NEARSHORE HYDRODYNAMICS AND THE BEHAVIOUR OF GROYNES ON SANDY BEACHES
by
D.J. Walker, B.E.(Hons.), M.Eng.Sc., M.I.E.Aust.
August, 1987
A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of the Imperial College
Hydraulics Section, Department of Civil Engineering Imperial College of Science and Technology DEDICATION
To my wife, Adrienne, and sons, James, Philip and Robert who all helped, although in different ways. 3 ABSTRACT
This thesis describes solutions to the nearshore dynamics in an attempt to understand the behaviour of groynes on sandy beaches.
The solution of the dynamics is achieved using three finite difference numerical models. The first solves a transient form of the Mild Slope Equation which allows the inclusion of wave diffraction, reflection and refraction by both bathymetry and currents. The second model solves the depth-averaged Navier-Stokes equations to determine local mean velocities, mean free surface elevation and turbulent eddy viscosity. The third model, using values calculated by the first two models, calculates the sediment transport and provides predictions of bathymetric changes caused by that transport.
The numerical models are checked for validity against known analytical solutions and, where possible, against field and laboratory data.
In addition to the numerical work a series of 1:36 scale model tests was undertaken to provide data with which to compare the numerical model results. Ten experiments were carried out under fixed wave conditions. The parameters varied included the number, spacing, length and height of the groynes.
The author was also involved in the collection of full scale data from sites in Norfolk and Lincolnshire, and responsible for its subsequent analysis. The data collected provides further corroboration of the numerical and physical models.
The thesis concludes with a number of runs of the numerical models and the prediction of likely accretion and erosion patterns caused by typical groyne arrangements. 4 ACKNOWLEDGEMENTS
The author wishes to express his thanks to his supervisor, Professor P. Holmes for the encouragement and help provided during the course of research and for the invaluable criticism of the original manuscript.
The author has also benefited from working with colleagues in the section, in particular Dr. Kostas Anastasiou and Mr. Dong Ping.
The laboratory staff involved in the project must also be thanked, Messrs. Geoff Thomas, H.(Clem) Clements, Stan Finch, John Audsley and Martin Roper who always reacted quickly, even to the most tedious requests, and came up with helpful suggestions and ideas (like,"that's an interesting idea, but wouldn't it be better to ..."). Thanks also to Mr. Greg Guthrie of CEEMAID Ltd. for all the work in generating the full scale data sets.
During the period at Imperial College the author has also had fruitful discussions and correspondance with Profs. J. Fredsoe (Technical University of Denmark), B. O'Connor (Univeristy of Liverpool) and R. Sternberg (University of Washington) and Drs. C. Fleming (Sir William Halcrow and Partners), J. Nicholson (Univeristy of Liverpool) and P. Nielsen (Public Works Dept., Sydney).
The library staff in the Civil Engineering Department, Mrs. Kay Crooks and Miss Jessica Underhill are also thanked for their help as is Kirsten Djorup at the Technical University of Denmark who responded quickly to a couple of urgent requests.
Finally the author would like to acknowledge the funding support of the Science and Engineering Research Council. 5 CONTENTS Pag' ABSTRACT 3 ACKNOWLEDGEMENTS 4 LIST OF TABLES 9 LIST OF FIGURES 10 LIST OF PLATES 13 LIST OF SYMBOLS 14 GLOSSARY OF TERMS 18
CHAPTER 1 - INTRODUCTION 19
CHAPTER 2 - LITERATURE REVIEW 21
2.1 Introduction 21 2.2 Groyne Behaviour and Design 22 2.3 Physical Models 23 2.3.1 Scaling Laws 23 2.3.2 Physical Model Tests 24 2.4 Numerical Models 29
CHAPTER 3 - PHYSICAL MODEL TESTS 32
3.1 Introduction 32 3.2 Basin Configuration 32 3.3 Measurement Techniques 37 3.4 Test Results 41 3.4.1 Introduction 41 3.4.2 Test No.l 42 3.4.3 Test No.2 42 3.4.4 Test No.3 42 3.4.5 Test No.4 42 3.4.6 Test No.5 47 3.4.7 Test No.6 47 3.4.8 Test No.7 47 3.4.9 Test No.8 50 3.4.10 Test No.9 50 3.4.11 Test No.10 50 3.5 Discussion of Results 50 6
Pago CHAPTER 4 - FULL SCALE MEASUREMENTS 56
4.1 Introduction 56 4.2 Site and Instruments 56 4.3 Data Recording Programme 65 4.4 Data Analysis 65 4.4.1 Introduction 65 4.4.2 Tidal Elevation and Mean Flows 65 4.4.3 Mean Flow Patterns 70 4.4.4 Surface Wave Energy Spectra 74 4.4.5 Wave Oscillatory Motion 74 4.5 Float Track Experiment 74 4.6 Beach Surveys 82 4.7 Discussion of Results 87
CHAPTER 5 - THE WAVE MODEL 88
5.1 Introduction 88 5.2 Literature Review 88 5.2.1 Introduction 88 5.2.2 Early Works 89 5.2.3 Mild Slope Equation 90 5.2.4 Assumptions and Approaches 92 5.2.5 Wave Breaking 93 5.3 Hyperbolic Approximation 93 5.3.1 Introduction 93 5.3.2 Theory 94 5.4 Finite Difference Equations 97 5.4.1 Introduction 97 5.4.2 Finite Difference Scheme 99 5.4.3 Initial Conditions 100 5.4.4 Driving Boundaries 102 5.4.5 Reflective and Transmissive Boundaries 104 5.4.6 Stability 105 5.4.7 Wave Height and Directions 105 5.4.8 Wave Breaking 106 5.4.9 Radiation Stresses 106 5.5 Model Verification 107 5.5.1 Introduction 107 5.5.2 Pure Refraction by Bathymetry 108 7
Page 5.5.3 Pure Diffraction around a Semi-Infinite Breakwater 108 5.5.4 Pure Diffraction through a Harbour Entrance 108 5.5.5 Pure Refraction by a Shear Current 108 5.5.6 Combined Refraction-Diffraction by a Submerged Shoal 119 5.5.7 Modelling of Arbitrary Configurations 131
CHAPTER 6 - THE CURRENT MODEL 133
6.1 Introduction 133 6.2 Literature Review 133 6.2.1 Introduction 133 6.2.2 Numerical Models 134 6.2.3 Circulating Flows 138 6.2.4 Turbulence Models 139 6.3 Current Model Equations 142 6.3.1 Depth-Averaged Navier-Stokes Equations 142 6.3.2 Radiation Stresses 143 6.3.3 Bottom Friction 143 6.3.4 Turbulence 143 6.4 Finite Difference Approximation 146 6.4.1 Introduction 146 6.4.2 ADI Scheme 147 6.4.3 Navier-Stokes Equations 147 6.4.4 Solution Technique 155 6.4.5 Turbulence Equations 157 6.4.6 Boundary Conditions 161 6.4.6.1 Navier-Stokes Equations 161 6.4.6.2 Turbulence Equations 161 6.5 Model Verification 162 6.5.1 Introduction 162 6.5.2 Comparison with Longuet-Higgins Profile 162 6.5.3 Comparison with Physical Model Results 162 6.5.4 Comparison with Full Scale Data 173
CHAPTER 7 - THE SEDIMENT MODEL 177
7.1 Introduction 177 7.2 Modes of Transport 177 7.3 Literature Review 178 7.3.1 Introduction 178 Pago 7.3.2 Longshore Sediment Transport 179 7.3.3 Cross Shore Sediment Transport 184 7.3.4 Experimental Studies 190 7.3.5 Vertical Velocity Profile 197 7.3.6 Bathymetric Evolution Models 197 7.4 Sediment Model Equations 200 7.4.1 Introduction 200 7.4.2 Longshore Transport 201 7.4.3 Cross Shore Transport 203 7.4.4 Vertical Velocity Profile 204 7.4.5 Bed Evolution Model 208 7.5 Finite Difference Equations 209 7.5.1 Introduction 209 7.5.2 Longshore Transport 209 7.5.3 Cross Shore Transport 211 7.5.4 Vertical Velocity Profile 214 7.5.5 Bed Evolution Model 218 7.6 Model Verification 218 7.6.1 Introduction 218 7.6.2 CERC Bulk Formula 219 7.6.3 Cross Shore Distribution of Transport 219 7.6.4 Comparison with Vertical Concentration Data 224 7.6.5 Onshore-Offshore Transport Mechanism 229
CHAPTER 8 - BATHYMETRIC MODEL RESULTS 231
8.1 Introduction 231 8.2 Numerical Prediction Results 232 8.2.1 Single Groyne, Surface Piercing 232 8.2.2 Single Groyne, Partially Submerged 238 8.2.3 2 Groynes, Surface Piercing, S/L=1.0 238 8.2.4 2 Groynes, Surface Piercing, S/L=2.0 244 8.3 Discussion of Results 244
CHAPTER 9 - SUMMARY, DISCUSSION AND CONCLUSIONS 253 9.1 Summary 253 9.2 Conclusions 254 9.3 Recommendations 257
REFERENCES 259 9
LIST OF TABLES Page 3.1 Physical Model Wave Parameters 43 3.2 Physical Model Test Details 43 6.1 k-e Model Empirical Constants 145 7.1 North Shore Wave Data 221 1 0
LIST OF FIGURES Page 2.1 Physical Model Results of Hulsgergen et al 26 2.2 Model Results of Price et al 30 3.1 Wave Basin Layout 33 3.2 Wave Heights in Wave Basin 38 3.3 Physical Model Results - Test 1 39 3.4 Signal from Current Velocity Probe 40 3.5 Physical Model Results - Test 2 44 3.6 Physical Model Results - Test 3 45 3.7 Physical Model Results - Test 4 46 3.8 Physical Model Results - Test 5 48 3.9 Physical Model Results - Test 6 49 3.10 Physical Model Results - Test 7 51 3.11 Physical Model Results - Test 8 52 3.12 Physical Model Results - Test 9 53 3.13 Physical Model Results - Test 10 54 4.1 Sea Palling - Location Plan 57 4.2 Sea Palling - Site Plan 58 4.3 Beach Cross Section - Open Beach 59 4.4 Beach Cross Section - Groyne Bay 60 4.5 Beach Sand Sieve Test Results 61 4.6 Beach "Pod" 63 4.7 Typical Data from Beach Pod 64 4.8 Mean Water Depth at Pod 3 66 4.9 Mean Water Depth at Pod B 67 4.10 Mean Velocity at Pod 3 68 4.11 Mean Velocity at Pod B 69 4.12 Wave Height Record for Experiment Duration 71 4.13 Mean Flow Patterns - Records 43-49 72 4.14 Wave Energy Spectra from Pod and Waverider Buoy 75 4.15 Bivariate Plot from Current Meter 76 4.16 Plot of Water Particle Displacement 77 4.17 Float used in Float Track Experiment 79 4.18 Float Track Records - May 1984 80 4.19 Float Track Records - October 1984 81 4.20 Total Beach Volume 83 4.21 Selected Beach Cross Sections - Section 1 84 4.22 Selected Beach Cross Sections - Section 2 85 4.23 Selected Beach Cross Sections - Section 3 86 5.1 SolutionGrid Layout for Wave Model 98 11
Page 5.2 Numerical Grid for Wave Model 101 5.3 Isometric Plot - Wave Refraction 109 5.4 Numerical Solution - Wave Refraction 110 5.5 Contour Plot - Semi-Infinite Breakwater 111 5.6 Isometric Plot - Semi-Infinite Breakwater 112 5.7 Wave Front Plot - Semi-Infinite Breakwater 113 5.8 Isometric Plot - Harbour Entrance 114 5.9 Analytical Solution - Harbour Entrance 115 5.10 Numerical Solution - Harbour Entrance 116 5.11 Wave Front Plot - Harbour Entrance 117 5.12 Refraction by Shear Current Test Schematic 118 5.13 Wave Angles in Shear Current Test 120 5.14 Submerged Elliptic Shoal Test Schematic 121 5.15 Depth Contours of Submerged Elliptic Shoal 122 5.16 Experimental Results - Submerged Elliptic Shoal 123 5.17 Numerical Results - Submerged Elliptic Shoal 124 5.18 Selected Section Results - Sections 1 and 2 125 5.19 Selected Section Results - Sections 3 and 4 126 5.20 Selected Section Results - Sections 5 and 6 127 5.21 Selected Section Results - Sections 7 and 8 128 5.22 Experimental Results of Wave Front Pattern - Elliptic Shoal 129 5.23 Numerical Prediction of Wave Front Pattern - Elliptic Shoal 130 5.24 Isometric Plot - General Harbour 132 6.1 Numerical Model Results of Thornton and Guza 137 6.2 Longuet-Higgins and k-e Model Eddy Viscosity 141 6.3 Numerical Solution Grid 1^8 6.4 Longuet-Higgins and k-e Model Velocity Profile 163 6.5 Comparison with Physical Model - Test 2 164 6.6 Comparison with Physical Model - Test 3 165 6.7 Comparison with Physical Model - Test 4 166 6.8 Comparison with Physical Model - Test 5 167 6.9 Comparison with Physical Model - Test 6 168 6.10 Comparison with Physical Model - Test 7 169 6.11 Comparison with Physical Model - Test 8 170 6.12 Comparison with Physical Model - Test 9 171 6.13 Comparison with Physical Model - Test 10 172 6.14 Revised Comparison with Physical Model - Test 3 174 6.15 Comparison with Data of Thornton and Guza - Feb.3 175 6.16 Comparison with Data of Thornton and Guza - Feb.4 176 7.1 Sediment Pick-Up Function of Nielsen 183 12
Pago 7.2 Eddy Viscosity Distribution in Vertical 185 7.3 Vertical Sediment Concentration Data of Deigaard et al 186 7.4 Longshore Transport Prediction of Deigaard et al 187 7.5 Bottom Concentration - Horikawa et al 193 7.6 Prediction of Bottom Concentration - Fredsoe et al 195 7.7 Bottom Concentration - Staub et al 196 7.8 Comparison of Bottom Concentration Prediction 198 7.9 Mean Velocity and Sediment Concentration Profiles 205 7.10 Temporal Variation of Sediment Concentration 215 7.11 Flow Chart of Velocity Profile Calculation 216 7.12 Comparison with Bulk Transport Formula 220 7.13 Distribution of Longshore Sediment Transport 222 7.14 North Shore Wave Record 223 7.15 Prediction of North Shore Sediment Transport 225 7.16 Prediction of Vertical Sediment ConcentrationProfiles 226 7.17 Cross Shore Sediment Transport Mechanism 230 8.1 Numerical Prediction - 1 Groyne, X^r=0.5L 233 8.2 Eddy Viscosity Around Groyne 234 8.3 Numerical Prediction - 1 Groyne, Xbr=1.0L 235 8.4 Numerical Prediction - 1 Groyne, X^r=1.5L 236 8.5 Longshore Continuity - 1 Groyne 237 8.6 Numerical Prediction - 1 Groyne, Part Submerged, Xj-^O.SL 239 8.7 Numerical Prediction - 1 Groyne, Part Submerged, X^r=1.0L 240 8.8 Numerical Prediction - 1 Groyne, Part Submerged, X^r=1.5L 241 8.9 Longshore Continuity - 1 Groyne, Part Submerged 242 8.10 Numerical Prediction - 2 Groynes, S/L=1.0, Xkr=0.5L 243 8.11 Numerical Prediction - 2 Groynes, S/L=1.0, Xbr=1.0L 245 8.12 Numerical Prediction - 2 Groynes, S/L=1.0, Xbr=1.5L 246 8.13 Longshore Continuity - 2 Groynes, S/L=1.0 247 8.14 Numerical Prediction - 2 Groynes, S/L=2.0, Xkr=0.5L 248 8.15 Numerical Prediction - 2 Groynes, S/L=2.0, Xbr=1.0L 249 8.16 Numerical Prediction - 2 Groynes, S/L=2.0, X^r=1.5L 250 8.17 Longshore Continuity - 2 Groynes, S/L=2.0 251 13 LIST OF PLATES Pago
Plate 1 Wave Basin - General View 34 Plate 2 Distribut ion System for Longshore Current 36 14 LIST OF SYMBOLS
The most common symbols used in this thesis are listed below. Occasionally one symbol may have a different meaning depending on where it is used. This should generally be obvious from the context.
a wave amplitude c wave celerity sediment concentration group velocity Cg Cf coefficent of friction constant in k-e model CP cle constant in k-e model c2e constant in k-e model c/* constant in k-e model c 0 bottom concentration c Courant number D total water depth particle diameter turbulence generation due to breaking E wave energy (= pgH^/8) g gravitational constant (9.81) h water depth H wave height Hb wave breaker height Hrms r.m.s. wave height Hsig significant wave height Hi wave height parameter i coordinate in x direction h immersed weight of longshore transport j coordinate in y direction k wave number (2t/L) turbulent kinetic energy bed roughness (2.5D) apparent bed roughness K constant in CERC equation for longshore transport L groyne length wave length p(t) - sediment pick-up function ph - turbulent production term in k-e model pkV ~ production term in k equation of k-e model p l ~ longshore flux of wave energy peV - production term in e equation of k-e model Q defined function in Chapter 5 total longshore transport
Tyy effective stress component u x velocity component ui steady component outside boundary layer unsteady component outside boundary layer u2 - uf - wave induced shear velocity ulm oscillatory wave velocity component U velocity = U(u,v) U+ - maximum forward velocity component U“ maximum backward velocity component
Ufc - current shear velocity Uf0 - current shear velocity inside boundary layer Ui - steady component inside boundary layer u2 - unsteady component inside boundary layer 16
V - y velocity component w - sediment fall velocity z - distance in vertical - parameter
<*b - breaker angle 0 - parameter 7 - wave breaking constant - angle between wave and current
8 - boundary layer thickness
A - relative density of sediment V - horizontal gradient operator e - rate of dissipation of turbulent kinet ic energy - eddy viscosity - error term eb - eddy viscosity component inside boundary layer es - eddy viscosity of sediment ew - eddy viscosity component due to wave breaking
V - surface elevation 8 - angular frequency - wave angle - Shields Parameter
8' - wave angle
K - von Karman's constant (0.4) X - parameter in Chapter 5 - bed porosity /* - ripple coefficent - ratio of max. forward to back, oscillatory veloc T - 3.1415926.... P - density of water or - absolute frequency - constant in k-e model - constant in k-e model T - time difference in Chapter 5 - bed shear stress rbx - bottom friction in x direction rby - bottom friction in y direction VV kinematic viscosity
v t eddy viscosity
'p phase angle in sediment pick-up function (j) intrinsic frequency Vt time step Vx space step in x
Vy - space step in y
in general superscripts refer to time and subscripts to space. An overbar refers to time mean quantity. GLOSSARY OF TERMS
Listed below are the common terms that are used in this thesis. The definitions are based on information provided in the Shore Protection Manual (1984) and Muir-Wood and Fleming (1981).
Deep water - where the waves are unaffected by bathymetry Intermediate Zone - where depth begins to influence the wave characteristics Surf Zone - region inside the breaker line Swash Zone - region in which there is an uprush and backwash as a result of breaking waves Nearshore Zone - where the forces of the sea react with the land Littoral Transport - movement of sediments in the nearshore zone by waves and currents Rip Currents ~ concentrated jets of water flowing seawards through the breaker zone Mean Velocity - depth and time averaged velocity over a wave period 19
CHAPTER 1 INTRODUCTION
"A groin is a shore protection structure designed to trap longshore drift for building a protective beach, retarding erosion of an existing beach, or preventing longshore drift from reaching some downdrift point, such as a harbor or inlet. Groynes are narrow structures of varying lengths and heights and are usually constructed perpendicular to the shoreline.
...The interaction between the coastal processes and a groyne or groyne system is complicated and poorly understood.”
So begins the section on groynes in the U.S. Army Corps of Engineers, Shore Protection Manual (1984). The quotation also forms a good starting point for this present work because it was this acknowledged lack of understanding of the behaviour of groynes that prompted the Construction Industry Research and Information Association (CIRIA) to embark on "Project 310, Effectiveness of Groyne Systems”.
The project was co-ordinated by Sir William Halcrow and Partners who also undertook the task of forming a data base of groyne usage in the U.K. The project included a physical model study undertaken by Hydraulics Research Limited and a full scale data collection programme which was the responsibility of Prof. P. Holmes of the Civil Engineering Department at Imperial College of Science and Technology. The present author was employed at Imperial College with the task, among others, of analysing the full scale data.
In setting up the project CIRIA saw the development of numerical bathymetric evolution models as fundamental research which should not be part of the research project (Summers and Fleming, 1983). The present author was interested in numerical models and the university in fundamental research so it was decided that this line should be followed, not as part of the CIRIA project, but as a supplement to it.
The work reported here falls into three main areas. Firstly the development of a numerical beach evolution model which in itself is split into a number of components. A wave model which solves the nearshore wave climate including the effects of diffraction, reflection, refraction by both bathymetry and currents and wave breaking. A circulation model which predicts not only the depth-averaged nearshore currents but also the distribution of turbulent eddy viscosity over the model area. The turbulence is important for the third component of the overall 20 model, the sediment model which solves the suspended sediment transport equations and allows predictions of the accretion and erosion patterns to be calculated.
Secondly, a set of physical model tests was carried out independantly of those at Hydraulics Research Limited, to provide proving data for the numerical circulation model.
Thirdly the full scale data set that was collected from sites in Norfolk and Lincolnshire is presented. It includes not only the measured nearshore wave and current field but also some measurements that were taken of beach profile changes that occurred during the measuring period.
Chapter 2 presents a review of the literature on groynes, including the recommended design approach, as well as articles on physical model tests and previous numerical models of groyne behaviour.
Chapter 3 concerns the physical model tests at Imperial College while the full scale data collection results are presented in Chapter 4.
The three numerical models are presented in Chapters 5 to 7. Each chapter includes a literature review relevant to the model concerned and the derivation, numerical representation and verification of that model.
In Chapter 8 a series on runs of the full bathymetric evolution model are undertaken and the results presented. These results together with the material from the other chapters in discussed in Chapter 9 which also includes the conclusions from the present work plus recommendations for further work in this field. 21 CHAPTER 2 LITERATURE REVIEW
2.1 INTRODUCTION
The literature specifically on the subject of groynes is somewhat limited but can be divided into three catagories: reviews of groyne behaviour and design procedures, reports on physical model tests and reports on numerical model tests.
Papers on groyne behaviour and design are often a distillation of field practise and experience, physical model results and numerical model predictions. Such publications often include a brief description of the assumed behaviour of groynes in the nearshore zone and guidelines for selecting design parameters such as groyne length, spacing, height and orientation. The Shore Protection Manual (1984) has such a section and the review paper by Tomlinson (1980) is another example.
Physical model studies have, in general, been carried out using mobile-bed models. The scaling laws, which will be discussed briefly in the next section, mean that the results of these studies must be treated with some caution when they are related to full scale conditions. It may be, in fact, that physical model studies can only show what will happen to sand or ground coal on a scaled beach under attack from scaled waves. Despite this, the papers are important because they represent a significant part of the history of groyne research. They have also influenced the way in which groynes have been designed and applied in the field
Numerical modelling in this topic began with Pelnard-Considere (1956) and has advanced and become more complex since then. The early models tended to predict the advance or recession of the shoreline adjacent to a total littoral barrier, neglecting such effects as wave refraction, diffraction or even currents. The more recent models allow for much greater flexibility. The potentially most accurate are general, nearshore, bathymetric evolution models capable of predicting changes not only from simple groynes but from any structure in the nearshore zone.
Since this thesis describes the development of such a model there is a great deal of literature relevant to individual model aspects such as wave refraction and diffraction, the specification of nearshore currents, and sediment dynamics. This literature will not be dealt with here but will appear at the beginning of the relevant chapters later in the thesis. 22 2.2 GROYNE BEHAVIOUR AND DESIGN
Although it would have been possible to go back through the years and investigate how the understanding of groynes and their design have changed, this will not be done. Instead a brief review will be carried out in an attempt to define the 'state of the art' thinking on groyne behaviour and design. For this purpose the Shore Protection Manual (1984) will be considered followed by the review article by Tomlinson (1980).
The latest Shore Protection Manual (1984) contains a much expanded section on groyne behaviour and design compared to earlier editions. More attention is given to flow patterns that result from the groynes interacting with the nearshore flows. It is suggested that groynes may cause the formation of rip currents which could take littoral material from the groyne bay and deposit it far offshore. Although these flow patterns are mentioned it appears that they are not used when the basics of groyne behaviour and design are being considered.
The general philosophy is that groynes trap part of the longshore littoral drift. This is held on the updrift side of the groyne near the shoreline. This in itself perhaps presumes that the swash zone transport forms an important part of the longshore transport. The build-up of material causes the shoreline to move seaward, whereas on the downdrift side of the groyne the subsequent loss of littoral supply causes erosion and a shoreline recession. If this process is repeated in all groyne bays a series of small beaches is formed which are in local equilibrium with the local wave conditions.
From the design point of view, the length of the groyne is therefore related to how much of the littoral drift is to be trapped. The spacing between groynes is determined by considering how the beach between groynes would re-align due to the trapped material, and the groyne height is also related to how much sediment is to be trapped before over-topping.
Tomlinson (1980) proposed that groynes alter the local topography by reducing littoral drift and/or accreting beach material inside the groyne bay. The idea that groynes may reduce littoral drift is probably based on physical model tests carried out at the Hydraulics Research Station and reviewed in the next section.
The selection of design parameters is similar to that presented in the Shore Protection Manual although more emphasis is given to groyne height in relation to its effect on local scour. The spacing is determined from experience and a spacing to length ratio (S/L) of between 1 and 4 is recommended. 23 In summary, the understanding of groyne behaviour and consequently their design is based almost entirely on ideas inferred from field and physical model experience. Although it is realized that circulatory flows may form, the effect of these circulations on erosion and accretion patterns is not considered. The assumption with regard to beach re-orientation applies well to long high groynes extending far enough offshore to alter the local wave environment. The situation where groynes are shorter is not so clearly defined.
Finally, there seems to be a lack of knowledge on exactly how groynes cause sediment to be deposited in certain areas and eroded from others. An understanding here should relate to the actual physical processes of sediment redistribution rather than the intuitive idea that waves approaching at an angle ’push' sediment into this corner and out of that.
2.3 PHYSICAL MODELS
2.3.1 SCALING LAWS
Physical models have been used widely in hydraulics for many years. If proper consideration is given to the laws of scaling the results can be extremely accurate. There are cases, however, when if the scaling laws are to be satisfied the only accurate scale is full-size. A movable-bed, short-wave model is such a case.
The wave field is governed by gravity as the principle force, and if diffraction is to be modelled correctly an undistorted model must be used with quantities scaled according to the Froude law. In this case the physical dimensions of the model are reduced according to the model scale, with time and velocities being scaled to the square root of the model scale.
The addition of sediment means that three features of sediment behaviour must also be scaled correctly: the initiation of sediment motion, bed load transport and suspended load transport, each requiring separate, and different, scales. The compromise that is often made is to reduce the size of the sediment particles, but by less than the other physical dimensions, and to use materials of lower specific gravity than beach sand. Coal dust, bakelite and ground telephone receivers have all been used in the past.
In the light of these difficulties the approach, often, has been to scale the sediment as well as possible and then to look for qualitative trends rather that to try and scale the sediment exactly and produce quantitative results. Kemp (1962), for example, chose not to work to any particular scale in a physical model study 24 but selected the sediment such that it behaved like shingle at full scale. This of course presupposes that the full scale sediment dynamics are fully understood, especially with respect to the relative importance of suspended versus bed load. This is probably not the case, as will be discussed further in Chapter 7.
2.3.2 PHYSICAL MODEL TESTS
Two different types of groyne structure have been tested in physical models. Barcelo (1968) and Hulsbergen et al (1976) considered surface-piercing groynes constructed of rock and rubble where the length of the groynes meant that in most cases they would trap all or nearly all of the littoral drift. Price and Tomlinson (1968) and Hydraulics Research (1986) on the other hand considered lower, smooth, vertical sided groynes more in keeping with the type of structure used extensively on the East Anglian coast. The results from the latter tests are of more direct relevance to the present study, however, the former are also of interest.
Many of the early numerical models of groyne behaviour which will be reviewed in the next section assume a total barrier to waves and currents and the predictions from such models have been compared with total barrier groyne physical models.
The problems associated with scaling the sediment mentioned in the previous section, together with the general lack of regard for the proper longshore current, mean that many of the results must be used cautiously. However, their study is rewarding.
Barcelo (1968) deliberaterly chose large groynes so that the beaches could be classified as "independant physiographical units, i.e., stretches of beach located between groynes long enough to prevent the transposition of mobile material at their edge". The tests were not carried out to any particular scale although the wave heights and periods used suggest a scale between 1:30 and 1:50 might have been appropriate. Different sediments were used in an effort to determine the effect of high suspended loads on beach evolution. It would appear from the tests that much of the transport took place in the swash zone although mention is made of the wave generated longshore currents. The tests have been used for comparison by authors of numerical models that consider groynes as a total littoral barrier.
Hulsbergen et al (1976) report some physical model experiments carried out at the Delft Hydraulics Laboratory (1976) in an attempt to verify Bakker's (1968) numerical model. Once again the groynes were surface-piercing, constructed from rock and rubble. Their length } however was such that they did not always 25 completely cut off the longshore transport. An external recirculation system was used in an attempt to model the longshore current correctly.
The authors classify the tests as either 'good' or otherwise. It appears that during the 'good' tests the groynes trapped the total longshore transport leading to a favourable comparison with Bakker’s numerical model. During these tests the rearrangement of material is that predicted by the model, namely, accretion on the updrift side of the groyne and erosion on the downdrift side. One such test result is shown in Fig. 2.1
During the 'other' tests, the groynes, for one reason or another, did not trap the total littoral drift. The resulting bathymetric changes led to secondary wave effects and subsequent 'odd' behaviour induced by the groynes. In one of the tests there was in fact erosion updrift of the groyne and accretion downdrift. This is also shown in Fig. 2.1. The authors concede that the variation in the wave field had "a strong negative effect on the homogeneous conditions sought".
Whereas the previous papers reported studies of massive surface piercing groynes the works discussed below relate to narrow, smooth, vertical-sided groynes that will normally be partially submerged for a proportion of their operation in a tidal regime.
Price and Tomlinson (1968) carried out tests on a series of groynes on a beach at spacing varying from L to 2L where L is the groyne length. The tidal height was varied during the tests and the groynes extended to the low-water mark (as is recommended practice in the field). Coal dust was used as the sediment and the model scale is not mentioned. There is no mention of longshore currents being generated or maintained by external recirculation.
The authors found that the groynes reduced the total littoral transport by 30% but that the beach between the groynes did not accrete material, rather, bars were formed off the ends of the groyne tips. The authors concluded that this build up was perhaps the most significant fact that had emerged from the tests. On the equilibrium open beach 60% of the littoral drift occurred between high and low water. This was reduced to 30% by the addition of the groynes. The authors postulated that should the littoral drift return to its open beach value the groynes would cause a redistribution of the littoral drift profile with less being transported landward of the groyne tips and more off the ends.
Although it is tempting to take the results of these tests and apply them in the field it must be remembered that without proper consideration being given to 26
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c)
FIGURE Physical Model Results of Hu 1sbergen e t al 2.1 a) Test 22, a "good” test (1976) b) and c) Test 18, an "odd" test the two factors which govern littoral drift, the longshore current and the sediment dynamics, the results must be treated cautiously. The two conclusions of the tests: that groynes reduce littoral drift, but do not result in accretion inside the groyne bays, are part of the folk-lore of groyne design. It is difficult to tell how much of the folk-lore is based on such tests.
The experiments of Brater and Ponce-Campos (1976), again using vertical sided groynes with sand as the movable bed material, also showed a reduction of total littoral transport on a beach after the addition of groynes. The same comments made above regarding longshore currents and sediment scaling also apply to this series of tests.
