numerical solution of laminar flow generated

7/28/2019 Numerical Solution of Laminar Flow Generated

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Ac t a Mechaniea 83, 9--24 (1990) A C T A M E C H A N I C A@ by Springer-Verlag i999

N u m e r i c a l s o lu t io n o f l a m i n a r f l o w g e n e r a t e di n a n a n n u l u s b y r o t a t i n g s c r e e n sE. Kit and E. Mazor, Tel-Aviv, Israel(I~eeeived August 3, 1989; revised September 9, 1989)

Summary. The governing equation and the appropriate boundary condition describing stationarylaminar flow in a curved channel and in an annulus with one (upper) and two (upper and lower)rotating screens, were solved numerically by finite-difference method. In the curved channel multiplesolutions were obtained in accordance with the predictions of previous theoretical and experimentalinvestigations. In contrast to that , no multiple solutions were found for the flow in an annulus, neitherwith two nor with one rotating screen. The numerically computed axial velocity distributions inannulus were compared to the corresponding experimental profiles measured in a turbulent flow ofa homogeneous fluid created in annulus by one or two rotat ing screens. The qualitative agreementbetween the results was unexpectedly good.

1 IntroductionStarting from the pioneering work of Kato and Phillips [1], annular tank (annulus) waswidely used for investigation of entr ainm ent across the den sity interface. Shear stress to theflow was imparted by a rotating screen; it was assumed that the flow in the mixed layeris basically one-dimensional. However, in the papers by Scranton and Lindberg [2] andDeardorf f and Yoon [3] it was established t ha t curved walls geomet ry of the annul us affectsconsiderably the structure of the flow and leads to radial stratification of turbulence. Thisstratification of turbulence in the annulus may be caused by two sources: a) rapid de-crease of the velocity near the walls, resulting in a stabilizing distribution of angular mo-mentmn near the inner wall, which dumps the turbulence in that region, and destabilizingstratification close to the outer wall, enhancing turbulence there; b) the secondary eireu-Iation which Mways presents in the mixed layer due to curvature. This circulation preventsa downwards vertical transport of shear stress and adveets turbulence towards the outerwall.

In the recent work of Chaiet al. [4] the semi-equilibrium turbulent energy model wasused to describe the flow pattern in the mixed layer. The effects of the radial stratificationof turbulence were taken in account in this model, but the advective transport by secondaryflow was neglected. The results concerning the entrainment, obtained from this two-dimensional model, were in a satisfactory agreement with experimental results. The model,however, was unable to catch the most striking features of the velocity field neither in themixed layer of density stratified flow nor in homogeneous flow in annulus, as they wereobtained in the experiments . The experimental profiles of the axial velocity in the radialdirection demonstrate higher level of asymmetry, with an appearance of so called velocity"ton gue" in the vicinity of the outer layer and a ccompanying inflection point.


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10 E . K i t and E . MazorI t i s n a t u r a l t o a t t r i b u t e t h e a b s e n c e o f t h e s e f e a t u r e s in t h e v e l o c i t y f i el d , c a l c u l a t e d

f r o m t h e m o d e l , t o t h e f a c t t h a t i t n e g l e c t s t h e s e c o n d a r y c i r c u l a ti o n . I n c o r p o r a t i o n o f t h i ss e c o n d a r y c i r c u l a t i o n i n t o t h e t u r b u l e n t m o d e l m a k e s i t e s s e n t i a l l y m o r e c o m p l i c a t e d a n dr e n d e r s t h e i n t e r p r e t a t i o n o f r e s u l t s v e r y d i f fi c u l t . A t t h e f i r s t s t a g e i t w o u l d t h u s b e i n -s t r u c t i v e t o s t a r t w i t h a c o n s t a n t v i s c o s i t y , i n o t h e r w o r d s , w i t h a l a m i n a r f lo w i n t h e a n n u -l u s. H e r e w e c a n t a k e a d v a n t a g e o f a v e r y s im i l a r c a s e o f f u l l y d e v e l o p e d l a m i n a r f l o w i nc u r v e d c h a n n e l , w h i ch w a s in v e s t ig a t e d v e r y c a r e f u ll y a n d n u m e r o u s e x p e r i m e n t a l a n dn u m e r i c a l p a p e r s w e r e d e v o t e d t o t h i s is s ue , s t a r t i n g f r o m t h e w o r k o f D e a n [ 5] . H e r e w ew i ll o n l y r e f e r t o t h o s e p a p e r s w h i c h a r e d i r e c t l y 1 el a te d t o t h e p r e s e n t w o r k .

S t a r t i n g f r o m t h e w o r k o f D e a n , t h e e x i s t e n c e o f a s e c o n d a r y f l o w p a t t e r n c o n s i s t in go f a p a i r o f c o u n t e r r o t a t i n g v o r t i c e s i n a c u r v e d d u c t i s w e l l e s t a b li s h e d . J o s e p h e t a l . [6 ]w e r e t h e f i r s t t o r e p o r t o n a n e w s e c o n d a r y f l o w r e g i m e w i t h f o u r v o r t i ce s , w h i c h w a so b s e r v e d a t D e a n n u m b e r s a b o v e 1 00 .

I n t h e w o r k o f C h e n g e t a l. [ 7] n u m e r i c a l s o l u ti o n s in c u r v e d r e c t a n g u l a r c h a n n e l w e r eo b t a i n e d f o r a s p e c t r a t io s 0 . 5 , 1, 2 , a n d 5 , a n d D e a n n u m b e r s r a n g i n g f r o m 5 t o 7 1 5 . T h e s ea u t h o r s a l so f o u n d t h a t a n a d d i t io n a l c o u n t e r - r o t a t in g p a i r o f vo r t ic e s m a y a p p e a r n e a rt h e c e n t r a l o u t e r r e g i o n o f t h e c h a n n e l , i n a d d i t i o n t o t h e f a m i l i a r s e c o n d a r y f lo w , d e p e n d i n go n t h e a s p e c t r a ti o . T h e y r e l a t ed t h i s p h e n o m e n o n t o D e a n ' s c e n t r i fu g a l in s t a b i li t y p r o b -l e m .

S i m i l a r r e s u l t s w e r e o b t a i n e d b y G h i a a n d S o k h e y [ 8] i n t h e i r s t u d y o f d e v e l o p i n gl a m i n a r f l ow in r e c t a n g u l a r c u r v e d d u c t . T h e D e a n n u m b e r f o r t h e t r a n s i ti o n t o t h es e c o n d a r y f lo w w i t h f o u r v o r t i c e s w a s f o u n d t o b e i n t h i s w o r k a b o u t 1 43 .

