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ADVANCES IN APPLIED MECHANICS

VOLUME VI

This Page Intentionally Left Blank

ADVANCES IN APPLIED MECHANICS Editors

H. L. DRYDEN

TH. VON

KARMAN

Managing Editor

G. KUERTI Case Institute of Twhnology, Cleveland, Ohio

Associate Editors

F. H.

VAN DEN

DUNGEN L. HOWARTH J. PEREs

VOLUME VI

1960 ACADEMIC PRESS

NEW YORK AND LONDON

COPYRIGHT0 1960, ACADEMICPRESSINC. ALL RIGHTS RESERVED NO P A R T O F T H I S BOOK M A Y B E R E P R O D U C E D I N A N Y FORM, B Y PHOTOSTAT, MICROFILM, O R A N Y O T H E R MEANS, WITHOUT WRITTEN PERMISSION

FROM T H E P U B L I S H E R S .

ACADEMIC PRESS

INC.

111 FIFTHAVENUE

NEW YORK3, N. Y .

United Kingdom Edition Published by ACADEMIC PRESS INC. (LONDON) LTD. 17 OLD QUEEN STREET,LONDON S.W. 1

Library of Congress Catalog Card Number: 48-8503

P R I N T E D I N T H E U N I T E D S T A T E S O F AMERICA

CONTRIBUTORS TO VOLUMEVI

W. CHESTER, University

of

Bristol, Bristol, England

M. HEIL, Institut fiir Theoretische Ph ysik der Freien Universitat Berlin, Berlin, Germany G. LUDWIG,Institut fur Theoretische Physik der Freien Universitat Berlin, Berlin, Germany

KLAUSOSWATITSCH, Deutsche Versuchsanstalt fiir Luftfahrt, Aachen, Germany

K. STEWARTSON, The Durham Colleges i n the University of Durham, Durham, England R. WILLE, Technische Universitat Berlin, Berlin, Germany

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Preface The sixth volume of Advances in Applied Mechanics includes five contributions in the field of fluid mechanics. They range from a short review of recent experimental work on vortex streets to a treatment of the flow of a gas in a boundary layer under extreme conditions which give rise to dissociation of the gas. In order to describe such a flow adequately, it is necessary to consider the underlying physical principles of the kinetics of gases and to apply the concepts of statistical mechanics. The major article in the volume is an extensive survey of similarity methods in aerodynamics, constituting a textbook in miniature on this important subject. Other papers deal with unsteady boundary layers and with shock waves in ducts of varying cross-section. Contributions to the Advances are, in general, by invitation, but suggestions of topics for review and offers of special contributions are very welcome and will receive careful consideration.

THE EDITORS January, 1960

vii

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Contents CONTRIBUTORS TO VOLUME VI PREFACE ..........

...................... ......................

v vii

The Theory of Unsteady Laminar Boundary Layers BY K . STEWARTSON. The Durham Colleges in the University of Durham. Durham. England Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rayleigh Problems for an Incompressible Fluid . . . . . . . . . . . . Rayleigh's Problem for a Compressible Fluid . . . . . . . . . . . . . Boundary Layer Growth in an Incompressible Fluid . . . . . . . . . . V Fluctuating Boundary Layers . . . . . . . . . . . . . . . . . . . . VI . Unsteady Compressible Boundary Layers . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. I1. I11. IV.

.

1

3 8 18 25

29 34 35

Boundary-Layer Theory with Dissociation and Ionization BY G. LUDWIG A N D M . HEIL.Institut f u r Theoretische Physik der Freien Universitat Berlin. Berlin. Germany Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 The Collision Equations . . . . . . . . . . . . . . . . . . . . . . . 42 The Equations of Transport for Molecular Properties of the Particles A , . 48 The Solution of the Collision Equations . . . . . . . . . . . . . . . . 54 The Collision Cross Section for the Dissociation of a Diatomic Molecule by Collision with an Atom . . . . . . . . . . . . . . . . . . . . . . . 87 V. The Boundary-Layer Equations for a Dissociating Gas A, . . . . . . . 93 VI . The Solution of the Laminar Boundary-Layer Equations for a Dissociating Gas 101 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11f3

I. I1. I11. IV.

.

The Propagation of Shock Waves along Ducts of Varying Cross Section BY W . CHESTER.University of Bristol. Bristol. England

.

I General Introduction . . . . . . . . . . . . I1. The Steady State Theory . . . . . . . . . . I11. Chisnell's Theory . . . . . . . . . . . . . . IV. Comparison of the Two Theories . . . . . . V. Steady Flow Regime Ahead of the Shock . . References . . . . . . . . . . . . . . . . . . . . ix

. . . . . . . . . . . .

.. . . . . . . ..

. . . . .

. . . . .

120

........ 123 . . . . . . . . 133 . . . . . . . . . 143 . . . . . . . . . 144 . . . . . . . . 162

CONTENTS

X

SImUarity and Equivalence in Compressible Flow

BY KLAUS OSWATITSCH. Deutsche Versuchsanstalt fur Luftfahrt

. Aachen. Germany

I . Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . I1. Applications of the Linear Theory . . . . . . . . . . . . . . . . . I11. Higher Approximations IV Transonic Similarity V . Hypersonic Similarity . . . . . . . . . . . . . . . . . . . . . . . . VI Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Bodies of Low Aspect Ratio . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. ....................... .........................

.

.

154 178 198 215 236 242 248 269

KPrmBn Vortex Streets

.

BY R WILLE.Technische Universitdt Berlin. Berlin. Germany 1.Introduction

............................ .......................... ...................

2. Stability Theory 3 Other Theories on Vortex Streets 4 Experimental Investigations of Vortex Streets 5 Related Problems 6. Summary

. . .

273 276 277 279 283 285 286

............. .......................... .............................. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AUTHOR INDEX ............................. 289 SUBJECT INDEX ............................. 293

The Theory of Unsteady Laminar Boundary Layers

BY K. STEWARTSON The Durham Colleges in the University of Durham, Durham, England Page

. . . . . . . . . . . . . . . . . . . . 2. Rotational Motion . . . . . . . . . . . . . . 111. Rayleigh’s Problem for a Compressible Fluid . . 1. Continuum Theory . . . . . . . . . . . . . . I. Introduction

. . . . . . . . . . . . . . . . . . . . . .

11. Rayleigh Problems for a n Incompressible Fluid 1. Translational Motion . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

2. Kinetic Theory . . . . . . . . . IV. Boundary Layer Growth in a n Incompressible Fluid . . . . . . . . . 1. Stagnation Boundary Layers . . . . . . . . . . . . . . . . . . . 2. Leading-Edge Boundary Layers . . . . . . . . . . . . . . . . . . V. Fluctuating Boundary Layers . . . . . . . . . . . . . . . . . . . . VI. Unsteady Compressible Boundary Layers . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

3 4

6 8 7

13 16 18

21 25 29 34 35

1. INTRODUCTION In many problems of fluid flow past solid bodies it may be assumed that a t a general point in the fluid the viscous stresses may be neglected. This means that almost everywhere the motion of the fluid is governed by Euler’s equations of inviscid flow and, consequently, that the fluid has a velocity of slip past the body. Since the fluid in contact with the body must be at rest relative to it, it follows that there must be a thin layer of fluid, adjacent to the body and called the boundary layer, in which the viscous stresses cannot be neglected. In his classical paper, initiating its study, Prandtl [l] assumed that changes in velocity occurred much more rapidly across the layer than along it. As a result he was able to reduce the Navier-Stokes equations for viscous flow to a much simpler form opening the way to the study of an important branch of fluid motion. Not only does its study provide information about local fluid properties such as skin friction, heat transfer between body and fluid, and the surface temperature of the body, but it also leads to a greater understanding of such large-scale phenomena as circulation, lift and the drag of bluff bodies. 1

2

K. STEWARTSON

There is no need at the present time to enter into a discussion of the general theory of boundary layers. The state of the theory for an incompressible fluid in 1938 has been discussed by Goldstein [2] and for a compressible fluid in 1953 by Howarth [3] while a large book solely on the subject has been written by Schlichting [4]. It has in fact long been recognized that the subject has grown so large that is is no longer convenient to review it all in a single article. Thus in earlier volumes of this series Kuerti [5] has reviewed compressible and Moore [6] three-dimensional boundary layer theory. The purpose of the present article is to consider certain aspects of unsteady boundary layers. All boundary layers occurring in practice are in a sense unsteady. Either the time froni the start of the experiment is small, or there are fluctuations in the velocity of slip of the inviscid flow outside the boundary layer, or the boundary layer is unstable (leading to turbulence), or there is some combination of these. Instability is the most important of these manifestations of unsteadiness and is usually considered on its own. The interested reader is referred to books by Lin [7] and Schlichting [a] for discussions of laminar instability and turbulence. However, other aspects of the theory are of considerable interest and importance, from a mathematical standpoint, for practical problems, and as an aid in understanding the behaviour of steady boundary layers. The present review of these aspects is divided into six chapters of which this introduction is the first. Chapters I1 and I11 are concerned with exact solutions of the Navier-Stokes equations. In steady flow such solutions are rare and, apart from three famous ones, trivial. In order to gain an insight into the role played by viscous stresses in the motion of real fluids and in boundary layer flows in particular, the exact solutions discussed here are therefore of importance. A common characteristic of all the solutions is that the solid boundaries move parallel to themselves so that, were it not for viscous effects, the fluid would not be disturbed. In the examples studied in Chapter I1 the fluid is supposed to be incompressible and usually each particle moves parallel to the boundary. I t is found that, so long as vt is small, where t is measured from the start of the relative motion, the solution has the character of the boundary layers envisaged by Prandtl. The mode of formation of boundary layers and their subsequent development can therefore be traced in these exact solutions. Further by using Rayleigh’s transformation [8] a considerable insight into the properties of steady boundary layers can be obtained. This is particularly true of the work described in Chapter I11 where the fluid is supposed compressible. Here the frictional heating in the shear flow induces temperature variations and hence, via the equations of state and of continuity, a component of fluid velocity normal to the boundary. Van Dyke’s solution of the flat plate problem [9] provided one of the earliest insights into the nature of steady boundary

UNSTEADY LAMINAR BOUNDARY LAYERS

3

layers in hypersonic flow. Again at the present time the study of this problem at very small values of t is one of the spearheads of the attack on determining the range of validity of the Navier-Stokes equations and deciding when kinetic theory can provide a more accurate answer. Chapter IV is concerned with boundary layer growth in an incompressible fluid when the solid boundaries do not move parallel to themselves, so that the fluid is disturbed everywhere from the time at which relative motion begins. The theory of the boundary layer depends on making the usual assumptions, that it is thin and that the flow outside it is given by the inviscid equations, but their consistency may be verified a posteriori. Two kinds of problem are considered. The first is the theory of boundary-layer growth on bluff bodies, in which the main interest is to determine the onset of separation and the subsequent behaviour of the fluid in the boundary layer. The second is the theory of boundary-layer growth on sharp bodies, in which the intrinsic interest is the elucidation of mathematical difficulties in the solution which are not present when the body is bluff. The effect of fluctuations in the velocity of the solid body or of the irrotational flow around it on the boundary layer is discussed in Chapter V. These problems are of practical interest, for example in the determination of the virtual mass of slender bodies and the theory of rotating stall of turbines. Finally in Chapter VI the theory of unsteady compressible boundary layers is discussed with special reference to shock tubes. The study is of importance for the determination of the temperature rise on and the heat transfer rate from the wall of the tube. It is also relevant to the theory of shock wave attenuation.

11. RAYLEIGHPROBLEMS FOR

AN

INCOMPRESSIBLE FLC'ID

The original problem considered by Rayleigh [8] is that of an infinite flat plate immersed in an incompressible fluid, .which is given impulsively at time t = 0 a velocity U in its own plane, thereafter moving with the same velocity and in the same direction. He showed that the fluid moved in the same direction as the plate with a velocity

where y measures distance from the plate. His solution is of great interest since it illustrates the way in which viscous effects, which are concentrated a t the solid boundary at t = 0 in a vortex sheet, subsequently diffuse outwards. Further, the basic assumption of boundary layer theory, that viscous effects are confined to the immediate vicinity of the plate is

4

K. STEWARTSON

confirmed by this example provided only that in some sense (vt)Yzis small. Rayleigh suggested that (2.1) could be used as a model to describe the steady flow past a semi-infinite flat plate. For, just as in the case of an infinite plate, the vortex sheet formed on the plate diffuses outwards, but now it is also convected parallel to the plate by the fluid stream. A model can therefore be constructed by assuming that the velocity of convection is U , the main stream velocity outside the zone of intense vorticity, and writing x = Ut where x measures distance from the leading edge along the plate. This model, admittedly approximate, is extremely useful, since it exhibits a number of salient features of the steady problem. Either in the form we have just described or as Oseen’s approximation to the boundary layer equations, its value as a model only ceases when non-linear effects, such as separation, become important. Accordingly a number of related problems have been studied, and we shall discuss them here giving them the name of Ra yleigh problems. They are all characterized by the feature that the solid boundaries move parallel to themselves so that, were it not for the viscous boundary condition, the fluid would not move. Hence as in Rayleigh’s original problem a vortex sheet is formed at t = 0 on the solid boundaries subsequently diffusing outwards. Two main problems have been studied, the translational motion of a cylinder in a direction parallel to its generators, and the rotational motion of a body about its axis of symmetry. 1. Translational Motion Problems of this kind are relatively simple because the fluid also moves parallel to the generators of the cylinder. Further, if u is the fluid velocity, Oy, Oz are fixed axes in the plane of a cross-section of the cylinder and O x is in the direction of motion, w is independent of x and satisfies

at

The boundary conditions are that u = U on the cylinder, where U is its constant velocity when t > 0; u + O as y 2 z2 + 00; and u = 0 when t < 0. The problem is thus reduced to a well-known form being the equation which describes the diffusion of temperature from a heated cylinder. Further, on applying the Laplace transform with respect to t the equation reduces to the harmonic wave equation in two-dimensions. Looked at from either point of view there are powerful and well-known techniques available for the solution of specific problems, and accordingly it is not necessary to discuss the details here.

+

UNSTEADY LAMINAR BOUNDARY LAYERS

5

First consider Rayleigh's problem for the half-plane y = 0, z 3 0 for which the solution was given by Howarth [lo]. The velocity u is of the form

-.+

it tends to (2.1) as z/(v~)'/~ 0 and to zero as z/(yt)li2 -+ - oo. In the region y2 z2 = O ( v t ) , which may be called the edge boundary layer, the behaviour of u is complicated but the skin friction is of a moderately simple form. Howarth was able to show that it is augmented and behaves like z-ll2 as z 0,. Rayleigh's transformation x = U t may now be applied to deduce certain qualitative features of the steady boundary layer on the 0, y = 0 in a uniform stream. In particular it may be quarter plate x inferred that the velocity changes of order U which depend on z occur within a distance 0(vx/U)'12 of the edge y = z = 0.However the value of the model is limited because the secondary flow, which is an important factor in controlling the quantitative behaviour of u near y = z = 0, is neglected. I t may also be inferred that the contribution from the side-edge boundary layer to the drag on the plate is O(pU1) but again the model neglects the potential flow, induced by the whole boundary layer, whose contribution to the skin friction is as important as that of the edge boundary layer. In a similar way cylinders whose cross-sections are infinite wedges have been studied (Sowerby [ l l ] , Hasimoto [12], Sowerby and Cooke [13]). The simplest and most illuminating of the solutions is for a wedge consisting of the two perpendicular planes y = 0, z > 0 and y > 0, z = 0; it is

+

4

which clearly shows the influence of the corner. Rayleigh's problem for a circular cylinder of radius a has been studied by Batchelor [14], among others (Hasimoto [15], Cooke [16]). From it he was able to infer important properties of the flows around more general cylinders. The new feature in his study is that the cylinder has a represent/ ~1 the ative length 1, equal to 2a if the cylinder is circular. When ( ~ t ) l<( boundary layer, in which viscous effects are important, is thin. The fluid velocity is therefore given by (2.1) except near a comer where a wedge solution is appropriate. When ( ~ t ) ">>~ E the boundary layer is much thicker than the cylinder. Now the precise shape of the cylinder is only important within a distance O(1) of it, which is a vanishingly small fraction of the boundary layer thickness. Hence in this region &/at may be neglected so that u is harmonic and known apart from a function of 1. On the other hand at distances from the cylinder 0(vt)'l2 its shape is not significant, and the solution has radial symmetry. Batchelor showed how the flows in these two regions may be matched and the solution at large times completed in a relatively simple way.

6

K. STEWARTSON

<

He found the drag on the plane strip y = 0, Iz[ a at small and large times as an example of his theory: a more complete study of this problem has been made by Hasimoto [l?] and Wilkinson [18] while Levine [19] has investigated the solution at small times. Batchelor's study has been further generalized by Lagerstrom and Cole [20] to include expansion of the circular cylinder as well as sliding (they supposed that a a t"; n, t > 0). They used the solution as a starting point of a thorough investigation of limiting processes and the theory of matching which has been of great use in certain problems of steady flow. 2. Rotational Motion

The flow due to an infinite circular cylinder of radius a which is given impulsively an angular velocity Q about its axis has been studied by Goldstein [21], Lighthill [22], and Mallick [23]. The fluid develops only an angular velocity which satisfies an equation of the same form as (2.2). The formal solution is straightforward and the motion can be traced right through to the well-known steady solution, reached in a time O(a2/v),in which all the fluid is rotating as if solid. A more complicated problem is the flow due to an infinite disc, which is given impulsively an angular velocity in its own plane (Thiriot [24], Nigam [25], Dolidze [26], Probstein [27]). Here initially there is only a sheared angular velocity in the fluid but the centrifugal force which it produces causes radial and axial motions as well. Only a couple of terms of the series for the velocity components have been worked out but it is fairly clear that the solution approaches the steady state solution of von KArmAn and Cochran [28], reaching it in a time O(i2-l). This new time scale appears because the boundary engendered a t the start of the motion does not increase indefinitely in thickness: viscous stresses are always confined within a distance ( Y / Q ) ~ / ' of the disc. A similar problem in which a sphere is given an angular velocity about a diameter has been discussed by Nigam [29]. Thiriot also gave the solution at a small time after an infinite disc, originally rotating about its axis with the fluid as if solid, was suddenly brought to rest. The modification to the solution, which is necessary when the disc has a finite radius, is discussed at the end of Chapter 4. It is noted that in the first problem considered by Thiriot the boundary layer on the disc is always independent of the radius a of the disc if a2i2 >> Y .

111. RAYLEIGH'S PROBLEM

FOR A COMPRESSIBLE

FLUID

The generalisation of the results of the previous section, to include the effect of compressibility, is very difficult because the shearing motion gives

UNSTEADY LAMINAR BOUNDARY LAYERS

7

rise t o temperature and density variations and hence, via the equation of continuity, to motion in a direction perpendicular to the solid body. Many of the new features in the flow may however be illustrated from a study of the motion engendered by an infinite flat plate which is immersed in a viscous compressible fluid, and which is set in motion at time t = 0 with a constant velocity U in its own plane. The attention of investigators in this field has in fact been mainly concentrated on this specific problem and solutions of the Navier-Stokes equations have been found in a number of limiting cases from which a picture of the flow pattern may be built up. In addition, the motion of the fluid has been examined a t times of the order of the relaxation time of the fluid molecules, when the validity of the Navier-Stokes equation has been questioned.

1. Continuum Theory

To begin with we shall suppose that the Navier Stokes equations are strictly valid and discuss the flow pattern which they imply. Let 0 be a fixed point in the plane of the plate, let O x , O y be measured parallel t o the direction of motion and perpendicular to the plate respectively. Then, since all x-derivatives are clearly zero, the equations of motion are

(3.2)

(:

p -+v-

(3.3)

:;)

=--+ ap - -

aP

-

at

ay

( i;),

4 a p3 ay

+ aya (pv)= 0 --

the validity of Stokes hypothesis, connecting the two coefficients of viscosity being assumed. There is no great advantage in using the hypothesis here but it is convenient to follow the original authors. To these equations must be added the equation of state (3.4) and the energy equation

p

=%

pT

8

K. STEWARTSON

On assuming the plate thermally insulating and taking conditions in the fluid at rest as standard, the boundary conditions are

I

u=u,

v=o,

~

aT =O aY

at

y=O;

and a t t = OK, just before motion begins. These equations are considerably more complicated than those for an incompressible fluid. The reason is that there is frictional heating in the fluid due to the shearing motion. This produces a temperature variation via (3.5), which causes a variation both of viscosity and of density. The density changes in turn produce a component of velocity normal to the wall through (3.3). Since fluid is being displaced in a direction perpendicular to the plate even if the viscous effects are confined to its neighbourhood, a compression wave will be sent out into the inviscid region beyond. In virtue of the assumption of a continuum, a t the instant when motion begins there will be an infinite dissipation a t the plate. Consequently the temperature will rise instantaneously at the plate carrying with it the pressure since the fluid has not had time to disperse. On the assumption that viscosity and conductivity are constant Howarth [30] has shown that at the plate

when t = 0, immediately after motion has started. The component of velocity parallel to the plate is the same as in the incompressible problem since v = O(tl”) when t is small. Again Stewartson [31] has shown that if u = y while ,u is an arbitrary function, a t the plate

-T- _To

Po

-1

+ +y(y - 1)M2

when t = 0,. The component of velocity parallel to the plate, u , now depends on ,u from (3.1). However ,u is a prescribed function of T and hence can be expressed in terms of u when t = O , . Thus as in the incompressible problem u is a function of y/t1I2 and the skin friction is O(t- l/’) when t is small. There now follows an interval of time O(vo/ao2)in which the motion is complicated, all terms being approximately of the same order of magnitude. The flow has been studied by Howarth [30], who supposed that M = U/ao<<1 so that squares and products of all perturbations, except the dissipation of

9

UNSTEADY LAMINAR BOUNDARY LAYERS

the shear flow may be neglected. A complete formal solution was given in terms of Heaviside’s operator in the particular case (r = 3/4,although it may actually be obtained for any value of u if desired. Howarth found that u

(3.9)

Y

= erfc-

2 v q’

while fi satisfies (3.10)

4 voy ay2at asp

+a

O

asp ~

ay2

-

azp ~

=

-

(Y - 1)PO2 (t-1 7E

at2

e-y’/vat)

at

and the boundary conditions

P=

POI

-

aP = O

at

t=O-,

dp = O

at

y=0.

aY

aY

y>O

and as y - m ,

From his formal solution Howarth deduced the pressure which agreed with (3.7)when t = 0 , and

(3.11)

Pw

-

Po -0.223

M2

p,

on the plate

(gy

as ao2t/vo m. It follows from (3.9),(3.10) that if M is small there is a region of thickness O(v0t)”’, corresponding to the boundary layer in Rayleigh’s problem for an incompressible fluid, in which frictional heating is important. The dissipation acts as a source inducing pressure variation and therefore motion in a direction perpendicular to the plate. Its effect may be seen on noting that (3.10) is the equation of sound propagation in a viscous fluid. Thus the pressure source in addition to producing changes in the region y = O(v,t)”’ also sends out a sound wave, with velocity a,, announcing the disturbance a t the plate. Of course the diffusive character of (3.10) means that the onset of the motion of the plate is known everywhere at t = 0, but the disturbance in the fluid is exponentially small ahead of the sound wave. The sound wave may be distinguished from the boundary layer when its distance from the plate (sot) >> ( ~ r ~ t ) ’ ’ i.e. ~ , when ao2t >> vo. If this condition is satisfied viscous effects may be neglected in the region between the sound wave and the zone of frictional heating. These considerations, which stem from the properties of (3.10),may be used t o describe the flow a t a general value of M provided that ao2t >> v,,. The fundamental difference is that the disturbances are no longer necessarily 4

10

K. STEWARTSON

small; the sound wave may now be a shock-wave and temperature variations will modify the shear boundary layer. The flow was first discussed by Illingworth [32] who obtained the properties of u,T when in addition (ao2t/M4vO) >> 1. He supposed in fact that t was large enough for the pressure variation in the fluid to be neglected and was then able to reduce the equations for u , T to a pair of ordinary dfferential equations with independent variable $/2t1’2. Here $ is defined by

(3.12)

-a*_ aY

,

whence

a*-_ - - PV

-

at

Po

Po

using (3.3). In particular, if the viscosity is proportional to the absolute temperature, explicit integrals were given from which u,T may be found in terms of $/2t1I2. I n the same paper he considered, among other problems (i) the diffusion of a vortex sheet separating gases at the same pressure but a t different temperatures and with different velocity components parallel to the sheet. (ii) flow near a plate moving with a variable velocity but at a constant temperature, neglecting dissipation. (iii) some effects of gravity. Illingworth’s solution of the impulsive problem has been used by Van Dyke [9] as a starting point in an iteration procedure alternating between the dissipation zone and the inviscid flow beyond. The solution has also been obtained when

by Stewartson [31]. Both of these new solutions are strictly only valid in the double limit M + ce, vo -+O so that M4v0 is finite. In them it is assumed that the first warning at any point of the impulse at the plate is a shock-wave of which the sound wave mentioned above is the limiting form as its strength tends to zero. Let its equation be y = Y ( t ) . Then $ ( Y , t ) = Y and the problem is essentially to find Y . Just behind the shock the fluid is moving in a direction perpendicular to the plate, conditions being known in terms of Y . It is also assumed that the flow in the region behind the shock is inviscid when the equations (3.1) - (3.5) reduce to

(3.14)

p=RpT,

ar

y - i r a p Y P at’

--- --_.

at

11

UNSTEADY LAMINAR BOUNDARY LAYERS

from (3.14)

where S is a function of t,h only. It is related in a simple way to the entropy and determined by conditions just behind the shock-wave where both the entropy and t,4 are known in terms of Y(t). Now S ( # ) 2 1 and is singular only a t # = 0 behaving like # - ' j 8 when # is small. Thus (3.13) reduces to a pair of equations for v,p whose coefficients are bounded except when $ = 0. Hence if the assumption that viscosity may be neglected is justified anywhere in 0 < t,h < Y it must be justified everywhere in that region. Therefore viscous effects must be confined to the immediate neighbourhood of t,h = 0 where they modify the effect of the singularity in S and enable v,zc,T to 0 to the change from their values according to the inviscid solution as i,b values on the plate which ensure that the appropriate boundary conditions are satisfied. Although this boundary layer is narrow in terms of t,b, that does not mean that it is thin (i.e. in terms of y) in comparison with Y . From the definition of #, it merely means that the mass of fluid in the boundary layer is small in comparison with the mass of fluid in the inviscid layer between it and the shock. In the boundary layer, aP/at,h is bounded from above since ii is bounded a t its outer edge, falling to zero at i+h = 0. Hence is constant across the layer and equal to its value according to (3.13) as # + O . If for simplicity the viscosity is taken to be proportional to the absolute temperature, the equations of the motion here now reduce to

-.

* (3.15)

with boundary conditions u=

u,

aT/a+=o,

+=o

u+ 0 ; v , T-+inviscid solution outside the boundary layer. The equation for u has solution

(3.17)

u = Uerfc-

*

2(voe)"2

1 t

where

8=

0

dt,

12

K. STEWARTSON

whence (3.18) Equation (3.16) may now be integrated with respect to # to give v at the outer edge of the boundary layer in terms of p/po without explicitly evaluating T. The matching of the boundary layer with the inviscid region is done by setting the value of v at # = 0, according to the inviscid solution, equal to the value of v a t # = 00 according to the boundary layer solution. Since, in the boundary-layer solution, T - r 0 as i,h+ 00 the match of T is automatic. The point about the matching of v is that any value #o > 0 of #, no matter how small it may be in comparison with Y,is large in comparison with (vofl)1/2 in the limit vo+O. In the case of the temperature it turns out that TIT, is an order of magnitude larger in the boundary layer than in the inviscid region, the orders being O(M2),

(3.19)

0(M2(vo/ao2t)1/2)

respectively in the two regions. Hence, to match the solutions, either T + 00 as t k - 0 in the inviscid solution or T + 0 as #-+ 00 in the boundary-layer solution or both, which is in fact what happens. The differential equations in the inviscid layer and the matching condition are both too complicated to permit of a solution in closed form. They make it clear however that the significant parameter is (3.20) When

x

is large it is found that, with y ' w-

=

1.4,

+ 0.64 + . . .,

0.3U441/2

Po

Y(t)= 0.879 aot(x'f4+ 1.01 x

- ~+ / ~. . .),

d(t) = 0.521 aot(x'14 - 1.25 x

- ~+ / ~. . .),

where fl, is the pressure a t the plate and d ( t ) is the sharply defined thickness of the boundary layer in terms of y. The reason for 6/Y being 0 ( 1 ) is that although the boundary layer contains little fluid relative to the inviscid layer it is relatively very hot according to (3.19), which in turn leads to a greatly reduced density and a greatly increased viscosity.

UNSTEADY LAMINAR BOUNDARY LAYERS

When

x

is small, Van Dyke

_ pw - 1 (3.21)

"31

+ 0.223

XI/*

13

found that, with y = 1.4,

+ 0.021 x + . . .,

Po d ( t ) = 0.45 u,~x'/'

+ . . ..

The next term in the series for the pressure p , on the plate is O(x3/' log x), and no more terms can be found explicitly by Van Dyke's method. The reason is twofold. In the first place the solution of the inviscid equations depends on knowing S ( $ ) in 0 < (CI < Y . In the calculation of (3.21)however S ( $ ) - 1 is neglected: it may be shown that the assumption is justified so long as terms O(x3") are neglected [33]. The iteration procedure however only determines E' when x is small and, hence, only S when (CI is large, whereas the term O(x"') depends on knowing S ( # ) everywhere. Secondly although the boundary-layer equations may be integrated to give the value of v used in the match, the formula depends upon 8 defined in (3.17). However, the iteration procedure only determines fi, when t is large, and this is clearly not sufficient to give 8 . Since the equations of motion in the inviscid layer are identical with those describing steady two-dimensional hypersonic flow, these two expansions, valid when x is large and small respectively, may be joined up if desired by any of the approximate techniques developed for this related problem. Of these the tangent wedge and shock expansion methods are perhaps the most convenient. For details of their use and range of validity the reader is referred to [34]. The theory sketched above is valid in the sense that the terms assumed negligible in it are confirmed to be so, provided only that v,,t >> uo8 (the condition given in [31] is slightly in error). I t is to be regarded therefore as the form which the boundary-layer solution of the title problem takes as M + w . 2. Kinetic Theory

Let us now consider the effect on the theory, given above, of the molecular nature of the fluid. According to the kinetic theory the mean time between successive collisions of a molecule is t = 4u2/3v. Hence we cannot expect the continuum theory to be correct, in an interval of time O ( r ) after the start of the motion. For this reason Howarth [30] expressed some doubt as to the validity of his solution, in which M << 1, when it differed significantly from Illingworth's asymptotic solution. Subsequently efforts have been made to use the kinetic theory to obtain a more accurate picture of the flow at these small times. The importance of the problem is that it may help to lay down limits within which the Navier-Stockes equations can be

14

K. STEWARTSON

used with confidence and in particular to decide whether it is meaningful to improve on the boundary layer solution within their frame work. At t = 0, the molecules of the gas are unaffected by the motion of the plate and they will have a mean velocity U of slip past it. The skin friction a t this time may be calculated because the distribution of velocities of the molecules is the same as when t < 0. Let us suppose it is Maxwellian and further let all molecules which strike the plate be absorbed by it and then re-emitted with a Maxwellian distribution at the temperature of the wall. Then [35] the skin friction coefficient at t = 0, is (3.22)

in contrast to Hawarth’s value (3.23)

for all t > 0. I t is noted that in the “free molecule” flow the skin friction depends on the absorption properties of the plate which is not the case in the “continuum’’ flow. Unfortunately, further progress on these lines is very difficult. A somewhat crude approximation but one which leads to some of the flow properties which might be expected to occur is due to Schaaf [36]. He retains the Navier-Stokes equation for u when M is small (3.24)

and the conditions u -+ 0 as y-, 00, zc = 0 a t y > 0, t = 0 but on the plate he uses a slip boundary condition u - U = L &lay where L is the mean free path of the molecules. He finds that (3.25)

which agrees with (3.23) when t is large and is of the same form as (3.22) when t = 0,. Further, by retaining the diffusion equation, he ensures that the general character of the flow is reasonable. A much more comprehensive attempt to bridge the gap between (3.22) and (3.23)using Grad’s equations [75] has been made by Lees and Yang [37]. Again they concentrate attention on the motion parallel to the plate, which is unaffected by the motion a t right angles t o the plate since M << 1. Grad’s equations, in the particular case of an insulated plate, then reduce to

15

UNSTEADY LAMINAR BOUNDARY LAYERS

(3.26)

where ex,,is the component in the x-direction of the stress across a plane perpendicular to the y-direction, qx = - 1 aT/ax and 1 is the coefficient of heat conductivity. There are in addition certain boundary conditions which depend on the reflective powers of the plate. A solution may now be obtained in terms of the Laplace transform; in particular if the fluid is monatomic and all the atoms which strike the plate are absorbed and reemitted randomly

Mc,= Mpxy(t'o) IP0U2

= 0.683

1 - 0.091 @

+ . . .] ,

U02t ~

VO

=

small,

YO

l . 1 2 8 ( $ 1 ~ ' z [ l -0.892-+V

...

a,%

These results agree with (3.23) when 1 is large and overestimate (3.22) by about 10% when t = 0,. The solution however cannot be regarded as a success. In the first place Grad's equations have been criticized on general grounds [38] and in one problem at least, the strong normal shock, the equations lead to a contradiction. Secondly, I'ang and Lees draw attention to some unsatisfactory features in the flow, among whch are (a) the distribution of the atomic velocities, which is Maxwellian at t < 0, changes discontinuously at t = 0 to another form even though no atom can have been affected by the motion of the plate, (b) the character of the equations (3.26) is hyperbolic implying the existence of a wave-front beyond which the atoms are unaffected by the motion of the plate. The velocity of the wave front is less than the speed of sound a, while the velocities of the majority of the atoms is very much greater than a,. The wave-front if set up therefore would immediately be smoothed out by the fast moving atoms near the plate. The essential character of the equations must in fact be parabolic and therefore diffusive, as in Schaaf's approximation. Thirdly, there is a curious feature about Grad's equations. All dependent variables should be independent of x as indeed has been the case hitherto. However qx is associated with the temperature gradient in the x direction, being proportional to it in the limit when the equations reduce to the NavierStokes equations, and yet it cannot be set equal to zero without a contradic tion.

16

K. STEWARTSON

In view of these criticisms the importance of the solution is mainly negative in that it throws further doubt on the value of Grad’s equations as a bridge between the solution of the Navier-Stokes equations and the “free-molecule” flow: it appears that the only hope of bridging the gap is by investigating Rayleigh’s problem with the help of the linearised MaxwellBoltzmann equation.

IV. BOUNDARY LAYERGROWTHI N

AN

INCOMPRESSIBLE FLUID

If a solid body, immersed in an incompressible fluid a t rest, is set in motion at t = 0 the initial motion of the fluid is irrotational. This may be seen experimentally and has been established theoretically in Chapter 2 when the surface of the body moves parallel to itself. I t may also be established in general if it is assumed that non-linear and viscous terms in the equations of motion may be neglected at t = O , , the assumption being justified a posteriori. Again as in Chapter 2 the fluid in contact with the surface of the body is at relative rest while adjacent fluid is now slipping past with a velocity determined by the ideal fluid theory. There is thus initially a vortex sheet in the fluid coincident with the surface of the body, which then diffuses into the fluid and is convected by the stream setting up a boundary layer. So long as the layer is thin the equations describing the flow in it may be derived from the Navier-Stokes equations on making Prandtl‘s classical assumption that the gradient of any component of the velocity in a direction perpendicular to the surface of the body is much greater than its gradient in a direction parallel to the surface of the body. Further since the boundary layer is thin we may assume to begin with that the irrotational motion outside is substantially unaffected by it and given by ideal fluid theory. Suppose for example that the flow is two-dimensional. Let the velocity of the body have components (0,P) in the (x,y) directions where x,y are measured parallel and perpendicular to its surface from an origin 0 fixed and the Navieron it. Then Prandtl’s assumption is that &/ax << Stokes equations reduce to

(4.2)

(4.3)

a% av -+-=o, ax ay

17

UNSTEADY LAMINAR BOUNDARY LAYERS

where K is the curvature of the surface of the body. Hence the change in the pressure p across the boundary layer is small and p may be determined from the conditions in the irrotational main stream just outside the boundary layer. If the fluid velocity there is L',(x,t) then

and (4.1) reduces to (4.5) The boundary conditions to be used with these equations are usually, but not always (4.6)

u=v=0,

x20,

y=o,

t>o;

(4.7)

u = 0,

x30,

y20,

t
(4.8)

u

y

t > 0.

4

U,(x,t),

x

3 0,

+

m,

I n addition there is a condition expressing the fact that the velocity is known a t some station x for all y,t > 0. If the station is x = 0 it means that the origin of coordinates must be either a stagnation point of the relative inviscid flow or the leading edge of the body. The condition there may be (4.9)

=

U,(O,t)

at

x =0

for all y ,

t

>0

or u may have a certain limiting behaviour. In view of its subsequent importance the condition a t x = 0 will be denoted by The form of the differential equation (4.5) is of interest. Regarding zc as a function of x,y or of t,y it is diffusive, while regarding u as a function of x,t it is wave-like with velocity of propagation u. Hence if the boundary layer is slightly disturbed a t ( x o , y o ) when t = to it is perturbed everywhere along the line x = xo immediately afterwards although it is exponentially small [- exp - ( y - y0)2/4v ( t - t o ) ] to begin with. However if u > 0 it also spreads in the direction of increasing x but not with velocity zc. The velocity of propagation is Max ( u ) which as is generally the case we shall assume to be U,. The mode is as follows. The disturbance travels u p the line x = xo immediately after it has been made and then travels through the outer part of the boundary layer, where u = U,, in the direction of x increasing with velocity U,. As soon as the disturbance reaches any new station x it immediately diffuses to all values of y a t that station. Although the disturbances do travel with a finite velocity there is no discontinuity at the wave-front because of the devious route by which the signal is transmit-

(n).

18

K. STEWARTSON

ted and because of the role played by diffusion. The disturbance a t a station downstream of (xo,yo) is exponentially small to begin with, thus ensuring continuity of all derivatives, and only gradually assumes its ultimate strength. If 1c < 0, i.e. if the boundary layer has separated, then in addition t o the mode of propagation just described there is also a similar mode of propagation upstream, with velocity Max (- u) as far as the start of the separated flow. This property of the governing differential equation means that unsteady boundary layers may be divided into two kinds. First there are the boundary layers in which it takes an infinite time for a signal from the line x = 0, travelling with the velocity U , of the main stream to reach any point downstream. These are the ones usually met with and are exemplified by the stagnation point flow in which U , x near x = 0 ; accordingly we shall refer to them as stagnation boundary layers. In them the fluid in x > 0 is never aware of conditions at x = 0 so that the condition (17)is irrelevant. Hence the straightforward iterative method of solution, to be described below, in which this condition is never used, is correct. Second there are boundary layers in which a signal from the line x = 0 takes only a finite time T to reach any point ( x , y ) , x > 0. The boundary layer a t (x,y) is independent of when t < T but is affected by (17)when t > T. This second kind will be called a leading edge boundary layer; an example is the uniform motion, after an impulsive start of a semi-infinite flat plate in which U , is a constant. In this case at ( x , y ) the fluid is unaware of the existence of the leading edge x = 0 if Ut < x so that the velocity is independent of x and given by (2.1). Subsequently however this can no longer be true.

-

(n)

1. Stagnation Boundary Layers The condition that a signal from the line x = 0, travelling with velocity U,(x,t) cannot reach any x > 0 in a finite time is that (4.10) x

-Po

Let us consider in detail flows in which U,is independent of t, t > 0. If (4.10) is satisfied we have a stagnation boundary layer and the condition (17) may be disregarded. These boundary layers may however be further subdivided, according as to whether separation (i.e. &lay = 0 at y = 0 for some x , t > 0) does or does not occur. If separation does not occur there is no difficulty: we may expect the steady-state boundary layer to be reached in a time O(Z/U),where 1 is a representative length on the body and U a representative velocity. An example of such a boundary layer, namely the flow engendered by a rotating disc in a fluid otherwise a t rest has already

19

UNSTEADY LAMINAR BOUNDARY LAYERS

been given in Chapter 2 : other examples may be constructed if desired and solved using the method to be described below. If separation does occur, then upstream of the point of separation steady state conditions will also be reached in a time O ( l / U ) . At separation however the steady two-dimensional boundary layer breaks down in general. I t appears that break-down can only be avoided if the main stream velocity satisfies some special condition, whose form is not known at present. From experimental and theoretical considerations it is known however that the U , derived from ideal fluid theory is not of this special kind. On the other hand it may easily be shown that the unsteady boundary layer does not break down a t separation. Downstream therefore it still exists but continues to grow in thickness until the assumption on which the governing equations are derived, that the boundary layer is thin, is no longer valid. This takes a time 0 ( l 2 / u ) . Subsequently we know from observation that the effect of viscosity is no longer confined to the neighbourhood of the wall downstream of separation, the main stream just outside the boundary layer upstream of separation is no longer given by the ideal fluid theory and there is usually an unsteady eddying wake to the rear of the body. The main interest in the investigation of boundary layer growth to date has been to find where separation first occurs and, occasionally, to discuss the subsequent growth of the boundary layer. Here the technique used by all the contributors is indicated for a particular example and a brief reference made to other cases considered. Suppose, following Blasius [39] that a t time t = 0 a cylinder is set in motion with a velocity 0, which is subsequently maintained, in a direction perpendicular to its generators. Let x be measured from the forward stagnation point. U,, which is known from ideal fluid theory and is independent of t, satisfies (4.10). Introduce a stream function JJ defined by u = a+iay,

v = - a+lax

and write (4.11)

Then

I/J =

4

2(vt)'/2U1(x)+(x,q,t), q = y/2(vt)1/2.

satisfies

(4.12)

with boundary conditions (4.13)

+= a+/aq = o

at

= 0,

a4laq-1

as

q-

00

20

the condition

K. STEWARTSON

(n)being disregarded. The solution is found by writing

(4.14)

On substituting into (4.12) it is found that the coefficients c$,,satisfy ordinary differential equations in q, x appearing only as a parameter, and can in principle be determined successively in order. Thus (4.15)

where j1 is known. One further term only has actually been worked out [40]. To a first approximation separation occurs at any particular place when (4.16)

1

3

+ (1 + -

U,'(x)t= 0,

first occurring when V,'(x)is a minimum. For a circular cylinder of radius a, U,= 2 0 sin x/u, and according to (4.16) separation first occurs at the rear stagnation point when Ot = 0.35 a, subsequently moving upstream. I t is noted that according to (4.14) v + 00 as q + 00 for fixed x,t, being of the form - yU,'(x) G ( x , t ) , but this does not imply any failure in the solution, the basic assumption of the boundary layer still holding. Continuity considerations in fact require v to be of this form when q is large and it makes possible a match between the boundary layer and the inviscid flow outside. The reader is referred to [2], pp. 181-190 for an account of solutions obtained up to the year 1938. A discussion is also given of the boundary layer of a uniformly accelerated cylinder: in this case only odd powers of t occur in (4.14) and (4.19) is slightly different. I n addition there is in [3], p. 60 a diagram of the stream lines round the rear of a uniformly accelerated circular cylinder shortly after separation has first occurred. The thickening of the boundary layer behind the point of separation is clearly shown. More recently the method described above has been used to determine the initial structure of the boundary layers occumng in the following problems :

+

(i) A body of revolution is given, simultaneously, an axial component of velocity and a n angular velocity (Illingworth [41]). (ii) A body of revolution is given, simultaneously, an axial component of acceleration and an angular acceleration (Wadhwa [a?]).

(iii) The impulsive motion of a general three-dimensional body (Squire [43]).

21

UNSTEADY LAMINAR BOUNDARY LAYERS

(iv) A cylinder is given a velocity At"-' or Aed, where A,a, c are positive constants, in a direction perpendicular to its generators (Watson [44], Gijrtler [45]). (v) The same problem as (iv) except that in addition the cylinder has an axial component of velocity (Wundt [46]). I n these papers interest has been centred on the determination, to a first approximation, of the onset of separation.

2. Leading-Edge Boundary Layers We now consider boundary layers in which (4.10) is not satisfied. They occur whenever the solid body has a sharp leading edge, the simplest example being, as already mentioned, the flat plate. From the general discussion it is clear that if T is the time it takes a signal travelling with velocity U , to reach ( x , y ) from the leading edge, the flow a t ( x , y ) when t < T is independent of (n).Hence the formal method sketched in the previous section is appropriate and in principle the formal solution may be written down. Once t > T however this straight forward solution is no longer sufficient and extra terms which depend in some way on because it ignores must be added. The precise dependence of these terms on is not known at present: the reason is partly that the governing equations are nonlinear and partly that it takes a time T for a signal from x = 0 to reach ( x , y ) . The nature of the flow and the difficulties involved in finding it is exemplified by the problem of the uniform motion, after an impulsive start of a semiinfinite flat plate. This problem has the advantage that many of the extraneous features of unsteady boundary layers are not present, so that attention can be concentrated on the special features requiring elucidation. The boundary conditions are (4.6) - (4.8) and (17)where U , = U , a constant. On dimensional grounds we may write

(n)

(4.17)

where (4.18)

and

+ satisfies

(4.19)

3 at3

(n)

(n)

22

K. STEWARTSON

with boundary conditions (4.20)

(4.21) The conditions a t t = do is

(n).

5 = 00 includes both (4.7), (4.8) while the condition a t

First consider the solution in t < 1, i.e. before the signal from the leading edge at t = 0 can reach ( x , y ) . Physically the fluid is not yet aware that the leading edge exists and will move as if the plate were infinite. The appropriate solution is given therefore by (2.1). Mathematically the character of equation (4.19) is determined by the highest order derivative with respect to each variable i.e. by its left hand side which is an equation of the heat conducting type with coefficient of conductivity (t- t aa+/a[). Hence if t < 1 the coefficient is positive, disturbances travel in the direction of t increasing and the condition at t = m is not applicable. The solution is therefore given by (2.1).

If t > 1 however the coefficient is partly positive and partly negative. Disturbances then travel in the direction of increasing t near the plate where 1 > t &$/a< and in the direction of decreasing t in the outer part of the boundary layer where 1 < t %#/a[. At the same time however they are diffused right across the boundary layer so that a disturbance at a n y point in t > 1 will eventually be felt at any other point in t > I . In particular the solution in t > 1 depends on the boundary conditions at t = 1 where u is given by (2.1) and at t = bo. Physically the reason is obvious. When Ut > x , the fluid knows about the existence of the leading edge (t= 00); further its motion must always depend to some extent on the previous history and in particular of the motion a t Ut = x (t= 1). Mathematically it is of interest to consider how # changes from being a function of ( y , t ) only in Ut < x to being a function of ( y , t , x )in Ut > x . Since the signal from x = 0 which reaches ( x , y ) when U t = x + partly as a result of diffusion first to and then from the outermost part of the boundary layer, it will be very weak, and an essential singularity, in which all derivatives with respect to t are zero at t = 1+, seems called for. The only attempt to find the way in which x does enter into a t t = 1, was made by Stewartson [47]. It was not however completely successful, although he was able to show that it could not be via a power series. At t = m a similar difficulty occurs. For then the steady state solution has ’ ’[t”’, ~ that is the Blasius been reached and I,!Iis a function of ~ ( U / V X )= function. The question is then how does the dependence on t finally disappear.

+

UNSTEADY LAMINAR BOUNDARY LAYERS

23

Stewartson [47] also considered this problem and showed first that the tenns depending on t could not be algebraically small when t was large and, second, that they could be exponentially small. There are apparently an infinite number of such terms each independent of the others when t is large and each containing an arbitrary constant, depending in some way, at present unknown, on the motion when Ut = x . The related problem, when U , = At", A and n > 0, has been considered by Cheng [48] both from a strictly mathematical and from an "engineering" point of view. The governing differential equation can be put into a form which is nearly the same as (4.19) on writing (4.22)

the only significant difference being that there is a constant forcing term as a result of the acceleration of the origin of the frame of reference. He came up against the same difficulty at t = 1 as we have been considering, but when t is large, because of the forcing term, he is able to obtain a formal solution in which is a function of Ct''' and t without including any of the exponentially small terms of the kind mentioned above and without making any reference to the boundary condition at t = 1 i.e. the condition that the flow as t - - r 1, must be the same as the known flow as z- 1-. Consider however the boundary layer when

+

U,=At",

z>l

= A F C ( t ) ,t

<1

where G is an arbitrary function of z except that C(1) = 1. According to Cheng the solution in z > 1 is independent of G(z)! In point of fact the dependence on G ( t ) when t is large occurs through the exponentially small terms which he ignored and for which the method of calculation was given in [47]. Two other problems, in which these difficulties emerge, have been studied. The first is the boundary layer formed, behind an advancing shock wave in a shock tube, which we shall consider separately in Chapter 6. The second is the flow engendered on suddenly stopping a disc of radius a , originally rotating together with the fluid surrounding it, with angular velocity SZ about the axis of symmetry. Thiriot [24] determined the initial stages of the motion at a finite value of r , the distance from the axis of symmetry, when the radius of the disc is infinite. If a is finite Thiriot's method, which is closely related to Blasius' [39], must be modified. Let x = a - Y , let y measure distance from the plane of the disc and let (u,v)be the components of the fluid velocity in the directions of x,y increasing, respectively. The equation of continuity is

24

K. STEWARTSON

a

- [%(a- x ) ] ax

a +[.(a aY

- x ) ] = 0,

and the equation of momentum in the x direction reduces to (4.23)

au at

+

au

24-

ax

au + 21 = R2(a - x ) [ l - w2] +

ay

Y-

a221

aY2

on applying the usual assumptions of the boundary layer theory, where o is the angular velocity of the fluid, and the boundary conditions on u,v are as given in (4.6) - (4.8) and except that U , r O . There is also an equation for w but for our purpose it is sufficient to consider (4.23) only. From the discussion in this chapter it is clear that the signal from the edge of the disc ( x = 0) travels as a wave in the direction of increasing x with velocity c towards the axis of symmetry. The wave front is a coaxial cylinder and the signal velocity c is the maximum value of u on Inside the fluid does not know of the existence of the leading edge so that Thiriot’s solution [24] is appropriate. Hence the signal velocity

(n)

r

r.

c = (a - x ) W ’ ( S Z t ) ,

r

F(0) = 0

where F is a function of Rt only, (SZt)2 when SZt is small and SZt when Rt is large, determined from [12]. One new feature is that the maximum value of u on C is not achieved in the outermost region of the boundary layer since u+ 0 there, but somewhere in the middle. The signal from x = 0 N

at t

=0

(4.24)

N

reaches any point in a time t given by

F(SZt) = log-

a

a-x

and subsequently [24] is inadequate. The effect of the edge of the disc is therefore felt at every x < a in a time (SZ-l), which is the time it would have taken Thiriot’s solution to have reached its ultimate steady state. It follows that that steady state, which has been calculated by Bodewadt [49] is never achieved in x < a , the boundary layer being dependent on a. However, according to (4.24) the edge effect never reaches x = a so that [49] is appropriate there when SZt 00. This implies that, as x + 0, the quantities u / ( a - x ) , w are bounded for all Rt. -+

This conclusion is of interest in connection with the theory of the steady state boundary layer on a finite disc. This originates at the edge of the disc entraining fluid as it grows towards the axis. Further on it loses fluid but unless it can lose it all by the time the axis is reached the boundary layer must erupt there. The plausible arguments in [50] against eruption are reinforced by the present discussion but still not made conclusive.

UNSTEADY LAMINAR BOUNDARY LAYERS

25

V. FLUCTUATING BOUNDARY LAYERS The motion induced by an infinite lamina, oscillating in its own plane, in an infinite incompressible fluid was first considered by Stokes [51]. He showed that when the frequency of the oscillation was large, the flow induced in the fluid was confined to the immediate neighbourhood of the lamina. Later Rayleigh [52] examined the influence of a rigid boundary on a standing wave, noted the existence of a thin frictional layer when the frequency of oscillation was high and pointed out that outside this layer there is a steady second order flow whose magnitude is independent of viscosity. A full discussion of this and other work of a similar kind is given by Goldstein [2], p. 187, Lamb [53] and Schlichting [4],p. 180. More recently the effect of small fluctuations in the main stream on the boundary layer have been studied. One of the originators of this study, Lighthill [54], points out that it is of interest from a number of points of view. First, if a thin body is moving with variable velocity through a fiuid it enables us to find "the frictional component of the virtual mass" which may be important since the virtual mass due to the irrotational flow is small. Second, the effect on heat transfer from a hot wire in a fluctuating stream may be examined. Third, it is of interest in connection with 'Rijke tube' phenomena and, when in addition the direction of the main stream is allowed to vary, flutter problems. Lighthill considered the boundary layer associated with a body whose velocity relative to the incompressible fluid surrounding it had a small sinusoidal component superposed on a non-zero mean value. Thus he had to solve (4.5) with a main stream (5.1)

Ul(%J)= U&) [I

+

Ee'l''l,

where E , o are constant and E is small, together with the boundary conditions (4.6) - (4.8)and He first supposed that o was small and wrote

(n).

(5.2)

u

= uo

+

E(U1

+ iou,)e'"

where u,, is the undisturbed value of u and (u,, the main stream is U,(x) (1 E ) . Hence

+

+

EUJ

is the value of u when

In additon up satisfied a differential equation which could only be integrated numerically. However provided that o was not too large he was able to argue, using the methods of the KArmQn-Pohlhausen approximation, that

(5.3)

(2)

y=o

. 1 =- UOdO* -2v

26

K. STEWARTSON

where So* is the displacement thickness of the undisturbed boundary layer, and inferred that u2 is then independent of w. This meant in particular that the skin friction increased with w and its phase advanced. The reason is that in the inner part of the boundary layer the tendency to respond more quickly than the main stream to the pressure gradient outweighs the inertial lag. At large values of w the non-linear terms may be neglected altogether whence it may easily be shown that (5.4)

= No

+ E u o ( x ) e i w t (1 - 8-

Y(W~)*'*

1.

