Nonlinear conical flow

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Nonlinear Conical Flow

B.M. Bulakh

translated from the Russian

by:

J.W. Reyn

f

and W.J. 8annink

Delft Unwetslty Press

;.

,

<

Mijnbouwplein 11

2628 RT Delft

(0) 15 783254

First published in the Russian language under the title:

,.';

Nelineynyye Konicheskiye Techeniya Gaza

Copyright

©

1970 (Russian Edition) by:

Izdatel'stvo Nauka, Moscow

Translated at Delft University of Technology by

J

.

W. Reyn, Department of Mathematics and Informatics, and

W.J. Bannink, Department of Aerospace Engineering

Typing:

Mrs

.

M.J. Schillemans-van Tuyl

CIP, Koninklijke Bibliotheek, Den Haag

Copyright

©

1985 by Delft University Press, Delft, The Netherlands

All rights reserved. No part of th is book may be reproduced in any form, by print, photoprint,

microfilm or any other means without written permission from the publisher: Delft University

Press

Printed in The Netherlands

ISBN 90 6275 163 6

'

..

, I . ~

Foreword to the English edition

Flow of compressible fluids past conical bodies was introduced as a subject of aerodynamics some fifty years ago by A. Busemann, who studied the flow past an axially symmetrie cone at zero angle of attack, placed in a supersonic strearn. For a long time the subject was studied experimentally and theoretically, using linear theory although the conical flow equations are nonlinear. The study of these nonlinear equations with analytical techniques obtained its main impetus in the fifties and sixties of this century, thereupon followed by a still continuing effort to solve these equations nurnerically with the aid of modern computers.

The present book compiled by B.M. Bulakh, an outs tanding Soviet research worker in this field, is an impressively complete description of the analytical techniques which could be used to understand and to study nonlinear flows of compressible fluids past conical bodies. As su eh i t is uniquely filling a gap in the l i terature in gasdynamics.

With this English edition the translators hope to make the book available to a wider scientific public.

Delft University of Technology November, 1984

J.W. Reyn W.J. Bannink

CONTENTS

Preface

Chapter 1. General properties and some particular types of conical flow

§ 1. Basic assumptions and equations 1.1. Introduction 1. 2. 1. 3. 1.4. 1. 5. 1. 6. 1. 7. 1.8. 1. 9. Spherical coordinates Cartesian coordinates

Generalized spherical coordinates Irrotational flows

Hodograph transformation Canon ic al variables

Characteristic equations for conical flows Shock waves

§ 2. Some special types of conical flows

a) Axially symmetric flow around a circular cone 2.1.

X

2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. b) 2.10. 2.11. c) 2.12. 2.13. Introductory remarks Hodograph method

Methods, using spherical and cylindrical coordinates Transonic flow around cones

Analysis of numerical and experimental results on flow around cones

Approximate analytical solutions Similarity laws

Semi-empirical formulae

Steady flow around a cone in a detonating gas flow Axisymmetric conical flows

Introductory remarks

Compression nozzle. Flows aro~nd boat tails. Channel flow

Flow around pyramidical bodies Introductory remarks

Flow around pyramidical bodies having a cross-section of a regular starshaped polygon

2.14. The flow around pyramidical bodies having a cross-section of regular polygon of composite star shape 2.15. Possible flow regimes around pyramidical bodies and

experimental results 3 3 3 3 6 8 9 10 12 13 15 18 18 18 20 29 30 35 45 51 52 53 56 56 56 59 59 60 62 66

dl Conical simple waves 2.16. Introductory remarks

2.17. Properties of conical simple waves

2.18. Conical flows bordering a uniform flow along a Mach cone

§ 3. Singular points in solutions of the equations for conical flow 3.l. 3.2. 3.3. 3.4. 3.5. Introductory remarks The Ferri singularity

The singularity in the meeting point of the Mach cone and a shock wave of vanishing strength The singular point where the simple wave flow and the flow behind the Mach cone meet

Other types of singular points

Chapter 2. Supersonic conical gas flows

A. Flow around conical bodies, lying completely inside the Mach cone of the undisturbed flow

§ 4. Classification and patterns of flow around conical bodies 4.l. 4.2. 4.3. 4.4. Introductory remarks Slender bodies Thin bodies

Methods of solution for problems of group A

§ 5. Conditions of the bow shock wave and Mach cone in the small perturbation theory

5.1. Introduction 5.2.

5.3.

Expansion in series with respect to small values of the conical potential F in a neighbourhood of the bow shock wave or of the Mach cone of the undisturbed flow

Location of the bow shock wave § 6. Second order theories for thin bodies

6.l. 6.2. 6.3. 6.4. 6.5. Introduction

The equations for the first and second order approximation approximations for thin bodies

Boundary conditions in first and second order theories Leading edges in the theory of thin bodies

Results of the second order theory for specific bodies 69 69 70 86 89 89 90 96 109 113 115 115 115 115 115 116 117 119 119 120 130 132 132 133 140 142 145

§ 7. Higher order approximations for slender bodies and the method of linearized characteristics

7.1. 7.2.

Higher order approximations for slender bodies The method of linearized characteristics § 8. Flow around a circular cone at angle of attack (Theory

of Stone) 8.l. 8.2. 8.3. 8.4. Introduction

The first order theory of Stone

First order solution in the "vortical layer" Concluding remarks

§ 9. Separated flows around conical bodies 9.1. Introductory remarks

9.2. Results of the slender body theory for a triangular plate

9.3. Results of the linear theory for a delta plate 9.4. Concluding remarks

§ 10. Numerical methods for the inverse problem 10.1. Introductory remarks

10.2. The method of two stream functions § 11. Numerical methods for the direct problem

11.1. Introduction

B.

11.2. The method of integral relations

11.3. Finite difference method (method of establishment) 11.4. Method of lines

Flows around conical bodies, lying outside the Mach cone of the undisturbed flow

§ 12. Classification and methods of solution for problems of flow around conical bodies

12.1. Introductory remarks

12.2. The method of characteristics 12.3. Thin wings with sharp leading edges

C. Flows around conical bodies, partly lying outside the Mach cone of the undisturbed flow

§ 13. Classification and remarks on flow patterns around conical bodies

13.1. Classification

13.2. Remarks on flow patterns around conical bodies § 14. Flow pattern around a triangular plate (flat delta wing)

14.1. Introduction 146 146 148 154 154 158 159 166 169 169 175 181 186 186 186 189 195 195 196 201 205 207 207 207 208 213 215 215 215 216 217 217

§ _15.

