the secondary flow in curved pipes

5/21/2018 The Secondary Flow in Curved Pipes

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S u m m a r y .- - T h e N a v i e r S t o k e s e q u a t i o n s f o r t h e f l o w o f a v i sc o u s i n c o m p r e s s ib l e f lu i d t h r o u g h c u r v e d p i p e s o f

d i f f e r e n t s e c t i o n s a r e s o lv e d i n p o w e r s e ri e s o f t h e c u r v a t u r e o f t h e p i p e . T h e s o l u t i o n i s g i v e n a s f a r a s t h e f i r s t p o w e r

o f t h e c u r v a t u r e f o r t h e c a s e o f a n e l l i p ti c s e c t io n a n d a d i s c u s s i o n g i v e n o f t h e e f f e c t o f t h e a s p e c t r a t i o o f t h e p i p e o n

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

e f fe c t. F o r v a l u e s o f t h e a s p e c t r a t i o n e a r u n i t y t i l e fi r st o f t h e s e p r e d o m i n a t e s a n d t h e r e s u l t a n t e f f ec t i s a n i n c r e a se

o f v e l o c i t y in t h e o u t e r h a l f o f t h e b e n d a n d a d e c r e as e i n t h e i n n e r : f o r l a r g e v a l u e s o f t h e a s p e c t r a t i o t h e s e c o n d t e r m

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

i n c r e a s e i n t h e i n n e r h a l f . T h e s o l u t io n i s al so g i v e n to t h e f i r st p o w e r o f t h e c u r v a t u r e f o r t h e c a s e o f a s q u a r e

s e c t io n . T h i s s h o w s t h a t t h e i n t e n s i t y o f t h e s e c o n d a r y f lo w i n a p i p e o f s q u a r e s e c t io n i s g r e a t e r t h a n t h a t i n a p i p e

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

a c u r v e d p i p e o f c i r c u l ar s e ct i o n w h e n s u c t io n p r o p o r t i o n a l t o t h e c u r v a t u r e i s a p p l i e d a t t h e w a l ls . T h e r e s u l t s h o w s

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

e n t i r e l y e l i m i n a t e d .

1. I n t r o d u c t i o n . W h e n

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

t h a t a s e c o n d a r y f l o w o c c ur s i n p l a n e s p e r p e n d i c u l a r t o t h e c u r v e d c e n t r a l a x is o f t h e p ip e . T h e

t h e o r e t i c a l e x p l a n a t i o n o f t h i s s e c o n d a r y f lo w w a s f ir s t g i v e n b y T h o m p s o n 1. T h e r e m u s t b e a

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

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

t h e i n n e r w a l l. T h e f l u id n e a r t h e t o p a n d b o t t o m w a i ls of t h e p i p e is m o v i n g m o r e s l o w l y t h a n

t h a t n e a r t h e c e n t r a l p l a n e d u e t o v i s c o s i t y a n d t h e r e f o r e r eq u i r e s a s m a l l e r p r e ss u r e g r a d i e n t

t o b a l a n c e i ts r e d u c e d c e n t r i f u g a l f o rc e . C o n s e q u e n t l y a s e c o n d a r y f lo w o c c u r s i n w h i c h t h e

f lu i d n e a r t h e t o p a n d b o t t o m w a l ls of t h e p i p e m o v e s i n w a r d s t o w a r d s t h e c e n t r e o f c u r v a t u r e

o f t h e c e n t r a l a x is a n d t h e f lu i d n e a r t h e c e n t r a l p l a n e m o v e s o u t w a r d s . T h i s in t u r n m o d if ie s

t h e a x i a l v e lo c i ty . T h e f a s t e r - m o v i n g f l ui d n e a r t h e c e n t r a l p l a n e p u s h e s t h e f lu i d i n th e b o u n d a r y

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

w a l l s w h e r e i t i s r e t a r d e d d u e t o i t s p r o x i m i t y to t h e s e w a l ls ) t o w a r d s t h e in n e r w a l l . F a s t e r -

m o v i n g f l u id is th e r e f o re c o n s t a n t l y t r a n s p o r t e d t o t h e o u t e r w a l l a n d r e t a r d e d f lu i d i s c a r ri e d

t o t h e i n n e r w al l. T h e a c c u m u l a t i o n o f t h e r e t a r d e d l a y e r a t t h e i n n e r w a l l r e s u l ts in a d i m i n u t i o n

o f f l u x t h r o u g h t h e p i p e .

E x p e r i m e n t a l i n v e s t i g a t i o n s o f t h e s e c o n d a r y f lo w w e r e m a d e b y E u s t a c e ~, W h i t e a a n d T a y l o P

a n d t h e f i r st t h e o r e t i c a l a n a l y s i s g i v e n b y D e a n 5 f o r t h e c a s e o f a n i n c o m p r e s s i b l e f l u id i n s t e a d y

m o t i o n t h r o u g h a p i p e o f c i r c u l a r c r o s s - s e c t i o n w h o s e a x i s i s b e n t t o t h e f o r m o f a c i r c u l a r a r c

o f s e v e r a l r e v o l u t i o n s . D e a n e x p a n d e d t h e v e l o c i t ie s a n d p r e s s u r e a s a p o w e r s er i es o f t h e

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

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

L he v e l o c i t ie s a l o n g t h e p i p e t o v a n i s h .

