qN /(γ − 1) = 9.6 at threshold. Adapted from Daou R., Clavin P., 2003, J. Fluid Mech., 482, 181–206, with permission.

in frequency at κ ≈ 0.75 in Fig. 4.32b. A simplification occurs for a purely ‘hydrodynamic instability’, βN (γ − 1) ≈ 0, for which Equation (4.5.31) corresponds to a single branch of solution. Such an example of bifurcation is shown in Fig. 4.33. The comparison with the branch of solution computed numerically shows a good accuracy of the analytical result. To summarise, near the instability threshold, the multidimensional instability of overdriven gaseous detonations is mainly due to a quasi-isobaric mechanism that is hydrodynamic in nature. In contrast to galloping detonations, the instability develops even in the absence of sensitivity of the induction length to temperature variations because the induced transverse velocity modifies the longitudinal distribution of heat release rate. Compressible

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phenomena have a stabilising effect that is essential to stabilise the disturbances at small wavelengths. The competition of these two mechanisms, quasi-isobaric and compressible, results in a Poincar´e–Andronov (Hopf) bifurcation at a wavelength larger than the detonation thickness with an oscillatory frequency of the order of the inverse of the transit time across the inner detonation structure. Nonlinear Model Equation for Cellular Detonations (2002) A weakly nonlinear analysis,[1,2] performed for values of parameters in (4.5.28) that are at the instability threshold, leads to an equation for the evolution of the detonation front that combines the nonlinear term in (4.4.28), responsible for singularity formation on an inert shock front (representative of Mach stems), studied in Section 4.4.2, and the linear integraldifferential equation corresponding to the dispersion relation (4.5.31)–(4.5.34). Using the same reduced coordinates as in (4.5.1) with nondimensional transverse coordinates reduced by uN tN /, as in (4.5.29), and a nondimensional amplitude of wrinkling reduced by uN tN , α = α/(u ˆ N tN ), the equation takes the form in the limit (4.5.28) ∂ 2α ∂|∇α|2 √ ∂ 2 2 = −2M N qN L(a) (α) + qN L(i) (α), − c ∇ α + 2 ∂t ∂t ∂t

(4.5.36)

where c2 = 1 + 3(γ − 1)/2, L(a) (α) is a linear term corresponding to the damping term (4.5.34) due to compressible effects, and L(i) (α) is a linear term corresponding to the quasiisobaric instability, represented by S (i) (κ) in (4.5.32), (i)

L(i) (α) = βN (γ − 1)lβN (α) + lq(i) (α), (4.5.37)  ∞  ∞ 2 ∂ (i) lβN (α) = 2 N (x)α(t − x)dx, lq(i) (α) = ∇ 2 !(x)α(t − x)/dx, ∂t 0 0 where !(x) ≡ (x) + d(x)/dx, and where the transverse variable of α(t, y) is not written (i) explicitly. The second derivative with respect to time in lβN (α) can be well replaced by ∇ 2 since to leading order, ∂ 2 α/∂t2 = ∇ 2 α. The form (4.5.37) is obtained directly from (12.2.86) when ∂α(t − x)/∂t is replaced by −∂α(t − x)/∂x and using integration by parts. The linear operator L(a) (.) is defined in Fourier space as κS (a) (κ)/2 in (4.5.34), so that the small wavelength limit yields L(a) (.) → κ/2 in Fourier space. The left-hand side of (4.5.36) corresponds to the nonlinear equation for Mach stem formation (4.4.28). Without the nonlinear term, third term in the left-hand side, Equation (4.5.36) yields the dispersion relation (4.5.31)–(4.5.34). In one-dimensional geometry this linear equation reduces to (i) (4.5.23). The term qN lq (α) describes a hydrodynamic instability that is produced even in the absence of temperature sensitivity. The numerical studies of (4.5.36) using periodic boundary conditions in the transverse direction have solutions that are periodic in time, with pulsating cells delimited by cusps propagating at constant velocity on the front; see Fig. 4.34. In large boxes with many [1] [2]

Clavin P., 2002, In H. Berestycki, Y. Pomeau, eds., Nonlinear PDE’s in condensed matter and reactive flows, 49–97. Kluwer Academic Publishers. Clavin P., Denet B., 2002, Phys. Rev. Lett., 88(4), 044502–1–4.

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299

(b) 0.6

Growth rate

0.4 0.2 0.0 –0.2 –0.4 0

2

4

6

8

10 12 14 16 18 20 n

Figure 4.34 (a) Growth rate as a function of mode number n in a periodic box of size L, k = 2π n/L, for a detonation with Arrhenius kinetics, βN = 6, γ = 1.2, q = 0.25, 2π/L = 0.5. The curves under the graph show the two-dimensional solutions for the front at three reduced times, τ ≡ t/tN . (b) One period of propagation of a cell of a two-dimensional detonation front with the same kinetics. Propagation is from bottom to top; the grey lines are the trajectories of the cusps. Reproduced with permission from Clavin P., Denet B., Physical Review Letters, 88(4), 044502–1–4. Copyright 2002 by the American Physical Society.

unstable modes, the final periodic solution is obtained after a long transient time and presents a cell size larger than the most amplified wavelength. The case presented in Fig. 4.34 corresponds to a box with 12 unstable modes, L = 4π . The wavelength selected in the final solution corresponds to a wavenumber kn = 2π n/L with n = 6. The band of linearly unstable modes ranges from n = 4 to n = 16 and the most unstable mode corresponds to n = 8. The nonlinear selection of the final cell size is illustrated by the numerical solution for an initial condition of a sinusoidal perturbation of small amplitude at n = 10, plus a much smaller level of noise. After a few periods of oscillation, well-ordered pulsating cells with a small size, corresponding to n = 10, are first observed. This regime lasts roughly 50 tN and is followed by a chaotic phase during which the number of cells decreases down to the final stable state of six pulsating periodic cells. The relaxation time towards the final solution is typically 2 × 102 tN . The pattern constituted by the trajectory of the cusps on the front in the final solution is similar to the ‘diamond’ pattern observed in experiments; see Fig. 4.34. Because of the nonlinear terms, the shape of the averaged front is not planar, and a mean streaming flow is associated with the mean front.[3] To summarise, the model equation (4.5.36) reproduces experiments and direct numerical simulations qualitatively well. Based on the ZND structure, the model reduces to a simple superposition of the linear oscillatory instability (due to the heat release) and the nonlinear dynamics of the lead shock [3]

Clavin P., 2002, Int. J. Bifurcation & Chaos, 12(11), 2535–2546.

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that leads to the formation of triple points. This is true for weakly unstable detonations in sense that they are still one-dimensionally stable (no galloping instability). Strongly Unstable Detonations Strongly unstable detonations are more complicated. A galloping instability can be superimposed on the multidimensional instability so that pockets of fresh mixtures can be engulfed in the shocked gas and reignited when the triple points cross each other. Experiments show that this phenomenon can take the form of local micro-explosions with an eventual intermittent character.[1,2,3] It should be possible to simply model this phenomenon by introducing the nonlinear equation (4.5.21) for dynamical quenching into (4.5.36). Model Equation Near the CJ Regime A complementary study of weakly unstable gaseous detonations near the CJ regime has also been performed. These conditions concern small heat release[4] and the analysis is presented in Sections 12.2.2 and 12.2.3. In such extreme (and nonrealistic) conditions the situation is contrary to the previous case; the entropy–vorticity wave plays a negligible role and the multidimensional instability results from the coupling of acoustic waves to the heat release rate. The results have similarities but also differences with the preceding study. The form of the result for galloping detonations (one-dimensional oscillatory instability) is similar, namely an integral equation involving a function of space N (x) but with a delay due to the upstream running acoustic waves; see Section 12.2.2. The multidimensional study presented in Section 12.2.3 also shows the existence of a Poincar´e–Andronov (Hopf) bifurcation at a finite wavelength. The instability is promoted by increasing the sensitivity of the heat release rate to temperature or by approaching the CJ condition. But at the leading order of the perturbation analysis, in contrast to the strongly overdriven cases, the nonlinear model does not involve the dynamics of the weak lead shock. As a consequence of the transonic character of the flow, the multidimensional dynamics of the detonation wave is fully dominated by the modification of the heat release rate; see the end of Section 12.2.3. 4.6 Appendix 4.6.1 Gaseous Shock Waves and Boltzmann’s Equation Consider a steady planar shock wave perpendicular to the x-axis and propagating at constant velocity. In the reference frame attached to the shock wave (Galilean frame) the shock structure is given by the steady solution to Boltzmann’s equation (13.3.2), (p1x /m)(∂f1 /∂x1 ) = [1] [2] [3] [4]

Radulescu M., et al., 2007, J. Fluid Mech., 580, 31–81. Shepherd J., 2009, Proc. Comb. Inst., 32, 83–98. Bhayyacharjee R., et al., 2013, Proc. Comb. Inst., 34, 1893–1901. Clavin P., Williams F., 2009, J. Fluid Mech., 624, 125–150.

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301

B(f1 ), where B(f1 ) is given in (13.3.5) and f1 ≡ f (r1 , p1 ), where r1 = (x1 , 0, 0) and p1 = (p1x , p1y , p1z ). Rankine–Hugoniot Relations The Rankine–Hugoniot relations, studied in Section 4.2.1, are derived here directly from Boltzmann’s equation using the notations of Section 13.3.1. According to (13.3.15), the divergence of the flux of conserved quantities, mass, longitudinal momentum and energy, is zero:   ∂ ∂ (4.6.1) d3 p1 p1x f1 = 0, d3 p1 p21x f1 = 0, ∂x1 ∂x1  ∂ (4.6.2) d3 p1 p1x (p21x + p21y + p21z )f1 = 0. ∂x1 Far from the shock wave, for x1 → ±∞, the gas is at equilibrium so that the distribution (0) in (13.3.19) with a set of paramfunction is the Maxwell–Boltzmann distribution f1±∞ eters (n±∞ , u±∞ , T±∞ ) corresponding to the equilibrium states at x1 → ±∞, u±∞ = (u±∞ , 0, 0). Integrating the first equation in (4.6.1) from x1 = −∞  to x1 = +∞, using isotropy and the normalisation of f (0) , d3 p(px − mu)f (0) = 0, d3 pf (0) = n, yields the mass conservation (4.2.1) in the form n+∞ u+∞ = n−∞ u−∞ .

(4.6.3)

Integrating second from x1 = −∞ to x1 = +∞ and using the  3equation in (4.6.1)  3 the 2 (0) 2 (0) relation d ppx f = d p(px − mu) f + n(mu)2 yields the momentum conservation (4.2.3) in the form n+∞ kB T+∞ + n+∞ mu2+∞ = n−∞ kB T−∞ + n−∞ mu2−∞ . Proceeding with (4.6.2) in the same way as before and using the relation  d3 p px (p2x + p2y + p2z )f (0)    3 2 (0) 2 3 2 2 (0) , = mu 3 d p(px − mu) f + n(mu) + d p(py + pz )f

(4.6.4)

(4.6.5)

which is obtained using isotropy of f (0) after replacing px by (px −mu)+mu, (4.6.2) yields, according to (4.6.3) and (13.3.20), 5kB T+∞ + mu2+∞ = 5kB T−∞ + mu2−∞ . This relation corresponds to (4.2.4) where h = cp T and cp = (5kB /2m).

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(4.6.6)

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Gaseous Shocks and Detonations

Irreversibility We show now that the entropy must increase across a gaseous shock wave, which is the proof that steady planar rarefaction shocks cannot exist. Consider the quantity   ∂ ∂f1 −1 3 −1 I≡m [f1 ln f1 ] = m [1 + ln f1 ]. (4.6.7) d p1 p1x d3 p1 p1x ∂x1 ∂x1 Using Boltzmann’s equation, the H theorem (13.3.12) then shows that I cannot be positive,  p1x ∂f1 = B(f1 ) ⇒ I = d3 p1 B(f1 )[1 + ln f1 ]  0. (4.6.8) m ∂x1 By definition, the variables x1 and p1 are independent variables so that the derivative with respect to the space coordinate x1 can be commuted with the integral over p1 , mI =  3 (∂J/∂x1 ) with J(x1 ) ≡ d p1 p1x [f1 ln f1 ]. Integrating I  0 from x1 = −∞  to x1 = +∞ leads to J+∞  J−∞ , where J±∞ ≡ J(x1 = ±∞). Isotropy of f (0) , d3 p(px − mu)[f (0) ln f (0) ] = 0, d3 p px [f (0) ln f (0) ] = mu d3 p[f (0) ln f (0) ], shows  (0) (0) J±∞ = mu±∞ d3 p1 [f1±∞ ln f1±∞ ]. (4.6.9) x1 → ±∞:  According to the definition of entropy s in (13.3.14), ns = −kB d3 p[f (0) ln f (0) ] + kB n, the relations J+∞  J−∞ and (4.6.3), n+∞ u+∞ = n−∞ u−∞ , then imply s+∞  s−∞ if u±∞ > 0. In other words the entropy of the gas behind the shock cannot be smaller than the entropy of the gas into which the shock propagates. The detonation thickness d is estimated from the definition (4.6.7) of I, kB I = −∂(nus)/∂x, and from the order of magnitude of the collision operator, B(f1 ) = O(δf1 /τcoll ), where 1/τcoll is the collision frequency; see (13.3.24). Equation (4.6.8) leads to −kB I = O(nδs/τcoll ), yielding u∂δs/∂x = O(δs/τcoll ) and d = O(uτcoll ). This corresponds to a thickness of order of the mean free path d/ = O(1),  = vτcoll , because the mean velocity of molecules v is of order of the sound speed, v/a = O(1), and because the flow velocity relative to the shock front is also of order of the sound speed, u/a = O(1), u > a on the upstream side of the shock and u < a on the downstream side. The large thickness (4.2.40) of weak shocks, d/ = O(1/(M − 1)) for a Mach number M = u/a close to unity (M − 1) 1, can be explained by the small variations of the thermodynamic parameters across a weak shock. For example, according to the Rankine– Hugoniot relation (4.2.15), δp/p = O(M − 1), the nondimensional gradient, reduced by the detonation thickness d, is small p−1 d ∂p/∂x = O((M − 1)), and not of order unity p−1 d ∂p/∂x = O(1), as in an ordinary shock. 4.6.2 Reference Frame Attached to the Front The nonlinear analysis is more conveniently carried out by working in a system of coordinates attached to the wrinkled front, ξ = x − α(y, t), ∂ ∂ → , ∂x ∂ξ

∂ ∂ ∂ → − α˙ t , ∂t ∂t ∂ξ

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4.6 Appendix

303

the boundary condition (4.4.23)–(4.4.24) being applied at ξ = 0. It is also convenient to introduce the flow splitting, p = pN + δp(a) + p˜ ,

u = uN + δu(a) + u˜ ,

w = δw(a) + w, ˜

where, according to the text below (4.4.16), the flow velocity of the acoustic waves (δu(a) , ˜ by a factor  2 . The continuity and Euler δw(a) ) is smaller than the linear part of (˜u, w) equations may then be written for H = 0 in the form ∂ w˜ ∂ u˜ ˜ + = X, ∂ξ ∂y

∂ u˜ 1 ∂ p˜ ∂ u˜ ˜ − , + uN =U ρ N ∂ξ ∂t ∂ξ

∂w ˜ 1 ∂ p˜ ∂ w˜ ˜ − , + uN =W ρ N ∂y ∂t ∂ξ

where the residual variation of density can be neglected since it introduces nonlinear corrections that are smaller than ε. When the linear solution (4.4.18) is introduced into the ˜ U, ˜ W, ˜ they take the form of source terms that introduce nonlinear corrections of terms X, order ε to the linear dynamics, D   α (y, t)α˙ ty (y, t − ξ/uN ), uN y D α˙ (y, t)α¨ tt (y, t − ξ/uN )  ˜ =− t ˜ ≡ − α˙ t (y, t)α˙ ty U , W (y, t − ξ/uN ), uN uN X˜ = −

˜ ˜ ˜ ˜ + ∂ W/∂y = ∂ X/∂t + uN ∂ X/∂ξ , where, using the relation DuN ≈ a2N and (4.4.15), ∂ U/∂ξ 2 2 2 2 so that the pressure term is a solution to Laplace’s equation ∂ p˜ /∂ξ + ∂ p˜ /∂y = 0. The acoustic pressure being excluded from the pressure term, p˜ is a quadratic term which must (i) be at least of order ρ N w˜ 2 ≈ ρ N (w2 )2 to be nonnegligible. The boundary condition (4.4.23) 

(i)

2



introduces a term of order ρ N a2N αy2 , which is smaller than ρ N (w2 )2 ≈ ρ N D αy2 by a factor of  2 . The bounded solution of Laplace’s equation with a zero boundary condition at ξ = 0 is null, p˜ = 0. Therefore, retaining the correction of order ε and neglecting terms of ˜ is a solution to order  2 , the flow (˜u, w) ∂ w˜ ∂ u˜ ˜ + = X, ∂ξ ∂y

∂ u˜ ∂ u˜ ˜ + uN = U, ∂t ∂ξ

∂w ˜ ∂ w˜ ˜ + uN = W. ∂t ∂ξ

(4.6.10)

The solution to the second and third equations in (4.6.10) subject to the boundary condition (4.4.25) is obtained by noticing that every function of t and ξ through the grouping (t − ξ/uN ) is a solution of the homogeneous equations (without second member), ˜ , y, t), u˜ = u˜ f (y, t − ξ/uN ) + U(ξ

where

˜ , y, t), w˜ = w˜ f (y, t − ξ/uN ) + W(ξ

[α(y, t − ξ/uN ) − α(y, t)]α¨ tt (y, t − ξ/uN ) , U˜ ≡ uN ˜ ≡ D [α(y, t − ξ/uN ) − α(y, t)]α˙  (y, t − ξ/uN ), W yt uN

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(4.6.11)

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Gaseous Shocks and Detonations

˜ U ˜ and W, ˜ and where, according to the definitions of U˜ , W, ∂ U˜ ∂ U˜ ˜ + uN = U, ∂t ∂ξ

˜ ˜ ∂W ∂W ˜ + uN = W, ∂t ∂ξ

ξ = 0:

˜ = 0. U˜ = W

˜ is found to satisfy the relation ˜ W) Using (4.4.15) and H = 0, the flow (U, ˜ ∂ U˜ ∂W ˜ + = X. ∂ξ ∂y 4.6.3 Mass-Weighted Coordinate In unsteady one-dimensional flow problems with variable density and when the boundary condition is given at a moving boundary, x = α(t), x = α(t):

ρ = ρf (t),

u = uf (t),

(4.6.12)

it is convenient to introduce a system of coordinates (x, t), based on the nondimensional mass-weighted coordinate, x, and also a velocity u(t) depending only on time,   ρf (t) dα 1 x uf (t) − , (4.6.13) ρ(x , t)dx and u(t) ≡ x≡ ρ α(t) ρ dt ∂ ∂ ∂ ∂ ρ ∂ ∂ , + u(x, t) → + u(t) , (4.6.14) → so that ρ ∂x ∂t ∂x ∂t ∂x ∂x where the last relation is obtained by using continuity, the first equation in (15.1.33), ∂ρ/∂t + ∂(ρu)/∂x = 0,  x  x ∂ ∂(ρu)  ρ(x , t)dx = − dx = ρ(x, t)u(x, t) − ρf (t)uf (t). (4.6.15)  ∂t α(t) α(t) ∂x The advantage of the mass-weighted coordinate is to eliminate the variable coefficient ρ(x, t)u(x, t) in favour of a coefficient depending only on time, u(t). The equation for conservation of mass (continuity), ∂ρ/∂t + u∂ρ/∂x = −ρ∂u/∂x, takes the form  ∂(1/ρ) ∂u ∂(1/ρ) + u(t) = . (4.6.16) ρ ∂t ∂x ∂x

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5 Chemical Kinetics of Combustion

Nomenclature Dimensional Quantities B c c˜ E kB ki n n˜ p Q T T∗ τ ω

Description Pre-exponential factor for reaction Molecular concentration Molar concentration Activation energy of reaction Boltzmann’s constant Reaction constant for reaction i Number density Moles per unit volume Pressure Heat of reaction Temperature Crossover temperature Characteristic reaction time Reaction rate

S.I. Units Depends on the order of reaction molecule m−3 mole m−3 J molecule−1 J K−1 Depends on the order of reaction molecule m−3 mole m−3 Pa J molecule−1 m−3 K K s molecule m−3 s−1

Nondimensional Quantities and Abbreviations M N P R X α ν CJ ZFK

Nonreacting species (third-body) Avogadro’s number Product of a reaction Reactive species Intermediate species Combination of H2 O reaction prefactors, see (5.3.10) Stoichiometric coefficient Chapman–Jouguet Zeldovich and Frank-Kamenetskii 305

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Chemical Kinetics of Combustion

Superscripts, Subscripts and Math Accents aqs ab aB ai aind aIn ao , ao aR aX a˜

Quasi steady-state Burnt gas Chain branching Chemical species i Induction Initiation Initial or reference state Chain breaking (recombination) Intermediate species, X In molar units

5.1 Introduction The elementary background of chemistry, concerning essentially the properties of chemical equilibrium in combustion, is recalled in Chapter 14. The chemical kinetics of combustion are complex. The rate of energy release is controlled by a large number of elementary reactions involving many intermediate species.[1,2] Fortunately, systematic reductions of the reaction schemes have been obtained, opening new perspectives in flame theory.[3] 5.1.1 Role of the Multiple-Step Chemistry Some phenomena cannot be described by the thermal runaway of the ZFK model. Important notions, such as the induction delay and the flammability limits, are controlled by chain reactions in the detailed reaction scheme. The flammability limits are defined by the critical compositions beyond which adiabatic flames can no longer propagate. They are due to a brutal transition in the combustion regime which cannot be represented in the ZFK model without modification. For a given initial condition, the induction delay is the time needed for the energy release rate to increase to a high value. The delay is defined without ambiguity because the transition from an incipient rate of heat release to thermal runaway is sharp. For ambient conditions the delay is infinite, but it becomes short for temperatures above 1300 K, decreasing to a few milliseconds at 2500 K. The induction delay and its temperature dependence control the inner structure and the dynamics of gaseous detonations. Their expressions depend on the rates of initiation and chain-branching reactions. Another challenging problem is to decipher how complex chemical kinetics influence the propagation and the dynamics of flame and detonation fronts. The ZFK model, for example, predicts a single Markstein number, but in most real flames there are two different [1] [2] [3]

Lewis B., von Elbe G., 1961, Combustion flames and explosions of gases. Academic Press. Law C., 2006, Combustion physics. Cambridge University Press. Peters N., 1997, Prog. Astronaut. Aeronaut., 173, 73–91.

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307

Markstein numbers, as mentioned in Section 2.3. Direct numerical simulations are now available for planar flames and detonations sustained by a complex chemistry. This is not yet the case for the dynamics of wrinkled or cellular fronts. However, analytical studies of simplified cases help improve the understanding of combustion phenomena. Reduced Schemes A great deal of effort has been deployed in the last 35 years to obtain, in a more or less systematic way, reduced kinetic schemes that can be handled in analytical and numerical studies of hydrogen and methane flames.[4,5,6] We will not attempt to present a review of the state of the art of this topic, it can be found in the specialised literature.[3,7] For the purpose of analytical studies, our objective is limited to obtaining the simplest model, beyond the ZFK model, that reproduces, at least qualitatively, the new phenomena introduced by chain reactions. Such two-step chain-branching models will be used in the analytical studies in the second part of the book. They are modified versions of the Zeldovich–Li˜nan model,[8,9] constituted by an autocatalytic reaction (B) producing an intermediate species X and consuming the reactant R, followed by an exothermic reaction (R) transforming X into a stable product P, (B)

R + X → 2X,

ωB = cR cX BB e−E/kB T , E  kB T,

(R)

M + X → P + Q,

ωR = h(T)cX nBR ,

(5.1.1)

where the concentration of species and the reaction rates are denoted by c and ω, respectively. Two versions (M) and (H) will be considered, depending on the definition of h(T), (M):

T < T ∗ : h = 0 and

(H):

h = 1 ∀T,

T > T ∗ : h = 1,

(5.1.2) (5.1.3)

where T ∗ is the temperature of the thin reaction zone in which the reaction (B) is confined, ωB = 0. The rest of this chapter is devoted to show to what extent these two-step schemes can represent qualitatively well the essential characteristics of the combustion of methane and hydrogen. The (M) version is the simplest model for methane flames. The (H) version is more appropriate for rich hydrogen flames with a modification of the reaction rate of step (R); see (5.3.17). 5.2 Basics Concepts in Chemical Kinetics The elementary reactions involved in gaseous combustion correspond to inelastic collisions between atoms, molecules or radicals. [4] [5] [6] [7] [8] [9]

Peters N., Williams F., 1987, Combust. Flame, 68(2), 185–207. Seshadri K., Peters N., 1990, Combust. Flame, 81, 96–118. Seshadri K., et al., 1994, Combust. Flame, 96, 407–427. Peters N., Rogg B., eds., 1993, Reduced kinetic mechanisms for applications in combustion systems. Springer-Verlag. Zeldovich Y., 1961, Kinetika i Katalis, 2, 305–313. Li˜nan A., 1971, A theoretical analysis of premixed flame propagation with an isothermal chain reaction. AFOSR Contract No. E00AR68-0031 1, INTA Madrid.

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5.2.1 Initiation, Chain Branching and Chain Breaking The elementary reactions may be classified into four categories. Chain-Branching Steps In addition to thermal runaway, which was understood during the nineteenth century, combustion involves autocatalytic reactions identified by Semenov around 1930. Such elementary reactions, called chain-branching reactions, consume reactants and produce highly reactive intermediate species (atoms or radicals) at a rate proportional to their concentration, leading to an exponential growth of radicals. They are often endothermic and have a large activation energy. Letting R denote the initial species (usually oxygen and a hydrocarbon or hydrogen), and X the intermediate species (atoms or radicals, H, O, OH, etc.), a chain-branching step may be represented by (B)

R + X → 2X − QB ,

(5.2.1)

and the production rate of radicals is dcX /dt = ωB with ωB = cR cX kB (t), where kB (T)1 is the reaction constant, kB (T) = BB e−EB /kB T , c denotes concentration, EB is the activation energy, EB /kB T > 1 and Q is the heat of reaction. Initiation Steps The intermediate species, X, are absent from the initial composition of the combustible. They are produced initially from the reactive species by initiation reactions, which are frozen at room temperature because of their high activation energy. Letting M denote any species, an initiation step may be schematically represented as: (In)

R + M → νX X + M.

(5.2.2)

The production rate of radicals is dcX /dt = νX ωIn with ωIn = ncR kIn and kIn (T) = BIn e−EIn /kB T , where n is the density, EIn a large activation energy, EIn /kB T  1, EIn > EB and νX is a stoichiometric coefficient, νX  1. Such steps are usually endothermic and their reaction rate is sufficiently slow to not significantly influence the global balance of energy and mass of reactants. They do not play any role in flame propagation since the concentration of intermediate species in the preheated zone is dominated by molecular diffusion from the reaction zone. However, they control the induction delay, and thus the structure of gaseous detonations. Chain-Breaking Steps The intermediate species are finally transformed into stable products, Pi , (CO2 , H2 O) by strongly exothermic chain-breaking reactions that are usually very fast (zero activation energy) and only weakly temperature dependent. There are two types of such steps. One is 1 k (T) should not be confused with Boltzmann’s constant, k . B B

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309

the recombination of radicals by a triple collision (Ri )

M + X + X → M + Pi + Qi ,

(5.2.3)

dcX /dt = −2ωRi with ωRi = nc2X BRi . The second type is also produced by a triple collision involving one of the initial reactants, (Rii )

M + R + X → M + Pii + Qii .

(5.2.4)

The consumption rate of radicals is dcX /dt = −ωRii with ωRii = ncR cX BRii . The reactions (Rii ) are usually faster than (Ri ). There is a fourth kind of elementary reaction, called a chain-propagating reaction, that transforms one intermediate species into another. They play an essential role in the production of NO, or in the combustion of halogens,[1] for example, but they are not important in the following. The elementary reactions (B), (In), (Ri ) and (Rii ) have backwards reactions which may be neglected for simplicity to illustrate the mechanisms controlling the flammability limits and the induction delay. 5.2.2 Two-Step Model for the Flammability Limits Some important characteristics of combustion kinetics in gases are well described by the simple two-step model constituted by (5.2.1) and (5.2.4). This model differs only from case (H) in (5.1.1) and (5.1.3) by the fact that the chain-breaking reaction (Rii ) involves a reactant R, while it does not in the chain-breaking reaction (R). This difference is not important for the following. In homogeneous mixtures the corresponding equations yield dcX = νX ncR kIn (T) + cX cR [kB (T) − nBRii ], (5.2.5) dt dT = cX cR [nBRii Qii − kB (T)QB ], (5.2.6) ncp dt where for simplicity the specific heat cp has been taken constant, and (Ri ) is assumed to be negligible compared with (Rii ), 2cX BRi cR BRii . This simplification is no longer valid for flames near the flammability limits; see Section 8.5.5. Crossover Temperature and Flammability Limits The competition between (B) and (Rii ) introduces a crossover temperature T ∗ that is in the range 950–1400 K at atmospheric pressure, depending on the reactive mixture, [kB (T ∗ )−nBRii ] = 0,

∗

BB e−EB /kB T = nBRii .

(5.2.7)

The crossover temperature may reach lower values at low pressure, typically T ∗ ≈ 700 K for p ≈ 10−2 atm. Below the crossover temperature, T < T ∗ , nBRii > kB (T), the chain-breaking reaction (Rii ) dominates the chain branching (B). The transition is sharp [1]

Li˜nan A., Clavin P., 1987, Combust. Flame, 70, 137–159.

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because EB /kB T ≈ 8–10 so that the reaction constant kB (T) becomes quickly negligible in front of nBRii when T is decreased below T ∗ . According to (5.2.5), the intermediate species reaches a very low concentration νX kIn (T)/BRii , too low to produce any effect when incipient losses (e.g. heat loss to the walls or by radiation) are taken into account. No flame can propagate with a combustion temperature smaller than the crossover temperature, Tb < T ∗ . For the same reason, no detonation can propagate if the temperature at the Neumann state (just downstream from the lead shock) is below crossover, TN < T ∗ . This defines the composition limits on both rich and lean sides, beyond which flames or CJ detonations cannot propagate. In real hydrocarbon–air mixtures, the crossover temperature denotes a transition between a fast exothermic reaction (T > T ∗ ) and a slow and weakly exothermic reaction (T < T ∗ ). For 500 K  T  T ∗ , incomplete combustion develops, producing peroxides and aldehydes with a reaction time of the order of 1 s. In these conditions, travelling waves, called cool flames, have been observed in carefully controlled laboratory experiments.[1] Their nature is different from ordinary flames. Below 400–500 K the reaction rate is totally negligible. Induction Delay Above crossover, T > T ∗ , [kB (T ∗ ) − nBRii ] > 0, a runaway occurs after an induction delay during which the heat release and the consumption of reactants are negligible. For T > T ∗ , the second term in (5.2.5) leads to an exponential growth of intermediate species cX ≈ cXo et/τB , dcX /dt = cX /τB ,

1/τB ≡ cR [kB (T) − nBRii ] > 0.

(5.2.8)

The initial condition cXo of the exponential growth is provided by the initiation reaction, the first term on the right-hand side of (5.2.5). Taking this term into account, Equation (5.2.5) yields dcX /dt = n/τIn + cX /τB ,

1/τIn ≡ νX cR kIn (T),

(5.2.9)

where the induction reaction is so slow that, because of the large activation energy, EB > kB T, one has τIn  τB , even just above crossover, T > T ∗ . For the initial condition t = 0: cX = 0, the solution is

cX /n = et/τB − 1 τB /τIn , (5.2.10) valid during the induction delay where cX /n is so small, kIn kB , that the temperature rise and the consumption of reactant are negligible. The induction delay τind is evaluated from (5.2.10) by writing t = τind : cX /n = O(1). Anticipating that τind > τB (eτind /τB  1), meaning that the runaway occurring after the induction delay involves a shorter time scale, Equation (5.2.10) gives

[1]

Lewis B., von Elbe G., 1961, Combustion flames and explosions of gases. Academic Press.

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5.3 Combustion of Hydrogen

τind ≈ τB ln(τIn /τB ).

311

(5.2.11)

Typical profiles of the reaction rate downstream from the lead shock, computed with a detailed chemical kinetic scheme of hydrogen–oxygen overdriven detonations, are plotted in Fig. 4.29 for different detonation velocities (different Neumann temperature TN ).[2] The three stages, induction, runaway and relaxation are clearly visible in this figure; see also the results obtained more recently with a systematically reduced kinetic scheme.[3] Limitations According to (5.2.6), a thermal runaway is triggered by the autocatalytic runaway. In hydrocarbon flames the runaway is followed by an exothermic relaxation stage, lasting a time scale τr of the same order as τind . This last stage is not represented by the simple scheme (5.2.1) plus (5.2.4). The chemical kinetics of combustion is more complex. There are many elementary reactions of each of the types mentioned in Section 5.2.1. Many different intermediate species X, two reactants R (oxygen and fuel) and two stable products, H2 O and CO2 , plus two other important species in the burnt gas of rich flames, CO and H2 , are also involved. However, as we will see in the next sections, the results obtained with the schemes (5.2.1) plus (5.2.4) or (5.1.1) plus (5.1.3) are useful. In particular, the induction mechanism described here works well for hydrogen combustion. It is different for hydrocarbons; see Section 5.4.

5.3 Combustion of Hydrogen An extensive review of hydrogen combustion is available.[4] The detailed mechanism of hydrogen combustion is relatively well known,[5] even if the expressions for the constants of some important reactions have been modified over the last 10 years, as shown by the comparison of data in the specialised literature.[4,5,6] It involves 42 elementary reactions, eight species, including the two reactants H2 and O2 , one product H2 O, five intermediate species, H, O and OH, plus two others, hydroperoxyl HO2 and hydrogen peroxide H2 O2 . Although much simpler than that of hydrocarbons, this detailed scheme is still too complicated for analytical studies of flames.

5.3.1 Basic Scheme Below 5 atm, namely below what is called the third limit of explosion,[1] the species H2 O2 may be neglected. The 12 elementary reactions given in Table 5.1[5,7] are sufficient to

[2] [3] [4] [5] [6] [7]

Clavin P., He L., 1996, J. Fluid Mech., 306, 353–378. Boivin P., et al., 2013, Combust. Flame, 160, 76–82. Sanchez A., Williams F., 2014, Prog. Energy Combust. Sci., 41, 1–55. Saxena P., Williams F., 2006, Combust. Flame, 145, 316–323. Law C., 2006, Combustion physics. Cambridge University Press. Fernandez-Galisteo D., et al., 2009, Combust. Flame, 156, 985–996.

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Table 5.1 Main elementary reactions for hydrogen combustion. Reaction rate constants are k˜ j = B˜ j T νj e−Taj /T . Units are mole cm−3 , seconds and Kelvin. Taj ≡ Ej /kB is the activation energy in Kelvin. Label

Reaction

k˜ j

B˜ j

k˜ 1f k˜ 1b k˜ 2f k˜ 2b k˜ 3f k˜ 3b

3.52 × 1016 7.04 × 1013 1.17 × 109 1.29 × 1010 5.06 × 104 3.03 × 104

−0.7 −0.264 1.3 1.196 2.67 2.63

νj

1

O2 + H  OH + O

2

H2 + OH  H2 O + H

3

H2 + O  OH + H

4f 5f 6f

O2 + H + M → HO2 + M H + H + M → H2 + M H + OH + M → H2 O + M

k˜ 4f k˜ 5f k˜ 6f

5.79 × 1019 1.30 × 1018 4.00 × 1022

−1.4 −1.0 −2.0

7f 8f 9f

HO2 + H → OH + OH HO2 + H → H2 + O2 HO2 + OH → H2 O + O2

k˜ 7f k˜ 8f k˜ 9f

7.08 × 1013 1.66 × 1013 2.89 × 1013

0 0 0

Taj 8590 72 1825 9412 3165 2433 0 0 0 148 414 −250

Source: Saxena P., Williams F., Combust. Flame, 145, 316–323, 2006.

describe atmospheric hydrogen flames throughout the domain of flammability. They include: • Three autocatalytic reactions (1f)–(3f), called shuffle reactions, in which the reactants H2 and O2 are attacked by intermediate species, the atoms H and O and the radical OH, to produce more intermediate species • Three reverse shuffle reactions (1b)–(3b) • Three recombination reactions: (4f) producing HO2 , (5f) and (6f) • Three hydroperoxyl consumption reactions (7f), (8f) and (9f). When the radical concentrations are small, the two recombination reactions (5f) and (6f) are negligible compared with (4f). When computing the induction delay it is necessary to add the reverse reaction (8b) which is the main initiation step (In), (8b)

H2 + O2 → HO2 + H.

This reaction plays no role in flame propagation.

5.3.2 Second Limit of Explosion The competition between chain branching and chain breaking is illustrated by the second limit of explosion[1] of hydrogen–oxygen mixtures around T ≈ 750–850 K and [1]

Lewis B., von Elbe G., 1961, Combustion flames and explosions of gases. Academic Press.

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313

p ≈ 0.1 atm.: the explosion is favoured by a decrease of pressure. In such conditions, chain branching is the result of the three shuffle reactions, (1f), (2f) and (3f): (1f) O2 + H → OH + O, k1f = B1 e−E1 /kB T , ω1f = k1f cH cO2 , (2f) H2 + OH → H2 O + H, k2f = B2 e−E2 /kB T , ω2f = k2f cOH cH2 , (3f) H2 + O → OH + H, k3f = B3 e−E3 /kB T , ω3f = k3f cO cH2 . The reverse reactions (1b)–(3b) and the trimolecular reactions (5f) and (6f) are negligible because the concentrations of H, O, OH and H2 O are too small, at least in the first stage of runaway. The only additional reaction to consider is (4f), (4f) M + H + O2 → M + HO2 ,

ω4f = cH cO2 nB4f .

(5.3.1)

This is a chain-breaking reaction consuming H. In a first step, the species HO2 may be considered as a quasi-stable species. This simplification will be removed below. In Table 5.1 the reaction rates are written for concentrations expressed in moles c˜ i = ci /N , ci = Ni /V, N being the Avogadro number (see Chapter 14), and with an explicit temperature dependence of the prefactors Bi , i = 1, 2, 3: T νi B˜ if = N Bif ,

T ν4 B˜ 4f = N 2 B4f .

(5.3.2)

This system of four reactions (1f)–(4f) describes the essential feature of hydrogen combustion and explains the second limit of explosion. Using a steady-state approximation for the intermediate species, it may be reduced to a two-step model similar to (B) and (Rii ) in (5.2.2) and (5.2.4), as shown next. Steady-State Approximation for O and OH In a homogeneous mixture, the evolution equations for the concentrations of O and O2 are, according to the reactions (1f) and (3f), dcO /dt = cH cO2 k1f − cO cH2 k3f ,

dcO2 /dt = −cH cO2 k1f .

(5.3.3)

Consider the case where the consumption of O involves a characteristic time shorter than that of the consumption of O2 , 1/cH2 k3f 1/cH k1f , which is valid before thermal runaway, when cH cH2 . Introducing the small parameter  ≡ cH k1f /cH2 k3f 1 and the time τ , reduced by the characteristic time for the consumption of O2 , dτ = cH k1f dt, Equation (5.3.3) takes the form dc/dτ = f (τ ) − c/. The solution quickly relaxes to c ≈ f (τ ), as shown by the limit  → 0 of the solution  τ  eτ / f (τ  )dτ  , c(τ ) = c(0)e−τ/ + e−τ/ 0

− τ  )/

is used. This approximation thus imposes that the when the integration variable (τ production and consumption rates of the intermediate species are always equal: the species is in steady state. This is valid for any intermediate species that is consumed on a time

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scale much shorter than the evolution of the time scale of its production. The steady-state approximation for O obtained from (5.3.3) takes the form ω1f ≈ ω3f ,

cO /cO2 ≈ (cH k1f )/(cH2 k3f ) 1.

(5.3.4)

The two reactions (1f) and (3f) proceed at the same rate to consume 1 mole of H2 and O2 , each of them producing 1 mole of OH, dcH2 /dτ ≈ dcO2 /dτ = −ω1f , dcOH /dτ ≈ 2ω1f . This results in the following overall step proceeding at the rate of (1f), O2 + H2 → 2OH

rate: ω1f ≡ cH cO2 k1f .

(5.3.5)

When the relation ω1f ≈ ω3f is used, the evolution of the radical OH in the elementary reactions (1f), (2f) and (3f), dcOH /dt = ω1f + ω3f − ω2f , becomes dcOH /dt = 2cH cO2 k1f − cOH cH2 k2f . If cH2 k2f  2cH k1f , the radical OH is also in steady-state equilibrium, ω2f ≈ 2ω1f ,

cOH /cO2 ≈ (2cH k1f )/(cH2 k2f ) 1.

(5.3.6)

To summarise, with the above assumptions, O2 is consumed by (1f), dcO2 /dτ = −ω1f , H2 is consumed by (2f) and (3f), dcH2 /dτ = −ω2f − ω3f ≈ −3ω1f , and H is produced by (2f) and (3f) and consumed by (1f), dcH /dτ = +ω2f + ω3f − ω1f ≈ 2ω1f . The three shuffle reactions (1f)–(3f) result in a global chain-branching step proceeding at rate ω1f , O2 + 3H2 → 2H2 O + 2H,

rate: ω1f ≡ cH cO2 k1f ,

(5.3.7)

describing the production of H atoms at a rate proportional to cH as the autocatalytic step (B) (5.2.1), in competition with (4f) as the chain-breaking step (Rii ) (5.2.4). In contrast to an elementary reaction, the step (5.3.7) is a global reaction whose rate does not correspond to collisions between the reactants. Crossover Temperature The two conditions, cH k1f cH2 k3f and 2cH k1f cH2 k2f in (5.3.4) and (5.3.6), are both necessary to ensure the validity of (5.3.7). In these conditions, the chain-branching step (5.3.7) is in competition with the chain-breaking step (4f), in the same way that (B) is in competition with (Rii ) in (5.2.1) and (5.2.4). The induction delay τind can thus be computed as in (5.2.11) with (5.2.8). The crossover temperature T ∗ (τB = 0, τind → ∞) is the solution to the equation 2k1f ≈ nB4f , ∗

2T ∗(ν1 −ν4 ) B˜ 1f e−Ta1 /T ≈ n˜ B˜ 4f ,

(5.3.8)

where, in the notation of (5.3.2), n˜ ≡ n/N is the molar density. The crossover temperature decreases as the density is decreased, explaining the second limit of explosion. This is due to the ternary character of the chain-breaking step (4f), while (1f) is binary. In standard ∗ conditions (po = 1 atm, To = 298 K), T ∗ is the solution to T 0.7 e−8500/T ≈ 11, where the values of Table 5.1 have been used along with the perfect gas law n˜ T ∗ = n˜ o To and the definition of a mole, 22.4 l, n˜ o = 4.464 × 10−5 moles/cm3 . This gives T ∗ ≈ 923 K. A more accurate result is obtained when the kinetics of HO2 are taken into account.

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315

Effect of the Consumption of the Hydroxyl Radical The radical OH is in steady state, and according to (5.3.6), its concentration is smaller than that of H. Therefore the consumption of HO2 by (9f) is negligible compared with (7f) and (8f), dcHO2 /dt ≈ ω4f − (ω7f + ω8f ) = cH [ncO2 B4f − (B7f + B8f )cHO2 ]. −1 At temperatures around 1000 K the consumption time of HO2 , c−1 H (B7f + B8f ) , is shorter −1 −1 than that of O2 given by (5.3.3), cH k1f , k1f  B7f + B8f . In such conditions HO2 is in steady state,

ω4f ≈ ω7f + ω8f ,

cHO2 ≈ ncO2 B4f /(B7f + B8f ),

(5.3.9)

and the rates (7f) and (8f) take the form ω7f ≈ αω4f ,

ω8f ≈ (1 − α)ω4f ,

where

α ≡ B7f /(B7f + B8f ).

(5.3.10)

The reactions (7f) and (8f) modify the balance of H, OH, O2 and H2 . In particular, the steady-state approximation for OH becomes ω2f ≈ 2(ω1f + ω7f ). One then gets dcH /dt = ω2f − (ω7f + ω8f ) − ω4f ≈ 2(ω1f + ω7f ) − 2ω4f , dcO2 /dt = −ω1f + ω8f − ω4f ≈ −(ω1f + ω7f ), dcH2 /dt = −ω2f − ω3f + ω8f ≈ −3(ω1f + ω7f ) + ω4f , dcH2 O /dt = ω2f ≈ 2(ω1f + ω7f ), leading to a two-step mechanism of chain branching and chain breaking, (I)

O2 + 3H2 → 2H2 O + 2H + QI , rate: ωI = (ω1f + ω7f ) ≈ (k1f + αnB4f )cH cO2 ,

(II)

H + H → H2 + QII , rate: ωII = ω4f ≈ ncH cO2 B4f ,

(5.3.11) (5.3.12)

where α is given by (5.3.10). Both steps are global reactions that are exothermic, the first only weakly, QI ≈ 11.4 kcal, and the second strongly, QII ≈ 103 kcal. They are in competition for the production of H, as (5.3.7) and (4f). The crossover temperature is not ∗ very different from the preceding case, B1f e−E1 /kB T ≈ (1 − α)nB4f .

5.3.3 Two-Step Flame Model The reduced two-step scheme (I)–(II) has to be modified at higher temperatures such as in flames. Partial Equilibrium of Shuffle Reactions When the reactions proceed at sufficiently high temperature, the concentrations of intermediate species are increased so that the backwards and forwards rates of the shuffle reactions

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(1)–(3) become of same order. The detailed analytical studies of hydrogen flames[1,2,3,4] are based on steady-state approximations for some intermediate species, depending on the conditions. For stoichiometric and lean hydrogen–air flames, the steady-state approximations for O, H2 O and OH are usually assumed. In order to give a flavour of what is going on, we limit our presentation to a simplified version. We focus attention on the four elementary steps (1)–(4f) in Table 5.1. The consumption of HO2 may be taken into account in a second step, as in Section 5.3.2. The shuffle reactions compete with reaction (5.3.1), (1)

O2 + H  OH + O,

K1 ≡ k1f /k1b ,

ω1 ≡ ω1f − ω1b ,

(2)

H2 + OH  H2 O + H,

K2 ≡ k2f /k2b ,

ω2 ≡ ω2f − ω2b ,

(3)

H2 + O  OH + H,

K3 ≡ k3f /k3b ,

ω3 ≡ ω3f − ω3b ,

where Ki and ωi denote the equilibrium constants and the reaction rates including the backwards reactions. The steady-state approximations for O and OH yield ω1 ≈ ω3 and ω1 + ω3 ≈ ω2 , so that dcO2 /dτ = −ω1 , dcH2 /dτ = −(ω2f + ω3f ) ≈ −3ω1 , dcH2 O /dτ = ω2 = 2ω1 and dcH /dτ = ω2f + ω3f − ω1 = 2ω1 . This leads to the overall reaction (I)

O2 + 3H2  2H2 O + 2H,

rate: ω1 = ω1f − ω1b ,

(5.3.13)

ω1f ≡ cH cO2 k1f , ω1b ≡ cO cOH k1b , where the concentrations of O and OH may be computed in terms of the concentrations of H, O2 , H2 and H2 O from the two equations ω1 ≈ ω3 and ω1 + ω3 ≈ ω2 , cO k1f + cOH k3b cO ≈ 2 , cH cOH k1b + cH2 k3f

2cO2 k1f + cH2 O k2b cOH ≈ . cH 2cO k1b + cH2 k2f

(5.3.14)

These solutions for cO and cOH are still difficult to handle in analytical studies, and further approximations, which are more or less controlled, are introduced. The most popular is the partial equilibrium of the shuffle reactions (2) and (3), ω2 ≈ 0, ω3 ≈ 0. This leads to cO /cH ≈ cOH /(cH2 K3 ) and cOH /cH ≈ cH2 O /(cH2 K2 ), instead of (5.3.14), and the reaction rate ω1 then takes the form . / c2H2 O c2H ω1 ≈ ωI ≡ cO2 cH k1f 1 − 3 (5.3.15) , KI = 755.8 e1668/T , cH2 cO2 KI where KI ≡ K1 K2 K3 is the equilibrium constant of the overall reaction (I). The compatibility of the steady-state approximations for O and OH, and the partial equilibrium of reactions (2) and (3) requires that ω1 should also be small, meaning that the reaction rate (5.3.15) is valid in the vicinity of the chemical equilibrium of (I), ωI  0. However, it is also valid far from equilibrium in initial mixtures where the concentrations of H and H2 O are infinitesimal, ω1 ≈ cO2 cH k1f ; see (5.3.7). Therefore (5.3.15) may be considered as a [1] [2] [3] [4]

He L., Clavin P., 1993, Combust. Flame, 93, 391–407. Seshadri K., et al., 1994, Combust. Flame, 96, 407–427. Fernandez-Galisteo D., et al., 2009, Combust. Flame, 156, 985–996. Fernandez-Galisteo D., et al., 2009, Combust. Theor. Model., 13(4), 741–761.

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317

convenient interpolation for the reaction rate of step (I). The main limitation is that the steady-state approximation is not very good for OH in lean hydrogen flames.[1,2] Hydrogen-Rich Flame Model When approaching the hydrogen-rich flammability limit, the flame temperature decreases and the forwards reaction may be neglected in (5.3.13), ωI ≈ ω1f . This approximation is valid for Tb  1600 K. However, in such conditions, the ternary recombination reaction (5f) in Table 5.1 cannot be neglected,[1] so that ω5f = nc2H B5f should be added to ωII in (5.3.12). Still neglecting the effects of hydroxyl consumption for simplicity (α = 0), we are left with the two-step model (I) and (II) but with modified expressions for the reaction rates ωI = cH cO2 B1f e−E1 /kB T ,

ωII = cH cO2 nB4f + nc2H B5f ,

(5.3.16)

where O2 is the limiting reactant. The autocatalytic production of H radicals occurs only when the temperature is above the crossover temperature T > T ∗ , where T ∗ is a fixed ∗ temperature, the solution to B1f e−E1 /kB T = B4f . Since the activation energy is high, E1  kB T ∗ , H production and O2 consumption are both concentrated in a thin temperature range just above T ∗ . The temperature is further increased, up to Tb , by consuming the H radical with the rate nc2H B5f . The analytical study of flame structure is still difficult to perform without additional approximations. An ad hoc approximation is to replace the quadratic term c2H of ω5f by a linear term, ω5f → cH nBR , ωI = cH cO2 B1f e−E1 /kB T ,

ωII = cH cO2 nB4f + cH nBR .

(5.3.17)

The (H) version (5.1.3) of the two-step model (5.1.1) is then recovered when cO2 nB4f nBR . However, very close to the flammability limit, it will be more appropriate to use (5.3.16) with a quasi steady-state approximation for the H radical; see Section 8.5.5.

5.4 Combustion of Lean Methane Mixtures This objective of this section is to set up the simplest two-step model useful for methane on the basis of the reduction of the chemical kinetics that has been proposed for lean methane– air flames.[5,6] 5.4.1 Basic Mechanisms C1 Chain The detailed kinetics of methane–air flames contain more than 100 species and 1000 elementary reactions;[7] however, for lean flames most recombination reactions, including all recombination of radicals containing carbon, can be neglected. The important reactions [5] [6] [7]

Peters N., Williams F., 1987, Combust. Flame, 68(2), 185–207. Peters N., 1997, Prog. Astronaut. Aeronaut., 173, 73–91. Konnov A., 2009, Combust. Flame, 156, 2093–2105.

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Table 5.2 Essential elementary reactions for methane combustion, in addition to those for hydrogen given in Table 5.1. Reaction rate constants are k˜ j = B˜ j T νj e−Taj /T . Units are mole cm−3 , seconds and Kelvin. Taj ≡ Ej /kB is the activation energy in Kelvin. Reaction

k˜ j

B˜ j

νj

10f 11f 12f 13f

CH4 + H → CH3 + H2 CH3 + O → CH2 O + H CH2 O + H → CHO + H2 CHO + M → CO + H + M

k˜ 10f k˜ 11f k˜ 12f k˜ 13f

1.30 × 104 8.43 × 1013 1.26 × 108 2.85 × 1014

3.0 0.0 1.62 0.0

4040 0 1090 8455

14f 14b

CO + OH  CO2 + H

k˜ 14f k˜ 14b

4.40 × 106 4.97 × 108

1.5 1.5

0 10800

15f

CH4 + M → CH3 + H + M

k˜ 15f

6.59 × 1025

−1.8

52800

Label

Taj

Source: Hewson J., Williams F., Combust. Flame, 117, 441–476, 1999.

then reduce to those already given for hydrogen, Table 5.1, plus a small set of reactions for the oxidation of CH4 , given in Table 5.2.[1] Reaction (15f) is necessary for initiation but plays no role in the propagation of methane flames. The oxidation of CH4 to CO is achieved mainly through a chain of elementary reactions, (10f)–(13f), in which CH4 , and the intermediate radicals CH3 , CH2 O and CHO are attacked by atomic H and O. This results in an overall step, CH4 + O → CO + 2H2 ,

(5.4.1)

whose rate is controlled by the first elementary reaction[1] which is the slowest, (10f)

CH4 + H → CH3 + H2 ,

ω10f = cCH4 cH k10f .

(5.4.2)

The radicals CH3 , CH2 O and CHO are in quasi-steady state. The species O appearing in (5.4.1) is produced mainly from the destruction of H2 O by H through the chain of reverse reactions (2b) and (3b), (2b) + (3b):

2H + H2 O → 2H2 + O.

(5.4.3)

The two steps (5.4.1) and (5.4.3) lead to the following overall step, controlled by the rate of (10f), (III) CH4 + 2H + H2 O → CO + 4H2 ,

ωIII = cCH4 cH k10f ,

(5.4.4)

where H is produced by the chain-branching step (I) in (5.3.11), whose reaction rate is given by (5.3.15) in competition with the chain-breaking step (II) in (5.3.12).

[1]

Hewson J., Williams F., 1999, Combust. Flame, 117, 441–476.

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319

Oxidation of CO The oxidation of CO is the final step in the combustion of most hydrocarbons. It proceeds on the hot side of the flame and contributes to further increase the temperature. It is controlled by the elementary reactions (14f) and (14b), where the radical OH is produced from H2 O by the elementary reaction (2) in partial equilibrium: H + H2 O  H2 + OH. This leads to the water–gas shift whose rate is given by (14f) and (14b) in Table 5.2: CO + H2 O  CO2 + H2 ,

(IV) ωIV = k14f

cH cH2 K2

(5.4.5)

  1 cCO2 cH2 , cCO cH2 O − KIV

where KIV ≡ K14 /K2 is the equilibrium constant of the water–gas shift.

5.4.2 Three-Step Scheme To summarise, the oxidation of methane can be described by the four-step mechanism, (I), (II), (III) and (IV) with reaction rates (5.3.15), (5.3.12), (5.4.4) and (5.4.5). The radical H is produced in step (I) and consumed in steps (II) and (III). This scheme may be reduced to a three-step scheme when the steady-state approximation for H holds, ωI = ωII + ωIII . Introducing coH , (5.4.6) coH ≡ (KI c3H2 cO2 )1/2 /cH2 O ,   the reaction rate (5.3.15) can be written ωI = cO2 cH k1f 1 − (cH /coH )2 , and the steadyqs state solution, ωI = ωII + ωIII , is cH = cH , where  nB4f cCH4 k10f 1/2 qs cH ≡ coH 1 − − , (5.4.7) k1f cO2 k1f qs

when the value in the bracket is positive, and cH = 0 when it is negative. At temperatures below the crossover temperature (5.3.8), T  T ∗ ≈ 1000 K, the second term in the bracket is larger than unity, so that cH = 0. The reaction rates of all four steps (I)–(IV) are proportional to cH , so the mixture is inert below this temperature. However, the ratio k10f /k1f is an increasing function of the temperature for T > 1229 K. Therefore, in order to have non zero reaction rates, cH = 0, the ratio cCH4 /cO2 must decrease quickly to small values when T increases. The reaction rate ωIII in (5.4.4), which is proportional to the concentrations of CH4 and H, is thus confined to a thin reaction layer in the flame. The three-step scheme is obtained by eliminating the radical H. The balance equations for species shows that this is done by adding (III) to (I) to give (I ): O2 + CH4 = CO + H2 + H2 O with the reaction rate ωIII , and by adding (II) to (I) to give (II ): O2 + 2H2 = 2H2 O with the reaction rate ωII , (I ) O2 + CH4 → CO + H2 + H2 O + QI  , 

(II ) O2 + 2H2 → 2H2 O + QII  ,

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(5.4.8) (5.4.9)

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Chemical Kinetics of Combustion

(IV) CO + H2 O  CO2 + H2 + QIV , ωI  = ωII  = ωIV =

qs cH cCH4 k10f , qs cH cO2 nB4f ,  qs cH k14f cCO cH2 O cH2 K2

(5.4.10) (5.4.11) 

−

1 cCO2 cH2 . KIV

(5.4.12) (5.4.13)

All three steps are exothermic, QI  ≈ 66.4 kcal, QII  ≈ 115.6 kcal and QIV ≈ 9.8 kcal. The scheme (I ), (II ) and (IV) is similar in nature to (I) and (II) with (5.3.15) and (5.3.12) for hydrogen. According to (5.4.6), (5.4.7) and (5.4.11), step (I ) is an auto catalytic reaction of order 3/2 producing H2 from the reactants O2 and CH4 . Step (II ) consumes H2 in the same way as step (II) consumes H produced by (I). The other species produced by (I ), CO, is consumed by the water–gas shift (IV). There are two significant differences with the two-step scheme presented for hydrogen: the reactions rates ωI  and ωII  are proportional to 3/2 cH2 and they both cancel below the crossover temperature. The final equilibrium is reached at high temperature when cH2 = 0, so that, according to the step (IV), CO disappears in parallel with H2 in the burnt gas, as expected for lean flames. Flame Structure lean flames, step (I ) stops when all the CH

In lean CH4 4 has been consumed and its reaction rate ωI  is confined to a thin reaction layer at a high temperature above the crossover temperature. Downstream from this thin layer where H2 and CO are produced, the temperature is further increased by the exothermic steps (IV) and (II ). Upstream from the thin layer there is an inert preheated zone into which H2 and CO diffuse, as shown in Fig. 5.1.

Figure 5.1 Structure of lean methane flame, (equivalence ratio = 0.65) showing the temperature and concentration profiles of the stable (solid lines) and intermediate (dotted lines) species retained in the four-step model. Concentrations are in mole fractions.

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321

Induction Delay Consider a CH4 –air mixture brought suddenly to a high temperature, as when crossing an inert shock wave, for example. The most efficient path for the initial production of the radical H is the thermal decomposition of CH4 , (15f)

CH4 + M → CH3 + H + M.

Let τIn be the characteristic time for consumption of CH4 by (15f), ω15f = cCH4 /τIn . This initiation reaction is in competition with the reaction (10f) which consumes CH4 and H. Consider for simplicity the case where the production of H by the overall reaction (5.3.7) is negligible compared with the consumption of H by (10f), cO2 k1f cCH4 k10f ; the equation for cH takes the form dcH /dt = cCH4 /τIn − cCH4 cH k10f , where, typically, k10f  1/τIn . In such conditions, H quickly reaches steady state, cH ≈ k10f /τIn 1, and the concentration of CH4 decreases with the time on a characteristic time scale τIn /2, dcCH4 /dt = −cCH4 /τIn − cCH4 cH k10f ≈ −2cCH4 /τIn . The concentration of H is maintained at a small value during this decay. The induction delay is then the time necessary to consume the methane, τIn , because the chain-branching reaction (I ) cannot start before cCH4 reaches a very small value. Further Reduction Three relations are obtained when the reaction rate ωIV is eliminated from the balance equations for the four species CO, H2 , H2 O and CO2 in the three-step scheme (5.4.8)– (5.4.13): c˙ CO + c˙ H2 = 2(ωI  − ωII  ), c˙ H2 O + c˙ H2 = 2ωI  , c˙ CO2 + c˙ CO = ωI  . A reduction of the three-step model scheme can be obtained when the water–gas shift reaction (5.4.5) is in quasi-equilibrium. This is the case when |ωIVf | ≈ |ωIVb |  ωI  > ωII  with |ωIV | ≡ |ωIVf − ωIVb | of the order of |ωI  |, meaning that the equilibrium of the water– gas shift is reached on a time scale much shorter than that of the reaction steps (I ) and (II ). The equations for the evolution of each species may then be expressed in terms of ωI  and ωII  by adding the quasi-equilibrium relation, cCO2 cH2 ≈ KIV cCO cH2 O , to the three relations above. Unfortunately, this cannot be put in a form as simple as a two-step scheme controlled by the rates ωI  and ωII  . However, a simple form can be obtained if it is assumed that the ratio a ≡ cCO /cH2 is constant in the reaction zone (cH = 0)[1,2] , instead of assuming quasi-equilibrium for the water–gas shift. The above balance equations then yield [1] [2]

Peters N., Williams F., 1987, Combust. Flame, 68(2), 185–207. Peters N., 1997, Prog. Astronaut. Aeronaut., 173, 73–91.

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c˙ CO = 2a(ωI  − ωII  )/(1 + a),

c˙ H2 = 2(ωI  − ωII  )/(1 + a),

showing that the sum H2 + aCO plays the role of an intermediate species X, produced by 3/2 the step (I ) at a rate proportional to cH2 and consumed by the step (II ). Unfortunately, in real flames, the ratio a is not really constant in the domain where cH = 0. For example, it varies between 3 and 6.5 in Fig. 5.1. However, the relation a ≈ cst. is not so bad in the thin reaction zone where ωI  = 0; the quantity a varies only between 4.9 and 6.4 in this layer of Fig. 5.1. A good strategy would be to use the quasi-equilibrium of the water–gas shift at high temperature where ωI  = 0 and to use the ratio a constant in the thin reaction zone where ωI  = 0. In conclusion, the main characteristics of lean methane flames may be qualitatively represented by the two-step model (5.1.1). Since the preheated zone is inert because of the absence of atomic hydrogen, the (M) version, (5.1.2), is more appropriate for methane flames.

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6 Laser-Driven Ablation Front in ICF

Nomenclature Dimensional Quantities a cp d DT g I Jq k m n p r t T u u ua UL x y α λ  ρ σ

Description Sound speed Specific heat at constant pressure Thickness Thermal diffusivity Acceleration Radiative energy flux Heat flux Wavenumber Mass flux Number density Pressure Coordinate (vector) Time Temperature Flow velocity (scalar) Flow velocity (vector) Ablation velocity Laminar flame speed Streamwise coordinate Transverse coordinate Local position of front Thermal conductivity Wavelength Density Growth rate

S.I. Units m s−1 J K−1 kg−1 m m2 s−1 m s−2 J s−1 m−2 J s−1 m−2 m−1 kg m−2 s−1 m−3 Pa m s K m s−1 m s−1 m s−1 m s−1 m m m J s−1 m−1 K−1 m kg m−3 s−1

323

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Laser-Driven Ablation Front in ICF

Nondimensional Quantities and Abbreviations ex Fr Le n O(.) ξ ν θ cst. ICF ZFK

Unit vector along x-axis Froude number Lewis number Unit vector normal to the front Of the order of Reduced longitudinal coordinate Temperature exponent of thermal conductivity Reduced temperature Constant Inertial confinement fusion Zeldovich and Frank-Kamenetskii

Fr−2 = gda /u2a DT /D

x/da ≈ 2.5 T/Ta

Superscripts, Subscripts and Math Accents aa ab ac af am amax au atot aT a a˜ aˆ

Cold side of ablation front Burnt (hot) gas Critical (absorption) surface of ablation wave (hot side) Hot side of ablation font, but cold side of absorption layer, Ta < Tf Tc At cut-off Maximum value Unburnt (cold) gas Total Thermal Value on the unperturbed (planar) front Fourier component of any a, a(x, y, t) = a˜ (x)eik y+σ t Value reduced by units of length and time at cold side of ablation front

The ablation wave in inertial confinement fusion (ICF),[1,2] presented in Section 1.3.3, has similarities but also differences with flames. Due to the implosion, the thermal wave is subjected to a very strong Rayleigh-Taylor instability which is counteracted by conduction damping. Because of a strong variation of the electron thermal conductivity with temperature, the dynamics of the ablation front in ICF differs from that of flame. The most striking result is that the conduction damping is stronger than the Darrieus–Landau instability in the full range of relevant wavelengths. This topic is analysed in this chapter. 6.1 Approximations and Constitutive Equations When the material of the external surface of the target is suddenly heated by the powerful radiative flux of a laser beam, the thermal wave is decomposed into a shock wave [1] [2]

Lindl J., 1998, Inertial confinement fusion. Springer. Atzeni S., Meyer-Ter-Vehn J., 2004, The physics of inertial fusion. Clarendon Press–Oxford Science Publications, 1st ed.

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325

(b) (a)

Figure 6.1 (a) Profiles of temperature (grey line) and density (black line) in an ICF ablation front. (b) Velocities and fluxes in the frame of the ablation front.

propagating inwards followed by a wave in which energy is transported upstream by electron-driven thermal conduction; see Fig. 6.1. The latter wave, called the ablation wave, is followed by a quasi-isothermal expanding wave of hot plasma which is transparent to the laser light. Attention is focused below on the ablation wave.

6.1.1 Structure of the Ablation Wave An energy flux I ≈ 1015 W/cm2 of radiation transforms the material into a fully ionised plasma at high temperature and pressure, Tc ≈ 108 K, pc ≈ 108 bar. The plasma becomes strongly absorbing and opaque to the laser light when the electron density reaches a critical value nc at which the plasma frequency is equal to the light frequency of the laser. For a laser wavelength of 1 μm the critical density is typically nc ≈ 1021 /cm3 , that is, 10–100 smaller than the initial electron density of the ablated material. For a density smaller than nc the plasma is transparent to the laser light. The surface in the plasma at which the electron density is close to nc defines a thin critical surface in which the laser energy is absorbed. In this layer the temperature is high and the density small, respectively, much higher and smaller than in the upstream material. Downstream from the critical surface the plasma is expelled outwards through an unsteady spherical rarefaction wave.[3] The latter cannot influence the ablation wave because the flow is sonic at the critical surface. Knowing I and nc , the critical conditions (Tc ≈ 108 K, pc ≈ 108 bar) can be roughly estimated from the energy flux I ≈ ρc a3c , where ac is the sound speed at Tc , and by assuming that nc , Tc and pc are approximately linked by the ideal gas law. Once the laser energy is locally deposited at high temperature and low density, the energy is carried towards the cold material by hot electrons and also photons in the form of a diffusive transport that can be approximated by a Fourier law, (15.1.28) Jq = −λ(T)∇T. The one-dimensional structure of this thermal wave, called the ablation wave, can be maintained steady by a constant propagation velocity, called the ablation velocity. [3]

Sanz J., et al., 1981, Phys. Fluids, 24(11), 2098–2106.

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Laser-Driven Ablation Front in ICF

Absorption layer

Absorption layer

Cold D-T

Cold D-T

(b)

(a)

Figure 6.2 (a) Density and (b) temperature profiles for a Spitzer-type model for a temperature ratio Tc /Ta = 50 and temperature dependency exponent, ν = 2.5. For comparison the profiles for ν = 1 are plotted as dashed lines.

The main differences with combustion waves are the large temperature difference between the hot and cold sides, T/T ≈ 105 −107 , and the strong variation of thermal conductivity with temperature λ = (T/Ta )ν λa , ν > 1, where the subscript a denotes the conditions at the low temperature in the cold material. For heat conduction by electrons, the thermal conductivity is approximated by the Spitzer formula, ν = 5/2, while ν ≈ 3 for radiative transport. In contrast with ordinary flames and detonations, the conduction length d = λ/(ρucp ) varies by many orders of magnitude across the wave, da /dc 1, so that the relative density variation (ρa − ρ)/ρa is concentrated on the cold side of the ablation wave in a thin layer, called the ablation front; see the structure of the planar wave in Fig. 6.2a. The large temperature ratio, Tc /Ta  1, has important effects on the dynamics. First, the density at the cold side ρa is much larger than at the hot side, ρa  ρc , so that the flow velocity ua at the cold side is not only much smaller than the flow velocity at the hot side ua ac , but also smaller than the sound speed at the cold side, aa , so that the ablation velocity is very subsonic, ua aa . This is because the variation of the sound speed with temperature is weaker than that of the flow velocity, as illustrated by the ideal gas law for a √ planar steady wave in the quasi-isobaric approximation, a/ac ≈ T/Tc , uT=cst.; see text below (6.1.8). A quasi-isobaric approximation is thus valid near to the ablation front. Another consequence of the large temperature ratio, Tc /Ta  1, is that the Rayleigh– Taylor instability in ICF conditions generates transverse disturbances with wavelengths much smaller than the total thickness of the thermal wave dc ; see (6.2.4). Then, because of the localisation of the density change at the ablation front, Rayleigh–Taylor unstable disturbances are generated at the cold side with a sharply peaked amplitude; see Fig. 11.1. The disturbances are quickly damped downstream from the ablation front under the effect of the strong increase of the thermal conductivity λ(T) with temperature, leading to a drastic increase of the diffusive damping rate. To summarise, the dynamics are fully described by the solutions to quasi-isobaric equations with boundary conditions corresponding to disturbances vanishing both downstream and upstream from the ablation front.

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327

6.1.2 Formulation. Definition of the Ablation Front The ideal gas law is a good approximation for the plasma, and the viscous effects are not essential in the dynamics of the ablation front; they are neglected for simplicity. Since the thickness of the thermal wave is smaller than the radius of the capsule, curvature effects can also be neglected in the unperturbed solution. Quasi-isobaric Model and Boundary Conditions The quasi-isobaric equations (15.2.2)–(15.2.5) for the conservation of mass, momentum and energy, written in the reference frame attached to the unperturbed wave, yield ∇.u = −

ρT = ρa Ta , ρcp

DT = ∇.(λ∇T), Dt λ = (T/Ta )ν λa ,

ρ

1 DT 1 Dρ = , ρ Dt T Dt

Du = −∇p + ρgex , Dt

(6.1.1) (6.1.2) (6.1.3)

in which ρa and Ta are the constant density and temperature at the upstream boundary condition, D/Dt ≡ ∂/∂t + u.∇ is the material derivative, g is the acceleration due to implosion and ex is a unit vector normal to the unperturbed ablation front propagating in the negative x direction. For simplicity the specific heat per unit mass cp is assumed constant. When the balance equation for mass (6.1.1) is introduced into the thermal equation in (6.1.2), the equation for energy conservation in the quasi-isobaric approximation takes a useful form relating the divergence of the flow velocity to the thermal gradient, ∇.[u − (λ/ρa cp )∇(T/Ta )] = 0.

(6.1.4)

Considering disturbances with a wavelength smaller than the radius of the capsule, the upstream boundary condition is x → −∞:

u = ua ,

T = Ta ,

p = pa ,

(6.1.5)

where Ta and pa are fixed prescribed quantities and ua is the ablation velocity of the unperturbed planar wave sustained by a constant energy flux I along the x-axis and oriented in the negative x-direction. According to what has been said before, the disturbances vanish at the downstream boundary, x → +∞. For the purpose of comparison with flames, it is more convenient to extend the quasi-isobaric approximation up to the critical surface. The downstream boundary then concerns the conduction heat flux at the isotherm T = Tc with Tc  Ta . Let x = αc (y, t) be the equation of the isotherm T = Tc , and n the normal to this surface; the downstream boundary condition is  x = αc (y, t): λc n.∇T = I/ 1 + (∂αc /∂y)2 , (6.1.6) corresponding to the conservation of energy across the thin absorbing sheet. In the linear approximation this flux is constant and the problem reduces to solving the ZFK model of a flame propagating upwards with unity Lewis number. However, for Tc  Ta , ν > 1,

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Laser-Driven Ablation Front in ICF

and for the accelerations involved in ICF, the dynamics of the ablation front differs from that of ordinary flames obtained in the limit of large wavelengths; see Section 2.2. In ICF conditions, in contrast to flames, the wavelengths of the disturbances are smaller than the total thickness of the wave. However, they are larger than the thickness of the sharp ablation front on the cold side of the structure of the planar ablation wave. Steady, Planar Solution. Ablation Velocity. Ablation Front In the steady, planar case the problem reduces to solving a simple thermal equation: d dT − m dx dx

.

λ dT cp dx

/

 = 0,

T = Ta ,

x → −∞:

m = ρa ua , T = Tc :

λ≡ λ

T Ta

ν λa ,

dT = I, dx

ν > 1,

Tc  Ta .

(6.1.7) (6.1.8)

The problem is the same as in the preheated zone of a flame. Assuming that the temperature gradient is negligible downstream from the thin critical layer, x > αc : dT/dx = 0, the ablation velocity is obtained by integrating the first equation in (6.1.7) from x = −∞ to a point just downstream of the critical layer, m = I/cp (Tc − Ta ) corresponding to ua ≈ I/(cp Tc ρa ). The difference with flame is that the energy flux I results here from absorption of the laser beam, while in the ZFK flame model it results from the chemical reaction rate at high temperature and is obtained from the analysis of the thin reaction layer. Because the compressible effects have been neglected at high temperature, this expression for the ablation velocity ua ≈ I/(cp Tc ρa ) is not accurate in ICF, but the order of magnitude is correct. The low Mach number approximation of the ablation velocity is easily verified for Tc  Ta by using the expressions for the sound speed, a ∝ T 1/2 , and for the laser flux, I ≈ ρc a3c , yielding ua /aa ∝ (Ta /Tc )1/2 1. Introducing the reduced temperature θ ≡ T/Ta and the nondimensional coordinate ξ , reduced by the small diffusion length at low temperature da ≡ λa /(cp m) = DTa /ua where DTa is the thermal diffusion coefficient at Ta , the thermal equation yields

d dθ − dξ dξ

. θ

ν dθ

dξ

/

θ ≡ T/Ta , = 0,

ξ = x/da , ξ → −∞: θ = 1,

(6.1.9) θ

ν dθ

dξ

= θ − 1.

(6.1.10)

The solution for θ − 1 is proportional to eξ for θ close to unity but the slope increases ν very strongly as soon as θ increases. At high temperature θ  1, one gets θ /ν ≈ ξ + cst., where the constant depends on the choice of origin. Choosing the origin at the critical (absorption) surface ξ = 0: θ = θc  1, the solution at high temperature takes the form θ  1:

θ ≈ θc



1/ν ξ +1 . θcν /ν

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(6.1.11)

6.2 Dynamics of the Ablation Front

329

For ν > 1 it has a sharp transition around ξ = −θcν /ν and the total nondimensional thickness dtot of the thermal wave is of the order of the diffusion length at high temperature, dtot /da = θcν /ν; see Fig. 6.2a. The inset shows the evolution for θ − 1 1. The existence of a thin ablation front is even more spectacular when the density ratio ρ/ρa = 1/θ is plotted versus the nondimensional distance, ξ ν/θcν , reduced by the conduction length at high temperature dc ≡ λc /(cp m), dc /da = θcν  1. A sharp drop from unity to small values, nearly to zero, is observed around ξ/θcν = −1/ν; see Fig. 6.2b, explaining why the transverse disturbances produced by the Rayleigh–Taylor instability are localised at the cold side. Delimiting the ablation front at the hot side by a temperature θf sufficiently larger than Ta but much smaller than Tc , the thickness of the ablation front df is, according to (6.1.11), larger than da but negligible compared with the total thickness dtot , 1 < θf θc :

df /da ≈ θfν /ν,

1 < df /da θcν /ν.

(6.1.12)

6.2 Dynamics of the Ablation Front In contrast to ordinary flames, the specificity of the ablation wave in ICF is the large domain of length scales, ranging from da to dc , dc /da ≈ 104 –105 . When length, velocity and time are reduced by, respectively, the diffusion length at low temperature da , the ablation velocity ua and the unit time da /ua , Equations (6.1.1)–(6.1.3) for θ ≡ T/Ta , u/ua and p/ρa u2a involve three nondimensional parameters, the temperature ratio θc , the exponent ν of the conductivity and the inverse of the Froude number, Fr, measuring the strength of the Rayleigh–Taylor instability, θc ≡ Tc /Ta  1,

ν > 1,

Fr−2 ≡ gda /u2a ,

(6.2.1)

with typical values in ICF: θc = 50, ν = 5/2, Fr−2 = 0.2. Attention should be paid to the definition of the Froude number in (6.2.1) since a different definition is often used in the ICF literature (Fr−1 = gda /u2a ). Using the small length scale da in the definition of the Froude number, values of Fr−2 of order 10−1 correspond to an acceleration much stronger than that experienced by flames propagating upwards.

6.2.1 Linear Growth Rate of the Ablative Rayleigh–Taylor Instability The linear dynamics of ablation fronts are analysed from the linearised version of (6.1.1)– (6.1.3) using the usual normal mode decomposition of the disturbances of the fields δT(r, t), δu(r, t) and δp(r, t) with the same notation as in (4.4.1), δf (x, y, t) = f˜ (x)eiky+σ t ,

(6.2.2)

in which a two-dimensional geometry is used for simplicity r = (x, y), x and y being, respectively, the longitudinal and transverse coordinates. In the following k will denote the modulus of the transverse wave vector |k|. With this notation, the results are valid for

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Laser-Driven Ablation Front in ICF

planar waves perturbed in two transverse directions (y, z). An extensive list of references concerning the linear analysis of ablation waves in ICF can be found in a book[1] for results before 2004 and also in a recent review article[2] stressing how the comparison with flames has improved the understanding. Attention is focused below on physical insights; the detailed analyses are performed in Chapter 11. Numerical Results: Comparison with Ordinary Flames The dispersion relation, σ (k), was investigated by early numerical solutions[3,4] to Equations (6.1.1)–(6.1.3), linearised around the planar wave and considering disturbances vanishing at x → ∞. In ICF conditions and for ν = 2.5, the best fits to the numerical results led to an empirical relation, known as the Takabe formula,  with a = 0.84 – 0.94, b = 2.5 – 3.5, (6.2.3) σ = a gk − bua k, in agreement with an early semi-phenomenological analysis[5] and also with experiments. The first term in the right-hand side of (6.2.3) is the growth rate of the Taylor instability, and the second term shows the stabilising effect of ablation. The instability is suppressed for disturbances with a wavelength smaller than a cut-off length lm = 2π/km , where km = (a/b)2 g/u2a . The linear growth rate is maximum for a wavelength four times larger, l = 4lm , 4σmax = (a2 /b)g/ua , with typical values km = 0.09g/u2a and σmax = 0.07g/ua . In nondimensional notation, this corresponds to km da = 0.09 Fr−2 and (da /ua )σmax = 0.07 Fr−2 , that is, a wavelength some 102 times larger than da for Fr−2 = 0.2. The relevant unstable wavelength clearly belongs to intermediate values, larger than da but smaller than dc = 5/2 θc da . Using the hat symbol (ˆ) in the following to indicate nondimensional quantities built with the units of length and time at the cold side, da and da /ua , Equation (6.2.3) yields ˆ σˆ = aFr−1 kˆ 1/2 − bk,

with

kˆ ≡ kda ,

σˆ ≡ σ da /ua .

(6.2.4)

The expression for the growth rate in (6.2.3) is surprising when compared with the result for unstable flames subjected to the Rayleigh–Taylor instability; see (2.2.18) in which −|g| is replaced by |g|. For the comparison we first recall the difference of notation; the temperatures of the hot (burnt) and cold (unburnt) gas were denoted Tb and Tu in flames, whereas in the ablation wave, the temperature at the (hot) absorption surface and at the cold side (upstream from the ablation front) are Tc and Ta , respectively. The flame velocity was denoted UL (or Ub when measured in the burnt gas), while ua is the ablation velocity. A first difference concerns the propagation velocity. The ablation velocity ua ≈ I/(cp ρa Tc ) is independent of the thermal diffusivity DT = λ/ρcp , while, according to (2.1.9), the flame velocity varies as the square root of DT . The most striking outcome of (6.2.3) is that the [1] [2] [3] [4] [5]

Atzeni S., Meyer-Ter-Vehn J., 2004, The physics of inertial fusion. Clarendon Press–Oxford Science Publications, 1st ed. Bychkov V., et al., 2015, Prog. Energy Combust. Sci., 47, 32–59. Takabe H., et al., 1985, Phys. Fluids, 28, 3676–82. Kull H., 1989, Phys. Fluids, B1, 170–82. Bodner S., 1974, Phys. Rev. Lett., 33, 761–764.

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stabilising term at short wavelength, −bua k, has a sign opposite to the Darrieus–Landau 1/2 instability σ = θc ua k, written here for θc  1 in the notation used for the ablation wave; see (2.2.3) and text below (2.2.5). Even more surprising, the thermal diffusivity DT does not appear in the damping term −bua k, while the damping in conditions of ordinary flames is σ ∝ −DT k2 . Two intriguing physical problems have to be deciphered: • What is the physical mechanism behind the so-called ablation damping in ICF conditions? • How can the Darrieus–Landau instability be fully suppressed? The answers come from the multiple-scale nature of the problem, da dc , and comparison with flames helps the physical understanding of the dynamics of ablation fronts.[6,7] The results of the analyses that are presented in the second part of the book, in Chapter 10 for flames and Chapter 11 for ablation waves in ICF conditions, are summarised below. Variable Thermal Diffusivity in Flame Dynamics Let us first revisit the linear dynamics of ordinary flames. Limiting attention to wrinkles that are larger than the total flame thickness, the result (2.2.18) has been extended to take account of the variation of thermal diffusivity with temperature.[8] For the purpose of comparison with ablation waves, the ICF notation will be used: UL → ua ,

Tu → Ta ,

Tb /Tu → θc ≡ Tc /Ta ,

dL → da ,

db → dc .

(6.2.5)

The dispersion relation of a Rayleigh–Taylor unstable flame with a unity Lewis number, Le = 1, and λ = (T/Ta )ν λa , ν > 1, has been obtained by a perturbation analysis for kdc ≡ θcν kˆ 1, taken up to first order, following the method used in Section 10.3.4. ˆ defined in (6.2.4), the result takes the nondimensional form[8] Written in terms of σˆ and k,      1 νˆ ˆ 2ν + 1 ν ˆ νˆ 2 −2 θ k θc kˆ kˆ = 0, (6.2.6) θc k 1: σˆ + 2 1 + θc k kσˆ − Fr + 1 − ν (ν + 1)ν c where, for the sake of comparison with ICF, terms that are negligible in the limit of a large temperature ratio, Tc /Ta , θc  1, have been omitted in the expressions for the coefficients. The origin of the correction terms, θcν kˆ 1, is clearly understood by considering the thermo-diffusive model of flames, ρ = cst., u = (UL , 0, 0), ∂T ∂T + UL = ∇.(DT ∇T), ∂t ∂x  ν T DTu , DT = Tu x → −∞: T = Tu ,

[6] [7] [8]

T = Tb > Tu : DTb n.∇T = (Tb − Ta )UL ,

Clavin P., Masse L., 2004, Phys. Plasmas, 11, 690–705. Sanz J., et al., 2006, Phys. Plasmas, 13, 102702. Clavin P., Garcia P., 1983, J. M´ec. Th´eor. Appl., 2(2), 245–263.

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(6.2.7) (6.2.8) (6.2.9)

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where n is the normal to the isotherm T = Tb and DTb ≡ (Tb /Tu )ν DTu is the thermal diffusivity at Tb . The second boundary condition in (6.2.9) corresponds to the limit of an infinitely large activation energy: the heat flux normal to the thin reaction zone is constant and equal to the unperturbed planar solution when Le = 1 (fixed flame temperature); see (8.2.52). When the coefficients of the differential equations are constant, the stability analysis is straightforward. This is the case for a constant thermal diffusivity, ν = 0, DT = DTu , and the result σ = −DT k2 is valid with no limitation on the wavelength (see Section 11.1.1), ˆ (ν = 0: σˆ = −kˆ 2 , ∀ k).

DT = cst.: σ = −DT k2 , ∀ k

(6.2.10)

This is the classical damping described by the diffusion equation in (13.3.38). The analysis is also straightforward for a variable diffusivity when attention is restricted to the long wavelength limit, kDTb /UL 1. The perturbation analysis, presented in Section 11.1.1, is straightforward because the temperature disturbances are small and vanish on both sides, at x → −∞ and at the reaction sheet, T = Tb . The result is   Tb 1 σ =− DT  dT  k2 Tb − Tu Tu and takes the nondimensional form, written in the notation of (6.2.5), θcν kˆ 1:

ν+1

σˆ ≈ −

1 θ c − 1 ˆ2 1 θ ν kˆ 2 , k ≈− ν + 1 θc − 1 ν+1 c

(6.2.11)

where the last relation holds when θc  1. As explained in Sections 2.2.3, 2.2.4 and 10.1.3, such a k2 diffusive relaxation rate introduces a k3 term in the last term of the quadratic equation (6.2.6) for the linear growth rate when the hydrodynamic instabilities due to the change of density are coupled to the transverse diffusion across the wrinkled wave. To summarise, the correction terms θcν kˆ 1 in (6.2.6) are introduced by diffusive damping in the presence of a density variation. When the diffusive relaxation mechanisms are neglected, Equation (6.2.6) reduces to (2.2.16) describing the superposition of two hydrodynamical instabilities, the Darrieus– Landau and the Rayleigh–Taylor instabilities for a flame considered as a hydrodynamic discontinuity (no thickness effect). For a large density jump, ρu /ρb  1, the expression for the growth rate of these two coupled hydrodynamic instabilities simplifies to  νˆ θc k → 0 ⇒ σˆ ≈ Fr−2 kˆ + θc kˆ 2 , (6.2.12) θc = ρu /ρb  1, where the term proportional to kˆ σˆ in (6.2.6) has been dropped since it is negligible in front of σˆ 2 , kˆ < σˆ . The first term is the Rayleigh–Taylor growth rate of an interface and the second is the Darrieus–Landau growth rate of a flame in the absence of acceleration Fr−2 = 0. The diffusion damping decreases the growth rate. The maximum growth rate, given by the maximum of the bracket [...] in (6.2.6), corresponds to wavelengths that are too small,

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θcν kˆ = O(1), beyond the limit of validity of the perturbation analysis. Disregarding this limitation and extrapolating (6.2.6) to disturbances with wavelengths of the order of dc , ˆ σˆ goes to zero as θc increases, and the second term in (6.2.6) is θcν kˆ = O(1), the ratio k/ negligible. The linear growth rate then takes the form  (2ν + 1) (ν+1) ˆ 3 k , σˆ ≈ Fr−2 kˆ + θc kˆ 2 − (6.2.13) θc  1, θc (ν + 1)ν in which the diffusive damping introduces a kˆ 3 term, similar to the surface tension effect in the Rayleigh–Taylor instability of an interface; see Section 10.1.3. This damping mechanism counteracts the Darrieus–Landau kˆ 2 term but cannot explain (6.2.4). Growth Rate of the Ablative Rayleigh–Taylor Instability in ICF Linear analyses leading to a growth rate of the ablative Rayleigh–Taylor instability in better agreement with (6.2.3)–(6.2.4) have been performed[1,2,3,4] in a limit opposite to (6.2.6), namely for wavelengths that are smaller than the total thickness of the thermal wave, kdc  1, 

(6.2.14) σˆ 2 + 4kˆ σˆ − Fr−2 − c(ν) ν 1/ν kˆ 1−1/ν kˆ = 0, θc  1, θcν kˆ  1: where c(ν) is a quantity of order unity (see (11.2.56)), c = 0.672 . . . for ν = 5/2. With no reference to the analyses of flame dynamics, carried out 10 years before, the original analyses[1,2,3,4] are difficult to follow. Neither the conditions of validity of the result nor the physical interpretation can be easily extracted from these pioneering works. To facilitate the physical understanding and the presentation of this tough problem we will proceed as follows. After a discussion of the result (6.2.14) we first consider exact solutions of much simpler models that shed light on the underlying physical mechanisms. Comparison with (6.2.14) then indicates the asymptotic method needed to solve the full problem. The tricky asymptotic analysis,[5] leading to the linear growth rate, is postponed to Sections 11.2.2 and 11.2.3. The order of magnitude of the wavelengths of both of the most unstable disturbances (maximum value of σ ) and the marginal wavelength (σ = 0) is, according to (6.2.14), ν/(ν−1)  ˆ ≈ Fr−2 /ν . (6.2.15) k/ν Limiting attention to an acceleration that is not too strong, Fr−2 < ν, the wavelengths of the unstable disturbances are larger than the diffusion length scale on the cold side, kˆ ≡ kda < ˆ σˆ < 1 holds, (k/ ˆ σˆ )ν−1 = 1. Moreover, near to the maximum growth rate, the relation k/ √ −1 O(Fr / ν), so that the second term in (6.2.14), 4kˆ σˆ , is negligible. The dispersion relation [1] [2] [3] [4] [5]

Sanz J., 1994, Phys. Rev. Lett., 73, 2700–2703. Bychkov V., et al., 1994, Phys. Plasmas, 1, 2976–2986. Sanz J., 1996, Phys. Rev. E, 53, 4026–45. Goncharov V., et al., 1996, Phys. Plasmas, 3, 1402–14. Sanz J., et al., 2006, Phys. Plasmas, 13, 102702.

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in (6.2.14) concerns an intermediate range of wavelengths da 1/k dc = θcν da and takes the simplified form  ν ˆ σˆ ≈ Fr−2 kˆ − c(ν) ν 1/ν kˆ 2−1/ν . (6.2.16) ν > 1, 1 1/k θc : The result (6.2.16) concerns the intermediate range of parameters 1/θcν−1 Fr−2 /ν 1,

(6.2.17)

where the lower and upper bounds in (6.2.17) ensure that the wavelengths (6.2.15) are, respectively, larger than da , kda 1, and smaller than dc = da θcν , kdc  1. The conditions (6.2.17) are usually satisfied in ICF experiments (ν > 1, θc  1), and the corresponding asymptotic analysis[1] is presented in Section 11.2. For a sufficiently strong variation of heat conductivity with temperature, that is, when 1/ν is negligible in front of 2, the damping term in (6.2.16) effectively takes a form similar to an anti-Darrieus–Landau term, as shown by comparison with (6.2.12). The transitional regime between the dynamics of flames in (6.2.6) and the dynamics of ablation waves in (6.2.14) should concern conditions in which the marginal wavelength (σ = 0) is of order of the total thickness, kdc = O(1). Numerical studies[2,3] show that this is effectively the case and corresponds to a Froude number of order Fr−2 /ν ≈ 1/θcν−1 . Also, an asymptotic analysis[1] performed in the limit Fr−2 /ν → 0 succeeded in covering both results (6.2.13) and (6.2.16). This confirms that the damping term in (6.2.16) results from a competition between thermal diffusion and the Darrieus–Landau instability for intermediate wavelengths, da 1/k dc . This unexpected behaviour for a diffusive relaxation can be understood by considering simplified models of the ablation wave. Results for Simplified Models Consider first the thermo-diffusive model (6.2.7)–(6.2.9) when the temperature difference is sufficiently large so that there is a wide range of length scales across the wave structure. The diffusive relaxation rate of disturbances with intermediate wavelengths, solution to (6.2.7)–(6.2.9), is proved to take an anti-Darrieus–Landau form, du 1/k db :

σ = −UL k,

(6.2.18)

or in the nondimensional form (6.2.5), θc  1,

ν > 1,

1 1/kˆ θcν :

−kˆ 2

ˆ σˆ = −k,

(6.2.19)

in (6.2.10) when there is only one length scale in the problem, namely instead of σˆ = for DT =cst. The mathematical proof is technically difficult[4] for the power law (6.2.8). A simpler but less rigorous analysis is presented in Section 11.1.1. Using a discontinuous but piecewise constant diffusivity, T < Tf : DT = DTu , T > Tf : DT = DTb , with [1] [2] [3] [4]

Sanz J., et al., 2006, Phys. Plasmas, 13, 102702. Clavin P., Masse L., 2004, Phys. Plasmas, 11, 690–705. Clavin P., et al., 2005, Combust. Sci. Technol., 177, 979–989. Clavin P., et al., 2011, Comm. Math. Sci., 9(1), 127–141.

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335

Tu < Tf < Tb , it is relatively easy to prove[2] that, in the limit du /db → 0 (du ≡ DTu /UL db ≡ DTb /UL ), the exact solution to (6.2.7), verifying the boundary conditions (6.2.9), takes the same form as (6.2.18)–(6.2.19). An extended discontinuous model is even more instructive. By adding a density jump T < Tf : ρ = ρa , T > Tf : ρ = ρf , an exact solution to the linearised equations (6.1.1)–(6.1.2) can also be obtained;[2] see the analysis in Section 11.1.2. The corresponding expression for the linear growth rate simplifies in the intermediate range of wavelengths where the damping term takes the form of an anti-Darrieus–Landau term. For a sufficiently large density ratio, θf ≡ ρa /ρf > 1, in the parameter range da /dc < Fr−2 /θf < 1, the term kˆ σˆ in the quadratic equation for σˆ becomes negligible (σˆ /kˆ  1) and the result takes a form similar to (6.2.16) in the limit ν → ∞, 1 1/kˆ dc /da :

 σˆ = Fr−2 kˆ − θf kˆ 2 .

(6.2.20)

The anti-Darrieus–Landau relaxation term −θf kˆ 2 results from the coupling of hydrodynamics and thermal diffusion. Expressions similar to (6.2.20) were obtained previously by semi-phenomenological analyses[5,6] using the so-called sharp boundary model in the ICF literature. The temperature θf that delimits the thin ablation front on the hot side is not determined in the analysis. The physical arguments presented below fill the gap between (6.2.20) and (6.2.16). Physical Insights The results presented above all concern disturbances with intermediate wavelengths when the diffusion coefficient DT (T) varies strongly across the wave. They can be interpreted in simple physical terms on the basis of classical diffusion damping (6.2.10), σ = −DT k2 . Introducing the variable diffusion length d(T) ≡ DT (T)/UL into the diffusion relaxation rate, σ = −UL d(T)k2 , the anti-Darrieus–Landau form (6.2.18) is recovered if the temperature to be considered in the diffusion coefficient is the one for which the diffusion length is equal to the wavelength, kd(T) ≈ 1. This relation expresses the fact that, inside the wide range of length scales [du , db ], du 1/k db , there is no other natural length scale than the wavelength  = 2π/k, since the situation corresponds to the double limit du / → 0 and db / → ∞. Such a dimensional analysis was used in turbulence for the inertial range of the Kolmogorov cascade; see Section 3.4.1. However, attention should be paid to the fact that such a simple argument is not as universal as it seems at first sight. In some cases, as for self-similarities of the second kind,[7] the effect of an extreme length scale, for example, du in the limit du / → 0, is not negligible and must be obtained by an asymptotic analysis. An example can be found in the mathematical analysis[4] of a modified version of the thermo-diffusive model [5] [6] [7]

Piriz A., et al., 1997, Phys. Plasmas, 4, 1117–26. Piriz A., 2001, Phys. Plasmas, 8, 997–1002. Barenblatt G., 1996, Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press.

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Laser-Driven Ablation Front in ICF

(6.2.7)–(6.2.9) in which a term Tu /T is introduced into the coefficient of the unsteady term ∂T/∂t to represent the variation of density in the thermal equation (6.1.2). In this case, the result differs from (6.2.18)–(6.2.19); the reduced growth rate σˆ of disturbances with an ˆ intermediate wavelength involves the power law kˆ 1−1/ν and also a logarithmic term ln k. The link between (6.2.20) and (6.2.16) is found by using the same type of universal argument claiming that there is no other length scale than the wavelength in the regime of intermediate wavelengths. Equation (6.2.16) is then recovered from (6.2.20) when the temperature delimiting the ablation front, θf , is taken at the point of the temperature profile where the diffusion length is of the order of the wavelength, see Section 11.2.2, ˆ 1/ν . θf ∝ (ν/k)

(6.2.21)

The coefficient of proportionality is determined by asymptotic analysis in Section 11.2.

6.2.2 Nonlinear Dynamics of the Ablation Front The asymptotic analysis[1] performed in the limit (6.2.17) and presented in Section 11.2, leading to the linear growth rate (6.2.16), has been extended to the nonlinear regime in the limit of a large power index, ν  1. The analysis is presented in Section 11.3. The essential feature is that, in the limit 1/ν → 0, the vorticity is localised inside a boundary layer adjacent to the ablation front, so that a new front is defined by incorporating the two layers. The problem then reduces to a study of the dynamics of a sheet separating two potential flows. Using the non–dimensional quantities defined in Sections 11.2 and 11.3 (see (11.2.14) and (11.2.15)), the nonlinear dynamics of the front in two-dimensional geometry (coordinates ζ and η, time τ ) is described by a free boundary problem controlled by two potentials, ψ− and ψ+ , satisfying the following boundary conditions[2] ψ− = 0,

ψ+ = 0,

lim ψ− = 0,

ξ →−∞

 nf .∇ψ− |f = Un ,

ψ+ |f = 0,

lim ψ+ = 1,

ξ →+∞

|∇ψ− |2 |∇ψ+ |2 ∂ψ− + + ∂τ 2 2

(6.2.22)

= ζf ,

(6.2.23)

f

where the subscript f indicates the point on the front having coordinates (ζf , ηf ) and where Un is the normal velocity of front and nf the normal to the front and the tildes have been suppressed to simplify the presentation. The ablation velocity is negligible compared with the velocity of the perturbation of the cold flow represented by the potential ψ− . The gradient of the potential ψ+ represents both the hot flow and the thermal gradient; see (11.2.16) and (11.2.23). Equations (6.2.22)–(6.2.23) have also been derived with a discontinuous model of ablation front.[3] For a periodic solution, the only free parameter in (6.2.22)–(6.2.23) is the size of the box. According to the scaling in (11.2.14) the nondimensional size of the box corresponds [1] [2] [3]

Sanz J., et al., 2006, Phys. Plasmas, 13, 102702. Almarcha C., et al., 2007, J. Fluid Mech., 579, 481–492. Clavin P., Almarcha C., 2005, C. R. M´ecanique, 333(379-388).

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337

(a) Without surface tension With surface tension

(b)

Figure 6.3 Numerical simulations of growth of ablative Rayleigh–Taylor instability in ICF using a boundary integral method. (a) Growth of a single mode in a small box, k = 0.5, showing the formation of a singularity of curvature at the inflexion points and regularisation by the addition of a weak surface tension (reproduced from Almarcha C. et al., 2007, J. Fluid. Mech. 579, 481–492, with permission) (b) Numerical simulation of growth of multiple modes in a large box. Courtesy of C.Almarcha, IRPHE Marseilles.

to the ratio of the size to the most amplified wavelength. The larger the nondimensional size of the box used to solve (6.2.22)–(6.2.23), the larger is the number of unstable modes. A single unstable mode can be obtained for a sufficiently small box. Accurate solutions to (6.2.22)–(6.2.23) have been obtained numerically for periodic conditions in the transverse direction by using a boundary integral method following the pioneering analysis of the Rayleigh–Taylor instability.[4] The method used to solve (6.2.22)– (6.2.23) is described in the original paper.[2] Typical results are shown in Figs. 6.3 for different sizes of the box. A singularity of curvature develops on the front in a finite time, after the amplitude of wrinkling has reached the order of magnitude of the wavelength, or slightly larger. The singularity concerns the inflexion points of the front. The two points of curvature extrema (of opposite sign), one on each side of each inflexion point, coalesce and the absolute value of the curvature goes to infinity. The formation of this singularity of curvature has been studied with great accuracy by the boundary integral method. The study shows a selfsimilar behaviour.[2] Denoting τ0 the critical time at which the singularity develops, the absolute values of the two extrema of curvature increase as (τ0 − τ )−1 and the arc-length distance between the two points decreases as (τ0 − τ )4/3 , leading to a derivative of the curvature with respect to the arc-length increasing as (τ0 − τ )7/3 at the inflexion point (zero curvature). These exponents are accurate within a few percent and do not depend on the wavelength.

[4]

Baker G., et al., 1980, Phys. Fluids, 23, 1485–1490.

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Laser-Driven Ablation Front in ICF

The singularity is smoothed out by the slightest surface tension–like term, introduced into the right-hand side of (6.2.23) to model the effect of finite thickness of the front. The singularity is a local event that seems to have no noticeable effect on the shape of the rest of the front; see Fig. 6.3a. However, it could correspond to the origin of a vortex sheet developing in the burnt gas, bringing into question the validity of the analysis as soon as the absolute value of the curvature becomes large. This requires further study.

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7 Explosion of Massive Stars

Nomenclature Dimensional Quantities a cp cv D/Dt e e E g G h Jq kB K L m m Ms M MCh ni p Q(j) r r Rs s

Description S.I. Units Speed of sound m s−1 Specific heat at constant pressure J K−1 kg−1 Specific heat at constant volume J K−1 kg−1 Material derivative, ∂/∂t + u.∇ s−1 Energy density J kg−1 Energy per particle J Total energy J Acceleration of gravity m s−2 Gravitational constant ≈ 6.67 × 10−11 m3 kg−1 s−2 Enthalpy J kg−1 Heat flux J s−1 m−2 Boltzmann’s constant J K−1 The constant in the polytopic equation of state, see (7.1.5) Length m Mass kg Molecular mass kg Mass of a star kg Mass of the sun ≈ 2 × 1030 kg Chandrasekhar mass ≈ 1.4 M kg Number density of species i m−3 Pressure Pa Heat release of elementary reaction j J Radial distance m Position (x, y, z) m Radius of a star m Entropy density J K−1 kg−1 339

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340

S t T TF u u v ˙ (j) W ρ τ φ(r) (r) ϕ(r, t)

Explosion of Massive Stars

Entropy Time Temperature Fermi temperature, see (13.2.33) (Radial) velocity Velocity of fluid (u, v, w) Specific volume 1/ρ Reaction rate of elementary reaction j Density A characteristic time Radial gravitational potential Generalised gravitational potential Time-dependent gravitational potential

J K−1 s K K m s−1 m s−1 m3 kg−1 m−3 s−1 kg m−3 s m2 s−2 m2 s−2 m2 s−2

Nondimensional Quantities and Abbreviations er uˆ rˆ R(X) ˆt U(X) Xi X Yi γ η ρˆ σ τ

Unit radial vector Nondimensional velocity, see (7.2.19) and (7.3.1) Nondimensional radius, see (7.2.19) and (7.3.1) Self-similar function for density variation, see (7.3.23) Nondimensional time, see (7.2.19) and (7.3.1) Self-similar function for velocity variation, see (7.3.23) Molecular fraction of particles i ni /nnuc Similarity variable, see (7.3.23) Mass fraction of species i ρi /ρ Adiabatic index (ratio of specific heats) cp /cv Mass-weighted coordinate Nondimensional density, see (7.2.19) and (7.3.1) Reduced growth rate ˆt Reduced time

Superscripts, Subscripts and Math Accents ac ae aeq aff aF aFe ag aint

Centre Electron Equilibrium Free fall Fermi Iron Gravitational Internal

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7.1 Constitutive Equations of Stars

anuc ar aref as atot aT a aˆ

341

Nucleus Inside a radius r Reference value or state Star Total Thermal Unperturbed value or equilibrium value Nondimensional quantity, see (7.2.19)

The current views of stellar evolution and explosion at the end of their lifetime (supernovae) are briefly summarised in Section 1.3.2. The formulation of the problem is presented in Section 7.1, and the hydrostatic equilibrium of star for a constant value of the adiabatic index γ is recalled in Section 7.2. The classical theory of stellar structure was developed a long time ago, before the Second World War, with important contributions at the end of the nineteenth century. The references to these early works can be found in the book of Chandrasekhar,[1] originally published in 1939. Dynamical problems have been investigated more recently; they are discussed in Section 7.3. The presentation in this book follows and summarises the classical monographs[1,2,3,4] with additional insights concerning the dynamics. The objective is to present in simple terms the shock formation and the so-called rebound during gravitational collapse. The presentation is limited essentially to the fluid mechanical aspects in spherical geometry when gravitation is taken into account. The drastic effect of thermal loss by the strong neutrino flux emitted at the end of the collapse of the iron core of massive stars is discussed in Section 7.3.3, where personal views of the mechanism of explosion are also presented. Nuclear physics at high density and temperature is not discussed in detail and only simple thermodynamic laws are used. The purpose is to provide the physical background to nonspecialists in astrophysics who are interested in the explosion of stars. Generally speaking, during a quasi-steady evolution, an abrupt transition can occur when a critical value of the parameter is reached, corresponding either to a loss of stability or to the disappearance of the solution, as in flame theory for the ‘C’-shaped curve describing flame quenching; see Fig. 8.16. The catastrophic collapse of the iron core of massive stars is an example of the first case and is presented in Section 7.3.1. 7.1 Constitutive Equations of Stars A star can be considered as a sphere of fluid. The differences with ordinary reacting fluids on earth are the gravitational force and the extreme thermodynamic conditions that imply the presence of nuclear physics. The fluid is a plasma since atoms are dissociated into nuclei [1] [2] [3] [4]

Chandrasekhar S., 1967, An introduction to the study of stellar structure. Dover. Zeldovich Y., Novikov I., 1971, Stars and relativity. Dover. Cox J., 1980, Theory of stellar pulsation. Princeton University Press. Kippenhahn R., Weigert A., 1994, Stellar structure and evolution. Springer-Verlag, 3rd ed.

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and electrons; however, plasma physics is not an essential ingredient in the explosion of stars, even though some recent studies consider magnetohydrodynamic effects. Also, the effects of relativity on the fluid dynamics are not considered to be essential even though the maximum flow velocity in the gravitational collapse is a fraction of the speed of light, typically one-tenth. The presentation in this chapter is carried out in the framework of classical fluid mechanics and the thermodynamic relations resulting from the Boltzmann and Fermi statistics.

7.1.1 Euler–Poisson Equation In a fluid constituted of particles interacting through the long-range force of gravitation, the macroscopic equation for the conservation of momentum is an extension of the Euler equation in which the gravitational force per unit mass is added to the pressure term. The latter is due to the short-range interaction between the microscopic particles, while the former results from a long-range force. The macroscopic equation then results from a separation between the short- and long-range forces. The expression for the macroscopic gravitational force per unit mass is derived from the microscopic equations using the same approximation as for the Vlassov equation in plasma physics:[1] it is assumed that there is no long-range correlation, meaning that the probability to find a couple of microscopic particles in two different positions r1 and r2 is the product of probability to find a particle at r1 and a particle at r2 . Introducing the material time derivative D/Dt = ∂/∂t + u.∇, the result takes the form of the so-called Euler–Poisson equation  1 ρ(r , t) 3  Du = − ∇p − ∇,  = 4π Gρ, (r, t) = −G d r, (7.1.1) Dt ρ |r − r | where G ≈ 6.67 × 10−11 m3 kg−1 s−2 is the gravitational constant. The Poisson equation (second equation) for the gravitational potential[2] (r, t) results from Gauss’s law for a force varying with the distance r as 1/r2 , (|r|−1 ) = 4π δ(r). The equation for the conservation of mass (15.1.3) is still valid. In the general case, the problem is closed by an equation of state p(ρ, T), where the temperature is the solution of a thermal equation including heat transfer, heat losses and heat release by nuclear reactions. In spherical geometry the acceleration g due to gravity is radial and takes the Newton form, g = Gmr (r, t)/r2 , where mr (r, t) is the mass inside the radius r:  r Dmr = 0. (7.1.2) mr (r, t) ≡ 4π ρr2 dr, with, according to continuity, Dt 0 The Euler–Poisson equation takes a form similar to (15.1.18), where g(r, t) ≡ ∂φ/∂r = er .∇ is minus the acceleration (oriented towards the origin) of a fluid particle under the gravitation force due to the interaction with the other fluid particles. The mass conservation and the Euler–Poisson equations read [1] [2]

Balescu R., 1975, Equilibrium and nonequilibrium statistical mechanics. John Wiley and Sons. Binney J., Tremaine S., 1994, Galactic dynamics. Princeton University Press.

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1 ∂(r2 ρu) ∂ρ + 2 = 0, ⇔ ∂t ∂r r ∂u ∂u 1 ∂p ∂φ +u =− − , ∂t ∂r ρ ∂r ∂r

∂ρ ∂ρ 1 ∂(r2 u) +u +ρ 2 =0 ∂t ∂r r r ∂r ∂φ 1 ρr2 dr. = 4π G 2 ∂r r 0

343

(7.1.3) (7.1.4)

However, the acceleration g(r, t) ≡ ∂φ/∂r in (7.1.4) is not an external time-dependent force. It results from the gravitational interaction between the massive microscopic particles constituting the fluid, essentially nuclei and/or ions (the gravitational contribution of electrons is negligible since their mass is too small). Equations (7.1.3) and (7.1.4) form a closed system for ρ and u if the pressure can be expressed in terms of the density, p(ρ), for example through a polytropic relation p = Kρ γ

(7.1.5)

with K and γ constant. Such a relation is valid for isentropic processes, s ≡ S/N = cst., of a polytropic Boltzmann gas (constant specific heat, γ = cp /cv ) (see (13.2.8)), and for isentropic processes of a degenerate electron gas, γ = 5/3; see (13.2.31). Moreover, the entropy of a degenerate electron gas is negligible for temperatures below the Fermi temperature, T TF ; see (13.2.35). According to (13.2.45)–(13.2.47), this is also the case at very high density for a degenerate gas of ultrarelativistic electrons but with a different value of the polytropic index, γ = 4/3; see (13.2.45)–(13.2.47).

7.1.2 Energy Equation Consider first the elementary problem of a point particle (centre of mass r) in an unsteady force field f(r, t) = −∇(r, t) (force per unit mass). According to Newton’s equation, d2 r/dt2 = −∇(r, t), the energy is not constant, d(u2 /2 + )/dt = ∂/∂t,

(7.1.6)

where u ≡ dr/dt is the particle velocity and u2 ≡ u.u is the kinetic energy. Equation (7.1.6) is simply obtained after scalar multiplication by u. When the radiative and neutrino losses are negligible, the star is an isolated system. The total energy is then a conserved scalar and the energy equation takes the local form in (15.1.21)–(15.1.22) when the potential energy due to gravitation, is included in the total energy per unit mass etot . Spherical Stars of Polytropic Gas In spherical geometry, as already known by Newton, the gravitational energy per unit mass at distance r from the centre is expressed in term of the total mass mr (r, t) inside the sphere of radius r in (7.1.2), ϕ(r, t) = −G

mr (r, t) , r

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(7.1.7)

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where ϕ is the time-dependent gravitational energy. By convention gravitational energy is zero at infinity so that it is negative at finite distance. According to the continuity equation (7.1.3) and (7.1.7), r2 ∂ρ/∂t = −∂(r2 ρu)/∂r, ∂ϕ/∂t = 4πρur, ∂ϕ/∂r = Gmr (r, t)/r2 − 4πρr, Dϕ ∂ϕ ∂ϕ ∂φ ≡ +u =u , Dt ∂t ∂r ∂r

where

∂φ Gmr (r, t) = ∂r r2

(7.1.8)

and (7.1.2) has been used. Multiplying the Euler–Poisson equation in (7.1.4) by u then yields the balance of mechanical energy in the form   ∂p 1 ∂(r2 up) 1 ∂(r2 u) D u2 + ϕ = −u =− 2 +p 2 , (7.1.9) ρ Dt 2 ∂r ∂r ∂r r r where, according to (7.1.3), (1/r2 )∂(r2 u)/∂r = −(1/ρ)Dρ/Dt, the last term represents work done by pressure, −(p/ρ)(Dρ/Dt) = ρp(Dv/Dt), where v ≡ 1/ρ. For the polytropic law (7.1.5), (p/ρ)(Dρ/Dt) = K(γ − 1)−1 ρ(Dρ γ −1 /Dt), Equation (7.1.9) can then be written in the following conservative form   K 1 ∂(r2 up) D u2 γ −1 + ρ , (7.1.10) +ϕ =− 2 ρ Dt 2 γ −1 ∂r r where the right-hand side is the divergence of the flux pu. According to the left-hand side of (7.1.10), the conserved energy is the sum of a mechanical energy per unit of mass, u2 /2 + ϕ, ϕ < 0, similar to that of a massive point particle in an external field, plus a term Kρ γ −1 /(γ − 1) coming from the work done by the pressure. More General Case In a way similar to (15.1.23), the Euler–Poisson equation in (7.1.4) yields ρ

D(u2 /2) = −∇.(up) + p∇.u − ρu.∇φ. Dt

(7.1.11)

Using Equation (7.1.8), Dϕ/Dt = u.∇φ, Equation (7.1.11) then takes the form ρ

D(u2 /2 + ϕ) = −∇.(up) + p∇.u, Dt

(7.1.12)

where the expression (7.1.7) for ϕis valid in spherical geometry. In nonspherical geometry the total potential energy, W = 12 ρd3 r, is the trace of a potential energy tensor.[1] The total energy per unit mass etot is defined in a way similar to (15.1.20), etot = eint + u2 /2 + ϕ,

(7.1.13)

as the sum of the internal energy eint and the mechanical energy u2 /2 + ϕ (kinetic energy plus potential energy). Equations (15.1.21) are still valid for the conservation of the total

[1]

Binney J., Tremaine S., 1994, Galactic dynamics. Princeton University Press.

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345

energy etot , and, according to (7.1.12)–(7.1.13), the long-range force of gravitation does not appear explicitly in the local balance of the internal energy,   Detot ∂(ρetot ) = − ∇.Jetot , ρ = −∇. Jetot − ρetot u , (7.1.14) ∂t Dt     D(1/ρ) Deint p +p = − ∇.J, J ≡ Jetot − ρu etot + , (7.1.15) ρ Dt Dt ρ where the equation for conservation of mass (15.1.3)–(15.1.5) has been used. Equation (15.1.28), derived for gaseous combustion, is a particular case of (7.1.15) in which the flux J corresponds to the sum of the diffusive fluxes (heat conduction and molecular diffusion). In stars J includes also the flux of energy lost by radiation and/or through neutrinos escaping from the star. The local thermodynamic equilibrium is due to the short-range forces (collisions) between the microscopic particles, including the Coulomb interaction between charged particles, which is screened by the Debye length.[2] When the potential energy due to the short-range interactions is neglected, the internal energy  takes an additive form eint = i Yi ei , where the mass fractions Yi (of species i) satisfy an equation of the same type as (15.1.12). Photons require a particular treatment yielding Planck’s distribution and black-body radiation at equilibrium. In the expressions for eint only the particles participating in the local thermodynamic equilibrium should be retained. If the matter is transparent to a species, as for example neutrinos that are generated by neutronisation (electron capture), a loss of energy is associated with the escaping flux. The mass of stars is essentially constituted by nuclei and ions, the mass of electrons being negligible. The nuclei and ions, denoted by the subscript nuc, usually form a Boltzmann gas and their internal energy per particle depends only on the temperature, enuc (T). The internal energy of a star is essentially due to electrons that form a degenerate gas, so that their energy depends also on their density ni , ei (T, ni ); see (13.2.29). Moreover, if T is well below the Fermi temperature, as is usually the case, the temperature dependence of ei is negligible, ei (ni ); see (13.2.35) and (13.2.47). Therefore the equation for the temperature is not as simple as the thermal equation (15.1.31) for ordinary gaseous reactive mixtures. A simplified version is presented in Section 7.3.3. Rankine–Hugoniot Conditions The Rankine–Hugoniot conditions across a shock wave (reactive or not), viewed as a hydrodynamical discontinuity, are obtained from the conservative form of the total   energy, namely the first equation in (7.1.14) where, according to (7.1.15), Jetot = J+ρu etot + ρp . Neglecting the flux J, which is of a purely diffusive nature in ordinary combustion, and using the conservation of mass yields the same Rankine–Hugoniot conditions as in (15.1.51) for the jump condition of energy, [etot + p/ρ]+ − = 0. However, this jump relation deserves further comment. As shown by dimensional analysis, diffusive fluxes are negligible on both sides of a shock in ordinary conditions; see the discussion below [2]

Landau L., Lifchitz E., 1982, Statistical physics. Part I. Oxford: Pergamon Press, 3rd ed.

