8/8/2019 Evolution of Vorticity

1/5

ELSEVIER

E v o l u t i o n o f V o r t ic i ty i n P e r t u r b e d F l o wi n a P i p eJ . T . S t u a r tDepartment o f Mathem atics, Imperial College,London, SW 7 2BZ, Uni ted Kingdom

Flows in long pipes often include "slugs" or " puffs" of complex or chaoticpatterns interspersed with regions of relative calm. A similar phenomenoncan occur in boundary layers, where "turbulent spots" can exist betweenrather calm regions of flow, especially in regions of transition from alaminar flow to a turbulent form. The vorticity field plays a significant rolein these phenomena, and it is an aim of this paper to examine this featurefrom a point of view that might be thought to be unusual.

Some forty years ago, G. 1. Taylor initiated his theory of dispersion of acon tam inan t in the flow in a long pipe. He showed that a combin ation oflongit udinal (axial) convection by the flow coupled with radial diffusion bymolecu lar (or by turbul ent) action produces an effective longitud inal diffu-sion process, which is governed approximately by the usual one-dimen-sional diffusion equa tion. Astonishingly, the effective longit udinal diffusiv-ity coefficient depends inversely on the molecular (or turbulent or eddy)diffusivity. This remarkable phenomenon is now known as Taylor diffusionand has been much studied both experimentally and theoretically over theintervening forty years.

The object of this paper is to study the development of vorticity in theflow in a pipe, with the vortieity treated as if it were a passive cont amin anton a longitu dinal flow through the pipe. I n general, this requires anapproximation, because the interaction between the velocity and vorticityfields is ignored. But, in a particular case--namely, that of a small swirlingvortex perturbation on a flow in a pipe--a theoretical construction be-comes possible in which the major approximation is simply that of lin-earization for small perturbation amplitudes.

As a result, it is found that the swirl amplitude satisfies a one-dimen-sional diffusion equation, just as it does for a scalar contaminant in Taylor'sclassical work. Ther e is, however, a significant difference in o ur proble m ofthe evolution of vorticity or swirl: the effective longitudinal diffusioncoefficient, which, like Taylor's, depends inversely on the molecular diffu-sivity, changes sign at a particular value of the Reynolds number. Thus, forReynolds numbers lower than the critical value, the effective longitudinaldiffusion coefficient is positive; in contrast, it is negative for Reynoldsnumbers larger than the critical value. The latter result implies that afocusing takes place instead of dispersion. The implications of this arediscussed. Elsecier Science Inc., 1996Keywords: Vortic ity, p i pe flow, s lug, puf f , turbulen t spot, Taylordi f fus ion, swir l

Address correspondence to J. T. Stuart, Department of Mathematics, Imperial College, London, SW7 2BZ, United Kingdom.Experimental Thermal and F luid Science 1996; 13:206-210 Elsevier Science Inc,, 1996655 Avenue of the Americas, New York, NY 10010 0894-1777/96/$15.00PII S0894-1777(96)00081-7

8/8/2019 Evolution of Vorticity

2/5

V o r t i c i t y i n P e r t u r b e d P i p e F l o w 2 0 7I N T R O D U C T I O N

O s b o r n e R e y n o l d s [1 ] o b s e r v e d t u r b u l e n t p h e n o m e n a i nt h e f l o w i n a p i p e m o r e t h a n a c e n t u r y a g o a n d w a s t h ef i rs t t o s e t t h e s t u d y o f t u r b u l e n c e i n t o a r a t i o n a l f r a m e -w o r k b o t h e x p e r i m e n t a l l y a n d t h e o re t i c al l y H e n o t i c e de s p e c i a l ly th e o c c u r r e n c e o f s lu g s o f t u r b u l e n c e i n t e r -s p e r s e d w i t h r e g i o n s o f c a l m e r ( l a m i n a r ) f l o w . O v e r t h ed e c a d e s , m u c h w o r k o n s u ch p h e n o m e n a h a s b e e n a s s o c i -a t e d w i t h t h e n a m e s o f T a y l o r [2 ], W y g n a n s k i a n dC h a m p a g n e [3 ], C o l e s [ 4], B a n d y o p a d h y a y [5 ], a n d o t h e r s A n a p p a r e n t l y r e l a t e d p h e n o m e n o n i s t h a t o f a tu r b u -l e n t s p o t i n a b o u n d a r y l a y e r, a s m a d e e v i d e n t b y t he w o r ko f E m m o n s [ 6 ], S c h u b a u e r a n d K l e b a n o f f [7 ], P e r r y e t a l.[ 8] , H e a d a n d B a n d y o p a d h y a y [9 ], C a n t w e l l e t a l . [1 0], a n do t h e r s .P h e n o m e n a o f t h e t y p e d e s c r i b e d h a v e i n c o m m o n ap r o p e r t y o f c o h e r e n c e w h e r e b y t h e v o r t ic i ty , w h i c h i sa s s o c i a t e d w i th t h e t u r b u l e n t s l u g o r s p o t , is s o m e h o wc o m p a c t e d a n d p r e v e n t e d f r o m d i f f u s i n g t o g r e a t e r l e n g t hs ca le s H o w d o e s t h i s h a p p e n ? W h a t a r e t h e p r o c e s s e st h a t c a n e n c o u r a g e c o h e r e n c e a n d c o n f i n e m e n t ? I t is t h isp r o b l e m t h a t i s a d d r e s s e d h e r e . A p a r t i c u l a r m e c h a n i s mw i ll b e d i s c u s s e d , o n e t h a t i s r e l a t e d t o t h e p h e n o m e n o no f T a y l o r d i f f u s i o n o f a c o n t a m i n a n t i n fl o w i n a p i p e . T ob e s u r e , o u r p r o b l e m i s t r u ly a n o n l i n e a r o n e , w h e r e a sT a y l o r ' s is l i n e a r F o r t h e t i m e b e i n g , w e s h a l l i g n o r e t h en o n l i n e a r i t y b y c o n s i d e r in g t h e c a s e o f a p a r t i c u l a r c o m -p o n e n t o f v o rt i c i ty , w h i c h i s s u p p o s e d t o b e i n f i n i t e s i m a li n m a g n i t u d e B e f o r e g o i n g o v e r t o o u r p r o b l e m , w e n e e d t o g i v e b r i e fa t t e n t i o n t o a n d e x p l a n a t i o n o f T a y l o r d i ff u si o n In s u b s e -q u e n t s e c t i o n s , w e s h a l l r e t u r n t o a s tu d y o f t h e e v o l u t i o no f v o r t i c i ty .