8~1) A


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T h e f ir st p a r t o f t h i s p a p e r e x t e n d s t h e s a m e m e t h o d t o d e a l w i t h p i p e s o f e l li p t i c s e c t i o n a n d

a d i s c u s s i o n is g i v e n o f t h e e f fe c t o f a s p e c t r a t i o o f t h e p i p e o n t h e i n t e n s i t y o f t h e s e c o n d a r y

f lo w . T h e s e c o n d p a r t a p p l ie s t h e m e t h o d t o p i p es o f r e c t a n g u l a r s e ct io n . A n a p p r o x i m a t e

s o l u t i o n f o r t h e p a r t i c u l a r c a s e of a s q u a r e s e c t i o n i s g i v e n e x p l i c i t l y a n d s h o w s t h a t t h e f lo w

is m o r e i n t e n s e t h a n i n a p i p e o f c i r c u l a r s e c ti o n . T h e l a s t p a r t i n v e s t i g a t e s t h e e f fe c t o f a

p a r t i c u l a r d i s t r i b u t i o n o f s u c t i o n a r o u n d t h e w a l l s o f a c i rc u l a r p i p e o n t h e f lu x t h r o u g h i t.

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

o v e r c o m e . B o t h t h e f ir s t a n d l a s t p a r t s c o n t a i n D e a n ' s r e s u l t s a s s p e c i a l c a se s .

2 . T h e S e c o n d a ry F l o w i n a P i p e o f E l l i p t i c S e c E o n . - - 2 .2 . T h e S tr e a m F u n c t i o n . - -

Ou~ er

A B , -

t

T h e f o l l o w in g r i g h t - h a n d e d , o r t h o g o n a l s y s t e m o f a x e s i s e m p l o y e d : 0 5 a l o n g th e m a j o r a x i s

o f a s e c t i o n ; 0 v a l o n g t h e m i n o r a x is a n d 0 a l o n g t h e c u r v e d c e n t r a l ax i s . L e t t h e v e l o c i t y

c o m p o n e n t s a l o n g th e s e a x e s b e (u , v , w ) ; C t h e c e n t r e o f c u r v a t u r e o f t h e a x i s 0 ~ ; O C = 2 / ~

w h e r e n i s t h e c u r v a t u r e o f 0 ; 2 a , 2 b t h e m a j o r a n d m i n o r a x es o f a s e c t io n . L e t b / a ---- ~, th is

w e s h a l l c a l l t h e a s p e c t r a t i o o f t h e p i p e .

T h e f lo w is d e t e r m i n e d b y t h e N a v i e r - S t o k e s ~ e q u a t i o n s o f m o t i o n

Ov

~-~ - - v c u r l v = - - g r a d

( { v ~ + p i p ) - - ~

c u r l c u r l v .

a n d t h e e q u a t i o n o f c o n t i n u i t y

d i v v = 0 . . . . . . . . . .

t o g e t h e r w i t h t h e b o u n d a r y c o n d i t i o n t h a t v = 0 a t t h e w a il s.

2 )

(2)

M a k i n g t h e n o n - d i m e n s i o n a l s u b s t i tu t i o n s x =

~ la , y = n / Z a , z = /a ; u = v # /a , v - = v ~ la ,

w = vz~ /a , p = pv '~ /a 2 . ~ a n d p u t t i n g Ou/Oz = Ov /Oz = Ow/Oz = 0 w e o b t a i n f r o m e q u a t i o n s

( 1 ) a nd ( 2 )

Og f~ ~ ~ a N 2

~ - x + x ~ y -

Off; ~ ~z~

a y d X ~ l

~ a~ w

~ax

- ~ { 2 +

~x

~z -t - ,~ ~y2 ~ ~x ~y

1 @ ~2~ 1 02 ~ a ( 0 ~ 1 ~ / )

Z ~ y + o x ~ ~ ~ x ~ y + l + ~ a x g-x ~ g }

1 @ 02 f f 1 02 z~ ~ a ~ n2a2 ~

l + ~ a x O z ~ y d s + z ~ y ~ + l + ~ a x o x

l + ~ a x )

~

~ax a + ~ ~ l + ~ a x ) ~ = 0 .

T h e b o u n d a r y c o n d i t i o n is t h a t # ---- ~ = z~ = 0 f o r x + y ~ = 1 .

2

. . 3 )


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D i f f e r e n t i a t i n g t h e t h i r d e q u a t i o n i n ( 3) w i t h r e s p e c t t o z w e h a v e

a ' ~ / a z ~

= 0 ; i t f o l l o w s f r o m

t h e f ir st a n d s e c o n d e q u a t i o n s t h a t

a } / a x

a n d

@ / a y

a r e i n d e p e n d e n t o f

z . @ / a z

i s t h e r e f o r e

c o n s t a n t . T o s o l v e e q u a t i o n s (3) t o t h e fi rs t p o w e r o f ~ w e p u t

w h e r e

~1 = U o + n a R ~

= ~o + ~ a R

R - -

1 q - 4 2 0 z

= Vo -Jr- nctR ~v~ l

. . . . ( 4 )

T h e t e r m s i n d e p e n d e n t o f ~ y i e ld t h e e q u a t i o n s o f m o t i o n f o r fl ow t h r o u g h a s t r a i g h t e l li p ti c

p i p e ,

v i z . ,

~Po ~Po

U o = V o = 0 ; ~ x - - B y - - 0

a n d ~Wo 1 ~wo 2(1 + 1 /4 ~)

8x 2 7~ 4 2 8 y ~ - -

w o = 1 - - x ~ - y ~ .

