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Page i Acknowledgement Page ii Page iii Table of Contents Page iv Page v Page vi List of Tables Page vii List of Figures Page viii Page ix Page x Page xi Page xii Page xiii Page xiv List of symbols Page xv Page xvi Page xvii Page xviii Page xix Abstract Page xx Page xxi Chapter 1. Introduction Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Chapter 2. Background study Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Page 41 Page 42 Page 43 Chapter 3. Numerical onedimensional Bingham fluid model Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Chapter 4. Twodimensional multilayered hydrodynamic model Page 57 Page 58 Page 59 Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Chapter 5. Viscoelastic property tests Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 Page 99 Page 100 Page 101 Page 102 Page 103 Page 104 Chapter 6. Erosion tests: Facility and procedure Page 105 Page 106 Page 107 Page 108 Page 109 Page 110 Page 111 Page 112 Page 113 Page 114 Page 115 Page 116 Page 117 Page 118 Page 119 Page 120 Page 121 Page 122 Page 123 Page 124 Page 125 Page 126 Page 127 Page 128 Page 129 Page 130 Chapter 7. Erosion tests: Results and discussion Page 131 Page 132 Page 133 Page 134 Page 135 Page 136 Page 137 Page 138 Page 139 Page 140 Page 141 Page 142 Page 143 Page 144 Page 145 Page 146 Page 147 Page 148 Page 149 Page 150 Page 151 Page 152 Page 153 Page 154 Page 155 Page 156 Page 157 Page 158 Page 159 Page 160 Page 161 Page 162 Page 163 Page 164 Page 165 Page 166 Page 167 Page 168 Page 169 Page 170 Page 171 Page 172 Page 173 Page 174 Page 175 Page 176 Page 177 Page 178 Page 179 Page 180 Page 181 Page 182 Page 183 Page 184 Page 185 Page 186 Page 187 Page 188 Page 189 Chapter 8. Summary, conclusions and recommendations Page 190 Page 191 Page 192 Page 193 Page 194 Page 195 Page 196 Page 197 Page 198 Page 199 Appendix A. Boundary conditions for multilayered model Page 200 Page 201 Page 202 Page 203 Page 204 Page 205 Page 206 Page 207 Appendix B. Comparison between prediction and measurement: Dynamic pressure and horizontal velocity Page 208 Page 209 Page 210 Page 211 Page 212 Page 213 Page 214 Page 215 Page 216 Page 217 Page 218 Page 219 Page 220 Page 221 Appendix C. Miscellaneous considerations for the wave Page 222 Page 223 Page 224 Page 225 Page 226 Page 227 Page 228 Page 229 Page 230 Page 231 Page 232 Page 233 Page 234 Page 235 Page 236 Page 237 Appendix D. Data on erosion tests Page 238 Page 239 Page 240 Page 241 Page 242 Page 243 Page 244 Page 245 Page 246 Page 247 Page 248 Page 249 Page 250 Page 251 Page 252 Page 253 Page 254 Page 255 Page 256 Page 257 Page 258 Page 259 Page 260 Page 261 Page 262 Page 263 Page 264 Page 265 Page 266 Page 267 Page 268 Page 269 References Page 270 Page 271 Page 272 Page 273 Page 274 Page 275 Page 276 Biographical sketch Page 277 Page 278 Page 279 Page 280 
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EROSION OF SOFT MUDS BY WAVES BY P.Y. IMAA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1986 ACKNOWLEDGEMENTS The author would like to express his sincerest appreciation to his research advisor and supervisory committee chairman, Dr. A.J. Mehta, Associate Professor of Civil Engineering and of Coastal and Oceano graphic Engineering, for his continuous guidance and encouragement throughout this research. Appreciation is also extended for the valu able advice and suggestions of the supervisory committee cochairman, Dr. B. A. Christensen, Professor of Civil Engineering, as well as the guidance received from the other committee members: Dr. R. G. Dean, Graduate Research Professor of Coastal and Oceanographic Engineering, Dr. B. A. Benedict, Professor of Civil Engineering, and Dr. A. K. Varma, Professor of Mathematics. Sincere thanks also go to Dr. R. A. Dalrymple, Dr. J. T. Kirby, Dr. M. C. McVay, Dr. L. E. Malvern and Mr. M. Ross for their sugges tions and help in this study. Special thanks go to Mr. Vernon Sparkman, Mr. C. Broward, and other staff of the Coastal Engineering Laboratory for their assistance with the experiments performed during this research. In addition, the author thanks Ms. L. Lehmann and Ms. H. Twedell of the Coastal Engine ering Archives for their assistance. Gratitude is due to the U.S. Army Corps of Engineers Waterway Experiment Station, Vicksburg, Mississippi, for their financial support of this research under Contract DACW3984CO013. Finally, the author thanks his wife, TaiFang, for her love, encouragement and patience, and his parents for their love and support. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ............................................. ii LIST OF TABLES ............................................... vii LIST OF FIGURES .............................................. viii LIST OF SYMBOLS .............................................. xv ABSTRACT ..................................................... xx CHAPTERS 1 INTRODUCTION ......................................... 1 1.1 Significance of Wave Erosion of Soft Mud ........ 1 1.2 Factors Characterizing the Wave Erosion Process. 3 1.3 Objectives and Scope of the Present Study ....... 7 1.4 Outline of Presentation ......................... 8 2 BACKGROUND STUDY ...................................... 10 2.1 Introduction .................................... 10 2.2 Wave Erosion/Resuspension ....................... 10 2.3 Constitutive Models for Soft Mud ................ 17 2.3.1 Nature of Clayey Soil .................... 18 2.3.2 Elastic Model ............................ 21 2.3.3 Poroelastic Model with Coulomb Friction. 21 2.3.4 Viscous Fluid Model ...................... 22 2.3.5 Viscoplastic Model ....................... 26 2.3.6 Viscoelastic Model ....................... 