14.2. Flow along the upper surface of the triangular flat wing

14.3. Flow on the lower side of the triangular flat wing 14.4. Flow around the triangular plate at large angles of

attack

Flow patterns around other conical bodies 15.!. Flow pattern around triangular wings

15.2. Flow pattern around the edge of a rectangular plate 15.3. Flow pattern around a cruciform wing

15.4. The flow pattern around a triangular plate with a half-cone as fuselage

§ 16. Methods of solution for the flow around conical bodies 16.1. Introduction

16.2. Review of results, obtained with the method of small parameters and other analytical methods

16.3. Numerical methods of solution of the flow problem around conical bodies

Chapter 3. Hypersonic conical flows of a gas

A. Flow around conical bodies in the case that the bow shock wave is only attached to the apex of the body

§ 17. General characteristic of the properties of hypersonic conical flows and methods of solution of the flow problem for conical bodies

17.1. Classification of flow problems

17.2. General characteristic of solution methods for flow problems around conical bodies

§ 18. Shock layer method for circular cones 18.1. Introductory remarks

18.2. Solution of the flow problem for a circular cone, not taking the vortical layer into account

18.3. Solution of the problem of the flow around a circular cone taking the vortical layer into account

18.4. Comparison of theoretical and experimental results § 19. The shock layer method in the general.case

19.1. Introductory remarks

19.2. Solution of the problem retaining the leading terms in the shock layer

19.3. Comparison of theoretical and experimental results for elliptic cones

218 224 225 227 227 228 229 231 232 232 233 241 251 251 251 251 252 254 254 256 267 27 273 273 275 283

§ 20. The shock layer method for triangular wings 20.1. Introduction

20.2. Shock layer method for a triangular plate 20.3. Similarity laws

§ 21. A survey of results obtained by other methods 21.1. Introductory remarks

B.

21.2. The method of linearized characteristics

21.3. Similarity laws for slender and thin conical bodies at large angles of attack

21.4. Method of the equivalent cone

Flow around conical wings, when the bow shock wave is attached to the leading edges

§ 22. Survey of results obtained by different methods 22.1. Numerical methods

22.2. Shock layer method

22.3. The method of small perturbations 22.4. Concluding remarks Appendices References 284 284 287 297 299 299 299 300 304 305 305 305 305 306 307 308 312

In the axisymmetric flow around a circular cone situated in a supersonic uniform gas flow the bow shock wave near the apex of the cone is also a circular cone (if the flow behind the shock wave is supersonic), but the flow behind the shock wave is not uniform.

Since in supersonic gas flow small disturbances do not propagate upstream, we restrict our attention to the flow near a cone of finite length which will be just the same as the flow around a semi-infinite cone. Then, as follows from the fact that in the problem there is no characteristic length, the flow quantities are constant along each ray emanating from the apex of the cone. Such a flow field is called conical, and is the subject of the present book. The cone problem, considered by A. Busemann [1] in 1929, was the first problem in the theory of conical gas flow. A long time i t remained the only one. Only in 1943 A. Busemann [2] derived the basic formulae for the linear theory of conical flow, with the aid of which today many important problems are solved

(see for instance [3], [4]).

The theory of non-linear conical gas flows (based on the exact equations of motion for an inviscid gas) essentially started to develop in the fifties. In the then following ten years the interest in conical flow increased consider-ably. Of course, although the flow quantities in a conical flow are functions of two angular independent variables these flows belong complètely to three-dimensional flows. Thus in the framework of the theory of conical flows such fundamenta'l problems of gasdynamics as the flow around a cone, around a tri-angular plate, etc. may be solved. Conical flows serve as a starting point for the solution of problems of spatial flow around non-conical bodies and are also interesting from the point of view of mathematical physics, since in conical flows many boundary value problems belong to the mixed elliptic-hyperbolic type, which are scarcely considered in the mathematical literature.

In recent years flow patterns around conical bodies were given by the formulation of corresponding boundary value pioblems, the development of analytical and numerical methods of solution for problems in the non-linear conical flow theory, both for supersonic and hypersonic speeds and by carrying out experimental investigations. As aresult, the non-linear theory of

conical flow has reached, to a certain extent, a final form. However, until now, books specially dedicated to this problem, do not exist neither in the Soviet Union nor abroad. The present book means to fill this gap. The author

hopes that all existing problems on the non-linear theory of conical gas flows will be reflected in this book, although the book is not a reference book on the non-linear theory of conical gas flows.

Therefore many theories are presented concisely. (For a detailed investigation of a problem one is referred to the literature.)

Experimental data are only taken into consideration in order to verify the theoretical results. Semi-empirical theory is not given much attention. Nowa-days, many problems of the non-linear theory of conical gas flows can be solved with a preassigned accuracy on a computer, and therefore analytical methods of solution of such problems are given relatively little attention. In those cases where the numerical methods of solution are not yet developed or not appropriate, the analytica I theory is expounded thoroughly.

In principle, technica I accounts of the various companies and special insti tu-tions, inaccessible to Soviet readers, are not listed in the bibliography. Unfortunately, in this book a unique notation could not be maintained and occassionally in different paragraphs the same letters are used for different quantities.

In conclusion I would like to remark, that this book emerged as a result of many years of work of the author in the field of non-linear conical gas flow, started under the guidance of S.V. Fal'kovich, whom the author is sincerely grateful.

CHAPTER 1. GENE RAL PROPERTIES AND SOME PARTICULAR TYPES OF CONICAL FLOW

§ 1. Basic assumptions and equations

1.1. - Introduction

In this book we consider the steady mot ion of a gas, assuming that the gas is inviscid, non-heat conducting and is locally in thermodynamic equilibrium

(or in "frozen" state), i.e. there exists an equation of state of the gas. Generally in the equations of motion of such a gas the unknown functions are the three components of the velocity vector V of a gas particle, the pressure p, the density p (or also two other thermodynamic functions of the gas), all depending on the three spatial coordinates. Under certain restrictions conical bodies placed in a supersonic uniform gas flow generate a conical flow, which is characterized by the fact that the velocity V and the thermodynamic

variables of the gas are constant on rays, emanating from the apex of the body, being the center of the conical flow (the occurrence of a flow of such a type depends on the fact that in a supersonic flow around a conical body this body can be considered to extend infinitely far downstream, i.e. there is no characteristic length) .

1.2. - Spherical coordinates

If a spherical coordinate system R, ~, 8 (fig. 1) is introduced, with origin in the apex of the body, then the unknown quantities will only depend on the angular coordinates ~ and 8, but will not depend on R, i.e.

di

dR

o ,

where S and i are the specific entropy and enthalpy, respectively.

z

Fig. 1

In the chosen coordinate system (fig. 1) the equation of continuity div(p~)

=

0, the E l ' dV _{u er equatlon}

_dt

_{= -}

_p

1 _{grad (}d .

P dt lS ~he substantial derivative with respect to time),

dS

and the energy equation dt

= 0,

can for a conical flow be written in the form:

(pv sin 9)9 + (PW)~

=

-2~p sin 9 , 2 vU 9 + w cosec 9u~ - v 2 w

o ,

2 vV 9 + w cosec 9v~ + uv - w cot 9 1 P . P9 ' (VW

9 + W cosec 9 w~) sin 9 + w(u sin 9 + v cos 9)

VS₉ + W cosec 9S~

o .