i t h t h e s o l u t i o n

E q u a t i n g c o e ff ic i en t s o f ~ w e o b t a i n

Z o ~ P i

1 ~2ul 1 ~v l

4 - - ~x + ~ ~y-~ - - 2 ~x ~y

1 ~ ib~ ~v ~ 1 8~ul

o = - ~ +

~x~ 4 ox ~y

. . . . . . . . . . 5 )

. . . . . . . . 6)

OW6 Vl 8Wo 3 ~ w l 3~Wl

u ~ 4 ~y - ~x~ 4~ ~y~

0 = - - 2 1 + x + ~ + 4 2 ~y~ + ~x

Ox + 4 a y - 0

T h e l a s t e q u a t i o n m a y b e s a t is f i e d i d e n t i c a l l y b y a s t r e a m f u n c t i o n ~o d e f in e d b y t h e e q u a t i o n s

1 ~ p

u~ - - t2 ~y

. . . . . . . . . . . . . . . . . . (7)

v~ - - 8x

E l i m i n a t i n g P l b e t w e e n t h e f i rs t a n d s e c o n d e q u a t i o n s o f (6) w e o b t a i n

~4~0 2 3~o 1 3 ~ _ (1

- - x ~ - - y 2 ) y . . . . . . . . .

(8)

~x--~ + 4~ 0 Z g y ~ + 4~ ~y~

w i t h t h e b o u n d a r y C o n d i t i o n s ~ o / a x = ~ / O y = 0 f o r x ~ - F Y~ = 1 . T h e s o l u t i o n i s

~o = (1 - - x - -y2 )2(B ~ + B 2 x 2 + B 3 y 2 ) y . . . . . . . . . . (9)

W her e Bz = 44(375 + 820 2 ~ + 1 ,1142 ~ + 21248 + 3948 ) /360(5 -- 24 ~ + 4~)G(2)

B ~ = - - 4~ 7S + 24 ~ + 34~) / 360G 4)

B ~ = - - 4~ 1S + 264 ~ + 394~) / 360G 4)

a n d G (4 ) = 3 5 + 8 4 4 ~ -t- 1 14 4 ~ + 2046 -l- 348 . . . . . . . . . . . (10)

3

371) A 2


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2.2 . N a t u r e o f t he S e c o n da r y F l o w . - - I t a p p e a r s f r o m e q u a t i o n s ( 4 ) t h a t ~ a R ~ is th e d y n a m i c a l

p a r a m e t e r o f t h e s e c o n d a r y f lo w . I f W 0 b e t h e v e l o c i t y a l o n g t h e c e n t r a l a x i s f or t h e f l ow

t h r o u g h a s t r a i g h t e l l i p t ic pi p e , f r o m e q u a t i o n (5)

W o - v R / 2 a ,

i .e . , R =

2 a W o / v . R

m a y t h e r e f o re

b e i n t e r p r e t e d a s t h e R e y n o l d s n u m b e r f o r s t e a d y f lo w t h r o u g h t h e u n b e n t p i p e . F r o m e q u a t i o n s

(7) and (9)

~ 4 ~ R 2

u =-= Z~ ( 1 -- x ~ - - y2)[(1 - - x ~ - - y 2 ) ( B , + B2 x ~ + 3 B . y ~)

4Y~(B~ + B2 x~ + B~Y2)] (11)

2 ~ v R ~

v - - ,~2 ( 1 7 x 2 - y ~ ) [ 2 ( B ~ + B ~ x 2 + B ~ y 2 ) - B ~ ( 1 - x ~ - y ~ ) ] x y

v v a n i s h e s a l o n g b o t h a x e s : t h e r e f o r e a f lu i d p a r t i c l e o r i g i n a l l y in t h e u p p e r ( o r l o w e r) h a l f o f

t h e p i p e w i l l r e m a i n i n i t d u r i n g t h e s u b s e q u e n t m o t i o n . A l s o

~ v R ~

[ u T l ~ o , o =

z~ - B > 0

a n d

[u ] (0 ,~-~/= - - , t8 - R2(B~ + Ba)0 < 0 fo r ~ sm a l l and p os i t iv e .

T h e f l u i d n e a r t h e t o p a n d b o t t o m w a l ls is t h e r e fo r e m o v i n g i n w a r d s w h i l s t t h a t i n t h e c e n t r a l

p l a n e is m o v i n g o u t w a r d s . T h e r e m u s t e x i s t t w o p o i n t s, s a y S a n d S , o n t h e m i n o r a x is a t w h i c h

u v a n i s h es . S i n ce v v a n i s h e s e v e r y w h e r e a l o n g t h e m i n o r a x i s t h e s e m u s t b e s t a g n a t i o n p o i n t s

o f t h e s e c o n d a r y f lo w . T h e f o l l o w i n g d i a g r a m s h o w s t h e e s s e n t i a l f e a t u r e s o f t h e f l ow c o n s i s t i n g

o f t w o o p p o s e d v o r t e x m o t i o n s c e n t r e d a b o u t t h e s t a g n a t i o n p o i n t s S a n d S .

uber

nner

A n o b s e r v a t i o n o f

a x i s. S u b s t i t u t i o n o f

i s t h e v a l u e o f p f o r x

h a v e

( p l ) - p l - -

i .e ., P I

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

u l a n d v l i n t h e f ir s t t w o e q u a t i o n s o f (6) sh o w s t h a t p~ - - P l ( w h e r e P l

----- 0 ) i s odd in x an d e ven in y . P u t t in g y = 0 and x = : J: 1 in tu rn we

- - P , ]

P l ) A + P l ) B .

t h e p r e s s u re a l o n g th e m i n o r a x i s is e q u a l t o t h e a r i t h m e t i c m e a n of t h e p r e s s u re s a t A a n d B .

