Andritos & Hanratty

8/10/2019 Andritos & Hanratty

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Int. J. Multiphase FlowVol. 13, No . 5. pp. 583--603, 1987 0301-9322/87 S3 .00 +0 .00Printed in Gre at Bri tain. All r ights reserved Cop yright "~ 1987 Pergam on Journals/Elsevier

I N T E R F A C I A L I N S T A B I L I T I E S F O R H O R I Z O N T A L

G A S - L I Q U I D F L O W S I N P I P E L I N E S

N . ANDRITSOS an d T. J. HANRATTYU n i v e r s i t y o f Il l in o i s , U r b a n a , I L 6 1 8 0 1 , U .S . A .

(Rece ived 6 Au gus t1985; in ret 'ised form 15 October 1986)

A b s t r a c t - - E x p e r i m e n t s w e r e c o n d u c t e d w i t h a ir a n d l i q u id f lo w i n g in h o r i z o n t a l p i p e l in e s , 2 .5 2 a n d9 . 5 3 c m d i a , t o d e t e r m i n e t h e i n t e r f a c i a l i n s t a b i l i t i e s t h a t e x i s t i n a s t r a t i fi e d f l o w. T h e l i q u i d v i s c o s i t y w a sv a r i e d f r o m l t o 8 0 c P. T h r e e t y p e s o f in s t a b i l i t i e s a r e d e f i n e d : r e g u l a r 2 - D w a v e s a r e a s s o c i a t e d w i t hp r e s s u r e v a r i a t i o n s i n p h a s e w i t h t h e w a v e s lo p e , i r r e g u l a r l a r g e - a m p l i t u d e w a v e s a n d a t o m i z a t i o n o f th el i q u id a r e a s s o c i a t e d w i t h p r e s s u r e v a r i a t i o n s i n p h a s e w i t h t h e w a v e h e i g h t . L i n e a r s t a b i li t y t h e o r y i s u s e dt o p r o v i d e a p h y s i c a l i n t e r p r e t a t i o n a n d t o p r e d i c t c o n d i t i o n s f o r t h e i n i ti a t i o n o f t h e s e i n s ta b i l it i e s.

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

A t l o w f l ows o f ga s and l iqu id i n a ho r i zon t a l p ipe l i ne a s tr a t if i ed r eg im e ex is t s w he re by t he l i qm o v e s a l o n g t h e b o t t o m o f th e p i p e a n d t h e g a s , c o n c u r r e n t l y w i t h it. A l t h o u g h t h is p a t t e r n cb e c o n s i d e r e d t h e s i m p l e s t o n e f o r g a s - l i q u i d f lo w s , la rg e d i ff e re n c e s a r e o b s e r v e d b e t w em e a s u r e m e n t s a n d p r e s e n t ly a v a il a b l e c o r r e l a ti o n s f o r p r e s su r e d r o p a n d h o l d - u p . T h e m a i n r e a sis t h a t w a v e s c a n c a u s e t h e d r a g o f t h e g a s o n t h e l iq u i d t o b e g r e a t e r t h a n i f t h e i n t er f a c e w es m o o t h . T h i s c a u s e s la rg e r p r e s s u r e d r o p s a n d s m a l l er h o l d - u p s t h a n a r e p r e d i c t e d i f t h e B l a se q ua t i on i s u sed t o c a l cu l a t e t he i n t e r f ac i a l s t r e s s .

T h i s p r o b l e m is r ec o g n i z e d in f l o w m a p s ( B a k e r 1 9 5 4 ; M a n d h a n ee t a l . 1974 ; Ta i te l & Du k le r1 9 7 6a ) b y t h e d e f i n it io n o f s m o o t h - s t ra t i fi e d a n d w a v y - s t ra t if i e d p a tt e r n s . H o w e v e r, t h e se m ah a v e s h o r t c o m i n g s s in c e th e y d o n o t a c c u r a t e l y c h a ra c t e ri z e t h e t y p e o f w a v e s a n d s in c e ths u g g e s t e d t r a n s it io n s d o n o t r e fl ec t o b s e r v e d e f f e ct s o f c h a n g e s o f l i q u id v i s c o s it y a n d o f p id i am e t e r.

M o s t o f t h e s tu d i e s o f w a v e g e n e r a t i o n i n c o n c u r r e n t g a s - l i q u i d f lo w s h a v e b e e n c o n d u c t e d fr e c t a n g u l a r c h a n n e l s w i th a la rg e a s p e c t r a t i o ( H a n r a t t y & E n g e n 1 9 5 7 ; H a n r a t t y & H e r s h m a1961 ; Ha n r a t t y 1983 ) . Fo r a i r -w a t e r f l ows no w aves a r e obs e rved a t su f f i c ien t ly l ow gas ve loc it iT h e f i rs t w a v e s o b s e r v e d w i t h i n c r e as i n g g a s v e l o c i ty a r e 2 - D c a p i l la r y - g r a v i t y w a v e s . T h e s e a p pt o b e u n s t a b l e t o 3 - D d i s t u r b a n c e s , s o t h a t t h e y e x i st o v e r a n a r r o w r a n g e o f g a s v e l o c it ie s . T3 - D w a v e s h a v e a p e b b l e d a p p e a r a n c e a n d c h a r a c t er i s ti c w a v e l e n g t h s w h i c h d e c r e a s e w i

i nc r ea s ing ga s ve loc i t y. A t su ff ic i en tl y h igh ga s ve loc i ti e s , l ong -w ave l eng th l a rge - am p l i t ude w avw i t h a h i g h ly r o u g h e n e d i n te r f ac e a p p e a r. T h e s e " r o l l w a v e s " c a r r y l arg e m a s s e s o f f lu i d a n d ma c t u a l l y b e l ooke d u po n a s f low su rges in t he l iqu id l aye r. A t ve ry h igh ga s ve loc i t ie s a t om iza t io c c u r s t h r o u g h t h e r e m o v a l , b y t h e g a s , o f w a v e l e t s r id i n g o n t h e t o p o f t h e r o ll w a v e s .

On e o f t he m a in t heo re t i c a l p rob l em s i n in t e rp r e t i ng t he se r e su lt s is t he p r ed i c t i on o f t he spa tv a r i a t io n o f t h e g a s - p h a s e p r e s s u r e a n d s h e a r s t re s s in d u c e d a t t h e i n t e rf a c e b y t h e p r e s e n c e o f w a v e s . Th e p r i n c ip a l t h e o r e ti c a l t o o l h a s b e e n l in e a r s t a b i li t y th e o r y. A s u m m a r y o f p r o g r e s s t h i s d i r e c t i o n i s g iven in a r e cen t r ev iew a r ti c le by Ha n r a t t y ( 1983 ). The 2 -D and peb b l ed w ava r e i n t e rp r e t e d a s re s u l ti n g f r o m a n i m b a l a n c e b e t w e e n t h e e n e rg y f e d to t h e w a v e s b y w a v e - i n d u cp r e s s u r e v a r i a t io n s i n p h a s e w i th t h e w a v e s l o p e a n d v i s c o u s d i s s ip a t io n . R o l l w a v e s o c c u r w ht h e de s t ab i l iz i ng e f f ec ts o f l i qu id i ne r ti a and l ong -w ave l eng th p r e s su re va r i a t i on s i n phase w i th

w ave h e i g h t ov e rco m e the s t ab i li z ing e f f ec t o f g r av i t y. A tom iza t i on i s sugges t ed t o b e the r e suo f a n i m b a l a n c e b e t w e e n t h e d e s ta b i li z in g e f fe c t o f p r e s s u r e v a r i a ti o n s o v e r s m a l l- w a v e l e n g th w aa nd t he s t ab i li z ing e f f ec t o f su r f ace t en s ion .

T h e o b j e c ti v e o f t h is p a p e r is t o c h a r a c t e ri z e t h e d i f fe r e n t ty p e s o f w a v e p a t t e r n s o b s e r v e d fga s - l i q u id f l o w in a ho r i zo n t a l c i r cu l a r p ipe . The r e su l t s a r e i n t e rp r e t ed u s ing t heo re t i c aa p p r o a c h e s a l r e a d y a p p l ie d t o c h a n n e l f lo w s . P a r t i c u l a r e m p h a s i s i s p l a c e d o n t h e p r e d i c ti o n t h e e f fe c t o f c h a n g e s o f l i q u id v i s c o s it y a n d o f p i p e d i a m e t e r.

M.r t35--,~ 58 3


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584 N , ANDRITSOS andT . J. HANRA'I-I'Y

2 . T H E O R Y

( a ) D ef in i ti ons o f t he p rob lem

T h e o b s e r v e d t r a n s i ti o n s a r e i n t e r p r e t e d b y e x a m i n i n g t h e c o n d i t i o n s u n d e r w h i c h s m a la m p l i t u d e s i n u s o i d a l w a v e s w i ll g r o w o r d e c a y. I t i s r e c o g n i z e d t h a t t h is a p p r o a c h h a s l im i t a ti os i n ce n o t a l l o f th e o b s e r v e d i n t e r f a c ia l s t r u c t u r e s o r i g i n a t e f r o m t h e g r o w t h o f s m a l l d i s t u r b a n c

A m o r e c o m p l e t e d e s c r i p t i o n w o u l d r e q u i r e t h e c o n s i d e r a t i o n o f n o n - l i n e a r e f f ec t s.I n o r d e r t o s i m p l i f y t h e a n a l y s is , t h e 2 - D s y s t e m s h o w n i n fi g u re 1 is c o n s i d e r e d . T h e f lo w

a s s u m e d t o b e s te a d y. T h e a v e r a g e v e l o c i t ie s i n t h e l i q u id a n d g a s a re d e s i g n a t e d b y U L a n d Ua n d t h e a v e r a g e h e i g h t o f t h e l iq u i d , b y h -

A d i s t u r b a n c e h ' = h - / ~ o f th e f o l l o w i n g f o r m is i n t r o d u c e d a t t h e i n te r fa c e :

h" = J~ exp [ /k (x - C t) ] , [1]

w i t h x b e i n g t h e d i s t a n c e i n t h e d i r e c t i o n o f f lo w a n d t , t h e t i m e . T h e a m p l i t u d e , a , a n d tw a v e n u m b e r , k = 2 n / 2 , a r e c o n s i d e r e d t o b e r e al . T h e w a v e v e l o c it y i s c o m p l e x ,

C = C R + iC I [2]

T h e w a v e w i l l g r o w o r d e c a y i n t im e d e p e n d i n g o n t h e s i g n o f._ C ,. T h e c o n d i t i o n C t = 0 d e f inn e u t r a l s t a b i li t y. T h e c a l c u l a t io n s g i v e t h e c o n d i t i o n s( U L , U a , h )f o r w h i c h C ~ i s p o s i t i v e a n d f o rw h i c h t h e d i s t u r b a n c e i s g r o w i n g t h e f a s t e s t (k C , is a m a x i m u m ) .

