2004 [heyuan wang] numerical simulation of spherical couette flow

7/26/2019 2004 [Heyuan Wang] Numerical Simulation of Spherical Couette Flow

1/8

E L S E VI E R

An intemational Journal

A v a i la b l e o n l i n e a t w w w s d e n c e d i r e c t c o m

c om p u t e r s

o , = . c = ~ o , . ~ T , m a t h e m a t i c s

w i t h a p p l i c a t i o n s

C o m p u t e r s a n d M a t h e m a t i c s w i t h A p p l i c a t i o n s 4 8 ( 20 0 4 ) 1 0 9 -1 1 6

w w w . e ls e v ie r . c o m / l o c a t e / c a m w a

N u m e r i c a l S i m u l a t i o n o f

S p h e r i c a l o u e t t e F l o w

H E Y U A N W A N G

D e p a r t m e n t o f M a t h e m a t i c s a n d P h y s i c s

L i a o n i n g I n s t i t u t e o f T e c h n o l o g y

J i n ' z h o u , 1 2 1 0 0 1 , P . R . C h i n a

a n d

S c h o o l o f S c ie n ce s , X i ' a n J i a o t o n g U n i v e r s i t y

X i ' a n , 7 1 0 0 4 9 , P . R . C h i n a

w a n g h e y u a n 6 4 0 0 s i n a o m

K A I T A I L I

S c h o o l of S ci e nc e s, X i ' a n J i a o t o n g U n i v e r s i t y

X i ' a n , 7 1 0 0 4 9 , P . R . C h i n a

Received Janu ary 2 003 ; revised and accepted December 2003)

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

s p h e r e s . An a l y t i c a l e x p r e s s i o n s o f e i g e n f u n c t i o n s o f a S t o k e s o p e r a t o r i n t h e s p h e r i c a l g a p r e g i o n

a r e p r e s e n t e d , n u m e r i c a l e x p e r i m e n t s a r e c a r r ie d o u t b y a p p l y i n g t h e s e e x p r e s s io n s t o t h e s p e c t r a l

a p p r o x i m a t i o n o f s p h e r i c a l Co u e t t e f lo w , a n d n u m e r i c a l r e s u l t s a r e g i v e n . Q 2 0 0 4 E l s e v i e r L t d . All

r i g h t s r e s e r v e d .

K e y w o r d s - - N a v i e r - S t o k e s e q u a ti o n s, S p h er i ca l -C o u e t te flo w, S t ok e s o p e r at o r.

N O M E N C L T U R E

2

( r , ~b, O) s p h e r i c a l p o l a r c o o r d i n a t e s w = - - , ~ t h e c o e f fi c i en t o f k i n e m a t i c

wl v i s c o s i t y

R 1 , R 2 r a d i u s o f t h e i n n e r a n d

o u t e r s p h e r e s R e

= R ~ w l / v

R e y n o l ds n u m b e r

/ / 2

r ] = R--~I' wl , w2 an gu l a r ve loc i ty o f the u = (u~ , u s , u 0 ) , p ve loc i ty o f the f lu ids an d

i n n e r a n d o u t e r s p h e r e s p r e s s u r e

, ~ s t r e a m a n d v e l o c it y

f u n c t i o n s

1 I N T R O D U C T I O N

I n t h i s p a p e r , w e n u m e r i c a l l y s t u d y t h e s t a t i o n a r y a x i s y m m e t r i c i n c o m p r e s s i b l e f l o w b e t w e e n

t w o c o n c e n t r i c a l l y r o t a t i n g s p h e r e s . T h i s f l o w i s c a l l e d t h e s p h e r i c a l C o u e t t e f l o w S C F ) . I n

m a n y w a y s , s p h e r i c a l C o u e t t e f l o w r e s e m b l e s t h e w e l l - k n o w n T a y l o r - C o u e t t e f l o w

T C F )

b e t w e e n

S u b s i d i z e d b y t h e s p e c i a l f u n d s f o r m a j o r b a s i c r e s e a r c h p r o j e c t s o f Ch i n a G1 9 9 9 0 3 2 8 0 1 - 0 7 a n d NS F C ( 1 0 1 0 1 0 2 0 ) .

