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## Material Information- Title:
- Wave field modifications and shoreline response due to offshore borrow areas
- Series Title:
- Wave field modifications and shoreline response due to offshore borrow areas
- Creator:
- Bender, Christopher
- Place of Publication:
- Gainesville, Fla.
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- Coastal & Oceanographic Engineering Dept. of Civil & Coastal Engineering, University of Florida1
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- English
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- University of Florida
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- University of Florida
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UFL/COEL -2001/005
WAVE FIELD MODIFICATIONS AND SHORELINE RESPONSE DUE TO OFFSHORE BORROW AREAS by Christopher J. Bender Thesis 2001 WAVE FIELD MODIFICATIONS AND SHORELINE RESPONSE DUE TO OFFSHORE BORROW AREAS By CHRISTOPHER J. BENDER A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2001 ACKNOWLEDGMENTS Many individuals at the University of Florida made the completion of this work possible. I first must thank my advisor and committee chairman Dr. Robert G. Dean for his abundant help and tireless work during the course of this study. His ideas and suggestions assisted me in countless ways. I would like to thank the other members of my committee, Dr. Daniel M. Hanes and Dr. Robert J. Thieke, for their help throughout the study. The staff at the Coastal Engineering Laboratory including Sidney Schofield, Jim Joiner, Vic Adams and Vernon Sparkman solved many of the problems I experienced during the laboratory experiments and made my research possible. I would like to acknowledge the Coastal Engineering Archives staff at the University of Florida including Helen Twedell and Kimberly Hunt for their help during the last two years. I thank Becky Hudson for answering my questions covering many topics in the graduate catalog. The Office of Beaches and Coastal Systems of the Florida Department of Environmental Protection provided partial funding for this study through a contract on the causes of erosional hot spots. I would like to thank my wife for her love, constant support and confidence in me and my endeavors. In addition I would like to thank my parents, family and friends for their encouragement and all that they have done to make me who I am. TABLE OF CONTENTS pA&e ACKNOW LEDGM EN TS ............................................................................................... 11 L IS T O F T A B L E S ......................................................................................................... v i L IS T O F F IG U R E S ....................................................................................................... v ii A B S T R A C T ................................................................................................................. x iv I IN T R O D U C T IO N ....................................................................................................... I 1 1 P ro b le m S tatem e n t ................................................................................................. 1 1.2 Objectives and Scope ............................................................................................. 2 2 LITERATURE REVIEW ............................................................................................ 4 2 .1 In tro d u ctio n ........................................................................................................... 4 2.2 Discussion of Previous Studies .............................................................................. 4 2.2.1 Rectangular Pits of Infinite Length .................................................................. 4 2.2.2 Pits of Arbitrary Shape .................................................................................... 6 2.2.3 Shoreline Change by Offshore Pits .................................................................. 7 2.2.4 Laboratory and Field Studies ........................................................................... 8 3 T H E O R Y .................................................................................................................... 9 3 .1 In tro d u ctio n ........................................................................................................... 9 3.2 Governing Equations ............................................................................................. 9 3 .2 .1 F lo w O v er A P it .............................................................................................. 9 3.2.2 Energy Considerations .................................................................................. 11 3.2.3 Shoreline Change .......................................................................................... 12 4 M O D E L S .................................................................................................................. 1 7 4 .1 In tro d u c tio n ......................................................................................................... 1 7 4.2. Num erical Solution M odel of W illiam s (1990) ................................................... 17 4.2.1 Green's Function Solution of W illiam s (1990) .............................................. 17 4.2.2 W ave H eight Reduction for Pit of Arbitrary Shape ........................................ 20 4.2.3 Reflection from Pit ........................................................................................ 22 4.2.4 Shoreline Change Induced by Pit ................................................................... 24 4 2.5 M odels for Solid Structure of Finite D im ensions ........................................... 27 4.3 Analytical Solution M odel for Circular Pit ........................................................... 28 4.3.1 W ave H eight Reduction for Circular Pit ........................................................ 28 4.3.2 Reflection from Circular Pit .......................................................................... 30 4.3.3 Shoreline Change Induced by Pit ................................................................... 31 4.4 M acCam y and Fuchs Solution M odel for Cylinder ............................................... 32 5 RESU LTS AN D D ISCU SSION ................................................................................ 34 5 .1 In tro d u ctio n ......................................................................................................... 3 4 5.2 W ave H eight Reduction M odels .......................................................................... 34 5.2.1 Comparison of Numerical Model Results with Results of Williams (1990) ... 34 5.2.2 Comparison of Numerical Model with Analytic Solution Model ................... 39 5.2.3 Results for M odels w ith Solid Cylinder ......................................................... 50 5.2.4 Plots of W ave Fronts in Fluid D om ain .......................................................... 54 5.2.5 M odel Sensitivity .......................................................................................... 56 5.3 Energy Reflection Caused by Pit .......................................................................... 59 5.3.1 A nalytic Solution M odel ............................................................................... 59 5.3.2 N um erical M odel .......................................................................................... 68 5.3.3 M acCam y and Fuchs Solution M odel for Solid Cylinder ............................... 72 5.4 Shoreline Change Induced by Pit ......................................................................... 74 5.4.1 Com parison of M odel Results ....................................................................... 74 5.4 2 Shoreline Evolution w ith Tim e ...................................................................... 85 5.4.3 Case H istories ............................................................................................... 90 5.4.3.1 G rand Isle, Louisiana ............................................................................. 90 5.4.3.2 M artin County, Florida ........................................................................... 96 6 LABORATO RY RESU LTS AN D D ISCU SSION ................................................... 102 6 .1 In tro d u ctio n ....................................................................................................... 10 2 6.2 Experim ental Setup and Equipm ent ................................................................... 102 6 .3 E x p e rim e n ts ....................................................................................................... 10 4 6.3.1 W ave H eight Reduction .............................................................................. 104 6.3.2 Shoreline Change ........................................................................................ 104 6.4 Experim ent Results and D iscussion ................................................................... 105 6.4.1 W ave H eight Reduction ............................................................................. 105 6.4.2 Shoreline Change ........................................................................................ 106 6.5 Com parison w ith N um erical M odels .................................................................. 115 7 SU M M ARY AN D CON CLU SION S ....................................................................... 122 LIST OF REFEREN CES ............................................................................................. 127 BIOGRAPHICAL SKETCH ....................................................................................... 129 LIST OF TABLES Table Page 5. 1: Coefficients and constants used in the shoreline change models .... ............. 75 6. 1: Values of parameters used in laboratory model for different trials in shoreline chang e exp erim ent ................................................................... 106 6.2: Dimensions of different pits used in shoreline change ex p e rim en ts ......................................................... ................. 10 7 LIST OF FIGURES Figure Page 3. 1: Definition sketch for flow over pit ............................................... 9 3.2: Definition sketch for shoreline change problem .....................13 5. 1: Contour plot of relative amplitude from numerical model for 8 by 8 mn pit and base conditions with location of pit drawn ... ....................... 35 5.2: Contour plot of relative amplitude for P3 = 0, kid = 7c/10, k2d = 7t/10'1, hid = 0.5, b/a = 1.0 and a/d = 2.0 equal to 8 by 8 mn pit with base conditions;, from W illiams (1990) .......... ................36 5.3: Contour plot of relative amplitude from numerical model for 8 by 24 m pit with f3= 450 and other base conditions with location of pit drawn ......... 37 5.4: Contour plot of relative amplitude for P3 = 450, kid = 7E/10, k2d = iRAM~i, h/d = 0.5, b/a =3.0 and a/d = 2.0 equal to 8 by 24 m pit with P3 = 45' and other base conditions; from Williams (1990) .................38 5.5: Contour plot of relative amplitude from numerical model for circular pit with r = 12 m, 200 points defining the pit boundary and base conditions with location of pit drawn................................................ 40 5.6: Contour plot of relative amplitude from analytic solution model for circular pit with r = 12 m, 80 terms taken in the series summation and base conditions with location of pit drawn .................................... 41 5.7: Contour plot of percent difference for numerical model and analytic solution model for circular pit with r = 12 m, 200 points defining the pit boundary in the numerical model, 80 terms taken in the series summation in the analytic solution and base conditions with location of pit drawn ...........41 5.8: Plot of transect taken parallel to the x-axis at y = 0 for pit of radius 12 m showing wave direction (03 = 0') ......................................... 42 5.9: Relative amplitude for numerical model and analytic solution model for transect show in Figure 5.8 for pit of radius = 12 m with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with pit drawn ...............43 5. 10: Relative amplitude for numerical model and analytic solution model for 3 transects parallel to the y-axis at X = 0, 24 and 100 mn for pit of radius = 12 mn with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with pit drawn ........................................ 44 5.11: Contour plot of relative amplitude from analytic solution model for circular pit with r = 75 mn, 1 10 terms taken in the series summation and base conditions with pit drawn .......................................45 5.12: Contour plot of percent difference for numerical model and analytic solution model for circular pit with r = 75 mn, 600 points defining the pit boundary in the numerical model, 1 10 terms taken in the series summation in the analytic solution and base conditions with pit drawn...46 5.13:- Relative amplitude for numerical model and analytic solution model for transect parallel to the x-axis at y = 0 mn for pit of radius = 75 m+ with 600 points defining the pit boundary in the numerical model and 100 terms taken in the series summation for analytic solution with pit drawn ....47 5.14: Relative amplitude for numerical model and analytic solution model for transect parallel to the y-axis at X = 100 m for pit of radius =75 mn with 600 points defining the pit boundary in the numerical model and 100 terms taken in the series summation for analytic solution with pit drawn ....48 5.15: Relative amplitude for numerical model and analytic solution model for transect parallel to the y-axis at X = 500 mn for pit of radius =75 mn with 600 points defining the pit boundary in the numerical model and 100 terms taken in the series summation for analytic solution with pit drawn ....49 5.16: Contour plot of relative amplitude from MacCamy and Fuchs solution model for a circular cylinder with r = 12 mn, 80 terms taken in the series summation and base conditions with pit drawn ...............50 5.17: Contour plot of percent from from solid numerical model and MacCamy and Fuchs solution model for a circular cylinder with r = 12 mn, 200 points defining the pit boundary, 80 terms taken in the series summation and base conditions with pit drawn............................52 5.18: Relative amplitude for solid numerical model and for MacCamy and Fuchs solution model for transect parallel to the x-axis at y = 0 mn for pit of radius = 12 mn with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with analytic solution model for circular pit shown....... 52 5.19: Relative amplitude for solid numerical model and for MacCamy and Fuchs solution model for transect parallel to the y-axis at x = 100 mn for pit of radius =12 mn with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with analytic solution model for circular pit of the same radius shown and with pit drawn................................................ 53 5.20: Contour plot showing wave fronts as lines of constant phase from analytic solution model for circular pit with r = 12 mn, 80 terms taken in the series summation base conditions with pit drawn................................ 54 5.21: Contour plot of showing wave fronts as lines of constant phase from MacCamy and Fuchs solution model for circular pit with r =12 mn, 80 terms taken in the series summation base conditions with pit drawn ....55 5.22: Plot of numerical model values of maximum and minimum relative amplitude versus the number of points defining the pit boundary for a grid of points defining the fluid domain with a circular pit of radius 12 mn and other base conditions with analytic model solution for same conditions drawn ...56 5.23: Plot of analytic solution model values of relative amplitude versus the number terms taken in the series summation for five points on the x-axis with a circular pit of radius 12 mn and other base conditions ...................... 57 5.24: Plot of numerical model values of relative amplitude for two sections of a transect parallel to the y-axis at x equal 100 mn for five different numbers of points defining the pit boundary for a circular pit of radius 12 mn and other base conditions with the analytic solution model results for the same conditions drawn ............................ 61 5.25: Plot of transect taken parallel to the y-axis for energy flux calculations with pit of radius 12 mn showing wave direction................................ 60 5.26: The four energy flux terms from the analytical solution model for transect shown in Figure 5.25 with base conditions and 90 points taken in the summation .............................................................. 