(b ) Influence o f wav e- ind uced var ia t ions o f in ter fac ia l s t resses

T h e p r e s e n c e o f t h e w a v e s i n d u c e s f l o w d i s t u r b a n c e s i n th e g a s . T h e s e g iv e ri s e t o p r e s s u r e as h e a r s t re s s v a r i a ti o n s a t t h e in t e r f a c e, w h i c h c a n f e e d o r e x t r a c t e n e rg y f r o m t h e d i s t u r b a n c e . Tw a v e s a r e c o n s i d e r e d t o b e o f s m a l l e n o u g h a m p l i t u d e t h a t a l in e a r r e s p o n s e is o b t a i n e d , w h e r e

r ( P~ exp [ ik (x C t ) ] . [3]

T h e a m p l i t u d e s , ~+ a n d / ~ i a r e c o m p l e x a n d l i n e a rl y d e p e n d e n t o n / z ,

"~i ~--""~iR + iTil [4]

a n d

P + =P i . + i / ~ , . I S ]

T h e r e a l p a r t s o f [4] a n d [5 ] c a n b e w r i t t e n a s

r( = fir COSk ( x - C t )- Q sink ( x - C t ) , [6]

a n d

e~ = ~l~iRO S x - - C t ) - - JDil sink ( x C t ) . [7]

T h u s -eia a n d / s i r a r e t h e a m p l i tu d e s o f c o m p o n e n t s i n p h a s e w i t h t h e w a v e h e ig h t a n d - ' ~ a n/ ~ i ta r e th e a m p l i tu d e s o f c o m p o n e n t s i n p h a s e w i t h t h e w a v e s lo p e .

T h e v e l o c i t y f i e l d i n t h e l i q u i d i s g i v e n a s

u ~ = ~+ u ; [ 8 ]

I I I I I I I I I I

G a s

HG " UG

U L .

I i t 7 1 I I I I I I I I I I I 1 1 1 1 t lIY t X

Figure I. System to wh ich stability analysis is applied.


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I N T E R F A C I A L I N STA BI L IT I ES N H O R I Z O N T A L G A S L I Q U I D F L O W 585

a n dJ

Uy = uy . [9]

T h e g a s - p h a s e p r e s s u re f ie ld f e e d s e n e rg y t o t h e l i q u i d b y c o u p l i n g w i t h v e l o c it y c o m p o n e n t A nega t ive P~R i s des t ab i l i z ing s ince i t g ives a suc t ion a t t he c r e s t w he re u~ is pos i t ive a ndc o m p r e s s i o n a t t h e t r o u g h w h e r e u ~ is n e g a t iv e . T h i s c a u s e s, w h a t i s c a l le d , a K e l v i n - H e l m h oins t ab i li t y. S imi l a r ly, t he des t ab i l i z ing e ff ec t o f a pos i t i ve P~I i s a s soc i a t ed w i th t h e sh e l t e rm e c h a n i s m f o r w a v e g e n e r a t i o n , d e s c r i b e d b y J e f fr e y s (1 9 2 5) .

T h e g a s - p h a s e s h e a r s tr e ss f ie ld f e e d s e n e rg y i n t o t h e l i q u id t h r o u g h v e l o c i ty c o m p o n e n t u~, ai s de s tab i l i z ing fo r po s i t ive ZiR o r nega t ive fi l. Bec ause o f t he sm a l l ga s v i s cos i ty r, i s u sua l ly ms m a l l e r t h a n P ; . O n e c a n , t h e r e f o r e , e x p e c t z[ to p l a y a n i m p o r t a n t r o l e i n w a v e g e n e r a t i o n ofo r t h in f i lms fo r wh ichu'~>>u'y.

(c) Stabi l i ty analys is

T h e s p a ti al a n d t e m p o r a l v a r i a ti o n o f u~ a n d u ~ a r e d e f i n e d b y t h e l i n e a r m o m e n t u m b a l a ne q u a t i o n s f o r t h e l i q u i d s i n c e u ~ a n d u ~ a r e c o n s i d e r e d s m a l l c o m p a r e d t o f t. T h e s o l u t i o n o f the q u a t i o n s is c o m p l i c a t e d b e c a u s e o f t h e v a r i a t i o n o f f f w i t h y. C o n s e q u e n t l y, i t is d i ff ic u l t t o o b t

r e su l t s va l id fo r a w ide r ange o f Reyn o lds num ber s a nd o f (z7 - CR) . Th i s d i f f icu l ty w i ll be avo ib y a s s u m i n g a p l u g f l o w in t h e l iq u i d , w i t h a v e l o c i ty o f U L . T h e m a i n l i m i t a t io n o f t h e s o l u t ii s t h a t t h e y d o n o t c o r r e c t l y d e s c r i b e si t u a ti o n s w h e r e t h e w a v e l e n g t h , 2 , is v e r y la rg e c o m p at o t h e h e i g h t o f t h e l i q u id l a y e r,/ 7 . T h e s e c a se s a r e b e s t c o n s i d e r e d b y u s i n g i n t e g r a l f o r m s o f m o m e n t u m e q u a t i o n s ( Ha n ra t_ ty 1 9 8 3) , C o n s e q u e n t l y th e s o l u t i o n s d e v e l o p e d b y L i n & H a n r a(1986) wi l l be use d w he n 2 >>h an d 2 >>/7.

The eq ua t ion s de sc r ib ing u~ and u~ a re , t he re fo re , g iven a s

gi + u~ o ; = - pS Ox + VL O~2 +-Gry~ , t lo]

a n d

t 2 P

_ _ _ f O , u ; 0 u qOuy Ouy 1 Op'+VL + - - g [11]0 -7 + UL Ox = PL Oy \ Ox 2 Oy 2 )

0u~ 0u~~x + Tr y = o. [121

T h e s e a r e s o l v e d u n d e r t h e c o n d i t i o n s t h a t u ~ a n duy van i sh a t y = 0 an d th a t t he s tr e sse s i n t hel iqu id a t t he i n t e r f ace , a ,~ and a , , equ a l t he s t re s se s i n t he gas , ~ i an d P~:

a n d

w i t h

a ; x = ~ ; , [ 1 3 ]

t Ia , = P~ + P i , [14]

~r =ULkOy + OX ]' [ 1 5 ]

a n d

= - p [ 1 6 ]a y -4- 2/aL oy

02h 'P~ = ~ 0x ~ . [17]

T h e s e a r e s o lv e d b y t h e m e t h o d s o u t l i n e d b y L a m b (1 9 45 ), L e v i c h (1 9 62 ) a n d Ta y l o r (1 9 6 3) . Tsolu t ions to [10] - [12] a re g iven as

0 0 u,, = Ox Oy ' [18]


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586 s. ANDRITSOS and T. 1. HANRATTY

a n d

p r o v i d e d

a n d

. , - e - 7

pP L

at U - g y - f i ) ,

a : ~ . a - ' ~c x + - ~ y 0

~ g ~ F ~ ~ 0 2 0 7

The so lu t ions o f [21] and [22] a re t aken as

~b = [A sinh(ky ) + Bc o s h ( k y ) ] e x p [ i k ( x - C t ) ]a n d

w i t h

0 = [D cosh ( /y) + Es i n h ( l y ) ] e x p [ i k ( x - C t ) ] ,

12= k: ik(C - UL)Vk

The requ i rement o f ze ro ve loc i ty a t y = 0 g ives

lA = i D , B = i ~ E .

Bo und ary co nd i t ions [13] an d [14] and the k inem at i c con d i t ion a t the in te r face ,

~ h ' a h '~-7 + u~ ~ 7 = u'~@y = L

d e f in e c o n s t a n t s D a n d E a n d p r o v i d e t h e f o l lo w i n g r e la t io n b e t w e e n k a n d C :

~ -(l-' + k : + f ) I ~ c o t h ( k h ) c o t h ( / h ) - 11

( l 2 - k a ) I ~c o t h ( l h ) - c o t h ( k h ) l

l- 21 k + ?~

kh (C - UL)2 s in h (k h ) si n h (I/~--

- lT F 2k 2-qc ) .-} - 12 dr-k2 - .cc )v I _ S k _ _ _ _ _ = _ _ _ _ _ _ _ _ - - A

+ i 2 ~ (k f2 ): (C - U , ) ~ _ _s i n h ( k h ) s i n h ( / h ) - - -- - -=

h [ (12 - k- ' ) I ~ coth(l /Tt - co th (kh )J]

= , 12 , l ~ , ,- I ( l + k'-q)+ 7.2(2k'-f)k I K F ( 2 k - - f ) + ( l - + k - - / ) l c o t h ( k / T t ) c o t h (

[191

[201

[211

[22]

[231

[24]

[25]

[26]

[271

[281


5/21

I N T E R FA C I A L I N S TA B I L I T IE S I N H O R I Z O N TA L G A S - L I Q U I D F L O W 587

w h e r e

= [291~ZL(C -- UL)"

B y s e p a r a t e l y e q u a t i n g t h e r e a l a n d i m a g i n a r y p a r t s o f [2 8 ] t o z e r o , t w o e q u a t i o n s d e f in i n g CR C~ a re ob t a ine d .

