H-S Research

NUMERICAL MODEL SIMULATION OF ENTRAINMENT OF SAND BED MATERIAL

Report No SR 75 March 1986

Registered Office: Hydraulics Research Limited, Wallingford, Oxfordshire OX l 0 8BA. Telephone: 0491 35381. Telex: 848552

T h i s r e p o r t d e s c r i b e s work funded by t h e Depar tment

o f t h e Environment under Resea rch C o n t r a c t PECD

7 / 6 / 5 6 , f o r which t h e DOE nominated o f f i c e r was

D r R P Thorogood. I t i s p u b l i s h e d on b e h a l f of t h e

Department of t h e Environment b u t any o p i n i o n s

e x p r e s s e d i n t h i s r e p o r t a r e n o t n e c e s s a r i l y t h o s e of

t h e f u n d i n g Depar tment . The ~ o r k was c a r r i e d o u t by

D r G V Mi l e s i n t h e T i d a l E n g i n e e r i n g Depar tment of

H y d r a u l i c s Resea rch , W a l l i n g f o r d , unde r t h e

manageaent of idr L4 F C Thorn.

@ Crown c o p y r i g h t 1986

P u b l i s h e d by p e r m i s s i o n of t h e C o n t r o l l e r of

Her M a j e s t y ' s S t a t i o n e r y O f f i c e

ABSTRACT

The study of sediment transport generally is very difficult but more so in the case of estuaries because:

- the water movements are continually changing with the rise and fall of the tide

- a wide range of sediment exists on the bed and in suspension

- certain sediments are not found in some parts leading to unsaturated loads in the water.

In recent years sediment transport models have been developed and used for making engineering assessments of the impact of works on the sediment regime. At present the full potential of the models cannot be realised because of the lack of calibration and verification data, and gaps in our understanding of the fundamental sedimentation processes.

The basic aim of this research was to improve the representation of sand transporting processes in computer models. It is relevant to the sand transport consequences of civil engineering works on the operation of ports and harbours and on environment aspects, and will ultimately benefit the industry by helping to minimise maintenance dredging of ports and navigation channels.

The first phase of the project, covered in this report, concentrates on finding the best numerical model representation of the exchange of sand between the bed and the flow, based on assessments of the available data and theoretical analyses. The report also contains descriptions of the fundamental physical processes affecting sand transport in estuaries and a brief review of existing numerical models of sand transport.

A sand transport model, based on theoretical and empirical relationships, has been tested to verify that it simulates the relevant physical processes. It was found that the most appropriate formulation for entrainment of sand at the bed was in terms of an entrainment rate. The connection between this entrainment rate and the associated sand transport law has been considered and it was concluded that strictly only one of these should be specified.

The model was compared with some flume data to test its response to a change in the sediment load. It was shown that the model simulation could be calibrated by adjusting the settling velocity and vertical diffusivity parameters.

The implications on the suspended load due to the unsteadiness in accelerating and decelerating flow and the numerical simulation of these effects will be studied in the second phase of the project. It is also intended to study the behaviour of mixtures of different sand sizes and consider how this might be represented in a computer model. These aspects will be described in a later report.

CONTENTS

P a g e

INTRODUCTION

1.1 E s t u a r i n e s e d i m e n t s 1 .2 S e s e a r c h o b j e c t i v e s

POTEVTI AL LOAD MODELS

SUSPENDED LOAD MODELS

3 .1 :.lodels of v e r t i c a l p r o f i l e s 3.2 E v o l u t i o n of suspended l o a d 3.3 Boundary c o n d i t i o n s 3.4 Numer ica l sand t r a n s p o r t a o d e l s

4 .1 E n t r a i n m e n t rates 4.2 Boundary c o n d i t i o n s a t t h e bed 4.3 C a l i b r a t i o n of e n t r a i n m e n t

SIMULATION OF SEDIMENT TXMTSPORT

5 .1 E v o l u t i o n of s ed imen t l o a d

CONCLUSIONS

LIST OF SYMBOLS

FIGURES :

1. Sed imen t exchange r e l a t i o n s a t t h e bed 2. Computed and measured e v o l u t i o n s of s ed imen t l o a d

1 INTRODUCTION

1.1 E s t u a r i n e s e d i m e n t s An e s t u a r y is a p a r t l y e n c l o s e d body of t i d a l w a t e r where r i v e r w a t e r i s mixed w i t h and d i l u t e d by s e a w a t e r . I n a g e n e r a l s e n s e t h e e s t u a r i n e env i ronmen t i s d e f i n e d by s a l i n i t y b o u n d a r i e s r a t h e r t han by g e o g r a p h i c a l o n e s , b u t a l t h o u g h t h e s a l i n i t y has i n f l u e n c e on t h e c l a y s ed imen t f r a c t i o n s i t i s t h e c u r r e n t s g e n e r a t e d by t h e t i d a l volume f l o w i n g i n and o u t of t h e e s t u a r y which domina te t h e aovement and d i s t r i ' o u t i o n of s e d i m e n t s . The s e d i m e n t s t h e m s e l v e s may have o r i g i n a t e d from n a t u r a l e r o s i o n i n l a n d o r f rom seawards . They c o n s i s t of m a t e r i a l s r a n g i n g from t h e f i n e s t c l a y p a r t i c l e s t o c o a r s e s and and g r a v e l s . A c o n v e n i e n t c l a s s i f i c a t i o n of s e d i m e n t s a s e s a g e o m e t r i c s c a l e of s i z e s .

mm p h i units

Very c o a r s e sand 1 . 0 - 2.0 C o a r s e s and 0.5 - 1.0 Xedium sand 0.25 - 0.5 F i n e s and 0.125 - 0.25 Very f i n e s a n d 0.064 - 0.125

C o a r s e s i l t 0.332 - 0.064 i4edium s i l t 0.016 - 0.032 F i n e s i l t 0.008 - 0.016 Very f i n e s i l t 0.004 - 0.008

C o a r s e c l a y 0.002 - 0.004 Medium c l a y 0 .001 - 0.002

TABLE 1 SEDIMENT G I A D I N G S

A s i g n i f i c a n t f e a t u r e of e s t u a r i e s i s t h e wide r a n g e of s ed imen t s i z e s found i n them. T h e s e s e d i m e n t s are s i f t e d and s o r t e d by t h e t i d a l c u r r e n t s .

I n t h e main c h a n n e l s bed s t r e s s e s a r e u s u a l l y t o o h i g h t o a l l o w t h e f i n e r m a t e r i a l s t o a c c u m u l a t e a l t h o u g h t h e y nay s e t t l e t e m p o r a r i l y a t s l a c k w a t e r . an17 c o a r s e sand and ; r a v e l c a n e x i s t a s permanent d e p o s i t s i n t h e s e h i g h e n e r g y r e g i o n s . Along t h e s h a l l o w marg ins of t h e e s t u a r y , and f u r t h e r d p s t r e a m , t h e t i d a l c u r r e n t s a r e t o o weak t o move t h e s and and e i t h e r no sand i s t r a n s p o r t e d t h e r e o r i t is cove red by s i l t o r c l a y t o p roduce c h a r a c t e r i s t i c mud f l a t s . These mud f l a t s a r e c o l o n i s e d by v a r i o u s forms of mar ine l i f e and become t h e f e e d i n g g rounds of b i r d s . I f c o n d i t i o n s a r e s u i t a b l e t h e l e v e l of t h e mud f l a t s r i s e s and e v e n t u a l l y a s a l t marsh d e v e l o p s .

