Sediment concentration due to wave action

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Sediment concentration

due to wave action

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SEDIMENT CONCENTRATION

DUE TO

WAVE ACTION

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus,

prof.drs. P.A. Schenck

in het openbaar te verdedigen

ten overstaan van een commissie,

aangewezen door het College van Dekanen

op donderdag 22 september 1988 te 16.00 uur door

< ^ N , S C ^

*-«? ^.

JAN VAN DE GRAAFF

geboren te Hengelo

civiel ingenieur

3?

Q Prometheusplein 1 'iL cni 2623 ZC

DELFT

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Dit proefschrift is goedgekeurd door de promotor ,'

prof.dr.ir. E.W. Bijker j

;l

1

Cover: waves during T.O.W .-field campaign 1982-83 Photo-service Delft Hydraulics

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V

ACKNOWLEDGEMENTS

The research project described in this thesis was a part of the Netherlands Applied Research Program Rijkswaterstaat — Coastal Research (TOW). I am greatly indebted to the members of the 'Sediment transport' working group of that program for their inspiring support.

Mr. J.A. Roelvink and Mr. R.C. Steijn carried out most of the experimental work. I would like to express my gratitude to them.

I wish to thank Mrs. G. Boone for her excellent typing work and her care of the printing of the thesis.

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.) VI

SAMENVATTING

SEDIMENT CONCENTRATIE ONDER GOLVEN

Enkele details van het gecompliceerde sediment transportproces als gevolg van golven en stroom zijn in deze studie onderzocht. Een cruciaal punt in dit transportproces is de concentratieverdeling over de waterdiepte onder golfwerking. De min of meer theoretische concepten die voor de beschrijving hiervan in het verleden zijn voorgesteld, blijken in een confrontatie met werkelijk uitgevoerde metingen vaak uitermate slecht te voldoen.

Als, ook onder golfwerking, ervan wordt uitgegaan dat een relatief eenvoudig mengings- of diffusieachtig concept kan worden gebruikt voor de beschrijving van de verdeling van de sedimentconcentratie over de waterdiepte, dan is uit gemeten concentratievertikalen de kennelijk aanwezige verdeling over de hoogte van de mengingscoëfficiënten te bepalen. Omdat in de praktijk zelden werkelijk uniform bodemmateriaal voorkomt, moet in de analyse van gemeten concentratievertikalen terdege rekening worden gehouden met de werkelijke korrelverdeling van het bodemmateriaal. Enkele rekenvoorbeelden hebben aangetoond dat als deze gradering niet op de juiste wijze in rekening wordt gebracht, er grote afwijkingen in de grootte van de te

berekenen mengingscoëfficiënten zijn te verwachten. Omdat niet bij voorbaat vaststaat dat verschillende korreldiameters voorspelbaar (dat wil zeggen bijvoorbeeld afhankelijk van de valsnelheid in stil water) reageren op een gelijke mate van mengingsactiviteit in het water, is dat aspect uitvoerig experimenteel onderzocht. [e(s) = (3 e(f); e(s): mengingscoëfficiënt voor het sediment; e(f): idem voor het wateren J3: factor]. Het gedrag van p is dus nagegaan. In een golfgoot is daartoe de concentratieverdeling bij diverse bodemmaterialen (5 tot 8 verschillende diameters) gemeten onder verschillende randvoorwaarden met betrekking tot de ingestelde golfhoogten en golfperioden. Voor het doel van deze proeven was het gewenst dat de horizontale bodem van de golfgoot van kunstmatige ruwheidselementen was voorzien. Uit de analyse van deze metingen is gebleken dat P inderdaad afhangt van de korreldiameter, maar ook, en dat maakt één en ander extra gecompliceerd, kennelijk van het 'niveau' van de mengingsactiviteit in het water. De proefresultaten waren niet in alle opzichten werkelijk

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IX SAMENVATTING

eenduidig; er bleek tamelijk veel spreiding voor te komen. Niettemin is geconcludeerd dat de grootte van P zowel een functie van de valsnelheid van het bodemmateriaal als van de mate van menging in het water is. Zowel p waarden kleiner dan 1 als groter dan 1 blijken voor te kunnen komen.

In de studie is het uit de experimenten gevonden gedrag van p in rekening gebracht om de kennelijke e(f) verdeling onder golf werking te bepalen voor verschillende golfcondities. Er zijn daarbij opmerkelijke e(f) verdelingen gevonden. (Overigens zouden die resultaten nauwelijks minder opmerkelijk zijn geweest als het complicerende effect van P niet apart in rekening zou zijn gebracht; bijvoorbeeld door P eenvoudigweg gelijk aan 1 te stellen en uit de proeven met verschillende korreldiameters gewoonweg een gemiddelde mengingscoëfficiëntenverdeling te berekenen). Het opmerkelijke van de gevonden e(f)-verdelingen is dat, zelfs onder niet-brekende golven over een horizontale bodem, relatief zeer hoge mengingsactiviteiten blijken voor te komen over een zeer groot gedeelte van de waterkolom. Dat er vlakbij de bodem een hoge mate van menging voorkomt, is wellicht, mede gezien de kunstmatige ruwheid van de bodem, niet zo vreemd. Het blijkt echter dat die hoge mate van menging niet beperkt is tot een dunne laag vlak bij de bodem, maar in veel gevallen zelfs toeneemt hogerop in de vertikaal. Enkele geanalyseerde gevallen met natuurlijke bodems (kleine schaal en grote schaal in de grote golfgoot van de Universiteit van Hannover), laten zien dat dit effect vermoedelijk niet

(uitsluitend) wordt veroorzaakt door de kunstmatige ruwheid. Ook in dergelijke gevallen is er een zeer sterke toename van de mengingsactiviteit over de hoogte te zien.

Dat het gekozen concept met mengingscoëfficiënten ook onder golfwerking een goed werkend concept is, kon bevredigend worden aangetoond aan de hand van een vergelijking tussen gemeten en berekende korrelverdelingskarakteristieken over de waterdiepte. Gegeven de korrelverdeling van het bodemmateriaal is met het mengingsconcept immers eenvoudig uit te rekenen hoe de ontmenging van het materiaal over de hoogte zal verlopen.

Zelfs regelmatige, niet-brekende golven over een horizontale bodem zijn kennelijk in staat relatief hoge sedimentconcentraties in de waterkolom te handhaven. Dat proces kan worden beschreven met een mengingscoëfficiënten concept. Een logische volgende gedachte is dan dat dergelijke golven eveneens een belangrijk effect op de overdracht van interne schuifspanningen in de waterkolom kunnen hebben in geval een combinatie van golven en stroom wordt

beschouwd. In de meeste sedimenttransport beschouwingen voor de combinatie van golven en stroom, die zijn gebaseerd op een 'snelheid' maal 'concentratie' concept, wordt voor de snelheidscomponent gewoonlijk een logaritmische snelheidsverdeling over de hoogte aangehouden. Een dergelijke logaritmische verdeling blijkt in het geval van uniforme stroom

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SAMENVATTING X

vaak goed te voldoen. Bij de beschrijving van de overdracht van de interne schuifspanningen in de waterkolom met een diffusiecoëfficiëntenverdeling over de hoogte, resulteert in dat geval een parabolische verdeling. Uit het onderzoek is gebleken dat de mengingscoëfficiënten voor het sediment zeker niet parabolisch over de hoogte zijn verdeeld. Als de voor golven geldende coëfficiënten tevens worden gebruikt om een snelheidsverdeling uit te rekenen, resulteert een verdeling die, zeker nabij de bodem, behoorlijk afwijkt (lagere waarden geeft) van een logaritmische verdeling met een zelfde gemiddelde snelheid. (Overigens staat nog niet onomstoteüjk vast hoe de afzonderlijke bijdragen van de golven en de stroom aan de resulterende e(f) verdeling in rekening dienen te worden gebracht). Enige experimentele resultaten laten inderdaad zien dat de snelheidsverdelingen nabij de bodem duidelijk van een logaritmische verdeling afwijken. Eén en ander kan belangrijke gevolgen voor het te berekenen zandtransport hebben.