-

The condition on o in order that (5.4) be valid is that ( O / Y ) ~ / ~>> So*. Hence w1/8 and its phase lead on the as w -+ 00, the increase in the skin friction main-stream fluctuations tends to n/4. When o = oo,where

and to is the skin friction in the unperturbed boundary layer, the phase lead of the skin friction according to the low frequency approximation, in which 2cz is independent of w ,is n/4. Lighthill noticed that at this value of w the amplitude of the skin friction in the low-frequency approximation was in good agreement with its value according to (5.4) and the corresponding velocity profiles were close to each other. He suggested therefore that the low-frequency approximation should be used if o < wo and (5.4)if w > q,. In the paper a discussion was also given of the temperature fluctuation induced by the velocity fluctuations. The theory is similar to that discussed above, the most notable difference being that the phase of the heat-transfer rate lags behind that of the fluctuation in the main stream velocity by as much asn/2 when w is large. This is because the temperature fluctuations arise from the inertia terms in the energy equation. Two particular cases of Lighthill's theory have been examined further. First Ghibelleto [55] and Illingworth [56] have studied the boundary layer on a flat plate placed end-on in a uniform stream and made to oscillate in its own plane. Illingworth's paper, which is the later of the two, is concerned with a slightly compressible fluid and in part it seeks to improve on Lighthill's solution at large frequencies by an iterative process started by substituting (5.4) back into the governing equations. Although such a procedure will formally give the leading terms in the expansion it must be used with caution. The reason is that the high-frequency solution is applicable when o x / U o is large and so may be regarded as an asymptotic expansion valid when x is large. Now since a signal from the leading edge travelling with the velocity of the main stream can reach any point in the boundary layer in a finite time it follows that the complete solution must take notice

UNSTEADY LAMINAR BOUNDARY LAYERS

27

of the boundary condition there, which Illingworth’s expansion by its very nature cannot do. There must therefore be additional terms present in the complete solution which are not known yet and whose influence a t moderate values of o x / U , cannot be assessed. Illingworth also considered the solution a t small values of wx/CJ, and finds strong support for Lighthill‘s approximate ~ be neglected. The subject of Illingworth’s inethod when ( O X / U , )may paper is the effect of a sound wave on a compressible boundary layer; the results just mentioned are in fact limiting cases in his study. He also obtains corresponding results when terms O ( M ) can no longer be neglected and when there is heat transfer across the plate. Secondly, Rott [57] and Glauert [68] have studied the two dimensional boundary layer on a flat plate placed perpendicular to a steady stream and made to oscillate in its own plane. Here the problem may be reduced to the solution of an ordinary differential equation; it proved possible to obtain a complete description of the flow a t all values of w using series either on ascending or descending powers of o. The difficulty mentioned earlier in connection with Illingworth’s solution does not apply here since the main stream is of the stagnation type discussed in Chapter IV. It appears, as may perhaps have been expected, that Lighthill’s approximate method is only of qualitative value near w = oowhere the two halves of the solution are joined together. The related problem, in which the direction of oscillation of the plate is perpendicular to the phase of the steady motion of the fluid, had been considered earlier in an independent investigation by Wuest [59]. Here the flow in the plane of the steady motion is unaffected by the oscillation, and Wuest gave numerical solutions of the equations, describing the flow in a direction perpendicular to this plane, for two values of o which should prove valuable as a test of approximate methods. A further check on Lighthill’s theory is provided by a solution in closed form of the equations when there is suction a t the wall. Stuart [60] considered a uniform stream flowing past an infinite plane wall into which fluid was being drawn a t a uniform rate and which was oscillating in its own plane. The boundary layer equations are now independent of x , and he showed that a simple solution valid for all o would be found, which may be compared with Lighthill’s theory. Again the approximate theory is only of qualitative value near o = 0,;thus at w = oothe correct value of the phase lead of the skin friction is only about one-half Lighthill’s value of n/4. Carrier and Diprima [61] have examined the flow in the neighbourhood of the leading edge of an oscillating semi-infinite flat plate in a uniform stream where the assumptions of the boundary-layer approximation are no longer valid. They base their analysis on Oseen’s modification of the full equations of motion and deduce a qualitative description of the flow near the leading edge. In particular they show that the phase lead in the skin friction at the leading edge varies from 0 a t w = 0 to n/8 as o+ OD.

28

K. STEWARTSON

Previously to Lighthill’s paper, Moore [62] had discussed the unsteady motion, with velocity U ( t ) ,of a semi-infinite flat plate in its own plane, when surrounded by a viscous fluid. Although he supposed that the fluid was compressible he was able to show (see Chapter \‘I below), that the governing equations may formally be reduced to those for a n incompressible fluid. As a first approximation he assumed that U(t)was independent of t, obtaining a “quasi-steady’’ boundary layer with a Blasius velocity profile. He then wrote down a formal expansion for the solution using this as a leading term and introducing an infinite number of nondimensional parameters, associated with U ( t ) , of which the first was xU’(t)/U2. Thus the assumption of the quasisteadiness of a boundary layer in unsteady flow is only justified if XU’ << U 2 , and Moore pointed out that this is equivalent to requiring the time of diffusion across the boundary layer to be much less than the characteristic time in the velocity fluctuations. In the opposite case, which Lighthill called the high frequency regime the velocity fluctuations all occur near the plate and are of the kind described by Stokes [51], while the outer parts of the boundary layer remain substantially unaffected. Moore’s expansion is not complete, for, by the nature of his method, no notice can be taken of the conditions at the start of the motion. For example it gives the wrong answer in the simple case U(t) = U,,1 > 0; U ( t ) = 0, t < 0. However since it may be shown, following [47], that the effect of this condition is exponentially small when t is large, Moore’s expansion is then probably a good approximation. The theory has been extended and combined with Lighthill’s (Ostrach [63], Moore and Ostrach [64]). I n particular by evaluating second order terms in the solution the effect on mean heat transfer rates of fluctuations in the plate velocity has been calculated. Wuest [69] and Moore [65] have also discussed the boundary layer when the main stream velocity U , = Axm (A,m constants), while the plate is oscillating in its own plane. In particular Moore considered the special case m = - 0.0904, a t which the mean profile is the separation profile, and discussed the implication in the theory of unsteady separation. It is admittedly speculative but of great interest as a first step in a study of the boundary layer aspects of such problems as rotating stall and stall flutter. One difficulty with this question is that, although it is likely that all steady boundary layers which occur in practice are regular a t separation, none of those calculated are regular there, while there is no evidence that a general unsteady boundary layer can be singular. Again the rapid thickening, which is almost always a characteristic of the steady boundary layer downstream of separation, is possible because the fluid in the separated region has not, ultimately, come from upstream of the plate as has the fluid in the rest of the layer. Presumably the same applies to unsteady separation, but deciding where the fluid has come from is a much more formidable

29

UNSTEADY LAMINAR BOUNDARY LAYERS

problem. In steady flow we can decide where the fluid comes from quite easily by studying the instantaneous flow pattern, i.e. the stream lines. In unsteady flow this is not sufficient and the whole history, i.e. the paths of the fluid particles, must be considered in framing a general criterion. In particular cases of course this may not be necessary.

VI. UNSTEADY COMPRESSIBLEBOUNDARY LAYERS As in the theory of incompressible boundary layers the basic assumption here is that the boundary layer is thin, so that changes across the layer occur much more rapidly than those along the layer. The derivation of the equation from the Navier-Stokes equations also follows similar lines and the interested reader is referred to Howarth [3] for a full discussion. The equations are (6.1)

{

as

p aau t+Udx+u--

au]

aY

a0

=pat--+-

pKu2 = p

a~

--

at

-

ap ax

aay

( ); p-

I

ap

-,

ay

together with the equation of energy

and the equation of state

In these equations cp and a are usually taken to be constant while p is known complicated function of the temperature T a close and convenient approximation to which is given by Chapman’s law [66],

a

T

where C is chosen so that the viscosity is correct at the wall and is therefore a function of x,t. Frequently it is assumed in the theory that a = C = 1. The boundary conditions on the velocity are the same as in (4.6) - (4.9). In addition however the density and temperature must tend to prescribed values as y+ 00 and as x,t tend separately to certain initial values. At the

30

K. STEWARTSON

wall however it is only necessary to prescribe either the temperature or a heat transfer property of the wall. The pressure may be assumed independent of y from (6.2) since the boundary layer is thin, while outside the boundary layer the entropy of any fluid element is constant in continuous flow. The main differences between compressible and incompressible boundary layers stem from thermal effects. As a result of the shearing flow in the boundary layer there is a dissipation of energy, represented by the last term of (6.4) which in turn modifies the temperature and density since the pressure is prescribed. Hence finally the velocity and profiles are affected not only because p enters into (6.1) but also because the viscosity depends on temperature. In general therefore the governing equations present a formidable problem necessitating the use of high-speed calculating machines for their solution. Some simplification is possible however by transforming the y coordinate. The method was originally used by Howarth [67] in steady flow and adapted by Stewartson [47] and Moore [62] independently to unsteady flow. Write (6.7) and use Y as an independent variable instead of y. Then it may be shown that the equations of momentum and energy reduce to

(6.9) together with certain boundary conditions which may be inferred from those given above in terms of y. In steady flow a further simplification is possible on adopting a model fluid in which a = C = 1 and supposing that the wall is thermally insulating, when there is a correlation between the compressible and an incompressible boundary layer. In unsteady flow however further simplification is only possible with particular main streams. Thus if fi, = fro, TI= To, U,= a constant U,,,which corresponds to a flow in which the wall is a semi-infinite flat plate, given impulsively a velocity V,, the solution is formally independent of whether the fluid is incompressible, or compressible and obeying Chapman's law (6.6). The differences in the actual flows arise because of the temperature effect in (6.7).

UNSTEADY LAMINAR BOUNDARY LAYERS

31

Two main problems have been considered in the theory of unsteady compressible boundary layers. It has proved convenient to discuss the first, i.e. fluctuating boundary layers, in Chapter V, because in virtue of (6.8) and (6.9) the formal theory is practically independent of compressibility effects. The second problem, the boundary layer behind an advancing shock, we shall discuss here. Consider a thin diaphragm at x = 0 in an infinitely long cylindrical shock tube lying parallel to the x axis. Let the fluid in the tube in x < 0, x > 0 he at rest in conditions characterised by a suffix 0 and an asterisk respectively, and let Po> P*. The diaphragm is broken at t = 0, causing a shock wave to advance with constant velocity C* in the direction of increasing x. Behind the shock the fluid has come from x > 0 and passed through the shock acquiring constant velocity U,, pressure $, and temperature T2*. The boundary between fluid originally in n > 0 and originally in x < 0 is a plane advancing with velocity U,. This plane is a contact discontinuity across which the temperature and density may jump while pressure and velocity are continuous. Between the contact discontinuity and a plane x = c,t, where c2 may be either positive or negative, is a second region of uniform flow. Beyond the plane x = c& is a centred rarefaction wave bounded on the other side by the plane x a$ = 0, which may be regarded as a sound wave advancing into the fluid to the left of x = 0. In the rarefaction wave the flow is homentropic with

+

(6.10)

and beyond the fluid is at rest. The present interest in this problem is to find the boundary layer on the walls engendered by the flow just described. For this purpose it is not necessary to specify c*, c2, P,, U 2 , T , further although in fact they may all be expressed in terms of the initial states of the fluid in x > 0, x < 0. Looking a t the problem from a physical standpoint it is useful to note that the velocity U , of the fluid in the main stream is positive. Thus a signal sent out from the line x = 0 at t = 0 and travelling with velocity U , will not overtake the contact discontinuity nor will it penetrate the region x < 0. Hence the fluid in the boundary layer in U& < x < c*t is independent of the condition a t x = 0, t = 0 and indeed of the flow in x < U,t for a similar reason. By the same argument it follows that the boundary layer in x < 0 is independent of the flow in x > 0. However the boundary layer in 0 < x < U& is not independent of the flow in x < 0 or in U,t < x < c*t and it may be expected that completing the solution in this region will be much harder than elsewhere.

32

K. STEWARTSON

The situation is reminiscent of the boundary layer engendered by a semiinfinite flat plate which was discussed a t length in Chapter IV and the analogy may be made mathematically clearer on selecting as origin of coordinates a point on the curve where the shock meets the wall of the tube. The complexity of the problem can be reduced to some extent by making use of the absence of a dimension of length in the flow. Write (6.11)

$ = (~'~oc*)~'*F(5,~) T = T(t,q)

(6.12)

Then (6.8) reduces to

(6.13)

and there is a similar form for (6.9). An appropriate set of boundary conditions is

(6.15) (6.16)

(6.17)

_ aF -0,

T=To

at

E = l + - ,a0 C*

a3

q>O,

In the discussion to follow we shall make the unexceptional assumption that

12 iJF/aq2 I - U,(t)/c*

in

0

i.e. that the velocity profile does not overshoot the limits at either end. There are then three regimes of flow (a) 0 < E

< 1 - U2/c*, (b) 1 - U,/c* < 5 < 1

and (c) 1 < E

< 1 + uo/c*.

33

UNSTEADY LAMINAR BOUNDARY LAYERS

I n region (a), which lies between the contact discontinuity and the shock, the leading term involving differentiation with respect to 6 has a positive coefficient and so disturbances propagate in the direction of increasing t . The only relevant boundary conditions therefore are those a t 7 = 0, bo and a t t = 0. Further [TI([), p,(t)are constant so that there is a solution in which F , T are independent of 6. The flow in region (a) has been studied by Mirels [68,691 and Demyanov [70] for general values of U , and by Nigam [71] when C’, = c*. All three authors assumed, explicitly or implicitly, that (6.6) held with C = 1 whence the equation for F is formally independent of T and is closely related to the Blasius equation, only the boundary conditions being different. Mirels also considered the temperature variation if u = 0.72 or 1 and if the wall is maintained a t a constant temperature. In region (c), to the left of the diaphragm, the coefficient of a2F/ayat is negative since aF/ay 1. Hence disturbances propagate in the direction of decreasing 6,and only the conditions a t y = 0,00 and at 6 = 1 a,/c* are relevant. The flow is more complicated than that in region (a) since U,, and therefore also F , is dependent on t. It has been studied by Cohen [72] who assumed the validity of (6.6). and that the temperature of the wall is constant. He expanded F , T as power series in 6 whose coefficients were functions of y and whose leading terms are associated with the boundary layer on a uniformly accelerated plate. In principle, either by using more terms of the series or a step-by-step numerical integration, his solution can be continued as far as 6 = 1.

<

+

I n region (b), between the diaphragm and the contact discontinuity, the critical coefficient is partly positive and partly negative so that, as in the leading edge flows described in Chapter IV, propagation takes place in the directions of 6 both increasing and decreasing. Hence there is no a priori reason why the conditions on aFjay a t t = 1 - U2/c* and a t E = 1 cannot be prescribed arbitrarily: they are in fact known independently of the flow in (b) and of each other. Nothing else is known however about the flow in regions such as (b) beyond the perhaps tentative remarks in 1471, and the determination of flow properties here remains one of the outstanding problems in the theory of unsteady laminar boundary layers. I t would be very helpful if a numerical calculation could be carried out in one instance to have a check on approximate theories and deciding between conflicting views of the flow. Two other interesting problems, as yet unsolved, may be mentioned. First, what is the effect of the discontinuity in the temperature in the main stream at the contact discontinuity E = 1 - U,/c* on the flow in the boundary layer. It is known that a temperature discontinuity at the wall is diffused and convected so that the temperature changes in the fluid occur smoothly. If instead the discontinuity occurs at q = 00, in general the

34

K. STEWARTSON

temperature does not change smoothly at any finite value of 7 either. However in the shock-tube problem the jump occurs at the contact discontinuity where the governing equations are in some sense singular, and special considerations may apply. If the temperature is also discontinuous in the boundary layer, it follows from (6.7) that the velocity in the boundary layer would have to be discontinuous even though in the main stream it is continuous. Secondly, at the end of the rarefaction wave, where 6 = 1 - cn/c*, the pressure gradient changes discontinuously. I t is unlikely that separation occurs however for the pressure gradient is not unfavourable, but it would be interesting to know what happens. As the boundary layer behind the shock expands, it will, in the way described in Chapter 111, send out a shock wave itself which however will not be parallel to the wall of the tube. The first steps towards finding its effect on the main shock and hence of developing a theory of shock attenuation has been made by Mirels [74] and by Huber and McFarland [73].

NOTATION Time from the start of the motion of the body relative to the fluid. Origin of coordinate system fixed relative to the body. X Distance from origin along surface of the body. Distance from the surface of the body. Y Components of velocity in the direction of x,y increasing of fluid in the w. lJ boundary layer. Components of the velocity of the body in the directions of x , y increasing. V Curvature of surface of body. 2 Representative length. U Representative velocity. a Velocity of sound. M = Ulao Mach number. P Pressure. P Density. T Absolute temperature. Specific heat a t constant pressure. CP a Prandtl number. P Coefficient of viscosity. V Kinematic viscosity. Ratio of specific heats of the fluid. Y 1

0

-

SUBSCRIPTS

Value at the body. Value at a standard position. Value in the main stream just outside the boundary layer. 1, 2 Other notation is defined in context. W

0

UNSTEADY LAMINAR BOUNDARY LAYERS

35

References L., Uber Fliissigkeitsbewegung bei sehr kleiner Reibung. Verhandl. 1. PRANDTL, 3. Intern. Math. Kongr., Heidelberg, 1904, 484-491. S., (editor) “Modern Developments in Fluid Dynamics”. Oxford, 1938. 2. GOLDSTEIN, L., (editor) “Modern Developments in Fluid Dynamics : High Speed 3. HOWARTH, Flow”. Oxford, 1953. 4. SCHLICHTING, H., “Grenzschicht-Theorie”. G. Braun, Karlsruhe, 1951. 5. KUERTI.G., The laminar boundary layer in compressible flow, in Advances i n Applied Mechanics 2 , 23-91 (1951). 6. MOORE,F. K., Three dimensional boundary layer theory, in Aduances i n Applied Mechanics 4, 159-228 (1956). 7 . LIN. C. C., ”The theory of Hydrodynamic Instability”. Cambridge U.P., 1955. 8. LORDRAYLEIGH, On the motion of solid bodies through viscous liquids, Phil. Mag. 21. (6), 697-711 (1911). 9. VANDYKE,M. D.. Impulsive motion of an infinite plate in a viscous compressible fluid, 2 . f. angew. Math. u. Phys. 8, 343-353 (1952). L., Rayleigh’s problem for a semi-infinite plate, Proc. Camb. Phil. Soc. 10. HOWARTH, 46, 127-140 (1956). 11. SOWERBY, L., The unsteady motion of a viscous fluid inside an infinite channel, Phil. Mag. 42, (7), 176 (1951). 12. HASIMOTO, H., Note on Rayleigh‘s Problem for a bent flat plate, J . Phys. Soc. Japan 6, 400-401 (1951). 13. SOWERBY. L. and COOKE,J . C., The flow of fluids along corners and edges, Quart. J . Mech. A p p l . Math. 6, 50-70 (19.53). 14. BATCHELOR, G. K., The skin friction on infinite cylinders moving parallel to their length, Quart. J . Mech. A p p l . Math. 7 , 179-192 (1954). 15. HASIMOTO, H., Rayleigh’s problem for a cylinder of arbitrary shape, J . Phys. S O C . Japan 9. 611-619 (1954). 16. COOKE.J. C., On Rayleigh’s problem for a general cylinder, J . P h y s . Soc. Japan 11. 1181-1184 (1956). 17. HASIMOTO, H., Rayleigh’s problem for a plate of finite breadth, Proceedings of the 1st Japan National Congress for Applied Mechanics, 1951, 447452. 18. WILKINSON, J., Dissertation for the degree of Doctor of Philosophy, Cambridge, England, 1955. 19. LEVINE,H., Skin friction on a strip of finite width moving parallel to its length, J . Fluid Mech. 2, 145-158 (1958). 20. LAGERSTROM, P. A . and COLE, J . D., Examples illustrating expansion procedures for the Navier-Stokes equations, J . Rat. Mech. Analysis 4, 817-882 (1955). 21. GOLDSTEIN, S., Some two-dimensional diffusion problems with circular symmetry. Proc. London Math. Sac. 84 (2) pp. 51-88 (1932). 22. LIGHTHILL, M. J .. Note on the ultimate form of rotary fluid motion inside a cylinder, Quart. J . Math. 10, 6 6 6 6 (1948). 23. MALLICK,D. D., Non-uniform rotation of an infinite circular cylinder in an infinite viscous liquid, 2. f . angew. Mafh. u. Mech. 87, 385-393 (1957). 24. THIRIOT.K. H., Grenzschichtstromung kurz nach dem Anlauf bzw. Abstoppen eines rotierenden Bodens, 2. 1. angew. Math. u. Mech. 80, 380-393 (1950). 25. NIGAM,S. D., Rotation of an infinite plane lamina: boundary layer growth: motion started impulsively from rest, Quart. A p p l . Math. 9. 89-91 (1951). D. E., Unsteady motion of a viscous liquid created by a rotating disc 26. DOLIDZE, (in Russian), Prikl. Mat. a . Mekh. 18. 371-378 (1954). 27. PROBSTEIN, R. F.. On a solution of the energy equation for a rotating plane started impulsively from rest, Quart. A p p l . Math. 11, 240-244 (1953).

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K. STEWARTSON

28. COCHRAN, W. G.. The flow due to a rotating disc, Proc. Camb. Phil. SOC.SO, 365-375 (1934). 29. NIGAM, S. D. and RANGASAMI, K. S. I., Growth of boundary layer on a rotating sphere, 2. f . angew. Math. u. Phys. 4, 221-223 (1953). 30. HOWARTH, L., Some aspects of Rayleighs problem for a compressible fluid, Quart. J . Mech. A p p l . Math. 4, 157-169 (1951). 31. STEWARTSON, K., On the motion of a flat plate a t high speed in a viscous compressible fluid. I. Impulsive motion, Proc. Camb. Phil. SOC.51, 202-219 (1955). 32. ILLINGWORTH, C. R., Unsteady laminar flow of gas near an infinite flat plate, Proc. Camb. Phil. Sac. 46, 603-613 (1950). 33. STEWARTSON, K., On the motion of a flat plate at high speed in a viscous compressible fluid. 11. Steady motion, J . Aeronaut. SOC.22, 303-309 (1955). 34. LEES, L., Hypersonic Flow. Fifth International Aeronautical Congress, Los Angeles. 1955. 35. HEINEMANN, M., Theory of drag in highly rarefied gases. Comnz. Pure and A p p l . Math. 1, 259-273 (1948). 36. SCHAAF,S. A., A note on the flat plate drag coefficient, Univ. of Calif., Inst. of Eng. Res.. Report No. He-15CL66 (1950). 37. YANG,H. T., and LEES, L., Rayleigh’s problem a t low Mach Number according to kinetic theory of gases, J. of Math. and Phys. 86, 197-235 (1956). 38. LAITONE, E. V., On the equations of motion for a compressible viscous gas, J . Aeronaut. Sci. 28, 846-854 (1956). 39. BLASIUS,L., 2. f . Math. u. Phys. 66, 20-37 (1908). S. and ROSENHEAD, L., Boundary layer growth, Proc. Camb. Phil. SOC. 4,O. GOLDSTEIN, 82, 392-401 (1936). 41. ILLINGWORTH, C. R., Boundary layer growth on a spinning body, Phil. Mag. 46, (7), 1-8 (1954). 42. WADHWA, Y. D., Boundary layer growth on a spinning body: Accelerated motion, Phil. Mag. 8, (a), 152-158 (1958). 43. SQUIRE,L. C., Boundary layer growth in three dimensions, Phil. Mag. 46, (7). 1272-1283 (1954). 44. WATSON,E. J., Boundary layer growth, Proc. Roy. SOC. 231, 104-116 (1955). H., , Ingen. Archiu 14, 286-305 (1946). 45. G ~ R T L E R 40. WUNDT,H., Wachstum der laminaren Crenzschicht an schrag angestromten Zylindern bei Anfahrt aus der Ruhe, Ingen. Archiv 28, 212 (1955). 47. STEWARTSON, K., On the impulsive motion of a flat plate in a viscous fluid,. Quart. Appl. Math. and Mech. 4, 182-198 (1951). 48. CHENG,SIN-I, Some aspects of unsteady boundary layer theory, Quart. A p p l . Math. 14, 337-352 (1956). 49. BODEWADT, U. T., Die Drehstromung iiber festem Grunde, Z. 1. angew. Math. u. Mech. 20, 241-253 (1940). 50. STEWARTSON, K., On rotating laminar boundary layers. I n “Boundary Layer Research” (H. artier, ed.) pp. 59-70. Springer, Berlin, (1958). 51. STOKES,G. G., On the effect of the internal friction of fluids on the motion of pendulums, Trans. Camb. Phil. SOC.9 (1851). 52. LORDRAYLEIGH,Phil. Trans. Roy. SOC.London, A, 176, 1-21 (1883). 53. LAMB,H., “Hydrodynamics”. Cambridge, 1932, 619-620. M. J., The response of laminar skin friction and heat transfer to fluctua54. LIGHTHILL, tions in the stream velocity, Proc. Roy. SOC. A, 224, 1-23 (1954). 55. GHIBELLATO, S., Atti Acad. Sci. Torino, CZ. Sci. Fis. Mat. Nut. 89, 180 (1955). 58. ILLINGWORTH, S., The effects of a sound wave on the compressible boundary layer on a flat plate, J . Ff. Mech. 8, 471-493 (1958).

UNSTEADY LAMINAR BOUNDARY LAYERS

37

57. ROTT,N., Unsteady Viscous flow in the vicinity of a stagnation point, Quart. A p p l . Math. 18, 444451 (1955). 58. GLAUERT. M. B., The laminar boundary layer on oscillating plates and cylinders, J . Fl. Mech. 1. 97-110 (1956). 59. WUEST,W., Grenzschichten an zylindrischen Korpern mit nichtstationarer Querbewegung. Z. /. ungew. Math. u. Mech. 82. 172-178 (1952). 60. STUART, J . T., A solution of the Navier-Stokes equations illustrating the response of skin friction and temperature of an infinite plate thermometer to fluctuations in the stream velocity. Proc. Roy. Soc. A. %31, 116-130 (1955). 61. CARRIER, G. F. and DIPRIMA,R. C., On the unsteady motion of a VISCOUS fluid past a semi-infinite flat plate, Quart. of ApPI. M a f h . 8K. 359-383 (1957). 62. MOORE,F. K., Unsteady boundary layer flow, Natl. Advisory Comm. Aeronaut. Tech. Note No. 2471 (1951). 63. OSTRACH. S., Compressible laminar boundary layer and heat transfer for unsteady motionsof a flat plate, Natl. Advisory Comm. Aeronaut. Tech. Note No. 3569 (1955). 64. MOORE,F. K. and OSTRACH, S., Average properties of compressible laminar boundary laver on a flat plate with unsteady flight velocity, Natl. Advisory C m m . ,4eronaict. Tech. Note No. 3886 (1956). 65. MOORE, F. K., Aerodynamic effects of boundary layer unsteadiness, Sixth AngloAmerican Aeronautscal Conference, Royal Aeronautical Society, 1957. 66. CHAPMAN, D. R. and RUBESIN. M. W., Temperature and velocity profiles in the Compressible Laminar boundary layer with arbitrary distribution of surface temperature, J. Aeronaut. Sci. 16, 547-565 (1949). 67. HOWARTH, L., Concerning the effect of compressibility on laminar boundary layers and their separation, Proc. Roy. Soc. A, 194. 1&42 (19t8). 68. MIRELS,H., Laminar Boundary layer behind shock advancing into stationary fluid, Natl. Advisory Comm. Aeronaut. Tech. Note No. 3401 (1955). 69. MIRELS, H., Boundary layer behind shock or thin expansion wave moving into stationary fluid, Natl. Advisory Comm. Aeronaut. Tech. Note No. 3712 (1956). 70. DEMYANOV, Y. A., Unsteady boundary layer theory of a compressible gas (in Russian), Prikl. M d . i Mekh. 18, 760-761 (1955). 71. NIGAM.S. D., Advancement of Fluid over an infinite plate, Bulletin o/ the Calcutta Mathematical Society 48, 149-152 (1951). 72. COHEN.N. B., A Power series solution for the unsteady laminar boundary layer flow in an expansion wave of finite width moving through a gas initially at rest, Null. Advisory Comm. Aeronaut. Tech. Note No. 3943 (1954). 73. HUBER.P. W. and MCFARLAND, D. R . . Boundary layer growth and shock attenuation in a shock tube with roughness, Natl. Advisory Comm. Aeronaut. Tech. Note No. 3627 (1956). W . H., Non-uniformities in shock tube flow due to unsteady 74. MIRELS,H. and BRAUN, layer action, Natl. Advisory Comm. Aeronaut. Tech. Note No. 4021 (1944). 75. GRAD,H.. On the kinetic theory of rarefied gases, Comm. Pure and A p p l . Math. 4. 331-404 (1949).

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Boundary-Layer Theory with Dissociation and Ionization BY G. LUDWIG

AND

M. HEIL

Institut f u r Theoretische Physak der Freien Universitat Berlin. Berfin. Germany Page Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. The Collision Equations . . . . . . . . . . . . . . . . . . . . . . . 11. The Equations of Transport for Molecular Properties of the Particles A , . . 111. The Solution of the Collision Equations . . . . . . . . . . . . . . . . 1. The First Approximation . . . . . . . . . . . . . . . . . . . . . 2. The Second Approximation . . . . . . . . . . . . . . . . . . . . 3. The Second Order Equations of Transport . . . . . . . . . . . . . 4. The Approximative Calculation of the Coefficient of Viscosity p . . . 5 . The Coefficients yp, u. I , cp . . . . . . . . . . . . . . . . . . . . IV. The Collision Cross Section for the Dissociation of a Diatomic Molecule by Collision with an Atom . . . . . . . . . . . . . . . . . . . . . V. The Boundary-Layer Equations for a Dissociating Gas A , . . . . . . . VI. The Solution of the Laminar Boundary-Layer Equations for a Dissociating Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. The Flat Plate Flow . . . . . . . . . . . . . . . . . . . . . . . 2. The Stagnation Point Flow . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 42 48 54 54 60 68

78 83 87 93

101 101 106 113 116

INTRODUCTION* The development of ballistic missiles has made rapid progress during the last ten years. One of the consequences of this development is that classical aerodynamics was found insufficient for the description of flows with very high velocities, and today we must take recourse to almost all branches of physics in order to treat certain problems connected with those very high velocities. While the results obtained in completely different fields of inquiry could be partly taken over, the technical development has been so rapid that we meet physical situations which we do not yet correctly understand. I t has become a matter of course that missiles fly at high altitudes, but up to the present we know very little about the flow pattern at these altitudes, where the mean free path of the air molecules is comparable with some characteristic dimension of the projectile. A satisfactory theory of this flow regime is A list of notations appears on page 113.

39

40

G. LUDWIG AND M. HEIL

still lacking, particularly a proof of the validity of the stochastic equations used there (e.g. the Maxwell-Boltzmann collision equation) 11, 2, 3, 41. A new starting point for the foundation of the stochastic equations has been given in the work of Ludwig [5]. For purely practical purposes such theoretical studies might not be so important. On the other hand, the theoretical aerodynaniicist wants to make quantitative statements that should agree with the experiment. For him it is important to know how far the flow equations which he uses are still applicable to the cases confronting him. In this review we shall restrict our studies to gas flows in which the mean free path of the gas molecules is much smaller than the characteristic dimensions of the projectile. Experience shows that in this domain the NavierStokes equations explain the experimental results sufficiently well, a t least for not too high gas temperature and for not too large gradients in the flow. However, at very high velocities the air near the body surface becomes quite hot and its state may change considerably. Dissociation and ionization can appear, i.e. phenomena that are not provided for by the usual form of the Navier-Stokes equations for a compressible fluid [6, 7 , 81. Moreover, the coefficients of transport, which occur in these equations, cannot be assumed constant. These coefficients reflect certain properties of the molecules constituting the gas, i.e. their values follow from the microscopic structure of the gas, but they enter the phenomenological flow theory as a priovi known constants or functions of the state. Unless they are known experimentally they must be calculated by kinetic methods. But at high gas temperatures direct measurements of the transport coefficients are very difficult, and it would be a great advantage if one could calculate them. In principle, the kinetic theory of gases can do that, although admittedly the kinetic calculations are generally not easy. Gas theory, however, gives not only the coefficients of transport, but also the macroscopic equations of motion, i.e. the flow equations for the gas considered, and of course also the equilibrium properties of the gas such as the internal energy and the pressure. Gas theory represents therefore a uniform starting point for the entire problem, well suited for the purpose of this review. Its method consists in stating the fundamental kinetic equations for the gas considered (Maxwell-Boltzmann equations) which are to be solved at least approximately. To state these equations we follow physical ideas of which the most essential one is the socalled “Stosszahlansatz” of Boltzmann. A rigorous justification of this rule by the methods of classical or quantum mechanics does not exist. Since it has worked well for simple gases in flows where the mean free path of the gas molecules is negligible compared to a characteristic dimension of the projectile, we may hope that it holds also in our case. We solve the kinetic equations by following Enskog-Chapman’s approximation method [9] and restrict our considerations to the second approximation. The resulting flow

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

41

equations with the corresponding transport coefficients are the NavierStokes equations for the gas under consideration. In the discussion of these equations the existence of “similar” boundary-layer flows has particular importance, since it is so much easier to solve ordinary differential equations than partial ones. Dissociation in the boundary layer was first treated by Moore [lo] who applied the usual compressible boundary-layer equations for the flow around a flat plate and used semi-phenomenological formulae for the coefficients of transport. H e solved these equations in the case of dissociation equilibrium. The influence of the dissociation is then only expressed by the variable coefficients p p , CJ and cp. However, Moore did not consider the effect of the strong increase of the heat conductivity in the domain of dissociation. The increase is a result of the diffusion o f atoms and molecules and is due to the transport of dissociation energy. This effect, which had already been calculated by Nernst [ l l ] in 1‘304 and later by several authors [12-141, is obtained automatically if one starts from kinetic gas theory [15, 16, 17, 181. In the papers of Moore [lo], Kuo 1191, and Metzdorf [16, 171 the boundarylayer equations have been solved in the case of dissociation equilibrium. The influence of a finite dissociation rate on heat transfer has been treated in detail by Fay and Riddel [20, 211. The present review is divided into two parts, Chs. I-IV and V-VI, which can be read independently. I n the first part, Chs. 1-111 bring the formal kinetic theory for a pure diatomic gas A , which is capable of dissociation and whose atoms can ionize. A generalization for the case of pure dissociation, studied by Heil [ 2 2 ] , is presented. However, the numerical evaluation of the kinetic formulae for the transport coefficients of reacting gases is still in its beginning. This is partly due t o the fact that the cross sections which enter into the kinetic equations are not at all or only very inexactly known. Moreover, the formulae are very complicated so that electronic digital computers must be used. The case of pure dissociation should be of interest since here the calculation is particularly meaningful and also some experimental results exist. The collision cross section for the dissociation of a molecule A , by collision with an atom A has been calculated by Petzold 1231 by a method of approximation. In Ch. IV some results of this work are given. In the second part the boundary layer equations for a dissociating gas A , are given (Ch. V). These equations have been solved in two special cases (flat plate flow and stagnation point flow) by Heil, Metzdorf, Kuo, Fay and Riddel [15--211. Some results are given in Ch. VI. Inasmuch as in this review ionization is discussed, we have assumed that the ionized gas is not influenced by magnetic fields. We thus have not considered magneto-hydrodynamical effects, which are of interest in the problem of boundary layer control [24--271. We have moreover assumed

42

G . LUDWIG AND

M. HEIL

that an isothermal plasma exists. In absence of external fields this is a useful approximation. The influence of ionization then consists mainly in additional terms in the internal energy and the heat flow vector. Since we were not able to find quantitative studies of these effects on boundary-layer flow, we had to restrict our considerations in Ch. V and VI to boundary layers in a dissociated gas.

I. THECOLLISIONEQUATIONS In order to avoid too complicated calculations we consider a pure diatomic gas A, which can dissociate into its atoms; the atoms A can be ionized by collision. To make the dissociation and ionization equilibrium possible the respective recombination collisions must be considered. For the sake of simplicity we neglect all such collisions where light quanta take part. Moreover, only a single type of ionization of the atoms will be considered. Dissociation of molecules into two ions A + , A-; formation of negative ions by capture of electrons and ionization of molecules are neglected. Thus the following collisions are considered, where A, characterizes the atoms, A, the molecules, A, the positive atomic ions and A, the electrons:

43 (1.15) (1.16) (1.17) (1.18) (1.19)

(1 20) (1.21) (1.22) (1.23) (1.24)

The collisions (1.1-7) were examined by Heil [22]. To give the collision equations we use a semi-quantum-mechanical description [28J. For the translational motion the classical description is retained, but for the internal motion we consider the atoms and molecules as being able to exist in different internal energy states. The internal energy we denote with F . where p stands Y' for 1, 2, 3, 4 and i is an abbreviation for i = n,Z (atoms), z = N',L' (ions), i = N , L (molecules). n,N' are total quantum numbers, N is the oscillation quantum number and i,L',L are the quantum numbers for rotation. Of course E ~ K, 0. A , ( U # ~ , ) characterizes, for instance, a molecule with velocity u2 and internal energy cZi. Our description of the state of a particle has the great advantage that there exist total inverse collisions even with inelastic collisions. The existence of total inverse collisions is one of the main assumptions in the derivation of the classical Boltzmann-Maxwell collision equation. In classical mechanics these total inverse collisions are related to very special molecular models, for instance to spherical molecules. Because of the importance of this point we shall illustrate it by an example. As is well known, the Maxwell-Boltzmann equation for particles A (u) is given by [29] (1.25)

f(u,r,t) is the velocity distribution function with the property + W

(1.26) -m

44

G. LUDWIG AND M. HEIL

n ( r , t ) is the particle density of A(u) a t the position r = {x,y,z} in a fixed coordinate system a t the time t . mF, where m is the mass of A @ ) , is an

external force which acts on the particle A and may depend on the position r and time t. a,f/at is an abbreviation for the collision term. It denotes the rate a t which the velocity distribution function f is being altered by encounters and consists of two parts. The collision with a particle A ( i ) (127)

A(u)

+A ( 6 )-A(u’) + A(6’)

leads to a diminution of f(u,r,t) since the particle A(u) leaves the velocity range u, u du by the collision process (1.27). If we denote by u(u,i/ u’,i’) the transition probability per unit time for the process (1.27), the total number of such collisions causing a diminution of f is given by

+

(1.28)

1

f(u,r,l)f(6,r,t)u(u,6 / u’,6‘)dfidu’d6’.

Consider now the collision obtained from (1.27) by taking the final state as initial state and allow that collision to take place (corresponding collision): (129)

A(d)

+A(6’)-A(u*) + A(6*).

I t is now a very important question whether U* and i* in (1.29) can be considered as u and with the same probability as in (1.27), that means that (1.30)

u(u’,6’ / U j ) = U ( U , V

/ U’j’).

The classical collision theory shows (see e.g. [9]) that (1.30) is true for spherical atoms. To show more generally the connection between the “direct” collision (1.27) and the corresponding one we define the inverse collision. Let P denote the space-reflection operator which transforms the space coordinates and the velocities into those with the opposite sign. Let T denote timereflection or, better, reflection of motion. The operator T transforms the velocities into those with opposite sign but does not transform the space coordinates (the position). The operator I = Pi”,the inversion, transforms therefore the spacecoordinates into those with opposite sign but does not transform the velocities. It is known that classical mechanics as well a s quantum mechanics is invariant (in the absence of external fields) against the operators P, T , and I. Hence also the quantum-mechanical scatteringmatrix S is invariant against the operator I = PT (see e.g. [30. 311). The consequence of this fact is: If xi,,denotes the initial state and xr the final state, the “direct” collision (1.31) Xin - X / has the same probability as the “inverse” collision (1.32)

BOUNDARY LAYER W I T H DISSOCIATION AND IONIZATION

45

If (1.33)

1x1 I= ~t

and

Ixtn

= Xznr

the corresponding collision (1.34)

Xi

--*XI*

is equivalent to the inverse collision and has the same probability (“total inverse” collision). As I does not change the velocities, (1.30) follows immediately. Thus the total number of collisions producing an increase of f, is given by

s

(1.35)

f(u’,r,t)f(G’,r,t)a(o’, V’ / o,fi)dVdu’dG‘,

and, together with (1.28), we have

(1.36)

By (1.30) this can be simplified to

at

-

[f(u’,r,t)f( fi’,r,t) - f ( o , r , t ) ffi,r,t)]o(u,fi ( / u’,fi’)dfidu‘dG’.

But in general (1.33) does not hold. As the Hamiltonian H is invariant against P , and as P is a unitary operator, all eigenstates p of H (and the states before and after the collision are eigenstates) have the property Pp = p. However, since T i s an anti-unitary operator, that relation does not hold for T , although H is also invariant against T . But T must transform the eigenstates of a degenerate eigenvalue of H into eigenstates of the same eigenvalue. For instance, the eigenstate f(r)P,’(6)e s p (imp) of the hydrogen atom, where f ( r ) and Pm’(S)are real functions, is trans) (imp) = f(r)P,’(S) exp (- i m q ) . This formed as follows: T f ( r ) P m ’ ( 8exp equation is the consequence of the fact that the angular momentum changes it sign b y application of the operator T . Therefore m goes over into - m. The same is true for the angular momentum in classical mechanics. If the cross sections were calculated according to the classical theory, it would be necessary to take non-spherical molecule models. The orientation of each such molecule would enter into the Boltzmann equation, and it would not be possible to use a symmetrical form similar to (1.37). But in the semiquantum-mechanical description we have the advantage that with q also Tp

46

G . LUDWIG A N D M. HEIL

is an eigenstate of the same eigenvalue of H . Therefore one can take suitable linear combinations of the eigenstates, so that for this system Tp, = p, holds. In this way one can see that e.g. the probability u(v,ezj,ulI vlf,Ulf,Ulf) is equal to (m1/2h)~u(v,’,~,’,~,’ / u2ezi,u1)The factor (m,/2h)3comes from the number of states in phase space. I t follows directly by quantum-mechanical calculations, but we shall use later the law of mass action for its calculation. All collisions listed in (1.l)-(1.24) are divided into two classes, according to (1.27) and (1.30). The collision terms belonging to the (+) class get a sign, the others a - sign. With the abbreviation

+

the Maxwell-Boltzmann equations can be written (1.39)

For the collision terms we introduce the following abbreviations (1.40)

with (1.41)

( 1.42)

(1.45)

47

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

(150) $97

(16 1 )

Jlt((%)t

3

2

1

(fl,fi,

- k,fik’f3llfpl)all3:dVldu~d~~d~L

(1)

etc. Z means that the summation has be taken over all indices except i, and (%)

fl, = j l , ( ~ l , r , ~ , ; i l ffat’ ) =

f 3 d ~ ~ ‘ , t , t , ;~ ~etc. ~ ’ ) The quantities

have the following properties: (132)

Jp:{St

(a)’

= - Jpt(a) ,

(a)

Jpr(~)’=

J,$lf,

Ia)

Jrr(Bft

(W - Jpr(a) *

In the equations (1.45) to (1.52)we have used the symmetric relations between the transition probabilites of &rect and total inverse processes :

(153)

(1.54)

(1.55)

48

G . LUDWIG AND M. HEIL

11. THE EQUATIONS OF TRANSPORT FOR MOLECULARPROPERTIES OF THE PARTICLES A, Let bpi be some property of the particle A,,: +,j(up,r,t,e,j);p = 1,2,3,4. The mean value of the properties is defined by

+,,

where

are the particle densities. n,,t$,, are the densities of the property in r - space. The equations of transport for the mean values $, follow from the equations (1.43), if these equations are multiplied by +pidu,, integrated over up and summed over i :

+,

The left hand sides of these equations can be transformed. With the condition + 0 for ]upl 00 it follows by partial integration that that --+

The mass-average velocity u, is defined by

in which p(r,t) = Zm,,n, is the overall density of the gas a t a particular point. c

The peculiar velocity of a particle of species p is defined as the velocity of the particle with respect to a frame moving with the mass-average velocity u,, thus (2.6)

V,,(u,,r,t) = up- u,,.

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

49

The diffusion velocity of species p is the rate of flow of particles p with respect to the mass-average velocity of the gas,

-

V,tr,t)

(2.7)

=

0, - uo

or

From (2.5) and (2.7) one gets

2 m,n,V,

(2.9)

=

o

cc

We now consider the equations of transport for the particle properties given as functions of V,,r,t instead of up,r,t.We introduce distribution functions with the following properties :

(2.11)

The connection between the derivatives of

fpi

and

fHi

is given by the relations

(2.12)

(2.13)

(2.14)

Written with the velocities V , the collision equations are thus

(2.15)

where D/Dt = alat

+ uoaiat.

50

G . LUDWIG AND M. HEIL

Analogous to equations (2.1) we have now

”

c

and form (2.17)

-

-

0, q?,i{,iV, -+0 IV,l do, it follows by With the condition that partial integration of the left hand side of (2.17) and use of (2.16) that

where the dyadic product ab : alarm c is defined by (2.20) The equations of transport for the gas as a whole follow from (2.21)

If we identify

$,

(2.22)

m,;

with

m,Vp;

I?,;

m

= P V,Z

2

+ + w,, ~,i

the right hand side of (2.21)must vanish since at an encounter the total mass, total (linear) momentum and total energy are conserved. The quantities (2.22) are the so-called invariants of the encounter (summational invariants). They have the following properties a) for binary collisions where the number of the particles is conserved:

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

51

b) for collisions where the number of particles changes (reaction collisions):

(2.24)

dGi

The relations (2.23), (2.24) are not only valid for but, of course, for the too. quantities I t is easy to show that the right hand side of (2.21) indeed vanishes for the summational invariants. For this purpose the collision integrals are symmetrized with respect to +,, or +,,. If one introduces abbreviations such as

(2.26)

the right hand side of (2.21)can be written in such a form that terms such as

62

G . LUDWIG AND M. HEIL

and

appear which vanish for the summational invariants. Thus the equations of transport for the gas as a whole follow from (2.27)

With (2.18) we obtain

Let us now begin by setting we get

4;,

= m,.

+ p ara

DP -

(2.29)

Dt

-'

With the help of (2.28) and (2.9)

u, = 0

or

aP + a * (pu,) = 0. -

(2.30)

For

at

at

= m,V,,

and with the abbreviation (definition of the pressure tensor)

we obtain (2.33)

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

If we put (2.34)

q

=

JP,

= EM,, we get, with

qt,

mPnP + qtnt 2 7 v,2vP+ 2’ nr(EP+ w P )V ,

53

~

=

P

P

(definition of the heat flux vector) and with

(density of the total internal energy),

or (2.37) P

The kinetic temperature T of the mixture is defined by (2.38) P

P

and the temperature T , for the single component is given by (2.39)

3

- kT, =

2

m -

-2’ VP2.

Besides the equation of continuity for the mass average density p we have still equations for the single particle components p. If one substitutes = 1 into (2.18) there results

Cpi

(2.40)

TP

with (2.41) as reaction terms. Only the collisions (2.24) give a contribution to r, r

(2.42)

54

G . LUDWIG A N D M. HEIL

The equations (2.40) are not independent since Zm,n,

=p

must hold. With

)1

the help of (2.30) and (2.9) one gets

2 mPrP

(2.43)

= 0.

P

For rl there results

;;;;1

r1 = 2 [ A

-A

[:if!] + 2 [ A[f:ii - A \i;:!] + 2 [;A; ;1 2[A;:3; - A;;;;!]

(2.4)

-A

+ [A;::;" - A(11) (134)'l

iii31 +

+ [A/::;)'

- A[!&]

-k

Besides the equations of transport for the (linear) momentum, energy, and particle density, an equation of transport for the charge density appears. Let e, be the electrical charge of the particle p. I n OUT case we have to take el = ez = 0 e3 = e ; e4 = - e. With = e, and

4,;

(2.45)

pc =

2 n,e, = 2 pc, P

= e(n3- n4).

rl

I t follows that (2.46) pc is known automatically with the solution of (2.40). For a calculation of rp, p, Q, V,, zc the distribution functions fpj must be known. In the following chapter we give the solution of the collision equations (1.43) by EnskogChapman's method of successive approximation (cf. e.g. [9]).

111. THE SOLUTION OF

THE

COLLISIONEQUATIONS

1. The First Approximation

In the equations (1.45-52) we treated the collisions between charged particles as collisions between neutral particles. Because of the long-range Coulomb forces one should consider the distant encounters which are not binary. If the Coulomb forces are dominant, the molecular fields interpenetrate one another to such an extent that all encounters might be regarded as multiple. For not too high a degree of ionization we may replace the action of the Coulomb forces by an averaged electric field which adds to the

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

55

external forces F,. If the gas is highly ionized the treatment of the distant encounters by the method of the Fokker-Planck equation [32] seems to be more adequate. In the present report we w ill restrict our discussion to the assumption of an averaged electric field [33, 34, 351. More extensive theoretical investigations are necessary to clear this point. The equations (1.43) shall be solved by the method of Chapman-Enskog. This method gives only the so-called "normal" solutions which vary with position and time only through their dependence on nu(r,t),uo(r,t), T(r,t) [9]. Supposed that the solution f@is expressible in the form of an infinite series which might converge uniformly : W

(3.1.1)

fu, =

2 f$' .

a=o

According to (3.1.1) the operators BJ{,, are similarly subdivided. The subdivision is made in such a way that the system of integro-differential equations (3.1.2) a

a

separates into systems of (inhomogeneous) integral equations for the functions This is attained by the following breaking-up of the operators:

fs).

(3.1.3)

gpsfpa

=

2 9;';

(0) 9:) = gfi(4 s (ffi8 .,/Fa(A-1) ) ;

L O

c), ,fc-

fc).

note that 9;' depends only on . .. '), but not on The operators Jrl are split in such a way that the quantities (')JPi also depend on f:: (3.1.4)

Jut =

2 "'J,, (f:), . . .,f,,(4).

a=o

The system to be solved can be written in the form

1

a

a

A

1

The equations (3.12) are satisfied if the functions f$' are solutions of the system

a

for instance.

a

56

G . LUDWIG AND M. HEIL

The breaking-up of gfljinto 9;) cannot be performed arbitrarily since, according to (3.1.6), we have

for the summational invariants (2.22). The calculation of the second approximation to f P j requires the solution of

a

a

a

a

It is usual t o put

g(? =0 w -

(3.1.9)

and to begin with the solution of

U

or a

The quantities

(l)J p j

a

are defined by

for the summational invariants collision terms with respect to

$Hi;

the proof consists in symmetrizing the Because of the similar relation

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

57

= 0,

(3.1.15) a

the necessary condition for (3.1.8) to be soluble is that

(3.1.18)

The quantities /?, are constants. One can assume that

PI = Pz,

(3.1.19)

P 3 =P4.

It is sufficient t o examine (3.1.18) if a/,,/at is replaced by a,/,,,/at. rizing this expression with respect to

+ log

(3.1.20)

(hut

(3.1.21)

eP1 = k , ..

=1

Symmet-

(P,fW).

eP3 = k,;

loge = 1

gives the result

aH --,to at

(3.1.22)

aH/at = 0 is possible if and only if each integrand of aH/at vanishes. This condition gives (3.123)

/W'fY*'

= /,J*

;

p,v

=

1,2,3,4

and

f11f/21r = klfltjltLrr fir'f3k'

= k2f3kf3kf4,

f21'/3k'

=klfbjiifsk,

/211f2,1 = klfl~fllf21J fit'fit'

=kzfirf~d4,

(3.1.24)

flt'f4' = kZf3kf4f4.

f2j'fa'

=kifiJiJ4.

58

G. LUDWIG

AND M. HEIL

(3.1.23), (3.1.24) express that

are summational invariants. Thus the equations (3.1.23) are the solutions of L’JPi{$

= 0.

The equa-

a

tions (3.1.24) state that with aHia.4 = 0 the laws of mass action for the dissociation and ionization must hold. They give relations between the particle numbers n,. The equations (3.1.20-21) and (3.1.23) give (3.1.25)

log

12)=):ro + a(’) m u + c J 3 ) ~ , , *

P P

or after simple calculations

With the known distribution functions mass action. (3.1.24) gives

1s’ we

can formt-ate the .,ws

of

(3.1.28)

The quantities W, and W , are the ionization and dissociation energies.

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

59

According to Fowler [36] there holds

We compare (3.1.28) with (3.1.30), (3.1.29) with (3.1.31) and obtain (3.1.32)

k,

Because of m,

=

g)”,

<< m3 we 1 2

w --

(3.1.33)

l-

k, =

(*)

3

m3m4

;

m, = m3

+ m4.

have k, w (h/mJ3. We put

w,;

w 2 = w 3 = 0;

w4=

w,.

The form of the distribution functions /$I given in (3.1.26-27) is important for the evaluation of the second approximation f;’. If no external fields are present, F,consists only of the contribution of the averaged field of the electrons and ions:

Fl

(3.1.34)

= 0,

F2 = 0,

m3F3 = - m4F4= eE.

For E the following relation holds: (3.1.35)

a E = &v E = 4ne(n3- n4)

at.

-

In the presence of a strong (external) electric field a slowly moving electron will, in average, gain far more energy from the field during a free path than it can lose at an encounter. As a consequence, the mean energy of electrons grows until the small fraction of it which is lost in collisions balances the energy gained during a free path. The mean energy in the steady state is thus much larger than the thermal energy # k T . Therefore the calculation

60

G. LUDWIG AND

M. HEIL

of the distribution functions for the steady state must be performed in a somewhat different way. To take account of the influence of strong electric fields, one must introduce different kinetic temperatures T , [28, 371. In our case it would be sufficient to put Tl = T , = T3 = T*, T , # T* that is, equation (3.1.27) is to be replaced by

In this report, however, we will only consider an isothermal plasma (T4 = T* = T ) . The results discussed here, hold then for weak electric fields. 2. The Second Approximation

If one starts with (3.1.10), (reaction non-equilibrium), ,Z(*)JP$)shall a

be deleted in the following equations. This means that the coefficients of transport will not be influenced by the reaction. Near equilibrium at high temperatures it may be, however, that the reaction collisions are impartant. This case will be described by (3.1.11). For sake of simplicity both cases will be treated formally by one system. Thus the following linear system of inhomogeneous integral equations for the functions is to be solved:

ft:

(3.2.1) a

a

U

The solution of (3.2.1) is the sum of the general solution of the homogeneous system (3.2.2) a

a

and a particular integral of the system (3.2.1). The conditions of solubility of (3.2.1) are given by (3.1.16). We put (3.2.3)

f(!l - $ .f(Y PI - Pa f i r

where /$) is given by (3.1.26-27). Now we form

and seek the solution of S

= 0;

this gives the solution of (3.2.2).

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

If S is symmetrized with respect to

61

and divided into

#&,

(3.2.5) a

a

one can prove the relations - si$

(3.2.6)

< 0.

The equality sign holds if and only if the following equations are satisfied: (3.2.7)

$pz

+

#pi

=

+

#pi'

$Mi'*

There is, for instance,

S{& would be "completely symmetrized" and therefore 3 0 only (0) (0)- k flo)'ffo)'f(o)' holds. if fij i 2 , - 1 1s 11 2m To have S negative semi-definite also for the reaction nonequilibrium,

S

< 0, we assume

(3.2.10) a

a

If the general solution of (3.2.1) is known the validity of (3.2.10) may be checked. The general solution of S = 0 and therefore of (3.2.2) is given by (3.2.7) and (3.2.11)

tCfp f #pi

= #qk' f

#el'

f #&n',

where (3.2.11) is an abbreviation for (2.24) if we substitute there # for +. Hence the general solution of (3.2.2)is the most general summational invariant

62

G . LUDWIG AND M. HEIL

Let us now calculate a particular integral of (3.2.1). According to the method of Enskog-Chapman [9] we have

or with

f$’ instead

of f;’

(3.2.13)

The time derivatives a,/at are replaced by the macroscopic equations for the first approximation /$I. With this choice the conditions of solubility for are fulfilled since these first order macroscopic equations follow from

92)

i.e. an, -&-

(3.2.15)

(3.2.17) (3.2.18)

+a

n,u, = r,(0)

9

pat

a - u, = 0, + pu, .-arda o ) + fi ar

3 2

+ 2 n,$) + 2 n,w,;

atdo)

p d 0 ) = - nkT

P

IZ, = w,.

P -(O)

These equations follow directly from (2.29 - 37) for V, = 0; q ( O ) = 0; p(O)= 1 p . With (3.2.19)

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

we have (3.2.20)

a

EF' = kT2 -IogZ,;

2, = 1.

aT

Let us introduce the abbreviations (3.2.21)

(3.2.22)

(3.2.23)

=

a3 kT + EJ0) +- w,, ar

(3.2.25)

1 VPoV , = VPVP- - V P 21 ; 3

PP

1

V

= unit

Then one obtains by a straight-forward calculation

V

tensor.

63

64

G . LUDWIG A N D M. HEIL

with

, are scalars. Therefore it suffices to consider only the scalar solutions the $ of (3.2.28). Now the quantities ( l ) J P i ( + ) are linear in $ and the left hand side is linear in the space-derivatives of T , u,,, n,, and p. Hence the most general scalar solution is the sum of six parts: (1) a linear combination of the components of a log T/ar; for this to be a scalar, it must begwen by thescalar product of a log Tlar and another vector, ( 2 ) a linear combination of the components of a/aruo;this must be the scalar product of a/aruo with another tensor,

(3) a linear combination of the components of d p ; this must be the scalar product of d, with another vector, (4) a scalar function which takes into account the non-vanishing of the reaction term Y,,(O), (5) a scalar function which takes into account a/ar-uo,

(6) the most general scalar solution of the homogeneous system (3.2.2).