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Explosion of Massive Stars

(15.1.51). The situation in stars is less clear: neutrinos carry a tremendous amount of energy and have a mean free path that varies drastically with density and temperature, so that the flux J in the compressed side could be different from its value on the upstream side. It is also often mentioned in the specialised literature that the shock is strongly weakened by the dissociation of heavy nuclei (mainly Fe) into nucleons (protons and neutrons) consuming about 8.8 MeV per nucleon.[1] The endothermic effect is included in the difference of etot on both sides of the shock front, and the Hugoniot jump relation describes the shock weakening. However, this assumes that the characteristic time of dissociation is shorter than the transit time across the shock. If this is not true, the endothermic dissociation reaction takes place in the compressed medium downstream from the shock. Thermodynamics of the Nuclear Matter in Stars The system of equations, continuity, momentum, species and energy is closed by the equation of state p(ρ, T, ..Yi ..) and the expressions for eint (ρ, T, ..Yi ..), the reaction rate and the flux J. As usual, the validity of the thermodynamic relations is limited by the approximation of local equilibrium (outside the inner structure of the shock waves). The macroscopic equations are thus limited to phenomena evolving on a characteristic time scale longer than that for local relaxation to equilibrium. The difficulties in star theory, especially during the gravitational collapse of the iron core of massive star, concern the rate of reactions (nuclear photodisintegration, electron capture, neutrino-electron scattering, etc.), the transport properties, controlling the fluxes of energy and species (especially the neutrinos, whose mean free path varies from 20 km for ρ = 1017 kg/m3 to 2 × 109 km for ρ = 109 kg/m3 ) and also the equation of state of nuclear matter at high density and temperature.[1,2] The derivation of these laws requires complicated analyses in the statistical physics of equilibrium and nonequilibrium nuclear matter in extreme conditions. In the absence of experimental data, these laws are not fully known. Unfortunately, the explosion of stars is sensitive to these details. A promising approach to improve physical insight into the explosion of stars is to use simplified models of the nuclear physics and to perform parametric studies of the sensitivity to each parameter in the models. 7.2 Stellar Equilibrium In this section we consider stellar structure in the framework of simplified models. 7.2.1 Preliminary Considerations on Stability As already mentioned, a star is a sphere in hydrostatic and thermal equilibrium: the gravitational force is balanced by the pressure, and the radiated heat losses are compensated by [1] [2]

Bethe H., 1990, Rev. Mod. Phys., 62(4), 801–866. Kippenhahn R., Weigert A., 1994, Stellar structure and evolution. Springer-Verlag, 3rd ed.

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347

the release of energy in nuclear reactions. These two processes involve very different time scales. Time Scales and Orders of Magnitude According to (7.1.4), in the absence of a pressure term, a fluid particle in the external layers of the star would acquire an acceleration of order gr ≈ GMs /R2s , where Ms and Rs are the mass and the radius of the star, respectively. Taking Rs as the characteristic length scale, the characteristic hydrodynamic time scale, tff ≡ (Rs /gr )1/2 , and the characteristic flow velocity, uff ≡ Rs /tff (free fall velocity), take the form 1/2 , tff = R3/2 s / (Ms G)

uff = (Ms G/Rs )1/2 .

(7.2.1)

In a stable configuration, these quantities characterise the response of the system to restore hydrostatic equilibrium. They also characterise the gravitational collapse when the star becomes unstable. For a stable star, such as the sun, Ms = M ≈ 2 × 1030 kg, Rs = R ≈ 7 × 108 m, the hydrodynamical time is tff ≈ 1.6 × 103 s. For the iron core of a massive star just before the gravitational collapse, Ms ≈ 1.5 M , Rs ≈ 3 × 106 m, Equation (7.2.1) yields tff ≈ 0.36 s and uff ≈ 8×106 m/s, in rough agreement with the numerical simulations (typically few 10−1 s and uff ≈ 80 × 106 m/s). The mass density throughout the star in steady state increases from zero at the surface, r = Rs : ρ = 0, to a high value at the centre, r = 0: ρ = ρc . The latter is typically two orders of magnitude larger than the mean density ρ ≡ 3Ms /(4π R3s ), ρc ≈ 102 ρ. In quasi-steady equilibrium the density at the centre of the iron core mentioned above is typically ρc ≈ 1013 kg/m3 . For comparison the density at the centre of the sun is ρc ≈ 1.62 × 105 kg/m3 . At the end of core collapse the density at the centre reaches the nuclear density ρc ≈ 1017 kg/m3 . The characteristic time tT for the evolution to thermal equilibrium is evaluated as the ratio of the thermal energy of the star to the rate of radiative losses (luminosity). For the sun tT ≈ 3 × 107 years. The lifetime of the star is defined as the ratio of the stock of available nuclear energy to the luminosity. This gives ≈ 1010 years for the sun. To conclude, the relation tff tT indicates that understanding the hydrostatic equilibrium is the primary task in the theory of stellar structure and its stability is of great importance. In a stable situation, the modifications of the stellar matter through nuclear reactions produces a very slow quasi-steady evolution. A catastrophic event occurs when an unstable hydrostatic equilibrium (or a turning point in the phase space of the equilibrium states) is reached. Variational Principle To begin it is worth showing that, for isentropic variations, the equation for hydrostatic equilibrium (u=0) in (7.1.4), mr (r) 1 dp = −G 2 , ρ dr r

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(7.2.2)

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corresponds to an extremum of the energy, E ≡ Eint + Eg , the sum of the internal energy, R R Eint ≡ 4π 0 s ρeint r2 dr, and the gravitational energy, Eg ≡ 4π 0 s ρϕr2 dr < 0,  Ms  Ms mr Eint = eint (s, ρ)dmr , Eg = −G (7.2.3) E = Eint + Eg , dmr , r 0 0 where (7.1.2) and (7.1.7) have been used. The equilibrium stellar structure is characterised by a distribution of density ρ(r) that corresponds to the mass distribution mr (r). The expressions in (7.2.3) suggest that it is more convenient to perform the variation calculus by using mr as the integration variable and the function r(mr ), the inverse of mr (r). The variation of the thermal energy for an isentropic transformation is given by the first law, δeint = −pδ(1/ρ). The variation of E then takes the form  Ms    Ms mr 1 dmr + G δE = − pδ δr dmr . (7.2.4) ρ r2 0 0 The variation pδ(1/ρ) is computed from (7.1.2), 1/ρ = 4π r2 (dr/dmr ), and using the relation δ(dr/dmr ) = d(δr)/dmr gives   dr d(δr) 1 = 8π rp δr + 4π r2 p (7.2.5) pδ ρ dmr dmr = 4π

d(r2 pδr) dp − 4π r2 δr. dmr dmr

(7.2.6)

Considering the variation in (7.2.4) at Ms and Rs fixed, the first term in (7.2.6) yields zero so that, using the relation dp/dmr = (4πρr2 )−1 dp/dr, Equation (7.2.4) yields  Ms  1 dp mr (r) δE = δr dmr . +G 2 (7.2.7) ρ dr r 0 The hydrostatic equilibrium, the solution to (7.2.2), thus corresponds to an extremum of E for a frozen distribution of specific entropy and distribution of concentration of species. Virial Theorem Multiplying the equation of equilibrium (7.2.2) by 4π r3 ρdr = r dmr , and integrating over the entire volume of the star, the right-hand side yields Eg . An integration by parts of the R  R left-hand side yields 4π r3 p 0 s − 3 0 s 4π pr2 dr. One then obtains  Ms 3 (p/ρ)dmr = −Eg . (7.2.8) 0

For an ideal (Boltzmann or degenerate) gas, the ratio of the two quantities P/ρ and eint is constant. Equation (7.2.8) then shows that the ratio of the thermal energy to the gravitational energy is a constant negative number, −Eint /Eg = cst., and the ratio E/Eg is also a constant. The corresponding sign of this ratio determines the sign of E that is of primary importance for the stability of the hydrostatic equilibrium. For a mono-atomic Boltzmann gas, p/ρ = kB T/m, eint = (3/2)kB T/m, where m is the molecular mass, one has p/ρ = (2/3)eint .

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349

The same relation is also verified for a nonrelativistic degenerate gas with a temperature lower than the Fermi temperature kB T F ; see (13.2.34). Equation (7.2.8) then yields 2Eint = −Eg

⇒

Eint + Eg = −Eint .

(7.2.9)

This is the simplest version of the virial theorem, which states that the average total kinetic energy of interacting point particles is equal to minus one half their total potential energy. For a more general case such as a polytropic (Boltzmann) gas with constant specific heats, cv > 3/2, eint = cv T/m, cp = cv + kB , γ ≡ cp /cv , p/ρ = (γ − 1)eint , Equation (7.2.8) leads to a more general form of the virial theorem, Eint + Eg = −(3γ − 4)Eint .

(7.2.10)

The virial theorem provides a simple physical argument to anticipate the stability of stellar equilibrium. When the total energy of the star Eint + Eg is negative (γ > 4/3), then, according to (7.2.10), the thermal energy Eint increases when the total for energy of the star decreases. Therefore when the nuclear reactions are insufficient to compensate for energy loss, the energy Eint + Eg will decrease, but the thermal energy, Eint , and hence T increase with the result of restoring the equilibrium by increasing the reaction rate. The opposite is true for γ < 3/4 and the star is unstable, as will be proved by a stability analysis in Section 7.3.1. It is worth noticing that, according to (7.2.8), a degenerate gas of ultrarelativistic electrons has p/ρ = eint /3 (see (13.2.44)), and corresponds to the marginal case Eint + Eg = 0.

(7.2.11)

Back to the Variational Problem The rough argument of stability, mentioned just above, can be illustrated by the following phenomenological considerations.[1] Introducing the mass-weighted quantities eint  and ρ ≡ 3Ms /(4π R3s ), the thermal energy and the gravitational energy of the whole star in (7.2.3) can be written Eint = eint  Ms ,

Eg ∝ −GMs 2 /Rs = −GMs 5/3 ρ1/3 ,

(7.2.12)

where the radius of the star, Rs , has been eliminated from the second relation in favour of ρ and Ms . The nondimensional coefficient of proportionality in the expression for Eg is of order unity and depends on the distribution of mass in the star. According to (13.2.7), (13.2.33)–(13.2.35) and (13.2.46)–(13.2.47), the thermal energy of an ideal gas, either a Boltzmann gas or a degenerate gas of electrons (relativist or nonrelativist), takes the form eint (s, ρ) = f (s)ρ γ −1 + a2 ,

[1]

Zeldovich Y., Novikov I., 1971, Stars and relativity. Dover.

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(7.2.13)

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where f (s) is a function of the specific entropy and the constant a2 depends on the composition. Equations (7.2.12)–(7.2.13) suggest phenomenological relations in the form[1] Ms eint  = Ms A1 ργ −1 + Ms A2 , E = A1 (S) Ms ρ

(γ −1)

(7.2.14)

+ A2 (S) Ms − A3 (S) Ms

5/3

ρ

1/3

,

(7.2.15)

where A1 , A2 and A3 are positive (dimensional) coefficients that depend on the structure and the entropy S of the star. At fixed entropy S and mass Ms , the energy E in (7.2.15) is a function of density ρ. A stable equilibrium state should correspond to a minimum. The function E(ρ) has effectively a minimum for γ > 4/3, but this is not the case when γ < 4/3 since the function E(ρ) has a maximum, corresponding to an unstable equilibrium state. The link between the density at equilibrium ρeq and the mass of the star Ms is given by the zero of the derivative of (7.2.15) with respect to ρ, ργeq−4/3 =

A3 /3 Ms 2/3 . (γ − 1)A1

(7.2.16)

In stable (unstable) equilibrium, γ > 4/3, the density increases (decreases) and thus the volume decreases (increases) when the mass of the star is increased, in agreement with the previous rough analysis since the pressure and the temperature are expected to increase when the volume decreases. The marginal equilibrium, γ = 4/3, is possible only for a finite mass Ms 2/3 = 3(γ − 1)A1 /A3 . This qualitative result is confirmed below. 7.2.2 Lane–Emden Equilibrium (Polytropic Gas) In this paragraph we consider the hydrostatic equilibrium of stars for the polytropic law in (7.1.5). This approximation is not valid throughout the entire structure of a massive star. However, the validity is not too bad for the iron core. At the onset of collapse, the conditions in the core, ρ ≈ 1011 –1013 kg/m3 , T ≈ 4 × 109 −1010 K, are such that matter is fully ionised and the electrons form a relativistic degenerate gas, while the nuclei form a Boltzmann gas whose contribution to the pressure nFe kB T is negligible compared with the pressure of the electron gas. According to (13.2.46)–(13.2.47), to a good approximation the matter then follows a polytropic law (7.1.5) with K = cst. and γ = 4/3. Lane–Emden Solution For a polytropic material, γ = cst., the hydrostatic equation in (7.2.2) can be put in the form of a second-order nonlinear ordinary differential equation for the density by multiplying (7.2.2) by r2 , taking the derivative with respect to r and using (7.1.5):   γ γ −1  1 d 4π G 1 d (γ − 1) 4π G 2 1 dρ 2 dρ r =− ρ ⇒ r =− ρ. (7.2.17) 2 2 ρ dr K dr γ K r dr r dr Assuming that the density at the surface of the star is negligible, the relation between the mass Ms and the radius Rs of the star is obtained by solving (7.2.17) with the boundary [1]

Zeldovich Y., Novikov I., 1971, Stars and relativity. Dover.

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351

 Rs

conditions r = 0: dρ/dr = 0; r = Rs : ρ = 0 and 4π 0 ρr2 dr = Ms . The density ρc at the centre (r = 0) is an outcome of the solution. Equation (7.2.17) can be written in a nondimensional form by introducing the length scale L, built with the dimensional quantities G, K and ρc ,   1 d 2 d γ −1 rˆ ρˆ = −ρˆ (7.2.18) dˆr rˆ 2 dˆr ρ 1 r (γ − 1) 4π G 1 ρˆ ≡ , with rˆ ≡ ≡ . (7.2.19) ρc L γ K ρcγ −2 L2 In the original formulation of the Lane–Emden equation, ρˆ γ −1 is used as the unknown function. Equation (7.2.18) is integrated numerically using the boundary conditions rˆ = 0:

ρˆ = 1,

dρ/dˆ ˆ r = 0.

(7.2.20)

The solution ρ(ˆ ˆ r) is a decreasing function of rˆ and the numerical value rˆs (γ ) for which the solution crosses zero, rˆ = rˆs : ρˆ = 0, defines the radius of the star, Rs = rˆs (γ )L; see Fig. 7.1. The mass of the star, Ms , is expressed in terms of the solution by computing the rˆ ˆ r) and ˆ r) ≡ 4π 0 ρ(ˆ ˆ r )ˆr2 dˆr , so that, according to (7.1.2), mr (r) = ρc L3 m(ˆ integral m(ˆ 3 ˆ s (γ ), where m ˆ s (γ ) ≡ m(ˆ ˆ rs ). When rˆs (γ ) and m ˆ s (γ ) are obtained numerically Ms = ρc L m for a given γ (see Table 7.1), the radius Rs and the mass Ms of the star can be expressed in terms of the density at the centre ρc , 3/2 3  4    ) K 1/2 − (2−γ K γ γ 2 γ−3 ˆ s (γ ) Rs = rˆs (γ ) ρc 2 , Ms = m ρc . γ − 1 4π G γ − 1 4π G (7.2.21) The radius of the star as a function of the mass, Rs (Ms ), is obtained by eliminating ρc . For 4/3 < γ < 2 the radius Rs decreases when the mass Ms increases. For γ < 4/3,

(a)

(b)

Figure 7.1 Numerical solutions to (7.2.18) for the reduced density with γ = 3/2, 1.4, 4/3, 5/4 and ˆ r) as a function of rˆ . Note that the ordering of the 6/5. Panels a and b show, respectively, ρˆ γ −1 and ρ(ˆ curves is inverted. Although it is not evident from (b), the radius Rs of the star is infinite for γ = 6/5; see Table 7.1.

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Table 7.1 Lane–Emden solutions for the reduced radius, mass and density of a stars as a function of the adiabatic index. γ γ γ γ γ

= 3/2: = 1.4: = 4/3: = 5/4: = 6/5:

rˆs rˆs rˆs rˆs rˆs

= 4.35, = 5.36, = 6.90, = 14.9, = ∞,

ˆ s /4π m ˆ s /4π m ˆ s /4π m ˆ s /4π m ˆ s /4π m

= 2.41, = 2.19, = 2.02, = 1.80, = 1.73,

ρ˜c /4π ρ˜c /4π ρ˜c /4π ρ˜c /4π ρ˜c /4π

= 11.38 = 23.44 = 54.21 = 622.50 =∞

Rs increases when Ms increases. The density at the centre is given by ρ˜c ≡ ρc / ρ = ˆ s ). (4π/3)(ˆrs3 /m Chandrasekhar Mass and White Dwarfs For the marginal value γ = 4/3, the second equation in (7.2.21) shows that there is only a single value, called the Chandrasekhar mass, MCh , of the mass Ms for which equilibrium is possible, but neither the central density nor the radius is determined, γ = 4/3:

MCh ≈ 2.28 (K/G)3/2 .

(7.2.22)

According to the virial theorem in (7.2.10)–(7.2.11), the energy of the star E is negative (positive) for γ > 4/3 (γ < 4/3) so that the equilibrium is stable (unstable). For the marginal case, γ = 4/3, E = 0, it is conjectured[1] that the star will contract indefinitely if Ms > MCh leading to gravitational collapse, while it will expand indefinitely for Ms < MCh . An exact analytical solution of these phenomena is given in Section 7.3.2 for particular initial conditions. According to (7.2.21) for γ > 4/3, the relation between the mass and the radius of a star in equilibrium, Ms (Rs ), is such that for large radii the density is relatively small and the degenerate electron gas is nonrelativistic, γ = 5/3. Therefore, according to (7.2.21), the function Ms (Rs ) decreases as 1/R3s . When the mass Ms increases the radius Rs can become sufficiently small and the density sufficiently large that the electrons become ultrarelativistic, γ = 4/3. Then Ms cannot take values above MCh . Real stars are more complicated; the density is not uniform and the electron gas becomes ultrarelativistic near the centre but not in the rest of the star. Roughly speaking the effective polytropic index decreases continuously when the mass of the central core filled with ultrarelativistic electrons increases. However this central mass cannot increase above MCh . This explains why cold white dwarfs cannot have a mass larger than MCh . 7.3 Instability and Gravitational Collapse In a massive star the size of the high density inner core where γ = 4/3 increases during the last step of nuclear combustion. As long as the mass of this core is smaller than MCh , [1]

Landau L., Lifchitz E., 1982, Statistical physics. Part I. Oxford: Pergamon Press, 3rd ed.

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353

the effective polytropic index of the whole star is larger than 4/3 and the star is stable. A catastrophic gravitational collapse of the core occurs as soon as its mass reaches MCh .

7.3.1 Linear Stability of the Lane–Emden Solution Nondimensional Equations  Introducing the reference time tref , 1/tref ≡ 4π ρ c G, and the same reference length L as in (7.2.19) but where ρc is replaced by the unperturbed value of the mass density at the centre of the star in equilibrium, ρ c , the nondimensional time ˆt and velocity uˆ are 6  t L γ 1 u γ −1 ˆt ≡ Kρ c . , ≡ 4π ρ c G and uˆ ≡ , uref ≡ = (7.3.1) tref tref uref tref γ −1 √ The reference velocity uref is proportional to the speed of sound dp/dρ at the centre of √ the star in equilibrium uref = ac / γ − 1. According to (7.2.19) and (7.2.21), the reference length L is proportional to the radius of the unperturbed star at equilibrium Rs and the reference time tref and the reference velocity uref are proportional to the free fall time tff and the free fall velocity uff , defined in (7.2.1) for the equilibrium state, Rs = Rs , . L = Rs /ˆrs ,

tref = tff

3

rˆ s

ˆ s /4π m

.

/1/2 ,

uref = uff

/1/2

rˆ s ˆ s /4π m

ac =√ , γ −1

(7.3.2)

ˆ s are the values rˆs and m ˆ s given in Table 7.1 for γ = 4/3. Multiplying (7.1.3) where rˆ s and m by L/(ρ c uref ) and (7.1.4) by 1/(4π Gρ c L) and using (7.1.5) the nondimensional equations take the form ∂ ρˆ 1 ∂(ˆr2 ρˆ uˆ ) = 0, + 2 ∂ rˆ rˆ ∂ ˆt    rˆ 1 2 ∂ ρˆ γ −1 ∂ uˆ ∂ uˆ 2  = − 2 rˆ + ρˆ ˆ r dˆr , + uˆ ∂ rˆ ∂ rˆ rˆ ∂ ˆt 0 where

rˆ = r/L,

ˆt = t/tref ,

ρˆ = ρ/ρ c ,

(7.3.3) (7.3.4)

uˆ = u/uref .

(7.3.5) If diffusive energy fluxes cannot be neglected during the dynamics, the isentropic approximation is no longer valid and the first term in the right-hand side of (7.3.4) must be replaced by the original pressure term    1 ∂ pˆ 1 rˆ 2  ∂ uˆ ∂ uˆ =− + 2 ρˆ ˆ r dˆr , + uˆ (7.3.6) ∂ rˆ ρˆ ∂ rˆ rˆ 0 ∂ ˆt γ

where the pressure is reduced by ρ c (L/tref )2 = [γ /(γ −1)]Kρ c , where the overbar denotes a steady Lane–Emden solution chosen as a reference. For a polytropic gas the reduced

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pressure is pˆ = [(γ − 1)/γ ]ρˆ γ , and Equation (7.3.4) is recovered. In the general case, Equation (7.3.6) is completed by an equation of state and the energy equation. Linear Stability Analysis The stability analysis is performed here in the isentropic approximation. As for the acoustic modes in spherical geometry, the analysis is performed in a Lagrangian formulation by ˆ r, ˆt) ≡ using (η, τ ) where η is the mass-weighted coordinate, η = m(ˆ  rˆ the  coordinates ˆ ˆt) ˆ r , ˆt)ˆr2 dˆr , and τ is the time, τ = ˆt. Introducing the radius R(η, τ ) = rˆ (m, 4π 0 ρ(ˆ ˆ ˆ the velocity field U(η, τ ) = uˆ (m, ˆ t) and the mass density (η, τ ) = enclosing the mass m, ˆ ˆt), expressed in terms of (η, τ ), we have for any field Y(η, τ ) = yˆ (ˆr, τˆ ) ρ( ˆ m,      ∂ yˆ  1 ∂Y  ∂ yˆ  ∂ yˆ  ∂Y  = , = + uˆ  , (7.3.7) ∂η τ ∂τ η ∂ rˆ ˆt 4π R2  ∂ rˆ ˆt ∂ ˆt rˆ   ∂R  ∂R  1 , = = U(η, τ ), (7.3.8) ∂η  ∂τ  4π R2  τ

η

where the continuity equation (7.3.3) has been used in the second equation of (7.3.7). In the coordinates (η, τ ), Equation (7.3.4) takes the form of Newton’s equation,  γ η (γ − 1) ∂ 2 R  2 ∂ 4π R − = − , (7.3.9)  2 γ ∂η ∂τ η 4π R2 where the first term in the right-hand side is the force due to the pressure and the last term is the gravitation force. The first equation in (7.3.8) and (7.3.9) form a closed system for R(η, τ ) and (η, τ ). For radial disturbances the linear solution is sought in the form     (7.3.10) R(η, τ ) = R(η) 1 + χ (η)eσ τ , (η, τ ) = (η) 1 + μ(η)eσ τ , where R(η) and (η) are the unperturbed solutions of the Lane–Emden equation. Eliminating μ(η) from the linearised equation (7.3.9) by using the first equation in (7.3.8), a second-order differential equation with variable coefficients that involves σ 2 is obtained for the disturbance of the radius χ (η). In the absence of the last term in (7.3.9), which describes the force of gravitation, this equation reduces to that for the acoustic modes of an inhomogeneous sphere. The presence of gravity does not change the nature of the problem. Using the boundary conditions at the centre and at the surface, the calculation of the eigenmodes is straightforward. The details are not reproduced here and can be found elsewhere.[1] There is an infinite set of acoustic modes corresponding to a series of eigenvalues, σn , n = 1, 2, ..., where σn2 are real and σn2 < σ12 ∀n > 1. The fundamental mode takes the form σ12 = − (γ − 4/3) H,

H > 0,

(7.3.11)

where H > 0 is the nondimensional eigenvalue and H = 1 when the variation of the density across the unperturbed star is neglected. This clearly shows instability for γ < 4/3 [1]

Kippenhahn R., Weigert A., 1994, Stellar structure and evolution. Springer-Verlag, 3rd ed.

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and oscillatory acoustic modes for γ > 4/3. According to (7.3.1) the characteristic  time scale for both the linear growth rate of the instability or the acoustic period is 1/ 4π ρ c G, which is of the order of the free fall time in (7.2.1), leading to the famous frequency–density relation ω2 /(G ρ) = cst. Thermo-acoustic Instability and Stellar Pulsation The case of real stars is more complex because the dynamics are not isentropic due to the energy liberated by nuclear reactions, to energy transfer by heat conduction and to heat losses at the surface. The linear analysis is more easily extended to take account of these effects if the entropy changes are supposed to be small. Dissipative effects, such as heat conduction, are expected to introduce damping, Re(σ )< 0. However, modulation of the heat release and of heat losses can introduce a positive feedback and thus a positive linear growth rate, Re(σ ) > 0, as in ordinary combustion; see Sections 2.5 and 15.2.4. For sufficiently small perturbations to acoustics, the modifications of the eigenvalues are small and a weakly nonlinear analysis can be performed in unstable situations, showing the existence of a limit cycle that corresponds to stellar pulsation. The pulsation frequency is approximately the acoustic frequency, ω2 /(G ρ) = cst. In real stars the polytropic index is not constant, nevertheless the frequency–density relation still holds approximately. Computing the constant for a known pulsating star[1] (M = 7 M , R = 80 R , period 11 days), this gives 220 days for the acoustic period of a supergiant with ρ ≈ 5 × 1011 kg/m3 and 4 s for a white dwarf, ρ ≈ 109 kg/m3 . However, attention should be paid to possible strongly nonlinear phenomena leading either to shock formation (see Section 15.3) or to subcritical instabilities. Such nonlinear mechanisms are at work during the gravitational collapse of the iron core of a massive star at the end of its life; see Section 7.3.3.

7.3.2 Nonlinear Dynamics Near to the Chandrasekhar Mass The objective of this section is to derive an exact solution for the dynamics in the marginal case γ = 4/3 when the initial mass Mr is slightly different from MCh . In the following we go back to the original reduced variables (ˆr, ˆt). Invariance Property of the Marginal Solution (γ = 4/3) For an equilibrium solution, uˆ = 0, the term in square brackets  rˆ 2 in (7.3.4) yields the ˆ r dˆr has the invariance Lane–Emden equation (7.2.18). The integral term Iˆ(ρ, ˆ rˆ ) ≡ 0 ρˆ −3 ˆ ˆ ˆ For γ = 4/3, the pressure term property I (ρ, ˆ rˆ ) = I (ρˆα , rˆ /α), where ρˆα (ˆr) ≡ α ρ(r/α). rˆ 2 ∂ ρˆ 1/3 /∂ rˆ has the same property. Let ρˆ1 (ˆr) denote the solution to the Lane–Emden equation (7.2.18) that satisfies the boundary condition (7.2.20), and consider the equilibrium solution ρ1 (r) ≡ ρc1 ρˆ1 (ˆr) that corresponds to the central density ρc1 , r = 0: ρ1 = ρc1 . Thanks to the invariance property, the equilibrium solution ρα (r) corresponding to a

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different central density, r = 0: ρα = ρc1 /α 3 (α is a positive number), takes the form ρα (r) = ρc1 ρˆα (ˆr), where ρˆα (ˆr) = α −3 ρˆ1 (ˆr/α) is the solution to (7.2.18) that satisfies the boundary conditions  rˆs /α ˆ s , (7.3.12) ρˆα (ˆr )ˆr2 dˆr = m rˆ = 0: ρˆα = 1/α 3 , dρˆα /dˆr = 0, ρˆα (αˆrs ) = 0 ⇒ 0

ˆ s are fixed numbers given in Table 7.1 for γ = 4/3. In agreement with where rˆs and m (7.2.21), this corresponds to the fact that there is only one mass, MCh given in (7.2.22), for a star in equilibrium with γ = 4/3, but there exist many solutions with different radii Rs and densities at the centre ρc such that the product ρc R3s is constant, MCh /(ρc R3s ) ˆ s /ˆrs3 . =m It is therefore tempting to look for unsteady solutions in the form    rˆs rˆ 1 ˆ s, ρˆ1 (ˆr, )ˆr2 dˆr = m (7.3.13) with ρ(ˆ ˆ r, ˆt) = 3 ρˆ1 α (ˆt) α(ˆt) 0

where ρˆ1 rˆ is the reduced mass distribution of a star of mass MCh in equilibrium ˆ s /ˆrs3 . The distribution with a radius Rs1 and a central density ρc1 , MCh /(ρc1 R3s1 ) = m in (7.3.13) describes a star of mass MCh in quasi-steady evolution (gravitation and pressure are balanced ∀ ˆt ) with a radius and a central density evolving as Rs (t) = α(t)Rs1 and ρc (t) = ρc1 /α(t)3 . An equation for α(t) is obtained when (7.3.13) is introduced into (7.3.3)–(7.3.4). Continuity in (7.3.3) yields  rˆ  rˆ/α(ˆt) ∂ 1 dα 1 ∂ α 3 (ˆt) 2 2 ˆ

ρ(η, ˆ t)η dη = − ρˆ1 (x)x2 dx = rˆ 3 , (7.3.14) rˆ uˆ = − ˆ ˆ ρˆ ∂ ˆt 0 α dˆt ρˆ1 rˆ /α(t) ∂ t 0 which is a homologous flow,[1] uˆ ∝ r, with a coefficient that is a function of time,  2    1d α ∂ uˆ ∂ uˆ 1 dα rˆ . (7.3.15) = rˆ ⇒ + uˆ uˆ (ˆr, ˆt) = α dˆt ∂ rˆ α dˆt2 ∂ ˆt Because of the invariance property, the term in the bracket in (7.3.4) is zero since ρ(ˆ ˆ r, ˆt) is the solution of the steady-state equation (7.2.18). Equation (7.3.15) then shows that d2 α/dˆt2 = 0 and α(ˆt) is a linear function of time, α(ˆt) = λˆt + 1, where the origin of time is chosen such that for ˆt = 0: Rs = Rs1 and α = 1. According to the first equation in (7.3.15) the flow field corresponds to a gravitational collapse for λ < 0 with a singularity at ˆt = −1/λ, and to an expansion for λ > 0, limˆt→∞ uˆ = 0 ∀ rˆ , uˆ (ˆr, ˆt) =

[1]

λ rˆ . λt + 1

Goldreich P., Weber S., 1980, Astrophys. J., 238, 991–997.

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(7.3.16)

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357

Mass Different from the Chandrasekhar Mass An exact solution for the evolution of a star having γ = 4/3, but with a mass different from the Chandrasekhar mass, Ms = MCh , can be obtained for particular distribution of the mass at the initial condition t = 0. Solutions are sought in the same form as (7.3.13)    rˆf rˆ 1 ˆ s, ρˆi (ˆr)ˆr2 dˆr = m (7.3.17) but with ρ(ˆ ˆ r, ˆt) = 3 ρˆi α (ˆt) α(ˆt) 0 so that ρˆi (ˆr) is not a equilibrium solution. Following the same calculations as in (7.3.14), continuity in (7.3.3) yields a homologous solution of the same form as in (7.3.15),   2    ∂ uˆ 1d α 3 ∂ uˆ 1 dα rˆ . = (7.3.18) rˆ ⇒ rˆ 2 + uˆ uˆ (ˆr, ˆt) = α dˆt ∂ rˆ α dˆt2 ∂ ˆt But now the bracket in the right-hand side of (7.3.4) is not zero since ρˆi (ˆr) is not a equilibrium solution. Consider a particular initial condition for which this bracket varies with the radius as rˆ 3 ,    rˆ 1/3 ∂ ρˆi 2 2  ˆt = 0: α = 1, ρ(ˆ ˆ r) = ρˆi (ˆr), + rˆ ρˆi rˆ dˆr = bˆr3 , (7.3.19) ∂ rˆ 0 where b is a given constant. Because of the invariance mentioned above for each term in ˆ r) = α −3 ρˆi (r/α), the bracket in the right-hand side of (7.3.4), Iˆ(ρ, ˆ rˆ ) = Iˆ(ρˆi , rˆ /α) where ρ(ˆ the value of the term in the bracket at any time is    1/3  rˆ ˆ r, ˆt) rˆ 3 2 ∂ ρ(ˆ 2  + ρ(ˆ ˆ r, ˆt)ˆr dˆr = b 3 . (7.3.20) ∀t: rˆ ∂ rˆ α 0 Introducing (7.3.18) and (7.3.20) into the Euler–Poisson equation (7.3.4) yields   2  d2 α dα 1 −1 , α 2 (ˆt ) 2 = −b ⇒ = 2b α dˆt dˆt

(7.3.21)

with the last equation corresponding to the initial condition ˆt = 0: α = 1, dα/dˆt = 0, uˆ = 0. The second equation in (7.3.21), when solved at short time (|α − 1| 1), ˆt 1: α ≈ −bˆt2 /2, shows that α(ˆt ) increases initially with time when b < 0 and decreases when b > 0. Moreover, according to the first equation in (7.3.21), the second derivative d2 α/dˆt2 cannot change sign. For b < 0 (α > 1), the star expands indefinitely, limˆt→∞ α = ∞, while a gravitational collapse develops for b > 0 (α < 1), limˆt→∞ α = 0. In the latter case a singularity develops at a finite time ˆt = ˆt0 . Near to the centre, as the singularity is approached (α 1), according to (7.3.21), the asymptotic behaviour,   2/3 rˆ 2 rˆ 1 2 3 (7.3.22) , ρˆ ≈ 3 1 − 2 , b1/3 (ˆt0 − ˆt)2/3 , uˆ ≈ − α≈ 4 3 (ˆt0 − ˆt) α 2α presents the same type of singularity for the flow velocity as in the previous solution (7.3.16) for a star with the Chandrasekhar mass. In contrast to the density, the asymptotic

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behaviour of the homologous flow (7.3.18) is universal, meaning that it does not depend on the initial condition through b. The link between the coefficient b and the mass of the star Ms is obtained from the initial condition (7.3.19) where the excess or deficit of mass relative to MCh is measured by ˆ s , then the integral term, taken at the external radius rˆ = rˆs . If it is larger (smaller) than m Ms > MCh (Ms < MCh ). Considering an initial modification of the density distribution (relative to the equilibrium state) that does not change the value of first term at rˆ = rˆs , the sign of b is the same as the difference Ms − MCh . Thus, the fact that a star with Ms > MCh (Ms < MCh ) contracts (expands) is proved for such initial conditions. Self-Similar Solutions The asymptotic behaviour in (7.3.22) corresponds to a a self-similar solution which can be written in the general form, ρˆ =

1 R (X) , (ˆt0 − ˆt)β

uˆ =

1 U (X) , (ˆt0 − ˆt)δ

X≡

rˆ , (ˆt0 − ˆt)η

(7.3.23)

with, in the particular case (7.3.22), γ = 4/3, δ = 1/3,

β = 2,

η = 2/3

and

U = −2X/3,

lim R = (4/3)2 /b,

X→0

(7.3.24)

where this asymptotic behaviour still depends on the initial condition through b. The homologous collapsing core, obtained as a self-similar solution for γ = 4/3, has been proved to be stable,[1] The self-similar analysis has been extended to γ < 4/3.[2] The main difference with the result for γ = 4/3 is that the homologous solution holds only in a central part, sufficiently close to the origin, while the far field is in steady state with a flow velocity decreasing toward the free fall velocity. The analysis is summarised as follows. Looking for a solution to (7.3.3)–(7.3.4), in the form (7.3.23), the exponents are determined by requiring that all the terms in these equations vary in the same way in the limit ˆt → ˆt0 , δ = (γ − 1),

β = 2,

η = (2 − γ ),

(7.3.25)

reducing to (7.3.24) for γ = 4/3. Introducing the corresponding expressions for ρˆ and uˆ into these equations, they are transformed into a set of two ordinary differential equations for R(X) and U(X),  U R + U  = −2 1 + , (7.3.26) [U + (2 − γ )X] R X  X R 1   (γ − 1)U + (2 − γ )XU + UU = −(γ − 1) (2−γ ) − 2 X 2 R dX, (7.3.27) X 0 R where R  ≡ dR/dX, U  ≡ dU/dX and X ≡ rˆ /(ˆt0 − ˆt)(2−γ ) . After multiplication by X 2 R (7.3.26) can be integrated once to give [1] [2]

Goldreich P., Weber S., 1980, Astrophys. J., 238, 991–997. Yahil A., 1983, Astrophys. J., 265, 1047–1055.

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X

[U + (2 − γ )X] RX 2 = (4 − 3γ )

359

X 2 R dX.

(7.3.28)

0

The solution near the centre is determined in the limit X → 0 where lim X→0 U = 0, and lim X→0 R = R c = 0, R c < ∞. Equation (7.3.28) shows that the flow becomes homologous, that is, the velocity varies linearly with the radius in the same way as in (7.3.22): X 1:

[U + (2 − γ )X] ≈ (4/3 − γ ) X,

U ≈ −2X/3,

(7.3.29)

and Equation (7.3.27) gives the relation between R c and R c ≡ d2 R/dX 2 | X=0 ,   1 2 X 1: R − R c ≈ −R c X 2 /2, Rc − R c(2−γ ) , R c = 3(γ − 1) 3

(7.3.30)

in full agreement with (7.3.22) for γ = 4/3. Since the density is expected to decrease when the radius increases, one must have R c  2/3. For R c = 2/3 the pressure would be negligible near the centre. Anticipating that the divergence of the collapse is limited to a central region, X = O(1), the density and the velocity should not diverge in the limit ˆt → 0 for X  1. Since the divergence comes from the unbounded factors, (ˆt0 − ˆt)−β and (ˆt0 − ˆt)−α in (7.3.23), they must be balanced by the implicit time dependence of R and U. This yields a far field in steady state X  1:

R∝X

2 − (2−γ )

,

ρˆ ∝ rˆ

2 − (2−γ )

,

U ∝X

−1) − (γ (2−γ )

,

uˆ ∝ rˆ

−1) − (γ (2−γ )

,

(7.3.31)

where the coefficients of proportionality of ρˆ and uˆ are time independent. Compared with the exact solution of (7.3.3)–(7.3.4) for an initial condition corresponding to a massive star with γ  4/3, the time-independent solution (7.3.31) could correspond to a characteristic time scale for the evolution of the external layers that is much longer than in the central core, X = O(1). However, the second equation in (7.3.31) corresponds to a total mass of the star (radius defined to be where ρ = 0) which is infinite (4−3γ )  rˆ 2 ρˆ ˆ r dˆr ∝ rˆ (2−γ ) . Therefore the self-similar solution cannot accurately for γ  4/3, describe the evolution of a star with a finite mass in a vacuum. The guess is that the selfsimilar solution works not too far from the centre, X = O(1), when approaching the time of collapse. In such circumstances the initial conditions could be forgotten and the external layers could be in quasi-steady state, decoupled from the core collapse. Nothing guarantees these conjectures since the exact solution (7.3.22) for γ = 4 shows an asymptotic behaviour of the density that still depends on the initial conditions. However, imposing a regularity criterion when crossing the sonic point, a numerical integration of (7.3.26)–(7.3.27) led to a unique solution[2] matching expressions (7.3.30) and (7.3.31); see Fig. 7.2. This self-similar solution presents a homologous inner core and a maximum infall velocity with a sonic point located at a radius rˆsonic smaller than that of the radius of maximum infall velocity, rˆsonic < rˆmax , the flow being subsonic (supersonic) for rˆ < rˆsonic (ˆr > rˆsonic ). The velocity in the far field decreases to zero  as the radius ˆ r)/ˆr at large increases and approaches a fraction (0.66) of the free fall velocity ∝ m(ˆ

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Figure 7.2 Numerical  solution of the self-similar equations (7.3.26)–(7.3.27) for γ = 1.3, X ≡ X/ , V ≡ −U/ ,  ≡ (4π )γ −1 (γ − 1)/γ and A is a dimensionless speed of sound A ≡ a/(uref ), where uref is defined in (7.3.1); the equation V = A denotes a sonic point. The self-similar solution is shown for t ≡ (ˆt − ˆt0 ) < 0 and also after the postcatastrophe collapse for t > 0, obtained by the formal transformation t → −t: V → −V, X → −X. From Yahil A., 1983, Astrophys. J., 265, 1047–1055, reproduced with permission.

 rˆ 2 radius, as shown by (7.3.31), rˆ −1 ρˆ ˆ r dˆr ∝ rˆ −2(γ −1)/(2−γ ) ∝ uˆ 2 . In the same limit, the Mach number approaches 2.94. The self-similar solution is in good qualitative agreement with the numerical solutions of the iron core during the collapse, before the formation of the shock wave that was briefly discussed in Section 1.3.2. The quantitative agreement is also satisfactory in the internal region of the collapsing core.

7.3.3 Shock Outburst At the onset of the gravitational collapse of a star of total mass 11.4 M the typical orders of magnitude at the centre of the iron core (mass ≈ 1.5 M , initial radius R ≈ 3000 km) are ρc ≈ 1013 − 1014 kg/m3 ,

Tc ≈ 1010 K ≈ 1 MeV.

(7.3.32)

The thermodynamic conditions at the edge are those of silicon burning into iron (Si→Fe), ρ ≈ 1011 kg/m3 ,

T ≈ 4 × 109 K.

(7.3.33)

During the collapse, the mass density at the centre increases by a factor of order 103 –104 . The temperature increase is not so large; at the centre it is multiplied typically by a factor of 10, and Tc does not exceed 11 MeV. In such conditions the centre of the star becomes a fluid of nuclear matter, and the equation of state stiffens dramatically.[1,2,3,4,5] The material can no longer be easily compressed and the ions forms a crystal between 1016 –1017 kg/m3 , before reaching the density of the nuclear matter, a few 1017 kg/m3 . Therefore, a sharp rise in the polytropic exponent occurs when the equation of state begins to stiffen. [1] [2] [3] [4] [5]

Bethe H., 1990, Rev. Mod. Phys., 62(4), 801–866. Cooperstein J., Baron E., 1990, In A. Petschek, ed., Supernovae, chap. 9, 213–266, Springer-Verlag. Woosley S., et al., 2002, Rev. Mod. Phys., 74, 1015–1071. Janka H.T., et al., 2007, Phys. Rep., 442, 38–74. Burrows A., 2013, Rev. Mod. Phys., 85, 245–261.

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When the density at the centre reaches ≈ 1017 kg/m3 , the homology is broken and a shock is formed approximately at the sonic point, near to the maximum infall velocity. The shock is formed at a radius of a few tens of kilometres, delimiting a mass of 0.6 M . This shock formation was observed in numerical simulations half a century ago for a relatively soft equation of state. The numerical simulations of the 1970s to 1980s showed the so-called prompt shock scenario; see Section 1.3.2. This is no longer the case for the state-of-the-art numerical simulations.[1,3,4,5] Since the end of the 1980s the numerical simulations show that supernovae explosions cannot be triggered by a prompt shock. After the core bounce the shock stalls inside the collapsing core and turns into an accretion shock at a radius typically between 100 and 200 km at a density not much larger than 1013 kg/m3 , and no explosion of the star is observed because the shock is not sufficiently strong to escape from the infall of material inside the iron core. One of the reasons often mentioned is that the shock is weakened by the dissociation of a very small percentage of nuclei into nucleons (at the expense of approximately 9 MeV per nucleon) and also by emission of neutrinos. Moreover, a failure of the shock outburst is also produced for sufficiently stiff equation of state, without weakening the shock by dissociation and/or neutrino losses. This point is not sufficiently stressed in the literature, and a simple example is given below. New physical mechanisms, not yet fully understood, must be involved to produce a revival of the shock in order to expel the envelope of the star. The real problem is extremely complex. In view of the uncertainties involved, attention is limited here to simple models with the hope that this approach can be helpful to decipher the explosion of massive star.

Sensitivity to Compressibility An important point is that the formation and the strength of the shock wave are very sensitive to variations of the compressibility associated with a change of density. This is illustrated by numerical simulations of (7.3.3)–(7.3.4) using phenomenological relations for pˆ (ρ). ˆ Consider an initial condition constituted by an unstable solution of the Lane– Emden equation, corresponding to an adiabatic index that is constant and smaller than 4/3 in the initial state, γ  4/3, pˆ = [(γ −1)/γ ]ρ, ˆ in the notation of Section 7.2.2. The relation pˆ (ρ) ˆ is assumed to change for densities higher than the central density of the initial state, ρˆ > 1, as, for example   1 + tanh (ρˆ − ρˆ ∗ )/ρˆ , pˆ (ρ) ˆ = pˆ 1 + (ˆp2 − pˆ 1 ) 2

where

pˆ i ≡

γi − 1 γi ρˆ , γi

i = 1, 2, (7.3.34)

and where γ1  4/3, γ2  4/3, ρˆ ∗  1 and ρˆ ρˆ ∗ . The initial state is the solution of the Lane–Emden equation for γ = γ1 , pˆ = [(γ1 − 1)/γ1 ]ρˆ γ1 , while γ = γ2 at high density,

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(a)

(b)

0

2 0

–2

–2 –4

–4 –6

–6 0

0.1

0

0.1

(c) 0 –1 –2 –3 –4 –5

0

0.1

Figure 7.3 Numerical solution of (7.3.3)–(7.3.4) with (7.3.34) for γ1 = 1.25 and γ2 = 4/3 showing shock outburst. (a) Formation of the shock. (b) Increase of shock strength when reaching the edge of the star where the density decreases to zero. (c) Solution for γ1 = 1 and γ2 = 4/3 showing the shock stalled inside the collapsing matter and unable to escape from the star. Courtesy of Bruno Denet, IRPHE Marseilles.

pˆ = [(γ2 − 1)/γ2 ]ρˆ γ2 . The numerical results of gravitational collapse using (7.3.34) can be summarised as follows: • No shock wave formation is observed for a constant adiabatic index, γ = γ1  4/3. This confirms that the self-similar solutions, presented in Section 7.3.2, are stable. • A strong shock is formed for a small variation of γ , as shown in Fig. 7.3a for γ1 = 1.25 and γ2 = 4/3. Even more interesting, the prompt shock scenario is easily observed in this case; see Fig. 7.3b. In agreement with a self-similar solution,[1] the flow velocity and the temperature increase when the shock emerges at the edge of the collapsing star where the density decreases to zero. • A sufficiently large variation of γ with the density leads to a different result. An example is shown in Fig. 7.3c for γ1 = 1 and γ2 = 4/3. The shock is still formed but it stalls and cannot escape from the collapsing core so that the star does not explode.

[1]

Zeldovich Y., Raizer Y., 1967, Physics of shock waves and high-temperature hydrodynamic phenomena II. Academic Press.

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A rough explanation is usually given, saying that when the matter of the inner part of the core is sufficiently compressible, it is first compressed and acts much like a spring, pushing back the in-falling matter with a stronger intensity than for a rigid inner core. The situation is not so simple since no shock is observed for γ = cst. The criteria of shock formation deserve further study, investigating the characteristic curves, as in the planar case presented in Section 15.3.4. Fundamental studies of the nonlinear acoustic waves propagating outwards from the centre remain to be done by using simplified models. Isentropic Model Under the thermodynamic conditions in (7.3.32)–(7.3.33) just before the gravitational collapse (ρ ≈ 1011 − 1013 kg/m3 , T ≈ 4 × 109 − 1010 K) the matter is ionised. Introducing the density of iron nuclei nnuc the molecular fraction of electrons is Xe ≡ ne /nnuc = 26 when the matter is fully ionised. The mass of iron nuclei mnuc ≈ 9.3 × 10−26 kg is 105 times larger than the electron mass, me ≈ 9 × 10−31 kg, so that the mass of the infalling matter is essentially that of heavy nuclei.[2] The Fermi energy (13.2.46) of the electron gas, F ≈ 9.76 × 10−26 (Xe nnuc )1/3 J, is sufficiently large for the electron gas to be considered as a relativistic Fermi gas at zero temperature. For example, for ρ = 1013 kg/m3 (nnuc ≈ 1038 m−3 ) and T = 1010 K, the Fermi energy, F ≈ 1.34 × 10−12 J ≈ 8 Mev is larger than kB T ≈ 1.38 × 10−13 J and than me c2 ≈ 8.2 × 10−14 J, by a factor of 10 and 16, respectively, F /kB T ≈ 10, F /me c2 ≈ 16. The approximation of a relativistic degenerate electron gas is less accurate at the edge of the iron core, just before the gravitational collapse. For simplicity, we will still use it throughout the core. Only a few iron nuclei are 4 dissociated because the photodissociation 56 26 Fe → 13 He + 4n requires too much energy: [3] 124 MeV per nucleus. This phenomenon will be neglected in the following. Moreover, in agreement with Debye shielding, the interaction between nuclei will be also neglected. −1/3 To fix ideas, the mean distance between iron nuclei, ne , is more than 40 times the radius of the nuclei (5.3 × 10−15 m) for ρ ≈ 1013 kg/m3 (nnuc ≈ 1038 m−3 ). Since the Fermi energy (13.2.33)–(13.2.34) of the iron nuclei at this density, F ≈ 10−17 J, is much smaller than kB T ≈ 10−13 J, F /kB T ≈ 10−4 , the nuclei form a Boltzmann gas, whose pressure at T ≈ 1010 K, p = nnuc kB T ≈ 1025 N/m2 , is negligible compared with the pressure of the degenerate gas of relativistic electrons, p = ne F /4 = 2.44 × 10−26 (Xe nnuc )4/3 ≈ 8.72 × 1026 N/m2 for Xe ≡ ne /nnuc = 26 and nnuc ≈ 1038 m−3 . To summarise, at the beginning of the core collapse, the pressure is due to the relativistic degenerated electron gas, while the mass density is that of the gas of iron nuclei. The nuclear contribution to the pressure is no longer negligible when the density approaches that of the nuclear matter, ≈ 2.6 × 1017 kg/m3 , reached in the centre at the end of the gravitational collapse. The compressibility decreases drastically to nearly zero at such high densities. Due to the stiffening of the matter, the shock wave that is formed during the gravitational collapse is too weak to escape from the collapsing core (stalled shock). Different equations of state of the stellar matter have been proposed.[2] To facilitate [2] [3]

Bethe H., 1990, Rev. Mod. Phys., 62(4), 801–866. Phillips A., 1994, The physics of stars. John Wiley and Sons.

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parametric studies involving the smallest number of parameters we will consider here a simplified model in which the nuclear matter at high density is considered as a gas of hard spheres of finite radius. Neglecting the internal degrees of freedom, the gas of nuclei can be modelled by an equation of the Van der Waals type. The internal energy eint (ρ, T) and the equation of state p(ρ, T) are 3 eint = 3bXe4/3 n1/3 + kB T, 2

p = bXe4/3 n4/3 +

nkB T , 1 − n/n0

(7.3.35)

where, from now on, n ≡ nnuc denotes the density of nuclei, ρ ≈ nmnuc , Xe ≡ ne /n = 26 is the molecular fraction of electrons, eint is the internal energy per nucleus (the internal energies per unit mass and per unit volume are, respectively, eint = eint /mnuc and neint = ρeint ), n0 is the saturation density of iron nuclei, n0 ≈ 1042 m−3 for ρ0 ≈ 1017 kg/m3 and b is a dimensional constant b ≡ (1/4)(3π 2 )1/3 h¯ c ≈ 2.44 × 10−26 , J m. The stiffening of the matter is described here by a single parameter, n0 . When dissipative transport is neglected, there is no entropy production and the equation to be added to (7.3.3)–(7.3.6) to solve the dynamical problem is Ds/Dt = 0, yielding Deint /Dt + pD(1/n)/Dt = 0, in agreement with the conservation of the total energy (7.1.14)–(7.1.15). Assuming a uniform distribution of entropy in the initial state, a simple calculation presented in Section 7.4.2 leads to the isentropic law to be introduced into (7.3.6),   5/3   ρˆ 4/3 ρ/ ˆ ρˆ0 1 4/3 + Kˆ , (7.3.36) pˆ (ρ) ˆ = ρˆ0 4 ρˆ0 1 − ρ/ ˆ ρˆ0 where ρˆ0 = ρ0 /ρ c ≈ 103 –104 is the ratio of the saturation mass density to the mass density ρ c at the centre of the core just prior the implosion, and the constant

1/3 , ranging from 0.1 to 10, is the ratio of the nuclei pressure Kˆ ≡ (pnuced /peed ) need /n0 to the electronic pressure times the cube root of the ratio of the saturation density to the density, both factors being evaluated at the edge of the core in the initial condition. The assumption of a uniform entropy is probably not accurate since the distribution of entropy depends on the way the iron core is formed. This illustrates that the gravitational collapse could well depend on the details of the evolution of the whole star structure prior to collapse. Simplified Model Including the Energy Loss by a Neutrino Flux Electron capture on nuclei by inverse beta decay in the stellar core produces neutrinos: e− +

56

Fe →

56

Mn + νe .

(7.3.37)

Neutrinos are trapped at very high density, but escape almost freely at lower density from the so-called neutrino sphere, located inside the sphere delimited by the shock front. The copious energy loss, due to neutrinos carrying away part of the gravitational binding energy of the stellar core, plays an essential role in gravitational collapse. The electronic pressure decreases as the electronic density to the power of 4/3, so that the collapse is facilitated by electron capture and the main problem left is to decipher a mechanism of explosion.

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The rest of this section addresses this problem with a minimal model taking into account both the neutrino flux and the stiffening of the equation of state at high density. Despite the fact that the model is obtained with approximations that are not quantitatively accurate, the results provide interesting physical insights into the problem. Consider a gaseous mixture of heavy nuclei, Fe and Mn considered as hard spheres (no internal degrees of freedom), electrons and neutrinos. In the following they are denoted by the subscripts Fe and Mn (or simply nuc for nuclei), e and νe , respectively. The internal energy of the mixture is decomposed into three parts: the purely thermal energy of a classical gas, (3/2)kB T per nucleus, the sum of the internal energy of two degenerate gases, ee per electron plus eνe per neutrino that is trapped, and the sum of the rest energies of the nuclei, mFe c2 or mMn c2 per nucleus. The latter is introduced to take into account the difference of mass in the reaction (7.3.37). The mass–energy difference between a neutron and a proton–electron pair is 0.78 MeV and (mMn − mFe − me )c2 = qm > 0, where c is the speed of light, qm ≈ 3.2 Mev. In this respect electron capture (7.3.37) is an endothermic reaction. The internal energy per unit volume takes the form nFe mFe c2 + nMn mMn c2 + ne (ee + me c2 ) + nνe eνe + (3/2)nkB T, where n denotes the density of nuclei and the mass density is ρ = nmnuc as before. Introducing the molecular fractions, Xi ≡ ni /n, n = nFe + nMn , the internal energy per nucleus is 3 eint = XFe mFe c2 + XMn mMn c2 + Xe (ee + me c2 ) + Xνe eνe + kB T, ρ ≈ nmnuc , (7.3.38) 2 and the equation for the conservation of energy (7.1.15) takes the form    D  D 1 e +p = −∇.J. (7.3.39) n Dt int Dt n To begin we briefly recall current views in neutrino physics.[1] The reaction of inverse beta decay (7.3.37) can proceed only when the electron energy is larger than the qm . This occurs for a sufficiently large internal energy of the electron gas, ee > qm , implying that the mass density of the contracting core must be greater than 1012 kg/m3 . However, the reaction rate becomes significant only at a higher density,[2] typically ρ  1013 − 1014 kg/m3 . At such a density, beta decay (neutron decomposition, the inverse of electron capture) is not possible[2] because the quantum states of electrons are fully occupied (Pauli exclusion principle). Therefore, in such conditions, electron capture (7.3.37) can be considered as an irreversible reaction, w˙ > 0, with a cutoff density ρ ∗ about 1013 kg/m3 , below which w˙ becomes very small. An important change occurs as the density reaches ρ ∗∗ ≈ 1015 kg/m3 . The neutrinos become trapped in the core because they do not have time to escape from the sphere delimited by ρ ∗∗ during the core collapse. Moreover, at such a high density, ρ  ρ ∗∗ , electron capture is balanced by neutrino capture to establish chemical equilibrium. As a consequence, during a time sufficiently long compared with the core collapse, the fraction of electrons Xe in the central part is restored to high values, slightly smaller than [1] [2]

Burrows A., 2013, Rev. Mod. Phys., 85, 245–261. Phillips A., 1994, The physics of stars. John Wiley and Sons.

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(but not far from) the initial value before the neutrino capture became important. In other words, neutrino trapping halts the short-term decrease in Xe observed during collapse.[1] This trough in the Xe profile concerns a slice of semi-transparent stellar matter, located behind the stalled shock at intermediate densities, between 1013 kg/m3 (transparent core) and 1015 kg/m3 (opaque core), where the rate of electron capture is first positive and then becomes negative on the high-density side. In the following we will simply assume that this rate varies with the density and is zero at chemical equilibrium, ρ  ρ ∗∗ ≈ 1015 kg/m3 , and also at low density, ρ  ρ ∗ ≈ 1013 kg/m3 (frozen reaction). We focus attention on the flux of neutrinos that escape freely from the stellar matter; all other transport and dissipative mechanisms such as radiation, molecular diffusion, heat conduction and photodisintegration are neglected. The approximation of negligible diffusion fluxes is valid because the characteristic time of gravitational collapse is small compared with the diffusion time. Introducing the rate of electron capture, w, ˙ by inverse beta decay (7.3.37), the equations for the molecular fractions Xi take the form DXe DXFe DXMn 1 DXνe = =− = −Xe w, = − ∇.jνe + Xe w. ˙ ˙ (7.3.40) Dt Dt Dt Dt n The determination of jνe is a difficult nonlocal transport problem because the neutrinos that escape freely from the point r where they are produced can be reabsorbed at a point that can be at a large distance from r. In the notation of (7.3.40) the value of the field 1n ∇.jνe at a point r contains two contributions: • The loss rate of neutrinos escaping freely from r, amongst those produced at the same ˙ point r by electron capture (rate Xe w) The absorption rate of neutrinos coming from points r = r and transported by jνe . • In a transparent medium neutrino trapping is negligible: the neutrinos that are produced ˙ and neutrinos do not contribute to the internal energy of the escape freely, ∇.jνe = nXe w, matter, Xνe = 0. Moreover, the density is usually sufficiently small for the reaction rate to be small, ω˙  0, consuming few electrons and increasing the neutrino flux only slowly. In such a region electron capture produces a small energy loss. In an opaque medium the density is sufficiently high for the neutrinos to be fully trapped and in chemical equilibrium. There is very little escaping neutrino flux jνe = 0, and both Xe and Xνe are fixed at their equilibrium value w˙ ≈ 0. The difficulty is to compute ∇.jνe in a semi-transparent medium. In the absence of reaction, the flux of neutrinos is the solution of an equation of the type ∇.jνe = −|jνe |/labs , where labs is called the absorption length. In general this length depends not only on the local properties of the matter but also on the energy of the neutrinos in the flux. The determination of the energy flux J associated with neutrino flux, jνe , is an even more difficult transport problem than for jνe , essentially because the energy that is absorbed depends also on the place where the neutrino has been emitted. This problem is not addressed here. Qualitative results can be obtained by the ad hoc relation

[1]

Burrows A., 2013, Rev. Mod. Phys., 85, 245–261.

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∇.J = eνe ∇.jνe ,

367

(7.3.41)

which is justified when the energy flux absorbed at r concerns neutrinos that have the same energy as the internal energy of the local neutrino gas. This is obviously the case if the neutrinos have a constant energy, eνe = cst., which is not true in the collapsing core of a massive star. For simplicity, both the electrons and the trapped neutrinos will be assumed to form a relativistic degenerate Fermi gas, whose Fermi energy is larger than the thermal energy, F  kB T; see (13.2.46). This approximation is not accurate for the neutrinos, especially when their concentration is small, as is the case when the mass density is not sufficiently high. This is not essential for the following. The internal energy and the pressure then take the form   nkB T 4/3 4/3 4/3 X + , p = b n + X , ee = 3b n1/3 Xe1/3 , eνe = 3b n1/3 Xν1/3 e νe e 1 − n/n0 (7.3.42)

   Deint D 1/3 4/3 3 DT DXe = 3b n + kB , (7.3.43) Xe + Xν4/3 − qm e Dt Dt Dt 2 Dt where b and n0 are the same as in (7.3.35). The Euler–Poisson equations (7.1.3)–(7.1.4) are solved by using the expression for p in (7.3.42) in which the molecular fractions Xe , Xνe and T are solutions of (7.3.40) and (7.3.39), respectively. When the expressions for w˙ and jνe are known, the system of equations (7.1.3)–(7.1.4) and (7.3.38)–(7.3.43) constitutes a consistent model for the gravitational collapse of the stellar core in the presence of electron capture. Neutrino-Driven Acoustic Instability? The coupling of acoustic waves to chemical reactions is known to produce strong instabilities in combustion. Could this also be the case for neutrino production during gravitational collapse? Acoustic mechanisms have been investigated relatively recently, in two-dimensional simulations; see references in the review article of Burrows.[1] The preliminary comments presented here are limited to spherical geometry using the simple model described above. Physical insight is gained when the temperature is eliminated from the energy equation to yield a relation between the variations of pressure and density similar to (15.2.18) (see the details of the calculations in Section 7.4.1), Dn Dp − a2 mnuc = q˙ , Dt Dt

q˙ = q˙ rr + q˙ νf ,

(7.3.44)

where the frozen speed of sound a is a2 mnuc =

4p 1 (1 + 4n/n0 ) + kB T , 3n 3 (1 − n/n0 )2

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(7.3.45)

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and the source term q˙ (sink if q˙ < 0) is decomposed into two parts q˙ rr and q˙ νf proportional respectively to the reaction rate ω˙ and the divergence of the fluxes ∇.J and ∇.jνe ,   n 1 4 3  1+ (ee − eνe ) − qm Xe nω, ˙ (7.3.46) q˙ rr =  9 1− n n0 2 n0   n    1 2 2 2 1 − 2 n0 n  ∇.J −  eνe ∇.jνe , (7.3.47) q˙ νf = −  eνe ∇.jνe = −  1+ 3 1− n 3 n0 9 1− n n0

n0

where the last relation is obtained using the relation (7.3.41). Through a mechanism similar to that recalled in Section 15.2.4, if the fluctuations of δ q˙ are in phase with the pressure fluctuations δp, an acoustic instability can develop. The situation is not easy to analyse, even in the framework of the simplified model. If the reaction rate nω˙ is an increasing function of the density, its fluctuations are expected to be in phase with δp. The contribution of q˙ rr will depend on the coefficient in front of nω. ˙ Since the reaction becomes relevant when the internal energy of the electron gas ee is large compared with the energy absorbed by electron capture, qm , the coefficient is positive, at least when eν remains well below ee , which is systematically the case at the beginning of the collapse and at small density. The contribution of q˙ rr is then expected to promote an instability if nω˙ is positive and increases with the pressure. The contribution of q˙ νf depends on δ∇.jνe and is not easy to evaluate. This quantity varies across the collapsing core that contains different regions. The neutrinos are trapped at a density above ρ ∗∗ ≈ 1015 kg/m3 , which is greater than the density at which the reaction rate becomes significant. However, the mean free path of the neutrinos, ν , increases quickly with decreasing density and becomes larger than the size of the stellar core for density below ρ ∗ that is slightly less than 1014 kg/m3 . Therefore the flux of escaping neutrinos varies strongly in a slice of stellar matter, between approximately 1014 kg/m3 and 1015 kg/m3 . Since the density decreases with increasing radius, the collapsing matter inside the sphere delimited by the shock wave can be decomposed into three parts: ∗ ∗ • An outer region, r > r , on the low density side, ρ < ρ , from which neutrinos escape ∗ freely. The sphere of radius r is called the neutrino sphere. In this transparent region the neutrinos do not contribute to the internal energy of the stellar matter, Xνe = 0, eνe = 0. According to (7.3.40), the neutrino flux satisfies the relation ∇.jνe = nXe w˙ > 0. Equations (7.3.46)–(7.3.47) yield   4 3 2 ee − qm Xe nω˙ − ∇.J, (7.3.48) q˙ ≈ 9 2 3

since n n0 . For such a small density, ρ < ρ ∗ , the reaction rate is small on the scale of the hydrodynamic time tff , Xe w˙ tff 1. The neutrino flux, jνe , and the energy flux, J, are important, but both ∇.jνe and ∇.J are small. This zone where both the absorption of neutrino and the rate of electron capture are negligible is not expected to be responsible for an acoustic instability.

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∗∗ • A spherical region around the centre, r < r , where the density is quasi-uniform and sufficiently high for neutrinos to be trapped, ρ > ρ ∗∗ . Obviously the orderings ρ ∗∗ > ρ ∗ and r∗∗ < r∗ hold. The fluxes associated with the escaping neutrinos are zero, jνe = 0, and J = 0,   n 4 1 + n0  (ee − eνe )Xe ω, ˙ (7.3.49) q˙ = q˙ rr ≈  9 1− n n0

where the simplification of q˙ rr , defined in (7.3.46), is valid if ee  qm , which is the case because ρ ∗∗ is greater than the threshold density at which the reaction starts to proceed, (ee ≈ qm ). Moreover, assuming chemical equilibrium, the global reaction rate is negligible, Xe ω˙ ≈ 0 (direct and inverse reaction are in equilibrium). Therefore, except for a purely mechanical mechanism associated with a large speed of sound (small compressibility of the matter at high density, close to the nuclear matter), this inner zone is not expected to play an important role in an acoustic instability. ∗∗ ∗ ∗∗ ∗ • A semi-transparent intermediate region, r < r < r (ρ > ρ > ρ ) from which escape a nonnegligible fraction of the neutrinos that are produced by electron capture, q˙ = q˙ rr + q˙ νf .

(7.3.50)

The fluxes jνe and J are oriented outwards along the radial axis and increase from zero at r = r∗∗ to positive values at r = r∗ where Xνe = 0. Therefore the quantities ∇.jνe and ∇.J are positive in this region and their spatial distributions have a bell-shaped form. The reaction rate w˙ varies from positive values (corresponding to a rapid electron capture producing a decrease of Xe ) to negative values that restore a higher value of Xe corresponding to the chemical equilibrium at high density. This intermediate region is the heart of the collapsing core that controls the coupling of neutrino physics with acoustics since the fluctuations of q˙ could be large. Unfortunately its net effect on the acoustic waves (amplification or damping) cannot be easily anticipated because it is very sensitive to the details of the deformation of the spatial distributions of w˙ and ∇.J when the pressure varies. This is illustrated by a simple example in Section 7.4.2. The acoustic mechanism discussed here is an unsteady phenomenon, based on fluctuations ˙ It is different from the neutrino heating mechanism for of ∇.jνe > 0, ∇.J > 0 and w. revival of the shock by reabsorption of neutrinos, ∇.J < 0 for r > r∗ , the so-called delayed shock,[1,2,3,4] introduced more than 30 years ago. To summarise, the development of an acoustic instability is quite possible, essentially due to the intermediate region. However, in order to produce an explosion of the star, the strength of the acoustic instability must be sufficiently strong to revive the shock. Due to the huge amount of energy carried by the neutrinos, this is likely. Unfortunately, as in ordinary [1] [2] [3] [4]

Bethe H., 1990, Rev. Mod. Phys., 62(4), 801–866. Janka H.T., et al., 2007, Phys. Rep., 442, 38–74. Janka H., 2012, Annu. Rev. Nucl. Part. Sci., 62, 407–451. Burrows A., 2013, Rev. Mod. Phys., 85, 245–261.

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combustion, the threshold and strength of this type of instability are very sensitive to details of the configuration and to small variations of the parameters. It is not clear that the current knowledge of neutrino physics at high density and temperature is sufficient to provide a definitive answer. To conclude, a necessary requirement to determine whether or not the scenario of neutrino-driven explosion of a massive star is relevant is a better knowledge of neutrino physics. Moreover, analytical studies and carefully controlled numerical simulations of simplified models of the collapsing stellar core, performed in association with the huge numerical simulations involving advanced nuclear physics, are necessary to decipher the supernovae.

7.4 Appendix 7.4.1 Simplified Model of Core Collapse Introducing the notation   , B ≡ b Xe4/3 + Xν4/3 e

p = B n4/3 +

nkB T , 1 − n/n0

(7.4.1)

the particulate derivative of the internal energy reads, according to (7.3.43), Deint B Dn 3 DT DB DXe = 3n1/3 + 2/3 − qm + kB , Dt Dt Dt Dt 2 Dt n and, using the second equation in (7.4.1),    D 1 B Dn kB T p = − 2/3 + . Dt n (1 − n/n0 )n Dt n The left-hand side of Equation (7.3.39) then takes the form    Deint Dn DB DXe D 1 3 DT kB T = 3n1/3 +p − qm + kB − . Dt Dt n Dt Dt 2 Dt (1 − n/n0 )n Dt

(7.4.2)

(7.4.3)

(7.4.4)

Isentropic Condition 4/3

In the isentropic case (no electron capture, B = bXe = cst., Xe = cst., Xνe = 0, J = 0), Equations (7.3.39) and (7.4.4) yield the following relation between n and T:  n/n0 D D 3/2 = 0. (7.4.5) ln T − ln Dt Dt 1 − n/n0 This relation can be integrated to give n/n0 = cst. × T 3/2 , 1 − n/n0

(7.4.6)

where the constant can be evaluated using the known conditions at the edge of the iron core, nFedge ≈ 1036 m−3 , Tedge ≈ 4 × 109 K. Using (7.4.6) to eliminate T in the second equation

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7.4 Appendix

371

of (7.4.1) leads to the isentropic relation p(n). Neglecting nedge /n0 in front of unity, one gets  4/3    5/3 n n0 2/3 n/n0 4/3 4/3 + n0 kB Tedge , (7.4.7) p = bXe n0 n0 nedge 1 − n/n0     5/3   nedge kB Tedge n/n0 n 4/3 n0 1/3 4/3 4/3 ˆ ˆ p = bXe n0 +K , , K≡ 4/3 4/3 n0 1 − n/n0 nedge bXe n edge

(7.4.8) which can also be written as p = pe0



ρ ρ0

4/3

+ Kˆ



ρ/ρ0 1 − ρ/ρ0

5/3  ,

(7.4.9)

4/3 4/3

where pe0 ≡ bXe n0 is the electronic pressure at the saturation mass density ρ0 . The value of the constant in (7.4.9) is Kˆ ≈ 2.9 in typical conditions; Xe = 26, nedge ≈ 1036 m−3 , Tedge ≈ 4 × 109 K at the edge of the iron core and n0 ≈ 1042 m−3 for the saturation density, n0 /nedge , ≈ 106 . This corresponds to a very small value, of order 3 × 10−10 , for ratio of the nucleon pressure at the edge, nedge kB Tedge = 5.3 × 1022 N/m2 , 4/3 4/3 to the maximum electron pressure at the centre, pe0 ≡ bXe n0 ≈ 2 × 1032 N/m2 . Model Including Neutrinos An equation relating p and n can be obtained from (7.3.39) using (7.4.2) from which T is eliminated using the second equation in (7.4.1), written in the form,    1  1 − p − B n4/3 . (7.4.10) kB T = n n0 According to the first equation in (7.4.1) and (7.3.40), two of the terms in (7.4.2) take the form 1/3

   DB DXe bXνe − qm = − 4n1/3 b Xe1/3 − Xν1/3 3n1/3 − q X w ˙ − 4 ∇.jνe . (7.4.11) m e e Dt Dt n2/3 Moreover, Equation (7.4.10) yields ⎧   ⎨ 5 p Dn − 2 n2 Dt + D 1 3 DT  kB +p = ⎩+ 2 n1/3 − 2 Dt Dt n n0





3 1 1 Dp 2 n −  n0 Dt 1 1 Dn 3 2 n2/3 B Dt − 2



1 n

−

1 n0



n2/3 DB Dt .

Introducing (7.4.11) and (7.4.12) into (7.4.2) leads to ⎧   5 p Dn 3 1 1 Dp ⎪ − + − ˙ ⎪ n n0 Dt +qm Xe w ⎪ 2n2 Dt 2    ⎪ ⎨ 4/3 4/3 n 1 1 Dn D  D 1 e +p = + 2 n0 + 2 b Xe + Xνe n2/3 Dt ⎪    Dt int Dt n ⎪  ⎪ 1/3 1/3 ⎪ ˙+ ⎩−2 1 + nn0 n1/3 b Xe − Xνe Xe w

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1/3

bXνe n2/3

(7.4.12)

∇.jνe .

372

Explosion of Massive Stars

Equations (7.3.39) and (7.3.41) then lead to (7.3.44)–(7.3.47):      n 2 n n1/3 b Xe 1/3 − Xνe 1/3 Xe ω, qm − 2 1 + q˙ rr = − ˙ 3 (1 − n/n0 ) n0 2 (1 − 2n/n0 ) 1/3 1/3 n bXνe ∇.jνe . q˙ νf = − 3 (1 − n/n0 )

(7.4.13) (7.4.14)

7.4.2 Sensitivity of Acoustic Instabilities to Energy Losses The fluctuations of density and pressure in acoustic waves not only produce fluctuations of the rate of energy loss per unit volume, |∇.J|, they also make the position of the neutrino sphere oscillate (i.e. displace the slice where |∇.J| is localised). Depending on the phase shift of these two mechanisms with respect to pressure, an acoustic instability can develop. This can be investigated in a systematic way by extending the stability analysis presented in Section 7.3.1 to the nonisentropic case, using for example the model presented in Section 7.3.3; however, the calculations are tedious. In order to illustrate the counter–intuitive idea that an energy loss due to a flux J of energy escaping freely from the matter can produce an instability, depending very sensitively on the way the distribution ∇.J varies with the pressure, we will limit our attention here to a simple acoustic model in planar geometry neglecting the variation of the sound speed; see Section 15.2.4. The starting point is the relation (15.2.21): 2 ∂ q˙ γ ∂ 2p 2∂ p . − a = ∂t ∂t2 ∂r2

(7.4.15)

According to (15.2.19) where the heat flux Jq is replaced by an energy flux J, the source term in the wave equation is ∂ q˙ γ /∂t = − (γ − 1) ∂∇.J/∂t. In the model for the collapsing core presented in Section 7.3.3, the coefficient is different from (γ − 1) and is still of order unity; see (7.3.47). The instability growth rate can be evaluated from   d (7.4.16) (δp)2 dr ≈ δ q˙ γ δp dr, where δ q˙ γ = − (γ − 1) δ(∇.J). dt The onset and the linear growth rate of instability depends on the way the field ∇.J is modified by the acoustic pressure. Assuming that the flux increases with the distance from zero at the inner edge r = r∗∗ (neutrino trapping sphere) to a uniform value J at the outer edge, r = r∗ (neutrino sphere), the term ∇.J is positive. More generally it takes the form of a bell-shaped curve   (r − rν ) J (7.4.17) ∇.J = h dν dν with a maximum at r = rν ∈ [r∗∗ , r∗ ] and a thickness dν ≈ (r∗ − r∗∗ ). The nondimensional  +∞ function h(η), describing the spatial distribution of ∇.J, is normalised to unity ˙ γ /∂t takes the form −∞ h(η)dη = 1 and the source term ∂ q

17:08:52 .009

7.4 Appendix

 δ q˙ γ = −(γ − 1)δ

373

 J h . dν

(7.4.18)

   An instability develops if δ dJν h δp dr < 0, or in other words if the absolute values of the losses are some way in phase opposition with the pressure. The strength of the instability is evaluated in the following examples, retaining planar geometry for simplicity: 1. Consider first the case for which the total flux J escaping freely from the matter is constant and the distribution ∇.J fluctuates with the density. Both quantities dν (t) and rν (t) fluctuate in time around their mean value dν (t) = dν + δdν (t), rν (t) = rν + δrν (t). According to the linear approximation of (7.4.17), the fluctuation of (∇.J)/J yields         r − rν (t)  1 1 h δ =δ h, h+ δ dν (t) dν dν (t) dν



where h ≡ h (r − rν )/dν , h ≡ h (r − rν )/dν and h (η) ≡ dh(η)/dη. Introducing the quantity S ≡ 1/(γ − 1)J δ q˙ γ δp dr (which is proportional to the growth rate), Equations (7.4.16)–(7.4.17), to leading order in small fluctuations, yield the quadratic expression ⎧

     ⎨ δdν hδp dr + (r−rν ) h δp dr 1 dν dν (7.4.19) δ q˙ γ δp dr = dνδr   dνdr S≡ ⎩+ ν h δp . (γ − 1)J dν

dν

This equation illustrates the difficulty of anticipating the result and in particular the sign of the linear growth rate! A simpler expression is obtained for the square-wave model in which ∇.J is constant in the domain [r∗∗ , r∗ ] and zero outside: - (δr∗ −δr∗∗ )  r∗ ∗∗ δp(r, t)dr ∗ ∗∗ 2 (7.4.20) S = (r −r1 ) r ∗  − (r∗ −r∗∗ ) δp(r , t)δr∗ − δp(r∗∗ , t)δr∗∗ . In the more general case, Equation (7.4.19) simplifies if dν , namely the extension in space of ∇.J, is smaller than the acoustic wavelength . Limiting attention to the first order in an expansion in powers of dν /, and using the expansion δp(r, t) = δpν (t) + (r − rν )δpν (t) + · · · , where δpν (t) ≡ δp(rν , t) and δpν (t) ≡ ∂δp/∂r|r=rν , yields   (7.4.21) S = − δdν ηh(η)dη + δrν δpν ,       2  where  the relations h(η)dη = 1, h (η)dη = 0, ηh (η)dη = −1 and η h (η)dη = −2 ηh(η)dη have been used. The first term in the right-hand side describes the effect of the deformation of the distribution. This term is zero if the distribution h(η) is symmetric. The second is the effect of the fluctuating shift in position. It is in agreement with the square-wave model (7.4.20) at the first order in the limit dν / 1, S ≈ −[(δr∗ + δr∗∗ )/2]∂δp/∂r|r=r∗∗ , where at this approximation δr∗ ≈ δr∗∗ ≈ δrν and r∗ ≈ r∗∗ ≈ rν .

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Explosion of Massive Stars

From now on attention is limited to the symmetric case in the limit dν / 1 in order to obtain analytical expressions. Assuming that the position of the slice rν (t) fluctuates with the pressure δpν (t), Equation (7.4.21) yields the growth rate in the form  δpν (t)δpν (t) δrν (t) δpν (t) d =b ⇒ , (7.4.22) (δp)2 dr = −(γ − 1)b J rν rν pν dt pν where the nondimensional b is a given parameter. A thermo-acoustic insta  coefficient  bility develops when b δpν δpν < 0, where the brackets denote time averaging over an acoustic period. The first relation in (7.4.22) is a model in which the position of the neutrino slice rν is determined by the density profile in the stellar core, ρ(rν ) ≡ ρν = cst., dp/dr|rν δrν + δρν = 0. Since the density and pressure in the spherical core both decrease strongly as the radius increases, and since the density and pressure fluctuations are in phase in an acoustic wave, the coefficient b is positive, 0 < b < 1. The instability then develops if the neutrino  slice is localised at a position where the acoustic wave satisfies the relation δpν δpν < 0. This is possible in some portions of a standing acoustic wave, and, in this case, the acoustic waves are fed by the fluctuations of position, δrν . The order of magnitude of the growth rate 1/tinsta can be evaluated by noticing that the quantity 3J/(p rν ) is of the order of the cooling time tcool of the inner part of the core, r  rν , since p is of the order of the thermal energy per unit volume. The lefthand side of the second equation in (7.4.22) is of the order (δp)2 /tinsta , where δp is the amplitude of the pressure fluctuation in the standing acoustic wave and  maximum  δpν δpν is of the order (δp)2 /, where  is the acoustic wavelength, of the order of the radius of the inner core, /rν ≈ 1:  2   rν (δpν )2 tcool 1  δpν δpν < 0 ⇒ ≈ (γ − 1)b , (7.4.23) tinsta 3  (δp)2 so that the nondimensional growth rate tcool /tinsta is of the order of (γ − 1)b/3. 2. Neglecting the displacement and the deformation of the neutrino slice but retaining the variation of the flux with the pressure, δ q˙ γ = −(γ − 1)δJh/dν , the thermo-acoustic instability develops if δJ and δpν are in phase opposition. This is possible, since inside the neutrino sphere, increasing the pressure reduces the mean free path of the neutrinos and vice versa. The linear growth rate is evaluated as before:   tcool rν (δpν )2 1 δpν δJ , c>0 ⇒ ≈ (γ − 1)c , (7.4.24) = −c p tinsta 3  (δp)2 J showing that tcool /tinsta is of the order of (γ − 1)c/3.

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Part Two Detailed Analytical Studies

17:10:07

17:10:07

8 Planar Flames

Nomenclature Dimensional Quantities ci cp dL dr D DT D/Dt E H(x) kB m M N P qm Q(j) R t T u u U Ub Uf UL V

Description Concentration of species i Specific heat at constant pressure Scale of laminar flame thickness Scale of reaction zone thickness Molecular diffusivity Thermal diffusivity Material derivative, ∂/∂t + u.∇ Activation energy Evolution of a viscous stagnation point flow Boltzmann’s constant Mass flux Molecular mass Number of molecules of species i Products Heat of combustion per unit mass Heat release of elementary reaction j Reactants Time Temperature Longitudinal velocity component Velocity fluid at front (u, w) Flow velocity at the reaction sheet Flame w.r.t. burnt gas (Stretched) flame speed Laminar flame speed Volume

S.I. Units molecules m−3 J K−1 kg−1 m m m2 s−1 m2 s−1 s−1 J mole−1 m s−1 J K−1 kg m−2 s−1 kg molecule−1 molecules mole J kg−1 J mole s K m s−1 m s−1 m s−1 m s−1 m s−1 m s−1 m3 377

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378

w ˙ W ˙ (j) W x y, z λ ρ τ τL τrb τs

Planar Flames

Transverse velocity component Global mass based reaction rate Reaction rate of elementary reaction j Streamwise coordinate Transverse coordinates Thermal conductivity Density A characteristic time Transit time of laminar flame ≡ dL /UL Reaction time at burnt gas temperature Time scale of stretch rate

m s−1 kg m−3 s−1 reactions m−3 s−1 m m J s−1 m−1 K−1 kg m−3 s s s s

Nondimensional Quantities and Abbreviations b F (ξ ) g h H (z) I J k Le ˙ M(z) O(.) Pr s w w X X, Y Yi YLe Y1 z z β θ  1,2,...

Prefactor in Arrhenius law Scaled evolution of stagnation point flow with gas expansion, see (8.5.35) A function defined in (8.5.17) A measure of reduced heat loss Scaled evolution of a viscous stagnation flow An integral defined in (8.5.12) An integral defined in (8.5.15) Eigenvector in phase space Lewis number √ sF Of the order of Prandtl number Reduced strain rate Reduced reaction rate Normalised reduced reaction rate, see (8.2.13) Internal reduced length Phase space variables Mass fraction of species i Ratio of reduced mass flux to strain YLe=1 Scaled coordinate in viscous stagnation flow Scaled coordinate in stagnation point flow with gas expansion Zeldovich number Reduced temperature Terms in asymptotic expansion of θ

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√ H/ ντs

DT /D √ ≡ −ρu/( sρu UL ) ν/DT τL /τs (β ϑ+1 /2ϑ! )ψ ϑ e−β(1−θ) ≡ βξ Y ≡ μdθ/dξ ρ √ √i /ρ Le√μf / 2s μf / 2s √ x/ ντs x √ s/dL 0 (ρ(x)/ρu )dx E(Tb − To )/kB Tb2 (T − Tu )/(Tb − Tu ) β(1 − θ )

8.1 General Formulation

ϑ μ ξ ξ τ υ  ψ !1,2,... aKPP aZFK

Reaction order Reduced mass flux Reduced length Mass-weighted reduced length in Section 8.5.4 Reduced time Reduced gas density Reaction potential Reduced mass fraction Terms in asymptotic expansion of ψ Kolmogorov, Petrovskii and Piskunov Zeldovich and Frank-Kamenetskii

379

m/(ρu UL ) x/dL  x 1/dL 0 (ρ(x)/ρu )dx t/τL ≡ t dL /Ub ρ/ρb (= Tb /T) w(θ ) = −d/dθ Y/Yu

Superscripts, Subscripts and Math Accents a1 a+ a− ai ab ac acoll af ai a(j) aL am ar as au

Reactant (in ZFK model) On the downstream side of the reaction sheet On the upstream side of the reaction sheet Derivative of a w.r.t. i Burnt gas Critical value of a Collision Value at flame front Pertaining to species i Relative to reaction j Pertaining to a laminar flame Marginal value of a Reaction. Reaction layer Stretch Unburnt gas

In this chapter we present different analytical studies of the internal structure of planar premixed flames. We will be particularly interested in the laminar flame velocity, the flammability limits and flame quenching. These phenomena are related to the high-temperature sensitivity of the reaction rate in contrast with the reaction–diffusion waves found in biophysics. The asymptotic analysis in the limit of large activation energy will be presented in a form that can be extended to wrinkled flames. We will study the effects of thermal self-acceleration, chemical kinetics, heat losses and stretch.

8.1 General Formulation The flame speed is clearly defined for a plane wave propagating in a uniform mixture at rest; see Fig. 1.1. But, as mentioned in Sections 2.1 and 2.3, the concept of flame velocity

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380

Planar Flames

in wrinkled flames, studied in Chapter 10, is associated with that of a flame seen as a propagating front. 8.1.1 Laminar Flame Velocity The laminar flame velocity UL is that of the simplest possible configuration. This configuration is not easy to obtain experimentally. As mentioned in Sections 2.2 and 2.4, the flame is subject to mechanisms that rapidly destabilise a planar front to produce cellular structures. Precise measurements of flame velocity is a difficult and delicate operation. The analytical calculation of UL in planar geometry is the first step in the study of the dynamics of flame fronts. 8.1.2 System of Equations for Planar Flames When the overall energy release rate is characterised by an activation energy that is large compared with the thermal energy, the Mach number of a flame propagating by thermal diffusion is much smaller than the sound speed; see (1.2.5). The equations are thus those of the quasi-isobar approximation discussed in Section 2.1.1 and 15.2.1. Since the propagation speed of the flame is constant, the problem is stationary in a reference frame attached to the planar flame front, ∂/∂t = 0, and the mass conservation equation, ∂ρ/∂t + ∇.(ρu) = 0, reduces to ∂(ρu)/∂x = 0, so that the mass flux through the flame m ≡ ρu = ρu UL = ρb Ub is an unknown constant, an eigenvalue of the problem, that determines the laminar flame velocity, UL = m/ρu , where u and b refer to the unburnt gas at initial conditions and to the burnt gas, respectively. Since ρD/Dt reduces to md/dx, Equations (15.2.3) and (15.2.4) for the balance of species i and energy conservation in reaction j form a system of ordinary differential equations:    d dT dT ˙ (j) (T, ..Yi ..), − λ = Q(j) W (8.1.1) mcp dx dx dx j    d dYi dYi (j) ˙ (j) (T, ..Yi ..), − ρDi = ϑi Mi W (8.1.2) m dx dx dx j

written in the notation of Chapter 15. For simplicity, we will suppose that λ and ρDi are constant. This simplification may easily be removed. The boundary conditions are: x = −∞: x = +∞:

T = Tu , dT/dx = 0,

Yi = Yiu , Yi = Yib ,

(8.1.3)

where Tu and Yiu are the temperature and mass fractions of the initial mixture frozen in a ˙ (j) (Tu , ..Yi ..) = 0, and where Yib are the equilibrium mass fracnonequilibrium condition W ˙ (j) (Tb , ..Yib ..) = 0, at the adiabatic temperature of combustion Tb . This latter can tions, W be calculated directly from the thermodynamic equilibrium as explained in Section 14.2. When the Equations (8.1.1)–(8.1.3) are solved numerically on modern computers, the details of the complex chain of elementary chemical reactions can be included; see Fig. 2.2. However, they cannot be handled in analytical studies that have the useful

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8.2 Thermal Propagation

381

advantage of describing parametric dependencies explicitly. The essential phenomena can be described analytically using simplified models for the chemical kinetics, presented in Chapter 5.

8.2 Thermal Propagation An essential characteristic of flames is the high sensitivity of their energy release rate to temperature, responsible for a violent thermal self-acceleration. This phenomenon is well captured by the one-step model (2.1.3). In the simplest case, the planar flame may then be described by the single equation (2.1.5) that will be derived below; see (8.2.11)–(8.2.14). It is a particular case of a reaction–diffusion wave. A comparison between flames and other reaction–diffusion waves, such as those found in biophysics, shows that the strong nonlinearity of the reaction term in flames modifies the nature of the solution describing the travelling wave; see Section 8.3. In the one-step model (2.1.3), the chemistry is reduced to an irreversible exothermic decomposition of a single reactant, R, into a product, P, controlled by an Arrhenius law (1.2.2) with an activation energy that is large compared with the energy of thermal agitation in the burnt gas. The mixture is diluted in an inert substance which is in abundance. Thanks to the high activation energy, the reaction rate is negligible at the temperature Tu of the fresh mixture, which is quasi-frozen in a composition far from equilibrium. This model is the minimal model demonstrating the role of thermal feedback in flame propagation.

8.2.1 Formulation For a reaction of order ϑ, the reaction rate of (2.1.3) in a homogeneous medium is, in the notation of (14.2.1) and (14.2.12), cϑ1 e−E/kB T dc1 = − ϑ−1 b , dt τcoll c1u where c1 = N1 /V is the concentration of reactant R, c1u is its initial concentration, 1/τcoll is the frequency of elastic collisions and E is the activation energy E  kB T. The term cϑ−1 1u has been introduced in the denominator in order to make the prefactor b dimensionless and of order unity. Introducing the mass fraction of the reactant, Y1 = M1 c1 /ρ, the energy liberated per unit mass of reactant R, qm1 = Q1 /M1 , where M1 is the molecular weight of ˙ the adiabatic evolution at constant volume is the reactant, and the reaction rate W, dY1 ˙ = −W, dt

d(cp T) ˙ = qm1 W, dt

˙ ≡ W

ρ ϑ−1 Y1ϑ

b ϑ−1

ρuϑ−1 Y1u

e−E/kB T , τcoll

(8.2.1)

where ρu and Y1u are the initial density and mass fraction. The specific heat per unit mass of the mixture, cp , is supposed constant in order to simplify the presentation. For the case of a flame propagating at constant speed, the system of equations (8.1.1)–(8.1.2) reduces to

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Planar Flames

  d dT dT ˙ − λ = ρqm1 W, mcp dx dx dx

  d dY1 dY1 ˙ − ρD1 = −ρ W. m dx dx dx

(8.2.2)

Integrating these equations from x = −∞ (fresh gas) to x = +∞ (burnt gas) with boundary conditions (8.1.3), and using the irreversibility condition Y1 (x → +∞) ≡ Y1b = 0,  +∞ ˙ the second equation yields mY1u = −∞ ρ Wdx. Integrating the first equation gives the temperature of the burnt gas (energy conservation in the quasi-isobar approximation): cp (Tb − Tu ) = qm ≡ qm1 Y1u ,

(8.2.3)

where qm = qm1 Y1u is the chemical energy liberated by unit mass of fresh mixture. Reduced Temperature We now introduce the thermal diffusivity DT ≡ λ/ρcp , the Lewis number Le ≡ DT /D1 (ratio of the thermal diffusivity to molecular diffusivity), the reduced mass fraction, ψ ≡ Y1 /Y1u and the reduced temperature, θ ≡ (T − Tu )/(Tb − Tu ). Using ρDT = constant for simplicity, the two equations (8.2.2) can be written in the following form m

˙ d2 θ W dθ − ρDT 2 = ρ , dx Y1u dx

m

˙ ρDT d2 ψ dψ W − = −ρ , dx Le dx2 Y1u

(8.2.4)

x = +∞: θ = 1, ψ = 0.

(8.2.5)

and the boundary conditions are: x = −∞: θ = 0, ψ = 1,

The reaction rate is characterised by the reaction time τr (T) at temperature T,  ϑ  ϑ−1 −E/kB T ˙ ρ ρb 1 1 e W ϑ ≡ . b , ρ = ρb ψ τr (T) ρu τcoll Y1u ρb τr (T) The temperature T in the Arrhenius term, e−E/kB T , can be expressed in terms of the reduced temperature, θ , with the help of the characteristic reaction time τrb at the temperature of the burnt gas Tb ,  ϑ−1  −E/kB Tb  1 ρb e , (8.2.6) ≡ b τrb ρu τcoll     1 1 E 1 β(1 − θ ) 1 1 = , (8.2.7) exp − exp − = − τr (T) τrb kB T Tb τrb 1 − (1 − Tu /Tb )(1 − θ ) where β is the reduced activation energy, β≡

E (1 − Tu /Tb ), kB Tb

(8.2.8)

also called the Zeldovich number, and we have used the identity, Tb 1 = . T [1 − (1 − Tu /Tb )(1 − θ )]

17:10:28 .010

(8.2.9)

8.2 Thermal Propagation

With this notation, the nonlinear term in Equations (8.2.4) is  ϑ−1  ˙ ρ 1 β(1 − θ ) W ϑ . = ρψ exp − ρ Y1u ρb τrb 1 − (1 − Tu /Tb )(1 − θ )

383

(8.2.10)

Unity Lewis Number In the particular case of unit Lewis number, Le = 1, the two equations (8.2.4) have exactly the same form. Eliminating the nonlinear chemical production term leads to m

d(θ + ψ) d2 (θ + ψ) − ρDT = 0. dx dx2

The only solution that satisfies the boundary conditions (8.2.5) is ψ = 1−θ , and the system reduces to a single equation: Le = 1

⇒

m

d2 θ dθ ρb  − ρDT 2 = w (θ ), dx τrb dx

(8.2.11)

with the boundary conditions x = −∞: θ = 0,

x = +∞: θ = 1.

(8.2.12)

Using ρb /ρ = T/Tb and Equation (8.2.9), the production term w (θ ) takes the form 

β(1−θ) exp − 1−(1−T /T )(1−θ) u b . (8.2.13) w (θ ) ≡ (1 − θ )ϑ [1 − (1 − Tu /Tb )(1 − θ )]ϑ To first order in the limit β  1, and for temperatures sufficiently close to that of the burnt gas, β(1 − θ ) = O(1), this expression simplifies to w (θ ) ≈ (1 − θ )ϑ e−β(1−θ) ,

(8.2.14)

and Equation (8.2.11) reduces to (2.1.5) for ϑ = 1. From a mathematical point of view, the problem in (8.2.11) is ill-posed. The production term w (θ ) is nonzero in the unburnt gas at x = −∞, w (θ = 0) = (Tb /Tu )e−βTb /Tu = 0. The flame will not propagate at constant speed in a mixture whose temperature evolves in time, so there can be no solution to Equation (8.2.11). However, as already mentioned in Chapter 1, this cold boundary difficulty is only formal. In reality, the production term is infinitesimally small in the unburnt gas and can be neglected: typically w (θ = 0) ≈ 10−25 at room temperature. The characteristic propagation time of the flame is very much shorter than the time scale on which the temperature of the fresh gas evolves, and the evolution of the cold unburnt gas can be completely neglected. In order to make the problem mathematically well posed, it suffices to introduce a fictitious cut-off temperature, higher than Tu but much lower than Tb , below which we impose that the reaction rate is strictly zero, w (θ < θc ) = 0. In the limit of very large values of the activation energy, the solution is independent of the value of the cut-off temperature, as we will see in the next section.

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384

Planar Flames (a)

(b)

Figure 8.1 Profiles of (a) reduced temperature and (b) reduced reaction rate, in the premixed flame model (2.1.3) for an Arrhenius law with a large activation energy.

8.2.2 ZFK Analysis for Le = 1 In 1938, Zeldovich and Frank-Kamenetskii (ZFK)[1] performed an analysis of the problem (8.2.11) in the limit β → ∞. The temperature profile through the flame, θ (x), is shown schematically in Fig. 8.1, along with the reduced reaction rate as a function of the reduced temperature. The main idea is that, when β is large, the reaction rate, w (θ ), is nonnegligible only when the temperature θ is close to unity in a thermal layer defined by (1 − θ ) = O(1/β). For β  1, the term −β(1 − θ ) in the exponential of (8.2.13) and (8.2.14) makes the reaction term w (θ ) negligible as soon as θ is significantly smaller than unity. The reaction rate also goes to zero when θ = 1 because of the presence of the prefactor (1 − θ ), equal to ψ, meaning that all the reactants have been consumed. The limit β → +∞ is singular since w (θ ) → 0 almost everywhere, except close to θ = 1, when (1 − θ ) = O(1/β). In other words, the reaction is confined to a vanishingly thin boundary layer of thickness ≈ 1/β located at θ = 1 in the ω − θ plane. The system can then be separated into three zones: • A thin reaction layer, located at high temperature • A cooler upstream region, called the preheat zone, where the temperature and species mass fraction may evolve, but where there are no chemical reactions • A hot and uniform downstream region in thermodynamic equilibrium θ = 1. Outside the thin reaction layer, Equation (8.2.11) reduces to a second-order linear equation with constant coefficients mdθ /dx − ρDT d2 θ /dx2 = 0,

(8.2.15)

easily integrated using the boundary conditions (8.2.12) mθ = ρDT dθ /dx, [1]

θ = emx/ρDT ,

Zeldovich Y., Frank-Kamenetskii D., 1938, Acta Phys. Chim., IX, 341–350.

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(8.2.16)

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where we have chosen to place the origin x = 0 at θ = 1 inside the thin reaction layer. This equation shows that the length scale of this preheat zone is dL ≡ ρDT /m = DTb /Ub = DTu /UL ,

(8.2.17)

ρD being assumed constant. Equation (8.2.15) describes the preheat zone in which a nonreacting gas is heated by thermal conduction from the heat liberated in the reaction layer. In the preheat zone, the convective energy flux, mθ , is everywhere compensated by the conductive flux, ρDT dθ/dx. At the hot side, x = 0, θ = 1, the conductive heat flux from the thin reaction layer into the preheat zone is ρDT dθ /dx = m.

(8.2.18)

In the limit of an infinite activation energy, β → +∞, the magnitude of the reaction term (8.2.13) in the thin reaction layer, where (1 − θ ) = O(1/β), is of the order of 1/β ϑ : w (θ ) ≈ (1 − θ )ϑ e−β(1−θ) = O(1/β ϑ ).

(8.2.19)

Let dr be the thickness of this reaction layer and let δθ = O(1/β) be the change of temperature within the reaction layer. Using (8.2.17) in the form m = ρb DTb /dL , the orders of magnitude of the three terms of (8.2.11) in the thin reaction layer are: δθ ρb DTb 1 dθ ≈m , ≈ dx dr dr dL β d2 θ δθ ρb DTb 1 ρb DTb 2 ≈ ρb DTb 2 ≈ , dx dr dr2 β ρb  ρb ϑ ρb 1 w (θ ) ≈ δθ ≈ . τrb τrb τrb β ϑ m

(8.2.20)

A comparison of the first two terms shows that, inside the reaction layer, the convective term is smaller than the conductive term by the ratio dr /dL . Anticipating that dr dL , the heat released by the reaction is here balanced by conduction: − ρb DTb

d2 θ ρb ≈ (1 − θ )ϑ e−β(1−θ) . τrb dx2

A comparison of the last two terms in (8.2.20) gives  dr ≈ β ϑ−1 DTb τrb .

(8.2.21)

(8.2.22)

Equation (8.2.21) multiplied by dθ/dx gives   DTb d dθ 2 dθ 1 − ≈ (1 − θ )ϑ e−β(1−θ) . 2 dx dx τrb dx Integrating from x to +∞ with the boundary condition (8.2.12) in the burnt gas, x = +∞: θ = 1, yields    1  β(1−θ) DTb dθ 2 1 ϑ −β(1−θ) τrb ≈ (1 − θ ) e dθ = ϑ+1 ϑ e− d, (8.2.23) 2 dx β θ 0

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Planar Flames

where the change of variable,  = β(1 − θ ), has been used. In the limit β → +∞, the upper bound β(1 − θ ) goes to infinity. So, for a large activation energy, the temperature gradient at the downstream entry to the reaction layer goes to an asymptotic limit  (8.2.24) β(1 − θ ) → ∞: DTb dθ /dx → 2(ϑ! /β ϑ+1 )DTb /τrb ,  ∞ ϑ − where we have used 0  e d = ϑ!. The unknown mass flux, m, and thus the laminar flame speed, UL , can now be found by matching the heat flux entering the preheat zone (8.2.18) to the heat flux leaving the thin reaction layer (8.2.24) to give  (8.2.25) m = ρb 2(ϑ! /β ϑ+1 )DTb /τrb , UL = m/ρu , Ub = m/ρb . Comparison of (8.2.17) and 8.2.22) shows that the thickness of the preheat zone dL is β times larger than that of the reaction layer,  dr /dL = 2(ϑ! )/β, (8.2.26) validating the initial assumption. Equations (8.2.17) and (8.2.25) provide expressions for the flame thickness, dL ≡ DTu /UL ,  dL = DTb τb , where τb ≡ (ϑ! )−1 β ϑ+1 τrb /2 = dL /Ub , (8.2.27) and the flame transit time, τL , defined by τL ≡ dL /UL , is τL ≡ dL /UL = (Tb /Tu )τb ,

⇒

dL =

 DTu τL ,

(8.2.28)

where we have used ρDT = constant. These results are in agreement with experimental measurements. For example, taking DTb = 3 × 10−4 m2 /s, τrb = 10−6 s, β = 10, ρu /ρb = Tb /Tu = 8 and ϑ = 1, which are typical values, we obtain UL ≈ 0.3 m/s, which is a typical laminar flame speed in these conditions. The characteristic bell-shaped curves for the laminar flame speed as a function of the equivalence ratio, with a maximum close to the stoichiometric composition, such as that shown in Fig. 8.2, can be explained simply by the variation of the burnt gas temperature

Figure 8.2 Typical laminar flame speed of a hydrocarbon–air mixture as a function of equivalence ratio.

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with equivalence ratio. However, as we have already remarked in Section 5.2, a description of the flammability limits can be obtained only with the help of more complicated chemical kinetics, such as that describing the competition between two reactions (5.2.1) and (5.2.4). 8.2.3 Laminar Flame Speed for Le = 1 To extend the method to more general cases, it will be useful to reformulate the problem in terms of reduced variables, scaled by the characteristic length (8.2.27) and velocity of the adiabatic flame with Le = 1, μ ≡ m/ρu UL ,

ξ ≡ x/dL ,

(8.2.29)

where ρu UL = ρb Ub . For more general cases, such as Le = 1, the unknown eigenvalue is μ, namely the ratio of the flame velocity to that for Le = 1. Using (8.2.17) and (8.2.27), Equations (8.2.4) multiplied by dL /ρb Ub =τb /ρb , yield μ

where

dθ d2 θ − 2 = w(θ , ψ), dξ dξ w(θ , ψ) ≡

μ

dψ 1 d2 ψ = −w(θ , ψ), − dξ Le dξ 2

β ϑ+1 ϑ −β(1−θ) ψ e 2(ϑ! )

(8.2.30)

(8.2.31)

is obtained from the dominant order of the reaction term (8.2.10) with ρ ≈ ρb and (1−θ ) = O(1/β). The boundary conditions are given by (8.2.5). The profiles of θ (ξ ) and ψ(ξ ) are shown schematically in Fig. 8.3. For the case Le = 1, the system reduces to that solved previously, giving μ → 1 in the limit β → ∞. Jump Relations across the Reaction Zone Eliminating the reaction term, w, in Equations (8.2.30) yields    2  dψ d θ 1 d2 ψ dθ = 0. + − + μ dξ dξ Le dξ 2 dξ 2

Unburnt gas

(8.2.32)

Burnt gas

Figure 8.3 Schematic representation of reduced temperature and mass fraction profiles for the flame model of Fig. 8.1 for ϑ = 1, μ = Le1/2 ; see (8.2.39).

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Planar Flames

Integrating this relation across the thin (internal) reaction zone gives a jump relation for the solutions in the nonreactive external zones,    1 dψ 0+ dθ + = 0, (8.2.33) μ(θ + ψ) − dξ Le dξ 0− where ξ = 0− and ξ = 0+ denote the upstream and downstream boundaries of the reaction sheet. This is a relation for conservation of energy (thermal energy plus chemical energy, θ + ψ), stating that the convective flux, μ(θ + ψ), is balanced by the diffusive flux, −[dθ/dξ + (1/Le)dψ/dξ ]. For the same reasons as in the preceding section, to leading order, convective flux is negligible compared with diffusive fluxes in the reaction zone. Equations (8.2.30) and (8.2.32) yield −

β ϑ+2 d2 θ ≈ ψ exp[−β(1 − θ )], 2(ϑ! ) dξ 2

1 d2 ψ d2 θ + ≈ 0. Le dξ 2 dξ 2

(8.2.34)

Integrating the second relation across the internal reaction layer, again gives a jump relation  dθ 1 dψ 0+ ≈ 0, (8.2.35) + dξ Le dξ 0− which is the dominant order of (8.2.33) when the convective fluxes are negligible compared with the diffusive flux (reaction layer). This is true at the leading order (1/β)0 of an asymptotic expansion in powers of 1/β. We will see in Section 8.2.4 to what extent, and under which conditions, relation (8.2.35) remains valid at the following order. Analysis for Le = 1 As previously, in the external zones, the reaction term is negligible at all orders in an asymptotic development in powers of 1/β. In the preheat zone, ξ < 0, the solutions to (8.2.30) satisfying the upstream boundary condition (8.2.5) are ξ < 0:

ψ(ξ ) = 1 − eμLeξ ,

θ (ξ ) = θf eμξ ,

(8.2.36)

where θf is the temperature at the origin, ξ = 0, where by definition, the reduced mass fraction, ψ, goes to zero, ψ(ξ = 0) = 0. In the downstream external region, the solutions to (8.2.30) which satisfy the upstream boundary condition (8.2.5) are simply ξ > 0:

θ (ξ ) = 1,

ψ(ξ ) = 0.

(8.2.37)

Relation (8.2.35) then imposes the value θf = 1. Double integration of the second equation in (8.2.34), using the boundary conditions (8.2.37), provides a relation between θ and ψ that is valid to leading order in the reaction zone, (1 − θ ) − ψ/Le ≈ 0. Eliminating ψ in the first equation of (8.2.34) yields an equation for the temperature in the reaction zone: −

β ϑ+1 ϑ d2 θ Le (1 − θ )ϑ exp[−β(1 − θ )]. ≈ 2(ϑ! ) dξ 2

(8.2.38)

This equation can be integrated to obtain the leading order of the temperature gradient at the upstream boundary of the reaction layer. The procedure is identical to that used to obtain

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(8.2.24) from (8.2.21), lim β(1 − θ ) → ∞: dθ /dξ → Leϑ/2 . The unknown mass flux, μ, is found by matching this gradient to dθ/dξ |ξ =0− = μ obtained from (8.2.36) in the preheat zone with θf = 1,  (8.2.39) μ = Leϑ/2 , ⇒ m = ρb 2(ϑ! Leϑ /β ϑ+1 )DTb /τrb , dL ≡ DTu /UL = DTb /Ub =



DTb τb ,

τb ≡ β ϑ+1 τrb /(2Leϑ ϑ! ),

(8.2.40)

where dL is the flame thickness for Le = 1, the transit time being τL ≡ dL /UL = (DTb /DTu )τb = (ρu /ρb )τb . Infinite Lewis Number Equation (8.2.36) shows that the concentration profile becomes stiffer as the Lewis number increases. Very large Lewis numbers require a specific study because the change in concentration of the reactant, ψ(x) is confined inside the reaction layer; see Fig. 8.4. For example, in an initially solid reactive mixture, the molecular diffusivity of the reactant is negligible compared with the heat diffusivity, Le = ∞. This is also the case for combustion of porous media or for polymerisation. In this case and for a large Zeldovich number, as suggested √ by the study of Section 10.2.1 for Le > 1, β(Le − 1) > 4(1 + 3), the combustion front is unsteady. The study of steady planar solution is preliminary to that of pulsating fronts. The reduced coordinates (8.2.29) based on the reference length and time scales of the case Le = 1 are not useful for Le = ∞. It is more convenient to use the dimensional coordinates. For a first-order reaction, ϑ = 1, Equations (8.2.4)–(8.2.11) yield Le = ∞

⇒

−m

d2 θ ρb −β(1−θ) dθ dψ ψe , =m − ρDT 2 = dx dx τrb dx

x = −∞: θ = 0, ψ = 1;

(8.2.41)

x = +∞: θ = 1, ψ = 0.

Using arguments similar to those of Section 8.2.2, the thickness of the reaction zone is dL /β, where dL = ρDT /m is the thickness of the preheat zone. The solutions in the external zones are

1

0

Figure 8.4 Reduced temperature and concentration profiles of the model of infinite Lewis number, (8.2.41).

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Planar Flames

x < 0: ψ = 1,

θ = ex/dL ;

x > 0: ψ = 0,

θ = 1,

(8.2.42)

Inside the reaction layer, mdθ/dx ρDT d2 θ/dx2 . Introducing  ≡ β(1 − θ ) = O(1), the equations of the reaction layer are −

dL d2  ρb dψ ≈ ≈ ψe− . dx β dx2 mτrb

(8.2.43)

Using the reduced internal variable, X ≡ βx/dL , these equations may be rewritten as −

d2  dψ ≈ , dX dX 2

d2  ρDT ρb − ≈ ψe . dX 2 βm2 τrb

(8.2.44)

Integration across the reaction layer with the second boundary condition in (8.2.42) yields  d ρDT ρb  d  d e− . , ≈− ψ ≈− 2 dX dX βm τrb 0 Matching with the preheat zone occurs for  → ∞,   dθ  d  ρDT ρb ≈− ≈ , dL   dx x=0 dX =∞ βm2 τrb where, according to the first equation in (8.2.42), dL (dθ/dx)|x=0 = 1. The flame velocity for infinite Le is thus  (8.2.45) m = ρb DTb /(βτrb ), to be compared with (8.2.25) for ϑ = 1.

8.2.4 Generalisation of the ZFK Method The energy balance can be modified by different phenomena: thermal losses, unsteady effects and/or wrinkling of the flame front (multidimensional effects). In the asymptotic limit of a large activation energy when considering wavelengths and/or characteristic times of the disturbances larger than the flame thickness dL and the transit time τL , the reaction zone remains quasi-planar and quasi-steady. But due to the modifications of the external layers (preheated zone and burnt gas), the temperature at the reaction layer, θf , (the flame temperature) is no longer equal to the adiabatic flame temperature, θf = θb = 1. In order to study these phenomena with the ZFK model, the asymptotic method for large activation energies is easily generalised under two conditions:[1] (i) The difference between the local flame temperature and the adiabatic temperature must remain small. To be more precise, the reaction rate in (8.2.30)–(8.2.31) must remain of order β ϑ+1 ψ in the limit β → ∞ or, in other words, β(1 − θf ) = O(1). This generally requires that the Lewis number remains sufficiently close to unity, β(Le − 1) = O(1).

[1]

Joulin G., Clavin P., 1979, Combust. Flame, 35, 139–153.

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(ii) The gradient of the reduced temperature must remain small in the burnt gas. The order of magnitude of the gradient must not be greater than 1/β; ξ > 0: dθ /dξ ≤ O(β −1 ). For nonadiabatic combustion, the later condition corresponds to relatively small heat losses. Under these conditions, the asymptotic method, in the limit β → ∞, reduces to solving the nonreactive equations in the outer zones, and then treating the reaction layer as a discontinuity with two jump relations that must be satisfied by the external solutions. The first jump relation is (8.2.35), the second is a kinetic relation that will be obtained now. Kinetic Relation For the moment we will ignore the constraint concerning the Lewis number, β(Le − 1) = O(1). We will come back to this point later. Using the reduced coordinates (8.2.29), the reaction layer is centred on ξ = 0 and has a thickness of order 1/β. By definition, the origin is the point at which the external solution reaches zero, ψ(ξ , τ ) = 0. Using an internal variable, X ≡ βξ , of order unity in the reaction layer, δξ = O (1/β), δX = O(1), d/dξ = βd/dX, an asymptotic expansion for the reduced temperature and mass fraction is introduced inside the reaction layer,     !1 (X) !2 (X) 1 (X) 2 (X) 1 1 , ψ= , − + +O +O θ = θf − β β β2 β3 β2 β3 (8.2.46) where the quantities X, β(1 − θf ), i (X) and !i (X) are all of order unity, i = 1, 2. There is an additional difficulty in the presence of heat losses when ϑ > 1. Traces of reactants may appear in the burnt gases due to a partial quenching,[2] X → ∞: ! = 0. The expansion of ψ may be different from (8.2.46). We will not consider this case here. In terms of the temperature difference θf − θ , Equations (8.2.34) take the form −

d2 θ β ϑ+1 −β(1−θf ) ϑ −β(θf −θ) e ≈ ψ e , 2(ϑ! ) dξ 2

d2 (θ − θf ) 1 d2 ψ + ≈ 0, Le dξ 2 dξ 2

(8.2.47)

where the relation e−β(1−θ) = e−β(1−θf ) e−β(θf −θ) has been used in the first equation. Thus, to leading order, the equations in the reaction layer become d2 1 1 e−β(1−θf ) !1ϑ e−1 , = 2(ϑ! ) dX 2

−

d2 1 1 d2 !1 + = 0. Le dX 2 dX 2

(8.2.48)

According to the expansion dθ/dξ = d1 /dX +β −1 d2 /dX + · · · , a downstream gradient dθ/dξ of the order 1/β (condition (ii)) will not change the dominant order of the gradient in the reaction layer, where both 1 ≈ β(θf − θ ) and d1 /dX are of order unity. The downstream boundary conditions for 1 and !1 are thus X → +∞:

[2]

1 = 0,

!1 = 0,

d1 /dX = 0,

Joulin G., Clavin P., 1976, Acta Astronaut., 3, 223–240.