32 2.4 Bed Shear Stress ................................ 36 2.5 Wave Diffusion Coefficient ...................... 39 3 NUMERICAL ONEDIMENSIONAL BINGHAM FLUID MODEL ........ 44 3.1 Introduction .................................... 44 3.2 Problem Formulation ............................. 44 3.3 Results ......................................... 50 3.4 Conclusion ...................................... 55 4 TWODIMENSIONAL MULTILAYERED HYDRODYNAMIC MODEL ..... 57 4.1 Introduction .................................... 57 4.2 Formulation ..................................... 57 4.3 Solution Technique .............................. 64 4.4 Input Data ...................................... 66 4.5 Model Results ................................... 69 4.5.1 Velocity ................................. 70 4.5.2 Pressure ................................. 73 4.5.3 Shear Stress ............................. 75 4.5.4 Water Wave Decay ......................... 77 4.5.5 Interfacial Wave Amplitude ............... 78 4.6 Resonance Characteristics ....................... 80 5 VISCOELASTIC PROPERTY TESTS .......................... 87 5.1 Introduction .................................... 87 5.2 Relaxation Test ................................. 87 5.2.1 Apparatus ................................ 88 5.2.2 Procedure ................................ 92 5.2.3 Results .................................. 93 5.3 Tests for Material Constants of Voigt Element... 97 5.4 Tests for Rheological Properties of Bingham Fluid Model ..................................... 102 6 EROSION TESTS: FACILITY AND PROCEDURE ................ 105 6.1 Introduction .................................... 105 6.2 Wave Flume ...................................... 105 6.2.1 Wave Reflection .......................... 107 6.2.2 Wave Horizontal Velocity ................. 108 6.2.3 Wave Decay for Rigid Bed ................. 108 6.3 Sediment ........................................ 109 6.3.1 Kaolinite ................................ 109 6.3.2 Cedar Key Mud ............................ 110 6.4 Eroding Fluid ................................... 113 6.5 Instrumentation ................................. 116 6.5.1 Wave Gauges .............................. 116 6.5.2 Pressure Transducers ..................... 116 6.5.3 Light Meter .............................. 118 6.5.4 Data Acquisition System .................. 120 6.5.5 Suspended Sediment Sampler ............... 122 6.5.6 Bed Sampler .............................. 124 6.5.7 Current Meter ............................ 124 6.6 Procedure ....................................... 125 6.6.1 Process of the Digital Data .............. 126 6.6.2 Suspension Samples ....................... 128 6.6.3 Waveaveraged Mud Surface Elevation ...... 130 7 EROSION TESTS: RESULTS AND DISCUSSION ................ 131 7.1 Introductory Note ............................... 131 7.2 Results ......................................... 131 7.2.1 Bed Density .............................. 131 7.2.2 Pressure Response ........................ 137 7.2.3 Instantaneous Sediment Concentration ..... 142 7.2.4 Timeaveraged Sediment Concentration ..... 146 7.2.5 Waveaveraged Mud Surface Elevation ...... 151 7.3 Wave Erosion and Entrainment .................... 156 7.3.1 Erosion .................................. 156 7.3.2 Erosion Rate ............................. 165 7.3.3 Entrainment .............................. 174 7.4 Dye Diffusion Tests ............................. 182 7.5 Concluding Comments ............................. 185 8 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ............. 190 8.1 Summary and Conclusions ......................... 190 8.2 Recommendations for Future Study ............... 197 APPENDICES A BOUNDARY CONDITIONS FOR MULTILAYERED MODEL .............. 200 B COMPARISON BETWEEN PREDICTION AND MEASUREMENT: DYNAMIC PRESSURE AND HORIZONTAL VELOCITY ................. 208 C MISCELLANEOUS CONSIDERATIONS FOR THE WAVE EROSION EXPERIMENT ....................................... 222 D DATA ON EROSION TESTS .................................... 238 REFERENCES ................................................... 270 BIOGRAPHICAL SKETCH .......................................... 277 LIST OF TABLES Page 21 Diffusion Coefficients in Wave Flow ...................... 42 41 Input Thickness for Each Layer .......................... 68 42 Interfacial Shear Stresses and Mud Wave Amplitudes ...... 78 51 Vane Dimensions ......................................... 91 52 Coefficients of the Correlation Equations ............... 101 61 Sediment Properties ..................................... 113 62 Composition of Clay Fraction ............................ 113 63 Chemical Composition of the Eroding Fluid ............... 115 64 Eroding Fluid Properties ................................ 115 65 Wave Conditions for Erosion Tests ....................... 129 71 Average Erosion and Entrainment Rates ................... 163 72 Coefficients for the Erosion Function ................... 171 73 Measured Diffusion Coefficient in the Upper Layer ....... 179 74 Constants for Steady Flow Erosion Function .............. 187 LIST OF FIGURES Page 11 Schematic Depiction of Response of WaveMud Interaction ............................................. 5 21 A Plot of the Suspended Solids and Bed Shear Stress for a Wave Resuspension Test ............................ 12 22 Measured Suspended Sediment Concentration Profiles from a Wave Resuspension Test ........................... 14 23 Shear StressStrain Loop for Clay under Cyclic Load ..... 19 214 Energy Dissipation Ratio X versus Shear Strain .......... 19 25 Shear StressStrain Loops and Effects of Cyclic Load .... 