supersonic flow blunt body angle of attack

7/30/2019 Supersonic Flow Blunt Body Angle of Attack

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High Temperature, Vol. 38, No . 3, 200 0, pp. 444--452. Translated ro m Teplofizika Vysokikh Temperatur, Vol. 38, No . 3, 20 00, pp. 468--476.Original Russian Text Copyright 9 2000 by Borodin. Peigin.

H E A T A N D M A S S T R A N S F E RA N D P H Y S I C A L G A S D Y N A M I C S

N um erica l Invest igat ion of Supersonic Flow p ast Blun t Bod iesof Intricate Sh ape at an An gle of A ttack and Sl ip Ang leA . I . B o r o d i n a n d S . V . P e i g i n

R e s e a r c h I n s t it u t e o f A p p l i e d M a t h e m a t i c s a n d M e c h a n i cs , T o m s k, R u s s i aReceivedApril 15, 1999

A bs tra ct- -A supersonic f low of viscous homogeneous gas past blunt bodies of intr icate shape at an angle o fattack and slip angle is investigated numerically within the model o f complete three-dimensional viscous shocklayer using the time relaxation method. The main regularities are studied of the general structure of flow and ofthe distribution o f pressure and heat flux along the surface. An analysis is performed o f their dependence o n theshape of the body, angle o f attack and slip angle, Mach and Reynolds num bers, and on other determining param-eters o f the problem. The accuracy and range of validity of a number of approximate approaches to the solutionof the problem are estimated.

I N T R O D U C T I O NN umer ica l inves t iga t ions o f th r ee -d imens iona ls uper s on ic f low s o f v i s cous homogeneous gas in thev ic in i ty o f b lun t bod ies u s ing d i f f e r en t gas dynamicm o d e l s o f f l o w w e r e p r e v i o u s l y p e r f o r m e d i n [ 1 - 9 ] .T h e s o l u ti o n o f e q u a t i o n s o f t h r e e -d i m e n s i o n a l l a m i n a rboun dary l ay e r under cond i t ions o f f low pas t b lun t bod -i e s w i th tw o s ymmet ry p lanes a t ze ro ang le o f a t t ackwas der ived in [1] ; a t an angle of a t tack , in [2 , 3] ; andfo r the genera l cas e o f f low a t an an g le o f a t t ack and s l ip

ang le , i n [4 ] . A n inves t iga t ion w i th in the mode l o fth r ee -d im ens iona l hyper s on ic ( th in ) v i s cous s hocklaye r u s ing a num ber o f s imp l i fy ing as s umpt ions o f thed i s t ribu t ion o f long i tud ina l p r es s u re g r ad ien t s w as pe r -fo rme d in [5 , 6 ]. Wi thou t s imp l i fy ing as s umpt ions fo rt h e g e n e r a l c a s e o f t h e a b s e n c e o f s y m m e t r y p l a n e s i nthe f low , th i s p rob le m w as t r ea ted in [7 ] and , bas ed onthe s o lu t ion o f equa t ions o f th r ee -d im ens iona l pa rabo -l i zed v i s cous s h ock l aye r , i n [8 ]. A s o lu t ion o f s imp l i -f i ed N a v i e r - S t o k e s e q u a t io n s f o r f lo w s w i t h t w o s y m -me t ry p lanes w i th p r eas s igned d i s t r ibu t ion o f long i tu -d ina l p r es s u re g r ad ien t s w as de r ived in [9 ] . A f a i r lyde ta i l ed r ev iew i s f oun d in [10 ].F O R M U L A T I O N O F T H E P R O B L E M

W e w i l l t r ea t a th r ee -d imen s iona l s upe r son ic f low o fv i s cous homogeneous gas in the v ic in i ty o f a s moo thb lun t body s ub jec ted to f low a t an ang le o f a t tack ands l ip a ng le . We w i l l i nves t iga te th i s f low in cu rv i l inea rcoo rd ina tes [7] { x /} , in w h ich the coo rd ina te x 3 i s r eck -oned o n a no rm al to the body s u r f ace , and the coo rd i -nate s {x , x 2} a r e s e l ec ted on the bo dy s u r f ace and a r epo la r w i th the o r ig in a t the s t agna t ion po in t ; i n s odo ing , w e w i l l a s s ume , fo r de f in i tenes s s ake , tha t x 1 i sa m arch ing co o rd ina te , and x 2 i s an angu la r coo rd ina te .

W e w i l l u s e t h e m o d e l o f c o m p l e t e v i s c o u s s h o c klaye r [11] a s the inpu t gas dynam ic mode l . A s i s dem on-s t ra ted by the resul ts of the analys is made in [12] , th ism o d e l m a y b e u s e d i n a w i d e r a n g e o f f lo w g e o m e t r ya n d M a c h a n d R e y n o l d s n u m b e r s a n d e n a b l e s o n e t ope r fo rm ca lcu la t ions o f f low pas t bod ies a t modera tes uper s on ic ve loc i ty o f inc iden t f low and fo r the cas esw h en the s hock l aye r i s no t th in in the en t ir e w indw ardpar t o f the body s ub jec ted to f low [ 10 ].D im ens ion les s uns teady - s t a t e equa t ions fo r a th ree -d i m e n s i o n a l c o m p l e t e v i s c o u s s h o c k l a y e r i n t h eabove- iden t i f i ed cu rv i l inea r coo rd ina tes { x i } w i l l bewri t ten as [10]

p ( D u ~ + A j k u u ) = - 4 g ~ a ~ ) g

Ro '

p ( D u 3 3 S k O P+ A j k u u ) = - ~ X 3 '( 1 )

0018-151X/00/3803-0444525.00 9 2000 MAIK "Nauka/Interperiodica"


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N U M E R I C A L I N V E S T I G A T I O N O F S U P E R S O N I C F L O W 4 4 5p = ? - l p T , i t = T O)

a u s a + u 3 ~ 3 , R eD - ~ + - - - -s = p = V . . R~ t ( v ~ / rH e r e , a l l o f t h e l i n e a r d i m e n s i o n s a r e r e l a t e d t o t h e

c h a r a c t e r i s t ic l i n e a r d i m e n s i o n R ; V ~. a n d u i a r e p h y s i -c a l 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 ; p = V ~ P , p . @ ,V~ T/cp~, I t ( V2~/Cp~)l .t ,o , a n d ~ /, r e s p e c t i v e l y , d e n o t e t h e

