Symmetric flow about blunt bodies in a supersonic stream of a perfect and real gas

SY~l~l~TRIC FLW AROUT RLUNT ROnIES IN A SUPERSONIC STREAM OF A PERFECT ANI) REAL GAS*

O.M. BELOTSERKOVSKII (Moscow) (Received

7

June

1962)

Introduction A. A. ~ro~its~’ s method of integral relations 111 has proved to be effective in the solution of systems of non-linear partial differential equations of mixed type. Equations of this kind describe, in particular, supersonic flow about blunt bodies when a shock wave leaves the surface of the body. Over the course of a number of years a method has been developed at the Computational Centre of the USSR Academy of Sciences with the help of integral relations to calculate the flow about plane ;Llid axisymmetric blunt bodies of different shanes [zI - [4]. A certain amount of experience has been acquired in the solution of such problems. A large number of calculations have been done, and various solution schemes have been examined. In this article we shall discuss particular features of the application of the method to this problem, we shall construct a more rational calculation scheme for flow about blunt bodies in a supersonic stream of a perfect and real gas, and we shall al so give some results of the calculations to illustrate the potentialities of the schemes. We shall not give here the gas dynamical analysis of the results we obtain. A detailed review of the method of integral relations has been made in [51. The method of integral relations has proved to be suitable for the calculation of flow from an outgoing shock wave firstly because it enables the direct problem to be solved in an exact form, i. e. the flow near the given body to be constructed. Despite their relative simplicity. the inverse methods of [1;3 -181and other works, which construct the body +

Zh. vyck.

mat. 2. No. 6.

1062-1085, 1242

1962.

Symmetric

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bodies

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corresponding to a given shape of shock wave, are troublesome in that the problem of determining the body from the shock wave is improper, that the computational process is unstable, and that difficulties r.:ise in calculating bodies of specific shapes (particularly for bodies with sharply changing curvature, or with a break). Further, with the I.:ethor: of integral relations the solution is found with a different degree of accuracy from a series of approximations, where in each approximation the complete picture of flow in the region of influence is formed, This makes it possible to estimate the accuracy of the calculations, and when we have insufficient info~ation about tire equations themselves or about the coefficients in them (which happens, for example, when the effects of a real gas are taken into account), calculation using approximations enables us to select appropriately the necessary number of approximations. In serial calculations, it is sufficient only to perform separate (control) variants with a large number of approximations. And, finally, we note that the programs for doing these calculations on high-speed computers are quite simple with this method, do not require a large machine store and the same program can be used to calculate flow about plane and axisymmetric bodies of various shapes (smooth, with a break in curvature, in the contour) in a supersonic flow of Perfect or real gas. Thus, the method we have developed enables us to examine the problem of the flow about bodies with an outgoing shock wave as a whole, from the same standpoint.

1. The initial equations Let us consider stationary flow with an outgoing shock wave about blunt bodies possessing an axis of symmetry. Suppose that the supersonic

stream of an ideal (non-viscous)

gas flows with constant velocity

wmon

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O.M.

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to such a body under a zero angle of attack. It is required to calculate the mixed turbulent flow in the region of influence bounded by the shock wave, the axis of symmetry, the surface of the body and the boundary characteristic (or characteristics) passing between the wave and the body. The complete system of equations describing this phenomenon consists of partial differential equations (written in divergent form), ordinary differential equations and finite relations. As we know, the initial partial differential equations are replaced in the method of integral relations by some approximating system of ordinary differential eqnations. It is sensible to introduce into the initial system of equations its kno;ill integrals (1 ike the Rernoull i integral, etc.) so that we can reduce the number of partial differential equations, reduce the amoun.t of computation and also facilitate the solution of the boundary problem for the approximate system. It is necessary to bear in mind that tile introduction of integrals at once into the approximation system can 1 ead to incorrect results. For the independent variables it is most convenient to use orthogonal coordinates s, n, where s is the length of arc measured along the contour of the body, n is the normal to the body (Fig. 1). This system can be used tJOth for bodies with a smooth contour and for bodies with a sharply curved Eontour, as well as when there are corners. In this system of coordinates, the boundary conditions on the body and the integral relations themselves are written in their most simple form. In addition, it is easy to transfer from one shape to another, and this enables t!ie calculation to be done for a wide family of bodies using the same proto gram. The system s, R is also convenient in the use of the results calculate the boundary layer and heat exchange. It is sometimes advisable to introduce coordinates s, < or 4, n, where < = n/E(s), ~1 = s/sl(n). Here n = 0 is the equation of the contour of the body, n = E(S) the equathe equation of the boundary charactertion of the sl,ock wave, s = s,(n) istic. If only one of the four boundaries of the region of influence in the coordinates s, < is curvilinear, then in the system <, q the region The polar system of coordinates I-, of integration becomes rectangular. for the calculation u which we used before in [31, [41 is less suitable of bodies of a complex shape. ;,s we noted in [51, in choosing the direction of approximation it is necessary to start from the nature of change of the functions in various directions and the possibility of representing them by interpolation formulae of a given form. Moreover, it is necessary to bear in mind that the values of the functions in the direction of inte&ration of the approximate sys;en are obtained more exactly than in the direction of their approximations. In the case examined, multinomial interpolation formulae using the values of the functions on the boundaries of the

Symmetric

f lor

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1275

strips were used. In the framework of the same approximation it is possible to make the results somewhatmore exact by using the derivatives on the boundaries of the region in the interpolation formulae, as was done, for instance, in [91. Such approximations increase the accuracy of the approximations without essentially complicating the approximating system, al though at the same time the accuracy of the next “ordinaryn approximation is much higher. Introducing dimensionless quantities from now on, we shall relate the velocity w to the ~~irnurn value of the velocity w,.,, the density p to the density of the incident stream pm, the pressure p to cmwz,X, the temperature T to 7~’ I SW, the specific enthalpy ii to the drag enthalpy 1: = 70’ /2 thewiiecific entropy I‘? to .%mf2, the linear dimensions &‘%e chz:cteristic dimensions of the body (for example, to half the middle transverse section of the body). Jtere .%‘, = S/m,, where .!8 is the universal gas constant, IQ, the molecular weight of the gas in the incident stream. We note that it is not suitable to relate the pressure and density to the corresponding drag quantities UP to the shock wave, as was done in [31, since behind the shock wave some dimensionless quantities do not remain bounded when the value of the Mach number of the incident stream I&, tends to infinity, Let us give the initial system of equations and the boundary conditions of the prob!em of symmetric flow about bodies with an outgoing shock wave in a steady adiabatic gas flow. We shall ignore viscosity, diffusion, heat conductivity and the effect of irradiation of the gases everywhere.

bet us consider in succession the fool1owing three cases: 1. A perfect

gas.

