Transonic Integration technique for shocks

7/30/2019 Transonic Integration technique for shocks

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Acta Meehanica 32, 141--151 (1979) A C T A M E C H A N I C A| by Springer-Verlag 1979

T h e T r a n s o n i c I n t e g r a l E q u a t i o n M e t h o d W i t h C u r v e d S h o c k W a v e sBy

D. Nixon*, London, EnglandWith 2 Figures

(Received Septemb er 1, 197 7)S u m m a r y m Z u s a m m e n f a s s u n g

T h e T r a n s o n i c I n t e g r a l E q u a t i o n M e t h o d W i t h C u r v e d S h o c k W a v e s . In the existingsolutions of the extended integral equation method for transonic flows the shock wave hasbeen assumed normal to the freestream. In this paper the effect of removing this assumptionis investigated and flows with curved shock waves are examined.D i e t r a n s o n i s c h e I n t e g r a l g l e i c h u n g s m e t h o d e b e i g e k r i i m m t e n Stoflwellen. In den bis-her verSffentlichten LSsungen der erweiterten Integralgleichung fiir transonische StrSmungenwurde ein gerader Stol3 senkrecht zur AnstrSmung angenommen. In der vorliegenden Arbeitwird diese Annahme fallengelassen, und StrSmungen mit gekrfimmten StSI3en werden be-trachtet.

I n t r o d u c t i o nAn alternativ e to t he commo nly used finite difference method s [1] for calcu-

lating transonic flows, namely the extended integral equation method, hasrecently appeared in the literature [2]. In this approach the basic non-linearpotential equation ;[or transonic flow is written in integral form using Green'stheorem and the resulting non-linear integral equation is then solved numerically.If shock waves are present in the flow then the integral equation must be solvedsubject to certain regularity conditions which ensure that the acceleration every-where except across the shock wave is finite and continuous; this effectivelyexcludes "expansion shocks". Because of the use of a particular method ofsatisfying these regul arity conditions during the numerical solution of the integralequati on it is possible to make the simplifying assumption th at an y shock waves inthe flow are normal to the ffeestream. In Ref. [2] the pressure distributio n ar ounda parabolic arc aerofoil is calculated using the extended integral equation methodand is compared to a result computed by the accurate "conservative" finitedifference method of Murman [1]. In this comparison the basic potential equationand the boundary conditions are identical in both methods but in the integralequation meth od the simplifying assumpt ion of a normal (to the ffeestream) shock

* Senior Research Fellow. Presently at Computational Fluid Dynamics Branch NASAAmes Research Center, Moffett Field, CA 94035, U.S.A.

0001-5970/79/0032/0141/$02.20


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142 D. Nixon:w a v e is m a d e ; a f u r t h e r s i m p l i fi c a ti o n is t h a t t h e v e l o c i t y a h e a d o f a n d b e h i n d t h ea e r o fo i l c a n b e n e g l e c te d i n th e c o m p u t a t i o n . T h e p r e s s u r e d i s t r i b u t i o n c a l c u l a t e db y t h e i n te g r a l e q u a t i o n m e t h o d a g re e s f a ir l y w e ll w i t h t h e r e su l t c o m p u t e d b yM u r m a n [1 ] b u t t h e r e a r e d i s cr e p a n c ie s a t t h e l e a d in g e d g e o f t h e a e r o f o i l a n d i nt h e r e g io n o f t h e s h o c k w a v e . I t i s t h e a i m o f th e p r e s e n t e x e r c is e to i n v e s t i g a t e t h ee f f e c t o f r e m o v i n g t h e t w o s i m p l i fy i n g a s s u m p t i o n s m a d e i n R e f . [2] p a r t i c u l a r l yt h e a s s u m p t i o n o f a n o r m a l s h o c k w a v e . T h e m e t h o d p r e s e n t e d i n th i s p a p e r i so n l y v a li d w h e n t h e s h o c k g e o m e t r y d o es n o t d e v i a t e s u b s t a n t i a ll y f r o m b e i n gn o r m a l t o t h e f r e e s t r e a m . T h i s a s s u m p t i o n i s t r u e f o r m o s t t r a n s o n i c f l o w s .

A l t h o u g h t h e e x t e n d e d i n t e g r a l e q u a t i o n m e t h o d h a s b e e n a p p l i e d t o l i f t i n gf l o w s [3 ], o n l y t h e f l o w a r o u n d a n o n - l i f t i n g p a r a b o l i c a r c a e r o f o i l i s e x a m i n e d i nt h i s n o t e , m a i n l y t o s i m p l i f y t h e p r e s e n t a t i o n a l t h o u g h a n a d d i t io n a l r e a s o n i s t h a tf o r t h is p a r t i c u l a r p r o b l e m t h e b a s ic p o t e n t i a l e q u a t i o n a n d t r e a t m e n t o f t h eb o u n d a r y c o n d i ti o n s a r e i d e n t ic a l t o t h a t s o l v e d b y M u r m a n [1 ], a n d t h u s a ne x a c t c o m p a r i s o n i s a v a i la b l e .

