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A I A A-84- 0093 Multigrid Solution of the Euler Equations for Aircraft Configurations A. Jameson and T.J. Baker, Princeton Univ., Princeton, NJ
AlAA 22nd Aerospace Sciences Meeting January 9-12, 1984/Reno, Nevada
I-
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1633 Broadway, New York, NY 10019
I'
MULTIGRID SOLUTION OF THE EULER EQUATIONS FOR AIRCRAFT CONFIGURATIONS
Antony Jameson'and Timothy 3 . Baker Department of Mechanical and Aerospace Engineering Princeton University P r i n c e t o n , N.J. 08544
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f o r a f i x e d r e g i o n Q W i t h boundary a R . Here W r e p r e s e n t s t h e c o n s e r v e d q u a n t i t y and is the c o r r e s p o n d i n g f l u x term. I f we w r i t e p , p, u, v, w , E and H f o r t h e p r e s s u r e , d e n s i t y , C a r t e s i a n v e l o c i t y components, t o t a l e n e r g y and t o t a l e n t h a l p y , t h e n e q u a t i o n 1 1 ) r e p r e s e n t s mass conserv a t i o n i f we p u t
Abstract
A m u l t i g r i d scheme f o r s o l v i n g t h e E u l e r equations i s presented. The method he5 been succ e s s f u i l y a p p l i e d t o two-dimensional a i r f o i l c a l c u l a t i o n s on b o t h & t y p e and C-type meshes. In three d i m e n s i o n s t h e scheme has p r o v e d e q u a l l y e f f e c t i v e and c a l c u l a t i o n s of f l o w s o v e r wing/body cornb i n a t i o n s a r e p o s s i b l e w i t h conve-gence a c h i e v e d i n l e s s t h a n 100 cycles.
W
For c o n s e r v a t i o n have I.
V
introduction
w =
M u l t i g r i d has emerged as a p a r t i c u l a r l y e f f i c i e n t t e c h n i q u e f o r a c c e l e r a t i n g t h e convergence o f numerical c a l c u l a t i o n s . Although i t was a r g i n a l l y d e v i s e d and a p p l i e d t o s o l v e e l l i p t i c e q u a t i o n s , c a n s i d e r a S i e i n t e r e s t has been g e n e r a t e d by t h e p o s s i b i l i t y o f e m p l o y i n g t h i s t e c h n i q u e f o r The o t h e r t y p e s o f p a r t i a l d i f f e r e n t i a l equationi,'. application of m u l t i g r i d t o solve transonic flow p r o b l e m s u s i n g t h e p o t e n t i a l e q u a t i o n ha3 been an area o f p a r t i c u l a r l y a c t i v e research 3,4*5. Here t h e e q u a t i o n i s o f mixed t y p e and t h e p r e s e n c e o f Shockwaves l e a d s t o n u m e r i c a l d i f f i c u l t i e s n o t n o r mal l y e n c o u n t e r e d w i t h el I i p t i c e q u a t i o n s where t h e s o l u t i o n s are e x p e c t e d t o possess a r e a s o n a b l e d e g r e e o f smoothness. F u r t h e r d i f f i c u l t i e s were caused by t h e w i d e V a r i a t i o n i n mesh c e l l a s p e c t r a t i o which i s t y p i c a l o f t h e weshes used f o r c a l c u l a t i n g t h e f l o w around aerodynamic shapes. T h i s p a r t i c u l a r p r o b l e m a r o s e because t h e d r i v i n g r o u t i n e used i n a m u l t i g r i d Scheme must p r o v i d e heavy damping o f a l l h i g h f r e q u e n c y rrodes. I n two and t h r e e d i m e n s i o n s t h e a m p l i f i c a t i o n f a c t o r depends 0- t h e r e s ? c e l l a s p e c t ' a - i c z".? i t i s hy no means easy t o d e v i s e Schemes w i t n good higbm f r e quency damping f o r a w i d e v a r i a t i o n i n t h e shape and s i z e o f mesh c e l l s . For two-dimensional f l o w c a l c u l a t i o n s using t h e p o t e n t i a l equation t h i s d i f f i c u l t y was s u c c e s s f u i l y r e s o l v e d by Jameson4 by t h e i n t r o d u c t i o n Of an a l t e r n a t i n g d i r e c t i o n method. Numerous c a l c u l a t i o n s by t h i s t e c h n i q u e have dem o n s t r a t e d t h a t m u l t i g r i d c a r indeed a c h i e v e v e r y r a p i d convergence f o r t r a n s o n i c f l o w c a l c u l a t i o n s .
of r r o m e n t m u,
= (P"
2
i n t h e x d i r e c t i o n we
+ p,
PU",
OUWI
i n e q u a t i o n ( I ) . C o n s e r v a t i o n of rromentum i n t h e y and z d i r e c t i o n s i s S i m i l a r l y d e f i n e d . Finally e n e r g y c o n s e r v a t i o n i s g i v e n by W = pE,
f=
(pH",
pHv,
pHw1
.
W h i l e t h e a v a i l a b l e theorems i n t h e t h e o r y of m u l t i g r i d methods g e n e r a l l y assume e l l i p t i c i t y , it seems t h a t i t o u g h t t o be p o s s i b l e t o a c c e l e r a t e t h e e v o l u t i o n o f a h y p e r b o l i c system t o a s t e a d y s t a t e by u s i n g l a r g e t i m e s t e p s on c o a r s e g r i d s , so t h a t d i s t u r b a n c e s a r e rrore r a p i d l y e x p e l l e d t h r o u g h t h e o u t e r boundary. The i n t e r p o l a t i o n o f c o r r e c t i o n s back t o t h e f i n e g r i d w i l I i n t r o d u c e e r r o r s , however, which c a n n o t be r a p i d l y e x p e l l e d from t h e f i n e g r i d , and o u g h t t o be l o c a l l y damped, i f a f a s t r a t e o f c o n v e r g e n c e i s t o be a t t a i n e d . Thus i t r e m a i n s i m p o r t a n t t h a t t h e d r i v i n g scheme s h o u l d h a v e t h e p r o p e r t y o f r a p i d l y damping Out h i g h f r e quency m d e s .