2. Equif ibrium flow. 3. Non-equilibrium flow. The equations of motion, of continuity, and the condition adiabatic flow for a13 three cases will have the form (WV) w +

F = 0,

V(pw)=O,

dh dt-

for

fJ$=O,

in dimensionless for-n, where Ju, r:, p, h are the dimensionless values of and the velocity, pressure, density and specific enthalny respectively. t the dimensionless time. If, the coorfiinstes s, or in divergent equation of motion on n and s, and also

foxx tf!e projections of tfie the equation of continuity, are

1276

O.M.

Be Zotserkovskii

written:

(1)

(2)

~+q+o.

ai% i7e have used the

following

notation:

z = r”pz4v ,

A = rvpv ,

IfI = p” (P + w”)

X II -4

(3)

L = rvpu ,

G = ry (p + Pug) 1

>

+ (Apsini3)‘,

Y = -$

+ (Apcosti)‘;

A = 1 f n/l?,

r = ro(s) f n cos 0 is the distance along the normal from the axis of symmetry of the body, r,,(s) is the radius of the body perpendicular to the axis of synmetry, 0 is the angle of inclination of the body to the direction of the incident stream, U, u are the components of the velocity in the directions n, s, P =: - cks/tiB is t!:e radius of curvacases reture of the body, v = 0 or 1 for the plane or axis~etri~ spectively. We shall denote quantities in front of the shock wGaveby the suffix CD. I. In the case of the flow of a perfect gas with constant specific heat capacities, the condition for adiabatic flow can be written in the form

V(pSw)=O

or S = S(g),

where S = &

In-$

+ const.

is the

is the adiabatic index, c speci fit dimension1 ess entropy, x = c, /c, P’ are the specific heat capacities for constant Pressure and volume cv respectively. Instead of the equation of motion (2) let us introduce the Bernoulli integral m2 + h = I. so that the comftlete system of equations (taking the equation of state into account) for the gas in question will be written in dimensionless form as follows:

wa-+-Ir=l, Ic=$~T,

S=--&ln-5,

p,pKP

(4)

Symmetric

f lov

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1277

stream

Here s, is the value of the entropy immediately behind the shock wave, y is a function of the current, the speed of sound

ca=

(ap @18

e=

q

(1 -

w’) .

From the last four equations of system (4) it is easy to express the values of p, p and T in terms of the squares of the velocity and entropy. The problem reduces to the integration of two differential equations with partial derivatives of the unknown functions u and V. The initial system of equations taken in this form is in some sense the best for constructing the approximating system corresponding to it, since it contains in all two partial differential equations, the entropy on the surface of the body is always constant, and the boundary problem for the approximating system will be the simplest. If we introduce the Bernoulli integral instead of the condition for adiabatic flow, the initial system becomes more complicated, and we forfeit the advantages we have just mentioned. The boundary conditions case. On the surface

of the problem are written

simply

of the body n = 0, u = 0, y = 0, A’3= s,(O)

for this

= con&.

On the shock wave all the unknowns are expressed in a known way in terms of the angle of inclination of the wave to the direction of the incident stream CL On the axis of symmetry s = 0, u = 0, y = 0, u = a/Z, S = .?,(O). The unknown boundary conditions here will be the magnitude Ed of the wave and the distribution law of the velocity u(n). We shall

give

the conditions

on the boundary characteristic

later.

2. Suppose that behind the shock wave adiabatic flow takes Place of a mixture of reacting gases which is thermodynamically in equilibrium. \ve shall take into account the excitation of oscillations, the equilibrium dissociation and ionisation of the mixture. Thermodynamic equilibrium is a good approximation if the characteristic time of flow about the ho& of *the stream of gas (fig/u, where ,qE is the radius of curvature of the wave for s = 0) is large in comparison with the characteristic time necessary for the excitation of oscillations, dissociation or ionisation. The condition for adiabatic flow in this taken in the form S = .q1(y), which is valid versible processes in the flow.

case too is conveniently due to the absence of irre-

1278

O.M. Be lotserkovskii

The initial system of equations here will have a form similar but instead of the last three equations we have

to (4),

TdS=dh--ZT,

h = h (A T),

(6)

P=P(PI

(71

T).

Equation (5) is the well-known thermodynamic relation, written in dimensionless form. The dependences (6)‘ (7) for h and p and also their partial derivatives with respect to p and T can be found from the approximations of the tables for the given mixture of gases, or by integrating the system of thermodynamic equations numerically. This form of the initial system of equations is convenient in that it enables us to construct a numeric31 scheme for the solution of the problem without specifying exactly the actual composition of the gas mixture. Of most interest is the ~al~ulation of flow about bodies in 3 current of air. There now exist tables of thermodynamic functions of air calculated to take into account equilibrium dissociation and ionisation over a wide interval of pressures and temperatures. It is difficult (although Possible) to use them directly in calculating gas dynamical problems, and so we have recourse to the analytic representation of the necessary dependences. 1.X. Naumova [lOI has made approximations of the functions h and p with respect to the two variables fj and T for temperatures up to 16800°K and for pressures from 0,001 to 1000 atmospheres. The relative error of the approximations in comparison with the tables we have mentioned is less than 1 70. In calculating flow about bodies in air we have used these representations. Another method of calculating the parameters of tile mixture for given p and T consists in the direct numerical integration of the system of equations of chemical equilibrium (for air, see ill] ). but with a preliminary approximation of the constants of equilibrium and of the molar enthalpies of the components of the mixture, depending only on the temperature. For gas dynamical calculations, apart from 11 and p, it is also necessary to know their derivatives. Therefore it is useful to differentiate the initial system of thermodynamic equations (taking into account the approximations introduced) and the resulting ordinary differential equations are added to the approximate system. This releases us from iterative methods of solving the equations of chemical equilibrium and enables us to do the calculation for an arbitrary mixture of gases. To firinl: the initial system of equations to a form which is the mOSt suitable for the construction of the approximate system corresponding to it, in the case of the flow of a perfect gas with constant heat

Symmetric

capacity

flow

about

we have expressed

For a gas with variable made from equations (5)-(7)

blunt

bodies

in

o supersonic

1279

stream

11, p and T in terms of w2 and ,C. similar re?Wesentations heat capacity, only in differential form

can be

dp=-+(dw2+TdS), dT = dp=--$ where

D = pphT -

pThp.