C o h e n & H a n r a t t y ( 1 9 65 ) a n d C r a i k ( 1 9 68 ) d e r i v e d a n e q u a t i o n s i m i l a r t o [2 8 ]. T h e y d i d nas sum e i f (y ) i s de sc r ibed b y a p lug f l ow and p re se n t a so lu t ion to [10 ]- [12 ] w h ich i s va l id fo r l al iq u i d R e y n o l d s n u m b e r s . E q u a t i o n [4 .8 ] f r o m C r a i k ' s p a p e r is i d e n ti c a l to [2 8 ] a t l a rg e R e y n on u m b e r, i f t h e s u r f a c e c o n t a m i n a t i o n t e r m s i n C r a i k ' s r e s u l t a r e i g n o r e d .

(d) Estimation of the surface stresses

The p r inc ipa l p ro b lem in ca l cu l a t i ng CR an d C t f rom [28 ] i s t he spec i f ica t i on o f /~ iR , /~a , fi e af ~ . T h e s e a r e o b t a i n e d b y s o l v i n g m o m e n t u m b a l a n c e e q u a t i o n s f o r t h e g a s , w h i c h a r e l i n e a r ia ro un d a n ave rage ve loc i ty p ro f i le . I f t h i s p ro f il e is a s sum ed to be a p lug f l ow and i f v i s cous e f

a r e neg lec t ed ,

a n d

In add i t i on , i f k /7 - -. 0 ,

/Si___~R= k /7 P~ (U a - C ) 2 [30]/ ~ t a n h ( k / 7 ) / 7

/~il = fi r = ~il = O. [31]

P i R P G - t U/~ = ~ o - C ) 2 - [ 3 2 ]

T h e i n c l u si o n o f v i s c o u s e ff e c ts a n d t h e l i n e a r i z a t io n a r o u n d a m e a s u r e d v e l o c i ty p r o fi le f o r fove r a f l a t p l a t e g ives m or e r ea l i s ti c r e su l ts , w i th n on -ze ro va lues o f /S i l, f i r an d c i r. Th i s pa pe r t h i s t ype so lu t ion to eva lua t e t he su r f ace s t r e s s t e rms in [28 ] .

A n a d d i t io n a l p r o b l e m a r is e s in f o r m u l a t i n g th e l in e a r m o m e n t u m e q u a t i o n s b e c a u s e th e g a s fis t u r b u l e n t . T h e p r e s e n c e o f w a v e s a t t h e i n t e r f a c e i n d u c e s v a r i a t i o n s o f t h e R e y n o l d s st re ssw h i c h n e e d t o b e s p e ci fie d . A c o m p l e t e l y s a t i s fa c t o r y m e t h o d f o r d o i n g t h is i s n o t y e t a v a i la[S e e H a n r a t t y (1 9 8 3) f o r a s u m m a r y o f p ro g r e s s. ]

The so lu t ion fo r/S IR , / 5 il , ~iR an d fix u se s a m od e l fo r t he w ave - indu ced Re yno lds s t re s se s w hd o e s a g o o d jo b i n d e s c r i b in g m e a s u r e m e n t s o f t h e s h e a r s t re s s v a r i a t io n a l o n g s o li d w a v y w aT h i s is t h e M o d e l D * d e s c r i b e d b y T h o r s n e s set al. (1978) and by Z i lke ret al. (1 9 7 7 ). T h e m e t h o do f c a lc u l a t i o n m a y b e f o u n d i n t h e a b o v e p a p e r s a n d , i n m u c h g r e a t e r d e ta i l, i n a r e c e n t t he si sA bra m s (1984) . I t is l im i t ed t o ca se s fo r w h ich 2 / t7 < 2 re .

L i n & H a n r a t t y (1 9 86 ), i n t h e i r r e c e n t s tu d y o f t h e i n i ti a t io n o f s lu g s, u s e d i n t e g r a l m e t h o d s bfo r t he gas an d the l i qu id , t o ana lyze t h e s t ab i l i ty o f t he l i qu id fo r t he ca se o f ). >> i t . P re s sam pl i t ude , P iR , w as ca l cu l a t ed us ing an eq ua t ion v e ry c lo se t o [32 ]. Th i s so lu t ion wi ll be u sed hwi th /S iR g iven by [30 ] i n s t ead o f [32 ], f o r l a rge 2 / /7 .

(e) E xam ples of s tabi li ty calculat ions

T h e s t a b i l i t y a n a l y s i s f o r a c h a n n e l w a s a p p l i e d t o a p i p e f l o w b y a s s u m i n g t h e l i q u i d l a yt h i c kn e s s u s e d i n t h e c h a n n e l f lo w c a l c u l a t i o n s is e q u a l t o a l iq u i d h e i g h t d e f i n e d a s t h e a ro c c u p i e d b y t h e l i q u i d d i v i d e d b y t h e w i d t h o f t h e i n t e r f a c e ( C h o w 1 9 5 9 ) .

F i g u r e s 2 a n d 3 il lu s t r a te t h e e f fe c t o f c h a n g i n g t h e r a t i o o f t h e h e i g h t o f th e l i q u id a l o n g p ipe ce n t e r l i ne t o t h e p ipe d i a m e te r f rom 0 .4 t o 0 .2 . F igu re 4 i l lu s t r a t e s t he e ffec t o f inc reasthe v i scos i ty f rom 1 to 80 cP. Th e o rd ina t e i n t he se p lo t s i s t he w ave leng th . The absc i s sa is supe r f ic i a l ga s ve loc i ty, c a l cu l a t ed a s i f no l i qu id we re fl owing . The ave ra ge ve loc i ty o f t he l iqw a s o b t a i n e d f r o m a m o m e n t u m b a l a n c e f o r a f u ll y d e v e l o p e d c o n d i t io n , a s o u t l in e d b y Ta i teD u k l e r ( 1 97 6 b ). T h e i n t e r fa c i a l f r ic t i o n f a c t o r w a s c a l c u l a t e d u s i n g t h e B l a si u s l a w a n d t h e c o n co f a h y d r a u l i c d i a m e t e r. I n a c t u a l f lo w s t h e i n t e r fa c i a l f r i c ti o n f a c t o r c o u l d b e l a rg e r t h a n p r e d i


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5 8 8 N. A N D R I T S O S a n dT, J . HANRAI-I'Y

100 ' I /

b-$o~utionfor l ~ r g e X / H . / 'In te rpo la t ion -~ 1 / ' /

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F i g u r e 2 . C a l c u l a t i o n o f n e u t r a l s t a b i li t y c u r v e a n d m a x i m u m g r o w t h r a t e l o c u s ( D = 9 . 5 3 c r n , U L = 1 c P,7,/o = o.4/.

by the B la s iu s equa t ion b ecause o f t he ex i s t ence o f a rou ghen ed in t e r f ace . Th i s la rge r fr i c ti on f ac to rdoe s no t a f f ec t app rec i ab ly t he neu t r a l st ab i li t y cu rve o r the l oca t ion o f t he cu rve o f m ax im umgrow th , bu t it does a ff ec t s ign i fi can t ly t he m agn i tude o f t he g ro wth r a te .

The so l id cu rve spann ing the l ower w ave leng ths i s t he neu t r a l s t ab il i ty cu rve ( t he l oc i o f Cz = 0 )ca l cu l a t ed f rom [28 ] fo r 2 / t i < 2n . The so l id cu rve a t la rge wave leng th s i s t he neu t r a l s t ab i li t y cu rveca l cu l a t ed f rom L in s ana lys is (wh ich i s i nva l id fo r sma l l 2 ) . T he da shed cu rve r ep resen ts a pos s ib l ein t e rpo la t i on b e tween these two r e su lt s . Th e r eg ion in s ide and to t he r i gh t o f t he se cu rves is a r eg ionof i n s t ab i li t y ( (7 , > 0 ) . Th e con d i t i on s fo r ma x im um g row th a re a l so i nd i ca t ed by the l oc i o f t hem a x i m a i n k C 1

The r e su lt s sho wn in f i gu re s 2 - 4 a r e e s sen t i a ll y un chan ged i f t he cond i t i on s f ir = r, = 0 a r e u sedin t he ca l cu l a t ions , p ro v ided 2 i s no t t oo la rge . Th i s i nd i ca t es t ha t on ly t he wave - indu ced va r i a t i ono f t he gas -phase p re s su re i s a f f ec t ing the ca l cu l a t i on o f t he min imum c r i t i ca l ga s ve loc i ty and theca l cu l a t i on o f t he f a s t e s t g rowing wave .

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F i g u r e 3 . C a l c u l a t i o n o f n e u t r a l s t a b i l i ty c u r v e _ a n d m a x i m u m g r o w t h r a t e l o c u s ( D = 9 , 5 3 c m , L = 1 c P,h D = 0 . 2 ) .


7/21

I N T E R FA C I A L I N S TA B I L IT I E S I N H O R I Z O N TA L G A S - L I Q U I D F L O W 589

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F i g u r e 4 . C a l c u l a t io n o f n e u t ra l s t ab i li ty c u r v e a n d m a x i m u m g r o w t h r a te l oc u s(Z) = 9.53 cm, #L = 80 cP,h It) = o . 2 ) .

