General Theory of High Speed Aerodynamics 9781400877553

Table of contents :
Foreword
Preface
Contents
A. On the Foundation of High Speed Aerodynamics
B. Mathematical Aspects of Flow Problems of Hyperbolic Type
C. Small Perturbation Theory
D. Supersonic and Transonic Small Perturbation Theory
E. Higher Approximations
F. Plane Subsonic and Transonic Potential Flows
G. The Method of Characteristics
H. Supersonic Flows with Shock Waves
Index

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GENERAL THEORY OF HIGH SPEED AERODYNAMICS

BOARD OF EDITORS THEODORE VON KXRMAN, Chairman HUGH L. DRYDEN HUGH S. TATLOE JOSEPH V. CHAKTK1 General Editor, 1951-

Assoeiate Editor, 1949-1951

MAHTIN SUMMEKFIELD, General Editor, 1949-1951

I. Thermodynamics and Physics of Matter. Editor: F. D. Rossini II. Combustion Processes. Editors: B. Lewis, R. N. Pease, H. S. Taylor III. Fundamentals of Gas Dynamics. Editor: H. W. Emmons IV. Laminar Flows and Transition to Turbulence. Editors: L. Lees, C. C.Lin V. Turbulent Flows and Heat Transfer. Editors: L. Lees, C. C. Lin VI. General Theory of High Speed Aerodynamics. Editor: W. R. Sears VII. Aerodynamic Components of Aircraft at High Speeds. Editors: A. F. Donovan, H. R. Lawrence VIII. High Speed Problems of Aircraft and Experimental Methods. Editors: A. F. Donovan, H. R. Lawrence, F. Goddard, A. E. Puckett, R. R. Gilruth IX. Physical Measurements in Gas Dynamics and Combustion. Editors: R. W. Ladenburg, B. Lewis, R. N. Pease, H. S. Taylor X. Aerodynamics of Turbines and Compressors. Editor: W. R. Hawthorne XI. Design and Performance of Gas Turbine Power Plants. Editor: W. R. Hawthorne XII. Jet Propulsion Engines. Editor: 0. E. Lancaster

VOLUME VI HIGH SPEED AERODYNAMICS AND JET PROPULSION

GENERAL THEORY OF HIGH SPEED AERODYNAMICS — I EDITOR: W. R. SEARS

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1954

32101 040276246 COPYRIGHT, 1954, by PRINCETON UNIVERSITY PRESS

London: GEOFFREY CUMBERLEGE, OXFORD UNIVERSITY PRESS L. c. CAHD 52-8779

Reproduction, translation, publication, use, and disposal by and for the United States Government and its officers, agents, and employees acting within the scope of their official duties, for Government use only, is permitted. At the expiration of ten years from the date of publication, all rights in material contained herein first produced under contract Nonr-03201 shall be in the public domain.

PRINTED IN THE UNITED STATES OP AMERICA BY THE MAPLE PRESS INC., YORK, PENNSYLVANIA

FOREWORD On behalf of the Editorial Board, I would like to make an acknowledgement to those branches of our military establishment whose interest and whose financial support were instrumental in the initiation of this publication program. It is noteworthy that this assistance has included all three branches of our Services. The Department of the Air Force through the Air Research and Development Command, the Department of the Army through the Office of the Chief of Ordnance, and the Department of the Navy through the Bureau of Aeronautics, Bureau of Ships, Bureau of Ordnance, and the Office of Naval Research made signifioant contributions. In particular, the Power Branch of the Office of Naval Research has carried the burden of responsibilities of the contractual administration and processing of all manuscripts from a security standpoint. The administration, operation, and editorial functions of the program have been centered at Princeton University. In addition, the University has contributed financially to the support of the undertaking. It is appropriate that special appreciation be expressed to Princeton University for its important over-all role in this effort. The Editorial Board is confident that the present series which this support has made possible will have far-reaching beneficial effects on the further development of the aeronautical sciences. Theodore von Kdrmdn

