Small Perturbation Theory 9781400879021

Table of contents :
Preface
CONTENTS
C. Small Perturbation Theory
1. Introduction
2. The Equation of Sound Propagation
3. The Prandtl-Glauert Equation
4. Application to Wing Theory
5. Application to Other Bodies
6. Pressure-Correction Formulas for Steady Flow
7. The Critical Mach Number
8. Experimental Confirmation
9. The Slender Body Theory of Munk and Jones
10. Rotational Small Perturbation Flow
11. The Stream Function in Rotational Steady Flow
12. Unsteady and Periodic Flow Problems. Fundamental Solutions
13. Cited References

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PRINCETON AERONAUTICAL PAPERBACKS 1. LIQUID PROPELLANT ROCKETS David Altman, James M. Carter, S. S. Penner, Martin Summerfleld. High Temperature Equilibrium, Expansion Processes, Combustion of Liquid Propellants, The Liquid Propellant Rocket Engine. 196 pages. $2.95 2. SOLID PROPELLANT ROCKETS Clayton Huggett, C. E. Bartley and Mark M. Mills. Combustion of Solid Propellants, Solid Propellant Rockets. 176 pages. $2.45 3. GASDYNAMIC DISCONTINUITIES Wallace D. Hayes. 76 pages. $1.45 4. SMALL PERTURBATION THEORY W. R. Sears. 72 pages. $1.45 5. HIGHER APPROXIMATIONS IN AERODYNAMIC THEORY. M. J. Lighthill.

156 pages. $1.95 6. HIGH SPEED WING THEORY Robert T. Jones and Doris Cohen.

248 pages. $2.95 PRINCETON UNIVERSITY PRESS · PRINCETON, N. J.

NUMBER 4 PRINCETON AERONAUTICAL PAPERBACKS COLEMAN duP. DONALDSON, GENERAL EDITOR

SMALL PER TURBA TIOJV THEOR T

BY W. R. SEARS

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

1960

© COPYRIGHT, 1954, 1960,

BY PRINCETON UNIVERSITY PRESS

L . C. CARD

60-12051

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 OF AMERICA

HIGH SPEED AERODYNAMICS AND JET PROPULSION

BOARD OF EDITORS THEODORE VON KARMAN, Chairman HUGH L. DRYDEN HUGH S. TAYLOR COLEMAN DUP. DONALDSON, General Editor, 1956-

Associate Editor, 1955-1956 JOSEPH V. CHARYK, General Editor, 1952-

Associate Editor, 1949-1952 MARTIN SUMMERFIELD, General Editor, 1949-1952 RICHARD S. SNEDEKER, Associate Editor, 1955-

I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

Thermodynamics and Physics of Matter. Editor: F. D. Rossini Combustion Processes. Editors: B. Lewis, R. N. Pease, Η. S. Taylor Fundamentals of Gas Dynamics. Editor: H. W. Emmons Theory of Laminar Flows. Editor: F. K. Moore Turbulent Flows and Heat Transfer. Editor: C. C. Lin General Theory of High Speed Aerodynamics. Editor: W. R. Sears Aerodynamic Components of Aircraft at High Speeds. Editors: A. F. Donovan, H. R. Lawrence High Speed Problems of Aircraft and Experimental Methods. Editors: A. F. Donovan, H. R. Lawrence, F. Goddard, R. R. Gilruth Physical Measurements in Gas Dynamics and Combustion. Editors: R. W. Ladenburg, B. Lewis, R. N. Pease, H. S. Taylor Aerodynamics of Turbines and Compressors. Editor: W. R. Hawthorne Design and Performance of Gas Turbine Power Plants. Editors: W. R. Hawthorne, W. T. Olson Jet Propulsion Engines. Editor: Ο. E. Lancaster PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

