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Scaling Relations from Scale Model Experiments on Equilibrium
Accretionary Beach Profiles
A Thesis
Submitted to the Faculty
of
Drexel University
by
Muhammad Shah Alam Khan
in partial fulfillment of the
requirements for the degree
of
Doctor of Philosophy
December 2002
ii
DEDICATIONS
To humanity, and the human race.
iii
ACKNOWLEDGEMENTS
Guru, in Bangla, means more than a teacher. Guru is a symbol of guidance that
directs the disciples; an open source from where the disciples gain knowledge and
wisdom. Dr. J. Richard Weggel is my Guru in Coastal Engineering. Dr. Weggel
systematically and patiently introduced this subject to me. Even outside the academic
area, Dr. Weggel has been the best advisor I have known. If I have accomplished
anything in this dissertation, that is because I have an exceptional Guru; if I have failed,
that is because I was unable to absorb the knowledge from my Guru.
I am thankful to have Dr. James Feir, Dr. Robert Sorensen, Dr. Joseph Martin and
Dr. Michael Piasecki in the supervisory committee. Their objective suggestions have led
this study to a successful completion. I must also thank Dr. Martin for his kind support as
the Department Chair, and Dr. Jonathan Cheng for his compassionate role as the
Graduate Advisor. My stay at Drexel University as a foreign student has been easy
because of these two extraordinary persons.
Expedient construction of the laboratory setup was possible because a number of
individuals. I express my gratitude for their contributions. Dr. Robin Carr provided his
involved assistance in data acquisition system design. Dr. Robert Koerner of the
Geosynthetic Institute generously contributed the geotextile materials for beach lining.
iv
Mr. Jerry Leva and Mr. Greg Ciliberto of Drexel University Facilities Department were
instrumental in design and construction of the beach support frame. ‘Nick’ and his crew
of Drexel University Machine Shop meticulously put together the beach profiling system.
My son, Tausif Raiyan Khan (Arnab), and wife, Sakina Sharmin Khan, patiently
waited long nights of experimentation and writing, and forgave many broken promises. I
share my degree with these two special people in my life.
v
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3. Scope of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1. Laboratory Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1. Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2. Wave Reflection and Absorption . . . . . . . . . . . . . . . . . . . . . 9
2.2. Beach Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1. Profile Types and Zones . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2. Forcing Functions for Profile Change . . . . . . . . . . . . . . . . . . 13
2.2.3. Equilibrium Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.4. Cross Shore Sediment Transport Variation . . . . . . . . . . . . . . . 17
2.3. Initiation of Sediment Motion . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4. Fall Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5. Scaling Relations in Movable Bed Scale Modeling . . . . . . . . . . . . . . . 23
2.6. Beach Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.1. Scale Model Experiments . . . . . . . . . . . . . . . . . . . . . . . 29
vi
2.6.2. Prototype Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.3. Field Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3. LABORATORY SETUP AND DATA ACQUISITION PROCEDURE . . . . . . 34
3.1. Wave Tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2. Beach and Beach Support Frame . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3. Wave Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4. Instrumentation for Data Acquisition . . . . . . . . . . . . . . . . . . . . . 41
3.4.1. Wave Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.2. Beach Profiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.3. Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.4. Measurement Accuracy and Errors . . . . . . . . . . . . . . . . . . . 48
3.5. Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.1. Beach and Wave Tank Preparation . . . . . . . . . . . . . . . . . . . 54
3.5.2. Wave Gauge Calibration . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.3. Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.4. Video and Still Photographs . . . . . . . . . . . . . . . . . . . . . . 56
3.5.5. Profile Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4. PHYSICAL MODEL EXPERIMENTS AND RESULTS . . . . . . . . . . . . . 57
4.1. Description of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2. Data Processing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.1. Wave Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.2. Beach Profile Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3. Data Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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4.3.1. Transport Rate and Beach Equilibrium . . . . . . . . . . . . . . . . . 69
4.3.2. Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5. DISCUSSION ON EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . 85
5.1. General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.1. Critical Criteria for Eroding or Accreting Profile . . . . . . . . . . . . 85
5.1.2. Reflection Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.3. Beach Face Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2. Sediment Transport Processes in Scaling Zones . . . . . . . . . . . . . . . . 89
5.3. Scale Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.1. Relative Magnitude of Scale Effect . . . . . . . . . . . . . . . . . . . 92
5.3.2. Example of Profile Prediction . . . . . . . . . . . . . . . . . . . . . 94
5.3.3. Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.4. Role of Shear Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4. Applications and Limitations of Study . . . . . . . . . . . . . . . . . . . . 104
6. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1. Summary of Model Experiments . . . . . . . . . . . . . . . . . . . . . . . 105
6.2. Conclusions of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3. Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
APPENDIX A: List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
APPENDIX B: Permeability Test Results . . . . . . . . . . . . . . . . . . . . . . 118
APPENDIX C: Details of Prototype and Model Tests . . . . . . . . . . . . . . . . 123
APPENDIX D: Variation of Water Surface Elevation in Wave Tank . . . . . . . . 128
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APPENDIX E: Measured Beach Profiles . . . . . . . . . . . . . . . . . . . . . . . 131
APPENDIX F: Dimensionless Beach Profiles . . . . . . . . . . . . . . . . . . . . 140
APPENDIX G: Transport Rate Variation . . . . . . . . . . . . . . . . . . . . . . . 149
APPENDIX H: Scaling Relations of Dimensionless Variables . . . . . . . . . . . 158
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
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LIST OF TABLES
2.1: Summary of theoretical scaling relations proposed by Noda (1972) . . . . . . . . 25 4.1: Summary of model and prototype variables . . . . . . . . . . . . . . . . . . . . 59 4.2: Wave gauge separation distances . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3: Identification of profile features . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1: Comparison of critical criteria for eroding or accreting profile . . . . . . . . . . 85 5.2: Summary of slope coefficients indicating magnitude of scale effect . . . . . . . 94 5.3: Summary of dimensionless coordinate relationship in four zones . . . . . . . . 95 B1: Experimental data for Test P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 B2: Experimental data for Test P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 C1: Prototype Test ST10_1 (Equilibrium erosion, Random waves) . . . . . . . . . 123 C2: Prototype Test STi0_1 (Equilibrium accretion, Sinusoidal waves) . . . . . . . . 123 C3: Model Test DST10_1 (Equilibrium erosion, Random waves) . . . . . . . . . . 124 C4: Model Test DSTi0_1 (Equilibrium accretion, Sinusoidal waves) . . . . . . . . 124 C5: Model Test DSTi0_2 (Equilibrium accretion, Cnoidal waves) . . . . . . . . . . 125 C6: Model Test DSTi0_3 (Equilibrium accretion, Sinusoidal waves) . . . . . . . . 125 C7: Model Test DSTi0_4 (Equilibrium accretion, Cnoidal waves) . . . . . . . . . . 126 C8: Model Test DSTi0_5 (Equilibrium accretion, Sinusoidal waves) . . . . . . . . 126 C9: Model Test DSTi0_6 (Equilibrium accretion, Cnoidal waves) . . . . . . . . . . 127 D1: Water surface elevation in wave tank during and after Run DSTi04_8 . . . . . 129
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LIST OF FIGURES
2.1: Nearshore regions (Dean 2002; after US Army Corps of Engineers 1984) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2: Zones of interest on the wave-beach interface (Wang 1985) . . . . . . . . . . . 11 2.3: Shields criterion and oscillatory flow data (Madsen and Grant 1975) . . . . . . . 20 2.4: Graphical representation of Noda’s (1972) scaling relations . . . . . . . . . . . 24 2.5: Sediment transport surface based on Shields diagram (Kamphuis 1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1: Cross-section of wave tank showing details of beach support frame and placement of geotextile layer . . . . . . . . . . . . . . . . . . . . . . 36 3.2: Grain size distribution of sand considered for construction of beach and that of sand used in the SUPERTANK tests . . . . . . . . . . . . . 38 3.3: Details of a typical wave gauge . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4: Details of beach profiler carriage and point gauge assembly . . . . . . . . . . . 44 3.5: Details of profiler sensor rod attachment and operating circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6: Sample record from data acquisition system demonstrating wave gauge calibration range and drift in voltage signal . . . . . . . . . . . . . 50 3.7: Difference in pre- and post-calibration of wave gauges . . . . . . . . . . . . . . 51 3.8: Comparison of profile measurements by point gauge and calibrated resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1: Changes in wave form during Test DSTi0_1 . . . . . . . . . . . . . . . . . . . 62 4.2: Changes in wave form during Test DSTi0_2 . . . . . . . . . . . . . . . . . . . 63 4.3: Incident and reflected spectra for Test DSTi0_1 . . . . . . . . . . . . . . . . . 65 4.4: Incident and reflected spectra for Test DSTi0_2 . . . . . . . . . . . . . . . . . 66 4.5: Measured profiles with interpolated surface . . . . . . . . . . . . . . . . . . . . 68
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4.6: Individual profiles without interpolated surface . . . . . . . . . . . . . . . . . . 68 4.7: Profile closure error and correction . . . . . . . . . . . . . . . . . . . . . . . . 70 4.8: Variation of net transport rate across the beach during a run . . . . . . . . . . . 71 4.9: Surface plot showing variation of transport rate during a test . . . . . . . . . . . 71 4.10: Migration of measured location of SWL . . . . . . . . . . . . . . . . . . . . . 73 4.11: Migration of SWL with respect to equilibrium SWL . . . . . . . . . . . . . . 73 4.12: Variation of RMSE between consecutive profiles for sinusoidal wave tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.13: Variation of RMSE between consecutive profiles for cnoidal wave tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.14: Selected equilibrium profiles in measured coordinates . . . . . . . . . . . . . . 76 4.15: Selected equilibrium profiles in dimensionless coordinates . . . . . . . . . . . 76 4.16: Distinct features and scaling zones on an equilibrium profile . . . . . . . . . . . 78 4.17: Variation of horizontal length scale ratio with horizontal distance . . . . . . . . 80 4.18: Variation of vertical length scale ratio with horizontal distance . . . . . . . . . 80 4.19: Variation of dimensionless horizontal length coordinate ratio . . . . . . . . . . 81 4.20: Variation of dimensionless vertical length coordinate ratio . . . . . . . . . . . 81 4.21: Relationship between model and prototype dimensionless horizontal length coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.22: Relationship between model and prototype dimensionless vertical length coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.23: Prototype profile calculated from DSTi01_13 profile . . . . . . . . . . . . . . 84 4.24: Prototype profile calculated from DSTi03_13 profile . . . . . . . . . . . . . . 84 4.25: Prototype profile calculated from DSTi05_11 profile . . . . . . . . . . . . . . 84 5.1: Reflection bar and wave crest envelope . . . . . . . . . . . . . . . . . . . . . 88
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5.2: Temporal variation of net transport rate during Test DSTi0_1 . . . . . . . . . . 91 5.3: Temporal variation of net transport rate during Test DSTi0_3 . . . . . . . . . . 91 5.4: Temporal variation of net transport rate during Test DSTi0_5 . . . . . . . . . . 91 5.5: Scale effect on dimensionless horizontal length coordinates . . . . . . . . . . . 93 5.6: Scale effect on dimensionless vertical length coordinates . . . . . . . . . . . . 93 5.7: Initia l and equilibrium profiles of the prototype . . . . . . . . . . . . . . . . . 96 5.8: Initial and equilibrium profiles of Test DSTi0_1 . . . . . . . . . . . . . . . . . 96 5.9: Prototype equilibrium profile predicted from DSTi0_1 . . . . . . . . . . . . . 97 5.10: Prototype equilibrium profile predicted from DSTi0_3 . . . . . . . . . . . . 97 5.11: Prototype equilibrium profile predicted from DSTi0_5 . . . . . . . . . . . . . 97 B1: Variation of infiltration rate with time during Test P1 . . . . . . . . . . . . . . 122 B2: Variation of infiltration rate with time during Test P2 . . . . . . . . . . . . . . 122 D1: Variation of water surface elevation during wave action . . . . . . . . . . . . . 130 D2: Variation of water surface elevation after test . . . . . . . . . . . . . . . . . . 130 E1: Measured beach profiles for Test ST10_1 . . . . . . . . . . . . . . . . . . . . 131 E2: Measured beach profiles for Test STi0_1 . . . . . . . . . . . . . . . . . . . . . 132 E3: Measured beach profiles for Test DST10 . . . . . . . . . . . . . . . . . . . . . 133 E4: Measured beach profiles for Test DSTi0_1 . . . . . . . . . . . . . . . . . . . . 134 E5: Measured beach profiles for Test DSTi0_2 . . . . . . . . . . . . . . . . . . . . 135 E6: Measured beach profiles for Test DSTi0_3 . . . . . . . . . . . . . . . . . . . . 136 E7: Measured beach profiles for Test DSTi0_4 . . . . . . . . . . . . . . . . . . . . 137 E8: Measured beach profiles for Test DSTi0_5 . . . . . . . . . . . . . . . . . . . . 138 E9: Measured beach profiles for Test DSTi0_6 . . . . . . . . . . . . . . . . . . . . 139
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F1: Dimensionless beach profiles for Test ST10_1 . . . . . . . . . . . . . . . . . . 140 F2: Dimensionless beach profiles for Test STi0_1 . . . . . . . . . . . . . . . . . . 141 F3: Dimensionless beach profiles for Test DST10 . . . . . . . . . . . . . . . . . . 142 F4: Dimensionless beach profiles for Test DSTi0_1 . . . . . . . . . . . . . . . . . 143 F5: Dimensionless beach profiles for Test DSTi0_2 . . . . . . . . . . . . . . . . . 144 F6: Dimensionless beach profiles for Test DSTi0_3 . . . . . . . . . . . . . . . . . 145 F7: Dimensionless beach profiles for Test DSTi0_4 . . . . . . . . . . . . . . . . . 146 F8: Dimensionless beach profiles for Test DSTi0_5 . . . . . . . . . . . . . . . . . 147 F9: Dimensionless beach profiles for Test DSTi0_6 . . . . . . . . . . . . . . . . . 148 G1: Transport rate variation for Test ST10_1 . . . . . . . . . . . . . . . . . . . . . 149 G2: Transport rate variation for Test STi0_1 . . . . . . . . . . . . . . . . . . . . . 150 G3: Transport rate variation for Test DST10 . . . . . . . . . . . . . . . . . . . . . 151 G4: Transport rate variation for Test DSTi0_1 . . . . . . . . . . . . . . . . . . . . 152 G5: Transport rate variation for Test DSTi0_2 . . . . . . . . . . . . . . . . . . . . 153 G6: Transport rate variation for Test DSTi0_3 . . . . . . . . . . . . . . . . . . . . 154 G7: Transport rate variation for Test DSTi0_4 . . . . . . . . . . . . . . . . . . . . 155 G8: Transport rate variation for Test DSTi0_5 . . . . . . . . . . . . . . . . . . . . 156 G9: Transport rate variation for Test DSTi0_6 . . . . . . . . . . . . . . . . . . . . 157 H1: Horizontal scaling relations for Test DSTi0_1, Zone I . . . . . . . . . . . . . . 158 H2: Horizontal scaling relations for Test DSTi0_1, Zone II . . . . . . . . . . . . . 158 H3: Horizontal scaling relations for Test DSTi0_1, Zone III . . . . . . . . . . . . . 159 H4: Horizontal scaling relations for Test DSTi0_1, Zone IV . . . . . . . . . . . . . 159 H5: Vertical scaling relations for Test DSTi0_1, Zone I . . . . . . . . . . . . . . . 160
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H6: Vertical scaling relations for Test DSTi0_1, Zone II . . . . . . . . . . . . . . . 160 H7: Vertical scaling relations for Test DSTi0_1, Zone III . . . . . . . . . . . . . . 161 H8: Vertical scaling relations for Test DSTi0_1, Zone IV . . . . . . . . . . . . . . 161 H9: Horizontal scaling relations for Test DSTi0_1, all zones . . . . . . . . . . . . . 162 H10: Vertical scaling relations for Test DSTi0_1, all zones . . . . . . . . . . . . . 162 H11: Horizontal scaling relations for Test DSTi0_3, Zone I . . . . . . . . . . . . . 163 H12: Horizontal scaling relations for Test DSTi0_3, Zone II . . . . . . . . . . . . . 163 H13: Horizontal scaling relations for Test DSTi0_3, Zone III . . . . . . . . . . . . 164 H14: Horizontal scaling relations for Test DSTi0_3, Zone IV . . . . . . . . . . . . 164 H15: Vertical scaling relations for Test DSTi0_3, Zone I . . . . . . . . . . . . . . . 165 H16: Vertical scaling relations for Test DSTi0_3, Zone II . . . . . . . . . . . . . . 165 H17: Vertical scaling relations for Test DSTi0_3, Zone III . . . . . . . . . . . . . . 166 H18: Vertical scaling relations for Test DSTi0_3, Zone IV . . . . . . . . . . . . . . 166 H19: Horizontal scaling relations for Test DSTi0_3, all zones . . . . . . . . . . . . 167 H20: Vertical scaling relations for Test DSTi0_3, all zones . . . . . . . . . . . . . 167 H21: Horizontal scaling relations for Test DSTi0_5, Zone I . . . . . . . . . . . . . 168 H22: Horizontal scaling relations for Test DSTi0_5, Zone II . . . . . . . . . . . . . 168 H23: Horizontal scaling relations for Test DSTi0_5, Zone III . . . . . . . . . . . . 169 H24: Horizontal scaling relations for Test DSTi0_5, Zone IV . . . . . . . . . . . . 169 H25: Vertical scaling relations for Test DSTi0_5, Zone I . . . . . . . . . . . . . . . 170 H26: Vertical scaling relations for Test DSTi0_5, Zone II . . . . . . . . . . . . . . 170 H27: Vertical scaling relations for Test DSTi0_5, Zone III . . . . . . . . . . . . . . 171 H28: Vertical scaling relations for Test DSTi0_5, Zone IV . . . . . . . . . . . . . . 171
xv
H29: Horizontal scaling relations for Test DSTi0_5, all zones . . . . . . . . . . . . 172 H30: Vertical scaling relations for Test DSTi0_5, all zones . . . . . . . . . . . . . 172
xvi
ABSTRACT Scaling Relations from Scale Model Experiments on Equilibrium
Accretionary Beach Profiles Muhammad Shah Alam Khan J. Richard Weggel, Ph.D., P.E.
Movable bed, scale model experiments were conducted at three length scales,
1/8.5, 1/10 and 1/11, in a 90 ft- long wave tank to study scale effect and equilibrium
profile characteristics under sinusoidal, accreting wave action. Geometric similarity, deep
water wave steepness, wave Froude number, densimetric Froude number, and particle
Reynolds number were preserved by selecting the same sediment and fluid in the model
and prototype. Wave height was measured with parallel-wire resistance gauges while a
programmable wave generator with wave absorption capability produced waves.
Placement of beach on a permeable frame simulated the natural groundwater. The
equilibrium endpoint of a test was indicated by a relatively small net transport rate at all
points and no significant change of the profile shape.
Profile shapes and transport rates were expressed in dimensionless forms with the
origin of coordinates at the equilibrium still water line. After an initial, relatively fast and
large change, the net transport rate asymptotically decayed to equilibrium. The maximum
transport rate occurred about when the initial foreshore slope was established. Net
transport rate asymptotically varied offshore from the wave breakpoint. Equilibrium
foreshore slope was about the same in the model and the prototype. Bottom roughness
was distorted because of exaggerated bedforms. Near equilibrium, high ‘reflection bars’
xvii
formed at the antinodes of a standing wave system, affecting the shoaling and local
transport processes.
Comparison of distinct features of the model and prototype equilibrium profiles
indicates that the scale effects differ in four zones where the transport processes are
significantly different. Assuming the same scale effect within each zone, empirically
derived equations satisfactorily predict the prototype equilibrium profile. Profile
prediction near the reflection bars is not satisfactory.
1
CHAPTER 1: INTRODUCTION
1.1. Background
Seasonal changes in the wave climate affect the beach, resulting in typical ‘storm
profiles’ or ‘normal profiles’. Storm waves erode the beach to form an offshore bar,
whereas relatively long period accreting waves build the beach during the normal season.
