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Journal of Wind Engineering and Industrial Aerodynamics, 33 (1990) 139-152 139Elsevier Science Publishers B.V., Amst erda m -- Printe d in The Nethe rlands

nvited Paper

N U M E R I C A L S I M U L AT I O N O F T U R B U L E N T F L O W F I EL D A R O U N D C U B ICM O D E L C U R R E N T S T AT U S A N D A P P L IC AT IO N S O F k M O D E L A N D L E S

S. MURAKAMI

Institute of lndustrial Science, University of Tokyo, Tokyo Japan)

Summaff

The state of the art of numerical meth od is reviewed briefly from the viewpoi nt of wind engineering.Next the diagnostic system for assessing the results of numerical simul ation is described and errorestimati on and mes h resolution are discussed. Lastly the time-d ependent flowfield is predicted b y 3DLarge Eddy Simulation and the results are illustrated using the techniques of animate d graphics.

KeywordsNumerical simulation, cubic model k-e model, LES, computer graphics

1. INTRODUCTION

\ m et ho d o f n u m e r i c a l s i m u l a t i o n f o r p r e d i c t i n g t h e v e l o c i t y f i e l d a ndp r e s s u r e f i e l d a r o u n d a c u b i c - s h a p e d b l u f f b od y i s i n v e s t i g a t e d i n t h i s p a p e r.

I n t h e f i e l d o f e n g i n e e r i n g , n ew a p p r o a c h e s b a s e d o n n u m e r i c a l m e t h od s h a veb e en d e v e l o p e d r a p i d l y i n r e c e n t y e a r s . T h e s e a p p r o a c h e s w er e made p o s s i b l e b yt h, , r e c e n t , r a p i d a d v a n ce of c o m p u t e r t e c h n o l o g y.

lhe a i r f l o w a r o u n d a b l u f f b od y p l a c e d w i t h i n a s u r f a c e b o u n d a r y l a y e r i sf u l l 3 t u r b u l e n t a n d v e r y c o m p l i c a t e d . I t i s c o mp o se d o f s e p a r a t i o n s a t t h ew i nd w ar d c o r n e r s o f t h e b l u f f b o d y, t h e c i r c u l a t i o n s b e h i n d i t , e t c . T h e r e a r emany d i f f i c u l t i e s i n c l a r i f y i n g t h e s e c om p l ex f l o w f i e l d s u s i n g u s u a l w in d t u n n e lt e c h n i q u e s . T h e r e f o r e t h e e s t a b l i s h m e n t o f a t h r e e - d i m e n s i o n a l n u m e r i c a l m e t h o df o r s i m u l a t i n g t u r b u l e n t f l o w i s r e q u i r e d f o r u s e i n t h e f i e l d o f w i nde n g i n e e r i n g .

2. EXPERIMENTAL STUDY AND NUMERICAL SIMULATION

T he r e s u l t s o f n u m e r i c a l s i m u l a t i o n c a n n o t b e f r e e f r o m v a r i o u s t y p e s o fn u m e r i c a l e r r o r s . T h e r e a r e m a ny f a c t o r s w h i c h p r o d u c e n u m e r i c a l e r r o r s f o re x a m p l e u n f i tn e s s o f t h e t u r b u l e n c e m o d e l l i n g i n a c c u r a c y o f t h e f i n i t ed i f fe r e n c e s c he m e t h e m a c r o s c o p i c t r e a tm e n t o f v i s c o u s s u b l a y e r a t s o l i d w a l lb o u n d a r y e t c . T h e r e f o r e i t i s i n d i s p e n s a b l e t h a t t h e a c c u r a c y o f n u m e r i c a ls i m u l a t i o n b e e x a m i n e d b y c o m p a r i n g t h e n u m e r i c a l r e s u l t s w i t h t h o s e f r o m w i n dt u n n e l t e s t s o r f i e ld e x p e r i m e n t s . T h e n u m e r i c a l m e t h o d c a n n o t be d e v e l o p e da n d r e f i n e d w i t h o u t t h e s u p p o r t o f a c c u r a t e e x p e r i m e n t s .

O n t h e o t h e r h a nd t h e g r o w i n g p r e c i s i o n o f p r e d i c t i o n b y n u m e r i c a l

s i m u l a t i o n a l s o i n c r e a s e s t h e i n c e n t i v e f o r n e w r e s e a r c h i n t o e x p e r i m e n t a lm e t h od s . T h e r e f o r e t h e t wo d i f f e r e n t m e t h od s o f r e s e a r c h s h o u l d p r o c e e d i nc o n c e , ' t a nd i n c o o p e r a t i o n w i t h e ac h o t h e r.

0167-6105/90/$03.50 1990 Elsevier Science Publ ishe rs B.V.

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3. STATE OF THE ART OF NUMERICAL METHODS

3 , 1 R e y n o l d s Av e r a g i n g , F i l t e r i n g a n d Tu r b u l e n c e M o d e l l i n g

l h e N a v i e r - S t o k e s e q u a t i o n f o r i n c o m p r e s s i b l e f l o w i s a s f o l l o w s ._~ u , _ + a a , , ~ , 1 a + a_ _ ( a u , + a u ~a t a x , /~ 8 x , ~ a x , a x j 8 x , ) ( 1 )

I t i s g e n e r a l l y i m p o s s i b l e t o d i r e c t l y a n a l y s e e q u a t i o n ( 1 ) i n i t s o r i g i n a lf o r m b y t h e n u m e r i c a l m e t h o d f o r a f l o w f i e l d w i t h l a r g o R e y n o l d s n u m b e r b e c a u s et h a t t y p e o f s i m u l a t i o n , c a l l e d d i r e c t s i m u l a t i o n , c a l l s f o r a n u n r e a s o n a b l yl a r g e n u m b e r o f g r i d p o i n t s w h i c h c a n h a r d l y b e a c o o m o d a t e d b y e x i s t i n gc o m p u t e r s . T h e r e f o r e , s o m e t y p e o f a v e r a g e d v e r s i o n o f e q u a t i o n ( l ) i s u s u a l l yu s e d f o r n u m e r i c a l a n a l y s i s .

