NUMERICAL MODELLING OF SEDIMENT TRANSPORT UNDER WAVES AND CURRENTS IN ESTUARIES

Report No SR 24 February 1985

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2 7 MAR 1985 CLASS No. ......................... *CC N1.

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

T h i s r e p o r t d e s c r i b e s r e s e a r c h c a r r i e d o u t u n d e r C o n t r a c t No DGR/465/35, f u n d e d by t h e Depa r tmen t o f T r a n s p o r t f r om A p r i l 1 9 8 2 t o March 1984 and t h e r e a f t e r by t h e Depa r tmen t o f t h e E n v i r o n m e n t . Any o p i n i o n s e x p r e s s e d a r e n o t n e c e s s a r l y t h o s e o f t h e f u n d i n g D e p a r t m e n t s . The work was c a r r i e d o u t i n t h e R i v e r E n g i n e e r i g D e p a r t m e n t o f HR by D r R B e t t e s s , Mr R W P e t h i c k a n d Miss H J M e l l o r f o r t h e T i d a l E n g i n e e r i n g D e p a r t m e n t . The DOE (ESPU) nomina t ed o f f i c e r was M r A J M H a r r i s o n and t h e P r o j e c t Manager f o r HR was M r M F C T h o r n . T h i s r e p o r t i s p u b l i s h e d by p e r m i s s i o n o f t h e D e p a r t m e n t o f t h e E n v i r o n m e n t .

Crown C o p y r i g h t 1985

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ABSTRACT

The movement of sediments in estuaries frequently causes problems. Any engineering vorks undertaken in any estuary may affect the pattern of sediment erosion or deposition and lead to beneficial or deleterious affects at locations throughout the estuary, ocassionally at places remote from the site of the original engineering works. Thus the construction of docks, quays or jetties may lead to increased sediment deposition or alternatively erosion. The mainteuance or enlargement of existing shipping channels may have an impact on sediment movement and any alterations in fluvial flow or tidal storage may affect patterns of sediment erosion or deposition. Before embarking upon such engineering works it is therefore important for their impact on an estuary to be assessed to enable rational decisions to be made about the advisability of carrying out proposed schemes and to assess which schemes will achieve the required aims. Because of the complexity of the physical system representing an estuary recourse has frequently to be made to numerical models to make such predictions. Numerical models have been successfully used in the past to predict the behaviour of both cohesive and non-cohesive sediments in estuaries but these models have ignored the effects of waves which in certain cases are significant. In this report the effects of waves in these situations are discussed and methods of including these in a numerical model are considered. Much is known separately about the behaviour of waves and currents and also the interactions between the two. The chief constraint at the moment on modelling sediment transport under waves and currents is the lack of a reliable theory for predicting sediment movement in wave dominated regions. The work should aid in the development of numerical models to predict the behaviour in estuaries under the action of both waves and currents.

1 INTRODUCTION The movement of sediment i n e s t u a r i e s f requent ly causes problems. Any engineering works undertaken i n an e s tua ry may a f f e c t the pa t t e rn of sediment e ros ion o r depos i t ion and lead to b e n e f i c i a l o r d e l e t e r i o u s a f f e c t s a t l oca t ions throughout the e s tua ry , occas ional ly a t places remote from the s i t e of the o r i g i n a l engineering works. Thus the cons t ruc t ion of docks, quays o r j e t t i e s may lead t o increased r a t e s of sediment depos i t ion or a l t e r n a t i v e l y eros ion . The maintenance o r enlargement of e x i s t i n g shipping channels may have an impact on sediment movement and any a l t e r a t i o n s i n f l u v i a l flow o r t i d a l s to rage may a f f e c t pa t t e rns of sediment e ros ion or depos i t ion . Before embarking upon such engineering works i t i s the re fo re important f o r t h e i r impact on an e s tua ry t o be assessed t o enable r a t i o n a l dec is ions t o be made about the a d v i s a b i l i t y of carrying out proposed schemes and to a s s e s s which schemes w i l l achieve the required aims. Because of the complexity of the physical system represent ing an e s tua ry recourse has f requent ly to be made t o numerical models t o make such p red ic t ions .