I t i s f o u n d t h a t t h e i n c lu s i o n o f a c u r v e d s h o c k w a v e in t h e a n a l y s i s a n d t h ec o n s i d e r a t io n o f v e l o c it ie s a h e a d o f a n d b e h i n d t h e a e r o f o i l i n t h e c o m p u t a t i o nl e a d t o r e s u l ts t h a t a r e i n e x c e ll e n t a g r e e m e n t w i t h t h e r e s u l t o f M u r m a n [1 ]. T h ee f f e c t o f i n t r o d u c i n g a c u r v e d s h o c k w a v e i s o n l y s i g n i f i c a n t i n t h e l o c a l r e g i o nc l o s e t o t h e s h o c k w a v e ; a l o c a l e f f e c t i s a l s o e x i b i t e d b y t h e i n c l u s i o n o f t h ea d d i t i o n a l v e l o ci ti e s a h e a d o f a n d b e h i n d t h e a e r o f o il .

B a s i c E q u a t i o n sT h e b a s ic i n t e g r a l e q u a t i o n f o r t h e t r a n s o n i c f l o w a r o u n d a p a r a b o l i c a r c

a e r o f o il is g i v e n in R e f . [2 ]. F o r a f r e e s t r e a m M a c h n u m b e r M ~ a n d a t r a n s o n i cp a r a m e t e r k t h e i n te g r a l e q u a t io n is

~2(~ , /~~ ( x , z ) ~ - - ' - ~ L ( ~ , ~ ) + I T ( ~ , 5 , ~ ) ( 1 )2w h e r e ( t ( 2 , 2 ) is r e l a t e d to t h e s t r e a m w i s e p e r t u r b a t i o n v e l o c i t y u ( x , z ) b y

a n d

w h e r e

~ ( ~ , ~ ) = ( 7 + 1 ) ~ u ( x , z ) (2 )( 1 - M s1

o

(y~- 1) kZT(X) - - (1 -- M s 3 /~ ZT(X); Z = Z T ( X )

(3 )

d e n o t e ss y s t e m (2 , 5) i s r e l a t e d t o t h e p h y s i c a l s y s t e m (x , z ) b y t h e r e l a t i o n

t h e t h i c k n e s s d i s t r i b u t i o n o f t h e a e r o f o il . T h e c a r t e s i a n c o - o r d i n a t e

~-- x , ~ ----- (1 - - M 2 ) 112 2~


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T h e T r a n s o ni c I n t e g r a l E q u a t i o n M e th o d W i t h C m ~ e d S h o c k W a v e s 1 4 3a n d t h e f ie l d i n t e g r a l I T ( ' 2 , ~ , " 2 , s ) i s d e f i n e d b yI t ( e , ~ , ~ s) 1 l i r a ~ 9= - 4 - ~ ~ - .0 ~ , ~ ( ~ , 2 ; ~ , $ ) ~2 (~ , ~ ) d $ d ~

6--+0

+ f 2 ; ~ , a ~ (; :, d $ d ~

w h e r e 2r = 2% (~.) i s t h e s h o c k l o c a t i o n a n dT ( 2 , ~ ; ~ , $ ) = 1 i n [ ( 2 - - ~ )e ~ _ ( ~ _ $ ) 2 ] .2

T h e s h o c k j u m p r e l a t i o n f o r E q . (1 ) c a n b e w r i t t e n a s[ ~ - - ~ ] + ~ - [ ~ ] + t a n 2 0 s = 0 (5 )

w h e r e [ ]+_ d e n o t e s a j u m p a c r o s s t h e s h o c k w a v e w h i c h l o c a l l y h a s a n i n c l i n a t i o n0~ t o t h e ~ - a x is . I n t h e d e r i v a t i o n o f E q . ( 5) t h e r e l a t i o n

[ e ] +_ = t a n Osi s u s e d w h e r e N ( 2 , 2) i s r e l a t e d t o th e p e r t u r b a t i o n v e l o c i t y n o r m a l t o t h e f r e e -s t r e a m , w( x , z ) , b y

~ (2 , ~) - - (~ + 1 ) k(1 - - - :M~-s w (~, z ) .I f E q . (1 ) i s r e a r r a n g e d s u c h t h a t

~ ( ~ , ~ ) se e ~ 0~ ~ ( ~ ' ~ ) - - ~ T L( ~ , ~ ) + { I T ( 2 , ~ , 2~s) + t a n 2 0~ ~ ( ~ , ~)} ( 6 )2t h e n t h e r i g h t h a n d s i de o f E q . ( 6) i s c o n t i n u o u s t h r o u g h a s h o c k a n d a d i s c o n t i n -u o u s j u m p b e t w e e n t h e t w o r o o t s o f t h e s e c o n d o r d e r e q u a t io n , E q . (6 ), a u t o -m a t i e a l l y s a t i s f y t h e s h o c k r e l a t i o n E q . (5 ).