Summers and Fleming (1983) conclude that since "the problems associated with scaling the properties of materials transported in water are considerable, whereas the problems associated with scaling of water circulation are not so forbidding ... the idea is to concentrate on fixed bed models of beaches ...". These recommendations led to experiments at Hydraulics Research Limited (1986) on a fixed bed model.
The work at Hydraulics Research Limited (1986) was designed to model conditions at Sea Palling in Norfolk. For this reason the majority of the tests were carried out using a typical winter beach profile that had been measured at the site. Only a few early tests were carried out on a plane sloping beach. The scope for direct comparison between these tests and those undertaken in the present study, described in Chapter 3, is limited and will not be pursued in detail.
Observation of the flow patterns led the authors to the following conclusions:
1. The effect of a groyne on the flow was to divert part of it offshore and to create a circulation in the lee of the groyne. The extent to which this happened depended on the height of the groyne.
2. Currents in the lee of a groyne often have a reverse flow near the shoreline and a seaward flowing 'rip' on the downstream side of the groyne.
3. The magnitude of velocities landward of the groyne tips is generally reduced below the open beach value and there is a corresponding increase seaward of the groyne tips. 28 4. The flow within the groyne bays was highly stratified. It has been noted in an interim report, Hydraulics Research Limited (1984), that there was often a significant difference between the near bed velocity and that at the free surface. This would be very significant in relation to sediment transport.
5. Varying the spacing-to-length ratio from 1 to 2 seemed to have little effect on the circulation patterns.
6. There seemed to be no advantage in varying groyne inclination by ±10* from the perpendicular to the beach contours.
7. Random waves did not cause any major changes in the flow patterns.
A review of the experimental results confirms these conclusions and, in addition, the following observations by the present author are put forward.
8. For the multiple groyne tests there is a general lack of repeatibility from one groyne bay to the next.
9. One effect of the flow stratification is to make the true mean flow difficult to measure. Many of the multiple groyne tests show groyne bays with an apparent net outflow. This was also observed in the present author's series of tests and will be discussed further in Chapter 3.
The important factors for an understanding of groyne behaviour are, therefore, the marked change in flow pattern inside the groyne bays and the effect that the groynes have on diverting the longshore flow passing the groynes.
It may be that the circulation inside the groyne bay would tend to hold sediment in place. The longshore flows may be diverted offshore to where the wave-induced pick-up of sediment is reduced, leading to an actual reduction of littoral drift.
It is not possible to test these ideas here. However, reference will be made to them in Chapter 8 when the bathymetric evolution model is evaluated. 29
2.4 NUMERICAL MODELS
The earliest numerical models of a beach with a groyne system follow the work of Pelnard-Considere (1956). By assuming an equilibrium beach profile, no currents and constant wave direction at a small angle of incidence the author was able to calculate changes in the orientation of the shoreline as a consequence of longshore transport. This was effectively a ’one-line shoreline model'.
Bakker (1968) extended the work by adding a second contour thus creating a two-line model so that a more realistic profile could be achieved. Bakker assumed that the beach would modify itself to maintain an equilibrium profile. Any deviation from equilibrium would be restored by subsequent onshore or offshore transport. Bakker suggested that other factors such as rip currents and a variable wave angle should be taken into account. Wave diffraction was thought to be of only minor importance.
Huslbergen et al (1976) conducted some physical model experiments and compared the results with the predictions of Bakker's theory. The authors found large variations between theory and experiment, but part of this must have been due to deficiencies in the latter. The conclusion reached was that the theory was good in cases with a stable well-defined longshore current system and with groynes that interrupt a substantial part of the longshore sand transport. In more complex systems the theory was inadequate. The authors recommended that currents should be taken into account.
Price et al (1972), following the work of Bakker, solved the sediment continuity equation using an explicit finite difference model. The results were compared against those of a physical model study conducted at Hydraulics Research Station. The physical model reproduced a simple wave field with the groyne interrupting the total longshore sediment transport. The comparisons between the physical and numerical model were very good, as can be seen in Fig. 2.2.
Le Mehaute and Soldate (1978) took the theory of Pelnard-Considere to what was believed to be its limit by considering refraction, diffraction and beach slope. Refraction was found to be particularly important. It was postulated that an n-Iine model could be formulated to solve the complete nearshore bathymetry.
Perlin and Dean (1978) also proposed that an n-line model should be possible and such a model was reported by Perlin and Dean (1985). The model included refraction, shoaling and diffraction as well as both the components of sediment transport i.e. longshore and cross shore. The model could not however accomodate
31 offshore bars and the cross shore transport was based on the assumption and knowledge of an equilibrium beach profile.
The resulting calculations for a groyne field gave unrealistic contours, particularly off the ends of the groynes, where the assumptions about the equilibrium profile led to disturbances in the bathymetry extending far past the end of the groynes. The authors concluded that improvements would be achieved with the use of better longshore and cross shore sediment transport relationships as well as an improved wave-field model.
The early work of Pelnard-Considere and Bakker provided important new tools for numerical simulation, however, the present author feels that this approach reached its limit around 1978 with the model of LeMehaute and Soldate. Since then the availability of powerful computational facilities makes a more general model quite feasible.
Fleming and Hunt (1976b), for example, give details of such a model - a general two dimensional depth-averaged bathymetric evolution model. The total model was solved in steps with a wave-field stage, longshore current stage and finally a sediment transport stage. The solution was iterative in that bathymetric changes were allowed to affect the wave field, which in turn affected the prediction of the bathymetric changes. The resulting predictions were in good agreement with physical model tests.
The authors acknowledged some of the weak points in the model. The sediment transport formulae required empirical site data and did not include the important cross shore transport. The diffraction model could only be used for simple breakwater cases. Despite these limitations the model represented a large step forward from the n-line model approaches that were still being pursued.
Once this basic framework is adopted, improvements in the overall model performance are possible by modification of the individual modules. Details of the many bathymetric evolution models that have been reported will be given later in Chapter 7. 32 CHAPTER 3 PHYSICAL MODEL TESTS
3.1 INTRODUCTION
Although, as noted in Chapter 2, there have been several series of physical model tests involving groynes, the only one, to the author's knowledge, to concentrate on achieving a proper longshore current and to measure in any detail the resulting flow patterns is reported by Hydraulics Research Limited (1986).
That series of tests was concerned with modelling the beach and a set of three groynes at Sea Palling in Norfolk where a set of full scale data had been collected as part of a research program on the 'Effectiveness of Groynes' under the auspices of the Construction Industry Research and Information Association, London. The full scale data was intended for use in calibrating the physical model and the model tests therefore concentrated on the same three groyne arrangement and local bathymetry.
It was decided that an independant set of scale tests should be carried out as part of the present work. No attempt would be made to reproduce the Sea Palling site conditions, rather, a more general groyne layout would be investigated with a view to proving the numerical models under development. The results could also be compared qualitatively with some of the early Hydraulics Research tests carried out on a plane beach.
3.2 BASIN CONFIGURATION
Previous model tests that have concentrated on setting up a proper longshore current, for example, Hydraulics Research (1986) and Visser (1982), have had the advantage of a large basin for the tests. The basins used in the above-mentioned tests were at least 30 metres wide with more than 15 metres from the wave paddle to the shoreline. In these situations the influence of the 'end effects', that is, the disturbances caused by the side walls, wave guides and the longshore pumping apparatus could be minimized during the tests by conducting the experiments sufficently far from each end.
The hydraulics laboratory at Imperial College is quite large but not large enough to accomodate a basin of such dimensions. The tests were therefore carried out in a basin approximately 5 metres wide and 6 metres from the wave paddle to the shoreline. The basin is shown in Plate 1 and the general layout is given in Fig. 3.1. The beach has a slope of 1:29 and at a scale of 1:36 breaking wave heights
PLATE 1 WAVE BASIN - GENERAL VIEW 35 of 2.0 metres can be generated with a period around 8 seconds. The bed of the model is constructed from sheets of galvinized iron and is quite smooth. No attempt was made to model the bed roughness found in prototype. The wave paddle is a plunger type mechanism with an approximately triangular cross-section. The recirculation system is driven by two 0.55 kW electric motors capable of delivering up to 10 litres per second. The downstream intake is a simple baffle while the important inflow is regulated by a series of 20 gate valves. The valves and manifold are shown in Plate 2.
In setting up the basin the end effects were found to be very important and every care was taken to ensure that they were minimized. After much effort it was found that a proper longshore velocity profile could be maintained to within 1.0 metre of the lateral boundaries leaving a 2.0 metre wide section of the model in which to carry out the tests.
The recommendations of Visser (1982) were of great assistance during the setting up procedure and were complied with as far as possible. Visser suggested an opening in the upstream wave guide, Fig. 3.1, for the current distribution system with a width of 1.7 to 2.0 times the expected surf zone width. The figure chosen for the present work was 1.3 times the surf zone width which, although smaller than ideal, was as large as the basin could sensibly accomodate. Visser also suggested an opening in the downstream wave guide of 1.2 times the expected surf zone width. In extensive preliminary tests a figure of 1.3 times the surf zone width was found to give good results.
The external recirculation system, used to simulate the longshore currents, was controlled by the 20 gate valves discharging into the basin through baffles 100 mm wide and 25 mm high. The total width of the distribution system was therefore 2.0 metres. Valves were used rather than weirs, as had been employed at Hydraulics Research and by Visser, because it was felt that the flow could be controlled more accurately. The flow was measured at each valve using a calibrated orifice plate; hence an accurate and controllable longshore profile could be generated. The flow at the downstream end was not controlled to any degree.
One of the problems encountered by Visser (1982) was that of a global circulation being set up in the basin. Visser concluded that the longshore current is uniform and optimized if the circulation in the basin is minimal. Not only does the global circulation affect the longshore current profile, it also affects the wave field, particularly near the downstream wave guide, due to wave-current interactions.
In an effort to reduce this global circulation in the basin, the intake to the PLATE 2 DISTRIBUTION SYSTEM FOR LONGSHORE CURRENTS p 3 7 recirculation system was isolated from the main body of the basin. This meant that water extracted from the downstream position had to come from the longshore flow. After a number of trials the flow from two of the gate valves was diverted such that they did not discharge into the longshore current. The overall effect of this modification was that the recirculation system was pumping less water into the longshore current than it was taking from the system. It was found that this reduced dramatically the global circulation in the basin and produced streamlines that were parallel to the shoreline as recommended by Visser (1982).
Wave heights were measured across the basin and are plotted in Fig. 3.2. The variation was measured at ±12% from the mean value which is close to the figure of ±10% quoted by Visser (1982).
The resulting longshore velocity profile is shown in Fig. 3.3. It will be noticed that the velocities increase slightly in the downstream direction. This is partly due to the external recirculation system but mainly due to the end effects and is not considered as being excessive.
3.3 MEASUREMENT TECHNIQUES
The author followed the well trodden path of reviewing techniques for measuring a mean velocity in an oscillatory environment, an environment that is also very shallow. The depth at mid-groyne distance from the shore was only 15mm. Hydraulics Research (1984) used dye injected into the flow and a video camera. The recording system included an electronic device that added a digital stopwatch to the recording. This allowed elapsed time to be calculated and the dye position to be plotted when the video-tape was played back frame by frame. Visser (1982) preferred to time dye manually over a fixed distance. Although the present author eventually adopted the method used by Visser an attempt was made to construct an electronic device to measure the mean velocity.
The concept, which is mentioned in literature on flow measurement, is to inject a small 'slug' of salt solution into the flow and to time it electronically over a fixed distance using two conductivity probes. A 'Churchill' wave gauge unit of resistance type was adapted to measure the conductivity and convert it to a voltage. This voltage was sampled using an eight channel analogue to digital converter which fed results directly to a BBC microcomputer. By sampling at a sufficently high frequency (100 Hz.) it was possible to calculate the travel time of the salt 'slug' between the probes and hence calculate the mean velocity. A typical output from the unit is shown in Fig. 3.4. FIGURE Wave Heights in Wave Basin 3.2 The results are for a position 2.0 metres from the shore line.
FIGURE Signal from Current Velocity Probe 3.4 u The advantage of the meter lies in the fact that the velocity is averaged over only a very short distance (20mm in this case) and that the measurement is not affected by human subjectivity. The major disadvantage is that the meter only gave good results where the diffusion was small and where the oscillatory motion was small compared to the mean motion. It worked well in the very shallow water but did not give consistent results in the deeper water. Eventually its development was abandoned in favour of manual timing.
The velocity at a point was taken as the consistent average over a number of readings (at least three) with the dye being timed over a distance of 150mm within the groyne bays and close to the shoreline, and 400mm outside the line of the groynes where the velocities were higher and the velocity gradients lower. The resulting elapsed times usually varied between 2.0 and 4.0 seconds.
The experimenter is faced with a dilemma when measuring velocities in this way. On the one hand it is best to time over as long an interval as possible to minimize the errors due to hand timing. On the other hand the velocity should ideally be measured at a point, especially where the velocity field is changing quickly in a spatial sense. The selected distances of 150mm and 400mm represent the best compromise taking the above-mentioned facts into consideration.
Measuring velocities in this manner often led to significant variations between individual measurements at a point. In this case the best average was taken and although the confidence in each individual value may be low the overall pattern forms a consistent picture of the flow. Errors are estimated as +10%.
3.4 TEST RESULTS
3.4.1 INTRODUCTION
The wave parameters, height, angle and period, used during the tests were kept at constant values. Although it would have been instructive to be able to vary them, the size of the basin and the way that it was set up dictated that they should remain constant. The wave paddle itself was not designed for ease of movement and a change in offshore wave angle would have required virtually reconstructing the entire wave basin. The wave height was easily variable but was set to the maximum that the dimensions of the recirculation system would allow (see Section 3.2). A smaller value would have reduced velocities and brought additional scale effects into play including reduced accuracy in the velocity data. With wave height and angle fixed it was not considered instructive to vary wave period which was set to 1.1 seconds or 6.6 seconds in prototype. The wave -♦ 42
parameters used are listed in Table 3.1.
In addition to the simple longshore velocity test already mentioned nine tests were run under different groyne configurations. Single, two and three groyne arrangements were tested as well as the effects of groyne height and spacing. Details of the tests are listed in Table 3.2 and each test will now be discussed in some detail.
3.4.2 TEST NO.l - PLANE BEACH
The first test, the results of which are shown in Fig 3.3, was to determine the baseline velocity conditions. As noted previously there is a slight increase in velocity towards the downstream boundary which was regarded as reasonable. The streamlines were parallel to the beach, or very nearly, indicating that the global circulation in the basin was small.
3.4.3 TEST NO.2 - 1 GROYNE, SURFACE PIERCING
The major characteristics of this test, shown in Fig. 3.5 are the circulation pattern, with dimensions approximately equal to the length of the groyne, that forms in the lee of the groyne and the offshore component that the groyne imparts to the mean longshore velocities. It should also be noted that the largest velocities are in the region of the groyne tip.
3.4.4 TEST NO.3 - 2 GROYNES, SURFACE PIERCING, S/L=1.0
This test shows one of the standard groyne bay configurations where the spacing between the groynes is equal to their length. The resulting flow pattern is plotted in Fig. 3.6. The upstream groyne behaves in a similar fashion to the single groyne with, again, large velocities near the tip. There is a well developed flow pattern inside the groyne bay with a strong counter-clockwise flow along the shore boundary and outward along the lee side of the updrift groyne. It is interesting to note, however, that there is no corresponding flow inwards along the inside of the downdrift groyne. More will be said about this later in Section 3.4.7.
3.4.5 TEST NO.4 - 2 GROYNES, SURFACE PIERCING, S/L=1.7
The flow pattern for this test, plotted in Fig. 3.7 shows the effect of an increase in groyne spacing. The pattern in the vicinity of the updrift groyne is similar to Test No.3. However the second groyne in this test is suffiently far from the first to allow the shore parallel flow to start re-establishing before being Deep Water Depth 0.35m Period 1.lOsec Wave Height 0.040m Wave Angle 8.0*
At Breaking Wave Height 0.050m Wave Angle 4.4* Surf Zone Width 1.80m
Table 3.1 Physical Model Wave Parameters
Test No. No. of Groynes Length Spacing Type (m) (m)
1 0 —
2 1 0.875 - P 3 2 0.875 0.875 P 4 2 0.875 1.500 P 5 3 0.875 0.875 P 6 2 0.875 0.750 P
7 1 0.875 - S 8 2 0.875 0.875 S 9 3 0.875 0.875 s 10 2 1.125 0.875 s
Groyne Type P: Fully Surface Piercing S: Partially Submerged
Table 3.2 Physical Model Test Details
« diverted by the groyne. The size of the circulation in the groyne bay would therefore appear to be controlled by the length of the updrift groyne and at a spacing to length ratio greater than 2.0 the groynes tend to act as separate structures. This is significant, since the often quoted ratios of spacing to length vary between 1.0 and 4.0. (e.g. Tomlinson, 1980)
3.4.6 TEST NO.5 - 3 GROYNES, SURFACE PIERCING, S/L=1.0
It seems from the plot in Fig. 3.8 that each groyne bay in a two bay field behaves as a separate two groyne bay. The flow past the mouth of each bay is similar and although there are diffences between the flow within bays the major trends are very much the same.
3.4.7 TEST NO.6 - 2 GROYNES, SURFACE PIERICNG, S/L=0.86
The flow pattern for this test, shown in Fig. 3.9, poses an interesting question with regards to continuity and the assumption that depth-averaged flow is being measured. It appears that there is a net flow out of the groyne bay - a situation that could not exist for long!
The explanation for this phenomenen, based on careful observation, is as follows. Although it is assumed that the flow being measured is uniform over the water depth, in fact the flow will be very much three-dimensional. The surface waves, travelling in as a bore, bring in fluid which eventually returns offshore as a mean undertow. Svendsen (1984) suggests that the discharge per unit length of crest can be estimated as the height of the bore above the mean water level times the bore celerity. The small thickness of this bore layer together with its relatively high velocity make it difficult to track using conventional dye tracking. The flow that the dye follows is the slower return flow over a greater depth combined with any horizontal circulation patterns that may be present.
An approximate model of this effect is given in Chapter 6. One important consequence of this observation is a forceful suggestion that to model nearshore flows accurately, especially where sediment is concerned, a three-dimensional model should be used, even if it only a two or three layer model.
3.4.8 TEST NO.7 - 1 GROYNE, PARTIALLY SUBMERGED
Since the groyne modelled here is a constant height above a sloping bed (12mm) its effectiveness as a barrier to the flow gradually decreases in the offshore direction. At the shoreline it is a fully surface piercing groyne and at its toe it can figure Physical Model Results - Test 6 3.9 Velocity Scale 1cm. = 0.3 m/s
VO 50 be but a minor disturbance if the water depth to groyne height ratio is sufficently large. It would therefore be expected to behave, to an extent, like a shorter length surface piercing groyne.
This experiment, the flow pattern for which is shown in Fig. 3.10, illustrates this point. The size of the circulation cell behind the groyne is smaller than the groyne length by as much as 25%. The pattern also shows less deviation of the flow offshore and also smaller velocities at the groyne tip compared to the fully surface piercing groyne.
3.4.9 TEST NO.8 - 2 GROYNES, PARTIALLY SUBMERGED, S/L=1.0
This test, plotted in Fig. 3.11, allows comparison with Test No.3, the fully surface piercing equivalent. As with Test No.7 the effects of a reduced barrier to the flow at the groyne tip are evident as are the differences in the velocities. Some flow non-uniformity is observed. 3.4.10 TEST NO.9 - 2 GROYNES, PARTIALLY SUBMERGED, S/L=0.78
This test was identical to Test No.8 except that the groyne lengths were increased by 0.25 metres or 30%. Due to the increased immersion of the submerged section of the groyne tip the flow pattern given in Fig. 3.12 differs only marginally from the previous example. In this test the water depth to groyne height ratio at the groyne tip was approximately 2.5:1.
3.4.11 TEST NO.10 - 3 GROYNES, PARTIALLY SUBMERGED, S/L=1.0
The final test, illustrated in Fig. 3.13, behaves as would be expected from the results of the fully surface piercing case. The updrift groyne bay has less flow along the shoreline than might be expected, however, the other main features are represented.
3.5 DISCUSSION OF RESULTS
The physical model tests are important for two reasons. Firstly they are required to help validate a numerical model developed as part of the work and described in Chapter 6. Secondly they are important as a data set in their own right. As such, the results give a good picture, albeit a scaled one, of how groynes affect the local flow pattern.
The results indicate that a circulation can be expected downstream of a groyne. The circulation will be stronger and more pronounced behind a surface
1.0 m breaker♦ line ♦
«*• 41 < N N N <«, N \ *, N N \ «s \ \ \ *\ *. ^ \ N \ N f f ^ % % » X . t t . N *. . - r / t t \ i •. - t t \ . \ N wave run-up
FIGURE Physical Model Results Test 8 3.11 Velocity Scale 1cm 0.3 m j s
Ui ho
55 piercing groyne. The magnitude of the velocities will generally be smaller than the open beach values at a similar distance offshore, except near the tips of the groynes.
The results also indicate that there is a strong case for modelling in three dimensions since it appears that the resulting flows are quite different from a depth-averaged model. The velocity profile in the vertical plane is particularly important for sediment transport inside a groyne bay.
The tests show that it is possible to obtain reasonable results in a small basin, however, for greater flexibility a larger basin is essential. 56
CHAPTER 4 FULL SCALE MEASUREMENTS
4.1 INTRODUCTION
The collection of a set of full scale data was one of the aims of the CIRIA
Project 310 ’The Effectiveness of Groynes'. It was intended that the full scale data would provide a means of calibrating the physical model experiments that were to be carried out at Hydraulics Research Limited. Due to unforseen circumstances this was not fulfilled but the data are still very important as a data set. The circumstances that prevented any calibration will be discussed in due course.
Sites in Lincolnshire and Norfolk were sought that would provide a suitable arrangement of groynes and open beach to allow the open beach alongshore velocity profile to be measured together with the effect of the groynes on that profile.
Three sites were selected. Sea Palling, in Norfolk, consisted of a stretch of open beach, a series of three groynes followed by more open beach. The beach was predominantly sand, backed by natural dunes. The other two sites, Anderby Creek and Sandilands, both in Lincolnshire, provided a continuous groyne field, in one case backed by sand dunes and in the other by a sea wall.
Although data were collected at all three sites the major effort was at Sea Palling and the results from this site will be presented in detail.
The data were collected by CEEMAID Ltd., a firm specializing in nearshore data collection. All analysis, and some aspects of the data collection, for example the float tracking experiment, were carried out by the present author.
4.2 SITE AND INSTRUMENTS
Sea Palling is located some 10 miles north of Great Yarmouth on the East Anglian coast. Fig. 4.1 shows the general location of the site and Fig. 4.2 a more detailed site plan. Beach cross sections of the open beach section and inside the northerly groyne bay are shown in Figs. 4.3 and 4.4.
The beach is composed mainly of sand although there was some shingle and course shell grit in places. Figure 4.5 shows the results of a sieve test that was carried out on the beach material. It appears a well graded sand with a mean size between 250pm and 500pm. 57 ♦
P beach ''pod* SEM surface elevation meter oPA RUG run-up gauge oPB
ABC 7 j% V •* marram hills
FIGURE Sea Palling Site Plan 4.2
Ln CO BEACH PROFILE 2 0 - 1 0 - 8 4
78
59
FIGURE Beach Cross Section Open Beach 4.3 LONG SECTION CENTRE GROYNES A & B
4.0
A.O.D.) , (m
d 2.0 O SWLRUN 7 8
ID > SWL RUN 5 7 a) 0.0 "aj ‘'g r o y n e A -2.0
-4.0 20 40 60 80 100 120 140 160 180
chainage from baseline (m )
FIGURE Beach Cross Section - Groyne Bay 4.4 0.125 0.25 0.5 1.0 2.0 m m .
particle size
FIGURE Beach Sand Sieve Test Results 4.5 62
The groyne field consists of three groynes approximately 80 metres long and spaced at 125 metres. The groynes are approximately at right angles to the beach contours with angles varying from 90* to 104*. Adjacent to the groyne field in both a northerly and southerly direction is open beach. The groynes are of sheet pile construction with timber beach heads.
Details of the instruments deployed at the site are also given in Fig 4.2. The instruments consisted of what were referred to as beach 'pods' together with a wire resistance wave gauge (Surface Elevation Meter), a number of calibrated wave poles for the visual estimation of wave height, an offshore wave-rider buoy and a number of wire resistance run-up gauges. A radar unit capable of detecting ocean surface waves was also used at the site to measure wave direction.
The beach pod consisted of a pressure transducer together with a two component electromagnetic current meter. The layout of the pod is shown in Fig. 4.6. Deployment consisted of burying the device to the level of the pressure transducer so that the current meter extended some 0.5 metres above the bed. The pressure transducer measured absolute pressure so that mean water level could also be calculated. The signal from the current meter consisted of two orthogonal horizontal components of velocity made up of the mean current plus the wave oscillatory component. The alignment of the instrument was measured at deployment so that the velocity components could be reduced to true bearings.
Eight pods were deployed with data being collected from all but P2 (see Fig. 4.2) for most of the recording period. A sample of the data from the pods is shown in Fig 4.7. The correlation between the pressure and the wave induced oscillatory motion is clearly visible. The data plotted has been adjusted to mean zero.
The data from the run-up gauges will not be reported in this work. The gauges were secured to poles in the beach at 5 metre centres over 50 metres in the cross- shore direction and this allowed beach profile changes to be measured and recorded. This will be reported in a later section.
The offshore wave-rider buoy telemetered data to shore during the recording period. The data allowed comparison of offshore and inshore wave energy spectra which will also be shown later in the chapter. 63
cD e.m. current meter
E LT> O pressure transducer
777--- 777 VT / 777 TTT TTT N- data & .) power
FIGURE Beach "Pod" 4.6 u
POD 3 v
u
P O D A v
u
P O D B v
60 sec.
u, v = orthogonal velocity components (Ws) p = pressure transducer output ( m)
FIGURE Typical Data from Beach Pod 4.7 4.3 DATA RECORDING PROGRAMME
Data were recorded at the site between 10th October and 26th October 1984. It had been decided by the CIRIA Steering Group that the period around high water would be of the greatest interest and hence data collection was concentrated at that time. During the daytime a 20 minute data set was recorded at high water and at hourly intervals up to three hours before and after. At night only the high water record was collected.
The instruments were sampled at 10 readings per second for 20 minutes giving 12000 data points per channel per record. On collection, the data set was subjected to a brief analysis on site to determine the minimim, maximum and mean voltage on each channel. These values were printed and used in subsequent analysis.
4.4 D A T A ANALYSIS
4.4.1 INTRODUCTION
The analysis of the data was carried out at two levels. Initially^ the data was analysed using simple techniques to check its validity and to provide information on the behaviour of mean water level, the levels of wave activity and the mean flow pattern near and around the groynes. Once this had been carried out the data was analysed in a more detailed fashion to check such features as the directional spread of the oscillatory wave data, the comparison of offshore and inshore wave energy spectra and the comparison of spectra calculated from pressure and velocity records.
4.4.2 TIDAL ELEVATION AND MEAN FLOWS
The mean value of the pressure channel allowed the mean water depth and, therefore, mean water level to be calculated for each record. Plots of the mean water level measured at two of the pods are given in Figs. 4.8 and 4.9. Since consecutive hourly data were only recorded during the daytime high water, only part of the harmonic variation in tidal elevation can be seen. The progression from springs tides to neap tides and back again is quite obvious.
The mean velocity measured at two of the open beach pods is shown in Figs. 4.10 and 4.11. The direction of the open beach has been broadly classified as running north-south. The mean velocities illustrate the tidally dominated flow; the velocity plots over two weeks all follow the same pattern. In fact it proved impossible to discern any wave-induced current in the total signal at all. This was the major reason for the data not being used to calibrate the Hydraulics Research MEAN WATER DEPTH POD - 3
FIGURE Mean Water Depth at Pod 3 4.8 FIGURE Mean Water Depth at Pod B 4.9 north current velocity (m/s) 0.3 0.2 October 4 8 9 1 CO ON FIGURE 1 1 . 4 Mean Velocity at Pod B north current velocity (m/ s ) south O. 0.2 0.1 0.0 0.1 0.2 0.3 0.4 3
I 12 14 - n B n P T N F R R U C E R O H S G N O L N A E M 16 18 October 4 8 9 1 20 22 4 2 6 2 70 model. That model assumed that longshore currents were wave generated. Since no wave generated currents could be detected on site during the recording period, calibration was impossible.
The lack of wave generated currents must be due in part to the lack of any real storms. From the initial analysis of pressure records it was possible to calculate a wave height parameter which was equal to the sum of the highest crest and the lowest trough. This is of course similar to a parameter used by Draper (1966) in the calculation of significant wave height. It is possible to give an approximate estimate of significant wave height: (based on Draper, 1966)
Hsig - 0.6 HX (4.1)
H| is plotted in Fig 4.12 and it can be seen that H sjg probably never exceeded 1.5 metres and would generally have been below 1.2 metres during the recording period.
4.4.3 MEAN FLOW PATTERNS
Although some 116 data sets were collected and analysed for mean flow the results shown in Fig. 4.13 are typical of most of the sets. The main features are:
1. Reasonable correlation between the three open beach measurements. If anything the pod furthest from the shore does have a slightly higher magnitude as would be expected in the case of a tidal current.
2. The three pods off the ends of the groynes show lower velocities than the open beach values even though the distances from the shoreline are similar. The directions of the mean flow seem to be influenced by the presence of the groynes.
3. A much reduced velocity at the pod inside the updrift groyne bay (P5). The positioning of the pod meant that it was sheltered to a large extent by the groyne next to it. The mean flow at this location also tends to be directed out along the groyne rather than alongshore as at the other pods.
4. The dramatic change in tidal flow in the course of an hour (between runs 47 and 48) and in two hours (between 47 and 49). During this time almost complete reversal has occurred. OFFSHORE WAVE HEIGHT
FIGURE Wave Height Record for Experiment Duration 4.12 t: r r r- ■' r r. r \ r- ......
FIGURE Mean Flow Patterns Records 43-49 4.13 r j_ l , " " - *- " i i - " / /.
FIGURE 4.13con t. 5. The comments of points 1 to 3 above also apply to the later records in this tide. The pod inside the groyne bay is now adjacent to a downdrift groyne. It still shows a flow out along the groyne direction.
4.4.4 SURFACE WAVE ENERGY SPECTRA
Spectral analysis of selected records was used to compare the wave energy spectrum calculated from the pressure record with that calculated from the wave-rider buoy. Fig. 4.14 shows one such comparison. The agreement in this case is good and other comparisons were generally also good although the wave-rider signal suffered from high frequency noise and this end of the spectrum was suppressed numerically.
4.4.5 WAVE OSCILLATORY MOTION
By plotting on orthogonal axes the simultaneous currents from both channels of the electromagnetic current meter it is possible to determine both the main axis of oscillation and the variation from this preferred direction. Fig. 4.15 shows such a bivariate plot. The scale interval is 0.1 m/s. Further calculation revealed that the distribution was bivariate Gaussian. By integrating numerically such a data set it is possible to calculate the path which a water particle might follow. The path shown in Fig. 4.16 is approximate only but shows that the sea was quite unidirectional. The data set plotted was the same as that analysed in Fig. 4.15.
4.5 FLOAT TRACK EXPERIMENT
In addition to the collection of wave and current data using electronic transducers an additional 'low level’ data collection program was carried out involving a series of float tracking exercises. A preliminary set of tracks was recorded in May 1984 and a second set during October 1984. In all 37 float tracks were recorded.