T b e v a r i a b l e t~ ~ i s d e c o m p o s e d i n t o t w o p a r t s , m e a n v a l u e a n d d e v i a t i o n .u , : u ~ u , ( 2 )

T h e n o n l i n e a r t e r m u , u ~ i n e q u a t i o n ( 1 ) i s e x p r e s s e d a s b e l o w b y a v e r a g i n go r f i l t e r i n g .

. , u , : u , u , + u , u ~ ' + { u , u , - u , u ~ } + { ~ h - , u , ' + . , ' ~ - , } ( 3 )I I t h e o p e r a t o r o v e r b a r i s r e g a r d e d a s t i m e a v e r a g i n g ( o r R e y n o l d s

a v e r a g i n g ) , t h e t h i r d a n d f o r t h t e r m s o f e q u a t i o n ( 3 ) b e c o m e z e r o . B u t i n t h ec a s e o f f i I t e r i n g t h e y a r e n o t z e r o . T h e n t h e f o l l o w i n g e q u a t i o n s a r e g i v e n .

1) in c a s e o f R e y n o l d s a v e r a g i n g :

$ ~ ~ + 8 u , u /__1 8 P + ao4-7 ~ ~ , a x , a x , { ~ (

2 ) i n c a s e o f f i l t e r i n g :8 ~ , + a u , u , 1 a P + a {~8 l a x p a x , a x ,

h r R (' , , , = - u , ' u , ' ,

L , = - ( ~ - , u ~-u , u ,) .

C , ' . . = - ( ~ - , u , "+u , ' ~ - , )

E q u a t i o n ( 4 ) i s u s e d f o r O - e q u a t i o n - - = 0o r t w o - e q u a t i o n t u r b u l e n c e m o d e l s , a x ,E q u a t i o n ( 5 ) i s u s e d for" L a r g e E d d y a u , 8 u , u ,S i m u l a t i o n . a t a x ,

i n t h e c a s e o f d i r e c t s i m u l a t i o n ( o rf u l l s i m u l a t i o n ) , e q u a t i o n ( I ) i s u s e d .D i r e c t s i m u l a t i o n i s a p p l i e d t o ak al-~,f l o w f i e l d s w i t h a r e l a t i v e l y l o w R e y n o l d s ~ - + 8 x ,n u m b e r ( b e l o w 5 x l O * o r s o ) .

~ 8~ ,

3 . 2 C l o s u r e P r o b l e m s o f Tu r b u l e n c e a t a x ,

k '= C ~ - -6

Mode I 1 i ng

Ira t h e c a s e o f R e y n o l d s a v e r a g i n g ,- u , ' /x ~ ' i s mode I I ed by me an s of an ed dyv i s c o s i t x m o d e l i n o r d e r t o c l o s e t h ee q u a t i o n (/+).- u , u . = v , ( 8 u - , + 8 ~ - , ) _ 3 ~ ( 6 )

a x / a x , Jh e r e ~ : e d d y v is c o s i t y 1 - -

b : t u r b u l e n c e k i n e t i c e n er g y= ~u , . t~ ~

A n e w. unknown va r i ab le vt a p p e a r si n e q u a t i o n ( 6 ) . I n t h e c a s e o f a k - et y p e t w o - e q u a t i o n t u r b u l e n c e m o d e l

( h e r e a f t e r a b b r e v i a t e d a s k - m o d el ) ,e q u a t i o n ( 6 ) i s c l o s e d b y i n t r o d u c i n g n e wv a r i a b l e s k a n d c , whe re v i s de f in ed

k ~a s C 0 - - ] 'h e s e t o f e q u a t i o n s i s

6s h o w n i n Ta b l e i ( L a u n d e r e t a l [ 1 97 2 ] ) .

a u ,+ a u ~ _u , u ,}X j X l

m

8 u , 8 u J + R e , , + L , j + C ; , ( 5 )8 x , 8 x ,

Ta b l e 1 M od el e q u a t i o ns f o r k c

a 4 ,x+ 8 ~ , I a ~ , a ~ ,

a k

a x , ~ , - ~ x , ) + C 7 * S - C T - ( ~

here S = (-~---- ~ t'-~m-~)-z--:-dX ~X, . ~j

a : =1.0. a 2=1.3.Co=0.09. C,=1.4 4. C,=1.92

T a b l e 2 M o d e l e q u a t i o n s f o r L E S

8 &m = o @

a t & r s = - 8 ~ ' z , ~ z s, v , u se s s e u

/ ~ : = \L i t :h e r e ~ , ~ = ( C z 1 Y- ~ - - ~ - ) , c = o .1

e ah~ au ju = ' ~ z j + a z,-

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I n t h e c a s e o f L a r g e E d d y S i m u l a t i o n ( h e r e a f t e r L E S ) , e q u a t i o n ( 5 ) i s u s e df o r t il e a n a l y s is . - u ~ i s m o d e l l e d b y m e a n s o f t h e S m a g o r i n s k y e d d yv i s c o s i t y m o d e l .