T h e d e v e l o p m e n t o f l a m i n a r f l o w i n a 1 8 0 ~ s e c t io n o f a c u r v e d s q u a r e d u c t w a s s t u d i e du s i n g L a s e r - D o p p l e r A n e m o m e t r y b y H i l l e e t a l. [ 9] . I n t h e i r v e r y d e t a il e d e x p e r i m e n t s i tw a s f o u n d t h a t i n a d d i ti o n t o a l w a y s a p p e a r in g v o r t e x p a ir i n s e c o n d a r y f lo w f o r D e a n n u m -b e r s b e t w e e n 1 5 0 a n d 3 0 0 th e r e e x i s ts a s e c o n d v o r t e x p a i r w i t h o p p o s i n g s e n s e o f c i r c u la -t i o n n e a r t h e o u t e r w a l l .

I n a r e c e n t w o r k o f W i n t e r s [ 10 ] , a b i f u r c a t i o n s t u d y o f a la m i n a r f l o w i n a c u r v e d c h a n -n e l o f r e c t a n g u l a r c r o s s- s e c ti o n w a s p e r f o r m e d . T h i s p a p e r s h o w s t h a t f o r a s q u a r e c r o s ss e c t io n t h e t r a n s i t i o n i s a r e s u l t o f a c o m p l e x s t r u c t u r e o f m u l t i p le , s y m m e t r i c a l a n da s y m m e t r i c a l s o l u ti o n s . B o t h t w o a n d f o u r c e ll s w e r e f o u n d i n t w o d i s t i n c t r a n g e s o f a x i a lp r e s s u r e g r a d i e n t f o r a s q u a r e c r o s s- s e c ti o n o f a c u r v e d c h a n n e l .

I n t h e p r e s e n t w o r k t h e a l g o r i t h m s u g g e s t e d b y C h e n g e t a l. [ 7] w a s a d o p t e d f o r t h en u m e r i c a l s o lu t i o n o f t h e f l o w i n c u r v e d c h a n n e l a n d i n a n n u l u s . I n o r d e r t o f i n d t h e m u l -t i p l e so l u t i o n s in t h e c a s e o f t h e f lo w i n c u r v e d c h a n n e l , s p e c i a l t r e a t m e n t b e c a m e n e c e s s a r y .T h e r e s u l ts w e r e c o m p a r e d w i t h t h e e x i s t in g n u m e r i c a l a n d e x p e r i m e n t a l d a t a . T h e n u m e r -i c a l ly c o m p u t e d a x i a l v e l o c i t y p r o f i le s i n a n n u l u s w e r e c o m p a r e d w i t h t u r b u l e n t v e l o c i t yf r o m a r e a l an n u l u s e x p e r i m e n t . F r o m t h e q u a l i t a t i v e p o i n t o f v ie w , t h e a g r e e m e n t b e -t w e e n t h e e x p e r i m e n t a l a n d t h e n u m e r i c a l r e s u l t s a p p e a r s t o b e s u r p r i s i n g l y g o o d .

2 Governing equat ions and descr ipt ion of algor i thmF o l l o w i n g C h e n g e t a l . [7 ], w e c o n s id e r a s e t o f n o n d i m e n s i o n a l e q u a t i o n s o f s t a t i o n a r ym o t i o n i n a c u r v e d c h a n n e l w i t h a r e c t a n g u l a r c r o s s -s e c t io n o f w i d t h a , h e i g h t b , a s p e c tr a t i o y = b / a a n d r a d i u s o f c u r v a t u r e R ( F i g . 1 ) . T h e h y d r a u l i c d i a m e t e r D = 2 a b / ( a ~ , b )i s c h o s e n a s a c h a r a c t e r i s t i c l e n g t h f o r n o r m a l i z a t i o n ; v e l o c it i es a r e n o r m a l i z e d b y v / Da n d p r e s s u r e b y ~ 2 / D ~ w h e r e u i s t h e k i n e m a t i c v i s c o s i ty a n d Q s t h e d e n s i t y o f t h e f l u i d .


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Flow gen erated in an annulu s by rota t in g screens ~[1

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F ig . 1 . Coor d ina te sys tem f or cur ved channe l andannulus : ~ is the angular veloci ty of screens , ws isthe i r l inear ve loc i ty

T h e n , t h e g o v e r n i n g e q u a t i o n s a r e a s f o ll o w s :V o r t i c i ty t r a n s p o r t e q u a t i o n f o r s e c o n d a r y f lo w

u - - ~ + v 8 y r ( l + x / r ) ~ x + r ( l + x / r ) 2 w ~Y - - ~x-- + ~y~ (1 )w h e r e u , v , w a r e d i m e n s i o n l e s s v e l o c i t y c o m p o n e n t s i n x , y a n d z d i r e c t i o n c o r r e s p o n d i n g l y ,a n d v o r t i c i t y ~ = ( ~ v / ~ x - ~ u / ~ y ) / ( 1 - [ -x / r ) ; r = R / D i s t h e d i m e n s i o n l e s s r a d i u s o fc u r v a t u r e .S t r e a m f u n c t i o n - - v o r t i c i t y e q u a t i o n

(1 + x / r ) ~ ~ = ~ x 2 + ~ y 2 ] + 1 ~ br (1 + x/r) ~--'x (2 )w h e r e t h e r e l a t i o n s b e t w e e n t h e s e c o n d a r y f lo w v e l o c i t y c o m p o n e n t s a n d s t r e a m f u n c t i o na r e

u 1 + x / r ~ y a n d v = - - 1 + x/r ~--x"A x i a l m o m e n t u m e q u a t i o n

~ w 3 w u w 1 ~ p ~ w ~ 2wu - - ~ + V - ~ y + r ( l + x / r ) - - 1 + x / r ~ z + ~ x- - ~ + ~ y~

1 ~ w w+ r ( 1 + x / r ) 9 x r2(1 + x/ r ) e 9 (3 )

T h e b o u n d a r y c o n d i t io n a t f ix e d w a ll s

- - ~ = w = o . ( 4 )~ x ~ yT h e b o u n d a r y c o n d i t io n a t t h e a x i s o f s y m m e t r y , y : 0

~x ~y~ ~y : ~ = O. (5)


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12 E . Ki t and E . MazorT h e b o u n d a r y c o n d i t i o n a t r o t a t i n g s c r ee n s ( u p p e r a n d l o w e r ) i n a n n u l u s d i ff e rs fr o m E q . 4o n l y b y n o n z e r o v a l u e o f a x i a l v e l o c i ty :w = ~ 9 . r ( 6 )

w h e r e X2 i s t h e n o n d i m e n s i o n a l a n g u l a r v e l o c i t y o f e a c h s c r e e n .D u e t o t h e a d o p t e d p r o c e d u r e o f r e n d e r i n g t h e v a r i a b l e s d i m e n s i o n le s s t h e m e a n v a l u eo f a x i a l n o n d i m e n s i o n a l v e l o c i t y w a c r o s s t h e s e ct i o n re p r e s e n t s t h e f l o w R e y n o l d s n u m -b e r , R e = @ ~ - W D / v , w h e r e W i s th e m e a n v a l u e o f t h e d i m e n s i o n a l a x i a l v e l o c i t y Wa c r o ss th e s e c t io n ; t h e D e a n n u m b e r K w a s d e fi n e d a s R e / r ~ T h e t o r q u e d i s t r i b u t i o n a td i f f e r e n t w a l l s a n d f l i e t i o n f a c t o r / w e r e o b t a i n e d b y e a l cu l a t. io n o f s h e a r s t r e s s a l o n g t h ew a l ls . N o t e t h a t i n c a l c u l a t io n s o f t h e f ri c t io n f a c t o r i n a n n u l u s o n l y fi x e d w a l ls w e r e t a k e ni n t o a c c o u n t .

