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A model study of longshore transport rate Ametepe, Joseph Kwaku 1991

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A MODEL STUDY OF LONGSHORE TRANSPORT RATE By J O S E P H K W A K U A M E T E P E B.Sc.(Civil) Eng., University of Science and Technology, Kumasi - Ghana A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE' REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1991 © J O S E P H K W A K U A M E T E P E , 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of O I V I J- E ^ & f ^ -The University of British Columbia Vancouver, Canada Date Jut-y 3D, ;??/ DE-6 (2/88) Abstract Theory for across-shore transport as a function of beach slope and sediment size is extended to longshore transport. To test the theory, experimental measurements of longshore transport are required for a range of beach slopes and sediment sizes. Measurements were made of the sand transport caused by waves of different char-acteristics approaching the toe of an inclined model test beach of variable slope. The initial beach slope was 1 on 9. Three different tests sands of median diameters 0.50mm, 0.85mm and 2.0mm were used as beach material to investigate the probable influence of grain size on the longshore transport rate. Long crested waves generated in a constant depth of water travelled over the beach, shoaled and were refracted before breaking near the shoreline. The breaking action caused the sand to be trans-ported along the shore in the direction of the longshore component of the wave energy flux. The laboratory measurements of the longshore transport are described, and it is shown that, for a given wave energy, transport increases with beach slope. Also, the distribution of the longshore sediment transport across the beach face is shown to be a function of beach slope and sediment size as higher transport rates were recorded for the coarser sands in zones of maximum wave energy dissipation. Based on the streampower approach and the radiation stress concepts, a theoretical model is de-veloped for the estimation of the longshore transport as a function of incident wave height and direction, sediment characteristics and beach slope. The predictions of this model are shown to be in agreement with the experimental measurements. i i Table of Contents Abstract ii Table of Contents /-/7 List of Tables VJf List of Figures IX List of Symbols xii Acknowledgements xvii 1 I N T R O D U C T I O N 1 2 A R E V I E W O F C U R R E N T A N D T R A N S P O R T M O D E L S 4 2.1 Introduction 4 2.2 Nearshore Currents 4 2.2.1 Rip Currents 5 2.2.2 Longshore Currents 6 2.2.2.1 Radiation Stress Approach 7 2.2.2.2 Semi-Empirical Approach 13 2.3 Longshore Transport 16 2.3.1 Wave Power (Energy Flux) Models 17 2.4 Detailed Predictor Models 24 2.5 Summary 34 in 3 WAVE BREAKING CHARACTERISTICS 37 3.1 Introduction 37 3.2 Breaker Type Classification 38 3.3 Breaker Classification Using Breaking Indices 39 3.4 Laboratory Studies of Breaker Types 47 3.4.1 Preliminary Experimental Results 48 3.5 Breaker Height Prediction 49 3.6 Beach Permeability Effects 56 3.6.1 Modifications of forces due to permeability 57 3.6.2 Wave energy loss due to permeabilty 59 3.7 Discussion 60 4 LABORATORY STUDIES OF LONGSHORE TRANSPORT 62 4.1 Introduction 62 4.2 Previous Studies of Longshore Transport 63 4.2.1 Laboratory Studies 63 4.2.2 Field Studies 65 4.3 Experimental Design and Procedures 69 4.3.1 Calibration of Wave Probe 75 4.3.2 Longshore Current Measurement 76 4.3.3 Efficiency of Sediment Collector 78 4.4 Factors Affecting Transport Rate Studies 79 4.4.1 The Beach Characteristics 79 4.4.2 The Wave Characteristics 80 4.4.3 The Characteristics of Facilities 81 4.5 Experimental Results 81 4.6 Discussion 97 iv 5 D E V E L O P M E N T OF T H E LONGSHORE MODEL 100 5.1 Introduction 100 5.2 Formulation of the Model 101 5.2.1 Model Assumptions 101 5.3 Transport and Roughness Coefficients 105 5.3.1 Transport Coefficient K 106 5.3.2 Roughness Coefficient C 109 5.4 Effect of Permeability on the Model 115 5.5 Onshore-Offshore Transport 115 5.6 Discussion 117 6 TESTING OF T H E LONGSHORE MODEL 121 6.1 Introduction 121 6.2 Longshore Transport Model Predictions 122 6.3 Comparison of Model and Wave Flux Model 129 6.4 Sensitivity of the Model 132 6.5 Discussion 140 7 SUMMARY AND CONCLUSIONS 143 Bibliography 153 APPENDICES 163 A Grain Size Distribution of Test Sands 163 B Calibration of Wave Probe 167 C Wave Reflection Data 169 v D Longshore Transport Results E Longshore Current Measurement List o f Tables 2.1 Values and Units of Coefficients A and B 21 2.2 Detailed Predictor Models 25 3.1 Transition values using deepwater wave steepness (Iversen (1953)) . . 42 3.2 Galvin's transition values for offshore and inshore wave parameters . 45 3.3 Transition values using deepwater wave steepness (Galvin (1968)) . . 46 3.4 Battjes' transition values for offshore and inshore wave parameters . . 46 3.5 Values for the surf similarity parameter for breaker types 47 3.6 Recorded wave data for wave breaker types on 1:9 slope 48 4.1 Range of experimental variables tested 74 4.2 Summary of transport results for wave period 1.41sec 82 4.3 Summary of transport results for wave period 1.50sec 82 4.4 Comparison of measured and predicted current velocities 96 6.1 Predictions of transport model for wave period of 1.41sec 124 6.2 Predictions of transport model for wave period of 1.50sec 125 6.3 Predicted and measured local peak transport rates for 1.41sec. wave period 128 6.4 Predicted and measured local peak transport rates for 1.50sec. wave period 128 D . l Experimental results on 1 : 9 slope for wave period of 1.41sec 172 D.2 Experimental results on 1 : 5.2 slope for wave period of 1.41 sec. . . . 173 vii D.3 Experimental results on 1 : 3.6 slope for wave period of 1.41sec. . . . 174 D.4 Experimental results on 1 : 9 slope for wave period of 1.50sec 175 D.5 Experimental results on 1 : 5.2 slope for wave period of 1.50sec. . . . 176 D. 6 Experimental results on 1 : 3.6 slope for wave period of 1.50sec. . . . 177 E. l Velocity data on 1 : 9 beach slope for wave period of 1.41sec 179 E.2 Velocity data on 1 : 5.2 beach slope for wave period of 1.41sec 180 E.3 Velocity data on 1 : 3.6 beach slope for wave period of 1.41sec 181 vm List of Figures 2.1 Classification of longshore current velocity models 10 2.2 Comparison of the longshore velocities measured by Galvin and Eagle-son (1965) with theoretical profiles. [After Longuet-Higgins 1970b] . . . 11 2.3 Classification of longshore transport models 18 3.1 Physical description of principal breaker types 40 3.2 Breaker classification using breaker type index H0/T2 (Iversen 1953) . 41 3.3 Variation of breaker type with H0/L0 and beach slope (Iversen 1953) 41 3.4 Breaker classification according to Galvin (1968) 43 3.5 Breaker type as a function offshore parameter (Galvin 1968) 44 3.6 Breaker type as a function of onshore parameter (Galvin 1968) . . . . 44 3.7 Plan view of experimental set-up 50 3.8 Experimental results of breaker type studies 51 3.9 Comparison of breaker height prediction theory with experimental results 54 3.10 Breaker height index versus deepwater wave steepness S P M (1984) . . 55 3.11 Non-dimensional depth at breaking versus breaker steepness S P M (1984) 55 4.1 Plan view typical of the test set-ups 72 4.2 Section of the variable beach face slope 73 4.3 Measured longshore transport rate on 1 : 9 slope 83 4.4 Measured longshore transport rate on 1 : 5.2 slope 84 4.5 Measured longshore transport rate on 1 : 3.6 slope 84 4.6 Measured longshore transport rate on 1 : 9 slope 85 ix 4.7 Measured longshore transport rate on 1 : 5.2 slope 85 4.8 Measured longshore transport rate on 1 : 3.6 slope 86 4.9 The effect of beach slope on fine sand transport 87 4.10 The effect of beach slope on coarse sand transport 87 4.11 The effect of beach slope on fine sand transport 88 4.12 The effect of beach slope on medium sand transport 88 4.13 The effect of beach slope on coarse sand transport 89 4.14 Measured longshore transport rate as a function of beach slope . . . . 91 4.15 Measured longshore transport rate as a function of beach slope . . . . 91 4.16 Reflection coefficient as a function of beach slope 92 4.17 Reflection coefficient as a function of beach slope 93 4.18 Cross-shore distribution of longshore velocity on 1:9 slope 94 4.19 Cross-shore distribution of longshore velocity on 1:5.2 slope 94 4.20 Cross-shore distribution of longshore velocity on 1:3.6 slope 95 5.1 Values of theoretical bedload efficiency factors (Bagnold (1966)) . . . 107 5.2 Increase of beach stress as a function of sediment size [Quick, 1991] . 112 5.3 The distribution of bottom stress TX, exerted by the waves on the beach face [Komar, 1975] 113 6.1 Comparison of model predictions and experimental results 126 6.2 Comparison of model predictions and experimental results 127 6.3 Comparison of predictions of both models and experimental measure-ments 131 6.4 Sensitivity of longshore transport model due to variation in H^. . . . 135 6.5 Sensitivity of longshore transport model due to variation in C 136 6.6 Sensitivity of longshore transport model due to variation in Oj, . . . . 137 x 6.7 Sensitivity of longshore transport model due to variation in C& 138 6.8 Sensitivity of longshore transport model due to variation in K 139 A . l Grain size distribution of original test sands 164 A.2 Distribution of original and transported test sands for 1.41sec. wave period 165 A . 3 Distribution of original and transported test sands for 1.50sec. wave period 166 B . l Calibration of wave probe for wave height recording 168 x i List of Symbols a - function of beach slope a0 - wave motion excursion A - coefficient b - function of beach slope B,B1,B2 - coefficients c - wave speed C,Cf - roughness coefficient Ch - Chezy coefficient of resistance eg - coarse grain c9 - wave group velocity c0 - deepwater wave speed d,D - grain diameter di,d2 - representative grain sizes db - water depth at breaking Dw - 10% finer sediment size D50 - median grain size - 60% finer sediment size DgT - dimensional grain size e - efficiency factor or void ratio - bedload efficiency factor - suspended load efficiency factor E - energy density xii Ea, Ei - alongshore wave energy flux Eb - energy density at breaking E0 - deepwater wave energy density / - friction coefficient fg - fine grain fw - Jonnson wave friction factor Fgr - sediment mobility number Fgrc - critical sediment mobility number Fgr(i) - instantaneous sediment mobility number Fwc - mean mobility number g - acceleration due to gravity h - water depth hb - water depth at breaking H - wave height Hb - wave breaker height Hi - incident wave height HL - total lateral thrust Hmax - maximum incident wave height Hmin - minimum incident wave height H0 - deepwater wave height Hsb - significant wave height i - sediment transport per unit width of beach ic - critical hydraulic gradient IL - total sediment transport rate k - surface roughness K - dimensionless transport coefficient or permeability xii i - reflection coefficient Krb - parameter given in the form of reflection coefficient K* - dimensional coefficient I - horizontal excursion of water particle L - length of beach L0 - deepwater wave length m, s, - beach slope n - ratio of wave group velocity to wave celerity N - numerical constant P - local longshore transport power per unit width PUP2 - exponents P - pressure P - power per unit area to move coarse sediment Pf. - power per unit area to move fine sediment Pi, Pis - longshore component of wave energy flux P J- w - available power Q,Qi - volume rate of sand transport Q - net sediment flux r - bed roughness Re - Reynolds Number s - specific gravity or beach slope - beach slopes SbB - bedload transport SL - longshore volume transport rate - gross longshore transport rate SSB - suspended load transport xiv radiation stress component time or wave period resisting shear stress horizontal velocity of flow mean velocity of flow velocity of boundary horizontal orbital velocity maximum local horizontal velocity of flow current velocity mean longshore current velocity longshore current velocit at breaker line shear velocity fluidizing velocity empirical coefficient distance up beach face mass flux per unit mass flow rate distance from the shoreline to the breaker line ratio wave height to water depth angle of beach to the horizontal similarity criterion relative apparent density eddy diffusivity wave angle of attack deepwater wave angle of attack wave breaker angle sediment friction coefficient xv - ripple coefficient V - kinematic viscosity of fluid P - density of fluid Ps - density of particle i - surf similarity parameter - inshore wave parameter io - offshore wave parameter cr - radial frequency &<(, - measure of sorting T - wave shear stress Tx - local alongshore wave stress Twc - modified bed shear stress <t> - friction angle U) - available streampower xvi Acknowledgements The author is very grateful to Professor M . C . Quick for his financial support and for the encouragement, guidance and supervision given by him which led to the suc-cessful completion of this work. The author would also like to express his gratitude to Kurt Neilsen who assisted in the design of the laboratory model and subsequent assitance in helping change beach slope conditions. Finally, the author would like to thank The Mighty One God, The Lord for might-ily inspiring in him strength and wisdom for this work. xvii Chapter 1 INTRODUCTION It has been recognized for many years that, when waves break at angle to the shoreline, they cause a transport of sand along the beach. The mechanics of the long-shore transport are not precisely known, but it is noted that when longshore material is moved by wave action, there is also an onshore and offshore movement of the ma-terial. Generally, depending on the waves prevailing at the time, either an onshore or an offshore movement of sand w i l l predominate. Therefore, even for the net longshore transport, the dominant sediment transport mechanism is the onshore-offshore move-ment. Thus, the various stress modifying factors considered for the onshore-offshore transport should be of similar importance for the longshore transport. The longshore transport rate and direction are the result of the summation of the forces imposed on the beach by the sea, and the reaction of the beach to these forces; thus, they are good indicators of the physical phenomena present i n the area. For this reason the transport rate and the direction are important to the design of shore protection, navigation and coastal erosion projects. In shore protection prob-lems, the transport rate and direction indicate where and to what extent erosion or accretion w i l l occur under given circumstances and thus provide information neces-sary i n determining the position and magnitude of remedial measures. In many of the problems faced by the coastal engineer, the rate and direction of the longshore transport are of prime importance i n indicating the problem as well as the required 1 Chapter 1. INTRODUCTION 2 corrective measures. Thus the economic and functional impact of a valid quantitative predictor of transport certainly justifies a need for laboratory and field investigations. Although, the main focus of this research is on the wave-induced longshore cur-rents and the longshore transport, many questions regarding the most widely used predictors which relate the longshore transport to the longshore component of wave energy flux at breaking are also considered. Among the most pressing ones are, 1. Is the longshore transport rate actually independent of sediment characteristics? 2. How does the longshore transport rate change for changing wave and beach conditions? 3. Does the breaker type influence longshore transport mechanisms in the surf zone? 4. What influence has the beach slope on the longshore current velocity? 5. What is the effect of beach permeability on the transport rate? This thesis will attempt to answer some of these questions, remembering that the complexity and the interconnection of the nearshore processes are still not properly understood. However, with the existing knowledge on beach changes, nearshore cur-rents and longshore transport an attempt to develop a simplified theoretical model for the longshore transport relationship can be made. The use of such a model will provide an insight into some of the above questions. Presented in this thesis is a review of events leading to the development of the theoretical model. Chapter two gives a brief review of both the longshore current ve-locity and transport rate models currently in use. The emphasis in this review is on Chapter 1. INTRODUCTION 3 the wave-induced longshore currents and the wave energy flux models for estimating the longshore transport potential. Since the fluid motions at breaking cause most of the sediment transport in the surf zone, the amount of sediment transported is partly determined by the breaker type. Chapter three therefore describes the wave breaking characteristics of the dif-ferent breaker types and the relationship between the breaker type, beach slope, deepwater wave steepness and the surf similarity parameter. Wave breaker height prediction theory and the influence of beach permeability on the breaking process are also discussed. Experimental measurements are compared with those found by other researchers. Chapter four describes the previous investigations of the longshore transport and the major experimental studies carried out in the present study. The experimental results are also presented. Chapter five describes the development of the theoretical longshore transport model incorporating the influence of beach slope and sediment characteristics based upon Bagnold's (1966) streampower theory and Longuet-Higgins' (1970) local along-shore wave stress and the mean longshore current velocity formulated using the ra-diation stress concepts. In chapter six, the theoretical model is tested using the experimental results found for beach slope changes and different sediment character-istics presented in chapter four. Finally, conclusions drawn from the results are presented in chapter seven. Chapter 2 A REVIEW OF CURRENT AND TRANSPORT MODELS 2.1 Introduction The purpose of this chapter is to briefly review the theoretical background on the generation and evaluation of nearshore currents as well as the resulting sand transport. In this review, attention will be mainly focused on the wave-induced longshore currents and the longshore transport. A survey of the various predictive longshore sediment transport equations is also presented. 2.2 Nearshore Currents Currents in the nearshore zone are important in several aspects and are of par-ticular interest to the coastal engineers. The major reason is that these currents are believed to be the chief cause of littoral transport. Nearshore currents in the littoral zone are predominantly wind and wave-induced motions superimposed on the wave-induced oscillatory motion of the water. Of these, the wave-induced current is the most dominant and most important. This is because waves affect sediment motion in the littoral zone in two ways;- they initiate sediment movement and they drive current systems that transport the sediment once motion is initiated. The net motions of the currents generally have low velocities, but because they transport whatever sand is 4 Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 5 moved by the wave-induced water motions, they are important in determining littoral transport. There are two wave-induced current systems in the nearshore zone which dominate the water movements in addition to the oscillatory motions produced by the waves directly. These are rip currents which are associated with the cell circulation system, and the longshore currents produced by an oblique wave approach to the shoreline. 2.2.1 Rip Currents Rip currents are strong, narrow currents that flow seaward from the surf zone and as such substantial quantities of sediment can be transported offshore and deposited at the rip head. On an infinitely long beach, the sediment transport never reaches a steady state, because of the build up of rip currents. These rip currents carry some sediment offshore through the breaker zone, so that the next downstream section of the beach is partly deprived of its longshore sediment transport. There have been many studies (Bowen, 1969a; Bowen and Inman, 1969; Hino, 1974) on rip currents since it was first studied by Shepard et al (1941). Hino (1974) found that rip currents form with a preferred longshore spacing that is about 4 times the distance from the shoreline to the breaker line, a ratio of rip spacing to surf zone width that is commonly observed on natural beaches. Although, this finding shows promise, it requires experimental and field testing. Reviewing recent theoreti-cal approaches on rip currents, Dalrymple (1978) found that the various mechanisms proposed to explain rip currents can be conveniently grouped into two broad generic Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 6 categories, the wave interaction models and the boundary interaction models. The dis-tinction between the two is that the first category mechanisms can occur on uniformly planar beaches (having no longshore variability), while the second group cannot. 2.2.2 Longshore Currents There has been considerable interest in the generation of longshore currents by waves breaking with an angle to the shoreline. This interest is reflected in the num-ber of theories that have been put forward to account for the generation of such currents and to predict their magnitudes. Longshore currents interest engineers and geologists because they erode, transport and deposit sediment. Along the beaches bordering oceans and large lakes these currents are capable of transporting hundreds of thousands of cubic yards of sands past a given point during an average year. The major aim of studies of longshore currents is the quantitative prediction of the rate of sediment transport by longshore currents. Because sediment transport is primar-ily related to the velocity of the transporting current, it is first necessary to predict this fluid velocity before approaching the more difficult task of predicting sediment transport. The theories put forward in the prediction of the longshore current were based on the following: • Energy flux, • Conservation of mass, • Momentum flux, • Empirical approach. Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 7 Figure (2.1) shows the various models and the investigators. In this review, only the models based on the conservation of momentum and empirical approach will be briefly presented. Galvin (1967), giving a complete review of the early theories arrived at the con-clusion that a proven prediction of longshore current velocity is not available, and reliable data on longshore currents are lacking over a significant range of possible flows. These early theories can be categorised according to the general approaches taken in their derivations. Since then, there has been considerable progress princi-pally through the studies of Bowen (1969b), Longuet-Higgins (1970a, 1970b), Komar and Inman (1970) and Thorton (1970). Bowen, Longuet-Higgins and Thorton have used the radiation stress concept introduced by Longuet-Higgins and Stewart (1964), to evaluate the momentum flux associated with the waves in examining the longshore current generation. It is pointed out here that while Longuet-Higgins' paper was in preparation, his attention was drawn to a then unpublished paper by Bowen (1969b) in which the concept of radiation stress was applied to the same problem taking into account bottom friction and horizontal mixing, though in a somewhat different man-ner. Komar and Inman's (1970) approach was semi-empirical. Rather interestingly, these two approaches arrived at similar relationships and will be presented below. Figure 2.1 shows the classification of the various approaches taken in the derivation of the predictive equations for the longshore current. 2.2.2.1 Radiation Stress Approach Longuet-Higgins (1970a) re-introduced the momentum flux approach which had previously been attempted by Putnam et al (1949). Putnam et al suggested that Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 8 the magnitude of the longshore current was related in some way to the energy or the momentum of the incoming waves. Of these two approaches, that employing mo-mentum is the more promising since momentum is conserved, whereas energy can be dissipated by breaking and other processes not immediately associated with sediment. The model of Putnam was found to have been based on erroneous assumptions, be-cause in the model's conservation of momentum equation, he equated the longshore component of the breaker velocity to the current velocity. This is not correct and it can be shown that the longshore current velocity exceeds the longshore component of the breaker velocity, Galvin (1967). In the approach, Longuet-Higgins (1970a) proposed a new method of calculating the momentum flux of the waves using the concept of radiation stress introduced by Longuet-Higgins and Stewart (1964). The radiation stress is defined as "the excess flow of momentum due to the presence of the waves." The longshore component of the radiation stress is given as Sxy = E 1 kh 2 + sinh2kh cos 9 sin 9 (2.1) The derivation of Longuet-Higgins' longshore current velocity equation was based on the assumption that the radiation stress produces a longshore force which is primarily equal to the force resisting the longshore current. The resulting equation for the mean longshore current velocity, V;, is c, 5TT tan/? Vi = —a T r i = -g-a-cr(eh) sin0 (2.2) where a = is a constant Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 9 tan j3 = beach slope Cf = the beach bottom drag coefficient g = acceleration due to gravity h = local water depth 6 = the wave angle of approach in deepwater c = the wave celerity in deepwater By Snell's law, the last factor in equation (2.2) is a constant. Equation (2.2), then states that in a given wave situation, if both Cf and tan/? are constant, the mean longshore current velocity is simply proportional to the local depth, h, which is assumed to be some function of the coordinate x normal to the shoreline and is independent of the distance y along the shore. The shoreline itself is at x = 0. Outside the surf zone, it is assumed that there will be no wave breaking and hence no energy dissipation and no driving force for the current. Thus the distribution of the current across the surf zone is just triangular with a discontinuity at the breaker line. However, laboratory and field data indicated no such discontinuity at the breaker line (figure 2.2). One of the major reasons for this discrepancy between theory and measurement is found to be the exclusion of the horizontal eddy viscosity, a horizontal mixing ef-fect which produces an onshore-offshore transfer of the momentum. In a real fluid, a discontinuity as shown in figure (2.2) cannot exist. The presence of any horizontal mixing, as well as any variability in wave height and position of breaker line, will tend to smooth out the discontinuity at the breaker line and produce a smoother velocity distribution profile. The maximum value of the longshore current is then no longer at the breaker line, but a little shoreward of that line as in figure (2.2), which is found to be the case in the field (Komar, 1971). Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 10 LONGSHORE CURRENT VELOCITY MODELS Energy Flux Momentum Flux Conservation of Empirical Mass Correlation Putnam et al Putnam et al Inman & Bagnold Brebner and (1949) (1949) (1963) Kamphuis (1963) Eagleson (1965) Bowen (1969b) Brunn(1963) Harrison and Krumbein (1964) Longuet-Higgins Galvin & Eagleson Harrison (1968) (1970a,b) (1965) Figure 2.1: Classification of longshore current velocity models Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 11 Breaker Line V Locus of peak velocities for different P values f P = 0.1 y / - P = 0.4 y / / / / / • y / Equation [2.3] • Experimental data V =v/vo ' vo = longshore current velocity at breaker line for P = 0.0 X = x/Xb / ' . • Xb = width of surf zone s y y s # _ s • i t t i 1 i i i • 1 1 1 1 1 X Figure 2.2: Comparison of the longshore velocities measured by Galvin and Eagleson (1965) with theoretical profiles. [After Longuet-Higgins 1970b] Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 12 Bowen (1969b), Longuet-Higgins (1970b) and Thorton (1970) have examined the variations of the longshore current across the width of the nearshore region again using the radiation stress approach. To do this they consider the local dSxy/dx as the driving thrust, and include a horizontal eddy viscosity. This leads to the solution f AX + B i X P l 0 < X < 1 (inside surf zone) V = I V ' (2.3) [ B2XP2 1 < X < oo (outside surf zone) where X = x/Xb V = v/v0 Xb = distance from shoreline to the breaker line v = longshore current velocity at the distance x from the shoreline 57r l£2 rr-tan/? . a a v ° = fe (i + 7 ) i/2 Vgh-^f sm eb cos eb p _ ITN tan 0 n - - 3 , 9 , 1 A = (1-1^) B 2 = i ^ A 7 = ratio of the wave to water depth Cf = bottom friction factor Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 13 The dimensionless parameter P reflects the significance of the horizontal eddy transfer; the larger the value of P the more important this effect. The term N is a numerical constant with limits 0 < N < 0.016 in the expression for the horizontal eddy viscosity, pNx(gh)1/2, used by Longuet-Higgins (1970b). If there is no horizon-tal mixing, P=0, a simple solution for equation (2.3) is obtained. That is to say, the current increases linearly from the shoreline to the breaker line. Beyond the breaker line it is zero. At the breaker line itself the current velocity is discontinuous and the velocity distribution profile tends to the triangular form as shown in figure (2.2). Comparison of the solutions with the laboratory measurements of Galvin and Eagle-son (1965) of the variations in velocity across the nearshore suggest a range of P from 0.1 (little mixing) up to 0.5 (complete mixing) but is commonly about 0.2. 2.2.2.2 Semi-Empirical Approach The sand transport studies of Komar and Inman (1970) had suggested a longshore current velocity prediction equation similar to equation (2.2) V| = 2 .7U m s in0 b (2.4) where Um is the maximum value of the horizontal orbital velocity evaluated at the breaker line with (2.5) Equation (2.4) gives the longshore current velocity at the mid-surf position and was obtained by a different approach. Komar and Inman (1970) found that in labora-tory tests much greater angles of wave breaking are achieved and the results indicate the relationship should be modified to V, = 2 .7U m s in0 b cos0 b (2.6) U r 2EL PK Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 14 Breaker angles in the field are generally small so that cosfy, « 1. Komar and Inman (1970) initially developed two models for the longshore sand transport rate. They found that both models agree well with field data. Prompted by the agreement, they equated the two transport equations for these two seemingly independent models. This leads to the relationship which they have shown to be supported by field data. 5n tan 3 f— , . / n _N V| = y a - ^ V g h b S i n f l b (2.7) The similarity of equations (2.2) and (2.4) can be seen if the breaker line conditions Cj, = y/ghb and Um = cty/ghb are substituted in equation(2.2). Both equation (2.2) and equation (2.4) are similar , except that equation (2.4) implies that the ratio tan 3/Cf must be a constant. Komar and Inman presented field and and laboratory data in a plot of V\ against Um sin 6b cos 9b, and showed that the slope of the best fit line is indeed 2.7 . From this they concluded the near-constancy of the ratio tan 3/Cf. However, Cf was shown to have an approximate value of 0.01. Therefore, the ratio taxi 3/Cf should not be a constant for different beach slope. Longuet-Higgins (1970) suggested that this apparent constancy can be explained. Wi th an increase in the slope tan 3, there would be an increase in the level of turbulence. This would seem likely to affect the horizontal mixing which will indirectly lessen the value of V] and maintain an apparent constancy for the ratio tan 0/Cf in equation (2.7). Losada et al (1986) also explained the slow variation of tan 3/Cf by considering that for spilling breakers, occurring on mild slopes with tan/? of order O(10 - 2 ) , the bottom will be rippled or flat with low sediment load with a value of Cf of order O(10 - 2 ) . Plunging breakers, on the other hand, will be more likely found on intermediate slopes (tan/? of order O(10 - 1 )) with an increased sediment load and Cf of order O(10 _ 1 ) . This can Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 15 be physically understood considering that the bottom friction coefficient depends on bed material and forms which are closely related to the flow properties in the area which in turn vary with breaker type. To date, the actual relationship between the longshore current velocity and the beach slope is still relatively unknown. However, Galvin and Eagleson (1965), using the conservation of mass in the surf zone derived an expression for the mean velocity of the longshore current in terms of the beach slope, m, wave period, T, and wave breaker angle, 60 which enjoyed little success. V = gmTsin20 b (2.8) The Shore Protection Manual (1984) also suggested using a modified Longuet-Higgins' (1970) equation to calculate the maximum longshore current velocity, Vm, in the surf zone given as, V m = 20.7 tan / S y ^ s i n 20 b (2.9) where hb = the water depth at the breaker line 6b = the wave breaker angle to the shoreline It should be noted that the above equation will only give the maximun longshore current value and not the whole variation across the nearshore. The constant 20.7 in equation (2.9) was obtained from the calibration of the equation with the laboratory data of Galvin and Eagleson (1965) and the field data of Putnam et al (1949). Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 16 2.3 Longshore Transport Longshore transport is the movement of beach material in the littoral zone by waves and currents in the direction parallel to the shoreline. For this report, the littoral zone is a strip which follows the shoreline bounded by the runup limit on the landward side and by a depth on the seaward side at which large waves begin to move bottom sediment in significant quantities. Because most wave energy is dissipated in the littoral zone, this zone is where beach changes are most rapid. These changes may be short term due to seasonal changes in wave conditions and to occurrence of intermittent storms separated by intervals of low waves, or long term due to an overall imbalance between the added and eroded sand. The rate at which littoral material is moved parallel to the shoreline is the long-shore transport rate. Since this rate is directed parallel to the shoreline, there are two possible directions of motion, which may be called the right and left directions, if defined for an observer standing on the shore and looking out to sea. A gross longshore transport rate is defined for a given point on the shoreline as the sum of the amounts of littoral material transported to the right and to the left, past that point on the shoreline in a given time period, (Galvin,1972). Similarly, a net long-shore transport rate is defined as the difference between the amounts of littoral drift transported to the right and to the left past a point on the shoreline in a given period. Figure 2.3 gives a classification of the longshore transport models. Basically, there are two categories: those based on wave power (energy flux) approach, commonly called overall predictors and those based on sediment equation approach, also called Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 17 detail predictors. A brief description of the equations in of these categories will be presented. 2.3.1 Wave Power (Energy Flux) Models A long-standing desire of the coastal engineers has been to be able to estimate the longshore transport rate from a knowledge of the wave and current parameters that cause the sand transport. The wave flux approach recognizes that energy is required to transport the sediment. To attempt this, empirical correlations between the rate of sediment transport and the amount of wave power available for this transport were established. The general form for the transport rate equations established based on the wave power approach is S| = AP,B (2.10) where Si = the longshore volume transport rate of sand Pi — the longshore component of the wave energy flux and A and B are coefficients. The value of Pi is calculated at the breaker line. This is because most of the longshore transport occurs within the surf zone. The approximation for P; at the breaker line is based on the following assumptions: • Conservation of wave energy flux in shoaling waves, • Linear or small amplitude wave theory, Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 18 LONGSHORE TRANSPORT MODELS WAVE ENERGY FLUX DETAIL PREDICTOR Wave Power or Estimation From Energy Flux Mean Wave Breaker Height Scripp Institution Oceanography (1947) Galvin (1972) Watts (1953b) Caldwell (1956) Savage (1959) Thorton (1969) Komar & Inman (1970) Fairchild (1972) Dean (1989) White and Inman (1989) Energetic Model Models Based on Mechanics of Transport Inman & Bagnold (1963) Bijker (1971) Fleming etal (1976) Nielsen etal (1978) •Willis (1978) * Swart etal (1980) * Adaptation of Ackers and White's (1973) formula Figure 2.3: Classification of longshore transport models Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 19 t Shallow water conditions. The longshore component of the wave energy flux is given as Pl = (ECn) b s in0 b cos0 b (2.11) From linear wave theory, Cg — Cn where , r i + , , ^ , J (2.12) 1 " = 2 smh(2kh) The wave celerity, C, is given by the relationship C = u>/k where u is the wave radian frequency and k is the wave number. The wave number, k, is determined from the linear dispersion relationship, J1 = gktanh(kh) (2.13) For shallow water conditions n is approximately equal to 1.0 and hence Cg — C and the energy density, Eb, evaluated at the breaker line is given as E b = « £ l (2.14) where p = mass density of water. Making substitution and shallow water approximations, equation (2.11) becomes P| = ^ C b s i n 2 0 b (2.15) where Cb is the wave speed at breaking evaluated in a depth equal to 1.28.H&. It is important to note that equation (2.15) is valid only if there is a single wave train with one period and one height. However, most ocean wave conditions are characterised by a variety of heights with a distribution usually described by a Rayleigh distribution. For a Rayleigh distribution, the correct wave height to use in equation (2.15) is the root-mean-square height. However, most wave data are available as significant Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 20 heights, therefore the significant wave height, Hsf,, is substituted into equation (2.15) to produce P|s = ^ C b s i n 2 0 b (2.16) The value of P[s computed using significant wave height is approximately twice the value of the exact energy flux for sinusoidal wave heights with a Rayleigh distribution. It is often necessary to use the offshore wave conditions due to lack of any other information. In this case, Cg = Of- = and the wave refraction coefficient Kr is used to allow for inshore conditions assuming refraction by straight parallel bottom contours so that equation (2.15) becomes P| = ^ ( H o K r ) 2 s i n 2 0 b (2.17) where H0 is the offshore wave height or the deep water wave height. Scripps Institution of Oceanography (1947) first suggested that the work rate (power or energy flux) performed by the waves in the nearshore zone might be a useful parameter for predicting the longshore transport rate from the wave action. Following this suggestion, there has been considerable progress made by a number of investigators using the same approach, but each obtained different values for the coefficients A and B of equation (2.10). Table (2.1) lists the various investigators and the values of the coefficients as well as the units in which they were obtained. Inman and Bagnold (1963) have pointed out that the longshore transport rate would be better expressed as an immersed-weight transport rate 7/ rather than as a volume transport rate Si. The two are related by I, = (Ps - P)giS| (2.18) Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 21 Table 2.1: Values and Units of Coefficients A and B Si = APf Coefficients Units of A B Si Pl Watts (1953b) 0.0289 0.9 m3/day Watt/meter Caldwell (1956) 0.0626 0.8 m3/day Watt/meter Inman & Bagnold (1963) 0.0046 1.0 m3/day Watt/meter S P M (1977, 84) 0.0709 1.0 m3/day Watt/meter Komar & Inman (1970)* * J, = APf 0.77 1.0 Ii = Newton/sec Pi = Newton/sec where ps = mass density of sand p — mass density of water g = acceleration due to gravity a = a correction factor for the pore space of the beach sand The use of the immersed-weight transport is based on consideration of the problem of sediment transport in general from an energetics point of view by Bagnold (1963). There are two distinct advantages in relating 7/ and P/, rather than Si and P/. The first is that both have units of force per unit time and hence can be related by I, = KP| = K(ECn)b sin 0 b cos 0 b (2.19) where K is a dimensionless proportionality coefficient. The second advantage in using the immersed-weight transport rate is that as can be seen from equation (2.18), it takes into consideration the density of the sediment Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 22 grains. Therefore, a correlation between J; and Pi could be used for beaches composed of coral sand and so forth, as well as the more usual quartz sand. It is noteworthy that the above equations for the longshore sediment transport do not account for sediment size and beach slope. The study of Komar and Inman (1970) suggests a lack of significant dependence of the transport rate on the beach slope. However, from the present study, it will be shown that the predominant beach slope and sediment size do have effects on the longshore transport rate. Galvin (1972) using intuitive reasoning observed that the longshore transport rate must be closely related to the breaker height, since as breaker height increases, more energy is delivered to the surf zone. At the same time, as breaker height increases, breaker position moves offshore widening the surf zone and increasing the cross-section area through which sediment is moved. Galvin showed that when field values of longshore transport rate are plotted against mean annual breaker height from the same locality a curve S g = 16.5xl05Hjj m3/year (2.20) S g = 2.0xl0 5H^ yds3/year (2.21) forms an envelope above almost all known pairs of (Sg,Hb). In equation (2.20) Sg is gross longshore transport, given in cubic meters per year and Hb is in meters, while in equation (2.21) Sg is given in units of cubic yards per year and Hb is in feet. Galvin's plot provides an upper limit on the estimate of the longshore transport rate. In equations (2.20, 2.21), wave height is the only independent variable, and the physical explanation assumes that waves are the predominant cause of the transport. Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 23 Also Galvin's empirical relationship does not take into account the wave angle of attack. This suggests that equations (2.20, 2.21) may only provide a first approxi-mation of engineering predictions and the longshore transport capability rather than the actual transport. The simplicity of equations (2.20, 2.21) and the unexpected, but empirically good fit of the data to the equations prompted Galvin to attempt a physical explanation. Assuming the longshore transport occurs mostly as suspended load, and using the law of conservation of mass or continuity and some approximations, Galvin obtained the relationship S g = [DKg/32cT sin 0b] H 2 (2.22) where c = is a mean sediment concentration in the surf zone j3 = Breaker depth-to-height ratio for a given slope K = is the ratio of the annual mean of individual H2 to the square of annual mean D = is a factor added to keep the units correct. If Sg is in yd3/yr and g is in ft/sec2, H is in ft and T is in seconds, then D has units of (sec — yd3/yr — ft3). If equation (2.22) is an explanation of the empirically derived equation (2.21), then the factors on the right side of equation (2.22) in the square brackets should be equivalent to or less than the factor on the right hand side of equation (2.20). That is, DKg/3 2cTsin0 b < 2 (2.23) Galvin tested this continuity hypothesis and arrived at values which were all even less than unity. This result is not too encouraging for his hypothesis. Furthermore, the Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 24 assumption that the longshore transport occurs mostly as suspended load is subject to great dispute, Komar (1976). 2.4 Detailed Predictor Models The longshore transport rates are also computed from the detailed sediment equa-tion transport formulae. Unlike the overall wave energy flux predictors, all the models of the detailed predictors in this category are based on two relationships, one for the longshore current velocity and a sediment transport relationship which requires the longshore current as an input. Most of these models have their origin from the work on sediment transport under oscillatory waves or in riverine conditions. As detailed presentation of all the models have been reported in literature, the details of their formulation will be omitted. However, a brief description of their approaches will be presented here. Table 2.2 shows the various so called detailed predictor models and their equations. From table 2.2, it is evident that most of the equations calculate the transport rate separately as bedload and suspended load with the notable exceptions of Inman and Bagnold (1963) whose equation is for total load, and Nielsen et al (1978) where the transport is assumed to be totally suspended. Classic treatments of the subject of suspended and bedload define suspended load as that part of sediment load that is supported by fluid turbulence, while bedload is material that is placed in motion by the tangential shear stress of the fluid over the bottom and has a upward dispersion maintained as a result of grain-to-grain contact and by the lift forces on the bed material (Bagnold, 1954, 1966). Although, there is a clear distinction between bedload and suspended load following the concepts of Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 25 Table 2.2: Detailed Predictor Models Investigator(s) Longshore Transport Rate Remarks Inman&Bagnold (1963) h = K{ECn)b cos 0bvi/Um £ = 0 . 2 8 (Komar et al, 1970) U - = J % Bijker (1971) SB — SbB + $sB SbB = Bedload SSB=Suspended load Fleming et al (1976) S, = /0e Cudz + Jeh Ccudz C, C c = Sediment concentration of the bed and suspended load regions respectively u, uc = Current velocity distribution of the bed and suspended load regions respectively. e = reference point from bed Nielsen et al (1978) Q(t) = tfu(z,t)c(z,t)dz Q t = instantaneous flux of sediment u(z,t) = vertical distribution of the horizontal water distribution c(z,t) = vertical distribution of the suspended sediment Q = Net sediment flux Modifications of Ackers et al (1973) 1) Willis (1978) 2) Swart et al (1980) p* = p \ £ v + w*caf*u* L " FWC Fgr(t) 7p JQ Fgr(t)dt Transport Load By Eqn (2.38) Pw = Available power' Fwc = Mean mobility no. Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 26 Bagnold (1966), from a practical point of measurement they are nearly inseparable. The question that has been studied a great deal by many investigators (Watts, 1953a; Komar and Inman, 1970; Fairchild, 1972,1977; Thorton, 1972; Benninkmeyer, 1974; Komar, 1976 and Nielsen et al, 1978) is, which is more significant, the suspended load or the bedload? Theories and empirical studies of longshore sediment transport rates within the surf zone have generally assumed or have estimated in some ad hoc manner, the contribution of suspended load to total load. However, there has been broad dis-agreement upon the importance of suspended load transport, ranging from dominant (Watts, 1953a; Fairchild, 1972,1977; Galvin, 1972; Thorton, 1972; Kana, 1977,1978 and Nielsen et al, 1978) to less than one-half of the bed load transport (Komar and Inman, 1970; Brenninkmeyer, 1974; Komar, 1976; and Inman et al, 1980). Fairchild (1977), Kana (1977, 1978) and Inman et al (1980) suggest that the ratio of suspended load to total load is not constant, but that the importance of suspended load varies for different surf zone conditions (e.g., breaker type) and wave intensities. Suspended sediment concentration have been measured in a number of field in-vestigations using a variety of sampling methods. For sufficiently intense flows, a region of high sediment concentration exists well above the at-rest surface of the bed. The motion of the grains in this near bed layer appears to involve elements of both grain dispersive pressure and turbulence. This leads to some confusion in estimates of suspended load transport since some workers have assumed that all sediment above the level of the at-rest bed is suspended load (Watts, 1953; Thorton, 1972; Kana, 1977,1978; Fairchild, 1972, 1977) while others have included the near bed high sedi-ment concentration with bedload (Komar and Inman, 1970; Komar, 1976) or specified Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 27 a level above which sediment is considered to be suspended (Brenninkmeyer, 1974; and Inman et al, 1980). For practical purposes, Zampol and Inman (1989) have chosen arbitrarily that suspended load occurs 10cm or more above the at-rest sand bed. Within the constraints imposed, Zampol et al find that the suspended load long-shore transport rate is roughly 10 to 30% of the total longshore transport rate of sand. In part the contradictions may result from different sampling devices and the na-ture of their deployment, and from the different concepts of bedload and suspended load which lead to different methods of separating the two transport modes. Inman and Bagnold's (1963) transport model was based on the coupling action of the wave motions and the superimposed longshore current. The wave orbital mo-tion provides a stress that moves the sediment back and forth following the orbital water but yields no net transport even though wave energy is expended. Inman et al assumed that if some portion of the energy flux (ECg)0 cos8b is dissipated in placing the sediment in motion, the mean stress applied to the beach face is proportional to (ECg)b cos 0b/uo, where uQ is the mean frictional velocity relative to the bed within the surf zone. It is assumed that u0 is proportional to u m , the maximum horizontal component of the orbital velocity near the bottom just before the wave breaks. Once sediment is in motion, it becomes available for transport by the superimposed long-shore current, vi, which imparts a longshore drift to the sand already supported and suspended by the waves. The resulting immersed-weight transport rate past a section of beach becomes Il = k(ECn) b cos0 b -^- (2.24) where K is a dimensionless factor of proportionality. Inman et al (1963) left the origin of the current uj unspecified so that the model permits a longshore transport Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 28 to result from a wave train arriving parallel to the shoreline (6b = 0) in the pres-ence of a longshore current whose generation is not dependent on an oblique wave approach and hence equation (2.24) could conceivably be used to evaluate the trans-port of sediment under tidal or local wind-generated longshore current. The equation also relates the total longshore sediment transport in the nearshore zone to the wave parameters evaluated at the breaker line. The above model was based on Bagnold's (1963) energetic model in which he stated that the available power from the wave motion, u> supports the sediment above the bottom. Bijker (1971) and Fleming et al (1976), adopted the approach that the longshore transport consists of two components, namely, a bedload component and a suspended load component. Bijker's (1971) bedload formula was adapted from the Frijlink formula for sediment under riverine conditions by modifying the bed shear stress term resulting from the combined motion of waves and currents. The modified bed shear stress due to waves and currents, TWC, is given as (2.25) where, Ch = Chezy coefficient of resistance £ — coefficient v = mean current velocity u0 = amplitude of orbital velocity at the bed Bijker assumed that the bedload takes place within a certain layer and developed the 2 Twc = ^Fl Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 29 formula for the bedload for the combined motion of waves and currents as 1/2 -0.27 AsgD . /*(W/>) S b B = 5D exp (2.26) where D = grain diameter \i = ripple coefficient As = relative apparent density The suspended load component was found from the Rouse-Einstein distribution of suspended material from the bed to the free surface and integrating the product of the instantaneous velocity and the concentration of bed material with depth, d, resulting in where i i and I2 are elliptic integrals and r is the bed roughness. The sum of the bedload and the suspended load then gives the total transport rate in any given water depth From Bijker's (1971) formulation, it is noted that there is no physical separation of regions between the bedload and the suspended load. Bijker's sediment transport model contains no incipient motion criterion. However, the influence of bed rough-ness is considered in his computational procedure. In Fleming et al's (1976) model, two regions of transport are identified; the bed load region in which grains are assumed to be supported by grain-grain interactions and a suspended load region in which grain particles are kept in suspension by fluid turbulence. Fleming et al, defined a reference depth 'e' (bedload thickness) that (2.27) SB = SbB + SSB (2.28) Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 30 separates the two regions. They used the force balance of bed particles to derive a theoretical expression for the concentration close to the bed which contains an incipient motion criterion. Fleming et al then assumed that the concentration at the bed cannot exceed 0.52 and that the eddy diffusivity is constant over the whole water mass above the elevation z = e. They also assumed the continuity of sediment concentration and velocity between the two regions. For both conditions Fleming et al used a one-seventh power law to determine the velocity at any given depth, d, as where z is the distance above the bed. Fleming et al then presented equations for finding Jm and e. The sum of the two numerical integrations from the bed to the ref-erence depth 'e' and from 'e' to the free surface yielded the total longshore transport rate. Nielsen et al (1978) developed their sediment transport model with a