PREFACE Rapid advances made during the past decade on problems associated with high speed flight have brought into ever sharper focus the need for a comprehensive and competent treatment of the fundamental aspects of the aerodynamic and propulsion problems of high speed flight, together with a survey of those aspects of the underlying basic sciences cognate to such problems. The need for a treatment of this type has been long felt in research institutions, universities, and private industry and its potential reflected importance in the advanced training of nascent aeronautical scientists has also been an important motivation in this undertaking. The entire program is the cumulative work of over one hundred scientists and engineers, representing many different branches of engineering and fiettis of science both in this country and abroad. The work consists of twelve volumes treating in sequence elements of the properties of gases, liquids, and solids; combustion processes and chemical kinetics; fundamentals of gas dynamics; viscous phenomena; turbulence; heat transfer; theoretical methods in high speed aerodynamics; applications to wings, bodies and complete aircraft; nonsteady aerodynamics; principles of physical measurements; experimental methods in high speed aerodynamics and combustion; aerodynamic problems of turbomachines; the combination of aerodynamic and combustion principles in combustor design; and finally, problems of complete power plants. The intent has been to emphasize the fundamental aspects of jet propulsion and high speed aerodynamics, to develop the theoretical tools for attack on these problems, and to seek to highlight the directions in which research may be potentially most fruitful. Preliminary discussions, which ultimately led to the foundation of the present program, were held in 1947 and 1948 and, in large measure, by virtue of the enthusiasm, inspiration, and encouragement of Dr. Theodore von Kdrmdn and later the invaluable assistance of Dr. Hugh L. Dryden and Dean Hugh Taylor as members of an Editorial Board, these discussions ultimately saw their fruition in the formal establishment of the Aeronautics Publication Program at Princeton University in the fall of 1949. The contributing authors and, in particular, the volume editors, have sacrificed generously of their spare time under present-day emergency conditions where continuing demands on their energies have been great. The program is also indebted to the work of Dr. Martin Summerfield who guided the planning work as General Editor from 1949-1952. The cooperation and assistance of the personnel of the Princeton University Press and of the staff of this office has been noteworthy. In particular, Mr. H. S. Bailey, Jr., the Director of the Press, and Mr. R. S. Snedeker,

PREFACE TO VOLUME Vl

who has supervised the project at the Press and drawn all the figures, have been of great help. Special mention is also due to Mrs. Η. Ε. H. Lewis of this office who has handled the bulk of the detailed editorial work for the program from its inception. Joseph V. Charyk General Editor

PREFACE TO VOLUME VI This volume summarizes present-day knowledge in the theory of high speed aerodynamics. It begins with an over-all sketch of the subject by Theodore von K&rm&n. Largely nonmathematical, Professor von Kdrm^n's section touches on the major features of subsonic, transonic, supersonic, and hypersonic flows and emphasizes both important recent developments and some problems still unsolved. K. 0. Friedrichs presents a new study of the mathematical nature of hyperbolic flows, i.e. supersonic or unsteady compressible flows. Following these introductory sections, attention is directed especially toward the theories based on the small disturbance hypothesis, which have proved so useful. W. R. Sears presents and summarizes subsonic small perturbation theory, with emphasis on recent studies that unify and clarify this subject, and on rotational and unsteady flows. The largest section of the volume is by Max. A. Heaslet and Harvard Lomax, who present in a unified manner the whole subject of linearized supersonic theory, especially with application to wings. Their section also includes the transonic and hypersonic nonlinear approximations. M. J. Lighthill's contribution concerns the methods of higher approximation which are used to improve the first order theories and to explain and eliminate their shortcomings. This piece of work is unique in aeronautical literature and should be an especially valuable feature of the volume. Departing now from the ideas of perturbation theories, Υ. H. Kuo and W. R. Sears collaborate in a section presenting a review of a highly complicated and confused subject: the theory of potential subsonic and mixed supersonic-subsonic plane flows based on the hodograph method. Finally, in two sections, Antonio Ferri writes on the method of character istics applied to steady two- and three-dimensional flows and flow patterns characterized by the presence of shock waves. These sections include Professor Ferri'e recent work on the linearized methods of characteristics. The seven authors of this volume have cooperated admirably with the editors throughout the preparation of the work. To them, to the General Editor and his staff, and to Princeton University Press the volume editor expresses his sincere thanks. W. R. Sears Volume Editor