$1(2 T

(8)

- phr) dw2- pTh, dS]

1(2p, + DP) dw2 + !?e have denoted

,

(9)

DpTdSl ,

the partial

(10)

derivatives

h@, 7’) and p(p, 7) with resnect to ;J and T Iv hp, h,, p,, pT ively. The velocity of sound is defined h.ere as:

of respect-

phT

DP + 2P; Thus, tions of

in the case of equilibrium flow instead of the last three system (4) we must take equations (7)-(9) or (8)-(10).

equ a-

The boundary conditions on the surface of the body are written as in the previous case, where the pressure and the temperature on the body are determined from equations (E)-(9), with 6 = 0. On the shock wave, the explicit expressions of the unknown functions in terms OP the angle of inclination of the wave CTdo not here exist. The values of the functions immediately behind the shock wave are found from
wt=w~eoso,

pw, = u’, sin 6, ~=(r/2)Yrwoot dS = $dq where

dw,,= ‘fda p =

poo -

h=1--2, dT= gds,

on the shock wave can be

p pu;Zn+ (woosin a)2,

(12)

0.1.

1280

dT -!do=hT

and

?Ut,

wn

are the velocity

6X

I?e lotserkovskii

-hp$+2w,,2),

components tangential

and normal to the wave.

The system (12) allows us to determine all the unknown functions in succession immediately behind the surface of discontinuity. The values of the quantities w,,, 11 and T at the point where the sh.ock wave intersects the axis of s~et~ are found !uitl; the help of iterations. It follows equation

from (R)-(9)

that along

the lines

w = const.

;He have the

As ip the previous case, along the axis of s_ymmetry (S = 0, y = 0) the values of the quantities u(n) and Em are unknown. If some law of equation (13) al one distribution of LI(I~) is given then, integrating ‘P = O from the surface of the wave to the body, we obtain the boundary conditions on the axis of s~etry for the remaining unknot functions. 3-e Suppose that the flow behind the front of the shock wave is nonequilibrium. This happens when the characteristic time of the flow about the body in a gas is of the same order as the characteristic time of the excitation of the internal degrees of freedom of the molecules, their dissociation and ionisation. As before, we shall assume that the components of the mixture are thermo~n~ically perfect gases, and we shall ignore diffusion, thermal conductivity and radiation. Also we shall restrict the range of values of density and temperature to that in which the excitation of electronic levels is small and the process of ionisation has practically no effect on the flow parameters. Generally speaking, there are no fund~ental difficulties involved in taking account of ionised relaxation, but the theoretical and experimental data on the speed of non-equilibrium ionisation is very scanty. 7e shall

only

consider

oscillatory

an_4 dissociation

relaxation

from

Symmetric

about

flotp

blunt

in (I supersonic

bodies

stream

1781

now on, assuming that the progressive and rotational degrees of freedom are always in equilibrium and for them the shock wave is a. surface of discontinuity. In order to use a macroscopic description of the system we shall assume that inside each oscillatory degree of freedom there exists local equilibrium which is described by its oscillatory temperature. If we take the oscillatory and dissociation we must introduce the new unknown functions T.i j-th oscillatory degree of freedom of the i-t i1 mass concentration of the i-th component) into equilibrium

flow all

the expressions

relaxation into account, (the temperature of the components and Ci (the

the systen. For nonbe given in dimensional form.

will

Let us consider a p-component gas mixture. Let Iri and cpi be respectively the specific enthalpy and the specific thermal capacity for constant pressure of the i-th component. Then T

h; =

hi=&+&,

s

CfidT,

(14)

0

where “g is the specific enthafpy of the formation of the i-th extrapolated to absolute zero (heat of fo~ation).

component,

enthalpy of the mixture is defined as:

The specific

(1.5) The equation

of state

for

the mixture has the form

i=1,2 ,..., p. p=pWT+, 9

(16)

f

where Ci = pi f p, p = Zip+; Ci, pi, ??Ziare respectively tion, the density and the molecular the universal gas constant.

the mass concentraweight of tf?e i-th component, .!$! is

‘Vhen the system car: tains non-equil ihrium processes, t?;e course of which is associated with the increase in entropy, t!;e conrlition for adiabatic flow in tl-!e form .C z .C,(I~) is no longer valid. Tn particular, 01~1~ for dissociation degrees of freedora m-e in eq:iil ikriurs, ir?g to

C,ibW

equat~ion

L13’

!,avr? the

relaxation,

wIten the oscillatory

cloer, tl F c!~a?r.~;ttin wtropy forh

TdS = dh -$--~iidCi,

nccord-

(f’7f

1282

0.M.

where A, is the specific

Relotserkovskii

chemical

potential

of the

i-th

component.

It follows from equation (17) and the condition for adiabatic flow that when there js no chemical equilibrium the entropy S changes along the current lines. I,et h’ = IZiCih: denote the specific enthalpy of the mixture without taking the heat of formation into account, and let +“P be the specific entropy corresponding to the degrees of freedom which are in equilibrium. BY definition TVS = Vh’ - Vp/p; ti,en the condition for adiabatic flow can be written in tb,e following form C133: ‘

v (ps’w)

= --&zWih; i

s

(18)

Yere oi is the mass velocity (per u..it volume) of the formation of the i-th component as a result of all the chemical reactions (Zioi = 0). Jt follows from equation (18), in particular, that only for th.e case of a perfect gtis in equilibriun CC= .c’ and ,? = ,c,(,,). Thus, when examining nor-equilibrium flow it is convenient to introduce the Bernoulli ittegral instead of the condition for adiabatic flow. In this case the complete system of initijl equations will consist of two parts: the gas ~dyriamic equations (equations of motion, continuity, the f,erEoulli integral and the equation of state) and tile kinetic (relaxation) equatiocs describing the course of the irreversible Processes in the flow. If we assume that the oscillations are in equilibrium, and consider then the kinetic system will consist only non-equil ihrium dissociation, of CI equations of the fon

dC$ df for

-5

“i

or

V(PCjW) = oj,

i = 12

,..-I

P*

These equations can be obtained by using the eq~Iations of continuity each component and for the whole mixture of gases.

,P;uppose that the incident stream consisting of a p-com>orent gas mixture contains r~ different forms of atoms. Then to reduce the number of partial differential equations of the kinetic system we must replace some equations of (19) by Dalton’s equation xici = 1 and n - 1 equations of material balance. Let IIS determine the mass velocities o1 enterin:: in equations (19). The chemical processes taking place in a reacting medium 2x-e described They can be writter, s3mhoI icall;i in the by the equations of reactions. following way :141:

Symnretiic

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about

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bodies

in

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strean

1283

where “Ii denotes the chemical component i; and pij, yij are the stoichiometric coefficients of tP.e original substances and the products of the jth reaction respectively. Let the constant velocities for the j-th direct and inverse reactions be equal to Kjjc’l) and I<;(7). Then the resultant velocity of the j-th reaction will have the form

or

Qj

=

Kj

CT)($)

xi l+j +j$!j..