Thus we can write

#;,

=

- A,;

*

a log T

__ -

ar

a

B,, : -U, at

+ 2 D,i,

*

d, -

V

Since the quantities Lpi and GMi are scalars they can be only functions of V,2, n,, T , E,,. In consequence of our identification of n,,uo, and T as particle densities, mass-average velocity, and kinetic temperature, the following relations must hold: (3.2.30) i

J

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

65

resp. (3.2.33) The difference between (3.2.33) and (3.2.32) consists in the following: If we put

T

(3.2.34) (3.2.35)

.#

=

=

T'O) + T(1),

+ u(1),

equation (3.2.32) gives (3.2.36)

u(1)= 0,

T " ) # 0,

# 0;

# 0,

T"' = 0,

# 0.

and (3.2.33) gives (3.2.37)

u(1)

In j$) the symbol T means T(') and (3.2.32) expresses the fact that = .$'( T ) . If different translational and internal temperatures must be considered (non-isothermal plasma), the equations (3.2.32), (3.2.33) should be replaced by

):?t

(3.2.38)

(3.2.39) According to (3.2.18) and (3.2.20),we prefer (3.2.32). The constants a,,('), a(')), a(3)are to be chosen such that the equations (3.2.30) to (3.2.32) are satisfied. One can show that the constants may be absorbed into A , , G,,,,L , . Because of (3.2.40)

z d , = O P

66

G. LUDWIG

AND M. HEIL

the quantities DPivcannot be independent, so that, for instance, (3.2.41)

D,irg

=0

can be chosen. If we substitute in (3.2.28),linear inhomogeneous integral equations L,,, GMiare obtained: for A,,, BMi, DMiV,

(3.2.42)

a

a

(3.2.45)

a

a

The conditions of solubility for (3.2.42) to (3.2.46) are satisfied. The only Hence variables involved in (3.2.42) to (3.2.46) are V,, fi,(r,t),T ( r , f ) , the A,i, Bpi, DPivmust be functions of V,, n,, T , E,'. The only vector which can be formed from these elements is V, itself, multiplied by some function of TZ, T, IV,l and E,,. Hence we can write (3.2.47) (3.2.48)

Api = A,i(Vp',&lri~,t)Vp. DFiV

= D p i v ( V,',&,i,r,t)

vp.

One can prove [22, 91 that B,; is a symmetrical and non-divergent tensor. Now Bpi depends only on np, T , Vp,ell, and the only symmetrical and non-

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

67

divergent tensors which can be formed from these elements are multiples of VPoV,, the factors being functions of n,, T , VF2,E ~ , . Hence we can write for the solution of (3.2.43)

and for

(3.2.50)

For (3.2.30)- (3.2.32) to be satisfied A,,i, DPiv,LPi,Gfii must be chosen such that (3.2.51)

j l

1

f$)Awim,VP2dV,= 0,

cC,l

(3.2.54)

(3.2.55)

(3.2.56)

From the given by (3.2.50) the quantities p, q , V,,, u, re can be calculated. Thus we get the wanted second order macroscopic equations of transport (generalized Navier-Stokes equations). In this chapter we have

68

G. LUDWIG AND M. HEIL

supposed that the external fields F,(r,t) do not depend on up. Thus we excluded the influence of magnetic fields. In treating magnetic fields we must modify the calculation of the second approximation /,(,?. This modification can be performed according to Chapman-Cowling [9, chapter 181. It might be important for a general theory of boundary layer influenced by magnetic fields [24 - 271 which will not be discussed here.

3. The Second Order Equations of Transport Let us begin with the determination of pu, the density of the total internal energy. With

l4.i

and

we get (3.3.1)

3 2

+ 2 n,(E;(O) + W P ) .

pu'0' = - IZkT(0)

P

Because of (3.2.32) we have (3.3.2)

p d l ' = 0.

Since only T(O)enters into the following equations, we omit the index (0) and write T for T(O). If the partition functions 2, are known the quantities EP(O) can be calculated by

(3.3.3) 2, = 1.

For r,, we obtain (3.3.4)

rP = rJ0) + rP(l).

69

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

If we substitute):/

into (2.44), rl(”)is given by

The quantities g, are the reaction velocities, for instance

K, and K, are the “constants” of the laws of mass action, defined by (3.1.28 - 31).

Only the terms with L,, and G,,, of $,,* contribute to higher terms of $, are neglected)

r p ( l ) (quadratic

and

(3.3.7)

rF2 denotes a correction of the reaction velocities if no reaction equilibrium

YF~

is the result of the excitation of the internal energies. holds [38]. Now let us calculate the pressure tensor p. I t was defined by c

p = p‘0’ + p‘l,

(3.3.9)

one obtains (3.3.10)

Y

~ ~ ’ v , , v=, ~I ~v ~, T ( o )

P,*

or (3.3.11)

p‘0’

r

1p;

p=nkT

70

G. LUDWIG AND M. HEIL

Contributions to

p(l)

come from BFi, GPi,LPi

With the help of the relation (see [9, Ch. 1.4211)

a __

a

is defined in (3.3.17). pa1) can be written

The tensor-uo

ar

0 __ __

(3.3.14)

i a p1(" = - uo 5 ar

2 m, psi

s

{$)(II,~ V,O V, :

V,O

V)N,

and with the bracket symbol (3.3.15)

{B,B)=z p,i

21i]["J,&B) +

l , i ((a) ~ , ~ ( B ) B,idv, l:

(1)

a

we get for plfl):

__ 0

(3.3.16)

1 a ~ 1 " )= - - kT{ 6, B} -0,. 5 ar

0 __ __

S

=

a

-uo is the symmetrical and non-divergent tensor ar

a,B = 1,2,3;

x1 = x ;

x2 = y ;

x, = z.

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

71

The bracket (3.3.15) can be symmetrized with respect to BMi. The bracket {B, B} is positive, in general it is {+, +} > 0 for 4 $ 0. The coefficient of viscosity p is defined by (3.3.18)

1 kT{B,B}. 10

,U = -

Therefore ~

(3.3.19)

p,("

a

= - 2p - 0

ar

Let us now calculate pZ(l), pa(l). We have

and because of (3.2.32)

There results

Here the factor

(3.32 3 )

O'

72

G . LUDWIG AND M. HEIL

is the so-called second coefficient of viscosity. If E,(O) = 0, the term

It vanishes for

E,(O)

= 0.

must be calculated in a somewhat different manner. For q one gets

(3.3.24)

2 n,(E,,co)+ w,)V,- - -13 k T { A , A ) a logar T + 1 n k T 2 (A,D,)d,. ___

M

V

It is usual to eliminate here the quantities d,. To do this we calculate

(3.3.25) With the coefficient of diffusion [29]

(3.3.26) and the coefficient of thermo-diffusion

DPT= ~ ~ ~ f $ ) A , i V p 2 d V P ,

(3.3.27)

i

-

we get for V ,

To eliminate d, we must solve the equations (3.3.29)

2 mrDnPrdr= nJ$ n f', + V

P

__

n%

a log T DMT ar ~

f,:

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

73

for d , which we will not do here (see e.g. [29]). The effect of this elimination on q is that new terms multiplied by a log T/arresult. Often these terms can be neglected so that

1 A' = k{A,A} 3

(3.3.30)

is approximately the coefficient of heat conduction. For the important case of a binary mixture (e.g. dissociation only, n3 = n4 = 0) the equations (3.3.29) can be solved a t once. According to (3.2.41) we have for Y' = p : (3.3.31)

D,,

= D2, = 0 ,

D, = D,,

D,

D2,.

1

The equations (3.2.50)and (3.2.44) reduce in this case to

One obtains

1-

{ D B } nd, + {D,A}

(3.3.34)

a log T

and, with tlUl2,D T , and kT given according to [9] by (3.3.35)

(3.3.36)

(3.3.37)

tlu12 =

DT

33 { D , D } , 3n

nn 3n

=L : {D,A),

at

74

G. LUDWIG AND M. HEIL

one has (3.3.38)

n n -ar

a 2

d,=-d (3.3.39)

*-

n , p - n m n1 nP

+

alogp

I--,

at

+ vt2n2V2= 0;

m p , q1 and finally (3.3.40) (3.3.41)

If we compare (3.3.40) with (3.3.28) we find for a binary mixture that (3.3.42) With (3.3.43)

q can be written for a binary mixture

P

P

(3.3.44) (3.3.46)

1 is the coefficient of heat conduction. We can prove that 1 > 0. Let N be defined by N = (R,R}Q - {R,Q}R. Then {NJ'J}

{QJ}I>,

= { R B } [ { R R }{Q,Q} - { K Q }

0

and, since {R,R}> 0, it follows that {R,R}{Q,Q) >, {R,Q} {Q,R). equality sign holds only for R = Q, therefore we have A > 0.

The

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

75

We collect here the second order equations of transport:

a

a

(3.3.47)

P+

c

PPFPI

P

For a binary mixture

(3.3.50)

__

~

a

(3.3.51)

at

3 pu = -. nkT 2

(3.3.52)

+ 2 n,,(EP(O)+ w P ) , f

5 aT 1 Q = - 2 k T Z % i P 4- ~ n P ( E ; ‘ o ~ + ~ P ) ~ Par- A ’ --3n+k T C { A , D , } d , . P

P

L

(3.3.53)

For { A P , }

=

{D,,A} we can write

-1n k T Z { A , D , } d , 3

=

P

For a binary mixture q is given by

(3.3.54)

- n k T c - - DPT d,. JnPmP

76

G. LUDWIG AND M. HEIL

where

(3.3.55)

a-B P

(3.3.56)

dc =

ar +

ar

PP

PFc -

c

PJV) ’

V

(3.3.57)

If reaction equilibrium and quasineutrality (n3 = ~z,) hold, we have nc = nP(p,T), andit can be assumed rlGG 0. Equation (3.2.45)must be altered. The condition of reaction equilibrium gives m2 = K,(p,T)n12,K,( T,p)n3n, = nl. Together with p = Zn&T we have only 3 equations for the 4 unknowns n!,. V

If n, = n4 (quasineutrality) holds the particle densities nP can be determined as functions of p and T only. We get, however, no such condition which establishes the quasineutrality. Because of the diffusion and thermo-diffusion of the electrons and ions quasineutrality cannot hold exactly. Thus in the case of a plasma the equations (3.3.46) must be considered even if the reaction equilibrium holds. This is due to the fact that with ionization of the atom A two different particles, an ion and an electron, appear (corresponding to a dissociation of a molecule A B A B ) . Because of the strong electric forces between the charged particles, however, which is not expressed in equation (3.1.35) the assumption of quasineutrality seems to be a good approximation if the gradients of the flow are not too large. In this approximation the equations of continuity for nP become superfluous for the reaction equilibrium since in this case they define only the reaction terms re.

-. +

77

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

The calculation of the coefficients of transport can be simplified if the inelastic collisions in the brackets are neglected. This means that with regard to the coefficients of transport the reacting gas is treated as a non-reacting mixture of elastic molecules. The influence of the reactions on the equations of transport consists then in the terms L'nP(ffl(o) w P ) ,

+

+

-

P

ZnP(.FT) w,)V, and r,,. Since even for a non-reacting gas the coefficients P

of transport according to the kinetic theory give too complicated expressions one usually uses semi-phenomenological formulas [16, 101. I t should, however, be studied which influence the inelastic reaction collisions have on the coefficients of transport. The discussion of the boundary layer equations suggests the study of the coefficient of viscosity ,u and the coefficients of diffusion. We give here a brief account of ,u for a binary partly dissociated gas [22]. Let us once consider the velocity of diffusion for the electrons. It may be assumed that only electrical fields act on the gas. If moreover quasineutrality is supposed, d, and d, contain no external fields since m3F, = - m4 F 4 -- eE. Let us consider the term of p4 which arises from the electric forces. One obtains from (3.3.28),(3.2.24)and (3.2.41)for v' = ,u (3.3.59)

V4el=

ne

--

~

PkT

m3Dn4,E.

The electrical conduction current density j e is given by (3.3.60)

Because of (3.3.61)

j e = en3V, - en4V,.

V3<< i4 we may

write

-

j e = - en4V4

and thus (3.3.62)

( j e ) e i ==

The factor (3.3.63)

is the electrical conductivity of the plasma.

78

G. LUDWIG AND M. HEIL

4. The Approximative Calclrlation of the Coefficient of Viscosity p

To calculate ,u for a partly dissociated gas A, we must study the following inhomogeneous system of integral equations for Bpi:

a

a

It is usual to introduce C,,. defined by (3.4.2)

The conditions of solubility for (3.4.1) are satisfied automatically. For Bpi we make the Ansatz m

Bpi =

(3.4.3)

2 b,b:i

r=-m

with the same coefficients b, for each series. Let the quantities /?, be defined by the equations C, : bt:)dV,

(3.4.4)

and let b,, = { b(‘),b(‘)}.

(3.4.6)

Then we get by (3.4.3) and (3.4.6)

B,b} = 2

2

f$)CPoC, : b,,dV,

p7i

the relation W

(3.4.7)

br = {B,b(’)} =

2 b,b,,,

r = - 00 , . . .,

+

00.

S=-W

Here we have defined the brackets { } in a somewhat different manner in order to agree with reference [9]. If the functions b$) are known, p, and b, may be regarded as known, so that the infinite set of equations obtained

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

from (3.4.7) serves to determine the infinite set of coefficients b,. substitute (3.4.3) into

79

If we

c

we get m

{B,B} = ZP,b,.

(3.4.9)

r=--m

For the quantities'b: (3.4.10)

we use the same orthogonal system as in reference [9] :

=

bl:'= 0, b'') -

(,-I)

b.$Fr) 0,

(r

< 0).

(c12)c10 c1,

(r > 0)

b.$Tr)-S$"(CZz)C," C,,

(r > 0)

IS

= s5/2

(3.4.11)

are Sonine polynomials. With these polynomials the The quantities S& quantities p, can be calculated. It follows that

The formal solution of (3.4.7) is given by (3.4.13)

/B1 denotes the determinant of which the r,s element is b,, while /a,[ is obtained from lBl by replacing the elements of the rth column by p,. The diagonal elements of IBI are positive. As, however, neither of the infinite determinants IBI, IB,( in general converges, one considers only the system r = - m to r = + m, calculates (3.4.14)

and assumes

6,

lim bJm)

=

m-m

(3.4.15) m

B,,

=

lim BL:); m-m

BZ) = 2 6Jm)b::, ,=-m

80

M. HEIL

G. LUDWIG AND

where [B,(m)l, IB(")I are the determinants composed of the first m rows and columns of lB,l, lBl. Let us consider the first approximation m = 1, that is

(3.4.16)

the solution of which is given by

Now

,M

is given by p = 1/10 n,n,kT{B,B}

+ 5 b,,

n

b] - - k T n 2

(3.4.18)

l-

2

* or in

2 b-l-l

a first approximation by

- bl-,

- b-11

n1 bJ-1-1

-~l-l~-ll

The matrix element b, is defined by

+

+

+

~z,n,b,~ = n,b2{b('),b(s)}= n12[b(v),b(s)]11 n,n2[b('),b(s)]12 n22[b"),b(s)]22

+

n13[b(r),b(S)]:i!4-n,%, [b('),b(s)]ia? n1n2[b(y),b(s)]l::,

+ n2,[b(y),b(~)]l;i

(3.4.19)

where, for instance,

kl

The transition probabilities which enter into the collision integrals can be referred to the coordinates of the center of mass of the colliding particles. Thus a , , ~ v , , ~1 u, ~ & ' ) depends on u = v, - u,. Moreover, the four velocities u1,61,u1'u1' resp. U,U' cannot be independent, since linear momentum and energy must be conserved in a collision. Thus all is proportional to the product of two &functions which express this fact. The form of the brackets shows that the following relations hold:

*

n,ne{B,B) stands for our former bracket

(B,BI.

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

81

in general, however, there is (if n2 # kn,,) (3.4.21)

[b(r),b(s)]::, # [b(s),b(r)],:; etc.

Let us split the matrix elements b,, into two parts: (3.4.22)

b,, = b k )

+ b::',

where

n,n,b!P) = n13[b(r),b(s)]i:!

+ n,%,

+

[b('),b(s)]&?

+

fi1fi2[b(r),b(S)]:fl, n22[b('),b(s)]]::?.

(3.4.24)

For the quantities b;), the known values (3.4.10), (3.4.11) are to be substituted. According to the special choice of b/! one gets [9]

(3.4.26) (3.4.27)

b'"' I P - [b;),bg)],,;

Y

> 0 > s.

If we neglect all inelastic collisions the equation (3.4.18) can be transformed into the equation (9.8.4.1) of reference [9]. In order to study the must be considered. For influence of the reaction collisions on [p],, sake of simplicity the excitation of the internal degrees of freedom without dissociation may be neglected and in the other terms the energy states are assumed to be so compact that the summations over the energy states may be replaced by integrals. Furthermore it might be assumed that the transition probabilities for the processes A , + A , + 3A,, A , + A , + 2A, + A , are nearly the same so that only one inelastic transition probability appears. The evaluation of the collision integrals for the reactions, however, requires the use of an electronic digital computer. For the calculation of [pI1 according to (3.4.18)it is expedient to split up the "reaction" brackets which constitute b!:'. We replace (3.4.24) by

82

G. LUDWIG AND M. HEIL

The brackets (b(‘),b($))are now independent of the densities nl,nz. Moreover (b(‘),b(”),,y = - (b(.),b(dr)(do Because of (3.4.10- 11) we have (3.4.29)

(b(’),b(’))a% = 0;

(3.4.30)

(b(yJ,b(s))lll = 0;

> 0, r < 0.

7

The brackets (b(‘),b(s)) are only functions of the temperature. we get thus n1

(3.4.31)

cull =

-Qi ”2

Qa ++

ni Qi -~ 82

bill

12,
@211

+-- ! k l l 12,

Q3

Bl

@Il

+ Qz klI1+ + A , Q4

Qz +

For

+ Q S + 4

b111@2ll

with

(3.4.33)

A, =

n a

n

P, + 2 P, n2

n +$ P3 + nzP4+ n P

1 Ii+P8-

1

The quantities Q and P are functions of T only. Culll and [,u,]~ are the viscosities of the pure gases A , and A , in first approximation. A , is too long to be given here. If all P are zero, we have dl = 0 and A , = 0, and (3.4.31)reduces to the viscosity of a non reacting binary mixture. A , and A , have not been calculated yet. There is, for instance

(3.4.34)

pi

2 (bz(-l),bz(-9)112 - 2(bl(l),b&-1))111

with

Since even the viscosity for a binary non reacting mixture (dl = A , = 0) is very complicated one usually applies more simple mixing formulae for boundary layer calculations. In the following section we give therefore such simplified expressions for the coefficients used in section VI.1.

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

5. The Coefficients pp,

0,

83

1, cp

In this section the coefficients which were used for the calculation of the curves shown in Figs. 7 and 8 are given. Because the formulae which follow from the kinetic theory of Chapter 3.3 are very complicated, simplified expressions are used. are the viscosity and heat conductivity of atoms and If plJ, and molecules, respectively, we use as formulae of the mixture

These formulae which were also used by Moore [lo], are derived in [39] with the help of free-path considerations based on a Maxwellian velocity distribution of the particles. If li’ stands for the mean free path of kind i in the mixed gas, whereas 1, is what the mean free path of this kind would be if it alone were present at its actual density pi, pi is given by

with (3.5.4) is the cross section for collisions of particles i with each other while 3, is the mutual cross section when they collide with kind i. For our binary mixture of atoms and molecules we have m, = 2m1 and for the model of rigid elastic spheres 5; = nda2. One usually takes

Thus we have (3.5.6)

(3.5.7)

84

G . LUDWIG AND M. HEIL

For the Sutherland model the kinetic theory gives in first approximation [9] : (3.5.8)

(3.5.9)

(3.5.10)

A

(3.5.11)

1

25

km,T

'- l+-?; S232d; (

F",.

___

I n (3.5.11) C,, is the specific heat at constant volume. For the Sutherland constants S, and S,, approximately S,= S, = 102.7 O K was taken. For rigid elastic spheres the coefficient of diffusion in first approximation [9] is (3.5.12)

The specific heat cp can be calculated according to (dissociation equilibrium) (3.5.13)

cp = cp,

+

C~D,

with (3.5.14) cpor= acpl

(3.5.15) C (3.5.16)

~ = D

+ (1 - a)+,,

a(1- a2)wp2 , 2m,k T 2

cp, = --

2 m1

for

5 2

= -kT

+ 2El'O) + W2 - E2(O),

E1(o)= 0.

The specific heat cp, of the molecules may be taken from reference [40]. With known cp, the internal energy E2(') follows from T

(3.5.17)

5 EJo)(T)= m21cp,dT- --kT 2 0

85

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

The degree of dissociation a is given by

K,=

(3.5.18)

+baa

1 - a2

~

The equilibrium constant K , for the reaction A,= 2 A , has been calculated by Krieger and White [40], for instance. With (3.5.19)

AD = $J'@sl2cpD

the Prandtl number (3.5.20) can be calculated. The following figures show some results of the mentioned calculations for nitrogen in the case of dissociation equilibrium.

1000 2000 3000 4000 5000 6000 1000 OK

. .-

Temperature T

1000

2000

3000 4000 sooo 6000 7000 r u

FIG. 1. The product g = 2 p p for = 3.0. l @ dyn cm-2 (from reference 17).

p

FIG. 2.

*

Viscosity of nitrogen (from reference 17).

16.erg cm sec O K x

.-2 .->

6.0

+ 0

4.0 c u c

2.0

m

I

L I 1000 2000 M O O 4000 S O 0 0 6000 7000 OK Temperature T e

FIG.3. H e a t conductivity of nitrogen without diffusion (from reference 17).

Temperature T

FIG. 4. Heat conductivity of nitrogen for dissociation equilibrium (from reference 17).

rWW.Jl",&

FIG.5. Specific heat of nitrogen at constant pressure for dissociation equilibrium (from reference 17).

-l%

FIG. 6. The Prandtl number for nitrogen with and without diffusion for dissociation

-

equilibrium. p = 3.0 l@dyn cm-* (from reference 58).

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

IV. THECOLLISIONCROSS SECTION FOR

THE DISSOCIATION OF A AN ATOM [23, 42-47]

87

DIATOMIC

MOLECULEBY COLLISIONWITH

Even if we use the approximation that with regard to the transport coefficients p, A, K , a,,,, k , the reacting gas is treated as a non-reacting mixture of elastic particles, we must calculate the reaction velocities g, which enter into r,,. For a dissociating diatomic gas A , we have to consider g, and g,. To calculate g, we need the transition probability respectively the cross section ai:I(v,ENL) where v is the magnitude of the relative velocity u2 - ul. We follow the calculation of Petzold [23] and consider the atoms as being without internal eneigy, = 0. Both atoms are assumed to be bound by a potential U with the property -

U={

b 0

for for

s< R, s > R,

that is, the interaction of the atoms of the molecule is approximated by a square-well potential. s is the vector from the atom 2 to atom 1 : (4.2)

s = r, - r,,

s = Is(,

r1 = coordinate of the atom 1 of the molecule, r,

= coordinate

of the atom 2 of the molecule.

The fact that both atoms are bound is described by a discrete eigenvalue E , * and eigenstate t,bdm of the Hamilton operator H, belonging to the molecule.

ml,m,' are the masses of the atoms in the molecule. I t was put: h/2n = 1; c = 1 ( h = Planck's constant, c = velocity of light). If the atoms of the molecule are free, they must be represented by an eigenfunction of the continuous spectrum of H M . The problem corisists in

*

In this chapter we use E for the internal energy of the molecule.

88

C . LUDWIG AND M. HEIL

calculating the transition probabilities from a discrete state to the continuous spectrum of the Hamilton operator H M . Let the molecule collide with an atom such that the internal energy of the molecule is excited and the molecule dissociates. The interaction between the molecule and the atom cannot be spherically symmetric, since this would mean that the atom would only be scattered elastically. Therefore the interaction between the molecule and the atom is described by a tensor force. Petzold puts (44

(4.7)

v = v1+v2; + co for v, = 0

r < R, r > R,

for

r is the vector from the center of mass of the molecule t o the colliding atom (4.9)

?'= RA

- RM;

Y

=

IZI,

(4.10)

V , accounts for a strong spherically symmetric potential since the elastic scattering prevails over the elastic one. The complete Hamilton operator of the problem is given by

(4.11)

mA is the mass of the colliding atom. One seeks eigensolutions of the operator (4.11), that is, of H Y With the Ansatz

= EY.

the kinetic energy E , of the center of mass can be separated; thus the following eigenvalue problem is to be solved (4.13)

(4.14)

89

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

One seeks solutions of the form (asymptotically for large values of

x

= J + ~ N L Mexp

(ikNL* I)

T)

+ outgoing scattered wave

(4.15) nim (+NLM) a,/iNLM is the bound state and EN, the internal energy of the molecule before the collision, (Clnrm and En, the corresponding quantities after the collision. l l m k N Lis the relative velocity u between the atom and the molecule before the collision, l / m knl after the collision. N is a normalization factor of the eigenfunctions $I. kd is given by

l / r fnrm exp (ikn1- r ) is the amplitude of the inelastic scattered atom where the molecule changed its internal state from # N L M t O #nlm. The cross section for this process is given by (in center of mass coordinates)

(4.17)

The total cross section for dissociation is thus

the integration ranges over the continuous spectrum of H M . Since each value of the azimuthal quantum number M has the same statistical weight (at fixed L and N ) the average of a12 over M is given by + L

90

G. LUDWIG A N D M. HEIL

The cross section E l 2 has been calculated for the case that the molecule has the internal angular moment L = 0 before the collision. Petzold [23] obtained approximately for m, = m,’ = mA

4” = 0.236 Re

~

B2R W , h E

(f m1v2 - E r

[ w2

+ Y2 EETR 2] v9m14

(4.20)

x

1

W , ( w b) is the dissociation energy. E is the binding energy of the molecule before the collision. v = k,/m = 3kNL/2m,is the relative velocity between the atom and the molecule before the collision, R is the molecule radius. B is the interaction energy between the colliding atom and the molecule according to (4.8). For f m,v2< E one must put E,-,12 = 0. In Chs. 1-111 we defined the energy in a somewhat different manner, but the difference consists only in a displacement of the energy scale by W,: The ground state is now defined by E = W,, that is E , = - W , instead of E , = 0. The cross section %l2 tends to zero for large values of v with l/v2. Its maximal value is about 1/27 mlv2 = E . This means that the relative kinetic energy mv2/2 = f mlva must have the magnitude of the binding energy. This might be a resonance effect. In obtaining (4.20) the condition

>> 1 was supposed to hold.

m1ER2 ti2

With

& mlv2 = [ E

(4.20) gives thus

m,E R2 (4.21)

91

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

For a binding energy of E = kT and T = 300

O K

one obtains

TABLE1

H* m,ERa 8'

3.57

J*

N*

0,

148

109

2.93

12.06

4.47

5.08

9.8

1.87

26.78

23.42

0.75

1.2

1.1

5780

0 . 0 4 1 3 - 10' 1.54 212.4

*

10-24

2.66 * 10-8

If we put R = W , , 4 l 2 reaches the magnitude of 10-14 cm2. For the lower excited states of the molecules %12 decreases and reaches a magnitude of 10-20cm2 for the ground states. With the cross section hl,we can calculate the velocity of dissociation g,. Since depends on therelativevelocityv (and the binding energy E = IENLI) we get for (3.3.6) (4.22)

g, =

2 i

I

~ ~ ' O ) ( V , E ~ ~ ) V ~ ' ~ (EN= V,E E ~Ei~ ) ~ U ,

with

The connection between E, which was used in the Chs. 1-111 and E, is given by lEll = W, - el. If we suppose that the energy values E , = E N , are dense in the range E , [Ell W , the sum 2 can be replaced by an

<

<

i

integral; (4.22) then gives approximately

(4.24)

g,

5.5 * 10-6R3W,2 = ___

ti

e

wz -_ kT

92

G . LUDWIG AND M. HEIL

For a temperature To which is defined by kTo = t W , the velocity of dissociation becomes TABLE2 H* g1

1.2

10-14

13.000'K

=ll

18

Oa

Na

5.7. 10-14

8.9.10-14

15.000'K

28.400"K

1.8 *

[cm-s sec-l]

4.480"K

The relaxation time (4.25) is the time between two successive dissociations of a molecule by collisions with an atom. The total relaxation time is then given by

(4.26)

If we assume g2 M g, we get 1

t~ w -; g1n

(4.27)

For To and n

=

n = n1

+ n,.

lo1' ~ m we - get ~ for t ~ . TABLE3

If v g is the velocity of the dissociating gas the length 1, = v g * t R is needed for reaching equilibrium. For vg = 105 cm sec-l one obtains for 0, and N, with t R w 1.5 10-4 sec; I, M 15 cm (complete dissociation). This seems to have the right magnitude as compared with recent experimental studies [47]. The quantity ( l / i R ) enters into the equation of continuity for nl. l / t R is the average number of dissociations per second of a molecule by collisions with other molecules and atoms. The quantum mechanical model which

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

93

was used in [23] to derive t&l2has some defects. The internal potential of the molecule was approximated by a square-well potential, where most energy values are close to the ground state, contrary to the experience. The molecular spectra show that the energy terms accumulate a t the dissociation energy. Moreover, E l 2 was only calculated for L = 0. But in the range of ~ n,) the angular moments are completely excited. Furdissociation ( N w thermore the excitation of the electron shell was neglected. The turning of the spins of the binding electrons and the change of the interaction potential of the atoms of the molecules connected with it, seems to give a large contribution to the dissociation. Instead of a square-well potential for the molecule a spin-dependent potential should therefore be considered.

v. THE BOUNDARY-LAYER EQLJATIONS FOR A

DISSOCIATING G.4S A ,

The effect of chemical reactions on boundary layer flows has been studied for the flows around a flat plate and near a stagnation point [15-211. For these flow processes the boundary layer equations can be transformed into a system of ordinary differential equations. For the stagnation-point flow this transformation gives an ordinary system even if no dissociation equilibrium holds. Moreover, the stagnation point flow is of practical importance, because it is realized a t the blunt nose of a high velocity projectile and governs the heat transfer to the projectile. The strong shock which forms in front of the body absorbs much energy from the flow, making it available for the dissociation of the molecules [as]. Let us assume that the flow gradients parallel to the body surface have the magnitude Re-1/2, normal to the surface, however, the magnitude 1. Re is the Reynolds-number. With this approximation the two-dimensional boundary layer equations follow from (3.3.46 - 58) for Fp 0 :

(5.4)

3- 0. aY

94

G. LUDWIG AND M. HEIL

(5.5)

The terms with the second coefficient of viscosity do not appear, and p 3 ( l ) was neglected. (5.1) to (5.5) are the usual two-dimensional boundary layer equations for a dissociating pure diatomic gas A,. We have neglected hypersonic effects. If the projectile has a sharp leading point or edge the equations (5.2-5) fail, and improved equations for hypersonic flow must be stated [49,50]. In particular, the interaction between the shock wave and the boundary layer near the sharp leading edge must be taken into account. If, however, the shock front is detached, the projectile h a s a stagnation point, and a boundary layer will exist independently of the detached bow shock wave, if the boundary layer thickness is much less than the shock detachment distance. Since the boundary layer thickness changes with R e d 2 and the shock detachment distance is independent of Re, there is a minimum Reynolds number below which the shock wave and the boundary layer merge. At hypersonic velocities such low Reynolds Number can only be attained by reducing the density, and this limit may not be reached before the flow regime changes into free-molecule flow. It thus appears that stagnationpoint boundary layers are quite relevant to hypersonic continuum flows [2Q, 211. With the equations (5.3-5) we can also treat the flow around an infinitely long and symmetrical body of revolution, if (6.1) and (5.2) are replaced by

I

art, -

at

a ( n p J +a (nlrvy)= rr, + +ax aY

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

95

here x is the distance along the meridian profile and y the distance normal to the surface. r ( x ) is the cylindrical radius of the body. Let us neglect kT and introduce the degree of dissociation a by

and the enthalpy i by a = u + - .P P

(5.9)

The stationary two-dimensional boundary layer equations can thus be writ ten

a

(5.10)

(PVX)

a +aY

b y )

=

0,

or, for axisymmetric boundary layers,

a

(5.11)

- (pru,) ax

ai

pvx ax (5.13)

(5.14)

(5.16)

(5.17)

ai + P"Y aY

ax

a

4(prvx) = 0; aY

96

G . LUDWIG AND M. HEIL

The equations (5.13) and (5.14) can be transformed. We introduce cpband cpT F18, 511 which are defined by

With (5.20)

t =-

.

m1

-(I

+ a)KT +

I 2 (1 - q ) i 2 ( 0 ) + d l , O )

-

+ a E22 ]

we obtain 1

(5.21)

cpT = - Wp, m2

cp, and cp, are the specific heats of the atoms and molecules. We introduce the following non-dimensional quantities

(5.24)

The equation of enthalpy (5.13) can now be written

ai

ai

ap

ax

ay

ax

pvZ--+pvy--v~-=p

(;y)z

2

p ai +-a~a [-+ -aapc + ~ - aaa~ (Le - 1) ,

I

a a a ~

(5.25)

a, is the Prandtl-number for the non-reacting mixture Schmidt-number and

Al,A2,

aD is the

(5.26)

the Lewis-number. In the equation of diffusion (5.14) we express the reaction term mlil by a. Instead of K , we use K p which is defined by

p ~ , K p= p:,,

$AP

= nPkT

(p= 1 3 ) .

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

97

Thus

(5.27) and K , can be calculated by (5.28)

If we assume g,

mlil can be written as

= g, = g ,

(5.29) Thus the equation of diffusion becomes

Let us summary the final equations:

a

(5.31)

-(pox)

ax

+ aYa (pvy) = 0, -

or, for axisymmetric boundary layers,

a

(5.32)

- (prv,)

ax

avx pvx ax

(5.33)

ai ax

pvx -

+ p"y

ai aY

av + pvy -aY

--

x

v x ap -=p ax

a +(P'") aY

= 0;

=--

-

a

+-a~

- - + - c , , - (PL e - l )aa

[0,ay p

ou

a~

(5.34)

The parabolic system (5.31-35) must be completed by initial- and boundary conditions which determine the quantities vx,vy,u,T,i uniquely. Let US

98

G. LUDWIG AND M. HEIL

first consider the problem of non-vanishing heat transfer to the wall; it is defined by (qy)y= # 0. The boundary conditions are given by

y = 0 :v,(x,O) = 0, (5.36)

a(x,O) = 0

T = T,(x);

vY(x,O) = 0,

for a catalytic wall, for a non-catalytic wall;

(5.37)

y = 00 :

v x ( x , a o )= v , ( n ) ,

T ( x , w )= T,(x),

a(x,aJ)= a,(x).

However, see also [59]. In addition t o these boundary conditions we must consider the initial conditions (5.38)

T(x0,y) = TO(Y)P

.x(xotY) = %,(Y),

a(xo,Y) = ao(Y).

The problem consists now in a continuation of the functions v,(y), T o ( y ) , ~ ( yfor ) x 2 xo in accordance with (5.31-35). The functions v,(y), T o ( y ) , ~ ( ycannot ) be arbitrary. At least lim v,(y) = v,(xo); lim To(y) = T,(xo) and lirn ~ ( y=) a,(xo)

Y+,

must hold.

Y-w

The functions urn(%),T,(x), a,(x)

Y-+,

are obtained by the solution of Euler's equations which are identical with the first order equations (3.2.15-18). The thermometer problem is characterized by

takes into account the radiation of the body surface. The The term &Tw4(x) temperature T,(x) is one of the unknowns. The equations (5.31-35) characterize the dissociation non-equilibrium. Let us now perform the transition to the dissociation equilibrium. It corresponds to

2-- -( a-a )+aTax

aT

ax

- ( a a ) ap

ap

ax

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

99

The enthalpy equation (5.34) changes into pwz

ai ax

-

+

PVY

ai aY

-

- vz ax

(5.42)

Equation (5.28) and the relation 1

-

a2 = (4$a2/Kp) give

(5.43)

(5.44)

CpD =

a(1 - a2)wp2 2m2kT 2

Thus (5.45)

and

where (5.47)

is the Prandtl-number for the dissociation equilibrium, and

are the specific heat at constant pressure and the heat conductivity for the dissociation equilibrium. For a w 4 the term (5.49)

is up to ten times larger than the heat conductivity of the non-reacting mixture. Because of (5.49) the Prandtl-number is still close t o unity.

100

G . LUDWIG AND M. HEIL

The diffusion equation (5.35) degenerates into a definition equation for the reaction term rl. Thus for the dissociation equilibrium only the following equations remain : (5.50) or, for axisymmetric boundary layers, (5.51) (5.52) (5.53)

ai

a;

According to (5.46) the heat transfer to the wall is given by (5.54)

and for the non-equilibrium by (5.55)

Suppose now that a, be constant and ,up be independent of ct. Thus for Le = 1 the heat transfer for the non-equilibrium is independent of the diffusion equation since (5.34) is not coupled with (5.35). Whether atoms recombine in the boundary layer or at the wall makes no great difference since the energy is conducted about as readily by normal conduction as by diffusion when the Lewis number is approximately one. Near the stagnation point the value ct,(x) at the boundary-layer edge can be calculated by the solution of the normal shock equations which follow from the first order macroscopic equations (3.2.15-18) (see also [52]). The boundary layer equations for an ionized gas have been studied by Neuringer, McIlroy, Rossow, Resler, Sears, Rosa, Patrik [24-27, 53, 541 for the presence of an external magnetic field, which produces an additional term in the energy equation (2.36) and in the equation of motion (2.33). The solution of the collision equations must be modified (see e.g. [9]). In [24-271 it is shown that the heat transfer may be lowered by application of magnetic fields. The analysis given in [24-271 differs from the classical equations of transport only by the two additional terms mentioned, the influence of reactions and of diffusion are neglected and the coefficients are assumed constant.

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

VI. THE SOLUTION OF

THE

101

LAMINAR BOUNDARY-LAYER EQUATIONS FOR A

DISSOCIATING GAS 1. The Flut Plate Flow [15-191

To study the influence of the reaction and of variable fluid properties the laminar boundary layer equations (5.31-35) have been solved for the flat-plate flow and the flow near a stagnation point of a blunt-nosed body. For these two cases a similarity variable can be found such that the equations can be reduced to ordinary differential equations. Fay and Riddel pointed out that the stagnation point appeared to be the only case in which the boundary layer equations admitted this great simplification without further approximation. Even for the flat plate this reduction is only possible for the extreme case of a “frozen” boundary layer or with reaction equilibrium. Since there are regimes where the boundary layer will remain laminar €or some distance away from the stagnation point, one is interested in an extension of the theory away from the stagnation point. This was done by Kemp, Rose and Detra [ 5 5 ] , suggested by Lees’ [48] assumption of “local similarity”. We begin with the flat plate flow. By this we mean a flow around a flat plate such that the transformation to an ordinary system is possible. This is true for ap/ax = 0 and constant boundary values at the wall and a t the outer edge of the boundary layer. The transformation of Dorodnizyn,

(6.1.1)

(6.1.2)

a

a

transforms (5.31-35) into

a

_ a P a ---, aY

Pw

a7

102

G. LUDWIG AND M. HEIL

where we used the abbreviations

With the stream function (6.1.8)

vx

defined by = *.qt

Vy = - *,€,

the equation of continuity is automatically satisfied and the system (6.1.4- 6) transforms into (6.1.10)

*.11+?l,t

- *,€+,11.11

= VW(rU*,11,11).11l

(6.1.13) If we introduce the similarity variable T (6.1.14) and the Blasius function (6.1.15)

((t),

defined by

+

= VSVwueE c.(T),

we obtain with (6.1.16) and the assumption that i and a depend only on (for E # 0): (6.1.17)

(6.1.19)

(GC,?,?),T

+

C(,Z,T

= O,

t

the following equations

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

103

The last equation shows that a can depend only on t, if wl = 0 (frozen flow) or if reaction equilibrium holds, since the diffusion equation degenerates then to a defining equation for w l . If one replaces wl by bw,/E similarity would hold for finite w1 ( b denotes a constant parameter with the dimension of a length). The boundary conditions which must be added to (6.1.17-19) follow from (5.36-37) : (6.1.20) t=m :

t =0

:

t(0) = 0,

C,,(O) = 0,

a(0)= 0,

resp.

C,,(m)

= 1,

i(0)= iw

a,(O)

T ( 0 )= T,,

= 0;

T ( m )= T,,

or

i ( m ) = i,

or

a( m) = at.

(6.1.21) The subscripts e and w denote external flow and wall conditions, respectively. These conditions define the heat transfer problem. If we wish to account for the radiation of the wall, the thermometer problem is defined by (6.1.22)

(4y)y= 0

+ d-u14(x,0)

= 0,

or accordmg to (5.55) by (6.1.23) This simplifies to (6.1.24) for the dissociation equilibrium and to (6.1.25) for a non-catalytic wall. According to (6.1.26)

i,y =CpaT,y

f

cpTX,y

the enthalpy i can be replaced by T and a. Since i,a,T should only depend on t, the conditions (6.1.23-25) would violate the assumed similarity. Therefore we replace (6.1.22) by an averaged equation L

(6.1.27)

104

G. LUDWIG AND Ed. HEIL

and obtain instead of (6.1.23)

This is a boundary condition which satisfies the condition of similarity. The wall temperature T(0) which follows by the solution of (6.1.17-19) for (6.1.28) is a mean (constant) wall temperature. Such a mean temperature has a physical meaning only if the true wall temperature does not vary too much with the cordinate x . For the non-equilibrium case it is usual to eliminate a,z,rin the energy equation (6.1.18):

The quantity c9T.T can be calculated to be

5 k

(6.1.30)

C ~ T , T = --

2 m1

- C p , = CpI - Cp,,

which follows from (5.18), (5.21) and (6.1.31)

With

(6.1.32)

il = cp,dT

W = 5 k T + -; w2 +2 2m, 2 ml 2m1

E,(O) = 0,

--

0

1 T

(6.1.33)

2

(O)

i2 = cp,dT = 2 0

m2

+

5 k

--

T,

2m2

we obtain

For the dissociation equilibrium (6.1.18) can be replaced by (6.1.35)

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

105

The system (6.1.17- 19) may be solved b y a method of successive approximation or by a Runge-Kutta process [15, 161.

FIG.7. The curves 1 and 2 show the temperature in the boundary layer depending on

q* =

1 li, ( p ( z , y ' ) dy'

without and with consideration of diffusion, respectively.

0

The la and 'a curves represent the corresponding velocity profiles

c'

= uX

[cm sec-11.

FIG.8 shows the temperature and velocity profiles for the case of heat transfer with diffusion taken into account. The wall temperature was assumed t o be 1000 "K. The diffusion lowers t h e maximum boundary layer temperature. This can also be seen in the results of Kuo [19]. Kuo. however, assumes t h a t pp may be regarded as constant. Fig. I shows t h a t this is not the case. The coefficients of transport ,upf u, 1 used in this chapter are calculated in chapter 3.5.

The thermometer problem without wall radiation for the equilibrium boundary layer with and without consideration of the diffusion was treated

106

G. LUDWIG AND M. HEIL

by Metzdorf [16]. The boundary values have been chosen in such a way that the temperature of the wall and the external flow take the same values in both cases. A result is that the diffusion allows higher free-stream velocity values for the same temperature increase across the boundary layer (see Fig. 7). The temperature profiles, however, are not realistic since the wall radiation has been neglected. 2. The Stagnation Point Flow

In an extensive paper Fay and Riddel [20] studied the influence of variable fluid properties and the effect of a finite reaction rate on the heat transfer in a partly dissociated gas. The results were verified by the experimental work of Rose and Stark [56]. They considered the flow near the stagnation point of a blunt nosed body. Only in this case, the nonequilibrium boundary layer equations with a finite reaction rate can be transformed into an ordinary system by a similarity transformation. The stagnation point flow for the two extreme cases of frozen flow and equilibrium flow has already been studied by Lees [48] and Mark [57]. Here we will give only a short account of the papers of Fay and Riddel. Moreover, we consider onIy a pure dissociating gas A, and neglect the thermodiffusion. The basic equations are given by (5.31-35):

(6.2.4)

The following transformation of the independent variables x and y is chosen, as proposed by Lees [48]: Y

(6.2.5)

With the stream function /, defined by

(6.2.7)

X

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

107

the system (6.2.1-4) transforms into

For the stagnation point flow the external velocity u, may be written as follows : (6.2.12)

we = x

( 4 s ;

the subscripts denotes the stagnation-point conditions of the external flow. Thus we get with (6.2.6) (6.2.13)

2F-L UeC

u,

=

1,

2E %C,X

1

-

(ue,x)s

Since now the left hand sides of (6.2.9-11) do not depend on [ explicitely, we may regard /, i and a as functions of 11' only: (6.2.15)

(6.2.17)

The equations (6.2.9-11) describe also the flow around a cone, which is defined by u ~=, 0.~ For this case, however, similarity holds only for wl = 0 (fronzen flow) or reaction equilibrium. If in (6.2.5-6) T is replaced by with k = 1 and k = 0, then k = 0 gives the flow around an infinitely long cylindrical body, which degenerates to the flat plate case for u ~=, 0.~ Thus the results of Ch. 6.1 can be obtained.

108

G . LUDWIG AND M. HEIL

The boundary conditions which must be added to (6.2.15-17) are given by (heat transfer problem)

q = 0:

f ( 0 ) = 0,

a(0)= 0,

f,,(O) = 0,

i(0)= iu;

for catalytic wall;

(6.2.18) a,?(O)= 0, ?j=w : With

p = nkT

f,q(m)=l,

= n,kTs

for non catalytic wall;

i(..)=i,,

u(w)=a,.

and p = mlnl f m2n2one obtains

(6.2.19) The reaction term wl was defined by (6.1.13). If we define a reaction rate parameter C, by (6.2.20) the diffusion equation can be written

The term l/(w8,Js is approximately the time for a particle in the free stream to move a distance equal to the nose radius, and thus also the time for a particle to diffuse through the boundary layer at the stagnation point. The factor gRn is, as we have seen in Ch. IV, the reciprocal of the lifetime of an atom, so that the reaction rate parameter is the ratio of the diffusion time to the lifetime of an atom. Because the diffusion time contains the nose radius while the lifetime does not, a scale effect is introduced by the chemical change which is not accounted for in the Reynolds number. Thus similar flows require equal Reynolds numbers and equal recombination rate parameters [20, 211. In (6.2.21) the equilibrium constant K , can be replaced by (6.2.22) where uE denotes the value of a at equilibrium. Thus (6.2.23)

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

109

The energy equation (6.2.16) can still be written in terms of the temperature according to (6.1.26)

Since for the stagnation point flow (6.2.25)

ue2

<< cp,(O) Ts,

the terms with ue2are neglected in [ Z O ] . In this case

(6.2.26) with (6.2.27) The system (6.2.15),(6.2.23)and (6.2.26)has been solved for the condition (6.2.18) by an electronic digital computer [20, 211. The reaction term w, differs somewhat from that used by Fay and Riddel. The calculations have been carried out for the following values of 0,Le, T,, Tw,(Il, u, (a) u

=

0.71.

(b) Le

=

1.0; Le = 1.4; Le = 2.0,

(c) 0

< C, <

m;

C,

=

0 = frozen flow; C, =

do

= equilibrium.

(d) Stagnation point conditions corresponding to thermodynamic equilibrium at velocities between 5,800 ft. per sec. and 22,XOO ft. per sec. arid at altitudes of 25,000 ft. to 120,000 ft. (e) Wall temperatures from 300°K to 3,000"K.

For the values of ,up,

CP,

- CP, , E,,

___~-~

see [HI.

CP,(O)

The heat transfer at the stagnation point may be written as (6.2.28)

110

G . LUDWIG AND M. HEIL

Nu and Re are the local Nusselt and Reynolds numbers, defined by Nu =

(6.2.29)

(QYh

&(is

x

CP.&

For the reaction equilibrium (C, = (6.2.31)

Nu

VRe ~

= 0.67

,

-iw)

R e = -UCX . l'W

oo),

(Ey{ 1

Nu/v% is approximately given by

+ (LeO.52 - 1) T (cPT)s } ,

for the frozen flow (catalytic wall) by

and for the non-catalytic wall by (6.2.33)

FIG.9. Comparison of enthalpy distributions for an equilibrium and a frozen stagnationpoint boundary-layer with the same external flow and wall conditions, g , = .0123, C A = ~ .499 (from reference 21).

I n Figs. 9-12 some results calculated by Fay and Riddel [20, 211 are given. g is defined by

(6.2.34)

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

111

cA is identical with the degree of dissociation a. If the wall catalyzes atomic recombinations, the total heat transfer is not much affected by a nonequilibrium state of the boundary layer, provided the Lewis number is near unity. If the wall is non-catalytic, the heat transfer may be appreciably LO

09 08

a7 06

a5 04

02

01

' 0

02

04

06

08

10

I2

14

I6

I0

20

2.2

24

26

5

FIG. 10. Comparison of temperature and atom concentration distributions for an equilibrium and a frozen boundary layer with the same external flow and wall conditions, g, = ,0123, C A = ~ ,499 (from reference 21). a5 04

RI

0

RECOMBINATION RATE W E T E R C,

FIG.1 1 . Heat transfer parameter N u / v E for a boundary layer with finite recombination rates, i.e., various values of the recombination rate parameter C,. g, = ,0123. C A = ~ ,536 (from reference 21).

reduced when the boundary layer is frozen throughout, as can be seen by (6.2.33). The major deviation in the heat-transfer parameter from the lowtemperature perfect-gas value is due to the variation of ,up across the boundary layer. To get better values for ,up, the viscosity p should be calculated according to the kinetic-theory formula (3.4.31).

112

G. LUDWIG AND M. HEIL

If one is interested in the boundary-layer characteristics away from the stagnation point one must go back to the full equations (6.2.9-11). I t would be, of course, very useful to reduce these equations again to ordinary differential equations even for regimes away from the stagnation point. Such an approximation is that of local similarity, as discussed by Lees [48], Fay and Riddel [21] and recently by Kemp, Rose and Detra [55]. The local similarity approximation assumes that at any point x the dependence of the unknowns f , i, T , a on [is such that their derivatives with respect to [may be neglected.

FIG.12. Distributions of atom mass fraction in a stagnation point boundary layer on a non-catalytic wall for several values of the recombination rate parameter C,. g, = ,0123, C A = ~ .536 (from reference 21).

Therefore, the right hand sides of equations (6.2.9-11) are taken to be zero. The terms on the left which depend on [, arising from external flow or wall conditions, are assumed to have their local values. Since w,,, depends upon the pressure gradient and hence body curvature in the meridian plane, an extra parameter will occur in the locally similar solutions. The equations again become ordinary differential equations in fj a t any point on the body, with parameters depending on the local external and wall conditions. The validity of this approximation depends on the fact that the external flow properties vary slowly with [ and the terms neglected in the differential equation are really negligible compared to those retained. A method for calculating the laminar heat-transfer on blunt bodies of revolution in axisymmetric, dissociating flow, based on the local similarity assumption, is presented by Kemp, Rose and Detra [MI. One of the results of the calculation is given by (dissociation equilibrium and frozen flow) (6.2.35)

(q,)s is given by (6.2.38) and (6.2.31), (6.2.32). I t is shown that is different from the stagnation point value only because of the local value

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

113

the external parameter which enters into u,,, and is not influenced by the local pp ratio. This means that only the pressure and external velocity must be calculated around the body, but not the external density or viscosity. The former are comparatively easy to obtain while the latter require use of thermodynamic properties and transport coefficients at very high temperatures. Kemp, Rose and Detra estimated the size of the non-similar terms for the hemisphere cylinder case. They showed that at 70" the nonsimilar term in the energy integral was about 15% of the similar terms. For actual boundary layer calculations all external stream values must be known. Thus at hypersonic velocities the inviscid gas dynamics must be generalized to include the effects of variable specific heats and reaction nonequilibrium. For flows around blunt bodies one must consider, moreover, mixed flows (transonic flows), which presents additional mathematical difficulties.

NOTATION Chapters 1-111 A, P

Particle of kind p 1: 2 ; 3; 4 ; 1 = atoms: 2 = molecules; 3 = positive atomic ions: 4 = electrons Coefficient of viscosity, equs. (3.3.18). (3.3.19). (3.3.51). (3.4.18) First approximation of the coefficient of viscosity according to (3.4.18), (3.4.31) for a binary gas mixture Defined by (3.4.32) Internal energy of a particle A, Velocity vector of the particle A, in a fixed Cartesian coordinate system Position vector in a fixed Cartesian coordinate system Time Density of particles A, at the position r in a fixed coordinate system at time 1, defined by (2.2) dv,, du,, dv,,; v , ~ , vpV, uPz are t h e n,y,z=components of v, fp(uP,E,,, r, t) distribution function for particles A, with internal energy t,, Distribution function for particles A, at local equilibrium, defined in 111.1 (first approximation t o /,i) Mass of p t h particle Collision term, defined by (1.40) Abbreviation for a sum of collision integrals belonging to the Boltzmann equation for particles A, where the number of particles at encounters is unaltered, defined by (1.41 - 48) A s before, but where the number of particles a t encounters changes due to reactions Collision integrals for particles A , being functions of (1.49 - 51)

I,, ,

defined by

114

G. LUDWIG AND

M. HEIL

As before with f$) substituted for f,,

- -

Transition probability a(ul,ul,ul / ul’, uI’.e2i’)

-

A,

for the process

+ A, + 2,-441‘ + A,’

(2hlml)a (mihlma4 Planck’s constant Over all density of the gas, defined by p

= Zin,n, P

Mass-average velocity of the gas, defined by (2.5) = &(Up, &,i. r, f ) , some property of the particle A, as m,+ E,i does not Mean value of A,, defined by (2.1). In (2.24 - 26) , , r, 1) signify mean values but stands for ~ $ ~ ( u4;. Peculiar velocity of a particle A,, defined by (2.6) Mean value of the peculiar velocity V,, usually denoted as diffusion velocity for particles A p ; defined by (2.7 - 9) V,(u,,r.t) = peculiar velocity of a particle A, with velocity u, Tensor with components avoZa/axfi; 0r.p = 1.2.3 x1 = x. xp = y. x, = I div U, = avoxiax avor/aY avo,iaz a&, a$,, ad8 (scalar product of two vectors a,b) Integral operators defined by (2.25 - 26) External force acting on particles A, Pressure tensor, defined by (2.32) Energy of particle A,, E,,i = mP/2 up4 E,; w, Heat flux vector, defined by (2.34). (3.3.54) Internal energy per unit mass defined by (2.35), (3.3.1 - 2) Constants: w, = i W a , wp = wJ = 0, w4 = W, Ionization energy = energy being necessary to ionize an atom A, at T = 0°K Dissociation energy = energy being necessary to dissociate a molecule at T = 0°K Temperature, defined by (2.38) The Boltzmann constant is)#,,= second approximation of the distribution function 6, defined in 111.2 Hydrostatic pressure Mean internal energy of a particle in a first approximation Vectors defined by (3.2.29), (3.2.47), (3.2.48) Tensor defined by (3.2.29). (3.2.49) Partition function, defined by (3.3.3) Reaction term, defined by (2.42). (2.43). (3.3.4 - 5) Second viscosity coefficient, defined by (3.3.23) Heat conductivity, defined by (3.3.45) Coefficient of thermodiffusion, defined by (3.3.36) Coefficient of diffusion for a binary gas mixture, defined by (3.3.36) Electrical conductivity defined by (3.3.63)

6,

-

- -

-

+

+

+

+

+

b

+

-

BOUNDARY LAYER WITH DISSOCIATION AND IONIZATION

ZZ

5

[

")J,,&(v)+ '')Jq&),,(p)]xp,dV,.