p r e s s u r e , d e n s i t y , t e m p e r a t u r e , v i s c o s i t y , P r a n d t l n u m -b e r , a n d t h e a d i a b a t i c e x p o n e n t ; T h e L a t i n s u p e r s c r i p t st a k e t h e v a l u e s o f 1 , 2 , 3 , a n d t h e G r e e k o n e s , t h e v a l u e so f 1 , 2 . S u m m a t i o n i s p e r f o r m e d f o r a p a ir o f r e p e a t in gs u p e r s c r i p t s , a n d n o s u m m a t i o n i s p e r f o r m e d f o r t h es u p e r s c r i p t s i n p a r e n t h e s e s . H e r e a n d b e l o w , t h e s u b -s c r i p ts w , ~ , a n d s i n d i c a t e t h e v a l u e s o n t h e b o d y s u r -f a c e , i n t h e i n c i d e n t f l o w , a n d b e h i n d t h e s h o c k w a v e ,r e s p ec t iv e l y . T h e c o v a r i a n t c o m p o n e n t s o f m e t r i c a l t en -so r are gsl~ = a~13 2bs[~x3 + a~'Xb~b~'f~(x3) , g3 s = g s 3 = O ,g3 3 = 1 , an d g = de t ] lgu[I = gl lg2 2 - g l E g 2 1 ; t h e c o m p o -n e n t s o f s y m m e t r i c a l t e n s o r s a s l~ a n d b s l3, r e s p e c t i v e l y ,d e n o t e t h e f ir s t a n d s e c o n d q u a d r a t i c f o r m s o f th e s u r -f a c e ; a n d t h e c o e f f i c i e n t s A ~k a re k n o w n f u n c t io n s o f t h em e t r i c a l t e n s o r c o m p o n e n t s a n d t h e i r d e r iv a t i v e s [ 1 0 ].

T h e s e t o f e q u a t i o n s ( 1 ) i s s o l v e d w i t h t h e b o u n d a r yc o n d i t i o n s o n t h e s h o c k w a v e , o n t h e b o d y s u r f ac e , o nt h e s t a g n a t i o n l i n e , a n d o n t h e o u t l e t b o u n d a r y x 1 =

1x , = c o n s t. T h e b o u n d a r y c o n d i t i o n s o n t h e s h o c kw a v e a r e p r o v i d e d b y t h e u n s t e a d y - s t a t e g e n e r a l i z e dR a n k i n e - H u g o n i o t c o n d i t i o n s , w h i c h m a y b e m o s ts i m p l y f o r m u l a t e d i n c o o r d i n a t e s w h i c h a r e n a t u r a l l yr e la t e d t o th e s h o c k w a v e s u r f a c e a n d h a v e t h e f o r m

p ( V 3 -O - ~ ) = V 3 - c ~ ,( v 3 - ~ ) ( v s - v~ ) = r t a v ~

R e ~ . a n '

P + p ( V 3 - C ~ ) 2 = P o o + ( V 3 o o - ~ ) 2 , (2 )a T = ( V 3 _ O ~ ) [ T _ T o . + ( V 37 ~ ) 2o R e ~ a n

( v ~ - ~ ) ~ ~ s o(~ _ vD ( v~ _ __ _ v b ]

H e r e , ~ ) i s t h e d i m e n s i o n l e s s v e l o c i ty o f t h e s h o c kw a v e f r o n t o n a n o r m a l t o i ts s u r f a c e , a n d n i s th e c o o r -d i n a t e r e c k o n e d a l o n g t h i s n o r m a l .O n t h e a s s u m p t i o n t h a t t h e s t a g n a t i o n l i n e is s t r a ig h ta n d c o i n c i d e s w i t h t h e n o r m a l t o t h e b o d y s u r f a c e a t th e

s t a g n a t io n p o i n t , th e b o u n d a r y c o n d i t i o n s o n t h e b o d ys u r f a c e a n d o n t h e s t a g n a t i o n l i n e h a v e t h e f o r m3 U sx = 0 : = 0 , p u 3 G ( x l ,x 2 ) ,

T = T ( o ( x l , x 2 ) . ( 3 )

l a p l a x 1= 0 : a u 3 l a x I = a T l O x 1 = = O ,I 2U = U = 0 . ( 4 )

T h e u p p e r b o u n d a r y X l - - X1 w a s s e l e c t e d f a r d o w n -s t r e a m o f t h e f lo w , s o t h a t t h e p e r t u r b a t i o n s d u e t o t h i sb o u n d a r y w o u l d n o t h a v e a s i g n i f i c a n t e f f e c t o n t h ef l o w i n t h e r e g i o n b e i n g t r e a t e d . T h e s o - c a l l e d s o f tb o u n d a r y c o n d i t i o n s w e r e p r e a s s i g n e d o n t h i s b o u n d -a r y, f o r w h i c h t h e s e c o n d d e r i v a t iv e s w i t h r e s p e c t to t h em a r c h i n g c o o r d i n a t e x I o f a l l s o u g h t f u n c t i o n s o n t h isb o u n d a r y w e r e z e r o .

N U M E R I C A L S O L U T I O N O F T H E P R O B L E MI n v i e w o f t h e s e l e c te d c o o r d i n a t e s a n d f o r r e s o l v i n gt h e s i n g u l a r i t i e s i n t h e i n p u t e q u a t i o n s a n d b o u n d a r yc o n d i t i o n s o n t h e s t a g n a t i o n l i n e , t h e n u m e r i c a l s o l u -t i o n o f t h e i n i ti a ll y b o u n d a r y p r o b l e m ( 1 ) i n v o l v e d t h et r a n si t io n t o n e w d e p e n d e n t a n d i n d e p e n d e n t v a r ia b l e so f t h e D o r o d n i t s y n t y p e , w h i c h f u r t h e r e n a b l e d o n e t oo b t a i n s u c h p r o f i l e s o f th e s o u g h t f u n c t i o n s i n t h e c r o s ss e c t i o n o f t h e l a y e r t h a t w o u l d b e c l o s e t o s e l f - s i m i l a r

o n e s a n d w e a k l y d e p e n d e n t o n t he m a r c h in g c o o r d i -n a t e ,

s ! X r p & . x 39 : , , r : x , U ,

P,4t-gd x l-tP 2,A : ( - G - l : R -3X

p u 3 / - g = a ~ , a u aa x ' a x 2 - N ; p ' l - g d x 3o

p u S / gq g ( s s )

a v ia ~ 'a y sa X 3 '

a w ~ t( ~x ~, )s_ s ,, & - ~ - ~ c ~ f s

U ------ - -

( 5 )

f s = W s b s - g 4 ~ ,j ( ~ , ) - s 'HIGH TEMPERATURE Vol. 38 No . 3 20 00


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4 4 6 BO R O D I N , P E I G I N~ l O f ~ + ( 2 , 10 1 n A '~ . Of 2A = o 0~ , + ~ - ~ r ) S l + 0 ~

0 1 n A . r 0 1 n A O x3+ - - ~ - J 2 + . ~ 0 - -7 " , a a = O I '~ .

