AIAA - 87 - 0600 An LU - SSOR Scheme for the Euler and Navier - Stokes Equations

A p p l i c a t i o n t o t r a n s o n i c f low 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 a r e 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 uns tab le , r e s u l t i n g i n t u r b u l e n t f l o w over iribst o f t h e sur face of t h e a i r p l a n e . However, t h e computa t iona l r e q u i r e ments f o r t h e s i m u l a t i o n of 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 of c u r r e n t computers. f i r s t l e v e l o f approx imat ion t i m e averag inq t h e 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 J Reynolds-averaged Navier-Stokes equat ions, which r e q u i r e a t u r b u l e n c e model f o r c l o s u r e . A t t h e p r e s e n t t i m e n o t much i s known about t h e behav io r of t u r b u l e n c e i n separa ted r e g i o n s , and t h i s has impeded t h e development o f t u r b u l e n c e models f o r complex th ree-d imens iona l f lows. S ince a u n i v e r s a l l v s a t i s f a c t o r v t u r b u l e n c e model has v e t t o be The f o u n i , current t u r b u l e n c e models have t o " b e t a i l o r e d t o t h e p a r t i c r i l a r f low. D u r i n g t h e l a s t decade t h e f e a s i b i l i t y o f s o l v i n g t h e Navier-Stokes equat ions has been e x p l o r e d b u t t h e methods developed so f a r have been t o o expens ive t o use i n a r o u t i n e prodirct i o n mode. l l e c e n 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 m i l l t i g r i d method f o r s teadyt t e s o l u t i o n o f t h e unsteadv E u l e r eauations.?,' A l thouah t h e m u l t i g r i d method i n two dimensions, ' i t s i n h e r e n t l i m i t a t i o n s i n t h r e e dimensions sugqest a l t e r n a t i v e approaches.? 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 d imensions i s based o n lower-upper (LU) f a c t o r i z a t i o n . The LU i m p l i c i t scheme proved t o be r o h u s t and e f f i c i e n t f o r h ighspeed f l ws up t o Mach 20 as w e l l as f o r t r a n s o n i c flow^.^-^ I t was a l s o shown t h a t a symmetric *Sen ior Engineer , Sverdrup Technology, I n c . **Professor, Department o f Mechanical and Member A I A A -' Aerospace Eng ineer ing Member A l A A 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 of t h e i l l 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 ga ted t o s o l e t h e s teady E u l e r o r Navier -Stokes 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 s o l v e d i n d ' r e c t l y . Gauss-Seid 1 method t o t h e E u l e r equat ions , w h i l e MacCormack 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 approx imate Newton i t e r a t i o n . The new lower upper 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 i a g o n a l 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 CU 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 n v e r s i o n s . The use of s c a l a r d iagona l 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 o rder of-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 so1 ved, f o r example, f o r hyperson ic f l o w s w i t h f i n i t e r a t e chemis t ry . 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 inc reases t h e computa t iona l work p e r c y c l e . 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 Is. Because of t h e r a p i d g rowth o f t h e Jespersen6 and Hemker and S p e k r e i j s e 3 a p p l i e d t h e symmetric a p p l i e d t h e l i n e Gauss-Seidel method