1 . 2 K e s e a r c h O b j e c t i v e s The s t u d y of s ed imen t t r a n s p o r t g e n e r a l l y is v e r y d i f f i c u l t b u t more s o i n t h e c a s e of e s t u a r i e s b e c a u s e

- t h e w a t e r movements a r e c o n t i n u a l l y c h a n g i n g g i t h the r i s e of t h e t i d e

- a wide r a n g e of s e d i m e n t s e x i s t on t h e bed and i n s u s p e n s i o n

- c e r t a i n s e d i m e n t s a r e n o t found i n some p a r t s l e a d i n g t o u n s a t u r a t e d l o a d s i n t h e w a t e r

I n r e c e n t y e a r s s e d i m e n t t r a n s p o r t models h a v e been d e v e l o p e d and used f o r inaking e n g i n e e r i n g a s s e s s m e n t s of t h e impac t of works on t h e s e d i m e n t reg ime. A t p r e s e n t t h e f u l l p o t e n t i a l o f t h e models c a n n o t b e r e a l i s e d b e c a u s e of t h e l a c k of c a l i b r a t i o n and v e r i f i c a t i o n d a t a , and g a p s i n o u r u n d e r s t a n d i n g of t h e fundamen ta l s e d i m e n t a t i o n p r o c e s s e s .

The b a s i c a i m of t h i s r e s e a r c h was t o improve t h e r e p r e s e n t a t i o n of sand t r a n s p o r t i n g p r o c e s s e s i n computer models which i n v o l v e d some work t o g a i n a b e t t e r u n d e r s t a n d i n g of t h e u n d e r l y i n g p h y s i c s .

The f i r s t phase of t h e p r o j e c t , c o v e r e d i n t h i s r e p o r t , c o n c e n t r a t e s on f i n d i n g t h e b e s t n u m e r i c a l model r e p r e s e n t a t i o n of t h e exchange of s a n d be tween t h e bed and t h e f l o w , b a s e d on a s s e s s m e n t s of t h e a v a i l a b l e d a t a and t h e o r e t i c a l a n a l y s e s . The r e p o r t a l s o c o n t a i n s d e s c r i p t i o n s of t h e f u n d a m e n t a l p h y s i c a l p r o c e s s e s a f f e c t i n g sand t r a n s p o r t i n e s t u a r i e s and a b r i e f r e v i e w of e x i s t i n g n u m e r i c a l models of s and t r a n s p o r t .

The i m p l i c a t i o n s on t h e suspended l o a d due t o t h e u n s t e a d i n e s s i n a c c e l e r a t i n g and d e c e l e r a t i n g f l o w and t h e n u m e r i c a l s i m u l a t i o n of t h e s e e f f e c t s w i l l b e s t u d i e d i n t h e second phase of t h e p r o j e c t . I t i s a l s o i n t e n d e d t o s t u d y t h e b e h a v i o u r of m i x t u r e s of d i f f e r e n t s and s i z e s and c o n s i d e r how t h i s might b e r e p r e s e n t e d i n a computer ~ n o d e l . These a s p e c t s w i l l b e d e s c r i b e d i n a l a te r r e p o r t .

2 POTENTIAL LOAD MODELS

The s i n p l e s t t y p e of s e d i m e n t t r a n s p o r t model i s e s s e n t i a l l y a s i n g l e e q u a t i o n r e p r e s e n t i n g c o n s e r v a t i o n of bed m a t e r i a l .

wherc! 'l (kg/m2) i s t h e q u a n t i t y of s e d i n e n t on t h e bed and Ts ( k g / s e c / a w i d t h ) is a p r e s c r i b e d sand t r a n s p o r t fo rmula . The b a s i c assumptiorl f o r t h i s t y p e of model i s t h a t t h e f l o ~ i s s a t u r a t e d w i t h s e d i m e n t , which means t h a t t h e £104 i s c a r r y i n g t h e rnaxirnum sand t r a n s p o r t t h a t can b e m a i n t a i n e d Eor t h e g i v e n h y d r a u l i c and sed i lnen ta ry c o n d i t i o n s . I lnder s a t u r a t e d c o n d i t i o n s t h e t r a a s p o r t can be c a l c u l a t e d f rom one of t h e rmny sed imen t t r a n s p o r t l a u s t o be found i n t h e l i t e r a t u r e . An a p p r a i s a l o f a v a i l a b l e methods i s g i v e n by van X i jn (1984) . The f l ow p a r a m e t e r s ( w a t e r d e p t h , mean v e l o c i t y and s h e a r v e l o c i t y ) r e q u i r e d f o r t h e t r a n s p o r t c a l c u l a t i o n c o u l d be o b t a i n e d from measurements i n a p h y s i c a l n o d e l b u t i t i s u s u a l l y q u i c k e r and c h e a p e r t o g e n e r a t e t h i s d a t a on a r e g u l a r z r i d from a s e p a r a t e n u m e r i c a l model of w a t e r movements.

The s e d i m e n t c a r r y i n g c a p a c i t y of f low i n c r e a s e s s i g n i f i c a n t l y f o r h i g h w a t e r v e l o c i t i e s - t y p i c a l l y i n p r o p o r t i o n t o t h e f o u r t h power. T h i s means t h a t t h e f l ow w i l l t e n d t o p i c k up m a t e r i a l f rom t h e bed when i t a c c e l e r a t e s and t o d e p o s i t e x c e s s m a t e r i a l when i t d e c e l e r a t e s . I f t h e f low i s a lways s a t u r a t e d w i t h s e d i m e n t t h e d i f f e r e n c e i n t r a n s p o r t i n g c a p a c i t y must d e f i n e t h e q u a n t i t y of m a t e r i a l p i c k e d up o r d e p o s i t e d on t h e bed. T h i s i s t h e b a s i s f o r t h e p o t e n t i a l l o a d model.

The p o t e n t i a l l o a d model i s n a t u r a l l y most s u i t e d t o s i t u a t i o n s where t h e bed m a t e r i a l i s n a r r o w l y g r a d e d and here t h e r e i s a n a d e q u a t e s u p p l y of e r o d i b l e m a t e r i a l on t h e bed t o m a i n t a i n t h e s a t u r a t e d l o a d . These c o n d i t i o n s a r e more o f t e n m e t i n r i v e r s and i t i s i n such s i t u a t i o n s t h a t p o t e n t i a l l o a d models have been found most s u c c e s s f u l . S e e f o r example Cunge and Yerd reau ( 1 9 7 3 ) , Thomas and P r a s u h n (1977) and B e t t e s s and Whi te (1979) .

L e p e t i t and S a g u e l (1978) ex t ended t h e m o d e l l i n g a p p r o a c h used i n r i v e r s t u d i e s , t o s i m u l a t e 2-d imens ional l o c a l s c o u r round a j e t t y i n a s t e a d y f low. The model i s q u a s i - s t e a d y and u s e s a p e r t u r b a t i o n t e c h n i q u e t o f e e d t h e changes i n d e p t h b a c k i n t o t h e £104. T r a n s p o r t i s c a l c u l a t e d fr:>rn a s a t u r a t e d bed l o a d sed imen t law and bed changes c a l c u l a t e d f r o ~ n t h e 2-d iiaens i o n a l form of e q u a t i o n 1, f o r c o n s e r v a t i o n of s e d i m e n t . The model r e s u l t s we re shown t o a g r e e q u a l i t a t i v e l y w i t h s c o u r p a t t e r n s measured i n a mob i l e bed p h y s i c a l model.

The p r e v i o u s l y ment ioned models have t h e cornnon f e a t u r e t h a t t h e w a t e r f l o ~ i s e i t h e r s t e a d y o r v a r y i n g v e r y s l o w l y . iJnder t h o s e c i r c ~ l ; n s t a n c z s t h e p o t e n t i a l l o a d model can be a p p l i e d t o t o t a l (bed p l u s suspended) l o a d s . I n e s t u a r i e s , cllhere l a g

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t h e a c t u a l sediment load of t h e approaching water i s i n s u f f i c i e n t t o s a t u r a t e even t h e slower f low;

3. e r o s i o n may i n f a c t occur i n an a r e a of p o t e n t i a l d e p o s i t i o n i f t h e sediment load of t h e approaching flow i s very low f o r example a f t e r f lowing over an a r e a of rock bed.