PREFACE The favorable response of many engineers and scientists throughout the world to those volumes of the Princeton Series on High Speed Aerody­ namics and Jet Propulsion that have already been published has been most gratifying to those of us who have labored to accomplish its completion. As must happen in gathering together a large number of separate contributions from many authors, the general editor's task is brightened occasionally by the receipt of a particularly outstanding manuscript. The receipt of such a manuscript for inclusion in the Prince­ ton Series was always an event which, while extremely gratifying to the editors in one respect, was nevertheless, in certain particular cases, a cause of some concern. In the case of some outstanding manuscripts, namely those which seemed to form a complete and self-sufficient entity within themselves, it seemed a shame to restrict their distribution by their inclusion in one of the large and hence expensive volumes of the Princeton Series. In the last year or so, both Princeton University Press, as publishers of the Princeton Series, and I, as General Editor, have received many enquiries from persons engaged in research and from professors at some of our leading universities concerning the possibility of making available at paperback prices certain portions of the original series. Among those who actively campaigned for a wider distribution of certain portions of the Princeton Series, special mention should be made of Professor Irving Glassman of Princeton University, who made a number of helpful sug­ gestions concerning those portions of the Series which might be of use to students were the material available at a lower price. In answer to this demand for a wider distribution of certain portions of the Princeton Series, and because it was felt desirable to introduce the Series to a wider audience, the present Princeton Aeronautical Paperbacks series has been launched. This series will make available in small paper­ backed volumes those portions of the larger Princeton Series which it is felt will be most useful to both students and research engineers. It should be pointed out that these paperbacks constitute but a very small part of the original series, the first seven published volumes of which have averaged more than 750 pages per volume. For the sake of economy, these small books have been prepared by direct reproduction of the text from the original Princeton Series, and no attempt has been made to provide introductory material or to eliminate cross references to other portions of the original volumes. It is hoped that these editorial omissions will be more than offset by the utility and quality of the individual contributions themselves. Coleman duP. Donaldson1 General Editor PUBLISHER'S NOTE: Other articles from later volumes of the clothbound

series, High Speed Aerodynamics and Jet Propulsion, may be issued in similar paperback form upon completion of the original series in 1961.

CONTENTS C. Small Perturbation Theory

3

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.

Introduction The Equation of Sound Propagation The Prandtl-Glauert Equation Application to Wing Theory Application to Other Bodies Pressure-Correction 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 Solu­ tions 13. Cited References

3 5 6 8 16 24 27 30 38 42 52 58 61

SECTION C

SMALL PERTURBATION THEORY W. R. SEARS C,l. Introduction. There is an important class of problems in aero­ dynamics in which the flow pattern consists of a small perturbation of a simple and well-understood flow. The most obvious case is the classical one of acoustics, where the unperturbed situation is simply a gas at rest. A more useful case in aeronautical applications, perhaps, is that of a slightly perturbed parallel uniform gas stream. In this section we consider these cases and some others. We shall find, as has already been suggested in Sec. A, that the mathematical problem of compressible fluid flow is greatly simplified by the small perturbation approximations, and that the resulting equations still exhibit some of the most significant features of the complete equations and are adequate to describe some of the important phenomena of high speed aerodynamics. The idea of attacking nonlinear problems of mechanics by the small perturbation method has been used many times in the history of the subject and in applications much different from gas flows. If we try to trace the history of this technique in gas dynamics, we must first recog­ nize that the "basic concept is implicit in any estimation of the velocity of sound. Sir Isaac Newton (1642-1727) probably made the first such estimation, and in doing so made a famous error, for he assumed that a small disturbance in a gas would compress the gas isothermally. His error was rectified by Laplace, who calculated the sound speed for the correct adiabatic process. In the eighteenth century, problems of acoustics were treated by Euler, D. Bernoulli, and Lagrange. The second order differential equation of sound propagation was known to them and to d'Alembert. The subject reached a state of great development in the nineteenth century, in the work of Lord Rayleigh, Helmholtz, Stokes, Kirchhoff, and others. In his famous treatise on The Theory of Sound [i] Rayleigh proceeded from a small perturbation hypothesis to the partial differential equation of sound propagation, in a manner substantially the same as we shall do later in this section. Turning now to the application of the same basic idea in the realm of aerodynamics, we find that apparently this did not evolve directly