T h e b o u n d a r y c o n d i t i o n s o n t h e s h o c k w a v e a t 4 = 1h a v e t h e f o r mlA g b t m ( O : f a O f ~ O l n g t m ~)

T h e c o n t i n u i t y e q u a t i o n i n th i s c a s e w i l l b e s a t i s f ie di d e n t i c a l l y , a n d t h e r e m a i n i n g e q u a t i o n s ( 1 ) i n n e wv a r i a b l es m a y b e t r a n s f o r m e d t o : < u .

O F l g 0 2 f a p g a l~ ~ - _ g O x ] = g ~ f l O P+ a , J ( ~ ' ) 2 - = p O ~ a

1 O (b ~ Of~)

I A g O T - 3 F + A 2 g ( O x '~2= (u| T -T **~3 - -o , - - 1u . - ~ ) 2 g~ (u - f i~ ) ( f i~ fi~)

2 2b(m/~(l~) J '(7 )

+ ~ r ~~ l ~ 0 ~ a ( ( ~ 1 ) 2 - U b c m ' ~ -~A O ( ~ 0 fc ~ a 3 l~(~-I~b(13)Ofl~b T . ) ~ t , " ( " c a ( ) + 2AallU (~ ' b , , ) 04

+ A ~ s ( ~ l ) 2 + ~ - f l - sbca)bcs)Of~Ofsbr 04 040 gOtl5 ,q /~ X 3

0 [ - ! g 0 T Y - 1 A T l n p ]O T , .2 - a O f a O T l O P ~ 2 -a O f a O P= ~ +~ ~5 - ~ pO~ p o4 o~~

- A ( 1 - Y-Yl(l-lnp))OT~ y - 1 y _T~OAlnp02 aO2 ~,- l g ( ~ ' ) 2 - a - ~ g a ~ 0 4 2 /)4

~ [ A u 3 _ ~.@ glp1 = 3 0 A y - 1 T 0 , fgA~ J u ~ ; v ~ 0 x 'AotBb(a)b(B)O aO Bu ' ( ~ I ) 2 - a 0 f a 0 U 3 3

+~+ ~ ~ + , 4 04 0 4 '3 . 0 X 3 ( ~ l ) 2 - a 0 f a 0 x 3 0 X 3U = - a - ~ ( + ~ ' ( ~ - ' ~ + 0 X '

A ~ I ( 0 x 3 " ) 1 y - 1P : ~t ~) ' P : Y pr.

(6 )

w h e r e

p ~ m 1y M 2~ g g = B 0 x ~ A ~ I A ,- u - t - - &

0xa '~a 2 g 1 2 - s i g 2 2 1 w _ 9c, ~' t o~ ~J 'a l g l 2 - - a 2 g l l ( O X 3 _ 9 O X 3 ]C 2 - - =g t o ~ A ~ j ,

t h e t il d e c o r r e s p o n d s t o t h e p a r a m e t e r s c a l c u l a t e d i n t h ec o o r d i n a t e s r e l a te d t o t h e s h o c k w a v e s u r f a c e, a n d M . .i s t h e M a c h n u m b e r .O n t h e b o d y s u r f a c e a t 4 = 0 , t h e b o u n d a r y c o n d i -t i o n s h a v e t h e f o r m

0 f a = 0 , f a = 0 , T = T w ( ~ 1 , ~ 2 ) . ( 8 )04T h e r e f o r e , t h e i n i ti a l p r o b l e m r e d u c e d t o f i n d i n g th es o l u t i o n o f t h e s e t o f d i f f e r e n t i a l n o n l i n e a r e q u a t i o n sr e l a t i v e t o t h e f u n c t i o n s o f c u r r e n t f a , t e m p e r a t u r e T ,a n d t r a n s v e r s e c o o r d i n a t e x 3 , t h e e q u a t i o n f o r w h i c h i sa c o r o l la r y o f t h e e q u a ti o n o f m o m e n t u m i n p r o je c t io n

o n a n o r m a l t o t h e b o d y s u r f ac e , a n d o f t h e e q u a ti o n s o fc o n t i n u i t y a n d s t a t e .A s t e a d y - st a t e s o lu t i o n o f th i s p r o b l e m w a s f o u n dn u m e r i c a l l y u s i n g t h e t i m e r e l a x a t i o n m e t h o d o n t h eb a s i s o f m o d i f i c a t i o n o f a n i m p l i c i t ( r e l a t i v e t o t h e c o o r -d i n a t e 4 ) s c h e m e [ 13 ] w i t h t h e o r d e r o f a p p r o x i m a t i o n

0 ( 5 4 ) 4 + O ( ~ 2 ) 2 + O ( ~ l ) + O ( ~ '~ ). I n t h e c o n v e c t i o no p e r a t o r , t h e d e r i v a t i v e s w i t h r e s p e c t t o t i m e x a n dm a r c h i n g c o o r d i n a t e ~ 1 w e r e a p p r o x i m a t e d b y f i n it eu n i l a t e r a l b a c k w a r d d i f f e r e n c e s , a n d o n t h e c i r c u m f e r -e n t i a l c o o r d i n a t e g 2 , b y c e n t r a l d i f f e r e n c e s c a l c u l a t e do n t h e b a s i s o f p r e v i o u s g l o b a l i t e r a t i o n o n t h e r u n n i n g1 2c i r c u m f e r e n c e ~ + ~ i~ = c o n s t .H I G H T E M P E RA T U RE V o l .38 No. 3 20 00


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NUM ERICAL INVESTIGATION OF SUPERSONIC FLOW 447T he de r iva t i ve s o f p r e s sur e a nd c oo r d ina t e x 3 a longthe m a r c h ing a nd c i r c um f e r e n t i a l d i r e c t i ons ( r e qu i r e dto de t e r m ine t he so - c a l l e d " e ll i p t i c " t e r m s i n t he se t o fe qua t ions ( 6 ) ) we r e c a l c u l a t e d on t he ba s i s o f c e n t r a ld i f f e r e nc e s w i th the a c c ur a c y o f o ( ~ l ) 2 + O ( ~ 2 ) 2 a n ds to r e d f o r t he e n t i r e f l ow f i eld . I n so d o ing , i n o r de r t oinc r e a se t he s t a b i l i ty o f t he c a l c u l a t i on r e su l t s , t he t huso b t a i n e d d i s tr i b u ti o n s w e r e s m o o t h e d i n t h e n e i g h b o r -h o o d o f t h e s t a g n a t i o n p o i n t w i t h d u e r e g a r d f o r t h eb e h a v i o r o f t h e s e f u n c t i o n s in t h e v i c i n it y o f t h e s t ag n a -t ion l ine .I n v i e w of t he non l ine a r i t y o f t he i n i t i a l p r ob l e m ,i t e r a t i ons we r e e m ploye d , na m e ly , l oc a l i t e r a t i ons i nd e r i v i n g a s o l u ti o n b e t w e e n t h e b o d y s u r f a c e a n d t h es h o c k w a v e a t t h e r u n n i n g p o i n t ~ l = c o n s t l a n d ~2 =c o n s t2 a nd g loba l i t e r a t ions i n d e r iv ing a s o lu t i on i n t hes h o c k l a y e r o n t h e r u n n i n g c i r c u m f e r e n c e E1 = c o n s h .Af t e r c om ple t i on o f e a c h g loba l i t e ra t i on , the va lue o fA( ~ l , ~2) wa s de t e r m in e d on t he r unn ing c i r c u m f e r e nc eus ing t he c yc l i c swe e p a lgor i t hm [ 14] , a nd t he t r a ns i -