T h e c o r r e c t s h o c k l o c a t io n i s d e t e r m i n e d b y e n s u r i n g t h a t t h e r e i s a f i n i tec o n t i n u o u s a c c e l e r a t i o n e v e r y w h e r e e x c e p t a t t h e s h o c k w a v e , a r e q u i r e m e n tw h i c h l e a d s t o t h e r e g u l a r i t y c o n d i t i o n s ,

1

ex {a r~ (2 , ~) + ~7v(2 , 2 , 2%)}~=~o(~ = 0 (7 b )


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144 D. Nixon:w h e r e f - - ~ 0 ( ~ ) r e p r e s e n t s t h e l in e ( ~ ( f, 3 ) = 1 ). E q . (1 , 7 ) t h e n c o m p l e t e l yd e t e r m i n e t h e f l o w .

F o r e a s e o f n u m e r i c a l c o m p u t a t i o n h o w e v e r , t h e p r o b l e m i s m o d i f i e d s l i g h t l yb y t h e i n t r o d u c t i o n o f a p a r a m e t e r e (~ ) s u c h t h a t E q . (1 , 7 ) a r e r e p l a c e d b y t h ee q u a t i o n s

~ ( f , 5 ) ~ ( ~ ' ~ )2 - ~ T L ( ~ , 5 ) + 4 ~ ) I ~ ( f , Z , ~ ) ( S )a n d

1{~rL(~ , 5 ) + ~(3) IT (f , ~ , ~ )}~=~, (~) = y (9 a)~--- {~TZ(f, ~) @ ~(~,) I T ( f . , 3 , Xs)}]~=~(~) = 0 (9 b)aTr e s p e c t i v e l y .

B e f o r e t h e p r o b l e m d e f i n e d b y E q . (8 , 9 ) c a n b e s o l v e d t h e f ie l d i n t e g r a lI T ( f , 5 , f ~ ) m u s t b e e v a l u a t e d . I n R e f . [ 2 ] t h i s i n t e g r a l i s e v a l u a t e d b y d i v i d i n gt h e f l o w fi e ld i n t o s t r ip s p a r a l le l to t h e f - a x i s a n d a p p r o x i m a t i n g t h e d i s t r i b u t io no f t h e s q u a r e s o f t h e v e l o c i t y i n e a c h s t r ip i n t e r m s o f v a lu e s o n e a c h s t ri p e d g e b yl i n e a r i n t e r p o l a t i o n . S i n c e t h e v a l u e o f ~ r L ( Y , S ) o n e a c h s t r i p e d g e i s e a s i l y o b -t a i n e d , E q . ( 1, 7 ) r e d u c e t o a s e t o f e q u a t i o n s f o r t h e v e l o c i t y o n t h e s t r i p e d g e s .I t i s s u g g e s t e d i n l ~ ef . [2 ] t h a t t h e e f f e c t o f t h e v e l o c i t y i n t h e o u t e r s t r ip s o n t h ev a l u e o f 4 ( f , 5 ) o n t h e a e r o f o i l s u r f a c e is s m a l l a n d c o n s e q u e n t l y t h e v e l o c i t ie s i nt h e o u t e r s t r i p s a r e e s t i m a t e d i n t e r m s o f v a l u e s o n t h e i n n e r s t r i p e d g e s. T h e p a r to f t h e f l o w f ie l d s o e s t i m a t e d i s c h o s e n s o t h a t t h e s u r f a c e v a l u e s o f a r e n o t s ig n i f-i c a n t l y a ff e c t ed b y t h e a p p r o x i m a t i o n .

T h e s o l u t i o n p r o c e d u r e g i v e n i n R e f . [ 2] i s a s f o l lo w s :F i r s t a n o r m a l s h o c k l o c a t i o n i s a s s u m e d , t o g e t h e r w i t h a n i n i t ia l g u es s f o r t h e

f i e ld v e l o c i t y . T h e f i e l d i n t e g r a l IT (Z ,, S, x s) c a n t h e n b e e v a l u a t e d a n d , k e e p i n gt h e s h o c k l o c a t i o n c o n s t a n t , E q . (8 , 9 ) a r e s o l v e d f o r n e w v a l u e s o f g ( f , 5 ); t h ep a r a m e t e r s ( S ) i s c h o s e n s o t h g g E q . ( 9 ) a r e s a t i s f i e d . T h i s i t e r a t i v e p r o c e d u r e i sr e p e a t e d u n t i l t h e s ( ~ ) a r e c o n v e r g e d . T h e n o r m a l s h o c k l o c a t i o n i s t h e n m o v e d

u n t i l t h e e ( S ) a r e c l o se t o u n i t y ( a n d d e ~ 0 ). S in c e a n o r m a l s h o c k w a v e i sdSa s s u m e d i t is im p o s s i b l e t o g e t s (~ ) e x a c t l y u n i t y ( o r d e = 0) .I t is t h e p r i n c i p le a i m o f t h e p r e s e n t a n a ly s i s to s t u d y t h e e f f e c t o f f i t ti n g ac u r v e d s h o c k w a v e i n t o t h i s s o l u ti o n . A s e c o n d o b j e c t i v e i s t o i n c l u d e t h e v e l o c i t ya h e a d o f a n d b e h i n d t h e a e r o f o i l i n t h e c o m p u t a t i o n o f IT (S :, 5 , ~ ) w h i c h a r eneg lec ted in I~e f . [2 ] .