T A Y L O R D I F F U S I O NT a y l o r [ 11 ] c o n s i d e r e d f l o w i n a l o n g p i p e o f c i r c u l a r c r o s ss e c t io n , w i th a n a p p l i c a t i o n i n m i n d o f t h e t r a n s p o r t o f o ilo v e r d i s t a n c e s o f th e o r d e r o f t h o u s a n d s o f k i lo m e t e r s H o w d o e s a n i n t e r f a c e e v o l v e b e t w e e n t w o g r a d e s o r ty p e so f o il ? H o w d o t h e y m e r g e o v e r l o n g d i s ta n c e s ?C o n s i d e r a p i p e o f r a d i u s a w i t h a l a m i n a r f l o w U 0( 1 -r e ) d o w n t h e p i p e , w h e r e l e n g t h s a r e s c a l e d o n a a n d t i m eo n a / U o. ( I n p r a c t i c e , a s T a y l o r [1 2] o b s e r v e d , a t u r b u l e n tv e l o c i t y f i e l d w o u l d b e m o r e r e a l i s t i c a n d w o u l d i n v o l v e at u r b u l e n t o r e d d y d i f fu s i vi t y r a t h e r t h a n t h e m o l e c u l a rd i f f u si v i ty ) . T h e e q u a t i o n f o r t r a n s p o r t a n d d i f f u s i o n o ft h e c o n t a m i n a n t C(r , t ) i s

(~ C 0 C ( 6 ~ 2 C 1 0 C 0 2 C t- - + ( 1 - - r 2 ) - = 0 - - - + - - - - + ( 1 )Ot Oz Or 2 r O r O Z 2 ] '

w i t h c o n d i t i o n sO C = 0 , r = 1 ; C i s r e g u l a r . ( 2 )O r

A l s o t h e f l u x is g iv e n i n a n a p p r o p r i a t e m o v i n g r e f e r e n c ef r a m e , a n d a n i n i t i a l c o n d i t i o n i s n e e d e d a t t = 0 . I n E q .( 1) , t h e s y m b o l s t , r , a n d z r e p r e s e n t n o n d i m e n s i o n a l t i m ea n d r a d i a l a n d a x i a l d i s t a n c e s , w h i l e0 - = D / U o a , ( 3 )

D b e i n g t h e m o l e c u l a r d i ff u s i v it y .

W e n o t e t h a t a n e i g e n s o l u t i o n i sC = C = c o n s t a n t ( 4 )

O u r o b j e c t n o w i s t o c o n s i d e r a s i tu a t i o n i n w h i c h C i st h o u g h t t o v a r y s l o w l y i n s p a c e a n d t i m e a c c o r d i n g t o E q s .( 1 ) a n d ( 2 ) T o t h i s e n d , a n d n o t i n g t h e n a t u r a l d i f f u s i v ea s p e c t o f t h e p r o b l e m , w e d e f in e

7" = a 2t, (5 ) = 6 ( z - c t ) , ( 6 )

w h e r e c5 i s a s m a l l p a r a m e t e r a n d c i s a p r o p a g a t i o ns p e e d , w h i c h m u s t b e d e t e r m i n e d . N o w C w i l l b e r e q u i r e dt o d e p e n d o n 7" a n d f .E q u a t i o n ( 1 ) b e c o m e sO C O C6 2 + 6 ( 1 - r e - c ) - -Or Of

0 2C 1 OC 2 ~T02C )= 0 - , -- ' - -- ; - + - - - - + 6 , ( 7 )r O ra n d w e e x p a n d t h e s o l u t i o n a s f o l l o w s :