0 8 9 8 - 1 2 2 1 / 0 4 / - s e e f r o n t m a t t e r @ 2 0 0 4 E l s e v i e r L t d . A l l r i g h t s r e s e r v e d . Ty p e s e t b y .AA.~ S-TEX

doi:10.1016/j .camwa.2003.12.004


2/8

110 H . WANG

N D

K. LI

t w o d i f f e r e n t i a l l y r o t a t i n g c o n c e n t r i c c y l in d e r s , e s p e c i a l l y i n t h e e q u a t o r i a l r e g i o n s o f t h e s p h e r e

w h e r e t h e c e n t r i f u g a l f o r c e o n t h e f l o w i s t h e g r e a t e s t . H o w e v e r , t h e n u m b e r o f v o r t i c e s a r e a lw a y s

c o n f in e d t o t h e e q u a t o r i a l r e g io n s i n t h e T C F .

P l e a s e s e e t h e N o m e n c l a t u r e f o r d e f in i t io n s u s ed t h r o u g h o u t t h i s p a p e r .

A t a l ow R e y n o l d s n u m b e r , t h e S C F i s e x p e r i m e n t a l l y f o u n d t o b e b o t h a x i s y m m e t r i c a n d

r e fl e c ti o n s y m m e t r i c a b o u t t h e e q u a t o r [ 1] . T h r e e t y p e s o f th e S C F t h a t e x is t a t m e d i u m g a p

g e o m e t r i e s ( 0 . 12 < ~7 _ Re c , t h e f lo w l o s es s t a b i l i t y a n d T a y l o r v o r t i c e s

a p p e a r s . Sc h r a u f [6] a n d M a r c u s a n d T u c k e r ma n [ 3] d i s c o v e r t h a t R e e l = 6 4 5 + 0 .0 5 , Re ~2 =

7 4 0 q - 0 . 05 [ 3 ,6 ] f o r ~7 = 0 . 1 8 f r o m c a l c u l a t i o n s u s i n g d i f f e r e n c e me t h o d s , f o r e x a m p l e , a r c l e n g t h

c o n t i n u a t i o n m e t h o d s w i t h d if f e re n t m e t h o d , p s e u d o s p e c t r a l m e t h o d s , c o l l o c a t i o n m e t h o d s , o r

f i n i t e e l e m e n t - a r c l e n g t h c o n t i n u a t i o n s m e t h o d s w i t h c o n j u g a t e g r a d i e n t f o r m u l a t i o n s , r e s p e c -

t ive ly .

I n t h i s p a p e r , w e n u m e r i c a l l y s t u d y f lo w b e t w e e n t w o c o n c e n t r i c r o t a t i n g s p h e r e s b y u s in g a

s p e c t r a l m e t h o d . O u r m a i n w o r k is t o p r e s e n t a n a n a l y t i c a l e x p r e ss i o n o f e i g e n f u n c ti o n s o f a

S t o k e s o p e r a t o r i n t h e s p h e r i c a l g a p r e g i o n , b y a p p l y i n g e i g e n f u n c t i o n s o f t h e S t o k e s o p e r a t o r

t o a p p r o x i m a t e f lo w o f r a n d 0 d i r e c ti o n s , r e sp e c t iv e l y , w e e x a m i n e p r e v i o u s e x p e r i m e n t r e s u l ts

a g a i n , o u r r e s u l t s a r e i n a g r e e m e n t w i t h t h e r e s u l t s o f M a r c u s a n d T u c k e r m a n , a n d o u r s p e c t r a l

m e t h o d h a s a n o b v i o u s p h y s i c a l m e a n i n g . I n f a c t , i f a n e i g e n f u n c t i o n o f t h e S t o k e s o p e r a t o r i s

r e g a r d e d a s b a s i c f l o w b a s i c v o r t e x ) , t o t a l f l o w i s a c c u m u l a t i o n o f v a r i o u s v o r t i c e s .

T h e c o n t e n t o f t h e p a p e r i s a r r a n g e d a s f o l l o w s . F i r s t , w e i n t r o d u c e a v e l o c i t y - s t r e a m f u n c t i o n

f o r m o f t h e N a v i e r - S t o k e s e q u a t i o n s i n t h e s p h e r i c a l p o l a r c o o r d i n a t e s i n S e c t i o n 2 . S e c o n d ,

w e d i s c u s s a n e i g e n v a l u e p r o b l e m o f t h i s S t o k e s o p e r a t o r i n t h e S e c t i o n 3 , a n d e i g e n v a l u e s a n d

e i g e n f u n c t i o n s o f t h e S t o k e s o p e r a t o r i n t h e s p h e r i c a l g a p r e g i o n a r e g i v e n . F i n a l l y , w e d i s c u s s a

c o m p u t i n g s c h e m e i n S e c t i o n 4 , a n d w e p r e s e n t n u m e r i c a l r e s u l t s i n S e c t i o n 5 .