20 26 A Plot of the Horizontal Velocity Amplitude Profiles by Using the TwoLayered Viscous Fluid Model ............ 25 27 Relationship between Yield Strength and Sediment Concentration .................................. 28 28 Relationship between Apparent Viscosity and Sediment Concentration .................................. 29 29 System Diagram for Simple Viscoelastic Models and Response under Constant Loading ......................... 33 210 Viscoelastic Behavior of Clay ........................... 33 211 Wave Friction Factor .................................... 38 31 System Diagram for the 1D Numerical Bingham Fluid Model ............................................. 45 32 Horizontal Velocity Profiles for Two Steady Flows. (a) TwoLayerd Viscous Fluid; (b) Bingham Fluid with Exponentially Increasing K and v ........................ 51 33 Comparison of the Wave Velocity Amplitude Profiles from 1D model and Dalrymple and Liu's Model with P, 1000 kg/m3, p2 1160 kg/m3, v, lxl06 and v22x10 m2/s... 52 34 Wave Velocity Profiles for the Water Bingham Fluid System with T 3 sec. (a) H 2.5 cm and kim 0.0011 m1; (b) H 5 cm and kim:O.016 m1 .............. 54 35 Shear Stress versus Shear Strain Rate for Kaolinite Bed with TwoDay Consolidation Period ................... 56 36 Wave Velocity Profiles for WaterBingham Fluid System with Measured Yield Strength and Viscosity ........ 56 41 Schematic Figure for the MultiLayered Model ............ 58 42 Layout of the Coefficient Matrix for the MultiLayered Model ................................................... 65 43 Comparison of Computed Bed Shear Stress (model and Kamphuis) at the Interface to Determine the Eddy Viscosity of Water ...................................... 68 44 Comparison of the Model Prediction and Measurement for Run 12. (a) Velocity; (b) Pressure ................. 71 45 Comparison of the Model Prediction and Measurement for Run 52. (a) Velocity; (b) Pressure ................. 72 46 An Example of the ModelPredicted Shear Stress Profile.. 76 47 Comparison of the Measured and Predicted Wave Decay Coefficient ............................................. 79 48 NonDimensional Interfacial Wave Amplitude versus Complex Reynolds Number ................................. 82 49 Resonance Phenomenon of WaterMud System. (a) with d2 = 0.05 m; (b) with d2 = 0.14 m .............. 84 410 Frequency Response of the WaterMud System. Mud is Assumed as (a) Viscoelastic Material; (b) Viscous Fluid. 85 51 Apparatus for Relaxation Tests and Viscoelastic Constant Measurement. (a) System Diagram; (b) Top View.. 88 52 Structure of Brookfield Viscometer ...................... 89 53 Miniature Vanes and Sample Container .................... 90 54 A Plot of the Residual Torque and Angular Displacement versus Elapsed Time for Run 1, Kaolinite with OneWeek Consolidation Period .................................... 94 55 A Plot of the Residual Torque and Angular Displacement versus Elapsed Time for Run 4, Cedar Key Mud with OneWeek Consolidation Period ........................... 95 56 A Plot of the Residual Torque and Angular Displacement versus Elapsed Time for Run 5, Cedar Key Mud with TwoDay Consolidation Period ............................ 96 57 A Plot of Shear Stress against Shear Strain for Determination of Viscoelastic Constant .................. 99 5'8 Relationship between Viscoelastic Constants and Mud Dry Density. (a) Viscosity; (b) Shear Modulus ........... 100 59 Shear Stress versus Shear Strain Rate obtained from Kaolinite Bed, Run 1, 6.2 cm below Mud Surface .......... 104 61 Wave Flume for the Erosion Experiments .................. 106 62 WaterMud Interface Elevations during Deposition and Consolidation of the Kaolinite Slurry ............... 111 63 Dispersed Grain Size Distribution for Sediment in Runs 4, 5, and 6 ..................................... 114 64 Pressure Transducer, Druck Model PDCR 135/A/F ........... 117 6&5 System Diagram of the PhotoSensing Light Meter ......... 119 66 Suspended Sediment Sampler .............................. 123 67 Wave Loading for the Erosion Test, Run 3 ................ 127 68 A Plot of Wave Height Decay in the Wave Flume ........... 129 71 Measured Mud Density Variation due to Wave Action, Run 1 ................................................... 132 7f 2 Volumetric Swelling of Mud Bed Caused by Wave Action. (a) Run 1; (b) Run 4 .................................... 134 73 DepthAveraged Bed Density Variation Caused by Wave Action ............................................. 135 74 Dimensionless Mud Dry Density Profiles. (a) Kaolinite; (b) Cedar Key Mud ....................................... 136 75 Instantaneous Pressure Response in the Kaolinite Bed, Run 1. (a) Wave Profile; (b) Pressure ................... 138 7,6 Average Pressure Response in the Kaolinite Bed, Run 1. (a) Dynamic Pressure Fluctuation; (b) WaveAveraged Pressure ................................................ 140 77 Variation of Dimensionless Apparent Bulk Density Caused by Wave Action, Run 1, Normalized by the Initial Value..141 78 Instantaneous Sediment Concentration Response in the Water Column. (a) Run 3; (b) Run 5 ..................143 79 WaveAveraged Response from Light Meter during Wave Erosion Process .................................... 144 710 Average Sediment Concentration Fluctuation Detected by the Light Meter during Wave Erosion Process .......... 145 711 Sample Profiles for the Suspended Sediment Concentration for Run 1. (a) at STA. B; (b) at STA. D... 147 712 Sample Profiles for the Suspended Sediment Concentration for Run 5. (a) at STA. B; (b) at STA. D ...148 713 Longitudinal Variation of the DepthAveraged Sediment Concentration, Run 4 ........................... 