8 ~ + 8 ~ J ) - ~ k ~ ~ ( 7 )- i t , u ~'= ~sos a x , a x , - - t / 2= ~ a TF O T F, + a uiPSQS= Cs~) {O x ~ ( O x j O x ,h e r e C s : O . 1 ( t h e S m a g o r i n s k y c o n s t a n t )

: ( ~ x . b y . A z ) , / a

C r o s s t e r m C j j i n e q u a t i o n ( 5) i s u s u a l l y n e g l e c t e d . L e o n a r d t e r m Li j i sr e g a r d e d a s p r o gr a m me d i m p l i c i t l y b y t h e s e c o n d o r d e r c e n t r a l d i f f e r e n c e o fcon v e c t i o n t e r m . The s e t o f equa t i ons fo r LES i s shown in Tab l e 2 (Sma gor in sky[ 1 9 6 3 1 ) .

T he t r a n s p o r t e q u a t i o n f o r u ' ~ u ' i i s s o l v e d d i r e c t l y i n t h e c a s e o ft h e d i l l e r e n t i a l s t r e s s m od el o r t h e a l g e b r a i c s t r e s s m o d el . T h e r e f o r e a n e d d yv i s c o s i L y m o d e l i s n o t n e e d e d i n t h o s e m e t h o d s .

3 . 3 E v a l u a t i o n o f Tu r b u l e n c e M o de l s

The va r i ous t ype s o f t u rb u l enc e mode l s shown above a r e compa red ande v a l u a L e d f r om t h e v i e w p o i n t o f w i n d e n g i n e e r i n g .

( 1 ) D i r e c t s i m u l a t i o nT h i s m od el i s a p p l i e d t o p r i m i t i v e , s i m p l e t y p e s of f l o w f i e l d s w i t h

a r e l a t i ve l y l ow Reyno ld s number, f o r examp le , chan ne l f l o ws , e t c (KimeL a l [ 1987 [ ) . I n a s t r i c t s ens e , i t i s no t e a sy t o app ly t h i s me thodLo p r o b l ems o f w ind eng i nee r i ng , whe re t he Rey no ld s number i s a lwaysx e r y l a rg e . S o m e t i m e s q u a s i - d i r e c t s i m u l a t i o n i s c o n d u c t e d u s i n g t h econv e c t i o n t e rm o f t h i rd o rde r upwind s cheme (Tamura e t a l [ 1987 ] ) .

2) LES

T h i s m e t h o d m a y b e a p p l i e d t o b a s i c f l o w s i n t h e f i e l d o f w i n deng ine er i ng (De ar dor ff [19701 , Schumann [ 1 9 7 5 1 H o r i u t i [ 1 9 8 2 1 Main e t~ l [1982] . Murakami e t a l [198 1 , Murakami e t a l [198 7]) . Thi s modelh a s t h e a d v a n t a g e o f g e n e r a t i n g a t i m e - d e p e n d e n t f l o w f i e l d . I t sd i s a dva n t age i s t ha t i t r equ i r e s a g r ea t d ea l o f CPU t ime .(3 ) O - equ a t i on mode l

T h i s m o d e l, w h i c h i s r a t h e r c r u d e , ma y b e u s e d v e r y c o n v e n i e n t l yt o g i v e a r o u g h p r e d i c t i o n a t a n e a r l y s t a g e o f r e s e a r c h . I t i s h i g h l y~ ff i c i en t i n t ha t i t n eed s ve ry l i t t l e CPU t ime .( 4 ) k - m o d e l

T h i s mo d e l i s m o s t w i d e l y a p p l i e d t o e n g i n e e r i n g p r o b l e m s . T h i sm o d el h a s a g o o d r e p u t a t i o n f o r r e l i a b i l i t y a n d s e e m s t o b e m o s tp r o m i s i n g f o r p r e s e n t a p p l i c a t i o n t o m a n y p r o b l e m s i n t h e f i e l d o f w i n deng ine er i ng (Murakami a t a l [1983] , Young e t a l 119851, Pa te rso n e t a l11986] , Mathews e t a l [1987a , 1987b] , Bae tke e t a l [19871 , Murakami e t a l[1988b, 1988e, 1988f] ) .( 5 ) D i f f e r e n t i a l s t r e s s m o d e l, a l g e b r a i c s t r e s s m od el

These mode l s a r e more soph i s t i c a t ed t han t hos e shown above . AsL h e s e m o d e l s a r e e x p e c t e d t o b e a b l e t o a n a l y s e t h e t u r b u l e n t f l o w f i e l dv e r y p r e c i s e l y, i t i s h o p e d t o a p p l y t h e s e m o d e l s t o t h e p r o b l e m s i n w i nde n g i n e e r i n g i n t h e f u t u r e ( Mu ra k am i e t a l [ 1 9 8 8 g ] ) .