2 . T H E N A V I E R - S T O K E S E Q U A T I O N

W I T H V O R T I C I T Y - S T R E A M F O R M

T h e N a v i e r - S t o k e s e q u a t i o n b e t w e e n t w o c o n c e n t r i c r o t a t i n g s p h e r e s a r e g i v e n b y

0 _ ~ u

+ ( u . V ) u + V p - 1 V 2 u = 0 , ( i . i )

0 t

V . u = 0 , ( 1 .2 )

a n d t h e b o u n d a r y c o n d i t i o n s o n t h e s p h e r i c a l s u r f a c e s a r e

u[~=l ----s in 0g, u ] . = n = w r I s in 0g, (1 .3)

w h e r e ( ~'~ , f ie , g 0) i s a l o c a l c o o r d i n a t e f r a m e o f t h e s p h e r i c a l p o l a r c o o r d i n a t e s .

B y a p p l y i n g t h e i d e n t i t y

( 1 .1 ) c a n b e r e w r i t t e n b y

O u

O t

1 g r a d l u [ 2 u ( V u ) + ( u . V ) u ,

2


3/8

Num erical Simu lation 111

T a k i n g r o t o f b o t h s i de s , a n d d e n o t i n g ( = V x u , w e h a v e

0 i v

5 7 - v x ( ~ x ( v x u ) ) - R e x ( V ~ ) = 0.

S i n c e u = u . ~' a n d ( = ( . ~ ' , b y a x i s y m m e t r y , w e o b t a i n t h e e q u a t i o n f o r t h e C - c o m p o n e n t s

o f v e l o c i t y a n d v o r t i c i t y

O u

( u x ) . ~ - ~ e V 2 U . f ie = O , ( 1 . 4)

1 (V 2u ) f ie O. (1 .5)

a ,p a t v x ~

) . ~ ,~

- f i - T e v x

O n e c a n d e f i n e t h e s t r e a m f u n c t i o n s u c h t h a t t h e m e r i d i o n a l f l o w i s

1

u r = r 2 s in1 0 0-00 r s in 0 ) ,

uo - r

s i n 0 ( r s i n 0 ~ ) . ( 1 . 6)

U s i n g a t e n s o r m e t h o d a n d s o m e c a l c u l a t io n s , w e o b t a i n

- 1 O O

- - 0 - ~ ( r s i n 0 u ) , 0 - r s i n 0 0 - r

( r s in O u ) ( = L 2

(1 .7)

(~ r 2 s in 0

w h e r e

- ~ r( r ' ) r + ~ - 2 0- 0 s i -~ O - ' 0 ( s i n 0 . ) .

S u b s t i t u t i n g ( 1. 6) a n d ( 1. 7) i n t o ( 1 .4 ) a n d ( 1 .5 ) , w e o b t a i n t h e v o r t i c i t y - s t r e a m f u n c t i o n f o r m u -

l a t io n o f t h e N a v i e r -S t o k es e q u a t i o n w i t h a x i s y m m e t r y

O u 1 2 1 O ( r

s i n

Ou , r

s i n 0 ~ )

0 t ~ e e L u + r3 s i n 2

O(r ,

0) = 0 , (1 .8)

1

O ( r s i n O ( , r s i n O )

+-2 ( e+ u * ) Y ( U + u * e ) + 2 ( Y - ~ - n 2 ( = 0 , (1 .9)

O t

r 3 s i n 2 0

O(r , O)

L 2 = , ( 1 .1 0 )

w h e r e N = ( c o t O / r ) - ( l / r 2 ) . B o u n d a r y c o n d i t io n s (1 .3 ) ca n b e r e w r i t t e n b y

u ] ~ = l = s i n 0 ,

u [ ~ =

= r ] w s i n 0 ,

~ { ~ = 1 = ~ 1 ~ = ~ = 0-~r r = l 0 l = 0 , ( 1 .1 1 )

1 0 = o = V - , I o = . = 0 .

