theory of the high-entropy layer in hypersonic flows

THEORY OFTHE HIGH-ENTROPY LAYER IN HYPERSONIC FLOWS*

0. S. RYZHOV and E. 0. TERENT’EV

Moscow

(Received 9 September 1970)

WE consider the inverse problem of hypersonic flow with an infinitely great Mach

number of the oncoming flow at a condensation jump differing little from the jump

obtained by using the explosive analogy. By joining the external and internal asymptotic

expansions it has been possible to construct the body and study the flow ~medi~tely

adjacent to the body, nameiy the flow in the entropy layer.

The very concept of the theory of the high-entropy layer formed when a

hypersonic flow flows past blunt bodies, is due to the papers of Cheng [ 11, V. V. Sychev

[2-41 and Yakyra [S] . Comprehensive reviews of the advances of this theory were made

first by Mirels [6], and later by Guiraud, Vallee and Zolver [7], to whom are also due

a whole series of original results. A thorough approach to the investigation of the flow

in the entropy layer was made by Freeman [8]. It was developed in greater detail

from the mathematical point of view in the paper by Hornung [9], who aho

compared the theoretical and experimental data [IO]. An analysis recently undertaken

by the authors [ 1 1, 121 led to simple qua~jtative results on the structure of the

velocity field in the inverse problem where the form of the shock wave is defined

by a power function of the coordinates.

The questions discussed below are closely connected with those studied in papers

[8- 121. Their solution is based on the assumption that the front shock differs little

from the jump obtained by the use of the explosive analogy [j3- 171 to calculate

the hypersonic flows. However, precisely this small difference makes it possible to

construct streamlined bodies the contours of which deviate from the particle trajectories

in the problem of a strong explosion [I%201 the more strongly, the further the points

chosen for the comparison are situated from the nose. Both the equation of the contour

of the streamlined body and also the equation of the particle trajectory in the explosion

are expressed asymptotically in terms of power functions, but the exponents of these

functions differ by a finite quantity. *LX. vjkhisl. Mat. mat. Fiz. 11, 2, 462-480, 1971.

197

198 0. S. Ryzhov and E. D. Terent’ev

1. In the study of the flow in the highentropy layer the Mises variables are the

most convenient. Let x and r denote the axes of a Cartesian or cylindrical coordinate

system, UX and U, the projections of the velocity vector on these axes, J/ the stream

function, p the pressure, p the density, and x the ratio of the specific heats. We ascribe

to the parameter u the value 1 if the flow is plane-parallel, and 2 when it possesses axial

symmetry. With this notation the equations of motion of a perfect gas have the form

dP (JY”-‘P - = (v-1)u do au

w VCI z-z (1.1) g--=0, g=-$

ao vu --=--, ax v,

f(v2 + v,2) + &-$ = f v-2.

Here the unknown functions u and u are connected with the transverse coordinate

r and the component ur of the velocity vector corresponding to it by the

transformation

which is an identity for plane parallel flows. The Mach number at infinity is assumed to

be infinitely great, so the constant on the right side of the Bernoulli integral is

determined only by the velocity V, of the flow in the unperturbed state.

As x--f 00 we define the front of the compression shock by the expansion

(1.2) r = rs (x) = Ca-2/P+z) (1 + Clx-2m’(v+2) + C25-2q1(v+z) + . . .)

with the arbitrary coefficients C, C, , C, . In what follows we will impose on the factor

m of the exponent of the first correction term constraints which will emerge naturally

in the course of the investigation. As for the factor 4 of the exponent of the next term,

to simplify the calculations we initially subject its values to the inequality

(1.3) 2m < q.

Necessary additional conditions will be clear from the subsequent analysis. The

case where the inequality (1.3) is not satisfied will be discussed at the end

If the coefficients Ci , Ca ,... vanish in formula (1.2), the function

rs = C12’(~‘+2)

of the paper.

also determines the front of a compression shock which is obtained by applying the

explosive analogy [13-171 to the calculation of the hypersonic flows. The mathematical

analysis of this problem is due to V. V. Sychev [2, 41 and Yakura ([5], pp. 296-324),

fundamental results of a qualitative order were formulated in [l I] . The use of the

expansion (1.2) to define the position of the compression shock implies that it differs

Theory of the high-entropy kzyer 199

little in its shape from the shock wave generated by the explosion of a plane or filamentary charge [ 18-201.