CONTENTS A. On the Foundation of High Speed Aerodynamics Th. von Kdrmdn, Air Force Scientific Advisory Board, Washington, D.C. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Incompressible Flow Propagation of Pressure Supersonic vs. Subsonic Flow Linearized Theory of Supersonic Flow Finite Disturbances in Compressible Flow. Shock Waves The Hypersonic Range The Transonic Range Entropy and Vorticity Entropy Rise and Drag The Role of Viscosity Limitations of the Theory of Continuous Fluids CitediReferences

B. Mathematical Aspects of Flow Problems of Hyperbolic Type

3 3 4 4 6 9 12 16 21 23 24 27 29 31

K. 0. Friedrichs, Institute of Mathematics and Mechanics, New York University, New York, N.Y. 1. 2. 3. 4. 5. 6. 7. 8. 9.

One-dimensional Flow Domain of Dependence, Range of Influence Uniqueness Proofs The Characteristic Equation. Existence Method of Finite Differences Nonisentropic Gas Motion Gas Flow Involving Shock and Reaction Fronts Remarks About Two-dimensional Steady Flow Cited References

C. Small Perturbation Theory

33 35 40 47 50 52 55 58 60 61

W. R. Sears, Graduate School of Aeronautical Engineering, Cornell University, Ithaca, New York 1. 2. 3. 4. 5.

Introduction The Equation of Sound Propagation The Prandtl-Glauert Equation Application to· Wing Theory Application to Other Bodies

61 63 64 66 74

CONTENTS

6. 7. 8. 9. 10. 11. 12.

82 85 88 96 100 110

Pressure-Correotion Formulas for Steady Flow The Critical Mach Number Experimental Confirmation The Slender Body Theory of Munk and Jones Rotational Small Perturbation Flow The Stream Function in Rotational Steady Flow Unsteady and Periodic Flow Problems. Fundamental Solutions 13. Cited References

116 119

. Supersonic and Transonic Small Perturbation Theory

122

Max. A. Heaslet, Ames Aeronautical Laboratory, National Advisory Committee for Aeronautics, Moffett Field, California Harvard Lomax, Ames Aeronautical Laboratory, National Advisory Committee for Aeronautics, Moffett Field, California Chapter 1. The Wave Equation and Methods of Solution 1. 2. 3. 4.

Introductory Remarks Partial Differential Equation of Fluid Flow Steady State Boundary Conditions and Methods of Solution Concluding Remarks Chapter 2.

5. 6. 7. 8. 9.

122 123 125 131

Two-dimensional Supersonic Steady State Flow

Basic Potential Equation and Solutions Airfoil Pressure Distributions Airfoil Characteristics Plane Waves and Applications Momentum Relations and Applications

133 137 139 140 143

Chapter S. Three-dimensional Supersonic Steady State Flow 10. Basic Potential Equation and Solutions 11. Direct Problems 12. Inverse Problems 13. Inverse Problems: Special Methods 14. Evaluation of Aerodynamic Characteristics 14.1 Appendix The Finite Part of an Integral The Generalized Principal Part of an Integral Chapter 4-

148 157 162 186 218 229 229 233

Axially Symmetric Supersonic Steady State Flow

15. Basic Potential Equation and Solutions 16. Slender Pointed Bodies of Revolution ( x )

235 238

CONTENTS

17. Slender Pointed Bodies of Revolution: Angle-of-Attaek Effects 239 18. Slender Pointed Bodies of Revolution: Thickness Effects 240 Chapter 5. 19. 20. 21. 22. 23. 24.

Slender Airplane Theory

Basic Potential Equation for Slender Airplanes Discussion of Boundary Conditions EvaluationofLiftingForces Examples of Planar Problems Examples of Interference Problems Concluding Remarks Chapter 6.

25. 26. 27. 28.

249 251 254 256 264 274

Unsteady Lift

Basic Equations and Wing Boundary-Value Problems The Indicial Functions Discussion of Some Two-dimensional Unsteady Problems Examples of Some Three-dimensional Unsteady Problems

275 292 298 304

Chapter 7. Reciprocity Relations and Reverse Flow Theorems in Aerodynamics 29. Introduction 30. Reverse Flow Theorems in Wing Theory 31. Applications

314 316 322

Chapter 8. Small Perturbation Theories of Transonic and Hypersonic Flows 32. 33. 34. 35.