I n a n o v e i m u l t i g r i d scheme proposed by N i 6 , t h e f l o w v a r i a b l e s are s t o r e d a t mesh nodes, and t h e r a t e s o f change o f mass, mmentum and e n e r g y i n each mesh c e l l a r e e s t i m a t e d f r o m t h e f l u x i n t e g r a l (11. The c o r r e s p o n d i n g change 6W0 i n t h e s o l u t i o n a t t h e c e l l center i s then d i s t r i b u t e d unequally between t h e nodes a t I t s f o u r c o r n e r s by t h e r u l e
Sw,
= 1/4 (6W0 f A6W0 t B6W0)
where 6W, i s t h e c o r r e c t i o n a t a c o r n e r , and A and B a r e t h e Jacaobian m a t r i c e s . The s i g n s a r e v a r i e d i n such a way t h a t t h e a c c u m u l a t e d c o r r e c t i o n a t e a c h m d e c o r r e s p o n d s t o t h e f i r s t two t e r m s o f a T a y l o r S e r i e s i n t i m e , l i k e a Lax Wendroff scheme. A S it s t a n d s , t h i s scheme does not damp o d d and N i introduces a r t i f i c i a l even p o i n t o 5 C i l l a t i o n s . v i s c o s i t y by a d d i n g a f u r t h e r c o r r e c t i o n p r o p o r t i o n a l t o t h e d i f f e r e n c e between t h e v a l u e a t each c o r n e r and t h e a v e r a g e o f t h e v a l u e s a t t h e f o u r R e s i d u a l s on t h e c o a r s e g r i d corners of the cel I. a r e formed by t a k i n g w e i g h t e d a v e r a g e s o f t h e c o r r e c t i o n s a t n e i g h b o r i n g nodes o f t h e f i n e g r i d , and c o r r e c t i o n s a r e t h e n a s s i g n e d to t h e c o r n e r s o f c o a r s e g r i d c e l i s by t h e s a w d i s t r i b u t i o n r u l e .
The a s s u m p t i o n of p o t e n t i a l f l o w i m p l i e s t h a t t h e flow i s i r r o t a t i o n a l . This i s not S t r i c t l y c o r r e c t when shock waves a r e p r e s e n t and an e x a c t d e s c r i p t i o n o f t r a n s o n i c i n v i s c i d flow r e q u i r e s t h e s o l u t i o n o f t h e Euler equations. T h i s i s a system o f h y p e r b o l i c e q u a t i o n s Which i n i n t e g r a l f o r m can be w r i t t e n as
Cop)riphl 'C American lnslilul~01 Aeronauticsand Arlronsutirr. Inr.. 1984. All dghlr resencd.
= p, F = (pu, p v , p w l .
1
Here D i j & i s t h e d i s s i p a t i o n which
When s e v e r a l g r i d l e v e l s a r e used, t h e d i s t r i b u t i o n r u l e i s a p p l i e d once on each g r i d down t o t h e m a r s e s t g r i d , and t h e c o r r e c t i o n s a r e t h e n i n t e r p o l a t e d back t o t h e f i n e g r i d .
fiow. form
U s i n g 3 or 4 g r i d l e v e l s , N i O b t a i n e d a mean e r r o r r e d u c t i o n o f a b o u t .95 p e r m u l t i g r i d c y c l e , l e a d i n g t o a f a i r l y we1 I converged s o l u t i o n i n 100-200 c y c l e s . The a c c u r a c y of t h e r e s u l t s which c a n be o b t a i n e d by t h e a l g o r i t h m i n I t s pub1 i s h e d f o r m , however, appears to be l i m i t e d by t h e a c t i o n of t h e a r t i f i c i a l v i s c o s i t y , which i n t r o d u c e s an e r r o r o f f i r s t order.
-
di-i/2,j,k di,j,k+1/2
+
-
dl.j+l/2,k 'i,j,k-I/2
'i,j-l/2,k
and A x
i s t h e forward d i f f e r e n c e Operator defined
by
Ax P i j k = P i + l , j , k
P'I J k
The c o e f f i c i e n t ~ ( ~ ) i + l /. 2k i s made p r o p o r t i o n a l t o t h e normal i r e d second d. Jl .f f e r e n c e o f t h e p r e s s u r e
I n adjacent c e l l s . T h i s q u a n t i t y i s of seco'id o r d e r except i n r e g i o n s c o n t a i n i n g a steep pressure gradient. The f o u r t h d i f f e r e n c e s p r o v i d e b a c k g r o u n d d i s s i p a t i o n t h r o u g h o u t t h e domain. In t h e n e i g h b o r h o o d o f a Shock wave. however, v i j k i s o f o r d e r one and t h e second d i f f e r e n c e s become t h e d o m i n a n t d i s s i p a t i v e terms. The d i s s i p a t i v e t e r m s f o r t h e o t h e r e q u a t i o n s a r e c o n s t r u c t e d from s i m i l a r f o r m u l a s w i t h t h e e x c e p t i o n o f t h e energy e q u a t i o n where t h e d i f f e r e n c e s a r e o f pH r a t h e r t h a n pE. The p u r p o s e o f t h i s i s t o a l l o w a s t e a d y s t a t e s o l u t i o n f o r which H r e m a i n s c o n s t a n t . The cel I volume V i j k we can w r i t e ( 3 ) as
3 + dt
i s i n d e p e n d e n t o f t i m e so
R(W1 = 0
(7)
The C o m p u t a t i o n a l d o m a i i i s d i v i d e : i n t o heX3hecral cel I S . A s s u i i n g t h a t t h e depende-,? v a r i a b l e s a r e known a t t h e center o f each c e i I , a s y s t e m of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i s o b t a i n e d by a p p l y i n g e q u a t i o n ( 1 ) s e p a r a t e l y t o e a c h Cel I. These have t h e form d (2) l V i j k Wijk) + Qtwl.. = 0 IJk
.