3 . 4 B a s i c E q u a t i o n s a n d B a s i c Va r i a b l e s f o r N u m e r i c a l S i m u l a t i o n

Two t y p e s o f v a r i a b l e - s e t s a r e u s e d i n n u m e r i c a l s i m u l a t i o n . T he f i r s t o n eu s e s t h e p r i m i t i v e v a r i a b l e s o f u ( v e l o o i t y ) - P ( p r e s s u r e ) . T he s e c o n d o n e i s t h e

( s t r e a m f u n c t i o n ) - ~ ( v o r t i c i t y ) s y s t e m . T he p r i m i t i v e e q u a t i o n s ( 1 ) - ( 5 ) u s et h e p r i m i t i v e v a r i a b l e s o f u - P. I n c a s e o f t h e ff - ~ s y s t e m , t h e e q u a t i o n o fw , w h i c h i s g i ven by t ak in g t he ro t a t i on o f equa t i on (1 ) and t he Po i s so ne q u a t i o n o f , w h e r e g e n e r a t i o n t e r m i s ~ , a r e u s e d .

q he u - P s y s t e m h a s b e c o m e v e r y p o p u l a r a n d n o w r e p r e s e n t s t h e m a i n s t r e a mo f n u m er i c a l m e t h o d o l o g y s i n c e t he M A C m e t h o d h a s be e n d e v e l o p e d. T h e r e l a t i v ea d v a n t a g e o f t he t t - P s y s t e m o v e r t h e - w s y s t e m l i e s in t h e e a s e w i t h w h i c h

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i t t r e a t s b o u n d a r y c o n d i t i o n s a nd i s a p p l i e d t 3D flow p r o b l e m s . The -~,,s y s t e m i s m or e a d v a n t a g e o u s i n t h e s e n s e t h a t t h e c o n t i n u i t y e q u a t i o n i ss a t i s t i e d e x a c t l y i n i t s c a s e .

3 . 5 M e t h o d o f D i s c r e t i z a t i o n f o r S p a c e D e r i v a t i v e

\ ~ r i o u s m e th o ds o f s p a c e d i s c r e t i z a t i o n a r e u s ed i n c o m p u t a t i o n a l f l u i dd y n a m i c s .

(1)FDM (fini te d i f f e r e n c e m e t h o d )( 2) FE M ( f i n i t e e l e m e n t m e t h o d )(3)CVM (con t ro l vo lume meth od)( 4 ) s p e c t r a l m e t h o d

l,~ t h e c a s e o f F DM, i t i s e a s y t o a s s e s s t h e a c c u r a c y o f d i s c r e t i z a t i o n ;t h e n a f o r e i t i s a l s o e a s y t o d e v e l o p a s ch em e w i t h a h i g h e r o r d e r a c c u r a c y i nth i s method . Th i s method i s mos t pop ula r.

FEM i s g o od i n m a t c h i n g t h e d i s c r e t i m a t i o n t o a c o m p l e x - s h a p e d b o u n d a r y. I nt h e c a s e o f FEM, l ar ge CPU memory st or ag e is u s u a l l y req u i r ed when compared wi t hFDM.

CVM may be reg ard ed as an in t egra ted ver s io n o f FDM. In CVM, the b as i cc o n c e p t o f t h e c o n s e r v a t i o n o f p h y s i c a l q u a n t i t y i s v e r y c l e a r. B e c au s e o f t h i sadv an t age th i s method has become the l ead ing method in the f i e ld o f CFD.

The spe c t r a l method i s good in sav ing CPU t im e and a l s o in acc ura cy. But i ti s d i f t i c u l t t o a p p l y t h i s m e t h o d t o p r o b l e m s i n w i n d e n g i n e e r i n g b e c a u s e i t i ss u i t a h l e o n l y f o r f l o w f i e l d s w i t h v e r y s i m p l e b o u n d a r y c o n d i t i o n s .

3 . 6 Method fo r D i s c r e t l z a t l o n fo r Ti m e D e r i v a t i v e

l im e d i s c r e t i z a t i o n m ay b e e x p r e s s e d i n g e n e r a l f o r m a s f o l l o w s .

. . . . ~ + ~ t ( 0 , . f ( ~ ) + ( i - 0 , ) f ( . . . . ) ) ( 8 )

here a, =l : e x p l i c i t methoO~a)

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3 . 7 F i n i t e D i f f e r e n c e S ch em e f o r C o n v e c t i o n Ter m a n d N u m e r i c a l Vi s c o s i t y

I n t h e s p a c e d i s c r e t i z a t i o n o f e a c h t e r m o f t h e b a s i c e q u a t i o n , t h ed i s c r e t i z a t i o n o f t h e n o n l i n e a r c o n v e c t i o n t e r m i s m os t i m p o r t a n t b e c a u s e t h et r u n c a t i o n t e r m s o f t h e d i s c r e t i z e d e x p r e s s i o n f o r t h e c o n v e c t i o n t e r m w o r k o f t e na s t h e n u m e r i c a l v i s c o s i t y . T h e y h a v e g r e a t i n f l u e n c e o n t h e r e s u l t s o fn u m e r i c a l s i m u l a t i o n .

T he w e l l - k n o w n f o r m s o f t h e f i n i t e d i f f e r e n c e s ch e m e f o r c o n v e c t i o n t e r m a r ea s f o l l o w s ( h e r e v e l o c i t y U i s a s s um e d t o b e p o s i t i v e ) .