115

For reaction equilibrium

~. ai

or

(1)

J,,(.),(a)

{v,v}> 0. {v.v}= 0

= 0 holds {px} = { x . ~ } , -

a

for

q=o

CP

P

U V

H

Reaction velocities, Y = 1 , . . ., 7, defined by (3.3.5). (3.3.6) Reaction equilibrium constants, defined by (3.3.5), (3.1.30- 31) Sonine polynomials (eq. 3.4.11) Degree of dissociation for a reacting binary mixture, defined by a = (ml%)/p Specific heat a t constant pressure of a binary mixture of atoms A , and molecules A , without dissociation, defined by (3.5.14) Specific heat a t constant pressure of a binary mixture in dissociation equilibrium, defined by (3.5.13) Contribution to cp resulting from dissociation equilibrium A , 2 2 A , , defined by (3.5.15) Prandtl number (reaction equilibrium) Contribution to 1 in dissociation equilibrium, defined by (3.5.19)

Hamilton operator of the molecule, defined by (4.4) Eigenstate of the Hamilton operator Internal energy (eigenvalue of Hu) of the molecule Oscillation-, angular momentum-, and azimuthal quantum numbers Distance of the atoms constituting the molecule Reduced masses of the molecule atoms, defined by (4.5) Internal molecule potential, defined by (4.1) Interaction potential of the molecule and the colliding atom, defined by (4.6 - 8) Hamilton operator for the system: molecule colliding atom, defined by (4.11) total cross section for dissociation of the molecule when the initial state of the molecule before collision is characterized by M . L, N, defined by (4.18) Average of ale over M, defined by (4.19) all for L = 0, given in (4.20) Magnitude of the relative velocity IJ = v~ - IJA before collision Defined by (4.16)

+

l / h

+ m,') + 1b.4

Masses of the atoms of the molecule Mass of the colliding atom h/2n Relaxation times, defined by (4.25 - 26)

Coefficient of viscosity Heat conductivity D,,12, coefficient of diffusion for a binary gas mixture, defined by (3.3.35) Thermal-diffusion ratio, defined by (3.3.37) Hydrostatic pressure

116

G. LUDWIG AND M. HEIL Gas density Reynolds-number. L is a characteristical length, u is a characteristical velocity Reaction term, defined by (5.15) or (5.29) Particle densities of atoms A , and molecules A , Temperature nl nz Enthalpy, defined by (5.9), (5.16), (5.20) Internal energy per unit mass, defined by (2.35), (3.3.1 - 2), (5.16) x. y-components of the average mass-velocity u,, Defined by (5.18), (5.28) F’randtl-number for the non-reacting mixture A I A , , defined by (5.23) Schmidt-number, defined by (5.24) Specific heats. defined by (5.19). (5.21 - 22), (6.1.34) Degree of dissociation, defined by (5.8)

+

Le 6‘R

%Dim Dissociation equilibrium constant at constant pressure, defined by (5.27). (5.20) Lewis-number, defined by (5.26) 2a Mean reaction velocity, g R = gl(n,/n) g, ( n 2 / n )= g1 l+a

+

+

1-a

Gi

grn

Transformed x,y-coordinates, defined by (6.1.1 - 2) Stream function defined by (6.1.8) plp = kinematic viscosity Similarity variable defined by (6.1.14) YllP

Blasius function, defined by (6.1.15) y-component of the heat flux vector, defined by (5.54), (5.55) Mean internal energies of a n atom resp. a molecule ks’. k is the Boltzmann constant, s‘ is the emissivity Transformed x.y-coordinates defined by (6.25) Stream function defined by (6.2.7) Reaction rate parameter, defined by (6.2-20) Local Nusselt-number defined by (6.2.29)

References 1. ADAMS,Mac C., and PROBSTEIN, R. F., On the validity of continuum theory for satellite and hypersonic flight problems a t high altitudes, Jet Propulsion 28, 86-89 (1958). 2. WANGCHANG.C . S., and UHLENBECK, G. E., The Couette flow between two parallel plates as a function of the Knudsen number, Eng. Res. Inst., Univ. of Michigan, Project 1999 June 1954. 3. GRAD,H., On the kinetic theory of rarefied gases, Comm. on Pure and APPlied Math. 2, 331-407 (1949). 4. SCHAAF,S. A., and SHERMAN, S. F.. Skin friction in slip flow, J. Aeron. Sci. 21, 8&90, 144 (L954). 5. LUDWIG, G., Zum Ergodensatz und zum Begriff der makroskopischen Observablen I, Z . f . Physik 160, 346-374 (1958).

BOUNDARY LAYER WITH DISSOCIATION A N D IONIZATION

117

6. KUERTI,G., The laminar boundary layer in compressible flow, A d v . in AppI. Mech.. Vol. 11, Academic Press Inc.. New York, 1951. 7 . PAI, S. I., “Viscous Flow Theory”, Part I. D. Van Nostrand Inc., New York. 1956. 8. SCHLICHTING, H., “Grenzschicht-Theorie”, G. Braun. Karlsruhe, 1951. S., and COWLING, T . G . , “The Mathematical Theory of Non-uniform 9. CHAPMAN, Gases”. Cambridge. 1953. 10. MOORE,L. L., A solution of the laminar boundary layer equations for a compressible fluid with variable properties, J . Aeron. Sci. 19, 50&518 (1952): 11. NERNST,W.. “Boltzmann-Festschrift”, p. 904. Leipzig, 1904. 12. MEIXNER,J., Zur Theorie der Warmeleitfahigkeit reagierender fluider Mischungen, Zeitschr. fur Naturforschung 711, 5 5 b 5 5 9 (1952). M . , Der F’lasmazustand der Gase, Ergebn. der exakten 13. ROMPE,R., and STEENBECK, Natuvwissenschaften 1H, 257-276 (1939). W.. and MAECKER,H . , “Elektrische Bogen und thermisches Plasma” 14. FINKELNBURG, i n Handbuch der Physik, Band X X I I , Gasentladungen I1 (S. Fliigge Ed.), pp. 254-44. Springer-Verlag, Berlin, 1956. 15. HEIL, M . , and METZDORF, J ., Aerodynamic heating and dissociation a t hypersonic speeds, part 11, Martin Report, May 1956. 16. METZDORF, J., Flows in partly dissociated gases, J. Aeron. Sci. 26, 200 (1958). 17. METZDORF,J., Stromungen in teilweise dissoziierten Gasen, Doctoral Thesis, Freie Universitat Berlin, 19.58. 18. HEIL, M., Flows in partly dissociated gases, J . Aeron. Scz. 26, 459 (1958). 19. Kuo. Y . H . , Dissociation effects in hypersonic viscous flows, J . Aeron. Scz. 24, 3 4 S 3 5 0 (1957). 20. FAY,J . A., and RIDDEL,F. R , Theory of stagnation point heat transfer in dissociated air, Avco Res. Lab., Res. Note 18, April 1957. 21. FAY, J . A , , a n d RIDDEL, F. R . . Theory of stagnation point heat transfer in dissociated air, J. Aeron. Sct. %. 7%85 (1958). 22. HEIL, M., Zur kinetischen Theorie dissoziierender Gase A,, Doctoral Thesis. Freie Universitat Berlin, 1958. 23. PETZOLD,J., Ein Modell zur Dissoziation zweiatomiger Molekiile, Z. Phys. Chem., Neue Folge 12, 77-95 (1957). 24. Rossow, V. J., On flow of electrically conducting fluids over a flat plate in the presence of a transverse magnetic field, Null. Advisovy Comm. Aeronaut.. Techn. N o f e , No. 3971 (1957). J . L., and MCILROY, W . , Incompressible two-dimensional stagnation 25. NEURINGER, point flow of an electrically conducting viscous fluid in the presence of a magnetic field, /. Aeron. Sci. 26, 1 9 6 1 9 8 (19.58). 26. RESLER, E. L., and SEARS,W. R.. The prospects for magneto-aerodynamics, J. Aeron. Sci. 26, 235-245 (1958). R. M., Magneto-hydrodynamics of compressible fluids, Ph. D. Thesis, 27. PATRIK, Cornell University, 1956. 28. WANGCHANG,C. S.. and LJHLENBECK, G. E., Transport Phenomena in Polyatomic Gases, Eng. Res. Inst.. C’niv. Michagan, Report CM 681, July 1951. 29. HIRSCHFELDER, J . 0. CURTISS.C. F., and BIRD, R. B.. “Molecular Theory of Gases and Liquids”, Part 11. Wiley. New York, 1954. 30. HEITLER.W., “Quantum Theory of Radiation”. Second ed., Oxford, 1954. 31. GRAWERT,G., and ROLLNIK,H.. Das CPT-Theorem. Institut fur theoretische Physik. Freie Universitat Berlin. 1958. 32. COHEN,R. S., SPITZER.L., and ROUTLY,P.. The electrical conductivity of a n ionized gas, Phys. Rev. 110, 2 3 S 2 3 8 (1950). W. L., “Der elektrische Strom im Gas”. Akademie-Verlag. Berlin 1955. 33. GRANOWSKI,

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G . LUDWIG A N D M. HEIL

34. WATSON,K. M., Use of the Boltzmann equation for the study of ionized gases of low density I, Phys. Rev. 102, 12-19 (1956). 35. KAEPPELER,H. J., Stochastic foundation of generalized macroscopic equations of change in a reacting plasma, Paper presented at the AFCRC conference on extremely high temperatures, Boston, Mass., March 1958. 36. FOWLER, R. H., “Statistical Mechanics”. Second edition, Cambridge, 1936. 37. BRAGINSKII, S . I., Transport phenomena in a completely ionized two-temperature plasma, Soujet Physics 8 (33) (JETP), 358-369 (1958). 38. PRIGOGINE. I., and MAHIEU. M., Sur la pertubation de la distribution de Maxwell par des rhctions chrmiques en phase gazeuse, Physica 16, 51-64 (1950). 39. KENNARD, E. H., “Kinetic Theory of Gases”, Chapter IV, McGraw-Hill Book Comp. Inc., New York. London, 1938. 40. KRIEGBR.F. J., and WHITE,W. B., The composition and thermodynamic properties of air at temperatures from 500 to 8000°K and pressures from 0.00001 to 100 Atmospheres. The Rand Corporation, Report R-149, April 1949. 41. HILSENRATH, J., and BECKETT,C., Tables of thermodynamic properties of argonfree air to 15,000”K. AEDC-TN-5612, Arnold Engineering Development Center, USAF. Sept. 1956. 42. BAUER,E., Method of calculating cross sections for molecular collisions. J . Chem. Phys. 98, 1087-1094 (1955). 43. BECKEL,C.. Improved vibrational potential for diatomic molecules, J . Chew. Phys. 24, 553-658 (1956). 44. MEAL, J. H., and POLO, S. R., Vibration-rotation interaction in polyatomic molecules, J. Chem. Phys. %4, 1119-1138 (1956). 45. SCHWARTZ, R. N., and HERZFBLD,K. F., Vibrational relaxation times in gases, J. Chem. Phys. 22. 767-773 (1954). 46. BROUT.R., Rotational energy transfer in H,, J. Chem. Phys. 22, 934-939 (1954). 47. CAMAC.M., CAMM,J., KECK, J., and PETTY,C., Relaxation phenomena in air between 3000 and 8000°K. Avco Res. Lab., Res. Report 22, March 1958. 48. LE s, L., Laminar heat transfer over blunt-nosed bodies a t hypersonic flight speeds, Jet Propulsion 26, 497-499 (1956). 49. LEES, L., Recent developments in hypersonic flow, Jet ProPulszon 27. 1162-1178 (1957). 50. Kuo, Y. H., Viscous flow along a flat plate moving at fight supersonic speeds, J. Aeron. Sci. 28, 125-136 (1956). 51. HEIL, M., Aerodynamic heating and dissociation a t hypersonic speeds, Part IV, Martin Report, 1957. 52. BOND,J. W., Structure of a shock front in Argon, Phys. Rev. 106, 1683-1694 (1957). 53. NEURINCER, J. L., and MCILROY,W.. Hydromagnetic effects on stagnation-point heat transfer, J . Aeron. Sca. 86, 332 (1958). 54. ROSA, R. J., Engineering Magneto-Hydrodynamics, Part 2 of Ph. D. Thesis, Cornell University, 1956. 55. KEMP,N. H., ROSE,P. H., and DETRA,R. W., Laminar heat transfer around blunt bodies in dissociated air, Avco Res. Lab., Res. Report 15, May 1958. 56. ROSE, P. H., and STARK,W. I., Stagnation point heat-transfer measurements in dissociated air, J. Aeron. Scd. E6, 8C97 (1968). 57. MARK,R., Compressible laminar heat transfer near the stagnation point of blunt bodies of revolution, Convair Report No. ZA-7-016, San Diego, Calif., 1955. 58. HEIL, M., Kinetische Theorie der Stramungen dissoziierender Gase, Brennstoff, Warme, Kraft (BWK) 10, 298 (1958). 59. SCALA,S . M., Hypersonic heat transfer t o catalytic surfaces, J. Aeron. Sci. S6, 273-275 (1958).

The Propagation of Shock Waves along Ducts of Varying Cross Section BY W. CHESTER Unaverstty of Bristol, Bristol, England

Page

.

.

11. The Steady State Theory 1. Discussion of the Model

.

. . . . . . 2. Discussion of the Results . . 111. Chisnell's Theory . . . . . . . 1. Introduction . . . . . . . .

.

. . . . . .

. . . . . . ... ... . . . . . . . . . . . . . . . . 3. Solution for a n Arbitrary Area Change . . . . . . 4. Weak and Strong Shocks . . . . . . . . . . . . 5. Cylindrical and Spherical Shocks . . . . . . . . . 6. Assessment of the Error . . . . . . . . . . . . . 7. Whitham's Approach . . . . . . . . . . . . . . IV. Comparison of the Two Theories . . . . . . . . . . V. Steady Flow Regime Ahead of the Shock . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . 2. Basic Equations . . . . . . . . . . . . . . . . 3. The Strong Shock . . . . . . . . . . . . . . . 4. The Weak Shock . . . . . . . . . . . . . . . . 5. The Generalization of the Steady State Theory . . References . . . . . . . . . . . . . . . . . . . . . . . I. General Introduction .

.

. . . . . .

. , . . . . . . . . . . 2. Solution for a Small Area Change .

NOTATION Cross-sectional area of the duct Sonic speed See equation (3.33) Characteristics Distance travelled by shock See equations (3.25), (3.26). (3.27) See equation (3.5) See equations (3.2). (3.10) Mach number Characteristic Pressure Fluid speed See equation (3.6)

119

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

. ,120

. . . . . .

. 123

.

. . . . . . . . . . . . . . . . . . . . . . .

. 123 .121 .133 ,133 ,134 .I34 . 136 . 136 ,137 .139 ,143 .144 . 144 . 145 . . . 146 . . . 148 . . . 151 . . . 152

120 Y

1

U x z

a. P Y 1+6 1 S E

4

1 P P t

W. CHESTER

Radial co-ordinate Time Shock. velocity Co-ordinate along the axis of the duct Pressure ratio across the shock See equations (5.10). (5.11) Adiabatic index Mach number of weak shock Mach number of the flow ahead of the shock See equation (3.14) See equation (3.32) See equation (3.3) Density See equation (3.38)

Subscripts: C

d

i Y

t T U

1-5

Characteristic quantity Flow quantities downstream of area change Incident shock Reflected shock Transmitted shock Stagnation conditions Flow quantities upstream of area change Except for Section V these subscripts refer to the regions of flow shown in Figs. 2 and 3. In Section V subscripts 1 and 2 refer respectively to flow quantities immediately in front of, and immediately behind, the shock

I. GENERALINTRODUCTION When a shock wave travels along a duct whose cross section changes continuously, there is an interaction between the shock and the changing cross section. The flow behind the shock is disturbed and there is a modification of the shock strength. A number of writers have considered the effect of the interaction on the flow behind the shock; this work has been noted in the bibliography and reference is made to some of it in the following pages. The present article will, however, mainly consider the modifying effect on the shock wave itself. This problem is of particular interest, not least because of the success of the relatively simple techniques which have been evolved to deal with it. By way of introduction consider the propagation of a shock wave between two walls of infinite extent, forming a two-dimensional channel of uniform width connected to a channel of uniformly increasing width. As the shock passes into the diverging section two independent diffracted pulses are produced, one at each corner (Fig. 1). Eventually, as the pattern expands, the two pulses will interact and suffer repeated reflection from the two walls. The whole process is too complex to attempt a complete mathematical description, but if the angle of divergence is small the problem becomes a

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

121

linear one. The interaction of the pulses can then be replaced by a simple superposition and the effect on reflection of the small divergence of the walls can be ignored, for this will be of the second order for a pulse in which the perturbation is already small. T h e diffraction problem for a single corner was solved by Lighthill [l], and the image system which simulates repeated reflection between two walls was studied by Chester [Z]. Furthermore, since the problem is linear, the solution for arbitrary (though small) variations in channel width can be obtained by summation over all the elementary variations in slope. Thus if the channel consists of two sections of uniform width joined by a transition section of finite length the asymptotic behaviour of the flow can be deduced, and consists essentially of a iiniform but modified transmitted shock and a plane reflected pulse. The theory of shock propagation described in Section I11 stems from the solution of this problem.

Ft

Shock

FIG.1 . The diffraction pattern prvducrd by the interaction of a shock wave with a diverging channel.

Freeman [3] has also investigated the image system in the problem of the last paragraph and has deduced therefrom that the perturbations behind the shock are oscillatory in character and decay asymptotically like d W 2 , where d is the distance travelled by the shock wave from the transition producing the perturbations. Freeman also shows that the rate at which the perturbations decay depends, in addition, on the strength of the shock, being strongest when the shock Mach number is about 1.15 and decreasing sharply for both strong and weak shocks. Another simple example which serves to introduce the subject matter of this review is the problem of an acoustic pulse travelling along a tube of varying cross section. This is again a linear problem because the weak shock can only produce a small disturbance. The restriction to small variations in cross-sectional area is, however, no longer necessary. Here two interesting cases arise, both of which are discussed by Rayleigh [4]. The first is that of a sharp fronted pulse proceeding along a duct for which the cross section varies continuously. At the front of the pulse the energy reflected is negligible on a linear theory, so that the transmitted energy is invariant. Since the latter is proportional to the square of the amplitude multiplied by the cross section, it follows that the amplitude varies inversely as the square root of the cross section. The result is not true in the rear part

122

W. CHESTER

of the disturbance because the second order effects may accumulate to a significant modification over a finite change of cross section. But this integrated effect cannot modify the front of the pulse since disturbances to the rear travel no faster than the front itself. Such disturbances therefore never reach the front. The other case is the problem of reflection and transmission at an abrupt change in cross section. Here the argument of the previous paragraph is no longer valid, for a significant proportion of energy will now be reflected by the discontinuity even at the head of the wave. But a simple argument gives the ultimate behaviour of the transmitted and reflected pulses. Let the cross section change discontinuously from A , to A , and let the Mach numbers of the incident, reflected, and transmitted pulses be respectively (1 d,), (1 6,) and (1 6J. Then continuity of pressure and mass flux demands that

+

(1.1) (1-2)

+

+

6i

+ 6,

A,(& - 6 7 )

=

4,

= A,&

since the perturbations in flow velocity and pressure are proportional to 6 on a linear theory. These equations imply that

which agrees with the previous result only when IA, - A,\ << Az*. These solutions for the acoustic pulse illustrate two extreme situations in the problem of shock propagation along ducts. In the first the interactions behind the front of the pulse have no effect whatever on the front itself. In the second they are allowed to exert their utmost effect. For a shock wave of finite strength travelling along a duct whose cross section varies continuously, but by a significant fraction of itself, the answer lies somewhere beteeen these two extremes. The disturbances created behind the shock front will now be continually overtaking the shock with consequent modification of its strength. If these effects continue to be ignored, notwithstanding the lack of rigorous justification, a simple solution can be obtained which has been studied in detail, originally by Chisnell [5] and later by Whitham [6, 71. This theory is the main theme of Section 111, and evidence is offered which suggests that Chisnell’s hypothesis is sound and in certain circumstances predicts the variation in strength of the shock wave, as it progresses, to good accuracy.

* The subscripts have been chosen to be consistent with the numbering of the various flow regions in Figs. 2 and 3, and hence with later equations which refer to more general models. Note that the subscript 2 refers to the upstream part of the duct, and the subscript 1 to the downstream part.

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

123

On the other hand, suppose the duct consists of two uniform sections connected by a transition section of finite length. The shock is then initially of uniform strength in the upstream section. Ultimately it tends to become a shock wave of uniform (though different) strength in the downstream section. When this happens the modifying effects of the disturbances produced by the change in cross section will have been completed, and the only nonsteadiness will be the continual growth of the uniform region behind the shock. This is the generalization of the second of the acoustic problems. The mode1 has been used by Kahane, Warren, Griffith and Marino [8], Laporte [9] and Parks [lo]. I t is described in Section 11. I t is naturally of interest to compare the two theories directly, and this is done in Section IV. Such a comparison gives some idea of the extent to which upstream disturbances are capable of modifying the shock strength. However it should be pointed out that, in application, they are meant to describe different situations. Chisnell intended that his solution should be used only to describe the progressive modification of a shock as it travelled along a duct whose cross section varied ‘not too rapidly’. On the other hand, the method of Section I1 gives the ultimate strength, when the shock has again become uniform, after passing through a transition section of finite length. Section V contains some previously unpublished results obtained by extending the ideas of the previous sections to deal with the progress of a shock wave along a duct in which a steady flow is already established. There it is found, for example, that the pressure difference across the shock has a maximum where the steady flow is subsonic, the precise cross section depending on the shock strength. The behaviour of a weak disturbance facing upstream, in the neighbourhood of a sonic throat, is also deducible from the theory of Section V. I t is shown that if such a disturbance is initially moving upstream it continues to do so and decreases in strength. If it is initially moving downstream it continues in that direction, but its strength may increase or decrease. 11. THE STEADY-STATE THEORY 1. Discussion of the Model

In the theory described here the channel will always consist of two sections, each of uniform cross-sectional area, joined by a transition section of finite length. Only the ultimate flow will be considered, when conditions behind the shock are independent of the time and the shock itself is moving along the downstream section with uniform speed and strength. To set up a model which describes such a flow it would seem natural to generalise the argument used for an acoustic pulse. This would imply a

I24

W. CHESTER

transmitted shock wave, and a reflected shock wave or rarefaction wave according as the transition is a contraction or expansion. The flow between the two waves would follow the known behaviour for steady flow in a duct.

MI

-

I, M,

>I

(d)

MI

>I.

M,

>

I

FIG.2. Interaction of a shock wave with a contraction: steady state theory.

Such a model, however, is not sufficiently flexible in the general case, and will not allow the transition relations across the two waves to be satisfied simultaneously with the relations which the flow must satisfy through the area change. I t is in fact necessary to include a contact discontinuity in the downstream section. The necessary matching of the different parts of the flow is then possible. The existence of such a discontinuity is well supported by experimental evidence. The above argument suggest the models pictured in Figs. 2(a) and 3(a) for a contraction and an expansion respectively. But these two models are still not sufficient to cover the whole range of possibilities. Thus for a contraction it is not possible to exceed unit Mach number immediately downstream of the contraction if the upstream flow is subsonic, and calculations show that for sufficiently strong shock waves the model shown in Fig. 2(a) is not consistent with this fact. Laporte [9] proposes model 2(b) in such cases. Here a simple wave begins at the end of the contraction, where sonic velocity is just reached. Supersonic flow upstream of the transmitted shock is then achieved through this expansion wave. Of course, it is possible for supersonic flow to exist behind the incident shock, and for this flow to remain supersonic even after the contraction. Then no reflected disturbance is produced which is strong enough to move upstream. In addition to the transmitted shock and the contact discontinuity, this situation is completed by an expansion wave facing upstream, but convected

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

125

downstream as in Fig. 2(d). Fig. 2(c) illustrates an intermediate case in which a stationary shock wave appears in the transition section itself. The situation in the downstream section is qualitatively similar to that in Fig. (2b).

(c)

M2

<

I,

MI > I

td)

M2 > I .

My >I

FIG. 3. Interaction of a shock wave with an expansion; steady state theory.

The corresponding possibilities in the flow through an expansion are shown in Fig. 3. For incident shock strengths below a certain critical value, the model differs from Fig. 2(a) only in that a rarefaction wave is reflected instead of a shock. As the strength of the incident shock increases, the Mach number of the flow downstream of the reflected wave also increases until finally it becomes sonic at the beginning of the expansion, and supersonic by further expansion until it reaches a stationary shock wave. Beyond this shock the flow remains subsonic [Fig. 2(b)]. For stronger incident shocks the stationary shock moves out of the transition section and is convected downstream [Fig. 2(c)]. Finally the expansion wave in the upstream section can disappear entirely giving Fig. 2(d). Not all of the flow regimes which have been described are necessarily attainable for any given transition. For example, that shown in Fig. 2(c) will first appear when the incident shock is just strong enough to produce a reflected shock which remains stationary at the upstream end of the transition section. Let the Mach numbers upstream and downstream of such a stationary shock be M , and M , respectively. Then, b y the shock transition relations

where y is the adiabatic index. I t follows that M , is a monotonic decreasing function of M,. Since M , cannot exceed { Z / y ( y- 1)}*’2,which is the limiting

120

W. CHESTER

value of M , as the strength of the incident shock tends to infinity, (2.1) implies that

Now the area, Mach-number relation for steady flow through the contraction is

where M , is the Mach number at the downstream end of the transition section and must satisfy the inequality M6 1. It follows that the maximum contraction which allows the possibility of model 2(c) is given by (2.3) with M, = {2(y - l)/y(3 - y)}ll2 and M , = 1. This gives

<

(2.4)

= 1.191

for

y = 7/5,

1.046

for

y

=

= 5/3.

For contraction ratios which exceed this value, models 2(a) and 2(b) appear t o cover the whole range of incident shock strengths, and there is always some upstream influence. It is, however, interesting that the critical condition for model 2(d) is slightly less restrictive than (2.4). For this model the area, Mach-number relation for steady flow through the transition section is similar t o (2.3) but with M , replaced by M,. Substitution of the limiting value of M,, namely {2/y(y - 1)}1/2,in this relation, together with M , = 1, gives 1

A,/A, (2.5)

=

(y - 1)1/2(2/y)3,

= 1.543

for

y

= 7/5,

= 1.073

for

y

= 5/3.

The implication is that under certain conditions there are two possibilities, 2(b) or 2(d). In practice it is probable that 2(b) will prevail; if it does not there is certainly no continuous transition from 2(b) to 2(d) unless the contraction ratio is less than the value given by (2.4). The various models contained in Figs. 2 and 3 seem to cover all the possibilities for a monotonic area change apart from minor modifications (for example, in the exparision it is possible for the reflected wave to disappear before the stationary shock has passed into the downstream

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

127

section). They are also calculable from, and consistent with, the initial conditions and the known physical limitations for such flows. The procedure for calculating the strength of the transmitted shock is illustrated by taking model 2(a) as a typical example. The flow in region 2 is that produced behind the incident shock, and is therefore calculable in terms of the strength of that shock, the transition relations which hold across it, and conditions in region 1. Similarly the flow in region 5 depends on that in region 2 and one other parameter - say the strength of the transmitted shock. The flow in region 4, which is deducible from that in region 5 and the relations for steady flow through a duct, is also dependent on this parameter. By its elimination, a relation can be obtained connecting the pressure and velocity in region 4. Since the pressure and velocity are continuous through the contact discontinuity, this relation persists in region 3. Together with the known flow ahead of it, this is just sufficient information t o determine the strength of the transmitted shock. The actual computation requires the use of simple numerical techniques. The details are omitted.

2. Discussion of the Results The results of some numerical calculations are shown in Figs. 4-7. Fig. 4 was constructed from calculations made by Laporte [Y]for contraction ratios of 2 : 1, 5 : 1 and co : 1 , and a value of 5/3 for the adiabatic index. It shows the variation of the ratio of the pressure differences across the transmitted and incident shocks with the Mach number of the incident shock.* Fig. 4 also shows the asymptotic values for strong shocks, and the tangent to the curve at the weak shock end of the scale. The latter is obtained from model 2(a), or 3(a), by expanding all the flow variables as power series in the strength of the incident shock. The first three terms are given by

(2.6)

* The pressure difference across the shock, rather than the pressure ratio, has been chosen as a measure of its strength since the ratio is unsuitable for weak shocks. The Mach number of the incident shock is. however. used as the abscissa, because it gives more prominence than the pressure difference to t h a t part of the curve which describes the weaker shocks. This is where most of the variation takes place. There is, of course, a simple relation, (2.7). between Mach number and pressure difference.

128

W. CHESTER

where 2

- 1 = p,/p,

- 1 = ___ 2Y

(M2- l),

Y f l and the term in square brackets is to be included only for an expansion, i.e. A , < A,. I n these equations (fi, - 9,) and (+, - p,) are respectively the pressure differences across the transmitted and incident shocks, and M is the Mach number of the latter.

FIG.4. The ratio of the pressure differences across the transmitted and incident shocks as a function of the Mach number of the incident shock; steady state theory. (a) y = 5/3 and 2 : 1 contraction ratio; (b) y = 5/3 and 5 : 1 contraction ratio; (c) y = 5/3 and ca : 1 contraction ratio.

The contraction and expansion have different analytical expressions for the third term in (2.6) because of the entropy change (which is of the

T H E PROPAGATION OF SHOCK W A V E S ALONG DUCTS

129

third order in the pressure difference) across the shock reflected by a contraction. Actually the difference is zero for y = 513 and quite small for y = 715. Reference to Fig. 4 shows that the series in (2.6) converges slowly except for very weak shocks. Even for a 2 : 1 contraction the tangent (obtained from the first two terms of (2.6)]very quickly ceases to be a reliable approximation. The addition of another term improves the approximation as far as M = 1.1. Beyond this it overestimates and even gives the wrong trend - the right hand side of (2.6) has a minimum a t roughly this value of M . For stronger contractions the convergence is still slower, and the series solution fails completely for very large contraction ratios. Fig. 4(c) shows the limiting behaviour, according to Laporte’s calculations, as the contraction ratio tends to infinity. The transmitted shock is still of finite strength, and it is not difficult to see that this must be so. Laporte uses model I ( b ) with the conditions behind the reflected shock equivalent to those for reflection from a solid wall. In particular, if P, and a, are the pressure and the speed of sound behind the reflected shock, then it follows that (2.8) (2.9)

a, ( ( 3 y- I ) P , / P I Y{(Y a12

+ (Y- 111{(Y -

+ l)P,/Pl + (Y

-

1)fiZlPI

+ 11 .

1))

The fluid velocity q6, and speed of sound a5 immediately upstream of the expansion are connected by (2.10) and since q5 = a5 this implies that (2.11) Also, across the expansion wave,

or, equivalently,

(2.13)

2

130

W. CHESTER

Now the velocity and pressure are continuous across the contact discontinuity so that

I

4

5

2 3 M FIG.5. The ratio of the pressures behind the transmitted and incident shocks as a function of the Mach number of the incident shock; steady state theory. y = 5/3 and CQ : 1 contraction ratio.

I n (2.14), q3/a, can be obtained in terms of $,/#J,from the shock transition relations. I t is in fact given by

Also $,/$, and a*/a, are expressible in terms of P2/P1 by (2.8) and (2.9) as a respectively. It follows that (2.14) can be used to determine function,of #J,/P1. Fig. 4(c) shows the result of such a calculation. A further interesting property of this limiting solution for infinite contraction ratio is that a shock of substantial strength is transmitted, according to this model, even when the incident shock is weak. For in the limit of a weak incident shock a, a,, #J, + p1 and so equation (2.14) gives

-.

131

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

(2.16) When y = 5/3, (2.16) gives $,/PI = 1.453. This feature of the solution is not brought out in Fig. 4 and accordingly P3/p, is shown in Fig, 5 as a function of the Mach number of the incident shock. A comparison of the three curves in Fig. 4 shows that the ratio of the pressure differences across the transmitted and incident shocks initially decreases with increasing shock strength, and reaches a minimum at roughly M = 5 for the 2 : 1 contraction and M = 4 for the 5 : 1 and bo : 1 contraction. After the minimum the ratio rises to its asymptotic value shown by the horizontal broken line in the figures. This increase is quite small. In fact for M > 3 the ratio changes by only 0.4% of itself for the bo : 1 contraction, and the change is even smaller in the other two cases. The strength of the transmitted shock is very sensitive to the contraction ratio for weak shocks. There is, however, a marked decrease in sensitivity when the shock is strong. This can be seen from Table 1, where the values of ( p , - p , ) / ( p , - p,) are shown, for various contraction ratios, in the limit as M + 00. DIFTABLE1. THE LIMITING VALUES,AS M + cw, OF THE RATIOOF THE PRESSURE FERENCES ACROSS T H E TRANSMITTED AND INCIDENT SHOCKS.STEADY STATETHEORY

(P, - P,)/(P, - P I ) Contraction Ratio

__ y

2 : l 5 : l a0:l

=

5/3

1.260 1.492 1.706

y

=

7 /5

1.246 1.374 1.511

Laporte's detailed calculations also show that there is a transition from model 2(a) to model 2(b) when M is about 2.4 for the 2 : 1 contraction, and 2.1 for the 5 : 1 contraction (Laporte's calculations are all based on these two models). Finally it appears that the contact discontinuity between regions 3 and 4 is insignificant except for very strong shocks. In the range 2 < M < 7 the density ratio (p4/p3) across this discontinuity varies between 1.03 and 1.06

132

W. CHESTER

for the 2 : 1 contraction, between 1.05 and 1.12 for the 5 : 1 contraction and between 1.06 and 1.16 for the 03 : 1 contraction. Laporte considered only a monatomic gas for which y = 513. A good idea of the effect of y is, however, obtained by noticing that, in Fig. 4, the asymptotic value of the relative strength of the shock is a good approxima3 the relative strength decreases from tion for 3 M < 00. For 1 6 M its value at M = 1. Now the value at M = 1 is independent of y , and the asymptotic values as M + 00, for y = 5/3 and 7/5 are given in Table 1. The effect of a change in y on (p, - p,)/(#, - p,) is relatively more important for the larger contraction ratios.

<

<

FIG.6. The ratio of the pressure differences across the transmitted and incident shocks as a function of the Mach number of the incident shock; y = 7 / 5 and IL : 4 expansion ratio. Upper curve - Chisnell theory; lower curve - steady state theory.

Fig. 6 is the result of a calculation for a 1 : 4 expansion and y = 1.4. As for the contraction, the change in cross-sectional area has least effect on the relative strength of the transmitted shock when the incident shock is strong. But unlike the contraction, the change in area has its maximum effect when the strength of the incident shock is substantial ( M = 1.5) and not in the limiting case of a weak shock. At M = 1.5 the relative strength

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

in region 5 , and that at M = 1.5 with the entry of the stationary shock into the downstream section. There is, however, no conclusive evidence that they are in fact connected with the transition between the models of Fig. 3. The results agree well with Parks's experiments using a 1 : 4 expansion and incident shock strengths within the range 3 < M < 5 [lo]. Note that (P,- P l ) / ( P 2- Pl) approaches its asymptotic value for large M much more slowly than for a contraction. Fig. 7 shows the limiting value of (P, - P , ) 4 / t P , - P l M , as 4 I A 2 O0It was assumed in the calculations that the stationary shock in the transition sec-

133

4

3

I

2

-

111. CHISNELL'STHEORY 1. Introduction

The approach used in this section is quite different from the steady state theory described in Section 11. Although the results of the latter theory are in fact verified when the area change is small (see for example [ 5 ] ) ,in its generality Chisnell's theory deals more appropriately with the continuously changing shock strength as a function of that cross-sectional area of the duct occupied by the shock a t any particular instant. The problem is first considered for a small change in cross-sectional area. Chisnell's extension to arbitrary area changes is then explained and criticized. Two estimates of the accuracy of the result, by Chisnell [5] and by Whitham [7], are discussed and a comparison is made with a known

134

W. CHESTER

problem, namely the converging cylindrical (or spherical) shock, first considered by Guderley [Ill and later by Butler [12]. 2. Solution for a Sntall Area Change Consider first a duct in which the cross-sectional area changes only by a small fraction of itself. Then the perturbations produced in the flow behind the shock are small and the problem can be linearized. Associated with a small change SA in cross-sectional area A is a small change 6M in the Mach number M of the shock given by the formula

SA _

- 2MSM A - (M2- 1 ) K ( M )

where (3.2)

K(M)=2

[(

l+--

r+l

1 - p 2 ) (2p

P

+1+

M-2)I-l

and (3.3) The result was first derived by Chester [2, 131 who solved the linearized equations of motion for the flow behind the shock. I t can also be obtained from the steady state theory of the previous section [5]. Both approaches predict, essentially, that a small change in area produces a transmitted shock of modified strength, and a weak reflected pulse which propagates upstream without further modification. Chester’s analysis shows also that the result is true, when the flow variables are averaged over the cross-section, during the actual motion of the shock through the transition and not only when the shock is again uniform. The simplest derivation of (3.1), which is due to Whitham, will be described later (see sub-section 111.6) as part of Whitham’s contribution to the discussion of Chisnell’s result. 3. Solution for an Arbitrary Area Change Chisnell’s suggestion is that the integrated form of (3.1) should be used as the relation between Mach number and cross-sectional area when the latter varies continuously, but without restriction to small perturbations. Since this suggestion is much easier to implement than to justify, the results will first be given and then criticized. When equation (3.1) is integrated, the result is [5] (3.4)

A f ( M ) = const.

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

135

-

The parameter z measures the pressure ratio across the shock.

FIG.8. The variation of log,,f

with Mach number according to (3.5). Chisnell's theory; y = 715.

Fig. 8 shows the variation of log,, f with Mach number. A rough idea of the relationship between cross-sectional area and Mach number can be obtained quite simply b y using the fact that the function K ( M ) in (3.1) is a monotonic decreasing function of M and vanes between 0.5, for weak

136

W. CHESTER

shocks, and 0.3941 (for y = 1.4), which is the limiting value for strong shocks. If this small variation in K is ignored, equation (3.1) integrates to

A K ( M 2- 1) = const.

(3.8)

Chisnell suggests that this relation may be adequate if used with a suitable mean value for K , especially when the variations in shock strength are not too large. Tables giving both f and K will be found in Chisnell.’~ original paper [5]. 4. Weak and Strong Shocks

In the extreme cases of weak and strong shocks, equation (3.4) reduces anyway to the simpler form (3.8) with the appropriate limiting values for K. These results are of particular interest, for they can be compared with known solutions and so give some idea of the accuracy involved when using Chisnell’s hypothesis. As M + 1, equations (3.2) and (3.3) show that K + & and so, from (3.8), we get

( M - 1) cc A-112,

(3-9)

and this is just the law of variation derived in the introduction for an acoustic pulse. That the correct result should be obtained here is not surprising. For although in general the disturbances produced behind the shock can overtake it, and so cause modifications in strength ignored by Chisnell’s hypothesis, when the shock is weak it travels with the same speed as the disturbances and so cannot be overtaken. As M -,co, p2 ( y - 1)/2y and --+

Equation (3.8) then gives (3.11)

M

o(

A-Km/2.

5. Cylindrical and Spherical Shocks Chisnell applied (3.11) to converging cylindrical and spherical shocks, which are special cases of propagation in a tube with cross-sectional area proportional to the distance from the axis (wedge-shaped tube) and proportional to the square of the distance from the centre (conical tube) respectively. Thus, if r denotes the distance of the shock either from its axis or

137

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

its point of symmetry, equation (3.11)gives, near the origin where the shock is strong,

M a

(3.12)

r-KK,/2

for a cylindrical shock, and

M cc r - K m

(3.13)

for the spherical shock. The problem of converging cylindrical and spherical shocks has been considered independently by Guderley [ll] who uses a similarity solution of the full equations of motion. A comparison with Guderley’s solution gives a striking check on the accuracy of (3.11). In Table 2 the indices K,/2 and K , are compared with the corresponding values according to Guderley’s solution. The values marked Guderley are the results of calculations by Butler [ l d ] . TABLE2. T H E EXPONENT n I N T H E MAGH-NUMBER, DISTANCE RELATION, M a r-“, FOR CONVERGING CYLINDRICAL A N D SPHERICAL SHOCKS. A COMPARISON OF T H E RESULTS FROM CHISNELL’S THEORYA N D GUDERLEY’SSIMILARITY SOLUTION OBTAINED Cylindrical Shock

Spherical Shock

Y

6/5 7/5 5/3

Chisnell

Guderley

Chisnell

Guderley

0.1631 12 0.197070 0.265425

0.161220 0.197294 0.226054

0.326223 0.394141 0.450850

0.320756 0.394364 0.452692

The similarity solution referred to above applies only when the shock is strong. Payne [14], however, has solved the equations of motion numerically for the converging cylindrical shock with initial pressure ratios across the shock of 1.9 and 8 and two values of the adiabatic index ( y = 715 and 5/3). In all cases Payne finds good agreement with Chisnell’s formula (3.4). 6. Assessment of the Error

Chisnell also assesses the error involved in the step from equation (3.1) to (3.4). Essentially two assumptions are made in this step. The first is that the variation in the strength of the shock over its front can be ignored, so that the shock can be adequately described by its overall strength (this

138

W. CHESTER

assumption does not apply to cylindrical or spherical shocks). It is in fact true in the linearized problem that (3.1) is valid at all times during the propagation of the shock, and not only when the shock is again uniform, provided the Mach number is calculated from the average pressure immediately behind it. Thus one may hope that in the general situation this is a minor limitation provided the variations in cross section are not too rapid. In any case the method can only be used in situations where the actual shock can usefully be replaced by an ‘equivalent’ mean shock. The second assumption is that the effect on the shock of the interactions which take place in the flow behind it can be ignored. As the shock passes through an elementary area change two disturbances, a reflected acoustic pulse and a contact discontinuity, are generated [see for example, Fig. 2(a) or 3(a)]. Of course, in a duct whose cross section is changing continuously these disturbances will in fact be elements of extended waves in which the transitions from one state to another are continuous. There will be a continuous and complex interplay between these waves and the changing area of the duct, resulting in the nonlinear disturbances which are the source of error in Chisnell’s theory. For the moment, however, it is convenient to think of the two elementary disturbances as discrete discontinuities. The reflected pulse may, for example, interact with a small area change upstream of the one at which the pulse originated. The effect is to modify the pulse itself, and also to produce a re-reflected wave which travels downstream. This re-reflected wave will eventually overtake, and thereupon modify, the shock. Note, however, that the cumulative effect of a substantial change in cross-sectional area is necessary before such modification is significant. For the interaction of a weak pulse with a small area change can only produce a re-reflected wave whose strength is of the second order of small quantities. Re-reflected waves play no part in the linearized theory. This is why the change in Mach number, according to (3.1), depends only on the current change in cross-sectional area, and not on the conditions upstream. It is therefore possible to reason that the interactions behind the shock may play a minor role in the propagation of the shock itself, provided the area changes are not too large. This, however, does not justify their neglect in all cases. It does not explain, for example, their manifestly minor role in the converging shock problem described above. For this latter problem Chisnell has made an estimate of the modifying effect of the re-reflected waves on the assumption that once re-reflection has occurred the resulting disturbance propagates unchanged until it reaches the shock. Higher order disturbances (such as reflections of the re-reflected wave) are not considered. There are three contributions to the re-reflected wave. Two of these originate from the interactions of the reflected pulse and contact discontinuity, respectively, with area changes upstream of the shock (these interactions resemble that of the original shock with the changes in area).

139

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

The third comes from the interaction of the reflected pulse with a contact discontinuity generated at an earlier stage in the progress of the shock. When these contributions have been calculated for a small area change, the results can be integrated to get the accumulated effect for an interval in which A changes appreciably. In particular, for the strong shock, one can write as a better approximation to (3.11) (3.14) TABLE3. THE VALUESOF 7 I N T H E RELATION M a c A - ( 1 + ’ ) K m ‘ 2 CALCULATED WAVE (i) FROM T H E MODIFYINGEFFECTOF THE RE-REFLECTED (ii) FROM G U D E R L E Y ’ S SIMILARITY SOLUTION. COLUMN(iii) GIVES T H E LARGEST SINGLECONTRIBUTION TO 17 I N T H E RE-REFLECTED WAVE Cylindrical Shock

Spherical Shock

Y

6/5 715 513

-0.016 +0.00042 +0.0036

-0.012 +0.0011 +0.0027

-0.029 +0.00072

+0.11 -0.13 -0.20

+0.0081

-0.017 +0.00055 +0.0042

+O.l9 -0.22 -0.34

Table 3 gives the values of q calculated by Chisnell, and shows that the re-reflected waves produce the expected small correction. The smallness seems, however, to be somewhat fortuitous. For of the three contributions to q from the three interactions described above, only one is of the same order as 7. The other two are more significant, but are of opposite sign and happen to cancel each other almost entirely. Table 3 illustrates this in more detail. 7 . Whitham’s Approach

In Whitham’s analysis [7] of the Chisnell hypothesis, the flow behind the shock is described in terms of the simplified one-dimensional equations, so that (3.16) (3.17) (3.18)

ap

-

at

a4 at

-

a (pqA) = 0, + A1 -~ ax -

a4 + 1 aP = 0, +q ax ax --

140

W. CHESTER

where p,p,q are the pressure, density, and fluid velocity, respectively, x is the co-ordinate measured along the axis of the tube, and t is the time. These equations can be put in the following characteristic form,

+ padq + qpa2q +a

(3.19)

dp

(3.20)

dp -pad9

~

~

pa2q + ---=O q-a

dA -0 A

on

C+:

dl=q+a,

dA A

on

C-:

-

on

P:

--4

dp-a2dp=0

(3.21)

dx

dx dt = q - - a ,

dx dt

where a is the sonic velocity. Consider first the linearized problem. Suppose that initially the crosssectional area is uniform ( A , ) and that the uniform flow behind the incident shock is described by the suffix 2. Then the perturbation produced by small changes in area must be such that, correct to the first order,

(3.24)

dp

- a22dp=

0

on

dx

dt = 9 2 .

These equations can be integrated to give (3.25)

(3.27)

p

- p, =

p

- p2

P2"224z -

4z2

- a22

A ( x ) - A2

+

A2

+ H ( x - q2t)

= a2

where F , G, and H are certain functions to be determined by the boundary conditions. In particular, since F represents a disturbance travelling

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

141

downstream, and since on a linear theory there is no mechanism for the production of such a disturbance, this function must be identically zero. The functions G and H are determined by the shock conditions (3.28)

(3.29) =

(3.30)

(Y

2

+ 1)M2

+ (y

-

1)MZ'

The elimination of M between (3.28) and (3.29) gives a relation to be satisfied between p and q at x = U t , where U is the velocity of the shock. Substitution of the expressions for p and q given by (3.25) and (3.26) in this relation determines G. The result is (3.31) where (3.32)

(3.33)

+ (7 i - 1)p2/(1

+

H = - ~1 - - M-'

~

1

+ M - 2 + 2/1

-

p2)

and p is given by (3.3). These results determine the reflected wave, and in a similar manner the function H can be found. Our main concern, however, is with the behaviour of the shock, and this is independent of the functions G and H . The function F arises from the integration of (3.22),and to say that F is identically zero is equivalent to saying that (3.22) holds not only on a positive characteristic but in the whole flow; in particular at the shock itself. Together with the shock relations this is sufficient to determine a relation between A4 and A which is just (3.1). The disturbances represented by G and H are, of course, by-products of the shock modification. They are not mechanisms which contribute to this modification; such a mechanism would be represented by F , and here F = 0. Whitham's interpretation of Chisnell's rule now follows immediately. If, in the non-linear problem, (3.1) is still applied a t the shock as a differential relation, then it is equivalent to applying the characteristic condi-

142

W. CHESTER

tion (3.19) at the shock. For the substitution of the shock transition relations, (3.28), (3.29) and (3.30) in (3.19) gives just

dA - 2MdM _ A - ( M 2 - l)K(M)

(3.34)

as before. The arguments already given explain why the rule holds for a small area change. That it must be true for an acoustic pulse is also made clear, for then the wave front is, in fact, a characteristic on which (3.19) must hold. In general, however, the strict interpretation of (3.19) is

If (3.19) is applied a t the shock, it is equivalent to assuming that, approximately, (3.36)

a@+-+ppa

-_

u at

ax

--+(hz

z)

+---pa2q

1 dA q + a A d x -0

a t the shock, where U is the shock velocity. Thus, by subtraction of (3.35) and (3.36) (3.37) must hold approximately at the shock. Now although the first factor in (3.37)is zero for an acoustic pulse, it tends to {VZy(y - 1) - ( y - l)}/(y 1) = 0.274 (for y = 1.4) as M m. Since this is too large to explain the close agreement between Chisnell’s and Guderley’s results for a converging shock, Whitham’s conclusion is that the second factor must be small at the shock. That this factor is zero is well known for a progressive wave on acoustic theory. It also follows, from (3.25) and (3.26) with F = 0, that the same is true for the flow behind a shock of any strength provided the area change is small. One can only deduce, however, that p, pug, differs from zero by a small quantity of the second order; for an arbitrary area change such small quantities may accumulate and become significant. To investigate this possibility, Whitham considered the problem of the converging shock using an expansion in the neighbourhood of that characteristic which meets the shock on the axis (of the cylindrical shock) or the centre (of the spherical shock). On this characteristic let the flow quantities be denoted by the suffix c, then the characteristic is given by z = 0, where

+

-.

+

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

143

(3.38)

(to be consistent with the previous equations, x should be measured in the direction of -propagation of the shock. The axis, or centre must then be - represented by, say, x = x,, - see Fig. 9). All the flow quantities are now expanded in powers of t with coefficients as functions of x , and the first approximation is obtained by putting t = 0. The lowest order terms then satisfy both the characteristic condition and the boundary conditions a t the shock. Chisnell’s rule is therefore obtained, together with an estimate of the individual flow quantities to this order of approximation. flow behind a converging cylinHigher order approximat~ons F I G . 9. The drical or spherical shock. can then be obtained by iteration. The success of the method depends on the smallness of T in the region between the limiting characteristic, where z = 0 , and the shock, where

(3.39)

In other words it depends on the smallness of the first factor of (3.37). In spite of this Whitham found the expected small correction to Chisnell’s result between the first and second approximations, though there were relatively much larger changes in the individual flow quantities.

IV. Comparison of the Two Theories The results of calculations for a 1 : 4 expansion, on the assumption that the initial and final conditions are connected by (3.4), are shown in Fig. 6. These calculations are not offered as an alternative to those obtained according to the steady-state theory, also shown in Fig. 6. However, in spite of the different situations envisaged in the two theories a direct compar-

144

W. CHESTER

ison of the two curves is interesting in that one can see immediately the ultimate modifying effect on the shock of the interactions behind it. It shows that the ratio of the final to the initial strength of the shock would increase monotonically with the incident Mach number if these interactions could be ignored. Hence the initial oscillation in the lower curve would seem to be due entirely to the effect of such interactions. The error in Chisnell’s theory (compared with the steady-state theory) is 25% for a weak shock and 5.7% in the limiting case of a strong shock. Chisnell’s result is least accurate in the region of M = 1.5, where the error is 48%. No detailed calculations have been made for a contraction using Chisnell’s theory. The variation of ($3 - @,)/(fi2 - @J with M is, however, much smoother for a contraction than for an expansion, and a reasonable comparison can be made from the limiting values of the pressure ratio for small and large Mach number. Table 4 shows the limiting values according to the two theories. TABLE4. THE RATIOOF

THE

INITIAL AND FINALSTRENGTHS OF THE SHOCK, THE Two THEORIES

OF (ps - pl)/(pz- pl). A COMPARISON

2 : 1 Contraction

5 : 1 Contraction

M

1

* (Y

5/3) (Y = 7/51 =

v.

Steady State

Chisnell

Steady State

Chisnell

1.333 1.200 1.246

1.414 1.367 1.314

1.067 1.492 1.374

2.230 2.000 1.886

STEADY

FLOWREGIME

AHEAD OF THE SHOCK

1. Ilztroduction Since Chisnell’s hypothesis seems so promising when there is no flow ahead of the shock, it seems reasonable to investigate the propagation of a shock wave along a duct in which a steady flow has been established, using similar ideas. There is not, of course, the same evidence for the reliability of the hypothesis when applied to this more general problem. In the absence of experimental confirmation, justification of the results must rest to some extent on the success of the special case considered by Chisnell. There is indeed some evidence that the re-reflected waves may be important under certain circumstances. Chisnell [15] has considered the propagation of a

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

145

shock wave through a quiescent gas in which the density (and hence the entropy) is non-uniform. A hypothesis similar to that described in Section 111 is used, that is to say all re-reflected waves are initially ignored. Their effect on the strength of the emergent shock for an incident shock Mach number of 1.35 is then assessed. I t is found t o be significant for a density ratio across the ends of the non-uniform region of the order of 80 : 1. On the other hand, for density ratios of 5 : 1 or less the result obtained by neglecting the re-reflected waves differs by less than 10% from the result obtained by reflection and transmission at a contact discontinuity with the same density ratio across it (this last calculation is analogous to the steady state theory of section I1 in that it takes into account all the modifying effects of the flow behind the shock). In the steady flow regime of the problem considered in this section there are also density variations, and it may be that when these are large the results are unreliable. Note, however, that the conditions in the problem here considered differ from those of Chisnell's problem, for there are no entropy variations in the steady flow regime. Furthermore the results must be valid, as before, in the limiting case of a weak shock, since the latter is still a characteristic. This includes the interesting case of a weak disturbance travelling upstream towards a sonic line.

2 . Basic Equations There is a minor change in notation. The suffixes 1 and 2 will henceforth refer respectively to the flow parameter immediately in front of, and immediately behind, the shock. The required equations are easily obtained from previous results. By Whitham's rule, the characteristic condition (3.19) is to be applied a t the shock. I t may be written in the following form:

The shock-transition relations yield the equations (5.2)

146

W. CHESTER

Here Ml and M , are the Mach numbers of the flow immediately in front of, and behind, the shock, while M now denotes the Mach number of the shock relative to the flow in front (relative to a fixed origin the Mach number of the shock is then M Ml). The above equations hold provided that the flow ahead of the shock is in the positive x-direction. For flow in the negative x-direction the change in sign of q, affects the sign of Ml. For the steady flow ahead of the shock the following relations also hold:

+

(5.5)

- 1/2 ,

y1=M1{1+-M Y -, ~1} @T 2

(5.7) where the suffix T refers to stagnation conditions, and A* is the value of A for M , = 1. By the substitution of (5.2) - (5.7) in (5.1) a first order differential relation may be obtained between M , and M (or between MI and any one of the quantities defining the strength Lf the shock). We consider in det,ail the two limiting cases of strong and weak shocks. 3. The Strong Shock

As M

+ M,

M , + { 2 / y ( y - 1)}1/2 and so, from (5.1), iM,

(5.8)

q2Mq211YA1+ X I =

constant.

With the help of (5.2) - (5.7), this result may be written

where

(5.10)

a=

2v

[ + {Y(Y ;I;") [ + {-3J] 1

2

'

147

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

The result is independent of the direction of flow ahead of the shock. In Fig. 10 is shown the variation of (9, - p,)/p, with the cross-sectional area, which is easily deduced from (5.7) and (5.9). The pressure ratio is normalized so that ( p , - p,)/$, = 1 when the flow speed in front of the shock is sonic. The values of a and ,Ll used are those corresponding to y = 1.4, namely /I= 0.893826.

a = 0.394140, 4

3

0 2

2 5

M,

04

2

0 6

IS

0 8

I0

A1IA

FIG. 10. The pressure ratio across weak and strong shocks as a function of the inverse of the cross-sectional area. The upper scale shows the Mach number of the steady flow ahead of the shock (MI). The pressure ratio IS normalized so that ( p e - p , ) / p , = 1 when M i = 1.

I t appears that the pressure ratio across the shock is a monotonic increasing function of the Mach number of the flow ahead of it. But perhaps a more realistic measure of the shock strength in this particular problem is the pressure difference, rather than the pressure ratio (for example, the destructive capacity of a shock depends on the pressure difference). In Fig. 11 the variation of (p, - p , ) / p , is shown, where p , is the stagnation pressure for the flow ahead of the shock. The variation of this quantity with M I is obtained from (5.5)and (5.9)which give

148

W. CHESTER

Fig. 11 shows that the pressure difference actually tends t o zero as --+ 00 as well as for M , + 0, and has a maximum when Ml = 0.840 corresponding to A * / A = 0.874. This is the place where the maximum effect of the shock will be felt.

M,

FIG. 11. The pressure difference across weak and strong shocks as a function of the inverse of the cross-sectional area. The upper scale shows the Mach number of the steady flow ahead of the shock ( M I ) . The pressure difference is normalized so that (Pa - p , ) / p = ~ 1 when M I = 1.