Q i j k r e p r e s e n t s t h e n e t f l u x o u t o f t h e c e l I which 1s b a l a n c e d by t h e r a t e of change Of W i n t h e c e l l whose volume i s V. I n order to s u p p r e s s t h e t e n dency for odd and even p o i n t o s c i I i a t i o n s and to i i m i t o v e r s h o o t s near shock wave5. t h e scheme i s augmented by a d i s s i p a t i v e term so t h a t e q u a t i o n (21 becomes d ( V , . w.. I t 0.. dt 0.. = 0 (31 I j k I j k tjk ,jk
-
-
i/
(4)
where
F i n i t e Volume Scheme and Time S t e p p i n g
5
,.
For t h e d e n s i t y e q u a t i o n D i j k C p ) has t h e
di+1/2,j,k
The m u l t i s t a g e scheme p r e v i o u s l y d e v e l o p e d by t h e a u t h o r s ' forms t h e b a s i s o f a m u l t i g r i d method This f i n i t e r e c e n t l y i n t r o d u c e d by Jameson8. volume d i s c r e t i z a t i o n has a number o f a t t r a c t i v e features. C e n t r a l d i f f e r e n c i n g i s used f o r t h e s p a t i a l d e r i v a t i v e s which l e a d s to second o r d e r a c c u r a c y i n r e g i o n s where t h e f l o w i s s m o o t h l y varying. The c o n t r o l l e d a d d i t i o n o f a r t i f i c i a l d i s s i p a t i o n permit5 t h e capture o f Sharply defined shocks. Away from shocks where t h e f l o w v a r i e s smoothly, a t h i r d o r d e r d i s s i p a t i o n i s a p p l i e d t o p r e v e n t t h e ~ ~ c ~ r r e nofc eodd-even p o i n t o s c i l l a tions. I n t h e n e i g h b o r h o o d o f a shockwave, however, a s w i t c h , s e n s i t i v e to second d i f f e r e n c e s i n t h e p r e s s u r e , t u r n s on a f i r s t o r d e r d i s s i p a t i o n t h a t w c C e s s f u I l y i n h i b i t s preshock o s c i l l a t i o n s . U n l i k e a s i n g l e s t a g e scheme, t n e use o f m u l t i p l e stages permits a s t a b l e time stepping operation when c e n t r a l d i f f e r e n c i n g i s used to e v a l u a t e t h e f l u x term. F u r t h e r m o r e t h e C o n s t r u c t i o n of t h e s e m u l t i s t a g e schemes e n s u r e s t h a t t h e s t e a d y state S o l u t i o n i s Independent o f t h e t i m e s t e p A t , a n a d v a n t a g e not s h a r e d by several o t h e r schemes. I n t h i s paper it i s shown t h a t t h e m u l t i s t a g e t i m e s t e p p i n g scheme can be a d a p t e d for use i n conj u n c t i o n w i t h a m u l t i g r i d t e c h n i q u e t o produce a r a p i d l y c o n v e r g e n t a l g o r i t h m for c a l c u l a t i n g s t e a d y s t a t e s o l u t i o n s of t h e Euler equations. Several r e s u l t s a r e p r e s e n t e d showing t h e p e r f o r m a n c e of t h i s a l g o r i t h m i n two and t h r e e d i m e n s i o n s and f o r flows Over a i r c r a f t - l i k e c o n f i g u r a t i o n s .
2.
i s constructed
so t h a t It i s o f t h i r d o r d e r i n Smooth r e g i o n s of
.
H e r e W" and W"+l a r e t h e v a l u e s a t t h e b e g i n n i n g and end of t h e n t h t i m e s t e p , and i n t h e ( q + l ) s t s t a a e R ( q ) i s e v a l u a t e d as
2
v
50 t h a t t h e d i s s i p a t i v e t e r m s ark o n l y e v a l u a t e d once i n each t i m e s t e p . The s t a n d a r d f o u r s t a g e scheme has t h e c o e f f i c i e n t s
aI = 1/4,
02 = 1/3,
a3 = 1/2.
W i t h c e n t e r e d d i f f e r e n c e s t h e r e s i d u a l h25 t h e form A t R ( u ) . = -A ( u . u. ) J 2 JCI J-1
-
-
+
(9)
+ 6u- - 4u. 1
4uj+l
J
+ uj-2)
J-
(13) 3.