( l ) c e n t r a l d i f f e r e n c e s c he m e( 8 U ) , = - U [- ' " - ' - ~

8 x h ]8 ~ I ~ I

= U [ ( ~ - ) + ~ ' h + 1 -E 5 h ~ + . . . . . ) ( 9 )

( 2 ) u p w i n d d i f f e r e n c e s ch em e ( f i r s t o r d e r )a L _ ~ . ~ , - - 6

- ( - - g ~ - - ) , - - c t h8 ~

: - u l ( ~ ) - ~ " h+ . . . . . ) (1o)( 3 )Q U I C K sc h e m e ( s e c o n d o r d e r u p w i n d s c h e m e )

U 1 39 ) = - U [ ~ ( ~ , . ,+ 3 -7 _ , ~ , -= ) ]T f x

8 1 ,,, 1: - U 1 ( -~ - - -7 - -) + ~ - ~ h ~ + - i - -6 h~ . . . . . ) ( i l )

The de r i v a t i ve o f even tb o rde r , f o r examp le ' , ~ . . . . , wo rks we l l a s t hen u m e r i c a l v i s c o s i t y b e c a u s e t h e o v o n t h o r d e r d e r i v a t i v e s p o s s e s s t h e s a mef u n c t i o n a s t h e d i f f u s i o n t e r m w h i c h i s c o m po s e d o f s e c o n d d e r i v a t i v e . T hec e n t r a l d i f f e r e n c e s c h e me h a s n o n u m e r i c a l v i s c o s i t y o f o v e n t h o r d e r. T he f i r s to r d e r u p w i n d s cheme has t he l a r ge numer i ca l v i s c os i t y o f ~ ' h / 2 . QUICK schemeh a s t h e t - e l a t i v o l y s m a l l n u m e r i c a l v i s c o s i t y o f - . . . . h 3 / 1 6 .

F i g . l c o m p a r e s n u m e r i c a l a n a l y s i s o f a n i n d o o r t u r b u l e n t f l o w f i o l d u s i n gd i f f e r e n t t y p e s of conv ec t i o n s chemes ba se d on a k - mode l .

F i g . l ( 2 ) (Murakami o t a l [ 1986} ) i s exac t l y s im i l a r t o t he exp e r ime n t a lr e s u l t s . T he c o m p l e t l y d i f f e r e n t f l o w f i o l d r e p r o d u c e d i n F i g . l ( 1 ) ( I s h i z u[1985 i ) shows t he e f f ec t o f numer i ca l v i s c os i t y c aus ed by t he f i r s t o rde r upwind

scheme w i t h c o a r s e m e sh .F i g . 2 s h o w s a n o t h e r e x a m p l e o f i n d o o r l a m i n a r f l o w f i e l d b a s e d o n d i r e c t

s i m u l a t i o n K a t e e t a l [ 19 80 ]) . I n t h i s c a s e , F i g . 2 l ) i s t h e c o r r e c t r e s u l t .

T h e a p p a r e n t t u r b u l e n t p h e n o m e n a i n F i g . 2 2 ) a n d ( 3 ) r e p r e s e n t n u m e r i c a li n s t a b i l i t y c a u s e d b y t r u n c a t i o n e r r o r s d u e t o o d d t h o r d e r d e r i v a t i v e s . When t h eR e y n o l d s n um be r i s s m a l l , t h e l a m i n a r f l o w f i o l d i s r e p r o d u c e d w e l l b e c a u s e t h ep h y s i c a l v i s c o s i t y i s domin an t . Bu t when t he Reyno ld s number becomes l a rg e , t hef l o w f i e l d b e co m es u n s t a b l e a s t h e p h y s i c a l v i s c o s i t y b e c o me s r e l a t i v e l y s m a l l a n dt h e t r u n c a t i o n e r r o r s a r e d o m i n a n t .

s u p p l y o u t l e t1 . I :~

---~-~ J ' , ,k . . . . ~

. , . . . . ~ .~ _ , , , , , .~ .P.

~ %iL'~ T, ' ' g ~ & ~ , , . , ~ . . .; _ - ~ - . m l ~ - . ~ .

I t -~ - ~ - - - . i ,~ , ; 2~~--.~.~.--.~-~

I ) R e = I O ( 2 ) R e = I O 0

F i g . 2

I L ~ I

( 3 ) R e = 2 0 0

I n d o o r l a m i n a r f l o w f i o l dw i t h d i f f e r e n t R e y n o l d s n u m b e r

a s e d o n 2 D d i r e c t s i m u l a t i o n .e s u l t s o f R o = IO i s c o r r e c t . Th o s e)f R e = lO 0 e n d 2 0 0 a r e i n c o r r e c t . .

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4 . D I A G N O S T I C SY S T EM FO R N U M E R IC A L S I M U L AT I O N

i ' h e n u m e r J o a J 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 c o n s t i t u te s a l a r g e s y s t e m

c o m p o s e d e l a g r e a t m a n y s u b s y s t e m s w h i c h r a n g er o m c o m p u t a t i o n a l m a t h e m a t i c s t ot h e f l u i d d y n a m i c s o f t u r b u l e n c e s t a t i s t i c s . I t i s v e r y d i f f i c u l t t o j u d g e t h e

d e g c e ~ o f a c c u r a c y o f t h e s i m u l a t i o n r e s u l t b e c a u s e t h e m a ny s u b s y s t e m s o f t h es i m u l a t i o n a r e u s u a l l y l e f t a s b l a c k b o x e s . H e r e , i t i s t h u s m o s t i m p o r t a n t t od e w ~ l~ , p a m ~ t h o d l o t e x a m i n i n g t h e a c c u r a c y o f t h e s i m u l a t i o n r e s u l t . T h i se x a m i n a t i o n m e t h o d i s h e r e c a l l e d t h e " d i a g n o s t i c s y s t e m " . i n t h i s p a p e r , w eg i x e tw o e x a m p l e s o f s u c h d i a g n o s e s . F i r s t i s a n e s t i m a t i o n o f e r r o r s c a u s e d b y'f i n i te d i f fe r e n c e . T h e s e c o n d i s t h e e f f e c t o f t h e f i n e n e s s o f m e s h r e s o l u t i o n

o r ] t h e s i m u l a t i o n , - e s u l t s . N u m e r i c a l s i m u l a t i o n s a r e h e r e c o n d u c t e d b y 3 D / e - ~

m o d e l ( M m - a k a m i e t a l [1988b] . S t a g g e r e d g r i d i s u s e d (M ACmethod, I i a r l o w e t a l[ 1 96 5 ] 1. A s i m u l t a n e o u s i t e r a t i o n m e t h o d f o r v e l o c i t y a n d p r e s s u r e i s a d o p t e d(,\B M .\(" m e t h o d , V i e c e l I i [ 1 9 7 11 ) .