T h i s r e s u l t i s v e r if i ed e x p e r i m e n t a l l y b y R i c h t e r 7 f o r t u r b u l e n t f lo w a n d a s s u m e d f o r l a m i n a r

f l o w b y K e u l e g a n a n d ] 3e ij s.

2 .3 . Ef fec t o f the Aspe c t Ra t io o74 the I74 te74s ity o f the F lo w . - - I t a p p e a r s f r o m e q u a t i o n s ( 1 0 )

a n d ( 1 1 ) t h a t f o r s m a l l Z, u = O (Z 2) a n d v = 0 ( a3 ) . F o r p i p e s o f s m a l l a s p e c t r a t i o t h e s e c o n d a r y

f l o w i s t h e r e f o r e g r e a t l y r e d u c e d . F o r l a r g e a , u - - 0 ( 1 / a ~) a n d v ---- 0 ( 1 / ~ ) ; t h e i n t e n s i t y i s

t h e r e f o r e d i m i n i s h e d f o r l a r g e a s p e c t r a t i o s . I n t h e c a s e o f ,~ = co, t h a t i s f o r t h e f l o w a l o n g t h e

c h a n n e l b e t w e e n t w o i n f in i t e c o n c e n t r i c c y l in d e r s , t h e s e c o n d a r y f lo w v a n i s h e s a l t o g e th e r .

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

4


6/19

n e a r t h e m a n d t h u s i n d u c e t h e s e c o n d a r y fl ow , a r e n o w a b s e n t . T h e f o ll o w i n g t a b l e s h o w s t h e

v a r i a t i o n o f ( ~) y = 0 a c r o s s , th e c e n t r a l p l a n e f o r v a r i o u s v a l u e s o f t h e a s p e c t r a t i o i n t h e c a s e w h e n

~ a = 0 . 2 a n d R = 1 00 . T h e r e s u l t s a r e s h o w n g r a p h i c a l l y i n F i g . l .

T A B L E 1

Ua=l/2

q ~ . = 1

5a=2

--1 2 / 3

0.6

1.9

2.4

1 / a

1-9

5.3

6.3

2.6

6.9

8.1

2/a

1/a

1 9

5 3

(; '3

4 3

2 9

0 0

0 '6

1 9

2 4

uz=a 0 1 7 4 3 5-5 1 7 0

5a=4 0 1 1 2- 9 3 7 1-1 0

5a=~ 0 0 0 0 0

I t i s n a t u r a l t o t a k e a s a m e a s u r e o f t h e i n t e n s i t y o f t h e s e c o n d a r y f l ow t h e t o t a l v o r t i c i t y f~

i n e i t h e r t h e u p p e r o r l o w e r h a l f o f t h e p i p e ( t h es e a r e e q u a l a n d o p p o s i te ) .

; i

4 ~ a v R ).2(125 + 3102,, -- 42 82 * -~- 82). 6 + 15,l s)

- -

52 5 (5 + 22, , + 24)G(2)

I f P ~ i s t h e v a l u e o f f~ f o r 2 = 1 , i .e . f2 is t h e t o t a l v o r t i c i t y i n t h e u p p e r h a l f o f a p i p e o f c i r c u la r

s e c t i o n

Y2 12826(125 + 3102, , + 428 2 ~ q- 8216 q- 1528)

0-~ -- 15( 5 + 22 + 24)(1 + 22)2(35 + 842,, + 11424 + 20 2 G + 32 s)

1 2 )

a n d f21 = 1 1 20 \ 0z / . . . . . . . . . . . . . . .

T h e v a r i a t i o n o f ~ /L )~ w i t h 2 is s h o w n g r a p h i c a l l y i n F i g . 2 f ro m w h i c h i t a p p e a r s t h a t L) a t t a i n s

a m a x i m u m v a l u e o f 3 . 1 ~ w h e n 2 = 2 . 2 . T h e d i m i n u t i o n o f f lu x t h r o u g h a c u r v e d pi p e i s d u e

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

s e c o n d a r y f l o w ; i t t h e r e f o r e a p p e a r s p r o b a b l e t h a t t h e d i m i n u t i o n is g r e a t e s t f o r v a l u e s o f 2

i n t h e n e i g h b o u r h o o d of 2 . F o r 2 = 6 t h e f l o w i s o f t h e s a m e s t r e n g t h a s f o r 2 = 1 ; f u r t h e r

i n c r e a s e in 2 d i m i n i s h e s t h e i n t e n s i t y o f t h e f l o w a s y m p t o t i c a l l y t o z e r o :

2 .4 . T h e A x i a l V d o c i t y . - - S u b s t i t n t i n g

f o r w0 i n t h e f o u r t h e q u a t i o n o f (6 ) w e h a v e

~2ze'2 1 ~2w , , 2 ( 2 + 1 )

~x,, + 2,, 3y,, ~ x

t h e s o l u t i o n o f w h i c h is

1 2 2 , ,

= - 1 + 3 2 8 1 - x , , - y , , ) x . . . . . . . 1 3 )

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

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

t h i r d e q u a t i o n i n ( 6), i s a p o l y n o m i a l of d e g r e e n i n e i n x a n d y t o g e t h e r , o d d i n x a n d e v e n i n y .

5


7/19

Since w~ is also odd in x and even in y, it follows th at the dim inut ion in flux is zero to th e first

order of the curva ture of the pipe. This polynomi al represents an increase of veloci ty in the

outside of the b end an d a decrease in the inside of the bend. The effect of curva ture of the

pipe on the axial velo city is therefore repres ented b y two terms w~ and w2 of opposite sign.