There has been a h i s t o r y of the successfu l use of numerical models to p red ic t the behaviour of both cohesive and non-cohesive sediments i n e s t u a r i e s under the ac t ion of t i d a l cu r ren t s (Abbott, 1979; Hydraulics Research S t a t i o n 1982 and Verwey 1983). I n these models, however, only the e f f e c t s of c u r r e n t s on sediment t r anspor t i s ca lcula ted and, a t present , no allowance i s made f o r the e f f e c t of waves. Though i n most e s t u a r i a l cases the movement of sediment i s dominated by the cu r ren t s , i n c e r t a i n circumstances waves may have a s i g n i f i c a n t e f f e c t . Numerical models have been developed a t HR f o r the ca lcu la t ion of waves (Southgate, 1981 and 1984). The physics of the i n t e r a c t i o n of waves and cu r ren t s has a l r eady been described elsewhere (Longuet-Higgins, 1970; Noda, 1974 and Noda e t a1 1974). This repor t considers aspects of the problems of combining both cu r ren t and wave numerical models t o allow f o r these i n t e r a c t i o n s and of determining any subsequent sediment movement. I t i s not intended to cover i n t h i s r epor t e i t h e r the numerical modelling of cu r ren t s o r waves sepa ra te ly o r the physics of the i n t e r a c t i o n s between the two a s these have been described extens ive ly elsewhere.

Though i t i s known t h a t cu r ren t s and waves i n t e r a c t and equations have been derived descr ib ing such i n t e r a c t i o n s and a l s o t h a t waves a f f e c t the t r anspor t of sediment by c u r r e n t s , the r e l a t i v e magnitude of the

various effects in practical situations remains unclear. A general assessment of their impact in common estuarial situations must await the development and implementation of the sort of numerical model discussed in this report.

2 SEDIMENT TRANSPORT UNDER WAVES AND CURRENTS

Sediment transport in estuaries may depend upon both the waves and the currents. To simulate the movement of non-cohesive sediments in an estuary it is necessary to have a theory which will predict the local sediment transport rate. Many of the theories of sediment transport under waves and currents have been developed from theories of sediment transport under steady uni-directional currents. In the limit of zero waves these theories become identical to the equivalent, steady uni-directional flow equations which produce satisfactory predictions. The waves and currents theories can be regarded as extrapolations of the steady uni-directional flow theories. A comparison of such theories with field and laboratory data has indicated that, providing the effect of the current dominates that of the waves, they provide reasonable predictions but that when waves dominate the results are less satisfactory (Hydraulics Research, 1985). The predictions in the case where waves predominate could be improved if a reliable predictor were available for the case of waves alone. The main findings of the comparison are summarised in Appendix 1.

The sensitivity of sediment transport rate to both waves and currents is demonstrated by Figure 1 which shows the sediment transport rate calculated using Ackers and White-Swart equations. This sensitivity implies that to effectively include both waves and currents it is necessary to model both adequately.

Thus one is lead to the conclusion that to include the effects of sediment transport under waves and currents one should have a model which has both a flow component and a wave component.

When the waves and currents are not in the same direction consideration must also be given to the direction of the resulting sediment transport. The net movement of the sediment may not necessarily be in the direction of the waves or the current. Since the direction of the net shear may be different at different levels throughout the depth it is possible

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tidal problems. The finite-difference scheme used in such models may be implicit or explicit. In general explicit schemes involve fewer calculations per timestep but stability constraints limit the length of timestep that can be used. Implicit schemes can utilise larger timesteps but each timestep involves more computational effort. Explicit schemes can readily be adapted to take advantage of the DAY available at HR.

In combining the wave and current models there would be advantages if they could use the same numerical grid or if one was a subset of the other.

