K.W. Olesen
L a b o r a t o r y o f F l u i d Mechanics Department o f C i v i l E n g i n e e r i n g D e l f t U n i v e r s i t y o f Technology
L i s t o f symbols Summary 1. I n t r o d u c t i o n j 1.1. General I 1.2. P r e v i o u s w o r k 2 1.3. The p r e s e n t i n v e s t i g a t i o n 3 2. M a t h e m a t i c a l m o d e l l i n g 5 2.1. Steady f l o w model 5 2.2. Bed shear s t r e s s g 2.3. E q u a t i o n o f c o n t i n u i t y f o r t h e sediment 12 2.4. Sediment t r a n s p o r t 14 3. S o l u t i o n f o r a d o u b l e harmonic p e r t u r b a t i o n ]7 3.1. S o l u t i o n o f t h e f l o w model 17 3.2. The s t a b i l i t y a n a l y s i s 21 4. D i s c u s s i o n on b a s i c a s s u m p t i o n 23 4.1. L i n e a r i z a t i o n o f t h e f l o w model 24 4.2. L i n e a r i z a t i o n o f t h e sediment model 24 5. The complex c e l e r i t y 26 5.1. I n f l u e n c e o f c o n s t i t u t i v e r e l a t i o n s 28 5.2. I n f l u e n c e o f t h e f l o w p a r a m e t e r s 29 6. Comparison w i t h f l u m e e x p e r i m e n t s 3 3 7. Conclusions^ and f u r t h e r r e s e a r c h 36 References 3 9 Appendix 42 T a b l e 46 F i g u r e s 47
I . Sand f l u m e d a t a 46 F i g u r e s : A l . Decay o f secondary f l o w i n t e n s i t y 45 1. Bed p a t t e r n s 47 2. Phase l a g and a m p l i t u d e f o r d i f f e r e n t e 48 3. Phase l a g and a m p l i t u d e f o r d i f f e r e n t F 49 4. L i n e a r and n o n - l i n e a r f l o w c o m p u t a t i o n 50 5. A m p l i f i c a t i o n f a c t o r . Reference example 51 6. A m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f a 52 7. A m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f b 52 8. A m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f 53 9. A m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f M2 53 10. A m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f Nj 54 I I . A m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f 54 12. A m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f t r a n s v e r s e bed shear s t r e s s i n f l o w model 55 13. Max. a m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f F 56 14. Max. a m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f f 57 15. Max. a m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f k 58 w 16. Max. a m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f k^. No secondary f l o w 59 17. Max. a m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f k^. No secondary f l o w i n e r t i a 60 18. Max. a m p l i f i c a t i o n f a c t o r . I n f l u e n c e o f k . No g r a v i t a t i o n a l ' w f o r c e 61 19. Roughness c o e f f i c i e n t . Flume d a t a 62 20. T r a n s p o r t r a t e . Flume d a t a 63 21. A m p l i f i c a t i o n f a c t o r . Flume e x p e r i m e n t s 64
22. A m p l i f i c a t i o n f a c t o r . Flume e x p e r i m e n t s . Adapted secondary f l o w
c o e f f i c i e n t s 65 23. H e i g h t o f a l t e r n a t e b a r s as a f u n c t i o n o f k^. Flume d a t a 67
24. H e i g h t and l e n g t h o f a l t e r n a t e b a r s as a f u n c t i o n o f f .
jar b c o e f f i c i e n t i n model f o r secondary f l o w i n e r t i a c c o e f f i c i e n t i n model f o r g r a v i t a t i o n a l f o r c e on g r a i n s f ~ 02" • Roughness c o e f f i c i e n t g a c c e l e r a t i o n due t o g r a v i t y h d e p t h o f f l o w i = /-T^. I m a g i n a r y u n i t k wave number i : i l o n g i t u d i n a l d i r e c t i o n k wave number i n t r a n s v e r s e d i r e c t i o n w 1 = k/k . R e l a t i v e wave number w 1 = 1 f o r di>./dl = 0 max 1 m number o f submerged b a r s i n a c r o s s - s e c t i o n max[4i_j^] = (j)^ f o r d^^/dl = 0 t t i m e c o o r d i n a t e u l o n g i t u d i n a l f l o w v e l o c i t y V t r a n s v e r s e f l o w v e l o c i t y X l o n g i t u d i n a l c o o r d i n a t e y t r a n s v e r s e c o o r d i n a t e
z v e r t i c a l c o o r d i n a t e and d i m e n s i o n i e s s p e r t u r b a t i o n parameter o f t h e bed z, bed l e v e l b C Chezy roughness c o e f f i c i e n t D d e t e r m i n a n t . Eq. (3.13) F Froude number F_ d e n s i m e t r i c Froude number h e i g h t o f b a r s X , h e i g h t o f dunes dun I e q u i l i b r i u m bed s l o p e L, l e n g t h o f b a r s D ar L, l e n g t h o f dunes dunes
M j , c o e f f i c i e n t s i n l i n e a r i z e d bed shear s t r e s s model N j , ^2 c o e f f i c i e n t s i n l i n e a r i z e d t r a n s p o r t r a t e model Q f l o w d i s c h a r g e
S sediment t r a n s p o r t r a t e per u n i t e w i d t h W w i d t h o f c h a n n e l
e
e
K p x' y A $ X S u b s c r i p t s i (*•) r (i>^) ' ( h ' ) * ( h * ) w d i m e n s i o n i e s s shear s t r e s s . S h i e l d parameter ^ 0.4 . v o n Karman c o n s t a n t mass d e n s i t y o f f l u i dbed shear s t r e s s i n x and y d i r e c t i o n , r e s p e c t i v e l y complex c e l e r i t y
d i r e c t i o n o f sediment t r a n s p o r t r e l a t i v e d e n s i t y o f sediment dynamic f r i c t i o n a n g l e
degree o f development o f t h e secondary f l o w
r e f e r t o zero o r d e r s o l u t i o n
r e f e r t o t h e i m a g i n a r y p a r t o f a complex number r e f e r t o t h e r e a l p a r t o f a complex number p e r t u r b a t i o n parameter
d i m e n s i o n i e s s p a r a m e t e r . The b u l k y s t a r i s o m i t t e d when t h a t does n o t l e a d t o c o n f u s i o n
topography in straight alluvial channels is treated. The flow is described by a horizontal two-dimensional model, but secondary flmv due to curvature of the streamlines is included. Further more knowledge about secondary flow inertia achieved in recentY~ars is incorporated.
The analysis suggests that secondary flow plays an important role for the development of meander bends in relatively narrow channels.
The results of the stability analysis are compared with some sandflume data. The agreement is unsatisfactory, but the discrepancy can be explained by insufficient knowledge about the secondary flow properties. However, the sandflume data and the results of the stability analysis exhibit the same trends with respect to dependence of width-depth ratio and alluvial roughness, i.e. increasing width-depth ratio as -.;;rell as increasing roughness coefficient promotes the formation of alternate bars.
J_^^ I n t r o d u c t i o n 1.1. General
The f l o w and bed t o p o g r a p h y i n c u r v e d a l l u v i a l r i v e r s p l a y an i m p o r t a n t p a r t i n s e v e r a l a s p e c t s o f r i v e r e n g i n e e r i n g , such as n a v i g a b i l i t y , bank p r o t e c t i o n and r i v e r r e g u l a t i o n . E n g i n e e r i n g problems c o n c e r n i n g t h i s
t o p i c a r e m o s t l y so c o m p l i c a t e d t h a t t h e y must be i n v e s t i g a t e d u s i n g p h y s i c a l s c a l e models o r n u m e r i c a l models, as t h e n o n - l i n e a r c h a r a c t e r o f t h e g o v e r n i n g e q u a t i o n s i n most cases makes an a n a l y t i c a l approach i m p o s s i b l e . However, t h e t h o r o u g h l i n e a r i z a t i o n o f t h e e q u a t i o n s , i n o r d e r t o make an a n a l y t i c a l s o l u t i o n f e a s i b l e , i s j u s t i f i e d f o r a s m a l l group o f problems. The r i v e r morphology p r o b l e m c o n c e r n i n g t h e f o r m a t i o n o f a l t e r n a t e b a r s i n s t r a i g h t a l l u v i a l channels may b e l o n g t o t h i s group o f problems.
S t a b i l i t y o f s t r a i g h t channels can be o f g r e a t i m p o r t a n c e : U n f o r e s e e n s t a b i l i t y problems i n f o r i n s t a n c e a n a v i g a t i o n channel can l e a d t o l a r g e c o s t s f o r
d r e d g i n g . A l t e r n a t e b a r s can d e v e l o p i n t o t r u e bends w h i c h b r i n g about a d d i t i o n a l roughness w h i c h i n t u r n can cause i n u n d a t i o n o f l o w s i t u a t e d a r e a s . A l t e r n a t e b a r s i n f l u m e e x p e r i m e n t s may impede t h e i n t e r p r e t a t i o n o f t h e measured d a t a . And many o t h e r s .
The p r e s e n t l i n e a r s t a b i l i t y a n a l y s i s was i n i t i a t e d by i n s t a b i l i t y o c c u r r i n g i n a n u m e r i c a l model f o r t h e f l o w and bed t o p o g r a p h y i n c u r v e d a l l u v i a l r i v e r s . The o r i g i n a l a i m was t o i n v e s t i g a t e w h e t h e r t h e s e o s c i l l a t i o n s had a p h y s i c a l cause o r w h e t h e r t h e y were o f pure n u m e r i c a l c h a r a c t e r . The o s c i l l a t i o n o c c u r r e d i n t h e s t r a i g h t r e a c h b e f o r e t h e e n t r a n c e o f a bend, and i t was t h e r e f o r e t h o u g h t t h a t a l i n e a r s t a b i l i t y a n a l y s i s f o r a s t r a i g h t c h a n n e l would s e r v e as a good f i r s t approach.
The s t u d y i s c a r r i e d o u t a t t h e L a b o r a t o r y o f F l u i d Mechanics a t t h e D e l f t U n i v e r s i t y o f Technology w i t h i n t h e framework o f t h e r i v e r bend p r o j e c t o f t h e j o i n t h y d r a u l i c r e s e a r c h programme T.Ü.W. (Toegepast Onderzoek W a t e r s t a a t ) i n w h i c h R i j k s w a t e r s t a a t (Governmental Water C o n t r o l and P u b l i c Works d e p a r t m e n t ) 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 and t h e D e l f t U n i v e r s i t y o f Technology p a r t i c i p a t e .