On the shock front the stream function must remain continuous, hence as z + 00 its equation is written as follows in Mises variables:

+ vtv ; I’ ~12X-6~lP+2) f.. _] ,

and Hugoniot’s conditions for the unknown parameters of the flow field assume the form 8

P = (v + 2)‘(x + 1) p,v,zc22-2v’(‘+2~j(

x I

1 + c, (v - ,).-Z~‘PV + C,‘( v - I) (~_l)l)I.I::,(..+v)i...l

8 Ux = v, 1- (v + 2)2(x. _ 1)C:5--PV’(z+V) x

X[~+2(~-~m)C1X-2”it2f”~+(1-~~)~~*2~--6~~*(~+~)_j-

From this the entropy is found directly

(1.5) s = ; = (v +8$;2(x-*)“” sc2x a3

x [q,-’ + (v + 2 - 2m)C&-(‘+“)” +

+ (1 + “ifv + ‘/N2 - vm - ?rzz)CiZq?t-fv+zmf’v + . . .] (

$I= v p,VcoC’

q. If the quantities on the left and right sides of the inequality (1.3) are equated,

the last formula is complicated, because there enters into it a term proportional to the constant Cz of the expansion (1.2).

2. We will seek a solution of the Cauchy problem posed by the system of equations (l.l), by means of the perturbation method; this solution is defined in the domain 1


shown in Fig. 1, and adjacent to the front of the shock wave. The initial data (1.4)

suggest the form of the expansions for the stream parameters, namely:

8 P = (y + 2)2(x + 1) pmVm2C2z-2V~(V+2)X

x [p,l (q) + Cix-2m’~v+*)p12(q) + C12~--(m(v+2)P13(q) + . . . I,

x+1 p=----_ X-

1 P~[[Pli(q)+~i~-2”‘~v+2)Pi2(r))+

(2.1)

+ C12X -bmj(v+2)pi3 (7l) + . . .])

0 = cvz~~‘(~+s~ [ crii (11) + CiZ-2m’(v+%si2 (q) +

+ ci2x- 4m’(v+2hJ (q) + . . .])

4 u=

(v -I- 2)2(x + 1) V~~~-P-w’+~) x

x [Ui, (q) + Cix-2m’(v+2)Ui2(q) + C~23r’.~~‘(v+2)Ui2(~) + . . .],

x [Ull (q) + cix- 2*‘(v+2bi2 (q) + ci2x- Lm’(v+2)Ui2(q) + . . .I} . The selfsimilar variable

rl=x -2v/(v+2) $1.

acts as the argument of the coefficients of the

generalised power series in the coordinate x.

It is clear that to the first approximation

the so-called plane section hypothesis [2 l-241 .r

must be satisfied, and the functions of the first FIG. 1

approximation themselves satisfy a system of

non-linear differential equations determining the solution of the powerful explosion

problem. This solution, which for brevity is not reproduced, was studied with exhaustive

completeness by L. I. Sedov [18, 191 and Taylor [20]. The system of the second

approximation consists of two finite relations and three homogeneous linear equations.

2y (sl(;w~ duiz 2vqoii - +

drl

+ [ 2 (3v - 2) o$-i)/” _!!C.$ _ du,,

2Wl---- drl

+

+

2vm-v2-2m 1 1 doi, Uli cI,2+ 2(v - l)rl-----

-(v + 2&+, = 0, dq

Thtqv of the high-entropy hzyer

In the system of the third approximation

+ 2vqof* % + T2 +\2nz (sl?UiL!,

(2.3)


2(V - - 1) ai;(3v-2Yv

U,i2&3 + 2cT11qy-1)‘v Uilk3 V I

besides TWO finite relations there are one homogeneous and two inhomogeneous equations. We notice immediately that in the systems (2.2) and (2.3) the last relations are used to determine the functions Y f 2 and pl a after the corrections for all the other flow parameters have been found in the corresponding approximation.