A n a l y s i s f o r a C u rv e d S h o c k W a v eT h e e f f e c t o f a n a l t e r a t i o n f r o m n o r m a l i n t h e s h o c k w a v e g e o m e t r y is t r a n s -

m i t t e d t h r o u g h t h e c h a n g e in t h e v a l u e o f t h e f i e ld i n t e g r a l I T ( Z , 5, f~) an d isp r i n c i p a l l y d u e t o a s i g n i f ic a n t j u m p i n v e l o c i t y b e ca u s e o f t h e a l t e r e d s h o c kl o c a ti o n . T h e c o n s e q u e n t c h a n g e s i n v e l o c i t y o u ts i d e o f t h e e n v e l o p e o f th e i n i ti a la n d f i n a l s h o c k l o c a t i o n s w il l a l so h a v e a n e f f e c t o n t h e s o l u t io n . T h e m o s t i m -p o r t a n t e f f e c t , t h a t i s t h e c h a n g e i n I ~ ( f , 2, 28) d u e t o c h a n g e s i n t h e v e l o c i t yr e s u l t i n g d i r e c t l y f r o m t h e d i f f e r e n t l o c a t i o n o f t h e s h o c k j u m p , i s c o n s i d e r e d f i r s t.


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The Transonic Integra l Equ at ion Method W ith Curved Shock W aves 145R e f e r r i n g t o F i g . 1 , t h e m a i n c h a n g e o f t h e c o n t r i b u t i o n t o t h e f i e ld i n t e g r a l in

t h e s t r i p ~1 ~ ~ ~ ~1+1 i s d u e t o t h e j u m p i n v e l o c i t y i n t h e r e g i o n B f r o m ap r e - s h o c k v a l u e t o a p o s t - s h o c k v a l u e .

_ r.~...--NORMAL ~HOCK

x ~ . z , ( - e lFig. L Sketch of shoe~ geometry in the J~ str ip

L e t th e v a l u e o f t h e e l e m e n t a r y f ie l d i n t e g r a l in t h e r e g io n A + B b e d e n o t e db y I j a n d

I~ = I ~ ( ~ ~) + I ~ ( ~ ) , 0 0 )w h e r e

G (~ 2 ) = - 4 - ~A

B

(11)

a n d th e " ~ - " a n d " - - " s u p e r s c r i p t s d e n o t e V a lu e s o f 4 ( ~ , B) w h i c h e x i s t i n A a n d Bi f t h e p o i n t i n q u e s t i o n i s a h e a d o f o r b e h i n d t h e s h o c k r e s p e c t iv e l y .A s i n t h e n o r m a l s h o c k a n a l y s i s t h e s q u a r e s o f th e v e l o c i t y i n t h e ~ th s t r ip a r ee s t i m a t e d b y l i n e a r i n t e r p o l a t i o n i n t e r m s o f v a l u e s o n t h e s t r i p e d g e s a n d ,a d d i t i o n a l l y i n t h i s c as e , o f v a l u e s a t t h e c u r v e d s h o c k w a v e . I n o r d e r t o s i m p l i f yt h e a n a l y s i s i t is a s s u m e d t h a t i n th e e v a l u a t i o n o f t h e e l e m e n t a r y f ie l d i n t e g r a lso v e r A a n d B t h e v e l o c i t y in A o r B d o e s n o t v a r y s i g n i f ic a n t l y i n t h e 5 - d i re c t io n .T h i s im p l ie s t h a t t h e s h o c k g e o m e t r y d o e s n o t d i ff e r s u b s t a n t i a l ly f r o m t h a t o f an o r m a l s h o c k w a v e ; w h i ch s h o u l ff b e t r u e f o r m o s t t r a n s o n i c f lo w s. T h i s m e a n st h a t t h e e l e m e n t s A a n d B a r e r e la t i v e l y s m a l l.

I f th e v e l o c i ty o n t h e u p s t r e a m s id e of t h e s h o c k w a v e i s d e n o t e d b y u + ( G , ~ )a n d t h e v e l o c i t y o n t h e d o w n - s t r e a m si de o f t h e s h o c k w a v e is d e n o t e d u s G )t h e n t h e v e l o c i t y v a r i a t i o n i n A i s g i v e n b y

~ (~ , C) = ~+~(~, Cj) + [u W (G q~) - u+'~(~,Ci)] (C _ Ci)G ( ~ ) - C~) ( 1 2 a )1 0 A c t a M e c b . 3 2/ 1 - -3


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146 D. Nixon:a n d i n B b y

9 ( ~ + , - ~ ( ~ ) )w he r e ( }s , -~ ) a r e t he c o - o r d i na t e s o f t he s hoc k w a ve i n t he e l e m e n t ( A B) . I f ,i n t h e ?'*~ s t ri p , t h e s h o c k w a v e c a n b e a p p r o x i m a t e d b y a s t r a i g h t l in e o f s lo p ecot O~ th e n

q, = ~. + (~e 2 ,) co t 0~, (13)w h e r e 0 s i s t he a ng l e t he s hoc k m a ke s w i t h t he 2 - ax i s a n d ~ , i i s t he l oc a t i on o~ t heshoc k o n th e ~'~us t r i p e dge .