The float tracking system was designed to be able to be carried out from the shore without the aid of a boat. This decision was taken since it was felt that the necessity to launch a boat may have inhibited the ability to collect on 'interesting' days. The system employed consisted of two polar plotters, surveying instruments mounted on a plane table that allowed bearings to be sighted and marked straight to paper, floats and a float launching device. p o d ------
buoy ------
FIGURE Wave Energy Spectra from Pod and Waverider Buoy 4.14 SEA PALLING RUN 073 POD 3
U>V HISTOGRAM
0 0 0 0 0 0 0 0 0 0 0 / 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 y1 0 0 0 0 0 0 0 0 0 "0 0 0 0 0 0 2 2 3 0 1 0 0 0 0 0 0 0 0 0 0 2 0 V <5 3 1 1 0 0 0 0 0 0 0 0 0 2 Q 21 122 23 5 0 0 0 0 0
0 0 0 0 0 2 7 13 45 j 52 34 5 1 0 0 0 0 0 0 0 0 0 1 15 34 W 63 16 4 1 0 0 0 0 0 0 0 0 c 6 13 65 54 7 4 0 0 0 0 0 0 0 0 0 0 5 44 30 108 44 10 4 0 0 0 0 0
FIGURE Bivariate Plot from Current Meter A.15 W A T E R PARTICLE DISPLACEMENT
FIGURE Plot of Water Particle Displacement 4.16 78
The floats, illustrated in Fig. 4.17, were launched from a converted cross-bow and designed to give maximum drag in the water and minimum drag in the air. It was possible to fire the floats some 40-50 metres from the shore. The floats themselves were visible to the naked eye to a range of about 200 metres.
Sample float tracks from two days, the 9th of May and the 12th of October are plotted in Figs 4.18 and 4.19. These days were chosen to illustrate the two nearshore current regimes that can exist at the site. The earlier data includes wave induced currents as there was a good sea running with breaking waves of 2.0 metres height. The surf zone was some 50 metres wide inside the groyne bays and further north the surf zone width was estimated at 100 metres. By contrast the latter date had a breaking wave height of between 0.5 and 0.75 metres with a negligible surf zone.
From experience on site and with reference to the float track plots a number of observations are possible. With reference to Fig. 4.18 (May 9th):
1. The floats are diverted by the leading groyne and once diverted continue parallel to the shore without returning inshore.
2. Eddies are formed near the tip of the leading groyne. Their position probably varies. On a number of occasions floats hovered in one place for a number of minutes before continuing alongshore. Examples of such behaviour can be seen in tracks 6 and 8.
3. The longshore flow within the groyne bay has a lower velocity than that outside. This may be due partly to the normal longshore velocity profile although it could also be influenced by the groynes.
With reference to Fig. 4.19 (12th October)
4. The tidal current profile is evident from tracks 3, 1 and 4 with the velocity increasing in the offshore direction.
5. Where the groyne is submerged the water that flows over the top has a much increased velocity. This can be seen in track 3. The effect of this is to reduce the amount of water diverted by the partially submerged groyne.
6. The speed at which the tidal flow reverses direction is impressive. Track 8 was taken only 1 hour after track 4, during which time almost 79
CURRENT TRACKING FLOAT
FIGURE Float used in Float Track Experiment 4.17 F L O A T T R A C K S 9-5-84
FIGURE Float Track Records May 1984 4.18 F L O A T TRACKS 12-10-84
20 m scale
FIGURE Float Track Records October 1984 4.19 82
complete reversal has occurred.
4.6 BEACH SURVEYS
The operation of the run-up gauges required them to be fixed on poles on the beach at approximately 5 metre intervals and these points were surveyed at regular intervals to determine the level of the gauges. During these surveys the level of the beach over the length of the run-up gauges (50 metres) was also taken. From this data it is possible to monitor the behaviour of the beach both on open beach conditions (RUG 1) and inside the groyne bay (RUGs 2,3,4). Unfortunately the total sand movement cannot be plotted since the surveys only extended to some 60 metres from the baseline. Despite this drawback some interesting features can be observed.
Fig. 4.20 shows the volume of sand above ordinance datum per unit width of beach for the period 10th October, 1984 to the 24th October, 1984. The main features of the plot are a gentle deposition of sand on the beach over the period extending until 0800 20th October. The tide on 20th October with high water at 1700, with increased wave action, then caused the beach to drop, both on the open coast and also inside the groyne bay. The subsequent sea conditions, including further storms, moved sand in and out of the section being measured with little net long-term change.
While much of the early changes in beach profiles are general throughout all sections, the storm associated with the high water at 0500 21st October built up Sections 2 and 4 while reducing levels on Section 3 - an indication of a remoulding of the beach inside the first groyne bay.
It is interesting to note that the tide with high water at 1800 on 22nd October built Sections 2 and 3 while the open beach section dropped dramatically.
Figures 4.21 to 4.23 plot the beach profiles at three interesting times which illustrate how changes in the toal beach volume are achieved. The building of the beach up to 0800 20th October is achieved by adding essentially a uniform depth of sand to the majority of the profile - this is evident on all three plots. Although not shown, the subsequent drop after the high water at 1800 on 20th October was the result of the beach returning to near its initial state as plotted at 1200 on 11th October.
The substantial remoulding of the beach by the tide with high water at 0500 on 21st October can also be seen in the plots. VOLUME OF BEACH AT R.U.G.'s
October 1 9 8 4
FIGURE Total Beach Volume 4.20 BEACH PROFILE AT R.U.G. 1
11-10-84 20- 10-84 21- 10-84
FIGURE Selected Beach Cross Sections Section 1 4.21 BEACH PROFILE AT R.U.G. 2
d 11-10-84 d < 2 0 - 10-84
je 21 - 10-84
c o ro > aj ai
FIGURE Selected Beach Cross Sections Section 2 4.22 BEACH PROFILE AT R.U.G. 3
FIGURE Selected Beach Cross Sections - Section 3 4.23
CO 0 ~ N 87 4.7 DISCUSSION OF RESULTS
One of the main aims of the data collection experiment was to provide calibration data for the physical model at Hydraulics Research Limited. Due to the low level of wave activity during the recording period the detection of wave induced currents was not possible. The data set does, however } provide valuable information about the flow patterns and beach behaviour in and around groyne bays.
One of the main features of the physical and numerical model studies is the prediction of a large eddy in the lee of the groynes. This was not observed in the field although that is not to say that it did not exist. The instruments and recording programme did not allow readings to be taken in the surf zone adjacent to the shore, exactly the place where the return flow would have been expected.
The beach surveys indicate some of the ways that groynes might behave on the coast. There is evidence that at times the groynes are unable to prevent erosion when material is removed by what is presumably pure offshore transport. At other times the groyne bay acts as a separate unit shifting sand around in response to the local wave conditions. It has also been seen that the groyne can hold material on the beach while adjacent open beaches are dropping, see. 4.20. 88 CHAPTER 5 THE WAVE MODEL
5.1 INTRODUCTION
The solution {or the nearshore wave field is required for two aspects of the general nearshore model. Firstly, the solution allows the calculation of the radiation stresses which produce the main driving force for the longshore currents and are therefore an important part of the circulation model. Secondly, the local wave heights and directions are required as they form the 'stirring function' for the sediment model in the solution of the vertical concentration profile.
Under ideal circumstances it would be possible to solve the wave field taking account of all the major processes that occur in the nearshore zone. In this case the wave non-linearities together with effects such as wave refraction, diffraction, reflection, bottom friction, wave breaking and the wave-current interactions would all be considered. Although such a solution is desirable it is neither possible nor feasible at this stage. A number of simplifying assumptions must therefore be made.
Section 5.2 discusses these assumptions as well as outlining briefly the history of developments that led to the wave model used in this work. The derivation of the model which follows the work of Copeland (1985c) and Dong (1987) is outlined in Section 5.3. Section 5.4 gives details of the finite difference representation of the model equations and the general solution technique wkile Section 5.5 deals with verification of the model.
5.2 LITERATURE REVIEW
5.2.1 INTRODUCTION
It is not intended that this section provides a complete review of all works relating to the problem of wave propogation in the nearshore zone. The wave model described in this chapter follows the theory developed by Berkhoff (1972) and the review concentrates on material that has followed this original contribution.
The review concludes with a brief discussion on the topic of wave breaking which, although it forms a crucial part of the nearshore hydrodynamics, will be treated in a very simple manner. 89 5.2.2 EARLY WORKS
The early work on wave refraction and diffraction treated the effects separately and was based on the theory of waves in optics.
Munk and Traylor (1947) produced the first method of calculating wave refraction due to bathymetry. Keller (1958) was able to show that the intuitive approach that had been adopted did in fact have a sound theoretical basis. Among the works on refraction by a mean current, Arthur (1950) derived a solution based on the theory of minimum flight time. A complete review of wave refraction by bathymetry is available in Meyer (1979), and by currents in Peregrine (1976).
In the solution of wave diffraction Penney and Price (1952) extended the optical work of Sommerfeld (1896) to produce a solution to the problem of diffraction around a semi-infinite breakwater. Wiegel (1962) generalized the solution and tabulated the results for use in engineering design.
Early attempts were also made at solving the problem of combined wave refraction and diffraction. Battjes (1968) produced one such model but, as Berkhoff (1976) was able to show, the solution was not applicable to the whole wave field from deep to shallow water.
The early wave refraction models solved the problem by a ray - following technique, a ray being defined as orthogonal, at all points, to the wave crest. Rays were started in deep water and followed to the shoreline. The information about wave angles and height had then to be extracted manually by measuring the angle and separation of rays at a particular point. Battjes (1968) improved this by calculating the refraction over a predetermined grid of points thus avoiding the problems of ray following. Noda (1972) and Noda et al (1974) also adopted such a technique. The resulting algorithms were used by many researchers in the development of nearshore circulation models, for example Ebersole and Dalrymple (1980), Birkemeier and Dalrymple (1975) and da Silva Lima (1981).
Noda et al (1974) included the important wave-current interactions in the formulation of the governing equations. The authors found that the inclusion of even small currents had a significant effect on the resulting flow patterns especially where rip currents were involved.
Perlin and Dean (1983) improved the numerical algorithm derived in the Noda model to calculate wave heights and angles. 90 The calculation of refraction alone,however, has numerous drawbacks. The most serious occurs where the bottom bathymetry causes focussing of wave energy. Ignoring diffraction can lead to an over-estimation of wave heights in this case. Diffraction effects are of course very important where structures are involved. Therefore, for a general nearshore model, the combined effects of refraction and diffraction should be calculated.
5.2.3 MILD SLOPE EQUATION
The development in 1972 of the so-called Mild Slope Equation (Berkhoff, 1972 and simultaneously Schonfeld, 1972) introduced the possibility of calculating the effects of combined refraction and diffraction. By assuming that waves were linear, simple harmonic, Berkhoff was able to derive an equation that was valid for all water depths, that reduced to the Helmholtz equation for diffraction in constant depth or deep water and to the refraction equations when diffractive effects were ignored.
The equation, as derived, was elliptic and therefore posed a boundary value problem. To solve this Berkhoff employed a finite element technique. The main disadvantage with the solution of an elliptic equation is the amount of computing time required. Booij (1981) pointed out that for a grid of M by M elements the solution of an elliptic problem requires of the order of M^ operations. The problem is therefore large and time-consuming.
Radder (1979) derived a parabolic approximation to the problem by splitting the wave field into two components, a transmitted field and a reflected field. The reflected field was then ignored and a solution produced based on the transmitted field. This reduced the number of operations required for solution to the order of M^. A significant saving, but at a cost. Since the transmitted field only is being solved it follows that the model cannot handle reflections. This precludes the use of the model where structures are present that could cause reflections, or even where reflections from bathymetry or the coastline might be significant. A further disadvantage is that the incoming wave field must be aligned with the major axis of the solution grid. If a number of runs are required at different incident wave angles a new grid must be generated for each.
Ebersole (1985) points out that the parabolic approximation would be unsuitable for complex bathymetry where wave direction could vary from the preferred grid direction.
Booij (1981) found that using the parabolic approximation led to difficulties at 91 the lateral boundaries. It was not possible to have waves entering through the lateral boundaries, nor to have reflections travelling through them, so that a buffer zone was required adjacent to the boundary where the solution was often 'spoiled'.
The parabolic approximation does> however; have a number of advantages besides the speed of solution. The method can be solved using finite differences and since it becomes a stepwise solution, effects such as wave breaking can be incorporated easily. This is obviously important for a nearshore model.
Berkhoff et al (1982) calculated results from three numerical models, one based on the full elliptic equations, one on the parabolic approximation and a third which considered refraction only. The results were compared to laboratory data of waves passing over a submerged elliptic shoal. The authors found that all three models gave reasonable results for use in engineering practice. The results from the refraction-only model were good in the region near the shoal, but agreement became less good behind the shoal where diffractive effects were important. Both the full elliptic and the parabolic approximation models gave good results in all regions of the test area although the most accurate results came from the elliptic model. The present author has run the present numerical model on the same test. The results of this, together with details of the experimental set-up, will be discussed in some detail in Section 5.5.
Booij (1981) derived the mild slope equation in such a way that a variable current field could be superimposed over the solution domain. Restrictions were imposed such that the current should be slowly varying in a spatial sense and that only the effect of the current on the wave could be calclated. That is, the current was assumed to be known a priori.
Since the mild slope equation is developed assuming potential theory the addition of the current violates the irrotationality assumption. Booij however justified its inclusion by proposing that the vorticity caused by the current would be restricted to a thin layer near the bed and would not affect the main body of the water column. The equation reduced to that derived by Berkhoff (1972) with zero current velocity.
Liu (1983) and Kirby (1984) also derived parabolic approximations that included wave-current interaction but based on slightly different assumptions in the derivation of the governing equation. Dong (1987) has shown that the differences are superficial only.
Copeland (1985c) developed a hyperbolic approximation to the mild slope 92 equation in a method similar to that employed by Ito and Tanimoto (1972) to solve the long wave equation of Lamb (1932). Ito and Tanimoto split the long wave equation into two first order equations which were solved using a simple finite difference technique. The results, compared against analytical solutions of diffraction around a semi-infinite breakwater, were impressive.
Copeland (1985c) found that the mild slope equation could be split in a similar fashion. Boundaries were developed such that reflections could pass out of the model without affecting the driving conditions and a method for calculating partial reflections from structures was also included. The resulting solution was a time stepping one which required of the order of operations. The hyperbolic model was therefore more time consuming than the parabolic approximation but included reflections, like the full elliptic solution. As in the parabolic approximation wave breaking could also be included.
Dong (1987) extended Copeland's hyperbolic approximation to include the wave-current interactions. As with Booij's (1981) model the equations reduced to the no-current equations with zero current velocity. The same restrictions regarding spatial gradients of velocity apply.
The theory developed by Dong represents the most general approximation to the mild slope equation to date. It requires an order less computations than the full elliptic model yet is not restricted by assumptions regarding reflections. Unlike the elliptic model, wave breaking can also be included easily.
5.2.4 ASSUMPTIONS AND APPROACHES
The problem of the nearshore wave climate can be solved using the mild slope equation, where assumptions are a mild bottom slope and a linear simple harmonic wave. The problem could also be solved using the Boussinesq approximation to the long wave equations which, although long waves are assumed, has been estimated by Abbott et al (1978) to be suitable up to a depth to deepwater wavelength ratio of 0 .2 , that is a six second wave in ten metres of water.
Each particular approximation has inherent advantages and disadvantages. The Boussinesq equations have the advantage that higher order wave theories can be used, the assumptions about water depth mean that the solution is suitable for most nearshore problems (Warren et al, 1985) and such effects as wave breaking and bottom friction can be included easily. On the other hand the method requires a high, at least 3 rc* order solution technique, (Abbott et al, 1978) and although 93 higher order wave theories can be used these would pose problems in the definition of open boundaries when reflections occur in the solution area.
The main advantage of the mild slope equation is its suitability for all water depths. In fact Lozano and Meyer (1976) showed that it reduced to the long wave equation if kh >> 1. Although it is based on potential theory, Booij (1981) has shown that it is suitable for effects such as wave breaking and bottom friction. The major disadvantage is the assumption of linear simple harmonic wave theory. This means that wave profiles are not represented well in shallow water, however, Booij (1981) concludes that overall conditions such as energy per unit surface area and wave direction are still calculated accurately.
On the basis of the assumptions and restrictions the author concludes that for the problems encountered in this work the mild slope equation provides the best vehicle for a solution. The hyperbolic approximation developed by Copeland (1985c) and extended by Dong (1987) provides the best method of solving that equation.
5.2.5 WAVE BREAKING
As mentioned previously the action of wave breaking is crucial to the solution of the nearshore problem. Svendsen (1984) and Svendsen et al (1978) give details of models designed to describe the wave breaking process, but it is not possible to go into such detail here. See also the work of Yoo and O'Connor (1986).
For all its complexity it has been found that breaking is very much a function of local depth. Numerous experiments, for example those by Svendsen et al (1978), confirm that a reasonable estimate of wave height can be obtained from:
H - 7D (5.1) where y 1S an empirical constant around 0.8 and D is the water depth. A value of y - 0.78 will be used in the model reported in this work for full scale simulations. A value of y = 0.88 was found to give better agreement with the physical model study of Chapter 3.
5.3 HYPERBOLIC APPROXIMATION
5.3.1 INTRODUCTION
As mentioned previously the hyperbolic approximation to the mild slope equation allows the full wave field, both transmitted and reflected, to be calculated. 94 It also offers the advantage of using an order less computing time than the full elliptic solution.
In the following section the derivation of the hyperbolic approximation is given. It follows the work of Dong (1987). The mild slope equation itself will not be derived. Details of that derivation can found in Booij (1981), Liu (1983) and Kirby (1984).
5.3.2 THEORY
The mild slope equation, including currents, can be written (Kirby, 1984)
-g(|f + V(U,,)) - V(ccgVp) + (
c - gktanh(kh) (5.3) o) = a + k.U (5.4) c=(r/k (5.5)
9 ( 7 cg“ 3k (5.6) where c is the wave celerity, Cg the group velocity, k the wave number, The linearized free surface boundary condition can be written BT + 8,1 “0 (5.7) or since Dt“ - 3t 2 - + u.v (5.8) + O.Vp + sv ~ 0 (5.9) 95 Now if the surface potential is assumed harmonic = ae*^ ■= ae“'wt (5.10) then 3tp (5.11) and \N Inserting (5.11) and (5.12) into (5.9) gives - [io)r . TT/ Va + iaV0 N.. V _ I [ Jo> - U( ^2 + iVfl )] v (5.14) Assuming Va/a < < 1, i.e., the wave amplitude modulation is small, and using the relation k - V0 (5.15) then V = ~ (cj - Uk) ip o (5.16) Now, using (5.4) i a T) = —g 3 r\ 3t " '1“’J (5.19) So 5.18 gives 1 3tj (5.20) * " s 5? 3F Substitution of (5.20) and (5.18) into (5.2) gives -<|f + v V( - i ccgV ( 2)+Uij)-x£j2-0 (5.22) where c j2 _ ]f2c c X = ------s - 1 (5.23) GXT It is now possible to split (5.22) into two first order equations. Define a function Q such that Q - -iccgV(2) (5.24) then (5.22) can be written VQ + \7(Uij) - X |2 _ 0 (5.25) and from (5.24) |2 + uccgV(2) - 0 (5.26) Thus the mild slope equation of Kirby (1984) has been split into two first order equations forming a hyperbolic approximation to the original equation. To show that the approximation reduces to Copeland's (1985c) solution put U=0 and therefore o)=a Then X — "cg/c anc* (5.25) and (5.26) become VQ + —cc — - 0 (5.27) c 8 t (5.28) <3t + ccs Vt? “0 which are Copeland's (1985c) equations 3.113 and 3.114. The effect of the current is introduced explicitly in the second term of (5.25) but also implicitly through the calculation of the wave number field in Eqn. 5.4. The solution to (5.25) and (5.26) is in terms of two variables 77 and Q. 77 is the surface elevation and Q a vertically integrated function of particle velocity. Q is in fact a dummy variable used in the solution, however, its form must be known at the boundaries. It can be shown that (5.29) is a solution to (5.25) and (5.26) provided that VCg = 0. This will be true in deep water or in areas of uniform depth. To calculate Q at the boundaries it must be assumed therefore that these conditions hold. It will be true at the offshore boundary which will usually be in relatively deep water. It will not be strictly true on the lateral boundaries, but the important gradient in this case is (9/3y)Cg which can usually be arranged to be small. 5.4 FINITE DIFFERENCE EQUATIONS 5.4.1 INTRODUCTION In this section the hyperbolic approximation to the mild slope equation is discretized in time and space and solved as a set of finite difference equations. The equations are solved over an area subdivided into a grid of points. A typical solution grid is shown in Fig. 5.1. The input data required includes offshore wave 98 shoreline__ m JLL offshore driving boundary FIGURE Solution Grid Layout for Wave Model 5.1 99 height, period and direction and bottom bathymetry. The output from the model includes wave height and direction, surface elevation and the radiation stress components at each point over the grid area. The general solution procedure is to calculate an approximate starting surface profile and Q function, usually based on simple refraction. The solution is then marched forward in time for sufficent wave periods for the correct solution to develop. This time is determined largely by the size of the grid in terms of wavelengths and the existance and type of structures in the model. 5.4.2 FINITE DIFFERENCE SCHEME Equations (5.25) and (5.26) which are repeated here VQ + V(U>,) - x ^2 (5.25) |2 + 0,ccgV(2) _ 0 (5.26) can be written in finite difference form t+At t At v l-iY }2 (5.30) Qxi,j " Qxi,j" (CCS*i,j W A* <“^77j ” ^i-l,j) t+At . t+At/2 t+At/2 4t .’’i.J Qv . . - Qv - • - (cCp.). .oo -7— (— --- — — ) (5.31) y!,J y!.J s i.J q,j t+At/2 t-At/2 At y . . + t Qxi+i,j ~ Qxi,j .Qyj,j+i Qyt,j 'i.j i, J ^y Ax 4y , t-At/2 t-At/2 v , t-At/2 t-A t/2 + ( y . , * . + y . . ' ) u.l+ , 1 l,j .- ( 17 'i,j. .______+ y.*1-1,J 1 \ ) u. i, .J 2Ax t-At/2 , t-At/2 + ( y ., + . y.t-At/2 , 77 t-At/2 N i,J+l• 1 y 'i,J > vi.,i+i ~ ( . + . . ' ) v. . 2Ay (5.32) # 100 The variables are defined on a staggered grid as shown in Fig. 5.2. The i subscript refers to the x direction and the j subscript to the y direction. The superscripts refer to the time step. The solution is also staggered in time with r\ being evaluated a half time step ahead of Qx and Qy. 5.4.3.INITIAL CONDITIONS In order to start the solution procedure some initial values are required for Qx, Qy and 17. As the solution develops these will propogate out of the model area, but a reasonable initial guess in the starting profile can aid the speed of solution. The initial conditions used in the model are based on a simple refraction technique using Snell's Law. This is used to calculate both the wave heights and directions used to generate the initial profile. The general form of 77 can be expressed T] = a sin(kcos( 0 )x + ksin( 0 )y -wt) (5.33) The calculation of Q is based on Eq. (5.29) (5.29) therefore Qx =* acgsin(kcos(0)x + ksin(0)y - a)t)cos(0) (5.34) Qy = aCgSin(kcos(0)x + ksin(0)y - cJt)sin(0) (5.35) The initial 17 values are calculated at t=-At/2 while Qx and Qy are calculated at t = 0 The variables in a shadow zone of any solid structure are set to Qx = Q y = 0 # 101 o 4. A hj o K> A o A f • o o A I r ° o f i A A ZJ i X i C p O t=- O O o ZD t X I---J------1— -4 — ' ZD X + j= 1 1 2 2 riy Hy ny+1 n Q a, FIGURE Numerical Grid for Wave Model 5.2 10 2 5.4.4 DRIVING BOUNDARIES The extremities of the grid fall on Q points. Lateral boundaries fall on Qy points and onshore and offshore boundaries on Qx points. This is illustrated in Fig. 5.2. It is therefore necessary to provide boundary conditions for the solution at these points. The boundaries that have incoming waves are called driving boundaries of which there will usually be two: a driving lateral boundary and the offshore boundary. The major problem with the driving boundaries is to allow reflections to pass out of the model at these boundaries without interfering with the driving function. Ito and Tanimoto (1972) overcame this problem by setting the driving boundaries sufficently far away such that the reflections did not reach them before the internal solution was attained. This method is generally not feasible due to the model sizes that would result. Copeland (1985c) adopted the simplest absorbing boundary condition from Engquist and Majda (1977) where the driving function is modified by the reflections that are present at the boundary. This approach will also be adopted here. The procedure is as follows. At the boundary, the Q value is considered to be composed of an incoming component plus a reflected component. The total Q value is calculated therefore as the incoming component modified by the reflected component. The derivation for the x direction is now given: The radiation condition at the boundary in terms of the total Q can be written: ^Qx ^ 9Qx ----- + ------— « 0 (5.36) 8 t cos( 0 ') 8 x where 0 ' is the wave angle at the boundary including the effect of the reflected wave. For the incoming component 8 qx c 8 qx ----- + ------= 0 (5.37) 8 t cos( 0 ) 8 x where 0 is the incoming wave angle and qx the incoming component of Q. Assuming that |cos 0 | - |cos0 I Eqns. 5.36 and 5.37 can be written in finite difference form o t+4t - o 1 yxi+l,j__ + (Q*i+l,j ~ Q*i,j } - 0 (5.38) At cos(0) Ax and t+At qxi+i,j qxi+ l.j (qxi+l,j ~ qxi,j ^ 0 (5.39) At cos(0) Ax hence t+At t+At *xi+l ,j " q*i+l,j + ( Q*i+l,j ' qxi+l, , ) + c ^ r(q t cos(0)Ax L v,v The situation for the y boundary is similar but the celerity is now c/sin( 0) rather that c/cos( 0). This means that for small offshore angles of incidence (0 near 180*) the celerity in the y direction can be extremely large. This can lead to Courant instability in the finite difference representation and to overcome this it is necessary to check the Courant Number. c At C sin ( 0 ) Ax (5.41) where C is the Courant Number and c is the wave celerity. If the Courant Number is larger than 1 .0 the solution will become unstable causing spurious oscillations at the boundary. The present author overcame this problem by effectively increasing Ax locally by taking the difference equation over a sufficent number of grids rather than one as would normally be done. For example, if the Courant number was 1.5 the differences calculated in Eq. 5.40 would be taken over two grids, giving an effective Courant number of 0.75. This procedure was found to give satisfactory results. Generally this problem is associated with the lateral boundaries but it is possible that for large angles of incidence (0 near 90*) problems could occur at 104 the offshore driving boundary. A similar procedure could be used in that case. The problem caused by differences between 0, the incoming wave angle, and 0' the total wave angle including reflections, is not insignificant. However, under conditions of weak reflections there will be only minor differences between the two values. Under strong reflections it is possible that errors could be caused. In this case standing waves will often form which make the wave angle (of an assumed progressive wave) impossible to calculate. The assumption that 8 = 0 ' is therefore a convenient way to a solution. 5.4.5 REFLECTIVE AND TRANSMISSIVE BOUNDARIES It is necessary, both for downwave boundaries and for internal structures, to be able to calculate the effects of either transmission or reflection of the waves at a boundary. For downwave boundaries it is necessary to allow waves to pass through them while for internal structures it is desirable to be able to specify reflection coefficents between 1.0 and 0.0. The method employed here was developed by Ito and Tanimoto (1975) and used by Copeland (1985c). Copeland (1985c) provides a full derivation which shows how Q values at a barrier or boundary can be calculated in terms of previous Q values at an adjacent up-wave point. Only the results will be quoted here as the derivation is quite long. For example, in the x direction at a boundary where i=N and i=N+l is upwave QxN,j ~ AF-QN+l,j (5.42) where d-r) AF------[(l+r)^ sin^(kAxcos( 0 ) ) + (1-r)^ cos^(kAxcos (8))] tan(cor) = y—p tan(kAxcos( 0 )) For full transmission r = 0 AF = 1.0 Ax T c co s(8) 105 For full reflection r = 1 AF = 0 In the y direction the derivations are similar with Ay replacing Ax and sin(0) replacing cos(0) throughout. 5.4.6 STABILITY The stability of the governing equations (5.25) and (5.26) is governed by a Courant criteria. Copeland (1985c) states that: cAt < 1 and cAt < l (5.43) “Ax “Sy In practice it has been found that the sum of the two is also important. cAt cAt Ax + Ay (5.44) A typical case will illustrate the limits. A 6.0 second wave in 2.0 metres of water with Ax = 2.5 metres limits At to 0.2 seconds. That is 30 time steps per period and 10 space steps per wavelength. 5.4.7 WAVE HEIGHTS AND DIRECTIONS Wave height is averaged over one period H - 2(2tj2) V 2 (5.45) Wave angle is also calculated by averaging over one period but its calculation is more detailed. Copeland (1985) calculated the angle: 6 = arctan[(Qy / Qx )V2] (5.46) The problem with this technique is that all information on the quadrant that the angle occupies is lost when the values of Qx and Qy are squared and numerous calculations are required to obtain the correct result. The method employed in this work is to calculate the angle at each time step 6 = arctan(Qx/Qy) (5.47) 106 and to vector sum and average them over one period. It was found that when Qx and Qy were small the angle calculated was unreliable so that at each time step the angle calculated was weighted by the magnitude of the Q value. Qsy 0 — arctan(----- ) (5.48) Qsx where n Qyi Qsx “ lQlcos(arctan(^— )) (5.49) n Qyi Qsy " iQl sin(arctan(^—- )) (5.50) where n is the number of timesteps per period. For added reliability 0 was averaged over each point and its 8 immediate neighbours (9 points in all). As was pointed out earlier, wave angles calculated in standing waves are not reliable measures of crest angle. 