t i on wa s m a de t o t he ne x t g loba l i t e r a t i on on t h i s c i r -c u m f e r e n c e . I n o r d e r t o i m p r o v e t h e a c c u r a c y o f c a l cu -l a ti o n s , u s e w a s m a d e o f a n o n u n i f o r m ( o n th e c o o r d i -na t e ~ ) d i ff e r e nc e ne t whic h e na b l e d one t o i de n t i f y i nthe so lu t i on t he bound a r y l a ye r a t t he bod y sur f a c e . Ani t e r a t i on a lgor i t hm a na logous t o t ha t o f [ 15] wa s use dto c ons t r uc t t h i s ne t .T h e s o l u t i o n w a s a s s u m e d t o b e c o n v e r g i n g i f t h em a x i m u m d i f f e r e n c e b e t w e e n a l l s o u g h t f u n c t i o n s i na l l nod e s o f t he d i f f e r e nc e ne t in t he r unn ing a nd p r e v i -o u s t i m e s t e p s d i d n o t e x c e e d 10-2~'LA n i m p o r t a n t f e a tu r e o f t h e s u g g e s t e d n u m e r i c a l

m e tho d i s t ha t i t doe s no t r e qu i r e f o r i t s re a l i z a t ion t hep r e s e n c e o f s y m m e t r y p l a n e s i n t h e f l o w ( an d , i n v i e wo f t h i s, e n a b l e s o n e t o p e r f o r m c a l c u l a t io n s f o r t h e m o s tge ne r a l c a se o f fl ow pa s t bod i e s a t a n a t t a c k a ng l e a nds l i p a ng l e ) a nd t ha t , i n c a l c u l a t i ng t he c oe f f i c i e n t s o ff r i c t i on a nd he a t t r a ns f e r on t he body sur f a c e , t he r e i sno ne e d t o num e r i c a l l y d i f f e r e n t i a t e t he ob t a ine d p r o-f il e s o f ve loc i t y a nd t e m pe r a tu r e i n t he c r oss se c t i on o fthe sh oc k l a ye r .A l o n g w i t h t h e c a s e w h e n t h e b o d y s u r fa c e w a s p r e -a s s igne d i n a n a na ly t i c a l f o r m , a c a se o f p r a c t i c a li m p o r t a n c e w a s t r e a te d i n w h i c h t h e b o d y s u r fa c e w a spr e a ss igne d i n a t a bu l a r f o r m .T h e f o l l o w i n g a p p r o a c h w a s u s e d t o s i m u l a t e th el a t t e r c a se . Assu m e tha t t he v a lue s o f t he r a d ius ve c to ro f t h e b o d y s u r fa c e a r e k n o w n o n l y a t t h e p o i n t s o f th e

d i f f e re n c e n e t rg(X I , x~ ) . Be c a u se , i n r e a l i ty , t he va lue so f r g ( X l , x ~ ) a r e k n o w n w i t h s o m e r a n d o m e r r o r , w ec a n wr i t e

r g ( x l , x ~ ) = r a (X l , X ~ ) + A ( 2 S - 1),w h e r e rg (XJ , x~ ) i s the e xa c t va lue o f t he r a d ius ve c to rr ( x l , x 2 ) o f t he bod y sur f a c e a t t he po in t o f n e t ( x~ , x~ ) ,

S i s a r a nd om n um be r f r om th e i n t e rva l [ 0 , 1 ], a nd A i st he c ha r a c t e r is t i c va lue o f t he e r r o r a m pl i t ude .I n o r de r t o c a l c u l a t e t he c oe f f i c i e n t s i n t he e qua -t i o n s, o n e n e e d s t o k n o w t h e v a l u e o f th e f u n c t i o n r a n di t s par t ia l der iva t ives Om+nr/Omxl~nX up to t he t h i r dor de r i nc lus ive ( m + n < 3 ) .L e t F( i , j ) = O m+n rJOmx~On X2 (X~ , X~) . T h e n , f o rsm ooth ing t he se f unc t i ons , t he f o l l owing two- s t e p i te r -a t i on p r oc e dur e i s use d , ba se d on c ub i c B- sp l i ne s [ 16]( k is t he i t e r a t ion n um be r ) .

S t e p 1T he pe r iod ic i ty o f t he f unc t ions be ing sm o othe d on t hec i rcumferent ia l coordina te ~2 ( subscr ipt j ) ( F ( i , J + 1) =F ( i , 1)) f o r a ll 1 < i < I a nd 1 < j < J wa s use d t o sm o othF ( i , j ) on the c o or d ina t e x 2 a s f o l l ows :

F k + l ( i , j ) = F k ( i , j )1 .+ ~ ( F k ( t , j - 1) - 2 E l ( i , j ) + F ~ ( i , j + 1) ) .

T h e i te r a ti o n s w e r e t e r m i n a t e d w h e n t h e n u m b e r o fc h a n g e s o f s i g n o f t h e s e c o n d d e r i v at iv e o f t h e f u n c t i o nF on t he c oo r d ina t e x 2 c e a se d t o va r y .

S t e p 2T h e b o u n d a r y c o n d i t i o n s a t th e s t a g n a t io n p o i n t

F ( 1 , j ) = 0 ( f o r m = 0 a nd n # 0 ) ,F ( 1 , j ) = 0 , F ( O , j ) = - F ( 2 , j ) ( f o r m = 1 , 3) ,

F ( 0 , j ) = F ( 2 , j ) ( f o r m = 0 , 2 )a n d t h e c o n d i t i o n o f s m o o t h n e s s a t th e r i g h t - h a n d e n do f F ( I + 1, j) = 2 F ( L j ) - F ( I - 1 , j ) for al l 1 < i < I and1 < j < J w e r e u s e d t o s m o o t h t h e f u n c ti o n F ( i , j ) o n t h em a r c h i n g c o o r d i n a t e a s f o l lo w s :

Fk+ l (i , j )1= F k ( i , j ) + ~ ( F k ( i - 1 , j ) - 2 F k ( i , j ) + F k ( i + 1, j ) ) .