Of these, w~ represe nts the effect generally associated wit h th e flow thr oug h a curve d pipe a nd

w2 the effect generally associated with the flow along a curved channel. In fact, bot h effects are

always present together. For values of ~ near unity the w~ term pr edomina tes; for large

w~ is o5 order 1/~ ~ bu t w2 tends to the limiti ng value -- x(1 -- x~). The following tab le gives

the va riat ion in (z~)y=0 across the majo r axis when R = 100, xa = 0.2 .

TABLE 2

'Wo

~ a = l

w a= 2

wa=4

w a = m

- - 1 - - 2 / 3

2 7 8

2 1 7

1 5 4

1 9 3

2 2 . 9

3 0 . 3

- - 1 / 3

4 4 5

3 6 0

3 0 6

3 4 5

3 8 3

4 6 5

5 0

5 0

5 0

5 0

5 0

5 0

4 4 - 5

5 3 0

5 8 4

5 4 5

5 0 7

4 2 . 5

2 / 3

2 7 8

3 3 9

4 0 2

3 6 3

32 7

2 5 3

0

0

0

0

0

0

These results are shown graphically in Fig. 3. As Z increases the po int of ma xi mum veloci ty

shifts from the outside of the bend over to the inside of the bend. The transition from one type

of flow to the other may be delayed by increasing R since w~ is preceded by an extra factor RL

2.5. T h e S p e c i a l C a s e ~ ---- 1 ; t h e C i r c u l a r S e c t i o n . - - D e a n s results for a pipe of circular section

are readily obtained by put ting ~ = 1 and transformi ng to polar co-ordinates by means of the

substitutions

x = r c o s 0 u , = , O 0

= = r s i n 0 u o = D r

.. .. (14)

where

u , , U o

refer to velocities in the radial and transverse directions respectively.

R i = - - ~ / ~ z from eq ua ti ons (10), B1 ~ B., B~

Then

~5,, and from equation (9)

~P - - 1 1 5 2 ( 1 - -

# ) 2 ( 4 - - r ~ ) r

s i n 0 .

1 5 )

Therefore

~ R12

u . - 1 - - r c o s 0

g4 ? R I 2

u o - = - - 1152 (1 -- r~)(7 -- 23r ~ + 4) sin 0

m)

6


8/19

3. The Secondary Flow in a Pipe of Rectangular Section.--3.1. The Stream Fun ct ion. --

Ouger wa l l

t

~ ~ n n e r

~ ] a

wall

~-- 2

U s i n g t h e s a m e n o t a t i o n a s in t h e p r e v i o u s se c t io n , e q u a t i o n s 1) to

i d e n t i c a ll y w h i l st t h e b o u n d a r y c o n d i ti o n s a re n o w

= ~ = @ = 0 f o r x = 4 - 1 a l l y )

W e p u t

y = 4 - 1 (a l l x )

3) m a y b e a p p l ie d

= Uo @ xaRR~uz ~ = Vo + naRR2vz

= Po + xaR~fl~ ~ = RR Wo + xaRR~w~ + ~aw~)

1 6 ~ s ec h ~ @ . . . . . 17)

w h e r e R ~ = - - x-w- 2 ~ az . . . . . . . .

T h e t e r m s i n d e p e n d e n t o f ~ y i e l d t h e e q u a t i o n s o f m o t i o n f o r f lo w t h r o u g h a s t r a i g h t r e c t a n g u l a r

p i pe , viz.,

a P o a P o

U o = V o = 0 ; ax ~ y - - 0

a n d a~Wo 1 ~w o ~3

ax ~ + ~ ~y~ _ 1 6 ~ co sh 2-~

w i t h t h e s o l u t i o n9

w 0 = ~ P ,~ c o s h 2 n + 1 ) ~ - - c o s h 2 n + 1 ) ~ c o s 2 n + 1 ) ~ y

- - 1 ) ~ ~

w h e r e P ,, = i2 n + 1)3 c o s h ~ s e c h 2n + 1) ~ . . . . . . . . . 18)

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

t e r m , viz.,

Wo = c o s h 2~ - - cos ~- . . . . . . . . . . .

E q u a t i n g c o e f f i ci e n ts o f ~ w e o b t a i n

ap~ 1 a~u~ 1 a~vl

w ax k ~ a y e. ~

dx dy

~ul 1 av~ _ 0

x k ~ y

1 a p l a ~ v l 1 a ~ u l

Z ~y + ax z a ~x ~y

( 2 0 )


9/19

T h e l a s t e q u a t i o n m a y b e s a t is f i e d i d e n t i c a l l y b y a s t r e a m f u n c t i o n ,/; d e f i n e d b y t h e e q u a t i o n s

U~ = ~

O~ . . . . . . . . . . . . . . 2 1 )

E l i m i n a t i n g 151 b e t w e e n t h e f i rs t a n d s e c o n d e q u a t i o n s w e o b t a i n

a ~0 2 O ~o + ~ - - c o s h - - c o s h ~ X ~ ~

Ox + ,V ax ~ Oy~ Oy~ ~ 2 ; t . / s i n ~ y

(22)

w h e n c e

7 4 a r ~ 9 4 : N ~ .