4 . 1 E s t i m a t i o n o f T r u n c a t i o n E r r o r ~ ' ~ a n d S o l u t i o n E r r o r - ~

' [h e d i s t r ib u t i o n o f t w o t y p e s o f e r r o r s , n a m e l y , s o l u t io n e r r o r s a n dt r u n c a t i o n e r r o r s , a r e e s t i m a t e d b y R i c h a r d s o n e x t r a p o l a t i o n .

' [h ~ s o l u t i o n o H - o r i s t h e r e s i d u a l b e t w e e n t h e e x a c t s o l u t i o n a n d t h e

s o l t ~ t io n 8 i x e n b \ t h e f i n i t e d i f fe r e n c e m e t h o d . I n t h e m e t h o d u s e d h e r e , w e

I o H b

Iqqq~; J I I J ] J I l l i i ', ', ; k, d . . . . . . . . . : . . . . . ., . . . . iJJi~ll LlliUups=~ eam ~ U ~ l I I I I I b i g i i ~ i 4 ~ 2 . - ? , , P ? r ,~ o s n ~ a ~ [ ' downst ream , P i + I - i- H + H - H + } + ~H +I+ t'I-t+ H+ ~H -~ b bb bH +l +l +l +l -b H : ', ', ', I : ', ' , : ,~ ~~H~un~a ry`~T~ ~ ~; ~`~ ~ ~:~~g~7 ~`: ~::~ ~ ::~` ~ ~b~undar~

[ { l i i . . . . . . I . . . . . . . . , N 4 4 ~ . . . . ~ , , . . . . . . . . . ~ + . . . . . . . . . . . .. . . . . . . . . . . I I ' , ' , I ' , : ' ,~ ' ' , , , I ' , ' , ' , I ' , ' , ,q . . . . ' , ' , I ' , I ' , : ' , ' , ' , ' ,

c ub i c / g r o u n dm o d e l

F i g . 3 M e s h i ( v e r t i c a l p l a n e )

1 ) t r n n c a ~ i o n e r r o r T

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::~...~:. el ~ . . . . . . .

~ ' ~ \ \ \ \ ~ ~ . . : : : : i : i i :. . . . . . . . . . . . . : : ii " 'i i i i iiiiiii i iii i...

:: ~ : '..' t }~ ~ , ~ ~ ~ ' '- '

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assume that the solution error can be expressed as a Taylor series. If thefinite difference scheme has a second order accuracy, the solution error e A ofmesh size h can be easily estimated with a small amount of algebra as follows(Caruso et al 119861 , Murakami et al [1988a] ).

T7,= U,-L 21 /3. (12)where 1 h is the solution of mesh size h . and Uzl is the solution of mesh size

2h.

Tl-uncation errors arise in finite difference approximation. In operator

form. the differential equation to be solved is expressed as below.

L [CT ,X] =.f (13)whet-e C; c\ is the exact solution and L is the spatial differential operator.The truncation error T I is defined as the residual obtained by substituting theexact solution of the differential equation CJEX into the finite differenceequation. i.e..

T r=L 1 lUF.x]-L ICI,,]= I. L [G ,,I -f I (14)

where L h is tho spatial difference operator. Here the exact solution II,, isestimated h>,

G PXLI;..,=c: ,+e,. 15).

is used in place of exact solution UEx in eq.(14) to estimate thetruncation error:

r = L h [E ..X] -f 13)If a \ery accurate solution is obtained with mesh size h , the solution error7 L is estimated by eq. (12) to be very small at al 1 mesh points. Truncationerrors are regarded as the source of solution errors and are supposed to beconcentrated in the region where large gradients of variables exist. If we candecrease truncation error in this region, solution error will become smaller inthe whole computational domain.

Fig. 4 illustrates distributions of errors with Mesh 1 (h ,=Hb/6) presentedin Fig.3 (Murakami et al [1988c] ). Figure 4(l) shows the distribution of

truncation error T *. r, centers in the region in front of the model. Thedistribution of solution error 7, is given in Fig.4(2). High values oftruncation error arising around the windward corner are convected and diffused:

mode

(1) Mesh 2

ii, ,=H b /6, Ir45(x)x37(y)x2l(z)=34.965

, is the mesh\adjaoant- to the model. /

2) Mesh 3

5O s)~49 y)~28 z)=68,600,The

h ,=24/Hbmesh distribution is concentrated

near the windward corner.

Fig.5 Mesh dividings (vertical plane)

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; ~ o ~ e s u l t : . ~ - at h er h i g h v a l u e s f o r s o l u t i o n e r r o r s a r e d i s t r i b u t e d i n t h e w h o l el i e u , l i ~ l d a ~ -o u nd t h e m o d e l . I t o w o v e r , t h e m a x i m u m v a l u e o f " ~ d o e s n o t e x c e e dI ()Z of t. he v+ loc i ty va l ue . I t i s c la r i f i ed f rom the se res u l t s tha t th e mesh~m oI ut i ,m al*otmd the ~i [~dward COl'eel- o f t h model i s one of th e most: i mp or ta ntf a c t . o r s i n p r e d i c : t i e g t h e f l o w f i e l d w i t h s m a l l e r r o r s .