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d!1 /dX = 0.

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Planar Flames

Double integration of the second line of Equation (8.2.48), along with the above boundary conditions at X → +∞, gives 1 = !1 /Le. Substituting this result into the first line of Equation (8.2.48), this latter can then be integrated to obtain the value of the temperature gradient at X → −∞ (the upstream boundary of the reaction zone), d1 /dX|X=−∞ = Leϑ/2 e−β(1−θf )/2 . This gradient can now be matched with the gradient, dθ/dξ , at the downstream limit of the preheat zone to obtain dθ /dξ |ξ =0− ≈ Leϑ/2 e−β(1−θf )/2 .

(8.2.49)

This relation is valid to leading order in the limit β → ∞ and shows that dθ/dξ = O(1) when β(1−θf ) = O(1) (condition (i)). It is a kinetic equation, relating the change of energy release rate with combustion temperature θf to the heat flux that is evacuated upstream by thermal conduction to heat the fresh gas. Jump Conditions The quantity β(1 − θf ) in (8.2.49) is obtained from the jump condition (8.2.35) which remains valid up to order 1/β when conditions (i) and (ii) are fulfilled. This is because the difference between total thermal flux entering and leaving the reaction zone in (8.2.33) is  0+ of second order, μ(θ + ψ) 0− = O(1/β 2 ), when the Lewis number is sufficiently close to unity, β(Le − 1) = O(1). This can be shown by calculating the two leading orders in the expansion of Equation (8.2.32) in the reaction zone. Using the expansions (8.2.46), along with Le = 1 + l1 /β + O(1/β 2 ), one obtains at order β, O(β):

−

d2 !1 d2 1 + = 0, dX 2 dX 2

and at order unity (following order):   2   d!1 d 2 d!1 d1 d2 !2 + − − = 0. O(1): μ − + − l1 dX dX dX dX 2 dX 2

(8.2.50)

(8.2.51)

Spatial integration of (8.2.50) across the reaction zone leads to the equality !1 = 1 , showing that the first term is zero in (8.2.33) and is of second order in (8.2.51). Spatial integration of (8.2.51) across the reaction zone then shows that the jump relation (8.2.35) is valid up to order 1/β. Summary In summary, when conditions (i) and (ii) are satisfied, an asymptotic analysis of the internal reaction zone shows that, for large values of the activation energy, β  1, the ZFK flame can be studied by treating the reaction zone as a thin discontinuity along with the jump conditions (8.2.35) and (8.2.49). These relations have been obtained by using the reduced coordinate (8.2.29), where the reference length and time are that of the planar flame for Le = 1. It is more convenient in the following to use that for Le = 1, given in (8.2.39)– (8.2.40). With these definitions the jump conditions across the reaction layer are obtained from (8.2.35) and (8.2.49) by the substitution ξ → ξ Leϑ/2

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8.3 Reaction–Diffusion Waves



dθ 1 dψ + dξ Le dξ

0+ ≈ 0, 0−

393

 dθ  ≈ e−β(1−θf )/2 . dξ ξ =0−

(8.2.52)

They are both valid for β(1 − θf ) = O(1) and the first is valid up to the first order 1/β. 8.3 Reaction–Diffusion Waves The objective of this section is to highlight the specificity of the thermal propagation of flames, compared with other reaction–diffusion waves, such as those found in population dynamics and/or in biophysics.[1] We will limit ourselves to a planar geometry. Equations (8.2.30)–(8.2.31) with Le = 1 are a particular case of reaction–diffusion waves of a scalar field θ (x, t) propagating in an infinite medium. The generic prototype obeys a diffusion equation coupled to a nonlinear production term, w(θ ), ∂ 2θ ∂θ − 2 = w(θ ), θ ≥ 0, x = ±∞: θ = 0. (8.3.1) ∂t ∂x The initial condition is a stationary state: θ = 0, w(θ = 0) = 0. This state can either be unstable, dw/dθ |θ=0 > 0, or metastable, dw/dθ |θ=0 < 0. We will further suppose the existence of a second stationary state: θ = 1, w(θ = 1) = 0, which is a stable equilibrium state, dw(θ )/dθ |θ=1 < 0, with (θ = 1) ≤ (θ ) ∀θ = 1, where (θ ) is the potential defined by d/dθ = −w(θ ). The stationary states, w = 0, correspond to the extrema of (θ ), d/dθ = 0. The equilibrium state, θ = 1, is the absolute minimum of (θ ). A metastable state corresponds to a local minimum, and an unstable state to a local maximum; see Fig. 8.5. For the homogeneous case, the initial evolution of an infinitesimal perturbation is found by linearising (8.3.1), θ = θo + θ  , ∂θ  /∂t = ωo θ  , with ωo ≡ ∂w/∂θ |θo = − ∂ 2 /∂θ 2 θ . If the initial state is stable or metastable, ωo < 0, the perturbation decreases o exponentially, and if it is unstable, ωo > 0, the perturbation increases exponentially. (a)

Metastable state

(b)

Stable equilibrium state

Unstable state

Stable equilibrium state

Figure 8.5 (a) Potential corresponding to a metastable state at θ = 0 and an equilibrium state at θ = 1. (b) Potential corresponding to an unstable state at θ = 0 and an equilibrium state at θ = 1. [1]

Murray J., 1993, Mathematical biology, Biomathematics, vol. 19. Springer.

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394

Upstream unstable state

Downstream equilibrium state

Figure 8.6 Reaction–diffusion wave propagating from right to left at speed μ, transforming an unstable state, θ = 0, into an equilibrium state, θ = 1.

8.3.1 Phase Trajectories When the initial state is unstable, an inhomogeneous local perturbation will give rise to two fronts that propagate in opposite directions, progressively transforming the unstable system to a stable equilibrium state. The same will also happen if the initial state is metastable provided that the initial perturbation is of sufficient amplitude and size. After an initial transient, these fronts eventually reach a stationary regime described by a propagating wave equation θ (x, t) → θ (ξ ), ξ ≡ x ± μt. Consider a wave propagating from right to left, ξ = x + μt, with an initial unstable or metastable state upstream, ξ = −∞, and an equilibrium state downstream, ξ = +∞; see Fig. 8.6. Equation (8.3.1) for θ (ξ ) then takes the form of the plane wave equation previously studied (see (8.2.30)) with Le = 1, μ

d2 θ dθ − 2 = w(θ ), dξ dξ

(8.3.2)

with the following boundary conditions, ξ = −∞:

θ = 0, w = 0,

ξ = +∞:

θ = 1, w = 0.

(8.3.3)

When the production term is positive for all values of θ between 0 and 1, w(θ ) > 0, 0 ≤ θ ≤ 1, the solution θ (x, t) of (8.3.1) can never become negative if the initial perturbation is positive, θ (x, t = 0) ≥ 0 ⇒ θ (x, t > 0) ≥ 0. The diffusion term tends to flatten and spread the initial profile, and the production term tends to increase θ . For the case of an unstable initial state we will limit our attention to positive production terms, w(θ ) > 0, 0 ≤ θ ≤ 1, and we will look only for positive solutions of (8.3.2). We may question the physical pertinence of a thermal wave propagating in an unstable medium, dw/dθ |θ=0 > 0 since the slightest fluctuation will cause the whole upstream fluid to evolve in time. Historically, this problem was first introduced in biophysics[1] to model the evolution of a population totally absent in the initial state, for which there can be no fluctuations of θ . In combustion, in the limit of a large activation energy, the unstable character of the initial state is not physically relevant and should be considered as neutral; see Section 8.2.1. [1]

Murray J., 1993, Mathematical biology, Biomathematics, vol. 19. Springer.

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Equations (8.3.2)–(8.3.3) can have a large variety of solutions depending on the form of the potential (θ ). Several questions arise: what are the values of μ for which there is a solution to (8.3.2)–(8.3.3), and how many solutions are there? Which solutions are the propagating long-time asymptotic solutions to (8.3.1)? For an initially metastable or neutral state, what are the critical conditions for an initial perturbation to achieve a stationary propagating state? Phase Space The easiest way to obtain the answers to these questions is to treat the problem as a dynamical system and examine the solutions in phase space. Let us write X ≡ θ , Y ≡ μdθ/dξ . Equation (8.3.2) then takes the form of a dynamical system in two dimensions, dX/dξ = Y/μ,

dY/dξ = μ[Y − w(X)].

(8.3.4)

The boundary conditions (8.3.3) correspond to singular points in phase space, (X = 0, Y = 0) and (X = 1, Y = 0), where the right-hand side of equations (8.3.4) is null. This system of equations can also be written as a first-order differential equation: dY/dX = μ2 [Y − w(X)]/Y.

(8.3.5)

For each value of the constant μ, this equation describes an ensemble of trajectories in phase space. Since the equation is first order, the trajectories cannot cross, and each point (X, Y) belongs to a single trajectory, except for two singular points, namely fixed points of (8.3.4), dX/dξ = dY/dξ = 0, at which dY/dX is of the form 0/0. For each value of μ, a solution to (8.3.2) that also satisfies the boundary conditions (8.3.3) corresponds to a trajectory Y(X), solution to (8.3.5), that goes through the two singular points (X = 0, Y = 0) for ξ → −∞ and (X = 1, Y = 0) for ξ → +∞. The behaviour of these trajectories close to the singular points can be found by linearising Equations (8.3.4) around these points, d2 Y/dξ 2 − μdY/dξ + YwX = 0, where wX is a shorthand notation for ∂w/∂θ|θ=0,1 . The linearised solutions are linear combinations of eξ r+ and eξ r− corresponding to two eigenvectors δY = k± δX,      2   2 2 k± ≡ μr± = μ 1 ± 1 − 4wX /μ /2. (8.3.6) r± = μ ± μ − 4wX /2, The form of the solutions to (8.3.2)–(8.3.3) depends on the form of the potential (θ ). They can be grouped into different classes depending on the topology of the phase trajectories. Coexistence of Two Phases in Equilibrium We start by examining the case of two phases in equilibrium, separated by an unstable state, (θ = 0) = (θ = 1) < (θ ), ∀θ , in the range 0 < θ < 1; see Fig. 8.7. We first note that the two singular points (X = 0, Y = 0) and (X = 1, Y = 0) are saddle points in phase space: wX ≡ ∂w/∂θ |θ=0,1 < 0, and so r+ and r− are real with r+ > 0, r− < 0. Close to these points, for a given value of μ, there can be only one trajectory Y(X) in the domain 0 < θ < 1, dθ/dξ > 0, that goes through the singular point. At the point

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Stable equilibrium state

(b)

Stable equilibrium state

Stable equilibrium state

Stable equilibrium state

Figure 8.7 (a) Reaction potential with two coexisting equilibrium states. (b) Corresponding reaction rate.

Saddle

Saddle

Equilibrium state

Metastable or equilibrium state

Figure 8.8 Saddle point in phase space corresponding to the existence of two coexisting equilibrium states, or the propagation of an equilibrium state in a metastable state. The arrows show the direction of the trajectories as ξ goes from +∞ to −∞.

(X = 0, Y = 0) this trajectory is tangent to k+ = μr+ > 0, and at the other equilibrium point, (X = 1, Y = 0), it is tangent to k− = μr− < 0; see Fig. 8.8. For these conditions, the problem defined by Equations (8.3.2)–(8.3.3) is easily solved. There is a solution θ (ξ ) having zero velocity, μ = 0, −d2 θ/dξ 2 = w(θ ), and which is continuous between the two equilibrium states θ = 0 at ξ = −∞, and θ = 1 at ξ = +∞,  (8.3.7) dθ/dξ = 2[(θ ) − (0)]. This solution is unique, as we will see below. It is a simple model for a system with two coexisting phases, θ = 0 and θ = 1. The families of trajectories in phase space depend on only one parameter, μ. For an arbitrary value of μ, there is no possible chance that a trajectory passing through one singular point will also pass through the other. This can happen only for a particular value of μ. Since the solution (8.3.7) for μ = 0 does this, then this is the unique solution, by virtue of the topological argument given above.

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Propagation Through a Metastable Medium In the case where the propagation of the front transforms a metastable state into an equilibrium state, the solution can be obtained by progressively deforming the potential. The singular points (X = 0, Y = 0) and (X = 1, Y = 0) keep their character of saddle points, and the shape of the trajectories around these points remains qualitatively the same; see again Fig. 8.8. By the same topological argument as above, it can be seen that the solution is unique, and it is not difficult to show that the sign of μ is such that the equilibrium state propagates into the metastable state.

8.3.2 Unstable Medium, KPP Scenario The case of the transformation of an unstable state into a stable one is quite different. It was studied at the end of the 1930s,[1,2] in the context of biophysics, for the particular case of a quadratic reaction term of the form w(θ ) = θ (1 − θ ). In this case Equation (8.3.1) is called the Fisher equation. The solution was later generalised to other forms of reaction satisfying w(θ ) > 0 for 0 < θ < 1. The equilibrium point (X = 1, Y = 0) is still a ‘saddle’, meaning that, for a given μ, there is a single trajectory leaving this point with θ  0, as in Fig. 8.7. However, the unstable state (X = 0, Y = 0), ∂w/∂θ|θ=0 = 0, can be either a ‘node’ or a ‘spiral point’. If the propagation velocity, μ, is large, μ2 > 4 ∂w/∂θ |θ=0 , then r+ and r− are real and positive (r+ , r− > 0) (see Fig. 8.9a), and the singular point is a ‘node’. In the opposite case, the eigenvalues are complex, and the singular point is a ‘spiral point’; see Fig. 8.9b. The critical value of μ, called μKPP in tribute to the work of Kolmogorov et al.,[2] ene rate Pa tra rtic je ul ct ar or y

(a)

Non

deg

(b)

e rat

Saddle

Focus

e en

D

Saddle

eg

Unstable state

Equilibrium state

Unstable state

Equilibrium state

Figure 8.9 Phase trajectories for unstable initial states. (a) Propagation velocity μ > μKPP . The unstable state is a node with two principal directions, of which one is degenerate. These two directions collapse for μ = μKPP . (b) Propagation velocity μ < μKPP . The unstable state is a node with two principal directions that are complex and the singular point is a spiral point. The arrows are oriented in the direction ξ = +∞ → ξ = −∞.

[1] [2]

Fisher R., 1937, Annals of Eugenics, 7, 355–369. Kolmogorov A., et al., 1937, Bjul. Moskovskovo Gos. Univ, 1(7), 1–72.

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 μKPP ≡ 2 dw/dθ |θ=0 .

(8.3.8)

corresponds to a marginal solution of the Fisher equation, as explained below. For μ > μKPP the two eigenvalues are real and positive, r+ > r− > 0, so that there are two distinct eigenvectors at the node, both with positive slopes, k+ = μr+ > k− = μr− > 0. According to (8.3.4), the phase trajectories in the neighbourhood of the node take the form δX = A+ eξ r+ + A− eξ r− ,

δY = k+ A+ eξ r+ + k− A− eξ r− ,

where A+ and A− are constants of integration. For a given value of μ, the linearised solutions around the initial unstable state form a continuous family controlled by a single parameter, A− /A+ . In the limit ξ → −∞, the term eξ r+ is negligible compared with eξ r− . Therefore the direction δY = k− δX is degenerate in the sense that there are an infinite number of phase trajectories indexed by the parameter A− /A+ = 0 that are tangent to to this direction when ξ → −∞. But there is a unique phase trajectory corresponding to A− = 0, which reaches the node with the slope k+ ; see Fig. 8.9a. For the critical value μ = μKPP , the two eigenvectors (directions) collapse onto the same direction k+ = k− = μ2KPP /2, and for smaller values, μ < μKPP , the vectors are complex. The unstable singular point becomes a ‘spiral point’ and the trajectories oscillate in a spiral around (X = 0, Y = 0); see Fig. 8.9b. These oscillating solutions are physically unacceptable since they take X (≡ θ ) through negative values, in contradiction with the solutions θ (x, t) of (8.3.1), which remain positive at all times when the initial condition is positive. Thus there is no physically acceptable solution for velocities lower than the critical velocity μ = μKPP . For μ > μKPP , a solution is represented by a phase trajectory linking the saddle to the node. Since there are many phase trajectories that reach the node through the degenerate direction in Fig. 8.9a, it may be argued that a solution exists for all values of μ greater than μKPP . This result was first obtained by Kolmogorov et al.,[1] called KPP in the following, for the particular case of the Fisher equation, w(θ ) = θ (1 − θ ). More precisely, they proved the following results: (i) There are as many solutions to Equations (8.3.2)–(8.3.3), satisfying θ ≥ 0, as there are values of μ greater than μKPP . (ii) The long-time solution for an initial condition θ (x, t = 0) > 0 tends towards the marginal solution with μ = μKPP = 2. A simple argument in favour of point (ii) can be found by linearising (8.3.1) around θ = 0: ∂θ/∂t − ∂ 2 θ /∂ξ 2 = ωo θ ,

where

ωo ≡ ∂w/∂θ |θ=0 > 0.

 Z(ξ , t)e+ωo t

(8.3.9)

Let θ (ξ , t) ≡ be a solution to (8.3.9), then Z(ξ , t) is a solution of the diffusion √ 2 2 equation ∂Z/∂t − ∂ Z/∂ξ 2 = 0, whose Green’s function is the Gaussian Z = e−ξ /4t / t. The solution to the linear (8.3.9) for θ must then have the form: θ ∝ exp[−ξ 2 /4t + ωo t − ln(t)/2]. [1]

Kolmogorov A., et al., 1937, Bjul. Moskovskovo Gos. Univ, 1(7), 1–72.

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The term ln(t) becomes negligible as time increases, and if the solution is stationary, then the first two terms must balance, ξ 2 = 4ωo t2 , implying that the stationary long-time solution propagates with the velocity 2 ωo , in agreement with (8.3.8). This argument shows that the KPP solution is selected by the leading edge of the front, close to the initial state θ = 0, in contrast to the ZFK solution. The KPP result can be generalised to other reaction terms, w(θ ) = θ (1−θ ), that have the same general properties, w(θ ) > 0 for θ ∈]0, 1[ , provided that they are not too nonlinear. By this last statement, we mean that the slope of the reaction rate at the leading edge, dw/dθ |θ=0 , is not too small compared with the modulus of the slope at the trailing edge dw/dθ |θ=1 , or in other terms, that the growth time of the instability in the unstable state is not too long compared with the relaxation time towards the equilibrium state. When these conditions are fulfilled, the propagation speed of the planar front at long times is μ = μKPP given by (8.3.8). The explanation[2,3,4] of the striking difference between the ZFK solution (8.2.23)– (8.2.25)   1 w(θ )dθ μZFK ∝ 2 0

and the KPP solution (8.3.8), both obtained by Russian scientists in 1938 for reaction– diffusion waves travelling in unstable media remained elusive for a long time.

8.3.3 The KPP-ZFK Transition The comparison of the KPP and ZFK solutions suggests that a transition should appear in the solutions to (8.3.2)–(8.3.3) when the stiffness of the nonlinear term w(θ ) is increased. Shooting Method This transition has been

investigated[2]

for a reaction term of the form

w(θ , β) = (β 2 /2)θ (1 − θ )e−β(1−θ) .

(8.3.10)

The parameter β controls the nonlinearity and allows a continuous transition from √ a weakly nonlinear term of the KPP type for β 1, w(θ ) ≈ (β 2 /2)θ (1 − θ ), μKPP = 2βe−β/2 , to a strongly nonlinear term of the ZFK type for β  1, limβ→∞ μ = μZFK = 1, obtained by the same asymptotic method as in (8.2.23)–(8.2.25). The KPP-ZFK transition can be observed through the change of the phase trajectory for different values of β. The trajectories are easily obtained using a numerical shooting method, integrating (8.3.5) ‘backwards’ from the saddle (X = 1, Y = 0), where the relevant trajectory is unique for fixed values of β and μ, with an initial slope k− given by (8.3.6), |k− | < |dw/dθ |θ=1 . [2] [3] [4]

Clavin P., Li˜nan A., 1984, In M. Velarde, ed., Nonequilibrium cooperative phenomena in physics and related fields, NATO ASI Series B. Physics, vol. 116, 291–338, Plenum Press. Zeldovich Y., et al., 1985, The mathematical theory of combustion and explosions. New York: Plenum. Ostriker J., ed., 1992, Selected works of Ya.B. Zeldovich, vol. I, p. 193. Princeton University Press.

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Figure 8.10 (a) Phase trajectory for moderate nonlinearity and large velocity. (b) Nonphysical phase trajectory for weak nonlinearity and velocity < μKPP .

It is then sufficient to follow the trajectory to see if, and how, it reaches the node (X = 0, Y = 0). Trajectories that go through negative values of X before reaching the node are nonphysical and must be rejected. We start with a large value for μ and progressively reduce it. According to (8.3.5), the trajectory is simply Y = w(X) for μ = ∞. For large values of μ, the trajectory remains close to Y = w(X), and always reaches the node on the positive X side, through the degenerate direction k− > dw/dθ |θ=1 > 0; see Fig. 8.10a. The slope of the trajectory is zero at the point of intersection with the curve Y = w(X), in agreement with (8.3.5). The results may be summarised as follows. • For all values of β there is a continuum of values of μ leading to satisfactory solutions in the range 0  θ  1. This continuum has a lower bound μ  μmin (β). There is no solution to (8.3.2)–(8.3.3) satisfying 0  θ  1 for μ < μmin . An example of a nonphysical solution is shown in Fig. 8.10b. However, there is always a physical solution for values of μ  μmin . ∗ • When β is varied, the lower bound μmin undergoes a transition for a critical value β , ∗ ∗ where β = 1.64 √ for the reaction model (8.3.10). For β  β , the lower bound is μmin = μKPP ≡ 2βe−β/2 . However, for β > β ∗ , the lower bound, μmin , is greater than μKPP . Furthermore, limβ→∞ μmin = μZFK = 1.

• For all values of β, the phase trajectories corresponding to μ > μmin reach the node (X = 0, Y = 0), tangent to the ‘degenerate’ direction, δY = k− δX. ∗ ∗ • The marginal phase trajectory (μ = μmin ) changes nature for β = β . For β  β = 1.64, the lower bound μmin = μKPP corresponds to the fusion of the two eigenvectors, k+ = k− transforming the node into a ‘focus’; see Fig. 8.9b. For β > β ∗ , μmin > μKPP , there are two distinct eigenvectors k+ = k− at the node, and the marginal phase trajectory is different from all the nonmarginal ones (μ > μmin ). It is the only phase trajectory that arrives at the node tangent to the nondegenerate direction δY = k+ δX; see Fig. 8.9a. The non physical trajectories (μ < μmin ) again arrive tangent to the degenerate direction δY = k− δX, but only after passing through negative values of θ ; see Fig. 8.11b.

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401

(b)

–0.2 –0.2

Figure 8.11 Phase trajectories for β = 5 and for μ very close to μmin ≈ 1.0825. (a) μ = 1.085. (b) Details of the trajectories near the node; μ = 1.080 is just below the lower bound (non physical solution) and μ = 1.085 and 1.090 are just above the lower bound.

Figure 8.12 Piecewise linear reaction model of (8.3.11), drawn here for  = 0.2, h = 0.2.

An Analytical Solution to the KPP-ZFK Transition The transition in the lower bound μmin can be found analytically for a particularly simple model.[1] This model is characterised by a reaction rate, w(θ ), that is piecewise linear with a discontinuity at θ = 1 − ; see Fig. 8.12. It has two parameters, h > 0 and , 0 <  < 1, that control the transition to the high-temperature region, θ = (1 − ), and the maximum of the reaction rate, h/: =θ : 0 (1 −  2 ) :

μmin = μ∗ > 2,

h < (1 −  2 ) :

μmin = μKPP ≡ 2.

(8.3.13)

Note that, in the KPP scenario, h < (1 −  2 ), the marginal phase trajectory is no longer linear in the low-temperature portion (0 < X < 1 − ), as shown in Fig. 8.13b. (a)

(b)

Figure 8.13 (a) Marginal phase trajectory at the ZFK-KPP transition (h = 1 −  2 ) for  = 0.2, h = 0.96, μ = 2. The marginal trajectories are also piecewise linear for h > 1 −  2 . (b) Marginal phase trajectory for  = 0.2, h = 0.2, μ = 2, illustrating that the low-temperature portion is no longer linear when h < 1 −  2 .

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403

Concluding Remarks In conclusion, the propagation of a reaction–diffusion wave in an unstable medium has a different nature according to the degree of nonlinearity of the reaction term. If the reaction term is strongly nonlinear, so that the linear growth rate of the instability is long compared with the relaxation time in the vicinity of the final equilibrium state (as for an Arrhenius reaction law with a large activation energy), the propagation speed is controlled by the behaviour of the reaction term close to the equilibrium state. However, if the reaction term is only weakly nonlinear (generalising the case of a quadratic reaction term) such that the growth rate in the unstable state is of the same order as the relaxation rate towards the stable state, the propagation speed is completely controlled by the linear growth rate of the instability in the initial state. For a large activation energy, the long-time solution to the problem tends towards the plane wave solution found by the asymptotic method of ZFK. This clearly emphasises the ‘fragility’ of the linear arguments used above to justify the selection of the lower limit, μKPP ; see (8.3.9). The argument which is right for ‘weak nonlinearity’ is false when the growth rate of the unstable medium is much smaller than the damping rate of the equilibrium state. The ZFK solution also clearly illustrates that the slow low-temperature chemistry, which characterises the so-called cool flames of certain hydrocarbons, has no effect on the propagation speed of classical high-temperature flames (Tb  1000K), which is entirely controlled by the fast high-temperature chemistry. 8.4 Chain Branching and Flame Propagation As mentioned in Section 5.2, the self-amplification of the reaction rate in combustion is not only thermal but also involves autocatalytic reactions, called chain-branching reactions. The purpose of this section is to extend the asymptotic analysis to include their effect on flame propagation. In order to shed light on the role of a multiple-step chemical scheme and on intermediate species, such as radicals that are highly reactive and have a large molecular diffusion rate, attention is limited to the simplified framework of the two-step chain-branching models (5.1.1). The chain-branching reaction (B) is thermally sensitive, E/kB T  1. In this reaction an intermediate species X attacks a reactant species R to produce more X at a rate proportional to the X concentration, cX . The chain-breaking reaction (R) is a first-order reaction that converts the intermediate species into products P and heat Q by collision with any species M. Its rate is usually larger than the chainbranching rate and not thermally sensitive. For simplicity, the chain-branching step (B) is treated as being nonexothermic. This limitation can be easily removed in the analysis. The analysis is performed in the limit E/kB T → ∞, considering the quantities BB and BR in (5.1.1) to be constant for simplicity. The basic assumption is that the quantity (BB /BR )e−E/kB T varies strongly with the temperature. The result will be also valid for moderate values of E/kB T when the ratio (BB /BR ) follows a power law like T m , provided that m + (E/kB T)  1. A consequence of the thermal sensitivity of (B) is that its reaction rate ωB is concentrated in a thin reaction zone at a temperature T ∗ , Tu < T ∗ < Tb , where

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0

Figure 8.14 Profiles of reduced concentrations and temperature in the two-step model.

the concentration of the reactant R is small and vanishes on the downstream side; see Fig. 8.14. In contrast to the crossover temperature, defined in (5.2.7), the temperature of the thin reaction zone T ∗ is here an unknown; it will be obtained by asymptotic analysis in the limit E/kB T → ∞ (see Section 8.4.3). The model is sufficiently simple to be fully studied by analytic methods. Its M-version (5.1.2) (the exothermic reaction is frozen in the preheat zone, T < T ∗ ) may be used for methane flames. The H-version (5.1.3) is more representative of hydrogen flames, although the chain-breaking step should be modified; see Section 5.1. 8.4.1 Formulation Introducing the characteristic reaction times τR and τB , 1/τR ≡ nBR , 1/τB ≡ nBB , and the molecular weights Mi , Equations (8.1.1)–(8.1.2) for a flame sustained by (5.1.1) take the form   d dT (Q/MX ) YX dT − h, (8.4.1) ρDT =ρ m dx dx dx cp τR   d ρ2 dYR e−E/kB T dYR − ρDR =− m YX YR , (8.4.2) dx dx dx nMX τB    ρ d dYX e−E/kB T 1 dYX − ρDX = ρYX YR − h , (8.4.3) m dx dx dx nMR τB τR where m = ρu UL and h is defined in (5.1.2) or (5.1.3). The boundary conditions are x → −∞:

T = Tu ,

YR = YRu ,

YX = 0,

(8.4.4)

x → +∞:

T = Tb ,

YR = 0,

YX = 0.

(8.4.5)

The flame temperature is obtained by combining the three Equations (8.4.1)–(8.4.3) to eliminate the reaction rates and form the group cp T + Q[(YR /MR ) + (YX /MX )], cp (Tb − Tu ) = (Q/MR )YRu .

(8.4.6)

As in the ZFK model, the mathematical problem is well posed in the limit E/kB T → ∞. Introducing the reduced temperature, θ ≡ (T − Tu )/(Tb − Tu ), the reduced mass fractions

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405

√ ψR ≡ YR /YRu , ψX ≡ (MR /MX )(YX /YRu ), the dimensionless coordinate ξ ≡ x/ DTu τR √ and dimensionless flame velocity μ ≡ UL / (DTu /τR ), Equations (8.4.1)–(8.4.5) yield dθ ρ d2 θ − 2 = ψX h, dξ ρu dξ 1 d2 ψR ρ − E (1− 1 ) − = − ψX ψR Be kB T T ∗ , 2 LeR dξ ρu    1 d2 ψX ρ − kE T1 − T1∗ B ψ − = ψ Be − ψ h , X R X LeX dξ 2 ρu μ

dψR dξ dψX μ dξ μ

(8.4.7) (8.4.8) (8.4.9)

ξ → −∞:

θ = 0,

ψR = 1,

ψX = 0,

(8.4.10)

ξ → +∞:

θ = 1,

dψR = 0,

ψX = 0,

(8.4.11)

where we have introduced the quantity −k

B ≡ (τR /τB )XRu e

E ∗ BT

;

(8.4.12)

XRu ≡ ρYRu /(MR n) is the molar fraction of reactant R in the initial mixture. Assuming that all diffusion coefficients verify the relation ρ 2 D = cst., the term ρ/ρu in the right-hand side of (8.4.7)–(8.4.9) can be eliminated by using a mass-weighted coordinate,  x ρ(x )  1 dx , (8.4.13) ξ=√ ρu DTu τR √ instead of x/ DTu τR . This system of equations is considered in the following. The analysis is detailed here for the case (5.1.2) where the reactions are frozen in the preheat zone. The case (5.1.3), h = 1 ∀T, is treated in a similar way. The eigenvalue μ and the temperature T ∗ are unknown. In addition to the Lewis numbers LeR and LeX , three nondimensional parameters are involved: Tb /Tu ,

E/kB Tb ,

(τR /τB )XRu . (8.4.14)   When E/kB T ∗  1, the reduced reaction rate ψR exp − kEB ( T1 − T1∗ ) is concentrated in a thin chain-branching reaction layer; see Fig. 8.14. This layer may be considered as a discontinuity for the gradients of mass fraction, dψX /dξ and dψR /dξ . The asymptotic analysis in the limit β∗ ≡

E → +∞ kB T ∗

(8.4.15)

is an extension of the ZFK analysis of Section 8.2.2. Choosing the origin ξ = 0 inside the chain-branching reaction layer, the flame structure consists of the three following zones: • Two external zones: the inert preheated zone, ξ < 0, and the downstream zone ξ > 0 where the chemical heat is released by the first-order chain-breaking reaction (R).

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• A thin layer where the chain-branching reaction (B) proceeds with a small value of ψR , as in the ZFK model. In this layer, T and ψX are close to values T ∗ and ψX∗ , which are outcomes of the analysis.

8.4.2 Analysis Choosing the origin to be the point at which the external profile ψR (ξ ) meets the ξ -axis, the solutions to (8.4.7)–(8.4.10) in the preheated zone are ξ < 0:

ψR = 1 − eμLeR ξ ,

θ = θ ∗ eμξ ,

ψX = ψX∗ eμLeX ξ .

(8.4.16)

In the downstream zone, the solutions to (8.4.7)–(8.4.11) are ξ > 0:

where

ψR = 0, χ≡

ψ∗ dθ = μ(θ − 1) + X e−χ ξ , dξ χ

ψX = ψX∗ e−χ ξ ,

(8.4.17)

  μ2 + 4/LeX − μ LeX /2 > 0.

(8.4.18)

Assuming continuity of ψX (ξ ), to leading order the constant ψX∗ is the same in (8.4.16) and (8.4.17). Solutions in the Thin Chain-Branching Layer (Internal Layer) To leading order in the limit (8.4.15), integration of (8.4.7) across the inner layer shows that the heat flux is constant inside this layer, where ξ = O(1/β ∗ ), and to first order the temperature varies linearly, [dT/dξ ]0+ 0− ≈ 0,

dT/dξ ≈ μT ∗ ,

(T − T ∗ )/T ∗ ≈ μξ .

(8.4.19)

Chain-branching dominates chain-breaking inside this thin inner layer, and diffusion also dominates convection, as in the ZFK analysis, giving, according to (8.4.8)–(8.4.9), 1 d2 ψX 1 d2 ψR −E ≈− ≈ ψX∗ ψR Be kB 2 2 LeR dξ LeX dξ



1 1 T − T∗



.

(8.4.20)

This gives the jump of the external solutions across the inner layer,  1 dψR 1 dψX 0+ + = 0, LeR dξ LeX dξ 0−

(8.4.21) −

E



1

−

1



∗

valid to leading order in the limit (8.4.15). According to (8.4.19), e kB T T ∗ ≈ eμβ ξ , and Equation (8.4.20) for ψR , written in terms of the internal variable η ≡ μβ ∗ ξ , takes the form LeR ψX∗ B d2 Y η ∗ = Ye , where Y ≡ β ψ and  ≡ . R dη2 β ∗2 μ2

(8.4.22)

The boundary conditions are obtained by matching the mass flux to the external solutions (8.4.16) and (8.4.17),

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dY → −LeR , dη

η → −∞:

η → +∞: Y → 0.

407

(8.4.23)

The problem (8.4.22)–(8.4.23) has been already encountered in the studies of diffusion flames[1] and premixed hydrogen flames.[2] Using the variable τ = 2e(η+ln )/2 ∈ [0, +∞[, Equation (8.4.22) reads 1 dY d2 Y + − Y = 0. 2 τ dτ dτ The solution satisfying τ → +∞: Y → 0 is the modified Bessel function of zeroth order,[3] Ko (τ ), multiplied by a constant, τ → 0+ :

Ko (τ ) = −[ln(τ/2) + γE + O(τ 2 )],

where γE ≈ 0.577 is Euler’s constant. The solution satisfying the two boundary conditions (8.4.23) thus behaves as   η 1 e , (8.4.24) lim ψR = −LeR μξ + ∗ (ln  + 2γE ) + O η→−∞ β β∗ where the term of order 1/β ∗ in the square brackets is significant. Matching The origin ξ = 0 has been chosen in order to make the expression of ψR (ξ ) in (8.4.16) exact to all orders in the expansion in powers of 1/β ∗ . Matching the internal solution with the solution of the external preheated zone then shows that the coefficient of order 1/β ∗ in the square bracket of (8.4.24) is zero,  = e−2γE ≈ 0.315, so, using (8.4.22), LeR ψX∗ B = 0.315(μβ ∗ )2 .

(8.4.25)

Conditions of Validity According to (8.4.13) with = ρu2 DTu , the reference length scale has the value √ ∗ DT ∗ τR at temperature T . The thicknesses dR and dX of the external upstream and downstream zones are given by the external solutions (8.4.16) and (8.4.17),   (8.4.26) dR = DT ∗ τR /(μLeR ), and dX = DT ∗ τR /χ . ρT2 ∗ DT ∗

According to (8.4.20), the thickness dB of the inner layer is   dB = DT ∗ τR / LeR ψX∗ B.

(8.4.27)

Two conditions are required for the validity of the boundary layer type of analysis performed above, namely dB /dR 1 and dB /dX 1:   dB /dR = μLeR / LR ψX∗ B 1, dB /dX = χ / LeR ψX∗ B 1. [1] [2] [3]

Li˜nan A., 1974, Acta Astronaut., 1(7-8), 1007–1039. Seshadri K., et al., 1994, Combust. Flame, 96, 407–427. Abramowitz M., Stegun I., 1972, Handbook of mathematical functions. New York: Dover, 9th ed.

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Using (8.4.25), these conditions can be written as dB /dR = 1.78LeR /β ∗ 1,

dB /dX = 1.78χ /(μβ ∗ ) 1.

(8.4.28)

As anticipated in (8.4.20), the first condition is automatically satisfied in the limit (8.4.15). The second has to be verified after obtaining the eigenvalue μ which depends on the flame regime characterised by the set of parameters in (8.4.14).

8.4.3 Results The three unknowns μ, ψX∗ and T ∗ are obtained from three relations. The first is given by introducing (8.4.12) into (8.4.25), ψX∗ μ2



τR XRu LeR τB



−

E

e kB T ∗ = 0.315. (E/kB T ∗ )2

(8.4.29)

The second is obtained when the external solutions (8.4.16) and (8.4.17) are introduced into one of the jumps (8.4.19) or (8.4.21), both yielding the same result (see (8.4.18)), χ 2 /LeX + μχ − 1 = 0, μ ψX∗ = χ μ = . (8.4.30) μ + (χ /LeX ) The third relation is obtained by integrating the first equation in (8.4.17) from ξ = 0, where T = T ∗ , to ξ = +∞, where T = Tb : (1 − θ ∗ ) =

ψX∗ 1 = , χ (μ + χ ) 1 + (χ /μ)

(8.4.31)

where (8.4.30) has been used. According to (8.4.18), this leads to an expression for the nondimensional flame velocity μ in terms of T ∗ :    LeX 4 1 LeX − 1 − > 0; (8.4.32) 1+ 2 = 2 (1 − θ ∗ ) 2 μ LeX the sign is positive because T ∗ lies between Tu and Tb . Equations (8.4.30) and (8.4.31) then yield ψX∗ /μ2 = χ /μ = θ ∗ /(1 − θ ∗ ).

(8.4.33)

An equation for T ∗ is obtained by introducing (8.4.33) into (8.4.29), ∗

e−E/kB T θ∗ = P, (1 − θ ∗ ) (E/kB T ∗ )2

P ≡ 0.315

τB 1 , XRu LeR τR

(8.4.34)

where P > 0 is a known parameter. The left-hand side of the first equation in (8.4.34) is an increasing function of T ∗ that diverges for T ∗ = Tb , θ ∗ = 1. Therefore there is a unique solution for T ∗ , 0 < T ∗  Tb , which increases as P increases, and reaches Tb in the limit P → ∞. Once T ∗ is known, the laminar flame speed, μ, and the maximum

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409

mass fraction of the intermediate species, ψX∗ , are obtained from (8.4.32) and (8.4.33), respectively. Different regimes exist, depending of the value of the parameter P > 0. We will limit our attention to regimes for which P is small. Slow Chain-Breaking Regime Consider first a regime corresponding to a sufficiently small parameter P 1 (small ratio τB /τR ), such that T ∗ is sufficiently far from Tb for the quantity (1 − θ ∗ ) ≡ (Tb − T ∗ )/(Tb − Tu ) to be not small. The order of magnitude of β ∗ ≡ E/kb T ∗  1 is found ∗ as follows. According to (8.4.34), the large parameter 1/P is of order β ∗2 eβ , ln(1/P) = β ∗ + 2 ln β ∗ + O(1). This expansion can be inverted to obtain the expansion of β ∗ : 1/P  1,

E/kB T ∗ = ln(1/P) − 2 ln[ln(1/P)] + O(1).

(8.4.35)

The unknowns μ and ψX∗ obtained from (8.4.32) and (8.4.33) are of order unity. Therefore the conditions (8.4.28) are satisfied, and, according to (8.4.16)–(8.4.17), the thickness of the downstream external zone is of same order of magnitude as that of the preheated zone, dX /dR = O(1). In contrast to the ZFK model and to ordinary flames, the laminar flame velocity in the slow chain-breaking regime is not very sensitive to the combustion temperature. Therefore this regime does not well represent hydrocarbon flames. Fast Chain-Breaking Regime and Temperature Sensitivity Consider now a regime with a chain-breaking rate faster than before (larger ratio τB /τR , larger P), such that T ∗ , defined in (8.4.34), becomes close to Tb so that (1 − θ ∗ ) is small:  ≡ (1 − θ ∗ ) 1.

(8.4.36)

According to (8.4.32) and (8.4.33), μ and ψX∗ are also small quantities of order ,  μ ≈ LeX (1 − θ ∗ ) 1, ψX∗ ≈ LeX (1 − θ ∗ ) 1, (8.4.37) √ and, from (8.4.18), χ is of order unity, χ ≈ LeX . Equation (8.4.28) shows that the domain of validity of this regime is limited to β ∗  1, 1/β ∗  1,

namely

Tb − T ∗ kB T ∗ 1. E Tb − Tu

(8.4.38)

According to (8.4.26), and (8.4.28), the ordering of the thickness of the three layers is now dB dX dR , dR /dX = O(1/),

dX /dB = O(β ∗ )

dR /dB = O(β ∗ ).

The laminar flame velocity is sensitive to the combustion temperature, Tb , as shown next. The variation in δT ∗ resulting from a variation δTb , for fixed values of the other quantities, is obtained from (8.4.34). In the limit (8.4.38), β ∗  1, this gives δT ∗ /T ∗ ≈ (1/β ∗ )δTb /(Tb − Tu ), showing that δT ∗ is negligible compared with δTb . By definition θ ∗ ≡ (T ∗ − Tu )/(Tb − Tu ), so that δθ ∗ = (δT ∗ − θ ∗ δTb )/(Tb − Tu ) and δθ ∗ ≈ −δTb /(Tb − Tu ). Since, according to the first equation in (8.4.37), δμ/μ = −δθ ∗ /, we finally obtain

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Planar Flames

δUL Tb δTb δμ 1 = = . μ UL  (Tb − Tu ) Tb The temperature sensitivity is similar to that of the ZFK model, when the large reduced activation energy is replaced by 1/. A similar conclusion holds for the three-step model of √ the methane flame.[1] The leading order of the laminar flame velocity UL ≡ μ (DTu /τR ) is  DTu Tb − T ∗ UL = LX , (8.4.39) Tb − Tu τR where (Tb − T ∗ )/(Tb − Tu ) ≡ (1 − θ ∗ ) is a small parameter obtained from T ∗ , the solution to (8.4.34). The low Mach number approximation is verified since, according to (8.4.34), ∗  2 τB /τR is a transcendentally small parameter of the order e−β /β ∗2 in the limit β ∗ → ∞. To summarise, the laminar flame speed in the fast breaking regime of the two-step models (5.1.1) is similar to that of the ZFK model. However, the modification to the normal burning velocity of wrinkled flames will be shown to be different for the M-version; see Section 10.3.6. Application to Lean Methane–Air Flames According to the discussion at the end of Section 5.4.2, lean methane–air flames may be roughly represented by a model similar to the M-version of the two-step model (5.1.1) with the reaction rates ωB = ωI  and ωR = 2ωII  given by (5.4.11) and (5.4.12), respectively, where k10f and B4f are given in Table 5.1; see also (5.4.2) and (5.3.2). The difference with (5.1.1) is that the reaction order is 3/2 for X and also that the coefficients BB and BR have a power law temperature dependence, B˜ 10f BB 10−14 5.4 = = T , BR 2XO∗ 2 2˜nB˜ 4f XO∗ 2 where XO∗ 2 is the mole fraction of oxygen in the thin chain-branching reaction zone. When this temperature dependence is taken into account, as in the beginning of Section 8.4, the temperature T ∗ of the thin reaction zone is the solution to (8.4.34) in the form ∗

10−14 T ∗5.4 e−4000/T Tb − T ∗ XRu LeR = ∗ . ∗ 2 2XO2 T − Tu 0.315 (5.4 + 4000/T ∗ )

(8.4.40)

For the conditions of Fig. 2.2, Tb = 1725 K, equivalence ratio φ = 0.65, and Lewis number LeCH4 ≈ 1, this gives T ∗ ≈ 1300 K. Despite the drastic simplifications that have been used, this is in reasonable agreement with the value T ∗ ≈ 1400 K (the temperature where the concentration of CH4 reaches zero and/or where the concentration of H2 is maximum) in Fig. 2.2. Notice that extinction, which is predicted for a flame temperature that makes the bracket of (5.4.7) go to zero, is not far from that of hydrogen flames in (5.3.8), 2k1f ≈ nB4f .

[1]

Peters N., 1997, Prog. Astronaut. Aeronaut., 173, 73–91.

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411

8.5 Flame Quenching Beside the flammability limits, which are chemical kinetic in nature (see Section 8.5.5), there are other quenching mechanisms. Abrupt extinctions correspond to turning points of the branch of solutions for flames subjected to external disturbances that influence the flame temperature, such as thermal losses or stretch rates. These nonlinear effects result from the strong sensitivity of reaction rate to flame temperature. 8.5.1 Extinction Through Thermal Loss The loss of a small fraction of the thermal energy, of the order of the inverse of the reduced activation energy, is sufficient to stop flame propagation. It is for this reason that flames cannot propagate in narrow tubes; the ratio of surface to volume increases as the inverse of the radius and thermal losses to the walls become too strong below a critical radius, of the order of β times the flame thickness. For the same reason, flames cannot propagate through a metal grid when the holes are too small; this is the principle of the miner’s safety lamp invented by Sir Humphry Davy in 1816. This quenching mechanism is a nonlinear phenomenon which was roughly explained in physical terms more than a century later.[2] Its analytical study was performed later.[3,4] Formulation Let us suppose that for some unspecified reason (heat conduction to walls, heat radiation, etc.) the hot gas cools down with a characteristic time scale τcool . For a planar flame, this can be represented by adding a volumetric loss (energy per unit volume and unit time) −ρcp (T − Tu )/τcool to the right-hand side of Equation (8.1.1). This term, which describes the cooling of a homogeneous medium on a time scale τcool , introduces an additional term −θ τL /τcool in the reduced equation for conservation of energy (8.2.30). For heat loss due to heat conduction towards the wall in a tube of radius R, the order of magnitude of the cooling rate can be roughly estimated, 1/τcool ≈ DT /R2 (without taking into account the thermal boundary layer). According to (8.2.28), τL ≈ dL2 /DT ≈ DT /UL2 , one gets   d2 τL DT 2 ≈ L2 ≈ ; (8.5.1) τcool R UL R this is a very small number in ordinary conditions. Using the one-step ZFK model for simplicity, the equations for the nonadiabatic thermal propagation of flames are then μ

dθ d2 θ τL − 2 =w− θ, dξ τcool dξ

μ

1 d2 ψ dψ − = −w, dξ Le dξ 2

(8.5.2)

where, according to (8.2.31) for a reaction order unity, ϑ = 1, the reaction rate is w(θ , ψ) = (β 2 /2)ψ exp[−β(1 − θ )]. The ratio τL /τcool is a measure of the strength of the heat loss, [2] [3] [4]

Zeldovich Y., 1941, Zh. Eksp. Teor. Fiz., 11(1), 159–169. Joulin G., Clavin P., 1976, Acta Astronaut., 3, 223–240. Buckmaster J., 1976, Combust. Flame, 26, 151–162.

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Figure 8.15 Schematic representation of the temperature profile through a flame with thermal losses.

which affects the external zones (upstream preheat zone and burnt gas zone) and which causes the maximum flame temperature to be lower than the adiabatic value, θf < 1. Because of thermal losses, the temperature decreases in the burnt gas to the fresh gas value over a characteristic distance given by (τL /τcool )dL , and the boundary conditions are ξ = −∞: θ = 0, ψ = 1,

ξ = +∞: θ = 0, ψ = 0.

(8.5.3)

The temperature profile is shown schematically in Fig. 8.15. The dimensionless mass flux μ ≡ m/ρu UL , where UL is the adiabatic laminar flame speed, is an eigenvalue of the problem. It determines the laminar flame speed in the presence of thermal losses. Solution Flame propagation is no longer possible when the strength of heat losses, τL /τcool , exceeds a critical value that becomes small when the reduced activation energy is large, β  1. Anticipating that this critical value is of order 1/β we write τL /τcool = h/β,

h = O(1).

(8.5.4)

The solution to (8.5.2) that satisfies the boundary conditions (8.5.3) determines the eigenvalue μ(h), or, in other words, the value of the laminar flame speed as a function of the thermal loss in the limit β → ∞. The equations in the external zones are linear and are√easily integrated using the upstream and downstream boundary conditions, θ =

θf e[μ± μ +4h/β]/2 . Retaining only the term of order 1/β in the exponent, the solutions to (8.5.2)–(8.5.3) can be written: θ− (ξ ) = θf e[μ+h/(βμ)]ξ , θ+ (ξ ) = θf e−[h/(βμ)]ξ , ξ < 0: ξ > 0: ψ− (ξ ) = 1 − eLe μξ , ψ+ (ξ ) = 0, 2

where the condition of nondivergence at ± infinity has been used. The jump relation (8.2.35) provides an equation for the flame temperature, θf , as a function of the heat loss, h: − (h/βμ)θf − (μ + h/βμ)θf + μ = 0,

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(8.5.5)

8.5 Flame Quenching

413

which, to first order in 1/β, can be rewritten β(1 − θf ) = 2h/μ2 , showing that the difference between the flame temperature and its adiabatic value is of order 1/β, β(1 − θf ) = O(1). The kinetic relation (8.2.49) can now be used to find the required relation between the reduced flame speed, μ, and the heat loss parameter, h: √ (8.5.6) μ = Le exp(−h/μ2 ) or (μ2 /Le) ln(μ2 /Le) = −2h/Le. The solution for the flame speed, μ(h), in (8.5.6) decreases as the heat loss is increased and has a turning point for a critical value hc of the heat loss parameter, with a minimum value of the flame speed μc ,  μc = Le/e; (8.5.7) hc = Le/2e,

Nondimensional flame speed

see Fig. 8.16. For h > hc , there is no solution, and the plane flame cannot propagate. If the heat loss parameter is smaller than the critical value, 0 < h < hc , there are two possible solutions. Only the upper branch of solution is stable (the solid line in Fig. 8.16) the lower (dashed line) is unstable to one-dimensional perturbation and is nonphysical. Negative values of the heat loss parameter correspond to external heating, in which case the flame is accelerated. In conclusion, when the heat release rate is very sensitive to temperature, a small relative heat loss of the order of 1/(2βe) is sufficient to quench flame propagation. Moreover, the flame speed at the extinction limit is finite (nonzero) and of the same order of magnitude as √ the adiabatic flame speed, μc = Le/e. Near to the flammability limit, studied in Section 8.5.5, thermal quenching can couple to chemical extinction. However, since thermal losses do not change the order of magnitude of the flame temperature or the flame speed at the quenching point, these quantities remain essentially determined by the chemistry.

Nondimensional heat loss

√ Figure 8.16 Nondimensional flame speed, μ/ Le, as a function of the nondimensional heat loss. Beyond a critical heat loss, hc = Le/2e, there is no solution for a stationary plane flame.

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8.5.2 Quenching by Stretch in the Thermo-diffusive Approximation As mentioned earlier in Section 2.3.4, flames propagating in a nonuniform flow may be quenched when the stretch rate of the flame front 1/τs is of the order of the inverse of the transit time 1/τL (Damk¨ohler number τs /τL of order unity). Quenching is a nonlinear effect, produced by modifications to the flame temperature when Le > 1, beyond the weak modifications to the reaction rate described in Section 2.3 and studied in Section 10.3. The simplest configuration to understand flame quenching by stretch is an adiabatic planar flame stabilised in a stagnation point flow with a constant strain rate 1/τs ; see Fig. 8.17. The basic ingredients of stability in a stagnation flow are recalled in Section 2.6.4. The simplest analysis was done for a two-dimensional planar flow (u, w), using the thermo-diffusive approximation in the limit β → ∞ of the one-step ZFK flame model .[1,2] In this approximation the modifications to the flow induced by gas expansion are neglected. The simplest model is an irrotational flow near a stagnation point, with a longitudinal velocity decreasing linearly with the normal coordinate, du/dx = −dw/dy = −1/τs .

(8.5.8)

As sketched in Fig. 8.17, such a flow is locally produced either in a stream of fresh mixture impacting a solid wall or in turbulent premixed flames at a mixing layer of fresh and burnt gases (counterflowing streams of reactants and products). The analysis of this simplified model can be performed analytically and is presented in a first step. The extension to axisymmetric flows in cylindrical geometry is straightforward. The effects of viscosity at the wall and of gas expansion are studied in a second step in Sections 8.5.3 and 8.5.4 where numerical calculations of the flow are coupled to the asymptotic analysis of the flame structure. (a)

(b)

Figure 8.17 Sketch of stagnation point flows. (a) Stagnation point flow against a solid wall. (b) Counterflow stagnation point showing location of a strained flame with negative mass flux (see text). The sloping line shows the longitudinal flow velocity u(x).

[1] [2]

Zeldovich Y., et al., 1985, The mathematical theory of combustion and explosions. New York: Plenum. Buckmaster J., Mikolaitis D., 1982, Combust. Flame, 47, 191–204.

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Formulation For a planar flame parallel to the y-axis, in the absence of transverse gradients of mass fraction and temperature, the governing equations (15.2.3) and (15.2.4), written with the notation of (8.2.4)–(8.2.10) for a reaction of order unity, ϑ = 1, for simplicity, take the form   d2 θ 1 x dθ ψeβ(1−θ) , U− − DT 2 = τs dx τ dx rb





dψ DT d2 ψ 1 = − ψeβ(1−θ) , − dx Le dx2 τrb (8.5.9) where U is the flow velocity at the reaction sheet and the reaction sheet is located at x = 0: u = U, θ = θf . The temperature is uniform in the burnt gas, x  0: θ = θf , ψ = 0. Due to the inhomogeneity of the flow, the flame temperature depends on the Lewis number, θf = 1 when Le = 1. As usual, in the limit β → ∞, the reaction rate is negligible in the preheated zone x < 0 and the boundary conditions in the fresh mixture are x → −∞: θ = 0, ψ = 1. Using the same reduced variable as in (8.2.29), ξ ≡ x/dL , introducing the reduced mass flux through the reaction sheet, μf ≡ U/UL , and the given reduced strain rate, s ≡ τL /τs > 0 (the inverse of the Damk¨ohler number), the equations take a form similar to (8.2.30) but with an extra term (stretching) in the preheated zone:

ξ < 0:

d2 θ dθ − 2 = 0, dξ dξ θ = 0, ψ = 1,

(μf − sξ )

ξ → −∞:

x U− τs

(μf − sξ )

1 d2 ψ dψ − = 0, dξ Le dξ 2

(8.5.10)

θ = θf , ψ = 0.

(8.5.11)

ξ  0:

For s = O(1) and β(Le − 1) = O(1) in the limit β → ∞, the convective fluxes are negligible in the thin reaction zone, the same as in Section 8.2.4, which is thus replaced by the jump conditions (8.2.52). The objective is to obtain μf in terms of s. Analysis Integration of (8.5.10) from ξ = −∞ to ξ = 0− using (8.5.11) yields

where

  θf 1 dψ  1 dθ  1 , , = − =   Le dξ ξ =0− Le ILe (μf , s) dξ ξ =0− I1 (μf , s)  0 2 ILe ≡ eLe(μf ξ −sξ /2) dξ , I1 ≡ ILe=1 . −∞

The unperturbed flame corresponds to s = 0: ILe = (μf Le)−1 . Using the identity 

sξ 2 − μf ξ Le 2



/2 . μ2f Le μ2f Le s = Leξ − − μ f s 2s 2μ2f Le

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(8.5.12) (8.5.13)

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Planar Flames

a simple calculation of ILe puts (8.5.12) in the form  √ μf μf 1 dψ  , where Y = − ≡ Le √ , Le Le dξ ξ =0− J (YLe ) 2s  −Y 1 3 2 2 e−Z dZ, lim J ≈ 1 − + ..., and J (Y) ≡ 2Ye Y 2 4 2Y 4Y Y→∞ −∞  μf θf μf dθ  , where Y1 ≡ √ ; = dξ ξ =0− J (Y1 ) 2s

(8.5.14) (8.5.15) (8.5.16)

see References [1] , page 298, for the asymptotic expansion of J in (8.5.15), which will be useful in the small stretch limit. In the limit β → ∞, β(Le − 1) = O(1), s = O(1), one has to compute the first term in the expansion of J (YLe ) in powers of (Le − 1) around J (Y1 ). This involves the derivative JY ≡ dJ /dY. The final result is obtained by using the two jump conditions (8.2.52) at the reaction sheet, ξ = 0, the first yielding the flame temperature and the second the mass flux, β(Le − 1) g(Y1 ), 2 2Y12 , g(Y1 ) = 1 + 2Y12 − J (Y1 ) β(1 − θf ) =

g(Y1 ) ≡

Y1 JY (Y1 )

, (8.5.17) J (Y1 ) 1 1 lim g(Y) = 2 + O( 4 ), (8.5.18) Y Y Y→∞  β(Le − 1) μf = f (Y1 ), f (Y1 ) ≡ J (Y1 ) exp − (8.5.19) g(Y1 ) . 4 √ √ Introducing Y1 ≡ μf / 2s into (8.5.19) yields μf in terms of μf / 2s, that is, an implicit relation for μf in terms of s, for different values of β(Le − 1). where

Discussion of the Results The solution in √ the limit of a weak strain rate, s 1, is obtained at the leading order in the limit Y1 ≡ μf / 2s → ∞ of (8.5.19). The asymptotic limit in (8.5.15) and (8.5.18) yields √ 2s 1: μf ≈ 1 − [1 + β(Le − 1)/2]s, (8.5.20) in agreement with the linear dynamics of wrinkled flames of Chapter 10; see the diffusive relaxation (10.2.25) and the Markstein number (10.3.36) for υb = 1. When the stagnation point flow is obtained by a flow impacting on a solid planar wall, the physical domain is limited to μf > 0; see Fig. 8.17a. For counterflowing streams of reactants and products, the reaction zone can be located on the side of the product stream corresponding to μf < 0; see Fig. 8.17b and the discussion below. A set of numerical solutions to (8.5.17)–(8.5.19) for β(Le − 1) = 6, 4, 2.1, 2, 0, −4 are plotted in Fig. 8.18. For β(Le − 1) > 2, the plot of mass flux at the reaction sheet, μf , as a function of strain s has a turning point, dμf /ds = ∞, for a critical value of strain, s = sc (Le). For s > sc there is no solution for μf and the flame is quenched. The critical strain [1]

Abramowitz M., Stegun I., 1972, Handbook of mathematical functions. New York: Dover, 9th ed.

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417

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8.18 Numerical solutions for mass fluxes at the reaction sheet for strained flames, calculated in the thermo-diffusive approximation for β(Le − 1) = 6, 4, 2.1, 2, 0, −4. Note the changes in scales on the axes.

increases with decreasing β(Le−1). For β(Le−1) > 4, Fig. 8.18a, the turning point occurs for μfc > 0 so that the flame is quenched at a finite mass flux oriented towards the burnt gas, as for freely propagating flames. For β(Le − 1) = 4, Fig. 8.18b, the flame is extinguished at zero mass flux (at the stagnation plane). For smaller values, 4 > β(Le − 1) > 2, Fig. 8.18c, the turning point is in the domain of negative mass flux, μfc < 0, so that the reaction sheet is located on the burnt stream side of a counterflow stagnation point; see Fig. 8.17b. The physical structure of such an unusual flame with a mass flux through the reaction zone oriented towards the fresh mixture can be explained as follows. The reactants enter the reaction sheet by molecular diffusion, as in ordinary flames. The convective fluxes of products leaving the preheated zone in the transverse direction is larger than that entering at the burnt side of the reaction sheet. The difference corresponds to the production rate of the chemical reaction irreversibly transforming reactants into products. As β(Le − 1) approaches the value 2 from above, the turning point goes to μfc = −∞ at a finite value of strain sc ≈ 2.3115; see Fig. 8.18d. For β(Le − 1) < 2 there is no turning point, meaning no flame quenching: the gradient dμf /ds remains negative and finite for all large s; see Fig. 8.18e and f. Finally for negative values of β(Le − 1) < −2 the reduced flame speed first increases with strain in agreement with (8.5.20), reaches a maximum for s = O(1), and then decreases monotonically; see Fig. 8.18f. For reduced strain rates much larger than unity, typically for s  5, these results are inaccurate; the internal structure of the thin reaction zone is modified and the jump relations (8.2.52) are no longer valid when s is of order β.

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√ The gradient dμf /ds on the axis μf = 0 can be found analytically. According to (8.5.19), 2s = f (Y1 )/Y1 , so that the limit Y1 → 0, (μf → 0), yields √ √ μf = 0: 2s = π e−β(Le−1)/4 . (8.5.21) The quantity dμf /ds is obtained from (8.5.19), dμf Y1 f  , = − √ √ ds 2s 2s − f  where f  ≡ df /dY1 . For small values of Y1 one has √ √ √ Y1 1: 2s = f /Y1 = ( π − 2Y1 + π Y12 · · · )e−β(Le−1)g/4   4 2 Y12 + · · · g = 1 − √ Y1 + 2 − π π

(8.5.22)

(8.5.23) (8.5.24)

and, more particularly, Y1 = 0:

f =

√

π e−β(Le−1)/4 ,

√  2s − f  β(Le − 1) −β(Le−1)/4 e =2 1− , Y1 4

so that, with the help of (8.5.21) and (8.5.22),  dμf  e−β(Le−1)/4 = 2 . ds μ=0 β(Le − 1) − 4 This slope is effectively infinite for β(Le − 1) = 4 and positive (negative) for larger (smaller) values in agreement with the numerical solutions plotted in Fig. 8.18. The cur√ vature of the plot of s as a function of μf for β(Le − 1) = 4 may also be computed at Y1 = 0 and shown to be negative. The extension to axisymmetric flows is straightforward by writing the divergence of the flow in cylindrical geometry, assuming dw/dy = 1/τs as before, du 1 d(yw) + = 0, dx y dy

⇒

du 2 =− . dx τs

(8.5.25)

8.5.3 Viscous Effects on Flame Quenching in a Stagnation Point Flow Viscosity and gas expansion do not introduce qualitative changes, but they modify the critical values of Le and s. The critical conditions for flame quenching must now be obtained by coupling numeric and asymptotic analyses. Viscous Flows Near a Stagnation Point When a viscous fluid is blown towards a rigid wall, a boundary layer develops near the wall (no-slip condition x = 0: u = w = 0). For a constant density flow, the solution is presented

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in text books.[1] Far from the wall viscous effects are negligible and the flow is given by (8.5.8) in planar geometry. Introducing Hx (x) ≡ dH/dx, we look for a solution in the form w = yHx (x), x = 0 : H = 0,

Hx

u = −H(x),

= 0,

(8.5.26)

x → −∞ : H = x/τs ,

(8.5.27)

where the first boundary condition is the no-slip condition at the rigid wall. The boundary conditions (8.5.27) are modified for counterflowing streams; see (8.5.31). Continuity is satisfied by (8.5.26). The Navier–Stokes equations (15.1.17) yield  , ρHHx = −∂p/∂x − ρνHxx



  yρ[−HHxx + Hx2 ] = −∂p/∂y + yνρHxxx ,

where ν is the diffusion coefficient associated with shear viscosity. The first equation shows that ∂p/∂x is independent of the transverse coordinate y, so that the coefficient of the term proportional to y in the second equation must be a constant (independent of x). This yields an equation for H(x), 

  + Hx2 − νHxxx = 1/τs2 , − HHxx

(8.5.28)

where the second boundary condition in (8.5.27) has been used. The thickness of the boundary layer at the wall where the viscous effects cannot be neglected (nonzero vorticity) √ √ √ is ντs and H scales as ν/τs . Introducing the notations z ≡ x/ ντs and H (z) ≡ √ H/ ν/τs , both Equation (8.5.28) and the boundary conditions (8.5.27) take a parameterfree form, 

 = 1, − H Hzz + Hz 2 − Hzzz

rigid wall: z = 0:

H = 0,

z = 0:

H = 0,

counterflowing stream:

Hz

= 0,

(8.5.29)

z → −∞:

z → ∓∞:

H = z,

H = ±z.

(8.5.30) (8.5.31)

The parameterless function H (z) is obtained by a numerical integration of these equations.[1] For a rigid wall, viscosity ceases to play a significant role as soon as z > 2.4, where the irrotational flow (8.5.8) is recovered. Viscous Effects on Flame Quenching In gaseous mixtures, the Prandtl number, Pr ≡ ν/DT , is close to unity. When the difference Pr −1 is of order 1/β it may be neglected in flame theory in the limit β → ∞, so that ν is replaced by the thermal diffusion coefficient DT . Introducing the reduced distance from the √ wall ξ ≡ x/dL = z/ s and the unknown position of the thin reaction zone ξf , the equations in the preheated zone become ξ < ξf :

−

√

√ dθ d2 θ sH (ξ s) − 2 = 0, dξ dξ

√ √ dψ 1 d2 ψ − sH (ξ s) = 0. (8.5.32) − dξ Le dξ 2

In this notation, the unknown mass flux through the reaction sheet μf ≡ U/UL is μf = √ √ − sH (ξf s) > 0, with, for a rigid wall, ξf < 0, H < 0 and μf > 0. The slopes of [1]

Batchelor G., 1967, An introduction to fluid dynamics. Cambridge University Press.

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the external solution at ξ = ξf− have the same form as in (8.5.12) at ξ = 0− , but with a different expression for ILe  ξf √s z √ √ −Le ξ √s H (z )dz 1 f ILe ≡ √ GLe (ξf s), GLe (ξf s) ≡ dz e . (8.5.33) s −∞ The problem can be solved in the limit β → ∞, β(Le − 1) = O(1), s = O(1) using the same methodology as in the preceding section, by using the two jump conditions (8.2.52) √ at the reaction sheet, ξ = ξf , but after having computed GLe=1 (ξf s) and ∂ GLe /∂Le|Le=1 numerically from (8.5.29)–(8.5.31). When the reaction sheet stands outside the boundary layer, the results are the same as those obtained without viscosity. This is the case when the strain rate is not too strong, s < 0.4, as in Fig. 8.18a and b. Significant modifications appear only at stronger strain rates.

8.5.4 Stretched Flame in the Presence of Gas Expansion A similar analysis may also be performed when the modifications to the flow induced by gas expansion are taken into account.[1,2] General Formulation Looking for solutions in which the density varies in the normal direction only, ρ(x), the equation for conservation of mass can be written 1 ∂(ρu) ∂w + = 0. ρ ∂x ∂y

(8.5.34)

It is convenient to introduce a mass-weighted variable ξ , reduced by the flame thickness dL , as in flame theory, and to consider a flow with a constant strain rate s = τL /τs . We then look for a solution for the flow in a form similar to (8.5.26),  x ρ(x) w(ξ , η) ρu 1 y = sηFξ (ξ ), = −sF (ξ ), ξ ≡ dx, η ≡ . (8.5.35) UL ρu UL dL 0 ρu dL Such flows automatically satisfy (8.5.34). Far from the stagnation point, the incoming flow of fresh mixture is assumed to be the same as before, ξ → −∞:

F = ξ.

(8.5.36)

It is assumed that all the diffusion coefficients D differ from the thermal diffusion coefficient DT at most by a small quantity of order 1/β, and also that the diffusion coefficients D satisfy the relation ρ 2 D = cst. The expressions (8.2.39)–(8.2.40) for the laminar flame speed UL and the flame thickness dL are still valid in this case. The results of this section are restricted to ρ 2 D = cst. They change quantitatively but not qualitatively for a different law [1] [2]

Libby P., Williams F., 1982, Combust. Flame, 44(1-3), 287–303. Libby P., et al., 1983, Combust. Sci. Technol., 34.

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421

ρ n D = cst., n = 2. Using Du = UL dL the Navier–Stokes equations then take the following reduced form, ∂(p/ρu UL2 ) d2 (u/UL ) d(u/UL ) = − − Pr , dξ ∂ξ dξ 2  ∂(p/ρu UL2 ) ρ  sη −sF (ξ )Fξξ (ξ ) + sFξ 2 (ξ ) − Pr Fξ , ξ ξ (ξ ) = − ρu ∂η −sF (ξ )

(8.5.37) (8.5.38)

where Pr = 1 since modifications of order 1/β to the flow do not matter in flame theory in the limit β → ∞. For the same reason as that leading to (8.5.28), self-consistency of these Euler equations yields 

F Fξξ − Fξ 2 + s−1 Fξ ξ ξ = −1 − (υb − 1)θ (ξ ),

(8.5.39)

where (8.5.36) and the relation ρu /ρ = 1 + (υb − 1)θ , valid in the quasi-isobaric approximation of perfect gas flows, have been used. The gas expansion parameter υb ≡ ρu /ρb is constant in the analysis with a typical value υb = 8. The equations for θ and ψ in the preheated zone take the form d2 θ 1 d2 ψ dθ dψ − 2 = 0, −sF (ξ ) − = 0, dξ dξ Le dξ 2 dξ ξ → −∞: θ = 0, ψ = 1, ξ  ξf : θ = θf , ψ = 0, ξ < ξf :

− sF (ξ )

(8.5.40) (8.5.41)

where both ξf and θf − 1 = O(1/β) are unknown. The flame position ξf is an eigenvalue that must be found as a function of the reduced strain rate s for different values of β(Le − 1). Attention will be focused to the existence of turning points of the curves ξf (s), denoting flame quenching. The solution is obtained by solving (8.5.39)–(8.5.41) with the three boundary conditions for F (ξ ) coming from (8.5.30) or (8.5.31) and with the two additional jump conditions (8.2.52) for θ (ξ ) and ψ(ξ ) at the reaction sheet, ξ = ξf . Methodology The problem may be decomposed into two parts: the first concerns the flow and the second the structure of the stretched flame. The key point in the analysis of Libby, Williams and Li˜nan[2] is to note that θ (ξ ) in the right-hand side of (8.5.39) is the solution to the first equation in (8.5.40) with the boundary condition ξ = ξf : θ = 1. The reason is that, as mentioned before for Pr = 1, modifications to the flow of order 1/β do not matter: the small quantity θf − 1 resulting from β(Le − 1) = 0 is negligible in this part of the analysis. √ √ ˙ ≡ sF , Introducing the reduced coordinate z ≡ sξ and the function M(z) √  x s ρ(x) dx, z≡ dL 0 ρu

1 ρu ˙ ≡ −√ M , s ρu UL

17:10:28 .010

w ˙ z , = sηM UL

(8.5.42)

422

Planar Flames

the system of nondimensional equations controlling the flow for a given flame position √ zf ≡ sξf takes the form, according to (8.5.39)–(8.5.41), 

˙ z2 + M ˙  ˙M ˙ zz − M M zzz = −1 − (υb − 1)θo (z),   ˙ z  zf : θo = 1, z < zf : − Mθoz − θozz = 0, √ ˙ = s x/dL ⇒ M ˙ z = 1, z → −∞: θo = 0, M √ ˙ = 0, ˙ z = υb , z = 0: M z → +∞: M

(8.5.43) (8.5.44) (8.5.45) (8.5.46)

where υb = ρu /ρb . The boundary conditions (8.5.46) are valid for counterflowing streams. ˙ = 0, M ˙ z = 0. The boundary For a rigid wall at z = 0 they must be replaced by z = 0: M condition (8.5.45) corresponds to the prescribed upstream flow, far from the stagnation ˙ are continuous in the range z ∈ (−∞, +∞). This plane. The functions θo (z) and M(z) is not the case for the derivative of θo (z) with respect to z which is discontinuous at the reaction sheet, z = zf , in the limit β → ∞. According to (8.5.44) the expression for θo (z) to be introduced into the r.h.s (8.5.43) takes the form z  zf : θo (z) =

z

−

 zf

− dz e

 −∞ dz e

−∞

 z zf

 z zf

˙  )dz M(z M(z )dz

z  zf :

,

θo (z) = 1.

(8.5.47)

√ For a given value of υb , no parameter other than zf ≡ sξf in (8.5.47) is involved in √ the solution to (8.5.43), the strain rate s being involved only through the grouping sξf . Also the Lewis number does not appear at this stage. Equation (8.5.43) has to be solved numerically for different values of zf by extending the method used to solve (8.5.29)– ˙ zf ). The numerical procedure is (8.5.30). The solution is a function of two variables, M(z, [1] not free of difficulties, except at small stretch rate for which an analytical solution may be obtained; see below. In the general case, one way to proceed is to work by iteration, √ starting with the expression θo = e(z−zf )/ s , valid in the limit of zero stretch. √ √ ˙ zf ) is known, the relation linking zf = sξf and s is obtained by solving Once M(z, the flame structure in the limit β → ∞, β(Le − 1) = O(1), using the method of Section 8.2.4. The equations in the preheated zone are ˙ zf )θz − θzz = 0, −M(z, z → −∞: θ = 0, ψ = 1,

 ˙ zf )ψz − ψzz −LeM(z, = 0,

(8.5.48)

z = zf : θ = θf , ψ = 0,

(8.5.49)

so that the derivatives θz and ψz /Le at the upstream side of the reaction sheet z = zf take the same form as in (8.5.12), z = zf : ψz = 1/ILe , θz = θf /ILe=1 , but with a more general expression for ILe ,  zf  z ˙  ,zf )dz −Le z M(z f ILe (zf ) = dz e . (8.5.50) −∞

[1]

Libby P., et al., 1983, Combust. Sci. Technol., 34.

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423

√ The unknowns θf and zf are then determined in terms of s by the two jump conditions at the reaction sheet (8.2.52), written with the variable z, √  θz + ψz /Le = 0, sθz = e−β(1−θf )/2 . (8.5.51) z = zf : As in Section 8.5.2, the first equation yields an expression for β(1 − θf ), proportional to √ √ β(Le − 1), and the second an expression for s corresponding to zf ≡ sξf . Small Stretch Analytical results may be obtained in the limit of small stretch, s 1. When the flame stands far from the rigid wall, viscosity may be neglected. Viscosity is also not important for counterflowing streams. In the small stretch limit, the viscous term in (8.5.43), that is, ˙  and only two boundary the third-order derivative term M zzz , will be neglected for simplicity, √ ˙ = 0, and z → −∞: M ˙ = s x/dL . According to (8.5.42), conditions will be used: z = 0: M ˙ zf ) takes large values inside the preheated zone for (x − xf )/dL = O(1): the function M(z, √ ˙ ˙M ˙ zz , is M = O(1/ s), since ρu/ρu UL = O(1). Thus the first term of Equation (8.5.43), M ˙ z is unity to leading order. This gives M ˙ zz ≈ 0, and dominant in the preheated zone and M ˙ ˙ f ≡ M(z ˙ = zf , zf ), which is M(z) is approximated by a linear function of z. Introducing M √ ˙ f , the solution to related to the reduced mass flux across the reaction zone, μf = − sM (8.5.42) satisfying (8.5.45) takes the form √  x  √ (x − xf ) ρ s ˙ ≈M ˙ f + (z − zf ) = M ˙f + s + − 1 dx, (8.5.52) z  zf : M dL dL xf ρu where the expression in terms of x is obtained from (8.5.42). The integral is computed using √ the variable z and the unperturbed solution 1−(ρu /ρ) = (υb −1)θo (z) with θo = e(z−zf )/ s , √ √ ˙ f | = O(1/ s), the term (z−zf ) = O( s) to give (υb −1)(θo −1). In the preheated zone, |M ˙ ˙ f of order s, so that the expression for M(z) in (8.5.52) is the is a relative correction to M solution to the equation 

˙ z2 = −1. ˙M ˙ zz − M M The source term in (8.5.43), −(υb − 1)θo (z), introduces smaller corrections to the approx˙ imate solution of M(z) in (8.5.52), of relative order s2 . This can be seen from the equa ˙ ˙ tion Mf Mzz ≈ −(υb − 1)θo (z) by successive integrations from −∞ to z, each of them √ introducing a factor s. This is an illustration of the two-length scale problem for an upstream flow varying on a distance larger than the flame thickness, s 1. Introducing ˙ ≈M ˙ f + (z − zf ) into (8.5.50), and proceeding as in Section the approximation (8.5.52) M 8.5.2, Equation (8.5.51), yields  M˙ f /√2 √ 2 ˙ 2f /2 M −β(1−θf )/2 ˙ ˙ , J ≡ 2Mf e e−X dX (8.5.53) s 1: μf = −J(Mf )e 

−∞

 ˙ 2f M β(Le − 1) 2 ˙ β(1 − θf ) = 1 + Mf + , ˙ f) 2 J(M

17:10:28 .010

(8.5.54)

424

Planar Flames

√ ˙ f / 2. For a given value which are the same equations as in (8.5.17)–(8.5.19) for Y1 = −M of β(Le − 1), the expression of the reduced mass flux at the reaction zone μf in terms √ ˙ f = −μf / s is thus the same as for the thermo-diffusive model, υb = 1, and the of M critical stretch for sudden quenching is also the same. For υb > 1, this result is valid only ˙ f . The turning point of Fig. 8.18a in the region if s is small, that is, for large values of M where s 1 corresponds to a value of β(Le − 1) that is too large to be easily accessible in ordinary gaseous mixtures. Discussion of the Results Numerical integration of (8.5.45)–(8.5.47) in the general case[1] show that the turning points correspond systematically to values of β(Le − 1) that are even larger than that in Fig. 8.18a. Therefore, when the influence of variable density (associated with heat release) is taken into account, abrupt extinction of stretched planar flames cannot be explained by the ZFK model as a consequence of the Lewis number alone. Thermal losses, studied in Section 8.5.1, appear to be always of a great importance in causing abrupt transitions of real flames. For flames stretched by inhomogeneities of the upstream flow, the value of the Markstein number M, defined in (2.3.2) as the coefficient in the linear relation between the modification to the flame velocity and the strain rate, depends on the definition of the flame surface inside the flame structure; see the discussion in Section 2.3.3. For the one-step ZFK model in the limit β → ∞, the velocity of the stretched flame, Un− , is usually defined as the extrapolation of the upstream flow u− (x) to the reaction sheet, Un− ≡ u− (xf ). For a planar flame in a stagnation flow, the upstream flow is a given datum of the problem, u− (x)/UL = −s x/dL . The flame velocity Un− is simply related to the position of the reaction zone xf , ˙ = Un− /UL = −s xf /dL . Matching the flow with the flow prescribed upstream, limx→−∞ M √ s x/dL (see (8.5.45)), the limit x → −∞ of (8.5.52) provides the relation between the √ ˙ f, flame velocity Un− and the mass flux through the reaction zone, μf = − sM Un− /UL = μf − s(υb − 1),

(8.5.55)

where the second term in the right-hand side comes from the integral in (8.5.52). The ˙f = linear approximation for small stretch is obtained from (8.5.53) in the limit M √ −μf / s → −∞. The limits (8.5.15) and (8.5.18) lead to (8.5.20), as in the thermodiffusive approximation. Then, according to (8.5.55), the Markstein number of the one-step ZFK for ρ 2 D = cst. is M = υb + β(Le − 1)/2,

(8.5.56)

in agreement with the more general result (2.9.44) for λ = T/Tu corresponding to ρ 2 D = cst. By comparison with the thermo-diffusive approximation υb = 1, Equation (8.5.56) shows that the Markstein number is increased by a large quantity υb − 1 ≈ 4– 8 when a realistic gas expansion is taken into account. Consequently the critical Lewis number of the ZFK model with M = 0 is small, Le ≈ 1 − 2υb /β, too small to be easily [1]

Libby P., et al., 1983, Combust. Sci. Technol., 34.

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425

accessible in ordinary gaseous combustible mixtures, except for lean hydrogen–air flames. Thermal loss and multiple-step chemistry are of great importance for thermo-diffusive instabilities (M < 0) since they tend to decrease the critical Lewis number at M = 0. The effect of thermal loss on the linear dynamics of flames is studied in Section 10.2.2 in the thermo-diffusive approximation. Also, multiple-step chemistry has a strong influence on flame dynamics near the flammability limits that are considered in the next section, Section 8.5.5. The computation of M for wrinkled flames, including the effects of curvature and gas expansion, is presented in Section 10.3.

8.5.5 Flame Speed Near Flammability Limits Detailed studies of hydrogen flames[2,3,4,5] show that the chemical kinetics can be reduced to a one-step scheme near the rich and lean flammability limits. In this section we present a simplified version of hydrogen flame near the rich flammability limit, focusing our attention on the basic mechanisms only. The reduced mechanism so obtained presents generic characteristics of the flammability limits for other flames. It is also a preliminary step before studying the ignition problem in the same conditions. Flame Model A simple flame model, useful near the flammability limits, can be obtained from a reduction of hydrogen kinetics. As already mentioned in Section 5.3, when the flame temperature is not very high the backwards shuffle reactions, (1b)–(3b) in Table 5.1, can be neglected, and the basic scheme for rich hydrogen combustion can be reduced to the two-step scheme (I) and (II) in (5.3.11) and (5.3.12), where H is the intermediate species. In the following, the consumption of hydroperoxyl, HO2 , will be neglected for simplicity since it does not play an essential role from a qualitative point of view. Therefore α = 0 in the expression (5.3.11) for ωI , near the flammability limits, the expression for ωII in (5.3.12) must also be modified to take into account the consumption of H by the trimolecular recombination reaction (5f) in Table 5.1. The reduced two step scheme is then (I)

O2 + 3H2 → 2H2 O + 2H, ωI = ω1f = cH cO2 k1f (T),

(II)

k1f (T) = B1f e−E1 /kB T .

H + H → H2 + Q, ωII = ω4f + ω5f = ncH cO2 B4f + nc2H B5f .

For simplicity, we have neglected the heat release of (I) which is 10 times smaller than that of (II). The resulting scheme is similar to the three elementary steps (B), (Ri ) and (Rii ) in (5.2.1)–(5.2.4) with BB = B1f , BRi = B5f , BRii = B4f , QB = 0, Qi = Qii ≡ Q, where H [2] [3] [4] [5]

He L., Clavin P., 1993, Combust. Flame, 93, 391–407. Seshadri K., et al., 1994, Combust. Flame, 96, 407–427. Fernandez-Galisteo D., et al., 2009, Combust. Flame, 156, 985–996. Fernandez-Galisteo D., et al., 2009, Combust. Theor. Model., 13(4), 741–761.

17:10:28 .010

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Planar Flames

is the intermediate species and (B) here has a net production of two moles of H instead of one. This two-step scheme for hydrogen leads to a flame model whose equations are  nB5f nB4f d2 ρ2Q d , YH YO2 + YH m T − ρDT 2 T = dx cp MH MO2 MH dx m

d d2 ρ2 YO2 − ρDO2 2 YO2 = − YH YO2 k1f (T), dx MH dx d2 d ρ2 ρ2 2 m YH − ρDH 2 YH = 2 YH YO2 [k1f (T) − nB4f ] − 2 Y nB5f , dx MO2 MH H dx x → −∞:

T = Tu ,

YO2 = YO2 u ,

YH = 0,

x → +∞:

T = Tb ,

YO2 = 0,

YH = 0.

(8.5.57) (8.5.58) (8.5.59) (8.5.60)

The flame temperature Tb is obtained by eliminating the reaction rates, 2cp T +Q[(YH /MH )+2(YO2 /MO2 )] = cst.

⇒

cp (Tb −Tu ) = (Q/MO2 )YO2 u . (8.5.61)

Since H cannot be produced if the bracket in the right-hand side of the equation for YH in (8.5.59) is negative, the flammability limit is given by the crossover temperature T ∗ (5.2.7), ∗

k1f (T ∗ ) ≡ B1f e−E1 /kB T = nB4f .

(8.5.62)

The exothermic reaction cannot proceed for a flame temperature Tb below T ∗ , and the critical composition YO2 = YO∗ 2 u beyond which the flame propagation is no longer possible is Tb = T ∗

⇒ cp (T ∗ − Tu ) = (Q/MO2 )YO∗ 2 u ,

(8.5.63)

since YO2 u ≤ YO∗ 2 u ⇒ Tb ≤ T ∗ . This simple model for hydrogen-rich flames may be adapted to describe, at least qualitatively, hydrocarbon flames near the flammability limits. One-Step Model. Vanishing Flame Speed Near the Flammability Limit Focusing the attention on the small flammable domain, near the flammability limit, k1f (T) − nB4f 1, nB4f

k1f (T) ≡ B1f e−E1 /kB T ,

(8.5.64)

the term in brackets on the right-hand side of (8.5.59) is small. For the corresponding compositions, the radical H then satisfies a steady-state approximation, as in Section 5.3.2, T  T ∗: T < T ∗:

[k1f (T) − nB4f ] MH , nB5f MO2 YH ≈ 0, frozen preheated zone. YH ≈ YO2

17:10:28 .010

(8.5.65) (8.5.66)

8.5 Flame Quenching

427

Retaining the leading order terms in the limit (k1f − nB4f )/nB4f → 0+ , Equations (8.5.57)– (8.5.59) reduce to two equations for T and YO2 , m

k1f (T) d2 YO2 dYO2 ≈ −ρ 2 YO2 2 [k1f (T) − nB4f ], − ρDO2 2 dx MO2 nB5f dx

(8.5.67)

B4f d2 T Q dT [k1f (T) − nB4f ]. − ρDT 2 ≈ ρ 2 YO2 2 2 dx cp dx MO B5f

(8.5.68)

m

2

Introducing the reduced variables ψ ≡ YO2 /YO2 u and θ ≡ (T − Tu )/(Tb − Tu ), these equations take the form m

d2 ψ ρ  dψ − ρDO2 2 ≈ − ∗ ωψ , dx τ dx dθ d2 θ ρ − ρDT 2 ≈ ∗ ωθ , m dx τ dx

− kE ( T1 − T1∗ )

 ωψ ≡ ψ 2e

B

J(T),

ωθ ≡ ψ 2 J(T),

(8.5.69) (8.5.70)

where J(T) and the characteristic reaction time τ ∗ are defined by T > T ∗: T < T ∗:

Tu − kE ( T1 − T1∗ ) − 1], [e B T J(T) = 0, J(T) ≡

∗

1/τ ≡

(nB24f c∗O2 u )/B5f ,

(8.5.71) (8.5.72) (8.5.73)

where cO2 u = ρu YO2 u /MO2 is the concentration of the limiting species in the initial composition. To leading order in the limit (8.5.64), cO2 u in the expression of τ ∗ is taken at the flammability limit, c∗O2 u ≡ ρu YO∗ 2 u /MO2 . The boundary conditions for θ and ψ at infinity are the same as in (8.2.5). Very close to the flammability limits, Equation (8.5.71) can be simplified as (Tb − T ∗ )/T ∗ kB T ∗ /E, E 1 1 − E (1− 1 )  [e kB T T ∗ − 1] ≈ ( ∗ − ) 1, ⇒ ωψ ≈ ωθ = ψ 2 J(T), kB T T

(8.5.74)

valid to leading order. As for the ZFK model, the reaction rate is localised in a thin reaction zone at high temperature, T ∈ (T ∗ , Tb ), where DO2 d2 ψ/dx2 ≈ −DT d2 θ /dx2 ≈ −ωθ /τ ∗ , so that ψ ≈ LeO2 (1 − θ ), LeO2 ≡ DT /DO2 , DT d2 θ/dx2 ≈ ωθ /τ ∗ ,

ωθ ≈ Le2O2 (1 − θ )2 j(θ ),

(8.5.75)

with, according to (8.5.72) and (8.5.74) θ > θ ∗: ∗

θ T ∗: W ρu τrb ∗ ˙  T < T : W = 0,

where

1 1 ≡ ∗e τrb τ

E 1 1 kB ( T ∗ − Tb )

=

B4f B1f e B5f

− k ET B b

c∗O2 u ,

(9.1.1) (9.1.2) (9.1.3)

and where the definitions of the crossover temperature T ∗ in (5.2.7) and (8.5.62) have been used. This model reduces to the ZFK model (8.2.31) far from the stability limits. We introduce the Zeldovich number β and a parameter ε that measures the distance from the flammability limit,   Tu (Tb − T ∗ ) E 1− , ε≡β , (9.1.4) β≡ kB Tb Tb (Tb − Tu ) where Tb is the adiabatic flame temperature of the planar flame (8.5.61). The parameter ε varies from 0 at the flammability limit to +∞ far from this limit. For large β, the parameter ε is of order unity when (Tb − T ∗ )/T ∗ is of order 1/β. With the same approximation as that used in (8.2.14), near the flammability limits, where the relative difference between T, Tb and T ∗ is not more than 1/β, the reaction rate (9.1.1) may be rewritten T > T ∗:

 2 ˙  (ψ, θ ) = ρ ψ 2 1 e−β(1−θ) − e−ε , W ρu τrb

(9.1.5)

where θ ≡ (T − Tu )/(Tb − Tu ) and ψ = YR /YRu are the reduced temperature and the mass fraction of the limiting species. The ZFK model is recovered when the adiabatic flame temperature Tb is far from the crossover temperature T ∗ , ε → +∞. Constitutive Equations and Asymptotic Method Solutions are obtained in this section for both adiabatic and nonadiabatic cases. As already stated in Section 2.4.2, a steady-state spherical solution exists in the absence of a convective flux. Using the reference length and time scales (8.2.27)–(8.2.28), dL and τL , associated with the flame thickness and the transit time of the ZFK model (ε → ∞) for Le = 1 and ϑ = 2 but where τrb is given in (9.1.3), the reduced equations for flame kernels (steady spherical solutions without flow) take the form θ = −w ˙ + h, ψ = Le w, ˙   d2 θ d2 ψ 1 d 2 dθ 2 dψ 2 dθ r = = − w ˙ + h(θ ), = Le w, ˙ (9.1.6) + + dr r dr r dr r2 dr dr2 dr2

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433

where r = R/dL and h(θ ) is a small quantity, of order 1/β, representing the nondimensional radiative energy lost per unit time and unit mass (the form used for studying the thermal quenching in the planar case in Section 8.5.1 was h(θ ) = hθ/β). According to (9.1.5), the reduced reaction rate takes a form that generalises (8.2.31), β(1 − θ ) < ε ⇔ θ > θ ∗ : β(1 − θ ) > ε ⇔ θ < θ ∗ : where

θ∗ ≡

(T ∗

− Tu ) (Tb − Tu )

and

β 3 2 β(θ−1) ψ [e − e−ε ], 4 w(θ ˙ , ψ) = 0, w(θ ˙ , ψ) ≈

ε ≡ β(1 − θ ∗ ) = β

(9.1.7)

− T ∗)

(Tb . (Tb − Tu )

Flammable mixtures (in which planar flames can propagate), Tb > T ∗ , correspond to θ ∗ < 1 (ε > 0), and nonflammable mixtures, Tb < T ∗ , correspond to θ ∗ > 1 (ε < 0). The burnt gas and the fresh mixture are, respectively, inside and outside the sphere delimited by the flame surface. Assuming that there is no heat source at the origin, the boundary conditions are r = 0: ψ = 0, r2 dθ/dr = 0,

r → +∞: θ = 0, ψ = 1.

(9.1.8)

We look for an asymptotic solution in the distinguished limit β → ∞,

ε = O(1),

β[(1/Le) − 1] = O(1).

(9.1.9)

In this limit the reaction rate is concentrated in a thin spherical sheet whose the nondimensional radius rf ≡ Rf /dL is an unknown eigenvalue. Simplified Model for the Radiative Losses For simplicity, thermal quenching by small radiative losses will be studied here using a simplified mechanism[1] considering only the radiative losses from the hot burnt gas, neglecting the loss in the preheated zone where the gas is considered as transparent to the radiation coming from the hot side. Introducing the reduced flame temperature θf , the ˙ = 0) takes the same solutions to (9.1.6)–(9.1.8) in the preheated zone (r  rf : h = 0, w form as in the Zeldovich solution (2.4.13): rf rf (9.1.10) r  rf : θ (r) = θf , ψ(r) = 1 − . r r Inner Solution and Kinetic Relation In the thin reaction zone, the radiative heat losses are negligible and the first-order derivative terms in (9.1.6) are, as usual, β times smaller than the second-order derivative; see (8.2.21). To leading order in the limit β → ∞, the equations in the reaction layer are ˙ , ψ), − d2 θ/dr2 = w(θ

[1]

(1/Le)d2 ψ/dr2 = w(θ ˙ , ψ).

Buckmaster J., et al., 1990, Combust. Flame, 79, 381–392.

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(9.1.11)

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Flame Kernels and Flame Balls

This leads to a relation between θ and ψ inside the reaction layer, d2 (θ + ψ/Le)/dr2 = 0

⇒

ψ = Le(θf − θ ).

(9.1.12)

Therefore, using (9.1.7), 7 8 −d2 θ/dr2 = (β 3 /4)Le2 (θf − θ )2 eβ(θf −1) eβ(θ−θf ) − e−ε−β(θf −1) . The boundary conditions, θ → θf : dθ/dr = 0, and θ  1 − ε/β: w ˙ = 0, yield 

dθ dr

2

Le2 β(θf −1) e = 2

θf

7 8 β 3 (θf − θ )2 eβ(θ−θf ) − e−ε−β(θf −1) dθ ,

(9.1.13)

θ ∗ =1−ε/β

where, in the limit (9.1.9), β(θf − 1) = O(1); see (9.1.19). Introducing f = O(1), a quantity of order unity that measures the difference between the crossover temperature and the flame temperature (see the definitions in (9.1.7)),   Tf − T ∗ (9.1.14) = β(θf − θ ∗ ) = ε + β(θf − 1)  0, f ≡ β Tb − Tu  and also introducing the function J(f ) ≡ 0 f 2 (e− − e−f )d/2 > 0, where  = β(θf − θ ) is the reduced integration variable, the slope of the temperature at the exit from the inner layer (9.1.13) can be written (dθ /dr)2r=rf + = eβ(θf −1) Le2 J(f ). The function J(f ) > 0 is an increasing function of f , . / 3f 2f −f J(f ) = 1 − e + 1 + f + , 2! 3!

(9.1.15)

(9.1.16)

that varies from J = 0 at the critical condition Tf = T ∗ , f = 0, to J = 1 far from the flammability limits, in the domain of flammable mixtures f  1. Near the critical condition, Tf close to T ∗ , 0 < f 1, this expression reduces to J(f ) ≈ 4f /4!.

(9.1.17)

Matching (9.1.15) with the external solution (9.1.10) yields the nondimensional radius rf of the spherical solution in terms of the flame temperature θf ,   β β (9.1.18) 1/rf = (Le/θf )e 2 (θf −1) J(f ) ≈ Le2 e 2 (θf −1) J(f ). This kinetic relation is valid to leading order in the limit (9.1.9). The flame temperature θf remains to be determined.

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435

9.1.2 Adiabatic Flame Kernel Adiabatic Flame Temperature and Limiting Composition In the adiabatic case (h = 0), the flame temperature Tf of the flame kernel is the same as in the Zeldovich kernel (2.4.7). Eliminating w ˙ from (9.1.6),    ψ d 2d r θ+ = 0, dr dr Le and integrating from r = 0 to r = ∞ gives  d(θ + ψ/Le)/dr = 0 since, in the absence of a source at the origin, r2 d(θ + ψ/Le)/drr=0 = 0. Integrating once more using the boundary condition (9.1.8) at r → ∞ (θ = 0, ψ = 1) yields θ+

1 ψ = Le Le

⇒

θf ≡

Tf − Tu 1 , = Tb − Tu Le

β(θf − 1) = β(

1 − 1), Le

(9.1.19)

where (2.4.7), θf = 1/Le, is obtained using (9.1.8) at r = 0, (θ = θf , ψ = 0). Then, according to (9.1.14), the reduced difference between the flame temperature Tf of an adiabatic flame and the crossover temperature T ∗ can be expressed in terms of the composition through ε, defined in (9.1.7), in the form     1 1 − θ∗ = ε + β −1 . (9.1.20) f = β Le Le The Lewis number dependence of Tf in (9.1.19) is of great importance for the critical composition beyond which a mixture cannot be ignited. The adiabatic flame temperature in spherical geometry is smaller (larger) than the adiabatic flame temperature in planar geometry, Tf < Tb (Tf > Tb ), when the thermal diffusivity is larger (smaller) than the molecular diffusivity of the species limiting the reaction, Le > 1 (Le < 1). According to (9.1.7), steady spherical solutions cannot exist for Tf  T ∗ . The relation (8.5.61) between the mass fraction of the limiting species YRu and the adiabatic flame temperature Tb takes the general form Q YRu = cp (Tb − Tu ), MR

(9.1.21)

where R denotes the limiting component, namely O2 for the rich flames of Section 8.5.5, or the fuel for lean flames. For Le = 1, it is convenient to introduce the limiting mass fraction s∗ for which the flame temperature of the kernel is equal to the crossover temperature, YRu Tf = T ∗ . According to (9.1.19), the adiabatic laminar flame temperature Tb of the mixs∗ is given by (T − T ) = Le (T ∗ − T ), (Q/M )Y s∗ = ture with initial composition YRu b u u R Ru ∗ , Le cp (T ∗ − Tu ). It is different from that of the composition at the flammability limit, YRu ∗ given by Tb = T , Q s∗ Q ∗ s∗ ∗ Y = cp (T ∗ − Tu ), Y = Le cp (T ∗ − Tu ) ⇒ YRu = Le YRu . MR Ru MR Ru

(9.1.22)

For Le = 1, the domains of existence of steady solutions for flames in spherical and in s∗ , the flame kernel (steady spherical solution) planar geometry are different: if YRu < YRu

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Flame Kernels and Flame Balls

∗ , the does not exist and isobaric ignition is impossible. On the other hand, if YRu < YRu planar wave does not exist (the mixture is nonflammable). According to (9.1.22), the critical s∗ , is inside (outside) the flammable domain, composition for existence of a flame kernel, YRu s∗ > Y ∗ (Y s∗ < Y ∗ ), if Le > 1 (Le < 1). A key point for ignition near the flammability YRu Ru Ru Ru s∗ , as shown limit is that the radius of the flame kernel diverges at the critical composition, YRu next.

Divergence of the Radius of the Adiabatic Kernel The radius of the adiabatic flame kernel is given by (9.1.18), valid to leading order in the limit (9.1.9),  β 1 (9.1.23) 1/rf ≈ Le2 e 2 ( Le −1) J(f ). It can be expressed in terms of θ ∗ ≡ (T ∗ − Tu )/(Tb − Tu ) or in terms of the initial composition YRu by using the definition of f in (9.1.19)–(9.1.22),   s∗ ) β(YRu − YRu 1 − θ∗ = > 0. (9.1.24) f = β Le Le YRu For J of order unity, the radius Rf = rf dL of the flame kernel is of the same order as the thickness of the adiabatic ZFK flame, dL , rf = O(1). According to (9.1.24), for mixtures far from the flammability limits, β(Tb − T ∗ )/(Tb − Tu )  1 ⇒ f  1, so according to (9.1.16), J → 1 and the Zeldovich result RfZ is recovered, s∗ s∗ )  YRu β(YRu − YRu

⇒

β

1

dL /RfZ ≈ Le2 e 2 ( Le −1) ,

(9.1.25)

where the difference with (2.4.11) concerning the exponent of Le comes from the reaction order ϑ = 2 in (9.1.7). To summarise, far from the flammability limits, the radius of the flame kernel Rf is of the order of but larger than the laminar flame thickness dL for Le ≡ DT /D > 1 and smaller for Le < 1. According to (9.1.17)–(9.1.18), the reduced radius of the flame kernel (9.1.23), rf , diverges when the composition of the initial mixture approaches the critical condition s∗ ,  → 0+ , YRu → YRu f √ 2 6 s∗ s∗ − β ( 1 −1) (9.1.26) 0 < β(YRu − YRu ) YRu ⇒ rf ≈

 2 e 2 Le , 1 Le2 β Le − θ∗ where Le2 in the prefactor can be replaced by 1 in the limit (9.1.9). For compositions s∗ there is no steady spherical solution less energetic than the critical condition YRu < YRu and the reactive mixture cannot be ignited by quasi-isobaric means. When the thermal diffusivity is larger than the molecular diffusivity of the limiting species, Le ≡ DT /D > 1, the divergence of the radius occurs in the domain of flammable mixtures, since, according to (9.1.22), the mixture is more energetic at the critical composition than at the flammability s∗ > Y ∗ . Therefore, when Le > 1, there are mixtures in which a planar flame can limit, YRu Ru s∗ ∗ 1, Tf < Tb . (b) Le < 1, Tf > Tb .

explains the ignition difficulties of lean hydrocarbon and rich hydrogen mixtures. In the opposite case, for rich hydrocarbon or lean hydrogen mixtures, the divergence occurs in s∗ < Y ∗ . Therefore, when Le < 1, steady spherical nonflammable mixtures, Le < 1 ⇒, YRu Ru s∗ < Y ∗ solutions exist in nonflammable mixtures, YRu Ru < YRu . Moreover, in the domain of flammable mixtures with Le < 1, the radius of the kernel is smaller than the flame thickness, explaining why these mixtures are easily ignited. These results are summarised in Fig. 9.1. Consider the flame speed Uε and the flame thickness dε ≡ DTu /Uε of a planar flame that is at a distance ε from the crossover condition; see (9.1.4) for ε as a function of Tb . As the flammability limit is approached, Tb → T ∗ (θ ∗ → 1), the parameter ε decreases to zero, the flame speed Uε decreases to zero (see (8.5.80)), and the ratio dε /dL increases to infinity as UL /Uε , where dL and UL are the thickness and the velocity of the planar flame far from the flammability limits (ε → ∞),   dε 1 (Tb − Tu )2 ∝ lim ∗ 2 ≈ 2 . (9.1.27) lim ∗ Tb →T Tb →T  dL β (Tb − T ∗ )2 f According to (9.1.26), the divergence of the spherical flame kernel, rf ≡ Rf /dL , occurs at the critical condition, namely for Tf → T ∗ , and according to (9.1.19), this critical condition 1 )(Tb − Tu ). For Le > 1 this is a positive quantity and corresponds to (Tb − T ∗ ) → (1 − Le so, according to (9.1.9) and (9.1.27), the thickness of the planar flame dε remains finite at the critical condition (Tf = T ∗ , θ ∗ = Le−1 ) and not much larger than its value dL far from the flammability limit; see Fig. 9.1a. For Le < 1, at the flammability limit (Tb = T ∗ ) where the thickness of the planar flame diverges (dε /dL → ∞), the radius of the flame kernel Rf remains smaller than dL . The critical condition where Rf /dL → ∞ corresponds to a composition beyond the flammability limit; see Fig. 9.1b.

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Flame Kernels and Flame Balls

9.1.3 Nonadiabatic Flame Kernels In the same way as for planar flames, flame kernels are also quenched before the flammability limit by radiative losses (thermal quenching). Near the flammability limit, where the radius of the adiabatic flame kernel increases, infinitesimal losses are sufficient to quench the flame kernel at a finite radius. The quenching mechanism can be studied by considering small radiative losses from the hot burnt gas, neglecting radiation from the preheated zone for simplicity.[1] Assuming small losses, the temperature distribution in the burnt gas is nearly uniform and, to first approximation, the quantity h in (9.1.6) can be considered constant in the burnt gas, and zero in the preheated zone, r < rf : h = cst. ≡ hb ,

r > rf :

h = 0.

(9.1.28)

The equation for the reduced temperature in the burnt gas then takes the form θ = hb , d2 (rθ )/dr2 = rhb . Using the boundary condition (9.2.2) at r = 0, r2 ∂θ/∂r = 0, the solution is r < rf : θ ≈ (r2 − rf2 )hb /6 + θf ,

dθ/dr|r− = rf hb /3. f

(9.1.29)

The analysis is again carried out in the limit (9.1.9) with the following order of magnitude for the small radiative loss, β → ∞:

hb = O(1/β),

rf = O(1),

(9.1.30)

so that, according to (9.1.29) the temperature gradient in the burnt gas is of order 1/β, dθ/dr|r− = O(1/β), and the asymptotic method presented in Section 8.2.4 may be used. f The solutions in the preheated region, r > rf , are the same as in (9.1.10). At r = rf , the first relation in (8.2.52) yields the flame temperature,

(9.1.31) rf hb /3 = 1/Le − θf /rf ⇒ β(θf − 1) = β(1/Le − 1) − rf2 βhb /3. The analysis of the thin reaction layer is similar to that leading to (9.1.18) where f is still given by (9.1.14) but the nondimensional flame temperature, β(θf − 1), given by (9.1.31), is smaller than its adiabatic value (9.1.19). Denoting f 0 the adiabatic value (9.1.24) of f , Equations (9.1.14) and (9.1.31) yield   s∗ ) (YRu − YRu 1 2 βhb ∗ (9.1.32) , where f 0 ≡ β =β −θ f =  f 0 − r f 3 (LeYRu ) Le s∗ = Le Y ∗ ; see (9.1.22). According to (9.1.14),  is always positive, θ > θ ∗ ⇒ and YRu f f Ru f > 0, rf2  3f 0 /(βhb ), and, in contrast to the adiabatic case (hb = 0), the kernel radius cannot diverge in the presence of nonzero heat loss (hb = 0). Equation (9.1.18) provides an implicit relation for rf expressed in terms of the initial composition (through f 0 ) and the heat loss coefficient βhb = O(1),  β 1 −r2 βh /6 1/rf = e 2 ( Le −1) e f b Le2 J(f 0 − rf2 βhb /3), (9.1.33)

[1]

Buckmaster J., et al., 1990, Combust. Flame, 79, 381–392.

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439

where Le2 in the prefactor can be replaced by 1 in the limit (9.1.9). This equation has no solution for a large intensity of heat loss βhb . It simplifies in two extreme situations, far from and near to the flammability limit. Critical Radius Far from the Flammability Limit, T ∗ Tb Consider flame kernels in mixtures far from the flammability limits, θ ∗ < 1, such that β(1 − θ ∗ )  1. For simplicity, consider the case for which the flame temperature Tf is not too far from Tb (θf not too far from 1) so that f ≡ β(θf − θ ∗ )  1. Then, according −rf2 βhb /6

to (9.1.16), J ≈ 1, and Equation (9.1.33) reduces to rf 0 /rf ≈ e 1 − β2 ( Le −1)

, where, according

to (9.1.25), rf 0 = e is the nondimensional radius of the adiabatic kernel far from the flammability limit, rf 0 = O(1) for β(Le − 1) = O(1). Equation (9.1.33) for rf can be written 6 6 βhb βhb −X 2 , H∞ ≡ rf 0 , (9.1.34) = H∞ , where X ≡ rf Xe 6 6 /Le2

2

X = (rf /rf 0 )H∞ , and the radius increases with X, rf /rf 0 = eX . The result is plotted in Fig. 9.2a. The flame kernel extinguishes for an intensity √ of heat loss and a critical radius √ ∗ ≡ 1/ 2e and X = X ∗ ≡ 1/ 2, respectively. This yields a critical rf∗ given by H∞ = H∞ √ radius rf∗ at thermal quenching of the order of that of the adiabatic kernel, rf∗ /rf 0 = e, βh∗b = 1/(e rf20 ). For a smaller intensity of heat loss, hb < h∗b , there are two branches of solutions: the first (X < X ∗ ) corresponds to rf 0 < rf < rf∗ , and tends to the adiabatic solution as the heat loss decreases to zero, H∞ → 0: X/H∞ → 1, so that rf → rf 0 ; the second (X > X ∗ ) corresponds to rf > rf∗ . The tail of this latter branch seems nonphysical since rf /rf 0 → ∞ when the heat loss vanishes βhb → 0 (H∞ → 0: X → ∞). On the first √ branch rf increases with hb , from rf 0 up to rf∗ = e rf 0 , so that the flame kernel is never much larger than the thickness of the planar adiabatic flame. (a)

(b)

3.0

1.0

2.0 0.5 1.0

0.0 0.0

0.2

0.4

0.6

0.0 0.0

0.1

0.2

0.3

0.4

Figure 9.2 Turning points of nonadiabatic flame kernels. Nondimensional flame radius H∞ versus nondimensional heat loss. (a) Far from the kinetic flammability limit. (b) Close to the flammability limit.

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Flame Kernels and Flame Balls

Critical Radius Near to the Flammability Limit, β(1 − θ ∗ ) 1 Generally speaking, the flame temperature Tf must be larger than the crossover temperature 1 − θ ∗ )  rf2 βhb /3. Consider mixtures near to the flammability limits T ∗ , f  0, β( Le 1 β(1 − θ ∗ ) 1 for β|Le − 1| 1 so that 0 < f 0 ≡ β( Le − θ ∗ ) 1. The quantity f = 2 f 0 − rf βhb /3 is small, f 1, and, according to (9.1.17), J ≈ 4f /4!. Equation (9.1.33) yields an equation for rf ≡ Rf /dL in the form   2f 1 1 Le βhb Le 1 (9.1.35) ⇒ √ + √ = √ − θ∗ , = Le2 √ rf 2 β rf rf 3 Le 4! (2 6)1/2 (2 6)1/2

where β(1 − θ ∗ ) 1 has been used. In a nondimensional form (9.1.35) yields   βh βhb b Z(1 − Z 2 )2 = H∞ , where Z ≡ rf , H∞ = rf 0 , (9.1.36) 3f 0 3f 0 √ and where rf 0 = 2 6/(Le2 2f 0 ) is the large nondimensional radius of the adiabatic flame kernel near the flammability limit (see (9.1.26)), f 0 is given in (9.1.32) and Z = (rf /rf 0 )H∞  1. The result is plotted in Fig. 9.2b. According to (9.1.36), rf /rf 0 = 1/(1 − Z 2 )2 is an increasing function of Z. The physical domain of variation of Z is Z ∈ [0, 1], where the boundary values Z = 0 and Z = 1 correspond to the adiabatic case βhb = 0 with, of solutions and flame respectively, rf = rf 0 and rf → ∞. As before there are two branches √ ∗ ≡ 42 /55/2 for Z = Z ∗ ≡ 1/ 5. The corresponding critical quenching occurs at H∞ = H∞ radius rf∗ is of the order of the radius of the adiabatic kernel, rf∗ /rf 0 = (5/4)2 . The heat 1 loss intensity that produces the flame quenching, βh∗b = ( 25 )5 Le4 [β( Le − θ ∗ )]5 , decreases quickly and becomes very small as the critical condition is approached, θ ∗ → 1/Le. The flame temperature at quenching Tf∗ is larger than but close to the crossover temperature T ∗ ,

 1 θ ∗ < θf∗ < 1/Le; it is given by β(θf∗ − θ ∗ ) = (4/5)Le β( Le − θ ∗ ) . The first branch of solution, Z < Z ∗ , rf < rf∗ , tends to the adiabatic solution when Z → 0, H∞ → 0, namely when βhb → 0, since rf /rf 0 = Z/H∞ → 1. The tail of the second branch, Z > Z ∗ , rf > rf∗ , does not seem physical since it corresponds to an infinite radius of the flame

1 kernel in the absence of heat losses, rf /rf 0 → ∞. For Le < 1, β( Le − 1) 1, these results ∗ concern both flammable (θ < 1) and also nonflammable mixtures (θ ∗ > 1); see Fig. 9.1b.

9.2 Stability Analysis of Spherical Flame Kernels Focusing attention on the diffusive effects, in this section we analyse the stability of flame kernels in the thermo-diffusive approximation, neglecting hydrodynamic effects associated with the change of density. In the conditions of slow evolution, the convection terms become negligible in front of the diffusion terms.[1,2] Ignoring hydrodynamic effects, the [1] [2]

Joulin G., 1985, Combust. Sci. Technol., 43, 99–113. Joulin G., 1987, SIAM J. Appl. Math., 47(5), 998–1016.

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441

dynamics of flame expansion is controlled by the unsteady version of the reaction–diffusion equations (9.1.6). Attention is focused on the vicinity of the flame kernels studied in the preceding section. The stability analysis of flame kernels differs from that of planar flame because the flame structure is different; the temperature and species concentration decrease with the radius R as Rf /R and not as e−(R−Rf )/dL .

9.2.1 One-Dimensional Instability of Adiabatic Flame Kernels In order to study the initiation of flames in reactive mixtures that are beyond the flammability limit, and for which the experimental results are more interesting, the stability analysis is performed here using a model (9.1.5) that includes kinetic extinction. It is an extension of a previous analysis[3] based on a simple Arrhenius law. Formulation Using the reference length and time scales (8.2.27)–(8.2.28) of the ZFK model for Le = 1, r = R/dL , τ = t/τL , the equations of the thermo-diffusive model (2.4.1) take the form     1 ∂ ∂ψ 1 1 ∂ ∂θ 2 ∂θ 2 ∂ψ − 2 r = w, ˙ − r = −w, ˙ ∂τ ∂r ∂τ Le r2 ∂r ∂r r ∂r which may be written ∂(rθ ) ∂ 2 (rθ ) 1 ∂ 2 (rψ) ∂(rψ) − − = r w, ˙ = −rw, ˙ (9.2.1) ∂τ ∂τ Le ∂r2 ∂r2 where, when a crossover temperature is taken into account, the reaction rate w(θ ˙ , ψ) is given in (9.1.7). The boundary conditions are the same as in (9.1.8), r = 0: ψ = 0, r2 ∂θ/∂r = 0,

r → +∞: θ = 0, ψ = 1.

(9.2.2)

The study will be carried out in the limit (9.1.9). The reaction rate is negligible outside the spherical reaction sheet of radius rf (τ ) ≡ Rf (τ )/dL . The problem reduces to solving the linear equations 1 ∂ 2 (rψ) ∂(rψ) ∂(rθ ) ∂ 2 (rθ ) − − = 0, = 0, (9.2.3) ∂τ ∂τ Le ∂r2 ∂r2 in two external regions: in the burnt gases, r < rf (τ ), where ψ = 0, with the boundary condition at r = 0, and in the preheated zone, r > rf (τ ), with the boundary condition at r → ∞. The solution is given by the jump conditions at r = rf , obtained by solving the inner Equations (9.1.11). Anticipating that the slope of the temperature at the burnt gas side of the inner region, (∂θ/∂r)rf − , is much smaller than that at the other side, (∂θ/∂r)rf + , (∂θ/∂r)rf − (∂θ/∂r)rf + [3]

= O(1/β),

Deshaies B., Joulin G., 1984, Combust. Sci. Technol., 37, 99–116.