5.4.8 WAVE BREAKING The simple limiting condition H=-yD is used to control the height of breakers in the model. This is implimented at each time step by checking the new absolute value of 77, which if it exceeds 0.5yD, is set to that value. This effectively limits the crests and troughs of the sinusoidal wave. It is found that as this 'chopped' profile propogates the finite difference scheme tends to round the corners and the profile looks and behaves like a reduced amplitude sinusoid. 5.4.9 RADIATION STRESSES The radiation stresses, first derived by Longuet-Higgins and Stewart (1964), are defined as the excess flow of momentum due to the presence of waves. Copeland (1985c) derived expressions for the componments of the radiation stress in terms of the numerical values of 77, Qx and Qy Dong (1987) has shown that these derivations are correct for the case including wave-current interactions. The full derivation of the terms can be found in Copeland (1985c) and will not be repeated here as it is quite lengthy. The stresses can be written: ♦ 107 2 2 9 3r^ sXx - Rx A ' I -g j + -gy 1 B + 3 5 [Rx(-®; + "B^)D1 Q 3RX SRy -- + ^ [Ry(3^" + 5 F )d] + °*5pst?2 (5.51) A similar expression holds for Syy, interchanging x and y in (5.51). Sxy RX Ry .A where Rx “ Qxc/ cg R y - QyC/Cg A ------^ ---- [sinh(2kh)+2kh] 4 sinh2(kh) B ------E------[sinh(2kh)-2kh] 4k sinh2(kh) D = -----—------[—— sinh(2kh)- cosh(2kh)] 4 sinh2(kh) 2kh 5.5 MODEL VERIFICATION 5.5.1 INTRODUCTION A series of numerical tests was carried out to verify the performance of the wave model. The tests were chosen to highlight a particular feature of the model and the results were compared with known analytical solutions or experimental results. The tests that have been carried out are: 1. pure refraction on a plane beach 2. pure diffraction around a semi-infinite breakwater 3. pure diffraction through a harbour entrance 4. pure wave refraction by a shear current 5. combined refraction-diffraction by a submerged shoal For most of the tests a nominal period of 6.0 secs and wave height of 1.0 m were used. The grid spacing was chosen to give not less that 10 grids per wavelength and the timestep selected gave approximately 50 steps per period. The models were run for 10 periods. 108 5.5.2 PURE REFRACTION BY BATHYMETRY The solution to the problem where waves approach a plane beach at an angle is well known. The waves change angle and the height changes due to shoaling. Fig. 5.3 shows an isometric view of waves approaching the shore with the wave height reducing due to breaking. Figure 5.4 shows the comparison between the theoretical and calculated wave heights and angles. The agreement between the theoretical and numerical results is seen to be excellent. The effect of the inexact down-wave boundary can be seen in the small amplitude standing wave that forms, causing an oscillation in the offshore wave heights with a corresponding effect on the wave angle. 5.5.3 PURE DIFFRACTION AROUND A SEMI-INFINITE BREAKWATER The solution of diffraction by an infinitely thin breakwater was first given by Penney and Price (1952). Fig. 5.5 shows the theoretical contours of wave height in the vicinity of the breakwater and the results obtained from the numerical model. Fig 5.6 shows an isometric plot of the water surface after the model has been run for sufficent time to generate the solution. There is some noise in the calculated solution particularly further away from the main area of interest but the results are generally good. As the solution proceeds in time the accuracy deteriorates due to noise being introduced mainly from spurious reflections at the boundaries. Fig 5.7 plots the contours of zero elevation and therefore gives an idea of wave direction. 5.5.4 PURE DIFFRACTION THROUGH A HARBOUR ENTRANCE An analytical solution exists to this problem and the numerical results are compared to it for one particular gap, in this case equal to two wavelengths. Fig. 5.8 shows the waves propogating through the gap and radiating out into the harbour. The theoretical wave height contours taken from the Shore Protection Manual are plotted in Fig. 5.9 and the numerical results in Fig. 5.10. Again the agreement is good and best near the entrance where the gradients in wave height are the greatest. Fig. 5.11 shows the wave fronts radiating into the harbour. 5.5.5 PURE REFRACTION BY A SHEAR CURRENT Mei (1983) gives the analytical solution for refraction by a shear current. A severe test was posed for the numerical model where the shear current had a triangular distribution with a maximum velocity at the mid-point of the flow. A schematic drawing is shown in Fig. 5.12. Once the waves had been refracted by 1 0 9 1 1 0 FIGURE Numerical Solution Wave Refraction 5 . A 113 ll direction of wave approach FIGURE Wave Front Plot Semi-Infinite Breakwater 5.7 <7 IT 115 FIGURE Numerical Solution Harbour Entrance 5.10 117 I, direction of wave approach FIGURE Wave Front Plot Harbour Entrance 5.11 118 PLAN VIEW shear current incoming wave FIGURE Refraction by Shear Current Test Schemat i c 5.12 # 119 the current in a constant depth section of the test they were refracted by bathymetry on a section of plane beach. Fig. 5.13 compares the theoretical and calculated wave angles. The comparison shows good agreement although the numerical model could not match the sharp changes in direction that occurred due to the nature of the current being used. A more realistic current profile would probably have resulted in a closer agreement. 5.5.6 COMBINED REFRACTION-DIFFRACTION BY A SUBMERGED SHOAL Berkhoff et al (1982) report a comparison between a physical model experiment carried out at Delft Hydraulics Laboratory and a number of numerical models. As mentioned earlier, the experiment concerned combined refraction and diffraction by a submerged elliptic shoal. The layout of the physical model is shown in Fig. 5.14 and the resulting depth contours and grid used in the present comparison are plotted in Fig. 5.15. In the original numerical comparison the models involved included the full elliptic solution, a parabolic approximation and a pure refraction model. The present author has run a numerical test using the hyperbolic wave model reported here. The comparison includes an examination of wave heights along eight section lines shown in Fig. 5.14, together with an overall wave height contour plot and the calculation of the wave crest and trough patterns. The wave height contours measured in the physical model are shown in Fig. 5.16. The contours calculated by the numerical model are illustrated in Fig. 5.17 and show good agreement with the main features. Fig. 5.16 is saturated with 1.0 contour lines which were not plotted in the numerical comparison for the sake of clarity. Figs. 5.18 to 5.21 show the comparison between the experimental results and the numerical predictions along sections 1 to 8. The agreement is generally very good although there is some deviation at the end of Section 6. This also occurred in the parabolic model of Berkhoff et al (1982) although the elliptic model gave better results. Finally the measured wave front patterns are shown in Fig. 5.22 for comparison with the numerical predictions in Fig. 5.23. Again the comparison is favourable. The agreement between the physical measurements and the numerical predictions is generally excellent. A detailed comparison shows that the model FIGURE5.13 Wave Angles in Shear Current Test 121 FIGURE Submerged E l l i p t i c Shoal Test Schematic 5.14 122 0 ------► x 20 FIGURE Depth Contour sof Submerged E l l i p t i c Shoal 5.15 123 incident- wave direction FIGURE Experimental Results - Submerged Elliptic 5.16 Shoal taken from Delft Hydraulics Lab. (1982) 124 ------1.5 ------1.1 ------0.5 FIGURE Numerical R esults - Submerged E l l i p t i c Shoal 5.17 125 250 5? 200 cn 150 o fD 50 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 distance along section (m) SECTION 1 250 3? 200 SZ C71 150 o1 / < CU o > OoSs JZ 100 o s \ . o<>/ >0 o o / < ' o '-0 O CL» o ~ FIGURE Selected Section Results - Sections 1 and 2 .5.18 250 i #/ n ° \ 0s 200 »/ u / "\\ / &\ 150 in \ CD i \ CD \ 100 i N D \ X 1 V o CD l S' 1° 50 o""o—o- --°-x NO /r\o' i J_____ ro> V o £ -5 -4 -3 -2 -1 0 1 2 3 4 5 distance along section (m) SECTION 3 250 /“ \ vO /O cr* 200 > i6 \ / 7 ? 150 1 \ CD 1 N» ___ / i >* < / ■ 6 CU *0 / o\ \ ° \ / <£ ov \ /Q ___ > 50 \ r ro D\ / V c* -5 -4 -3 -2 -1 0 1 2 3 4 5 distance along section (m) SECTION 4 measurement o computation----- FIGURE Selected Section Results Sections 3 and 4 5.19 250 i—\ v° / \ oN 200 / \ /o°o' \ i 150 |Q 7 ~\ CD / b / \ OJ p \ 100 N. / to t \ /"\o o o X o o / \ / \ '6 OJ > o > 50 n nj O o r N o 5 o S o -5 -4 -3 -2 -1 0 1 2 3 4 5 distance along section (m) SECTION 5 250 S? 200 cn 150 CD ---e — 1 _0_ _5 ___ / 100 /c > / > o / _ / < CL) o > 50 o ro c» o < > 0 1 2 3 4 5 6 7 8 9 10 distance along section (m) SECTION 6 measurement o computation FIGURE Selected Section Results Sections 5 and 6 5.20 128 250 s ' s*S' 1 g 200 / O < / » 0 ,5 o ~~ / / o 1> o^c £ 150 ' o cn r ' ( 3 2 100 >--cr ^ ro> 50 5 01 23A 567 89 10 distance along section (m) SECTION 7 distance along section (m) SECTION 8 measurement o computation----- FIGURE Selected Section Results Sections 7 and 8 5.21 129 incident wave direction FIGURE Experimental Results of Wave Front Pattern 5.22 - Elliptic Shoal taken from Delft Hydraulics Laboratory (1982) 130 FIGURE Numerical Prediction of Wave Front Pattern 5.23 - Elliptic Shoal 131 performs at least as well as the parabolic approximation model but not as well as the full elliptic solution. It should be said, however, that the elliptic solution of Berkhoff, Booij and Radder was calculated over a reduced area so that a solution could be generated in a reasonable time. It is anticipated that in a case where reflections were present the hyperbolic solution would be superior to the parabolic model predictions. 5.5.7 MODELLING OF ARBITRARY CONFIGURATIONS Thus far the model has been shown operating under simple or idealized geometries but the formulation of the solution allows for a varied geometry and bathymetry to be implemented. Fig. 5.24 shows one such example of a harbour. No attempt has been made to compare predictions with known solutions, the inclusion of this example is purely illustrative. CHAPTER 6 THE CURRENT MODEL 6.1 INTROUCTION The determination of nearshore currents and wave-induced circulation patterns has been a problem facing coastal engineers for some sixty years. Early researchers attempted, with limited success, to predict the mean longshore current. Although the importance of wave height and angle of approach was recognized it was not until the radiation stress theory of Longuet-Higgins and Stewart (1960,-1,-2,-3,-4) that the correct physical framework could be assembled. With this framework available the emphasis shifted to the accurate representation of the physical processes involved, and to an efficient and accurate numerical solution technique. It has been found by numerous researchers that turbulence and bottom friction play an important role in determining the nearshore circulation patterns. Correct representation of the non-linear convective terms is also essential if accurate results are to be obtained. This chapter details the development of a finite difference model capable of solving the nearshore hydrodynamic equations, where particular importance is attached to the presence of structures which could cause secondary flows and circulations. Section 6.2 reviews the literature on the subject over the last forty years. The details of the model equations are given in Section 6.3 and the finite difference representation of the equations in Section 6.4. The verification of the model in Section 6.5 is based mainly on comparisons with field and laboratory experiments. 6.2 LITERATURE REVIEW 6.2.1 INTRODUCTION Galvin (1967) began a review of longshore velocity prediction methods stating ” 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." Galvin then went on to review the numerical methods that were available at the time. These were broken down into three approaches, one based on the conservation of energy or momentum (Putnam et al, 1949, Eagleson, 1965), one on 134 the conservation of mass (Bruun, 1963, Inman and Bagnold, 1963) and the third based on an empirical correlation with field and laboratory data (Inman and Quinn, 1951) The methods attempted to predict the mean longshore current, but all were unreliable and able to predict only about half of the laboratory and field data sets. Longuet-Higgins (1970) presented an approach based on the radiation stress concept which was able to predict the cross-shore distribution of longshore velocity as well as the mean velocity. The solution was an analytical one, assuming a plane beach and a small angle of wave approach. The solution ignored any lateral mixing and predicted a maximum velocity at the breaker-line. In the second paper of 1970 Longuet-Higgins included lateral mixing using a simple eddy viscosity turbulence model where the mixing length was assumed proportional to the distance from the shoreline. This allowed a smoothed longshore velocity profile to be calculated which was in reasonable agreement with earlier experimental studies. Although the mixing length assumption has been criticized as being unrealistic, particularly outside the breaker-line where turbulence is known to be small, the concept of the radiation stress being the main driving force of longshore current is now well established. All subsequent numerical models have used this theory and the next section reviews some of these models. 6.2.2 NUMERICAL MODELS Noda (1972) and Noda et al (1974) studied the problem of nearshore currents on a coast with a rhythmic bathymetry. The work followed the field work of Sonu (1972) investigating the formation of circulation cells and rip currents. It was found that an existing current could modify the wave pattern to a significant extent and great emphasis was placed on the wave-current interactions. The solution technique for the wave field was quite sophisticated and was used by many subsequent workers. The solution of the currents on the other hand was rather crude, neglecting both the non-linear convective terms and lateral mixing, both of which have been found to be important particularly where rip currents are concerned. The solution technique involved writing the depth averaged Navier-Stokes equations in terms of a stream function which was solved using a Guass-Seidel relaxation method. Birkemeier and Dalrymple (1975) solved a similar set of equations to that of Noda et al (1974) in that neither the convective terms nor the lateral mixing were included. The solution technique involved an explicit finite difference scheme. Results of predicted set-up and set-down were compared with laboratory results of Bowen et al (1968) and gave close agreement. 135 Allender et al (1978) used the model of Birkemeier and Dalrymple (1975) in a comparison with field data collected at a beach on Lake Michigan. Although the agreement between measured and predicted wave heights across the surf-zone was reasonable there was very poor agreement between the predicted and observed currents. The authors attributed this lack of agreement to uncertainties in the model input data such as wave angle, bathymetry and still water level. The absence of horizontal mixing in the model was also cited as a possible source of error. In addition to these suggestions the present author feels that the field conditions may have been too complicated and variable for comparison with a numerical model that assumed a stationary, unidirectional and monochromatic wave field. Measurements showed that the wave field was in fact changing rapidly, indicating the possiblity of multiple wave trains. The assumption of longshore periodicity as one of the boundary conditions led to some unrealistic flow patterns and neglecting the convective accelerations would have reduced the likelihood of agreement being reached. Basco (1983) cites these experiments as a reason for distrusting numerical modelling results. However, as pointed out by Kirby and Dalrymple (1983) in a discussion of Basco's paper, it was unfair to compare results of this primitive model when more sophisticated ones were available. Ebersole and Dalrymple (1980) extended the earlier model of Birkemeier and Dalrymple (1975) to include both the convective accelerations and lateral mixing terms. The resulting model was shown to give good agreement with the Longuet-Higgins' type profile. In the case of rip current formation studied by Noda et al (1974) it was found that the inclusion of horizontal mixing reduced the spread of the rip. The effect of the convective terms was not evident. While the majority of the numerical models solve the equations using finite differences there have been a number employing the finite element method. Bettess et al (1978) solved both the wave field and the nearshore circulation patterns around a structure using finite elements. The lateral mixing terms were included using a simple eddy viscosity model similar to the mixing length assumption of Longuet-Higgins (1970). A constant eddy viscosity model was also used to test the sensitivity of the solution to this parameter. The predicted flow patterns were compared with physical model experiments which indicated that the shape and distribution of the circulation was modelled accurately. There was however some discrepancy in the magnitude of some of the velocities. 136 The solution using finite elements has the advantage of allowing irregular shapes to be modelled easily and accurately. This is particularly evident in the Bettess et al paper where the shape of the harbour is well represented. Liu and Lennon (1978) also employed finite elements in a solution of nearshore currents. The authors suggest that the state of nearshore circulation theory had reached a plateau. The model described by the authors was, however, below that plateau in that it did not include lateral mixing or the convective terms. Vemulakonda et al (1982) extended the finite difference solution technique by employing an accurate three time level ADI (Alternating Direction Implicit) scheme. In fact the ADI scheme had been used previoulsy in hydrodynamics. Leendertse (1967) had modified the ADI scheme of Peaceman and Rachford (1955) to study the problem of long waves in the nearshore environment. Kuipers and Vreugdenhil (1973) had used Leendertse's scheme to study open channel flows. The main advantage of the implicit scheme is that it exhibits extra stability over the explicit schemes. The lack of stability is probably the reason that early researchers who used an explicit scheme (Noda et al, 1974, Birkemeier and Dalrymple, 1975 etc.) did not include the convective terms. Wu et al (1985) and Thornton and Guza (1986) show that by employing a wave model where dissipation due to breaking is modelled accurately it is possible to obtain good agreement between field data and numerical predictions. The field data chosen for comparison was selected such that it approximated (as closely as nature would permit) a single frequency uni-directional wave. The local topography was also selected to approximate plane beach conditions. Under these conditions the comparison of wave heights and nearshore currents is extremely good. Some results are shown in Fig. 6.1. The wave model employed by the above authors generated a progression of breakers at different points rather than the usual case with monochromatic waves where all the waves break at the same point. The result of this was that the longshore driving radiation stress SXy was smoothed at the break point. The authors found that it was not necessary to include lateral mixing to obtain a smooth velocity profile. On the basis of this Thornton and Guza (1986) conclude that eddy viscosity is not important, at least for the nearly planar bathymetry used. The present author feels that the implication, that lateral mixing (and turbulence) are only a useful way of obtaining a smooth profile of longshore velocities is wrong. Turbulence is a real physical phenomenon which should be included if accurate results are to be obtained. The fact that under certain (a) (b) FIGURE Numerica 1 Model Results of Th ornton and Guza 6 .1 (a) the wave mo del results Cl9 86 ) (b) the current model results 138 circumstances its effect is difficult to observe does not detract from its importance. This point will be pursued further during a discussion on the importance of turbulence in suspended sediment transport, in Chapter 7. Wind and Vreugdenhil (1986) underline the importance of lateral mixing. It was found to be important for its effect on the overall velocity field. The authors studied the generation of rip currents in the laboratory and made comparisons with a numerical model. The convective terms were found to be particulary important in the description of the rip, although the magnitude of the velocity was largely determined by bottom friction. Discrepancies in earlier numerical models were attributed to unrealistically high viscosities, particularly outside the breaker line. To overcome this the authors employed a two equation (k-e) turbulence model in the calculation of the lateral mixing. More will be said about the turbulence modelling in a later section. 6.2.3 CIRCULATING FLOWS The models reviewed to date have been designed to operate in the nearshore zone under a variety of conditions. Some have been used for plane beach conditions, Bettess et al (1978) looked at circulations behind a structure, others concentrated on the formation of rip currents. It is understandable therefore that the researchers will have found that different terms or physical effects were important. Since the purpose of this present study is to model nearshore flows and circulations in the presence of structures a review will now be made to determine the parameters that should be included. Flokstra (1977) investigated the vorticity balance in a fluid and found that secondary circulation cannot occur without the inclusion of the so-called effective stresses. These stresses are made up of viscous stresses, turbulent stresses and a stress which arises from the fact that the depth-averaged equations being solved have been integrated over the water depth. Flokstra suggested that to model the turbulent stresses at least a two equation model, for example the k-e model, should be used. Kuipers and Vreugdenhil (1973) had previously carried out numerical experiments to show that circulating flow could not be generated without the convective acceleration terms. This was also confirmed by Ponce and Yabusaki (1980) who studied secondary circulation in open channel flow where there was a sudden expansion or a side wall cavity. Lean and Weare (1979) carried out numerical experiments with a groyne in an 139 open channel to confirm Flokstra's (1977) proposal. It was found that both the convective terms and the effective stresses had to be included for circulations to form. The authors showed it was possible to produce a circulation without these terms; however, the "spurious" circulations were shown to be caused by the finite size of the numerical grid employed rather than by any physical means. Basco (1983) also concluded that if rip currents and nearshore circulations are to be modelled some high order turbulence model should be used. In fact the turbulence is important for two aspects of the total nearshore model. It has been stressed by many authors how important it is in the solution of the nearhsore hydrodynamics. The turbulence also plays a vital role in the sediment transport as it is largely responsible for maintaining sediment in suspension. The use of a high order turbulence model can therefore lead to a much more accurate picture of the nearshore sediment dynamics assuming other parameters are adequately modelled. 6.2.4 TURBULENCE MODELS As shown by Flokstra (1977), among others, the depth-averaged Navier-Stokes equations contain effective stresses which are expressed in terms other than the principle velocities and mean surface elevation. Therefore if a closed set of equations is to be found it is necessary to approximate these stresses in terms of the known velocities. This section deals with the turbulent or Reynold's stresses. Boussinesq proposed that the turbulent stresses could be modelled in a similar fashion to the viscous stresses. In that case the turbulent stresses are formulated in terms of the mean velocity gradients and an eddy viscosity which replaces the kinematic viscosity. Unlike the kinematic viscosity, which is a function of the fluid and can always be measured, the eddy viscosity is a function of position and of the flow itself and is therefore more difficult to determine. Prandtl (1925) suggested that the eddy viscosity should be a function of a velocity characterizing the flow and a length which was called the mixing length. For simple flows it is possible to specify this mixing length, usually in terms of the distance to an adjacent boundary, and the subsequent results can be quite accurate. For more complicated flows the simple assumptions break down. Longuet-Higgins (1970) and numerous subsequent workers have used the Prandtl mixing length hypothesis to model the turbulent stresses. Longuet-Higgins chose the shallow water celerity as the characteristic velocity and the distance to the shore as the mixing length. The resulting eddy viscosity therefore increases from the shoreline and leads to unrealistically high viscosities offshore. Several researchers (e.g. Inman et al, 1971) studying dye diffusion in the laboratory and field found that turbulence virtually disappears outside the breaker-line. The distribution, although unrealistic outside the breaker line, does however allow the prediction of a sensible longshore velocity profile. Numerical comparisons between the Longuet-Higgins model and the authors k-e model show good agreement on the turbulence inside the breaker zone. See for example Fig. 6.2. The Prandtl mixing length hypothesis follows a very simple approach to the problem of determining the eddy viscosity. The fields of aeronautics and fluid dynamics have spawned a whole range of more detailed and therfore more complicated models. The two equation k-e model is such an example. A full description of the model and its derivation would be out of place here and for that the reader is referred to excellent articles by Launder and Spalding (1972) and Rodi (1980). The model is so named because it estimates the eddy viscosity at a point in terms of two quantities, k, the turbulent kinetic energy and e, the rate of dissipation of the turbulent kinetic energy. The model is called a two equation model because it solves two equations, one for k and one for e where each equation accounts for the convection, diffusion production and dissipation of the relevant quantity. To solve the two equations it is necessary to be able to specify the processes which either add to or detract from the turbulent kinetic energy or its rate of dissipation. For many hydraulics problems the major production of turbulence is through shear. However, in the case of breaking waves there is considerable generation by breaking. Battjes (1975), for example, assumed that all energy lost through breaking went into the production of turbulence. It is therefore nesessary to modify the standard k-e model to take account of this important process. Dong (1987) modified the depth-averaged equations of Rodi (1980) to formulate them as depth-integrated equations and to take account of the breaking process. The resulting two equation model has been solved by the present author and will be reported in a later section. Wind and Vreugdenhil (1986) also report the use of a k-e model in the nearshore zone. The model equations use results proposed by Battjes (1975) and although similar to those proposed by Dong they have slightly different source term coefficents. The determination of the coefficents is, to a degree, arbitrary at FIGURE Longuet-Higgins and k-e Model Eddy Viscosity 6.2 0 equation - Longuet-Higgins 2 equation - k-e Model 141 142 present until measurements allow proper calibration of the eddy viscosity in the nearshore zone. 6.3 CURRENT MODEL EQUATIONS 6.3.1 DEPTH-AVERAGED NAV3ER-STOKES EQUATIONS The time mean, depth averaged equations of motion in the nearshore zone can be written (Mei, 1983) 3t + (U'D) + 3^ as XX BSxy 3F n 9rj 1 8Txx 9T*y _ 1 - ® Bx p ^ Bx By ^ p r ^x (6.2) 9s as xy yy If (V.D) + I j (U.V.D) + ^ n Bv , i /0Txy . dTyy s _ i - ® By p Bx By ^ p rby (6.3) where U and V are the x and y velocities respectively, D is the total depth and rj the water surface elevation. Sxx, SXy and Syy are the radiation stress components, Txx, TXy and Tyy the effective stresses and an overbar indicates time mean. Equations (6.2) and (6.3) can be rearranged to read (dropping the overbars) au , TT au , „ au an , 1 0Sxx 0Sxy . 1 Bt" Bx By + ® Bx pD ^ Bx + By ^ + pD T^x 1 9(DTxx) a(DTxy) (6.4) pD ^ Bx h By ^ BSXy 9Syy J 8 v . it 9v . v 9V d r j ^ 1 . 3F + u 3^ + v 3 ^ + s 3^ + pD ^“3x + “37" } + Tby a(DTxy) 3(DTvy) 1 ' * + ) = o (6.5) pD ( Bx 9 7 In order to solve (6.1), (6.4) and (6.5) it is necessary to specify the form of 143 bottom friction, the radiation stresses and the effective stresses in terms of known values and the unknown variables U, V and rj. 6.3.2 RADIATION STRESSES The three components of radiation stress, Sxx, SXy and Syy are calculated during the solution of the wave field, as detailed in Section 5.4.9. 6.3.3 BOTTOM FRICTION Longuet-Higgins (1970) showed that in a wave dominated environment bottom friction is due mainly to the orbital velocity. Following Birkemeier and Dalrymple (1975) (and others) the bottom friction components are isotropic and can be written: Cf rbx = 4-°P— umax u ( 6. 6) Cf Tby " 2.Op— Umax V (6.7) X where xH Umax = Tsinh(kD) (6.8) Cf £ 0.010 Various researchers quote different values for C f based on field and laboratory studies. It is generally agreed that C f can vary depending on local conditions. This particular formulation for bottom friction would not be ideal for situations where the climate is not wave dominated, for example in the presence of strong rip currents. However, in general, the circulations studied in this thesis should conform with the general assumption. 6.3.4 TURBULENCE Flokstra (1977) derived expressions for the effective stresses. These can be written Txx = j (2pu^ " Pu'2- p(u-u)2 } dz (6.9) 144 8u + dv pu v - p(u-u)(v-v)} dz xy iJ{pv(3? 3^} " (6.10) Tyy “ 5 | (2Pug^ - Pv'2- P(v-v)2 ) dz (6.11) where u and v are the instantaneous velocities, the overbar indicates a time averaged quantity, and the prime a fluctuation. The effective stresses are composed of three components. The first in each equation is due to viscous effects. The second is due to turbulence and the third arises from the fact that the equations have been integrated over depth. The viscous stresses can be neglected as being much smaller that the other two. If it is assumed that the flow is reasonably uniform in the vertical the third component would also be small compared with the turbulence term. Using the Boussinesq eddy viscosity approximation the effective stresses due to turbulence can be written ~ 0 8 u T X X ” 2PVt 0 ^ (6.12) Txy - PH (g^ + > (6.13) Tyy - 2 pu t | (6.14) It is necessary therefore to be able to specify the eddy viscosity ut which will be based on the k-e turbulence model. The depth averaged k-e equations can be written (Rodi, 1980) 8k 8k 0 ut 3k 0 ut U + V ^ ~ 3^ )+ gy > + ph + pkv - € (6.15) 3 6 3e a ,ut 8e .. a ut 9e ^ 8x + ^ 8y 8x ^(re 8x 8y ^ ut " c/i 7 (6.18) The constants c^, c je, C2e, a^ and o e are empirical values which on the basis of experiments, usually take the values (Rodi, 1980): cn C1 e c2e ^k 0.09 1.44 1.92 1.00 1.30 Table 6.1 k-e Model Empirical Constants This model takes no account of the important source of turbulence caused by the waves breaking. Dong (1987) modified the above equations to take account of wave breaking. The modified equations read: 0(Dk) , TT 0 Bt Bx By 3x By oy^ cle*^>h'^*jc c2e ^ + ®tcp D (6.20) where Dt is the production term due to breaking and o^, ae> c^ e, C2e take the standard values as defined above. A new constant Cp was derived by Dong to have a value of the order of 0.2. Numerical experiments by the present author indicate that a value of 0.33 gave reasonable comparisons with published cross shore velocity profiles. Dj, the production term due to wave breaking, can be written D d (6.21) (5.cgx " 3y gy * r where E is the wave energy and CgX> Cgy are the components of group celerity. 146 There is some uncertainty about the value of that should be used in the model when applied to the nearshore zone. Wind and Vreugdenhil (1986) used the standard value (0.09) however Visser (1984), on the basis of experiments, recommended a value of the order of 3.0. A significant difference! Dong recommends a value of 2.5, based on Visser's work, and numerical experiments carried out by the present author verify that this value allows the prediction of reasonable longshore velocity profiles. In fact all calculations of circulations in this thesis have been carried out with this value. The main problem is that there is no satisfactory method of calculating ut at present and inferring it from velocity profiles is unsatisfactory in that the velocity profiles are not very sensitive to the magnitude of eddy viscosity. The prediction of vertical sediment concentration profiles, which will be dealt with in the next chapter, requires an estimate of eddy viscosity in the vertical plane, and the accuracy of this estimate can be tested by comparing measured concentration profiles with those predictions. In the course of this work the present author found that the eddy viscosity predicted using a value for of 2.50 gave concentration profiles which indicated that the eddy viscosity was orders of magnitude too high. More reasonable values were obtained if an equivalent c^ value of 0.09 was used. This will be explained in more detail, with reference to experiments, in the next chapter. 