4 . 2 E f f e c t :o f S2n ~ t i _ v i ty o f Mesh Reso lu t_ in__ n S im ul_ a t ion Res_u lt_ s_

Xumeric:al s i m u l a t i o n i s c o n d u c t e d u s i n g t w o d i f f e r e n t t y p e s o f mes hd iv id i ngs and compared wi t h wind tun ne l exp er i men ts (Murakami e t a t [1988b] } .The mesh d iv id i ngs used here a re shown in F ig .5 . The mesh in t e rv a l h ~ ad j ace n tto th~~ model i s Hb/6 wi th Mesh 2. With Mesh 3, th e mesh in te rv al h ~ aro und th ewindwa~ 'd cor ner i s Hb/2 /+ and the va l ue a roun d the l ee war d cor ner i s I | b /6 . Thex e l o c l t V v e c t o t - s a r o u n d t h e m od el a r e c o m pa r e d i n F i g . 6 ( l ) , ( 2 ) . T he e n t i r e f l o wp a t t e l - n i n t i l e v e r t i c a l s e c t i o n i s w e l l r e p r o d u c e d i n t h e r e s u l t u s i n g Mesh 2 .I l o w e x e r, t h e r e v e r s e f l o w o n t h e r o o f a n d n ea t- t h e s i d e w a l l s w h i c h e x i s t s i n t h ee x p e r i m e n t a l d a t a d o es n o t a p p e a r i n t h e r e s u l t s o f Me sh 2 ,

T h e p ~ e s s m - e c o e f f i c ie n t s C p a r e c o m p a r e d i n P i g . 7 . I n t h e c a s e o f M e s h 2 ,

t h e p o s i t i v e p r e s s u r e o n t h e w i n d w a r d w e l l i s o v e r e s t i m a t e d a n d t h e n e g a t i v ep r e s s u r e o n t h e to p a n d o n t h e s i d e w a l l s i s u n d e r e s t i m a t e d i n c o m p a r i s o n w i t ht h e e x p e r i m e n t a l d a t a .

' [' he I -e suI t s us i ng Mesh 3 a re a l so p re sen ted in F igs .6 and 7 . Wi th Mesh 3 ,t h e mesh d i s t r i b u t i o n i s c o n c e n t r a t e d n e a r t h e w i n d w a r d c o r n e r. T he r e v e r s ef l o ~ , s cn~ t h e r o o f a nd n e a r t h e s i d e w a l l s a r e c l e a r l y r e p r o d u c e d . S u r f a c ep r e s s u r e d i s t r i b u t i o n a l s o c o r r e s p o n d s v e r y w e l l t o t h e e x p e r i m e n t s . B ut t h e r es t i l l e x i s t : s ome d i f f e r e n c e s i n t h e f l o w f i e l d o f t h e w ak e w h i c h s h o u l d b ei m l ) r o v e d .

5. P R E D I C T I O N O F T I M E D E P E N D E N T F L O W F I E L D B Y L E S A N D V I S U A L A N I M A T I O N

I n t h i s s ec t. i o n , t h e t i m e - d e p e n d e n t f l o w f i e l d p r e d i c t e d b y 3D LES i s

presen t . ed f i r s t . Some res u l t s o f v i s u a l a n i m a t i o n a r e i l l u s t r a t e d i n t h e l a t t e r "p a r t .

, ~, e x p e r i m e n t(Murakami et a l . , Re=7xlO'*)

~ h e x p e t " i m e r i t

f~) Mesh 2

2 : : ~ ~ - > > ) ' - > .> > > .~,-->---e

. . . . . . .

_~) Mesh 3

( l ) v e ~ ' t J c a l p l a n ea t c e n t e r o f t h e m o d e l

F i g 6

Mesh 2

~ ) M e s h 3

( 2 ) h o ~ i z o n t a p l a n ea t Z = H ~ / 2

C o m p a r i s o n o f v e l o c i t y v e c t o r s

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~/~ - . - . . p . , - i m..~//A~ 7

, .L 0 . : 0 .0 ~ / / / / / y / j / / / S / / / / / ~ 0 . 0 - ; . ~ - L 0

( 1 ) v e r t i c a l p l a n e

+LO 0.5 o.o ~ --0.5 -LO

--LO

( 2 ) h o r i z o n t a l p l a n ea t l = H b 2

C L

F i g . 7 C o m p ar is o n o f s u r f a c e p r e s s u r ec o e f f i c i e n t s b a s e d o n k - ~

5 . 1 B u i l d i n g M o d e l a n d O u t l i n e o f C o m p u t a t i o n a l M e t h o d

T h e a r c a n g e m e n t o f t h e b u i l d i n g m o d e l i s i l l u s t r a t e d i n F i g . 8. T h e c u b i cs h a p e d b l o c k s a r e s e t r e g u l a r l y. Some t y p e s o f s t r e e t c a n y o n s o c c u r b e t w e e nthe b lo cks (Murakami e t a l 11987bl ) .