Veel meer onderzoek is noodzakelijk voordat een totaal nieuwe beschrijving voor het zandtransportproces door golven en stroom kan worden voorgesteld. Deze studie geeft belangrijke aanknopingspunten voor dat verdere onderzoek. Naar verwachting is echter nog een lange, maar uiterst boeiende weg te gaan voordat een samenhangend totaal concept kan worden geformuleerd.

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1

CHAPTER 1 INTRODUCTION

1.1 GENERAL

A wide range of practical coastal engineering problems can be properly solved only if clear insights are available into the sediment transports involved. Examples of these problems are: • Sedimentation of (navigation) channels

• Erosion / sedimentation near harbour breakwaters • Gradual erosion of coasts

• Beach and dune erosion during severe storm surges • Natural development of coastlines with time

The sediment transport phenomenon in practical coastal engineering problems is essentially affected by currents as well as by waves. That transport mechanism is rather complicated, as will be pointed out further on in this section.

The limited knowledge of the fundamentals of the transport processes has obliged coastal engineers in the past to use more or less integral solution approaches to their practical problems. Smart and clever methods have been developed and used, frequently with great successes, e.g.: • Development of CERC formula

• Pelnard-Considere coastline computation method

• Equilibrium profile approaches to describe cross-shore sediment transport rates [Swart (1974)]

• Dune erosion computation methods [Dean (1982); Vellinga (1986)]

Nowadays, however, our problems are frequendy more complicated; they call for physically conceived solutions. This means that the transport processes and the transport rates involved have to be known in detail.

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2 1. INTRODUCTION

In quasi-uniform flow cases (e.g. in rivers) the transport rate through a cross-section can, in principle, be simply described as the product of velocity and average concentration (S = v * c), to be integrated over the water depth.

In cases where waves play an important role in the transport process, things become far more complicated. It seems clear that in the transport direction parallel to the wave propagation direction the time-dependence of the sediment concentration has to be taken into consideration. (To achieve a net transport, an integration over time has to be worked out, besides the integration over the water depth).

For a transport direction more or less perpendicular to the orbital motion plane of the waves it seems permissible to take time-averaged sediment concentrations into account.

Therefore that transport direction is far easier to deal with than the other.

Since our knowledge of time-dependent sediment concentrations (on a wave period scale) is insufficient to carry out the appropriate integrations, the fundamental approach of the transport in the wave propagation direction cannot be used at this moment.

Although the transport in the direction of wave propagation is hardly amenable to treatment by a proper physical procedure, it should be mentioned that very promising results are achieved with simplified methods [e.g. Stive & Battjes (1984)]. In their method they take into account time-averaged sediment concentrations and the net flow circulation pattern in a cross-section as a result of breaking waves.

However, sediment transports in the wave propagation direction will not further be specifically discussed in the present study.

In the present study the sediment transport direction perpendicular to the orbital plane is in fact the main research subject. Thus time-averaged sediment concentrations can be used. Bijker (1971) developed a coherent computation method for this mode of sediment transport. With the help of a bottom transport formula he calculates a reference sediment concentration close to the bed. Next, an entire sediment concentration distribution over the water depth is calculated. That calculation can be carried out, since Bijker 'knows' the mixing or diffusion coefficients for the sediment in the water column. According to his assumptions, diffusion activity is affected by the mere current velocity and bed roughness, but also, to a large extent, by the wave action. At the time of Bijker's pioneering work the experimental tools for adequately verifying the implications of his assumptions were lacking. In recent years experimental work has revealed that some of Bijker's assumptions should be replaced by better ones. However, up to now no clear evidence as to dependable alternatives has emerged.

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1.2 SCOPE OF THE STUDY 3

In studying sediment concentration distributions over the water depth under wave action, two aspects are encountered which attract sometimes some attention in the literature, but are in the present author's opinion of such importance that they should always be fully taken into account. These aspects are:

• Effect of non-uniform bottom material

• Different apparent diffusion activities for different particle sizes

In the present study the effects of both aspects are discussed. It will become apparent that our ideas concerning the mechanism which holds the particles in suspension, together forming a sediment concentration distribution, have to be changed. An important fact like the magnitude of the (near) bottom concentration as a function of the boundary conditions — such as wave height, wave period, current velocity and particle size characteristics — is not a part of the present study.

1.2 SCOPE OF THE STUDY

The ultimate purpose of the present and further studies is to obtain a reliable knowledge of time-averaged sediment concentration distributions over the water depth under current and wave action. One should be able to predict the distribution characteristics as a function of the boundary conditions. In conjunction with a current velocity distribution description over the water depth, the resulting sediment transports can then be calculated.

The non-uniformity of the bottom sediment particles affects the shape of a (measured) sediment concentration distribution over the water depth quite considerably. In Chapter 2 these effects will be clarified and illustrated with the help of a series of simplified numerical examples. From many examples it has become clear that the shape of the concentration distribution over the water depth, compared with well-known 'classical' distributions, is often quite different from the 'classical' ones. An entirely 'new' distribution shape seems to occur. From a series of experimental cases it emerges that in fact a reliable description of measured concentration distributions cannot be found with any of the 'classical' distributions.

In Chapter 3 details of the suspension process will be discussed. It will be concluded that some simple small-scale tests will be helpful to clarify some important details of the mechanism. Chapter 4 deals with the results and analysis of a series of small-scale tests. The mechanism assumed and the method adopted for calculating the effects of non-uniform bottom material will be shown to be valid.

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4 _{1. INTRODUCTION}

With the help of the ideas developed, arbitrary concentration distribution measurements can next be analysed. An entirely new mixing or diffusion coefficient distribution over the water depth due to wave action for the water (and sediments) is found. That diffusion activity distribution looks quite different from the distributions according to existing ideas. In Chapter 5 some evidence is discussed for the distributions that have been found. Also, the theoretical consequences are indicated for the current velocity distribution over the water depth if a combination of waves and currents is present.

Chapter 6, finally gives a discussion of the results. Furthermore, subjects for further research will be pointed out.

1.3 SUMMARY AND CONCLUSIONS

To be able to solve practical coastal engineering problems, it is important to unravel the mechanism of sediment transport under the combined action of waves and currents.

For a better understanding of the mechanism of the sediment transport the underlying processes should be studied (viz.: time-averaged concentration c(z) and velocity v(z) distribution). Time-averaged parameters can be used, since only limited types of problems (viz.: longshore sediment transport problems) are mainly considered in the present study.