M u l t i q r i d Method where ,4 = A t / h i s t h e Couran+numb+r. I f we c o n s i d e r a F o u r i e r node u = e ' p x t h e d i s c r e t i z a t i o n i n space y i e l d s
The new m u l t i g r i d scheme p r e s e r v e s t h e second o r d e r a c c u r a c y and e f f e c t i v e shock c a p t u r i n g capab i l i t y o f t h e o r i g i n a l m u l t i - s t a g e scheme. A u x i l i a r y meshes a r e I n t r o d u c e d by d o u b l i n g t h e mesh s p a c i n g a n d t h e v a l u e s o f t h e f l o w v a r i a b l e s a r e t r a n s f e r r e d t o a c o a r s e r g r i d by t h e r u l e
At
dc = dt
2;
where I i s t h e F o u r i e r symbol of t h e r e s i d u a l . When t h e r e s i d u a l i s g i v e n by e q u a t i o n ( 1 3 ) we obtain
where t h e s u b s c r i p t s d e n o t e v a l u e s of t h e mesh I n t h r e e d i m e n s i o n s t h e sum i s s p a c i n g parameter. over t h e 8 c e l 1s on t h e f i n e g r i d composing each c e l I on t h e c o a r s e g r i d . T h i s r u l e conserves mass, momentum and energy. A f o r c i n g f u n c t i o n i s then d e f i n e d as
z =
-
A i s i n t - 4 1 u ( 1 - cos < ) 2
(14)
where 5 = P A X . The i m a g i n a r y p a r t c o r r e s p o n d s t o t h e centered d i f f e r e n c e approximation t o t h e f l u x t e r m and t h e r e a l p a r t c o r r e s p o n d s t o t h e d i s s i p a t i v e term. i f we e v a l u a t e t h e d i s s i p a t i o n a t e v e r y s t a g e , t h e a m p l i f i c a t i o n f a c t o r g ( z ) Of t h e scheme w i t I be a f u n c t i o n o f t h e s i n g l e complex v a r i a b l e Z. For a n e x p l i c i t scheme t h i s f u n c t i o n w i l l be a n a l y t i c It f o l l o w s f r o m t h e -ximum i n any f i n i t e domain. modulus t h e o r e m t h a t t h e maximum v a l u e o f I g ( z ) over a f i n i t e domain wi I I occur on t h e boundary. then Thus, i f we c o n s i d e r C o n t o u r s o f I g ( r ) w i t h i n any c o n t o u r , say [ g ( z ) l = l , t h e v a l u e o f The r e g i o n I g ( r ) wi I 1 a l w a y s be l e s s t h a n one. bounded by t h i s p a r t i c u l a r c o n t o u r c o r r e s p o n d s t o v a l u e s o f L f o r which t h e scheme w i I I be s t a b l e . W i t h t h e c o e f f i c i e n t s q i v e n by ( 9 ) we o b t a i n
where R i s t h e r e s i d u a l o f t h e d i f f e r e n c e scheme. To u p d a t e t h e s o l u t i o c on a c o a r s e g r i d t h e m u l t i s t a g e scheme i s r e f o r m u l a t e d as
1
1,
I
... where R ( q ) i s t h e r e s i d u a l o f t h e qth s t a g e . In t h e f i r s t s t a e of t h e scheme, t h e a d d i t i o n o f Pzh cancels and r e p l a c e s it by L Rh(Wh), w i t h t h e r e s u l t t h a t t h e e v o l u t i o n on t h e c o a r s e g r i d i s d r i v e n by t h e r e s i d u a l s on t h e f i n e g r i d . T h i s p r o c e s s i s r e p e a t e d on s u c c e s s i v e l y c o a r s e r F i n a l l y t h e c o r r e c t i o n c a l c u l a t e d on each grids. g r i d i s passed back t o t h e n e x t f i n e r g r i d by b i i i near i n t e r p o l e t i o r .
The s t a b i l i t y r e g i o n Of t h i s scheme i s show? i n F i g u r e ) ( a ) . The F i g u r e a i s 0 shows t h e locus Of L a s t h e wave number 5 i s v a r i e d between 0 and 277 f o r a C o u r a n t number A = 2.8 and a d i s s i p a t i o n c o e f f i c i e n t u = 1/32. The c o r r e s p o n d i n g v a r i a + i o n o f 191 W i t h 5 i s p r e s e n t e d i n F i g u r e l ( b ) . I f we do n o t e v a l u a + e t h e d i s s i p a t i o n a t eve-y t i m e s t e p , t h e a m p i i f i c s t i o n f a c t o r and hence t h e s t a b i I i t y r e g i o n Of t h e scheme undergoes an interesting nodification. Suppose we have a k s t a g e scheme and we e v a l u a t e t h e d i s s i p a t i o n once a t t h e f i r s t stage, Then f o r t h e l a s t s t a g e we have
S i n c e t h e e v o l u t i o n OP a c o a r s e g r l d i s d r i v e n by r e s i d u a l s c o l l e c t e d from t h e n e x t f i n e r g r i d , t h e f i n a l s o l u t i o n on t h e f i n e g r i d i s i n d e p e n d e n t o f t h e c h o i c e Of boundary c o n d i t i o n on t h e c o a r s e grids. The s u r f a c e boundary c o n d i t i o n i s t r e a t e d i n t h e same way on e v e r y g r i d by u s i n g t h e normal pressure gradient to extrapolate the surface p r e s s u r e from t h e p r e s s u r e i n t h e c e l l s a d j a c e n t t o the wall. The f a r f i e l d c o n d i t i o n s can e i t h e r be t r a n s f e r r e d f r o m t h e f i n e g r i d or r e c a l c u l a t e d by t h e procedure described i n t h e next section.
,(k)
NOW w r i t i n g
The success o f a m u l t i g r i d scheme i s c r i t i c a l l y d e p e n d e n t on t h e shape o f t h e a m p l i f i c a t i o n f a c t o r . I n p a r t i c u l a r i t i s n e c e s s a r y f o r t h e scheme t o p r o v i d e r a p i d damping o f h i g h f r e q u e n c y nodes. To s e e how t h e m u l t i s t a g e schemes can be a d a p t e d t o m e + t h i s r e q u i r e m e n t we c o n s i d e r t h e node1 p r o b l e m
u t + ux
+
uAx3 u x x y y = 0
.
I
"(0)
-
3
~ " ( 0 ) )
z = x + i y we f i n d t h a t g(x,y)
(12)
Atc~u(k-l)
= 1
+
x + iy f(x,y)
(16)
.