Introduction Similarity Rules for Transonic Flow Hypersonic Flow Cited References

E. Higher Approximations

327 333 335 340 345

M. J. Lighthill, Department of Mathematics, The University of Manchester, Manchester, England 1. Introduction 2. Subsonic Flows 3. Supersonic Two-dimensional Airfoil Theory: Surface Pressures 4. Supersonic Two-dimensional Airfoil Theory: Complete Flow Pattern 5. Supersonic Two-dimensional Airfoil Theory: Influence of a Blunt Leading Edge 6. Supersonic Projectile Theory: Complete Flow Pattern

345 352 373 396 416 425

CONTENTS

7. Supersonic Projectile Theory: Surface Pressures 8. Supersonic Three-dimensional Wing Theory 9. Cited References F. Plane Subsonic and Transonic Potential Flows

451 477 487 490

Y. H. Kuo, Graduate School of Aeronautical Engineering, Cornell University, Ithaca, New York W. R. Sears, Graduate School of Aeronautical Engineering, Cornell University, Ithaca, New York 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Introduction The Hodograph Transformation The Particular Solutions of the Chaplygin Equation Particular Solutions of the Equations Resulting from the Legendre Transformation The Chaplygin-Kdrmdn-Tsien Approximation Examples of Simple Flows Two-dimensional Gas Jets Flow Around a Closed Body Two-dimensional Laval Nozzle Approximate Solutions for Transonic Flows: Channel Flows Approximate Solution for Transonic Flow Past an Airfoil Singularities of the Hodograph Transformation Conflicts Between Theory and Experiment Theory of Stability of Transonic Flows Theory of Nonexistence of Smooth Transonic Flows Properties of the Hypergeometric Functions Cited References and Bibliography

G. The Method of Characteristics

490 492 496 499 501 510 516 521 532 540 546 553 561 565 567 572 577 583

Antonio Ferri, Department of Aeronautical Engineering, Polytechnic Institute of Brooklyn, Brooklyn, New York

Chapter 1. Introductory Considerations 1. Introduction 2. The Equations of Characteristics for Two-dimensional and Axially Symmetric Flow

583 586

Chapter 2. Two-dimensional Potential Flow 3. The Equations of Characteristics for Two-dimensional Potential Flow 4. The Characteristic Lines in the Hodograph Plane

591 593

CONTENTS

5. Prandtl-Meyer Flow 6. Flow Along a Curved Surface 7. The Linearized Theory Derived from the Characteristics Theory 8. Numerical and Graphical Methods for General Two-dimensional Potential Flow Problems 9. Boundary Conditions: Solid Boundary 10. Boundary Conditions: Free Boundary

600 605 607 608 611 613

Chapter 3. Axially Symmetric Potential Flow 11. 12. 13. 14. 15. 16. 17.

Numerical Methods Points on the Boundaries and Points on the Axis Numerical Method Using Tabulated Values Graphical Methods Conical Flow Flow Around Corners in Axially Symmetric Flow The Shape of the Characteristic Lines at Infinity

616 620 624 626 628 630 632

Chapter 4- The Characteristics Method for Two-dimensional or Axially Symmetric Rotational Flow 18. Numerical and Graphical Methods 19. Reflections from an Entropy Discontinuity or Enthalpy Discontinuity

634 638

Chapter 5. Characteristics Methods for Problems in Three Independent Variables 20. 21. 22. 23.

The Equations of Motion Along Characteristic Surfaces The Characteristic Surfaces Numerical Methods of Characteristics for General Problems Two-dimensional Unsteady Flow

642 646 649 654

Chapter 6, Linearized Characteristics Methods 24. Linearized Characteristic Equations 25. The Linearized Characteristics Method for Axially Symmetric Flows 26. Bodies of Revolution at Small Angles of Attack 27. Quasi-two-dimensional Flow Fields 28. Cited References H. Supersonic Flows with Shock Waves Antonio Ferrij Department of Aeronautical Engineering, Polytechnic Institute of Brooklyn, Brooklyn, New York 1. Introductory Remarks

657 659 663 666 668 670

670

CONTENTS

Chapter 1. Two-dimensional and Axially Symmetric Flow with Shock Waves 2. Shape of the Shock Wave in Two-dimensional Flow and Axially Symmetric Flow 3. Intensity of the Reflected Waves Due to the Presence of a Curved Shock Wave 4. Relations Between the Curvature of the Head Shock Wave and the Curvature of the Streamlines 5. The Curvature of the Head Shock Wave at Infinity and the Determination of the Head Shock Drag

671 677 678 683

Chapter 2. Transonic Flow 6. 7. 8. 9. 10. 11. 12. 13.