Sim i l a r ly t o S t e p 1 , t he i t e r a t i ons we r e t e r m ina t e dw h e n t h e n u m b e r o f c h a n g e s o f s i g n o f t h e s e c o n dde r iva t i ve o f t he f unc t i on F on t he c oo r d ina t e x I c e a se dto vary.T h i s p r oc e dur e wa s f i r s t a pp l i e d t o t he f unc t i on

srg(X , x 2 ), a f te r w h i c h t h e s m o o t h e d f u n c t i o n rg (x l , x 2)w a s d e t e r m i n e d , a n d c e n t ra l d i f f er e n c e s w e r e u s e d t o

a p p r o x i m a t e $ )e t e r m i n e t h e v a l u e s o f O r g / O x a ( x l , x ~w h i c h w e r e s m o o t h e d b y t h e f o r e g o i n g p ro c e d u r e . T h es e c o n d - a n d t h i r d - o r d e r d e r i v a t i v e s w e r e s m o o t h e da n a l o g o u s l y .

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448 BOROD IN, PEIG INp 0 , p o , x1.00 . 9 :0 .80. 70. 60. 50. 40. 30. 20.1 2

10 20 30 40 50 60 70 80 909 , degF i g . 1 .

D I S C U S S I O N O F T H E R E S U L T SW e t r e a t e d a f l o w p a s t a t r i a x i a l e l li p s o i d a t a n a n g l eo f a t t a c k t t a n d s l ip a n g l e ~ , w h o s e e q u a t i o n i n C a r t e -s i a n c o o r d i n a t e s { z ' } h a d t h e f o r m ( z l / a - 1)2 + ( z2 / b ) 2 +( z3 / c ) 2 = 1 . T h e d e t e r m i n i n g p a r a m e t e r s o f t h e p r o b l e ma s s u m e d t h e f o l l o w i n g v a l u e s :to = 0 .5 , 0 = 0 .71 , a = 1 , T = 1 .2 -1 .4 ,

T w = 0 . 0 5 - 0 . 3 , M ** = 4 - 1 5 ,R e = 1 02 -1 06 , 0 . 3 < b < 3 , 0 . 3 < c < 3 ,

0 ~ ~ 0 ~ ~T h e p r o c e s s o f s o l u t i o n i n v o l v e d d e t e r m i n i n g t h ec o m p o n e n t s o f t h e v e l o c it y v e c t o r a n d t h e p r e s s u r e a n dt e m p e r a t u r e i n t h e c r o s s s e c t i o n o f t h e s h o c k l a y e r, a s

w e l l a s t h e a b s o l u t e , P w a n d C q , a n d r e l a t i v e , p O a n dC~q, d i s t r ib u t i o n s o f p r e s s u r e a n d h e a t f l u x a l o n g t h es u r f a c e o f t h e b o d y s u b j e c t e d t o f l o w ( r e l a t e d t o t h e i rv a l u e s a t t h e s t a g n a t i o n p o i n t ) .

T h e n u m e r i c a l a l g o r i t h m w a s t e s t e d b y c o m p a r i n gt h e c a l c u l a t i o n r e s u l t s w i t h t h e d a t a o f [ 1 7 ] o n t h e s h a p eo f t h e s h o c k w a v e a n d o n t h e d i s t ri b u ti o n o f p r e s s u r ea l o n g t h e s u r f a c e o f t h e b o d y a n d s h o c k w a v e . S o m er e s u lt s o f c o m p a r i s o n a r e g i v e n i n F i g . 1 , w h e r e c u r v e s1 , 2, a n d 3 i n d i c a te t h e d e p e n d e n c e o n t h e c e n t r a l a n g l e( t h e s p h e r i c a l c o o r d i n a t e s w e r e d e f i n e d i n [ 1 7 ] ) o f t h ed i s tr i b u ti o n o f p r e s s u r e a l o n g t h e s h o c k w a v e a n d a l o n gt h e b o d y s u r f a c e, a s w e l l a s t h e s h o c k w a v e d e p a r t u r e .H e re , a = l , b = c = 0 . 5 , R e = 1 0 S , M * * = 6 , T w = 0 . 1 ,T = 1 .4 , a n d t~ = 0 ; t h e c u r v e s i n d i c a t e t h e r e s u l t s o f o u rc a l c u l a t i o n s , a n d t h e p o i n t s i n d i c a t e t h e d a t a o f [ 1 7 ].

I n a d d i t i o n , a c o m p a r i s o n w a s m a d e b e t w e e n t h er e s u lt s o b t a i n e d i n t h e s e q u e n c e o f n e s t e d n e t s H M N rf o r d i f f e r e n t s t e p s w i t h r e s p e c t t o t i me ( M, N , a n d Kd e n o t e t h e n u m b e r o f p o i n t s o n t h e c o o r d i n a t e s ~ l , ~ 2,a n d 4 , r e s p e c t iv e l y ) . S o m e c o m p a r i s o n r e s u l ts a r e g i v e ni n T a b l e 1 , w h i c h c o n t a i n s t h e v a l u e s o f t h e h e a t - tr a n s -f e r c o e f f i c i e n t C q, o f t h e p r e ss u r e o n t h e s u r f a c e P ~ , a n do f t h e s h o c k w a v e d e p a r t u r e x ~ a t t h e s t a g n a t io n p o i n ta n d a t o n e o f t h e p o i n t s o f e l l i p s o i d m i d s e c t i o n a tR e = 1 0 4 , o~ = O , b = O . 7 , c = O .3 , a n d T w = O . 1 5 .

A n ana lys i s o f the resu l t s reve a l s tha t an H20 32 sn e t i s s u f f i c i e n t t o d e r i v e a s o l u t i o n w i t h a n a c c u r a c y o fu p t o 1 % , a n d K = 5 i s s u f f i c i e n t f o r mo d e r a t e v a l u e s o fR e . T h e n u m b e r o f s t e ps w i t h r e s p e c t t o ti m e r e q u i r e dt o d e r i v e a c o n v e r g i n g s o l u t i o n a t R e = l 0 s w a s1 3 0 - 1 8 0 o n t h e a v e r a g e , w h i c h i s s o m e w h a t w o r s e th a nt h e r e s p e c t i v e r e s u lt s o b t a i n e d u s i n g a s c h e m e o f a h i g ho r d e r o f a c c u r a c y f o r t w o - d i m e n s i o n a l f l ow s [ 1 5 ], b u tm u c h l e ss t h a n t h e n u m b e r o f i te r a t io n s i n t i m e t h a t i sr e q u i r e d t o d e r i v e a s t e a d y - s t a t e s o l u t i o n u s i n g n u me r -i c a l m e t h o d s o f t h e s e c o n d o r d e r o f a p p r o x i m a t i o n o nt h e t r a n s v e r s e c o o r d i n a t e [ 1 8, 1 9 ].