Figu re 5 e xa m i n es t he r e l a t i v e im p o r t a n ce o f P~, and /~ iR fo r a wa t e r f l ow. Th eJ c u r v e p r e s e n t st h e m a x i m u m v a lu e s o fk C l . T h e - - - c u r v e i s t h e s a m e c a l c u l a t i o n w i t h /~ i R -----0 . T h e - . - c u r v ei s t h e ca l cu l a t i o n w i t h P il = 0 . I t i s n o t e d t ha t /5~R has n o e f f ec t on t he ca l cu l a t ed m ax im a i nk C la t s m a l l g a s v e l o c i t ie s . T h i s s u g g e s t s t h a t t h e c r it ic a l g a s v e l o c i ty n e e d e d t o g e n e r a t e w a v e s o n w a t e rf l o w s is d e t e r m i n e d c o m p l e t e l y b y a s h e l te r i n g m e c h a n i s m , w h e r e b y e n e rg y i s f e d t o th e w a v e s b ya p r e ss u r e c o m p o n e n t w h i c h i s i n p h a s e w i t h t h e w a v e s l o p e . I t i s a l s o n o t e d t h a t p r e ss u r e v a r i a t io n si n p h a s e w i t h t h e w a v e h e i g h t ( / ~ R ) s t a r t t o b e c o m e i m p o r t a n t a t a s u p e r f i c i a l g a s v e l o c i t y o fUGS -----7 m / s a n d b e c o m e d o m i n a n t a t U o s ~ 9 m / s .

Ve r y - l a rg e - w a v e l e n g t h d i s t u r b a n c e s a r e n o t p o s s i b l e f o r s m a l l g a s v e l o c i ti e s . H o w e v e r , i t i s n o t e dtha t a t a c e r t a i n c r i t i c a l ga s ve lo c i t y, wh ich i s s t r ong ly dependen t onh / D t h e g e n e r a t i o n o fve r y - l a rge -w ave l e n g th w a v es i s p os s i b l e . L in & Han ra t t y ( 1986 ) i n t e rp r e t t h i s a s the c r i t ic a lc o n d i t i o n f o r t h e in i t i a ti o n o f s l u g s. I n t h i s r e g i o n a l l o f t h e i n t e r fa c i a l s tr e s s c o m p o n e n t s a r econ t r i b u t i n g , bu t P~R i s t he d om inan t e f f ec t.

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UGs(m/s)F i g u r e 6 C a l c u l a t i o n o f g r o w t h r a t ea s a f u n c t i o no f s u p e r f i c ia l g a s v e l o c i t y( D = 9 . 5 3 c m , / ~ L = 8 0 c P,

h / Z ) = 0 . 2 ) .


8/21

5 9 0 N A N D R I T S O S a n d T I . H A N R AT T Y

The e f f ec t o f an i nc r ea se in v i s cos i t y c an be s een by com par in g f i gu re s 4 and 6 w i th fi gu rea nd 5 . A s w ou ld be expec t ed , t he c r it ic a l ve loc i t y nece s sa ry t o gene ra t e w aves inc r ea se s winc r e a s ing l i q u id v i s cos i t y. A c om pa r i so n o f f igu re s 5 and 6 fo r t he ca se o f /5~R = 0 i nd i ca te s ta s h e l t e r i ng hy po the s i s by i t se l f (P~ I # 0 , P,R = 0 ) w ou ld p r ed i c t t he i n i t i a t ion o f w aves a t a gv e l o c i t y w h ich i s t oo h igh . Th e d i s c r epa ncy bec om es g r ea t e r w i th inc r ea s ing l iqu id v i s cos i t y. Ti l lu s t r a t e s wh a t had a l r ead y been p o in t ed ou t by F ranc i s ( 1954, 1956 ) and by Mi l es ( 1959 ) t ht h e in i t ia t io n o f w a v e s o n h i g h - v i s co s i ty l iq u i d s is c o n t r o l l e d b y a K e l v i n - H e l m h o l t z ( K - Hm ech anis m (P iR # 0 , /S il = 0) .

T h e c u r v e s in f ig u r e s 2 - 4 m a r k e d i n v is c id K - H t h e o r y r e p r e s e n t c a lc u l a t io n s t h a t a s s u m e i nv i sp l u g f lo w s b o t h i n th e g a s a n d t h e li q u id . T h e y a r e o b t a i n e d u s i n g t h e K - H i n s ta b i li ty e q u a t i o

_ . , > . . . . . H F ta n h (k /7 ) p G t a n h ( k h )(UG --U a)' ' tK'a +(PL--PC)gJ-~L --ff P - '~ J [33]

F o r t h e c a s e o f w a t e r f l o w s , i n v is c id K - H t h e o r y p r e d i c t s a n i n i t ia t io n o f w a v e s a t g as v e l o cit h a t a r e t o o h i g h . T h i s is b e c a u s e t h e i n f lu e n c e o f p r e s s u r e c o m p o n e n t s i n p h a s e w i th t h e w a v e s

i s i gno r e d ( J e f f r eys 1925 ) . Fo r v i s cous fl u id s t he i nv is c id K- H theo ry p r ed i c t s t he in i t i a ti on o f w aa t a l ower ga s ve loc i t y t han i s c a l cu l a t ed f rom the fu l l v i s cous equa t i ons , because i t i gno re s v i s cda mping i n t he f l u id .

A s h a s a l r e a d y b e e n p o i n t e d o u t b y L in & H a n r a t t y (1 9 8 6) , i n v is c id K - H t h e o r y is f o u n d p r e d i c t g a s v e loc i t i e s f o r t he i n i ti a t ion o f l a rge -w ave l eng th w aves wh ich a r e t o o h igh ( see f ig2 ) . T h i s i s b e ca use t he i nv i s cid t heo ry u nde re s t im a t e s t he de s t ab i l i z ing e f f ec ts o f i ne r t ia i n l iqu id .

(f ) Effect of pressureThe e f f ec t o f ga s -ph ase p r e s su re o n t he s t ab i l i ty c a l cu l a t i ons is i l lu s t r a t ed i n f i gu re 7 f o r t he c

o f a 1 2 c P l i q u i d f l o w i n g i n a 9 .5 3 c m p i p e . I t is n o t e d t h a t t h e g r o w t h r a t e i s a p p r o x i m a t ei n d e p e n d e n t o f p r e s s u r e i f i t is p l o t te d a g a i n s t U G S X / ~ . T h i s s u g g e s ts t h a t t h e i n it ia t io n l a rg e - a m p l i tu d e w a v e s a n d o f a t o m i z a t i o n w i ll o c c u r a t l o w e r g a s v e l o c it ie s i n h i g h - p r e ss u r e f l o

I t is a l s o n o t e d t h a t t h e v a l u e o f U G S X /~ G n e e d e d t o i n it ia t e r e g u l a r w a v e s i n c r ea s e s w ii n cr e as in g p r e s su r e a n d a p p r o a c h e s t h e v al u e th a t w o u l d b e c a lc u l a te d b y a s s u m i n g / 5 = 0 . Ts u g g e s t s t h a t t h e r a n g e o f g a s v e l o c it ie s o v e r w h i c h r e g u l a r w a v e s e x is t b e c o m e s s m a l le r winc r e a s ing p r e s su re .

1 0 0

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F i g u r e 7 . E f f e c t o f p r e s s u r e o n g r o w t h r a t e(hiD= 0 . 2 , D = 9 . 5 3 c m , # = 1 2 c P ) .


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INTERFACIAL INSTABILITIES IN HORIZONTAL GAS-LIQUID FLOW 591

3 . D E S C R I P T IO N O F T H E E X P E R I M E N T S

T h e e x p e r i m e n t s w e r e c a r ri e d o u t i n t h e g a s - l iq u i d f lo w s y s t e m d e s c r ib e d b y L a u r i n a tet al.(1 9 8 4). Te s t s w e r e m a d e w i t h t w o P l e x i g l a s s e c t i o n s w i t h i .d . s o f 2 .5 2 a n d 9 .5 3 c m , l o c a t e d 1 5a n d 2 4 .6 m f r o m t h e e n t r y. S t u d i e s w i t h h i g h - v i s c o s i ty fl u id s in t h e 2 .5 2 c m p i p e w e r e c a r r ie d ow i t h a s m a l l e r t e s t l e n g t h o f 1 0 m i n o r d e r t o r e d u c e f r i c t i o n a l l o s se s . T h e p i p e l i n e s w e r e c a r e f u

l ev e le d b y i n t r o d u c i n g w a t e r i n t o t h e m , c lo s i n g b o t h e n d s a n d a d j u s t i n g t h e s u p p o r t s s o t h a t tl e v e l w a s t h e s a m e a l o n g t h e p i p e l i n e .

T h e p i p e s d is c h a r g e d i n t o a s e p a r a t o r w h i c h w a s a t a t m o s p h e r i c p r e s s u re . A s i m p l e p i p e te e wu s e d t o c o n t a c t t h e p h a s e s a t t h e e n t ry. T h e l i q u i d w a s i n t r o d u c e d i n th e r u n o f t h e te e a n d t hg a s , i n t h e b r a n c h .

T h e l i q u i d v i s c o s i ty w a s v a r i e d b y u s i n g s o l u t i o n s o f w a t e r a n d g l y c er in e . S o l u t i o n s w i tv i s c o s it ie s o f 1 2 a n d 8 0 c P w e r e u s e d i n t h e 9 . 5 3 c m p i p e o f 4 .5 , 1 6 a n d 7 0 c P i n t h e 2 . 5 2 c m p iB e c a u s e o f li m i t a t i o n s in t h e r e c i r c u l a t io n p u m p i t w a s n o t p o s s ib l e t o c a r r y o u t s t u d i es a t h il i q u i d v e l o c i ti e s w i t h 1 2 a n d 8 0 c P s o l u t i o n s i n t h e 9 . 53 c m p i p e .