17:10:09 .011

(9.2.4)

442

Flame Kernels and Flame Balls

the inner solution is the same as for the steady-state solution. Therefore the jump relations are the first equation in (8.2.52) and (9.1.15), where f = β(θf − θ ∗ ),     ∂θ 1 ∂ψ rf + ∂θ + = 0, = eβ(θf −1)/2 Le J(f ), (9.2.5) ∂r Le ∂r rf − ∂r r=rf + where the second equation is valid to leading order in the limit β → ∞ and the first holds up to first order.[1] The steady solution for spherical kernels will be denoted by an overbar, and the external solutions (9.1.10) are written as r  rf :

θ (r) = θ f rf /r,

r < rf :

θ (r) = θ f ,

ψ(r) − 1 = −rf /r,

(9.2.6)

ψ(r) = 0,

(9.2.7)

where θ f = 1/Le for an adiabatic kernel (see (9.1.19)), and rf is given by (9.1.18). Considering the stability of spherical kernels, we look for linear solutions in the form rf (τ ) = rf + δrf (τ ), δrf (τ ) = eσ τ rˆf , θf (τ ) ≡ θ (rf , τ ) = θ f + eσ τ θ˜f rˆf ,

rf (τ ) = rf + eσ τ rˆf , ˜ rf , θ (r, τ ) = θ (r) + e θ(r)ˆ

˜ rf , ψ(r, τ ) = ψ(r) + e ψ(r)ˆ

στ

στ

(9.2.8) (9.2.9)

where the perturbed terms are written as quantities that are proportional to the perturbation to the radius, rˆf , so that the latter will disappear from the linear analysis in which the complex number σ is an eigenvalue of the problem. The flame kernel is unstable if Re(σ ) > 0. External Solutions According to (9.2.3), the linear equations in the external zones are ˜ 1 d2 (rψ) d2 (rθ˜ ) = 0, σ rψ˜ − = 0. 2 2 Le dr dr The solutions that satisfy (9.2.2) take the form σ rθ˜ −

(9.2.10)

√

√

e−r Leσ e−r σ ˜ r  rf : θ˜ (r) = A+ , ψ(r) =B , (9.2.11) r√ r √ e+r σ − e−r σ ˜ , ψ(r) = 0, (9.2.12) r < rf : θ˜ (r) = A− r √ where, by definition, Re( σ )  0. The constant of integration B is obtained from the continuity of ψ at the reaction sheet, ˜ f ) = 0, ψ(rf , τ ) = ψ(rf ) + eσ τ rˆf (dψ/dr)r=rf + eσ τ rˆf ψ(r √

obtained from (9.2.9) where ψ(rf ) = 0, yielding B = erf Le σ . In the same way, the two other constants of integration, A+ and A− , are expressed in terms of the perturbation of the temperature at the reaction sheet θ˜f , by using the relation (dθ /dr)|rf + θ˜ (rf ) = θ˜f valid [1]

Joulin G., Clavin P., 1979, Combust. Flame, 35, 139–153.

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443

on both sides of the reaction sheet and obtained from (9.2.8)–(9.2.9). Equations (9.2.11)– (9.2.12) yield √

r  rf : r < rf :

e−(r−rf ) σ ˜ = (θ f + rf θ˜f ) , θ(r) r √ √ r θ˜ (e+r σ − e−r σ ) ˜ = f f √ √ , θ(r) r (e+rf σ − e−rf σ )

e−(r−rf ) ˜ ψ(r) =− r

√

˜ ψ(r) = 0.

Le σ

,

(9.2.13) (9.2.14)

Jump Relations Using the general definition (9.1.14) of f , the linear approximation of the jump relations (9.2.5) at rf = rf + rˆf eσ τ takes the form 

1 d2 ψ d2 θ + Le dr2 dr2

r=rf +

(dθ˜ /dr + d2 θ /dr2 )|r=rf + dθ /dr|r=rf +

r=rf −



1 dψ˜ dθ˜ + + dr Le dr

β θ˜f , =b 2

 b≡ 1+

r=rf + = 0,

(9.2.15)

r=rf −

dJ/df J(f )

 ,

(9.2.16)

where f and J(f ) are given in (9.1.32) and (9.1.16), J > 0, dJ/df > 0, b  1. Far from the critical condition, f  1, J ≈ 1, Rf /dL = O(1). Near the critical condition where the 4

radius of the flame kernel Rf becomes much larger than dL , 0 < f 1, J ≈ f /(4! ). The coefficient b in (9.2.16) takes the form f 1: b ≈ 4/f ;

f  1: b ≈ 1.

(9.2.17)

Results for Adiabatic Kernels According to (9.2.6)–(9.2.7), the first term in (9.2.15) is zero in the adiabatic case, and Equations (9.2.15) and (9.2.16) yield (dθ˜ /dr)|rf + +

1 ˜ (dψ/dr)| rf + = (dθ˜ /dr)|rf − , Le bθ f 2θ f (dθ˜ /dr)|rf + = − β θ˜f − 2 , 2rf rf

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(9.2.18) (9.2.19)

444

Flame Kernels and Flame Balls

where the external solution (9.2.6) for θ(r) has been used. The slopes of the unsteady profiles are computed from (9.2.13) and (9.2.14),  √ 

σ + 1/rf , (dθ˜ /dr)|rf + = − θ˜f + θ f /rf    √     √ dθ˜  1 σ 1 dψ˜  1 1 1 ˜ , + θ − θ σ + − − θf √  +  = f f 2 dr  Le dr  Le rf rf r Le f rf + rf + . √ √ / √ θ˜f e+rf σ + e−rf σ √ √ (dθ˜ /dr)|rf − = − + σ θ˜f . rf e+rf σ − e−rf σ (9.2.20) The two first equations and (9.2.19) are still valid in the nonadiabatic case. For adiabatic √ kernels, θ f = 1/Le. Equation (9.2.18) gives rf θ˜f in terms of rf σ , √ √  β( Le − 1) ˜ 1 − e−2rf σ , βrf θf = (9.2.21) 2Le where rf is given by (9.1.18). Notice that the basic assumption for validity of the asymptotic √ analysis in the limit (9.1.9), β θ˜f = O(1) (see Section 8.2.4), is valid for Re( σ ) > 0, since rf is of order unity and diverges when f → ∞. Equation (9.2.19) gives a second relation √ between β θ˜f and rf σ . To leading order in the limit (9.1.9), this relation is √  √ 2Le rf σ b 1− βrf θ˜f , (9.2.22) (rf σ − 1) = 2 bβ √ where the second term in the brackets is negligible when rf σ is anticipated to be of order √ unity. The equation for rf σ that is obtained by introducing (9.2.21) into (9.2.22) involves two nondimensional parameters, rf and a, defined below, √

√  √ b β(1 − Le) , (9.2.23) a≡ (1 − rf σ ) = a 1 − e−2rf σ , 4 Le where b is defined in (9.2.16) and |a| = O(b) in the limit (9.1.9). Equation (9.2.23) has a positive root whatever be the sign of a. √ • For a Lewis number unity, Le = 1, a = 0, and rf σ = 1.  1: b ≈ 1 (see (9.2.17)), |a| = O(1), and the • Far from the critical condition, f √ positive root is of order unity, 0 < rf σ = O(1). • Near the critical condition f 1: b ≈ 4/f  1, and in the limit (9.1.9), |a|  1, |a| = O(b). The order of magnitude of the positive root then depends on the sign of a: ◦ for Le < 1, a  1, r √σ is small, r √σ ≈ 1/2a, 0 < r √σ = O(1/b), f f f ◦ for Le > 1, a < 0, the positive root is large, r √σ ≈ |a|, r √σ = O(b). f f The results show that, as was anticipated at the beginning of Section 9.1, the adiabatic flame kernel is always unstable, δrf (τ ) = eσ τ rˆf , σ > 0. Any initial perturbation to the kernel radius is amplified; the size of the flame increases (decreases) with time if its initial radius

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9.2 Stability Analysis of Spherical Flame Kernels

445

is larger (smaller) than that of the steady-state solution. In all cases, the approximation √ rf σ bβ, used to obtain (9.2.23), is satisfied in the limit (9.1.9). Finally, the results √ for rf σ and for rf in (9.1.18) show that the slopes of the reduced temperature given in (9.2.20) satisfy the condition (9.2.4) for the validity of the asymptotic method.

9.2.2 One-Dimensional Stability of Nonadiabatic Flame Kernels. Flame Balls The stability analysis of the nonadiabatic flame kernel presented here is similar to the preceding analysis. It is an extension of the original analysis[1] modified to include chemical kinetic extinction. In principle, small radiative heat losses, h, of order 1/β introduce a term of the form θ˜ (dh/dθ )|θ=θ into the right-hand side of the linear approximation of the unsteady version of the first equation in (9.1.6). Under the approximation of zero radiative losses in the preheated zone (r > rf ), this additional term is negligible in the burnt gas (r < rf ) because it introduces a correction of order 1/β 2 . The external equations are the same as (9.2.10) for r < rf , and the solutions (9.2.11)–(9.2.12) are still valid to leading order. The neglected term would introduce a nonuseful correction of order 1/β to the nondimensional linear growth rate σ , which is of order unity in the present analysis, as we will see. The effects of the heat loss come from perturbation terms induced by nonadiabatic solutions of the steady state (9.1.29)–(9.1.31), dθ /dr|r− = rf hb /3, f

d2 θ /dr2 |r− = hb /3, f

θ f = 1/Le − r2f hb /3.

The two jump relations (9.2.15)–(9.2.16) are still valid, where, according to (9.1.14) and (9.1.32), f = f 0 − r2f βhb /3 should be used in the expression for b in (9.2.16). The expressions for θ and ψ in terms of θ f for r > rf are the same as in (9.1.10). The first term in (9.2.15) is now equal to −hb so that (dθ˜ /dr)|rf + +

1 ˜ (dψ/dr)| rf + = (dθ˜ /dr)|rf − + hb Le

(9.2.24)

replaces (9.2.18), where the left-hand side is the same as previously, but (dθ˜ /dr)|rf − = −

   +z  θ˜f e + e−z hb hb √ ˜ , + σ θf − rf + rf 3 3 e+z − e−z

where for simplicity we have introduced the notation z ≡ rf provides an expression for θ˜f

√

σ . Equation (9.2.24) then

√ √    √ 1 + e−2z √ hb r f ( Le − 1) σ θ˜f − − hb , σ + σ = rf 3 Le 1 − e−2z

[1]

Buckmaster J., et al., 1990, Combust. Flame, 79, 381–392.

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(9.2.25)

(9.2.26)

446

Flame Kernels and Flame Balls

which is of order 1/β in the limit (9.1.9) with (9.1.30). Equation (9.2.26) is an extension of (9.2.21) to the nonadiabatic case, r2f βhb = 0. It can be rewritten 2

[rf β θ˜f − r2f βhb /3] [1 − e−2z ]

√ 2 β( Le − 1) rf βhb = − . Le z

(9.2.27)

The kinetic condition (9.2.22), obtained from (9.2.16) and (9.2.19), is still valid at the leading order in the limit (9.1.9), (z − 1) = brf β θ˜f /2. The two equations (9.2.26) and (9.2.28) provide an equation for σ .    3η/2  z − (η + 1) = − a + 1 − e−2z , z √ √ 2 where z = rf σ , η ≡ (b/6)rf βhb > 0, a ≡ (b/4)β(1 − Le)/Le,

(9.2.28)

(9.2.29)

which is an extension of (9.2.23), obtained by introducing a new parameter η measuring the √ intensity of the heat loss. Recalling that Re( σ )  0 by definition, the root with Re(z) < 0 must be rejected, so that the stable domain corresponding to exponential decays of the disturbances is defined by two conditions, Re(z2 ) < 0 and Re(z)  0. Relation (9.2.29) must be solved numerically; however, it is possible to obtain an analytical expression for the stability limits near the point (a = 1/4, η = 1/2) where the solution to (9.2.29) is small, |z| 1. Expanding the term e−2z in powers of z leads to a quadratic equation for z, 2(η − a)z2 + (1 + 2a − 3η)z + 2η − 1 ≈ 0, whose solution is  (9.2.30) z ≈ 3(η − 1/2) − 2(a − 1/4) ± 2 (a − 1/4)2 − (η − 1/2), where only the leading order terms have been retained in the double limit |a − 1/4| 1 and |η − 1/2| 1. The case when z is real is unstable. When z is a complex number, (η − 1/2)  (a − 1/4)2 , the condition Re(z)  0 is satisfied for (η − 1/2)  (2/3)(a − 1/4),

(9.2.31)

and the stability condition Re(z2 ) < 0 is (η − 1/2) > 2(a − 1/4)2 .

(9.2.32)

The stability domain delimited by (9.2.31) and (9.2.32) is plotted in Fig. 9.3a. The stability domain obtained numerically from (9.2.29) is plotted in Fig. 9.3b for a large domain of parameters. It fits the analytical result near the point (a = 1/4, η = 1/2). Stable flame balls are obtained only for η  1/2. Examples of flame balls observed by Ronney[1,2] in lean hydrogen mixtures in a microgravity environment are shown in Fig. 2.23.

[1] [2]

Ronney P., et al., 1998, AIAA J., 36, 1361–1368. Kwon O., et al., 2004, In 42nd AIAA Aerospace Sciences Meeting, Reno, Paper No. 2004–0289.

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9.3 Flame Expansion at Lewis Number Smaller Than Unity (a)

447

(b)

1.0

10

0.9

8

0.8

6

0.7

4

Unstable

Stable

Stable 0.6

2 Unstable

0.5 0.0

Unstable 0

0.2

0.4

0.6

0

0.8

2

4

6

8

10

Figure 9.3 Flame ball stability limits. Unstable means that there is no exponential decay of the disturbances. (a) Analytical result (9.2.31) and (9.2.32). (b) Numerical result obtained from (9.2.29). η is a measure of the heat loss, defined below (9.2.29), and a is a function of the Lewis number, a > 0 for Le < 1; see (9.2.23) and (9.2.17).

Discussion According to the numerical √ solution plotted in Fig. 9.3b, there is a minimal value am ≈ 0.03 of a ≡ (b/4)β(1 − Le)/Le below which all the spherical solutions for nonadiabatic flame kernels are one-dimensionally unstable. This is the case for Le > 1. For Le < 1 and a > am there is a range of stable nonadiabatic flame kernels. The corresponding range of η is located above η =√1/2. The turning points denoting thermal quenching in Fig. √ 9.2 are located at X = 1/ 2 far from the flammability limit and Z = 1/ 5 near to the flammability limit. Both cases correspond to η = 1/2. The stable domain being located at η > 1/2, it concerns a part of the second branch of solutions if a > 0, namely for Le < 1. The entire first branch is unstable.

9.3 Flame Expansion at Lewis Number Smaller Than Unity The expansion of spherical flames was considered in Section 2.4.4 using a quasi-steadystate approximation. Generally speaking, this approximation is not verified in expanding flame kernels. The problem is reconsidered here. Except for the stability analysis of flame kernels presented in Section 9.2 and also the cases of flame structure close to that of planar flames, unsteady flame expansion is a tough mathematical problem to resolve analytically, even for the simplified version of spherical reaction–diffusion waves in the limit of a large activation energy where the flame front is defined by the position of the reaction sheet Rf .

17:10:09 .011

448

Flame Kernels and Flame Balls

Approximate solutions are presented in this section for expanding spherical flames whose structure does not approach that of planar flames in the long time limit (when the radius becomes much larger than the planar flame thickness). This is the case near to the flammability limits and especially in nonflammable mixtures. Attention is limited to the thermo-diffusive model (2.4.2)–(2.4.1) since it contains the basic physical ingredients of the self-extinguishing flames observed in experiments.[1] The key point for the dynamics of the front is that the temperature and species distribution of slowly growing flames in mixtures near to the flammability limits for Le < 1 have a long upstream tail reminiscent of the 1/R profiles of the steady-state solution in spherical geometry; see (2.4.13). The √ long-range thermal perturbations, DT (t − t ) at time t, generated by the unsteady motion of the flame front at an earlier time t , make the reaction zone propagate in an unsteady medium that is perturbed by the previous motion of the front. The analyses, performed so far in the context of flame initiation,[2,3] indicate that, for Le < 1, this feedback mechanism produces a specific regime of propagation quite different from that of the planar flame. Joulin’s analysis concerns the slow evolution of spherical flames governed by an Arrhenius law for small Lewis number, near to the Zeldovich kernel, Rf /RfZ = O(1), when 0
0 ∀t > 0, the solutions to Equation (9.3.24) do not tend to that of a planar wave (propagating at constant velocity) when the radius increases √ This is easily seen from the integral in (9.3.24), written in the form  τ rf → ∞. J ≡ 0 dτ  ˙rf (τ  )/ τ − τ  , because J would go to infinity as τ 1/2 if limτ →∞ ˙rf (τ ) = cst., leading to a contradiction in (9.3.24). In agreement with another study,[2] the asymptotic √ behaviour τ → ∞: rf (τ ) → (1/π ) τ (ln τ + · · · ) seems to be a better approximation. 1 √ Using the relation 0 du/ u(1 − u) = π this leads to J → (1/2)(ln τ + ...), which is not in contradiction with the leading order of (9.3.24), written in the form ln rf = J, when ln(ln τ ) is negligible in front of ln τ . However, the terms of following order in the limit τ → ∞ do not match. In any case, Equations (9.3.24)–(9.3.25) cannot be relevant in the long time limit because the flame temperature would decrease with no lower bound, rf → ∞: β(θf − 1/Le) → −∞.

[1] [2]

Joulin G., 1985, Combust. Sci. Technol., 43, 99–113. Buckmaster J., Joulin G., 1989, Combust. Flame, 78(275-289).

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Reduced radius

9.3 Flame Expansion at Lewis Number Smaller Than Unity

S =10 K=5

2

6

5

455

4

E=Ec

1 1 0

10

2 20

30

3 40

50

60

s

Reduced time

Figure 9.4 Evolution of a flame kernel ignited by an energy pulse of varying energy above and below the critical energy Ec . S and K are parameters controlling the duration and shape of the energy pulse. Reproduced from G. Joulin, 1985, Combustion Science and Technology, 43, 99–113, with permission from Taylor and Francis Ltd., www.informaworld.com.

An extension of (9.3.24), taking into account the short pulse of an energy source, yields relevant results for the ignition threshold.[1] Following (2.4.16), this extension consists in adding a time-dependent point-source term to the flame temperature in (9.3.25), and thus to the term in the exponent of (9.3.24). Near to the critical condition for flame ignition by a short pulse of sufficient energy, according to the numerical results of the extended version of (9.3.24)–(9.3.25) the transition between failure and ignition occurs effectively during a slow motion of the flame radius near to the Zeldovich radius[1] (rf = 1); see Fig. 9.4. This indicates that, although both the short and the long time limits are not accurately captured, Equations (9.3.24)–(9.3.25) contain the essential features of ignition dynamics. As mentioned in a similar analysis,[2] Equations (9.3.24)–(9.3.25) could well describe an intermediate regime preceding planar flame propagation at constant velocity. The case beyond the flammability limits, θf∗ > 1 (in nonflammable mixtures, widely used in the microgravity experiments of Ronney[3,4] ), is different because planar propagation is not possible. Flame expansion in such mixtures is studied now. Expansion of Adiabatic Flames Near to the Flammability Limit for Le < 1 The preceding analysis is extended to take into account the proximity of the crossover 1 − θ ∗ ) 1. The attention is focused on temperature in the limiting case 0  β( Le ∗ nonflammable mixtures, 1 < θ < 1/Le; the corresponding compositions are given in (9.1.24). The simplest results are obtained in the limiting conditions,   (1/Le − θ ∗ ) 1 ∗ − θ 1, = O(1), Le < 1. (9.3.26) 0 ρ + . By convention, the initially nonperturbed planar front is perpendicular to the axis Ox, and the direction of propagation (if nonzero) will be in the negative x direction. We will use a reference frame in which the nonperturbed front is placed at x = 0. The flow is thus oriented in the direction x > 0. The upstream, x < 0, and downstream, x > 0, parts of the fluid will be denoted by superscripts − and +, respectively; see Fig. 10.1. Euler Equations In the presence of an external field perpendicular to the unperturbed front, such as gravity for flames propagating upwards or downwards, the equations of motion for an incompressible inviscid fluid are Euler’s (15.2.6): ∇.u± = 0,

ρ ± Du± /Dt = −∇p± + ρ ± g(t),

(10.1.1)

g+g  (t), is a uniform, time-dependent, external field parallel to

where g(t) = g(t)ex , g(t) = the x-axis, g is the acceleration of gravity and ex the unit vector oriented along the positive

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10.1 Hydrodynamics

465

Figure 10.2 System of axes and local normal to front.

x-axis. The acceleration of gravity g is positive if it is oriented towards the burnt gas, x > 0, and negative in the opposite direction (flame propagating downwards). A fluctuating acceleration g  (t) = ua ω cos(ωt) is experienced by a flame propagating in a tube in the presence of a longitudinal standing acoustic wave of frequency ω. Assuming that the mass flux crossing the flame is not modified, the position of the flame front oscillates with the acoustic velocity −ua sin(ωt) involving a displacement da cos(ωt), whose amplitude da = ua /ω is negligible compared with the acoustic wavelength. The fluctuating velocity thus depends only on time, not on position, and Equations (10.1.1) are valid in the moving reference frame attached to the planar flame, replacing x by x − da cos(ωt) and u by u − ua sin(ωt), so that the linear acoustic flow appears only through g  (t). The solutions to Equations (10.1.1) have to satisfy two types of boundary conditions: 1. The boundary conditions on the front (15.1.45)–(15.1.47) 2. The boundary conditions at plus and minus infinity x → −∞: no disturbances,

δu− = 0,

x → +∞: disturbances remain finite,

(10.1.2) (10.1.3)

where δu is the perturbed part of u. The boundary condition (10.1.3) allows vorticity generated at the front to be convected downstream. Jump Relations at the Front Let x = α(y, t) be the equation for the perturbed front at time t; see Fig. 10.1. Let nf be the local unit vector perpendicular to the front, directed towards x > 0 (see Fig. 10.2), and let Df be the local normal velocity of the perturbed front in the reference frame of the planar front,

17:10:10 .012

466

Wrinkled Flames

⎞

⎛ αy

1 ⎟ ⎜ nf = ⎝  , − ⎠, 2 2 1 + αy 1 + αy

Df = 

α˙ t 1 + αy2

,

(10.1.4)

where we have used the notation αy = ∂α/∂y and α˙ t = ∂α/∂t. In order to simplify the presentation, we will used two-dimensional Cartesian geometry most of the time. The results are easily generalised to three dimensions. Let uf = (uf , wf ) and pf be the flow velocity and the pressure at the front. The velocity components normal and tangential to the front, un = uf .nf and wtg , can be written       un = uf − αy wf / 1 + αy2 , wtg = wf + αy uf / 1 + αy2 . (10.1.5) The normal and tangential components of the fluid velocity relative to the front, Un = un − Df and Wtg , can then be written    Un = uf − α˙ t − αy wf / 1 + αy2 , Wtg = wtg . (10.1.6) The mass flux, m− f , entering the propagating front is    − − − u− / 1 + αy2 . m− ˙ t − αy w− f = ρ Un = ρ f −α f

(10.1.7)

Conservation of mass and momentum at the front give three jump relations. To the leading order in the limit kdL → 0 they are the following. 1. Mass conservation (15.1.45), ρ − Un− = ρ + Un+ , yields     = ρ + u+ . ρ − u− ˙ t − αy w− ˙ t − αy w+ f −α f f −α f

(10.1.8)

If the front does not propagate, it is simply convected by the flow Un = 0, and Equation (10.1.8) then leads to two equations:     u− = u+ = 0. (10.1.9) ˙ t − αy w− ˙ t − αy w+ f −α f f −α f These are the usual kinematic conditions for passive interfaces.  + 2. Conservation of normal and tangential momentum, (15.1.46), (15.1.47), p + ρUn2 − = +  0, Wtg − = 0 give − p− f +ρ

 2  w− u− − α ˙ − α t y f f

 2  w+ u+ − α ˙ − α t y f f

+ = p+ f +ρ 1 + αy2 1 + αy2      −  + w− = w+ . f + αy uf f + αy uf

,

(10.1.10) (10.1.11)

A fourth condition at the front is given by the kinematic relation describing how the mass flux (flame speed) is modified by the wrinkling of the front. To leading order in the limit

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10.1 Hydrodynamics

467

kdL → 0, the front is locally planar and the normal burning velocity is the laminar flame speed UL , Un− = UL ,

+ − m− f = mf = mf ≡ ρ UL .

(10.1.12)

The conditions for passive interfaces are slightly different. If the front does not propa− gate, Un = 0, there is neither mass nor momentum flux through the front, m− f = 0, pf =   + p+ f . There is then no constraint on the tangential velocity since the relation Wtg − = 0 was obtained after simplification by ρUn . Equation (10.1.11) no longer applies and there are only three jump relations left. However, there is one less unknown. The vorticity produced at the front cannot be convected away from the front in the absence of normal velocity, so that the downstream flow remains potential and the front is a vorticity sheet.

10.1.2 Linear Analysis The linear analysis consists in studying fronts that are only slightly different from the planar solution, neglecting all nonlinear terms in the equations. For any physical quantity a, we introduce the decomposition a = a + δa, where a represents the unperturbed planar front and δa is the small perturbation to a. The position of the unperturbed planar flame is taken at x = 0, α = 0, α = δα. Linear Equations ρ − u−

ρ + u+

Let mf = = be the mass flux through the unperturbed planar front and let δu± ± and δw be the perturbations to the x and (y, z) components of the flow velocity; linearising Equations (10.1.1) in three-dimensional geometry yields ∂ ∂ ± δu + δw± = 0, ∂x ∂y   ∂ ∂ ∂ δu± = − δπ ± , ρ ± + mf ∂t ∂x ∂x   ∂ ∂ ∂ δw± = − δπ ± , ρ ± + mf ∂t ∂x ∂y with

π ± ≡ p± − ρ ± g(t)x,

(10.1.13)

δπf ± = δpf ± − ρ ± g(t)δα(y, t),

where y = (y, z) are the transverse coordinates and π is the pressure including the external field. Since the unperturbed flame front is planar and perpendicular to the flow, the unperturbed tangential velocity is zero, w± = 0. Combining Equations (10.1.13) shows that the pressure satisfies Laplace’s equation,   2 ∂ ∂2 ∂2 δπ ± = 0. + + (10.1.14) ∂x2 ∂y2 ∂z2 For small perturbations, the front is single valued, so we can write its position in the form x = α(y, t).

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Flow Perturbation In the linear approximation, there is no coupling between different modes of wrinkling, and it is sufficient to analyse the evolution of an arbitrary harmonic perturbation of a variable f by writing δf (x, y, t) = f˜ (x, t)eik.y ,

ik.y α(y, t) = α(t)e ˜ ,

where k is the transverse wave vector of the problem. Equation (10.1.14) for the pressure gives ∂ 2 π˜ ± − |k|2 π˜ ± = 0. ∂x2

(10.1.15)

Using the boundary conditions (10.1.2) and (10.1.3), the solutions to Equation (10.1.15) are π˜ ± (x, t) = π˜ f± (t)e∓|k|x .

(10.1.16)

The two quantities π˜ f± (t) are constants of integration which are independent of x but are functions of t and k. Equations (10.1.13) for the flow velocity give ρ

±



 ∂ ± ∂ +u u˜ ± (x, t) = ±|k|π˜ f± (t)e∓|k|x , ∂t ∂x ∂ u˜ ± ˜ ± = 0. + ik.w ∂x

(10.1.17) (10.1.18)

From now on k will denote the modulus of the wave vector, k = |k| = 2π/, where  is the wavelength of the perturbation. The pressure amplitudes π˜ ± (x, t) appear as second members in (10.1.17) for the x-component of velocities, u˜ ± (x, t). The general solution to (10.1.17) can then be written as the sum of a particular solution plus the general solution of the homogeneous equation with no second member. The particular solution to (10.1.17) is ∓kx , with the potential part of the velocity field, which satisfies Laplace’s equation, u˜ ± p (t)e ρ±



 d ∓ u± k u˜ ± ˜ f± (t). p (t) = ±kπ dt

(10.1.19)

Without the second member, (10.1.17) is the transport equation (∂/∂t + u± ∂/∂x)˜u± = 0, ± whose solution has the form u˜ ± r (t − x/u ). It represents the x-component of a shear flow (rotational flow) convected downstream by the mean flow. Since there is no perturba˜− = tion at minus infinity (x → −∞), the upstream flow is potential, u˜ − r (t) = 0, u − ∓kx u˜ f (t)e . However, downstream vorticity can be created by gas expansion through a propagating curved front, u˜ + r (t)  = 0. The upstream and downstream flow perturbations are thus

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characterised by three unknown functions of time, u˜ − ˜+ ˜+ p (t) and u r (t): f (t), u ⎧ − − +kx ⎪ ⎨ u˜ (x, t) = u˜ f (t)e ,   x0 d −kx ⎪ − u+ k u˜ + , ⎩ kπ˜ + (x, t) = ρ + p (t)e dt

(10.1.20)

(10.1.21)

and the transverse velocities are then given by Equation (10.1.18): ˜ − (x, t) = −k˜u− (x, t), ik.w

˜ + (x, t) = − ik.w

∂ + u˜ (x, t). ∂x

(10.1.22)

Linear Equation for the Evolution of Flame Fronts The three unknowns, u˜ − ˜+ ˜+ p (t) and u r (t), may be expressed in terms of the amplitude f (t), u of wrinkling of the front, α(t), ˜ by using three boundary conditions at the front, x = α. The linearised version of the jump relation for mass conservation, (10.1.8), along with (10.1.12) (δm− f = 0), is     + + = δu = 0, (10.1.23) ρ − δu− − α ˙ ρ − α ˙ t t f f + yielding the relation δu− ˙ t . Equations (10.1.20) and (10.1.21) then give f = δuf = α

u˜ − ˜+ ˜+ p (t) + u r (t) = f (t) = u

dα˜ . dt

(10.1.24)

The transverse derivative of the linearised equation for conservation of tangential momentum, (10.1.11), along with flow equations (10.1.20)–(10.1.22), yield   1 1 d˜u+ 1 r (t) − k2 α(t). + k˜ u k˜u+ (t) + (t) = m − ˜ (10.1.25) f p f ρ+ ρ− u+ dt Linearising (10.1.10) for conservation of normal momentum gives − + − − + + δp− ˙ t ) = δp+ ˙ t ). f + 2ρ uf (δuf − α f + 2ρ uf (δuf − α

(10.1.26)

− Using (10.1.23) in (10.1.26) leads to δp+ f = δpf , that is,

π˜ f+ − π˜ f− = (ρ − − ρ + )g(t)α(t), ˜

(10.1.27)

where π˜ f± ≡ π˜ ± (x = 0, t). Multiplying (10.1.27) by k, and using the expressions for kπ˜ ± in (10.1.20) and (10.1.21), gives     1 d 1 d − + + + k u˜ − − k u˜ p + mf ˜ (10.1.28) mf f = (ρ − ρ )kg(t)α(t). u− dt u+ dt

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Wrinkled Flames

According to (10.1.24), the quantity d˜u+ r /dt takes the form d˜u+ d2 α˜ d˜u+ p r =− + 2. dt dt dt Equation (10.1.25) then leads to an expression for the first term in the left-hand side of (10.1.28),     1 1 d2 α˜ 1 1 d + (10.1.29) − − k2 α˜ + + 2 + k˜u− − k u˜ p = −mf f , + + ρ ρ u dt u dt where, according to (10.1.24), u˜ − ˜ A linear equation for the evolution of the f (t) = dα/dt. amplitude of wrinkling, α(t), ˜ is then obtained by introducing (10.1.29) into (10.1.28), d2 α˜ dα˜ − k[(ρ − − ρ + )g(t) + (u+ − u− )mf k]α˜ = 0. + 2mf k (10.1.30) 2 dt dt When the notations of part 1 are used, ρ − = ρu , ρ + = ρb , u− = UL , u+ = (ρu /ρb )UL and mf = ρu UL , Equation (10.1.30) is identical to Equation (2.2.16), which was introduced in the first part of the book. (ρ − + ρ + )

Passive Interfaces, Rayleigh–Taylor and Parametric Instabilities For passive interfaces, as already mentioned, the flow is potential everywhere, u˜ r = 0. The interface is a vortex sheet, and there is no condition for the jump of the tangential velocity; Equations (10.1.11) and (10.1.25) are suppressed. The equation for the evolution of the interface is obtained from (10.1.30) by setting u+ = u− = mf = 0, d2 α˜ = k(ρ − − ρ + )g(t)α. ˜ (10.1.31) dt2 When g(t) = g with (ρ − − ρ + )g > 0, this equation describes the Rayleigh–Taylor (RT) instability; see (2.2.14) and Section 2.6.1. In the opposite case, (ρ − −ρ + )g < 0, it describes the oscillating waves at the surface of a liquid, known to Newton. For an oscillating force, g(t) = g +ω2 da cos(ωt), Equation (10.1.31) describes the parametric instability discovered by Faraday when he covered a horizontal plate with water and vibrated the plate.[1,2] This last phenomenon is well explained by the Mathieu equation (2.5.24); see Section 2.9.4. (ρ − + ρ + )

Darrieus–Landau (DL) Instability In the absence of external forces, g = 0, Equation (10.1.30) reduces to (2.2.4) for the amplitude of wrinkling and gives the DL dispersion relation (2.2.5) for the growth rate σ ,   1 σ −1 =A≡ −1 ± , (10.1.32) 1 + υ − υ b b UL k 1 + υb−1 where we have introduced the gas expansion ratio υb ≡ ρu /ρb = Tb /Tu . The negative root is not interesting since it corresponds a transient part of the solution that relaxes rapidly to [1] [2]

Faraday M., 1831, Philos. Trans. R. Soc. London, 121, 299–338. Drazin P., Reid W., 1982, Hydrodynamic instability. Cambridge University Press.

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zero. The growth rate is thus linearly proportional to UL k and increases as the density ratio increases. In the limit of high- and low-density contrasts, the DL growth rate reduces to  ρu  ρb : σ = Ub UL k, (ρu − ρb )/ρu 1 : σ = (Ub − UL )k/2, where Ub ≡ (ρu /ρb )UL is the burnt gas velocity. For a flame propagating upward, g = g > 0, (ρ u − ρ b )g > 0, the RT instability is superimposed on that of DL. In the limit of a high-density contrast, ρu  ρb , Ub  UL , the dominant order of Equation (10.1.30) gives (10.1.33) σ 2 + 2UL kσ − (gk + Ub UL k2 ) = 0,  leading to a growth rate σ = gk + Ub UL k2 − UL k, where the last term, −UL k, is a √ stabilising term that is weaker than the destabilising DL term, UL UL Ub . The RT √ instability is dominant at large wavelengths, k g/UL Ub , σ ≈ gk, and the DL instability √ dominates at small wavelengths, k  g/UL Ub , where σ ≈ UL Ub k. 10.1.3 Curvature Effects: A Simplified Approach The previous analysis is based on the jump relations (10.1.8)–(10.1.12) across the flame, valid to leading order in the limit of long wavelengths,  ≡ dL / → 0. Finite thickness effects may be captured by a perturbation analysis of the inner structure of wrinkled fronts, given in Section 10.3 for flames. This analysis is rather involved. Curvature effects are more easily described when gas expansion is neglected; see Section 10.2. The coupling of curvature effects and hydrodynamics may be understood in a simple but incomplete manner. This rough analysis is presented before the detailed analyses of Sections 10.2 and 10.3. Passive Interfaces The situation is simple for passive interfaces. The inner structure of the interface is microscopic, and the only modification due to curvature is the pressure jump, related to the mean radius of curvature of the front R, 1/R = 1/R1 + 1/R2 , by Laplace’s law, − (p+ f − pf ) = /R,

(10.1.34)

where  > 0 is the surface tension. The equation is written using the conventions of Section 2.3.2: the radii of curvature are defined as positive when the fluid with the + label forms a locally convex volume. In the linear approximation 1/R reduces to the Laplacian of α, 1/R ≈ ∂ 2 α/∂y2 in two-dimensional geometry, so that Equation (10.1.27) becomes 

˜ (10.1.35) π˜ f+ (t) − π˜ f− (t) = (ρ − − ρ + )g(t) − k2 α(t). Following the same method as in the previous section, the equation for evolution of the front is 

˜ (10.1.36) ˜ 2 = (ρ − − ρ + )g(t) − k2 kα. (ρ − + ρ + )d2 α/dt

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Wrinkled Flames

This equation  of evolution reduces to (10.1.31) for sufficiently long wavelengths, k kc , where kc ≡ |(ρ − − ρ + )g|/ . In the case of the RT instability, g(t) = g, (ρ − − ρ + )g > 0, the instability is suppressed at sufficiently small wavelengths, k > kc , where neutral oscillating modes appear. In the absence of an acceleration term, Equation (10.1.36) describes capillarity waves whose frequency ω and phase velocity vφ = ω/k are   ω = k3/2 /(ρ − + ρ + ), vφ = k1/2 /(ρ − + ρ + ). Flames A qualitative physical insight can be obtained using a simple approach, retaining the change in flame speed produced by curvature of the front. The mass flux is no longer constant along the wrinkled flame front, − m− f = mf + δmf (y, t),

δm− ˜ f (t)eik.y + c.c., f (y, t) = m

where δm− f (y, t) is a function of the displacement of the front α(y, t), so that, in the linear ˜ approximation, the quantity m ˜− f (t) depends on the wavelength and is proportional to α(t). [1] The simplest idea is to assume that the change in mass flux is proportional to the mean curvature of the front, as suggested by (2.3.10), retaining only the first term in 1/R for simplicity, M = Mc : 2 δm− m ˜− ˜ (10.1.37) f /mf ≈ −MdL /R, f (t)/mf ≈ MdL k α(t).   − δu− − α Since δm− ˙ t , this equation shows that a diffusion-induced motion of the f = ρ f flame front, similar to (2.2.7), is added to the convective motion. Diffusive transport inside the wrinkled flame structure is involved in this mechanism. The jump relation for mass conservation (10.1.23) is replaced by     − + + − + δu = δm δu ≡ δmf = 0. = ρ − α ˙ = ρ − α ˙ (10.1.38) δm− t t f f f f

Consequently, Equation (10.1.24) is replaced by the two equations u˜ − f (t) =

m ˜ f (t) dα˜ , + dt ρ−

u˜ + ˜+ p (t) + u r (t) =

m ˜ f (t) dα˜ . + dt ρ+

(10.1.39)

Equation (10.1.25) for tangential momentum is not changed. However, the modification to the mass flux, (10.1.38), introduces a new term in the Bernoulli equation (10.1.26). As a result Equation (10.1.27) for the pressure jump is replaced by   1 1 + − ˜ f (t) + (ρ − − ρ + )g(t)α(t). π˜ f − π˜ f = −2mf − − m ˜ (10.1.40) ρ+ ρ

[1]

Markstein G., 1964, Nonsteady flame propagation. New York: Pergamon.

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When (10.1.37) is introduced into (10.1.40), the extra term is similar to a surface tension term, (10.1.34). Following the same method as in Section 10.1.2, it is found that extra terms  −2mf

1 1 − − ρ ρ+

 km ˜ f (t),

˜f 1 dm mf dt

and

m ˜ f (t) ρ−

appear in the right-hand sides of Equations (10.1.28), (10.1.29) and (10.1.24), respectively. The linear equation for the evolution of the flame front (10.1.30) is modified,   dm ˜f d2 α˜ dα˜ + (ρ + ρ ) 2 + 2 mf k dt dt dt    1 1 − + + − ˜f , = k (ρ − ρ )g(t)α˜ + (u − u )mf kα˜ − 2mf − − m ρ+ ρ −

+

(10.1.41)

where the last term in the right-hand side comes from the first part of the surface tension– like term in (10.1.40). Introducing (10.1.37) into Equation (10.1.41) shows that corrections of order kdL appear when compared with (10.1.30), (ρ − + ρ + )

d2 α˜ dα˜ + 2mf k (1 + kdL M) dt dt2   = kα(ρ ˜ − − ρ + ) g(t) + u− u+ k (1 − 2kdL M) .

(10.1.42)

This is the same form of equation as (2.2.18) and (2.5.13), which were used in Chapter 2. The disturbances at small wavelengths are stabilised by a surface tension–like term if M > 0. The description is qualitatively good. However, the numerical factors in front of the kdL corrections in (10.1.42) are different from those obtained by the systematic perturbation analysis of the ZFK model in Section 10.3.4.

10.2 Thermo-Diffusive Instabilities of Planar Flames As already mentioned in Section 2.4, planar flames also experience thermo-diffusive instabilities. They result from the competition between the molecular diffusion of the chemical species limiting the reaction rate and the diffusion of heat, that is, the same mechanism involved in the flame kernels and flame balls studied in Chapter 9. In this mechanism, contrary to the case of hydrodynamic instabilities, it is the modification of the internal structure of the flame that drives the instability of the planar wave.

10.2.1 Stability Analysis The comprehension of thermo-diffusive instability is facilitated by a study of the simplified model (2.4.1) in which the hydrodynamic effects generated by gas expansion are neglected, ρu = ρb , Tu = Tb .

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Wrinkled Flames

Reduced Equations of the Thermo-diffusive Model The simplest model showing thermo-diffusive instabilities is based on the ZFK model. It involves the two reaction–diffusion equations in (2.4.1) that are the time-dependant multidimensional versions of Equations (8.2.30). They are conveniently written using the reduced space and time variables, ξ ≡ x/dL ,

η = y/dL ,

τ ≡ t/τL ,

(10.2.1)

where the length and time scales dL and τL are here those of the unperturbed planar flame for Le = 1. The term μ = Leϑ/2 in the left-hand side of (8.2.30) is replaced by unity, and the reaction rate w(θ , ψ) is (8.2.31) divided by Leϑ . In the reference frame attached to the unperturbed flame, the equations of the thermo-diffusive approximation (2.4.1) are ∂θ ∂θ + − θ = w(θ , ψ), ∂τ ∂ξ

∂ψ ∂ψ 1 + − ψ = −w(θ , ψ), ∂τ ∂ξ Le

(10.2.2)

where  is the Laplacian operator,  ≡ ∂ 2 /∂ξ 2 +∂ 2 /∂η2 and θ and ψ are, respectively, the reduced temperature and mass fraction. In order to simplify the presentation only a single transverse coordinate is used, but the results are valid in three dimensions. The streamwise boundary conditions at infinity are ξ = −∞: θ = 0, ψ = 1,

ξ = +∞: θ = 1, ψ = 0.

(10.2.3)

The boundary conditions in the transverse direction will be taken as periodic. In the limit of high values of the reduced activation energy, the reaction zone is replaced by the surface ψ = 0, whose equation can be written in the form ξ = a(η, τ ),

a ≡ α/dL .

(10.2.4)

According to the asymptotic analysis in Section 8.2.4, for a large activation energy it is sufficient to solve Equations (10.2.2) in the external (upstream and downstream) regions (w = 0), using (10.2.3) and the jump relations at the reaction sheet (8.2.52). Reference Frame Attached to the Wrinkled Flame It is convenient to work in a frame of reference attached to the wrinkled reaction sheet, using (ζ , η, τ ) as independent variables, where ζ ≡ ξ − a(η, τ ).

(10.2.5)

The working equations are obtained by the change of variables. ∂ ∂ = , ∂ξ ∂ζ

∂ ∂ ∂a ∂ → − , ∂η ∂η ∂η ∂ζ

∂ ∂ ∂a ∂ → − , ∂τ ∂τ ∂τ ∂ζ

and the jump relations (8.2.52) are applied at ζ = 0.

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(10.2.6)

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475

Linearised Equations The field variables are decomposed, in the moving reference frame, into the part containing the nonperturbed solution θ (ζ ), ψ(ζ ), and the perturbations δθ (ζ , η, τ ), δψ(ζ , η, τ ), δθf (η, τ ) with θf ≡ θ (ζ = 0, η, τ ), θ = θ (ζ ) + δθ ,

θf = 1 + δθf ,

ψ = ψ(ζ ) + δψ.

In the linear approximation, the equations in the external zone are      2 ∂ ∂a ∂ ∂ ∂ 2 a dθ ∂2 δθ = + − − 2 , + 2 ∂τ ∂ζ ∂τ dζ ∂ζ 2 ∂η ∂η      2 ∂ ∂a ∂ 1 ∂ 1 ∂ 2 a dψ ∂2 + − − . + δψ = ∂τ ∂ζ Le ∂ζ 2 ∂τ Le ∂η2 dζ ∂η2

(10.2.7) (10.2.8)

As in the study of hydrodynamic instabilities, we use harmonic analysis with notations similar to those of (2.2.1) and (2.2.2), δθf (η, τ ) = θ˜f aˆ e(iκη+ςτ ) + c.c.,

a(η, τ ) = aˆ e(iκη+ςτ ) + c.c., δθ = θ˜ (ζ )ˆae(iκη+ςτ ) + c.c.,

˜ )ˆae(iκη+ςτ ) + c.c. δψ = ψ(ζ

where ς and κ are the dimensionless growth rate and wavenumber, reduced respectively by the transit time and the thickness of the planar laminar flame, ς ≡ σ τL ,

κ ≡ kdL .

(10.2.9)

In order to simplify the notation, the κ-dependence of the harmonic amplitudes is implicit in the preceding definitions, and we use κ to denote the modulus of the reduced wavevector, |κ| → κ. This latter is a given real number, whereas ς is an unknown complex value whose real part is the growth rate, Re(ς ) > 0 (or relaxation rate if Re(ς ) < 0), of the perturbation whose reduced wavelength is 2π/κ. Equations (10.2.7) and (10.2.8) then reduce to ordinary differential equations with constant coefficients,      dθ d2 d − 2 θ˜ (ζ ) + ς + κ 2 θ˜ (ζ ) = ς + κ 2 , (10.2.10) dζ dζ dζ      d 1 d2 κ2 κ 2 dψ ˜ )+ ς + ˜ )= ς+ − , (10.2.11) ψ(ζ ψ(ζ dζ Le dζ 2 Le Le dζ where, according to (10.2.3), the boundary conditions at infinity are ζ = −∞: θ˜ = 0, ψ˜ = 0,

ζ = +∞: θ˜ = 0, ψ˜ = 0.

(10.2.12)

Analysis The particular solutions to Equations (10.2.10) and (10.2.11), θ˜ = dθ /dζ , ψ˜ = dψ/dζ , are easily found using the nonperturbed equations. Using the boundary conditions (10.2.12), along with the definition of the reaction zone, ψ(ζ = 0) = 0, the solutions to the external

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equations (10.2.10)–(10.2.11) are

   ± ˜θ ± = dθ ± /dζ + θ˜f − dθ ± /dζ  er ζ , ζ =0    ± ± ψ˜ ± = dψ /dζ − dψ ± /dζ ζ =0 es ζ ,

where r and s stand for   1 ± 2 r = 1 ∓ 1 + 4(ς + κ ) , 2 −

−

(10.2.13) (10.2.14)

⎡ ⎤    2 κ Le 4 ⎣1 ∓ 1 + ⎦ , (10.2.15) ς+ s± = 2 Le Le +

+

and θ = eζ , ψ = 1 − eLeζ , θ = 1, ψ = 0. The linear growth rate, ς , and the perturbation, θ˜f , to the flame temperature are found by using the jump conditions (8.2.52) at the reaction sheet, ζ = 0. In the linear approximation, the second jump condition is dθ˜ − /dζ = β θ˜f /2. This gives the leading order of the kinetic relation 1 − r− = r+ = β θ˜f /2.

(10.2.16)

The first jump condition then leads to an expression for the amplitude of the temperature variation of the reaction sheet θ˜f θ˜f (r+ − r− ) = (s− − r− ) + (1 − Le), where, according to the definitions (10.2.15),  (r+ − r− ) = − 1 + 4(ς + κ 2 ).

(10.2.17)

(10.2.18)

The right-hand side of Equation (10.2.17) goes to zero for Le = 1, and θf is of order 1/β when Le differs from unity by the same amount, Le = 1 + l/β,

l ≡ β(Le − 1) = O(1) in the limit β  1.

(10.2.19)

This is the basic hypothesis for the validity of the asymptotic method in the limit β  1 and (Le − 1) 1; see Section 8.2.4 for more detail. Developing the root in the expression for s± , to leading order in small values of (Le − 1), gives    4 1 κ2 ς + 2κ 2 2 . 1+ (ς + ) ≈ 1 + 4(ς + κ ) 1 + 2( − 1) Le Le Le [1 + 4(ς + κ 2 )] The definitions (10.2.15) then lead to    2 2ς + 4κ (Le − 1) − − (s − r ) ≈ . 1 + 1 + 4(ς + κ 2 ) −  2 1 + 4(ς + κ 2 ) Introducing this result into Equation (10.2.17), along with (10.2.18), gives an expression for the perturbation to the flame temperature,   1 (Le − 1) 2ς + 4κ 2 ˜θf ≈ −1+ , (10.2.20)  2 1 + 4(ς + κ 2 ) 1 + 4(ς + κ 2 )

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showing that θ˜f is of the same order of magnitude as (Le − 1).

10.2.2 Cellular and Oscillatory Instabilities The dispersion relation for ς (κ) is found by injecting the result for the flame temperature, (10.2.20), into the kinetic relation (10.2.16):      l 2 2 − 1 − 1 + 4(ς + κ ) + 2ς = 1 − 1 + 4(ς + κ ) 1 + 4(ς + κ 2 ) . 2 (10.2.21) This equation was obtained by Sivashinsky[1] and generalised to flames with thermal losses[2] studied in Section 8.5.1. For such flames, the dispersion relation has a form similar to that of (10.2.21), but in which growth rate, ς , and wave vector, κ, are reduced by the transit time and thickness of the flame in the presence of heat loss. The only significant difference is the presence of an additional term on the right-hand side of (10.2.21) containing the heat loss,[2] 2(ς + κ 2 )H,

(10.2.22)

with, in the notation of (8.5.2) and (8.5.4), H ≡ 2h (UL /U)2 = ln (UL /U) ,

(10.2.23)

where UL and U are the speeds of the adiabatic and nonadiabatic flames. This relation is valid to leading order in the limit β → ∞. With this notation, H = 1 represents a flame at the extinction limit, U = U ∗ ≡ UL /e; see (8.5.6). Cellular Instability for Le < 1 Consider first an adiabatic flame, H = 0. It can be seen from (10.2.21) that for κ = 0:ς (κ) = 0, so on this branch planar perturbations are neutrally stable (translational invariance). The behaviour for small wavenumbers can be found by developing Equations (10.2.20) and (10.2.21) for small κ and ς . The leading order of these expansions gives θ˜f = (Le − 1)κ 2 ,

ς = −(l + 2)κ 2 /2,

(10.2.24)

where the growth rate ς is a real number, positive when l < −2 and negative in the opposite case. When l > −2, the second expression in (10.2.24) corresponds to a diffusive relaxation equation for the flame front x = α(y, t),  β(Le − 1) + 2 ∂ 2α ∂α = DT 2 , (10.2.25) ∂t 2 ∂y

[1] [2]

Sivashinsky G., 1977, Combust. Sci. Technol., 15, 137–146. Joulin G., Clavin P., 1979, Combust. Flame, 35, 139–153.

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Wrinkled Flames

Figure 10.3 Thermo-diffusive approximation: shape of the dispersion relation in the vicinity of the cellular instability.

which can easily be generalised to three dimensions for x = α(y, z, t),  2   ∂ α DT β(Le − 1) + 2 ∂α ∂ 2α = DT , + 2 = [β(Le − 1) + 2] 2 ∂t 2 R ∂y ∂z

(10.2.26)

where 2/R = 1/R1 + 1/R2 is the mean curvature. This result corresponds to (2.2.8) with 2B = β(Le−1)+2. When β(Le−1) < −2, Equation (10.2.25) describes a purely thermodiffusive instability, whose driving mechanism was explained in Section 2.4 and is apparent in the cellular flame in Fig. 2.19b. This instability appears when the molecular diffusivity of the species limiting the reaction rate is greater than the thermal diffusivity of the mixture (Le = DT /D < 1). This is typically the case for rich mixtures of heavy hydrocarbon fuels where oxygen molecules diffuse faster than the abundant hydrocarbon. The next order of the development (10.2.24) becomes significant close to the stability limit, |l + 2| 1, ς = −(l + 2)κ 2 /2 − 8κ 4 .

(10.2.27)

This result shows that the next order in the expansion of the growth rate in powers of κ is negative. It stabilises perturbations at small wavelengths (large κ); see Fig. 10.3. Oscillatory Instability for Le > 1 A numerical study of Equation (10.2.21) shows that another bifurcation exists, Re(ς ) = 0, for a finite value of κ, κ ∗ = 0, and a large positive value of l ≥ l∗ ≈ 10.67. For a typical value β ≈ 10, this corresponds to a Lewis number close to 2. Contrary to the cellular instability, the imaginary part of the growth rate is nonzero at the bifurcation (Poincar´e[1] – Andronov bifurcation) describing an oscillatory instability. The form of this branch of the solution to (10.2.21) is sketched in Fig. 10.4. The stable domain is thus defined by −2 < β(Le − 1) < l∗ ≈ 10.5. According to (10.2.21) a Poincar´e–Andronov bifurcation also exists in the planar case, κ = 0, for a critical value value l∗∗ of l slightly greater than l∗ . When κ = 0, the quantity √ 1− 1 + 4ς factorises and ς is the root of a quadratic equation that can be solved explicitly. [1]

Poincar´e H., 1908, Revue d’´electricit´e, 387.

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479

Figure 10.4 Thermo-diffusive approximation: shape of the dispersion relation in the vicinity of the oscillatory instability.

Adiabatic

Extinction

Oscillating instability

Stable

Cellular instability

Figure 10.5 Stability limits with heat loss, in the β(Le − 1) – H plane.

√ The bifurcation (Re(ς ) = 0) occurs for l∗∗ = 4(1 + 3) ≈ 10.928 with Im(ς ) ≈ 0.6356 corresponding to a period of oscillation of the order of 10 times the transit time τL . It is almost impossible to obtain a Lewis number of 2 with usual gaseous fuel mixtures. However, pulsating reaction fronts have been observed in solid–solid combustion (no gas), during polymerisation and for combustion in porous media where the thermal conduction is provided by the solid matrix and molecular diffusion is greatly reduced by the porous geometry. In these cases Le  1 and the flame model is similar to that presented at the end of Section 8.2.3 with Le = ∞. Effect of Heat Loss The presence of thermal loss reduces the domain of stability;[2] see the sketch of stability limits in the β(Le − 1) – H plane, shown in Fig. 10.5. The stability limit for oscillating instability (Le > 1) is obtained from a numerical evaluation of Equation (10.2.21) to which the heat loss term (10.2.22) has been added. The limit for cellular flames (Le < 1) can be obtained analytically. For a reduced heat loss of order unity, H = 0, the leading order in [2]

Joulin G., Clavin P., 1979, Combust. Flame, 35, 139–153.

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Wrinkled Flames

Unstable

Stable

Figure 10.6 Sketch of dispersion relation for a freely propagating planar flame in the absence of gravity.

the expansion of (10.2.21) – (10.2.22) for small values of ς and κ leads to the relation

ς =−

[β(Le − 1) + 2(1 − H)] 2 κ , 2(1 − H)

(10.2.28)

showing that the stability limit (ς = 0) is given by the straight line having the equation β(Le − 1) + 2(1 − H) = 0. Let us finally remark that the value β(Le − 1) = 18/5 for the oscillating stability limit at thermal extinction (H = 1) is more easily accessible to gaseous hydrocarbon mixtures than in the adiabatic case. We also note that the factor 1 − H in the denominator of Equation (10.2.28) makes the instability more violent near to the extinction limit.

Limitations The neglect of hydrodynamic effects created by the density difference between fresh and burnt gas implies that the results the thermo-diffusive model cannot be applied directly to flames. However, this model shows the existence of a new instability mechanism that must be coupled to the hydrodynamics in order to obtain reliable results. For long wavelength (small wavenumber) perturbations, the DL instability (10.1.32) has a growth rate σ proportional to k and dominates thermo-diffusive effects whose growth rate is proportional to k2 . In the absence of gravity, the bifurcation of the cellular thermo-diffusive instability (Le < 1) produces a change of sign of the coefficient B in Equation (2.2.9). The term of next order, k4 , remains negative, as suggested by Equation (10.2.27), and ensures stabilisation at sufficiently large wavenumbers. When coupled to hydrodynamics, the thermo-diffusive instability, in the absence of gravity, g = 0, produces a change in the shape of the dispersion relation as sketched in Fig. 10.6, obtained by combining Figs. 2.9 and 10.3. However, the expression for B is different from that of the thermo-diffusive model since it includes the expansion parameter υb ≡ ρu /ρb , as shown in the next section.

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481

10.3 Hydrodynamics and Diffusion Coupling In this section we carry out an analysis taking into account both nonlocal hydrodynamic effects (ρu = ρb ) and diffusive mechanisms. We limit our attention to weakly wrinkled flames (long wavelengths) for which the inner structure of the flame is only weakly modified from the planar case. A perturbation analysis using the small parameter  = dL / 1 is taken one order higher than in Section 10.1. For simplicity, the stability analysis of the ZFK model[1] is presented for negligible thermal variation of the diffusivities. The analysis including such thermal variations[2,3] is technically more elaborate but follows the same methodology. Such more detailed results are useful for comparison with experiments; they are given in Section 2.9.5. The structure of wrinkled flames in also studied in this section for finite amplitudes of wrinkles of the front. The analysis is first presented for the ZFK model,[4,5] leading to an expression for the first Markstein number. For didactic reasons the first part of the analysis is carried out in a pedestrian way, using fixed Cartesian coordinates. The more general case of a multiple-step chemistry[6] is presented in a second step in a more synthetic and more elegant way, using local coordinates attached to the flame. This second part of the analysis points out the nongeneric character of the local laws of wrinkled flames obtained with the one-step ZFK model. It leads to an expression for the second Markstein number, introduced in Section 2.3.

10.3.1 Formulation for the ZFK Model In the limit of a large reduced activation energy, the reaction sheet is used to define the flame surface, as in Section 10.2, and we use the same nondimensional coordinate system (10.2.5) attached to the flame. Constitutive Equations Using the notations $ = ρ/ρu , u = u/UL , w = w/UL and a = α/dL for the reduced density, the reduced longitudinal and transverse velocity components and the reduced flame position, we introduce the reduced quantities s and μ:  s ≡ $(u − ∂a/∂τ − w∂a/∂η). (10.3.1) μ = s/ 1 + (∂a/∂η)2 , The reduced mass flux crossing the flame is μ = μf ≡ mf /(ρu UL ); see (10.1.7). Using the change of variables (10.2.6), the equations for conservation of mass and momentum (15.1.33), written in the reference frame of a downwards-propagating flame, take

[1] [2] [3] [4] [5] [6]

Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237. Clavin P., Garcia P., 1983, J. M´ec. Th´eor. Appl., 2(2), 245–263. Garcia P., et al., 1984, Combust. Sci. Technol., 42, 87–109. Matalon M., Matkowsky B., 1982, J. Fluid Mech., 124, 239–259. Clavin P., Joulin G., 1983, J. Phys. Lett., 44, L–1– L–12. Clavin P., Gra˜na-Otero J., 2011, J. Fluid Mech., 686, 187–217.

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Wrinkled Flames

the form,

 ∂$ ∂($w) ∂s =− + , ∂ζ ∂τ ∂η  ∂u ∂u ∂p ∂u s + + $Fr−2 = − $ + $w , ∂ζ ∂ζ ∂τ ∂η  ∂a ∂p ∂w ∂w ∂p ∂w − =− $ + $w + , s ∂ζ ∂η ∂ζ ∂τ ∂η ∂η

(10.3.2)

where p = p/ρu UL2 is the reduced pressure, Fr2 ≡ UL2 /|g|dL > 0 is the square of the Froude number and |g| is the acceleration of gravity. Viscous effects have been neglected for simplicity because it has been shown that they do not appear in the final result when the Prandtl numbers and the heat conductivity are constant.[1] This assumption may be removed.[2] Equation (15.1.35) for the conservation of the species limiting the reaction in the ZFK model, written in the preheat zone, takes the form ∂ψ ∂ψ (∂ 2 a/∂η2 ) ∂ψ [1 + (∂a/∂η)2 ] ∂ 2 ψ ∂ψ = $ + $w + − s Le ∂ζ ∂t ∂η Le ∂ζ ∂ζ 2  1 ∂ 2ψ 1 ∂ ∂a ∂ψ − +2 . Le ∂ζ ∂η ∂η Le ∂η2

(10.3.3)

The equation for the temperature (15.1.34) is obtained by replacing ψ by the reduced temperature θ and Le by 1. In the isobaric approximation the reduced density $ can be expressed in terms of θ : 1/$ = T/Tu = 1 + (υb − 1)θ ,

$b = Tu /Tb = 1/υb < 1.

(10.3.4)

Boundary Conditions The boundary conditions at infinity are

-

Fresh mixture, ζ → −∞: Burntgas, ζ → +∞:

u = uT , w = 0, p = pu , ψ = 1, θ = 0,

(10.3.5)

u, w and p are bounded, ψ = 0, θ = 1,

where in the nonlinear study uT = 1 is unknown, resulting from the increase of surface area of the flame front, uT = S /So ; see (3.1.11). In the linear approximation the flow is unperturbed far upstream, uT = 1. To leading order, in the distinguished limit β(Le − 1) = O(1), β → ∞, the jump conditions (8.2.52) at the reaction sheet ζ = 0 (ψ = 0) are verified in the normal direction, +  1 ∂ψ ζ =0 eβ(θf −1)/2 ∂θ ∂θ  + = , = 0, (10.3.6)  ∂ζ ζ =0− ∂ζ Le ∂ζ ζ =0− 1 + (∂a/∂η)2 [1] [2]

Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237. Clavin P., Garcia P., 1983, J. M´ec. Th´eor. Appl., 2(2), 245–263.

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483

Figure 10.7 Wrinkled flame in the reference frame of planar flame. The thick line represents the reaction sheet x = α(y, t) and the thin line the upstream limit of the preheat zone.

where θf ≡ θ (ζ = 0) is the temperature at the reaction sheet, with β(θf − 1) = O(1); see Section 8.2.4. Temperature, density, mass fraction, flow velocity and pressure are continuous at the reaction sheet (see (15.1.45)–(15.1.47)), ζ = 0:

ζ =0+

ζ =0+

ζ =0+

ψ|ζ =0 = 0, [u]ζ =0− = 0, [w]ζ =0− = 0, [p]ζ =0− = 0.

(10.3.7)

Multiple-Scale Method, Expansion in Gradients A wrinkled flame of finite thickness is sketched in Fig. 10.7, which generalises Fig. 2.7. The problem has two different length scales whose ratio is  1: the flame thickness dL and the long scale of wrinkling,  = dL /, which is also the scale of the inhomogeneities in the external flow (outside the flame structure where $ is constant). Singular points, such as the tip of the Bunsen burner, are outside the scope of this analysis. Far from the flame, the external flow is also supposed to evolve on a time scale larger by at least a factor 1/ than the transit time across the flame, τL . Due to gas expansion, the flow velocity changes on the small length scale, dL , within the flame thickness (deflection of the stream lines). Due to hydrodynamic effects, this induces modifications to the external flow on the large length scale, . The deflection of the streamlines modifies the convective transport terms in the mass and energy fluxes inside the flame. Coupled with the diffusive fluxes, this modifies the internal structure of the flame and the local propagation speed. Since the wrinkling is weak, the transverse fluxes are small and so are the changes to the internal structure and the flame speed. The front thus evolves on a time scale that is longer than τL , at least by a factor 1/, and the terms on the right-hand side of Equations (10.3.2)–(10.3.3) are small, of order . The solution to (10.3.2)–(10.3.3) is then obtained by a perturbation analysis limited to first order in an expansion in powers of . The flow is decomposed into two parts, u=u− + u˘ . The first part, u− , represents the external flow in the upstream region outside the flame thickness, and the second, u˘ , is the change across the flame structure.

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Wrinkled Flames

A similar decomposition is introduced for the quantity s in (10.3.1), s = s− (x, y, t) + s˘(ζ , y, t), u − uT = u− (x, y, t) + u˘ (ζ , y, t), ˘ (ζ , y, t), w = w− (x, y, t) + w

(10.3.8)

p = p− (x, y, t) + p˘ (ζ , y, t), where

x = x/ ≡ (a + ζ ),

y = y/ ≡ η,

t = UL t/ ≡ τ .

(10.3.9)

The origin, ζ = 0, is attached to the reaction sheet and we have omitted the superscript − in s˘, u˘ , w ˘ and p˘ to simplify the notation. The external flow, u− , w− , p− , satisfies Euler’s equations with $ = 1. Each of the terms on the right-hand side of Equations (10.3.8) is expanded in powers of ; see (10.3.15). The method works well because the variables s˘(ζ ), u˘ (ζ ), w ˘ (ζ ) and p˘ (ζ ) tend exponentially fast to zero as ζ → −∞. The constant uT in (10.3.8) stands for the speed of the uniform flow far upstream that holds the mean position of the front stationary. Far upstream, x → −∞, the fields u− (x, y, t), w− (x, y, t), p− (x, y, t) are the given boundary conditions for the problem. For the case of a flame in a turbulent flow field, these fields are the fluctuations (whose time average is zero) of the incoming flow field and uT is the turbulent flame speed. As in the DL analysis, the fields u− , w− , p− are modified near to the front over a distance of the order of . For the simple case of a uniform incident flow perpendicular to the unperturbed flame, u− and w− tend to zero far upstream, x → −∞. If the planar flame is unstable, the cellular flame speed is greater than UL , uT > 1, because of the increase in surface area of the flame; see (10.3.19). In order not to be limited to small amplitudes of wrinkling we will write a(y, t) = A/,

x = A(y, t) + ζ ,

with A(y, t) = O(1),

(10.3.10)

A˙ ≡ ∂A/∂t = ∂a/∂τ = O(1).

(10.3.11)

so that the slope of the front is of order unity: A ≡ ∂A/∂y = ∂a/∂η = O(1),

Since $ = 1 upstream of the flame, (10.3.1) and (10.3.8) yield s− = (uT + u− − A˙ − w− A ),

s˘ = ($ − 1)s− + $(˘u − w ˘ A ).

(10.3.12)

The quantity s− , taken on the reaction sheet at x = A, is related to the reduced mass flux of fresh mixture, μf ≡ Un− /UL , where Un− is the normal propagation speed defined in (10.1.6),  − 2 μf (y, t) = s− where s− (10.3.13) f (y, t) ≡ s (A, y, t). f (y, t)/ 1 + A , We also introduce the upstream flow field at the reaction sheet, − u− f (y, t) ≡ u (A, y, t),

− w− f (y, t) ≡ w (A, y, t),

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− p− f (y, t) ≡ p (A, y, t). (10.3.14)

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485

We look for the solutions to Equations (10.3.2)–(10.3.3) in the form of perturbations to the planar solution by expanding in powers of : − − − − − − − − − − − 2 (s− f , uf , wf , pf ) = (sf 0 , uf 0 , wf 0 , pf 0 ) + (sf 1 , uf 1 , wf 1 , pf 1 ) + O( ),

(˘s, u˘ , w ˘ , p˘ ) = (˘s0 , u˘ 0 , w ˘ 0 , p˘ 0 ) + (˘s1 , u˘ 1 , w ˘ 1 , p˘ 1 ) + O( 2 ), θ (ζ , y, t) = θ0 + θ1 + O( ), 2

(10.3.15)

ψ(ζ , y, t) = 1 + ψ0 + ψ1 + O( ). 2

10.3.2 Curved Flame Speed for the ZFK Model The objective here is to find an expression for the normal burning velocity μf in terms of the local properties of the flame surface and of the upstream flow for the ZFK flame model. The particularity of this model is that the reaction sheet delimits the flame thickness. Leading Order To leading order in the limit  → 0, the internal structure of the flame is that of an inclined planar front. The terms on the right-hand side of Equations (10.3.2)–(10.3.3) being at least of order , at the leading order the first equation of (10.3.2) reduces to ∂ s˘0 /∂ζ = 0, and the boundary condition limξ →−∞ s˘0 = 0 then gives s˘0 = 0. The leading order of Equations (10.3.3) then yields 2 Le ζ s− f 0 /(1+A )

ζ < 0:

ψ0 = −e

ζ > 0:

ψ0 = −1,

,

2 ζ s− f 0 /(1+A )

θ0 = θf 0 e θ0 = θf 0 ,

and the jump relations (10.3.6) give θf 0 = 1,

,

s− f 0 (y, t) =



1 + A2 ,

(10.3.16)

(10.3.17)

so that, according to (10.3.13), the local flame speed is not modified and is equal to that of the planar flame, μf 0 = 1, Un− = UL , in agreement with the basic hypothesis of the DL analysis. Using the first definition in (10.3.12), the second equation in (10.3.17) gives the leading order for the local equation of evolution,  −  2 ˙ (10.3.18) (UT + u− f 0 − A − wf 0 A ) = 1 + A . ˙ = 0, where . means time averaged, and the The mean front is planar and stationary, A −  mean mass flux is constant. Since u0 − w− 0 A = ∂φ/∂y, where φ is the stream function in the laboratory frame, integrating Equation (10.3.18) over the transverse coordinate and taking the time average shows that the increase in the speed of the flame brush is equal to the increase in surface area (3.1.11),   1 L dy 1 + A2 . (10.3.19) UT0 = L 0 We will also limit our attention to weak acceleration, Fr−2 = G,

G = O(1),

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Go = G/υb ,

(10.3.20)

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Wrinkled Flames

where Go was introduced in (2.2.20), so that, at the leading order, it changes only the external flow field on the long length scale but not the flow inside the flame thickness. The second equation in (10.3.12) and the last two equations in (10.3.2) then lead to the following system,

u0 − w ˘ 0 A ) = 0, ($0 − 1)s− f 0 + $0 (˘ s− ˘ 0 /∂ζ + ∂ p˘ 0 /∂ζ = 0, f 0∂ u

s− ˘ 0 /∂ζ − A ∂ p˘ 0 /∂ζ = 0, f 0∂ w

(10.3.21)

where, according to (10.3.4), 1/$0 = 1 + (υb − 1)θ0 , and s− f 0 is given by (10.3.17). The solutions to (10.3.21) satisfying the three upstream boundary conditions, ζ → −∞: u˘ 0 = w ˘ 0 = p˘ 0 = 0 are (υb − 1) 1 u˘ 0 = √ θ0 , w ˘ 0 = −A u˘ 0 , p˘ 0 = − √ u˘ 0 . (10.3.22) 1 + A2 1 + A2 These solutions are simply those of the inclined planar flame. The flow is deflected as shown in Fig. 2.6, but as implied by (10.3.20), the internal structure of the flame is not modified by gravity. A technical difficulty is that the leading order terms depend on time and space through the tangent of the angle of the front with the y-axis, A (y, t). This difficulty disappears when local coordinates are used, as in Section 10.3.5. However, as a first step, it is instructive to work with the coordinate system (10.2.5)–(10.2.6). Method The changes to the internal structure and velocity of the flame are given by the next order in the expansions, s− f 1 and μf 1 ; see (10.3.13). This necessitates calculation of the change in mass flux and flame temperature,

s˘f 1 ≡ s˘1 (ζ = 0, y, t),

θf 1 ≡ θ1 (ζ = 0, y, t).

(10.3.23)

The final result is obtained using the first jump relation (10.3.6) where the heat flux leaving the reaction zone, ∂θ/∂ζ |ζ =0− , is expressed in terms of s− f 1 and s˘ f 1 and where θf 1 will be given by the second jump relation of (10.3.6). Modification of Mass Flux Through the Flame According to the first equation in (10.3.2), the continuity equation in the upstream flow at the flame front, written with the variables (ζ , y, t), ∂s− /∂ζ = −∂w− /∂y, and with the notation (10.3.14), yields  − ∂s− /∂ζ ζ =0 = −∂w− s− = s− (10.3.24) f /∂y, f − ζ (∂wf /∂y) + · · · , where the expansion of s− in powers of ζ around ζ = 0 has been used. In the preheat zone, the next order in the expansion of the continuity equation is ˘ 0 )/∂y − ∂[($0 − 1)w− ∂ s˘1 /∂ζ = −∂($0 − 1)/∂t − ∂($0 w f 0 ]/∂y.

(10.3.25)

˘ 0 tend exponentially to zero as the inner scale Note that the quantities ($0 − 1) and w ζ → −∞. Using (10.3.16)–(10.3.22), the integral over ζ from −∞ to ζ of (10.3.25)

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10.3 Hydrodynamics and Diffusion Coupling

487

with the boundary condition limζ →−∞ s˘1 = 0 yields s˘1 (ζ , y,√ t). Since the right-hand side 2

of (10.3.25) depends on ζ only through  ≡ (υb − 1)eζ / 1+A , the integrals can be calculated by permuting the derivatives with respect to y or t against the integral over ζ , and using (ζ ) as the integration variable: 

ζ

−∞

(1 − $0 )dζ  =



 1 + A2 ln (1 + ) ,

ζ

−∞

$0 w˘0 dζ  = −A ln (1 + ) .

The local change in mass flux thus obtained can be written

s˘1 (ζ ) =

   ∂  ∂  ∂ − A ln(1 + ) + wf 0 1 + A2 ln(1 + ) , 1 + A2 ln(1 + ) + ∂t ∂y ∂y (10.3.26)

yielding on the reaction sheet, ζ = 0,

s˘f 1 = S˙ ln υb ,

(10.3.27)

where, using the notation A ≡ ∂ 2 A/∂y2 ,  ∂ ∂ (w− S˙ ≡ 1 + A2 + 1 + A2 ) + A . (10.3.28) f 0 ∂t ∂y √ ˙ 1 + A2 is in fact the stretch rate of an element of flame As shown in Section 10.4.1,  S/ surface δ 2 s (2.3.9),  S˙ τL τL d 2 (δ s). = ≡ 2 √ τs (δ s) dt 1 + A2

(10.3.29)

Modification of Conductive Heat Flux The next order in Equation (10.3.3) can be written using the expansions (10.3.15) and (10.3.24) in the expression for s− , ⎧  ∂w− ⎪ ∂ψ0 f0 − ⎪ ˘ −ζ + s + s (ζ ) ⎪ 1 f1 ∂y ∂ζ ⎨ (1 + A2 ) ∂ 2 ψ1  ∂ψ 1 2 ) ∂ψ0 ∂ψ0 ∂ψ0 − = (10.3.30) − (1 + A +$ + w $ + w ˘ 0 0 0 $0 ∂y f0 ∂t ∂y ⎪ Le ∂ζ ∂ζ 2 ⎪ 2 ⎪ ⎩+A 1 ∂ψ0 + 2 1 A ∂ ψ0 . Le ∂ζ Le ∂ζ ∂y The corresponding equation for θ1 is obtained from (10.3.30) by replacing ψ1 by θ1 and Le by 1. The change in heat flux ∂θ1 /∂ζ at ζ = 0− is found by integrating the equation for θ1 across the preheated region, using the boundary conditions limζ →−∞ θ1 = 0 and ζ = 0: θ1 = θf 1 = O(1/β). At the leading order in the limit β → ∞, the boundary condition at ζ = 0 is simply θf 0 = 1. The calculation can be simplified by integrating over ζ from −∞ to 0− the difference between the equation for θ1 and the product of (10.3.25) and θ0 . This yields the quantity ∂(θ0 s˘1 )/∂ζ . The integration can also be performed directly

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Wrinkled Flames

on the nonperturbed Equations (10.3.2) and (10.3.3), yielding ⎧

 ⎨s− + s˘f 1 + ∂ w−  0 $0 θ0 dζ + A ∂θ 1 f 0 −∞ ∂y = f1  0 ζ = 0− : (1 + A2 ) 0 ⎩+ ∂ ∂ζ $0 θ0 dζ + ∂ $0 w ˘ 0 θ0 dζ . ∂t −∞

Using the results from the leading order,  0  ln υb , $0 θ0 dζ = 1 + A2 υb − 1 −∞

∂y −∞



0

−∞

$0 w ˘ 0 θ0 dζ = A

ln υb − A ; υb − 1

along with expression (10.3.27) for s˘f 1 , one obtains ζ = 0− : (1 + A2 )

∂θ1 υb ln υb ˙ = s− S(y, t), + f 1 ∂ζ υb − 1

(10.3.31)

where S˙ is the quantity defined in (10.3.28)–(10.3.29). Modification to Flame Temperature In the burnt gas, ζ > 0, the right-hand side of Equations (10.3.30) for ψ1 (ζ ) and θ1 (ζ ) is null, and one has ζ > 0: ψ1 = 0, ∂θ1 /∂ζ = 0. Thermal relaxation by diffusion appears only at the next order,  2 , in the expansion in , since the temperature change is of order  and its spatial relaxation occurs over a distance of order 1/ greater than the flame thickness. Consequently the first two orders in the second jump relation of (10.3.6) are simply ζ = 0− :

∂θ0 1 ∂ψ0 + = 0, ∂ζ Le ∂ζ

∂θ1 1 ∂ψ1 + = 0. ∂ζ Le ∂ζ

(10.3.32)

The unknown quantity s− f 1 , in the right-hand side of Equations (10.3.30) is eliminated in the calculation of θf 1 (y, t) by integrating over ζ from −∞ to 0 the sum of the two Equations (10.3.30) for ψ1 (ζ ) and θ1 (ζ ). Integrating by parts the term containing s˘1 (ζ ), and using the boundary conditions θ0 = 1, θ1 = θf 1 , θ0 = 0, θ1 = 0, − ζ =0 : ζ = −∞: ψ0 = −1, ψ1 = 0, ψ0 = 0, ψ1 = 0, and (10.3.32), one obtains the change in flame temperature, θf 1 ,   0  ∂ 0 ∂ 2 (1 + A )θf 1 = − $0 (θ0 + ψ0 )dζ − $0 w ˘ 0 (θ0 + ψ0 )dζ ∂t −∞ ∂y −∞   (Le − 1) ∂ − 0 wf 0 A . − $0 (θ0 + ψ0 )dζ − ∂y Le −∞ According to (10.3.16) and (10.3.17), Le √ = 1: ψ0 (ζ ) + θ0 (ζ ) = 0, and |Le − 1| 1: √ ζ / 1+A2 . Since θ must be of order 1/β in the limit θ0 + ψ0 ≈ −(Le − 1)(ζ / 1 + A2 )e f1

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β → 0 for the asymptotic method of Section 8.2.4 to be valid, the quantity (Le − 1) must be of order 1/β, so that, performing the integral as above, one obtains  υb −1  ln(1 + ) 1 2 ˙ (1 + A )θf 1 = −(Le − 1)S d, (10.3.33) υb − 1 0  ˙ t) is given by (10.3.27). where the expression for S(y, Flame Speed s− f 1,

is obtained by introducing (10.3.31) and (10.3.33) into the order  of The final result, the first equation of (10.3.6),  βθf 1 ∂θ1 = , (10.3.34) 1 + A2 ζ = 0− : ∂ζ 2   2 2 ˙ s− (10.3.35) f 1 / 1 + A = −M S(y, t)/ 1 + A ,  υb −1 υb ln υb β(Le − 1) ln(1 + ) M= + (υb − 1)−1 d, (10.3.36) υb − 1 2  0 and the reduced flame speed, μf ≡ Un /UL , is obtained from (10.3.13) and (10.3.17), μf = √ 2 . According to (10.3.29), this corresponds to (2.3.2) with an expression / 1 + A 1 + s− f1 for the first Markstein number given by (10.3.36). In the thermo-diffusive approximation υb = 1, this result reduces to (8.5.20). When the thermal variation of the diffusivities are taken into account, the expression of the Markstein number M is given by (2.9.44)– (2.9.45), which reduces to (10.3.36) for λ = 1.

10.3.3 Linear Hydrodynamic Jumps in the ZFK Model The objective of this subsection is to evaluate the first-order correction to the hydrodynamic jumps (15.1.45)–(15.1.47) across a wrinkled flame, when the finite flame thickness is taken into account. This is a preliminary step before studying the stability limits of a flame propagating downwards in the next Section, Section 10.3.4. In the burnt gas, we use notations similar to (10.3.14) for the far field at the flame front (ζ = 0), + u+ f ≡ u (A, y, t),

+ w+ f ≡ w (A, y, t),

+ p+ f ≡ p (A, y, t).

(10.3.37)

Using the ZFK model, the first-order correction to the jumps was first obtained in the linear approximation[1] and then extended to finite amplitudes of wrinkling.[2] In this section we limit our attention to the linear case using the same decomposition as in (10.3.8) and the same coordinates (10.3.9). In the preheat zone,

u˘ (ζ ) = u˘ 0 + δ u˘ 1 + · · · ,

[1] [2]

δw ˘ (ζ ) = δ w ˘ 0 + δ w ˘1 +··· ,

Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237. Matalon M., Matkowsky B., 1982, J. Fluid Mech., 124, 239–259.

17:10:10 .012

p˘ (ζ ) = u˘ 0 + δ p˘ 1 + · · · ,

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Wrinkled Flames

and the quantities of order unity u˘ 0 (ζ ) and p˘ 0 (ζ ) are, according to (10.3.22), the modifications to the longitudinal flow across the planar flame, w = 0. When calculating the jumps across the flame, we recall that the downstream boundary of the ZFK flame is delimited ˘ f ≡ δw ˘ (ζ = 0), by the reaction sheet (ζ = 0). Using the notations δ u˘ f ≡ δ u˘ (ζ = 0), δ w + − + ˘f, and δ p˘ f ≡ δ p˘ (ζ = 0), Equations (10.3.7) yield δuf = δuf + δ u˘ f , δwf = δw− f + δw − ˘ = δp + δ p . Therefore the jumps of the external flows across the flame take the form δp+ f f f ⎧ + + ⎪ ⎨ [δu]− = [δuf ]− = δ u˘ f = δ u˘ f 1 + · · · , + ˘ f = δw ˘ f 0 + δ w ˘ f1 + · · · , (10.3.38) ζ = 0 : [δw]+ − = [δwf ]− = δ w ⎪ ⎩ + + [δp]− = [δpf ]− = δ p˘ f = δ p˘ f 1 + · · · . For small amplitudes of wrinkling, δa 1, the perturbation δA = δa is of order  in the limit  → 0, in contrast to (10.3.10), A = 0,

A˙ = O(),

A = δA = O(),

A = O().

(10.3.39)

Therefore, the perturbations to the external flows are of order , δu± = O(),

δw± = O(),

δp± = O(),

(10.3.40)

as in the DL analysis. We are interested in the corrections for finite flame thickness, which are here, according to (10.3.39), of the order  2 : δw ˘ 0 = O(),

δ w ˘ 1 = O( 2 ),

δ u˘ 1 = O( 2 ),

δθ = δθ1 + · · · = O( ),

δ p˘ 1 = O( 2 ),

δψ = δψ1 + · · · = O( ).

2

2

(10.3.41) (10.3.42)

Longitudinal Velocity Jump The linear approximation of the stretch rate is, according to (10.3.28), δ S˙ = A − u− xf ,

− u− xf ≡ ∂δu /∂x|x=0 = O().

where

(10.3.43)

Mass conservation (10.3.12) yields

 δ u˘ 1 (ζ ) = $−1 (ζ )δ˘s1 (ζ ) − 1 − $−1 (ζ ) δs− f 1,

(10.3.44)

where, according to (10.3.26), (10.3.27) and (10.3.35), ζ δ˘s1 (ζ ) = (A − u− xf ) ln[1 + (υb − 1)e ],

(10.3.45)

δ˘sf 1 = (A − u− xf ) ln υb ,

(10.3.46)

δs− f1

(10.3.47)



= −(A

− u− xf )M.

Both quantities A and u− xf should be understood here as their leading order in the expansion in powers of . The longitudinal velocity jump is given by (10.3.44) and (10.3.38): −  ζ = 0: [δuf ]+ − = (A − uxf ) [υb ln υb − (υb − 1)M] + · · · .

(10.3.48)

It is a correction of order  2 to the DL result (10.1.24), [δuf ]+ − = 0. According to (10.3.36), this jump is proportional to β(Le − 1).

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Kinetic Relation − ˙ According to (10.3.12) and (10.3.13), δs− f = [δuf − A], and the linear approximation (10.3.47) takes the form −  A˙ − δu− f = M(A − uxf ) + · · · .

(10.3.49)

This equation is an order  correction to (10.1.24), which described a front simply con− vected by the flow A˙ ≈ δu− f . The stretch term uxf in (10.3.49) was not taken into account in the simplified approach (10.1.37). Pressure and Transverse Velocity Jumps The pressure and transverse velocity jumps are obtained from the second and third equations in (10.3.2). The planar solution reduces to s = 1,

dp− + G = 0, dζ

dp du + + rG = 0, dζ dζ

(10.3.50)

where G is defined in (10.3.20). Since δ$ = O( 2 ), the acceleration of gravity in the second equation of (10.3.2) is negligible in the linear approximation when attention is limited to order  2 , ∂δu ∂δu du ∂δp + δs + = −$ , ∂ζ dζ ∂ζ ∂τ ∂δw ∂δw ∂δp ∂a dp − = −$ − . ∂ζ ∂η dζ ∂τ ∂η

(10.3.51) (10.3.52)

In the far upstream field these equations reduce to ∂δp− ∂δu− ∂δu− + =− , ∂ζ ∂ζ ∂τ ∂δw− ∂a dp− ∂δw− ∂δp− − =− − . ∂ζ ∂η dζ ∂τ ∂η

(10.3.53) (10.3.54)

Introducing these equations and (10.3.8) into (10.3.51) and (10.3.52), the following equations are obtained in the preheated zone: ∂δ u˘ du ∂δ p˘ ∂δu− ∂δ u˘ + δs + = −($ − 1) −$ , (10.3.55) ∂ζ dζ ∂ζ ∂τ ∂τ  ∂δ p˘ ∂δ w ˘ du ∂a ∂δw− ∂δ w ˘ + ($ − 1)G + = −($ − 1) −$ − , ∂ζ dζ ∂η ∂τ ∂τ ∂η where, according to (10.3.24) and continuity,

 − δs =  δs− + ζ u + δ˘ s (ζ ) = O( 2 ) 1 f1 xf and where δ˘s1 (ζ ) and δs− f 1 are given in (10.3.45) and (10.3.47). The leading order of the second equation of (10.3.55) gives (10.3.22), δw ˘ 0 (ζ ) = −[(υb − 1)eζ ]∂a/∂y,

δw ˘ f 0 = −(υb − 1)∂a/∂y.

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Wrinkled Flames

According to (10.3.41), the last term in the right-hand sides of the two Equations (10.3.55) is negligible when attention is limited to order  2 . Integration of (10.3.55) across the preheated zone from ζ = −∞ to ζ = 0 yields 0 du −δ u˘ f 1 − (υb − 1)δs− f 1 − −∞ δ˘s1 dζ dζ δ p˘ f 1 =   0 −  0 − du −uxf −∞ ζ dζ dζ − u˙ tf −∞ ($ − 1)dζ , (10.3.56)  0  0  0 $dζ −  Gay + w ˙− ($ − 1)dζ , δ wf 1 =  2 a˙ ty tf −∞

−∞

− where the quantities ay ≡ ∂a/∂y = O(1), a˙ ty ≡ ∂ 2 a/∂t∂y = O(1), u˙ − tf ≡ ∂δu /∂t|x=0 = − O() and w ˙− tf ≡ ∂δw /∂t|x=0 = O() have to be understood here as their leading order in the expansion in powers of  and where, according to (10.3.22), 0 = (υb − 1)eζ , $ = (1 + 0 )−1 . Equations (10.3.56) then lead to the jumps of pressure and velocity,

-

ζ =0:

+  [δpf ]+ ˙− − = −2[δuf ]− + (υb − 1)A +  u tf ln υb + · · · , +    ˙− [δwf ]− = −(υb − 1)A + (A˙ + GA + w tf ) ln υb + · · · ,

(10.3.57)

˙ where [δuf ]+ − is given by (10.3.48) and where we have introduced the notation A ≡ 2 2 ∂ A/∂y∂t = O(). The right-hand side of the first equation is of the order  . It is the + correction to the DL analysis, [δpf ]+ − = 0 in (10.1.27). The term −2[δuf ]− comes from the Bernoulli equation (10.1.10). It also appears in the simplified analysis of Section 10.1.3, but with an expression different from (10.3.48); see (10.1.39) and (10.1.40). The extra terms in the pressure jump show that, because of transverse and unsteady effects inside the wrinkled flame structure, the Bernoulli equation (10.1.10) is no longer valid in the conservation of longitudinal momentum, when the finite thickness of the flame is taken into account. The term (υb −1)A is a stabilising surface tension–like effect due to gas expansion. It reinforces the effect introduced by the Bernoulli equation which is proportional to Le − 1 and which stabilises the flame only if Le > 1; see (10.3.48). 10.3.4 Stability Analysis of the ZFK Model The external flows are given by (10.1.20) and (10.1.21). We will use notations similar to (10.1.13) for the pressure, %± ≡ p± + $± Gx,

± ± δ%± f = δpf + $ GδA,

(10.3.58)

− + ± where %± f ≡ % (x = A), $ = 1 and $ = 1/υb . We will consider flames propagating downwards and use the Fourier decomposition, f = f + δf , δf(ξ , η, τ ) = fˆ (ξ )ˆaeiκη+ςτ , (10.3.59) δA =  aˆ eiκη+ςτ ,

where, according to (10.3.39) and using the notation of (10.2.9), aˆ = O(1),

fˆ = O(),

A = −k2 aˆ eiκη+ςτ ,

κ ≡ kdL = O(), A = −κ 2 aˆ eiκη+ςτ ,

17:10:10 .012

ς ≡ σ τL = O(), A˙ = ς aˆ eiκη+ςτ .

(10.3.60)

10.3 Hydrodynamics and Diffusion Coupling

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Equations (10.1.20)–(10.1.22) for the external flow, written using the notations (10.3.14) and (10.3.37), take the form +|κ|ξ , = uˆ − uˆ − f e ξ 0 (10.3.62) −|κ|ς , ˆ + = (ς/υb − |κ|) uˆ + |κ|% pe iκ w ˆ ± = −∂ uˆ ± /∂ξ , uˆ − f = O(),

uˆ + p = O(),

uˆ + r = O(),

(10.3.63) uˆ + ˆ+ ˆ+ p +u r =u f .

ˆ+ ˆ+ The three quantities uˆ − p, u r and the reduced linear growth rate ς = ς0 + ς1 are f , p obtained with the help of the four linear jumps of the preceding section. Using (10.3.61), ˆ−  uˆ − f x = κu f , and (10.3.62), the longitudinal velocity jump (10.3.48) and the kinetic equation (10.3.49) yield 2 uˆ + ˆ+ ˆ− ˆ− p +u r −u f = −(κ + κ u f ) [υb ln υb − (υb − 1)M] ,

(10.3.64)

2 (ˆu− ˆ− f − ς ) = (κ + κ u f )M,

where from now on κ denotes |κ|. Using (10.3.63), (10.3.61) and (10.3.62), differentiating the second equation in (10.3.57) with respect to η yields 2 2 ˆ+ ˆ− ˆ− κ uˆ + p + ςu r /υb + κ u f − κ (υb − 1) = −κ (ς + G) ln υb − κς u f ln υb .

(10.3.65)

Using the second equations of (10.3.61) and of (10.3.62) with the notation (10.3.58), the pressure jump in (10.3.57) yields −1 u− (υb−1 ς − κ)ˆu+ p + (ς + κ)ˆ f − κ(υb − 1)G

= 2κ(κ

2

+ κ uˆ − f ) [υb ln υb

− (υb − 1)M] − (υb − 1)κ

(10.3.66) 3

+ κς uˆ − f ln υb .

Limiting our attention to the order  2 , the leading order of the second equation in (10.3.64), uˆ − f ≈ ς , may be introduced into the right-hand sides of Equations (10.3.64)–(10.3.66), which are smaller by an order  than the left-hand sides. The dispersion relation ς (κ) is obtained by following the same procedure as in the DL analysis of Section 10.1.2. The −1 ˆ+ u+ quantity uˆ + r is computed in terms of u p from (10.3.64). The quantity (υb ς − κ)ˆ p is + obtained from (10.3.65) with the help of preceding expression for uˆ r , 2 −1 2 (υb−1 ς − κ)ˆu+ p = ς (υb + Mκ) + ς κ(1 + 2Mκ) − κ [(υb − 1) − (Mκ + G ln υb )].

When this result is introduced into (10.3.65) with uˆ − f given by (10.3.64), one gets a quadratic equation for the reduced growth rate ς , a(κ)ς 2 + 2b(κ)κς − (υb − 1)κn(κ) = 0,

17:10:10 .012

(10.3.67)

494

Wrinkled Flames

a(κ) ≡ (1 + υb−1 ) + κ[2M − ln υb ], b(κ) ≡ 1 + κ[(1 + υb )M − υb ln υb ],   υb (υb ln υb ) Go − κ 2 1 + 2 (M − ln υb ) . n(κ) ≡ −Go + κ 1 − (υb − 1) υb − 1

(10.3.68)

The 1982 result[1] corresponds to (10.3.67) multiplied by (1 − Mκ). Since κ and ς are of order , the last terms in the expressions of a(κ) and b(κ) are corrections of order . For consistency of the perturbation analysis, Go in n(κ) should also be of order , Go ≡ υb−1 |g|dL /UL2 = O(), as was assumed in (10.3.20). To leading order Equation (10.3.67) reduces to the DL equation with gravity in (2.2.16), (1 + υb−1 )ς02 + 2κς0 − (υb − 1)κ (−Go + κ) = 0.

(10.3.69)

Introducing the nondimensional growth rate reduced by the hydrodynamical time (UL k)−1 , σ˜ ≡ σ/(UL k) = ς/κ, Equation (10.3.67) may be written in a form similar to (2.2.18), σ˜ 2 + 2B(κ)σ˜ − D(κ) = 0,

(10.3.70)

where B = b/a and D = (υb − 1)κn/a. To first order in the κ-expansion, the coefficients B and D, expressed in terms of the Markstein number (10.3.36) and the density contrast υb , are . / . / 0 υb2 + 1 υb2 ln υb υb B= 1+ M− κ , (10.3.71) υb + 1 υb + 1 υb + 1   N υb − 1 υb , (10.3.72) D= υb + 1 κ   υb ln υb 2υb κ2 N ≡ −Go + κ + M− Go κ − , (10.3.73) υb + 1 υb − 1 κm . / . / 4υb2 3υb + 1 1 ≡1+ (10.3.74) M− υb ln υb . κm υb2 − 1 υb2 − 1 Equation (10.3.70) should be used to compute the order  correction ς1 to the DL linear growth rate, ς = ς0 + ς1 . When the temperature dependence of the diffusion coefficients is taken into account, the expression for the Markstein number and the coefficients B, D, N and κm are given in (2.9.45) and (2.9.50)–(2.9.54). Thanks to large numerical factors the quantity 1/κm is large, typically 10, so that the κ-correction to D, namely the last term in (10.3.73) obtained by the perturbation analysis, yields a significant diffusive effect. The stability limits of a planar flame propagating downwards in the absence of acoustics is defined by N = 0 and dN/dκ = 0. The third term in the right-hand side of (10.3.73) is typically negligible, so that the expressions (2.2.22) for the critical quantities Gc and kc in terms of km are still valid. [1]

Pelc´e P., Clavin P., 1982, J. Fluid Mech., 124, 219–237.

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Table 10.1 Reduced marginal wavenumber κm as a function of Markstein number M, for expansion ratios, υb = 5 (upper) and υb = 8 (lower). M

1.8

κm (constant diffusivities) κm (variable diffusivities)

0.319 –

κm (constant diffusivities) κm (variable diffusivities)

0.584 –

2

3

4

5

6

υb = 5 0.252 1.299

0.123 0.203

0.081 0.110

0.061 0.075

0.048 0.057

υb = 8 0.396 –

0.152 0.379

0.094 0.149

0.068 0.093

0.053 0.067

Figure 10.8 Reduced marginal wavenumber κm as a function of Markstein number M, plotted for two values of the expansion ratio, υb = 5 (open symbols) and υb = 8 (solid symbols) for constant diffusivities (10.3.74) (dotted lines) and temperature-dependent diffusivities (2.9.54) (solid lines).

Typical values of the reduced marginal wavenumber κm are shown in Fig. 10.8 and Table 10.1 as a function of the first Markstein number M and for two expansion coefficients, υb = 5 and υb = 8. The values are calculated in the approximation of constant diffusivities (10.3.74) and for temperature-dependent diffusivities (2.9.54). For small values of M, diffusive effects are not sufficient to overcome the DL hydrodynamic instability at small wavelengths and the marginal wavenumber apparently diverges, indicating that the perturbation analysis must be extended to include terms of higher order in κ.

10.3.5 The Case of Multiple-Step Chemistry As mentioned in Section 8.4 (see Fig. 8.14), the relevant surface defining a flame front sustained by multiple-step chemistry is not necessarily the surface delimiting the hot side of the flame zone, as in Fig. 8.1 for the ZFK model. Once the surface that defines the front is chosen, the flame structure consists of two regions, one ahead of the front and the other downstream. This requires a new analysis.

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Wrinkled Flames

Figure 10.9 Coordinate systems.

Formulation in Local Coordinates The equations for the balance of species and energy in wrinkled flames take a simpler form when written in local coordinates attached to the flame front and when attention is limited to order . In this case, transverse effects appear only through the mean radius of curvature R of the front, and unsteady terms are negligible. To be more specific, using the notation of Sections 2.3.1 and 2.3.2, consider a point rf (t) on the front, moving with the velocity drf /dt = nf Df , where Df and nf are, respectively, the normal velocity of the front and the unit vector normal to the front at the point rf (t); see Fig. 10.9. In two-dimensional geometry, the Cartesian system of normal and tangential coordinates is xn ≡ (r − rf ).nf ,

yt ≡ (r − rf ).tf ,

tf .nf = 0,

|tf | = 1.

Excluding singular points (R = 0) and inflexion points (R = ∞), near rf (t) the equation of the front takes the form xn = αn (yt , t),

αn ≈ (1/2)y2t /R(t),

dL /R = O().

(10.3.75)

The equations for the balance of species and energy, written in the system of coordinates attached to the front at rf (t), (z, yt , t), where z ≡ (xn − αn ), take the following form, valid up to order ,   ρDT ∂ 2 Yi ∂Yi 1 ∂Yi − = i (..Yj , ..); − (10.3.76) yt = 0: m ∂z Lei R ∂z ∂z2 see Section 10.4.2. The boundary conditions are z → −∞:

Yi = Yiu ,

z → +∞:

∂Yi /∂z = 0.

(10.3.77)

Here Yi denotes either a species mass fraction or the temperature (i=1), Lei = DT /D √ i , i is the chemical term and m the normal component of the mass flux, m/(ρ u UL ) = s/ 1 + A2 (see (10.3.1)), m(r, t) ≡ ρ(un − Df ),

17:10:10 .012

(10.3.78)

10.3 Hydrodynamics and Diffusion Coupling

497

where un ≡ nf .u(r, t) is the normal component of the flow velocity, u = un nf + wt tf , wt being the tangential component, wt ≡ tf .u. Equations (10.3.76) are similar to those for the steady-state approximation of the diffusive thermal model, except that due to the deflection of the streamlines, the mass flux m(r, t) varies across the flame and has to be obtained by solving the fluid mechanic problem. The problem is made easier because, up to order , the equation for mass conservation in the moving frame takes a quasi-steady form (see Section 10.4.2): yt = 0:

∂wt ∂m +ρ = 0. ∂z ∂yt

(10.3.79)

η ≡ yt /d,

(10.3.80)

 Let t and d ≡ DTu t be the characteristic time and length scales of the planar flame. We introduce the reduced normal and tangential coordinates, and also the reduced time, ζ ≡ z/d,

τ ≡ t/t.

The precise definitions of t and d depend on the flame model. They are of the same order of magnitude as the transit time τL and the flame thickness dL , respectively. The slowly varying reduced coordinates, r/ = xnf + ytf , are introduced in a way similar to (10.3.9), x = (an + ζ ),

y = yt / = η,

t = τ ,

(10.3.81)

where, according to (10.3.75), an (η, t) = αn /d ≈ (d/R)η2 /2. The flow field is decomposed on both sides of the front (denoted by ± according to the sign of ζ ) in a way similar to (10.3.8), with the difference that here we use dimensional quantities m, un , wt and p, and nondimensional normal and tangential coordinates, m = m± (x, y, t) + m ˘ ± (ζ , y, t), un = u± (x, y, t) + u˘ ± (ζ , y, t), wt = w± (x, y, t) + w˘ ± (ζ , y, t),

(10.3.82)

p = p± (x, y, t) + p˘ ± (ζ , y, t). Each quantity m, un , wt and p is continuous at the front. This is not the case for the quantities m± , u± , w± and p± , which refer to the external flows where the density ρ is constant and equal to ρ ± , ρ + ≡ ρb < ρ − ≡ ρu . On both sides, the terms describing the modifications through the flame decrease quickly to zero, ˘ ± , u˘ ± , w˘ ± , p˘ ± ) = 0. lim (m

ζ →±∞

(10.3.83)

The flame velocities Un± of a wrinkled flame are defined locally on both sides by extrapolating the external flows to the front (ζ = η = 0), ± Un± = m± f /ρ ,

where

± ± ± m± f ≡ m (0, 0, t) = ρ (uf − Df ),

(10.3.84)

± ± and u± f ≡ uf (0, 0, t). The quantities Un reduce to UL and Ub in the planar case, but, in − − + + general, neither ρ Un nor ρ Un corresponds exactly to the mass flux crossing the front

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498

Wrinkled Flames

at rf (t), mf ≡ m(0, 0, t). Introducing (10.3.82) into (10.3.78) yields m± = ρ ± (u± − Df ),

m ˘ ± = ρ u˘ ± + (ρ − ρ ± )(u± − Df ),

(10.3.85)

and Equation (10.3.79) for mass conservation splits into two parts at yt = 0: continuity of the external flows, ∂u± /∂x + ∂w± /∂y = 0, and  ∂m ˘± ∂ ∂ +  (ρ − ρ ± ) w± + ρ w˘ ± = 0, yt = 0: (10.3.86) ∂ζ ∂y ∂y ˘ ± is also of order which shows that ∂ m ˘ ± /∂ζ is of order , so that, according to (10.3.83), m . Similarly to the ZFK model, the solution is sought in the form of a power expansion in the small parameter  ≡ d/, Yi = Yi0 (ζ ) + Yi1 (ζ , y, t) + · · · , ± ± m± f = mf 0 + mf 1 + · · · ,

m ˘ ± = m ˘± 1 (ζ ) + · · · ,

(10.3.87)

± ± ± ± ± (u , w , p ) = (u± 0 , w0 , p0 ) + (u1 , w1 , p1 ) + · · · , ˘± ˘± u± ˘± ˘± (˘u± , w˘ ± , p˘ ± ) = (˘u± 0 ,w 0 ,p 0 ) + (˘ 1 ,w 1 ,p 1 ) + ··· . ±

±

±

Introducing, in a way similar to (10.3.43), the strain rates of the external flows at the front, which here are quantities of order unity, 



uxf± ≡ ∂u± /∂x|x=y=0 ,





uxf± = −wyf± ,

wyf± ≡ ∂w± /∂y|x=y=0 ,

(10.3.88)

the expansion of m is obtained by using the Taylor expansion of m± : 

± yt = 0: m± = m± f + ζ uxf + · · · , 

± ± ± ˘± m = m± 1 (ζ )] + · · · , f 0 + [mf 1 + ζ ρ uxf + m

mf =

m± f0

+ [m± f1

+m ˘± 1 (0)] + · · ·

(10.3.89)

.

Leading Order Solution To leading order, introducing (10.3.89) into (10.3.76), ρDT ∂ 2 Yi0 ∂Yi0 − = i (..Yj0 ..), ∂z Lei ∂z2 shows that Yi0 (ζ ) is the steady-state solution of the planar flame, m± f0

yt = 0:

Yi0 (ζ ) = Y i (ζ ),

±

± m± f 0 = mf ≡ ρ U ,

−

+

U ≡ UL ,

U ≡ Ub .

(10.3.90)

(10.3.91)

In the quasi-isobaric approximation, the density can be decomposed in the same way as the temperature θ ≡ Yi=1 , ρ = ρ(ζ ) + ρ1 (ζ , y, t) + · · · ,

(10.3.92)

where ρ(ζ ) is the density profile across the planar flame which is related to the temperature profile by (10.3.4). According to (10.3.85), yt = 0:

±

± ρ(ζ )˘u± 0 (ζ ) = −[ρ(ζ ) − ρ ]U ,

17:10:10 .012

(10.3.93)

10.3 Hydrodynamics and Diffusion Coupling

499

where we have used m ˘ 0 = 0. Near rf (t), the modification to the upstream and downstream flows across the flame has the form of stagnation flows: the normal and tangential compo˘ ± (ζ , y, t), are even and odd functions of y, respectively. To leading nents, u˘ ± (ζ , y, t) and w order, the Euler equations in the vicinity of rf (t) are mf d˘u± p± 0 /dζ = −d˘ 0 /dζ ,

mf dw˘ ± p± 0 /dζ ≈ (∂αn /∂yt )d˘ 0 /dζ ,

where the first equation, valid at yt = 0, corresponds to the planar case, and the second is limited to the term proportional to y; see Section 10.4.2. Using (10.3.83) and (10.3.75), the deflection of the streamlines takes the form u± u± w˘ ± 0 (ζ , y, t) ≈ −(∂αn /∂yt )˘ 0 (ζ ) ≈ −(/R)y˘ 0 (ζ ), yt = 0:

(10.3.94)

∂ w ˘± u± 0 /∂y = −(d/R)˘ 0 (ζ ).

(10.3.95)

Viscous effects do not change the result. Mass Fluxes and Stretch Rate The perturbation to the mass flux is given by (10.3.86),   ζ   ±   ± yt = 0: m − ˘± (ζ ) = −w dζ ρ(ζ ) − ρ 1 yf ±∞

ζ ±∞



±  dζ  ρ(ζ  )w˘ 0y (ζ ),

±

˘± where we have used (10.3.83) and the notation w˘ 0y ≡ ∂ w 0 /∂y. Using (10.3.93) and (10.3.95) in the above equation yields  ζ

   ± ± (10.3.96) ˘± (ζ ) = − w + (d/R)U dζ  ρ(ζ  ) − ρ ± . yt = 0:  m 1 yf ±∞

Continuity of the tangential velocity wt at the front, in the vicinity of rf (t), leads to the − + ˘− ˘+ relation ζ = yt = 0: [∂wt /∂y]+ − = 0, namely wyf + ∂ w 0 /∂y = wyf + ∂ w 0 /∂y. According to Equations (10.3.93) and (10.3.95) written at ξ = 0, this shows that the total flame stretch (2.3.9) has the same form on both sides of the front,   

 − + (10.3.97) wyf− + (d/R)U = wyf+ + (d/R)U = d/τs ,  ζ   d yt = 0:  m (10.3.98) ˘± dζ  ρ(ζ  ) − ρ ± , 1 (ζ ) = − τ s ±∞ 



+ where wyf± = −uxf± = −dnf .∇u± f .nf . Continuity of the mass flux, ζ = yt = 0: [m]− = 0, leads to the difference of external mass fluxes, / .  +∞ 0  −    d +     + , (10.3.99) dζ ρ − ρ(ζ ) − dζ ρ(ζ ) − ρ [mf ]− = τs −∞ 0 + − where, to first order, [mf ]+ − = (mf 1 − mf 1 ).

17:10:10 .012

500

Wrinkled Flames

10.3.6 Markstein Numbers for the Chain-Branching Model Introducing (10.3.89) into (10.3.76) yields mf

± ± dYi1 ρDT d2 Yi1 − dz Li dz2 

=−

m± f1

(10.3.100)

+ ζρ

±

±

uxf + m ˘± 1 (ζ ) +

mf d Li R



 dY io  ± ∂i + Yj1 , dz ∂Yj Yi =Y i0 j

where d = ρDT /mf , d/R = O() and where m ˘± 1 (ζ ) is given by (10.3.98). The eigenvalues ± mf 1 are obtained by solving these linear equations with the boundary conditions (10.3.77) and the conditions at the front z = 0. This has be done explicitly[1] for the two-step chainbranching model (8.4.1)–(8.4.12). The result for the fast chain-breaking regime (8.4.36)– (8.4.38), (1 − θf∗ ) 1, takes a simple form in the distinguished limit (LeR − 1) = O(1 − θf∗ ), LeX = O(1). This yields expressions for the two Markstein numbers appearing in (2.3.11), both of order unity in the limit (1 − θf∗ ) → 0. For the model (5.1.2) with a frozen preheated zone, representative of lean methane flames, one gets  (LeR − 1) (υb − 1)−1 υb −1 ln(1 + ) υb − 1 − d, M = ln υb + 2LeX − 1 (1 − θf∗ ) (2LeX − 1) 0  M2 =

1

. (10.3.101) 2LeX − 1 A different result is obtained for the model (5.1.3), h = 1, ∀ T, representative of rich hydrogen flames,  υb −1 (LeR − 1) ln(1 + ) υb ln υb −1 + (υ d, − 1) M= b ∗ υb − 1 (1 − θf )  0 M2 = 48(LeX − 1)(1 − θf∗ )2 .

(10.3.102)

The first Markstein number M in (10.3.102) has the same form as that of the ZFK model (10.3.36), where β is replaced by 2/(1 − θf∗ ). 10.4 Appendix 10.4.1 Curvature and Stretch in R2 This section is devoted to the proof of (10.3.29). Attention is restricted to two-dimensional geometry for simplicity; extension to three dimensions is straightforward. The first term in (10.3.28), (1 + A2 )−1/2 ∂(1 + A2 )1/2 /∂t = (1 + A2 )−1 A (∂A /∂t), is computed using (10.3.18) and the definitions in (10.3.14), √     ∂ 2 A uy− + A2 ux− − A2 wy− − A3 wx− − A A w− A2 A ∂t 1 + A − = , (10.4.1) √ (1 + A2 ) (1 + A2 )3/2 1 + A2 [1]

Clavin P., Gra˜na-Otero J., 2011, J. Fluid Mech., 686, 187–217.

17:10:10 .012

10.4 Appendix

501

where all the quantities involving the upstream flow velocity and its derivative should be taken at the flame front and at leading order. The last terms yield √ ∂ − 1 + A2 ) + A A A A w ∂y (w + = w− + A w− + . (10.4.2) √ √ y x (1 + A2 ) 1 + A2 1 + A2 Putting together (10.4.1) and (10.4.2) and comparing with (2.9.19) in Section 2.9.1 yields the required result.

10.4.2 Equations in the Local Coordinate System in R2 Consider a moving surface in two-dimensional geometry having equation x = α(y, t) in the laboratory frame. The normal velocity Df , the unit normal vector nf and the unit tangent vector tf , tf .nf = 0, in (10.1.4) are functions of αy ≡ ∂α/∂y, nf (αy ), so that any derivative of nf is proportional to tf : ∂nf = −tf ∂αy /(1 + αy2 ),

∂tf = nf ∂αy /(1 + αy2 ).

(10.4.3)

Consider a point rf (t) = {xf (t), yf (t)} on the surface, xf = α(yf , t), moving with the normal velocity of the front, drf /dt = Df (t)nf (t). The subscript f denotes a quantity attached to rf (t). The time derivative  of nf (t) can be computed from (10.4.3) using the relation

 / 1 + α 2 , with dy /dt given by (2.9.1), dα  /dt = ∂αy /∂t − αy αyy f y

dnf = tf dt

.

∂αy /∂t Df  αy − R 1 + αy2

/ ,

(10.4.4)

y=yf

 /(1 + α 2 )3/2 is the curvature. The derivative dt /dt is similar to (10.4.4) where 1/R ≡ αyy f y with tf → −nf in the right-hand side. Using  ∂α/∂t ∂αy /∂t αy αyy ∂ Df (y, t) = − , ∂y (1 + αy2 )1/2 (1 + αy2 )3/2

obtained from (10.1.4), Equation (10.4.4) yields dnf (t) ∂ Df (s, t) = −tf (t) , dt ∂s where we have introduced the arc length ds =

dtf (t) ∂ Df (s, t) = nf (t) , dt ∂s

(10.4.5)

 1 + αy2 dy. Consider two Cartesian systems

of coordinates attached to rf (t), with r = rf (t) + r :    • (x , y , t) in the moving frame C , having the fixed unit vectors of the laboratory frame,    r = x ex + y ey  • The normal and tangential coordinates (xn , yt , t) in the local frame Cn , r = xn nf (t) + yt tf (t) (see Fig. 10.9),

17:10:10 .012

502

Wrinkled Flames

∂ ∂ + wt , ∂xn ∂yt ∂un ∂wt ∇.u = + , ∂xn ∂yt

∂ ∂ + tf (t) , ∂xn ∂yt ∂2 ∂2  ≡ ∇.∇ = 2 + 2 , ∂xn ∂yt

∇ = nf (t)

u.∇ = un

(10.4.6) (10.4.7)

where un ≡ u.nf and wt ≡ u.tf are the normal and tangential components of u(xn , yt , t) = un nf (t) + wt tf (t). The expression for the material derivative is D/Dt = ∂/∂t + u.∇ in the laboratory frame and D/Dt = ∂/∂t + (u − Df nf ).∇ in the moving frame C  . The expression for D/Dt in the local frame Cn is found by using (10.4.6) and the change of variables (x , y , t) → (xn , yt , t), dnf dtf ∂ ∂ → − xn .∇ − yt .∇, ∂t ∂t dt dt ∂ Df ∂ ∂ Df ∂ D ∂ ∂ ∂ = + (un − Df ) + wt + xn − yt , Dt ∂t ∂xn ∂yt ∂s ∂yt ∂s ∂xn

(10.4.8)

(10.4.9)

where (10.4.5) has been introduced into (10.4.8). Equations for Conservation of Species and Energy In the notation of (10.3.76), the equations for conservation of species or energy in the low Mach number approximation are ρ

ρDT DYi − Yi = i (Y1 , Y2 , . . . YN ), Dt Lei

(10.4.10)

where for simplicity ρDT and Lei are considered constant. In order to compute Yi (r, t) in terms of the unperturbed planar profile Yio (ζ ), it is convenient to use coordinates fully attached to the flame front (z, yt , t), where z ≡ xn − αn (yt , t). The equations in this system of coordinates are obtained by the change of variables (xn , yt , t) → (z, yt , t), ∂ ∂ → , ∂xn ∂z

∂ ∂  ∂ , → − αny ∂yt ∂yt ∂z

∂ ∂ ∂ → − α˙ nt , ∂t ∂t ∂z

(10.4.11)

 ≡ ∂α /∂y and α where αny ˙ nt ≡ ∂αn /∂t. In the local frame Cn , the equation of the front, n t  = 0, written in the vicinity of rf (t), takes the simple form (10.3.75), so that yt = 0: αny α˙ nt = 0, ∂ 2 αn /∂y2t = 1/R. Therefore, according to (10.4.6) and (10.4.11), the Laplacian operator in two-dimensional geometry  = ∂ 2 /∂xn2 + ∂ 2 /∂y2t , written at yt = 0, reduces to

yt = 0:

=

∂2 ∂2 1 ∂ . + 2− 2 R ∂z ∂z ∂yt

(10.4.12)

In three-dimensional geometry the result has the same form if R is the mean radius of the front, 1/R = 1/R1 +1/R2 . The expansions (10.3.87) and (10.3.92) of Yi and ρ show that, in the reference frame (z, yt , t), the tangential and time derivatives ∂Yi /∂yt , ∂ρ/∂yt , ∂Yi /∂t and ∂ρ/∂t are smaller than the derivatives with respect to z (in the normal direction) by at least an order  2 . Therefore, according to (10.4.9) and (10.4.12), Equation (10.4.10), written at yt = 0, reduces to (10.3.76) when the attention is limited to the first-order correction O().