(1. 1.) (1.2. ) (1. 3.) (1.4. ) (1.5. )

Here u, v, ware the components of the velocity vector V in the direction of increasing R, 9,

~,

respectively; where the notation f

9

-=

~~

,

f~

=

~!

is used. In order to complete the system of equations (1.1.) - (1.5.) i t is necessary to specify the dependency

S S(p,p) , (1.6. )

or to consider pand p as known functions of S and another thermodynamic quantity, for example the specific enthalpy i:

p p(S,i), P p(S,i) . (1. 7.)

[Also the functional relationships T = T(S,i), e = e(S,i), a = a(S,i) may be used where T is the absolute temperature, e is the specific internal energy, a is the speed of sound. The graphical representation of relation (1.7.) is called the Mollier diagram (see for example [5]).]

f d th 1 . dV

o ~, an e Eu er equatLon

dt

1 grad p,

re-p

Taking the inner product placing

!

dp by di (di

=

P

1

P

dp for dS

=

0) and integrating along a streamline, we obtain the Bernoulli equation

1 2 + ;

2.

V ~ i

o (1.8. )

where

v

2

=

1~12

=

u2 + v2 + w2, and i is thé stagnation enthalpy or total

o

enthalpy. If a conical flow emerges from a uniform flow,i

o is constant through-out the conical flowfield, since io does not change when the gas passes through a shock wave.

One of the equations (1.2.) - (1.4.) may be replaced by the integral (1.8.). With the aid of eq. (1.8.) we now simplify the system of equations (1.1.)

-(1.5.), (1.7.) (in this connection see also [6]).

Temporarily, in order not to interfere with the notations used in thermo-dynamics, the derivatives will be written out fully. Upon differentiating eq.

(1.7.) we obtain:

3p (3

P)

3s (3

P)

3i

~

=

as

i 3<1> + 3i S 3<1>

Multiplying eq. (1.9.) with v sin

e,

eq. (1.10.) with w, adding the two results and taking into account (1.5.) we obtain:

3p

ae

v sin

_e

₊

~~

_{w =}

(~~)S (~~

_{v sin}

_e

₊

~~

_w)

From the second law of thermodynamics for reversible processes

TdS di -

P

1 dp

it follows dlat, for S

where const., di P

2 '

_a 1 - dp P

(a is the local speed of sound of the gas.) Differentiating (1.8.) with respect to

e

and <I> we obtain:

di (u 3u dV _3w) as as+ v as + w a s ' 3i - (u au av dW) ~= a<l> + v 3<1> + w 3<1> . (1.9. ) (1.10. ) (1.11.) (1.12. ) (1.13. ) (1.14. )

We eliminate p from (1.1.) making use of (1.11.), (1.13.), (1.14.) and (1.2.); thus the equation of continuity takes the form (1.15.)

(1. 15.)

(The derivatives are indicated again with subscripts.)

From (1.12.) it follows that:

_.!.(~)

=T

P

as.

'

~

1 ,

therefore, taking into account (1.14.)

and eq. (1.4.) can be written in the form

v sin 8w₈ - uU<I> - vv<l> + sin 8w(u + v cot 8) . (1. 16.)

As a consequence of (1.5.), (1.16.), (1.2.) Iollows the equation

2

- uu

s

-

ww S + W cosec Sv <I> + uv - w cot S . (1. 16a. ) As ~ result of the transformation we obtain instead of the system (1.1.)

-(1.5.), (1.7.) the much simpier system of equations (1.15.), (1.2.), (1.16.), (1.5.) for u, v, w, S [T

=

T(S,i), a

=

a(S,il; i is determined fr om (1.8.)].

1.3. - Cartesian coordinates

Fig. 2

In a number of problems (for example in the case of plane wings) it is convenient to use a cartesian coordinate system 0lXYz (fig. 2). Then, the parameters of the gas in the conical flow only depend on the coordinate ratios

x y '

ç

=

z'

n

=

z·

The

Ç

,n

plane has a simple physical meaning: it is the plane z = 1 in the xyz space, and

Ç

,n

are the corresponding coordinates x and y of points in this plane. The equations of

continuitYI momentum (Euler) and energy written in cartesian coordinates , when using the variables ~ and n, take the form:

P(u~ + v

n E,wE, - TlWn) + (u - E,w) PE, + (v - nw) Pn

o

I (u - ~w) _u~ + (v - nw) u

_{- "i)}

1 _p~ n (u - ~w) vE, + (v - nw) v

_{- "i)}

1 _Pn n (u - E,w) w~ + (v - nw) w n

o

Here u, v, ware the components of the vector V in the X, y, z directions , respectivelYI using the subscript notation for the derivatives.

From these equations using transformations , similar to those in the case of sphe'rical coordinates I we obtain the system of equations

~w)

(u 2 2 + w2) nw) (u 2 + 2 2) (u - + v + (v - v 2 + w 2 ~ 2 _u - V ) = O , + a (E,w~ + nW n - ~ n + (v - nw) vnJ + (u - E,w)

w~

+ (v - nw) w n (u - E,w) wE, + (v - nw) w n + E, (u 2

o .

2 + v

o

1 + n (1. 17.)

o

I (1.18. ) (1. 19.) (1.20. )

[a

=

a(S,i), T

=

T(S,i), i is determined by (1.8.).] For a perfect gas p PRT

(where. R is the "characteristic" gas constant) I c the specific heat at p

constant pressure and Cv the specific heat at constant volume are constant (R

=

Cp - cv) I and

S

=

c In (PP-Y) + const I v

c y ...l: c v 2 i c T a p y - 1

,

2 Y E.

=

yRT a p

From this i t follows, that T

2

a and the integral (1.8.) takes the form

yR

2 a2

~V + - - 1

=

i , that means

y - 0 that in the system of equations (1.15.), (1.16.), T and a2 are known (linear) functions of v2

=

I

(1. 5.) and (1. 17.) - (1. 20.)

u2 + v2 + w2•

1.4. - Generalized spherical coordinates

For the study of the flow field around conical bodies in hypersonic gas flows i t is convenient to use a system of orthogonal generalized spherical

coordinates R,

e,

~, in which one family of coordinate surfaces consists of 2 2 2 2

the spheres R

=

x + Y + z

=

constant; the second and the third family is constructed in the following way. On the surface of the sphere R lie two families of orthogonal coordinate lines such that the surface of the body corresponds with one of the coordinate lines. Then from these coordinate lines conical surfaces are formed which are also taken as the coordinate surfaces

e

=

const., ~

=

const. (see fig. 3).

Fig. 3

v w

If we indicate by u, v, w the components of the velocity vector V in the directions of increasing R,

e,

~, respectively, then the equations of continuity, momenturn and energy can be written as [7] : (1.22. ) - - v + - -v + uv + vw Al

e

A 2 ~ (1. 23.) v

C

u 2 _{+ v 2} + w 2

)e

w

C

u 2 - - 1- + + - - 1- + Al 2 A₂ (1.24. ) ~ S + Al

e

~S A 2 ~

o .

(1. 25.)

Here 2 +

Ye

[ 2 2 R x~ + y~ + z~

2

]

~

are the coefficients of Lamé, to be calculated on the surface of the sphere R

=

1 whereas one of the equations of Euler is replaced by the Bernoulli equation in differential form (1.24.). We note the important special case when the surfaces ~

=

constant are taken to be a family of planes normal to the surface of the wetted body.