61 5.27: Energy flux at each point in transect shown in Figure 5.25 with incident value (dotted) for base conditions and 90 terms taken in the series summation ....62 5.28: Reflection coefficients determined as the average over one energy flux oscillation for each upcrossing in the transect ............................ 63 5.29: Reflection coefficients for different transect locations with several incident wave periods for the transect shown in Figure 5.25 taking the reflection coefficient as the average of the last energy flux oscillation in the transect......... ...................................................64 5.30: Reflection coefficients versus pit diameter divided by wavelength outside the pit and pit diameter divided by wavelength inside the pit for different pit radii for the transect shown in Figure 5.25 taking the reflection coefficient as the average of the last energy flux oscillation in th e tra n se c t ........................................................................... 6 5 5.31: Reflection coefficients versus water depth divided by pit depth for different pit radii for the transect shown in Figure 5.25 .................................. 66 5.32: Reflection coefficients versus pit diameter divided by wavelength inside pit with different pit radii for a pit of radius equal to 12 m and the transect show n in Figure 5.25 ........................................................ 67 5.33: Reflection coefficients versus the number of points defining the pit boundary for the numerical model with a pit of radius equal to 12 m, a period of 12 s and the other base conditions with the analytic solution m o del resu lt show n .................................................................... 69 5.34: Percent difference in relative amplitude, percent difference in energy flux values and difference in energy flux values for analytic solution model and numerical model for a pit of radius equal to 12 m, 500 points on the pit boundary in the numerical model, and the other base conditions along a transect at x equal to 100 m ............................................. 70 5.35: Reflection coefficients versus the pit diameter divided by the wavelength outside the pit for the numerical model for different pit radii along a transect at x equal to 100 m and the other base conditions with the analytic solution model results shown ........................................... 72 5.36: Reflection coefficients versus the cylinder diameter divided by the wavelength outside the pit with the MacCamy and Fuchs solution model for different pit radii along a transect at x equal to 0 m and the other base conditions ...... 73 5.37: Relative Amplitude along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for this shoreline change model ...... 76 5.38: Longshore transport value along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining th e p it b o u n d ary ........................................................................ 7 7 5.39: Shoreline change from the full transport equation for a 300 s time step along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change model ........................ 78 5.40: Shoreline change for a 300 s time step along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline ch a n g e m o d e l ........................................................................... 7 9 5.41: Shoreline change for first and second transport terms for a 300 s time step along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change model ............................ 80 5.42: Longshore transport along the transect for the analytic solution model and the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change m odel ...................................................... 81 5.43: Shoreline change for a 300 s time step along the transect for the analytic solution model and the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change model ........................ 82 5.44: Relative amplitude along transect for numerical model with a rectangular pit (40 x 200 m) at x = 1500 m with 800 points defining the pit boundary and the other base conditions for this shoreline change model .................. 83 5.45: Shoreline change for 300 second time step along transect for numerical model with a rectangular pit (40 x 200 m) at x = 1500 m with 800 points defining the pit boundary and the other base conditions for this shoreline change m odel ....................................................... 84 5.46: Total shoreline change for 300 second time step along transect for numerical model with a rectangular pit (40 x 200 m) at x = 1500 m with 800 points defining the pit boundary and the other base conditions for this shoreline change m odel ............................................................ 85 5.47: Shoreline evolution for analytic solution model for 2 hour time step with pit radius of 6 m, water depth of 2 m, pit depth of 3 m and 50 terms taken in the series summation using full transport equation ..................... 87 5.48: Shoreline evolution for analytic solution model for 2 hour time step with pit radius of 6 m, water depth of 2 m, pit depth of 3 m and 50 terms taken in the series summation using each transport term separately ...... 88 5.49: Shoreline change over each time step for analytic solution model for 2 hour time steps with pit radius of 6 m, water depth of 2 m, pit depth of 3 m and 50 terms taken in the series summation ....................................... 89 5.50: Aerial photograph showing salients shoreward of borrow area looking East to West in August, 1985 (Combe and Soileau, 1987) ....................... 91 5.51:- Relative amplitude along the transect for beta equal to 6.5 degrees and averaged over five betas from numerical model for the Grand Isle, LA case study............. ...................................................93 5.52: Shoreline change for one 300 s time step and longshore transport (filtered and unfiltered) for Grand Isle, LA case study with 50 terms taken in the moving average...................................................... 94 5.53: Longshore transport (filtered and unfiltered) from numerical model for Grand Isle, LA case study with 25 and 75 terms taken in the moving average ....95 5.54: Shoreline change for one 300 s time step from numerical model for Grand Isle, LA case study with 25 and 75 terms taken in the moving average ....96 5.55: Shape of borrow area and transect location for Martin County, FL numerical model ......................................................... 98 5.56: Relative amplitude along transect for bathymetry pre-dredging and post-dredging for Martin County, FL numerical model ..............98 5.57: Change in shoreline position during one time step for original bathymetry and post-dredging for Martin County, FL numerical model ............99 5.58: Difference in shoreline change during one 300 second time step from original bathymetry to shoreline change after dredging of shoal for Martin County, FL numerical model ............... .............100 6. 1: Schematic layout of fixed-bed model used in the laboratory experiments........ 103 6.2: Change in dry beach width with time for Pit 1 for Trial 1 from initial equilibrium without pit, to shoreline after 3 hours with pit ...........108 6.3: Change in dry beach width with time for Pit 2 for Trial 3 from initial equilibrium without pit, to shoreline after 3 hours with pit ...........109 6.4 Change in shoreline position from equilibrium without a pit to after 1 hour with a pit present for 3 pit sizes and wave conditions of Trial 1 ....... 109 6.5: Change in shoreline position from equilibrium without a pit to after 3 hours with a pit present for 3 pit sizes and wave conditions of Trial 1........... 110 6.6: Change in shoreline position from equilibrium without a pit to after 1 hour with a pit present for 3 pit sizes and wave conditions of Trial 2...........111I 6.7: Change in shoreline position from equilibrium without a pit to after 3 hours with a pit present for 3 pit sizes and wave conditions of Trial 2 ...........112 6.8: Change in shoreline position from equilibrium without a pit to after I hour with a pit present for 2 pit sizes and wave conditions of Trial 3 ............... 113 6.9: Change in shoreline position from equilibrium without a pit to after 3 hours with a pit present for 2 pit sizes and wave conditions of Trial 3 ............... 113 6.10: Change in shoreline position from equilibrium with no cylinder to after 3 hours with a solid cylinder of diameter equal to 32 cm present for the wave conditions of Trial 2 with the pit results drawn ....................... 114 6.11: Contour plot of relative amplitude from numerical model for Pit I and incident wave conditions of Trial I with wave guide drawn and x = 0 representing the baseline for the experiment ..................................... 115 6.12: Relative amplitudes determined from the numerical model for experiment Trials I and 3 with 3 pits at x equal to -0.4 in with 150 points on the pit boundary and the location of the wave-guides used in the exp erim ent d raw n ..................................................................... 1 16 6.13: Change in shoreline position from equilibrium from the numerical model for experiment Trials I and 3 with 3 pits at x equal to -0.4 m with 150 points on the pit boundary, one 300 second time step, and the location of the wave-guides used in the experiment drawn ..................... 117 6.14: Change in shoreline position by each transport term from the numerical model for experiment Trials I and 3 with and Pit I at x equal to -0.4 in with 150 points on the pit boundary, one 300 second time step ............... 119 6.15: Change in shoreline position by each transport term from the numerical model for parameters from Horikawa et al. (1977) at x equal to -0.4 in with 150 points on the pit boundary, one 300 second time step ............... 120 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science WAVE FIELD MODIFICATIONS AND SHORELINE RESPONSE DUE TO OFFSHORE BORROW AREAS By Christopher J. Bender May 2001 Chairman: Dr. Robert G. Dean Major Department: Civil and Coastal Engineering This study was undertaken to form a better understanding of the wave field modifications and resulting shoreline changes caused by offshore borrow areas. These offshore areas can cause wave refraction, diffraction, reflection and dissipation resulting in a shoreline that is modified to a new equilibrium by the pit altered wave field. In the context of a beach nourishment project, this new equilibrium can lead to the presence of erosional hot spots that reflect negatively on the overall project performance. Numerical and analytic solutions are used to model the wave conditions in and around the pit using shallow water wave theory. The numerical solution employs the formulation by Neil Williams in his paper on wave diffraction by a pit, published in 1990. This method uses Green's Theorem and suitable Green's functions on a pit boundary of arbitrary shape. The analytic solution uses a series expansion of Bessel functions at each point in the defined fluid domain for a circular pit. These models determine the complex velocity potential found anywhere in the fluid domain, allowing quantities such as the velocity and pressure to be solved. This allows modeling of the energy reflected and shoreline change induced by a pit. The wave field modification caused by a pit is compared to that due to a solid structure using numerical and analytic solutions. A laboratory experiment was conducted to determine the wave field modification and shoreline change caused by pits of various sizes under several incident wave conditions. The wave field modification caused by a pit of finite dimensions is found to be significant, with a partial standing wave pattern of increased and decreased wave amplitude in front of the pit, and an area of reduced relative amplitude flanked by two lobes of increased amplitude behind the pit. The numerical and analytic solutions for a circular pit compare well, with larger pit radii leading to greater differences in the results. The energy reflected by a pit is found to be considerable and to depend on the dimensionless pit diameter and other parameters. The shoreline change shoreward of the pit shows accretion directly behind the pit bordered by areas of erosion, with both transport terms of importance. The laboratory results for the shoreline change behind a pit show either accretion or erosion directly behind the pit; trends that vary in their correlation with the computer models. In summary, an offshore pit is shown to modify the incident wave field, which can lead to energy being reflected and diffracted, and associated changes to the shoreline near the pit. CHAPTER I INTRODUCTION 1. 1 Problem Statement Beach erosion is a primary concern for many coastal communities. The beach system provides natural storm protection, recreation area and wildlife habitat. When significant erosion occurs, the benefits provided by the beach system are diminished. Beach nourishment has emerged as a "natural" technique for restoring the equilibrium of the beach system. By adding sand to the dynamic sediment system in the nearshore zone, beach nourishment replaces the material lost to erosion; an advantage over "hard" solutions such as seawalls, breakwaters and groins. Beach nourishment projects require sediment for the beach fill that is placed at the nourishment location. The fill material needs to be carefully chosen to have sediment characteristics that combine with the native sediment to provide successful performance. The fill sediment can be dredged from local offshore locations or nearby inlets, transported from distant offshore locations or mined from inland areas. When the fill sediment is dredged from offshore, borrow pits are usually created. These borrow pits range in size and location offshore depending on the amount of material needed and the location where suitable fill material is found. Recent cases of erosional hot spots (EHS) associated with beach nourishment projects have increased interest in the prediction of the mechanisms by which borrow pits alter the local wave field. An offshore borrow pit can cause four wave transformation processes: wave refraction, wave diffraction, wave reflection and wave dissipation. A better understanding of the interaction of the incident wave field and borrow pit characteristics such as size and depth is needed to determine the effect of the pits on the nourished beach and to anticipate the effects of various borrow pit designs. The problem of waves propagating over a pit of uniform depth has been studied in the past by various methods. Some studies have investigated the reflection and transmission characteristics of a constant depth trench of infinite length located in water of constant depth. More recently the diffraction caused by a rectangular pit of finite shape and constant depth has been studied numerically. Laboratory as well as field studies have been performed to determine the shoreline response caused by offshore dredge pits. 1.2 Objectives and Scope The present study employs numerical, theoretical and laboratory approaches. The numerical analysis uses a boundary element approach for a pit of finite dimensions in uniform depth as in Williams (1990). The fluid is considered incompressible and the flow irrotational allowing the application of potential flow and linearized shallow water wave theory. The fluid regime is divided into two regions; one representing the pit area and the other, the region outside the pit. Using Green's second identity and appropriate Green's functions, the velocity potential and derivative of the potential normal to the pit are found at the interface of the two regions using standard matrix techniques. The velocity potential at any location in the fluid domain can then be determined after a reapplication of Green's Law with the values determined on the pit boundary. An analytical solution to the pit problem is obtained using the general method of MacCamy and Fuchs (1954) with a circular pit instead of a solid cylinder by allowing a solution inside the pit with appropriate boundary conditions. The solution inside the pit is of the same general form as the incident solution. As in the numerical solution the boundary conditions match the pressure and flow across the pit boundary. This analytical solution allows for the determination of the potential at any location in the fluid domain containing a circular pit. For both the numerical and analytical solutions, the free surface elevation is obtained based on the known velocity potential at any point. This provides the basis for the development of a diffraction diagram of relative amplitude over the entire field of interest. The velocity potential can also be used with sediment transport and sediment conservation equations to calculate the shoreline change landward of the pit, as well as the energy reflection caused by the presence of the pit. The numerical solutions to these cases are compared to the analytic solutions for both a pit and a solid structure. Laboratory experiments were carried out to study the wave field modification around a pit and the shoreline evolution behind pits of several platform shapes in uniform water depth. The wave height alteration and shoreline change were compared to the computer model results for different pit depths and sizes as well as different incident wave characteristics. In addition to comparison with the laboratory results, the methodology is illustrated by application to two beach nourishment projects: Grand Isle, LA and Martin County, FL. CHAPTER 2 LITERATURE REVIEW 2. 