1.2. P r e v i o u s work
The f o r m a t i o n o f meanders has been s t u d i e d by many s c i e n t i s t s and f r o m s e v e r a l p o i n t o f v i e w s (see C a l l a n d e r , 1978). I t i s a w i d e l y accepted a s s u m p t i o n t h a t d e f o r m a t i o n o f t h e bed i s t h e f u n d a m e n t a l cause o f meandering and t h a t e r o s i o n o f t h e banks f o l l o w s a t a l a t e r s t a g e .
The bed i s c o n s i d e r e d t o be deformed due t o u n s t a b l e response o f a s m a l l p e r -t u r b a -t i o n o f -t h e bed. From -t h i s approach s e v e r a l s c i e n -t i s -t s have
c a r r i e d o u t l i n e a r s t a b i l i t y a n a l y s i s i n o r d e r t o f i n d t h e o r i g i n and i n i t i a l w a v e l e n g t h o f meandering and b r a i d i n g i n a l l u v i a l s t r e a m s . I n the f o l l o w i n g a b r i e f r e v u e o f some i m p o r t a n t p u b l i c a t i o n s w i t h i n t h i s t o p i c x \ f i l l be g i v e n .
The l i n e a r s t a b i l i t y a n a l y s i s a r e c a r r i e d o u t by Engelund and Skovgaard, 1973; P a r k e r , 1976 and Fredsc^e, 1978. A common b a s i s f o r these t h r e e a n a l y s i s i s t h a t t h e c o n s i d e r e d channels a r e w i d e i n o r d e r t o be a b l e t o n e g l e c t any w a l l e f f e c t , t h a t t h e channels have a r e c t a n g u l a r c r o s s -s e c t i o n a l -shape and t h a t t h e bank-s a r e n o n - e r o d i b l e . I n a l l t h r e e a n a l y s i s t h e s t a b i l i t y o f a double p e r i o d i c and h a r m o n i c p e r t u r b a t i o n o f t h e bed i s i n v e s t i g a t e d .
The a n a l y s i s c a r r i e d o u t by Engelund and Skovgaard i s r e m a r k a b l e because a t h r e e - d i m e n s i o n a l f l o w model i s a p p l i e d . A p a r a b o l i c d i s t r i b u t i o n o f the l o n g i t u d i n a l f l o w v e l o c i t y i n c o m b i n a t i o n x ^ i t h a f i n i t e b o t t o m s l i p v e l o c i t y i s assumed. T h i s n o n - u n i f o r m v e r t i c a l d i s t r i b u t i o n o f v e l o c i t y p r o v i d e s f o r secondary c u r r e n t due t o f l o w c u r v a t u r e , w h i c h i s a s i g n i f i c a n t advantage o f t h i s a n a l y s i s . The f l o w i s c o n s i d e r e d t o be q u a s i - s t e a d y , w h i c h i s j u s t i f i e d w i t h t h e c l a s s i c a l a s s u m p t i o n f o r a bed l e v e l model; i . e . d i s t u r b a n c e s o f t h e f l o w t r a v e l a t much h i g h e r c e l e r i t y t h a n d i s t u r b a n c e s o f t h e bed. Engelund and Skovgaard a r e u s i n g a model f o r t h e d i r e c t i o n
o f t h e sediment t r a n s p o r t , w h i c h i s n o t o n l y t a k i n g t h e d i r e c t i o n o f t h e f l o w c l o s e t o t h e bed i n t o c o n s i d e r a t i o n , b u t a l s o t h e g r a v i t a t i o n a l
f o r c e a c t i n g on t h e g r a i n s a l o n g a s l o p i n g bed. The a n a l y s i s e x p l a i n s why some channels t e n d t o b r a i d , o t h e r t e n d t o meander and why a t h i r d
group remair^ s t r a i g h t .
P a r k e r (1976) a p p l i e d a t w o - d i m e n s i o n a l f l o w model i n h i s a n a l y s i s , t h u s secondary f l o w i s n o t t a k e n i n t o c o n s i d e r a t i o n . P a r k e r i s u s i n g t h e e q u a t i o n s f o r u n s t e a d y f l o w , and he o b t a i n e d a complex f o u r t h degree a l g e b r a r i c e q u a t i o n f o r t h e a m p l i f i c a t i o n f a c t o r . A p p l y i n g a s s y m t o t i c e x p a n s i o n he f o u n d an a p p r o x i m a t i o n f o r t h e a m p l i f i c a t i o n f a c t o r ; w h i c h
i s i d e n t i c a l t o t h e e x p r e s s i o n w h i c h can be o b t a i n e d i f t h e d e r i v a t i v e s w i t h r e s p e c t t o t im e i n t h e f l o w e q u a t i o n are n e g l e c t e d . The a n a l y s i s y i e l d s t h a t a l l streams are u n s t a b l e because P a r k e r n e g l e c t e d t h e g r a v i t a t i o n a l f o r c e a c t i n g on t h e g r a i n s , w h i c h i n d e e d i s a v e r y i m p o r t a n t s t a b i l i m n g e f f e c t .
I n t h e a n a l y s i s c a r r i e d o u t by Freds^fe (1978) t h e ( s t e a d y ) f l o w i s
e s s e n t i a l l y d e s c r i b e d by t h e same two d i m e n s i o n a l f l o w model as P a r k e r used. So a l s o i n t h i s a n a l y s i s secondary f l o w due t o c u r v a t u r e o f t h e s t r e a m l i n e s i s n e g l e c t e d . The a n a l y s i s d i f f e r s f r o m t h e p r e v i o u s ones by
a c c o u n t i n g f o r t h e g r a v i t a t i o n a l f o r c e f r o m t h e t r a n s v e r s e s l o p e o f t h e bed and by d i v i d i n g t h e sediment t r a n s p o r t i n t o bed l o a d and suspended
l o a d , w h i c h makes i t p o s s i b l e t o take a c c o u n t o f t h e phase l a g between t h e bed s h e a r s t r e s s and t h e t r a n s p o r t i n s u s p e n s i o n .
R e c e n t l y I k e d a , P a r k e r and Sawai (1981) c a r r i e d o u t a l i n e a r 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 i t h e r o d i b l e banks. The a n a l y s i s y i e l d s wave l e n g t h s o f t h e same o r d e r o f magnitude as t h e more t r a d i t i o n a l s t a b i l i t y a n a l y s i s f o r a l l u v i a l s t r e a m s . T h i s s u p p o r t s t h e a s s u m p t i o n t h a t t h e a l t e r n a t e b a r s e v o l u t e i n t o t r u e bends.
F i n a l l y , i t s h o u l d be m e n t i o n e d t h a t s e v e r a l a u t h o r s have a t t e m p t e d t o e x p l a i n t h e f o r m a t i o n o f a l t e r n a t e b a r s and meanders f r o m o t h e r approaches. For i n s t a n c e , E i n s t e i n and Shen (1964) q u a l i t a t i v e l y e x p l a i n e d t h e f o r m a t i o n o f a l t e r n a t e b a r s by secondary f l o w i n d u c e d by d i f f e r e n t shear s t r e s s a t
the s i d e w a l l s or i n d u c e d by an a s y m m e t r i c a l c r o s s - s e c t i o n a l shape.
1.3. The p r e s e n t i n v e s t i g a t i o n
T h i s r e p o r t concerns a l i n e a r s t a b i l i t y a n a l y s i s o f t h e same t y p e as t h e ones c a r r i e d o u t by among o t h e r s Engelund and Skovgaard ( 1 9 7 3 ) , P a r k e r
(1976) and Freds^^e ( 1 9 7 8 ) , i . e . t h e s t a b i l i t y o f a d o u b l e h a r m o n i c p e r -t u r b a -t i o n o f -t h e bed i n a w i d e r e c -t a n g u l a r c h a n n e l w i -t h non e r o d i b l e banks i s i n v e s t i g a t e d . As i n t h e a n a l y s i s by P a r k e r and F r e d s ^ e a two d i m e n s i o n a l f l o w model i s employed. The main d i f f e r e n c e f r o m t h e s e two a n a l y s i s i s t h a t t h e bed shear s t r e s s i s n o t p a r a l l e l t o t h e main f l o w d i r e c t i o n , b u t t h e d e v i a t i o n f r o m t h e main f l o w d i r e c t i o n due t o c u r v a t u r e o f t h e s t r e a m l i n e s i s t a k e n i n t o a c c o u n t ; i . e . , secondary f l o w i s
c o n s i d e r e d a l t h o u g h a two d i m e n s i o n a l f l o w model i s employed. T h i s i s a s i m p l e r approach t h a n o b t a i n i n g t h e secondary f l o w f i e l d d i r e c t l y f r o m t h e ( l i n e a r i z e d ) t h r e e - d i m e n s i o n a l e q u a t i o n (see Engelund and S k o v g a a r d ) . Hox^ever, t h e employment o f t h e p r e s e n t approach i s more t r a n s p a r a n t and i t has t h e advantage t h a t i t i s p o s s i b l e t o i d e n t i f y the i n f l u e n c e s o f t h e f l o w and sediment t r a n s p o r t w h i c h i s i m p o r t a n t f o r t h e development o f a l t e r n a t e b a r s . F u r t h e r on, i n t h i s a n a l y s i s knowledge a c h i e v e d i n r e c e n t y e a r s about t h e d i r e c t i o n o f t h e sediment
2. M a t h e m a t i c a l m o d e l l i n g
The b a s i c assumption u n d e r l y i n g t h e m a t h e m a t i c a l model f o r t h e f l o w and bed t o p o g r a p h y i n a l l u v i a l channels i s t h a t t h e f l o w can be c o n s i d e r e d q u a s i - s t e a d y , i . e . t h e f l o w i s assumed t o adapt much f a s t e r t o a change i n bed l e v e l t h a n t h e bed l e v e l change i t s e l f . T h e r e f o r e t h e c o m p u t a t i o n o f t h e bed l e v e l development can be d i v i d e d i n t o s m a l l t i m e s t e p s , d u r i n g w h i c h t h e bed i s k e p t f i x e d and t h e f l o w f i e l d i s c o n s i d e r e d t o be s t e a d y . The bed l e v e l a t t h e f o l l o w i n g t i m e s t e p can now be computed by means of t h e e q u a t i o n o f c o n t i n u i t y f o r t h e s e d i m e n t . T h i s c o n v e n i e n t d i v i s i o n between f l o w c o m p u t a t i o n and bed l e v e l c o m p u t a t i o n w i l l be m a i n t a i n e d i n t h e f o l l o w i n g .
