I am supervision.

grateful

to N.

B.

Ii'inskii

for

suggesting

LITERATURE

the

problem

and

for

scientific

CITED

I. M. A. Lavrent'ev and B. V. Shabat, Problems of Hydrodynamics and Mathematical Models for Then [in Russian], Nauka, Moscow (1977). 2. V. M. Kuznetsov, "Explosion on the surface of a plate," Zh. Prikl. Mekh. Tekh. Fiz., No. 3 (1962). 3. N. B. II'inskii, A. V. Potashev, A. V. Rubinovskii, and P. A. Fishchenko, "Solution of some problems of the theory of explosions in the impulse--hydrodynamic formulation," in: Proc. Seminar on Boundary-Value Problems, No. 14 [in Russian], Izd. Un. Kazan' (1977). 4. V. M. Kuznetsov, E. B. Polyak, and E. N. Sher, "Hydrodynamic interaction between fuse charges," Zh. Prikl. Mekh. Tekh. Fiz., No. 5 (1975). 5. N. B. II'inskii and N. D. Yakimov, "Interaction between two flat charges exploding on the ground," Zh. Prikl. Mekh. Tekh. Fiz., No. 1 (1977). 6. M. I. Gurevich, Theory of Jets of an Ideal Fluid [in Russian], Nauka, Moscow (1979).

APPROXI~.~TE THE

RADIATION N.

N.

I~THOD

FOR

CALCULATING

INTENSITY

IN A FAR

Pilyugin,

S.

G.

THE

AIR

PAR~TERS

AND

WAKE

Tikhomirov,

and

S.

Yu.

Chernyaskii

UDC 5 3 5 . 3 7 : 5 3 3 . 6 . 0 1

In the present paper an approximate method is proposed for solving wake problems that makes it possible to take into account the main features of a flow with complicated chemical kinetics. Analytic dependences are obtained for the distributions of the gas parameters and the radiation intensity, and also the determining dimensionless numbers. Satisfactory agreement is obtained between the theoretical and experimental results for all parameters that have been measured.

The determination of the parameters in the far viscous and chemically nonequilibrium wake behind a supersonic body is of interest for interpreting experimental data. These parameters have been calculated by finite-difference [i, 2] and integral [3, 4] methods of solution of the equations of an axisymmetric boundary layer. Approximation of the results of numerical solution of the problem in the form of characteristic dependences on the determining dimensionless parameters is very difficult. In addition, because there is no complete solution to the problem of the n o n e q u i l i b r i u m flow in the near wake behind a body, the distribution of the parameters in the initial section of the far wake is known with low accuracy. To investigate the far wake, it is therefore expedient to develop approximate methods of solution. In the present paper an approximate method is proposed for solving wake problems that makes it possible to t a k e into account the main features of a flow with complicated chemical kinetics. Analytic dependences are obtained for the distributions of the gas parameters and the radiation intensity, and also the determining dimensionless numbers. Satisfactory agreement is obtained between the theoretical and experimental results for all parameters that have been measured. I. The system of dimensionless boundary layer equations describing the flow of a gas with n o n e q u i l i b r i u m chemical reactions in an axisymmetric viscous wake behind a body has the form

-~-~z(Par) <-~r(pVr)=O'

6, pp.

Pa~-x+P.v0--~= r

Or ~| r~--~-r !~

Moscow. T r a n s l a t e d from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 165-175, November-December, 1980. Original article submitted July 25, 1978.

0015-4628/80/1506-0925507.50

9 1981 Plenum Publishing

Corporation

925

OH OH t 0 [I,r[OH . . . . . &all pa-~7 x + 9v ar ' = T T r / . ~ r L - ~ r - r ~ r r - l ~ = g ]

Pr=" ~tC"' H = Z h ' c ' + - - 2 - : h + 2 - '

Or -----7"-~'rt ) S c ,Or +W~ (1.1)

h':I6";'dT+h'~

t:i

h:t

J:t

'

c, ppo.,C~, --~-7c,=0, [X~l=-~-7'

,

_ Iz c,C~., SC'--oD''