4 IRIERACTIORS OF WAVES AND CURBEATS

Under certain circumstances the mutual interaction of waves and currents may be significant (Noda et a1 1974). In these cases it is not sufficient to calculate the currents and waves independently. Even in the absence of tidal currents local variations in wave height result in wave-induced Reynolds. stresses - or radiation stresses which drive currents (Longuet-Higgins, 1970 and Noda 1974). Thus in these situations the wave field must be determined first, the Reynolds stresses calculated and their effect included when determining the current field. A summary of the equations describing wave-induced Reynolds - stresses is given in Appendix 3.

Variations in the spatial velocity distribution cause wave refraction. Thus where this is significant the current field must be known first and then the wave field may be calculated.

The magnitude of these current-wave interactions will vary with the situation and conditions and may frequently be insignificant. Initially some effort should be expended on characterising those conditions where such interactions may be ignored and those where it is important that they be included.

To include all these interactions it is necessary that the part of the model that calculates the wave refraction should be capable of including the effect of currents and that the flow part should be capable of including the Reynolds stresses induced by variations in wave height. If the waves and currents elements are separate in the model then some iterative technique is required in which the wave and current fields are solved for successively.

If the flow element is solved using a time-stepping procedure consideration should be given to the frequency with which the wave-field needs to be calculated. In many tidal situations, provided the waves are only weakly dependent upon the currents, the timescale for variations in the current field will be very much smaller than that for variations in the wave field. Consideration may then be given to whether different timesteps could be used for the calculation of the wave and current fields so that the wave field is up-dated less frequently than the currents.

5 EFFECT OF WAVES AND CURRENTS ON BED FEATURES

The presence of waves affects the bed features that are developed on a mobile bed and hence influences the hydraulic roughness; a knowledge of which is important in the determination of the flow field. Swart (1976) relates the hydraulic roughness to the bed form steepness though as he says this does not solve the problem of the determination ofthe bed roughness, itjust shifts i t . Swart presents equations to predict bed form steepness and hence hydraulic roughness under wave dominated conditions but it is unclear how reliable such equations are.

6 HOVERENT OF COBESIVE SEDIMENTS

In the current study we have restricted attention to the movement of non-cohesive sediments and have not considered cohesive sediments but the work that has been done does have some bearing upon the cohesive behaviour. The presence of waves leads to an increase in the maximum shear stress developed in the flow and this may lead to hindered settling of cohesive material in suspension or to the erosion of material from the bed if the magnitudes of the waves and currents are sufficiently large. Expressions for the bed shear stress under waves and currents are discussed in Appendix 2.

The presence of waves in a channel induces a non-zero mean velocity which, close to the bed, is in the direction of the waves. This can induce a steady motion in mud layers on the bed in the direction of the waves (Lhermitte, 1958 and Migniot 1977). For muds of viscosity ranging from 3 X 10-~m~/s up to 7 X 10-~m~/s Lhermitte (1958) found the following expression for the drift velocity us at the surface of

the mud bed,

- = 0.25 X maximum drift velocity in neighbourhood of

2 d

Migniot then quotes this as:

- us = 0.25 X 0.18 U, l.6 d-Ov6 in cgs units though his expression for the maximum drift velocity is curious.

7 CONCLUSIONS To include the effect of sediment transport under waves and currents in numerical estuary models the Ackers and White uni-directional sediment transport equations can be simply adapted by altering the effective shear. This should produce satisfactory predictions of transport rates provided the effect of the currents dominates that of the waves. The predictions will be less satisfactory where the waves dominate the currents. Since the sediment transport rate is sensitive to the wave height it is suggested that consideration should be given to the accuracy with which the wave field needs to be modelled and the possible effect of iteractions between the waves and currents.

The equations for the bed shear under waves and currents may also be used to include the effect of waves on the behaviour of cohesive sediments.

It is suggested that more work is directed at:

(a) a better expression for the bed shear under waves and currents,

(b) the direction of sediment transport when waves and currents are not in the same direction,

(c) the prediction of the hydraulic roughness of the bed under waves and currents,

(d) the development of a reliable theory for sediment transport under waves alone so that improvements could be made to predictions in the wave dominated area.