While much attention is paid to the eroding beach processes, the hydrodynamics and
transport processes for accreting waves are still relatively unexplored.
Movable bed scale models are often utilized in the study of nearshore processes to
address topics such as impact of coastal structures on shoreline changes, and effect of
storm waves on beach erosion. However, in the nearshore region the fluid and sediment
movement are complicated by several factors including uncertain variable properties,
nonlinear processes and complex boundary conditions, whereas the dominant physical
phenomena include wave breaking, turbulence and bottom friction effects. Our
understanding of these phenomena is not yet fully developed. Consequently,
determination of scaling criteria and interpretation of model results depend largely on
personal experience and intuition (LeMehaute 1990) along with the known theories of
similitude.
2
Most movable bed scale models are subject to both laboratory and scale effects.
Laboratory effects are nonsimilarities of model and prototype response resulting from
limitations and adverse conditions of the laboratory facilities. These include, for example,
inability of the wave generator to produce specified waves, and support structure
extensions inside the wave tank that affect the flow regime. Scale effects are
nonsimilarities that occur when relative importance of the dominant forces in the
prototype is not appropriately represented in the model. For example, in a small scale
model using the same fluid as in the prototype, the effect of surface tension relative to
gravity may be overrepresented, resulting in nonsimilarity of wave propagation. While
laboratory effects can be minimized with better equipment and material, the scale effects
have to be interpreted and quantified from the model results.
A more prevalent problem in scale models occurs from inappropriate scaling of
the sediment size. If the densimetric Froude number is preserved in a model having the
same fluid as in the prototype, the required sediment size scale ratio is proportional to the
vertical length scale ratio. Consequently, the model may require sediment that is too fine
and cohesive. On the other hand, if the particle Reynolds number is preserved, the model
requires a larger sediment size than that in the prototype. To overcome these conflicting
requirements, researchers adopt several methods including use of empirical scaling
relations, use of lightweight model sediment, and preserving a dimens ionless fall velocity
parameter, or the Dean number, Ho/wT. Although limited success has been reported by
preserving the Dean number, there are, however, scale effects in each case. For example,
although both the densimetric Froude number and the particle Reynolds number may be
3
preserved by using lightweight sediment, the relatively large model sediment size results
in a high bed porosity. Moreover, the sediment to fluid density ratio is not preserved. As
a result, serious nonsimilarity of sediment movement may occur, especially near the still
water line.
Shear velocity due to oscillatory flow in the bottom boundary layer has a
significant role in sediment transport process. The bottom friction is comprised of skin
friction around individual grains, and form drag due to bedform. However, the overall
friction coefficient can be assumed to be a function of the bedform height and the orbital
amplitude of water particles in the bottom boundary layer. Therefore, the shear velocity
changes with water depth and bedform height across the profile. If the sediment size and
the vertical length are not properly scaled in the model, it is likely that the resulting scale
effects will also vary cross shore.
Despite their complex nature, appropriately designed movable bed scale models
are expected to reproduce the dominant physical processes of the prototype, and can be
utilized as useful tools for studying the beach evolution toward equilibrium. Comparison
of the model and prototype equilibrium profiles provides important information regarding
the scale effects and scaling relations. An equilibrium profile indicates that all accreting
and eroding forces on the beach have reached a balance. Consequently, the net transport
rate at every point on the beach is zero, and there is no temporal change of the beach
profile geometry. Based on this similarity of the model and the prototype, and the model
scaling relations, the scale effects can be indirectly quantified. A series of model tests
4
conducted at different scales will indicate the trend in scale effect. The prototype profile
can be predicted from this indirect measure of cross shore variation in scale effects.
1.2. Objectives
The physical model experiments were conducted to reproduce the prototype
equilibrium profile in a scale model. In addition to the observation of profile evolution
toward an equilibrium shape for specified wave characteristics, the following more
specific topics are addressed in the study:
(a) cross shore variation in scale effect, and
(b) prototype equilibrium profile prediction from model results.
1.3. Scope of Dissertation
Movable bed, scale model experiments were conducted at three length scales,
1/8.5, 1/10 and 1/11, to generally investigate the profile evolution process as compared to
the prototype. A SUPERTANK (US Army Corps of Engineers 1994) test, conducted with
sinusoidal, accretionary waves toward equilibrium, was selected as the prototype. The
sediment used in the model was approximately identical to that used in the prototype.
Geometric similarity, deep water wave steepness, and wave Froude number were
5
preserved in the model. This dissertation summarizes the model selection criteria,
experimental procedure, and test results.
Chapter 3 includes a description of the equipment, materials and methods of the
experiments, and the accuracy and reliability of the collected data. Sinusoidal waves were
generated in a 90 ft- long wave tank using a programmable wave generator equipped with
dynamic wave absorption capability. However, sinusoidal wave paddle motion produces
‘free secondary waves’ resulting in wave form modification. Wave height was measured
with four resistance gauges placed outside the offshore end of the beach. The beach was
placed on a wedge-shaped frame covered with geotextile. This arrangement allowed
water to flow through the beach face; somewhat simulating the natural groundwater table.
Beach profile was measured with a semi-automatic profiler several times during each test
to adequately describe the profile evolution. Video and still photographs of the profile
and important features were obtained.
Chapter 4 presents the model selection criteria, a detailed description of the model
and prototype tests, and a summary of the processed and analyzed data. Chapter 4 also
includes the analysis results of profile evolution, equilibrium profile, and transport rate
variation. Lengths and transport rates are expressed in dimensionless forms with the
origin of the coordinate system shifted to the equilibrium still water line. Equilibrium is
specified by relatively small net transport rate, and no change in still water line location
and overall profile shape. Comparison of distinct features of the model and prototype
equilibrium profiles in dimensionless coordinates indicates that the scale effects are
6
different in four zones across the profile. Assuming a uniform scale effect within each
zone, equations are derived to predict the prototype equilibrium profile. Equilibrium
profile prediction from model results indicates generally good agreement with the
prototype.
Chapter 5 includes discussion on the experimental results in the context of scale
effect variation and bottom roughness distortion, limitations of the experiments, and
application of the results. Dominant transport processes in the four scaling zones are also
discussed. Net transport rate asymptotically decays offshore from the wave breakpoint,
and temporally after the initial foreshore slope is established. Bottom roughness
distortion because of exaggerated bedforms is found to be an important cause of scale
effect. The shoaling and local transport processes are affected by high ‘reflection bars’
formed at the antinodes of a partial standing wave system. A profile prediction example
shows that although prediction is not satisfactory near the reflection bars, a single scaling
relation can satisfactorily predict the prototype equilibrium profile within the present
model range.
Chapter 6 presents a summary of this study, and the related conclusions and
recommendations. Additional experiments conducted at other scales are required to
verify the applicability of the predictive equations derived herein. Experiments conducted
with other sediment sizes, specifically those preserving the Dean number, Ho/wT, will
provide more insight into scale effects.
7
CHAPTER 2: LITERATURE REVIEW
The present study investigates scale effects and scaling relations for sinusoidal,
accretionary waves. While laboratory waves differ from ocean waves, wave generators
can reasonably simulate the regular prototype waves in the model. The beach profile
evolves due to wave- induced sediment transport, and assumes a characteristic
equilibrium shape for accretionary or erosive conditions.
Scale effects occur in models because of the inability to represent relative
importance of dominant forces in the prototype. In movable bed scale modeling, the most
challenging task is to select practical model sediment material and size while preserving
important prototype variables. These variables include the densimetric Froude number,
the particle Reynolds number and the dimensionless fall velocity parameter. Since the
cross shore sediment transport processes vary across the profile, scale effects also differ
in different regions on the profile. Appropriate scaling relations can minimize these scale
effects.
8
2.1. Laboratory Waves
2.1.1. Wave Generation
Dean and Dalrymple (1984) and Hughes (1993) presented the general first order
wave generator solution for the wave height to stroke ratio,
( )
+
−+
+=
ldkkdcosh1
kdsinhkd2kd2sinh
kdsinh4SH
(2.1)
where H = wave height, S = wave paddle stroke, d = water depth, k = 2π/L = wave
number and L = wavelength; l = 0 for flap-type wave paddles hinged at the bottom and
∞→l for piston-type paddles. Dean and Dalrymple (1984) showed that movement of
the piston-type wave paddle more closely follows the water particle trajectories under
waves in shallow water.
Periodic waves generated by a sinusoidally oscillating wave paddle include free
secondary waves that propagate at a speed slower than the primary waves. The combined
wave form changes spatially and temporally in the wave tank. The wave form
modification is more pronounced as the wave steepness, H/L, increases or relative depth,
d/L, decreases. Goda (1967) appears to be the first to explicitly describe these ‘free
second harmonic’ waves.
Madsen (1970 and 1971) developed a wave generation theory for relatively long
second order Stokes waves where the prescribed wave paddle motion suppresses the free
second harmonic disturbances. Madsen’s approximate theory is limited to relatively
9
shallow water. Flick and Guza (1980) presented a method to suppress the second
harmonic waves by generating waves in relatively deep water by sinusoidal paddle
motion, and then shoaling them to shallow water.
In theoretical and experimental studies conducted on propagation of tsunamis
onto continental shelves, Goring (1978) showed that the Cnoidal theory approximates
shallow water waves better than the Stokes theory when the Ursell number, U = HL2/d3,
is more than 20. Goring developed wave generation theory for cnoidal waves of
permanent form and experimentally verified the theory using a piston-type wave
generator.
2.1.2. Wave Reflection and Absorption
Waves reflected from the beach change the incident wave characteristics. Hughes
(1993) stated that while wave reflection is correctly represented in a properly scaled
model, re-reflected waves from the wave paddle modify the wave field in the absence of
an active wave absorption system. Hughes presented methods to separate the incident and
reflected waves. Goda and Suzuki (1976) and Goda (1985) developed a method,
originally for irregular waves, for estimation of incident and reflected wave spectra. This
method uses wave record from two gauges placed along a line parallel to the travel
direction of waves. The gauge separation distance, ∆l, is constrained to, 0.05<∆l/L<0.45.
Based on Goda and Suzuki’s method, Hughes (1992) developed a computer program that
separates the incident and reflected waves derived from wave records from three gauges.
10
Dean and Dalrymple (1984) indicated that first order wave theory can be used to
predict the wave paddle motion that would absorb the reflected regular waves. Hughes
(1993) argued that the paddle velocity would not exactly match the wave velocity
resulting in re-reflection; therefore, wave height measurement at the paddle and an active
control feedback system is required. Most wave generators equipped with a wave
absorption system use this technique.
2.2. Beach Profile
2.2.1. Profile Types and Zones
The ‘Shore Protection Manual’ (US Army Corps of Engineers 1984) defined
various zones and features of the wave-beach interface in descriptive terms, as shown in
Figure 2.1. The surf zone and the beach face exhibit the most dynamic response to wave
action. The beach profile assumes characteristic shapes during ‘storm’ and ‘normal’ wave
conditions, as shown in Figure 2.2. A storm profile generally contains a bar, whereas the
unbarred normal profile is identifiable from the berm. Figure 2.2 also identifies the major
zones of the wave-beach interface in terms of wave energy level.
Depending on the relative magnitude of the dominant forces affecting beach
processes, the beach may either erode to produce a bar at the wave breakpoint, or may
accrete resulting in an unbarred profile. Waters (1939) and Rector (1954) attempted to
11
Figure 2.1: Nearshore regions (Dean 2002; after US Army Corps of Engineers 1984).
Figure 2.2: Zones of interest on the wave-beach interface (Wang 1985).
12
establish the criteria based on the deep water wave steepness, Ho/Lo, that define these
profile shapes, where Ho = deep water wave height and Lo = deep water wavelength.
Dean (1973), based on field observation and small scale laboratory experiments,
proposed that offshore bars form if,
85.0wTHo ≥ (2.2)
where w = sediment fall velocity and T = wave period. Based on prototype and large
scale laboratory experiments, Kriebel et al. (1986) suggested that small scale tests can be
described if the constant on the right hand side of Equation 2.2 ranges between 2 and 2.5
with lower values recommended for smaller scales. Kraus et al. (1991) examined large
scale laboratory data and proposed a set of dimensionless curves to separate barred from
unbarred profiles. Their criteria for bar fo rmation are,
5.1
o
o
gTw
115LH
π≥ (2.3)
and 3
o
o
o
wTH
00070.0LH
≥ . (2.4)
Dalrymple (1992) combined and rearranged Equations 2.3 and 2.4, and proposed
a ‘Profile parameter’, given by,
TwHg
P3
2o= . (2.5)
If P exceeds 10,400, the profile is expected to be barred.
13
2.2.2. Forcing Functions for Profile Change
Dean and Dalrymple (2002) described ‘constructive’ and ‘destructive’ forces
responsible for profile change in the nearshore region. Gravity tries to make the beach
profile horizontal, and is the most important destructive force. In some instances, for
example on the shoreward slope of a bar, gravity can act constructively by contributing to
onshore transport. Other destructive forces include high level turbulent fluctuations in the
surf zone due to breaking waves. Turbulent fluctuations transport sediment offshore in
suspension and along the bed. Undertow or the seaward return of mass transport is
another important destructive force. These return flows induce shear stress in the seaward
direction and cause seaward bedload transport.
The most important constructive force is the net onshore-directed bottom shear
stress due to the nonlinearity of the water particle velocities caused by shallow water
waves. In shallow water waves, higher velocities occur under the crest over a shorter
period of time than the offshore-directed velocities under the trough. The mean bottom
shear stress caused by this nonlinearity is given by,
bbb UU8fρ
=τ (2.6)
where f = bed roughness coefficient, ρ = density of water and Ub = near-bottom velocity.
The steady onshore-directed velocity in the bottom boundary layer is another constructive
force. This mean streaming velocity was first observed in the laboratory by Bagnold
(1947) and was further evaluated by Longuet-Higgins (1953). Although the streaming
14
velocity was believed to be the result of viscous effect, Longuet-Higgin’s theoretical
definition of the near-bottom velocity is independent of viscosity, and is given by,
kdsinh16Hks3
U2
2
b = (2.7)
where σ = 2π/T = angular frequency and k = 2π/L = the wave number.
Another constructive force is the suspension and transport of sediment by the
shoreward directed crest velocities. If the settling time of the suspended particles is less
than one half wave period, the particles are deposited shoreward of the location from
which they were suspended (Dean and Dalrymple 2002).
2.2.3. Equilibrium Profile
While a ‘true’ equilibrium is never attained in nature due to the constantly
changing forcing functions, laboratory profiles assume equilibrium shapes for specified
wave conditions. Keulegan (1945) found that a beach with a gentler initial slope takes
longer to reach equilibrium. Scott (1954) concluded that the rate of initial profile change
is greater if the beach is further away from equilibrium.
Sunamura and Horikawa (1974) argued that the equilibrium profile shape is a
function of the initial beach slope, tan β , and proposed three principal profile types.
Collins and Chesnutt (1975) and Chesnutt (1975) suggested that the initial beach slope
influences the final stable profile shape. Kriebel et al. (1986) concluded in small scale
15
experiments that the equilibrium shape for an initial concave profile is significantly
different than that obtained for an initial planar profile.
Rector (1954) suggested that two dimensionless variables, deep water wave
steepness, Ho/Lo, and sediment particle size normalized by deep water wavelength, D/Lo,
are important in equilibrium shape prediction. Eagleson et al. (1963) predicted
equilibrium profile shape seaward of the wave breakpoint from the deep water wave
characteristics, and sediment and fluid properties. Bedload transport was important and
profiles were classified as accreting or eroding based on the deep water wave steepness.
Bruun (1954) concluded that two factors determine equilibrium. First, the onshore
components of the shear stress and wave energy gradient are constant. Second, energy
dissipation occurs only due to bottom friction, and dissipation per unit area is constant.
Based on field observations, Bruun predicted equilibrium shape given by,
32Axd = (2.8)
where d = water depth, x = horizontal distance and A = a sediment particle shape
parameter; coarser sediment gives a larger value of A and a steeper slope.
Nayak (1971) concluded that the dimensionless variable, H/wT, is a significant
parameter in determining the wave reflection coefficient and profile slope. Specific
gravity of the sediment is more important than the grain size in determining the
equilibrium profile slope at the still water line.
16
Based on field observations, Dean (1977) concluded that the power law suggested
by Bruun (1954) is an important criterion to define the equilibrium profile shape.
Considering the energy dissipation per unit volume, D*, equilibrium shape is given by,
( ) 32
32
2*32 x
gg5
D24Axxd
κρ== (2.9)
where xd
dg165
D 21223* ∂
∂κρ= , A = a profile scale factor dependent on energy
dissipation and grain size, and κ = breaker height index. After an empirical relationship
between the grain size and equilibrium energy dissipation developed by Moore (1982),
Dean (1987) expressed the profile scale factor in terms of the fall velocity as,
44.0w067.0A = (2.10)
where A is in m1/3 and w is in cm/sec.
Based on Bagnold’s (1963) transport equations, Bowen (1980) proposed an
equilibrium profile shape equation. Bowen assumed that the net transport at every point
on the beach is zero at equilibrium. Larson (1988) found that sediment transport occurs in
laboratory experiments even when the profile reaches equilibrium. This transport occurs
due to unsteadiness in the experimental conditions, turbulent fluctuations and random
sediment properties while the profile fluctuates about an average shape. In experimental
and numerical studies, Kobayashi and Tega (2002) found non-zero net cross shore
transport rates, and a non-zero difference between the time-averaged sand suspension and
settling rates at equilibrium.
17
Larson et al. (1999) developed theoretical models to predict equilibrium profile
shapes under breaking and non-breaking waves. For breaking waves, the offshore-
directed transport by undertow is balanced by a net sedimentation. Three different models
are proposed for non-breaking waves in the offshore region. The first model assumes
minimum energy dissipation at equilibrium. The second model integrates a shear stress
based transport rate equation over one wave cycle, and assumes zero net transport at
equilibrium. The third model assumes that at equilibrium a balance exists between the
onshore transport caused by wave asymmetry, and the offshore transport by gravity.
2.2.4. Cross Shore Sediment Transport Variation
Larson and Kraus (1989) assumed in a numerical model, SBEACH, that transport
processes are different in four zones on the beach profile. In Zone I, the ‘prebreaking
zone’ or the region offshore of the wave breakpoint, the transport rate at a location x is
given by,
( )bxxbeqq −λ−= (2.11)
where qb = transport rate at the wave breakpoint, xb = horizontal location of the
breakpoint and λ = a spatial transport decay coefficient. Zone II, the ‘breaker transition
zone’ or the region between the breakpoint and the ‘plunge point’, is introduced to link
the computations of Zone I and Zone III, the ‘broken wave zone’. The transport equation
in Zone II is of the same form as in Zone I with a different value of λ. Width of Zone II is
18
approximately 3Hb. Transport in Zone III, between the plunge point and the mean water
line, is given by,
xd
KDDfor
xd
KDDKq eq**eq** ∂
∂ε−>
∂∂ε
+−= (2.12)
and xd
KDDfor0q eq** ∂
∂ε−<= (2.13)
where K = an empirical transport rate coefficient, ε = an empirical coefficient for the
slope dependent term, and 2323* Ag
245
D κρ= = equilibrium energy dissipation per unit
water volume where A = the equilibrium profile shape parameter described by Dean
(1977). The calibration parameter K is found to range between 6101.1 −× m4/N and
6107.8 −× m4/N, while ε is approximately 0.0006. The transport direction in Zone III is
determined by,
3o
o
o
wTH
MLH
> (2.14)
for onshore transport where M = 0.0007 for regular waves. In Zone IV, the ‘swash zone’
or the zone above the mean water line, the transport rate is estimated by a linear decay,
rs
rs xx
xxqq
−−
= (2.15)
where qs = transport rate at the shoreward boundary of the surf zone located at xs, and xr =
horizontal location of the runup limit. The runup height, zR, is given by,
79.0
ooo
R
LHtan
47.1Hz
β= (2.16)
where tanβ = foreshore slope.