W e w i l l d w e l l o n s o me a n a l y t i c a l re s u l t s . N o t e , f i r s to f a l l , th a t t h e a b s o l u t e v a l u e s o f t h e c o e f f i c i e n t s o f f r i c -

T a b l e 1Variants

Q u a n t i t i e s H 2 0 x 3 2 x 8 ~r 3 2 x 8 ~ 2 0 _ x 3 2 x 8 ~ .4 0 3 2 x 8 H 4 0 x 6 4 x S H 4 0 x 3 2 x 6= 0.1 0.0 5 0.0 1 _____x.05 8r = 0.02 5 8r = 0.05Cq(O, O)P ~ ( O , O )x~ (0, O)Cq(1, O)Pro(l, 0)3x~ (1, 0 )

0.045920.96020.033660.0022200.031640.3132

0.045920.96020.033660.0021720.030850.31477

0.045930.96010.033660.0020840.028980.3162

0.046300.96020.033390.0023800.033270.3087

0.046340.96020.033390.0023610.032840.3112

0.046320.96020.033390.0024930.033010.3087

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7/9

450 BOR ODIN, PEIGIN

1.0~0.90.80 .70.60.50 .40 .30.20.1

0 I I | I I I I | | |0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig . 3 .

I t was previou sly obse rved [7 , 8] that , und er cond i-t i ons o f hyper son ic f l ow o f v i scous hom ogene ous gaspas t smo o th b lun t bod ies wi th in the f r amew ork o f moreapprox imate mode l s o f t h in and pa r abo l i zed v i scousshock l aye r (TVSL and PV SL mode l s ) , the r e l a tive d i s -t r ibution of h eat f lux C~q in the neigh borho od of b lunt -ness o f the body is largely conservat ive wi th respect tothe Reyno lds number and the quan t i t y T w ( for thecoo led su r face o f t he body ) .

The r e su lt s o f ca l cu la t ions pe r fo rmed by us us ingthe mo de l o f comple t e v i scous shock l aye r have dem -onst ra ted that the qu ant i ty C~q is inde ed ra ther we aklydepen den t on the above- iden t if i ed pa r amete rs and ,therefore , the inference abou t the conse rvat ism of C~q( including that wi th respect to the var ia t ion of the M achnumber ) i s va l id wi th in the mode l o f v i scous shocklayer as wel l , th is being t rue o f the ent i re f ronta l sur faceo f t he body up to i t s m idsec t ion .

The r e su l t s o f a sympto t i c ana lys i s o f t he Nav ie r -Stokes equat ion s [20, 21] indicate that wi th in the TVSLand PV SL mod e l s , because o f t he pa r abo l ic behav io r o fthe respect ive se ts of equat ions, the solu t ion of thep rob lem in the ne ighborh ood o f the s t agna t ion li ne maybe found i r re spec t ive o f t he so lu t ion in t he ne ighbor ingreg ions . I n so do ing , t he on ly geom et r i c pa r amete r o f

simi lar i ty of f low in th is region is the ra t io o f the radi iof the pr incipal curvatures of the body sur face a t thes t agna tion po in t. I n o rde r t o che ck the accu racy o f t h i sapproxim at ion, def ine the range of i t s val id i ty , and est i -ma te t he deg ree to w h ich the shape o f t he b ody a f fec t sthe so lu t ion ups tr eam o f t he f low, we h ave pe r fo rme dcalcula t ions of f low, a t zero angle o f a t tack and zero s l ipangle , past a family of e l l ipsoids of d i f ferent shapesp reass igned by the cond i t i on b = Cl ,~ and c = C2 4~ .On e can readi ly see that, b ecause the radi i of the pr in-cipal curvatures of the e l l ipsoid sur face a t the s tagna-t ion point , Rl and RE, are de term ined by the form ulasR 1 = b2 / c a n d R2 - c2/a, al l e l l ipsoids of th is fam ily havethe s am e valu es of R l = C~ and R 2 = C22 at the stagna-t ion point .

An ana lys i s o f numer i ca l so lu t ions has r evea l edthat, for a l l t reated ranges o f var ia tion of the R eyno ldsand M ach number s , t h is app rox imat ion i s cha r ac t e ri zedby a fa i r y goo d accuracy. On e can assume that the f lowin the ne ighbo rhood o f b lun tness o f t he body i s i ndeedlocal ly se l f - s imi lar . For a l l t reated e l l ipsoids of thegiven family ( the longi tudinal axis a var ied f rom 0.25to 4 .0 ), t he ex t en t o f depar tu re o f t he shock w ave andthe p ro f il e s o f ve loc i ty and t empera tu r e o f t he com po-nen t s i n t he c ross sec t ion o f t he l aye r depende d weak lyon the para meter a . As i s s een in Table 2 , the var ia tionof the absolute values o f the heat - t ransfer coef f ic ient C., . . i /a t t he s t agna tion pomt fo r t he above-des cnbed f amdyof sur faces was l ikewise insigni f icant (Re = 103, M~. =l O , T w = O . l , ~ l = 1.2, o~ = 0).

We fu r the r made a compar i son o f t he r e su l t s o fnumer i ca l ca l cu la t ions fo r d i f f e r en t me thods o f p r eas -signing the body sur face , nam ely , in analyt ical and tab-ular forms. F igures 3 and 4 i l lust ra te exam ples of suchcompar i son fo r t he case o f f l ow pas t an e l l ip so id wi thaxes b = 0.7 an d c = 0.3 with r = 3 0 ~ and ~ = 15 ~ ( thedist r ibut ion of cOo along the marching coordinate ~ l(0 < ~ l < 1), F ig . 3 , and that a lon g the c i rcumferen t ia lcoord inate ~2 (0 < ~2 < 2~), Fig. 4) . H ere, the curve s indi-cate the case of ana lyt ical preassigning of the b ody sur-face , and the po ints indicate the case of tabular preas-s ign ing o f t he su r f ace wi th the r andom e r ro r ampl i tudeA = 0.05; in Fig. 3, l ines 1 a nd 2 indic ate ~2 = 0, 3.14;in F ig . 4 , l ines I and 2 indicate ~ l = 0 .2 , 0 .55. An a nal -ys i s o f t hese r e su l ts has demo ns t r a t ed tha t t he smo o th -ing procedure suggested by us i s fa i r ly ef f ic ient andenab les one to de r ive a num er i ca l so lu t ion to t he p rob -lem wi th an accuracy suf f ic ient for pract ical appl ica-t i ons in a wide r ange o f va r i a ti on o f t he am pl i tude o f

T a b l e 2Ellipsoid axes a= 4 , b= 1 .4, c = 0 .6 a = 1 , b= 0.7 , c = 0 .3 a = O . 2 5 , b = 0 . 3 5 , c = O . 1 5Cq 0.071 0.073 0.076

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NUMER ICAL INVESTIGATION OF SUPERSONIC FLOW 451c ~1.00.9~0.80.70.60.50.40.30.20.1

I I t I t1.26 2.51 3.77 5.02 6.28

c O0.210.180.150.12i:..__ ~ \ 100 9 .i \0.L ~ ~ ~ , ~ 0 3 :5 ' ~ ' ~ "*,,, ~

0 0.2 0.4 0.6 0.8 1

F i g . 4 . F ig . 5 .

r andom e r ro r in p reas s ign ing the r ad ius vec to r o f thebody s u r face .W e h a v e f u r t h er a n a l y z e d t h e e f fe c t o f t h e M a c h a n dReyn o lds num ber s and o f the pa ramete r s co and T w onthe s hape o f s hock w ave depar tu re and on the d i s t r ibu -t ion o f the long i tud ina l and c i r cum feren t i a l p r es s u reg rad ien t s unde r cond i t ions o f f low pas t a bod y o f f ixed

s hape fo r the fo l low ing r ange o f va r i a t ion o f the pa ram -eters :102 < Re < 10 6, 0.5 < CO < 1.0, (9 )0.05 < T w < 0 .4 .