I n o rder t o overcome t h e s e l i m i t a t i o n s a d i f f e r e n t s o r t of model i s r e q u i r e d based on c o n s e r v a t i o n p r i n c i p l e s which s i m u l a t e s the sediment t r a n s p o r t i n terms of a suspended s o l i d s c o n c e n t r a t i o n . The e r o s i o n o r d e p o s i t i o n of m a t e r i a l on t h e bed can t h e n be assumed i n t h e model depending on whether t h e a c t u a l load i s l e s s o r g r e a t e r than t h e s a t u r a t e d load which would o b t a i n under s t e a d y , uniform flow c o n d i t i o n s a t t h e sane v a l u e s a s t h e i n s t a n t a n e o u s flow. Under t h e s e c i rcumstances t h e suspended s o l i d s c o n c e n t r a t i o n , c , (kg/m 3, s a t i s f i e s , (eg Graf (1971) )

a ac a ac a a2 = -(D -) + -(D -) + -(D -) h x a x a y y a y ? E z ? E

where

U , v , W a r e t h e v e l o c i t y components (m/s) X , y, z a r e space co-ord ina tes , w i t h z v e r t i c a l l y

upwards (m) ws i s t h e s e t t l i n g v e l o c i t y (mjs) t i s t ime D x , D y , D Z a r e d i f f u s i o n c o e f f i c i e n t s (m2/s).

3.1 :.lodels of v e r t i c a l p r o f i l e s

Xost s o l u t i o n s t o be found i n t h e l i t e r a t u r e a r e f o r s p e c i a l c a s e s of t h i s equa t ion . The e a r l i e r s o l u t i o n s by S c h n i d t (1925) and Lane e t a 1 (1941) , and l a t e r Hunt (1965) provide i n s i g h t i n t o t h e v e r t i c a l s t r u c t u r e of t h e suspended s o l i d s p r o f i l e . These assume one-dimensional, uniform, s t e a d y flow c o n d i t i o n s f o r which equa t ion 2 reduces t o

T h i s equa t ion r e p r e s e n t s t h e e q u i l i b r i u m p r o f i l e ob ta ined a s a ba lance between s e t t l i n g and v e r t i c a l d i f f u s i o n due t o t h e tu rbu lence . I t i s impor tan t t o a p p r e c i a t e t h a t e q u i l i b r i u m d e f i n e d i n t h i s way does

n o t mean s a t u r a t i o n . I n d e e d t h e sed imen t l o a d can b e i n e q u i l i b r i u m i f t h e bed i s n o t mob i l e , even when t h e f low is n o t s a t u r a t e d w i t h sed imen t . I n t e g r a t i o n w i t h r e s p e c t t o z and t h e a p p l i c a t i o n of a boundary c o n d i t i o n of z e r o f l u x of sed imen t a t t h e f r e e s u r f a c e (and i m p l i c i t l y a l s o a t t h e bed) y i e l d s t h e govern ing e q u a t i o n

T h i s can be i n t e g r a t e d f u r t h e r i f t h e v e r t i c a l s t r u c t u r e of t h e d i f f u s i v i t y i s p r e s c r i b e d . These p r o f i l e s have proved v a l u a b l e i n t h e u n d e r s t a n d i n g of sediment t r a n s p o r t b u t t hey a r e n o t r e l e v a n t t o t h e u n s t e a d y , u n s a t u r a t e d f low c o n d i t i o n s which a r e t h e main conce rn h e r e , s o t h e s e s p e c i a l s o l u t i o n s a r e n o t c o n s i d e r e d f u r t h e r . Graf (1971) i s a good s o u r c e of a d d i t i o n a l i n f o r m a t i o n on t h e s e s o l u t i o n s .

3.2 E v o l u t i o n of suspended l o a d

A c l a s s of s o l u t i o n s which have more r e l e v a n c e t o e s t u a r i e s have been p r e s e n t e d by K a l i n s k e ( 1 9 4 0 ) , Dobbins (1943) , Mei (1969) and Lean (1980) , f o r u n s t e a d y and un i fo rm o r s t e a d y and non-uniform c o n d i t i o n s governed r e s p e c t i v e l y by t h e e q u a t i o n s

ac a a= ac - = -(D -) + W - at a? Z a ? S a?

These e q u a t i o n s are m a t h e m a t i c a l l y t h e same when t h e a d v e c t i o n v e l o c i t y uo is c o n s t a n t . The f i r s t r e p r e s e n t s a c o n c e n t r a t i o n changing w i t h t ime f o l l o w i n g a change i n t h e magni tude of a u n i f o r m f low, w h i l e t h e second r e p r e s e n t s t h e c o n c e n t r a t i o n changing as a f u n c t i o n of p o s i t i o n as might occur f o r example when c l e a r w a t e r f l o w s f rom an a r e a w i t h an i n e r o d i b l e bed i n t o an a r e a where e r o s i o n can c o m e n c e . S o l u t i o n s of t h e s e e q u a t i o n s p r o v i d e i n f o r m a t i o n a b o u t t h e t ime o r d i s t a n c e of t r a v e l r e q u i r e d f o r t h e sed imen t c o n c e n t r a t i o n t o a d a p t t o changes i n t h e f low c o n d i t i o n s .

The assumpt ion of c o n s t a n t eddy d i f f u s i v i t y p e r m i t s a n a l y t i c s o l u t i o n of t h e s e e q u a t i o n s . The d i f f u s i v i t y no rma l ly used i s

which is t h e depth averaged v a l u e of t h e p a r a b o l i c eddy v i s c o s i t y

c o n s i s t e n t w i t h the 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 . K

i s t h e Von Karman c o n s t a n t . Apmann and iiumer (1970) p resen t exper imenta l evidence t h a t s u p p o r t s t h i s assumption. The exac t s o l u t i o n s , which invo lve t h e use of Laplace Transforms, o r s i m i l a r , may be expressed i n t h e form of i n f i n i t e s e r i e s . Mei (1969) recognised t h a t an approximate s o l u t i o n , v a l i d f o r smal l t imes of d i s t a n c e s of t r a v e l , could be ob ta ined from t h e expansion of t h e Laplace Transform f o r l a r g e v a l u e s of t h e t r ans form parameter. Under most c o n d i t i o n s t h i s expansion i s v a l i d f o r d i s t a n c e s of t h e o rder of twenty water dep ths o r t h e e q u i v a l e n t i n a t i n e dependent s i t u a t i o n . Lean (1980) proposed an a l t e r n a t i v e bed boundary c o n d i t i o n and t h e p r e s e n t au thor has reworked :4ei1s s o l u t i o n f o r t h i s case . Th is s o l u t i o n i s used i n t h e sediment t r a n s p o r t model proposed i n t h e nex t s e c t i o n .

3 . 3 Boundary c o n d i t i o n s The s o l u t i o n of the unsteady e q u a t i o n 5 r e q u i r e s an i n i t i a l c o n d i t i o n a t say t = 0 and boundary c o n d i t i o n z = 0 ( t h e bed) and z = d ( t h e f r e e s u r f a c e ) . The s u r f a c e boundary i s c l e a r l y z e r o v e r t i c a l f l u x of sediment v i z

There a r e two p o s s i b l e c o n d i t i o n s a t t h e bed which admit a n a l y t i c a l s o l u t i o n , f i r s t l y one could assume (Mei 1969) t h a t t h e c o n c e n t r a t i o n c ( o , t ) a t t h e bed responds i n s t a n t a n e o u s l y t o t h e changing flow c o n d i t i o n s . That i s

where

c s ( o , t ) i s t h e c o n c e n t r a t i o n of t h e e q u i l i b r i u m p r o f i l e a t t h e bed when t h e flow is

- s a t u r a t e d w i t h sediment c s ( t ) i s t h e dep th averaged v a l u e of t h i s p r o f i l e

PS = c s ( o , t ) / c s ( t ) i s a p r o f i l e f a c t o r

Th is i s a much more r e a l i s t i c c o n d i t i o n than t h a t i m p l i c i t i n a p o t e n t i a l load model which assumes t h a t the f u l l load responds i n s t a n t a n e o u s l y . Xowever, t h e c o n d i t i o n s t i l l i m p l i e s an i n f i n i t e r a t e of exchange

of m a t e r i a l a t t h e bed a t t = 0 ( o r a t X = 0 i n t h e non-uniform v e r s i o n ) .