6.4 FINITE DIFFERENCE APPROXIMATION 6.4.1 INTRODUCTION Both the depth-averaged Navier-Stokes equations and the two equation turbulence model are solved using an ADI finite difference scheme. The scheme was first proposed in companion papers by Peaceman and Rachford (1955) and Douglas (1955). Its properties regarding stability are well established and will not be repeated here. The solution of implicit finite difference schemes usually involves the solution of a large number of simultaneous equations, employing a Gaussian elimination scheme. However, if the number of unknowns can be restricted to three in each equation the resulting set of equations can be formulated as a tri-diagonal matrix and solved very efficently using the Thomas algorithm. The equations as written contain more that three unknowns but this problem is overcome by using an iterative procedure described by Roache (1972). Before the finite difference representation of the equations is given a brief 147 description of the principle of the ADI scheme is presented. 6.4.2 ADI SCHEME Following Roache (1972), if the equation to be solved can be written 0£ 9$ 0$ 02£ 02£ — =. -u — - v — + a ------h a ----- (6.22) 0t 0x 0y 0x2 0y2 then the solution is achieved over two steps. In the first step the derivatives in x are written at the new time level, in the second step the derivatives in y are now written at the new time level. The two steps can be written: £n+l/2_ £n g£n+l/2 s£n 52{n+l/ 2 {2£n ------= - u ------v — + a ------+ a ----- (6.23) 1/2 At bx by <5x2 $y2 ^n+l _ ^n+1/2 ^n+1/2 5£n+1 82^n+1/ 2 8^ n+1 ------= - u ------v ----- + a ------+ a ------(6.24) 1/2 At 5x $y 5x2 5y2 where b/bx and 52/Sx2 are the space centered approximations to 0/0x and 02/0x2. On summation of the two half time steps one obtains £n+1_ 5$n+1/2 v 5£“n n+1 5£ 52£n+1/ 2 ------= - u — ) + ot ------At 5x 2 by by bXJ a + _ (---- + ---— ) (6.25) 2 5y2 5y2 By inspection the resulting equation is space and time centered and therefore second-order accurate in both space and time. 6.4.3 NAVIER-STOKES EQUATIONS The variables u, v and r\ are defined on a staggered mesh as shown in Fig. 6.3. The solution proceeds following the method set out in the previous section. In the first step the x momentum and continuity equations are solved together followed by the y momentum equation. In the second step the y momentum and continuity -c=» y ------y i i r o cm o n=» o o o T K5 i i IX) i y y y y i i cm c» Cm L UU i 0 0 0 0 j i V y y y i UJ i I i o c» o c=» o Cm o i t i l y y y y i i c=» e» Cm +■ r 0 0 0 i i i j y y y y i L 0 c» 0 0 Cm 0 i i lI i__ — — - y - — -y-- -?• XZ5 1 2 2 3 3 ny ° 1 i SXX; SXy Syy, k ,6 «=- V v y FIGURE Numerical Solution Grid 6 .3 The Current Model 149 equations are solved together followed by the x momentum equation. The equations can be written in finite difference form as follows: STEP 1 - X MOMENTUM The individual terms of equation (6.4) can be written n+1/2 n “ V I‘ I -— Uf 1.J 3t 1/2 At n+1/2 n+1/2 3u n+l/2.u^+^»j u --- ui,j <------> 3x 2 Ax n n 3u 1 . n n n n . ui>j+l — - T (vi.J + vi+l.J + vi, j-1 + vi+l, j-l) ------< ;------> 3y 4 2 Ay n+1/2 n+1/2 3tj , 771+1 >J‘ " 771 > J g — - g (------3x Ax i ^^xx ^^xy Sxxi+l,j- SXXi)j — (--- + --- ) pD 3x 3y P(Di+l,j + Di,j) Ax + Sxyj.j+i + Sxyj+i,j+i ~ Sxyj,j-i ' Sxyi+i,j-i i 4 Ay n Tbx 4 cf H u 4 °f Hiij ui,j pD pDTsinh(kD) p(Djj + D i+l,j) Di,j + D i+l,j —------— T sinh ( k — ------— ) n+1/2 n+1/2 1 3(DTxx) 'XX' l r ui+lJ " ui,j\ “ Fd ( 3x )== p(Dif j + Di+1> j) l2Di+1 J (' (Ax) ui,j " ui-1, j i " 2Di,j ^i,j ( }] (Ax)z 150 \ 9(DTxy) 1 3 n 3u v 1_ ,3 3v p S ^ 3^ > pD (sp ^ } 1 ,3 n3uN pD^3y p(Dif j+DI+if j)’ 2 Di, j+Di , j+l+Di+l, j+Di+l, j+1 , j+H , j+l+^i+l, j+/M+l, j+1 4 ' 4 n n ui t j+1 - ui ,j (Ay)2 Di , j +Di+ l, j +Di , j - l +Di+ l, j-1 H ,j+^i+l,j+^i,j-l+^i+l,j-1 4 * 4 * (Ay)2 J . (3 MD^ ) ______1______pD ^3y '“W p(Dif j + Di+lfj) ' 2 Di , j +Di , j+ l+Di+ l, j +Di+ l, j+1 H , j+^i, j+l+^i+l, j+Pi+l, j+1 4 * 4 n v i+ l. j j Ay Ax Di , j +Di+ l, j +Di , j - l +Di+ l,j-1 H ,j+^i+l,j+/*i,j-l+/*i+l,j-1 4 • 4 n n v i+ l,j- l “ V i,J-1 Ay Ax Therefore we can write Equation (6.4) as 151 n+1/2 _ n Ui»J Ui’J n+1/2 _ g . n+1/2 n+1/2 ------+ u; UD1 + VA.UD2 + (r?i+1/ ] - r]{ + SX + SY + F - T1 - T2 - T3 - 0 (6.26) where n+1/2 n+1/2 ui+l,J " ui-lJ UD1 2 Ax n n n VA 4 ^vi.j + v i+1»j + + v i+l,j-l > n n ui,j+l “ ui,j-1 UD2 2 Ay S^ i + l,j - SX X jJ SX - ( ------) / DDE 4x DDE - p(Di j + Di+1j )/2.0 Sx y i,j +1 + Sxyi+l,j+l Sxyi,j -1 " Sxyi+ 1 fj_i SY - ( ------)/DDE 4 Ay F = 4cfH i ju 7 j / (DDE.T.sinh(kifj .DDE )) on / n+1/2 n+1/2 . n+1/2 n+1/2 . 2Di+l,j-Mi+l,j-(ui+l,j ui,j ) " 2Di, jH , j(ui, j " ui-l,j ) T1 DDE. (Ax) 2 D5.E5.(uf j +1 - Ui j )- D6 .E6 .(Ui j - Ujj.j ) 12 = DDE . (Ay) 2 _ D i > j + D i+l,j + D i,j+1 + Di+l,j+l D5 4 152 Di ,j + Di+ 1,j + Di,j-1 + Di+1»j~l D6 4 ^i,j + /*i+l, j + ^i , j+1 + AM+1J+1 E5 4 M i.j + ^i+ 1, j + , j-1 + H + l.j-1 E6 4 D5.E5.(vi+1> j - VJ j ) - D6.E6.(vi+1j_ 1 - vj j.i ) T3 DDE.Ax.Ay Collecting terms Eq. (6.26) can be written -gAt n+1/2 /1 n 1 A Tm1N n+1/2 gAt n+1/2 ’H.j + (1-0 + “ At.UDl^ij' + r) i+i'j 2 Ax 2 2Ax u "j + i At (-VA. UD2 - SX - SY - F + T1 + T2 + T3 ) (6.27) STEP 1 - Y MOMENTUM In a similar fashion the y momentum equation can be formulated. The equation reduces to: n+1/2 n n+1/2 n+1/2 vi,j - vi,j vi+l,j ------+ UA. + WD + g.ED + SX + SY 1/2 At 2 Ax + F - T1 - T2 - T3 = 0 (6.28) where UA - -1 (uj— . n+1/2j ' j + n+1/2 + Ul>j' n+1/2 + n+1/2 ) n n n n n mj+l ~ ^i.J ED Ay Sxyi+i,j + sxyi+iij+i - sxyi-ij - sxyi_1>j+i sx DDE.4Ax syyi,j+i ' syyj,j SY DDE. Ay F - 4cfHi,jv i,j / (DDE.T.sinhCkj^j.DDE)) n n n n ui,j+l ui,j ui-l,j+l ui-l,j Tl = D5.E5.------D7.E7.------DDE(AxAy) DDE(AxAy) D7 " 4 ^Di.j + D i»j+1 + D i-l,j + Di-l,j+l ) E7 = 4 (^i,j + AM,j+l + Mi —1 , j + Mi — 1, j-f-1 ) n n n n vi+l,j " viJ vi,j " vi-l,j T2 - D5.E5. ------D7.E7. ------— DDE.(Ax) 2 DDE.(Ax) 2 n n n n vi,j+1 “ vi,j vi,j vi,j—1 T3 - 2 Di,j+W , j+1 ------— i - 2^i, jMi, j DDE.(Ay) 2 DDE.(Ay) 2 Collecting terms Eq. (6.28) can be written . -UA.At . n+1/2 , n+1/2 , UA.At N n+1/2 ( 4Ax” } + Vi J + ( "TZ5E } v” f j - ( W D + g.ED + SX + SY + F - Tl - T2 - T3 ) (6.29) 154 STEP 1 - CONTINUITY Equation (6.1) can be written term by term as: n+1/2 n 8rj _ ~ ^i.j 3t 1/2 At 8 (UP) n+1/2 , D>.j + Di+l.j, n+1/2 „ Di.j + 8x ui,j < —2 — .Ax-----> - ui-i,j < ------2------.Ax > 3(VD) n , DiJ + Di J+1 n Di,j + Di,j-1 vi, j < > - vi,j-i( 8y 2 Ay 2 Ay writing in full n+1/2 n - ’H.j u f ]/2.EDl - u t Y , ] -ED2 + VED = 0 (6.30) 1/2 At where Di,j + Di+l,j EDI 2 Ax Di,j + Di-1,j ED2 2 Ax lfro n VED = Vi>j . ----Di,j +Di,j+1— ------vif n j.! . Di,j------+ Di J - l 2 Ay 2 Ay Collecting terms , At 0 n+1/2 n+1/2 At . n+1/2 n (-J.ED2) u t.l'j + !!,/ + (-j.EDI) At VED (6.31) A similar set of equations can be generated for Step 2. In this case the y derivatives are taken at the new time level (n+1) and the x derivatives at the old time level (n+1/2) 155 6.4.4 SOLUTION TECHNIQUE The final equations for the x momentum and continuity for Step 1 (Eqs. 6.27 and 6.31) can be written: n+1 /2 n+1 /2 n+1/2 Aiui - 1 . j + ’H.J (6.32) n+1/2 TrHT«n+V ,j 2 +Diui,j + n-u ?+1./2 - T „ +1'j (6.33) where Aj — (- —^ • ED2)i - (^| . EDI)i T - ~5 At 2 Ax Dj = (1.0 + j A t . UDl)i Ci - (’M.j - T • VED )t Ei “ (u",j + j At( -VA.UD2 - SX - SY - F + T1 + T2 + T3)); and where j is constant. The pair of equations can be written in matrix form if considered over a range of i from 0 to n where 0 and n are the boundary values where u and t] are assumed known. - " 1 Bj VI Cl - A1U0 T T>i -T U1 El A2 1 B2 V2 C2 T D2 -T u 2 E2 a3 1 b3 V3 — C3 An-1 1 V n -1 En-1 ” Bnun-1 dk « - (6.34) The pair of equations form a tridiagonal matrix which can be solved using the Thomas algorithm. The solution gives the values of u and 77 at the new time level. Having solved for u and 77 it is possible to fomulate Eq (6.29) in tridiagonal form and solve for v. Equation 6.29 can be written: n+1/2 n+1/2 n+1/2 Hi F iv i - 1 1j vi,j +Gi vi+l,j (6.35) where UA.At 4 A x >i UA A tN = ( 4 Ax'* WD + g.ED + SX + SY + F - Tl - T2 - T3 )) { Hi = ( v 1 . J - -2 (^ 157 As before, these can be formulated as a tridiagonal matrix: 1 Gj V1 - FXV0 F2 1 C2 v2 h2 F3 1 C3 v3 h3 ^n-1 ^ vn-l ^n-1 ^n-lvn (6.36) and solved in a similar fashion. Inspection of the continuity and X momentum equations reveals terms on the right hand side of the equations which are at the new time level and therefore unknown. For an accurate solution the procedure must be iterated a number of times. The author has found from experience that two iterations are sufficent. The Y momentum equation on the other hand is solved in one iteration as there are only three unknowns per equation. It should be noted that the diffusion terms contain cross derivatives and cannot be centered correctly in time. They are always written at the old time level. In the second time step the 17 and v terms form a tridiagonal matrix when considered together and must be iterated to form a solution. Then the u velocities can be calculated from a single solution of the matrix. 6.4.5 TURBULENCE EQUATIONS The solution technique for the k and e turbulence equations is similar to that for the Navier-Stokes equations. The solution is obtained over two steps using the ADI scheme. Step 1 - k equation Equation 6.19 can be written in finite difference form 0t 1/2 At A? z XV Z ------)&,- ( ------f o - = i-p i 3 - i+r4ia p x -i3 . ri+i3 r XV P P i-Iu a - x-P x-I u a - Px+xA u + 1-T‘I+I u a A y f )+ 1 -f‘Jnu X-PX'Inu - X+Pln u + X+Px-ln u Ay xy F 7 F ------F T 5 X-Pi,u ~rTy- i u • a °'z + ^I X u In - i u in °'z ViaVln ~ (Ay) ^ Z X -P l, + Pb| I-r ‘ xa + Pla X"Pln + Pin (Ay) o z - _ ( & „ !L£) £e P b | - i+P !■>{ f ‘ Xq + X+P Iq r * X a + I+Pln ^ ® (XV) z p j - !^ + ,p i^ T'i- iq + r*!q p x-in + Pin z/l+u z/l+u (xy) z ^-0 Z xg ^Id xg r* i. - P.X+I* P la + Px+Ia Pin +P X+!n ( Q *a 6 Z/X+u z / i + u ^V Z i-piv-x-r'jfTu x-i.-f1 -a -- x+r‘^ u iv•1 x+r*Fiin u P IA +X-P! u a (>ia)e XV Z xe n f‘l-!vPx-!a - P x + I^ -P x +Iq ) ( / ‘In + r ‘ x- In (Ma)e Z/l+u Z/l+u Z/l+u Z/l+u 851 » 159 Writing in full n+1/2 n n+1/2 , n+1/2 k i,j " ki’J+ul (°i+1.j'ki+1J D i-^,j k i-l,j ’i. J 1/2 At 2 Ax vn+1/ 2 Ki+l,ikn + l/2 “ kn+l/2 Ki,i kn+] i-l,i / 2 +V1.DKDY - Fl.(------—) + F2.(------) (Ax) (Ax) n n - F3.XK1 + F4.XK2 - PH + Dt - D i, j-ei,j (6.37) where n+1 /2 + n+1/2 U1 ui-1,j 2 n n VI vi,j + viij-1 2 DKDY D i,j+l-k i,j+1 " D i,j-1*ki,j-1 2 Ay FI - ui+l,j + ui,j Di+l,j + Di,j 2 F2 - + ui .j Di —1» j + Di i j 2 a k 2 F3 - uiij+1 + ui,j Di ,j+1 + Di,j 2 a k 2 F4 = ui.j-1 + ui.j Di, j - i+ Di,j 2 - n . n XK1 ki,j+l ki,j (Ay)2 160 1 n i n XK2 ki,j ki,j-l (Ay)2 PH - Ph.D Collecting terms and knowns and unknowns the equation can be expressed in the form A* k^+i^^ + B* k^+^^ + C- = D' (6.38) where 1/2 At. U1 1/2 A t F2 2 Ax ®i,j Dj^j ^Ax)^ 1/2 At (FI + F2 ) Bi - ( 1.0 + ------— )i Dj j.(Ax)2 1/2 A t U1 Dt.i i 1/2 At FI q - ( ------:------) t 2 Ax Dj' j Di' j(Ax)1 1/2 At ----- (PH + Dt - Di,j*ei,j + F3-^1 " F4.XK2 - Vl.DKDY) + It is readily seen that this forms a tridiagonal set of equations which can be solved using the Thomas algorithm. Step 1 - e equation The formulation of this equation is similar to the k equation with e replacing k throughout and it should be noted that the source terms are slightly different. The e equation is solved in the same way. The procedure is repeated in Step 2 except that the y derivatives are written at the new time level and the solver sweeps in the y direction. 161 6.4.6 BOUNDARY CONDITIONS 6.4.6.1 NAVIER-STOKES EQUATIONS There are generally four model boundaries, a shore boundary, an offshore boundary and two lateral boundaries. At the offshore boundary it is assumed that u = v = rj = 0 since the boundary is selected such that this would be true. At the shoreline there is no normal velocity (u = 0). The no-slip condition has been chosen (v = 0) and a convenient value for rj is that of the adjacent value, inside the model. Experience has shown that this set of conditions, although not rigorously correct, gives acceptable results. At the lateral boundaries periodicity is assumed so that if the range of the points in the y direction is j = 1 , m then 50 “ * m-1 51 = S m Sm+1 = S2 where £ represents any of the variables u, v or 77. Velocities normal to internal boundaries are set to zero. 6.4.6.2 TURBULENCE EQUATIONS Zero eddy viscosity can cause spurious velocities to grow and for this reason base values of k and e are chosen such that there is always sufficent eddy viscosity to damp out such oscillations. From the point of view of accuracy this base value is chosen to be quite small (0.005), especially compared to the values that will normally be present in the surf zone and nearby. At the offshore boundary k = k base e ~ c base 162 At the shore boundary the adjacent values inside the model are used and periodicity is assumed again on the lateral boundaries. At the start of the solution procedure a simple eddy viscosity model similar to that used by Longuet-Higgins (1970) is employed to allow the solution to begin to develop in a stable turbulence condition. After a number of time steps (10% of the total solution time) the solution changes to the calculation of turbulence using the k-e model. There have been no problems due to instabilities using this technique. 6.5 MODEL VERIFICATION 6.5.1 .INTRODUCTION Unlike the situation with the wave model, where a host of analytical solutions exist with which to compare the numerical model, there are few analytical solutions for the depth-averaged Navier-Stokes equations. The model verification therefore relies heavily on comparisons with physical model studies and field work. As a preliminary test the numerical model is compared with the analytical solution of Longuet-Higgins (1970). Following this, a comparison is made with the physical model experiments described in Chapter 3. Finally the model is compared with a set of field data collected by Thornton and Guza (1986) 6.5.2 COMPARISON WITH LONGUET-HIGGINS PROFILE Fig. 6.4 illustrates a comparison between a velocity profile calculated using Longuet-Higgins' eddy viscosity model and the two equation turbulence model. It is evident that despite the differences in the eddy viscosity the profiles are quite similar. This emphasizes that fact that the velocity profile is not very sensitive to the distribution of eddy viscosity. 6.5.3 COMPARISON WITH PHYSICAL MODEL RESULTS Figs. 6.5 through 6.13 show the comparisons between the physical model tests and the numerical predictions. The results are reasonably good although the physical model generally shows a larger return flow along the shoreline towards the updrift groyne and a correspondingly larger outflow on the downstream side of that groyne. This is particularly evident in Figs. 6.5, 6.6, 6.7, 6.8 and 6.9. The physical model velocities beyond the groyne tips show a deviation in the offshore direction. This may be due in part to the limited size of the physical model and the tendency for FIGURE Longuet-Higgins and k-e Model Velocity Profile 6.4 0 equation - Longuet-Higgins 2 equation - k-e Model 163 164 TEST No. 2 0 N \ * \ s -* - \ N \ J s ' **• Si "• \ >* \ 1 - 1 ' * / — \ ' ^ M i 1 ' \ ^ \ \ \ N >» _• \ s \ N N ^ N \ ^ ^ N ^ (a) experimental results *> *H --- *------!■ (b) numerical prediction FIGURE Comparison with Physical Model - Test 2 6.5 165 TEST No. 3 ■ * - \ \ - / s ^ — •* ■'• l i -* ^ 1 * is*'-' - ^ - y / V \ \ 1 \ ^ / • \ - ; y - \ \ \ \ \ i * \ \ * ; * ' \ '\\\ 1 "• % ^ \ \ - - \ s \ *■ -* —► (a) experimental results % \ * f f N S \ % » | » I I I . + <• X \ \ ' * • t . l - - —* ^ n s s \\\N n*“ (b) numerical prediction FIGURE Comparison with Physical Model T est 3 6.6 166 TEST No. 4 * * * j * ✓ • l ✓ * ✓ •» • \ I i / / * * N \ ^ •* \ 1 » i ' * \ \ \ 1 \ \ v« ~ v. - N, -^ \ ^ — ^ N V\»\ \ ^ ^ (a) experimental results -» *• ^ "* “♦ -H S S % » I * * * 'I S S \ % I I * * * ^ N \ \ I ' ^ ^ ^ ^ \ \ \ ' " (b) numerical prediction FIGURE Comparison wi th Physical Model Test 4 6.7 167 TEST No. 5 / ✓ - - * • / ✓ / - - - / ✓ - " " \ l ' ' ' i \ J ' \ ' J l • 1 \ 1 \ \ * ' \ \ \ • * ' \ \ ^ \ \ N \ ^ (a) experimental results "* ** N % \ * | / * N S S \ ^ | t • N S \ \ \ » % - . # / ^ >* ** ** ** ^'s‘S \ W s ~ * ' "> N S, S .-s,-*--*- (b) numerical prediction FIGURE Compar i son wi th Phys ical Model Test 5 6.8 168 TEST No. 6 ' l l * * \ / l ^ l J 1 ' ' I l \ M M\W NN\N\ (a) experimental results —» —* —► —>• ) / r *■ —V \ \ J I * * "» "* N \ \ » ' ' " N S \ \ \ ' - - ' »‘^'*NNS \ WN - * - * ’ (b) numerical prediction FIGURE Comparison wi th Physical Model Test 6 6.9 16 TEST No. 7 -i------i i \ -» \ \ \ / / - — - * — ^ N \ \ j ; / " fc N % \ \ ; \ N — \ u \ ; \ \ N - \ \ \ \ 1 Nk ^ -* —► —* —*■ Ni \ ^ (a) experimental results (b) numerical prediction FIGURE Comp ar ison with Physical Model Test 7 6.10 TEST No. 8 — -* \ 1 • n ; / - " 1i ' \ i i i J • ' ' 1 ^ % * l \ * ' ’ • S \ \ i i i - \ \ \ 1 \ — \ \ ^ \ S. 1 s -* \ \ i \ \ \ —* ► - ► —*■— — (a) experimental results (b) numerical prediction Comparison with Physical Model - Test 8 TEST No. 9 ------r- i \ \ \ \ \ H \ t I l s ~ v ' ! ** >■ \ i » i ' - * * i m* \ \ ; \ \ \ • \ • i / \ \ ; \ \ \ ^ \ ^ ! \ Nil \ ^ \ \ ^ ! N . \ ! ^ \ - * ' • * { n * \ 1 vSk'Si j . . i •V - --- ^ ' — ► (a) experimental results (b) numerical prediction FIGURE Comparison with Physical Model - Test 9 6. 12 172 TEST No. 10 1 t • / l » * * 1 t 1 \ ' K ' * / • » 1 • \ l / “ * * ^ ! J \ * \ j l 1 l ' \ \ \ * ■* \ \ i \ 1 \ \ \ • \ 1 S. \ N % \ \ i \ \ \ \ \ \ \ \ N, >■> \ \ i ^ \ J 1 *** \ \ s s s s “—*■— (a) experimental results (b) numerical prediction FIGURE Comparison with Physical Model - Test 10 6.13 global circulations to form. Figs. 6.10 to 6.13 show the comparison with the partially submerged groynes. The reduction of the size of the eddy in the lee of the groyne is well represented; however, it would appear that the physical model groyne proved to be more of a barrier to the flow than its numerical counterpart. This may have been due to slight measurement errors in the physical model which were then scaled by a factor of 36. The scale of the physical model may also have contributed to this, where very small water depths could behave differently from their prototype values. As has been mentioned in an earlier section it is believed by the author that one of the reasons for the discrepancy between the physical and numerical results is due to the mass transport of fluid into the groyne bay by the incoming waves. A simple model of this was developed and a revised prediction for Test 3 was run. It was assumed that the wave profile could be represented by a solitary wave allowing a calculation of the mass of water being transported into the groyne bay at the wave celerity. This water was distributed over the groyne bay and along the downdrift groyne where the waves impinge, as an additional inflow. The results of the revised prediction are shown in Fig. 6.14 and show a marked improvement, especially with regard to the return flow along the shoreline in the lee of the leading groyne. 6.5.4 COMPARISON WITH FULL SCALE DATA The experimental work of Thornton and Guza (1986) provides a good set of data to test both the wave and current models. As has already been seen in Fig. 6.1, Thornton and Guza obtained good agreement with both their wave and current model and the measumements. In fact the agreement in part was due to an optimization of the parameters of wave breaking index and bottom friction. Using the values that were obtained for these parameters, two of the tests were run using the present wave and current model. The results are shown in Figs. 6.15 and 6.16. Once again the agreement is very good, given reasonable estimates of the breaking index and friction factor. 174 TEST No. 3 - — - \ % - f j ^ *- N -» \ > -* >• » • J S S ' --* - 1 * - \ \ \ 1 \' i ' l * - \\\1 \ / i \ \ ' I • ' \ \ \ \ 1 \ \ - ** M\\ \ ^ ^ V»S\ —► —*—*• N 's ^ ^ \ — ► >*■ ^ \ ^ ^ — — (a) experimental results (b) numerical prediction FIGURE Revised Comparison with Physical Model 6.14 - Test 3 175 FEB 3 0.8- ''tKN & \ s o / V 0.6 o o. l/) O / E / o o / / y O / 0.4 / / measured ° <7 / predicted / 0.2- 0 20 40 60 80 FIGURE Comparison with Data of Thornton and Guza 6.15 - Feb. 3 176 FEB 4 0.8 ^ s \ / / o \ O N 0.6- / o 10 e / / 0.4 p V measured o predicted — 0.2 0 20 40 60 80 0.6 00 / ss \ / O o 0"s E 00 JOO \ 0.4- \ > I/ ° o 0.2- b\ 0 20 40 60 80 distance offshore (m) FIGURE Comparison with Data of Thornton and Guza 6.16 - Feb. 4 177 CHAPTER 7 THE SEDIMENT MODEL 7.1 INTRODUCTION The way in which sediment moves and the prediction of that movement have been of considerable interest to engineers, oceanographers and geographers for some time. From an engineering point of view sediment can be a problem either due to its presence or its absence. Its presence in harbour mouths, river mouths or around coastal structures can cause problems of navigation and adversely affect the structures' performance. Its absence on recreational beaches or on the foreshore in front of important capital investment can be equally serious. It is for these reasons that sediment transport has been studied so intensely for so long. The early work on sediment transport was carried out in unidirectional flow for applications in river hydraulics. More recently the work has progressed into oscillatory flow following the work of Bagnold (1963). The literature of interest to the present work is reviewed in Section 3 after a brief discussion of the modes of sediment transport in Section 2. In Section 4 the differential equations that describe the sediment transport are given and the finite difference solution technique is discussed in Section 5. The final section deals with the verification of the model. The verification draws on as large a range of field and numerical experiments as possible. Wherever possible quantitative comparisons are carried out; however, in some cases, only a qualitative comparison is feasible. It will be seen that the model predicts not only the bulk formulae for longshore transport but also the distribution of the transport. The accuracy of the vertical distribution of suspended concentration will also be seen to be quite good. On the basis of these comparisons it is concluded that the model developed can be used with confidence in the surf zone. 7.2 MODES OF TRANSPORT In the discussion of sediment transport a distinction is often made between bed load and suspended load. While these terms should refer to quite separate forms of transport they are often used loosely, so loosely in fact that they lose all meaning. Strictly speaking bed load transport occurs where the particles move whilst being suported by grain to grain interactions. Movement is by a combination of 178 rolling, sliding and jumping. Suspended load transport occurs where the grains are supported by the flm'd (Graf, 1971, Yalin, 1977) The distinction was first noted in unidirectional flow and it was natural that when researchers ventured into oscillatory flow the terminology should be carried over. Oscillatory flow is quite different from steady flow. There is generally a large oscillatory velocity which sets the particles in motion and a smaller mean current that is then able to transport the particles. Thornton (1972) discussed bed load transport and in a series of experiments measured it using 'bed load traps'. These traps were 200mm high and designed, as Thornton put it, "to allow the sediment to fall from suspension". In this case bed load actually refers to sediment near the bed. Unfortunately these and other figures are used by researchers to calculate the ratio of bed load to suspended load resulting in, not surprisingly, wildly varying estimates. More recent experiments have been able to specify the suspended load more accurately and have identified it as the major process in the surf zone. Sternberg et al (1984) and Kana and Ward (1980), for example, measured longshore transport by suspended load only and found that it would account for all that predicted by the CERC bulk formula (Shore Protection Manual, 1984). See also the work of O'Connor et al (1983). Nielsen et al (1978) having studied high speed films of sediment behaviour on a rippled bed concluded that "the mechanisms by which sediment is moved show little resemblance with unidirectional flow patterns" and "actual bed load transport as known from unidirectional flow hardly occurs". On the basis of these results and conclusions, only suspended load transport will be covered in this thesis. This will mean results are only applicable to fine sands, 7.3 LITERATURE REVIEW 7.3.1 INTRODUCTION There is a great wealth of papers on sediment transport in the literature, so great in fact that it would be impossible to review even a reasonable proportion of them. Instead, effort is concentrated on those papers referring to the surf zone where high bed stresses result in plane beds, any ripples that may have occurred having been washed out. As mentioned in the previous section only suspended load is considered. 179 Sediment transport in the nearshore zone falls into two catagories, the general longshore transport where sediment is stirred by the incoming waves and then transported by the mean longshore currents, and the onshore-offshore transport which is believed to be carried out by a combination of wave action and mean currents. The long ore transport is reviewed in Section 2 and the cross-shore transport in Section 3. Experimental studies, both in the laboratory and in the field that have added to the total knowledge are discussed in Section 4. The papers specifically on the formulation of the vertical velocity profile, which forms an important part of the suspended transport, are reviewed in Section 5 and finally Section 6 looks briefly at the development of bathymetric evolution models. 7.3.2 LONGSHORE SEDIMENT TRANSPORT Bagnold (1963) carried out early work on both unidirectional and oscillatory flows and the subsequent transport of material. The work led to the developoment of the so-called energetics approach where the sediment transport is calculated in terms of the work involved in moving it. The flowing water is viewed as a 'transporting machine' and the quantity of sediment moved is a function of the power of the stream, the slope of the bed and the efficency of the water in transporting the sediment. The last term, the efficency factor, is a purely empirical factor. Bagnold extended the concept to include both bed load and suspended load leading to an expression for total load. Both components relied on separate efficency factors but it was not possible to determine them theoretically, nor could they be calculated separately from experiments. From the unidirectional model Bagnold was able to derive a model to describe transport by a mean current in an oscillatory environment. Komar and Inman (1970) carried out a series of experiments using sand tracers to determine the correct relationship for longshore transport. The authors tested Bagnold's model as well as an intuitive one based on the longshore component of energy flux of the waves. The models were calibrated and shown to be equivalent if one assumed that the waves were responsible for the generation of the longshore currents, and that the latter were not due to tidal effects. Komar (1971, 1977) continued the work on longshore transport by including further data sets in the calibration. The work led to the commonly accepted expression used in the U.S. Army Corps of Engineers, Shore Protection Manual II " K p l (7.1) 180 where Ij is the immersed weight of longshore transport and P| is the longshore flux of wave energy. The value of the non-dimensional constant K is generally taken equal to 0.77 as determined by Komar. The value does of course change depending on the units used. Bailard and Inman (1981) extended the bed load model of Bagnold (1963) by calculating a time-averaged rate of sediment transport from a consideration of the instantaneous time-dependant velocities. The authors concluded that Bagnold's theory would only be valid for near normal wave angles and weak currents. The solution allowed not only the longshore transport to be calculated but also the shore normal component. This latter aspect will be dealt with in the next section. Bailard (1981) extended the earlier work with the formulation of a total load model. However, the energetics based model suffers a number of serious drawbacks especially when applied to the nearshore coastal zone. The basis of transport, that a percentage of the fluid power is used to move the sediment, does not allow for any real physical understanding of the particle movements. The lack of theoretical basis for the efficency factor is also a disadvantage. Finally, it has been shown in field studies (Jaffe et al, 1984) that the vertical distribution of suspended sediment and especially its variation in time are important in understanding many processes. No such information can be obtained using the energetics approach. An alternative approach to that taken by Bagnold is to look at the sediment transport in terms of the shear stress on the sediment at the bed. Shields (1936) developed an expression for the initiation of sediment transport based on the bed shear. From that came the definition of the Shields Parameter: T e p(s-l).g.D (7.2) where r is the bed shear stress, s the relative sediment density and D the particle diameter. Bijker (1968) took the work of Frijlink (1952) and compared the results of laboratory experiments with the theoretical prediction. Frijlink (1952), following earlier work by Kalinske, had suggested that the transport should be of the form a ADpg = b . e fir (7.3) D( V-t / p ) 1 / 2 181 where r is again the bed shear stress, ^ is a ripple coefficent, S is the sediment transport, a and b are empirical coefficents, A is the relative density of the sediment and D is the particle diameter. Based on laboratory data Bijker was able to calculate values for the empirical coefficents a and b. Taking the value for b as -0.27, in agreement with Frijlink, Bijker found a = 0.74. This compared with the value a = 5.0 that Frijlink had found. Bijker ascribed the difference to the fact that Frijlink had used prototype data whereas Bijker had used laboratory data. Bijker (1968) went on to predict the total longshore transport using an expression for the mean longshore velocity derived by Eagleson (1965). The calculation also included an allowance for suspended load which was assumed to have an exponential decrease of concentration from the known bed value. The calculation was in fact quite crude in that the velocity used was a mean value with no limit given on its extent in the offshore direction. The principle advantage of the method was that it allowed transport to be calculated where the mean current was not necessarily wave induced. The Bijker model has subsequently been used by many researchers. Coeffe and Pechon (1982) used it to calculate transport as part of a bathymetric evolution model. Comparison with the CERC total transport figure gave order of magnitude agreement although the coefficent value b=5.0 had been used as originally suggested by Frijlink (1952). Fleming and Hunt (1976) derived a transport model based on bed shear in an application to the design of a cooling water intake. By assuming that the shear in excess of the critical value must be dispersed within the bed the authors were able to derive the concentration and extent of the bed load layer. The suspended concentration was obtained by solving the vertical transport equation 8c 8 3 t = + cw) (7.4) where c is the sediment concentration, w the fall velocity and e the turbulent eddy viscosity. In fact the authors also assumed steady state conditions such that 8c/8t=0. The subsequent bed evolution model was tested against measured beach volume changes and shown to give good agreement. The accuracy of the suspended concentration profile was also shown to be good although the model required as input a bottom concentration value that had to be derived from field measurements. 182 The model therefore was not universal, however, if such data was available the results could be expected to be very good. Nieslen et al (1978) and Nielsen (1979) studied in some detail the behaviour of sediment on a rippled bed under oscillatory flow. It was found that the sediment tended to be thrown up into suspension twice every period as the direction of the bottom velocity changed. This led to the specification of a time dependant pick-up function which formed the bottom boundary condition in the solution of the vertical concentration profile. The boundary condition can be written "e 35 “ at z - zb (7.5) Various forms of pick-up function are possible, examples of which are shown in Fig. 7.1. In this figure the ratio of the forward and backward peak velocities determines the ratio of the heights of the two peaks and the phase angles \p+ and are, in the case of a rippled bed, a function of the order of the wave theory used. The angles are a measure of the phase where the velocity changes sign. Under a pure sinusoidal wave the height of the two peaks would be equal and = x/2 \p~ = 3x/2 Nielsen (1979), using estimates of bottom generated eddy viscosity, was able to show good agreement between predicted and laboratory measured vertical concentration profiles. Further, by including an allowance for an increased eddy viscosity due to wave breaking, Nielsen was able to obtain good agreement with laboratory results measured under breaking wave conditions. Nielsen assumed a parabolic distribution of eddy viscosity due to breaking with the maximum value at the surface. High in the water column the eddy viscosity tended to be dominated by that due to breaking whereas close to the bed that influence was reduced. Fredsoe et al (1985) considered the vertical distribution of sediment taking special account of the distribution of eddy viscosity. The model was quite detailed in that it included the time dependant boundary layer thickness and its effect on eddy viscosity. The numerical predictions were compared with full scale field data and the agreement was good. One of the difficult aspects for suspended load models is the specification of the bed level concentration. Nielsen (1979) and Nielsen et al (1978), although using 183 FIGURE Sediment Pick-Up Function of Nielsen (.197 9) 7 . 