NOVEMBRE 1 9 6 3 - № 7 L A H O U I L L E B L A N C H E 7 7 5
N O T U L E H Y D R A U L I Q U E
A new method of predicting the flow in a 90° branch channel
(FLOW IN THE MAIN CHANNEL IS SUBCRITICAL AND THE BRANCH CHANNEL IS SUPERCRITICAL)
B Y
Gr. K E I S H N A P P A , AND K S E E T H A B A M I A H ,
C.S.I.R. RESEARCH FELLOW, CIVIL AND HYDRAULIC ASSISTANT PROFESSOR, CIVIL AND HYDRAULIC ENGINEERING SECTION, INDIAN INSTITUTE OF SCIENCE, ENGINEERING SECTION INDIAN INSTITUTE OF SCIENCE,
BANGALORE 12, INDIA. BANGALORE 12, INDIA.
Bused on dimensional analysis, an experimental relation for predicting the flow in a 90-degree branch channel, with subcritical flow in the main and supercritical flow in the branch, has been developed. The discharge distribution Qb/Qi, where Qb is the discharge in the branch channel and Qt is the discharge in the main channel, varies linearly with the F0, the Froude number in ihe main channel upstream of the branch, for a constant LIB ratio in the range LIB = 1/4 to 1 and Fo = 0.2 to 0.S, where L is the width of the branch and B is the width of the main channel. The discharge distribution decreases as the Froude number increases.
I N T R O D U C T I O N
T h e p r o b l e m of b r a n c h c h a n n e l flow is defined a s p r e d i c t i n g t h e flow i n a b r a n c h c h a n n e l for a g i v e n d i s c h a r g e i n t h e m a i n c h a n n e l , k n o w i n g t h e flow c o n d i t i o n a n d f e a t u r e s of t h e m a i n c h a n n e l a n d t h e f e a t u r e s of t h e b r a n c h c h a n n e l (like s l o p e , r o u g h n e s s coefficient, a n g l e of off t a k e , e t c . ) . T h i s is a p r o b l e m of s p a t i a l l y v a r i e d flow i n o p e n c h a n n e l h y d r a u l i c s . I t c a n also be c o n s i d e r e d a s a p a r t i c u l a r c a s e of t h e w e l l - k n o w n p r o b l e m , of s i d e w e i r flow, w h e n t h e h e i g h t of t h e w e i r i s r e d u c e d t o z e r o [ 1 ] .
C o n s i d e r i n g t h e n u m b e r of v a r i a b l e s t h a t a r e i n v o l v e d , i t w o u l d n o t b e p o s s i b l e for a n y s i n g l e
i n v e s t i g a t i o n t o give a c o m p l e t e s o l u t i o n of t h i s p r o b l e m .
T h i s p r o b l e m c a n be solved b y t h e w e l l - k n o w n m e t h o d of c o n f o r m a i r e p r e s e n t a t i o n [ 2 ] . In t h i s c a s e t h e level of t h e fluid b o t h a l o n g t h e m a i n c h a n n e l a n d t h e b r a n c h is a s s u m e d t o be t h e s a m e . As s u c h , w h e n a p p l i e d t o p r a c t i c a l p r o b l e m s , t h i s i n t r o d u c e s c o n s i d e r a b l e e r r o r .
S o l u t i o n g i v e n b y T a y l o r [ 3 ] for r e c t a n g u l a r c h a n n e l j u n c t i o n s is o n l y for a p a r t i c u l a r c a s e of e q u a l w i d t h s of c h a n n e l a n d s u b c r i t i c a l c o n d i t i o n s of flow in b o t h t h e m a i n a n d t h e b r a n c h c h a n n e l s . T h e s o l u t i o n is a g r a p h i c a l o n e i n v o l v i n g a l a b o r i o u s t r i a l a n d e r r o r p r o c e s s .
K. P a t t a b h i r a m i a h a n d N. R a j a r a t n a m [ 1 ] h a v e solved t h i s p r o b l e m for s u p e r c r i t i c a l flows
Article published by SHF and available athttp://www.shf-lhb.orgorhttp://dx.doi.org/10.1051/lhb/1963055
7 7 6 L A H O U I L L E B L A N C H E № 7 - NOVEMBRE 1 9 6 3
FIG. 4
Fio. 1
FIG. 5
F IG. 2
FIG. 6
FIG. 3
FIG. 7
NOVEMBRE 1963 - № 7 K R I S H N A P P A AND S E E T H A R A M I A H 777
b y t h e c o n c e p t of c o n s t a n t specific e n e r g y . H e n c e , t h e p r e s e n t i n v e s t i g a t i o n m a i n t a i n i n g s u b c r i t i c a l flow i n t h e m a i n a n d s u p e r c r i t i c a l flow
i n t h e b r a n c h c h a n n e l h a s b e e n t a k e n u p .