Lean (1983) assumes t h a t t h e r a t e E ( k g / m 2 / s ) a t which m a t e r i a l i s e n t r a i n e d i n t o t h e f low i s t h e q u a n t i t y which r e s p o n d s most r e a d i l y t o changes i n f low. I n t h i s c a s e t h e boundary c o n d i t i o n a t t h e bed would be

o r , f rom e q u a t i o n 4 , t h e n e t v e r t i c a l f l u x FZ (kg /m2/ s ) a t t h e bed i s p r e s c r i b e d a s

T h i s p r o v i d e s an a l t e r n a t i v e boundary c o n d i t i o n t o c o n d i t i o n 10. The a s y m p t o t i c form of s o l u t i o n s t o e q u a t i o n 5 f o r i n i t i a l c o n c e n t r a t i o n c0 a r e

f o r t h e bed c o n c e n t r a t i o n boundary c o n d i t i o n (Mei) and

f o r t h e bed e n t r a i n m e n t boundary c o n d i t i o n ( L e a n ) , where

These s o l u t i o n s a r e v a l i d when t 1, i e f o r t h e t i m e needed f o r t h e w a t e r t o f low ove r a d i s t a n c e e q u a l t o a b o u t 1 0 t o 100 w a t e r d e p t h s .

3.4 Numer ica l s and t r a n s p o r t models

Al though t h e s p e c i a l s o l u t i o n s d e s c r i b e d above p r o v i d e i n s i g h t i n t o t h e sed imen t t r a n s p o r t p r o c e s s e s t h e y s t i l l l a c k many of t h e f a c t o r s which a r e i m p o r t a n t i n e s t u a r i e s , namely

1. t h e combina t ion of non u n i f o r m i t y w i t h u n s t e a d i n e s s

2. v a r i a b l e s u p p l y of e r o d i b l e m a t e r i a l

3. l a te ra l as w e l l a s l o n g i t u d i n a l v a r i a t i o n .

The i n c l u s i o n of t h e s e f a c t o r s c o m p l e t e l y p r e c l u d e s of a n a l y t i c s o l u t i o n and l e a d s t o t h e need f o r n u m e r i c a l models.

The s i m p l e s t form of n u m e r i c a l s ed imen t t r a n s p o r t model t a k e s t h e form of f i n i t e d i f f e r e n c e o r f i n i t e e l emen t s o l u t i o n s of t h e approx ima te e q u a t i o n s 5 and 6 . Apmann and Aumer (1970) and Y a l i n and F i n l a y s o n (1973) p r e s e n t models of t h i s t ype . The a d v a n t a g e o f s e e k i n g n u m e r i c a l s o l u t i o n s i s t h a t more r ea l i s t i c eddy d i f f u s i v i t i e s and v e l o c i t y p r o f i l e s can be i n c o r p o r a t e d . Y a l i n employs a n eddy d i f f u s i v i t y e q u a l t o t h e p a r a b o l i c eddy v i s c o s i t y ( e q 8) c o n s i s t e n t w i t h t h e l o g a r i t h m i c f low p r o f i l e . The model was t e s t e d a g a i n s t e x p e r i m e n t a l measurements ( F i g 5 ) . Though t h e model i s n o t immedia t e ly r e l e v a n t t o e s t u a r i e s t h e r e i s no i n h e r e n t d i f f i c u l t y i n e x t e n d i n g t h e n u m e r i c a l t e c h n i q u e s t o non-uniform, u n s t e a d y c o n d i t i o n s .

K e r r s e n s e t a1 (1979) have deve loped a m u l t i - l a y e r , l - D model of t h i s t y p e which a l l o w s non-uniform c r o s s - s e c t i o n s t o be c o n s i d e r e d b u t i t h a s a p p a r e n t l y o n l y been a p p l i e d under s t e a d y f low c o n d i t i o n s . A l o g a r i t h m i c p r o f i l e i s assumed f o r t h e v e r t i c a l s t r u c t u r e of t h e f low. The suspended s o l i d s c o n c e n t r a t i o n i s computed from t h e l - d i m e n s i o n a l fo rm o f t h e sed imen t c o n c e n t r a t i o n e q u a t i o n 2 u s i n g non- un i fo rm v e r t i c a l g r i d t o g i v e g r e a t e r a c c u r a c y n e a r t h e bed. K e r r s e n s e t a1 (1979) assumed t h e t u r b u l e n t d i f f u s i v i t y , D, t o e q u a l t h e p a r a b o l i c eddy v i s c o s i t y ( e q 8) a p p r o p r i a t e f o r l o g a r i t h m i c f low. The boundary c o n d i t i o n a t t h e bed i s t a k e n t o be t h e e q u i l i b r i u m c o n c e n t r a t i o n , e q u a t i o n 1 0 , which would occu r a t t h e i n s t a n t a n e o u s f low c o n d i t i o n s . The s o l u t i o n s i m u l a t e s t h e t r a n s i e n t e v o l u t i o n of t h e e q u i l i b r i u m p r o f i l e f rom a n o n - e q u i l i b r i u m c o n d i t i o n .

The model was t e s t e d a g a i n s t i n f i l l r a t e s measured i n a g a s p i p e l i n e t r e n c h i n t h e Western S c h e l d t . T h i s model has been developed f u r t h e r t o i n c l u d e a g i t a t i o n by waves (van R i j n (1985) ) .

G a l a p p a t t i and Vredgdenh i l (1985) approached t h e problem i n a d i f f e r e n t way. They f o r u u l a t e d t h e i r model on a s e r i e s expans ion i n which t h e v e r t i c a l d imension i s e l i i n i n a t e d by means of an a s y m p t o t i c s o l u t i o n . The r e s u l t i n g depth-averaged model was t e s t e d f o r a s t e a d y , u n i d i r e c t i o n a l f low c a s e u s i n g a p r e s c r i b e d c o n c e n t r a t i o n f o r t h e boundary c o n d i t i o n a t t h e bed. I t i s n o t c l e a r whe the r t h i s t y p e of boundary c o n d i t i o n was used i n p r e f e r e n c e o r d i c t a t e d by t h e n a t u r e of t h e method. Al though t h e model had l i m i t e d a p p l i c a b i l i t y , t h e a u t h o r s concluded t h a t t h e t e c h n i q u e i s a s t e p towards b r i d g i n g t h e gap between 2D and 3D models.

Although none of t h e models d e s c r i b e d s o f a r have been a p p l i e d t o r e a l e s t u a r y c o n d i t i o n s t h e r e i s no t e c h n i c a l r e a s o n why t h i s shou ld n o t be done. The main problems p r e v e n t i n g t h i s a t p r e s e n t seem t o be t h e h igh expense of runn ing 3-dimensional models and d e f i c i e n c i e s i n our knowledge of sed imen t t r a n s p o r t p r o c e s s e s i n e s t u a r i e s . I n a n a t t e m p t t o g a i n an u n d e r s t a n d i n g of t h e consequences of u n s a t u r a t e d f low i n e s t u a r i e s Yiles e t a 1 (1980) proposed a 2-dimensional , depth-averaged model. T h i s t y p e of model r e q u i r e s s p e c i a l p r o v i s i o n t o t a k e i n t o a c c o u n t t h e v e r t i c a l p r o f i l e e f f e c t s of t h e sed imen t c o n c e n t r a t i o n .

