Multigrid solution of the Euler equations for aircraft configurations

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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

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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 .

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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=

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pHv,

pHw1

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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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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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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

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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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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

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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

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. 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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