C = C ( r , r , f ) + 6 C l ( r , r , f ) + 6 2 C 2 ( r , r , f ) + . ..(8 )

S u b s t i t u t i n g E q . ( 8 ) i n E q . (7 ) a n d s e p a r a t i n g t e r m s o fo r d e r s 6 o , 6 , c5 , w e o b t a i n a s e q u e n c e o f p r o b l e m s .02C 1 0 CA t O ( 6 ) : 0 = + - - - , ( 9 )Or 2 r Or

o f w h i c h t h e s o l u t i o n s u b j e c t t o E q . ( 2 ) i sC = C ( r , f ) ; ( 10 )

t h u s C d o e s n o t d e p e n d o n r .O C [ 0 2 C 1 1 0 C a ]

A t 0 ( 6 ) : ( 1 - r 2 - c ) O ff = ' t 7 + -r O r ] " ( 1 1 )0 C 1 = 0 ; r = 1 . ( 1 2 )0 r

P r o b l e m E q . ( 1 1 ) s u b j e c t t o E q . ( 12 ) h a s a s o l u t i o n o n l y i fa n o r t h o g o n a l i t y c o n d i t i o n is s a t i s f i e d - - n a m e l y ,0 Cf 0 1(1 - r 2 - c ) r d r = 0 . ( 1 3 )0 f

B e c a u s e C i s i n d e p e n d e n t o f r , w e h a v e f r o m E q . ( 13 )c = 1 / 2 , ( 1 4 )

t h e p r o p a g a t i o n s p e e d b e i n g e q u a l t o t h e m e a n v e l o c i t y o ft h e g i v e n f l o w .B e c a u s e t h e f l ux i s g i v e n a n d w e a l l o w i t t o b e g i v e n b yC ' (7 - , f ) , we ha ve

o l rC1 dr = 0 ; ( 1 5 )t h e n E q s . ( 1 1 ) , (1 2 ) , a n d ( 1 5 ) i m p l y t h a t

C 1 = ( 8 0 -) 1 ( - 1 / 3 + r 2 - 1 / 2 r 4 ) 0 C ( 1 6 )oCS o f a r, w e h a v e n o t o b t a i n e d a n y c o n d i t i o n o n i f ( r , f ) , b u tt h i s w i l l a r i s e a t n e x t o r d e r

8/8/2019 Evolution of Vorticity

3/5

2 0 8 J . T . S t u a r t

A t O ( 6 2 ) , a n e q u a t i o n a n a l o g o u s t o E q . (1 1 ) a r i s e s , b u tw i t h C 2 i n p l a c e o f C a _ a n d w i t h a l e f t - h a n d s i d e d e p e n -d e n t o n r , O f f / O ~ , 0 2 C / 0 ~ 2. A n o r t h o g o n a l i t y c o n d i t i o ni s r e q u i r e d t o b e s a t i s f i e d a n a l o g o u s t o E q . ( 1 3 ) i n w h i c h Ca p p e a r s a s a p a r a m e t e r . A s a r e s u l t , a d i f f e r e n t i a l e q u a -t i o n f o r C ( z , ~" ) a p p e a r s :o e ( 1 t o g a~r + - - = 0 . ( 1 7)0~" 1 9 2 o - ] 0 ~ "2

E q u a t i o n ( 1 7 ) is a cl a s s i c a l d i f f u s i o n e q u a t i o n ; i t s e f f e c -t i v e d i f f u s i o n c o e f f i c i e n t i s c o m p o s e d o f t w o t e r m s , o n er e p r e s e n t i n g m o l e c u l a r d i f f u si o n a n d t h e o t h e r T a y l o rd i f f u s io n . I n t h e l a t t e r t e r m , w e n o t e t h a t t h e m o l e c u l a rd i f f u s iv i t y a r i s e s i n a n i n v e r t e d w a y ; t h is i s a c o n s e q u e n c eo f l o n g i t u d i n a l ( a x i a l ) c o n v e c t i o n a l l i e d s i m u l t a n e o u s l y t or a d i a l m o l e c u l a r d i f f u s i o n ( T a y l o r , R e f . 1 1 ) . T h e f o r e g o i n gf o r m o f t h e m a t h e m a t i c a l e x p l a n a t i o n i s t a k e n f r o m t h et h e s i s o f J e f f e r s o n [ 13 ]. M u c h w o r k h a s b e e n d o n e b yo t h e r s i n t h e i n t e r v e n i n g y e a r s .I f w e c h o o s e t o w r i t e E q . ( 1 7 ) i n d i m e n s i o n a l t e r m s , w eh a v eO ff ( U~2a 2 ] 0 2CD + = 0 , ( 1 8 )O T 1 9 2 D ]

w h e r eZ = a z - 1 / 2 U o T , T = a t / U o . ( 1 9 )

I n E q s . ( 1 7 ) o r ( 1 8 ) , w e s e e t h a t t h e T a y l o r d i f f u s i o n t e r mi s b i g g e r a n d m o r e i m p o r t a n t i fUoacr I = > 14 . (20 )D