.I
g e n e r a t e s a scheme w i t h t h e s t a b i i i t y r e g i o n shown i n Figure 5 b ) . W i t h 1 = 3 and J! = 1/25 we o b t a i n an a m p l i f i c a t i o n f a c t o r whose magnitude v a r i e s w i t h 5 as shown i n F i g u r e 5 ( b ) .
where f ( x , y ) a r i s e s from t h e F o u r i e r t r a n s f o r m o f u ( ~ - ' ) . We now f i n d t h e g i s no l o n g e r a f u n c t i o n o f t h e s i n g l e m m p i e x v a r i a b l e z and i t t o i l o w s t h a t ( g l may exceed t h e v a l u e one a t i n t e r i o r p o i n t s o f t h e s t a b i i i t y region. A l t h o u g h c a r e must be t a k e n when d e r i v i n g schemes W i t h o u t e v a l u a t i o n o f t h e d i s s i p a t i o n a t every stage, it i s n o t d i f f i c u l t t o d e v l s e schemes which s t i l l have w e l i behaved s t a b i l i t y r e g i o n s . For example, t h e stand a r d c o e f f i c i e n t s ( 9 ) now produce a scheme w i t h t h e s t a b i l i t y r e g i o n Shown i n F i g u r e 2 ( a ) . This a i m Shows t h a t t h i s form o f scheme i s s t a b l e f o r a C o u r a n t number o f 2.6 w i t h t h e same amount o f dissipation. The c o r r e s p o n d i n g v a r i a t i o n o f t h e a m p l i f i c a t i o n f a c t o r 191 w i t h E. i s shown i n 2 ( b ) .
I n a d d i t i o n we can a l s o i n c l u d e i m p l i c i t smoothing' which p e r m i t s s t a b l e o p e r a t i o n a t C o u r a n t numbers beyond t h e o r d i n a r y C o u r a n t number I i m i t o f t h e e x p l i c i t scheme. i n f a c t , the r e s u l t s p r e s e n t e d i n S e c t i o n 5 have been c a l c u l a t e d u s i n g t h e f i v e s t a g e scheme w i t h c o e f f i c i e n t s g i v e n by E q u a t i o n (191, two e v a l u a t i o n s o f d i s s i p a t i o n and a C o u r a n t number o f around seven. The Smoothing c o e f f i c i e n t d e s c r i b e d i n R e f e r e n c e 7 h a s been g e n e r a l i s e d t o a l l o w d i f f e r e n t amounts o f smoothing i n each c o o r d i n a t e d i r e c t i o n . The v a l u e s used t o r u n a t a C o u r a n t number of around seven were c i = 0.5 and E . = 0.3 where t h e s u b s c r i p t i r e f e r s J t o t h e m s h l i n e which wraps around t h e a i r f o i l and j r e f e r s t o the r a d i a l m s h line. In t h e t h r e e d i m e n s i o n a l c a l c u l a t i o n s , t h e a d d i t i o n a l smoothing c o e f f i c i e n t ck f o r t h e spanwise d i r e c t i o n was g i v e n ' t h e v a l u e 0.2. The r a n g e o f p o s s i b l e schemes i s c o n s i d e r a b l e and t h e b e s t c h o i c e f o r m u l t i g r i d i s s t i I i under i n v e s t i g a t i o n .
We n o t e f r o m (16) t h a t g ( x , y ) has a z e r o on t h e r e a l a x i s a t x = -1. It f o l l o w s t h a t i f we a d j u s t t h e d i s s i p a t i o n or C o u r a n t number we can a r r a n g e for t h e l o c u s of z t o pass t h r o u g h t h i s zero. S i n c e t h e i n t e r c e p t on t h e r e a l a x i s Corresponds to C = n, we t h u s a c h i e v e a scheme w i t h good damping p r o p e r t i e s f o r t h e h i g h frequency components. From equation (14) it i s apparent t h a t t h e choice a = 2, p = 1/32 w i l i cause t h e locu5 o f I t o pass t h r o u g h t h i s zero.
4. However we a r e s t i i i f r e e t o a d j u s t the c o e f f i c i e n t s o t t h e scheme and can t h e r e f o r e f u r t h e r improve i t s h i g h frequency damping b e h a v i o u r . For example, When t h e f o u r s t a g e scheme i s used w i t h a s i n g l e e v a l u a t i o n o f t h e d i s s i p a t i v e t e r m s . a good c h o i c e of t h e c o e f f i c i e n t s i s a 1 = .25,
a2 = .5,
a3 = .55
(17)
The r a t e o f convergence t o a s t e a d y state w i l l !be i m p a i r e d if outgoing waves ace r e f l e c t e d back i n t o t h e f l o w from t h e o u t e r b o u n d a r i e s . The ' t r e a t m e n t o f t h e f a r f i e l d boundary c o n d i t i o n i s :based on t h e I n t r o d u c t i o n o f Riemann i n v a r i a n t s f o r a one d i m e n s i o n a l flow normal t o t h e boundary. Let 'subscripts and e denote f r e e s t r e a m values, and v a l u e s e x t r a p o l a t e d from t h e i n t e r i o r c e l l s a d j a c e n t t o t h e boundary, and l e t q, and c be t h e v e i o C i t y component normal t o t h e boundary and t h e speed of sound. Assuming t h a t t h e f l o w i s s u b s o n i c a t i n f i n i t y , we i n t r o d u c e f i x e d and e x t r a p o l a t e d Riemann i n v a r i a n t s
lgI
a2 = .6
-
(181
g e n e r a t e The s t a b i l i t y r e g i o n s h m n i n F i g u r e L ! e > . W i t h 1 = 1.5 and I ; = 1/25 we o b t a i n an a m p l i f i c a t i o n f a c t o r whose magnitude v a r i a t i o n w i t h wave number i s shown i n F i g u r e 4 ( b ) .