19
2.3. Initiation of Sediment Motion
Initiation of sediment particle movement has an important role in transport
process and profile change. While the initiation of motion on a plane bed can be predicted
by the critical criterion established by Shields (1936), additional considerations are
necessary for oscillatory flows.
Madsen and Grant (1976) compared experimental oscillatory flow data with
Shields critical criterion for unidirectional steady flow, as shown in Figure 2.3 where s =
specific gravity of sediment, 2*om vf
21
ρ=τ = maximum bottom shear stress, v* = bottom
shear velocity, and f = bottom friction factor. The general trend of the experimental data,
shown by the vertical range bars, indicates a slightly higher Shields parameter,
Dg)1s(4D
S* −ν
= , than that for unidirectional steady flow. Hallermeier (1980) showed
that for a rough turbulent boundary layer, the critical values of the friction factor, f, and
the densimetric Froude number, D
vF
s
2*
* γρ
= , can be assumed to be 0.01 and 0.04,
respectively, where g)( ss ρ−ρ=γ = submerged unit weight of sediment and ρs = density
of sediment. For a laminar boundary assumption, the estimates of the Shields criterion are
lower than the experimental data.
20
Figure 2.3: Shields criterion and oscillatory flow data (Madsen and Grant 1976).
21
You (1998) developed an empirical equation for predicting near-bottom velocity
amplitude at the initiation of sediment movement under oscillatory flow, given by,
−
ν=ω=
δδ a
DB1
DKau o (2.17)
where dg
4pHT
a d = = bottom orbital amplitude, and K and B are dimensionless
variables given by,
87.0*S053.0K −= (2.18)
and 67.0*S280B −= . (2.19)
The estimates reasonably agree with experimental data and Hallermeier’s results.
2.4. Fall Velocity
Fall velocity, directly related to the particle size, is an important variable in
sediment transport estimation and scaling relations. Considering the viscous and impact
resistance on a particle falling in a still fluid, Rubey (1933) estimated the fall velocity of
a spherical particle. Rouse (1937) expressed the fall velocity of a single particle as a
function of size and temperature. McNown and Lin (1952) included the effect of nearby
particles on the fall velocity given by,
+=
sD
1.31ww o (2.20)
where w = fall velocity of a particle falling in a group, wo = fall velocity of a single
particle in still water, D = particle diameter, and s = distance between adjacent particles.
22
Maude and Whitmore (1958) included the effect of volumetric sediment concentration,
C, on the fall velocity given by,
( )ßo C1ww −= (2.21)
where β = a function of particle shape and size distribution given by,
1.0Rfor65.4 * <=β , (2.22)
3* 10Rfor2.35ß >= , (2.23)
and 3*
0.129* 10R0.1for7.478Dß <<= − (2.24)
where ?Dw
R * = = particle Reynolds number, 31
2
3
* ??gD
D
= , ∆ = (ρs – ρ)/ρ, ρ =
density of water, ρs = density of particle, ν = kinematic viscosity, and g = acceleration of
gravity.
Hallermeier (1981) developed empirical equations to calculate the terminal
settling velocity, w, of commonly occurring quartz and other sand given by,
39Afor18ADw
<=ν
, (2.25)
47.0
10A39for6
ADw<<=
ν, (2.26)
and 645.0 103A10forA05.1Dw
×<<=ν
(2.27)
where 23 ?DgA ∆= .
23
2.5. Scaling Relations in Movable Bed Scale Modeling
Based on scaling laws for river models, LeMehaute (1970) presented a set of
scaling relations for coastal movable bed scale models. LeMehaute argued that since the
goal of a movable bed model is to reproduce the bottom evolution, it is not required to
achieve that through exact similitude of water particle motion. Assuming a turbulent
boundary layer, a ‘natural distortion’ is given by, z21
x NN = , with scaling relations,
21zxT NNN −= , 21
zv NN = and 211D
23z
2xt NNNNN γ′
−−= , where Nx = horizontal length
ratio, Nz = vertical length ratio, NT = wave period ratio, ND = sediment size ratio, γ′N =
submerged unit weight ratio, Nv = flow velocity ratio, and Nt = profile evolution time
ratio. LeMehaute concluded that lighter model sediment provides less distortion, and use
of sand in the model leads to relatively large scale effect.
Noda (1972) presented theoretical scaling relations, given in Table 2.1, which
were tested in laboratory experiments, and the following general laws are proposed:
55.0z
85.1D NNN =γ ′ (2.28)
and 386.032.1zx NNN −
γ′= (2.29)
where γ ′N = 1 if sand is used in both the model and the prototype. Figure 2.4 shows
Noda’s scaling relations for various sediment material and size. Noda suggested that a
lighter material is preferable to sand, and that there is an inherently required distortion of
the length scale indicated by LeMehaute (1970). Noda’s scaling laws require a larger
model sediment size than that required by densimetric Froude number similarity.
24
Figure 2.4: Graphical representation of Noda’s (1972) scaling relations.
25
Table 2.1: Summary of theoretical scaling relations proposed by Noda (1972).
Similitude criteria Similitude variable Proposed scaling relation Froude number and Wave steepness ( ) 21gd
vF =
Ho/Lo
21zv NN =
zL NN = 21
zT NN = Densimetric Froude number
( ) 21*
*Dg
vF
γ′=
2vD *
NNN =γ′
Particle Reynolds number
ν=
DvR *
* 1NN Dv*
=
Bed shear velocity Ufv 21* ∝ 21
f21
zv NNN*
= Chezy coefficient 21
fg8
C
=
1xzf NNN −=
1zx
2C NNN −=
Ratio of horizontal and vertical displacements
v/w = x/y 1zx
w
v NNNN −=
Fall velocity
νγ′
=gD
181
w2
86.050Dw γ′∝
γ′= NNN 2Dw
Table 2.1 indicates that two different sediment sizes are required to preserve the
densimetric Froude number and the particle Reynolds number if the same sediment is
used in the model and the prototype. Also Kamphuis (1972) described this conflicting
sediment size requirement and presented a three-dimensional transport surface, shown in
Figure 2.5, to combine the Shields parameter for initiation of sediment movement, and
the particle Reynolds number. The model and prototype variables should represent the
same point on this surface for no scale effects. The scaling variable α is the proportion of
the bottom shear stress that induces sediment transport.
26
Figure 2.5: Sediment transport surface based on Shields diagram (Kamphuis 1972).
27
Paul et al. (1972) concluded that sediment to fluid property ratios, including
ρs/ρ and D50/Lo, should be preserved to avoid scale effects. Also, the sediment size should
be scaled geometrically at the vertical length scale ratio to meet the Reynolds and Froude
similarity requirements. In almost all scale models, this requires a very small sediment
size in the clay or silt range. Equilibrium profile experiments conducted with sand and
lightweight materials showed that exact similarity between a sand prototype and its
lightweight sediment model is not achieved. Improved similarity is observed when
geometric similarity is preserved and sand is used in the model. Kamphuis (1975b)
concluded that lightweight material is undesirable as model sediment due to local scale
effect caused by incorrect particle acceleration in the model. Since the particle size is
scaled such that the submerged weight scale is equal to the shear force scale to preserve
the underwater particle movement similarity, lightweight sediment particles are relatively
too heavy above water and tend to accumulate near the shoreline.
Mogridge (1974) concluded in laboratory experiments that scaling of bedforms is
important in modeling wave propagation and sediment transport, and sediment of the
same density scaled geometrically should be used in the model. If the required sediment
is in the cohesive range, sediment with a lower density should be used.
Dalrymple and Thompson (1976) showed that although a model law preserving
the dimensionless fall velocity, Ho/wT, is not experimentally proven, the foreshore slope
is uniquely related to this variable. Dalrymple and Thompson suggested that the model
28
should preserve geometric similarity, Ho/wT and Ho/Lo. The same material should be used
in the model and the prototype.
Based on laboratory experiments on dune erosion, Vellinga (1982) developed
scaling relations given by,
( )0.282wzzx NNNN = , (2.30)
( )0.5zt NN = (2.31)
and z2TH NNN == . (2.32)
The exponents are determined from correlation analysis of experimentally found erosion
quantities. The scaling relations for a distorted coastal dune erosion model developed by
Hughes (1983) are given by,
1wz
21zxT NNNNN −− == , (2.33)
21zxT NNN −= , (2.34)
and 1w
23zx NNN −= (2.35)
where Ho/wT is preserved and the model distortion,
1zx NN −=Ω . (2.36)
The model was verified from limited prototype data with satisfactory results.
Dean (1985) suggested that the model should: (a) preserve the Froude number for
waves, (b) preserve geometric similarity, (c) preserve the dimensionless fall velocity
parameter, Ho/wT, and (d) be large so that scale effects due to viscosity and surface
tension are negligible. These criteria follow the relations,
29
21xT NN = (2.37)
and 21x
1Txw NNNN == − . (2.38)
For 1.0250Dw ≅ , 0.49
zD NN50
= , which is in good agreement with Noda’s (1972) empirical
relation for 1N =γ′ . Kriebel et al. (1986) verified Dean’s (1985) model laws in small
scale laboratory experiments using Saville’s (1957) data as the prototype. The model is in
good agreement with the prototype for erosive cases. For acretionary cases, model
predictions are moderately successful only in the offshore region. Multiple bars formed at
the antinodes of a standing wave system. The difference in model prediction occurred due
to the presence of these ‘reflection bars’, and scale effects.
2.6. Beach Experiments
Numerous studies have been conducted to investigate waves, sediment transport,
profile evolution, and equilibrium profile characteristics. Selected studies are reviewed in
this section. Relevant details of other experiments are discussed earlier.
2.6.1. Scale Model Experiments
Meyer (1936), Waters (1939) and Bagnold (1940) were among the first to conduct
systematic laboratory model experiments. These studies indicated scale effects in models
and the importance of preserving several variables including the wave steepness.
30
Rector (1954) conducted experiments in an 85 ft long, 14 ft wide and 4 ft deep
concrete wave tank. The width was divided in four channels so that sand of four different
sizes could be tested simultaneously at the same wave condition. The beach sand, varying
from 0.21 to 3.44 mm in diameter, was placed on several initial slopes including 1:30,
1:20 and 1:15. The deep water wave height varied from 0.3 ft to 0.6 ft while the wave
period ranged from 0.69 to 2.45 sec. Ripples formed in almost all cases. Empirical
equations were developed that related the profile shape and surf zone width.
Eagleson et al. (1963) conducted experiments in a 90 ft long, 2.5 ft wide and 3 ft
deep glass-walled wave tank. Well sorted Ottawa sand with 0.37 mm median diameter
was used to construct the planar initial beach. Wave height was measured with parallel
platinum wire resistance gauges while the wave period was calculated by timing the wave
paddle motion. Equilibrium was determined by visual observation of changes in profile
shape. The beach profile was measured with a point gauge at approximately 0.5 ft
intervals. Empirical relations between the equilibrium beach slope and deep water wave
steepness were proposed.
2.6.2. Prototype Experiments
Saville (1957) conducted experiments in a large wave tank to study profile
evolution toward equilibrium. Nine ‘cases’ or ‘tests’ were conducted with an initially
plane beach constructed of 0.22 mm median diameter sand in an outdoor concrete tank,
635 ft long, 15 ft wide and 20 ft deep. In a similar series of tests conducted at the same
31
facility in 1962 (Kraus and Larson 1988), 0.4 mm median diameter sand was used.
Identical wave conditions were applied in the similar tests of each series. A sinusoidally-
moving, piston-type wave generator was used to produce waves. Initial beach slope was
1:15 in almost all cases. The beach profile was measured at approximately 4 ft intervals
in the first series of tests, and at 0.1 ft intervals in the second series. Wave height was
measured with stepped resistance gauges and by visual observation on the side walls.
Wave reflection was assumed to be insignificant. Equilibrium profile was determined
from successive profile plots while test duration varied between 30 and 100 hours. Each
case was qualitatively identified by characteristic profile features including an inshore
step, a bar or a berm. Porosity of beach sand sand varied across the profile, while the sand
was more compacted near the shoreline.
One of the most significant prototype experiments was conducted by US Army
Corps of Engineers (1994) to investigate the cross-shore hydrodynamics and sediment
transport processes. Details of the 8 week long data collection project, called
SUPERTANK, were summarized by Kraus et al. (1992). Tests were conducted in a 104
m long, 3.7 m wide and 4.6 m deep wave tank with a 76 m long beach constructed of
uniform quartz sand with 0.22 mm median diameter. Monochromatic, and broad and
narrow band random waves were generated by a wave generator equipped with wave
absorption capability. The zero-moment significant wave height varied from 0.2m to
1.0m, while the peak spectral wave period varied from 3 sec to 10 sec. TMA spectral
shape, with a spectral width parameter between 1 and 100, was used to generate the
random waves.
32
Extensive instrumentation was deployed to collect data in three basic areas:
hydrodynamics, sediment transport and beach profile change. The measurement
equipment included 16 resistance wave gauges, 10 capacitance wave gauges, 18 two-
component electromagnetic current meters, 34 optical backscatter sensors, 10 pore
pressure gauges, sediment concentration profilers, and laser Doppler velocimeters. Most
tests started with an initial profile resulting from the previous test. Each test was
conducted in a series of several wave runs or ‘bursts’ with a typical duration of 10, 20, 40
or 70 min. The core measurements for each run included the wave and current data, and
the beach profile survey. The beach profile data were collected with an auto-tracking
infra-red geodimeter. Sixty-six different wave conditions were tested for 129 hrs. Seventy
percent of these waves were random waves. The tests were categorized in 20 major
groups. Five of these groups directly or indirectly studied the equilibrium conditions.
2.6.3. Field Experiments
The Nearshore Sediment Transport Study (NSTS), conducted at Torrey Pines
Beach and Leadbetter Beach, California, is one of the first field experiments (Seymour
1989). The experiments aimed at improving sediment transport prediction in the surf
zone. Measurements included deep water wave characteristics, wave shoaling, near-
bottom velocity across the surf zone, near-surface wind velocity, and beach profile
survey. Predictive capability of analytical models was tested with field data. None of the
models was able to predict transport with reasonable agreement.
33
Field experiments have been conducted at a 560-meter- long pier operated by the
US Army Corps of Engineers at Duck, North Carolina since 1977 (Dean and Dalrymple
2002). Several observations on hydrodynamics and sediment transport processes have
been conducted at this facility. These include shear waves and bar migration.
34
CHAPTER 3: LABORATORY SETUP AND DATA ACQUISITION PROCEDURE
A successful laboratory study requires an appropriate setup of equipment and
instrumentation. It is also necessary to ensure that the quality of acquired data be
acceptable for analysis and interpretation. The equipment and instrumentation for the
physical model experiments discussed in this thesis were carefully selected or constructed
to meet the objectives of the research. This chapter includes a description of the
laboratory equipment and instrumentation, and relevant data acquisition procedures.
3.1. Wave Tank
The physical model was set up in the Hydraulic Research Laboratory at Drexel
University. The 90 ft- long, 3 ft wide and 2.5 ft deep wave tank is constructed of 5 ft long
panels. The side walls and bottom of the tank are made of 3/8 inch thick tempered glass.
Two additional aluminum 5 ft- long segments are attached to the tank ends. The tank is
elevated approximately 2.5 ft from the ground with a braced aluminum support structure.
Because the tank was required to hold sand, additional load bearing capacity for the
bottom glass panels was provided by placing two 2 inch by 4 inch pressure treated
wooden posts below each bottom glass panel across the width at each third point. Each of
these horizontal posts was supported on the floor by a similar vertical post attached to the
middle of the horizontal post. The contact surfaces between the glass and wood were
separated by 0.25 inch thick rubber pads.
35
A recirculating filtration system is installed to clean the water in the wave tank.
The filtration capacity of the unit is approximately 40 gpm. Water is withdrawn from the
wave generator end of the tank, and after filtration released through PVC pipes and
flexible hoses at the beach end of the tank. An additional coarse filter made of wire mesh
and fiber glass fabric was placed at the outlet of the tank from where water was
withdrawn.
3.2. Beach and Beach Support Frame
The sand beach was placed at one end of the wave tank on a 25 ft long, wedge-
shaped frame. This frame reduced the amount of sand required and thus also reduced the
total load in the tank. Figure 3.1 shows a cross-section of the wave tank, and illustrates
important features of the beach support frame. The frame was constructed by joining two
10 ft long segments and one 5 ft long segment made of pressure treated wood. A deck
made of the same type of wood was placed on the frame. The top of the deck was
approximately 21 inch high from the bottom of the tank at the most shoreward end, and
had a slope of about 0.06. A 0.75 inch clearance was maintained between the side of the
decking and glass wall on either side of the frame. This clearance was filled with sponge
packing at several places to avoid direct contact between the glass and decking. Rubber
pads, approximately 0.25 inches thick, were placed between the frame and the bottom
glass at every joint of the aluminum support structure. In order to account for the buoyant
force on the wooden frame, the frame was anchored by 0.25 inch thick and 2 inch wide
aluminum struts from the tank flanges at three places on either side. These struts were
36
Figure 3.1: Cross-section of wave tank showing details of beach support frame and placement of geotextile layer.
37
placed close to the side walls with their width parallel to the wall. The edges were
beveled to minimize their effect on the waves. The frame was also anchored horizontally
into the end tank by aluminum angle bars.
A layer of geotextile was placed on the decking to retain the sand, and to allow
water to flow through the beach. The geotextile layer was attached to the wooden frame
with screws at several places, and was extended approximately 2 ft onto the bottom glass
beyond the offshore end of the frame. All contact lines between the geotextile margin and
the wave tank were sealed with marine grade caulking to prevent sand losses.
Initially two slightly different gradations of sand were considered for beach
construction. The aim was to select sand with the same median grain diameter and size
distribution as used in the prototype SUPERTANK (US Army Corps of Engineers 1994)
tests. Both of these gradations of commercially available sand, ‘Ottawa F-55’ and
‘Ottawa F-52’ silica sand (U.S. Silica Company 1999), had a median grain diameter of
0.212 mm, approximately equal to the median grain diameter of 0.22 mm used in the
SUPERTANK tests. Figure 3.2 shows the grain size distributions of these two sand
gradations along with that of the SUPERTANK tests. The grain size distribution of the
SUPERTANK sand was inferred from the settling velocity data. Since the overall
distribution of ‘Ottawa F-55’ was more similar to that of the SUPERTANK sand,
‘Ottawa F-55’ was used in all model experiments.
38
Figure 3.2: Grain size distribution of sand considered for construction of beach and that of sand used in the SUPERTANK tests.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.010 0.100 1.000
Sediment diameter (mm)
Fra
ctio
n f
iner
SUPERTANK
Ottawa F-55
Ottawa F-52
39
During initial construction of the beach, relatively small frequency waves were
generated in the wave tank while sand was spread on the beach support frame. It was
anticipated that the wave agitation would allow the sand particles to settle on the beach in
the same way as occurs in nature. This procedure of beach construction also eliminated
the possibility of air entrapment within the sand, and rendered a realistic porosity of the
beach. During preparation of beach for the subsequent tests, the sand was thoroughly
cleaned with a hose and the particles were allowed to settle. The beach was then graded
to the specified profile.
3.3. Wave Generator
A hydraulically-driven, piston-type wave generator is installed at one end of the
wave tank. The wave paddle moves on two sets of horizontal tracks supported by a frame
above the tank. A wave absorbing porous screen, approximately 1 inch thick, is mounted
behind the wave paddle to attenuate the waves created by the backward movement of the
paddle. Additional absorption is provided by a PVC pipe wave absorber inside the tank
end.