Measured c(z) distributions under several wave action conditions may yield a first set of 'facts' from which possible effects of boundary conditions can be seen and probably be understood. In analysing measured c(z) distributions over the water depth, however, the rate of sorting of the bottom material must be fully taken into account. Since different rates of sorting occur in practice and during tests, quite erroneous conclusions may be drawn if the sorting effects are not properly taken into account.

The c(z) distribution over the water depth under wave action can be described in principle with a mixing or diffusion coefficient approach. [e(s) distribution].

With this mixing coefficient approach the distribution of particle size distribution parameters over the water depth can also be calculated. Comparisons between measured and calculated distributions yield quite good agreement.

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1.4 RECOMMENDATIONS 5

The characteristics of derived e(s) distributions under wave action do not agree with these of proposed distributions as found frequently in the literature.

In the literature often a relationship between e(s) (mixing coefficient for sediment) and e(f) (coefficient for fluid) is assumed. Viz.: e(s) = P e (f); p being a factor.

A further complicating fact in analysing measured c(z) test results is that the same wave induced mixing activity affects different particle sizes differently (in some cases to quite a considerable degree). Different P values have to be taken into account for different particles.

Series of tests have been carried out and have been analysed to clarify that P effect. Fairly distinct trends have been found experimentally. Theoretical explanations have so far not been fully convincing.

Even apart from possible p effects, it can be stated that waves are very effective in maintaining particles in suspension. Relatively high mixing coefficients are found up to high levels in the water column.

Under normal small-scale model test conditions, even without breaking waves, the mixing activity is by no means restricted to a thin layer close to the bed. In many theoretical approaches diffusion activities are indeed confined to that small layer.

In a combination of waves and currents it can be assumed that the relatively high wave-generated mixing coefficients will also affect the v(z) distribution over the water depth. Test results and calculations prove these effects. Especially a reduction of the current velocities close to the bed in comparison with a logarithmic distribution is found.

The consequent effects on the resulting sediment transport are considerable.

Further research on this subject still has to clarify many facts before real practical applications can be tackled with the present approach.

1.4 RECOMMENDATIONS

Since it has turned out that with the help of a mixing or diffusion coefficient e(s) the

concentration and the particle size distributions under wave action can properly be described, it is worthwhile to relate the characteristics of the underlying e(s) distributions to the (wave)

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6 1. INTRODUCTION

boundary conditions. This should be done very carefully. The urgent need for direct applications calls probably for ad hoc solutions. However, too many of these approaches already exist at present. Series of tests have to be carried out under a wide variety of conditions (e.g. horizontal bottoms and actual slopes; regular and random waves; breaking and non breaking waves; sheetflow and non-sheetflow conditions). Particle sorting and (J effects have to be taken into account.

Besides carrying out careful experiments, convincing 'theoretical' explanations of the trends that will be found have to be given.

The experimental approach is intentionally mentioned before a theoretical approach. It is sincerely felt by the author that many aspects of the complicated mechanism of maintaining sediments in suspension under wave action still primarily call for 'fact-finding' studies. Too many quite simple facts are still unknown.

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7

CHAPTER 2 RELEVANCY

2.1 GENERAL

In this chapter the appearance of measured sediment concentration distributions over the water depth, due to wave action, will first be outlined. The scatter of the measuring points is discussed. More or less 'classical' diffusion activity distributions over the water depth (e distributions) will be reviewed. It will be shown that none of these distributions is generally able to predict successfully the position of the measuring points.

In the commonly adopted 'theoretical' approach, uniform bottom material is assumed. In practical cases, however, non-uniform material is present. The effect of this sorting of the bed material on the concentration and the particle size distribution over the water depth is explained. The computations in this chapter start with the assumption that the e value for the sediment equals the e value for the water; (P = 1). In Section 2.5 some numerical examples will clarify the effect of sediment size dependent P values.

2.2 SEDIMENT CONCENTRATION MEASUREMENTS

Measurements of sediment concentration due to wave action or a combination of wave action and currents, have been carried out for many years now. The desire to understand the

fascinating mechanism by which the sediment particles are held in suspension has been a major reason. Many aspects of that mechanism were unknown for many years, and the first

measurements had been carried out mainly to gain some preliminary answers to questions such as:

• What does the shape of the distribution over the water depth look like? • What happens with different wave heights?

• What is the effect of the bottom particle size?

Among many others, for example Shinohara et al. (1958); Fairchild (1959); Homma et al. (1965); Das (1971); Bhattacharya (1971); MacDonald (1977); Nielsen (1979); Bosman (1982)

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8 2. RELEVANCY

and Kos'yan (1985) have carried out and analysed concentration measurements under wave action. Fruitful contributions have been made. However, final answers to the above-mentioned clear and fairly simple questions could not be given. The 'empirical' approach of the problem has received, and is still receiving, much attention from researchers. In recent years purely 'theoretical' approaches have also been used to clarify some aspects of the problem [e.g. Nielsen (1984b)].

In Fig. 1 some arbitrary measured sediment concentration distributions over the water depth under wave action are given. Time- and bed-averaged values are plotted in Figs, la and b. Figs. lc and d refer to only time-averaged values. 'Time'-averaged values are sufficient for the purpose of the present study: sediment transports more or less perpendicular to the orbital plane. 'Bed'-averaged values result from Bosnian's (1982) study which deals with the accuracy and reproducibility of this kind of measurements. It turned out that the scatter of the measuring points could be drastically reduced by the bed-averaging procedure. In that procedure the

still water level

i

still water level "=3=-_ - ~*^^—-^__^ i ' h = 0.30m ( b ) H = 0.09m ^ T = 1.7s D50= 97um [ v a n de Graaft a n d Roelvink I1984I] i " — - 1 0,1 ► concent ration (kg/tn3)

still water level

h=1.5m ( d ) H,= 0.7m T = 9.3s D5 0=200um [ N i e l s e n l l 9 8 £ a l ] 1 concentration ( k g / m3)

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2.3 CONCENTRATION DISTRIBUTION 9

concentration at a certain level above the bed is obtained by moving the measuring device to and fro over a number of ripples by a slowly moving carriage.

In this study, results of measurements on samples obtained by suction from the masses of water will mainly be discussed. The direction of suction was in most cases perpendicular to the orbital plane of the waves. Bosman, Van der Velden & Hulsbergen (1987) describe the effectiveness of this method of sampling. Because in the present study the distribution of the concentration receives more attention than the actual precise values, the relative errors introduced by the suction method are of minor importance.

In concentration measurements mostly great differences in concentrations are found between positions close to the bed and positions at higher levels [factor 0' (1000)].

Therefore the use of a logarithmic horizontal scale in diagrams like Fig. 1 seems appropriate. The vertical z-axis is linear; z = 0 is the bed level, which is usually defined as the average level after flattening of the possible bed ripples.

In Fig. 1 smooth curves have been fitted by hand through the measuring points. Sometimes a straight line represents the position of the points quite well; in other cases obviously curved lines are needed for an appropriate fit. In spite of the time- and bed-averaged measuring procedure, some scatter of the measuring points around the fitted lines can still be found. It seems impossible to avoid that for the time being. Errors up to approximately 10% are frequently found with the present measuring technique.

Up to now the 'concentration' has been discussed in quite general terms. In this study the concentration will be further expressed in kg/m3; this links up with the direct results of analyses of concentration measurements. Using these concentrations in sediment transport formulae (S expressed in m3/sm) calls for additional conversion factors.