In this coordinate system e and ~ are the lengths of the curves, measured on the sphere R

=

1, normal to the surface of the body and along the body, respectively; Al

=

1, A

2 cos e - ~' (~) sin e, where ~' (~) is equal to R times the curvature of the body surface, positive when the surface is concave towards increasing values of e (see

[8]).

1.5. - Irrotational flows

If the motion of the gas is irrotational, then there exists a velocity potential <P (~

= grad

<p) and S

=

So

=

constant.

For conical flows <p can be represented in the form <p are spherical coordinates.

RF(e,~), if R, e and ~

We shall call the function F the conical potential. The velocity components are determined by the formulae

u

=

F , _v

₌

_R

1 _<Pe

₌

_{Fe '} w

R sin (1.26. )

and the system of equations (1.15.), (1.2.), (1.16.), (1.5.) reduces to one equation for the determination of F (see [20]).

sin 2 e(a 2 - v2) _Fee

_-

2vw sin e _Fe~ + (a 2

-

w2) _{F~~ +} + sin 2 8(2a 2 - v 2 w2) F + sin 28(a 2 w2) 0

2 + Fe

,

(1.27. ) [a

=

a(i,So) 2 2 2 (1.8.) ] u + v + W

,

2 + i i 0

In cartesian coordinates follows

lP = zF(!;,n) ,

v = lP = F Y

n'

( 1.28)

The system of equations (1.17.) - (1.20.) is reduced to equation (1.29.) (see [9]) : L[F] _{AF!;!; + 2BF!;n +} CF

o

,

nn (1.29. ) where a2(1 + !;2) (u 2 A - - !;w) , B a2!;n (u !;w) (v - nw) (1. 30.) . C a2 (1 + n2)

-

(v - nw) 2 u2 2 2 [a = a(i,So) + v + w + i 2 i 0 see (1.8.)].

If in the !;,n plane we introduce polar coordinates by means of the formulae

2 2 ~

r= (1; + n ) , tan 8 = n/i;, then equation (1.29.) transforms into ([10]):

+ 2 [F - (r +

~)

Fr] F8 +

(a

2 and (1.28.) in u = cos 8 F sin 8 ! F r r 8 sin 8 8 1 v = F + cos - F r r 8 1.6. - Hodograph transformation

Because for conical flows we have

+

(!

r F r8

_! F ) +

r 8, w = F - rF r (1.31.) (1. 32.)

u(Cll) , v v(i;,I1) , w w(Cll) _{\Pz '}

we can write

I; (u,v) , 11 11 (u,v)

and from this w w(u,v) (provided that the Jacobian of the transformation

D(u,v)

=

F F -

F~n2

~

0) 0(1;.11) 1;1; 1111 ~'I

which means that to any conical flow there corresponds some surface in the

2 _ )

hodograph space u, v, w (the case FI;~l111 - Fi;l1 = 0, will be considered later. The differential equation, governing w

=

w(u,v), can be obtained in a simpler way, applying the Legendre transformation [11, 12, 13] on the conical

potential F(I;,I1):

x(u,v) + F(I;,I1) uI; + Vil ,

(1. 33.)

V ₁₁

x

.

V

From (1.28.) it follows, that the Legendre potential X(u,v) in our case is simply -w(u,v), therefore

-w

u 11 -w V (1. 34.)

The equation for w(u,v) can be obtained from (1.29.), if (1.34.) is substituted in (1.30.), and the second derivatives of F with respect to I; and 11 are re-placed by the familiar expression (see [13])

where

-w g ,

uu

Upon dividing by g, equation (1.29.) takes the form:

where Al a2(1 + w 2) (u + ww )2 u u B 1 2 (u +ww) (v + ww ) aww u v u v (1. 36.) Cl a2 (1 + w )2 _v

-

(v + ww _v )2 1. 7. - Canonical variables

If AC - B2 > 0 equation (1.29.) can be reduced to a canonical system of

equations, containing as unknown functions ~, n, u, v, w [see (1.28.)], if as

new independent variables p, cr are introduced using equations (see [14]):

o ,

(1. 37.)

o •

(1.38. )

The remaining equations of the canonical system have the form:

Au + Bv + [AC - B2] ~ v

o ,

p p cr (1. 39.)

2 ~

Au + Bv - [AC - B ] v

o ,

cr cr P (1.40. )

IJ.w + nIJ.v + ~IJ.u

o ,

(1.41.)

( a2

a

2

. )

IJ.

= -

-

+ - -1.S the operator of Laplace .

ap2 a i

Moreover, on the boundary of the region, wherein ~, n, u, v, w as functions

of p, cr are defined, the condition

dw + ndv + ~du

o

,

(1.42. )

must be satisfied.

From the system of equations (1.37.) - (1.41.) it is possible by

cross-differ-entiation to obtain a system containing the highest derivatives of ~, n, u, v, w with respect to p and cr in the form of the operator of Laplace (see [14]). The system of equations (1.37.) - (1.42.) is invariant with respect to a

conformal transformation in the plane of the independent variables (another method to reduce equation (1.29.) into a canonical system for the case AC - B2 < 0 and AC - B2 > 0 is given in [15]).

1.8. - Characteristic equations for conical flows

In order to examine the type of the system of equations describing the motion of the gas i t is possible to use an arbitrary system of coordinates. Let us consider, for example, a cartesian coordinate system, wherein these equations [(1.17.) - (1.20.)] have the most symmetrical form.

From eq. (1.18.) and (1.20.) i t immediately follows, that the lines of constant entropy, which coincide with the streamlines in the ~,n plane, are

-1 -1

determined by the equation (u - ~w) d~

=

(v - nw) dn, and are "double" characteristics of the system of eq. (1.17.) - (1.20.). Multiplying the equations of this system with as yet undetermined factors, adding the

resulting equations and requiring that the combination obtained only contain derivatives of the unknown functions in one direction (or considering that a characteristic is a line of weak discontinuities), we find that the two re-maining characteristics can be determined by equations, which also apply in the case of irrotational motion

o ,

where A, B, Care given by (1.30.). Relations on the characteristics are derived in par. 12.2.

(1.43. )

Equations (1.29.) is of elliptic (hyperbolic) type in a point of the x, y, z space, if the projection of the velocity ~ on the plane, normal to the radius vector is smaller (larger) than the local speed of sound a [this can be se en in a very simple manner from eq. (1.27.)].

The corresponding system of equations (1.17.) - (1.20.) will have two coin-ciding real and two complex characteristics in the elliptic case (in the hyperbolic case all four characteristics are real).

We will show that the characteristics of the system (1.17.) - (1.20.) manifest themselves as intersections of conical characteristic surfaces with the plane z = 1. Characteristic surfaces for steady flow of a gas are determined from the condition, that the magnitude of the projection V

f

on the normal to the characteristic surface is equal to the local speed of sound

a,

that means V

n + a.