In order to formulate the problem completely it is necessary to indicate the initial values of the unknown functions. They are easily found from Hugoniot’s conditions (1.4), which have already been written down in the standard form for the pert~bation method. Since to a first approximation the value 77 = 1 corresponds to a shock wave, taking away the initial data at this point, we have for the functions in the system (2.2)

Pi2 dpii

= 2(1-mm)-S--, @ii

drl pi2 =

-vT-7

duti l&i?, = v - 2(1 -m)- “5. -m-vdrl’ v12- drl

The initial values for the functions of the third approximation are established in the same way

-

The last equation of (2.4) and (2.5) can in fact be omitted since the quantities u12

and Y I3 are uniquely determined throughout the whole range of variation of q after

finding the remaining functions to the required approximation. Naturally these equations

do not contradict the formulas for Y 12 and y13 which occur in the systems (2.2) and

(2.32, as is easily verified by a direct check. If both the quantities in the inequa~ty (1.3)

are equated, terms proportional to the ratio C,/C: occur in the initial data (2.5) for the

functions of the third approximation.

3. In the subsequent analysis principal interest is not in the complete solution of

the Cauchy problem formulated above, which could, for example, be constructed by the

numerical integration of the ordinary differential equations (2.2) and (2.3), but

knowledge of the asymptotic behaviour of the unknown functions as q + 0. indeed,

by the def~njtion of the selfsimilar variable 71 for a fixed coordinate x in this case the

reduced stream function q41 --f I). Hence, small values of 77 describe the stream region

situated close to the streamlined body. The asymptotic values of the unknown functions

2lS q -+ 0 make it possible to estimate the orders of magnitude of the various

characteristics of the gas in this region and determine the contour of the body itself.

In the analysis of the system of linear equations (2.2) it is convenient first to

eliminate from them the correction pi2 to the density. As a result we obtain two

differential equations and one finite relation connecting the functions plr, ~~1 and ~~2.

It is obvious from this that two arbitrary constants must occur in the general solution

of the system (2.2). Denoting them by A 12 and Bt2 we write the asymptotic expansions

in the form PI? = Arr + . l . ,

k 1 = v(x - 1) - 3%x, h, = (v+w(x+q k

8 2*


The coefficient kz occurs in the theory of a powerful explosion, an expression for

it is given in the monograph by L. I. Sedov [25], where it is denoted by the same

symbol.

In the expansions of the functions u12 and ur2 the second terms are less than the

first terms only in the case where the value of the parameter m satisfies the inequality

0.2) m<m,= v(x- 1)

I FE

which, as is obvious from what follws, is equivalent to the coI~dition imposed by

Freeman [8] and Hornung [9, lo] on the exponent in the equation of the contour of

the streamlined body. The constant ml also occurs as a fundamental characteristic in

the theory of one-dimensional unsteady motions of a gas [25]. In the subseq~let~t

analysis the inequality (3.2) is assumed to be satisfied; then the coefficient kr > 0.

In nrrbr tn nhtain the asvmntotir be!lavingr of the fun&~!?s ~gc_‘~rrjntr in thP .*. ” .__. __ ____1* L . ..I ---, --.r ----_ D ... ...- system of linear equations (2.3) it is convenient to proceed by analogy with the

prescription for investigating the system (2.2) given above. Eliminating from (2.3) the

correction p 13 to the density, we obtain two differential equations and one finite

relation, containing only the functions p is, ~3,~ and u r3. If we denote the arbitrary

constants by A i3 and B r3, the unknown asymptotic expansions can be represented in

the form

p,s = it,, + ‘ . ‘ ,

ho’!xk3

p13 = ---T---q (V-2 ,,lX)/W

+ . ..)

/r

k 3=m(v+/t)-i--+-v-3m~

k i = m(v j-4) -1-~r-am’+~~~~-i_~-2m)2,

h-5 = 2mx - v(x - 1).

Theory of the high-entropy layer 205

The second terms in the expansions of the functions u13 and u13 are less than the

first terms only provided that

(3.4) v(x - 1)

2% = m2 = ‘/zmi < m.

If the sign of the inequality in (3.4) is reversed, the first and second terms in the

formulas (3.3) for u13 and u13 must be interchanged. For m = m2 logarithmic terms

occur in the expansion of these functions. Integration of equations (2.2) and (2.3) using

the Cauchy date (2.4) and (2.5) enables the constants A 12, B12 and A 13, B13 to be

determined.

