ADVANCES IN APPLIED MECHANICS, Volume 6

ADVANCES IN APPLIED MECHANICS

VOLUME VI

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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 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> ( ~ 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 o;

(4.7)

u = 0,

x30,

y20,

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 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’ 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

,

= { 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 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,

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

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

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


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.


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 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 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 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. 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 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 > 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