T h e g o v e r n i n g e q u a t i o n s a n d t h e c o r r e s p o n d i n g b o u n d a r y c o n d i t io n s E q s . ( 1 )- - (6 ) w e r es o l v e d b y f i n i t e -d i f fe r e n c e m e t h o d i n a m a n n e r v e r y s i m i l a r t o t h a t s u g g e s t e d b y C h e n ge t a l . [ 7 ]. I n o r d e r t o l o o k f o r m u l t i p l e s o l u t i o n s , w e u s e d t h e s o c a l l ed " c o l d " a n d" h o t " s t a r t i n g c o n d i t i o n s . [ [h e s e c o n d i t i o n s d e s c r i b e t h e i n i ti a l v e l o c i t y fi e l d n e c c s s a r y t os t a r t t h e i t e r a t i o n s f o r s o l v in g t h e E q s . ( 1 ) -- ( 3 ). " C o l d " s t a r t i n g c o n d i ti o n s w e r e o b t a i n e db y s o l v i n g t h e a x i a l m o m e n t u m E q . (3 ) f o r a c h o se n p r e ss u r e g r a d i e n t ~p/~z a s s u m i n g t h a tt h e s e c o n d a r y v e l o c i t i e s u a n d v a r e z er o . A s a " h o t " s t a r t i n g c o n d i t i o n f o r a n e w v a l u e o fp r e s s u r e g r a d i e n t ~p/O z ~ A(Op/Oz) w e c h o o s e th e v e l o c i t y f ie l d w h i c h w a s o b t a i n e d a s af i n a l s o l u t io n f o r p r e s s u r e g r a d i e n t ~p/~z . T h e d e c r e m e n t A ( ~p /~ z) o f t h e p r e s s u r e g r a d i e n tc o u l d b e c h o s en e i t h e r p o s i t i v e o r n e g a t i v e d e p e n d i n g o n i ts ' s t a r t i n g v a l u e . I n m o s t o fo u r c o m p u t a t i o n s t h e m e s h s i z e o f 4 0 2 0 w a s c h o s e n fo r a s q u a r e c r o s s -s e c t io n . T h e c o n -v e r g e n c e c r i te r i o n d e fi n e d a s a m a x i m u m v a l u e o f t h e d i f f er e n c e b e t w e e n t h e i n p u t a n d o u t -p u t v a l u e s o f w a n d ~b f o r e a c h i t e r a t i o n c y c l e w a s 1 0 - 4 , c o m p a r e d t o 1 0 - a i n t h e w o r k o fC h e n g e t a l.

3 Numerical computation results3 . 1 F l ow i n a c ur ve d c hanne lS t a t e d i a g r a m p r e s e n t i n g t h e d e p e n d e n c e o f R e y n o l d s n u m b e r o n t h e p re s s u r e g r a d i e n t f o ra c u r v e d c h a n n e l w i t h a s q u a r e c r o ss - se c t io n o b t a i n e d b y a p r o c e d m e d e s c r ib e d i n t h ep r e v i o u s s e c ti o n , is s h o w n i n F i g . 2 . S o l u t i o n s o b t a i n e d u s i n g t h e " c o l d " s t a r t i n g c o n d i -t i o n s a r e p r e s e n t e d b y a s o li d l in e , w h il e so l u t io n s c o m p u t e d w h e n t h e " h o t " s t a r t i n g c o n -d i t io n w e r e ap p l i e d , a re d r a w n b y a d a s h e d l i n e. I t i s i m p o r t a n t t o n o t e t h a t t h e t r a n s i t i o nf r o m a s o l u t i o n w i t h t w o v o r t i c e s i n t h e f l o w t o a s o l u t i o n w i t h f o u r v o r t i c e s o c c u r s f o r t h ef i rs t t i m e a t D e a n n u m b e r K = 1 4 7, c o m p a r e d t o K = 1 50 o b t a i n e d i n t h e e x p e r i m e n t s o fH i l l e e t a l. [ 8] . T h e t r a n s i t i o n b a c k t o a f l o w w i t h t w o v o r t i c e s o c c u r s i n o u r c o m p u t a t i o na t K ~ 2 7 0 , w h i l e H i ] l e e t a l . r e p o r t e d t h a t i n t h e i r e x p e r i m e n t s t h i s c r i t i c a l v a l u e o fK = 3 0 0 . S i n c e i t i s u n c l e a r f r o m t h e i r d e s c r i p t i o n o f t h e e x p e r i m e n t s w h a t t y p e o f s t a r t -i n g c o n d i t io n s t h e y u s e d t h e p o s s i b le v a r i a t i o n o f t h i s c o n d i t io n m a y e x p l a i n t h is s m a l ld i s c r e p a n c y b e t w e e n t h e c r it i ca l v a l u es o f D e a n n u m b e r . A n o t h e r r e a s o n m a y b e a t t r i b u t e dt o d i f f e r e n t v a l u e s o f c u r v a t u r e : r = 4 i n o u r c o m p u t a t i o n s a n d r = 6 . 5 i n t h e i r e x p e r i -m e n t s .

F i v e r e g i o n s o f m u l t i p l e s o l u t io n s ( h y s t er e si s ) w e r e o b t a i n e d w h e n " h o t " c o n d i t io n sw e r e u s e d in o u r n u m e r i c a l c a l c u l a t i o n s c f . F i g . 2 . T h r e e o f t h e m ( in t h e p r e s s u r e g r a d i e n t


5/16

F low gener a ted in an annulus by r o ta t ing sc reens 138 - -

0o

r = 47 - 7 -=1

6 / // z //J- / 72 / , t , I

1 2 39pz / 1 0 0 0 0

4 5

F ig . 2 . The va r ia t ion o i Reynolds nu m be r wi thpr es sur e g r ad ien t f o r cur ved channe l wi th squa r ecross-section; r = 4~

r a n g e s 1 3 5 0 0 < - -Op/~z < 1 4 0 0 0 , 3 0 0 0 0 < - - ~p / ~z < 4 1 5 0 0 , 4 1 5 0 0 < - - ~ p / '~ z < 4 3 0 0 0 )w e r e o b t a i n e d w i t h i n c r e a s i n g p r e s s u r e g r a d i e n t a n d t h e o t h e r t w o , i n t h e r a n g e 2 3 5 0 0< - -~p /Oz < 3 0 0 0 0 a n d 1 1 0 0 0 < - - ~p / ~z < 1 3 5 0 0 , w i t h t h e d e c r e a s i n g p r e s s u r e g r a d i e n t .I t i s i n t e r e s t in g t o n o t e t h a t i n t h e r a n g e 3 0 0 0 0 < - - @ / ~ z < 4 1 5 0 0 m u l t i p l e s o l u t i o n s w e r eo b t a i n e d w i t h " h o t " s t a r t i n g c o n d i t i o n s fo r b o t h i n c r e a si n g a n d d e c r e a s in g p r e s s u re g r a -d i e n t s .