S i n ce i t is a s s u m e d t h a t u ( ~ , ~ ) d o e s n o t v a r y w i t h ~ i n A o r B t h e n r e f e rr i n gt o F i g . 1W ( ~ , ~ ) = ~+( ~ . ~ , r~ ( ~ , r = a ( ~~ ~ ) (~4)

w he r e x s 0 i s t he l oc a t i on o f t he no r m a l s hoc k w a ve .Us ing l inea r in te rp ola t ion fo r z~z(~ , ~) in the ~ s t r ip

a n d,~+= ~ r ~+2 -

- [ ( ~ , - ( % + " r ( ~ ( ~ ) - r ( 1 5 a )22 ( ~ , , ~ ) : a+" ( x ., ,+ , , ~ ) + ( r _ r

S u b s t i t u t i o n o f E q . ( 1 2 , 1 3) in t o E q . ( 1 1 ) g i ve s , a f t e r s o m e m a n i p u l a t i o n ,

A

+ [g+~(~+*, r -- ~+~(~*+,,~)] ( r _ ~) } d ~ d }(r - r1IB(~2/~+~ )- 4 7 z / f gt (g., }; ~, ~-){~2_~(g.,o, ~ )

B

( 1 5 b )

U s i n g E q . ( 1 6 ) t h e n , E q . ( 1 0 ) c a n b e w r i t t e n a s

( 1 6 a )

(16b)

I i - - , ~ ~ ~ . 1 , ~)A+B

(r e j ) -B

( 1 7 )


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The Transonic Integral Equ at ion Method W ith Curved Shock W~ves 147w h e re [~ i 2 ]+ i s t h e s h o c k j u m p o n t h e j t~ s t r i p e d g e w h i c h i s g i v e n b y E q . ( 5) a s

[ ~ j2 ] _ + = 4 s e e 2 0 ~ [ ~ 2 ( ~ , ~ j ) - s e c ~ 0 ~ ] . ( 1 8 )S i n c e i t h a s b e e n a s s u m e d t h a t i n t h e e v a l u a t i o n o f t h e i n t e g r a l o v e r t h ee l e m e n t A + B t h e v e l o c i t y d o e s n o t v a r y s i g n i f i c a n t l y w i t h Z" t h e n , t o a f i rs t

a p p r o x i m a t i o n~ + ( z " , . . , ~ j + l ) = ~ - ( z " ~ 0 , r

w h e re u -(z "s0 , ~ j+ l) , u +(x~ o, ~i) a r e k n o w n f ro m t h e n o rm a l s h o c k s o l u t i o n a n d i t c a nt h e n b e s e en f r o m E q . (17) t h a t t h e p e r t u r b a t i o n t o I i , I ~ d u e t o a c h a n g e i ns h o c k g e o m e t r y i s g i v en b y

/ / { }[u j+~ ]_- [~ ]2 ) (~ r de d~ (19)BT h e e v a l u a t i o n o f t h e e l e m e n t a r y i n te g r a l I ~ i n E q . (1 9) is s im p l i fe d c o n -

s i d e r a b l y if t h e s h o c k i n t h e ]th s t r ip is a p p r o x i m a t e d b y a s t r a i g h t l in e a s inE q . ( 13 ). T h e t o t a l p e r t u r b a t i o n , I ~ d u e t o a c h a n g e i n s h o c k g e o m e t r y is o b t a i n e db y s u m m i n g t h e c o n t r i b u t i o n o f e a c h s tr ip ; t h u s

Nb = 2 : ~ , ~ , ( 2 0 )i= ~w h e r e 2V i s t h e n u m b e r o f st r ip s a n d I ~ is g i v e n b y E q . (1 9) . T h e g o v e r n i n ge q u a t i o n f o r a c u r v e d s h o c k w a v e i s t h e n

~ ( z " , 2 ) ~ ( ~ ' ~ )2 - - 7 s , 2 ) - ~ I T ( Z " , z , Z " s.) + I ~ , (21)w h e r e I T ( Z " , 2 , Z " 8o )i s e v a l u a t e d a s s u m i n g a n o rm a l s h o c k a t Z " S o -

T h u s t h e i m m e d i a t e c h a n g e i n t h e f l o w d u e t o a c h a n g e in t h e s h o c k g e o m e t r yis t r a n s n f i t t e d t h r o u g h t h e a d d i t i v e t e r m I p g i v e n b y E q . (1 9, 2 0 ). A p a r t f r o mt h i s i m m e d i a t e c h a n g e d u e t o I p i t s h o u l d b e r e m e m b e r e d t h a t t h e r e i s a s e c o n d a r yc h a n g e d u e t o t h e a l t e r a t i o n o f t h e v e l o c it ie s o u ts i d e t h e e n v e l o p e o f t h e i n i ti a la n d f i n a l sh o c k l o c a t i o n s.