I n o r d e r t o o b t a i n t h e N a v i e r - S t o k e s e q u a t i o n s w i t h t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n , w e

i n t r o d u c e t h e S t o k e s f lo w u *,

u . . . . = ( )

= u e ,

u o ~ r + /3 r - 2

s i n O ,

w h e r e c~ = ( @ w - 1 ) / ( ~ 3 - 1 ) a n d / 3 = r ~3 (1 - w ) / ( ~ 3 - 1 ) .

L e t u = U + u ~ , t h e n u ~ s a t i s f ie s

~ ;1 ~ = , = s i n0 , ~ ; 1 ~ = ,

= r / w s i n 0 ,

a n d

L2u*

= 0 . A c c o r d i n g l y , w e o b t a i n t h e e q u a t i o n f o r U a n d

0__UU+ 1 0 ( r s i n 0 ( U + u ; ) , r s in 0 )

O t

?-3 sin 2 0

O(r, O)

0 L 2

O t

wh r

1 L 2 U = O ,

(1 .1 2 )

R e

_ _ 0 ( L 2 ,r s i n O )

+ ( 2

( U+ u * ) N U + 2 U N u * c + L 2 N ) - ~---~ L49 = f ,

(1.13;)

r 2 s in 0

O( r , O)

f = - 2 u* c N u * = - 6 / 3 ( a t - a + / 3 r - ~ ) s i n 0 c o s 0 .

T h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s a r e g i v e n b y

U[~=I = U[ , -= , = O ,

0 = O .

a = t r = = ~ r =

P = 1 = b 7 ~ = 1

1.14)


4/8

112 H. WANGAND K. LI

3 . E I G E N V A L U E S A N D E I G E N F U N C T I O N S

O F S T O K E S O P E R A T O R

In this section, we stu dy the following eigenvalue problem of the Stokes operator: find (i , U, ~)

E R x H01(ft) x H3 (fl ) such th at

1

L 2 U = ~ U,

Re

U[o~ = O,

R e L

AL2,

C[ oa = 0-~r~ oa = 0,

( 2 . 1 )

where f~ = {( r, ,0 ); 1 < r _< , , 0 _< < 27r, 0 < 0 < 7r}. Eigen value prob lem (2.1) c an be wri tte n

as fol lows: f ind (A, U, ~;) e R x H~ ([}) x H 2( ~) su ch tha t

L 2 U + AU = 0 , 2 . 2 )

U[o~ = 0,

and

L4 q- IL 2 = O,

-~r o~ (2.3)

{ o ~ = 0 = 0 ,

where A = Re .A.

Below, we discuss eigenvalue problems (2.2) and (2.3).

Let U = w r ) O O ) , the n prob lem (2.2) is decomposed into the following Sturm-L iouville prob-

lem:

1 d2rw w

r dr---5-

+ Aw - tt~-~ = 0, (2.4)

wlr=l = wl r= ~ = O,

and

d--O (sin 0 0 ) + 0 = O, (2.5)

0 [ o = o < + o o .

It is well know n tha t problem (2.5) has a nonzero solution Ol(O) = Pzl(cosO) if and only if

= z (1 + 1 ) , l = 1 , 2 , . . . , w h e r e p ~ ( x ) = ( 1 - x 2 ) 1 / = ~ ; ~ ( ~ ) a n d ; ~ ( ~ ) = ( 1 / 2 ~ Z ) ~ ( ~ 2 - 1 ) ~ i s

a genera l Legendre polynomial . Subst i tu t ing t t =

l 1 +

1) into (2.4), according to the general

solution the ory o f the Sturm-L iouville problem [7], we obtai n a special solution of equation (2.4)

as follows:

~ z ( ~ , r ) =

where j l z ) = ~ J t + , / 2 z ) and y z z ) = - 1 ) ~ + l ~ J - z - 1 / 2 z ) are the first kind and

second-kind spherical Bessel 's function, respectively, and J~+l /2 z ) is Bessel 's function of half-

integer order.

Together with the boundary condition of (2.4), we have

bz l , 1 ) = 0 2 . 6 )

while eigenvalues oil, n l, n = 1, 2 ,. .. ) of (2.2) are the r oots of prob lem (2.6) and sati sfy [7]

~t,n > 0, al +l ,n > (~ l, n, ~l,~+1 > c~l,n- (2.7)


5/8

Nu m e r ic a l S imu la t io n 1 13

L e t

bl, ,~(r) = {) l(a l ,n ,

r ) , l , n = 1 , 2 , . . . , t h e n e i g e n f u n c t i o n s

o

2 . 2) c o r r e s p o n d i n g t o t h e e ig e n -

v l u e s

O~l, a r e

U~,,~ = b~,,~(r)O~(O ), l n = 1 , 2 , . . . . 2 .8 )

S i m i l a r l y , w e c o n s i d e r e i g e n v a l u e p r o b l e m 2 . 3 ).

