Rivers
Influence of a sloping bed on the sediment
transport direction
M.C.L.M. van Mierlo
Report on experimental investigations
A44
R 657-XXIX
Q 186
IW-August 1986
toegepast onderzoek
1. I n t r g d u c t i o n
An i m p o r t a n t element of a m a t h e m a t i c a l model f o r t h e p r e d i c t i o n of two dimen-s i o n a l bed d e f o r m a t i o n i n a l l u v i a l r i v e r dimen-s i dimen-s t h e f o r m u l a t i o n of t h e dimen-s e d i m e n t t r a n s p o r t d i r e c t i o n ^ A common a p p r o a c h i s to d i v i d e t h e g o v e r n i n g phenomena i n t o two components: - t h e i n f l u e n c e of t h e s p i r a l w a t e r m o t i o n due t o c u r v e d f l o w , - t h e i n f l u e n c e of t h e s l o p i n g bed. A r e v i e w of t h i s s u b j e c t i s g i v e n by Odgaard ( 1 9 8 1 ) ,
From S t r u i k s m a e t a l ( 1 9 8 5 ) i t can be c o n c l u d e d t h a t more e x p e r i m e n t a l e v i -dence i s needed t o a c h i e v e a c o r r e c t formulation» G e n e r a l l y t h e a v a i l a b l e d a t a c a n o n l y be used f o r an i n t e g r a l c h e c k a s , f o r i n s t a n c e , was done by Zimmerman and Kennedy ( 1 9 7 8 ) . More u s e f u l i n f o r m a t i o n c o u l d be o b t a i n e d by c a r r y i n g out e x p e r i m e n t s d i r e c t e d a t t h e i n d i v i d u a l components. The f i r s t e x p e r i m e n t a l a t -tempts i n s t r a i g h t f l u m e s t o s t u d y the i n f l u e n c e of t h e s l o p i n g bed component a l o n e were c a r r i e d out by F r e d s j i e ( 1978) and Wan ( 1 9 8 1 ) . However, t h e y d i d not a r r i v e a t u n i f o r m c o n d i t i o n s i n t h e e x p e r i m e n t s and t h e r e f o r e t h e i n t e r p r e t a -t i o n of -t h e r e s u l -t s was d i f f i c u l -t . I n a d d i -t i o n , -the range of f l o w c o n d i -t i o n s u s e d was r e l a t i v e l y s m a l l . The e x p e r i m e n t s p r e s e n t e d i n t h i s r e p o r t c a n be c o n s i d e r e d a s a c o n t i n u a t i o n of t h e s e e x p e r i m e n t s . Much a t t e n t i o n was p a i d t o t h e e s t a b l i s h m e n t of u n i f o r m c o n d i t i o n s i n the flume and a w i d e r a n g e of f l o w c o n d i t i o n s h a s been u s e d .
The p r e s e n t e x p e r i m e n t s were c a r r i e d out i n a s t r a i g h t flume under p r e v a i l i n g bed l o a d c o n d i t i o n s i n w h i c h i n s i g n i f i c a n t g r a i n s o r t i n g took p l a c e . B e f o r e t h e s t a r t of e a c h e x p e r i m e n t a c o n s t a n t t r a n s v e r s e s l o p i n g bed p r o f i l e and a c o n s t a n t l o n g i t u d i n a l bed s l o p e were p r e p a r e d o v e r t h e e n t i r e l e n g t h of t h e f l u m e . The i n f l u e n c e of t h e t r a n s v e r s e s l o p i n g bed p r o f i l e on t h e d i r e c t i o n of the s e d i m e n t t r a n s p o r t was d e t e r m i n e d from the e x p e r i m e n t s by means of t h e f i r s t - o r d e r s o l u t i o n of t h e s e t of e q u a t i o n s , w h i c h d e s c r i b e t h e t i l t i n g of t h e t r a n s v e r s e bed p r o f i l e a s a f u n c t i o n of t i m e .