R, = Q,, - & Y-1
F u r t h e r v a r i a t i o n s on t h i s theme a r e p o s s i b l e . F o r example, w i t h e v a l u a t i o n of t h e d i s s i p a t i o n a t t h e f i r s t and second s t a g e s o n l y , we o b t a i n t h e f o l l o w i n g form f o r t h e a m p l i f i c a t i o n f a c t o r
g(x,y)
c o r r e s p o n d i n g t o incoming and o u t g o i n g waves. These my be added and s u b t r a c t e d t o g i v e
2
= I + x t a1x + iy if(x,y)+elxl and
where a g a i n f ( x , y ) a r i s e s from t h e F o u r i e r t r a n s i n t h i s case t h e r e a r e two zeros f o r m Of U(k-l). on t h e r e a l a x i s . The p a r t i c u l a r c h o i c e e l = 1/4 l e a d s to a d o u b l e z e r o a t x = -2 which m i g h t be e x p e c t e d t o produce a scheme w i t h good damping over
c =
a1
=
1/4.
a2 = i / 6 ,
a , = 3 / 8 , a4 = 1/2
$ (Re
-
%)
where q, and c a r e +he a c t u a l normal v e l o c i t y cornponent and speed o f sound t o be s p e c i f i e d i n t h e far field. A t an o u t f l o w boundary. t h e t a n g e n t i a l v e l o c i t y component and e n t r o p y a r e e x t r a p o l a t e d from t h e i n t e r i o r , w h i l e a t an i n f l o w boundary t h e y a r e s p e c i f i e d as h a v i n g f r e e s t r e a m v a i u e s . These four q u a n t i t i e s Provide a complete d e t i n i t i o n o f
a b r o a d neighborhood o f t h i s zero. For exampie t h e 5 s t a g e Scheme w i t h two e v a l u a t i o n s o f d i s s i p a t i o n and c o e f f i c i e n t s 1191
9
u
Boundary C o n d i t i o n s
A t a s o l i d boundary t h e o n l y c o n t r i b u t i o n t o t h e f l u x b a l a n c e comes f r o m t h e p r e s s u r e . The normal p r e s s u r e g r a d i e n t ap/an a t t h e w a l i can be e s t i m a t e d from t h e c o n d i t i o n t h a t a/at(pq,) = 0, w h e r e qn i s t h e normal v e l o c i t y component. The ! p r e s s u r e a t t h e w a i l i s t h e n e s t i m a t e d by e x t r a p o l a t i o n f r o m t h e p r e s s u r e a t t h e a d j a c e n t c s l i =en' t e r s , u s i n g t h e known v a l u e o f ap/an.
The r e s u l t i n g s t a b i l i t y r e g i o n f o r t h e m d e l p r o b l e m ( 1 2 ) Is shown i n F i g u r e 3 t a ) . From F i g u r e 3 ( b ) it can a l s o be seen t h a t w i t h a C o u r a n t number 1 = 2 and a d i s s i p a t i o n y = 1/32, t h e m a g n i t u d e of t h e a m p l i f i c a t i o n f a c t o r i s l e s s t h a n .35 for a l I wave numbers i n t h e r a n g e 1.0 < 5 < n. Another Scheme which has p r o v e d very e f f e c t i v e has t h r e e stages w i t h a s i n g l e e v a l u a t i o n o f t h e d i s s i For t h i s scheme t h e c o e f f i c i e n t s pation. a 1 = .6,
/.
'
: ' W
c
'the f l o w i n t h e f a r f i e l d . i f t h e f l o w i s supersonic i n t h e f a r f i e l d , a l I t h e f l o w q u a n t i t i e s a r e s p e c i f i e d a t an i n f l o w boundary and t h e y a r e e x t r a p o l a t e d f r o m t h e I n t e r i o r a t an O U t t l O w Doundary.
v
5.
Our l a s t example I s t h e b e i n g 747 w i t h o u t n a c e l l e s ; t h e d i s t r i b u t i o n of mesh c e l I f a c e s on t h e a i r c r a f t i s shown i n F i g u r e 13. The c o n s t r u c t i o n o f t h e C-H mesh, which has 9 6 x 1 6 ~ 1 6c e i i s , i s d e s c r i b e d In R e f e r e n c e 7. i n F i g u r e 14 t h e wing p r e s s u r e s o b t a i n e d from a u i n g / b o d y / c a l c u l a t i o n a r e shown f o r Mach number o f 0.84 and an a n g l e o f a t t a c k of 2.44 degrees. The convergence h i s t o r y for t h i s case i s p r e s e n t e d i n F i g u r e 15. The mean e r r o r r e d u c t i o n r a t e Over 200 cycles i s .95 and t h e s u p e r s o n i c zone has s e t t i e d a f t e r o n l y 100 c y c l e s .
Results
The m u l t i g r i d scheme has been coded f o r t w o d i m e n s i o n a l a i r f o i l c a l c u l a t i o n s i n b o t h an C-mesh and a C-mesh v e r s i o n and i n t h r e e d i m e n s i o n s f o r c a l c u l a t i n g f l o w s o v e r wings and w i n g / b o d y / t a i l / f i n combinations. The a i r f o i l C a i c u i a t i o n s were a i l c a r r i e d o u t u s i n g a m s h w i t h 160 i n t e r v a l s i n t h e d i r e c t i o n around t h e a i r f o i l and 32 i n t e r v a l s i n the radial direction. In each case two i n d i c a t o r s o f t h e convergence r a t e a r e p r e s e n t e d . The f i r s t i s t h e decay o f t h e l o g a r i t h m o f t h e e r r o r where t h e e r r o r i s w a s u r e d by t h e root m a n square r a t e o f change o f d e n s i t y on t h e f i n e g r i d . The second i s t h e b u i l d - u p o f t h e number o f p o i n t s i n t h e supersonic zone. For a t r a n s o n i c f l o w t h i s i n d i c a t e s how q u i c k l y t h e S u p e r s o n i c zone d e v e l o p s and I s a u s e f u l m a s u r e o f t h e g l o b a l convergence o f t h e flowfield.