A s s u m i n g t h a t 4 =

w ( r ) @ ( O ) ,

p r o b l e m 2 .3 ) m a y b e d e c o m p o s e d i n t o th e f o ll o w i n g e q u a t i o n :

G 4 w = A G 2 w ' 2 . 9 )

~ l ~ - - 1 = ~ l ~ = , = ~ I ~ = ~ = ~ 1 ~ = ~ = o

d 2

a n d p r o b l e m 2 . 5) , w h e r e G 2 w = - (1 /r )- aT ~ ( r w ) + ( / r 2 ) w a n d G 4 w = G 2 ( G 2 w ) . T h e f o l l o w i n g :

s p e c i a l s o l u t i o n o f p r o b l e m 2 . 9) is o b t a i n e d :

R l ( ~ , ~ ) =

f l l ? ~ - - I -- 1 j l x / - f ? ~ )

w h e r e j [ a n d y ~ a r e d e r i v a t i v e s o f j l a n d Y l, r e s p e c t i v e l y .

E i g e n v a i u e s / 3 1 , n t , n = 1 , 2 , . . . ) o f 2 . 3) a r e t h e r o o t s o f t h e f o l lo w i n g e q u a t i o n :

R ~ ( ~ , 1 ) = 0 ( 2 . 1 0 )

a n d s a t i s f 5 [7 ]

~ z , ,~ > 0 , ~ l + i , , ~ > f l l , ,~ , ~ l , , ~ + i > / ~ l , , ~ - ( 2 .1 1 . I

L e t R t , n ( r ) = R z / 3 l, n, r ) , 1, n = 1 , 2 , . . . , t h e n e i g e n f u n c t i o n s o f e i g e n v a l u e p r o b l e m 2 . 3 ) co r re --

s p o n d i n g t o t h e e i g e n v a lu e s /~ l ,~ c a n b e w r i t t e n b y

Cz ,,~ = Rt ,~ ( r )6 ) l (O ) , l , n = 1 , 2 , . . . . 2 . 1 2 )

4 . T H E C O M P U T I N G S C H E M E

L e t

Lo No Lo No

i= i= i= i=

C l e a r l y ,

U ( r , O )

a n d ~ b r , 0 ) s a t i s f y t h e b o u n d a r y c o n d i t i o n 1 . 1 4) .

N o w , w e d i s c r e t i z e 1 . 12 ) a n d 1 . 13 ) i n t i m e , t h e n w e o b t a i n

U ~+1 - U ~ 1

A t - t r a s in 2 0

3.21)

[ O ( ~ rr s in O U ' ~ + l, rs in O g 2 ~ ) O ( r s i n O ( ( 1 - a ) U ' ~ + u * , r s i n O ~ b n + l ) l 1 L 2 U . + I

. o r , o ) + o r , o ) ~

~ ~

L 2 4 n + l - L 2 n 1 0 ( L 2 9 n + l , r s i n 0 ~ )

+

A t r 2 s i n e

O(r, O) 3 . 3 )

( ( U n - l - u * ) N u n + 1 - l - 2 U n + 1 N u * + L 2 4 n + l N ~ , b n ) - R - - -- ~ L 4 ~ n + l = f , n = O , 1 , 2 , . . . ,

2

w h e r e c r i s a p a r a m e t e r , 0 < a < 1 , a n d t h e s u p e r s c r i p t n r e p r e s e n t s t h e s o l u t i o n o f t =

n a t .

W h e n n = 0 , U a n d 4 r e p r e s e n t t h e i n i ti a l c o n d i t i o n s i n t h i s p r o b l e m , w e t a k e

U = V 0 ) = 0 , 4 0 = 4 0 ) = 0 , L 0 = 4 8 , N 0 = 5 .

I n t h e s t e a d y - s t a t e c a s e , t h e f ir s t t e r m o f 3 . 2 ) a n d 3 . 3 ) w i l l n o t o c c u r . I n t h i s c a s e, t h e

s u p e r s c r i p t n r e p r e s e n t s t h e s o l u t i o n o f t h e i t e r a t i o n

n t

s t e p . E x i s t e n c e a n d u n i q u e n c e o f t h e

s o l u t i o n o f p r o b l e m ( 3 . 2 ) a n d 3 . 3 ) h a v e b e e n p r o v e n i n [8 ].