I t has been de mo ns t r a ted tha t , f o r a ll t r ea ted s hapeso f the body , the va r i a tion o f the ab ove- iden t i f i ed pa ram-eters with in the l imits of (9) causes l i t t le var ia t ion ofthes e quan t i ti e s . O n the o the r hand , i t has been dem on-s tra ted in [21] that , i f the shap e of the sho ck w ave andthe d i s t ribu t ion o f the long i tud ina l and c i r cum feren t i a lp res s u re g rad ien t s a re a s s um ed to be know n func t ions ,t h e m a r c h i n g m e t h o d s o f c a l c u la t i o n s m a y b e u s e d t oadvan tage in s o lv ing s t eady - s ta te equa t ions o f v i s couss hock l aye r.

A l l th i s made i t pos s ib le to s ugges t the fo l low ing(cons i s t ing o f tw o s t ages ) app rox imate app roach to theca lcu la t ion o f f low pas t a body o f p reas s igned s hape fo ra w i d e r a n g e o f v a ri a ti o n o f t h e R e a n d M , n u m b e r sand o f the pa ramete r s y and T w . In the firs t s tage, thet ime r e laxa t ion m e thod i s u s ed to f ind a num er ica l s o lu -t ion o f the inpu t uns teady - s ta te equa t ions o f v i s couss hock l aye r fo r a f ixed s et o f pa ramete r s Re* , M * , T* ,and T* ( in ou r case , thes e a r e Re* = 103, M * = 10,

= 1 .2 , and T* = 0 .1 ) and to de te rmine the s hape o fs hock w av e depar tu re and the long i tud ina l and c i r cum -fe ren t i a l p r es s u re g rad ien t s . I n the s econd s t age , fo r anyo the r s e t o f pa ramete r s R e , M . , co, and T w , a so lu t ion tos teady - sta te equa t ions o f v i s cous s hock l ay e r i s foundf rom the r ange o f (9 ) u s ing the march ing (on the coo r -d ina te ~ l ) numer ica l me thod on the a s s umpt ion tha t thes hape o f the s hock w ave and the long i tud ina l and c i r -cumferen t i a l p r es s u re g rad ien t s a r e de te rmined in thef ir s t s tage o f s o lv ing the p rob lem .

F igu re 5 i l lu s t r a te s a compar i s on fo r r e l a t ive hea tf lux be tw ee n the s o lu t ions o f the p rob lem in comple tefo rmula t ion ( cu rves ) and the r e s u lt s o f ca lcu la t ions pe r -fo rmed us ing the s ugges ted app rox imate app roach(po in ts ) . H ere , the fo l low ing pa rame te r s o f the p rob lemw ere va r i ed a s compared w i th the bas ic va r i an t :(1) wi th R e = 102, (2) wi th M ~ = 6 , (3) wi th 7 = 1 .4,(4) with T w = 0 .25 , (5) w i th Re = 105 (curves 2 - 5 indi-cate the values of C~q multiplied by five).

A s demons t r a ted by compar i s on w i th the r e s u l t s o fca lcu la t ions pe r fo rmed fo r exac t fo rmu la t ion o f thep rob lem , the s ugges ted app roach i s cha rac te r i zed by agood a ccu racy and m ay be u s ed to advan tage fo r p rac -t i ca l app l i ca tions . I n add i t ion , the s ugges ted app roachi s econo mica l , becaus e , in the cas e o f pa ramet r i c ca lcu -la t ions o f a f low pas t a body o f p reas s igned ge om et ryfo r a l l va lues o f Re , M ~ , y , and T w f rom the r ange o f (9 ),i t makes s u f f i c i en t a s ing le app l i ca t ion o f the t imere laxa t ion me thod , w h ich , f rom the compu ta t iona ls t andpo in t , r equ i r es cons ide rab ly more compu te rHIGH TEMPERATURE Vol. 38 No. 3 20 00


9/9

4 5 2 B O R O D I N , P E I G I Nr e s o u r c e s f o r i t s re a l iz a t io n t h a n t h e m a r c h i n g n u m e r i -c a l a l g o r i t h m s .

C O N C L U S I O NT h e t i m e r e l a x a t i o n m e t h o d i s u s e d t o p e r f o r m an u m e r i c a l i n v e s t i g a ti o n o f a s u p e r s o n i c f lo w o f v i s c o u s

h o m o g e n e o u s g a s a t a n a n g l e o f a tt a c k a n d s l ip a n g l ep a s t t r i a x i a l e l l i p s o i d s o f d i f f e r e n t s h a p e s . A n e f f i c i e n ts m o o t h i n g p r o c e d u r e i s s u g g e s t e d t h a t e n a b l e s o n e t od e r i v e , w i t h a g o o d a c c u r a c y , a s o l u t i o n t o t h e p r o b l e mf o r a c a s e o f p r a c ti c a l i m p o r t a n c e i n w h i c h t h e s h a p e o ft h e b o d y s u r f a c e i s p r e a s s i g n e d i n a t a b u l a r f o r m . A na n a l y s is i s p e r f o r m e d o f t h e e f f e c t o f th e d e t e r m i n i n gp a r a m e t e rs o f t h e p r o b l e m o n t h e s h a p e o f s h o c k w a v ea n d o n t h e d i s t r i b u t i o n o f t h e c o e f f i c i e n t s o f f r i c t io n a n dh e a t t ra n s fe r . A n a p p r o x i m a t e a p p r o a c h i s s u g g e s t e d t ot h e c a l c u la t i o n o f f l o w p a s t b o d i e s o f p r e a s s i g n e d s h a p ei n a w i d e r a n g e o f v a r i a ti o n o f t h e R e a n d M ** n u m b e r sa n d o f t h e p a r a m e t e r s T a n d T w. T h e a c c u r a c y a n d r a n g eo f v a l i d i t y a r e e s t i m a t e d f o r a n u m b e r o f a p p r o x i m a t ea p p r o a c h e s t o th e s o l u t i o n o f th e p r o b l e m .