Simplified Basic Equations The Shock Polar Diagram for Transonic Flow Boundary Conditions Transition from Subsonic to Supersonic Flow Around Corners Transition from Attached to Detached Shock for Sharpnosed Two-dimensional Bodies Formation of Lambda Shocks Transonic Properties of Two-dimensional Double Wedge Profiles Two-dimensional Bodies of Arbitrary Shape

690 694 697 700 706 710 714 717

Chapter 8. Conical Flow 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. Index

Conical Flow Fields Singular Points in Conical Flows Conical Flow in the Hodograph Plane Circular Conical Bodies at Zero Angle of Attack The Hodograph Diagram for an Axially Symmetric Conical Flow The Flow Field Around Circular Cones at Zero Angle of Attack Other Problems Involving Axially Symmetric Conical Flow Circular Cones at Small Angles of Attack The Entropy Field for Circular Cones at Small Angles of Attack Numerical Determination of the Flow Field Conical Flow Without Axial Symmetry Cited References

721 723 726 727 729 730 732 734 737 738 741 746 749

GENERAL THEORY OF HIGH SPEED AERODYNAMICS

SECTION A

ON THE FOUNDATION OF HIGH SPEED AERODYNAMICS TH. YON KARMAN A,l. Incompressible Flow. Looking back on fifty years of aerodynamics research during the first half of this century, it appears to me most remarkable that the crude approximation which considers the air as an incompressible nonviscous fluid has proved itself so valuable in solving many practical problems of aircraft design. In certain cases, for example in the performance calculations of airplanes and prfipellers, useful results were obtained even by a further simplification: by approximating the actual flow by infinitely small perturbations of a uniform and parallel airflow. Of course it was recognized at the time that certain phenomena, especially those connected with drag and stalling, require the consideration of viscosity. However, even for such problems PrandtPs classical idea of restricting the influence of viscosity to the neighborhood of solid walls—i.e. the concept of the boundary layer—proved to be sufficiently exact for the description, and in some cases for the prediction, of the phenomena. This state of affairs was fundamentally changed by the advent of high speed aircraft. First it seemed that compressibility troubles could be met by "compressibility corrections." However, very soon it became clear that the engineer needed a full grasp and knowledge of fluid mechanics over the entire speed range, extending from incompressible flow to flows with velocities large in comparison to the velocity of sound, and over a density range extending almost to a complete vacuum. The concept of an incompressible fluid ignores the fact that pressure variations are propagated in a fluid with a finite velocity. The assumption of instantaneous propagation of pressure introduces essential mathematical simplifications which make possible the application of Laplace's equation and the techniques of eonformal transformation,^,e. the"simplest and most popular methods of mathematical physics. It also justifies the concept of an apparent mass in the study of nonsteady phenomena. It is one of the fundamental theorems of the mechanics of incompressible nonviscous fluids that an arbitrary continuous sequence of irrotational