I n p r a c t i c e , a s T a y l o r o b s e r v e d , a t u r b u l e n t v e l o c i t yp r o f i le w o u l d b e r e q u i r e d t o g e t h e r w i t h a n e d d y ( o r t u r b u -l e n t ) d i f f u s iv i t y ( D ) . T h i s w o u l d a l t e r t h e c o e f f i c i e n t s inE q s . ( 1 7 ) a n d ( 1 8 ) b u t n o t t h e e s s e n t i a l s o f th e s t r u c t u r e .E V O L U T I O N O F V O R T I C I T Y I N P I P EF L O W

E v o l u t i o n o f a f l u i d f ie l d in v o l v e s t h e v e l o c i t y a n d t h ev o r t i c i ty , b o t h o f w h i c h a r e v e c t o r f i e l d s , a n d t h e p r e s s u r ea n d t h e d e n s i t y , b o t h o f w h i c h a r e s c a l a r s . I f t h e f l u i d i si n c o m p r e s s i b l e a n d i f t h e p r e s s u r e i s e l i m i n a t e d f r o m t h eN a v i e r - S t o k e s e q u a t i o n s , w e h a v e0o- - + U " V w = t o " V u + R - 1 V 2 w , ( 2 1 )O t

V . u = 0 , ( 2 2 )~o = curl u . ( 2 3 )

H e r e , u i s t h e n o n d i m e n s i o n a l v e l o c i t y f i e l d ( s c a l e d o nU 0 ) t o i s t h e v o r t i c i t y f i e l d ( U o / a ) , t i s t h e t i m e ( a / U o ) ,a n dR = U o a / V ( 2 4 )

i s t h e R e y n o l d s n u m b e r , w h e r e v i s t h e k i n e m a t i c v is c o s it y( m o l e c u l a r ) a n d a i s a c h a r a c t e r i s t i c l e n g t h .I t i s c l e a r t h a t t h e v o r t i c i t y , ~ o, i s n o t t r a n s p o r t e dp a s s i v e l y f o r t w o r e a s o n s : ( 1 ) E q . ( 2 3 ) c o n n e c t s to w i t h t h ev e l o c i t y f i e l d u ; a n d ( 2 ) t h e f i r s t t e r m o n t h e r i g h t - h a n ds i d e o f E q . ( 2 1 ) - - n a m e l y , ~ o- V u - - g i v e s a n e n h a n c e m e n t

( o r d i m i n u t i o n ) o f v o r t i c i ty b y p r o c e s s e s o f v o r t e x s t r e t c h -i n g a n d t i l t i n g , b o t h o f w h i c h a r e a s s o c i a t e d w i t h c o n s e r v a -t i o n o f a n g u l a r m o m e n t u m i n a g y ro s c o p i c s e n se .T o m a k e s o m e p r o g r e s s a n d t o i l l u s tr a t e t h e p o i n t, w es h a l l n o w c o n s i d e r a r a t h e r s p e c i a l p r o b l e m f o r w h i c hl i n e a r i z a t i o n i s a c c e p t a b l e f o r a p e r t u r b a t i o n o f s m a l la m p l i t u d e .W e c o n s i d e r f l o w t h r o u g h a s t r a i g h t p i p e o f u n i f o r mc i r c u la r cr o s s s e c t io n a n d r a d i u s a - - n a m e l y , H a g e n -P o i s e u i l l e f l ow . U p o n t h i s fl o w , a s w i rl p e r t u r b a t i o n i si m p o s e d , a n d t h is i s a s s u m e d t o b e o f s m a l l a m p l i t u d e . I fd e n o t e s t h e c i r c u l a t i o n , w h i c h c a n b e a f u n c t io n o fr a d i a l d i s t a n c e ( r ) , a x i a l d i s t a n c e ( z ) , a n d t i m e , t h e n 1~s a t i s f i e s t h e l i n e a r d i f f e r e n t i a l e q u a t i o n

0~ '~ 0~ '~ ( 02 ~ 1 0~ ~02~ 1- - + ( 1 - r 2 ) = R 1 _ _ _3 t O z 3 r 2 r ~F -~- O z 2 )"

( 2 5 )B o u n d a r y c o n d i t i o n s a re t h a t

1 ~ = 0 , r = 1 , ( 2 6 )a n d 1~ is r e g u l a r . A n i n i t ia l c o n d i t i o n i s a l s o n e e d e d a tt = 0 .A s f o r t h e c a s e o f T a y l o r d i f f u s io n , w e n o w c o n s i d e rl o n g t i m e ( o f o r d e r 8 - 2 ) a n d a x i a l le n g t h ( o f o r d e r 8 - 1 )s c a l e s , w h e r e 8 i s a s m a l l p a r a m e t e r . T h u s , w e d e f i n e

7" = 8 2 t , ~ = 8 ( Z - c t ) , ( 2 7 )w h e r e c i s a p r o p a g a t i o n s p e e d .A n e i g e n s o l u t i o n f o r 1 ~ i s