i=i

a~ A~ ~ b~ ~=1 1 ~ bj~A~

H e r e , N a n d Nr a r e , r e s p e c t i v e l y , the numbers of components and reactions; xd a n d r d are the axial and radial coordinates; uV~ a n d uV~ a r e t h e c o m p o n e n t s o f t h e v e l o c i t y v e c t o r a l o n g x a n d r ; pp~, hV= 2, HV= ~, TT~, pp| are the density, enthalpy, total enthalpy, temperature, a n d p r e s s u r e o f t h e g a s ; c~, M~, h~V~ 2, e~ a r e t h e m a s s c o n c e n t r a t i o n , molecular weight, enthalpy, a n d c h a r g e o f c o m p o n e n t i ; l~9=V~d, ~9~V~o~d/T| a r e t h e c o e f f i c i e n t s of viscosity and thermal conductivity; C~V| is the specific h e a t ; D~Vood a r e t h e e f f e c tive ambipolar diffusion coefficients; h~~ ~, C~Y| are the enthalpy of formation and specific heat of component i; P r a n d Sc i a r e t h e P r a n d t l a n d S c h m i d t n u m b e r s ; kfk and kbk a r e t h e r a t e s o f t h e f o r w a r d a n d b a c k w a r d r e a c t i o n k ; aik and bik are the stoichiometric coefficients; [Xi] a n d W i ( o ~ V ~ / d ) a r e t h e m o l e - v o l u m e c o n c e n t r a t i o n and rate of formation of component i; d is the dimensional diameter of the body; p ~ , V~, 9~, T~, M~ are the pressure, velocity, density, temperature, a n d mean m o l e c u l a r w e i g h t o f t h e o n coming gas stream; a n d Ai a r e t h e c h e m i c a l s y m b o l s o f t h e c o m p o n e n t s . In accordance with the calculations of [1-3], the pressure in the region of the far wake x > 50 i s v i r t u a l l y constant. Therefore, i n E q s . ( 1 . 1 ) we h a v e o m i t t e d t h e t e r m c o n taining the pressure gradient. On t h e r i g h t - h a n d s i d e o f t h e e q u a t i o n f o r H we h a v e omitted the term associated With heat transport by the diffusing components, which is small for the far wake. " I n . t h e r e g i o n of t h e wake, P r and Sc i w e r e t a k e n t o be c o n s t a n t , basically b e c a u s e o f t h e weak t e m p e r a t u r e d e p e n d e n c e . In the turbulent f l o w r e g i m e , we considered the averaged characteristics o f t h e m e d i u m , a n d t h e f u n c t i o n s Wi w e r e c a l culated for averaged values of the temperature and the concentrations. In the system (1.1), there is no equation for the electromagnetic field, since the condition of quasineutrality is satisfied in this region of the wake. Separating t h e i n d e p e n d e n t c o m p o n e n t s ( w h i c h we c a n t a k e t o b e t h e Ng c h e m i c a l elements), we c a n e x p r e s s a l l t h e r e m a i n i n g c o m p o n e n t s ( t h e r e a c t i o n p r o d u c t s ) Ai ( i = 1,..., NL, NL = N -- Ng) i n t e r m s o f t h e c o m p o n e n t s Aj ( j = NL + 1 , . . . , N) i n t h e f o r m [5]

A/-----s voA,, $--.s'z.+i where vii are the stoichiometric coefficients. Multiplying each diffusion equation for t h e c o m p o n e n t s i n ( 1 . 1 ) b y ~M~/M, (i=i,...,N, ]=-N~-F]. . . . . N, ~o=~o f o r i > E L ) , a d d i n g t h e results, and using the conservation of the elements in the chemical reactions: N

s

voMtWJl~,:O

t--!

we o b t a i n

the

diffusion

equations

8cj*_

0r

of the

i

elements:

8 [

O / c~* \ 1

Or N

,

,V_.. o,.=t, SmN/~+t

926

c,--,-c~ (i=t .... N,.), cj*-+e~|

(1.2)

N

Y', 0r

=~

$=NI.+ I

I n t h e r e g i o n o f t h e f a r w a k e , t h e s h o c k w a v e i s weak a n d h a s the parameters of the uniform exterior flow. Therefore, we c a n s e t

r-,-% u-,.t, H-,-H|

M~

'

Nr.

i~l

eL

little

(]=N,.+t, . . . N).

influence

on

On t h e w a k e a x i s ~

r=0, 8u/Or=Og/Or=ac/Or=v=O.

The i n i t i a l conditions a r e x=xr~, u=uz(r), H=Hz(r), c~=c~(r), w h e r e t h e s u b s c r i p t H corresponds to the initial section. For the further analysis, we o b t a i n i n t e g r a l relations from the system (1.1). Integrating E q s . ( 1 . 1 ) o v e r r f r o m r = 0 t o cr u s i n g t h e conditions

Ou Or and using the laws of conservation surrounding t h e b o d y , we o b t a i n

"of t h e

OH Or

mass,

Oc~ Or momentum,

and energy

for

a control

surface

pa(t--a)rdr

(1.3)

pa(H~-H) r dr

(1.4)

C==16 ~ 0

C n = I 6 .[ 0

where

C x and C H are Similarly,

the

from

drag

the

and h e a t

diffusion

transfer

equation

coefficients

(1.2)

~ pa(cs*-cj~*)rdr=C~j*,

for

for

the body.

the e l e m e n t s

we o b t a i n

]=NL+I,...N

(1.5)

o

determined w h e r e C*~_- i s a c o n s t a n t e f f i c i e n ~r of mass transfer for the we o b t a i n

by t h e body.

initial conditions From the diffusion

and proportional to the coequations for the components

O-~-;pa (ci-c~ ) r dr= S W~r dr Ox o

In the

case

of frozen

chemical

(1.6)

o

reactions,

(1.6)

gives

eo

~ ptt(ci--c~)r dr=C3~

(1.7)

0

where

C3i

is

the

constant

We i n t r o d u c e

the

for

component

new i n d e p e n d e n t

i

determined

from the

initial

variables r

re

s=x, z : r . R:(x) ' R(x)=(~o prdr)'/~ where

r e is the From

(1.8)

We s e e k

characteristic there

follow

a solution

half-width

(1.8)

of the wake.

the r e l a t i o n s

of Eqs.

rz Y=R

(x) o

dz

(1.1)

in

the

re" =R 2(x) 2

(1.9) o

form

o

[3]

u=t--Uo(Z)]iz), h=h.+ho(x)~(z), c,=c,=+$~(x)F~(z). In the far wake, coming stream: u0(x) substituting (1.10) in tion (1.10) to satisfy Then from (1.3), (1.4),

conditions.