. -K;(T)(&)

Ei

Yij

g%y..,

where zi = tli/tt is the molar concentration, !ti the number of moles of the i-th component in unit volume tt = &fti = p/_!i$?T. The overall velocity of formation of the i-th component as a result of all the chemical reactions is equal to

From this, in particular, for oi = 0 we can determine the equilibrium composition of the mixture as a function of the pressure and temperature. The quantities Ki(7J and f<‘.(l) in (20) are defined only for a small class of reactions and then wI th a large error. They are usually taken from experiments. The velocity of the direct reaction is sometimes conputed from Arrenius’ formula [141, and then I{;(?) is eliminated b:r using the equilibrium constant. Now let us write out (in dimensional form) the initial system of equations for the case of flow with non-equilibrium dissociations:

aA x+aq=o,

-$ + h = hToP ,

[J./f.

1284

Re lotserkouskii

-J@”

xci=1, i

Atkf,

Ci =

V(pCjW)

=

Oj

3

t

i = I., 2, * . , , p;

.,a-I;

k=1,2,..

i = a + 1, a + 2, . . , p.

This system contains c; + v unknown functions ‘4, v, r, T, ik, p, Ci. L and A The quantities 2, E, I;, A, L, ,Y, Y were determined earlier, are the constant coefficients of t!:e equations of material h~ance which are the ratio of the molecular weights and the composition of the mixture, Before discussing the hcundary conditions give the equations of the characteristics.

the Problem,

let

us

directions are here three characteristic and the lines of flow). J!owever, in of sound plays the Dart of the velocity

As for a perfect gas, there (two families of characteristics this case the “frozen” velocity of sound: c2=

of

PkT

(+ 1s, ci =L)p-t *

where

D .= pphT-

P&,

,

3P pp = ( ap ) r, ci

% = ($)

P, ci 2

8P ’

pT==

(n

1P

Cj

Along the characteristics

dn = A tan@fa)ds,

dPlt: -f$dp+Qds=O.

(23)

wJiere

‘ma& angle (sin a = c/w), v = 0, the “frozen’ the plane s,ntnd axisg.metric cases (the llppermost sia characteristics of the fix-at fattily). a is

1

respectively refers to

for tliC?

Symmetric

flow

about

blunt

bodies

in

a supersonic

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1285

The boundary conditions of the problem are: on the surface of the body u = 9; moreover, for simplicity we assume that the surface of the body does not possess catalytic properties and, consequently, the flow does not interact chemically with it. Let us also assume that on intersection of tbe shock wave in the flow rotational (rotational and oscillatory) degrees of freedom are completely excited and we examine the oscillatory and dissociation (or only dissociation) relaxation. Due to this, the composition of the concentrations immediately behind the shock wave will be the same as that in the incident stream, and the tangential and normal components of the velothe enth‘alpy and the temperature are city, the pressure. the density, determined by means of iterations or from the differential relations by using the known relations on the wa.2. On the axis (24) hold.

of

symmetry u = 0, +I = 0 and, consequently,

On the boundary characteristic

conditions

equations

(23) must be satisfied.

If we also consider the oscillatory relaxation, then, assuming that the different oscillatory degrees of freedom are excited independently, it is necessary to add to the kinetic system equations defining the change of the specific internal energy eji:

deji (TjJ

dt=

iji (7’)- eji (Tgi) tit (P, T)

i = 1, 2, . . . , p;

’

i = 1, 2, . . . , 1.

Here T. ., e. ., T are respectively the temperature, *specific internal energy an~‘tim~‘of Jibe relaxation of the j-th oscillatory degree of freedom of the i-th component, Tii is the equilibrium value of the quantity ej i. The specific dom of the i-th

internal energy of all the oscillatory component is equal to ei = x:eji (Tji),

degrees of and hence

free-

f hi = & + ei +

where tional

f$ T + 6 ,

8i is the specific internal energy of the progressive degrees of freedom on the i-th component.

For single-atom components e i = ei = ei(T,) and hi = h,(T, T,). In number of oscillatory temperatures degrees of freedom and ei = ei(T,,

and rota-

for two-atom components 0, 11; = h,(7), the case of multi-atom components the is equal to the number of oscillatory T2, . . . . T,), hi = h,(T, T,, T2,. . .,T,).

O..Y.

I.286

Re lotscrkovskii

With the help of theoretical calculations we can obtain various pressions for 7ji (for two-atom molecules they are given in [151).

ex-

It is convenient to write the differential equations of the kinetic system (19) or (25) along the current 1 ines. Then, from (24)) we shall have a system of ordinary differential equations determining the change of ci along the different current lines. Similar equations can be obtained for e also. This form is suitable in that in the approximating system the re“I axation equations are taken in their exact form. Moreover, the law of change of the quantities Ci and e .i along the body will be determined from the corresponding exact equua i.*ions written along ty = 0, while for the kinetic system taken in the form (19) these equations can only be used for control. By way of example, let us give the kinetic system of equations for a two-atom gas when the oscillatory and dissociation relaxation are taken into account (u = 2, 1 = 1, a = 1). Denoting the products of disintegra“2” tion (of atoms) hy the suffix Ill”, and of molecules by the suffix along the current 1 ines we shall have

tide, (Tk) =

pudC, = qdrr , dn

=

A

tan

eZCO- ed Tk) x (P* T)

c, =

pds ,

1-

da,

c, .

P-3

where u, = I

P2(2%-t-CJ

z (P, T) = uu$- exp (cxp (T”/

e,(T)=%

1

K (T) C,

me

h 0-9 r,>

(NT-““)

T) -

I)_“,

- 4K’ T; “:j,

(1 - exp (-

(27)

P/T))-“,

hl (T) = 3

$

(28) T -+ h; )

(29) (30)

=gp+em;

C,, Cz are the mass concentrations of the atoms and molecules, ‘fi; is the temperature of the oscillatory degrees of freedom, TX is the characteristic temperature, and .& and N are const.ants determined by the mechanism of inter-molecular interaction f151.

3

a*

T!re

solution of

scltmes

the

and some

Cilletllati0n

examples

Symmetric

Jlota

about

blunt

bodies

in

a

supersonic

1287

stream

and also numerous calculations, have shown that for flow about blunt bodies of various shapes the regions of influence shown in Fig. 2 can occur. For plane bodies the regions o f influence of type 2a (for Pm < !!*) and 3b (for A&,> !!,) occur, where in the case of the flow of a perfect gas with constant heat capacity the value of the bouncjary Path number f;!* depends only on the arliabatic index K. In the case of axisymmetric

bodies

all

three

types

of regions

of

b Fig.