1 Introduction Wave field modification by an offshore pit has been studied numerically, analytically, and in laboratory and field studies. The purpose of this chapter is to present a broad review of the studies undertaken on the wave transformation caused by pits and the resulting nearshore changes. As stated before, the pit can induce four transformation processes: wave refraction, wave diffraction, wave reflection and wave dissipation. The studies differed in their methodology, the transformation process(es) studied and their conclusions. 2.2 Discussion of Previous Studies 2.2.1 Rectangular Pits of Infinite Length A general analysis of wave propagation over variable depth geometries was performed by Kreisel (1949). The procedure involved mapping the fluid domain to a rectangular strip, which for suitable geometries allows a linear integral equation to resolve the velocity potential through iteration. Matching the solution along the geometrical boundary separating the regions of different depths is a method used by Bartholomneitz (1958) and Miles (1967). The reflection and transmission coefficients for a trench of uniform depth with water depth upwave and downwave from the trench that is uniform, although not necessarily equal, was studied by Lassiter (1972) for the case of monochromatic plane progressive waves. Complementary variational integrals were used to solve for the 4 velocity potential after applying matching conditions on the trench boundaries. Results show reflection coefficients demonstrating a dependency on the dimensionless wave frequency, k*h. The analysis is performed for different wave and trench conditions and shows complete transmission at certain k*h values. Lee and Ayer (198 1) developed an analytical solution for normally incident waves over a rectangular trench of infinite length. The analytic solutions for the region outside the pit and inside the pit are matched and solved numerically. The region outside the pit is considered one continuous strip from negative to positive infinity with the pit region occurring below this for a specified distance and depth. Reflection and transmission coefficients found in the study show an infinite number of discrete frequencies at which waves are completely transmitted, while at other frequencies reflection coefficients of 0.45 can be obtained. The reflection coefficient was defined as the ratio of the reflected wave amplitude to the incident wave amplitude. The results compare reasonably well with those of Lassiter (1972) and show that for a deep trench, the transmission coefficient is a minimum when the pit length divided by the incident wavelength is equal to one-half The transmission coefficient can reach one when the pit length is equal to the wavelength. Miles (1982) studied the diffraction of normally incident, long waves over a trench of infinite length using the mapping procedure of Kreisel (1949). A correction was applied to determine the reflection and transmission coefficients for oblique waves. The ability to determine results for long waves is a feature that Lassiter (1972) and Lee and Ayer (198 1) do not include. The diffraction of waves with large incident angles was studied by Kirby and Dalrymple (1983) using the boundary matching conditions discussed previously and a modified form of the solution used by Takano (1960). For normally incident waves the reflection coefficient results nearly match those of Lee and Ayer (198 1), but show some difference from Lassiter (1972). Much lower values of transmission coefficients are found for an incident angle value of 45* as opposed to normal incidence. This study was extended to include a current flowing along the trench in Kirby et al. (1987). Ting and Raichlen (1986) applied the theory of Lee and Ayer (198 1) to examine the dependence of the kinetic energy in the pit region to the incident frequency and pit characteristics. The goal was to determine the frequency-wise dependence of the induced motion in the trench relating to navigational channels. 2.2.2 Pits of Arbitrary Shape A Green's function approach was used by Willliams (1990) to determine the diffraction of long waves by a rectangular pit. Long waves were studied rendering the problem two dimensional in two horizontal coordinates, where the previous studies were two dimensional in one horizontal and one vertical coordinate. Using the potential and derivative of the potential on the pit boundary and applying Green's second identity allows for the solution of the velocity potential at any point in the fluid domain. The results show a shadow zone created behind the pit with two bands of increased wave height flanking the shadow zone. A partial standing wave system develops seaward of the pit defined by alternating bands of increased and decreased wave amplitudes. For the cases presented in the paper a reduction of ten percent in the wave amplitude is found landward of the pit and increases of 10 % occur seaward of the pit. The long wave restriction of the previous study was removed in Williams and Vasquez (199 1) using a three-dimension Green's function and a double Fourier series expansion as the solution inside the pit boundary. The method of Williams (1990) was expanded to include diffraction by multiple pits in McDougal et al. (1996). The effect of pit width, pit depth, pit length, and angle of incidence on the relative amplitude and location of the shadow zone are investigated for a single pit. The minimum diffraction coefficient is much lower than for the infinite length trench studied by Lee and Ayer (1981) and Kirby and Dalrymple (1983) with the possible explanation that the finite width pit results in a refraction divergence in the lee of the pit. 2.2.3 Shoreline Change by Offshore Pits The shoreline response due to wave refraction over dredged holes was investigated by Motyka and Willis (1974) with a mathematical model. The model used wave characteristics typical of the English Channel coasts of Britain with deepwater angles ranging from 200 to -10'. The beach was found to erode directly behind the dredged hole with increasing erosion for an increasing pit depth and decreasing original water depth. This finding is opposite to that of Horikawa et al. (1977) who found shoreline advancement in the lee of a dredge pit in their model. It is interesting to note that in Horikawa et al. It is stated that the results concur with those of Motyka and Willis (1974), but they clearly do not. The model uses wave data defined for different seasons with six different pit configurations. The results show a salient forming behind the pit with the size dependent on the incident wave values, pit characteristics and duration of simulation. It is suggested that the sand accumulates behind the pit due to the relatively calm water occurring there as a result of the reduced wave action. A study by Gravens and Rosati (1994) examined the EHSs at the Grand Isle, Louisiana Nourishment Project and concluded that refraction around an offshore borrow pit can cause low energy immediately landward of the pit and higher energy at the ends of the pit. This situation results in a salient forming behind the pit as a current, and therefore sediment flow, is created by the difference in wave setup at the edge of the pit and behind the pit. Their approach used the full CERC transport equation with an unusually high value of 2.4 for the coefficient K2 needed to achieve the salient behind the pit. 2.2.4 Laboratory and Field Studies As part of Horikawa et al. (1977) a laboratory experiment was conducted to see if a salient formed behind an offshore borrow pit as in the numerical model of the same paper. Using a fixed-bed model in a small wave basin, an equilibrium beach profile was created under monochromatic, shore-normal waves with no pit and then with the presence of a rectangular pit. The results show the formation of a salient behind the pit for the conditions tested. The shoreline change results compare fairly well to those of their numerical model with larger values of accretion and erosion occurring in the laboratory tests. The reasons for this difference in offshore displacement are theorized as resonant waves particular to the wave tank and wave refraction over the inshore seabed, which was not considered in the model. Two other studies that investigated the effects of offshore dredge pits on the shoreline are Price (1978) and Kojima et al. (1986). Price (1978) examined locations off the coast of England through rods driven into the seabed, tracer studies, and numerical modeling and concluded that sand mining in water depths greater than 46 ft (14 mn) to 59 ft (18 mn) caused little effect to the coast. Kojima et al. (1986) concluded that offshore mining likely played a role in observed beach erosion off the coast of Japan. This analysis was based on examining wind and wave data, shoreline histories, offshore mining activities, sand tracers, and monitoring of dredge pits. CHAPTER 3 THEORY 3. 1 Introduction This chapter presents the fundamental equations used as the basis for this study. These governing equations are used in their respective sections of the study as the foundation of the models for the numerical and analytic solutions. The models employ these fundamental equations to solve the problems of flow over a pit, reflection from a pit and shoreline change induced by a pit. 3.2 Governing Equations 3.2.1 Flow Over A Pit A uniform wave field encounters a region of constant water depth containing a pit of finite dimensions and uniform depth as shown in Figure 3. 1. y a. b Figure 3. 1: Definition sketch for flow over pit Long waves will be studied as the greatest influence of the pit occurs in the presence of these waves. The area of study is divided into two regions. The pit, of depth d, and its projection comprise Region 1 while the rest of the fluid domain, of depth h, is contained in Region 2. The fluid is taken to be inviscid and incompressible, and the flow is taken to be irrotational. Linear shallow water wave theory is applicable and Laplace's solution is valid: V2o = a- 0 +22 (3.1) &2 ay2 aZ 2 with the bottom boundary condition: a_ =0 (3.2) az on z = -h in Region 2 or on z = -d in Region 1. Solving, using separation of variables and 0(x,y, z, t) = X(x)Y(y)Z(z)T(t), a valid solution is Z(z) = cosh(k(h+z)). Inserting this into the Laplace equation leads to: 2 + (3.3) &2 ay2 The depth averaged velocity potential can be defined as: -D = Re(q(x,y)e-i"t) (3.4) where co is the wave frequency. The wave numbers in the two fluid regions are: k, O(3.5) Vgd O k 2 = g- (3.6) The boundary conditions for the interface, F, between the two regions are: 0b1 = 02 (3.7) on F, which equates pressure across the boundary and: d a0_ h 02 (3.8) On On on F, which equates discharge across the boundary. The solution must also meet the radiation condition for large r: rim r,. Jr ik2 )(02 02' (3.9) where the complex incident potential is defined as 0'2 The value of instantaneous free surface elevation may be found using the equations: 1/= i O Ae l(coE) (3.10) g at g A Jre, + 0ib, ag (3.11) tan OiMag (3.12) Real for the 0 values found inside and outside of the pit. 3.2.2 Energy Considerations The amount of energy reflected by a pit is determined using a time-averaged energy flux approach. At each point along a transect running parallel to the incident wave fronts, the depth integrated, time-averaged energy flux and incident energy flux are determined. The time averaging is obtained by taking the conjugate of one of the complex variables: EFuxT =PTrUT =PT~ (* means conjugate) (3.13) where p is the pressure and u is the velocity in the direction perpendicular to the transect: PT = a-P-T (3.14) UT (3.15) vT = -- (3.16) y The total potential is defined as the sum of the incident potential and the reflected potential due to the presence of the pit: OT = 01 + OR (3.17) Therefore the equation to determine the time average energy flux, in the x direction, at a single point due to the incident potential and the reflected potential is: EFluxXT =PTUT =(PI +PR)(Ul +UR) = Pll* + pU* + PRu +PRUR (3.18) The EFluxX T values at each transect point are multiplied by the spacing between the transect points and the water depth to determine the depth integrated, time-averaged energy flux in the x direction through the transect. The time-averaged energy flux in the y direction can be determined using Eq. 3.16 instead of Eq. 3.15 with a transect parallel to the x-axis. 3.2.3 Shoreline Change The principle of an equilibrium beach profile is the foundation for all planform evolution concepts. The continuity equation is used to allow for a relationship between the shoreline position and the change in volume for the profile. The assumptions of straight and parallel bottom contours and no refraction are made. These concepts are used to determine the longshore transport and resulting shoreline change. The wave angle incident on the shore must be known to use the transport equation. A sketch of shoreline change problem is shown in Figure 3.2. N Y P i "lI , /...... / I SShoreline I Shioreline Figure 3.2: Definition sketch for shoreline change problem Two methods were developed to determine the wave angle at the shoreline. The first method uses the energy concepts discussed in the previous section. At each point on the transect the time-averaged energy flux in the x and y directions are determined using Eqs. 3.13 to 3.18. The angle of wave approach at each point is then solved in this way: a-tan EFlUXy (3.19) a tn- ,EFlux,, 14 Using the definition of the depth average velocity in Equation 3.4 the wave angle is found in an alternative way using the following methodology and equations: S= real + i imag = A + iB B A + iB = C cos(a) + iC sin(a) -> A = C cos(a) & B = C sin(a) tan(a) = B A where a is the angle of wave approach. To track a constant phase: a0~ c9 &+ aa y a Oy a = x ay -= -- (3.20) (3.21) (3.22) (3.23) Taking the derivative of t defined in Equation (3.21) with respect to x and y: AoB aA A -B - x Ox A2 AB BOA A -B 2oa Oy O and sec2 a SA2 (3.24,3.25) and combining: C BA A a -B (3.26) where: A = 0,real OA 8_= -" real OA re = "I real B = 1 0imag =B "O imag OB 8 # Ox imag OB imag 2 a sec2 a &a (3.27,3.28) (3.29,3.30) (3.31,3.32) The wave angle is then determined using the formula: a =tan-' d ) (3.33) The value of the longshore transport angle is equal to a +c for the previous derivation. From the coordinate system in Figure 3.2 it is seen that when a is greater than 0, the longshore transport is positive and the transport is negative when c is less than 0. This method provides an angle, but the quadrant of the angle is not determined due to the negative sign in Eq. 3.26. Due to this limitation, only the magnitude of the angle from the first method can be checked. The wave angle and wave amplitude at each point of a transect representing the shoreline are used to determine the longshore transport with the full CERC transport equation: -KH H15 gsin(O-ab)cos(O-ab) -K H'- g1 cos(O -a) d Q= F9K4 2 b K b(3.34 8(S-1X P) 8(s- 1X1- p)tan(y) dy (3.34) where Hb equals the wave height at breaking, ab equals the wave angle at breaking, 0 equals the shoreline orientation, s equals the specific gravity ratio of sand to water, p equals the porosity of the sediment, Kc equals the breaking criteria, K1 and K2 equal the sediment transport coefficients, and tan(y) equals the slope from nearshore to depth of closure. The first transport term is driven by the waves approaching the shore at an angle and the second term, which is sometimes not included, is due to the gradient in the wave heights or setup. Bakker (1971) developed a transport term based on the variation in longshore wave height. This method is outlined in Ozasa and Brampton (1979). The shoreline change is calculated using the formula: ax- Ax = I At (3.35) at y h + B ay h. + B where h, is the closure depth, B is the berm height and a positive change in x-direction indicates erosion. The time step At needed for model stability is determined using: Ay2 Atmx where: Kv f251Z G b H I2 (3.37) 8(s IXI p) h. + B CHAPTER 4 MODELS 4.1 Introduction In this study the problem of long wave propagation over a pit is solved with a numerical solution and an analytic solution. Both the numerical and analytic solution models are compared to solutions for a solid structure. This chapter provides details of the solution method for each model. The numerical model uses the solution of Williams (1990), which applies Green's second identity to points defined on a pit boundary of arbitrary shape. The analytic solution model solves the problem in a manner similar to the MacCamy and Fuchs solution for diffraction by a cylinder, but with a circular pit instead of a cylinder. 