T h e h e i g h t o f t h e li q u i d la y e r w a s m e a s u r e d b y t w o p a r a l l e l w i r e c o n d u c t a n c e p r o b e s t h ae x t e n d e d o v e r t h e e n t i re c r o s s s e c ti o n o f t h e p i p e . T h i s t e c h n i q u e i s d e s c r i b e d b y L a u r i n a tet al.(1 9 8 4) a n d b y L i n & H a n r a t t y (1 9 87 ). F o r t h e 9 .5 3 c m p i p e , t h e p r o b e w a s l o c a t e d 2 2 0 p i pd i a m e t e r s f r o m t h e e n t ra n c e . F o r t h e 2 .5 2 c m p i p e , t h e p r o b e w a s l o c a t e d 5 0 0 p i p e d ia m e t e r s f r ot h e e n t r a n c e f o r w a t e r a n d , 3 5 0 p i p e d i a m e t e r s f o r g l y c e r i n e - w a t e r s o lu t i o n s . W a v e v e l o c it ie s w em e a s u r e d b y l o c a t i n g a s e co n d p a i r o f c o n d u c t a n c e p r o b e s 2 6 . 7 c m d o w n s t r e a m i n th e 9 .5 3 c m p i2 5 . 4 c m d o w n s t e a m f o r w a t e r fl o w s i n t h e 2 . 5 2 c m p i p e a n d 1 0 . 2 c m d o w n s t r e a m f o rw a t e r - g l y c e r i n e f l o w s i n t h e 2 . 5 2 c m p i p e .

E x p e r i m e n t s w e r e a ls o c a r r ie d o u t i h t h e 9 .5 3 c m p i p e i n w h i c h t h e l i q u i d l a y e r h e ig h t w am e a s u r e d 8 0 a n d 1 6 0 p i p e d i a m e t e r s f r o m t h e e n t r a n c e . T h i s w a s d o n e i n o r d e r t o d e t e r m i n e t hh y d r a u l i c g r a d i e n t s t h a t e x i s t a t l o w g a s v e l o c i t i e s .

4 . D E S C R I P T I O N O F T H E WAV E S

(a ) Low- visco sity l iquids

T h e t y p e o f w a v e s o b s e r v e d o n a l o w - v i s c o s i t y f l u i d a r e i l l u s t r a t e d b y t h e t r a c i n g s i n f ig u r ef o r a 4 .5 c P l i q u i d . F o r a f i x e d l i q u i d f l o w t h e f i r s t w a v e s o b s e r v e d w i t h i n c r e a s i n g g a s v e l o ca r e s m a l l - a m p l i tu d e 2 - D w a v e s o f w a v e l e n g t h 2 - 4 c m t h a t c a n o n l y b e o b s e r v e d v i s u a ll y b y l i gr e fl e ct io n f r o m t h e l i q u id s u r f a c e. T h e s e d i s t u r b a n c e s f i rs t a p p e a r a t t h e d o w n s t r e a m e n d o f tp i p e. A n i n c r e a s e in t h e g a s v e l o c i ty c a u s e s t h e p o i n t o f i n i ti a t io n t o m o v e u p s t r e a m . T h e s e w a vi n cr ea s e in a m p l i tu d e a n d i n w a v e l e n g t h a s t h e y p r o p a g a t e d o w n s t r e a m . T h e t r o u g h s o f t h e w a vo b s e r v e d i n t h e 9 .5 3 c m p i p e w e r e c o v e r e d w i t h a n i n t e r f e re n c e w a v e p a t t e r n o f s m a l l b r o a d c r e sc a p i l l a r y w a v e s . T h i s w a s n o t t h e c a s e i n t h e 2 . 5 2 c m p i p e .

U p o n i n c r e a s i n g t h e g a s f lo w r a t e , t h e a m p l i t u d e s o f t h e w a v e s b e c o m e l a r g e r a n d t h e w a v e l e n gd e c r ea s e . T h i s is i ll u s t ra t e d b y c o m p a r i n g t h e w a v e t r a c i n g s i n f ig u r e s 8 (a ,b ) . T h e p e b b l y w a vp a t t e r n o b s e r v e d b y H a n r a t t y & E n g e n (1 95 7 ) w a s e n c o u n t e r e d o n l y f o r w a t e r f l o w s i n th e 9 .5 3 p i p e. I t o c c u r r e d o v e r a n a r r o w r a n g e o f fl o w c o n d i t i o n s . A c u r v e d w a v e f r o n t w a s a l s o o b s e r vf o r l o w l i q u i d v e l o c i t i e s i n t h e 9 . 5 3 c m p i p e .

A t h i g h e n o u g h g a s v e lo c i ti e s a " l a r g e a m p l i t u d e " w a v e w i t h a st e ep f r o n t a n d a g r a d u a l l y s lo p ib a c k f o r m s c l o s e t o t h e e n t r y. T r a c in g s f o r t h i s t y p e o f w a v e a r e s h o w n i n f ig u r e s 9 (b ,c ). T h e s" l a r g e - a m p l i t u d e " w a v e s a r e l e ss re g u l a r t h a n t h e 2 - D w a v e s o b s e r v e d a t l o w e r g a s v e lo c it ie s . T hc a n b e s e e n f r o m t h e t ra c i n g s b u t i t c a n a l so b e d e t e c t e d f r o m t h e c r o s s c o r re l a t io n s o f th e f ilh e i g h t si g n a ls f ro m t w o l o c a t i o n s s e p a r a t e d i n t h e f lo w d i re c t i o n . A m a x i m u m i n th e c o r r e l a t ii s o b s e r v e d f o r a c e r t a i n ti m e d e l a y. F o r t h e 2 - D s m a l l - a m p l i t u d e w a v e s t h is m a x i m u m i s c l o se u n i t y. H o w e v e r , i ts v a l u e i s < 1 f o r t h e " l a r g e - a m p l i t u d e " w a v e s a n d c o n t i n u e s t o d e c r e a s e w ii n c re a s i n g g a s v e l o c it y. A n o t h e r s i g n a t u r e o f t h e " l a r g e - a m p l i t u d e " w a v e s i s t h a t t h e d i s t r ib u t if u n c t i o n f o r f i l m h e i g h t h a s a f l a t n e s s f a c t o r > 3 .

A t c o n d i t i o n s c lo s e to t h e p s e u d o - s l u g t r a n s i t i o n d e f i n e d b y L i n & H a n r a t t y ( 19 87 ) a w a vp a t t e r n s i m i la r to t h e r o ll w a v e s d e s c r ib e d b y H a n r a t t y & E n g e n (1 95 7) f o r g a s - l i q u i d f l o w in


10/21

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F i g u r e 8 F i l m t h i c k n e s s t r a c i n g s fo r c o n s t a n t s u p e r f i c ia l l iq u i d v e l o c i t y( D = 2 . 5 2 c m , # t = 4 . 5 c P ,UL~= 0.03 m/s).

chann e l i s obse rved in the 2 .52 cm p ipe . In t h i s ca se , som e o f t he l a rge -ampl i tude wav es coa l e sceto fo rm an even l a rger acce l e r a ti ng w ave . F igu re 8 (c ) , o r 9 (d ) , g ives an exam ple o f th i s t ype o fint er rac ia l d i s tu rbance . I t i s t o be no ted tha t it i s s imi la r t o t he ro l l wav e obse rved by Ha nra t ty& E n g e n ( 1 9 5 7 ) a n d b y M i y ae t a l (1971) i n a channe l f l ow.

Above a ce r t a in gas ve loc i ty t he l i qu id s t a r t s t o c l imb up the wa l l s o f t he p ipe and the ave rageshape i s no longe r app rox im a ted by a fi a t ho r i zon ta l p l ane . In add i t i on , d rop le t s o r li qu id f il amen t sa re t o rn f rom the l i qu id phase and depos i t ed on the p ipe wa l l s . The in i t i a t i on o f a tomiza t ion i sde f ined a s t he gas ve loc i ty a t wh ich d rop le t s f i r s t h i t t he t op o f t he p ipe . A t h ighe r gas ve loc i t i e senough d rop le t s a r e depos i t ed so t ha t l i qu id s t r eaks fo rm. Th i s d rop le t depos i t i on p roces s i s t hem echa n i sm by wh ich annu la r f l ow i s i n i t ia t ed in l a rge-d i ame te r p ipes (L in & Hanra t ty 1987) a t ga sve loc i t i e s rough ly tw ice t ha t needed to i n i t i a t e a tomiza t ion . Wave we t t i ng can be an add i t i ona lm echan i sm fo r t he in i t i a t ion o f annu la r f low in sma l l -d i ame te r p ipes (L in & Ha nra t ty 1987) ,

A t l a rge gas ve loc it i e s in t he 2 .52 cm p ipe the l a rge -am pl i tude w aves t ake the fo rm o f ae ra t edr ing waves (o r d i s tu rbances ) t ha t wrap a round the p ipe c i r cumfe rence . Fo r even h ighe r gasve loc i t ie s ( > 10 0m /s ) t he se d is tu rbances d i sappea r. I n t he 9 .53 cm p ipe they appea r a s b rokenp ieces o f a r i ng . They do no t wrap a round the en t i r e c i r cumfe rence un t i l ve ry h igh gas ve loc i t i e s ,say 70 m /s , a re reached.


11/21

INTERFAC IAL INSTA BILITIES IN HORIZONTA L G AS--LIQUID FLOW

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T h e f l o w p a t t e r n s o b s e r v e d o v e r t h e r a n g e o f l iq u i d v i s c o s i ti e s o f 1 - 1 4 c P a r e s im i l a r. T h e m a i nd i ff e r en c e i s t h a t t h e r eg u l a r 2 - D w a v e s a p p e a r a t h i g h e r g a s v e lo c i t ie s a n d e x i st o v e r a n a r r o w e rr a n g e o f f l o w c o n d i t i o n s a s t h e l i q u i d v i s c o si t y i n c r e a se s .