150 714 Mud Surface Profiles at Selected Times. (a) Run 1 ............................................... 152 714 (continued). (b) Run 2; (c) Run 3 ....................... 153 715 Response of the SpatiallyAveraged Mud Surface Elevation. (a) Kaolinite Bed; (b) Cedar Key Mud Bed ..... 155 716 Wave Erosion/Entrainment Behavior for Kaolinite. (a) Run 1 ............................................... 157 716 (continued). (b) Run 2 .................................. 158 716 (continued). (c) Run 3 .................................. 159 717 Wave Erosion/Entrainment Behavior for Cedar Key Mud. (a) Run 4; (b) Run 5 .................................... 160 717 (continued). (b) Run 6 .................................. 161 718 Shear Strength Profiles for Resistance of Wave Erosion. (a) Kaolinite Bed; (b) Cedar Key Mud Bed ................ 167 719 Influence of the Period of Consolidation on Erosion Resistance. (a) Kaolinite; (b) Cedar Key Mud ............ 168 720 Erosion Rate versus Dimensionless Excess Shear Stress. (a) Kaolinite; (b) Cedar Key Mud ........................ 170 721 Dimensionless Erosion Rate versus Dimensionless Excess Shear Stress ..................................... 172 xi 722 Erosion Rate versus Bed Shear Stress, Run 5 ............. 173 723 Suspended Sediment Concentration Profiles at Steady State. (a) Run 1; (b) Run 2 ...................... 175 723 (continued). (c) Run 3; (d) Run 4 ....................... 176 723 (continued). (e) Run 5; (f) Run 6 ....................... 177 724 Settling Velocity versus Sediment Concentration for Kaolinite ............................................... 179 725 SpatiallyAveraged Variation of (a) the Distance between the Lowest Tap and Mud Surface; (b) Sediment Concentration at the Lowest Tap ............ 181 726 Diffusion of a LineSource Dye in a Wave Flow with T = 1.2s, H 7 cm. (a) t = 0.5s; (b) t 5s; (c) t 14 s; (d) t 20 s .............................. 183 BI Comparison between Observed and Predicted Horizontal Velocity Amplitudes in the Bed .......................... 210 B,2 Comparison of Model Prediction and Measurement for Run 11. (a) Velocity; (b) Pressure ................. 211 B3 Comparison of Model Prediction and Measurement for Run 21. (a) Velocity; (b) Pressure ................. 212 B4 Comparison of Model Prediction and Measurement for Run 22. (a) Velocity; (b) Pressure ................. 213 BA5 Comparison of Model Prediction and Measurement for Run 31. (a) Velocity; (b) Pressure ................. 214 B6 Comparison of Model Prediction and Measurement for Run 32. (a) Velocity; (b) Pressure ................. 215 B7 Comparison of Model Prediction and Measurement for Run 41. (a) Velocity; (b) Pressure ................. 216 B8 Comparison of Model Prediction and Measurement for Run 42. (a) Velocity; (b) Pressure ................. 217 B,9 Comparison of Model Prediction and Measurement for Run 51. (a) Velocity; (b) Pressure ................. 218 B10 Comparison of Model Prediction and Measurement for Run 61. (a) Velocity; (b) Pressure ................. 219 B,11 Comparison of Model Prediction and Measurement for Run 62. (a) Velocity; (b) Pressure ................. 220 xii B12 Comparison of Model Prediction and Measurement for Run 63. (a) Velocity; (b) Pressure ................. 221 Ci Normalized Frequency Response of the Current Meter......225 C2 Circuit Diagram for the LED Driver of Light Meter ......227 C3 Light Meter Probe ....................................... 228 C4 Circuit Diagram for the PreAmplifier of Light Meter .... 230 C5 Circuit Diagram for the Main Processor of Light Meter.. 231 C16 Light Meter Calibration Device .......................... 232 C7 Light Meter Calibration Curves for Kaolinite ............ 233 C8 Light Meter Calibration Curves for Cedar Key Mud ........ 234 C9 Response Function of the LowPass Filter ................ 236 C.I10 Comparison of Data Record with/without the LowPass Filter ......................................... 237 D1 Wave Loading for the Wave Erosion Test, Run 1 ........... 253 D2 Wave Loading for the Wave Erosion Test. (a) Run 2; (b) Run 4 .................................... 254 D3 Wave Loading for the Wave Erosion Test. (a) Run 5; (b) Run 6 .................................... 255 D4 WaveAveraged Pressure Response in the Mud Bed. (a) Run 2; (b) Run 3 .................................... 256 D5 WaveAveraged Pressure Response in the Mud Bed. (a) Run 4; (b) Run 5 .................................... 257 D6 WaveAveraged Pressure Response in the Mud Bed, Run 6.. .258 D7 Average Dynamic Pressure Fluctuation in the Mud Bed, Run 2 ................................................... 258 D8 Average Dynamic Pressure Fluctuation in the Mud Bed. (a) Run 3; (b) Run 4 .................................... 259 D9 Average Dynamic Pressure Fluctuation in the Mud Bed. (a) Run 5; (b) Run 6 .................................... 260 D10 Average Apparent Bulk Density Variation. (a) Run 2; (b) Run 3 .................................... 261 D11 Average Apparent Bulk Density Variation. (a) Run 4; (b) Run 5 .................................... 262 D12 Average Apparent Bulk Density Variation, Run 6 .......... 263 D13 Variation of Kaolinite Bed Density Profile Due to Wave Action, Run 2 ...................................... 264 D14 Variation of Kaolinite Bed Density Profile Due to Wave Action, Run 3 ...................................... 265 D15 Variation of Cedar Key Mud Bed Density Profile Due to Wave Action, Run 4 ............................... 266 D16 Variation of Cedar Key Mud Bed Density Profile Due to Wave Action, Run 5 ............................... 267 D17 Variation of Cedar Key Mud Bed Density Profile Due to Wave Action, Run 6 ............................... 