In the following we shall be interested only in conical characteristic sur-faces. Their equation can be written in the form:

r

(~,

; )

=

0 or

r(~,n)

The unit normal n on such a surface in the xyz space has the components:

n

=

N

whereas the velocity vector ~ is (u, v, w). The condition v n can now be written in the form

v

n

=

+ a

+ a (1.44. )

Upon squaring relation (1.44.), substituting r~ by dn, r

n by -d~ (because

-1 -1

(r~) dn

=

-(r

n) d~ along a curve r

=

0) and performing some simple trans-formations we obtain eq. (1.43.). Reversing the arguments mentioned above we obtain that every characteristic in the ~,n plane corresponds to a conical characteristic surface in the xyz space. Streamlines are the intersections of conical streamsurfaces, also appearing as characteristic surfaces of steady motion of a gas, with the plane z

=

1. In fact, if the equation of a conical streamsurface is r(~,n)

=

0, then the projection of the velocity V on the normal ~ is equal to zero, that means:

[see (1.44.)

1

which, af ter replacing r~ by dn, and r

n by -d~, respectively, can be written in the form (u -

~w)-l d~

= (v - nw)-l dn (in spherical coordinates the characteristics are the intersections of conical characteristic surfaces with the sphere R

=

1).

The system of equations (1.17.) - (1.20.) is very similar to the equations of plane rotational gas flow.

In fact, if for plane rotational flow as unknown functions are taken the com-ponents u, v of the velocity vector along the cartesian coordinate axes x and

-1

y and s = S[cvY(Y - 1)] ,where S is the specific entropy, then the equations of motion take the form

(a 2 _ u2 ) u uv(u + v ) x y x us + vs 0 x y v(u v ) a s 2 0

,

y x x

where a is the speed of sound;

2 a (fX = df dX' f y df\ dY) + (a 2 _ _{v2 ) v} y 0

,

(1.45. )

The streamlines, determined by dx =

~

appear as characteristics of the system u v

of eqs. (1.45.) (along them S

=

constant, therefore these characteristics are cal led lines of constant entropy) •

The type of flow (supersonic, subsonic) and the regions of influence are

completely determined by the two remaining characteristics of the system

(1.45.), of which the equation has the same form as in the irrotational case. Exactly the same we have for the system of eqs. (1.17.) - (1.20.), with the difference only that here is one more unknown function; therefore another equation is added, and the streamlines in the ~,n plane (lines of constant

entropy), are determined by theequation (u -

~w)

-l

d~

= (v - nw)-l dn, and are

appearing as double characteristics. The types of flow and regions of influence

(in the ~,n plane) are completely determined by the remaining two characteris-tics; see (1.43.). Because of this, in points where the latter characteristics

are real (complex), the flow will be cal led conical supersonic (conical sub-sonic) .

The mentioned analogy between the gasdynamics of plane flow and the theory of

conical flows in the ~,n plane (or on the unit sphere) permits in many cases

to predict correctly the proper ties of conical flows from the known facts of

plane flow.

1.9. - Shock waves

condition that the flow is tangent to the body reads V

n

=

~

.

~

=

0, that

means that the surface of the body must be a stream surface (in particular, in the ç,n plane along the line, corresponding to the surface of the body,

the condition (u - çw)-l dç

=

(v - nw)-l dn must be satisfied).

On the surface of the conical shock wave the laws of conservation of mass,

energy and momenturn, are satisfied which, if we indicate the gas parameters in front of the shock wave by the index 1, and behind ~~e shock wave by the index 2, can be written in the form

V 2

2 2

(1. 46.)

Here V_n, V'l' V'2 are the projections of the velocity ~ on the unit normal

vector, and two arbitrary directions tangent to the shock wave, respectively

In the following the relation (1.46.) will be written in a form convenient for the solution of the problems in a particular section. Here we will consider

only the case, where the shock wave is given by the equation n that the gas is a perfect gas, which means i

=

y/(y - 1) (pip).

We assume, that the shock wave borders a flow, having velocity components ul'

vl,w

l along the axes of the cartesian coordinate system 0lXYz and speed of

sound al. The velocity components u

2' v2' w2 behind the shock wave can easily be obtained fr om the condition that the velocity components tangent to the shock are continuous, and from the condition of Prandtl for the normal velocity components across an oblique shock. Omitting simple calculations, we may write

for the final formulae:

where:

P (1.47. )

n

_s

'

dn

s di;

-1

The increment of s

=

S

[

y(y

-

1) c ] across the shock is determined by means

v

of the usual formula for the increment of entropy through a shock wave

2

1

{[

2Y~

-

Y

+

1]

[2

+

(y

--Y-(Y--=---l'-) ln Y + 1 - Y ln

(y + 1)

(1.48. )

where qn is the component of the non-dimensional velocity normal to the shock wave; it is determined by the relation

We now assume that _{u 1 ' v l'} w₁ are the velocity components of an irrotational conical flow; then the relation dW₁ + _{ns dV 1} + E, dU

1 = 0, is valid, which holds according to (1.28.) . We also assume, that the flow behind the shock

wave is irrotational, then u

=

FE,' v

=

Fn' w

=

F - i;FE, - nFn, where F is the conical potentialof the flow. According to (1.47.) for n = ns(E,) the relations

n

'p s ' (1.50. ) hold. The functions F₂, (Fi;) 2" (F

n)2' determined from (1.50.), satisfy the strip conditions, whereas the potential F is continuous through the shock wave.

Indeed, if we multiply the first equation of (1.50.) by E" the second by ns

and add the third, we obtain as aresult:

Upon differentiating F

2 along the shock wave we find:

(1.51. )

Multiplying the first eq. (1.50.) by dE" the second by dn

s and adding them we obtain as aresult:

(1. 52.)

From (1.51.), (1.52.) follows the strip condition:

dF

2 (1.53. )

This means, that in the study of flows behind shock waves in the irrotational approach the conditions for the velocity components on the shock wave can be satisfied exactly, and i t is not necessary to deduce approximate expressions, valid only for irrotational flows.

In a series of problems i t is needed to consider contact discontinuities on conical surfaces; on such surfaces are satisfied the usual conditions:

(1.54. )

(here n is the normal to the contact surface) .

§ 2. Some special types of conical flows

In § 2 will be considered mainly the simplest types of conical flows, for which the equations of motion re duce to ordinary differential equations or to a system of algebraic equations.

a) Axially symmetrie flow around a circular cone

2.1. - Introductory remarks

The problem of finding the axially symmetrie supersonic flow around a circular cone was considered for the first time by A. Busemann in 1929 in a report from which only a small abstract [1] was published.

Af ter that, a very short note on the solution was published by M. Bourquard [16], and almost at the same time the work of G. Taylor and J. Maccoll [17], appeared, where the problem of the cone was investigated with great complete-ness both theoretically and experimentally (still later J. Maccoll [18] extended and calculated more accurately the results of [17], and obtained shadowgraphs of flying projectiles with conical head, from which one is repro-duced in fig. 4).