( b ) H i g h - r i s c o s i t y l iq u i d sW a v e p a t t e rn s w i t h a li q u i d o f h i g h v i s c o s i t y, 8 0 c P, d i ff e r f r o m t h o s e w i t h a l o w - v i s c o s i t y l i q u i d

i n t h a t th e r e g i o n o f r e g u l a r 2 - D w a v e s b a r e l y ex i s ts a n d i n t h a t t h e in t e r f a c e a p p e a r s l e s s r o u g h e n e de v e n w h e n w a v e s a r e p r e s en t . E x a m p l e s o f t h e s e w a v e s a r e g i v e n i n f ig u r e 1 0 . T h e f ir st d i st u r b a n c e st h a t ar e o b s e r v e d w i t h i n c r e a s i n g g a s v e l o c it y a r e s m a l l - a m p l i t u d e , s m a l l - w a v e l e n g t h , r a t h er r e g u la r2 - D w a v e s . H o w e v e r , w i t h j u s t a s li g h t i n c re a s e i n g a s v e lo c i t y, t h e s e g i v e w a y t o a f e wl a rg e - a m p l i t u d e w a v e s w i t h s t ee p f r o n t s a n d s m o o t h t r o u g h s , a n d w i t h s p a c i n g s t h a t c a n v ar y f r o ma f e w c e n t i m e t e r s t o a m e t er ; s e e f i g u r e 1 0 a ) . O c c a s i o n a l l y, s e v e r a l s m a l l 2 - D w a v e s c a n b e s e e ni n f r o n t o f t h e l a rge w a v e s . T h i s s e q u e n c e o f e v e n t s t h a t a c c o m p a n i e s t h e i n i t i a t io n o f w a v e s o na v e r y v i s c o u s l iq u i d i s t h e s a m e a s h a d b e e n o b s e r v e d s o m e y e a r s a g o b y F r a n c i s 1 9 5 4 , 1 9 5 6 )i n h i s s t u d i e s o f g a s f l o w o v e r a d e e p l i q u i d .

Wi t h i n c r e a s i n g g a s f l o w t h e s p a c i n g b e t w e e n t h e w a v e s d e c r e a s e s a n d t h e c r e s t s b e c o m e c u r v e d ;s e e f ig u r e t 0 b ). E v e n t u a l l y t h e w a v e s b e c o m e c e l l- l ik e a n d e x t e n d o n l y o v e r a p a r t o f t h e li q u i ds u r f a ce ; s e e fi g u r e s 1 0 c , d ) . U n d e r t h e s e c o n d i t i o n s t h e s p a c i n g i s 1 - 2 c m .


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15/21

INTERFAC IAL INSTA BILITIES IN HORIZONTA L GA S-LIQUID FLOW 597

A s t h e c r es t b e c o m e s m o r e c u r v e d t h e l iq u i d s t a r ts t o c l i m b a r o u n d t h e p ip e c i rc u m f e r e n c e a na t o m i z a t i o n is i n i ti a t e d . T h e d r o p l e t s f o r m e d w i t h th e 8 0 c P l i q u i d a p p e a r e d t o b e m u c h s m a lt h a n t h o s e fo r m e d w i th w a t e r f lo w s . T h e y a r e t h o u g h t t o o r i g i n a te b y t h e s p ew i n g o f l iq u i d f r ot h e c re s t o f t h e la r g e a m p l i t u d e w a v e s r a t h e r t h a n b y t h e r e m o v a l o f a w a v e l e t, a s h a s b e es u g g e s t e d f o r w a t e r f l o w s .

T h e a n n u l a r f l o w t h a t e v e n t u a l l y f o r m s a t h i g h e n o u g h g a s v e l o ci ti e s (s ee f ig u r e 11 ) h a s s o mm a j o r d i f fe r e n ce s f r o m t h a t o b s e r v e d f o r w a t e r f lo w s ; c f. fi g u re s (1 l a ,c ) . N o d i s t u r b a n c e w a v e s wo b s e r v e d f o r t h e h i g h - v i s c o s it y l iq u i d f o r t h e r a n g e o f f lo w s s t u d i e d . T h e w a v e s a re m o r e b r oc r e st e d a n d h a v e a l a r g e r s p a c i n g . In f a c t , s o m e o f th e s e w a v e s r i n g a r o u n d t h e p i p e c i r cu m f e r enA n i n t e r e s t i n g f e a t u r e i s t h a t t h e s e r i p p l e s a r e u s u a l l y o r i e n t e d a t a n a n g l e o f < 9 0 t o t h e f loT h i s s u g g e s ts t h e p o s s i b il i ty t h a t a l i f ti n g fo r c e o n t h e se w a v e s b y t h e g a s f lo w m a y b e a n i m p o r tm e c h a n i s m t o c o u n t e r b a l a n c e t h e t e n d e n c y o f th e l i q u id t o d r a in t o t h e b o t t o m o f t h e p ip e b e c a uo f g r a v i ty.

5 . F L O W R E G I M E T R A N S I T I O N S

M a p s s h o w i n g t h e r a n g e o f f lo w c o n d i t i o n s f o r t h e d i ff e r e n t o b s e r v e d p a t t e r n s a r e g i v en b y tc u r v e s i n M a n d h a n e - t y p e p l o ts i n f ig u r e s 1 2 -1 7 . T h e m e t h o d u s e d t o d e t e c t s l u g s a n d p s e u d o - s l uis th e s a m e a s d e s c ri b e d b y L i n & H a n r a t t y (1 98 7). I n t h i s p r e v i o u s p a p e r c o n s i d e r a b l e a t t e n t iw a s g i v e n t o t h e t r a n s i t i o n s t o a n n u l a r f lo w, t o s l u g f lo w a n d t o p s e u d o - s l u g f l o w. In f a c t, tt r a n s i ti o n s t o t h e s e r e g im e s f o r w a t e r s h o w n i n f ig u r e s 1 2 a n d 1 5 a r e t h e s a m e a s r e p o r t e d b y L& H a n r a t t y (1 98 7). C o n s e q u e n t l y, th i s p r e s e n t a t i o n w i l l f o c u s m a i n l y o n t h e c h a n g e o f w a v e p a t t ei n th e s t ra t i f ie d fl o w r e g im e . T h r e e p a t t e r n s a r e d e f i n e d : 2 - D r e g u l a r w a v e s , la rg e - a m p l i t u di r r e g u l a r w a v e s a n d a t o m i z i n g f l o w s .

A s m e n t i o n e d i n s e c t io n 3 , t h e f l o w i n t h e p i p e li n e w a s n o t f u l l y d e v e l o p e d a t l o w g a s v e l o ci tiw i t h t h e c o n s e q u e n c e t h a t h y d r a u l i c g r a d ie n t s e x is te d . E x p e r i m e n t s m a d e u n d e r s u c h c o n d i t ioa r e i n d i c a t e d b y t h e - - - c u r v e s i n f i g u r es 1 2 - 1 5 .

I t is n o t e d f r o m f ig u r e s 1 2 - 1 4 t h a t l i q u i d s o f l o w e r v i s c o s it y r e q u i re a l o w e r l i q u i d t h r o u g h pf o r t h e g e n e r a t i o n o f s lu g s a t l o w g a s v e l o c it ie s . T h i s a p p e a r s t o b e r e s p o n s i b l e f o r t h e c h a n g et h e s h a p e o f t h e p s e u d o - s l u g r e g i o n w i t h l iq u i d v i s c o s i ty.

T h e o t h e r m a i n e f fe c t o f v i s c o s it y is t o i n c re a s e t h e g a s v e l o c i t y r e q u i r e d f o r t h e i n i t ia t i o n r e g u l a r 2 - D w a v e s a n d t o d e c r e a s e t h e r a n g e o f f lo w c o n d i t i o n s f o r w h i c h t h e y e x i s t. In f a c t, fa 7 0 c P l i q u i d ( se e fi g u r e s 1 4 a n d 1 7 ) t h e s e r e g u l a r w a v e s a r e n o t o b s e r v e d a t a l l.

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16/21

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I t i s n o t e d t h a t o n a M a n d h a n e p l o t a l l o f t h e o t h e r t r a n s i ti o n s b e s i d e s t h e st r a ti fi ed t op s e u d o - s l u g , t h e s tr a ti fi ed t o s l u g a t l o w g a s v e l o c it i e s, a n d t h e s m o o t h - s t r a t i f i e d t o r e g u la r w a v e )

a r e n o t s t r o n g l y s e n s it i v e e i th e r t o c h a n g e s i n p i p e d i a m e t e r o r in l i q u i d v i s c o si t y.F i g u r e 1 8 s h o w s t h e s a m e r e s u l t s g i v e n i n fi g u r es 1 2 a n d 1 4 p l o t t e d a shi s t h e s u p e r f i c i a l g a s

v e l o c i ty, w h e r e h i s th e h e i g h t o f t h e l iq u i d l a y er a t t h e b o t t o m o f t h e p i p e . I t i s t o b e n o t e d t h a ti n th i s t y p e o f p l o t w h i c h s h o u l d b e le s s se n s i t iv e t o w h e t h e r t h e f lo w i s f u l ly d e v e l o p e d ) t h e ef f ec t

0 . 5 , , 0 . 5 ,

0.2

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n~lar

i i

o . o o ~ l i

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0 , 0 0 1 - - k C i " l "VWCI IO 1-8C~

I

I 2 5 I 2 0 5 0 1 0 0

U G S ( m / s )

F i g u r e 1 6 . F l o w r e g i m e m a p a n d c o m p a r i s o n w i thl i n e a r t h e o r y f o r a 9 . 5 3 c m p i p e a n d a 1 2 c P l i q u i d :- - e x p e r i m e n t a l o b s e r v a t i o n s ; . . . . . l i n e a r t h e o r y.

O.OZE

D 0 , 0 1

0 . 0 0 0 5

0 . 2

0.1

0 . 0 5

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k C i . O

~rorn. n n u l o r

~C~,80k C , , 1 0

t l T ~ i1 5 1 0 2 0 5 0 1 0 0

U G s (m I s )

F i g u r e 1 7 . F l o w r e g i m e m a p a n d c o m p a r i s o n w i t hl i n e a r t h e o r y f o r a 9 .5 3 c m p i p e a n d a n 8 0 c P l i q u i d :- - e x p e r i m e n t a l o b s e r v a t i o n s ; . . . . . l i n ea r t h e o ry.