E s t i m a t i o n o f t h e S h o c k C h a n g eI n t h e p r e c e d i n g a n a l y s is i t is a ss u m e d t h a t t h e n e c e s s a r y c h a n g e .i n t h e s h o c k

g e o m e t r y f r o m a c o n v e r g e d n o r m a l s h o c k l o c a t i o n t o m a k e s (~ ) e q u a l t o u n i t y i sk n o w n . I n t h i s s e c t io n i t is s u g g e s t e d h o w t h e s e c h a n g e s c a n b e e s t i m a t e d .

I n g e n e r a l t h e f i e l d i n t e g r a l I T ( Z " , 2 , Z " 8 ) a n d t h e p a r a m e t e r e ( 2 ) a r e d e p e n d e n to n t h e s h o c k l o c a t i o n x ~(2 ) ; t h e ~ (~ ) a r e f i x e d b y t h e r e g u l a r i t y c o n d i t i o n s , E q . (9 ).I f t h e r e is a n a r b i t r a r y c h a n g e in E q . ( 9 a) d u e t o a c h a n g e o f s h o c k g e o m e t r y t h e n

DS--~ {s (2) IT(Z " 2 , Z"~)} [s = 0 (2 2 )10 "


8/11

148 D. Nixon:w h e r e ( ~ X o ( 2 ) an d (~x~(2) a re th e p e r t ur ba t io ns of th e l ine ( f i@ , ~) = 1) an d th es hoc k l oc a t i on r e s pe c t i ve l y . U s i ng E q . ( 9 ), E q . ( 22 ) r e duc e s t o g i ve

or( ~ x s ( 2 ) {e(~)IT~ @ , ~ , Y ~ s) @ S Z ~ ( Z ) I T ( Y , 2 , x~)}IZ=~0(~) = 0

(23)T h e f u n c t i o n I T S , ( 2 , Z , X s) i s e v a l u a t e d f o r t h e i n i t i a l n o r m a l s h o c k l o c a t i o n .5 s, a n d i s f o u n d t o g i v e

oo

- r r ~ . ( ~ , e , : % ) = ~ [ a ~ (e ~ ~ r ~ r ~ ( ~ , ~ ~ ~ , ~ ) d ~o ( 2 4 )

0

- - o o

I n a s o l u t i on f o r a no r m a l s hoc k w a ve s (~ ) i s c oi~ver ge d b u t no t n e c e s s a r i l y t ou n i t y . I f e 0(2 d e n o t e s t h e c o n v e r g e d v a l u e o f e (2 ) f o r a n o r m a l s h o c k t h e n t h ene c e s s a r y c ha nge i n t he s hoc k g e om e t r y t o g i ve e ( 2 ) = 0 ( a n d d~e _ . 0 ) c a n h ef ou nd a s f o l low s , d~

E x p a n s i o n o f e (~ ) i n a T a y l 0 r ' s s e ri es a b o u t ~ , g i v es4 ~ ) = ~ 0(~ ) + ~ x ~ ( ~ ) ( ~ M ) ) ~ . ~ ~ + ( 25 )

w her e (~xs(2) i s a 6h an ge : in sho ck lo ca t io n a n d (e,~(2))~,=~so is giv en b y E q. (23, 24) .D i f f e r e n t i a t i on o f E q . ( 25) w i t h r e s pe c t t o 2 g i ve s

w h e r e 0~ = a r c t a n ( ~ ) g i v e s t h e a n g l e o f t h e s h o c k r e l a ti v e t o t h e 2-a xis. O np u t t i n gas(~ ) _ 0og

a n d o n u s i n g E q . ( 25) , E q . ( 26 ) g i ve s~a~0(~) ( i - ~o(~)___~)a 1tan 0~ = ( -~ -~+ (~(~))~=~' ~ (~(~)!~=~'" (27)

that is, the shock slope relative to the ~-axis. The movement of the ~oot of thes hoc k , ~x~o(0 , is f oun d f r om E q . ( 25 ) by pu t t i n g e (0 ) e qua l t o u n i t y ; t h us

(1 - ~o(O)) (2 8)~ z ~0 (0 ) _ ( ~ ( ~ ) ) ~ o = ~ 0 .


