Navier-Stokes computations for a transonic projectile at angle of attack and comparison with experimental data

The o b j e c t i v e o f t h i s work i s t o i n v e s t i g a t e t h e a b i l i t y o f an i m p l i c i t Nav ier -Stokes code t o p r e d i c t t h e t r a n s o n i c , v i scous f l o w about a p r o j e c t i l e , body o f r e v o l u t i o n a t sma l l ang les o f a t t a c k . The computa t ions have been per formed f o r t h e geometry and t e s t c o n d i t i o n s o f w ind t u n n e l exper iments wh ich i n c l u d e s u r f a c e p r e s s u r e d i s t r i b u t i o n s and boundary l a y e r su rveys a t Mach numbers o f 0.94 and 0.97 and a t 0 and 4 degrees a n g l e o f a t t a c k . P a r t i c u l a r a t t e n t i o n i s focused on t h e b o a t t a i l a f t e r b o d y r e g i o n because o f t h e s t r o n g shock wave boundary 1 aye r i n t e r a c t i o n s and near wake i n f l u e n c e s which o c c u r t h e r e . The r e s u l t s show good q u a n t i t a t i v e agreement between p r e d i c t i o n and exper iment f o r b o t h p r e s s u r e d i s t r i b u t i o n and boundary l a y e r p r o f i l e s a t zero ang le o f a t t ack . To a c c u r a t e l y p r e d i c t t h e b o a t t a i l c h a r a c t e r i s t i c s i n t h e f i n a l 1-2 pe rcen t o f t h e body l e n g t h , i t i s necessary t o c o r r e c t l y model t h e base and nea r wake f low. At f o u r deqrees ang le o f a t t a c k , t h e agreement i s l e s s good because: a) t h e c i r cumf e r e n t i a l v a r i a t i o n o f t h e p ressu res a r e l a r g e r than i n d i c a t e d by t h e exper iments ; b) s l i g h t d i f f e r e n c e s i n shock wave l o c a t i o n e x i s t between p r e d i c t i o n s and measurements and between t h e two exper iments cons idered; c ) t h e v e l o c i t y p r o f i l e s a r e qua1 i t a t i v e l y p r e d i c t e d b u t t h e r e a r e d i s c r e p a n c i e s i n some areas because o f t h e missmatch i n t h e p r e s s u r e d i s t r i b u t i o n s and because t h e measured v e l o c i t y g r a d i e n t s near t h e model s u r f a c e a re unde r -p red i c ted . These r e s u l t s suggest t h a t iqprovements c o u l d be made i n t h e a l g e b r a i c t u r b u l ence model used. I. I n t r o d u c t i o n The p r imary o b j e c t i v e o f t h i s work i s t o i n v e s t i g a t e t h e t h e a b i l i t y o f a numer ica l compu ta t i ona l t echn ique t o p r e d i c t t h e t r a n s o n i c , v i scous f l o w f i e l d around an ax i symmet r i c p r o j e c t i l e a t z e r o and smal l ang le o f a t t a c k . The q u a l i t y o f t h e p r e d i c t i o n i s assessed by comparison w i t h w ind t u n n e l 3easurements i n c l u d i n g boundary l a y e r surveys. A l though t h e b a s i c t h e o r y and a l g o r i t h m s needed f o r such computa t ions have been a v a i l a b l e f o r seven o r e i g h t yea rs , o n l y r e c e n t l y have computer 's memory and speed improved t o t h e p o i n t where t h r e e d imens iona l , t r a n s o n i c, Nav ier -Stokes computa t ions w i t h adequate r e s o l u t i o n a r e f e a s i b l e . The r e s u l t s desc r i bed he re were per formed on t h e US Army B a l l i s t i c Research L a b o r a t o r y (BRL) Cray 2 computer u s i n g a v e r s i o n o f t h e P u l l i a m and Steger l impl i c i t , f i n i t e d i f f e r e n c e code. These computa t ions co r respond t o t h e t e s t c o n d i t i o n s and model c o n f i g u r a t i o n t e s t e d i n t h e NASA Lang ley T ranson i c Pressure Tunnel i n 197919802 bu t n o t p r e v i o u s l y r e p o r t e d i n d e t a i 1. P r e l i m i n a r y r e s u l t s f rom t h e s e t e s t s were g i v e n i n Reference 3, i n wh i ch a l i m i t e d compar ison was made w i t h a p o t e n t i a l f l o w theory . The exper iment i n v o l v e d measurements o f t h e p ressu re d