(1.10)

the velocity defect is small compared with the velocity of the on<< 1. We t r a n s f o r m E q s . ( 1 . 1 ) a n d t h e i n t e g r a l relations (1.3)-(1.7), them and r e t a i n i n g the principal t e r m s i n u 0. We r e q u i r e the soluthe transformed equations f o r r = 0 and a l s o t h e i n t e g r a l relations. and (1.7), we obtain

Cx

16R (x)Z,

1,--

(l.ll) 0

ho(X)

_ cC:] = Jz' J:= f: o?(z)dz

(1.12)

0

927

xF,(x)= R2(x)j, Differentiation

of

(1.11)-(1.13)

with zt0'R

Js,=

,

respect

h0"R

,(z)dz

(1.13)

0 to x gives

r

= -2

(1.14)

CoB" = ~h0R'= ~ R ' Here, the prime denotes e q u a t i o n f o r r = 0, a n d u s i n g

differentiation with respect ( 1 . 9 ) a nd ( 1 . 1 0 ) , we o b t a i n

\'-'-~z ' / ,=o'~" R ' x=xm

t o x.

Writing

down t h e

RH=R(X~)

momentum

(1.15)

From t h i s e q u a t i o n f o r known f u n c t i o n D 0 ( x ) we f i n d t h e d e p e n d e n c e R ( x ) f o r b o t h l a m i n a r and t u r b u l e n t flow regimes. I n t h e f a r v i s c o u s w a k e ( x > 10 2 ) t h e t u r b u l e n t reg i m e c o m m e n c e s when Re > 10 5 [ 6 - 8 ] , w h e r e t h e R e y n o l d s n u m b e r i s c a l c u l a t e d using the parameters of the Oncoming stream and the diameter of the body. In this case, the coefficient of viscosity h a s t h e f o r m [3] ~ = ~ 0 ( x ) = kRu 0 (k i s an e m p i r i c a l constant) and (1.15) gives

R (x) = [R,3+3kC=B (x-x,)/t6I,

],/,

(1.16)

From the expressions f o r ~0 a n d t h e r e l a t i o n s (1.11) and (1.16) it can be seen that 90 § 0 a s x § oo. T h e r e f o r e , strictly speaking, t h e a s y m p t o t i c b e h a v i o r o f an a x i s y m m e t r i c w a k e i n t h e l i m i t x § oo f o r a n y f i n i t e v a l u e o f Re i s d e t e r m i n e d b y t h e m o l e c u l a r v i s c o s i t y . H o w e v e r , f o r Re > 5 " 1 0 5 up t o d i s t a n c e s x < 104 f r o m t h e b o d y m o l e c u l a r v i s c o s i t y is negligibly small compared with the turbulent viscosity. At even greater distances from the body the far wake becomes laminar. In what follows, we s h a l l c o n s i d e r the solution to the problem for the turbulent flow regime. O ( u 0)

Going over from (1.1)

to

the

new c o o r d i n a t e s

(1.8)

Oz x

and u s i n g

(1.9)-(1.14),

we o b t a i n

to

terms

az d z .,-"~1

~-z ( B q ) z + P r

=0

(1.18)

0

- 2~We w r i t e

the

boundary

70-

-r- = 2-~,

conditions

in

the

z + 0% ~ = 0, Integrating

where

r is

any o f

(1.17)-(1.19)

the

over

z~0.

functions

z (for

(1.19)

(1.20)

],%F~.

Wi = 0) S.

using

(1.20),

we o b t a i n

iF

P~dZ-~'t]'-z'dZO )'o

r where ~ =

+ - -W, P

form

rdr as

P-c; dF,

i, Pr, Sci, r e s p e c t i v e l y .

N e a r the w a k e

axis,

p ~ Po(X),

and the s o l u t i o n s b e c o m e

r =exp

(--~Bz), z=rVr;

(1.2 I)

Using (1.21), we o b t a i n t h e v a l u e s o f t h e i n t e g r a l s J1 = B - l ' J 2 = ( B P r ) - t ' J3 i = (BSci)-l. I n t h e r e g i o n o f t h e f a r w a k e a s x § oo t h e i n i t i a l conditions have little inf l u e n c e on t h e d i s t r i b u t i o n of the parameters and t h e g r o w t h o f t h e wake w i d t h R ( x ) , w h i c h e n a b l e s u s t o i g n o r e RH a n d x H ( 1 . 1 6 ) . U s i n g t h e o b t a i n e d v a l u e o f J l ' we f i n d

.(.)=

928

(---~[' 3kC~B2'

) ".,,,,

a0(x)= /

C,,

~v, .-,,.