2.3 CONCENTRATION DISTRIBUTION

The mechanism by which the particles are held in suspension, has not yet been completely elucidated. Turbulent diffusion processes as well as convective processes are felt to play important parts. In Section 3.3 these aspects are discussed in more detail. For the purpose of the discussion in this chapter a quite simple mixing or diffusion process will be assumed to take place. Assuming a steady-state condition without gradients in the x- and y-direction, the vertical distribution of the sediment concentration can be described with the help of:

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10 2. RELEVANCY where: w c(z) e(z) z dc(z) wc(z) + e ( z ) - ^ = 0 (2.1)

fall velocity of the particles

average sediment concentration at height z above the bed diffusion coefficient at height z

vertical ordinate; z = 0 being the bottom

Two parameters in Eq. (2.1) require some special attention, viz.: w and e(z).

If the bottom material is perfectly sorted (all particles have exactly the same size) and if the particles have the same shape, w can be assumed to be constant for all particles. In practice, however, well-sorted or well-graded bottom samples will be found. (Well-sorted: all particles have sizes which are close to the typical size; well-graded: the particles are evenly distributed over a wide range of sizes). This means in general that each particle of a sample has a different fall velocity.

Normally w in Eq. (2.1) is related to the D50 value of the bottom samples, D50 being that diameter where 50% of the weight of the sample consists of particles that are finer. In Section 2.4 it will be outlined what happens if non-constant w values are taken into consideration. To derive an expression for the c(z) distribution over the water depth, Eq. (2.1) has to be integrated, yielding:

c(z) = c(a) exp < -w J [ 1 / e(z)] d z > (2.2) where:

c(a) : concentration at reference height z = a

The reference height z = a can be arbitrarily chosen. The fall velocity w in Eq. (2.2) is provisionally held constant.

The solution of Eq. (2.2) depends on the distribution of e(z) over the water depth. For wave (and mere current) conditions three more or less 'classical' distributions are used frequently, viz.:

a) Rouse-type [Rouse (1937)] b) Coleman-type [Coleman (1970)] c) Bhattacharya-type [Bhattacharya (1971)]

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Rouse

The e(z) distribution over the water depth h has a parabolic shape:

* » - * W ^ (2-3)

where:

emax '• maximum value of e(z) at height z = 0.5 h above the bed . (2.3) inserted into Eq. (2.2), yiek Is:

wh c(z) = c(a) h - z a

z h - a 4 e

max (2.4)

At the bed an infinite concentration is found according to this c(z) description; near the water surface the concentration approaches 0.

Coleman

In the Coleman-type e distribution, e(z) is constant throughout the water depth {e(z) = e}. Eq. (2.2) results in:

c(z) = c ( o ) e x p j — j (2.5) where:

c(o) : bottom concentration (at level z = 0)

In the Coleman distribution, both the bed concentration and the water surface concentration are finite values.

Bhattacharya

In the Bhattacharya-type distribution of e(z), a triangular distribution over the water depth is assumed, with e(z) = 0 for z = 0. This results in an e(z) distribution according to:

e(Z> = -hLemax (2.6)

where:

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12 2. RELEVANCY

A distribution according to Eq. (2.6), inserted into Eq. (2.2), yields: w h

c(z) = c(a) _(2.7)

According to Eq. (2.7) the bed concentration becomes infinite; at a value of z equal to the water depth a finite concentration is found.

Each of the 'classical' c(z) distributions Eqs. (2.4), (2.5) and (2.7) is occasionally adopted for use under wave conditions. Bijker (1971), for example, uses in his sediment transport formula a Rouse-type c(z) distribution. In the latter case even a mathematical description for the characteristic parameter of the distribution [emax in Eq. (2.4)] has been implicitly given by Bijker. emax can be expressed as a function of the boundary conditions.

-gO.30 T3 X I 4J SO.20 O x: JÜ 0.10 t

1

s t i l l water Level '-=-w = 0 . 0 K m / s Emax =0.000962 m2/ s

- v ©

• \ • \ . ' ^ ^ _ 0.1 1 10 • c o n c e n t r a t i o n ( k g / m3) 0.20 0.10 s t i l l w a t e r

®

level w = 0.0095 m / s E „ „ = 0.000260 m2/s 0.01 0.1 1 — * ■ c o n c e n t r a t i o n { k g / m3) 10 8 E

1*

4) > O ï ( x: V

V

1

still water level w = 0.0067 m / s em o x =0.046229 m .

\' ®

\ \ \ 1 • " J

still water level

0.1 1

concentration ( k g / m3) concentration ( k g / m3)

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In this stage of the study, however, the possible relationship between e values and the boundary conditions, is somewhat premature. First, the general shapes of the 'classical' c(z) distributions are compared with the measured arbitrary distributions as given in Fig. 1. Figs. 2 and 3 show the results. To achieve the relatively 'best' results in each of the cases, two fit parameters (a characteristic e value and a characteristic concentration) have been derived with the help of a least square approximation method. The characteristic parameters that have been found are indicated in Figs. 2 and 3. As only four cases are compared in these figures, it is of course not a convincing proof, but it can be stated that in fact none of the 'classical' distributions holds for an arbitrary series of concentration measurements. (In fact this statement is based on a far greater number of comparisons).

Fig. 4 shows the resulting e(z) distributions, which holds for the various cases of Figs. 2 and 3. Quite different orders of magnitude for the 'classical' distributions are found in each of the four cases.

still water level = 0.000729 mVs max = 0.002777 mVs 0.20 — - — - ^ \ § '

still water level

"^=" C: E =0.000136 m2/ s B: £max= 0.000884 m2/s

©

i i " — 1 1 10 • concentration ( k g / m3) 0.01 0.1 concentration ( k g / m3)

still water level

= 0.024670 m2/s m0,,= 0.157949 m2/s 1.5 0.1 1 ■ concentration ( k g / m3) S>0.5 \B c""-\ 1

still water level ■'s»-C: E = 0.004718 m2/s B: Em,,, = 0.031056 mVs

®

• i ^**=C——• i 0.1 1 concentration ( k g / m3 Fig. 3 Coleman and Bhattacharya approximations.

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14 2. RELEVANCY 0.30 0.20 0.10 NP \ c 1 -/ B

®

₁

0.30 0.20 0.001 0.002 0.003 ■ E ( Z ) ( m2/ s ) 0,001 0.002 • E ( z ) ( m2/ s ) 10 8 6 U 2 \ -/ C \ R — /B

©

1.5 1.0 jE

V

0.05 0.1 - * e ( z ) ( m V s ) 0.01 0.02 0.03 • e ( z ) ( m V s )

Fig. 4 Diffusion coefficient distributions.

In Section 2.6 it will be indicated that in the further analyses none of the 'classical' distributions with respect to the shape of e(z) will be used. The shape will be free in the first stages of that analysis.

As mentioned before, a constant fall velocity w value has provisionally been taken into consideration in the present calculations. In the next section the effect of varying w values will be illustrated with the help of some numerical examples.

2.4 EFFECT OF VARYING FALL VELOCITIES

Natural beach sediments are not perfectly sorted (perfectly sorted: Dc^ = D50 = D10), but can mostly be classified as well-sorted (D90/D10 < approximately 2.5). If the ratio Dgo/Djo is greater than approximately 2.5, then the sediments are classified as well-graded.