4. The Weak Shock As before the Chisnell hypothesis is substantiated by acoustic theoq, for the wave front is, in fact, a characteristic. The result, however, now depends on whether the shock is facing upstream or downstream. We consider first the downstream-facing shock. If M = 1 6 we have, from (5.2) - (5.6),

+

(5.13)

M,=M,+-

26 {2 - ( y - l)M1}, Y+1

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

149

correct to the first order. When these results, together with equation (5.7), are substituted in (5.1) we get

or, since (5.17) it follows from (5.16) that

The following relation for

( p , - pl)/P,

follows at once from (5.5)and (5.18)

The results are shown in Figs. 10 and 11 and are qualitatively similar to those for a strong shock. Thus (P, - f , ) / p , increases monotonically with Mach number (though the proportionate increase is smaller than in the case of the strong shock), whereas ( p , - P , ) / f i , has a maximum when M , = 0.402 corresponding t o A * / A = 0.631. The result for an upstream facing shock may be deduced from (5.18) and (5.19) by changing the sign of M , . Thus, for example,

(5.20)

{-)

P z - PI ( 1

-

M,)M,-l~z

Mlz)

= constant.

PT

In this case (p2 - P , ) / p , is monotonic increasing for M , < 1 and monotonic decreasing for M , > 1. But in the region of M , = 1 the approximation fails, as in so many other problems. Equation (5.1),however, remains valid, and it is only through the subsequent linearization that the approximation fails. I t is not difficult to refine the approximation in the neighbourhood of M I = 1.

150

W. CHESTER

+

+

If we put M = 1 6, M I = 1 E , and remember that the sign of M , should now be changed in (5.3) - (5.6), we get (5.21)

M2 = - (1

+

E

- 26} + O ( E 2 + @),

(5.22)

(5.23) (5.24) Substitution in (5.1), and neglect of all but the leading terms, gives (5.25)

SdE

+

(E

- 26)dS = 0,

or (5.26)

d{6(6 - E ) }

=0

which integrates to (5.27)

6(6 - E )

= constant.

The constant is zero if initially S = 0 (no disturbance), or 6 = E when the relative Mach number of the shock is equal to the Mach number of the flow ahead, and hence the shock remains stationary. If, however, 6 >: E initially then by (5.27) it remains so and the shock travels upstream. In a subsonic-supersonic nozzle this means that E decreases and so the shock strength decreases as is shown b y Fig. 12, where the variation of 6 with E when 6(6 - E ) = 1 is shown. Conversely 6 is always less than E if this is so initially, and so the shock is convected downstream. This, of course, is only possible in the supersonic section where E > 0, for 6 cannot be negative. For a subsonic-supersonic nozzle, Fig. 12 shows that, in this case, 6 can either increase or decrease according as 26 2 E . This is an interesting example of a situation in which stability depends on the magnitude of the initial disturbance. For a converging-diverging nozzle with supersonic flow in the upstream section, a disturbance for which 6 > E will always move upstream and ultimately increase in strength, evidently causing a breakdown of the flow (it may initially decrease in strength depending on whether or not it starts in supersonic flow downstream of the throat). On the other hand if 6 i:E the disturbance must move downstream. A solution of this type may be possible if the downstream flow is also supersonic; if it is possible then

THE PROPAGATION OF SHOCK WAVES ALONG DUCTS

151

ultimately 6 increases or decreases according as 26 28 as before. But in certain circumstances the solution itself breaks down. Suppose the disturbance begins in the upstream section and that 6 < E . Then the shock moves downstream and E decreases. Fig. 12 shows, however, that if a stage is reached for which E = 26, then E cannot decrease further. In particular, when conditions a t the throat are sonic, the solution must always break down. Presumably a more sophisticated model is required here, though it is interesting to compare these conclusions with those of experiment.

FIG. 12. The Mach number of a weak upstream-facing shock as a function of the Mach number of the steady flow ahead of i t ; Mach number of shock = 1 + kb, Mach number of steady flow = 1 + kc, where k is a normalizing factor.

In supersonic tunnels with two throats, the flow through the second throat is supersonic-supersonic after starting, with a stationary shock downstream. When this regime is established the area of the second throat is reduced. This moves the stationary shock upstream and, since its strength also decreases, improves the pressure recovery. Ideally the area is reduced until the shock has moved to the throat; then its strength is zero and the diffuser is isentropic. Experimentally it is found that the flow ‘breaks down’ before this stage is reached. The breakdown is usually ascribed to boundary layer effects which produce choking. The present analysis suggests that there may be other contributory mechanisms. 5. The Generalization of the Steady State Theory

No detailed study has been made of the generalization of the steady state theory which appears in section 1. The calculations will be correspondingly more complex, though in principle there is no difficulty. The only simple result which the author has been able to obtain is the analogue of equation (1.3) for the change in strength of a weak shock after passing through an area change in which there is initially a steady flow. The result is

162

W. CHESTER

+

where 1 at, 1 + 6, are respectively the Mach numbers of the incident and transmitted shocks relative t o the flow ahead, and Mu,Md are respectively the Mach numbers of the steady flow upstream and downstream of the area change. This formula should also be compared with (5.18). As before there is agreement between the two theories only when the area change is small.

References 1. LIGHTHILL, M. J., The diffraction of Blast, Part I, Proc. Roy. Soc. 198, 455 (1949). 2. CHESTER,W., The propagation of shock waves in a channel of non-uniform width, Quart. Journ. Mech. A p p . Math. 6, 440 (1953). N. C., On the stability of plane shock waves, J . Fl. Mechs. 2, 397 (1957). 3. FREEMAN, 4. LORDRAYLEIGH,Theory of Sound, 2nd ed. London (1896). 5. CHISNELL,R. F., The motion of a shock wave in a channel, with applications t o cylindrical and spherical shock waves, J . FZ. Mech. 2, 286 (1957). 6. WHITHAM,G. B., A new approach to problems of shock dynamics Pt. I - Twodimensional problems, J. Fl. Mech. 2, 145 (1957). 7. WHITHAM,G. B., On the propagation of shock waves through regions of nonuniform area or flow, J . FZ. Mech. 4, 337 (1958). 8. KAHANE,A., WARREN. W. R.. GRIFFITH,W. C., and MARINO,A. A., A theoretical and experimental study of finite amplitude wave interactions with channels of varying area, J . Aeronaut. Sci. 21, 505 (1954). 9. LAPORTE,0.. O n the interaction of a shock with a constriction, University of California, Los Alamos Scientific Lab. Report LA-1 740. 10. PARKS,E. K., Supersonic flow in a shock tube of divergent cross-section, University of Toronto U.T.I.A. Report No. 18 (1952). 11. GUDERLEY, G., Starke kugelige und zylindrische VerdichtungsstoBe in der Nahe des Kugelmittelpunktes bzw. der Zylinderachse, Luftfahrtforsch. 19, 302 (1942). 12. BUTLER,D. S., Converging spherical and cylindrical shocks, Ministry of Supply A . R . D . E . Report No. 54 (1954). 13. CHESTER,W., The quasi-cylindrical shock tube, Phil. Mag. 46, 1293 (1954). 14. PAYNE,R. B., A numerical method for a converging cylindrical shock, J . FZ. Mech. 2, 185 (1957). 15. CHISNELL. R. F., The normal motion of a shock wave through a non-uniform onedimensional medium, Proc. Roy. Soc. 282, 350 (1955). 16. FREEMAN, N. C., A theory of the stability of plane shock waves, Proc. Roy. Soc. 228, 341 (1955). 17. GUDERLEY,G., Non-stationary gas flow in thin pipes of variable cross-section, Natl. Advisory Comm. Aeronaut., Tech. Mem. No. 1196 (1948). 18. SAUER,R., Theory of non-stationary gas flow I11 - Laminar flow in tubes of variable cross-section, A r m y Air Force Translation No. F-TS-770-RE (1946). 19. WARREN,W. R., Interaction of plane waves of finite amplitude with channels of varying cross-section, Princeton University Aero. Eng. Lab. Report No. 206 (1952).

Similarity and Equivalence in Compressible Flow BY KLAUS OSWATITSCH Dezctsche Versuchsanstalt fur Luftfahrt. Aachen. Germany Page

.

I Basic Considerations . . . . . . . . . . . . . . . . . . . . . . . . 154 1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 154 2 . Differential Equations . . . . . . . . . . . . . . . . . . . . . . 155 3 . Approximations for the Speed. the Direction of the Velocity. and the Pressure Coefficient . . . . . . . . . . . . . . . . . . . . . . . 159 4 . Shock Equations for Small Disturbances . . . . . . . . . . . . . 162 5 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . 163 6 . Simplification of the Boundary Conditions for Thin Profiles a n d Wings 164 7 . Simplification of the Boundary Conditions for Bodies of Revolution . 168 8 . Linearization of the Gas-dynamic Equation: Corresponding Points . . 172 9 . Transformation of the Velocity Components . . . . . . . . . . . . 176

.

I1. Applications of the Linear Theory . . . . . . . . . . . . . . . . . . 178 10. The Prandtl-Glauert Analogy . . . . . . . . . . . . . . . . . . 178 11. The Effect of Cpmpressibility for Bodies of Revolution a t Zero Incidence 183 12. Application of the Prandtl Rule: Limits of the Domain of Linearization 189 13. Mach-number Dependence of the Aerodynamic Forces Acting on a Wing 195 I11. Higher Approximations . . . . . . . . . . . . . . . . . . . . . . . 198 14. Higher Approximations for the Gas-dynamic Relations . . . . . . . 198 15. The Shock Equation in Non-parametric Representation . . . . . . 206 16. Reduction of the Differential Equations . . . . . . . . . . . . . 211 IV . Transonic Similarity . . . . . . . . . . . . . . . . . . . . . . 17 . Similarity Laws for Profiles and Wings in Transonic Flow . . . 18 . Transonic Flow past Profiles and Wings at Non-zero Incidence 19. Bodies of Revolution in Transonic Flow . . . . . . . . . . 20. Transonic Flow past a Circular Cone . . . . . . . . . . . . V . Hypersonic Similarity . . . . . . . . . . . 21 . Similarity Laws in Hypersonic Flow . 22 . Hypersonic Flow at Non-zero Incidence

. . . . . . . . . .

. . . .

215 215 228 231 234

. . . . . . . . . . . . . 236 . . . . . . . . . . . . . . 236 . . . . . . . . . . . . . 240

VI . Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . .

242 242

VII . Bodies of Low Aspect Ratio . . . . . . . . . . . . . . . . . . . . 248 24. Bodies of Low Aspect Ratio at Non-zero Incidence . . . . . . . . 248 25. Bodies of Low Aspect Ratio a t Zero Incidence: Law of Equivalence . 253 26. Mach-number Dependence of Wings with Low Aspect Ratio . . . . 259 264 27 . Area Rule and Similarity . . . . . . . . . . . . . . . . . . . . References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

269

164

KLAUS OSWATITSCH

I. BASICCONSIDERATIONS 1. Introduction

The theory of mechanical similarity was almost fully developed when the aerodynamic theory of compressible flow began to take shape. The classical theory established the conditions under which experimental results obtained with small-scale models would allow to predict the behavior of fullsize bodies, the only relation admitted between body and model being complete geometric similarity. For the steady inviscid flow of an ideal gas it followed from the governing system of differential equations that the Mach number and the ratio of the specific heats had to be the same at a certain pair of points in the flows to be compared. The additional requirement for viscous flows was equal Reynolds number, and for oscillatory motions equal ,,reduced frequency”. If the considerations are restricted to thin wings and slender bodies the classical theory of similarity can be extended so that the flows past bodies of different thickness and at different Mach numbers may be compared. The first similarity law of this kind was formulated in 1922 by Prandtl in his lectures on theoretical aerodynamics ; it was independently discovered and published by Glauert in 1927. This law, generalized in the following decades, is now known under the name of Prandtl-Rule or Prandtl-Glauert-analogy. I n addition, similarity rules have been developed for transonic and hypersonic flows. All these similarity rules are based on affine relations between body and model. In general, the bodies to be compared are no longer geometrically similar, but their dimensions are affinely distorted. The disturbances of the velocity components differ not only by one factor from the real disturbances, but in general each velocity component is multiplied by a different factor. Comparable flows are mapped into each other in the hodograph plane by such an affine transformation. An example for the application of these similarity laws is the flow over a flat plate at some angle of attack and Mach number which is compared with the flow around the same flat pla.te a t a different angle of attack and a different Mach number. All affine rules mentioned so far apply in steady inviscid flows. For wing-body combinations of low aspect ratio, e.g. for delta and swept-back wings, the restriction to affinely distorted bodies is no longer necessary. A comparison of the flows is here possible if only the bodies have cross sections of the same area distribution. Such bodies are called equivalent bodies. The rules that allow a comparison of the flows around these bodies are called rules of equivalence. Unsteady flows occur only in Section 23, where similarity laws for certain limiting cases of flutter are considered. Viscous flows are not treated a t all. Inclusion of viscous effects would have gone beyond the

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

155

scope of this article. Here it will only be mentioned that in similarity considerations of compressible viscous flows the Reynolds number must be changed such that the displacement thickness varies as the thickness ratio or the angle of attack, see e.g. [57] and [58]. This implies certain restrictions for the application of the analogies to thin bodies a t very high Mach numbers. Some of the recent results reviewed here include the analogies for transonic and hypersonic flows at very small incidence and for unsteady aerodynamic forces. I wish to express my sincere appreciation to the managing editor of this series, Prof. G. Kuerti, for reading the whole manuscript and to my assistant M. Fiebig for the translation of the article and assistance with the figures.

2. Differential Eqzlations The basis for the developement of the following analogies and equivalence theorems is the simplification of the differential equations, shock relations, and boundary conditions for flows around slender bodies. The expression slender here means that all surface elements have only a slight inclination with respect to the free stream direction. This causes only small disturbances in direction and magnitude of the free stream conditions. What may still be called a slight inclination of the surface elements or a small disturbance depends somewhat on the body and the speed regime. For instance, analogies still hold for bodies of revolution with quite large inclinations of the surface elements, and quite large velocity disturbances are allowable at transonic speed. For flows around slender bodies at hypersonic speed the velocity is nearly undisturbed, but there occur considerable changes in pressure. Thus it is not possible to speak of small disturbances in general. I t is important to know, whether it is a disturbance of the velocity, the mass flow density, the flow direction, or of a function of state. The continuity equation for inviscid steady flow can be written in the form of the gas-dynamic equation:

(2.1) Here, U , V , W are the components of the velocity vector of magnitude q, c is the velocity of sound, and X , Y , Z are Cartesian coordinates. As shown in Fig. 1, X will always be taken as the body axis, Y the coordinate in the direction of the thickness of the body, and Z the coordinate in the direction of the span,

166

KLAUS OSWATITSCH

The component of the velocity in the spanwise direction, W , is certainly always small in comparison with the velocity of sound. Therefore the last two terms of (2.1) may be neglected, because they are multiplied by the factor W << c. By the same reasoning we neglect W2/c2in comparison to 1 in the third term of (2.1),but we must retain aW/aZ. Even if W itsell' is small the change of W in the spanwise direction may be quite large. This term is of special importance for wings of small aspect ratio. With the above simplifications (2.1) reduces to

&au + ayav + aw US

(2.2)

(1 -

=

&

v2av uv av 2aY + +

g).

The terms on the right-hand side have the factors V2/c2or V/c. .At low subsonic velocities V is always small compared with c. So the righthand side of (2.2) may be neglected even for bodies with blunt front portions. With the free stream Mach number increasing towards and beyond unity one has to require sharper leading edges or pointed bodies so that V << c is still true. I/' and c become of the same order of magnitude when the apex semi angle of the body or wing becomes cornFIG. 1. Coordinate system and parable with the Mach angle of the velocity components. free stream. Quite instructive for this case is the exact solution for the flow around a wedge. For the limiting case M , do (where M , is the free stream Mach number), the Mach angle a t the wedge is only about half the wedge semi angle (Fig. 2). This implies that the V component of the velocity is twice the value of the local velocity of sound. The domain of Mach number Y

-

(2.3)

1 ~

M

= sin a, -tan

6

W

(where a is the Mach angle, 6 the flow angle, and 00 denotes the free stream condition) is called hypersonic; the limiting case 1 ---tan6=--

Ma

sin a, -0 tan 9

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

157

is called the hypersonic limit. The two terms on the right-hand side of (2.2) will therefore be called the hypersonic terms of the gas-dynamic equation. They are no longer negligible at hypersonic speed. On the other hand, they are not so large that the first or second term of the left-hand side of (2.2) could be neglected.

FIG.2. Flow over a wedge for M ,

6

=

+ m, 8 = wedge semi angle, u = shock angle, Mach angle behind the shock.

Thus, excepting the hypersonic regime, the gas-dynamic equation may be simplified to read for slender bodies (1 -

s)g+ ayav + aw

= 0.

In addition, Crocco's vortex theorem

q x curl q = - T grad s

(2.6)

has to be satisfied by the components of the velocity, where q is the velocity vector, T the absolute temperature and s the entropy per unit mass. Since friction and heat transfer have been neglected entropy changes can be caused only by shock waves. If the free stream is irrotational then at subsonic speeds the flow stays always irrotational, hence

aw

av

-0;

au az

aw ax

--__

-0;

au = 0. -- -

av ax

ay

168

KLAUS OSWATITSCH

For sonic flows with shock waves and sufficiently thin profiles, G. Guderley 1241 has shown that even then the flow may still be called irrotational. This result is in accordance with two dimensional supersonic first and second order theory (Ackeret theory and Busemann theory). Corresponding results exist for axisymmetric flows. Thus the equation of irrotationality (2.7) may be applied without hesitation to any three-dimensional flow at moda1 erate supersonic velocity. At hypersonic velocities, however, this is no longer valid. To show this, we restrict ourselves to the simple case of the flow around a profile a t M, + 00 (hypersonic limit, Fig. 3). According to equation (2.6) vorticity is produced by any entropy gradient. For a 0.1 4 straight wedge-like leading edge we FIG. 3. Flow over the front part of get a straight shock front. But the a curved profile for M , m. entropy behind a straight shock is constant, hence the flow is irrotational in the wake of the shock. However, for a profile with the curvature K p the shock front has also a curvature, K,. The relations between these two quantities are known [l, Ch. 8, Eq. 691; for the hypersonic limit they are

‘I

-

---t

If y = 1.40 is taken there results a shock curvature from (2.8) which is 80% of the profile curvature. Because the entropy is constant in the stream direction the entropy gradient can be deduced from the entropy gradient along the shockfront. A t the hypersonic limit the density is constant behind the shock, therefore the entropy change is proportional to the pressure change. From this the following relation between entropy gradient and shock curvature follows for M , bo ( M = local Mach number) : --f

1

-lgradsl= CP

__

Y+l

M2KS.

Crocco’s vortex theorem reads in two-dimensional flow, for the Y-direction, (2.10)

1 iias _i a_v- _ i-a-u u ax u ay- 3 W G Z ’

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

159

This equation is invariant in a rotation of the system of reference. Now, if X is taken along the direction of the profile surface and the curvatures are defined as positive it follows that (2.11) On the other hand, with (2.8) and (2.9), the right-hand side of (2.10) may be written as

The absolute magnitude of (2.12) is no longer small in comparison with (2.11) as it would have to be in nearly irrotational flow. For y = 1.40 the ratio of (2.11) to (2.12) is 3 : 10. Thus at hypersonic speed the curvature of the leading edge of the profile leads to strong vortices. Therefore the right-hand side of (2.6), as earlier the right-hand side of (2.2), cannot be neglected for flows around slender bodies. These terms are characteristic for hypersonic flows. Summarizing the results of the last section, one may say that the differential equations can be simplified strongly for slender bodies at speeds below the hypersonic flow regime. The error arising from these simplifications will be estimated later. Neglecting the last term of the right-hand side of (2.2) results in an error which vanishes as the thickness ratio of the body under consideration. This has been shown in [28] in a way similar to Guderley's proof of irrotationality. 3. Approximations for the Speed, the Direction of the Velocity, and the Pressure Coefficient . Let us assume that the flow differs only slightly from a parallel flow with the components of velocity (U,,O,O). Then it is easy to express the disturbance of the flow speed by the disturbances of the velocity components: let

then

160

KLAUS OSWATITSCH

TABLE1 TYPICAL VALUES

i

Flow

OF

u =

subsonic, at maximum thickness; profile, spindle 4 _.

two-dimensional

R

U/U, -

1

supersonic; wedge, cone

t

v m 2

- tans 8,In

axisymmetric

v s )

(tan 8,

TABLE2 t v1

- M,z

-

tan 6, VMm2- 1 : 0.20

0.10

0.05

0.02

In [2/tan@,.VMWa - 13 : 2.30

3.00

3.70

4.60

21n [ 2 / t -V l - M m a ] - 3 : 1.60

3.00

4.40

8.20

=

The neglected terms are at least of fourth order in 21, v , and w . For an estimate of the magnitude of the different terms, typical maximum values of the disturbance 21 are given in Table 1. The values obtained by the linear theory for a profile and a spindle in subsonic flow [l, p. 301 and 3131 are compared with the values for a wedge and a cone in supersonic flow [l, p. 346 and 3661. The thickness ratio t in the formulas for subsonic flow corresponds to the tangent of the apex semi-angle 19,of the wedge and cone in supersonic flow. Both quantities t and 6, are closely connected with v . For the wedge and cone v = tan 8, except for a small error which will be considered later. For the two-dimensional or axisymmetric spindle v = r for x equalling - 11 is of order one, as long as we do not one fourth of the chord. 1/(MW2 consider the transonic or hypersonic speed regime for which the linear theories do not hold anyway. Therefore, for considerations of order of magnitude we take only the two standard Mach numbers M , = 0 and M , = VT Then we get the simplification IMmZ- 11 = 1. Table 1 shows that for two-dimensional sub- and supersonic flows 26 is of the same order of magnitude as t or tan6, and, therefore, of the same order of magnitude as v . The three terms of the right-hand side of (3.2) have the order of magnitude v , v 2 , and v3. For axisymmetric flows the dominant quantity is the factor t 2or tan2 6,. The other factor is of the order unity, even though it contains the logarithm of a large number. Table 2 gives some examples for this statement.

161

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

- 11 corresponds to a spindle The value 0.02 of t or tan 6, times of 5% thickness a t M , = 0.92 or a cone of 3" apex semi-angle a t M , = 1.08. This shows that the logarithmic term becomes large only for extremely slender bodies at nearly transonic speeds. For all practical purposes something between 2v2 and 4v2 is a good estimate for the possible values of u for bodies of revolution. Hence ;1(u2 w 2 )may not be neglected in comparison to u ; &(v2 w2) may, however, always be neglected. Another way of expressing the u disturbance is to say that, except for the transonic and the hypersonic regime, u is proportional to the crosssection in two-dimensional and axisymmetric flow. The following approximative relation holds between the flow direction 6 and the v-velocity for two-dimensional and axisymmetric flows :

+

+

u + d -

(3.3)

..

For tan 6 often v is substituted in boundary conditions. In two-dimensional flow within the domain of linearized theory, this substitution is of the same accuracy as the approximation of the speed disturbance by u ; for axisymmetric flows it is as accurate as the approximation of the speed $(u2 u2). disturbance by t4 The expansion of the pressure coefficient cp in terms of the speed disturbance is given b y the well-known formula e.g. [ l , p. 471

+

+

c p = -P- - Pa2 - - 2

(3.4)

3P m Y w

tW1 --1

-(l-M,Z)

4 - 1

((Iu

Y +-...,

where h , denotes the pressure and p the density. Without considering the assumptions made in the expansion (3.4), it can be seen that (3.4) becomes invalid somewhere in the hypersonic regime. To get an estimate where the limit of validity of (3.4) lies, the linearized flow over a wedge is considered. The u-disturbance follows from Table 1, with

(3.5)

cota=

V M ~ -1

where a is the Mach angle. The pressure coefficient becomes c, =

-

2 tan 6, +tan2@,+ V M m 2- 1

... r

1 tan6,

+...I.

+ 2 E

I t is evident from (3.6) that the convergence of the series becomes doubtful when the free-stream Mach angle, a,, becomes equal to the apex semi-angle 8, of the wedge. In the hypersonic regime therefore (3.4) may not be used without considerable caution.

162

KLAUS OSWATITSCH

4. Shock Equations for Small Disturbalzces

Quantities immediately in front of an oblique shock (shock angle 0) will be left unmarked, and quantities immediately behind the shock will be marked by a circumflex. Then the following relation (e.g. [l, p. 366, Eq. (38)]) holds:

Because of its simple form, (4.1) will be used also for small disturbances. We get another shock relation, which is valid for small disturbances only, if we use the fact that the normal component of the mass-flow density (subscript a) is not changed by the shock: pqn = P4n.

(4.2)

Because p, (4.2a)

=q

sin cr and

gn = 4 sin (a - 6) it follows that 8

PQ

= cos6 - cot asin 6 = 1 - cot a s i n 6 - 2sin2 - .

2

By (4.1), the last term in (4.2 a) is always smaller by $(u- Zi) than the term before. Therefore the relation (4.3)

cotrr=-;

1 0

(

1--

I; +

...

holds for small values of u - 12. From (4.1) and (4.3) the shock polar for small disturbances is obtained: (4.4)

62=(U--zi)

( 2)f.... 1--,

For transonic speeds this relation has been used before [ l , p. 4581. general case this relation seems to be new. It is not assumed in (4.4) disturbance of the mass-flow density (= second factor) is small. contrary, we shall find that in hypersonic flow that disturbance large. Otherwise the gas could not flow through the small region the shock and body (Fig. 2). For the general case it is important is in the denominator. Even for small u-disturbances

For the that the On the is quite between that 64

The important shock equations are (4.1) and (4.4). As in the case of the differential-equations, the error implied by these relations will be estimated later.

163

SIMILARITY AND EQUIVALENCE I N COMPRESSIBLE FLOW

5. Boundary and Initial Conditions

The conditions far upstream are given b y the following relation for subsonic flows :

v X 2 + Y 2 + Z 2 + ~ :

M,<1;

U=U,;

V=V,;

W=O.

(6.1) Lifting problems are included by giving an inclination to the free stream ( V , # 0). Bodies in yaw will not be considered ( W , = 0). Sometimes it is of advantage to assume a u-disturbance in the free stream. Then the subscript m does not mean the conditions far upstream but different flow conditions. In general, however, U , is the velocity far upstream, which implies u, = 0. In linearized supersonic flow the initial condition is taken along the characteristic surfaces originating at the leading edge of the body. The following condition at 2 = 0 holds for the two-dimensional and axisymmetric case when the leading edge (or point) of the body passes through (or coincides with) the origin of the coordinate system:

(5.2)

M,

> 1;

v X 2 - ( M W2 1)Y2= 0 :

u =0;

ZI = 0.

Here we can disregard the possibility of a disturbance in the free stream. If, however, head waves of finite strength exist, as it is the case a t small supersonic (transonic) and hypersonic speed, the boundary and initial conditions have to be satisfied immediately behind the head wave. Usually the position of the head wave is unknown because of the influence of the body shape. This is not only true for head waves; it holds in general for all shock fronts in the flow. Mathematically every shock represents a boundary value problem where the boundary values of the flow are coupled on both sides of the shock front, and the shock relations have necessarily to be satisfied. Therefore the similarity considerations have t o include (4.l ) , which enables us to calculate the position of the shock, and (4.4), which couples the flow conditions before and behind the shock front. The boundary conditions a t the body are the same for all speed regimes. If the body contour is given in the upper half plane by Y = H(X,Z),

(5.3)

the boundary conditions takes on the following form: (5.4)

Y

= H(X,Z):

HKU - V

+ HZW = 0.

(The scalar product of the velocity and the vector (Hx, - 1, Hz), i.e. the surface normal of the body (5.3),vanishes on the body surface.)

162

KLAUS OSWATITSCH

For bodies asymmetric in Y , an analogous condition holds for the lower half plane, which would however not demand new similarity conditions. Eq. (5.4) for W = 0 includes (3.3). If we neglect u with respect to 1, (5.4) may be written in the form

Y =H(X,Z):

v

Hx

1

+ WHZ.

For wings with large aspect ratio, W as well as Hz are small. The last term of (5.5) may then be neglected, but this is not always valid for low aspect ratio. 6 . Simplifications of the Boundary Conditions for Thin Profiles and Wings.

The w-component of the velocity of a delta wing with low aspect ratio is not difficult to determine for subsonic speed or supersonic speed such that the wing lies well within the forward Mach cone (e.g. [IS]).The w-disturbance is proportional to the thickness ratio t. This still holds for the body of revolution, which is the limiting case of bodies of low aspect ratio. Physically it is evident that the order of magnitudeof the ratio of Hxto Hzis proportional to the ratio of the half span, s, to the chord, L. From this it follows that

wHz-(.~/s)Hx,

(6.1)

where L has been taken as unity. Thus, to justify the omission of wHz in ( 5 4 , z/s should be very small compared with one. This is really the case as may be seen from Fig. 4 for a delta wing with the thickness distribution (6.2)

0

<X < 1 ;

0

H = 2 t ( l - X ) ( X -Z / S ) ,

where H x / t (left in Fig. 4) and w s H z / t 2 (right in Fig. 4) are of the order of magnitude one. The values of H x / t are larger than one for a large region of the wing; the values of wsHz/z2 however are larger than one only for a narrow region near the leading edge and the apex of the wing. In this region wHz is negligible compared with Hx only for very small values of z/s. For sufficiently low thickness ratio the last term of (5.5) can be neglected even for wings of small aspect ratio and much more so for wings of medium and large aspect ratio. This result holds also in the transonic region. Moreover, the v-component of the velocity will not be prescribed on the body surface but on the projection of the body surface. The simplified boundary condition is then (6.3)

Y

=0 :

21 = V(X,O,Z)= H x ( X , Z ) .

(The word linearization is often used for the reduction of the boundary conditions (5.4) to the form of (6.3). This is not quite correct, since the boundary condition (5.4) is already linear in the components of the velocity.

165

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

Furthermore, this simplification of the boundary condition has nothing to do with the linearization of the gas-dynamic equation. It is not possible for thick bodies even in incompressible flow and serves best in the non-linear transonic region, but leads to considerable errors at higher supersonic Mach

--

FIG.4. Comparison of H x / t (left) and w s H z / a Z(right) for a delta wing of the thickness distribution 0 < X < 1; 0 < Z S sX : H = 2a(l - X ) ( X - Zjs).

(6.4) v ( X , H , Z )= v ( X , O , Z ) 0 +

($)x,o,z*H+

. . . I

to be cut off after the second term. The ratio of the two terms is a

_.

-5 -

-10 -

a5

\

,s&

a

x

166

KLAUS OSWATITSCH

For the simplified theories of bodies with low aspect ratio it is important that the first term of the gas-dynamic equation (2.5) is small with respect to the other two terms. This is illustrated in Figs. 5 and 6. Fig. 5 shows aU/aX for Y = 2 = 0 for the halfspan s = 1/9; 1/6; 1/3 a t the Mach numbers 0 and Here the coordinates are already the reduced quantities (8.9), (9.1) and (17.7) For M , = 0 they can be replaced by their non-reduced counterparts. The necessary formulas can be deduced from areport by F.Keune [18]. The value for ( a U / a X ) / s tis obtained by measuring the curve of the corresponding Mach number M , = 0 or VTfrom the value corresponding to the specific halfspan s. For example, for M , = s = 1/3 there results for this simple wing the constant value ( a U / a X ) / s t= 8.2 - 2.6 = 5.6.

1/%

2 10

P8

0.1

0.4

91

0

- 0

FIG.6.

(12

011

Q6

M

1 D X

0

0.2

p1

06

0.8

10

x

( s / t ) aW/aZ (left) and ( s / t a ) Ha V / a Y (right) for the delta wing of Fig. 4;

Y=Z=O.

U and aU/aX are essentially proportional to the product st. aW/aZ however is of the order of magnitude of t / s (left side of Fig. 6) and is on the average by 1 / 3 9 larger than aU/aX. So we may indeed neglect the first term of the gas-dynamic equation close to the surface of bodies of small span. The differential equation for flows around such bodies reads

By means of (3.1) and (6.5) the error of the v-component due to the transfer of the boundary condition is seen to be the second term on the right side

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

167

of (6.4). I t can be calculated as the derivative of the w-component in the plane Y = 0 : (6.6)

Hav/aY

= -H a q a z

Fig. 6 gives on the left the derivative aw/aZ,on the right the error (6.6). The comparison with H , in Fig. 4 shows then that the second term of the expansion (6.4) is on the average smaller than the first term by a factor 3 s / t . So the error due the transfer of the boundary condition depends on the ratio of thickness to span. Typical for slender-body theory is that the results are independent of Mach number. They are valid not only for M , = 0 and but also for all Mach numbers in between and sometimes even beyond M , = v2 as long as the wing lies well within the forward Mach cone. At higher supersonic Mach numbers the omission of the first term of (2.5) is no longer permissible. For any body under 4 Y 2.Q consideration aU/aX is still decreasing as we go to higher Mach numto' bers but the increase of the factor (U2/c2- 1) is much stronger. In addition we have seen that (2.5) is not valid a t hypersonic speeds. It has to be replaced by (2.2). For two-dimensional flow the -2a. estimate of H a V / a Y follows from (2.5) and (3.1): FIG. 7 . [ ~ / t z V i- ~ , a ] a v / afor ~ a

vz

parabolic arc in subsonic and supersonic flow.

HavIaY is easy to calculate for the domain where the gas-dynamic equation may be linearized. This domain will be estimated in the next chapter. In the case of a parabolic profile the results are given in Fig. 7 for subsonic and supersonic flow. v is proportional to t times a factor which lies between f 2 . &/ax is proportional to t times 1/v11 - Mm2I. This we had seen already from Table 1 . By (6.7), &/ax has to be multiplied by 1(1 - Mm2)I and by H which is proportional to t ; so the second term of the expansion (6.4) is proportional to t2v11- M a 2 / . The error produced by the transfer of the boundary conditions from the body surface to the body axis is for incompressible flow of the same order of magnitude as the substitution of the tangent of the flow angle by v , see (3.3). At transonic speeds the approximation is best because of the low variability of the mass-flow density.

168

KLAUS OSWATITSCH

At high supersonic Mach numbers this approximation is no longer permissible. At hypersonic flows the boundary condition has to be taken a t the body surface. Physically this is plausible because the disturbances take place only in the small region between head shock and the body surface (Fig. 2). The disturbances may change therefore quite rapidly in the Y-direction. In supersonic steady flow all disturbances propagate along Mach lines. To get a certain inclination of a surface element this surface element would have to be projected upstream along the Mach line to the axis. It would be wrong to project the surface element in the Y-direction. Despite the good qualities of the boundary condition (6.3) a t transonic speed the accuracy of the approximations is not even there sufficient for all cases. Higher practical standards can make it necessary to distinguish between the v-component of velocity at the body surface and the disturbance on the projection surface. This may be seen as well from the simplified transonic equation (2.6) as from Fig. 7. This last result of the linear theory shows that the second term of the expansion (6.4) becomes small only with 1111 - M a 2 / . It is not completely negligible at the limits of the region of linearization, as they will be found later in Figs. 18 and 19. In simplifying the boundary conditions for two-dimensional flow, the main error arises from putting the flow direction equal to the v-component of the velocity. This corresponds to using the first term of the expansion (3.3) only. The assumption of small disturbances - which is postulated throughout this article - holds for the u-components of the velocity a t transonic speeds only for very thin profiles or wings of medium and small span. F. Keune [20] gives a thorough investigation of the errors arising from the simplification of the boundary conditions especially in the domain of linearization of the gas-dynamic equation. 7 . Simplification of the Bozcndary Conditions for Bodies of Revolution In this section it will always be assumed that the flow far upstream is parallel to the axis of the body of revolution. The omission of the first term of (2.5) is then even more justified and for the same basic reason as in the case of wings with low aspect ratio. If we limit our considerations to the plane 2 = 0 the last term in (2.5), aW/aZ, may be replaced by

The error which arises from the omission of the first term of (2.5) can be expressed by the magnitude of the ratio of the neglected term to the expression (7.1). For the incompressible flow over a parabolic spindle there results at the point where u = 0

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

169

a multiple of the maximum u-disturbance in Table 1. At the point where ) u = 0, Z J / Y = 2v3. So the ratio ( a u / a X ) / ( v / Ybecomes (7.3) which for t = 0.16 becomes 0.08. For wings with low aspect ratio we know from experience that the error considered becomes even smaller at transonic speeds and is still negligible a t low supersonic speeds. With (6.5) and (7.1), it follows for the region near the axis that av -+ aY

(7.4)

-=0

or

Y

vY=f(x)

where the value of V Y on the axis can be set equal to the value on the body. If in addition the flow direction is sct equal to v we get the boundary condition dH 1 vY=H-=--Q' ax. - 2n

Y=O:

(7.5)

with

Q = H2z.

The last approximation is always appropriate for slender bodies of revolution because the u-disturbance is very small. Relation (7.5) replaces (5.5) for slender bodies of revolution and not too high supersonic Mach numbers. The flow conditions for M , > 1 can be studied well in the example of the circular cone. If the exact boundary condition a t the body, (5.5), is taken the following equation results from the linearized gas-dynamic equation (8.2):

-.-U 1

+

1 2~

tan2 6,

-~

-

~

- 1.