4.2. Numerical Solution Model of Williams (1990) The solution model of Williams (1990) determines the velocity potential at any point inside or outside of a pit of arbitrary shape in the fluid domain. The ability to determine the velocity potential at different points is utilized in the models to determine the reflection from the pit and the shoreline change induced by the pit. Also, the Williams solution is modified to allow for the study of solid barriers. 4.2.1 Green's Function Solution of Williams (1990) The governing equations and definition sketch of Section 3.1 provide the basis for this solution method. The complex incident potential is defined as: 0b' (x,y,t) igH e k2(xcosP+ysin 8-wt) (4.1) 2co where H is the incident wave height and 3 is the incident wave angle. Suitable Green's functions for the two regions are defined as: i-H '(kR) and G2 =-H (kzR) (4.2,4.3) 2 2- 0 2 2 0 for Region 1 and Region 2, respectively where H' is the Hankel function of the first kind, zero order equal to Jo (kR) + iYo (kR). The distance between any point in the fluid domain and one on the pit boundary is defined as: R = (x ,) +(y_ .y,) (4.4) with (x,y) defining the location of a point on the pit boundary and (x',y') defining the location of a point not on the pit boundary. At locations of r = r' the Green's functions each have a singularity resulting from Yo(O) = -oo, which must be accounted for. The second Green's function is seen to satisfy the radiation condition. The divergence theorem and Green's theorem are used to achieve the desired solution allowing for determination of the potential at any point in Region 1 or Region 2. The divergence theorem is: J v. V = Vd V -v. ndS (4.5) and is true if V and its partial derivatives are continuous in v and on S. To apply Green's theorem we substitute: V = 4,V2 (4.6) where 0 has been defined previously and 02 is equal to G, both of which are scalar functions of position. Using the first form of Green's theorem: fv[ 01V202 + (V1). (V02)]dV = n.10V2dS (4.7) By interchanging 04 and 02: [V 02V201 + (V02). (V01)]dV = Sn. 02V IdS (4.8) Subtracting the Eq. 4.8 from Eq. 4.7 gives the second form of Green's theorem: JJ.[01V242 -02 V20l]dV = n (01V02 2V01)dS (4.9) If 401 and 02 are both solutions of (V2 + k2)0 = 0 then the left hand side of the previous equation vanishes and we are left with: f n. (01V02 -02V01)dS= 0 (4.10) Accounting for the residue found at the singularity on the pit boundary, F, over Region 1 and using the above equation with 01 and G, leads to: a (r') + (r) (r, r') G, (r, r') (r) 0 (4.11) with c1 = 2 if r' is inside F, a, = 1 ifr' is on a smooth portion off and a, = 1/2 ifr' is on a corner point ofF. Applying 02 and G2 to the same equation gives the following equation: a27r2 (r') = [02(r) ~2-(r,r') G2(r,r') -O (r) + 27r (r') (4.12) with 82 = 2 ifr' is inside F, a2 = 1 ifr' is on a smooth portion off and U2 = 3/2 ifr' is on a corner point ofF. In these equations r is the location of a point on the pit boundary and r' is a point inside or outside of the pit. In order to solve numerically for the potential inside and outside of the pit, 4 and LO on the pit must be determined. This is achieved by setting r' on the pit boundary and an using the boundary conditions stated earlier. This procedure leads to two equations that can be solved numerically with standard matrix procedures after discretizing the pit boundary into a specified number of segments: al52 (r') + 02 (r) -C (r, r') hG, (r, r') -(r)lF 0 (4.13) 7C I- a d an a2, 02 (r') -l 2(r)"G2-(rr) -G2(rr') 2 (r)]dF 20 (r') (4.14) where the integrals are replaced by summations. The above equations lead to values of 0 and LO on F that are assumed constant for the length of each segment. Knowing the an values of 02 and 02 on F allows for the potential at any point in the fluid to be ,On determined with the previous solutions. The singularities inherent in the Green's functions are accounted for in the a values which must be determined for each point, r', in the fluid domain. The value of instantaneous free surface elevation may be found using the Eqs. 3. 10 through 3.12. 4.2.2 Wave Height Reduction for Pit of Arbitrary Shape A numerical model was developed to solve the problem of diffraction around a pit of uniform depth, in water depth which is otherwise uniform assuming linearized shallow water wave theory. The model follows the solution of Williams (1990) and obtains results that are very similar. The basic program results in plots of relative amplitude in and around the pit. The user defines values for: beta angle of incident wave approach in radians h water depth around pit in meters d depth of pit in meters T period of incident waves in seconds H uniform height of incident waves in meters x x values defining pit in meters, entered in counterclockwise motion around pit y -y values defining pit in meters, entered in counterclockwise motion around pit nump number of points defining pit in calculations, greater number gives better accuracy totalpoints number of points in grid where potential is found in fluid domain, must be a square number C multiplier for grid boundary, grid size is C*wavelength (out of pit) on each side of pit The program calculates values of wavelength and wavenumber inside and outside of the pit using linearized shallow water theory. Four sub-programs are used in the main MATLAB program during the execution of the models; these programs can be obtained by contacting the author. The program outputs the velocity potential at each point defined in the fluid domain. Knowing the complex value of the potential at each point in the grid allows for the calculation of the instantaneous free surface elevation at each location using Eqs. 3. 10 through 3.12. The relative amplitude (diffraction coefficient) of the grid locations is taken as: Are R (4.15) H1 Several checks were performed on the numerical model. The values obtained are very similar to those of Williams (1990) and McDougal, et al (1996), both of which use the same theory and primary equations as this model. The values of 0 and on the pit an boundary and the diffraction pattern are found to be symmetric with the x axis for a symmetric pit with normal incident waves, as would be expected. A pit depth, d, equal to the surrounding water depth, h, is found to create free surface elevations that are found with no pit present. The program results check with the boundary conditions. 4.2.3 Reflection from Pit This model uses the values of potential along a transect parallel to the y-axis to calculate the time-averaged energy flux in the x direction at each point along the transect and the sum of the energy flux along the transect. The methodology of Section 3.2.2 is followed for these calculations. Using the definition for the velocity potential in Eq. 4.1 the real values of the incident pressure and velocities in the x and y directions are determined with: S= pgH cos[kxcos(O) + kysin(O) cot] (4.16) at 2 Using the identity cos(a f) = cos(a)cos(f) + sin(a)sin(f,8), P, = pgH {cos[kxcos(f) + ky sin(f)]cos(wcot) + sin [kxcos(f,) + ky sin(f8)]sin(COt)} (4.17) 2 The final expression for the real value of the incident pressure is: P, = PIcos(ot-sC1) (4.18) with IP, = S2 +S and p, = tan' ( where: \Sl S, = gH {cos[kx cos(O) + ky sin(O)]} and S2 p gH {sin [kx cos(O) + ky sin(O)]} (4.19,4.20) 2 2 The real value of the incident velocity in the x direction obtained using: U gH k cos(0) cos[kx cos(0) + ky sin(0) cot] (4.21) ~x 2 o Using the same methodology as in the pressure calculation the final expression for the real value of the incident velocity is: U1 = |U1 cos(t Cu) (4.22) with UI = SJ +S2 and E~u, =tan-' i2 ) where S1 and S2 are determined using Eqs. 4.19 and 4.20 with a coefficient of gHk. Taking the derivative of the velocity 2c potential at each point with y instead of x and following the procedure just described determines the value of velocity in the y direction. The model returns values of complex velocity potential at each point along the transect parallel to the y-axis. The reflected pressure and velocity at each location are determined using the value of the total velocity potential at each point. ORtotal = A,, +i B,-? '" (4.23) where An and Bn are the values determined from each point on the pit boundary plus the incident potential if the point is outside the pit. PR = -P O = imp- A, + i B,, "' = oP(- B, + i A,, (4.24) The final expression for the real value of the reflected pressure is: PR = CCOp cos(Ct eRP) (4.25) where C= ZB2 + A2 and eRP = tan Z, The real value of the reflected velocity in the x direction is determined as: UR n_ (real)+ i (imag)ei (4.26) 8x 8x 8x where the summation of the complex derivative of the velocity potential with x at each point is taken. The equations used to determine "L_ are shown in Section 4.2.4. The 8x final expression for the real value of the reflected velocity is: UR = C cos(ot 8RU) (4.27) 22 a (imag) where C = (real) + (imag) and eu = tan' a ao (real) The total velocity potential is determined from Eq 3.17 and the time-averaged energy flux for each of the four quantities in Eq. 3.18 is determined from: UP = cos(c', P) (4.28) 2 4.2.4 Shoreline Change Induced by Pit This model uses the values of potential along a transect parallel to the y axis to calculate the relative amplitude, wave angle, longshore transport, and shoreline change along the transect. The methodology and inputs from the main numerical model are used except that there is no grid, just a transect of points, and instead of only calculating the potential, 0, values of ,- and .- are determined for each point along the transect. ax Sy The calculation of 0, and are performed in a new program. This program calculates and at each point along the transect. The same method as the main ax y program is used for calculating 0, with the same results. The values of are found ax starting with the Eq. 4.12: 02 (r')= f 2(r)G2 (r, r') G2(r r') 2 (r)]dF + 27rff (r') (4.29) an \I jn an Taking the partial derivative with respect to x' gives: c8 1 8O SG 0 (r') = 0 ,2(r)- (r, r')-G2 (rr') (r)]dF + 27r02 (r') ax a2 7C JL'r an I a2 (r) a2 a(r) G2 (r, r') G2 (rr') 2 (r)]dF + T0 1 (r') x a2 L- a2, "ax n (4.30) This equation can be broken into three parts, with part A being the partial derivative with respect to x' of the incident potential: A: a (r')= ik2 cos(8)q 0(r') (4.31) The second part is the partial derivative with respect to x' of the Green's function: BG i7 (x x') B: -(k)[J (kR) + iY (kR)] (4.32) ax 2 R The last part is the partial derivative with respect to x' of the derivative of the Green's function normal to the pit boundary. This derivative must be expanded to accommodate any pit orientation: a aG a( aG aG aa G a aG -a = a +b = a L +b I (4.33) -x' Ln y' B yx &' y. where a and b are direction coefficients dependent on the angle the pit boundary makes with the x axis at that point: 8_G_ _G BR 8 br 8 ax a a aG Rand b G = b a G) OR (4.34,4.35) ax x' R x )-x x -S0-y bR y &' 8 G is k( 2 x' 4.6 a( a i k2(x-a x')[J2(kR)+ iY2(kR)] (4.36) KR x' 2 R2 aR (x- x') - (4.37) ax R b =-b k2(yy')[J2(kR)+ iY (kR)] (4.38) bR y 2 R 2 The resulting equation for last part is: a aG is k 2 (xx' C: a ka2( 2 X [J2 (kR) + iY2 (kR)][a(x x') + b(y y')] (4.39) x' an 2 R' The values for are found in the same manner with the resulting equations being: 8 1 8 BG 2 # a02 (2 G2L (r, r') G2 (r, r') -(r)]dF]r + 2-0 (r') (4.40) cy a 2 r ay an A: a 6 (r') = k2 sin() 0 (r') (4.41) B: G (k)[J (kR) + iY(kR)](y-y' (442) ay' 2 R C: G ixz k 2 Y)[J2(kR) + iY2(kR)Ia(x- x')+b(y- y')] (4.43) y' an 2 R With these equations the program computes the integral as the summation of the contribution from each point on the pit times the spacing between the points. The program returns the value of 0 the potential at each point in the transect, the derivative of the potential with x' at each point, and -, the derivative of the potential with y' at each point. The shoreline change is determined using these values and the equations in Section 3.2.3. 4.2.5 Models for Solid Structure of Finite Dimensions A solid structure is modeled by changing the boundary conditions at the pit to have a no flow condition at the pit border. By making c- = 0 on the border the equation c "n to solve for 02 and the matrix solution from the original numerical model is changed. The equation used to solve for q02 is, 02(r') = L [02(r) (n(rr') ]d+ 27rb(r') (4.44) In order to solve numerically for the potential outside of the pit, 0 on the pit must be determined. This is achieved by setting r' on the pit boundary and using the boundary conditions stated earlier. This procedure leads to two equations that can be solved numerically with standard matrix procedures after discretizing the pit boundary into a specified number of segments: a102(r')+ I 02(r)-O1- (r, r')j 0 (4.45) a2 02(r')-l[ f02(r) 2(r,r')-frF: 20'(r') (4.46) "n 2 This leads to values of 0 on F that are assumed uniform for the length of each segment. Knowing the values of 0, on r' allows for the potential at any point in the fluid to be determined with the previous solutions in the same procedure as in the main numerical model for the pit. 4.3 Analytical Solution Model for Circular Pit An analytical solution was developed for the problem of diffraction by a circular pit of finite dimensions. This problem and solution are similar in form to that of diffraction around a cylinder solved by MacCamy and Fuchs (1954). The solution method involves defining an incident and reflected velocity potential outside the pit as well as one inside the pit. An analytical solution model is developed that determines the velocity potential at any point inside or outside of the pit. Using these values of velocity potential, the reflection from the pit and the shoreline change induced by the pit are determined using other models. 4.3.1 Wave Height Reduction for Circular Pit The setup for the problem is the same as shown in Figure 1, but with a circular pit. The methodology and inputs from the numerical model are used except that there is no real pit boundary with points defined on it and therefore no calculations of 0 and LOon the pit. The value of 0 is calculated directly from the final equation for each point on the grid and the free surface elevation is then determined. This solution is for a circular pit of radius, a, and uniform depth, d, in water of otherwise uniform depth, h. The velocity potential outside the pit is the addition of the incident wave, 0, and the reflected wave, 0,R 0, = M1 L,8. cos(mO)Jm(kr)e (4.47) Im=0I - IgH where Pm = 1 for m = 0 and 2im otherwise, M r is the distance from the point in 2wo the fluid domain to the center of the pit and 0 is the angle between the two points measured clockwise from the positive x axis. The reflected velocity potential is: = Amcos(mO)[Jm(kr)+iY,,m(kr)] e-1't (4.48) Im=0 o The solution inside the pit is defined as: -,, = IBm cos(mO)Jm(kr) e-It (4.49) The boundary conditions that must be met are the same as Equations 3.7 and 3.8, except on r equal to the pit radius, a. Ains = out r=a (3.7) h aOJ = d o= (3.8) B^r B r r~a T=Gr Using this boundary condition, the solutions for Am and Bm for m equals 0 to oo: k2h J' (k2 a)Jm(ka) MI,8 J,, J. (k 2a Am M-Jk~d J(ka) (4.50) Am = (4.50) [Jm(k2a) +iY,(k2a)- Jm (k a) k2h [m(k2a)+ iYm(k2a)] J.'