A · FOUNDATION OF HIGH SPEED AERODYNAMICS

flow patterns always represents a dynamically correct transient flow. It is evident that this means an enormous reduction of difficulties in dealing with nonsteady phenomena. A„2. Propagation of Pressure. I believe the first calculation of the propagation of a pressure wave through air was made by Sir Isaac Newton. Since he could not know the difference between isothermal and isentropic compression, his formula for the velocity of sound has a wrong numerical factor. He says quite clearly, however, in Theorem XXXVIII of his second book [i]: "The velocities of pulses propagated in an elastic fluid are in a ratio compounded of the square root of the ratio of the elastic force directly, and the square root of the ratio of the density inversely; supposing the elastic force of the fluid to be proportional to its condensation." In the same book he gives the following description of the mechanism of the resistance of a solid body moving in an elastic fluid: "Projectiles excite a motion in fluids as they pass through them, and this motion arises from the excess of the pressure of the fluid at the fore parts of the projectile above the pressure of the same at the hinder parts; and cannot be less in mediums infinitely fluid than it is in air, water, and quicksilver, in proportion to the density of matter in each. Now this excess of pressure does, in proportion to its quantity, not only excite a motion in the fluid, but also acts upon the projectile so as to retard its motion; and therefore the resistance in every fluid is as the motion excited by the'projectile in the fluid; and cannot be less in the most subtle ether in proportion to the density of that ether, than it is in air, water, and quicksilver, in proportion to the densities of those fluids." We shall see later that this concept of the drag was correct for the case of the supersonic motion of projectiles, whereas in the subsonic case—at least for rounded bodies—his description is at variance with d'Alembert's theorem. A,3. Supersonic vs. Subsonic Flow. From Newton's concept of the propagation of pressure (Fig. A,3a) in an elastic fluid, one arrives directly at the well-known picture of subsonic and supersonic flows. For the following considerations we restrict ourselves to flows produced by small disturbances and neglect viscosity, i.e. the absorption of energy in the air. Since a slight pressure change is propagated at sound velocity, it is evident that the effect of pressure changes produced in the air by a body moving faster than sound cannot reach points ahead of the body. It may be said that the body is unable to send signals ahead. It is seen that there is a fundamental difference between subsonic and supersonic motion. Consider the case of subsonic steady motion—for example, the uniform level flight of an airplane. Here a pressure signal travels ahead at sound velocity minus flight velocity, relative to the airplane, whereas a signal

A,3 • SUPERSONIC VS. SUBSONIC FLOW

Fig. A,3a. Newton's illustration of sound propagation in the case of diffraction through an opening. This illustration is used here merely to show his concept of the propagation of pressure disturbances in the shape of spherical waves.

Fig. A,3b.

Propagation of sound signals from stationary and moving sources.

(

5 >

A · FOUNDATION OF HIGH SPEED AERODYNAMICS

travels backward at a speed equal to the sum of flight and sound velocities. So, although the mechanism of pressure propagation is no longer symmetric, every point in space is reached by a signal, provided the flight started from an infinitely remote point. As can easily be seen, this is not the case in supersonic flight. Take the simplest case of a point source (Fig.*A,3b). The upper left diagram shows the spherical surfaces reached by the pressure effect in equal time intervals in the caseof a point source at rest. The upper right shows the same surfaces relative to the point source moving with a speed less than that of the sound. The lower left represents the case of a point source moving with sonic velocity, and the lower right, the case of a source moving faster than sound. It is seen that in the last case all action is restricted to the interior of a cone that includes all the spheres emitted by the source before the instant considered. The region outside of this cone can be called the zone of silence. It is easily seen that the sine of the half vertex angle of the cone is equal to the reciprocal of the Mach number. This angle is called the Mach angle. The cone that separates the zone of action from the zone of silence is called the Mack cone. The points plotted in Fig. A,3b show the location of mass points that are supposed to be emitted from the source and to move at sound velocity in all directions. They illustrate qualitatively the distribution of the density of action in the various cases. In the subsonic case one finds that the pressure effect not only decreases with increasing distance from the source, but is also dispersed in all directions. In the case of a body moving at supersonic velocity, the bulk of the effect is concentrated in the neighborhood of the Mach cone that forms the outer limit of the zone of action. A,4. Linearized Theory of Supersonic Flow. The so-called linearized theory of supersonic flow builds up the flow produced by the motion of a body by superposition of small disturbances such as considered in the last paragraph. One can develop in this way relatively simple methods for the computation of velocity and pressure distributions in the field and also for the computation of the forces, lift and drag, acting on moving bodies. In the last decade very extensive analytical work has been done using the linearized theory of supersonic flow. This grew out of such modest beginnings as Ackeret's work on the lift of a two-dimensional thin airfoil, published in 1925 [#], the computation of the drag of slender bodies, published by N. B. Moore and the writer in 1932 [S], the article on "Problems of resistance in compressible fluids," presented by the writer at the Volta Congress for high speed in 1935 in Rome [43, and Busemann's work of 1935 [β]. In the case of irrotational flow, the equations of motion can be reduced to equations analogous to the wave equation. The coordinate parallel to the direction of the main flow or the motion of the missile, wing, or