_

_

~~~~~

r l - tan2G0(M,2 - 1)

1 _ In

+ Vl - tan2 6 , ( ~ , 2 -1) _

tan ?90VMm2- 1

(7.6)

which relates the u-disturbance, the apex semi-angle and the free stream Mach number t o each other. The graph of relation (7.6) is drawn in Fig. 8 over the abscissa tan 6,1/Mm2- 1. With a denoting the Mach angle,

170

KLAUS OSWATITSCH

the abscissa is the ratio of the inclination of the cone surface to the inclination of the Mach lines. If this ratio becomes equal to one, i.e.

-.1+u U

1

tans

6,

-4

-3

\

---

x

8, = 10"

0

6,=

- In

exact 5 O

2 t a n @ vM%

- In 1 + V1-

-1 2

tanz@(M, - 1) 1- I

tan

- 1

-2

8

Asymptote

-I

I

VML -

tan 8,,

1

FIG. 8. Comparison of the flow around a circular cone (a) by the linear theory with the exact boundary condition (solid curve) and with the simplified boundary condition for Y = 0: Yu = (1 u) tanP6,, (dot-and-dash curve) (b) by exact nonlinear theory taken from the MIT tables for semi apex-angle 8,, == 5" and 10' (dots and crosses). The dashed curve is the approximation for tan2 6,cot2urn 1.

+

<<

(7.6)gives a limiting value which is about 10% lower than the limiting value of the exact non-linear theory for Mm -+ 00. The points and crosses correspond to the values of the ordinate for half apex-angles of 5" and 10" according to the well-known tables of 2. Kopal. (The asymptotes are a

SIMILARITY A N D EQUIVALENCE IN COMPRESSIBLE FLOW

171

little different for the two angles as M , -,m ; this cannot be seen from Fig. 8 because of the small scale.) The difference between the results of the linear and the exact theory can have two reasons which mathematically must be sharply distinguished. One reason is given by the linearization of the differential equation, the other by the change of the initial conditions which have to be satisfied for the exact problem behind the shock, for the linearized problem at the head Mach line. Fig. 8 indicates clearly that a substantially better agreement of the linear theory with the exact theory could be achieved if a more suitable linearization and a more exact boundary condition had been taken. If we consider the flow over a wedge we have exact parallel flow behind the shock in a different direction and at a different Mach number. Now, let us linearize with respect to the flow condition behind the shock. Then the flow over a wedge is represented exactly up to M , 00 if we fulfill the boundary condtion behind the shock. For the flow over a cone it is not possible to get an exact agreement between the linear and the exact theory by linearizing with respect to the flow condition behind the shock. The reason is the small variation of flow direction and Mach number behind the shock. But one still gets an approximation of high accuracy by this procedure. The dashed curve of Fig. 8 is the approximation of the solid curve above and the dot-and-dash curve below for tan2 6, cot2a, << 1. This assumption is the basis of the linearization with respect to the conditions far upstream. Because of its frequent application it has been put into Fig. 8. It is purely accidental that the curve for the approximation tan2 6, cot2 a, << 1 agrees better with the solid curve than the dot-and-dash curve. It is also accidental that it may be continued beyond the value of (7.8). Finally, the dot-and-dash curve is the solution of the linear problem with the boundary condition ---f

Y =0 :

YTJ= (1

+ u) tanz6,,

which is for slender cones nearly identical with (7.5). The difference between the solid and the dot-and-dash curve consists only in the projection of the boundary condition on the axis. This is the decisive error at high Mach numbers. At a value of 0.90 for the abscissa the upper curve gives 76% of of the exact value where the lower curve gives only 34%. The upper curve also gives still a reasonable limiting value where the lower curve gives a completely wrong limiting value. So it is inadmissible to simplify, a t hypersonic speeds, the boundary conditions by projecting them on the axis. The physical reason is that in supersonic flow disturbances are propagated along Mach lines. When the inclination of the Mach lines becomes comparable to the inclination of the stream lines, the projection of flow properties on the Y-direction becomes necessarily wrong. This effect has clearly to be distinguished from the

172

KLAUS OSWATITSCH

linearization of the differential equation. I t happens in the same way when the differential equation is linearized with respect to the local flow condition. The wedge flow would be an unsuitable object for the preceding considerations because i t corresponds to a source distribution of constant strength on the axis; therefore it is unimportant whether the source strength is projected along the Y-axis or along a Mach line on the axis. The flow over a cone corresponds, however, to a source whose strength is equal to the value of VY on the axis and increases linearly with x . 8. Linearization of the Gas-dynamic Equation ; Corresponding Points

If the variations of the U-component of the velocity and the velocity of sound c are sufficiently small the gas-dynamic equation (2.5) may be “linearized”. The first factor in (2.5) may be replaced by a constant. In most cases this is done in the following manner:

Here it is assumed that the flow a t infinity is parallel to the x-direction, and that the flow condition at infinity gives the most suitable approximation to the non-constant value of this factor. The last assumption is not always valid. There are cases where better results can be achieved by other approximations of the term (1 - U2,1c2). In general, M , in (8.1) could be another Mach number than the one at infinity, but we shall not make use of this possibility. In lifting problems it is often of advantage to give the free stream an inclination rather than to incline the body; U W / c , # M , if this is done, because the free stream has now a V-component. Hut we shall consider only small angles of attack, and if approximation (8.1) may be used at all, i t can be used also for small inclinations of the free stream. If U passes through the value of c, the coefficient (1 - U2/c2)changes its sign; thus if (8.1) is still to be valid as M , approaches one, the changes of the flow conditions due to the presence of the body must tend to zero, which requires that the thickness of the body becomes infinitesimal. For transonic flows therefore the linearization becomes usually invalid. The same is true for hypersonic flows where (2.5) and (2.7) by themselves are invalid. In the following sections we shall use the system of linear differential equations

SIMILARITY AND EQUIVALENCE I N COMPRESSIBLE FLOW

173

where u , v , and w are defined by (3.1),but we have always to bear in mind the limits of application of this system. In Section 12 we shall approximately determine these limits. I t is convenient to introduce for subsonic flows the Pradtl-factor

In supersonic flow, (1 - M a 2 ) can be expressed in terms of the cotangent of the free-stream Mach angle. If we introduce for subsonic and supersonic flow separately the affine transformations

(8.4)

M,<1:

x=X,

V=py,

z = pz;

M,>1:

x=X,

y=cota,Y,

z=cota,Z;

the system (8.2) becomes

-

a(ucota,) ax

aw + avay + = 0, az -

a(ucota,)

av

aY

ax

= 0;

a(ucot am)- aw az

ax

= 0.

Sometimes it is suitable to transform all three coordinates X , Y , Z instead of only two as in (8.4). Nothing substantially new is added, because the more general case can be obtained from (8.4) by the addition of a similarity transformation. The latter, however is a trivial change in scale of the body under consideration. The system (8.5) for subsonic flow is reducible to the form of (8.2) in the case M, = 0. The components of the velocity disturbance for M, = 0 will be marked by the subscript a :

(8.7)

M,

= 0:

174

KLAUS OGWATITSCH

Here, x,y,z denote the coordinates in the incompressible flow. The system (8.6), however, is reducible to (8.2) for M, =

v?:

_ _auvu _ (8.8)

ax

M, = V2:

avvz +-+-ay

- _ _ _ avV.=,; ay

ax

M,= 0.80

'1 -

aq,

-

o,

az

auyz

ayE - 0. ax

az

M,=O

FIG. 9. Corresponding points in subsonic and supersonic flow.

vz,

Here x,y,z denote the coordinates for the flow with M , = and u v z , V are ~ the corresponding components of the velocity disturbance. The affine transformation (8.4) is the first step in reducing the system (8.2) to its 'normalized form' (8.7). By this transformation a point P(X,I',Z) of any subsonic flow corresponds to a transformed point P,(x,y,z) of an incompressible flow. In the same way a point P ( X , Y , Z ) of any supersonic flow corresponds to a transformed point P,(x,y,z) of the flow at M, = Of course, the correspondence could be chosen in such a way that a subsonic flow at M,, is compared with a subsonic flow at Mm2,and the same is irue for flows with M > 1, but it is more suitable to refer to the normalized Mach numbers M, = 0 and M , = v5. However, it is not possible to compare a flow a t M , < 1 with a flow at M , > 1 by an affine transformation. Points ( X , Y , Z ) in flow fields with different M,-values will be called corresponding, if they are mapped on the same point (x,y,z) by (8.4). Fig. 9 V V ~W,

v<.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

175

gives examples for their position. The distance of corresponding points from the x-axis becomes always larger as the free stream Mach number approaches one. Because M , is always larger than the normalized M , = 0, the distance from the x-axis of corresponding points in subsonic flow is always larger than the distance of their common image Pi. In supersonic flow, however, where the free stream Mach number can be larger or smaller than the norcorresponding points may be situated nearer or farther malized M , = from the x-axis than the points P V ~ . Because the position of the corresponding points depends on M,, the same holds for the planforms of corresponding wings. Fig. 10 shows two examples. The half-span of a "corresponding wing" becomes Mm=1.17 indefinitely large as the "corresponding Mach number" approaches one, since the half-span transforms as the point-coordinates in (8.4) :

12,

-

FIG. 10. Corresponding planforms in subsonic and supersonic flow.

For supersonic flow this implies that the span changes as tan a,. Therefore any point which originally lies outside the forward Mach cone will again lie outside the forward Mach cone after the transformation (Fig. 10). A comparison of two supersonic flows would be impossible, unless the transformation preserves domains of dependence and determinacy. In general the transformation (8.9) does not permit statements about the Mach-number influence for a given wing of finite span except, of course, in the two-dimensional problem. Other exceptions are bodies of revolution and low-aspect-ratio wings, as will be shown later. We may now distinguish between two fundamental tasks. The first is the task of the engineer: A wing of certain span is given; determine its aerodynamic properties for lfferent free-stream Mach numbers. Fig. 11 shows the case of a delta wing at three different M , < 1. The calculation can be reduced to the incompressible case with one of the analogies given

176

KLAUS OSWATITSCH

in section 10, but for every free-stream Mach number M , a different span sd must be taken at the normalized Mach number M , = 0. Here s is given ~ smaller as M , approaches one. and sd has to be determined; s , becomes The second fundamental task consists in finding the “corresponding flows” for a given solution. In corresponding flows the normalized half-span is the S const. same. Here.smd is given and s is determined by (8.9); it becomes indefinitely larger as M , approaches one. It should be noted that the matching of correspondMe= 0 %=0.60 &.=0.80 ing points (or the transformation of the span) is S,, ‘const. independent of the transformation of the velocity components. In the following sections several possible transformations of the velocity components will be found, M-2 0 M,=0.60 Mg0.80 each resulting in a different form of the Prandtl-Glauert analogy. But all are conFIG. 11. Upper half: half-span s is given, sred nected with the same afEine is required for different Mach numbers. Lower transforniation of the cooris given, s is half: reduced half-span Sd, required for different Mach, numbers. dinates.

8

- -

- - -

9. Transformation of the Velocity Components

If we compare the systems (8.5) and (8.6) for M , 2 1 with the corresponding normalized systems (8.7) and (8.8) we see that we have still a constant factor A at our disposal for the transformation of the velocity components : A@(X,Y,Z) = ~red(x,y,z), (9.1)

A cota,u(X,Y,Z)

= a(x,y,z),

for

M m <

for

M, >l;

1,

AV(X,Y,Z) = vred(x,y,z),

A w ( X , Y , Z )= &cl(x,y,z). This upwash factor A, as it will be called, relates the upwash vd a t the normalized Mach number to the upwash v at the corresponding subsonic

SIMILARITY A N D EQUIVALENCE IN COMPRESSIBLE FLOW

177

or supersonic flow. (Such a factor, if applied in the coordinate transformation, would result only in a change of scale.) The upwash factor has still a meaning when we keep the Mach number constant. If we change the upwash by a constant factor a t some fixed M , then the other two velocity components change by the same factor. In this case it is not necessary to speak of corresponding points because the coordinates are not changed. For the lifting problem of an infinitely thin plate in the plane Y = 0 in a free stream u, = 0, II,, w, = 0, this implies that u,v,w are proportional to v , at all points. For the thickness problem of a body symmetrical about the plane Y = 0, it implies that the disturbances are proportional to the source strength u ( x , O , z ) . Thus for thin profiles and wings, all disturbances of the velocity components ,are proportional to the thickness ratio within the accuracy of the simplified boundary condition (6.3).

For bodies of revolution, thick profiles or wing-body combinations the proportionality of u,v , and w to the source or dipole strength is not selfevident. The maximum of the disturbance velocity for a sphere, for instance, is always 4, and for a circular cylinder it is 1, independent of the strength of the dipole. In addition, Table 1 shows that for a spindle of revolution u at the thickness maximum is not proportional to the source strength because this would imply, by ( 7 4 , proportionality to t2. The proportionality holds indeed, but only for fixed points in space and not for fixed points on the surface of the body under consideration, and if the strength of the sources or dipoles is changed, the surface of the body changes too. For bodies of revolution and thick profiles the rate of change of the disturbances in the direction perpendicular to the body surface is large. Thus the shift of the surface due to the multiplication of v by the upwash factor A must be taken into account together with the change of all dsturbances at all space points. The change of the u-disturbance perpendcular to the x-direction follows for irrotational flow from (2.7) and (3.1):

Because u is essentially the inclination of a surface element along a section 2 = const, it follows that is approximately the curvature of a surface element along that section; thus is determined by the body shape. In analogy to (6.4) we may use (9.2) and write

(9.3)

u ( X , Y , Z )= u(X,O,Z)

av + -.ax

H

+ ...

178

KLAUS OSWATITSCH

to find the error due to the shift of the u-disturbance from the body surface to the plane Y = 0. This error is independent of the Mach number and proportional to 9. By (6.3) we have for thin profiles and wings (9.4)

u(X,O,z) = u(X,H,Z) - HHXX.

This error is a higher order effect because u itself is of the order of 7 for profiles and wings of not too low aspect ratio. In axisymmetric flows (9.2) remains valid. Thus the order of magnitude of the error due to shifting the u-disturbance from the body surface to the axis for axisymmetric flows is of the same order of magnitude as for wings and profiles. However, the u-disturbance for bodies of revolution is itself much smaller, namely of the order of t 2 , as can be seen from Table 1. Here the u-disturbance is therefore of the same order of magnitude as the shifting error. Hence flows past bodies of revolution and wing-body combinations must be recalculated at corresponding points even near the body axis. For thin wings and profiles this is only necessary far from the body.

11. APPLICATIONSOF

THE

LINEARTHEORY

10. The Prandtl-Glazcert Analogy Although the various forms of the Prandtl-Glauert analogy would permit a more precise formulation provided that the statements are restriced to upwash and u-disturbance in the plane Y = 0, such a formulation seems doubtful in view of the linearization of the gas-dynamic equation. We therefore prefer to speak of the u-disturbance on the profile or wing and of the thickness ratio or angle of attack, which are substantially proportional to the upwash factor. This formulation has the advantage of greater physical immediateness, but it must always be kept in mind that, in addition to errors not yet comidered due to the linearization of the gas-dynamic equations, errors arise from the shift of the boundary conditions (Section 6) and from the shift of the %-disturbance (9.4). - Bodies of revolution and wings with low aspect ratio will be considered separately in a later section. Prandtl’s original formulation applied to a wing of unchanged thickness ratio and angle of attack. Hence A = 1 and the first form of the Prandtl rule follows from (9.1): The u-disturbance in a subsonic (supersonic) flow has the value of l//l(tan a,) times the u-disturbance at the “corresponding point” in the flow at 11.1, = 0 ( M , = v2) past the “corresponding wing” of the same thickness ratio and angle of attack. This formulation can be carried over from the u-disturbance t o the speed disturbance by (3.2) and hence to the pressure coefficient by (3.4). The

179

SIMILARITY A N D EQUIVALENCE I N COMPRESSIBLE FLOW

latter is the formulation which Prandtl gave in 1922 in his lectures on aerodynamics: The pressure coefficient on a profile increases with the Mach number proportionally to lip. H. Glauert [4], considering the compressibility effect at the tips of propeller blades, rediscovered the Prandtl rule. He chose a distortion of the abscissa rather than the transformation used in our representation. Both the terms Prandtl rule and Prandtl-Glauert analogy may be used in speaking of the similarity rules deduced from affine transformations of linearized subsonic or supersonic flows.

-110

0.9

08

0.7 0.6

0.5 0.4

03

02

0.l

0

PO

FIG. 12. Curve of the mass flow-density.

The following remarks refer to the physical content of the original form of the analogy. The linearization of the gas-dynamic equation corresponds essentially to a linearization of the curve of the mass-flow density (Fig. 12). The disturbance of the mass-flow density is p2 times the disturbance of the speed [l,p. 471. The transformation (8.4)increases the distance of corresponding streamlines in the same way as the Y-coordinate, namely by the factor l/p, if compared with M , = 0. Therefore the variations of the stream tube diameters have become smaller, taking the factor p. This implies, however, that the u-disturbances of the corresponding flows have become larger, taking the factor 1/p. In this way the larger speed variations may be explained from the continuity of the flow, but they can also be viewed from the equation of irrotationality (9.2). The curvature of the corresponding stream lines at the distance Y = y / P is still the same as the curvature at the distance y for M , = 0. The integration of u with respect to Y results in a u-disturbance on the axis, larger by the factor lip. A point on the body surface will only map into a point on the body surface in the transformed space if the body surface is distorted in the same way

180

KLAUS OSWATITSCH

as the coordinates. This requires that the inclination of the surface elements, substantially the two velocity components and w , be transformed as the coordinates. A comparison of the point transformation (8.4) with the velocity component transformation (9.1) shows that the upwash factor must be chosen as (10.1)

A

= /?

for

M,<

1

A

and

= cotaa

for

M , >1.

This is the second form of the Prandtl-Glauert analogy, often called stream-line analogy because of the pointwise transformation of the streamlines. For M , < 1 and (9.1) and (10.1) we can express the stream-line analogy in the following way: The u-disturbance on the surface of a thin body (e.g. a wing body combination) in subsonic flow is l/P2 times the u-disturbance at a corresponding point on the surface of a body in incompressible flow of the same length, /?-fold span, thickness-ratio and angle of attack. The “incompressible prototype” is thinner by the factor P. Therefore the velocity disturbances on the prototype surface are smaller, and it is physically clear, that u, has now to be multiplied by a factor l/P2 to get the right 21-disturbance a t M , > 0. The restriction that it should be permissible for the bodies under consideration to shift the boundary condition and the results from the body surface to the plane Y = 0 can now be dropped because “corresponding points” remain on the body surface before and after the transformation. A direct comparison of the disturbances on the body is now possible. In spite of this advantage and the very broad field of applications the stream-line analogy cannot be called the best form of the Prandtl-Glauert analogy. We shall show later that the stream-line analogy is the only right form when hypersonic conditions are approached, but it is useless at transonic speed. In the United States the stream-line analogy is often called Goethert rule. Goethert [6] was the first who mentioned that the stream-line analogy can be applied to arbitrary thin bodies. E. Krahn mentions in [S] that the stream-line analogy has already been used by A. Busemann in a paper published in 1928 [5], the year Glauert published his paper on the effect of compressibility on the lift of airfoils. The only difference between this representation and that of Busemann is that Busemann transformed the x-coordinate instead of the y - and z-coordinates. Goethert interpretes the stream-Iine analogy by a transformation of the free stream velocity. Of course, one can always rearrange the factors in (9.1) but that has little to do with the physics involved. Before considering the third form of the Prandtl rule it is useful to insert some preliminary considerations about two-dimensional subsonic flows. From the well-known definitions of the stream function ‘P and the stream function for the disturbances I,4 (10.2)

- I,4x = v

and

-

t,kZ= v,

181

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

it follows that

[ I

Y ( X , Y )= Ump, Y -

1

v(X,Y)dX

--m

9

(10.3)

The integration usually starts far upstream. Stream lines are the curves Y = const, and not the curves y = const. From (10.3)i t is seen once more that (8.4) transforms stream lines into stream lines only if v is transformed in the same way as Y . This requires A = p. Only then the factor 1/p can be put in front of the bracket, and

P’Y(X,Y) = Y , ( x , y ) .

(10.4)

For the coordinate transformation ,8x = X,y = Y the upwash factor A again equals p, if a stream-line analogy is desired. In this case the factor ,8 does not appear in (10.4) and stream lines with equal ‘Y-values now have the same distance from the x-axis. The distortion takes only place in the x-direction. Similar considerations apply to the potential @. With the usual definition of the disturbance potential (10.5)

4 x

= u;

$,z

=u,

we get for three-dimensional obstacle flows (10.6)

[

@(X,Y,Z)= U m X

+

5

udX1 ;

@&Y,z)

=

U,

[ X

1

+

uidx].

-m

--cL

If we want to transform the potential surfaces @ ( X , Y , Z )= const into potential surfaces @ , ( x , y,z) = const of an incompressible or supersonic prototype flow we have to demand, by (8.4) and (9.1), that for (10.7) M m < l :

AP=l;

and

M,>l:

Acotcr,=l.

This potential analogy can be expressed for M , < 1 as follows: The udisturbance of a compressible flow is equal t o the u-disturbance at the corresponding point of an incompressible flow past a wing of (l/P)-fold thickness ratio and angle of attack, and P-fold span. This time, the “incompressible prototype” is thicker by the factor 1/p in order to produce the same u-disturbance as in compressible flow. This third form of the Prandtl rule was formulated by Prandtl in connexion with Busemann’s W.G.L.-lecture in 1928 [5].

182

KLAUS OSWATITSCH

In two-dimensional or axisymmetric incompressible or compressible flow stream lines and potential lines are perpendicular to each other; they form a grid of infinitesimal squares only in two-dimensional incompressible flow. Other flows require grids of infinitesimal rectangles. An affine transformation is not a conformal mapping of the grids. Thus a transformation of the type (8.4) cannot achieve simultaneous correspondence of stream lines and potential lines. However, the stream lines are essentially parallel to the x-axis, hence the pointwise transformation of the two grids does not result in a strong deviation from the orthogonality. In the presentation of theoretical and expenmental results it is in general helpful to use reduced quantities of the order of unity. For wings and profiles this is achieved by choosing the upwash factor A in (9.1) equal to l / z in the speed regime where the linearized gas-dynamic equation is valid. For this choice of A

Here again only corresponding points may be compared, which calls for equal reduced coordinates x , y,z. The simplified boundary condition (6.3) for vrd becomes independent of the specific p but is still dependent on t: (10.9) The boundary condition (10.9) and the systems (8.5) and (8.6) written in terms of ured,vd,wrdnow contain only quantities of the order of unity. Therefore the reduced velocity components are in general of the same order of magnitude. I n the representation (10.8) there is no difference between the three forms of the Prandtl-Glauert analogy. The difference comes in only by the different coupling of the thickness ratio to the Mach number. Let us introduce a similarity parameter which is not changed by the similarity transformation. In view of its connection with the thickness ratio t, it may be called reduced thickness ratio. For the three different forms of the PrandtlGlauert analogy we now get the Prandtl-rule for constant thickness ratio: (10.10) the stream line analogy: the potential analogy:

t = td,

zp = Zred, = zred.

For all three cases rrd has the meaning of the thickness ratio of the prototype wing a t M w = 0 ( M , = v2). In general tM is small compared to unity in the domain of linearization of the gas-dynamic equation.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

183

Fig. 13 shows as example profile measurements by Stack, Lindsey, and Littell [66].

0.12

0.10

a08

a06 0.04 0.02

0

o

ai

0.2

a3

04

a5

a6

a7

ae as

1.0

FIG. 13. Profile measurements b y Stack, Lindsey, and Littell.

11. The Effect of Compressibility for Bodies of Revolution at Zero Incidence

For bodies of revolution the shifting of boundary conditions is not possible as shown, and only the stream-line analogy would seem to be applicable in this and similar cases, where the flow on the body surface must be compared a t different Mach numbers. However, in applying the streamline analogy the thickness ratio of the body is changed. If, for example, the cross section Q ( x ) ir required to remain unchanged, the product V Y has to be independent of M , within the accuracy of the simplified boundary condition (7.5). Consider now the product of the corresponding reduced quantities: From (8.4) and (9.1) for M , < I it follows that (11.1)

V

~ = YA

1 ~ Yv= AD -Q’. 2n

Invariant Q thus requires constant AD. This was also the condition for the potential analogy (10.7),which thus requires the same T for comparable bodies of revolution. Although the u-component at corresponding points is independent of Mach number for this artalogy, this is not true for points on the body surface. (It would be true in the stream-line analogy). If we want to say something about the flow on the body surface we have to recalculate “corresponding points” into surface points. This requires a consideration of the flow near

184

KLAUS OSWATITSCH

the axis and makes it necessary to consider w ( X , Y ) . From (7.6)and (9.1) it follows that in the neighborhood of the axis:

u can be integrated over Y,and the additional function of X may be expressed by the unknown u-component on the body surface u ( X , H ) . Then, a,fter division by t2,

Mm=Y2,

x cot a,

02

0.4

x = l

0.6

Q8

1.0

FIG. 14. Supersonic flow field near a circular cone.

Relation (11.2) is only valid near the body axis. It is equivalent to ( 7 4 , which shows the behavior of v near the axis. We may conclude: There exists near the axis a combination of u(X,Y) and Y,which depends on X only, and is therefore constant in planes X = const. It was seen earlier that v is O ( Y - l ) as Y approaches 0, u,however, has only a logarithmic infinity on the axis. Fig. 14 presents as example the supersonic flow over a cone which shows that VY and the expression (11.2) do not change very much

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

185

even in some distance from the axis. The expression (11.2) is the finite part of u ( X , Y ) for Y = 0. It is called the “spatial influencr”, because it is not determined by the local cross section. The reduced product vr4y must be the same in corresponding cases. This implies by (11.1) that (11.3)

‘

A B 2 -- t,,d

is a similarity parameter for bodies of revolution. Furthermore, for corressponding points ured= A & by (9.1), hence (11.4)

and it follows that u is proportional to r 2 but iiidepeltdent of Mach number. In (11.4) urd and T~~ are u-disturbance and thickness ratio in the prototype flows a t M , =-o or M , =

VZ.

Y

%=(180

X

FIG. 1.5. Point transformation for bodies of revolution.

The effect of compressibility for bodies of revolution in the linearized domain of the gas-dynamic equation can be easily determined, since within the accuracy of (7.5), it does not make any difference whether the “spatial influence” is taken on the axis, on the body, or near the body surface. Thus Y in (11.2),for M , < 1, may be taken as H = HJP, the ordinate of a surface point of the prototype (Fig. 15). By (11.4),

U(X,H,/P)- u,(x,H,) T2 t.f2 ’ and by (11.2) with Q l t 2 = Ql/t,2,

186

KLAUS OSWATITSCH

Analogous results hold for supersonic flow. It is useful to write the results with H / H , = t / z , in the following symmetric form:

(11.5)

Relation (11.5) is independent of M , and t. By means of (3.2) it is not difficult to extend the expression (11.5) to the speed disturbance by adding v 2 / 2 t z = vi2/2zi2 on both sides of the equation

(11.6)

Equations (11.5) and (11.6) presuppose that Q can be differentiated twice. This is not satisfied for a body composed of a cone and cylinder a t the section where the two halfbodies are joined. But a t those points the simplified boundary conditions (7.5) which are the basis for conventional linear methods are also invalid. An application of (11.6) for the cone is found in Fig. 8 with

Q = nXztan2 tlo;

1

-Q" 2n

= tanZ8,,,

where t is replaced by tan tlo. The dashed curve gives just the Mach nurnber influence for a body of specified thickness, t = t i or t = t~

(11.7)

The same relations hold for zc, if we accept the small error which arises from the neglect of u in the denominator of the ordinate. As would be expected, (11.5), (11.6), and (11.7) are invalid for hypersonic flow, unless we replace

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

187

the free-stream Mach number by an average Mach number on the body surface. Generally, equations (11.5), (11.6), and (11.7) are better than Fig. 8 would indicate. Table 3 gives some numerical examples for the cone. The exact speed disturbances divided by tan2i3.0are given in column 4. Those expressions that should be independent of M , and t by (11.6) are given in column 6. At first sight, the results in the sixth column seem to indicate a considerable error. However, to estimate the accuracy, the differences in the last column have to be related to the values of the fourth column. Column G is the difference of the nearly equal values of columns 4 and 5. For moderate supersonic Mach numbers a conversion corresponding to (11.6) gives only an error of a few percent. The linear theory [ l , p. 3561 gives for the last column the value 0.50 - In 2 = - 0.193. Therefore it is suitable in praxis to make the conversion at somewhat higher values. The agreement seems to be expecially good for a comparison a t equal values of tan 6,1/Mrn2- 1 (3rd column). This is the case of the stream-line analogy (10.10). We shall come back to this point later. Mach number influences can be found in all papers about axisymmetric flow past slender bodies. The agreement of (11.7) with the results of Schmieden and Kawalki [9] for ellipsoids of revolution in subsonic flow has been shown earlier in [l, p. 3321. The logarithmic term in the expansion of the velocity disturbance for M , < 1 can also be found in Laitone's paper [lo]. The explicit form of (11.7) for bodies of revolution of arbitrary shape and constant t in subsonic and supersonic flow has only been known since 1950 [ l l ] [12]. The more general formulation for arbitrary T (11.5), (11.6)' and (11.7) seems not yet be published. The drag coefficient cD with respect to the cross section t 2 n / 4 can be expressed in first order approximation by the speed distribution and (3.4) in the following way: I.

L is the body length. The body shall have a pointed nose, Q ' ( 0 ) = 0. With (11.6) and the relation Q / t 2 = Q , / T ~ it ~ , follows for M , < 1, that

c

TABLE3

00

00

MACH N U M B E R

5"

10"

I N F L U E N C E FOR T H E CIRCULAR C O N E

1.155

0.050563

-3.188

-2.985

- 0.203

1.506

0.098519

- 2.561

-2.317

- 0.244

2.247

0.176047

- 2.038

- 1.737

-0.301

3.749

0.3161 13

- 1.568

-1.152

-0.416

4.860

0.416092

- 1.359

-0.877

- 0.482

1.116

0.087359

- 2.756

- 2.438

-0.318

1.403

0.173520

- 2.039

- 1.752

-0.287

2.075

0.320587

- 1.547

-1.138

- 0.409

2.179

0.457188

- 1.277

-0.783

- 0.494

3.304

0.555259

- 1.155

-0.589

- 0.566

SIMILARITY A N D E Q U I V A L E N C E I N COMPRESSIBLE FLOW

This can be written in the following symmetric form (the formula for M, has been added without separate proof) :

189

>1

The logarithmic term containing the Mach number influence on the drag vanishes only for bodies ending in a point or cylinder. For the cone in supersonic flow cD = cr, and ( 1 1.8) can be reduced directly to ( 1 1.6).

12. Application of the Prandtl Rule; Limits of tJ?.eDomain

of

Linearization

We shall now calculate the maximum velocity of an elliptic wing at zero angle of attack as an example for the subsonic flow. The thickness distribution 212 is given by x2

22

-+-
a2

(12.1)

h(x,O,z) ~-

a

=.I1-.-?], a2

b2

Longitudinal as well as cross sections of the wing are parabolic arcs (Fig. 16). The well known formula for the velocity disturbance [l, p. 4991 gives a t the maximum thickness and M , = 0 - 2nuj(O,O,O) FIG. 16. Elliptical wing with parabolic longitudinal and cross section.

(12.2) The result of this integration can be reduced to the complete elliptic integrals B and D [2, p. 741. The following final forms result for the wing of large aspect ratio (a < b) and small aspect ratio ( a > b ) :

190

KLAUS OSWATITSCH

(12.3)

For a = b, B(0) = D(0) = 4 4 . For a/b -+ 0 the upper formula gives with B(1)the result of the profile theory for a parabolic arc. For small values of bla, the expansion of D has the logarithmic term typical for wings of low aspect ratio.

Fxc. 17. Rhombic cone of length 1.

If we introduce the non-dimensional halfspan s = b/2a, it follows from the first form of the Prandtl-Glauert analogy that u equals 1/P times its value for a wing in incompressible flow with the halfspan /Is, that is

2ps

<1 :

4t

u(O,O,O) = - D ( k ) ZP

*

2sp,

K

= v1 - 4s2p.

In the second formula, enters only through D ( k ) ; for small halfspan, p occurs only in the logarithmic term of the expansion of D(h).The application of any of the other forms of the Prandtl rule would have given the same result. ui is always divided by the same upwash factor A , by which 't is multiplied. For supersonic flow let us take a cone of rhombic cross section as example (Fig. 17). The surface inclination of such a cone is given by the relation

(12.6)

IzJ

<sx :

h,(x,O,z) = tant3,,,

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

191

where #lois the apex semi-angle of the longitudinal section at y = 0. One must now distinguish between subsonic and supersonic leading edges. For M , = 1/2one gets the following results [22; 1, p. 6241 on the axis:

(12.6)

Again the agreement of both results for SVY= 1 can be shown. For s ; ~<< 1 the upper formula takes a form closely related to that of the circular cone. For sfT -+ 00 the lower formula gives the result for the flow past a wedge. If we use again the first form of the Prandtl rule, (12.6) has to be divided ~ am. The result is by cot a, and svy has to be substituted by s v cot

1

2

<1:

u(x,O,O) = - -

< scot am:

u(x,O,O) = - -

scot a,

2 x

s tana0

V1- sa cot2a,

In

s tan80

VSZ cotza,

1

+ Vl - szcot2a, s cot a,

arc tan Vszcot2am - 1 .

-1

(12.7) Within the validty of the linearization it should not make any difference whether one linarizes with respect to the free-stream Mach number, the local Mach number or any value in between. We get a measure for the accuracy of the linearization, if we determine the change of the component of the velocity disturbance du produced by a change of the Mach number by which one linearizes, assuming that the change in Mach number is only caused by a disturbance of the u-component of the velocity. (For an elliptic wing the influence of the v-component vanishes anyway; only for the rhombic cone it is of some importance.) (14.19) gives the connection of dM2and dq. From (14.19) i t follows:

(12.8)

192

KLAUS OSWATITSCH

To determine the du caused by a change of p or cot amit is only necessary to differentiate (12.4) with respect to p and (12.7) with respect to cot a, according to (12.8). The results can be written in the following form for the elliptic wing of Fig. 16:

(12.9)

For the rhombic cone

s3 cot3 a, (12.10)

[

I n ( l + V l - ~ ~ ~ ~ t ~ as cot m a, scot am 1 - S*COt2(%,

1

s3 cot3 am

s cot a, arc tan Vs2cot2am - 1 s2 cot2 a, - 11* l/s2cot2am- l3

The results have the simple form of a product of a function of the Mach number times a function of sp or s cot a,. The left hand sides dultu and dzl/u tan 6, give the relative zc variation, caused by the change of the Mach number of linearization, divided by the thickness ratio or the tangent of the apex semi-angle of the longitudinal section at 2 = 0. For example, dulut = 2.0 for du/u = 0.2 and z = 0.1. Fig. 18 shows the limits of the validity of the linearization for three values of dzc/ut = 0.5, 1.0 and 2.0. The free stream Mach number and the non-dimensional halfspan with respect to the body length have been chosen as coordinates. The dashed curves give the limits for the strip theory, the dot-dash-curves give the limits for the low aspect-ratio problem. The domain of linearization for the planar case is the upper left region of the intersection of the full and dashed curves, the domain of linearization for the low aspect-ratio case the region below the intersection of the full and dot-dash-curves. The trend is as we expected; if less accuracy is demanded (larger values of dzllut), the limit of linearization lies closer to M, = 1. Bodies of revolution correspond to a wing of halfspan s = t / 2 . I t is also readily seen that the limit of the linearization for bodies of revolution lies much closer to M , = 1 than for wings of large aspect

193

SIMILARITY A N D EQUIVALENCE IN COMPRESSIBLE FLOW

ratio (two-dimensional case). For the rhombic cone we use tan t90 instead of t, but these quantities should not be compared without some caution. For a closed body such as a wing of double parabolic cross section, the tangent of the apex semi-angle a,, equals 27. For a cut-off cone tan ?Yo = tt.

2.0

\ a5 \ \

20

\

\

\

19 0

0.475

Mm--

1.0

FIG. 18. Limits of linearization for the elliptic wing of Fig. 16 in subsonic f l o w .

1.0

15 1.73

2.0 3.5

MZM-4

695 8

FIG. LO. Limits of linearization for the rhombic rone of Fig. 1 7 in supersonic flow.

A subsonic flow problem should not be compared with a supersonic one without extreme caution. For M , < 1 the main interest is directed t o the point of maximum velocity, which is the point of maximum thickness. For M , > 1 the velocity disturbance is nearly zero at this point. Therefore from the start two different bodies where chosen for subsonic and supersonic flow. Fig. 19 shows the limits for linearization of the rhombic cone for the values of dulzc tan tlo = 0.5, 1.0, and 2.0. Coordinates and curves have the same meaning as in Fig. 18. The dashed and dot-dash-curves are similar to those of Fig. 18, but the curves for the limits of linearization have changed very much. The domains of linearization have become very narrow for the case dulu tan 8, = 0.5 and 1.0. For M , > 1 there is a strong dependence of the limit of linearization on the value of du/u tan 6,. This is typical for the supersonic linearization, which is more problematic than the subsonic one.

194

KLAUS OSWATITSCH

For M, > 1 the limits of linearization depend a good deal on the way in which the linearization has been achieved : the relation between the velocity components may have been linearized (Ackeret theory), or the relation between speed and flow angle, or that between speed and pressure coefficient. In supersonic flow it should always be said in which way the linearization has been achieved. In the transonic regime the variation of the factor 1 - M Zin the first term of the gas-dynamic equation is the main reason for the failure of the linearization. Near the hypersonic region the neglect of the ‘hypersonic terms’ in the gas-dynamic equation and Crocco’s vortex theorem are the main cause which renders the linearization invalid. Equation (12.4) can be written in the form 1<2&,

k=

fxz

zr(O,O,O)

:

1--

=

42

-. nB

[l

+ B(k) - 11.

For large span, B ( k ) - 1 << 1. It is then easy to determine for what values of s/3 the deviation from the two-dimensional flow is 5, 10 or 20%. The corresponding formula for small span is

-

(

u(x,O,O) = - 7 s In - - 1

n

i s

D(k)

)[1+(ln(2/ps-

1)

- l)] .

The last bracket becomes small only for s << 1. Similarly one can proceed in the case of the rhombic cone. Figs. 18 and 19 show that the limits of the strip theory depend more strongly on the desired accuracy than the limits of the low aspect-ratio theory. Therefore we may expect that these limits depend also more strongly on the specific problem under consideration, for instance on the plan-form, or whether it is a thickness or a lifting problem, whether the whole wing or only one point is considered. The strip theory is always wrong at the side edges. In the thickness problem the true disturbance calculated by threedimensional theory at the side edge is just half the value given by the strip theory. In general we may say: In transonic flow the theory of low aspect-ratio wings is favored. The theory of large aspect-ratio wings is relatively unimportant because of the large side effects in transonic flow. The region of high Mach numbers however is a domain of the strip theory. The side effects decrease with increasing Mach number. This will be confirmed once more in the hypersonic section.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

195

13. Mach-number Dependence of the AerodyNamic Forces on a Wing

In applying the Prandtl rule to the aerodynamic forces on a wing it is indispensable to examine the necessary premises on which the formulas for those forces rest. If F is the lifting surface and n its outward normal (Fig. 20), the lift and drag coefficients, cL and cD, with respect to the surface I; can be written in terms of the pressure coefficient cp:

CL

=

CI,

=-

-

'{{cficos I;

F

(n,Y)dF;

51

cp cos (n,X)dF.

(13.1)

We will only consider infinitely thin lifting surfaces and also assume that the form of the leading edge excludes strong disturbances there. This assumption avoids the problematic situation in the immediate neighborhood of FIG.20. Lifting surface S and outward normal the nose. For infinitely thin PZ of a lifting wing. surfaces the direction cosines are equal and opposite on the upper and lower side of the surface. Thus, with the subscript (-) referring to the upper (lower) side, 4

+

FCL = -

1s

(cp+

- cp-) cos (n,Y)dF,

F

(13.2) J J F

where the integration extends over the upper surface only. By (3.2) and (3.4), we have for c p + and cp cp+ = - But

-

i',

-

2 2 w: - (1 - M,)%+

(13.3) 2 2 cp- = - 2%-- u-2 - w'_ - (1 - M,)u-

+ . . ., + .. .

I96

KLAUS OSWATITSCH

Within the accuracy obtainable by a distribution of vortices in the plane Y = 0, u ( X , Y , Z ) = - .(X, - Y,Z),

+

(13.4)

2,

+

(X, Y , Z ) =

w(X,

v

(X, - YJ),

+ Y , Z ) = - w ( X , - Y,Z).

While these relations are not exact on the upper and lower side of the wing, it can be shown that up to the third order cp+ - cp- = - 2(.+ - u - )

(13.5)

+ . .. .

For small disturbances and nearly plane wings, cos (n,Y) = 1 -1 . , . ; cos ( n , X ) = - ‘u . . .. Thus without any assumption about the linearization of the differential equation, the aerodynamic forces are

+

The integration can either be taken over the actual wing or its projection on Y = 0. In these formulas appear the velocity disturbances and the geometric quantities F and d F . The introduction of ‘reduced‘ geometric quantities by (8.4) or the transformation of the span (8.9) changes the surface element cZF in exactly the same way as the surface F , but F is transformed as the halfspan because the x-direction remains unchanged. Thus (13.7)

M,
F,=pF;

M,>1:

Fv-=Fcota,. e

This means that the transformation of the geometric quantities in (13.6) has no effect on cL and cD, hence cL transforms as u and cD as the product z1v. But v changes in the same way as the angle of attack. So the angle of attack

E

(13.8)

may be substituted for v in (9.1). With the notation Mm
AE=E~; M,>l:

AE=EVF

we have

(13.9)

and ei can be chosen freely in the region of linearization. The relation between s and si, however, is determined by the transformation (8.9).

E

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

197

The forniulas for the lift coefficient were first given by Goethert [6] by means of the stream-line analogy. At first, only the stream-line analogy seemed to handle adequately the problem of the induced drag. For the induced drag is an effect of second order in the angle of attack, hence second order quantities should be included in the expansion of the speed disturbances and the stream-line analogy should be used since it is the only form which gives correct conversion of the pressure coefficient up to terms of second order. The preceding deduction shows, however, that the quadratic terms of the disturbance components cancel when the pressure difference between suction and pressure side of the wing is taken (13.5).So only terms of first order in (u+ - u-) and 71 are needed for the determination of the drag coefficient cD (13.6). For wings of large span and elliptic lift distribution the following formula is known for M , = 0:

(13.10)

By means of (13.9) the lift coefficient at the same angle of attack but different Mach number can be determined as 2Z&

1

-

2xs

p 1 + 2 F / p s 2 - j3 + 2Fls2'

CL(&,S) = -

~

Because the influence of the aspect ratio is given by (13.10),cL at M , > 0 can be reduced to cL, at the same angle of attack and the same halfspan s = s i :

Since this formula was developed by expressing cL by cLs of the corresponding wing, s has to be so large that (13.10) still holds for si = ,8s. For instance, if we are still satisfied with the accuracy of (13.10)down to s, = 2.5, then (13.11)is applicable only down to s = s i / p = 3.5 for M , = 0.70. The application of (13.11)for M , l,,8 + 0 is not justified, even though (13.11) has a finite value for this limit. The well-known formula for the induced drag at M , = 0 for elliptic lift distribution independent of aspect ratio, -+

(13.12) can be written easily for M , < 1 by means of (13.9):

(13.13)

198

KLAUS OSWATITSCH

While (13.9) is the complete expression of the Prandtl rule for the lifting problem, a momentum consideration may give a deeper insight into the analogy. The set (8.2) yields easily

Now Gauss theorem is applied to this formula for a control surface situated immediately above and below the plane Y = 0 (plane of vortex distribution) and closed by a plane perpendicular to the x-axis a t x = 00, but everywhere else in the finite domain. In addition, v+ may be different from v - , removing the restriction to infinitely thin wings. The result is FCD= - 2

11

[u+v+ - u-v-jdXdZ

Y=O

(13.14) X =const

By means of (8.4)and (Y.l),the integral relation (13.14)can immediately be brought into a form which contains reduced quantities only:

-

ZsS Y=O

[ u , ~ + v & +- u & - v , d - ] d ~ d z =

11 H4.d+

V L

+ WLIdYdZ.

x = const

(13.15)

I t is instructive to note that here the Mach number influence appears only in the reduced quantities.

111. HIGHERAPPROXIMATIONS 14. Higher Approximations for the Gas-dynamic Relations

In the preceding sections the different forms of the Prandtl-Glauert analogy were completely equivalent within the limits of the linearization, as long as they were applied to corresponding points. This can be seen particularly well from (14.1)which was first given by Ackeret [3] as early as 1925 for two-dimensional supersonic flows: (14.1)

u ( X , Y )= - v ( X , Y )tan am.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

199

I t can immediately be written in reduced quantities for arbitrary upwash factor A :

ureci(x,y) =

(14.2)

-

vred(X,y).

The second-order theory for two-dimensional Busemann can be written in u and v in the form

u ( X , Y )= - v ( X , Y )tan am- v 2 ( X , Y )

(14.3)

4

supersonic flows by

M L tan4a,.

Here a reduced form can only be obtained for the value of

A

(14.4)

= (y

+ I)ML tansa,.

By means of (14.4) the reduced form is (14.5) This form of the Prandtl-Glauert analogy goes beyond the linearization for two-dimensional supersonic flow. A is no longer a power of cot a, only; but at the limits of the domain of linearization the upwash factor still takes the forms

~ 2 -, 1 GK 1 :

(14.6)

IGKM::

(14.7)

A

= (y

+ 1)c0t-3aCa,+ . . .

~=(y+l)cota,+

... .

In (14.5) y does not appear. Therefore a comparison of mediums with different values of y is possible within the domain of Busemanns approximation Eq. (14.7) corresponds to the stream-line analogy except for the factor ( y 1). If we omit this factor in (14.4), M m 2 > 1 leads exactly to the stream-line analogy; but then a factor y would appear in (14.5), restricting the results to a certain ratio of the specific heats. Mm2+ 1 leads to a new form of the Prandtl rule. Eq. (14.4) gives a law of similarity which reaches beyond the limits of linearization on both sides of the supersonic domain in Fig. 19. Contrary to an earlier law of similarity by Pack and Pai [23] no variation in the power of cot am has been introduced in (14.6) and (14.7). The upwash factor is simply determined by the compatibility conditions at the Mach lines. An example is shown in Fig. 21. The exact solution for each apex semiangle consists of two branches. That part of the upper branch where M , < 1 is here of little interest because (14.5) is valid only for supersonic flow. For the transition from (14.4) to (14.6) and (14.7) the following rule holds, which will be later confirmed frequently: The condition M a 2 >> 1 is often satisfied at high Mach numbers still within the domain of linearization

+

200

KLAUS OSWATITSCH

for thin bodies. However, M m 2- 1 <( 1 can only hold very close to M , = 1 and it may be applied throughout the whole transonic regime only to bodies of revolution, but not to wings of medium and large aspect ratios. Therefore we have to expect that at transonic speeds the specific choice of A is important but that it does not make much difference at hypersonic speed whether the form (14.4) or (14.7)is chosen.

I

-"red

0'

0.5

I.o

I :5

2.5

2.0

FIG.21. Flow past wedge: solid curve through origin (14.5); A,

0 ,0

"red

exact solution.

The choice of a single function of Mach number for A throughout the whole supersonic regime as given for example by (14.4) is advantageous but it is of course impossible to convert a hypersonic flow, where the Mach lines are nearly parallel to the body surface into a transonic flow, where it must be assumed that the Mach lines start vertically from the x-axis. On the following pages a number of functional relations are examined in order to decide which form of the Prandtl rule they satisfy best. We start from (3.2), which connects the speed and the velocity components. It can be written by means of (9.1) for M > 1 in the following way:

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

With A = cot a,, be written as

20 1

which holds only for the stream-line analopv, this may

(14.8)

where the quadratic terms of the disturbances u,v,w are included. An analogous result holds for M , < 1. Especially for wing-body combinations the quadratic terms in v and zei cannot be neglected in the approximation for q. Therefore it is important that the stream-line analogy holds for the u-disturbance as well as for the q-disturbance, including the quadratic terms. If we apply (9.1) to the flow inclination, (3.3), we get for M , < 1:

Here the analogy can be extended only by means of the potenttal analogy, A = l/P, which yields tan 6 = P tan 8,.

(14.9)

The extension of the analogy to regions of strong stream-line inclination is of importance in the neighborhood of stagnation points. In general, the potential analogy seems to be the form of the Prandtl rule most suitable for stagnation-point flow. At the stagnation point it gives for incompressible as well as compressible flows the true value u = u, = - 1 and maps a thin profile a t higher subsonic velocities into a thicker profile at M , = 0 where the condition of continuity is always linear in u and v . By nieans of the relation corresponding to (14.8) in subsonic flow we may rewrite (3.4) : (14.10)

p2cp

= -2

(2

-

1) -

(6

- 1)

2

+ ... =

Cp,,

and a corresponding relation for supersonic flow exists also: (14.10) shows that the relation for the pressure coefficient holds for the stream-line analogy up to the quadratic terms in the speed disturbance and of course also u p to the quadratic terms in the disturbances of the velocity components. As already mentioned in the context of (3.4),the validity of the simple connection between pressure and velocity disturbance is lost in the hypersonic speed regime, and it has to be replaced by a more complicated relation.

202

KLAUS OSWATITSCH

With the exception of the hypersonic domain, the mass-flow density can be represented by the following power series in q / U , - 1 of the absolute value of the velocity [l, p. 471:

~PQ

pm urn

-ML[3-((2--)M2,] 21

(14.11)

--1 (Jm

+... .

Up to second order in u, v , and w , (14.11) yields

-p4 p m urn

1 = (1 - M 2, ) u

+ (1 1 2

- -&[3

(14.12)

- (2 - y ) M : ] u 2 +

...

or, written in the reduced quantities,

(14.13)

-

The left side of (14.13) can be considered as the reduced mass-flow density. On the right side only one of the two factors containing A and M , can be made equal to one by a suitable choice of A . If (Mm2 - l)'''/A is set equal to one (corresponding to the stream-line analogy), the second factor becomes nearly independent of M , only if M, is very large, but the approximation (14.11) is then no longer valid. In the transonic regime, however, the first and second term of (14.11) have the same order of magnitude and represent the parabolic character of the massflow density a t transonic speed. The second term in (14.12) becomes small compared to the other two terms and can now be neglected. The choice of (14.14)

A = M , [3 - ( 2 - y ) M2m ]tan3a,

allows the following reduced representation near the speed of sound:

Again the left hand side can be considered as a reduced mass-flow density.

SIMILARITY A N D EQUIVALENCE I N COMPRESSIBLE FLOW

203

A comparison of (14.14)with (14.4)shows that both upwash factors have the same limiting value for M , + 1 ; the dominant factor tan3 a, is the same in both equations. Both basic equations, the compatibility condition (14.3)and the equation for the mass-flow density (14.11), are approximations and disagreements in the coefficients of the quadratic term are permissible as long as they are of the same order as the neglected higher-order terms. The conditions of linearization for the mass-flow density and for the pressure coefficient are often contradictory. A linear approximation of p q by q is useless at transonic speed, but such an approximation of cp by q is especially good in this speed regime. The reason is that the mass flow density is the first derivative of the pressure coefficient with respect to the speed, as follows from the general Bernoulli equation (14.16) The nonlinearity arises from the factor (1 - M2) in addition to the hypersonic terms at high Mach number. This factor is connected with the derivative of the mass-flow density by the following relation which is exact for isentropic flow:

(14.17) The assumption of constant Mach number in (8.1)corresponds only approximately to the replacement of the mass-flow density curve by a straight line. For the representation of the Mach-number factor it is important to note that a small variation in q does not necessarily imply small variations in Mach number. Under the weaker assumption of isoenergetic flow the following relation holds:

(14.18) where c* is the critical velocity, or, in differential form

I t can be seen that in the transonic regime the relative variation of the Mach number dMIM is about equal to the relative variation of the speed dqlq. The expression 1 - M 2 therefore fluctuates around zero at transonic speeds. The Mach-number factor can only be replaced by a constant if the solution of the problem is not essentially influenced by its value and sign. Otherwise the Mach number factor must be equal in corresponding points. A better

204

KLAUS OSWATITSCH

approximation of 1 - M2 is to assume a linear dependence on u (which now stands for the q-disturbance, (1 - M 2 ) = ( 1 - M L ) - K ( M , ) u ,

(14.20)

where K(M,) is a suitable function of M,. If we demand that for M = M,, u = 0, the derivative dM2ldu, which corresponds to dM2/dq, will be still correct, this would imply by (14.18) and (14.19)

This choice is suitable near M = M,, u = 0. If however the transition from the elliptic to the hyperbolic differential equation is of main interest, one must demand (14.22)

for

M=l

U

1

C*

g=--l=--l=-U, U,

1,

M:

hence (14.23)

K

In order to represent the Mach-number variation in a reduced form, the ratio I-MZ K(Mm) = I - K(Mm) u = l Ured 2 1-M, 1 -Ma A ( l -M2,) Vll must be made independent of M , , hence the coefficient of urd for M , 5 1 should equal & 1 ; A is then always positive. For transonic flow we have (14.24)

if we choose for A the following values:

Eq. (14.21) (neighborhood M,) : A

=

M: p

+ (y - 1)~:1/Vp-

~ 2~

(14.25)

Eq. (14.23) (critical velocity) :

1

3

,

SIMILARITY A N D EQUIVALENCE I N COMPRESSIBLE FLOW

205

Again we see that both representations of A in (14.25) approach the value of (14.6) as M , -+ 1 . The velocity variations stay small for slender bodies even for M , .+ do as can be seen from Fig. 2, but this does not imply that the Mach number variations have to stay small too. As can be seen from (14.19),small velocity disturbances result in strong variations of M 2 for large enough Mach numbers. Therefore the expansion of the Mach number a t hypersonic speeds in terms of small disturbances becomes invalid. Because of the small variations of q we may conclude from (14.18) (14.26)

From (14.26) it can be seen that M m 2>> 1 implies also M2 >> 1 , which in turn implies by (14.18)

and

Thus, for the stream-line analogy at M m 2>> 1, the following reduced form is possible: ML-1 (14.28)

2

M2-1

Here as in (14.7) we do not get a new form of the analogy a t high Mach numbers but again the stream-line analogy. But it will have to be shown later that the stream-line analogy satisfies all necessary conditions. Already van Dyke [48] pointed out that the stream-line analogy is applicable up to high Mach numbers. Let us mention here again that our definition of hypersonic flow (2.3) is not identical with the requirement of high Mach numbers (M2 >> l ) , which is a much broader conception in the case of slender bodies. The assumption M 2 >> 1 used in the preceding equations is already satisfied at 3 < M, yet the linearization may still be valid for a sufficiently slender body. This may be seen for instance from Fig. 8. Hypersonics in the sense used here implies that Mach angle and thickness ratio are of the same order of magnitude. Then the linearization is in general invalid. Let us finally consider the relation between the thermodynamic state and the speed, choosing the absolute temperature as state variable. For constant

206

KLAUS OSWATITSCH

stagnation pressure, that is along a stream line between shocks, it is possible to change to another state variable by the isentropic relations. The exact relation between the variations of temperature and speed is

(14.29) indicating that a small speed disturbance can only produce a small ternperature disturbance as long as M , is not too large. For h42 >> 1, we may simplify (14.29):

(14.30)

T T,

~- 1 =

L

,;(

( y - 1) cot2am

~

-

1)+

..-

*

In complete analogy with (14.28)for the Mach number variations, the temperature ratio TIT,, by (14.8)is equal in corresponding points of two flows relatedbythe stream-line analogy. This is necessary because the principal cause of the strong variations of the Mach number in the hypersonic domain is the variation of the thermodynamic state, whereas the velocity variations are very small. Also in the transonic regime the analogy can be extended to the thermodynamic quantities. Here we can simplify (14.29)to

(14.31)

--l=-(y-l) T T,

because M , - 1 << 1. Eq. (14.31)expresses that the thermodynamic variations in corresponding points transform as the speed disturbances.

15. Th.e Shock Equation in Non-parametric RePresentation The equation for the shock angle u (4.1)transforms by (9.1) into the form

(15.1)

y%

Here urd is the shock-front angle in the prototype flow at M , = Whether that flow is again past a slender body is not answered by (15.1),which says only how the shock front changes. involves the density. Therefore The equation of the shock polar, (4.4), we should first get some insight into the changes of the thermodynamic state, especially in those cases where the linearization of the gas-dynamic equation becomes invalid (transonic and hypersonic regime).

207

SIMILARITY A N D EQUIVALENCE I N COMPRESSIBLE FLOW

The changes of all state functions in a shock transition depend only on the combination

(15.2)

M2

sin2a

sin2u/sin2a.

The density, e.g., is

r

(15.3)

2 d

sin2 +in2 a), l

-

and similar (but more complicated) relations hold for the temperature and stagnation-pressure ratios. Under the assumption of small Mach- and shock angles, as they appear in the hypersonic domain for slender bodies, it follows by (15.1) that tan a,

1/M2-11

(15.4)

In the case of the stream-line analogy (14.28) and (15.1) yield

Again, the reduced quantities ur4,vrd are not small when the velocity disturbances become small. Eq. (15.5) states: The ratio of all thermodynamic quantities before and behind the shock remains the same a t corresponding points in the application of the stream-line analogy for M 2 >> 1. This is not only so for $/p,fi/p, and FIT, but also for the ratio of the stagnation pressures. For this to be true, the state disturbances need not be small. On the contrary, they can be fairly large, as the jump in pressure across a head shock actually is. The conditions in the transonic regime are quite dfferent. Here the change of state is always small. The relative density disturbance for instance is nearly equal to the relative speed disturbance; a as well as c are nearly 4 2 ; density, pressure, and temperature disturbance can be written in the same form as the following relation for the density:

(15.6) Because sin u and sin ci are nearly one, we get by means of (15.1) (15.7)

sin2 u - sin2ci = cosza - cosZu = cot2am sin2u

1

cot 2 Uld

208

KLAUS OSWATITSCH

We see from (14.24) and (15.1) that the square bracket depends only on the reduced components. This implies that the density disturbances for similar flows must be proportional to cot2a,. Let us introduce a reduced density predin analogy to urd, which may be interpreted as the density in the prototype flow at M , = So we get

vz

(15.8)

Corresponding relations could be written for pressure and temperature. These considerations are not concerned with higher approximations, they only show how the shock equations fit into the general picture. Eq. (15.8) expresses that a t transonic speed the density disturbance has to be converted as the velocity disturbance in corresponding flows. If A is chosen in correspondence with (14.6), then, by (9.1), also the u-disturbance takes the conversion factor cota a,. By (3.4), the pressure coefficient transforms as w in first approximation. The velocity pressure, however, is essentially constant at transonic speed. Thus the relative pressure disturbance fi/fi, - 1 transforms as u. This again is in accordance with the shock equations. The change in stagnation pressure, however, is proportional to the 3rd power of the change in pressure; therefore the stagnation pressures of similar transonic flows are proportional to cots a,. The consideration of the shock polar w ill again be restricted to the case where the free stream is parallel to the x-axis (v, = 0). Since for M 2 > 1 the density ratio across the shock is always large compared with the speed ratio, a very good approximation at hypersonic speed is (15.9)

The shock polar (4.4) now becomes (15.10)

when we apply the stream-line analogy. Eq. (15.10) is the reduced shock polar for hypersonic flow, because the density ratio across the shock remains the same by application of the stream-line analogy. By means of (15.3) and (15.5) the shock polar can be written in the velocity components alone. For ured= vred = 0 in front of the shock one gets after a short calculation (15.11)

SIMILARITY AND EQU1VALENC.E IN COMPRESSIBLE FLOW

For the limiting case M , (15.12)

+

co

sin2 a << sin2 a :

209

the simple form &d

=

2 ~

+

(%red - &red)

which is shown in Fig. 22, follows directly from (15.10) and (15.3). Eq. (15.12) becomes false for weak disturbances, as can also be seen from Fig. 22. There the assumption sin2 a << sin2 (T breaks down.

321c

i o -crrd

5

FIG. 22. Shock polar for hypersonic flow.

In the transonic domain the influence of the entropy change on the mass-flow density is negligible. For greater clarity let us denote the condition just before the shock by the subscript do. This does not imply any loss of generality because the choice of the condition co is completely a t our disposal, and the analogy can be shown to hold for every point separately. The disturbance of the mass-flow density is small, hence approximately A "