6/8

114 H. WANG AND K. LI

5 . N U M E R I C A L R E S U L T S

In th is sec t ion , we present the numerica l resu l t s o f the ax isym metr ic 0-vo rtex flow and 1-vortex

flOWS.

a) b)

Figure 1 . r , 0 )-pro jec t ion of the merid iona l s t reamline and co ntours of constan t

angular azimuthal velocity.

Fig ure 1 depicts the th ree-d ime nsion al 0-vo rtex flow at Re = 600, ~ = 1.18, and w = 0. Bo th

figures are two-d imensional p ro jec t ion of the f low on to r , 0 )-p lane a t f ixed .

In these f igures and a l l o ther f igures of r , 0 )-pro jec t ions in th i s paper) , the gap wid th be-

tween the inner and outer spheres i s exagger a ted for c lar i ty of the fe a tures the rad ia l in terval

[1,1.18] is ma ppin g l inear ly to [1,2]). Figu re l a shows the str eam lines of the m eridion al flow or

r , 0 )-com ponent of the f low), i .e . , contours of constan t r s in t~ r , t ?) . The merid ional ve loc i ty i s

everywh ere tange nt to these contours . Th e f low is c lear ly ref lec t ion-symm etr ic abou t the equ ator .

Two s t reaml ines loca t ing exact ly a t the equator a re the ou t f low boundary be tween the two large

vort ices.

The qual i ta t ive be haviour of the f low in Figure 1 can be unde rs too d physica l ly by considering

the f low near th e po les , where the geom etry resembles tha t be tween th e para l le l d ifferen t ia l ly

ro ta t in g d isk inner sphere) and pul led from the cen t re of the s ta t ion ary d isk outer sphere) .

This m ot ion forms the in flow rad ia l bou ndar y je t s a t the po les . Th e f lu id moving down from

the north po le a long the inner sphere meets f lu id moving up from the south pole a t the equator

and forms the equatoria l rad ia l ou tf low boundary . The f lu ids in the nor thern and southern

hemispheres do not mix .

Figure lb shows the contou rs of constan t angular ve loc i ty u/ r sin 8) for the flow in Figur e la.

Th e angular ve loc i ty decreases mon otonica l ly from the inner to the ou ter spheres . For low Re

flows, the contours of constan t angular ve loc i ty are nearly para l le l to the spherica l boundaries .

In fac t , the y are nearly equal to th e angular ve loc i ty contours of the Stokes f low, bu t we can see

the s l igh t wiggles of the contours near the equator in Figure lb .

As the R eynolds num ber increases , the basic f low develops wha t B onne t and Alz iary de Roque-

fort 1976) have ca l led a p inching of s t reaml ines . This p inching i s i l lus t ra ted in Figure 2a which

is a p lo t o f the m erid ional s t reaml ines in the r , 0 )-p lane using the convent ions of Figure 1) of

our num erica l ly calcu lated flow at Re = 650, U = 1.18, and ~ = 0. Th e pinching is chara cter-

ized by a s tagnat ion poin t i .e ., c ross ing of s t reaml ines) in the r , 0 )-com ponent of the ve loc ity .

Al though i t i s no t ex pl ic i t ly shown in Figure 2a , i t i s obvious tha t there must be a cross ing of

s t reaml ines in each hemisphere somewhere the c losed s t reaml ines of the la rge basic vortex and

the c losed s t reaml ine of the sm al l d iameter s i tu a ted in the p inched par t o f the f low near the


7/8

Numerica l Simulat ion 115

a ) b )

Figure 2

r ,O) -pro jec t ion

of the meridional streamline and contours of constant

angular azimuthal velocity

a ) b )

F i g u r e 3. r , ) - p r o j e c t i o n o f t h e m e r i d i o n a l s t r e a m l i n e a n d c o n t o u r s o f c o n s t a n t

a n g u l a r a z i m u t h a l v e l o c i t y .

e q u a t o r . I n a p i n c h e d 0 - v o r t e x f l o w , t h e r e a r e a t l e a s t t w o l o c a l m a x i m a i n t h e s t r e a m f u n c t i o n

i n e a c h h e m i s p h e r e , b u t t h e c i r c u l a t i o n i n t h e c l o s e d s t r e a m l i n e s o f t h e 0 - v o r t e x p i n c h a l w a y s h a s

t h e s a m e s i g n a s t h a t i n t h e l a r g e b a s i c v o r t e x , b e c a u s e i t i s n o t s e p a r a t e d f r o m t h e l a r g e b a s i c

v o r t e x b y a n i n f l o w o r o u t f l o w b o u n d a r y t h a t e x t e n d s f r o m t h e i n n e r t o t h e o u t e r s p h e r e a n d i t

i s n o t a T a y l o r v o r t e x .