2. -----.-

influence czf occur, now on the curvature

characteristic characteristic

family; of the first of the second family.

the values of the houndafy Mach numbers depending of the wave.

It is clear from an examination of the structure of the regions of influence that there always exists a characteristic of one family passing between the wave and the body and bounding the region of influence above. We shall take this characteristic from now on to be a bounding one, and a correct representation of the possible types of regions of influence will help us to construct rational solution schemes for the problem. With the help of these schemes we have to obtain the numerical solution of the initial system of equations in the minimal region of influence. Further, in each ~proximation all the boundary conditions of the problem must be satisfied. In addition, the approximating system must be clpsed. i.e. the number of equations, unknown functions, arbitrary parameters and conditions for determining them must be strictly in agreement. Suppose it is required to construct the numerical solution of a system of k differential equations with partial derivatives of the first In solving t!le problem by the order, containing k unknown functions. metl,od of integral relations the coordinate system is chosen so that two opposite boundaries of the region are coordinate lines. In this direction the integral relations are formed, If on opposite tiunltaries of the region k Loul~dary conditions are given iK the sum, then the correspor~ding

1288

O.M.

Re lotserkovskii

approximating system of ordinary differential closed. In the case when one of the caddies

equations is completely of the region is not

/0=6

Fig.

3.

known beforehand, an additional boundary condition is required. When there are singular points on the boundaries, the corresponding boundary conditions can be omitted, and they can be replaced by the conditions for the regularity of the solution.

Fig.

4

Let us discuss the schemes for the numerical solution of the resulting systems of equations for all the cases examined (Figs. 3 & 4). The technique of constructing the corresponding approximating systems is described in c2! - [51.

The initial syst.em of equations is written in the coordinates s, R. The region of integration is divided into strips which are equidistant functions, 3s well as the intein II, as shown in Fig. 3. The integrated grals themselves, are represented by interpolation polynomials in n with interpolation nodes on the boundaries of the strips. lire examined two variants

of the scheme.

In the first of them, the region of integration is bounded above lg the licitiny ray .Y = s, = const. (Fig. 3a). Integrating the partial

Symmetric

flow

about

blunt

bodies

in

a supersonic

stream

1289

from which we find the N unknown parameters for s = 0. The approximating system is thus closed, and the solution of the problem unique. As we see from (32), on the body the singular and sonic points coincide, while in the flow field the singular points lie in the supersonic region. In the first variant of the scheme the approximating system is integrated numerically from s = 0 to some supersonic ray s = s the position of which must ensure that conditions (32) are satisfied.*Nere, in all the cases we have examined, the region of influence of the initial system of equations is inside the region of integration. In the second variant of this scheme the region of integration is bounded above by some boundary characteristic s = sl(n) which is picked out during the calculation. The position of the characteristic is not fixed beforehand. The point of its intersection with the body sT = ~~(0) is selected as near as possible to the sonic, the characteristic itself having to reach the shock wave and bound the minimal region of influence. This makes it possible to carry out the calculation with an exact examination of the region of influence. Up to the ray s = sT the integral relations are formed from the body to the wave, and for s > sT from the boundary characteristic (i = 0) to the wave. For s = sT a continuous matching of the corresponding functions and derivatives must be performed. In practice we construct a single approximating system with different boundary conditions on the lines with a zero suffix. Schematically the approximating system in this case also can be written in the form (31) to which it is only necessary to add equations (23) for the boundary characteristic. The second of the equations (23) can be used to determine u,,. Thus, the scheme we have described ensures the construction of the numerical solution in the minimal region of influence of the initial system of equations. Calculation according to the two variants of the scheme gave close results. Scheme 1 was used for the calculation of both smooth bodies and those with a break in their contour or in the curvature of the generator. In the case of smooth bodies or bodies with a break in the curvature of their contour for s = 0 the conditions for determining the unknown parameters are given by the regularity of the solution (32).

If we now make a break in the generator on such a body in the region of influence (for-s = d) and the flow picture is changed so that the sonic point on the body lies at the point of break, then ir, a supersonic neighhourhood of this point Prandtl-leier type. flow will take place. For bodies with a break in contour the conditions for determining the parameters are:

0.M. Rc totserbovskii

1290

differential equations across each strip from the surface of the body to the shock wave and replacing the integrals by the corresponding interpolation polynomials we obtain an approximate system of ordinary differential equations. For example, for the case of equilibrium flow, t5e approximate system in tile N-th approximation (the region is divided into A! strips) schematically looks like: de

ds= d.f

dSt du.

1=

cis

dVi

ui t

’

ds- =-f

dT,

d9i -=;:

ds= Tt,

ds

i =0,2,3 ,..., The enthalpy, density and velocity corresponding values of the pressure t ions.

dTt

rD

ds=ds

dT_1 rf, ’

ds=drs

dpi

Ei

(31)

P.1,

v:- ca 1

x=

yi ,

si(q%)=

81(qh),

N.

of sounci zre here found for the and temperature from the approxima-

In the system (31) the quantities on the wave are denoted by the suffix Ml”, those on the !mdy (i = 0) by the suffix *On, and on the intermediate 1 ines, where R = :, .E(s) and 2 = (A’ - j + 1)/Z by the with respect to sufPix j = 2, 3, . . . . I’/. The valies of the Derivatives u are determined from (12). and the velocity of sound c from the formula (11). The boundary conditions on the body and the shock wave are satisfied automatically in each approximation. On the axis of symmetry s = 0 311 the boundary conditions are known, apart from E@ and ui (j = 2.3,. . . ,nr>. The right-hand sides of the system contain the known homomorphic functions 0, Ut, Es, Pi, T*, Yi. However, as in tte case of a perfect gas with constant heat capacity, in the ceighbourhood of a sonic line (with (vi = ci) system (31) has N motionless singular points of regular type, the position of which, by the way, does not depend on tl,e actual choice of unknown functions. From the condition points we must have

for Ore regularity

E:‘i = 0 for

of t,Fe solution

Vi = Ci, i = 0,2,3, ‘

l

l

, N*

at t::e sinylar

132)

Symmetric

flow

270 =

about

Cl-J

Et =O

blunt

bodies

in

(I supersonic

stream

for S = 3, for

Vi =

Ct

1291

(33) (i =

2, 3,...,A’).