4. The asymptotic behaviour of the solution of the problem of a powerful

explosion as 7 -+ 0 was established by L. I. Sedov [25], the form of the second terms _I? &L,. ̂ _.__-^:_.._ _l____:l_:_- L,-_ c A,~. ___ .,_.. I_ r,~. ~,._. .A? r r, cu LILC txpdu~~um uescriumg ine pwmiei~is 01 me gas cme Lu me place 01 origin 01 rne

shock wave, can be found in the book by V. P. Korobeinikov. N. S. Mel’nikova and

E. V. Rayazanov [27]. By the results given there we have v+z (x-1) --

PI1 = 11, + 11,q yx + . . . ,

hR(x-l)/xhl 2 (y-1)

Pll = I#" p _j_

x rl + . . . .

X-l v (x - 1) @+1)‘x I1

vx+2 (x-1)

t4*l) Gll=;g+l-- (x$._~)[yx'2(x_-;), rl YK +..., /

. . .,

-2(V-1)/7 [Y + x (v + 2) (x - 1)] hi21VX x

4(x-!-1)~~+2(X--)l

Y, il,= - ( ! x+1

’ -2 (~+l)i” vx + 2 (x - 1) h -(2-,,)jvx

4[1+-2(x.--1)1 ” - We now compare the principal terms of the expansions of the functions u12 and

u,~ given by equations (3.1), with the principal terms of the expansions of the

corresoondine functions of the fundam_enta! solution of the nroblem- of a nowerflul -~..~-~~~~-.~~o ~... _.._.._

explosion. It is easy to see that as Y-+0 the product

C1&@m/(V+2) -(=1)/x 77

may be a quantity of order unity even if 3-+iu. This condition determines the selfsimilar variable

(4.2) 5 _2 52mx/(~+2) (x-1) q = 2-2.U(~+2) (x-1) ql

in the middle of the stream domain, marked in Fig. 1 by the digit 2. Direct use of

formulas (2.1) to calculate the parameters of the gas in this region is impossible, since

the various terms in the expansions of the functions u and u becomes quantities of

206 0. S. Ryshov and E. D. Teren t ‘ev

comparable order, which contradicts the fundamental principle of perturbation theory.

The singularities in the other functions p r2, plz and v12, ocurring in the second approximation, are not so strong as to affect the choice of the variable {.

To construct the solution in the domain 2 we join external and internal asymptotic

expansions. Starting from the definition (4.2) of the selfsimilar variable in this region of

motion of the gas and assuming that the inequality (3.4) is satisfied, we write

(4.3)

8

C

.2v

P = @ + 2)’ (% + 1) f%p2 hdr: - y+i P2l (5) -t

2 (v+m) 2 (v+2m) -- --

+ G-4125 “+’ pm (5) + G24,z “+’ P23(:) + *. * , 1

1

P x+1 --iqi2&

=-p, h,“x L Pa1 (5) -

(Y + 2 - 2m) hi’” x--l

X x

am (v+kt) 1

x c,x-

2m tv+2w

” (“+2) tx-l) pz2 (5) + hFk,C12x- ’ (v+a) (‘-l) ps3 (5) + . . . I ,

2 (v-m) - -

ff = C” [

*fp

x+l” v-t:! 021(C) + v (x - 1) (v -+- 2 - 2m) hi”” x

(x -t- 1) k, 2 [v (V---m) (x-1)-m&]

x ClX ” (v+a) (x-11,

*22 (5) - v (x - 1) /pL

(x t 1) ks

“4 x

2 [v (v-m) (x-I)-2mk,] I -

x c,2x -J (vf2) (x-1) 023 (5) + * * * 1 I

u = (v + af (X + 1) VcJ’

,;l.k v-2 (m+l)

2 x

d-2 U2L (5) +

+ v (x - 1) (v + 2 _ zrn) h;l’” c

2xkl 1

xy ‘-;;“;;;;-+-;:]-2mk’ u22 (5) -

v (% _ 1) &l’Xk4 -

2xk,

c

1

2xv (x-1,[~v~~~1~~~~-4mk1 u23 (5) + * * * I 1

I- (v + 2)28(n + 1) c2

&$-l)/x 2 [m-v (x-l)]

- 5 (v+z)(x-l)

KS-1 v21(5) +

+ (v + 2 - 2m) hp-l)‘X 2 {v [m-v (x-I)]-mk,) _-

ClX ” (v+2) (x-1)

x+1 v22 (5) +

+ hr1”” k4 2 {v [m-v (x-l)l-2mk,)

x+l x

v (v+2) (X-1) I 7723(C) + * * * l

If the inequality sign in (3.4) is reversed the representation of the parameters of the gas in the region 2 becomes somewhat different; this case will be considered below.