S o l u t i o n s w e r e u s u a l l y o b t a i n e d f o r h a l f e l o s s - s e e t i o n , a s s u m i n g s y m m e t r y a b o u t t h eh o r i z o n t a l a x i s . S o m e o f t h e c o m p u t a t i o n s w e r e m a d e f o r t h e f u l l cr o s s - se c t i o n , e s p e c i a l l y int h e r a n g e o f 1 5 0 0 0 < - - @ / ~ z < 2 3 0 0 0 , w h e r e a s o l u t i o n w i t h f o u r v o r t i c e s o n l y e x i s t s .N o s y m m e t r y b r e a k i n g w a s o b t a i n e d i n t h e s e c a l c u l a t i o n s .

I n o r d e r t o f a c i l i ta t e c o m p a r i s o n w i t h o t h e r a u t h o r s , s o m e i n t e g r a l p a r a m e t e r s o f th ef lo w , l i ke f r ic t i o n f a c to r / m u l t i p l i e d b y R e / . I {e , a n d t o r q u e d i s t r i b u t io n , w e r e c o m p u t e da n d p r e s e n t e d o n F i g . 3 a n d F i g . 4 a s a f u n c t i o n o f D e a n n u m b e r . T h e c o l l ap s e o f t h e r e s u l t sf o l / . R e i n F i g . 3 , o b t a i n e d f o r d i f f e r e n t v a l u e s o f a s p e c t r a t i o a n d c u r v a t u r e s , i s s u r -p r i s i n g l y g o o d a n d p r o v e s t h a t e v e n f o r h i g h v a l u e s o f K i t c a n b e u se d a s a n a p p r o p r i a t er e p r e s e n t a t i v e p a r a m e t e r f o r t h e fl o w i n a c u r v e d c h a n n e l. F o r a w e a k c u r v a t u r e r - - 1 00 ,t h e t o r q u e d i s t r i b u t io n i s cl o se t o u n i f o r m f o r lo w v a l u e s o f K . W i t h i n c r e as i n g K , t h e t o r q u ea t t h e i n n e r w a l l d e c r e a s e s w h i l e i n c r e a s i n g a t t h e o u t e r w a l l , so t h a t t h e i r r a t i o a p p r o a c h e st h e v a l u e o f a b o u t 4 ( F ig . 4) . F o r a s t r ) n g c u r v a t u r e , r = 4 , t h e t o r q u e a t t h e i n n e r w a l l i sn o t a b l y s m a l l e r t h a n a t t h e o u t e r w a i l ' e v e n f o r l o w K a n d t h e r a t i o b e t w e e n t h e m i n c r e as e sw i t h i n c r e a s i n g D e a n n u m b e r s .

T h e a x i a l v e l o c i t y d i s t r i b u t i o n f o r t h e c r o s s - s e c ti o n is s h o w n i n F i g s . 5 a a n d 5 b f o r t h es a m e n o r m a l i z e d p r e s s u r e g r a d i e n t ~p/~z = - - 3 0 0 0 0 , b u t f o r d i f f e re n t n u m b e r o f v o r t i c e si n t h e s e c o n d a r y f l o w , 2 v o r t i c e s in F i g . 5 a a n d 4 v o r t i c e s i n F i g . 5 b . I t c a n b e s e e n f r o mt h e s e F i g u r e s t h a t i n t h e f l o w w i t h 4 v o r t i c e s t h e r e i s a p r o n o u n c e d v a l l e y i n v e l o c i t y d i s t ri -


6/16

1 4 E . K i t a n d E . M a z o r

n. -

"- 2 5

4 55 5

r S y m b o l

15~

r S y m b o L4 5 -6 - x 4 - x1 5 - - A 0 - - A xx

1 5 0 - n xX 3 5 - I O 0 _ n --~ x,~ xx~ r ~ / ' ~ ' -~ - 2 5

( a ) I ( b )

5 0 _ I I I I ~ IZ O O 4 0 0 6 0 0 o 2 o o 4 o o 6 0 0K K

F i g . 3 . T h e v a r i a t i o n o f / . R e w i t h D e a n n u m b e r K f o r c u r v e d c h a n n e l w i t h c r o s s - s e c t i o n s of d i f fe r -e n t a s p e c t r a t i o s : a y = 1 ; b y = 0 .5

r ~ P ~ : b o o c~3 ~ ~J X x xx X x ~ X ~

F - 2 5 ~xxx xx xxX

F-

W0nr0F-

-o 176176176176176 ~176x xxxX XlXX-- x i ' ~

(a) (b)I~ ~- - -A

a ~ ~ aa,~a,~~, [ I I y I50 200 400 600 0 200 400K K

~,~ (c)i i I 1600 0 200 400 600K

F i g . 4 . T o r q u e b a l a n c e a t d i f f e r e n t w a l l s : / k - - i n n e r w a ll , - - o u t e r w a l l, [ ] - - u p p e r a n d l o w e rw a l l s o f c u r v e d c h a n n e l a s a f u n c t i o n o f D e a n n u m b e r f o r t h r e e v a l u e s o f r : a r = 4 , b r = 1 0 , e r ~ 10 0

A~o f

(o) (b )

F i g . 5 . A x i a l v e l o c i t y d i s t r i b u t i o n i n a s q u a r e c r o s s - s e c t i o n o f c u r v e d c h a n n e l a t a p / ~ z = - - 3 0 0 0 0 ,l e a d i n g t o m u l t i p l e s o l u t i o n s w i t h d i f f e r e n t n u m b e r o f v o r t i c e s in s e c o n d a r y f l o w : a t w o v o r t i c e s ,b f o u r v o r t i c e s


7/16

Flow generated in an annulus by rotating screens 15bution in tile central band of cross-section in the vicinity of the outer wall. A more detailedpicture of the velocity field ma y be obtaine d f rom I~'ig. 6, where axial velocity profilesalong the horizontal and the vertical lines are presented. Results are shown at differentvertical and horizontal locations for five values of pressure gradient in the case of strongcurv atur e (r = 4). In Fig. 7 the str eam funct ions of the secondary flow are plot:ted for thesame flow conditions. The mo st striking result here is a very high level of asymmetJ:y of thevelocity profile in the radial direction. W hen the sec ondary flow becomes stronger, theasymmetry of the axial velocity profile increases, leading to a formation of the so called"tongue" in the region near to the outer wall, where most of the velocity discharge occurs.In the major part of the cross-section the flow velocity is decaying very slowly towards theinner wall, Fig. 6, and at the transitmn between the "tongue" region and the remainingpar t of the velocity profile appears an reflection point. The veloeity profile in the vertic aldirection for high values of K (at least in central par t of the cross-section) is much mo rehomogeneous than for small values of K, Fig. 6. All these results may be explained by thefact th at the secondary flow leads to a very high momen tum transport, so that away fromthe solid walls it becomes dominant and takes over the transport caused by viscosity.