The depth-averaged c o n c e n t r a t i o n c ( x , y , t ) s a t i s f i e s t h e d e p t h - i n t e g r a t e d form of e q u a t i o n 2 which may be w r i t t e n

where

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

D n i s t h e l a t e r a l ( t u r b u l e n t ) d i f f u s i o n c o e f f i c i e n t

( s , n ) a r e n a t u r a l c o - o r d i n a t e s i n t h e d i r e c t i o n and normal t o t h e f low

The pa ramete r s a and PS a r e i n t r o d u c e d t o a c c o u n t f o r

t h e v e r t i c a l c o n c e n t r a t i o n and v e l o c i t y p r o f i l e s . For example

d a = J q c d ~ = T'iqcd -

qcd o

r e p r e s e n t s t h e f a c t o r r e q u i r e d t o r ecover t h e t r u e t r a n s p o r t of sediment from t h e product of d e p t h averaged q u a n t i t i e s , where

L q = (E + 7 is t h e h o r i z o n t a l w a t e r speed , and T = t h e sand t r a n s p o r t (kg/m w i d t h / s )

S i n c e high c o n c e n t r a t i o n s occur n e a r t h e bed i t fo l lows t h a t a l , and PS 2 1.

H i l e s e t a 1 (1980) ana lysed sediment t r a n s p o r t measurements from t h e Conwy e s t u a r y i n terms of a and

PS. The mean v a l u e f o r a w a s found t o be i n good agreement w i t h t h e o r e t i c a l v a l u e s from Sumer (1977) and t h e mean v a l u e f o r f3 was i n good agreement w i t h t h e o r e t i c a l v a l u e s

from t h e e x p o n e n t i a l e q u i l i b r i u m p r o f i l e of Lane e t a 1 (1941) f o r c o n d i t i o n s t y p i c a l of t h e f low i n t h e Conwy Es tua ry .

The r e s u l t s show t h a t i t i s p o s s i b l e t o o b t a i n s e n s i b l e p r o f i l e f a c t o r s from f i e l d measurements and, a l though , t h e r e l a t i o n s found a r e v a l i d o n l y f o r t h e Conwy, t h e t echn iques involved could be a p p l i e d t o any s i t e . With t h i s e m p i r i c a l method of p r e s c r i b i n g a and PS t h i s model has t h e b a s i c f e a t u r e s f o r s t u d y i n g sediment t r a n s p o r t i n e s t u a r i e s . I t has advec t ion by c u r r e n t s , d i s p e r s i o n i n t h e d i r e c t i o n of f low due t o t h e v e r t i c a l p r o f i l e and l a t e r a l d i f f u s i o n by tu rbu lence . D e p o s i t i o n o r e r o s i o n t a k e s p lace depending a s t o whether t h e i n s t a n t a n e o u s sediment load exceeds o r f a l l s s h o r t of t h e s a t u r a t e d l o a d , and, i f r e q u i r e d , e r o s i o n may be prevented i f t h e r e i s no sediment of t h e a p p r o p r i a t e s i z e a v a i l a b l e on t h e bed. A s h o r t a g e of m a t e r i a l on t h e bed would be r e f l e c t e d i n a l o ~ c o n c e n t r a t i o n of suspended s o l i d s being advected away by t h e f low.

McAnally e t a 1 (1984) fo l low a s i m i l a r approach b u t they use a d i f f e r e n t f o r m u l a t i o n f o r d e p o s i t i o n and e r o s i o n . The d e p o s i t i o n r a t e i s based on a r e p r e s e n t a t i v e s e t t l i n g v e l o c i t y and t h e e r o s i o n r a t e i n c l u d e s a r e sponse time c o e f f i c i e n t bu t no d e t a i l s of i t s form a r e g iven.

The most u n s a t i s f a c t o r y a s p e c t of t h e s e models i s t h e u s e of t h e s e t t l i n g v e l o c i t y a s t h e main s c a l i n g f a c t o r f o r t h e exchange r a t e of m a t e r i a l between t h e f low and t h e bed. A b e t t e r approximat ion can be ob ta ined from t h e a n a l y t i c a l s o l u t i o n s (14) o r (15) f o r t r a n s i e n t c o n d i t i o n s . The r a t e s of exchange of sediment a t t h e bed

a r e

f o r t h e boundary c o n d i t i o n s of Mei and Lean r e s p e c t i v e l y . The n a t u r e of t h e s e r e l a t i o n s i s shown i n F i g l a f o r a t y p i c a l f low c o n d i t i o n . S i n c e r e l a t i o n 24 i m p l i e s an i n f i n i t e f l u x of sediment a t t = 0 t h e second fo rmula t ion is favoured i n t h e fo l lowing . I f p r e f e r r e d , t h e a n a l y s i s could be r e p e a t e d f o r t h e o t h e r case .

F i g l a shows t h a t t h e r a t e of exchange of sediment a t t h e bed v a r i e s c o n s i d e r a b l y over t imes of t h e o r d e r of t i m e s t e p s normal ly used i n numer ica l models. Accordingly i t is a d v i s a b l e t o i n t e g r a t e e q u a t i o n 25 over a model t imes tep . That is t h e v e r t i c a l f l u x of sediment , S S ( 7 ) , dur ing t h e i n t e r v a l ( 0 , t ) i s

S ( T ) = w s ~ ( - c ) ( C - CO) (26)

where

W - 2 ( 4 ~ ~ ( 1 + ~ 2 ) e r f c ( ~ ) + e r f ( ~ ) - P( d = BsDz

i s t h e bed exchange s c a l i n g f a c t o r t h a t i n c o r p o r a t e s t h e e f f e c t s of t h e v e r t i c a l s t r u c t u r e and t h e l a g t ime f o r t h e c o n c e n t r a t i o n p r o f i l e t o a d j u s t t o t h e changing flow c o n d i t i o n s . The n a t u r e of @(S) i s

shown i n F i g l b f o r a t y p i c a l s ed imen t f r a c t i o n and t i m e s t e p .

A new sand t r a n s p o r t model h a s been deve loped by t h e a u t h o r ( Y i l e s (1981) ) u s i n g S ( z) g i v e n i n e q u a t i o n 26 t o r e p l a c e t h e s o u r c e / s i n k term on t h e r i g h t hand s i d e of e q u a t i o n 20. T h i s i s e q u i v a l e n t t o a s s u a i n g t h a t t h e sed imen t l o a d i s i n e q u i l i b r i u m w i t h t h e f l o w , which i s r e a s o n a b l e provided t h e f low i s s l o w l y v a r y i n g i n s p a c e and t ime. However i t f o l l o w s t h a t t h e new method canno t be a b s o l u t e l y p r e c i s e b e c a u s e t h e t h e o r e t i c a l s o l u t i o n s 14 and 15 have a g r a d u a l t r a n s i t i o n w h i l e t h e approx ima te computa t ion i s c a l c u l a t e d from t h e assumed e q u i l i b r i u m p r o f i l e a t t h e s t a r t of each t i m e s t e p . Yowever, r e s u l t s unde r u n i f o r m f low c o n d i t i o n s shown i n F i g l b i n d i c a t e t h a t t h e e r r o r S i nvo lved a r e l e s s t h a n t h e d i f f e r e n c e s which r e s u l t from a p p l y i n g t h e two a l t e r n a t i v e boundary c o n d i t i o n s 1 0 and 12.