C · SMALL PERTURBATION THEORY from the science of acoustics. Ludwig Prandtl, who had already made a series of contributions of essential importance in fluid mechanics, intro­ duced the small perturbation theory of steady flows in his lectures at Gottingen in 1922. It was applied to both subsonic and supersonic cases by his student, J. Ackeret, in a publication of 1928 [#]. Independently, the same method was proposed and applied to the subsonic case by H. Glauert in 1928 [5]. Apparently all these writers began their deduc­ tions having in mind the typical aeronautical case mentioned above, i.e. the slightly perturbed parallel stream. This is also true of an earlier work of Ackeret [4], where he considered small perturbations of a super­ sonic flow on the basis of the exact solution, already known, for super­ sonic flow around a corner. In fact, the identity of these methods with those of classical acoustics—to which they are reduced by a simple translation of coordinate axes—seems not to have been noticed by aerodynamicists until somewhat later. It is not difficult to see how this would occur, first because of the aeronautical preoccupation with parallel streams and also because, in reality, the idea of small perturbations of a parallel stream was already well known, having been exploited for somewhat different reasons in the Uterature of airfoil theory. Prandtl's famous lifting-line wing theory (1918), although it treated wings in incompressible flow, was a small perturbation theory, as were the thin airfoil theories of Munk (1922), Birnbaum, Glauert, and others. Consequently, it was more natural to proceed from this background toward an approximate theory of com­ pressible fluids than to return to the equations of acoustics. To be sure, there are undoubtedly prior investigations in the field of acoustics that anticipated salient features of the Prandtl-Glauert or Ackeret theories, and could be thought of as linking the acoustic and aeronautical versions of small perturbation theory. One of these is a 1907 paper by von KdrmiLn [5] concerning standing waves in supersonic jets. Nevertheless, the main course of the idea, so far as aeronautical engineers are concerned, has clearly been through airfoil theory in the work of Prandtl and Glauert, rather than from Rayleigh directly. In spite of this, our treatment of small perturbation flows in the present section will begin with the acoustical situation of a gas at rest and pro­ ceed to Prandtl's and Ackeret's problems as subcases. This procedure simplifies and clarifies certain matters, especially pertaining to unsteady flows. After introducing the basic ideas in this way, and obtaining the funda­ mental equations in several different forms, we shall devote ourselves in this section principally to their exploitation in subsonic flow situations. In Sec. D the applications to supersonic flight conditions are treated. This constitutes a report on the vast and rapidly growing body of liter­ ature on the aerodynamics of thin wings at supersonic speeds, and on

C,2 · E Q U A T I O N O F S O U N D P R O P A G A T I O N

some important aspects of the mathematical techniques found useful for these applications. Sec. D closes with a consideration of those special situations (transonic and hypersonic) where more careful analysis is required and nonlinear terms remain, even in the small perturbation equations. Finally, in Sec. E, we complete our study of small perturbation meth­ ods by considering their shortcomings and methods that have been pro­ posed to improve them. Actually, these considerations will bring us to a much clearer understanding of the first order perturbation theories themselves; consequently the authors of Sec. C and D present these theories without complete justification of the approximations involved, depending upon Sec. E to make up these deficiencies. C,2. The Equation of Sound Propagation. As in most of the sec­ tions of this volume, the equations of inviscid fluid flow provide the basis for our considerations. These are satisfactory to describe the flow of real gases in regions where neither friction nor heat transfer are important: namely, as we have seen (III,A,6), where velocity and temperature gradients are not large. It has already been noted (III, A,7) that, in the absence of friction and heat transfer, each particle of the fluid undergoes reversible adiabatic changes of state. Consequently, if the entropy of the gas was initially uniform, as in the case of a homogeneous gas at rest or flowing in a uni­ form stream, it must remain constant and uniformly distributed through­ out the flow. The equations of the flow are Dq

-ψ— 1

Dp

=

1, grad

p;

pp~i

= const (2-1)

P

1 Dp

j.

-^ + pdlvq=^-^ + pd1vq = 0 n

where D / D t is the usual "convective" or "substantive" derivative that measures a rate of change for an individual particle of gas, and a2 denotes the quantity dp/dp for reversible adiabatic compression, i.e. the square of the speed of sound. Clearly, the replacement of Dp/Dt by ar Dp/Dt can only be made on the basis stated above, that each particle moves about under adiabatic conditions. Both of the statements in Eq. 2-1 are non­ linear, and consequently have been solved only for a few special cases and under rather severe limitations. In the particular case of sound propagation through a body of gas uniformly at rest, we proceed to make the small perturbation approxima­ tion in Eq. 2-1. We assume the velocity and its derivatives to be small, and the corresponding density and pressure to differ only slightly from the values pM, px, which pertain to the undisturbed gas. Denoting these small departures from the steady values by p', p', and neglecting second 2

