Transonic Flows about Oscillating Airfoils Using the Euler Equations

T r a n s o n i c f l ow o v e r a h a r m o n i c a l l y o s c i l l a t i n g a i r f o i l is computed by s o l v i n g t h e two-dimens ional E u l e r e q u a t i o n s i n i n t e g r a l form u s i n g t h e f i n i t e volume scheme. The d i s s i p a t i o n t e rms a r e c o n s t r u c t e d a c c o r d i n g t o t h e o r y of t o t a l v a r i a t i o n d i m i n i s h i n g schemes i n o r d e r t o o b t a i n good r e s o l u t i o n of shock f r o n t s . N o n r e f l e c t i n g boundary cond i t i o n s have been u t i l i z e d i n t h e f a r f i e l d . It is s e e n t h a t t o t a l v a r i a t i o n d i m i n i s h i n g schemes a r e w e l l s u i t e d f o r u n s t e a d y f l o w problems, w h i l e t h e f i n i t e volume f o r m u l a t i o n a l l o w s a ve ry s i m p l e t r e a t m e n t of de fo rming meshes. 1. I n t r o d u c t i o n The problem of u n s t e a d y t r a n s o n i c f l o w a b o u t o s c i l l a t i n g a i r f o i l s ha s r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n t h e l a s t decade . Many of t h e numeric a l schemes f o r s o l v i n g u n s t e a d y problems a r e based on t h e t r a n s o n i c s m a l l d i s t u r b a n c e p o t e n t i a l e q u a t i o n , which is e i t h e r l i n e a r i z e d w i t h r e s p e c t t o a s t e a d y f l ow o r is s o l v e d d i r e c t l y . B a l l h a u s and G o o r j i a n [ l ] have s o l v e d t h e n o n l i n e a r s m a l l d i s t u r b a n c e p o t e n t i a l e q u a t i o n w i t h low f r e q u e n c y a p p r o x i m a t i o n u s i n g a n a l t e r n a t i n g d i r e c t i o n e x p l i c i t scheme. To d e s c r i b e i n v i s c i d t r a n s o n i c f l ow c o r r e c t l y , t h e E u l e r e q u a t i o n s must be s o l v e d . The n u m e r i c a l s o l u t i o n of t h e E u l e r e q u a t i o n s s h o u l d p r o v i d e a n a c c u r a t e p r e d i c t i o n of t h e l o c a t i o n and s t r e n g t h of t h e shock and t h e a s s o c i a t e d wave d r ag . T h i s c a n be p a r t i c u l a r l y i m p o r t a n t i n u n s t e a d y t r a n s o n i c f l o w , where t h e changes i n a n g l e of a t t a c k c a u s e t h e shock t o move a p p r e c i a b l y . The u n s t e a d y E u l e r e q u a t i o n s have been s o l v e d by Magnus & Yosh iha r a [ 2 ] , L e r a t and S i d e s [ 3 ] and S t e g e r ( 4 1 . Magnus & Y o s h i h a r a [ 2 ] u sed a f i n i t e d i f f e r e n c e scheme of t h e Lax-Wendroff t ype . L e r a t and S i d e s [ 3 ] s o l v e d . t h e E u l e r e q u a t i o n s i n i n t e g r a l c o n s e r v a t i o n law fo rm by u s i n g t h e f i n i t e volume method of MacCormack. S t e g e r [ 4 ] u s e d a n i m p l i c i t f i n i t e d i f f e r e n c e a l g o r i t h m t o s o l v e b o t h i n v i s c i d and v i s c o u s f l ow a b o u t a i r f o i l s . D e s p i t e t h e i n t e n s i v e e f f o r t s of numerous i n v e s t i g a t o r s , t h e o b j e c t i v e of combining 1 ) High a c c u r a c y 2 ) R e s o l u t i o n of shock waves and c o n t a c t d i s c o n t i n u i t i e s 3 ) E l i m i n a t i o n of s p u r i o u s o s c i l l a t i o n s Copyright O American Institute of Aeronautics and Astronautics, Inc., 1985. All rights reserved. c o n t i n u e s t o be a n e l u s i v e g o a l . It has l o n g been r e c o g n i z e d t h a t upwind d i f f e r e n c i n g can e l i m i n a t e s p u r i o u s o s c i l l a t i o n s i n t h e ne ighborhood of shock waves a t t h e expense of low a c c u r a c y i n r e g i o n s where t h e f l ow is