A,4 · LINEARIZED THEORY

body plays the role of the time coordinate. Hence, the well-established methods of finding solutions of the wave equation can be used in a great number of problems of practical importance, similarly to the capabilities of the methods for the solution of Laplace's equation in the case of incompressible flow. Such methods constitute the subject of Sec. D of the present volume. Wave drag. The concept of the linearized supersonic flow theory also reveals the existence of a novel kind of drag, which we do not encounter in subsonic motion and which we designate as "wave drag." Plane surface control

Mach Iinp

Δ

(1)-

Incompressible flow

Subsonic compressible Supersonic flow (Moo= 0.707) compressible flow (M= 1.414) Drag

HEfa. Thrust

Fig. A,4.

Comparisons of disturbance fields of airfoils in various speed regimes.

As before, we neglect viscosity and assume that the motion of the body produces disturbances that can be considered small. At a certain distance from the moving body this second assumption will, in general, be satisfied. Now consider the body and the surrounding air inside a cylindrical control surface as one mechanical system. Then one finds that, because of the concentrated action that characterizes the propagation of pressure from a source moving at supersonic velocity, the total flux of momentum of the air masses entering and leaving the cylindrical boundary remains finite even when the boundary is removed to an arbitrarily large distance. Fig. A, 4 refers to the case of a two-dimensional symmetric airfoil, with sharp leading edge, moving through air initially at rest. Let us consider the flow through a plane parallel to the plane of symmetry at a certain distance from the body. The diagram first shows the distribution of induced velocities (1) and the horizontal component of momentum transfer (2) along this plane for three cases. It is evident that the reaction

A · FOUNDATION OF HIGH SPEED AERODYNAMICS of outgoing flow having a horizontal component opposite to the flight direction and incoming flow with a horizontal component in the flight direction is equivalent to a propulsive thrust acting on the body. Conversely, outgoing flow with a component in the flight direction and incoming flow directed opposite to the flight direction give rise to a drag. In the two subsonic cases (Mv —*• 0 and Mco = 0.707), thrust and drag contributions are balanced, and the total horizontal momentum transfer is equal to zero. This is in accordance with d'Alembert's theorem. The influence of increasing Mach number is essentially to increase the magnitude of the induced velocities and to increase the concentration of the disturbance in the region extending laterally outward from the body. The increase in the concentration of action is also illustrated by the pressure distribution (3) on the control surface. In the supersonic case (Maa = 1.414), the disturbance is restricted to two strips bounded by two Mach lines. These lines are the intersections of planes that are envelopes of the Mach cones starting from points of the leading and trailing edges of the airfoil. The horizontal component of the outward flow is in the flight direction; that of the inward flow is the opposite. Hence, both represent drag on the body; this is called wave drag. Linearized theory limitations. The linearized theory, however, has serious limitations. First, it gives only a first approximation, since all deviations from the uniform parallel flow are considered infinitely small and therefore additive. If the disturbance caused by the body ca'nnot be considered small, the linearized theory needs important corrections. To this end considerable progress has recently been made in the development of methods of higher approximation (see Sec E). Such methods permit taking into account the existence of discontinuities (shock waves) in supersonic flow. Second, there are speed ranges in which the linearization of the equation of motion, even for small disturbances, is not justified. The conditions for the validity of the linearized theory of steady flow are twofold: 1. The perturbation velocities must be small in comparison with both the main-stream velocity and the velocity of sound. 2. The perturbation velocities must be small in comparison with the difference between the mainstream velocity and the sound velocity. Condition (1) excludes the case of very high velocities; evidently, if the main-stream velocity is several times larger than the sound velocity, disturbances which are small relative to the main-stream velocity may be of the same order of magnitude as the sound velocity. This speed range is called the hypersonic range. On the other hand, condition (2) excludes the range near M x = 1 from the scope of the linearized theory. We call this range the transonic range. In these two cases, even the study of small perturbation flows requires solution of nonlinear equations (D,32-D,34).