We now use formulas (4.1), (3.1) and (3.3) to deduce the conditions which must be satisfied by the unknown functions as 5 --, co. Following the standard procedure for joining the external and internal asymptotic expansions, we substitute these formulas in

Themy of the h~h~~tropy layer 207

(2.1), specifying the gas-dynamical characteristics in the region the boundary of which is

given by the front of the shock wave. From this as 6-+ 00 we have

pzi --f 1, pzz --t 1, p23 * 1,

pzi --t 5’lX, pzz * ,$‘- mwx, pzy --f iyW’~~,

zJ23 + g*t-~w~,

z&i - iy, uz2 * p+wVX, u23 _+ 5-(VC2rnX,!W*

It is obvious from the definition (4.2) that for ki > 0 and as LC --t 00 the

values of the given stream function J/i increase without limit, provided that the value of

the seifsimilar variable 5 is not chosen too small. In the domain 2, obviously, 5 - 1, hence equation (1.5) establishing a ~nnection between the stream function and the entropy, still appfies. We use it and substitute the relations (4.2) in the original Eulerian

equations. We finally obtain a simple system of ordinary differential equations the integration of which enables all the unknown functions to be found explicitly. It is easy to show that for any values of the variable { they are identical with their limiting values (4.4). Therefore in the band considered the perturbed stream field can be constructed by merely

starting from the asymptotic representations of the gasdynamical parameters in the domain 1 as 9 + 0, whatever new qualitative properties the flow may possess here.

5. On comparing the various terms in the expansion of the density it is obvious that in the interior stream region, marked in Fig. 1 by the digit 3 and occupied by the highentropy layer, the product

C*5-2mk,;v(o+z)(X-f) c-“”

may be a quantity of unit order. This implies that the reduced stream function 3/i must

be chosen as the interior variable. The singularities in the function-corrections pzz, uz2,

us2 and u22 cannot affect the choice of the variable in the entropic layer, since the singularities contained in them are not too strong as 5 -+ 0.

It is essential to emphasize that the direct continuation of the solution (2.1) from

the domain 1 into the domain 3 is impossible. Indeed, if we rewrite all the terms of the series occurring in it, using the stream function as the fundamental variable, in the

expansions of the functions o and u quantities of the second order of smallness will be

greater than the fundamental quantities. Naturally, such a situation contradicts the fundamental principle of perturbation theory. The introduction of the intermediate region 2 is necessary, since writing the solution in the form (4.3) enables this difficulty to

be avoided, although it does not lead to the appearance of qualitatively new stream

properties. In the functions u2i and zdzl the principal terms of the asymptotic expansions of the solutions of the problem of a powerful explosion and of the solution of the


variational problem occur equivalently as 5 --f 0 they can change places. Further con-

tinuation of the solution of equations (1 .I) from the region 2 into the region 3 is again

achieved by a standard application of the method of joining the external and internal

asymptotic expansions. In the last of these domains we seek the representation of the gas

parameters in the form 8

$-)= (Y+z)a(3C+1) p_J:C2 Jw-=P,, ($Q -/- A,,x c

2.9 2 (“+nr) -- *+2 x

XP32NJl)f. f * 7 I

x-1 x 2 (x-11

8x& --

+- 2)2(x + I)2 c2x x (~4-2)

u32 (41) + * ’ * ,

where p3r = I. Formulas (4.3) and (4.4) facilitate the calculation of the limiting values,

which the new unknown functions must satisfy as 9, + co. At the external boundary

of the external boundary of the high-entropy layer we have

15.2)