At the break we I.avc essentially a singular point, since v;(8) = m (the dash denotes the derivative with respect to s). In the construction of the computing scheme of t!le neigl,bourhoo:i of such a point definite difficulties arise: the approximating system may turn out to be overdefined, it is difficult to establish beforehand tY!e lower boundary of the flow region of Prandtl-Veier t,ype, it is necessary to “turr14’ the stream along the break. Tl.e variants a break,

of the scheme we have described

are also

for

bodies

with

1. Up to the point of break s < r* the solution is done by the usual method in the coordinates s, n, and for s > s^ in a polar coordinate system with a pole at the point of !,reak (,I’ = 1 in the system s, n). If we use the well-known Prandtl-B!eier solution for s > d, then we must reject the differential equation for 11;. Otherwise, in forming the integral relations from the point of break to the wave, it is necessary, in order to determine u,,, to take into account the relation along the characteristics of t,he second family, which here degenerates to a point. The approximating system is then completely retained. For tile whole region of integration we construct one approximating system with different boundary conditions for u0 on the body. The calculation is done with one program. 3. The solution is constructed in the coordinates n, TJ= s/sl(n), where s = sl(n) is the equation of the boundary characteristic coming from the hrenk point. Here the calculation is done at once in the whole region of integration without matching the solution at the break point. It must be noted that in the neighbourhood of the point s = d the unknown functions will be discontinuous. Let us turn briefly to the construction of the solution in the neighhourhood of singular points (32). In any approximation through such point there pass only two solutions, both of which are regular.

each

For, at the singular points .s = s i, from (32), ui(sl) = O/O. However it follows from physical considerations that in the region of integration 0 < V;(s) < m. Therefore among all the possible solutions in the neighbourhood of singular points of the form Zli = 0 ](s -Si)a],where u is any real number the conditions listed ahove are satisfied only by regular solutions with a = I. If we immediately line up these solutions

1292

0.3.

Be lotserkovskti

it is not difficult to see that for the equations vi = Ei/(uH-CT), there exist two solutions of this type, only one of which satisfies the Physical sense and is matched with the solution of the system up to the singular point (the number of parameters and conditions for matching are equal). Experience of finding the numerical solution has shown that in the neighhourhood of singular points it is not necessary to line up special series. lith the help of the nsual methods of the numerical integration of ordinary differential equations we can practically enter the singular point itself, or approach it without any essential loss of accuracy at a distance of one or two computing steps. Thus we can satisfy the necessary condition at the singular point and determine the corresponding parameter. me magnitude of the derivative u[ in the neighbourhood of the singular Point s = si is weakly changing, and in continuing the calculation it is sufficient to extrapolate ui using two or three of its values for s < si over several computing steps after the singular point. To control the accuracy of the crossing the magnitude of the derivative vi immediately after the singular point must be found by means of extrapolation, from the corresponding formula, it being necessary to discontinue the extrapolation where both values practically coincide. The construction of the numerical solution in the nei~~I]rii~d of singular points is easily standardised by this method and does not cause any difficulties during the computation. The process of finding the parameters for s = 0 from the conditions at the singular points is done by the machine itself, for example, with the help of interpolations with respect to the parameter, using the continuous dependence of the solution on the parameter. It must be noted that in computing smooth bodies the dimensionless values of the derivatives in the region of influence are bounded, 0 < v: < I, and so the region of possible values of the parameters lies between Ri = 0 and E. = A. where A. = V? - ~2. It is convenient to use conditions (32) in tie folfn /?. = Af = O’which enables us to remain in the class of continuous derivaiives: I’ie can also carry out the process of finding the parsmeters with the help of a method similar to that of Newton for the solution of functional equations. V.F. Ivanov has worked out such a scheme for /V = 2. Ye consider now an iterational scheme which uses the information of t,he Drevious approximation, for each iteration only one singular point being traversed. Ye can also use here the Poincar&-Lighthill-kuo method in which the equations are lineariser! at the same time as the independent variahles are somewhat liefo~ed. The nppl ication of ttiis method to the solutiori of

f lols about

boundary problems in Id.

for

blunt

approximate

bodies

in

o supersonic

system of the type

(31)

stream

1293

is described

Using Scheme I the calculation of a large class of bodies, both plane and axisymmetric, in a stream of perfect gas with constant heat capacity, has been and is being done [171, [181, in the case of equilibrium flow al 50. Scheme I is suitable for the calculation of flow with large values of the I”aci: number of the incident stream #,, when the shock layer is comparatively thin and the distribution law of the parameters across the lalver is of a smooth nature. TO calculate flow for &, >3 it is usually sufficient to take two approximations (P: = 2) and for hypersonic values of “b we can even restrict ourselves to one (?! = 1). In this scheme the parluneters of flow on the shock wave and the body are found more exactly, and in the case of the second variant of the scheme on the boundary characteristic also. Good results are obtained also in the calculation for bodies of complex shape. For N = 1, f! the scheme gives the data which are necessary to calculate the supersonic region to sufficient accuracy. Using the results of the calculations of nose regions, P. I. Chushkin and N. P. ~hulishnina have calculated, by the method of characteristics, a large series of blunt plane and axisymmetric bodies (K = 1. 4) UP to considerable dist3nces along the length of the body fl91, [zd . V.P. Ivanov has carried

0

Fig.

5.

0.4

0.6

0.8

method of integral relations AAA - method of characteristics.

(iy = 2):

O.M.

1294

Re Zotaerkovskii

out a series of calculations of both a mixed and a supersonic region of flow for blunt bodies of various shapes in a perfect gas, with various values of the adiabatic index K. Figs.

5-11 illustrate

some results

Fig.

of the calculations.

6.

In Fig. 5 MUZ give the pressure distribution along ellipsoids of revolution (S = 0.5 and 1.5) a sphere (6 = 1) and a cylinder with a plane nose (6 = (0) for the flow of a Perfect gas (K = 1.4) with r!,,, = 4 and co. i,, is the ratio of the pressure on the body to the pressure at the critical point, 6 = b/a is the ratio of the vertical and horizontal semiaxis of the ellipsoid, y is the axis of ordinates. The calculation by the method of integral relations was continued outside the region of influence into the supersonic zone, although it is clear that as we move through this region the accuracy of the calculations will decrease. In Fig. 5 we give a comparison between these calculations and the characteristics method [zo]. In Fig. 6 we have set out the shock wave and sonic lines which arise in the flow about a sphere in a perfect gas (!r, = 5) with various values of the adiabatic index K (the calculations are by V.F. Ivanov and N.A. Yegorova). In Fig. 7 we show how the shape of the shock wave changes and the law of pressure distribution along axisymmetric bodies of various shapes gas. K = 1.4) : 1 - sphere with a break in the contour (&, = 4, perfect at the point .? = 0.5, 2 - sphere without a break. 3 - cylinder with a without plane nose and with rounding (rroundlng = 0.5)) 4 - a cylinder

Symmetric

flow

about

blunt

bodies

in

a supersonic

stream

1295

ion rounding. The 1 inear dimensions are related to ha1 f the mid-cross-sect of the body, E is the distance along the ray s = const. from the surface of the body to the wave. The calculations were done by ?‘. k!. Golomazovi i.