Fig. 4

A. Busemann solved the problem in the hodograph space, G. Taylor and J. Maccoll used spherical coordinates in the physical space. Since then for the problem of the axially symmetrie flow around a circular cone extensive tables were compiled, experiments were carried out, the possible regimes of flow around the cone were defined more precisely, numerous

approximate solutions were obtained, empi -rical formulae for the distinct regimes in the flow around the cone we re found. Many of these results are mentioned in text books on gasdynamics. For this reason the weil known facts will be considered here only briefly and most attention will be paid to facts con -cerning the main features of transonic and hypersonic flows around the cone, which are less generally known.

Let us consider the circular cone, with semi cone angle E, in a uniform supersonic flow, having a density P1' a pressure P1' a temperature Tl' a specific entropy Sl and a specific enthalpy i₁, a Mach numb~r M₁ and velocity ~1 in the direction of the co~e axis (fig. 5).

z

If the quantities of the uniform flow and the magnitude of E are such, that in the region in between the bow wave and the surface of the cone, the flow is supersonic, then the cone can be considered to be infinitely far extended downstream (because small disturb-ances in supersonic flow do not propagate upstream) .

Since under these condi tJ:~ . ..-!h~_.RLQ):J)&~do not

_{----_..---_.-}

contain a S.llê-x.aç,ter.istri·Q··-l-eFlEj·t-h; me·

-Fig. 5 parameters of the gas will de end onl on the

;

~g

le·

e~

-

:

~

·

:-:

Ph

~;l

-

co

9~sl

~

.

n~

~~

em

..i

s

_ used, or s2 +

n

_ _ _ 2 =

1.

3 -Vx2 +

i

-

.

.