Most n a t u r a l a l l u v i a l streams have a l a r g e w i d t h - d e p t h r a t i o w h i c h s u g g e s t s a t w o - d i m e n s i o n a l d e s c r i p t i o n o f t h e f l o w . T h i s approach has p r o v e d t o d e s c r i b e t h e main f l o w f i e l d r a t h e r good when main f l o w i n e r t i a and bed f r i c t i o n d o m i n a t e , e x c e p t i n a n a r r o w r e g i o n c l o s e t o t h e banks where t h e w a l l f r i c t i o n has d i r e c t i n f l u e n c e . However, when t h e secondary f l o w has l a r g e i n f l u e n c e on t h e main f l o w d i s t r i b u t i o n , f o r i n s t a n c e i n a r e c t a n g u l a r channel w i t h c u r v e d a l i g n m e n t , t h e t w o - d i m e n s i o n a l model f a i l s .
A m a t h e m a t i c a l model f o r t i m e dependent change o f t h e bed l e v e l i n a l l u v i a l channels c o n s i s t s i n p r i n c i p l e o f momentum e q u a t i o n f o r each c o n s i d e r e d d i r e c t i o n f o r t h e f l o w and an e q u a t i o n o f c o n t i n u i t y f o r b o t h t h e sediment and t h e f l o w . I n t h e p r e s e n t case o f a d e p t h a v e r a g e d approach t h e r e
w i l l be f o u r e q u a t i o n s , w h i c h c o n s e q u e n t l y can r e l a t e o n l y f o u r v a r i a b l e s . Since t h e r e a r e more v a r i a b l e s i n t h i s p r o b l e m a d d i t i o n a l c o n s t i t u t i v e r e l a t i o n s ( f o r bed shear s t r e s s and sediment t r a n s p o r t ) must be i n t r o d u c e d
i n o r d e r t o c l o s e t h e system o f e q u a t i o n . I n t h e f o l l o w i n g f i r s t t h e s t e a d y f l o w w i l l be t r e a t e d . The complete d e p t h averaged f l o w e q u a t i o n s w i l l be g i v e n , whereupon t h e u n p e r t u r b e d ( z e r o o r d e r ) s o l u t i o n and t h e l i n e a r i z e d ( f i r s t o r d e r ) e q u a t i o n w i l l be d e r i v e d . Much a t t e n t i o n x v f i l l be p a i d t o t h e d e s c r i p t i o n o f t h e bed shear s t r e s s . F i n a l t h e l i n e a r i z e d e q u a t i o n o f c o n t i n u i t y f o r t h e
sediment w i l l be d e r i v e d , and t h e d i r e c t i o n and amount o f sediment t r a n s p o r t w i l l be c o n s i d e r e d , 2.1, Steady f l o w model For t h e m a t h e m a t i c a l d e s c r i p t i o n o f t h e f l o w a c o o r d i n a t e system i s a p p l i e d w h i c h has t h e x - a x i s c o i n c i d i n g w i t h t h e c h a n n e l a x i s and p o s i t i v e i n t h e f l o w d i r e c t i o n , t h e y - a x i s h o r i z o n t a l and p e r p e n d i c u l a r t o t h e X- a x i s and t h e z - a x i s v e r t i c a l upwards. I n t h i s c o o r d i n a t e system t h e d e p t h averaged f l o x j i s d e s c r i b e d by ^ . i i i + ^ -H g ( ^ + ^ ) + ^ = 0 ( 2 . 1 ) 3x ^ ^^9x 3x
'
ph 3v , 3v ^ .3h ^ ^^b., ''•y „ in n\ u — +v —
+ e ( — + T — ) + ~T- - 0 \^'^) ^ 3y ^^3y 3y ' ph K l i H l + i C h y l = 0 ( 2 . 3 ) 9x 3y i n w h i c h g a c c e l e r a t i o n due t o g r a v i t y h d e p t h o f f l o wu,v d e p t h averaged f l o w v e l o c i t y i n x and y d i r e c t i o n , r e s p e c t i v e l y z, ( g i v e n ) bed l e v e l
b
p mass d e n s i t y o f f l u i d
T , T bed shear s t r e s s i n x and y d i r e c t i o n , r e s p e c t i v e l y , x' y
E q u a t i o n (2.3) i s e x a c t , whereas t h e l o n g i t u d i n a l and t r a n s v e r s e momentum e q u a t i o n s , eq. (2.1) and eq. (2.2) r e s p e c t i v e l y , h o l d good under t h e
a s s u m p t i o n t h a t v e r t i c a l a c c e l e r a t i o n s a r e n e g l i g i b l e ( h y d r o s t a t i c
p r e s s u r e ) , and t h a t d e p t h averaged p r o d u c t terms o f t h e h o r i z o n t a l v e l o c i t y components e q u a l t h e p r o d u c t o f t h e d e p t h averaged v e l o c i t y . T h i s i m p l i e s t h a t t h e main f l o w i s u n a f f e c t e d by t h e h o r i z o n t a l component o f t h e
secondary f l o w . I n case o f a c o n s i d e r a b l e c u r v a t u r e o f t h e f l o w and/or a l o n g bend i t i s a r a t h e r r o u g h approach t o n e g l e c t t h e c o n v e c t i v e i n f l u e n c e f r o m t h e secondary f l o w . However, no g e n e r a l a p p l i c a b l e and adequate way t o account f o r t h i s e f f e c t i n a two d i m e n s i o n a l model i s
a v a i l a b l e . Only f o r t h e case o f m i l d l y s l o p i n g banks and b o t t o m a model, w h i c h accounts f o r secondary f l o w c o n v e c t i o n , i s developed ( K a l k w i j k
& De V r i e n d , 1980). N e v e r t h e l e s s , t h e r e d i s t r i b u t i o n o f t h e main f l o w due t o t h e secondary f l o w i s a t l e a s t as pronounced i n r e c t a n g u l a r c h a n n e l s as i n channels w i t h m i l d l y s l o p i n g banks (De V r i e n d , 1981b).
I n t h e f i r s t p l a c e knowledge about t h e unperturbed s o l u t i o n must be o b t a i n e d . The unpertutbed bed l e v e l i s g i v e n by
\ = ^0 - ^0 • ^ ( 2 . 4 ) i n w h i c h i s a r e f e r e n c e l e v e l ( a t x=0) and IQ i s the e q u i l i b r i u m s l o p e o f t h e bed. E q u a t i o n (2.4) i n s e r t e d i n e q s . ( 2 . 1 ) , (2.2) and (2.3) i n c o m b i n a t i o n w i t h impermeable s i d e w a l l , w h i c h p r o v i d e s t h e boundary c o n d i t i o n s , y i e l d s t h e f o l l o w i n g zero o r d e r s o l u t i o n h = hQ u = u = 0 WhQ V = 0 = TQ = pg I^hQ T = 0 y ( 2 . 5 )
where Q i s t h e t o t a l d i s c h a r g e and W t h e w i d t h o f t h e channel
I n o r d e r t o make t h e s t a b i l i t y a n a l y s i s f e a s i b l e t h e f l o w model, eqs. ( 2 . 1 ) , (2.2) and ( 2 . 3 ) , must be l i n e a r i z e d . T h i s i s done by superimpose a s m a l l p e r t u r b a t i o n t o t h e zero o r d e r s o l u t i o n i n t h e f o r m : h = hp + h 1 h' << U = Uy + u u' < < ^0 V = 0 + v' v' << ^0 "b = "o " z' << ^0 T ' X T ' X « -^0 T = 0 + y T ' y T ' y « ^0 (2.6)
I n s e r t i n g t h i s p e r t u r b e d s o l u t i o n i n t o t h e f l o w e q u a t i o n ( 2 , 1 ) , ( 2 . 2 ) and (2.3) and n e g l e c t i n g second and h i g h e r o r d e r terms l e a d s t o t h e l i n e a r i z e d f l o w e q u a t i o n s ^ .9h' 3z; ^ "^0 / x h's „ "0 9^ §^9^ " 9^) " ^ - h^) = ° ^'-'^
f L \ , i f - \ f - ] . \
( 2 . 8 )
0 9x 9y 9y ph. T Qu ÏB\
„„^\ .
0 (2.9) 0 9x 0 9z 0 9y For convenience d i m e n s i o n i e s s v a r i a b l e s x j i l l be i n t r o d u c e d . F o r t h i s t r a n s f o r m a t i o n t h e d e p t h o f f l o w h^ w i l l be used as t h e c h a r a c t e r i s t i c l e n g t h s c a l e and ^Q/^Q as c h a r a c t e r i s t i c t i m e s c a l e . For i n s t a n c e h = h'/hQ, z* = z'/hQ, V * = V ' / U Q e t c . The a s t e r i s k s i n d i c a t e d i m e n s i o n i e s s p e r t u r b a t i o n v a r i a b l e s , b u t x \ r i l l be o m i t t e d when t h a t does n o t g i v e r i s e t o c o n f u s i o n . The d i m e n s i o n i e s s s e t o f e q u a t i o n s now becomes 9x 9y 9y y + 3Z + = 0 (2.12) 9x 9y 9x i n w h i c h F i s t h e Froude number h o l d i n g f o r t h e u n p e r t u r b e d f l o x j s i t u a t i o n p = (2.13) and f a roughness c o e f f i c i e n t d e f i n e d as f = = -S (2.14) P - 0 i n w h i c h C i s t h e w e l l k n o w n Chezy roughness c o e f f i c i e n t .2.2 Bed shear s t r e s s
For two reasons t h e bed shear s t r e s s deserves some e x t r a a t t e n t i o n . F i r s t , t h e shear s t r e s s must be e l i m i n a t e d by a c o n s t i t u t i v e r e l a t i o n i n o r d e r t o c l o s e t h e system o f e q u a t i o n s ( 2 . 1 ) , ( 2 . 2 ) and ( 2 . 3 ) .
Second, t h e d i r e c t i o n o f t h e bed shear s t r e s s , w h i c h g e n e r a l l y w i l l d e v i a t e f r o m t h e main f l o w d i r e c t i o n due t o secondary f l o w , has a l a r g e i n f l u e n c e on t h e d i r e c t i o n o f t h e sediment t r a n s p o r t .