Conclusion These r e s u l t s c l e a r l y d e m o n s t r a t e t h a t t h e ionv e r g e n c e of a t i m e dependent h y p e r b o l i c system t o a s t e a d y s t a t e can be s u b s t a n t i a l l y a c c e l e r a t e d by t h e introduction of m u l t i p l e grids. The s t e a d y s t a t e i s d e t e r m i n e d by t h e space d i s c r e t i z a t i o n on t h e f i n e g r i d , and i s independent o f b o t h t h e t i m e s t e p p i n g scheme and t h e d i s c r e t i z a t i o n p r o c e d u r e The p r e s e n t f i n e g r i d used on the c o a r s e g r i d s . d i s c r e t i z a t i o n scheme i s second o r d e r a c c u r a t e i n smooth r e g i o n s o f f l o w and t h e C o n t r o l l e d a d d i t i o n o f f i r s t o r d e r d i s s i p a t i o n i n t h e neighborhood o f shockwaves leads t o good shock c a p t u r i n g p r o p e r ties. The a b i l i t y of t h e scheme t o a c h i e v e a shock-free t r a n s o n i c flow a t t h e design c o n d i t i o n f o r a s h o c k f r e e a i r f o i I and t h e a p p a r e n t insens i t i v i t y t o m s h S t r u c t u r e . d e m o n s t r a t e d by C-mesh and C-mesh comparisons, p r o v i d e s c l e a r e v i d e n c e of t h e a c c u r a c y of t h i s method. C a l c u l a t i o n s of t h e f i o v over t h r e e dimensional geometries i n c l u d i n g wing/body c o n f i g u r a t i o n s show t h a t t h e convergence speed o f t h e m u l t i g r i d t e c h n i q u e combined w i t h t h e a c c u r a c y o f t h e space d i s c r e t i z a t i o n e x t e n d s t o t h e t r e a t m e n t o f complex aerodynamic shapes.
F i g u r e 6 p r e s e n t s t h e p r e s s u r e d i s t r i b u t i o n for t h e NACA 0012 a i r f o i i a t Mach .8 and an a n g l e o f a t t a c k o f i.25O. T h i s p a r t i c u l a r c a s e was c a l c u l a t e d on a C-mesh. However t h e C-mesh r e s . u l t o v e r p l o t s it and g i v e s a d r a g c o e f f i c i e n t which d i f f e r s by o n l y two d r a g c o u n t s from t h e C-mesh c a i c u i a tion. The convergence h i s t o r y on t h e 0-mesh f o r a r u n o f 300 c y c i e s i s p r e s e n t e d i n F i g u r e 7 and t h e c o r r e s p o n d i n g convergence i n f o r m a t i o n f o r t h e Cmesh i s shown i n F i g u r e 8. In b o t h cases t h e r e i s a s t e a d y r e d u c t i o n i n t h e e r r o r and t h e s u p e r s o n i c zone I s e v i d e n t l y f r o z e n i n l e s s t h a n 100 cycles. Cooveraence on t h e C-mesh i s D a r t i c u i a r l v f a s t a m a n convergence r a t e o f u n d e r ' .92 o v e r
showing
\c/
Acknowledqments
.\
300 cycles w i t h t h e number o f s u p e r s o n i c p o i n t s s t e a d y a f t e r no m r e t h a n 50 c y c l e s .
,
S u p p o r t for t h i s work has been p r o v i d e d by t h e NASA Langley Research C e n t e r under G r a n t NAG-1-186, t h e NASA Ames Research C e n t e r under G r a n t NAG-2-96 and the of Naval Research under Grant N00014-81-K-0379.
The n e x t example i s t h e KORN a i r o i i a t a shock-free condition. T h i s 1s known t o be a sens i t i v e t e s t f o r f l o w f i e l d methods s l n c e s m a l l chanqes i n t h e boundary c o n d i t i o n s can i n t r o d u c e a shockwave. Moreover, what appears to be a s m a l l n u m e r i c a l e f f e c t can a i s 0 r a d i c a l l y a l t e r t h e p r e s s u r e d i s t r i b u t i o n , and r e l 2 - i v e i y m a i l amouits o f a r t i f i c i a l d i s s i p a t i o n can p r e v e n t t h e appearance of a s h o c k - f r e e f l o w . The e s s e n t i a l l y shock-free c h a r a c t e r o f t h e pressure d i s t r i b u t i o n shown I n F i a u r e 9 w i t h a p r e d i c t e d d r w c o e f f i c i e n t of only f i v e c o u n t s 1 5 a g r a t i f y i n g i n d i c a t i o n o f The convergence t h e a c c u r a c y t h a t can be o b t a l n e d . h i s t o r y i s p r e s e n t i n F i g u r e 10 and t h e m a n conv e r g e n c e r a t e o f t h i s C-mesh c a l c u l a t i o n i s below .93 o v e r 300 c y c l e s . Over t h e f l r s t 100 c v c l e s . however, t h e &an convergence r a t e I s b e t t e r t h a n .9 for b o t h t h e NACA 0012 and t h e KORN a i r f o i l calculations.