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INTERFAC IAL INSTAB ILITIES IN HORIZONTAL GA S--LIQUID FLOW 5 9 9

I D=2.520 .8 - -'~L '7O eP

Slug

~ N ,s.~Z,h ' , '7", , . P~ uao-0,4 _ ~ r o ~ P I " . " - . . ~ slug

0.2 - ' X . \ / " - . , ~s o o , - . r . 2 - 0 , o ,( l o P ) ~ - ' ~ TM

" ~ . ~ ' , l" , - / . ~ L = 7 0 C l

o t \ 1 I '2 5 1 2

UGs m/s)

Figure 18. Liquid heigh ts at the transit ions for 1 and 70 cP liquids: - - transit ion to slug s andpseudo-slugs; - - - transit ion from 2-D w aves to large-amplitude waves; . . . . . init iation of 2-D waves.

o f a n i n c re a s e o f v i s c o si ty is t o m a k e t h e s y s t e m m o r e s t a b l e t o t h e i n i t i a t io n o f sl u gs . A s t r os t a b i l i z in g e f fe c t o f v i s c o s it y w i t h r e s p e c t t o t h e i n i t i a t i o n o f r e g u l a r w a v e s a n d a w e a k s t a b i l i ze f f e c t w i t h r e s p e c t to t h e i n i t i a t i o n o f la rg e a m p l i t u d e i r r e g u l a r w a v e s a r e a l s o n o t e d .

6 . C O M P A R I S O N O F WAV E T R A N S I T IO N S T O L I N E A R T H E O R Y

O b s e r v e d t r a n s i ti o n s a r e m a r k e d o n t h e p l o ts o f r e s u lt s c a l c u la t e d f r o m l i n e a r t h e o r y if i g u r e s 2 - 6 .

F i g u r e s 2 a n d 3 i n d i c a te t h a t t h e o b s e r v e d i n i t ia t i o n o f 2 - D w a v e s o n l o w - v i s c o s it y l iq u ic o r r e s p o n d s r o u g h l y w i t h t h e e x t r e m u m o f t h e C I = 0 c u rv e . A m o r e c a r e f u l c o m p a r i s o n is g i vi n f i g u r es 1 2 - 1 6 . I t is n o t e d t h a t t h e t r a n s i t i o n i s o b s e r v e d t o o c c u r a t a l a rg e r g a s v e l o c i ty t hi s p r e d i c t e d ( t h ekCi= 0 c u rv e s ) . A p o s s i b l e e x p l a n a t i o n f o r t h i s c a n b e s e e n i n fi g u r e 5 , w h i c h s h o w st h a t g r o w t h f a c t o rkCIm a i n t a i n s a r e l a t iv e l y s m a l l v a l u e o v e r a r a n g e o f g a s v e lo c i t ie s . T h i s c o u l dm e a n t h a t b e t t e r a g r e e m e n t w o u l d b e o b t a i n e d i f a l o n g e r p ip e l in e w e r e u s ed . S o m e s u p p o r t ft h is i n t e r p r e t a t i o n c o m e s f r o m t h e o b s e r v a t i o n t h a t t h e f ir s t w a v e s a p p e a r a t t h e o u t l e t o f tp i p e li n e . A s c a n b e s e e n f r o m f ig u r e s 1 2 -1 6 , a s o m e w h a t b e t t e r a v e r a g e a g r e e m e n t b e t w e e n l i ns t a b i li t y t h e o r y a n d o b s e r v a t i o n s i s o b t a i n e d i fkC1= 1 s - l i s u s e d a s a c r i t e ri o n .

I t is n o t e d f r o m f ig u r es 2 a n d 3 t h a t t h e in i t i a t io n o f " l a r g e - a m p l i t u d e w a v e s c o r r e s p o n d s r o u g ht o t h e e x t r e m e o f t h e i n v i s c i d K - H a n a l y s i s f o r l o w - v i s c o s i t y f l u id s (/a = 1 c P ) . F o r a l i q u i d w i8 0 c P v i s c o s i t y t h e i n v i s c id K - H a n a l y s i s s li g h t l y u n d e r p r e d i c t s t h e c r it i c a l g a s v e l o c i t y. T h i s at h e r es u l ts s h o w n i n f ig u r e s 5 a n d 6 i n d i c a t e th a t t h e s e " l a r g e - a m p l i t u d e " w a v e s a p p e a r w hp r e s s u r e v a r i a ti o n s i n p h a s e w i t h t h e w a v e h e i g h t s t a r t t o p l a y a n i m p o r t a n t r o le in d e t e r m i n it h e g r o w t h f a c t o rkCi.I n f a c t , i t is n o t e d f r o m f i g u r e 5 t h a t g o o d a g r e e m e n t i s o b t a i n e d b e t w e e nt h e c r it i c a l g a s v e l o c i t y c a l c u l a t e d f o r / 5 = 0 a n d o b s e r v a t i o n s o f t h e c r i ti c a l v e l o c i t y f o r t hi n i t i a t i o n o f " l a r g e - a m p l i t u d e " w a v e s .

I t i s a l s o n o t e d t h a t w h e n # m i s i m p o r t a n t t h e g r o w t h f a c t o r ,kCl ,i n c re a s e s v e r y r a p i d l y w i t hi n c r e a si n g g a s v e l o c i ty. T h u s , t h e c r it ic a l g a s v e lo c i ty s h o u l d n o t d e p e n d v e r y s t ro n g l y o n t h e l e no f t h e p ip e . T h i s e x p l a in s w h y " l a r g e - a m p l i t u d e " w a v e s a r e i n i t ia t e d c l o se t o t h e e n t r y.

F o r t h e c a se o f a n 8 0 c P l i q u i d , s h o w n i n f i g u re 6 , t h e i n i t i a t io n o f " l a r g e - a m p l i t u d e " w a v eo c c u r s a t a g a s v e l o c i t y o n l y s l ig h t l y l o w e r t h a n t h e c a l c u l a t e d c r i t i c a l(kC~= 0 ). M u c h c l o s e ra g r e e m e n t i s o b t a i n e d i f tr a n s i t i o n i s d e f i n e d b y a g r o w t h f a c t o r o fkCj= 1 0 s - L F i g u r e s 1 2 - 1 7c o m p a r e o b s e r v e d c r it ic a l c o n d i t i o n s f o r t h e in i t i a ti o n o f " l a r g e - a m p l i t u d e " w a v e s w i t h c a l c u l a ti of r o m l i n e a r s t a b i l i t y t h e o r y f o rkCi= 1 0 s - L


18/21

6 0 0 N . A N D R I T $ O S a n d T . J . H A N R AT T Y

T h e i n i ti a ti o n o f a t o m i z a t io n o c c u r s a t g a s v e l o c it ie s a p p r o x i m a t e l y t w i c e t h a t n e e d e d f o r ti n it ia t io n o f la rg e - a m p l i tu d e w a v e s f o r w h i c h t h e g r o w t h r a te s p r e d i c t e d b y l i n ea r t h e o r y a r e ql arg e. F i g u r e s 12 -17 com pa re ob se rve d c r i ti c a l ga s ve loc i t ie s f o r a t om iza t i on w i th ti nea r t hec a l cu l a t i ons fo rkC~= 80 s -~.

7 . D E S I G N C R I T E R I A

The c h a rac t e r o f t he wav es a t t he i n t e r f ace i n s t r a ti f ied f l ow i s o f c r it ic a l imp or t an ce d e t e r m i n i n g p r e s s u r e d r o p a n d h o l d - u p a n d , c o n s e q u e n t l y, in t h e d e v e l o p m e n t o f d e s i g n e q u a t i o

T h e e x p e r i m e n t a l r e s u l t s p r e s e n t e d h e r e i n d i c a t e t h e M a n d h a n eet al. (1974) c r i t e r ion for thei n i ti a ti o n o f w a v y - s t r a t if i e d fl o w r o u g h l y c o r r e s p o n d s t o t h e i n it ia t io n o f l a rg e - a m p l i tu d e w aT h e t r a n s i ti o n t o w a v y - s t r a ti f ie d f lo w p r o p o s e d b y Ta i te l & D u k l e r ( 1 9 7 6 b ) c o r r e s p o n d s t o i n i ti a ti o n o f r e g u l a r 2 - D w a v e s . O n t h e b a s i s o f t h e s h e l te r in g h y p o t h e s i s o f J e f fr e y s t h e y s u g g ethe fo l l o wing r e l a t i on :

i

i I 3 4U o >tL spo UL '

w h e r e s is a she l te r i ng coe ff i c ien t . Th i s b eco m es mea n ing l e s s f o r h igh -v i s cos i t y l iqu id s , s i nce r eg2 - D w a v e a r e o b s e r v e d o n l y f o r l i q u id s w i th v i s c o s it y le ss t h a n 2 0 c P. T h e d a t a p r e s e n t e d f o r i n i ti a ti o n o f t h e s e w a v e s a r e r e p r e s e n t e d r e a s o n a b l y b y t h e c r i t e ri o n s u g g e s t e d b y Ta i te l & D u ke x cep t t ha t a sh e l t e ri ng coe ff i c i en t equ a l t o 0 . 06 sho u ld be u sed .

A s i n d i c a te d i n t h e p r e v i o u s d i s c u s s io n t h e in v i sc i d K - H e q u a t i o n i s a g o o d f ir s t a p p r o x i m a tf o r t he g a s ve l oc i t y r equ i r ed t o i n i ti a t e i r r egu l a r l a rg e - am p l i t ude wav es . W e des igna t e t h i s a s Ua n d d e f in e i t a s f o l l o w s f o r g a s - l i q u i d f lo w s :

k p P L g l t a n h ( k H ) . [35]u K _ > I + p o k l

F o r t h e r a n g e o f / 7 u s e d in t h e ex p e r im e n t s , in v is c id K - H t h e o r y p r e d ic t s th a t t h e m i n i m u m v ao f U K _~ is a t ta i n e d f o r a w a v e n u m b e r o f

k i n = P ~ ' g . [3 6]x/ 0.