+ E C , , i n h

n, t=y

+ D , , y co s h ~2 .~y ) c o s n = x

~ I 7 ~ X ~ 7 ~ X

+ 1 -~ 4 74 9 ~ x 2 c o s h ~ - - - 5 1 2 ,a ~ c o s h ~ c o s h ~ + 1 4 4 ~ '

+ 7 2Z = c o s h i J s i n u y . . . . . . . . . . . .. . (23)

I n s e r t i o n o f t h e b o u n d a r y c o n d i t i o n s le a d s t o t h e f o l l ow i n g e q u a t i o n s

- 2 A , , s i n h T q - B ,, s i n h T q - T c o s h

G , f l I 1 8 ~ 2

. + 1 4 4 a ~

C ,, s i n h n , ~ + D , , c o s h n , ~ = 0

~ A , , c o s h q - B , s i n h - - -4,

,,=, T 1

q - E ( -- 1 ) { C f lc s h n ;b r Y + D ( l c s h n ' ~ r Y +

( - - 1 ) ~ A ,~ c o s h ~ - - [- B ,~ x s i n h

c o s h I + ( 9 ~ = - - 1 2 8 ,t =) s i n h x = 0 . . . .

o . . . . . . . . .

1 4 4 n ~ ( 9 a = - 1 8 4,t ~) c o s h ~ - 1 1 2,t

~ 0 o

c o s

n ~ y

i )

i i )

(i i i )

w h e r e G p

~.' [

1 4 4 ~ 9 ~ % ~ c o s h 7 - - 5 1 2 't 2 c o s h ~ c o s h

=1 p= l

q - 1 4 4 i + 7 2 ~- ', c o s h - ~ l

= 0 , p ~ - 1 .

= 0

( iv)

8


10/19

S o l v i n g e q u a t i o n s (i) t o ( i v ) w e f in d

( 2 2 p ~ ) 4 ; t p pz~ ~ s i n h n 2 ~

1 -{- 2 ~ s in h ~ - Bp - - ~ s in h ~- ,~=~ ( - - 1) +p (p~ + ~%~2)2D,,

1 4 4 ~ 4 2 8 a s i n h ~ - + 1 1 2 s i n h ~ + 9 ~ ( 1 + c o s h

a n d

s inh 2pZ

2 p ~

4X s i n h p ~ ~ ( - - 1) + p

D p - - p u 2 ~=1

1) p s inh p2 ~ ~5

+ - ~ [ 3 6 ~ ~

(24)

n 3

s inh ~- B,~

n 2 + ~p~)~

7 3 + 1 0 1 ~ p ~ + 6 4 z~ p ~

(1 + z~p~)~ s i n h ~

6 4 ~ s i n h ~ ~ ' c o s h ~ 1

2 5 )

9 ~ 1 + 4 x ~ p ~ 2 ~ ( 1 + z p ~ ) 2 j

3 .2 .

T h e S p e c i a l C a s e 2 = 1 ; t he S q u a r e S e c t i o n . - - W h e n ,t ----- 1 we h av e th e ca se o f a sq ua re

S o l v i n g e q u a t i o n s (2 4) a n d (2 5) a n d r e t a i n i n g o n l y A ~, B , C~ a n d D , as ae c t i o n .

f ir s t a p p r o x i m a t { o n w e f in d

A 1 = 0 0 3 3 3 7

s

L e t U ~ b e t h e v e l o c i t y a t

U s

C1 = - - 0 0 0 1 8 6 4

- - 0 0 2 6 2 1 D 1 = 0 0 0 1 8 5 8

. . . . . . . (26)

RR)~=I

= - - s ec h ~.

t h e c e n t r e o f a p ip e o f s q u a r e s e c t i o n ; t h e n

= 0 0 5 0 6 2 ~

v R s 2 .. . . . . . . . . . . . . . .

(27)

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

e q u a t i o n ( 1 6 )

v R 2

U c - 2 8 8

T h e r e f o re U . ( ~ f

U ~ = 1 4 . 5 8

= 2 . 4 7 . . . . . . . . . . . . . . . . . (28)

T h e s e c o n d a r y f lo w is t h e r e f o r e m o r e i n t e n s e i n a p ip e o f sq u a r e s e c t io n t h a n i n a p i p e o f c i r c u l a r

s e c t i o n .

4 . T h e E f f e c t o f S u c t i o n o n t he F l u x t h r o u g h a C u r v e d P i p e o f C i r c u l a r S e c t i o n . - - 4 . 1 . F i r s t

A p p r o x i m a t i o n to th e A x i a l V e l o c i t y .- -

Outer ~ ~

wal l ~ ' ~ ~

_ . A 13

nne r

wal l

~ C

I

W e t a k e a s c o - o r d i n a t e s y s t e m p o l a r c o - o r d i n a t e s i n t h e p l a n e o f a s e c t io n a n d 0 a l o n g t h e

c e n t r a l a x i s t h e n o r m a l s u c t io n v e l o c i t y a t t h e w a l l is t a k e n t o b e ~ a U c o s 0 , t h a t i s,

9

371) A*


11/19

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

x = r / a , z / a ; u = v ~ / a , v = v ~ / a , w = ~ z ~ / a , 25 = p ~ / a ~ . ~

a n d p u t t i n g

~ u /O z = a v / a z : = a w / a z = o

w e o b t a i n f r o m e q u a t i o n s (1 ) a n d (2 )