To d e t e r m i n e the p a r a m e t e r B, w e u s e d m e a s u r e m e n t s of the radial v e l o c i t y d i s t r i b u tion of the gas in the t u r b u l e n t w a k e b e h i n d a s p h e r e [9] at M a e h n u m b e r s M = 3 . 5 - 1 2 . 8 and R e y n o l d s n u m b e r s Re = ( 0 . 1 - 1 ) ' 1 0 6 . F o r d i s t a n c e s x = 300-2000, these m e a s u r e m e n t s can be approximated [9] w i t h e r r o r less than 10% by the curve

/ = e x p ( - - r 2 / r , 2) , r v = 0 . 3 0 5 x v' Here,

r y is

Comparing

the

radius

(1.21)

with

at which (1.23),

the

velocity

we f i n d

for

B----3.36. t 0 -s

defect

(1.23) is

reduced

by e t i m e s .

r = rV

k -s, re=~Brv relation ao(x)=i.27C=V'x -v',

(1.24)

Substituting B in (1.22), we o b t a i n t h e which agrees satisfactorily with the experimental d a t a o f [9] f o r x /> 103 . Introducing by a n a l o g y w i t h r V the wake radii for the defects of the enthalpy, r h, and the concentration of component i , rci, a n d u s i n g ( 1 . 2 4 ) we o b t a i n f r o m ( 1 . 2 1 ) t h e r e s u l t rh=rv(Pr) -v', r r -v'. The equation of the chemical kinetics on t h e w a k e a x i s h a s t h e f o r m

&p, _ 2R~ ,, ( dF, ~ + W,o dx R BSd' / ==0 po S u b s t i t u t i n g (1.25) integrands, we o b t a i n

W, p

in

(i.19),

(1.25)

r e g r o u p i n g the terms and s e t t i n g p = P0(x)

W~OFi -2Rqp(= p0

BSc(\

dz !~=o

-z

dz

~

BSc~

in the

(1.26)

~ z '!

The l e f t - h a n d s i d e o f ( 1 . 2 6 ) d e p e n d s on x a n d z , a n d t h e r i g h t - h a n d s i d e o n l y on z. It is therefore necessary to equate both sides to zero, which gives the radial distribution of the components in the form (1.21) and the relation W~=WiopF/po. Th e o b t a i n e d s o l u tion makes it possible to reduce (1.25) to the equation

a,, +

R"

(1.27)

The i n t e g r a l relation (1.6) is then satisfied automatically. This result reflects a physical feature of the investigated flow, namely, the concentration defects of the c h e m i c a l c o m p o n e n t s and the t e m p e r a t u r e d e c r e a s e in the t r a n s v e r s e d i r e c t i o n much m o r e r a p i d l y than in the axial [10-12]. T h e r e f o r e , the c h e m i c a l r e a c t i o n s are b a s i c a l l y r e s t r i c t e d to the w a k e axis [7]. By c o m p a r i s o n w i t h e x p e r i m e n t a l l y m e a s u r e d r a d i a l d i s t r i b u t i o n s of the flow p a r a m eters in the far wake, the e x p r e s s i o n s (1.21) e n a b l e us to f i n d the v a l u e s of the t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s Sc i and Pr of the gas. F r o m [i0] w e o b t a i n the v a l u e of the eff e c t i v e a m b i p o l a r S c h m i d t n u m b e r of the i o n i c component: Sc = 0.4. By v i r t u e of the quasin e u t r a l i t y , Sc = 0.4 for the e l e c t r o n s . F o r the n e u t r a l c o m p o n e n t s , the S c h m i d t number, c a l c u l a t e d u s i n g the e f f e c t i v e a m b i p o l a r d i f f u s i o n c o e f f i c i e n t , is then Sc = 0.8 [5]. U s i n g the d a t a of [ii], we d e t e r m i n e the t u r b u l e n t P r a n d t l number: Pr = 0.8. The solutions o b t a i n e d for R, a0, h0,/, q~, F~ m a k e it p o s s i b l e to r e d u c e the p r o b l e m of the flow of the m u l t i c o m p o n e n t , c h e m i c a l l y n o n e q u i l i b r i u m v i s c o u s gas in the far t u r b u l e n t w a k e b e h i n d the b o d y to the s y s t e m (1.27) of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s of first o r d e r for R ' / R = 1/3 and i n i t i a l c o n d i t i o n s x = x H and ~i = ~iHThe c o n c e n t r a t i o n s of the c h e m i c a l e l e m e n t s on the w a k e axis can be r e p r e s e n t e d in the f o r m cj = cj~ + ~j (j = N L + 1 .... N). F r o m (1.27), we o b t a i n the d i f f u s i o n e q u a t i o n s of the e l e m e n t s on the w a k e axis:

~J*=0,

de j* 3 dx'~ 2 x

x=x,,, ~j*=~p~,* (]=NL+i,...N)

The a n a l y t i c solution ~j*=~jH* (x/x,) -'/' o f t h e s e e q u a t i o n s makes it possible to reduce the number of differential equations i n ( 1 . 2 7 ) t o NL . We f i n d t h e c o n c e n t r a t i o n s o f t h e r e m a i n i n g Ng c o m p o n e n t s f r o m t h e r e l a t i o n s NL

*'= r

-2=__

M,

(1.28)

'l=i

In the g e n e r a l case, the d i s t r i b u t i o n of the t e m p e r a t u r e be d e t e r m i n e d by m e a n s of (1.12) f r o m the r e l a t i o n

on the W a k e

axis, TO,

can

929

TABLE

i k{ (T)

No.