(15.13)

1

-

PQ

84

~

1

- p..'I-

Pi

=

p4 - 1. pmqw

The mass-flow density disturbance being quadratic in u, it can be written (cf. (14.13)) as

but it is necessary to choose A according to (14.14). A different choke of A corresponds only to a different approximation of the curvature of the

210

KLAUS OSWATITSCH

mass-flow density curve. It has little meaning to replace the factor 4 in front of by a different but constant value. This would just change A by a constant factor. We cannot deduce (14.24) from (15.14)by means of (14.15). We would get

UL

(15.15)

M, > 1 :

1-M2 --

2

1-Mm

- (1

+

%d)

[ + y? 1

l)] *

Now wrd is of the order of unity, hence both terms in the first bracket are of the same order. But p,/p - 1 is not the reduced, but the real density disturbance. It gives therefore a term of higher order which corresponds to the ones already neglected in (15.14). This contradiction is not serious, the reason is simply that the massflow density (15.14) and the Mach number factor (14.24) 2.0 cannot be reduced simultaneously when also the exact relation (14.17) is to be satisfied. With regard to the meaning of the subscript 00 (u = 0) and with (15.14) applied behind the FIG. 23. Transonic shock polar. shock, (4.4) yields

-

(16.16) and after transformation by (9.1) we obtain (15.17) the reduced transonic shock polar. The normal shock is not described by the hypersonic shock polar (15.10) and (15.1l ) , because it produces strong velocity variations in contradiction to our assumption. Eq. (15.17),however, represents the complete transonic shock polar (Fig. 23). The maximum value of fired results for (15.18) and the normal shock for (15.10)

d&=

-2.

SIMILARITY A N D E Q U I V A L E N C E IN COMPRESSIBLE FLOW

211

The correct representation of the normal shock and its neighborhood is especially important for certain transonic flow problems, for example the local subsonic region in front of a wedge with detached shock. For this purpose Spreiter [36] proposed to choose the upwash factor A so that the normal shock is represented exactly. This requires 0 . U = c * ~ ,and it follows that

hence by means of (14.18) a new form for the upwash factor in transonic flow: (15.20)

A=

.

2 c*2

cot am = ( y

+1)~h.n3a,.

This new formula (15.20) is closely related to (14.4) deduced from the compatibility condition. But great accuracy in satisfying the compatibility condition corresponds only to a good representation of that part of the + 0 ; (15.20) however corresponds to a good representashock polar where ti, tion of strong shocks. The last two sections have taught us that the stream-line analogy ( A = cot a,) or the modified stream-line analogy according to (14.7) is the best form of the Prandtl-Glauert analogy for high Mach-numbers ( M 2>> 1). For transonic flows however where the assumption M 2 - 1 << 1 is in general not too well satisfied, a number of upwash factors are at our disposal: (14.4), (14.6), (14.14), (14.25), and (15.20). All these upwash factors have the same limiting value as M , -+ 1. For bodies of revolution and wings of low aspect ratio the typical transonic domain is rather small, see Figs. 18 and 19. Here it will not make a great difference which form of A is chosen, but for wings of large aspect ratio it is important to choose the right form of the upwash factor. 16. Reduction of the Differential Equations

For an extension of the I’randtl-Glauert analogy beyond the domain of linearization it is necessary to rewrite the non-linear differential equation in terms of the reduced quantities only. We shall start with the gas-dynamic equation in the form (2.2). Rut first the coefficient 1 - U 2 / c 2needs some attention. I t may be written as

212

KLAUS OSWATITSCH

With the exception of the transonic domain the underlined term is dominant because V2 << q 2 and W 2<< y2 for slender bodies. For the transonic domain (16.2)

(1

-

f )= (1 - + ) ( 1 +

):

shows that (1 - y2/c2) becomes of the same order of magnitude as the u-disturbance. For two-dimensional flows and flows around wings of medium and large aspect ratio, (1 - q2/c2) is on the average still by one order of magnitude larger than V 2 / c 2 and W2/c2. Only for bodies of revolution and related body shapes the term V2/c2 may become as large as 1 - y2/c2 near the body surface. But we saw already in Section 7, in discussing the boundary conditions for such bodies, that the first term of (2.2) is unimportant near the surface of bodies of revolution at transonic speed. This will be confirmed again by the equivalence rule in a later section. At some distance from the body the underlined term in (16.1) is always dominant. The following simplifications are thus valid also for bodies of revolution at transonic speed. Because the u-variations a t hypersonic speed become very small it is always a good approximation to replace the factor U by U , and vice versa in the hypersonic terms on the right-hand side of (2.2). So the followi.ng form evolves from (2.2) for M , > 1 by means of (9.1):

Let us inspect (16.3) first for transonic speeds. Independent of the upwash factor chosen the first Mach-number factor has always to be presented by means of (14.24). Thus the left-hand side of (16.3) is already in the reduced form. We shall find that the right-hand side is always small, hence the particular choice of A does not matter; the simplest form of A is the one given in (14.6). Setting the factor M 2 equal to 1 in (16.3), we get for M , > 1

(16.4) On the right-hand side, the reduced quantities are multiplied by cot2u, or coteam. For equal reduced components, that is for similar solutions,

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

213

omission of the hypersonic term will be the better justified the smaller cot2 a m , i.e. the closer sonic flow is approached. The last term may always be neglected, for it is multiplied by coteam. A criterion when the right-hand side may be neglected can be got from the flow around a profile at zero angle of attack. This example was treated by G. Guderley and can be found e.g. in [l, p. 492, Fig. 2501. In applying (16.3) all disturbances must be referred to a supersonic velocity. In Fig. 24 the velocity at x = 1 has been chosen, consequently the free-stream velocity at M , = 1 is a constant negative disturbance. Guderley's example shows that the differential expression vr&(avredax a u r e d a ~ ) l ( ~ 1) multiplied by cot2 a , is on the average as large as the first term on the left-hand ax. a, corresponds side (1 urd)a@red/ to the Mach angle a t x = 1 which FIG. 24. Transonic flow over the becomes here

+

+

+

Guderley profile.

(16.5)

cotza,

.

= 0.81 t 2 I 3 ,

where t is the thickness ratio as given in Fig. 24. If we make it a rule that the neglected term should not exceed on the average 20% of a main term, then cot2am should be smaller than 0.20, and by (16.5), t < 12%. This indicates a kind of limit for the thickness ratios of profiles, if a reduced form of the gas-dynamic equation is to be used for the calculation of the flow past it. Small thickness ratios are necessary for wings of large and medium aspect ratio. For bodies of revolution and wings of low aspect ratio the condition of small velocity variations is much better fulfilled as can be seen from the solutions in the domain of the linear theories. >> 1) the factor M 2 on the right-hand side For high Mach-numbers ( M 2 of (16.3) may be replaced by ( M 2 - 1 ) . The stream line analogy ( A = cot a,) and (14.28) now permit to write (16.3) in the reduced form

(16.6)

where the Mach-number is eliminated, but the ratio of the specific heats appears.

214

KLAUS OSWATITSCH

The equation of vorticity will be examined only in the two-dimensional case, (2.10). In reduced quantities it takes the form (16.7)

In transonic flow the losses and thus the entropy differences are proportional to the third power of the pressure or density differences. If we compare the entropy changes with those of a flow at M , = 1/2we get in analogy to (15.8) s - s,

(16.8)

- cots a,

CP

S V-~S m

CP

Here the free-stream entropy is assumed to be the same for all similar flows. This is no restriction because a constant may always be added to the entropy. Since the entropy is constant along streamlines behind the shock, the circumflex has been omitted. Hence (16.8) applies to any corresponding points. If the upwash factor of (14.6)is chosen and l/MZis replaced by one, then (16.7) takes the form (16.9)

]/a

To estimate the influence of vorticity the flow over a wedge at M , = will be considered for an apex semi-angle of loo; the shock is then still attached, but the u-disturbance is $4 = - 0.29. This is no longer a weak disturbance by a thin body, but for the intended estimate the example serves well. It is a genuine transonic flow problem: the Mach number behind the shock is already subsonic. By means of (9.1) ud

=

(y

+ 1) cot2aa;u = - 2.4.0.29 = - 0.69.

The stagnation pressure ratio fio/fi, = 0.98. In transonic flow, this is an appreciable loss; it corresponds to a normal shock a t M = 1.22. With this value

-

‘ 2 0.020 = 0.034. Y

This is only 6% of the value of ured just calculated. The u-disturbance attenuates in the y-direction in about the same way as the shock, so the right-hand side of (16.9) amounts to about only 5 % of one term of the left hand-side a t M , = 1/2. As M m approaches unity the neglect of the entropy term becomes still better justified.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

215

Along these lines G. Guderley [24] has shown first that transonic flows may be considered as irrotational flows. However, neglecting the entropy vortices in (16.7) does not mean that the flow may be considered as exactly homentropic. The entropy increase has to be taken into account as an integral value over the total flow in the calculation of the drag from the momentum losses or the irreversible heating of the air. >> 1) we may again substitute M a - 1 for For high Mach-numbers ( M 2 M a in (16.7). By the stream-line analogy and (14.28), it follows from (16.7):

(16.10) We saw in Section 14 by means of the stream-line analogy for high Machnumbers that the change in state and therefore the entropy increase is the same a t corresponding points of the shock for similar flows. Since the entropy is constant along stream lines and stream lines now correspond to each other, the derivative of the entropy in (16.10) is the same for all comparable flows. Hence (16.10) represents the reduced form for M 2>> 1. In our reduction of the gas-dynamic and vorticity equations, in the transonic domain, it was necessary to demand for the two-dimensional case sufficiently thin profiles so that the hypersonic term in the gas-dynamic equation with the factor cot2a, could be neglected. Thls condition was independent of the choice of the upwash factor A . Also these flows can always be considered irrotational. For high Mach numbers the stream-line analogy alone achieves the reduction of the differential equations which includes the hypersonic terms in the gas-dynamic equation as well as the entropy term in the equation of vorticity.

IV. TRANSONIC SIMILARITY 17. Similarity Laws for Profiles and Wings in Transonic Flow For transonic speed, we have from Section 16 the gas-dynamic equation (17.1)

-

and the equations of vorticity ; (17.2)

216

KLAUS OSWATITSCH

TABLE4

-

* %XI

1.

.

(Y

+ _. 1) .

..

.

-

(14.6)

2.

3.

Y + l (l+&)

4.

5.

6.

(Y +

1)dm

(15.20)

-2 1 1+,

-

M m

In addition to the boundary conditions a t infinity and at the body, the compatibility conditions a t the shock must be fulfilled. At supersonic freestream velocities the condition a t infinity is replaced by the condition a t the head shock, and in Section 15 the formulas for the shock inclination (15.1) and the shock polar (15.17) were expressed in reduced quantities. It remains to investigate how the boundary conditions a t the body and the free-stream conditions behave for similar flows. For the conversion of the reduced components urd,V-, wrd into velocity disturbances and for the calculation of aerodynamic forces the upwash factor A is essential. Six different forms of A were found for transonic flows. They are tabulated with their main properties for M , > 1 in Table 4. For

217

SIMILARITY A N D E Q U I V A L E N C E IN COMPRESSIBLE FLOW UPWASHFACTORS FOR TRANSONIC FLOWS (M,>

1)

for

~

1Vlm = 1.20

*

Ured

.*

Remark

"red

~~

-2

- 0.74

- 1.39

limiting case M ,

1.54

- 2.88

compatibility condition

- 0.95

- 1.78

mass-flow density

- 1.15

-2.15

Mach-number factor

- 1.00

- 1.86

critical velocity

- 1.07

- 2.00

normal shock

+

1

M2,

-2M i

- 2

-

Y+l

- 2

M , < 1, cot a, is replaced by p. The fact that the forms 2 and 6 have been deduced from the equations for supersonic flow does not necessarily prohibit their application to flows with M a < 1. In that case the form 2 will soon give a poor approximation for Mach numbers smaller than unity because of the factor M,4. The factor M m 2which occurs in all forms 3 to 6 is typical for the non-linear terms of the gas-dynamic equation. The Mach-number expression for A ~ 0 t 3 a , in 4 is identical with the Mach-number factor in (12.8) and (12.9) because of (17.3)

218

KLAUS OSWATITSCH

The fourth and fifth column give the values of the reduced a-component for the critical velocity and the normal shock. By (9.1) and use of (14.18), (17.4)

(17.5)

A cot2 am

u s = A cota,

Zid=Acotam

[zz: ']=-m 2

~-

A cot3#,

M%

'

To estimate how good the approximations are, values for M a = 1.20 have been calculated in the other columns. By (14.24) the value of should always be minus one at any point v = 0 for all M , > 1; this is only the should always be minus two for the case for form 5. Also by (15.19),Zi normal shock; this is the case only for form 6. From this point of view the last two forms show the smallest errors. The forms 2 , 3 and 4 have Ma4 in the expression for A cot3 a, and reduce to the stream-line analogy for high Mach-numbers ( M , >> 1) ; so they can be used in the whole supersonic domain. Some other upwash factors could still be added. The main difference between the forms given in Table 4 and those of the Prandtl-Glauert analogy of Section 10 is this: For the forms of Section 10 the reduced quantities are identical with the incompressible flow for M , < 1 and with the flow at M , = 1/2 for M , > 1 which is not the case here. But M , = lies in general already outside the typical transonic domain, so there is no need for comparison with such a flow. If this is desired the upwash factor can be chosen, with the same accuracy as form 1, to be

fi

(17.6)

14 = llVIM2,

3

- 1) .

The possibility of conversion to other values of y would then be lost. In the form of (17.6) the close relation to the three forms of Section 10 shows up, but the form of the gas-dynamic equation (17.1) and the shock polar (15.17) would no longer be valid. ( y would appear as a factor.) The carrying-over of the analogy to other values of y is important with respect to the shallow-liquid analogy with y = 2.00 [7]. We now turn to the main purpose of this section: to find the conditions which the actual aerodynamic and geometric quantities should fulfill .so that the assumption of transonic similarity, namely equal reduced quantities, is not violated. By (6.3), t is proportional to v (x,O,z) for profiles and wings. A 'reduced thickness ratio' can thus be formed in analogy to (9.1) for all Mach-numbers : (17.7)

t d =A t .

SIMILARITY AND EQUIVALENCE I N COMPRESSIBLE FLOW

219

Eq.(17.7) is one of the weakest points of the transonic similarity laws for profiles and wings of large aspect ratio because u << 1 has been used in deriving (6.3); but just in the transonic regime large u-disturbances occur. The error which arises from this neglect is somewhat eased by the fact that the maximum values of u generally do not occur at the points of maximum values of z). Eq. (17.7) determines how the Mach-number M , should change with t to satisfy the condition of constant trd. It is practical to exhibit the factor COP^,. Then (17.8)

tred = t.

B/V~M: - 11

3

results for M , 2 1. Often it is useful to take a certain power of similarity parameter : 2

(17.9)

Mm

'

M,-1 (t.@2/3

tred as

1 =

2/3 * rred

As td may be considered as a reduced thickness ratio, so l/$Z may be considered as a reduced free-stream Mach-number. tred in (17.9) becomes infinite as M a + 1, but that does not matter. The disadvantage of these representations of the transonic similarity laws is that also the reduced u-component and, hence, the reduced pressure coefficient and reduced aerodynamic force coefficients become infinite as M , + 1, which makes a useful representation of these coefficients at M , = 1 impossible. The reason for this is that A becomes infinite according to Table 4 or (17.6). This difficulty can be easily resolved. One possibility would be to denote by M , a flow condition sufficiently different from the sonic condition. Then u # 0 in the free stream, but this would be no drawback. Another possibility would be to choose a value for A which does not vanish for M , + 1. This is always possible. Since for similar flows the similarity parameter rred remains the same, new upwash factors A can be formed with td which avoid the drawback. These combinations of reduced quantities could of course be given new names, but here we will not introduce new notations. The %-disturbance (and hence the pressure coefficient) follow from (Y.l), which can be rewritten to read (17.10)

If we eliminate [Ma2- 11 from (17.8) and (17.10), we get

220

KLAUS OSWATITSCH

The left-hand side of (17.11), composed of reduced quantities, is the same for comparable flows, i.e., the w-disturbances of similar flows have to be converted as @.23-l'. For M , = 1, B is always equal to (y I) (see, Fig. 26). Eq. (17.9) and (17.11) represent a transonic similarity law which includes the special case of M , = 1. It was published with B = y 1 in separate papers by G. Guderley [24] in 1946, and simultaneously by Th. v. K i r m i n [26] and K. Oswatitsch [27] in 1947. For the special case M , = 1 it was given by v. Kirmin at the 6th International Congress for Applied Mechanics in 1946 [25]. Several different formulations have been given later, some of them have been mentioned earlier ([28], [36], [23]).

+

+

a7

W p9 1.0 11 12 1.3 14 15 FIG. 25. The values o f B for different Mach-numbers near M , = 1.

Fig. 26 shows measurements by Michel, Marchand, and Le Gallo for a parabolic arc profile [35] of various thickness ratios. The results are given in the usual manner and in the reduced quantities. The u-disturbance has been set equal to the q-disturbance.

22 1

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

In Figs. 27a to 27f the different forms of the transonic analogy are applied to wedge flow. This may provide an estimate of the speed regimes for which

"L

FIG. 26. Transonic measurements by Michel, Marchand and Le Gallo for a parabolic arc profile of various thickness.

the different forms of the analogy are suitable. In [36] Spreiter gives for the first time form 6 with the exact representation of the normal shock.

222

KLAUS OSWATITSCH

a

0

1.0

2

l+ln-

Mw (t.*

-0.5

0

0.5

1.0

0.5

-1

B)2/3

0

FIG.27a and 27b. Six different forms of the transonic similarity laws according to Table 4 as applied to wedge flow.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

223

For the wedge with shock detachment this form is very good (Fig. 27f). I t exceeds by far the accuracy of form 1, but it has to be noticed that the exact representation of the normal shock is achieved only, if M , is taken as the Mach-number just in front of the shock. For a head shock wave this is always fulfilled, but in the example [ l pp. 4701 with a normal shock at the profile, M , would have to be the Mach-number at the profile. For wings of medium and low aspect ratio normal shocks do not have so much importance as for two-dimensional flows, and other requirements determine the choice of A cot3a,:

FIG.27c. Six different forms of the transonic similarity laws according to Table 4 as applied to wedge flow.

The conversion of the coordinates of corresponding points and of the span follows directly from the Prandtl-Glauert analogy (8.4) and (8.9) for M , # 1. For M , = 1 a form is used which, in analogy to (17.11), does not contain 11 - M,21: (17.12)

T~&J

= d3B1/3Y,

T$ :

= 21/3B1/3z,

1/3 Tr&Sd

= Tl/3B1/3S.

FIG.27d. Six different forms of the transonic similarity laws according to Table 4 as applied to wedge flow.

LD

I

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

I

0

225

226 KLAUS OSWATITSCH

FIG.27f. S i x different forms of the transonic similarity laws according to Table 4 as applied to wedge flow.

227

SIMILARITY A N D E Q U I V A L E N C E IN COMPRESSIBLE FLOW

For sonic free-stream velocity M , = 1, B = y + 1, and the coordinates Y and Z and the halfspan s transform as t - l l 3 . Again an impulse consideration is instructive [27]. The impulse losses in the wake or the heating of the wake are proportional to the entropy increase. The entropy varies as the 3rd power of the velocity disturbance or, by (17.11), as t2. The extent of the shock and, hence, the extent of the wake is proportional to t-1/3by (17.12). The total drag therefore is proportional to t - 1 / 3t2 . = t 5 I 3 for similar flows. The same holds for the drag coefficient as will be seen later in (17.16). A simple interpretation of uredin (17.10) is possible if forni 5 of Table 4 is used. For M , # 1 (17.13)

ecrd is just the ratio of the U-disturbance to the difference between freestream velocity and critical velocity. Also vred

V

=

Ic* -

___ ;

U,I V l l - M2,I

W wed =

Ic*

- U,I VIl

- M2,I

.

(17.14) From (17.9), (17.11), and (17.12) the similarity laws for other quantities can be deduced. This will be done here for the aerodynamic forces. In (3.4), the equation for cp, the quadratic term of the velocity disturbance is multiplied by l - M W 2 . This term may be neglected in comparison to the first term a t transonic speeds. Since U-disturbances are quite large at transonic speeds even for slender wings and thin profiles, cp = - 2u is a good approximation. Therefore, (17.15)

-213 -

c p redtred

- cpt-2/3B1/3

is a reduced quantity by (17.11). The drag coefficient with respect to the thickness ratio t for twodimensional flow is also given by (17.15) if cp is replaced by c,; but usually cD is given with respect to the area unit in Y = 0. Then c, is proportional to cp T, and the transonic drag parameter for profiles is (17.16)

C,)rdtr2/3

= CDt--5/3B1/3.

The above expression is also the transonic drag parameter for wings of finite span, if the drag coefficient is referred to the planform. The change in the spanwise coordinate cancels out.

228

KLAUS OSWATITSCH

18. Transonic Flow past Profiles and Wings at Nan-zero Incidence

As indicated in Section 5, it is more practical to give the free stream an inclination E with respect to the x-axis than to consider the wing or profile at an angle of attack. Then, by (3.3), (18.1)

In general u, is zero or at least very small. For small angles of attack the approximation (18.2)

&=ZI,+

...

is then well justified. Therefore E , v-component, and thickness ratio transform alike. For all forms of the Prandtl-Glauert analogy the ratio of the angles of attack must be equal to the ratio of the thickness ratios for comparable flows. In the domain of linearization of the gas-dynamic equation the lifting problem can usually be separated from the thickness problem, permitting an independent variation of E and t. This is not possible for a nonlinear problem. Only in the cases of “small” and “large” angles of attack is i.t possible to avoid the proportionality of E and t. Of course, a “large angle of attack” does not mean “near or beyond stalling” (which would be prohibited already by the assumption of small disturbances). It only means that the angle of attack is large in comparison to the thickness ratio, in which case the flow is essentially determined by the angle of attack and not by the thickness. In transonic flow with small disturbances this is the case only for very thin wings, and can be realized in aeronautical design only for low aspect ratio. The limiting case, the lifting problem of an infinitely thin plate, has been treated theoretically several times. From a practical point of view the transonic flow over a wing a t a small angle of attack is more important. Here again, “small” means with respect to z. This corresponds to the calculation of dcLILIE a t E = 0. For sufficiently small E the pressure coefficient of the wing is determined by the thickness distribution. A problem of mixed elliptic-hyperbolic type results, but the effects of angle of attack are linear in E. Returning to the assumption of E >> t, we introduce in analogy to (9.1) a reduced angle of attack (18.3)

EA = E B~ ~ t - ~ =a ~,d ,.

In analogy to (17.9) the reduced Mach-number becomes (18.4)

SIMILARITY -4ND EQUIVALENCE IN COMPRESSIBLE FLOW

229

For comparable flows, M a 2 - 1 changes as ( E - B)2/3.At M , = 1, all flows are comparable. Contrary to (17.9),Eq. (18.4)is valid for an infinitely thin are replaced plate. Eqs. (17.11),(17.12),and (17.15)hold also if t and tred by E and for instance,

Ec’’ = c p .

(18.5)

E-2/3.

B1/3.

This holds also for the normal force coefficient,

(18.6)

cnred.

Er=/3 = C, . &-2/3.

and implies that c, grows as &’Is for a profile at M , = 1. The infinite gradient at the orign (Fig. 28) is in contradiction with all experimental evidence ; obviously, the assumption of “large angles of attack” breaks down. One may conclude only that the normal force coefficient grows less than linearly with E for large angles of attack. In an experimental verification this result may be observed at large E , but caused by partial stall and similar effects that will be present in- and outside the transonic domain. For the examination of the transonic flow at small angles of attack the reduced gas-dynamic equation (17.1) is differentiated once with respect to E , ~ . Denote this differentiation by a prime. Then

-lo

-5

FIG. 28. Normal force coefficient for a profile at M , = 1 under the assumption E << t.

(18.7) The same could be done with the equation of irrotationality, but because of the linearity of these equations this would not give rise to any new considerations. 0, (18.7)may be considered as the gas-dynamic equaIn the limit tion for the rate of change of the velocity components with respect to the angle of attack urd, vrd, wid. In these new dependent variables (18.7)is linear, but it involves terms that depend on the solution of the flow at zero angle makes (18.7)identical in t y p e with of attack. The coefficient - (1 + ,

I

230

KLAUS OSWATITSCH

the actual flow equation a t = 0. For the comparison of two transonic flows at small angles of attack, the flows at zero angle of attack must be identical. The characteristic quantities (17.9) for Mach-number and thickness ratio and the characteristic quantities (17.12) for span and thickness ratio must then be the same for the comparison of two transonic flows at small E . The derivation of zlrd with respect to &,d is by (9.1) and (18.3) (18.8)

because the constant factor A appears in the numerator as well as in the denominator. Eq. (18.8) still holds for all upwash factors, therefore for all forms of the Prandtl rule. The special conditions of transonic flow appear only in the dependance of cot a, on t. By going from u to cp, it follows that (18.9)

and, on eliminating cot a,

by means of (17.8), the final form '

1/3

cfieti"B1/' = Cgr&tred

(18.10)

has been achieved. In the special case M , = 1, cp changes as t - l I 3 . Very thin wings are therefore desirable. The normal force coefficient c, and moment coefficient c, result by integration of the pressure coefficient over the wing. The resulting equations are completely analogous to (18.10) (18.11)

c,,

.

~ 1 / 3+/3

.

(18.12)

c,

*

113 BlP. t 1 / 3 = cm red * zred.

'

= &red

1/3 tredp

Measurements on wings with a double-wedge section of 4.5" wedge semiangle and different span by W. G . Vincenti [38] show that up to E = 2" and often beyond that value the lift coefficient is linear in E. This implies that in this region C,/E may be substituted by c,. Using this fact we conclude from (18.6) that c:de:$

= ~,.+/3Blla.

Since the angles of attack fulfill the same condition as the thickness ratios, namely tr& = (18.11) can be deduced directly from the last equation. In (18.10)to (18.12) the angle of attack does not appear as an additional parameter, but it has to be kept in mind that the characteristic quantity (18.11) depends on two additional characteristic quantities, namely the reduced Mach-number (17.9) and the reduced halfspan s(t * B)'IS, (17.12). Therefore a three dimensional plot would be necessary. The right-hand side of Fig. 29 shows the cL-parameter over the reduced Mach-number for s = bo.

231

SIMILARITY A N D EQUIVALENCE I-N COMPRESSIBLE FLOW

+

B has been chosen as ( y 1)Mm2. I t can be seen that, very near to sonic speed, cL is practically independent of the reduced Mach number. This conclusion can be drawn also from the calculations of T. Gullstrand [33]. 0.08-

0.08

cLs[t. (y + 1)~:11/3

I

0.06‘

* .

.+

0.06-

c-. 0.04.

. ;; (M,

0.02.-

0.04.. -

~ ) ( T * B ) - ’ / ~1.40 <

o 8

1

s

00787

1 2

00787 0 100

s[(y

2 + 1)M,t]”3

=

r = 00787

P

FIG.29. Right-hand side : characteristic lift coefficient over reduced Mach-number for a profile of t = 0.0787. Left-hand side: characteristic lift coefficient over reduced halfspan for small reduced Mach-numbers.

For wings of low aspect ratio this follows also from Jones’ theory. It can therefore be assumed that for all aspect ratios the lift does not depend very strongly on the reduced Mach-number near sonic velocity. This is confirmed by Vincenti‘s measurements. The left-hand side of Fig. 29 shows the characteristic lift coefficient over the reduced halfspan for sufficiently small reduced Mach-numbers. The British experiments [39] are well in line with the American ones [38]. Fig. 29 shows also that the lift coefficient of wings of aspect ratio 4 is about 20% lower than the value for the corresponding profile. 19. Bodies of Revolution in Transonic Flow The transonic similarity law for wings and profiles has been found to be a special form of the similarity laws in the domain of linearization; hence it may be expected that the transonic similarity law for bodies of revolution is also a special form of the similarity laws of Section 11 for bodies of revolution in the domain of linearization. Eq. (11.3) holds only if a particular relation exists between A and p. (The same is true for M , > 1.) In

232

KLAUS OSWATITSCH

Section 17 the permissible relations for M , > 1 between upwash factor A and Mach-number were deduced in full generality for three-dimensional flows. The results of Table 4 are also valid for axisymmetric flows because they assume only a reduction of the first term of the gas-dynamic equation (17.1). The gas-dynamic equation for axisymmetric flows differs from (17.1) only in the last term where awred/az has to be replaced by vrd/y. On changing from the upwash factor A by means of Table 4 to the factor B which is always different from zero, the special form of (11.3) valid for transonic flows M , 5 1 follows: (19.1) For corresponding points (11.4) is also valid in transonic flow because it does not include A . The transonic similarity laws can be deduced from those valid in the linearized domain of the gas-dynamic equation, if in (11.5), (11.6), and (11.8) the Mach-number expression p or cot 01, are replaced by z with the help of (19.1). The Mach-number appears only as argument of the logarithm in the above mentioned equations, for M , > 1 in the form

The term with In tWd appears now on both sides of (11.5), (11.6), and (11.8), if (19.2) is inserted in these equations (note that Q ” ( x ) / t 2 = Q i f ( x ) / t i 2 ) . Contrary to the considerations in Section 11, now two arbitrary flows a t Mach-numbers M , , and M,, past bodies with tl and t2 are compared; by (19.1), they have the same reduced thickness ratio or the same reduced Mach-number, (19.3)

M,51:

M 2, - 1 t2B

1

==FT Zrd

In addition, both flows are transonic. It now follows immediately from (11.5) by means of (19.2) that

Instead of (11.6) we give the corresponding expression for the pressure coefficient cp :

This form is extremely accurate in the transonic regime.

233

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

In the paper by Oswatitsch and Berndt [29] where this similarity law was deduced, the exact flow over a circular cone was taken for comparison.

1 0.2 0.I

0 -0.I

-0.2 0

20 40

80 100 120 140 160

-Ee --

Xmm

60

-

FIG.30. Pressure measurements at M , = 1 by G. Drougge for two spindles of different thickness ratio, B = ( y 1) M,2.

+

Fig. 30 shows pressure measurements by G. Drougge [41] for two spindles V2 at M , = I . The results are given in the usual with T, = 6 and t, = Q 1)Mm2is applied; ( y 1) in and in the reduced coordinates. B = ( y the logarithmic term can be omitted. The same paper contains another good example, the drag variation with Mach-number. For the formdrag coefficient with respect to the cross section t 2 n / 4 the following relation holds by (11.8) and (19.2):

+

+

(19.6) Here Q'(L) is the derivative of the cross section at the end of the body; it vanishes if the body ends at the maximum thickness or if it ends conically with zero thickness. In both cases the logarithmic term vanishes, and the drag coefficient as well as the pressure coefficient of similar flows is proportional to t 2or to the cross section. Surface points of two comparable bodies correspond to each other only in the stream-line analogy. For flows which satisfy the transonic similarity

234

KLAUS OSWATITSCH

laws the state at the surface point of one wing may therefore be subsonic, whereas at the surface point with corresponding x- and z-coordinates of the comparable wing the state may be supersonic. For wings of large aspect ratio and profiles this paradoxon is resolved immediately : the transonic similarity laws presuppose very slender bodies. In addition, the flow does not change very much with y, hence the states at two surface points with corresponding projections cannot be very different. One point cannot lie far in the supersonic domain when the other lies in the subsonic domain. For bodies of revolution, however, the transonic similarity laws are still valid for appreciable thickness ratios, and the change of the flow perpendicular to the axis is not so small. Surface points of comparable bodies a t the same abscissa may we11 lie on either side of the sonic limit. As an example, the flow over a circular cone will be considered later. The line where U = c* is now no longer very significant, provided it is near the body axis. The flow conditions near the axis are essentially determined by the Laplace equation for the cross section (see (6.5)). Whether the first term of the gas-dynamic equation (17.1), changes the type of the equation somewhat t o the elliptic or hyperbolic side is not so important. At larger distances from the axis or body such considerations become false, and the three terms of (17.1) are of equal importance. 20. Transonic Flow past a Circular Cone In order to give a one-parameter representation of the transonic flow past a circular cone, we introduce again combinations of coordinates and velocity disturbances which remain finite on the axis of the cone. The coordinate to be chosen for conical flow is of course (20.1)

q = - =Y

x

-cotam, Y

x

and it seems reasonable to take instead of v the product v Y , which stays firfite on the axis. Written in the reduced form, we have (20.2)

Y qwd(q) = A cota=-v X

After introduction of the apex semi-angle

=

B

Y

-V-

cot2am X

~

a0 we

find for the limit Y - + O

(20.3) In conical flow, tan 6, is the quantity corresponding to the thickness ratio, and by (19.1) the above expression is equal to the square of the reduced thickness ratio zd2.

SIMILARITY A N D E Q U I V A L E N C E IN COMPRESSIBLE FLOW

235

In complete analogy to (19.4) the expression

can be formed, which is finite for Y = 0.

tan

D

1.8 cot a m ’ 1.6

1.4

1.2

1D 0

0.2

0.4

0.6

\OB

FIG.31. Reduced shock-angle aced over reduced vertex semi-angle tAOred for cones after Oswatitsch and Sjodin, B = Ma* ( M a 2 - l)/(M,* - 1 ) .

Now the differential equations and shock relations can be rewritten in the new variables given by (20.2) and (80.4). A system of ordinary differential equations results which can be solved in the same way as in the case of the exact supersonic flow past a circular cone: starting from the conditions behind the shock, one proceeds by numerical or graphical integration towards the cone surface. This has been done in the paper by Oswatitsch and Sjodin [37]. Fig. 31 shows the reduced shock angle urd over the reduced apex semi-angle Gored. The maximum angle 6, for which the shock may still be attached is

(20.5)

B tan2 6, tan2a,

= 0.74.

+

With B not much different from ( y l ) , the apex semi-angle is proportional to cot a,. If the tangent is replaced by the angle, (20.6)

[2“

6,=0.55 --a,

1+

...,

quite accurately up to M , = 1.20. Fig. 32 shows in reduced form, what corresponds in our treatment of the problem to Busemann’s “apple curve”. In addition, the apple curve for the linear theory, the sonic curve, and the shock polar are given. The connecting curves present the conditions in the flow field. Starting from the

236

KLAUS OSWATITSCH

x-axis, represented by the apple curve, the value of qvred increases until the critical velocity is reached; it then decreases toward the shock polar. For a certain value of M , and 6, the ordinate of the apple curve is known by (20.3), and Fig. 32 gives the abscissa. For different values of q one gets then different values of the reduced zc-component.

_. X

-1.8 -1.6 -1.4 -12 -1.0 -0.8 -0.6 -0.4 4 2

\ Y

FIG.32. Reduced ‘apple curve’ for a cone after Oswatitsch and Sjodin, B = M a * ( M a 2 - l ) / ( M m *- 1).

Thus a representation of the axisymrnetric flow field is possible without difficulties, but the body contour cannot be drawn into the plane of the reduced flow because, in transonic flow, the body contour cannot be expressed in reduced coordinates. This is in contrast to hypersonic flow. The ordinate q of a certain state of the flow between shock and apple curve can be represented by the inclination of the connecting curve, (20.7)

tabulated in the original paper.

V. HYPERSONIC SIMILARITY 21. Similarity Laws in Hypersonic Flow.

The reduced gas-dynamic and vorticity equations for high Mach-numbers (MmZ>> l ) , (16.6) and (16.10) are repeated here for convenience. In twodimensional flow

237

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

2 vred

avred

-ar ’

(21.1) 2

I n Section 15 it was shown that the stagnation pressure losses and therefore entropy increases are the same in corresponding points of the shocks. According to Section 16, Eqs. (21.1) and (21.2) present the reduced system of differential equations, which could only be supplemented by the equation for the entropy. From the value of the upwash factor in the stream-line analogy and (9.1), v, but also t and E satisfy (21.3)

z~ = t c o t a,;

= E cot a,.

The reduced thickness ratio was first introduced by H. S. Tsien [46] who called it the hypersonic parameter. In this first paper about hypersonic similarity and also in the following ones VMm2- 1 was replaced by M,, which is always justified; but that increases unnecessarily the number of similarity laws and restricts the domain of application. The reduced u-component follows from the stream-line analogy. In Section 14 it was shown that an analogous relation holds for the velocity disturbance when the quadratic terms are included (see (14.8)). The expansions of the pressure coefficient .cp in terms of the velocity disturbances for instance (3.4),are not applicable without due caution, because convergence is not certain at high Mach-numbers. Eq. (14.30) shows, however, that the temperature disturbances are the same at corresponding points, and since the entropy is the same, the pressure disturbances must also be the same. For high Mach-numbers the pressure coefficient is approximately

(This was The pressure coefficient is therefore proportional to tan2 u,. also found to be true for smaller Mach-numbers in the stream-line analogy.) The reduced pressure coefficient thus becomes (21.4) becomes infinite for M, 4 bo. (Recall that the pressure increase for wedge-flow becomes also infinite if M , --* m.) Therefore the same device

cgd

238

KLAUS OSWATITSCH

as in the transonic domain is used: M , is eliminated from (21.3) and (21.4), and reducing the pressure coefficient with t we get (21.5)

which is comparable to (17.15). Choosing the upwash factor 1 A =-tanam

(21.6)

t2

would only mean a change in name for the reduced quantities; in particular, the combination (215 ) would reappear as the reduced pressure coefficient. Already Fig. 8 uses hypersonic characteristics as coordinates, if the denominator (1 u) in the ordinate is disregarded. But this denominator can be put equal to one without serious errors in view of the small u-disturbances. .4s always for wedge and cone the tangent of the apex semi-angle plays the role of t. The abscissa tan 6,/tan u, is then the ratio of the surface inclination to that of the Mach lines. Fig. 33 shows the characteristic quantity (21.5) over the hypersonic parameter (21.3) evaluated from the exact theories for wedge and cone. I t can be seen very well that this form of the hypersonic similarity is valid far down, close to the transonic range (see also Van Dyke [49]). For large values of the hypersonic parameter, two-dimensional and axisymmehric flows are nearly the same. In addition, the range of thickness ratios, in which the similarity laws still apply, may be estimated. Measurements by McLellan [50] show that the influence of the boundary layer is small. He also performed experiments with a diamond profile and finite wings at M , = 6.9. Additional measurements for bodies of revolution were carried out by Eggers, Savin, and Syvertson [51]. The transformation of the coordinates and the halfspan s follows from @.a), (8.9), and (21.3):

+

(21.7)

_ S M_ -

5 .

t

’

-Y _ -y .

- _z - . -

z

tred

t,ed

t

(Once more the stream-line analogy proves valid also for bodies of revolution because the aspect ratio is transformed as the thickness ratio.) The drag coefficient referred to the cross section transforms as cp in (21.5). The drag coefficient referred to the planform c, follows the rule (21.8) The limiting case M, eration; it corresponds to

+ 00

tred --+

(hypersonic limit) requires special considM. Because the thickness ratio t stays

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

239

usually finite, the inclination of the Mach lines in the free stream must be small compared to the surface inclinations of the body. The limiting case

6

5 4

3 2 I

0 FIG.33. Hypersonic flow past cone and wedge.

zd -+ 00 has more than mere theoretical importance, for it gives the limiting values for cp, c, and c, etc. for the body under consideration when M , -,m.

240

-

KLAUS OSWATITSCH

For thick bodies (T 1) this limit is already reached a t medium supersonic Mach-numbers. Fig. 34 shows the pressure coefficient, calculated by exact theory, of a circular cone with an apex semi-angle of 40" and axis parallel to the free stream; it does not vary much between M , = 3 and M , = 4. The hypersonic limit has been considered by K. Oswatitsch [47] without the restriction to slender bodies. For all bodies, then, finite values result for the aerodynamic force coefficients and the pressure distribution. The hypersonic limit ( M , + w) is in a sense the counterpart of M , -+0, but also of M, + 1. At these three free-stream conditions similar flows exist at the same Mach-number, which makes it possible to consider the dependence of the aerodynamic force coefficients on the angle of attack for an infinitely thin flat plate at M , -+ 00; this will be done in the next section. However, in contrast to the transonic limit, the ratio of the specific heats, y , cannot be eliminated irom the equations; thus only gases with equal y-values can be compared. Also, there is no exact analogy between hypersonic and Newtonian flow [53] because the Newtonian flow theory corresponds to y = 1 which cannot occur in a real gas. At very high Mach-numbers the boundary layer has a definite influence on the flow past slender bodies, which is especially strong for two dimensional flow. The heat produced close to the wall diminishes the density there considerably and FIG.34. Pressure coefficient for a thick increases the displacement effect cone 1 9 = ~ 40' at Mach-numbers 2 to 4 after Hantzsche and Wendt. of the boundary layer. Under such circumstances the boundary layer may fill out a considerable part of the space between the wall and the shock. Its influence may be of primary importance as was shown by the experiments of Bogdonoff and Hammit [62]. 22. H y p e r s o k c Flow at Non-zero Incideme

The essential point for flows at angles of attack is the same for all similarity laws considered in this review: the angle of attack changes as the thickness ratio (as follows also from (21.3)):

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

241

(22.1) In (21.5) t 2may therefore be replaced by E ~ . This is of special importance if the flow past an infinitely thin plate is considered (t= 0). Eq. (21.5) then becomes

‘“t

and for the normal force coefficient (22.3)

cn

I /

- %red

2 - __ 2 &red

holds in full analogy to (18.5) and (18.6) for transonic flow. As .+ o,the left-hand sides of (22.2) . . and (22.3) do not vanish. At sufficiently high FIG.35. Quadratic variation of the lift Mach-numbers the lift ‘Oefficoefficient with angle of attack at high cient then varies with the supersonic speed. square of the angle of attack (Fig. 35). Eq. (22.3) is valid for hypersonic Mach-numbers and even in the lower part of the domain of linearization. But only for very high Mach-numbers, more exactly for very high values of the parameter (21.3) is it possible to compare the flows at different E for the same M , . In transonic flow cp was , as in proportional to in hypersonic flow cp is proportional to E ~ but, transonic flow, E >> t,if (22.3) is to be applied to a body of constant thickness. The exact value of t is only unimportant, if (22.1) is very large. In general the case (22.4) In this case the normal force coefficient is proportional to the angle of attack:

has more practical importance.

&

-‘G< 1

(22.6)

I

:

cp=

cpp;

Cpred

= CpredEred,

242

KLAUS OSWATITSCH

If (21.5) is written by means of (22.1) as

I

cfired ZredEred = c p / Z E ,

we have by (22.5) (22.6)

Elt

<< 1 :

C;red/Zred

= CpJt

and obtain for the derivative of the normal force coefficient

4 __ c75, tan 8, 30$0

20-

.

x

5'

+

10'

0

20"

7

6

* 10.-

i

i

0-

FIG.36. Hypersonic flow over a wedge at small angles of attack.

Contrary to the considerations of Section 18, the last two sections are based on the stream-line analogy and hold also for bodies of revolution. At hypersonic speeds the influence of the aspect ratio is small (see Fig. 19). Fig. 36 shows the results for the wedge a t small angles of attack in hypersonic flow.

VI. UNSTEADY FLOWS 23. Unsteady Flows

Here we restrict ourselves mainly to the domain of linearization of the two-dimensional gas-dynamic equation, assuming limits of the domain of linearization to be the same as for steady flow. Large accelerations, however, require special considerations.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

243

The linearized gas-dynamic equation for the velocity potential @, which satisfies

U=@x,

1

V=@y;

-

2

(UZ

+ v2

-

U2,)

+

1

(9- C L )

~

= - @t,

r+l

(23.1)

is (see [l, Chap. lo]) (23.2)

(1 - M L ) @ x x

+

2

1

@yy

- 1@u

-

C,

- M M , @ x i = 0. Cm

I n addition to the two steady terms there appear now two unsteady terms, but only one new parameter is at our disposal. For accelerated motion such a parameter is for instance the time in which the body doubles its velocity. For periodic phenomena (flutter) it is the circular frequency (0. I n what follows, periodic unsteady flows will be considered. If we introduce in (23.2) a disturbance potential which includes the periodic time dependance explicitly,

+

(23.3)

@ ( X , Y , t )- U,X

=

U,+(x,y)eiot,

we get for supersonic flow in reduced coordinates (8.4) (23.4)

M <1

+Iz

- +yy -

M2,

w2

~~

-1

u:

M2 w ++2ipm-+dr=o. M2,-1U,

Again the chord has been set equal to unity so that o/U, may be called the redclced frequency. Similarity of the flow can only be achieved if for comparable flows the coefficients of and c $ ~ are the same. But this is only the case for equal Mach-number M , and equal reduced frequency w / U , . Ths is no progress beyond the demand of the general laws for mechanical similarity and the affine distortion of the axes has not brought any advantage. In the earlier sections it was of main importance that the upwash factor A could be chosen freely. However, only one additional, possibly nonlinear, relation can be imposed on A . Multiplication of by A would not influence the coefficients of $C and in (23.4). Therefore affine similarity laws for unsteady flows are only possible in such limiting cases, where the two new coefficients may essentially be reduced to one. Even so, (23.4) is valid for M , > 1 only, but the considerations may be extended to M , < 1. The simplest limiting case is M 2 , >> 1. The two new coefficients of (23.4) then simplify t o ( W / U ~and ) ~( w / U , ) , respectively. The reduced frequency alone is essential. Because the assumption M m 2>> 1 is already satisfied within the domain of linearization of the gas-dynamic equation no contradictions will result to earlier assumptions.

+

+

244

KLAUS OSWATITSCH

The limiting case M m 2<< 1 is more problematic. The coefficients reduce to ( C U / C , ) ~ and (M,w/c,). If (w/c,J is small, the last two terms may be neglected. Then the problem has been reduced to the steady gas-dynamic equation with unsteady boundary conditions. For moderate values of w / c , and small values of M , the neglect of the last term of (23.4) alone would be appropriate, and w/c, would remain as the only parameter. Such an approximation cannot be made without a closer estimate of the magnitude of +xx,+yy,+x, in comparison with 4. This is also the reason why some caution has to be exercised for the limiting case (1 - 1 / M m Z ) / ( w / U ,<< ) 1. Here the variation of the v-disturbance will be much stronger than the variation of the u-disturbance, and the first term of (23.2) may be neglected. If the affine coordinate transformation

M,Y

(23.6)

=y

is introduced, (23.2) becomes by means of (23.3) and (23.5) (23.6)

independent of M , . As in the case of steady flow, it is useful to refer the disturbances to the upwash on the x-axis v ( x , O ) . For flutter phenomena we may set

-

v ( x , O ) = a f(x)eiut

(23.7)

where a is a definite amplitude, usually the maximum value, and f ( x ) is the amplitude distribution over the x-axis. For flutter phenomena a is the counterpart of t or E in steady flow. By means of (23.1) and (23.31, (8.4) or (23.5), the boundary condition is 1<M , :

(23.8)

or

cot a,

- #y(x,O) = a/(%)

M,+r(x,O)

=a/@).

For similar solutions the dstribution of the amplitudes f ( x ) must be the same. The differential equation (23.4) for the above mentioned limiting cases and the boundary condition (23.8) may be written in a form independent of M , if a redlcced potential

(23.9)

or

is introduced.

245

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

The representation chosen here differs slightly from the Prandtl-Glauert analogy. Now the constant amplitude a may still be chosen as a suitable function of M , , and different forms of the analogy result from the specific choice of a ; even an adjustment beyond the domain of linearization is ill not be discussed here. possible. This w

P1

p2

P3

M

p5

06

01

Q8 09

1

1.1

U

U

k=--

1.4 1 w

3

urn

FIG.37. Two-dimensional flutter at different Mach-numbers and reduced frequencies.

For unsteady flow the pressure coefficient is proportional to the variation of the sound speed up to the first order disturbances. By (23.1), then, cr and c, are:

246

KLAUS OSWATITSCH

where only the real part is taken. If one changes to the reduced potential (23.9), a reduced normal force coefficient csd results which will be written here with the symbols used in the paper by Garrick and Rubinow [54]:

(23.12)

or 1

1 w

l o

L , and L, are real quantities; c , , ~is the real part of the right-hand side. In [54], which is the source of Fig. 37, the reduced frequency is taken as k = $olU,. For the above mentioned limiting cases, the quantities 1

and

OT

and

SIMILARITY A N D EQUIVALENCE IN COMPRESSIBLE FLOW

247

have to be essentially the same for all M , . In Fig. 37 the results of the calculations [54] have been presented as if they were measurements for different Mach-numbers and reduced frequencies. This form of presentation has been chosen because the points do not lie close enough to draw a curve through them. The wavelike variations of the ordinate correspond to reality and are by no means inaccuracies. By (23.13) the ordinates of Fig. 37 are and o l U , could of course reduced quantities. Different combinations of have been chosen, e.g. the amplitude and phase of cSred.

FIG.38. Two-dimensional flutter a t M ,

%

1 and large values of the reduced frequency.

For the limiting case of high Mach-number represented by M , = 5, 10/3 and even 512 the ordinates agree well for all values of k . In addition,

248

KLAUS OSWATITSCH

they are nearly independent of k , but that does not matter here. The limiting case M m 2<< 1 can of course not be studied in supersonic flow, but the case of k - 0 can be discussed. The gas-dynamic equation reduces then to its steady form and the similarity laws of steady flow hold, as can be seen also from Fig. 37. Fig. 38 shows the results of Garrick and Rubinow for the limiting case (1 - 1/Mm2)/(w/U,)<< 1. For values of the reduced frequency larger than unity the agreement is quite good for all Mach-numbers, even though the assumption (1 - 1/Mm2)/(w/Um) << 1 is not well satisfied for the Machnumbers M , = 6, 10/3 and 5/2.

VII. BODIESOF Low ASPECT RATIO 24. Bodies of Low Aspect Ratio at Non-zero Incidence

In this section it will be shown that, beyond the laws for bodies of revolution at incidence, statements can be made for the larger class of bodies where the span is small compared to the chord. I n all such theories for bodies of low aspect ratio, bodies of the same aspect ratio can be compared. This corresponds to two-dimensional flow (infinite aspect ratio). Already in connection with (6.5) it was mentioned that in the vicinity of wings of low aspect ratio the Laplace equation is approximately valid in planes x = const. The gas-dynamic equation (2.5) written for disturbance components reads av

(24.1)

-.

aY

+ -aw= az

-

azl

(I - M2)-.

ax

Here X , Y , Z are not the reduced coordinates. For bodies of revoluti.on without incidence aw/aZ was set equal to vjY, and the right-hand side was neglected. Now the right-hand side is always small for the bodies considered in this section. The easiest way to estimate the order of magnitude of the different terms in (24.1) is to consider a delta wing in supersonic flow with subsonic leading edges. For such a wing ES

(24.2)

21 =

scot a,)

zeJ=

)‘I- (Z/SX)Z’

&

z/sx

E’(s cot am) y1 - ( Z / s X ) 2 ’

where E‘ is a complete elliptic integral (e.g. [l, p. 5291. The ratio of both terms becomes at the wing surface (24.3)

(&a

-

= - s2 cot2am(Z/sX)2.

SIMILARITY A N D E Q U I V A L E N C E IN COMPRESSIBLE FLOW

249

Because z / s X is always smaller than unity at the wing the right-hand side of (24.1) becomes small as s2 cot2 a,. If is the non-reduced disturbance potential where

+

(24.4)

v=+y,

u=+x>

W=$zz,

the two-dimensional Laplace equation (24.5)

holds in the vicinity of the body. By (24.5) and the boundary conditions at the wing 4 is only determined up to an additive function* F ( x ) . For the cross section x of the delta wing with the local halfspan - s ( x ) Z s ( x ) , the solution

< <

+ six1

would satisfy the differential equation (24.5) as well as the boundary condition for Y = 0, v = 0. The boundary condition a t infinity 4 y = E for v Y 2 Z2 -+ 00 requires F ( x ) G 0, but (24.6) is only valid in the vicinity of the wing, hence a closer investigation is necessary. Now for a symmetrical body a t small angle of attack the condition

+

(24.7)

holds. (24.8)

+(x,Y,Z)= - + ( x , - Y . Z ) But then it follows from (24.6) that, indeed,

F ( x ) G 0.

Within a certain accuracy, the lift of symmetrical bodies of low aspect ratio can be determined in this way and is found to be independent of Mach-number. This result is due to R. T. Jones [15] and includes also bodies of revolution. To estimate the error involved, the same boundary-value problem will now be treated with the complete differential equation (24.9)

The right-hand side can be thought of as a source distribution of a Poisson equation. The well known solution of the Poisson equation is (24.10)

*

We write now x instead of X.

250

KLAUS OSWATITSCH

where $L is the solution of the Laplace equation for the flat plate a t incidence, i.e. the solution of R. T. Jones (24.6)with F ( x ) = 0. The "source distribution" (1 - M2)au/ax is a function of the variables of integration. In the domain of linearization of the gas-dynamic equation (1 - W ) is replaced by (1 - Mm2). The effect of angle of attack for a symmetrical body implies that u and azc/ax are odd functions of Y . Therefore the contributions of awlax and u at Y = 0 from the upper ( Y > 0) and lower (Y < 0 ) sides cancel each other, and the vortex distribution at Y = 0 is not influenced by the disturbance integral in (24.10). But still $ differs from $L, because the boundary condition at Y = 0 is not satisfied. Differentiation of (24.10) with respect to Y gives a t

FIG.39. Inner and outer domain of integration.

Because the odd function (1 - M2)azl/ax of 1;1 is multiplied by an odd function of 17 the values of the upper and lower halfplane add to each other. We shall estimate the error only on the axis ( Y = Z = 0) and divide for this purpose the domain of integration into an inner and outer domain, separated by a circle of diameter equal to the local span 2s(x) (Fig. 39). From the solution for conical flows (24.2) it can be seen that u = O(ES) for small aspect ratios, since the complete elliptic integral E'(0) = 1. Because & / a x is of the same order of magnitude as u and does not change sign in the upper halfplane it may be taken in front of the integral. The result for the upper inner domains is:

(24.12)

-

(ML - 1 ) ~ s ~ .

>s

of (24.6) leads to

cos #ad# r=O

*,-n 2

For the outer domain an expansion in Y for

Y

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

251

The order of magnitude of u is the same as in the inner domain for Y = sx namely u = ~ ( E S ) . Far outside a vanishes only as l/r. If & / a x of (24.13) would be substituted into the disturbance integral (24.12) with the limits s ,< r 00 a logarithmic infinity would result. The reason for this is that a u j a x was taken from (24.6) which is only correct in the neighborhood of the wing, say r x . Far from the wing u actually vanishes stronger than 1/r.

<

<

M

a2

Of 0

02

a4

0.6

0.8

1.0

FIG. 40. Comparison of the Laplace and exact solution for a Delta wing with subsonic leading edges at incidence, s = 0.50.

Fig. 40 shows a comparison of the solution of the Laplace equation with the exact solution .away from the wing. The latter has to vanish outside r cot a = x . A t this line however & / a x has an integrable infinity. A more careful consideration shows that the integral (24.12) has an order of magnitude larger by the factor In (sVMm2- 1) because of the factor l/r. This is unimportant in most cases because usually the aspect ratios are not so small that this term would have values far from unity.

262

KLAUS OSWATITSCH

The neglect of the right-hand side of (24.9) a t Y error in the v-component of the order of magnitude: (24.14)

Y=O:

4y~~IM:-lls21n(sVM:-

=0

gives rise to an

1)r~d.

6 is the error, referred to the v-component in the free stream $ y = E . The value of u a t the wing contains the same error. From (24.6) with F ( x ) = 0 the solution of the Laplace equation is for Y = 0 (24.16)

I$L

=& Vs2(x) - 22.

By means of (24.14) it follows that for

An error of the same order of magnitude should be obtained by an expansion of E’ in the exact formula for linearized supersonic flow around delta wings with subsonic leading edges, (24.2). Indeed (24.18)

E’(s cot a,)

=1

+

s2 cot2am

+ .. . ,

which gives rise to an error of the same order of magnitude as the approximate theory based on (24.6). Of course, (24.18) is better than (24.17) for the determination of the error of the conical flow around delta wings, but (24.17) is also valid for arbitrary wings and also for subsonic flows. The error becomes large for large values of the Mach-number. This can be seen already from Fig. 19. But a t Mm = 1 the error would become zero for all finite values of s, if the results of the linear theory could be extended into the transonic domain. I t is not difficult to get an estimate of the accuracy of the Jones theory a t M , = 1. As in Section 18 we consider only a wing symmetrical in Y at very small angles of attack. If the gas-dynamic equation (24.1) is differentiated with respect to E the following equation results: (24.19)

&8 -

ay

+ __ = - (1 az

M2)

- + 2MM,-, au8

ax

ax

where the right-hand side is again odd in Y for E = 0 because the functions M and u in the symmetrical problem are even in Y . For sonic free-stream velocity the following relations are valid: Mm=l, (24.20)

M2-1=(y+1)u+

**.;

ZMe=(y+l)~lE+ ... .

SIMILARITY AND EQUIVALENCE I N COMPRESSIBLE FLOW

253

These relations are of the right order of magnitude throughout the transonic domain ( M , - 1 << 1 ) . The order of magnitude of u, is the same for the whole region where (24.5) is \.slid, namely u, SE. The estimates of the integrals will not change very much, only the order of magnitude of the source terms has changed to

-

(24.21)

(1 - M2) ax

+ 2MM,

-au

ax

(y

+ l)us(x).

The expression (24.17) for the error is also correct in the ransonic region where (24.22)

Here u denotes the average velocity disturbance for the wing a t zero angle of attack. Of course, ( y + 1) could be omitted for the estimation of the error; it has been included here because this factor is typical for the transonic domain. The order of magnitude of zd will be considered in the next sections (see e.g. (26.5)). Eq. (24.6) corresponds to the formula of R. T. Jones for the lifting problem of a flat plate. This result goes beyond the usual similarity theory, because it gives information about the complete velocity distribution. Only a t the trailing edge certain conditions have to be met which will be simply summarized : d s l d x has to become zero before or at the trailing edge. Within the above mentioned accuracy the following similarity theorem can be established : For wings of low aspect ratio the u-disturbance is proportional to ES in corresponding points that is for points of the same ratio Z / s ( x ) . The same proportionality holds also for the normal force coefficient with respect to the planform. This similarity theorem, which is independent of M , , is the better satisfied the closer M , approaches one. If the formulas of the linear conicalflow theory are applied for M , = 1, which is wrong, the result of R. T. Jones would arise exactly. The error would be zero contrary to the Prandtl rule where the error goes to infinity when M , approaches one. 25. Bodies of Low Aspect Ratio ut Zero Incidence; La&,0 1 Equivalence

The Laplace equation (24.5) for the “cross sectional flow” with the boundary condition for a symmetrical body a t zero angle of attack, which is (25.1)

has the solution

Y 40- :

+y(x,O,Z) =vn(x,Z),

254

KLAUS OSWATITSCH

where F ( x ) can no longer be determined by a simple consideration. The difficulties would not be resolved even by the assumption that (24.5) is valid for large Y 2 Z2 so that the conditions at infinity could be used. The expansion corresponding to (24.13) for r > s ( x ) leads to

+

S(X)

:

q3 = F ( x )

+ -In 1 jT

r -- s

1 cos2$

j.vot,,,

+ . .. .

--s

(26.3)

and

FIG.41. Conical flow field for a rhombic cone after Hjelte.

The potential function has the logarithmic infinity of a two-dimensional source distribution for r = 00. u = t#z and aulax diverge in the same way for r = d~ because the differentiation with respect to x in (25.3) affects only the function F ( x ) and the two integrals which depend on x through 'vo;

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

255

and the expansion (25.3) is not correct either for large T because it has been obtained from the solution of the Laplace equation valid for Y2

+

2 2

-

s2.

The logarithmic term is purely axisymmetric. Only the last term in (25.3) reflects the extension of the body in the spanwise direction in the harmonic function cos 2 4 which has its maximum on the Y and Z-axis and is zero for =n/4. But this term is much smaller than the logarithmic term for moderate Y , essentially so by a factor because 5 s. Therefore a wing of low aspect ratio at no incidence acts at a moderate distance from the wing very much like a body of revolution. OnIy far away from the body the propagation of the disturbances in the x-direction becomes important. In the whole subsonic regime of linearization all bodies act like dipoles at large distances; in supersonic flow all disturbances are zero outside the head shock. Fig. 41 shows the flow conditions for a rhombic cone in supersonic flow; the values are taken from the paper by HjeIte [22]. After some calculations the following formula results for M , = VT and s ( x ) = s x :

<

@ = -

s tan6

2n v1--2 (26.4)

This expression can be expanded for small s, and the result is

(25.5)

-

Bodies of low aspect ratio with the same cross-sectional distribution Q ( x ) will turn out to be of special interest. They will be called equivalent

bodies. In particular the body of revolution which corresponds to a wing of the same Q ( x ) is called eqnivalent body of revolution. The cross-sectional area is here only a geometrical picturization of the source strength a t the cross section. Equivalent bodies have the same

(25.6)

256

KLAUS OSWATITSCH

The circular cone of apex semi-angle 6, which is the equivalent body of revolution to a rhombic cone of apex semi-angle t90 is given in Fig. 42 with (25.7)

2s tan 8, = z tan2 8,.

The u-distribution around this equivalent cone at M , [I, Eq. 8.221),

=

1/2

(see e.g.

(25.8) is not completely the same as the axisymmetric term of the rhombic cone, but the difference is of higher order than s2. This can be understood in the following way: The disturbances of a body of low aspect ratio become approximately a.xisymmetric before they vanish at the front shock in supersonic or at infinity in subsonic flow. The differences between the flows past equivalent bodies are given for the inner domain by (24.10), FIG.42. Rhombic cone and its equivalent body of revolution. for the outer region by an additional source distribution which alters the equivalent body of revolution slightly. For M , = v2 this additional source distribution has the strength of u and the extension s2x. This accounts for the order of magnitude of this effect. For the deviation of 4 from the axisymmetric flow, (25.3) gives t'he following value for the rhombic cone by means of (12.5)

