Instabilities in the wake/mixing-layer region of a splitter plate separating two supersonic streams

The s t a b i l i t y o f t h e l a m i n a r wake o f a s p l i t t e r p l a t e i s i n v e s t i g a t e d u s i n g l i n e a r i n v i s c i d s p a t i a l t heo ry . The two streams a r e supe rson i c b u t have d i f f e r e n t v e l o c i t i e s and temperatures , t h u s t h e wake i s n o t symmetr ic. The b a s i c s t a t e i s o b t a i n e d by s o l v i n g t h e Nav ie r Stokes equa t i ons u s i n g a f i n i t e d i f f e r e n c e code. The equa t i ons a r e s o l v e d s t a r t i n g f rom an assumed i n i t i a l s t a t e , which i s g i v e n by a s u r f a c e o f d i s c o n t i n u i t y ( i .e . a v o r t e x shee t ) , and a s teady l a m i n a r f l o w i s o b t a i n e d f o r t h e t h r e e cases cons idered. The v a r i a t i o n o f g row th r a t e s and phase speeds w i t h f r equency a r e determined. The r e s u l t s show a s t r o n g s t a b i l i z i n g e f f e c t o f t h e Mach number. 1. I n t r o d u c t i o n M i x i n g o f two supe rson i c streams i s o f p r i m a r y impor tance t o t h e d e s i g n e r o f p r o p u l s i v e systems which emply supe rson i c combust ion. Exper imen ta l e f f o r t s on m i x i n g l a y e r s have e s t a b l i s h e d t h a t t h e sp read ing r a t e s o f supe rson i c f r e e shear l a y e r s a r e subs a t i a l l y l ower t h a n t h e r a t e s o f subson ic l a y e r s f 9'. The e f f e c t o f c o m p r e s s i b i l i t y on t h e sp read ing r a t e was found t o c o r r e l a t e v e r y w e l l w i t h t h e c o n v e c t i v e Mach number j n t r o d u c e d by ~ o ~ d a n o f f ~ and Papamoschou and Roshko . Theoret i c a l s u b s t a n t i a t i o l o f t h a t c o r r e l a t i o n i s p r o v i d ed by Ragab and Wu u s i n g l i n e a r s t a b i l i t y t h e o r y o f m i x i n g l a y e r s . A s p l i t t e r p l a t e i s u s u a l l y used i n exper imen ta l work on t h e m i x i n g o f two p a r a l l e l streams. The re fo re , a wake-1 i ke r e g i o n wi 11 always e x i s t downstream o f t h e t r a i l i n g edge. I n t h e case o f l a m i n a r boundary l a y e r s on t h e s p l i t t e r p l a t e , i n s t a b i l i t y waves p r e v a i l i n t h e f i r s t p a r t o f t h e w a k e l m i x i n g l a y e r reg ion . Amp1 i f i c a t i o n o f t h e s e waves e v e n t u a l l y l eads t o t r a n s i t i o n t o t u r b u l e n c e and t h e f o r m a t i o n o f l a r g e s c a l e s t r u c t u r e s . It i s o f i n t e r e s t t o t h e e x p e r i m e n t a l i s t t o know how f a r downstream t h e w a k e l i k e p r o f i l e p e r s i s t s . Fur thermore, when t h e t u r b u l e n t m i x i n g l a y e r i s o f p r i m a r y i n t e r e s t , one must be su re t h a t t h e t u r b u l e n t shear l a y e r has had enough l e n g t h t o deve lop so t h a t i t s independent o f t h e i n i t i a l p r o f i l e . Papemoschou3 used a c o n s e r v a t i v e e s t i m a t e o f t h e d i s t a n c e f o r a f u l y deve loped Z i . . t u r b u l e n t f l o w based on Bradshaw's c r i t e r i o n o f 1000 momentum th i cknesses . Bradshaw's c r i t e r i o n , however, i s f o r a s i n g l e s t ream shear l a y e r a t l ow speed. An e m p i r i c a l c o r r e l a t i o n between l i n e a r s t a b i l i t t h e o r y and t h e onse t o f t r a n s i t i o n known iG as t h e e method has been w i d e l y used i n boundary* A s s o c i a t e P ro fesso r , Member AIAA C o p y r i g h t 1988 by S. A. Ragab Pub1 w i t h permiss ion. l a y e r research7$' . A c r i t e r i o n f o r t r a n s i t i o n f r o m l a m i n a r t o t u r b u l e n t f l o w s i n wakes based on hydrodynamic s t a b i l i t y c a l c u l a t i o n s n ds t o be developed. The work o f Lees and Goldeg i s a s t e p i n t h a t d i r e c t i o n . The o b j e c t i v e o f t h e p r e s e n t paper i s t o i n v e s t i g a t e t h e l i n e a r i n s t a b i l i t y o f t h e l a m i n a r wake formed downstream o f a s p l i t t e r p l a t e when t h e two streams a r e superson ic . A comprehensive r e v i e w o f t h e l i t e r a t u r e on wake i n s t a b i l i t i e s i s beyond t h e scope o f t h