if

-

~

~~~

t;

Si

;

~

"'

;~

'

~;;~~

ates

are - -!...-used that means _ _ _ _ _ _ the flow must be conical (other flow regimes will be discussed later) .

---.~~---All elements of the bow wave, are inclined with respect to the direction of the undisturbed flow with the same angle, therefore, behind the shock S = S2

constant, and the flow is irrotational.

2.2. - Hodograph method

J

Follo~ing A. Busemann [1, 11] we consider the solution of the problem in the

hodograph space.

Since the flow around the cone is axially symmetric, the surface in the

hodo-graph space corresponding to this flow must be a surface of revolution'around

the w axis, and its equation can be written in the form

w w(w) , w (2.1. )

where w is the velocity component, normal to the axis of symmetry.

Because the surface of revolution is determined by an arbitrary meridional

section, we consider the intersection of the surface with the plane u

= o.

Differentiating (2.1.) with respect to u and v, and then taking u

=

0, we

obtain the following relations:

0, dw

=

dw w v, w w

,

u v dw dv d2w d2w (2.2. ) 1 dw dw _0, w

_W

_dw

₌

_{;- dv'} w w _{- 2 - 2} uu uv vv dw dv

According to (2.2.), equation (1.35.) can now be written in the following form:

(2.3. )

If we inter change the dependent and independent variables as a result of which v is now a function of w, then equation (2.3.) can be written in the form:

vvn 1 + (v') 2 - (w + vv,)2

2

a

-;

(2.4. )

(Primes indicate differentiation with respect to w.) In (2.4.) the speed of

sound a is a known function of S and i (diagram of Mollier); S

=

S2

=

constant

behind the. bow shock wave, and i, as a function of

v

2

=

v2 + w2 (for the cross-section u

=

0), is determined from Bernoulli's integral (1.8.):

2 2

v + w

2 + i i o (2.5. )

with the aid of eq. (1.34.) which for the cross-section u =

°

is obtained in the form: tan 8

n

Y.. z dw dv v' ' ~=o z (2.6. )

From (2.6.) i t follows, that the direction of the ray in the y01z plane on

which the velocity components are equal to 0, v, w, respectively, coincides with the direction of the normal to the meridional curve in point A(v,w)

(fig. 6).

Fig. 6

(In this figure the cross-section u

=

0 is represented.)

For the graphical integration of (2.4.) i t is more convenient to write

(2.4.) in a different form. If we introduce the notations:

R [1 + (v')1 3 /2 v" U Iw + VVl

I

11

+ (v') 2 N v

11

+ (v') ~

then equation (2.4.) is obtained in the form:

R - - -N

-u

2 1 -2 a

The quantities R, N, U have a simple geometrical meaning (see fig. 6).

(2.7. )

A(v,w) ,~ where, if R > 0, the curve is convex in the direction of increasing v, if R < 0 the opposite is true; N (for v > 0) is the length of the segment AB

of the normal to the curve in A; U is the length of the normal OC, drawn fr om the origin of coordinates to the normal AC. Let us further indicate by W the

length of the segment of the normal AC; W is the component of the velocity vector, in the direction of the radius, having an angle

e

with the axis, and

U is the velocity component normal to this direction.

Let us now consider the tangency condition for the flow at the cone surface

and the condition on the bow shock wave. On the surface of the cone we have tan

e

tan E = ~ or, U = 0, because here the velocity vector has a direction

w

along the corresponding ray on·the surface of the cone.

The bow shock wave in the present problem is a circular cone, of which the

semi-apex angle will be indivated by

B

.

Formula (1.46.) can be written in the form: for

e

=

B:

W_l W 2, PlU 1 P2U2 U 1 2 _{U 2} i 1 2 2 2 (2.8. ) + -2- = i₂ + -2-' Pl + P₁U_{1 = P2} + P 2U2 Here i 1 used] . i(P2'P

2) [also therelation p= p(S,i), P =p(S,i) maybe

In addition, it is necessary to satisfy the kinematical relations:

V

l cos

B

,

V 2 cos

(B

-

cS) , (2.8a)

o

where cS is the deflection angle of the velocity vector at the shock wave.

For a perfect gas

i c T P P RTp 2 a y - 1 o ~E. Y - 1 P ,

c y = --.E = c const.

,

v 5 c In (PP-y) + 5 v 0

and the relation (2.8.) can be written (see for example [19]) in the

following form: p -2 P 1 P 1 2 T 2 a2 Tl 2 al 5 2 - 51 R where

~

(M 2 sin 2

8

-

1) , Y + 1 1 1 + 2(y - 1) 2 (y + 1) 2 . 2 D M 1 s~n i J -2 . 2 D M 1 s~n IJ In

{[1

2y 2 ]l/(Y-l) + --- (M 1 sin 2

₈

-1) x

y

+ 1 2 x [ (y + 1)

M~

(y - 1) M 1 (2.9. ) (2.10. ) (2.11. ) (2.12. )

is the Mach number of the undisturbed flow; POl' P0

2 the stagnation pressure in front of and behind the shock wave, respectively,

(2.13. )

the relation between cr and 8 is given in the formula

1

2 sin 2 8 1 M 1 -tan

0

2 cot 8 2 M 1 (y + cos 28) + 2 (2.14. )

We now have obtained in the formulae (2.8.) - (2.14.) a complete set of relations.

In the following i t is convenient to relate the velocity components to the

..

' 0"

limit (maximum) velocity V

max' which corresponds to zero enthalpy (expansion into vacuum) in Bernoulli's equation (1.8. )

v

2 2 V i i max - + _{= -2} -2 0

From formula (2.8a.) i t follows, that

V max V 1 - - cos V max

s

.

For a perfect gas eq. (2.15.) may be written in the form

From 2 M eqs. U 2 U₁ (2.8a. ) U 2 V max V₁ - - sin V _max

and (2.9.) follows the formula

= Y : 1 ( 2 _{. 2}

_S

+-Y - --2 1)

S

M1 s~n V (2.15. ) (2.16. ) (7. .17.) (2.18. )

Because the quantity M

1 is related to ~ V by eq. (2.17.), (2.18.) may be 1

written in the form:

U 2 V1 - - = - - sin V V max max y - 1 1 - -- - - + 2 sin2

S

y

]}

.

Substituting (2.16.) in this relation we obtain (2.19.)

(2.19. )

For a given ratio V

1/V max (that means for a given value of M1) the components of the velocoty vector ~2(0, _{v 2 , w2}) behind the shock wave satisfy the known equation of the shock polar, which may be written (see for example [20]) in the form:

(2.20. )

the problem is solved numerically, where two methods ~re possi~le. In the ~-~~

first method, applicable only for a perfect' gas, the semi-apex angle E and the value of V

3/V are given, where V3 is the velocity,on the cone surface. __ I-...--~ ___ ,_ max

From these data point A3 in the hodograph diagram is constructed (see fig. 6 where all velocities are related to Vma). Equation (2.15.), for a pl"rfect gas, may be written in the form:

~

2 (2.21. )

For the value V

3/Vmax from (2.21.) one finds (a3/vmax)2. Moreover for point A3 the following equalities hold: u

3/Vmax

=

0 (tangency condition for the flow around the cone) and N

3/Vmax = V3/Vmax. With these data from eq. (2.7.), written in the form R V max N V max _{(2.22. )} may be determined R

3/Vmax' and on the extension of segment OA3 the centre of curvature for the point A3 (point D

3) of the meridional curve (A3D3 = IR31/vmax) is constructed.

The end point A (fig. 6) of a small circular arc with centre in D3 and radius I

equal to IR31/v , determines a point which lies approximately on the max

meridional curve. When we construct in point A the normal to the meridional curve (normal to the corresponding circle), we find the lengths of the segments AB, AC, OC, OA (fig. 6), determining N/V

max' W/Vmax' U/Vmax' V/Vmax' and further, with the aid of eqs. (2.21.), (2.22.), R/V

max in point A is deter-mined and the center of curvature in point D is constructed.

The end of a small arc of the circle with centre in D having the radius IRI/v , determines the following point of the meridional curve etc. The

max

calculation is carried out as long as the flow does not satisfy condition (2.19.), valid on the bow shock~~ (point A

2 in fig. 6).

A practical method to find

B

consists therein that for a series of values of 8, u/v max in the hodograph plane is constructed until UO /V max' satisfying the formula

L=

_v

max

:L:..J. _W cot

8 [(V max)2 -

11

Y + 1 V _max W

-is reached, where the magnitude of the right hand side also can be found fr om the hodograph diagram. The point of intersection of these two curves

determines

S.

From the known values of

S

and W

2/V max , by means of (2.16.),

-V₁/V can be determined and from (2.17.) the Mach number of the undisturbed max

flow. From (2.9.), (2.10.) then Pi P1-' Pi P1 can be determined, 50 that the pressure and the density in all points of the flow behind the bow shock wave can be found from the condition of isentropic flow

and Bernoulli's integral (2.15.), written in the form

( V ) 2 2 (V 1 )2 1

P

1 V max +

~

V max M 2

~

P

= 1 • 1 (2.23. ) (2.24. )

(We remark, that the discussed method of the construction of the meridional

curve need not be carried out graphically but mayalso be done numerically.)

(see[17]).

In the second method, which can be used for a perfect as weil as for an imper-fect gas, the angle

S

-

being the semi-apex angle of the bow wave - and the parameters of the undisturbed flow are given; the semi-apex angle E of the conical body is then determined.

Let us first consider the case of the perfect gas. For a given M

l, from formula (2.17.) can be determined V

1

/v

max

,

then, from (2.8a.), U1/V max , Wl/V max and furthermore from (2.9.), (2.13.), (2.14.), u

2/V max ,

w

2

/v

maX

,

tan Ó. With the aid of the known value

and Ó the point A

2 in the hodograph diagram (fig. 6) can be constructed. The direction of the normal to the meridional curve is given by 'the angle

S.

The subsequent numerical method does not differ from the way of calculation in the first method. The end point A

3, corresponding to the surface of the cone, is determined by the condition U

3

=

0 (the normal to the meridional curve in ₍ point A3 passes through the origin 0 of the coordinate system, fig. 6).

The graphical representation of the results of integration can be obtained with the aid of the shock pol ar curve (see [11]).