Fig.

‘7.

In Figs. 8-11 we give the results of the calculations for flow about dissociation and ionisetion a sphere (f? = I) in air taking equilibrium into account for various values of the Mach numbers film, the pressure pm and the temperature T, in the incident stream (the results of the calculations are given by the continuous lines and by lines with triangles). To calculate the thermodynamic functions the approximations of [lo! were used. In the figures we also give a comparison with a perfect gas for K = 1.4 (dotted lines). w 6 10 10 10 20 30

Es 300 3w 308 300 216.65 216.65

0.1 1 0.01 1

0.1 0.M 0.~54546 0.054546

2152 2i50 2147 4347 403 1 31.86 5419

10336

1296

O.M.

Be lotserkovskii

In Fig. 8 we give the flow Picture for I$, = 10, and in Figs. g-11 the pressure distrib~ltionPO along the sphere related to the pressure at the critical Point, and the shape of the shock wave E for various values of !I,, pm, T,. The calculations were done by N.P. Shulishnina and M.M. Golomazovii. Ke can see from Pigs. 8-10 that the change in the

Fig. ---x=1.4;

-pm

[CITM]-

AAA poo Values Of pm from 0.01 atn. no effect on the results.

8.

[~TM] =:

1.0 Ta3[“K]=300 0.01

to 1 atm. in these

examples has practically

The values of the temperature Tdrag and the pressure pdra at the critical point for the cases considered are given in the tab f e. .%heme

IT.

The initidl system of equations is written in the coordinates s, < = ~L/E(s), in which the contour of the body (< = 0) and the shock wave (5 = 1) become straight. As in the second variant of the previous scheme, the region of integration is bounded above by the characteristic s = ~~(5) passing between the wave and the body. The intermediate lines are drawn at equal distances in s between the axis of symmetry and the bounc!ary characteristic (Fig. 4). The partial differential equations are integrated across each strip, the integrals being approximated with the help of interpolation DOZY nomials in s, using the symmetry of the integrated functions with respect to s = 0 (the shock wave is a polynomial of even degree).

Synrmetrrc

{low

nhout

blunt

hodies

in

R supersonic

streaa

1297

In this case the approximate system has no singular Points, and the corresponding boundary Problem in the !li-th approximation contain A’ + 1 parameters on the ‘Nave (the values of E on the boundaries of the strips) on the body (no-flow condition) to determine them. and ,1’ + 1 conditions

I

t

0

02

, 17.4

I

f&s

Fi:> 9. Iv; .= 6; --- x = 1.4; -

T,

["Kl-300,poo[aThi]

=

1;o.t;

1

0.8S

0.01

on the shock wave and the surface of the body The houndary conditions only at separate points (on the boundaries of the strips) are satisfied as the approximation becomes higher. the number of which increases It follows from the construction of the solution scheme itself that the region of influence is taken into account exactly in any approximation, the approximating system being completely closed (although the shape of the boundary characteristic is not known beforehand, an additional boundary condition on it closes the system}. For example, in the case of a perfect gas with constant heat capacity (system (4)) the Problem reduces in essence to the solution of two differential equations in partial derivatives with two unwon functions u and u (k = 2). Applying Scheme II we shall have one boundary condition on the axis of symmetry (V = 0) and two on the boundary characteristic (conditions (23)). Thus we shall have in all three (h + 1) conditions. iVe note that in this scheme it is not possible to bound the region of integration above by a ray or any other non-characteristic line, since in this case the region of influence is incorrectly accounted for and on the upper boundary of the region there is no sl~pplementar~~ boundary

condition

which would close

the system.

The tecbniyue of constructing the approximate systems remains as before, IJUt in the consideration Of axiSYmmetriC flOW (U = 1) it is necessary to represent functions of the type I; = r(s, <)/(u, u, . ..) (r is the distance along the normal from the axis of symmetry of the body) which are always zero on the axis of symmetry. It is not possible to approximate the whole expression at once (as was done before) since in this case the values of the unknown functions On the lower tmundary of the region of integration will not enter in it. Therefore for v = 1 the functions r(s, 5) and /(u, v, . . . ) are interpolated separately.

Fig.

Fin. lo. M, = 10; ----x

---MMco=oo,

= 1.4;

-T,_J°K]=3~,p,]aTM]=

-

1; 0.1;0.01

The calculation of the approximating the body. The equation for the entropy, integrated along the current lines.

M,

= 211

11. x=1.4:

7’,

[“K] = 216.65

AAA ICI, = 30 i pot iitTA4f

=

0.054546

system is done from tlie wave to and also t!le kinetic system, is

Scheme 11 is suited to those cases When the represented functions change more weakly in. s than n. The scheme is convenient for the calculation of the flow about blunt bodies for small values of the f.!ac!l flow, where numbers of the incident stream, and also for non-equilibrium the distribution of the flow parameters across tile shock wave is not of a smooth nature. Scheme II was used hy P. I. %;shkin {al to calculate the limiting case, flow about bodies in a sonic stream of gas, when the shock wave leaves the body to an infinitely large distance. We have examined the example of a weak outgoing shock wave (t!le flow bebind it

Symmetric

f tow about

blunt

in a supersonic

bodies

tt rea*

1299

is assumed to be non-vertical and the wave to be curved). ‘&is scheme is being used at present to calculate non-equilibrium flow. To carry out series of calculations of the flow about various bodies with an outgoing shobk wave it is convenient to have standard programs for both schemes. Experience of the calculations has shown that for Scheme I it is sufficient to have a program of the second approximation for the equilibrium case. The calculation of the flow of a Perfect gas with K = const. with this progr~ does not in Practice require any increase in machine time compared with the calculation according to a special program, provided the calculations of the thermodynamics1 functions are done according to the formulae for a perfect gas. The calculation for N = 1 can also be done with the program for the second approximation. In this case instead of the corresponding differential equations of the approximating system on the middle line we use dependences which are obtained as a result of equating the coefficients of the last texms of the interpolation polynomials (I, = (/e + fr)/2) to zero. It would anpear to be advisable flow according to Scheme II.