A C K N O W L E D G M E N T ST h i s s t u d y r e c e i v e d s u p p o r t f r o m t h e R u s s i a n F o u n -d a t i o n f o r B a s i c R e s e a r c h ( p r o j e c t n o . 9 8 - 0 1 - 0 0 2 9 8 ) .

R E F E R E N C E S1. Bryk ina , I .G . , Ge r shbe in , E .A. , a nd Pe ig in , S .V. , Izv.

A k a d . N a u k S S S R M e k h . Z h i d k. G a z a , 1982, no. 3 , p . 49.2 . And re e v , G.N. , Tr. Inst . Mekh. Mo sk. G os. U niv., 1970,no. 5 , p . 50.3 . Ale ks in , V .A. a nd She ve l e v , Yu .D., l zv . Akad . N auk SS SRMekh . Zh idk . Gaza , 1983, no. 2 , p . 39.4 . Borod in , A. I . a nd Pe ig in , S .V. , Teplofi z . Vys . Temp. ,1987, vol . 25, no. 3 , p . 509.5 . G ershb ein, E .A. , S hche l in , V.S. , and Yuni tski i, S.A. ,N u m e r i c a l a n d A p p r o x i m a t e A n a l y t i c a l S o l u t i o n s o fH y p e r s o n i c T h r e e - D i m e n s i o n a l V i s c o u s S h o c k L a y e rw i t h M o d e r a t e l y S m a l l R e y n o l d s N u m b e r s , i nGiperzvukovye pros t rans t vennye t e chen i ya pr i na l i ch i i

f i z iko -kh imichesk i kh prev ra shchen i i (Hype rson i c T hre e -D i m e n s i o n a l F l o w s i n t h e P r e s e n c e o f P h y s i c o c h e m i c a lT r a n s f o r m a t io n s ) , M o s c o w : M o s k . G o s . U n iv . ( M o s c o wSta te Univ.), 1981, p . 72.6 . Ge rshb e in , E .A. , Krupa , V.G. , a nd Shc h e l i n , V .S ., Prikl .Mat . Mekh . , 1986, vol . 50, no. 1 , p . 102.

7 . Borod in , A. I . a nd Pe ig in , S .V. , I z v . A k a d . N a u k S S S RMekh . Zh idk . Gaza , 1989, no. 2 , p . 150.8 . Borod in , A. I . a nd Pe ig in , S .V. , Teplofi z . Vy s. Temp. ,1993, vol . 31, no. 6 , p . 925 ( H i g h T e m p . (Engl . t ransl . ) ,vol . 31, no. 6 , p . 853) .9 . Ge rshb e in , E .A. a nd Shc he rb a k , V.G. , I z v . A k a d . N a u kSSSR Mekh . Zh idk . Gaza , 1987, no. 4 , p . 134.

10. Pe igin , S.V. and Tirski i , G.A. , I t og i Nauk i Tekh . V INITISer . Mekh . Zh idk . G aza , 1988, vol . 22, p . 62.11. Davis , R.T. , A A I A J . , 1970, vol . 8 , no. 5 , p . 843.12. Ti rski i , G.A. , I zv . Vyssh . Uchebn . Zaved . F i z . , 1993,vol . 36, no. 4 , p . 15.13 . Pe tukhov , I.V ., Nu me r i c a l Ana lys i s o f T w o-D ime ns iona lF l o w s i n B o u n d a r y L a y e r , in Chi s l ennye me tody re sh -en i ya d i f f e ren t s ia l ' nykh i i n t egra l ' nykh uravnen i i ik v a d r a tu r n y e f o r m u l y ( N u m e r i c a l M e t h o d s o f S o l v i n gDi f fe re n t i a l a nd In t e g ra l E qua t ions a nd Q ua dra tu re For -m u l a s ) , M o s c o w : A N S S S R ( U S S R A c a d . S c i . ) , 1 9 6 4 ,p . 305.14 . Abra mov , A.A. a nd Andre e v , V.B . , Zh. Vychisl . Mat .

Mat . F i z . , 1963, vol . 3 , no. 2 , p . 377.15 . Borod in , A. I ., Iva nov , V.A. , a nd Pe ig in , S .V. , Zh. Vychisl .Mat . Mat . F i z , 1996, vol . 36, no. 8 , p . 158.16 . Gre be nn ikov , A. I . , M etod sp la inov d l ya re shen i ya neko-rrek tnykh za dach teori i pribl i zh eni i ( T h e M e t h o d o fS p l i n e s f o r S o l v i n g I l l -D e f i n e d P r o b l e m s o f A p p r o x i m a -t i o n T h e o ry ) , M o s c o w : M o s k . G o s . U n iv . ( M o s c o w S t a t eUniv.), 1983.17 . L yub im ov , A.N. a n d Rusa no v , V.V. , Techenie gaza okolotupykh te l ( G a s F l o w p a s t B l u n t B o d i e s ) , M o s c o w :Na uk a , 1970 , vo l . 2 .18. Anderson, J . , Tannehi l l , J . , and Ple tcher , R. , C o m p u t a -

t i ona l Hydrodynamic s and Hea t Trans f e r , N e w Y o r k :He misphe re , 1984 . T ra ns l a t e d unde r t he t i t l e Vychisli-t e l ' naya g idrod inamika i t ep loobmen , M o s c o w : M i r ,1990, vol . 2 .19 . Kove nya , V.M. a nd Ya ne nko , N.N. , M e t o d r a s s h c h e p -l en i ya v zadachakh gazovo i d inamik i (T he Sp l i t t i ngM e t h o d i n P r o b l e m s i n G a s D y n a m i c s ) , N o v o s i b i r s k :Na uka , 1981 .2 0 . G e r s h b e i n , E . A . , A n A s y m p t o t i c S t u d y i n t o th e P r o b l e mo f H y p e r s o n i c T h r e e- D i m e n s i o n a l F l o w o f V i sc o u s G a spa s t B lun t Bod ie s wi th a Pe rme a b le Sur fa c e , i nGiperzvukovye pros t rans t vennye t e chen i ya pr i na l i ch i if i z iko -kh imichesk i kh prev rashch en i i ( H y p e r s o n i c T h r e e -D i m e n s i o n a l F l o w s i n t h e P r e s e n c e o f P h y s i c o c h e m i c a l

T r a n s f o rm a t i o n s ), M o s c o w : M o s k . G o s . U n i v . ( M o s c o wSta te Univ.), 1981, p . 29.21 . Pe ig in , S .V. , Pa ra bo l i c Vi sc ous S hoc k L a y e r T he ory fo r3 D H y p e r s o n i c G a s F l o w : S h o c k W a v e I , Proc. 19th Int .Symp . on Shock Waves , Marse i l le , 1993, Ber l in :Spring er , 1995, vol . 1 , p . 139.

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supersonic flow blunt body angle of attack

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