(kza) kd Bm = Jm (k2 a)Mfi + Am [Jm (k2a) + iYm(k2a)] (4.51) Jm (ka) The values for Am and Bm are inserted into the equations for j,, and 0,, equal to + ,bR. Knowing the complex value of the potential at each point in the grid allows for the calculation of the instantaneous free surface elevation at each location using equations 3.10 through 3.12. The relative amplitude of the grid locations with the pit and without the pit is taken as: H Arel = HR (4.15) HI 4.3.2 Reflection from Circular Pit The energy reflection from the circular pit follows the methodology of Section 3.2.2 and equations 3.13 through 3.18. From the definition of the incident and reflected velocity potentials, the pressure at any point outside the pit can be defined as: pl = pdI (iW) L ,. cos(mO)J,(k2r) e(-i't) (4.52) PR = p(iCO) Am cos(mO)(Jm (k2r) + iY (k2r e(-itw) (4.53) Im= o A polar coordinate system is used in this solution, which makes some algebra necessary in order to determine the velocity in the x-direction. The velocity in the xdirection is obtained using the following equations: u = vel, cos(0) velo sin(O) (4.54) where velr = and velo 1= I Or r 0O Using these equations the equation for the incident and reflected velocities in the u-direction are determined as: u, = Mk2 cos(0) m cos(mO) Jm+1 (k2r) + Jm(k2r e(-'iwt) Smm sin(mO)J (k2r) + sin(0) m8 sin(m)J(k2 e(-iot) (4.55) r ,m=o Im (J (- +1 uR =k2 COs(0) A. cos(m) Jm,, (k2 r) i'+l+ (k2 r) (k2 n)+ (k2 (r)) e( -it) + sin(O) inmA,, sin(mO)(Jm (k2r) + iYm(k2r) e(-iat) (4.56) r Er= o The depth integrated, time-averaged energy flux in the x direction at any point along the transect is obtained from: EFluxT = PTu = (pI +ps)(u1 +us) = PIuIu +PIZS +psU +psUs (3.17) The EFluxT values at each transect point are multiplied by the spacing between the transect points and the water depth to determine the x-directed energy flux through the transect. A different solution for the energy flux is found inside the pit due to the lack of an incident velocity potential there. The energy flux inside the pit is obtained using: p,,, = p(i CO Bm cos(mnO)Jm (kr) e( "t) (4.57) m=0t ui = k cos(O) B. cos(nO) -Jm+ (kr)+ J,(kir) e(i) + sin(0) L1m sin(mO)J (k,r) e(-it) (4.58) r ,,=o The value of the velocity in the y direction is found using: v = vel, sin(O) + velo cos(O) (4.59) 4.3.3 Shoreline Change Induced by Pit The shoreline change induced by a circular pit follows the methodology of section 3.2.3 and equations 3.19 through 3.37. Values of and L are needed for this model &x Sy for equations 3.29 through 3.32. In the preceding section the equations for ui and uR are given in equations 4.54 through 4.56. Equation 3.15 is used to determine from u. Ox The following equations are used to determine the velocity in the y direction: si()I~ rmjje 1] t v1 =MIk2 sin(O) o t cos(nO) -m+J, (k2r) + J,(k2r) e(-i -cos(O)M' 1 sin(mO)J.(k2r) e(-i t) (4.60) Sm=O VR =k2 sin(O) Am cos(m9)- J,,, (k2m) + -i (k2r)+ (k2r) iYm(k2r))e( t VR "1=0k,r cos(0) mA sin(mO)(J(kr)+iY,,(k2r)) e(-i ) (4.61) r m=0 The value of -9 is determined as: ay 0 = v (3.16) 4.4 MacCamy and Fuchs Solution Model for Cylinder This model uses the solution for the diffraction around a cylindrical object first solved by MacCamy and Fuchs (1954) for general depths. The setup for the problem is the same as shown in Figure 1, but with a solid cylinder. The methodology and inputs from the analytic solution model are used. The value of 0 is calculated directly from the final equation for each point on the grid and the free surface elevation is then determined. This solution is for a circular cylinder of radius, a, in water of otherwise uniform depth, h. ('AehT/lo .1,,,(k~r)(J,,(k~r)+iY,,(k2r))]" (. ic + ref = 0 = MI If,, cos(mO) J,, (k2r)- (Jm(k) +iim(k2)) (4.62) n= o(1m (k2a) + iYm(k2a)) 1) where r, 0, and M, have been previously defined. Knowing the complex value of the potential at each point in the grid allows for the calculation of the instantaneous free surface elevation at each location using equations 3. 10 through 3.12. The relative amplitude of the grid locations with the pit and without the pit is taken as.A rel -_ HR (4.15) HI Values for the reflection caused by a cylinder and the shoreline change induced by a cylinder can be obtained using the same methodology used with the analytic model. CHAPTER 5 RESULTS AND DISCUSSION 5.1 Introduction Many trials were run to verify and analyze the models that were developed. The numerical model for an arbitrary shaped pit based on the theory in Williams(1 990) was compared to the results of that paper to verify the current model. After demonstrating good agreement between the developed numerical model and that of Williams (1990) the analytic solution model developed for a circular pit was compared to the results found with a circular pit using the numerical model. These results for a circular pit are then compared to the wave field alteration by a solid cylinder of the same dimensions based on the theory of MacCamy and Fuchs (1954). The wave field alteration by a solid cylinder is also found using a modified form of the numerical model with a no-flow boundary condition allowing for the modeling of any arbitrary shape as a solid entity. For some of these models the energy reflected by the pit or cylinder as well as the shoreline change induced by the pit is determined and compared for the different models and for different incident wave and pit considerations. 5.2 Wave Heigzht Reduction Models 5.2.1 Comparison of Numerical Model Results with Results of Williams (1990) The relative amplitude in the fluid domain surrounding a pit of arbitrary shape is determined in a numerical model based on the solution of Williams (1990). The base conditions for the following trials are a water depth, h, of 2 m, a pit depth, d, of 4 mn and incident waves with 3 equal to zero and a period of 12.77 s (kh = 0.22). For this trial the pit is comprised of 120 points and a grid of 1600 points defines the fluid domain. Figure 5.1 shows a contour plot of the relative amplitude for these conditions and a square pit (8x8 in). A partial standing wave is seen to develop in front of the pit with a significant area of wave sheltering behind the pit. Two "lobes" of increased wave amplitude are seen to project out behind the pit at an approximate angle of 30 degrees. . .... ..:.... . . 4010 ............... ..... .......... 30~ ~ ~ .... .1. . ' I:::i::i::ii:i::iiiiiiii ii 0 99; iiilliiiiiiiii ~~~~ ~i : .:;..! ....................................... . .........................: .o N : : .:.: .. . . 20 ......... ... 20 -20 .1.0..0.10..20 30 ~.9 ..:.:...:.:. ,... .......... 4 -2 0~ ........ ...... .. :-- .N . ''':::::::::::: .. .:.:.:.: ..: ...: :.......... .- .:: ::: ::::,,,.+ . ...,.. . ? "............. .......... i i i i i i......... -20 ..........2. 0 4 X-Direction (in) Figure 5.1: Contour plot of relative amplitude from numerical model for 8 by 8 m pit and base conditions with location of pit drawn The maximum increase in relative amplitude is found directly in front of the pit and is on the order of 10%. The shadow zone directly behind the pit encompasses the minimum relative amplitude, which is also on the order of 10%. Figure 5.2 contains the plot from Williams (1990) for the same model parameters. ...... .........~r~.:: ... : Figure 5.2: Contour plot of relative amplitude for =0', kid 76m10, k2d 7L/1IM, h/d = 0.5, b/a = 1.0 and a/d = 2.0 equal to 8 by 8 m pit with base conditions; from Williams (1990) The two figures are in close agreement, as they should be since the same theory and formulation were used in the development of both models. Comparing Figures 5.3 and 5.4 provides further verification of the numerical model. Figure 5.3 is a contour plot with the base conditions except that the pit is 8 by 24 m and the incident wave angle is now 45 degrees. Once again the alternating bands of increased and decreased relative amplitude are found seaward of the pit, although offset by the incident angle for this trial. As in the first trial, there are two lobes of increased wave amplitude that border the shadow zone behind the pit. The magnitudes of the maximum and minimum relative amplitude are seen to be larger for this trial, than for the previous one due to the increased pit size. .......................... ..0 .. ... .. .. ... .. .. ...............~ . 40 11*1 20.......... 20 ........ ......... 1 ... .. ...... 40. 20-2x4 0 8 .N .~ ~ ~ .......... 11 .... Figue 5. Conour lot f reativ ampitud fro numrica modl.fo.8.b. 24 pi with 3 = 50 ad-oter Se odtoswt oaino i rw The rsult fromWillams (990)for tese odelparamtersare.sen.i.Figue.5.,.an one gan oo areMen ewe h eut seiet A ~ ~ ~ ~ ~ ~ ~ ~~~~~~~...... thruhinetgtino.heefc.f.h.i.dmninsadicietwv condiionson te difracton ceffiient(relaive mpliude)is.fond.i.Mc.uga.et.a (1996. Inthispape themethd ofWillams (990)and he crren.numrica.modl.ar used o detrmin the ariaion o the inimm difracton.cofficint.wth.th.non ~1.05 0 ~3 dimensional values of pit width, pit depth, pit length and angle of wave incidence. The variation in the location of a defined wave shadow zone is also determined for these four parameters. The minimum diffraction coefficient is found to generally decrease with increasing dimensionless pit width (a/L) and dimensionless pit depth (d/L). The minimum diffraction coefficient is found to decrease for dimensionless pit lengths (b/L) of 0.1 to 0.55 and then increase for values of b/L near 0.6 and then fall again as the dimensionless pit length increases to 1. T'1 .............:..:,..+.:::: Figure 5A4 Contour plot of relative amplitude for j3450, kid 7c/10, k2d nA/M,1 hid = 0.5, b/a = 3.0 and aid = 2.0 equal to 8 by 24 mn pit with 13=450 and other base conditions; from Williams (1990) Only a slight variation in the diffraction coefficient is found when varying the angle incidence between 0 and 45 degrees. McDougal et al (1996) point out that the diffraction coefficients found with the numerical models are much smaller than those found in the two-dimensional cases of an infinitely long trench as in Lee and Ayer (1981), Kirby and Dalrymple (1983) and Furukawa(1991). A possible explanation is given that the finite with pit results in a refraction divergence in the lee of the structure; with the divergence still present as the pit width increases. 5.2.2 Comparison of Numerical Model with Analytic Solution Model With confidence that the numerical model accurately determined the velocity potential and therefore wave amplitude in the presence of a pit, it was next compared the analytic solution model for further verification. Defining a circular pit in the numerical model allowed for the direct comparison of results for the numerical model and the analytical solution model. For the numerical model a pit was created with 72 points at equal arc length spacing. Increasing this to 360 points resulted in almost no difference in the model results. These points defined the skeleton of the circular shape and next a certain number of equally spaced points were placed between these to more accurately represent a circular boundary. Figure 5.5 shows the results for the numerical model of a circular pit with a radius of 12 in, 200 points defining the pit boundary and 1600 points in the grid. A period of 12 s is now used as the base period in the following model results. The center of the pit is the origin of the coordinate system and the base conditions for the water depth, pit depth and wave criteria are used. The features seen in Figure 5.1 are repeated with the partial standing wave in front of the pit, and the lobes of increased amplitude behind the pit bordering the shadow zone of decreased relative amplitude. The 40 wave field alteration due to the pit in Figure 5.5 is seen to be larger than in Figure 5.1 partly due to the larger pit area of the circular pit. ... .... I : .... I:: ......../ .. ~............. 0....... ... ......... . ..... ......:......::i iii- . .... A11.4 _0 ]ii:.... ..: :::::::::::::: ............=: .... ... ...... 40:-::':' . .. 14 . ............................ ........ x ........... ..... ..7 100 80 60 40 20 0 20 40 60 80 100 X, Di rectlion (mn) Figure 5.5: Contour plot of relative amplitude from numerical model for circular pit with r = 12 m, 200 points defining the pit boundary and base conditions with location of pit drawn The analytic solution model results for this trial are found in Figure 5.6 with the series summation taken as the first 80 terms. The relative amplitude values are found to be very close to those of the numerical model. The percent difference between the two ReA rn l m ena, Re lAmpan,y 1 0 Th models results is shown in Figure 5.7 as 'MalO* The Re lAmpnuenca, percent error is seen to vary from 1% to -6% with the error decreasing as the number of points defining the pit in the numerical solution is increased. These low values of percent error indicate good agreement between the two models for this trial. [I .... 1 X-Dtrection rn Figure 5.6: Contour plot of relative amplitude from analytic solution model for circular pit with r = 12 m, 80 terms taken in the series summation and base conditions with location of pit drawn 44) "" -20 -. -0 .Go -410 420 0 20V X-0Dg-ecion (m2 Figure 5.7: Contour plot of percent difference for numerical model and analytic solution model for circular pit with r = 12 m, 200 points defining the pit boundary in the numerical model, 80 terms taken in the series summation in the analytic solution and base conditions with location of pit drawn Another way of comparing the two model results is by taking a transect across the fluid domain and comparing the relative amplitude values for the different models along the transect. Figure 5.8 shows a representative transect taken parallel to the x-axis for the pit with a radius of 12 m. The spacing along the transect is defined and the velocity potential and therefore free surface elevation can be determined at each point along the transect. -150 -100 -50 0 X-Direction (n) 50 100 150 200 Figure 5.8: Plot of transect taken parallel to the x-axis at y = 0 for pit of radius 12 m showing wave direction (3 = 0) A comparison of the relative amplitude for the previous case is shown in Figure 5.9 for the transect in Figure 5.8. The results of the numerical model and the analytic solution model are found to be in good agreement with the numerical model values )0 50. Wave Direction 0 -50 -200 ....... -- ----------- z ---- --- .... --- ...... ...... ------------- - -- -- .... -- -- -------- I ---- --1-1.1 ... ... -------- slightly higher than those for the analytic solution. The numerical model produces scattered values for points very near the pit boundary, due to the singularity that occurs there in the solution method. 1 .3 ...... .......... --------------------.. --------------------..I ....... ............ ..................... --------------------.... ................ ..................... solid Analytic Model 1 .dashed Numerical Model Pit 0.8J 0.7 - I I I) 00 -150 -100 -50 0 5 100 150 200 X-Direiction (m) Figure 5.9: Relative amplitude for numerical model and analytic solution model for transect show in Figure 5.8 for pit of radius = 12 m with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with pit drawn The models are further compared in Figure 10 with 3 transects taken perpendicular to the direction of wave propagation at X equal to 0, 24 and 100 m. The relative amplitude values for the two models are very similar along all 3 transects with the numerical model having noticeably smaller values directly behind the pit for the X= 0 m transect and also to a lesser extent for the X =100 m transect. In this plot the primary 44 lobes of increased wave amplitude are seen to spread and increase as x increases with the oscillations of increased and decreased relative amplitude growing smaller as the ydistance from the pit increases. X = 0 (m) 300 ........--------------- ................. 200100 io:o " Pit -100~ -200. -300: 0.8 1 1.2 Relative Amplitude -100 -200 X = 24 (m) .. . .r . . . . ." .................. ( - -- - t 5' 0.8 1 1,2 Relative Amplitude X = 100 (M) 3 0 0 ......... ................--------- .............. 300: ..) 200 100.. 0 -100 (.......... -200::-300: ( 08 1 1.2 Relative Amplitude Figure 5.10: Relative amplitude for numerical model and analytic solution model for 3 transects parallel to the y-axis at X = 0, 24 and 100 m for pit of radius = 12 m with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with pit drawn The reduced amplitude behind the pit is seen to decrease as the distance behind the pit increases. With the good agreement between the models for a representative small pit, a second trial was conducted with a pit radius of 75 m to better represent the size of a pit found in the coastal environment. The wave field modification caused by a pit of this size with the predefined base conditions is seen in Figure 5.11 with the summation of the first 110 terms taken in the analytic solution. The magnitudes of the maximum and minimum relative amplitudes are 1.71 and 0.058 respectively. The relative amplitude increase and reduction are substantially larger than those found with a circular pit of 12 mn radius. 1 4 -300 -200 -100 0 Y-Direction (in) -100 2100 300 Figure 5. 