1 184 a pick-up function, still required a bed level concentration to calibrate that function. This was obtained from experiments and expressed as a function of the Shields parameter. Fredsoe et al (1985) on the other hand used a theoretical relationship between concentration and the Shields parameter calculated by Engelund and Fredsoe (1976) based on work by Bagnold. The latter work gives much higher concentrations for the same Shields number although no attempt has been made by the present author to investigate this discrepancy. It is believed by the present author that Nielsen's expression, based on experiments, may be the most reliable. The most detailed treatment of eddy viscosity is given by Deigaard et al (1986a). A one-equation turbulence model was solved to calculate the distribution of eddy viscosity caused by wave breaking. The production term was based on energy losses in a hydraulic jump or propogating bore. Although there are problems with determining eddy viscosity from a one-equation model, in that it is still necessary to estimate a mixing length, the resulting distribution shows some interesting features. Fig. 7.2a shows the non-dimensional turbulent kinetic energy calculated by Deigaard et al while Fig. 7.2b shows an eddy viscosity distribution calculated using typical values of wave height and period, water depth and particle size. It can be seen that eddy viscosity is approximately constant in the top 80% of the water column while in the lower 20% a parabolic distribution fits reasonably well. The approximate distribution is in fact the one used by the present author in the sediment model. Deigaard et al (1986a) tested the model by comparing it against both laboratory and field data and in both cases the agreement is satisfactory. Fig. 7.3 shows the comparison with some field data collected by Nielsen (1984). Deigaard et al (1986b) used the model for a littoral drift calculation. Field data were available from a 1600m long trench that had been dredged across a coastline in the North Sea. (The trench was part of a scheme to bring gas and oil pipelines ashore.) The subsequent natural refilling had been closely monitored and in fact much of it had occurred over a single storm period. The results show order of magnitude agreement as seen in Fig. 7.4. 7.3.3 CROSS-SHORE SEDIMENT TRANSPORT The problem of specifying the factors involved in the transport of sediment in the longshore direction, as seen in the previous section, is not a trivial one. The mean current profile and the vertical concentration of the sediment must be represented accurately or quite large errors can occur. The question of cross-shore transport, however, is an order of magnitude more difficult. 185 186 FIGURE Vertical Sediment Concentration Data of 7.3 Deigaard et al (1986a) 187 FIGURE Longshore Transport Prediction of Deigaard 7.4 et al (1986b) 188 Many of the models described in the last section are general sediment models that could predict both the longshore and cross - shore components of sediment transport. These models will be looked at again briefly in this section together with the models that have been designed specifically for onshore-offshore transport. Bailard (1982), following the energetics approach of Bagnold produced a model that would predict the cross shore movement based on wave asymmetries and a specified mean current profile. The model was subsequently tested by Seymour and King (1982) who found that although it performed well at times in predicting profile changes that had been measured in the field, its performance was not consistently good. Seymour and King point out that since the net transport is the difference between two numbers, which are themselves cubes, the results are likely to be sensitive to noise in the input data. Seymour and King also looked at a number of other on-offshore models in the study. Some were simple models based on parameters such as offshore wave steepness or tidal current and local wind speed. None was able to predict all the bed changes. One that did work reasonably well was based on an heuristic model by Dean. Dean (1973) argued that if sediment were thrown into suspension as the crest of a wave passed, the net direction of transport would depend on the fall time of the sediment. If it fell in less than half a wave period the sediment would have been under the influence of a net forward velocity and moved forward or onshore. On the other hand, if it fell in over half a period the net movement would have been offshore. This of course neglects any effects due to mean currents that may be present. Based on this argument and with the aid of laboratory data Dean was able to derive a parameter, the value of which would predict whether a storm profile (net offshore) or a normal profile (net onshore) would form. The parameter can be written TW H0 onshore 0gT (7.6) LO < offshore where w is the fall velocity, T the wave period, Hq and Lq the offshore wave height and length respectively and /3 a dimensionless constant related to the height that the sediment is thrown. Dean found that |3 = 0.6. 189 The model, although crude to the extent that it ignores mean currents and does not go into detail about the actual pick-up and deposition of sediment, is appealingly simple. The present author has investigated the model Lk some detail and some results will be shown in a later section. Dally and Dean (1984) extended the model to include mean currents in the onshore-offshore direction. These were calculated taking into account the radiation stresses. A simple exponential distribution of sediment concentration was assumed and the problem of calculating a proper bed concentration was left unsolved. The model was compared with laboratory and field data and although the predictions looked reasonable it was not possible to obtain good agreement between the model predictions and observations. The model was criticized by Seymour (1984) on the basis that field observations had shown that maximum concentrations in the surf zone often occurred as a result of low frequency offshore surges rather than periodically at the wave frequency. This may be a valid criticism, but, as will be seen in the next section, there have been field investigations that have found suspension events at the wave frequencies. Nielsen (1979) and Nielsen et al (1978) attempted to use the numerical model, described in the previous section, to predict the cross-shore transport in a wave flume. The results were generally not good, due in part perhaps to the wave conditions in the flume which included a partial standing wave pattern. Nielsen (1979) was able to show that no net transport was possible under first order Airy waves. Hedegaard (1985) studied net cross-shore transport due to a combination of wave asymmetry, mean flows and Lagrangian motion. The conclusion reached was that only transport in the direction of wave propagation was possible. The present author has not studied the mean current or Lagrangian aspect of the transport but the part due to the wave asymmetry does deserve some comment. The sediment model used followed Fredsoe (1984) and Fredsoe et al (1985) assuming a bed concentration in phase with, and a function of, the Shields parameter. In this case only transport in the direction of wave propagation is possible. However, as will be seen in the next section, experimental work by various authors suggests a phase lag between bottom velocity and the bed concentration. This lag is crucial to the net direction of transport and its inclusion can allow sediment to move in either direction depending on the ratio of fall velocity to wave period as suggested by Dean (1973). 190 7.3.4 EXPERIMENTAL STUDIES Despite the enormous difficulties involved in measuring the behaviour of sand particles numerous experiments have been carried out providing an important contribution to the overall knowledge of sediment transport. Komar and Inman (1970) measured longshore transport using dyed sand. By measuring the depth of disturbance of the sand and the rate of movement of the centre of gravity of the sand, estimates of the longshore transport rate were obtained. Thornton (1972) used a special design of 'bed load' trap to estimate longshore transport rates and the distribution in the cross shore direction. The traps sat on the bottom and were 200mm high and 400mm wide. Thornton estimated that the efficency in terms of the percentage of sediment trapped was between 40% and 100%. Although referred to as bed load traps the devices would have in fact given a good estimate of total load as much of the suspended transport is in the vicinity of the bed. As the emphasis changed to recognize the importance of suspended load, transport researchers turned their attention to measuring suspended load concentrations through the water column. Coakley et al (1978) used a pumping system where samples of fluid were collected at three different elevations over the depth. The system was mounted on a sled that could be towed in and out across the surf zone providing information on the whole zone. The sled carried instruments to measure velocity and could therefore provide an estimate of total suspended transport. Kana (1978) and Kana and Ward (1980) describe a series of experiments using an instantaneous sampling system. Whereas a pumping system takes some time to collect a sample, around 30 seconds for Coakley et al (1978), and therefore gives a time-mean concentration, the system allowed samples to be collected more quickly. This would be an advantage if the temporal behaviour was being studied but could give spurious results otherwise. The authors noted that the suspended sediment load increased dramatically inside the breaker line, as did Fairchild (1972). The results from Kana (1978) showed that plunging breakers entrain an order more sediment than spilling breakers do. This present study, however, is confined to spilling breakers where entrainment is assumed due to bed shear stress rather than any turbulent effects. 191 Kana and Ward (1980) measured suspended sediment during and after a storm. The authors found that as much as five times more sediment had been in suspension during the storm and that during these conditions the total CERC-predicted littoral transport rate was transported in suspension above 50mm from the bed. During the post-storm condition only a third of the predicted rate was transported in this manner. Sternberg et al (1984) used an optical device to measure instantaneous suspended sediment concentrations. The authors found that individual suspension events were in phase with bores propogating across the surf zone and that the frequency and duration of suspension events were strongly correlated to the incident wave conditions. The authors also noted that there were generally high suspended sediment concentrations in the swash and backwash zone. The distribution of the longshore transport was also measured with the maximum ocurring at around 60% of the surf zone width. It is believed that in this case the surf zone width refers to the total width over which waves were breaking. Had the sea been monochromatic, with equivalent wave conditions, the width would have probably been reduced. This would move the maximum transport closer to the 'monochromatic' breaker line. The study also estimated the total longshore transport and like Kana and Ward (1980) showed that the total transport rate predicted by the CERC formula could be accounted for by the suspended load transport. Jaffe et al (1984) using a similar optical device took measurements of suspended sediment concentrations with a view to studying the cross-shore transport. Again the instrument, together with an electromagnetic current meter and a pressure transducer, was mounted on a sled that was towed across the surf zone. The records show long periods of low concentrations with occasional bursts of high suspended sediment concentrations. The bursts appear to be related to incoming waves but the reason for the variable response to what appear to be similar waves is unknown. The most important finding of the study was the possibility of net transport of sediment onshore even under a net offshore mean current. The authors found that the moments of high suspended sediment concentration were strongly correlated to the onshore component of the wave cycle resulting in net onshore transport. This is significant because it shows the importance of wave induced net transport in the surf zone. 192 Murray (1967) also provides evidence for the importance of wave action for cross-shore transport. The author studied the behaviour of coloured sand of three different sizes (and colours) under the same wave conditions. The conclusion reached was that under identical conditions finer particles had a greater tendency to move offshore. This is in fact in agreement with Dean's hypothesis based on fall velocity. Richmond and Sallenger (1984) studying changes in cross-shore beach profiles and sediment size distributions also provide evidence for different size particles moving in different directions under identical wave and current conditions. Since waves have been shown to be important in net transport in the cross shore direction the phase relationship between bottom sediment concentration and velocity becomes critical. Dean (1973) assumed that sediment was thrown up instantaneously as the crest passed but it is more likely that the suspension event occurs over some finite time. Hedegaard (1985) assumed that velocity and concentration were in phase and on the basis of this postulated that sediment could move in the direction of wave propogation only. The experimental studies cited above have shown this to be incorrect. Horikawa et al (1982) studied bed concentration under sheet flow conditions in an oscillatory water tunnel. Although the eddy viscosity and turbulence would be different under waves Nielsen (1979) has shown that bed conditions inside the oscillatory bed layer are due mainly to the oscillatory motion and not to other effects such as surface turbulence. The authors first tested the inception criteria for sheet flow conditions. Using sand of various sizes ranging from 0.2mm to 0.7mm a Shields parameter of between 0.5 and 0.8 gave a good approximation to the point of inception. The agreement was at least as good as that predicted by the formulae of Dingier and Inman (1976), Manohar (1955) and Komar and Miller (1974). Using a special device that measured concentrations near the bed the authors recorded concentration, during one wave period at levels between 1 mm and 25mm above the bed. The authors concluded that the sediment concentration variation in the vicinity of the bed was almost equal to that of the velocity. Inspection of the results in Fig. 7.5 reveals that the velocity tends to lead the concentration by a phase angle of the order of 30 . Naturally the phase difference increases away from the bed showing the time taken for the sediment to diffuse upwards. The other interesting feature of the concentration is the rapid increase and the more gradual decrease in the full cycle - a feature predicted by Fredsoe et al (1985) and related to the formation and collapse of the boundary layer. Fredsoe's 193 194 prediction is shown in Fig. 7.6. Staub et al (1983) also measured the sediment concentration near the bed under sheet flow conditions in an oscillating water tunnel. Fig. 7.7 shows one set of results. Again the phase lag is evident although it must be remembered that these results are measured 18mm above the bed. At this level the lag is of the order of 40*. The measuring apparatus in these tests involved pumping 16 separate samples per period and it was noted that the orientation of the intake nozzle had a strong effect on the results. The results shown in Fig. 7.7 are for the sampling nozzle at right angles to the main flow. It is believed by the present author that this should give the most reliable readings. The previous experiments were carried out under controlled laboratory conditions. It is always possible that the important forces in the field may be quite different from those assumed for the purposes of laboratory simulation. Brenninkmeyer (1974) measured suspended sediment concentrations in the field using an array of light sensitive diodes opposite a light source. The data was digitized and spectrally analysed to reveal the predominant frequencies of suspension events. The results indicate that the majority occur in the low frequencies, occasionally coincident with the swell period, but often much longer, around 30 seconds. The behaviour was also position dependant within the surfzone with some positions showing frequency of suspension events equal to the incoming waves and swell. The conclusion to be drawn is that in the field, although waves are obviously important for stirring up sediment, there are additional forces which influence sediment suspension. Since it is not possible to identify and model these at present the models developed will only be of limited validity. It would be hoped ) however, that the qualitative features of the model are reasonably accurate. Numerical models of suspended sediment need to be able to predict not only the phase but also the magnitude of the bottom concentration. Nielsen (1986) analysed data for both sheet flow conditions and over rippled beds. The result was a simple relationship between mean bed concentration and the maximum Shields parameter over a period. For sheet flow conditions this can be written: C0 - 0.005.03 (7.7) 195 - c/cb.ma*. * 10° ~ K)'1 KTJ K)*3 0 it ut FIGURE Prediction of’Bottom Concentration - 7.6 Fredsoe et al (.1985) 196 FIGURE Bottom Concentration Staub et al (1983) 7.7 197 where 01 is in this case the maximum Shields parameter over one wave period. As mentioned earlier^ this formulation by Nielsen differs from that of Fredsoe et al (1985). A comparison is shown in Fig. 7.8. For the present work the method of Nielsen will be adopted. 7.3.5 VERTICAL VELOCITY PROFILE The total longshore transport is the product of velocity and sediment concentration integrated over depth and time. Since the sediment concentration is greatest at the bed and decreases rapidly higher in the water column it is important that the velocity profile is specified correctly. Previous workers, for example, Fleming and Hunt (1976), applied a power-law profile over the total depth, of the form U - u' ( ^ ) 1 / 7 + constant (7.8) where U' is the value of the mean current at the surface and h is the total depth. However it is known that the profile is affected to a large extent by wave-current interactions and the effects of the bottom boundary layer. These effects should be taken into account as far as possible. The most recent model of the vertical velocity profile is due to Fredsoe (1984). By assuming a logarithmic velocity profile within the main flow as well as in the boundary layer Fredsoe was able to calculate the instantaneous velocity throughout the entire water column. Included in the model were the effects of a growing boundary layer and its effect on eddy viscosity and the resulting profile. It was also possible to model the increase in apparent roughness of the bed caused by the oscillatory wave motion. Taking averages over a wave cycle, Fredsoe was able to obtain good agreement with the experimental work of Bakker and Doom (1978) in combined waves and currents. In the case of pure waves good agreement was found with predicted and measured friction factors over both rough and smooth beds. The model was subsequantly used by Fredsoe et al (1985), Deigaard et al (1986a) and Hedegaard (1985). 7.3.6 BATHYMETRIC EVOLUTION MODELS Although there has been some mention of models to predict bed changes 198 FIGURE Comparison of Bottom Concentration 7.8 Prediction 199 around groynes earlier in this thesis, it is appropriate to make further mention of some more general bathymetric evolution models. De Vriend (1987) in a study of bathymetric evolution models postulates that three points must be considered in the design of a model. 1 . each module must be able to perform a specific task. e.g. to predict waves or currents. 2 . the combination of the modules should be well balanced, i.e. there should be no weak or over-strong links. 3. the combination of the modules must not give rise to spurious interactions. The third point is perhaps the most difficult condition to prove or satisfy but in general most evolution models to date have obeyed the first two conditions. Fleming and Hunt (1976b), as mentioned in Chapter 2, developed an evolution model in the study of a cooling water intake harbour. The wave model used a simple refraction method and the current model was based on the radiation stress concept of Longuet-Higgins. The sediment model, developed for the task, has already been discussed in the previous section. Coeffe and Pechon (1982) also used pure refraction and a simple current model combined with the Bijker (1968) sediment model. The resulting model was used to predict bed changes in a semi-circular harbour. The changes appeared reasonable but were not compared with field or laboratory experiments. O'Connor et al (1981) give details of a model where improvement was sought by enhanced current modelling facilities and a more accurate sediment transport model. The model was not designed for use with nearshore structures as the wave model did not include diffraction effects, however the current model allowed tidal and mass transport currents to be taken into account. The sediment model followed the work of Bijker (1968) and Swart (1974) and account was taken of the increases in both bed and suspended load that occur under the influence of waves. The basic assumption of the sediment transport stage, that local equilibrium in the sediment suspension exists, was also discussed in some detail. The authors argue that local equilibrium is a valid assumption in the case of coarse sediments, such as sand, but would not be valid in the case of fine silts. 200 Yamaguchi and Nishioka (1984) introduced enhanced wave and current modelling by including simple diffraction in the wave model and the non-linear convective terms in the current model. It is believed that, particularly in the vicinity of coastal structures, these two phenomena will be very important. The authors used the model to predict bed changes around common coastal structures such as offshore breakwaters and various groyne configurations. No experimental confirmation was given although the authors claimed that the model produced the general characteristics of erosioh and deposition caused by the presence of coastal structures. Watanabe (1985) reported the most detailed wave and current model to date. The wave model solved the mild slope equation using an hyperbolic approximation. In this case refraction, diffraction and reflection would all be calculated accurately. No wave-current interactions were calculated. The author noted that one of the problems of the increased accuracy of the wave and current model was the increased execution time involved. Consequently it was not possible to iterate the wave and current model with the predicted bed changes, which has been a standard feature of earlier and simpler evolution models. This restriction also applies to the model developed by the present author. It is felt that the increased accuracy obtained justifies this shortcoming and that it may soon be overcome as computers become faster and more powerful. One of the major difficulties in models of this type is the accurate inclusion of the important cross-shore transport. Yamaguchi and Nishioka (1984) mention this as being important but apparently do not include it in their model. Watanabe (1985) included it by estimating two parameters which would give the direction of transport and calculating the magnitude of that transport independantly. The other approach possible is that adopted by Perlin and Dean (1985) which was to assume an equilibrium profile and to use the presence of cross- shore transport to achieve this equilibrium. This had also been assumed earlier by Bakker (1968) in the two-line model of the coastline near a groyne. 7.4 SEDIMENT MODEL EQUATIONS 7.4.1 INTRODUCTION Although the transport of sediment in the longshore and cross-shore direction is caused by the combined wave and current field and should therefore be treated together, it is convenient to treat them separately. 201 The longshore transport is dealt with as a time-averaged transport of suspended material by the mean currents. The waves are responsible for the suspension by firstly stirring the material at the bed and then the eddy viscosity caused by a combination of wave breaking and the mean current holds the sediment in suspension. The currents then transport the material. One mechanism for cross-shore transport is considered which relates to the wave asymmetry. It will be shown that sediment can move in either the onshore or offshore direction depending on the ratio of fall velocity to wave period. Since this model forms only a part of the total cross-shore transport it will not be included in the final bathymetric model. In its place a simple diffusion term is used which, although not able to predict actual cross-shore transport in the short term, would, the present author believes, give a reasonable representation of the long term effects of this transport. The vertical velocity profile under combined waves and currents is calculated following the method derived by Fredsoe (1984). Finally the bathymetric evolution model is presented. 7.4.2 LONGSHORE TRANSPORT The equation governing the vertical sediment concentration is well known and given by many authors (Deigaard et al, 1986, Fleming and Hunt, 1976 and Hunt, 1954) It can be written: 3c 3c 3 . 3c. 3 t = w3 5 + (7.9) where c is the concentration, w the fall velocity and es the eddy viscosity of the sediment. As is commonly assumed the eddy viscosity of the sediment will be assumed equal to the eddy viscosity of the surrounding fluid. Two boundary conditions are needed, one at the water surface where it is convenient and reasonable for most of the surf zone to assume that: c (z)z=h - 0 and the other at the bed where, following Fredsoe et al (1985) and others it is assumed that: c (z ) z=0 = c ( 0) The maximum value of 0 is used 202 where 0 is the Shields parameter. It is further assumed that the relationship between bed concentration and Shields parameter can be written, after Nielsen (1986) c(z)z=o “ 0.005 Having specified the boundary conditions the only remaining unknown is the magnitude and distribution of the eddy viscosity. At a point in the water column the total eddy viscosity is made up of a number of components. There is generally an important contribution from wave breaking at the surface. There will also be contributions from the wave and current interaction at the bed plus an additional component throughout the depth from the mean current. The use of the k-e turbulence model in the solution of the current field gives an estimate of the eddy viscosity over the whole field. This figure is depth-averaged and therefore the distribution in the vertical must be determined. Nielsen (1979) suggested a parabolic distribution but the one equation turbulence model of Deigaard et al (1986a) shows that this will only be true near the bed. See Fig. 7.2. Therefore eddy viscosity due to breaking will be written: e '(^h/5; Z ) 2 z< 0 . 2 h (7.10) f 6 z ^ 0 . 2 h where e' is the surface value of wave-induced eddy viscosity. Although the eddy viscosity away from the bed is dominated by the contribution due to breaking, the eddy viscosity at and near the bed is of prime importance in that it determines how much of the sediment is thrown up from the bed to a higher level in the water column. The eddy viscosity due to the waves and currents follows the work of Fredsoe et al (1985) who suggested a parabolic distribution similar to that established in open channel flows. Inside the wave boundary layer z Ufc €b = K.Z.Uf[l - - (1 - _ ) ] (7.11) 203 where k is the von Karman constant (0.4), Uf is the wave-induced shear velocity, Ufc the current shear velocity and 5 the wave boundary layer thickness. The wave-induced velocity is period averaged. Outside the wave boundary layer eb - KzUfc( 1 - 2 ) (7.12) where h is the water depth. The total eddy viscosity at a point is the sum of the values due to breaking and to the waves and currents. es =* + ew (7.13) 7.4.3 CROSS-SHORE TRANSPORT The basic equation for suspended transport in the cross-shore direction is identical to that in the longshore direction. For completeness it is repeated here. 0C 0C + 0C, 0F " w05 (7.9) The distribution of eddy viscosity, es, is also the same but the bed boundary conditions will be specified in a manner that allows the phase differences between the bed concentration and the velocity to be taken into account. In the above equation c is instantaneous but es is period averaged. At the bed "es0z = p(t) at 2=0 (7.14) where p(t) is the pickup function specifying the amount of sediment thrown into suspension at the bed. After Nielsen (1979) the pickup function can be written: p(t) ~ “T + jT ~(2m -1 )! ! t cos2m j ( w t - ^ + ) + p cos2m I(cot-^-)] (7 .1 5 ) where n - (U- / U + ) 2 204 and U and are the extreme return and forward velocities respectively. As mentioned previously, \p~ and ft allow the phase relationship between velocity and pickup at the bed to be taken into account. The values used in the model at present mean that the pickup lags the velocity by 30* as determined by laboratory experiments. Although it would have been more correct to calculate the growth and decay of the boundary layer and the oscillatory velocity profile it has been assumed for the purposes of simplicity that a uniform oscillatory motion extends to the bed. It is not anticipated that this will affect the qualitative results of the model. 7.4.4 VERTICAL VELOCITY PROFILE For the transport by a mean current the vertical distribution of that current becomes critically important. Fig. 7.9 illustrates approximate profiles of velocity and sediment concentration. It can be seen that the maximum sediment concentration is at the bed, where the velocity is at a minumum. This concentration can be orders of magnitude higher than the concentration higher in the water column. Therefore it is imperative that the velocity be modelled correctly at this level. A convenient theory of velocity distribution in combined waves and currents is due to Fredsoe (1984). The velocity profile is assumed to be composed of two components: a steady component Uj due to the mean current and an unsteady component U 2 due to the waves. Inside the boundary layer the unsteady component is assumed to have a logarithmic profile. Uf y U2 =— Ink730 <7-16> and the steady component is assumed to be given by UfO y Ul= — lnk730 <7-17) where k is the grain roughness (k=2.5D) and D is the particle diameter, k is the von Karman constant. # 205 cone, (p.p.m) 400 800 1200 1600 2000 FIGURE Mean Velocity and Sediment Concentration 7.9 Profiles Values calculated by present author. 206 Outside the boundary layer the wave component is assumed to be given by potential theory u2 - ulm sin(w t) (7.18) and the steady component by UfC y ------In (7.19) where kw is the apparent roughness of the bed. At the top of the wave boundary layer the vectorial sum of the potential flow and the mean current profile is equal to the instantaneous value of the velocity given by Eq. 7.16. If y is the angle between mean current direction and the direction of wave propogation then Uf 5+k/30 2 [ k ln ( k/30 + [Uq siiTy]^ (7.20) A quantity z is defined kUq (7.21) UfTT * where (7.22) Therefore an expression for the instantaneous boundary layer thickness can be obtained 5 (7.23) By applying the momentum equation on the boundary layer in the direction perpendicular to the mean current it is possible to obtain a differential equation for z. z(l/1 + z - ez) d U nu 3 0 k k 2Uq2 + z 2U f 02 + 2zKUf()Uo cos 7 dz ------4- ' J (7.24) dt (ez(z-l)+l)U q dt k ez (z-l)+ l This equation can be solved; but to start the solution a relationship between z and t at t=0 must be derived since Eq. 7.24 is singular at that point. For small t and z Eq. 7.24 reduces to dz l K2 ulm „ (w t> Uim ut [~ + ( tytt )' --- ] n ------cosy) - — (7.25) d(o)t) z 2 Ufo .3 uf 0 ^2 7 «t where 60 k U p Q 0 o)k ( 7 . 2 6 ) This has the solution 2 / ■ 1 * ^ (3 J u t (7.27) For larger values of z and t Equation 7.24 must be solved numerically. Having calculated the variation of z and Uf in time it is possible to solve for the mean bed shear stress r = pUfc2 = f j Tb cosP dt = i J pUf2 cos u o k c o s 7 + U f Q z c o s kw 30 5m l-Ufo/Ufc > <7-30> 5m is taken as the mean value of 5 at cot = x/2 and cot = 3 rl2 . Hence it is possible, knowing the mean velocity, water depth and wave and bed characteristics, to calculate the mean velocity profile. 7.4.5 BED EVOLUTION MODEL The bed evolution model is based on the equation relating bed changes to the transport quantities locally. 