NASA Contractor Report 179556 AIAA - 87 - 0600 An LU - SSOR Scheme for the Euler

A p p l i c a t i o n t o t r a n s o n i c f l o w shows I n t r o d u c t i o n The Reynolds numbers f o r a l a r q e a i r p l a n e are o f t h e o r d e r o f t h i r t y m i l l i o n . Therefore, laminar f l o w i n t h e boundary l a y e r becomes unstable, r e w l t i n g i n t u r b u l e n t f l o w over most o f t h e su r face o f t h e a i r p l a n e . However, t h e computat ional requ i re ments f o r t h e s i m u l a t i o n o f t u r b u l e n t f l o w a r e c l e a r l y beyond t h e reach o f c u r r e n t computers. The f i r s t l e v e l o f approx imat ion t i m e averaginq the r a p i d l y f l u c t u a t i n g components y i e l d s t h e Reynolds-averaged Navier-Stokes equat ions, which r e q u i r e a tu rbu lence model f o r c losu re . p resen t t i m e n o t much i s known about t h e behavior o f t u r b u l e n c e i n separated reg ions, and t h i s has impeded t h e development o f t u rbu lence models f o r complex three-dimensional f l ows . S ince a univers a l l y s a t i s f a c t o r y tu rbu lence model has y e t t o be found, c u r r e n t t u rbu lence models have t o be t a i l o r e d t o t h e p a r t i c u l a r f l ow . Dur ing t h e l a s t decade t h e f e a s i b i l i t y of s o l v i n g t h e Navier-Stokes equations has been exp lo red b u t t h e methods developed so f a r have been t o o expensive t o use i n a r o u t i n e product i o n mode. A t t h e Recen t l y severa l i m p l i c i t schemes have been developed s u c c e s s f u l l y i n c o n j u n c t i o n w i t h a mult i g r i d method f o r steadyt t e s o l u t i o n o f t h e unsteady E u l e r equations.',' Al though t h e a l t e r n a t i n g d i r e c t i o n i m p l i c i t scheme cou ld be improved t o achieve t h e expected e f f i c i e n c y o f t h e m u l t i g r i d method i n two dimensions, l i t s inherent 1 i m i t a t i o n s i n t h r e e dimensions suggest a l t e r n a t i v e approaches.2 An a l t e r n a t i v e i m p l i c i t scheme t h a t i s s t a b l e i n any number of space dimensions i s based on lower-upper (LU) f a c t o r i z a t i o n . scheme proved t o be r o b u s t and e f f i c i e n t f o r highspeed f l o w s up t o Mach 20 as w e l l as f o r t ransonic flows.3-5 The LU i m p l i c i t I t was a l s o shown t h a t a symmetric *Senior Engineer, Sverdrup Technology, Inc. **Professor, Department o f Mechanical and Member AIAA Aerospace Engineer ing Member A I A A This paper is declared P work of the U.S. Covcrnmenl and i 5 not subject to copyright Protection in (he United States. Gauss-Seidel r e l a x a t i o n method f o r s o l v i n g t h e u n f a c t o r e d i m p l i c i t scheme was a v a r i a n t o f t h e LU i m p l i c i t scheme. The Newton i t e r a t i o n method has been i n v e s t i gated t o s o l e t h e s teady E u l e r o r Navier-Stokes equations.6-E Because o f t h e r a p i d growth o f t h e o p e r a t i o n count w i t h t h e number o f mesh c e l l s , t h e system was so l ved i n d ' r e c t l y . Gauss-Seid 1 method t o t h e Eu le r equat ions, w h i l e t o t h e Navier-Stokes equat ions. I n t h i s paper an e f f i c i e n t m u l t i g r i d r e l a x a t i o n scheme i s developed f o r approximate Newton i t e r a t i o n . The new lowerupper symmetric success ive o v e r r e l a x a t i o n (LU-SSOR) scheme r e q u i r e s s c a l a r d iagonal i n v e r s i o n s w h i l e t h e Gauss-Seidel method and t h e LU i m p l i c i t scheme r e q u i r e b l o c k m a t r i x i nve rs ions . The use o f s c a l a r d iagonal i n v e r s i o n s o f f e r s t h e p o t e n t i a l f o r orderof-magnitude speedups when l a r g e systems o f p a r t i a l d i f f e r e n t i a l equat ions must be solved, f o r example, f o r hypersonic f l o w s w i t h f i n i t e r a t e chemis t r y . i s d e s i r a b l e t h a t t h e m a t r i x be d i a g o n a l l y dominant t o assure t h e convergence o f a r e l a x a t i o n method. The new method based on c e n t r a l d i f f e r e n c e s achieves t h i s w i t h o u t t h e f l u x s p l i t t i n g t h a t s u b s t a n t i a l l y i nc reases t h e computat ional work p e r cyc le . Numeric a l examples i n c l u d e i n v i s c i d and v iscous t r a n s o n i c a i r f o i 1 s. Jespersen6 and Hemker and Spek re i j se J a p p l i e d t h e symmetric MacCormack 8 a p p l i e d t h e l i n e Gauss-Seidel method