I n o r d e r t o meet n a t u r a l c o n d i t i o n s , i . e , a dune c o v e r e d bed, i n a l l t e s t s r e -p o r t e d a r t i f i c i c a l dunes were s u -p e r i m -p o s e d on t h e a l r e a d y d e s c r i b e d i n i t i a l
2
-bed topography» T h i s was done b e c a u s e t h e time r e q u i r e d f o r t h e development of t h e dunes a p p e a r e d t o be i n t h e same o r d e r of m a g n i t u d e a s t h e t i m e r e q u i r e d f o r t h e t i l t i n g of t h e t r a n s v e r s e s l o p i n g bed p r o f i l e .
F o r two e x p e r i m e n t s t h e i n f l u e n c e of t h e t r a n s v e r s e s l o p i n g bed p r o f i l e c o u l d not be d e t e r m i n e d by means of t h e f i r s t - o r d e r s o l u t i o n . The i n f l u e n c e of t h e h i g h e r - o r d e r t e r m s a p p e a r e d t o be s i g n i f i c a n t and t h e r e f o r e a m o r p h o l o g i c a l computer program was made f o r t h e i n t e r p r e t a t i o n of t h e s e two e x p e r i m e n t s .
The e x p e r i m e n t s w e r e c a r r i e d out from O c t o b e r 1985 t o F e b r u a r y 1986 i n t h e s t r a i g h t , w e l l e q u i p p e d . Sand Flume of t h e D e l f t H y d r a u l i c s L a b o r a t o r y . E l e v e n t e s t s were c a r r i e d out w i t h S h i e l d s v a l u e s r a n g i n g from 0.1 t o 0.8.
M.C.L.M. v a n M i e r l o , P r o j e c t E n g i n e e r w i t h HASKONING, Nijmegen, The N e t h e r -l a n d s .
2.1 Sediment t r a n s p o r t d i r e c t i o n f o r m u l a t i o n The f o r m u l a t i o n of t h e d i r e c t i o n of t h e s e d i m e n t t r a n s p o r t c a n be d i v i d e d i n t o two components, s e e F i g u r e 2.1: - t h e i n f l u e n c e of t h e s p i r a l w a t e r m o t i o n , due t o t h e c u r v e d f l o w , - t h e i n f l u e n c e of t h e s l o p i n g bed. Van Bendegom
(1947)
g i v e s t h e f o l l o w i n g d i r e c t i o n f o r m u l a f o r s e d i m e n t g r a i n s moving a l o n g a s l o p i n g bed, s e e F i g u r e 2.1: K s i n ( 6 ) - G 8z^/9y t a n ( a ) = ^ cos ( 6 ) - G 3z^/ 8x or . r , s i n ( 6 ) - ( W e ) ) 8z^/8y '''' = c o s ( 6 ) - ( l / f ( 9 ) ) 8 z 7 ï 7 ^'-'^ i n w h i c h : d i m e n s i o nf ( 9 )
= K^/G = w e i g h f u n c t i o n-e
=u2/c2
AD^Q
= S h i e l d s p a r a m e t e r-c
= Chezy c o e f f i c i e n t mO«Vs = median g r a i n s i z e mu
= v e l o c i t y of main f l o w i n d i r e c t i o n of K^g> Fig.,2.1
m/s A = ( p ^ - p ) / p r e l a t i v e submerged d e n s i t y of s e d i m e n t -p s = s p e c i f i c d e n s i t y of s e d i m e n t kg/m^ P = s p e c i f i c d e n s i t y of w a t e r kg/m3 G = submerged w e i g h t of s e d i m e n t g r a i nN
Kb = r e s u l t i n g d r a g f o r c e on g r a i n s N x,y = h o r i z o n t a l s p a t i a l c o o r d i n a t e s m zb = bed e l e v a t i o n m a = d i r e c t i o n of s e d i m e n t t r a n s p o r t r a d . 6 = a n g l e of d i r e c t i o n of r e s u l t i n g main f l o w r a d .-4-y
A
- > X
F i g u r e 2.1 D i r e c t i o n of t h e s e d i m e n t t r a n s p o r t when i n f l u e n c e d by t h e s p i r a l w a t e r m o t i o n and the s l o p i n g bed
d i m e n s i o n K-bn bs = d r a g f o r c e on g r a i n s p e r p e n d i c u l a r to t h e d i r e c t i o n of t h e m a i n f l o w , due t o s p i r a l f l o w = d r a g f o r c e on g r a i n s i n main f l o w d i r e c t i o n = r e s u l t i n g f o r c e on g r a i n s , i . e . d i r e c t i o n of s e d i m e n t t r a n s p o r t 2.2 Aim of t h e e x p e r i m e n t s
The aim of t h e e x p e r i m e n t s i s t o d e t e r m i n e the w e i g h - f u n c t i o n f ( 9 ) , w h i c h de-s c r i b e de-s t h e r a t i o between t h e e f f e c t i v e f l o w d r a g and t h e de-submerged w e i g h t of the s e d i m e n t i n E q u a t i o n ( 2 . 1 ) . The v a l u e of the w e i g h - f u n c t i o n i s assumed to depend o n l y on t h e S h i e l d s p a r a m e t e r .