11:- Contour plot of relative amplitude from analytic solution model for circular pit with r = 75 mn, 110 terms taken in the series summation and base conditions with pit drawn The maximum relative amplitudes are found in the lobes of increased amplitude behind the pit. The extremely low values of relative amplitude are found just inside these lobes in the shadow region behind the pit where an area with relative amplitude values less than 0.2 occurs. The partial standing wave pattern that develops in front of the pit has oscillations of large magnitude, but they are not strongly evident in the contour plot due to the even larger values occurring behind the pit. They will be shown later through transect plots. The contour plot of the numerical model results with 600 points defining the pit is similar to that in Figure 5.11. The increased number of points on the pit was used to allow for spacing between points on the pit that was similar, although larger, to that in the previous trial of 200 points defining a pit with a radius of 12 m. Once again, the relative difference was calculated to compare the values from the two models at each grid point; this can be found in Figure 5.12. 300~~ ~ ----------------:i -. ; -'* -- ----------, ---Y ---"-:: -- .... .-------.. ... ......".........-.. ... -20 2ooI 1 100\ -3 0 ;9 3 YeD ....... '57 -6 .... -.............. ..... -200.': 'N, ::..k,\ \.: ,. u : :::. : 2 .k O -200 -100 0 300 X-DIrection (m) Figure 5.12: Contour plot of percent difference for numerical model and analytic solution model for circular pit with r = 75 mn, 600 points defining the pit boundary in the numerical model, 1 10 terms taken in the series summation in the analytic solution and base conditions with pit drawn The most noticeable feature of this plot is the large percent error values occurring on the edge of the shadow zone behind the pit. At one of these two points the value of the relative amplitude from the analytic model is 0.05 8, while for the numerical model it is 0.029, which results in a percent error over 100%; however, the large error is due to division by a very small number (0.058) and the validity of the value there is questionable. For most of the plot the percent error is found to be less than 5% and only 16 points in the grid of 1600 points show a percent error great than 10%, which indicates good agreement between the two models. As shown before, transects through the fluid domain are another tool useful in comparing the results from the two models. A transect of relative amplitude values taken parallel to the x-axis is shown in Figure 5.13. 1.2 ....................... ....................... r ................. .. r ... .................... r ....................1.j solid = Analytic Model S 1 dashed -NumericalModel T. T't] o0.7 0 .4 ------------------------ ------------- .................. ............... L ..................... L .............. ". .. ................... ; ........................ -400 -30 -200 -100 0 100 200 300 X Direction (m) Figure 5.13: Relative amplitude for numerical model and analytic solution model for transect parallel to the x-axis at y = 0 m for pit of radius = 75 m with 600 points defining the pit boundary in the numerical model and 100 terms taken in the series summation for analytic solution with pit drawn The numerical model solution is seen to result in smaller values in front of the pit that are slightly out of phase with the analytic model, larger values inside the pit, and good agreement between the models behind the pit. Two noticeable spikes in the numerical solution are seen at the pit border and occur due to the proximity of the point to the pit boundary where a singularity occurs. The large size of the pit leads to waves forming inside the pit where the relative amplitude is less than 1, but large oscillations are seen. Taking a transect parallel to the y-axis located just behind the pit at X equal to 100 m results in Figure 5.14. a) 1.2 <1 r 0.8 ' i I II I iJ ~ .I 11 -400 -300 -200 -100 0 Y Direction (m) 100 200 300 400 500 Figure 5.14: Relative amplitude for numerical model and analytic solution model for transect parallel to the y-axis at X = 100 m for pit of radius = 75 m with 600 points defining the pit boundary in the numerical model and 100 terms taken in the series summation for analytic solution with pit drawn 0.21 -50( 0 This plot shows good agreement between the models and highlights the large gradient in the relative wave amplitude occurring behind the pit where the lobes of increased amplitude border the shadow zone. The shadow zone shows two spikes of very low wave amplitude (- 0.2) bordering the shadow zone where relative amplitudes of 0.6 are seen. The rapid decrease in the oscillations of the relative amplitude with distance from the pit is also noticeable. Taking a transect at a greater distance behind the pit shows the spreading of the lobes on increased wave amplitude concurrent with spreading in the wavelength of the oscillations of relative amplitude. Figure 5.15 shows a transect taken parallel to the y axis at X equal to 500 mn, with almost no difference noticeable between the two models. 1.41 JI 12 P I n 0.6II 1 I f .4~ 1 ___0 -------------............. ... ....................... ... ... ... ...... ... ..... ... .......... Y6 Diet ( m) Figue 515:Reltiv amlitue fr nmercalmodl ad anlytc sluton ode fo transect ~ ~ ~ ~ ~ ~ paale to th -xsa 0 nfrptofrdu 5i ih60pit deiig h i budryi h nmrcl oe ad10tem a0ni4hesre Fiure515meati ve amltdfornmrclroe n analytic solution model fordaw A decrease in the maximum and minimum values of the relative amplitude are seen in this transect when compared to Figure 5.14. 5.2.3 Results for Models with Solid Cylinder The wave field modification by a solid structure was analyzed by employing the solution of MacCamy and Fuchs (1954) for the diffraction caused by a circular cylinder. This model is similar to that for the analytic solution model for the circular pit, except with a no-flow constraint for the boundary condition. Figure 5.16 shows a contour plot of the relative amplitude for a cylinder with a radius of 12 m and the base conditions without a pit depth, d. 1.6 15 14 1.3 1.2 1.1 .. .. . -20 0 20 x Direction (m) Figure 5.16: Contour plot of relative amplitude from MacCamy and Fuchs solution model for a circular cylinder with r = 12 m, 80 terms taken in the series summation and base conditions with pit drawn A partial standing wave developed in front of the cylinder, as occurred for the case of the pit, however these oscillations are 90 degrees out of phase with that of a pit. The contour pattern behind the cylinder is seen to differ from the case of the pit as the areas of minimum relative amplitude are not directed out from the pit along the x-axis, but project out at an angle. The maximum relative amplitude shows a 60% increase in the wave amplitude in front of the cylinder and the minimum relative amplitude indicates a 40% reduction both behind the pit and in the first trough in front of the pit. These relative amplitude values are much larger than those found for a pit of the same radius. By taking a no-flow boundary condition in the numerical model, a solid structure numerical model was developed. The results of this model for the previous case compare very well with those of the MacCamy and Fuchs solution. The percent difference between these two models for this trial can be seen in Figure 5.17. In this numerical model, 200 points define the cylinder boundary. The contour plots shows less than 1% error for each point in the grid with the largest errors occurring directly behind the pit and in two lobes of increased amplitude projecting out behind the pit. Transects taken through the fluid domain compare the values obtained by the two models and illustrate the effect of the cylinder on the wave field. Figure 5.18 shows the relative amplitude for the two models for a transect taken parallel to the x-axis. Included in this plot are the relative amplitude values for the analytic solution for a circular pit of the same radius, seen in Figure 5.9. The results of the two cylinder models are so close that they appear as a single line except for the numerical model values directly in front of and behind the pit. The results show a large value of relative amplitude directly in front of the cylinder and a reduction in relative amplitude, which diminishes rapidly behind the /..,........, .. .. . o 0.1 -20 -40 ... .. .............. -0 -0015 ~::::::.. .- .... ... -I O0 .0 .. 60 -4. -. .. .. 40 -100 -80O -60 -40 -20 0 20 4C 60 80 104y x Direction (m) Figure 5.17: Contour plot of percent from from solid numerical model and MacCamy and Fuchs solution model for a circular cylinder with r = 12 m, 200 points defining the pit boundary, 80 terms taken in the series summation and base conditions with pit drawn -----------------..------------------------------------------------------------------------------------. ----.. --------------------- ---------------------------------........... dashed Cimru ar Pit *. .4 -200 -150 -100 -0 0 50 1--0 1 --50 200 X DiIOCO:Jon (-) Figure 5.18: Relative amplitude for solid numerical model and for MacCamy and Fuchs solution model for transect parallel to the x-axis at y = 0 m for pit of radius = 12 m with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with analytic solution model for circular pit shown pit. The plot shows the phase difference between the case of a pit and that of a cylinder. The pit is seen to result in a greater sheltering effect directly behind the pit. The large value right at the cylinder boundary due to the no-flow condition is contrasted by the reduction in wave amplitude found at this location for the pit. A transect taken parallel to the y-axis at x equal to 100 meters is shown in Figure 5.19. Once again the values of the two cylinder models are so close they appear as one line. 0 Y Direction (mn) Figure 5.19: Relative amplitude for solid numerical model and for MacCamy and Fuchs solution model for transect parallel to the y-axis at x =100 mn for pit of radius = 12 mn with 200 points defining the pit boundary in the numerical model and 80 terms taken in the series summation for analytic solution with analytic solution model for circular pit of the same radius shown and with pit drawn To further compare the solution to that of a pit of the same radius, the solution of the analytic model is also included (Figure 5. 10). This plot shows that the minimum relative amplitude for the case of a cylinder does not lie directly behind the pit, but in two areas project out behind the pit. The sheltering in these areas is found to be greater than in the case of the circular pit. The relative amplitude oscillations are found to die off much more quickly for a pit. 5.2.4 Plots of Wave Fronts in Fluid Domain Another way to consider the wave modification caused by a pit is to plot the wave fronts as lines of constant phase. The complex value of the velocity potential was used to determine the phase at each point in the grid and from this a contour plot of phase was created. Figure 5.20 shows the contour line wave fronts for the case of a pit with a radius of 12 mn and the base conditions using the numerical model results, which are very close to those of the analytic solution model. .......~ ....... ........ ............ ILL _100 -B0 .60 *0 2 0 20 40 so EI 0 0~ X- DIr&.fion (MI Figure 5.20: Contour plot showing wave fronts as lines of constant phase from analytic solution model for circular pit with r = 12 mn, 80 terms taken in the series summation base conditions with pit drawn The plot shows lines of constant phase, which illustrate the wave fronts at a point in time. The phase was calculated as c" = atan- K1r'iag The asymptotes of the tangent function cause the closely spaced contour lines observed in the figure. The other contour lines indicate the wave fronts and clearly show an increase in wave speed as the front travels through the pit, which results in a divergence of the wave front at the rear of the pit. Further behind the pit this "bulge" is seen to spread laterally, which at large distances directly behind the pit results in a concave shape; a result verified in the shoreline change model. The contour plot of phase associated with a solid cylinder is shown in Figure 5.21. 401-!::< -40 1 260 1i 80 .8 -100 -80 NUMFRICA, contour plot of phase ; ...... ":r / ............ ...: ......... ........ *, 'i 0 ', i ii 7 -40 -20 0 2)0 40 60 80 X Oirertion (mn) Figure 5.21: Contour plot of showing wave fronts as lines of constant phase from MacCamy and Fuchs solution model for circular pit with r = 12 m, 80 terms taken in the series summation base conditions with pit drawn This plot of the results of the solid numerical model shows wave crests that diffract around the cylinder, as would be expected. A greater disturbance to the entire wave field is seen in this plot as compared to Figure 5.20. 5.2.5 Model Sensitivity The numerical model is based on the assumption that the pit can be represented by a number of equally spaced points defining its boundary. The value of 0b and .a- are determined at each point and using these values, the velocity potential can be determined at any point in the fluid domain. Increasing the number of points that define the pit should result in more accurate results for the model, as a better representation of the pit should result. Figure 5.22 shows the value of the maximum and minimum relative amplitudes for the parameters used in Figure 5.5 with 900 points in the fluid domain. 0.7 dotted~ emllaytic sofution 0.43 0.5:--- --- -------------- -------........... .. ................ .............. :.............. j............... .......... 0 100 200 300 400 500 1300 700 130 900 1000 1 100) Number of Points Defining Pit Boundary Figure 5.22: Plot of numerical model values of maximum and minimum relative amplitude versus the number of points defining the pit boundary for a grid of points defining the fluid domain with a circular pit of radius 12 mn and other base conditions with analytic model solution for same conditions drawn. The values are seen to approach a near constant value after the number of points on the pit is 200. Increasing the number of points on the pit boundary from 200 to 1000 results in a 0.4% change in maximum relative amplitude for this case; 1. 1555 to 1. 15 1. For a pit with a 12 m radius, 200 points on the boundary results in a spacing of 0.38 m between the points on the pit. The analytic solution model and the MacCamy and Fuchs model are based on a series summation to obtain accurate results. The relative amplitude values found with the analytic solution for several points are plotted against the number of terms taken in Figure 5.23. 1.3..............r ................i..... I J /l 'N I, I' --------------------........................................... ................. ..... 0 ....... -....(800.0) (150.0) 20 40 '30 s0 100 Terms Taken in Series Summation 120 140 160 Figure 5.23: Plot of analytic solution model values of relative amplitude versus the number terms taken in the series summation for five points on the x-axis with a circular pit of radius 12 m and other base conditions ............. - I ---- 1 ----------- . ........ --- L .................. -L ..... 08 The relative amplitude is seen to become constant for each distance from the pit with less than 140 terms taken in the summation, except for 1000 mn. At 140 terms taken in the summation, a value of NAN or not-a-niumber was returned by the MATLAB program for unknown reasons. The number of summation terms needed to achieve a constant relative amplitude value (within 5 decimal places) is seen to increase as the distance from the pit increases. The radius was not found to influence this value, but decreasing the period was found to result in more terms required in the summation to achieve a constant value. The same results were found for the MacCamy and Fuchs model. In each analytic model trial, the limit used was one that ensured the value of the velocity potential had reached a constant value. This means that in the models run, the analytic solution results in a constant relative amplitude value at each point, but increasing the number of points on the pit can change the numerical solution values. Figure 5.24 compares the relative amplitude values found along two sections for a transect taken parallel to the y-axis at x equal to 100 mn. The base conditions were used with a pit radius of 12 m. The effect of increasing the number of points on the pit boundary is seen to result in numerical model results that approach those of the analytical solution model. The top plot in the figure is located at the first peak of increased amplitude outside the pit and shows numerical model values that are larger than those for the analytic solution. The bottom plot is for the section directly behind the pit and shows numerical model values less than the analytic values. In both figures a less than 3% change in relative amplitude is observed by increasing the number of points on the pit from 100 to 1000. 1165 ~nump O 116. ....nump 200 n.m400 S ........................ . W 1.145 - ..-" .....:?.'C :.7.::-:....... ," num p = 10 0 ......':: : :..... . +.= Analytic Solution __,....... ... I '.......... ........ ... -74 -72 -70 -68 -66 -64 -62 -6( Y Direction (m) 0.84 ................ T ................. "- ...............