i s t r i b u t i o n and boundary l a y e r surveys on t h e a f t e r b o d y o f an a r t i l l e r y p r o j e c t i l e based on a s l i g h t l y m o d i f i e d v e r s i o n o f t h e US Army's M549. S ince t h e s e measurements a r e r e s t r i c t e d t o t h e a f t e r b o d y , t h e p r e s s u r e d i s t r i b u t i o n d a t a over t h e e n t i r e i d e n t i c a l body o b t a i n e d by M i l l e r and Mo lna r4 i s a l s o used f o r comparison w i t h t h e numer i ca l r e s u l t s . Other t r a n s o n i c s u r f a c e p r e s s u r e d i s t r i b u t i o n measurements on a s i m i l a r shape have been o b t a i n e d by ~ a y s e r ~ and Kayser and Whiton. P r a c t i c a l Nav ier -Stokes computa t ions o f t r a n s o n i c f l o w s were s i g n i f i c a n t l y advanced t h r o u g h t h e development o f t h e impl i c i t , app rox ima te l y f a c t o r i zed method based on t h e work o f Beam and warming7 and subsequent ly i n c o r p o r a t e d i n t h e code o f P u l l i a m and Steger. The equa t i ons o f mo t i on , i n t h i s code, a r e w r i t t e n i n g e n e r a l i z e d body o r i e n t e d coo rd ina tes . The t h i n shear l a y e r a p p r o x i m a t i o n i s used t o s i m p l i f y t h e v i scous terms and t h e r e s u l t i n g equa t i ons a r e l i n e a r i z e d . Approx imate f a c t o r i z a t i o n i s c a r r i e d o u t i n f o r m u l a t i o n o f t h e a l g o r i t h m so t h a t b l o c k i n v e r s i o n techn iques can be used s e p a r a t e l y i n t h e s p a t i a l c o o r d i n a t e d i r e c t i o n s . The Bal dwin-Lomax a1 g e b r a i c t u r b u l e n c e n o d e l 8 was i n c o r p o r a t e d f o r h i g h Reynolds number f lows. T h i s t echn ique was a p p l i e d , f o r t h e f i r s t t i m e , t o t h e t r a n s o n i c b o a t t a i l a f t e r b o d y f l o w f i e l d by ~ e i w e r t . ~ He compared h i s r e s u l t s w i t h t h e expe r imen ta l d a t a o f ~ e u b u s h , ' ~ Benek l1 and shrewsbury12 on a f t e r b o d i e s w i t h plume s i m u l a t o r s . He found t h a t q u a l i t a t i v e agreement between t h e computa t ions and expe r imen t was e x c e l l e n t b u t q u a n t i t a t i v e l y t h e r e appeared t o be areas where f u r t h e r s tudy was needed. The techn ique was s p e c i a l i z e d t o ax i symmet r i c bod ies w i t h s p i n i n t h e e t a i n v a r i a n t code as w e l l as g e n e r a l i z e d t o t h r e e dimensions. These codes were a p p l i e d by N i e t u b i cz, e t a1 13-16 t o p r o j e c t i l e s and comparisons were made w i t h t h e d a t a o f Danberg, and Kayser. I n Reference 16, N i e t u b i c z conc luded t h a t t h e r e was inadequate v i scous r e s o l ut i o n p a r t i c u l a r l y f o r s p i n n i n g bod ies because o f l i m i t a t i o n s i n c a p a c i t y o f t h e computers (e.g., Cray 1s). The most r e c e n t r e s u l t s p r e s e n t e d by Sahu and N ie tub i cz ,17 Sahu and Steger18 o v e r come t h e s e o b j e c t i o n s by u s i n g t h e c a p a c i t y of t h e Cray *Na t i ona l Research Counci 1 A s s o c i a t e **Aerospace Eng ineer , Assoc ia te Fe l l ow , AIAA 2 and by u s i n g more advanced a1 g o r i thms i nco rpo ra t i n g v e c t o r s p l i t t i n g and upwind d i f f e r e n c i n g . l 8 Improved g r i d r e s o l u t i o n a1 lowed exce l l e n t p r e d i ct i o n s o f t h e 0 and 4 degrees a n g l e a t t a c k p r e s s u r e d i s t r i b u t i o n s o f Kayser and Whiton. The c u r r e n t work i s a c o n t i n u a t i o n o f t h e s e v a l i d a t i o n s t u d i e s o f t h e P u l l i a m and S tege r t ech n ique b u t w i t h t h e emphasis d i r e c t e d a t e v a l u a t i n g c u r r e n t a b i l i t i e s t o p r e d i c t v i scous e f f e c t s a t ang le o f a t t a c k u s i n g t h e g r i d r e s o l u t i o n c a p a b i l it i e s o f t h e Cray 2.