. 2 . 0

1 . 5

l . O

0 . 5 0

0- - 0 .

V f . 0 -i i . 5 -t . 0 -0 . 5 0 -

0- 0 . 5 0

y=5 /8

5 0 - 0 . 2 5 0 . 0 0 . 2 5 0 . 5 0

y = 1 14

I , , x ^>~Is "~- 0 . 2 5 O . 0 O. 25 O. 50

2 . 0 '

> 1.5'~ t . O 84

x = 5 / 4

0 t I Y AXIIS0.0 0.125 0.25 0.375 0.50i i o_; x= 112~ '" " ' ~ " ' 0 , : "

,~.o. - . % . . . . ~0.50 - "" ""x.~i

0 . 0 0 . ~ 2 5 0 . 2 5 0 . ~ 7 5 0 . 5 0

_] 2.O7y=18 .~ ._ . ~2"~ x=l1 4t I f t " l

I 0 / ' . . " . . ~ \ '.~1 H /" i t . . . . _ _ _ _0 l i 0 1 ,

--0.50 --0 . ~5 0 . 0 0"2 5 0"5 0 1 O: 0 l 0 "1~25 l 0"2 5 0.3 75 0"5 0Fig. 6. Axial velocity profiles in the radial and vertical directions at various positions of cross-sectionof curved channel with r = 4 for the following values of --gp/gz" 500, - - - - - - ; 6000, . .......... ;20000, - . . . . . . ; 30000, ,30000, -- 9 --; 90000, .. .. .


8/16

16 E. Kit and E. MazorVery similar results were obtained for smaller curva ture (r = 10 and r = 100), at least

at high values of K. At low values of K, for r = 100, the axial v elocity profiles have verysymmetrical form.

3 . 2 N u m e r i c a l s o l ut i o n s m a n n u l u s

Since the chosen algorithm has been very successful in resolving the laminar flow in acurved channel, we applied it to co mpute th e flow in annulus with one o1' two rot atingscreens. Note that experiments in an annulus with two rotating screens were performed forthe first time in the Laboratory for Stratified Flow Studies, Tel-Aviv University; theseresults were report ed in Chai [11] and in Chai and K it [12]. As it was alrea dy stressed in theprevious chapter, the govelning equations for curved channel and for annulus are the same,and only the boundary conditions for these two types of flow are different. It was thereforenatural to look for multiple solutions and hysteresis in this case too. However, as it can be

0 . 5

y

0 -

- 0 . 5

" ~_ _E _ 5 0 0~ ) zI F

I I

0 0 . 5

0 . 5

0 . 5

X

O P = - 2 0 0 0 09 z

-0.5 0 0.5X

~ P = - 3 o o o o~ z

I I I

0 i ~ l- 0 . 5 0 0 . 5

-0 .5

"~.__EP - 6 O O O9 zI 1 I

1 I

0 0.5

"~P = - 3 0 0 0 09 zI

D0.5 0 0.5X

O._E= - 9 0 0 0 0~z

I 1

I i i

-0.5 0 0.5 X

Fig. 7. Secondary flow streamlines in curved channel; r = 4


9/16

Flow generated in an annulus by rotating screens 17seen from the state diagram presented at the Fig. 8, no multiple solutions were obtainedneither for the flow in annulus with one rotating screen, nor for the flow ia annulus withtwo rotating screens. If the appearance of an additional vortex pair in the curved channelat higher values of Dean number may be interpreted as some kind of instability, the lack ofsuch an instability in annulus reminds the situation with the flow stability in straightchannels. It is well known that linear stability analysis shows that the plane Poiseuilleflow is unstable, in con trast to Co uette flow, which is absolutely stable (cf. Schlichting [13],p. 465 and p. 480). The m axi mum value of nondimensional an gular velocity, obtainedbefore the solution started to diverge, was 500 for the flow with two rotating screens, thusbeing essentially higher than the corresponding critical value of 250 for the flow with onerotatin g screen.

The friction coefficients for the flow in annulus with one and two rotating screens arepresen ted in ]?ig. 9 as a function of Dean numbe r. Again, the collapse of the results o btainedfor different flow condition, like curvatule and aspect ratio, for the both cases of annulusflow was more th an reasonable (Figs. 9a, b). As in the case of the flow in curved channel, itindicates that Dean number remains an appropriate parameter for the annulus flow evenat high values of K. As it can be seen from Figs. 10, 11, the redi stribution of torque with in-creasing K in annulus appears to be even more dramatic then in the curved channel. Atsmall values of K, the torque at the inner wall is almost equal to that at the outer wall, atleast in the ann ulus with weak curvatur e, r = 100 for annulus wit h two rotat ing screensand r = 150 for annulus with one rotatin g screen. At high Dean numbers, however, theratio between these two values of torque becomes very high and approaches the value of8. This result implies that the momentum transport in annulus, which is caused by theseco ndar y flow vortices, is even more impo rta nt tha n t hat in the case of the flow in a curvedchannel.

ooa::

1"2F Two ro ta tin g screens , / /1.0 r= 5 , Y=] ~ /

One rotating screen //0 . 8 r =Z 5 =u /

/.60 .4 /0 .2 z /I I I I Jo I 2 3 4 5

, G / I O 0

Fig. 8. The variation of Reynolds number withangular velocity of one rotating screen, r = 7.5,y ~ 0.5 and two rotating screens, r = 5, y ~ 1 ;the width a and the dimensional radius of curvatureR are the same in both cases


10/16

18 E. Ki t and E. Mazor

CDI:K

r Symbol .4 5 7 . 5 - - x

1 5 - z ~5 5 _ 1 5 0 - a

X

2 5X

5 0 I I2 0 0 4 0 0

K

S y m b o l .- x- - A AZ~ x-- [3 Z~

r54 5 l O

I 0 03 5

2 5 ~

( o ) ( b )15I 5 I I I6 0 0 0 2 0 0 4 0 0 6 0 0

KYig . 9 . The var i at io n of ] 9 R e with Dea n numbe r K for annul us wi th a one ro ta t ing screen and b tworo t a t i n g sc r eens