17:10:10 .012

10.4 Appendix

503

Continuity For the same reasons, when the attention is limited to order , Dρ/Dt at yt = 0 reduces to (un − Df )∂ρ/∂z, and the equation for the conservation of mass, Dρ/Dt + ρ∇.u = 0, reduces to (10.3.79). Momentum Written in terms of the reference frame (z, yt , t), Euler’s equation ρDu/Dt = −∇p, where u = un nf (t) + wt tf (t), becomes ∂ Df ∂p Dun + ρwt =− , Dt ∂s ∂z ∂ Df Dwtg ∂p  ∂p ρ − ρun =− , + αny Dt ∂s ∂yt ∂z ρ

(10.4.13) (10.4.14)

where Equations (10.4.5) have been used and where the differential operator D/Dt is given by (10.4.9) and (10.4.11). To leading order in the expansion in gradients (see Section 10.3.5), the terms proportional to ∂ Df /∂s are negligible since they are smaller by an order . Moreover, to leading order near rf (t), Dun /Dt ≈ (un − Df )∂un /∂z and Dwt /Dt ≈ (un − Df )∂wt /∂z, the first equation being valid at yt = 0, and the second being limited to  ≈ y /R. the terms proportional to yt ; see the discussion above (10.3.94), αny t

17:10:10 .012

11 Ablative Rayleigh–Taylor Instability

Nomenclature Dimensional Quantities cp d dm DT Df g k m p r t T u u ua w x y α λ  ρ σ τm

Description Specific heat at constant pressure Thickness Reference length scale da /(ν 2ν ), see (11.2.14) Thermal diffusivity Normal propagation velocity of ablation front Acceleration Wavenumber Mass flux Pressure Coordinate (vector) Time Temperature Longitudinal flow velocity Flow velocity (vector) Ablation velocity Transverse flow velocity Stream wise coordinate Transverse coordinate Local position of front Thermal conductivity Wavelength Density Growth rate Reference time scale dm /ua , see (11.2.14)

504

17:10:11 .013

S.I. Units J K−1 kg−1 m m m2 s−1 m s−1 m s−2 m−1 kg m−2 s−1 Pa m s K m s−1 m s−1 m s−1 m s−1 m m m J s−1 m−1 K−1 m kg m−3 s−1 s

Ablative Rayleigh–Taylor Instability

Nondimensional Quantities and Abbreviations a˜ ex Fr kˆ Le n P t u u, w Uf x X, Y ˜ ∇  ζ η η θ κ μ ν ξ π σˆ ς τ τ φ ψ  cst.

Reduced Fourier component of front wrinkling, see (11.1.16) Unit vector along x-axis Froude number Fr−2 = gda /u2a Reduced wavenumber kda Lewis number DT /D Unit vector normal to the front A reduced pressure, see (11.3.4) Unit vector tangential to the front Order unity flow velocity, see (11.2.18) and (11.2.23) Components of u Order unity propagation velocity, see (11.2.18) Df /ua A scaled reduced coordinate x = κζ = kx Curvilinear coordinates based on isotherms and streamlines of potential flow Reduced gradient, see (11.2.15) dm ∇ 1 −2 2(ν−1) A small parameter, see (11.2.14) Fr /ν Order unity longitudinal coordinate, see (11.2.15) x/dm Reduced transverse coordinate y/da Order unity transverse coordinate, see (11.2.15) y/dm Reduced temperature T/Ta Order unity reduced wavenumber kdm Reduced mass flux m/(ρa ua ) Temperature exponent of thermal conductivity ≈ 2.5 Reduced longitudinal coordinate (x − αc )/da Order unity pressure, see (11.2.18) and (11.2.23)  2 p/(ρa u2a ) Reduced growth rate σ da /ua Order unity reduced growth rate σ τm Reduced time tua /da Order unity reduced time, see (11.2.15) t/τm Nondimensional flow potential, see (11.2.3) and (11.2.6) Order unity flow potential, see (11.2.16) and (11.2.19) Nondimensional vorticity, see (11.3.5) Constant

Superscripts, Subscripts and Math Accents a+ a− aa ac af

Downstream side of ablation front Upstream side of ablation front Cold side of ablation front Critical (absorption) layer of ablation wave (hot side) Hot side of ablation font, but cold side of absorption layer, Ta < Tf Tc

17:10:11 .013

505

506

Ablative Rayleigh–Taylor Instability

a(p) a(r) atot a aˆ aˆ a˜

Potential part (of flow) Rotational part (of flow) Total Value on the unperturbed (planar) front (Scalar) Value reduced by units of length and time at cold side of ablation front (Field) Reduced Fourier component, see (11.1.16) (Field) Reduced time-dependent Fourier component, see (11.1.16)

11.1 Linear Analyses of Simplified Models The stability analyses of simplified models, discussed in Section 6.2.1, are presented in this section as a first step. They are helpful to understand the dynamics of the ablation front and they serve as a guide to develop the asymptotic analysis in Section 11.2.

11.1.1 Thermo-diffusive Model To begin, consider the thermo-diffusive equation in (6.2.7) with the boundary conditions (6.2.9). This corresponds to the thermo-diffusive flame model in the limit of an infinite activation energy for Le = 1. The stability analysis of the corresponding ZFK model is investigated in Section 10.2.1 for Le = 1 and a constant thermal diffusivity DT . It is extended here to the case of a variable diffusivity DT (T). The simplification for Le = 1 in flame theory comes from the constant flame temperature in (10.2.20). This is because the equations for temperature and concentration are the same in the preheated zone. We will use the notation defined in (6.2.5) for ablation fronts and the same reduced temperature θ ≡ T/Ta as in Chapter 6, different from that used for flames in Chapters 8–10. Using the reference frame attached to the critical isotherm T = Tc , whose equation is x = αc (y, t), ξ ≡ (x − αc )/da , and the notation δαc (y, t) = α˜ c (t)eiky = αˆ c ei(ky+σ t) ,

˜ ) δθ ≡ θ − θ (x) = θ(ξ

αˆ c i(ky+σ t) e , da

(11.1.1)

where the overbar denotes the unperturbed solution, the linearised versions of (6.2.7) and (6.2.9) take a nondimensional form similar to (10.2.10) for flames (but with different notation). Written using the units of time and length defined at the cold side in (6.1.9), σˆ ≡ σ da /ua , kˆ ≡ kda , they take the form d2 (θ θ˜ ) dθ dθ˜ ν ν − + (σˆ + θ kˆ 2 )θ˜ = (σˆ + θ kˆ 2 ) 2 dξ dξ dξ ξ = 0: θ˜ = 0, dθ˜ /dξ = 0 ξ → −∞: θ˜ = 0, ν

ν

(11.1.2) (11.1.3)

where the relation δ(θ ν dθ/dξ ) = d(θ δθ )/dξ has been used. According to the steady-state solution (6.1.10), the function dθ /dξ is a particular solution of (11.1.2). The problem is then reduced to finding the general solution to the homogeneous equation.

17:10:11 .013

11.1 Linear Analyses of Simplified Models

507

(a) (b)

Figure 11.1 (a) Reduced temperature gradient from (6.1.10) and (b) temperature perturbation (11.1.4) for θ c = 50 and ν = 2.5.

Constant Thermal Diffusivity For a constant thermal diffusivity, DT = cst., namely for ν = 0, the problem is a particular case of the solution in Section 10.2.1, The solution θ˜ (ξ ) is given by θ˜ − (ξ ) in (10.2.13)

1/2 where 2r− = 1 + 1 + 4(σˆ + kˆ 2 ) with, according to (10.2.16), r− = 1. This corresponds to the classical diffusion damping (6.2.10), σˆ = −kˆ 2 , σ = −DT k2 , valid for any perturbation. Long Wavelength Limit The effect of a variable conductivity, ν = 0, is easily described in the long wavelength limit, for perturbations of wavelength much larger than the thickness of the wave, kdc 1, ν θ c kˆ 1 with θc > 1 (Tb > Tu , in notation used for flames). In this limit the quantities ν ν θ c kˆ 2 and σˆ are small so that the term (σˆ + θ kˆ 2 ) θ˜ in (11.1.2) can be neglected to leading order. The result is then obtained directly by integration with respect to ξ from ξ = −∞ to ξ = 0. The boundary conditions (11.1.3) yield the result in (6.2.11) describing the same type of relaxation as before but with a temperature-weighted thermal diffusivity. For a large temperature ratio, θc  1, the relaxation is controlled by the hot side with an effective diffusion coefficient DTc /(ν+1), DTc = θcν DTa . However, due to the sharp temperature drop ˜ ), the solution to (11.1.2)–(11.1.3), is at the ablation front, the temperature perturbation θ(ξ sharply peaked at the cold side:   ν+1  ξ ν+1 kˆ 2 θc − 1 θ − 1  dθ ; (11.1.4) θ˜ (ξ ) = ξ− dξ ν+1 dξ θc − 1 θ −1 0 see Fig. 11.1. This solution of the  ξ is readily obtained by noticing that any function ν form Y(ξ ) ≡ (dθ /dξ ) dξ  f (ξ  ) verifies the relation (dY/dξ ) − (d2 (θ Y)/dξ 2 ) = −(d/dξ )[(θ − 1)f ], as can be checked using Equation (6.1.10) for the steady state, ν (θ − 1) = θ dθ /dξ .

17:10:11 .013

508

Ablative Rayleigh–Taylor Instability

Diffusive Relaxation for Intermediate Wavelengths Consider now perturbations of intermediate wavelength in the case of a large density ratio, discussed at the end of Section 6.2.1 (see (6.2.16)), θc  1,

ν > 1,

1 1/kˆ θcν .

(11.1.5)

ν

Anticipating that the terms involving θ in the left-hand side of (11.1.2) are dominant near ν 2 + kˆ 2 (θ ν θ) ˜ ˜ ≈ 0. to the hot side, the homogeneous equation reduces there to −d2 (θ θ)/dξ In the limit (11.1.5) the solution to (11.1.2) satisfying the two boundary conditions on the hot side (11.1.3), ξ = 0, takes the form / / .  . (θc − 1) 2 2 ν ˆ ˆ − kξ kξ e − 1− e θ θ˜ = − (11.1.6) + (θ − 1) 1+ 2 θcν kˆ θcν kˆ  (θc − 1) kξ ˆ ˆ e − e−kξ + (θ − 1), (11.1.7) ≈− 2 ν

where, according to the steady-state equation (θ − 1) = θ dθ /dξ , the term (θ − 1) corresponds to the particular solution, θ˜ = dθ /dξ . The approximate expression (11.1.7) ˆ cν  1. An expression for θ˜ (ξ ), matching the solution (11.1.7) at the hot side is valid for kθ and verifying the upstream boundary condition (ξ → −∞) in (11.1.3), is obtained when (θc − 1) in (11.1.7) is replaced by (θ(ξ ) − 1), θ˜ ≈ Z(ξ ) +

dθ , dξ

Z(ξ ) ≡ −

 1 dθ kξ ˆ ˆ e − e−kξ , 2 dξ

(11.1.8)

ν

where dθ /dξ = (θ − 1)/θ has been used. This expression simplifies upstream from the critical region at a distance from the critical surface larger than the wavelength ˆ −ξ  1/k:

θ˜ ≈

1 dθ −kξ dθ ˆ + e . dξ 2 dξ

(11.1.9)

The expression (11.1.8) is an approximate solution to (11.1.2)–(11.1.3) if Z(ξ ) is an approximate solution of the homogeneous equation. Putting Z(ξ ) into the left-hand side of (11.1.2) leads to the expression    1 dθ ˆ  kξ ˆ ˆ ˆ ˆ k e + e−kξ − σˆ ekξ − e−kξ . 2 dξ

(11.1.10)

ˆ cν  1, this quantity is effectively negligible in front of k2 θ Z on the hot side, near to For kθ ξ = 0. The expression for Z(ξ ) in (11.1.8) is thus a good approximation of the solution to ˆ On the cold the homogeneous equation on the hot side provided that σˆ is not larger than k. ˆ ν − kξ become very side, near to the ablation front ξ ≈ −θc /ν, the terms proportional to e ˆ This condition large and are dominant. Expression (11.1.10) is negligible only if σˆ = −k. yields the anti-Darrieus–Landau relaxation rate (6.2.18) discussed in Section 6.2.1, ν

da 1/k dc :

σ = −ua k

17:10:11 .013

⇔

σ = −k.

(11.1.11)

11.1 Linear Analyses of Simplified Models

509

ˆ ˆ which is negligible in front of The residual term (11.1.10) then reduces to (dθ /dξ )ekξ k, ν ν σˆ Z on the cold side, ξ ≈ −θc /ν, and also in front of k2 θ Z on the hot side when ξ is close to zero. Moreover, for kˆ 1, the residual term (11.1.10) is also negligible in an intermediate range between the ablation front and the critical surface at the intermediate ˆ In the limit (11.1.5), Equation (11.1.8) is then a good distance, θcν /ν > −ξ > 1/k. candidate for a composite solution leading to an anti-Darrieus–Landau linear growth rate.

11.1.2 Discontinuous Model Consider now an ablation wave, the solution to the full quasi-isobaric problem (6.1.1)– (6.1.3) for Tc  Ta with the boundary condition (6.1.6) at the critical surface. As in flames the variation of density, associated with the variation of temperature through the quasiisobaric approximation, introduces strong hydrodynamical effects. According to the steadystate solution sketched in Fig. 6.1a, the main variation of the reduced density ρ/ρa = Ta /T is localised at the ablation front. This suggests that the residual variation of density downstream from the ablation front is not essential in the dynamics. Therefore, the main features of the hydrodynamics can be described using a discontinuous model for the density in which the quasi-isobaric approximation is used at low temperature but is replaced by a thermo-diffusive approximation at high temperature, T < Tf : ρ/ρa = Ta /T,

T > Tf : ρ = ρf ,

(11.1.12)

where ρf /ρa = Ta /Tf and where the fixed temperature Tf lies in an intermediate range Ta Tf Tc , closer to Ta than Tc . Let df be the thickness of the temperature profile below Tf , df ≡ λ(Tf )/(cp ρa ua ), where ua is the unperturbed ablation velocity, df = θfν da dc = θcν da ; see (6.1.12). Consider perturbations with intermediate wavelengths, df 1/k dc

⇔

θfν 1/kˆ θcν .

(11.1.13)

In these conditions the isotherm T = Tf defines the location of a thermal front of negligible thickness, kdf 1, namely the ablation front, across which there is a jump of density (ρa − ρf ) and of temperature (Tf − Ta ). The equation of this ablation front will be denoted x = αa (y, t) in the following. For the same reasons as in flames, longitudinal and transverse flow fields, u± (x, y, t), w± (x, y, t), are associated with the wrinkled ablation front, the superscripts − and + denoting the upstream and downstream sides T < Tf , T > Tf , ± respectively. Let (u± f , wf ) be the values of these flows at the ablation front; the mass flux ± mf (y, t) crossing the ablation front can be defined in terms of u± f and wf in the same way as in (10.1.7)–(10.1.8). In the linear approximation this flux can be written, using notation similar to (10.1.38), δmf (y, t) ≡ ρa [δu− f (y, t) − ∂αa /∂t].

(11.1.14)

This is also the perturbation of the ablation velocity. For perturbations satisfying (11.1.13), the difference with flames, studied in Section 10.1.3, concerns the modification δmf . In

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Ablative Rayleigh–Taylor Instability

ablation fronts the modification of the ablation velocity is produced by the transverse perturbations of the heat flux on the hot side, T > Tf , and not across the ablation front, T < Tf , as in flames, the latter effect being negligible (kdf 1) in front of the former (kdc  1). Neglecting curvature effects on the internal structure of the thin ablation front, which is approximately planar and steady, integration of the thermal equation (6.1.2) across the front, along the normal direction, yields the mass flux mf in terms of the temperature gradient on the downstream side denoted by the superscript +, mf cp (Tf − Ta ) = λ(Tf )nf .∇T|T + =Tf ,

(11.1.15)

where nf is the normal to the ablation front and where the temperature gradient at the front ∇T|T + =Tf comes from the perturbations in the downstream external layer Tf  T  Tc . We will come back to this expression few lines below. Flow Field In the linear analysis, the flow field can be computed as for flames in Section 10.1.2. Proceeding in the same way, the same second-order differential equation as (10.1.41) is obtained relating the dynamics of the wrinkled ablation front to the perturbation of the mass flux δmf (y, t), defined in (11.1.14). Introducing the nondimensional variables τ ≡ ua t/da , η ≡ y/da , the nondimensional quantities μf (η, τ ) ≡ mf (y, t)/(ρa ua ), θf ≡ Tf /Ta and the following notation in the normal mode analysis, δαa αˆ a ikη+ ˆ ˆ σˆ τ ˆ τ )eikη = a˜ (k, = e , da da

ˆ

ˆ τ )eikη = μˆ f δμf = μ˜ f (k,

αˆ a ikη+ ˆ σˆ τ e , da

ˆ the nondimensional form Equation (10.1.41) takes, for a given k,

 d2 a˜ ˆ d˜a + 2 dμ˜ f − Fr−2 kˆ + θf kˆ 2 − 2θf kˆ μ˜ f = 0, + 2 k dτ dτ dτ 2 

2 ˆ σˆ + 2(k + μ) ˆ σˆ − Fr−2 kˆ + θf kˆ 2 − 2θf kˆ μˆ f = 0,

(11.1.16)

(11.1.17) (11.1.18)

where Fr−2 is defined in (6.2.1) and where, for simplicity, the coefficients in the bracket ˆ is [...] have been written to leading order in the limit θf  1. The dispersion relation σˆ (k) ˆ σˆ ), is known. obtained from (11.1.18) when the perturbation to the mass flux, μˆ f (k, Heat and Mass Fluxes The perturbation of the mass flux across the ablation front is obtained from the linearised version of (11.1.15), written in the nondimensional form and in the limit θf  1, μf = 1,

δμf = θfν−1 δ(∂θ/∂ξ |θ=θ + ). f

(11.1.19)

Anticipating that, in the hot part of the ablation wave, both the unsteady term and the convective energy flux are negligible compared with heat conduction, the temperature gradient at the ablation front in (11.1.15) and (11.1.19) is computed using the thermo-diffusive solution (11.1.9). In the reference frame attached to the critical surface,

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11.2 Asymptotic Analysis of the Quasi-Isobaric Model

511

ξ = (x − αc )/da , the position of the perturbed ablation front is determined by the isotherm T = Tf , ξ = (αa − αc )/da : θ (ξ ) = θf , θ (ξ ) + δθ (ξ ) = θf , where αa = α a + δαa , αc = α c + δαc and θ |ξ =(αa −α c )/da = θf . In the linear approximation, using the notations (11.1.1) and (11.1.16), the following relation is obtained at the ablation front: ξ=

(α a − α c ) : da

(αˆ a − αˆ c ) dθ αˆ c (δαa − δαc ) dθ + δθ = 0 ⇔ + θ˜ = 0. (11.1.20) da dξ da dξ da

Introducing (11.1.9) into (11.1.20) and simplifying by dθ /dξ yields αˆ c −k(α ˆ a −α c )/da e , (11.1.21) 2 showing that the amplitude of the wrinkles on the ablation front is much greater than the amplitude on the critical surface in conditions (11.1.13) since (α c − α a ) is of order dc and kdc  1. The linear approximation of the temperature gradient ∂θ/∂ξ |θ=θ + in (11.1.19) is f      ∂θ  ∂δθ  (δαa − δαc ) d2 θ  δ + , =   ∂ξ ξ =(αa −αc )/da da ∂ξ ξ =(α a −α c )/da dξ 2  αˆ a = −

ξ =(α a −α c )/da

ˆ ˆ /dξ )e−kξ /2 and, accordwhere, according to (11.1.9), ∂ θ˜ /∂ξ = d2 θ/dξ 2 + (d2 θ/dξ 2 − kdθ ing to (11.1.1), δθ is proportional to θ˜ αˆ c /da . With the notations (11.1.1) and (11.1.16), introducing (11.1.21) into these expressions, Equation (11.1.19) yields

ˆ ν−1 dθ /dξ |ξ =(αa −α c )/da αˆ a , μˆ f αˆ a = kθ f ˆ μˆ f = k,

(11.1.22) (11.1.23)

where the relation θfν−1 dθ /dξ |θ=θf ≈ 1, valid for θf  1, has been used in (11.1.23). The result (11.1.23) is the analogue of the diffusive relaxation (11.1.11). Dispersion Relation When (11.1.23) is introduced into (11.1.18), one gets a dispersion relation in which heat diffusion, coupled to hydrodynamics, introduces a damping mechanism stronger than the hydrodynamic instability of Darrieus–Landau, as discussed at length in Section 6.2.1,  

σˆ ≈ Fr−2 kˆ − θf kˆ 2 , (11.1.24) σˆ 2 + 4kˆ σˆ − Fr−2 kˆ − θf kˆ 2 = 0, where the last relation is valid for θf  1. The temperature θf delimiting the hot side of the ablation front is arbitrary in this discontinuous density model. An expression for θf in ˆ similar to (6.2.21), is obtained by the analysis presented in the next section. terms of k,

11.2 Asymptotic Analysis of the Quasi-Isobaric Model The discontinuous model sheds light on the diffusive relaxation mechanism of the ablation front, but the drastic assumption of constant density downstream from the ablation front

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Ablative Rayleigh–Taylor Instability

does not satisfy the quasi-isobaric approximation. An asymptotic analysis is performed in this section without introducing additional simplifications to the quasi-isobaric model in (6.1.1)–(6.1.6). This section is limited to a linear analysis. An extension to nonlinear dynamics is presented in Section 11.3 for a sufficiently large thermal sensitivity of heat conduction.

11.2.1 Dynamics of the Ablation Front. General Formulation The ablation front is still defined as a thin layer (of thickness df ) separating two wide regions: a cold region of constant density and temperature (ρ = ρa , T = Ta ) and an external hot region where θ  1. The matching of the temperature profiles in the hot region and in the thin layer occurs in the vicinity of the isotherm T = Tf . The key point of the asymptotic analysis for the intermediate regimes under investigation, namely those involving wrinkles of intermediate wavelength , df  dc , is that the order of magnitude of Tf is smaller than that of the temperature in the hot external zone. We will come back to this point in Section 11.2.2. First, it is convenient to formulate the problem in a form suitable for the analysis, including the nonlinear dynamics. Flow Splitting and Equations in the Hot Region The equation for the conservation of energy (6.1.4) suggests to split the flow into two parts, u = u(p) + u(r) ,

u(p) ≡ DTa θ ν ∇θ ,

∇.u(r) = 0,

(11.2.1)

where the expression (6.1.3) for the thermal diffusivity DT ≡ λ/ρa cp has been used, DT = DTa θ ν , where DTa is the thermal diffusivity on the cold side and θ ≡ T/Ta . Downstream from the ablation front, on the hot side, denoted by the subscript +, the flow splitting yields (p)

(r)

u+ = u+ + u+ ,

(p)

u+ = DTa θ ∇φ+ ,

(r)

∇.u+ = 0,

(11.2.2)

where the notation φ+ ≡ has been introduced for later use. Mass conservation (6.1.1) then shows that the temperature is coupled only to u(r) , θ ν /ν

−1 ∂θ −1 /∂t + u(r) + DTa φ+ = 0, + .∇θ

φ+ ≡ θ ν /ν,

(11.2.3)

where θ −1 = ρ/ρa . The drastic simplification of the previous analysis was to use the approximation φ+ ≈ 0 downstream from the ablation front. We will see that the term u(r) .∇θ −1 is not negligible and introduces a correction that becomes small for large ν. Equations (11.2.1)–(11.2.3) are completed on the hot side by the Euler equation (6.1.3), θ −1 ∂u+ /∂t + θ −1 u+ .∇u+ = −∇p+ /ρu + θ −1 gex .

(11.2.4)

For perturbations of wavelength smaller than the total thickness dc , sufficiently far away from the ablation front, the downstream temperature is not perturbed from the steady state ν ν solution θ dθ /dξ ≈ θ , θ /ν ≈ ξ + cst. an the downstream boundary condition can then be written x → ∞:

φ+ = x/da ,

(r)

u+ is bounded.

17:10:11 .013

(11.2.5)

11.2 Asymptotic Analysis of the Quasi-Isobaric Model

513

Upstream Region Upstream from the ablation front, on the cold side denoted by the subscript −, the flow is incompressible and potential as in flames or for the Rayleigh–Taylor bubble (see (2.6.1)), u− = ua da ∇φ− ,

φ− = 0,

x → −∞: φ− = x/da ,

(11.2.6)

where φ− is a nondimensional potential. For   df the layer separating the cold flow from the hot flow (ablation front of thickness df ) is a considered as a surface of equation x = αa (y, t). Introducing the reference time τa ≡ da /ua , the Bernoulli equation (15.2.9) gives the condition at the ablation front r = rf , written in a nondimensional form,   2 ∂φ− p− 2 |∇φ− | = (xf /da )Fr−2 , (11.2.7) τa + da + ∂t 2 ρa u2a r=rf

where xf = αa (yf , t) is the longitudinal coordinate of the ablation front, rf = (xf , yf ) and the Froude number, Fr, is defined in (6.2.1). Conditions at the Ablation Front The ablation front is considered as planar and steady, meaning that the curvature effects of the front are neglected, df / 1. Introducing the normal propagation velocity Df of the ablation front, in the same way as in (10.1.4) where α → αa , mass conservation (15.1.45) yields   μf ≡ da nf .∇φ− |f − Df /ua = θf−1 nf .u+ |f − Df /ua , (11.2.8) where the subscript f denotes the value at the ablation front, θf ≡ Tf /Ta is the reduced temperature on the hot side of the ablation front and μf denotes the nondimensional mass flux, reduced by ρa ua . The conservation of energy (15.1.48), written in nondimensional form for an inert gas in the quasi-isobaric approximation (see (11.1.15)), yields μf =

1 1 − θf−1

da nf .∇φ+ |f ,

φ+ ≡ θ ν /ν,

(11.2.9)

where the expression da = DTa /ua for the length scale has been used. The conservation of longitudinal momentum (15.1.46) yields (p− |f − p+ |f )/ρa u2a = (θf − 1)μ2f .

(11.2.10)

The conservation of transverse momentum (15.1.47), namely equality of the tangential components, tf .u− |f = tf .u+ |f , reduces to tf .u− |f = tf .u(r) + |f because, according (p) (p) to the definition of u in (11.2.1), tf .u+ |f = 0 since the ablation front is viewed from the hot side as the isotherm θ = θf . Moreover, the normal component of u(r) is continuous across the front since the flow u(r) is divergence free, ∇.u(r) = 0. This leads (r) to the relation nf .u− |f = nf .u+ |f because the temperature is uniform on the cold side,

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514

Ablative Rayleigh–Taylor Instability

(p)

u− = 0, u(r) − = u− . Both the transverse and the longitudinal components being equal, the conservation of the transverse momentum reads (r)

u− |f = u+ |f .

(11.2.11)

The relations (11.2.8)–(11.2.10) are valid for any isotherm θf satisfying df 

⇔

θfν /ν /da ;

more precisely they are valid to leading order in the limit

(11.2.12)

θfν da /(ν)

→ 0.

11.2.2 Scaling Laws and Leading Order The asymptotic analysis is carried out for the range of parameters (6.2.17), namely for conditions such that the quantity Fr−2 /ν is small. The lower bound in (6.2.17) is automatically taken into account by the downstream boundary condition (11.2.5), saying that the critical surface is not perturbed. Following the discussions in Section 6.2, the relevant wavelengths (the most amplified and the neutral ones) are then anticipated to belong to an intermediate range, df  dc , in which a kind of universal behaviour is expected to hold. Scaling Laws in the Asymptotic Limit The orders of magnitude of the most amplified wavelength and of the relevant temperature at the hot side are determined using the same reasoning as for the evaluation of θf in the discontinuous model in Section 6.2. On one hand, the orders of magnitude of the quantities ν θ and 1/kˆ are linked by the steady-state solution, θ /ν ≈ ξ +cst. Assuming that there is no other natural length scale in the intermediate range of wavelength than the size of the ˆ 1/ν , see (6.2.21). On the other hand, another relation wrinkles, this gives θ of order (ν/k) is suggested by the dispersion relation of the discontinuous model (11.1.24) through the most amplified mode (or also the marginal mode σ = 0). This gives kˆ of order of Fr−2 /θ . Putting together these two relations leads to the ordering  1   ν  ν−1 ν−1 ˆ , (k/ν) of order Fr−2 /ν . (11.2.13) θ of order ν/Fr−2 Assuming that the temperature in the hot layer is higher than on the cold side, θ  1, we introduce a small parameter , and also a reference length and time, dm and τm , defined by   1 ν−1 θ = O(1/ 2 ),  2 ≡ Fr−2 /ν , (11.2.14) 2ν 2ν−1 ) = dm /ua . dm ≡ da /(ν ), τm ≡ τa /(ν The length scale dm characterises the relevant wavelength; it follows from the ordering of kˆ ≡ kda in (11.2.13). The characteristic time scale of the evolution of the ablation front τm ˆ with k ≈ 1/dm . We follows from the dispersion relation, σ 2 = O(gk), σˆ 2 = O(Fr−2 k), ˜ next introduce the nondimensional position r˜, gradient ∇ and time τ , r˜ ≡ r/dm = (ζ , η),

˜ ≡ dm ∇ = (∂/∂ζ , ∂/∂η), ∇

17:10:11 .013

τ ≡ t/τm .

(11.2.15)

11.2 Asymptotic Analysis of the Quasi-Isobaric Model

515

The asymptotic analysis is performed in the limit  → 0. According to (11.2.13)–(11.2.14), the reduced temperature θ is of order 1/ 2 in the hot layer. This leads to the ordering of φ+ and of the reduced mass flux (11.2.9) in the limit  → 0. We then introduce the nondimensional quantity ψ+ of order unity ψ+ ≡ ( 2 θ )ν = (da /dm )φ+ ,

˜ + |f /(1 − θ −1 ), μf = nf .∇ψ f

(11.2.16)

where lim→0 ψ+ = O(1) and lim→0 μf = O(1). However, in order to verify the quasi-isobaric approximation, according to Section 2.1.1, the increase of the flow velocity and consequently of the temperature cannot be too large. Specifically,  has to be larger than the square of the propagation Mach number of the ablation front. Moreover, the lower bound in (6.2.17) should be taken into account. In other words the limit  → 0 has to be understood as an intermediate asymptotic. Scaling of the Cold Flow The scaling of the flow velocity in the cold layer, θ = 1, is suggested by the linear analysis of the discontinuous model. Equations (11.1.14) and (11.1.23),   (11.2.17) δμf ≡ δu− |f − ∂αa /∂t /ua = (αa /da )kda , and the dispersion relation of the discontinuous model in (11.1.24) show that, for θf  1, the front is mainly convected by the upstream flow, δu− f ≈ ∂αa /∂t. This is because, according to (11.2.17) and (11.1.24), the modification of the mass flux δμf introduces a 1/2 ˆ σˆ ≈ small correction to (∂αa /∂t)/ua , of order 1/θf , σˆ ≈ Fr−1 kˆ 1/2 , kˆ ≈ Fr−2 /θf , k/

1/θf . It can be anticipated that, for the full quasi-isobaric problem, the relation δu− f ≈ ∂αa /∂t is also valid at the leading order of the limit  → 0. Considering amplitudes of the wrinkles of the ablation front of order dm , ∂αa /∂t = O(dm /τm ), that is, according to (11.2.14), ∂αa /∂t = O(ua /), the relation δu− f ≈ ∂αa /∂t leads to a flow velocity on the cold side u− larger than the ablation velocity ua by a factor 1/, u− /ua = O(1/). In the same way, the reduced normal velocity of the ablation front Df /ua is of order 1/. We thus introduce Uf , u− and π− , the nondimensional velocities and pressure of order unity in the limit  → 0, 1/2

Uf ≡ Df /ua ,

u− ≡ u− /ua ,

π− ≡  2 p− /(ρa u2a ),

(11.2.18)

and, according to (11.2.6), the nondimensional flow potential ψ− of order unity is ψ− ≡ (da /dm )φ− ,

˜ −, u− = da ∇φ− = ∇ψ

˜ − = 0, ψ

(11.2.19)

where the Laplace equation has to be solved with the boundary condition ζ ≡ x/dm → −∞:

ψ− − ψ − = 0,

ψ − = ζ .

(11.2.20)

Choosing the origin at the ablation front and introducing the nondimensional displacement of order unity of the ablation front, ζf ≡ αa /dm , the Bernoulli equation (11.2.7) at the front takes the form

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516

Ablative Rayleigh–Taylor Instability



˜ − |2 ∂ψ− |∇ψ + + π− ∂τ 2

 = ζf .

(11.2.21)

f

Scaling of the Hot Flow The scaling of the flow velocity in the hot layer, θ > θf , is given by (11.2.2) and (11.2.15)– (11.2.16), (p) ˜ +, u+ /ua = θ da ∇φ+ = θ ∇ψ

(11.2.22) (r)

showing that u+ /ua is of order 1/ 2 since, according to (11.2.11), u+ is not expected (p) (p) to be larger than u+ . We thus introduce the nondimensional flow velocities u+ , u+ and pressure π+ of order unity in the limit  → 0, ˜ +, u+ ≡  2 θ ∇ψ (p)

u+ ≡ ε2 u+ /ua ,

π+ ≡  2 p+ /(ρa u2a ).

(11.2.23)

1/ν

Using the definition of ψ+ in (11.2.16),  2 θ = ψ+ , Euler’s equation (11.2.4) yields

 −1/ν −1/ν 1/ν ˜ + +  2 ψ+ ˜ + = −∇π (∂u+ /∂τ ) + (u+ .∇)u ex ,  2 θ = ψ+ , (11.2.24) ψ+ where, according to the flow splitting (11.2.2) with (11.2.22), ˜ (r) ∇.u + = 0,

1/ν ˜ (r) u+ = ψ+ ∇ψ + + u+ , −1/ν

∂ψ+

(r) ˜ −1/ν ˜ + = 0; /∂τ + u+ .∇ψ + ψ +

(11.2.25)

the last equation is continuity (11.2.3). The conservation of mass and momentum across the ablation front, (11.2.8) and (11.2.10)–(11.2.11), leads to the boundary conditions μf = [nf .u− |f − Uf ] = (θf )−1 [nf .u+ |f − Uf ], π− |f − π+ |f =

 2 μ2f (θf

(r)

u+ |f = u− |f ,

− 1),

(11.2.26) (11.2.27)

where, according to (11.2.16), μf is of order unity. According to (11.2.5), expressing that the heat flux is not perturbed far downstream from the ablation front, the downstream (r) boundary conditions for ψ+ and u+ , written in the notations (11.2.15)–(11.2.16), are ζ → ∞:

ψ+ → ζ ,

(r)

u+ is bounded.

(11.2.28)

Leading Order Equations Working in the limit  → 0, the matching of the external hot layer and the ablation front (inner layer) concerns an intermediate range of temperature, 1 θ 1/ 2 .

(11.2.29)

In this range if, for example, θf is selected to be of order 1/, θ = O(1/), according to (11.2.26)–(11.2.27), the specific value of θf is not involved in the leading order terms in the

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11.2 Asymptotic Analysis of the Quasi-Isobaric Model

517

limit  → 0 as shown now. With this choice, according to (11.2.14), the thickness df of the ablation front in (6.1.12) is smaller than the wavelength of the perturbation by a factor  ν , θf = 1/:

df /da = 1/ν ν ,

df /dm =  ν .

(11.2.30)

In the limit  → 0, the boundary conditions at the ablation front (11.2.26)–(11.2.27) yield Uf = nf .u− |f ,

(r)

u+ |f = 0,

(p)

u+ |f = 0,

π− |f = π+ |f ,

ψ+ |f = 0,

(11.2.31)

where the last relation results from the definition of ψ+ in (11.2.16), ψ+ |f =  ν . The first relation is the same as for a passive interface since the mass flux (11.2.16), ˜ + |f , μf = nf .∇ψ

(11.2.32)

is of order unity and thus negligible in front of the upstream velocity that is large, of the order of 1/. The problem is reduced to solving the Laplace problem (11.2.19)–(11.2.20) in the cold region and applying the free boundary condition (11.2.21) in the form   ˜ − |2 |∇ψ ∂ψ− + + π+ = ζf . (11.2.33) ∂τ 2 f

An analogous equation is obtained for the Rayleigh–Taylor bubble above a vacuum; see Section 2.6.1. The difference comes from the pressure term π+ |f that has to be obtained from the solution in the hot region of the ablation wave, where π+ and ψ+ are of order unity. According to (11.2.24)–(11.2.25), to leading order in the limit  → 0, the flow in the hot zone is the solution to the following system of equations for ψ+ and u(r) + that have to be solved with the boundary conditions (11.2.28) and (11.2.31), (r) ˜ −1/ν ˜ (r) ˜ + = 0, + ψ ∇.u u+ .∇ψ + + = 0,     −1/ν 1/ν ˜ 1/ν ˜ (r) ˜ u(r) ˜ +. ψ+ u+ + ψ+ ∇ψ = −∇π + ψ ∇ψ + .∇ + + +

(11.2.34) (11.2.35)

The system of equations (11.2.34)–(11.2.35) represents a hot flow in the quasi-steady-state approximation. This is due to a density that is much smaller than in the cold region. The steady-state solution to (11.2.34)–(11.2.35) is ψ+ = ζ ,

u(r) + = 0,

(p)

u+ = u+ = ζ 1/ν ex ,

(11.2.36)

dπ + /dζ = −dζ 1/ν /dζ = −1/(νζ 1−1/ν ), ν

ν

in agreement with the temperature profile for θ  1, θ dθ /dξ ≈ θ , θ /ν ≈ ξ . Notice the tricky matching of the pressure with the ablation front. Inside this thin inner layer, the relative variation of the pressure is negligible, (pa − p)/ρa u2a = u/ua − 1 = θ − 1, and, using the notation π ≡  2 p/ρa u2a , one gets (π − πa ) <  2 θf so that (π − πa ) <  since θf = O(1/). In the limit  → 0 the last equation in (11.2.36) then yields π + − πa = −ζ 1/ν ,

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(11.2.37)

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Ablative Rayleigh–Taylor Instability

showing a power law for the pressure and for the longitudinal velocity near to ζ = 0 (ablation front). An intermediate inner layer is required for a regular matching of the hot layer with the ablation front. The following results are obtained without such a refinement. This is possible because, as we shall see, the perturbations at the moving ablation front are not singular (in contrast to dπ + /dζ at the unperturbed front).

11.2.3 Linear Analysis The stability analysis is performed using a normal mode decomposition of the wrinkles of the ablation front ζ = ζf (η, τ ) = αa /dm and of any field f (ζ , η, τ ) written in the nondimensional form with the variables of order unity ζ , η and τ , defined in (11.2.15), δf = f˜ (ζ )ζˆf eiκη+ςτ , (11.2.38) where the overbar denotes the unperturbed solution, and κ and ς are, respectively, the nondimensional wavenumber and the linear growth rate of order unity in the limit  → 0, ζf (η, τ ) = ζˆf eiκη+ςτ ,

f (ζ , η, τ ) = f (ζ ) + δf (ζ , η, τ ),

κ ≡ kdm = O(1),

ς ≡ σ τm = O(1).

(11.2.39)

The unperturbed ablation front is at the origin, ζ f = 0. From now on κ will denote the modulus of the wave vector. Solution in the Cold Region In the cold region, the velocity of the unperturbed flow is negligible in front of the perturbed flow, u− /ua = O(1/), ψ − = ζ , so that the solution to (11.2.19)–(11.2.20) yields ˆ f = ς ζˆf , and ψ˜ − (ζ ) = ψ˜ − (0)eκζ . In the linear approximation, the normal velocity is U the first relation in (11.2.31) yields ς ζˆf = κ ψ˜ − (0). The first term in the left-hand side of (11.2.33) reads ς ψ˜ − (0) = ς 2 ζˆf /κ. The second term is quadratic and thus vanishes in the linear approximation since   the unperturbed value is negligible. The third term takes the form ζˆf dπ + /dζ + π˜ + ζ =0 and Equation (11.2.33) yields   ς 2 + κ dπ + /dζ + π˜ + ζ =0 = κ,

(11.2.40)

so that the dispersion relation is obtained as soon as π˜ + (0) is known, provided that the sum dπ + /dζ + π˜ + remains bounded at the origin ζ = 0 in the limit  → 0, in contrast to dπ + /dζ |ζ =0 in (11.2.36). The linear perturbation of the pressure at the ablation front, [dπ + /dζ + π˜ + ]ζ =0 , is obtained from the linear solution of the flow in the hot region.

With the notation −

1

νζ

(r) u˜ +

Solution in the Hot Region   (r) (r) = u˜ + , w˜ + , the linearised thermal equation (11.2.34) yields

(r)

u˜ + 1+1/ν +

d2 ψ˜ + − κ 2 ψ˜ + = 0, dζ 2

d˜u(r) (r) + = −iκ w ˜+ , dζ

17:10:11 .013

(11.2.41)

11.2 Asymptotic Analysis of the Quasi-Isobaric Model

519

where the steady-state solution (11.2.36), ψ = ζ , has been used. The two components of the linearised Euler equations in (11.2.35) yield d˜u+ 1 (r) d2 (ζ 1/ν ψ˜ + ) dψ˜ + 1 dπ˜ + + u˜ + + 1−1/ν + , =− 2 dζ νζ dζ dζ νζ dζ (r) 1 d2 u˜ + d(ζ 1/ν ψ˜ + ) = −π˜ + , + dζ κ 2 dζ 2 (r)

(11.2.42) (11.2.43)

(r)

where the second equation in (11.2.41) has been used to eliminate w˜ + . Eliminating the pressure term yields a third-order differential equation for u˜ (r) + , d˜u dψ˜ + 1 1 d3 u˜ + 1 (r) + + + u˜ + = − 1−1/ν , 2 3 dζ νζ dζ κ dζ νζ (r)

−

(r)

(11.2.44)

that has to be solved together with the first equation in (11.2.41) using the boundary conditions at the ablation front (11.2.31), (r)

u˜ + = 0,

ζ = 0:

(r)

d˜u+ /dζ = 0,

ψ˜ + = −1,

(11.2.45)

(r)

where the steady-state solutions (11.2.36), u+ = 0 and ψ + = ζ , have been used as well as the second equation in (11.2.41). The downstream boundary conditions are given by (11.2.28); the temperature is not perturbed far downstream from the ablation front: ζ → ∞:

ψ˜ + = 0,

(r)

u˜ + is bounded.

(11.2.46)

Linear Growth Rate The limit limζ →0 π˜ + can be computed from (11.2.43). Anticipating that dψ˜ + /dζ |ζ =0 does not diverge and using (11.2.37), dζ 1/ν /dζ = − limζ →0 (dπ + /dζ ), the perturbation of the pressure at the moving ablation front takes the form   (r) (11.2.47) dπ + /dζ + π˜ + ζ =0 = (1/κ 2 )d2 u˜ + /dζ 2 |ζ =0 . The only parameter left in the first equation in (11.2.41) and in (11.2.44)–(11.2.46) is κ 2 . The form (6.2.16) of the linear growth rate is obtained when noticing that, after introducing (r) the change of variable x ≡ κζ = kx and the function u˜ + (x)/(κ 1−1/ν ), the parameter κ disappears from Equations (11.2.41) and (11.2.44):  2  (r) u˜ + d 1 = − 1 ψ˜ + , (11.2.48) x ≡ κζ : νx1+1/ν κ 1−1/ν dx2   (r) (r) (r) u˜ + d3 u˜ + d˜u+ 1 dψ˜ + 1 + . (11.2.49) − + = − 1−1/ν 3 1−1/ν dx νx κ dx dx νx (r)

(r)

(r)

The quantity (1/κ 2 )d2 u˜ + /dζ 2 |ζ =0 = d2 u˜ + /dx2 |ζ =0 then takes the form d2 u˜ + /dx2 |ζ =0 = −c(ν)κ 1−1/ν with a scalar coefficient c(ν) > 0 depending only on ν. According to (11.2.40) and (11.2.47), the linear growth rate σˆ then takes the form (6.2.16), where the

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520

Ablative Rayleigh–Taylor Instability

coefficient c(ν) in front of ν 1/ν kˆ 2−1/ν is obtained by solving the system (11.2.48)–(11.2.49) 1−1/ν ) and ψ ˜ + (x) using the boundary conditions (11.2.45)–(11.2.46). for u˜ (r) + (x)/(κ The Case of a Large Power Index An analytical expression can be obtained for a large power index, ν  1. In conditions (6.2.17), for  → 0, the limit of a large power index implies that the critical temperature is sufficiently large, θc  2 > 1; see the definition of  in (11.2.14). The left-hand side of (11.2.48) and the third term in the left-hand side of (11.2.49) are both small and introduce (r) a correction of order 1/ν to u˜ + (x)/κ 1−1/ν . This will be shown later. When these terms are neglected, Equations (11.2.48)–(11.2.49) reduce to  2  d x ≡ κζ : − 1 ψ˜ + = 0, (11.2.50) dx2   (r) (r) d3 u˜ + d˜u+ 1 dψ˜ + 1 , (11.2.51) − + = − 1−1/ν 3 1−1/ν dx κ dx dx νx and the solution satisfying the boundary conditions (11.2.45)–(11.2.46) takes the form ν  1:

ψ˜ + ≈ −e−x , (11.2.52)  ∞  ∞ (r) d˜u+ /dx ≈ e−x x1/ν e−2x dx − ex x1/ν e−2x dx (11.2.53) κ 1−1/ν 0 x  x 1 x1/ν e−2x dx, (11.2.54) ≈ (e−x − ex ) 1+1/ν (1 + 1/ν) + ex 2 0  ∞  ∞ (r) d2 u˜ + /dx2 −x 1/ν −2x x ≈ −e x e dx − e x1/ν e−2x dx + e−x x1/ν , κ 1−1/ν 0 x (11.2.55) (r)

valid up to the first order 1/ν. The coefficient c(ν) = −(1/κ 1−1/ν )d2 u˜ + /dx2 |ζ =0 takes the form  ∞ 1 x1/ν e−2x dx = 1/ν (1 + 1/ν), (11.2.56) ν  1: c(ν) ≈ 2 2 0 where (x) is the gamma function. The validity of the asymptotic expressions (11.2.52)–(11.2.56) can be checked by intro(r) ducing the expression for u˜ + /(κ 1−1/ν ), obtained from the integration of (11.2.53) using 1−1/ν ) the boundary condition (11.2.45), into (11.2.48)–(11.2.49). An expansion of u˜ (r) + /(κ near the origin x = 0 yields (r)

ν  1,

x 1:

u˜ +

κ 1−1/ν

≈−

x2

x2+1/ν + ··· , (1 + 1/ν) − 1 + 1/ν 21+1/ν

(11.2.57)

showing that the left-hand side of (11.2.48) and the third term in the left-hand side of (11.2.49) are both effectively negligible even near the origin. Downstream from the ablation

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11.2 Asymptotic Analysis of the Quasi-Isobaric Model

521

(b)

(a)

Pressure Vorticity

Pressure Vorticity

ˆ k/ν) ˆ 1/ν and vorticity. (a) Solutions Figure 11.2 Linear solutions for the reduced pressure p/(ρa u2a k)( for power index ν = 2.5 and wavenumber k = 5/dtot . (b) Solutions for an infinite power index. The bounds correspond to the ablation front (x = 0) and to the critical absorption layer (x = dtot ). From C. Almarcha, Ph.D. thesis, Universit´e de Provence, Marseilles, 2007, with permission.

front, the velocity u˜ (r) + becomes quasi-constant with a small value of order 1/ν: ν  1,

x  1:

u˜ + /(κ 1−1/ν ) ≈ (2−1/ν − 1)(1 + 1/ν) + x1/ν e−x /2 + · · · , (r)

(11.2.58) (r) lim u˜ /(κ 1−1/ν ) x→∞ +

1 = − (ln 2)(1 + 1/ν). ν (r)

(11.2.59)

(r)

The vorticity, which is proportional to d2 u˜ + /dx2 −κ 2 u˜ + , becomes quickly small downstream from the ablation front, of order 1/ν, since, according to (11.2.55), the second (r) (r) derivative of u˜ + decreases to zero as x increases, d2 u˜ + /dx2 ∝ x1/γ e−x . Boundary Layer for Pressure Both the vorticity and the pressure vary strongly in the hot gas near to the ablation front 2 1/ν . For a large power index, the since, according to (11.2.55), limx→0 d2 u˜ (r) + /dx ∝ x gradient of vorticity diverges at the ablation front. However, the longitudinal component (r) u˜ + /(κ 1−1/ν ) is of order 1/ν everywhere in the hot gas and its gradient is a smooth function in the limit ν → ∞. Numerical solutions[1] to the system of equations (11.2.48)–(11.2.49) (r) for u˜ + (x)/(κ 1−1/ν ) and ψ˜ + (x), using the boundary conditions (11.2.45)–(11.2.46), show very little difference with the asymptotic solution (11.2.52)–(11.2.56) for ν = 2.5, and the difference is not perceptible to the eye for ν = 10. The most striking feature is that the vorticity is concentrated in a thin boundary layer across which the perturbation of pressure varies abruptly and changes sign, from π˜ +f at the ablation front to −π˜ +f at the exit of the boundary layer; see Fig. 11.2. This thin layer is adjacent to the hot side of the ablation front. The ratio of its thickness to the characteristic wavelength of the most unstable perturbation [1]

Sanz J., et al., 2006, Phys. Plasmas, 13, 102702.

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522

Ablative Rayleigh–Taylor Instability

vanishes in the limit ν  1, δx = O(1/ν). Outside this boundary layer the hot gas flow is −1 dψ+ /dx. potential and the density quasi-constant, θ −1 dθ/dx = (1/ν)ψ+ 11.3 Nonlinear Dynamics in the Limit of a Large Power Index An analysis of the nonlinear dynamics of the ablation front is performed here on the basis of the linear analysis presented in the preceding section. The objective is to obtain a model equation that can be easily solved numerically.

11.3.1 Formulation The basic assumption is that the main properties of the linear solutions in the limit  → 0 with a large power index, ν  1, also characterise the nonlinear dynamics in the same (r) limit. The part u+ of flow velocity in the splitting (11.2.25) is assumed to remain small, of order 1/ν, and the evolution of the hot gas is assumed to remain quasi-steady. This implies that, at the leading order in the limit  → 0 and ν  1, the vorticity remains confined in a boundary layer adjacent to the ablation front, as found in the linear analysis. The validity of (r) ˜ (r) this assumption is discussed in Section 11.3.3. Neglecting the quadratic terms u+ .∇u + , Equations (11.2.23)–(11.2.25) in the hot gas, and, more specifically, (11.2.34)–(11.2.35), written with the notations (11.2.15) and (11.2.16), reduce to ˜ (r) ∇.u + = 0,

˜ + = 0, ψ (r) ˜ ˜ (u+ .∇) ∇ψ+

˜ + .∇u ˜ (r) + ∇ψ +

=

(11.3.1)

1/ν ˜ ˜ + − ∇ψ+ .∇(ψ ˜ + −∇π ∇ψ+ ).

(11.3.2)

These equations are the nonlinear versions of (11.2.50)–(11.2.51) before the pressure term (r) is eliminated. As in the linear approximation, the small terms proportional to u+ have to be retained because they produce a jump of order unity in the pressure and vorticity across the boundary layer adjacent to the ablation front; see the end of the preceding subsection. The boundary conditions at the ablation front are given in (11.2.31); the pressure π+ |f is unknown. According to (11.2.28), the temperature is not perturbed far downstream from (r) the ablation front and the flow u+ becomes parallel, ζ → ∞:

ψ+ → ζ ,

w(r) + = 0,

∂u(r) + /∂ζ = 0,

π+ → −ζ 1/ν + cst.,

(11.3.3)

where the behaviour of the pressure results from (11.3.2). In two-dimensional geometry, Equations (11.3.1)–(11.3.2), written introducing the vorticity  and the quantity P, take the form (r)

1/ν ˜ (r) ˜ 2 P ≡ π+ + u+ .∇ψ + + ψ+ |∇ψ + | /2

(r)

 ≡ ∂w+ /∂ζ − ∂u+ /∂η,

(11.3.4) ˜ + |2 ∂ψ+ ∂P |∇ψ ∂ψ+ = + , ∂η ∂ζ 2 ∂ζ 1/ν



˜ + |2 ∂ψ+ ∂ψ+ ∂P |∇ψ =− − . ∂ζ ∂η 2 ∂η 1/ν



17:10:11 .013

(11.3.5)

11.3 Nonlinear Dynamics in the Limit of a Large Power Index

523

11.3.2 Solution It is convenient to introduce the curvilinear and orthogonal coordinates (X, Y) based on (r) the isotherms, X = ψ+ (ζ , η) = ( 2 θ )ν , and the streamlines of the potential flow, u+ , ⊥ (ζ , η), u(r) = ∇( ⊥ /∂ζ = −∂ψ /∂η, ∂ψ ⊥ /∂η = ∂ψ /∂ζ . ˜ 2 θ )ν+1 ν/(ν + 1), ∂ψ+ Y = ψ+ + + + + ⊥ ˜ + |, one gets the differential relations ˜ + | = |∇ψ Introducing the additional constraint |∇ψ ∂ψ+ ∂ ∂ψ+ ∂ ∂ = − , ∂ζ ∂ζ ∂X ∂η ∂Y

∂ ∂ψ+ ∂ ∂ψ+ ∂ = + , (11.3.6) ∂η ∂η ∂X ∂ζ ∂Y ˜ = |∇ψ ˜ + |[nψ+ ∂/∂X + tψ+ ∂/∂Y], nψ+ .∇ ˜ = |∇ψ ˜ + |∂/∂X, ˜ = |∇ψ ˜ + |∂/∂Y ∇ tψ+ .∇ where nψ+ and tψ+ are, respectively, the unit vectors normal and tangent to the isotherms: ˜ + | nψ+ = (∂ψ+ /∂ζ , ∂ψ+ /∂η), |∇ψ

˜ + | tψ+ = (−∂ψ+ /∂η, ∂ψ+ /∂ζ ). |∇ψ

Using the first equation in (11.3.1), the following relations of differential geometry hold,  2  ˜ ∂ ∂2 1 2 ˜ ˜ ψ+ = − ∂|∇ψ+ | , ˜ ˜ ψ+ = 0 ⇒  = |∇ψ+ | + = ∇.n , Rψ+ ∂X ∂X 2 ∂Y 2 (11.3.7) where 1/Rψ+ is the curvature of the isotherm ψ+ (ζ , η) = X. Multiplying the first equation in (11.3.5) by ∂ψ+ /∂ζ and adding the second equation in (11.3.5), multiplied by −∂ψ+ /∂η, yields an equation for P. Adding the first, multiplied by ∂ψ+ /∂η, to the second, multiplied by ∂ψ+ /∂ζ , yields a relation between P and : . /  1/ν  ˜  1/ν  ˜ + |2 ∂X ∂X |∇ψ+ |2 ∂P ∂ ∂ |∇ψ ∂P =− , =− , = . ∂X ∂X 2 ∂Y ∂X ∂X ∂Y 2 (11.3.8) According to (11.2.31) and (11.3.4), P at the ablation front, denoted Pf , is equal to the pressure, X = 0: P ≡ Pf = π+ |f . From now on the subscripts f and ∞ will denote, respectively, the values at the ablation front, X = 0, and at infinity, X → ∞. Integrating the first equation by parts from X = 0 in (11.3.8) yields 1 ˜ 1 1/ν 2 ˜ + |2 + 1 − 1)|∇ψ P = π+ |f − |∇ψ + |f − (X 2 2 2



X



dX (X 0

1/ν



˜ ψ+ |2 ∂|∇ − 1) , (11.3.9) ∂X 

where, according to (11.3.7), the last term involves a curvature effect. According to (11.3.3) (r) and (11.3.4) the boundary condition at infinity takes the form limX→∞ P = π+∞ +u∞ (Y)+ X 1/ν /2, where π+∞ = −X 1/ν + cst., and since X = ζ at infinity, “one gets” (r)

lim P = −X 1/ν /2 + u+∞ (Y),

X→∞

X = 0: P = π+ |f .

Taking the limit X → ∞ in (11.3.9) then yields  ˜ + |2 1 ˜ 1 ∞ ∂|∇ψ (r) 2 u+∞ (Y) = π+ |f − |∇ψ+ |f + dX (X 1/ν − 1) + cst. 2 2 0 ∂X

17:10:11 .013

(11.3.10)

(11.3.11)

524

Ablative Rayleigh–Taylor Instability (r)

If u+∞ (Y) is of order 1/ν in the limit 1/ν → 0, as is the case in the linear analysis (see (11.2.59)), then the terms of order unity must balance, π+ | f =

1 ˜ |∇ψ+ |2f , 2

(11.3.12)

since the integral in the last term in the right-hand side is also of order 1/ν. This can be checked by using the expansion (X 1/ν −1) ≈ (ln X)/ν+ · · · , which leads to a non-divergent ˜ + |2 /∂X vanishes sufficiently quickly integral term of order 1/ν if the quantity (ln X)∂|∇ψ in the limit X → ∞, namely for ln X ν. There is no divergence near to X = 0 since in the limit 1/ν → 0 the integral is proportional to (1/ν)X ln X.

11.3.3 Discussion To summarise, in the limit 1/ν → 0 and in conditions (6.2.17), namely for  → 0 and for θc  2 > 1 (see the definition of  in (11.2.14)), according to (11.2.33) and (11.3.12), the nonlinear dynamics is simply described by the parameter-free system (6.2.22)–(6.2.23) that is studied numerically in Section 6.2.2. The validity of Equation (11.3.12) at the leading order in the limits  → 0 and 1/ν → 0 is checked now. Validity of the Assumptions The flow field in the hot gas is computed by introducing the potential ,  2  ∂ ∂2 (r) (r) 2 ˜ ˜ , + u = ∂/∂η, w = −∂/∂ζ ,  = − = |∇ψ+ | ∂X 2 ∂Y 2 (11.3.13) (r) (r) and, according to the second equation in (11.3.8), the vorticity  ≡ ∂w+ /∂ζ − ∂u+ /∂η satisfies Poisson’s equation   2 1 ∂P ∂ ∂2  = G, G≡ , (11.3.14) + 2 2 ˜ + |2 ∂Y ∂X ∂Y |∇ψ (r)

with the boundary conditions at the front (11.2.31), u+ |f = 0, ∂/∂ζ = 0, ∂/∂η = 0, and at infinity (11.2.46), w(r) = −∂/∂ζ = 0, ∂ψ+ /∂η = 0, ∂ψ+ /∂ζ = 1, X = 0:

∂/∂X = 0,

∂/∂Y = 0,

X → ∞:

∂/∂X = 0,

∂/∂Y is bounded,

(11.3.15)

(r)

˜ + | = 1, |∇ψ

G = −∞ (Y), (11.3.16)

where −∞ = ∂u+∞ /∂Y. The last relation in (11.3.16) comes from (11.3.10). Considering periodic conditions in the transverse direction, we introduce the Fourier series  iKY ˜ , (11.3.17) A(X, Y) = A(X)e K

17:10:11 .013

11.3 Nonlinear Dynamics in the Limit of a Large Power Index

525

where K denotes the modulus of the wave vector. It is also more convenient to introduce ˜  ˜ ≡ − ˜  ˜ ∞ /K 2 , ˜ ∞ that vanishes at infinity, and also the potential ! ˜ ≡ G+ the quantity H so that the system of equations (11.3.14)–(11.3.16) reduces to   2 d 2 ˜ = H, ˜ ! (11.3.18) − K dX 2 ˜ ˜ = − ˜ ∞ /K 2 , X = 0: d!/dX = 0, ! (11.3.19) X → ∞:

˜ d!/dX = 0,

˜ is bounded. !

(11.3.20)

The solution satisfying the last condition in (11.3.19) and also the bounded condition at infinity takes the form  X  ∞   ˜ = − e−KX ˜  )dX  − eKX ˜  )dX  2K ! eKX H(X e−KX H(X X (11.3.21) 0  ∞  −KX KX ˜ ∞ /K + ˜  )dX  . −2 +e e H(X 0

The first relation in (11.3.19) then yields  ∞  KX  ˜   ˜ ˜ X = 0: 2Kd!/dX = 2∞ − K e H(X )dX − K 0

∞

 ˜  )dX  = 0. e−KX H(X

0

(11.3.22) (r) When the expressions (11.3.9)–(11.3.9) for P and ∞ = −∂u+∞ /∂Y are introduced into the compatibility condition (11.3.2) one gets a very complicated integral equation for ˜ + |2 ) in terms of |∇ψ ˜ + |. Following the same argument as Z(Y) ≡ (d/dY)(π+ |f − 12 |∇ψ f developed below Equation (11.3.12), the terms that do not involve Z are all of order 1/ν. The compatibility condition then takes the form  ∞  ∞ KX ˜ e Z1 dX + K e−KX Z˜ 2 dX + Z˜ = O(1/ν), (11.3.23) K 0 0   1 1 where Z1 ≡ − 1 Z, Z2 ≡ Z, (11.3.24) 2 ˜ ˜ |∇ψ+ | |∇ψ+ |2 which provides a strong argument in favour of Z(Y) being of order 1/ν and thus of the validity of (11.3.12) at the leading order in the asymptotic analysis. Comparison with the Discontinuous Model The result (6.2.23) was also obtained by using the discontinuous model (11.1.12) in the limit of an infinitely large density ratio,[1] keeping the mass flux finite. The density and temperature jumps across the sharp discontinuity are larger than across the ablation front of the preceding analysis, ρ+ =  2 ρa . For a sufficiently large density jump, the hydrodynamic discontinuity in the model (11.1.12) incorporates the boundary layer for the vorticity. The vorticity of the flow vanishes on both sides of the sharp discontinuity, as is the case in the preceding analysis, outside the boundary layer adjacent to the ablation front. The upstream [1]

Clavin P., Almarcha C., 2005, C. R. M´ecanique, 333(379-388).

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526

Ablative Rayleigh–Taylor Instability

˜ − , u+ = ∇ψ ˜ +, and downstream flows are thus incompressible and potential, u− = ∇ψ ˜ ˜ ψ− = 0, ψ+ = 0. The Bernoulli equation at the sharp discontinuity is the same as in (11.2.21). However, the term π− |f is here different from the value of the pressure of ˜ + |2 /2. The term π− |f has to be computed the hot flow at the discontinuity π+ |f = −|∇ψ f by using the pressure jump (11.2.27) in which the reduced temperature θf is now equal to ˜ + |2 . This leads to π− |f = 1/ 2 and where μf is given by (11.2.32), π− |f − π+ |f = |∇ψ f 1 ˜ |∇ψ+ |2 and thus to the same result (6.2.22)–(6.2.23) as obtained in the previous analysis. 2

f

The pressure has a sharp transition across the discontinuity in the same way as across the vorticity boundary layer adjacent to the ablation front in the linear analysis.

17:10:11 .013

12 Shock Waves and Detonations

Nomenclature Dimensional Quantities a A cp cv D D/Dt D e E k kB l m p qm s t tN T T∗ TN u u v w x

Description Speed of sound Position of front in frame of mean position Specific heat at constant pressure Specific heat at constant volume Molecular diffusivity Material derivative, ∂/∂t + u.∇ Propagation speed Energy density Activation energy Transverse wavenumber Boltzmann’s constant Longitudinal wavenumber, see (12.1.11) Mass flux Pressure Heat of combustion per unit mass Entropy Time Reaction time at Neumann state, τr (T N ) Temperature Crossover temperature Temperature of Neumann state Longitudinal velocity component Velocity of fluid (u, v, w) Velocity Transverse velocity component Streamwise coordinate

S.I. Units m s−1 m J K−1 kg−1 J K−1 kg−1 m2 s−1 s−1 m s−1 J kg−1 J mole−1 m−1 J K−1 m−1 kg m−2 s−1 Pa J kg−1 ≡ (m/s)2 J K−1 kg−1 s s K K K m s−1 m s−1 m s−1 m s−1 m 527

17:10:14 .014

528

y, z α θ ρ σ τ ω

Shock Waves and Detonations

Transverse coordinates Local position of the front Local slope of front Density (Complex) growth rate of perturbation A characteristic time Angular frequency

m m radian kg m−3 s−1 s s−1

Nondimensional Quantities and Abbreviations a, b, c f h h H m(t) M n P Q r r

Nondimensional coefficients, see (12.1.27) and (12.1.34) 2 2 M u /M uCJ Overdrive factor A measure of (inverse) reduced heat release (γ − 1)/Q 1/2 See (12.2.32) 2hβe A measure of temperature sensitivity (n + 1)h/u¯ Reduced mass flux 1 − (∂ α/∂ ˆ ˆt)/D Mach number D/a Parameter characterising strength of shock, see (4.4.4) Reduced pressure, see (4.2.11) Reduced heat of reaction (γ + 1)qm /(2cp Tu ) Parameter characterising the fluid, see (4.4.4) ρ N /ρ (In Section 12.2.4) Reduced specific volume 

S

Reduced complex growth rate, see (12.1.12)

t u˘ u¯ V w ˙ x x y y α βe γ ε  ζ η θ θ˘ κ

Reduced time t/tN Reduced velocity, see (12.2.11) u/au Reduced velocity, see (12.2.27) −μ Reduced specific volume, see (4.2.11) Reaction rate reduced by the reaction time, tN (In Section 12.2.1) Reduced coordinate x/(au tN ) (In Section 12.2.4) Mass-weighted reduced coordinate, see (4.5.1) Reduced transverse coordinate y/(au tN ) Reduced transverse coordinate y/uN tN Reduced position of lead shock, see (12.2.14) Order unity Zeldovich number ε2 E/(kB TN ) Ratio of specific heats cp /cv A small quantity (small heat release), see (12.2.5) A second small quantity, see (12.2.54) ξ See (12.2.28) (u (ξ  ))−1 dξ  √0 Order unity reduced transverse coordinate εy ˘ 2 Order unity reduced temperature, see (12.2.16) θ/ε

2

σ/ aN |k|(1 − M N )1/2

(T − Tu )/Tu √ kau ¯tN / ε

Reduced temperature, see (12.2.11) Reduced wavenumber

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12.1 Linear Dynamics of Wrinkled Shocks

μ ν ν˘ ξ π π˘ σ τ υ φ± ϕ χ ψ

Order unity reduced flow velocity, see (12.2.16) Order unity reduced transverse velocity Reduced transverse velocity Reduced distance, see (12.2.17) Order unity reduced pressure, see (12.2.16) Reduced pressure, see (12.2.11) Reduced complex growth rate, see (12.2.41) Reduced time, see (12.2.14) Scalar field of transverse velocity gradient Particular solutions, see (12.2.48) See (12.2.39) A measure of overdrive, see (12.1.49) Reaction progress variable, see (15.1.42)



Reduced frequency, Im(S)

(ξ , τ ) O(.) CJ ZND

Distribution of heat release rate, w(ψ, ˙ θ) Of the order of Chapmann–Jouget Zeldovich, von Neumann and D¨oring

529

(˘u − 1)/ε ν˘ /ε3/2 w/au x−α π˘ /ε γ −1 ln(p/pu ) σ tN /ε εt x  0 ∇.vdx ∂ν/∂η − αη ∂ν/∂ξ   2 ω/ aN |k|(1 − M N )1/2

Superscripts, Subscripts and Math Accents a∗ a(a) ab aCJ a(i) aN ao ar au ay a˙ t a a˜ aˆ a˘

A critical or particular value Acoustic (compressible) part of flow Burnt gas Chapman–Jouget solution Incompressible part of flow Neumann state Reference or base state Reaction Unburnt gas ∂a/∂y ∂a/∂t Average, unperturbed or steady-state value Fourier component, a(x, y, t) = a˜ (x)eiky ± σ t (In Section 12.2) A dimensional quantity (In Section 12.2) A nondimensional quantity, see (12.2.11)

12.1 Linear Dynamics of Wrinkled Shocks To include both shock waves and detonations (when modifications to the inner structure are neglected), we consider the general case for arbitrary media, first investigated during the

17:10:14 .014

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Shock Waves and Detonations

1950s.[1,2] Particular attention is paid to the case of a polytropic gas (ideal gas with constant specific heats cp and cv ). The formulation of the problem, the method of solution and the discussion of the results are presented in Section 4.4.1. The present section is devoted to the technical details of the analysis.

12.1.1 Acoustic Waves and Entropy–Vorticity Wave Using the notations of Section 4.4.1, the linearised Euler equations (15.1.4) (15.1.18) and (15.1.59), written in two-dimensional geometry for simplicity, take the form ∂ ∂ 1 D δρ + δu + δw = 0, (12.1.1) ρ N Dt ∂x ∂y ∂ ∂ D D (12.1.2) ρ N δu = − δp, ρ N δw = − δp, Dt ∂x Dt ∂y D D D δs = 0 ⇒ δp = a2N δρ, (12.1.3) Dt Dt Dt where the overbar denotes the unperturbed planar wave, and the material derivative D/Dt in the linear approximation takes the simple form D/Dt = ∂/∂t + uN ∂/∂x, since the unperturbed flow is uniform. Assuming that the wavelengths of the wrinkles are much larger than the thickness of the shock, the boundary conditions at the shock (x = 0), labelled by the subscript N (Neumann state) are obtained from the Rankine– Hugoniot relations; see (12.1.25)–(12.1.26). A boundedness condition is used at infinity in the shocked gases, x → ∞. Flow Field Eliminating δρ from (12.1.1) and (12.1.3) gives 1 ρ N a2N

∂ ∂ D δp + δu + δw = 0. Dt ∂x ∂y

(12.1.4)

Eliminating δu and δw from (12.1.2) and (12.1.4) yields the wave equation (d’Alembert’s equation) for acoustics in a fluid moving at constant velocity uN ,  2  D2 ∂ ∂2 2 δp = 0. (12.1.5) δp − a + N Dt2 ∂x2 ∂y2 In the linear approximation around a uniform flow, the pressure fluctuations are fully incorporated in the acoustic waves. The acoustic waves propagating in the shocked gas are generated by the pressure fluctuations δpN (y, t) at the Neumann state x = 0 that are produced by wrinkling of the front. Entropy fluctuations are also generated at x = 0 [1] [2]

D’yakov S., 1954, Zh. Eksp. Teor. Fiz., 27, 288. Kontorovich V., 1957, Zh. Eksp. Teor. Fiz., 33, 1525.

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531

since the Hugoniot curve is not an isentropic curve. The solution to (12.1.2) may then be decomposed into acoustic waves, superscripted (a), and an entropy–vorticity wave, superscripted (i). The flow velocity in the acoustic waves is obtained from (12.1.2), where the pressure is the solution to (12.1.5) with the boundary condition at x = 0, δp = δpN (y, t). The flow in the entropy–vorticity wave (δu(i) , δw(i) ) is the solution to (12.1.2) with δp = 0. It is an isobaric rotational flow generated at the Neumann state, x = 0, propagating with the unperturbed flow, Dδu(i) /Dt = 0,

Dδw(i) /Dt = 0.

which is incompressible, according to (12.1.4), ∂ ∂ (i) ∂ ∂ δu + δw(i) = 0 ⇒ δu(i) = uN δw(i) . ∂x ∂y ∂t ∂y

(12.1.6)

The flow disturbances may thus be written as δp = δp(a) ,

δu = δu(a) (x, y, t) + δu(i) (y, t − x/uN ),

δw = δw (x, y, t) + δw (y, t − x/uN ), (a)

(i)

(12.1.7) (12.1.8)

where (δu(a) , δw(a) ) is the flow velocity in the acoustic wave. The functions δu(i) (y, t) and δw(i) (y, t) are obtained by subtracting the flow velocity of the acoustic wave from the flow at x = 0, δu = δuN (y, t), δw = δwN (y, t) in (12.1.26). The density fluctuations δρ(x, y, t) will be eliminated from the analysis. They can be obtained from the pressure fluctuations δp(x, y, t) propagating with acoustic waves and from the propagation (12.1.3) of the entropy fluctuations, which are generated at x = 0 by the wrinkled shock, δsN = 0, δpN = a2N δρN , δp − a2 δρ = δpN (y, t − x/uN ) − a2N δρN (y, t − x/uN ).

(12.1.9)

Normal-Mode Analysis Using a normal mode decomposition (4.4.1), the objective is to obtain the growth rate σ for a given wavenumber of wrinkling k. The two pressure eigenmodes, the solution to (12.1.5), take the form δp = p˜ N exp (il± x + iky + σ t) ,

(12.1.10)

where p˜ N is given by the boundary condition x = 0: δp = δpN (y, t) = p˜ N eiky+σ t , and where the quantities l± are solutions to the second-order algebraic equation associated with (12.1.5), 2 + k2 ) = 0, (σ + il± uN )2 + a2N (l±

(12.1.11)

resulting in √ M N S ± 1 + S2 l± = , i  |k| 2 1 − MN

with S ≡

1 σ ,  aN |k| 2 1 − MN

17:10:14 .014

(12.1.12)

532

Shock Waves and Detonations

expressing il± /|k| in terms of σ/(aN |k|). For unstable modes in the sense Re(S) > 0, the ± sign in (12.1.12) must be chosen so as to enforce boundedness of the acoustic waves downstream, x → ∞, thereby requiring that the real part of il± be nonpositive,    Re M N S ± 1 + S2  0. (12.1.13) For neutral modes, Re(S) = 0 with 1 + S2 < 0, as for shock waves in a polytropic gas studied in the next section, the sign in (12.1.12) is determined in a different way; see (12.1.37) below. A more intuitive insight is obtained when the quantity σ/(aN |k|) is expressed in terms of l± /|k| from (12.1.11) or (12.1.12). In the particular case of acoustic modes, σ = ±iω with ω > 0 and l± real, this leads to the well-known result of the frequency shift by the Doppler effect,  2 + k2 ∓ u l  0. (12.1.14) ω = aN l± N ± In the limit of infinite sound speed, M N → 0, σ/aN |k| → 0, Equation (12.1.12) gives the low Mach number approximation il± = ±|k| of (10.1.16). The velocity field associated with the acoustic modes is obtained from the pressure by using the Euler equations (12.1.2), u˜ (a) = −

p˜ N il± x il± uN e , σ + il± uN ρ N uN

w˜ (a) = −

p˜ N il± x ikuN e , σ + il± uN ρ N uN

and the entropy–vorticity wave then takes the form  p˜ N il± uN (i) e−σ x/uN , u˜ = u˜ N + σ + il± uN ρ N uN  p˜ N ikuN (i) e−σ x/uN , w˜ = w˜ N + σ + il± uN ρ N uN

(12.1.15)

(12.1.16) (12.1.17)

where p˜ N (k, σ ), u˜ N (k, σ ) and w ˜ N (k, σ ) are obtained from the Rankine–Hugoniot relations; see (12.1.25)–(12.1.26) below. More precisely, u˜ N (k, σ ) and w˜ N (k, σ ) are the Fourier transforms of the components in the Galilean laboratory frame of the flow velocity in the shocked gas at x = 0. For unstable cases with s ≡ Re(σ ) > 0, the entropy–vorticity wave is damped as x increases. For stable cases with s < 0 there is no real divergence since, according to the initial value problem, the quantities δu(i) (y, t−x/uN ) and δw(i) (y, t−x/uN ) are bounded for t > 0 and x  uN t by the initial condition at the front, t = 0, x = 0, δu(i) (y, 0) and δw(i) (y, 0). Compatibility Condition The entropy–vorticity wave must satisfy the incompressibility condition (12.1.6), −

(il± σ + uN k2 ) p˜ N u˜ N −σ + ikw˜ N = 0. (σ + il± uN ) ρ N uN uN

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(12.1.18)

12.1 Linear Dynamics of Wrinkled Shocks

533

The fraction in the first term on the left-hand side may be written in a simpler form as follows. When the first term in (12.1.11) is developed, the numerator may be written as 2 + k2 ) −(il± uN σ + u2N k2 ) = σ 2 + il± σ uN + (a2N − u2N )(l±   

2 2 = −(σ + il± uN ) il± uN 1 − M N − σ M N ,

where the term in the square brackets is proportional to the term equation of (12.1.12). Equation (12.1.18) may then be written   p˜  σ2 u˜ N 2 N +σ − ikw˜ N = 0 ± |k| 2 + 1 − M N 2 ρ N uN uN aN k

√

1 + S2 in the first

(12.1.19)

or, in a nondimensional form,[1,2] ±



S2 + 1

MN p˜ N u˜ N ikw˜ N +S − = 0,  ρ N aN uN uN |k|uN 2 1 − MN

(12.1.20)

where the ± sign is the same as in (12.1.12) and has to be chosen to satisfy (12.1.13). Equation (12.1.19) is valid for any supersonic discontinuity when modifications to the inner structure are neglected. Equations (12.1.19) or (12.1.20) lead to an equation for the reduced complex growth rate σ/(aN |k|), when the quantities p˜ N /(ρ N uN aN ), u˜ N /uN and w˜ N /uN are expressed in terms of σ and k using the Rankine–Hugoniot relations at the shock, x = 0.

12.1.2 Dispersion Relation for General Materials The flow velocity at the Neumann state is given by the jump conditions for conservation of mass (15.1.45) and transverse momentum (15.1.47), ρN (uN − ∂α/∂t − wN ∂α/∂y) = ρu (D − ∂α/∂t),

(12.1.21)

(D − ∂α/∂t) cos θ = δwN sin θ + (uN − ∂α/∂t) cos θ , where using the normal and tangential velocity of the flow relative to the front in (10.1.5)– (10.1.6), cot θ = ∂α/∂y, the linear approximation yields δρN uN + ρ N (δuN − ∂α/∂t) = −ρu ∂α/∂t, δwN = (D − uN )∂α/∂y.

(12.1.22) (12.1.23)

The Hugoniot relation (4.2.5) provides the slope of the Hugoniot curve at the Neumann state dpN /dρ −1 N . The medium and the shock strength are characterised by two positive nondimensional parameters r and n defined in (4.4.4). The perturbation of the mass flux δm obtained from (12.1.21) and the slope of the Hugoniot curve in (4.4.4) yield

[1] [2]

Buckmaster J., Ludford G., 1988, Proc. Comb. Inst., 21, 1669–1676. Clavin P., et al., 1997, Phys. Fluids, 9(12), 3764–3785.

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Shock Waves and Detonations

δm = −ρu

∂α , ∂t

δpN =

1 r



ρu ρN

2

2

D δρN .

(12.1.24)

Equation (4.2.3) then leads to the variations of pressure and density at the Neumann state in terms of the variation of shock velocity, ∂α/∂t,

  1 − pu /pN ∂α/∂t ρN r ∂α/∂t δρN δpN = −2 = −2 −1 , . (12.1.25) pN (1 − r) ρ ρ (1 − r) D D u N Using the density variations in (12.1.25), Equations (12.1.22)–(12.1.23) yield the variation of the flow velocity at the Neumann state,     ρN ρN 1 + r ∂α/∂t ∂α δwN δuN . (12.1.26) = −1 = −1 , uN ρu (1 − r) D uN ρu ∂y Introducing (12.1.25) and (12.1.26) into (12.1.20) yields the dispersion relation (4.4.2), where the ± sign is the same as in (12.1.12). The complex linear growth rate of the normal modes, σ/(aN |k|), is expressed in terms of r, n and M N using the roots of the quadratic equation (4.4.2) for S2 : aS4 + 2bS2 + c = 0, 2

a ≡ (1 + r)2 − 4M N ,

b≡

S2 ≡

σ2

a2N k2 2 (1 − r2 )n − 2M N ,

1 2

(1 − M N )

,

(12.1.27)

c ≡ (1 − r)2 n2 > 0.

The roots to be retained are those which satisfy (12.1.13) and (4.4.2). The discussion of the general case is straightforward but tedious. Details are given for polytropic gases in the next sections. The general results are summarised in Fig. 4.17. The most striking feature is that the domain of stability with Re(σ ) < 0 is separated from the domain of instability with Re(σ ) > 0, by a wide domain of neutral modes in which at least one of the roots of (12.1.27) is negative, S2 < 0, Re(σ ) = 0, Im(σ ) = 0. Depending on the sign of the quantity a one may have either a single neutral mode and a stable mode with Re(σ ) < 0 or a pair of neutral modes, as discussed in detail in Section 12.1.3 for polytropic gases. The critical values −(1 + 2M N ) and (n − 1)/(n + 1) for the parameter r in Fig. 4.17 were obtained in the 1950s.[1,2] They play an important role in the stability limits. Later, different authors[3,4] introduced the Gruneisen coefficient  instead of r. The above-mentioned critical values for r correspond to the critical values (1 + M N )/  and 2 1/(1 + ) for the parameter (ρ N /ρu − 1)M N . The critical value r∗ separating the domain of neutral modes from the domain of exponential relaxation (Re(σ ) < 0) corresponds to the double roots S2 = −b/a < 0, b2 = ac, 

2 n − (1 − M N ) 1 − ρu /ρ N ∗ r = . (12.1.28) n+1 [1] [2] [3] [4]

D’yakov S., 1954, Zh. Eksp. Teor. Fiz., 27, 288. Kontorovich V., 1957, Zh. Eksp. Teor. Fiz., 33, 1525. Fickett W., Davis W., 1979, Detonation. University of California Press. Majda A., Rosales R., 1983, SIAM J. Appl. Math., 43(6), 1310–1334.

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535

To the best of our knowledge, the above expression for r∗ was obtained only recently.[5] In the range of parameters −(1 + 2M N )  r  r∗ corresponding to neutral modes, the stability limit is given by the critical value r = (n − 1)/(n + 1), separating the cases of nonradiating and radiating acoustic modes; see Section 12.1.4 for a polytropic gas. For (n − 1)/(n + 1) < r < r∗ (nonradiating condition), the amplitude of an initial disturbance relaxes with a power law in time; see the discussion in Section 4.4.1.

12.1.3 Polytropic Gases For a polytropic gas (γ = cst.) the parameters r and n are related simply to the propagation 2 2 2 Mach number M u , r = 1/M u and n = M u /(M u − 1), (1 − r)n = 1 (c = 1). Equation (4.4.2) then reduces to    −2 (12.1.29) ± 2SM N 1 + S2 = S2 1 + M u + 1, where the ± sign must be chosen to satisfy condition (12.1.13). The following relations are readily obtained using (4.2.14); M u > 1,

γ >1

⇒

(n − 1)/(n + 1) < r < r ∗ .

So, according to Fig. 4.17, shock waves in polytropic gases have neutral normal modes with nonradiating acoustic waves. This problem is not clearly presented in the literature. The terms ‘structural stability’ and ‘strong stability’[4] used for r > (n−1)/(n+1) do not mean ‘exponential relaxation’, since for (n − 1)/(n + 1) < r < r∗ the normal modes are neutral with nonradiating conditions (see Fig. 4.17) and correspond to a relaxation of initial disturbances with power laws such as t−3/2 , as is the case for gaseous shock waves; see the discussion in Section 4.4.1. In order to clarify this question and to avoid misunderstanding, it is worth reconsidering the normal mode analysis and the stability of gaseous shock waves from the beginning. This is done now. The density and pressure fluctuations at the Neumann state are obtained from the linearised versions of (4.2.14) and (4.2.15) using δMu = −(∂α/∂t)/au , 1 4 δρN

=− ρN (γ + 1) M u 1 + δpN = − pN 1+

4γ (γ +1) M u 2 2γ (γ +1) (M u

1 2 (γ −1) (γ +1) (M u

∂α/∂t , − 1) au

∂α/∂t , − 1) au 



(12.1.30)

(12.1.31)

in agreement with (12.1.25). The velocity fluctuations at the Neumann state are obtained from (12.1.22) and (12.1.23) using (4.2.14), (4.2.15) and (12.1.30),

[5]

Clavin P., Williams F., 2012, Philos. Trans. R. Soc. London Ser. A, 370, 597–624.

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Shock Waves and Detonations

δuN = uN δwN = uN

2 2 (γ +1) [2 + (M u − 1)] ∂α/∂t , 2 uN Mu 2 2 (γ +1) (M u − 1)

 ∂α/∂y, 2 −1) 1 + (γ (M − 1) u (γ +1)

(12.1.32)

(12.1.33)

in agreement with (12.1.26). Equation (12.1.29) is obtained directly by introducing (12.1.31)–(12.1.33) into (12.1.20). Equation (12.1.27) then reduces to aS4 + 2bS2 + 1 = 0,     −2 2 2 −2 2 − 4M N , b ≡ 1 + M u − 2M N , a ≡ 1 + Mu

(12.1.34)

  2 2 −2 2 2 b2 − a = 4M N M N − M u , where M N is expressed in terms of M u and (γ − 1) using

 √ 2 = −b ± b2 − a /a, are real numbers since (4.2.17). The two roots of (12.1.34), S± 2

−2

b2 > a, M N > M u , as can be checked using (4.2.17). It is also easy to see that b > 0, since the sign of b is the same as the expression    4 2 2 2 2M u − (3 − γ )M u − (γ − 1) = M u − 1 2M u + (γ − 1) , 2

which is positive for M u > 1 and γ > 1. Furthermore, one has a < b since the sign of (b − a) is the same as the expression 8 7 4 2 2 2(γ − 1)M u + 2(2 − γ )M u + (γ − 1) > 0, for M u > 1. 

2 2 2 The signs of a and of Mu2 − 1 2(2 − γ )M u − (γ − 1) are identical for M u > 1, so that the condition a > 0 is verified for γ  5/3 and a < 0 for γ  2. In the interval 5/3 < γ < 2 the quantity a is positive for sufficiently fast shocks, 2

5/3 < γ < 2: M u >

(γ − 1) > 1: a > 0; 2(2 − γ )

2

1 < Mu
0 in gaseous shocks. Noticing that the signs of the quantities a and −b + b2 − a are opposite, one finds a > 0:

2 S− < 0,

2 S+ < 0,

a < 0:

2 S− > 0,

2 S+ < 0.

The relevant solutions for S obtained from (12.1.34) are those that are the solutions of Equation (12.1.29) satisfying condition (12.1.13). Consider first the case a > 0. The two solutions S± are purely imaginary,

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12.1 Linear Dynamics of Wrinkled Shocks

537

S− = ±i2 , 2 > 1 > 1, √ 2 b+ b −a 22 = . (12.1.35) a  2 is also real. This has The right-hand side of (12.1.29) being real, the quantity ±S± 1 + S± two consequences. a > 0:

S+ = ±i1 , √ b − b2 − a 2 , 1 = a

2 • First one must have (1 + S )  0, S = ±i with   1. This condition is verified by the two roots 2 > 1 > 1, as can be checked directly from (12.1.35) by using the expressions for a and b in (12.1.34). This implies that the right-hand side of (12.1.29) is negative. √ √ • Second, with the choice S = +i one must take ± 1 + S2 = i 2 −√1 in Equa2 tions √ (12.1.29) and (12.1.12), while for S = −i one must write ± 1 + S = 2 −i  − 1, in order to get a negative contribution from the left-hand side of (12.1.29),    −2 (12.1.36) − 2M N  2 − 1 = −2 1 + M u + 1 < 0.

A single absolute value of il± in (12.1.12) corresponds to each , il± = +il for S = i ( > 0, σ = iω, ω > 0), while il± = −il for S = −i, where ⎡ ⎤ √ 2 l 1 ω ⎢ MN  +  − 1 ⎥ =⎣ ≡ > 0. (12.1.37)   ⎦ > 0, |k| aN |k| 2 2 1 − MN 1 − MN The frequency ω > 0 may be expressed in terms of l and k from (12.1.37),  ω = aN l2 + k2 − uN l,

(12.1.38)

which satisfies the inequality ω2 /(aN |k|)2 < 1 + (l/k)2 as it should according to (12.1.36). This result corresponds to one of the two possibilities in (12.1.14). To summarise, when a > 0 there are two neutral oscillatory modes associated with every wavelength 2π/|k|. Each of them involves an acoustic wave whose frequency ω is given by (12.1.35) and (12.1.37) and satisfies (12.1.38). This is the case for ordinary shock waves in gases. Consider now the nonusual case a < 0. The solution S+ is still purely imaginary while the other, S− , is real, a < 0:

S+ = ±i1 , 21 =

√

1 > 1,

−b + b2 − a > 1, (−a)

S− = ±'2 , '2 > 1 , √ b + b2 − a '22 = > 21 . (−a)

(12.1.39)

The S+ normal mode is neutral and oscillatory. The associated acoustic wave has the same properties (12.1.37)–(12.1.38) as the previous ones. The solution S− = +'2 > 0 has to be rejected because it would correspond to the + sign in (12.1.29) and the boundedness condition (12.1.13) would not be satisfied. The second normal mode is thus stable and damped exponentially, S− = −'2 < 0.

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Shock Waves and Detonations

This stability of gaseous shock waves is discussed in the next section by considering the response to an acoustic wave impinging on the shock front from the shocked gas.

12.1.4 Spontaneous Emission of Sound: Radiation Condition The conclusions of Equations (12.1.36)–(12.1.38) are quite general for neutral modes in any material. Introducing the wave vector K ≡ lex + key , the unit wave vector eK ≡ K/|K| and the position r ≡ xex +yey , and using (12.1.37) and (12.1.38), the Fourier representation (12.1.10) with il± = il for σ = iω, yields δp = p˜ N exp [iK.(r − uN ex t + aN eK t)] .

(12.1.40)

This shows simply that the propagation velocity of the sound waves relative to the gas flow of is aN . In the reference frame of the planar front of the unperturbed shock, the velocity √ 2 the sound wave is uN ex − aN eK and its component along the x-axis is uN − aN l/ l + k2 . Radiating or Nonradiating Acoustic Waves There are two possibilities, depending on whether the acoustic wave propagates downstream from the shock towards infinity in the shocked gases (radiating wave, also called spontaneous sound emission) or propagates upstream from infinity to impinge on the shock (upstream running modes, also called incoming waves or nonradiating waves). Equation (12.1.40) leads to the following: radiating waves: uN − aN √ nonradiating waves: uN − aN √

l l2

+ k2 l

l2 + k 2

>0 < 0.

The radiating condition is always fulfilled for l < 0, whereas for l > 0 one has  l 2 radiating waves: 1 − MN < MN , |k|  l 2 nonradiating waves: 1 − MN > MN . |k|

(12.1.41) (12.1.42)

The nonradiating condition is verified by (12.1.37) since  > 1. This is confirmed by (12.1.36); see (12.1.46) below. In conclusion, the acoustic waves associated with the neutral oscillatory modes of a shock in polytropic gases are nonradiating waves (running upstream), or in other words, there is no spontaneous sound emission. Linear Stability Criterion for Neutral Modes Coming back to the general case, the classification given by (12.1.41) and (12.1.42) plays an important role in the linear stability analysis of shock waves that are characterised by neutral oscillatory modes, as explained now. For an arbitrary medium, the nature of the

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539

acoustic waves (radiating or nonradiating) associated with neutral modes, S = i,  > 1, depends on the form taken by the dispersion relation (4.4.2) when expressed in term of :  

 2 (12.1.43) radiating waves: l/|k| = M N  − 2 − 1 / 1 − M N ,  2M N  2 − 1 = −2 (1 + r) + (1 − r)n > 0, (12.1.44)  

 2 (12.1.45) nonradiating waves: l/|k| = M N  + 2 − 1 / 1 − M N ,  −2M N  2 − 1 = −2 (1 + r) + (1 − r)n < 0. (12.1.46) The outcome depends on the values of the coefficients r and n, which control the existence and the nature of the modes. The critical value r = (n − 1)/(n + 1) separating radiating and nonradiating acoustic waves in Fig. 4.17 corresponds to the solution  = 1 since 1 + r = (1 − r)n. In this case, according to (12.1.13) and (4.4.2), the other mode, if any, is necessarily stable with S < 0. Consider a planar acoustic wave impinging on the shock from the shocked gas (incoming wave). It is reflected by the shock. The reflected wave (radiating wave) has the same values of k and ω as the incident wave. Following an analysis similar to the linear analysis described above, the complex amplitudes, p˜ r , of the reflected wave and of the wrinkles of the shock, α, ˜ may be calculated in terms of the amplitude of the incident acoustic wave p˜ i ,

 √ 2 − 1 − 2 (1 + r) + (1 − r)n 2M   N p˜ r , = − √ p˜ i −2M N  2 − 1 − 2 (1 + r) + (1 − r)n  √ 2 − 1/ 1 − M 2 −i  (ρ N − ρu )/ρ N N .

|k|α˜ = √ 2 2(1 − r) p˜ i /ρ N aN −2M N  2 − 1 − 2 (1 + r) + (1 − r)n These equations show that the response of the shock diverges when one of its normal modes is radiating. This occurs when the reflected wave matches the radiating eigenmode. This is why a shock is considered as unstable when at least one of its eigenmodes is radiating (spontaneous emission of sound), and stable in the opposite case. Shock waves in a polytropic gas are thus stable in that sense.

12.1.5 Reacting Shocks In this section we present the stability analysis of reacting shocks. Overdriven detonations in polytropic gases can be considered as a discontinuities of zero thickness if the modifications to the inner structure can be neglected. In contrast with shock waves in inert fluids, this approximation is not valid for ordinary gaseous detonations. This is because unsteadiness of the induction zone cannot be neglected, even in the long wavelength limit if the planar detonation is pulsating, a phenomenon called ‘galloping detonation’ studied in Section 4.5.1. Nevertheless, it is instructive to perform a stability analysis of reacting shocks by

17:10:14 .014

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Shock Waves and Detonations

comparison with inert shocks. This analysis is a particular case of the one presented in Section 12.1.2 when the parameters r and n are conveniently expressed in terms of the propagation Mach number M u and the reduced heat release Q ≡ [(γ + 1)/2]qm /cp Tu , and when M N is replaced by M b . The parameter r, defined in a way similar to (4.4.4) when the Neumann state is replaced by the burnt gas state, is obtained from (4.2.43), r≡−

ρu D 2 dpb /dρb−1

2 Vb

= Mu

+1

Pb + 1

1 + Vb

2

= Mu

2

1 − MuV b

,

(12.1.47)

2

where V b and P b = −M u V b are defined in (4.2.11), and can be expressed in terms of Q 2 and M u using the smallest root of the quadratic equation 4.2.43 for V b since the burnt gas state is represented by the point B in Fig. 4.4,

 −2 V b = M u − (1 + χ ) /2, (12.1.48) 6   −2 2 −2 1 − Mu where χ≡ − 4QM u . (12.1.49) According to (4.2.44), the heat release Q can be expressed in terms of the Chapmann–   −2 2 2 Jouget (CJ) Mach number of propagation, Q = 1 − M uCJ M uCJ /4 and the parameter χ takes the form 6     −2 2 −2 2 2 −2 1 − Mu − 1 − M uCJ M uCJ M u . (12.1.50) χ= −1

−1

Using the relations 1 < M uCJ  M u , (M u − M u ) > (M uCJ − M uCJ ), it is easy to prove that the parameter χ is in the range [0, 1], 0  χ < 1,

(12.1.51)

where χ = 0 corresponds to the CJ waves, M u = M uCJ . It is also useful to introduce the parameter n as in (4.4.4), 

2 2 n ≡ ρ b /ρu M b /(1 − M b ) .

(12.1.52)

2

The calculation of n requires an explicit expression for M b ≡ u2b /a2b . Using the relations a2 = γ p/ρ, ρ b ub = ρu D and the definitions in (4.2.11) with Pb = −Mu2 Vb , one gets 2

Mb =

ρu pu 2 1 + 2V b /(γ + 1) , M u = −2 ρ b pb M u − 2V b γ /(γ + 1) 2

1 − Mb =

ρb 2 1 , M b = −2 ρu M u − 2V b γ /(γ + 1)

−2

M u − 1 − 2V b

−2

M u − 2V b γ /(γ + 1)

.

17:10:14 .014

(12.1.53)

12.1 Linear Dynamics of Wrinkled Shocks

541

Introducing these expressions into (12.1.52) gives n=

1 −2 Mu

− 1 − 2V b

=

1 , χ

(12.1.54)

where the last relation comes from (12.1.48). Using (12.1.54), and (12.1.47)–(12.1.48) then yields ⎤ ⎡  1+ 2 1  (1 − χ ) + 12 n−1 ⎣ M u (1−χ ) Mu ⎦. r= = (12.1.55) n+1 1+ 2 1 (1 + χ ) + 12 M u (1+χ )

Mu

According to the general result plotted in Fig. 4.17, neutral modes, if they exist, have nonradiating acoustic waves since, according to (12.1.51), the bracket in the right-hand side of (12.1.55) is larger than unity, r > (n − 1)/(n + 1) for 0 < χ < 1. This conclusion is in contradiction with the basic assumption of the pioneering nonlinear analyses.[1] Notice however, that, according to (12.1.50), χ = 0 in a CJ wave, so that the term in brackets in the right-hand side of (12.1.55) is unity, and r has the critical value of (n+1)/(n−1) (equal here to unity, since n = 1/χ ), separating radiating modes from nonradiating modes. This means that, on a CJ detonation, the acoustic wave propagates in the direction parallel to the unperturbed front. In order to clarify the situation, it is worth reconsidering the normal-mode analysis from the beginning, but in the limit of a large Mach number of propagation in order to simplify the presentation. This is done below. Normal Mode Analysis in the Large Mach Number Limit The configuration is the same as in the preceding sections for inert shocks, and x = α(y, t) represents the position of the front. Introducing the parameter r defined in (12.1.47) and using (12.1.24), the equations for the conservation of mass and momentum yield the equivalent of (12.1.25) in which the label N is replaced by b for burnt gas,   1 α˙ t r ρ b α˙ t δρb δpb = −2 = −2 , (pb − pu ) 1−rD (ρ b − ρu ) 1 − r ρu D      ρu 1+r ρu α˙ t , αy , 1− δub = δwb = D 1 − 1−r ρb ρb where the notations α˙ t ≡ ∂α/∂t and αy ≡ ∂α/∂y have been introduced. Notice that the above expressions diverge in the limit r → 1 (χ → 0), namely for CJ waves. The analysis is limited to overdriven detonations, r > 1. The normal-mode analysis is obtained from the roots of (4.4.2), where M N is replaced by M b . It is useful to express r in terms of the overdrive factor, 2

2

f ≡ M u /M uCJ > 1. [1]

Majda A., Rosales R., 1983, SIAM J. Appl. Math., 43(6), 1310–1334.

17:10:14 .014

(12.1.56)

542

Shock Waves and Detonations 2

2

In ordinary situations Q is large, M uCJ ≈ 4Q + 2  1. In the limit M u  1, an expansion 2

2

in powers of 1/M u = 1/(f M uCJ ) may be used,  (γ − χo ) −4 2 + ··· , χ = χo + O(M u ), χo = 1 − f −1 < 1, M b = γ (1 + χo )   (1 − χo ) 2 2f χo (1 − χo ) 2f χo + ··· , r= + ··· , 1 + r = + 1+ 2 2 (1 + χo ) 1 + χ o Mu M u (1 + χo ) 2 2f (1 − χo ) 2f (1 − χo ) (1 − r)n = + · · · , (1 + r) − (1 − r)n = − 2 + ··· , 2 1 + χo (1 + χ ) o M M u

u

−2

where the term of order M u in the expansion of χ is zero and where we have limited the 2 expansion of M b to the leading order term (order unity). It is useful to express the dispersion relation (4.4.2) in which MN is replaced by Mb as an equation for S2 + 1 in the form  2 −2 M u  1: 2M b S(± 1 + S2 ) = (1 + r)(1 + S2 ) − 2f (1 − χo )M u , (12.1.57) where the last term is small and the coefficients 2M b and (1 + r) are of order unity. The two solutions for S that satisfy the boundedness condition (12.1.13) are the following: −4

2 • A first root that is purely imaginary and corresponds to 1 + S small, of order M u , −4 S = i1 , 21 = 1 + O(M u ), so that the first term on the right-hand side of (12.1.57) is negligible compared with the left-hand side:  −2 (12.1.58) 2M b S(± 1 + S2 ) ≈ −2f (1 − χo )M u < 0.

This solution corresponds to a neutral mode, 21 = 1 + f

γ (1 − χo ) −4 M > 1, 2 (γ − χo ) u

−2

M b 1 + O(M u ) l ≈ ,  |k| 2 1 − Mb

(12.1.59)

with, according to (12.1.46), a nonradiating acoustic wave since the right-hand side of (12.1.57) is negative; see also (12.1.42) with M N replaced by M b . 2 • A second root with S of order√unity so that the last term on the right-hand side of (12.1.57) is negligible, 2M b S(± 1 + S2 ) ≈ (1 + r)(1 + S2 ),  2M b S(± 1 + S2 ) ≈ (1 + r)(1 + S2 ),

S2 =

(1 + r)2 2

4M b − (1 + r)2

+ ··· .

(12.1.60)

The nature of the second mode is different depending on the sign of the denominator in S2 . When it is negative, the mode is neutral S = i2 with 22 > 1, so that (1 + S2 ) < 0 and the right-hand side of the first equation in (12.1.60) is negative. Therefore, according to (12.1.46), this mode is nonradiating. This can be checked in a different manner as follows.

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12.2 Dynamics of Detonation Fronts

543

√ According to (12.1.60), S and ± 1 + S2 are two imaginary numbers having the same sign. Therefore, according to (12.1.12),  M  + 22 − 1 b 2 l ≈ , (12.1.61)  |k| 2 1 − Mb and the acoustic wave is nonradiating; see (12.1.42) and (12.1.46) with M N replaced by M b . 2 The condition for the existence of a second neutral mode, 4M b < (1 + r)2 , is verified when γ < 2 if the overdrive is sufficiently large, 1/f < γ (2 − γ ). For γ > 2 (whatever be f ) 2 or for γ < 2 at a sufficiently small overdrive (1/f > γ (2 − γ )), one gets 4M b > (1 + r)2 so that S2 > 0. In this case the normal mode is damped exponentially, S < 0, since the minus sign in front of the square root must be chosen in order to satisfy the boundedness condition (12.1.13). To summarise, when the inner structure is disregarded, reacting shock waves in the regime of overdriven detonations always have a neutral mode with a nonradiating acoustic wave and a second mode that either is damped exponentially or is also neutral with a nonradiating acoustic wave. There is no sound radiation, and the wave is stable in that sense. However, the presence of an oscillatory instability of the inner structure strongly limits the interest of these results for the stability properties of real detonations.

12.2 Dynamics of Detonation Fronts As explained in Section 1.2.5 and also in the text below Equation (4.2.45), the flow velocity in the shocked gas of detonation waves is sufficiently large and the reaction rate sufficiently low compared with the time between elastic collisions for the diffusive transport to be localised in the lead (inert) shock and negligible in the shocked gas (downstream from the shock) including the exothermic reaction zone. Constitutive Equations According to (15.1.33)–(15.1.35) the reactive Euler equations can be written Du 1 Dρ = −∇.u, ρ = −∇p, ρ Dt Dt 1 DT (γ − 1) 1 Dp qm w ˙ − = , T Dt γ p Dt cp T tN

p = (cp − cv )ρT, w ˙ Dψ = , Dt tN

(12.2.1) (12.2.2)

where D/Dt ≡ ∂/∂t + u.∇ is the material derivative, ψ is the reaction progress variable (fraction of heat release), tN ≡ τr (T N ) is the reaction time at the Neumann state (just downstream from the lead shock) of the unperturbed detonation wave and w ˙ is the reduced reaction rate that is a function of T and ψ in the simplest one-step model w(ψ, ˙ T). The energy equation, written in the entropy form in (12.2.2), can also be written by use of (12.2.1)

17:10:14 .014

544

Shock Waves and Detonations

qm w ˙ 1 Dp + ∇.u = . γ p Dt cp T tN

(12.2.3)

The unsteady propagation of unstable gaseous detonation is a hyperbolic problem that is too complicated for general solutions to be obtained analytically, even in the framework of the one-step model. Analytical results have been obtained for conditions near to the stability limits where the instability is weak in such a way that the cellular detonation can be described by a weakly nonlinear analysis. Ordinary detonations are strongly unstable so that the analytical results are obtained for values of the parameters that are not very realistic. The hope is that, in the absence of a secondary bifurcation, the essential nonlinear mechanisms controlling real detonations are picked up by the weakly nonlinear analysis. Two limiting cases have been considered to describe cellular detonations. A bifurcation analysis was first performed[1] for strongly overdriven detonations M u  1 in the Newtonian limit (γ − 1) 1. The stability limits concern a heat release smaller than the enthalpy of the gas just behind the shock,[2] qm /cp T N 1. The analysis is presented in Section 4.5 and details of the calculations are given in Section 12.2.4.. The opposite limit of detonations near to CJ regimes has also been investigated.[3] In the Newtonian approximation the stability limit of this second case also concerns small heat release, now smaller than the enthalpy of the initial gas, qm /cp Tu 1, and the flow is transonic throughout the entire wave (M − 1) 1. These two extreme cases are complementary. In the former, the flow of compressed gas is quasi-isobaric and the dynamics of the waves are controlled by the entropy–vorticity wave, while compressible effects (reactive acoustic waves) dominate in the latter. The constitutive equations can be put in a useful form that generalises the characteristic equations (15.3.41) for simple one-dimensional waves. They are obtained when the equation for the conservation of longitudinal momentum in (12.2.1) is multiplied by √ a/(γ p) = 1/(aρ), where a = γ p/ρ is the sound speed, added to and subtracted from (12.2.3), qm w ˙ ∂w 1 D± p 1 D± u ± = , − γ p Dt a Dt cp T tN ∂y

Dw 1 ∂p =− , Dt ρ ∂y

(12.2.4)

written in two-dimensional geometry u = (u, w) for simplicity, where the differential operator D± /Dt ≡ ∂/∂t ± (a ± u)(∂/∂x) + w(∂/∂y) has been introduced. The form (12.2.4) is useful when both the transverse component of the flow and the transverse derivative are small. This is the case when the heat release is small. 12.2.1 Small Heat Release Approximation Near to CJ Condition For small heat release, Equation (4.2.44) shows that the propagation Mach number of the CJ wave is close to unity, [1] [2] [3]

Clavin P., Denet B., 2002, Phys. Rev. Lett., 88(4), 044502–1–4. Daou R., Clavin P., 2003, J. Fluid Mech., 482, 181–206. Clavin P., Williams F., 2009, J. Fluid Mech., 624, 125–150.

17:10:14 .014

12.2 Dynamics of Detonation Fronts

ε2 ≡

(γ + 1) qm 1, 2 cp Tu

(MuCJ − 1) ≈ ε.

545

(12.2.5)

According to (4.2.13), (4.2.42) and (4.2.44), overdriven detonations correspond to f ≡ √ (Mu − Mu−1 )2 /4ε2 > 1, (Mu /MuCJ )2 ≈ 1 + 2ε( f − 1), so that a small degree of overdrive corresponds to values of f of order unity, f = 1, corresponding to the CJ wave. In the limit ε → 0, assuming (γ − 1) small, of order ε so that (γ + 1) ≈ 2 to leading order, the Rankine–Hugoniot relations in (4.2.14)–(4.2.17) lead to  (Mu2 − 1) ≈ (1 − MN2 ) ≈ 2ε f , TN ≈ 1 + (γ − 1)(Mu2 − 1), Tu

 (1 − Mb2 ) ≈ 2ε f − 1, pN ρN ≈ ≈ 1 + (Mu2 − 1), pu ρu

(12.2.6) (12.2.7)

and the ZND structure of detonations presented in Section 4.2.3 yields   T − TN ≈ ε2 ψ − (γ − 1)ε f [1 − 1 − (ψ/f )], Tu   (γ + 1) (p − pu ) (γ + 1) 2 (D − u) Mu ≈ ε f Mu [1 + 1 − (ψ/f )], = 2γ pu 2 D

(12.2.8) (12.2.9)

where ψ = 0 corresponds to the Neumann state. Equations (12.2.7) and (12.2.8) show that the temperature variations are much smaller than the pressure variations if γ − 1 is of order ε. In the following we assume h ≡ (γ − 1)/ε = O(1).

(12.2.10)

According to (12.2.5) and (12.2.10), temperature variations are seen to be of order ε2 , while variations of pressure and therefore of density are of order ε. The temperature changes have the same general characteristics as in ordinary detonations; due to compressible effects the √ √ temperature has a maximum. The restriction ε/(γ − 1) = 1/h > f − f − 1 is desirable, however, to prevent the final temperature from being less than the Neumann temperature. Other restrictions are required to ensure that the physical constraints of ordinary detonations are not violated. The shock thickness must be negligible compared with the detonation thickness. In other words, the elastic collision time τcoll must be shorter than the reaction time tN . More precisely, according to the inner shock structure studied in Section 4.2.2, the condition (τcoll /tN )/(Mu − 1) 1 must be fulfilled. According to the Arrhenius law (1.2.2), this is the case when e−E/kB TN ε. This condition is easily satisfied for sufficiently large activation energies. In order to ensure that the reaction rate is zero in the fresh mixture we impose an Arrhenius reaction rate modified by a crossover temperature that lies between ˙ = 0. With these conditions the detonation structure Tu and TN , Tu < T ∗ < TN , T < T ∗ : w is, as in the ordinary case, constituted by an inert shock wave followed by a much thicker reaction zone through which the diffusive fluxes are negligible.

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12.2.2 Galloping Detonations Near CJ Condition It is convenient to investigate the distinguished limit (12.2.5) and (12.2.10) in the planar case first, in order to highlight the simplification of the dynamics near to the CJ condition. Nondimensional Equations In the following, in order to simplify the presentation, the notation will be different from that in the rest of the book. We will introduce the hat (ˆ) symbol to denote dimensional quantities and the breve (˘) symbol to denote nondimensional quantities. The reduced variables for time and length will be denoted by roman, rather than italic, characters. The nondimensional variables are then defined as follows:   ˆt (Tˆ − Tˆ u ) pˆ xˆ uˆ 1 , θ˘ ≡ , u˘ ≡ , π˘ ≡ ln , (12.2.11) t≡ , x≡ tN aˆ u tN aˆ u γ pˆ u Tˆ u where tN is the dimensional reaction time at the unperturbed Neumann state. According to (12.2.5), the variation of temperature is small, θ˘ = O(ε2 ). Anticipating that 1 − u˘ and π˘ are small quantities of same order, 1 − u˘ = O(ε), π˘ = O(ε), the variation of sound speed, which is of order ε2 , a/au = 1 + O(ε2 ), is negligible in Equations (12.2.2) and (12.2.4). To leading order in the limit ε → 0, these latter can then be written in a nondimensional form to yield   ∂ ∂ ∂ ∂ 2 + (1 + u˘ ) (π˘ + u˘ ) = ε w, − (1 − u˘ ) (π˘ − u˘ ) = ε2 w, ˙ ˙ (12.2.12) ∂t ∂x ∂t ∂x   ∂ ∂ ∂ ∂ + u˘ [θ˘ − (γ − 1)π˘ ] = ε2 w, + u˘ ψ = w, ˙ (12.2.13) ˙ ∂t ∂x ∂t ∂x valid up to order ε2 . Equations (12.2.12) describe two acoustic modes propagating in the reacting gas. The first one runs downstream and the second runs upstream. Equations (12.2.13) represent the entropy wave, which propagates downstream with the flow. Slow Time Scale The dynamics of the lead shock wave is the result of a feedback loop. Modifications of temperature, pressure and flow velocity at the Neumann state, created by fluctuations of the shock velocity, propagate downstream by both the downstream-running acoustic mode and the entropy wave. The resulting perturbations to the reaction zone generate pressure fluctuations that are sent back upstream to the lead shock by the upstream-running acoustic wave and close the feedback loop. In the transonic flow of compressed gas, the streamline velocity is of the order of the sound speed aˆ u . The downstream running acoustic wave and the entropy wave propagate at velocities of order 2ˆau and aˆ u , respectively, but the upstreamrunning acoustic wave is much slower and propagates relative to the lead shock at a velocity of order εaˆ u . This wave controls the longest time in the feedback loop, tN /ε, so it is also the time scale of the dynamics of the lead shock. The fluctuations of the position of the lead shock can then be written in the reference frame of the mean position of the front in the form ˆ xˆ = A(εt/t N ),

x = α(τ ),

where

ˆ au tN ), α ≡ A/(ˆ

17:10:14 .014

τ ≡ εt = εˆt/tN ,

(12.2.14)

12.2 Dynamics of Detonation Fronts

547

and the modification to reduced propagation velocity of the lead shock is 1 dAˆ 1 dAˆ =ε = εα˙ τ , aˆ u dˆt aˆ u tN dτ

α˙ τ ≡

dα = O(1), dτ

α = O(1).

(12.2.15)

Here α˙ τ denotes the modification to the reduced detonation velocity. Introducing the nondimensional variables of order unity μ, π and θ , suggested by (12.2.5)–(12.2.10), u˘ = 1 + εμ,

π˘ = επ ,

θ˘ = ε2 θ ,

(12.2.16)

and working in the moving frame of the lead shock with, according to (12.2.15),   ∂ ∂ ∂ ∂ ∂ = , =ε − α˙ τ , (12.2.17) τ ≡ εt, ξ ≡ x − α(τ ), ∂x ∂ξ ∂t ∂τ ∂ξ Equations (12.2.12)–(12.2.13), to leading order, take the form  ∂(π + μ) ∂ ∂ = 0, + (μ − α˙ τ ) (π − μ) = w, ˙ ∂ξ ∂τ ∂ξ ∂(θ − hπ − ψ) ∂ψ = 0, = w. ˙ ∂ξ ∂ξ

(12.2.18) (12.2.19)

For the Arrhenius law in (1.2.2), the reduced reaction rate takes the form w(ψ, ˙ θ ) = (1 − ψ)eβe (θ−θ N ) ,

with

βe ≡

E 2 ε = O(1), kB TN

(12.2.20)

where the overbar denotes the planar wave in steady state. For a more general reaction rate, the last relation has to interpreted as the scaling for the temperature sensitivity to be used in the asymptotic analysis. Boundary Conditions and Relations between Temperature, Pressure and Velocity The boundary conditions for π and θ at ξ = 0 are given by the Rankine–Hugoniot conditions (12.2.7) with Mu = M u − εα˙ τ , Mu2 − 1 ≈ 2(M u − 1) − 2εα˙ τ , where, according to √ (12.2.6), (M u − 1) ≈ ε f . To leading order in the limit ε → 0 one gets     ξ = 0: θ ≡ θN = 2h (12.2.21) f − α˙ τ , π ≡ πN = 2 f − α˙ τ , so that θN − hπN = 0, and, according to the first equation in (12.2.19), θ = hπ + ψ.