T H E O R E T I C A L A N A L Y S I S O F T H E P R O B L E M
I n c a s e of s u b c r i t i c a l flow i n t h e m a i n c h a n n e l a n d s u p e r c r i t i c a l flow i n t h e b r a n c h c h a n n e l , t h e profile a l o n g t h e m a i n c h a n n e l is s h o w n i n F i g u r e 1. T h i s is s i m i l a r t o t h e c a s e of s i d e Weirs, w h e n t h e flow i n t h e m a i n c h a n n e l is s u b c r i t i c a l a s g i v e n b y d e M a r c h ! [ 4 ] .
T h e d i s c h a r g e i n t h e b r a n c h c h a n n e l Q6 d e p e n d s o n d0, V0, B , L , 0, g, y.
w h e r e
d0 = d e p t h a t t h e u n i f o r m flow i n t h e m a i n c h a n n e l ;
V0 = v e l o c i t y a t t h e u n i f o r m flow i n t h e m a i n c h a n n e l ;
6 = a n g l e of off t a k e of t h e b r a n c h t o t h e m a i n ;
g = a c c e l e r a t i o n d u e t o g r a v i t y ; Y = specific w e i g h t of w a t e r .
E x p r e s s i n g i n f u n c t i o n a l r e l a t i o n s h i p :
U (Q». <*o> V0, B, L, 9, g, y) = 0 (1)
B y d i m e n s i o n a l a n a l y s i s :
Q » / Q i = / s < F0, L / B , d0/ B , e ) (2) w h e r e :
Q1 = d0V0B a n d F0 = VoVgyi) F r o m e x p e r i m e n t s i t h a s b e e n f o u n d t h a t df0/B is n o t a s i g n i f i c a n t p a r a m e t e r (in t h e r a n g e 0,1 ^ d0/ B < 0,5). So, f o r a p a r t i c u l a r v a l u e of 9 = 9 0 ° :
Q » / Q i = M F o , L / B ) (3)
E X P E R I M E N T A L D E T A I L S
I n t h e e x p e r i m e n t a l s e t - u p , t h e m a i n c h a n n e l is of 2 ft w i d e , 2 * ft d e e p a n d 50 ft l o n g . T h e b r a n c h c h a n n e l is of v a r i a b l e w i d t h a n d 10 ft
l o n g . T h e l o n g i t u d i n a l s l o p e of t h e b r a n c h c h a n n e l w a s k e p t t o m a i n t a i n s u p e r c r i t i c a l flow for t h e m a x i m u m d i s c h a r g e ( 1 / 3 0 ) .
T h e d i s c h a r g e i n t h e m a i n c h a n n e l w a s v a r i e d f r o m 0.5 t o 3 c u s e c s . E x p e r i m e n t s w e r e c o n d u c t e d f o r t h e 90° b r a n c h . T h e w i d t h of t h e b r a n c h c h a n n e l w a s k e p t a t 1/2', 3 / 4 ' , 1', 1 1 / 3 ' , a n d 1 2 / 3 ' a n d 2 ' so a s t o give a L / B r a t i o r a n g i n g f r o m 1/4 t o 1. T h e d e p t h a n d v e l o c i t y i n t h e m a i n c h a n n e l w e r e v a r i e d b y m e a n s of baffles.