8 REFERENCES Abbott M B, 1979, Computational hydraulics; Elements of the theory of free surface flows, Pitman.

Ackers P and White W R 1973, Sediment transport : new approach and analysis J Hydraul. Div. ASCE., 99, - HYll, 2041-2060.

Bakker W T 1974, Sand concentration in an oscillatory flow, Chapter 66 Coastal Engineering.

Bijker E W 1967, Some considerations about scales for coastal models with moveable beds, Delft Hyd. Lab Publ. No 50.

Fredsoe J, 1983, The turbulent boundary layer in combined wave-current motion, Institute of Hydrodynamics and Hyd. Eng., Tech. Univ of Denmark, Report No 259.

Hydraulics Research Station, 1982, HRS Tidal modelling system, FIR Report IT 238.

Hydraulics Research, 1985, Sediment transport under waves and currents, FIR Report SR 22.

Jonsson I G, 1967, Wave boundary layers and friction factors, Coastal Engineering, Tokyo, pp 127-149.

Lhermitte, P, 1958, Contribution 3 lbtude de la couche limite des houles progressivcs, COEC No 136, Miniatere de la Dgfense Nationale, Paris.

Longuet-Higgins M S, 1970, On the longshore currents generated by obliquely incident sea waves. J. Geophys Res 75. - Migniot, C, 1977, Action des courants de la houle et du vent sur lea sbdiments, La Houille Blanche, pp9-47.

Noda E K, 1974, Wave-induced nearshore circulation, J Geophy Res 79 - pp 4097-4100. Noda E K et al, 1974, Nearshore circulations under sea breeze conditions and wave-current interactions in the surf zone, Rep. TETRA-P-72-149-4, ppl-216, Tetra Tech, Pasadena, Calif.

Southgate H N, 1981, Ray methods for combined refraction and diffraction problems. Hydraulics Research Station Report, IT 214.

Southgate H N. 1984, Techniques of ray averaging, Int J. Num Methods in Fluids 4 pp 725-747. -

Swart D H , 1976, Computation of longshore t r a n s p o r t , D e l f t Hyd Lab Report R 968 P a r t 1.

Van de G r a a f f , J and Van Overeem J, 1979, E v a l u a t i o n of sediment t r a n s p o r t formulae i n c o a s t a l e n g i n e e r i n g p r a c t i c e , C o a s t a l E n g i n e e r i n g , 3 , - pp 1-32.

Verwey A , 1983, Computational h y d r a u l i c s i n h y d r a u l i c d e s i g n , i n Developments i n h y d r a u l i c e n g i n e e r i n g - e d i t e d by P Novak, Applied Sc ience P u b l i s h e r s .

Willis D H , 1978, Sediment l o a d under waves and c u r r e n t s . P r o c . 1 6 t h C o a s t a l Engineer ing Conference, Hamburg, L, pp 1626-1637.

APPENDIX 1

SEDIMENT TRANSPORT UNDER WAVES AND CURRENTS

The f o l l o w i n g t h e o r i e s have been compared w i t h f i e l d and f lume o b s e r v a t i o n s :

1. F r i j l i n k - B i j k e r

2. Ackers and White-Swart

3. Ackers and White-Will is

4 . Ackers and White-van d e Graaf and van Overeem

5. Engelund and Hansen-Swart

The above sediment t r a n s p o r t e q u a t i o n s have been d e r i v e d from e q u a t i o n s f rom u n i - d i r e c t i o n a l sediment t r a n s p o r t . A s a n i l l u s t r a t i o n of t h e methods involved we d e s c r i b e t h e Ackers and White-Swart method.