The wave generator is capable of producing regular and random waves by
accepting paddle position signals sent by a computer program, called ‘WaveGen’ (HR
Wallingford 1996), capable of producing a number of wave spectra including
J.O.N.S.W.A.P and Pierson-Moskowitz spectra. Additionally, random waves can be
generated either by filtered white noise or Fourie r method. Ramp-up and ramp-down
40
periods are included in the program to smoothly start and stop the wave paddle. Real-
time animation displays the paddle position and the wave characteristics on the computer
monitor. ‘WaveGen’ is also capable of accepting user-defined sequences of paddle
position. The signals generated by ‘WaveGen’ are processed in a remotely located
control unit before they are sent to the hydraulic unit. In addition to the ‘WaveGen’
signals, the control unit is capable of accepting direct input signals generated externally
by a signal generator.
The wave generator includes a dynamic wave absorption system to absorb wave
reflection from the beach or other reflective structures. Two wave probes attached to the
front side of the wave paddle measures the instantaneous wave height at the paddle. The
wave absorption module of the control unit compares this signal with the paddle position
signal specified by ‘WaveGen’ to check if the wave paddle is at the correct position. A
corrected paddle position signal is sent from the control unit to the hydraulic unit to
adjust the paddle position backward or forward to match the paddle position with the
signal demanded by ‘WaveGen’.
The wave generator can be operated with or without the wave absorption module.
Although no formal experiment was conducted to test the efficiency of the absorption
system, the performance of the system was occasionally checked immediately after wave
generation was discontinued while the absorption system was still in operation. At this
time, the data acquisition system recorded only one noticeable group of reflected waves
from the beach. Re-reflection of waves from the paddle was found to be relatively
41
insignificant. The reflected and re-reflected wave signals from the data acquisition system
were identified by visual observation of the waves in the tank. Performance of the
absorption system was also checked by visually observing the paddle motion in response
to the in-coming reflected waves. From these observations, the absorption capability of
the system was inferred to be adequate. Characteristics of laboratory waves and
separation of incident and reflected spectra are discussed in Section 4.2.1.
3.4. Instrumentation for Data Acquisition
To acquire accurate and reliable data, a significant part of the research work was
devoted to design, select and devise appropriate data acquisition instrumentation. The
following sections describe the instrumentation set up for data acquisition.
3.4.1. Wave Gauge
Wave height was measured with parallel-wire resistance gauges. Figure 3.3 shows
a typical wave gauge. The gauges were constructed by attaching two 32 gage stainless
steel wires 0.25 inch apart on an L-shaped bracket made of 0.25 inch diameter brass rod.
One end of each wire was connected to a plastic insulator block attached to the end of the
horizontal arm of the brass rod. The other ends of the wires were attached to a plexiglass
arm that extended horizontally from the vertical arm of the brass rod. The vertical arm
was attached to the top of the tank with a bracket so that the gauge could be raised or
lowered in the wave tank as required. The ends of the stainless steel wires attached to the
42
Figure 3.3: Details of a typical wave gauge.
43
plexiglass arm were soldered to RG-174/U shie lded cables which were connected to a
signal processing unit that contained Wheatstone bridges and other signal preprocessing
circuits. The output voltage signal from the signal processing unit was sent to a Data
Acquisition System.
When the gauge is immersed in water, the resistance between the two parallel
wires varies with the depth of immersion with lower resistance for greater immersion.
Resistance is converted to an equivalent voltage in the signal processing unit. To consider
the nonlinearity of immersion depth and voltage relationship, and to compensate for the
signal drift, calibration of the gauges was performed before and after each wave run.
3.4.2. Beach Profiler
The beach profile was measured with a motor-driven point gauge assembly
mounted on a mobile carriage as shown in Figure 3.4. The carriage moves on V-grooved
wheels along 40 ft- long tracks placed on tank flanges. The tracks are made of inverted
aluminum angles elevated from the flanges by spacer blocks. The wave gauge brackets
can be placed through this clearance between the tracks and the tank flanges so that the
profiler can move uninterrupted along the beach.
The point gauge moved upward or downward with an electric motor operated by a
remote control unit. The gauge was modified by replacing the pointer with a lightweight
sensor rod suspended by a spring. Main features of the sensor rod attachment and related
44
Figure 3.4: Details of beach profiler carriage and point gauge assembly.
45
operating circuits are shown in Figure 3.5. The top of the sensor rod triggers a switch as
soon as the bottom of the rod comes in contact with the beach. This switch cuts off power
only to the downward motion of the motor through a relay circuit. As a result, the bottom
of the rod remains in contact with the beach until the point gauge is raised upward.
Sensitivity of the sensor rod can be increased by adjusting the tension of the spring.
Because of the inertia of downward motion, bottom of the rod penetrated loose sand by
relatively small amounts, mostly at the crest of ripples.
The vertical position of the bottom of the sensor rod relative to the bottom of the
wave tank was recorded manually from the Vernier scale on the point gauge. The
horizontal position was recorded manually from a plastic-coated steel tape attached to the
profiler tracks. In addition to the manual measurement of the profile data, the profiler is
equipped to send position coordinate signals directly to the Data Acquisition System. The
vertical position of the point gauge can be recorded by a variable resistor that rotated with
the upward or downward movement of the point gauge. The horizontal position can be
recorded by the change in resistance between two separated stainless steel wires placed
along the tracks. A contact wire attached to the carriage shorts these two wires. As a
result, the resistance value of the wires changes as the carriage moves along the tracks.
The variable resistor and the resistance wires are connected to the Data Acquisition
System with RG-174/U shielded cables. After placing the sensor rod on the beach, these
two resistance values can be sent to the Data Acquisition System by pressing a button on
the remote control unit. Calibration of these two resistors is required before each set of
46
Figure 3.5: Details of profiler sensor rod attachment and operating circuit diagram.
47
profile measurements. However, to save the additional time required for calibration and
data processing, profile data were obtained manually.
The point gauge assembly is placed on the carriage to measure the profile along
the middle of the tank; however, the assembly can be shifted on the carriage to measure
the profile along any other line. During initial tests, the variability of profiles across the
width was checked, and was found to be relatively insignificant in the offshore region.
However, significant variability was observed in the surf zone at the early stages of
profile evolution. This variability diminished as the test progressed. Measurements along
two additional lines on either side of the mid- line were obtained when significant
variability was observed during a profile survey. In addition to the motor-driven point
gauge, a manually operated point gauge is also mounted on the carriage. This point gauge
was used to obtain profile measurements along additional lines over a short distance
without shifting the main point gauge assembly.
3.4.3. Data Acquisition System
The voltage signals from the signal processing unit, and the resistance signals
from direct profile measurements were sent to a computer via a Data Acquisition System
(Agilent Technologies, Inc. 1999). The Data Acquisition Unit (model HP34970A) has a
16-channel multiplexer (model HP34902A) with a maximum scanning speed of 250
channels per second. The maximum open/close speed of channels is 120 per second.
48
The Data Acquisition Unit has an internal storage capacity of 50,000 time-
stamped readings. However, the experimental data were directly recorded in a computer
using a computer program called ‘Agilent BenchLink Data Logger’ (Agilent
Technologies, Inc. 1999). The ‘BenchLink’ program also provides programming
capability of the Data Acquisition Unit.
3.4.4. Measurement Accuracy and Errors
The Data Acquisition System was programmed for 4 wave gauges for a scan
interval of 0.001 sec and a channel delay of 0.0001 sec to achieve the maximum possible
resolution. However, the actual scan interval and channel delay in most cases were found
to be approximately 0.15 sec and 0.04 sec, respectively. The scan interval indicates the
time interval between two consecutive readings of the same gauge while a channel delay
means the time interval between the readings of two consecutive gauges. A scan interval
of 0.15 sec indicates that approximately 20 wave gauge readings are obtained to define a
wave having a period of 3 sec, which is assumed to be adequate. A channel delay of 0.04
sec indicates that there is a time lag of about 0.12 sec between the readings of the first
and the fourth gauges. This time lag was unaccounted for during processing of the wave
data.
Because of possible temperature and material changes in the laboratory and
equipment, a drift of the mean signal level was observed in almost all the wave records.
Figure 3.6 shows a sample of such drift in a wave record. In this sample the voltage
49
signal corresponding to the still water level varied from 7.289 volt at 360 sec to 7.137
volt at 1660 sec. In most cases this drift in the voltage signal was relatively small and
linear, and was compensated by linear interpolation during data processing. In addition to
signal drift, modification of wave form occurred because of ‘free secondary waves’ from
sinusoidal wave paddle motion, and reflected waves from the beach. Temporal change in
wave form is discussed in Section 4.2.1.
Calibration of the gauges was performed before and after each wave run, and the
average of the two calibrations was applied to the entire wave record. The stepped signal
parts of Figure 3.6 shows the change in voltage during calibration of the gauges. Voltage
values for the same immersion depths of the gauge were slightly different during the pre-
and post-calibrations. Figure 3.7 shows the difference in calibration before and after the
wave run. The calibration relation was determined as a nonlinear regression line fitted to
the average values of the two calibrations.
The point gauge Vernier scale allowed measurement of the vertical position of the
beach profile to 0.001 ft. However, because of the reduced sensitivity of the sensor rod on
loose sand, a maximum error of approximately 0.005 ft could occur if the position of the
sensor rod tip was not visually checked and manually adjusted. The horizontal position
tape allowed measurement to 0.01 ft. An estimate to 0.001 ft was recorded while
measuring the horizontal position manually.
50
Figure 3.6: Sample record from data acquisition system demonstrating wave gauge calibration range and drift in voltage signal.
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (sec)
Wav
e g
aug
e si
gn
al (
volt
)
51
Figure 3.7: Difference in pre- and post-calibration of wave gauges.
y = 0.1174x2 - 5.488x + 45.131R2 = 0.9961
0.0
5.0
10.0
15.0
20.0
25.0
4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0
Wave gauge signal (volt)
Gau
ge h
eigh
t (c
m)
Pre-calibrationPost-calibrationAverageRegression
52
Profile measurement by resistance signals provided reasonably accurate data.
Accuracy of estimate in the vertical position was better than that in the horizontal
position in general. Figure 3.8 shows profiles measured by point gauge and by calibrated
resistance. Correlation coefficients for these measurements in the horizontal and vertical
directions were found to be 0.99993 and 0.99955, respectively. In this comparison the
profiles significantly differed at two points.
3.5. Test Procedure
Each model test was carried out in several steps to maximize the consistency and
accuracy of the data. Other data were collected before, during and after each wave run in
addition to the wave and profile data. These data included water temperature, location
and height of selected nodal points of the standing waves, breaker location and height,
wave runup limit, video and still photographs, and measurement of wave height on the
side wall glass at the wave gauge locations. Two rulers were attached permanently to one
side of the wave tank to measure the water level, one near the wave paddle and another
near the beach. A thermometer was also placed permanently on one side of the wave tank
to read the room temperature. A multi-colored grid was drawn on glass side wall of the
tank to define the profile and wave dimensions from the video and still photographs.
53
Figure 3.8: Comparison of profile measurement s by point gauge and calibrated resistance.
-0.5
0.0
0.5
1.0
1.5
2.0
10.015.020.025.030.035.040.0
X (ft)
Y (
ft)
Point gauge
Calibrated resistance
Correlation coefficientsX : 0.99993Y : 0.99955
54
3.5.1. Beach and Wave Tank Preparation
The wave tank was thoroughly cleaned before each ‘test’ or series of ‘wave runs’
to remove the algae and unwanted foreign materials from the tank. The wave probes on
the wave paddle were also cleaned. The initial profile was drawn on the grid on one side
of the tank. While the tank was being filled up with water, a ‘beach grader’ was used to
level the sand along the profile marked on the side wall glass. The ‘beach grader’ was
made of two flat pieces of wood separated in parallel and joined by two perpendicular
pieces of wood. One flat piece of wood was placed on the beach while the other was
placed on top of the tank. The ‘beach grader’ was pulled several times along the beach to
match the sand level in the tank with the profile marked on the side. Relatively less effort
was required to grade the submerged part of the beach. For the part of the beach above
the water level, water was sprayed on the beach while the ‘beach grader’ was being used.
The slope of the beach face was extended to raise the crest to prevent overwash. Before
starting each wave run, an identification label of the run was attached to the side of the
wave tank.
Before starting all subsequent runs, the wave tank was filled with water up to the
required level and any floating material removed. The starting beach profile was marked
on the glass for comparison with the ending profile.
55
3.5.2. Wave Gauge Calibration
The wave gauges were cleaned before each wave run to remove any dirt from the
wires. The gauges were placed in water at their measuring depths for approximately 30
minutes while the Data Acquisition System and signal processing unit were on. This
delay allowed the voltage signal to reach a reasonably constant level. Starting from a
‘zero’ position on the gauge that corresponded to the maximum immersion depth of the
gauge, each gauge was raised in 3 cm steps while the ‘BenchLink’ program recorded the
voltage signals. During processing of wave data, these voltage values were used together
with the corresponding depth values to find the calibration relation. A second calibration
of the gauges was performed after completion of the wave run.
3.5.3. Wave Generation
Before starting each wave run, a trial run of the ‘WaveGen’ program was
performed without activating the hydraulic unit to check the performance of the program
and the wave absorption module of the wave generator. Adjustment of the dial settings of
the wave absorption module in the control unit was required during most of these trial
runs. These adjustments were applied to synchronize the ‘null’ position of the absorption
module and the zero demand signal of ‘WaveGen’.
56
Random and regular (sinusoidal) waves were generated by ‘WaveGen’. Cnoidal
waves were generated by a user-defined sequence of paddle positions. This paddle
position sequence was generated by a computer program (Hinis 2000).
3.5.4. Video and Still Photographs
Still photographs of the initial beach profile, and the profiles after each run were
obtained. All important features of the beach and waves including wave breaking
characteristics, and any irregularity in profile evolution were also photographed. Similar
records were acquired by video camera. Additional video were recorded from the side
and top of the wave tank during and after each run. All video recordings included the
time of recording and the identification label of the run.
3.5.5. Profile Measurement
Profiles were measured by manually recording the vertical and horizontal
positions of the point gauge. A sufficient number of points on the profile were measured
to define the profile adequately. At least three points were measured to define each ripple.
Profile data were recorded to a computer spreadsheet while the profile was being
measured. Measurement errors could be caught by looking at a real-time plot of the data.
All important features of the beach including any asymmetry of the beach across the
width, and ripple formation and irregularity were recorded on the spreadsheet.
57
CHAPTER 4: PHYSICAL MODEL EXPERIMENTS AND RESULTS
Profile evolution experiments were carried out in a movable bed scale model at
three scale ratios with a selected SUPERTANK (US Army Corps of Engineers 1994) test
as the prototype. Each set of model experiments was conducted with sinusoidal and
cnoidal waves. An additional test with random waves was also conducted. This chapter
includes brief descriptions of both prototype and model tests. Descriptions of the data
processing and analysis procedures, and the results of the analysis are also presented in
this chapter.
4.1. Description of Experiments
The scale model was designed by preserving the wave Froude number as follows:
LT2NNor,1
TgH
== (4.1)
where H = wave height, T = wave period, g = acceleration of gravity, p
mT T
TN = = wave
period scale ratio, p
mL L
LN = = length scale ratio, L = characteristic length, and subscripts
m and p denote model and prototype, respectively. The same scale ratios were selected
for wave height and horizontal length while model wave period was determined from
Equation 4.1. The vertical length scale ratio was initially attempted to be determined by
preserving the fall velocity parameter or Dean number,wTHo , which leads to the relation,
58
Twz NNN = (4.2)
where z = vertical length, w = sediment fall velocity, p
mz z
zN = = vertical length scale
ratio, and p
mw w
wN = = sediment fall velocity ratio. For the same sand in the model and
the prototype Nw = 1, and Nz = NT from Equation 4.2. However, the available vertical
dimension of the wave tank was insufficient to accommodate the model initial profile
height calculated from Equation 4.2. Alternatively, a geometric similarity was preserved
by maintaining xz NN = , where x = horizontal length and p
mx x
xN = = horizontal length
scale ratio. The scaling relations and their impact on the experimental results are
discussed in Chapter 5.
Table 4.1 summarizes the basic variables of the model and the prototype tests
while the model water depth was determined from the vertical length scale ratio. The
prefix ‘ST’ stands for SUPERTANK prototype tests and the prefix ‘DST’ indicates
Drexel scale model experiment of the corresponding prototype test identified by the two
alphanumeric characters following it. The last digit of the model test ID indicates the test
number in the series. Table 4.1 also indicates that the horizontal and vertical length scale
ratios were identical in four tests, and only approximately identical in three other tests.
Therefore, it was assumed that the model was geometrically similar to the prototype.
59
Table 4.1: Summary of model and prototype variables.
Test ID Duration (min)
Wave period (sec)
Wave height (cm)
Wave scale ratio
Horizontal length
scale ratio
Vertical length
scale ratio
Wave type
ST10_1 270 3 80 Prototype test Random STi0_1 570 8 50 Prototype test Sinusoidal DST10_1 311 1.05 8 1/10 1/10 1/9.5 Random DSTi0_1 578 2.53 5 1/10 1/10 1/9.5 Sinusoidal DSTi0_2 718 2.53 5 1/10 1/10 1/9.5 Cnoidal DSTi0_3 506 2.74 5.88 1/8.5 1/8.5 1/8.5 Sinusoidal DSTi0_4 638 2.74 5.88 1/8.5 1/8.5 1/8.5 Cnoidal DSTi0_5 536 2.41 4.55 1/11 1/11 1/11 Sinusoidal DSTi0_6 579 2.41 4.55 1/11 1/11 1/11 Cnoidal
During the prototype and model experiments, each ‘Test’ was conducted in a
sequence of several ‘Runs’ while the wave generator was continuously operated during a
run. Each run of the cnoidal wave tests, however, comprised of several time periods of
continuous operation of the wave generator, the usual duration of each period being 25
minutes.
STi0 was the primary prototype test for this study and was conducted to
investigate the accretionary behavior of a beach under sinusoidal wave action toward
equilibrium. An equilibrium condition was assumed to occur when the net sediment
transport at every point on the beach was relatively insignificant. However, the last run of
this test was carried out to examine the effect of a varying water level on the beach. The
prototype data excluding this run was designated STi0_1. Prototype test ST10 comprised
of several runs of different random and monochromatic wave conditions. One of the
objectives of this test was to investigate the profile’s approach to equilibrium under
60
erosive waves. The first 5 Runs of ST10 with random waves were selected for this study
and was designated ST10_1 herein.
Model tests DSTi0_1, DSTi0_3 and DSTi0_5 were conducted with sinusoidal
waves at three different scales. Tests DSTi0_2, DSTi0_4 and DSTi0_6 were conducted at
the same three scales but with cnoidal waves. DST10_1 was the model equivalent of the
prototype test ST10_1. The last run of each model test was conducted with a wave
condition different than that of the main test to observe the initial response of the existing
equilibrium profile to a different water level or wave type.
Appendix C describes the salient features of the prototype and model tests related
to wave generation along with location of the Still Water Line (SWL). Beach profile was
measured after each run except during Tests DSTi0_5 and DSTi0_6. During processing
of the data the origin of the coordinate system was shifted to the equilibrium SWL, or the
final SWL for tests in which reaching equilibrium was not the objective. In this
coordinate system a positive x indicates offshore, and a positive y indicates above the
equilibrium or final SWL. The SWL column in Appendix C indicates the water depth at
the wave paddle.
Appendix C also includes water temperature measured before each run of the
model tests. For the entire testing period the temperature ranged from 17 oC to 25 oC
while both the median and mean values were approximately 19.5 oC.
61
4.2. Data Processing Procedures
Wave and profile data were processed before their analysis. To compare results,
similar processing was performed for the prototype data. The following sections describe
the data processing procedures.
4.2.1. Wave Data
Wave height was measured with four wave gauges outside the offshore end of the
beach. The gauges were calibrated for each run as described in Sections 3.4.1, 3.4.4 and
3.5.2. Wave data were processed to account for the signal drift during the run by linearly
distributing the total drift over the record length.
Waves generated in the laboratory are not purely sinusoidal or cnoidal. Figure 4.1
shows how the wave form changes with time during Test DSTi0_1. Changes are more
obvious as the initial foreshore slope is established at about 185 minutes into the test, and
near the equilibrium at approximately 575 minutes. The specified wave period and wave
height for these sinusoidal waves are 2.53 sec and 5.0 cm, respectively. Figure 4.2 shows
similar changes during a test conducted with cnoidal waves and the same specified wave
period and height.