8 qx 3qy 3z + 3F The assumption, implicit in this equation, is that local equilibrium exists. This point has been discussed by O'Connor et al (1981) and implies that the sediment will respond quickly to changes in the capacity of the flow to transport sediment. With sand, in a marine environment, this is considered a reasonable assumption. The equation as written contains no diffusion term, save that due to numerical effects of the finite difference scheme, and can therefore predict unrealistically harsh bed changes. To overcome this some diffusion has been added deliberately in the form of spatial averaging of the predictions. This has been shown by Ponce and Yabusaki (1980) to be equivalent to calculating diffusion using second order derivatives in the transport equation. The diffusion in the longshore direction was chosen to be small so that it smoothed the predictions without affecting the results significantly. A higher diffusion coefficent was applied in the cross-shore direction to smooth the profiles and to simulate the long term effect of the cross-shore transport. This is a similar approach to imposing an equilibrium profile on the model as suggested by Perlin and Dean (1985) but the method adopted here is less formal. 209 7.5 FINITE DIFFERENCE EQUATIONS 7.5.1 INTRODUCTION Under certain circumstances analytical solutions are available for some of the differential equations listed in the previous section. Nielsen (1979) for example, by assuming a simple eddy viscosity distribution was able to obtain analytical solutions for the vertical sediment concentration. Under more general conditions analytical solutions are not available and numerical ones must be pursued. This section lists such numerical solutions. In many cases the techniques are quite standard and proofs of stability, convergence and consistency are not given. Details can be obtained from any number of standard mathematical texts. 7.5.2 LONGSHORE TRANSPORT When written in time-mean form Equation 7.9 is 8 c 8c w 35 (€ 0 (7.32) which reduces to 8c 0 wc + 3 1 (7.33) 8c wc (7.34) This equation can be solved by a variety of techniques from a simple Euler method, to a more accurate Runge-Kutta or, as in this case, by the Runge-Kutta—Merson method, (De Vahl Davis, 1986). This is an improved variation on the usual Runge-Kutta method in that it allows an estimate of the error at each time step to be calculated. The method can be written first in general terms. If a solution is sought to the equation y' - f(x,y) 210 then the solution and error term are based on the estimation and combination of five quantities. kl = h .f(x n ,yn) k2 = h.f(xn+h/3,yn+k|/3) k3 ~ h.f(xn+h/ 2 ,yn+(k 1 +k2 )/ 6 ) k4 = h.f(xn+h/ 2 ,yn+(k 1 +3 k3 )/ 6 ) k5 = h .f(x n+h,yn+ (k i- 3 k3+4k4 )/ 2 ) The new value can be written yn+l - yn + 1/6 (kl + 4k 4 + k5) + 0 (h5 ) (7 . 3 5 ) and the error term calculated as e - 1/15 (k! - 9/2 k3 + 4k4 - 1/2 k5) (7.36) If the error term is greater than some predetermined limit then the step size, h can be reduced and the calculation repeated. Since the scheme is fifth order accurate halving the step size reduces the error 32 times. The error calculation can also be used to increase the step size if the error is less than some predetermined value. The overall result is an efficent numerical solution that solves to a given accuracy as efficiently as possible. In the solution of the vertical concentration profile near the bed typical values of the space step used would be around 0.5mm whereas near the surface larger increments were possible. Strictly speaking it would have been more correct to solve the vertical concentration profile in time and then average the result over a period. This method, which is used for the cross-shore model, is very time consuming especially if a large spatial grid is to be considered. The time mean equation was therefore chosen as a viable alternative. 211 7.5.3 CROSS SHORE TRANSPORT Equation 7.9, which is repeated for the last time, 3c 3c 3 . 3c. 3F " w55 + 35(es3S> (7.9) can be expressed in finite difference form as either an explicit scheme or an implicit scheme. Since an implicit scheme affords greater stability and the possibility of a more efficent solution, the Crank-Nicolson implicit method has been used to solve this equation. In the scheme the equation is written at an intermediate time level: dco n+1/2 W/dc o n , odc n+1 N 1 9 . 9cn 0cn+1 +e ) - 0 —3t------2(35“ + “SS- ' dz The individual terms can be written n+1 n n+1 / 2 c j - cj 3c dt At n n n+1 n+1 o n o n+1 w , dc dc w °j+i °j-i cj+i ■ °j-i 2 ( g + ) ~ i ( 2 Az+ — 2 Az---- ) n 0n n+1 n+1 dc dc dc dc , ^ n ^ n+ 1 1 d, dc dc . 1 £35_j+l/2 ' ^ j - 1 / 2 ' dzj+l/2 ' dzj-l/2 2“3if_g s + e— a— > 2( Az + ------Az>------) n n n n c cj+i - c J_ j ~ Cj~l - fj+ l / 2 €j -/ l 2 2 (Az) 2 2 (Az) 2 n+1 n+1 n+1 n+1 cj+l “ CJ cj - cj-i + e j+ 1 / 2 ej - l / 2 2 (Az) 2 2 (Az) 2 212 Writing out in full gives , n+1 n . wAt , n n n n . (CJ ‘ °j > ' 445 (CJ+1 " Cj-1 + Cj+1 " °j-l > A t n\ / n n v , n+1 n+1 . tc j+ i/ 2 .(cj+ i c j ) ' £j - l2 / -(cj °j-l )+fj+l/2-(cj+]-Cj ) 2 (Az) 2 n+1 n+1 cj °j-l )) (7.37) Collecting terms n+1 cj . (1.0 + sej+i/2 + sej- 1/2 ) + Cj . ( -1.0 + sej+1/2 + sej - l / 2 > + n °j+i • (-d -scj+1/ 2 ) + c o r-< — 1 j / 2 (d _sej - l ) + n+1 cj+i • (-d -scj+1/ 2 ) + n+1 cj-l • (d-S6j _ 1 /2 ) - 0 where rl vfAt At 4Az 2(Az)2 Therefore (d-sej_ 1/ 2 )cj_i + ( 1 . 0+sej+1/ 2+sej_1/2)cj +(-d-sej+l/ 2 )cj+i = ("d+S€j-l/2> cj-l + (1.0-sej+ i/2 -sej-l/2)cj + (d+sej+i/2 )cj+l (7.38) 213 This can be written as a tridagonal matrix and solved using the Thomas algorithm. Ft Gj C1 Hi - Ejcq e 2 f 2 g 2 c 2 h2 e3 f 3 g3 c3 h3 En-1 Fn-1 cn-l ^n-1 “ Gn-1 -Cn (7.39) where Ej “ (d“S€j-1/ 2) Fj - (1.0+sej+1/2 +sej - 1/ 2) Gj - (-d -se j+1/2) Hj - (-d + s e j.^ C j.! + (1.0-sej+1/2-sej_1/2)cj +(d+sej+i / 2)"j+1 The two boundary conditions are also required. At the water surface (j=n) it is assumed that cn - 0 At the bed Eq. 7.14 describes the concentration in terms of the gradient. "es3i = P(t) at z = 0 (7.14) This is expressed in finite difference form C1 " c 0 = -€s .p(t) (7.40) Az, 214 or co ~ cj + Az.rs.p(t) (7.41) This is an explicit relationship which unfortunately cannot be calculated exactly at the boundary since cq is required for the calculation of cj which in turn is required for the calculation of cq. The author tried using the old time step value of c^ to calculate cq and then iterating this procedure a number of times. In fact a satisfactory answer was obtained with two iterations. A typical solution is shown in Fig. 7.10 where the vertical concentration is calculated at 1.0 second intervals for the first half of an 8.0 second wave. For this example a linear wave is used and the concentration in the second half of the period would be identical to the corresponding stage in the first half. The solution takes some time to develop from a ’cold' start. The author has found that usually one period is sufficent to allow the correct conditions to develop so that the solution for the second period is correct. 7.5.4 VERTICAL VELOCITY PROFILE The ordinary differential equation 7.24 which describes the behaviour of z in time can be solved using the Runge—Kutta—Merson technique described in detail in the previous section. In the case of the velocity profile, the use of this method produces an efficient solution technique because close to the bed the gradient is extremely large and very small steps must be taken. Away from the bed much larger steps can be used. The size of the first step used in the solution proved to be quite critical for the stability of the solution. Too small a value and the function in Eq. 7.24 would be too close to its singularity. Too large a step and the error term would be excessive. Eventually a value of tQ = 10 sec was found to give good results over the full range of wave and current conditions. The procedure is an iterative one, the flow chart for which is shown in Fig. 7.11. The solution generally closes to within 0.2% within five iterations. 215 FIGURE Temporal Variation of Sediment Concentration 7.10 * 216 FIGURE F low Chart of Velocity Profile Calculation 7.11 217 FIGURE 7.11 co n t. 7.5.5 BED EVOLUTION MODEL Writing Eq. 7.31 again 9cIx 3 q y 9z + 0 3F + This can be written in finite difference form t+1 t At . 1 w . At , 1 w . :i.j " Zi.j Ax (l-\ )(qxi,j“ qxi-l,j )_ Ay (1-X )(qyi,j " qyi,j-l } (7.42) The diffusion is estimated by spatial averaging. For example in the x direction: zi,j = (l“ax)’zi,j + ax^*^zi-l,j + ®-^zi+l,j) (7.43) where otx is the diffusion coefficent in the x direction. 7.6 MODEL VERIFICATION 7.6.1 INTRODUCTION Since the author was not able to conduct any experiments involving the collection of data on sediment behaviour the verification of the model is based on data and results collected from other sources. The verification will be quantitative wherever possible, but occasionally only a qualitative result will be possible. This is particularly so in the case of cross-shore transport. The verification is based on as wide a selection of data and results as possible. It will be shown that the model predicts reasonably well the CERC bulk formula. The longshore model is then compared with predictions from the Bijker sediment model. The prediction of the vertical concentration profiles is compared with field data collected by Nielsen (1984). Finally comparison is made with the longshore transport study reported by the Danish Hydraulic Institute (1984). 219 7.6.2 CERC BULK FORMULA The CERC bulk formula relates the longshore transport of sediment to the longshore component of wave energy flux. It is based on numerous field and laboratory data over a wide range of beach conditions, slopes and particle sizes. Although it is postulated that factors such as sediment size should be important in transport, only recently have attempts been made to include it. ( see e.g. O'Connor et al, 1983) Deigaard et al (1986b) calculated the predicted rate of longshore transport under different sediment sizes. The results are shown in Fig. 7.12. It can be seen that a particle size around 0.25 mm gives reasonable agreement between that predicted by the CERC formula and that by the model. The present author has run a number of tests, the results of which are also shown in Fig. 7.12. It can be seen that a particle size of 0.2 mm in this case gives reasonable agreement in general although other factors such as bed slope will also cause some variation. The CERC formula, with which the comparison is made, is taken from Muir-Wood and. Fleming (1981) and can be written Q - K' | | (Hb ) 2 c sin 2 a h (7.44) where K' = 2.55 x 10^ and Q is the transport in m^/year. It has not been possible to verify if these predictions of factors affecting the transport are correct. The verification is based on the fact that for reasonable values of bottom slope and particle size there is general agreement, and that variation of those parameters causes changes in the predictions which are consistent with previous works (e.g. Deigaard et al, 1986b) 7.6.3 CROSS-SHORE DISTRIBUTION OF TRANSPORT The CERC formula predicts the total littoral drift but gives no information on how that transport is distributed across the nearshore zone. There have been attempts both to predict and to measure the distribution. Thornton (1972) measured the distribution using bed load traps. The number of traps in the surf zone was limited but Thornton found that the maximum transport occurred at the breaker-line. Unfortunately insufficent wave and bathymetric data are available to try and reproduce those results. 220 a , (a) Deigaard et al. (1986 b) o d = 0.1mm • d = 0.2mm x d= 0.3mm (m^/s) FIGURE Comparison with Bulk Transport Formula 7 . 12 221 Bijker (1968) was able to predict the cross-shore distribution usiqg a numerical model. Fig. 7.13 shows one such prediction together with the prediction based on the present model. The maximum transports have been normalized so that the distributions can be compared. It can be seen that the distributions are quite similar despite the quite different theoretical backgrounds. Coeffe and Pechon (1982) have shown that the Bijker model gives order of magnitude agreement with the CERC bulk formula. The most detailed comparison comes from a 1600m long trench dredged across the entire surf zone in the North Sea. As already mentioned the trench was part of a scheme to bring oil and gas pipelines ashore and it was recognized that it would provide an ideal opportunity to monitor nearshore sediment transport. The trench was surveyed at regular intervals and a continuous record was kept of nearshore wave conditions. During the period March 22nd to April 15th, 1982 significant backfilling took place and most of it can be related to a single storm that occurred between April 7th and April 10th. Unfortunately, only the area in the region of the offshore bar was surveyed for backfilling during this period. During the storm significant wave heights in excess of 4.0 metres occurred, high enough to cause substantial breaking on the outer bar. (See Fig. 7.4) The wave record itself is shown in Fig. 7.14. For the purposes of reproducing the storm the event was split into five sections with individual wave heights, periods, angles and durations. These are listed in Table 7.1. By assumimg a breaking index of 0.8, using the significant wave height, it was possible to assume that waves below a height of 3.0 metres would not break on the outer bar and therefore would not cause longshore transport. Sect ion Hsig ^rms Tz 6 Durat ion (m) (m) (sec) C) (hrs) 1 4.6 3.3 6 .2 12.5 8.4 2 3.8 2.7 5.0 5.5 8.4 3 4.1 2.9 6 .0 5.5 4.2 4 3.4 2.4 5.0 5.5 4.2 5 3.6 2.5 5.2 5.5 2 1 .0 Table 7.1 North Shore Wave Data XJ CL) N ro E c _ o c distance from shore FIGURE Distribution of Longshore Sediment Transport 7.13 222 223 03/2800.00 00.0004/00 00.0004/12 00.0004/18 00.00 04/20 03/2800.00 00.00 04/00 04/1200.00 04/1800.00 00.00 04/20 1982 FIGURE North Shore Wave Record " DHI(1984) 7.14 224 The wave height and breaking index were converted to root mean square (r.m.s) values for use in the sediment model, using the relationship: 2.0 Hrms (7.45) therefore 1 Trms ” j 2 ^sig (7.46) Due to the size of the area to be modelled a simple refraction scheme replaced the full wave model. The current model with the k-e turbulence model was used to predict the currents and turbulence levels. The resulting transport prediction is shown in Fig. 7.15. It can be seen that there is order of magnitude agreement between the observed and predicted transport generally although the predicted distribution is much more concentrated near the breaker-line than that measured on site. This is not surprising as the numerical model did not include any onshore-offshore transport which would tend to spread the distribution nor did it have a progressive wave breaking model which would also tend to smooth the transport distribution. The distribution is in fact quite similar to that predicted by Deigaard et al (1986b) using the one equation turbulence model. 7.6.4 COMPARISON WITH VERTICAL CONCENTRATION DATA Nielsen (1984) made extensive measurements of suspended sediment concentrations in the field. The results of these measurements have already been used for comparison purposes by Deigaard et al (1986a) The results of those comparisons can be seen in Fig. 7.3. The present author has run the same tests so that the model can be compared not only with the field data but also with another similar numerical model. The author's predictions are shown in Fig. 7.16. It can be seen that the agreement between the prediction and the field data is satisfactory * The major difference between the two numerical models is in the bottom concentration value. Despite using a relationship that gives much lower bed concentrations the predictions of the present model are generally as good as those of Deigaard et al. 225 p r e d i c t e d m e a s u r e d P(m 3/m ) 1 (f 10l I I » * • t i i . . t i FIGURE Prediction of North Shore Sediment Transport 7.15 226 TEST No. 21 10^ 2_ • k 103 » o 0 • o 0 • 0 102 ---•-- o • m1 u l 10° 1(y6 10-5 10"4 10'3 10 -2 TEST No. 23 10^ z k o 103 0 0 ° • o o • 102 experiment o predicted • * • 101 10° -| 10-6 10-5 10-4 0-3 10-2 C FIGURE Prediction of Vertical Sediment Concentration 7.16 Profiles 227 TEST No. 39 •to** 2 Q° k 1 0 3 ■ • o • c o 1 0 2 • o • • 1 0 1 • 10° io"6 id5 10’4 _ 103 102 C (w3/m 3) C (\\ jrn3) FIGURE 7.16 co n t . 228 TEST No. 65 1 0 ^ Z k 1 0 3 c 9X3-- • O 4'o 10 2 • o « a 10 1 10° 1 0 6 10~5 10'4 10 '3 10"2 c {»?/»?) experimento predicted • FIGURE 7 .16cont 229 7.6.5 ONSHORE OFFSHORE TRANSPORT MECHANISM The heuristic model of Dean (1973) has been tested with regard to the dependance of the direction of transport on the ratio of fall velocity to wave period. Fig. 7.17 illustrates two conditions, one where w/T is large (indicating a relatively large or heavy particle) and the other where w/T is smaller (indicating a fine particle). The velocity profile was generated using the hyperbolic wave approximation to cnoidal waves proposed by Iwagaki (1968). The approximation covers only a certain range of the possible conditions but is suitable for most of the nearshore zone. The pick-up function peaks at 30 * after the velocity with the peak in the forward direction exceeding the peak in the return direction considerably. The effect of the different sediment sizes can be seen in the behaviour of the suspended load and the subsequent sediment transport. The finer particle diffuses higher into the water column and falls more slowly. The effect is to produce a net backward transport which is evident as the backward transport exceeds the forward. The behaviour of the heavier particle means that it falls from suspension more quickly and therefore moves in a net forward direction. This is in agreement with the work of Dean and the field observations of Murray (1967). It was not possible to reproduce all the data of Dean since the other important parameter Hq/Lq would have no effect on this aspect of the behaviour. Its effect is probably related to the generation of currents in the onshore-offshore direction which are not considered in the present work. 230 FIGURE Cr o s s Shore Sediment Transport Mechanism 7.17 231 CHAPTER 8 BATHYMETRIC MODEL RESULTS 8.1 INTRODUCTION Using results calculated by the wave, current and sediment models the bathymetric evolution model has been run on a series of standard groyne configurations. The predictions are presented and discussed in this chapter. Although it would have been possible to vary any number of the wave and groyne parameters it was decided to hold most constant and to vary only the ones which would affect the results to the greatest degree. The wave period and deep water angle were kept constant as was the plane beach slope and the groyne length. The wave height was varied so that the breaker line was at either 50%, 100% or 150% of the groyne length from the shoreline; the number of groynes was either one or two with varied spacing on the two groyne cases. The height of the groyne was also varied such that it acted either as a fully surface piercing barrier or was partially submerged from about its mid-point. Varying the number and spacing of the groynes gives an idea of how groyne geometry affects the behaviour of the groynes and changing the wave height models the change from a total littoral barrier that the groyne forms when the waves break at 50% of the groyne length to a partial littoral barrier when the waves break at or offshore of the groyne tips. Verification of the numerical predictions is difficult with few full scale or physical model results available. The predictions are verified qualitatively based on previous numerical predictions and knowledge W om experiments. The plots that are presented concern three main aspects: the circulation pattern, the pattern of erosion and accretion and the resulting changes in bathymetry. Since different wave heights were used the predictions of erosion and accretion patterns and the bathymetric changes have been normalized so that emphasis can be placed on the pattern of erosion and accretion rather than on the actual magnitudes. The bathymetric evolutions that are presented were calculated after sufficent time to show the changes that were occurring. Section 8.2 presents the results of the predictions and these are discussed in Section 8.3. 232 8.2 NUMERICAL PREDICTION RESULTS 8.2.1 SINGLE GROYNE, SURFACE PIERCING The results for a single surface piercing groyne are shown in Figs. 8.1, 8.3 and 8 .i where the breaker zone width was 50%, 100% and 150% of the groyne length respectively. Fig. 8.1 illustrates the classic example of a total littoral barrier. The longshore velocity is diverted offshore around the groyne and although it carries with it some turbulence (see Fig. 8.2) and therefore some sediment transporting potential, its sediment load is deposited on the updrift side of the groyne. The resulting changes in bathymetry agree well with the solutions of the early shoreline evolution models - as would be expected in the case of a total littoral barrier. In the case where the waves break at the groyne tip the results predicted are quite different. In this case the flow carries the sediment out past the groyne tip to where the transporting potential is reduced and hence deposition occurs. The groyne still traps some sediment but it tends to be trapped offshore giving a quite different bathymetric development. This is illustrated in Fig. 8.3. This trend is continued when the waves break well beyond the end of the groyne. As shown in Fig. 8.4 this results in a well developed circulation in the lee of the groyne. This however has little effect of the sediment transport as the potential for transport is reduced close to the shoreline. The net deposition pattern shows erosion on the updrift side of the groyne and accretion downdrift. This is contrary to what is generally expected although Hulsbergen et al (1976) recorded such behaviour, as mentioned previously, in a physical model test where the waves were breaking beyond the end of the groyne. See Fig. 2.2. Although this numerical test was not designed specifically to mirror the experiment it is interesting that a similar pattern should be predicted. The different behaviour of the three arrangements can be examined by considering the potential for longshore transport at each cross section along the shoreline. Fig. 8.5 plots this where the open beach potential is assigned a nominal value of 100%. It can be seen that the first two wave heights considered result in a drop in the longshore transport at the groyne which results in accretion upstream of the groyne and erosion downstream. The largest wave height on the other hand results in an increase of transport potential at the groyne as flow is diverted by the groyne to an area where the turbulence is greater. The pattern predicted therefore shows erosion on the upstream side and accretion downstream. 233 * * \ \ ■• - (a ) 1.0 2.0 3.0 4.0 5.0 (c) FIGURE Numerical Prediction - 1 Groyne, X, =0.5L 8.1 (a) flow field (b) deposition pattern (c) predicted bathymetric change s FIGURE Eddy Viscosity Around Groyne 8 .2 The pattern is for the flow field shown in Fig. 8.1 U)N3 N s - - - N S \ S S - ' * N N N •» *» •* (a) (c ) FIGURE Numerical Prediction - 1 Groyne , X, =1.0L 8.3 (a) flow field r (b) deposition pattern (c) predicted bathymetric changes 236 ( a ) FIGURE Numerical Prediction - 1 Groyne, X, =1.5L 8.4 (a) flow field r (b) deposition pattern (c) predicted bathymetric changes 237 (c) distance along shore FIGURE Longshore Continuity 1 Groyne 8.5 (a) X =0.5L (b) X, = 1. OL (c) X ^ = 1.5L 238 8.2.2 SINGLE GROYNE, PARTIALLY SUBMERGED The situation where the groyne is partially submerged is difficult to represent accurately using a depth-averaged model, since the flow and the behaviour of the sediment will be stratified. Despite this several runs were carried out on the single groyne configuration to investigate the deposition pattern that the model would predict. No acount was taken of form drag caused by the groyne. As might be expected, a number of significant differences occur although at the low wave height the prediction is quite similar. The model allows some of the sediment to pass over the submerged section of groyne so that a build-up of sediment occurs, not only on the updrift side but also on the downdrift side close to the groyne. Further downdrift the pattern of erosion is similar to that predicted in the case of the fully surface piercing groyne. The results for the partially submerged groyne are illustrated in Fig. 8.6. For the wave height when the breaker zone width is equal to the groyne length the reduction of depth-averaged flow over the submerged section of groyne again results in deposition in the region of the groyne, see Fig. 8.7. This is in contrast to the prediction for the fully surface piercing groyne shown earlier in Fig. 8.3. For the maximum wave height, the results of which are shown in Fig. 8.8, there is again deposition in the vicinity of the groyne but otherwise the general pattern of erosion and accretion is quite similar to the surface piercing case. The different behaviour exhibited by the partially submerged groyne is also evident in the consideration of the longshore transport potential plotted along the shoreline. This is shown in Fig. 8.9. The pattern for the low wave height is very similar to that shown in Fig. 8.5 which is not surprising since at this wave height the partially submerged groyne is still acting like a total littoral barrier. The situation as the wave height increases is different though. It is readily seen that the partially submerged groyne traps less material and also diverts the flow to a smaller degree. This is particularly important for the highest wave height where the increase in transport predicted for the surface piercing groyne is much reduced with the partial barrier. 8.2.3 TWO GROYNES, SURFACE PIERCING, S=L The results for a two groyne bay which are illustrated in Figs. 8.10 to 8.12 are similar to those predicted for the single groyne. Again the total littoral barrier, 239 ' i \ * i i \ - (a ) 1.0 2.0 3.0 4.0 5.0 (c ) FIGURE Numerical Prediction - 1 Groyne, X, =0.5L 8.6 Part Submerged r (a) flow field (b) deposition pattern (c) predicted bathymetric changes * 240 S N S N > la) (b) ( c ) FIGURE Numerical Prediction - 1 Groyne, X, =1.0L 8.7 Part Submerged r (a) flow field (b) deposition pattern (c) predicted bathymetric changes 241 (a) FIGURE Numerical Prediction - 1 Groyne, X, =1.5L 8.8 Part Submerged r (a) flow field (b) deposition pattern (c) predicted bathymetric changes distance along shore FIGURE Longshore Continuity - 1 Groyne, Part Submergec 8.9 (a) X =0.5L (b) x£j>1.0L (c) x “£-1.5L 243 ■ » ► ► >»-»■ ► >- » ► » T r r r r t ^ * J \ N - (a) (b) FIGURE Numerical Prediction - 2 Groynes, S/L=1.0 8.10 Xb =0.5L (ay flow field (b) deposition pattern (c) predicted bathymetric changes 244 illustrated in Fig. 8 .10, shows the classic bathymetric changes with accretion on the updrift side of the groynes and erosion on the downdrift. This results in a re-orientation of the beach inside the groyne bay as is generally observed in both physical model tests and in prototype. At this spacing the two groynes are acting together in that the behaviour of the second is modified by the first. This can be seen by the greater accretion in front of the first groyne compared to the second. This effect is magnified in the case where the waves break at the groyne tips, see Fig. 8.11. Here the second groyne has little effect on the deposition pattern. This is also the case where the waves break well beyond the groyne tips as illustrated in Fig. 8.12. Again the circulation inside the groyne bay has little effect on the depostion pattern and the model predicts extensive depostion outside and downstream from the groyne tips. The potential for longshore transport along the shoreline is illustrated in Fig. 8.13. Once again the lower wave heights result in reduced transport at the groynes while the largest wave height causes an increase in the transport. 8.2.4 TWO GROYNES, SURFACE PIERCING, S=2L The general effect of increasing the spacing of the groynes is to reduce the effect that the upstream one has on the other. This is evident when comparing Fig. 8.14 with Fig. 8.10. It can be seen that the downdrift groyne traps much more sediment when moved to a greater spacing. One could speculate that at a spacing to length ratio of more than 2 the groynes, at least for these wave conditions, would be acting quite independantly of each other. The independance of the groynes is reduced as the wave height is increased as seen in Fig. 8.15 and 8.16. Once again the potential for longshore transport is illustrated in Fig. 8.17 8.3 DISCUSSION OF RESULTS With reference to the runs completed a number of points can be made. The model prediction for the case of a total littoral barrier i.e. one where the waves break well within the length of the groyne corresponds well to the simple beach planform models developed earlier by Bakker ( 1968) and others. The model exhibits beach re-orientation as would be expected and which has been observed in both physical models and at full-scale. 245 » \ \ \ N *» -» (a ) (b) FIGURE Numerical Prediction - 2 Groynes, S/L=1,0 8.11 Xb =1.OL (a^ flow field (b) deposition pattern (c) predicted bathymetric changes (a) (b) FIGURE Numerical Prediction - 2 Groynes, S/L=l.0 8.12 Xb =1.5L (aj flow pattern (b) deposition pattern (c) predicted bathymetric changes 247 (C) distance along shore FIGURE Longshore Continuity - 2 Groynes, S/L=1.0 8.13 (a) X =0.5L (b) X?*=1.0L (c) x £M .5L 248 • \ \ " * * * (a) (b) (c) FIGURE Numerical Prediction - 2 Groynes, S/L=2.0 8.14 Xb =0.5L (aj flow field (b) deposition pattern (c) predicted bathymetric changes 249 %• %» \ NNN*'-\ \ > ^ ■* (a ) (b) FIGURE Numerical Prediction - 2 Groynes, S/L=2.0 8.15 X, =1.0L (a; flow field (b) depos i tion p at tern (c) predi cted bathyme trie changes 2 (a) (b ) FIGURE Numerical Prediction - 2 Groynes, S/L=2.0 8.16 Xb =1.5L (a^ flow field (b) deposition pattern (c) predicted bathymetric changes 251 CJ (C) distance along shore FIGURE Longshore Continuity - 2 Groynes, S/L= 2.0 8.17 (a) X =0.5L (b) X?*=1.0L (c) x£ = 1.5L + 252 Where the groyne length is of the same order as the width of the breaker zone the pattern is quite different. The model predicts an increase in local transport updrift and at the groyne as the flow is diverted to where it can carry more sediment. This diversion of the flow is greater with a fully surface piercing groyne and consequently so is the local transport. Although there is limited evidence for this a similar pattern has been observed in physical model results. It would appear that the circulating flow pattern that occurs with the higher wave heights has little effect on the sediment deposition pattern. This results from the fact that most of the sediment transport is occurring further offshore near the wave break point. This of course assumes that the suspension pattern predicted by the numerical model is correct. In fact it may be that waves plunging at the shoreline create a significant swash zone transport in which case the circulation pattern inside the groyne would have a much greater effect. Its apparent importance would also be magnified since even small changes at the shoreline are noticable as they cause shoreline re-orientation. 253 C H A P T E R 9 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 9.1 SUM M ARY The results of a detailed investigation into the behaviour of groynes on sandy beaches is described. The work includes extensive numerical modelling in the three main areas of hydrodynamics affecting groynes in the nearshore zone, the wave field, the pattern of mean circulations and the behaviour of sediment, a physical model study conducted in a fixed bed model where particular attention was paid to the setting up of the correct longshore currents and finally a full scale data collection programme at a groyne-field in Norfolk on the East Anglian coast. Although the work described has concentrated on the study of groynes the numerical models are in fact quite general and could be applied to any nearshore situation. The wave model solves a hyperbolic approximation to the mild slope equation which allows the inclusion of the effects of wave diffraction, reflection, refraction by both bathymetry and currents, and wave breaking. The model has been tested against both analytical solutions and the results of field investigations and shown to be excellent. j The current model solves the depth-averaged Navier-Stokes equations using an efficent and accurate ADI scheme where an iterative procedure is used to produce a scheme which is fully time and space centered (with the exception of the cross derivative diffusion terms which are lagged in time). The model also includes refined turbulence modelling in the form of a two equation k-e model which generates a solution for the eddy viscosity over the entire numerical grid. The model has been compared to laboratory and field data and shown to produce reasonable results. The sediment model solves the suspended sediment transport making use of the turbulent eddy viscosity calculated in the current model. The model has been shown to be able to predict not only the bulk formulae for littoral drift but also the distribution of the transport, a feature that is important for bathymetric evolution models. The accuracy of the suspended concentration profiles has also been shown to be reasonable in comparisons with measured field data. The physical model study was carried out in a 1 :36 scale fixed bed model. Considerable attention was paid to the problem of establishing the proper longshore current in the model and a series of ten tests was carried out under fixed wave 254 conditions with a variety of groyne configurations. The effects of groyne spacing and height were investigated as well as multiple groyne arrangements. The results were used for comparison with the numerical model predictions. The results from a full scale field data collection programme are also presented. The data was intended to verify a physical model study carried out at Hydraulics Research Limited however the low sea states encountered on site prevented this. Measurements were taken of the wave climate and the nearshore velocity field at the site which, although tidally dominated at the time of measurements, still allows conclusions to be drawn with regard to the effect of groynes on the nearshore flows. Several runs of the numerical bathymetric evolution model are reported. The runs were carried out using varying wave and groyne configurations and the importance of the breaker zone width in relation to groyne length and spacing is one important factor to emerge from the runs. This will be discussed further in the next section. 