A,5 · FINITE DISTURBANCES

A,5.

Finite Disturbances in Compressible Flow. Shock Waves.

The solution of the equations of motion in the case of finite disturbances of a uniform parallel stream of an ideal compressible fluid is, in general, a cumbersome mathematical problem. The problems of proceeding beyond the successive approximation techniques treated in Sec. E, are somewhat simpler for a supersonic than for a subsonic stream. In the former, at least in the domains where the flow remains supersonic, the method of characteristics (Sec. G) can be applied with good results. The transfer of the computation from the physical plane to the so-called hodograph plane, i.e. the replacement of space coordinates by velocity coordinates as independent variables (Sec. F), is helpful in many problems, since the equations in the hodograph plane become linear. As was mentioned on the preceding page, a new phenomenon has to be taken into account: from the mathematical point of view, the possibility of discontinuous solutions and, from the physical point of view, the existence of shock waves. Shock waves. In subsonic flow the only possible discontinuous change of the velocity is tangential to a surface consisting of streamlines. Such a discontinuity does not violate any mechanical law because of the nonexistence of shear stresses in the fluid. In supersonic flow, however, a discontinuous change of velocity along streamlines is also possible. Such a discontinuous change is called a shock. In fact, the laws of mechanics are satisfied provided the fluid element conserves its mass, momentum, and energy. The physical reason why a discontinuous change is possible only in supersonic flow can easily be seen. The theorem of conservation of mass calls for the equality of the so-called Fano number (product of density and normal velocity) on both sides of the surface. The question arises: Can this product have the same value for different individual values of velocity and density while at the same time the momentum and energy of the fluid element remain unchanged? Now, if we consider the expansion or compression process of a gas, we find that the Fano number has a maximum when the velocity of the gas is equal to the velocity of sound. Consequently, in the case of a shock, the velocity normal to the discontinuity surface has to be subsonic on one side and supersonic on the other side. Therefore no discontinuous change can occur in a purely subsonic steady flow. The rule, however, does not exclude a discontinuous change in a purely supersonic flow, since it refers to the components of the velocity normal to the discontinuity surface. Thus an oblique shock is possible also in a purely supersonic flow. One gains a certain insight into the physical relationships by the consideration of the following problem. Assume a gas flow in a cylindrical pipe of constant cross section starting with a given pressure po, velocity Fo1 temperature T0, and stagnation enthalpy h0. Compute the variation of these quantities when heat in the amount Q is introduced into the

A · FOUNDATION OF HIGH SPEED AERODYNAMICS

gas. One finds that, if we start with an arbitrary subsonic velocity Fo, the velocity can be increased only to the sonic limit by introduction of heat. If one computes the velocity as function of the introduced heat beyond the sonic limit, one finds that heat has to be taken out of the gas. The curve showing the velocity and the temperature as functions of the introduced heat is represented in Fig. A, 5. Evidently there are two states for every value of the total enthalpy, or the sum h0 + Q, such that both states correspond to the same values of mass flow and momentum.

max

M=I

Fig. A, 5.

Velocity and temperature produced in a gas flowing in a cylindrical pipe by addition of heat.

Mathematically, a sudden transition would be possible either from subsonic to supersonic velocity or from supersonic to subsonic velocity. However, if we consider the transition between the points (T) and ©, i.e. from high to low velocity along the continuous curve shown in Fig. A,5, we find that, in order to realize this transition, heat is introduced at a low temperature level and taken out at a high temperature level; whereas, if the process is carried out in the opposite sense, from low to high velocity, heat is taken out at a low temperature and introduced at a high temperature. Evidently, according to the second law of thermodynamics, the second process is impossible without applying mechanical

A,5 · FINITE DISTURBANCES

work. Hence, the only possible transition is the compression shock, i.e. the transition from supersonic to subsonic flow. Of course, the same answer can be obtained by computing the entropy corresponding to the two states; then we find that the compression shock involves increase of entropy whereas the expansion wave would mean decrease of entropy. No discontinuous change of state can occur without change of entropy, but in so-called weak shocks the entropy rise is small. This explains why the linearized theory gives reasonable results, even in cases in which weak shocks are involved. The concept of shock waves goes back almost a century to Riemann [