- SX

Affer differentiation with respect to x the deviation from the axisymmetric u-component uR follows as (25.9)

SIMILARITY A N D EQUIVALENCE IN COMPRESSIBLE FLOW

257

This result also differs from the last term of (25.5) by the factor v l - Y2/x2/v1- s2. The factor l / v l - s2 can be accounted for in the same manner as before in (25.5); the factor vl - r z / x 2makes all disturbances vanish a t the Mach cone; it enhances the circular symmetry. Even though it is not possible to calculate the flow for bodies of low aspect ratio a t zero incidence from the two-dimensional Laplace equation (24.5), it seems possible in this way to determine the difference between the flow around a wing and its equivalent body of revolution. This difference (25.9) attenuates faster than the u-disturbance of a flat plate with low aspect ratio per unit incidence, see (24.13). Therefore it will be assumed that the unknown F ( x ) in (25.2) is the same for two equivalent bodies except for an error of order 6 so that

$ - $R

(25.10)

=

($1. - $ R L )

(1

+O(6)).

Here the index L denotes again the solution of the two-dimensional Laplace equation. For a body represented by a source distribution on Y = 0, ( 2 5 . 2 ) , it follows that

(25.11)

Note that $RL and $L need not be taken a t the same point, but, of course, y2 = Y 2 Z2. The “spatial influence” of the potential F ( x ) is within the accuracy of the theory only a function of x as was true also for u in (11.2):

+

+ sf xl

(25.12)

where rl is any radius yl << 1. For the difference between the potential of the body of low aspect ratio and its equivalent body of revolution $ - $ R an equation completely analogous to (24.10) results where the double integral furnishes the error. For the linear gas-dynamic equation one gets on the axis

~ u R ) / a xor of (u - uR)is the same as, that The order of magnitude of a ( of $ - $ R . The integration over the inner region results in an error

258

KLAUS OSWATITSCH

In the outer regron u - tbR oscillates harmonically, and no additional error results. Therefore the error in (25.13) comes out to be

6 = 11 - M 2a p .

(25.14)

If the logarithmic factor in (24.14) is given any weight, which seems doubtful, the error would be smaller than the one of the Jones theory for wings at incidence. In the transonic domain, the variation of the Mach-number in the coefficient 1 - M 2in the gas-dynamic equation must be considered. Again the velocity disturbance u will be referred to the velocity of sound as the undisturbed velocity. Then (24.20) may be used as the relation between ( M 2- 1) and u. After introduction of (24.20) into (24.10) and subtraction of the corresponding axisymmetric solution + R , u R , the result is

(25.15) which is quite different from (25.13). By the same reasoning as before it can be seen that the integration over the inner domain involves an error of the order of magnitude

-

(25.16)

(y

+ 1)

U('$

- '$R)S2*

The outer domain needs a more careful consideration. For a free-stream Mach-number of unity, both zc and U~ vanish far from the body. Guderley and Yoshihara [30] calculated that the disturbance of a body of revolution attenuates far from the body according to

(25.17) If this relation is introduced into (25.15), one sees that the integral is at least finite. The quantities

(25.18)

a

U R -(U - u R )

ax

and

(U

a - UR) UR ax

have oscillatory character far from the body. Hence no contribution larger than u(+ - $ R ) is expected from the integration of these terms. The last term in the bracket of (25.15) is quadratic in (u - uR) and thus vanishes

259

SIMILARITY AND EQUIVALENCE I N COMPRESSIBLE FLOW

even faster. I n addition (u - uR) is by a factor s2 smaller than u. So no essential contributions are expected. Oswatitsch and Keune [40] have shown that also for the nonlinear gas-dynamic equation the essential contributions to the double integral come from the inner domain, while the outer domain is characterized by strong attenuation and mutual canceling of the positive and negative values of (u - uR). For the transonic domain the order of magnitude of the error in (25.13) is there fore

s = ( y + l)us2

(25.19)

where u denotes again the average velocity disturbance at the body. The error of the linear theory and the error in the transonic domain are related in the same way as in Jones’ theory, that is, the difference between (25.14) and (25.19)is the same as between (24.14)and (24.22): (1 - M m 2 ) is replaced by the average value of ( y 4-1)zc at the body. 26. Mach-number Dependence of Wilzgs with Low Aspect Ratio

In (25.10)and (25.11),the disturbance potential of wings with low aspect ratio a t zero lift has been related to the disturbance potential of the equivalent body of revolution, and the order of magnitude of the error was determined in (25.14)and (25.1!)).By differentiation of (25.10)and (25.11) with respect to x one gets the analytic form of the law of equivalence for u with a source distribution in the plane Y = 0 :

+s(x) u ( x , Y , Z )- u R ( x , r l )=

([ - z)2d5 [ 1 rl

y2+

ax

+ O(41.

(26.1) On the right-hand side of (26.1)the Mach-number occurs only in the error term. Thus the influence of the Mach-number M , on the u-disturbance of a body with low aspect ratio i s the same us for the equivalent body of revolution. I t follows in particular that (11.7) holds for all bodies of low aspect ratio in the domain of linearization of the gas-dynamicequation if only theargument x , H of q is replaced by an arbitrary point x , Y , Z in the neighborhood of or at the body. The cross section Q ( x ) equals the cross section of the wing by assumption. So a surprisingly simple rule for the influence of compressibility has been found. I t was derived by F. Keune and K. Oswatitsch and first published in the paper about wings of low aspect ratio by Keune [18]. The influence of compressibility depends for all points in the neighborhood of the body on x only. This can be seen a t once from (25.2)because u and 4 are determined except for an additive function of x .

260

KLAUS OSWATITSCH

A simple application of (26.1) can be given for the rhombic cone of small aspect ratio. We consider only the point x = 1, Y = Z = 0. By (25.7), 1 -

Q ( x ) = 2s tan 8,xZ = n tan2a0x2;

2 s tan 6, = tan2 8,.

I/

znQ

=n

For conical bodies Q” is a constant. For x = 1, Y = 2 = 0 the source distribution for a rhombic cone is

= 2s

1 tan 6, In (s * x ) = -Q” In s. 2

This integral for the cross-sectional flow would hold also for the subsonic flow of a body of the same local form. For the flow around a circular cone one has by (25.6) and substitution of the values from the linearized supersonic theory (e.g. [ l , Eq. 8, 221):

uR(x,~,)

1 z

--In7

a

-

lax

1%

1 2x

t l d C = ~ R ( ~ , r, ) -Q“lnr

-

1 ,, coturn In,-. A 2n

-Q

-s1

Substitution of this value into (26.1) leads to (26.2)

1

w(l,O,O) = - Q” In 2n

s cot a, [1 2 ~

+ W)l.

The exact solution for the rhombic cone (linearized theory) is (25.4). This solution may be expanded in terms of the halfspan s, which leads to the result given in [40]

(26.3)

A comparison of (26.3) with (26.2) and (25.14) shows that the estimate of the error was right; (26.3) gives also the next higher term. In [40] the error of the law of equivalence (26.1) has not only been estimated for the domain of linearization of the gas-dynamic equation by substitution of the cross-sectional flow into the neglected terms but the error has also been

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

261

calculated on the axis; hence the second term in (26.3). Of course, the results of other papers on this subject (for example [Zl]) could be used to calculate the error. The calculation of the error for the chosen example shows that O(6) of (26.1) is only 8 % for a rhombic cone with an aspect ratio of 50"/b of the Mach cone diameter (2s = tan am). The law of equivalence will therefore hold practically in the whole transonic domain. For high supersonic Machnumbers, however, it will be of little use, as can be seen from Fig. 18, but then the strip theory is a good approximation, and a new theory is not needed. In transonic flow one of the known approximate methods may be used to calculate the flow past the equivalent body. The flow around a circular cone in transonic flow was presented in Section 20 under the assumption of an attached shock. The shock detaches for conical bodies at the same Mach-number as for the equivalent circular cone. In that case the condition is (20.5). This relation has to be rewritten for the general conical body. The apex semi-angle 8,,of the circular cone has to be expressed by the crosssectional area Q ( x ) , to establish the relation to the equivalent general cone. The condition for the shock detachment becomes (26.4)

B

1 Q" tan2 a m = 0.74, 2n

-

-where B is given in Table 4 for the different forms of the transonic similarity laws. For the calculation of the u-disturbance in the transonic domain a special form of the law of equivalence is useful. Eq. (26.1) holds in the domain of linearization as well as in the transonic domain; only the error is slightly different. Denote a solution in the linearized domain by the subscript lin; it follows from (26.1) by a subtraction of a linear solution for the same body

that

This simple relation for the nonlinear influence of compressibility holds if the linearization is carried out at the same Mach-number for both equivalent bodies. The application of the general equation (26.5) to the special case of transonic conical flows is especially simple, because uR(x,rl) - uR,lin( x,rJ can be taken directly from Fig. 32. I t is simply the difference of the abscissas of the transonic and linear curve for the corresponding ordinate of the example. From this figure the difference 14 - u , ~is, known for the rhombic cone. The error of the equivalence rule (denoted by I in Fig. 43) can be given quite accurately because the error of the linear theory is known.

262

KLAUS OSWATITSCH

Fig. 43 shows the example calculated by Oswatitsch in [31] in connection with the transonic law of equivalence. The pressure coefficient is nearly equal to twice the negative u-disturbance. The two curves have a consta.nt difference of their ordinates which is not quite obvious. Since the flow past bodies of revolution at different Machnumbers may be compared by means of the similarity law (19.4), if cross section and freestream Mach-number yield the Equ. I 0.15 / same reduced thickness ratio / \ qedr (19.1), bodies of low aspect ratio may also be compared if cross section and Mach-number M; 1.10 are changed in a suitable way. Let us introduce the maximum cross section Qmax instead of z2 of the body of revolution in (19.1). The reduced maximum cross o.5 x section 0.5

zzz=-

FIG.43. Comparison of the pressure coefficient of a rhombic cone calculated by the equivalence rule and by linear theory.

B Q r d max = Qmax

IM? - 11

(26.6)

must then be thesame for the two bodies. In (26.6),Qmaxmay be the maximum cross section of any low aspect ratio body because it is equal to the maximum cross section of the equivalent body of revolution. In the following considerations the error quantity O(6) has been omitted in the law of equivalence (26.1), rl has been replaced by the height of the body, and Y has been set equal zero. Dividing (26.1) by Qmx one gets

(26.7)

The right-hand side of (26.7) is the same as the expression in the similarity law (19.4). For two bodies of low aspect ratio therefore the left-hand side must also be the same, if they have the same reduced maximum crosssections (26.6).

SIMILARITY AND EQUIVALENCE I N COMPRESSIBLE FLOW

263

Especially interesting is the case of cross-sections that are related by an affine transformation. The source distribution depends now only on the coordinate Z/s: (26.8)

-1

< z/s < + 1 :

"(X,Z) = T f ( X , Z / S )

and f(x,Z/s) is the same function for comparable bodies. The maximum cross section will be given in the form (26.9)

Qmax

= Cts

where the constant C depends on the body shape but is the same for comparable bodies. (For a rhombic cone C = 2.) Now the integral in (26.7) becomes

The first two terms on the right-hand side are the same for comparable bodies because the integral depends neither on s nor on T and the functions Q"/Qmax and H/Qmaxare the same for equivalent bodies. Only the third term is different and the following similarity law results for affine distorted bodies or wings of low aspect ratio (see Keune and Oswatitsch [42, Eq. 271) :

1 [u(x,o,z,) - Q2n In 1/B,t1s,3 J"

QI max

~

=

&

[24(x,0,Z2)- Q 2" In ~ / B , T , s , ~ .

2n

(26.10)

Again the comparison has to be made a t corresponding points ZJs, = Z,/s,. Two special cases will be mentioned: 1. The logarithmic term can be omitted if (26.11)

B,t,s13 = B , ~ , s , ~ ,

but this is the relation (17.12) which urbitrary comparable wings have to satisfy in transonic flow. In this case (26.10),divided b y the reduced crosssection (26.6), reestablishes by means of (26.11) the similarity law (17.10). 2. The comparison is possible for the same cross section and the same Mach-number if (26.12)

TISl = T z S 2 .

264

KLAUS OSWATITSCH

Then Qlmax equals Qemaxin (26.10) and can be omitted, and the argument of the logarithm reduces to svg. In (19.4), however, the argument of the logarithm is tzVBwhich has a quite different meaning: s ( x ) , and therefore s, is essentially the width of the source distribution while T in (19.4) is the thickness ratio of the body of revolution. For bodies of revolution the thickness ratio is small but finite, while the width of the source distribution is infinitely small. The similarity law has been presented here in a form where only the u and v component at Y = 0 were considered. Conclusions with respect to the actual body shape, using the simplified boundary condition (6.3) and the u-distribution on the body surface Y = H , are only possible if t l s << 1 is assumed. Otherwise correction terms must be added to eliminate the error, which may be much larger than the one of the usual application of the law of equivalence and transonic similarity laws. 27. Area Rule and Similarity

The comparison of the u-disturbance of a body with low aspect ratio and its equivalent body of revolution can be used to relate the form drag coefficients of the two bodies to each other. This was first done by G. N. Ward [17] in the supersonic domain of linearization. Ward calculated the flux of momentum across the plane x = const a t the end of the body and related it to the cross-sectional distribution Q ( x ) . This can also be done for transonic flow by showing that the Laplace equation for the cross-sectional flow (24.5) and hence the expansions (25.9) are valid up to a sufficiently large distance from the body. This problem has been considered by Berndt [43]. A different approach was taken by Keune and Oswatitsch [ 4 2 ] . They started from the law of equivalence for the u-component (26.1); the relation for the pressure coefficients of equivalent bodies can then be found, and integration over the body surface gives the relation between the form drag of a body with low aspect ratio and the equivalent body of revolution. These investigations lead to the same result

(27.1)

D[1

+ O(S)] = DR + ~2

4sr

+[I)

v,,(l,~) n

2=0

1=

v(1,t)In - S(1)

iEfi

dtdZ

lt--Zl

for the form drag up to the point x = 1, which is not necessarily the end of the body. The double integral is valid only under the special assumption that the sources are distributed in the plane Y = 0. For sufficiently thin wings of low aspect ratio, v o ( l , Z ) may be expressed by the simplified boundary condition (6.3). Not so thin cross sections may also be represented by the

265

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

same source distribution, but a different boundary condition. The double integral (27.1) is identical with the potential integral in the final cross section in Ward’s paper [17, Eq. (31)] or the paper of Lomax and Heaslet [44,Eq. (45)]. By means of (25.6) and an integration of the potentials 4 and + R for the cross-sectional flow of the two bodies

+s ( 4

(27.2)

’I

4 =-n

tlo(x,C)

In v Y 2 + (5 - Z ) V C ;

+R =

1 -9’ 2x

In r

- s(+)

one gets the result

5

5lz

1

+ SO)

S(1)

v,(l,z)

z=o

f =

v(1,t) In H ( l ) It - ZI ~

1 w z = -9yi) 2n

~n ~ ( 1 )

- s(1)

I

+s m (27.3)

-2

tl0(l,Z)~(1,0,Z)dZ = 2nH(%)

- s(1)

I

+SO)

4 ~ ( 1 , H) 2 x= 1

r=H

:MU,

- $0)

FIG.44. Contour of integration at the final cross section for a wing and its equivalent body of revolution.

where the path of integration for the wing has to be taken a t the final cross section x = 1 around the upper and lower side of the plane of the source distribution Y = 0 and for the equivalent body of revolution around the final cross section (Fig. 44).

266

KLAUS OSWATITSCH

The error 0(6), written on the left-hand side of (27.1) for formal simplicity, is the same as the one occurring in the law of equivalence for u, (26.1), where 6 is given by (25.14) in the domain of linearization and by (25.19) for the transonic regime. O(6) is the relative error in (26.1) and in (27.1). The relative error is the same in both expressions, because the error in the form drag D is caused solely by the error in u. Eq. (27.1) was first given by G . N. Ward except for the estimate of O(S), which in (27.1) extends also to the transonic speed regime, but differs from Ward’s estimate even in the linear domain. The applicability of (27.1) in the transonic domain was first established by Keune and Oswatitsch [42], and in a somewhat more restricted way by R. T. Whitcomb [32]. The difference between the form drag of a body with low aspect ratio and its equivalent body of revolution a t zero lift arises only from the source distribution at the final cross section, that is from the shape and change in shape of the end cross section. This difference is completely independent of Mach-number and the following relation exists for the form drag in the transonic domain [42, Eq. (45)]: (27.4)

which is completely analogous to (26.5). Base drag and friction drag are, of course, excluded. If the body at x = 1 changes into an axial cylinder of arbitrary shape or if it terminates with vanishing cross section but finite surface inclination, then its drag D is equal to the drag DR of the equivalent body of revolution. W. D. Hayes [16] showed earlier for the supersonic domain of linearization that under the above mentioned conditions the same drag formula holds for bodies of sufficiently low aspect ratio that was first given by von KArmGn for very slender bodies of revolution and later extended by Lighthill to not so slender bodies of revolution. About the same time when the transonic law of equivalence was given by Oswatitsch [31], R. T. Whitcomb [32] showed along more empirical lines that the zero-lift drag had to be the same a t transonic speeds for equivalent bodies. He found experimentally that the shock patterns of bodies with low aspect ratio and of their equivalent bodies of revolution are very much alike a t some distance from the axis. Since the shock waves are the reason for the momentuni losses and the essential momentum losses occur for bodies of revolution a t some distance from the axis, the conclusion can be drawn that equivalent bodies have nearly the same drag. Fig. 45 shows an impressive experiment by Whitcomb. After deduction of the friction drag which is practically independent of Mach-number but larger for the wing-body combination than for the body of revolution the remaining drag is very nearly equal for the two equivalent bodies shown.

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

267

This is the case even for the combination of a very slender fuselage with a wing of not very small aspect ratio. Let us assume the wing terminates with a trailing edge perpendicular to the free stream velocity and finite surface inclination. Then uo(l,Z)# 0 but H(1) = 0. The double integral in (27.1)would give a logarithmic infinity. This shows that the estimation of the error cannot be right for such a trailing edge. Indeed a t such a trailing edge 21 = O(ln(1- x ) ) and h / a x = 0(1/ (1 - x ) ) . A similar behavior is shown by u for a body of revolution, which starts or terminates conically. But here the singularity is restricted to one point and is therefore harmless, while the singularity of the trailing edge of a wing that extends over a finite length produces by (25.13) an infinite error which also has influence on the drag formula of (27.1). A similar trailing-edge effect 0.0 I 2 exists also for the delta wing in subsonic flow at an angle 0.008 of attack. A sudden start ACD, or ending of the wing con0.004 tradicts the assumption of 0 small changes in the x-direc0 .84 0.88 0.92 0.96 1.00 1.04 1.08 1.12 tion which is the basis of % the whole low aspect ratio theory. These edge effects FIG. 45. Measurements by R . T. Whitcomb comparing the drag of t w o equivalent bodies. are subject to special considerations and corrections not treated in this artick. Eq. (27.1) may also be written in the form

(27.6)

268

KLAUS OSWATITSCH

where cD is the drag coefficient with respect to Qmar. The right-hand side of (27.5) is practically the same as the transonic similarity law (19.6) except for the factor 1/t2. Under the assumption of affine wilzgs again a similarity law can be found for cD by using (26.8), (26.9) and (19.6) and omitting all parts of the double integral in (27.5), which are the same for affine wings. Because of the assumption of equal reduced maximum cross sections, (26.6), the following relation results for two affine wings distinguished by the subscript :

the error is of the same order of magnitude as in (27.5). This similarity relation is slightly more general than the one given by Keune and Oswatitsch [42, Eq. (49)], where B was set equal to ( y 1) and it was assumed that the comparison had to be done for the same y , the usual case. If the same approach is taken in the domain of linearization, then in (27.5) rlMmZ - 11 is the argument of the logarithm instead of Q m a x v . On the right-hand side of (27.5) appears exactly the form of (11.8). A further division by Qmaxis not necessary, the parts of the double integral in (27.5) independent of the affine distortion are omitted, and a new relation results for affine wings of the same maximum cross section Qmx:

+

(27.7)

The logarithmic term of (27.6) and (27.7) vanishes only for Q’(1) = 0. If two wings at the same Mach-number are considered the Mach-number can naturally be omitted. But (27.7) may also be used to consider a wing of constant t / s at different Mach-numbers. Then t / s can be omitted and the same Mach-number influence results for C, as for the equivalent body of revolution. A brief review of what is covered by the term “area rule” can be found in [46].

SIMILARITY AND EQUIVALENCE IN COMPRESSIBLE FLOW

269

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(1955). 41. DROUGGE, G., 42.

43. 44. 45. 46. 47.

Some measurements on bodies of revolution a t transonic speeds. IX. Int. Cong. Appl. Mech. Brussels, 1956 (in preparation). KEUNE,F. and OSWATITSCH, K., Aquivalenzsatz, Ahnlichkeitssatze fur schallnahe Geschwindigkeiten und Widerstand nicht angestellter Korper kleiner Spannweite. Z A M P 7 , 40-63 (1956). BERNDT,S. B., On the drag of slender bodies a t sonic speed. F F A Med. 70 (1956). LOMAX, H. and HEASLET, M., Recent developments in the theory of wing-body drag. J . Aeron. Sci. 28, 1061 (1956). OSWATITSCH, K., The Area Rule. A p p . Mech. Review 10, 543-545 (1957). TSIEN, H., Similarity laws of hypersonic flows. J. Math. Phys. 25, 247 (1946). OSWATITSCH. K . , Ahnlichkeitsgesetze fur Hyperschallstrtimung. Z A M P 249-264 (1951).

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48. VAN DYKE,M. D., The combined supersonic-hypersonic similarity rule. J. Aeron. Sci. 18, 499 (1951). 49. VANDYKE,M. D., Applications of hypersonic small-disturbance theory. J. Aeron. Sci. 21, 179-186 (1954). 50. MCLELLAN, C. H., Exploratory wind-tunnel investigations of wings and bodies at M a , = 6.9. J. Aeron. Sci. 18, 641 (1951). 51. EGGERS,A. J.. J R . , SAVIN, R. C., and SYVERTSON, C. A,, The generalized shockexpansion method and its application to bodies travelling at high supersonic air speeds. J . Aeron. Sci. 2 2 , 231-238 (1955). 52. BOGDONOFF, S. M., and HAMMITT. A. C., Fluid dynamics effects at speeds from M = 11 to 15. J. Aeron. Scr. 28, 108-116 (1956). 53. COLE, J. D., Newtonian flow theory of slender bodies. ,I. Aeron. Scz. 24. 448-455 ( 1957). 54. GARRICK, I. E. and RUBINOW, S. I., Flutter and oscillating airforce calculations for a n airfoil in a two-dimensional supersonic flow. N A C A Re@. No. 846 (1946). 55. LANDAHL, M., Mollo-Christensen, E. L.. and Ashley, H., Parametric studies of viscous and non-viscous unsteady flows. Mass. Inst. Techn. O f f i c e Sca. Research Tech. Rept. No. 55-13 (1955). 56. STACK,J., LINDSEY,W. F., and LITTELR. E., The compressibility burble and the effect of compressibility on pressures and forces acting on a n airfoil. N A C A . Rept. No. 646 (1938). H . T., Similar solutions of compressible boundary 57. TING-YILI,and NAGAMATSLJ, Layer equations. J. Aeron. Sci. 20, 653-655 (1953). 58. HAYES,W. D., and PROBSTEIN, R . F., Viscous hypersonic similitude. Pres. at the I. A. S. 27th Ann. Meet., New York, N. Y.,(1959), I. A. S. Rept. No. 59-63.

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KSrmSn Vortex Streets

BY R. WILLE Technische Universitat Berlin, Berlin, Germany

Page 1. Introduction . . . . . . . . . . . . . . . 2 . Stability Theory . . . . . . . . . . . . . 3 . Other Theories on Vortex Streets . . . . . . 4. Experimental Investigations of Vortex Streets 5. Related Problems . . . . . . . . . . . . . 6. Summary . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

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1. Introduction

The phenomenon of periodic vortex shedding from a symmetrical bluff body and the formation of the vortices in a street have been the concern of experimenters and theoreticians for 50 years. v. KBrmLn [ l ] himself reports how, by the unsuccessful endeavors of a Gottingen research associate to keep vortex shedding symmetrical, he had been confronted with the problem of treating the alternating vortex separation theoretically, thus linking the periodic asymmetric double-row vortex street with his name. L. Rosenhead [2], in 1953, surveyed the state of our knowledge of vortex streets and of related flow problems at that time, a paper which has been a most valuable guide for many workers in the field. Another review published in the same year by M. 2. Krzywoblocki [3] contains a list of 150 references to vortex streets and, in addition, quotes 41 items on related subjects. In the meantime new contributions have added further facets to our subject matter, and some of the older work has achieved new importance. Experimental results should be given full consideration as the properties of vortex streets have become important for many technical processes. Problems of flame stabilizers and of fuel sprays may be mentioned as examples. Experience has proved that flow processes which appear to be of the three-dimensional type very often are controlled by two-dimensional vortex patterns. The Kdrmin vortex street (see Fig. 1) is the object most easily accessible for the study of vortex characteristics ; moreover, the detailed examination of vortices leads to improved conceptions of turbulent flow.

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Thus the vortex street is in many places considered a research theme which proceeds from special questions, but may lead to the most far-reaching conclusions.

FIG.1. Kirmin vortex street behind a cylinder produced in a towing tank; cylinder moving from top to bottom. Photograph by A. Timme [33].

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The author is well aware that this survey cannot be complete, although the international channels of scientific documentation have improved appreciably. The limitation lies in the fact that even such a comparatively narrow field as vortex streets can be approached from too many different sides. 2 . Stability Theory

I t will be recalled that v. Karma, [4, 61 in his classical papers formulated the stability criterion as a part of the flow-resistance problem. In the course of time, only the stability problem has been reexamined in theoretical fluid mechanics, whereas no new contributions to v. Karmans conception of hydrodynamic drag have been found. Although boundary-layer theory has solved many intricate problems of wall flow, it cannot yet attack problems of separated flow. The difficulties outlined by v. Karma, and Rubach [6] still persist. The first part of this survey is devoted to theoretical papers on the stability of KBrm6n vortex streets. I t may be pointed out right away that the application of refined mathematical methods leads to the result that all two-dimensional vortex streets are unstable. C. Schmieden [7], introducing a special kind of “group disturbances”, proved that the double-row vortex street in which each vortex is opposite to the mid-point of the interval between two vortices in the other row, has a weak instability even if the Karmin coefficient h/l = 0.281, if in the equation of motion the second order terms are taken into account. N. J. Kotchin [8] found the same result based on a stability definition given by Ljapounoff. * Considerable elucidation of the problem may be found in a short brilliant paper by U. Domm [9]. Domm treats the two-parameter configuration introduced by A. Maue [lo] and 13. Dolaptschieff [ l l ] with regard to second order terms. Two-parameter configurations make use of the spacing ratio hll = K and of a dislocation ralio d l l = p , whereby in general the vortices of one row are not positioned midway between the vortices of the other row. With linearized equations Maue and Dolaptschieff indepentlv formulated the stability condition sin np

= sinh Z K .

Domm proves that all vortex streets satisfying the Maue-Dolaptschieff condition are unstable in the second order, whereas all other vortex arrangements are unstable in the first order. The special case of the Kirrnan vortex

* This reference I S quoted from the German translation of the Russian book: N. J. Kotchin. I. A. Kilbel, and N . W. Rose, Theoretische Hydrodynamik, Vol. 2.

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street, p = #, is the element of greatest width and of smallest translatory velocity within the class of vortex streets of minimum instability. With the foregoing, all modifications of the stability theory of vortex streets in ideal fluids have, in fact, been fully treated. The theory predict!; instability not only in the case of finite or three-dimensional disturbances, but also for infinitesimal two-dimensional disturbances. The question arises whether a theory based on ideal-fluid mechanics can have any bearing on the physical phenomenon. Experiments prove that in a towing tank 100 a formation of 20 to 30 vortices with sufficiently quiet water at Re can be produced. Now, if a weak instability of the second order should be felt only at such distances downstream, then the model of a line vortex is certainly no longer applicable because of the old age of the real vortices. On the other hand, the trajectories of disturbed line vortices calculated by Schmieden [12] permit an estimate of the number of vortices generated at the cylinder before a noticeable dislocation of the street pattern can be detected: the number is about 50. This again is a number of vortices which cannot be observed simultaneously in water, as the diffusion of vorticity gradually blurs the contours of a vortex. Outside the realm of the dynamics of incompressible fluids a paper of M. 2. v. Krzywoblocki [13] should be mentioned. The author introduces compressibility of the fluid in the immediate neighborhood of the vortex centers and finds instability in the first order. The problem of stability in the first order of a vortex street in a viscous incompressible fluid has been taken up by U. Domm [la, 151 in two papers. Domm investigates vortex streets in which the line vortices are replaced by vortices described by a solution of the Navier-Stokes equation according to Oseen [16] and Hamel [17]. The circumferential velocity is given by

-

The most significant result of the analysis is that for the stable vortex formation the spacing ratio h : I is a function of time. In Domm’s calculation all vortices are of the same age and begin their life at zero time with the velocity field of a potential vortex. Therefore the KQrmQncoefficient is a part of the solution. For times greater than zero the spacing ratio decreases, and this can be interpreted in such a way that the width of the street becomes smaller with growing distance from the cylinder. The theory of course describes the gradual approach of two parallel infinitely long vortex rows. Thus Domm for the first time gives a theoretical explanation for the fact, well known indeed since BCnard, that the photographs taken by different experimenters give different values of the spacing ratio hfl. The difference is accounted for by the age of the vortex group subject t o measurement. G. J. Richards [18] pointed out the time dependance of h : I from his

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experimental data, but the older stability theories based on the model of line vortices in an ideal fluid, could not account for this effect, as a potential vortex does not change with the passage of time. When comparing Domm’s theory with experimental results, one observes that real vortex streets do not become narrower but in fact wider with growing downstream distance. This discrepancy can be explained by the fact that Domm’s theory, although it introduces finite vortices growing in time, has to work with the simplifying assumptions that all vortices are growing older at the same rate throughout the process, that the street is of infinite length, and that in the vortex cores, too, the resultant velocity can be found by superposition as in potential flow. On the whole the part of the theory of vortex streets that comes under the heading stability is not yet satisfactory. One cannot easily understand the fact that the stability theory of the first order describes the physical phenomenon in very good approximation, whereas further mathematical analysis always predicts instability, which means that the vortices cannot remain in a regular pattern. Schmieden [7] did not fail to see this dilemma and remarks that in a real fluid viscosity could have a stabilizing effect. But this argument is not conclusive, as for example in boundary-layer theory the viscous forces precisely serve to destabilize the laminar wall boundary layer. An explanation for the disagreement between experiment and theory should perhaps be sought in the time lactor: Looking a t a vortex trail behind an obstacle towed along in a tank we see growing and decaying eddies, but the stability theory starts with a given permanent pattern of flow lines. I t may be added that the stability theory cannot give any information about the origin of the street with staggered rows of vortices; it is tacitly implied that the first unsymmetry leads to a pattern which by feed-back coupling, through a mechanism vet to be investigated, controls the shedding process a t the cylinder. 3. Other Thpories on Vortex Streets

We owe to C. C . Lin [19] a promising formulation, describing the properties of a vortex street in a viscous fluid. Oseen’s linearization of the NavierStokes equation is applied to the flow far downstream of the cylinder. Lin examines the linearized vorticity equation for a small disturbance superposed on a translatory flow of velocity U :

at

a; = vAC. + I/ ax

~-

at

In a periodic solution of this equation the points of maximum vorticity are interpreted as vortices, and by superposition a double-row vortex street

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is formed. The important result is that the spacing ratio is a function (of time. I t is true, that Lin used linearized equations and made no reference to the origin of the street ; he also left open an integration parameter the value of which cannot be found theoretically. But his solution provides a time-law for the increasing spacing ratio, which by a suitable choice of that parameter can be made to agree well with the experimental results of Wille and Timme [24]. Fig. 2 illustrates this result.

I

I ;ID

I 20

1

-

I

I

30

% 9 ‘

W

1

Distance fmm vortex soume

FIG. 4. Lateral spacing of vortex rows according to the theory of Lin “1, calculated by Timme [29]. Dots represent experimental results of Wille and Timme [24].

G. Birkhoff [ZO]establishes as a main invariance theorem that the moment of the vorticity hk* is constant for finite and infinite vortex streets. He concludes that the mean lateral spacing h and the mean longitudinal spacing I are dynamical invariants and that all spacing ratios in a non-viscous fluid are stable. Is this argument of Birkhoff’s conclusive ? In general, average findings valid for “mean” vorticity are not valid for special arrangements of discrete vortices. Moreover, Birkhoff‘s theorem holds for symmetrical vortex streets too, but no case is known in which a two-dimensional vortex street having vortices opposite one another could be observed experimentally. For a viscous fluid, it follows from the constancy of the vortex moment that the mean transverse spacing of the street increases with time, as the

so

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internal fluid friction absorbs the vorticity gradually. For the change of longitudinal spacing with time no conclusions can be drawn. Wille [23] applied the law of continuity in order to show the increase of h : I, not of h alone. In the same paper [20] Birkhoff discusses the problem of the spacing ratio in connection with the Strouhal number, whereby the order of magnitude of the Strouhal number itself is deduced from the motion and the area of the oscillating wake. Birkhoff’s analysis has to use many rough estimates, and the basic question remains open, whether the periodic generation of vortices a t an obstacle can be described as a wake whose oscillatory motion can be compared to a flapping flag. When looking at motion pictures of vortex streets, one does not get the impression that downstream of the cylinder a mass of fluid swings “from side to side, somewhat like the tail of a swimming fish”. The motion picture gives the impression that each separated vortex creates a flow pattern of its own thereby governing the entire width of the wake in the neighborhood of the cylinder. Parts of one vortex reach far over to the opposite side of the cylinder and show no tendency to return in a swinging motion, but gradually mix with the fluid outside the street. B. Dolaptschieff [21] formulated the laws of motion for any single fluid particle in a given vortex street. The evaluation of the integrals and the physical conclusion are yet to be published. 4. Experimental Investigations of Vortex Streets Since Rosenheads survey in 1953 some significant changes in experimental technique and in the interpretation of results have occurred. Until about 1960 all detailed information about vortex streets had been drawn primarily from photographs of particle paths. After that time the hot-wire technique provided new possibilities, which have not yet been exhausted. The use of the schlieren method in compressible-flow photography again has contributed new results. In connection with the proper utilization of particlepath photography we owe to S. G. Hooker [22] the significant point that on vortex pictures the locus of zero velocity is not identical with the center of vorticity. In a paper by R. Wille [23] the vortex street and the free jet are treated as examples of trunsitzon phenomena from regular to turbulent flow. Data on the time-dependence of the spacing ratio h : 1 are given, and it is found that in the range 2000 < Re < 4000 the Reynolds number is of no influence. By changing the surface tension of the water and by kinematography of the particle paths at and below the surface it is proved that the surface is an adequate indicator. In a second paper on the same subject Wille and Timme [24] present data evaluated from photographic and kinematographic

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flow pictures in the range 100 < Re < 200. The velocity distribution of a great number of vortices has been calculated from the length of the particle paths, and with the translatory velocity of the street, measured at the same time, the true centers of the vortices could be found. S. G. Hooker [22] supposed that the widening of the street, as taken from the centers of rotation in the photographs, “is more apparent than real”, but the method adopted by Wille and Timme [24] have proved that for the vortex street in water behind a circular cylinder a real increase of the lateral spacing h exists. The same measurements prove that with growing downstream distance the longitudinal spacing 1 increases too; however, this growth is weaker than that of h, so that the quotient h/l increases as a whole (see Fig. 3). The simultaneous growth of lz and 1 with the decrease of vortex

FIG.3. Increase of the spacing ratio of a KArmln vortex street with growing distance from the cylinder. (Wille and Timme [24]).

strength is understandable, it is assumed that in a real fluid, too, the translation of the street is due to the mutual induction of the vortices in both rows. The law of continuity explains the increase of transverse spacing with diminishing vortex strength, and again the growing lateral distance of the vortices explains the decrease of translatory velocity, which results in the growth of the longitudinal distance for a given number of vortices. This relation has not yet been considered in the theoretical analysis of the vortex street. We owe to R. Frimberger [25] numerous experimental data on vortex streets in air and water. The increase of h : 1 with time is substantiated, and it is proved by hot-wire measurements that in a downstream range oS 8 cylinder-diameters the longtudinal spacing of the vortices increases and the translatory velocity decreases. Further downstream these parameters tend to constant values.

S ‘U38WnN 1 V H f l O N l S

The first and in many respects fundamental work introducing hot-wire techniques into vortex-street researches was done by L. Kovasznay [26].

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A. Roshko [27] uses the advantages of this technique to investigate the mechanism of transition from a vortex street to a turbulent wake. In his paper many problems of the vortex street are treated in great detail. The frequency law of vortex shedding is investigated in the range 46 < Re < 7000, and the relation between the Strouhal number and the Reynolds number is expressed by empirical functions (see Fig. 4). Roshko describes a “stable range”, 40 < Re < 150 in which the Strouhal number rises strongly with the Reynolds number, and an “irregular range”, 300 < Re < 10,000, where the Strouhal number remains essentially constant, while irregular turbulent bursts are superimposed on the periodic vortex shedding. I n the intermediate range, called “transition range” by Roshko, with Reynolds numbers between 150 and 300, the vortex shedding has been found t o be highly irregular and a frequency law could not be established. Frimberger’s [25] measurements of the Strouhal number begin a t Re = 5 x lo3 and his data closely follow Kovasznay’s and Roshko’s results. Another part of Roshko’s work is devoted to the spectral distribution of frequency and energy in the wake. He finds that the regular KArmBn vortex street in the Reynolds-number range 40 < Re < 150 is formed by vortices whose energy is only destroyed by internal fluid friction. On the other hand, for vortices above Re = 300 a frequency spectrum already exists a t short distances behind the cylinder. The discrete frequencies disappear gradually as the distance from the cylinder increases, and at 50 diameters downstream the wake is fully turbulent. I t is worth noting that Kovasznay and Roshko were the first to make use of the frequency law in the stable range for the measurement of velocities below U = 400 cmjs. This method of measuring velocity by frequency is very practical as the hot-wire probe need not be calibrated. Data on single row vortex streets along a wall and data on the velocity distribution of the vortices are given in a paper by 0. Wehrmann [ZS]. The single-row vortex street was produced from a KBrmPn vortex street by splitting the well developed street approximately a t a distance of 5 diameters downstream of a cylinder. To effect the splitting a thin plate of 1/10 mm thickness was brought into the wake parallel to the direction of flow. The experiment proved that the two vortex rows traveled left and right of the plate in good order and that a small disturbance created by the leading edge of the plate was only of local importance. A Timme [29] calculated the velocity distribution for this case using a modified approximation due to S. G. Hooker [22]. Wehrmann’s hot-wire measurements contribute in a most interesting way to our knowledge of the velocity distribution of real vortices, and on the whole his experiment of splitting the KArmAn street throws new light on the problem of stability. In hydrodynamic theory, a single-row vortex street is unstable in the first order. The same holds for a double row symmetrical vortex street, the second row of which is the image

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of the first one with respect to the cutting plane. On the other hand Domm‘s theory [la, 151 of viscous vortex streets includes the possibility of a stable symmetrical double row arrangement, if the vortex pattern has “survived” an unstable “phase of life”. Presumably Wehrrnann’s experiments cannot decide on the stability or instability of the whole pattern, as the time for rolling along the wall can only be measured in fractions of seconds, but they show that each individual vortex is a rather stable system of moving fluid, able to rearrange its velocity field in a short time after the bump at the leading edge of the separating wall. A. Naumann and H. Pfeiffer 1301 investigated vortex streets behind cylinders in a Reynolds-number range of 8 x lo4< Re < 2 x lo6 and a Mach number range of 0.35 < M < 0.75; at the cylinder M = 1 was reached and surpassed. In the schlieren picture Fig. 5 the vortices appear as dark spots. All data concerning shedding frequency have been evaluated from high-speed film pictures taken at speeds up to 40,000 frames per second. One remarkable result is that the critical Mach number, which was M = 0.45 in the experimental setup used, did not influence the Strouhal number, and it is surprising that the periodic vortex generation can be observed even in the caSe where the shock due to FIG. 5. K i r m i n vortex street ~n highlocal supersonic areas reach from speed flow, M = 0.577; R~ = 1.16. 106. wall to wall in the flow channel. Schlieren film pictures bv Naumann [ 3 O ] . 5. Related Problems

Here a few papers are mentioned, related to the problems of Karman vortex streets inasmuch as the flow immediately at the cylinder or the phenomena of single vortices are treated. H. Drescher [31] investigated the pressure distribution and its periodic variations at a cylinder. The pressure changes were recorded at 12 bores

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distributed evenly along the cylinder diameter, and simultaneously kinematographic pictures of a dye filament, escaping from one of the bores, were taken by a camera running synchronously with the pressure recorder. Drescher's results complete the data concerning the variation of the Strouhal number in the range lo4 < Re < 2 x lo6. The numerical data of the periodic pressure distribution can be used as the base of a boundary layer calculation. By correlation of the flow pictures with the recorded pressure diagrams Drescher finds that the maximum lift occurs when the wake swings out maximally to the opposite side. At short intervals the lift is of the same value as the drag which is a new result as compared with the older ones of Schwabe [36]. In Drescher's as well as in Naumann's report of the experimental observations it is hinted that at certain high Reynolds numbers the periodic vortex shedding ceases through brief intervals and is replaced by other not clearly definable flow patterns. The reason for that behavior could not be given, and one might investigate whether it is caused by the finite channel width.

" Ln/nJ

(16

-5

-4

-3

-2

t

-I

FIG.6. Velocity distribution of a vortex in a KArmAn vortex street. velocity profile perpendicular to the axis of the street. - Comparison of the equation of Hooker 1221 and experiments of Timme [33].

W. Kaufmann [32] investigates the structure of the single vortices in a KAnnLn vortex street. For an ideal fluid the distribution of the vortex strength is calculated, which on the whole is not in disagreement with the physical reality, but cannot avoid the difficulty of infinite velocity at the center.

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A. Timme [33] in a comprehensive study deals exclusively with the velocity distribution of individual vortices of a KBrman vortex street. Whereas Hooker [22] had used particle paths under water, Timmc evaluates photographs of the surface; both methods give the same result. Timme proves that Hookers approximation, according to which only one vortex in an ideal KArnian street is replaced by an Oseen-Hamel vortex, gives a velocity distribution in close agreement with the observations (seeFig. 6). Thus it has been possible by the evaluation of two consecutive photographs of one and the same vortex not only to measure its strength and age as well as its translatory velocity, but also to calculate by comparison with the OseenHamel solution the active kinematic viscosity inside the vortex. At Re = 200 (based on the cylinder diameter), Timme found the viscosity equal to its laminar vaIue. Only for young vortices immediately behind the cylinder the active viscosity was three times higher, but the laminar value was quickly reached. However, in the range of Re == 1000 the active viscosity in the average proved to be ten times higher than the normal viscosity. Timme explains this result by the turbulent momentum exchange inside the vortices, which in a similar way was introduced by H. B. Squire [34] in his theoretical work on the vortex law in turbulent flow. Timme postulates a critical Reynolds number based on the characteristic dimension of a vortex, above which the vortex is no longer stable. Similar considerations are brought forward by Wille and Wehrmann [35] for annular vortices along the boundary of a free jet where the velocity distribution of the vortices could be measured near the orifice bv hot-wire technique. 6. Swmmary

The subject of vortex streets is a part of the wider problem of wakes behind obstacles. The term “vortex street” is usually restricted to two-dimensional flow patterns, and we speak of a “street” if a multitude of vortices in a regular arrangement can be observed simultaneously. The hydrodynamic theory for ideal and for viscous fluids would seem to indicate that these streets, cannot exist since they are unstable, but they can be produced experimentally in air and in water and their characteristics are being studied. Whether this is a contradiction between theory and experiment, so often lightheartedly referred to, or whether theory and experiment actually refer to two entirely different subjects is worthy of clarification. Resides the problem of the street there exists the problem of periodic vortex shedding. At high velocities, at which the vortices generated a t the obstacle decay rapidly, one still observes an alternating flow separation, even if the street is no longer visible. The third problem is that of vortex formation itself. Here the question is whether modern boundary layer theory, making use of experimental data

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on pressure distribution, is able to account for the circulation camed in one vortex. If this could be achieved, then the circle of investigations could be closed by returning to v. KQrmin’s conception of hydrodynamic drag. ACKNOWLEDGMENT

Dr. A. Timme gave most valuable assistance in the compilation of the papers reviewed and in drafting the text. For this and for many discussions on the subject I wish to express my thanks.

References 1. v. KARMAN.TH., “Aerodynamics.

Selected Topics in the Light of their Historical Development”. Cornell University Press, Ithaca, N. Y.,1954. 2. ROSENHEAD, L., Vortex systems in wakes, Advances i n Applied Mechanics 3, 185-195 (1953). 3. v. KRZYWOBLOCKI, M. Z., Vortex streets in incompressible media, Applied Mechanics Reviews 6, 393-397 (1953). 4. v. KARMAN, TH., uber den Mechanismus des Widerstands, den ein bewegter Korper in einer Fliissigkeit erfahrt, Gottinger Nachrichten, Math. Phys. Kl., pp. 509-519 (1911). 5. v. KARMAN, TH., Uber den Mechanismus des Widerstands, den ein bewegter Korper in einer Fliissigkeit erfahrt, Gottinger Nachrichten, Math. Phys. K1. pp. 547-556 (1911).

6. v. KARMAN,TH., und RUBACH, H., Uber den Mechanismus des Fliissigkeits- und Luftwiderstands, Phys. Zeitschrift 13, 49-59 (1912). 7. SCHMIEDEN, C., Zur Theorie der Kdrmdnschen WirbelstraBe, Zng. Arch. 7. 215-241 (1936). 8. KOTCHIN,N. J., Compt. Rend. (Doklady) de I’Academie des Sciences de L’SSSR 24, 18-22 (1939). 9. DOMM,U., Uber WirbelstraOen von genngster Instabilitat, Z . angew. Math. u. Mech. 36, 367-371 (1956). 10. MAUE, A., Zur Stabilitat der Kdrmdnschen WirbelstraOe, Z . angew. Math. u. Mech. 20. 129-137 (1940). 11. DOLAPTSCHIEFF, B., Compl. Rend. de I’Academie Bulgare des Sciences (1943). 12. SCHMIEDEN, C., Zur Theorie der Khrmdnschen WirbelstraBe 11, Zng. Archiv 7 , 337-341 (1936). 13. v. KRZYWOBLOCKI, M. Z., On the stability of BBnard-Kdrmdn vortex street, in Compressible Fluids, Acta Physica Austriaca 7 , 283-298 (1953). 14. DOMM. U., Ein Beitrag zur Stabilitatstheorie der WirbelstraBen unter Beriicksichtigung endlicher und zeitlich wachsender Wirbelkerndurchmesser, Zng. Arch 22,, 400-410 (1954). 15. DOMM,U., The stability of vortex streets with consideration of the spread of vorticity of the individual vortices, J . Aeronaut. Sci. 22, 750-754 (1955). 16. OSEEN,C. W., “Hydrodynamik”, Leipzig, 1927, and Ark. f . Mat., Astron. och Fys.. 7, 1-11 (1911). 17. HAMEL,G., Spiralfermige Bewegung zaher Fliissigkeiten, Jahresbericht d . Deutschen. Mathematiker- Vereinigung. 2s. 34-60 (1916). 18. RICHARDS,G . J., An experimental investigation of the wake behind an elliptic. cylinder, ARC R. & M . No. 1590 (1934). 19. LIN,C. C.. On periodically oscillating wakes in the Oseen approximation i n “Studies etc. presented to R. v. Mises”, pp. 170-176, Acad. Press, New York, 1939.

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VORTEX STREETS

287

20. BIRKHOFF, G., Formation of vortex streets, J . of APPL. Physics 24, 98-103 (1953). 21. DOLAPTSCHIEFF, B., Verallgemeinerte Fopplsche Kurven im Zusammenhang mit der Wirbelwiderstandsbestimmung, Z . angew. Math. u . Mech, 36. 427-434 (1955). 22. HOOKER,S. G.. On the action of viscosity in increasing t h e spacing ratio of a vortex street, Proc. R o y . SOC.,A , 164, 67-89 (1936). 23. WILLE, R., Uber Stromungserscheinuiigen im ubergangsgebiet von geordneter zu ungeordneter Bewegung, jahrhuch der Sckiffbautechn. Gesellsckafi 16, 176-187 (1952). 24. WILLE. R., und TIMME,A.. Uber das Verhalten von WirbelstraOen, ,lah.rbuch der Schiflbaulechn. Gesellschaff 51, 215-221 (1957). 25. FRIMBERGER, R . , Experimentelle Untersuchungen a n KBrmanschen U’irbelstraOen, Z. Flugwzss. 6. 355-359 (1957) 26. KOVASZNAY. L.. Hot-wire investigation of the wake behind cylinders a t low Reynolds Numbers, Proc. R o y . SOC.,A . I O N , 174-190 (1949). 27. ROSHKO,A., On the developrncnt of turbulent wakes from vortex streets, N a f l . .4dvisory Comm Aeronauf.. Tcch. Note No. 2913 (1953). O., , Hitzdrahtrnessungen in einer aufgespaltenen WirbelstraUe, DVL28. W E H R M A N N Bericht No. 43, 1957. 29. TIMME, A , , Mathematische Analyse und physikalische Interpretation von Hitzdrahtsignalen einer WirbelstraOe, DVL-Bericht No. 7 7 , 1958. 30. N A U M A N N A., und PFEIFFER, H . , Versuche a n WirbeJstraOen hinter Zylindern bei hohen Geschwindigkeiten, Forschungsberichte des Wirtschafts- und Verkehrsministeriums Nordrhein-Westfalen, No. 493, 1958. 31. DRESCHER, H.. Messung der auf querangestromte Zylinder ausgeiibten zeitlich veranderten Driicke, Z . FLugwiss. 4, 17-21 (1956). 32. KAUFMANN, W.. Uber den Mechanismus der Wirbelkerne einer KarmAnschen WirbelstraDe, I n g . Arch. 19, 192-199 (1951). 33. TIMME, A., Uber die Geschwindigkeitsverteilung in Wirbeln, Ing. Arch. 26, 205-225 (1957). 34. SQUIRE,H. B.. The Growth of a vortex in turbulent flow, Aeronautical Research Council 16.666, F.M. 2053, 1954. 35. WILLE, R., und WEHRMANN, O., Beitrag zur Phanomenologie des laminar-turbulenten Ubergangs bei kleinen Reynoldszahlen, i n IUT.4M-Symposium, Freiburg, 1957 (International Union for Theoretical and Applied Mechanics), 387-404, Springer, Berlin, 1958. 36. SCHWABE, M., Uber Druckermittlung in der nichtstationaren ebenen Stromung, Ing. Arch. 8 . 3&50 (1935).

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Author Index Numbers in parentheses are reference numbers and are included to assist in locating references when the authors’ names are not mentioned in the text. Numbers in italics refer to the page on which the reference is listed.

A

Cochran, W. G., 6, 36 Cohen, N. B., 33, 37 Cohen, R. S., 55(32), 777 Cole, J. D., 6, 35, 240(53), 277 Cooke, J. C . , 5, 35 Cowling, T. G., 40(9). 44(9), 54(9), 55(9). 62(9), 66(9), 68, 70(9), 73(9), 78(9), 81(9), 84(9), 100(9), 777 Curtiss, C. F., 43(29), 72(29), 73(29). 777

Ackeret, J.. 198, 269 Adams, Mac C., 40(l), 776, 261(21), 269 Ashley, H., 277

B Batchelor, G. K., 5, 35 Bauer, E., 87(42), 778 Beckel, C., 87(43), 778 Beckett, C., 85, 778 Berndt, S. B., 233, 264, 270 Bird, R. B.. 43(29), 72(29), 73(29), 7 7 7 Birkhoff, G., 278, 279, 287 Blasius, L., 19, 23, 36 Bodewadt, U. T., 24(49), 36 Bogdonoff, S. M., 240, 277 Bond, J. W., lOO(52). 718 Braginskii, S. I., 60(37), 778 Braun, W. H.. 34(74), 37 Brout, R., 87(46), 778 Bryson, E., Jr., 270 Busemann, A., 180, 181, 269 Butler, D. S., 134, 137, 752

D Demyanov, Y. A., 33, 37 Detra, R. W., 101, 112, 778 Diprima, R. C., 27, 37 Dolaptschieff, B., 275, 279. 286, 287 Dolidze, D. E., 6, 35 Domm, U., 275, 276, 283, 286 Drescher, H., 283, 287 Drougge, G., 233, 270

E Eggers, A. J., Jr., 238, 277 Emde, F., 189(2), 269

F

C Camac, M., 87(47), 92(47), 778 Camm, 3.. 87(47), 92(47). 778 Carrier, G. F., 27, 37 Chapman, D. R., 29, 37 Chapman, S., 40, 44(9), 54, 55(9), 62, 66(9), 68, 70(9), 73(9). 78(9), 81(9), 84(9), lOO(9). 177 Cheng, Sin-I, 23, 36 Chester, W., 121, 134, 752 Chisnell, R. F., 122, 133(5), 134(5), 136, 144, 152

289

Fay, J. A., 41, 93(20. 21), 94(20, 21), 106, lOS(20, 21). 109(20, 21), llO(21). l l l ( 2 1 ) . 112(21), 777 Finkelnburg, W., 41(14), 777 Fowler, R. H., 59, 778 Freeman, N. C . . 121, 752 Frimberger, R., 282, 287, 288

G Garrick, I. E., 246(54), 247(54), 277 Ghibellato, S., 26, 36

290

AUTHOR INDEX

Glauert, H., 179, 269 Glauert, M. B., 27, 37 Gortler, H., 21, 36 Gtithert, B., 180, 197, 269 Goldstein, S., 6, ZO(40). 35, 36 Grad, H., 14, 37, 40(3), 116 Granowski, W. L., 55(33), 177 Grawert, G., 44(31), 117 Griffith, W. C., 123, 152 Guderley, G.. 134, 137, 152. 158, 215, 220, 258, 270 Gullstrand, J.. 231, 270

Kemp, N. H., 101, 112, 118 Kennard, E. H., 83(39), 118 Keune, F., 164(18), 166, 168, 259, 260(40), 263, 264, 266(42), 268, 269, 270 Kotchin, N . J., 275, 286 Kovasznay, L., 281, 287 Krahn, E., 180, 269 Krieger. F. J., 84(40), 85, 118 Kuerti. G., 2, 35, 40(6), 117 Kuo, Y. H., 41, 93(19), 94(50), 101(19), 105, 117, 118

L H Hamel, G., 276, 286 Hammitt, A. C., 240, 271 Hasimoto, H.. 5, 6, 24(12), 35 Hayes, W. D., 155(68), 266, 269, 271 Heaslet, M., 265, 270 Heil, M., 41(15, 18), 43, 66(22), 77(22), 86(58). 93(15, 18). 96(18, 51). lOl(15, 18). 105(15), 117, 118 Heinemann, M., 14(35), 36 Heitler, W.. 44(30), 117 Herzfeld, K. F., 87(45), 118 Hilsenrath. J.. 85, 178 Hirschfelder, J. O., 43(2!)), 72(29), 73(29), 717 Hjelte, F., 191(22), 255, 269 Hooker, S. G., 279, 280. 282. 284, 285, 287 Howarth, L.. 5, 8, 13, 30, 35, 36, 37 Huber, P. W., 34, 37

I Illingworth, C. R., 10, 20. 26, 36

J Jahnke, E., 189(2), 269 Jones, R. T., 249, 269

K Kaeppeler, H. J., 55(35), 178 Kahane, A., 123, 152 Kaufmann, W., 284, 287 Kawalki, K. H., 187, 269 Keck, J., 87(47). 92(47). 178

Lagerstrom, P. A., 6, 35 Laitone, E. V., 15(38), 36, 187(12). 269 Lamb, H., 25, 36 Landahl, M . , 271 Laporte, O., 123, 124, 127, 152 Lees, L., 13(34), 14, 36, 93(48), 94(49), 101, 106, 112, 118 Le Gallo, J., 220. 270 Levine, H., 6, 35 Lighthill, M. J., 6, 25, 35, 36, 121, 152 Lin. C. C., 2, 35. 277, 278, 286 Lindsey, W. F., 183, 271 Littel, R. E., 183, 271 Lomax, H., 265, 270 Lord Rayleigh, 2, 3, 25, 35, 36, 121, 152 Ludwig, G.. 40, 116

M McFarland, D. R., 34, 37 McIlroy, W., 41(25), 68(25). 100(25), 117, 118 McLellan, C. H., 238, 271 Maecker, H., 41(14), 117 Mahieu, M.. 69(38), 118 Mallick, D. D., 6, 35 Marchand, F.. 220, 270 Marino, A. A., 123, 152 Mark, R., 106, 118 Maue, A., 275, 286 Meal, J. H., 87(44), 118 Meixner, J., 41(12), 117 Metzdorf, J., 41(15, 16, 17), 77(16), 85(17), 86(17), 93(15, 16, 17). lOl(15, 16, 17), 105(15, 16), 106, 117

291

AUTHOR INDEX Michel, R., 220, 270 Mirels, H.. 33, 34, 37 Mollo-Christensen, E. L., 271 Moore, F. K., 2, 28, 30, 35, 37 Moore, L. L., 41, 77(10), 83, 1 1 7 Munk. M. M., 269

N Nagamatsu, H. T., 155(57), 271 Naumann, A , , 283, 287 Nernst, W., 41, 117 Neuringer, J . L., 41(25), 68(25), 100(25), 117. 118 Nigam, S. D.. 6, 33, 35, 36, 37

0 Orman, P. I,., 231(39), 270 Oseen, C. W.. 276, 286 Ostrach, S., 28, 37 Oswatitsch, K., 158(1), 159(28), 160(1). 1 6 l ( l ) , 162(1), 178(1), 187(1, 1 1 ) . 189(1), 191(1), 202(1), 213(1), 220(28), 223(1), 227(27), 233. 233 240. 243(1). 248(1), 256(1), 259, 260(1, 40). 262, 263, 264, 266(42). 268(45), 269, 270

P Pack, I). C.. 199, 220(23), 270 Pai, S. D., 40(7), 117. 199, 220(23), 270 Parks, E. K., 123. 133(10), 152 Patrik, R. M., 41(27), 68(27), lOO(27). 117 Payne, K. B., 137, 152 Petty. C., 87(47), 92(47), 118 Petzold, J., 41, 87(23), 90, 93(23), 7 1 7 Pfeiffer, H., 283, 287 Polo, S. R., 87(44), 118 Prandtl. L., 1, 35 Preiswerk. E., 218(7), 269 X g o g i n e , I., 69(38), 118 Probstein, R. F.. 6, 35, 40(1). 176, lSTi(38). 271

It Rae, R . S., 231(39), 270 Rangasami. K. S. I., 6(29), 36 Resler, E. L., 41(26), 68(26), 100(26), 117 Richards, G. J . , 276. 286

Riddel, F. R., 41, 93(20. 21). 84(20, 21), 106, 108(20, 21), 109(20, 21), 110(21), 111(21), 112(21), 117 Rollnik, H., 44(31), 117 Rompe. R., 41(13), 177 Rosa, R . J., 118 Rose, P. H., 101, 106, 112. 718 Kosenhead, L., 20(40), 36, 273, 286 Roshko, A., 282, 287 Rossow, V. J., 41(24), 68(24), lOO(14). 717 Rott, N., 27, 37 Routly, P., 55(32), 117 Rubach, H., 275. 286 Rubesin, M. W., 29(66). 37 Rubinow, I.. 246(54). 247(54). 277

s.

s Sauer, R., 152 Savin, R. C., 238, 271 Scala, S. M., 98(59), 178 Schaaf, S. A., 14, 36, 40(4), 116 Schlichting. H., 2, 25, 35, 40(8), 117 Schmieden, C., 187, 269, 275, 176, 277, 286 Schwabe, M., 284, 287 Schwartz, R. N., 87(4d), 178 Sears, W. R., 41(26), 68(26), 100(26), 177, 261(21), 269 Sherman, S. F., 40(4), 716 Sjodin, L., 235, 270 Sowerby, L., Ti, 35 Spitzer, L., 55(32). 117 Spreiter, J. R., 211, 220(36), 221, 270 Squire, H. B., 285. 287 Squire, L. C., 20, 36 Stack, J., 183, 271 Stark, W. I., 106, 178 Steenbeck, M., 41(13), 1 1 7 Stewartson, K., 8, 10, 13(31, 33). 22, 23(47), 24(50). 28(47). 30, 33(47), 36 Stokes, G. G., 15, 28, 36 Stuart, J . T., 27, 37 Syvertson. C. A,. 238, 271

T I’hiriot, K. H., 6, 23, 24(24), 35 Timme, A.. 274, 278, 279, 282, 284, 285, 28 7

292

AUTHOR INDEX

Ting-Yi Li, 156(67), 277 Tsien, H., 237, 270

U Uhlenbeck, G. E., 40(2), 43(28), 60(28). 116

V v. Krzywoblocki, M. Z., 273, 276, 286 Van Dyke, M. D., 2, 10, 13, 35, 205, 238, 27 1

Vincenti, W. G., 230, 231(38). 270 von KArmAn, Th.. 220, 269, 273, 275, 286

W Wadhwa, Y. D., 20, 36

Wang Chang, C. S., 40(2), 43(28), 60(28), 176

Ward, G. N., 231(39), 264, 265, 269, 270 Warren, W. R., 123, 152 Watson, E. J., 21, 36 Watson, K. M., 55(34), 718 Wehrmann, O., 282, 285, 287 Whitcomb, R. T., 266, 270 White, W. B., 84(40), 85, 118 Whitham. G. B., 122, 133, 139, 152 Wilkinson, J., 6, 35 Wille, R., 278, 279, 285, 287 Wuest, W., 27. 28, 37 Wundt, H., 21, 36

Y Yang. H. T., 14, 36 Yoshihara, H., 258, 270

Subject Index A

F Flat-plate flow (b.1. with dissociation), 101 ff. Flutter (similarity), 24.5

Ackeret theory, 168 Acoustic pulse, 122

B

G:

Bodies of low aspect ratio (similarity, 248 f f . Boltzmann equation, 43 Boundary layer, unsteady, behind advanring shock, 31 ff. developing from leading-edge, 2 I developing from stagnation point, I X fluctuating, 25 in compressible fluid, 29 ff. incompressible fluid, I6 f f . Boundary conditions (lin. theory), I63 simplified, 164 ff. for bodies of revolution, 168 f . Busemann theory, I58

Gas-dynamic equation, linearization of, 172 ff. Cuderley profile, 2 13

H Hypersonic flow, cone and wedge in, 239 similarity, 236 f f . parameter, 237 Higher approximations (similarity), 198 ff. Hypersonic limit defined, 157

L Lewis-number defined, 96 Limits of linearization, 189 ff. for elliptic wing, 193 for rhombic cone, 193

C Circular cone (lin. and non-lin. thcory), 170, 185 Mach-number influence, 188 Collision cross section for dissociation, 87 ff. Collision equations, 42 ff. solution of, 54 ff. Compressibility effect for bodies of revolution, 183 ff. Corresponding points defined (Iin. theory), 174 Cylindrical and spherical shocks (Chisnell theory), 136

N Navier-Stokes equations, generalized, 67

P Prandtl-factor defined, 173 Prandtl-Glauert analogy defined, 178 Pressure-coefficient expansion, 161 Pressure ratio behind transmitted and incident shocks, 130, 132

Q E

Quasineutrality, 76

Enskog-Chapman method, 55, 62 compared with linear theory, 26’1 Equivalence, rule of, 253 f f .

R Reduced frequency, 243

293

294

SUBJECT INDEX

Rayleigh’s problem, 3 ff. for compressible fluid, 6 continuum theory of, 7 kinetic theory of, 13 ff.

S Shock equations in lin. theory, 162 f. in non-parametric representation.206 ff. Shock-polar, hypersonic, 209 in lin. theory, 162 transonic, 210 Shock wave interacting with contracting channel, 124 interacting with diverging channel, 121 Shock waves in ducts, models for, 123 ff. Chisnell’s theory, 133 ff. Witham’s theory, 139 Schmidt-number defined, 98 Stability theory (vortex street), 275 Stagnation-point flow (b.1. with dissociation), 106 ff. Stokes hypothesis, 7 Strouhal number, 279 f. Summation invariants defined, 50

T Thermometer problem, f. 98 Transonic flow, bodies of revolution in, 231 Transonic similarity for wedge flow, 22 ff. Transport equations, 48 ff., 68 ff., 75

U Upwash factor defined, 176 for transonic flow, 217 Unsteady flow (similarity), 242

V Viscosity coefficient, approximative calculation of, 7 8 ff. Vortex streets, 273 ff. experimental investigations of, 279 ff. invariance theorem for, 278

W Wedge flow (lin, theory and exact sol.:l, 200