= r J l ( A j r ) e - ~ t / n , ( 2 8 )w h e r e J l ( x ) i s t h e B e s s e l f u n c t i o n o f o r d e r 1 a n d A j( j = 1 , 2 , 3 . . . . . w ) r e p r e s e n t s t h e z e r o s o f t h a t f u n c t i o n :

J ~ ( x ) = 0 , j = 1 , 2 , 3 . . . . . o o. ( 2 9 )I n t e r m s o f t , ~ -, r , ~" v a r i a b l e s , E q . ( 2 5 ) b e c o m e s

R + 6 2 R + 8 R ( 1 - r 2 - c ) - -3 t 3 T 3 ~0 2 ~ 1 31 ) 321~

- - Jr- 8 2 - ( 3 0 )o~r 2 r OF oq~ 2 "B e a r i n g i n m i n d t h a t E q . ( 2 8 ) c a n b e m u l t i p l i e d b y a na r b i t r a r y c o n s t a n t S , w h i c h w e m a y t a k e t o b e a f u n c t i o no f t h e s l o w v a r i a b l e s T a n d ~ ', a n d n o t i n g o u r a n a l o g y w i t hT a y l o r d i f f u s i o n , w e e x p a n d 1~ a s f o l lo w s :

1 ) = S ( " c , ~ ) ~ ( t , r ) + 6 1 ~ l ( t , r; " r , ~ )+ 6 2 1 ) 2 ( t , r ; ~ - , ~ " ) + - - ' ( 3 1 )

W e n o w s e p a r a t e t e r m s o f in c r e a s in g o r d e r i n 6 a s in t h es e c t i o n o n T a y l o r d i f f u s i o n .A t o r d e r 6 0 , w e f i n d t h a t S d e p e n d s o n l y o n ~- a n d ~" a sa s s u m e d .A t o r d e r 6 , w e o b t a i n a n e q u a t i o n f o r 1 ) 1:3 1 ~ a _ 3 S c ) 2 ~ 1 0 1 ) 1R + R ( 1 - r 2 - c ) ~ - , ( 32 )3 t O f f Or 2 Or

1 ~ 1 = 0 , r = 1 ; 1 ) 1 r e g u l a r . ( 3 3 )

8/8/2019 Evolution of Vorticity

4/5

V o r t i c i t y i n P e r t u r b e d P i p e F l o w 2 0 9A n i n t e g r a b i l i t y c o n d i t i o n t h e n y i e l d s

c = 2 / 3 , ( 3 4 )w h i c h i s n o t t h e m e a n f lo w s p e e d o f T a y l o r d if f u si o n ;r a t h e r i t i s t h e g r o u p v e l o c i t y o f s m a l l p e r t u r b a t i o n s . As o l u t i o n f o r 1 ) 1 s u b j e c t t o E q . ( 3 3 ) t h e n f o l lo w s .A t o r d e r 6 2, w e o b t a i n a n e q u a t i o n f o r I I 2 , w h i c h i sa n a l o g o u s t o E q . ( 3 2 ) , a n d a n i n t e g r a b i l i t y c o n d i t i o n y i e l d sa n e q u a t i o n f o r S ( ~- , ~ ' ):

o~S o 2 S- - - K = 0 ; ( 3 5 )O r O ~ 2

RK = R - 1 + 1 5 A ; ( 8 - 1 / 3 t 2 ) . ( 3 6 )T h e e f f e c t i v e d i f fu s i o n c o e f f i c i e n t K , w h i c h i s a n a l o g o u st o t h e c o e f f i c i e n t i n E q . ( 1 7 ) , h e r e p l a y s a m o r e c o m p l e xr o l e t h a n i n t h e c a s e o f T a y l o r d i ff u s i o n :

1 . K d e p e n d s o n A j ( j = 1 , 2 , 3 . . . . ), o r n a m e l y , o n t h ee i g e n v a l u e o f t h e s w i r l.2 . K m a y c h a n g e s i g n , th u s a l l o w i n g t h e p o s s i b i l i ty o fn e g a t i v e d i f f u s i o n . T h i s c h a n g e o f si g n o c c u r s a t ac r i t i c a l v a l u e R c ( j ) o f t h e R e y n o l d s n u m b e r , a s T a b l e 1i n d i c a t e s . I n t h a t t a b l e , t h e v a l u e s o f A j h a v e b e e nt r u n c a t e d a t t w o d ec i m a l p l a c e s a n d t h o s e o f R e ( j ) a tt w o s i g n i f ic a n t f i g u re s . F o r a g i v e n v a l u e o f j , K i sn e g a t i v e f o r R > R e ( j ) a n d t h e e f f e c t i v e d i f f u s i o n p r o -c e s s i s n e g a t i v e .