S i n c e t h e DHL Sand Flume i s s t r a i g h t and t h e x c o o r d i n a t e i s i n the f l o w d i -r e c t i o n , t h e i n f l u e n c e of t h e s p i -r a l w a t e -r m o t i o n ( h e l i c a l f l o w ) i s e x c l u d e d ( 6 = 0 ) . I n a d d i t i o n , t h e i n f l u e n c e of t h e l o n g i t u d i n a l bed s l o p e c a n be neg-l e c t e d , s i n c e the neg-l o n g i t u d i n a neg-l bed s neg-l o p e i s s m a neg-l neg-l w i t h r e s p e c t to t h e magni-tude of t h e w e i g h - f u n c t i o n . T h e r e f o r e E q u a t i o n ( 2 . 1 ) r e d u c e s to E q u a t i o n ( 2 . 2 ) and F i g u r e 2.1 c a n be r e p l a c e d by F i g u r e 2.2.
t a n ( a ) = G 3z^/3y or t a n (cx) = ( 2 . 2 ) F i g u r e 2.2 D i r e c t i o n of s e d i m e n t t r a n s p o r t when i n f l u e n c e d by t h e t r a n s v e r s e s l o p i n g bed o n l y
The v a l u e s of t h e w e i g h f u n c t i o n f ( 9 ) have been c a l c u l a t e d from t h e e x p e r i m e n -t a l r e s u l -t s u s i n g E q u a -t i o n ( 2 . 1 1 ) , a s s u m i n g -t h a -t -t h e f i r s -t - o r d e r s o l u -t i o n des c r i b e des t h e t i l t i n g p r o c e des des of t h e t r a n des v e r des e bed des l o p e a d e q u a t e l y , des e e S e c -t i o n 2.3. A d i f f e r e n -t a p p r o a c h had -t o be f o l l o w e d f o r -t h e e x p e r i m e n -t s T2 and T 3 , s e e S e c t i o n 4.3. 2.3 T h e o r e t i c a l background t o t h e e x p e r i m e n t s ( i ) E q u a t i o n s
Water and s e d i m e n t movement a r e c o n s i d e r e d t o be u n i f o r m i n t h e l o n g i t u d i n a l d i r e c t i o n , t h a t means a l l l o n g i t u d i n a l d e r i v a t i v e s a r e n e g l i g i b l e . The t i l t i n g of a s l o p i n g bed l e v e l p r o f i l e c a n be d e s c r i b e d w i t h E q u a t i o n s ( 2 . 3 ) t o ( 2 . 6 ) , s e e a l s o F i g u r e 2.3.