- ------------------ r ---------.........T + Analytic Solution . .83F ... :::::::: :....~.. ..f .. .,-.......... . -.... ... nump =1000 S ................... ..... .. ... E-. ...........-. . < 0.82 .... nump = 4C.np= ..................... nn.=2. 0. ........ ------0 .8 : ___, . . -8 -7 -6 -5 -4 Y Direction (in) -3 -2 -1 0 Figure 5.24: Plot of numerical model values of relative amplitude for two sections of a transect parallel to the y-axis at x equal 100 m for five different numbers of points defining the pit boundary for a circular pit of radius 12 m and other base conditions with the analytic solution model results for the same conditions drawn. 5.3 Energy Reflection Caused by Pit 5.3.1 Analytic Solution Model The amount of energy reflected by a pit was determined using the time-averaged energy flux through a transect parallel to the y-axis (shore-normal wave fronts). Due to the symmetry of the problem for a symmetric pit with normally incident waves, a transect over only one side was used as can be seen in Figure 5.25. At each point along the transect the four energy flux terms described in Section 3.2.2 were calculated. The amount of energy reflected was determined as the difference between the amount incident 60 on the transect and the amount passing through the transect; the sum of the four energy flux terms. 1 0 [ ........ .......... .................. ....... .......... ................... .................. ................... r .................. I ................... 1 -100 2-200 wave direction -300 -400 -500 - 0 ................................................... .................. ............. ......................... -400 -300 -200 -100 0 100 200 304 400 X Direction (n) Figure 5.25: Plot of transect taken parallel to the y-axis for energy flux calculations with pit of radius 12 m showing wave direction Dividing this difference by the energy incident on half of the pit and taking the square root resulted in a reflection coefficient. The analytic solution with its exact, separate equation for the incident and reflected potentials provided for a more direct approach to the reflection problem. Using the equations shown in Section 4.3.2 the four terms of energy flux were determined at each transect point. Figure 5.26 shows the four terms for the representative transect shown in Figure 5.25. The first term consisting of the incident pressure and x-directed velocity, PiUi*, is constant at all points until the boundary of the pit is reached, as would be expected. The PiUr* term is seen to have a large negative value just outside the pit and then oscillate with quickly diminishing amplitude as the distance from the pit increases. 3000 -----------------........................... T -------..... ........................... 7 ----------- ........................- T ........................ i............................. ............................ .................................................. ............................................................. 5 2000: 0 -600 -500 -400 -300 -200 -100 0 ---------- ................................................................ ............. ..................... .... ... ............................. 0 ------- 11-........................."Il"----"-----------i..500w -1000........-----------j.. ..............L............................L--------------------------------................ ............ -600 -500 -400 -300 -200 -100 0 2 ........ ....... 2 0--------------- ...............T---------------------------- r----------------------------- ------ -.------ 0 ......................................... ........................................... S ........... ....... -600 -500 -400 -300 -200 -100 0 2 0 0 0 ; ............................ !............................ ............................ ................. ........... ............................ ............................ o1000 U.'. 0: -----------------...................------- -....... -------------------------- .......... -----------600 -500 -400 -300 -200 -100 0 Y Direction (im) Figure 5.26: The four energy flux terms from the analytical solution model for transect shown in Figure 5.25 with base conditions and 90 points taken in the summation This negative value near the pit indicates reflected energy caused by the pit. The PrUi* term follows the same form as PiUr*, except with a smaller negative value at the pit boundary and oscillations that do not dampen as quickly. The PrUr* term has a small value until very near the pit boundary where it rises steadily until inside the pit where it is the only term and shows more energy passing through the pit near the sides than at the center. This term shows energy passing through the pit, which is largest near the edge of the pit and decreases towards the center. Summing these four terms at each transect point results in Figure 5.27, which shows the total energy flux through the transect at each point, along with the constant incident value. 3000.. 2700 ... . ,600- :L E 2400g2200 .';S2000 1600 I 1400 1200 .................................................. ................ ......... 600 -500 -400 -300 -200 -100 0 Y Direction (m) Figure 5.27: Energy flux at each point in transect shown in Figure 5.25 with incident value (dotted) for base conditions and 90 terms taken in the series summation It is clear that the energy flux oscillates as the distance from the pit increases. This results in a fluctuating energy sum for the transect depending on where the transect ends. Ideally, the transect would extend out far enough to where the fluctuations reduce to zero, but the computing requirement for this limit is not practical for these models. For these energy models, the transect was taken out to y equal to -600 m with a value taken every meter. To solve the fluctuating energy problem, an average over 1 oscillation was made to determine the reflection coefficient for that oscillation in the transect. By comparing the reflection coefficients found by ending the transect at each upcrossing along an entire transect out to y equal to -600 m it was found that a stable value was reached by the end of the transect, and therefore 600 m was a long enough transect. Figure 5.28 shows the reflection coefficients for the transect shown in Figure 5.25 with the energy flux shown in Figure 5.27. The reflection coefficients is determined as the average over one oscillation in the energy flux from each upcrossing, with 1 being closest to the pit. 0.26 0.259 0.257 '; 0.256 .. 02 U 0.255110 o 0.256 0.253 0.252 L 0.251 025 + 4. A. 4 + ..... .. ............. ........ .. . ....... L .............. L -... .. . ............ . ........ j ........... t ...... ............. 2 3 4 5 6 7 8 9 10 11 Uperossing Number Figure 5.28: Reflection coefficients determined as the average over one energy flux oscillation for each upcrossing in the transect The reflections coefficients are seen to reach an almost steady value as the distance from the pit increases. Next, the location for the transect was investigated. Figure 5.29 shows reflection coefficient values for transects taken at different x locations; in front of, through and behind the pit. 0.24i- 0.23 O 0 0.22 0.21 - 0 Analytic Solution (h=2,d=4,a=12) for X=0(+) X=50(1 X=100(o) X=200(d) X=-100(x) + (X = 0) (X = 50) o (X = 100) > (X = 200) x (X = -100) ................................. L ................ J ................. L ................ ............... 12 13 14 15 16 Period [s] 17 18 19 20 21 Figure 5.29: Reflection coefficients for different transect locations with several incident wave periods for the transect shown in Figure 5.25 taking the reflection coefficient as the average of the last energy flux oscillation in the transect Different period waves were used for a pit of radius 12 m with the other base conditions. The reflection coefficients are seen to not depend on where the transect is taken. It 11 should be noted that at larger periods, fewer oscillations in the energy flux value occur before y equals -600 m and therefore these values might be slightly less accurate. A series of trials was made to determine how the reflection coefficient varied with the pit diameter and incident wavelength. Figure 5.30 shows how the reflection coefficient varies with D/L(h) and D/L(d) the wavelength outside and inside the pit, respectively. 0.24: 0.2: 0 L)0.1 6:r o -~ 021 O1,. 014018 0.12,0 .1 0.06 Pit Diameter~Navelength(inside pit.d) 0.2 0.4 0.6 0.8 1 1,2 1.4 1.6 1.8 2 1.5 2 Pit Diameter, avelength(outside pit.h) Figure 5.30: Reflection coefficients versus pit diameter divided by wavelength outside the pit and pit diameter divided by wavelength inside the pit for different pit radii for the transect shown in Figure 5.25 taking the reflection coefficient as the average of the last energy flux oscillation in the transect This figure indicates that the energy flux is highly dependent upon the pit diameter to wavelength ratios. A maximum reflection coefficient of 0.257 is seen at values of 0.5 + (radius = 6 m) : o (radius = 12 m) * (radius = 25 m) x (radius = 30 m ,::, (radius = 75 m) :.: <,:. : 4 .:. +0 4 *l: and 0.3 for the diameter over the wavelength outside the pit and diameter over the wavelength inside the pit, respectively. The minimum value of the reflection coefficient shown is 0.07, but this will approach zero as the pit size approaches zero. The oscillations in the relative amplitude have a period of D/L of approximately 0.8 outside the pit and approximately 0.6 inside the pit for these base conditions of pit depth equal to twice the water depth. For the same incident wave conditions, a deeper pit should result in greater reflection coefficients than a shallower pit of the same size. This is shown in Figure 5.31, which is plot of the reflection coefficient versus the pit depth for pits of radius equal to 12 m and 25 m with the other base conditions and the pit depths ranging from 3 m to 12 m. ......................................................... ------- ............................ ---- .. ..... .. . ....... ............ . . 0.55 + 0.45 + (radius =12 n) 0.4- o (radius =25 m) ") 0.35 " 0 o 0.3 .o ; 0.25 0.2 0.15+ 0.1 0.05 ................--- ..... ...... .----------------------- ----.-------- -- ...... ...... -...............0.1 0.2 0.3 0.4 0.5 0.6 0.7 Water Depth (h) I Pit Depth (d) Figure 5.31: Reflection coefficients versus water depth divided by pit depth for different pit radii for the transect shown in Figure 5.25 The reflection coefficient values for the pit of radius equal to 12 m are seen to approach those of the 25 m for the larger values of pit depth. Another way of examining the influence of the pit depth on the reflection coefficient is shown in Figure 5.32, which shows the reflection coefficient for two different pit depths over a range of pit diameter divided by wavelength values. o.4 0 0 0 2 02.. 0,15 0.05:-0 .2>* o (radius = 12 m) + + (radius = 25 m) '-" water depth / pit depth = 0.25 p .5 water depth / pit depth =0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pit Diameter / Wavelength(i inside pitd) Figure 5.32: Reflection coefficients versus pit diameter divided by wavelength inside pit with different pit radii for a pit of radius equal to 12 m and the transect shown in Figure 5.25 D The maximum reflection coefficient values are seen to occur for the same D values L(d) with the deeper pit having the higher reflection coefficient. 5.3.2 Numerical Model Determining the reflection coefficient for a pit using the numerical model solution is not as straight forward as using the analytic solution model. The numerical model presents some bookkeeping problems for transects through the pit due to the perturbations in the velocity potential values around the singularity at the pit boundary. As seen with the analytic solution in Figure 5.29, the reflection coefficient value is independent of where the transect is taken, and this holds for the numerical solution allowing the transect to be taken outside the pit at x equal to 100 mn. The number of points defining the pit boundary was found to have an effect on the reflection coefficient values determined by the numerical model; as was found for the relative amplitude values. This presents a larger problem than the singularity found at the pit boundary. Figure 5.33 shows the energy flux value determined as the average value from the last upcrossing versus the number of points defining the pit boundary for a pit radius of 12 mn with the other base conditions. The reflection coefficient value from the analytic solution model for this case has been included in the figure. At first, the numerical model values rise steadily as the number of point's increases, but then the increase in relative amplitude from increasing the number of points diminishes rapidly. With 1000 points on the pit boundary, a spacing of 0. 075 m between the points on the pit, the numerical solution is still around 9 percent lower than the analytic solution value. When modeling a pit with a much larger radius, the number of points defining the boundary needed to keep this small spacing leads to a very large computing requirement. Figure 5.34 shows a comparison for the results of the numerical and analytic solutions for a pit of radius equal to 12 mn, 500 points defining the pit boundary in the numerical model and the other base conditions for a transect taken at x equal to 100 m and 500 points defining the pit boundary in the numerical solution. 0.26 ---------r-------- .......I........7--------r--------F........r---------------r-------r........i--------------O 6 ... ... ...v ... ... ... r ... .. ... .. .. ... .. ... .. .... ....... .. .. ..... ......... .. .. .. .. .... .......r .... .. ... ..... .. .. .. .. -analytic solution 0.24 0.22 0,20.18 0.16 0.14 -------......... --- .. ... ..... ...... .L .......... ..... ... ... t. ... .. .... 0 100 200 300 400 500 600 700 800 900 1000 1100 Number of Points Defining Pit Boundary Figure 5.33: Reflection coefficients versus the number of points defining the pit boundary for the numerical model with a pit of radius equal to 12 m, a period of 12 s and the other base conditions with the analytic solution model result shown To try and identify where the difference in the energy flux values from the two models occurred, the relative amplitude and energy flux at each point in a transect were compared. The percent difference in the relative amplitude and energy flux energy were determined at each transect point and plotted, where the percent difference is equal to analtyic numerical *100. analytic00 U.b 0.4 D. 0: 0 .o2 7.." -. .,7 / k / -0.2 -600 -500 -400 -300 -200 -100 0 0.4 1 02 0 1 kh/ ('' LJ r,, / ,\ / i/ /i z / \ i I \\ i / *.' -0.2 ... - ,J b -0.4 -600 -500 -400 -300 -200 -100 0 10,,,. /\ , C- \ I, 1/ " 55 v j . -10 -600 -500 -400 -300 -200 -100 0 Y Direction (m) Figure 5.34: Percent difference in relative amplitude, percent difference in energy flux values and difference in energy flux values for analytic solution model and numerical model for a pit of radius equal to 12 m, 500 points on the pit boundary in the numerical model, and the other base conditions along a transect at x equal to 100 m The percent difference found for the relative amplitude and energy flux are seen to be less than 0.5 % with the largest error occurring behind the pit. Close examination of the energy flux errors shows a slight skewness in the error with the negative values larger than the positive values in the oscillations occurring as the distance from the pit increases. These larger negative values, along with the area directly behind the pit lead to a larger value for the total energy flux through the transect for the numerical model, and therefore a lower reflection coefficient. These small values in percent error are shown in another way as differences in the energy flux at each location (analytical numerical). These values are all less than 10 n-s/m2, but when multiplied by the distance along the transect, the total difference can become significant. For a 600 m transect with a pit of 12 m radius with a 12 s period incident wave and the base conditions, the analytic solution model gives an incident energy flux of 1.6295* 106 n-s/m2 through the transect, which with the reflection coefficient of 0.255 found in Figure 5.30, equates to an energy flux passing through the transect of 1.6252* 106 n-s/m2 or a difference of only 4230 n-s/m2. This difference in very small when compared the sum of the incident and actual values of energy flux along the entire transect. If a very large number of points is not used to define the pit boundary, this error can result in the numerical model producing results with energy flux values through the transect larger than those incident on the transect; a 'negative' reflection coefficient. Even with a large number of points defining the pit boundary the numerical model does not produce results the match well with those of the analytic solution model. Figure 5.35 shows the reflection coefficients from the numerical and analytic models versus the pit diameter divided the wavelength outside of the pit. Results were obtained using four different pit sizes all with the base conditions for pit depth, water depth and angle of incidence. For the numerical model, the pits of radius equal to 6, 12, 25, and 30 m where defined by 200, 400, 800, and 900 points on the pit boundary, respectively. For four trials, two each of pit radius equal to 25 m and 30 m, the sum of the energy flux through the transect was found to be greater than the incident sum through the transect, resulting in a zero value for the reflection coefficient. Even though the spacing between the points on the pit boundary were similar, the reflection coefficient values do not mesh together 72 for the different radius values with the same dimensionless diameter, as they do in the analytic solution results. analytic solution model O + (radius = 6m) o (radius = 12 m) o (radius = 25 m) x (radius = 30 m) numerical model (larger symbols) * * 0.4 0.6 0.8 Pit DiameterNVavelength(outside pith) Figure 5.35: Reflection coefficients versus the pit diameter divided by the wavelength outside the pit for the numerical model for different pit radii along a transect at x equal to 100 m and the other base conditions with the analytic solution model results shown This suggests that when radius in the numerical model is increased, increasing the spacing on the pit by the same factor does not result in the same accuracy when compared to the analytical results. Dependencies on the pit dimensions, and incident wave field characteristics must be involved. 