I f w e r e v e r t t o a d i m e n s i o n a l n o t a t i o n , E q s . ( 3 5 ) a n d( 3 6 ) c a n b e p u t i n t h e e q u i v a l e n t f o r mc 9 S ( U 2a2 ) 0 2Sv + ( 8 - 1 / 3 A 2 ) = 0 , ( 3 7 )o rw h e r e

Z = a z - 2 / 3 U o T , T = a t / U o . ( 3 8 )E q u a t i o n ( 3 7 ) i s a n a l o g o u s t o E q . ( 1 8 ) f o r T a y l o r d i f f u -s i o n . A s i n t h e l a t t e r c a s e , w e n o t e t h a t t h e e f f e c t iv ed i f f u s i o n c o e f f i c i e n t is c o m p o s e d o f t w o p a r t s , t h e f i r s tb e i n g t h a t o f k i n e m a t i c v i s co s it y a n d t h e s e c o n d b e i n gd e r i v e d f r o m a x i a l c o n v e c t i o n a l l i e d t o r a d i a l d i f f u s i o n . I nt h e s e c o n d t e r m , t h e k i n e m a t i c v i s c o s it y a p p e a r s i n a ni n v e r t e d f o r m , j u s t a s i t d o e s i n T a y l o r d i f fu s i o n .

N E G A T I V E D I F F U S I O N A N D F O C U S I N GH e r e , w e g iv e so m e a t t e n t i o n t o t h e p r o b l e m p o s e d b y E q .( 3 5 ) w h e n K i s n e g a t i v e . S u p p o s e w e i m p o s e

r = O , S = a e 2 ~ 2 , ( 3 9 )[~'[ ~ o % S - -* 0 , ( 4 0 )

T a b l e 1 . C r i ti c a l R e y n o l d s N u m b e r s f o r a C h a n g eo f S i g n o f t h e E f f e c t i v e D i f f u s i o n C o e f f i c ie n t K o f E q . ( 3 5) ,d e p e n d e n t o n t h e I n t e g e r j a n d t h e C o r r e s p o n d i n g Z e r oo f t h e B e s s e l F u n c t i o nj 1 2 3 4RAi ( J )c 3 .83 66 .02 780.17 963.32

Note that K < 0 for R > R e ( j ) .

a n d l e t K = - k < 0 a n d / x 2 > 0 . T h e n a s o l u t i o n t o E q .( 3 5 ) s u b j e c t t o E q s . ( 3 9 ) a n d ( 4 0 ) i sS = a ( 1 - 4ktz2"r)- l /2e-~Z~E/(1-4k~2r) ( 4 1 )

W e c o n s i d e r t w o p o s s i b l e l im i t s:1 . S u p p o s e t h a t s i s f i xe d a n d i s n o n z e r o , a n d l e t r( 4 k / x 2 ) - 1 f r o m b e l o w . T h e n S ~ 0 .2 . S u p p o s e t h a t ~ ' 2 / ( 1 - 4 k / ~ 2 ~ - ) is h e l d f i x ed a n d l e t~- - ~ (4 k / x 2 ) - 1 f r o m b e l o w . T h e n S ~ ~ .

T h e i m p l i c a t i o n s o f t h e s e t w o r e s u l ts a r e i m p o r t a n t a n df a r r e a c h in g . T h e y i m p l y a c o n c e n t r a t i o n a n d f o c u s in gw i t h a n a s s o c i a t e d a m p l i f i c a t i o n . T h i s i s a n a r t e f a c t o fw h a t is m a t h e m a t i c a l l y a n i l l - p o s e d p r o b l e m . I t d o e s p r o -v i d e o n e m e c h a n i s m , h o w e v e r , b y w h i c h a s l u g o f v o r t i c i ty( s i m p l i f i e d a s t h e s w i r l i n t h e p r e s e n t c a s e ) c a n b e m a d em o r e c o h e r e n t a n d a n y in t e r f ac e m a i n t a i n e d o r s h a r p e n e de v e n a g a i n s t t h e n a t u r a l d i f fu s i v e p ro c e s s e s .A p p l i c a t i o n o f t h i s ty p e o f t h e o r y t o t u r b u l e n t f l o ww o u l d r e q u i r e c h a n g e s i n t h e m e a n v e l o c i t y f i el d a n d i nt h e d i f fu s i v it y , w h i c h w o u l d n e e d t o t a k e o n a n " e d d y "f o r m , a n a l o g o u s t o t h a t o f T a y l o r [ 1 2 ] f o r T a y l o r d i f f u s i o n .T h e e f f e c t o f t h i s o n t h e c r i t ic a l R e y n o l d s n u m b e r R c ( j )f o r a r e v e r s a l s i g n o f t h e e f f e c ti v e d i f f u s io n p a r a m e t e r Kw o u l d b e o f s o m e i n t e r es t , a s w a s p o i n t e d o u t t o t h e w r i t e rb y P r o f e s s o r G . M . L i l le y .

P R A C T I C A L S I G N I F I C A N C ET h e r e r e m a i n s t h e c o m p l e x m a t t e r o f th e r e l e v a n c e o ft h e s e i d e a s t o a c t u a l t u r b u l e n t s l u g s o r p u f fs . W h y a r et h e y c o h e r e n t a n d c o n f i n e d ? W h a t a r e t h e s t r u c t u r e s? W en e e d t o k n o w m o r e o f s u ch p h e n o m e n a a n d t o b e tt e ru n d e r s t a n d t h e p o s s ib l e m e c h a n i c s b y w h i c h t h e s p a t i a lc o h e r e n c e a n d c o n f i n e m e n t i s r e a ll y a ch i e v e d . M u c h m o r ew o r k i s n e e d e d o f b o t h a n e x p e r i m e n t a l a n d a t h e o r e t i c aln a t u r e ; b u t i t i s h o p e d t h a t t h e r e l a ti v e l y s im p l e a n a l y s i so f th i s p a p e r p r o v i d e s a c o n t r i b u t i o n t o t hi s e n d e a v o r .