Ackers and White-Swart

The Ackers and White e q u a t i o n s f o r sediment t r a n s p o r t u t i l i s e f o u r parameters , n,A,m and C which depend upon t h e d imens ion less g r a i n s i z e D

g r of t h e sediment. Swart keeps t h e s e unchanged when a p p l y i n g t h e e q u a t i o n s t o sediment t r a n s p o r t under waves and c u r r e n t s . The m o b i l i t y i n t h e c a s e of c u r r e n t s above i s d e f i n e d by

and Swart d e f i n e s t h e cor responding v a r i a b l e f o r waves and c u r r e n t s , F W C by:

g r

(Al. 2 )

( s e e Appendix 2 f o r n o t a t i o n ) where v%c = ( l+( 5 )

The e q u a t i o n f o r sediment c o n c e n t r a t i o n i n t h e case of c u r r e n t s a l o n e i s

D V n X = Ggr S a (T) (A1.3)

I n t h e caae of c u r r e n t s and waves Swart r e p l a c e s t h i s by

Thus, deno t ing v*wc/v*c by p t h e e q u a t i o n f o r t h e sediment c o n c e n t r a t i o n

The comparisons w i t h o b s e r v a t i o n s i n d i c a t e d t h a t , i n g e n e r a l , t h e methods performed s a t i s f a c t o r i l y when t h e e f f e c t s of t h e c u r r e n t s dominated t h o s e of t h e waves bu t t h a t they uniformly over -p red ic ted the t r a n s p o r t r a t e when waves dominated, F i g 2 . T h i s was i n p a r t due t o t h e e x p r e s s i o n adopted by t h e v a r i o u s methods f o r t h e s h e a r s t r e s s under waves and c u r r e n t s . The comparisons seemed t o i n d i c a t e t h a t t h e p r e d i c t i o n s by t h e v a r i o u s t h e o r i e s depended very h e a v i l y on t h e e x p r e s s i o n used f o r t h e bed s h e a r s t r e s s developed under waves and c u r r e n t s , v-,. When e q u a t i o n f o r bed s h e a r s t r e s s ( A 2 . 6 ) was rep laced by e q u a t i o n ( A 2 . 3 ) then improved p r e d i c t i o n s r e s u l t e d though t r a n s p o r t r a t e s were s t i l l over -p red ic ted , F i g s A 1 and AZ. An ad-hoc c o r r e c t i o n t o t h e s h e a r s t r e s s w a s employed which improved t h e p r e d i c t i o n s .

I n g e n e r a l t h e methods based on Ackers and White sediment t r a n s p o r t t h e o r y performed b e t t e r than t h e o t h e r s . Within t h e group t h e r e were v a r i a t i o n s i n t h e behaviour of t h e d i f f e r e n t a d a p t i o n s b u t , t a k i n g i n t o account t h e inadequac ies of t h e d a t a upon which t h e s e methods were t e s t e d , i t would be d i f f i c u l t t o argue t h a t t h e s e v a r i a t i o n s were s i g n i f i c a n t .

It i s u n l i k e l y t h a t p r e d i c t i o n s i n t h e wave dominated r e g i o n can be s i g n i f i c a n t l y improved u n t i l more r e l i a b l e e q u a t i o n s a r e a v a i l a b l e f o r sediment t r a n s p o r t under waves a lone .

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D~screpancy rat10 Ipredlcted /observed)

L h! W 2 " - - I I I

0

8 0 0

'8 0 0

9 0

0 0

c3

0

0

0

0 0

0

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D~screpancy ratio (pred~rted /observed1

AF'PENDIX 2

SHEAR STRESS UNDER WAVES AND CURRENTS

B i j k e r (1967) assumed t h a t t h e v e l o c i t i e s under waves and c u r r e n t s were t h e sum of a u n i f o r m f low 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 ,

and t h e wave v e l o c i t y g i v e n by t h e f i r s t - o r d e r wave t h e o r y

where H is t h e wave h e i g h t , d is t h e d e p t h and k i s t h e wave number. B i j k e r t h e n c a l c u a l t e d t h e mean component of t h e r e s u l t a n t bed s h e a r i n t h e d i r e c t i o n of t h e c u r r e n t t o g i v e