rather dif-ferent approach. From direct visual observation and high speed movies of motion of the sand particles, they found that the actual bedload transport as known from unidirectional flow, hardly occurs and assumed that all sediment is moved in suspen-sion. Nielsen et al also assumed that sediment is brought into suspension by a pick-up mechanism p(t), defined as solid sediment being picked up per unit area of bottom per second. They further assumed that the balance between the agitating processes tending to keep the sediment in suspension, and the settling of sediment (described by settling velocity ui) has the nature of a diffusion process with eddy diffusivity e, for 0 < z < e for e < z < d (2.29) (2.30) Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 31 which is independent of time and horizontal coordinate x such that the concentration c(z,t) of suspended sediment satisfies the diffusion equation dc dc 8 dt "dz dz dc 'dz = 0 (2.31) z being the vertical coordinate. From their laboratory data, Nielsen et al determined empirically quantitative pre-dictors for the eddy diffusivity e, which they found is constant with distance from the bed to the free surface for non-breaking waves and increases strongly with dis-tance from the bed for gentle spilling breakers. The effect of spilling breakers was to increase the eddy diffusivity at the bed with two orders of magnitude. This is significant, since the Nielsen formula is the only one of the detail predictors which includes the effects of wave breaking in their formula. In Nielsen et al's model, the concentration of suspended material at the top of the bed forms contains an incipient motion criteria. Nielsen et al then obtained analytical solutions for the variation of the concentration profile, the instantaneous sediment flux and the net flux of sediment over a wave period. The last group of detail predictor transport models to be discussed is basically variations of the approach developed by Ackers and White (1973) for determining sediment transport rate in unidirectional flow over an alluvial bed. The technique of Ackers and White is based on the stream power approach in which the work done in moving sediment is the product of the power available to move sediment and the efficiency of the system. Although, Ackers and White's technique is a total load concept, their derivation does make a distinction between bedload and suspended load. But this distinction is made, not on the basis of position in the water column, Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 32 but rather on dimensionless grain size, Dgr, where D g r = D g(s -1) 1/3 (2.32) where v = kinematic viscosity of fluid s = specific gravity of grains D = typical grain diameter Coarse material, Dgr > 60, is considered to be moved as bedload and fine material,Dgr < 1, is moved as suspended load. An important feature of the method, and one which makes it almost unique, is the inclusion of a criterion for the beginning of sediment motion which is expressed by the sediment mobility number, Fgr as \/gD(s - 1) -, 1-n V 5.66 (2.33) where V* = shear velocity n = transition exponent depending on sediment size a = a numerical constant taken as 11 d = mean flow depth V = mean flow velocity For coarse sediment n = 0, and for fine sediment n = 1 and intermediate sizes of sediment may take values between 0 and 1. Modifications of this approach have been carried out by Willis (1978) and Swart et al (1980) for use in the coastal environment. The most notable modification made Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 33 is the change of the original shear stress relationship of Ackers and White (1973). Making vector addition of waves and currents velocities and assuming that the uni-directional Chezy friction factor can be applied to the unidirectional term only and the Jonnson wave friction factor fw/2 to the oscillatory term only and that /„, is independent of wave phase, and making a further modification to compensate for the differences in wave and current thresholds by adding an empirical coefficient, Wc, to the wave term, Willis (1978) obtained an expression for the combined bed shear stress due to waves and currents as V 2 . , \A/2fw_ij2 (2.34) The total power per unit bed area available to move sediment under waves and cur-rents was defined as Pw = P ^ V + W2 C g ^ U 2 (2.35) In equations (2.34) and (2.35), the first term is that for currents only, as in Ackers and White method, with the second term adding the wave effect. In the modification of Swart et al (1980), they defined an instantaneos sediment mobility number, Fgr(t), for waves and currents given as F 8 , ( , ) ^ > y - " (,36) where t denotes time variation and the subscripts fg and cg refers to fine grain and coarse grain sediments. Instead of adapting the fine and coarse grain components of the shear stress individually by integrating each separately with respect to time, Swart et al claimed that it is more logical to compute the average effect of the inclusion of waves on the mean mobility number by integrating the instantaneos mobility number with respect to time, Fwc = Mt) = i / q T Fgr(t)dt (2.37) Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 34 where Fwc is the mean sediment mobility number for combined wave and current action. The total sediment transport, mass flux per unit mass flow rate, X , is determined from the expression where X = C g rc m sD d g eg TcgV 1-n (2.38) C = coefficient in sediment transport function m = exponent in sediment transport function grc the critical value of the sediment mobility number n = transition exponent depending on sediment size eg power per unit area available to move coarse sediment Pfg — power per unit area to move fine sediment In the case of the modified approaches adopted by Willis and Swart et al, V in the above equation represents the resultant velocity of the waves and currents. The results of these modifications compared with other reliable predictors for the longshore transport rate gave quite a good agreement Swart et al (1980). 2.5 Summary It is clearly seen from the above survey that currents in the nearshore zone are important in several respects. Of these, the longshore currents are responsible for the longshore transport of sand on beaches. Of all the equations formulated for the generation of these currents, those based on radiation stress (momentum flux) Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 35 concepts have the firmest theoretical basis (Bowen, 1969b; Longuet-Higgins, 1970a, 1970b; Thorton, 1970). The equation formulated by Longuet-Higgins (1970a), equa-tion (2.2), is basically the same as that deduced by Komar and Inman (1970), equation (2.4), based on the equivalence of two models for longshore sand transport. The survey also reveals that there are many different approaches in the formula-tion of reliable predictors for the longshore transport rate. Swart et al (1980) and Bruno et al (1980) have compared the computed longshore transport rates from some of these predictors to field data. They concluded that most of these equations agree well with the data, and are fairly consistent with each other when the same method of predicting the input variables is used. Also, from all the predictive equations for the longshore sediment transport rate presented, it seems that no account of sediment and beach characteristics are taken into consideration. Since most of these relationships were based on beach sands, it is assumed that perhaps there was not enough variation in the grain size for its effect to be reflected in the predictive equations presented. However, it will be shown from the present study that the predominant beach slope and sediment size have some influence on the longshore transport rate. Furthermore, it should be noted that any calculation of the longshore transport rate using any of the predictors presented is only an estimate of the potential long-shore transport rate. If sand on the beach is limited in quantity, then the calculated rates may indicate more sand transport than there is sand available. Similarly, if sand is abundant but the shore is covered with ice for 2 months, then the calculated transport rates must be adjusted accordingly. Chapter 2. A REVIEW OF CURRENT AND TRANSPORT MODELS 36 In conclusion, it seems that the wave energy flux type are easier to use than the detailed predictor type i n that they require less variables as input. Al though, the wave energy flux method is currently widely used, direct correlation of the longshore transport potential to a single parameter of the longshore wave energy flux places a l imitat ion on its use when the other parameters such as beach slope and sediment size vary widely. The estimation of the longshore transport potential by the wave energy model should therefore be considered as a qualitative indicator of the actual transport. Chapter 3 WAVE BREAKING CHARACTERISTICS 3.1 Introduction The action of waves is the principal cause of most shoreline changes. Without wave action on a coast most of the coastal engineering problems involving littoral processes would not occur. The stress caused by the breaking waves on a beach face induced by turbulence and velocity gradients moves sediment in the surf zone with each passing breaker crest. This sediment motion is both bedload and suspended load transport. Sediment in motion oscillates back and forth with each passing wave and moves alongshore with the average longshore currents whose velocities are normally too low on their own to move the sediment. Since the fluid motions at breaking cause most of the sediment transport in the surf zone, the amount of sediment transported is partly determined by the breaker type. The type of breaker is also important with respect to the amount of energy reflected from a slope and the elevation of runup on a slope. For breaking waves, the shapes have been classified into spilling, plunging, collapsing and surging with each type dominating for certain values of deepwater wave steepness, beach slope, and surf similarity parameter. 37 Chapter 3. WAVE BREAKING CHARACTERISTICS 38 It has been observed that spilling breakers differ little in motion from unbro-ken waves and generate less bottom turbulence and thus tend to be less effective in transporting sediment than plunging and collapsing breakers (Divoky et al, 1970; Galvin, 1972a; and Fairchild, 1972, 1977). Kana (1977, 1978); Inman et al (1980); and White and Inman (1989) also report that plunging breakers suspend significantly more sediment than do spilling breakers. This is intuitively expected from the dif-ferent character of the bottom stresses associated with the two breaker types. The expected greater vertical velocities and stresses induced by the plunging process and its associated overturning bores should lead one to expect a thicker moving bed layer under plunging breakers. Since the flow dynamics vary for different breaker types it is important to under-stand the conditions associated with each breaker type. 3.2 Breaker Type Classification The pioneering work of breaker type classification was done by Iversen (1952, 1953). Since then, studies by Galvin (1968, 1972a); Weggel (1972); Battjes (1974); and Holmes (1982) have led to considerable progress in breaker type classification. Galvin (1968), stated that the breaker type can be generally classified into one of four principal types; spilling, plunging, collapsing or surging. The physical appearance of each breaker type is shown in figure 3.1 and the description of each type as defined by Galvin (1968) is as follows:-1. Spilling :- white turbulent water appears at the crest preceded by a small jet of water. The turbulence "spills" down the face of the wave. The wave shape as Chapter 3. WAVE BREAKING CHARACTERISTICS 39 a whole remains symmetric. 2. Plunging :- The front of the wave face reaches a vertical and the crest forms a jet which plunges ahead of the wave. 3. Collapsing :- The front of the wave face reaches a vertical and the lower portion of the wave acts as a truncated plunging breaker. 4. Surging :- The wave stays relatively smooth as it moves up the beach, except for minor turbulence at the wave - shoreline interface. Turbulence is generated by bottom boundary shear. There is a continuous sequence of possible breaker types running from spilling to plunging to collapsing to surging, and the transition from one breaker type to another is gradual without distinct dividing lines. 3.3 Breaker Classification Using Breaking Indices The pioneering work of Iversen (1953) on the classification breakers provided a graph of the variation of the breaker characteristics with beach slope and breaker type index, H0/T2 as shown in figure 3.2. The parameter H0/T2 is equivalent to the deepwater wave steepness, H0/L0, because LQ = gT2/2w, and is plotted in figure 3.3. Iversen gives only the ranges of H0/T2 over which each of three breaker types are observed, breaker types for individual waves are not available. From figure 3.3, the transition values of the different breaker types in terms of the deepwater wave steepness on the different beach slopes tested are tabulated in table 3.1. Chapter 3. WAVE BREAKING CHARACTERISTICS 40 (d) SURGING Figure 3.1: Physical description of principal breaker types Chapter 3. WAVE BREAKING CHARACTERISTICS 41 Beach Slope -1:10 SURGING Beach Slope -1:20 Beach Slope -1:50 PLUNGING PLUNGING PLUNGING SPILLING SPILLING SPILLING 0.01 0.02 0.03 0.05 0.1 0.2 0.3 Deepwater Wave Steepness, Ho/T2 o.s Figure 3.2: Breaker classification using breaker type index H0/T2 (Iversen 1953) Beach Slope -1:10 SURGING PLUNGING Beach Slope -1:20 PLUNGING Beach Slope -1:50 PLUNGING SPILLING SPILLING SPILLING 0.001 0.01 0.1 Deepwater Wave Steepness, Ho/Lo Figure 3.3: Variation of breaker type with H0/L0 and beach slope (Iversen 1953) Chapter 3. WAVE BREAKING CHARACTERISTICS 42 Table 3.1: Transition values using deepwater wave steepness (Iversen (1953)) Slope Surging Plunging Spilling 0.020 HJL0 < 0.00195 0.00195 < H0/L0 < 0.027 HJL0 > 0.027 0.033 0.050 H0/L0 < 0.00390 0.00390 < H0/L0 < 0.0390 H0/L0 > 0.0390 0.100 Ho/L0 < 0.00781 0.00781 < H0/Lo < 0.0684 H0/L0 > 0.0684 Having recognized that the breaker type generally depends on wave steepness and beach slope, Galvin (1968) formulated a quantitative classification of breaking types based upon laboratory studies which demonstrated a dependence upon beach slope, m, wave period, T , deepwater wavelength, L0, and either deepwater or breaker height, HQ or Hb. He formulated offshore and inshore wave parameters from these variables. The offshore wave parameter is, H 0 / L 0 m 2 (3.1) and the inshore wave parameter is, H b / g m T 2 (3.2) Both wave parameters are empirically determined. Initially, Galvin plotted (H0/L0) and (Hb/gT2)1/2 for three beach slopes (figure 3.4). The regularity evident on figure 3.4 suggests that it should be possible to combine the data for separate slopes on this figure by including the slope in the classifying parameters, which resulted in the plotting of figures 3.5 and 3.6. Chapter 3. WAVE BREAKING CHARACTERISTICS A Spilling O Plunging • Collapsing • Surging ® Plunging affected by reflection 0.201 0.10L -•—g-QO O — 9 Jo -ocL 0.051 O o 03 J * I I I t 1 J ' 1 I > I t I I I ' I ' I i 10 - 4 10" 10 -2 10 Computed Deep-water Steepness, H 0 / L 0 0.201 cxJfco-J c s V 0.10 I C D > ^ ^ - 0 O — 0 ok-0.05 I O—OO-O oLs £r& 1 I I I 1 1— 0.02 0.04 0.06 0.08 0.10 0J2 Brioker Stetpncss, ( H b / g T z )' Figure 3.4: Breaker classification according to Galvin (1968) Chapter 3. WAVE BREAKING CHARACTERISTICS 44 A Spilling O Plunging # Collapsing • Surging <g) Plunging affected by reflection Surging - Collapsing Plunging Spilling 10' 10 - 3 I0" 2 10"' H0/(L0mz) I0< 10' Figure 3.5: Breaker type as a function offshore parameter (Galvin 1968) Figure 3.6: Breaker type as a function of onshore parameter (Galvin 1968) Chapter 3. WAVE BREAKING CHARACTERISTICS 45 Table 3.2: Garvin's transition values for offshore and inshore wave parameters Parameter Surge-Plunge Plunge-Spill Offshore (H0/L0m2) 0.09 4.8 Inshore (H^KgrnT2)) 0.003 0.068 It is evident that the dimensionless parameters in equations (3.1) and (3.2) are very similar, since L0 — gT2/2ir. The offshore parameter is more dependent on the slope, since there it enters as the square, whereas in the inshore parameter, slope is present only to the first power. The experimental study of Iversen (1953) had earlier shown that the breaker height increases with beach slope which is in agreement with Galvin's inshore parameter which is dependent on the beach slope to the first power. Table 3.2 shows the transition values for the two wave parameters due to Galvin. Using the offshore wave parameter in conjunction with the beach slopes on which Galvin performed his experiments, a table of transition values for the different breaker types in terms of the deepwater wave steepness is obtained as shown in table 3.3. Battjes (1974), in the development of a general surf similarity parameter for the breaking behaviour, also presented a basis for breaker classification using offshore and inshore wave parameters define respectively as, (3.3) where m = beach slope = tan 9 Chapter 3. WAVE BREAKING CHARACTERISTICS 46 Table 3.3: Transition values using deepwater wave steepness (Galvin (1968)) Slope Surging Plunging Spilling 0.020 HJL0 < 0.000036 0.000036 < H0/Lo < 0.000192 H0/L0 > 0.000192 0.033 Ho/L0 < 0.000100 0.000100 < H0/L0 < 0.00520 H0/L0 > 0.00520 0.050 H0/L0 < 0.000225 0.000225 < Ho/L0 < 0.01200 H0/L0 > 0.01200 0.100 H0jL0 < 0.000900 0.000900 < H0/L0 < 0.04800 H0/Lo > 0.04800 Table 3.4: Battjes' transition values for offshore and inshore wave parameters Parameter Surge-Plunge Plunge-Spill Offshore £ 0 3.3 0.5 Inshore 2.0 0.4 As noted by Battjes, 7 % = Co2 (3-5) -O ' -2\ However, the inshore parameter used by Galvin, (Hb/gmT ), is not equivalent to the parameter £(,. Battjes re-examined Galvin's data in terms of his classification and suggested the transition values for the two wave parameters as listed in table 3.4. A further work by Holmes (1982), on the classification of wave breaker types using the surf similarity parameter gives the ranges for the different breaker types includ-ing collapsing breakers which are not classified by Battjes. It is obvious that as this Chapter 3. WAVE BREAKING CHARACTERISTICS 47 Table 3.5: Values for the surf similarity parameter for breaker types Spilling Plunging Collapsing Surging £ < 0 . 4 0.4 < £ < 2.3 2.3 < £ < 3.2 3.2 < £ parameter increases the breaker type goes from spilling, plunging, collapsing to surg-ing. It is also interesting to note that these values do not significantly differ from the transition values suggested by Battjes in table 3.2. 3.4 Laboratory Studies of Breaker Types The main objective of these preliminary experiments was to observe the wave breaking processes in the test wave basin, namely to investigate the relationship be-tween the breaker type, beach slope, and deepwater wave steepness, and to compare the findings of these experiments with those of other researchers. The experiments were performed in a small model wave basin which is 12.2 meters long, 4.7 meters wide and 0.61 meters deep. Figure 3.7 shows the plan view of the experimental set up used in the investigation. The beach was rigid, wooden, and impermeable and sloped at 1 : 9. This slope was chosen to provide different breaker types and nearshore conditions. The waves were generated by a flap-type wave paddle capable of varying the wave height and period by varying the paddle excursion and period. Chapter 3. WAVE BREAKING CHARACTERISTICS 48 Table 3.6: Recorded wave data for wave breaker types on 1:9 slope Run Depth Period Incident Deepwater Ho/Lo Breaker # Wave Height Wave Height Type (cm) (sec.) (cm) (cm) 1 40.0 1.50 4.88 5.28 0.0150 Plunger 2 44.0 1.50 7.01 7.63 0.0217 Plunger 3 45.0 1.50 7.31 7.97 0.0227 Plunger 4 47.0 1.50 6.93 7.56 0.0215 Plunger 5 50.0 1.50 9.75 10.66 0.0303 Plunger 6 41.0 1.70 4.88 5.18 0.0115 Plunger 7 44.0 1.70 5.22 5.22 0.0116 Plunger 8 46.0 1.70 5.49 5.90 0.0131 Plunger 9 50.0 1.70 5.79 6.26 0.0139 Plunger 10 41.0 1.19 5.94 6.49 0.0293 Plunger 11 44.0 1.19 8.53 9.29 0.0420 Plunger 12 41.5 1.09 14.02 15.17 0.0818 Spilling 13 41.5 1.00 11.58 12.35 0.0791 Spilling 14 45.0 1.00 13.11 13.87 0.0888 Spilling Wave breaking conditions were varied by changing the depth of the constant depth section, the wave period, and, the wave height. The wave height was measured 3 me-ters from the toe of the beach. The breaker type was recorded using the definitions found in section 3.2. 3.4.1 Preliminary Experimental Results The results from the preliminary experiments for the wave breaking types on a 1 : 9 slope are shown in table 3.4 and figure 3.8. From the recorded wave data, the value of HQ/L0 was back calculated from the intermediate depth conditions using Chapter 3. WAVE BREAKING CHARACTERISTICS 49 linear wave theory. The laboratory observation of the breaker types were checked with both Galvin's (1968) offshore parameter and Battjes' (1974) surf similarity pa-rameter. The results show good agreement with the findings of other researchers as shown in figure 3.8. Walsh (1989) also tested the breaker type classification on a 1 on 15 impermeable beach slope and found reasonable agreement with other researchers. However, Weishar and Byrne (1978) report from their field data for wave breaker type classification that the breaking indices do not seem to be useful in predicting breaker types in the ocean environment. This might presumably be due to the fact that the conditions in a natural beach are complex and are affected by composite slopes, bottom friction, and varying permeability on the beach face. It should also be noted from figure 3.8 that there are differences in the values of the deepwater wave steepness by Iversen (1953) and Galvin (1968). Iversen (1953) measures wave heights, which are affected by reflection and secondary waves, to cal-culate H0/L0. Galvin (1968) eliminates the effects of backrush and secondary waves by calculating Ho/L0 using the displacement of the wave generator. Values of H0/L0 calculated by considering the generator displacement are larger than those calculated by considering measured wave heights. However, Iversen's values plot above Galvin's. 3.5 Breaker Height Prediction Many engineering applications entail the estimation wave breaker height the deep-water wave characteristics. Komar and Gaughan (1972) tested three sets of laboratory data and set of field data from the California coast (Scripps pier) against a formula-tion for breaker height prediction using linear wave theory combined with a similarity criterion for relative breaking depth, Hb/db. The resulting relationship, empirically Chapter 3. WAVE BREAKING CHARACTERISTICS 50 4.70 m 2.44 m 3.66 m 14.5° Toe of Slope Wave Probe 12.2 m Carriage for Wave Instrumentation Wave Profile Recorder Incident Wave Wave Filters Wave Generator Wave Absorber (Drawing is not to scale) Figure 3.7: Plan view of experimental set-up Chapter 3. WAVE BREAKING CHARACTERISTICS 51 0.1 0.08 H 0.06 0.04 0.02 Deepwater steepness, Ho/Lo 0.05 O Spilling Breaker * Plunging Breaker $ Plunge ii 0.15 0.1 Slope (m) Figure 3.8: Experimental results of breaker type studies 0.2 Chapter 3. WAVE BREAKING CHARACTERISTICS 52 fitted with data, was found to be an adequate predictor for breaker height over the range from small laboratory waves to the field data set consisting of breaker heights from 1.2 to 3.5 meters. In the derivation of the breaker height predictor, Komar and Gaughan (1972) made use of the conservation of energy flux, (ECn)b = (ECn)o (3.6) From linear wave theory, E b = l /8pgH 2 b (3.7) and C b = y&~b (3-8) Substitution of deepwater characteristics and equation (3.7) and (3.8) in equation (3.6) yield with reduction, Hg(gd b) 1 / 2 = g / M T H 2 ) (3.9) Using the similarity criterion^ = Hb/db gives the results, H b = k g ^ c m 2 ) 2 ' 5 (3.io) where k = 7;,/5(47r)2/5. A plot of Hb against ^^(Ti? 2 ) 2 / 5 should obtain a straight line whose slope is dependent upon the value of jb. Since fb is known to vary with beach slope, one might expect a separate straight line for each beach slope. However, Komar et al found that the one-fifth power of 7^  is involved so that the expected variations in 75 should not produce a very marked change in the line slope. A best fit comparison with the experimental and field data yielded k = 0.39. It is to be noted that the effects of wave refraction and frictional dissipation are not incorporated, Chapter 3. WAVE BREAKING CHARACTERISTICS 53 consequently, the fit to the deduced relationship should be viewed as a relationship between breaker height and "apparent" deepwater wave height with refraction and frictional effects ignored. By using the relationship L0 = gT2/2ir between the deepwater wave length L0 and the period, T, equation (3.9) can be modified to the dimensionless form H b / H ° = (H o 0/L o 6)V 5 ( 3 ' U ) which indicates that Hi, is a function of the deepwater wave steepness, HQ/L0. Labora-tory tests run to test the breaker prediction theory reveal that Komar and Gaughan's (1972) equation slightly overpredicts the breaker heights as shown in figure 3.9. The Shore Protection Manual (1984) also provides two graphs which demonstrate the relationship between the breaker type, Hb/db, and Ho/L0 (figures 3.10 and 3.11). Figure 3.10 shows Goda's (1970) empirically derived relationship between Hb/H0 and H0/L0 for several beach slopes. Figure 3.11 shows the relationship between db/Hb and Hb/gT2 for various slopes. The curves in figure 3.11 are given by k = b - (aHb/gT 2) ( 3 - 1 2 ) where a and b are function of the beach slope, m, and may be approximated by 43.75 (l - exp- 1 9 m) (3.13) Chapter 3. WAVE BREAKING CHARACTERISTICS 54 14 0 10 2 0 3 0 4 0 l / 5(T#o)2 / 5 c m Figure 3.9: Comparison of breaker height prediction theory with experimental results Chapter 3. WAVE BREAKING CHARACTERISTICS 3-0 i 55 H i 0.02 0.03 (offer Godo, 1970) Figure 3.10: Breaker height index versus deepwater wave steepness SPM (1984) 0.020 Figure 3.11: Non-dimensional depth at breaking versus breaker steepness SPM (1984) Chapter 3. WAVE BREAKING CHARACTERISTICS 56 3.6 Beach Permeability Effects Beach permeability is one of the major factors controlling the rates of both onshore-offshore and longshore sediment transport since the extent of uprush and backrush and fluid forces are affected by it, (Sleath, 1984; and Quick, 1989, 1990). If a beach is permeable, the backrush will be reduced since the uprush can infiltrate into the beach. If the beach is impermeable, the backrush will be strong since all the uprush must return as backrush. Surging breakers generally occur closer to the shoreline. Therefore, a surging wave on a permeable beach can be "forced" to plunge on an impermeable beach. Waves in the spill-plunge transition generally break fur-ther offshore than surging breakers and are less vulnerable to the effect of backrush. Thus the beach permeability influences the wave breaking conditions in the surf zone. Permeability of sands and gravels is known to be a function of sediment size. Hazen (1911) proposed a formula that relates the permeability of sediment to the square of 10% finer material, DIQ, since the drainage that occurs in a material is controlled by the particles that fill the pores. The formula is given as, K = 10D?0 (3.15) where K is the permeability in millimeters per second. Also, from experiments on sands, Krumbein and Monk (1942) find the permeability K in darcys to be given by the formula K = 760(D50)2 exp- 1" 3 1** (3.16) where D 5 0 is the median diameter in phi units and <r^  is measure of sorting. Thus from equations (3.15 ) and (3.16), the permeability increases with the square of both D\o and D50. These formulae confirm the common assertion that coarser sediments are more permeable than finer sediments. The sand size as indicated above either by Chapter 3. WAVE BREAKING CHARACTERISTICS 57 the 10% diameter or the median diameter of a sample, will vary along a beach profile, particularly for beaches with coarse composite sizes. Coarse sand beaches are much steeper than those of fine sand. Steep beaches allow wave energy to be absorbed over a relatively narrow zone, and they are therefore more mobile. The cause of decreasing slope with decreasing grain size is the variation of the percolation rate through the beach material, (King 1972). Coarse sand is very permeable and a large proportion of the advancing swash infiltrates into the beach, and the backrush is reduced correspondingly in volume. However, on a fine sand beach only a relatively small volume of swash is lost by percolation, owing to the much reduced permeability. 