268 D18 WaveAveraged Response from Light Meter, Run 6. (a) Sediment Concentration; (b) Sediment Concentration Fluctuation ............................................. 269 LIST OF SYMBOLS a: Wave amplitude in the x wave propogation direction. a : Wave amplitude at x = 0. 0 ab: Semiexcursion distance of a water particle just outside 6. A/D: Analog to digital. boi: The ith interfacial mud wave amplitude at x 0, a complex no. C: Suspended sediment concentration (g/l). c: Complex coefficient matrix c(i,j) or wave celerity. cd: Damping coefficient in the diskshaft system. cg: Wave group velocity. d i: Thickness of ith layer. d : Total thickness of mud bed. D: Rate of deformation tensor, D I ( j )" S2 axj Dxi D': Deviator part of the rate of deformation tensor, Dj= D  D i ii 3 r Dm: Apparent Diffusion coefficient (molecular diffusion + turbulent diffusion + dispersion). D : Diameter of the miniature vane. v D90 Grain size through which 90% of the total soil mass is finer than this size. E': Deviator part of the small strain tensor, El = Eij  E 1 3a ij 3 rr E : Small strain tensor, Eij= + ( j ){. j: Rate of strain. j': Rate of shear strain. En: Wave energy. Ed : Timeaveraged energy dissipation per unit surface area. xv f w g: g/l: G: G: 0 h: H: H: v i: j J: k: kim: k: r K: L: p: MP: n M: r n: N: P: pO: pt: r: Wave friction factor. Acceleration due to gravity. gm/liter. Shear Modulus of elasticity. A constant coefficient. Water depth. Wave height. Height of the miniature vane. Free index. Free index or V T. Grid number at the water surface. Complex wave number, k kr + j kim Imaginary part of the complex wave number, also the wave decay coefficient, defined as a(x)= a exp(k imAx). The real part of the complex wave number, k r= 27/L. Yield strength of Bingham Fluid. Wave length. Constant coefficient, or free index. Sediment mass entrained into water column measured by water samples withdrawn from water column. Sediment mass eroded by waves. Constant coefficient. Number of layers in the hydrodynamic model presented in Cha Dynamic pressure. Correction term for the static pressure due to density diff Total pressure; Dynamic pressure amplitude. Radius, or free index. pter 4. erence. R: Complex Reynolds number, R Rr + j Rim. Rw: Wave Reynolds number, R = 2 / o w 0w 0V RMS: Root mean square. S : Undrained shear strength. u SWL: Still water level. t: Time. T: Wave period or stress tensor. T': Deviator part of the stress tensor, Tij= Tij Trr Tdc: Period of Consolidation of a sediment slurry. j': Rate of shear stress. ui Velocity in the xi direction; horizontal velocity for ith layer. Ub: Horizontal velocity amplitude just outside the boundary layer. u1: Horizontal velocity amplitude. w Vertical velocity. wi: Vertical velocity amplitude. wl: First derivative of wi with respect to z. i w": Second derivative of wi with respect to z. i w : Settling velocity of a sediment particle or floc. x: Longitudinal direction, or xI. xi: Cartesian coordinate using index notation, i 1, 2, 3. X: Column matrix containing Al, BI, .... CN, DN, and b.,N1. y: Laterial direction, or x2. z: Vertical coordinate, or x3, starting from the water surface, positive upward. z': Vertical coordinate, starting from mud surface, positive downward. z": Vertical coordinate, starting from bed, positive upward. z : An elevation where a pressure transducer was mounted. m xvii z : An elevation where a pressure transducer was mounted. n At: Increment in the time domain. Ax: Increment in the x direction. Az: Increment in the z direction. a: Constant coefficient. a : Constant coefficient. 0 B: Constant coefficient. a: Wave frequency, a = 2n/T. 6: Thickness of wave boundary layer. 6 L: Thickness of the viscous wave boundary layer. p: Material density. PB: Bulk density of mud. PD: Dry density of mud. P: Depthaveraged dry density of mud at a time. PDo: Depthaveraged dry density of mud before wave loading. Pe: Average apparent bulk density of mud between two sensors. Peo: Pe value at the beginning of wave loading. Ps: Density of sediment grain. pwt Density of water. p': Normalized apparent bulk density between two sensors, = p / p e e T: Shear stress. ;: Shear stress amplitude. Tb x): Bed shear stress amplitude at x=x. Tbol Bed shear stress amplitude at x0. xviii T Critical shear stress. The minimum bed shear stress that will c cause erosion of sediment bed. Te Excess shear stress, Te =T b Ts(z'). Ts: Erosion resistance or bed shear strength. Tao: Erosion resistance at the bed surface. Ts: Depthaveraged (top 5 mm) erosion resistance of mud bed. n: Water surface displacement. C: A constant eddy viscosity coefficient. c Erosion rate constant for wave flow. 0 Cf Erosion rate constant for steady flow. c n Rate of upward entrainment of sediment. E Erosion rate. r E: Interfacial wave displacement, 0 b J(kxat). Ci: Displacement in the x direction. U: Dynamic viscosity. Pe: Apparent dynamic viscosity, Pe = PBe" Vo: Constant coefficient. pw: Dynamic viscosity of water. Im: Dynamic viscosity of mud. V: Kinematic viscosity. v e: Apparent kinematic viscosity. e: Angular shear strain. de/dt or 6: Rate of angular shear strain. IID,: Second invariant of the deviator part of D'. : Diameter. 0: Angular speed. A: Energy dissipation ratio. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EROSION OF SOFT MUDS BY WAVES By P.AY. Maa August 1986 Chairman: Dr. A. J. Mehta Cochairman: Dr. B. A. Christensen Major Department: Civil Engineering The influence of water waves on soft muds can result in high turbidity and significant wave attenuation. A multilayered 2hD hydro dynamic model was developed and experimentally verified in order to evaluate the bed shear stress at the watermud interface, for inter preting results on interfacial sediment erosion. In the model, the mud bed was discretized into layers of constant density, viscosity and shear modulus of elasticity. The last two parameters characterize mud as a linear viscoelastic material. Layering was essential for simu lating the depthvarying bed properties. For given nonbreaking, regular waves, the model gives profiles of velocities, pressure and shear stress as well as wave attenuation coefficient. Laboratory flume tests of sediment erosion under waves were con ducted using a commercial kaolinite and an estuarine mud. Sediment concentration response in the water column and the variation of total eroded sediment mass with elapsed time under different wave loading were measured. Additional tests were conducted to evaluate the visco elastic constants of the selected sediments. xx Two types of erosion process were identified. The first is applicable to a partially consolidated bed, in which erosion resist ance, Ts(Z'), increases with depth, z', and the erosion rate, Er, decreases as erosion proceeds. The proposed erosion rate function is r Lb T, >? br sZ E = Or [ I ] r>r (z') E0 5 Z' where Tb is the bed shear stress, co is a rate constant and a is an empirical exponent. The average value of a for all test data was close to one, which reveals a simple, linear relationship. The second type of erosion process is for a newly deposited bed, in which the erosion resistance is a small constant, and the erosion proceeds at a constant rate. Measured suspended sediment concentration profiles indicated relatively low concentrations in the upper 80% of the water column and the formation of a high density fluid mud layer near the bed. Low upward diffusion, high longitudinal dispersion and advective transport characterized the nearbed layer. Bed softening and fluidmudgene rating capacity of waves appear to be significant features of the wave erosion process. CHAPTER 1 INTRODUCTION 1.1 Significance of Wave Erosion of Soft Muds Most estuaries and several reaches of shorelines have soft muddy bottoms consisting primarily of silt and clay comprising a cohesive sediment. With the rapid development of coastal and estuarine harbors and increasing concerns from the environmental protection point of view, the need to predict the erosion of soft mud has increased. Basically, in these environments, tidal currents and water waves are the two forces that cause erosion of sediments. Extensive studies of the erosion behavior of cohesive sediment under tides, in quasisteady flow, have been conducted by various researchers. Expressions of the erosion rate have been proposed (Partheniades, 1965; Krone, 1962; Mehta et al., 1982; Parchure, 1984). Mathematical models for cohesive sedi ment transport by current are also available (Odd and Owen, 1972; Ariathurai, 1974; Hayter, 1983). However, the influence of water waves on the erosion of cohesive sediment has by and large remained unex plored. Only a few important laboratory studies have been made (Alishahi and Krone, 1964; Anderson, 1972; Jackson, 1973; Thimakorn, 1980). However, data obtained have been scattered and incomplete. A comprehensive and systematic study of wave erosion behavior is neces sary to elucidate the mechanism of the wave erosion process. Here the physical processes which have been observed at muddy coasts are intro duced first. The most obvious feature of the muddy coastal zone is the lack of large waves due to the great damping ability of the soft mud beds (Wells, 1983). This has been noticed since the early 1800s in South west India and northeastern South America. The damping ability comes from the motion of the mud bed, a phenomenon observed in both labo ratory and field studies (Tubman and Suhayda, 1976; Schuckman and Yamamoto, 1982). An immediate benefit of this high damping ability is protection of shorelines. Fishermen in the Gulf of Mexico have taken refuge during stormy weather in waters over the "mud hole," a localized accumulation of soft mud bottom (Morgan et al., 1953). High suspended sediment concentration (0.1 2 g/l) is another feature of soft muddy coasts even in calm weather (wave height 10 cm). In severe wave conditions, the entire soft mud can be stirred up resulting in great turbidity. The mechanism of wave transformation when propagating over soft mud beds and the resulting erosion of mud are important to both coastal and environmental engineers because waves and the associated sediment transport are major factors for design of most coastal waterways and harbors, e.g., the maintenance of navigation channels, and the control of pollutants. However, studies on this subject are limited apparently because the change of characteristics of surface water waves caused by the dissipation of wave energy in the mud and resuspension of mud are interlinked and complex. Added complexities arise from the influences of physicochemical properties of the sediment and the fluid (Mehta, 3 1983), the effect of bed structure (Parchure, 1984), and the time dependence of bed structure (Schuckman and Yamamoto, 1982) on the overall problem. 