The t r a d i t i o n a l method t o e x p r e s s t h e bed shear s t r e s s by means o f t h e f l o w parameter, i s Chezy's l a w , w h i c h , f o r u n i f o r m f l o w , i m p l i e s t h a t t h e shear s t r e s s i s p r o p o r t i o n a l t o t h e square o f t h e mean f l o w v e l o c i t y . I n a h o r i z o n t a l t w o - d i m e n s i o n a l f l o w t h e bed shear s t r e s s i s o f t e n expressed by r f u /u^ + v ^ (2.15) T y p f V Vu^ + v^^
w h i c h holds good i n case o f no secondary f l o w . As a f i r s t a p p r o x i m a t i o n eq. (2.15) i s a p p l i e d i n t h e f l o w model, i . e . t h e secondary f l o w i s o n l y t a k e n i n t o account i n t h e model f o r t h e sediment movement.
As t h e roughness c o e f f i c i e n t i s n o t a c o n s t a n t , t h e v a r i a t i o n o f f must be expressed i n terms o f t h e f l o w p a r a m e t e r s . However no g e n e r a l a p p l i c a b l e and r e l i a b l e model i s a v a i l a b l e . S e v e r a l s c i e n t i s t s u s i n g d i f f e r e n t
approaches have d e a l t w i t h roughness p r e d i c t i o n , and g e n e r a l agreement.seems t o e x i s t about a f u n c t i o n a l r e l a t i o n s h i p f o r t h e roughness c o e f f i c i e n t l i k e f = f ( F ,1) (2,16) S' u. i n w h i c h F = — — i s t h e d e n s i m e t r i c Froude number, A i s t h e r e l a t i v e d e n s i t y o f t h e sediment compared t o t h e d e n s i t y o f t h e w a t e r and d i s a c h a r a c t e r i s t i c d i a m e t e r o f t h e s e d i m e n t . Thus t h e bed
shear s t r e s s a c t u a l l y i s a' f u n c t i o n o f t h e f l o w v e l o c i t y and t h e s l o p e . The c o n s t i t u t i v e r e l a t i o n can t h e n be l i n e a r i z e d b y means o f a d o u b l e T a y l o r s e r i e e x p a n s i o n i n t o t h e s e two v a r i a b l e s . The terms i n eqs.
(2.10) and ( 2 . 1 1 ) , w h i c h c o n t a i n s t h e shear s t r e s s , can t h e n be a p p r o x i m a t e d by (see a l s o P a r k e r , 1976)
f (T - h) = (M,u - M h) ( 2 . 1 7 ) f t = f , v ( 2 . 1 8 ) y O i n w h i c h Mj and a r e g i v e n by ^ 0 ^ ^ 0 I ^ % M = ( 2 + ^ ^ ) / ( 1 - I
- f )
1 fg 8UQ fg 9 1 I ^'o ^ ^ 2 = ' / - I g - 9 1 - )I n case o f a c o n s t a n t roughness c o e f f i c i e n t = 2 and Mj = 1 .
I n a c u r v e d f l o w t h e r e i s a t r a n s v e r s e c i r c u l a t i o n due t o t h e n o n - u n i f o r m v e r t i c a l d i s t r i b u t i o n o f t h e main f l o w . For t h a t r e a s o n t h e d i r e c t i o n o f t h e f l o w c l o s e t o t h e b o t t o m w i l l d e v i a t e f r o m t h e d i r e c t i o n o f t h e d e p t h averaged f l o w . I n case o f f u l l y developed f l o w i n a w i d e c u r v e d c h a n n e l t h e h o r i z o n t a l component o f t h i s secondary f l o w i s o f t e n f o u n d by s o l v i n g t h e momentum e q u a t i o n i n t r a n s v e r s e d i r e c t i o n , d i s r e g a r d i n g a l l l a t e r a l f r i c t i o n terms and a l l i n e r t i a terms e x c e p t t h e c e n t r i f u g a l ones. A p p l y i n g t h i s p r o c e d u r e t h e d i r e c t i o n o f t h e bed shear s t r e s s can be e x p r e s s e d l i k e
t a n 6 = - a | ( 2 - 1 9 )
i n w h i c h 6 i s t h e a n g l e between t h e shear s t r e s s and t h e d i r e c t i o n o f t h e c h a n n e l a x i s , R i s t h e r a d i u s o f c u r v a t u r e o f t h e c h a n n e l and a i s a c o n s t a n t depending on t h e model f o r t h e l o n g i t u d i n a l f l o w . Rozowski ( 1 9 5 7 ) f o u n d a = 1 0 1 2 f o r a l o g a r i t h m i c v e l o c i t y p r o f i l e . L a t e r De V r i e n d ( 1 9 7 7 ) m o d i f i e d Rozowski's t h e o r y and f o u n d a = 2 / K ^ ( 1 - — ) i n w h i c h K 0 , 4 i s t h e v o n Karman c o n s t a n t . Engelund ( 1 9 7 4 ) K o b t a i n e d a ^ 7 f o r a p a r a b o l i c d i s t r i b u t i o n o f t h e l o n g i t u d i n a l f l o w v e l o c i t y . For a p a r a b o l i c v e l o c i t y d i s t r i b u t i o n i n t h e upper p a r t o f t h e f l o w and a l o g a r i t h m i c i n t h e l o w e r p a r t Knudsen ( 1 9 8 1 ) f o u n d a ^ 1 0 1 1 . Thus a = 1 0 i s a r e p r e s e n t a t i v e t h e o r e t i c a l v a l u e .
So f a r t h e model f o r t h e d i r e c t i o n o f t h e bed shear s t r e s s i s based on pure t h e o r e t i c a l c o n s i d e r a t i o n s f o r an i d e a l i z e d c h a n n e l . E x p e r i m e n t s i n c u r v e d f l u m e s w i t h smooth bed and f i n i t e w i d t h show t h a t t h e t h e o r e t i c a l models t e n d t o u n d e r e s t i m a t e t h e m a g n i t u d e o f t h e shear s t r e s s a n g l e
o f u n c e r t a i n t y i n t h e bed shear s t r e s s a n g l e .
I n a d e v e l o p i n g f l o w ( e . g . e n t r a n c e and e x i t o f a bend) t h e s t r e a m l i n e c u r v a t u r e does n o t e q u a l t h e channel c u r v a t u r e . However i t i s assumed t h a t eq. (2.19) a l s o a p p l i e s f o r the a n g l e between t h e s t r e a m l i n e s and t h e bed shear s t r e s s i n a d e v e l o p i n g f l o w . I n t h i s case t h e a n g l e between t h e bed shear s t r e s s and t h e c h a n n e l a x i s becomes
t a n 6 = ^ - a I (2.20) s i n w h i c h R i s t h e r a d i u s o f c u r v a t u r e o f t h e s t r e a m l i n e . I n a s t r a i g h t s c h a n n e l t h e s t r e a m l i n e c u r v a t u r e can be a p p r o x i m a t e d by (De V r i e n d , 1978) 1 1 9v R = ~ u ^ (2.21) Combining eq. (2.20) and (2.21) l e a d s t o a model f o r t h e d i r e c t i o n
o f t h e bed shear s t r e s s i n terms o f t h e dependent v a r i a b l e s
t a n . - ^ . a i | ï (2.22,
This model a p p l i e s i f t h e secondary f l o w i s c o n s i d e r e d t o respond i m m e d i a t e l y t o a change i n f l o w c u r v a t u r e . As w e l l as t h e main f l o w , t h e secondary f l o w w i l l need a c e r t a i n l e n g t h a f t e r t h e b e g i n n i n g o f a bend b e f o r e i t i s f u l l y d e v e l o p e d , and a c e r t a i n l e n g t h t o decay beyond a bend. I t i s v e r y i m p o r t a n t t o have a d e s c r i p t i o n o f t h i s r e t a r d e d a d a p t i o n t o change i n c u r v a t u r e , because t h e s t r e a m l i n e c u r v a t u r e o f t h e f l o w over a l t e r n a t e b a r s r a p i d l y changes s i g n . De V r i e n d (1981) s u g g e s t e d t o d e s c r i b e t h i s secondary f l o w i n e r t i a by means o f a damped e x p o n e n t i a l f u n c t i o n l i k e h
1^
+ X = - — (2 23) i n w h i c h x i s a v a r i a b l e r e p r e s e n t i n g t h e degree o f development o f t h esecondary f l o w and b i s a d i m e n s i o n i e s s c o n s t a n t . A c c o r d i n g t o De V r i e n d b 1.3. The d i r e c t i o n o f t h e bed shear s t r e s s i s now g i v e n by
t a n 6 = - + a X (2.24) u
I n appendix A i t i s o u t l i n e d how De V r i e n d o b t a i n e d t h i s model. The model f o r t h e secondary flox7 i n e r t i a i s based on c o n s i d e r a t i o n s about t h e decay o f t h e h e l i c a l f l o w a f t e r a bend i n a v e r t i c a l p l a n e t h r o u g h a s t r a i g h t s t r e a m l i n e . F u r t h e r more d e r i v a t i v e s i n y - d i r e c t i o n
( e x c e p t t h e p r e s s u r e ) a r e c o n s i d e r e d much s m a l l e r t h a n d e r i v a t i v e s i n z - d i r e c t i o n and c o n s e q u e n t l y t h e y a r e n e g l e c t e d . T h i s i s o f c o u r s e a v e r y schematic approach, and i t may n o t be j u s t i f i e d t o employ t h e model t o t h e development o f t h e secondary f l o w i n case o f c u r v e d s t r e a m -l i n e s and/or i n n a r r o w c h a n n e -l s . F u r t h e r more so f a r o n -l y e x p e r i m e n t a -l v e r i f i c a t i o n f o r smooth bed have been c a r r i e d o u t (De V r i e n d , 1981).