smooth. C e n t r a l d i f f e r e n c e schemes, on t h e o t h e r hand, produce good s o l u t i o n s i n smooth r e g i o n s , b u t a r e p rone t o o s c i l l a t i o n s i n t h e ne ighborhood of shockwaves. Stemming f rom t h e m a t h e m a t i c a l t h e o r y of s c a l a r c o n s e r v a t i o n l aws [ 5 ] , H a r t e n p roposed t h e concep t of t o t a l v a r i a t i o n d i m i n i s h i n g (TVD) schemes [ 6 ] . H a r t e n a l s o d e v i s e d second o r d e r a c c u r a t e e x p l i c i t and i m p l i c i t TVD schemes, which i n c o r p o r a t e f l u x l i m i t e r s t o c o n t r o l t h e a c t i o n of a n a n t i d i f f u s i v e t e r m [ 7 ] . Jameson [ 8 ] ha s d e r i v e d a s e m i d i s c r e t e TVD scheme f o l l o w i n g a s i m i l a r app roach which is o u t l i n e d i n t h e n e x t s e c t i o n . Van Lee r had earlier u s e d f l u x l i m i t e r s t o produce a second o r d e r a c c u r a t e scheme which would p r e s e r v e t h e monotonic i t y of an i n i t i a l l y monotone p r o f i l e (91 . I m p o r t a n t c o n t r i b u t i o n s t o t h e t h e o r y of upwind schemes have a l s o been made by Roe [10 ,11] and Oshe r 112,131. Both have d e v i s e d second o r d e r a c c u r a t e upwind schemes u s i n g f l u x l i m i t e r s . Many of t h e i d e a s on f l u x l i m i t e r s have r e c e n t l y been u n i t e d by Sweby [14 ] . Jameson & Lax [15 j have d e r i v e d c o n d i t i o n s on t h e c o e f f i c i e n t s f o r a mu l t i p o i n t d i f f e r e n c e scheme t o have t h e TVD p r o p e r t y . Jameson (161 ha s a l s o c o n s t r u c t e d a s e m i d i s c r e t e TVD scheme which u s e s t h i r d o r d e r d i s s i p a t i v e t e rms m o d i f i e d by f l u x l i m i t e r s . I n t h i s work, t h e u n s t e a d y E u l e r e q u a t i o n s a r e s o l v e d on a r e g u l a r q u a d r i l a t e r a l g r i d abou t a n a i r f o i l i n two d imens ions . The f i n i t e volume scheme i n t h e form proposed by Jameson, Schmidt and T u r k e l [17] h a s been used . S e c t i o n 2 p r e s e n t s t h e s e m i d i s c r e t e TVD scheme f o r t h e s c a l a r conse r v a t i o n law. The i n t e g r a l form of t h e E u l e r e q u a t i o n s f o r a moving mesh and t h e f i n i t e volume scheme a r e p r e s e n t e d i n S e c t i o n 3 . S e c t i o n 4 d e a l s w i t h t h e e x t e n s i o n of TVD schemes t o a sy s t em of c o n s e r v a t i o n laws. S e c t i o n 5 d i s c u s s e s t h e t ime s t e p p i n g s t r a t e g y . I n S e c t i o n 6 we d i s c u s s t h e boundary c o n d i t i o n s i n t h e f a r f i e l d and on t h e a i r f o i l . F i n a l l y , S e c t i o n 7 p r e s e n t s t h e r e s u l t s f o r a v a r i e t y of one-d imens ional problems and t h e o s c i l l a t i n g a i r f o i l . 2. Total Variation Diminishing Scheme for a ScalarConservation Law Fi = fi + gi in which gi is an anti-diffusive flux which approximates a u A x a , ax and thus cancels the first order error. It is necessary to limit gi, however, to prevent the possibility of (gi+l-gi)/(ui+l-~i) becoming unbounded. For this purpose define For the scalar conservation law a * + f(u) = 0 at ax (1) it is well known that the total variation can never increase. Correspondingly it seems desirable that the discrete total variation m and where B is an averaging function which limits the magnitude attainable by (gi+l gi)/(ui+l ui) and satisfies the condition of a solution of a difference approximation to (1) should not increase. It can be shown that a semidiscretization will have the TVD property if it can be cast in the form