~ ~ ~7 ~ ~a co s 0z~~

~2 g -x + x ~ O x ~ 1 + ~ a x co s O - -

M a k i n g t h e n o n - d i m e n s i o n a l s u b s t i t U t i o n s

W e p u t

a x x 3 x 30 x ~ ~ 0 + x ~ 302

s n o

- ~ 1 4 - ~ 7 ~ o s o ~ + x - x g O

~ ~7 ~5 ~1~7 ~ a s i n 0z~2 @ a ~ 1 ~ ~ 1 O ~a

a g -x + x ~ + 7 + l ~ 2 ~ x ~ d ~ O = - x Oo + -g ~ + x b x~ - x ~ x O-~o

1 ~ ~ a co s 0 / ~ ~ ~7

+ x ~ O 0 q 1 + ~ a x c o s O i , , ~ + x - - - - -

~z~ ~5 ~z~ ~az~ ( ~ c o s 0 - - ~ si n 0) = - - 1 @ a~z~ 1 ~z~

g ~ + x ~ - - 1 + ~ a x c o s O

1 +

~ a x c o s O a z

~ - ~ 4 - x a x

+ x ~ aO = q - 1 + ~ a x c o s O

~ c o s 0 - - ~ 6 s i n 0 - - ( l + ~ a x c o s 0 ) -

. _ ~ I ( l + ~ a x c o s 0 ) x ~

+

( l + ~ a x c o s 0 ) ~ = 0

X / . . . . . . .

T h e b o u n d a r y c o n d i t i o n s a r e t h a t v / a . ~ = ~ a U c o s O, f f = ~ = 0 f o r x = 1 .

x ~ O

2 9 )

3 o )

1 a v l 1 a 2 u l

xm ~ x 2 a 0 ~

Wo~ 3p , a~vl 1 8v l v ,

4 s i nO = - - x ~ O .

q - ~ q x ~ x x ~

awo a=w~ 1 Ow~ 1 a=w,

u , a x - - a x ~ + ' ; 7 a T + x ~ a 0 =

~ 7 1

_ a x u ~ ) + - 0

1 a = u ~ 1 a u ~

x a x a 0 x 2 a 0

1 0

. . . . . . a ] )

. . a g ~ )

w i t h t h e a x i s y m m e t r i c s o lu t i o n

W 0 ~ 1 - - - X 2 . . . . .

E q u a t i n g c o e f f i c ie n t s o f ~ w e f i n d

Wo~ ~ P l 1 ~ v l

4 c o s 0 - - ~ x x ~ x ~ 0

o P o _ ~ P o _ 0

~ x ~ 0

o --=- vo = 0 ;

a n d O2Wo 1 aWo 1 a~Wo

O x~ + x ~ x + x ~ ~ 0 ~

= Po q - ~aR~2P~ + ~2a~R~4P~

w h e r e R 1 - - 2 ~ z ' . _

W e h a v e a p p r o x i m a t e d t o ~ b y n e g l e c t i n g a t e r m ~ a w , ' i n c o m p a r i s o n w i t h ~ a R d w , ; t h i s h a s

b e e n s h o w n v a l i d fo r an a s p e c t r a t i o o f u n i t y i n s e c t i o n 2 .4 . T h e t e r m s i n d e p e n d e n t o f ~ y i e l d

t h e e q u a t i o n s f o r t h e f l o w t h r o u g h a s t r a i g h t c i r c u la r c y l i n d e r w i t h t h e s u c t i o n v e l o c i t y e v e r y -

w h e r e z e r o ,

v i z . ,


12/19

T h e l a s t e q u a t i o n m a y b e s at is f ie d i d e n t i c a l l y b y a f u n c t i o n f x ) d e f in e d b y t h e e q u a t i o n s

= f * ) c o s 0

3 3 )

d

v , = - - -d x [ x f x ) ] s i n 0 .

E l i m i n a t i n g p l b e t w e e n t h e f ir st a n d s e c o n d e q u a t i o n s o f (3 2) w e h a v e , p u t t i n g d f / d x = h x ) ,

x ~ d3h d~h dh

) - ~ + 6x ~ ) - ~ + 3x )-~ - - 3 h = x ~ - - x ~ . . . . . . . . (34)

B C 1 1 (4x3 x~

w i t h t h e s o l u t i o n h x ) = A ~ x + x + -X + ~ - 2 - -

C 1

w h e n c e f x ) = A x 2 + B l o g x + x ~ + D + 1 ~ ( 6 x ' - - x 6) . . . . . (35)

T h e b o u n d a r y c o n d i t io n s a re t h a t v~ - -- - O , u z = a U / v R ~ 2 . c o s 0 f o r x = 1 ; i . e . , d x f ) / d x = O ,

f x ) = a U / v R ~ 2

f o r x = 1. S i n c e u~ a n d v~ a r e b o t h f i n it e a t x = 0 , B = C = 0 . P u t

7 = 1 9 2 a U / v R ~ 2, t h e n A = - - ~ v ~ (9 + 3 7 ), D - - ~ ( 4 + 9 7 ), w h e n c e

u~ = ~ ,~ [ (4 + 97) - - (9 + 37) x 2 + 6x ~ - - x ~] c os 0

3 6 )

]

v~ ~ , ~ [ (4 + 9 7 ) 9 ( 3 + 7 ) x ~ + 3 0 x ~ - - 7 x ~ s i n 0

P u t t i n g w ~ = g x ) c o s 0 i n t h e t h i r d o f e q u a t i o n s (3 2)

d ~ g d g

x~ -d~ + x ~ - - g = - - 2 x y x )

= v { v I x ~ - - 6 x 7 + 9 + 3 7 ) x ~ - - 4 + 9 7 ) x 3 ~ . . . 3 7 )

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

g x ) = 0

f o r x = I . S o l v i n g w e f i n d

g x ) ~

[ 1 9 + 8 0 7 ) - - 2 1 + 1 0 7 ) x 2 + 9 x * - x ~ ] l - -

x ~ ) x . . .