4 ENTRAINMENT

4.1 E n t r a i n m e n t r a t e Van X i j n (Ref 23) has d e v i s e d a n o v e l expe r imen t f o r measu r ing t h e e n t r a i n m e n t r a t e . A sed imen t l i f t c o n s t r u c t i o n w a s d e s i g n e d whereby t h e sed imen t p a r t i c l e s c o u l d be moved upwards a t a c o n s t a n t r a t e t h r o u g h a c i r c u l a r open ing i n t h e f lume bottom. The upward speed of t h e l i f t c o u l d be s e t a t d i f f e r e n t ( c o n s t a n t ) v a l u e s by u s e of a mechan ica l d r i v e system. To e s t a b l i s h a un i fo rm f l o w , t h e f lume w a s equ ipped w i t h a f a l s e f l o o r which was g i v e n a p r e - s e t s l o p e e q u a l t o t h e e x p e c t e d w a t e r s u r f a c e s l o p e i n e a c h expe r imen t . The f a l s e f l o o r w a s cove red w i t h p r e - f a b r i c a t e d wooden p l a t e s of which s e d i m e n t p a r t i c l e s of a s i z e e q u a l t o t h o s e i n t h e s e d i m e n t l i f t were g lued .

I n a l l , f i v e t y p e s of a l m o s t u n i f o r m sand m a t e r i a l were used w i t h d v a l u e s of 130, 190 , 360, 790 a n d 1500 p. The f low d e p t h was a b o u t 0.25m. The f l o w v e l o c i t i e s were v a r i e d i n t h e r a n g e of 0.5-lm/s.

The most well-known sed imen t pick-up f u n c t i o n s , a s r e p o r t e d i n t h e l i t e r a t u r e , were summarised and compared w i t h t h e e x p e r i m e n t a l r e s u l t s , and a n e m p i r i c a l s ed imen t pick-up f u n c t i o n r e p r e s e n t i n g t h e p r e s e n t e x p e r i m e n t a l r e s u l t s was used t o compute t h e t r a n s p o r t r a t e of t h e bed-load and t h e suspended l o a d p a r t i c l e s .

The main c o n c l u s i o n s of t h e s t u d y were :

1. The e x p e r i m e n t a l l y d e t e r m i n e pick-up r a t e f o r a f l a t bed c o u l d b e r e p r e s e n t e d by a s i m p l e

e m p i r i c a l pick-up f u n c t i o n f o r p a r t i c l e s i n t h e r a n g e 130-1500 p.

2 . The p r e d i c t i v e a b i l i t y of t h e t h e o r e t i c a l p ick-up f u n c t i o n s of E i n s t e i n and Y a l i n was r a t h e r poor; t h e t h e o r y of D e R u i t e r produced t h e b e s t r e s u l t s f o r t h e l a r g e p a r t i c l e s s i z e s (d > 1000 p) ; w h i l e t h e t h e o r i e s of Nagakawa-Tsujimoto and Fe rnandez Luque y i e l d e d t h e b e s t r e s u l t s f o r t h e s m a l l e r p a r t i c l e s i z e s ( Q00 p).

3. D e f i n i n g t h e bed-load t r a n s p o r t as t h e p r o d u c t o f t h e pick-up r a t e and t h e s a l t a t i o n o r jump l e n g t h , t h e bed-load t r a n s p o r t was computed f o r 553 f lume and f i e l d d a t a , r e s u l t i n g i n a s c o r e of 60% of t h e p r e d i c t e d v a l u e s i n t h e r a n g e 0.5 t o 2 t imes t h e measured v a l u e s ; t h e bed-load f o r m u l a s of I leyer -Pe ter - r l i i l l e r and F r i j l i n k produced similar r e s u l t s .

4 . Applying t h e proposed e m p i r i c a l s e d i m e n t p ick-up f u n c t i o n as a bed-boundary c o n d i t i o n i n a m a t h e m a t i c a l model f o r suspended sed imen t t r a n s p o r t , t h e a d j u s t m e n t of c o n c e n t r a t i o n p r o f i l e s i n a un i fo rm f low w i t h o u t i n i t i a l s ed imen t l o a d was computed and when compared w i t h e x p e r i m e n t a l r e s u l t s , showed a r e a s o n a b l e ag reemen t .

4.2 Boundary c o n d i t i o n s a t t h e bed

As d i s c u s s e d e a r l i e r p r e s c r i b i n g t h e e n t r a i n m e n t r a t e of s ed imen t f rom t h e bed i s c o n s i d e r e d t o b e s u p e r i o r t o s p e c i f y i n g t h e n e a r bed c o n c e n t r a t i o n . The magni tude of t h e s ed imen t e n t r a i n m e n t r a t e E ( k g j u n i t a r e a / s e c ) e s s e n t i a l l y d e t e r m i n e s t h e magni tude of t h e a s s o c i a t e d s e d i m e n t t r a n s p o r t Ts (kg/m w i d t h / s e c ) a n d , v i c e v e r s a , t h e u s e of a g i v e n s e d i m e n t t r a n s p o r t r e l a t i o n s Ts i m p l i e s a c o r r e s p o n d i n g e n t r a i n m e n t E. T h a t i s b o t h r e l a t i o n s c a n n o t be p r e s c r i b e d i n d e p e n d e n t l y i n a model.

F u r t h e r i n s i g h t i n t o t h e c o n n e c t i o n between E and Ts can be o b t a i n e d by c o n s i d e r i n g t h e r e q u i r e m e n t s of v a r i o u s models . At p r e s e n t ou r l i m i t e d u n d e r s t a n d i n g of t h e u n d e r l y i n g p h y s i c s h a s r e s u l t e d i n two b a s i c a l l y d i f f e r e n t a p p r o a c h e s b e i n g used. On one hand w e have methods t i e d t o s e d i m e n t t r a n s p o r t r e l a t i o n s , a s f r e q u e n t l y u sed i n u n i d i r e c t i o n a l f l ow and i n c r e a s i n g l y used i n e s t u a r i e s i n t h e form of p o t e n t i a l sand t r a n s p o r t models . X o d e l s f o r m u l a t e d i n t h i s way do n o t need t h e s e d i m e n t e n t r a i n m e n t ra te t o b e p r e s c r i b e d . On t h e c o n t r a r y t h e exchange of material a t t h e bed i s o b t a i n e d e x p l i c i t l y f rom t h e s p a t i a l v a r i a t i o n s i n t r a n s p o r t by assuming s a t u r a t e d f low c o n d i t i o n s and u s i n g c o n s e r v a t i o n of s e d i m e n t .

A t t h e o t h e r extreme a model c o n t a i n i n g a f u l l d e s c r i p t i o n of t h e p h y s i c a l p r o c e s s e s i n t h e v e r t i c a l d imension would o n l y need sediment e n t r a i n m e n t and s h e a r s t r e s s r e l a t i o n s a t t h e bed. No sed imen t t r a n s p o r t r e l a t i o n would be r e q u i r e d because t h e f low and suspended sediment p r o f i l e s would be g e n e r a t e d from t h e b a s i c e q u a t i o n s and t o g e t h e r t h e s e d e f i n e t h e sediment t r a n s p o r t . The model of Ker r sens e t a 1 (Ref 1 3 ) f a l l s i n t o t h i s second c a t e g o r y .

The depth-averaged model p r e s e n t e d i n t h i s r e p o r t i s i n t e r e s t i n g i n t h a t t h e a c t u a l r e l a t i o n between E and Ts can be d e r i v e d . Thus from e q u a t i o n s 12 , 21 and 2 2

I t s h o u l d be n o t e d t h a t t h i s r e l a t i o n a p p l i e s e x c l u s i v e l y t o t h e tE9 Sandflow-2D model f o r m u l a t i o n . O the r models have d i f f e r e n t r e l a t i o n s between E and Ts and i n many c a s e s i t i s n o t p o s s i b l e t o e x t r a c t t h e e x a c t form of i t from t h e e q u a t i o n s .