D I S C U S S I O N
O F T H E E X P E R I M E N T A L R E S U L T S
I n t h e t h e o r e t i c a l a n a l y s i s , t h e d e p t h c o n s i d e r e d is t h e d e p t h a t t h e u n i f o r m flow i n s t e a d of t h e d e p t h , d2 a t t h e u p s t r e a m e n d of t h e b r a n c h ( F i g . 1 ) . T h i s is i n o r d e r t o f a c i l i t a t e t h e p r e d i c t i o n of flow t o t h e b r a n c h k n o w i n g t h e u n i f o r m d e p t h of flow i n t h e m a i n c h a n n e l u p s t r e a m of t h e b r a n c h . T h e r e s u l t s t h a t a r e p l o t t e d i n F i g u r e s 2 t o 7 s h o w , i n e a c h c a s e t h e d i s c h a r g e d i s t r i b u t i o n Q6/ Q i v a r i e s l i n e a r l y w i t h F0 f o r a n y c o n s t a n t L / B r a t i o a n d t h e d i s c h a r g e d i s t r i b u t i o n d e c r e a s e s a s F r o u d e n u m b e r i n c r e a s e s . W h e n Fc w a s i n c r e a s e d to b e y o n d 0.8, t h e d e p t h dx i n t h e m a i n c h a n n e l a t t h e u p s t r e a m e n d of t h e b r a n c h a p p r o a c h e d t o c r i t i c a l c a u s i n g f l u c t u a t i n g flow.
T h e d i s c h a r g e d i s t r i b u t i o n e q u a t i o n c a n b e g i v e n a s C V Q i = m F0 + C.
T h e v a l u e s of m a n d C b e i n g p l o t t e d w i t h L / B r a t i o , t h e e q u a t i o n s of m a n d C a r e :
m = — ( 1 . 4 5 L / B + 0.32) C = (1.575 L / B + 0.16) T h e r e f o r e , e q u a t i o n (3) b e c o m e s :
= (1,545 — 1,45 F0) -g- + 0,16 (1 — 2 F0) (4)
C O N C L U S I O N S
I n c a s e of 90° b r a n c h c h a n n e l , i n t h e r a n g e L / B = 1/4 t o 1 a n d F0 = 0.2 t o 0.8.
1. T h e d i s c h a r g e d i s t r i b u t i o n Q,,/Qi is a l i n e a r f u n c t i o n of F0 for a c o n s t a n t L / B r a t i o .
778 LA HOUILLE BLANCHE № 7 - NOVEMBRE 1963
2. T h e d i s c h a r g e d i s t r i b u t i o n d e c r e a s e s a s t h e F r o u d e n u m b e r i n c r e a s e s .
A c k n o w l e d g e m e n t s
T h e a u t h o r s a r e d e e p l y i n d e b t e d t o P r o f e s s o r N . S . G o v i n d a R a o for h i s g u i d a n c e a n d k e e n i n t e r e s t d u r i n g t h e c o u r s e of t h i s i n v e s t i g a t i o n . T h e a u t h o r s a r e t h a n k f u l t o D r N. R a j a r a t n a m for h i s h e l p .
R e f e r e n c e s
[1] PATTABHIRAMIAH ( K . R . ) and RAJARATNAM (N.). — A new method to predict flow in a branch channel.
Irrigation & Power Journal, New Delhi, Jan. 1960.
[2] MILNE THOMPSON. •— Theoretical Hydrodynamics.
[3] Edward H. TAYLOR. — Flow characteristics at rec
tangular junctions. Trans. ASCE., 1944.
[ 4 ] COLLINGE ( V . K.). — The discharge capacity of side weirs. Proc. Inst, of Civil Engineers (London), vol. 6, 1957.
[5] VEN TE CHOW. — Open channel hydraulics.
RÉSUMÉ
Nouvelle méthode pour la détermination du débit dans un branchement à 90° sur un canal
(l'écoulement est fluvial dans le canal principal et torrentiel dans le branchement)
PAR G. K R I S H X A P P A ET K. S E E T H A R A M I À H
Les auteurs présentent des résultats expérimentaux valables dans le domaine suivant : L / B compris entre 0,25 et 1 F0 = V0 VffUo compris entre 0,2 et 0,8
(L est la largeur de la dérivation, B celle du canal p r i n c i p a l , V0 la vitesse et y0 le t i r a n t d'eau à l'amont du canal p r i n c i p a l ) .
Les figures 2 à 7 montrent les résultats obtenus, qui peuvent être représentés avec une ap- proximation suffisante p a r la formule :
= (1,545 —1,45 F0) - | - + 0,16 (1 — 2 F0) où Q„ et Qj sont les débits- dans la dérivation et dans le canal principal.
Il faut noter que, dans le domaine des expériences, le r a p p o r t d0/B (qui a varié entre 0,1 et 0,5) n'avait aucune influence sensible sur le débit dérivé.