6O

zo~ - 4 0mr r1 -5 91

2 0uJo rn*op-

0

X X X X X

LlA

D n

Z~ A ZI

( a )

X XX

A

X X X Xx

Z~AZ~ A

O O O D []T I I I I

5 0 1 50 2 5 0 0 2 5 0 2 5 0K

(b ) - - = (c )El Q []r'l D

I I l I5 0 1 5 0 0 5 0 1 5 0

K KFig . 10 . Torque balan ce at d i f feren t wal l s : [] - - i nner wal l , -- outer wal l , /k - - b o t t o m o f an n u l u swit h one ro ta tin g scree n as a func tio n of Dea n numbe r for thre e values of r: a r = 7,5, b r = 15,er = 150

70 - -z0l -m0 ~ 5 0 - -I'-

I,LIOfn" 30--0 XI--

[] rn

(Q )

D []

0

X

[3D

O O 0

(b )

n[]

- D 0

XXX

x_ _ x

(c )x

x x x x

iX X l X x l ! I I I I ;0 2 0 0 4 0 0 6 0 0 0 2 0 0 4 0 0 6 0 0 0 2 0 0 4 0 0 6 0 0K K K

Fig . 11. Torque balanc e at d i f feren t wal l s : X - - inner wal l , [ ] - - ou ter wal l o f annu lus wi th tw orot at ing scr een as a func tion of De an num ber for thre e valu es of r: a r = 5, b r = 10, e r = 100


11/16

F l o w g e n e r a t e d i n a n a n n u l u s b y r o t a t i n g s c re e n s 1 9

T h e a x i a l v e l o c i t y d i s t r i b u t i o n i n t h e c r o s s - s e c t i o n o f a n n u l u s w i t h o n e r o t a t i n g s c r e e ni s p r e s e n t e d i n t h e F i g . 1 2 a , a n d w i t h t w o r o t a t i n g s c r e e n s i n F i g . 1 2 b . I n t h e c a s e o f t w or o t a t i n g s c r e e n s , t h e v e l o c i t y d i s t r i b u t i o n i s a l m o s t h o m o g e n e o u s i n th e m o s t o f c r os s -

B( A:of

~n

F i g . 12 . A x i a l v e l o c i t y d i s t r i b u t i o n f o r a n a n n u l u s a t .(2 = 2 0 0 w i t h a o n e r o t a t i n g s c re e n , b t w or o t a t i n g s c r e e n s/~. 0

1 . 5 J1 . 0

0 . 5 0 -

/ . . : . : .~ , " i/ \

r / " , , x A~Is 0 l A S,I i-- 0 . 5 0 - 0 . 2 5 0 . 0 0 .2 5 .5 0

o.,o -I0 ~ , , x " v ~ \ 1- o . 5 0 - 0 . 2 5 0 .0 o .2 5 0 .5 0

0 . 0 0 . 2 5 0 . 5 0 0 . 7 5 i . 0

9 3 . 0

~ . 0

t . 0

00.0

x = l / 2 _ 7 . C ~ . ~ , ~ . .'. .

J I Y A ~ S " " ..+0 . 2 5 0 . 5 0 0 . 7 5 i. O

2 . 0 "

1 . 5 -

0 . 5 0 -0 Y

-0.50

y = I / 4 ~ ~ ~-o.~ o.o o.~ o.~o

II 3 . 0I x = 1 / 4

t 0 ] / / /% ~ . . ~ . . ~ ...".

o ~ , , YX-'-x~ J ,0.0 0.25 0.50 0.75 ~.0

F i g . 1 8 . A x i a l v e l o c i t y p r o f i l es i n t h e r a d i a l a n d t h e v e r t i c a l d i r e c t io n s a t v a r i o u s p o s i t i o n s o f c r os s -s e c t i o n o f a n n u l u s w i t h o n e r o t a t i n g s c r e e n , r ~ 7 : 5 , f o r f o l l o w i n g v a l u e s o f t 9 : 5 , - - - - ; 1 0 0 , 9 . . . . . . . . . . ;2 0 0 , - . . . . . ; 2 5 0 , - - . . . .


12/16

20 E . K i t and E . Mazor,9/ ,=5

I I I

tD

0 1 I I- 0 . 5 0

x

0 .5

9 , : I 0 0I I I

I I 1- 0 . 5 0 0 . 5

Q , : . 2 0 0 = 2 5 0

y _

0- 0 . 5

I I [

i I i0 0 . 5

I I I- 0 . 5 0 O . 5

x xF i g . 1 4 . S e c o n d a r y f l o w s t r e a m l i n e s i n a n n u l u s w i ~ h o n e r o t a t i n g s c r e e n , r = 7 . 5

s e c t io n , a n d o n l y n e a r t h e w a l l w e c a n a g a i n s ee a " t o n g u e " o f v e l o c i t y . O n t h e c o n t r a r y , i nt h e c a s e o f o n e r o t a t i n g s c r e e n t h e a x i a l v e l o c i t y i s g r a d u a l l y d e c r e a s i n g t o w a r d s t h e i n n e rw a l l , F i g . 1 2 a . M o~ e d e t a i l s a b o u t t h e a c t u a l a x i a l v e l o c i t y p r o f i l e s a n d c o r r e s p o n d i n g s t r e a mf u n c t i o n s o f t h e s e c o n d a r y f lo w c a n b e o b t a i n e d f r o m t h e F i g s. 1 3 , 1 4, d e s c r i b i n g t h e f l o w i na n n u l u s w i t h o n e r o t a t i n g s c r e e n , a n d f r o m F i g s . 1 5, 1 6 f o r t h e fl o w w i t h t w o r o t a t i n g s c r e en s .T h e v e l o c i t y " t o n g u e " a n d e x i s t e n c e o f a n i n f l e c ti o n p o i n t i n t h e v i c i n i t y o f t h e o u t e r w a l li n a n n u l u s , e s p e c i a l ly f o r t h e e a s e o f o n e r o t a t i n g s c r e e n , is e v e n m o r e p r o n o u n c e d t h a n i nt h e c u r v e d c h a n n e l f lo w . I t i s i n t e r e s t i n g t o n o t e t h a t t h e v e l o c i t y p r o f i le i n t h e v e r t i c a ld i r e c t i o n ( F i g . 1 6 , x = 0 ) i s n o n - m o n o t o n o u s a n d t h e r e i s a d i p i n t h e v e l o c i t y p r o f i l e i nt h e c l os e v i c in i t y o f t h e a x i s o f s y m m e t r y . A g a i n , s u c h b e h a v i o r o f t h e v e l o c i t y p r o f i l e c a no n l y b e e x p l a in e d b y t h e f a c t t h a t t h e m o m e n t u m t l a n s p o r t is m o s t l y d u e t o t h e s e c o n d a r yf l o w v o r t i c e s .