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2.4 EFFECT OF VARYING FALL VELOCITIES 15

The horizontal and vertical axes correspond respectively to 'logarithmic' and 'normal' distributions. As is frequendy found in similar cases, the sieve results in this example fit fairly well with a straight line. In the calculation examples to be discussed, a perfect log-normal distribution is assumed. Moreover, various DCX/DJQ ratios are adopted.

Beach sediment samples consist of various particle sizes, each with its own fall velocity w. Various relationships between particle size and fall velocity can be found in the literature. Here the relationships as proposed by the Delft Hydraulics (1983) will be used.

The general form of the formula is as follows:

where: A, B, C D w [A*log(D) + B*log(D) + C] 1 / w = 10

parameters depending on the particular case particle size (in m)

fall velocity (in m/s)

(2.8)

{Eq. (2.8) holds at least for 50 nm < D < 400 nm}.

O -qn a Ë™ c * 5 0 c >-> ™xn 5 5 8.10 O c i) Q. 2 , s *y Ss approximation—. /, s£S / 'x / /J <yf

yA

/ / \ — measuremen // // 125 150 175 200 225 250 275 300 325 ► diameter { ^ m ) (Logarithmic distribution )

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16 2. RELEVANCY

In Table 2.1 the parameter values A, B and C are given for three different cases. The freshwater situations may be encountered under laboratory conditions; the saltwater situation applies for instance to seawater during winter on the northern hemisphere.

water fresh fresh salt temperature 10 deg. C 18 deg. C 5 deg. C A 0.476 0.495 0.476 B 2.180 2.410 2.180 C 3.190 3.740 3.226

Table 2.1 Parameter values as a function of water condition.

A simple way to illustrate the effect of sorted bottom material on the concentration distribution starts by dividing the total weight of a bottom material sample into various classes. Each class contains particles between specified boundaries. Selecting for instance n = 10 classes, the class limits are D10, D2o, ••• D ^ . Class 1 contains all particles with sizes < D10; class 2 all particles between Di0 and D2o; etc. All particles in a class can be characterized by a characteristic value D (for instance the average diameter by weight in that class). The example with n = 10 then gives:

Dch(i) = D[ ( i o .i )- 5 ] (2-9)

where:

Dch(i) : characteristic diameter of the i-th class i : class number

Eq. (2.9) yields D5, D15, D2 5,... and D95 as characteristic diameters. With Eq. (2.8) a characteristic fall velocity w(i) can be computed which belongs to Dch(i).

In each of the three 'classical' distributions mentioned in Section 2.3, the fall velocity w plays an important role. To illustrate the sorting effect of the bottom material, it can be assumed that each of the 10 fractions 'builds up' its own vertical concentration distribution. The resulting concentration at an arbitrary level z above the bottom then equals the sum of the contributions of the 10 fractions. Since the Coleman-type concentration distribution is the simplest of the three 'classical' distributions, this distribution is used in the following calculation examples. Furthermore, a case with a dimensionless height above the bottom (z„ = z/h) is used in the examples. Fig. 6 shows the result of several calculations with different rates of sorting (expressed as the ratio D90/D10), but all with the same D50 value (D50 = 100 |im). The bottom concentration c(o) of 10 kg/m3 is arbitrarily chosen. The e value used in the set of calculations is given in Fig. 6. It can be seen from Fig. 6 that the rate of sorting affects the shape of the

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2.4 EFFECT OF VARYING FALL VELOCITIES 17 0.8 0.6 O.t 0.2

o©

-1

©

© ©

Fig. 6 Effect of Dgo/D10 ratio on concentration distribution.

distribution curve quite considerably. The straight line, always typical of a Coleman-type distribution, vanishes completely. The curves for the different DOQ/DJO ratios are all situated above the uniform size distribution line.

With the same system of 10 different partial concentration distributions it is also possible to calculate the resulting particle size distribution characteristics over the water depth. In Fig. 7 the

(p(D (!) © @Q

25) : 0 , 0 / 0 , 0 = 1 2 5

e ( s ) = 0.001000 m V s

20 40 6 0 100 120

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18 2. RELEVANCY

D50 distribution is given for the several rates of sorting. At higher levels in the water column a rapid decrease of the D50 values is found for high rates of sorting. Similar plots can be given for other characteristic diameters (D90 or D10).

Calculations with more subdivisions (e.g. n = 20 instead of n = 10) yield results that are hardly different; for the purpose of the present series of calculations the accuracy is considered to be sufficient.

With respect to the sediment transport phenomenon it is advantageous to quantify the increase in the sediment load in the vertical of the sorted cases in comparison with the uniform diameter case. Fig. 8 shows the result. For reasons of comparison some other e values are also taken into consideration. For normal beach sediments (Dgo/D^ ratio approximately 2-3), the increase of the load looks rather modest compared with the uniform sediment case (approximately

10-20%). This means therefore that to achieve an increase in the accuracy of sediment transport formulae, it does not at first sight seem very important to take sorting effects into account However, although the ultimate aim of the present study is to achieve better sediment transport formulae, it is necessary to understand the underlying concentration mechanism better. The shape of a distribution is affected quite considerably by the sorting rate of the bottom material. Since that shape can help us to obtain a better knowledge of the mechanism, it is advantageous to continue studying the sorting effects.

For D J J / D , , , = 1 ; Load = 100% x

* D9o/D,o ratio (-)

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2.4 EFFECT OF VARYING FALL VELOCITIES 19

In the previous calculation examples, the bottom concentration was split up into 10 fractions and each fraction formed its own vertical distribution, based on its (equal) partial bottom

concentration. Since the finer fractions go 'more easily' into suspension than the coarser ones, the ratio between the total mass of each fraction in the water column becomes quite different from the initial one. (I.e. the ratio between the bottom concentrations and also the ratio between the several fractions in the original bottom material; in our case, by definition, a 1 : 1 : 1 . . . : 1 ratio). The finer fractions are in fact 'over-represented' in the water column. It seems

questionable whether this is physically true.

On the other hand it can be assumed that the ratios between the total masses of each fraction in suspension are the same as those in the original bottom material. This would mean that, in achieving the same mass in suspension for a coarse fraction as for a finer one, quite different bottom concentrations should be taken into account. In Fig. 9 the resulting concentration distributions for several rates of sorting are given with the starting point as mentioned. Compared with Fig. 6 the differences with regard to the uniform size distribution case look somewhat smaller. Furthermore for zt heights up to approximately 0.35 the sediment

concentrations are slightly lower than in the uniform size case. According to this approach each case holds the same total amount of sand in the water column. Higher concentrations (higher compared with the uniform size case) close to the water level therefore have to be compensated by lower concentrations close to the bed.

Fig. 10 shows the resulting D50 distributions over the water depth. In this case higher D50 values are found close to the bed than the D50 value of the original bed material.

' 1.01 G>@ © © ® © ©: D* °/ D 1° -4

0.8- ^ ^ ^ N v \ W E(S) =0.001000 m2/s

0.001 0.01 0.1 1 10 * concentration ( k g / m3)

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20 2. RELEVANCY iT\f7\ 17\ O i ^^rrs © : Djo/Dm =1.5 1.0 0.8 0.6 0.4 0.2 20 40 60 80 100 120 KO 160 ► D5 0 ( p.m)

Fig. 10 D50 distribution assuming a constant mass approach.