3.6.1 Modifications of forces due to permeability Permeability of the beach modifies the forces acting on the beach face. The force required to move water through a permeable soil layer is normally called the seepage force. The interfacial bed particles experience an upward force whenever the water flows out of the bed, and a downward force whenever the water flows into the bed. Sediment motion may be enhanced or hindered by this force. Sleath (1984) finds that this seepage has two effects. First, the seepage will mod-ify the flow in the boundary layer above the bed, which will change the shear stress exerted by the flow on the bed. In general, seepage out of the bed reduces the shear stress while seepage into the bed increases it. However, these effects may be reversed if the boundary layer is near the laminar-turbulent transition since the seepage into the flow may keep the flow laminar and vice versa. Secondly, seepage exerts a vertical force on the grains of sediment. Oldenziel and Brink (1974) earlier observe for steady Chapter 3. WAVE BREAKING CHARACTERISTICS 58 flow in channels that seepage into the bed decreases the rate of sediment transport while seepage out of the bed increased it. Gourlay (1980) finds that Wf is a very important sediment property which deter-mines the shape of the beach profiles formed in relatively large lighweight sediments. Wf has been described as the fluidizing velocity, that is, it is the overall upward veloc-ity of flow through a bed of sediment which just causes the sediment to be fluidized or to become unstable. It is related to the permeability K by the relationship w f = K i c (3.17) where i c = ( s - l ) ( l - e ) (3.18) is the critical hydraulic gradient at fluidization, and s and e respectively the specific gravity of sediment and void ratio. The condition for bed fluidization is dP P l - p pgdy p where pi is the density of the saturated bed material. (3.19) Gourlay (1980) also finds that there are significant differences between the imper-meable sand formed by 0.22mm marine sand and the permeable 1.55mm crushed coal beach he used for his experiments, with regard to the beach and surf zone parameters. These effects are almost certainly related to the differences in permeability of the two beaches. In general the values of the wave uprush and other similar parameters of impermeable beaches are greater than those of permeable beaches. Gourlay then finds that the quantity with the least difference is the breaker height. He observes that the breaker type is seen to influence the form of the relationship investigated. For instance, the maximum berm height occurs when surging breakers are just changing Chapter 3. WAVE BREAKING CHARACTERISTICS 59 to collapsing breakers. Gourlay also finds that both plunging and spilling breakers dissipate more energy offshore of the beach with a consequent reduction in wave up-rush which determines the berm-dune foot height. Quick (1989) reports that the wave-induced shear stress acting on a beach is dependent upon sediment size, the beach slope and beach permeability; and that the beach permeability increases the shoreward wave-induced stress and decreases the offshore stress on the sediment. Quick (1989) observes that the infiltrating water has its momentum destroyed as it enters the beach, and this loss of momentum must therefore create an additional shoreward stress on the sediment. Quick also finds that steady infiltration and exfiltration can be associated with the time-averaged wave set up on the beach, as defined by Longuet-Higgins and Stewart (1964). The exfiltration occurs just outside of the wave break point where the wave set down is a maximum. It therefore appears that infiltration is strongest near the upper part of the beach, but occurs to some extent inshore of the breakpoint, so that on the whole beach face there will be an increase in shoreward stress on sediment. 3.6.2 Wave energy loss due to permeabilty The beach permeability does not only modify the fluid forces acting on the sea bed particles but also affects the mechanical energy of the progressive waves. Pres-sure variation at the sea bottom will induce currents in a permeable layer, and these currents will in turn dissipate some of the mechanical energy of the waves, Put-nam (1949). This loss in energy is reflected as reduction in the wave height as the waves move shoreward (Putnam, 1949; Bretschneider et al, 1954; Gourlay, 1980; and Holmes, 1982). Chapter 3. WAVE BREAKING CHARACTERISTICS 60 3.7 Discussion The action of waves is the principal cause of most shoreline changes. Fluid motions at breaking cause most of the sediment transport in the surf zone, because the bottom stresses and turbulence at breaking suspend more bottom sediment. The amount of sediment transported is therefore partly determined by the breaker type. Divoky et al (1970), Galvin (1972a), Fairchild (1972, 1977), Inman et al (1980) and White et al (1989) report that plunging breakers significantly cause more sediment transport than do any other breaker type due to the greater vertical velocities and stresses induced by the plunging process. Thus the amount of energy dissipation which is the motivating force for sediment transport is partly dependent on the breaker type. Some differences exist between the findings of researchers in the determination of breaker indices for the classification of breaker types as seen in figure (3.8). The differences in the value of H0/Lo might be due the method of calculation. Galvin (1968) eliminates the effects of wave reflection and secondary waves by calculating H0/L0 using the displacement of the wave generator while Iversen (1953) measures wave heights which are affected by reflection and secondary waves. In spite of these differences, experimental results presented in this thesis show that a reasonable pre-diction of the breaker type can be made given the deepwater wave steepness and the beach slope of an impermeable beach. However, from the field study of Weishar and Byrne (1978), the breaking indices do not seem to be useful in predicting conditions in the ocean environment which are complex. The prediction of the breaker height can also be made by the use of the deepwa-ter wave characteristics. The predictions of the equation formulated by Komar and Gaughan (1972) agree very well with the experimental results presented in figure 3.9, Chapter 3. WAVE BREAKING CHARACTERISTICS 61 even though the theory does not take into account the effects of wave refraction and and frictional dissipation. According to Quick (1989, 1990), the beach permeability is one of the major fac-tors controlling the rates of both onshore-offshore and longshore sediment transport, since it influences the extent of wave uprush and backrush and fluid forces. The breaking condition is also influenced by the beach permeability. For example, it is possible that a surging wave on a permeable beach could be "forced" to plunge on an impermeable beach due to the high volume of backrush. The formulae presented for the computation of beach permeability show that the permeability increases with the square of a representative grain size confirming the common assertion that coarser sediments are more permeable than finer sediments. The beach permeability also modifies the forces acting on the beach face. Oldenziel and Brink (1974), and Sleath (1984) find that in channel flow, seepage into the bed decreased sediment transport while seepage out of the bed increased it. Gourlay (1980) also finds from his experi-ments that there are significant differences between the impermeable beach and the permeable beach with regard to the beach and surf zone parameters such as berm-dune foot height, wave uprush and wave breaker heights. In general, the values of the wave uprush and other similar parameters of impermeable beaches are greater than those of permeable beaches. Quick (1989) also finds that the wave-induced shear stress acting on beach sediments is dependent upon sediment size, the beach slope and beach permeability. Chapter 4 LABORATORY STUDIES OF LONGSHORE TRANSPORT 4.1 Introduction Laboratory studies, both model and general, have proven to be a useful tool with which to study hydraulic problems. The scale relationships in fixed-bed studies have been well developed and, in general, confidence can be placed in the results obtained. Unfortunately, scale effects present in movable-bed studies have not been as well de-lineated and results obtained from these studies must be examined and interpreted with care in prototype results. Scale effects are definitely present when sands of pro-totype size and density are used in laboratory model test. Water waves impinging obliquely on a sandy shore scour and suspend materials causing them to move in the direction of the longshore component of the wave energy flux. The longshore transport and direction are the result of the summation of the forces imposed on the beach by the sea, and the reaction of the beach to these forces; thus, they are good indicators of the physical phenomena present in the area. For this reason, the transport rate and direction are important to the design of shore protection, navigation and coastal erosion projects. In many of the problems faced by the coastal engineer, the rate and direction of the longshore transport are of prime 62 Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 63 importance in indicating the problem as well as the corrective measures. Reliable field data on longshore transport is required in the design and economic evaluation of these projects. However, data, usually of questionable accuracy, is available for a few coastal areas and also field data is expensive and difficult to get. Therefore, the present study has designed a program to obtain laboratory data which would define the basic relationships and which might be used with field data. Perhaps the most valuable contribution of laboratory studies has been to emphasize the complexity of the problem, although their contribution in providing data on the action of small waves on ordinary beach sands should not be overlooked. These data become mean-ingful when compared with field data. 4.2 Previous Studies of Longshore Transport Since it is generally agreed that most of the longshore transport is caused by the waves which approach the shore obliquely, several investigations, both field and model have been carried out in an attempt to correlate the magnitude and direction of the longshore transport with the direction and energy of the motivating waves. These investigations show in general that there is a strong correlation. 4.2.1 Laboratory Studies Many laboratory studies have been made in an effort to determine the factors affecting the longshore transport rate. Among the first of these was one by Krumbein (1944), indicating that the longshore transport rate, Q, varies with the deepwater wave steepness, increasing continuously as the deepwater wave steepness increases. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 64 However, in view of later studies, for example that of Saville (1950), it appears that Krumbein's apparently linear relation of the longshore transport with the deepwater steepness may have been a coincidence because the wave energy he chose fell in a straight line when related to wave steepness. Also, his definition of the longshore transport rate is at variance with that commonly accepted in that he defined it as the rate at which the waves remove sand from a feeder beach rather than the rate of sand moving along, or off, the downdrift end of the beach. Saville's laboratory study showed that, for the waves tested, the longshore transport rate was a maximum for a deepwater wave steepness on the order of 0.025, and that for deepwater wave steep-ness both larger and smaller than 0.025, the transport rate decreases. Saville's work also showed that the longshore transport rate increases with increasing wave energy. A subsequent study by Shay and Johnson (1951) confirmed Saville's conclusions and further indicated that the transport rate varies with the angle of wave approach, ct0, increasing as a0 increases up to an angle of 43 degrees and decreasing thereafter. In all the foregoing model tests an initial beach slope on the order of 1 on 10 was used and the longshore transport was measured after it was felt that the beach had essentially reached equilibrium with the impinging waves. A later laboratory study conducted by Sauvage and Vincent (1954) further con-firms Saville's findings with deepwater steepness. Other laboratory studies by Savage (1959, 1962); Inman and Bowen (1963) and Fairchild (1970) have shown that there is some correlation between the longshore component of the wave energy flux although with great scatter of data points. Savage (1962) determined the distribution of the transport and also observed that the predominant beach slope has an effect on the transport rate. For all these studies a short beach section has been used, usually with Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 65 provisions to add sand at the up drift to simulate long beaches in nature. Dalrymple and Dean (1972) develop a technique for simulating an infinitely long beach in the laboratory with the objective of eliminating end effects usually present with short straight beach sections. This technique involves the use of the spiral wave-maker generating waves in the center of a circular wave basin with a circumferential beach. The waves, which propagate away from the wavemaker in a spiral pattern, impinge on the beach everywhere at the same angle. No quantitative tests were made to determine the longshore transport rates. A rock groin placed across the beach and offshore, showed after two hours of testing, the usual downdrift scour and updrift de-position. However, the current attributable to this stirring action of the wavemaker was measured as 1.52 cm/s approximately 0.61 meters from the beach, a velocity which is much lower than the incipient motion velocity of the sand. The author of this thesis recently visited the Oregon State University coastal engineering laboratory where a large scale model of the spiral wavemaker is being built for investigating longshore processes. At the time of the visit, (January, 1991), no quantitative investigations have been made. 4.2.2 Field Studies Watts (1953) made a study of the longshore transport rate at South Lake Worth Inlet, Florida. The longshore transport rates were based on quantities of sand trans-ferred by a permanent sand bypassing plant on the north jetty of the inlet. The wave characteristics were measured with a pressure-type wave gage and wave directions were measured with a sighting bar that was mounted on a transit-like scale grad-uated in degrees. This study yielded four data points with a sediment diameter of Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 66 approximately 0.40mm. Using this data, Watt (1953) developed an expression for the rate of nearshore littoral movement as: Q = 240E° 9 (4.1) where Ea is the wave energy flux associated with the alongshore component of incident waves in millions of foot-pounds per day foot of beach for those waves producing only southward littoral transport and Q is the rate of sand transport in cubic yards per day. A subsequent study was made by Caldwell (1956) at Anaheim Bay, California. Dredge material from the entrance to the bay was placed on the downdrift shore and repeated surveys of this area were conducted as the material was transported in a southerly direction. The changes in volume were interpreted as longshore transport rates and the estimates of the longshore component of wave energy flux were based on two wave gages measurements and wave directions based on the use of hindcasts. This study provided a total of five data points with sediment diameter approximately 0.40mm. Making use of this data set Caldwell developed a correlation of littoral drift with the alongshore component of wave energy expressed as: Qi = 210E?"80 (4.2) where Ei is the intensity of the net alongshore wave energy flux in millions of foot-pounds per foot of beach per day and Qi is the average rate of net littoral transport in cubic yards per day. Other field studies have been carried out by Thorton (1969); Komar and Inman (1970); Fairchild (1972); White and Inman (1989) and Dean (1989) to investigate the relationship between the longshore transport rate and the longshore component of the wave energy flux. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 67 Thorton's (1969) study was carried out at Fernadina Beach, Florida where me-chanical sand traps were placed at discrete locations across the surf zone to measure the bedload distribution. The traps which were operated from a pier could be closed and the trapped material pumped up to the pier for measurement. Wave data (height and direction) were based on two wave gages located at the end of the pier. The study yielded a total of 14 data points with a sediment diameter of approximately 0.20mm. With the use of this data, Thorton (1969) developed a relationship between the long-shore transport rate and the longshore energy flux as: l| = 0.048P| (4.3) Although the data points were many, the correlation expressed by the above relation-ship is weak. Komar and Inman's (1970) study were based on sand tracer measurements at two separate sites, Silver Strand, California, and E l Moreno, Baja California. Measure-ments of wave energy fluxes were determined from an array of wave sensors. Sand transport volumes were determined over a fraction of a tidal cycle as the product of the width of the surf zone, the longshore displacement of the center of gravity of the tracer and the thickness of tracer movement. The average sediment size at Silver Strand was 0.18mm and 0.60mm at E l Moreno. Analyzing this data, Komar et al developed the most widely used correlation of the immersed weight transport rate // with the longshore energy flux at breaking, P; s expressed as: I, = KP,S (4.4) where K is a dimensionless proportionality coefficient with a value of 0.77. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 68 Fairchild's (1972) study measured the longshore transport rate for suspended sedi-ment at two pier sites, Ventnor, New Jersey and Nags Head, North Carolina. Average sampling time was three minutes. Median size at Ventnor ranged from 0.12mm to 0.15mm and averaged about 0.20mm in depths of 4 feet and less at Nags Head. Fairchild summarized his results in a series of scatter plots which relate suspended sediment concentration to wave height, water depth and sampling distance from an observed wave breaker line. The wide scatter resulted in no good correlation between longshore transport and the corresponding longshore wave energy flux. The nearshore sediment study by Dean (1989) included two total trap experi-ments conducted at Rudee Inlet, Virginia and Santa Barbara, California to evaluate the longshore transport relationship. The Santa Barbara trap consists of the spit formed in the lee of the Santa Barbara breakwater. At Rudee Inlet, the trap is inside a weir of a weir-type jetty with a crest elevation at about mean sea level. The average sediment size at Santa Barbara was 0.22mm and 0.30mm at Rudee Inlet. Using data for seven intersurvey periods at Santa Barbara, Dean (1989) obtained a relationship between the immersed weight sediment transport rate J; and the longshore wave en-ergy flux Pi with a K value of 1.23 while the data for five intersurvey periods at Rudee Inlet resulted in a K value of 0.94. Dean (1989) carried a second correlation on the Santa Barbara data to determine Kt, where I, = K.S x y (4.5) where K+ is a dimensional coefficient with units (m/s) and Ii and SXY having units of (N/m). The value of K„ from the correlation is 2.63 (m/s). Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 69 Although, sediment and beach characteristics were investigated in all these field tests, no certain relationship was found between the longshore transport rate and the sediment and beach characteristics. Even though, some correlation is made between the longshore transport rate and the longshore component of the wave energy flux, the number of data points used are generally few and of questionable accuracy and hence such a correlation might be of limited validity and places a severe limitation on the use of the wave energy flux model. A direct correlation of longshore transport to the single parameter of longshore wave energy flux may therefore not be practicable where the other parameters vary widely. 4.3 Experimental Design and Procedures The longshore wave power or energy flux relationship has been shown to be corre-lated with the longshore transport. However, this relationship has been demonstrated for seasonal or annual estimates, and only for sand beaches. It should be noted that the presently used relationship assumes that a given wave power will produce the same transport rate, independent of beach slope and sediment characteristics (size). The present work has therefore set out to examine the transport processes in more detail with the following specific objectives, 1. to investigate the influence of sediment size on the longshore transport rate, 2. to examine the cross-shore distribution of the transport rate on the beach face, 3. to vary the beach slope and investigate how the transport rate and how the cross-shore distribution are influenced. A model test beach was built in a wave basin 12.2 meters long, 4.7 meters wide and 0.61 meters deep. The waves were generated in a constant depth of water by Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 70 a flap-type wave paddle capable of varying the wave height and period. Tests were made with waves approaching a beach at a horizontal angle of 14.5 degrees to the direction of incident wave attack. The beach face, built in two layers of plywood on a subframe, had an initial slope of 1 on 9. Wave breaking conditions were varied by changing the depth of the constant depth section, the period, and the wave height. A wave height was selected which would produce active transport and this height was kept constant for all tests on the various slopes so that the wave breaking re-gion would be reasonably constant. Once the breaking line has been determined, the upper part of the plywood beach was cut along the breaker line, so that the upper part of the beach inshore of the breaker line could be adjusted to a range of steeper slopes. This design maintained constant beach approach conditions offshore of the wave breaking line, but gave the freedom to change the beach slope inshore of the breaking waves. The plywood beach face was roughened by painting the beach face and sprinkling on a layer of medium sand. This roughening eliminated any sliding of the sand layer during tests. At the downdrift end of the beach a compartmentalized sand trap was installed so that the longshore transport rate could be measured as a function of its cross-shore distribution. Beyond the trap, on its downdrift end, slots were cut in the plywood beach face so that water driven alongshore by the waves could be absorbed into the beach. This prevented the development of an offshore current which tended otherwise to deflect sediment offshore, thus giving a distorted measurement of the cross-shore distribution of the sand transport. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 71 The most important feature of this experimental design is that the beach geom-etry outside of the wave breaker line is kept exactly the same for all the tests. In addition, the incident wave attack is also kept constant in size and direction for all tests. The only changes that are made are inside the breaker line, where the beach slope and sediment characteristics are changed systematically. Even with this exper-imental design, the incident wave attack is still slightly influenced, because, as will be seen, the beach reflection changes because of the changes of beach slope inside the breaker line. However, it is probable that this design gives the most constant incident and breaking wave conditions that can be achieved. Figure 4.1 is a plan view typical of the test set-ups used, showing the wave gen-erator, wave absorbers, wave instrumentation and the test beach with the compart-mentalized sand trap at the downdrift end. Figure 4.2 also shows a section of the variable beach face slope of the model beach. Three types of test sands were used in the experiments to investigate the proba-ble influence of sediment size on the longshore transport rate. The test sands were uniformly sized sands and the mechanical analysis of the test sands by sieves gave median diameters of 0.5mm, 0.85mm and 2.0mm. The test sands were laid over the rigid impermeable wooden base of the beach with an average thickness of 20mm. The grain size distribution of the test sands is shown in Appendix A . Once the test facilities and conditions have been arranged, an important part of the tests is the measurement of the test variables present. The most important variables are, • the basic wave conditions • the beach slope Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 72 3.66 m 4.70 m Shoreline Sediment Collector Beach 14.5° Toe of Slope 12.2 m Wave Probe Carriage for Wave Instrumentation Wave Profile Recorder Wave Filters Wave Generator (Note: Not to scale) Figure 4.1: Plan view typical of the test set-ups Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 73 Region of Transport Measurement Break Point Variable Slope Beach Figure 4.2: Section of the variable beach face slope Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 74 • the water depth • the transport rates • the longshore current velocity along the beach. Table 4.1 outlines the experimental variables and the range of these variables tested. For each selected beach slope, test were carried out firstly, with no sand on the beach, Table 4.1: Range of experimental variables tested Experimental Variables Range of Variables Wave Period 1.0 - 1.50 seconds Water Depth 0.40 - 0.50 meters Beach Slope 1 : 9 - 1 : 3.6 Beach Material 0.50-2.0 millimeters Angle of Beach to Waves 14.5°, constant for all tests Test Duration 85 seconds, constant for all tests by generating long crested waves which travelled from the wave generator to the toe of the beach slope, to establish the basic wave condition, breaker type and to measure wave reflection. As waves continued over the beach they shoaled and were refracted before break-ing. When the waves broke, part of their energy was dissipated in turbulence, scouring and suspending sand, part was transformed into a longshore current and part was re-flected from the beach. The wave action scoured the test sand and forced it into Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 75 suspension, to be carried along the shore by the longshore currents in a carpet-like sheet flow. Also, the swash and backwash of the waves caused the sand to move onshore and offshore with a resultant slantwise movement along the shore. The test sands were uniformly spread on the beach with an average thickness of 20mm to maintain the longshore flow of sand towards the sand trap for the duration of the test and hence no attempt was made to feed sand to the updrift end of the beach. At the end of all runs, mechanical analysis was made of the sand deposited in the compartment of the sand trap which recorded the maximum local peak transport rate to determine the grain size distribution of the transported sands. Appendix A shows the grain size distribution of the original and transported sands. The test sands were remixed after each run to eliminate any effects of sorting from the previous runs. Video pictures of each run were made so that the breaking and transport processes which are quick and complex could be analyzed at a slower speed. 