031- 1, 03: --f lglx-*)‘x +

Y(X - 1) (v f 2 -2m) c $% II -

xk, v(x--- I)& -

xks Cl?~~-w~X )

u3t * 1, u.32. -+ $1 b+i)/x +

v (x -1) (v + 2 - am) Ci~I<,:YX _

XlCl 1

V(% - l)k, - xks

Ci2$ ,-hs/= (

Theory of the high-entropy hyer 209

Within the region 3, obviously, $ - 1. Meanwhile, formula (1 S), connecting

the values of the entropy with the stream function, is valid only for a sufficiently great

value of J/r, since its choice is based on the assumption that as x---f. co the position

rs (z) of the compression shock is given by equation (1.2). The variation of the

entropy across the stream lines in the region 3 is determined not by the shape of the

shock wave at a great distance from the nose of the streamlined body, but by that part

of it where it is oriented at each point at a large angle to the direction of the velocity

of the oncoming flow. When passing through a condensation shock we have for the

entropy of a perfect gas

Here the combination on the left depends only on the values of $i, the derivative

dr, i clx is converted by the formula

(5.4) $,= ($

the arbitrariness in the choice of the function r,(x) being constrained only by the

condition (1.2).

We now turn to the system of equations (1.1). Substitution of the expansions (5.1)

in Euler’s equation, projected onto the $-axis, makes it possible to determine the

pressure immediately. Indeed.

which implies

(5.5) psi = p:? = 1.

It is convenient to take the condition of conservation of entropy along a streamline

in integrated form. Taking into account equations (5.5) we find from formula (5.3) the

density distribution

where the derivative dr,< / a..~’ is understood as an expression in terms of the reduced

stream function $,using the relation (5.4). As I#, --t w the function p3, possesses

the required asymptotic expansion which appears in the limiting conditions (5.2). This

statement follows directly from the preceding analysis which was based on a series

expansion of the right side of formula (5.3) for the entropy of a gas pinched by a shock

wave.

210 0. 5‘. Ryzhou and E. D. Terent’ev

It follows from the third equation of the system (1.1) that

do31 o da32 x-l 1 -=v -=--7 d% d$f x P31

from which we find by simple integration

1, PC--- 1

031 = cr32 = - P3i-*(qi)d$i + B32. x s

If the parameter m satisfies the inequality (3.4), the coefficient ks > CL As is

shown by the limiting condition (5.2) for the function u32, there must not be a constant

in its expansion as 9, --f 00 .This condition determines the value of Bx. Finally

where the corner above the improper divergent integral denotes its terminal part in

Hadamard’s sense.

The equation giving the slope of the streamlines immediately gives the final

relations

(5.7) &i = 1, us?. = (r32.

Finalfy, from Bernoulli’s integral we deduce the equation

(5.8) 1

v32 = -. P31

Since the function p31 possesses the required asymptotic behaviour as -+, --f 03,

the functions (~32, ~32 and ~32 which are expressed in terms of it by means of formulas

(5.6) - (S.8), also tend to the required limiting values on the externai boundary of the high- entropy layer. This statement is easily verified by a direct check.

6. From the point of view of applications it is of greatest interest to know the

equation of the contour rb(z) of the streamlined body. It is obtained by substituting

the value $ = 0 in the expansion for the function u. Finally,

rb’ = C”(Dz” + LdF + 0. *) , CZ= 2(17-m)

2v v+2 '

ai =

++q '

(6.1) D = C,Bl,, D1 = _ (X-i)h~'X jmp31-1(,#&jq

c x+1 ;, 1.

The principal term on the right side of this equation is uniquely determined by the

parameters of that stream domain which is immediately adjacent to the shock front.


Only the correction to it depends on the characteristics of the high-entropy layer, the

quantity p3, including a function ra (~1, giving the position of the compression shock

close to the nose of the streamlined body arbitrarily. In the approximation considered

this correction is connected with the shape of the shock front only due to the

dependence of the asymptotic expansion of p31 on the coefficient ‘C, as *I -+ 0~).

If in the expansion (1.2) for the shock front all the coefficients C1, C2 ,... vanish,

formula (6.1) implies that

(6.2) rb = CBi’:’ zz/x(W)*

ln this case, integrating the function 1 / PS, we have

d6.3) D,=

Equations (6.2) and (6.3) give the asymptotic representation of the trajectory of a

particle in the problem of a powerful explosion, the entropy of this particle corresponding

exactly to the pinching of a gas by a hypersonic flow at a normal shock [I I] _ It is clear

that, whatever the value of the constant Dr , the correction term on the right side of the

relation (6.1) is connected with the characteristics of the gas at an explosion, and its

coefficient depends essentially on the shape of the compression shock close to the nose

of the streamlined body. In particular, it vanishes if the position of the shock front

satisfies the condition

which plays a fundamental part in direct problems of the aerodynamics of the high-

entropy layer [7] .