Reaction

1

0+0+03 ~ Oz+Oz 0 + 0 + 0 -+ 0~+0

b 9

2 3

N+NO -~ N~+O N+Oz -~ O + N O

7 8

N

P0 f r o m t h e

equation

3(t~)

0

0 0 0 -3300

-5501 ' 0 0

T

i=i

density

1.8 (13) i.8(2t)

O+NO~ -+ NO+Os N0+,+e -+ N+O NO+O ~ N0z+h~

6

-2.7, -2.7 -2.7 0 t.5 ~ 0 -1.5' -t.55

5.07(21) t.5(13) t.8(8)

O+O+N2 ~ 02-kN2

4 5

a nd t h e

2(22) 6.16(22)

0

of state.

2. Th e m e a s u r e m e n t s o f [13] r e v e a l e d a slow decrease in the temperature with increasing distance x along the axis. Therefore, with a view to obtaining an a n a l y t i c solut i o n we a s s u m e d f o r w h a t f o l l o w s t h a t t h e t e m p e r a t u r e is constant. As an e x a m p l e , we c o n s i d e r t h e f l o w i n t h e wake b e h i n d a b o d y f l y i n g in the air. Using the analysis of the chemical kinetics in [ i , 2 , 6 ] , we i d e n t i f i e d the main reactions in the aeroballistic experiment making a contribution t o Wi0 o f n o t l e s s t h a n 10% a t t h e temperatures 1000-3000~ (see Table 1). The r e a c t i o n r a t e s h a v e t h e f o r m k ~ a T b e ~/~, i n which T is measured in ~ [ki] = (cm3/mole)q-lsec-1 an d q i s t h e o r d e r os t h e r e a c t i o n . F o r a , we h a v e u s e d t h e n o t a t i o n ~(n) = ~-10 n. Th e n i t r o g e n recombination reactions are absent, since the mass concentration of atomic nitrogen i n t h e wake i s low [ 6 ] . We i n t r o d u c e t h e new v a r i a b l e s @i = % i / ~ i H , w h e r e i = 1 - 8 c o r r e s p o n d t o O2, O, N2, NO2, NO, NO+ , e , N; y = X/Xl.I. We s i m p l i f y the expressions f o r Wi0 i n tions of the equations for the concentrations tribution of the e l e c t r o n s and the r a d i a t i o n

(1.27), of the in the

--~la 02--y {1-3F2[y 'I~-l+~z~(i _ _ y- ' h )]},

[kic,~. _bk,cs.. ] a.~MR.~2T~2M~ [ M~ M3 ] '

turbulent

u,--u~ . . . . .--y . . /',

F2=x. 2dp"~M|176

gas

omitting the small terms. components that influence the

form

O,=Y-V'[3F~YV~+I-3F6]-'

dp..M|162

~

k~r

[k,c,~ +k,c,~.:] -~ (2.1)

Fs=x, a..R.~T~M6M ' ~ = - M - ~ [ M~

M3 ]

p~p0NA~ H n,~:

M~

06=n,0~0~

w h e r e NA i s A v o g a d r o ' s n u m b e r , a n d t h e s u b s c r i p t e is scribe the behavior of the electron concentration. Th e f o r m u l a f o r t h e i n t e g r a l density w i t h a l l o w a n c e f o r ( 1 . 8 ) and ( 1 . 2 1 )

N~D~=d Since

p0~p~l,

we o b t a i n

i

N~D~,

which

appended is

n~dr=-~/2p~NA~fl(x)d ~ e -Bsr " M~ the

following

estimate

(2.2)

for

~ ~ r~fln~o O~/Y,po--~N~D~ u ~ rofln~o

to

measured

the in

IV]

dz" ]-' dz p j

the

integral

functions the

which

de-

experiments,

(2.3)

(2.3):

0Jp0

On t h e b a s i s o f t h e e x p e r i m e n t s os [ 1 4 ] , we d e t e r m i n e N e , concentration averaged over the transverse section of the wake. ( 1 . 2 1 ) , we o b t a i n

930

have

H e r e , a~ i s t h e v e l o c i t y o f s o u n d i n t h e o n c o m i n g g a s s t r e a m , a n d RA i s t h e u n i v e r s a l constant. The c o n c e n t r a t i o n he0 o f t h e e l e c t r o n s on t h e w a k e a x i s was f o u n d f r o m t h e

equat ion

takes

wake

The s o l u the dis-

(2.4) which is the electron Using (1.9), (1.10),

and

r n~cm- 3 I0 ~

\

fOiO

%,cm-3'~___~

"%. '

/09

-'

100

ZOO

Fig.

i

X

108Zt

JO0 i

~0 x'10-2 ~0

I0

2

Fig.

iO ,z

/ , , ......