C · SMALL PERTURBATION THEORY

order terms in these and their derivatives, we have qt = (2-2) i Pt + Ρ» div q = 0

U00

where a®, denotes yp x /p„, the square of the speed of sound in the undis­ turbed gas. Eq. 2-2 are linear. Moreover, in view of the initial condition of uni­ form rest, the first of Eq. 2-2 implies that q is the gradient of a scalar function, i.e. that a potential function φ exists. Eq. 2-2 can then be written as follows: (2-3) P*4 >t = - v ' φα

=

(2-4)

α%ν 2 φ

These are the classical equations of sound propagation [ί, p. 15]. (Eq. 2-4 is known as the wave equation.) They are useful in aerodynam­ ics to describe a great variety of small perturbation situations outside of boundary layers and where shock waves either do not exist or are so weak as to justify our approximations. They can sometimes be used to describe the flow behind stronger shock waves, which are nearly uniform in strength, but this case requires special care, as is made clear below (Art. 10). An essential assumption in the derivation of Eq. 2-3 and 2-4 is that the initial condition is one of uniform rest, which has been used to obtain q as the gradient of a potential function. Clearly, these equations must be discarded whenever there is evidence that the small disturbance hypothesis is violated; we shall see (Art. 4) that this sometimes occurs even when the flow is due to the motion of a slender body. The PrandtI-Glauert Equation. Suppose now that the undis­ turbed flow is the familiar one of aerodynamics, namely, a uniform steady stream. This is reduced to the preceding case by steady translation of the reference axes. Let the χ axis be chosen in the stream direction and the stream speed be called TJ. The coordinate transformation between the moving (¾', y', z', t') system and the stationary (χ, y, ζ, t) one is, of course, C,3.

χ = χ' — Ut',

y = y',

ζ = ζ',

t = ί'

(3-1)

Thus, Eq. 2-3 and 2-4 become ρΛΦι' + UΦ*) = -ρ' = P w - P

Φιί> + 21Ιφ^ + U φ= α^ν' φ 2

!

(3-2) (3-3)

where again p , a,, are values pertaining to the undisturbed stream, and V" involves derivatives with respect to x', y', z'. x

C,3 · THE PRANDTL-GLAUERT EQUATION The equations for the special case where the flow is steady when viewed from the moving coordinate system are called the Prandtl-Glauert equations [0,8]: P - Ρ» = -Ρ«,ϋφ* (3-4) (l —

Φ»ν + ΦυΊf + φζ'ζ' = O

(3-5)

Eq. 3-5 illustrates clearly the change of mathematical character of gas flow in the three major speed regimes, subsonic, transonic, and super­ sonic. This equation is elliptic, parabolic, or hyperbolic when the coeffi­ cient of the first term is positive, zero, or negative, i.e. when the stream speed falls into the three regimes mentioned. It is a result of our small perturbation hypothesis, of course, that the character of the equation depends only on the Mach number of the undisturbed stream; this will unavoidably restrict the usefulness of this approximate theory, especially in the transonic regime. Eq. 3-5 is also of value because it establishes the significance of incompressible flow theory in relation to the aerodynamics of compressible gases. It is apparent that there will be a range of Mach numbers near zero, at least for slender bodies, for which the compressible and incom­ pressible flow patterns will be nearly alike. There is another way to establish this fact, which is better because it is not restricted to slender bodies, namely, the expansion method of Janzen and Rayleigh, which is treated in E,2. The Janzen-Rayleigh theory confirms that the first departures from incompressible fluid theory, as the Mach number of steady flow, Mm is increased from a small value, appear in terms of order Af21, as might be inferred from Eq. 3-5. This equation is put into the form of Laplace's celebrated equation by a simple affine transformation. Dropping the primes in Eq. 3-4 and 3-5 and writing β2 for (1 — Z72/«2), which is a positive number ior sub­ sonic stream flow, we put χ = νιξ,

with

y = ηη,

ζ = πζ

(3-6)

= β. Eq. 3-4 and 3-5 are immediately transformed into P - P. = — ^ P-JJn

(3-7)