5.3.3 MacCamy and Fuchs Solution Model for Solid Cylinder The energy reflected by a solid cylinder was investigated using the MacCamy and Fuchs solution. This model was used to compare the wave field changes caused by a 0.25 P 0 0 . + 0.2 solid cylinder to those caused by pit in Section 5.2.3. Using the same procedure for the determining the energy flux through a transect as was previously discussed the reflection coefficient was determined for solid cylinders of different radius (Figure 5.36). 0.851-. * N * 4. 4 C) 4. o radius= 12 m) '(radius 25 m) x (radius = 50 m) + (radius 75 m) 0.65[ 0.5 1 1.5 2 2.5 Cylinder DiameterA^avelength(water depth h) Figure 5.36: Reflection coefficients versus the cylinder diameter divided by the wavelength outside the pit with the MacCamy and Fuchs solution model for different pit radii along a transect at x equal to 0 m and the other base conditions The reflection coefficient values are much larger than those found for a pit of the same dimensionless diameter, with a peak value for the cylinder of 0.85 and for the pit of 0.257. The maximum value of the reflection coefficient for the cylinder is seen to occur once at a dimensionless diameter of 0.35 and then again as the dimensionless diameter increases near 3. Slight oscillations in the reflection coefficient for the cylinder are 00.75 a: 0 ........................... 1 .......... ..... ...... ............................ I ............................ I .......... ............................................. apparent, but they are small as compared to the oscillations seen in Figure 5.30 for the pit. The large oscillations of the relative amplitude in the pit can be explained by resonance in the reflected waves occurring for certain dimensionless diameters, where this cannot happen for the case of a cylinder. 5.4 Shoreline Change Induced by Pit 5.4.1 Comparison of Model Results The preceding sections have documented the wave field modification caused by a pit, as well as quantified energy reflection by a pit. These changes to the incident wave field can impact the shoreline and result in changes there as well. This section presents the shoreline change induced by an offshore pit. In these models a transect parallel to the y-axis was considered to represent the shoreline. The models used to determine the energy flux were extended to calculate the wave direction and relative amplitude at each point along the transect. The wave direction was determined as a -: tan' Fux, (Eq. 3.19). Knowing the shoreline orientation and the values of wave direction and height at each location allowed for the longshore transport to be calculated with the CERC longshore transport equation with both transport terms, Eq. 3.34. The changes in shoreline position were determined using these values of longshore transport in the continuity equation (Eq. 3.35). The values of the coefficients and constants in these equations that were used in the models are listed in Table 5. 1. The assumptions used in these models are detailed in Section 3.2.3. For these models a situation more representative of the coastal environment was chosen. To compare the numerical and analytic models a circular pit was required, and a radius of 50 m was used. The pit is 8 m deep and located in 4 m of water. The transect was taken 1500 m behind the pit and the transect representing the shore extended 1600 m in the longshore direction with a point every 5 m. Table 5.1: Coefficients and constants used in the shoreline change models K,= 0.77 K2= 0.77 K= 0.78 s = 2.65 P = 0.5 tan(7) = 0.05 h. + B = 6 (m) At = 300 s Incident wave heights of 1 m were used with periods of 10, 12 and 14 s. These wave periods lead to h/L values that meet the shallow water condition of h/L less than 1/10. The shoreline change models were exercised for monochromatic waves with one direction resulting in changes in the shoreline position that oscillated with large amplitude to great distances from the pit. This effect is interpreted as the result of the diffraction pattern. This resulted in shoreline changes that were unrealistic for the coastal environment where waves of different period and different incident angle impact the shore. To account for this, an averaging procedure was employed where the models were run for incident angles from -10' to 100 every 20 and for 3 different periods. By taking the average of the relative amplitude, longshore transport, and shoreline change values over the different incident wave angles and then averaging those values for the 3 different periods, a better representation of a typical nearshore situation was made. The numerical model for this trial had 400 points defining the pit boundary. The relative amplitude for the situation described above is found in Figure 5.37 with the axes oriented in the same manner as the transects that were presented for the energy flux with the waves approaching from the left. 8 0 00 r-------------------------- ............... .......-- ........... -----........- --....... ...............---........................---.. ....................-- -...................... ! ... .. ... .. .. .... ....... 200 . ~ ~ ~ .. ... ... .. .. ... 4 0 0 1- .. ..... . solid (avg of wave direction and 3 periods) S:dotted (avg of wave direction for T = 12 ) dashed (value for beta = 0degaend T =12M .-80c0 .... L -. ............... ..,,------ --.- T ........... J. . ......... 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Relative Amplitude Figure 5.37: Relative Amplitude along the transect for the numerical model with a pit of radius equal to 50 -, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for this shoreline change model This figure shows the relative amplitude for the case of shore normal waves with a period of 12 s (dashed line), the average relative amplitude of the 11I wave directions for a period of 12 s (dotted line), and the average of the average relative amplitude of the 11I wave directions for periods of 10, 12, and 14 s (solid line). The values for the single wave direction and period are seen to oscillate very far from the pit while the averaged values do not. Further averaging over the wave direction for each period was found to not change the values near the pit, but slightly reduced the oscillations far from the pit. Ideally a large number of periods and wave directions would be used to approximate the coastal environment, resulting in even more smoothing of the oscillations; however due to the computational requirements this problem presented, an average over the wave direction for only one period was deemed satisfactory. The averaged values over the wave direction are seen to greatly reduce the maximum relative amplitude compared to the single direction and period, but only slightly reduce the minimum value. This is due to the larger values of wave direction moving the shadow zone from directly behind the pit to areas alongshore where the largest relative amplitudes were found for the more shore-normal waves; thereby reducing the average value at these locations. The same averaging procedure was performed on the longshore transport values determined with the numerical model for the same conditions, Figure 5.38. 800 ------------- r ---------- .. --- ......... _ __-..-TF 7 -...... ------ --- ........------ -- -------600 .. .' . .. . said i(:vg of Wave direction and 3 periods) 400 : dotted (avo of wave direction for T 12 S) dashed (vakie for bete 0 deg anc T 12 s) 200 :__ .. ..... -200 .'':: 7 " 60 ..200 .-- "'We"' -0.25 -0.2 -015 -0.1 -0.05 0 0.05 0.1 015 02 0.25 Longshore Trarsport (m 3) Figure 5.38: Longshore transport value along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary Once again the averaging is seen to result in a significant reduction in the oscillations far from the pit. The large peaks in the longshore transport behind the pit are also greatly reduced by the averaging. The second averaging has little effect, except far from the pit where the oscillations are slightly reduced. The shoreline change was calculated based on these values of longshore transport using the full transport equation for the three methods presented in the last two figures. Figure 3.39 shows the results for a 300 s time step with a negative change value indicating shoreline advancement. As seen previously the single direction and wave period model leads to great fluctuations in the shoreline change, even at large distances from the pit. -0.3 -0.2 -0.1 0 0.1 Change in Shoreline Position (m) 0.2 0.3 0.4 Figure 5.39: Shoreline change from the full transport equation for a 300 s time step along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change model 200 - -400 -600 -.VV -0.4 ... .... '. ... . . l : : .: : : . .. ..... .._ . 'N solid (avg'of Wave dif'ection and3 periods) ...... --dotted (avg of wave direction fo T = 12 s) .. dashed (value for beta =0 deg and T 12 s. S... ...... ...- . The averaging procedure is seen to remove these fluctuations and shows a salient forming behind the pit flanked by two areas of erosion and then oscillations in the shoreline change as distance from the pit increases. The salient directly behind the pit is found to be slightly smaller than the two immediately upshore and downshore from it. Figure 5.40 provides a closer view of these changes. 600" solid (avg of wave direction and 3 periods) 200 ' " dotted (avg of wave direction for T 12 s) .........-0 -200 . -4030 -800 : .---- ----- ...... ----... -0.2 -0.15 -0.1 -0,05 0 0.05 0.1 0.15 0.2 Charge in Shoreline Position (m) Figure 5.40: Shoreline change for a 300 s time step along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change model This plot clearly shows the increased reduction in the oscillating shoreline change values at large distances from the pit resulting from the extra averaging. 80 The magnitude of each transport term was investigated. Figure 5.42 shows the first and second terms of the transport equation plotted separately for the total transport shown in the previous plot. 600 F 2001- -200--400 -600 - -0.1 -0.05 0 0.05 Change in Shoreline Position (m) 0.1 0.15 0.2 Figure 5.41: Shoreline change for first and second transport terms for a 300 s time step along the transect for the numerical model with a pit of radius equal to 50 m, at x = 1500 mn with 400 points defining the pit boundary and the other base conditions for the shoreline change model The second transport term is seen to contribute greatly to the total transport, especially behind the pit where the gradients in the wave height are the greatest. This second transport term is also seen to oscillate at a higher frequency than the first transport term. It is clear that the two transport terms can oppose each other in some locations, such as behind the pit where refraction may cause the waves to diverge and result in little change -solid (I1st transport term) dashed (2nd transport term) 2 81 or erosion for the first transport term, but the gradient in the wave height of the second term leads to accretion behind the pit. The analytic solution model was run for the same pit and incident wave conditions with the same averaging techniques applied. The results compare well with those of the numerical model. Figure 5.42 shows for a wave period of 12 s, the wave direction averaged value of the longshore transport for the two models for half of the transect. ......... ----.............. solid (numericel mode dotted (analytic soluti -0.25 -0.2 -O ......... -----. . .... ... .......... ......... . -.. .........I.... .. onr o e; .....) .--- nroe......-... .............. . ............. .......... .... 0.15 -0.1 -0.05 0 0.05 Longshore Transport m3 ) Figure 5.42: Longshore transport along the transect for the analytic solution model and the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change model The magnitudes of the numerical values are found to be larger than those of the analytic solution model. Increasing the number of points defining the pit should bring the values - -400 -500 -600 -700 into closer agreement. These differences in longshore transport are carried into the shoreline change results, Figure 5.43. .......... .. ............. -- 2 ; SI; ........ ... dotd(nlisouonoe -0.04 -0,02 0 0.02 Change in Shoreline Position (m) 0.04 0.06 0.08 0.1 Figure 5.43: Shoreline change for a 300 s time step along the transect for the analytic solution model and the numerical model with a pit of radius equal to 50 m, at x = 1500 m with 400 points defining the pit boundary and the other base conditions for the shoreline change model The larger values in longshore transport for the numerical model found in the previous figure results larger gradients and therefore larger shoreline changes in Figure 5.43. A comparison was made of the shoreline change caused by a circular pit to that of a rectangular pit, which more accurately represents the shape of most borrow areas. For this test the pit dimensions were 40 m in the cross-shore direction and 200 m in the longshore direction; this results in an area almost equal to that for a circular pit with a 600- 400 -200 -400 -600 -800 -. -0.1 -0.08 .0.06 83 50 m radius. A 12 s incident period was used, with 800 points defining the pit boundary, and the same pit depth, water depth and transect location were used as in the trial for the circular pit of 50 mn radius. The relative amplitude found for the case of the large rectangular pit is shown in Figure 5.44. 2001- -200 k -600K 1.4 1.2 1 0.8 Relative Amplitude 0.6 0.4 0.2 Figure 5.44: Relative amplitude along transect for numerical model with a rectangular pit (40 x 200 m) at x = 1500 m with 800 points defining the pit boundary and the other base conditions for this shoreline change model The relative amplitude was found to be significantly smoothed by averaging over the five wave directions, beta equal to -5, -2, 0, 2, 5 degrees. The shadow zone was seen to be larger for this case of the long pit, as compared to the circular pit with the same area, which was expected. This will result in differences in the shoreline changes as well. ..dashed (value for beta 0. T =12 s) solid (avg of wave direction for T =12 s) -800u 1.6 The shoreline change for each of the two transport terms are shown in Figure 5.45. 600 I- 200 F- -200 -400 -600 - -0.03 -0.02 -0.01 0 0.01 Change in Shoreline Position (in) 0.02 0.03 0.04 Figure 5.45: Shoreline change for 300 second time step along transect for numerical model with a rectangular pit (40 x 200 mn) at x = 1500 m with 800 points defining the pit boundary and the other base conditions for this shoreline change model The two transport terms are shown to result in different shoreline changes directly behind the pit, as was seen in Figure 5.41 for the case of the circular pit. The 2 transport term leads to shoreline advancement directly behind the pit due to the large gradients in the local wave height. Both plots show erosion flanking the shadow area of the pit. The total shoreline change induced by the rectangular pit is shown in Figure 5.46 with the results for the circular pit drawn. ..- solid (1st transport term) dashed (2nd transport term) [both averaged for 5 directions] -800' -0.0 4 |