4.3 C a l i b r a t i o n of s n t r a i n m e n t and p r o f i l e e v o l u t i o n

Computed sand t r a n s p o r t s w i l l d i f f e r even from f l u x e s o b s e r v e d under c o n t r o l l e d l a b o r a t o r y c o n d i t i o n s b e c a u s e of approx ima t ions i n t h e models and u n c e r t a i n t i e s i n t h e p r e s c r i b e d e n t r a i n m e n t r a t e ( o r sand t r a n s p o r t r e l a t i o n ) . The normal p rocedure would be t o a d j u s t t h e sediment e n t r a i n m e n t r a t e ( o r sand t r a n s p o r t r e l a t i o n ) d u r i n g c a l i b r a t i o n t o t u n e t h e model t o a g r e e w i t h obse rved t r a n s p o r t r a t e s . Ilowever, due t o u n c e r t a i n t i e s i n t h e form of t h e t u r b u l e n t d i f f u s i v i t y , (DZ) and s e t t l i n g v e l o c i t y (ws) r e l a t i o n s i n sediment l o a d e d f l o w , some a d j u s t m e n t of t h e s e pa ramete r s w i l l a l s o be n e c e s s a r y t o o b t a i n r e a l i s t i c e v o l u t i o n s of t e sed imen t l o a d under u n s t e a d y of nonuniform f low c o n d i t i o n s .

5 SIMULATION OF SEDIMENT TRANSPORT

5 .l E v o l u t i o n of s e d i m e n t l o a d

The ma themat i ca l model r e s u l t s were compared w i t h t h e r e s u l t s of a l a b o r a t o r y expe r imen t performed i n a f lume u i t h a l e n g t h of 30n, a w i d t h of 0.5m and a d e p t h of 0.7m a 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 (de f 6 ) . The d i s c h a r g e was measured by a c i r c u l a r w e i r . The mean f low d e p t h w a s 0.25m and t h e mean f l o w v e l o c i t y was 0.67m/s. The bed m a t e r i a l had a

5 0 = 230 p and a d 9 0 = 320 p. The median d i a m e t e r of t h e p a r t i c l e s i n suspens ion was e s t i m a t e d t o be

-sassa3oad le3rsXqd 3ueAalaz ay3 saJelnmys 2r JeqJ Kjyla~ 02 paJsaJ uaaq seq 'sdrqsuoy~ela~ lez~r~ydrna

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(qz 8yd) suoy2eAzasqo q3r~ 3uamaa~Ze za~~aq uyeJqo OJ uoy~e~qyle3

%urznp s~a~ame;zed lapom ay2 Burun2 Kq amo3~a~o aq ue3 say~uye~za3un ay2 amy2ueam aq2 ul .aznJng ay2 uy sar~ln3yg3rp asaq2 aaTosaJ 02 dlaq Xlsnor~qo plnom

s~aadse asay2 uo qa~easaz zaq2lnd .sJuamaznseam m013 33e12Xa 10 aJemy3sa OJ 2ln3yjjrp X~l~ln3~3led S? q3?qM 63y3ola~ leaqs ay3 Xq pauyjap Xlley~uassa

S? LzaJ2el aqJ lapom XH aq3 UI -Krlyaysnjgyp le3y~la~ a43 pue X~y3ola~ 8uyl33as al3gzed

aqJ ale slaJame~ed JueAalaz aqL -Xl33a.1103 2as JOU azan s~a3amezed lapom aq2 asneaaq 10 amnlg aq3 uy

leyaa3em paq 30 Juamuyea2ua ay3 unop %uymols paq aq2 30 8uy~nomle zo/pue aloq ~no3s aqq 30 uoyJemao3 aqJ OJ anp Xl~~ed aq Xem syq~ .slapom q~oq uy Jsej 003

S? aseazauy 30 a2ez rey2yuy aq3 ~eln~y3~ed ul *peal

auamypas papuadsns aq3 30 uoyJnloAa pahzasqo ay2 moaj Xl'lqS?~s a2eyAap sJTnsaJ lapom m pue Jjlaa aq3 q~og

OJ papuaJxa aq X~oaq~ uy plno~ Tapom Jjlaa aqa qE?noq31v .uoy~aazyp ~e~uozyzoq auo uy m013 Xpeaas

03 pajrmyl ~uasa~d 212 sy lapom 23laa aqJ '~a~amo~ .(g bz) a3ueleq Juamypas TE~TJ.I~A aq2 jo uoyanlos

33azyp B sem ~y 3oq~ asuas ay2 uy pa~e3y~syqdos azom sem lapom 33~aa aq3 313142 8uy~apysuo~ JTnsaJ

%uyZe~no~ua XJ~A e sy q~yqm 'uoy2nlos 23laa aq3 02 Jauuen Lzelymys XJ~A B UJ saa-yoAa uoy2n.2.0~ y,, aq;l

.In = X %ursn sysoq amy2 E o3ul pam~ojsu~~2 uaaq aAeq sJlnsa1 3glaa ay3 Jeq2 a2oN '"1 2~d uy pa~uasa~d ale

s2lnsaJ aqJ pue 9 gay ur pa2emy2sa s/mLLBaO X~~3ola~ Lzeaqs TTelaAo aqJ 8uymnsse unz sem lapom a~-mol~pues

m aq;l -~~odsue~~ peol papuadsns ay2 say% 03 pa~ezZa3uy azam asaqJ pue paq a8e~a~e aqj aAoqe mzz-o pue 50.0 'SZO*O '5~0.0 ~noqe 30 7qZyaq

E ~e pa~3allo3 azam sa~dmes ahyj (alyjozd) uoy2e~ol q3ea 2v .SUO~~EJ~U~~UO~ pues aq2 30 uoy2nqr~2 srp

ley2eds aq2 aurwaJap 02 suoy~e3ol znoj 3e poy2am uoqd~s I? 30 swam 6q d~snoauej~nrn~s pal~a~~o~

aJam saldues 1a2e~ -62y3ola~ molj 30 uo~~nqy~2syp le3y~la~ aq3 auymza3ap 02 pasn aAam saqn2 203yd

llems 'm5~0.0 Jnoqe 30 2q%raq e pup m~-o 2noqe 30 q33ua~ E 4u~aeq smJog paq q~ym paza~oa SEM paq meaJJs

aq;l '(3 .6 sa~n2ezadma3 la2em) s/mzz-O 30 62y3ola~ 11~3 a~rle2uasalda;l B U? 8urJlnsaJ 'uri 00~ 3noqe

I t was found t h a t t h e b e s t f o r m u l a t i o n f o r e n t r a i n m e n t of sand a t t h e bed was i n terms of a n e n t r a i n m e n t r a t e . The c o n n e c t i o n between t h i s e n t r a i n m e n t r a t e and t h e a s s o c i a t e d sand t r a n s p o r t law has been c o n s i d e r e d and i t was conc luded t h a t s t r i c t l y o n l y one of t h e s e shou ld be s p e c i f i e d . P h y s i c a l l y t h e e n t r a i n m e n t ra te i s more fundamenta l because i t i s t h i s , t o g e t h e r w i t h t h e p h y s i c s of t h e sys t em, which d e f i n e s t h e t r a n s p o r t . However, s i n c e i t i s e a s i e r t o measure sand t r a n s p o r t , t h i s i s o f t e n p r e s c r i b e d i n p r a c t i c a l problems.

The model was compared w i t h some f lume d a t a t o t e s t i t s r e s p o n s e t o a change i n t h e sed imen t l o a d . I t was shown t h a t t h e model s i m u l a t i o n c o u l d be c a l i b r a t e d by a d j u s t i n g t h e s e t t l i n g v e l o c i t y and v e r t i c a l d i f f u s i v i t y p a r a n e t e r s . T h i s p rocedure i s j u s t i f i e d f o r p r a c t i c a l a p p l i c a t i o n s because i n n a t u r e t h e s e pa ramete r s a r e n o t w e l l d e f i n e d . F o r example t h e r e i s no un ique s e t t l i n g v e l o c i t y because t h e suspended l o a d would c o n t a i n a r a n g e of s e d i m e n t s i z e s and t h e t r u e n a t u r e of t h e v e r t i c a l d i f f u s i v i t y p r o f i l e i s n o t f u l l y u n d e r s t o o d . These a s p e c t s w i l l b e ' c o n s i d e r e d f u r t h e r i n l a t e r s t a g e s of t h e p r o j e c t .