w a t e r movement:
-=•6-s e d i m e n t movement: s ^ = F(e) ( 3 s ^ / 9 x = 0) ( 2 . 5 ) 1 9h
t a n cc = j-^-Q-^ -g-- = ( 2 . 6 )
X i n w h i c h : d i m e n s i o n h = w a t e r depth m = F ( 0 ) = s e d i m e n t t r a n s p o r t , i n c l u d i n g p o r e s , p e r u n i t w i d t h i n x - d i r e c t i o n m^/s Sy = s e d i m e n t t r a n s p o r t , i n c l u d i n g p o r e s , per u n i t w i d t h i n y - d i r e c t i o n m^/s t = t i m e s u = f l o w v e l o c i t y i n x - d i r e c t i o n m/s V = f l o w v e l o c i t y i n y - d i r e c t i o n m/s X = c o o r d i n a t e i n l o n g i t u d i n a l d i r e c t i o n m y = c o o r d i n a t e i n t r a n s v e r s e d i r e c t i o n m I^
— I
1 f
CO ¥////////////////////////////)(///////////, I 7TOP VIEW
CROSS SECTION I - I
F i g u r e 2.3 D e f i n i t i o n s k e t c h
By i n s e r t i n g E q u a t i o n s ( 2 . 3 ) , ( 2 . 5 ) and ( 2 . 6 ) i n t o E q u a t i o n ( 2 . 4 ) t h e f o l l o w -i n g d -i f f e r e n t -i a l e q u a t -i o n c a n be o b t a -i n e d , w h -i c h d e s c r -i b e s t h e t -i l t -i n g of t h e t r a n s v e r s e bed s l o p e .
r 1 3F(e) 1 8 f ( 9 ) i .8h>.2 , a^h f ( 9 ) 3h ^ „
^ F ( 0 ) 8h f ( 9 ) 3h J ^-SyJ " ^ 1 ^ F ( 9 ) 3t ^
( i i ) ïi2Ê_scale_for_the_tilting_£rocess
The time s c a l e T w h i c h c a n be d e r i v e d from E q u a t i o n ( 2 . 7 ) , y i e l d s :
B2
f ( e
) B2 f ( 9 )^ = - T T o f - = - T - ^ (2.8)
O x,o i n w h i c h d i m e n s i o n T = t i m e s c a l e o f t i l t i n g p r o c e s s s B = t h e w i d t h of t h e flume m f ( Q ) * = v a l u e o f w e i g h f u n c t i o n -F ( 0 )*=s = s e d i m e n t t r a n s p o r t , i n c l u d i n g p o r e s , p e r u n i t O X ^ O w i d t h i n x - d i r e c t i o n m^/s ( i i i ) Ei£5tZ2£ËSE_SE^_^^£2EÉZ2E^SE s o l u t i o n s The f i r s t - o r d e r and t h e s e c o n d - o r d e r s o l u t i o n of E q u a t i o n ( 2 . 7 ) a r e g i v e n i n E q u a t i o n ( 2 . 9 ) and ( 2 . 1 0 ) , r e s p e c t i v e l y , and a r e shown s c h e m a t i c a l l y i n F i g u r e2.4.
The f o l l o w i n g r e m a r k s c a n be made about b o t h s o l u t i o n s :
the i n i t i a l c o n d i t i o n i s t h e s u p e r i m p o s i t i o n of t h e i n i t i a l s e c o n d - o r d e r s o l u t i o n on t h e i n i t i a l f i r s t - o r d e r s o l u t i o n , - t h e boundary c o n d i t i o n n e a r t h e i m p e r m e a b l e s i d e w a l l s i s 3h/3y = 0, - h' = h-h^ i s a p e r t u r b a t i o n added t o t h e z e r o - o r d e r s o l u t i o n , - s = F ( 0 ) , and f ( 0 ) a r e assumed to be f u n c t i o n s of t h e f l o w v e l o c i t y , v i z . s = F ( 0 ) :: u and f ( 0 ) :: u i n w h i c h b and n a r e c o n s t a n t p a r a m e t e r s . So X no c r i t i c a l v a l u e f o r t h e s e d i m e n t t r a n s p o r t i s a p p l i e d ,
h = h^ + R s i n y ) e x p ( - t / T
^^p
( 2 . 9 )
h = h^+ R [ s i n( 1
y ) + - ^ | - e x p ( - t / T ^ ^ ^ ) c o s ( | ^ y ) ]e x p ( - t / T
^^p
( 2 . 1 0 ) o * t h e i n d e x 0 r e f e r s to t h e z e r o - o r d e r s o l u t i o n of E q u a t i o n ( 2 . 7 )8 -and i n w h i c h B2