A scheme of such a construction is given in fig. 7.

For a given Mach nurnber MI of the oncoming flow the end point of the

IJ

Y.:

Fig. 7

velocity vector behind the shock

wave lies on the shock polar curve

(the curve GA

₂

A

₂

A

2H in fig. 7), deterrnined by equation (2.20.), [MI and v I / v a r e related by

ma x .

(2.17.)]. On the shock polar curve the series of points A

2, A

2,

A

2,

is taken corresponding to various

values of the angle

B,

being the angle of inclination of the bow shock wave;

for each of them the meridional curve A

2A3, A

2

A3', ... is constructed. The end points A

3, A

3,

A

3, ...

of these curves corresponding to various semi-apex

angles f: of the cone, form a curve which was cal led "apple curve" by

A. Busemann. A series of "apple curves" for various Mach nurnbers MI is

con--~_ ....

-structed in [11].

Let us now consider the case of a imperfect gas. In the theory of conical

flows rnain interest lies in the deviation at high temperaturesof the properties

of the gas fr om those of a perfect gas.

Then the vibrational degrees of freedom of the molecules are excited, giving rise to dissociation and ionization, and new chemical combinations are formed (so, for instance in air, mainly consisting of a mixture of pitrogen, oxygen and argon, besides the reactions of dissociation and ionisation, nitrogenoxyd etc. is formed). If the relaxation time for these reactions is

small in cornparison with a characteristic time in an actual gasdynamical

problem, it is possible to consider the gas to be locally in thermodynamic

equilibrium. The other limiting case is that of so cal led "frozen"

thermo-dynamical equilibrium, when the relaxation time of the reactions is large

compared.to the characteristic time of the problem.

In both cases an equation of state of the gas exists and there is a possible

regime of conical flow around the conical body.

In the intermediate case, when the relaxiation time of the reactions that

oc~ur in the gas is comparable with the characteristic time of the problem,

determined from the equation of chemical kinetics (see, for example [22], [23]) .

Neither is the flow conical in the case where the radiation plays an essential

role in the energy balance between the hot gas and the wetted body.

In the case of thermodynamical equilibrium (or "frozen" equilibrium) the properties of the gas may be determined theoretically, using methods of

statistical physics in their quantum mechanical form, together with

spectro-scopical data.

Results of such calculations for air and values of the parameters of the gas

that are of interest in gasdynamics may be found in [24] [25]; and also in [26]. In these references the equations of state of the gas are in the form of various relations between the parameters of the gas and represented in

tables and diagrams. For the calculations on the computer, either by means of tables approximate expres si ons relating the parameters of the gas are

determined (see for example [27], [28]), or the equations of chemical kinetics

for the case of thermodynamical equilibrium are added to the equations of motion of the gas (see for example [29]).

For an imperfect gas the parameters of the gas in the region between the bow shock wave and the surface of the cone, and also the semi-apex angle E of the

co ne for given parameters of the undisturbed flow, are determined in two steps.

In the first step the parameters of the gas immediately behind the bow shock

wave are determined.

Since in the given case relation (2.8.) cannot be represented in the form of

simple formulae like (2.9.) - (2.12.), the parameters of the gas behind the

shock wave are determined by means of an iteration process or constructing an auxilary diagram and interpolations. Following reference [30], we consider

for instance, the calculation procedure in the latter case, when

v

₁' Tl' Pi and

8

are given. First the missing parameters of the undisturbed flow are

determined; subsequently, with the

constant Tand p have been drawn Pi is determined with the aid of i-p-a

aid of the i-p diagram where lines of

and it are determined, when al

(Mi

=

~~)

diagram, whereas with the i-S diagram and

the and

known values of Pi _{and i 1 '}

(2.8a.) are represented in 5

1 is obtain~d. Furthermore the relations (2.8.)

the form: U 2 V1 cos

8

tan

(8 -

0)

(2.25.) U 2 _ U 2 i 2 i1 1 2 + 2 (2.26. )

(2.27. )

(2.28. )

When the deflection angle 0 of the velocity vector behind the shock wave is

( )

~

. *

*

~

given, using formulae (2.25.) - 2.28. , the values of U₂ ' 1

2 ' P2 and P2 can be determined and the graph P2~ = P2~(0) can be constructed.

Then using the obtained values of

i2~'

P2* from the i-p-diagram the function P

₂

=

P

₂

(0) is determined and its diagram is constructed. The point of inter-section of these curves gives us 0 and P

2. Then with the value of 0, from formula (2.25.) M

2 is found, and fr om formulae (2.26.) and (2.27.) - corres-pondingly i

2 and P2. The temperature T2 is determined with the i-p-diagram and the known values of i

2, P2; the speed of sound a2 - with the i-p-a-diagram, and finally the entropy S2' from P2' T2 and the p-T-diagram. (With the

i-S-i 2

diagram and the values of S2 and io + ~Vl the parameters of the gas when isentropically brought to rest can be determined.)

, / 2

2'

With the obtained values of 0 and V

2

=

yU2 + W2 (where W2

= V

1 cos 8), point A₂ can be constructed (fig. 6); the normal to the rneridional curve includes an angle with the

8

axis. The second step of the calculation differs from the case of the perfect gas in the sen se that the speed of sound a in equation (2.7.) is determined with the a-i-diagram, wherein lines of constant entropy are drawn, and using the known values of S2 and i, which for each ray are determined from the Bernoulli equation:

Af ter the construction of the hodograph diagram, p, P, T in the flow field are determined by the S-i-diagram with the aid of the known values of S2 and i.

2.3. - Methods, using spherical and cylindrical coordinates

Following Taylor and Maccol l [17], we consider the solution of the cone problem in spherical coordinates. If the axis of the spherical coordinate system

coincides with the axis of symrnetry of the cone (and direction of the undis-turbed flow), then the parameters of the gas will depend only on the angle 8

(see fig. 5) and w

=

0 (see fig. 1). Moreover, in the flow S

=

S2

=

const., so that the equations (1.15.), (1.2.) take the form

2 2 dv 2 2 2

(a - v ) de

=

u(v - 2a ) - a v cot e , (2.29.)

du

de v , (2.30.1

and equations (1.16.), (1.5.) are identically satisfied.

In the case of a perfect gas the speed of sound a is directly determined from the Bernoulli equation (1.8.), written in the form (2.21.1:

For an imperfect gas a is determined with the aid of the i-8-diagram using the values 82 and i, and also i is found from (1. 8.1, if ilie value of v2

₌

u 2 + v 2

is known.

The boundary conditions on ilie bow shock wave are given in the formulae (2.8.) - (2.14.), where i t is necessary to take into account, that U

=

-v, W

=

u. The boundary condition on the cone surface reads v

=

0 for e

=

E.

In the case of a perfect gas the solution of the problem may be started either

by giving values of E and __ u __ for e = E and ilien determining

8

and Mi as was Vmax

done by G.J. Taylor and J.W. Maccoll [17], or by giving the values of

8

and the parameters of the undisturbed flow, finding E. For an imperfect gas only the second possibility exists. For ilie integration of the system of equations

(2.29.1, (2.30.1 eiilier the rneiliod of finite differences can be applied, without using ilie hodograph plane [17], of the hodograph plane may be used for a

graphical solution [20].

The cone problem was also solved wiili the rneiliod of establishment [29], which will be considered later on in relation to the solution of the problem of the circular cone at angle of attack, and also with the rnethod of integral

relations [32].

2.4. - Transonic flow around cones

The rnethods considered so far allow us in principle to obtain, within a given accuracy, the solution of the cone problem for an arbitrary value Mi > 1 (under the condition that ilie flow is conicall. However, if the Mach number Mi of the

components vary rapidly and the application of the described methods becomes difficult. For M

1 - 1

«

1 the conical flow is only possible for sufficiently small angles of the semi-apex angle E.

To overcome these difficulties K. Oswatitsch and L. Sjödin [33] introduced new unknown functions and independent variables.

If cartesian coordinates are used (see fig. 5) and the notations:

cot Cl 1

v'M/

w - w 1

x

w w 1 - a if

are introduced, where

a x 2 y - 1 2

- - - v

y + 1 max !t Y cot Cl l Y

,

if v v tan Cl 1

,

w 1 - a if

a is the critical speed of sound,

x

(2.31. )

then the equation of the axi-symmetric potential flow of a gas in the transonic approximation can be written in the form:

(1 + wif) dW if _{d (yifv}if₎ 0 - - + _dZ

,

if _dyif Y (2.32. ) dWx dVif 0 dyif

az

(2.33. )

conical flow x !t only depend or.

For v

,

w

if

x _{L ";L} II

Z Z cot al II cot al

,

(2.34. )

and the equations (2.32.), (2.33.) take the form:

(llX) 2 (l + w~) dw if _d(vifllif₎ 0 - - +

,

dllif dll* (2.35. ) dwif d (vifllif) ~ if - - + -~ 0 if if x dll dll II (2.36. )

From equation (2.35.) it follows, that if wif does not have a "strong"

singularity for llif + 0, vifllif + const. rapidly enough, if llif + O. Let us indicate by ~(llX) the product vXllx, whereas ~(llx) + ~(O), if llx + 0, then it