(12.2.22)

The boundary condition for μ is obtained from mass conservation across the moving lead ˆ ˆt) = ρˆN (ˆu| ˆ − dA/d ˆ ˆt), that is (M u − εα˙ τ ) = (ρˆN /ρˆu )(1 + εμ|ξ =0 − shock ρˆu (D − dA/d xˆ =A εα˙ τ ), where ρˆN /ρˆu is given in (12.2.7) where Mu2 − 1 ≈ 2(M u − 1) − 2εα˙ τ ,  ξ = 0: μ ≡ μN = − f + 2α˙ τ , and ψ = 0, (12.2.23) √ valid to leading order. The second equation in (12.2.21) then yields μN + πN = f , so that, according to the first equation in (12.2.18),  (12.2.24) μ + π = f.

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Nonlinear Equations for Galloping Detonations Near the CJ Condition The problem now reduces to solving two equations[1] for μ and ψ with the boundary conditions in (12.2.23):   w ˙ ∂ψ ∂ ∂ + (μ − α˙ τ ) μ=− , = w(ψ, ˙ θ ), θ = h f − hμ + ψ. (12.2.25) ∂τ ∂ξ 2 ∂ξ In steady state, the detonation structure is given in terms of ψ by a straightforward integration, μ2 = f − ψ, showing that the reduced flow velocity μ is negative and increases from √ √ − f at the Neumann state to − f − 1 in the burnt gas. In the unsteady case, the unknown function α˙ τ (τ ) is obtained by imposing that the flow velocity at the end of the reaction is that of the steady-state solution, μ2b = μ2b = μ2N − 1,  ξ → ∞: ψ = 1, μ = μb = − f − 1. (12.2.26) This downstream condition is natural for an overdriven detonation sustained by a piston propagating at a constant velocity. It is also valid for self-sustained CJ waves, since it corresponds to a radiation condition at the downstream end of the reaction zone; there can be no upstream-running acoustic wave crossing the end of the reaction in the burnt gas (sonic point for a CJ wave). Equations (12.2.23)–(12.2.26) constitute the nonlinear model for a planar CJ detonation[1] in the limit of small heat release (12.2.5) and (12.2.10). The first equation in (12.2.25) represents the upstream-running characteristic curve, while the second equation and the third relation result from the fast downstream-running acoustic and entropy waves (quasi-instantaneous propagation). The study of the nonlinear behaviour requires numerical solutions. However, the key mechanism of the linear instability is the modification of the reaction rate δ w ˙ and can be described analytically. The stability limit of planar CJ waves is investigated from the linear solution[1] μ = μ(ξ ) + δμ(ξ , τ ). We first show that the wave is stable in the absence of modifications of the reaction rate. Stability in the Absence of Modification of the Reaction Rate If δ w ˙ = 0, the first equation in (12.2.25) reduces to  ∂ dμ(ξ ) ∂ − u (ξ ) δμ = [α˙ τ (τ ) − δμ] , ∂τ ∂ξ dξ

u ≡ −μ(ξ ) > 0.

(12.2.27)

ξ After multiplication by μ, Equation (12.2.27), expressed in terms of ζ ≡ 0 dξ  /u (ξ  ),  ∂ dμ ∂ w ˙ dμ − (μδμ) = G(ζ )α˙ τ (τ ), G = μ = − , −G(ζ ) = , (12.2.28) ∂τ ∂ζ dξ 2 dζ

yields an equation for y ≡ μδμ, and the general solution takes the form  τ +ζ y(ζ , τ ) − y(0, τ + ζ ) = − G(ζ + τ − τ  )α˙ τ (τ  )dτ  . τ [1]

Clavin P., Williams F., 2002, Combust. Theor. Model., 6, 127–129.

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(12.2.29)

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549

In view of the boundary condition (12.2.26), ζ → ∞: y → 0 ∀τ , we can let ζ → ∞ with ζ + τ ≡ τo fixed, obtaining  τo  ∞ y(0, τo ) = G(τo − τ  )α˙ τ (τ  )dτ  = G(ζ  )α˙ τ (τo − ζ  )dζ  . (12.2.30) −∞

0

Applying the boundary condition (12.2.23) at ζ = 0 then yields the equation  ∞  2 f α˙ τ (τ ) = − G(ζ  )α˙ τ (τ − ζ  )dζ  ,

(12.2.31)

0

where G is given in (12.2.28). A similar equation was derived in (4.5.23) for high overdrive but with a different G. For G(ζ ) in (12.2.28) Equation (12.2.31) yields a stable state. This can be checked through a particular model having the same characteristics as dis ∞ordinary  n −ζ  n −ζ tributions of the chemical heat-release rate, −2G(ζ )/(μb − μN ) = ζ e / 0 ζ e dζ  , √ √ √ n  1. Substituting α˙ τ (τ ) = eσ τ gives (σ +1)n+1 = ( f − f − 1)/4 f < 1, the solution to which is always σ < 0. Instability Threshold for a Simplified Model of Reaction Rate ˙ = 0. A simple An equation similar to (12.2.31) is obtained for the full problem,[1] δ w model is sufficient to understand the instability mechanism. It is obtained using an ad hoc simplification, neglecting heating by compression in the energy equation (12.2.19), ∂θ/∂ξ = ∂ψ/∂ξ = w(ψ, ˙ θ ). The instantaneous distribution of the heat-release rate can ˙ = (βe θN (τ ), ξ ), then expressed in terms of the varying Neumann temperature, θN (τ ), w (βe θN , ξ ) is the steady-state distribution corresponding to a fixed value TN , where ∞  dξ = 1. In the linear approximation, δ w ˙ = ∂(βe θN , ξ )/∂θN |θN =θ N δθN (τ ), where, 0 according to (12.2.21), δθN = −2hα˙ τ , Equation (12.2.25) leads to an equation of the same form as (12.2.31) but with − 2G = (ξ ) + hu (ξ )N (ξ ),

h ≡ 2hβe ,

(12.2.32)

where  = (βe θ N , ξ ) and where, in view of (12.2.5), (12.2.10) and (12.2.20), h =  − 1) qm /cp Tu is a parameter of order unity, N (ξ ) ≡ ∂(, ξ )/∂|TN =T N 2(E/k  B∞TN )(γ and 0 N (ξ )dξ = 0 is a function of order unity measuring the deformation of the distribution of the heat-release rate when the Neumann temperature varies. For particular examples, the instability growth rate in (12.2.31) can be obtained analytically; see (12.2.55). When the second part of G in (12.2.32) is introduced into (12.2.31) one gets the same equation as (4.5.23). The instability mechanism is then similar to that at strong overdrive, but with a different bifurcation parameter, 6 E (γ − 1) qm h −1 , b = √ = √ kB TN 2 f cp Tu 4 f and a different origin of time lag, which is here associated with the upstream-running acoustic wave instead of the entropy wave for strong overdrive. As explained in Section 4.5.1 a Poincar´e–Andronov (Hopf) bifurcation occurs and the detonation becomes unstable

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for a given function N (ξ ) when b−1 is increased. The stiffer the function N (ξ ), the lower the critical value of b−1 .

12.2.3 Cellular Detonations Near the CJ Condition for ε 1 We still consider the detonation regimes in the limit (12.2.5), (12.2.10) and (12.2.20). ˆ au and Introducing the reduced transverse variable y ≡ yˆ /ˆau tN and velocity ν˘ ≡ w/ˆ anticipating, as we shall see later, that the transverse derivative w∂/∂ ˆ yˆ and modifications to aˆ N introduce negligible terms in the limit (12.2.5) and (12.2.10), Equations (12.2.4) written in the notation of (12.2.11) take the form   ∂ ∂ ∂ ∂ π˘ ∂ ∂ ν˘ ± (1 ± u˘ ) (π˘ ± u˘ ) = ε2 w , + u˘ ν˘ = − , (12.2.33) ˙ − ∂t ∂x ∂y ∂t ∂x ∂y and Equations (12.2.13) are still valid. The reactive sound waves are similar to the planar ˙ replaced by ε2 w ˙ − ∂ ν˘ /∂y. case with ε2 w Scaling in the Linear Approximation In the multidimensional case the boundary condition for the pressure and temperature at the Neumann state is obtained by introducing Mu =

ˆ ˆt D − ∂ A/∂   2 1/2 ˆ yˆ au 1 + ∂ A/∂

(12.2.34)

into (12.2.7). Using the notations in (12.2.14)–(12.2.16) and in the linear approximation this √ gives the same leading order expression Mu2 − 1 ≈ 2ε( f − α˙ τ ) as in the planar case. And using the notations in (12.2.16), the same conditions as in (12.2.21) are obtained. According to mass conservation (4.4.7), Equation (12.2.23) for the longitudinal flow velocity is still valid. Using the second equation in (12.2.7) to compute uN /ˆau , conservation of transverse momentum in (4.4.8) yields   (12.2.35) ξ = 0: ν˘ ≡ ν˘ N = 2ε f ∂α/∂y, ∂ ν˘ /∂y = 2ε f ∂ 2 α/∂y2 , where x = α(y) is the equation of the shock front at time t. For the same reasons as explained below Equations (12.2.12) and (12.2.13), the characteristic time of evolution is longer than the transit time tN , τ ≡ εt = O(1), so that the equation of the front is written in the form x = α(y, εt) = O(1). In order to make the transverse effect in the righthand side of the first equation (12.2.33) of same order as the reaction rate, one guesses from (12.2.35) that ∂ 2 α/∂y2 = O(ε), so that the characteristic length of variation in the √ transverse direction is expected to be longer than the detonation thickness by order ε. It is then convenient to introduce ν = O(1) and the transverse coordinate of order unity η and to write the equation of the front in the same notation as in (12.2.14)–(12.2.15), in the √ form x = α( εy, εt) = α(η, τ ) = O(1),

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η≡

√

εy,

ξ = 0:

∂ 2 α/∂y2 = εαη2 (η, τ ),  ν = νN ≡ 2 f αη (η, τ ),

ν˘ ≡ ε3/2 ν(ξ , η, τ ),

551

(12.2.36) (12.2.37)

where αη ≡ ∂α/∂η and αη2 ≡ ∂ 2 α/∂η2 are quantities of order unity. Equations in the Moving Frame √ In the moving frame ξ = x − α(η, τ ), the relation ∂/∂y = ε[∂/∂η − αη ∂/∂ξ ] is added to the relations (12.2.17), and the two last equations in (12.2.25) and the boundary conditions (12.2.21)–(12.2.23) at the Neumann state ξ = 0 are still valid to leading order. Written in the notation in (12.2.16), Euler’s equations (12.2.1) then yield ∂μ ∂π ≈− , ∂ξ ∂ξ

∂ν ∂π ∂π ≈− + αη , ∂ξ ∂η ∂ξ

so that

∂ν ∂μ ∂μ ≈ − αη , ∂ξ ∂η ∂ξ

(12.2.38)

where the last relation is obtained from the integration of the first equation using the √ boundary conditions (12.2.21) and (12.2.23). This leads to the same relation μ + π = f as in the planar case. The first equation (12.2.33) takes the form  w ˙ ϕ ∂ν ∂ν ∂ ∂ + (μ − α˙ τ ) μ = − + , where ϕ(ξ , η, τ ) ≡ − αη (12.2.39) ∂τ ∂ξ 2 2 ∂η ∂ξ and where ν(ξ , η, τ ) is obtained by integrating the last equation in (12.2.38) with the boundary conditions (12.2.23) and (12.2.37). Dispersion Relation for a Simplified Model of Reaction Rate In the linear approximation ∂ ν˘ /∂y ≈ ε2 ∂ν/∂η and ϕ ≈ ∂ν/∂η with ∂δμ dμ ∂ν ≈ − αη . ∂ξ ∂η dξ

(12.2.40)

Considering harmonic perturbations α = exp(iκη + σ τ )α˜ and a normal-mode decomposition written for any field function φ(ξ , η, τ ) in the form φ = φ(ξ ) + δφ,

˜ )eiκη+σ τ α, δφ(ξ , η, τ ) = φ(ξ ˜

(12.2.41)

where κ and σ denote here the reduced wavenumber and complex growth rate, one gets two ordinary differential equations for μ(ξ ˜ ) and ϕ(ξ ˜ ) = iκ ν˜ , σ μ˜ +

˜ ϕ˜ dμ  d (μμ) − + , ˜ =σ dξ dξ 2 2

dϕ˜ dμ = −κ 2 μ˜ + κ 2 , dξ dξ

(12.2.42)

where (ξ , η, τ ) denotes the spatial distribution of the instantaneous heat release rate ˜ = −hσ  (ξ ) for the simplified model of reaction rate in (12.2.32).  = w(ψ, ˙ θ ), and  N The formulation is written in two-dimensional geometry for simplicity. Extension to three dimensions is straightforward. The boundary conditions for the flow field at the Neumann state (12.2.23) and (12.2.35) take the form  (12.2.43) ξ = 0: μ˜ = 2σ , ϕ˜ = −2 f κ 2 .

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Shock Waves and Detonations

Introducing the function !,  ξ   !≡ μ(ξ ˜  )dξ  + f + f − 1,

μ˜ = d!/dξ ,

0

√ one gets ϕ˜ = κ 2 [−! + (μ − f − 1)], and  ξ (12.2.42) yields a second-order differential equation expressed using the variable ζ ≡ 0 dξ  /u (ξ  ) in the form  κ2 d! κ2 d2 ! − − σ u ! = σ G + u ( u − f − 1), dζ 2 2 dζ 2

(12.2.44)

where G is defined in (12.2.32), limξ →∞ G = 0, d/dζ = u d/dξ , u (ξ ) ≡ |μ| > 0, u (0) = √ √ f , u (∞) = f − 1. The dispersion relation is obtained[1] by requiring that the solution satisfies three boundary conditions,    d!/dζ = 2 f σ , ξ = 0: ! = f + f − 1, (12.2.45) ξ → ∞: ! = 0, where the second equation is given by u μ˜ in (12.2.23) and the third relation corresponds to limξ →∞ ϕ˜ = 0 (no upstream-running acoustic wave at the end of the reaction zone). For κ 2 = 0 the model reduces to that of the planar case studied in Section 12.2.2; see (12.2.28) and (12.2.32). We suppose in the following that the planar detonation is stable to planar disturbances and attention is focused on the effects of multidimensional disturbances. The

√ problem can be also formulated in terms of Y ≡ ϕ/κ ˜ 2 = −! + μ − f − 1 ,   κ2 d2 Y dY d − + σ hu N (ξ ) , − σ u (ζ )Y = S(ζ , σ ), 2S ≡ 2 dζ 2 dζ dζ   ζ = 0: Y = −2 f , dY/dζ = −2σ f + (0), ζ → ∞:

Y = 0.

(12.2.46) (12.2.47)

For κ = 0 the problem reduces to a first-order differential equation for dY/dζ = −u μ˜ + /2, which is exactly (12.2.28). We suppose in the following that the planar detonation is stable to planar disturbances, κ = 0: Re(σ ) < 0, and attention is focused on the effects of multidimensional disturbances. The last term in the left-hand side of the first equation in (12.2.46) describes how the detonation can become unstable to transverse disturbances. For overdriven detonations, f > 1,  the solutions of (12.2.46) in the burnt gas can be √ expressed in the form el± ζ , 2l± ≡ σ ± σ 2 + 2u b κ 2 , u b ≡ f − 1. In the unstable case, Re(l+ )¿0, the solution el+ ζ diverges. The solution of (12.2.46) that goes to zero at infinity is expressed in terms of two particular solutions φ+ and φ− of the homogeneous equations satisfying ζ → ∞: φ± ∝ el± ζ , ζ = 0: φ± = 1,   ∞  ∞  ζ Sφ− Sφ+ Sφ− Y = Y(0) + φ− (ζ ) − φ+ (ζ ) dζ dζ dζ − , (12.2.48) W W W 0 0 ζ

[1]

Clavin P., Williams F., 2009, J. Fluid Mech., 624, 125–150.

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553

 − φ φ  is the Wronskian (φ  ≡ dφ/dζ ), so that where W = φ− φ+ + −     ∞ Sφ− dY   ζ = 0: (0) + φ− (0) − φ+ (0) dζ = Y(0)φ− , (12.2.49) dζ W 0   (0) = σ ± σ 2 + 2u κ 2 and u ≡ where, neglecting du /dζ |ζ =0 ((0) neglected), 2φ± N N √ −μ(0) = f . The dispersion relation is then obtained when the boundary conditions (12.2.47) at ζ = 0 are used:       ∞ Sφ−  . (12.2.50) dζ Re(σ ) > 0: f σ + σ 2 + 2κ 2 f = σ 2 + 2κ 2 f W 0

For the unstable case, Re(σ ) > 0, the result is valid in the limits defined in (12.2.5) and (12.2.10) for any value of f  1, includingthe CJ wave (f = 1). The planar case described ∞ in (12.2.31)–(12.2.32), dY/dζ |ζ =0 = − 0 Se−σ ζ dζ , is recovered from the dispersion relation in the limit κ = 0, limκ→0 φ− = 1, limκ→0 φ+ = eσ ζ .   2 Looking for solutions of the √ homogeneous equation φ −σ φ − u (ζ )φ/ε = 0 with σ of order unity in the limit ε ≡ 2/κ → 0 (κ denoting the modulus of the wave vector, κ → ∞ n ∞), it is natural to seek a solution of the form φ/φ(0) = eF(ζ )/ε , F(ζ n=0 ε Fn (ζ ), in √ )='   [2] the limit ε → 0 (WKB method ). To the two first orders, F0 = ± u , 2F1 = σ − F0 /F0 , φ± (ζ ) takes the form   ζ f 1/8 κ σ 1/2   u (ζ )dζ + ζ + · · · , κ  1, σ = O(1): φ± ≈ 1/4 exp ± √ 2 2 0 u √ 1/4 σ ζ and W ≈ 2f κe . Introducing this result into (12.2.50) with S in (12.2.46),     ζ  ∞  1 σ d κ 1/2    + σ hu N exp − √ κ 2f ≈ dζ u (ζ )dζ − ζ 2 2 0 2f 1/8 u 1/4 dζ 0 shows that there is no solution for σ of order unity with Re(σ ) > 0 since the right-hand side goes to zero while the left-hand side diverges in the limit κ → ∞. As we shall see in a particular example, the detonation is strongly stable to disturbances at small wavelength with a relaxation rate increasing faster than κ. Instability Threshold. Dispersion Relation Physical insight can be obtained from the approximate solution corresponding to constant √ mass flux, u (ζ ) ≈ f = cst. This √ is a good approximation for√sufficiently large f . In this case φ± = el± ζ , 2l± ≡ σ ± σ 2 + 2u κ 2 , φ− /W = el+ ζ / σ 2 + 2u κ 2 and (12.2.50) yields  ∞   dξ d/dξ + σ hN e−ξ l+ /u . (12.2.51) 4u l+ = 0

[2]

Bender M., Orszag S., 1984, Advanced mathematical methods for scientists and engineers. McGraw-Hill.

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This equation for σ becomes polynomial when considering a particular example,  ∞  ∞ ξ n −ξ k = e , dξ = 1, dξ e−kξ d/dξ = , (12.2.52) n! (1 + k)n+1 0  ∞ 0  ∞ (n + 1 − ξ ) n −ξ (n + 1)k N = N dξ = 0, N e−kξ dξ = , ξ e , n! (1 + k)n+2 0 0 ¯ where, according to (4.5.26), N ≡ d(ξ )/dξ , . . / / √ √ n+2  + σ 2 + κ 2 1 σ  + σ 2 + κ 2 σ = Hσ  + 2 1 + 4 1+ , (12.2.53) 2 2 u √ √ where σ  ≡ σ/u¯ , κ  ≡ 2κ/ u¯ , H ≡ (n + 1)h/u¯ and n is a measure of the induction length at the steady-state; see the first equation in (12.2.52). For n and u fixed, an oscillatory instability (Re(σ ) > 0, Im(σ ) = 0), develops when H is increased. In the small wavelength limit (κ   1) the quantity √ 1 in the parentheses is negligible and it is convenient to introduce z ≡ σ  /κ  and Z ≡ z + z2 + 1, Re(Z) > 0, z = (Z 2 − 1)/2Z, and also the small parameter  ≡ 2n−1 H/κ n+1 ,    1 2n−1 H 1 n+2 Z . (12.2.54)  ≡ n+1 1: Z = − + 1+ 2 Z κ u H 1

This shows that Z is a small complex number, Z ≈ (−) n+3 , with, by definition Re(Z) > 0, 1 so that z ≈ −1/2Z is of order 1/ n+3 with a negative real part. Therefore the detonation is stable at large wavenumber κ with a large relaxation rate and a large frequency of n+1 n+1 oscillation, Re(σ  ) ∝ −κ 1+ n+3 , Im(σ  ) ∝ κ 1+ n+3 . Setting κ = 0 in (12.2.53) leads to the same equation as (12.2.31)–(12.2.32) for the stability analysis of planar disturbances presented in Section 12.2.2:

n+2

1 = Hσ  + 2 1 + σ  . 4 1 + σ

u

(12.2.55)

When the parameter H, which measures the sensitivity to temperature, is sufficiently small, the first term in the right-hand side is negligible, (1 + σ  )n+1 ≈ 1/(2u )2 < 1, and the detonation is stable to planar disturbances (Re(σ  ) < 0, no galloping instability). The bifurcation, Re(σ )=0, occurs for a sufficiently large sensitivity to temperature, h > hc , hc = u Hc /(n + 1), with a pulsating frequency ωc = Im(σ ) of order unity,     Hc 1 1 1 , ωc2 = 3 − 8 + 2 , n = 2: (12.2.56) = 8+ 2 − 2+ 2 16 4u 16u 4u √ where u = f  1. When h < hc , the detonation is stable to planar disturbances and also to disturbances with wavelength smaller than the detonation thickness κ  1. However, attention should be paid to the fact that the dispersion relation in (12.2.53) near to κ = 0 does not include the hydrodynamic branch limκ→0 σ = 0 associated with the neutral modes described in Section 12.1.5. This hydrodynamic branch for f > 1 is described at the

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12.2 Dynamics of Detonation Fronts (a)

555

(b)

Wavenumber

Wavenumber

Figure 12.1 Numerical solutions to (12.2.53) for the growth rate of cellular instability on a detonation wave.

following orders in the perturbation analysis in the limits (12.2.5) and (12.2.10). Therefore the present analysis is limited to detonations sufficiently stable to planar disturbances, namely for H sufficiently smaller than Hc . For a sufficiently large H, still smaller than Hc , Equation (12.2.53) has a Poincar´e– Andronov (Hopf) bifurcation at intermediate wavelengths for which κ  is of order unity, provided that the undisturbed profile of reaction rate is not too stiff. This can be shown analytically[1] when the last term proportional to 1/u 2 in the right-hand side of (12.2.53) is neglected. Fig. 12.1 shows the numerical solution for the real part of the reduced growth rate, σ  , in (12.2.53) for n = 2, two values of u and several values of H ranging from Hc down to the critical value of H for the threshold of cellular instability. To conclude, at small overdrive, including the CJ regime, the oscillatory instability responsible for the cellular structure is due to the temperature sensitivity of the reaction rate, in a way similar to strongly overdriven detonations that are discussed in Section 4.5 and detailed in Section 12.2.4. However, there are important differences, as shown by the linear dynamics of a weak shock in the distinguished limit in (12.2.5) and (12.2.10). Expending

2 2 2 (12.1.29) in powers of (1 − M N ) using (4.2.18), one gets S2 (1 − M N ) + 1 = O(ε3 ) for (γ − 1) = O(ε). According to the definition of S in (12.1.12), this corresponds to σˆ /aN kˆ = ±i, where σˆ and kˆ denote the original dimensional quantities. Expressed in terms of the reduced quantities σ and κ, introduced in (12.2.41), the relation σˆ /aN kˆ = ±i, valid for a weak shock, corresponds to σ/κ = O(1/ε1/2 ). This contradicts the scalings of time and length in (12.2.14) and (12.2.36), σ = O(1) and κ = O(1). Without heat ˜ = 0, the linear equation (12.2.42) has no solution satisfying the release, dμ/dξ = 0 and  boundary condition (12.2.43). A solution σ/κ = O(1/ε1/2 ) is obtained when considering the next order in the expansion in powers of ε. This shows that, in contrast to the study at large overdrive, the multidimensional dynamics of cellular detonations does not involve [1]

Clavin P., Williams F., 2009, J. Fluid Mech., 624, 125–150.

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the dynamics of the lead shock. The dynamics are fully controlled by the modifications of the heat release that introduce higher order terms than those associated with the dynamic of the lead shock. This is a consequence of the transonic character of the flow throughout the detonation structure. These conclusions are not valid for real CJ detonations because their flow is strongly subsonic at the Neumann state, typically MN ≈ 0.2, so that the model equation (4.5.36) is more appropriate. For the purpose of comparison, it is however instructive to derive a weakly nonlinear model for the CJ waves near the stability threshold in the Newtonian limit. Nonlinear Model for Cellular Detonations Near the CJ Condition for ε 1 The scalings in (12.2.14)–(12.2.16) and (12.2.36) are still valid for the nonlinear regime near the instability threshold in the distinguished limit (12.2.5), (12.2.10) and (12.2.20). Retaining only the quadratic terms, Equation (12.2.34) then yields

  (12.2.57) Mu2 − 1 ≈ 2ε f − α˙ τ − (αη )2 /2 . According to the conservation of transverse momentum in (10.1.11), ξ = 0: wˆ = Aˆ yˆ (D−ˆu), the boundary condition of the transverse component of the flow velocity yields  (12.2.58) ξ = 0: ν = ( f − μ)αη . Introducing (12.2.57) into (12.2.7), the boundary conditions (12.2.21) are replaced by   ξ = 0: θ = 2h[ f − α˙ τ − (αη )2 /2], π = 2[ f − α˙ τ − (αη )2 /2]. (12.2.59) The nonlinear terms associated with the transverse convective fluxes w∂/∂ ˆ yˆ introduce corrections smaller than the unsteady term ∂/∂ ˆt by a factor ε. Therefore equations in (12.2.19) are still valid, so that, according to (12.2.59) Equation (12.2.22) holds. The boundary condition for μ, obtained from mass conservation, yields the same result as in (12.2.23), √ ξ = 0: μ+π = f , so that Equation (12.2.24) holds as well as Euler’s equations (12.2.38). The leading order solution, namely the relation linking α˙ τ and αη , is obtained from the solution μ(ξ , η, τ ) of the nonlinear equation in (12.2.39) (upstream-propagating acoustic mode in the reacting media),   w ˙ 1 ∂ν ∂ν ∂μ ∂ν ∂μ ∂ ∂ + (μ − α˙ τ ) μ=− + − αη , where ≈ − αη , ∂τ ∂ξ 2 2 ∂η ∂ξ ∂ξ ∂η ∂ξ (12.2.60) satisfying the boundary conditions at the Neumann state (12.2.59)–(12.2.58) with √ μ = f − π,   ξ = 0: μ = − f + 2α˙ τ + (αη )2 , ν = 2( f − α˙ τ )αη , (12.2.61) limited to quadratic terms, and satisfying also the boundary condition at the end of the reaction zone, ξ → ∞, where the upstream-propagating acoustic waves should be zero:  ξ → ∞: μ = − f − 1. (12.2.62)

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557

The instantaneous distribution of heat release rate in the first equation (12.2.60) is obtained from the solution of the entropy wave, namely the second equation in (12.2.25),  ∂ψ (12.2.63) = w(ψ, ˙ θ ), where θ = h( f − μ) + ψ with ξ = 0: ψ = 0, ∂ξ where the sensitivity of the reaction rate to temperature was assumed to be of order 1/ε2 , as in the study of the galloping detonations; see (12.2.20). The solution of the two coupled nonlinear equations (12.2.60) and (12.2.63) has to be obtained numerically. Singularities of the slope of the detonation fronts are expected to be produced by the nonlinear term (αη )2 in the first equation in (12.2.61) coming from the geometrical term in (12.2.34).

12.2.4 Cellular Instability at Strong Overdrive This section is devoted to the details of the stability analysis of strongly overdriven detonations[1,2,3] discussed in Section 4.5.2. The problem is to solve the linearised reactive Euler equations (12.2.1)–(12.2.2) in the limits (4.5.10) and (4.5.28) for a wrinkled leading shock using the Rankine–Hugoniot conditions (4.4.13)–(4.4.14). A boundedness condition is used at infinity in the burnt gas. For linearly unstable modes this last condition is equivalent to a radiation condition. Scaling and Reduced Equations It is convenient to use generalised mass-weighted (reduced) coordinates (4.5.1), scaled with the time and length scales of the unperturbed solution and a reduced transverse coordinate y describing variations on a length scale larger than the detonation thickness, x,

t,

y ≡  yˆ /uN tN ,

where here yˆ denotes the dimensional coordinate. The scaling of y is explained at the beginning of Section 4.5.2 by the coupling of longitudinal pulsations at a frequency of order of the inverse of the transit time with transverse propagation of shock wrinkles at the sound speed. The nondimensional variables u, v, p and T, ˆ N , and α = α/u ˆ N tN , u ≡ uˆ /uN , v ≡  vˆ /uN , p ≡ pˆ /pN , T ≡ T/T are introduced, where uˆ , vˆ , pˆ , Tˆ and αˆ denote here the dimensional longitudinal and transverse velocities, the dimensional pressure, temperature and the position of the front. The dimensional quantities at the Neumann state of the unperturbed solution (planar wave) are simply denoted tN , uN , pN and T N to simplify the notation. The scaling for v results from the scaling of the transverse coordinate and from the large jump of density across the lead [1] [2] [3]

Clavin P., He L., 2001, C. R. Acad. Sci. Paris, 329(IIb), 463–471. Clavin P., 2002, In H. Berestycki, Y. Pomeau, eds., Nonlinear PDE’s in condensed matter and reactive flows, 49–97. Kluwer Academic Publishers. Daou R., Clavin P., 2003, J. Fluid Mech., 482, 181–206.

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Shock Waves and Detonations

shock, ρ N /ρu ≈ 1/ 2 . The reduced quantity ∇.v is of order unity, and it is convenient x to introduce the scalar field υ(x, y, t) ≡ 0 ∇.vdx associated with the transverse velocity generated by wrinkling of the front. In the linear approximation, the material derivative and the continuity equation (conservation of mass) take the form[1,2] ∂ ∂ D = + [m(t) − υ(x, y, t)] , Dt ∂t ∂x ∂r ∂r ∂ + m(t) = [u + r(x)υ(x, y, t)] , ∂t ∂x ∂x

(12.2.64) (12.2.65)

where m(t) ≡ 1 − (∂ α/∂ ˆ ˆt)/D is the reduced mass flux across the leading shock and r(x, y, t) ≡ ρ N /ρˆ is the reduced specific volume, r(x) denoting its value  x in the unperturbed solution. According to the above scaling, the scalar field υ(x, y, t) ≡ 0 ∇.vdx is of order unity, υ = O(1). Anticipating that the wrinkled wave evolves with the characteristic time ˆ ˆt are of order uN , the scale tN and that the fluctuations of the detonation velocity ∂ α/∂ 2 variation of m(t) is small, m(t) = 1 + O( ), since uN /D = O( 2 ). Every nondimensional quantity f is expanded in powers of  2 , f = f0 +  2 f2 + · · · . From now on, the reduced quantities in the unperturbed solution are denoted f , f (x, y, t) = f (x) + δf (x, y, t). Further simplifications are introduced in the limit (4.5.10) and (4.5.28). The variation of the reduced quantities across the unperturbed wave are small, at least of order  2 , f 0 = 1, u = 1 +  2 u2 (x) + δu, T = 1 +  2 T 2 (x) + δT, p = 1 +  4 p4 + δp. When terms of order  4 are neglected and when the density is eliminated, the reactive Euler equations, written in the linear approximation and in the limit (12.2.1)–(12.2.2), take the form    x ∂ ∂ du ∂δp + δu − υ =− , υ≡ ∇.vdx , (12.2.66) 2 ∂t ∂x dx ∂x 0   ∂ ∂ + (∇.v) = −u∇ 2 δp, (12.2.67) ∂t ∂x   1u ∂ ∂ ∂ + δp + (δu + uυ) = qN (δ w ˙ + υ w), ˙ (12.2.68) γ p ∂t ∂x ∂x where qN ≡ qm /cp T N and  2 in the first equation result from the unperturbed solution in which, written in dimensional form, one has ρ N u2N /pN =  2 . Also the relation r = u valid up to order  2 has been used. Introducing the notation α˙ t ≡ ∂α/∂t, α˙ t = O(1), the shock conditions (4.4.13)–(4.4.14) yield, up to O( 2 ),   1 (γ − 1) 1 (12.2.69) α˙ t , x = 0: δu ≈ 1 + 2 − v ≈ 1 − 2 ∇α, 2 Mu Mu δp ≈ −2 2 α˙ t ,

[1] [2]

δTN ≈ −(γ − 1)α˙ t .

(12.2.70)

Clavin P., 2002, In H. Berestycki, Y. Pomeau, eds., Nonlinear PDE’s in condensed matter and reactive flows, 49–97. Kluwer Academic Publishers. Clavin P., et al., 1997, Phys. Fluids, 9(12), 3764–3785.

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12.2 Dynamics of Detonation Fronts

559

Outer Flow in the Burnt Gas In contrast to the regimes near CJ, the flow in the burnt gas plays an important role in the dynamics of overdriven detonations. In the following the piston is assumed to be at infinity, x → ∞, where the linear disturbances should be bounded in the unstable case. In the burnt ˙ and δ w ˙ vanish, gas (when the exothermic reaction is completed) the reaction source terms w the unperturbed solution is uniform and an exact solution to the linear equations (12.2.66)– (12.2.68) is easily obtained. The solution is a superposition of a sound wave (superscript a) and an isobaric entropy–vorticity wave (superscript i) δu = δu(a) + δu(i) , v = v(a) + v(i) , δp = δp(a) and δp(i) = 0. Written in Fourier representation, α = α˜ exp(σ t + iκ.y), δf = f˜ (x)α˜ exp(σ t + iκ.y), the wave equation for the pressure (12.1.5) yields 0  2 1 ub d 1 d2 (12.2.71) + σ − 2 2 + u2b κ 2 p˜ (x) = 0, γ pb dx  dx where κ ≡ |κ| and ub and pb are the reduced values in the burnt gas of the steady solution. The solution p˜ (x) = p˜ b α˜ exp(ilx) of (12.2.71) may be written in a convenient form in terms of ε2 ≡ (ub /γ pb ) 2 ≈  2 and * 2 ≡ (γ ub pb )κ 2 , √ ε2 σ ± ε σ 2 + * 2 − ε2 * 2 il = . (12.2.72) 1 − ε2 When (σ√2 + * 2 ) is different from zero and of order unity, il is of order ε in the limit  → 0, il = ±ε σ 2 + * 2 + O(ε2 ). The situation is different when qN = O( 2 ), pb = 1 + O( 4 ) and ub = 1 + qN + O( 4 ). As as we will see below in (12.2.76), σ and * are of order unity but (σ 2 + * 2 ) is of order ε2 , leading to il = O(ε2 ), il =  2 il2 with il2 = O(1). The longitudinal length scale of the pressure is then larger than the detonation thickness by a factor of 1/ 2 , and the sound waves propagate in the burnt gas in a direction quasi-parallel to the unperturbed shock. Anticipating that the same scaling is also valid throughout the detonation structure, we find that the leading-order pressure may be expressed in terms of the postshock fluctuations (12.2.70), p˜ ( 2 x) = −2 2 σ exp(i 2 l2 x) + O( 4 ),

(12.2.73)

showing that the amplitude of the acoustic flow is small, of order  2 so that at leading order the flow field is fully controlled by the entropy–vorticity wave, the solution to Equations (12.2.66)–(12.2.67) with δp = 0 and u = ub = 1 + qN + O( 4 ),     ∂ ∂ ∂ ∂ + δu(i) = 0, + v(i) = 0, (12.2.74) ∂t ∂x ∂t ∂x ˙ = 0 and and Equation (12.2.68) is the continuity equation ∂δui /∂x + ub ∇.v(i) = 0, since w δw ˙ = 0. Every quantity a is expanded in powers of  2 , a = a0 +  2 a2 + · · · , and when the boundary conditions (12.2.69) are used, (i)

δu0 =

∂α (t − x, y), ∂t

(i)

v0 = ∇α(t − x, y).

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(12.2.75)

560

Shock Waves and Detonations (i)

(i)

The continuity equation ∂δu0 /∂x + ∇.v0 = 0 then yields the wave equation ∂ 2 α/∂t2 − ∇ 2 α = 0,

σ0 = ±iκ,

(12.2.76)

where the expansion σ = σ0 + 2 σ2 +· · · has been introduced. The leading-order frequency of the oscillatory modes is thus given by the isobaric entropy–vorticity wave of a strong inert shock. Introducing q2 = O(1), qN ≡  2 q2 , (γ − 1) =  2 h + · · · , * 2 = (1 +  2 (h + q2 ) + · · · )κ 2 in the limit (4.5.10), Equation (12.2.72) shows, using σ02 + κ 2 = 0, that the longitudinal pressure gradients is effectively of order  4 , 1/2

+ O( 2 ), (12.2.77) il2 = σ0 − 2σ0 σ2 + (h + q2 − 1) κ 2 where by definition Re[...]1/2 > 0, and the minus sign in front of [...]1/2 is chosen in order to satisfy the boundedness of the acoustic waves at x → +∞, Re(il2 ) < 0. Equation (12.2.74) for the vorticity wave is valid up to order  2 so that, using the leading order of the boundary conditions (12.2.69)–(12.2.68) and the acoustic flow (12.2.73) the flow velocity in the burnt gas can be written in the form  

(i) −σ x (i) −σ x e e , ∇.˜v(i) = −κ 2 +  2 ∇.˜vb2 , u˜ (i) = σ0 +  2 u˜ b2 (12.2.78) 2 2 u˜ (a) = 2 2 il2 ei l2 x , ∇.˜v(a) = −2 2 κ 2 ei l2 x , (i) (i) (i) valid up to order  2 , where u˜ b2 and ∇.˜vb2 = σ0 u˜ b2 are constants of integration linked together by the continuity equation. They are determined by the solution across the detonation thickness.

Inner Flow and Matching Using the notation δu = u˜ (x)α˜ exp(σ t + iκ.y), δv = v˜ (x)α˜ exp(σ t + iκ.y), the complete flow field throughout the detonation structure is obtained by introducing the splitting ˜ (i) (x) + u˜ (a) ( 2 x), u˜ ≡ U

˜ (i) (x) + v˜ (a) ( 2 x). v˜ ≡ V

(12.2.79)

After subtracting out the acoustics, Equation (12.2.68) leadsto the isobaric approximation  (i) ˜ (i) is balanced by the rate of gas ˜ of low Mach number. The divergence of the flow U , V expansion produced by heat release w, ˙  x

   d (i) ˜ (i) dx , (12.2.80) ˜˙ + υ˜ (i) w ˜ (i) + u(x)υ˜ (i) (x) ≈ qN w U ˙ , where υ ˜ ≡ ∇.V 0 dx 0 ˜ (i) ≈ qN w ˜˙ when the relation du/dx = qN w ˜ (i) /dx + u∇.V which is equivalent to dU ˙ is 2 used. Equation (12.2.80) is valid up to order  . The postshock fluctuations of pressure (12.2.70) being fully absorbed by the sound wave, the first two orders of the quantity ∇.V(i) are, according to (12.2.67), simply advected by the flow field from the shock, (∂/∂t + ∂/∂x)∇.V(i) = O( 4 ), leading at the two first orders to    1 (i) 2 ˜ κ 2 e−σ x + O( 4 ), ∇.V = −1 +  2 + 2 2 (12.2.81)  Mu

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12.2 Dynamics of Detonation Fronts

561

˜ (i) at x = 0 has been obtained from (12.2.69) by where the boundary condition for ∇.V subtracting (12.2.78). The leading order term yields  x

κ 2 −σ x  (i) e e−σ x dx = −1 , υ˜ 0 (x) = −κ 2 σ 0 (12.2.82) (i) ∂υ0 2 2 = −∇ α(t − x, y) + ∇ α(t, y), ∂t ˜ (i) is obtained in the same way, and in agreement with (12.2.75). The boundary condition U forward integration of Equation (12.2.80) with respect to x from the shock x = 0 then yields    x 1 γ − 1 (i) 2 ˜ (i) dx ˜ ∇.V U (x) − 1 + 2 − σ + 2 il2 + u(x) 2 MU 0 (12.2.83)  x  (i)  4 w ˙˜ + υ˜ w = qN ˙ dx + O( ), 0

0

into which the expansion σ = σ0 +  2 σ2 + · · · and (12.2.81) have to be introduced. The ˜ (i) (x) in (12.2.83) contains two kinds of terms at the end of the reaction, expression U x  1; the first does not vary with x while the second does, and depends on x, through exp(−σ x), exhibiting the same fast oscillations exp(±iκx) damped on a longer length scale as u˜ (i) in (12.2.78), exp(− 2 σ2 x), with Re(σ2 ) > 0 (unstable cases). Matching the inner and ˜ (i) and u˜ (i) , requires that the sum of all constant terms must be zero, the outer solutions, U   . /  κ2 1 γ −1 1 2 2 2 (il2 ) − σ 1 + 2 − + (1 + qN ) −1 +  2 + 2 2 σ M  2Mu (12.2.84)  ∞ u (i)  4 ˜˙ + υ˜ w)dx (w + O( ). = qN 0 ˙ 0

(i) Matching the oscillatory terms gives the constant of integration u˜ b2 in (12.2.78). According 2 to the ordering qN = O( ), only the leading order of the disturbance of the reaction rate has to be introduced into (12.2.84).

Inner Structure of the Wrinkled Detonation The method of solution is similar to that used in Section 4.5.1 for galloping detonations. The temperature and species profiles are modified by order unity by two effects: the transverse velocity induced by wrinkling (through υ0(i) = O(1)) and the sensitivity of the induction length to the Neumann temperature. Compressional heating being negligible, Equations (12.2.2) can be written to leading order in the limit  → 0 in terms of the reduced mass-weighted coordinates, ∂T ∂T (i) + = qN (1 + υ0 )w, ˙ ∂t ∂x x = 0: T = TN (y, t),

∂ψ ∂ψ (i) + = (1 + υ0 )w, ˙ ∂t ∂x ψ = 0,

17:10:14 .014

(12.2.85)

562

Shock Waves and Detonations

which is an extension of (4.5.19). A similar system holds for a complex chemical kinetic scheme. With a given expression for the reaction rate w(ψ, ˙ T), these equations form an autonomous system when the field υ0 (x, y, t) is prescribed. This system is solved using the solution (4.5.20) T = T (N , x) and ψ = Y(N , x) (T/T N → T) for the steady state ˙ dψ/dx = w ˙ satisfying x = 0: T = TN , where N = β(TN −1) and problem dT/dx = qN w, β  1 denotes the thermal sensitivity; see (4.5.17). For υ0(i) = 0, the solution to (12.2.85) is the retarded solution (4.5.20). In the general case, for υ0 = 0, the solution takes the form T = T (N (t − x), x + μ),

ψ = Y(N (t − x), x + μ),

where here the scalar field μ(x, y, t) is by definition the solution to

dμ˜ ∂μ ∂μ κ 2 −σ x e μ˜ + −1 , x = 0: μ = 0, + = υ0(i) , = ∂t ∂x dx σ  ∂μ κ2  (i) = υ0 − x∇ 2 α(t − x, y). μ˜ = − 2 1 − e−σ x − σ xe−σ x , ∂t σ Introducing (N , x), the distribution of reaction rate in the planar detonation propagating at constant speed for a Neumann temperature TN ,    (N , x) = w ˙ Y(N , x), T (N , x) , the distribution of reaction rate in the solution of (12.2.85) is (N (t−x), x+μ). Therefore the distribution of the perturbation of reaction rate, δ w, ˙ to be introduced in the right-hand side of (12.2.84) is N (x)δN (t−x)+μd/dx, where N (t) is given by TN (t) in (12.2.70), and where (x) = (0, x) is the distribution of the unperturbed detonation, TN = 1, and N (x) is defined in (4.5.23). This gives  ∞   ∂ ∞  (i) 2 δw ˙ + υ0 w ˙ dx = ∇ α(t) − β(γ − 1) N (x)∇ 2 α(t − x)dx ∂t 0 0   ∞ ∂ ∇ 2 α(t − x)dx + O( 2 ), − (x) 1 + x ∂t 0  ∞   κ2  (i) ˜˙ + υ˜ (i) w −1 + S w ˙ dx = (σ ) + O( 2 ), (12.2.86) 0 0 σ 0 where S (i) (σ0 ) is defined by (4.5.32)–(4.5.33) when  ∞iκ is replaced by σ0 . In the integral by ∞ (i) parts of the term 0 dx(d/dx)υ0 , the relation 0 N (x)dx = 0 has been used and the heat release rate at the Neumann state has been neglected, (x = 0) ≈ 0, as is the case in the induction zone of real detonations. Equation (12.2.86) is valid for any chemical kinetic scheme characterised by an induction delay at the Neumann state. Dispersion Relation An equation for the linear growth rate σ2 is obtained from (12.2.77) and (12.2.84) when (12.2.86) is introduced in the latter, 6 q2 σ0 σ2 σ0 σ2 3 σ0 2 2 + h + q2 − 1 − 2 + 1 − h = S (i) (σ0 ), (12.2.87) − κ 4 2 κ κ

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12.2 Dynamics of Detonation Fronts

563

√

where, by definition Re ... > 0. This yields a quadratic equation for σ2 and the two roots are given by 6 

3 q2 (i) h ± σ0 h+ S + q2 S (i) − 1 . (12.2.88) σ2 = σ0 4 2 2

The solution must satisfy the relation Re σ2 − σ0 (q2 /2)S (i) ≤ 0. This is shown when (12.2.87) is multiplied by σ0 , using σ 02 = −κ 2 . Therefore the only admissible root in √ (12.2.88) is that for which Re ±σ0 ... < 0. This yields the result (4.5.31)–(4.5.34) since √ √ q2 = qN /M N . The acoustic flow field in the burnt gas is obtained when the square root in (12.2.77) is expressed using (12.2.87) and when the term σ2 appearing in the resulting expression is replaced by (12.2.88), 6

2 h (12.2.89) il2 = ±2iκ − κ + q2 S (i) − 1 , 2 √

where Re ... is positive.

17:10:14 .014

17:10:14 .014

Part Three Complements

17:11:45

17:11:45

13 Statistical Physics

Nomenclature Dimensional Quantities A c cp cv e E f1 F hj h¯ H J Jq kB  m n p p p q q Q s S t

Description Helmholtz free energy Speed of light in vacuum Specific heat at constant pressure Specific heat at constant volume Energy density E/N Energy One-particle distribution function Helmholtz free energy Energy of free particle j Dirac’s constant h/2π Hamiltonian A flux Heat flux Boltzmann’s constant Mean free path Mass of particle Number density N/V Pressure Modulus of particle momentum of a particle Momentum of a particle Heat Position of centre of mass of a particle Heat Entropy density S/N Entropy Time

S.I. Units J ≈ 2.99792 × 108 m s−1 J K−1 mole−1 J K−1 mole−1 J J kg−3 m−6 s−3 J J ≈ 1.05457 × 10−34 J s J According to context J m−2 s−1 ≈ 1.38066 × 10−23 J K−1 m kg m−3 Pa kg m s−1 kg m s−1 J m J J K−1 J K−1 s 567

17:11:43 .015

568

T u v V W δf k k λ  μ τ φ χ

Statistical Physics

Temperature Flow velocity Particle velocity Volume Work Fluctuating part of a variable f Energy of particle with quantum numbers k Internal energy of particle with quantumnumbers k A length scale A macroscopic length scale Chemical potential A time scale Interaction potential Compressibility

K m s−1 m s−1 m3 J (same as variable) J J m m J s J m s2 kg−1

Nondimensional Quantities and Abbreviations cst. H k N N P Q Q R W l ηl {ηl } ρ  ωl 

Constant Set of points in phase space, with same energy Set of quantum numbers Number of atoms, molecules or particles Dimension of phase space Probability Partition function Grand partition function The set of real numbers Number of microscopic states corresponding to a macroscopic state Energy level of cell l Occupation number of cell l Set of occupation numbers, representing a macroscopic state Probability density of microscopic states in phase space A domain in phase space Number of quantum states in cell l A point in phase space

Superscripts, Subscripts and Math Accents acoll aeg aF ai

Collisions Equilibrium Fermi Species i

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13.1 Statistical Thermodynamics

ak amac amic ao atot

569

With set of quantum numbers k Macroscopic Microscopic A reference state Total

13.1 Statistical Thermodynamics The dynamics of N particles is given by the solution to Hamilton’s equations. For ordinary systems the Hamiltonian, expressing the energy of the system, takes the form H=

N  j=1

hj +



φj,j ,

(13.1.1)

j=j

where hj is the energy of the free (noninteracting) particle j and φj,j is the interaction potential between particles j and j . The potential φj,j introduces a microscopic length, r0 , corresponding to the short-range interaction between particles; r0 is typically of the order of 10−10 m for atoms and molecules. When the particles are enclosed in a macroscopic box of finite volume V, an interaction rule with the wall must be added, for example an elastic reflexion law in which hj is conserved. For a rectangular box, the confinement condition can be obtained in another way by imposing periodic conditions. The mean distance between particles is n−1/3 , where n ≡ N/V is the mean density. Discarding extreme conditions (such as in the centre of stars), the length n−1/3 cannot be much smaller than r0 in dense matter but it can be much larger; this is the case in ordinary gases n−1/3  r0 . A macroscopic system is characterised by a large separation of length scales, n−1/3 V 1/3 , corresponding to a large number N of particles in the box, for example 6 × 1023 in a cubic decimetre of gas at atmospheric pressure. The dynamics of microscopic particles in a macroscopic system is characterised by a small microscopic time scale τmic , as for example the inverse of collision frequency in gases, typically τmic ≈ 10−9 s at ordinary conditions. The matter at very high density, as in stars, is discussed in Section 13.2.4. When a macroscopic system is observed with spatial and temporal resolutions, λmac and τmac , that are much larger than required for a microscopic observation, V 1/3  λmac  n−1/3 and τmac  τmic , the dynamical state is represented by fewer degrees of freedom than for the microscopic states. The dynamics are then represented by macroscopic equations. In the long time limit, isolated systems, observed at the macroscopic scale, reach an equilibrium state, called thermodynamic equilibrium. Macroscopic dynamics are irreversible, in contrast to the dynamics of microscopic particles. Moreover the thermodynamic laws that govern the equilibrium states are universal. These macroscopic behaviours are experimental facts, well observed for microscopic particles interacting through forces with a short interaction range involving a repulsive core. It is also the case for plasmas in which the long-range interaction is screened by the electrical charges. For astrophysical objects

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570

Statistical Physics

that are not confined in a finite box and that interact with the long-range gravitational force, the situation is not so clear since long living clusters (galaxies) are formed. The mathematical proof of the macroscopic laws of the matter from the microscopic dynamics based on (13.1.1) is far from complete.

13.1.1 Laws of Thermodynamics As recalled in the introduction, the laws of thermodynamics were established during the nineteenth century, after the analysis of Carnot[1] and before the microscopic nature of the matter was firmly established at the end of that century. It is worth recalling these macroscopic laws before the statistical concepts from which they can be derived. The presentation of thermodynamics is limited here to a short outlook in a form adapted to the statistical approach discussed afterwards. More details can be found in textbooks.[2,3,4] In a restrictive sense, thermodynamics concerns only equilibrium states. In the absence of an external field, equilibrium states are uniform. For phases in equilibrium, each phase is in a uniform state. Equilibrium states are fully characterised by a small set of parameters. For a simple body, these parameters are the internal energy E, the volume V and the quantity of matter, represented for example by the number N of microscopic particles (molecules, atoms or electrons in a continuum background of positive charge, etc.). Irreversible thermodynamics is an extension to nonhomogeneous states in local equilibrium. The corresponding macroscopic equations are presented in Chapter 15. First Law The laws of thermodynamics may be formulated in terms of a single function of state, the entropy S, expressed in terms of the extensive parameters, E, V, N, S = S(E, V, N). The first law says that entropy is an extensive quantity, which means that the function S(E, V, N) is a homogeneous function of first order, S(λE, λV, λN) = λS(E, V, N),

∀λ.

(13.1.2)

The intensive parameters, temperature T, pressure p and chemical potential μ, are obtained by derivation, TdS = dE + pdV − μdN.

(13.1.3)

This equation is called the Gibbs formula; it is the usual formulation of the first law, δE = δQ + δW, since, for an elementary reversible process, one has δW = −pdV and δQ = TdS, which are the expressions for work and for the exchange of heat, respectively. It is worth recalling that δQ and δW are not variations of state functions, so that the total [1] [2] [3] [4]

Carnot S., 1824, R´eflexions sur la puissance motrice du feu et sur les machines propres a` d´evelopper cette puissance. Bachelier. Fermi E., 1956, Thermodynamics. New York: Dover. Callen H., 1985, Thermodynamics. New York: Wiley, 2nd ed. Prigogine I., Kondepudi D., 1999, Thermodynamique: Des moteurs thermiques aux structures dissipatives. Odile Jacob.

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571

heat and work ; ; exchanged with the external medium during a cycle are usually nonzero, δQ = 0, δW = 0; only their sum is zero for a closed system, N = cst., according to (13.1.3). The entropy being an extensive quantity, Equation (13.1.3) yields TS = E + pV − μN.

(13.1.4)

Denoting by Ni the number of particles of species i, Equations (13.1.3) and (13.1.4) can be generalised to mixtures:   μi dNi , TS = E + pV − μi Ni . (13.1.5) TdS = dE + pdV − i

i

Introducing the number density n = N/V, and the energy and entropy per particle e = E/N, s(e, 1/n) = S/N = S(e, 1/n, 1), S = Ns(e, 1/n), where (13.1.2) has been used, Equations (13.1.3) and (13.1.4) yield Tds = de + pd(1/n),

Td(ns) = d(ne) − μdn,

μ = e + p/n − Ts,

(13.1.6)

leading to the Gibbs–Duhem relation for the function μ(T, p), dμ = −sdT + (1/n)dp.

(13.1.7)

Second Law The modern formulation of the second law is ‘the entropy of an isolated system cannot decrease and is maximum at equilibrium’. The latter sentence is better understood when considering a composite system, constituted by different subsystems at internal (but not mutual) equilibrium, in contact with each other through boundaries. Defining the entropy of the composite system as the sum of the entropies of the subsystems, the second law shows that, at total equilibrium of the composite system (maximum entropy), two subsystems that can exchange energy through heat across their separation boundary have the same temperature, T1 = T2 , as can be checked by a variation around equilibrium, δS = 0, δS =

1 1 δE1 + δE2 , T1 T2

δE1 + δE2 = 0.

(13.1.8)

In a similar way, when the separation boundary is transparent to species j, the chemical potential of this species μj is the same in the two subsystems at mutual equilibrium, μj1 = μj2 . The case of chemical reactions is recalled in Section 14.2. The equivalence with the original formulation of the second law, ‘mechanical work cannot be furnished to the external medium during a cycle by using a single heat source’ of heat received by (Lord Kelvin’s postulate), is shown as follows. Let qs be the amount ;  a heat source s kept at a fixed temperature Ts > 0, − s qs = δQ, (qs < 0, the source  to the;external world, s loses energy). One must have s qs < 0 in order to furnish work ; ; − δW > 0, since, according to the first law during a cycle, δQ = − δW. Consider an isolated system constituted by the heat sources and the subsystem making a cycle. Its   entropy variation, s qs /Ts , cannot be negative, s qs /Ts  0. Therefore some qs must  be positive even when s qs < 0. Consider the case of two sources, with temperatures

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Figure 13.1 Sketch of entropy per particle, s, as a function of e and v ≡ 1/n. The solid lines show cuts at constant v and e. The secants (dashed lines) are systematically below the surface (convex surface).

TI > TII > 0. In order to furnish work to the external medium, −(qI + qII ) > 0, under the constraint of entropy production, qI /TI +qII /TII  0, one must have qI < 0 and qII > 0 (the hot source is cooled and the cold source is warmed). Moreover, the work −(qI + qII ) > 0 is maximum for qI /TI + qII /TII = 0, that is, for a reversible cycle. This statement that the reversible cycles are the most efficient was originally formulated by Carnot[1] (with the erroneous assumption qI = −qII ) in order to rule out the possibility of constructing a perpetual motion machine of the second kind. It is at the foundation of thermodynamics. Stability of Equilibrium States As a direct consequence of the second law, the function s(e, 1/n) is convex, meaning that a plane tangent to the surface s(e, 1/n) is above the surface (the secants are below the surface, (see Fig. 13.1), or in other words that the second term of the Taylor series s − s0 = δs+δ 2 s/2+· · · is quadratic and negative, δ 2 s < 0. Introducing the notation v ≡ 1/n = V/N for the specific volume, this condition takes the form    ∂ 2 s  ∂ 2 s  ∂ 2 s  2 (δe) + 2 δeδv + (δv)2 < 0, ∀(δe, δv). (13.1.9) δ2s = ∂e∂v o ∂e2 o ∂v2 o This is proved by decomposing an isolated system of a simple body into two subsystems defined by a fixed number of particles and by considering the division of energy and volume between these two subsystems. Equation (13.1.9) is obtained by imposing that the entropy is maximum at equilibrium, that is, for a division corresponding to the same density of energy and volume e and v in the two subsystems (uniformity of the equilibrium state). Using (13.1.6), ∂s/∂e|v = (1/T), ∂s/∂v|e = (p/T), the second variation may also be written in the form δ 2 s = δeδ(1/T) + δvδ(p/T), yielding 

(13.1.10) Tδ 2 s = −δsδT + δvδp = − δsδT + δpδn/n2 < 0.

[1]

Carnot S., 1824, R´eflexions sur la puissance motrice du feu et sur les machines propres a` d´evelopper cette puissance. Bachelier.

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This shows that the specific heats and the compressibility, χ , defined at either constant entropy or constant temperature, are positive, cp ≡ ∂s/∂T|p > 0, cv ≡ ∂s/∂T|v > 0, χs ≡ −(1/v)∂v/∂p|s > 0 and χT ≡ −(1/v)∂v/∂p|T > 0. Both entropy and specific volume v increase if the temperature increases. It can also be shown that (cp − cv ) > 0 and χT /χs = cp /cv > 1.

(13.1.11)

13.1.2 Foundations of Statistical Mechanics. Statistical Entropy The foundations of statistical mechanics were elaborated in the second half of the nineteenth century by Clausius, Maxwell and Boltzmann and at the beginning of the twentieth century by Gibbs. The scientific domain has been developed during the whole twentieth century. The principles of statistical mechanics[2] can be presented by following and extending the picture developed by Maxwell and Boltzmann for the ideal gas. We will come back to this fascinating topic in Section 13.3. The original papers, collected in two volumes,[3] are worth consulting. Statistical Entropy of Nonequilibrium States Due to the huge separation of scales between macroscopic observation and the microscopic world, many different microscopic states correspond to the same macroscopic state. Notice that in quantum mechanics the microscopic states form a discrete ensemble. It is also the same for the states of a macroscopic system, even though the separation between two neighboring states is too small to be relevant. Let {η} denote the set of parameters representing a macroscopic state. To be more concrete take a simple example of a macroscopic state {η}: the distribution of energy, momentum and mass in spatial cells inside the total volume V, with the restriction that the number of microscopic particles in each spatial cell is large. The number of microscopic states W{η} corresponding to a given macroscopic state {η} is called the number of complexions of {η} and is a very large number, W{η}  1. For an isolated system with fixed E, V and N, the entropy S{η} and the probability P{η} of any macroscopic state are defined by S{η} ≡ kB ln W{η} ,

P{η} ≡ 

W{η} , {η } W{η }

(13.1.12)

where kB is Boltzmann’s constant, so that the quantity TS{η} has the dimensions of energy. The number of complexions W{η} is related to the microscopic states corresponding to {η} in an energy shell containing all the microscopic states of the system whose energy is included in a band delimited by E and E + δE. We will come back to this point; see the discussion above (13.1.19). One can say for the moment that this is because it is impossible to know with precision the exact energy of a macroscopic system (probed with macroscopic diagnostics). Notice that the first equation in (13.1.12) introduces an entropy for any macroscopic state. [2] [3]

Uhlenbeck G., Ford G., 1963, Lectures in statistical mechanics. Lectures in applied mathematics. Providence, R.I.: American Mathematical Society. Brush S., 1966, Kinetic theory. Vols. 1 and 2. Pergamon Press.

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Nonequilibrium states

Equilibrium state

Figure 13.2 Sketch of phase space around equilibrium. Each point corresponds to a microscopic state. The macroscopic states are represented by domains of the phase space. The equilibrium state contains the largest number of microscopic states.

Equilibrium State as the Most Probable Macroscopic State For prescribed conditions, for example a fixed total energy E (isolated system), the basic principle of statistical thermodynamics is that thermodynamic equilibrium {ηeq } is the macroscopic state {η} that corresponds to the maximum value of W{η} , {ηeq } = {η}, where W{η} > W{η} ∀{η} = {η} (see Fig. 13.2) so that the statistical entropy, defined in (13.1.12), is maximum at equilibrium. The mean value f  of any function of macroscopic quantities, f ({η }), is defined by the ensemble average:  f  ≡ f ({η })P{η } . (13.1.13) {η }

The second equation in (13.1.12) corresponds to what is called the microcanonical ensemble. Roughly speaking it corresponds to the assumption that all the microscopic states of an isolated system have the same probability of occurring during evolution on a long time scale. More precisely the time spent by the system in any domain of phase space, that is, the space of the microscopic states, is proportional to the number of microscopic states in that domain. Therefore the time average of any  T macroscopic quantity f is equal to the ensemble average f  in (13.1.13), limT→∞ T −1 0 f dt = f . This is called the ergodic theorem. More details are given below Equation (13.1.20). Entropy and Probability There is another essential property of the macroscopic systems under consideration: in the thermodynamic limit, V → ∞, N → ∞, N/V = cst. < ∞, denoted lim∞ , the number of complexions of the equilibrium state becomes infinitely larger than that of all the other macroscopic states, W{η}  W{η} ∀{η} = {η}, lim P{η} = 1 ∞

⇒

lim f  = f ({η}). ∞

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(13.1.14)

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575

This has to be understood in the following way. Consider the root mean square fluctuation of the quantity f , 1/2  1/2   1/2  ≡ (f − f )2 = f 2 − f 2 ; (13.1.15) (f )2  1/2 Equation (13.1.14) implies that lim∞ (f )2 / f  = 0. To be more precise, for a quantity fs relative to a subsystem including Ns  microscopic particles on average, the relative √ fluctuation is proportional to 1/ Ns ,  1/2 (fs )2 1 . (13.1.16) ∝√ fs  Ns  This property is easy to demonstrate for a quantity that is the  sum of N  statistically indeN  s   s f  and s s = ∀i = j (statistical s , = s pendent quantities, f = N i i i j i j i=1 i=1 N    independence). Introducing f ≡ f −f  = i=1 si , where si ≡ si −si  and si sj = 0 ∀i =          2 j, the mean square f 2 increases in proportion to N, f 2 = N i=1 s , as does also the √ N mean value f  = i=1 si . The relative fluctuation is therefore inversely proportional N   1/2 √ as in (13.1.16), f 2 / f  ∝ 1/ N. Therefore, in the thermodynamic limit, disregarding the small fluctuations, we have f ≈ f.

(13.1.17)

Thermodynamic Entropy According to (13.1.14), the entropy of a macroscopic systems in an equilibrium state is S ≡ S{ηeq } = S{η} , (13.1.24), 

S ≡ kB ln W{η} ≈ kB ln Wtot ,

(13.1.18)

where Wtot ≡ {η } W{η } is the total number of microstates satisfying the macroscopic constraints on the energy E, the volume V and the number of particles N in the system. The equilibrium entropy defined in (13.1.18) satisfies the thermodynamic relations (13.1.2)– (13.1.4), as shown in (13.1.24) for example. Distribution Function In quantum mechanics, the microscopic states are represented by discrete points  in a phase space, considered as a continuous space of many dimensions, typically N ≈ 6N if, for example, one thinks of N point particles in classical mechanics,  ∈ R N . The number of microscopic states in an elementary domain δ ⊂ R N of phase space in the neighbourhood of  ∈ R N is proportional to the N -dimensional volume d N  of this elementary domain, d N /(2π h) ¯ N /2 . The coefficient (2π h) ¯ −N /2 is introduced for dimensional reasons and is computed here for N point particles in a box in quantum mechanics. Consider the evolution in phase space and let t and δt be the images at time t of  and δ at time t = 0. By definition d N  = d N t (the same number of microscopic

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states are in the image as in the initial condition). This property is true in the phase space with N dimensions but not in the hypersurface with N −1 dimensions containing the set of points defined by the equation H() = E; see Section 13.1.3. This is why the summation in (13.1.12) includes all the microscopic states {η } in the energy shell (E): E < H() < E + δE (E and δE E being fixed quantities). The entropy (13.1.18) may then be written  dN  , (13.1.19) S(E) = kB ln W(E), where W(E) ≡ ¯ N /2 (E) (2π h) and, in agreement with the thermodynamic relation (13.1.3), the temperature is given by 1/T = ∂S/∂E. Any macroscopic function of macroscopic states f ({η}) has an equivalent function defined in the phase space F() which is constant and equal to f ({η}) in the subdomain corresponding to the macroscopic state {η}. Quite generally, following Gibbs, one introduces the density ρ N () of microscopic states, in the phase space  under consideration, such that ρ N ()d N  is the fraction of microscopic states in the elementary domain d N ,  N  ρ N ()d  = 1, and we have  f  = F()ρ N ()d N . (13.1.20) 

For the microcanonical ensemble (13.1.12)–(13.1.14), defined on the energy shell (E), for which the equilibrium value  is given in (13.1.14), the distribution function ρ N () is constant in (E), ρ N = 1/ (E) d N . This is not the case if the ensemble is defined on the hypersurface of constant energy defined by the equation H() = E; see Section 13.1.3. By extension to what has been said for the macroscopic states, the distribution function ρ N () in (13.1.20) is considered as the probability density of a microscopic state . This has no physical meaning since, in classical mechanics, the evolution is deterministic at the microscopic level; the only stochastic variables are the macroscopic quantities (with small fluctuations). In quantum mechanics, the situation is even worse: an isolated system in a given quantum state does not evolve. However, the concept of distribution function is useful to help us understand the physical meaning of a microcanonical ensemble and the irreversible behaviour of the macroscopic state of an isolated system, which systematically approaches equilibrium in the long time limit. The reasoning is different in classical and quantum mechanics.[1,2] The topic has a long history throughout the twentieth century.[3] The energy difference between two successive quantum states shrinks to zero as the number of degrees of freedom increases. As a consequence, the slightest interaction of a macroscopic system with the external world can never be neglected. This interaction leads to a random mixing of quantum states, described by a uniform distribution in the energy shell (E) of the macroscopic system.

[1] [2] [3]

Huang K., 1987, Statistical mechanics. New York: Wiley, 2nd ed. Landau L., Lifchitz E., 1982, Statistical physics. Part I. Oxford: Pergamon Press, 3rd ed. Farquhar I., 1964, Ergodic theory in statistical mechanics, Monographs in statistical physics, vol. 7. Interscience.

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Figure 13.3 Sketch of the mixing process on the energy shell (E) as time progresses, illustrating the irreversible approach to equilibrium (uniformity of the distribution function). The microscopic evolution is represented by a trajectory of each point in (E). The image of a subdomain (in grey) of (E) is sketched at three different times.

Irreversibility in Classical Mechanics In classical mechanics, the interaction with the external world is not necessary to understand irreversibility and the microcanonical ensemble. Each macroscopic state {η} corresponds to a subdomain of the phase space {η} ⊂ (E) and can be represented by the distribution function that is uniform in {η} and equal to zero outside. Taking {η} as the initial macroscopic state, and denoting by {η} (t) the image at time t of {η} , one has  N  {η} (t) d  = cst. (Liouville theorem) since this quantity is proportional to the number of microscopic states whose trajectories in the phase space are considered. However, because of the instability of these trajectories (positive Lyapunov exponents), the shape of  {η} (t) will change drastically in time: two points in {η} which are originally close together will soon become far apart.[4] The irreversibility corresponds to the fact that, in the long time limit,  {η} (t) becomes a thin ribbon (of constant volume) that is distributed more and more uniformly in the energy shell (E); see Fig. 13.3. In a coarse-grained sense, this corresponds to a uniform distribution function on the energy shell (E) representing the equilibrium state characterised by the maximum of entropy (13.1.18). This ‘mixing property’ of the evolution in the phase space and the ‘ergodic theorem’, mentioned below Equation (13.1.13), are not easy to prove. They have been established only for a few particular cases. The Gibbs Distribution. Canonical Ensemble Let us now consider a system (of fixed volume and fixed number of particles) exchanging heat with a reservoir. The subsystem is not isolated; its energy fluctuates and cannot be considered as a fixed quantity. According to Gibbs, the distribution function ρ() of such a system is ρ() = 

[4]

e−H()/kB T  e−H( )/kB T d

⇔

wn =

e−En /kB T , 'n e−En /kB T

(13.1.21)

Uhlenbeck G., Ford G., 1963, Lectures in statistical mechanics. Lectures in applied mathematics. Providence, R.I.: American Mathematical Society.

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Figure 13.4 Energy shell (in grey) in the phase space of two subsystems.

where T is the temperature of the reservoir and wn is the equivalent probability of the quantum state labelled n. The sum 'n runs over all the microscopic quantum states of the system. This is called the canonical ensemble. It can be obtained as follows. The composite system, constituted by the system (label 1) and the reservoir (label 2), being isolated with a fixed total energy E, is described by the microcanonical ensemble (13.1.19) corresponding to the energy shell, E < E1 + E2 < E + 2δE. The number of complexions W(E) is evaluated by decomposing each phase space 1 and 2 into energy shells of ‘thickness’ δE; see Fig. 13.4. Obviously, the number of complexions of the composite system, corresponding to a distribution (E1 , E2 ) of total energy E, E2 = E − E1 , is W1 (E1 )W2 (E2 ), and the total number of complexions of the composite system is  W(E) = W1 (E1 )W2 (E − E1 ), E1

where the sum runs over all the energy shells in 1 . According to (13.1.14), the distribution of energy at equilibrium (E1eq , E2eq = E − E1eq ) corresponds to the largest term in the sum, which becomes infinitely larger than all the others in the thermodynamic limit, W(E) ≈ W1 (E1eq )W2 (E2eq ),

S(E) ≈ S1 (E1eq ) + S2 (E2eq ),

(13.1.22)

where (13.1.19) has been used. According to the principle of maximum entropy, any linear variation around the equilibrium state should be zero, δ[W1 (E1 )W2 (E2 )] = 0 for δE1 + δE2 = 0. This yields δ ln W1 (E1 ) + δ ln W2 (E2 ) = [∂ ln W1 /∂E1 |E1 − ∂ ln W2 /∂E2 |E2 ]δE1 = 0 ∀δE1 , implying to equality of temperature, T1 = T2 , since 1/Ti ≡ kB ∂ ln[Wi (Ei )]/∂Ei . The probability ρ1 (1 )d1 to find the system 1 around the microscopic state 1 , corresponding to an energy E1 , independently of the state of the reservoir whose energy is fixed to E − E1 , is obtained by integrating Wd1 d2 over the energy shell 2 (E − E1 ). This gives a result proportional to W2 (E − E1 )d1 = eS2 (E−E1 )/kB d1 . Assuming now that the reservoir is much larger than the system 1, E1 E, a Taylor expansion S2 (E − E1 ) ≈ S2 (E) − E1 /T2 + · · · yields, using T2 = T1 , ρ1 (1 ) ∝ −H(1 )/kB T1 , which corresponds to the distribution of the canonical ensemble (13.1.21).

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The denominator of (13.1.21) is called the partition function of a system of N particles,  d3N  − H() QN (V, T) ≡ e kB T ⇔ QN (V, T) = 'n e−En /kB T , (13.1.23) N! (2π h) ¯ 3N where N! is introduced here to take into account the indiscernibility of microscopic particles (correct Boltzmann counting). It can then be shown from (13.1.23) that the quantity −kB T ln QN (V, T) is the Helmholtz free energy F(N, V, T) F ≡ −kB T ln QN = E − TS = −pV + μN,

(13.1.24)

where the second relation follows from (13.1.4), and, according to (13.1.3), dF = −SdT − pdV + μdN, S = −(∂F/∂T)V,N ,

p = −(∂F/∂V)T,N ,

(13.1.25)

μ = (∂F/∂N)V,T ,

F = E + T(∂F/∂T)V,N .

(13.1.26) (13.1.27)

The relation F(N, V, T) = −kB T ln QN (V, T) is obtained by noticing that the function −kB T ln QN (V, T) satisfies the equation (13.1.27) F − T(∂F/∂T)V,N = E. The energy fluctuations are computed from the canonical ensemble in terms of the specific heat CV = Ncv ≡ (∂E/∂T)V > 0, E ≡ H,   H 2 − H2 = NkB T 2 cv , (13.1.28) showing that the energy fluctuations are small and satisfy the law (13.1.16). Grand Canonical Ensemble For a system with a fixed volume, in equilibrium with a reservoir, exchanging not only energy (heat) but also particles with the reservoir, the number of particles N is a stochastic variable. It can then be shown in a similar manner that the distribution function ρN () and the probability wnN for the system to contain N particles and to be in a quantum state whose energy is EnN takes the form[1,2] ρN () =

e−[HN ()−μN]/kB T N! (2π h) ¯ 3N Q(μ, V, T)

⇔

wnN =

e−[EnN −μN]/kB T , Q(μ, V, T)

(13.1.29)

 where Q is the grand partition function Q(μ, V, T) ≡ N zN QN (V, T) and z ≡ eμ/kB T . It is then shown[1,2] that the thermodynamic relations are given by pV/kB T = ln Q,

E/kB T = T(∂ ln Q/∂T)V,T ,

N = z(∂ ln Q/∂z)V,T .

The grand canonical ensemble is useful to calculate the density fluctuations in terms of the isothermal compressibility κT ≡ −[v(∂p/∂v]−1 > 0,   N 2 − N2 = N kB TκT /v, (13.1.30) showing that the density fluctuations are also small and satisfy the law (13.1.16). [1] [2]

Huang K., 1987, Statistical mechanics. New York: Wiley, 2nd ed. Landau L., Lifchitz E., 1982, Statistical physics. Part I. Oxford: Pergamon Press, 3rd ed.

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Figure 13.5 According to the Liouville theorem the ‘hypervolume’ of a subdomain (area of grey region) in the energy shell is a conserved quantity. This is not the case for the corresponding ‘hypersurface’ H = E (length of thick black line). The distribution function of the microcanonical ensemble is constant on the energy shell (E), but not on the energy ‘surface’ H = E.

13.1.3 Appendix to Section 13.1 According  to differential geometry, the hypervolume of the energy shell for δE E takes the form (E) d N  ≈ σ (E)δE, where σ (E) is obtained by an integral on the hypersurface  N −1 H = E, σ (E) = H=E |∇H|−1 d N −1 ', where  d N −1' denotes the element of surface, ', and the differential operator ∇ used to define the area of the hypersurface, H=E d is defined in R N . In general, the quantity |∇H| is not constant along H() = E, except for hyperspheres; see Fig. 13.5. According to what has been said before, the number of microscopic states per unit hypervolume of phase space is constant, equal to (2π h) ¯ −N /2 , N and the element of hypervolume d  is a conserved quantity of the dynamics in the phase space, d N  = d N t . However, the corresponding conserved quantity on the hypersurface H() = E is d N −1 '/|∇H|, and not the element of hypersurface. The element d N −1 '/|∇H| is proportional to the number of microstates and is a conserved quantity of the dynamics in the phase space. According to (13.1.12), (13.1.13) and (13.1.19), the microcanonical ensemble yields   dN  1 d N −1 ' 1 f  ≡ . (13.1.31) F() ≈ F() W(E) (E) σ (E) H=E |∇H| (2π h) ¯ N /2 In other words, the distribution function of the microcanonical ensemble defined on the energy surface H = E is ρ N −1 () ∝ 1/|∇H|, which is not constant in general, except for a hypersphere.

13.2 Ideal Gases The ideal gas is the state of matter in which the interaction between the particles is so weak as to be negligible, so that the energy in (13.1.1) can be approximated by the sum over  the particles H = N j=1 hj . This is the case in the conditions of classical mechanics when the gas is sufficiently rarefied. Surprisingly, in the conditions of quantum mechanics, the ideal gas approximation is valid in the opposite limit of large density; see Section 13.2.4.

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The states of a system of N particles can be expressed in terms of the states of a single particle. Let k and k be the set of quantum numbers that define, respectively, the state of a single particle and the energy of that state. The state of the system is defined by the number Nk of particles that are in the kth quantum state. Nk is called the occupation number, so   N = k Nk , E = k Nk k , where the sum runs over all the quantum states of a single particle. However, in quantum mechanics, the reconstruction of the quantum states of a system of N particles from the quantum states of a single particle is not trivial. The mean occupation number at equilibrium N k is different for fermions and bosons (see Section 13.2.2), except when N k 1 ∀k, corresponding to the limit of classical mechanics.

13.2.1 The Maxwell–Boltzmann Gas The Maxwell–Boltzmann Distribution (1877) Consider first the simplest case in classical mechanics of noninteracting identical particles with no internal degrees of freedom and in the absence of any external field, hj = p2j /2m, where m is the mass of the particles and pj = |pj | is the modulus of the momentum of the jth particle pj =(pjx , pjy , pjz ). Applying the Gibbs distribution (13.1.21) to the gas in equilibrium, the distribution of particles in the volume V of the system is uniform and the number of particles per unit volume δnp with momenta in the cell δ 3 p, centred on the momentum p, is δnp = f (0) (p)δ 3 p,

f (0) (p) ≡

  n p2 exp − , 2mkB T (2π mkB T)3/2

 d3 pf (0) (p) = n,

(13.2.1) where n ≡ N/V is the number density. The distribution of particles per unit volume with a modulus of momentum equal to p ± δp/2 is obtained from (13.2.1) by using δ 3 p = 4π p2 δp. In the presence of an external field, hl = p2l /2m + ϕ(ql ), where ql = (xl , yl , zl ) is the position of the centre of mass of the particle, the number density varies in space, n(q) = n0e−ϕ(q)/kB T , where n0 is the density at point φ = 0, obtained by the normalisation condition V n(q)d3 q = N. The Free Energy

 −E /k T  n B Coming back to the quantum statistics in (13.1.23), F = −kB T ln , the ne energy En of the system in the quantum state labelled by n (do not confuse with the number density) is, for an ideal gas, a sum of N values of k . In the Boltzmann case (N k 1 ∀k) they are all different; see the text above (13.2.22). Writing e−En /kB T as a product of N factors e−k /kB T , and summing independently over all the states of each particle, one would  − /k T N k B . However, two distributions of particles (on the energy levels k ) that obtain ke differ only by a permutation of particles represent the same quantum state (indiscernibility of particles). Therefore, in the limit of the Boltzmann gas, the partition function QN , the

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mean occupation numbers equilibrium

N N k and the free energy F take the following form:  at −  k /kB T e /N! , QN ≡ n e−En /kB T ≈ k    (μ−k )/kB T −k /kB T Nk ≈ e and F ≈ −NkB T ln (e/N) e , (13.2.2) k

where the approximation of a large number N has been used, N  1: ln N! ≈ N ln(N/e). Equation (13.2.2) becomes clear when considering the Boltzmann gas as the high temperature limit of the Fermi and/or Bose gases presented in Section 13.2.2. The first equation in (13.2.2) is called the Boltzmann distribution. Ideal Gas of Molecules The translational motion of the molecules in a gas is quasi-classical at ordinary conditions, but the internal structure of each molecule is in a quantum state. The energy of molecules may be written in the form k (p) = p2 /2m + k ,

(13.2.3)

where the internal energy of the molecule in the quantum state k, k , is a quantity independent of the velocity and the coordinates of the centre of mass of the molecule. The free energy in (13.2.2) then takes the form     V mkB T 3/2  −  /kB T e k , (13.2.4) F ≈ −NkB T ln e N 2π h¯ 2 k  −  /kB T is a function only of T, depending on the internal degrees of where the sum k e k freedom of the internal structure. According to (13.1.26), the equation of state and the chemical potential take the form pV = NkB T,

μ(T, p) = kB T ln(p/po ) + μ(o) (T),

(13.2.5)

where μ(o) (T) depends on the internal structure of the molecules through the internal   partition function of a molecule, k e−k /kB T , and po is the reference pressure at which μ(o) (T) is defined. The equation of state of an ideal gas is also recovered from the Maxwell– Boltzmann distribution (13.2.1) by computing the momentum transfer at the wall due to elastic collisions. For a Boltzmann gas of particles with no internal structure (elementary particles), and in the absence of an external field, the internal energy, the free energy F, the chemical potential μ = (∂F/∂N)V,T and the entropy S = −(∂F/∂T)V,N , s ≡ S/N reduce to E = 3NkB T/2, ⎡ .    / ⎤  2 3/2 V mkB T 3/2 N F/N 2π h μ ¯ ⎦, = − ln = ln ⎣ e , (13.2.6) kB T N 2π h¯ 2 kB T V mkB T     V mkB T 3/2 3 CV 5 5 = kB , s = kB ln cv ≡ + kB , Ts = kB T − μ. (13.2.7) 2 N 2 N 2π h¯ 2 2

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Introducing the equation of state, the first equation in (13.2.5), into the expression for entropy (13.2.7) leads to Poisson’s adiabatic law for an isentropic compression (or dilatation), pv5/3 = cst., Tv2/3 = cst., where v ≡ V/N. Introducing cp = cv + kB , a more general isentropic law for a gas of molecules with constant specific heats cv = d(E/N)/dT= cst. = 3/2, is pvγ =cst., where γ ≡ cp /cv . The equation for the entropy generalising (13.2.7) may be written pvγ = po vγo e(s−so )/(cv ) ,

(13.2.8)

where s ≡ S/N and the index o denotes a reference state.

13.2.2 The Fermi and Bose–Einstein Distributions If, for a given number density, the temperature of an ideal gas is sufficiently low, the mean occupation number at equilibrium N k is not a small number and the quantum statistics must be used. Quantum statistics differ according to the symmetry property of the wave function of N identical particles. For particles with half-integral spin, called fermions, the wave function is antisymmetric. In this case the Pauli’s principle says that there cannot be more than one particle in each quantum state, Nk = 0, 1 (Pauli exclusion principle) and the Fermi distribution is obtained. If the wave function is symmetric (particles with integral or zero spin, called bosons), Nk = 0, 1, 2, 3, ..., a different distribution, called the Bose–Einstein distribution (1925), is obtained. Here, we shall used the microcanonical ensemble to derive these distributions. Most Probable Distribution To avoid all the unnecessary complications illustrating the phase space of a single particle, we consider, to begin, a gas of N identical nonrelativistic particles of mass m with no internal structure in a cubic box of volume V. However, the methodology developed in this section is not limited to elementary particles. According to quantum mechanics, any quantum state k of an individual free particle in the box is determined by three signed integers k = (kx , ky , kz ), such that the momentum eigenvalue is p = (2π h/V ¯ 1/3 )k and the 2 energy of the particle in this quantum state is p = p /2m, where p ≡ |p|, p = (p2x + p2y + p2z )/2m.

(13.2.9)

Each quantum state of the ideal system is specified by a set of occupation numbers {Np } defined such that there are Np particles having the momentum p in the state under consideration. In other words, the distribution {Np } represents a microscopic quantum state of the system. The total energy E of the state and the total number of particles N are   Np p , N= Np . (13.2.10) E= p

p

For a macroscopic box, the difference of energy between two consecutive states is a microscopic quantity. The energy spectrum of an individual particle is divided into groups of

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Figure 13.6 Representation of the energy spectrum of a quantum gas. Fermions

Bosons

Figure 13.7 Distribution of fermions and bosons. There can be only zero or one fermion per quantum state, but the corresponding number of bosons is unlimited.

levels (cells) containing, respectively, ω1 , ω2 , ..ωl , ... quantum states of individual particles, such that ωl  1 ∀l. Moreover, the largest difference between the ωl energy levels in the same cell l is assumed to be a small quantity at the macroscopic scale (see Fig. 13.6), so that the energy of a cell, l is well defined. The set of occupation numbers on these cells, {ηl }, is considered as a macroscopic state. Within a small uncertainty E, whose value is unimportant, the total energy E can be written   η l l , N= ηl . (13.2.11) E= l

l

The number of complexions W{ηl } of a macroscopic distribution {ηl }, namely the number of microscopic distributions {Np } corresponding to {ηl }, is large, W{ηl }  1. Since interchanging particles in different cells does not lead to a new microscopic state, we have < W{ηl } = l wηl , where wηl is the number of ways ηl particles may be assigned to the lth macroscopic cell containing ωl energy levels. The number wηl differs according to the type of particles fermions or bosons (see fig. 13.7), Fermi gas: wηl =

ωl ! , ηl ! (ωl − ηl )!

Bose gas: wηl =

(ηl + ωl − 1)! . ηl ! (ωl − 1)!

(13.2.12)

According to (13.1.18), the equilibrium state is represented by the distribution {ηl } having the largest value of ln W{ηl } , but the maximisation must be carried out with the restriction in (13.2.11). This can be done by the method of Lagrange’s multipliers α and β,     δηl + β l δηl = 0, (13.2.13) δ ln W{ηl } − α l

l

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585

where the elementary variations δηl are considered as independent variations (∀l). The constants α and β are determined in a second step in terms of E and N using (13.2.11). Focusing attention to the cells in which ηl  1 and applying Stirling’s formula x! ≈ x ln x− x to all the factorials that appear in (13.2.12) (even though this approximation is not always well verified for (ωl − ηl )! in the Fermi case ωl  ηl ), the logarithm of the number of complexions may be written,      ωl ηl ηl ln , (13.2.14) ∓ 1 ∓ ωl ln 1 ∓ ln W{ηl } ≈ ηl ωl l

where the signs − and + in (13.2.14) are for the Fermi and Bose cases, respectively. According to (13.2.13), the equilibrium distribution {ηl } is obtained from     ωl ln ∓ 1 − α − βl δηl = 0, (13.2.15) ηl ηl =ηl l

and, in view of the arbitrariness of all the variations δηl , the equilibrium state is characterised by the following distributions Fermi gas:

ηl 1 , = β +α l ωl e +1

Bose gas:

ηl 1 , = β +α l ωl e −1

with, according to (13.1.18) and (13.2.14), 

  S/kb = ln W{ηl } ≈ ηl (α + βl ) ± ωl ln 1 ± e−α−βl , l

S/kb − βE − αN = ±



ωl ln 1 ± e−α−βl .

(13.2.16)

(13.2.17) (13.2.18)

l

The comparison of (13.2.18) with (13.1.3) and (13.1.4) then yields    μ pV 1 , α=− , =± β= ωl ln 1 ± e−(l −μ)/kB T , kB T kB T kB T

(13.2.19)

l

where the signs + and − in (13.2.18) and (13.2.19) are for the Fermi and Bose cases, respectively. Equation (13.2.16) can then be interpreted as the mean number of particles N k in the kth microscopic quantum state of a single particle, Fermi or Bose distribution:

Nk =

1 e(k −μ)/kB T

±1

,

(13.2.20)

where the sign + is for fermions and − for bosons. The expression for the chemical potential in terms of the temperature T and the particle density N/V is obtained by the normalisation  condition k N k = N, and, according to (13.2.19), the equation of state and the internal energy E can be written      pV =± ln 1 ± e−(k −μ)/kB T , N = Nk, E = N k k ; (13.2.21) kB T k

k

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the sums run over all the quantum states of individual particles in the volume V. Equations (13.2.20)–(13.2.21) are valid for any type of particle, not only for elementary particles. The Boltzmann distribution (13.2.2) is recovered from (13.2.20) in the limit N k = e−(l −μ)/kB T 1, which is an accurate approximation for the thermodynamic states satisfying the condition eμ/kB T 1. According to (13.2.6), the condition eμ/kB T 1 corresponds to . /3/2 N 2π h¯ 2 1, (13.2.22) V mkB T showing that the chemical potential of a Boltzmann gas is negative. In the limit of classical mechanics, ηl ωl , and according to (13.2.14), the entropy of a gas out of equilibrium, S = kB ln W, is given by    ηl + N. (13.2.23) S=− ηl ln ωl l

The condition (13.2.22) is fulfilled when the mean distance between particles (V/N)1/3 √ is much larger than the thermal de Broglie wavelength 2π h/ ¯ 3mkB T. In gases at ordinary conditions the range of the interaction force is larger than the thermal de Broglie wavelength, so this condition is systematically fulfilled at densities for which the interaction is negligible. However, the condition in (13.2.22) is violated for electrons in metals, described by the Fermi distribution (13.2.20), although the ions may be described by classical (not quantum) mechanics. The difference is explained by the ratio of the electron rest mass to the neutron rest mass, which is about 5.44 × 10−4 . 13.2.3 Fermi and Bose Gases of Elementary Particles The energy of elementary particles (without internal structure) is just the energy of their translational motion. In nonrelativistic conditions, the energy levels then take the form (13.2.9) and the number of quantum states in the cell d3 pd3 q, centred at the point (p, q) in the phase space of a single particle, is dωp = gd3 pd3 q/(2π h) ¯ 3 , where g = 2s + 1 (s being the spin of the particle, s = 1/2, g = 2 for an electron). This expression for dωp is also valid in the relativistic case, but the expression for the energy levels, p , is more general than in (13.2.9); see (13.2.36). Integrating d3 q over the volume V and using spherical coordinates in momentum space, the number of particles in the system dNp having the absolute magnitude of momentum in the range p ± dp/2 is obtained from (13.2.16) or (13.2.20), after integrating over all directions, dNp =

gVp2 dp  , 2π 2 h¯ 3 e(p −μ)/kB T ± 1

(13.2.24)

where, for simplicity, we have omitted the overbar to denote the equilibrium state. In conditions for which the elementary particles are nonrelativistic and in the absence of an external potential,

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587

p = p2 /2m,

(13.2.25)

the energy distribution (13.2.20), namely the number of particles in the volume V with an energy equal to  ± d/2, obtained from (13.2.24)–(13.2.25), takes the form dN =

gVm3/2

 1/2 d

21/2 π 2 h3

e(−μ)/kB T

¯

±1

.

(13.2.26)

This corresponds to a uniform distribution in space that generalises the Maxwell–Boltzmann distribution (13.2.1) when the effects of quantum mechanics are nonnegligible. The expression for the chemical potential μ in terms of T and n ≡ N/V is given by integrating  (13.2.26) from  = 0 to  = ∞ and using the normalisation condition N = dN . Using /kB T as the variable of integration, this condition shows that μ/kB T is a function of (N/V)(kB T)−3/2 , μ/kB T = F (ξ ) ,

ξ ≡ (kB T)3/2 V/N.

(13.2.27)

Using the same integration procedure, the equation of state and the internal energy are  obtained from (13.2.21) and E = dN , respectively: pV gVm3/2 =± kB T 21/2 π 2 h¯ 3 E gm3/2 V = N 21/2 π 2 h¯ 3 N



∞

  1/2 ln e−(−μ)/kB T ± 1 d,

(13.2.28)

0



∞ 0

 3/2 d e(−μ)/kB T ± 1

.

(13.2.29)

Integration by parts of (13.2.28) then leads to the same relation between E and pV as for the Maxwell–Boltzmann gas of particles without internal structure pV = 2E/3.

(13.2.30)

Using /kB T as the variable of integration, Equation (13.2.29) shows that, as for μ/kB T in (13.2.27), the quantity e/kB T is a function of ξ ≡ (kB T)3/2 v only, so that, according to (13.2.30), the same is true for pv/kB T. Then, according to the expression for the entropy in (13.1.4), Ts = e + pv − μ, the density of entropy s ≡ S/N is also a function of ξ only. Consequently, Poisson’s adiabatic law for isentropic compression (or dilatation) of a Boltzmann gas is also valid for a Fermi or Bose gas, s = cst.:

vT 3/2 = cst.,

pv5/3 = cst.,

(13.2.31)

where the second relation is obtained by multiplying the quantity pv/kB T, which is a function of ξ ≡ (kB T)3/2 v, by v3/2 . However, in contrast to the Boltzmann gas, for the ideal Fermi or Bose gas the exponent 5/3 is not the ratio of the specific heats.

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13.2.4 The Degenerate Electron Gas Low Temperature and High Density In some circumstances, the Coulomb interaction (electron–electron and electron–ion) is negligible, so that the electrons form a Fermi gas. This is the case for matter at high density and low temperature in white dwarf stars. A completely degenerated electron gas corresponds to a temperature of absolute zero, T = 0, that is, to the ground state (minimum total energy) of the system of N electrons in a volume V. However, as explained below, for a sufficiently high electronic density n ≡ N/V, the results at T = 0 are relevant at relatively high temperature. Since electrons are fermions, no more than one electron can be in each quantum state, so that for T = 0, all quantum states, from zero energy to some upper bound F , called the Fermi energy, are each occupied by one electron, k  F :Nk = 1; k > F : Nk = 0; see Fig. 13.8. In other words, in the limit T → 0, the Fermi distribution (13.2.20) becomes a step function. This requires T = 0:

μ = F ,

(13.2.32)

where the Fermi energy F depends on the number of electrons N in the electron gas  enclosed in the volume V and is obtained from the normalisation condition N = dN . The Nonrelativistic Degenerate Electron Gas Consider electrons (s = 1/2, g = 2) with a translational kinetic energy given in (13.2.9). According to the beginning of Section 13.2.3 for g = 2, the number of quantum states of an electron moving in a volume V with the absolute magnitude of momentum in the √ p = 2mF , the range p ± dp/2 is Vp2 dp/(π 2 h¯3 ). Introducing the Fermi momentum, F  pF 2 3 3 2 normalisation condition N = dN yields N = [V/(π h¯ )] 0 p dp = VpF /(3π 2 h¯ 3 ), expressing the Fermi energy F in terms of the number density n. It is also useful to introduce the degeneracy (or Fermi) temperature, TF , F = kB TF , . / h¯ 2 n2/3 h¯ 2 2 2/3 TF = (3π 2 )2/3 . (13.2.33) F = (3π ) n2/3 , 2m 2mkB The internal energy at T = 0 is the ground state energy of the system of N electrons p Eo = [V/(2mπ 2 h¯ 3 )] 0 F p4 dp = Vp5F /(10mπ 2 h¯ 3 ). Introducing the energy per particle e = eo ≡ Eo /N and the pressure p = po at T = 0, one gets

Figure 13.8 Filling of energy states for a Fermi gas. The thin line (step function) shows the distribution at T = 0, the thick line shows the effect of a nonzero temperature.

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13.2 Ideal Gases

T = 0:

eo = (3/5)F ,

po = (2/5)nF = (3π 2 )2/3 (h¯ 2 /5m)n5/3 ,

589

(13.2.34)

where the equation of state for T = 0 (the last equation), p ∝ n5/3 , is obtained from (13.2.30). It agrees with the Poisson adiabatic law (13.2.31) (v = 1/n), as it should do, since the entropy is kept fixed to zero at T = 0. The thermodynamic relations for the internal energy e and the pressure p in (13.2.34) for T = 0 are also accurate for 0 < T TF , as indicated by Fig. 13.8. The limit T TF corresponds to a thermal de Broglie wavelength of a particle much larger than the average √ interparticle separation, 2π h/ ¯ 3mkB T  n−1/3 . It is the opposite limit of the Boltzmann gas in (13.2.22). The higher the density, the better is the accuracy of the expressions for e and p in (13.2.34). However, this is true only if the Fermi distribution of an ideal Fermi gas is still valid, that is, if the interaction energy per electron due to the Coulomb force, q2e n1/3 (qe is the electronic charge and 1/n1/3 is the mean distance between electrons), is sufficiently small and negligible compared with the mean energy per particle of an ideal Fermi gas ≈ F . This condition, q2e n1/3 F , requires a sufficiently high density n  (q2e m/h¯ 2 )3 , where m is the electron mass and h¯ 2 /q2e m ≈ 5.29 × 10−11 m is the Bohr radius. According to (13.2.33), the critical degeneracy temperature TFc corresponding to the density (q2e m/h¯ 2 )3 is TFc ≈ 0.5 × 106 K. A system of electrons is then well approximated by an ideal Fermi gas in the intermediate temperature range TFc < T < TF . For T TF , the Fermi distribution differs appreciably from unity and/or zero only in a narrow range of the energy  close to F , the thickness of this region being kB T; see Fig. 13.8. A perturbation analysis[1,2] using T/TF as the small parameter then allows computation of the effect of temperature as a first-order correction to the thermodynamics quantities at T = 0. For a nonrelativistic degenerate Fermi gas one gets       5 2 T 2 p 5 2 T 2 s π2 T e , (13.2.35) ≈1+ π , ≈1+ π , ≈ eo 12 TF po 8 TF kB 2 TF where eo and po are the values of the energy per particle e ≡ E/N and of the pressure p in (13.2.34). The first equation shows that, for T/TF 1, the specific heat cv ≡ ∂e/∂T|n = (5/6)(T/TF )kB increases linearly with the temperature but is much smaller than that of a Boltzmann gas; see (13.2.7). The last equation in (13.2.35) shows that the entropy per electron, s ≡ S/N, is smaller than e/T ≈ F /T ≈ (TF /T)kB by a factor (T/TF )2 . Examples of degenerate electron gas are provided by white dwarf stars. They are inert stars composed of a fully ionised plasma of helium. For a typical mass equal to a solar mass ≈ 1030 kg, the mass density and temperature at the centre are approximately ≈ 107 g/cm3 and T ≈ 107 K. Therefore, the gas of electrons has a density of approximately n ≈ 1036 electrons/m3 , corresponding to a degeneracy temperature TF ≈ 1011 K. These electrons then form an ideal Fermi gas, TFc < T < TF . However, such electrons must be treated by relativistic dynamics since the energy per electron, kB TF ≈ 10−12 J, is larger than mc2 ≈ 8 × 10−14 J, where c is the speed of light. [1] [2]

Huang K., 1987, Statistical mechanics. New York: Wiley, 2nd ed. Landau L., Lifchitz E., 1982, Statistical physics. Part I. Oxford: Pergamon Press, 3rd ed.

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The Relativistic Degenerate Electron Gas At very high density, for an electronic density of the order of n ≈ m3 c3 /(3π 2 h¯ 3 ) or larger (see (13.2.38)), typically n  1036 electrons/m3 , which corresponds to a mass density  106 g/cm3 for one electron (namely for two nucleons if the number of protons and neutrons are equal, which is the case for white dwarfs and/or for the iron core in the centre of large stars at the end of their life), the degeneracy temperature TF becomes so high that the energy per particle e becomes comparable to mc2 . In this case the relativistic effects become important. The quantum states for a single electron are specified by the momentum p, but the singleparticle energy levels are expressed in terms of p by a relation that generalises (13.2.9) to  (13.2.36) p = (pc)2 + (mc2 )2 . In the limit p/mc 1 the expression for p in (13.2.36) reduces effectively to (13.2.9) plus a constant shift, the rest energy mc2 , that has no importance for nonrelativistic conditions. The normalisation condition N = dN leads to the  p same expression as before for pF in terms of the electronic density, N = [V/(π 2 h¯ 3 )] 0 F p2 dp ⇒ p3F = 3π 2 h¯ 3 n. One then  defines the Fermi energy F = (pF c)2 + (mc2 )2 and the degeneracy temperature TF , 

2 2 1/3 1/3 2 , F ≡ mc 1 + pF /mc , kB TF ≡ F . (13.2.37) pF = (3π ) hn ¯ The transition region where relativistic effects can no longer be neglected is then given by 1/3 ≈ mc. pF = (3π 2 )1/3 hn ¯

(13.2.38)

As before, the chemical potential at T = 0 is equal to the Fermi energy F . The expressions for these quantities and for the internal energy per particle eo (n) ≡ Eo /N p (where Eo = [V/(π 2 h¯ 3 )] 0 F p p2 dp is the ground state energy), each including the rest energy, are  pF  mc m4 c5 T = 0: μo (n) = F , eo (n) = x2 1 + x2 dx. (13.2.39) 3 2 π h¯ n 0 In general T = 0 and the expression for the thermodynamic quantities depends also on the nondimensional parameter pF /mc. Simplifications occur at very high densities, in the extreme regime of ultrarelativistic particles, (pF /mc)2  1, which for electrons corresponds to n1/3 > 1012 m−1 ,  p 2 F 1/3 > 1: p ≈ cp, F ≈ cpF = (3π 2 )1/3 hc (13.2.40) ¯ n , mc  ∞  2 d V N≈ , (13.2.41) π 2 c3 h¯ 3 0 e(−μ)/kB T + 1  ∞  3 d V , (13.2.42) E≈ π 2 c3 h¯ 3 0 e(−μ)/kB T + 1  ∞

 pV V 2 −(−μ)/kB T  ln e + 1 d, (13.2.43) ≈ kB T π 2 c3 h¯ 3 0

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obtained from (13.2.20)–(13.2.21) using (13.2.24) and (13.2.41), dp = d/c. For such high densities the Coulomb interaction is negligible, q2e n1/3 F , since q2e hc. ¯ Using the same method as in (13.2.28)–(13.2.29), the pressure and the internal energy of ultrarelativistic elementary particles are shown to be related by the relation  p 2 F > 1: pV = E/3, (13.2.44) mc instead of (13.2.30) in the opposite limit pF /mc 1, where relativistic effects are negligible, pV = 2E/3. Using the variable of integration /kB T, Equations (13.2.41)–(13.2.43) show that the three quantities μ/kB T, e/kB T and pv/kB T (where v ≡ 1/n) are functions of n and T through the same group of variables n/T 3 . According to (13.1.4), Ts = e + pv − μ, this is also true for the entropy per particle s ≡ S/N, so that in an isentropic process, the volume, pressure and temperature of an extremely relativistic Fermi gas are related by  p 2 F > 1, s = cst.: vT 3 = cst., pv4/3 = cst. (13.2.45) mc The internal energy at T = 0 of an ultrarelativistic Fermi of elementary particles  pgas 3 F 2 3 (ultrarelativistic degenerate Fermi gas) is Eo = [V/(π h¯ )] 0 cp dp = [Vcp4F /(4π 2 h¯ 3 )], so that the energy per particle eo and the pressure po are  p 2 3F F F 1/3 , po = n , F = (3π 2 )1/3 hc > 1, T = 0: eo = (13.2.46) ¯ n . mc 4 4 For T TF , where TF ≡ F /kB ≈ mc2 /kB ≈ 5.8 × 109 K, a perturbation analysis[1] using T/TF as the small parameter leads to an expression for the temperature effects as a first-order correction to the thermodynamics quantities in (13.2.46):  p 2 T F > 1, < 1: mc TF    2   e T T 2 2 T 2 p s 2 2 . (13.2.47) ≈1+ π , ≈ 1 + 2π , ≈π eo 3 TF po TF kB TF These approximations are valid for electronic densities above 1036 electrons/m3 and temperatures below 5.8 × 109 K (at this density the upper-bound temperature increases with density as n1/3 ) and are relevant in stars. 13.3 Physical Kinetics Based on the work of Clausius concerning the dynamical theory of gases, Maxwell was convinced, as early as 1860, that the irreversibility nature of the second law should result from statistical considerations. This was confirmed in 1872 by Boltzmann’s H-theorem. In the same vein, when considering the dissipation of energy, Thomson (Lord Kelvin) noted in 1874 that the contrast (paradox) between the reversible dynamics of microscopic particles and the irreversible dynamics of macroscopic systems may be explained by the [1]

Landau L., Lifchitz E., 1982, Statistical physics. Part I. Oxford: Pergamon Press, 3rd ed.

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large number of microscopic particles in a macroscopic system.[1] Boltzmann’s analysis took time to be accepted by the scientific community, as illustrated by his controversies at the end of the nineteenth century with Mach, Poincar´e and Zemerlo. Modern statistical physics, including the topics concerning dense matter, rests on Boltzmann’s views, which can be summarised by (13.1.12) and (13.1.18). We present here a summary of physical kinetics,[2,3] retaining only the simplest aspects in gases, useful in this book.

13.3.1 Boltzmann’s Equation and Irreversibility Consider a system of identical particles with no internal structure (hard spheres) for simplicity. Define the one-particle distribution function f1 ≡ f (r, p1 , t)  0 so that f1 d3 rd3 p1 represents the number of microscopic particles in the element of volume d3 r centred in r 3 which, at time  t, have momenta lying within an element d p1 about p1 . The particle density is n(r, t) = f1 dp1 . The one-particle distribution function f1 can be obtained by integration of ρN (, t) over all the particles except one. Dilute Gas. Boltzmann’s Equation. ‘Stosszahlansatz’ Boltzmann was looking for the evolution of f (r, p1 , t) in a dilute gas. When the interaction between the microscopic particles is neglected all the particles in the cell d3 rd3 p1 about (r, p1 ) at time t are, at time t = t + δt, in the cell d3 r d3 p about (r = r + v1 δt, p = p1 ), where v1 ≡ p1 /m is the velocity and d3 r d3 p = d3 rd3 p1 . Since the number of particles is a conserved scalar we have the equality f (r + v1 δt, p1 , t + δt) = f (r, p1 , t). Boltzmann assumed that the effects of interaction between particles can be represented by a local operator B(.) acting on the functions of p1 , so that, in the notation f1 = f (r, p1 , t), one gets (13.3.1) f (r + v1 δt, p1 , t + δt) = f (r, p1 , t) + B( f1 )δt, p1 ∂f1 + .∇f1 = B( f1 ). (13.3.2) ∂t m This is a nontrivial assumption. The exact evolution leads to the so-called BBGKY hierarchy of equations[4] linking the evolution of the one-particle distribution function f to the many-particle distribution functions, and the Boltzmann equation (13.3.2) corresponds to a subtle closure of the hierarchy which is valid at low density and for short-range interaction forces between the microscopic particles.[2] Assuming that the gas is sufficiently dilute, Boltzmann derived an expression for B( f1 ) by considering binary collisions only. The collisions involved in B( f1 ) can be classified [1] [2] [3] [4]

Brush S., 1966, Kinetic theory. Vols. 1 and 2. Pergamon Press. Balescu R., 1975, Equilibrium and nonequilibrium statistical mechanics. John Wiley and Sons. Lifshitz E., Pitaevskii L., 1999, Physical kinetics. Butterworth Heinemann. Yvon J., 1966, Les corr´elations et l’entropie. Dunod.

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593

Figure 13.9 Geometry of an elastic collision between an incoming particle with momentum p and a fixed scatterer at O showing the impact parameter b and the scattering angles φ, θ and d

into two categories: a first type of collision, called direct and labelled (d), in which one of the particles before collision is in d3 rd3 p1 and a second type, called inverse and labelled (i), in which one of the particles after collision is in d3 rd3 p1 . Therefore B( f1 ) is the difference of rate per unit volume between these two types of collisions, B( f1 ) = B (i) ( f1 ) − B (d) ( f1 ). He then made the crucial assumption that the probability of finding simultaneously two particles, labelled 1 and 2, the first in d3 rd3 p1 and the other in d3 rd3 p2 , is proportional to the product f1 f2 ≡ f (r, p1 , t)f (r, p2 , t). The assumption that the two particles 1 and 2 are uncorrelated before collision was called ‘Stosszahlansatz’ (molecular chaos). The contribution of direct collisions then takes the form of an integral over p2 , B (d) ( f1 ) =  3 d p2 G12 f1 f2 , where the kernel G12  0 can be measured by a scattering experiment. It  is expressed in terms of the differential cross section dσ , G12 = |v1 − v2 |(dσ/d)d, where the number of incident particles 2 scattered per second by the particle 1 into the solid angle element d about  is I(dσ/d)d and where I = n|v1 − v2 | is the incident flux; see textbooks.[5] For a given couple of impulsions (p1 , p2 ) of two incident particles, the momenta (p1 , p2 ) after collision depend on the distance between the straight lines forming the trajectories of the two incident particles (that are free flying before collision). This distance is called the impact parameter b, and geometrical considerations give a relation between b and the differential cross section dσ , I(dσ/d)d = Ib db dφ; see Fig. 13.9. When considering all the possible outcomes (p1 , p2 ) of an elastic collision between two collision, it is conincident particles, one with momentum p1 the other with p2 before  venient to introduce a transfer matrix T12→1 2 so that G12 = d3 p1 d3 p2 T12→1 2 . The elements T12→1 2 are proportional to delta functions corresponding to momentum and 2 energy conservation p1 + p2 = p1 + p2 , p21 + p22 = p2 1 + p2 . The contribution of the direct collisions can then be written in the form  (d) (13.3.3) B ( f1 ) = d3 p1 d3 p2 d3 p2 T12→1 2 f1 f2 .

[5]

Huang K., 1987, Statistical mechanics. New York: Wiley, 2nd ed.

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a

b

c

Figure 13.10 (a) A direct collision; (b) the inverse collision and (c) the rotational invariant.

The contribution of the inverse collisions takes a similar form,  (i) B ( f1 ) = d3 p1 d3 p2 d3 p2 T1 2 →12 f1 f2 ,

(13.3.4)

where fi ≡ f (r, pi , t). Because of reversibility and rotational invariance in mechanics (see Fig. 13.10), the relation T1 2 →12 = T12→1 2  0 holds and the Boltzmann collision operator B( f1 ) = B (i) ( f1 ) − B (d) ( f1 ) takes the form  B( f1 ) = d3 p1 d3 p2 d3 p2 T12→1 2 ( f1 f2 − f1 f2 ). (13.3.5) Equations (13.3.2) and (13.3.5) then yield a nonlinear integro-differential equation for the distribution function f (r, p, t), called Boltzmann’s equation. H-theorem, Boltzmann’s Entropy and Irreversibility Consider a state out of equilibrium which is uniform, ∇f = 0, f1 ≡ f (p1 , t). According to (13.3.5), the Boltzmann equation (13.3.2) then takes the form  (13.3.6) ∂f1 /∂t = d3 p1 d3 p2 d3 p2 T12→1 2 ( f1 f2 − f1 f2 ). This shows that a sufficient condition for an equilibrium distribution, ∂f (p)/∂t = 0, is f (p1 )f (p2 ) = f (p1 )f (p2 ) ⇒ B( f1 ) = 0. (13.3.7)  3 Consider the scalar H(t) ≡ d p1 f1 ln f1 and its derivative with respect to time,  (13.3.8) H(t) ≡ d3 p1 f1 ln f1 ,   dH/dt = d3 p1 (∂f1 /∂t)[1 + ln f1 ] = d3 p1 B( f1 )[1 + ln f1 ], (13.3.9)  dH/dt = d3 p1 d3 p2 d3 p2 d3 p1 T12→1 2 ( f1 f2 − f1 f2 )[1 + ln f1 ], (13.3.10)  dH/dt = (1/2) d3 p1 d3 p2 d3 p2 d3 p1 T12→1 2 ( f1 f2 − f1 f2 )[2 + ln( f1 f2 )], obtained by interchanging 1 and 2 using the invariance of the transfer matrix. Due to the reversibility of mechanical systems, the elements T12→1 2 are also invariant under the interchange of p1 , p2 and p1 , p2 ,

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 dH/dt = −(1/2)

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d3 p1 d3 p2 d3 p2 d3 p1 T12→1 2 ( f1 f2 − f1 f2 )[2 + ln( f1 f2 )].

Taking half the sum of the two last equations, we obtain    dH 1 f1 f2 = d3 p1 d3 p2 d3 p2 d3 p1 T12→1 2 ( f1 f2 − f1 f2 ) ln   . dt 4 f1 f2

(13.3.11)

The integral is never positive, (x − 1) ln x  0, so that H(t) is a decreasing function of time,  dH d3 p1 B( f1 )[1 + ln f1 ]  0,  0. (13.3.12) dt The function H(t) reaches its minimum (dH/dt = 0) for a distribution satisfying the condition (13.3.7) that implies ∂f (p)/∂t = 0. In agreement with (13.2.23), the statistical entropy of a Boltzmann gas out of equilibrium, uniformly distributed in space, called Boltzmann’s entropy SB (t) = NsB (t), is proportional to H in (13.3.8):  nsB (t) = −kB d3 p[f ln f ] + nkB . (13.3.13) The distribution satisfying both (13.3.7) and the conservation of momentum and energy, 2 p1 + p2 = p1 + p2 , p21 + p22 = p2 1 + p2 , is the Maxwell–Boltzmann distribution (13.2.1), where n and T are constant (systems at thermodynamic equilibrium are homogeneous). The 1867 analysis of Maxwell[1] to obtain (13.2.1) was based on similar considerations. The H-theorem is the proof that if, at a given instant, the assumption of molecular chaos is verified in a sufficiently dilute gas, then the distribution function f (p, t) approaches the Maxwell–Boltzmann distribution f (0) (p) and the Boltzmann entropy sB (t) reaches its maximum in the long time limit, limt→∞ f = f (0) , limt→∞ sB = s(0) , where  ns(0) ≡ −kB d3 p[f (0) ln f (0) ] + nkB . (13.3.14) It is easy to check that expression (13.3.14) for s(0) yields the thermodynamic expression (13.2.7) for the equilibrium entropy per microscopic particle, except for a numerical constant which is not important. Therefore the thermodynamic law (13.1.7) Tds(0) =   1 2 3 de + pd(1/n) is satisfied by (13.3.14) for e = 2 p /m = 2 kB T using the notation  n a = d3 pa(p)f (0) . To summarise, in a homogeneous system, Boltzmann’s equation shows that the Maxwell–Boltzmann distribution f (0) (p) is reached in the long time limit, and f (0) is the most probable distribution maximising Boltzmann’s entropy, that is, the most probable distribution corresponding to the largest number of complexions; see (13.2.23). Among the attempts of the twentieth century to systematically extend Boltzmann’s analysis to dense matter, the work of Prigogine’s Brussels school on the dynamics of correlations[2,3] is worth mentioning. [1] [2] [3]

Brush S., 1966, Kinetic theory. Vols. 1 and 2. Pergamon Press. Balescu R., 1975, Equilibrium and nonequilibrium statistical mechanics. John Wiley and Sons. Clavin P., 1972, C. R. Acad. Sci. A, 274(13), 1085.

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13.3.2 Local Equilibrium, Dissipation and Transport Coefficients We consider now inhomogeneous states of macroscopic systems. The macroscopic equations of conservation are detailed in Section 15.1. The present section aims to derive these equations and the expression for the transport coefficients from Boltzmann’s equation. Attention is limited to the simplest case of a gas of molecules with no internal degrees of freedom. An outline of the Chapman–Enskog method,[1] used to solve the problem in 1917, is given here in simple terms. The detailed analyses can be found in textbooks.[2,3] Conservation Laws In an inhomogeneous gas, the one-particle distribution function f depends also on space, f (r, p, t), and verifies the Boltzmann equation (13.3.2). Consider a quantity that  can be written in the form of a sum over the molecules, i χi . The local value of the macroscopic quantity χ , function of (r, p, t), is defined as n χ  ≡ d3 pχ f , 3 where n(r, t) ≡ d pf (r, p, t). If the quantity χi = χ (pi ) is invariant in a collision, χ1 + χ2 = χ1 + χ2 , it is not difficult to show using interchanges of integral variables[2] that d3 pχ B( f ) = 0. Multiplying (13.3.2) by χ and integrating over p yields the same conservation form as in (15.1.2),% ∂ (13.3.15) (n χ ) = −∇.Jχ , Jχ ≡ n vχ  , v ≡ p/m, ∂t where the r, p and t are independent variables. If χ is a vector, the flux Jχ is a tensor. Equations for the conservation of mass, momentum and energy are then obtained using χ = v, n(r, t) v ≡ m,  3χ = p/m and χ = p.p/(2m). Introducing the flow velocity u(r, t) ≡ d p(p/m)f , ρ(r, t) ≡ nm, Equation (13.3.15) for χ = m leads to the continuity equation, ∂ρ/∂t = −∇.(ρu).