4.3.1 Calibration of Wave Probe The accurate determination of the basic wave conditions requires an efficient wave instrumentation set up. Wave height recordings using strip chart recorders and oscil-loscope were made regularly in the tests using capacitance-type wave probe which is traversed in the direction of the wave propagation in order to measure the maximum and minimum wave heights of the combined wave field denoted by Hmax and Hmin. The incident wave height H and reflection coefficient KT are then obtained from the formulae H max min (4.6) H (4.7) Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 76 Spot recordings were also made at the breaker point of the approaching waves to determine the wave breaker heights. The wave probe was calibrated by lowering it into still water in one inch increments over a range of 7 inches, and recording on the oscilloscope the change in resistance between the wires at the different positions. A plot of gage elevation (in feet above an arbitrary datum against the recorded deflection in centimeters (see Appendix B)), is linear and constant. The slope of this curve, in feet per centimeters, is the calibration constant used to obtain wave height. During the tests of the wave probe, an important possible source of experimental error was noted. The stylus of the recorder is held by a spring against the recording paper, and if the spring tension is too high, friction between the stylus and paper can cause large errors, particularly when low paper speeds and small deflection of the stylus permit sticking to occur. For this reason most wave probe data were taken at reasonably higher speed (2.54mm/s) and about two-fifth of full scale deflection, under which conditions the rapid relatively motion between stylus and paper minimizes the friction effect. Recording data near the edges of the paper was avoided when possible because the responses of the stylus in this region is slightly nonlinear. 4.3.2 Longshore Current Measurement Estimates of the longshore currents were made by the use of current meters. Lab-oratory data suggest that the longshore current takes a longshore distance of about ten surf widths to become fully developed, Galvin and Eagleson (1965). Although this criterion was not fully met, a reasonable data on the longshore current was ob-tained on each beach slope. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 77 The longshore velocity data from the current meters provide a cross-shore distri-bution of the velocity, v(x). In addition to the Eulerian currents obtained by the use of the current meters, dye was released into the water during experiments to obtain a Lagrangian picture of the current system. The dye was squirted into the wave uprush at the updrift end of by a plastic "squeeze" bottle. Actual clockings observed the travel time of the leading edge of the dye trace, and not the travel time of the dye-patch center. Since the leading edge of the dye trace was timed, the rates estimated are probably maximum rates for the wave conditions tested. However, these values were not used in any analysis because of the rather fast spreading of the dye by tur-bulence in the surf zone. The dispersion of the dye during the experiments showed the presence of a "forced" offshore flow at the downdrift end of the beach. This flow was however minimized by slots cut in the plywood beach face to absorb this flow and prevent sediment being carried offshore. Two predictor equations were used to compare the theoretical values with the measured rates. The first predictor equation is due to Komar and Inman (1970), which has been shown to be equivalent to the longshore velocity of Longuet-Higgins (1970) derived based on the radiation concepts. The second predictor is that developed by Galvin and Eagleson (1965), using the conservation of mass concepts. The predicted values are compared with the mean values obtained by the use of current meters. V, = 2.7U m sin #b c o s $b (4.8) V| = gmT sin 26^ = 2gmT sin0b cos 0b (4.9) Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 78 It should be noted that the measuring of accurate current velocities in the presence of a wave field, such as the surf zone, has long been a problem. This has greatly limited the quantitative nature of the results collected pertaining to sediment transport. The major problem encountered in the surf zone with the use of the conventional propeller-type current meter is a poor response to rapid acceleration due to the inertia of the blades and the inability to distinguish between the inshore flux and the longshore flux. 4.3.3 Efficiency of Sediment Collector Trapping the longshore material at the downdrift end of the beach was accom-plished with as slight an effect on the natural processes present in the study as possible by installing the compartmentalized sand trap flush with the beach face. The material moving along the beach was collected, dried and weighed to obtain the dry weights. These dry weights, along with the test duration were used to compute the longshore transport rates. Prior to the actual experimentation program, trial runs were made for all the test sands on all slopes to access the efficiency of the designed sediment collector. After several trial runs, a duration of 85 seconds was selected in which the compartments of the trap efficiently held the transported material without an overflow. Even on the steepest slope the compartments did not overflow even though some of them were full. The trap extended seaward from the point of maximum uprush covering a distance of 1.40 meters with each compartment having dimensions of 15.6 cm long and 10 cm wide and 12.7 cm deep. In these laboratory tests of longshore transport in a small wave basin, it is very Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 79 important to note that this behaviour can occur purely under bedload transport conditions since the general turbulence levels in a small wave basin are not sufficient to cause high level of suspension. The designed trap can be regarded as a "total trap" since sufficient time is allowed for any small amount of suspended material to settle into the trap. The primary advantage of a total trap experiment is the potential of accumulating large, and therefore easy to measure quantities of sediment and yield much greater confidence in total volumes transported and ultimately the sediment transport efficient. 4.4 Factors Affecting Transport Rate Studies The factors affecting the longshore transport rate in laboratory studies may be divided into three main categories namely, 1. the beach characteristics, 2. the wave characteristics, 3. the characteristics of the facilities. 4.4.1 The Beach Characteristics The characteristics of the beach that have predominant effect on the reaction of the beach to the imposed wave energy are the characteristics of the beach material, including its size, size distribution and specific gravity. These characteristics, in conjunction with the imposed wave conditions, determine the profile of the beach and, thus, the predominant beach slope. However, in this study, the beach slopes were imposed. Finer sands generally have a flatter equilibrium slope than coarser material. Consequently, a fixed slope of test beach can only be at equilibrium value Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 80 for a particular size of material and for a selected wave attack. A further characteristic of the beach is its permeability which is one of the major factors controlling the rates of both onshore-offshore and longshore transport. The beach permeability depends on the sediment size distribution and on the nature of the flow, whether laminar or turbulent. 4.4.2 The Wave Characteristics The effect of the active forces or the waves on the longshore transport rate varies with the energy level, the steepness, and the angle of approach of the waves. In general, all investigations including the present study have shown that the longshore transport rate increases as the energy level of the imposed wave increases. Saville (1950); (Sauvage and Vincent (1954)) also found that the longshore transport rate varies with wave steepness, increasing to a critical steepness and decreasing there-after. Most previous investigators also agree that the the longshore transport rate increases with an increase in the angle of inclination of the crests of the waves to the beach to a critical angle, beyond which the transport rate decreases with any further increase in the wave crest inclination. Since the fluid motions at breaking cause most of the sediment transport in the surf zone, the amount of sediment transport is partly determined by the breaker type. It has been observed that spilling breakers generate less bottom turbulence and thus tend to be less effective in transporting sediment than plunging and collapsing breakers (Galvin, 1972a; Fairchild, 1972, 1977). Kamphuis and Readshaw (1978) also state that the rate of littoral transport is closely related to the rate of energy dissipation in the breaker zone which in turn is dependent on the wave breaker type. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 81 4.4.3 The Characteristics of Facilities The facility characteristics are the factors that are not ordinarily present in nature, but are unavoidable when conducting a laboratory study. These facilities necessary in conducting laboratory studies can affect the transport rate, and care should be given to assure that they are properly arranged. One possible source of difficulty is the waves that are reflected from the laboratory beach. In nature these would be returned to an infinite ocean, however, in the lab-oratory they are re-reflected from the training walls and the generating faces of the wave generators. In the present study however, re-reflection was not a major concern since the duration of the test was short and therefore the effect of re-reflection was greatly minimized. Another problem in some studies is the differential refraction that occurs over the relatively deeper water created by the traps at the downdrift end of the beach. Because waves travel faster in relatively deeper water over the traps, they refract in such a manner that an updrift component of the wave energy is created in the area immediately updrift of the traps and near the still-water line. This pattern is undesirable because it does not allow the material that is moving.along the beach to properly feed into the traps. However, this effect was not observed in the present study as the trap used is relatively shallow. 4.5 Experimental Results The summary of the transport results is listed in tables 4.2 and 4.3, which reveal that the total transport rate for a given beach slope and wave attack is of the same order of magnitude for the different test sands. This would suggest that the transport Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 82 rate is reasonably independent of sediment size. However, a gross indication of size effect is revealed at the zones of maximum wave energy dissipation where measured local peak transport rates are higher for the coarser sands than for the fine sand. This is a clear indication that coarser sediments attract or extract more stress than for fine sand. Both the total transport rate and the local peak transport rate increase considerably Table 4.2: Summary of transport results for wave period 1.41sec Beach Slope Total Transport Rate gm/sec Local Peak Transport Rate gm/m/sec Fine (0.5mm) Coarse (2mm) Fine (0.5mm) Coarse (2mm) 1 : 9.0 73 78 164 202 1 : 5.2 120 139 243 366 1 : 3.6 208 189 317 408 Table 4.3: Summary of transport results for wave period 1.50sec Beach Slope Total Transport Rate gm/sec Local Peak Transport Rate gm/m/sec (0.5mm) (0.85mm) (2mm) (0.5mm) (0.85mm) (2mm) 1 : 9.0 90 98 97 171 219 233 1 : 5.2 121 124 138 241 265 353 1 : 3.6 178 178 171 323 378 384 with beach slope. This increase of total sediment transport and the local peak trans-port rates with beach slope can be clearly seen on figures 4.3, 4.4, 4.5, 4.6, 4.7 and 4.8. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 83 500 400 300 « 200 100 co E E 3 c o I £ V> 5 o> fo oc •c o a. CO c a Wave Breaker Line Coarse Sand B Fine Sand — © — Test Conditions Wave Period = 1.41 sec. Water Depth = 0.47m Running Time = 85sec. Seaward -0.5 Shoreward 1.5 U) "5 - 0 . 5 0 0.5 1 Distance to Breaker Line X , meters Figure 4.3: Measured longshore transport rate on 1 : 9 slope Saville (1950) and Kamphuis and Readshaw (1978) also report that the predominant beach slope does have an effect on the longshore transport rate. Figures 4.3, 4.4, 4.5, 4.6, 4.7 and 4.8 also show that the distribution of sediment transport in the cross-shore direction is significantly different for the different test sands for a given beach slope and wave attack. It could be seen that the higher peak rate of transport for the coarser sediment is compensated by the lower width of beach over which sediment transport is active, indicating that the stress must fall off more rapidly for the coarser sediment. Also, a further characteristics of coarser sand transport compared with medium and finer sands is that the peak transport always occurs lower down the beach, closer to the wave breaker line. However, the position of the peak transport for the different test sand sands moves closer to the breaker line with increasing beach slope indicating net offshore movement of sediment flow for the steeper beaches. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 84 Distance to Breaker Line X , meters Figure 4.4: Measured longshore transport rate on 1 : 5.2 slope Distance to Breaker Line X , meters Figure 4.5: Measured longshore transport rate on 1 : 3.6 slope Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 0 0.5 1 Distance to Breaker Line Xb, meters Figure 4.6: Measured longshore transport rate on 1 : 9 slope to E 3 c o *c CO 5 d> +^  (0 OC •c o a to e 15 500 400 300 Breaker Line * | Test Conditions i Wave Period = 1.50s — j Water Depth = 0.47m ! Running Time = 85.0s ! Breaker Type = Plunger ! o' — :' S 200 100 1.5 0.5mm Sand -0.5 0 0.5 1 Distance to Breaker Line Xb, meters Figure 4.7: Measured longshore transport rate on 1 : 5.2 slope Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 86 0 0.5 1 Distance to Breaker Line Xb, meters Figure 4.8: Measured longshore transport rate on 1 : 3.6 slope An explanation can be given to the above observation by giving consideration to the net cross-shore transport which is associated with each test slope and each sediment size since it is expected that the cross-shore sediment flux will influence the longshore transport. Finer sands have a flatter equilibrium slope than coarser mate-rial. Consequently, a fixed slope of test beach can only be at equilibrium value for a particular size of material and for a selected wave attack. Figures 4.1 and 4.4 show that the cross-shore distribution of longshore transport of the test sands when the beach slope at 6.3°. The approximately 20mm sand depth permitted some limited infiltration into the beach, so that equilibrium slopes would tend to be flatter than for a deeper and more permeable beach. However, some onshore transport depicted on the beach face as material build up shoreward of the plunge point was observed, especially for the coarser sand, indicating that the slope was approximately at equi-librium value for the fine sand, but less than equilibrium slope for the coarse. When Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 500 87 E E 400 cn c o "S 300 *c to • mmm Q 0) 200 to oc •c Q. 100 CO c CO -0.5 Seaward Wave Breaker Line Wave period = 1.41 sec. Water Depth = 0.47m 1 in 9.0 1 in 5.2 1 in 3.6 Shoreward 0 0.5 1 Distance to Breaker Line X , meters Figure 4.9: The effect of beach slope on fine sand transport 500 co 1)400 c o "§ 300 CO Q O 200 to DC O 100 u. CO c CO -0.5 Seaward Wave Breaker Line 1 in 9.0 Wave Period = 1.41 sec. Water Depth = 0.47m 1 in 5.2 1 in 3.6 —> • <» \ • \ ' *. \ • % • X 0 0.5 1 Distance to Breaker Line X , meters Figure 4.10: The effect of beach slope on coarse sand transport Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 88 500 -0.5 0 0.5 1 1.5 Distance to Breaker Line Xb, meters Figure 4.11: The effect of beach slope on fine sand transport 500 -0.5 0 0.5 1 1.5 Distance to Breaker Line Xb, meters Figure 4.12: The effect of beach slope on medium sand transport Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 89 500 -0.5 0 0.5 1 1.5 Distance to Breaker Line Xb, meters Figure 4.13: The effect of beach slope on coarse sand transport the beach slope was increased to 10.9°, a net offshore was observed for the different test sands, indicating that the equilibrium slope for the coarser sands lay between these two limits of 6.3° and 10.9°. Comparing the longshore transport measurement for the 10.9° slope as shown on figures 4.4 and 4.7, it can be seen that the peak transport position for the test sands has moved further offshore which is consistent with the net offshore movement of sediment for the steeper beach. The strongest net offshore movement of sediment is observed on the steepest slope of 15.5" as shown on figures 4.5 and 4.8. It is interesting to note that the region of maximum transport has moved in the offshore direction to just inside the breaker zone, but the transport is now significant even slightly outside the breaker line es-pecially for the finer sand. On this slope it appears that the longshore flow is being carried slightly offshore of the breaker zone by the strong beach backflow caused by Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 90 the steep beach. It should also be noted that for this steeper beach, it is the fine sand transport which is slightly greater than the coarser sand transport. This increase in fine sand is reasonable because the offshore transport of fine sand would be very high on a beach so much steeper than the equilibrium value for such a beach. Figures 4.9, 4.10, 4.11, 4.12 and 4.13 clearly describes the change of position of the peak transports for the different test sands. However, it is observed that the position of the peak transport for the test sands on figures 4.10 and 4.13 occurs on the same place on the steepest slope. The linear relationship between the total transport rate and beach slope is well depicted in figures 4.14 and 4.15. This observation reveals that the longshore trans-port potential cannot be correlated to a single parameter of alongshore wave energy flux. The predominant equilibrium beach does have an influence on the longshore transport potential. Another sediment size effect investigated is the mechanical analysis of the material transported in the compartment of the sand trap which recorded the maximum local peak transport. The grain size analysis of the transported sand shows no appreciable change in grading from the original sand initially supplied to the beach (see Appendix A) . This material represents the longshore transport by the wave-induced longshore current. Hence there was no differential grading of sand longitudinally along the beach regardless of the incident waves. This would seem to mean that the size of the particles within the mixture might probably not influence the rate of transport, but that all particles are transported at the same rate along the beach. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 91 250 E B 200 o CC o Q. </> c (0 c © E "D O « o 150 100 50 Coarse Sand Fine Sand Test Conditions Wave Period = 1.41 sec. Water Depth = 0.47m Incident Wave Height = 6.60cm I I I I I I 0 0.1 0.2 Beach Slope Figure 4.14: Measured longshore transport rate as a function of beach slope 0.3 250 200 150 (0 E © oc •c o Q. (A C (0 •£ 100 © E •o © o 0.5mm Test Sand 0.85mm Test Sand • 2.0mm Test Sand • Test Conditions Wave Period = 1.50sec. Water Depth = 0.47m Incident Wave Height = 6.93cm • -r' 50 g I I i I i i i , , I i , i i , 0 0.1 0.2 0.3 Beach Slope Figure 4.15: Measured longshore transport rate as a function of beach slope Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 92 0.3 0.25 75 0.2 o> o £ 0) S 0.15 c o »mmm TS £ 0.1 « E 0.05 6 Wave Period = 1.41 sec. Water Depth = 0.47m 8 10 12 Beach Slope (degrees) 14 16 Figure 4.16: Reflection coefficient as a function of beach slope For each beach slope the reflection coefficient was measured both with and without sediment. Within experimental accuracy there was no difference in reflection with or without sediment. The results of the reflection coefficient are listed in Appendix C. Figures 4.16 and 4.17 clearly show that steeper beaches are more reflective. This is significant because the beach slope is only changed inside the breaking zone, so that the beach geometry outside of the breaking zone is unchanged. Therefore any changes in reflection which occurs must presumably be caused by the backrush of water from the beach face and any influence this may have on wave breaking. The steeper beaches are consequently subjected to a slightly lower wave attack because part of the wave energy is reflected. In spite of this lower wave attack, the sediment transport is highest on the steepest slope. This means that the imposed beach slope is a predominant factor controlling the transport process. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 93 0.3 0.25 .SJ 0.2 u *: o S 0.15 o ts c 0.1 OC 0.05 Wave Period = 1.50sec. Water Depth = 0.47m 4 8 12 16 Beach Slope (degrees) Figure 4.17: Reflection coefficient as a function of beach slope The results of the velocity measurements on the different beach slopes for the wave period of 1.41 seconds are listed in Appendix E. The longshore current velocity data provide a cross-shore distribution of the velocity, v(x). The longshore current distribution shown in figures 4.18, 4.19 and 4.20 for this wave condition exhibits the classical quasi-parabolic distribution with maximum velocities recorded shoreward of the observed breaker line. The most significant observation is that the average longshore current velocities on the different beach slopes are of the same order of magnitude suggesting that the long-shore current velocity is reasonably independent of beach slope. This result agrees with the findings of Komar and Inman (1970) and Komar (1979) that the longshore current velocity is independent of beach slope. Predictions were made of the longshore current velocity using equation (4.3) developed by Komar and Inman (1970) which has been shown to be equivalent to the longshore velocity of Longuet-Higgins (1970) Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 94 -0.5 0 0.5 1 Distance to Breaker Line , meters Figure 4.18: Cross-shore distribution of longshore velocity on 1:9 slope 0.5 «> "g 0.4 £ o o g 0.3 c 2 O 0.2 2 o CO O) 0.1 Breaker Line Av«. Velocity = 0.198 m/s Max. Velocity « 0.297 ml* | ] | / j \ i " • Shoreward • ~m%t— Seaward | i i i i i i i ' ' i i i i i i i ' ' ' -0.5 0 0.5 1 Distance to Breaker Line, meters 1.5 Figure 4.19: Cross-shore distribution of longshore velocity on 1:5.2 slope Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 95 0 0.5 1 Distance to Breaker Line, meters Figure 4.20: Cross-shore distribution of longshore velocity on 1:3.6 slope derived based on the radiation stress concepts. The results of this prediction with those equation (4.4) developed by Galvin and Eagleson (1965) using conservation of mass concept, are listed in table 4.4 alongside with the average experimental current velocities which shows that the predictions by Komar et al's longshore velocity pre-dictor agree reasonably well with the experimental measurements. From the above results, the slope dependency of the longshore current as described by Galvin et al is questionable. Observations made during and after the tests revealed several important charac-teristics of the longshore process. A process most apparent in the shaping of the tests sands on the beach face by the wave action throughout this series of tests has been the hydraulic sorting of the original bed material by the wave action. For the majority of these tests, the water depths, wave characteristics and bed materials were Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 96 Table 4.4: Comparison of measured and predicted current velocities Beach Slope Measured Average Velocity (m/sec) Predicted Average m/sec Velocity Equation (4.3) Equation (4.4) 1 : 9.0 0.185 0.23 1.47 1 : 5.2 0.198 0.23 2.54 1 : 3.6 0.212 0.23 3.67 such that some bed movement could be observed in the greatest depth at the seaward toe of the beach profile. The earliest manifestation of bed movement is the rapid formation of a ripple pattern over the bottom, seaward of the wave breaking zone. These ripples formed almost immediately (after the passage of very few waves) near the breaking zone and spread to greater depths. Wi th original test sands of median size 0.5 and 0.85mm, bottom ripples formed. However, with the original test sand of 2mm median size, no bottom ripples were ever formed with any of the test waves. For all test sands, observations also showed the formation of longshore bars which were generally formed some few centimeters outside the breaker line. The relatively short duration of the tests and the formation of these bed forms revealed that the level of turbulence is relatively high so that the fully rough turbulent flow conditions are achieved in these laboratory tests. Under these conditions, the magnitude of the Reynolds Number is large. Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 97 4.6 Discussion Several important results have been presented in this chapter from the small scale experimentation program of the longshore transport rate. The experimental findings reveal that the total transport rate for a given beach slope and wave attack is of the same order of magnitude for the different test sands tested. However, a gross indication of size effect is revealed at the zones of maximum wave energy dissipation where measured local peak transport rates are higher for the coarser sands than for the fine sand. This is a clear indication that coarser sediment attracts more wave stress than fine sediments. The total transport rate increase with increasing beach slope and this increase is approximately linear. Saville (1950) and (Kamphuis and Readshaw, 1978) also report that the beach slope influences the longshore transport. No relationship was however reported between the beach slope and the longshore transport rate by both authors. The distribution of sediment in the cross-shore direction is significantly different for the different test sands. The coarser sediments have higher peak rate of transport and this is compensated for by the lower width of beach over which the transport is active, indicating that the stress must fall off more rapidly for the coarser sediment. The areas under transport distribution curves (figures 4.3, 4.4, 4.5, 4.6, 4.7 and 4.8) multiplied by the width of the surf zone represent the measured values of the total longshore transport rates. Consideration must be given to the net cross-shore transport which is associated with each of test slope and each sediment size. As noted, finer sands have flatter equi-librium slope. Consequently a fixed slope of test beach can only be at the equilibrium value for a particular size of material and for a selected wave attack. The increase Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 98 of beach slope significantly influences the cross-shore distribution of the longshore transport. Net offshore transport is observed for all test sands on the steeper slopes of 1 on 5.2 and 1 on 3.6. However, the strongest net offshore movement of sediment is observed on the steepest slope of 1 on 3.6. This relatively steep slope might be considered unrealistically steep. Such a steep beach might represent a man-made protective structure. On such a beach, there would be very strong offshore sediment flow for both fine and coarser sands. The mechanical analysis of the material transported along the beach and deposited in the compartment of the sand trap which recorded the maximum local peak trans-port at the downdrift end of the beach show no appreciable change in grading from the original sand initially supplied to the the beach (see Appendix A) . Hence there was no differential grading of sand longitudinally along the beach regardless of the incident waves. This would seem to indicate that the size of the particles within the mixture might probably not influence the rate of transport, but that all particles are transported at the same rate along the beach. The findings of Krumbein (1944) and Saville (1950) corroborate this conclusion. The reflection coefficient is found to increase with the beach slope. This means that the steeper beaches are subjected to a slightly lower wave attack. However, in spite of this lower wave attack, the sediment transport is highest on the steepest beach. This means that the beach slope is a predominant factor influencing the long-shore transport process. The most significant point to note is that larger reflection from the steeper beach is caused solely by an increase of beach slope inside of the wave breaking region. This clearly demonstrates that processes on the beach face influence the wave breaking and it is probably the larger backrush of water from the Chapter 4. LABORATORY STUDIES OF LONGSHORE TRANSPORT 99 breaking waves which produce this higher reflection. The results of the longshore current velocity measurements also reveal an impor-tant finding, that the longshore velocity is reasonably independent of beach slope. The predictions of the longshore velocity using the equation developed by Komar et al (1970) which has been shown to be equivalent to the longshore velocity equation developed by Longuet-Higgins (1970), agree well with the experimental measure-ments. This agreement is very encouraging since the longshore transport model to be developed in the next chapter is partly based on the radiation stress concepts of Longuet-Higgins and Stewart (1964). Chapter 5 DEVELOPMENT OF THE LONGSHORE MODEL 5.1 Introduction The subject of longshore sediment transport has received a great deal of attention over several decades, from both experimental and theoretical points of view, because of the rapidly expanding field of coastal engineering and the need to find a reliable quantitative predictor of the longshore transport rate. The purpose of this thesis is to formulate a model to predict the longshore sediment transport rate by incorporating the influence of beach slope and sediment character-istics which have been neglected in all the previous predictors, and to compare the theoretical expression with some experimental results presented in chapter four. The absence of sediment and beach characteristics and the sole dependency of the long-shore sediment transport rate on a single wave parameter of longshore wave energy flux places some limitation on the validity of the wave power relationship for evalua-tion of the longshore transport rate. 100 er 5. DEVELOPMENT OF THE LONGSHORE MODEL 101 .2 Formulation of the Model .2.1 Model Assumptions The following assumptions are made in the development of the model. 1. Equilibrium slope conditions are assumed, where the equilibrium profile is de-fined by Gourlay (1980) as 'the profile shape which when subjected to a given wave condition dissipates and/or reflects all the wave energy reaching it in such a manner that no net transport of the beach/bottom sediment occurs anywhere along the profile.' In the laboratory the true state of equilibrium profile can be achieved under a constant one direction monochromatic wave. In the field, a true equilibrium profile cannot exist because of changing wave conditions. However, a dynamic equilibrium can be achieved if the time consideration is extended to a complete cycle of changing season. 2. Moderate slope conditions are assumed such that the beach slope tan /3 is equiv-alent to sin j3 since most natural beach slopes are mild. The range of natural beach slopes for most coastlands is 1 on 100 to 1 on 5. It should be noted that two of the beach slopes used in the experimental studies of the longshore transport fall within this range. 3. The breaking wave criterion is assumed such that the water depth at breaking is directly proportional to the wave height at breaking, 4. Linear wave theory conditions are assumed, 5. Impermeable beach conditions are assumed since the experimental studies were carried out on an impermeable beach. Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 102 Longuet-Higgins (1972) has pointed out that the use of the wave energy flux is in-correct although it fortuitously yields the correct grouping of terms. The theoretically sound approach is to calculate the flux towards the shore of momentum parallel to the coast, which is the radiation stress component, Sxy, defined by Longuet-Higgins and Stewart (1964). Considering the balance of momentum of the water between the breaker line and the shoreline, Longuet-Higgins (1970), developed an expression for the total lateral force per unit length of beach exerted by the waves on the water and sediment inside the breaker zone given by in which E0 is the deepwater incident wave energy density and 60 is the deepwater wave angle at breaking. It is interesting to note that the lateral thrust is a maximum for a given wave amplitude at infinity, when sin 20o = 1 or 0O = 45° and also constant for a given wave amplitude as well as independent of beach slope. Quick (1991), which is the original work of the longshore model to be presented and (Quick and Ametepe, 1991) find that the constancy and the independency of the lateral force of beach slope appears to be soundly based and any theory should therefore comply with this constraint. Since the development of the present longshore model is partly based on the radia-tion stress concepts, an attempt will be made to establish the fact the total longshore wave thrust is independent of beach slope and constant for a given wave attack within the constraints imposed on the model. Longuet-Higgins (1970) formulated equations for the local alongshore wave stress, r , per unit area exerted by the waves on the water in the surf zone and the mean longshore velocity, vi, given respectively as H L = -rE o sin20 o (5.1) r = -a /9ghssin0 (5.2) Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 103 v, = ^(eh)^ssin0 (5.3) in which h is the water depth, a is the ratio of wave height to water depth at breaking, C is the roughness coefficient, s is the beach slope, 6 is the wave angle of attack, and c is the wave speed assumed by linear wave theory to equal y/gh. Integration of equation (5.2) to obtain the total longshore wave force per unit length of beach can be carried out by assuming a linear dependency on both the wave height and beach slope given as L = H/si n /a (5.4) where L is the length dimension of the beach. Hence equation (5.2) becomes H L = L b j^Pg^mO-^ = ? « 2 P g H b - ^ s i n 0 (5-5) JO 4 sm p 8 sm p Applying the assumption of moderate beach slope conditions where s « sin /? we have H L = ^ g H ^ i n 0 (5.6) which is constant for a given wave attack and independent of beach slope and therefore satisfies the Longuet-Higgins' (1970) longshore wave thrust requirement developed us-ing the radiation stress concepts. The Bagnold's (1966) streampower approach, the wave-induced local alongshore stress and the mean longshore velocity formulated by Longuet-Higgins (1970) will now be used to develop a transport power relationship from which the longshore sediment transport potential can be estimated. For convenience, equations (5.2) and (5.3) are re-written respectively as r = \ a W h (5.7) Sir a , (sin0\ _ v, = T ^ ( — ) (5.8) Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 104 Bagnold (1966) postulates that the fluid power available to transport the sediment per unit width of beach, i , should be proportional to the product of the shear stress and flow velocity. Applying this criterion to the local alongshore wave stress, (equation 5.7) and mean longshore current velocity, (equation 5.8), we obtain the local longshore transport power per unit width of beach, p, as p = Tv, = ^ ^ ( g h ) 5 / V ( ^ ) 2 (5.9) where all the variables are as defined before. The fluid power expended therefore is due to the combined wave and current action. The term ^ s * g ^ , according to Snell's law would stay constant across the beach face, but for steeper beaches, where waves plunge and most of the uprush velocity is produced at this initial breakpoint, it is possible that little further refraction will occur. Bagnold (1966) has shown that the sediment transport rate per unit width of beach, i , is related to the available streampower, u>, by an efficiency factor e and a friction angle, <f>, as i tan <j> = eu (5.10) Using this idea, we have itan<£=Kp (5.11) where K is being defined as a dimensionless transport coefficient dependent on the na-ture of the breaking waves, wave reflection and the beach slope and p is the longshore wave power. The choice of such a transport coefficient factor is based on the fact that the wave breaking characteristics as discussed in chapter three, greatly deter-mine the amount of sediment transported. The relationship expressed by equations (5.10) and (5.11) is true for either bedload or suspended load, although for suspended load, Bagnold states that tan <f> be replaced by the ratio of the settling velocity to the Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 105 mean flow velocity. Hence for either bed or suspended load a proportionality between the transport rate per unit width, i, and the local longshore wave power, p, can be established as i oc Kp oc |^,(gh) 5 / 2s 2 ( ^ ) 2 (5.12) The total longshore transport of sediment, IT,, depends on the integration of equa-tion (5.13) across the beach width L. i t a n ^ K ^ ^ ( g h ) ^ s 2 ( ^ ) 2 (5.13) Applying the assumption of linear dependency on both the wave height and beach slope that is L = H/ sin 0, the integration yields, This gives l L t a n , = K | ^ H ^ ( ^ ) 2 (5.15) Finally application of the moderate slope condition where s « sin/3, the total long-shore model equation is written as 't = ^ ^ 3 ^ 5 / 2 ^ / 2 s^inA>| tan <j> 112a P g " b \ cb J § (5-16) For a given wave attack, Hb = 2ahb, and angle 6, the term in the square brackets is a constant, so that a first glance conclusion of the model should be that IT, should increase with beach slope, s but decrease with roughness factor C. The units of II is force/unit time. 5.3 Transport and Roughness Coefficients Before the predictions of the variation in the transport rate can be made, using the model developed (equation 5.16), further examination of the influence of the Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 106 transport and roughness coefficients must be made. The formulation of the transport coefficient is also presented. 5.3.1 Transport Coefficient K Bagnold (1966) proposed transport efficiency factors eb and es for bed and sus-pended load respectively based on the concept of a moving flow boundary and the available streampower. At the threshold of motion of the solids both eb and es are evidently zero. The solids are all stationary on the bed. As the available power is increased, however, more and more solids move over the bed as bedload and conse-quently more and more of the boundary shear stress is applied to the stationary bed indirectly in the form of solid-transmitted frictional stress via the moving bedload solids. Based on the resisting shear stress, T, the mean velocity of flow, u, and the transport velocity of the boundary, uc and the fluid shear stress r , Bagnold defines the maximum transport efficiency, ec, as follows ac = ^  = 4" (5.17) r u 1 + n where n is defined as 2 for fully turbulent flow and 1 for laminar flow. Thus the maximum transport efficiency is therefore 1/3 for fully turbulent flow and 1/2 for laminar flow. But only one-third of this is finally useful because two-third is locally dissipated in the carpet or sheet flow of moving sediment, as a result of the transfer of stress from fluid to solids. Hence the maximum transport bedload efficiency, eb is about 1/9 in a fully turbulent flow. He finally computed theoretical values for the transport efficiency factor for fully turbulent conditions for a range of arbitrary values of the mean flow velocity u and for a range of grain sizes D as shown in figure 5.1 and concluded that the bedload efficiency eb should range from 0.11 for large grains and large flows to 0.15 for very fine grains and low flow velocities. Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 107 0 - 1 1 1.5 2 3 4 5 6 8 10 MEAN FLOW VELOCITY OF FLUID (u). IN FEET PER SECOND Figure 5.1: Values of theoretical bedload efficiency factors (Bagnold (1966)) Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 108 It is very obvious that Bagnold's streamflow efficiency factors cannot be directly applied in the ocean environment where several factors including the nature of wave breaking characteristics, wave reflection, longshore current velocity and beach slope characteristics make flow conditions very complex. Hence an attempt to define a transport coefficient factor K taking into consideration most of these wave effects. White and Inman (1989) define a transport coefficient K based on their sand tracer experiments. White et al observe that the systematic range of the transport coefficient values indicates that transport coefficient is a variable dependent upon the nature of the breaking waves and the slopes of the beach. The most likely variable for parameterizing the breaker type and beach slope is a form of the dimensionless surf similarity parameter of Battjes (1974), here given in the form of a reflection coefficient, Krb, following Inman and Guza (1976) as 2gtan 2/3 L 0 t an 2 / ? K r b = —Tj — = ——Tj (5.18J where 0 is the slope of the beach, cr = 2TT/T is the radian frequency of the inci-dent waves. The term KTb is related to the Battjes' (1974) form & of the similarity parameter by the relation K r b = ^ (5-19) Laboratory studies by (Kamphuis and Readshaw, 1978) had earlier indicated a rela-tion between the transport coefficient and a form of the surf similarity parameter, £&, as K » 0 . 7 £ b for 0.4 < & < 1.4 (5.20) Expressing Kamphuis and Readshaw's equation in terms of K and KTi, White et al (1989) find that the results of their experiments become K = 2.16yK^b~ for 0.02 < K r b < 0.42 (5.21) Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 109 K = 0.74V/Krb for 0.48 < Krb < 0.62 (5.22) Equations (5.21) and (5.22) adequately describe the nature of the breaking waves, wave reflection and the beach slope and will therefore be used in the model developed for the evaluation of the longshore transport rate. It is observed from the above relationship that the transport coefficient K in-creases with increasing values of the surf similarity parameter Krb , implying that the transport is more efficient for plunging waves on steep beaches with narrow surf zones. This is probably because plunging waves produce more bottom stress and turbulence, and steeper beaches result in a higher concentration per unit area of surf zone. This observation agrees with the experimental findings presented in chapter four as the to-tal longshore transport increases with increasing beach steepness for plunging waves. It should be noted that the data used by (Kamphuis and Readshaw, 1978), (White and Inman, 1989) and the present experimental studies for the development of the transport coefficient relationship covers only spilling and plunging breakers, but not surging. In summary, the longshore transport coefficient K appears to be a variable which is related to easily measurable wave parameters such as a form of the surf similarity parameters KTb and and would therefore be a reliable parameter to be used in the evaluation of the longshore model developed. 5.3.2 Roughness Coefficient C The roughness coefficient C depends on bed material and forms which are closely related to the flow properties in the area which in turn vary with the breaker type. Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 110 Longuet-Higgins (1970) has pointed out that the roughness coefficient appears to de-pend on just two parameters. The first is the Reynolds Number Re — Ul/u where U denotes the horizontal velocity, 1 denotes a typical length scale and v is the kinematic viscosity. The boundary layer is fully developed rough turbulent, when the magnitude of the Reynolds Number is large. Under these conditions which normally occur when bed forms are present, the Reynolds Number has less effect and can be neglected (Henderson, 1966; Mogridge, 1974; and Noda, 1978). Under the above conditions the roughness coefficient is dependent only on a second parameter which is the ratio (1/k), where k denotes a length parameter characteristic of the surface roughness and 1 the horizontal excursion of water particle from its mean position. Further considerations must therefore be given to the behaviour of the longshore stress which acts on the sediment. This longshore stress is identical to the fluid stress defined by Longuet-Higgins (1970), namely where v is the longshore current, and C, is the friction coefficient, defined by the ratio of sediment size to wave motion excursion. From Prandtl's (1952) replotting of Nikuradse's classic work, Quick (1991) develops an approximate relation for the friction coefficient as where a0 is the wave motion excursion and d a representative sediment size. From equation (5.24), it is evident that the roughness coefficient increases with increasing sediment size. However, a rougher beach attracts a larger stress from the wave-induced fluid motion above the beach. With the usual assumption that most of the energy loss is attributable to the wave breaking process, and only a small part of the B y = C/»|Uorb|v (5.23) (5.24) Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 111 energy degradation depends on the beach stress, it can therefore be concluded that an increase in beach stress has a very small effect on the total energy degradation. Therefore, the beach stress induced by a given wave attack can vary considerably and yet still be consistent with equation (5.24). The local alongshore wave stress relationship, equation (5.3), developed by Longuet-Higgins (1970) can be re-written as The variational parameter in this expression is the water depth h and for a given wave angle of attack 0, and beach slope s, the local wave stress varies linearly with the water depth. From the above relation, it is evident that, for the longshore flow, the increase in stress on the beach will be greatest just inside the wave breaking point, but must decrease further inshore because the total longshore wave thrust must stay constant a condition which has already been established. Following from the above argument, Quick (1991) therefore states that this constancy of longshore wave thrust can be maintained by a simple, but realistic approximation which is sketched on figure 5.2, which depicts a simplified picture of the stress as a function of distance, x, across the beach face. Komar (1975) has earlier obtained a similar distribution of the hypothetical bottom stress exerted by the waves across the breaker and surf zones as shown in figure 5.3. In figure 5.2, ab represents a simplified linear longshore stress distribution associ-ated with a finer sediment size. The line, cd, represents the stress picture for a coarser sediment, but the integration of stress, represented by the area under the curves, will be the same for both stress distributions, that is, T x = fa2/>gh(ssin0) (5.25) ( * ) . \ = (Tx) f (5.26) Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 112 i Figure 5.2: Increase of beach stress as a function of sediment size [Quick, 1991] Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 113 Figure 5.3: The distribution of bottom stress T s , exerted by the waves on the beach face [Komar, 1975] Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 114 But (5.27) From equations (5.26) and (5.27) 0.25 lb (5.28) The beach steepness for the finer sediment Si is equal to h/lb, and it will effectively increase for the coarser sediment to s2, equal to h/ld- For the new beach steepness From the above relationship, it is therefore seen that the effect of larger sediment can be represented by artificially increasing the actual slope by the factor (c^ /^ i ) 0 ' 2 5 , which is the ratio of maximum longshore stresses on the beach. This increase of beach slope is therefore consistent with constant longshore thrust because equation It has been found that roughness coefficient generally varies from 0.01 to 0.10 in fully rough turbulent flow conditions (Henderson, 1966; and Streeter and Wylie, 1981). Losada (1986) explains that for spilling breakers, occurring on mild slopes with tan /? of order of magnitude 0.01, the bottom will be rippled or flat with low sediment load with a value of roughness coefficient of order O(10 - 2 ) . Plunging breakers on the other hand, will be more likely found on the intermediate slopes (tan /? of order O(10 - 1 )) with an increase sediment load and roughness coefficient of order O(10 _ 1 ) . Approximate estimates of the roughness coefficient are computed using the empirical relationship developed by Henderson (1966) given as (5.29) (5.6) shows that this thrust is independent of the slope. 1/3 (5.30) Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 115 5.4 Effect of Permeability on the Model The model assumes an impermeable beach conditions in its formulation. However, most natural beaches consist of sand and shingle and are therefore permeable. Due to their permeability, beaches change their slope and profile according to the intensity of the wave energy, and it is the ability of the beach to adjust itself in this way to the prevailing wave forces that makes it the most effective and efficient method of coast defence. Putnam (1949), Gourlay (1980) and Holmes (1982) find that the permeability affects the mechanical energy of the progressive waves due to the induced currents in a permeable layer which results in the dissipation of the wave energy. This loss in energy is reflected as a reduction in the wave height. The parameter which is therefore significantly affected in the model is the wave breaker height. However, Gourlay (1980) reports from his experiments on permeable and impermeable beaches that, the parameter with the least difference is the breaker height. Also, in the field, the wave breaker height can be fairly measured, and its value subsequently used in the model for the prediction of the longshore transport potential on permeable beaches. 5.5 Onshore-Offshore Transport The main aim of this thesis has been directed towards understanding the pro-cesses waves and beaches undergo when waves approach the shoreline obliquely. In actuality, the wave crests often travel perpendicular to the beach. The momentum flux approaching a beach can therefore be resolved into cross-shore and longshore components. Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 116 The cross-shore and longshore transport under oblique wave attack are linked because they are both produced by components of the wave-induced stresses on the beach face. The cross-shore analysis proposed by Quick (1989, 1990) for the cross-shore transport was a starting point for denning the beach slope, the sediment induced stress and the sediment flux for the onshore-offshore flow. The analysis was based on the time average behaviour of a control volume bounded by the wave breaking zone at the seaward end and the beach face. The analysis also considered the wave radiation stress, the wave set up, the shear stress on the beach face and the permeability. It was then used to define the stresses acting on the moving sediment, including the offshore weight component of the sediment. The end result was a relationship which defined beach slope, in terms of the incident wave attack, H , and a measure of the beach sediment size and grading expressed by D60 and Dw sediment sizes, namely, tan a oc A 1 / 2 ( D i o ) b / 2 ( D 6 o ) " a / 2 H " 1 / 4 (5.31) where A is the sediment friction coefficient and a and b are coefficients with estimated values of 1.4 and 0.8 but these values are subject to revision when more data becomes available. The success of the predictions equation (5.31) for the estimation of beach slope as a function of changing wave attack, or as a function of beach sediment com-position has led to the present study in which the effects of beach slope and sediment characteristics on the longshore transport processes are being investigated. Although, there is a great deal of interaction between the cross-shore and longshore transport processes, there are differences. This is because, even though produced by the same basic wave flux, the cross-shore sediment transport is strongly influenced by the beach slope, the offshore weight component of the moving sediment and the beach permeability, whereas the longshore transport is not only a function of the time varying wave stress component, but also of the longshore current induced by the Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 117 oblique wave attack. Thus the strength of the longshore transport will depend upon the product of the shear stresses and the current velocity throughout the surf zone, modified by changes in the shear stress caused by the beach permeability. 