I n t h e e x p e r i m e n t s i n a n n u l u s t h e f l o w i s u s u a l ly t u r b u l e n t , c o n s e q u e n t l y t h e t u r b u l e n tv i s c o s i t y a c r o s s t h e c r o s s - s e c t i o n d e p e n d s o n t h e f l o w r e g i m e . H o w e v e r , i f t h e m o m e n t u mt r a n s p o r t i s m a i n l y d u e t o t h e s e c o n d a r y fl o w , t h e a c t u a l d i s t r i b u t i o n o f t u r b u l e n t v i s c o s i t yw i ll p l a y o n l y & m i n o r r o l e in d e t e r m i n i n g t h e f i n a l s h a p e o f t h e a x i a l v e l o c i t y p r o f il e ina n n u l u s . T h u s , a q u a l i t a t i v e c o m p a r i s o n o f t h e v e l o c i t y p r o f i le s o b t a i n e d i n o u r n u m e r i c a lc o m p u t a t i o n s , w h i c h a r e b a s e d o n a n s s u m p t i o n o f c o n s t a n t v i s c o s it y , w i t h a c t u a l e x p e r i -m e n t a l v e l o c i t y p r o f i le s i n a n n u l u s c a n b e m a d e . R e s u l t s o f s u c h c o m p a r i s o n a r e p r e s e n t e di n F i g . 1 7, w h e r e b o t h e x p e r i m e n t a i ' :v e l o c i ti e s a n d t h o s e o b t a i n e d i n n u m e r i c a l c o m p u t a t i o n sw e r e n o r m a l i z e d b y t h e s c r e e n v e l o c i ty . T h e q u a n t i t a t i v e a g r e e m e n t i s n o t v e r y s a ri s-


13/16

F l o w g e n e r a t e d i n a n a n n u l u s b y r o t a t i n g s c r ee n s 2 1

. 2 . 0 y : 5 / 8~ . 5 i . ~ . . . . . 1 7 . : : ~ ; : Al . . . . . . . . . _ _ ~ 2 . ~ ".

0 " 5 0 V / , , X ^XiIS " l0 - 0 . 5 0 - 0 . 2 5 0 . 0 0 . 2 5 0 . 5 0

i 2 . o - i y : l / 4- ~ t . 5 1 . ~ - . ~ ~

t . 0 I . . -- ~ . '. - : -~ . - ~ - - ' - - ~ - - - . " . ,I , / ' / \ : ' ~0.50 I r do I " , , x ^ u , ~- 0 . 5 0 - 0 . 2 5 O . 0 O . 2 5 O . 5 0

2 . 5 4

~2 . 0 -

t . 5 -

i . o -0 . 5 0 -

00 . 0

x : 5/4. . ' /

~ . : . . . . ~ . ~i i Y A ~ S i

0 . t 2 5 0 . 2 5 0 . 3 7 5 0 . 5 0

. 5 X : l / 2

5 0i i Y AXrLS"" !

0 . 0 0 . 1 2 5 0 . 2 5 0 . 3 7 5 0 . 5 0

2 .0 ] Y : I / 8> 1 . 5 1

t . 0 r 7 ~ . ~ . ~ . , ~ Z . . . . , , ~< / / / . " / " ~ . " '.) }

o . 5o - I I / . . ' . / ~ . :. ,\o Y =

- 0 . 5 0 - 0 . 2 5 0 . 0 0 . 2 5 0 .5 0

j ] x : I / 42 . 5i i ~ 2 . 0 1

0 5 0 4 " : : : ~ - , ~ ~ 'Y i T

0 . 0 0 . 1 2 5 0 . 2 5 0 . 3 7 5 0 . 5 0F ig . 1 5 . A x i a l v e lo c i ty p ro f i l e s i n th e r a d ia l a n d th e v e r t i c a l d i r e c t io n s a t v a r io u s p o s i t i o n s o f c ro s s -s e c t io n o f a n n u lu s w i th tw o ro t a t in g s c re e n , r = 7 .5 , fo r fo llo w in g v a lu e s o f D : 5 , - - . - - ; 1 00 , 9 . . . . . . . ;2 00 . - . . . . . ; 5 0 0 , - - - - - -

f a c t o r y , b u t q u a l i t a t i v e l y , t h e m a i n f e a t u r e s o f th e v e l o c i t y p r o f il e , l i k e s t r o n g v e l o c i t ya s y m m e t r y , a p p e a r a n c e o f v e l o c i ty " t o n g u e " i n t h e v i c i n i t y o f t h e o u t e r w a l l a n d e x i s t e n c eo f i n f l e c t io n p o i n t , w e r e c a u g h t c o r r e c t l y b y t h e n u m e r i c a l m o d e l . E v e n n o n - m o n o t o n o u sb e h a v i o r o f t h e v e l o c i t y p r o f i l e i n t h e c e n t r a l p a r t o f c r o s s - s e c t i o n (F i g . 1 7 b ) w a s o b t a i n e d .

4 C o n c l u d i n g r e m a r k sT h e g o v e r n i n g e q u a t i o n s d e s c r i b i n g s t e a d y l a m i n a r f l o w i n a n n u l u s w i t h o n e o r t w o r o t a t i n gs c r e e n s , w e r e s o l v e d b y f i n i t e - d i f f e r e n c e m e t h o d . I n o r d e r t o c h e c k t h e a l g o r i t h m a p p l i e di n t h e p r e s e n t w o r k , a d v a n t a g e w a s t a k e n o f a l a r g e b u l k o f n u m e r i c a l a n d e x p e r i m e n t a ld a t a k n o w n f r o m t h e l i t e r a t u r e f o r a s im i l a r p r o b l e m o f t h e f lo w in c u r v e d c h a n n e l w i t h ar e c t a n g u l a r c r o s s- s e ct i o n .

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


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22 E . Ki t and E . Mazor

0 . 5

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Fig. 16. Secon dary flow s treamlines in annulus with tw o rota t in g screens , r = 5

c h a n n e l , i n a c c o r d a n c e w i t h e a r li e r t h e o r e ti c a l p r e d i c t io n s . D i f f e r e n t s e c o n d a r y f l o w p a t t e r n sw h e r e t h u s o b t a i n e d . T h e s e p a t t e r n s c o n s i s t o f e i t h e r t w o o r fo u r v o r t i c e s , f o r i d e n t i c a lp r e s s u r e g r a d ie n t s . I n c o n t r a s t t o t h a t , i n a n a n n u l u s , n o m u l t i p l e s o lu t i o n s w e r e o b t a i n e df o r t h e f u ll r a n g e o f a n g u l a r v e l o c i t ie s o f t h e s c r e e n s . W h e n t h e f l o w in a n a n n u l u s w a sg e n e r a t e d b y o n e s c re e n , t h e s e c o n d a r y f lo w c o n s i s te d o f o n e v o r t e x , w h i le in t h e e a s e o ft w o r o t a t i n g s c re e n s , o n l y t w o v o r t ic e s c o u ld b e o b t a i n e d .