9/11

The Transonic Integra l Eq uatio n Method W ith Curved Shock W aves 149E q . (2 7, 2 8) t h e n g i v e t h e a p p r o p r i a t e p e r t u r b a t i o n o f t h e s h o c k w a v e t o

o b t a i n a p r o p e r s o l u t i o n t o t h e t r a n s o n i c p r o b l e m , w h e n (e (~ ) = 1 ).A s n o t e d e a r li e r t h e v e l o c i t y o n t h e o u t e r s t ri p e d g e s c a n b e e s t i m a t e d i n

t e r m s o f t h e v a l u e s o n t h e o u t e r s t r ip e d g e s w i t h o u t s i g n i fi c a n t ly a f fe c t i n g t h ev a l u e o n t h e a e r o fo i l s u rf a c e. H o w e v e r t h i s a p p r o x i m a t i o n c a n l e a d t o s i g n if i ca n te r ro r s i n ~ (2 ) a n d i n p r a c t i c e t h e s h o c k w a v e g e o m e t r y i s r e p r e s e n t e d b y a s e c o n do r t h i r d o r d e r c u r v e w i t h t h e c o e f f i c ie n t s d e t e r m i n e d b y E q . ( 26 , 2 7 ) u s i n g 0% (~)3~o n t h e i n n e r s t r i p e d g e s o n l y , w h i c h a r e k n o w n t o b e a c c u r a t e . P r o v i d e d t h es h o c k c u r v a t u r e is n o t t o o g r e a t a d i s c r e p a n c y in t h e s h o c k lo c a t i o n i n t h e f a rf i e ld d o e s n o t s i g n i f i c a n t l y a f fe c t t h e s o l u t i o n o n t h e a e r o f o i l s u r f ac e . T h i s i sc o n s i s te n t w i t h t h e i d e as o f t h e o v e r a l l t h e o r y i n w h i c h i~ i s f o u n d t h a t t h e s u r fa c ev e l o c i t y i s n o t s e n s i t iv e t o m i n o r c h a n g e s i n t h e o u t e r f lo w fi e ld .

S o l u t i o n P r o c e d u r eT h e b a s i c e q u a t i o n f o r a fl o w w i t h a c u r v e d s h o c k i s E q . (2 1). I n o r d e r t o g i v e

a s m o o t h f lo w u p s t r e a m o f t h e s h o c k w h e n t h e f i n a l s h o c k g e o m e t r y is u n k n o w n ,i . e . i n a n i t e r a t i o n p r o c e s s , E q . ( 2 1 ) i s wr i t t e n a s

4 ( '2 , Z ) ~ 2 ( ' ~ , ) C t L (X , 2 ) - ]- e ( 2 ) { I T ( X Z , X s , ) @ I v } (29)2w i t h t h e r e g u l a r i t y c o n d i t i o n s

1 ( 3 0 a )

- ~ { ~ L ( ~ , ~ ) + ~ ( ~ ) [ z T ( ~ , ~ , ~ 0 ) + 4 ] }1 ~ = ~ .( ~ ) = 0 ( 3 0 b )T h e s o l u t i o n t o t h e p r o b l e m w i t h a c u r v e d s h o c k is f o u n d a s fo l lo w s .a ) F i r s t a c o n v e r g e d s o l u t i o n t o E q . (8 , 9 ) w i t h a n o r m a l S h o c k w a v e i s

o b t a i n e d , i n c lu d i n g v e l o c it ie s a h e a d o f a n d b e h i n d t h e a e r o fo i l i n t h e e v a l u a t i o nof I r (~ , 2 , ~0) .

b ) F r o m t h i s c o n v e r g e d s o l u ti o n th e v a l u e s o f I T ~ ( X , 2 , ~ 2 S o a n d ( e ~ ( 2 ) ) ~ = ~ 0a r e c a l c u l a t e d f r o m E q . ( 23 , 2 4 ).c ) A n e s t i m a t e o f t h e r e v i se d s h o c k g e o m e t r y is t h e n o b t a i n e d f r o m E q . (27 ,

2 8 ) w i t h t h e a p p r o x i m a t i o n t h a t i n e a c h s t r i p t h e s h o c k c a n b e r e p r e s e n t e d b y as t r a i g h t l i n e s e g m e n t , E q . ( 1 3 ) .d ) T h e f i e l d i n t e g r a l f o r a n o r m a l s h o c k , I t ( X , 2 , 2 ~ o ) i s a s s u m e d k n o w n f r o mt h e n o r m a l s h o c k s o l u t i o n .e ) T h e p e r t u r b a t i o n i n t e g r a l I v d e f i n e d b y E q . ( 19 , 2 0) f o r a c u r v e d s h o c k i s

c a l c u l a t e d u s i n g t h e p r e v i o u s l y c o m p u t e d v e l o c i t i e s a t t h e p r e - s h o c k l o c a t i o n .N e w v e l o c it ie s o v e r t h e f low f i e ld a r e t h e n c o m p u t e d u s i n g E q . (2 9, 3 0 ). T h e n e wp r e - s h o c k v e lo c i ti e s a r e t h e n u s e d t o e v a l u a t e a n e w v a l u e o f I p a n d t h e p r o c e d u r er e p e a t e d u n t i l t h e c o n v e r g e n c e o f t h e p a r a m e t e r s s(~).