The waves shown in Figures 4.1 and 4.2 are combinations of the incident waves
from the wave generator and the reflected waves from the beach. A computer program,
62
Figure 4.1: Changes in wave form during Test DSTi0_1.
-3-2-10123456
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Time (sec)
Mea
sure
d w
ave
heig
ht (
cm)
50 min
Sinusoidal
-3-2-10123456
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Time (sec)
Mea
sure
d w
ave
heig
ht (
cm)
185 min
Sinusoidal
-3-2-10123456
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Time (sec)
Mea
sure
d w
ave
heig
ht (
cm)
575 min
Sinusoidal
63
Figure 4.2: Changes in wave form during Test DSTi0_2.
-3-2-10123456
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Time (sec)
Mea
sure
d w
ave
heig
ht (
cm)
50 min
Cnoidal
-3-2-10123456
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Time (sec)
Mea
sure
d w
ave
heig
ht (
cm)
185 min
Cnoidal
-3-2-10123456
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Time (sec)
Mea
sure
d w
ave
heig
ht (
cm)
575 min
Cnoidal
64
called ‘PC Goda’ (Hughes 1992), separated the incident and reflected spectra, as shown
in Figures 4.3 and 4.4. While most incident waves are generated at the specified
frequency of 0.395, secondary wave components exist at other frequencies.
‘PCGoda’ separates the incident and reflected waves based on the method
proposed by Goda and Suzuki (1976). Since the results of ‘PCGoda’ were sensitive to
gauge spacing, guidelines recommended by Goda and Suzuki were followed carefully.
These guidelines recommend that the minimum and maximum spacing be 0.05 and 0.45
of the wavelength, respectively. Table 4.2 indicates the gauge separation distance for
each test. The reference gauge was closest to the wave paddle and located at least ten
times the water depth away.
Table 4.2: Wave gauge separation distances.
Distance from reference gauge (cm) Test ID Gauge 1 Gauge 2 Gauge 3 Gauge 4
DST10_1 0 25 50 80 DSTi0_1 0 15 40 60 DSTi0_2 0 15 40 60 DSTi0_3 0 15 40 60 DSTi0_4 0 15 40 60 DSTi0_5 0 25 50 70 DSTi0_6 0 25 50 70
During the tests the mean water surface elevations near the wave paddle and
behind the beach were found to be slightly different. Appendix D summarizes and
explains a set of measurements of the water surface elevations collected during a run. The
effect of this small difference in water surface elevations is deemed insignificant.
65
Figure 4.3: Incident and reflected spectra for Test DSTi0_1.
0
50
100
150
200
250
0 0.5 1 1.5 2Frequency (Hz)
Ene
rgy
(cm
2 -se
c)
IncidentReflected
Sinusoidal50 min
0
50
100
150
200
250
0 0.5 1 1.5 2Frequency (Hz)
Ene
rgy
(cm
2 -se
c)
IncidentReflected
Sinusoidal185 min
0
50
100
150
200
250
0 0.5 1 1.5 2Frequency (Hz)
Ene
rgy
(cm
2 -se
c)
IncidentReflected
Sinusoidal575 min
66
Figure 4.4: Incident and reflected spectra for Test DSTi0_2.
0
50
100
150
200
250
0 0.5 1 1.5 2Frequency (Hz)
Ene
rgy
(cm
2 -se
c)IncidentReflected
Cnoidal50 min
0
50
100
150
200
250
0 0.5 1 1.5 2
Frequency (Hz)
Ene
rgy
(cm
2 -se
c)
IncidentReflected
Cnoidal185 min
0
50
100
150
200
250
0 0.5 1 1.5 2Frequency (Hz)
Ene
rgy
(cm
2 -se
c)
IncidentReflected
Cnoidal575 min
67
4.2.2. Beach Profile Data
Beach profiles were initially measured with respect to a fixed location in the wave
tank. The origin of coordinates of the se measured data and the prototype profile data was
translated to the equilibrium SWL or the final SWL, whichever appropriate. Profile data
were further processed to obtain interpolated profile height, d, at equal horizontal
intervals.
To demonstrate the evolution of the beach profile with time, all profiles from a
test were plotted on a single graph. An interpolated surface was plotted from these profile
data using a surface mapping computer program ‘SURFER’ (Golden Software, Inc.
1994). Figure 4.5 is a sample plot that shows profile changes along the middle line of the
beach with time, t. A triangulation scheme with linear interpolation was selected in
‘SURFER’. Figure 4.6 shows the individual profiles without the interpolated surface.
Profiles for all tests are in Appendix E.
4.3. Data Analysis and Results
An important part of the tests was to establish the beach profile equilibrium
condition. The relevant variables of both model and prototype tests were cast in
comparable dimensionless forms. The profiles were expressed in terms of a
dimensionless horizontal distance, x/gT2, and a dimensionless depth, d/wT, where x and d
are the horizontal and vertical distances from the equilibrium SWL shoreline,
68
Figure 4.5: Measured profiles with interpolated surface.
Figure 4.6: Individual profiles without interpolated surface.
ST i0
STi0_1
69
respectively. Evolution of the dimensionless profiles is shown with surface plots prepared
using ‘SURFER’, and are included in Appendix F.
4.3.1. Transport Rate and Beach Equilibrium
Net cross shore sediment transport rate at a point on the beach was calculated
from the profile difference befo re and after a run. An eroded volume contributed to the
offshore transport and an accreted volume accounted for the onshore transport. The
volumetric transport rate was converted to a submerged weight transport rate and
expressed as the average transport rate for the run at that point. When sand is conserved
across the profile the total onshore and offshore volume changes should be equal, and the
cumulative volume change at the two ends of the profile should be zero. However, a non-
zero cumulative volume change at the offshore end resulted due to some offshore losses
and beach consolidation. To adjust the transport rate calculated from the volume change,
the closure error was distributed proportionately to the local volume changes across the
profile. Figure 4.7 shows the cumulative volume change across the profile before and
after a typical closure error correction.
The net transport rate was expressed in the dimensionless form, q/(ρs-ρ)gw4T3,
where q = net transport rate, sρ = sediment density, and ? = fluid density. Figure 4.8
shows cross shore variation in net transport rate where net offshore transport is positive.
Figure 4.8 also shows the corresponding profiles plotted in dimensionless coordinates.
Transport rates are minimum or maximum where the two profiles cross.
70
Figure 4.7: Profile closure error and correction.
-8
-6
-4
-2
0
2
4
-25 0 25 50 75 100 125 150 175
Distance (ft)
Cu
mu
lati
ve V
olu
me
Ch
ang
e (f
t3/f
t)
Uncorrected
Corrected
71
Figure 4.8: Variation of net transport rate across the beach during a run.
Figure 4.9: Surface plot showing variation of transport rate during a test.
-10.0
-5.0
0.0
5.0
-0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
x/gT2
d/w
T
-0.0016
-0.0014
-0.0012
-0.0010
-0.0008
-0.0006
-0.0004
-0.0002
0.0000
0.0002
0.0004
Net
tra
nsp
ort
rat
e, q
/ ∆gw
4 T3
Profile 1
Profile 2
Transport Rate (Corrected)
Positive implies offshore transport
DSTi0_1
72
To demonstrate the variation in transport rate with time during a test, the
dimensionless transport rate plots for all runs of the test were placed sequentially, and an
interpolated surface plotted using ‘SURFER’. Figure 4.9 shows the variation of transport
rate during a test. q_nd is the dimensionless net transport rate indicated earlier. Appendix
G presents transport rate surface plots for all tests. Initial net transport rates were
relatively high when the profile was far from equilibrium. Transport rates were relatively
low as the profile approached equilibrium. During an accretionary test, onshore transport
predominates for almost the entire test duration. Near equilibrium, the net transport
fluctuates between offshore and onshore directions. This transport direction fluctuation
and relatively low dimensionless transport determines whether the profile has reached
equilibrium. For all tests including the prototype tests the dimensionless net transport rate
on the beach was within approximately ± 0.01 at equilibrium.
Equilibrium was specified in two ways, the first when there was no longer any
change in the horizontal SWL location. Figure 4.10 shows how the SWL location moved
as it approached equilibrium. The horizontal distance increases offshore as the beach
accretes. The equilibrium SWL locations are different since the initial profile for each test
was at a slightly different location in the tank. Sinusoidal and cnoidal wave tests
conducted at the same scale started with the same profile at the same location. However,
Figure 4.10 shows that the equilibrium SWL locations for sinusoidal wave tests differed
from those of the cnoidal wave tests. Figure 4.11 shows how the SWL moved with
respect to the equilibrium SWL. While this relative SWL location approached zero at
73
Figure 4.10: Migration of measured location of SWL.
Figure 4.11: Migration of SWL with respect to equilibrium SWL.
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
0 100 200 300 400 500 600 700 800
Time (min)
Ho
rizo
nta
l lo
cati
on
of
SW
L (
ft)
DSTi0_1DSTi0_2DSTi0_3DSTi0_4DSTi0_5DSTi0_6
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
0 100 200 300 400 500 600 700 800
Time (min)
Ho
rizo
nta
l dis
tan
ce o
f S
WL
fro
m e
qu
il. S
WL
(ft
)
DSTi0_1DSTi0_2DSTi0_3DSTi0_4DSTi0_5DSTi0_6
74
equilibrium the SWL movement in Figures 4.10 and 4.11 show that the SWL moved
offshore for all tests indicating accretion.
The second specification of equilibrium was based on the root mean square error
(RMSE) of profile elevations between consecutive profiles. The RMSE was divided by
the corresponding run duration so the average RMSE in ft/sec indicates the overall
change on a profile during the run. Figures 4.12 and 4.13 show how the average RMSE
varies with time for sinusoidal and cnoidal tests, respectively. The dotted lines are
exponential trend lines fitted to the data assuming that the average RMSE reaches a non-
zero equilibrium asymptotically. A zero RMSE would be a ‘perfect’ equilibrium or no
profile change with time. Although the general shape of the profile is unchanged at
equilibrium, small movements of bed ripples contribute to the RMSE. The change in net
transport direction at equilibrium also causes relatively small changes on the profile
contributing to the non-zero RMSE. The magnitude of these changes was greater for the
prototype tests.
Based on the equilibrium criteria, one profile from each test was selected as the
equilibrium profile shape. Figure 4.14 shows the equilibrium profiles for tests conducted
with sinusoidal waves. Figure 4.15 shows the same profiles with the prototype
equilibrium profile in dimensionless coordinates. Although the model equilibrium
profiles were similar to one another, they were significantly different from the prototype
equilibrium profile. This difference is due to scale effects as discussed in Section 4.3.2
and Chapter 5.
75
Figure 4.12: Variation of RMSE between consecutive profiles for sinusoidal wave tests.
Figure 4.13: Variation of RMSE between consecutive profiles for cnoidal wave tests.
0.00000
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0 100 200 300 400 500 600
Time (min)
Ave
rag
e R
MS
E o
f mo
del
test
s (f
t/se
c)
0.00000
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0.00007
0.00008
0.00009
0.00010
Ave
rag
e R
MS
E o
f pro
toty
pe
test
(ft/
sec)
DST i0_1DST i0_3DST i0_5ST i0_1
0.00000
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0 100 200 300 400 500 600 700
Time (min)
Ave
rag
e R
MS
E o
f mo
del
test
s (f
t/se
c)
DST i0_2
DST i0_4
DST i0_6
76
Figure 4.14: Selected equilibrium profiles in measured coordinates.
Figure 4.15: Selected equilibrium profiles in dimensionless coordinates.
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
-5.0 0.0 5.0 10.0 15.0 20.0 25.0
x (ft)
d (f
t)DSTi01_13
DSTi03_13DSTi05_11
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
-0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
x/gT2
d/w
T
DSTi01_13
DSTi03_13
DSTi05_11STi01_10
77
4.3.2. Scaling Relations
Although the initial model profiles were determined from pre-selected length
scales for a geometrically similar model, because of scale effects the actual equilibrium
length scales were different. While the variation of this profile nonsimilarity represents
the scale effect, the actual length scales can be found by comparing the coordinates of
distinct features on the model and prototype equilibrium profiles. These features are
indicated in Figure 4.16 and listed in Table 4.3.
Table 4.3: Identification of profile features.
Feature number
Description
1 Runup limit 2 Still Water Line (SWL) 3 Foreshore inflection point 4 Slope discontinuity 5 Bottom of trough 6 Top of first bar 7 End of first bar 8 Top of second bar 9 Top of third bar 10 Top of fourth bar
The foreshore inflection point is the location where the foreshore slope changes
from concave to convex. The slope discontinuity indicates the location where the
foreshore return flow meets with the incident broken waves. Figure 4.16 also shows four
scaling zones within which the scale effects are fairly uniform as discussed later.
78
Figure 4.16: Distinct features and scaling zones on an equilibrium profile.
1
2
3
4
56 7 8 9 10
I II III IV
79
Figures 4.17 and 4.18 show how the scale ratios Nx and Nz vary across the profile,
as determined from equilibrium profile coordinates for the similar features. The
variations in all tests are almost identical in Zones I, II and III, and are within a relatively
small envelope in Zone IV. The ratios of the corresponding model and prototype
dimensionless length coordinates exhibit similar variation as shown in Figures 4.19 and
4.20, where Xr and Zr are the ratios of the model and prototype equilibrium profile
coordinates x/gT2 and d/wT, respectively. The smaller envelope of variation in Zone IV
indicates that scale effects in all tests are better represented in terms of the dimensionless
coordinates x/gT2 and d/wT. Figure 4.21 shows that the relationship between the model
and prototype horizontal length x/gT2 is approximately linear in the four zones of
similarity across the profile. Similarly, Figure 4.22 shows that the relationship between
model and prototype vertical length, d/wT, also can be assumed to be linear in the same
four zones. The sediment transport processes in the four zones are significantly different,
leading to the difference in scale effect. The implications of these four scaling zones with
respect to transport processes and scaling rela tions are discussed in Chapter 5.
80
Figure 4.17: Variation of horizontal length scale ratio with horizontal distance.
Figure 4.18: Variation of vertical length scale ratio with horizontal distance.
0.00
0.05
0.10
0.15
0.20
0.25
-5.0 0.0 5.0 10.0 15.0 20.0 25.0
x (ft)
Nx
DSTi01_13DSTi03_13DSTi05_11
0.00
0.05
0.10
0.15
0.20
0.25
0.30
-5.0 0.0 5.0 10.0 15.0 20.0 25.0
x (ft)
Nz
DSTi01_13DSTi03_13DSTi05_11
81
Figure 4.19: Variation of dimensionless horizontal length coordinate ratio.
Figure 4.20: Variation of dimensionless vertical length coordinate ratio.
0.0
0.5
1.0
1.5
2.0
2.5
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
x/gT2
Xr
DSTi01_13DSTi03_13DSTi05_11
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
x/gT2
Zr
DSTi01_13DSTi03_13DSTi05_11
82
Figure 4.21: Relationship between model and prototype dimensionless horizontal length coordinates.
Figure 4.22: Relationship between model and prototype dimensionless vertical length coordinates.
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
Model x/gT2
Pro
toty
pe
x/g
T2
DSTi01_13DSTi03_13DSTi05_11
I II III IV
-7.00
-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
Model d/wT
Pro
toty
pe
d/w
T
DSTi01_13DSTi03_13DSTi05_11
83
The linear relationship among the dimensionless length variables in the scaling
zones can be expressed as follows:
1m1p bxax +′=′ (4.3)
and 2m2p bdad +′=′ (4.4)
where x′ = the dimensionless length, x/gT2; d′ = the dimensionless length, d/wT; the
subscripts m and p denote model and prototype, respectively. The variables a1, a2, b1 and
b2 can be determined from the straight lines fitted to the profile coordinate data in each
scaling zone. See Appendix H for details. Equations 4.3 and 4.4 can be expanded and
rearranged as follows:
2p1m
2
T1p Tgbx
N1
ax +
= (4.5)
pp2mTw
2p TwbdNN
1ad +
= (4.6)
Since the same sediment was used in model and prototype, Nw = 1 in Equation 4.6.
Prototype equilibrium profile coordinates, xp and dp, can be predicted using
Equations 4.5 and 4.6 from the nominal scale ratios, the model equilibrium profile
coordinates xm and dm, and the straight line variables found from the scaling relations in
the four zones. Equations 4.5 and 4.6 are independent of the length scale ratios. Figures
4.23, 4.24 and 4.25 show the predicted prototype equilibrium profiles from model
variables. For all three tests the predicted profiles show reasonably good agreement with
the prototype.
84
Figure 4.23: Prototype profile calculated from DSTi01_13 profile.
Figure 4.24: Prototype profile calculated from DSTi03_13 profile.
Figure 4.25: Prototype profile calculated from DSTi05_11 profile.
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
-25.0 0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0
x (ft)
d (f
t)
Calculated from DSTi01_13
Measured STi01_10
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
-25.0 0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0
x (ft)
d (f
t)
Calculated from DSTi03_13
Measured STi01_10
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
-25.0 0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0
x (ft)
d (
ft)
Calculated from DSTi05_11
Measured STi01_10
85
CHAPTER 5: DISCUSSION ON EXPERIMENTAL RESULTS
5.1. General Considerations
5.1.1. Critical Criteria for Eroding or Accreting Profile
Eroding and accreting profiles are fundamentally different, and can be
distinguished by empirically derived critical criteria. Although the model and prototype
tests have accreting profiles, as shown later, three major critical criteria cited in literature
are tested with the test data. Table 5.1 shows the critical values calculated for the model
Tests DSTi0_1, DSTi0_3 and DSTi0_5, and the prototype Test STi0_1.
Table 5.1: Comparison of critical criteria for eroding or accreting profile.
Dean (1973)
Dalrymple (1992)
Kraus et al. (1991) Test
Ho/Lo
Ho/wT gHo2/w3T 115(πw/gT)1.5 0.0007(Ho/wT)3
DSTi0_1 0.0045 0.544 222.3 0.0311 0.00011 DSTi0_3 0.0045 0.580 274.1 0.0275 0.00014 DSTi0_5 0.0045 0.510 186.1 0.0334 0.00009 STi0_1 0.0045 1.689 6777.8 0.0055 0.00337
Accreting if < 0.85 < 10,400 > Ho/Lo > Ho/Lo
The test data agree with Dean’s (1973) criterion that Ho/wT should exceed 0.85
for eroding profiles to exist. The prototype data do not fit with this criterion. Both model
and prototype results agree with the modified criterion presented by Kriebel et al. (1986)
that Ho/wT should exceed a value between 2 and 2.5 for an eroding profile. Model and
86
prototype results also agree with Dalrymple’s (1992) criterion that the critical variable,
the ‘profile parameter’, gHo2/w3T, should exceed 10,400 for eroding profiles. The dual
criteria proposed by Kraus et al. (1991) are partially met by the model results. In both
cases, the deep water wave steepness, Ho/Lo, should be greater than critical variable for
eroding profiles. The prototype results show marginal agreement with the criteria.
5.1.2. Reflection Bars
A partial standing wave system developed as the beach approached equilibrium.
Figure 5.1 shows distinct nodes and antinodes of the system on the wave crest envelope.
A high point indicates a node and a low point indicates an antinode. Pronounced bars
offshore of the break point can be seen at the antinode locations. These ‘reflection bars’
developed because of the effect of the standing wave system on local transport. Sediment
movement and wave propagation were affected by these exaggerated features, thus
affecting the transport process in the offshore region and introducing scale effect.
Additional scale effects were introduced in the surf zone as the reflection bars altered the
shoaling properties.