T h e m o m e n t u m t r a n s p o r t i n t h e c e n t r a l p a r t o f t h e a n n u l u s c r o ss - se c t io n i s m a i n l yd u e t o t h e s e c o n d a r y fl o w . F o r t h i s r e as o n , t h e q u a l i t a t i v e a g r e e m e n t o b t a i n e d b e t w e e n t h en u m e r i c a l l y c a l c u la t e d l a m i n a r v e l o c i t y p r o f il e s w i t h t h o s e m e a s u r e d e x p e r i m e n t a l l y in at u r b u l e n t f l o w , w a s so g o o d. T h e i m p o r t a n t c o n c l u si o n fo l lo w i n g f r o m s u c h a n a g r e e m e n ti s t h a t a n y t u r b u l e n t m o d e l t o b e e m p l o y e d t o d e s c r ib e t h e fl o w i n a n a n n u l u s , s h o u l d a c c o u n tf o r t h e s e c o n d a r y fl o w . T h i s e x p l a in s w h y t h e r e l a t i v e l y s o p h i s t i c a t e d t u r b u l e n t m o d e lu s e d b y C h a i e t a l. [4 ], w h o n e g l ec t e d t h e s e c o n d a r y f lo w , w a s n o t a b l e to p r e d i c t s u c hi m p o r t a n t f e a t u r e s o f t h e f lo w a s t h e a p p e a r a n c e o f a " t o n g u e " i n t h e v i c i n i t y o f o u t e r w a lla n d t h e a c c o m p a n y i n g i n f l ec t i o n p o in t .

I n c o r p o r a t i o n o f a n a d v a n c e d t u r b u l e n t m o d e l in o r d e r t o d e s cr i b e t h e t u r b u l e n t v i s-c o s i t y i n to t h e p r e s e n t s e t o f e q u a t i o n w i ll h o p e f u l l y a ll o w u s t o c h a r a c t e r i z e m u c h b e t t e rt h e p r o c e s s e s i n a r e a l a n n u l u s . T h e n e x t s t e p i n t h e r e f i n i n g o f t h e t h e o r e t i c a l m o d e l s h o u l di n c l u d e t h e t u r b u l e n t d i f fu s i o n c o e f fi c ie n t in o r d e r t o e v a l u a t e t h e e n t r a i n m e n t p r o c e ss e s i na n a n n u l u s .


15/16

F l o w g e n e r a t e d i n a n a n n u l u s b y r o t a t i n g s c r ee n s 2 3

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o t t ~ / ~ l , r , r b /- 0 . 4 - 0 . 2 0 0 . 2 O .L

F i g . 1 7 . C o m p a r i s o n o f nu m e r i c a ll y c o m p u t e d ( s ol i d l in e s) a n d m e a s u r e d ( x ) v e l o c i t y p r of i le s i nr a d i a l d i r e c t i o n i n a n n u l u s w i t h a o n e r o t a t i n g s c r e e n , y ~ 1 / 2 a n d b t w o r o t a t i n g s c r e e n s , y = i / 4

R e f e r e n c e s

[ 1] K a t o , H . , P h i l l i p s , O . M . : O n t h e p e n e t r a t i o n o f a t u r b u l e n t l a y e r i n t o s t r a t i f i e d f lu i d . J . F l u i dM e c h . 3 7 , 6 4 3 - - 6 5 5 ( 1 96 9 ).[ 2] S c r a n t o n , D . R . , L i n d b c r g , W . R . : A n e x p e r i m e n t a l s t u d y o f e n t r a i n i n g , s t r e s s - d r i v e n , s t r a t i f i e d

f l o w i n a n a n n u l u s . P h y s . F l u i d s . 2 6 , 1 1 9 8 - - 1 2 0 5 ( 1 98 3 ).[ 3] D e a r d o r f f , J . W . , Y o o n , S. C . : O n t h e u s e o f a n a n n u l u s t o s t u d y m i x e d - l a y e r e n t r a i n m e n t . J .

l ~ l u id M e c h . 1 4 ~ , 9 7 - - t 2 0 ( 1 9 84 ) .[ 4] C h a i , A . , H a s s i d , S . , K i t , E . , T s i n o b e r , A . : A s t u d y o f a t w o - l a y e r s t r a t i f i e d f l o w i n a n a n n u l u s :

e x p e r i m e n t a n d m o d e l . P C H . 1 0 , 5 / 6 , 5 6 1 - - 5 7 8 ( 1 9 8 8 ) .[ 5] D e a n , W . R . : T h e s t r e a m - l i n e m o t i o n o f fl u i d in a c u r v e d p i p e . P h i l . N a g . 5 , 6 7 3 - - 6 9 5 ( 1 9 28 ) .[6 ] J o s e p h , B . , S m i t h , E . P . , A d l e r , R . J . : N u m e r i c a l t r e a t m e n t o f la m i n a r f l o w i n h e li c a l l y c oi l ed

t u b e s o f s q u a r e c r o s s - s e c t io n . A I C h E J . ~ 1 , 9 6 5 - - 9 7 4 ( 1 97 5 ).[ 7] C h c n g , K . C. , L i n , 1~ . C . , O u , J . W . : F u l l y - d e v e l o p e d l a m i n a r f l o w i n c u r v e d r e c t a n g u l a r c h a n -

n e l s. T r a n s . A S M E I : J . F l u i d s E n g n g . 9 8 , 4 1 - - 4 8 ( / 9 76 ) .


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24 E. Kit and E. Mazor: Flow generated in an annu lus by rotatin g screens[8] Ghia, K. I~., Sokhey, J. S.: Lamina r incompressible viscous flow in curve d ducts of regula r

cross-section. Trans. ASME I: J. Fluids Engng. 99, 640--648 (1977).[9] Hille, P., Vehrenkamp, 1~., Schulz-DuBois, E. O. : The development and structu re of pri mary

and secondary flow in a curved square duct. J. Fluid Mech. 151, 219--241 (1984).[10] Winters, K. H. : A bifurcation stud y of lamin ar flow in curved t ube of rectang ular cross-section.

J. Fluid Mech. 180, 343--369 (1987).[11] Chai, A.: Experi mental invest igation of mixing process in a two-layer stratified flow in an a nnu-lus. Ph. D. thesis, Tel-Aviv 1989.

[12] Chai, A., Kit, E.: Experiments on ent rainmen t in an annulus with an d without velocity gra-dient across the density interface. Sub mitted for pub licatio n (1989).

[13] Schlichting, H. : Boundary layer theory. McGraw-Hill 1979.

Author s' address: E. Ki t and E. Mazor, Tel-Aviv University, Israel

numerical solution of laminar flow generated

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