At this stage of the study it is not clear which approach is physically true. With the second approach (constant masses in suspension for each fraction) a total distinction could conceivably be made between the particles which join the suspension process and the particles which remain undisturbed in the bed. Compare a case with a concrete bottom with some loose particles on it. For a uniform flow condition it might be plausible that all loose particles indeed join the suspension process. However, for an oscillatory flow condition even that simple approach will not be true for all phases during the water motion. For near maximum velocities all particles are probably in suspension; for somewhat more moderate velocities settlement of (presumably the heavier?) particles will occur. Even for the simplest model (concrete bottom and some loose particles) serious complications arise. Without a sharp distinction between suspension-joining particles and undisturbed bottom particles the actual processes are still more complex. Assuming the first approach to be true (equal bottom concentration for each fraction), the differences in mass of the several fractions in suspension over the water depth can conceivably be explained by extra withdrawal of fine particles from the bottom layers. The coarser particles remain at the bottom and the so-called armouring of the bottom layer occurs. This does not, however, alter the fact that the basic assumption in this approach (equal bottom concentrations) is in fact probably questionable. Equal bottom concentrations can probably be made plausible by adopting the 'model' in which the orbital velocities near the bottom 'scrape' an entire bottom layer and bring that total layer into suspension. A more selective pick-up process is, however, also conceivable. In the latter case equal bottom concentrations for each fraction are not automatically true.

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2.5 EFFECT OF p VALUES 21

From the foregoing brief discussion it is evident that there are no convincing arguments as to whether one or the other approach is true or indeed neither of them. However, as far as the author knows, there is no empirical evidence up to now for a higher D50 value very close to the bed as compared with the D50 value of the bottom material. So the second approach is not likely to occur.

In the further examples in this chapter the first approach (equal bottom concentrations for the several fractions) is assumed to be true. In Sections 4.3 and 4.7 comparisons with empirical results are discussed.

In the examples given in this section only one particle size diameter D50 has been used. Similar results can, however, be found with different D50 values. Also only a few different e values have been used; other values will yield similar results.

2.5 EFFECT OF p VALUES

In the examples given in the preceding section a certain constant (constant; due to the assumed Coleman-type approach) diffusion activity e(s) has been assumed to be responsible for the distribution of the particles over the water depth. For each fraction, with sometimes quite different characteristic diameters, the same e value was assumed to hold. That assumption is in fact not generally true. Van Rijn (1982) argues that, at least for steady-state conditions, a higher £(s) value has to be taken into account if higher fall velocities occur. Some preliminary tests analysed and discussed by Van de Graaff & Roelvink (1984) indicated that also under a wave field higher e(s) values should be taken into account for higher fall velocities. A more

comprehensive discussion of this subject will follow in Section 4.5. Here it is sufficient to state that in general the following relation holds:

e(s) = pe(f) (2.10) where:

e(s) : diffusion coefficient for sediment e(f) : diffusion coefficient for fluid P : factor

The p factor is frequently assumed to be greater than 1. As stated, p seems to be particle size dependent. Probably P is also dependent on the level of diffusion activity.

The exact formulation of a probable fall velocity versus P relationship is not important at this moment Just to be able to calculate some examples, the following relationship is assumed to be

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22 2. RELEVANCY true: where: w 25 P = 1 + 25 w

: fall velocity of particles (in m/s)

: experimental, non-dimensionless factor (s/m)

(2.11)

The factor is chosen as 25 s/m and is close to the factor of 23 s/m mentioned by Van de Graaff & Roelvink as being valid for only one particular set of boundary conditions in a wave flume. In Section 4.3.1 these tests are re-analysed; in that section the structure of Eq. (2.11) is also discussed.

In Fig. 11 the resulting concentration distributions over the water depth are given for several D90/D10 ratios. In the calculations an e(f) value has been taken into account which leads, together with Eqs. (2.10) and (2.11), to a value of e(s) of 0.001 m2/s for a 100 |im particle (w = 0.0095 m/s). This value was also used in the examples of the preceding section. Compared with Fig. 6, the differences with respect to the uniform material case are somewhat smaller. However, the differences are still so large that a considerable effect results.

In Fig. 12 the D50 distribution has been given. In this case, too, far smaller D50 values than the D50 of the bottom material are found higher up in the water column if large D90/D10 ratios are considered. 08 0.6 0.2

©@ ©

^ -'

©

1

© ©

1 © : D,o/D,o =3 e(f) = 0.000808 m2/s I ^ S J 0.001 0.01 0.1 concentration { k g / m3

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2.6 DISCUSSION 23

Q © © @ ©Q

Fig. 12 Effect of P parameter on D50 distribution.

2.6 DISCUSSION

In this chapter several examples have been given of the effect of the non-uniformity of bottom sediments under wave action. It turned out that for normal beach sediments (ratio Dgg/Djo approximately 2 to 3) there is found to be a considerable effect upon the shape of the resulting concentration distribution and upon the distribution of the D50 value over the water depth. The examples have been calculated on the assumption of a constant e value over the water depth (Coleman-type distribution). On the assumption of an essentially different type of distribution (e.g. Rouse or Bhattacharya) different distributions will of course result, but the same phenomena can be observed as in the calculation examples given. (Relatively high

concentrations at large distances from the bed and a steep decrease of the D50 values there). From analyses of various real concentration measurements under wave action it was found that none of the three 'classical' distributions can be applied to describe the distributions properly. (If the sorting effects and the effect of the P factor are taken into account, the results do not become essentially better). See Figs. 1,2 and 3. This means that obviously an alternative E distribution over the water depth holds under these conditions. Since it is hard to derive in advance a 'theoretical' description of an e distribution, a different approach has in fact been chosen to obtain some insight into the phenomenon. Briefly, the following steps are to be distinguished:

a) start with an arbitrary concentration measurement

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24 _{2. RELEVANCY}

c) derive the obvious e(s) distribution

d) derive the obvious e(f) distribution; {e(f) = e(s) /13}

e) try to understand the e distributions that have been found, in terms of boundary conditions f) relate the characteristics of the e distribution and the boundary conditions

In steps c) and d) both effects discussed in this chapter have to be taken into account. However, step d) is hard to carry out as long as for instance a possible P versus w (or D instead of w) relationship [cf. Eq. (2.11)] is not known. Part of the present study is devoted to arriving at such a relationship.

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25

CHAPTER 3 THEORETICAL CONSIDERATIONS

3.1 GENERAL

The concentration distribution under current and wave action is in fact the main research item of the present study. However, from an experimental point of view (wave flumes instead of wave basins) it is easier to obtain distributions due to wave action only. In the present analysis therefore that case will receive most attention. In Section 3.2 some existing ideas with respect to a possible e field under wave action will be discussed. It will emerge that this is a rather under developed topic. Some quite different approaches can be found in the literature; many of them are of an entirely conflicting nature.

Section 3.3 deals with a discussion of the probable mechanisms which hold particles in suspension.