5.1.3. Beach Face Infiltration
Significant infiltration from wave runup occurred through the beach face. The infiltration
rate was higher than the recovery rate of water flow back to the beach. Consequently, the
mean water level behind the beach in the tank was higher than the level in front. A head
87
gradient between the two water levels simulated the seaward flow of water through a
beach due to natural groundwater table. Appendix D presents water level measurements
that demonstrate the varying conditions. An in situ permeability test was conducted to
measure the infiltration rate. The test results, described in Appendix B, indicate that the
coefficient of permeability of the beach and the geotextile layer combined was
approximately 0.13 cm/sec. Since both the model and the prototype used the same
sediment size, infiltration rates through their beach faces were of the same order. The
beach face infiltration rate would be lower in the model if smaller sediment size was
selected using other scaling laws.
88
Figure 5.1: Reflection bar and wave crest envelope.
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
-2.0 3.0 8.0 13.0 18.0
Horizontal distance (ft)
Bea
ch p
rofi
le e
leva
tio
n (f
t)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Wav
e cr
est e
leva
tio
n (f
t)
Beach profile
Wave crest
89
5.2. Sediment Transport Processes in Scaling Zones
Four distinct scaling zones are indicated in Chapter 4. Since the transport processes
are significantly different in each of these zones, different magnitude of scale effect is
expected in each zone. In Zone I, the ‘swash zone’, sediment transport occurs under
oscillatory sheet flow while onshore wave propagation on the beach face is roughly
approximated by movement of a bore. Broken incident waves propagate up the slope to
the runup limit and return under gravity. Assuming negligible wave setup, the mean
seaward limit of the return flow is approximately at the still water line (SWL). Transport
in Zone I is a function of sediment properties, local beach slope and incident wave
characteristics.
Below the SWL, in Zone II, the ‘surf zone’, the return flow from the beach face acts
in conjunction with the broken waves or bore. As the beach approaches equilibrium, the
return flow meets the broken incident waves causing intense turbulence at the seaward
limit of this zone. The turbulence creates a sharp discontinuity in slope, and suspends a
significant amount of sediment. The suspended sediment moves partially onshore with
incident waves and partially offshore by return flow or undertow.
In Zone III, the ‘breaker zone’, offshore transport occurs by undertow while
breaking waves carry the suspended sediment onshore by turbulence. Because of the
undertow both bedload and suspended transport are important, the latter dominating.
Significant transport in the breaker zone is observed even when the profile has reached
90
equilibrium. Since the net transport is zero at equilibrium, the onshore and offshore
transports balance over one wave cycle. The undertow moves sediment offshore as
bedload to the seaward limit of this zone (the break point ), where it is suspended and
transported onshore by incident waves. Some of the suspended sediment deposits while
in transit and an overall equilibrium exists in each wave cycle among suspended,
deposited and transported sediment.
In Zone IV, the ‘offshore zone’, bedload transport and bottom boundary layer
processes are more significant. As waves shoal, energy is dissipated by bottom friction in
the form of grain roughness or overall bedform resistance. Sediment motion occurs when
the velocity in the bottom boundary layer exceeds a critical value. Because of shallow
water transformations waves attain narrower and higher crests, and broader and shorter
troughs; the bottom velocity being onshore-directed under the crest and offshore-directed
under the troughs. The duration of the onshore- or offshore-directed velocity relative to
the duration of suspension of the sediment determines the net direction of transport.
Additionally, a mean onshore directed streaming velocity at the bottom boundary layer
transports sediment onshore while gravity transports sediment offshore.
Spatial and temporal variations of net transport rate are demonstrated by ‘surface
plots’ in Appendix G where offshore transport is positive. Most transport action occurred
in Zones I, II, and III while transport rates changed asymptotically offshore from a
maximum at the break point. Figures 5.2, 5.3 and 5.4 show the typical temporal changes
of net transport rate at a point in each of the four scaling zones. At the beginning of each
91
Figure 5.2: Temporal variation of net transport rate during Test DSTi0_1.
Figure 5.3: Temporal variation of net transport rate during Test DSTi0_3.
Figure 5.4: Temporal variation of net transport rate during Test DSTi0_5.
DST i0_1
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0 100 200 300 400 500 600
Time (min)
Net
tra
nsp
ort
rat
e, q
/g
w4 T
3
Zone I
Zone II
Zone III
Zone IV
DST i0_3
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0 50 100 150 200 250 300 350 400 450 500
Time (min)
Net
tra
nsp
ort
rat
e, q
/g
w4T
3
Zone IZone II
Zone IIIZone IV
DST i0_5
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0 100 200 300 400 500 600
Time (min)
Net
tra
nsp
ort
rat
e, q
/g
w4T
3
Zone IZone II
Zone IIIZone IV
92
test, relatively high transport rates occurred indicating that the initial profiles were far
from equilibrium. During this initial period, relatively fast profile shape changes were
observed. Transport rates moved asymptotically toward equilibrium after peaking
approximately when the initial foreshore slope was established.
5.3. Scale Effects
5.3.1. Relative Magnitude of Scale Effect
In the absence of scale effects, a linear relationship would exist at equilibrium
between the model and prototype dimensionless length coordinates. This relationship is
represented by a straight line with unit slope passing through the origin, as shown in
Figures 5.5 and 5.6. The relationships among the length coordinates deviate from
linearity because of scale effects; however, they can be assumed to be piecewise linear in
the four zones, suggesting similarity of the processes and uniform scale effect within each
zone. The slope of a straight line fitted to the coordinate data gives an indirect measure of
the scale effect. The coefficient a1 in Equations 4.3 and 4.5, and the coefficient a2 in
Equations 4.4 and 4.6 indicate these slopes. A slope less than unity means the model
overestimates the dimensionless length, x/gT2, or underestimates the dimensionless
depth, d/wT. Calculation of these slope coefficients is described in Section 4.3.2 and
Appendix H. A summary of their values is given in Table 5.2.
93
Figure 5.5: Scale effect on dimensionless horizontal length coordinates.
Figure 5.6: Scale effect on dimensionless vertical length coordinates.
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
Model x/gT2
Pro
toty
pe
x/g
T2
DSTi01_13DSTi03_13DSTi05_11No scale effect
I II III IV
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
Model d/wT
Pro
toty
pe
d/w
T
DSTi01_13DSTi03_13DSTi05_11No scale effect
IIIIIIIV
94
Table 5.2: Summary of slope coefficients indicating magnitude of scale effect.
Slope coefficient of x/gT2 Slope coefficient of d/wT Test Nx Zone
I Zone
II Zone III
Zone IV
Zone I
Zone II
Zone III
Zone IV
DSTi0_1 0.100 0.473 4.174 0.398 0.470 1.449 2.732 1.416 2.363 DSTi0_3 0.118 0.495 4.587 0.340 0.517 1.552 2.963 1.126 2.627 DSTi0_5 0.091 0.429 3.355 0.317 0.464 1.397 2.323 1.101 4.215
Figures 5.5 and 5.6 show that the relationship between the model and prototype
length coordinates for all tests can be represented by a straight line in each zone. The
slope of the straight lines fitted to the data represents a relative magnitude of the scale
effect. The slope and intercept can be used in Equations 4.5 and 4.6 to predict the
prototype equilibrium profile from model results, as shown with the following example.
5.3.2. Example of Profile Prediction
Sinusoidal waves were generated in the prototype with H = 50 cm and T = 8 sec.
Tests DSTi0_1, DSTi0_3 and DSTi0_5 were cond ucted at length scales 1/10, 1/8.5 and
1/11 with H = 5 cm, 5.88 cm and 4.55 cm, respectively. Froude scaling law gives,
xT NN = ; hence NT = 0.316, 0.343 and 0.302, and Tm = 2.53 sec, 2.74 sec and 2.41
sec were used in Tests DSTi0_1, DSTi0_3 and DSTi0_5, respectively. The model and
prototype both used 0.22 mm median diameter sand having a fall velocity, w = 0.1083
ft/sec, giving Nw = 1.
95
Slope and intercept of straight lines fitted to the dimensionless coordinate data in
four zones, shown in Figures 5.5 and 5.6, are summarized in Table 5.3. Figure 5.7 shows
the initial and equilibrium profiles of the prototype.
Table 5.3: Summary of dimensionless coordinate relationship in four zones.
Zone x/gT2 d/wT Slope, a1 Intercept, b1 Slope, a2 Intercept, b2 I 0.47 0 1.47 0 II 4.04 0 2.7 0.04 III 0.35 0.016 1.3 -0.8 IV 0.52 0.012 2.5 1.1
Figure 5.8 shows the initial and equilibrium profiles of Test DSTi0_1. Figure 5.9
shows the prototype profile coordinates calculated in Equations 4.5 and 4.6 from model
profile coordinates, xm and dm, and using the slopes and intercepts of Table 5.3. Figures
5.10 and 5.11 show similarly calculated prototype profiles from Tests DSTi0_3 and
DSTi0_5.
The predicted profiles show generally good agreement with the prototype in Zones
I, II and III. Although the average profile shape in Zone IV has been approximately
predicted, the predicted profiles significantly differ near the reflection bars.
96
Figure 5.7: Initial and equilibrium profiles of the prototype.
Figure 5.8: Initial and equilibrium profiles of Test DSTi0_1.
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-25.0 0.0 25.0 50.0 75.0 100.0 125.0 150.0
x (ft)
d (f
t)
Initial
Equilibrium
Prototype
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
x (ft)
d (f
t)
Initial
Equilibrium
DSTi0_1
97
Figure 5.9: Prototype equilibrium profile predicted from Test DSTi0_1.
Figure 5.10: Prototype equilibrium profile predicted from Test DSTi0_3.
Figure 5.11: Prototype equilibrium profile predicted from Test DSTi0_5.
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-25.0 0.0 25.0 50.0 75.0 100.0 125.0 150.0
x (ft)
d (
ft)
Prototype
Predicted by DSTi0_1
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-25.0 0.0 25.0 50.0 75.0 100.0 125.0 150.0
x (ft)
d (f
t)
Prototype
Predicted by DSTi0_3
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-25.0 0.0 25.0 50.0 75.0 100.0 125.0 150.0
x (ft)
d (f
t)
Prototype
Predicted by DSTi0_5
98
5.3.3. Model Properties
Scale effects result from the inability to model the relative importance of dominant
forcing functions and variables. While all important prototype properties can not be
preserved because of conflicting requirements, careful model selection ensures that most
properties are preserved. The present model preserves the following prototype properties:
(1) Geometric similarity: zx NN ≅
(2) Wave Froude number: 21z
21xT NNN ==
Deep water wave steepness: Ho/Lo = 0.0045
(3) Same sediment size and fall velocity: ;1N50D = 1N w =
(4) Same sediment and fluid: 1N;1N == νγ′
The effect of surface tension and viscosity on wave propagation is assumed to be
negligible because of the relatively large model. An important variable of similitude, the
dimensionless fall velocity, Ho/wT, is not preserved. To preserve this variable the
following should apply:
Twz NNN = ; and for 21zT NN = ,
21zw NN = . (5.1)
For relatively low Reynolds number, assuming a fall velocity in the Stokes range,
νγ′
=gD
181
w250 (5.2)
99
from which 12Dw NNNN
50
−νγ ′= , and for 1Nand 1N ?? ==′ ,
2Dw 50
NN = . (5.3)
From Equations 5.1 and 5.3,
41zD NN
50= . (5.4)
Equation 5.4 requires that a smaller sediment size be used in the model to preserve
Ho/wT. For example, for a prototype sediment size of 0.22 mm and 1.0N z = , a model
sediment size of 0.124 mm would be required. However, there will be scale effect
because other variables including the densimetric Froude number and the particle
Reynolds number described below would not be preserved.
The densimetric Froude number is given by,
50
2*
* gDv
Fγ′
= (5.5)
for 1N*F = ,
50* D2v NNN γ′= , and since 1N =γ′ ,
2vD *50
NN = . (5.6)
The particle Reynolds number is given by,
ν= 50*
*
DvR (5.7)
and for 1N*R = and 1N ? = ,
1vD *50
NN −= . (5.8)
100
Equations 5.6 and 5.8 indicate that both the densimetric Froude number and the
particle Reynolds number is preserved only if 1N50D = . However, the similarity of
Ho/wT is violated, and the bottom shear velocity is of the same order of the prototype
( 1N*v = ) indicating that the sediment moving force has not been reduced
proportionately in the model.
5.3.4. Role of Shear Velocity
Shear velocity is a function of bottom shear stress, which plays an important role in
sediment incipient motion and transport in the offshore zone, Zone IV. Bed roughness
comprises of form drag due to bedforms and skin friction due to individual particles. If
bedforms are present in the model the bottom roughness is distorted thereby introducing
nonsimilarity of shear velocity or scale effect. The scale effect varies with a change of the
local bedform height.
Investigators have shown that bedform roughness is more important than skin
friction for energy dissipation. At equilibrium the overall roughness is proportional to the
bedform height, which is also related to the bottom orbital amplitude and the shear
velocity. Therefore, a scaling criterion for shear velocity has been suggested by
Kamphuis (1996),
21zv NN
*= (5.9)
However, this criterion may require model sediment too fine in the cohesive range if the
densimetric Froude number is preserved (Equation 5.6), or larger than the prototype
101
sediment if the particle Reynolds number is preserved (Equation 5.8). Scale effects are
introduced in both cases.
Flow reversal occurs in the bottom boundary layer twice in each wave cycle causing
periods of low and high velocities resulting in laminar and turbulent flow conditions.
Scale effects are introduced if the bottom friction does not reflect the flow condition
variability correctly. The following discussion compares possible shear velocity scaling
relations under the present model conditions with laminar and turbulent boundary layer
assumptions.
Assuming a laminar boundary layer, the bed friction factor for gravity wave flow
can be written as (Jonsson 1966),
TV?
2p
f21
= (5.10)
for 1N ? = which leads to,
21T
-1Vf NNN −= . (5.11)
The bed shear velocity in terms of the friction factor can be written as (Henderson 1966),
V8
fv
21
* = (5.12)
which leads to,
V21
fv NNN*
= . (5.13)
Combining Equations 5.13 and 5.11,
41-T
21Vv NNN
*= . (5.14)
102
Obtaining 21zV NN = from wave Froude number similarity and substituting into
Equation 5.14,
41-T
41zv NNN
*= . (5.15)
For 21zT NN = , Equation 5.15 reduces to,
81zv NN
*= . (5.16)
Assuming a rough turbulent boundary layer, the bed friction factor for flat beds
(Kamphuis 1975a),
43
d
s
ak
0.4f
= (5.17)
where dg
4pHT
a d = = the bottom orbital amplitude and ks = sand grain roughness. The
following scaling relations can be derived from Equation 5.17:
43a
43kf ds
NNN −= (5.18)
and T21
za NNNd
= . (5.19)
Assuming the grain roughness to be on the order of the particle size, or 50s Dk NN ≅ ,
Equations 5.18 and 5.19 give,
4-3T
83z
43Df NNNN
50
−= . (5.20)
Using shear velocity and wave Froude number similarity as before, Equation 5.20 leads
to,
8-3T
165z
83Dv NNNN
50*= . (5.21)
For 1N50D = and 21
zT NN = , Equation 5.21 reduces to,
103
81zv NN
*= . (5.22)
Equations 5.16 and 5.22 both indicate that the shear velocity in the model should be
lower than that in the prototype. The model shear velocity should be even lower if
bedforms are present, as indicated by Equation 5.9.
Additional scale effect may arise through not preserving such variables as 50
d
Da
and
*vw
(Kamphuis 1985). The characteristic length ratio 50
d
Da
relates the boundary layer
wave and sediment properties while the vertical to horizontal velocity ratio *v
w links the
roughness and suspended transport properties of the sediment.
The scale effect of roughness distortion is relatively unimportant in Zones II and III
where turbulence dominated suspended transport is the primary process. As wave
breaking dominates the processes, failure to preserve the characteristic length ratio 50
b
DH
and the characteristic velocity ratio bgH
w introduces scale effects (Kamphuis 1991).
Preserving these ratios is not feasible under the present model conditions since for
1N50D = and 1N w = the only valid model is at prototype scale.
104
5.4. Applications and Limitations of Study
The equilibrium profile for the prototype (SUPERTANK) tests can be predicted
from the model results for given wave and sediment properties. The predictive empirical
equations can be used in the study of beach processes and the relative magnitude of scale
effects can be estimated in four zones across an accreting beach. This can be used to
identify important variables and processes that define the scaling relations. Model
experiments similar to the present study may be conducted to investigate the scale effects
for cnoidal or random waves, provided appropriate prototype data are available.
The results of the present study are limited to the range of model conditions
investigated. Scale effects in experiments conducted at other scales will likely differ.
Therefore, any extrapolation of results outside the model range must be done carefully.
The present study is also limited to accreting wave conditions. Scaling zone selection for
eroding waves based on dominant transport processes is also likely to be different.
Sinusoidal waves, generated using the first order wavemaker theory, are
contaminated because of the presence of free secondary waves. Wave reflection in the
laboratory is generally higher than on natural beaches where the slope is flatter.
‘Reflection bars’ developed at the antinodes of a standing wave system affected shoaling
and sediment movement. Prediction of prototype equilibrium profile from model results
near these bars is less satisfactory.
105
CHAPTER 6: SUMMARY
6.1. Summary of Model Experiments
A series of movable bed, scale model experiments were carried out to study scale
effects and equilibrium profile characteristics under accretion conditions. Along with
general observation of the profile evolution and transport processes, empirical equations
have been derived to predict prototype equilibrium profiles from the model results.
Experiments were conducted at three length scales, 1/8.5, 1/10 and 1/11, in a 90
ft- long, 3 ft-wide and 2.5 ft-deep wave tank. Geometric similarity, deep water wave
steepness, wave Froude number, densimetric Froude number, and particle Reynolds
number were preserved by selecting the same sediment size and density in model and
prototype, and the same fluid. A SUPERTANK (US Army Corps of Engineers 1994) test
conducted with 0.22 mm sand, and accretionary sinusoidal waves toward equilibrium was
considered as the prototype.
Wave heights were measured at four locations outside the offshore end of beach
using parallel-wire resistance gauges while a programmable wave generator with wave
absorption capability produced waves. Each ‘test’ was conducted in a sequence of several
‘runs’ with the wave generator operated continuously during a run. Beach profiles were
measured several times during a test using a semi-automatic profiler developed for the
106
present research. The equilibrium endpoint of a test was indicated when only a relatively
small net transport rate prevailed at all points on the profile and there was no significant
change in profile shape.
The beach was built on a permeable frame so that percolation through the beach
face simulated the natural groundwater. The model equilibrium foreshore slope was about
the same as that in the prototype. Bottom roughness was distorted because out of scale,
exaggerated bedforms occurred in the model. High bars developed at the antinodes of a
standing wave system as the profile approached equilibrium. These ‘reflection bars’
affected shoaling and local cross shore transport processes.
Profile coordinates and net transport rate were expressed in dimensionless form
with the origin at the equilibrium still water line (SWL). After an initial, relatively large,
and irregular change, the net transport rate asymptotically moved to equilibrium. The
maximum transport rate occurred about when the initial foreshore slope was established.
An asymptotic spatial variation of transport rate was also observed from the break point
offshore.
Comparison of similar equilibrium profile features of the model and prototype
indicates that the scale effects differ in the four zones where the transport processes
differ. Assuming the same scale effect within each zone, empirically derived equations
can predict prototype equilibrium profile from model results with satisfactory agreement.
However, profile prediction in the vicinity of reflection bars was not satisfactory.
107
6.2. Conclusions of the Study
The following conclusions are drawn from the study:
(1) Scale effects differ in the four distinct zones identified on an accreting profile.
Transport processes in each of those zones are significantly different. The scale
effect within each zone can be assumed to be the same.
(2) Normalizing horizontal length coordinates as x/gT2, and vertical length
coordinates as d/wT provides a better representation of the scale effect.
Equilibrium profiles for prototype scale can be predicted using the relationship
among the dimensionless coordinates of similar distinct features of model and
prototype profiles.
(3) The inability to model bottom shear velocity appears to have significant
contribution to the scale effect under the present model conditions.
108
6.3. Recommendations
The following further experiments should be undertaken to check the general
validity of the predictive empirical equations:
(1) Preserving the present model conditions and using the same sediment size and
density, conduct experiments at smaller and larger scales.