, i / / i //////////

'/////////

4

./o~~

9

10 x10 fO

Fig.

0.21 5

3

18-2 10 Fig.

4

rce

P0

0

which in

conjunction

with

(2.4)

gives

an e s t i m a t e

f o r Dp~NeDp/N,:

]/~Po ( 1 --e -i ) rvd/1/ See <~Dp ~<~r~ ( 1 - e -l) rvd/Y See

(2.6 )

Using the relation f r o m [ 1 5 ] , r v d = 0 . 4 d (~ i s t h e mean d i a m e t e r o f t h e wake m e a s u r e d by t h e s c h l i e r e n m e t h o d ) , we t r a n s f o r m ( 2 . 6 ) f o r S c e = 0 . 4 t o t h e f o r m 0.7081'p06<~Dp~<0.7086. This e s t i m a t e agrees s a t i s f a c t o r i l y with the e x p e r i m e n t a l d e p e n d e n c e [i0] D p = 0."75d. In the w a k e b e h i n d a sphere flying in air, for x > 200 by r e a c t i o n 8 [12, 16] in the table. per unit length of the wake we have

I = 2~d 2

I~rdril

the r e c o m b i n a t i o n r a d i a t i o n is d e t e r m i n e d For the total i n t e n s i t y of this r a d i a t i o n

I~---5"47"iO-'7-~

4n

~

w h e r e k 1 a n d k2 a r e t h e b o u n d a r i e s o f t h e s p e c t r a l interval, h is Planck's the velocity of light, [T]=~ [n0]=[nN0]=cm -3, and [I] = W-cm-l*sr -1 . From ( 2 . 7 ) ,

using

(1.21)

and (2.1)

I/IH=OoO~voU

~/s

for

TO = const, 'h

c is

we o b t a i n

=Oo=y- '{'J--31"2[U --'l+c~,(t--y- % ) 2/

constant,

(2.8)

It can be seen that the d i s t r i b u t i o n I/I H along the w a k e axis agrees with the dist r i b u t i o n %0(x) of a t o m i c oxygen, w h i c h agrees with the e x p e r i m e n t a l data [12]. The axial d i s t r i b u t i o n of the e l e c t r o n c o n c e n t r a t i o n and the r a d i a t i o n i n t e n s i t y depends on the d i m e n s i o n l e s s p a r a m e t e r s F 6 and F2, respectively. S i n c e the mean temperature of the gas in the w a k e is d e t e r m i n e d b a s i c a l l y by the M a c h n u m b e r M, to c o n s e r v e the