where

p = 0.45 i s t h e Chezy roughness c o e f f i c i e n t g i v e n by Ch = l 8 loglO r is t h e bed roughness and $ i s t h e a n g l e between t h e wave c r e s t s and t h e normal t o t h e c u r r e n t . B i j k e r e v a l u a t e d t h e e l l i p t i c i n t e g r a l i n (A2.3) f o r a range of v a l u e s of @,uo and V and f i t t e d a n approx ima t ion t o (A2.3) of t h e form

f o r v a r i o u s v a l u e s of $, s e e B i j k e r (1967) f o r d e t a i l s . These approx ima t ions a r e o n l y v a l i d i n p a r t i c u l a r p a r a m e t e r r a n g e s and s h o u l d on ly be used i n t h o s e r a n g e s . I n t h e c a s e where t h e component of t h e s h e a r i n t h e d i r e c t i o n of t h e c u r r e n t i s a lways p o s i t i v e then (A2.3) r e d u c e s t o

b u t t h i s shou ld o n l y be used i f uo < V.

The maximum bed s h e a r s t r e s s is g i v e n by

Swart (1976) suggested r e p l a c i n g e q u a t i o n (A2.4) by t h e e q u a t i o n

where f w is t h e Jonsson wave f r i c t i o n f a c t o r which can be approximated

by

where a, is t h e o r b i t a l amphitude a t t h e bed, t h a t i s

B i j k e r ' s model, us ing 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 f o r t h e c u r r e n t and a f i r s t o r d e r wave theory , is r e l a t i v e l y crude and must be seen a s a major weakness i n any theory of sediment t r a n s p o r t t h a t u s e s i t .

There a r e more s o p h i s t i c a t e d models bu t t h e s e f r e q u e n t l y i n v o l v e s o l v i n g a d i f f e r e n t i a l e q u a t i o n through t h e depth t o o b t a i n t h e flow s t r u c t u r e and hence o b t a i n i n g t h e s h e a r s t r e s s (Bakker, 1974 and Predsoe, 1983). Such models would be computa t iona l ly expensive t o implement. The p o s s i b i l i t y a r i s e s , however, of developing a p p r o p r i a t e ' look-up' t a b l e s which might l e a d t o s a v i n g s i n computation.

amosaq (S-CV) Pus (+?'€V) '(€'€V) suo~asnba uo~2ernyxo~dde zaaen nol~sqs aqa asn an $1

(9.CV) UTS

W+%=,

PUP STXE-X aq~l pue asazs ahsn aqj oa ~mzou aqa uaanjaq a18ue aqa SS 6

(+?'CV) [ez so^(%-U) + ez urs(%-uz)] HU = '(b

(E'CV) [~~u?s(%-~) + 62 so3(%-'-'~)] Z~% 1 =

TaAal Jaaen ueam aqj ar.oqs assjlns laqen aql 30 aqB?aq ~sso~ aq3 S? L alaqn

NOTATION

A Ackers and White parameter

constant

wave rbital amplitude at the bed

constant

Ackers and White parameter

Chezy coefficient

constant

sediment diameter

flow depth

sediment mobility, Ackers and White

sediment mobility under waves and currents

Jonsson friction factor

acceleration due to gravity

dimensionless sediment transport rate

wave amplitude

wave number, 2 dwavelength

Ackers and White parameter

Ackers and White parameter

constant

roughness of bed

specific gravity of sediment p,/ p

period of waves

maximum orbital velocity of waves in neighbourhood of the bed

drift velocity at surface of mud bed

slualln> aq3 03 lemxou aqa pue slsalJ ahen aql uaanlaq a~Sua

slualln:, pue sahen lapun ssaxls leaqs

slualln3 lapun ssaals leavs

luamypas 30 Kjysuap

lalen 30 djysuap

('*.+./'+A =) SauallnJ aapun d~yso~ah 1Eaqs 02 sluallnJ pus sahen lapun XlyJo~ah leaqs 30 oylel

SluallnJ pue sahen 02 anp XlpoTah leaqs

1ualinJ 02 anp KlyJoTaA xsaqs