(2) Repeat Series 1 experiments with at least two additional sediment sizes.
(3) Repeat Series 1 experiments while preserving the fall velocity parameter, Ho/wT.
This will require a different sediment size for each scale. (Sediment sizes less than
0.1 mm should be excluded from testing.)
The recommended experiments will describe scale effects more completely and
provide a better relationship between model and prototype equilibrium profiles. Similar
experiments should be conducted to develop predictive equations for eroding profiles.
More, and more closely spaced wave height measurements in the offshore and
breaker zones should be made to better define shoaling and breaking characteristics.
Velocity measurements in the bottom boundary layer would provide data on roughness.
109
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Goring, D.G. (1978). “Tsunamis – the propagation of long waves onto a shelf.” Rept. No. KH-R-38, W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena. Hallermeier, R.J. (1980). “Sand motion initiation by water waves: two asymptotes.” J. Waterw., Port, Coastal, Ocean Eng., 106(WW3), 299-318. Hallermeier, R.J. (1981). “Terminal settling velocity of commonly occurring sand grains.” Sedimentology, vol. 28, 859-865. Hanes, D.M., and Vincent, C.E. (1987). “Detailed dynamics of nearshore suspended sediment.” Proc., Coastal Sediments ‘87, ASCE, 285-299. Henderson, F.M. (1966). Open Channel Flow, McMillan Company, New York. Hinis, M. (2000). A computer program for laboratory cnoidal wave generation. Personal Communication. HR Wallingford (1996). “Flume wave generation system for Drexel University.” Operating Manual, Howbery Park, Wallingford, UK. Hughes, S.A. (1983). “Movable-bed modeling law for coastal dune erosion.” J. Waterw., Port, Coastal, Ocean Eng., 109(2), 164-179. Hughes, S.A. (1992). ‘PC Goda’, a computer program to separate incident and reflected spectra. Personal Communication. Hughes, S.A. (1993). Physical models and laboratory techniques in coastal engineering, World Scientific, Singapore. Kajima, R., Shimizu, T., Maruyama, K., and Saito, S. (1982). “Experiments on beach profile change with a large wave flume.” Proc., 18th Coastal Eng. Conf., ASCE, Cape Town. Jonsson, I.G. (1966). “Wave boundary layers and friction factors.” Proc., 10th Coastal Eng. Conf., ASCE, London. Kamphuis, J.W. (1972). “Similarity of equilibrium beach profiles.” Proc., 13th Coastal Eng. Conf., ASCE, Vancouver, 1173-1195. Kamphuis, J.W. (1975a). “Friction factor under oscillatory waves.” J. Waterw., Harbor, Coastal Eng., 101. Kamphuis, J.W. (1975b). “The coastal mobile bed model – does it work?” Proc., Modeling ’75, ASCE, San Francisco, 993-1009.
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Kamphuis, J.W. (1985). “On understanding scale effect in coastal mobile bed models.” In Physical Modelling in Coastal Engineering, R.A. Dalrymple, ed., 141-162, A.A. Balkema, Rotterdam. Kamphuis, J.W. (1991). “Physical modeling.” In Handbook of Coastal and Ocean Engineering, J.B. Herbich, ed., Gulf Publishing Company, Houston. Kamphuis, J.W. (1996). “Physical modeling of coastal processes.” In Advances in Coastal and Ocean Engineering, vol. 2, P.L. Liu, ed., 79-114, World Scientific Publishing Co., Singapore. Keulegan, G.H. (1945). “Depths of offshore bars.” Engineering Notes No. 8, Beach Erosion Board, U.S. Army Corps of Engineers. Kobayashi, N., and Tega, Y. (2002). “Sand suspension and transport on equilibrium beach.” J. Waterw., Port, Coastal, Ocean Eng., 128(6), 238-248. Komar, P.D., and McDougal, W.G. (1994). “The analysis of exponential beach profiles.” J. Coastal Res., 10(1), 59-69. Kraus, N.C., and Larson, M. (1988). “Beach profile change measured in the tank for large waves, 1956-1957 and 1962.” Tech. Rept. CERC-88-6, Coastal Engineering Research Center, U.S. Army Corps of Engineers. Kraus, N.C., Larson, M., and Kriebel, D.L. (1991). “Evaluation of beach erosion and accretion predictors.” Proc., Coastal Sediments, ASCE, 572-587. Kraus, N.C., Smith, J.M., and Sollitt, C.K. (1992). “SUPERTANK laboratory data collection project.” Proc., 23rd Coastal Eng. Conf., ASCE, 2191-2204. Kriebel, D.L. (1982). “Beach and dune response to hurricanes.” M.Sc. thesis, Univ. of Delaware. Kriebel, D.L., Dally, W.R., and Dean, R.G. (1986). “Undistorted Froude model for surf zone sediment transport.” Proc., 20th Coastal Eng. Conf., ASCE, Taipei, 1296-1310. Larson, M. (1988). “Quantification of beach profile change.” Rept. 1008, Dept. of Water Resources Eng., Univ. of Lund. Larson, M., and Kraus, N.C. (1989). “SBEACH: Numerical model for simulating storm induced beach changes.” Tech. Rept. CERC-89-9, U.S. Army Corps of Engineers. Larson, M., Kraus, N.C., and Wise, R.A. (1999). “Equilibrium beach profiles under breaking and non-breaking waves.” Coastal Eng., 36(1), 59-85.
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115
APPENDIX A: LIST OF SYMBOLS
Symbol Description
aδ Maximum horizontal bottom orbital amplitude d Local water depth db Water depth at break point d′ Dimensionless vertical length = d/wT f Friction factor g Acceleration of gravity k Wave number = 2π/L ks Grain roughness q Net sediment transport rate qb Bed load qs Suspended load um Maximum near-bottom orbital velocity
*v Bed shear velocity w Sediment fall velocity
ow Fall velocity of spherical particles x horizontal distance x′ Dimensionless horizontal length = x/gT2
116
z vertical distance A Shape parameter of sediment particle C Volumetric sediment concentration CD Drag coefficient D Sediment diameter D* Energy dissipation per unit water volume
*F Particle Froude number H Wave height Ho Deep water wave height Hb Breaker height L Wave length Lo Deep water wave length Nt Morphological time ratio Nx Horizontal length ratio Nz Vertical length ratio ND Sediment diameter ratio NT Wave period ratio Nv Wave velocity ratio
γ′N Submerged unit weight ratio R Runup height
*R Particle Reynolds number T Wave period Ub Near-bottom velocity
117
V Horizontal fluid particle velocity tan α Foreshore slope tanβ Bed slope β Particle shape and size distribution parameter
? Surf similarity parameter oLH
atan=
φ Internal friction angle for bedload ?′ Submerged unit weight of sediment ( )g?? s −= η Ripple height κ Breaker height index λ Ripple length µ Fluid dynamic viscosity ν Fluid kinematic viscosity ρ Fluid density
s? Sediment density bt Mean bottom shear stress
ω Wave angular frequency O Model distortion 1
zx NN −=
118
APPENDIX B: PERMEABILITY TEST RESULTS
Significant amount of infiltration occurred during the model tests through the
beach face above the Still Water Line (SWL). In order to obtain a quantitative measure of
this infiltration rate, in situ falling head permeability tests were conducted at two
locations on the beach face after Run DSTi04_14 had been completed. A plexiglass
cylinder having an inside diameter of 10 cm was vertically inserted into the sand up to the
geotextile layer underneath the sand. The cylinder was filled with water and the
decreasing water level with time was recorded.
The data obtained from the permeability tests are included in Tables B1 and B2.
Test P1 was conducted on the beach face, approximately in the middle of the runup limit.
At this location the sand was partially saturated. Test P2 was conducted behind the berm
crest where a pool of water formed due to overwash of the berm. At this location the
upper 6 inch layer of sand was unsaturated.
Figures B1 and B2 demonstrate the decay of the calculated infiltration rate with
time. A logarithmic trend line was fitted to the infiltration rate data resulting in
correlation coefficients of 0.831 and 0.834 for tests P1 and P2, respectively. For these
tests the coefficient of permeability, k was determined from the following equation:
119
t
o
HH
lntL
k = (B1)
where L is the height of sand in the cylinder, t is the elapsed time of test, Ho is the initial
head above sand, and Ht is the final head. From Equation B1 the coefficients of
permeability were calculated to be 0.140 cm/sec and 0.113 cm/sec for Tests P1 and P2,
respectively. The lower coefficient of permeability obtained from Test P2 indicates that
the air inside the voids of the upper unsaturated zone reduced the flow area through the
sand.
120
Table B1: Experimental data for Test P1.
Depth of sand in cylinder = 19.5 cm. Water level in cylinder
(cm)
Time (sec)
Time difference
(sec)
Infiltration rate, I
(cm/sec)
Head, H (cm)
59 0.0 39.50 58 8.5 8.5 0.118 38.50 57 15.0 6.5 0.154 37.50 56 25.0 10.0 0.100 36.50 55 33.0 8.0 0.125 35.50 54 42.5 9.5 0.105 34.50 53 51.5 9.0 0.111 33.50 52 61.5 10.0 0.100 32.50 51 70.5 9.0 0.111 31.50 50 80.0 9.5 0.105 30.50 49 90.5 10.5 0.095 29.50 48 101.0 10.5 0.095 28.50 47 112.0 11.0 0.091 27.50 46 123.0 11.0 0.091 26.50 45 134.0 11.0 0.091 25.50 44 146.0 12.0 0.083 24.50 43 156.0 10.0 0.100 23.50 42 168.0 12.0 0.083 22.50 41 179.0 11.0 0.091 21.50 40 190.0 11.0 0.091 20.50 39 204.0 14.0 0.071 19.50 38 217.0 13.0 0.077 18.50 37 230.0 13.0 0.077 17.50 36 244.0 14.0 0.071 16.50 35 257.0 13.0 0.077 15.50 34 273.0 16.0 0.063 14.50 33 288.0 15.0 0.067 13.50 32 301.0 13.0 0.077 12.50 31 318.0 17.0 0.059 11.50 30 333.0 15.0 0.067 10.50 29 352.0 19.0 0.053 9.50 28 368.0 16.0 0.063 8.50 27 385.0 17.0 0.059 7.50 26 403.0 18.0 0.056 6.50 25 421.0 18.0 0.056 5.50 24 445.0 24.0 0.042 4.50 23 462.0 17.0 0.059 3.50 22 482.0 20.0 0.050 2.50 21 506.0 24.0 0.042 1.50
121
Table B2: Experimental data for Test P2.
Depth of sand in cylinder = 18.0 cm. Water level in cylinder
(cm)
Time (sec)
Time difference
(sec)
Infiltration rate, I
(cm/sec)
Head, H (cm)
59 0.0 41.00 58 4.5 4.5 0.222 40.00 57 11.0 6.5 0.154 39.00 56 22.0 11.0 0.091 38.00 55 34.0 12.0 0.083 37.00 54 44.0 10.0 0.100 36.00 53 55.0 11.0 0.091 35.00 52 67.0 12.0 0.083 34.00 51 78.0 11.0 0.091 33.00 50 89.0 11.0 0.091 32.00 49 101.0 12.0 0.083 31.00 48 113.0 12.0 0.083 30.00 47 126.0 13.0 0.077 29.00 46 138.0 12.0 0.083 28.00 45 151.0 13.0 0.077 27.00 44 165.0 14.0 0.071 26.00 43 177.0 12.0 0.083 25.00 42 190.0 13.0 0.077 24.00 41 204.0 14.0 0.071 23.00 40 218.0 14.0 0.071 22.00 39 233.0 15.0 0.067 21.00 38 248.0 15.0 0.067 20.00 37 262.0 14.0 0.071 19.00 36 277.0 15.0 0.067 18.00 35 293.0 16.0 0.063 17.00 34 309.0 16.0 0.063 16.00 33 326.0 17.0 0.059 15.00 32 342.0 16.0 0.063 14.00 31 361.0 19.0 0.053 13.00 30 378.0 17.0 0.059 12.00 29 398.0 20.0 0.050 11.00 28 414.0 16.0 0.063 10.00 27 433.0 19.0 0.053 9.00 26 453.0 20.0 0.050 8.00 25 472.0 19.0 0.053 7.00 24 493.0 21.0 0.048 6.00 23 517.0 24.0 0.042 5.00 22 538.0 21.0 0.048 4.00 21 562.0 24.0 0.042 3.00 20 585.0 23.0 0.043 2.00
122
Figure B1: Variation of infiltration rate with time during Test P1.
Figure B2: Variation of infiltration rate with time during Test P2.
I = -0.022Ln(t) + 0.195R2 = 0.831
k = 0.140 cm/sec
0.00
0.05
0.10
0.15
0.20
0.25
0.0 100.0 200.0 300.0 400.0 500.0 600.0
Time, t (sec)
Infi
ltra
tio
n r
ate,
I (c
m/s
ec)
I = -0.026Ln(t) + 0.208R2 = 0.834
k = 0.113 cm/sec
0.00
0.05
0.10
0.15
0.20
0.25
0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0
Time, t (sec)
Infi
ltra
tio
n r
ate,
I (c
m/s
ec)
I = -0.022Ln(t) + 0.195R2 = 0.831
k = 0.140 cm/sec
0.00
0.05
0.10
0.15
0.20
0.25
0.0 100.0 200.0 300.0 400.0 500.0 600.0
Time, t (sec)
Infi
ltra
tio
n r
ate,
I (c
m/s
ec)
I = -0.026Ln(t) + 0.208R2 = 0.834
k = 0.113 cm/sec
0.00
0.05
0.10
0.15
0.20
0.25
0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0
Time, t (sec)
Infi
ltra
tio
n r
ate,
I (c
m/s
ec)
123
APPENDIX C: DETAILS OF PROTOTYPE AND MODEL TESTS
Table C1: Prototype Test ST10_1 (Equilibrium erosion, Random waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
Tp (sec)
Hmo (cm)
γ SWL (ft)
X (ft) Y (ft) Initial A0509 - - - - - - 8.510 0.000
1 A0509A 20 20 3 80 20 10 7.416 0.000 2 A0510A 40 60 3 80 20 10 4.912 0.000 3 A0512A 70 130 3 80 20 10 3.071 0.000 4 A0515A 70 200 3 80 20 10 1.206 0.000 5 A0517A 70 270 3 80 20 10 0.000 0.000
Table C2: Prototype Test STi0_1 (Equilibrium accretion, Sinusoidal waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
T (sec)
H (cm)
SWL (ft)
X (ft) Y (ft) Initial S0512A - - - - - -0.672 0.000
1 S0513A 20 20 8 50 10 -0.426 0.000 2 S0514A 20 40 8 50 10 0.093 0.000 3 S0515A 40 80 8 50 10 -0.886 0.000 4 S0516A 70 150 8 50 10 -1.608 0.000 5 S0517A 70 220 8 50 10 -2.004 0.000 6 S0607B 70 290 8 50 10 -2.504 0.000 7 S0609A 70 360 8 50 10 -2.254 0.000 8 S0610A 70 430 8 50 10 -2.214 0.000 9 S0612A 70 500 8 50 10 -2.457 0.000
10 S0614A 70 570 8 50 10 0.000 0.000
124
Table C3: Model Test DST10_1 (Equilibrium erosion, Random waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
Tp (sec)
Hmo
(cm) Water Temp. (oC)
SWL (cm)
X (ft) Y (ft) Initial - - - - - - - 0.016 0.027
1 DST101_1 30.5 30.5 1.05 8 19.5 33.40 -0.082 0.027 2 DST101_2 15.5 46.0 1.05 8 19.5 33.02 0.034 0.011 3 DST101_3 65.0 111.0 1.05 8 19.0 33.02 0.034 0.007 4 DST101_4 40.0 151.0 1.05 8 19.5 33.05 -0.014 0.011 5 DST101_5 40.0 191.0 1.05 8 19.5 33.10 0.028 0.008 6 DST101_6 40.0 231.0 1.05 8 19.6 33.05 0.007 0.010 7 DST101_7 40.0 271.0 1.05 8 19.0 33.10 -0.038 0.011 8 DST101_8 40.0 311.0 1.05 8 19.5 33.10 0.000 0.000 9 DST101_9 17.0 328.0 2.53 5 - 33.10 0.287 0.016
Table C4: Model Test DSTi0_1 (Equilibrium accretion, Sinusoidal waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
T (sec)
H (cm)
Water Temp. (oC)
SWL (cm)
X (ft) Y (ft) Initial - - - - - - - -1.743 0.001
1 DSTi01_1 17 17 2.53 5 18.5 33.20 -1.561 -0.004 2 DSTi01_2 11 28 2.53 5 18.5 33.10 -1.595 -0.005 3 DSTi01_3 9 37 2.53 5 19.0 33.10 -1.616 -0.003 4 DSTi01_4 13 50 2.53 5 19.5 33.10 -0.930 -0.002 5 DSTi01_5 30 80 2.53 5 19.5 33.10 -0.311 -0.003 6 DSTi01_6 15 95 2.53 5 19.5 33.10 -0.192 0.003 7 DSTi01_7 22 117 2.53 5 19.5 33.10 -0.125 -0.004 8 DSTi01_8 46 163 2.53 5 19.0 33.15 -0.102 0.000 9 DSTi01_9 23 186 2.53 5 19.0 33.10 -0.061 0.000
10 DSTi01_10 60 246 2.53 5 18.0 33.10 -0.060 -0.001 11 DSTi01_11 51 297 2.53 5 18.0 33.10 -0.062 -0.003 12 DSTi01_12 70 367 2.53 5 18.0 33.15 -0.040 0.000 13 DSTi01_13 71 438 2.53 5 18.0 33.15 0.000 0.000 14 DSTi01_14 70 508 2.53 5 17.0 33.15 0.030 0.000 15 DSTi01_15 70 578 2.53 5 18.0 33.10 0.074 0.000 16 DSTi01_16 70 648 2.53 5 18.0 33.15 0.108 0.000
125
Table C5: Model Test DSTi0_2 (Equilibrium accretion, Cnoidal waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
T (sec)
H (cm)
Water Temp. (oC)
SWL (cm)
X (ft) Y (ft) Initial - - - - - - - -1.555 0.000
1 DSTi02_1 10 10 2.53 5 18.0 33.15 -1.916 0.000 2 DSTi02_2 20 30 2.53 5 18.0 33.15 -0.849 0.000 3 DSTi02_3 20 50 2.53 5 17.0 33.15 -1.166 0.000 4 DSTi02_4 26 76 2.53 5 17.0 33.15 -0.565 0.000 5 DSTi02_5 50 126 2.53 5 19.0 33.15 -0.425 0.000 6 DSTi02_6 60 186 2.53 5 17.0 33.15 -0.233 0.000 7 DSTi02_7 60 246 2.53 5 17.0 33.15 -0.124 -0.001 8 DSTi02_8 61 307 2.53 5 17.0 33.10 -0.005 -0.002 9 DSTi02_9 60 367 2.53 5 17.0 33.10 -0.005 -0.001
10 DSTi02_10 70 437 2.53 5 19.0 33.10 -0.020 0.000 11 DSTi02_11 50 487 2.53 5 19.0 33.10 0.017 -0.001 12 DSTi02_12 60 547 2.53 5 19.0 33.10 -0.015 -0.001 13 DSTi02_13 31 578 2.53 5 19.0 33.10 -0.018 0.000 14 DSTi02_14 60 638 2.53 5 19.0 33.10 -0.030 0.000 15 DSTi02_15 60 698 2.53 5 20.0 33.10 0.000 0.000 16 DSTi02_16 20 718 2.53 5 20.0 33.10 0.020 0.000 17 DSTi02_17 70 788 2.74 5.88 21.0 33.10 0.086 0.000
Table C6: Model Test DSTi0_3 (Equilibrium accretion, Sinusoidal waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
T (sec)
H (cm)
Water Temp. (oC)
SWL (cm)
X (ft) Y (ft) Initial - - - - - - - -1.889 0.000
1 DSTi03_1 10 10 2.74 5.88 19.0 35.85 -1.878 -0.002 2 DSTi03_2 20 30 2.74 5.88 19.0 35.85 -1.508 0.000 3 DSTi03_3 20 50 2.74 5.88 17.0 35.85 -1.318 -0.001 4 DSTi03_4 30 80 2.74 5.88 19.0 35.85 -0.639 0.002 5 DSTi03_5 30 110 2.74 5.88 18.0 35.85 -0.310 0.002 6 DSTi03_6 30 140 2.74 5.88 19.0 35.85 -0.208 0.000 7 DSTi03_7 15 155 2.74 5.88 19.0 35.85 -0.150 0.000 8 DSTi03_8 46 201 2.74 5.88 18.0 35.85 -0.121 0.000 9 DSTi03_9 45 246 2.74 5.88 19.0 35.85 -0.085 0.000
10 DSTi03_10 60 306 2.74 5.88 19.5 35.85 -0.099 0.000 11 DSTi03_11 60 366 2.74 5.88 19.0 35.85 -0.087 0.000 12 DSTi03_12 70 436 2.74 5.88 20.0 35.85 -0.024 0.000 13 DSTi03_13 70 506 2.74 5.88 19.5 35.85 0.000 0.000 14 DSTi03_14 100 606 2.74 5.88 21.0 35.85 0.037 0.000
126
Table C7: Model Test DSTi0_4 (Equilibrium accretion, Cnoidal waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
T (sec)
H (cm)
Water Temp. (oC)
SWL (cm)
X (ft) Y (ft) Initial - - - - - - - -2.339 0.000
1 DSTi04_1 10 10 2.74 5.88 21.0 35.85 -2.331 0.000 2 DSTi04_2 20 30 2.74 5.88 20.0 35.85 -2.126 0.000 3 DSTi04_3 20 50 2.74 5.88 20.0 35.85 -1.144 0.000 4 DSTi04_4 30 80 2.74 5.88 20.0 35.85 -0.733 0.000 5 DSTi04_5 46 126 2.74 5.88 20.0 35.85 -0.404 0.000 6 DSTi04_6 60 186 2.74 5.88 19.0 35.85 -0.038 0.000 7 DSTi04_7 60 246 2.74 5.88 19.0 35.85 0.064 0.000 8 DSTi04_8 60 306 2.74 5.88 19.0 35.85 0.115 0.000 9 DSTi04_9 60 366 2.74 5.88 19.0 35.85 0.074 0.000
10 DSTi04_10 70 436 2.74 5.88 19.5 35.85 0.149 0.000 11 DSTi04_11 68 504 2.74 5.88 20.0 35.85 -0.019 0.000 12 DSTi04_12 74 578 2.74 5.88 21.0 35.85 0.000 0.000 13 DSTi04_13 60 638 2.74 5.88 21.0 35.85 -0.035 0.000 14 DSTi04_14 100 738 2.74 5.88 22.0 35.85 -0.089 0.000
Table C8: Model Test DSTi0_5 (Equilibrium accretion, Sinusoidal waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
T (sec)
H (cm)
Water Temp. (oC)
SWL (cm)
X (ft) Y (ft) Initial - - - - - - - -1.797 -0.011
1 DSTi05_1 30 30 2.41 4.55 23.0 27.7 -2.053 -0.006 2 DSTi05_2 83.5 113.5 2.41 4.55 24.0 27.7 - - 3 DSTi05_3 56.5 170 2.41 4.55 24.4 27.7 - - 4 DSTi05_4 31 201 2.41 4.55 25.0 27.7 -0.106 0.000 5 DSTi05_5 45 246 2.41 4.55 22.0 27.7 -0.047 -0.012 6 DSTi05_6 50 296 2.41 4.55 21.5 27.7 - - 7 DSTi05_7 50 346 2.41 4.55 21.5 27.7 -0.043 -0.004 8 DSTi05_8 60 406 2.41 4.55 20.0 27.7 -0.075 0.001 9 DSTi05_9 50 456 2.41 4.55 21.0 27.7 - -
10 DSTi05_10 20 476 2.41 4.55 19.0 27.7 -0.038 -0.003 11 DSTi05_11 60 536 2.41 4.55 20.0 27.7 0.000 0.000 12 DSTi05_12 100 636 2.41 4.55 - 27.7 -0.118 0.000
127
Table C9: Model Test DSTi0_6 (Equilibrium accretion, Cnoidal waves).