3.2 SOME EXISTING IDEAS

Bijker (1971) presents implicitly an entire description of a possible e(s) distribution for the combination of currents and waves. He assumes that for a uniform current velocity with logarithmic distribution over the water depth, the e(s) field equals the e(f) field. In fact a Rouse-type e distribution is assumed [cf. Eqs. (2.3) and (2.4)]. Eq. (2.4) was as follows:

wh c(z)=c(a) h - z * -4e max (3.1) = (2.4) where:

c(z) : concentration at level z above the bed

c(a) : concentration at reference level at z = a above the bed h : water depth

z : vertical ordinate (z = 0: bed level) a : reference level height

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26 3. THEORETICAL CONSIDERATIONS

w : fall velocity

emax • maximum value of diffusion coefficient (due to parabolic shape: occurring at z = 0.5 h)

Normally the exponent in Eq. (3.1) is expressed as: w z * = (3.2) where: w K v* c exponent fall velocity

Von Karman 's constant (app. 0.4) shear stress velocity; index c: current

For instance Einstein (1950) uses this parameter z» when he elaborates the Rouse distribution in order to derive a total sediment transport formula. From Eqs. (3.1) and (3.2) it can be deduced that:

e = 0.25 K h v,c (3.3)

max *c w - ' /

The parameter vtc can be expressed as:

v« = V V P

(3.4)

where:

xc : bottom shear stress due to current p : mass density of the water Eq.(3.3) can then be rewritten as:

emax = 0-2 5 K hV/V P (3-5)

It follows from Eq. (3.5) that emax, so according to Eq. (2.3), the whole e(s) field, further depends, apart from the water depth h, on the bottom shear stress only.

In his combined current - wave interaction study Bijker (1967) derives an increased bottom shear stress expression, yielding:

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3.2 SOME EXISTING IDEAS ₂₇

X = X _{wc c}1 + 0 . 5 [ ^ L _(3.6)

where:

xvlc : bottom shear stress due to waves and currents

% : parameter;^ = c y fw/ 2 g C : Chezy coefficient

fw : Jonsson's (1966) wave friction parameter

g : acceleration of gravity

u0 : maximum orbital velocity near the bed

v : mean current velocity over the water depth Eq. (3.6) can be rewritten as:

Xwc = Tc + 0-5 Xw (3-7)

where:

xw : maximum bottom shear stress due to mere wave action; xw = 0.5 p fw ü0 lü0l

Bijker's (1971) approach for the calculation of the sediment concentration distribution in the case of a combination of waves and currents was quite straightforward. In fact he changed the v.c parameter in Eq. (3.2) into vt w c.

:W

' P (3.8)

where:

v„wc : shear stress velocity belonging to a wave and current condition

That procedure yields far smaller z, values, which means that higher sediment concentrations high in the vertical will be reached. With respect to the present study, Bijker's approach leads implicitly to relatively high e(s) values. [Change xc in Eq. (3.5) by xwc]. According to this

approach the e(s) field under the combined current and wave action depends also directly on the (increased) bottom shear stress.

Although Bijker did not mention this explicitly, the mere wave condition can be seen as a limit case if v -> 0. With the help of Eqs. (3.7), (3.4), (3.3) and (2.3), a finite e(s) distribution can be found in that case for a wave alone condition.

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It is interesting to observe the orders of magnitude of the resulting e(s) distributions that can be found in a real example case.

Example water depth wave height wave period current velocity

bottom roughness height

h = 3.0 m H = 1 . 1 8 m T = 8.0 s v = 1.0 m/s r = 0.06 m

From the given conditions the maximum orbital velocity near the bed can be calculated, yielding ü0 = 1.0 m/s. Jonsson's friction coefficient becomes: fw = 0.045. The Chezy factor yields: C = 50 m1/2/s. The equations for TC and xw are:

Tc = Pg — (3.9)

and:

xw = 0.5pfwÜ0IÜ0l (3.10)

Eqs. (3.9) and (3.10) inserted into Eq. (3.7) and using Eqs. (3.8) and (3.3) and next using Eq. (2.3), yield the e(s) distributions for several cases (see Fig. 13).

Bijker

waves+current

0.01 0.02

• E ( Z ) ( m V s )

0.03 0.04

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3.2 SOME EXISTING IDEAS 29

a) mere current b) waves alone

c) waves and current combination

According to Bijker's approach, the effect of the wave action on the e(s) distribution turns out to be quite considerable; the differences between cases b) and c) are modest. In Bijker's approach the bottom shear stresses due to waves affect the e(s) values over the entire vertical. Obviously, diffusion activities, which most probably are generated near the bed, are ultimately diffused over the entire water depth. Whether this assumption is physically true is questionable. Bakker & Van Doom (1978) studied the interaction between waves and currents. In the case of waves alone they assume the internal shear stresses to be confined to a relatively small layer close to the bed. In their studies the shear stress at a height z above the bed is related to the diffusion coefficient e(z) according to:

where: x(z) P e(z) u z x(z) = p e(z) du dz (3.11)

shear stress at level z above the bed mass density of the water

diffusion coefficient at level z horizontal velocity component [f(t,z)] vertical ordinate

According to this approach, diffusion coefficients are also confined to that relatively small layer. An approximation of the boundary layer thickness according to Bakker & Van Doorn is as follows: where: ZBL K v» T ZBL = KV,T

boundary layer thickness Von Karman's constant maximum shear stress velocity wave period

(3.12)

With the boundary conditions of the present example ZBL becomes 0.48 m. Generally Bakker & Van Doom take 1.5 times ZBL as the height of layer where shear stresses and thus diffusion

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coefficients may occur. In the present example this means that wave-induced diffusion coefficients occur only over approximately 25% of the water depth. Compared with Bijker's approach this is only a relatively small section of the total water column.

According to Jonsson & Carlsen (1976), the layer where shear stresses occur should be taken as still smaller than according to Bakker & Van Doom. In the present example it should be taken as less than 0.10 m.

In his thesis Nielsen (1979) describes, among a number of other subjects, the suspension mechanism under wave action in detail. He also gives analytical functions of the effect of some special sorted materials on the c(z) distribution over the water depth. For a bottom material where the dimensionless fall velocities w' (w' = w / w0) have a T shaped density distribution, with a mean value 1 and variance V [V = Var(w) / w02], a surprisingly simple analytical function that describes the actual e(z) distribution was found by Nielsen. In the present notation that function is as follows:

^act^Us^^f

(3 13)

where:

e(z)act : actual e value at height z

e(z)meas : e value at height z if uniform material with fall velocity w0 is assumed c(z) : measured concentration

c(o) : concentration at reference level z(o); it is assumed that in z =z(o) the suspended material has the same composition as the original bottom material

V : variance of T density function of w' w0 : mean fall velocity

Since Eq. (3.13) holds only for a quite special case, this formula is not generally applicable. Nielsen also studied the obvious e(s) distribution under wave action in small-scale models. For non-breaking regular waves he found that a constant e(s) value over the water depth could be assumed. Taking his experimental relationship between e(s) and the boundary conditions as true, in the present example of this section an e(s) value of approximately 0.001 m2/s can be computed (this value is also given in Fig. 13), which is far smaller than the other e values given in Fig. 13.