The model f o r t h e d i r e c t i o n o f t h e shear s t r e s s and t h e secondary f l o w i n e r t i a y i e l d s , when l i n e a r i z e d and made d i m e n s i o n i e s s i n t h e same way as b e f o r e - w i t h o u t secondary f l o w i n e r t i a t a n 6 = V + a 1^ (2.25) 9x - w i t h secondary f l o w i n e r t i a 3^ + ^ = ^ (2.26) y|-9x ^ 3x t a n Ö = V + a X (2.27) i n w h i c h v, S and x are d i m e n s i o n i e s s p e r t u r b a t i o n p a r a m e t e r s . 2.3. E q u a t i o n o f c o n t i n u i t y f o r t h e sediment The mass b a l a n c e f o r t h e sediment y i e l d s
9z, 9s 9s
— A + _ J i + = 0 (2.28) 9 t 9x 9y
i n w h i c h s and s a r e sediment t r a n s p o r t s p e r u n i t w i d t h i n x and y
X y
d i r e c t i o n , r e s p e c t i v e l y . The d i r e c t i o n o f t h e sediment t r a n s p o r t \jj i s d e f i n e d as
s
t a n ijj = (2.29)
X
By means o f eq. (2.29) t h e t r a n s v e r s e sediment t r a n s p o r t i n eq. (2.28) can be e l i m i n a t e d . The e q u a t i o n o f c o n t i n u i t y t h e n becomes
3z, 9s 9s , b ^ X ^ ^ , 3 I t a n i ^ Q
9 t 9x 9y X 9y
The zero o r d e r s o l u t i o n o f eq. (2.30) i n c o m b i n a t i o n w i t h t h e u n p e r t u r b e d f l o w y i e l d s
Z ^ = Z Q - I x
''x " ^0 (2.31) t a n ijj = 0
Superimposing a s l i g h t p e r t u r b a t i o n t o t h e zero o r d e r s o l u t i o n , i n s e r t i n g i n eq. (2.30) and n e g l e c t i n g second and h i g h e r o r d e r terms l e a d s t o t h e l i n e a r i z e d e q u a t i o n o f c o n t i n u i t y 1 ^ ' + l £ ' + 3 s t a n d i ; ' ^ 9t 9x ^0 9y " ^^--^^^ D i m e n s i o n i e s s v a r i a b l e s w i l l be i n t r o d u c e d as f o l l o w s :
\ '
X = x/hg y"" = y/hg * 1/ t " = t Up/h 0 I n terms o f t h e s e d i m e n s i o n i e s s p e r t u r b a t i o n v a r i a b l e s t h e e q u a t i o n o f c o n t i n u i t y becomesi n w h i c h t h e a s t e r i s k s • have been o m i t t e d . 3 i s t h e r a t i o between t h e s p e c i f i c d i s c h a r g e s o f the u n p e r t u b e d s i t u a t i o n , i . e . g = t h e s p e c i f i c d i s c h a r g e s o f the f l o w o f t h e sediment e v a l u a t e d i n t h e SO " o % 2.4. Sediment t r a n s p o r t
The magnitude and d i r e c t i o n o f t h e sediment t r a n s p o r t must be e x p r e s s e d i n terms o f t h e dependent p a r a m e t e r s i n o r d e r t o c l o s e t h e system o f e q u a t i o n s . I n p r i n c i p l e t h e s t a b i l i t y a n a l y s i s i s based on eq. ( 2 . 3 3 ) , so a p r o p e r d e s c r i p t i o n o f t h e sediment t r a n s p o r t p r o p e r t i e s i s v e r y i m p o r t a n t . T h e r e f o r e t h i s p o i n t demands some a t t e n t i o n .
A l a r g e number o f f o r m u l a e w h i c h r e l a t e t h e amount o f s e d i m e n t t r a n s p o r t and t h e flox7 parameters a r e a v a i l a b l e . A good d e a l o f t h e t r a n s p o r t f o r m u l a e y i e l d s t h a t t h e t r a n s p o r t r a t e i s a f u n c t i o n o f t h e d i m e n s i o n i e s s shear s t r e s s ( S h i e l d p a r a m e t e r ) 9 = h l / A d and p o s s i b l y more p a r a m e t e r s , i . e . s = s(e, ) (2.34) By means o f a d o u b l e T a y l o r s e r i e e x p a n s i o n t h e p e r t u r b e d d i m e n s i o n i e s s sediment t r a n s p o r t can be a p p r o x i m a t e d by s = N ] U - N^h (2,35) i n w h i c h Nj and are g i v e n by ( P a r k e r , 1976) «2 s„ 3I„ «2
Mj and are d e f i n e d a t p.10. B o t h Nj and and and are e v a l u a t e d i n t h e u n p e r t u b e d s i t u a t i o n .
Models f o r t h e d i r e c t i o n o f t h e sediment t r a n s p o r t a r e s c a r c e . Three t h e o r e t i c a l models w i l l be c o n s i d e r e d h e r e . The models a r e based on t h e a s s u m p t i o n t h a t a g r a v i t a t i o n a l f o r c e a c t i n g a l o n g t h e i n c l i n e d bed causes a d e v i a t i o n o f t h e d i r e c t i o n o f t r a n s p o r t f r o m t h e d i r e c t i o n o f t h e bed shear s t r e s s . The models a r e
Engelund (1974) -tan$ 8y 1 3 2 tan \b = t a n ö - -— /o ^ tan$ 3 v {Z.3b) Koch (1980) -- I
If
t a n , i . - - — — ( 2 . 3 7 ) Engelund (1981) -t-n^ = tan6~-^ ^ (2.38) i n w h i c h $ i s t h e dynamic f r i c t i o n a n g l e ($ - 30° - 40°), y i s a f a c t o r o f t h e o r d e r o f magnitude o f u n i t y , 6 i s t h e S h i e l d s parameter and 6' i s the e f f e c t i v e S h i e l d s p a r a m e t e r , i . e . t h e shear s t r e s s r e l a t e d t o t h e s k i n f r i c t i o n . A c c o r d i n g t o E i n s t e i n (1950) 0' can be o b t a i n e d f r o m ^0 6 + 2.5 I n ^ /TT - ° ^ ^'^ ITSÓ-Q (2.39)e
Engelund (1974) used eq. (2.36) t o c a l c u l a t e t h e bed t o p o g r a p h y i n a meandering c h a n n e l . A t f i r s t s i g h t t h e good agreement between t h e o r y and e x p e r i m e n t s s u p p o r t s eq. ( 2 . 3 6 ) . However, i f t h e r e l e v a n t parameter f o r t h e e x p e r i m e n t s are i n s e r t e d i n eq. (2.37) and eq. ( 2 . 3 8 ) , t h e n t h e t h r e e models a r e a l m o s t i d e n t i c a l . Eq. (2.37) i s used t o c a l c u l a t e t h e bed p r o f i l e f o r a few f u l l y developed bends (Koch, 1980). The agreement between t h e o r y and e x p e r i m e n t a l d a t a was s a t i s f a c t o r y , b u t i t was n e c e s s a r y t o t u n e t h e model w i t h t h e f a c t o r y. Eq. (2.38) i s a t h e o r e t i c a l model i n w h i c h a c o n s t a n t (0.6) i s d e t e r m i n e d e m p i r i c a l l y . The model i s t e s t e d w i t h a l a r g e number o f e x p e r i m e n t a l d a t a o b t a i n e d f r o m t h r e e a l m o s t 360° bends ( d a t a : Zimmermann & Kennedy, 1978). The d a t a c o n f i r m t h e t h e o r e t i c a l model b u t t h e r e i s a r a t h e r l a r g e s c a t t e r i n t h e d a t a , e s p e c i a l l y f o r low S h i e l d p a r a m e t e r s .
F u r t h e r iiEire a l l bends i n t h e s e e x p e r i m e n t s had a r a t h e r s t e e p t r a n s v e r s e s l o p e , and t h e r e f o r e i t cannot be t a k e n f o r g r a n t e d t h a t t h e model and/or t h e e m p i r i c a l
c o n s t a n t do a p p l y i n case o f a weak t r a n s v e r s e s l o p e . C o n s e q u e n t l y t h e model may n o t a p p l y i n case o f a s m a l l p e r t u r b a t i o n . N e v e r t h e l e s s f r o m t h e t h r e e models eq. (2.3 8) seems most r e l i a b l e .
W i t h a l i n e a r i z a t i o n a c c o r d i n g t o t h e r u l e s o u t l i n e d i n c h a p t e r 2.1 a l l t h r e e models g e t t h e f o r m t a n (|J = t a n (S - c ||- (2.40) i n w h i c h ip, ö and z a g a i n a r e d i m e n s i o n i e s s p e r t u r b a t i o n p a r a m e t e r s . E q u a t i o n (2.40) w i l l be a p p l i e d i n t h e f o l l o w i n g w i t h c c a l c u l a t e d a c c o r d i n g t o eqs. (2.38) and ( 2 . 3 9 ) .
S o l u t i o n f o r a d o u b l e harmonic p e r t u r b a t i o n The s t a b i l i t y a n a l y s i s i s i n f a c t an a n a l y s i s o f t h e development i n t i m e o f a t w o - d i m e n s i o n a l s m a l l a m p l i t u d e wave superimposed on t h e e q u i l i b r i u m bed. T h i s p e r t u r b a t i o n can m a t h e m a t i c a l l y be e x p r e s s e d as z = z s i n ( k ^ y ) c o s ( k ^ y ) exp- i ( k x - (j)t) ( 3 . 1 ) ^ . ^ 0 . . . i n w h i c h z i s t h e a m p l i t u d e o f t h e p e r t u r b a t i o n , k = m-rr — i s t h e d i m e n s i o n i e s s w w
wavenumber i n t r a n s v e r s e d i r e c t i o n , m i s two times t h e number o f waves i n a c r o s s - s e c t i o n , k = 2TT ^ Q /L t h e d i m e n s i o n i e s s wavenumber i n l o n g i t u d i n a l d i r e c t i o n , L t h e wave l e n g t h , ^ t h e complex c e l e r i t y and /V = - 1 .
The v a r i a b l e s m d e t e r m i n e t h e bed p a t t e r n . For m = 1 t h e r e i s one sub-merged b a r i n a c r o s s - s e c t i o n , w h i c h c o r r e s p o n d s t o t h e e a r l y s t a g e o f meandering. For m = 2,3 e t c . t h e p e r t u r b a t i o n o f t h e bed c o n s i s t s o f an
i n c r e a s i n g number o f surbmerged b a r s , w h i c h g h a r a c t e r i z e r s t h e i n c i p i e n t b r a i d i n g r i v e r ( f i g u r e 1 ) .