1.2 Factors Characterizing the Wave Erosion Process A summary of the parameters which influence the wave erosion problem is presented in Fig. 11. Thick arrows show the dominant processes, and thin arrows are the less important processes. Dashed lines represent processes or parameters that are known to have influ ences on the dynamics of wavemud interaction, but which were not considered in detail in this study. The responses of this wavemud system, contained in the dashed rectangular box, subjected to dynamic wave loading, are dependent on the characteristics of both wave and mud. Pressure and shear stress at the watermud interface are the two forces that have linked the water and mud bed together. Two immediate responses are wave damping and erosion, shown in the circles. The contribution of pressure fluctuation at the interface is mainly res ponsible for the motion of the mud beds, and therefore, wave damping. Shear stresses mainly contribute to the erosion process, a conclusion from steady flow studies. Turbulence in the water flow influences the bed shear stress and the sediment transport process as well. Erosion is dependent on the properties related to waves, i.e., bed shear stress, as well as erosion resistance of the mud material, i.e., shear strength. Therefore, development of a hydrodynamic model that can simulate the water wave decay and the interfacial shear stress (also called bed shear stress, Tb) is necessary before an erosion rate 4 expression can be developed. With Tb known, the results from wave erosion tests can be better interpreted. Defining the properties of soft mud beds is in itself an inter esting subject. Such beds were assumed to be a viscous fluid by Gade (1958) and Dalrymple and Liu (1978); an elastic material by Mallard and Dalrymple (1977) and Dawson (1978); a poroelastic material by Yamamoto and Takahashi (1985); a viscoelastic material by MacPherson (1980) and Hsiao and Shemdin (1980); a Bingham fluid by Migniot (1968), Krone (1965), Pazwash and Robertson (1971), Gularte (1978), Vallejo (1979, 1980) and Englund and Wan (1984). However, what is important in this case is to find a model that can reasonably simulate the two important parameters: bed shear stress and mud motion. These two parameters are directly related to wave erosion and wave attenuation. This is the first step in studying the wave erosion process. Initially a 1D numerical Bingham fluid model was developed by the author to simulate these two observed phenomena. However, the results demonstrated that it could not appropriately simulate wavemud interac tions. A linearized multilayered model was then developed to predict the wave decay and the bed shear stress. In this model, the mud bed was arranged as layers of linear viscoelastic material to meet the depth variation of mud properties. The water layer was characterized by having a constant viscosity. Velocities, pressure, shear stress profiles and wave attenuation coefficient were the products. Following the selection of a model for the soft mud, an effort was made to measure the mud properties by using the viscometer and other auxiliary equipment. 5 Water Waves r  .. .__ _ Water: Turbulence Wave ' Tu uk cea q rre n4u Newtonian Fluid ' ..urr.... 4 ft I 2 1 T rso Sedirnm 1  Do>Tonspot<: 5 Soft Mud: NonNewtonian 'TmI I Material I ime Deperdent\ L Jl 1 Long Term Bed \ProPerty Change / 1. Pressure and shear stress at the watermud interface contribute to the motion and erosion of the soft mud. 2. Momentum transfer from waves to muds alters the kinematics of water waves. 3. Turbulence structure of the water determines the suspended sediment distribution. 4. Bed shear stress and the mud resistance to erosion determine the rate of erosion. 5. Mud motion and associated energy dissipation plus the minor energy loss from the water viscosity cause the attenuation of water waves. 6. Eroded sediment mass serves as a sediment source in the sediment transport. 7. Waveinduced second order current contributes to the sediment transport process. 8. Successive oscillatory shear deformation in the mud bed alters the mechanical properties of the mud, and therefore, the dynamic response of the wavemud system. Legend: Box with dashed line represents the system, ovals show the processes, circles show the results. Fig. 11. Schematic Depiction of Response of Wave Mud System. Because of the complexity of the erosion process itself, labora tory tests to observe wave erosion under clearly specified conditions were absolutely necessary. Alishahi and Krone (1964), Jackson (1973), and Thimakorn (1980) have all done some initial studies. However, their data were not complete enough to give an expression for the erosion rate function. Therefore, wave erosion tests were performed to provide information on the actual erosion and entrainment behavior. In the present study, quantities of sediment mass eroded were obtained and incorporated with the bed shear stress calculated from the 2D mathe matical model to develop an erosion rate function. The diffusion process in the wave flow field was studied by ob serving the spreading of dye and the suspended sediment concentration profiles. A conclusion concerning the possible profile of the diffusion coefficient is proposed. Marine sediment beds tend to change their physical properties under wave action. Alishahi and Krone (1964) reported a sudden loos ening of the bed. Schuckman and Yamamoto (1982) reported the reduction of shear strength of a bentonite bed. Both observations point to the degradation and possible liquefaction of mud bed under wave loading. This may result in a change of flow structure and a change in the interfacial shear stress and energy dissipation as well. Therefore, this longterm bed property change was also monitored in the wave erosion tests. In addition to the above, there are other complexities which need to be briefly addressed: the effects of salinity differences between 7 the eroding fluid and the pore water; the construction of a mattress like layer by organic material; the effects of waveinduced current, etc. Although these were not here studied, they may be considered in future investigations. 