f ) U s i n g t h e n e w v a l u e s o f 4 ( ~, 2) t h e n o r m a l s h o c k i n t e g r a l I T ( Z , ~ , Z , o ) ise v a l u a t e d . T h e p r e - s h o c k v e l o c i t i e s i n t h e ' s h o c k e n v e l o p e ' , r e g i o n B i n F i g . 1 ,


10/11

150 D. Nixon:a r e o b t a i n e d b y e x t r a p o l a t i o n . U s i n g th i s n e w v a l u e o f I t ( 2 , 2, ~ ) s t e p ( e) isr e p e a t e d .

g ) S t e p s ( e - - f ) a r e r e p e a t e d u n t i l c o n v e r g e n c e o f t h e ~ (~ ).h ) I f n e c e s s a r y s t e p s ( b - - g ) a r c r e p e a t e d u n t i l t h e v ( 5) a r e u n i t y o n a s p e c if ie dn u m b e r o f s t ri p e d g e s. T h i s s t e p i s u s u a l l y n o t n e c e s s a r y .

Re s u l t s

T h e p r e s s u r e d i s t r ib u t i o n a r o u n d a 6 o p a r a b o l i c a r c a e r o fo i l a t M ~ = 0 . 87 6 5i s c o m p u t e d w i t h a n d w i t h o u t a c u r v e d s h o c k w a v e a n d i s s h o w n i n F i g . 2 . T h er e s u l ts c a l c u l a t e d b y M u r m a n [1 ] u s i n g a c o n s e r v a t i v e fi n it e d i ff e r e n c e m e t h o da r e s h o w n f o r c o m p a r i s o n . T h e e f f e c t o f i n c lu d i n g t h e c u r v a t u r e o f t h e s h o c k w a v e

(,7M U R M A N- - - e - - - - I N T E G R A L M E T H O D (NORMALSHOCK)C l) i - - -+ - - - -IN T E G R A L E T H O D ( C U R V E DH O C K }

-0'6-4 i

~

Fig. 2. Pressure distr ibut ion arou nd a 60/o biconv ex aerofoi l. M ~ = 0.8715

i n t h e a n a l y s i s i s t o p r o d u c e a l o c a l v a r i a t i o n i n t h e v e l o c i t y n e a r t h e s h o c kg i v in g c lo se a g r e e m e n t w i th M u r m a n ' s r e s u lt . T h e b e t t e r a g r e e m e n t n e a r t h el e a d in g e d g e is d u e t o t h e c o n s i d e r a ti o n o f v e l o ci ti e s a h e a d o f a n d b e h i n d t h ea e r o f o i l 'i n t h e e v a l u a t i o n o f t h e f i e ld i n t e g r a l I ~ @ , 2, Y ~,). T h e r e a s o n f o r t h ed i s c r e p a n c y b e t w e e n t h e r e s u l t s j u s t b e h i n d t h e s h o c k w a v e is n o t a p p a r e n t .

T h e i n t r o d u c t i o n o f s h o c k c u r v a t u r e i n t o t h e e x t e n d e d i n t e g r a l e q u a t i o nm e t h o d r e m o v e s a n y t h e o r e t ic a l d if fe r en c e b e t w e e n t h e p r e s e n t m e t h o d a n dM u r m a n ' s [1 ] c o n s e r v a t i v e f in i te d if f e re n c e m e t h o d o t h e r t h a n n u m e r i c a l in -a c c u r a c y a n d t h is i l lu s t ra t e d b y t h e c o n s i d e ra b l y im p r o v e d a g r e e m e n t b e t w e e nt h e r e s u l t s c a l c u l a t e d b y b o t h m e t h o d s .


11/11

T h e T r a n so n i c I n t e g ra l E q u a t i o n M e t h o d W i t h C u r v e d S h o c k W a v e s ] 5 1

R e f e r e n c e s

[1 ] M u r m a n , E . : A n a l y s i s of E m b e d d e d S h o c k W a v e s c a l c u l a te d b y R R e l a x a t i o n ~ I e th o d s .A I A A J o u r n a l 1 2 , 6 2 6 - - 6 3 3 ( 19 74 ).[ 2] N i x o n , D . : E x t e n d e d I n t e g r a l E q u a t i o n M e t h o d f o r T r a n s o n i c F l o w s . A I A A J o u r n a l 1 8 ,9 3 4 - - 9 3 5 ( 1 9 7 5 ) .[ 3] N i x o n , D . : C a l c u l a t i o n o f T r a n s o n i c F l o w s u s i n g a n E x t e n d e d I n t e g r a l E q u a t i o n M e t h o d .A L ~ J o u r n a l 1 5 , 2 9 5 - - 2 9 6 (1 97 7) .D. N ixonDepartment o/Aeronautical EngineeringQueen Ma ry CollegeUniversity o/Lon donLondon, U.K.

Transonic Integration technique for shocks

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