The s o l u t i o n f o r t h e d o u b l e harmonic p e r t u r b a t i o n w i l l t a k e p l a c e i n two s t e p s . F i r s t t h e s o l u t i o n o f t h e f l o w model w i l l be d e r i v e d . Next t h i s s o l u t i o n w i l l be used t o o b t a i n an e x p r e s s i o n f o r t h e complex c e l e r i t y ; a c t u a l l y b e i n g t h e s o l u t i o n o f t h e e q u a t i o n o f c o n t i n u i t y .
3.1. S o l u t i o n o f t h e f l o w model
The l i n e a r i z e d f l o w model a r i s e s f r o m c o m b i n i n g t h e eqs. ( 2 . 1 0 ) , ( 2 . 1 1 ) , (2. 12), (2.17) and (2.18)
i ^ ^ ~ ' ( i ^ l f ) u - H ^ h ) = 0 ( 3 . 2 )
As t h e r e i s no t i m e dependence i n t h i s s e t o f e q u a t i o n s , i . e . s t e a d y f l o w , the complex c e l e r i t y i n eq. ( 3 . 1 ) will f o r t h e t i m e b e i n g be t a k e n e q u a l z e r o . I t i s e a s i l y seen t h a t t h e s o l u t i o n has t h e f o r m z = z s i n ( k ^ y ) c o s ( k ^ y ) exp i k X a. u = u h = h sin(k^^y) cos (k^^y) s i n ( k ^ y ) c o s ( k ^ y ) ^ exp i k X exp i k X ( 3 . 5 ) V = V c o s ( k ^ y ) -8in(k^y)_^ exp i k X
m w h i c h u, h and v a r e complex a m p l i t u d e s . The i n f l u e n c e o f a complex a m p l i t u d e can be d i s p l a y e d by r e p r e s e n t i n g f o r i n s t a n c e u i n p o l a r f i rm u = r exp i é u ^ ^u s i n ( k y ) c o s ( k ^ y ) w exp i k X = r s i n ( k ^ y > c o s ( k y ) w exp i ( k x + ij;^) (3.6) i n w h i c h r = / u ^ + u? u r 1 and t a n I|J : I . 1
So t h e complex a m p l i t u d e d e t e r m i n e s t h e phase and t h e a m p l i t u d e o f t h e wave.
Impermeable s i d e w a l l s p r o v i d e t h e boundary c o n d i t i o n s f o r t h e f l o w model, i . e .
V = 0 f o r y = + W ( 3 . 7 )
I n s e r t i n g eq. ( 3 . 7 ) i n t h e s o l u t i o n f o r v i n c o m b i n a t i o n w i t h t h e d e f i n i t i o n o f k y i e l d s
V c o s ( y I T) e x p i k X = 0 => m
V s i n (-J I T) e x p i k x = 0 =>in
1,3,5
2,4,6
( 3 . 8 )
Consequently t h e upper s o l u t i o n i n eq. ( 3 . 6 ) a p p l i e s f o r m ,podd and t h e l o w e r one f o r m even.
The complex a m p l i t u d e can noxj be f o u n d by s o l u t i o n o f a s e t o f s i m p l e a l g e b r a i c e q u a t i o n s . I n s e r t i n g eq. ( 3 . 6 ) i n t o t h e l i n e a r i z e d f l o w model, eqs. ( 3 . 2 ) , ( 3 . 3 ) and ( 3 . 4 ) , y i e l d s a f t e r r e d u c t i o n o f t h e h a r m o n i c p a r t i k u + F2(£]^li^ + i l ^ ^ ) + f M j u i k v + F ^ ( k ft!+k z ) + f v = 0 w w i k u + i k h - k v = 0 w f M2 1Ï = 0 (3.9) (3.10) (3.11)
After'.risairtrtamging and d i v i s i o n by k t h e t h r e e l i n e a r e q u a t i o n s can be w expressed i n m a t r i x - f o r m as F2 e M + i 1 -e M + i F ^ 1 -2 0 F i 1 i 1 0 e + i 1 -1 r 'V n " - i l U " - i l - 1 h = z - 1 - 0 L V - - 0 (3.12) i n w h i c h 1 = k/k and e = f / k . w w
The d e t e r m i n a n t o f t h i s complex m a t r i x can be e l a b o r a t e d t o
D = e[Mj + 1^(1 - F ^ ( l + Mj +
M^))-+ i [ 1 ( 1 M^))-+e?F^(Mj M^))-+ M^)) M^))-+ 1^(1 - F \
The s o l u t i o n o f t h e t h r e e complex v a r i a b l e s f i n a l l y a r e
u = ^ [ e ( l 2 - + i l3J (3.14)
h = f + M,) - i ( 1 + l 3 ) ] (3.15)
O,
V = ^ i 1 [-e (Mj + M^) - i l ] (3.16)
I n f i g u r e 2 t h e phase and t h e a m p l i t u d e o f u, h and v a r e d e p i c t e d as a f u n c t i o n o f t h e r e l a t i v e wavenumber 1 f o r two d i f f e r e n t v a l u e s o f e. I n f i g u r e 3 t h e i n f l u e n c e o f t h e Froude number i s d e p i c t e d . I n b o t h cases t h e roughness c o e f f i c i e n t i s c o n s i d e r e d t o be i n d e p e n d e n t o f t h e f l o w p a r a m e t e r s , so l i = 2, and Mj = 1.
The phase l a g o f t h e w a t e r d e p t h i s a l m o s t c o n s t a n t IT and t h e a m p l i t u d e i s c l o s e t o u n i t y , a t l e a s t f o r moderate Froude number and f o r l o n g waves i n l o n g i t u d i n a l d i r e c t i o n . I n t h i s case t h e r i g i d - l i d a p p r o x i m a t i o n a p p l i e s , i . e .
h = - z (3.17)
T h i s i s a v e r y a t t r a c t i v e a p p r o x i m a t i o n because a p e r t u r b a t i o n o f t h e d e p t h o f f l o w can be c o n s i d e r e d i n s t e a d o f a p e r t u r b a t i o n o f t h e bed l e v e l .
Consequently one dependent v a r i a b l e can be e l i m i n a t e d . Reducing z/D f r o m eqs. ( 3 . 1 4 ) , (3.15) and (3.16) y i e l d s . ^ e d l ^ - l ^ ) - i 13 u = h — — • ( 3 . 18) e ( M j + l 2 ) + i ( l + l 3 ) v = h i l - ' (3.19) e ( M j + l 2 ) + i ( l + l 3 )
The r e s u l t i s now i n d e p e n d e n t o f t h e Froude number. Note t h a t eqs. (3.18) and (3.19) always a p p l y i n d e p e n d e n t o f t h e r i g i d - l i d a p p r o x i m a t i o n .
F i g u r e 2 p r o v i d e s t h e p o s s i b i l i t y o f making some r e f l e c t i o n s about t h e cause o f t h e i n s t a b i l i t y . For a g i v e n wave l e n g t h t h e phase l a g between u and z w i l l be e q u a l t o — , w h i c h c o r r e s p o n d s t o maximum p o s i t i v e g r a d i e n t o f u a t
t h e same l o c a t i o n where z i s m i n i m a l . A s i m p l e one d i m e n s i o n a l e q u a t i o n o f c o n t i n u i t y f o r t h e sediment, i n w h i c h t h e sediment t r a n s p o r t i s c o n s i d e r e d t o be p r o p o r t i o n a l t o t h e f l o w v e l o c i t y , reads
_3z 3 t ^^0 _3u 8x = 0 (3.20) I n t h e case c o n s i d e r e d — w i l l be n e g a t i v e and t h e a m p l i t u d e o f t h e p e r t u r b a t i o n w i l l grow. I n t h e p r e s e n t a n a l y s i s t h e o c c u r r e n c e o f maximum i n s t a b i l i t y i s more c o m p l i c a t e d t h a n o u t l i n e d h e r e because f e a t u r e s l i k e t r a n s v e r s e t r a n s p o r t , secondary f l o x ^ e . t . c . a r e t a k e n i n t o c o n s i d e r a t i o n . However, t h e phase l a g between u and z o f about ^ i s s t i l l one o f t h e dominant f a c t o r s .
3.2. The s t a b i l i t y a n a l y s i s
To c a r r y o u t t h e s t a b i l i t y a n a l y s i s eq. ( 3 . 1 ) and s i m i l a r p e r t u r b a t i o n s f o r u, V and h a r e i n t r o d u c e d i n t o t h e l i n e a r i z e d e q u a t i o n s o f c o n t i n u i t y f o r t h e sediment. The r e a l p a r t o f t h e c e l e r i t y i n eq. ( 3 . 1 ) i s r e l a t e d t o t h e m i g r a t i o n v e l o c i t y o f t h e p e r t u r b a t i o n , whereas t h e i m a g i n e r y p a r t (j)^ i s t h e e x p o n e n t i a l g r o w t h r a t e as
z = z
z
s i n ( k ^ y )
cos (k^^y) exp i ( k x - (f)t) =
(3.21) exp s i n ( k ^ y ) c o s ( k ^ y ) exp i ( k x - <f,^t)
For (f)^ < 0 t h e a m p l i t u d e o f t h e p e r t u r b a t i o n w i l l d e c r e a s e , whereas i n s t a b i l i t y o c c u r s f o r > 0.