F o r t h e s y s t e m s s t u d i e d i n t h i s r e s e a r c h k~ , = 3 . 7 - 4 .3 c m - ~ .A b e t t e r c r i te r i o n t h a n [ 3 5 ] f o r t h e i n i ti a ti o n o f l a rg e - a m p l i tu d e w a v e s is o b t a i n e d f r o m [ 28 ] w

Pj~ = 0 . H o w e v e r, th i s is f a r to o c o m p l i c a t e d a n e q u a t i o n t o b e u s e d f o r d e s ig n p u r p o s e s . T h e r e f ot h e f o l lo w i n g e m p i r i c a l e q u a t i o n , w h i c h t a k e s i n t o a c c o u n t t h e s m a l l e f fe c ts o f li q u id h e i g h t al i q u i d v i s c o s i t y, i s r e c o m m e n d e d :

UGS UK-H~-~)0 025tanh~ ) ) l ( ~ ) ' [3 7]

H e re UK_ . i s de f i ned b y [35 ] w i th

k = k m = ~ .

D i m e n s i o n l e s s g r o u p 0 w a s u s e d b y Ta y l o r ( 19 6 3 ) t o a c c o u n t f o r t h e e f f e c ts o f l iq u i d v i s c o s it y i s def ined as

0 = Pl. 0"2, , . [38]pG # L U~

T he t e r m 0w i s the va lue o f 0 i f t he l i qu id i s w a t e r. V e loc i t y UGs i s t he supe r f i c ia l ga s ve loc i t y t h e i n i t ia t io n o f l a rg e - a m p l i tu d e w a v e s a n d =e is t h e f r a c t i o n o f t h e c r o s s s e c t i o n o f t h e p i p e o c c u pby g a s .


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INTERFACIALNSTA BILITIES IN HORIZONTAL G AS--LIQUID FLOW 601

O O J

E

0.002

0.1 0.1I '

0 . 0 5 ~ I 0 . 0 5

0 .0 2 ~ ~ f :~ i'~1/ 0.02L o r g e~ I A t o m .

P 0.01

0.002 -- 0.002

0.001 -- 0.001

o * o O O ~ I I ' II I 0 . 0 0 0 20 . 5 I ~ ~ 1 0 2 0 5 o 1 0 o

U G s m ls )Figure 19. C omparison between experimental transitions ( ) and predictions of design equations

( - - - ) f o r a 9 . 5 3 c mpipe and a 1 e P liquid.

i i i i i i

_ 2 - 0

m o o t h I~,~ 1

- , , 171 )0 . 5 I 2 5 I 0 20 50 1 0 0

U G s m / s )

Figure 20. Com parison between experimental tran-sitions ( ) and predictions of design equations

(- -- ) for a 9.53 cm p ipe and a 12 cP liquid.

0.1

0 0 5

0 . 0 2

o . o ~ -

E

0 . 0 0 5 -, ,-i

O . 0 0 2 -

0.001

0 . 0 0 0 50.5 2 5 0 1 0 0

\ o o , : , \

Sin . . ,k\ \ , i m

I I I5 I 0 2 0

U G s ( m / s )

Figure 21. Comparison between experimental transitions ( ) and predictions of design equations ( -- - )for a 2.52 cm pipe and a 70 cP liquid.

A n a p p r o x i m a t e c r i t e r io n f o r t h e in i t i a t i o n o f a t o m i z a t i o n i s a s f o ll o w s :

U G s - - 1 .8 U o s f r o m [ 3 7 ]) . [ 3 9 ]

I f t h e c o n c l u s i o n s p r e s e n t e d i n s u b s e c t i o n 2 ( f ) a r e c o r r e ct , o n e w o u l d e x p e c t t h a t t h e i n i t i a t io no f r e g u l ar w a v e s w o u l d o c c u r a t h i g h e r g a s v e l o c it i e s f o r h i g h p r e s s u r e s t h a n p r e d i c t e d b y [ 3 4] .E q u a t i o n s [ 37 ] a n d [ 3 9] , h o w e v e r , s h o u l d s t il l b e g o o d a p p r o x i m a t i o n s a t h i g h p r e s su r e s s i n c e ,a c c o r d i n g t o [ 3 5 ] , U K - H~ (PG ) -1;: .

C o m p a r i s o n s o f [ 34 ] ( w i t h s = 0 . 0 6 ) , [ 37 ] a n d [ 39 ] w i t h o b s e r v a t i o n s o f t h e g a s v e l o c i t y n e e d e dt o i n i ti a t e r e g u l a r 2 - D w a v e s , ir r e g u la r l a rg e - a m p l i t u d e w a v e s a n d a t o m i z a t i o n a r e g i v e n i nf i g u r e s 1 9 - 2 1 .

Acknowledgements This work i s be ing suppor ted by the She l l Companies Founda t ion and by theDepartment of Energy under Grant DOE FGO2-86E13556.


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602 N. ANDRITSOS and T. J. HAN1L-~TTY

N O M E N C L A T U R E

A = C o n s t a n tB = C o n s t a n tC = Wa v e v e l o c i tyD = C o n s t a n t

D = D i a m e t e rE = C o n s t a n tf = F r i c t i o n f a c t o rh = H e i g h t o f l i q u i d f i l m

H = H e i g h t o f g a s p h a s ek = Wa v e n u m b e rl = Wa v e n u m b e r , d e f i n e d i n [2 5]

p , P = L o c a l p r e s s u r e i n li q u i d a n d g a s p h a s e s , r e s p e c t i v e l ys = S h e l t e r i n g c o e f f i c i e n tt = T i m e

u = L o c a l v e l o c i t y o f li q u i d p h a s eU o s , ULS = S u p e r f i c i a l v e l o c i t i e s o f t h e g a s a n d l i q u i d , r e s p e c t i v e l y

U = L o c a l v e l o c i t y o f g a s p h a s eU K- H = G a s v e l o c i t y, d e f i n e d i n [ 3 5 ]

x = C o o r d i n a t e i n d i r e c t i o n o f f l o wy = C o o r d i n a t e p e r p e n d i c u l a r t o d i r e c ti o n o f f lo w

G reek letters

= Vo i d f r a c t i o n/~ = V i s c o s i t yv = K i n e m a t i c v i s c o s i t y

p = D e n s i t yo = S t r e s scr = S u r f a c e t e n s i o n

z = S h e a r s t r e s s

f c = M o d i f i e d a m p l i t u d e o f s h e a r s t r es s a t t h e i n t e r f a c e , d e f i n e d i n [2 9]4) = Ve l o c i t y p o t e n t i a l o f l i q u i d~b = F u n c t i o n , s o l u t i o n o f [ 22 ]

Subscripts

a = Q u a n t i t i e s s p a t i a l ly a v e r a g e d o v e r l e n g t hG = G a s p h a s e

i = A t g a s - l i q u i d i n t e r f a c eI = I m a g i n a r y p a r t

L = L i q u i d p h a s eR = R e a l p a r t

Superscripts

' = F lu c t u a t in g c o m p o n e n t o f p e r t u r b a t i o n= T i m e a v e r a g e d q u a n t i t i e s= A m p l i t u d e o f f lu c tu a t in g c o m p o n e n t o f p e r t u r b a t i o n

R E F E R E N C E S

A BRA MS , J . 19 8 4 T u r b u l e n t f l o w o v e r sm a l l a m p l i t u d e s o li d w a v e s . P h . D . D i s s e r t a t i o n , U n i v. oI l li n o i s , U r b a n a , I ll .

B A KE R, O . 1 9 5 4 S i m u l t a n e o u s f l o w o f o il a n d g a s .Oil Gas J. 5 3 , 1 8 5 - 1 9 5 .C H o w , V. T. 1 9 5 9Open-channel H ydraulics.M c G r a w - H i l l , N e w Yo r k .


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INTERFACIAL INSTABILITIES IN HORIZONTAL GA S-LIQUID FLOW 60 3

C ot- ra N , L . S . & HArCRATTY, T. J . 196 5 G ene ra t i on o f w aves i n t he con cu r r e n t f l ow o f a ir an dl iquid. A I C h E J l11, 138--144.

C RAm , A . D . D . 1968 Win d -ge ne ra t ed w aves i n con t am ina t ed l i qu id f ilms .J. Fluid Mech.31 ,141-161.

FRAN O S , J . R . D . 1954 W ave m o t ion s and t he ae rod yn am ic d r ag on a f r e e o il su r f ace .Phil . Mag.4 5 , 695 -7 0 2 .

FRANCIS, J . R . D . 1956 Wav e m ot io ns on a f ree o i l sur face .Phil . Mag. 1, 685--688.HANRATTY, T. J . 19 83 In te r fac ia l ins tab i l i t i es cau sed by a i r f low o ve r a th in l iqu id layer. InWaves

on Fluid Interfaces.A c a d e m i c P r e s s , N e w Yo r k .HAN RATTY, T. J . & ENGEN, J . M. 195 7 In t e r ac t i on be tw een a t u rbu l en t a i r s t r e am and a m ov in

w a t e r su r f ace .A I C h E J l3 , 299 -304 .HANRATTY, T. J . & HERSHMAN, A. 1961 In i t ia t i on o f r o l l wav es .A I C h E J l7 , 488 -497 .J E Fn ~ Y S, H . 1 9 2 5 O n t h e f o r m a t i o n o f w a t e r w a v e s b y w i n d .Proc. R. Soc.107, 189-206.L AMa, H . 1945Hydrodynamics,6 t h e d n . D o v e r, N e w Yo r k .LAURINAT, J . E. , HArCRATTY, T. J . & DALLMAN, J . C. 1984 P res su re d ro p an d f i lm heig ht

m e a s u r e m e n t s f o r a n n u l a r g a s - l i q u i d f l o w s .Int. J . Multiphase Flow10, 341-356 .

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