T he s t a g g e r e d g r i d s y s t e m i s u s e d h e r e . A l l s p a t i a l d e r i v a t i v e s a r ea p p r o x i m a t e d b y c e n t r a l d i f f e r e n c e ( s e c o n d o r d e r ) . T he A d a m s - B a s h f o r t h s c h em e( s e c o n d o r d e r ) i s u s e d f o r t i m e - m a r c h i n g . N u m e r i c a l i n t e g r a t i o n s a r e c o n d u c t e dfo l l o wing t he ABMAC me thod , a s imu l t aneo us i t e r a t i on me thod fo r p r e s su re andv e l o c i t i e s

The mesh d iv id ing i s shown in F ig .9 . The number o f g r i d po in t s amoun t s t o3 9 1 x ) X 3 6 ( y ) X 2 1 ( z ) = 4 3 , 5 2 # . T he m es h i n t e r v a l h ~ a d j a c e n t t o t h e w a l l s u r f a c e i sHb /20. One u n i t o f b lock a r ea (one bu i l d ing p lu s ha l f o f t he w id th o f t hes u r r o u n d i n g s t r e e t s ) i s s e l e c t e d f o r t h e s i m u l a t i o n s , a n d c y c l i c b o u n d a r y

c o n d i t i o n s a r e a d o p t e d h e r e .T he b o u n d a r y c o n d i t i o n s a r e a s f o l l o w s :1) D o w ns t r e am a n d s p a n w i s e d i r e c t i o n s : c y c l i c b o u n d a r y c o n d i t i o n s .

G r o s s d o w n s t r e a m p r e s s u r e g r a d i e n t i s i m p o s ed . H e r e ~ P = = O . O I I .LiP m e a n s t h e n o n - d i m e n s i o n a l i z e d p r e s s u r e d i f f e r e n c e b a s e d o n p u b 2 .Ub means t ime a ve ra ged ve loc i t y a t i n f l o w b o u n d a r y o f c o m p u t a t i o n a ldomain a t h e ig ht of Hb.

2 ) U pp er s u r f a c e o f c o m p u t a t i o n a l d o m a i n : f r e e - s l i p b o u n d a r y c o n d i t i o n .3 ) Wel l b o u n d a r y : t h e p r o f i l e s o f t h e t a n g e n t i a l v e l o c i t y c o m p o n e n t s

a r e a s sumed t o obey a power l aw exp r e s se d a s u~ z ~ nea r t he wa l l .I n t h i s s t ud y m=0 .5 . Normal ve loc i t y componen t a t t he boun da ry i sa s sumed t o b e z e r o

5 . 2 T i m e - D e p e n d e n t Ve l o c i t y F l u c t u a t i o n a n d Ve l o c i t y S p e c t r u m

F i g . lO i l l u s t r a t e s t h e t i m e - h i s t o r y o f t h e v e l o c i t y f l u c t u a t i o n s a t p o i n t 1p r e s e n t e d i n F i g . 9 . I n t h i s f i g u r e , t * d e n o t e s t h e n o n - d i m e n s i o n a l t i m e s c a l etUb /Hb . F i f i. l l s hows t he ve l oc i t y spec t rum o f u componen t a t t h e s ame po in t . Thep r e d i c t e d s p e c t r u m c o r r e s p o n d s v e r y w e l l w i t h t h a t o f t h e e x p e r i m e n t

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5 .3 M ean Ve l o c i t y F i e ld

Time-ave raged ve loc i ty vec to r s a r e i l l u s t r a t e d i n F ig . 12. i n gene ra l , t hee n t i r e f l o w p a t t e r n f r o m t h e n u m e r i c a l s i m u l a t i o n a g r e e s w e l l w i t h t h a t f r o m t h ee x p e r i m e n t a l r e s u l t s . B ut s l i g h t d i f f e r e n c e s d o e x i s t i n t h e w ak e b e h i n d t h emod el and a l so i n t he s t and ing vo r t ex i n f r on t o f i t . The wake r eg i on and t hes t a n d i n g v o r t e x i n t h e n u m e r i c a l s i m u l a t i o n a r e a l i t t l e l a rg e r t h a n t h o s e i n t h eo x p e r i m e n t a l r e s u l t s .

A t t he w indward co rn e r on t he ro o f , a r eve r s e f l ow shou ld appea r bu t i s no tr e p r oduce d i n F ig . 12 (2 ) . F ig . 13 shows mean ve loc i t y vec t o r s w i th f i ne r meshes(60 ( x ) x S l (y ) X46 (z )=140 ,760 ) , i n wh ich t he mesh i n t e r va l ad j ac en t t o t he mod01s u r f a c e i s H b/ 4 0. T he r e v e r s e f l o w o n t h e r o o f s u r f a c e i s h e r e w e l l r e p r o d u c e d .

5 . 4 D i s t r i b u t i o n o f Tu r b u l e n t K i n e t i c E n e rg y

l+he d is t r ibu t i ons of tur bul ent k in et i c energy k are compared in Fig . 14.Fig , 14(2} shows the r esu l t wi t h about 4xlO ~ gr i d poi nts and Fig . 14(3) shows theres ul t wi th 14xlO ~ gr i d poi nts , The res ul t wi t h 4xlO ~ gr i d poi nts shows somed i ff e ren ce when compared to t he expe r ime n ta l r e su l t , e spe c i a l l y fo r t he kdis t r i but ion in the v ic in i ty of the windward cor ner . The correspo ndence betweenthe resu l t obta ined wi th f i ner meshes (14xlO ~ gr i d poi nts ) and tha t f rom theexper iment i s conf i rmed to be very good,

milding model

wind; t reet

Y

t l ' 3Hb i: Hb ' 3Hb IF i g . 8 A r r a n g e m e n t o f b u i l d i n g m o d e l s

(Shape o f bu i l d ing i s cub i c )

~ ind

4 H b

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