The l i n e a r i z e d e q u a t i o n o f c o n t i n u i t y f o r t h e sediment can be expressed i n terms o f t h e dependent v a r i a b l e s by c o m b i n i n g eqs. ( 2 . 2 6 ) , ( 2 . 2 7 ) , (2.33) and (2.40)
3 3 t ^ 1 3x ^2 3x 3y 3y ~ (3.22)
w i t h X f r o m
b 8X _^ y 3
I n s e r t i n g eq, (3.1) and t h e s i m i l a r one f o r t h e r e m a i n i n g dependent v a r i a b l e s and r e d u c i n g t h e harmonic p a r t l e a d s t o t h e d i s p e r s i o n e q u a t i o n , w h i c h reads
^ i ( f . z = N, i k u - N „ i k h - k v - -~r^^— k V + c k2 z ( 3 . 2 4 ) g l 2 w . , b , w w 1 k — + 1 / • l b , ,.-1 8v as X = ( i k — + 1) — / f S u b s t i t u t i n g eqs. ( 3 . 1 4 ) , (3.15) and (3.16) i n t o t h e d i s p e r s i o n e q u a t i o n g i v e s , a f t e r a few m a n i p u l a t i o n s , an e x p r e s s i o n f o r t h e complex c e l e r i t y . k 1 I = - i c k j + ~ - [ e ( l 2 ( N j + N 2 ) + N2MJ - NjM2 + Mj + M2 + i ( l 3 ( N +N ) + 1(N + 1 ) ] (3.25) 'Ï "2 k^ + a ( i — k + 1 ) " ' [ - l 3 + i e D / f w 1- 1 2 - ^ I n w h i c h D i s g i v e n by eq. (3.13)
The t h r e e main terms i n e q . ( 3 , 2 5 ) c a n be a t t r i b u t e d t o d i f f e r e n t e f f e c t s . The t e r m "- i c k^ " i s due t o t h e g r a v i t a t i o n a l f o r c e on t h e g r a i n s a l o n g t h e
w
t r a n s v e r s e s l o p e o f t h e bed. The t e r m "k 1/D [,..1 " d e s c r i b e s t h e i n f l u e n c e w '¬
o f t h e main f l o w on t h e complex c e l e r i t y . T h i s t e r m i s i d e n t i c a l w i t h t h e e x p r e s s i o n P a r k e r (1976) based h i s s t a b i l i t y a n a l y s i s on. The r e m a i n i n g t e r m i s new i n t h i s t y p e o f a n a l y s i s . I t a c c o u n t s f o r t h e i n f l u e n c e o f t h e secondary f l o w on t h e s t a b i l i t y o f t h e b e d . The e x p r e s s i o n f o r t h e complex c e l e r i t y i s n o t v e r y t r a n s p a r a n t . A s h o r t s e n s i t i v i t y a n a l y s i s w i l l be c a r r i e d o u t i n o r d e r t o i l l u s t r a t e t h e i n f l u e n c e o f t h e d i f f e r e n t v a r i a b l e s and p a r a m e t e r s . However, f i r s t a d i s c u s s i o n o f t h e b a s i c a s s u m p t i o n s , w h i c h u n d e r l i e s t h i s a n a l y s i s , w i l l be p r e s e n t e d .
4. D i s c u s s i o n on b a s i c assumption
Always when d e a l i n g w i t h sediment t r a n s p o r t i n a l l u v i a l channels a g r e a t d e a l o f u n c e r t a i n t y i n t h e p r e d i c t i o n o f t h e t r a n s p o r t r a t e and t h e a l l u v i a l roughness i s p r e s e n t . Of course t h i s a l s o a p p l i e s t o t h i s a n a l y s i s o f t h e l i n e a r i z e d e q u a t i o n s . However, i n t h e p r e s e n t case o f a t w o - d i m e n s i o n a l approach, t h e major source o f u n c e r t a i n t y o r i g i n a t e s f r o m t h e model f o r t h e d i r e c t i o n o f t h e bed shear s t r e s s (secondary floxvr) and f r o m t h e model f o r t h e d i r e c t i o n o f t h e sediment t r a n s p o r t . I n c h a p t e r 2 t h e r e l i a b i l i t y o f t h e
models f o r t h e secondary f l o w and f o r t h e t r a n s p o r t d i r e c t i o n were b r i e f l y t r e a t e d . A t h o r o u g h d i s c u s s i o n o f t h e roughness and t r a n s p o r t r a t e p r e d i c t i o n i s o u t o f t h e scope o f t h i s r e p o r t . I n t h e f o l l o w i n g a b r i e f d i s c u s s i o n o f t h e problems w h i c h a r e s p e c i f i c f o r t h e p r e s e n t approach w i l l be g i v e n . A c o n d i t i o n f o r t h e v a l i d i t y o f t h e approach i s t h a t t h e c o n s i d e r e d submerged b a r s d i f f e r s i g n i f i c a n t l y f r o m t h e bed f o r m s . For i n s t a n c e i f t h e l e n g t h o f t h e a l t e r n a t e b a r s and t h e dunes a r e o f t h e same o r d e r o f m a g n i t u d e , t h e n the a m p l i t u d e o f t h e a l t e r n a t e b a r s must be much l a r g e r t h a n t h e dune h e i g h t i n o r d e r t o a v o i d any a p p r e c i a b l e i n f l u e n c e on t h e f l o w ( p h a s e l a g e t c . ) . The o t h e r way around, i f t h e a m p l i t u d e o f t h e a l t e r n a t e b a r s i s o f t h e same o r d e r o f magnitude as t h e dune h e i g h t , t h e n t h e i r l e n g t h must be much l a r g e r t h a n t h e l e n g t h o f t h e dunes i n o r d e r t o enable an a v e r a g i n g p r o c e d u r e over the l a r g e s c a l e b e d f o r m .
The h e i g h t o f t h e dunes i s t y p i c a l 10% - 20% o f t h e d e p t h o f f l o w , so t h e h e i g h t o f t h e a l t e r n a t e b a r s w i l l always be o f t h e same o r d e r o f m a g n i t u d e ,
f o r i n s t a n c e never a f a c t o r 10 l a r g e r . T h e r e f o r e t h i s a n a l y s i s a p p l i e s t o the cases where t h e a l t e r n a t e b a r s a r e much l o n g e r t h a n t h e dunes. S e v e r a l i n v e s t i g a t o r s have r e l a t e d t h e dune l e n g t h t o t h e d e p t h o f f l o w . Y a l i n (1964) suggested t h e dune l e n g t h - d e p t h o f f l o w r a t i o t o e q u a l f i v e . T h i s a n a l y s i s i n d i c a t e s a l e n g t h o f t h e a l t e r n a t e b a r s w h i c h i s o f t h e o r d e r o f m a g n i t u d e o f t h r e e t i m e s t h e w i d t h . An e x p r e s s i o n f o r t h e dune l e n g t h - a l t e r n a t e b a r l e n g t h r a t i o t h e n becomes 5 ^0 ^dunes ^ ^ a r s ~ 3" W~ (^'^^ Consequently t h i s a n a l y s i s o n l y a p p l i e s t o channels w i t h s m a l l d e p t h - w i d t h r a t i o s .
A n o t h e r q u e s t i o n a b l e p o i n t i n t h i s a n a l y s i s i s t h e l i n e a r i z a t i o n o f t h e model. Below, f i r s t a d i s c u s s i o n on t h e l i n e a r i z a t i o n o f t h e f l o w model w i l l be g i v e n , n e x t tUe l i n e a r i z a t i o n o f t h e bed l e v e l model w i l l be t r e a t e d
4.1. L i n e a r i z a t i o n o f t h e f l o w model
The f l o w model i s known t o be o n l y w e a k l y n o n - l i n e a r . A s u i t a b l e way t o d e m o n s t r a t e t h i s i s t o compare t h e s o l u t i o n o f t h e l i n e a r i z e d f l o w model w i t h t h e s o l u t i o n o b t a i n e d by a c o m p u t a t i o n a l model, w h i c h a l s o t a k e s t h e n o n - l i n e a r terms i n t o c o n s i d e r a t i o n .
I n f i g u r e 4 t h e r e s u l t o f such a comparison i s d e p i c t e d . The n o n - l i n e a r r e s u l t i s o b t a i n e d w i t h a c o m p u t a t i o n a l f l o w laodel w h i c h d i s r e g a r d s t h e t r a n s -v e r s e f r i c t i o n ( O l e s e n , 1982). The l i n e a r s o l u t i o n , a c c o r d i n g t o eqs. (3.18) and ( 3 . 1 9 ) , a r e c o r r e c t e d f o r t h i s o m i s s i o n .
I t i s e x p e c t e d t h a t a s h o r t wave i n l o n g i t u d i n a l d i r e c t i o n w i l l c o u r s e t h e l a r g e s t d i f f e r e n c e between t h e two models. I n f i g u r e 4 t h e wave l e n g t h i n l o n g i t u d i n a l d i r e c t i o n i s two t i m e s t h e w i d t h (1=1) and t h e a m p l i t u d e h=0.10 w h i c h c o r r e s p o n d s t o t h e o r d e r o f magnitude o f t h e dune h e i g h t . Even i n t h i s
case t h e l i n e a r i z a t i o n does n o t g i v e r i s e t o any a p p r e c i a b l e d i s c r e p a n c y . Thus t h e l i n e a r i z a t i o n o f t h e f l o w model does n o t c o n t e s t t h e v a l i d i t y o f t h i s s t a b i l i t y a n a l y s i s .
4.2. L i n e a r i z a t i o n o f t h e sediment model
The sediment t r a n s p o r t model has a s t r o n g l y n o n - l i n e a r c h a r a c t e r , w h i c h f i r s t o f a l l o r i g i n a t e s f r o m the n o n - l i n e a r dependence o f t h e t r a n s p o r t r a t e on t h e f l o w v e l o c i t y . The n o n - l i n e a r c h a r a c t e r causes a d e f o r m a t i o n o f an i n i t i a l s i n u s c o i d a l wave and f i n a l l y a shock w i l l be formed. The deformed wave can v e r y w e l l be d e s c r i b e d by a .Fourier s e r i e . U n f o r t u n a t e l y t h e a p p l i c a t i o n o f a F o u r i e r s e r i e w o u l d impede t h e a n a l y s i s c o n s i d e r a b l e . The a n a l y s i s would l e a d t o a complex c e l e r i t y w h i c h would depend on the l o n g i t u d i n a l c o o r d i n a t e X , i . e . t h e wave w o u l d d e f o r m and t h e i n i t i a l F o u r i e r s e r i e w o u l d no l o n g e r a p p l y .
However, i n case o f a s m a l l a m p l i t u d e o f t h e p e r t u r b a t i o n n o n - l i n e a r e f f e c t i s n e g l i g i b l e . The l i n e a r approach may t h e r e f o r e a t l e a s t g i v e a s a t i s f a c t o r y i n i t i a l g r o w t h or damping r a t e . Thus t h e l i n e a r i z a t i o n does n o t so much
r i v e r s , whereas t h e p r o p a g a t i o n and d a m p i n g / a m p l i f i c a t i o n o f a p e r t u r b a t i o n w i t h a c e r t a i n a m p l i t u d e may be t r e a t e d somewhat i n c o r r e c t l y due t o n o n - l i n e a r e f f e c t .
5. The complex c e l e r i t y
As mentioned b e f o r e t h e r e a l p a r t o f t h e complex c e l e r i t y g i v e s i n f o r m a t i o n about t h e m i g r a t i o n v e l o c i t y o f t h e p e r t u r b a t i o n s and t h e i m a g i n a r y p a r t shows t h e r a t e o f g r o w t h . The i m a g i n a r y p a r t o f t h e c e l e r i t y can t h e r e f o r e be used t o d i s t i n g u i s h between s t a b l e and u n s t a b l e r i v e r s as a p o s i t i v e ^. corresponds t o i n c r e a s i n g a m p l i t u d e o f t h e d i s t u r b a n c e and a n e g a t i v e one corresponds t o a d e c r e a s i n g a m p l i t u d e .