Equating the powers a and a, gives m = m 1. Therefore, at great distance from the

nose a streamlined body widens more quickly than the particles of a gas are moved in a

strong explosion only in the case where the inequality (3.2) is satisfied. A similar

situation also holds for the study of unsteady one-dimensional motions arising from the

action of an expanding piston [26].

In order to obtain the principal term in the equation of the contour of the

streamlined body, it is not necessary to consider the velocity field in a layer with high

entropy of the gas. The principal term can be obtained by using the results for the stream

parameters in the domain 2. Hornung 191 whose analysis was based on the inequality

(6.4) a < ui.

proceeded in this way.

212 0. S. Ryzhov and E. D. Terent ‘ev

As is clear from the above, in the formulation of the inverse problem the inequality

(6.4) follows from the condition (3.2) imposed on the magnitude of the exponent of the

first correction term in the equation of the shock front. The role of the inequality (6.4)

was first pointed out by Freeman [S] .

7. From the relations (1.3) and (3.4) we have

(7.1) mi < 4.

Assuming this condition to be satisfied, we repface the inequality (I -3) by the

opposite 2m > q. Variations in the parameters of the medium can mainly take place

because the form of the expansions for the entropy s and the functions u and u changes.

We will be primarily interested in the behaviour of the two latter quantities. In the region

1 we obviously obtain

(J = ~x?v,(r+z) [G,* (q) + C1x-2”‘~fV+2)c7~2 (q) + ci2x-zg’~~+sks*s (17) + . . .] ,

(7.2) L

U = (Y + 2) (X + 1) v,cv5(~-z)‘(~‘*)[u,l (q)+ ci3-~‘~‘(~+%z~2(~) +

+ c~~-zq’(viz)~~~ (q) + . * *I.

Formulas (2.1) for the remaining parameters are modified similarly. As before the

asymptotic behaviour of the function& of the second approximation will be given by the

expansions (3.1). As for the functions of the third approximation, they can be found

from the system of equations (2.2) if 4 is substituted for pn everywhere. From this it

immediately follows that as r~ -+ 0

(fis = - v(x- 1) tv + 2 - 2q)hl”” __R,;rx

(“x + 1)ks r +...,

Z.&s = - Y (x - 1) (Y + 2 - 2q)&-i’x R ,_

Zxk, q-6 +...,

ke = qx. - v(y. - 1).

In the middle of the stream domain

x~~lIx 2(v--ml

- -

o=C” -x [ x+4

v+2 % (5) -i- v (x - 1) (Y + 2 - 2m) h;“’ x

@ + 1) kl 2 iv fY--mf(x-l)-mk,]

XClX v (vi-2IG+-1) *22(5) -

Y (x - 1) (v + 2 - 2q) &‘” x

(x-t 1)ks

2 [u (V--m) (x-lk-qk,l

x c,x ” (v+2) P-1)

=29 (5) 4 . * . I r

Theory of the high-entropy dyer 213

(7.3)

* = (v + 2:(x + 1) vcac” v (x - I) (Y + 2 - 2m) ho-ii”

*(x---I) [v---2 (m3_lq--smk~

+ 2xk, 0 v (VfZ) (X-4)

u22(5) -

_ v (x - 1) (v f 2 - 2q) h,‘pr c’ 2xks

gs ‘“-‘1 ;;;;;“;;-2q”1 2

where the selfsimilar variable 5‘ is defined by formula (4.2), the functions of the first and

second approximation are given by the relations (4.4), and for the solution at the third

approx~ation we have

u23 = (323 = I;.+x*

We now recall the expressions (5.2) which prescribe the limiting values of the

stream parameters on the external boundary of the high-entropy layer. The first two

terms in the formulas for oz and usL remain unchanged, while the third term will be of

order $I~-‘& I VX. . As is shown by the inequality (7.1), the coefficient ks > 0. From

this we conclude that the rule for ~lculating the constant Bs2 remains true, hence

formula (5.6) and equation (6.1) for the contour of the unknown body, derived from

it, also remains true

8. We now discard condition (7.1), hitherto imposed on the exponent of the second

correction term in the expansion (1.2) for the compression shock, which is valid at a

great distance from the nose of the streamlined body. Let

As in the preceding paragraph we concentrate our attention mainly on the study of

the quantities CJ and U. As q -+O the functions u13 and u13 in (7.2) can be calculated

by using equations (3.1), if we replace m by 4 and use a new constant Br2. On the right

sides of formulas (7.3) for the middle domain of motion of the gas, new terms appear

which must be placed after the principal terms.