3. In conclusion we give successfully illustrates Copsider

to write

A model

a program for non-equilibrium

problem

an example suggested our problem ( L211).

by A.A. Dorodnitsyn

which

the equation

or the system equiv~ent

with the following y=o,

to it

boundary conditions

o
y= O,
v=o;

z=o,

in the elliptical

1, O
sub-region:

u=z;

The 1 ine x = I is a line of transition. The minimal region of influence is hounded by x = 0, O< y < 1 and also by the characteristics (5 coning

1)“s dy = f dz,

from the points

(1.0)

(5 -

1) da T (z - l)“* du + udx

and (I. 1).

= 0,

(35)

1300

0.M.

It is required to construct solution which is continuous, Using the Fourier

Be lotserkovskii

in the minimal region of influence together with its derivatives.

method we can find

u (5, y) = r: c, (1 -

x)-“1 ~“@)JI

this

a

solution: [

&n(i

-d’*

(36)

],

n

Here J,, Bessel

J,

function

are Bessel

functions,

a,, = 2 /- h,

J1(2{k)

= 0, C,

the coefficients

the zeros

of the

of the expansion

As we see, the boundary conditions on the elliptical nart of the boundary uniquely define inside the whole elliptical sub-region a regular solution which can be continued analytically in the hyperbolic part of the region of influence also. In the more general case, when the line of transition is curved, the solution which is continuous together with its derivatives in the region of influence will he determined also by the boundary conditions on the hyperbolic part of the boundary in the region of influence. The solution here is constructed as if in a region with an “open boundary” (1 ine of transition), although the requirement that the derivatives shall he bounded for x = 1 determines a unique solution of the prohl em. If in equation (34) the we have Tricomi’s equation. elliptical sub-region and hounded on the transition tion. For this we must be houndary characteristics.

singularity is transferred to another term, In this case the boundary conditions in the the requirement that the derivatives shall be line will no longer determine a unique solugiven the values u, for example, on one of the

Let us construct the approximating systems of the second tion (1 = 2) for (34’) according to both our schemes. Scheme

approxima-

I.

Consider the region 0
< 1, 0 < y < 1 (the 1) and draw the one y = 0 hy the suffix the suffix “2”. The

boundary ray coincides intermediate line y=#. *On, for y = 1 by the integrated functions

Symmetric

flow

about

blunt

bodies

a supersonic

in

f @l Y) = lo + (4 fz - #I - 3 fcJ y 4 2 (Forming integral ing system dvl -&-=

relations

in y,

we obtain

stream

1301

2fz + fl + fo) y? the following

approximat-

4 (5 + uo- 2u,f,

dva = + (x - su, + 4u,), --& dun dr= dua -z=

& 1-z -- Ez 1-x’

s

Eo = 1 -+- ug - 2 (x -

28, + 4u,),

(37)

E, =~[2u~--1~-5vl+2(s+2v,)].

The boundary conditions for x = 0 (uO = u2 = 0) and the requirement that the solution shall he bounded on the “open boundary” x = 1 (E, = I’, = 0) uniquely determine the regular solution in the elliptical sub-region, and this can be continued analytically for x > 1 in t!Ie minimal region of influence.

The integral relations are formed for x from x = n to x = xi (y) /? and x = x1 0). Flere x = xl(p) is t.te equation of t!le hounriary characterTlic :lnkl~own functions istic coming, for instance, from the point (1,1), will

be: GO (for

r = X,(Y))

z = 0); Gz, Uz (for n = x1(5)/3), Be use the nnproximstions:

The ap~roxilnating system consists characteristic equations:

51 -

d;r, ---=&dy d;t _ dy -

L,,

iI

(for

and r,(y).

c

of

four

inteyral

relations

8 (h + 6%) +

21(JUo+7v,--1Gu,!i-~~(11~r

-

iGu,f

I

,

and two

O.i.1.

1302

The boundary conditions

for

Go= El = ii,=

f,

Fe lotserbovskii

tbis

system are

vs=

0

z&=1

for

y=o,

for

y=l.

On all the boundaries of the region 0,’ influence here the boundary conditions are given and t!ie system proves to !)e completely closed without the additional requirement of continuity of t!Je derivatives (it is taken into account hy the corresponding intern01 ation formulae) .

Fig.

12.

The numerical solutions of the approximating systems (37) and (38) constructed hy Ye.S. Bogomolova and V.K. %shin do not differ in practice from the exact solution of (36). In Fig. 12 we show this comparison for values of u, v with y = !$ (the continuous line is the solution of system (37). the broken 1 ine that of (38), and the dotted line the exact solution (36)). In conclusion, the author expresses his deep gratitude to all the colleagues mentioned in this article, and also to Yu.P. Lun’ kin who has been of great assistance in the fo~ulation of the problem of calculating non-equilibrium flow.

Translated

by R. Feinstein

Symmetric

about

flow

bodies

blunt

in a supersonic

1303

stream

REFERENCES 1.

Dorodnitsyn, vo

A. A., Tr. III Vses. matem. s’ezda, Nauk SSSR, 447-453. MOSCOW. 1958.

Akad.

2.

Chushkin,

?.I.,

3.

Belotserkovskii.

prikl.

mekhan.,

21:

1956.

Vol.

3,

353-360.

No.

3,

Izd-

1957.

matem.

i

O.h!.,

Prikl.

natem.

i mekhan.,

22:

No.

2,

0.M..

Prikl.

aatem.

i mekhan.,

24:

NO.

3,. 511-517.

206-219,

1958. 4.

Belotserkovskii, 1960.

5.

Belotserkovskii, matem.

6.

Van Dyke,

7.

Vaglio-Laurin,

M.D.,

10.

Naumova, 300.

11.

J.

I.N.,

Aero/Space

vychisl.

Zh.

matea.

i

1962. 25,

Sci.,

and Ferri.

J.

A.,

Aero/Sapce uychisl.

Zh.

No.

8,

AerolSpace

485-496.

sci.,

1958.

25:

No.

12.

J.

H.M.,

Aeronaut.

Sci.,

25:

No.2,

27,

Sci.,

matem.

No.

5.

361-3’70,

fiz.,

i natem.

1:

1960.

NO.

2,

295-

1961.

Rozhdestvenskii,

I. 5..

gasodinamika).

De Groat.

In the

collection

Izd-vo

Akad.

S. R. , The Thermodynamics

dinamika 13.

Li Ting

14.

Hirschfelder. of

?.I.,

731-759.

1958. S. C.,

(Fiz. 12.

5,

and Liberstein,

P.R.

109-118. Traugoht,

No.

1958.

Garabedian,

9.

J.

R.

761-770. 8.

O.M. and Chushkin. 2.

fiz.,

neobratimykh

Yi.

gases

Paper,

ARS

J.,

of

Curtiss,

and liquids.

June,

No.

852-859.

1959.

The molecular

Moscow,

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