3 8 )

4 . 2 . T h e S e c o n d A l b ~ r o x i m a t i o n t o th e A x i a l V e l o d t y . - - S i n c e w ~ i s o f t h e f o r m g x ) c os 0 , i t

f o ll o w s t h a t t h is t e r m m a k e s n o c o n t r i b u t i o n t o t h e f l ux t h r o u g h t h e p i p e ; t h e d i m i n u t i o n .in

f l u x ( w h i c h w a s d e d u c e d o n p h y s i c a l g r o u n d s i n th e I n t r o d u c t i o n ) is t h e r e f o r e z e r o t o t h e f i r st

p o w e r o f t h e c u r v a t u r e . E q u a t i n g c o e ff ic ie n ts o f n~ i n e q u a t i o n s (32) w e f i nd t h a t t h e e q u a t i o n

o f c o n t i n u i t y m a y b e s a ti s fi e d b y a fu n c t i o n f ~ ( x ) d e f i n e d b y t h e e q u a t i o n s

= A x ) c o s 2 0

l d

v~ = - - 2 ~ [x f~ x)] s i n 20 .

Ow l OWo v~ Ow~ 3~ w ~ 1 3w~ 1 O2w~

a n d U l ~ + U s ~ - + x 00 - - ~ x 2 + x ~ + x ~ 005

~2w~ Ow~ 3~w~ d~x g d

~ x 2 + x ~ -x + x ~ 0 03 - f c o s ~ 0 + x ~ - x X f ) s i n ~ O - 2 x f l c o s 2 0

. e . ,

02w ~ 0w 2 102w 2 1 [ dg d f

i .e . , x - - ~ + - ~ + x 0 0 ~ - - 2 X f d x + X d x g + f g

1 x / - x / g 4 x y

+ ~ c os 20 ~ g - -

f r o m w h i c h i t f o l l o w s t h a t w 2 is o f th e f o r m

r e x ) + n x ) c o s

20 .

1

. . 3 9 )

T h e i n t e g r a l of t h e s e c o n d


13/19

t e r m o v e r a c r o s s - s e c t i o n i s ze r o , c o n s e q u e n t l y t o f in d t h e d i m i n u t i o n o f f lu x J t w i l l s u ff ic e t o

d e t e r m i n e r e (x ) . E q u a t i n g t e r m s i n d e p e n d e n t of 0 i n e q u a t i o n ( 39 )

i . e . ,

w h e n c e

d~m d m 1 [ _ddg

x - d ~ + d x - - 2 x f +

x ~ ) - - 2 d x

dm _ l fg

d x

X U x g +

t h e c o n s t a n t o f i n t e g r a t i o n b e i n g z e r o t o p r e s e r v e a f in i te v a l u e o f d m / d x a t x = 0 .

c o n d i t i o n o n m is t h a t

r e (x ) = 0

fo r x ---- 1.

1 ~

T h e r e f o r e

r e (x ) = ~2 , f ( t ) g ( t ) d t

4 0 )

T h e b o u n d a r y

1

cfj~/ 76 + 491 7 + 720 y ) t (331 + 149 7y + 1050 7 ) t 3

A 1

+ (594 + 17207 + 360y~) t5 - - ( 5 6 9 + 8 9 0 v + 3 0 y ~ ) f

+ (314 + 189 7)t 9 - - (99 + 13~)t ~1 + 16# 3 - - #5 }

d t

w h e r e A = 8 0 ( 1 1 52 ) L

L e t F s , F~ b e t h e f lu x e s t h r o u g h t h e s t r a i g h t a n d c u r v e d p i p e s r e s p e c t i v e l y ; th e n

~ I ~ , R ~ v }

F~ = 2~ a ~ f0 [ 2 a (1 - - x ~ )x + ~ - a . ~ a 2 R ~ S x m ( x ) d x

= 7 t a ~ R ~ + ~ 2 a R ~ 5 ~ o x m ( x ) d x

(41)

g

a n d Fs ----- ZI av R ~ .

T h e r e f o r e

F , _-- 1 + 4 ~l

F--~ 7~ z~a~R~4 o x m (x ) d x

= 1 - - \ 1 - 1 ~ /

(42)

4 .3 .

V a r i a t io n o f F l u x w i t h S u c l i o n . - - D e a n ' s 1

v a l u e f o r t h e f lu x it o b t a i n e d b y p u t t i n g 7 = 0 ,

t h a t i s t h e s u c t i o n is e v e r y w h e r e z er o. T h e n

F ~ _ 1 z a R ~ 2 ) 2

4 3 )

F , - 0 . 0 3 0 5 8 , , 1 1 5 2 2 . . . . . . . . . . .

P u t Y (7 ) = 0 . 0 3 0 5 8 + 0 . 3 5 1 8 7 + 1 .1 7 57 2 . T h i s e x p r e s s io n h a s a m i n i m u m v a l u e o f 0 - 0 0 4 2 4

w h e n 7 = - - 0 . 1 4 9 7.

T h e r e f o r e

2

a n d i s a t t a i n e d w h e n 7 -- -- - - 0 1 4 97 , i . e . , w h e n U = - - O . O 0 0 7 7 9 6 v R ~ / a . I t a p p e a r s f r o m

e q u a t i o n ( 43 ) t h a t t h e s e r i e s f o r

F ~ / F ,

c o n v e r g e s p r o v i d e d

n a r d 2

the secondary flow in curved pipes

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