W e h a v e n u m e r i c a l l y f o u n d t h a t p i n c h e s o c c u r f o r 6 5 0 _ R e ~ 7 5 0 a n d t h a t t h e i r d e v e l o p m e n t

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

R e y n o l d s n u m b e r i n c r e a s e s , t h e p i n c h e s o n l y b e c o m e d e e p e r . F i g u r e 2 b s h o w s t h e a z i m u t h a l

a n g u l a r v e l o c i t y c o n t o u r s o f t h e p i n c h e d f l o w i n F i g u r e 2 a . T h e w i g g l e s a t t h e e q u a t o r h a v e

i n c r e a s e d i n a m p l i t u d e . T h e i n w a r d a n d o u t w a r d b e n d i n g o f a z i m u t h a l c o n t o u r s c o r r e s p o n d t o

m e r i d i o n a l f l o w i n t h e p i n c h w h i c h i s r a d i a l l y i n w a r d o r o u t w a r d , r e s p e c t i v e l y .

A s t h e R e y n o l d s n u m b e r b e c o m e s l a r g e a b o u t R e _ > 7 5 0 ) , t h e b a s i c v o r t e x i n t h e 0 - v o r t e x f l o w

d e v e l o p s a p i n c h . F i g u r e 3 a d e p i c t s t h e f l o w a t R e - - - 7 5 0 , ~ / = 1 . 1 8 , a n d w - - 0 . I t s h o w s t h a t

t h e T a y l o r v o r t i c e s a r e s e p a r a t e d f r o m t h e l a r g e b a s i c v o r t i c e s b y n e a r l y s t r a i g h t r a d i a l s t r e a m s

a n o u t f l o w b o u n d a r y ) t h a t e x t e n d s f r o m t h e i n n e r t o o u t e r s p h e r e . F i g u r e 3 b s h o w s t h a t t h e


8/8

116 H. WANG AND K. LI

equatorial wiggles in the contours of the azimut hal velocity are mo re pr ono unc ed tha n those of

the pinched 0 vortex flow in Figure 2a.

R E F E R E N E S

1. O. Sawatzki an d J . Zie rep , Das Stromfe ld in Spa l t Zwischen Wei Konzentr ischen Kugelf iacechen von Denen

die Innore Rotiert .~ Acta Mechanica 9, 13-35~ (1970).

2. P.S. Mar cus and L.S. Tuckerm an, Sim ulation of flow between two concentric rotat ing spheres. Par t 1. Steady

sta tes , J. Fluid Mech. 185, 1-30, (1987).

3 . P .S. M arcus a nd L.S. Tuckerman, Simula t ion of f low be tween two concentr ic ro ta t i ng spheres. Par t 2 . Tran-

sitions, Y. Fluid Mech. 185, 31-66, (1987).

4 . M. W immer, Experim ents on a v iscous f lu id f low be tween two concentr ic ro ta t in g spheres, 9 . Fluid Mech.

73, 371-335, (1976).

5 . J . Z ie rep and O. Sawatzki , Three d imensiona l s tab i l i t ies and vort ices be tween two concentr ic ro ta t in g spheres,

In the Eighth Syrup. Naval Hydrodyn. pp. 275-288, (1970).

6. G. Schrauf, Branching of Navier-Stokes equations in a spherical gap, In Proc. Eighth Intl. Conf. Numerical

Methods in Fluid Dyn. pp. 474-480, Springer, Aachen, (1983).

7. S. Liu and S. Liu, The Special Function pp. 202-203, Atmosphere Press, Beijing, (1988).

8 . W . Feng and K. Li , The existence and uniquence of weak so lu t ion of flow be tween two concentr ic ro ta t ing

spheres,

Applied Mathematics and Mechanics

21 (1), 61-66, (2000).

2004 [heyuan wang] numerical simulation of spherical couette flow

Documents