931

p a r a m e t e r F 6 we m u s t h a v e t h e r e l a t i o n s p~d/a~=const, M=c0nst with the remaining initial conditions u n c h a n g e d (xR, ~ , T~,M,). This simulation law agrees with the experiments for pure air and a mixture of it with chemically inert xenon [17]. In the latter case, ins t e a d o f p~ i t i s n e c e s s a r y t o u s e t h e p a r t i a l pressure of the air. A s p e c i a l c a s e -- t h e binary similarity law p~d = c o n s t -- i s c o n f i r m e d e x p e r i m e n t a l l y f o r a i r i n [10, 1 8 ] . T h i s l a w c a n b e a p p l i e d t o t h e wake b e h i n d b o d i e s l a r g e r t h a n t h o s e u s e d i n a e r o b a l l i s t i c e x p e r i m e n t s a n d m o v i n g w i t h t h e same v e l o c i t y if binary similarity h o l d s f o r OiH" Assumi n g t h a t i n t h e n e a r wake t h e f l o w i s c h e m i c a l l y f r o z e n [ 1 ] , a n d u s i n g t h e r e s u l t s o f [19] f o r s h o r t b l u n t b o d i e s , we f i n d t h a t t h i s l a w f o r t h e e l e c t r o n concentration is satisfied i n t h e r a n g e V~ < 7 k m / s e c , p ~ d < 5 0 0 (mm Hg),mm. For the distribution of atomic oxygen, which determines the radiation intensity, the similarity criteria i n t h e f a r wake a r e p~d/a~ = const, M = const. The m e a s u r e m e n t s o f [16] c o n f i r m t h e s i m i l a r i t y of the distribution of the radiation intensity i n a c c o r d a n c e w i t h t h e p a r a m e t e r p~d w i t h c o n s e r v a t i o n o f t h e T~, ~.~, V v a l u e s . 3. I n F i g s . 1 - 4 , we c o m p a r e t h e o b t a i n e d d i s t r i b u t i o n s of the parameters in the far turbulent wake b e h i n d a n o n a b l a t i n g sphere moving in air with the experimental results. F i g u r e 1 shows t h e c a l c u l a t e d distribution n e ~ ( X ) f o r M = 12, p~d = 212 (mm Hg).mm, a n d t e m p e r a t u r e s T : 1500~ ( c u r v e 1) a n d T = 1000~K ( c u r v e 2) e s t i m a t e d on t h e b a s i s o f t h e data of [13]. Curve 3 corresponds to the measurements of [14]. F i g u r e 2 shows t h e t h e o retical (T = 1500~ and experimental [10] d i s t r i b u t i o n n e 0 ( X ) f o r p~d = 375 (mm Hg)-mm, V~ = 5 . 5 k m / s e c . The c a l c u l a t e d v a l u e s o f N~D~ a r e g i v e n i n F i g . 3 f o r M = 1 8 . 9 , p ~ d = 375 (mm Hg).mm, x H = 100, T = 1500~ n~0 = 7 . 7 - 1 0 1 1 , c m - 3 ( c u r v e 1 ) . The e x p e r i m e n t a l curve 2 is taken from [18]. It can be seen that the calculated and experimental values a g r e e w e l l e x c e p t f o r t h e r e g i o n x > 2 0 0 0 , w h e r e t h e f o r m a t i o n o f n e g a t i v e i o n s [6] h a s a significant influence on t h e d e c r e a s e i n t h e n u m b e r of e l e c t r o n s . F i g u r e 4 shows t h e axial distribution of the intensity of chemiluminescent radiation in the interval of w a v e l e n g t h ~1 = 0 . 6 a n d t 2 = 0 . 7 ~m f o r M = 1 8 , p ~ d = 375 (mm Hg) omm. C u r v e 1 i s o b t a i n e d by a calculation i n a c c o r d a n c e w i t h ( 2 . 8 ) a t T = 1500~ We a l s o g i v e t h e c a l c u l a t e d curve 2 and the experimental data (hatched region) from [12]. The r e s u l t s of the calculations in accordance with (2.8) are closer to the experimental data than the results obtained in [12]. It follows from comparison with the results o f t h e m e a s u r e m e n t s , some o f which a r e shown i n F i g s . 1 - 4 , that.the proposed theory is applicable for describing t h e f l o w i n t h e wake a n d t h e r a d i a t i o n i n t h e p a r a m e t e r r a n g e s 100 ~ x < 2 0 0 0 , V~ < 7 . 103 m / s e c , p ~ d < 500 (mm Hg).mm. O u t s i d e t h i s r e g i o n , i t i s n e c e s s a r y t o make t h e m o d e l of the chemical kinetics m o r e a c c u r a t e a n d t a k e i n t o a c c o u n t t h e c h a n g e i n t h e wake t e m perature. We a r e v e r y g r a t e f u l t o G. Yu. S t e p a n o v , discussing the results of the work.

V. P .

Shkadova,

a n d K. S. K h o r o s h k o f o r

LITERATURE CITED 1. K. S. K h o r o s h k o , " H y p e r s o n i c wake b e h i n d b l u n t b o d i e s , " I z v . A k a d . Nauk SSSR, Mekh. Z h i d k . G a z a , No. 2 ( 1 9 6 9 ) . 2. L. I . S k u r i n , " N u m e r i c a l i n v e s t i g a t i o n of the parameters of the far turbulent wake i n a compressible fluid," I z v . A k a d . Nauk SSSR, Mekh. Z h i d k . G a z a , No. 1 ( 1 9 7 4 ) . 3. L. I . S k u r i n , " A s y m p t o t i c b e h a v i o r o f s r e a c t i n g f a r w a k e , " i n : H e a t a n d Mass T r a n s T e r , V o l . 1, P a r t 3 [ i n R u s s i a n ] , M i n s k ( 1 9 7 2 ) . 4. L. I . S k u r i n , " D e t e r m i n a t i o n o f t h e p a r a m e t e r s o f a f a r t u r b u l e n t w a k e , " I z v . Akad. Nauk SSSR, Mekh. Z h i d k . G a z a , No. 6 ( 1 9 7 2 ) . 5. O. N. S u s l o v a n d G. A. T i r s k i i , "Determination, properties, and calculation of effective ambipolar diffusion coefficients in a laminar multicomponent ionized boundary layer," Zh. P r i k l . Mekh. T e k h . F i z . , No. 4 ( 1 9 7 0 ) . 6 . E. A. S u t t o n , " C h e m i s t r y o f e l e c t r o n s in pure-air h y p e r s o n i c w a k e s , " AIAA J . , 6 , No. 10 (1968). 7. F . L. F e r n a n d e z a n d E. S. L e v i n s k y , " A i r i o n i z a t i o n i n t h e h y p e r s o n i c l a m i n a r wake o f s h a r p c o n e s , " AIAA J . , 2 , No. 10 ( 1 9 6 4 ) . 8. L. N. W i l s o n , " F a r wake b e h a v i o r o f h y p e r s o n i c s p h e r e s , " AIAA J . , 5 , No. 7 ( 1 9 6 7 ) . 9. C. L a h a y e , " V e l o c i t y d i s t r i b u t i o n s o f s p h e r e s w a k e , " C a n . J . P h y s . , 5 2 , No. 12 ( 1 9 7 4 ) . 10. I . P . F r e n c h , T. E. A r n o l d , a n d R. A. H a y a m i , " I o n d i s t r i b u t i o n s in nitrogen and air wakes behind hypersonic spheres," AIAA P a p e r , No. 87 ( 1 9 7 0 ) . 11. A. D e m e t r i a d e s , " M e a n - f l o w m e a s u r e m e n t s i n an a x i s y m m e t r i c c o m p r e s s i b l e t u r b u l e n t wake,"