C O N C L U S I O N ST h e a p p l i ca t i o n o f t h e c o n c e p t o f T a y l o r d i f fu s i o n h a s l e dt o a n i n t e r e s ti n g r e s u l t w h e n a p p l i e d t o a d y n a m i c a lv a r i a b l e s u c h a s t h e c i r c u l a t i o n . T h e e f f e c t i v e d i f f u s i o np a r a m e t e r c a n b e c o m e n e g a t iv e i f t h e R e y n o l d s n u m b e r i sg r e a t e r t h a n a c r i t i c a l v a l u e . T h i s m e a n s t h a t a " f o c u s i n g "a n d c o n c e n t r a t i o n a r e d e v e l o p e d i n c o n t r a s t w i t h t h en o r m a l d i f f us i o n p r o c e ss o f s m o o t h i n g a n d s p r e a d i n g . W en o t e , h o w e v e r , t h a t i n t h e p r e s e n t c a s e t h e c i r c u l a t i o n i sn a t u r a l l y d e c a y i n g i n t im e , s o t h e n e g a t i v e d i f f u s i o n a c t ss o a s t o c a u s e a f o c u s i n g a n d c o n c e n t r a t i o n w i t h i n t h ed e c a y i n g p a t c h o f v o r t ic i t y o r c i r c u la t i o n .I t w o u l d b e d e s i r a b l e t o i m p r o v e a n d d e v e l o p t h e s t u d -i e s m a d e i n t h e p r e s e n t p a p e r s o as t o i n c l u d e n o n l i n e a re f f e c t s . O n e p o s s i b i l i t y t h a t i n t r i g u e s t h e p r e s e n t w r i t e rc o n c e r n s t h e t r e a t m e n t o f t h e n e u t r a l n o n a x i s y m m e t r i cf lo w p e r t u r b a t i o n o f H a g e n - P o i s e u i l l e f lo w (S m i t h a n dB o d o n y i , R e f . 1 4 ). W o u l d t h is b e a m e n a b l e t o th e p r e s e n tt y p e o f a n a l y s is ? T h i s r e m a i n s t o b e a n s w e r e d .N a t u r a l l y , a n a t t e m p t t o d i s c u ss a n y t u r b u l e n t s i t u a t i o nw o u l d a l s o r e q u i r e n o n l i n e a ri t y t o b e i n t r o d u c e d . A g a i n ,t h i s i s a q u e s t i o n t h a t m a y b e a n s w e r e d i n t h e f u t u r e .

8/8/2019 Evolution of Vorticity

5/5

2 1 0 J . T . S tu a r tI n g e n e r a l , h o w e v e r , w e n o t e t h a t d i r e c t n u m e r i c a ls i m u l a t i o n m a y p r o v e t o b e a p o w e r f u l a n d r e l e v a n t t o o li n t r e a t i n g a n y n o n l i n e a r a s p e c t o f th i s g e n e r a l p r o b l e m ,w h e t h e r t u r b u l e n t o r o t h e r w i s e . T h i s c o u l d a r i s e a n d b ev a l u a b l e i n t h e c a l c u l a t io n o f a n e u t r a l b u t n o n l i n e a re i g e n s o l u t i o n , a n e x a m p l e b e i n g t h e S m i t h - B o d o n y i s o lu -t i o n , m e n t i o n e d a b o v e . D i r e c t n u m e r i c a l s i m u l a t i o n c o u l df u r t h e r m o r e b e o f i m m e n s e v a lu e i n t h e t r e a t m e n t o fp e r t u r b a t i o n s f r o m t h e n o n l i n e a r e i g e n s o l u t i o n . I t w o u l d

b e v e r y i n t e r e s ti n g t o p u r s u e t h i s r o u t e o f d i re c t n u m e r i c a ls i m u l a t i o n .This paper is dedicated with pleasure to Peter Bradshaw, whose workon turbulence has been so influential and whose 60th birthday tookplace on December 26, 1995.