(13.3.16)

Introducing the velocity of the particle relative to the centre of mass (v − u) and the tensor (v − u)(v − u) = vv − uu, Equation (13.3.15) for χ = v leads to the momentum equation and to the expression for the pressure tensor, ρDu/Dt = −∇.%,

where

% = nm(v − u)(v − u).

(13.3.17)

Introducing  χ =  p.p/(2m) into (13.3.15) leads to an equation for the internal energy 1 2 e = 2 m |v − u| with the expression for the heat flux Jq ,   1 n De/Dt = −∇.Jq − % : (∇u)(s) , where Jq = nm (v − u)|v − u|2 . (13.3.18) 2 Equations (13.3.16)–(13.3.18) have the same form as those obtained in Chapter 15 for the conservation of the macroscopic quantities, mass, momentum and energy (see (15.1.3)– (15.1.5), (15.1.14) and (15.1.37)), but in addition they give a microscopic expression for the pressure tensor and the heat flux. [1] [2] [3]

Chapman S., Cowling T., 1939, The mathematical theory of non-uniform gases. Cambridge University Press. Huang K., 1987, Statistical mechanics. New York: Wiley, 2nd ed. Ferzigzer J., Kaper H., 1972, Mathematical theory of transport processes in gases. North-Holland.

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Local equilibrium approximation. Chapman–Enskog method (1917) When the centre of mass of the thermodynamic system moves with a constant velocity u the Maxwell–Boltzmann distribution (13.2.1) takes the form 

f  d3 p

(0)

(p) =

ne

−(p−mu)2 /(2mkB T)



(2π mkB T)3/2

p (0) f = n u, m

 d3 pf (0) = n,

,  d3 p

3 (p − mu)2 (0) f = nkB T. 2m 2

(13.3.19) (13.3.20)

Consider the local Maxwell–Boltzmann distribution, obtained by introducing the local values of the density n(r, t), the flow velocity u(r, t) and the temperature T(r, t), 

 n=

3

d pf ,

nu =

p d p f, m 3

3 nkB T = 2

 d3 p

(p − mu)2 f, 2m

(13.3.21)

into (13.3.19). From now on this distribution is denoted f (0) (p, r, t). Thanks to the conservation of momentum and energy during a binary collision, p1 + 2 p2 = p1 + p2 , p21 + p22 = p2 1 + p2 , the local Maxwell–Boltzmann distribution satisfies the equation B( f (0) ) = 0, as can be checked by using the relation (p/m − u)2 = p2 /m2 − 2u.p/m + u2 . The validity of the macroscopic equations in fluid mechanics, discussed in Chapter 15, rests on the so-called local equilibrium approximation. The basic assumption is that the one-particle distribution function f (p, r, t) quickly approaches the local Maxwell– Boltzmann distribution f (0) (p, r, t) after a time lapse of the order of the mean time between two successive collisions τcoll , defined in (1.2.3). The fluid particles (centre of mass located at r) are considered as macroscopic systems in quasi-thermodynamic equilibrium. This is meaningful if the macroscopic length scale , defined by the gradient of the local quantities n(r, t), u(r, t) and T(r, t), is larger than the mean free path , / 1. The local quantities then evolve on a time scale τevol longer than the relaxation time towards internal equilibrium, τevol  τcoll , because the rate of transfer of mass, momentum and energy between fluid particles is smaller than the local relaxation rate to internal equilibrium. In short the fluid particles in a flow in local equilibrium are macroscopic systems in internal equilibrium but not in mutual equilibrium. This is possible because small systems relax faster to full equilibrium than large ones. Considering systems in local equilibrium, the method of Chapman–Enskog consists in solving the two time-scale problem by a perturbation method using  ≡ / 1 as a small parameter. The expression for τcoll /τevol in terms of  is part of the solution. The basic assumption is that the solution to Boltzmann’s equation differs slightly from the local Maxwell–Boltzmann distribution (13.3.19) by an order , f = f (0) + h ,

where

h = φ(p, x, t)f (0) ,

and where x ≡ r/ and t ≡ t/τevol

 ≡ / 1,

φ(p, x, t) = O(1), (13.3.22) are reduced variables of order unity.

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In the linear approximation the Boltzmann equation (13.3.2) takes the form ∂f p + .∇f ≈ L B (h), ∂t m

(13.3.23)

where L B (h) denotes the linearised collision operator and where the relations B( f (0) ) = 0 and B( f ) = B( f (0) + h) ≈ B( f (0) ) + L B (h) = L B (h) have been used. Part of the  3 second term of the linear collision operator (13.3.5) takes the form − d p2 G12 f2(0) h.  2  2 Introducing the total cross section σtot = d σ , where σtot = d σ , the kernel G12 takes (0) the form G12 ≈ |p1 − p2 |σtot /m. Integration of |p1 − p2 |f2 over p2 yields a result of √ the order of nv, where v is the meanvelocity of themicroscopic particles, v = 3kB T/m,  3 so that the order of magnitude of d p2 G12 f2(0) is the collision frequency 1/τcoll ≈ nvσtot . Replacing L B (h) by −h/τcoll in (13.3.23) yields a simplified version of Boltzmann’s equation, called the BGK equation,[1] that provides useful insights into the problem with the right orders of magnitude,  ∂f p f − f (0) =− φf (0) , + .∇f ≈ − ∂t m τcoll τcoll

(13.3.24)

where (13.3.22) has been used. This equation yields the expression for h in the limit  → 0,   p ∂f (0) + .∇f (0) . h ≈ −τcoll (13.3.25) ∂t m The small value h = 0 indicates that the local Maxwell–Boltzmann distribution (13.3.19), f (0) , is not an exact solution of Boltzmann’s equation.  Introducing the approximation f ≈ f (0) , n a ≈ d3 pa(p, r, t)f (0) , into the expression (13.3.18) for the heat flux Jq yields zero because of the isotropy of f (0) . In the same way the pressure tensor % in (13.3.17) reduces to the pressure of an ideal gas, 2 ne, ρ ≡ nm, (13.3.26) 3 where I is the unit tensor and e denotes here the internal energy per particle. In this approximation the gas flow satisfies Euler’s equations, written using the particulate derivative Jq = 0,

% = pI,

p = nkB T =

Du De D(1/n) D ∂ 1 Dn = −∇.u, ρ = −∇p, = −p , ≡ + u.∇. (13.3.27) n Dt Dt Dt Dt Dt ∂t These equations, derived by Euler in 1752 from macroscopic considerations, are the particular form of the more general equations (15.1.33)–(15.1.34) when the diffusive fluxes are neglected (nondissipative approximation). It is the small term h in (13.3.22) that controls the diffusive fluxes Jq and π , % = pI + π . Therefore the divergence of the diffusive fluxes, −∇.Jq and −∇.π , introduces a small correction to the energy and momentum equations, ρ [1]

Du = −∇p − ∇.π , Dt

n

De = −p∇.u − ∇.Jq − π : (∇u)(s) . Dt

Bhatnagar P., et al., 1954, Phys. Rev., 94(3), 511–525.

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(13.3.28)

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The energy equation is extended to a reactive mixture in (15.1.37) where slightly different notations are used, the energies e being defined per unit of mass in Chapter 15. Introducing the expression (13.3.25) for h into (13.3.17)–(13.3.18) yields the viscous tensor in the form (15.1.15), leading to Navier–Stokes equations (15.1.17) and Fourier’s law (15.1.25), Jq = −λ∇T. The Chapman–Enskog analysis provides the expressions for heat conductivity λ and for the viscous coefficients η and ξ , after tedious algebra.[2,3] Elementary Kinetic Theory of Gases Fairly good expressions for the diffusion coefficients, DT ≡ (2/5)λ/(nkB ) or Dμ ≡ μ/ρ, can be obtained more simply from dimensional analysis of (13.3.17)–(13.3.18) using h = O(f (0) ). According to (13.3.18), the order of magnitude of Jq is nmv3 . Using v = √ 3kB T/m,  ≡ /, this yields Jq ≈ vnkB T/, to be compared with λ∇T ≈ λT/, √ kB T/m |Jq | = O(nmv3 ) ⇒ DT ∝ v ≈ , (13.3.29) nσtot where the expressions of the collision frequency 1/τcoll ≈ nσtot v and of the mean free path  ≈ vτcoll ≈ 1/(nσtot ) have been used. This shows that nD √ T is independent of density and √ varies with the mass and the temperature as 1/ m and T, respectively. This was known by Maxwell (1867). Similar expressions are obtained for the viscous diffusion coefficients. The equation for a conserved scalar Y, nDY/Dt = −∇JY with Fick’s law JY = −nD∇.Y takes, in the absence of flow, u = 0 and for nD = cst., the form of a diffusion equation ∂Y/∂t = DY. This equation is the archetype of irreversible processes. In a box with no flux across the walls the solution relaxes towards a uniform distribution. The irreversible relaxation is easily obtained for D = cst. and Fourier’s decomposition, 2 Y(r, t) = Y˜ k (t)ek.r , dY˜ k /dt = −Dk2 Y˜ k , Y˜ k (t) = Y˜ k (0)e−Dk t . The same expression as (13.3.29) can be obtained for the diffusion coefficient D using the elementary kinetic theory of gases. Consider a quantity Y(x) in one-dimensional geometry, the overall flux of Y crossing the plane orthogonal to the x-axis at x is the difference between the microscopic fluxes in the two opposed directions. Thanks to isotropy, the flux of microscopic particles in any direction is proportional to nv. Considering that the transported quantity is determined by the last collision before crossing the plane, JY ∝ [nvY(x−)−nvY(x+)], that is, to first order, JY ∝ −nv∂Y/∂x, yielding D ≈ v, in agreement with (13.3.29). Therefore, using 1/τcoll ≈ v/, D ≈ 2 /τcoll , the relaxation rate is Dk2 = O( 2 /τcoll ), where  ≡ /. Dissipation Process In a system in local equilibrium, the field of the so-called thermodynamic entropy, s0 (r, t), is defined by the thermodynamic expression (13.2.7) of the entropy of an ideal gas in full equilibrium, into which the local values T(r, t) and n(r, t), given in (13.3.21), are introduced. In a way similar to the full equilibrium state, the field s0 (r, t) is recovered [2] [3]

Chapman S., Cowling T., 1939, The mathematical theory of non-uniform gases. Cambridge University Press. Huang K., 1987, Statistical mechanics. New York: Wiley, 2nd ed.

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when the local Maxwell–Boltzmann distribution (13.3.19) f (0) (p, r, t) is introduced into (13.3.14). According to the thermodynamic relation (13.1.6), the field s0 (r, t) verifies the thermodynamic relation T Ds(0) /Dt = De/Dt + p D(1/n)/Dt. Using the continuity equation, the first relation in (13.3.27), and the Equations (13.3.28) for the conservation of momentum and energy leads to the local equation for the balance of the thermodynamic entropy. The full calculation is performed in Section 15.1.7. Consider the purely thermal problem, u = 0, ∇T = 0, for simplicity,   Jq Ds(0) 1 Ds(0)     . (13.3.30) = −∇.Jq ⇔ n = −∇.Js + w˙ s , Js = , w˙ s = Jq .∇ Tn Dt Dt T T This local equation for s0 (r, t) shows that the entropy flux is the heat flux divided by the temperature, as expected from thermodynamics, and also that, thanks to Fourier’s law, Jq = −λ∇T, λ > 0, the production of entropy is positive, w˙ s = λ|∇T|2 /T 2 > 0, in agreement with the second law of thermodynamics. This illustrates the so-called dissipation process: the diffusive fluxes are responsible for the increase of entropy and consequently for the irreversible approach towards the full equilibrium of isolated systems (no flux at the walls). The order of magnitude of the heat flux in (13.3.29), |Jq | = O(nmv3 ), then shows that the dissipation rate is of order  2 /τcoll , in agreement with the diffusive relaxation rate Dk2 evaluated just above from the elementary kinetic theory of gas, . / |Jq | w˙ s Ds(0) w˙ s =O = O( 2 /τcoll ). (13.3.31) = O(v/) ⇒ kB−1 ⇒ n nT nkB Dt For a sufficiently fast flow the damping rate by diffusive transport, 1/τvis ≈  2 /τcoll , is smaller than the evolution rate by convection, 1/τconv ≈ (u.∇A)/A = O(|u|/), evaluated by assuming that the relative variation on the distance  is of order unity, A/A = O(1), diffusion: 1/τvis =  2 /τcoll ,

convection: 1/τconv = M/τcoll ,

(13.3.32)

where  ≡ /, 1/τcoll ≡ v/ and M ≡ |u|/a is the Mach number, the speed of sound a being of the order of the mean velocity v, a/v = O(1). The evolution rate by convection 1/τconv = (M/τcoll )(A/A) becomes as small as the viscous rate rates 1/τvis either if the Mach number is small M = O() and A/A = O(1), as in flames (Zeldovich 1938), or if the flow is transonic and the variation small, M − 1 = O(), A/A = O(M − 1), as is the case for the weak shock structure studied in Section 4.2.2. Boundary layers (Prandtl 1904) are other examples for which diffusion and convection are balanced (in the transverse direction). To summarise, the evolution of a macroscopic system in local equilibrium is slow at the collision time scale, with a ratio τcoll /τevol of order (/)M. Thermodynamic Entropy versus Boltzmann’s Entropy In local equilibrium the difference between the thermodynamic entropy s(0) (r, t) and Boltzmann’s entropy sB (r, t) is of second order. This corresponds to the fact that, amongst

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the distributions satisfying the constraints (13.3.21), the Maxwell–Boltzmann distribution maximises Boltzmann’s entropy, as shown now. Using f (r, p, t) = f (0) + h with, according to (13.3.22), h = O(), the expansion of f ln f in powers of  ≡ / 1 yields f ln f = f (0) ln f (0) + [φf (0) ln f (0) + φf (0) ] +  2 φ 2 f (0) + · · · .



(13.3.33)

this expansion into (13.3.13), the constraints (13.3.21), d3 ph = 0 and Introducing 3 d p[(p − mu)2 /2m] h = 0, imply that the first-order term coming from the bracket in (13.3.33) vanishes. The difference sB − s(0) then takes the form  (0) 2 n[sB (r, t) − s (r, t)] = − kB d3 pφ 2 f (0) + · · · < 0. (13.3.34) Moreover, the quantities n, φ and f (0) vary on the long scales  and τevol (through the reduced variables of order unity x ≡ r/ and t ≡ t/τevol ). Therefore the particulate derivative (D/Dt) d3 pφ 2 f(0) is of order 1/τevol , so that the difference n(Ds(0) /Dt) − n(DsB /Dt) =  2 kB (D/Dt) d3 pφ 2 f (0) = O( 2 kB /τevol ) is smaller than n(Ds(0) /Dt) = O( 2 kB /τcoll ) by a small factor τcoll /τevol which, according to (13.3.32), is of order M. To conclude, the production of entropy w ˙ s of a flow in local equilibrium, computed using the thermodynamic entropy, as in (13.3.30), is valid in the limit  → 0, since, according to (13.3.34), the same result is obtained with the statistical Boltzmann entropy.

13.3.3 Brownian Motion. The Equations of Einstein and Langevin Brownian motion is the random displacement of a particle under the effect of collisions with the microscopic particles of the surrounding fluid. It was observed very early during the eighteenth century using charcoal particles and more systematically by the botanist Brown (1827) examining grains of pollen in water under a microscope. A sound theoretical analysis of this stochastic process was first given by Einstein in 1905 (Annus mirabilis papers). The experiments of Perrin (1908) on Brownian motion confirmed definitively the existence of atoms and molecules. A considerable literature is devoted to this fundamental topic, leading to the so-called fluctuation–dissipation theorem. Basic considerations that are useful in this book are recalled here; more details can be found in textbooks[1,2,3,4,5,6] and a historical presentation of the development of the scientific fields.[7] Random Walk and Diffusion. Einstein (1905), Smoluchowski (1906) Consider for simplicity a discrete-time random walk on a one-dimensional lattice of equal intervals (l). At each time step τ , the walker moves a distance l either to the right or [1] [2] [3] [4] [5] [6] [7]

Reif F., 1965, Fundamentals of statistical and thermal physics. McGraw-Hill. Pathria P., 1972, Statistical mechanics. Pergamon Press. McQuarrie D., 1973, Statistical mechanics. Harper and Row. Forster D., 1975, Hydrodynamic fluctuations, broken symmetry, and correlation functions. Benjamin Cummings. de Groot S., Mazur P., 1984, Non-equilibrium thermodynamics. Dover. Lifshitz E., Pitaevskii L., 1999, Physical kinetics. Butterworth Heinemann. M¨uller I., 2007, A history of thermodynamics. Springer.

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to the left with an equal probability 1/2. The probability w(x, t) of finding the walker at position x at time t satisfies the difference equation 1 1 w(x − l, t − τ ) + w(x + l, t − τ ). (13.3.35) 2 2 Assuming l x and τ t, the leading order of the expansion of (13.3.35) in a Taylor series gives the diffusion equation w(x, t) =

∂ 2 w(x, t) ∂w(x, t) (l)2 =D > 0. (13.3.36) , where D = ∂t 2τ ∂x2 Notice that this expression for the diffusion coefficient is in agreement with the elementary kinetic theory of gases D ≈ lv. The Fourier transform  ∞  ∞ 1 ikx w(k, ˜ t) = w(x, t)e dx, w(x, t) = w(k, ˜ t)e−ikx dk, (13.3.37) 2π −∞ −∞ recasts Equation (13.3.36) into ∂ w(k, ˜ t) 2 = −Dk2 w(k, ˜ t) with solution w(k, ˜ t) = w(k, ˜ 0)e−Dk t . (13.3.38) ∂t Assuming  ∞that the walker starts from x = 0 at t = 0, w(x, 0) = δ(x), where δ(x) = ˜ 0) = 1, (1/2π ) −∞ e−ikx dk is the Dirac function, one gets, according to (13.3.37), w(k, w(k, ˜ t) = e−Dk t , so that 2

w(x, t) =

1 2π



∞ −∞

e−Dk t e−ikx dk = √ 2

1 4π Dt

e−x

2 /4Dt

,

(13.3.39)

obtained by using the√relation Dtk2 + ikx = Dt [k + i(x/2Dt)]2 + x2 /4Dt and the variable √ 2 ∞+ia of integration K = Dt [k + i(x/2Dt)], −∞+ia e−K dK = π . The solution in threedimensional geometry is similar to (13.3.39). In this case the scalar√|x| is replaced by the distance from the origin |r|, r = (x, y, z), and the coefficient 1/ 4π Dt is replaced by 1/(4π Dt)3/2 , as is easily checked by the normalisation condition w(|r|, t)d3 r = 1, with w(|r|, t = 0) = δ(r), where δ(r) = δ(x)δ(y)δ(z). The Gaussian distribution in (13.3.39) is the Green function of the diffusion equation ∂Y/∂t = D∇ 2 Y; spatial convolution with the initial condition yields the solution to the diffusion equation in an infinite medium for any initial condition. For a time-dependent source considered in (2.4.24), the solution is given by time convolution. Equation (13.3.39) illustrates the irreversible nature of the solution to the diffusion equation in an unbounded medium,    |r|2 ≡ |r|2 w(|r|, t)d3 r = 6Dt, (13.3.40) D > 0: lim w = 0, t→∞

showing that diffusion is a dissipative mechanism. The second equation (13.3.40) describes the spreading of a spot of particles by self-diffusion into the surrounding fluid. An expression for the diffusion coefficient is obtained by considering the velocity v of the test particle as the stochastic variable instead of the position r

17:11:43 .015

13.3 Physical Kinetics

 t = 0:

r = 0;

r(t) =

603 t

v(t )dt .

(13.3.41)

0

Considering an ensemble of test particles and the ensemble average a(t) for any function a(t) relative to a test particle, and proceeding in the same way as in Section 3.1.2 for the turbulent diffusion, one gets the same behaviour in the long time limit as in (13.3.40) with an expression for the diffusion coefficient     1 ∞  lim |r|2 = 6Dt, where D= v(t ).v(0) dt . (13.3.42) t→∞ 3 0 This result is  obtained by assuming  ∞ of the autocorrelation function  a stationary property     v(t + t ).v(t) = v(t ).v(0) = |v|2 gv (t ) and 0 < 0 gv (t )dt < ∞. Assuming equipartition of energy |v|2 = 3kB T/m (see (13.2.1)), one gets  kB T ∞ D= gv (t )dt , (13.3.43) m 0 where m is the mass of the test particle. Similar results involving autocorrelation functions of fluxes, the so-called Green–Kubo relations (1950), can be obtained for all the transport coefficients. The time-correlation function formalism, based on the Fourier transform of gv (t) (or of its analogues for other fluxes), is useful in the interpretation of laser light scattering and inelastic neutron scattering that developed in the second half of twentieth century.[1] The Langevin Equation (1908). Fluctuation–Dissipation Theorem To describe the Brownian motion of a test particle of radius R and mass M, both larger than that of the microscopic particles composing the surrounding fluid, Langevin took account of the inertia arising from the laminar viscous drag exerted on the Brownian particle by the fluid. Langevin’s equation for the velocity v(t) of the Brownian particle takes the form v dv =− + f(t), dt τvis

(13.3.44)

where τvis is the viscous relaxation time due to the Stokes force (15.2.16), 1/τvis = 6π μR/m, and f is the stochastic acceleration ( force/m) due to the collisions with the microscopic particles. The ensemble average of f(t) is zero, so that the evolution of the mean velocity of the Brownian particle corresponding to a deterministic initial condition, t = 0: v = v0 =cst. (nonfluctuating quantity), is f(t) = 0

⇒

v(t) = v0 e−t/τvis .

(13.3.45)

A first result is obtained by considering the mean of r(t).dv/dt, assuming two conditions: first that the position of the Brownian particle r(t) and the stochastic force f(t) are not correlated r(t).f(t) = 0, and second that the Brownian  particle has attained thermal equilibrium equilibrium satisfying equipartition |v(t)|2 = 3kB T/m. Then, after simple [1]

McQuarrie D., 1973, Statistical mechanics. Harper and Row.

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604

Statistical Physics

algebra shown below, the stochastic equation (13.3.44) yields the famous Einstein relation (1905), verified by the experiments of Perrin (1908),   τvis kB T τvis kB T t, ⇒ D = . (13.3.46) lim |r|2 = 6 t→∞ m m r.dv/dt = (d/dt) r.dr/dt− This result   is obtained by using the relations v(t) = dr(t)/dt,   (dr/dt)2 , so that, according to (13.3.44), the quantity Y(t) ≡ 12 (d/dt) |r|2 − 3τvis kB T/m, satisfies the Equation dY/dt = −Y/τvis , so that (13.3.46) is obtained for t  τvis . In his 1905 paper, Einstein found this result in a different way, by balancing the downward flux due to Stokes’s law for a spherical particle under gravity with the upward diffusion flux associated with the density gradient given by the barometric formula.[1] Another important result, but concerning the fluctuating force mf(t), is also obtained from (13.3.44). Assuming that after a sufficiently large difference of time |t1 −t2 | the values f(t1 ) and f(t2 ) are not correlated, meaning that the autocorrelation function f(t1 ).f(t2 ) ≡ Kf (|t1 − t2 |) is nonzero only when |t1 − t2 | is not too large, one introduces the correlation time τf of the fluctuating force f(t), τ  τf : Kf (τ ) = 0. Due to the difference of mass between the Brownian particle and the microscopic particles in the fluid, the vector f(t) fluctuates on a time scale shorter than the evolution time of mean values of functions of v(t), τf τvis . The result concerning f(t) is then obtained from the formal solution to (13.3.44),  t  dt et /τvis f(t ), (13.3.47) v(t) = v0 e−t/τvis + e−t/τvis 



0

by computing the mean square velocity = v(t).v(t) in the long time limit t  τf , using f(t) = 0 and t  τf : Kf (t) = 0,  ∞   |v(t)|2 ≈ v20 e−2t/τvis + (1 − e−2t/τvis )τvis Kf (t )dt . (13.3.48) t  τf : |v(t)|2

0

t This is obtained from the relation t  τf : 0 dt1 0 dt2 e(t1 +t2 )/τvis Kf (t2 − t1 ) t ∞  ≈ 2 0 Kf (t )dt 0 dt e2t /τvis . By requiring that, in the long time limit t  τvis  τf , the Brownian particle is in thermal equilibrium, t  τvis : |v(t)|2 = 3kB T/m, Equation (13.3.48) yields the fluctuation–dissipation theorem  ∞ mkB T (kB T)2 , (13.3.49) m2 Kf (t )dt = 3 =3 τvis D 0 t

result[2]

relating the integral of the autocorrelation function of the fluctuating force to the transport coefficient. According to the difference of time scale, τf τvis , for t  τf one has v(0).f(t) = 0, so that the Langevin equation (13.3.44) implies that for t  τf : d v(0).v(t) /dt = − v(0).v(t) /τvis . The autocorrelation of flux relaxes exponentially with time on the macroscopic time scale and, neglecting small terms of order τf /τvis , yields [1] [2]

Einstein A., 1905, Ann. Phys. (Leipzig), 17, 549–560. Pathria P., 1972, Statistical mechanics. Pergamon Press.

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13.3 Physical Kinetics

605

kB T −t/τvis e , (13.3.50) m in agreement with (13.3.42)and (13.3.46). When considering the diffusion equation in the Fourier transform associated with (13.3.38), 1/τvis = Dk2 , the autocorrelation ˜ function of the   Fourier transform Y(k, t) of the concentration Y(r, t) takes the form ˜Y(k, t)Y(k, ˜ 0) ∝ e−Dk2 t , according to (13.3.50). When extended to the macroscopic equations in fluid mechanics recalled in Chapter 15, this relation is known as Onsager’s mean regression hypothesis and is at the root of the calculation of the laser or neutron scattering spectrum from the field equations. In experiments, when ensemble averaging is identified to time averaging, the spectral analysis of the fluctuations[3] is based on the Wiener–Kintchin theorem[4] relating the spectral density to the correlation function through Fourier transform. v(t).v(0) = 3

[3] [4]

McQuarrie D., 1973, Statistical mechanics. Harper and Row. de Groot S., Mazur P., 1984, Non-equilibrium thermodynamics. Dover.

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14 Chemistry

Nomenclature Dimensional Quantities ci Cp c˜ i eoi E EG G hei hoi J± k± M Mi n p Q S T v V μi σnr τnr

Description Molecular concentration of species i Specific heat at constant pressure Molar concentration of species i Bond energy Internal energy Gamow energy Gibbs free energy Thermal enthalpy Enthalpy of formation Forwards and reverse reaction rates Forwards and reverse rate constants Mass Molar mass of species i Atoms or molecules per unit volume Pressure Heat of reaction Entropy Temperature Impact velocity Volume Chemical potential of species i Collision cross section of nuclear reaction Characteristic time of nuclear reaction

606

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S.I. Units molecule m−3 J K−1 mole−1 mole m−3 J mole−1 J J J J mole−1 J mole−1 molecule s−1 m−3 s−1 kg kg mole−1 m−3 Pa J mole−1 m−3 J K−1 kg−1 K m s−1 m3 J m2 s

14.1 Elementary Combustion Chemistry

607

Nondimensional Quantities and Abbreviations K M N N Yi Xi Z ϑ ξ

Equilibrium constant An arbitrary (conserved) third body in a reaction Number of atoms or molecules Number of moles Mass fraction of species i Molecular (or molar) fraction of species i Atomic number (number of protons or electrons) Stoichiometric coefficient Progress variable

Superscripts, Subscripts and Math Accents ai a+ a− ao , ao ab aliq as asto au , au avap

Chemical species Forwards reaction Reverse reaction Reference state Burnt gas Liquid state Solid state Stoichiometric Unburnt gas Vapour state

14.1 Elementary Combustion Chemistry The main source of the energy used by mankind is combustion. It is the result of a chemical reaction between fuel molecules, such as hydrogen or hydrocarbons, and an oxidiser, usually the oxygen in ambient air. The source of energy lies in the chemical bonds between the individual atoms that constitute molecules: the atoms (oxygen, hydrogen, carbon) in combustion products (water, carbon dioxide) are more strongly bonded by their valence electrons than in the corresponding reactive species.

14.1.1 Orders of Magnitude Combustion provides a simple and efficient way to obtain usefully large quantities of energy from a manageable mass of reactants in conditions that are easy to use. For example an energy source of 10 kW (1 watt ≡ 1 J/s, 1 kW ≡ 103 watt) is provided by a boiler consuming ≈ 2 kg/h of wood and ≈ 13 m3 /h of air, equivalent to roughly 1 g/s of oxygen, in standard conditions. The power released in a rocket engine burning a mixture of liquid oxygen and hydrogen is of the same order as that of a nuclear reactor: several gigawatt

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Chemistry

(1 GW ≡ 109 W), and during the 10 min that such motors typically operate, roughly 200 tons of oxygen are consumed. By comparison, the average power dissipated by an active animal the size of a man, or in other words the quantity of energy that the animal must consume in food to keep alive, is a few hundred watts, corresponding to a food intake of the order of 2000 kcal/day. As already mentioned, the energy released by combustion (heat of reaction) comes from the modifications of chemical bonds. Energy is exchanged between different forms: atomic bonds, vibration and rotation of molecular structures, electronic excitation, thermal agitation and flow velocity. In gases, energy exchange occurs during collisions between molecules, free atoms and radicals. The total energy is always conserved and is distributed between the different degrees of freedom to establish a local equilibrium distribution. It turns out that the frequency of ‘reactive’ collisions is several orders of magnitude lower than that of the elastic collisions that lead to local equilibrium, as explained in Section 13.3.1. In these conditions the reaction rate is much smaller than the elastic collision rate and can be expressed as a function of temperature, which is locally well defined at each instant in time. In order to open or break a chemical bond, it is necessary to transfer energy into the molecular structure. As a result, the kinetic energy of the incident particles is greater than that of those leaving the collision. These decomposition reactions are endothermic and tend to cool the mixture. During a recombination collision, when a chemical bond is created, the chemical energy is released as mechanical energy: the kinetic energy of the particles after the collision is greater than before the collision; the reaction is exothermic and tends to heat the mixture. The chemical energy in the bonds of a molecule is necessarily lower than that of the free atoms that together form the molecule. It is necessary to provide energy to break bonds. By convention it is supposed that the energy of any atom is zero (the internal structure of atoms is neglected). The chemical energy of molecules is thus negative. The stronger the bonds, the lower the chemical energy, or in other words more stable molecules have a lower (more negative) chemical energy. In combustion, energy is released because the combustion products, CO2 and H2 O, are molecules that are more stable than the reactants, or in other words the chemical bonds between their elements are stronger. Combustion is a process that goes towards equilibrium by minimising the energy of chemical bonds. However, the chemical reaction between reactive molecules (fuel and oxidant) can occur (liberating energy) only if the initial temperature is sufficiently high. The simple explanation is that the molecules must have a sufficient reserve of kinetic energy to break open the bonds of the reactive species before the elements can recombine to form more stable products. Chemical bonds result from the sharing of valence electrons; see below. For this reason physicists use the electron-volt (eV) as a measure of binding energy. The absolute value of the energy of these electrons is smaller than but of the same order of magnitude as the energy of ionisation, which is roughly 13.6 eV for hydrogen. The binding energy of diatomic molecules is in the range 4–10 eV or approximately 100–250 kcal/mole. The calorie, introduced by chemists, is the amount of energy needed to heat 1 g of water by

17:11:49 .016

14.1 Elementary Combustion Chemistry

609

1◦ C in normal conditions (15◦ C, 1 bar) and is still frequently used in combustion. The correspondence between different units of energy is 1 cal = 4.18400 J, kB = 1.38066 × 10−23 J/K, R = 1.98719 cal/K, R = 8.31440 J/K,

1 eV = 1.60218 × 10−19 J, 1 eV/kB = 11,604.5 K, 1 eV/particle = 23.06 kcal/mole, N = 6.02205 × 1023 ,

where Boltzmann’s constant kB , Avogadro’s number N and the gas constant R ≡ N kB are part of the perfect gas law (13.2.5), pV = NkB T,

pV = NRT,

in which N is the number of molecules and N ≡ N/N is the number of moles. Using pV/N as a measure of the internal energy of 1 mole of gas gives 0.592 cal/mole at the reference temperature T = 298.15 K (25◦ C) Historically it was Boyle and Mariotte, the precursors of the idea that heat is related to molecular agitation, who in the seventeenth century used the product of pressure and volume of a gas to measure temperature. However, it was only at the beginning of the nineteenth century that Avogadro introduced the hypothesis, confirmed later by statistical thermodynamics, that identical volumes of different gases contain the same number of molecules.[1] This led to the definition of a mole, introducing Avogadro’s number N as the number of molecules in 1 mole. This number was chosen arbitrarily as being the number of molecules in 1 g of hydrogen with no isotopes. In normal conditions (15◦ C and 1 atm) one mole of gas occupies 22.4 L. Combustion involves temperatures of the order of 2000 K for fuel–air mixtures, and 3000 K for fuel–oxygen mixtures, so at atmospheric pressure combustion reactions take place almost exclusively in the gas phase. An evaluation based on an absolute value of 100 kcal/mole for bond energies, as mentioned above, would give temperatures three to five times higher for fuel–oxygen mixtures. However, this estimation neglects equilibrium chemistry and inverse reactions, such as the dissociation of combustion products, which are endothermic. On the other hand, the absolute value of the binding energies of particles (nucleons) in atomic nuclei are of the order of 1 MeV (1 MeV ≡ 106 eV). This explains why nuclear reactions involve temperatures of the order of tens to hundreds of millions Kelvin and generally take place in plasmas.

14.1.2 Elementary Atomic Physics In order to understand origins of heat of reaction, it is necessary to look into the nature of chemical bonds. The first ideas concerning the structure of matter date back to Empedocles and Democritus in the fifth century BCE. The distinction between composite chemicals [1]

Cox P., 1989, The elements, their origin, abundance and distribution. Oxford University Press.

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(molecules) and elements (atoms) was already formulated in the seventeenth century, by Boyle in particular; however, it was firmly established only a century later as a result of the work of Black, Priestley and Lavoisier. Some 30 elements had been identified in 1789 by Lavoisier, and more than 80 were known by the end of the nineteenth century.[1] The investigations of Mendeleev led him to propose his famous periodic classification of the elements in 1869. This table was fully justified in the twentieth century on the basis of the description of atomic structure provided by quantum mechanics, one of the major triumphs of this theory of the microscopic structure of matter.[2] In the following we will recall some of the useful results without going into the details. Mendeleev’s Periodic Classification of the Elements Atoms are composed of a positively charged nucleus surrounded by a cloud of electrons. The diameter of nuclei is of the order of ≈ 10−15 m, whilst that of atoms (the electron cloud) is very much larger, ≈ 10−10 m. The nuclei contain positively charged protons and also neutrons, which are electrically neutral. These two types of particles, called nucleons, have approximately the same mass, mp ≈ 1.7 × 10−24 g. The mass of an electron is very much smaller, me ≈ 9 × 10−28 g. The charge of a proton being equal and opposite to that of an electron, neutrality of the atom imposes an equal number of protons and electrons, called the atomic number, Z. The total number of protons and neutrons, A, A  Z, is an integer that determines the mass of the atom; the mass of the electrons is negligible. When a nucleus with a given number of protons can exist in a stable form with slightly different numbers of neutrons, they are called isotopes. They have quasi-identical chemical properties but different masses The first 18 elements listed by order of increasing atomic number is given in Table 14.1. Mendeleev’s classification is presented in Table 14.2. The elements are arranged horizontally on seven lines (called periods) in order of increasing atomic number. Each of the 18 vertical columns contains a group of elements with similar chemical properties. The main groups are in columns 1, 2 and 13–18, with the alkali metals in column 1, the halogens in column 17 and the noble gases in the last column. The elements in groups 3–12 are called the transition metals. The atomic number increases by unity between consecutive elements in a line, except in period 6 between lanthanum (La, Z = 57) and hafnium (Hf, Z = 72) and in period 7 between actinium (Ac, Z = 89) and rutherfordium (Rf, Z = 104). A group of 14 additional elements, shown in the lower block of Table 14.2, fits into each of these two ‘windows’: these groups have essentially the characteristics of group 3 and are called ‘rare earths’ (or lanthanides) between La and Hf and actinides between Ac and Rf. Periods 6 and 7 thus contain 32 elements. With the exception of the rare earths, nearly all nuclei with an atomic number greater than 83 (Bi) are unstable and thus radioactive. Beyond element 104, rutherfordium, the nuclei have half-lives so short that they can be synthesised only in such small quantities that their chemical properties are still unconfirmed. Their names are still [1] [2]

Cox P., 1989, The elements, their origin, abundance and distribution. Oxford University Press. Landau L., Lifchitz E., 1967, M´ecanique quantique. Mir.

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611

Table 14.1 The first 18 elements listed in order of increasing atomic number. The molar mass is given in grams. Z

Name

Symbol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...

Hydrogen Helium Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon Sodium Magnesium Aluminium Silicon Phosphorus Sulphur Chlorine Argon ...

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar ...

A 1 4 7 9 11 12 14 16 19 20 23 24 27 28 31 32 35 40 ...

Molar mass 1 4 6 9 10 12 14 15 18 20 22 24 26 28 30 32 35 39

008 0026 941 0122 81 011 0067 9994 9984 179 9898 305 9815 086 9738 06 453 948 ...

subject to discussion and they are designated here by their atomic numbers. This periodic structure can be explained by the electronic structure of atoms, as recalled below. Electronic Structure of Atoms The electronic structure of an atom can be described in a first approximation by considering the motion of each electron, carrying a negative charge −e, in an orbit around a fixed positive point charge (spherical symmetry). According to quantum mechanics, the dynamics of each electron is described by a wave function, called an orbital, whose squared modulus represents the probability of finding an electron at a given point in space. The orbitals are solutions to Schr¨odinger’s equation with a negative energy. The number of possible solutions is finite. Each orbital is characterised by its angular momentum and its energy. The possible values of angular momentum are discrete and are represented by an integer l = 0, 1, 2, 3, . . . . The values of energy are also discrete, represented by another integer n  l + 1, called the principal quantum number. For the simplest case, that of a single electron orbiting around a positive point charge (hydrogen), the energy of the orbitals is a function of n only and varies as −1/n2 with a constant proportional to the mass of the electron, me , to the

17:11:49 .016

612

Table 14.2 Periodic table of the elements. The column number denotes groups of elements with similar chemical properties. The row number is the period. Below the main table are the rare earths (or lanthanides) (first row) and the actinides (second row)

17:11:49 .016

1

2

3

4

1 2 3 4 5 6 7

H Li Na K Rb Cs Fr

Be Mg Ca Sr Ba Ra

Sc Y La Ac

Ti Zr Hf Rf

Ce Th

Pr Pa

Nd U

Pm Np

Sm Pu

5

V Nb Ta 105

Eu Am

6

Cr Mo W 106

Gd Cm

7

8

9

Mn Tc Re 107

Fe Ru Os 108

Co Rh Ir 109

Tb Bk

Dy Cf

Ho Es

10

Ni Pd Pt 110

Er Fm

11

Cu Ag Au 111

Tm Md

12

13

14

15

16

17

18

Zn Cd Hg 112

B Al Ga In Tl 113

C Si Ge Sn Pb 114

N P As Sb Bi 115

O S Se Te Po 116

F Cl Br I At 117

He Ne Ar Kr Xe Rn 118

Yb No

Lu Lr

14.1 Elementary Combustion Chemistry

613

Figure 14.1 Qualitative sketch of energy levels in an atom. The absolute spacing of energy levels is not to scale.

square of the electron charge, e2 , and to the square of the charge of the nucleus. The energy depends only weakly on the mass of the nucleus mn through a factor me /mn 1. The presence of more electrons complicates the problem. The energy levels then depend on both n and l. However, the states associated with a large principal quantum number n are similar to the previous case: the outer electron effectively orbits around a point charge equal to the charge of the nucleus minus the total charge of all electrons in the inner layers. This approximation is sufficient to provide a qualitative description of the structure of all electronic states. For historical reasons related to the visual aspect of spectral lines (‘sharp’, ‘principal’, ‘diffuse’, etc.), the orbitals are designated by the letters s, p, d, f, g, . . . associated, respectively, with l = 0, 1, 2, 3, 4, . . . . The orbitals are grouped by energy levels designated by the symbols 1s, 2s, 2p, 3s, 3p, 3d, . . . , where the number is the principal quantum number, n, and the letter is the orbital, 0  l  n − 1. The possible energy levels of an outer electron in the electric field of a nucleus, partially shielded by the inner electrons, are shown schematically in Fig. 14.1.[1] The absolute spacing between levels is only indicative. The spacing between the levels with consecutive principal quantum numbers does not follow the −1/n2 law, as in the hydrogen atom; the potential is not the same. The energy of the states is negative. The 1s state has the lowest energy; in this base state an electron is most strongly bound (and closest to) to the nucleus. According to Schr¨odinger’s equation, the angular momentum can take 2l + 1 different orientations. There are thus 2l + 1 different orbitals for a given pair of quantum numbers, n, l, and the level is said to be 2l + 1 degenerate. In other words for a given n there is one s orbital, three p orbitals, five d orbitals and so on. Electrons are fermions whose spin can take the values ± 12 . Pauli’s exclusion principle imposes that two identical fermions cannot occupy the same quantum state. It follows that each orbital can be occupied by at most two electrons having opposite spins. The 1s, 2s, 3s, . . . levels can thus be occupied by two

[1]

Cox P., 1989, The elements, their origin, abundance and distribution. Oxford University Press.

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electrons. The 2p, 3p, 3p, . . . orbitals can be occupied by six electrons, the 3d, 4d, 5d, . . . by 10 and the 4f, 5f, . . . by 14. Madelung’s Rule In Fig. 14.1 the levels are grouped into seven layers corresponding to the seven periods (rows) of Table 14.2. The maximum number of possible electrons in each layer is shown in parentheses on the right. The first layer has a single level that can hold at most two electrons. The second and third layers containing the levels (2s, 2p) and (3s, 3p) can each hold up to eight electrons, the next two layers (4s, 3d, 4p) and (5s, 4d, 5p) can hold 18, and the last two layers 32. This succession of numbers is exactly that of the number of elements in each row (period) of Mendeleev’s table. The successive levels (and thus successive layers) are filled starting with the lowest levels first. Madelung’s rule (1936) states that levels with smaller values of n + l have lower energy and are filled first. The hydrogen atom has a single electron which occupies the 1s level in the base state. The next atom in Mendeleev’s table is helium, He, Z = 2, with two electrons. The second electron fills the 1s level, completely filling the first layer. Since the first layer is saturated, the extra electron of the next element, Li, Z = 3, goes into the next layer in the 2s level. The electron of the next element, beryllium Be, Z = 4, completes the 2s level. The next six elements from boron, B, Z = 5, to neon, Ne, Z = 10, progressively fill the 2p level which is saturated for neon. The electrons of the next eight elements fill the 3s and 3p levels of the next layer. Sodium, Na, Z = 11, has one electron in the 3s level, and for argon, Ar, Z = 18, the third layer is completely saturated, after which the fourth layer starts to be filled and so on. With the exception of the last group (the noble gases) all elements in the same row in Mendeleev’s table have the same layers saturated with electrons. However, the number of electrons in the outer layer increases by unity going from one column to the next, until saturation for the noble gases. Except for elements heavier than lanthanum, each column groups elements that have the same number of electrons in the outer layer. For instance the first column groups the alkali metals, which have one electron in the outer layer, and the last column groups the noble gases, for which all the layers are saturated, leading to inert chemical properties; see below. It should be remembered that the energy levels change from one element to the next, according to the charge on the nucleus. For example the 1s level is lower (stronger binding) in helium than in hydrogen. The ionisation energy, which the energy required to pull an electron away from the nucleus, is ≈ 25 eV for He and ≈ 14 eV for H. Madelung’s rule for filling energy levels suffers some exceptions in the transition elements, as for example for chromium, Cr, which has five electrons in the 3d level and a single electron in the 4s level, and for copper, Cu, where the 3d level is filled before the 4s which again has only one electron. Similarly, starting with the fourth layer, there are other exceptions where the 3d level is filled before the 4s level. This arises from interactions between electrons, which have been neglected in Fig. 14.1, but does not change the model of successive layers.

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615

Valence Electrons The average distance between an electron and the nucleus increases with the principal quantum number n. The orbital can be grouped in shells, each corresponding to a value of n. The electrons in the outermost shell, are called the valence electrons. The alkali metals (group 1) have one valence electron. The metals in group 2 have two, those in group 13 have three, those in group 14 (carbon) have four, those of group 15 (nitrogen) have five, group 16 have six, the halogens (group 17) have seven and the noble gases (group 18) have eight. The chemical properties of an atom are related to the number of valence electrons, explaining Mendeleev’s grouping by columns.

14.1.3 Chemical Bonds and Heat of Reaction Chemical bonding is the result of a redistribution of electrons between the nuclei of atoms in a molecule. The electrons concerned are almost exclusively valence electrons. The electrons in the inner layers are too strongly bound to their own nucleus to be able to participate in chemical bonds. Covalent Bonds The chemical behaviour of an atom is associated with the notion of valence, which is the number of missing electrons needed to saturate the valence shell. This notion is useful to describe the chemical bonds between the principal elements but is less pertinent for the transition metals. The valence of the hydrogen atom H is unity, since it has one electron in the 1s level, saturated in the presence of two. Concerning the elements in the second period, those generally participating in combustion – carbon, C, nitrogen, N, and oxygen, O – have valences of 4, 3, and 2, respectively, since their electronic levels 1s and 2s are saturated and their 2p level can hold six electrons, but contain respectively two, three and four electrons. In a covalent bond between two elements, each element tends to complete its valence shell by sharing valence electrons. The sharing is done in pairs formed by one electron from each nucleus. The density of the shared electrons is distributed almost uniformly around the two nuclei, such that the dipole moment is weak or quasi-null. For the particular case of noble gases, the valence shell is already saturated; these elements are unable to share electrons and are (nearly) chemically inert. Other types of chemical bonds exist, but we will not discuss them here. We simply mention the existence of ionic bonds, which are similar to covalent bonds, but the electronic density distribution around the nuclei is not uniform, giving rise to a strong polarisation of the molecule. The distinction between covalent and ionic bonds is not sharp; the only true covalent bonds are those between identical molecules, such as H2 or O2 . In other cases the electron density distribution has an asymmetry leading to some polarisation of the molecule. The notion of ionic bond is related to the ionisation capacity of atoms. The elements on the left of Mendeleev’s table have a tendency to donate their valence electron and

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become positively charged. Those on the right of the table tend to form negative ions by incorporating an electron into their valence shell. These two types of elements have a particularly strong attraction to each other and tend to form ionic bonds. A covalent bond is said to be multiple when several pairs of electrons are involved. Except for unimportant exceptions, the number and the character of bonding are related to the valence of a element. For example H can have only one unique bond, whereas an element such a C, which has four valence electrons, can form up to four simple bonds, or three simple and one double, or two double or one triple and one single. When a molecular structure has fewer bonds than those corresponding to its valence, the resulting compound is called a radical. This is the case, for example, for the hydroxyl and methyl radicals, OH and CH3 , which have a global valence of unity, O and C having respectively only one and three simple covalent bonds with H in these radicals. Radicals are strongly reactive and tend either to establish a bond with a hydrogen atom or to replace a hydrogen atom in another molecule to create a new molecule. Molecular Structure The simplest covalent bond is that of the hydrogen molecule H–H. There are two simple bonds in the water molecule H–O–H. The oxygen molecule, O=O, has a double bond and carbon dioxide, O=C=O, has two double bonds. The nitrogen molecule, N≡N, is the result of a triple covalent bond between the two atoms. Simple, C–C, double, C=C, and triple bonds, C≡C, can occur between carbon atoms in the carbon chain of hydrocarbons. Open chains having only simple bonds are called alkanes (Ci H2i+2 , methane i = 1, ethane i = 2, propane i = 3, butane i = 4, pentane i = 5, . . . ). Carbon chains with double bonds are called alkenes (or olefins) (Ci H2i , ethylene, propylene, . . . ) and those with triple bonds are called alkynes (–C≡C–, acetylene, propyne, . . . ); see Fig. 14.2. The best known molecule with a closed ring is benzene. When one of the hydrogen atoms in an alkane is replaced by a hydroxyl radical, OH, the compound is an alcohol (methanol, ethanol, propanol, . . . ). Methylbutane, C5 H10 , is obtained from butane by replacing a hydrogen atom by a methyl group, CH3 , and trimethylpentane (also called iso-octane), well known for its antiknock properties, is obtained when three hydrogen atoms are replaced by three methyl groups. Bond Energies Some values of absolute bond energies, in kilocalories per mole, are given in Table 14.3. In complex molecules, the bond energy is also a function of the environment of the two atoms.[1] The values given in Table 14.3 for polyatomic molecules are typical values. For diatomic molecules it is the energy released (heat of reaction) when the molecule is formed from a collision between two free atoms and a third body, M, as shown in Table 14.4. These recombination reactions are necessarily ternary, or in other words they can take [1]

Law C., 2006, Combustion physics. Cambridge University Press.

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Table 14.3 Some absolute values of bond energies, in kilocalories per mole. Diatomic

Polyatomic

Bond

Energy

Bond

Energy

Bond

Energy

Bond

Energy

H–H H–C H–O H–N O=O

104.2 80.9 102.8 81.2 119.2

C=O C=C C≡N N=O N≡N

257.3 145.8 178.8 150.7 225.9

H–C H–O H–N C–N O–N N–N

103 104 96 71 49 66

O–O C–O C=O C–C C=C C≡C

37 60 127 90 174 231

Source: D.R. Lide ed., CRC handbook of chemistry and physics, 75th ed., CRC Press 2014–2015. H H

C

H H H

H

H Methane

C C H H Ethane

H C

H

H

H

H

H

C

C

C

H H H Propane

H

H

H

H

C

H

H H

H

H

H

C

C

C

C

H

H

H H H Butane

H H C

Ethylene

H

C C

H

H Propylene H

H H C C H Acetylene

H

C C

C

C H

C C

H

H Benzene

Figure 14.2 Structure of several hydrocarbon fuels. These figures do not show the geometry of the molecules, but only their covalent bonds.

place only during a triple collision; the presence of a third body (M in the table) is necessary to satisfy both conservation of energy and conservation of momentum. Triple collisions are uncommon and so diatomic molecules are created more efficiently (more rapidly) by reactions other than those of Table 14.4. For instance the formation of nitrogen monoxide, NO, by Zeldovich’s mechanism, or that of ‘prompt NO’,[1] is the result of elementary reactions different from the ternary collision between N, O and M. The bond energies in Table 14.4 show that the molecules CO, N2 and NO are very stable. However, carbon monoxide, CO, is reactive since it is less stable than carbon dioxide, CO2 formed from CO by addition of an oxygen atom. Similarly the radical OH is less stable than

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Table 14.4 Heats of reaction in kilocalories per mole of product, classified in order of decreasing value. C N N C O H H C

+ + + + + + + +

O N O C O H O H

+ + + + + + + +

M M M M M M M M

       

CO N2 NO C2 O2 H2 OH CH

+ + + + + + + +

M M M M M M M M

+ + + + + + + +

257.3 kcal 225.9 kcal 150.7 kcal 145.8 kcal 119.2 kcal 104.2 kcal 102.8 kcal 80.9 kcal

O H H

+ + +

CO OH OH

+ + +

M M M

→ ← → ← → ←

CO2 H2 Ovap H2 Oliq

+ + +

M M M

+ + +

127.2 kcal 119.2 kcal 129.7 kcal

Source: D.R. Lide ed., CRC handbook of chemistry and physics, 75th ed., CRC Press 2014–2015.

the water molecule, H2 O, formed by adding a hydrogen atom; see the bottom rows in Table 14.4. Notice, as shown in Table 14.3, that the bond energy between C and O is different in CO and CO2 . Similarly the bond energy between O and H is slightly different in OH and H2 O. The bond energy of solid carbon (graphite) is also very strong: 171.3 kcal/mole, meaning that it is necessary to provide this amount of energy to obtain free carbon atoms in the gas phase. By comparison, only 59.6 kcal/mole is required to liberate an oxygen atom from a molecule O2 . Heat of Reaction and Enthalpy of Formation Knowledge of bond energies permits the calculation of the heat liberated in a global reaction scheme transforming reactants into more stable reaction products. For example, the result of the incomplete oxidation of solid carbon by molecular oxygen liberates 26.4 kcal per mole of carbon consumed. The calculation proceeds as follows: +275.3 (formation of CO from C + O) −119.42/2 (1/2 the decomposition of O2 into O + O) −171.3 (bond energy of C in graphite): C(s) + (1/2)O2



CO + 26.4 kcal.

(14.1.1)

The heats of reaction are given here, and in the following, per mole of fuel consumed. Historically the above reaction was used to measure the bond energy of carbon in graphite. The heat of reaction for the oxidation of CO into CO2 , per mole of CO, CO + (1/2)O2



CO2 + 67.6 kcal,

(14.1.2)

is +127.2 (heat of formation of CO2 from O and CO) −119.2/2 (1/2 the decomposition of O2 ). The heat of reaction for the full oxidation of solid carbon into CO2 is obtained by

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Table 14.5 Heats of reaction for some common fuels Methane Acetylene Ethylene Propane

CH4vap C2 H2vap C2 H4vap C3 H8vap

+ + + +

 CO2  2CO2  2CO2  3CO2

2O2 5O 2 2 3O2 5O2

+ + + +

2H2 Ovap H2 Ovap 2H2 Ovap 4H2 Ovap

+ 191.8 kcal + 300.2 kcal + 316.2 kcal + 488.5 kcal

summing (14.1.1) and (14.1.2): C(s) + O2



CO2 + 94 kcal.

(14.1.3)

Similarly, the heat of reaction for the combustion of 1 mole of hydrogen with half a mole of oxygen to produce 1 mole of water vapour is 119.2 + 102.8 − 104.2 − 119.2/2 kcal: H2 + 1/2O2



H2 Ovap + 58.2 kcal.

(14.1.4)

However, the heat of reaction released by the combustion of hydrocarbon fuels cannot be obtained precisely using the bond energies for C–H and C–C in Table 14.3. This is because these energies change significantly according to the position of the bond within the hydrocarbon molecule. Table 14.5 gives the heat of reaction for some common molecules, Similarly, the heat of reaction released by the combustion of a mole of hydrogen to produce a mole of water is not given precisely by the bond energies of H–H and H–O in Table 14.4. To facilitate the calculation of the heat of reaction of any reaction, physico-chemists have introduced the enthalpy of formation, hoi . It is defined as the negative of the heat released when a molecule is formed from its elements in their standard state (1 bar, 25◦ C). The enthalpy of formation is directly related to the bond energy of the elements in the molecule, but the energy zero (origin) is chosen in a particular way: the enthalpy of formation of molecules of pure elements (H2 , O2 , N2 ) is defined to be zero in the standard state. Thus that of single atoms (H, O, N) in the gas phase is strictly positive and related to the bond energy of the corresponding molecule; see Tables 14.3 and 14.6. The enthalpy of liquid water molecules, H2 O, is smaller than that of water vapour, the difference being the latent heat of vaporisation. By definition the enthalpy of formation of solid carbon, graphite, C, is zero and that of gaseous carbon is 171.3 kcal/mole. The enthalpy of formation of some common molecules is given in Table 14.6; more extensive documentation can be found in specialised publications.[1] Any reaction, elementary or global, can be written formally as ϑ1+ A1 + · · · + ϑn+ An



ϑ1− A1 + · · · + ϑn− An + Q,

(14.1.5)

where Q is heat released when ϑ1+ moles of A1 are consumed. The coefficients ϑi± are called the stoichiometric coefficients. For example, the global reaction 2H2 + O2  2H2 O gives ϑH+2 = 2, ϑ0+2 = 1, ϑH+2 O = 0, ϑH−2 = 0, ϑ0−2 = 0, ϑH−2 O = 2. The heat of reaction is [1]

Law C., 2006, Combustion physics. Cambridge University Press.

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Table 14.6 Enthalpy of formation in the gaseous state at To = 298.15 K and p0 = 1 bar, of some common molecules and radicals, in kilocalories per mole. Substance

Formula

hoi (To )

Substance

Formula

hoi (To )

Atomic H Atomic C Atomic N Atomic O Hydroxyl Carbon monoxide Nitrogen monoxide Carbon dioxide Nitrogen dioxide

H C N O OH CO NO CO2 NO2

52.1 171.3 113.0 59.6 9.3 −26.4 21.8 −94.0 7.9

Ozone Water Hydrogen peroxide Methane Acetylene Ethylene Propane Butane Pentane

O3 H2 O H2 O2 CH4 C2 H2 C2 H4 C3 H8 C4 H10 C5 H12

34.1 −57.8 −32.6 −17.8 54.3 12.5 −24.8 −30.0 −35.1

Source: D.R. Lide ed., CRC handbook of chemistry and physics, 75th ed., CRC Press 2014–2015.

obtained from the enthalpies of formation in the reference state (po = 1 bar, To = 298.15 K) by the formula Q=

n  (ϑi+ − ϑi− )hoi (To ).

(14.1.6)

i=1

The relation between enthalpy of formation, hoi , and bond energy, eoi , is such that   (ϑi+ − ϑi− )eoi = (ϑi+ − ϑi− )hoi . (14.1.7) i

i

The heat of reaction is positive (exothermic forwards reaction) when the enthalpies of formation of the reactants are greater than that of the products. As a final remark, we point out that the chemical energy liberated by any combustion reaction is of the order of 100 kcal per mole of oxygen consumed; see Table 14.5. This rule of thumb provides an easy means to estimate the heat of reaction in combustion.

14.2 Chemical Equilibrium In the gaseous state, elementary chemical reactions occur during collisions between atoms or molecules. These reaction are reversible, or in other words they can proceed in both directions, but at a different rate when away from equilibrium. Chemical equilibrium occurs when the forwards and reverse reaction rates are equal. Global reactions, such as (14.1.4) or those describing the combustion of hydrocarbon fuels, represent the global balance of species and energy. They are the outcome of a multitude of elementary reactions produced by inelastic collisions between highly reactive intermediate species, such as atoms and free radicals. These intermediates do not appear in

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621

the global reaction. Only small quantities remain present in the combustion products and are generally considered as pollutants.

14.2.1 Law of Mass Action The laws of thermodynamics make it possible to calculate the composition of an equilibrium state without need to calculate all the details of a complete scheme of elementary chemical reactions. Elementary Reaction Rates We introduce the direct reaction and inverse reaction rates, J + and J − . They are the number of times that a given reaction occurs (in the forwards and reverse directions, respectively) per unit time and unit volume. By convention the forwards reaction, written from left to right as in (14.1.5), is exothermic, and the reverse reaction, proceeding from right to left, is endothermic. Again, by convention, the species on the left, ϑi+ = 0, ϑi− = 0, are called reactants, and those on the right, ϑi− = 0, ϑi+ = 0, are called products. The elementary reaction rate, forwards or reverse, is traditionally written in terms of atomic or molecular concentrations, ci , ci ≡ Ni /V (molar concentrations are c˜ i = ci /N ), and the stoichiometric coefficients, ϑi± . The reaction rate can then be written as = ± (14.2.1) J ± = k± (ci )ϑi , i

where the rate constants k± (T, p) are functions of temperature and pressure obeying a relation imposed by thermodynamics; see below. For a single elementary reaction in a homogeneous mixture, the evolution of the composition is described by a first-order differential equation, dci = (ϑi− − ϑi+ )(J + − J − ). dt

(14.2.2)

For global reaction schemes, which are the result of many elementary steps, the progress rate cannot be written in such a simple form. We will come back to this point in the following paragraphs. Uniqueness of Chemical Equilibrium in Gases Chemical equilibrium is a consequence of the second law of thermodynamics, which states that the entropy of an isolated system is maximal. Note that, for a given reaction, the variation in the number of each species is independent of the species, which leads us to

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introduce the progress variable ξ , δNj δNi = δξ . + = − − ϑi ) (ϑj − ϑj+ )

(ϑi−

Let us first consider the case of a single elementary reaction. Chemical equilibrium can be seen in two different ways: (a) First, the composition ceases to evolve when J + = J − . We introduce the progress variable ξ by the relation dξ/dt ≡ (J + − J − )V, where by convention ξ = 0 in the initial mixture with coi = Nio /V. So in a given volume, δNi = (ϑi− − ϑi+ )δξ

and

Ni = (ϑi− − ϑi+ )ξ + Nio .

(14.2.3)

When the expression for the forwards and reverse reaction rates (14.2.1) is known, the relation J + = J − gives an equation for ξ that determines its value at thermodynamic equilibrium in the given conditions ( p, T). (b) Second, the second law of thermodynamics requires that the entropy of an isolated system equilibrium is maximal. This allows us to show that the equilibrium composition in an ideal gas is unique. Effectively, again according to the second principle, the second derivative of entropy at constant temperature and volume in an arbitrary state (equilibrium or not), Tδ 2 S = −

n 

δμi δNi

i=1

(where μi is the chemical potential of species i), is quadratic and negative for all compositions since, for a perfect gas, δμi = kB TδNi /Ni

⇒

δ 2 S = −kB

n 

(δNi )2 /Ni ≤ 0.

i=1

The function S(N1 . . . Ni . . . Nn ) of a perfect gas is thus everywhere convex and the maximum (chemical equilibrium) is thus unique. Law of Mass Action According to the second law of thermodynamics, the entropy of an isolated system is maximal in the equilibrium state. Correspondingly, when the system is at a given temperature and pressure (imposed by a reservoir), the free energy G(T, p, . . . Ni , . . .) is minimum at equilibrium where G ≡ E − TS + pV =

n 

μi Ni , and dG = −SdT + Vdp +

i=1

n  i=1

17:11:49 .016

μi dNi .

14.2 Chemical Equilibrium

623

A small variation, at fixed temperature and pressure, around the equilibrium state gives n 

μi δNi = δξ

i=1

n 

μi (ϑi− − ϑi+ ) = 0.

i=1

The second law of thermodynamics thus imposes that the total chemical affinity of the reaction is zero at equilibrium: n 

μi (ϑi− − ϑi+ ) = 0.

(14.2.4)

i=1

The expression for the chemical potential in an ideal gas mixture is given by statistical thermodynamics, extending (13.2.5) to ideal mixtures (p → pi ), μi (p, T, Xi ) = kB T ln(Xi p/po ) + μoi (T),

(14.2.5)

where pi = Xi p is the partial pressure and Xi the molecular (or molar) fraction, Xi ≡

ci Ni = , N n

N≡

n 

Ni ,

n≡

i=1

N ; V

po is the reference pressure at which the chemical potential, μoi (T), is defined. An expression for the latter can be obtained from statistical thermodynamics. Using (14.2.4) and (14.2.5) we obtain a relation between the mole fractions of the species in the equilibrium composition, also called the law of mass action,  (ϑi− −ϑi+ ) = (ϑ − −ϑ + ) po i i i Xi = K(T) , p (14.2.6) i where

−

K(T) ≡ e

 i

(ϑi− −ϑi+ )μoi (T)/(kB T)

and where K(T) is called the equilibrium constant. The relation linking μoi and hoi in the  heat of reaction, Q = (ϑi+ − ϑi− )hoi (To ) (see (14.1.6)), is given by the thermodynamic relations (13.1.6)–(13.1.7), μoi (T) = hoi (T) − Tsoi (T), dμoi = −soi dT (at constant pressure), μoi (T) = hoi (T) − T and Equation (14.2.6) yields

 d ln K =− dT

i

dμoi dT

(14.2.7)

(ϑi− − ϑi+ )hoi (T) kB T 2

.

(14.2.8)

This relation simplifies when the temperature dependence of hoi (T) can be neglected, hoi ≈ μoi , which is the case when there is no internal degree of freedom or when they are frozen. This gives a simple relation between the equilibrium constant and the heat of reaction, K ≈ eQ/kB T ,

17:11:49 .016

(14.2.9)

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Chemistry

or K ≈ eQ/RT according to whether Q is expressed per molecule or per mole. The law of mass action applies both to elementary reactions and to global reactions. Using (14.2.3) it also gives an equation for the progress towards chemical equilibrium. The solution gives the equilibrium composition. For elementary reactions, the compatibility of Equation (14.2.6) with the equilibrium equation J + = J − gives a relation between the forwards and reverse rate constants, k+ and k− ,  − (ϑi− −ϑi+ )  = (ϑ − −ϑ + ) (ϑi −ϑi+ ) = (ϑ − −ϑ + ) p k+ o i i i i i = ci =ni Xi = K(T) . (14.2.10) k− kB T i

i

Arrhenius’ Law In the approximation where the enthalpies of formation are independent of temperature, μoi = hoi , the ratio of the rate constants is simply +

−

 + (ϑi −ϑi− )hoi /kB T

k /k ∝ e

i

= eQ/kB T ,

where the prefactor, specifying the algebraic dependence on temperature, has been omitted for simplicity. This implies that the difference between the activation energies of the forwards and reverse reactions is equal to the heat of reaction, k± ∝ exp(−E± /kB T),

with E− = Q + E+ .

(14.2.11)

Using relations (14.1.6) and (14.1.7), along with the above ratio for the reaction rates, it is tempting to express the rate constants in terms of the bond energies as follows,   k± ∝ exp ϑi± eoi /kB T , i

This leads to the interpretation of the activation energies E+ ≥ 0 and E− ≥ 0 as the sum of the absolute values of the bond energies that must be broken before the new molecule can be constructed, as shown schematically in Fig. 14.3. This is not always exact, but it is true for the recombination reactions of Table 14.4. The activation energies of the forwards reactions are effectively zero corresponding to zero binding energies of the reactants (free atoms). The reverse reactions (decomposition) have an activation energy equal to the absolute value of the binding energy of the diatomic molecules (heat of reaction of the direct reaction). The activation energy of an exothermic elementary reaction is in general nonzero, E+ = 0, but generally lower than the binding energies of reactants. It corresponds to the existence of what chemists call ‘activated complexes’, or intermediate states that the reactive species must reach during a collision before recombining as products. The physics of these collisions is not as well understood as that of chemical bonds. In practice, expressions for the reaction constants are sought in the form k = bT ν exp(−E/kB T),

17:11:49 .016

(14.2.12)

14.2 Chemical Equilibrium

625

Energy

Progress Reactants

Products

Figure 14.3 Schematic representation of the origin of the Arrhenius law for a reaction R → P + Q.

where the values of b, ν and E and are obtained by fitting the above expression to experimental curves of the evolution of reaction rate with temperature. Of course these fits are made in such a way that the thermodynamic relation (14.2.10) is verified (forwards rate = reverse rate at equilibrium). To retrieve the chemical energy released by the reaction, Q = E− − E+ , it is necessary have a sufficient reserve of energy, E+ , as shown in Fig. 14.3. In gases the chemical reaction proceeds only via the inelastic collisions that have enough kinetic energy to cross the E+ barrier. The reaction rate is proportional to their number per unit time and volume (the collision frequency). The enumeration of these collisions from the velocity distribution leads to the Maxwell–Boltzmann exponential factor of the Arrhenius law. As a consequence, when the activation energy is high, chemical equilibrium will be reached in a short time only when the temperature is sufficiently high. However, at low temperature, the composition remains frozen far from chemical equilibrium.

14.2.2 Molecular Dissociation To become familiar with these physico-chemical concepts, we start with the simplest case, the dissociation of a diatomic molecule A2 and its inverse, the recombination reaction (see Table 14.4), A + A + M



A2 + M + Q kcal/mole of A2 ,

which can be written in a condensed form as 2A



A2 + Q,

(14.2.13)

with J + = c2A nk+ , J − = cA2 nk− , where n = N/V is the total density. The dimension of k− is that of k+ multiplied by a density. The goal is to find the equilibrium concentrations for a given condition (p, T). The law of mass action (14.2.6), which is equivalent to J + = J − , (cA )2 /cA2 = k− /k+ , can be written   (XA )2 po −Q/RT e = , (14.2.14) XA2 p

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Chemistry

where the right-hand side is a small number in usual conditions. Indeed, from Table 14.4, the heat of reaction, Q, expressed per mole of A2 is of the order of 100 kcal, both for H2 and for O2 , which is large compared with RT: T = 300 K: RT ≈ 0.593 kcal/mole, T = 3000 K: RT ≈ 5.930 kcal/mole.

(14.2.15)

For mole fractions of A2 of order unity, XA2 = O(1), (14.2.14) gives equilibrium mole fractions XA of order 2 × 10−4 at 3000 K and p ≈ po , and totally negligible values at room temperature. The right-hand side of (14.2.14) is an increasing function of temperature and decreasing the pressure; it describes the well-known empirical law that a reaction moves in the direction that opposes the change in thermodynamic parameters p and T. Dissociation of Carbon Dioxide To illustrate how to proceed with complex scheme of elementary reactions, which is the general rule in combustion, we start with the still relatively simple case of the dissociation of carbon dioxide, CO2 , into CO, O and O2 , in the absence of water vapour and hydrogen. The reverse process, the oxidation of carbon monoxide, can be treated in the same way. This dissociation results from two elementary reactions involving the oxygen atom, (i) (ii)

CO + O2 CO + O + M

 

O + CO2 CO2 + M

+ 8 kcal/mole + 127.2 kcal/mole

to which it is necessary to add the elementary reaction for oxygen dissociation from Table 14.4, (iii)

O+O+M



O2 + M

+ 119.2 kcal/mole.

The three laws of mass action of these reactions are not independent. According to their definition (14.2.6), the equilibrium constants K(T) satisfy the relation Kiii = Kii /Ki

(14.2.16)

because the mass and energy balances of reaction (iii) can be obtained from (i) and (ii), written symbolically as (iii) = (ii) − (i). This leaves only two independent equations to determine the equilibrium composition. This is sufficient, since the expression for the final composition as a function of the three progress variables, ξi , ξii , ξiii , and the initial composition, contain only two groups of unknowns, (ξi − ξiii ) and (ξii + ξiii ), u NCO2 = NCO + (ξi + ξii ), 2 u NCO = NCO − (ξi + ξii ), N = N u − (ξii + ξiii ),

NO2 = NOu 2 − (ξi − ξiii ), NO = NOu + (ξi − ξii − 2ξiii ),

where the superscript u denotes the value in the initial mixture. The second law of thermodynamics ensures that the solution of two algebraic equations is unique (see above). It can be obtained numerically without difficulty.

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627

It is also possible to proceed in a different manner. When the species O is ignored, the oxidation of carbon monoxide (in the absence of H2 O and H) is represented by the overall reaction (14.1.2), which can be written in the equivalent form (A)

2 CO + O2



2 CO2

+ 135.2 kcal/mole,

obtained by eliminating O from the elementary reactions (i) and (ii), (A) = (i) + (ii). The equilibrium constant becomes KA = Ki Kii . The equilibrium composition, p and T, is given by the laws of mass action for reactions (A) and (iii), 2 X XCO O2 2 XCO 2

 =

po p



1 , KA (T)

2 XO = XO2



po p



1 , Kiii (T)

(14.2.17)

which correspond to the two algebraic equations for the two independent progress variables ξi and ξiii . The solution can also be obtained without need to introduce the progress variables; consider the four unknowns XCO2 , XCO , XO2 and XO , with the two relations (14.2.17) completed by two others. The first concerns the sum XCO2 + XCO + XO2 + XO , which is fixed by the dilution (1 when the mixture is not diluted); the other relation concerns the number of carbon and oxygen atoms in the initial condition. Thus, during dissociation of CO2 , no matter whether the initial carbon dioxide is pure or diluted, the total number of carbon atoms is half the number of oxygen atoms. The fourth relation is thus simply,

2XCO2

XCO2 + XCO 1 = . + XCO + 2XO2 + XO 2

(14.2.18)

At atmospheric pressure the equilibrium concentrations, XCO , are of the order of 10−4 at T = 1500 K, 10−2 at T = 2000 K, and 10−1 between 2500 and 3000 K. A simple estimation of the equilibrium concentration, valid only up to temperatures around 2000 K, can be obtained by noting that the right-hand side of Equations (14.2.17) are small numbers, as suggested by the approximation (14.2.9). Thus the mole fraction of oxygen XO2 is unaffected by the dissociation of O2 . The second equation can be used only to calculate the mole fraction of atomic oxygen XO . Those of molecular oxygen and carbon monoxide, XO2 and XCO , are given by the first equation, −1/3 2/3 XCO2 ,

XCO = 2XO2 ≈ (2po /p)1/3 KA

−1/2 1/2 XO2 ,

XO ≈ (2po /p)1/2 Kiii

where the molar fraction of CO2 is approximately equal to that of the initial state. The above method can be generalised to complex chemical schemes, the resolution of the algebraic equations being solved by standard numerical methods.

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The Water–Gas Reaction At high temperature and in the presence of water vapour and hydrogen, the oxidation of CO to CO2 is controlled almost exclusively by the so-called overall water–gas reaction (B)

CO + H2 O



CO2 + H2

+ 9.8 kcal/mole.

(14.2.19)

This equation is obtained from two elementary reactions in which the intermediate species H and OH participate, (11) (2)

CO + OH H2 + OH

 

CO2 + H H2 O + H

+ 24.8 kcal/mole, + 15 kcal/mole;

(14.2.20)

by eliminating H and OH simultaneously, (B) = (11) – (2). The law of mass action for (B) is XCO XH2 O 1 , = XCO2 XH2 KB (T)

where

KB =

K11 . K2

(14.2.21)

This reaction plays an important role in the composition of the steady-state combustion products of a rich hydrocarbon mixture, as explained below. It is also very important in metallurgy in the operation of blast furnaces. Some values of the equilibrium constant are given below: T = 500 K: KB = 138.30,

T = 1000 K: KB = 1.443,

T = 1500 K: KB = 0.3887,

T = 2000 K: KB = 0.220,

T = 2500 K: KB = 0.1635,

T = 3000 K: KB = 0.1378.

14.2.3 Adiabatic Combustion Temperature In this section we obtain the equilibrium temperature of an initial reactive mixture assuming that the reactions proceed adiabatically (without heat exchange to the outside) and at constant pressure. A Simple Example Consider an initial reactive mixture of species A diluted in an inert substance M. Suppose that it undergoes an exothermic chemical reaction 2A  A2 + Q, and suppose that A2 is absent in initial mixture at temperature Tu , XAu = NAu /N u ,

N u = NAu + NM and XAu2 = 0,

where u indicates the composition in the unburnt gas. If the activation energy is zero, this initial state is cannot be prepared experimentally because the reaction rate J + is too fast. It is therefore preferable for the feasibility of the experiment to consider a forwards reaction (exothermic direction) with a high activation energy. Let XA = NA /N,

XA2 = NA2 /N,

N = NM + NA2 + NA ,

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629

be the equilibrium composition and Tb be equilibrium temperature. To simplify the presentation we will assume that the heat of reaction and the specific heat per mole of mixture, Cp , are constants. This last approximation is valid when the reactive mixture is diluted in an inert substance, such as nitrogen in air when oxidation of the latter is negligible. In this approximation, energy conservation in the quasi-isobaric approximation (enthalpy conservation; see Section 2.1.1) is simply NCp (Tb − Tu ) = Q

NAu − NA , 2

(14.2.22)

and mass conservation gives NAu − NA . (14.2.23) 2 The temperature and composition at equilibrium are found by solving a system of three equations, (14.2.14), (14.2.22), (14.2.23), with three unknowns (NA , NA2 , Tb ). The same method can be applied to a more general reaction (14.1.5) and for an arbitrary initial composition Xiu = Niu /N, NA2 =

NCp (Tb − Tu ) = Q

N1u − N1

ϑ1+ − ϑ1− N1u − N1 Niu − Ni = . ϑi+ − ϑi− ϑ1+ − ϑ1−

,

(14.2.24) (14.2.25)

Case of Combustion In principle the calculation is easily generalised to cases where the enthalpies of formation vary with temperature and where the specific heats are different from one body to another and also vary with temperature. In gaseous mixtures, it suffices to replace NCp (Tb − Tu ) by  Ni [hei (Tb ) − hei (Tu )], i

where Ni are the equilibrium compositions and hei (T) are the thermal enthalpies of species i:  T dhei = Cpi (T), hei (T) = Cpi (T  )dT  . dT Tu The enthalpy of each species is the sum of a chemical contribution, the enthalpy of formation at a temperature Tu , hoi (Tu ) and a thermal contribution hei (T), hi (T) ≡ hei (T) + hoi (Tu ).

(14.2.26)

The values of these quantities can be found in specialised reference works;[1,2,3] some examples are given in Tables 14.6–14.8. The energy balance at constant pressure and [1] [2] [3]

Lide D.R., ed., 2014-2015, CRC Handbook of Chemistry and Physics. CRC Press, 75th ed. NIST, ed., NIST-JANAF Thermochemical Tables. http://kinetics.nist.gov/janaf/. Turns S., 2000, An introduction to combustion. McGraw-Hill, 2nd ed.

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Table 14.7 Some specific heats, Cp (T), at constant pressure as a function of temperature in units of calories per mole per degree. 298.15 K

500 K

1000 K

1500 K

2000 K

2500 K

3000 K

CH4

8.52

11.08

17.16

20.72

22.56

23.59

24.24

C3 H8

17.55

26.97

41.85

49.31

55.37

66.05

87.41

O2

7.02

7.43

8.33

8.74

9.02

9.29

9.53

N2

6.96

7.09

7.81

8.33

8.60

8.75

8.85

H2 Ovap

7.98

8.42

9.86

11.26

12.24

12.88

13.32

CO2

8.87

10.66

12.98

13.96

14.42

14.69

14.87

H2

6.89

6.99

7.22

7.72

8.19

8.57

8.86

O

5.24

5.08

5.00

4.98

4.98

4.98

5.00

H

4.97

4.97

4.97

4.97

4.97

4.97

4.97

Source: NIST-JANAF Tables, http://kinetics.nist.gov/janaf.

Table 14.8 Some thermal enthalpies he (T) as a function of temperature in kilocalories per mole for a reference temperature Tu = 298.15 K. 298.15 K

500 K

1000 K

1500 K

2000 K

2500 K

3000 K

CH4

0

1.96

9.13

18.68

29.54

41.11

53.06

C3 H8 O2

0

4.51

22.10

45.04

71.25

101.29

139.08

0

1.45

5.43

9.70

14.14

18.72

23.42

N2

0

1.41

5.13

9.18

13.42

17.76

22.16

H2 Ovap

0

1.65

6.21

11.51

17.40

23.69

30.23

CO2

0

1.99

7.98

14.75

21.85

29.13

36.52

H2

0

1.41

4.94

8.67

12.66

16.85

21.21

O

0

1.04

3.55

6.05

8.54

11.03

13.52

H

0

1.00

3.49

5.97

8.45

10.94

13.42

Source: NIST-JANAF Tables, http://kinetics.nist.gov/janaf.

in the quasi-isobaric approximation (see Section 2.1.1) takes the well-known form of the conservation of enthalpy of mixing, which for a perfect gas is written   Ni hi (Tb ) = Niu hi (Tu ). (14.2.27) i

i

The generalisation to a complex scheme of many elementary reactions, such as combustion, poses no difficulty in principle. Energy conservation (14.2.27) provides the relationship that must be added to the laws of mass action to determine the composition and equilibrium temperature from a given initial condition. Nevertheless, even for a simple

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631

hydrocarbon fuel, the system of algebraic equations is complicated, and the solution must be obtained numerically. Numeric codes are available in the literature for this purpose.[1] Equivalence Ratio A mixture with a stoichiometric composition is characterised by a ratio of quantities of fuel, Fu , and oxidiser, Ox , equal to the ratio of the stoichiometric coefficients. These two species are thus completely consumed after combustion, neither being in excess. Thus, for an overall reaction Fu + ϑO+x Ox = ϑP−1 P1 + ϑP−2 P2 + Q, the stoichiometric composition is  XFu  1 = + XOx stoi ϑOx

and

 YFu  MF = + u , YOx stoi ϑOx MOx

(14.2.28)

(14.2.29)

where MFu and MOx are the molar masses of fuel and oxidiser and Yi is the mass fraction of species i. The equivalence ratio, φ, is defined as the ratio of fuel to oxidiser divided by the corresponding stoichiometric ratio: φ≡

(YFo /YOx ) (XFo /XOx ) = . (XFo /XOx )stoi (YFo /YOx )stoi

(14.2.30)

A mixture is said to be rich, φ > 1, if the fuel is in excess, or lean, φ < 1, if it is not. Combustion in Oxygen At very high temperatures, decomposition of the products of combustion (reverse reactions) limits the combustion temperature. Consider a global reaction such as (14.1.4) and denote by the subscript 1 the reactant (ϑ1+ = 0, ϑ1− = 0) that limits the forwards reaction, that is, the reactant whose concentration tends to zero if the reaction proceeds irreversibly. The maximum temperature that could be achieved in principle after depletion of species 1 is obtained by setting N1 = 0 and ϑ1− = 0 in (14.2.24), Tmax = Tu +

QN1u

NCp ϑ1+

.

(14.2.31)

Assuming that the reaction consumes all the reactants to produce only water vapour, H2 O, and carbon dioxide, CO2 , and taking the specific heat per mole as Cp = 4R, Equation (14.2.31) would give Tb = 11, 750 K for the combustion of carbon in oxygen (14.1.3) and Tb = 7250 K for that of hydrogen in oxygen (14.1.4). Using Cp from Table 14.7 with (14.2.31) and (14.1.4) gives Tb = 6816 K for carbon combustion and Tb = 5036 K for hydrogen combustion. Actual combustion temperatures in oxygen are about 3000 K, Tb = 3076 K for an H2 -O2 mixture at stoichiometry. The difference arises mainly from the dissociation of combustion products; these reactions are endothermic and increase the number of moles. They play a significant role in the chemical balance when [1]

Turns S., 2000, An introduction to combustion. McGraw-Hill, 2nd ed.

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the final temperature exceeds 2500 K. In addition, the increase in the specific heat of the molecules with temperature also contributes to limit the combustion temperature. On the contrary, the adiabatic combustion temperature of mixtures of hydrocarbon fuels and air (O2 + 3.76 N2 ) under normal conditions (T = 298 K, p = 1 atm) does not exceed 2500 K. Under these conditions the calculation of the final state of combustion products is simplified. For temperatures below 2500 K, dissociation has little effect on the final result. In a first approximation, the concentrations of atoms such as O, H and N and radicals such as OH or NO or can be neglected in the composition of the burnt gases, as shown below. Combustion of Lean Hydrocarbon–Air Mixtures For lean hydrocarbon–air mixtures, the excess oxygen molecules are carried over into the burnt gas. The overall reaction  y y O2 → xCO2 + H2 O Cx Hy + x + 4 2 can be considered, in a first approximation, as irreversible, and expression (14.2.31) gives a reasonable estimate of the adiabatic temperature of combustion. The burnt gas contains mainly molecules of water, carbon dioxide and oxygen, in addition to those of nitrogen, considered to be inert. Knowing the initial compositions of the hydrocarbon fuel and air (O2 + 3.76 N2 ), conservation of the number of carbon, oxygen and hydrogen atoms gives (b) , NH(b)2 O and NO(b)2 , in the burnt gas. three relations to solve for the three unknowns, NCO 2 The nitrogen molecules in the air are considered to be inert, so NN(b)2 is given by the ini(b)

(u)

tial composition, NN2 = NN2 . The final composition being known, energy conservation (14.2.27) then gives the combustion temperature. Such a calculation gives good results for lean hydrocarbon–air mixtures and remains reasonable even for stoichiometric mixtures. Thus, for a stoichiometric mixture of CH4 –air under normal conditions Tu = 298 K, p = 1 atm and using constant specific heats for all species in order to simplify the calculation, evaluated from Table 14.7 at an intermediate temperature of 1250 K and with the enthalpies of formation from Table 14.6, we obtain an adiabatic combustion temperature (2308 K) that is overestimated, but only by about 3–4 % compared with that of a complete calculation (2226 K), which includes the chemical equilibria of all the elementary reactions, including dissociation.[1] Combustion of Rich Hydrocarbon Mixtures The evaluation of the combustion temperature of rich hydrocarbon–air mixtures is a little more complicated. The hydrocarbon fuel being in excess, atoms of hydrogen and carbon are present in the burnt gas in the form of hydrogen, H2 , and carbon monoxide, CO, molecules, the latter coming from the decomposition of carbon dioxide by reaction with water vapour (14.2.19). The exhaust gases then contain mainly, in addition to nitrogen, [1]

Turns S., 2000, An introduction to combustion. McGraw-Hill, 2nd ed.

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633

Figure 14.4 Equilibrium composition and temperature of a propane–air mixture as a function of equivalence ratio for initial temperature and pressure of 298 K and 1 atm.

the four species CO2 , H2 O, CO and H2 . Knowing the initial composition of the hydrocarbon fuel and air, the conservation of carbon, oxygen and hydrogen atoms gives three relations for the four unknowns NCO , NCO2 , NH2 and NH2 O . A fourth relation is given by the water gas equation (14.2.19), which involves the final temperature. Energy conservation (14.2.27) gives the additional equation required to solve the problem. The result shows that in rich hydrocarbon–air mixtures, the mole fraction of CO in the burnt gas can reach 10%. The final result can be improved by an iterative procedure in which the result of the decomposition reactions of H2 , O2 and H2 O are introduced into the compositions obtained in the previous step. An example of the temperature and composition of the equilibrium state of propane–air mixtures as a function of equivalence ratio, for an initial temperature and pressure of 298 K and 1 atm, is shown in Fig. 14.4. For combustion in pure oxygen, the adiabatic combustion temperature is much higher and the final concentrations of active species are high. It is necessary to take account of all the elementary dissociation reactions to obtain composition and temperature of the burnt gas in the initial calculation.

14.3 Elements of Thermonuclear Fusion Thermonuclear reactions and ordinary combustion have analogies and differences.

14.3.1 Gamow Tunnelling. Reaction Rates The calculation of nuclear reaction rates between two species involves quantum mechanics and, in particular, the phenomenon of quantum tunnelling discovered by Gamow (1928).

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Chemistry 10–21 10–22 10–23 10–24 10–25 10–26

Figure 14.5 Reaction rate of deuterium–tritium nuclear reaction. Adapted from https://commons .wikimedia.org/wiki/File:Fusion rxnrate.svg, by Dstrozzi, licensed under Creative Commons attribution 2.5 (CC BY 2.5).

This purely quantum effect describes the probability that a particle with an impact energy  can ‘tunnel’ through a potential barrier whose energy is much greater. This effect introduces √ − E / G in the probability for a nuclear reaction, where the Gamow an exponential factor e energy, EG , is of the order of 1 MeV, typically 1010 K. Therefore the cross section usually takes the form σnr () =

S −√EG / e , 

(14.3.1)

where  is the centre-of-mass energy of the two particles and S is a weakly varying function of the energy. The above expression is valid for  < EG . It leads to a reaction rate that initially increases very rapidly with temperature. However, at very high temperature, above a few hundred million degrees, a saturation occurs, followed by a slow decrease in the reaction rate. The characteristic time τnr of a nuclear reaction is related to an effective collision cross section, σnr , 1/τnr ≈ n σnr v ,

(14.3.2)

where v is the impact velocity and n the number density. The effective cross section is a strong function of the impact velocity v, and the brackets . indicate an average taken over the Maxwell–Boltzmann velocity distribution. This leads to a strong temperature dependence of the reaction rate, of the form σnr v ∝ (EG /kB T)2/3 S exp[−a(EG /kB T)1/3 ],

(14.3.3)

where a ≈ 1.89. An illustration of the strong temperature sensitivity of the reaction rate is shown in Fig. 14.5. Fusion reactions are negligible for temperatures below 107 K.

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14.3 Elements of Thermonuclear Fusion

635

14.3.2 Thermonuclear Burning of Hydrogen into Helium Most nuclear reactions are chain reactions. A synthetic presentation of some important reactions of thermonuclear fusion and a detailed bibliography can be found in the literature.[1] The most famous example of nuclear chain reactions is the p–p cycle in the sun to form 4 He (two neutrons, two protons) from four protons through two intermediate species in the form of nuclei, 3 He and 2 H, an isotope of hydrogen of atomic mass 2 (one neutron, one proton), called deuterium, denoted D and discovered in 1932. The first elementary step of the chain is the reaction between two protons to form 2 H. For charge conservation an antielectron e+ (with a positive electric charge +e), called a positron, is produced. In a plasma, the positron is quickly annihilated by an electron to produce a gamma photon. For conservation of the number of leptons, a neutrino, νe , is also produced, p + p → 2 H + e+ + νe .

(14.3.4)

The neutrino νe is an elementary particle of negligible mass (almost zero) that interacts only very weakly with the matter. Its energy is carried away (energy loss). The second step is the interaction between a proton and 2 H (deuterium) to form 3 He with emission of a gamma photon, p + 2 H → 3 He + γ .

(14.3.5)

The last step is the interaction of two 3 He to form 4 He and two protons: 3

He + 3 He → 4 He + 2p.

(14.3.6)

Running the two first steps twice, the global reaction of the p–p cycle can then be written 4p → 4 He + 2νe + 2e+ + 2γ + 26.46 Mev,

(14.3.7)

where the total energy liberated per 4 He produced is 26.2 MeV, excluding 0.26 MeV carried away in the form of neutrinos. The first step (14.3.4) is the slowest reaction of the chain (cross section σ ≈ 3.9 × 10−54 cm2 at 10 keV, σ ≈ 4.4 × 10−53 m2 at 100 keV) and it controls the global reaction rate of hydrogen burning (14.3.7). The characteristic time of the chain reaction, namely the lifetime of a proton, is typically 1.5 × 1010 years near the centre of the sun. This relatively ‘slow’ chain reaction is responsible for 90% of the ‘nuclear combustion’ of hydrogen in the sun (central temperature T ≈ 1.4 × 107 K, 1.3 keV), the other 10% resulting from the so-called CNO cycle (Bethe 1939) involving six intermediate species, 12 C, 13 C, 13 N, 14 N, 15 N and 15 O. The proportion is changed (in favour of the CNO cycle) at a higher temperature. The Gamow energy of the CNO cycle (G ≈ 40 MeV) is higher than that of the p–p cycle so that its reaction rate increases faster with temperature. The CNO cycle prevails over the p–p cycle at temperatures higher than 1.5 keV.

[1]

Atzeni S., Meyer-Ter-Vehn J., 2004, The physics of inertial fusion. Clarendon Press–Oxford Science Publications, 1st ed.

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14.3.3 Deuterium–Tritium Reaction For all possible fusion reactions of the lightest elements, the maximum value of the effective cross section occurs for impact energies above 60 keV. The most efficient fusion reaction (in the sense of a large cross section), much more efficient than the p–p chain, is that of deuterium, D, and tritium, T, two isotopes of hydrogen having atomic masses of 2 and 3, respectively, 2 H (one neutron and one proton) and 3 H (two neutrons and one proton). This reaction has a maximum rate for an impact energy of ≈ 64 keV (T ≈ 7.5 × 108 K), σ ≈ 5 × 10−28 m2 , σnr v ≈ 10−21 m3 /s; all other nuclear reactions have a smaller cross section and thus are much slower. This explains why the D-T nuclear reaction is chosen to address the goal of terrestrial power production using controlled fusion, either by magnetic confinement (ITER Tokamak at Cadarache) or by inertial confinement using powerful lasers (ICF at Livermore and Megajoule at Bordeaux). The D-T reaction is a onestep reaction that produces a stable nucleus of helium, 4 He, called an alpha particle and often denoted α, with a kinetic energy of 3.5 MeV and a neutron (n) with a kinetic energy of 14.1 MeV: D + T → 4 He (3.5 MeV) + n (14.1 MeV). The collision cross section between a neutron and an ion is small, and so, in eventual terrestrial installations, the neutrons will escape from the plasma. Their energy is lost from the reaction zone and does not contribute to heating of the plasma. Only the alpha particle contributes to heat production, so the effective heat of reaction is 3.5 MeV. The evolution of the reaction rate σnr v as a function of temperature is shown schematically in Fig. 14.5.

17:11:49 .016

15 Flows

Nomenclature Dimensional Quantities a A() ci cp cv d D D/Dt D e E g h j J− J+ J (j) Ja Ja k  L M Ni p qm

Description Speed of sound Acoustic transfer function Concentration of species i Specific heat at constant pressure Specific heat at constant volume Thickness (of a front) Molecular diffusivity Material derivative, ∂/∂t + u.∇ Propagation speed Energy density Total energy Acceleration of gravity Enthalpy See 15.3.7 Riemann invariant on −ve characteristic Riemann invariant on +ve characteristic Reaction rate of reaction j Total flux vector of some quantity a Diffusive flux vector of some quantity a Wavenumber Mean free path Length Molecular mass Number of molecules of species i Pressure Heat of combustion per unit mass

S.I. Units m s−1 s−1 molecules m−3 J K−1 kg−1 J K−1 kg−1 m m2 s−1 s−1 m s−1 J kg−1 J mole−1 m s−2 J kg−1 m2 s−2 m s−1 m s−1 reactions m−3 s−1 (units of a) m−2 s−1 (units of a) m−2 s−1 m−1 m m kg molecule−1 mole Pa J kg−1 ≡ (m/s)2 637

17:11:45 .017

638

q˙ m q˙ v q˙ γ Q(j) r s S t tb T u u U UL v V w˙ s W ˙ (j) W x y, z η λ  μ μi ξ ξ  ρ τ φ ω ˙ 

Flows

Heat release rate per unit mass Heat release rate per unit volume (γ − 1)˙qv Heat release of elementary reaction j Position (x, y, z) Entropy Surface area Time ‘Breaking’ time of a nonlinear wave Temperature Longitudinal velocity Velocity of fluid (u, v, w)) Normal velocity of fluid in frame of front Laminar flame speed Transverse velocity, or |u| Volume Entropy production rate Tangential velocity of fluid in frame of front Reaction rate of elementary reaction j Streamwise coordinate Transverse coordinates Shear viscosity Thermal conductivity Scale of distance for evolution of a flow Viscosity 4η/3 + ξ Chemical potential of species i Bulk viscosity Coordinate in mobile reference frame Distribution of reaction delays (4.3.28) Density A characteristic time A potential function Angular frequency Total instantaneous heat release rate

J kg−3 s−1 J m−3 s−1 J m−3 s−1 J m J K−1 kg−1 m2 s s K m s−1 m s−1 m s−1 m s−1 m s−1 m3 J K−1 m−3 s−1 m s−1 reactions m−3 s−1 m m Pa s J s−1 m−1 K−1 m Pa s J kg−1 Pa s ξ = x − Dt s−1 kg m−3 s (according to context) s−1 J s−1

Nondimensional Quantities and Abbreviations cst. C− C+ r

Constant Riemann negative characteristic trajectory Riemann positive characteristic trajectory Reduced mass density

17:11:45 .017

ρ/ρo

Flows

u Yi γ  ϑ (i) ϑ ξ ψ

639

Reduced flow velocity Mass fraction of species i Ratio of specific heats A small quantity Stoichiometric coefficient of species i Reaction order Reduced ratio of space-time Reaction progress variable, see (15.1.42)

u/ao ρi /ρ cp /cv D/

x/(ao t)

Subscripts, Superscripts and Math Accents a− a+ ab achem acoll ad ae af ai ains aint a(j) a(k) aL an aN ao aq as aT atot au

Low pressure/temperature side of a discontinuity High pressure/temperature side of a discontinuity Burnt gas Chemical Collision Delay Energy Value at flame or shock front Pertaining to species i Instability Internal Relative to reaction j Relative to wavenumber k Pertaining to a laminar flame Normal component of a Neumann state (just behind a shock) Reference state Heat Entropy Thermal Total Unburnt or unshocked gas

This chapter is broken down as follow. The equations for fluid mechanics with exothermic chemical reactions are recalled in Section 15.1. We will not attempt to present these equations in their most general form.[1,2] This would require a heavy formalism, not useful in this book. Only the simplest conditions that contain the dominant phenomena are considered. The jump relations through the zones of strong gradients are presented [1] [2]

de Groot S., Mazur P., 1984, Non-equilibrium thermodynamics. Dover. Poinsot T., Veynante D., 2005, Theoretical and numerical combustion. Edwards.

17:11:45 .017

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Flows

in Section 15.1.6 and entropy production is discussed in Section 15.1.7. The internal structure of shock waves is also discussed in this section. The basic approximations used in fluid mechanics and combustion are presented in Section 15.2. When the flow is strongly subsonic, combustion proceeds in the quasi-isobaric approximation; see Sections 2.1.1 and 15.2.1. This is the framework for the theory of flames (deflagrations) that are slow combustion waves studied in Chapters 2 and 8–10. Even in quasi-isobaric approximation, combustion can couple to compressible phenomena to generate sound and to develop thermo-acoustic instabilities; see Sections 2.5 and 3.3. The elementary theory of the coupling of heat release and acoustics is recalled in Section 15.2.4. Detonations are fast combustion waves propagating at a supersonic velocity. The background of compressible flows, useful for the study of shocks and detonations in Chapters 4 and 12, is given in Section 15.3. 15.1 Macroscopic Conservation Equations The relaxation time towards equilibrium decreases with the decreasing size of the system. A large macroscopic system can be considered as a continuum medium, constituted of small subsystems in internal equilibrium, when the characteristic time of evolution is much larger than the relaxation time of the quasi-punctual subsystems which is of the order of the collision time; see Section 13.3.2. This assumption of local equilibrium corresponds to gradients that are characterised by a length scale much larger than the microscopic length scale (the mean free path in gas). In this approximation the thermodynamic laws are valid locally and the evolution is governed by conservation equations in which the molecular transport is described by transport coefficients (viscosity, thermal conductivity, molecular diffusion, etc.). For gaseous mixtures, the corresponding macroscopic laws, Fick’s law, Fourier’s law and so on and the expressions for the transport coefficients have been derived from the kinetic theory of gases (Boltzmann’s equation) assuming that the local distribution of molecular velocity is near to equilibrium (the Maxwellian distribution). The so-called Chapman–Enskog method is outlined in Sections 13.3.1 and 13.3.2. Attention is focused below on macroscopic equations. 15.1.1 Continuity Equation In a continuous medium, physical quantities are fields whose values are a function of position, r, and time, t. Extensive quantities are defined by their density. An extensive quantity AV , relative to a fixed volume V, having a mass-weighted distribution a(r, t),  d3 rρa, (15.1.1) AV = V

where ρ is the mass density, is said to be conserved if its evolution is due only to its flux through the surface that delimits V, so that, according to the divergence theorem, the equation of evolution takes the form

17:11:45 .017

15.1 Macroscopic Conservation Equations

∂(ρa)/∂t = −∇.Ja ;

641

(15.1.2)

see (13.3.15) for the microscopic origin of these equations in a gas. Equation (15.1.2) is written here for a scalar, whose flux Ja is a vector. It can be generalised to a vector quantity, a, whose flux is a tensor Ja . For the total mass, a = 1, the mass flux is J = ρu, where u(r, t) is the flow field, that is, the velocity at time t of the centre of gravity of an element of fluid at the point r. The local balance for mass is given by the conservation equation ∂ρ/∂t = −∇.(ρu)

(15.1.3)

1 Dρ = −∇.u, ρ Dt

(15.1.4)

or, in its Lagrangian formulation,

where the operator D/Dt ≡ ∂/∂t + u.∇ is called the Lagrangian or material (or also particulate) derivative. According to (15.1.4), the rate of change of specific volume, 1/ρ, is ρ

D(1/ρ) = ∇.u, Dt

(15.1.5)

so that the divergence of the flow field is the expansion rate of the fluid. The Lagrangian derivative D/Dt is the derivative attached to an element of fluid in movement. Considering the volume V(t) each of whose points evolves with the flow velocity u, a simple geometrical calculation shows that   Da 3 d d r. ρad3 r = ρ (15.1.6) dt Dt V(t)

V(t)

Using (15.1.4), the conservation equation (15.1.2) of a conserved quantity can also be written ρDa/Dt = −∇.Ja ,

(15.1.7)

Ja ≡ ρau + Ja .

(15.1.8)

where the flux Ja is defined by

15.1.2 Fick’s Law In the absence of chemical reactions, the total mass of each chemical species is a conserved quantity. We introduce the mass fraction of a species i, Yi (r, t) = ρi /ρ = Mi ci /ρ, where Mi and ci = Ni /V are, respectively, the molecular (or molar) mass and concentration of the species i. In the gas phase, the partial density of each species is ρi = ρYi . In the absence of

17:11:45 .017

642

Flows

production or consumption of species (no chemical reaction) the conservation equation for the species i obtained from (15.1.7) is (15.1.9) ρDYi /Dt = −∇.Ji ,   where Ji is the molecular diffusion flux, and i Ji = 0 since i Yi = 1. In any mixture with more than two constituents, the diffusive flux of each species is a nontrivial function of the gradient of the mole fractions of each of the species present. This does not introduce a major difficulty, but technically it makes the calculation very difficult. It is possible to avoid this problem without losing the essential part of the phenomena. The situation greatly simplifies when one of the species is present in abundance. In this case the diffusion reduces to simple binary processes of the diffusion of each minority species through the major species, and the diffusive flux of each species is proportional to the gradient of its own mass fraction, Ji = −ρDi ∇Yi ,

(15.1.10)

where, for each minority species, Di is the binary diffusion coefficient of the species in the abundant diluent. This is Fick’s law. It is possible to introduce another simplification for perfect gases √ at constant pressure where the product ρDi changes only weakly with temperature as T. This weak variation has no essential consequence in flames and can be included without great difficulty. However, in order to simplify the presentation, this temperature dependence will be neglected in the following and Equations (15.1.9)–(15.1.10) simplify to the diffusion equation DYi /Dt = Di Yi .

(15.1.11)

The order of magnitude of the diffusion coefficient is the same for all species of comparable size, Di ≈ a2 τcoll , where a is the speed of sound. For example, Di = 1.56 × 10−3 m2 /s for CO2 molecules in ambient air. In a reactive mixture, the chemical reactions consume and produce different chemical species. The mass of individual species is no longer a conserved quantity. The reactions introduce volume sources that cannot be expressed as the divergence of a flux. Introducing the dummy index j to indicate the pairs of reactions (both forward and inverse), the conservation equation for species becomes  (j) ˙ (j) , ϑi Mi W (15.1.12) ρDYi /Dt = −∇.Ji + j

with, by definition, (j) (j) ˙ (j) ≡ (J+ − J− ) W

and

(j)

−(j)

ϑi ≡ (ϑi

(j)

+(j)

− ϑi

),

(15.1.13)

where J± are the reaction rates (number of times that the forwards or inverse reaction j occurs per unit volume and time), Mi is the molecular mass (or molar mass according to the ˙ (j) ) of the species i and ϑi(j) is its stoichiometric coefficient in the reaction j; definition of W see (14.1.5).

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15.1.3 Navier–Stokes Equation In the presence of terrestrial gravity the conservation of momentum takes the general form ρDu/Dt = −∇.% − ρgez ,

% = pI + π ,

(15.1.14)

where the thermodynamic pressure p has been singled out in the stress tensor, g is the terrestrial acceleration of gravity and ez is the unit vector in the vertical direction. In a Newtonian fluid the viscous tensor takes the form π ≡ −2η(∇u)(s) − I(ξ − 2η/3)∇.u,

(15.1.15)

where I is the identity tensor, (∇u)(s) is the symmetric part if the deformation tensor, (s)

(∇u)i,j = (∂i vj + ∂j vi )/2 and, η and ξ are, respectively, the shear and bulk viscosity. Equation (15.1.14) yields ρDu/Dt = −∇.[ p I − 2η(∇u)(s) − (ξ − 2η/3)(∇.u) I ] − ρgez

(15.1.16)

and takes the form of Navier–Stokes equation ρDu/Dt = −∇(p + ρgz) + ηu + (ξ + η/3)∇(∇.u),

(15.1.17)

where the contribution of the terrestrial gravity has been written for an incompressible fluid (∇ρ = 0), the z-axis being oriented upwards. When viscous effects can be neglected this yields the Euler equation ρDu/Dt = −∇p − ρgez = −∇(p + ρgz);

(15.1.18)

the last term has been written in the approximation ∇ρ = 0. 15.1.4 Energy Conservation Let etot be the total energy per unit mass of fluid. It is the sum of mechanical energy and internal energy. The mechanical energy density is the kinetic energy per unit mass of fluid, |u|2 /2, where u is the flow velocity, plus the potential energy ϕext when the fluid is submitted to an external force. For example, the terrestrial gravity yields ϕext (r) = gz. The internal energy density, eint , is the sum of thermal energy plus the energy of the chemical bonds in each of the constituents eint = eT + echem .

(15.1.19)

For a gas, the thermal energy per unit mass eT is the sum of energies of thermal agitation (kinetic energy of atoms and/or molecules with respect to the centre of gravity of an element of fluid moving with velocity u) and of internal degrees of freedom in molecules (vibration, rotation, etc.). In a liquid the interaction between atoms and/or molecules yields an additional contribution to the thermal energy eT of same order as the kinetic energy. For an external potential ϕext (r) constant in time, the total energy etot = |u|2 /2 + eint + ϕext

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(15.1.20)

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is a conserved scalar, so that, according to (15.1.2), (15.1.7) and (15.1.8), its balance may be written in terms of its fluxes in the form ∂(ρetot )/∂t = −∇.Jetot Jetot

where

or

ρ Detot /Dt = −∇.Jetot ,

≡ Jetot − ρetot u.

(15.1.21) (15.1.22)

In order to simplify the presentation and to show the main phenomena in gaseous combustion, different approximations will be used in the energy balance: constant specific heats, constant enthalpy of formation, no external forces ϕext =0, no viscous dissipation. The relation between Jetot and the heat flux, Jq , is found as follows. Inviscid Flow The kinetic energy, ρ|u|2 /2, is given directly by scalar multiplication of (15.1.18) by u, 1 D 2 ρ |u| = −u.∇p = −∇.(pu) + p∇.u, (15.1.23) 2 Dt where we have neglected both the acceleration of gravity and viscous dissipation (g = 0, η = ξ = 0). By subtracting (15.1.23) from (15.1.21) one gets ρ Deint /Dt ≡ ρ D(eT + echem )/Dt = −∇.Jq − p∇.u,

Jq ≡ Jetot − pu.

(15.1.24)

In a pure inert fluid, D(echem )/Dt = 0 and the heat flux, Jq , defined as the diffusive flux of thermal energy, eT , is simply Jq . It is assumed to satisfy Fourier’s law, Jq = −λ∇T,

(15.1.25)

where λ is the thermal conductivity of the mixture. The term p∇.u in Equation (15.1.24) is the work done by the pressure, as is seen from the equation for mass conservation (15.1.5) written in terms of the volume per unit mass ρ −1 , p∇.u = pρ(Dρ −1 /Dt), and Equation (15.1.24) is just the first law of thermodynamics saying that the change in internal energy of a unit volume of fluid is equal to the work plus the heat. Thermal Balance for Inviscid Flows of Ideal Gaseous Mixtures In ideal gaseous mixtures the chemical energy is the sum of the bond energy of each molecule; see Section 14.1.3. Neglecting the variation of the enthalpy of formation hoi for simplicity, hoi ≈ hoi (To ) ≈ μoi , defined here per unit mass of species i, the balance of chemical energy is obtained from (15.1.12) and is expressed in terms of the diffusive fluxes Ji and of the rate of chemical energy release,    ˙ (j) , (15.1.26) hoi (To )Yi , ρDechem /Dt = − ∇.(hoi Ji ) − Q(j) W echem ≈ i

i

Q(j)

j

where is the heat of reaction of the jth elementary reaction; see (14.1.6). The heat of reaction is defined as positive when the forwards reaction is exothermic. The minus sign

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in front of Q(j) in (15.1.26) arises because the energy of chemical bonds is negative; see Section 14.1. Then, according to (15.1.24) and (15.1.26),   ˙ (j) , Jq ≡ Jq − Q(j) W hoi Ji , (15.1.27) ρDeT /Dt = −p∇.u − ∇.Jq + j

i

where Jq is defined as before, Jq ≡ Jetot − pu. The heat flux Jq is the energy flux in (15.1.27) that contributes to the modification to the thermal energy eT in addition to the pressure work and the chemical heat release. This flux satisfies Fourier’s law (15.1.25). In the absence of a thermal gradient the diffusion of inert species does not modify the thermal energy eT . Putting together these definitions, the flux of total energy in (15.1.21) takes the self-explaining form  hoi Ji , Jq = −λ∇T. (15.1.28) Jetot = ρu [etot + p/ρ] + Jq + i

For an ideal gas the internal energy, eT , is a function of the temperature only, and by definition of the specific heat per unit mass at constant volume, cv , δeT = cv δT, yielding an equation for the temperature. When the specific heats per unit mass, cv and cp , are assumed to be constant for simplicity, ρDeT /Dt = ρcv DT/Dt, and according to the ideal gas law, p = (cp − cv )ρT.

(15.1.29)

The approximation of strong dilution of the reactant in an inert substance makes it possible to neglect the change in molar mass that can occur when the number of moles is not conserved in the chemical reaction. Using (15.1.4) and (15.1.29), the term −p∇.u can be written − p∇.u =

D D p D ρ= p − ρ [(cp − cv )T]. ρ Dt Dt Dt

Using (15.1.28) and (15.1.30), the energy equation (15.1.27) reduces to  ˙ (j) , ρcp DT/Dt = Dp/Dt − ∇.Jq + Q(j) W Jq = −λ∇T,

(15.1.30)

(15.1.31)

j

where the term containing the pressure reflects compressible effects. In the absence of heat conduction and chemical energy release, (15.1.31) shows up as a change in temperature during isentropic compression or expansion, [cp /(cp − cv )]δT/T = δp/p. Viscosity introduces viscous dissipation terms into (15.1.31) that are negligible in front of the chemical heat release. In a fluid at rest (u = 0), when the pressure term and the reaction rates are neglected, Equation (15.1.31) reduces to Fourier’s equation, ρcp ∂T/∂t = ∇.(λ∇T).

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(15.1.32)

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Constitutive Equations for Inviscid Flows of Reactive Gas In a gaseous mixture, when the reaction rates are written as functions of the temperature, the ˙ (j) (p, T, Y1 , . . . , Yi , . . . , Yn ), the equations pressure and the composition of the mixture, W for the conservation of mass (15.1.4), species (15.1.12), momentum (15.1.18) and energy (15.1.31) form with the ideal gas law (15.1.29) an autonomous system (as many equations as unknowns): Du 1 Dρ = −∇.u, ρ = −∇p, ρ Dt Dt  DT Dp ˙ (j) , Q(j) W ρcp = − ∇.Jq + Dt Dt

p = (cp − cv )ρT, Jq = −λ∇T,

(15.1.33) (15.1.34)

j

ρ

 (j) DYi ˙ (j) . = ∇.(ρDi ∇Yi ) + ϑi Mi W Dt

(15.1.35)

j

This system of equations controls all the mechanisms in a reacting gas when viscous effects can be neglected, as is the case in most of the phenomena involving flames, ablation fronts or detonations. However, viscous terms cannot be neglected in the inner structure of shock waves; see Section 4.2.2. Viscous Flow When the fluid viscosity is taken into account, according to (15.1.14), Equations (15.1.23) and (15.1.24) become 1 D 2 ρ |u| = −∇.[pu + u.π ] + p∇.u + π : (∇u)(s) , 2 Dt ρ Deint /Dt ≡ ρ D(eT + echem )/Dt = −∇.Jq − p∇.u − π : (∇u)(s) Jq ≡ Jetot − pu − u.π

where

and the heat flux Jq , defined as before as the flux of thermal energy, is  Jq ≡ Jq − hoi Ji = −λ∇T.

(15.1.36) (15.1.37) (15.1.38)

(15.1.39)

i

The last term in (15.1.36) and (15.1.37) describes the irreversible transfer of mechanical energy into internal energy through viscous dissipation since, according to the positive sign of the entropy production, the last term in the right-hand side of (15.1.36) should be negative; see (15.1.57). In combustion the viscous dissipation is negligible in (15.1.31).

15.1.5 Conservative Form The jump relations across a thin reaction zone (flames, detonations, etc.) are obtained from these equations when they are written in a conserved form. Using the definition of Jq in (15.1.38), the equation for conservation of total energy (15.1.21) becomes ∂(ρetot )/∂t = −∇.[ρuetot + u.% + Jq ] = −∇.[ρu (etot + p/ρ) + Jq + u.π ]. (15.1.40)

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Note that the nondiffusive part of the total energy flux is the convective flux of total enthalpy, ρu (etot + p/ρ). In the absence of dissipation and in a stationary regime, ∇.(ρu) = 0, (15.1.40) yields ∇.[ρu(etot + p/ρ)] = ρu.∇(etot + p/ρ) = 0.

(15.1.41)

In an inert flow at low Mach number in which the variation of thermal energy is negligible, Equation (15.1.41) is equivalent to Bernoulli’s equation, derived in Section 15.2.2, since, in the presence of gravity, we have δetot = δ(|u|2 /2 + gz), showing that p/ρ + |u|2 /2 + gz is a conserved quantity along a flow line. In a reacting system it is useful to introduce the progress variable, ψ, which goes from 0 for the composition of the initial fresh mixture (far from chemical equilibrium) to 1 in the burnt gas in thermodynamic equilibrium. It is defined by echem = −qm ψ + eochem ,

ψ ∈ [0, 1],

(15.1.42)

where qm is the total chemical energy released per unit mass of gas when the exothermic reaction goes to completion. Recalling the characteristics of a perfect gas with constant specific heats, eT = cv T, p/ρ = (cp − cv )T, and using (15.1.20), in the absence of gravity Equation (15.1.40) can be written ∂(ρetot )/∂t = −∇.[ρu(cp T + |u|2 /2 − qm ψ + eochem ) + Jq + u.π ].

(15.1.43)

The equations of continuity, (15.1.4), and momentum conservation in the absence of gravity, (15.1.14), can be written in the following conservative form: ∂ρ/∂t = −∇.(ρu),

∂(ρu)/∂t = −∇.(pI + ρuu + π ),

(15.1.44)

yielding for inviscid flows in two-dimensional Cartesian coordinates, r = (x, z), u = (u, w), ∂ρ/∂t = −∂(ρu)/∂x − ∂(ρw)/∂z, ∂(ρu)/∂t = −∂(p + ρu2 )/∂x − ∂(ρuw)/∂z, ∂(ρw)/∂t = −∂(ρuw)/∂x − ∂(p + ρw2 )/∂z.