The a n a l y s i s a l s o y i e l d s t h e p r e v a i l i n g wave l e n g t h . I n f i g u r e 5 t h e
a m p l i f i c a t i o n f a c t o r ( t h e i m a g i n a r y p a r t o f t h e complex c e l e r i t y ) i s d e p i c t e d a a f u n c t i o n o f 1 . The u s u a l assumption i s t h a t a l t e r n a t e b a r s , w i t h t h e wave l e n g t h f o r w h i c h i> ^ has i t s maximum, w i l l d e v e l o p i n t h e stream. T h i s i s an o b v i o u s assumption, b u t i t can be doubted w h e t h e r t h i s i s t h e o n l y c r i t e r i o n f o r t h e f i n a l wave l e n g t h o f t h e a l t e r n a t e b a r s . A p o s s i b i l i t y i s t h a t non-l i n e a r e f f e c t may s h i f t t h e maximum a m p non-l i f i c a t i o n f r o m one x^ave non-l e n g t h t o a n o t h e r , when t h e a m p l i t u d e o f t h e p e r t u r b a t i o n i n c r e a s e s . The i n i t i a l d i s t u r b a n c e , w h i c h may be r e l a t e d t o t h e bed f o r m s , may have i n f l u e n c e on t h e d e v e l o p i n g a l t e r n a t e b a r p a t t e r n . Hoxrever, as no model e x i s t s w h i c h accounts f o r t h e s e e f f e c t s , t h e p r e s e n t a n a l y s i s w i l l be based on t h e assumption t h a t the maximum o f (f>^ d e t e r m i n e s t h e wave l e n g t h o f t h e d e v e l o p i n g submerged b a r s .
The complex p r o p a g a t i o n f a c t o r eq. (3.25) i s an i n t r i c a t e f u n c t i o n o f a l a r g e number o f v a r i a b l e s . T h e r e f o r e i t i s d i f f i c u l t t o r e c o g n i z e , w h i c h e f f e c t
the d i f f e r e n t v a r i a b l e s and terms have on t h e b e h a v i o u r o f e q , ( 3 . 2 5 ) . Never-t h e l e s s Never-t h e e x p r e s s i o n f o r Never-t h e complex c e l e r i Never-t y g i v e s a l r e a d y r i s e Never-t o an i m p o r t a n t o b s e r v a t i o n . The t e r m a c c o u n t i n g f o r t h e g r a v i t a t i o n a l f o r c e i s alxrays complex and n e g a t i v e (independend o f 1), thus i t i s s t a b i l i z i n g . The magnitude i s p r o p o r t i o n a l t o t h e square o f k^, whereas t h e t e r m a c c o u n t i n g f o r t h e main f l o w i s p r o p o r t i o n a l t o k^. The secondary f l o w t e r m i n c r e a s e s l i n e a r l y w i t h k^ f o r l a r g e 1 and q u a d r a t i c l y f o r s m a l l 1 . A l l e f f e c t s
c o n s i d e r e d t h e r e i s a l a r g e s t a b i l i z i n g e f f e c t f o r l a r g e k^. T h i s c o r r e s p o n d s w i t h t h e o b s e r v a t i o n t h a t n a r r o w channels r e m a i n s t a b l e .
Regarding t h e i n t r i c a c y o f t h e e x p r e s s i o n f o r t h e c e l e r i t y a s e n s i t i v i t y
For t h i s aim i t i s c o n v e n i e n t t o d i v i d e t h e v a r i a b l e s i n t o t h r e e g r o u p s . 1. V a r i a b l e s f r o m l i n e a r i z e d c o n s t i t u t i v e r e l a t i o n s -M j , -M^, N j , a, b and c 2. V a r i a b l e s d e s c r i b i n g t h e u n d i s t u r b e d f l o w s i t u a t i o n -F, f , 3 and W ( k = m TT ^ ) w W 3. Wave number o f d i s t u r b a n c e s -ho k and m ( k = m ir — ) w W
The s e n s i t i v i t y a n a l y s i s w i l l have one example as a p o i n t o f r e f e r e n c e . The u n d i s t u r b e d e q u i l i b r i u m s i t u a t i o n f o r t h i s r e f e r e n c e example i s g i v e n by
F = 0.25 f = 0.01
k^ = 0.10 ( W = lO-TT^hg i n case o f i n c i p i e n t m e a n d e r i n g ) .
The v a r i a b l e s f r o m t h e c o n s t i t u t i v e g e l a t i o n s a r e assumed t o be g i v e n by
a = 10 (Rozowski, 1957; M. Knudsen, 1981; and many o t h e r s ) b = 1 . 3 (De V r i e n d , 1981a) c = 1.76(Engelund, 1981) Mj = 2.06 ( A c c o r d i n g t o Engelung-Hansen, J967. = 1.29 For t h e e l a b o r a t i o n o f Mj and Nj = 5 . 0 9 see P a r k e r , 1976) ^2 = 0 . 4 3 = 0.001 i s used f o r t h e e l a b o r a t i o n o f c, M^^.. N.^. I n f i g u r e 5 t h e a m p l i f i c a t i o n f a c t o r f o r t h e r e f e r e n c e example i s d e p i c t e d as a f u n c t i o n o f 1 . Here t h e c o n t r i b u t i o n f r o m t h e t h r e e main terms i n eq. (3.25) i s a l s o i n d i c a t e d . I n t h e s e n s i t i v i t y a n a l y s i s b e l o w t h e i n f l u e n c e o f one v a r i a b l e a t t h e t i m e w i l l be d i s p l a y e d , so no t e s t f o r m u t u a l i n t e r a c t i o n between t h e v a r i a b l e s w i l l be c a r r i e d o u t . F i r s t t h e i n f l u e n c e o f t h e v a r i a b l e s f r o m t h e c o n s t i -t u -t i v e r e l a -t i o n s and second -t h e v a r i a b l e s d e s c r i b i n g -t h e e q u i l i b r i u m s i -t u a -t i o n w i l l be t r e a t e d .
5.1. I n f l u e n c e o f c o n s t i t u t i v e r e l a t i o n s
The c o n s t i t u t i v e r e l a t i o n s a r e p r o b a b l y t h e main source o f u n c e r t a i n t y i n t h i s a n a l y s i s . I t t h e r e f o r e seems a p p r o p r i a t e t o g a i n some i n s i g h t i n t o t h i s p o i n t , b e f o r e a d i s c u s s i o n o f t h e i n f l u e n c e o f t h e
p h y s i c a l p a r a m e t e r s on the development o f submerged b a r s t a k e s p l a c e . To t h i s end a s h o r t s e n s i t i v i t y a n a l y s i s i s c a r r i e d o u t .
The i n f l u e n c e o f t h e secondary f l o w , i . e . t h e p a r a m e t e r s a and b, i s i l l u s t r a t e d i n f i g u r e 6 and 7. The magnitude o f t h e secondary c u r r e n t i n case o f f u l l y developed f l o w ( t h e parameter a) seems t o have o n l y l i t t l e i n f l u e n c e on t h e wave l e n g t h , f o r w h i c h the maximum <^ ^ o c c u r s , whereas t h e i n f l u e n c e on t h e maximum i t s e l f seems c o n s i d e r a b l e . The e f f e c t o f the secondary f l o w i n e r t i a
(b) i s v e r y s i g n i f i c a n t , and i t extends t o b o t h t h e magnitude o f t h e a m p l i -f i c a t i o n -f a c t o r and t o t h e wave l e n g t h 1 , -f o r w h i c h max .] o c c u r s .
E q u a t i o n (3.25) shows t h a t t h e t e r m a c c o u n t i n g f o r the g r a v i t a t i o n a l f o r c e i s independend o f 1 , and t h a t i t e q u a l s t h e a m p l i f i c a t i o n f a c t o r f o r 1 = 0 . T h e r e f o r e a change o f c e x c l u s i v e l y e f f e c t s t h e m g n i t u d e o f cj), as i t o n l y causes a v e r t i c a l d i s p l a c e m e n t o f t h e graph (see f i g u r e 5 ) .
The p a r a m e t e r s o r i g i n a t e f r o m t h e l i n e a r i z a t i o n d f t h e shear s t r e s s model (Mj and M^) h a r d l y i n f l u e n c e t h e magnitude o f the t e r m a c c o u n t i n g f o r
secondary f l o w , b u t t h e e f f e c t on t h e main f l o w t e r m i s r a t h e r s i g n i f i c a n t . Mj ( f i g u r e 8) has f i r s t o f a l l a c o n s i d e r a b l e i n f l u e n c e on t h e 'maximum' xvrave number 1 , whereas t h e e f f e c t on t h e a m p l i f i c a t i o n f a c t o r i t s e l f i s
max
r e l a t i v e l y modest. As suggested by f i g u r e 9 t h e c o n t r a r y a p p l i e s t o M^.
I n f i g u r e 10 and 11 t h e e f f e c t o f t h e v a r i a b l e s from, t h e l i n e a r i z e d model f o r t h e sediment t r a n s p o r t r a t e a r e i l l u s t r a t e d . The t e r m i n eq.
(3.25) w h i c h a c c o u n t s f o r secondary f l o w i s c o m p l e t e l y independend o f the p a r a m e t e r s Nj and N^, t h e r e f o r e o n l y t h e t o t a l a m p l i f i c a t i o n f a c t o r s a r e
d e p i c t e d . The magnitude o f max cj) ^ i s s t r o n g l y e f f e c t e d by change i n where as 1 i s a l m o s t i n d i f f e r e n t . F i g u r e 11 d i s p l a y s t h a t <l> . i s a l m o s t
max 1 U n e f f e c t e d by a change i n K^- Eq. (3.24) i n c o m b i n a t i o n w i t h t h e f a c t t h a t
t h e r i g i d - l i d a p p r o x i m a t i o n does a p p l y o f f e r s a n e x p l a n a t i o n o f t h i s . I n s e r t i n g a = - z i n eq. (3.24) shows t h a t o n l y has i n f l u e n c e on t h e r e a l p a r t o f t h e c e l e r i t y .