i s paper. A 1 i m i t e d r i e w w i l l be g i v e n here . M a t t i n y l y and Cr i rn ina leT ' i n v e s t i g a t e d e x p e r i m e n t a l l y and t h e o r e t i c a l l y t h e g row th o f smal l d i s t u r b a n c e s i n a symmetr ica l wake o f an NACA0003 a i r f o i l a t a l ow speed. Measured v e l o c i t y p r o f i l e s were ana l yzed u s i n g b o t h tempora l and s p a t i a l l i n e a r i n v i s c i d s t a b i l i t y t h e o r y o f quas i -para1 1 e l f l ows . They conc luded t h a t s t a b i l i t y s t u d i e s o f h i g h l y u n s t a b l e f l o w s shou ld be based on s p a t i a l t h e o r y wh ich was found t o c o r r e l a t e b e t t e r w i t h measurements. Lees and old" have shown t h a t t h e s i nuous mode i s more u n s t a e t h a n t h e v a r i c o s e mode. Kubota and BehrensPJ i n v e s t i g a t e d t h e l i n e a r i n v i s c i d s t a b i l i t y o f t h e f a r wake beh ind a c i r c u l a r c y l i n d e r a t Mach number 6. They showed t h a t t h e g row th r a t e s depend s t r o n y on t h e v e l o c i t y p r o f i l e s . Behrens and KoP' i n v e s t i g a t e d e x p e r i m e n t a l l y and t h e o r e t i c a l l y t h e t r a n s i t i o n r e g i o n i n wakes o f s l e n d e r wedges and a f l a t p l a t e a t Mach number 6. Measured v e l o c i t y and temperat u r e p r o f i l e s were used i n t h e l i n e a r i n s t a b i l i t y c a l c u l a t i o n s . Good agreement between t h e o r y and expe r imen ts i s o b t a i n e d f o r t h e maximum a m p l i f i c a t i o n r a t e and t h e most u n s t a b l e f requency. Concern ing l i n e a r s t a b i l i t y compu ta t i o asymmetr ic wakes, Koochesfahani and F r i e l e r v e s t i g a t e d t h e l i n e a r s p a t i a l i n s t a b i l i t y o f a v e l o c i t y p r o f i l e wh ich i s a l i n e a r s u p e r p o s i t i o n o f t h e h y p e r b o l i c t a n g e n t p r o f i l e and a Gaussian p r o f i l e ; wh i ch r e p r e s e n t s a wake l i ke component. They a l s o assumed a h y p e r b o l i c t angen t p r o f i l e f o r t h e d e n s i t y d i s t r i b u t i o n b u t t h e f l o w i s assumed t o be i ncompress ib le . T h e i r computa t ions i n d i c a t e d t h e presence o f two u n s t a b l e modes wh ich approach t h e s inuous and v a r i c o s e modes i n t h e l i m i t o f a p u r e wake f l ow . I n t h i s paper, i n v i s c i d i n s t a b i l i t i e s o f supe rson i c asymmetr ic wakes a r e cons idered. The mean f l o w p r o f i l e s a r e o b t a i n e d by s o l v i n g t h e Nav ie r -S tokes equa t i ons downstream o f a s p l i t t e r p l a t e t r a i l i n g edge. L i n e a r t h e o r y o f s p a t i a l l y a m p l i f i e d d i s t u r b a n c e s i s used, and t h e mean f l o w i s assumed t o be l o c a l l y para1 l e l . i s h e d by t h e American I n s t 1095 i t u t e o f Ae ronau t i cs and A s t r o n a u t i c s , I nc . 2. Mean Flow Computat ion The d e v e l o p i n g w a k e l m i x i n g l a y e r downstream o f a s p l i t t e r p l a t e i s o b t a i n e d by s o l v i n g t h e 1 ami n a r two-d imens iona l t h i n 1 ayer Nav ier -Stokes equa t i ons u s i n g t h e computer code "ARCPD" which has been deve loped a t NASA Ames. The computa t iona l domain i s bounded 1 a t e r a l l y by two p a r a l l e l s o l i d w a l l s sepa ra ted by a d i s t a n c e H = 23 mm. I n t r o d u c i n g C a r t e s i a n c o o r d i n a t e s (x ,y ) and u s i n g H as a r e f e r e n c e l e n g t h , we d e f i n e t h e computat i o n a l domain by t h e r e c t a n g l e 0 ( x ( 10 and t h e 0.5 ( y ( 0 . 5 . The s p l i t t e r p l a t e i s a l i g n e d w i t h t h e n e g a t i v e p a r t o f t h e x a x i s , w i t h x = 0, y = 0 b e i n g t h e t r a i l i n g edge. The re fo re , t h e s p l i t t e r p l a t e i s n o t p a r t o f t h e computat i o n a l domain. A nonun i f o rm C a r t e s i a n g r i d i s used. The s t e p s i z e i n t h e x d i r e c t i o n s t a r t s w i t h A X = 0.025 and geomet r i ca l l y s t r e t c h e s acco rd ing t o ax , = 1.05 Ax, , p r o v i d e d t h a t Ax < 0.2. w i t h a r e f e r e n c e tempera tu re o f 225 K. P e r f e c t gas i s a l s o assumed w i t h y = 1.4.