The combinatjon 2 (+-_e, --

C2B12’x ‘+’ ,

occurs in the representation of u, and the expression for u must be supplemented by the

quantity

(v - 4) (x -i- 1) 2v

c,B12Px?$P .

214 0. S. Ryzhov and E. ff. Terent’ev

These differences also affect the nature of the flow in the high-entropy layer. Here a (v--m) 2 PJ-9) - x~,l/x _._!L

C,B,,x v+z -/- C2B1at2 v+2 + - x-j- 1

xx (Y+2b&l)f.. * I

a = (v -j-a 2:(, + 1) vcac” [ (v - ml (x + 1) c B

2v 1 12 xv-2vy..) $-

+ (v - d (x + 1) 2V

C,B1,?x=t$P I h?‘K ,e Usa (+1> +

2 . . .

I each being identically equal, respectively, to the functions ~132 and ~32 defined by equation (5.6). Along the surface of the streamlined bady $ r = 0, whence

(8.2) rby = c” (DC?? + I&?? -j- f-l,33 + * . .) *

The parameters a, al and D, D1 occurring here are found from formulas (6.1) and the additional constants

P = ‘(’ - ‘) v+2 ’

Q, = C&a’.

Therefore, if the range of variation of the exponent q is bounded by the inequa~ties (8.1), a new term appears in the equation of the contour of the body. In order of magnitude it is greater than the correction term the coefficient D, of which depends essentially on the shape of the leading part of the compression shock. As mentioned above, the latter gives the law of motion of the gas particles in the powerful explosion problem.

9. Weakening of the constraints on the exponent q in practice indicates the introduction of additional terms into the equation of the shock front as x --f co. We now consider changes in the stream structure accompanying breakdown of the inequality

(3.4) which establishes minimal values for the parameter ~tl. Let

ms < m < m,, m3 = v(x-11)

3x ’

and q > mt. We seek a solution in the outer region adjacent to the front of the shock

wave, in a form similar to (2.1), but in all the expansions we retain terms up to the

fourth order of smallness inclusive. As in the two previous paragraphs we will be interested

only in the behaviour of the functions u and U. Their asymptctic bahaviour as q --+ 0 is established by the relations (3.3) on the right side of which the first and second terms

must now change places. As for the solution of the system of equations of the fourth

approximation, it is only necessary to estimate it. For this purpose we first notice that

the discarded term in the expression (1.5) for the entropy is of order t$-(v+3m)‘v, as

$t’=J. As rt -+ 0 we obtain correspondingly pi& - rj~v--3st~vx7whence it is easy

to establish that ui4. k or4 - rj-k~“‘x, where the constant k, = 3mx - v(x - 1).

Theory of the high-entropy layer 21s

The solution in the region 2 is constructed as in all the cases considered previously,

fundamentally by means of asymptotic expansions of the unknown functions as q .--+ r3,

In the layer of gas with high entropy

[

2 (v--m) 2 (v-24

0 = C’ C1B12n: “+’ + C12B,,x “+’ +

4 I"

u =(Y + 2)(x+ 1) vmc (y-n) (' +l)~lB,,~?$?+

2v

+ ;v - 2m) (x + 1)

2v

c *B

1 13

xv=J’+

1 -_ vx+z (x--Y)

hl x - --- f,X

x Pf2)

I u32(91) +. * * 9

where all the quantities retain their original definitions. Finally, the equation of the

contour of the streamlined body assumes the form (8.2) with the constants

DB = Ci’B,s.

Therefore, the consequences caused by decreasing m are similar to those from

decreasing 4: in the equation of the surface of the streamlined body a new term appears

which is greater in order of magnitude than the term due to pinching of the gas at the

leading part of the shock front. This reasoning can be extended by subjecting the

exponent m to the conditions

mi+l < m < mi, TiZi = v(x - 1)

iic ’

Each such successive decrease leads to an increase in the number of terms which occur

in the equation of the contour of the streamlined body.

Translated 6-v J. Berry

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theory of the high-entropy layer in hypersonic flows

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