9.2 CONCLUSIONS With regard to the numerical models used in this work a number of conclusions are possible: 1. The inclusion of the wave-current interaction is an important feature of a general nearshore wave model. It has been found during this present work that even moderate currents can alter the wave field and in cases where either the bathymetry or structures cause rip currents to form the effect would be quite significant. The model has been shown to be able to predict, at least in the case of a simple shear current, the change in direction calculated by theory. 2 . One of the interesting points to emerge from the numerical current model is the lack of sensitivity of the model to the size or distribution of the eddy viscosity. The size affects the degree of smoothing of the current profile and therefore the magnitude of the current, but the magnitude is controlled much more effectively by the bottom friction. That is not to say that the eddy viscosity is unimportant, completely the opposite is true, only that it is difficult to calibrate an eddy viscosity model from the velocity field. 3. The other interesting lesson learnt is that, in the case of a plane beach under the influence of monochromatic linear waves, the assumption made by Longuet-Higgins on eddy viscosity is remarkably good (at least inside the breaker line). The distribution calculated using the k-e model has been seen to be very similar to the simple mixing length model adopted earlier. Obviously the simple model breaks down at the breaker-line. However the longshore current magnitude tends to be much reduced there and the errors produced are small. 4. The mixing length model may be sufficent for simple nearshore cases but in the situation where the bathymetry is not regular or where there are structures, a more detailed eddy viscosity model is required. The k-e model has, to the authors knowledge, been used only once before in the nearshore zone (Wind and Vreugdenhil, 1986) when it was found to produce improved numerical predictions. 5. Although the depth-averaged currents are not very sensitive to eddy viscosity the suspension of sediment depends critically on it. The use of the k-e model therefore allows not only the prediction of accurate mean currents but also the possibility of an accurate suspended sediment model. It has been seen that the sediment model can predict the bulk rate of sediment transport as well as the distribution. This of course assumes that the vertical distribution can be assumed. With regard to the predictions of the numerical bathymetric evolution model a number of conclusions can be made: k. It must be recognized that the sediment model has severe limitations. However, •the model has been seen to reproduce the results of the simple one and two-line numerical shoreline evolution models for the case where the groynes act as a total littoral barrier (as assumed in the simple models). The results are also similar to those found in the field. The predictions made when the groynes do not form a complete littoral barrier are more interesting. The model predicts that in this case there may not be accretion upstream of the groyne and erosion downstream as would be expected intuitively. The effect of predicting maximum sediment suspension at the breaker-line is to produce accretion offshore of the tip of the groynes and in some cases offshore and downstream. Unfortunately the predictions cannot be taken too far since the model is not an iterative one and after a certain stage changes in the predicted bathymetry should be permitted to affect the wave and currents if accurate results are to be obtained. 256 With regard to the physical model experiments: 7 . The work in the laboratory has shown that although it is possible to conduct experiments of the nearshore zone in such a small basin, a more extensive physical model would make the work easier and more versatile. The most important fact to emerge from the experiments is the effect of the vertical structure of the mean currents on the circulation pattern. It was found that the groyne bays experienced a net outflow through a large proportion of the water column. 8. It was found in the laboratory that the groyne produced a circulation in the downstream field and the dimensions of the circulation were strongly related to the length of the groyne that was fully surface piercing. In the tests carried out it appears that at groyne spacing to length ratios greater that 2 the groynes, at least from the mean current point of view, were acting independantly. A number of conclusions can be drawn from the full scale data collection programme: 9. Although it was not possible to confirm the existance of large scale circulations inside the groyne bays there was certainly evidence of eddies forming at the tips of the groynes. One could conclude from this that local disturbances to the flow pattern could be significant in determining design parameters for groynes. In fact this has already been recognized as it is suggested practice to keep groyne heights to a certain maximum level to reduce local scour. 10. The monitoring of beach profiles, one on a stretch of open beach and three inside an adjacent groyne bay has given an insight into how groynes can affect the local beach levels. In some cases the groynes seem to have no effect as beach levels either rise or fall uniformly on a stretch of coast. At other times ^ however, it appears that the groynes are successful in holding material while open beach levels are falling. There is also evidence of the groyne bay acting as a separate unit with remoulding of the beach levels between groynes. 9.3 RECOMMENDATIONS FOR FUTURE WORK In the field of numerical modelling a number of areas need further work: 1. The wave model, as assembled, provides accurate predictions in the case of monochromatic simple harmonic linear waves. Although there is an obvious need to extend the model to be able to include a directional wave energy spectrum (early work in this area has been reported by Anastasiou, 1986) a more immediate need, the present author believes, is for a better wave breaking model to be incorporated. This is important, firstly for accurate calculation of the radiation stress components but more importantly (at least for the case where sediment is being considered) for the correct calculation of turbulence near the breaker-line. This should lead eventually to better predictions of the distribution of longshore sediment transport. 2. The k-e turbulence model which is used in this present work depends to an extent on a number of empirical constants which have been determined over many years of experiments, mainly in the laboratory. No tests have been carried out where the production of turbulence due to wave breaking has been included and since this is such an important factor effort should be made to measure it and calibrate the empirical constants in this environment. There has been a small number of experiments measuring diffusion of dye in the surf-zone both in the laboratory and in the field but many of them were carried out twenty or thirty years ago and in the light of recent advances, particularly in the field of velocity measurement (using laser doppler anemometry), new experiments would yield a wealth of information. In the field and the laboratory a number of areas need further work: 3. One of the most important predictions of the sediment model is the distribution of the transport across the surf zone. This is affected by two factors, the longshore velocity and the suspended sediment concentration. Both have been measured by many researchers but it would seem that consideration must be given to collecting data under conditions where the wave field and bathymetry are simple and well defined so that the sediment model could be tested under conditions where there are fewer unknowns. 4. Another simplifying assumption of the sediment model comes from the use of spilling breakers. This means that sediment is entrained purely by the oscillatory wave motion in the bottom boundary layer. It has been shown 258 that up to ten times as much sediment can be entrained by plunging breakers than the equivalent spilling breaker and this would be of prime importance especially close to the shore-line. Work on entrainment of sediment would be most rewarding. 5. The physical model study emphasized the importance of the vertical structure of the mean currents. The result of this is a strong recommendation to model numerically in three dimensions. Initial attempts using a multi-layer model should produce much better predictions of the flow pattern but more importantly, the sediment deposition pattern. 6. In the numerical models the important effects of bed friction were included in a simple manner and improvements could be expected with better representation of the phenomenon. 259 REFERENCES ABBOTT, M.B., PETERSEN, H.M. and SKOVGAARD, O. (1978) On the Numerical Modeling of Short Waves in Shallow Water Journ. Hydr. Res. 16 No. 3 pp 173-204 ABBOTT, M.B. and RASSMUSSEN, C.H. ( 1977) On the Numerical Modelling of Rapid Expansions and Contractions that are Two-Dimensional in Plan 17th Congress of the IAHR Vol. 2 pp 229-237 ALLENDER, J.H., DITMARS, J.D., HARRISON, W. and PADDOCK, R.A. ( 1978) Comparison of Model and Observed Nearshore Circulation Proc. 16th Coastal Eng. Conf. pp 810-827 ANASTASIOU, K. (1986) Nearshore, Random Wave Induced Circulation 3rd Indian Conf. on Coastal Eng. Vol. A pp 203-214 ANASTASIOU, K., DONG, P. and WALKER, D.J. (1987) Turbulence Modelling and the Effects of Directional Random Waves in Computations of Nearshore Circulation Proc. Int. Conf. Num. Methods in Eng.: Theory and Appl., Swansea Paper D 24 ARTHUR, R.S. (1950) Refraction of Shallow Water Waves: The Combined Effect of Current and Underwater Topography Trans. Am. Geophys. Un. Vol. 31 No. 4 pp 549-552 BAGNOLD, R.A. (1963) Mechanics of Marine Sedimentation in The Sea: Ideas and Observations, Vol. 3 Interscience pp 507-528 BAILARD, J.A. (1981) An Energetics Total Load Sediment Transport Model for a Plane Sloping Beach Journ. Geophys. Res. Vol. 86 No. C ll pp 10938-10954 BAILARD, J.A. (1982) Modeling On-Offshore Sediment Transport in the Surfzone Proc. 18th Coastal Eng. Conf. pp 1419-1438 260 BAILARD, J.A. (1984) A Simplified Model for Longshore Sediment Transport Proc. 19th Coastal Eng. Conf. pp 1454-1470 BAILARD, J.A. and INMAN, D.L. (1981) An Energetics Bedload Model for a Plane Sloping Beach: Local Transport Journ. Geophys. Res. Vol. 86 No. C 3 pp 2035-2043 BARKER , W.T. (1968) The Dynamics of a Coast with a Groyne System Proc. 11th Coastal Eng. Conf. pp 492-517 BARKER, W.T. and DOORN, T. ( 1978) Near-Bottom Velocities in Wave with a Current Proc. 16th Coastal Eng. Conf. pp 1394-1413 BARKER, W.T. , KLEIN BRETELER, E.H.T. and ROOS, A. (1970) The Dynamics of a Coast with a Groyne System Proc. 12th Coastal Eng. Conf. Chapter 64 B A R C E L O , J.P . (1968) Experimental Study of the Hydraulic Behaviour of Groyne Systems Proc. 11th Coastal Eng. Conf. pp 526-548 BASCO, D.R. (1983) Surfzone Currents Coastal Engineering 7 pp 331-355 BATTJES, J.A. (1968) Refraction of Water Waves ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 94 No. WW4 pp 437-451 BATTJES, J.A. (1975) Modeling of Turbulence in the Surf Zone Symp. on Modeling Techniques. San Francisco. Vol. 2 pp 1050-1061 BERKHOFF, J.C.W. (1972) Computation of Combined Refraction-Diffraction Proc. 13th Coastal Eng. Conf. pp 471-490 BERKHOFF, J.C.W. (1976) Mathematical Models for Simple Harmonic Linear Water Waves; Wave Diffraction and Refraction Delft Hydraulics Lab. Publ. No. 163 BERKHOFF, J.C.W., BOOY, N. and RADDER, A.C. (1982) Verification of Numerical Wave Propogation Models for Simple Harmonic Linear Water Waves. Coastal Engineering 6 pp 255-279 BETTESS, P., FLEMING, C.A., HEINRICK, J.C., ZIENKIEWICZ, O.C. and AUSTIN, D.I.(1978) Longshore Currents due to Surf Zone Barrier Proc. 16th Coastal Eng. Conf. pp 776-790 BIJKER, E.W. (1967) Some Considerations about Scales for Coastal Models with Movable Bed Delft Hydraulics Lab. Publ. 50 BIJKER, E.W. (1968) Littoral Drift as Function of Waves and Current Proc. 11th Coastal Eng. Conf. pp 415-435 BIRKEMEIER, W.A. and DALRYMPLE, R.A. (1975) Nearshore Water Circulation Induced by Wind and Waves. Proc. Symp. Modeling Techniques A.S.C.E. Vol. 2 California pp 1062-1081 BOOIJ, N. (1981) Gravity Waves on Water with Non-Uniform Depth and Current Report 81-1 Dept, of Civil Eng. Delft Univ. of Tech. BOWEN, A.J., INMAN, D.L. and SIMMONS, V.P. (1968) Wave Set-Down and Set-Up Journ. Geophys. Res. Vol. 73 No. 8 pp 2569-2577 BRATER, E.F. and PONCE-CAMPOS, D. (1976) Laboratory Investigation of Shore Erosion Processes Proc. 15th Coastal Eng. Conf. pp 1493-1511 BRUUN, P. (1963) Longshore Currents and Longshore Troughs Journ. Geophys. Res. Vol. 68 pp 1065-1078 BLUMBERG, A.F. (1977) Numerical Tidal Model of Chesapeake Bay ASCE J. Hyd. Div. Vol. 103 No. H Y 1 pp 1-10 BRENNIKMEYER, B.M. (1974) Mode and Period of Sand Transport in the Surf Zone Proc. 14th Coastal Eng. Conf. pp 812-827 COAKLEY, J.P., SAVILE, H.A., PEDROSA, M. and LAROCQUE, M. (1978) Sled System for Profiling Suspended Littoral Drift Proc. 16th Coastal Eng. Conf. pp 1764-1775 COAKLEY, J.P. and SKAFEL, M.G. (1982) Suspended Sediment Discharge on a Non-Tidal Coast Proc. 18th Coastal Eng. Conf. pp 1288-1304 COEFFE, Y. and PECHON, Ph. (1982) Modelling of Sea-Bed Evolution under Wave Action Proc. 18th Coastal Eng. Conf. pp 1149-1160 COPELAND, G.J.M. (1985a) A Practical Alternative to the "Mild-Slope" Wave Equation Coastal Engineering 9 pp 125-149 COPELAND, G.J.M. (1985b) Practical Radiation Stress Calculations Connected with Equations of Wave Propogation Coastal Engineering 9 pp 195-219 COPELAND, G.J.M. (1985c) A Numerical Model for the Propogation of Short Gravity Waves and the Resulting Circulation Around Nearshore Structures Ph.D Thesis, Civil Eng. Dept., University of Liverpool DALLY, W.R. and DEAN, R.G. (1984) Suspended Sediment Transport and Beach Profile Evolution ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 110 No. 1 pp 15-33 263 DANISH HYDRAULIC INSTITUTE (1984) Monitoring of Sedimentation in a Dredged Trench Final Report, Aug. 1984 DEAN, R.G. (1973) Heuristic Models of Sand Transport in the Surf Zone Conf. on Eng. Dynamics in the Coastal Zone. Sydney, pp 208-214 DE VAHL DAVIS, G. (1986) Numerical Methods in Engineering and Science Allen and Unwin, London DEIGAARD, R., FREDSOE, J. and HEDEGAARD, I.B. (1986a) Suspended Sediment in the Surf Zone ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 112 No. 1 pp 115-128 DEIGAARD, R., FREDSOE, J. and HEDEGAARD, I.B. (1986b) Mathematical Model for Littoral Drift ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 112 No. 3 pp 351-369 De VRIEND, H.J. (1987) 2DH Mathematical Modelling of Morphological Evolutions in Shallow Water Coastal Engineering. 11 pp 1-27 DELFT HYDRAULICS LABORATORY (1976) The Influence of Groynes on the Coast Report M918 Part 1 (experiments) ( in Dutch ) DELFT HYDRAULICS LABORATORY (1982) Refraction and Diffraction of Water Waves. Wave Deformation by a Shoal. Comparison between Computations and Measurements Report W154 part V III DINGLER, J.R. and INMAN, D.L. (1976) Wave-Formed Ripples in Nearshore Sands Proc. 15th Coastal Eng. Conf. pp 2109-2126 D O N G , P. (1987) The Computation of Wave-Induced Circulations with Wave-Current Interaction and Refined Turbulence Modelling Ph.D. Thesis. University of London DOUGLAS, J. (1955) On the Numerical Integration of cPu/dx^ + 9 ?i/9y ^ = 9u/9t by Implicit Methods J. Soc. Indust. Applied Maths Vol. 3 N o .l pp 42-65 DRAPER, L. (1966) The Analysis and Presentation of Wave Data - A Plea for Uniformity Proc. 10th Coastal Eng. Conf. pp 1-11 EAGLESON, P.S. (1965) Theoretical Structure of Longshore Currents on a Plane Beach MIT Hydrodynamics Lab. Tech. Rept. 82 pp 1-31 EBERSOLE, B.A. (1985) Refraction-Diffraction Model for Linear Water Waves ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. Ill No. 6 pp 939-953 EBERSOLE, B.A. and DALRYMPLE, R.A. (1980) Numerical Modelling of Nearshore Circulation Proc. 17th Coastal Eng. Conf. pp 2710-2725 ENGELUND, F. and FREDSOE, J. (1976) A Sediment Transport Model for Straight Alluvial Channels Nordic Hydrology Vol. 7 pp 293-306 ENGQUIST, B. and MAJDA, A. (1977) Absorbing Boundary Conditions for the Numerical Simulation of Waves Math. Comput. 31 pp 629-651 FAIRCHILD, J.C. (1972) Longshore Transport of Suspended Sediment Proc. 13th Coastal Eng. Conf. pp 1069-1088 FALCONER, R.A. (1976) Mathematical Modelling of Jet-Forced Circulation in Resevoirs and Harbours Ph.D Thesis University of London FALCONER, R.A. (1986) A Water Quality Simulation Study of a Natural Harbour ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 112 No. 1 pp 15-34 FLEMING, C.A. and HUNT, J.N. (1976a) A Mathematical Sediment Transport Model for Uni-Directional Flow Proc. ICE Vol. 61 pp 297-310 FLEMING, C.A. and HUNT, J.N. (1976b) Application of a Sediment Transport Model Proc. 15th Coastal Eng. Conf. pp 1184-1202 FLOKSTRA, C. (1977) The Closure Problem for Depth-Averaged Two-Dimensional Flow 17th Congress of the IAHR Vol. 2 pp 247-256 FREDSOE, J. (1984) Turbulent Boundary Layer in Wave-Current Motion ASCE Jnl. Hyd. Div. Vol. 110 No. 8 pp 1103-1120 FREDSOE, J., ANDERSEN, O.H. and SILBERG, S. (1985) Distribution of Suspended Sediment in Large Waves ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. Ill No. 6 pp 1041-1059 FRIJLINK, H.C. (1952) Discussion des formules de debit solide de Kalinske, Einstein et Meyer-Peter et Muller. Deuxieme journee de l'hydraulique. Transport hydraulique et decantation de materiaux solides. Grenoble GALVIN, C.J. (1967) Longshore Current Velocity: A Review of Theory and Data Rev. Geophys. 5(3) pp 287-304 GRAF, W.H. (1971) Hydraulics of Sediment Transport McGraw-Hill HEDEGAARD, I.B. (1985) Wave Generated Ripples and Resulting Sediment Transport in Waves ISVA Tech Univ. Denmark Series Paper 36 HORIKAWA, K., WATANABE, A. and KATORI, S. (1982) Sediment Transport under Sheet Flow Condition Proc. 18th Coastal Eng. Conf. pp 1335-1352 HOUSTON, J.R. (1981) Combined Refraction and Diffraction of Short Waves using the Finite Element Method Appl. Ocean Research Vol. 3 No. 4 pp 163-170 HULSBERGEN, C.H. , BARKER, W.T. and van BOCHOVE, G. (1976) Experimental Verification of Groyne Theory Proc. 15th Coastal Eng. Conf. pp 1439-1458 HUNT, J.N. (1954) The Turbulent Transport of Suspended Sediment in Open Channels Proc. Roy. Soc. Series A Vol. 224 pp 322-335 HYDRAULICS RESEARCH LIMITED (1984) The Effectiveness of Groyne Systems Report No. EX 1221 HYDRAULICS RESEARCH LIMITED (1986) The Effectiveness of Groyne Systems. Physical Model Study of Groynes a Beach Report No. EX 1351 HYDRAULICS RESEARCH STATION (1957) Hydraulics Research 1956 Dept. Scientific and Ind. Research HMSO London INMAN, D.L. and BAGNOLD, R.A. (1963) Littoral Processes in The Sea: Ideas and Observations Vol. 3 Interscience pp 529-553 INMAN, D.L. and QUINN, W.H. (1951) Currents in the Surf Zone Proc. Coastal Eng. Conf. pp 24-36 INMAN, D.L., TAIT, R.J. and NORDSTROM, C.E. (1971) Mixing in the Surf Zone Journ. Geophys. Res. Vol. 76 pp 3493-3514 ITO, Y. and TANIMOTO, K. (1972) A Method of Numerical Analysis of Wave Propogation - Application to Wave Diffraction and Refraction Proc. 13th Coastal Eng. Conf. pp 503-522 IWAGAKI, Y. (1968) Hyperbolic Waves and Their Shoaling Proc. 11th Coastal Eng. Conf. pp 124-144 JAFFE, B.E., STERNBERG, R.W. and SALLENGER, A.H. (1984) The Role of Suspended Sediment in Shore-Normal Beach Profile Changes Proc. 19th Coastal Eng. Conf. pp 1983-1996 JONSSON, I.G. (1963) Measurements in the Turbulent Wave Boundary Layer 10th Cong, of the IAHR London Vol. 1 pp 85-92 KANA, T.W. (1978) Surf Zone Measurements of Suspended Sediment Proc. 16th Coastal Eng. Conf. pp 1725-1743 KANA, T.W. and WARD, L.G. (1980) Nearshore Suspended Sediment Load During Storm and Post-Storm Conditions Proc. 17th Coastal Eng. Conf. pp 1158-1174 KELLER, J.B. (1958) Surface Waves on Water of Non-Uniform Depth Jour. Fluid Mech. 4 pp 607-614 KEMP, P.H. (1962) A Model Study of the Behaviour of Beaches and Groynes Proc. I.C.E. Vol. 22 pp 191-210 KIRBY, J.T. (1984) A Note on Linear Surface Wave-Current Interaction over Slowly Varying Topography Journ. Geophys. Res. Vol. 89 No. Cl pp 745-747 KIRBY, J.T. and DALRYMPLE, R.A. ( 1984) Surfzone Currents by D.R. Basco - Discussion Coastal Engineering 8 pp 387-392 * 268 KOMAR, P.D. (1971) The Mechanics of Sand Transport on Beaches Journ. Geophys. Res. Vol. 76 No. 3 pp 713-721 KOMAR, P.D. (1977) Beach Sand Transport: Distribution and Total Drift ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 103 No. WW2 pp 225-239 KOMAR, P.D. and INMAN, D.L. (1970) Longshore Sand Transport on Beaches Journ. Geophys. Res. Vol. 75 No. 30 pp 5914-5927 KOMAR, P.D. and MILLER, M.C. (1974) The Initiation of Oscillatory Ripple Marks and the Development of Plane-Bed at High Stresses Under Waves Journ. Sed. Petrology Vol. 45 No. 3 pp 697-703 KOUTITAS, C., KITOS, N. and KATOPODI, I. (1984) Sand Scouring Around Breakwaters Mathematical Models and 2 Cases of N. Greece Fishing Harbours Proc. Int. Symp. on Maritime Structures in Mediterranean Sea. Athens KUIPERS, J. and VREUGDENHIL, C.B. (1973) Calculations of Two-Dimensional Horizontal Flow Delft Hydraulics Lab. Rept. S 163-1 LAMB, Sir H. (1932) Hydrodynamics Cambridge Uni. Press. LAUNDER, B.E. and SPALDING, D.B. ( 1972) Mathematical Models of Turbulence Academic Press, London Le MEHAUTE, B. and SOLDATE, M. (1978) Mathematical Modeling of Shoreline Evolution Proc. 16th Coastal Eng. Conf. pp 1163-1179 LEAN, G.H. and WEARE, T.J. (1979) Modeling Two-Dimensional Circulating Flow ASCE Jnl. Hyd. Div. Vol. 105 H Y 1 pp 17-26 # 269 LHENDERTSE, J.J. (1967) Aspects of a Computational Model for Long-Period Water Wave Propogation RM-5294-PR The Rand Corp. Santa Monica Calif. LEENDERTSE, J.J. and LIU, S-K. (1982) Sensitivity of Parameters and Approximations in Models of Tidal Propogation and Circulation Proc. 18th Coastal Eng. Conf. pp 573-581 LIU, P.L-F. (1983) Wave-Current Interactions in a Slowly Varying Topography Journ. Geophys. Res. Vol. 88 No. C 7 pp 4421-4426 LIU,P. L-F. and LENNON, G.P. (1978) Finite Element Modeling of Nearshore Currents ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 104 WW2 pp 175-189 LIU, P.L-F. and TSAY, T-K. (1985) Numerical Prediction of Wave Transformation ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. Ill No. 5 pp 843-855 LONGUET-HIGGINS, M.S. (1970) Longshore Currents Generated by Obliquely Incident Sea Waves, 1 and 2 Journ. Geophys. Res. Vol. 75 No. 33 pp 6778-6801 LONGUET-HIGGINS, M.S. and STEWART, R.W. (1960) Changes in the Form of Short Gravity Waves on Long Waves and Tidal Currents Jour. Fluid Mechanics No. 8 pp 565-583 LONGUET-HIGGINS, M.S. and STEWART, R.W. (1961) The Changes in Amplitude of Short Gravity Waves on Steady Non-Uniform Currents Jour. Fluid Mechanics No. 10 pp 529-549 LONGUET-HIGGINS, M.S. and STEWART, R.W. (1962) Radiation Stress and Mass Transport in Gravity Waves with Application to Surf Beats Jour. Fluid Mechanics No. 13 pp 481-504 270 LONGUET-HIGGINS, M.S. and STEWART, R.W. (1963) A Note on Wave Set-Up Jnl. Marine Res. No. 21 pp 4-10 LONGUET-HIGGINS, M.S. and STEWART, R.W. (1964) Radiation Stress in Water Waves; A Physical Discussion with Applications Deep Sea Res. Vol. 11 pp 529-562 LOZANO, C. and LIU, P.L-F ( 1980) Refraction-Diffraction Model for Linear Surface Water Waves Jour. Fluid Mechanics Vol. 101 part 4 pp 705-720 LOZANO, C. and MEYER, R.E. (1976) Leakage and Response of Waves Trapped by Round Islands Phys. Fluids 19(8) pp 1075-1088 MANOHAR, M. (1955) Mechanics of Bottom Sediment Movement due to Wave Action Beach Erosion Board, Tech. Memo No. 75 121pp MASON, C., SALLENGER, A.H., HOLMAN, R.A., BIRKEMEIER, W.A. (1984) DUCK 82 - A Coastal Storm Processes Experiment Proc. 19th Coastal Eng. Conf. pp 1913-1928 MEI, C.C. (1983) The Applied Dynamics of Ocean Surface Waves Wiley - Interscience MEYER, R.E. (1979) Theory of Water Wave Refraction in Advances in App. Mechanics, ed. C-S. Yih Vol. 19 pp 53-141 MUIR-WOOD, A.M. and FLEMING, C.A. (1981) Coastal Hydraulics, 2nd edition Macmillan Press MUNK, W.H. and TRAYLOR, M.A. (1947) Refraction of Ocean Waves: A Process Linking Underwater Topography to Beach Erosion J . Geol. 40 pp 1-26 MURRAY, S.P. (1967) Control of Grain Dispersion by Particle Size and Wave State Jour. Geology Vol. 75 pp 612-634 NIELSEN, P. (1979) Some Basic Concepts of Wave Sediment Transport ISVA Tech. Univ. Denmark Series Paper 20 NIELSEN, P. (1984) Field Measurements of Time Averaged Suspended Sediment Concentrations under Waves Coastal Engineering 8 pp 51-72 NIELSEN, P. (1986) Suspended Sediment Concentrations Under Waves Coastal Engineering 10 pp 23-31 NIELSEN, P., SVENDSEN, I.A. and STAUB, C. (1978) Onshore-Offshore Sediment Movement on a Beach Proc. 16th Coastal Eng. Conf. pp 1475-1492 N O D A , E . (1972) Wave Induced Circulation and Longshore Current Patterns in the Coastal Zone Tetra Tech. Rept. TC- 149-3 NODA, E., SONU, C.J., RUPERT, V.C. and COLLINS, J.I. (1974) Nearshore Circulations Under Sea Breeze Conditions and Wave-Current Interactions in the Surf Zone Tetra Tech Rep. TC- 149-4 O'CONNOR, B.A., FANOS, A.M. and CATHERS, B. (1981) Simulation of Coastal Sediment Movements by Computer Model Proc. 2nd Int. Conf. on Eng. Software, London PEACEMAN, D.W. and RACHFORD, H.H. (1955) The Numerical Solution of Parabolic and Elliptic Differential Equations J. Soc. Indust. Applied Maths Vol. 3 M o.l pp 28-41 PELNARD-CONSIDERE, R. (1956) Essai de theorie de 1'evolution des formes de rivages en plages de sable et de galets. Quatrieme Journees de l'Hydraulique, Paris Les Energies de la Mer, Question III rapport 1 pp 289,298 PENNEY, W.G. and PRICE, A.T. (1952) The Diffraction Theory of Sea Waves and Shelter Afforded by Breakwaters Phil. Trans. Roy. Soc. Series A Vol. 244 pp 236-253 PEREGRINE, D.H. (1976) Interactions of Water Waves and Currents Adv. Appl. Mech. 16 pp 9-117 PERLIN, M. and DEAN, R.G. (1978) Prediction of Beach Planforms with Littoral Controls Proc. 16th Coastal Eng. Conf. pp 1818-1838 PERLIN, M. and DEAN, R.G. (1983) An Efficient Numerical Algorithm for Wave Refraction/Shoaling Problems ASCE Proc. Coastal Structures '83 pp 988-999 PERLIN, M. and DEAN, R.G. (1985) 3-D Model of Bathymetric Response to Structures ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol 111 No. 2 pp 153-170 PONCE, V.M. and YABUSAKI, S.B. (1980) Mathematical Modeling of Circulation in Two-Dimensional Plane Flow Report C E R 79- 80VMP-SBY 59 Civil Eng. Dept. Colorado State University PONCE, V.M. and YABUSAKI, S.B. (1981) Modeling Circulation in Depth-Averaged Flow ASCE Jnl. Hyd. Div. Vol. 107 No. H Y 11 pp 1501-1518 PRANDTL, L. (1925) Uber die Ausgebildete Turbulenz Z.A .M .M . 5 p i36 PRICE, W.A. and TOMLINSON, K.W. (1968) The Effect of Groynes on Stable Beaches Proc. 11th Coastal Eng. Conf. pp 518-525 273 PRICE, W.A., TOMLINSON, K.W. and WILLIS, D.H. (1972) Predicting Changes in the Plan Shape of Beaches Proc. 13th Coastal Eng. Conf. pp 1321-1329 PUTNAM, J.A. and ARTHUR, R.S. (1948) Diffraction of Water Waves by Breakwaters Trans. Am. Geog. Un. Vol. 29 No. 4 pp 481-490 PUTNAM, J.A., MUNK, W.H. and TRAYLOR, M.A. (1949) The Prediction of Longshore Currents Trans. Am. Geophys. Union Vol. 30 pp 337-345 RADDER, A.C. (1979) On the Parabolic Equation Method for Water Wave Propogation Jour. Fluid Mechanics Vol. 95 part 1 pp 159-176 RICHMOND, B.M. and SALLENGER, A.H. (1984) Cross-Shore Transport of Bimodal Sands Proc. 19th Coastal Eng. Conf. pp 1997-2008 ROACHE, P.J. (1972) Computational Fluid Dynamics Hermosa Publ. R O D I, W. (1980) Turbulence Models and their Application in Hydraulics Inti. Assoc. Hydraulic Res. State-of-Art-Paper SCHONFELD, J.Ch. (1972) Propogation of Two-Dimensional Short Waves Manuscript (in Dutch), Delft University of Tech. SEYMOUR, R.J. (1984) Discussion on - Suspended Sediment Transport and Beach Profile Evolution by Dally and Dean ( 1984) ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 113 No. 1 pp 85-86 SEYMOUR, R.J. and KING, D.B. (1982) Field Comparisons of Cross-Shore Transport Models ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. 108 No. WW2 pp 163-179 * 274 SHIELDS, A. (1936) Anwendung der aehnlichkeitsmechanik und der turbulenz forschung auf die geschiebebewegung Mitt. Preuss. Versuchsanstalt Wasserbau Schiffbau, Berlin, 26 da SILVA LIMA, S.S.L. (1981) Wave Induced Nearshore Currents Ph.D. Thesis Dept, of Civil Eng. University of Liverpool SMITH, R. and SPINKS, T. (1975) Scattering of Surface Waves by a Conical Island Jour. Fluid Mechanics Vol. 72 part 2 pp 373-384 SOMMERFELD, A. (1896) Mathematische Theorie der Diffraction Math. An. 47 pp 317 and Optics, Lectures on Theoretical Physics, Vol. IV Academic Press SONU, C.J. (1972) Field Observations of Nearshore Circulation and Meandering Currents Journ. Geophys. Res. Vol. 77 No. 18 pp 3232-3247 STAUB, C., SVENDSEN, I.A. and JONSSON, I.G. (1983) Measurements of the Instantaneous Sediment Suspension in Oscillatory Flow by a New Rotating-Wheel Apparatus ISVA Tech. Univ. Denmark Prog. Rept. 58 pp 41-49 STERNBERG, R.W., SHI, N.C. and DOWNING, J.P. (1984) Field Investigations of Suspended Sediment Transport in the Nearshore Zone Proc. 19th Coastal Eng. Conf. pp 1782-1798 STIVE, M .J.F.. and BATTJES, J.A. (1986) Random Wave Breaking and Induced Currents Int. Colloq. on Math. Modeling of Wave Breaking and Wave Induced Currents, Grenoble France SUMMERS, L. and FLEMING, C.A. ( 1983) Groynes in Coastal Engineering: A Review Construction Industry Research and Information Assoc. Tech. Note 111 SVENDSEN, I.A. (1984) Wave Heights and Set-Up in a Surf Zone Coastal Engineering 8 pp 303-329 SVENDSEN, I.A., MADSEN, P.A. and HANSEN, J.B. (1976) Wave Characteristics in the Surf Zone Proc. 16th Coastal Eng. Conf. pp 520-539 SWART, D.H. (1974) Offshore Transport and Equilibrium Beach Profiles Delft Hyd. Lab. Publ. 131 THORNTON, E.B. (1972) Distribution of Sediment Transport across the Surf Zone Proc. 13th Coastal Eng. Conf. pp 1049-1068 THORNTON, E.B. and GUZA, R.T. ( 1983) Transformation of Wave Height Distribution Joum. Geophys. Res. Vol. 88 No. CIO pp 5925-5938 THORNTON, E.B. and GUZA, R.T. (1986) Surf Zone Longshore Currents and Random Waves: Field Data and Models Journ. Phys. Oceanography Vol. 16 pp 1165-1178 TOMLINSON, J.H. (1980) Groynes in Coastal Engineering Hydraulics Research Station Report No. IT 199 U.S. ARMY COASTAL ENGINEERING RESEARCH CENTER Shore Protection Manual 4th edition , 1984 VEMULAKONDA, S.R., HOUSTON, J.R. and BUTLER, H.L. (1982) Modeling Longshore Currents for Field Situations Proc. 18th Coastal Eng. Conf. pp 1659-1676 VISSER, P.J. (1980) Longshore Current Flows in a Wave Basin Proc. 17th Coastal Eng. Conf. pp 462-479 » 276 VISSER, P.J. (1982) The Proper Longshore Current in a Wave Basin Comm, on Hydraulics Dept. Civil Eng. Delft Uni. of Tech. Report No. 82-1 VISSER, P.J. (1984) A Mathematical Model of Uniform Longshore Currents and the Comparison with Laboratory Data Comm, on Hydraulics Dept. Civil Eng. Delft Uni. of Tech. Report No. 84-2 WARREN, R., LARSEN, J. and MADSEN, P.A. (1985) Application of Short Wave Numerical Models to Harbour Design and Future Development of the Model Int. Conf. on Num. and Hydraulic Modelling of Ports and Harbours Birmingham England pp 303-308 WAT AN ABE, A. (1982) Numerical Models of Nearshore Currents and Beach Deformation Coastal Eng. in Japan Vol. 25 pp 147-161 WAT AN ABE, A. (1985) Three-Dimensional Predictive Model of Beach Evolution Around a Structure Proc. Symp on Water Wave Research Hanover pp 121-141 WEARE, T.J. (1976) Instability in Tidal Flow Computational Schemes ASCE Jnl. Hyd. Div. Vol. 102 No. HY5 pp 569-580 WIEGEL, R.L. (1962) Diffraction of Waves by Semi-Infinite Breakwater ASCE J. Hyd. Div. Vol. 88 pp 27-44 WIND, H.G. and VREUGDENHIL, C.B. (1986) Rip-Current Generation Near Structures Jour. Fluid Mechanics Vol. 171 pp 459-476 WU, C-S and LIU, P.L-F. (1984) Effects of Non-Linear Inertial Forces on Nearshore Currents Coastal Engineering 8 pp 15-32 WU, C-S and LIU, P.L-F. (1985) Finite Element Modeling of Nonlinear Coastal Currents ASCE Jnl. Water. Ports. Coastal Ocean Eng. Vol. Ill No. 2 pp 417-432 WU, C-S, THORNTON, E.B. and GUZA, R.T. (1985) Waves and Longshore Currents: Comparison of a Numerical Model with Field Data Joum. Geophys. Res. Vol. 90 No. C3 pp 4951-4958 YALIN, M.S. (1977) Mechanics of Sediment Transport Pergamon Press 2nd Edition YAMAGUCHI, M. and NISHIOKA, Y. (1984) Numerical Simulation on the Change of Bottom Topography by the Presence of Coastal Structures Proc. 19th Coastal Eng. Conf. pp 1732-1748 O'CONNOR, B.A., WAKELING, L., COX, N., and GOSH, A. (1983) Study of littoral drift at Paradip, India. Int. Conf. on Coastal and Port Engg. in Developing Countries, Sri Lanka O’CONNOR, B.A. (1985) Coastal Sediment Modelling Contribution to Offshore and Coastal Modelling. Dyke, P.G., Moscardini, A.O. and Robinson, E.H. (eds) Springer-Verlag, New York. O'CONNOR, B.A. and YOO, D. (1987) Bed friction model of wave-current interacted flow Coastal modelling Conference, University of Delaware YOO, D. and O’CONNOR, B.A. (1986) Mathematical Modelling of Wave-Induced Circulations Proc. 20th Coastal Eng. Conf.