15.1.6 Jumps across a Hydrodynamic Discontinuity Consider a plane wave propagating at constant velocity, held stationary and perpendicular to the x-axis by adjusting the upstream longitudinal gas velocity to be equal to the constant propagation velocity. In such a reference frame, jump relations can be found by direct integration of Equations (15.1.43)–(15.1.44) along the x-axis, expressing the conservation

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of mass, momentum and energy through the front. For an inviscid fluid they are Mass: Longitudinal momentum: Transverse momentum: Energy:

[cp T

[ρU]+ − = 0, +

p + ρU 2 = 0,

(15.1.45)

= 0,

(15.1.47)

− [W]+ − 2 + U /2 − qm ψ]+ −

=

(15.1.46)

[λ∂T/∂x]+ − /ρU,

(15.1.48)

where [..]+ − indicates the difference of a quantity between the upstream and downstream flows, and U and W are the normal and tangential components, respectively, of the fluid velocity in the reference frame of the wave. For thermal waves, such as ablation fronts studied in Chapters 6 and 11, the thermal flux has to be been retained in the right-hand side of (15.1.48). However, the diffusion fluxes are negligible in the jump conditions of shock waves because, as explained in Section 15.1.7, the thickness of the shock is of the order of the mean free path  and the flow velocity is of the order of the speed of sound, U ≈ a. Using the elementary kinetic theory of gas, molecular, viscous and thermal diffusion coefficients are of order D ≈ a (see (1.2.4)), so that the diffusion fluxes on both sides of the shock yield a negligible contribution to the jumps, of the order of / 1, where  is the length of variation of the macroscopic quantities outside the shock. For perfect gases with specific heats varying with the temperature, cp T is replaced by T o 0 cp dT. In the most general case, cp T − qm ψ + echem is replaced by the enthalpy htot ≡ eint + p/ρ; see (15.1.51). Across a pure shock wave (qm = 0), the jump conditions can be also obtained directly from Boltzmann’s equation; see (4.6.1). For a Newtonian fluid (15.1.17) in one-dimensional geometry, introducing the viscosity μ ≡ 4η/3 + ξ , Equations (15.1.43) and (15.1.44) yield  ∂ ∂(ρu) ∂ ∂u ∂ρ = − [ρu], =− p + ρu2 − μ , (15.1.49) ∂t ∂x ∂t ∂x ∂x     ∂ ∂(ρetot ) ∂T  o  ∂u 2 =− + hi Ji − μu ρu htot + u /2 − λ , (15.1.50) ∂t ∂x ∂x ∂x i

where htot ≡ eint +p/ρ = h+echem and h ≡ eT +p/ρ. The general form of jump conditions at a surface of discontinuity (Rankine–Hugoniot) is given by the brackets in the right-hand side. Neglecting the diffusive fluxes on both sides of the discontinuity, this gives +

+ p + ρU 2 = 0, htot + U 2 /2 = 0, (15.1.51) [ρU]+ − = 0, −

−

where U is the normal component of the relative flow velocity normal to the shock. For ordinary shock waves, dimensional analysis shows that the diffusive fluxes are effectively negligible on both sides of the front. Outside the inner structure of the shock the macroscopic length of variation L is larger than the thickness of the shock, which is of the order of the mean free path. For example the conduction flux λ∇T in (15.1.48) is smaller than ρUcp T by an order /L 1 since U ≈ a and λ/ρcp ≈ a.

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The equality of the transverse velocity components, (15.1.47), comes from the conservation of transverse momentum after dividing by the constant mass flux, ρU = 0. Relation (15.1.47) is not valid for an impermeable interface, U = 0; see below. Curved Fronts The jump relations across a curved front of negligible thickness that evolves only slowly in a nonuniform flow are always given, to a first approximation, by (15.1.45)–(15.1.48) when U is replaced by the normal component of the fluid velocity with respect to the front, U → Un ≡ (u.n − Dn ),

(15.1.52)

where n is the normal to the front and Dn is its normal velocity; see (10.1.6). This is easily understood as follows. When the ratio  ≡ d/ between the thickness d of the wave front (the distance over which the physical values change rapidly) and , the characteristic length over which the external flow field changes, usually the radius of the front, tends to zero,  → 0, all the terms, except the derivatives with respect to the normal coordinate, give contributions that tend to zero by integration along the normal direction. Similarly, when the evolution of the external field has a characteristic time scale much longer than the transit time through the front (quasi-steady-state approximation), integration of nonstationary terms give contributions that can be neglected. The jump relations (15.1.45)– (15.1.48) are thus valid at leading order in a perturbation expansion in terms of the small parameter . Correction terms of order , induced by curvature and strain, such as those mentioned for flames in Section 2.3, appear at the next orders in the expansion. The jump relations (15.1.45)–(15.1.48) apply with a good approximation to thin supersonic fronts such as shock waves (in a nonreactive flow qm = 0) and detonations (in reacting flows qm = 0). Particular attention must be paid to momentum conservation for the case of a passive front, that is, a front that does not propagate and is not traversed by a mass flux, Un = 0. The equality of the transverse velocity components, (15.1.47), comes from the conservation of transverse momentum after dividing by the mass flux, ρU = 0, which is a constant according to mass conservation (15.1.45). Relation (15.1.47) is not valid for an impermeable interface, ρU = 0. In this case the normal component of fluid velocity is zero, the pressure jump is null, [p]+ − = 0 and there is no constraint on the tangential velocity jump across the interface. To summarise, the flux of a conserved quantity is constant through a planar wave propagating at constant velocity. When the wave separates two homogeneous fluids, all diffusive fluxes of conserved quantities are negligible outside the discontinuity. This implies that the convective fluxes are equal on both sides of the wave, so that the jump relations are independent of the internal structure of the wave. This is also true for a curved zone of strong gradients, called hydrodynamic discontinuities, in the limit  ≡ d/ → 0 if the inner structure is quasi-steady.

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15.1.7 Entropy Production and Structure of Shock Waves In thermodynamics, entropy is a function of the equilibrium states; see Section 13.1.1. The jump of entropy across a shock wave that separates two equilibrium states is thus a function of the upstream and downstream states only and does not depend on the internal structure of the wave. Entropy was introduced in statistical mechanics for macroscopic systems that are not necessarily in thermodynamic equilibrium; see Section 13.1.2. In a gas out of equilibrium, considered in Section 13.3.1, this leads to the expression (13.3.13) for Boltzmann’s entropy. However, the thermodynamic expression for equilibrium entropy, as for example (13.2.7) in gas, can be extended to local equilibrium states. The resulting expression, s(0) (r, t), satisfies a local equation, also valid for the Boltzmann entropy in the limit  ≡ d/ → 0; see Section 13.3.2. From now on in the present section, the entropy per unit mass is denoted s(r, t) and is computed using local equilibrium. According to Boltzmann’s equation and in agreement with the second law, the entropy of an isolated system increases with time; see Section 13.1.1. Therefore the production of entropy must be positive, ∂(ρs)/∂t = −∇.Js + w˙ s ,

where

w˙ s ≥ 0 (second law);

(15.1.53)

more precisely, each term describing a dissipative mechanism in w˙ s must be positive. Entropy Balance for Inert Mixtures Extending the analysis (13.3.28) to all the dissipative mechanisms, the expressions for the ˙ s , are obtained from the balance equations entropy flux, Js , and entropy production rate, w for energy, momentum, and mass of each species, along with Gibb’s formula (13.1.5), Deint D(1/ρ)  DYi Ds = +p − , (15.1.54) μi T Dt Dt Dt Dt i

where μi is here the chemical potential per unit mass of species i. In the absence of chemical reaction, using continuity, (15.1.9) and (15.1.37), one gets  Ds = −∇.Jq + μi ∇.Ji − π : (∇u)(s) , (15.1.55) Tρ Dt i

which can be written in the form (15.1.53) with  μi 1 J , Js = ρus + Jq − T T i i    μ  1 1 i − − π : (∇u)(s) , w˙ s = Jq .∇ Ji .∇ T T T

(15.1.56) (15.1.57)

i

highlighting the fact that entropy production arises from phenomena related to molecular transport (dissipative phenomena). The production term (15.1.57) takes the form of a sum of products of a diffusion flux by the gradient of the intensive thermodynamic parameter. When the complete equilibrium is reached, both terms in each product are zero; the fluxes

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and gradients are zero. Equations (15.1.53) and (15.1.57) contain nothing new compared with the conservation equations for mass, momentum and energy. However, the production of entropy is useful for describing the dissipative phenomena.[1,2] According to second law of thermodynamics, each term in (15.1.57) should be positive. This implies that the transport coefficients relating the diffusion flux to the gradient should be positive. The positivity of the last term in (15.1.57) and comparison of (15.1.36) and (15.1.37) show the irreversible transfer of mechanical energy into internal energy by viscous effects. In a reacting flow, the reaction rates introduce an entropy production term[1,2] that will be discussed later. In a polytropic gases, when the variations of the specific heats per unit mass, cp and cv , are neglected, γ ≡ cp /cv = cst., the entropy takes, according to (13.2.8), the simple form   (s − so ) p/ρ γ (15.1.58) = ln γ . cv po /ρo If the dissipative phenomena are negligible, the right-hand side of (15.1.55) is zero and Equation (15.1.58) leads to the variations of pressure and density in the form Ds Dρ 1 Dp 1 Dρ Dp =0⇒ −γ =0⇒ − a2 = 0, Dt p Dt ρ Dt Dt Dt √ where a = γ p/ρ is the speed of sound in a perfect gas.

(15.1.59)

Simple Fluids The simplest case of an inert fluid containing a single species is sufficient to show the basic features of the inner structure of a shock wave. According to Fourier’s law (15.1.28) and to the viscous tensor in (15.1.14)–(15.1.16), the entropy flux Js and the entropy production rate w˙ s in (15.1.56) and (15.1.57) take the form     λ ∂T μ ∂u 2 λ ∂T 2 Js = ρus − , w˙ s = + 2 . (15.1.60) T ∂x T ∂x ∂x T According to the second law (H-theorem in Section 13.3.1), entropy cannot decrease in the absence of fluxes (isolated system) so that the thermal conductivity and viscosity must be positive, λ > 0 and μ > 0. The entropy balance (15.1.53) can also be written    2 ∂ ∂ ∂u Ds = λ T +μ . (15.1.61) ρT Dt ∂x ∂x ∂x This corresponds to the classical thermodynamic relation between the variation of entropy and the exchange of heat δS > δQ/T, a relation that was the nineteenth-century key for the introduction of the notion of entropy by Clausius in 1862 on the basis of the work of Carnot

[1] [2]

Prigogine I., 1967, Thermodynamics of irreversible processes. Interscience, 3rd ed. de Groot S., Mazur P., 1984, Non-equilibrium thermodynamics. Dover.

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(1824). Equations (15.1.53) with (15.1.60), or Equation (15.1.61), can be also written       Ds ∂ λ ∂T μ ∂u 2 λ ∂T 2 ρ = + + 2 , (15.1.62) Dt ∂x T ∂x T ∂x ∂x T where the two last terms (entropy production) are positive. The increase of entropy under the effect of heat conduction during the approach to a uniform temperature in an isolated system at rest (u = 0, n.∇T = 0 at the wall) can also be seen directly from Fourier’s equation (15.1.32) since, assuming for simplicity constant diffusivity, DT ≡ λ/(ρcp ) = cst., integration by parts shows that the quan thermal 3 r ln T is an increasing function of time, in agreement with entropy production d tity V (15.1.60),   λ ∂T/∂t |∇T|2 = d3 r d3 r 2 > 0, (15.1.63) T ρcp T V

V

illustrating the irreversible nature of the diffusion equation. Thickness of Shock Waves Shock waves are discussed in Section 4.2.1. They are thin sheets, separating two different thermodynamic states of the fluid, and which propagate at a supersonic velocity relative to the upstream flow. As explained in Section 15.3, they can be viewed as singularities formed after a finite time in the solutions of Euler equations. In real fluids their inner structure is controlled by dissipative mechanisms. A shock wave is an irreversible process fully out of equilibrium. For a propagation Mach number significantly higher than unity, the inner structure of a shock wave cannot be described using macroscopic equations such as (15.1.49) or (15.1.61). This is because the molecules crossing the front do not suffer a number of elastic collisions sufficient to establish the local equilibrium. In a way similar to the H-theorem in Section 13.3.1, the increase of entropy across the shock, illustrating its irreversible structure, is proved in Section 4.6.1 by using Boltzmann’s equation, and the thickness of the shock d is shown to be of the order of the mean free path , d/ = O(1). This order of magnitude of d can also be obtained from the macroscopic equations, even though these equations are not valid across strong shocks. Equating the normal convective momentum flux, ρu2 , to the diffusive flux, μ∂u/∂x, conservation of momentum (15.1.49) shows that d is of order μ/ρU, where the speed of the upstream flow into the shock, U, is the propagation speed of the shock in a gas at rest, D, d ≈ μ/ρD. The thickness of the shock decreases when the velocity D increases or when the diffusivity decreases. If we take the speed of sound as a characteristic value for the shock velocity, D ∼ a, expressions (1.2.4) or (13.3.29) for the viscous diffusivity, μ/ρ ≈ a, then show that d is of order . The energy equation (15.1.50) leads to the same result since the diffusion coefficients λ/ρcp and μ/ρ have the same order of magnitude in gas. According to thermodynamics, the entropy jump s through the shock wave cannot depend on the transport coefficients since s is a thermodynamic function. This can be

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653

checked by noticing that the term with the square of the gradient in the right-hand side of (15.1.61) is of the order of μU 2 /d2 and its integral through the shock yields μU 2 /d. Using the evaluation d ≈ μ/ρU, the integral of (15.1.61) then gives an expression for s from which the dissipative phenomena, responsible for the entropy production, disappear: ρUTs ≈ μU 2 /d ≈ ρU 3 ⇒ s = O(D 2 /T),

since

U/D = O(1).

The Rankine–Hugoniot jump conditions across a shock are computed in Section 4.2.1 by equating the convective fluxes of mass, momentum and total energy on both sides of the shock; see (4.2.14)–(4.2.17). This was done by Hugoniot[1] (1889), without need to solve the internal structure of the shock as Rankine[2] (1870) did for a weak shock using the macroscopic equations, neglecting viscosity and retaining only heat conduction. Viscous effects were considered much later by Lord Rayleigh.[3] The full calculation of the structure of a weak shock is presented in Section 4.2.2. It took some time to realise that local equilibrium is not valid inside the inner structure of strong shock, d/ = O(1), so that such a structure can be described only by Boltzmann’s equation (1872). This was not even mentioned in Rayleigh’s 1910 paper,[3] even though it is reported that ‘the thickness involved is finite and indeed extremely small’. It turns out that the approximation of local equilibrium, discussed at length in Section 13.3.2, becomes valid for weak shocks, when the shock velocity approaches the sound speed, (D − a)/a 1, and the macroscopic equations of fluid mechanics can be used in this limit. This is because, as shown in Section 4.2.2, the ratio d/ diverges as 1/(Mu − 1) in the limit Mu → 1. A simple explanation is given few lines below (13.3.32). Using the equations of fluid mechanics across a planar weak shock, (15.1.62), written in the reference frame attached to the shock, takes the form   d λ dT ds = + w˙ s , (15.1.64) ρU dx dx T dx where, according to (15.1.45), the mass flux is constant ρU = ρ− D > 0. Using the subscripts − and + to denote the uniform state of the initial and shocked gas, respectively, integration of (15.1.64) across the shock wave then confirms that the jump of entropy across the shock is positive, s+ > s− , since entropy production (15.1.60) is positive,  ∞ w˙ s dx > 0. (15.1.65) w ˙ s > 0 ⇒ ρ− D(s+ − s− ) = −∞

However, the full calculation, presented in Section 4.2.2, shows that the jump (s+ − s− )/cp is small, of order (Mu − 1)3 and /d = O(Mu − 1); see (4.2.37)–(4.2.40). From a historical point of view, let us recall that the method of characteristics to solve hyperbolic equations, which is recalled in Section 15.3, was introduced in a paper published [1] [2] [3]

´ Hugoniot P., 1889, Journal de l’Ecole Polytechnique, 58(1), 1–125. Rankine W., 1870, Philos. Trans. R. Soc. London, 160, 277–288. Rayleigh J., 1910, Proc. R. Soc. London, 84, 247–284.

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by Riemann[1] in 1860, in which the formation of shock waves is described as a singularity appearing after a finite time in the solutions to Euler’s equations. Once the shock had formed, Riemann calculated its propagation speed using the isentropic relation p/ρ γ = constant through the shock. This relation is not valid for strong shocks, (s+ − s− )/cp = O(1), and must be replaced by the conservation law for total energy (15.1.51). Entropy Production by Reactions When the reaction terms in (15.1.12) are introduced into (15.1.54), additional terms appear in the entropy production (15.1.57). They take the form of a sum over all the reactions, . / 1   (j) ˙ (j) , ϑi μi Mi W (15.1.66) − T j

i

˙ (j) is the reaction rate of the where μi is the chemical potential per unit mass of species i, W (j) jth reaction and ϑi is the stoichiometric coefficient of the ith particle in the jth reaction, defined in (15.1.13). According to the second law (15.1.53), the positivity of each term j in (15.1.66) indicates the sense of evolution of each reaction. When the chemical equilibrium of the jth reaction is reached, the rate of the direct reaction is balanced by the rate of (j) (j) ˙ (j) ≡ (J+ − J− ) = 0, and the so-called affinity of the reaction is zero, inverse reaction, W  (j) i ϑi μi Mi = 0; see (14.2.4) (the notations are slightly different; μi is here defined per unit mass).

15.2 Approximations In this section we present some approximations that are useful to describe the structure and dynamics of flame or detonation fronts and various combustion instabilities.

15.2.1 Exothermic Flows at small Mach Number In exothermic flows with small Mach numbers (velocity smaller than the sound speed, M ≡ u/a 1), which evolve on a time scale that is long compared with the acoustic time, the relative fluctuations in pressure are negligible compared with those of temperature; see Section 2.1.1. The pressure term can be neglected in the energy equation (15.1.34), which then reduces to a simple thermal balance,                1 Dp    1 DT  ⇒  Dp  ρcp DT  .  (15.2.1)  T Dt   Dt    p Dt  Dt  Since the temperature variations are large, the density variations are also large. The density is obtained from the temperature by supposing that the product of temperature and density [1]

Riemann B., 1860, Abhandl. Ges. Wiss. G¨ottingen, 8(43–65). (English translation Int. J. Fusion Energy 2, 1–23 1980).

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is constant. The equations governing combustion then reduce to the following system: ρT = ρo To , where

(15.2.2)

(ρo , To ) is a reference state,  ˙ (j) (T, . . . , Yk , . . .), Q(j) W ρcp DT/Dt = ∇.(λ∇T) + j

ρDYi /Dt = ∇.(ρDi ∇Yi ) +



˙ (j) (T, . . . , Yk , . . .), ϑi Mi W (j)

(15.2.3) (15.2.4)

j

ρDu/Dt = −∇p,

∇.u = −

1 DT 1 Dρ = , ρ Dt T Dt

(15.2.5)

where the Euler equation, the first equation in (15.2.5), is valid when viscous effects are negligible and in the absence of external forces such as gravity. These are the equations of quasi-isobaric combustion governing flames. The second equation of (15.2.5) shows that temperature variations introduce a source term in the velocity field. These hydrodynamic equations are similar to those of incompressible flow where ∇.u = 0, but with the notable difference that the combustion fluids dilate, ∇.u = 0, the energy release playing the role of a volume source. It is necessary to distinguish clearly between dilatation (density changes produced by changes in temperature) and compressibility (density changes produced by changes in pressure). In the approximation of quasi-isobaric combustion compressible effects are negligible, but density changes arising from changes in temperature are not. The flow field appears only in the total derivative D/Dt ≡ ∂/∂t + u.∇ in the two balance equations (15.2.3) and (15.2.4). Without that, these two equations alone would form an autonomous system. This is the case for stationary planar flames studied in Chapter 8. Equations (15.2.2)–(15.2.5) can be used to describe the dynamics of strongly subsonic wrinkled flames. In short, when the evolution is slow compared with the acoustic time, strongly subsonic flows are modified by combustion processes in a manner that is isobaric in the equation of state (15.2.2) and in the energy equation (15.2.3). This is not the case, however, neither for detonations, which are supersonic waves, nor for thermo-acoustic instabilities in which the energy release fluctuates on the same time scale as the acoustic waves. 15.2.2 Inviscid Inert Flows at Small Mach Number Consider an adiabatic, nonreactive nondissipative flow that is uniform at infinity. According to (15.1.61) with λ = 0, μ = 0, the production of entropy is zero, and the entropy is constant. Assume also that the Mach number is small, M 1. To leading order in the limit M → 0, the density is constant, since δρ/ρ = O(M 2 ) (see Section 2.1.1) and the fluid can be considered as incompressible, ρ = cst., ∇.u = 0. The flow is described by Euler’s equation (15.1.18), which in the presence of gravity can be written ∇.u = 0,

ρDu/Dt = −∇(p + ρgz).

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(15.2.6)

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where the z-axis is oriented upwards. The pressure in Euler’s equation should be considered as a functional of the velocity field; the thermodynamic relations are useful only to compute the negligibly small variations of density and temperature. Vorticity Using the relation u.∇u = ∇.u = 0,

∇|u|2 /2 − u × (∇ × u), Euler’s equation (15.2.6) takes the form

∂u/∂t − u × (∇ × u) = −∇(p/ρ + |u|2 /2 + gz).

(15.2.7)

In planar geometry, the vorticity ∇ × u is perpendicular to the flow, ∇ × u = e⊥ (e⊥ is a unit vector), and the rotational of the second term in the left-hand side of (15.2.7) takes the form u.∇e⊥ . Taking the rotational of (15.2.7) shows that the vorticity is conserved in an invisicid and incompressible two-dimensional flow, D(∇ × u)/Dt = 0,

(15.2.8)

since the rotational of a gradient is zero. The case of an incompressible viscous flow in three-dimensional geometry in given after (15.2.13). Using Stokes’s theorem to transform the integral of (constant) vorticity over a surface into the line integral of velocity around the boundary of the surface leads to Thomson’s theorem: the circulation of velocity along a closed loop of fluid is constant. Bernoulli Equation for Potential Flows If the flow is uniform (irrotational) at infinity, it follows from (15.2.8) that it is irrotational everywhere, ∇ × u = 0. In other terms the flow is potential, u = ∇φ(r, t), and Equations (15.2.7) take the form of Bernoulli’s equation, ∂φ/∂t + p/ρ + |u|2 /2 + gz = C(t),

φ = 0,

(15.2.9)

where C(t) is a gauge function that is usually determined by the boundary conditions for the pressure. Rotational Two-Dimensional Flows In two-dimensional flow, u = uex + wey , the vorticity is represented by a scalar field,  = ∂w/∂x − ∂u/∂y. It is useful to introduce the stream function ψ, u = −∂ψ/∂y,

w = ∂ψ/∂x,

ψ = .

(15.2.10)

For incompressible flows, Equations (15.2.7) then take a form which will be useful in the absence of gravity in Chapter 2,   1 ∂u 2 (15.2.11) − ρ∇ψ = −∇ p + ρ|u| . ∇.u = 0, ρ ∂t 2 For a steady flow, ∂u/∂t = 0, the iso-ψ curves being orthogonal to the streamlines, u.∇ψ = 0, Equation (15.2.11) shows that the Bernoulli equation reduces to ∇.u = 0,

∂u/∂t = 0,

1 p + ρ|u|2 = cst. along each streamline, 2

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(15.2.12)

15.2 Approximations

657

but, due to nonzero vorticity, the constant differs from streamline to streamline.

15.2.3 Reynolds Number. Stokes Flows In an inert viscous flow at small Mach number (incompressible flow ∇.u = 0), in the absence of external forces, the Navier–Stokes equation (15.1.17) reduces to 1 Du = − ∇p + νu, Dt ρ

∇.u = 0,

ν ≡ μ/ρ,

(15.2.13)

and the vorticity equation takes the form[1] D/Dt = .∇u+ν. Introducing the length L over which significant changes in the flow take place and letting U be a representative velocity in the flow, a typical convective transport term |u.∇u| is of order U 2 /L and the diffusion (viscous) term is of order νU/L2 . The Reynolds number is the ratio of these quantities: Re ≡

UL . ν

(15.2.14)

For a steady flow at small Reynolds number, Re 1, the convective transport term is negligible, so that (15.2.13) reduces to the so-called Stokes equations, Re 1, M ≡ U/a 1:

∇.u = 0,

∇p = μu,

⇒

p = 0.

(15.2.15)

The classical shear flows, Couette flows Poiseuille flows and other two-dimensional viscous flows are solved with this approximation. Viscous Drag. Stokes (1851) The drag F of a solid sphere of radius R propagating at constant velocity U into a viscous flow (Stokes drag), F = 6π μRU,

(15.2.16)

is computed from the pressure distribution of the stationary (laminar) flow around a sphere, the solution to (15.2.15). This classical problem in fluid mechanics is presented in textbooks.[1,2] The expression in (15.2.16) can be obtained by a dimensional analysis based on entropy production. As explained in Section 15.1.7, mechanical energy is transferred  into internal energy at the rate of entropy production, FU = T w˙ s d3 r. Assuming that the deformation of the flow is characterised by the length R, the order of magnitude of T w˙ s is, according to (15.1.60), μU 2 /R2 , and the volume concerned is of order R3 , so that F is of order μRU as in (15.2.16). In compressible flows, the damping of acoustic waves (Kirchhoff’s formulae) can be obtained in the same way using entropy production.[2]

[1] [2]

Batchelor G., 1967, An introduction to fluid dynamics. Cambridge University Press. Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed.

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15.2.4 Acoustic Coupling and Thermo-Acoustic Instabilities The emission of sound by combustion, studied in Section 3.3 for the particular case of a turbulent burner, is a subject of current interest in the fight against acoustic pollution, particularly in aeronautics. The study of the interaction between gaseous combustion and acoustics is facilitated by rewriting the energy conservation equation in a form where the temperature has been eliminated with the help of the equation of state for gases, p = (cp − cv )ρT, written as ρcp

cp D cv D D T= p− a2 ρ, Dt cp − cv Dt cp − cv Dt

(15.2.17)

where a is the sound speed, a2 = (cp /cv )(cp − cv )T and cp and cv are supposed constant to simplify the presentation. Using (15.2.17) to eliminate the temperature in (15.1.34), the energy equation takes the form Dp/Dt − a2 Dρ/Dt = q˙ γ , where

q˙ γ ≡ (γ − 1) [−∇.Jq +



˙ (j) ], Q(j) W

(15.2.18)

Jq = −λ∇T,

γ ≡ cp /cv . (15.2.19)

j

Equation (15.2.18) shows how local energy addition modifies adiabatic compression. The term q˙ γ is, to within a factor (γ − 1), the energy received per unit volume and unit time through thermal conduction and exothermic reaction. In the absence of this term, (15.2.18) describes the propagation of an isentropic compression wave in an inert gas. Neglecting the pressure, and using (15.2.3) and the equation of state (15.1.29), Equation (15.2.18) gives the standard expression for isobaric gas expansion, (1/ρ)Dρ/Dt = −(1/T)DT/Dt. The linearised versions of Equations (15.1.33), (15.2.18) and (15.2.19), ρ = ρ + ρ,

p = p + p ,

u = u + u ,

q˙ γ = q˙ γ + q˙ γ ,

describe the propagation of acoustic waves in a reactive medium. Neglecting field gradients and the Doppler effect of the mean flow to simplify the presentation, ∇a ≈ 0, u.∇ ≈ 0, gives ∂ρ  /∂t = −ρ∇.u ,

ρ∂u /∂t = −∇p ,

∂p /∂t − a2 ∂ρ  /∂t = q˙ γ .

(15.2.20)

Eliminating ρ  and u between these equations yields a relation in the form of the wave equation driven by a source term ∂ q˙ γ /∂t, ∂ 2 p /∂t2 − a2 p = ∂ q˙ γ /∂t.

(15.2.21)

In the study of pure acoustics with no source term, q˙ γ = 0 (the isentropic approximation), it is usual to express the unknowns as functions of a potential φ that also satisfies the

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659

wave equation, ∂ 2 φ/∂t2 − a2 φ = 0, u = ∇φ,

(15.2.22)

p = −ρ∂φ/∂t.

In the presence of a source term q˙ γ = 0, (15.2.22) becomes ∂ 2 φ/∂t2 − a2 φ = −˙qγ /ρ. Sound Emission Consider the free turbulent burner of Section 3.3 in open space. Let ω be the characteristic frequency of fluctuations of the heat release rate q˙ (r, t); the acoustic wave introduces a length scale equal to the wavelength λ = a/ω. On this scale of distance, the compressibility of the gas plays an essential role. However, for turbulent burners in free space, the size of the flame zone, L, is often much smaller than λ, pressure gradients are negligible across the flame and its dynamics are described by the equations of the quasi-isobaric approximation, that is, by a quasi-uniform pressure field whose time evolution has no significant effect on combustion. Moreover, in the acoustic equations, the source term in Equation (15.2.21), associated with energy release, can be considered to be localised at a point in space and can be written as ¨ ∂ q˙ γ (r, t)/∂t = δ(r)(t),

   ¨ ˙ ˙ where we have used the notation (t) = ∂ (t)/∂t and (t) = q˙ γ (r , t)d3 r denotes the total instantaneous heat release rate per unit time. The sound emitted from this point propagates in the nonreactive burnt gas. Green’s retarded propagator for the wave equation, G(r, t) = aδ(r − at)/4π r,

(1/a2 )∂ 2 G/∂t2 − G = δ(r)δ(t),

can be used to write the acoustic pressure field in spherical geometry, radiated by combustion as  ¨ − r/a) 1 (t 1 ∂   q˙ γ (r , t − r/a)d3 r = p (r, t) = , r = |r| . (15.2.23)  2 r ∂t 4π a 4π a2 r The power spectrum of the emitted sound can thus be related to the Fourier  transform  of ¨ (0) ¨ the autocorrelation function of the time derivative of the heat release rate, (t) . When the main source of fluctuation of the heat release rate is the fluctuation in total ˙ surface area of the flame, S(t), (t) = ρu UL cp (Tb − Tu )S(t), as for the wrinkled flame anchored to the exit of the turbulent burner in Section 3.3, the combustion noise is the signature of the dynamics of the flame front and thus also of the turbulence of the flow;[1] see Equation (3.3.2) and Section 3.4.3. In other cases it is possible for the intrinsic instabilities

[1]

Clavin P., Siggia E., 1991, Combust. Sci. Technol., 78, 147–155.

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of the flame front to play a dominant role, as for the oxygen–acetylene flame mentioned in Section 3.3.3. Another cause of sound generation occurs in combustion chambers when entropy spots of nonuniform temperature go though the nozzle. This source of acoustic disturbances has been analysed analytically in the 1970s[1] and is now considered as an important contribution to the noise emitted by modern turbo-reactors. This problem is not considered in this book. The Rayleigh Criterion (1896). Thermo-acoustic Instability When combustion takes place in a confined system, combustion noise can generate standing acoustic waves and thermo-acoustic instabilities can occur if the acoustic wave has a retroaction on the combustion rate. To illustrate this, consider a limiting situation contrary to the precedent where combustion is homogeneously distributed throughout a cavity (combustion chamber), represented by the approximation of a well-stirred reactor. We neglect the incoherent combustion fluctuations produced by turbulence and concentrate on perturbations produced by an acoustic wave. To do this, we introduce the response function A(τ ) that describes, in the linear approximation, the local response of the energy release rate q˙ v (x, t) to a fluctuation of the acoustic pressure,  t 1 A(t − t )δp (x, t )dt , (15.2.24) δ q˙ v (x, t) = τins −∞  where A(τ ) is a function of time, normalised to unity, A(τ )dτ = 1, and whose dimension is the inverse of time. The term 1/τins is a positive coefficient measuring the strength of the coupling and whose dimension is the inverse of time. The pressure fluctuation p has the dimensions of energy per unit volume, and q˙ v (x, t) is the fluctuation of energy release rate per unit volume. The transfer function, A(τ ), takes account of a possible delay between the pressure fluctuation and the response of the energy release rate. This can be the case, for example, in a rocket engine combustion chamber when the flow of reactants through the injectors is inversely related to the pressure in the chamber (with no delay), but the heat release occurs a finite time later, after a delay controlled by mixing of the reactants and/or by the chemical induction time. Many other mechanisms, such as liquid droplet evaporation, the response of combustion chemistry to pressure and temperature, velocity and acceleration induced deformations of flame shape, and so on, may contribute to thermo-acoustic couplings, each with a characteristic delay. The transfer function A(τ ) and the time τins characterise the system. They can be obtained from a detailed study of the mechanism(s) by which the pressure fluctuations modify the combustion processes. This key part of the study is the most difficult, and the results depend strongly on the system under study. [1]

Marble F., Candel S., 1977, J. Sound Vib., 55(2), 225–243.

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Homogeneous reactive mixture

Figure 15.1 One-dimensional homogeneous combustion chamber closed at left end and open at right. The line shows the acoustic pressure for the first harmonic (3λ/4 mode)

Let us suppose that A(τ ) and τins are known and look at the resulting evolution of acoustic waves in a reactive medium. For simplicity, we consider a simple one-dimensional case in which the combustion chamber is a long tube of length L, supporting planar longitudinal acoustic modes, boundary layers being neglected; see Fig. 15.1. The spatial Fourier transform of a longitudinal acoustic mode is  pk (t)eikx , k = nπ/L, (15.2.25) p (x, t) = k

where the selection of the integers, n, depends on the acoustic boundary conditions (node or antinode) at the extremities of the tube. As a first step consider the simple case when fluctuations of the heat release rate, q˙ v (x, t), are either in phase with the pressure, q˙ v (x, t) = p (x, t)/τins ,

A(τ ) = δ(τ ),

τins > 0,

or in phase opposition, q˙ v (x, t) = −p (x, t)/τins ,

A(τ ) = −δ(τ ),

τins > 0.

The source term in Equation (15.2.21) is then 1  d ∂  q˙ v (x, t) = ± pk (t)eikx . ∂t τins dt k

Equation (15.2.21) then takes the form of an oscillator that is amplified or damped at a rate 1/τins according to whether the fluctuations of pressure and heat release are in or out of phase: 1 dpk d2 pk + a2 k2 pk = 0. ∓ τins dt dt2

(15.2.26)

This is the Rayleigh criterion[2] for thermo-acoustic instability. In the general case, let τd (ω) be the delay so that ωτd is the phase lag between the two fluctuations at frequency ω,  +∞ A(τ ) = r(ω)eiωτd (ω) eiωτ dω + c.c., −∞

[2]

Rayleigh J., 1945, The theory of sound, vols. 1 and 2. New York: Dover.

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where r(ω) is a function of frequency, defined as positive, and c.c. stands for complex conjugate. A thermo-acoustic instability tends to occur on a mode k when the following phase condition is verified, −π/2 < ωk τd (ωk ) < +π/2, where ωk = ak is the acoustic frequency. In the opposite case, combustion damps the acoustic mode at this frequency. It is often the case that a given chamber is thermo-acoustically damped for some resonant frequencies but amplified for others, where the same delay produces a favourable phase shift. The stability of a combustion chamber also depends on the other sources of acoustic damping (boundary layers, radiative acoustic losses, etc.) that must be included in the analysis to obtain the stability limits. Obtaining the function A(τ ) and the coupling rate 1/τins , characterising the effect of acoustics on combustion in a given chamber, is a delicate and difficult operation. The most dangerous mechanisms, that is, those that have the highest coupling rate, are often ‘velocity couplings’ that modify the combustion rate through the periodic acceleration produced by acoustic waves. These mechanisms are generally stronger than ‘pressure couplings’ where the energy release rate is modulated by pressure fluctuations. Except for the case of the effect of pressure on injection rates, most pressure couplings have only a small relative effect, of the order of the propagation Mach number of the flame front. In general, the amplification rate 1/τins is often of the same order of magnitude as the rate of acoustic loss through dissipative and radiative mechanisms. This implies that the design of a stable high-power combustion chamber is a delicate undertaking; a change in a small detail can change the global stability. Long test campaigns are often necessary to validate approximate calculations and sometimes produce very disagreeable surprises. In particular, it is found that chambers can be linearly stable (i.e. stable for small acoustic amplitudes), but become strongly unstable through a nonlinear effect for higher acoustic amplitudes produced by ignition or other transient changes in the combustion regime. Instability of Rocket Engines Spontaneous acoustic instabilities are dangerous, particularly in the context of rocket engines, where the thermal power density is of the order of 10 gigawatts/m3 and the conversion of ≈ 0.5% of thermal energy into acoustic energy is sufficient maintain acoustic pressure oscillations exceeding half the main chamber pressure. Such an instability can lead to rapid destruction of the chamber, in a few hundred acoustic cycles, by breakdown of the thermal boundary layer isolating the metal chamber wall from the hot gas at ≈ 3000 K and/or by the effect of the intense mechanical vibrations produced by the violent pressure oscillations. Another famous instability of rocket engine, called the pogo effect, is a low-frequency instability related to the inertia of the high-pressure injection pumps, often also coupled with the ρgh pressure of the feed lines when the rocket accelerates or decelerates.

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663

Zeldovich Criterion for a Spontaneous Ignition of Detonations Another compressible phenomenon, reminiscent of Rayleigh’s criterion, is the spontaneous ignition of a detonation, presented in Section 4.3.4. It can be simply described with the help of Equations (15.2.21) for acoustic waves and (1.2.2) for the reaction time (Arrhenius’s law).

15.3 One-Dimensional Compressible Flows The theory of compressible flows in planar geometry, including planar shock waves and detonations propagating at constant speed, goes back to the nineteenth century. A review of the early work on planar shock waves was presented by Lord Rayleigh.[1] The full understanding of their formation and inner structure was completed in the mid-twentieth century. These classical results are recalled briefly in this section. They are useful to understand modern theory and the multidimensional dynamics of shock waves and detonations, presented in the main body of the book, including Mach stem formation on these fronts. The presentation here is a synthesis of that in classical books.[2,3,4] Planar Geometry When a piston is put into motion in a tube filled initially with gas at rest, its motion is transmitted into the adjacent layers of gas. If the formation of boundary layers at the walls is neglected, the resulting flow is one-dimensional in the first approximation. In planar geometry the progressive acceleration of successive layers of gas implies the presence of velocity gradients and thus compressible effects. When dissipative effects (viscosity, heat conduction) are neglected, the problem is described by Euler’s equations (15.1.44), ∂ρ/∂t + ∂(ρu)/∂x = 0,

∂(ρu)/∂t = −∂(p + ρu2 )/∂x,

(15.3.1)

where the second equation may also be written ∂u ∂u 1 ∂p +u =− . ∂t ∂x ρ ∂x

(15.3.2)

According to the laws of thermodynamics, the pressure may be expressed as a function of mass density and specific entropy, p = P(ρ, s). For a polytropic gas, this relation is, according to (15.1.58), po p = γ e(s−so )/cv , (15.3.3) γ ρ ρo where the subscript o denotes a reference state. Two cases should be considered: if dissipative transport (heat conduction, viscosity, etc.) is negligible everywhere so that entropy is constant throughout the flow field, s = cst., [1] [2] [3] [4]

Rayleigh J., 1910, Proc. R. Soc. London, 84, 247–284. Courant R., Friedrichs K., 1967, Supersonic flow and shock waves. John Wiley. Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed. Whitham G., 1974, Linear and nonlinear waves. John Wiley.

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Equations (15.3.1)–(15.3.3) constitute a closed system. However, if, due to boundary conditions, the entropy changes with time at some locations, as is the case on the compressed gas side of an unsteady shock wave, and if dissipative transport is negligible elsewhere, Equation (15.1.59) for the entropy balance with zero entropy production must be added to Equations (15.3.1),  ∂s ∂p ∂p ∂ρ ∂s 2 ∂ρ +u =0 ⇔ +u −a +u = 0, (15.3.4) ∂t ∂x ∂t ∂x ∂t ∂x √ where a = dp/dρ|s=cst. is the speed of sound, a2 = γ p/ρ for a polytropic gas; see (15.3.3). The linearised version of (15.3.1)–(15.3.3) in a uniform medium describes plane acoustic waves propagating without deformation at the speed of sound. The effects of the nonlinear terms in these hyperbolic equations were first studied by Poisson[1] and solved by Riemann.[2] For an initial condition associated with a localised perturbation in an infinite medium, the solution becomes stiff and finally multivalued in a finite time; see Fig. 15.4. From a mathematical point of view the nonlinear problem is ill-posed with the Euler equations (15.3.1)–(15.3.3). So what happens to the physical system? The answer is given by the Navier–Stokes equations for which Euler’s equations are an approximation in the limit of weak gradients. In the thin regions where the gradients introduce strong dissipative effects, viscosity and heat conduction become important and entropy production can no longer be neglected. For the piston problem, a thin zone of strong gradients, called a shock wave (or simply a shock), is formed after a finite time and then propagates at a supersonic velocity with no further deformation as long as the piston is maintained at constant speed. Thus the setting into motion of the gas is obtained ultimately by the propagation of a shock wave, considered in the first approximation to be a discontinuity that is formed at the moment when the velocity gradient becomes infinite. It turns out that the shock waves, considered as discontinuities, can be included into the solution of Euler’s equations (15.3.1) without recourse to Navier–Stokes equations, provided that the isentropic condition (p/ρ γ =cst.) is abandoned across the discontinuity and replaced by the equation for the conservation of total energy including the internal energy; see (15.1.48). As briefly explained at the end of Section 15.1.7, the dissipative effects control the inner structure of shock waves, but the transport coefficients do not appear in the relation linking the propagation velocity and the jumps across the discontinuity.[3,4] Even though the problem of shock formation was solved a long time ago, this nice and subtle piece of work is worth recalling for the understanding of the topics studied in the present book. The solution is presented below in a didactic way, starting from simple model equations to end up with those of fluid mechanics. However, we must first make a remark concerning nonplanar geometry. [1] [2] [3] [4]

´ Poisson S., 1808, Journal de l’Ecole Polytechnique, 14(7), 319–392. Riemann B., 1860, Abhandl. Ges. Wiss. G¨ottingen, 8(43–65). (English translation Int. J. Fusion Energy 2, 1–23 1980). Rankine W., 1870, Philos. Trans. R. Soc. London, 160, 277–288. ´ Hugoniot P., 1889, Journal de l’Ecole Polytechnique, 58(1), 1–125.

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Cylindrical and Spherical Geometry Using notations similar to that used for the planar case, with x and u denoting the radial coordinate and the radial component of the fluid velocity, respectively, only the first equation in (15.3.1) (continuity) is modified since ∇.u = (1/xj )(∂(xj u)/∂x), where j = 1, 2 for cylindrical and spherical waves, respectively. Equations (15.3.1) then become ∂ρ (∂ρu) j + + ρu = 0, ∂t ∂x x

∂u ∂u 1 ∂p +u =− , ∂t ∂x ρ ∂x

(15.3.5)

to be solved together with (15.3.4).

15.3.1 Model for the Formation of Discontinuities We start with a model equation simpler than Euler’s equations. The linear equation describing the propagation of a field u(x, t) at constant speed ao in a given direction (progressive wave) is ∂u/∂t + ao ∂u/∂x = 0. The solution is u = uo (x − ao t), where uo (x) is the initial condition, u(x, t = 0) = uo (x), and the constant propagation velocity ao is positive for propagation from left to right. The prototype of a nonlinear hyperbolic problem is obtained by generalising the above problem to the case where the propagation speed is a function of the propagating quantity, u, ∂u/∂t + a(u)∂u/∂x = 0.

(15.3.6)

Consider now the problem defined by an initial condition u(x, t = 0) = uo (x) (Cauchy’s problem). Method of Characteristics The method of characteristics is a technique for solving partial differential equations. The method is to look for trajectories (or characteristics) in parameter space along which the partial differential equation reduces to an ordinary differential equation. For (15.3.6), the characteristics are particularly simple. In the (x, t) plane, the quantity u is conserved along any trajectory obeying dx/dt = a(u), Du/Dt ≡ ∂u/∂t + a∂u/∂x = 0. Along this trajectory u keeps its initial value, u = uo , and thus the propagation speed a(u) is also constant, a = ao ≡ a(uo ). These characteristics are simply straight lines in the (x, t) plane, x = ao t + xo , shown in Fig. 15.2, and the solution may be written u = uo (x − ao t), where uo (x) denotes the initial velocity profile, uo (x) ≡ u(x, t = 0). If the propagation speed increases with increasing u, da/du > 0, the initial profile of propagation speed, ao (x) ≡ a(uo (x)), has the same qualitative shape as the profile of uo (x); see Fig. 15.3. High values of u propagate at a speed a(u) faster than small values. For an initial profile containing only positive values of u and having a region of decreasing values, duo /dx < 0,

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Figure 15.2 Characteristics of (15.3.6) with dao /dxo < 0 are drawn as dotted lines. As time advances the characteristics intersect. The envelope enclosing the intersections is drawn in bold, inside this envelope the solutions to (15.3.6) are multi-valued. The construction is the same as that of Huygens for caustics.

Figure 15.3 Initial conditions.

Figure 15.4 Transition to multivalued solutions, or wave breaking.

the profile u(x, t) becomes multivalued (or ‘breaks’, analogous to a wave breaking on the shore) after a finite time; see Fig. 15.4. Effectively the characteristics, x = xo + ao (xo )t, with dao /dxo < 0 (da/du > 0, duo /dx < 0), cross as time advances, each transporting a different value of the scalar u; see Fig. 15.2. The profile u(x, t) becomes steeper and a transition occurs at time t = tb when the slope becomes infinite, ∂u/∂x = −∞; see Fig. 15.4. Before this time, t < tb , each point in the plane (x, t) is traversed by a single characteristic, so that there is a single initial position that is associated with a point (x, t). The initial position xo can then be considered to be a function of x and t, xo (x, t). Since u is constant and equal to uo on a characteristic, x = xo + ao (xo )t, we can thus write u(x, t) = uo (xo (x, t)),

∂u/∂x = (∂uo /∂xo )(∂xo /∂x).

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Using the relation ∂x/∂xo = 1 + t(dao /dxo ), one obtains ∂u/∂x = (duo /dxo )/[1 + t(dao /dxo )]. For a given value of u, the slope becomes infinite after a time t = 1/(−dao /dxo ) with dao /dxo < 0. The time, tb , at which the wave ‘breaks’ is given by the value of uo for which the quantity |dao /dxo | = (da/duo )|duo /dxo | is maximal and which is the inflection point of the initial profile of propagation speed: d2 ao /dxo2 = 0. Discontinuous Solutions In contrast to waves on the sea, multivalued solutions are not acceptable as physical solutions for the speed of gas in a tube and the initial value problem has no continuous solution after the instant at which the wave ‘breaks’. Discontinuous solutions to (15.3.6) also exist. Let us look for those that have a jump between two constant uniform values on either side of the discontinuity, u+ = u− , and let us suppose that the discontinuity moves at constant speed. What is the propagation speed D of such a discontinuity? Writing (15.3.6) in a conservative form, ∂u/∂t + ∂j/∂x = 0,

where

d j(u) = a(u), du

(15.3.7)

and rewriting in the moving frame in which the discontinuity is supposed to be stationary u(ξ ), ξ = x − Dt, Equation (15.3.6) becomes −D

dj du + = 0. dξ dξ

This result shows that the quantity j − Du is conserved across the discontinuity and the propagation speed is a function of the values u+ and u− , D=

j(u+ ) − j(u− ) . u+ − u−

(15.3.8)

This solution, however, is not unique. Multiplying Equation (15.3.6) by an arbitrary function, f (u), and using the same procedure yields a different arbitrary propagation speed, G(u+ ) − G(u− ) , F(u+ ) − F(u− ) d d F(u) = f (u) and G(u) = f (u)a(u). du du D=

where

(15.3.9)

Which solution is effectively selected by the physical problem? In other words, which quantity is effectively conserved across the discontinuity? There is no answer to this question in the strict framework of Equation (15.3.6); the problem is ill-posed. To solve the problem, Equation (15.3.6) should be considered to be a simplified approximation of a

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more general equation including a dissipative term, as in fluid mechanics where Euler’s equations are obtained by neglecting viscosity and thermal conductivity. A dissipative term, no matter how weak, removes the degeneracy and selects the physical solution. Adding a weak dissipative term to (15.3.6), ∂u/∂t + a(u)∂u/∂x = ∂ 2 u/∂x2 ,

 > 0,

(15.3.10)

the conserved quantity becomes unique: ∂u/∂t + ∂J/∂x = 0,

where

J(u) = j(u) − ∂u/∂x.

(15.3.11)

For the case of a wave propagating at a constant speed and separating two constant uniform states, u− for x → +∞ and u+ for x → −∞,  d du −uD + j(u) −  = 0. (15.3.12) dx dx The internal structure of the jump is given by double integration of (15.3.12), 

du = j(u) − uD + C, dx

where C is an integration constant, x = 

 0

u

du , j(u) − uD + C

(15.3.13)

and the origin x = 0 is arbitrary. Equation (15.3.13) shows that the thickness of the front is proportional to . The unknown D and the constant C are given by the boundary conditions at x = ±∞: du/dx = 0, j(u+ ) − u+ D + C = 0,

j(u− ) − u− D + C = 0.

(15.3.14)

Eliminating the constant C, these relations yield a unique expression for the propagation speed D, independent of  and given by (15.3.8) when u+ and u− are prescribed. Law of Equal Areas. Shock Capture For a given initial condition, the trajectory of the discontinuity xc (t) can be found directly from the multivalued solution to (15.3.6) by a law of equal areas, shown in Fig. 15.5. This property is proved below for the particular case where the propagating quantity is the propagation speed, or in other words when the nonlinear term is quadratic, a(u) = u,

j(u) = u2 /2,

∂u/∂t + u∂u/∂x = 0,

(15.3.15)

and the propagation speed will be shown to be in agreement with (15.3.8), D=

u+ + u− . 2

(15.3.16)

According to the method of characteristics, u(x, t) = uo (xo ),

x = xo + uo (xo )t,

17:11:45 .017

(15.3.17)

15.3 One-Dimensional Compressible Flows

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Figure 15.5 Law of conservation of areas.

Figure 15.6 Conservation of areas.

the multivalued profile at an instant t > tb in Fig. 15.5 is obtained from the initial profile uo (xo ) by translating each point horizontally a distance tuo (xo ). Consider now a vertical secant at position xc (t) such that the two surfaces shown on the right of Fig. 15.5 have the same area. At each instant, this construction defines a unique position xc (t), and two values, u+ (t) and u− (t), that change in time. We will show that xc (t) is the position of the shock by verifying that xc (t) moves at the speed (15.3.16). By inverse transformation, this line of discontinuity corresponds to the sloping secant on the initial profile, uo (xo ), shown on the left of Fig. 15.5. For an arbitrary initial profile, the abscissa of the intersection points, xo+ (t) and xo− (t), change in time with the values of u± , u+ (t) ≡ uo (xo+ (t)) and u− (t) ≡ uo (xo− (t)). According to the equation of the characteristics, (15.3.17), the following relations hold: xc (t) = xo− (t) + tu− (t) = xo+ (t) + tu+ (t).

(15.3.18)

We now show that the secant on the curve uo (xo ) also obeys the law of equal areas by showing that the area enclosed between the curve u(x, t) and any secant is invariant by application of (15.3.17). This is done in Fig. 15.6 by verifying that the time derivative x (t) of the integral I(t) = x+−(t) u(x, t)dx and the time derivative of the area of ABCD, S(t) = [x− (t) − x+ (t)](u− + u+ )/2, are equal, so that the difference between the two is constant. The conservative form (15.3.7), along with relations (15.3.15), (15.3.17) and

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Flows

dx± /dt = u± , yields dx− dx+ 1 dS dI = u− − u+ − [j(u− ) − j(u+ )] = (u2− − u2+ ) = . (15.3.19) dt dt dt 2 dt The grey areas in the left of Fig. 15.5 are thus equal and the area under the secant is simply  xo− (t) 1 uo (xo )dxo = [xo− (t) − xo+ (t)][u− (t) + u+ (t)] (15.3.20) 2 xo+ (t) Equations (15.3.18) and (15.3.20) can be used to show that the propagation speed of xc is given by (15.3.16). The time derivatives of (15.3.18) give x˙ c = [1 + tuo (xo− )]˙xo− + u− , x˙ c = [1 + tuo (xo+ )]˙xo+ + u+ . Taking the half sum of the above two equations and replacing the time by t = −(xo− − xo+ )/(u− − u+ ),

(15.3.21)

obtained from (15.3.18), gives  (u− + u+ ) 1 (xo− − xo+ )   + (˙xo− + x˙ o+ ) − {˙xo− uo (xo− ) + x˙ o+ uo (xo+ )} . x˙ c = 2 2 (u− − u+ ) (15.3.22) The time derivative of (15.3.20) shows that the term in square brackets is null and so the propagation speed of the discontinuity is effectively equal to that of the shock, D in (15.3.16). Finally, from (15.3.21) we see that the time needed for the shock to reach a point xc is equal to the inverse of the slope of the secant imaged onto the initial profile. The time of formation of the shock (shortest time) is thus given by the slope of the tangent at the point of inflection on the leading edge of the initial profile, as already argued above.

15.3.2 Burgers’ Equation The above results can also be obtained directly from the solution of the time-dependent dissipative problem. For a given initial condition, uo (x), it is possible to solve Equation (15.3.10) in an infinite medium when the nonlinear term is quadratic (Burgers’ equation), ∂u/∂t + u∂u/∂x = ∂ 2 u/∂x2 ,

 > 0.

(15.3.23)

Hopf–Cole Transformation Burgers’ equation can be transformed  x into a linear diffusion equation by the Hopf–Cole transformation. Introducing φ ≡ 0 dx u(x , t), u = ∂φ/∂x, Equation (15.3.23) can be put in the form ∂φ/∂t + (∂φ/∂x)2 /2 = ∂ 2 φ/∂x2

17:11:45 .017

(15.3.24)

15.3 One-Dimensional Compressible Flows

671

by integrating along x. Introducing the quantity ψ ≡ e−φ/2 , it is easy to show that ψ verifies the diffusion equation, ∂ψ/∂t = ∂ 2 ψ/∂x2 .

(15.3.25)

Using Green’s function, G(x, t) ≡ (4π t)−1/2 e−x

2 /t

, ∂G/∂t = ∂ 2 G/∂x2 , G(x, t = 0) = δ(x),

where δ(x) is the Dirac distribution, the solution to the initial value problem with φ(x, t = 0) = φo (x) takes the form  " !  +∞ 1 1 (x − xo )2 ψ(x, t) = √ φo (xo ) + . (15.3.26) dxo exp − 2 2t 4π t −∞ The initial condition on u is u(x, t = 0) ≡ uo (x) = ∂φo /∂x, and the solution for u is obtained from the relation u(x, t) = ∂φ/∂x = −2(∂ψ/∂x)/ψ,

8 7  +∞ x−xo

(x−xo )2 1 φ exp − dx (x ) + o o o −∞ t 2 2t

8 7 u(x, t) = . (15.3.27)  +∞ (x−xo )2 1 −∞ dxo exp − 2 φo (xo ) + 2t In the following we will suppose that uo (x) > 0, or in other words that the direction of propagation is towards increasing x. WKB Method in the Limit of Weak Dissipation 0+ ,

the dominant contribution to the integrals in (15.3.27) comes from In the limit  → the region close to the value xo = xm (x, t), which minimises the square brackets in the exponents and is the solution to uo (xm ) −

(x − xm ) = 0. t

(15.3.28)

Let us suppose for the moment that there is only one minimum, xm . In the limit  → 0+ , the dominant order of an integral of the form  +∞   , dxo f (xo )e−g(xo )/2 is I ≈ f (xm )e−g(xm )/2 4π /gm (15.3.29) I= −∞

 gm

stands for the second derivative of g(xo ) at its minimum, xo = xm . This result is where found by developing g(xo ) to second order in a Taylor expansion around xm (WKB method, also called the real saddle method). Using (15.3.29), Equation (15.3.27) yields (x − xm ) , (15.3.30) t where xm (x, t) is given by (15.3.28). This corresponds to the solution to (15.3.6) for the particular case a(u) = u, since u is constant along the characteristics x = xm + uo (xm )t, u = uo (xm ). As mentioned above this solution becomes multivalued after the solution breaks, although the exact solution (15.3.27) remains continuous and single valued for all times u(x, t) ≈

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t > 0. This contradiction comes from the fact that there are in fact two distinct minima, that is, two solutions to Equation (15.3.28), xm1 (x, t) and xm1 (x, t), where by definition xm1 > xm2 , uo (xm1 ) −

(x − xm1 ) = 0, t

uo (xm2 ) −

(x − xm2 ) = 0. t

The dominant order of (15.3.27) then becomes x−xm1   −1/2 −g /2  x−xm2   −1/2 −g /2  (gm1 ) + (gm2 ) e m1 e m2 t t u≈ ,   −1/2 −g /2 −1/2 −g /2 (gm1 ) e m1 + (gm2 ) e m2

(15.3.31)

(15.3.32)

where gmi ≡ g(x, xo = xmi , t), i = 1, 2, with, by definition, g(x, xo , t) ≡ φo (xo ) + (x −  is the second derivative of g with respect to x taken at the points xo )2 /(2t), and where gmi o xo = xmi (x, t), which are solutions to (15.3.28) corresponding to the two minima of g at fixed x and t. In the limit  → 0+ , the first or the second term in the numerator and denominator dominates, according to the relative values gm2 > gm1 or gm2 < gm1 , to give  → 0+ ,

gm2 > gm1 : u ≈

x − xm1 , t

gm1 > gm2 : u ≈

x − xm2 . t

(15.3.33)

The transition occurs on the trajectory x(t) defined by gm1 (x, t) = gm2 (x, t)

or

φo (xm1 ) +

(x − xm1 )2 (x − xm2 )2 = φo (xm2 ) + . 2t 2t

(15.3.34)

Using (15.3.31) in the form (x − xm1 )2 = (x − xm1 )uo (xm1 ) = [tuo (xm2 ) + xm2 − xm1 ]uo (xm1 ), t (x − xm2 )2 = (x − xm2 )uo (xm2 ) = [tuo (xm1 ) + xm1 − xm2 ]uo (xm2 ), t Equation (15.3.34) can be written uo (xm1 ) + uo (xm2 ) (xm1 − xm2 ) = 2



xm1

dx uo (x ).

(15.3.35)

xm2

This is the law of equal areas (15.3.20), which has been shown to be equivalent to the expression for the propagation speed (15.3.16), D=

uo (xm1 ) + uo (xm2 ) . 2

(15.3.36)

15.3.3 Riemann Invariants The mechanism of formation of shock waves in fluid mechanics is similar to that of the discontinuities of the solution to (15.3.15). The main difference, the source of supplementary difficulty, is that in Euler’s equations there are two independent variables, u and ρ, instead

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of one. In planar geometry and under the condition of constant entropy, introducing the √ speed of sound, a ≡ dp/dρ|s , Equations (15.3.1)–(15.3.3) form an autonomous system a2 ∂ ∂ ∂ ρ + u + u u = 0, ρ ∂x ∂t ∂x

∂ ∂ ∂ ρ + u ρ + ρ u = 0, ∂t ∂x ∂x

(15.3.37)

where a2 = γ p/ρ for a perfect gas p/ρ = (cp − cV )T with a constant value of the ratio of specific heats (polytropic gas) γ ; see (15.3.3). Linearising these equations around a uniform state gives the equations of plane acoustic waves, satisfying d’Alembert’s equation (the wave equation) with a constant sound speed a. We are interested here in nonlinear solutions. Multiplying the first equation by a factor λ and summing the two gives   a2 ∂ ∂ ∂ ∂ ρ + u + (λρ + u) u = 0. (15.3.38) λ ρ + λu + ∂t ρ ∂x ∂t ∂x This equation can be written as a linear combination of total derivatives along a trajectory x(t), λDρ/Dt + Du/Dt = 0,

D/Dt ≡ ∂/∂t + (dx/dt)∂/∂x,

provided that λ is given by the equality λu + a2 /ρ = λ(λρ + u), λ = ±a/ρ.

(15.3.39)

These two values correspond to perturbations propagating along two families of trajectories, C+ and C− , called characteristics, whose equations are C+ :

dx = u + a, dt

C− :

dx = u − a. dt

(15.3.40)

These characteristics are waves that propagate with respect to the fluid at a speed equal to the local speed of sound, in the direction of increasing x for C+ and decreasing x for C− . With these two values (15.3.39) of λ, Equation (15.3.38) takes the form of two equations generalising those of linear acoustics,   ∂ ∂ ∂ ∂ p + (u ± a) p ± ρa u + (u ± a) u = 0, (15.3.41) ∂t ∂x ∂t ∂x where we have used the isentropic relation dp = a2 dρ. Different quantities are propagated along each of these two families of characteristics. According to (15.3.38), the following relations hold along the two characteristics, C+ :

a dρ + du = 0, ρ

C− :

a dρ − du = 0, ρ

(15.3.42)

C− :

dp − du = 0, ρa

(15.3.43)

which can also be written as C+ :

dp + du = 0, ρa

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Figure 15.7 Construction of the solution from the initial condition by propagating Riemann invariants along the characteristics.

in agreement with (15.3.41). In the linear approximation, these two equations correspond to the relations between the fluctuations of pressure and velocity in a progressive planar acoustic wave. During an isentropic transformation of a simple fluid, there is only one independent thermodynamic variable and Equations (15.3.42) and (15.3.43) can be written   dp dρ +u= + u, C+ : J+ = cst., J+ ≡ a ρ ρa   (15.3.44) dp dρ C− : J− = cst., J− ≡ a −u= − u. ρ ρa These are the Riemann invariants; they are constant along the characteristics. For a polytropic gas, p/ρ γ = cst., T/ρ (γ −1) = cst., a/ρ (γ −1)/2 = cst., ρ/a2/(γ −1) = 2/(γ −1) , dρ/ρ = [2/(γ − 1)] da/a, a(dρ/ρ) = [2/(γ − 1)] a, the expressions ρo /ao of J± are simply J+ =

2 a + u, γ −1

J− =

2 a − u. γ −1

(15.3.45)

The fields u(x, t) and a(x, t) are computed by propagating the Riemann invariants J+ and J− along the two characteristics C+ and C− , which intersect in the (x, t) plane; see Fig. 15.7. The solution is well defined on the condition that no more than two characteristics intersect at each point x, t. If this is not the case the solution is multivalued and shock waves must have been formed at some earlier time. We will come back to this point in the next section. If the entropy is not uniform in the flow field, as behind an unsteady shock wave, it is convenient to consider a third family of characteristics Co associated with (15.3.4), called entropy waves, that propagate the value of the entropy at the flow velocity, Co :

dx = u, dt

dp − a2 dρ = 0,

17:11:45 .017

(15.3.46)

15.3 One-Dimensional Compressible Flows

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s , called the Figure 15.8 Diagram showing a rarefaction wave in the (x, t) plane. The characteristic C+ separator, bounds the region of fluid at rest and is shown as a dot-dashed line.

where, according to (15.3.3), the sound speed is a function of ρ depending on the entropy γ so of the characteristic Co , a2 = γρ (γ −1) po /ρo . The quantity p/ρ γ is constant along each γ characteristic curve Co but its value po /ρo varies from one characteristic to another.

15.3.4 Simple Waves in a Polytropic Gas Consider a planar flow created by the motion of a piston starting from rest at an initial instant in an infinite tube filled on one side of the piston with a perfect (nonviscous) fluid initially at rest. We will suppose that the tube is sufficiently long that the presence of the end of the tube is not felt at the positions and times considered. At the initial instant the piston is at x = 0 and the fluid fills the half-space x > 0. Rarefaction Wave in Planar Geometry Let us first examine the case where the piston moves away from the fluid, creating a rarefaction wave. Let up (t) < 0 and xp (t) < 0 be the time-dependent speed and position of the piston. The region containing fluid at rest is on the right-hand side of the plane (x, t) shown in Fig. 15.8. In this region the characteristics C+ and C− are straight lines with slopes dx/dt = ±ao , where ao is the speed of sound in the fluid at rest. This region is s , called the separator and indicated bounded in the (x, t) plane by a linear characteristic, C+ by the dot-dashed line. This characteristic propagates information, at the speed of sound, from the initial position of the piston, putting the fluid into motion. The characteristics C− leaving the fluid at rest cross the separator and enter the region of moving fluid propagating the value of J− . According to (15.3.45), this value is the same for all characteristics and is equal to J− =

2 ao . γ −1

This relation holds until the separator reaches the end of the tube, after which time different information propagates back up the tube by the characteristics C− , modifying the analysis.

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Until this happens Equation (15.3.45) yields a relation between the local speed of sound and the fluid velocity that is valid everywhere, 2 2 a−u= ao . γ −1 γ −1

(15.3.47)

The characteristics C+ propagate the invariant J+ , which can be written, using (15.3.45) and (15.3.47), as J+ =

2 2 2 a+u= ao + 2u = (2a − ao ). γ −1 γ −1 γ −1

(15.3.48)

According to (15.3.48), the fluid velocity, u(x, t), and the sound speed, a(x, t), are constant along the characteristics C+ . Assuming that the speed of the fluid at the face of the piston is equal to that of the piston, one gets t > to :

u(x, t) = up (to ),

a(x, t) = ap (to ),

(15.3.49)

where up (to ) and ap (to ) are, respectively, the speed of the piston and the local sound speed of the gas on the piston at time to at which the characteristic C+ left the piston to be at position x at time t > to . The family of characteristics C+ , whose equations are dx/dt = u + a, are thus straight lines, t > to :

x = [ap (to ) + up (to )](t − to ) + xp (to ),

(15.3.50)

where xp (to ) is the position of the piston at time to . Using (15.3.47) at the piston, (γ − 1) up (to ), 2 (15.3.51) (γ + 1) [ap (to ) + up (to )] = ao + up (to ), 2 the equation of the characteristics C+ in (15.3.50) may be expressed in terms of the instantaneous position xp (to ) and velocity up (to ) of the piston:  (γ + 1) t > to : up (to ) (t − to ) + xp (to ). x = ao + (15.3.52) 2 ap (to ) = ao +

Ejection Velocity into a Vacuum The speed of sound ap (to ) cannot be negative and so it can be seen from the first equation in (15.3.51) that up (to ) < 0 has a negative value bounded by up (to ) = −2ao /(γ − 1). This is the velocity of ejection of the fluid into a vacuum. If the piston recedes at a greater velocity, a vacuum forms between the piston and the rarefied fluid and the piston has no effect on the motion of the fluid. Quite generally, waves, for which one of the families of characteristics are straight lines, are called simple waves. The rarefaction wave described above is an example. Moreover, if the absolute value |up (to )| of the piston velocity up (to ) < 0 never decreases in time, the

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15.3 One-Dimensional Compressible Flows

677

Figure 15.9 Compression diagram in the (x, t) plane.

rectilinear characteristics C+ never cross in the future (see Fig. 15.8) since according to the second equation (15.3.51), the slopes x/t of successive characteristics decrease in time. The velocity of the fluid remains everywhere continuous at all times. Compression Waves in Planar Geometry The situation is different for a simple compression wave, formed when the piston moves towards the fluid initially at rest, up > 0. The above relations and reasoning are still valid. However, when the velocity of the piston increases in time, the characteristics C+ intersect after a finite time; see Fig. 15.9. The physical explanation is quite simple: perturbations leaving the piston propagate faster when the velocity of the piston increases and overtake those emitted at a previous time. Simple Centred Waves in Planar Geometry According to (15.3.50) and (15.3.51), the slopes of the characteristics C+ leaving the piston change only when the piston accelerates or decelerates. They remain constant when the piston has a uniform velocity. Simple waves have a particularly simple form, called centred waves, when the velocity of the piston is changed instantaneously from zero to a constant velocity up , xo (t) = up t. In this case, there is neither time nor length scales involved in the problem, so that the solution for the flow takes a self-similar form in which space and time appear only through the ratio x/t shown in (15.3.54) below. A further discussion of this point is postponed to Section 15.3.5. According to (15.3.52), the characteristics C+ leaving the piston are parallel lines and their slope ao + (γ + 1)up /2 is different from the slope ao of the characteristics C+ whose origin is in the fluid at rest. Therefore there are two regions of fluid, one at rest ahead of the piston and another near the piston, that are both uniform but in a different thermodynamic state. They are separated by a transition region which is different for the case of a compression, up > 0, and a rarefaction, up < 0. For the case of a compression, up > 0 (see Fig. 15.10), the two regions of fluid are separated by a discontinuity (shock) that propagates at supersonic speed with respect to the fluid at rest. As explained in

17:11:45 .017

678

Flows

Figure 15.10 Self-similar solution for a compression shock wave generated by a piston travelling at constant speed towards the fluid at rest. Only the C+ characteristics are shown.

Sections 15.1.7 and 15.3.1, there is an entropy jump across the shock wave, and the shock speed is obtained by the conservation of total energy leading to the Rankine–Hugoniot conditions in Section 4.2.1. Simple Rarefaction Wave in Planar Geometry. Weak Discontinuities For the case of a rarefaction, up < 0 (see Fig. 15.11), the two uniform zones are separated by an expansion zone, called a rarefaction wave, whose width increases linearly in time, as explained now. Since the change of piston velocity is confined to the initial instant (the acceleration is infinite), up (to = 0) ∈ [0, up ], these two families of characteristics meet at the origin, x = 0, t = 0. In the intermediate region between these two families, all the characteristics C+ with intermediate slopes pass through the origin (x = 0, t = 0) and form a centred bundle (xo = 0, to = 0), x = (a + u)t,

a + u = ao +

γ +1 u, 2

u = up (to ) ∈ [0, up ],

(15.3.53)

bounded by the two characteristics whose equations are x = ao t (fluid at rest u = 0) and x = [ao +(γ +1)up /2]t for the fluid moving at piston speed, u = up < 0; see (15.3.52). The first equation in (15.3.53) results from (15.3.49) and from the equation for the characteristics C+ , dx/dt = a+u. The second equation in (15.3.53) results from (15.3.51) with u = up (to ), xo = 0 and to = 0. According to (15.3.53) the velocity field in the intermediate expansion

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679

Figure 15.11 Self-similar solution for a rarefaction wave generated by a piston travelling at constant speed away from fluid at rest. Only the C+ characteristics are shown. The dashed lines are weak discontinuities.

zone is u=

 2 x − ao . γ +1 t

(15.3.54)

At each instant t > 0 the speed of the fluid in the expansion zone changes linearly with position from u = 0 to u = up . According to (15.3.52), the boundary with the fluid at rest propagates at speed ao and the boundary with the fluid in uniform motion at the piston velocity up < 0 propagates in the laboratory frame at speed ao + up (γ + 1)/2 = ao − |up |(γ + 1)/2. This latter velocity corresponds to propagation at the local sound speed, ao + up (γ − 1)/2 (see (15.3.51)), with respect to the gas moving with the piston, |up | + ao − |up |(γ + 1)/2 = ao − |up |(γ − 1)/2. The width of the transition region, namely the rarefaction wave, increases linearly in time, |up t|(γ + 1)/2 < ao ; see Fig. 15.11. The rarefaction wave plays an essential role in the selection of the propagation speed of an autonomous detonation (CJ regime); see Section 4.2.3. Weak Discontinuities The two boundaries of the zone are discontinuities in the gradient of the fluid velocity. They are called weak discontinuities since the velocity of the fluid is continuous. Quite generally, weak discontinuities propagate at the local sound speed relative to the gas flow. The inner

17:11:45 .017

680

Flows

structure of weak discontinuities involves the dissipative transports. The study can be found in textbooks[1] and is not presented here. Cylindrical and Spherical Geometry In cylindrical or spherical geometry, the extra geometrical term in (15.3.5) invalidates the analysis used for simple waves in planar geometry. The characteristic equations become   d 2 j dr C± : = u ± a, a ± u − au = 0, (15.3.55) dt dt γ − 1 r where r is the radius and j = 1, 2, respectively. These equations can no longer be integrated once to give an algebraic relation between a and u on each characteristic, as was the case in (15.3.45). There are approximate analyses for weak disturbances[2] only. In more general cases, different approaches, based on self-similar solutions presented now, are then useful.

15.3.5 Self-Similar Solutions. Point Blast Explosion In some circumstances it is possible to reduce a system of partial differential equations to a system of ordinary differential equations. The corresponding solutions, called self-similar solutions, have been extensively investigated in different problems. A detailed review and many examples can be found in the book of Zeldovich and Raizer[3] and in that of Barenblatt.[4] We limit the presentation here to briefly recall some simple results that will be useful for the topics treated. Simple Waves as First Examples of Self-Similar Solutions Equation (15.3.54) is an example of a self-similar solution of Euler’s equations, since, in the limit of an infinitely short time for the piston initially at rest to reach the constant velocity up , the variables x and t appear only as a ratio x/t. The solution can then be obtained in a direct way, different from Sections 15.3.3 and 15.3.4. In the absence of a time scale characterising a finite acceleration of the piston, the only parameters defining the system are the constant speed of the piston, up , the initial density of the fluid, ρo , and the velocity of sound in the fluid at rest, ao . There is no combination of these dimensional quantities that has the dimensions of distance or time. Therefore the flow field, the solution to Euler’s equations (15.3.37), written in terms of nondimensional variables u ≡ u(x, t)/ao , r ≡ ρ(x, t)/ρo and a given function of r, a(r) ≡ a/ao , can be a function of space and time only through the ratio x/t, the only possible combination (having the dimension of velocity) that can be transformed into a nondimensional variable, ξ ≡ x/(ao t).

[1] [2] [3] [4]

Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed. Whitham G., 1974, Linear and nonlinear waves. John Wiley. Zeldovich Y., Raizer Y., 1967, Physics of shock waves and high-temperature hydrodynamic phenomena II. Academic Press. Barenblatt G., 1996, Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press.

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When looking for self-similar solutions, u(ξ ), r(ξ ), Euler’s equations take the nondimensional form (u − ξ )(r /r) + u = 0,

(r /r)a2 + (u − ξ )u = 0,

(15.3.56)

where the notations u ≡ du/dξ and r ≡ dr/dξ have been used. Nontrivial solutions (different from zero) can exist only if the determinant is zero, a2 = (u − ξ )2

⇒

ξ = u ± a.

This corresponds to the first equation in (15.3.53), x/t = u+a, if the positive sign is chosen. The first equation in (15.3.56) then yields u = a(r)r /r,  a(r) dr. (15.3.57) u= r √ Recalling that the sound speed a in a perfect gas is proportional to T, the isentropic condition of a polytropic gas yields a(dr/r) = 2/(γ − 1)da; see (15.3.45). Equation (15.3.57) then yields the relation u = 2/(γ − 1)(a − ao ), that is, the second equation in (15.3.53). Self-Similar Solutions of Euler’s Equations In the rest of this section we change notation for the flow velocity; v(r, t) will denote the radial component of the flow velocity in the laboratory frame so that u = D − v is now the component in the reference frame moving with the front (D is the front velocity). In spherical geometry, Euler’s equations (15.3.4)–(15.3.5) for a polytropic gas take the form   ∂ ∂v v 1 ∂ +v ρ+ + 2 = 0, (15.3.58) ρ ∂t ∂r ∂r r   ∂ ∂ 1 ∂p +v v=− , (15.3.59) ∂t ∂r ρ ∂r    ∂ ∂ p = 0; (15.3.60) +v ∂t ∂r ργ the coefficient 2 in the last term of (15.3.58) is replaced by 1 in cylindrical geometry. The pressure field p(r, t) is said to be self-similar if it can be expressed in terms of timedependent pressure and length scales pf (t), rf (t), in the form p = pf (t)P(ξ ), where ξ ≡ r/rf (t). The flow is said to be self-similar if the other variables, density ρ(r, t) and velocity v(r, t), can be expressed similarly using the velocity scale r˙f (t) ≡ drf /dt and the scale of mass density, ρf (t) ≡ pf /˙rf2 , v = r˙f (t)V(ξ ),

ρ = ρf (t)R(ξ ),

p = ρf (t)˙rf2 (t)P(ξ ),

where

ξ ≡ r/rf (t). (15.3.61)

A solution of the form (15.3.61) is possible only if, after introducing (15.3.61) into (15.3.58)–(15.3.60), the variables ξ and t can be separated. A straightforward calculation shows that this is the case only if the basic scales, rf (t) and ρf (t), satisfy power laws, rf (t) = Atα ,

ρf (t) = Btβ ,

17:11:45 .017

(15.3.62)

682

Flows

where the dimensional constants A, B and the exponents α and β are not determined at this stage. The functions V(ξ ), R(ξ ) and P(ξ ) then satisfy a system of ordinary differential equations involving the parameters A, B, α and β. The final solution, if it does exist, is obtained using the initial and boundary conditions. Euler’s equations (15.3.58)–(15.3.60) contain only the nondimensional parameter γ , but the initial and boundary conditions involve dimensional parameters. Dimensional analysis then plays an important role in the construction of self-similar solutions.[1] Self-similar solutions in the form (15.3.61) with the power laws (15.3.62) are meaningful only in limiting cases, when some of the initial dimensional parameters become irrelevant, so that only two dimensional parameters, A and B, can be constructed from the relevant parameters. There exist two different types of self-similar solutions. In the first type, both the similarity exponents (α, β) and the dimensional constants (A, B) in (15.3.62) are determined either by dimensional considerations or from conservation laws. Such solutions are considered in Sedov’s book.[1] To illustrate this type of solution, assume for example that, in a limiting case, the initial and/or boundary conditions involve only two-dimensional parameters, a constant density ρu and a parameter A having dimension length/timeα , but no other dimensional scales, and in particular no length or time scales. Therefore the only scales that can be constructed are the length scale Atα , the velocity scale At(α−1) and the pressure scale ρu A2 t2(α−1) . In this case, according to (15.3.62) for β = 0 and B = ρu , a self-similar solution (15.3.61) exists, v = αAtα−1 V(ξ ),

ρ = ρu R(ξ ),

p = ρu (αAtα−1 )2 P(ξ ),

ξ = r/Atα ,

(15.3.63)

so that r/t = ξ Atα−1 . A typical example is the strong blast wave generated by an intense explosion, presented below. In self-similar solutions of the second kind, the similarity exponent α or the parameter A is determined from the solution of the ordinary differential equations by requiring, for example, that the integral curve passes through a singular point in order to satisfy the boundary conditions. In other words α or A are eigenvalues of a nonlinear problem. The implosion of a spherical shock wave[2,3] is an example of a self-similar solution of the second kind. An example in planar geometry is the flame speed UL determined as an eigenvalue in Chapter 8, α = 1, A = UL . Strong Blast Wave Produced by an Intense Point Explosion The explosion resulting from a sudden release of an amount of energy E concentrated at a point was solved independently in 1941 by Taylor[4] and by von Neumann, and also in 1946 by Sedov in the context of the atomic bomb; see Barenblatt’s book[5] for historical notes. The solution is constituted by a strong spherical shock wave of radius r = rf (t), [1] [2] [3] [4] [5]

Sedov L., 1959, Similarity and dimensional methods in mechanics. Academic Press. Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed. Zeldovich Y., Raizer Y., 1967, Physics of shock waves and high-temperature hydrodynamic phenomena II. Academic Press. Taylor G., 1950, Proc. R. Soc. London Ser. A, 201(1065), 159–174. Barenblatt G., 1996, Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press.

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683

expanding in the quiescent initial medium at velocity D(t) ≡ drf (t)/dt > 0, followed by an unsteady inert and adiabatic flow, t > 0, r  rf (t): v(r, t) = 0. The boundary condition for the radial component of the flow velocity in the laboratory frame at the shock front, v, takes the form r = rf (t): v = vN ≡ D(t) − uN (t), where uN is given by (4.2.14), Mu = D(t)/au is the instantaneous Mach number of propagation and au is the speed of sound in ambient air at rest. Assuming that the shock is strong, D  au , so strong that the initial pressure pu is negligible compared with the pressure pN in the compressed flow at the shock (Neumann state just behind the shock), the Rankine–Hugoniot relations (4.2.14)– (4.2.15) for a polytropic gas in the limit Mu ≡ D/au → ∞ yield r = rf (t): v = vN (t), ρ = ρN , p = pN (t), with drf 2 γ +1 D(t), D(t) ≡ , ρN = ρu , vN = γ +1 dt γ −1

(15.3.64) pN =

2 ρu D 2 (t), γ +1

where the term 1/pu , pu = ρu a2u /γ , in factor in the expression for pN , has been simplified, so that neither the sound speed au nor the pressure pu of the ambient air appears in the formulation of the problem in the limit Mu → ∞. In other words the initial conditions reduce to t = 0,

r > 0:

v = 0,

ρ = ρu ,

p = 0.

(15.3.65)

There are four unknowns: D(t), v(r, t), ρ(r, t) and p(r, t). The three equations in (15.3.58)– (15.3.60) are solved for r  rf (t) with the boundary conditions (15.3.64). A fourth equation ∞ is provided by the conservation of total energy, 4π 0 ρetot r2 dr, where etot ≡ eint + v2 /2 p 1 and eint = cV T = γ −1 ρ, 

rf (t)

4π 0

v2 2 1 p + r dr = E. ρ γ −1ρ 2 

(15.3.66)

Since only two-dimensional parameters characterise the problem, E and ρu , a single nondimensional combination of length r and time t can be built using E and ρu : r(ρu /Et2 )1/5 . The flow field is then sought in the form (15.3.61)–(15.3.62), where A = η(γ ) (E/ρu )1/5 , α = 2/5, B = ρu and β = 1. The law of propagation of the leading shock is thus obtained by simple dimensional argument, 

E rf (t) = η(γ ) ρu

1/5 t

2/5

,

⇒

2η(γ ) D(t) ≡ r˙f (t) = 5



E ρu

1/5

t−3/5 ,

(15.3.67)

where η(γ ) is a dimensionless constant, unknown at this stage, and depends only on γ , the only dimensionless parameter in (15.3.58)–(15.3.60). Introducing the dimensionless variables ξ≡

r , rf (t)

V≡

v , D(t)

R≡

ρ , ρu

17:11:45 .017

P≡

p , ρu D(t)2

(15.3.68)

684

Flows

Figure 15.12 Self-similar solutions for the velocity, pressure and mass density inside an intense spherical blast wave, normalised by the values just behind the shock (Neumann state).

the flow field is sought in the form V(ξ ), R(ξ ) and P(ξ ), solutions to the system of three ordinary differential equations of first order, obtained when (15.3.68) is introduced into (15.3.58)–(15.3.60), with the boundary conditions (15.3.64), ξ = 1:

V=

2 , γ +1

R=

γ +1 , γ −1

P=

2 . γ +1

(15.3.69)

The constant η(γ ) in (15.3.67) is then obtained from the additional relation (15.3.66). The calculation is straightforward but fastidious and is not presented here, but can be found in textbooks.[1,2] The result for γ = 1.4 is η = 1.0033 and the functions V(ξ ), R(ξ ) and P(ξ ), normalised by their values at ξ = 1, are plotted in Fig. 15.12. Notice that the major parts of the variations of the pressure and density are located near the leading shock. In real explosions, the self-similar solution is accurate in an intermediate regime, at sufficiently long time, much longer than the duration of energy release so that the shock radius is much larger than the size of the region where the energy is deposited, but not too large so that the shock is still sufficiently strong for the validity of (15.3.64). In the initial stage of a nuclear explosion there exists also a self-similar solution obtained by Zeldovich and co-workers in the 1950s; see the reference in Barenblatt’s book,[3] describing a very intense thermal wave of a very hot plasma in which the thermal conductivity due to electrons varies strongly with the temperature as a power law. More recently, the finite time of heat deposition, of the order of the acoustic time, has been taken into account.[4]

[1] [2] [3] [4]

Landau L., Lifchitz E., 1986, Fluid mechanics. Pergamon, 1st ed. Whitham G., 1974, Linear and nonlinear waves. John Wiley. Barenblatt G., 1996, Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press. Kurdyumov V., et al., 2003, J. Fluid Mech., 491, 379–410.

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17:13:05 .018

Index

Ablation front, 8, 9, 15, 23, 40, 41, 43, 324–338, 506, 507, 509–523, 525, 646, 648 Ablation wave in ICF, 324–338 ablation velocity, 328 absorption layer, 325 comparison with flames, 330–333 density and temperature profiles, 326 linear growth rate, 329 quasi-isobaric model, 327–329 Ablative Rayleigh–Taylor instability, 329–338 anti-Darrieus–Landau relaxation, 334 dispersion relation, 330, 333–335 nonlinear dynamics, 336 Accretion shock, 39, 361 surface, 39 Acoustic (or sound) wave, 5, 266, 538, 559–561 Acoustic instability, see Thermo-acoustic instability Acoustic restabilisation, 116 Activation energy, 20 Admittance function, 106 d’Alembert’s equation, 106, 208, 244, 530, 673 Arrhenius law, 20–21, 23, 52, 110, 213, 225, 240, 283, 288–290, 381, 382, 441, 452, 545, 547, 624, 625 Autocatalytic, see Chain branching Avogadro number, 313, 609 Barr`ere–Borghi diagram, 180 Bernoulli equation, 656 Beta decay, 364 Bifurcation, 479, 480, 554 analysis, 544 Hopf, see Poincar´e–Andronov bifurcation parameter, 80, 549 secondary, 294, 544 subcritical, 176 Binding energy, 18, 30, 364, 608, 624 Blast, 41 energy, 38 explosion, 235, 237, 680 furnace, 2, 628 wave, 33, 233, 236, 241–243, 682, 684

Bohr radius, 589 Boltzmann, 573, 592 collision operator, 594 constant, 573, 609 counting, 579 entropy, 594, 595, 600, 601 equation, 19, 24, 218, 219, 222, 300, 302, 345, 350, 363, 592–601, 640, 648, 652 gas, 343, 348, 349, 363, 581–583, 586, 589 H theorem, 302, 591, 594, 595 linear equation, 598 statistic, 342 Bose–Einstein distribution, 583 de Broglie, 586, 589 Brownian motion, 178, 601, 603, 604 particle, 603 Bunsen flame, 16, 17, 58, 64, 96, 120, 181, 184, 201, 202 open tip, 76, 92, 98, 100, 101, 184 Burgers’ equation, 139, 172, 273, 274, 670 Burke–Schumann diffusion flame, 26 Candle, 3, 5, 17, 33, 91 candles of the universe, 8 Carnot, 4, 570, 572, 652 Cellular detonation, see Detonation, cellular Cellular flame, 5, 78–81, 90, 101, 103, 104, 107, 110–121, 134, 135, 140–151, 176, 190, 380, 477–480 Chain branching, 52, 308–309, 312–318 Chain breaking, 306–309, 313–315, 318, 409–410 Chandrasekhar mass, 34, 35, 37, 352, 355, 357 Chapman–Enskog method, 596, 597, 599, 640 Chapman–Jouguet, see under Detonation Chemical kinetic scheme, 15, 51, 55, 243, 283, 291, 292, 306, 307, 311, 315, 319–321, 425, 562, 621, 626, 627, 630, see also Four-step reaction model; Three-step reaction model; Two-step reaction model global, 618, 621 reduced, 307

704

17:13:11

Index Chemical kinetics, 8, 19–21, 50, 62, 63, 73, 231, 232, 237, 238, 248, 290, 295, 306–322, 379, 381, 387, 425 Chemical potential, 570, 571, 582, 585–587, 590, 622, 623, 650 Clausius, 4, 573, 591, 651 Cold boundary difficulty, 383 Combustion modes, 15–18 Combustion noise, 197–203, 659, 660 blowtorch, 201–203 intensity, 198–200 origin, 197 power spectrum, 200–201 Combustion temperature, 18–19, 21, 25, 79, 310, 409, 628–633 Conical flame in acoustic field, 120 Cool flames, 21–22 Core collapse, 35–39, 363 Correlation, 207 dynamics of, 595 function, 178, 179, 200, 203, 603–605, 659 length, 190 long range, 342 time, 604 Corrugated flamelet regime, 183, 190–196 Countergradient diffusion, 179 Cross section, 20, 41, 101, 103, 181, 251, 593, 598, 634, 636 Crossover temperature, 51, 82, 85, 88, 231, 259, 290, 309, 310, 314, 315, 317, 319, 320, 404, 426, 432, 434, 435, 440, 441, 448, 456 Curvature, 17, 61–64, 69–77, 93–101, 131–133, 156–157, 181–184, 233–243, 337–338 quenching by, see Quenching Curved flame front, 124–135, 485 approximate solution, 129 stability, 131 stretching , 132 Curved fronts, 121–135 Cusp formation, 63 Cusped flame regime, 135 Cutoff temperature, 21, 51 Damk¨ohler number, 414, 415 turbulent regime, 180 Darrieus–Landau instability, 56, 58 dispersion relation, 60, 149 experiments, 67, 146–152 integral operator, 136 linear analysis, 59 marginal wavenumber, 63 nonlinear dynamics, 149–152 small wavelength stability, 61 stability threshold, 66 DDT, 7, 15, 35, 250–261, see also DDT experiments by localised thermal explosion, 252–253

705

other acceleration mechanisms, 255–256 phenomenology, 250–251 runaway mechanism, 253–255 turbulence induced, 251 DDT experiments, 256–261 in capillary tubes, 258–259 in choked tubes, 258 in ordinary tubes, 256–257 Debye, 363 length, 345 Deflagration-to-detonation transition, see DDT; DDT experiments Deflection of streamlines, 58 Degenerate electron gas, 35, 343, 588 nonrelativistic, 352, 588 relativistic, 590 Detonability limits, 232, 257 Detonation, 27–29, 213–300, 543–563, see also Detonation, cellular; Detonation, galloping burnt gas state, 224–227 Chapman–Jouguet, 27–29, 213, 224, 229–231 dispersion relation, 296, 551–556, 562–563 inner structure of wrinkled detonation, 561 instability, see Detonation, cellular; Detonation galloping nonlinear equation, 287, 290, 298–300, 546–548, 556–557 overdriven, 223, 227, 235, 281, 282, 284–298, 311, 541–543, 557–563 propagation speed, 27 quenching, see Quenching spherical effect of curvature, 238–242 inner structure, 236–242 self-similar solution, 233–234 spinning, 257, 261 square-wave model, 239, 283, 288 stability analysis, 539–543 structure of, see ZND structure of detonation temperature, velocity, pressure profiles, 228–229 thickness, 29, 35, 233, 238, 239, 242, 246, 292, 296, 298, 545, 550, 554, 557, 559 transit time, 246, 247, 249, 298 Detonation, cellular, 292–300, 550–557 Chapman–Jouguet regime, 300, 550–557 diamond pattern, 279, 281, 292, 299 dispersion relation, 296 instability threshold, 294–298 nonlinear equation, 298–300, 546–557 at strong overdrive, 557–563 Detonation, galloping, 281–292, 539, 546–550 dispersion relation, 289 instability threshold, 290 modified Arrhenius model, 288 nonlinear dynamics, 287, 290 pulsating, 539, 554 Detonation ignition, see Detonation initiation

17:13:11

706

Index

Detonation initiation, 231–261 critical ignition energy, 242–243 direct initiation, 233–243 infinitely fast chemistry approximation, 235 in parallel flow, 255, 260 spontaneous initiation, 243–250, 663 in a temperature gradient, 244–250 Zeldovich criterion, 236 Detonation quenching, see Quenching Deuterium–tritium reaction, 40, 636 Diamond pattern, see under Detonation, cellular Diffusion coefficient, 32, 145, 166, 168, 225, 335, 405, 420, 494, 602, 603, 648 binary, 642 molecular, 25, 26, 78, 85, 118 thermal, 22, 62, 118, 328, 419, 420, 599, 652 turbulent, 178, 179 viscous, 118, 181, 204, 205, 260, 419, 599, 652 Diffusion flame, 17, 24–26 Direct drive, 42 Discontinuity, 39, 107, 401, 405, 525, 664 hydrodynamic, 55, 58, 69, 124, 125, 230–239, 262–332, 647–649, 665–678 reaction layer, 391, 392 supersonic, 533 surface of, 181 weak, 678–680 Dispersion relation, 61, 146, 148, 149, 263–265, 270, 294, 296, 298, 330, 331, 334, 470, 477, 480, 493, 511, 514, 518, 533, 534, 539, 542, 551–554, 562 ablative Rayleigh–Taylor instability, 330 Darrieus–Landau instability, 60 detonation, 551–556, 562–563 shock wave, 263–265, 533–535 Dissipation, 591 fluctuation–dissipation, 601, 603, 604 scale, see under Kolmogorov viscous, 185, 220, 252, 256, 260, 644–646 Doring, see ZND structure of detonation Einstein, 583, 601, 604 Brownian motion, 178 relation, 601, 604 Electron, 35–39, 363 Electron capture, 35–39, 346, 364 Entropy, 4, 37, 217–222, 230, 302, 343, 348, 350, 355, 364, 655, 663–664, 672–677 Boltzmann, 594–595, 600 flux, 600–601, 650–654 jump, 222, 652, 678 at local equilibrium, 599, 600 production, 260, 600–601, 650–654, 657, 664 spots, 660 statistical, 573–577, 586–591, 595 thermodynamic function, 570–573, 582, 621–622 wave, 262, 282, 283, 286, 292, 295, 300, 530–533, 543–549, 557–560

Equivalence ratio, 18, 19, 51, 90, 149, 168, 259, 386, 410, 631, 633 Euler constant, 407 Euler equations, 126, 127, 230, 233, 262, 271, 294, 303, 421, 464, 499, 519, 530, 532, 543, 598, 652, 664 reactive, 543, 557, 558 Euler–Poisson equation, 342, 344, 357 Explosion of stars, 8, 9, 15, 39, 339–375 Extinction, see Quenching Faraday instability, 103, 117, 470 Fermi, 342, 343, 349, 363, 582, 583, 586 distribution, 583–586 energy, 363 gas, 582, 586–591 statistic, 342 temperature, 343, 349, 588 Fick law, 23, 179, 640–642 Fisher equation, 397, 398 Fission, see under Nuclear Flame, wrinkled, see Wrinkled flame Flame balls, 77, 83, 85, 88–91, 431–459 Flame dynamics experiments, nonlinear dynamics, 149–152 linear, 55–137, 166–169 nonlinear, 137–146 sensitivity to noise, 141–146 Flame initiation, 82, 95–100 Flame instability cellular, see Cellular flame hydrodynamic, 55–69, 135–146, 463–473, 481–503 oscillatory, 478–480 parametric, 116–120, 148, 164–166, 169 pulsating, 77–81, 103, 389, 478–480 thermo-acoustic, see Thermo-acoustic instability thermo-diffusive, 55, 56, 63, 77–81, 90, 135, 138, 142–144, 425, 463, 473–480 vibrating, 103 vibratory, 103–105 Flame kernels, 81, 89–91, 431–459 Flame quenching, see Quenching Flame speed, 60, 70, 417 at quenching, see Quenching critical, 67, 68 curved, stretched, 16, 70, 472, 485 laminar, see Laminar flame speed local, 61, 69–70, 73, 136, 184, 196, 485 near flammability limits, 425–428, 437 turbulent, see Turbulent premixed flames wrinkled flame, 61, 69, 79, 184, 472, 483, 489 Flame stretch, 69–77, 158–160 at Bunsen tip, 76 finite thickness effects, 74 high-frequency response, 76 in spherical flames, 75 pure curvature, 73

17:13:11

Index pure strain, 72 quenching by, see Quenching stretch rate, 70, 158–160 Flame thickness, 24, 50, 54, 58, 61–64, 77, 80, 84, 88, 93, 94, 99, 124, 166, 179, 180, 331, 386, 389, 390, 420, 423, 431, 436, 448, 457, 463, 475, 483, 485, 488, 490, 497 Flame transit time, 16, 23, 24, 50, 54, 70, 76, 97, 108, 136, 181, 187, 236, 237, 386, 389, 390, 414, 432, 475, 479, 483, 497, 550 Flame, weakly wrinkled, see under Wrinkled flame Flammability limits, 21, 51, 309, 425–428, 437 Floquet theory, 117 Fluctuation–dissipation, see Dissipation Flux, 641–649 conductive, 380–393 convective, 177, 417, 432, 556 diffusive, 17, 24, 66, 79, 81–101, 346, 353 dissipative (diffusive), 598, 604, 642–649, 652 energy, 107, 108, 113, 205, 209, 325, 327, 328, 380–393 entropy, 650, 652 heat, 53, 54, 332, 380–393, 406, 596 incident, 593 mass, 50, 69, 75, 124, 127, 197, 215, 219, 247, 263, 282, 285, 344, 380–393, 406, 412–428, 465–558, 653 molecular, 642 momentum, 467, 652 neutrino, 33, 37, 39, 341, 364–375 neutron, 7 transverse, 61 Four-step reaction model, 319 Fourier coefficient, 171 decomposition, 205, 492 equation, 645, 652 law, 23, 32, 179, 325, 640, 644, 645, 651 representation, 106, 538, 559 series, 122, 129, 171, 524 space, 136, 298 transform, 60, 169, 170, 200, 203, 532, 602, 603, 605, 659, 661 Fractal dimension, 193, 206–207 geometry, 191, 206–207 structure, 143 Free energy, 581, 582 Gibbs, 622 Helmholtz, 579 Free fall, 122 time, 353, 355 velocity, 347, 353, 358, 359 Fronts, 30 Froude number, 56, 67, 78, 117, 131, 329, 334, 482, 513 Fusion, see under Nuclear

707

G equation, 182 Gamow energy, 30, 634 tunnelling, 634 Gibbs, 573, 576 chemical potential, 623 distribution, 577, 581 Gibbs–Duhem relation, 571 relation, 217, 570 Gibson scale, 191–195, 207, 208 Gravitational collapse, 8, 15, 34, 37–39, 341, 346, 347, 352–375 Green function, 92, 398, 448, 602, 671 retarded propagator, 659 Green–Kubo relations, 603 Helmholtz free energy, 579 Hopf bifurcation, 288, 296, 298, 300, 478, 549, 555 Hopf–Cole transformation, 670 Hugoniot curve, 215–219, 223, 225, 258, 263 relation, see Rankine–Hugoniot relations Huygens’ construction, 15, 64, 181, 182, 266, 274, 665 Hydrodynamic instability, 43, 55–69, 78, 118, 135, 143, 161, 162, 295, 298, 495, 511 Hydrogen kinetics, 311–317 crossover temperature, 314 effect of hydroxyl radical, 315 main elementary reactions, 311 rich flame model, 317 shuffle reactions, 313, 315 steady-state approximation for O and OH, 313 two-step model, 315–317 Ignition, 16, 42, 51, 76, 77, 81–99, 142, 188, 425, 437, 454, 662 by constant heat source, 86 effect of Lewis number, 84 energy, 90–91, 242 facilitated by turbulence, 85 near flammability limit, 88–89 quasi-isobaric, 81–88, 436, 449 quasi-steady state approximation, 92–97 self-ignition, 252 spontaneous, see Detonation initiation temperature, 31 thermonuclear, 8, 34, 40–43 time, 260 Zeldovich radius, 82 Implosion, laser-driven, 40 Induced flow, 59, 189, 251, 257, 260 Induction delay, 310 detonation, 232–244, 249, 252, 282, 288, 295, 562 kinetics, 306–321

17:13:11

708

Index

Inertial confinement fusion, 8, 9, 15, 23, 30, 40–43, 324 Initiation, 35 detonation, see Detonation initiation of flame, see Flame initiation process, 459 reaction, step, 308–312, 318, 321 spontaneous, see Detonation Instability threshold, see Detonation, cellular; Detonation, galloping; Flame instability; Stars; Thermo-acoustic instability Integral equation, 127, 130, 138, 284–291, 300, 525 integral-differential equation, 298 length, 179, 193 method, 124, 337 operator, 136 scale, 187, 188, 190, 192–194, 196, 203, 205–208, 279 spin, 583 time, 178, 179, 199 volume, 200, 207, 208 Irreversibility, 221, 302, 382, 577, 591, 592, 594 Kapitza pendulum, 117, 161 Kelvin–Helmholtz instability, 126 Kolmogorov cascade, 176, 178, 180, 190, 192–201, 203–206, 279, 335 dissipation laws, 192 dissipation scale, 181, 204, 206 Kolmogorov–Petrovskii–Piskunov (KPP), 397–403 Kolmogorov–Obukhov K41, 204 KPP-ZFK transition, 399–403 scaling laws, 187, 191–201, 204–206 Kuramoto–Sivashinsky equation, 80 Laminar flame speed, 16–17, 23–24, 50–51, 54, 68, 78, 88, 103–105, 120, 126, 420, 467 chain-branching reactions, 403–410 methane–air flame, 410 ZFK analysis, 384–390 Lane–Emden equation, 351, 354, 355 equilibrium, 350 solution, 350, 351, 353 stability of solution, 353, 354 Langevin equation, 142, 145, 601, 603, 604 Laser Megajoule, 41, 636 Law of mass action, 622–624 Lean mixture, 19, 78, 82, 84, 88, 232, 310, 425, 435, 437, 631 colour of lean flames, 78 Lewis number, 53, 54, 62, 63, 76–177, 327, 331, 382–489 Local equilibrium, 20, 219, 220, 222, 346, 570, 596–601, 608, 640–654

Mach, 592 number, 23–29, 49, 107–111, 216–248, 261–302, 328, 360, 380, 410, 463, 464, 515, 532–561, 600, 647, 652, 654–662, 683 stem, 261–274, 278, 279, 293, 298, 663 Markstein number, 67, 69–76, 98, 118, 147, 166–168, 182, 184, 306, 416, 424, 489, 494 in burnt gas, 74 first Markstein number, 69–70, 118, 489, 500 measure, 150 second Markstein number, 73, 481, 500 Mass-weighted average, 179 coordinate, 304, 354, 405, 557, 561 distance, 282 quantities, 349 variable, 420 Mathieu’s equation, 117, 118, 160–161, 470 Maxwell, 591 Maxwell–Boltzmann distribution, 20, 301, 581, 595, 597, 634 Maxwell–Boltzmann gas, see Boltzmann, gas Mean free path, 21–41, 213, 219, 220, 222, 302, 597, 599, 640, 652 neutrino, 346 Methane kinetics, 317–322 induction delay, 321 main elementary reactions, 317 three-step scheme, 319 two-step scheme, 321 Methane–air flame, 19, 24, 50, 51, 90, 261, 317, 320, 410 mixture, 29, 103, 459 Michelson–Rayleigh line, 215–219, 223–224, 258 Mixture fraction, 26 National Ignition Facility (NIF), 40 Navier–Stokes equations, 185, 204, 205, 419, 421, 599, 643, 657, 664 Neumann, 682, see also ZND structure of detonation boundary conditions, 27, 141 state, 27, 29, 214, 217, 223, 226, 236, 239, 247, 254, 257–300, 310, 530, 533–563, 683, 684 temperature, 29, 35, 232, 239, 246, 249, 257–300, 311, 533–563 Neutrino, 33, 35–39, 364, 365 outburst, 34 Neutrino-driven explosion, 39, 372–375 Neutron, 30, 31, 35–39, 41, 346, 365, 590, 610, 635, 636 Neutronisation, 35–39 Non-premixed combustion, 17–18 Normal burning velocity, see Flame speed Normal-mode analysis, 263–264, 531–532 at large Mach number, 541 Nuclear burning, 31–39

17:13:11

Index combustion, 31–39, 339–341 density, 347 energy, 41, 42, 339–341 explosion, 8, 40, 41 fission, 7 fusion, 7, 15, 40 matter, 35–39, 339–341 physics, 8, 15, 34–39, 339–341 power plant, 6, 7 reactions, 8, 9, 15, 31–39, 41, 42, 339–341 weapons, 7, 8, 40 Nucleon, 30, 31, 39, 346, 361, 590, 609, 610 Obukhov, see Kolmogorov Obukhov law, 178, 205 One-step Arrhenius model, see One-step reaction model One-step flame model, see One-step reaction model One-step reaction model, 52, 70, 76, 77, 83, 88, 98, 166, 225, 239, 283, 288, 290, 381, 411, 414, 424, 426, 432, 481, 543, 636 One-step ZFK model, see One-step reaction model Oxidation, 21, 629 of C, 618 of CH4 , 318, 319 of CH4 to CO, 318 of CO, 319, 618, 626–628 Parametric analysis, 39 instability, 103, 114, 116–166, 169, 470 resonance, 164 stabilisation, 120, 161–163, 168 Pauli exclusion principle, 365, 583, 613 Perrin experiment, 601, 604 Phase space, 395–396, 574–586 Phase trajectory, 398 Photodisintegration, 346 Photodissociation, 363 Photon, 32, 35–39, 78, 325, 635 Physical kinetics, 591–605 kinetic theory of gas, 599 Piston effect, 57, 296 Planck’s distribution, 345 Plasma, 30, 32, 40, 41, 325, 327, 341, 569, 589, 609, 635, 636 frequency, 325 physics, 138, 342 Poincar´e, 592 Poincar´e–Andronov bifurcation, 288, 296, 298, 300, 478, 549, 555 Poisson, 664, see also Euler–Poisson equation equation, 524 law, 583, 587, 589 Pole decomposition, 139, 140, 144, 151, 169–173 Positron, 35–39

709

Premixed combustion, 15, 18, 50–304, 379–503, 543–563 Prigogine school, 595 Prompt shock, see under Shock Proto-neutron star, 39 Proton, 30, 31, 35–39, 41, 346, 365, 590, 610, 635, 636 Pulsating cells, 104, 298, 299 detonation, 238, 250, 539 flame, 77, 79, 81, 235, 389, 479 membrane, 197 stars, 355 Quasi-isobaric, 16 analyses of the ablation front in ICF, 329–338, 512–526 analyses of flames, 379–503 approximation, 48, 49, 81, 239, 255, 284, 285, 292, 629, 630, 640, 659 combustion, 105, 655 equations, 49, 655 expansion, 8 flame ignition, 41, 81–88, 92 heat release, 107 instability of detonation, 294–300, 557–563 model of ablation front in ICF, 327–329 Quenching detonation quenching by curvature, 236–242 dynamic quenching of detonations, 290 flame quenching by curvature, 98 flame quenching by heat loss, 85, 411–413, 438 flame quenching by stretch, 77, 414–425 near flammability limits, 425–428 spontaneous quenching of detonations, 246–250 Radicals, 51, 109, 307–322, 426, 608, 616, 617, 619, 620, 632 loss, 232 steady-state approximation, 313 Random walk, 145, 178, 601 Rankine–Hugoniot relations, 27, 214–217, 220, 221, 233, 254, 260, 263, 269, 270, 283, 285, 287, 294, 301–302, 345, 530 Rarefaction wave, 218, 221, 229, 325, 675–679 Rayleigh criterion, 105, 106, 660, 661 Rayleigh–Taylor bubble, 122–124 rising velocity, 123 solution at spikes, 124 solution at vertex, 123 steady-state solution, 123 Rayleigh–Taylor instability, 32, 35, 43, 60, 65, 324, 326, 329, 330, 332, 333, 337, 472 Reaction–diffusion problem, 83 system, 81 turbulent wave, 177

17:13:11

710

Index

Reaction–diffusion (cont.) unsteady equations, 441–447, 474–480 wave, 21, 22, 54, 186, 379, 381, 393–403 Rebound, 38, 40, 341 Reynolds number, 204, 251, 259, 260, 657 tensor, 137 Rich mixture, 19, 56, 68, 78, 82, 84, 88, 92, 100, 101, 103, 232, 259, 307, 310, 311, 317, 425, 426, 435, 437, 478, 500, 628, 631, 632 colour of rich flames, 78 Richardson, 205 law, 178 Richtmyer–Meshkov instability, 152, 255 images, 153 Riemann, 654, 664 invariant, 259, 672–675 Rijke tube, 111 Saffman–Taylor finger, 135 Scheme, see Chemical kinetic scheme Self-extinguishing flame, 90, 431, 448, 459 Shear Couette, Poiseuille flows, 657 flow, propagation in, 183–185, 189 flow, vorticity wave, see Vorticity wave pure shear flow, 126 viscosity, 419, 643 Shock accretion, see Accretion, shock front, see Shock wave outburst, 360, 362 prompt, 38–40, 361, 362 stalled, 39, 362, 363 strong shocks, 269–274 Shock wave, 214–222 dispersion relations, 263–265, 533–538 inner structure, 219–222 irreversibility of, 302 isolated, 262 linear stability, 261–266 linear stability criterion, 538–539 neutral modes, 264–265 in polytropic gas, 216–217 sound radiation condition, 538 thickness of, 221–222 transit time, 346 weakly wrinkled, 265, 271 Shock–turbulence interaction, 278–279 Shock–vortex interaction, 274–279 Shooting method, 399 Singing flames, see Thermo-acoustic instability Singular point, 395–402, 483, 496, 682 focus, 400 node-spiral, 397–402 saddle, 395–402

Singularity, 124, 171, 181, 264, 273, 274, 298, 337, 338, 356, 357, 654 Sivashinsky equation, 135 nonperiodic solutions, 139 periodic solutions, 140 pole decomposition, see Pole decomposition pole dynamics, 139 sensitivity to noise, 141 Smoluchowski, 601 SOFBALL experiments, 91 Sound, 103–116 emission, 658, 659 generation, 274, 640, 660 intensity, 278 monopolar sound generation, 197–209 nonradiating, 538 radiation, 278, 538 speed of, 20, 22, 23, 29, 33, 41, 49, 59, 213–271, 325, 326, 328, 353, 380, 532, 544, 557, 642, 648, 651, 652, 654, 658, 664–684 spontaneous emission, 265, 538–539 velocity, see speed of wave, see Acoustic (or sound) wave Spitzer formula, 326 Spray flames, 114 Stagnation point flow, 72, 77, 91, 123, 131 quenching of flame at, see Quenching stability of flame in, 133 viscous effect, 418 Stars, 339–375, 569, 588, 590, 591 constitutive equations of, 341–346 dying, 8 energy, 349, 352 explosion, 8, 9, 15, 39, 341–374 hydrostatic equilibrium, 346, 350 instability, 352 lifetime, 347 mass, 350–352 pulsating, 355 in quasi-steady state, 15, 347 radius, 347, 349, 350, 352 stability analysis, 353–355 stable, 347 surface, 350 theory, 346 unstable, 349 Statistical thermodynamics, 569–591 Stellar equilibrium, 346 evolution, 31 pulsation, 355 structure, 32 Stochastic equation, 177, 182, 183, 190, 604 field, 145, 177, 183 force, 603 intrinsic stochasticity, 81

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Index motion, 178 process, 179, 190, 601, 602 variable, 576, 579 Stoichiometric coefficient, 308, 619, 631, 642, 654 composition, 19, 25, 29, 232, 386, 631 mixture, 19, 23, 24, 90, 103, 255, 256, 259, 291, 316, 632 ratio, 631 Stokes flow, 657 force, 115, 603, 657 theorem, 656 Stosszahlansatz, 592, 593 Strain, 70, 73, 75, 136, 649 critical strain rate, 414–425 pure, 72 rate, 72, 77, 91, 132, 134, 414, 415, 420, 421, 498 Stretch, see Flame stretch Stretch, of a passive surface, 71 Strong shocks, see Shock Sun, 15, 30, 32, 33, 347, 635 Supernovae, 8, 31, 33–40 Takabe formula, 330 Taylor, see also Rayleigh–Taylor instability; Saffman–Taylor finger analysis, 129 bubble, 131 diffusion coefficient, 178 expansion, 71, 217, 449, 450, 498, 572, 578, 602, 671 hypothesis, 189 instability, 330 scale, 178 self-similar solution, 243, 682 solution, 123 Test particle, 21, 602, 603 Thermal quenching, see Quenching Thermal waves, 30, 40, 42, 648 Thermo-acoustic instability, 40, 55, 56, 101–121, 291, 296, 355 acceleration coupling, 112 acceleration transfer function, 113 admittance function, 106 experiments, 114, 116, 118, 119 neutrino-driven, 372–375 pressure coupling, 109 two-phase (spray) flames, 114 velocity coupling, 111 velocity transfer function, 111 vibrating flat flame, 116–119 Thermo-diffusive approximation, 79, 177, 414–418, 424, 440 effects, 61, 480 model, 56, 79, 80, 92–101, 331–336, 420, 424, 441, 448, 463, 474–480, 506–509 phenomena, 77–101

711

Thermo-diffusive instabilities, see Flame instability Thermonuclear burning, 31 explosion, 34, 35 fusion, 30–31 ignition, 8, 42 reactions, 15, 30–31 Three-step reaction model, 291, 319, 321 Transport process coefficient, 596, 603, 604, 640, 651, 652 convective, 483, 657 dissipative (diffusive), 8, 22, 29, 56, 102, 219, 225, 325, 472, 543, 650, 663, 680 equation, 468 of heat and mass, 22, 69 molecular, 23, 282, 640 of neutrino, 39 radiative, 326 turbulent, 34, 177–179 Triple flame, 26 Triple point, 257, 261, 267–269, 279, see also Mach, stem Tulip flame, 152–156 auto acceleration, 154 geometrical model, 154 Turbulent diffusion, 177–179 limitations, 179 Taylor’s diffusion coefficient, 178 Turbulent premixed flames, 174–209 corrugated flamelet regime, 183, 190–196 covariant laws, 195 flame speed, 181, 187, 189, 195, 196, 251, 484 folded flame brush, 255 fractal dimension, 193 propagation velocity, 184, 187, 189, 193, 195 surface area fluctuations, 207 thickened flame regime, 183 well-stirred regime, 32, 180, 183 wrinkled flame regime, see Wrinkled flame Turning point, 85, 99, 135, 238, 248, 254, 347, 411–424, 439, 447 Two-step reaction model, 52, 55, 109, 307, 309–311, 313, 315, 317, 320, 321, 403, 405, 410, 425, 500 Variational principle, 347 problem, 349 Vibrating flat flame, see Thermo-acoustic instability Virial theorem, 348, 349 Vortex–flame interaction, 185–188 chaotic flow, 187 vortex array, 186 vortex tube, 185 Vorticity wave, 262, 267, 268, 270, 271, 468, 530–533, 559–560

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712 Water–gas shift reaction, 319–322, 628 Wave equation, 208, 270, 372, 394, 530, 559, 560, 658, 659, 673 Weakly nonlinear analysis, 271, 272, 274, 293, 294, 298, 300, 355, 544 Well-stirred reactor, 15, 32, 660 regime, 32, 180, 183 White dwarfs, 32, 35, 352, 588, 589 Wiener–Kintchin theorem, 605 WKB method, 671 Wrinkled flame, 463–503 speed, 61

Index turbulent regime, 180, 183, 185, 188 weakly, 55, 58, 185 weakly wrinkled flame regime, 189–190

Zeldovich, see also Detonation initiation; Ignition; Zeldovich mechanism; ZND structure of detonation KPP-ZFK transition, 399–403 mechanism, 243, 252, 259 Zemerlo, 592 ZFK analysis of flames, 52 ZND structure of detonation, 27, 222–231

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