Run SWL coordinates on beach
Number ID
Duration (min)
Total time (min)
T (sec)
H (cm)
Water Temp. (oC)
SWL (cm)
X (ft) Y (ft) Initial - - - - - - - -1.788 -0.003
1 DSTi06_1 30 30 2.41 4.55 20.0 27.7 -1.843 0.002 2 DSTi06_2 100 130 2.41 4.55 19.0 27.7 - - 3 DSTi06_3 60 190 2.41 4.55 19.5 27.7 - - 4 DSTi06_4 50 240 2.41 4.55 20.0 27.7 0.153 0.000 5 DSTi06_5 66 306 2.41 4.55 19.2 27.7 0.041 0.005 6 DSTi06_6 50 356 2.41 4.55 20.2 27.7 - - 7 DSTi06_7 66 422 2.41 4.55 20.5 27.7 - - 8 DSTi06_8 27 449 2.41 4.55 20.0 27.7 0.026 0.000 9 DSTi06_9 55 504 2.41 4.55 20.5 27.7 0.054 0.001
10 DSTi06_10 75 579 2.41 4.55 20.0 27.7 0.000 0.000 11 DSTi06_11 100 679 2.41 4.55 19.5 27.7 0.173 -0.003
128
APPENDIX D: VARIATION OF WATER SURFACE ELEVATION IN WAVE TANK
The wooden beach frame with a geotextile layer underneath the beach allowed
infiltration of water through the beach to the water body behind the beach. This
infiltration occurred mostly from wave runup above the Still Water Line (SWL).
However, during Runs where the berm was overwashed, water trapped behind the crest
of the berm formed a pool. This pool existed as long as the berm was being overwashed.
Water infiltrated from this pool to the back of the beach. It was observed that during each
Run the water depth behind the beach was generally higher than the initial water depth in
the wave tank. It was believed that this difference in water depth between the two sides of
the beach occurred as a result of relatively fast infiltration of water through the beach
face above the SWL, and relatively slow recovery of water back to the wave tank.
In order to evaluate the variation of water surface elevation in the tank, a series of
measurements was recorded during and after Run DSTi04_8. These measurements
included the water surface elevation in the wave tank and the pool behind the berm. The
recorded measurements are included in Table D1. The first group of measurements up to
22 minutes was recorded during the first segment of the Run. The second group of
measurements was recorded after the final segment of the Run when there was no wave
129
action in the tank. Figure D1 demonstrates the gradual rise of water surface elevation
behind the beach since the beginning of the Run. During this Run the berm overwashing
was reduced after approximately 16 minutes. The gradual decline of water surface
elevation in the pool is the result of this less overwash. Figure D2 demonstrates the
recovery of water surface in the wave tank after the wave action had been terminated.
During this time the front of the beach regained water from the back of the beach. Since
the length of the tank in front of the beach was approximately three times that behind the
beach, the water surface recovery rate behind the beach was higher.
Table D1: Water surface elevation in wave tank during and after Run DSTi04_8.
Water surface elevation (cm) in front of beach behind beach
Time (min)
at wave paddle
at 38 ft at 25 ft at 18 ft (pool)
at 3.5 ft
Comment
0.00 35.85 35.80 35.80 35.85 before run 4.00 48.10 36.75
10.00 48.15 38.00 15.83 47.75 38.55 less overwash 22.00 35.40 35.30 35.35 45.45 38.80 after first segment
60.00 35.25 35.10 35.15 46.60 39.15 after final segment 68.25 35.35 38.15 no water in pool 73.00 35.45 37.65 76.50 35.55 37.35 85.50 35.55 36.85 93.50 35.65 36.45
105.25 35.70 36.15 117.50 35.71 36.00
130
Figure D1: Variation of water surface elevation during wave action.
Figure D2: Variation of water surface elevation after test.
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0
Time since start of test (min)
Wat
er s
urf
ace
elev
atio
n (
cm)
at wave paddle
behind beach
30.0
35.0
40.0
45.0
50.0
55.0
0.0 5.0 10.0 15.0 20.0 25.0
Time since start of test (min)
Wat
er s
urf
ace
elev
atio
n (
cm)
at 18 ft (pool)
behind beach
131
APPENDIX E: MEASURED BEACH PROFILES
Figure E1: Measured beach profiles for Test ST10_1.
ST 10
132
Figure E2: Measured beach profiles for Test STi0_1.
ST i0
133
Figure E3: Measured beach profiles for Test DST10.
DST 10
134
Figure E4: Measured beach profiles for Test DSTi0_1.
DSTi0_1
135
Figure E5: Measured beach profiles for Test DSTi0_2.
DSTi0_2
136
Figure E6: Measured beach profiles for Test DSTi0_3.
DSTi0_3
137
Figure E7: Measured beach profiles for Test DSTi0_4.
DSTi0_4
138
Figure E8: Measured beach profiles for Test DSTi0_5.
DSTi0_5
139
Figure E9: Measured beach profiles for Test DSTi0_6.
DSTi0_6
140
APPENDIX F: DIMENSIONLESS BEACH PROFILES
Figure F1: Dimensionless beach profiles for Test ST10_1.
ST 10
141
Figure F2: Dimensionless beach profiles for Test STi0_1.
ST i0
142
Figure F3: Dimensionless beach profiles for Test DST10.
DST 10
143
Figure F4: Dimensionless beach profiles for Test DSTi0_1.
DSTi0_1
144
Figure F5: Dimensionless beach profiles for Test DSTi0_2.
DSTi0_2
145
Figure F6: Dimensionless beach profiles for Test DSTi0_3.
DSTi0_3
146
Figure F7: Dimensionless beach profiles for Test DSTi0_4.
DSTi0_4
147
Figure F8: Dimensionless beach profiles for Test DSTi0_5.
DSTi0_5
148
Figure F9: Dimensionless beach profiles for Test DSTi0_6.
DSTi0_6
149
APPENDIX G: TRANSPORT RATE VARIATION
Figure G1: Transport rate variation for Test ST10_1.
ST 10
150
Figure G2: Transport rate variation for Test STi0_1.
ST i0
151
Figure G3: Transport rate variation for Test DST10.
DST 10
152
Figure G4: Transport rate variation for Test DSTi0_1.
DSTi0_1
153
Figure G5: Transport rate variation for Test DSTi0_2.
DSTi0_2
154
Figure G6: Transport rate variation for Test DSTi0_3.
DSTi0_3
155
Figure G7: Transport rate variation for Test DSTi0_4.
DSTi0_4
156
Figure G8: Transport rate variation for Test DSTi0_5.
DSTi0_5
157
Figure G9: Transport rate variation for Test DSTi0_6.
DSTi0_6
158
APPENDIX H: SCALING RELATIONS OF DIMENSIONLESS VARIABLES
Figure H1: Horizontal scaling relations for Test DSTi0_1, Zone I.
Figure H2: Horizontal scaling relations for Test DSTi0_1, Zone II.
y = 0.4729157xR2 = 1.0000000
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0.000
-0.012 -0.010 -0.008 -0.006 -0.004 -0.002 0.000
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_1Zone I
y = 4.1737188x - 0.0000460R2 = 0.9998547
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.000 0.001 0.002 0.003 0.004 0.005
Model x/gT2
Pro
toty
pe
x/g
T2
DSTi0_1Zone II
159
Figure H3: Horizontal scaling relations for Test DSTi0_1, Zone III.
Figure H4: Horizontal scaling relations for Test DSTi0_1, Zone IV.
y = 0.3978342x + 0.0156249R2 = 0.9958198
0.000
0.005
0.010
0.015
0.020
0.025
0.000 0.005 0.010 0.015 0.020 0.025
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_1Zone III
y = 0.4699331x + 0.0141527R2 = 0.9965909
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.000 0.020 0.040 0.060 0.080 0.100
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_1Zone IV
160
Figure H5: Vertical scaling relations for Test DSTi0_1, Zone I.
Figure H6: Vertical scaling relations for Test DSTi0_1, Zone II.
y = 1.4493091x - 0.0000000R2 = 1.0000000
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_1Zone I
y = 2.7318438x + 0.0252631R2 = 0.9951732
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_1Zone II
161
Figure H7: Vertical scaling relations for Test DSTi0_1, Zone III.
Figure H8: Vertical scaling relations for Test DSTi0_1, Zone IV.
y = 1.4158923x - 0.6711068R2 = 0.6513250
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_1Zone III
y = 2.3633169x + 0.9969716R2 = 0.9925386
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_1Zone IV
162
Figure H9: Horizontal scaling relations for Test DSTi0_1, all zones.
Figure H10: Vertical scaling relations for Test DSTi0_1, all zones.
y = 0.5712661x + 0.0080318R2 = 0.9394692
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
Model x/gT2
Pro
toty
pe
x/g
T2
DSTi0_1All points
y = 1.8443948x - 0.3191740R2 = 0.9870252
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0
Model d/wT
Pro
toty
pe
d/w
T
DSTi0_1All points
163
Figure H11: Horizontal scaling relations for Test DSTi0_3, Zone I.
Figure H12: Horizontal scaling relations for Test DSTi0_3, Zone II.
y = 0.4951526xR2 = 1.0000000
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0.000
-0.012 -0.010 -0.008 -0.006 -0.004 -0.002 0.000
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_3Zone I
y = 4.5869115x - 0.0003248R2 = 0.9871129
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.000 0.001 0.001 0.002 0.002 0.003 0.003 0.004 0.004
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_3Zone II
164
Figure H13: Horizontal scaling relations for Test DSTi0_3, Zone III.
Figure H14: Horizontal scaling relations for Test DSTi0_3, Zone IV.
y = 0.3404431x + 0.0163250R2 = 0.9633183
0.000
0.005
0.010
0.015
0.020
0.025
0.000 0.005 0.010 0.015 0.020 0.025
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_3Zone III
y = 0.5170719x + 0.0125218R2 = 0.9974321
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.000 0.020 0.040 0.060 0.080 0.100
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_3Zone IV
165
Figure H15: Vertical scaling relations for Test DSTi0_3, Zone I.
Figure H16: Vertical scaling relations for Test DSTi0_3, Zone II.
y = 1.5518747xR2 = 1.0000000
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_3Zone I
y = 2.9632637x + 0.0486323R2 = 0.9673936
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_3Zone II
166
Figure H17: Vertical scaling relations for Test DSTi0_3, Zone III.
Figure H18: Vertical scaling relations for Test DSTi0_3, Zone IV.
y = 1.1258073x - 0.9300652R2 = 0.5998161
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
-2.0 -1.5 -1.0 -0.5 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_3Zone III
y = 2.6269665x + 1.3410089R2 = 0.9155173
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_3Zone IV
167
Figure H19: Horizontal scaling relations for Test DSTi0_3, all zones.
Figure H20: Vertical scaling relations for Test DSTi0_3, all zones.
y = 0.5998056x + 0.0078992R2 = 0.9405889
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
Model x/gT2
Pro
toty
pe
x/g
T2
DSTi0_3All points
y = 1.9426771x - 0.2707669R2 = 0.9732436
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0
Model d/wT
Pro
toty
pe
d/w
T
DSTi0_3All points
168
Figure H21: Horizontal scaling relations for Test DSTi0_5, Zone I.
Figure H22: Horizontal scaling relations for Test DSTi0_5, Zone II.
y = 0.4288388x + 0.0000000R2 = 1.0000000
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0.000
0.001
-0.014 -0.012 -0.010 -0.008 -0.006 -0.004 -0.002 0.000
Model x/gT2
Pro
toty
pe x
/gT
2
DSTi0_5Zone I
y = 3.3546360x - 0.0000808R2 = 0.9995295
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.000 0.001 0.002 0.003 0.004 0.005 0.006
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_5Zone II
169
Figure H23: Horizontal scaling relations for Test DSTi0_5, Zone III.
Figure H24: Horizontal scaling relations for Test DSTi0_5, Zone IV.
y = 0.3165962x + 0.0159986R2 = 0.9565501
0.000
0.005
0.010
0.015
0.020
0.025
0.000 0.005 0.010 0.015 0.020 0.025
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_5Zone III
y = 0.4643662x + 0.0155990R2 = 0.9392969
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.000 0.020 0.040 0.060 0.080 0.100
Model x/gT2
Pro
toty
pe x
/gT2
DSTi0_5Zone IV
170
Figure H25: Vertical scaling relations for Test DSTi0_5, Zone I.
Figure H26: Vertical scaling relations for Test DSTi0_5, Zone II.
y = 1.3967315x - 0.0000000R2 = 1.0000000
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0
Model d/wT
Pro
toty
pe
d/w
T
DSTi0_5Zone I
y = 2.3231904x + 0.0374176R2 = 0.9868045
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_5Zone II
171
Figure H27: Vertical scaling relations for Test DSTi0_5, Zone III.
Figure H28: Vertical scaling relations for Test DSTi0_5, Zone IV.
y = 1.1012072x - 0.7862612R2 = 0.5765978
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
-2.0 -1.5 -1.0 -0.5 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_5Zone III
y = 4.2153230x + 4.8225003R2 = 0.9967769
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
Model d/wT
Pro
toty
pe d
/wT
DSTi0_5Zone IV
172
Figure H29: Horizontal scaling relations for Test DSTi0_5, all zones.
Figure H30: Vertical scaling relations for Test DSTi0_5, all zones.
y = 0.5906999x + 0.0081354R2 = 0.9332500
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
-0.02 0.00 0.02 0.04 0.06 0.08 0.10
Model x/gT2
Pro
toty
pe
x/g
T2
DSTi0_5All points
y = 1.9388359x - 0.2822884R2 = 0.9444835
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
-3.0 -2.0 -1.0 0.0 1.0 2.0
Model d/wT
Pro
toty
pe
d/w
T
DSTi0_5All points
173
VITA
Muhammad Shah Alam Khan
EDUCATION
• Ph.D. Civil Engineering (2002). Drexel University. • M.S. Civil and Environmental Engineering (1991). University of Rhode Island. • B.Sc. Engineering (Civil) (1988). Bangladesh University of Engineering and Technology.
HONORS
• Graduate Student Research Award (2000-01). Drexel University. • George Hill, Jr. Endowed Fellowship (2000-01). Drexel University.
POSITIONS
• Adjunct Assistant Professor (Summer 2002). Department of Civil and Architectural Engineering, Drexel University.
• Teaching Assistant (1997 to 2002). Department of Civil and Architectural Engineering, Drexel University.
• Assistant Professor (1995 to date). Institute of Water and Flood Management, Bangladesh University of Engineering and Technology.
• Lecturer (1992-95). Institute of Water and Flood Management, Bangladesh University of Engineering and Technology.
• Teaching Assistant (1989-91). Department of Civil and Environmental Engineering, University of Rhode Island.
• Internee (Summer 1990). Rhode Island Department of Environmental Management. • Junior Engineer (1988-89). Geotex-Hydro Consultants, Bangladesh.
TRAINING
• Post Graduate Certificate Course on Analysis and Management of Geological Risks (1994). University of Geneva, Switzerland.
• Third United Nations Training Course on Remote Sensing Education for Educators (1993). Stockholm University, Sweden.
SELECTED PUBLICATIONS
• Khan, M.S., and Chowdhury, J.U. “Dhaka City Storm Water Quality Assessment, Technical Report No. 1: Analysis of 1996 Monsoon Data”. IFCDR, Bangladesh University of Engineering and Technology, May 1997.
• Rahman, M.R., and Khan, M.S. “Application of Remote Sensing Technology to Rainfall Forecasting”. Japan-Bangladesh joint study report, IFCDR, Bangladesh University of Engineering and Technology, 1997.
• Hoque, M.M., and Khan, M.S. “Post Farakka dry season surface and ground water conditions in the Ganges and vicinity”. In Women for Water Sharing, H.J. Moudud, ed., Academic Publishers, Dhaka, 1995.