Kos'yan (1985) also proposes an e(s) distribution under wave action for non-breaking waves. His resulting distribution consists of three components:

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3.3 SUSPENSION MECHANISM UNDER WAVES 31

1. Diffusion coefficient due to orbital motion 2. Turbulent diffusion in the waves

3. Turbulent diffusion in the bottom area

The first component is assumed to be zero at the bed, but reaches extremely high values near the still water level. This contribution depends, according to the assumptions, on the vertical water displacements in the orbital motion. The second component turns out to be rather small (negligible). The third component is confined to the region near the bed and there reaches rather high values. At a distance of approximately 10% of the water depth above the bed, the

contribution of that component becomes zero again. The relationships as proposed by Kos'yan for his three components are somewhat speculative. However, he reports some cases [among others, wave tunnel measurements of Nakato (1974) and prototype measurements of

Keremitchiev (1982)] where the results fit very well. If Kos'yan's relationships are applied to the example of the present section, above z = 0.25 m above the bed the e(s) values are far beyond the scale of Fig. 13 (see Fig. 13).

Concluding remarks

From the (non-exhaustive) review of some relevant literature it becomes clear that by no means standard ideas exist concerning the possible distribution of e(s) over the water depth. Almost every researcher proposes his 'own' approach. Applied to the example case in this section a variety of distributions is found (see Fig. 13). This is unacceptable for design purposes. Some unifying ideas are urgently needed. The present study tries to offer these.

3.3 SUSPENSION MECHANISM UNDER WAVES

The brief literature review in Section 3.2 has revealed that different researchers propose quite different diffusion coefficients (cf. Fig. 13). Although a diffusion type description was in fact only a simple help, it means in fact that therefore quite different theories are used. In the present section some possible mechanisms by which the particles are held in suspension are discussed. It will turn out that again no clear evidence can be found for the existence of one or more convincing mechanisms that can be widely applied.

In the present discussion a rippled bed is the starting point. A rippled bed is frequently found in small-scale model tests in a wave flume. One or another irregular bed shape may also be encountered under moderate prototype conditions. More severe prototype conditions will cause sheet flow. [See e.g. Dingier & Inman (1976); Horikawa, Watanabe & Katori (1982); Bakker & Van Kesteren (1986)].

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Under sheet flow conditions the bed is entirely flattened due to the strong orbital motions close to the bed. High concentrations will occur in relatively small layers close to the bed, However, sheet flow will not be considered in detail in the present discussion.

Under rippled bed conditions in principle three zones can be considered. Starting from the bed, there are successively:

• zone between the ripples; from the bed up to the top level of the ripples

• vortex ejection and dispersion zone; from the top level of the ripples up to a few ripple heights above the bed

• upper zone; from a few ripple heights above the bed up to the still water surface The processes with respect to the behaviour of the sand particles in each of the zones will be discussed.

Zone between the ripples

Fig. 14 shows schematically the current pattern for the (near maximum) orbital velocity condition close to die bed. The accelerated (accelerated: due to slight contraction) main current flows over ripple crest 1 and generates a vortex v behind the ripple crest. Sand particles are carried by the main current ('freshly' picked-up from the up-stream side u.s. from the bed ripple and 'returning' particles to the bottom, brought into suspension during a preceding wave period). These particles are partly caught by the vortex and partly carried over the top of the vortex in the direction of crest 2. (In the present description the term 'vortex' is used. This term is usually used also in the literature. In fact the water motion between the ripples is physically better characterized by the term 'eddy').

In a very simple description of the behaviour of a vortex (with a so-called standard vortex) the velocities increase linearly from the centre (v = 0) to the limits of the vortex. At the vortex limit the velocities are almost equal to the main current velocity.

An order of magnitude estimate of some standard vortex parameters under small-scale test conditions yields:

returning particles

— H •

* main current

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3.3 SUSPENSION MECHANISM UNDER WAVES 33

r üo

(0

radius of vortex = 0.01 m

main current maximum orbital velocity = 0.20 m/s angular frequency ~ 20 rad/s

If acceleration terms are neglected (remaining water motion within the vortex and fall velocity of the particles), it can be shown [see Nielsen, Green & Coffey (1982)] that the particles can be 'caught' by the vortex. The particles describe circles around a point in the vortex where the upward directed vortex velocity equals the fall velocity of the particles.

If accelerating terms are incorporated, the particles move in widening spiral orbits. An order of magnitude of the centrifugal velocity vr of a particle in a standard vortex can be found [see Nielsen (1979)] from: (3.14) where: vr CO w g R V 2 - 1 = 10-R w g : centrifugal velocity : angular frequency : fall velocity : acceleration of gravity : radius of the particle orbit

[Eq. (3.14) indicates that the centrifugal velocity vr depends in the same manner on the centrifugal acceleration (co2 R) as the fall velocity w depends on the acceleration of gravity g. According to Nielsen's derivation, Eq. (3.14) holds in fact for a Stokes case only].

With R = 0.005 m and co = 20 rad/s, the centrifugal velocity vr yields 1/5 of the fall velocity w. Different particle sizes caught in the same vortex show different behaviour (see Fig. 15 where two different particles pass the same point A). For a 'coarse' particle the radius R is smaller than for a 'fine' particle. For a 'coarse' particle the centrifugal velocity is, however, mostly higher than for a 'fine' particle. In a relative sense 'coarse' particles are hurled away more easily than 'fine' ones.

Up till now a standard vortex has been considered. In reality, however, a vortex moves more or less as a whole and accelerates and decelerates continuously due to the varying orbital velocities. For this reason the behaviour of different particles caught in a vortex is hardly amenable to mathematical treatment.

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Fig. IS Particle motion in a standard vortex. Vortex ejection and dispersion zone

Shortly after the reversal of the main current, the vortex is ejected towards the zone above the top of the ripples. For some time the vortex (though decelerating) maintains its shape and is carried away from its initial generation position between the ripples. In consequence, sand particles in the vortex are carried back in the direction from where they originally came. After some time the vortex loses its identity; the particles 'rain' down to the bottom again.

During the ejection phase of the vortex a convective type of sediment transport occurs

undoubtedly towards higher water levels. And also the particle motion within the vortex is then a convective rather than a diffusion type of motion. So convective transports are generally very important in the distribution of sediments. However, a mathematical description of the vortex behaviour, including acceleration, deceleration and fading-out processes, is hard to formulate reliably for the time being. Therefore an adequate quantification of the actual convective processes cannot be given at the moment. This is the main reason why in the present study a 'diffusion' or 'mixing' type concentration distribution approach is still assumed. In the future the proper approach (including correct convective terms) can probably be applied.

In the downward direction the layer under consideration is limited by the top level of the bottom ripples; the upper limit is a few ripple heights above the bed. In the various phases of the passage of the waves quite different types of water motion occur in that small layer. During near-maximum orbital motions almost horizontal velocities occur, later vertical (ejection) velocities also occur.

Sediment concentration due to wave action

Sediment concentration

due to wave action

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Jan van

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SEDIMENT CONCENTRATION

DUE TO

WAVE ACTION

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus,

prof.drs. P.A. Schenck

in het openbaar te verdedigen

ten overstaan van een commissie,

aangewezen door het College van Dekanen

op donderdag 22 september 1988 te 16.00 uur door

JAN VAN DE GRAAFF

geboren te Hengelo

civiel ingenieur

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Dit proefschrift is goedgekeurd door de promotor ,'

prof.dr.ir. E.W. Bijker j

ACKNOWLEDGEMENTS

CONTENTS

SAMENVATTING

SEDIMENT CONCENTRATIE ONDER GOLVEN

CHAPTER 1

INTRODUCTION

CHAPTER 2

RELEVANCY

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CHAPTER 3

THEORETICAL CONSIDERATIONS

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3.3 SUSPENSION MECHANISM UNDER WAVES

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