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Graf W H (1971) . H y d r a u l i c s of s e d i m e n t t r a n s p o r t . 14cGraw H i l l Book Company, New York.

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Hunt J N (1954) . The t u r b u l e n t t r a n s p o r t o f s e d i m e n t i n open c h a n n e l s . P roc . Roy Soc. A. 224. No 1158 pp 322-335.

K a l i n s k e A A (1940) . Suspended m a t e r i a l t r a n s p o r t a t i o n unde r n o n - e q u i l i b r i u m c o n d i t i o n s . T r a n s Am Geophys Union Vol 21.

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19. H i l e s G V , Ozasa H and Webb D G (1980). S e d i m e n t t r a n s p o r t r e l a t i o n s f o r e s t u a r y models . P a p e r p r e s e n t e d a t t h e SANDS M a r i n e S t u d i e s Group M e e t i n g , G e o l o g i c a l S o c i e t y , London, 3-4 Dec 1980. R e p r i n t e d by H y d r a u l i c s B e s e a r c h , W a l l i n g f o r d , UK.

20. Miles G V (1981). Sed imen t t r a n s p o r t models f o r e s t u a r i e s , d y d r a u l i c s R e s e a r c h , W a l l i n g f o r d , UK,

27 PP*

21. Odd N V M , Miles G V and Mann K (1976). H a t h e m a t i c a l and p h y s i c a l model s t u d i e s f o r t h e Wash Water S t o r a g e Scheme. P a p e r p r e s e n t e d a t j o i n t ICE/CWPU Symposium on Wash Water S t o r a g e Scheme, 9 Nov. 1976. i l y d r a u l i c s X e s e a r c h S t a t i o n R e p r i n t , W a l l i n g f o r d , UK.

22. X i j n L C Van (1384). Sed imen t pick-up f u n c t i o n s . J o u r n a l of H y d r a u l i c E n g i n e e r i n g Vol 110 (10) Oct 1984 ASCE pp 1494-1502.

23. R i j n L C van (1984). Sedi laent t r a n s p o r t 11 Suspended l o a d t r a n s p o r t . J o u r n a l of d y d r a u l i c E n g i n e e r i n g Vol 110 (11) Nov 1984 aSCE pp 1513-1641.

24. R i j n L C van (1985). : . l a thema t i ca l models f o r s e d i m e n t a t i o n of s h i p p i n g c h a n n e l s . I n t Conf on Num & Byd X o d e l l i n g of P o r t s and H a r b o u r s , Birmingham, England 23-25 Apr 1985 o r g a n i s e d by BHRA, Cranf i e l d , B e d f o r d , Eng land , UK.

25. Schmidt W (1925) . Der ~ ~ a s s e n a u s t a u s c h i n f r e i e r L u f t und ve rwand te E r sche inungen , i n "2robleme d e r k o s n i s c h e n P h y s i k " , Bd 7 Hamburg.

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8 LIST OF SYMBOLS

Ts (u ,v ,w) - - ( U , v 1 - U

c r i t

c o n c e n t r a t i o n of suspended s o l i d s (kg m 3,

depth-averaged c o n c e n t r a t i o n (kg mn- 3,

i n i t i a l c o n c e n t r a t i o n (kg m- 3,

c o n c e n t r a t i o n under s a t u r a t e d c o n d i t i o n s

depth-averaged s a t u r a t e d c o n d i t i o n s

w a t e r d e p t h ( m )

sediment g r a i n s i z e (mm)

l a t e r a l d i f f u s i o n c o e f f i c i e n t (m S- l )

l o n g i t u d i n a l d i s p e r s i o n c o e f f i c i e n t ( m S- )

d i f f u s i o n c o e f f i c i e n t s i n 3-D a x e s (m S- l )

sediment p i ck up r a t e (kg m- S- l )

n e t v e r t i c a l f l u x of sand (kg m- S-

a c c e l e r a t i o n of g r a v i t y (ms- 2,

q u a n t i t y of mobi le bed m a t e r i a l (kg m'2)

c u r r e n t speed , ( T 2 + ~ ~ 1 % (ms- l )

d i s c h a r g e per u n i t w i d t h (m S- l)

s e t t l i n g v e l o c i t y Reyno lds Number, wSd/IlZ

i n t r i n s i c c o o r d i n a t e s

s o u r c e / s i n k t e r m a t t h e bed (kg ih S- l)

t ime (S)

sand t r a n s p o r t (kg m- S- l )

components of v e l o c i t y v e c t o r (ms-l)

depth-averaged v e l o c i t y components (ms- l )

t h r e s h o l d v e l o c i t y f o r sand t r a n s p o r t (as- l)

uni form v e l o c i t y (ms- l )

s e t t l i n g v e l o c i t y (ms- l )

c a r t e s i a n c o - o r d i n a t e s (m)

p r o f i l e pa ramete r , ~ u c d z / d u c

bed exchange f a c t o r

p r o f i l e pa ramete r , c ( x , y , o , t ) / c ( x , y , t )

Von Karman c o n s t a n t

eddy v i s c o s i t y (m S' l)

s h e a r v e l o c i t y (ms'l) L

d i m e n s i o n l e s s v a r i a b l e , d / (4Dz t ) '

1.0

t e o f release o f sediment prescribed a t bed

Concentration prescribed at the bed

0 100 200 300 400

Time (secs)

( a ) Net sediment exchanges a t the bed I F z ) f o r 0-2mm sand 10.02rn/s se t t l i ng velocity). Water depth = 10m, depth-averaged velocity = Im/s and shear velocity = O.OSm/s,

20

18

16

14

- 12 E

. 10 C a W

0 8

6

4

2

0 0 . 2 0 . 4 0.6 0 .8 1.0 1.2 1 . 4 1.6 1.8 2.0

Depth averaged speed (m/sl

( b ) Bed exchange factor (B) f o r 0.2mm sand (0,02m/s set t l ing velocity) using Darcy- Weisbach f r ic t ion fac to r = 0.02

Fig. 1 Sediment exchange r e l a t i o n s a t t h e bed.

Fig. 2 Computed and measured evo lu t i ons o f sed iment l o a d

-

C 0 .-

L C

O Measured (Ref 6 1 Computed (Ref 6 1 - - Computed f o r var ious HR model t imes teps based on U + = 0 .0477m/s

-

I I

Q

0 5 10 15

Time (seconds)

( a ) Sens i t i v i t y t o model t imes tep

U s = O.OlBm/s

0 . 6 - @ Measured (Ref 6 ) - L C

C -- Standard case ( U s = 0 .022m/s ,E = 1 .0)

aI U =L= ) Sens i t i v i t y t o E (Ws = 0 - 022m/s l

- Cal ib ra ted case U s = 0.019m/s, E; = 1 .3)

0 5 10 15

Time (seconds)

Ib) Sens i t i v i t y t o s e t t l i n g ve loc i ty and ve r t i ca l d i f f us ion

Flume d a t a and t e s t condi t ions

Mean f l o w d e p t h 0.25111, mean f l ow ve loc i ty 0 .67m/s Pa r t i c l e d iameters ( d s o l 230ym b e d mater ia l , 200ym suspended S e t t l i n g ve loc i ty 0.022m/s, w a t e r t empera tu re 9OC Sediment dens i ty 2650kg/m3, f l u i d dens i t y 1000kg/m3 Ove ra l l bed shear ve loc i ty 0 .0477m/s