-
References
We now c o n s i d e r 3-0 C a i c u i a t i a n s and as a f i r s t example show t h e p r e s s u r e d i s t r i b u t i o n o v e r t h e at Mach .84 and an angle of attack of ONERA M6 3.06O. The r e s u l t which i s d i s p l a y e d i n F i g u r e 11 was c a l c u l a t e d on a mesh of 8 0 ~ 1 6 x 1 6cel I s . Convergence i n f o r m a t i o n appears i n F i g u r e 12 and although the e r r o r reduction r a t e deteriorates to some e x t e n t a f t e r t h e f i r s t 100 c y c l e s , it i s c l e a r t h a t t h e s u p e r s o n i c zone has f r o z e n i n j u s t o v e r 50 cycies.
1.
B r a n d t , A., " M u l t i - l e v e l Adaptive S o l u t i o n s t o Boundary-Value Problems", Math. Comp., VoI. 31, No. 138, pp. 333-390, 1977.
2.
B r a n d t , A., " M u l t i - l e v e l A d a p t i v e Computations i n F l u i d Dynamics", A i A A 4 t h C o m p u t a t i o n a l F l u i d Dynamlcs Conference, W i l l i a m s b u r g , VA, AlAA Paper 19-1455, J u l y , 1979.
3.
South, J. C., and B r a n d t , A., "The M u l t i g r i d Method: F a s t R e l a x a t i o n for T r a n s o n i c Flows", 1 3 t h Annual M e e t i n g of S o c i e t y o f E n g i n e e r i n g Science, Hampton, VA., November 1976.
4.
Y 5
Jameson, A.. *%ccelera*ion of Transonic P o t e n t i a i Flow C a l c u l a t i o n s on A r b i t r a r y Meshes by t h e M u l t i p l e G r i d Method", AlAA 4 t h C o m p u t a t i o n a l F l u i d Dynamics Conference, W i l l i a m s b u r g , VA, AlAA Paper 79-1458, J u l y 1979.
,,
1
I.
5.
Caughey, D.A., " M u l t i g r i d C a l c u l a t i o n of Three D i m e n s l o n a l T r a n s o n i c P o t e n t i a l F l o w s " , AlAA Paper 83-0374. 1983.
6.
N i , R. H., "A M u l t i p l e G r i d Scheme for S o l v i n g t h e E u l e r Equations". A I A A 5 t h Computational F l u i d Dynamics Conference, P a l o A l t o , 1981.
7.
Jameson, A. and B a k e r , T. J., "Solution of t h e E u l e r E q u a t i o n s f o r Complex C o n f i g u r a t i o n s " , A l A A 6 t h C o m D u t a t i o n a I F l u i d Dvnamics C o n f e r e n c e , banvers, MA, A I A A Paper 83-1929, J u l y 1983.
8.
Jameson. A . , " 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 for Two-Dimensional T r a n s o n i c F l o w by a M u l t i g r i d Method", p r e s e n t e d a t t h e I n t e r n a t i o n a l M u l t i g r i d C o n f e r e n c e , Copper M o u n t a i n , 1983 ( s e e a l s o P r i n c e t o n U n i v e r s i t y , MAE R e p o r t No. 1613. June 1983).
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1
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Figure l ( a )
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S t a b i l i t y Region of S t a n d a r d 4 Stage Scheme c o n t o u r lines 1.. .9, . 8 , plus locus o f z(F) for A = 2.8, u = 1 1 3 2 c o e f f i c i e n t s a1 114, 0 2 113, 0 3 = 112
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Figure l ( b )
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A m p l i f i c a t i o n F a c t o r lg o f S t a n d a r d 4 Stage Scheme for A 2.8, u 1132 C o e f f i c i e n t s a, 1 1 4 , a 2 = 1 1 3 , a3 112
6
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Figure 2 ( a ) Stability Region of 4 Stage Scheme with Single Evaluation of Dissipation Contour lines lgl = I . , . 9 , .8, plus 1 0 ~ of ~ 6z(:) for 1 2 . 6 , P 1132 Coefficients a1 114, 02 1 1 3 , a3 = 112
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--
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Figure 2 ( b ) Amplification Factor I g of 4 Stage Scheme with Single Evaluation of Dissipation for A = 2.6,
LJ = 1 / 3 2
Coefficients o 1 = 1 / 4 , a2 = 1 / 3 . a 3 = 1 / 2
v
Figure 3(a) Stability Region of 4 Stage Scheme with Single Evaluation of Dissipation and Modified Coefficients contour lines lg) = I . , .9, . R , plus locus of z ( 5 ) for A 2., P 1132 Coefficients a 1 .25, 02 = .5, 03 .55
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Figure 3 ( b ) Amplification Factor IgI of 4 Stage Scheme with Single Evaluation of Dissipation and Modified Coefficients f o r A
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2 , P = 1/32
Coefficients a l = . 2 5 , a2
. 5 , a 3 = .55
. J Figure 4 ( a ) Stability Region of 3 Stage Scheme with Single Evaluation of Dissipation contour lines J g / = 1.. .9, .8, p l u s locus of z(S) for A = 1.5, u = 1/25 Coefficients a l = .6, a2 = .6
Figure 5(a) Stahility Region of 5 Stage Scheme vith Tvo Evaluations of Dissipation Contour l i n e s l g I 1.. . 9 , .fl, plus locus of z ( 5 ) for A = 3, P = 1125 Coefficients o 1 114, a 2 116, a3 318, a4 112
Figure 4 ( b ) Amplification Factor l g I of 3 Stage Scheme vith Single Evaluation of Dissipation for k = 1.5. u = 1/25 Coefficients a1 .6, a2 = .6
Figure 5(h) Amplification Factor l g l of 5 Stage Scheme w i t h TWO Evaluations of Dissipation for a 3, u = 1/25 coefficients a l = 114, a2 1 1 6 , a3 = 3 1 8 , 0 4 = 112
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