aCCC~ a n d C 2cDJ l ( x )JKkRR c ( j )SrTtc l ou

xZz6A jtxu

N O M E N C L A T U R Er a d iu s o f p ip e , mc o n c e n t r a t i o n o f c o n t a m i n a n t , d i m e n s i o n l e s sm e a n o f C , 2 f l r C d r , d i m e n s i o n l e s sp e r t u r b a t i o n s o f C , d i m e n s i o n l e s sp r o p a g a t i o n s p e e d , d i m e n s i o n l e s sd i ff u s iv i t y o f C , m 2 / sB e s s e l f u n c t i o n o f a v a r i a b l e x ; o f o r d e r 1 ;d i m e n s i o n l e s si n t e g e r l , 2 , 3 , . . . , ~ , d i m e n s i o n l e s se f f e c t iv e d i f f u s iv i ty o f c i r c u la t io n , d ime n s io n -less- K , d i m e n s i o n l e s sR e y n o l d s n u m b e r = U o a / v , d i m e n s i o n l e s sc r i t i c a l R e y n o l d s n u m b e r , d e p e n d e n t o n j ,d i m e n s i o n l e s sa m p l i t u d e o f c i r c ul a t i on , d i m e n s i o n l e s sr a d i a l d i s t a n c e , d i m e n s i o n l e s st i m e , sT U o / a , t i m e , d i m e n s i o n l e s sm a x i m u m f lo w v e lo c it y , m / sv e l o c i ty v e c t o r , d i m e n s i o n l e s sf u n c t i o n v a r i a b l e i n B e s s e l f u n c t i o n , d i m e n -s io n le s sa z - c U o T , p r o p a g a t i o n v a r i a b l e , ma x ia l d i s t a n c e , d ime n s io n le s ss m a l l p a r a m e t e r , d i m e n s i o n l e s sj t h z e r o o f J l ( x ) : J 1 (A , ) = 0 , d im e n s io n le s sp a r a m e t e r , d i m e n s i o n le s sk i n e m a t i c v i s c o s i t y , m 2 / s8 ( z - c t ) , p r o p a g a t i o n v a r i a b l e , d i m e n s i o n -lessD / U o a , d i f f u s iv i ty o f C , d ime n s io n le s s

T12~ 1 , ~ 209V

6 2 r, s lo w t ime v a r i a b le , d im e n s io n le s sc i r c u la t io n , d ime n s io n le s se ig e n s o lu t io n f o r f ~ , d ime n s io n le s sp e r t u r b a t i o n s o f 1 2 , d i m e n s i o n l e s sv o r t ic i t y v e c t o r , d i m e n s i o n l e s sg r a d i e n t o p e r a t o r , d i m e n s i o n l e s sREFERENCES

1. Reynolds, O., An Experimental Investigation of the Circum-stances Which Determine Whether the Motion of the WaterShall Be Direct or Sinuous, and of the Law of Resistance inParallel Channels, Phi l . T rans . R oy . Soc . , Lond on A 174, 935-982,1883.2. Taylor, G. I., in M o de m De ve l opme nt s i n F l u i d Dy nami c s , S .Goldstein, Ed., pp. 324-325, Cl arendon Press, Oxford, 1938, andDover, New York, 1982.3. Wygnanski, I. J., and Champagne, F. H, On Transition in a Pipe:Part I, The Origin of Puffs and Slugs and the Flow in a TurbulentSlug, J . F l u i d M e c h . , 59, 281-335, 1973.4. Coles, D., Prospects for Useful Research on Cohere nt Structuresin Turbulent Shear Flow, Proc . India n Aca d. S c i . (Eng . Sc i .) , 4,111-127, 1981.5. Bandyopadhyay, P. R., Aspects of the Equilibrium Puff in Transi-tional Pipe Flow, J. F l u i d M e c h . , 163, 439 458 , 1986.6. Em mons, H. W., The Laminar- Turbulent Transit ion in a Bound-ary Layer, Part I , J . Aero. Sc i . , 18, 490-498, 1951.7. Schubauer, G. B., and Klebanoff, P. S., Contributions on theMechanics of Boundary-Layer Transition, N.A.C.A. Report 1289,Washington, 1955. (See also Rosenhe ad, L., Ed., L a m i n a r B o u n d -ary Layers, pp. 577-578, C larendon Press, Oxford, 1963).8. Perry, A. E., Lim, T. T., and Teh, E. W., A Visual Study ofTurbulent Spots, J . F l u i d M e c h ., 104, 387-405, 1981.9. Head, M. R., and Bandyopadhyay, P. R., New Aspects of Turbu-lent Boundary-Layer Structure, J . F luid Mech. , 107, 297-338,1981.10. Cantwell, B., Coles, D., and Dimotakis, P., Structure and En-trainment in the Plane of Symmetry of a Turbulent Spot, Report,Calif. Inst. Tech. 1977.11. Taylor, G. I., Dispersion of Soluble Matter in Solvent FlowingSlowly through a Tube, Proc . R oy . Soc . , LondonA 119, 186-203,1953.12. Taylor, G. I., The Di spersion of Matte r in Turbulent Flowthrough a Pipe, Proc. R oy . Soc . , L ond on A 123, 446-468, 1954.13. Jefferson, R. J., Some Aspects of Flow and the Diffusion ofContaminant in Pipes, Ph.d. Thesis, Mathematics Dept., ImperialCollege, London, 1977.14. Smith, F. T., and Bodonyi, R. J., Ampli tude- depen dent NeutralModes in the Hagen-Poiseuille Flow through a Circular Pipe,Proc . Roy . Soc . , London A 384, 463-489, 1982.

Received March 13, 1996; revised June 12, 1996