5.6 Discussion A model has been developed to estimate the longshore transport potential of sed-iment. The model formulation was based on the available wave power and radiation stress concepts. A major constraint on the development of this model is the con-dition that the longshore wave thrust be independent of beach slope and constant for a given wave attack. This condition appears to be soundly based and therefore any theory should comply with this constraint (Quick, 1991; and (Quick and Ame-tepe), 1991). Using the usual assumptions of linear dependency on both the wave height and beach slope and moderate beach slope conditions, it was shown that the Longuet-Higgins' (1970) longshore wave thrust requirement was satisfied within the constraints imposed on the model. The total model is described by equation (5.16). A longshore transport coefficient has also been developed based on a form of the dimensionless surf similarity parameter of Battjes (1974), given in the form of a reflec-tion coefficient. Although, the streampower approach has been used in the longshore model, the streamflow efficiency factors by Bagnold (1966) cannot be directly applied in the complex ocean environment where several factors including the nature of the breaking waves, wave stresses, wave reflection, longshore current velocity and beach characteristics influence the longshore transport process. The longshore transport coefficient, K , appears to be a suitable variable which is related to easily measurable wave parameters and can therefore be used as a reliable parameter in the evaluation of transport efficiency. It should be noted that the experimental results presented in Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 118 chapter four for both the total longshore transport rate and the reflection coefficient exhibit an approximate linear relationship with the beach slope lending a good sup-port for the transport coefficient. Before predictions of the variation in the transport rate can be made using the model developed, the roughness coefficient is examined since it is shown that this co-efficient depends on the bed material and forms which are closely related to the flow properties. Further consideration is therefore given to the behaviour of the longshore stress which acts on the sediment. The most important parameter which influences the roughness coefficient in a fully rough turbulent flow conditions is the ratio of sedi-ment size to wave motion excursion. It has been shown that the Reynolds Number has negligible effects on the roughness coefficient in fully rough turbulent flow (Henderson, 1966; Mogridge, 1974; and Noda, 1978). Using the fully rough turbulent conditions under the which the present experimental studies were carried, a relationship was developed for the roughness coefficient shown in equation (5.24). It is evident from this equation that the roughness coefficient increases with sediment size. However, it is known that rougher beaches attract or extract larger wave stress. The final result is that the effect of increased roughness coefficient due to increased sediment size is approximately balanced by the extraction of larger stress. The model shows that the longshore sediment transport is primarily dependent upon the magnitude and direction of wave characteristics, the beach slope and rough-ness coefficient. It should be noted that the most widely used wave energy flux model shows no dependency of beach and sediment characteristics on the longshore trans-port process. However, the present model shows that there is a dependency of beach slope on the longshore transport and that this dependency is approximately linear. It Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 119 appears though that the total longshore transport rate is reasonably independent of size. Based upon this model, simple relationship between beach slope, wave direction and magnitude and beach roughness are presented. Equation (5.12) shows that the local longshore transport is more dependent on the beach slope, being a function of slope squared. Several important conclusions based on these findings can be made about the longshore transport model developed. The first conclusion is that the beach slope does have an effect on the longshore sediment transport. Although, this finding is for an impermeable imposed beach slope, it is expected to hold true for permeable beaches with slight modifications. Steeper beaches formed by coarser sediments are more permeable than beaches formed by finer sediments. This means that the mod-ification will differ depending on sediment size. It appears from the development of the model that sediment size effect is fairly modelled. However, as discussed earlier, the total longshore transport is reasonably independent of sediment size. The local sediment transport per unit length of beach increase with the square of beach slope. This result should apply to the maximum measured transport rates. The results presented in chapter four show that the maximum local rate does not only increase with the square of the beach slope, but also with increased sediment size. A result which confirms that coarser sediments attract larger stress at the zone of maximum wave energy dissipation. The current approach of analyzing the evaluation of longshore transport potential, which is based on the wave power and radiation stress concepts answers to some extent the questions raised earlier in chapter one. It has now been found that the Chapter 5. DEVELOPMENT OF THE LONGSHORE MODEL 120 equilibrium beach slope does have an influence on the longshore transport potential. However, the total longshore transport appears to be independent of sediment size. It is also shown that the influence of the breaker type is important as reflected in the transport coefficient formulation. Although, the model answers some questions, it raises others. The major question raised is whether the present laboratory wave conditions represented by a single regular wave train with one period and height can be applied to a random wave condition as exhibited in the natural sea state. Chapter 6 TESTING OF THE LONGSHORE MODEL 6.1 Introduction In order to test the efficacy and reliability of the longshore transport model devel-oped in chapter five, comparison of the predictions of the model and the small scale experimental results presented in chapter four is made. The longshore transport rate is calculated using equation (5.16) developed for impermeable beach conditions. For each wave condition, the longshore transport rate is calculated for different beach slopes and sediment sizes. Since the beach slope is imposed and impermeable it is expected that the computed longshore transport is the largest possible for the wave condition. Comparison of the predictions of the model and those of the wave energy flux model which is currently used is also made. Finally, sensitivity analysis of the model is carried out to establish the extent of the influence of inaccuracies in the prime input variables (that is, wave height, wave angle, wave speed and roughness coefficient), on the longshore transport rate. 121 Chapter 6. TESTING OF THE LONGSHORE MODEL 122 6.2 Longshore Transport Model Predictions Predictions of the variations of the longshore transport is now made. For conve nience, the equation describing the total longshore model is repeated here, J L ^ o S .j/2H7/2 /sinAV1 an (£112 P g b V cb ) { (6-1) t  (j) For a given wave attack, the term in the square brackets is constant. However, the values of K which are dependent upon a form of the surf similarity parameter, Krb, changes with changing beach slopes. The values of K are calculated using the following equations, K = 2.16y/K^ for 0.02 < K r b < 0.42 (6.2) and L 0 tan 2 0 K rb = • ( 6- 3) 7 T H | j The values of Krb, calculated by the equation (6.3) are approximately equal to the values of the reflection coefficients obtained from the experimental results presented in chapter four. The similarity criterion, a, remains fairly constant for the different beach slope conditions since the wave breaking point is unchanged for a given wave attack on the different slopes. The term (^^^j > according to Snell's law would stay constant across the beach face, but for steeper beaches where waves plunge and most of the uprush velocity is produced at this initial break point, it is possible that little further refraction will occur. However, the term is being assumed to be constant on the steeper beaches. The wave speed at breaking c& is calculated using the shallow water approximation of the linear wave theory and is constant for a given wave condition. The wave breaking angle measured in the experimental study is 13.5° which remains constant for all the tests. For most sands, Bagnold (1966) observes that the friction angle (f> is 33° giving tan <f> to 0.649. Chapter 6. TESTING OF THE LONGSHORE MODEL 123 Since the value of K significantly varies with the beach slope, equation (6.1) could be re-written such that it is removed from the square brackets, 257T . a 3 p g 5/2 H 7/2^sinAy (6.4) 112 tan <f> where the term in the square brackets is represented by A in tables 6.1 and 6.2. As has been stated before, this term decreases slightly with beach slope due to increasing wave reflection. Before the predictions of the model are compared with the small scale experi-mental results, a re-examination of two earlier relationships, equations (5.12) and (5.16) is made. As stated earlier, for a given wave attack and angle, the only variable quantities are beach slope, s, and roughness coefficient, C. Suppose we compare the transport on a beach of specified slope but for two different sediment sizes, d\ and d2. To calculate the sediment transport for the coarser sediment, the effective beach (d \°'25 slope will by equation (5.29) increase by the factor ( , but the friction coeffi-cient C, will increase by exactly the same factor. The beach slope effect in equation (5.12) is represented by the square law. Consequently, equation (5.12) will increase 0.25 . This is an important conclusion, which indicates that the local peak (d \°'25 transport rate of the coarser sediment should increase by f ^  J compared with the finer sediment. Also, because the beach steepness has to be artificially increased, the implication is that transport on the steep beach will occur over a lesser length of the beach for the coarser sediment as depicted in figures 4.3, 4.4 and 4.5. Making a similar examination of equation (6.1) for the total longshore transport rate, it can be seen that the beach slope is represented by a linear law. In this case the (d \°'25 • increase in slope ( J for the coarser sediment will cancel with the friction factor. Consequently, equation (6.1) implies that the total sediment transport rate should be Chapter 6. TESTING OF THE LONGSHORE MODEL 124 Table 6.1: Predictions of transport model for wave period of 1.41sec. Beach Slope K A (N/sec) s/C Predicted Rates (N/sec) (gm/sec) 1 : 9.0 0.600 0.402 3.072 0.741 76 1 : 5.2 0.716 0.322 5.317 1.226 125 1 : 3.6 0.837 0.287 7.680 1.845 188 constant for any given slope and for any sediment size. However, the total transport rate for a given sediment size should increase with beach slope. With the findings from the re-examination of the model relationships, the friction coefficient C, will be averaged for the different sediment sizes so that the total model will predict a single transport rate for the different test sands. An average value of friction coefficient is computed from equation (5.30) and adopted for the model. Since the wave motion excursion a0 changes for different wave conditions the average roughness coefficient is recalculated. Data collected for two different wave conditions, for three different test sands on three different beach slope conditions is listed in Appendix D. The different wave conditions produced plunging waves. Tables 6.1 and 6.2 show the relevant terms in equation (6.4) which describes the total model as well as the predicted total transport rate. The total model values which are in units of force per unit time (N/sec) is divided by the acceleration due to gravity, g, and multiplied by 1000 to convert it to units of gram per second (gm/sec). Chapter 6. TESTING OF THE LONGSHORE MODEL 125 Table 6.2: Predictions of transport model for wave period of 1.50sec. Beach Slope K A (N/sec) s/C Predicted Rates II (N/sec) (gm/sec) 1 : 9.0 0.748 0.377 3.175 0.896 91 1 : 5.2 0.816 0.249 5.494 1.116 114 1 : 3.6 0.914 0.249 7.936 1.806 184 The results of the predictions are also plotted with the experimental results pre-sented in chapter four in figures 6.1 and 6.2. The predicted values and the measured values are of the same order of magnitude. This fact can be clearly seen on figures 6.1 and 6.2. Hence, it can be said that for the impermeable beach conditions, the model predictions agree reasonably well with the small scale experimental results. This will therefore suggest that the longshore transport rate evaluation using the wave power and radiation stress concepts is sound. A further comparison of predicted and measured local peak transport rate is made using equation (5.29) which indicates that the local peak transport rate of the coarser / j \ 0.25 sediment should increase by ( j 2 ") compared with the finer sediment. The pre-dicted and measured values of the local peak transport rates are listed in tables 6.3 and 6.4 for the different test sands based on the di size results. It can be seen from tables 6.3 and 6.4 that the predictions of the model for the local peak transport rates are also in good agreement with the small scale experimental results. One interesting observation to note is that predictions of the model are in close agreement for the low and medium slopes of 6.3° (1 : 9) and 10.9° (1 : 5.2) Chapter 6. TESTING OF THE LONGSHORE MODEL 126 250 O CD 200 CO E D) — 150 0 03 DC o Q.100 CO c 2 05 •+-» O r-50 Test Conditions Wave Period = 1.41 sec. Water Depth = 0.47m Roughness C = 0.036 Incident Wave Height = 6.60cm Measured rates are for coarse sand 0.1 0.2 0.3 Beach slope Figure 6.1: Comparison of model predictions and experimental results Chapter 6. TESTING OF THE LONGSHORE MODEL 127 250 Measured o a> 200 E Predicted • Test Conditions Wave Period = 1.50sec. Water Depth = 0.47m Roughness C = 0.035 Incident Wave Height = 6.93cm s DC o Q. 100 CO c 2 O 50 p-/ • • Measured rates are for coarse sand 0.1 0.2 Beach slope 0.3 Figure 6.2: Comparison of model predictions and experimental results Chapter 6. TESTING OF THE LONGSHORE MODEL 128 Table 6.3: Predicted and measured local peak transport rates for 1.41sec. wave period Beach Slope Predicted Peak Transport Rates (gm/m/sec) Measured Peak Transport Rates (gm/m/sec) d3 (2mm) di (0.5mm) d3 (2mm) 1 : 9.0 232 164 202 1 : 5.2 344 139 366 1 : 3.6 448 208 408 Table 6.4: Predicted and measured local peak transport rates for 1.50sec. wave period Beach Slope Predicted Peak Transport Rates (gm/m/sec) Measured Peak Transport Rates (gm/m/sec) d2 (0.85mm) d3 (2mm) di (0.5mm) d2 (0.85mm) d3 (2mm) 1 : 9.0 195 242 171 219 233 1 : 5.2 275 340 242 265 353 1 : 3.6 368 456 323 378 384 Chapter 6. TESTING OF THE LONGSHORE MODEL 129 respectively. The greatest error in prediction occurs on the steepest beach slope of 15.5° (1 : 3.6). This can be explained by using the equilibrium beach slope behaviour of sediments. It has already been pointed out that fine sands have flatter equilibrium slopes. On artificially imposed steep beaches therefore, the net offshore transport of the fine sand would be very high on a beach so much steeper than its equilibrium slope value. Since the predictions are based on the fine sand measurements, any deviation from its equilibrium slope will result in poor prediction of the model. It should be however recalled that one of the assumptions used in the development of the model is that beach slopes are mild. It is therefore expected that the predictions on steep beaches will be in some error. 6.3 Comparison of Model and Wave Flux Model The purpose of this section is to make comparison between predictions of the longshore model developed in chapter five and the predictions of the most widely used C E R C wave energy flux model given some arbitrary values of the wave, beach and sediment characteristics. The predictions of both models will then be compared with the experimental measurements of the total longshore transport rate taken on the three beach slopes under the wave conditions of wave period of 1.41 seconds. The equation of the wave energy flux model is repeated here for convenience. IL = K(ECn) b sin0 b cos0 b (6.5) where K = 0.77 and // has units of force per unit time (N/sec). The wave energy density at breaking, Eb and the wave speed at breaking, Cb are computed from the linear wave theory such that Eb = (6.6) C b = VST (6.7) Chapter 6. TESTING OF THE LONGSHORE MODEL 130 Cb is the wave speed evaluated in a depth equal to 1.28/fj. It should be noted that the coefficient 0.77 of equation (6.5) is based on data in terms of the root-mean-square wave breaker height. Since most wave data are available as significant heights, a correction factor is used to convert equation (6.5) such that K = 0.385 and equation (6.5) becomes l L =O.385(ECn)bsin0bcos0b (6.8) From the measured wave breaker heights and breaker angle and the computed wave energy density, Eb and speed, Cb at breaking, the total longshore transport rate predictions by the wave energy flux model are evaluated. These results are plotted in figure 6.3 alongside with the predictions of the longshore model and the experimental measurements. It is clearly observed from figure 6.3 that the wave energy flux model poorly predicts the total longshore transport rate on varying beach slopes. The wave flux model seems to be overestimating the longshore transport on low beach slopes and underestimating it on steeper beach slopes. This is a very significant observation. It should be noted that most beach slopes lie within the range of 1 in 100 to 1 in 10. It is therefore likely that the wave energy flux model will overestimate the longshore transport rate on these slopes. Kamphuis and Readshaw (1978) have reported that the wave energy flux model overpredicts the longshore transport rate. This observation supports the fact that a direct correlation of the longshore transport rate to the single parameter of the longshore wave energy flux may not be practicable where other parameters such as beach slope and sediment characteristics vary widely. Questions have been raised about the reliability of a number of data sets (discussed in section 4.2.2), which have been used for estimating the longshore transport rate. Probably, it is therefore timely to re-investigate some of these issues. Chapter 6. TESTING OF THE LONGSHORE MODEL 131 2 5 0 0 0.1 0.2 0.3 Beach Slope Figure 6.3: Comparison of predictions of both models and experimental measurements Chapter 6. TESTING OF THE LONGSHORE MODEL 132 6.4 Sensitivity of the Model Most of the experimental measurements made are likely to be inaccurate as in all laboratory studies. The wave height, H , which is a very significant parameter in the determination of every wave-induced process, was obtained from the wave probe recordings. It was noted that during the tests of the wave probe an important possi-ble source of experimental error in the measurement could occur. The stylus of the recorder is held by a spring against the recording paper, and if the tension is too high, friction between the stylus and paper can cause large errors, particularly when low paper speeds and small deflection of the stylus permit sticking to occur. Also, it is difficult to obtain highly accurate measurements of the breaker height in the surf zone. In spite of these shortcomings, reasonably accurate measurements of the wave heights were made. When the measured wave height itself, is only a few centime-ters, the sticking can cause significant error in the final measurement. This is also true for the measured wave angle, wave reflection, the wave speed and the transport coefficient which are predicted using the measured wave height. Also, the roughness coefficient C, is determined empirically. The validity of its value therefore depends on the reliability of the empirical relation. In the field, some of these input variables used for the prediction of the longshore transport rates are usually predicted, either because of difficulties in measuring the required parameters in the coastal environment, or because of the impracticability of measuring over a long time, say one year, to deduce seasonal trends from the input data. If the techniques used to predict the input variables, which are in turn to be used to predict sediment transport rates are not quite accurate, the computed long-shore transport rates could be in error by an appreciable margin. Chapter 6. TESTING OF THE LONGSHORE MODEL 133 It is therefore apparent that both in the laboratory and in the field, all input variables could be subject to possible inaccuracies, which could affect the longshore transport model predictions. For this reason a sensitivity analysis must be carried out to establish the extent of the influence of inaccuracies in the prime input vari-ables (wave height Hb, wave angle 9, wave speed cb, roughness coefficient C, and the transport coefficient K) on the longshore transport rate as predicted for the situation outlined in section 6.2 for the wave condition of period 1.41 seconds. This is done by varying the prime input variables within some range. These input variables were allowed to vary in the following ranges, namely, 0.025 < C < 0.055, 10° < 9° < 15°, 7.50cm <Hb< 12.5cm, 0.40 < K < 0.80 and 0.80m/s < Cb < 1.50m/s. Figures 6.4, 6.5, 6.6, 6.7 and 6.8 show the various responses of the model's sen-sitivity due to the variations of the prime input parameters. The sensitivity factor E , shown in these figures is defined as the ratio of the predicted longshore transport rate to the computed longshore transport rate due to the variations of the prime input parameters. From the results shown on these figures, the model appears to be sensitive to small inaccuracies in the measurements of the input variables. The figures also indicate that when the input variables correspond to their measured or predicted values, the sensitivity factor, E equals one. Obviously, the most significant variable is the wave breaker height. This seems reasonable since the wave height is a very significant parameter in the determination of every wave-induced process. Any inaccuracy in its measurement will either result in excessive underestimation of the longshore transport potential or its overprediction. The rather high sensitivity of the model due to small inaccuracies in the measurement of the wave breaker height might also be due to the seven-second power law of the breaker height in the model. It should also be remembered that there are difficulties in obtaining highly accurate Chapter 6. TESTING OF THE LONGSHORE MODEL 134 measurement in the surf zone. The least significant variable in the model appears to be the transport coefficient K . The contribution of the transport coefficient, although important is relatively small compared with the other prime input parameters. The sensitivity of the model with respect to the wave breaker height, Hb, and the wave speed at breaking, C j , can be combined since Cb is a function of Hb- In such a case, the term H^2 jC\ in the total longshore model equation, equation (6.1) will reduce to Hh^. This might result in a lesser response of the model's sensitivity due to the variation of the wave breaker height, Hb-Although, the model is sensitive to all the input variables, it is hoped that with the current refinements in instrumentation for measurement of wave characteristics in the ocean environment, fairly accurate measurements can be obtained leading to reasonably accurate prediction of the longshore transport model. er 6. TESTING OF THE LONGSHORE MODEL 135 Measured Breaker Height On Beach Slope 1:9 = 9.75 cm For Wave Period of 1.41 sec. i i i 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Variation of Breaker Height, H (cm) Figure 6.4: Sensitivity of longshore transport model due to variation in Hi,. Chapter 6. TESTING OF THE LONGSHORE MODEL 136 Roughness Coefficient = 0.038 Used For Prediction For Wave Period of 1.41 sec. 1.5 111 O (0 LL * 1 C CO 0.5 i i i i i i i i i i M i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 0.025 0.03 0.035 0.04 0.045 0.05 Variation of Roughness Coefficient, C Figure 6.5: Sensitivity of longshore transport model due to variation in C 0.055 Chapter 6. TESTING OF THE LONGSHORE MODEL 137 1.5 HI U CO •> mmmm +•> c CD 0.5 Measured Wave Angle = 13.5° Used For Prediction For Wave Period of 1.41 sec. minim i i i " i i i i i i ini 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 Variation of Breaker Angle, (degrees) Figure 6.6: Sensitivity of longshore transport model due to variation in 9b Chapter 6. TESTING OF THE LONGSHORE MODEL 138 1.75 1.5 W 1.25 O (0 LU & 1 > w I 0.75 0.5 0.25 Computed Wave Speed = 0.99m/s Used For Prediction For Wave Period of 1.41 sec. --- \ --- -- -0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Variation of Wave Speed, C. (m/s) Figure 6.7: Sensitivity of longshore transport model due to variation in Cb-Chapter 6. TESTING OF THE LONGSHORE MODEL 139 1.5 UJ l_ o Z (0 LL • • • • > • H i +•< '55 c 0) 0.5 Computed Transport Coefficient On Beach Slope 1 :9 •0.600 For Wave Period of 1.41 sec. 0 1111 i 111 i 11 11 111 11 11 i i f i i 11 i u 11 1111 u 111 111 111 1111 111 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Variation of Transport Coefficient, K Figure 6.8: Sensitivity of longshore transport model due to variation in K. Chapter 6. TESTING OF THE LONGSHORE MODEL 140 6.5 Discussion Re-examination of the relationships of the longshore transport model reveals that the total sediment transport rate should be constant for any given slope and for any sediment size. However, the total transport rate for a given sediment size should increase with beach slope. It is also noted from the re-examination of the model that the local peak transport rate of the coarser sediments should increase by (^/^I)0'25 compared with the finer sediments. Wi th the findings from the re-examination of the model relationships, predictions of the longshore transport were made of the total and local peak transport rates. Comparison of these predictions with the experimental results show reasonably good agreement. The predictions of the local peak transport rates show close agreement with the experimental results for the low and medium slopes of 6.3° (1 : 9) and 10.9° (1 :5.2). The greatest disagreement in prediction occurs on the steepest slope of 15.5° (1 : 3.6). This disagreement can be explained by using the equilibrium beach slope behaviour of sediments. It has already been pointed out that fine sands have natter equilibrium slopes. On imposed steep beaches therefore, the net offshore transport of fine sand would be very high on a beach so much steeper than its equilibrium slope. Since the predictions are based on the fine sand measurements, any deviation from its equilibrium slope value will result in poor prediction of the model. The comparison between the predictions of the longshore transport model and the wave energy flux model reveals that for low beach slopes, the wave energy flux model predicts very high transport rates although, it is expected that on such slopes breaker types will be predominantly spilling breakers and therefore less effective in transporting sediments, (Galvin, 1972; Fairchild, 1972, 1977; Losada et. al., 1986). Chapter 6. TESTING OF THE LONGSHORE MODEL 141 Kamphuis and Readshaw (1978) have also observed that the wave energy flux model overpredicts the longshore sediment transport. Questions have been earlier on raised about the reliability of a number of data sets which have been used for the develop-ment of relationships for estimating the longshore transport rate. From the findings of this study, probably, it is therefore timely to re-investigate some of these issues. The