932

AIAA J., 6, No. 3 (1968). 12. M. Steinberg, K.-S. Wen, T. Chen, and C. C. Yang, "Ballistic range and theoretical studies of chemiliminescent processes in hypersonic turbulent wake flows," AIAA Paper, No. 729 (1970). 13. H. Mach, "Spectroskopische Untersuchungen an Nachleuf von ablatierenden Modellen yon HyperschallflugkSrpern," Raumfahrtforschung, 18, 1 (1974). 14. N. N. B a u l i n , A. K. D m i t r i e v , N. N. I v a n c h i n o v - M a r i n s k i i , V. E. L o p a t i n , N. N. P i l y u g i n , a n d S. Yu. C h e r n y a v s k i i , "Investigation of flow behind a sphere in supersonic flight i n a i r by an open m i c r o w a v e r e s o n a t o r , " I z v . A k a d . Nauk SSSR, Mekh. Z h i d k . G a z a , No. 3 ( 1 9 7 6 ) . 15. S. Yu. C h e r n y a v s k i i , "Investigation o f t h e s i z e o f t h e wake a n d t h e r a d i a l d i s t r i b u tion of the gas velocity i n i t b e h i n d a h y p e r s o n i c b l u n t b o d y , " I z v . A k a d . Nauk SSSR, ~ e k h . Z h i d k . G a z a , No. 3 ( 1 9 7 6 ) . 16. V. H. R e i s , " C h e m i l u m i n e s c e n t r a d i a t i o n f r o m t h e f a r wake o f h y p e r s o n i c s p h e r e s , " AIAA J . , 5 , No. 11 ( 1 9 6 7 ) . 17. N. N. B a u l i n , A. K. D m i t r i e v , S. "E. Z a g i k , V. E. L o p a t i n , V. A. L y u t o m s k i i , N. N. Pilyugin, a n d S. Yu. C h e r n y a v s k i i , "Investigation into the radio physical parameters of ionized gas by the methods of radio diagnosis," in: Proc. Second All-Union Conf e r e n c e on t h e M e t h o d s o f A e r o p h y s i c a l Investigations, Novosibirsk, 1979 [ i n R u s s i a n ] , Novosibirsk (1979). 18. R. A. H a y a m i a n d R. I . P r i m i c h , "Wake e l e c t r o n density measurements behind hypersonic s p h e r e s a n d c o n e s , " AGARD C o n f . P r o c . , No. 19 ( 1 9 6 7 ) . 19. N o n e q u i l i b r i u m P h y s i c o c h e m i c a l P r o c e s s e s in Aerodynamics [in Russian], Mashinostroenie, Hoscow ( 1 9 7 2 ) .

I~ACT

OF A FIGURE OF REVOLUTION

INCO~PRESSIBLE

IN A STREAM OF

FLUID WITH SEPARATION OF A JET

UDC 5 3 2 . 5 . 0 1 1

D. G. Shimkovich

The impact interaction of bodies with a fluid in a flow with jet separation has been considered in [i-3], for example. This investigation w a s i n the two-dimensional formulation. The present paper considers the three-dimensional p r o b l e m of impact of a figure of revolution in a stream of an ideal incompressible fluid with separation of a jet in accordance with Kirchhoff's scheme. A boundary-value problem is formulated for the impact flow potential and solved by the Green's function method. A method for constructing the Green's function is described. Expressions are given for the coefficients of the apparent masses. The results are given of computer calculations of these coefficients in the case of a cone using the flow geometry of the corresponding two-dimensional problem.

I. We consider a fixed figure of revolution in an unbounded stream of an ideal incompressible fluid with separation of a jet (Kirchhoff's scheme), the stream having velocity V 0 at infinity (Fig. i). Suppose that after the impact the velocities of the points of the body become equal to V(M). We assume that directly after the impact the surface E = E I + E 2 has not changed (E l is the wetted part of the surface, and E 2 is the free surface). It follows from [4] that the flow due to the impact of the body on the incompressible fluid will have potential ~, which is related to the impulse pressure p arising in the fluid by p = --p~, where p is the density of the fluid. Since plz2=O on the free surface, @ix~=0 as well. condition on El, this leads to a m i x e d boundary-value

6,

pp.

Moscow. Translated from Izvestiya 176-180, November-December, 1980.

0015-4628/80/1506-0933507.50

In conjunction with the no-flow problem for the Laplace equation:

A k a d e m i i Nauk SSSR, M e k h a n i k a Z h i d k o s t i i Gaza, Original article s u b m i t t e d J u n e 11, 1 9 7 9 .

9 1981 Plenum Publishing Corporation

No.

933