f ( e )
B2 f( 0 ) tt2 Tr2F(e ) ïï2s o x.o ( 2 . 1 1 ) h^ = z e r o - o r d e r w a t e r d e p t h h = a m p l i t u d e of f i r s t - o r d e r s o l u t i o n of E q u a t i o n ( 2 . 7 ) •'"eff ^ t i l t i n g t i m e of t r a n s v e r s e bed l e v e l s l o p e , c o r r e s p o n d i n g w i t h a r e d u c t i o n of e x p ( - l ) of i n i t i t a l s l o p e d i m e n s i o n m mf i r s t - order
second - o r d e r
F i g u r e 2.4 T r a n s v e r s e bed l e v e l p r o f i l e , f i r s t and s e c o n d - o r d e r s o l u t i o n of E q u a t i o n ( 2 . 7 ) a t a c e r t a i n p o i n t of t i m e , 2.4 C o n d i t i o n s t o b e m e t by t h e e x p e r i m e n t s The w e i g h - f u n c t i o n f( e ) c a n be o b t a i n e d from t h e e x p e r i m e n t s u s i n g E q u a t i o n s ( 2 , 9 ) and ( 2 . 1 1 ) , p r o v i d e d t h a t t h e c o n d i t i o n s g i v e n below a r e f u l f i l l e d .- a c o n s t a n t l o n g i t u d i n a l bed s l o p e o v e r t h e e n t i r e l e n g t h of t h e movable bed i n t h e sand f l u m e t o p r o v i d e u n i f o r m c o n d i t i o n s i n the l o n g i t u d i n a l d i r e c -t i o n ,
- a n e g l i g i b l e i n f l u e n c e of t h e l o n g i t u d i n a l bed s l o p e on t h e d i r e c t i o n of t h e sediment t r a n s p o r t .
s i n e c u r v e o v e r the e n t i r e l e n g t h of t h e movable bed i n t h e Sand Flume i n o r d e r to meet t h e f i r s t - o r d e r s o l u t i o n a s n e a r l y a s p o s s i b l e ,
- s u p e r i m p o s i t i o n of a r t i f i c a l dunes on t h e s t r a i g h t l o n g i t u d i n a l bed p r o f i l e and t h e t r a n s v e r s e s i n e shaped bed l e v e l p r o f i l e , i n o r d e r to o b t a i n an a l m o s t c o n s t a n t bed r o u g h n e s s (Chezy v a l u e ) d u r i n g t h e t i l t i n g p r o c e s s . ( i i ) Boundary c o n d i t i o n s - c o n s t a n t d i s c h a r g e , - c o n s t a n t e n e r g y - l e v e l s l o p e , e q u a l to the i n i t i a l l o n g i t u d i n a l bed s l o p e , - r e c i r c u l a t i o n of t h e s e d i m e n t t r a n s p o r t e d i n t h e sand f l u m e , - c o n s e r v a t i o n of sediment i n y - d i r e c t i o n , - No c o n t r o l m e a s u r e s were t a k e n a t t h e u p s t r e a m boundary w i t h r e s p e c t t o f l o w and s e d i m e n t d i s t r i b u t i o n , be c a u s e t h e d i s t a n c e between t h i s boundary and t h e b e g i n n i n g of the m e a s u r i n g s e c t i o n p r o v i d e s a s u f f i c i e n t l y l o n g a d a p t a t i o n l e n g t h , - c o n s t a n t w a t e r t e m p e r a t u r e . ( i i i ) F l o w and s e d i m e n t t r a n s p o r t c o n d i t i o n s - c o n s t a n t bed r o u g h n e s s ( C h e z y v a l u e ) , - c o n s t a n t s e d i m e n t t r a n s p o r t i n t h e x - d i r e c t i o n and t i m e , - no s p i r a l w a t e r m o t i o n , - i n s i g n i f i c a n t suspended l o a d , - i n s i g n i f i c a n t g r a i n s o r t i n g e f f e c t s ( h a r d l y p o s s i b l e due t o t h e u n i f o r m i t y of t h e bed m a t e r i a l ) .