sensitivity analysis of the model shows that the model appears to be sensitive to small inaccuracies in the measurement of the input variables. The difficulty in obtaining highly accurate measurements in the surf zone will obviously affect the ac-curate prediction of the model. The wave height which is a very significant parameter in the determination of every wave-induced process is the most significant variable. Since the measured wave height is only a few centimeters itself, any inaccuracies can lead to significant errors in the final model prediction. Although, the model appears to be sensitive to all input variables, it is hoped that with the current refinements in instrumentation for measurement of wave characteristics in the ocean environment, fairly accurate measurements can be obtained leading to reasonably accurate predic-tion of the longshore transport model. Finally, it should be noted that these findings are for impermeable beach con-ditions. As discussed in chapter three, the beach permeability is one of the major factors controlling the rates of both onshore-offshore and longshore sediment trans-port since it modifies the fluid forces. From the experiments of Gourlay (1980), it has been found that there are significant differences between the impermeable and per-meable beaches, with regard to the beach and surf zone wave parameters. In general, the values of the parameters of the impermeable beaches are greater than those of permeable beaches. From the above findings therefore, for a similar wave conditions, Chapter 6. TESTING OF THE LONGSHORE MODEL 142 the total longshore transport of sediment occurring on an impermeable beach slope is expected to be greater than that occurring on a permeable beach slope. Although, the results of this model are promising, further experimentation on a larger scale should be done to confirm the validity of the predictions of the model. Chapter 7 SUMMARY AND CONCLUSIONS This thesis has presented a description of the events occurring in the nearshore zone and the development of a theoretical longshore transport model which predicts the longshore transport potential in terms of the wave direction and magnitude, beach and sediment characteristics. The following is a summary of the sequence of events leading to the further extension and testing of the model developed by Quick (1991), presented in chapters five and six. A brief review of the longshore and transport models currently in use was made before the development of the model. In this review, it was noted that currents in the nearshore zone are important in several respects. Of these, the longshore currents are responsible for the longshore transport of sand on beaches. Of all the equations formulated for the generation of the longshore currents, those based on radiation stress (momentum flux) concepts have the firmest theoretical basis. This approach is believed to be most suitable as momentum is conserved when waves break, whereas energy is not. The review also revealed that the equation rigorously formulated by Longuet-Higgins (1970), equation (2.2) is basically the same as that developed em-pirically by Komar et al (1970), equation (2.4). From the review of the transport models, it was found that none of them takes into consideration the beach and sediment characteristics in their formulation. The 143 Chapter 7. SUMMARY AND CONCLUSIONS 144 transport models were divided into two broad categories, namely the wave energy flux models and the so-called detailed predictor models which are based on various sediment transport relationships. It was found that most of the wave energy flux models have similar approaches yielding a general equation of the form: S, = AP,B (7.1) However, there was no such similarity in the formulation of the detailed predictor models. Even though predictions by the predictors were shown to be in agreement with some field data, there is still an apparent lack of agreement on the dominant mode of transport in the surf zone. Some workers have assumed that all sediment above the level of the at-rest bed is suspended load (Watts, 1953; Thorton, 1972; Fairchild, 1972, 1977; Kana, 1977, 1978) while others have included the near high sediment concentration region with the bedload (Komar and Inman, 1970; Komar, 1976) or specified a level above which sediment is considered to be suspended (Bren-ninkmeyer, 1974; Inman et al, 1980). In part the contradictions may result from different sampling devices and the nature of their deployment, and the different con-cepts of bedload and suspended load which lead to different methods of separating the two transport modes. In general, the application of the wave energy flux models are easier and require less inputs than the detailed predictor models. However, it was stated in the review that the estimation of the longshore transport potential by the use of wave energy flux model should only be considered as a qualitative indicator of the actual transport since direct correlation of the longshore transport potential to a single parameter of the longshore wave energy flux places a limitation on its use when other factors such beach profile and sediment size vary widely. Chapter 7. SUMMARY AND CONCLUSIONS 145 Since the fluid motions at breaking cause most of the sediment transport in the surf zone, dynamics of wave breaking characteristics were studied. Because the fluid motions cause most of the sediment transport, the amount of sediment transported is therefore partly determined by the breaker type. For breaking waves, the shapes are classified into spilling, plunging, collapsing and surging. There is a continuous sequence of possible breaker types running from spilling to plunging to collapsing to surging, and the transition from one breaker type to another is gradual without distinct dividing lines. Since the flow dynamics vary for different breaker types, it is important to understand the conditions associated with each breaker type. It has been observed that spilling breakers differ little in motion from unbroken waves and generate less bottom turbulence and thus tend to be less effective in transporting sediment than plunging and collapsing breakers (Divoky et al, 1970; Fairchild, 1972, 1977). Kana (1977, 1978) and Inman et al (1980) also report that plunging break-ers suspend significantly more sediment than do spilling breakers. This is intuitively expected from the different character of the bottom stresses associated with the two breaker types. Breaker type classification of the breaking waves using the deepwater wave steep-ness and the surf similarity parameter was tested. Various researches (Iversen, 1953; Galvin, 1968; and Battjes, 1974) have found out different transition values for the wave index parameters used in classifying the breaker type. In spite of the differences, which were probably due to differences in the method of calculation, experimental results presented in figure 3.8 showed that a reasonable prediction of the breaker type can be made given the deepwater steepness and the beach slope. However, Weishar and Byrne (1978) have shown that the breaking indices do not seem to be useful in predicting the breaker types in the ocean environment. This might presumably be Chapter 7. SUMMARY AND CONCLUSIONS 146 due to the fact that the conditions in the natural beach are complex and are effected by composite slopes, bottom friction and varying permeability. The prediction of the breaker height was also made by the use of deepwater char-acteristics. The prediction of the equation formulated by Komar and Gaughan (1972) agrees very well with the experimental results shown in figure 3.9 although the the-ory does not take into account the effects of wave refraction and frictional dissipation. It was noted that the beach permeability was one of the major factors controlling the rates of both cross-shore and longshore sediment transport since it modifies the fluid forces acting on the beach face, Quick (1989, 1990). However, in the present ex-perimental studies, permeability was only fairly modelled by the approximately 20cm thickness of sediment uniformly spread over the impermeable plywood beach. It is therefore expected that greater thickness of sediment be used to fully modelled the effects of beach permeability on the sediment transport. The investigations by Oldenziel and Brink (1974) and Sleath (1984) showed that seepage into the beach decreased sediment transport while seepage out of the bed increased it. Also the experiments of Gourlay (1980), showed that significant differ-ences exist between the impermeable and permeable beaches with regard to surf zone parameters such as the wave uprush and wave heights. In general, it was found that the values of the parameters of the impermeable beaches were greater than those of permeable beaches. Since laboratory studies, both model and general have proven to be a useful tool with which to study hydraulic problems, an experimental study on the longshore Chapter 7. SUMMARY AND CONCLUSIONS 147 transport rate was presented after studying the dynamics of the wave breaking char-acteristics in the surf zone. The specific objectives were to investigate the influence of sediment and beach slope effects and also to examine the cross-shore distribution of the transport rate on the beach face. Several important observations were made from the experimentation program. A theoretical longshore transport model developed by Quick (1991) and extended by the author of this thesis is presented. The development of the model was based on Bagnold's (1966) streampower approach and the Longuet-Higgins' (1970) alongshore wave stress and the mean longshore velocity formulated using the radiation stress concepts. Equations were then developed which define the longshore transport rate in terms of the beach slope and roughness coefficient for a given set of wave parame-ters. The following observations are made from the small scale experimental study carried on an impermeable beach slope condition. • A n important observation was that the longshore transport rate increases with beach slope and that this increase is approximately linear as shown in figures 4.14 and 4.15. For a given wave attack, increasing the beach slope increases the longshore transport. The predictions of the model using the total model, equa-tion (5.16) agreed reasonably well with the above finding as shown on figures 6.1 and 6.2. The present experimental finding is therefore in agreement with equation (5.16) and not with equation (2.19) which is the currently accepted relationship. Although it was observed that the reflection coefficient increases with beach slope implying a slightly lower wave attack on the steeper beaches, the longshore Chapter 7. SUMMARY AND CONCLUSIONS 148 transport rate increased with the beach slope. A relationship for a transport coefficient was developed, equations (5.21) and (5.22) which was related to a form of the dimensionless surf similarity parameter of Battjes (1974) given in a form of a reflection coefficient Kr\,. The increase of total transport with in-creasing Krb which is dependent on beach slope suggests that plunging waves on steep beaches are more efficient, because plunging waves produce more bottom stresses and turbulence, and steeper beaches result in a higher concentration per unit area of beach slope. This observation actually proved that the beach slope is a predominant factor in determining the longshore transport rate. • Based on the model, simple relationships between the local maximum transport rate and sediment size were computed. Examination of equation (5.12) predicts that the local peak rate of sediment transport for any given artificially imposed beach slope will be higher for a coarser sediment, approximately by a factor of (di/d,2)025. Experimental and modelling results presented in tables 6.4 and 6.5 showed a very good agreement. It was noted that the the predictions on the low and medium slopes slopes of 1 on 9 and 1 on 5.2 respectively were in close agreement with the experimental results. This is significantly interesting since the model assumed moderate beach slopes in its formulation. The greatest error in prediction occurred on the steepest slope of 1 on 3.6. This relatively steep slope might be considered unrealistically steep. On this steeper slope, it ap-peared that the longshore flow was being carried slightly offshore of the breaker zone by the strong down beach backflow. This large increase of transport is an extremely interesting observation with important implications for the design of protective works. It was noted that the fine sand transport was slightly greater than the coarser sands. This is reasonable because the offshore transport of Chapter 7. SUMMARY AND CONCLUSIONS 149 fine sand transport would be very high on a beach so much steeper than the equilibrium value for such a sand. This observation is very interesting since the model assumed equilibrium slope conditions in its development. Since the model prediction were based on finer sand peak transport rate, it is reasonable to expect greater error when the imposed slope is much higher than the equi-librium value the fine sand. The increasing local peak transport rate with increasing sediment size indicates that coarser materials attract more wave stress in the zones of maximum wave energy dissipation than finer sediments. • From the experimental study of the longshore transport rate, it was seen that for a given wave attack and beach slope, the total sediment transport is ap-proximately constant and reasonably independent of sediment size. At a first glance, the total model equation of the longshore transport rate appeared to suggest that the total sediment transport must increase with decreasing rough-ness coefficient (decreasing sediment size). However, a re-examination of the total model showed that this was not true. This was explained based on the fact that coarser sediment attracts more wave stress than finer sediment. The roughness coefficient also increases with increasing sediment size. These two effects exactly balanced each other with the result that sediment transport is reasonably independent of sediment size. Using therefore an average roughness coefficient for the test sands, the predictions of the model were compared with the experimental results as shown in figures 6.1 and 6.2. It could be seen that the prediction of the model agrees reasonably well with the experimental results. ex 7. SUMMARY AND CONCLUSIONS 150 The experimental study revealed that the mechanical analysis of transported sand along the beach showed no appreciable change in grading from the orig-inal sands initially supplied to the beach (see Appendix A) . Hence, there was no differential grading of sand longitudinally along the beach regardless of the incident waves. This would seem to indicate that the size of the particles within the mixture might not influence the rate of transport, but that all particles are transported at the same rate along the beach. The findings of Krumbein (1944) and Saville (1950) corroborate this conclusion. The experimental results also supported Komar et al's (1970) and Komar (1979) findings on the independent relationship between the mean longshore current velocity and the beach slope. This relationship (equation 4.3) was found to be equivalent to Longuet-Higgins' (1970) longshore velocity formulated based on the radiation stress concepts. This agreement was very encouraging since the longshore transport model developed in this thesis was partly based on the ra-diation stress concepts. The predictions of the longshore current velocity using Galvin et al's (1965) predictor resulted in high overestimation of the longshore velocity. This further concludes that the radiation stress concepts based on the the conservation of momentum is soundly based. A sensitivity analysis was carried out to establish the extent of influence of in-accuracies in the prime input variables on the longshore transport as predicted for the situation outlined in section 6.2. The results of the sensitivity analysis showed that the model is sensitive to small inaccuracies in the measurements of the input variables. It should be noted that it is very difficult to obtain highly Chapter 7. SUMMARY AND CONCLUSIONS 151 accurate measurements in the surf zone conditions. Although, the model is sen-sitive to small inaccuracies in all the input variables, it is hoped that with the current refinements in instrumentation for measurements of wave characteris-tics both in the laboratory and in the natural sea environment, fairly accurate measurements can be obtained leading to accurate prediction of the longshore transport model. The present work clearly demonstrates both experimentally and theoretically, that longshore sediment transport is not entirely independent of beach slope and sediment size as is often assumed. It is therefore desirable to carry out larger scale experiments for a broader range of conditions to check the validity of the present small scale ex-perimental results. It should however, be noted that these findings would probably hold for a limited range of sediment sizes, because the experimental results depend on very active, carpet-like flow as defined by Bagnold (1966). As sediment size in-creases, the decrease of carpet flow and the increase of critical shear stress will limit the transport, so that the efficiency and extent of transport would be limited. It must be stated that several factors such as the effect of rip currents, tidal action and scale effects were not taken into consideration. The effects of rip currents were not investigated due to the limitation on the laboratory beach length. On a long beach, the sediment transport never reaches a steady state, because the build up of longshore current becomes unstable and eventually produces offshore rip currents. These rip currents are known to carry some sediment offshore through the breaker zone, so that the next downstream section of the beach is partly deprived of its long-shore sediment supply. For an infinitely long beach therefore, on which there are rip currents, the model cannot account for the sediment lost due to effects of rip currents. Chapter 7. SUMMARY AND CONCLUSIONS 152 Scale effects were also not fully investigated. Due to laboratory limitations, wave dimensions in the experiments were necessarily small compared to those of waves nor-mally prevailing on natural beaches. Also, the investigation is limited since the size of beach materials used cannot be reduced in the same proportion as the wave dimen-sions. As the small laboratory waves acted upon sands of natural size, the laboratory results cannot necessarily be considered as model reproductions of any larger scale natural processes. In spite of these shortcomings, study under controlled conditions in the laboratory remains as the only feasible method for the initial investigations of the basic relationships. The results from the simplified experimentation program which reduces the com-plexity of the natural sea state conditions should therefore be used with caution. 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Development and field tests of a sampler for suspended sediment in wave action, U.S. Army Corps of Engineers, Beach Erosion Board, Technical Memo. No. 34. [88] Watts, G . M . (1953b). A study of sand movement at South Lake Worth Inlet, Florida, U.S. Army Corps of Engineers, Beach Erosion Board, Technical Memo. No. 42. [89] Weishar, L . L . , and Byrne, R . J . (1978). Field study of breaking wave character-istics, P roc , 16 t f c Coastal Engineering Conf., A S C E , pp. 487-506. [90] Willis, D . H . (1978). Sediment load under waves and currents, P roc , 16th Coastal Engineering Conf., A S C E , pp. 1626-1637. [91] Zampol, J .A. and Inman, D.L. (1989). Discrete measurements of suspended sed-iment, Nearshore Sediment Transport, edited by Richard J . Seymour, Plenum Press, New York and London. Appendix A Grain Size Distribution of Test Sands 163 Appendix A. Grain Size Distribution of Test Sands 164 Figure A. l : Grain size distribution of original test sands Appendix A. Grain Size Distribution of Test Sands 165 0.05 0.1 0.2 0.5 1 2 5 10 Sieve Size (mm) Figure A.2: Distribution of original and transported test sands for 1.41sec. wave period Appendix A. Grain Size Distribution of Test Sands 166 Cummulative Percentage Passing (%) - x r o c o ^ c n o - ^ i o o c D o 0 o o o o o o o o o o o o Figure A.3: Distribution of original and transported test sands for 1.50sec. wave period Appendix B Calibration of Wave Probe 167 Appendix B. Calibration of Wave Probe 168 1.2 E 1 CO > co 0.8 > O -Q CO C o I 0.6 LU 0.4 1 2 3 4 5 6 7 8 9 Recorded displacement on oscilloscope (cm) 10 Figure B.l: Calibration of wave probe for wave height recording Appendix C Wave Reflection Data 169 Appendix C. Wave Reflection Data Reflection Tests on a Plane Impermeable 1 on 9 Sloped Beach 170 Run Depth Period Hmax Hmin Ni Hb Db Lo Steepness Reflection Reflected # (cm) (sec.) (cm) (cm) (cm) (cm) (cm) (cm) (Hi/Lo) Coefficient Wave Ht.* (cm) 1.1 47.0 1.41 7.11 6.10 6.60 9.75 10.5 310.4 0.0212 0.077 0.508 2.1 47.0 1.50 7.77 6.10 6.93 9.50 10.0 351.3 0.0197 0.120 0.830 * The height of the reflected wave i s given by (Reflection Coefficient)*Hi Reflection Tests on a Plane Impermeable 1 on 5.2 Sloped Beach Run Depth Period Hmax Hmin Hi Hb Db Lo Steepness Reflection Reflected # (cm) (sec.) (cm) (cm) (cm) (cm) (cm) (cm) (Hi/Lo) Coefficient Wave Ht.* (cm) 1.2 47.0 1.41 7.62 6.10 6.86 9.15 10.5 310.4 0.0221 0.110 0.755 2.2 47.0 1.50 8.53 6.40 7.47 8.50 10.0 351.3 0.02126 0.143 1.068 * The height of the reflected wave i s given by (Reflection Coefficient)*Hi Reflection Tests on a Plane Impermeable 1 on 3.6 Sloped Beach Run Depth Period Hmax Hmin Hi Hb Db Lo Steepness Reflection Reflected # (cm) (sec.) (cm) (cm) (cm) (cm) (cm) (cm) (Hi/Lo) Coefficient Wave Ht.* (cm) 1.3 47.0 1.41 8.63 6.35 7.49 8.85 10.5 310.4 0.0241 0.150 1.12 2.3 47.0 1.50 8.53 5.94 7.23 8.50 10.0 351.3 0.0206 0.179 1.29 * The height of the reflected wave i s given by (Reflection Coefficient)*Hi Appendix D Longshore Transport Results 171 Appendix D. Longshore Transport Results 172 SET U P E X P E R I M E N T #1 Wave Period = 1.41sec. Water Depth = 0.47m Test Duration = 85sec. Breaker Type = Plunger Test Sand Sizes = 0.5mm and 2.0mm Incident Wave Height = 6.60cm Breaker Distance from Shoreline = 1.27m Table D . l : Experimental results on 1 : 9 slope for wave period of 1.41sec. Distance Measured Transport Rates to (gm/m/sec.) Breaker Line 0.5mm Sand 2.0mm Sand (m) (Fine Sand) (Coarse Sand) 1.12 0 0 1.02 13.76 9.88 0.92 31.76 25.65 0.82 77.76 43.29 0.72 144.47 61.88 0.62 163.76 127.76 0.52 105.88 201.53 0.42 69.53 133.53 0.32 51.41 78.59 0.22 40.71 53.65 0.12 15.65 19.41 0.02 5.29 8.47 -0.08 1.12 7.65 -0.18 2.47 7.06 -0.28 9.41 5.88 -0.38 0 0 Appendix D. Longshore Transport Results 173 Table D.2: Experimental results on 1 : 5.2 slope for wave period of 1.41sec. Distance Measured Transport Rates to (gm/m/sec.) Breaker Line 0.5mm Sand 2.0mm Sand (m) (Fine Sand) (Coarse Sand) 1.12 0 0 1.02 0 0 0.92 0 0 0.82 4.94 0.47 0.72 25.53 25.06 0.62 65.53 41.29 0.52 159.06 78.70 0.42 195.18 153.18 0.32 243.23 317.55 0.22 225.18 365.55 0.12 152.94 346.59 0.02 54.12 44.94 -0.08 33.76 10.59 -0.18 32.12 5.29 -0.28 31.53 0.82 -0.38 0 0 Appendix D. Longshore Transport Results 174 Table D.3: Experimental results on 1 : 3.6 slope for wave period of 1.41sec. Distance Measured Transport Rates to (gm/m/sec.) Breaker Line 0.5mm Sand 2.0mm Sand (m) (Fine Sand) (Coarse Sand) 1.12 0 0 1.02 0 0 0.92 0 0 0.82 0 0 0.72 4.35 7.65 0.62 17.53 18.82 0.52 93.41 38.23 0.42 150.71 91.06 0.32 220.12 169.65 0.22 287.06 311.76 0.12 316.71 407.65 0.02 276.12 398.59 -0.08 275.41 358.82 -0.18 227.29 77.53 -0.28 207.76 10.35 -0.38 0 0 Appendix D. Longshore Transport Results SET UP EXPERIMENT #2 Wave Period = 1.41sec. Water Depth = 0.47m Test Duration = 85sec. Breaker Type = Plunger Test Sand Sizes = 0.5mm and 2.0mm Incident Wave Height = 6.93cm Breaker Distance from Shoreline = 1.27m Table D.4: Experimental results on 1 : 9 slope for wave period of 1.50sec. Distance Measured Transport Rates to (gm/m/sec.) Breaker Line 0.5mm Sand 0.85mm Sand 2.0mm Sand (m) (Fine Sand) (Medium Sand) (Coarse Sand) 1.12 0 0 0 1.02 20.48 17.86 27.63 0.92 47.34 47.06 42.63 0.82 84.09 79.61 67.86 0.72 128.03 121.18 108.02 0.62 170.99 188.81 158.16 0.52 121.41 219.49 235.87 0.42 83.52 135.92 154.56 0.32 66.43 62.75 82.11 0.22 57.34 52.36 50.35 0.12 48.25 37.85 20.89 0.02 42.25 12.87 7.62 -0.08 16.66 5.27 6.27 -0.18 6.88 4.81 5.87 -0.28 5.88 2.05 2.74 -0.38 0 0 0 Appendix D. Longshore Transport Results 176 Table D.5: Experimental results on 1 : 5.2 slope for wave period of 1.50sec. Distance Measured Transport Rates to (gm/m/sec.) Breaker Line 0.5mm Sand 0.85mm Sand 2.0mm Sand (m) (Fine Sand) (Medium Sand) (Coarse Sand) 1.12 0 0 0 1.02 0 0 0 0.92 1.51 1.23 0.14 0.82 18.56 18.13 12.05 0.72 29.13 28.47 31.09 0.62 75.31 74.19 38.48 0.52 167.03 131.74 78.49 0.42 184.98 167.75 143.07 0.32 240.74 264.15 280.93 0.22 210.58 258.87 353.25 0.12 155.72 137.12 318.65 0.02 54.88 75.51 104.54 -0.08 33.89 24.83 11.71 . -0.18 19.28 9.19 7.53 -0.28 15.79 5.33 4.91 -0.38 0 0 0 Appendix D. Longshore Transport Results 177 Table D.6: Experimental results on 1 : 3.6 slope for wave period of 1.50sec. Distance Measured Transport Rates to (gm/m/sec.) Breaker Line 0.5mm Sand 0.85mm Sand 2.0mm Sand (m) (Fine Sand) (Medium Sand) (Coarse Sand) 1.12 0 0 0 1.02 0 0 0 0.92 0 0 0 0.82 0 0 0 0.72 12.45 8.35 6.33 0.62 22.28 19.65 19.42 0.52 79.51 79.25 29.13 0.42 141.52 127.51 87.51 0.32 201.91 192.75 183.13 0.22 280.11 278.45 262.55 0.12 300.55 339.47 344.75 0.02 322.51 378.48 383.79 -0.08 233.43 242.64 314.54 -0.18 113.20 68.16 75.89 -0.28 73.11 43.13 5.96 -0.38 0 0 0 Appendix E Longshore Current Measurement 178 Appendix E. Longshore Current Measurement SET UP EXPERIMENT # 1 Wave Period = 1.41sec. Water Depth = 0.47m Breaker Type = Plunger Incident Wave Height = 6.60cm Velocity Equation V = 0.042n + 0.076 (m/sec.) for n < 6.62 Table E. l : Velocity data on 1 : 9 beach slope for wave period of 1.41sec. Distance to Breaker Line (m) Number of Counts in 10 Seconds Average Count Per Second n Current Velocity (m/s) Average Velocity (m/s) 0.635 16, 16, 17, 16 1.65 0.1432 0.483 21, 23, 24, 22 2.25 0.1642 0.330 35, 37, 36, 34 3.55 0.2020 0.180 40, 35, 39, 36 3.75 0.2440 0.185 0.020 32, 30, 31, 33 3.15 0.2230 -0.13 25, 26, 26, 25 2.55 0.1810 -0.28 15, 14, 14, 15 1.45 0.1348 Appendix E. Longshore Current Measurement Table E.2: Velocity data on 1 : 5.2 beach slope for wave period of 1.41sec. Distance to Breaker Line (m) Number of Counts in 10 Seconds Average Count Per Second n Current Velocity (m/s) Average Velocity (m/s) 0.635 12, 11, 10, 13 1.15 0.1243 0.483 16, 17, 18, 15 1.65 0.1453 0.330 24, 25, 26, 24 2.47 0.1799 0.180 40, 45, 42, 41 4.20 0.2524 0.198 0.020 53, 53, 51, 54 5.27 0.2975 -0.13 34, 34, 36, 36 3.40 0.2188 -0.28 22, 21, 23, 23 2.22 0.1694 Appendix E. Longshore Current Measurement 181 Table E.3: Velocity data on 1 : 3.6 beach slope for wave period of 1.41sec. Distance to Breaker Line (m/s) Number of Counts in 10 Seconds Average Count Per Second n Current Velocity (m/s) Average Velocity (m/s) 0.635 12, 11, 16, 10, 13, 11 1.22 0.1264 0.483 24, 29 30, 30, 32, 28 2.88 0.1800 0.330 29, 31, 30, 31, 30, 33 3.07 0.2076 0.180 34, 35, 35, 37, 36, 37 3.57 0.2381 0.212 0.020 45, 47, 48, 46, 47, 48 4.68 0.2734 -0.13 41, 43, 44, 44, 40, 42 4.23 0.2461 -0.28 33, 32, 34, 31, 32, 33 3.25 0.2153 

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