1.3 Objectives and Scope of Study The objectives of this study were as follows: 1. to develop a hydrodynamic model that can be used to simulate the two important parameters of wavemud interaction: mud bed motion and bed shear stress, 2 to conduct laboratory experiments to obtain necessary data as input for the model, 3. to conduct laboratory experiments to verify model results, 4. to conduct laboratory experiments on the wave erosion and entrain ment of soft mud beds, and 5. to propose an erosion rate function. The scope of this study was defined as follows: 1. The hydrodynamic model developed was limited to the primary forcing mechanism only, i.e., linear waves; secondary effects were excluded. 2. Water and mud were considered immiscible layers in the model. 3. The equations of motion were linearized by neglecting the convective acceleration terms. 4. Verification of model results was limited to comparing measurements of wave attenuation, velocities, and pressures to calculated values. 5. Wave erosion tests were conducted in a wave flume to be described in Chapter 6. Mud bed thicknesses were limited to within 10 15 cm. Waves were restricted to regular (monochromatic), progressive and nonbreaking types. 6. The pore fluid and the eroding fluid were the same. This was achieved by equilibrating the sediment with the eroding fluid. 7. Vertical sediment concentration profiles were observed but no attempt was made to develop a theoretical approach for simulating the depthdistribution. 8. Three consolidation periods, 2 days, I week, and 2 weeks, for the mud beds were selected, and limited to selfweight consolidation. 9. Erosion rate studies were limited to two flocculated sediments and one fluid. Kaolinite and Cedar Key (Florida) mud were the two sediments. Tap water without added salt was the eroding fluid. 1.4 Outline of Presentation This study is presented in the following order. A brief review of previous studies on wave erosion of soft muds is given in Chapter 2, followed by a discussion of problems in those studies, including those related to constitutive models for the soft mud, bed shear stress, and diffusion features in the wave flow. Chapter 3 describes an effort to develop a 1D numerical Bingham fluid model to simulate the watermud system and the conclusion that the 1D model is not necessarily a good approach. Chapter 4 presents a 2D hydrodynamic model which treats muds as a viscoelastic material. Bed shear stresses needed for correlation with the erosion data from Chapter 6 are also presented. Chapter 5 describes the experimental procedures used to obtain the viscoelastic constants for the mud beds used in the wave 9 erosion tests. Experiments to measure the erosion of kaolinite and Cedar Key mud are given in Chapter 6 and Chapter 7. Erosion rates are also presented. Summary and conclusions of the study are given in Chapter 8. The derivation of boundary conditions for the multilayered hydrodynamic model is presented in Appendix A. In Appendix B the horizontal velocity and pressure predicted by the model are compared with measurements. Calibration of the current meter in an oscillatory flow, design of the light meter, and development of a numerical low pass filter are given in Appendix C. In Appendix D, data from the wave erosion experiments are presented. CHAPTER 2 STUDY BACKGROUND 2.1 Introduction Previous studies on the wave erosion, constitutive models for mud and their behaviors, and wave diffusion coefficients will be reviewed. These subjects are important in the understanding of the wavemud system and provide necessary information to simulate the dynamics of this system. 2.2 Wave Erosion/Resuspension Water waves apply periodic shear stresses at the watermud inter face. Whenever the shear stress exceeds the bonding force (cohesive forces and self weight) between particles, erosion occurs. The eroded material is transported and dispersed into the water column and high turbidity is subsequently observed. Investigations of wave erosion phenomena have been made primarily on cohesionless material, i.e., sand beds. Only a few experiments deal with cohesive sediment. In this section, erosion studies with cohesive sediment under nonbreaking wave conditions are discussed. Alishahi and Krone (1964) conducted the earliest wind wave erosion experiments for the San Francisco Bay mud. The test section, where the mud was placed, was 1.2 m in length and located at the downstream side of the wave flume (18 m in length). The average thickness of the mud beds was about 1 cm. Water samples were taken at several locations via a special device that gave only depthaveraged sediment concentra tion. Total suspended sediment mass was then obtained by summation of the product of sediment concentration and water volume. Figure 21 shows one of their two test results. Their most significant result was "the demonstration of the existence of critical wave induced shear for suspension of the bed surface." No or negligible erosion occurred for a bed shear stress less than the critical shear. See the response in the first 4 hr in Fig. 21. Another interesting response reported was "a sudden loosening of the bed and direct movement of sediment into suspension." This phenomenon occurred in both runs and was responsible for a significant increase of suspended sediment. See the response after 4.5 hr in Fig. 21. However, possible reasons for this sudden loosening were not given. Their Run 1 showed that an equilibrium state had been reached during the erosion process. Unfortunately, Run 2 was not sufficiently long for an equilibrium condition to have been reached, as was noted by the investigators. Anderson (1972) performed field experiments in The Great Bay estuarine system in New Hampshire. He took records of tide, wind speed, wind direction, wave height, and suspended sediment concentra tion. He found a linear relationship between suspended sediment con centration and water wave height at flood tide, while at ebb tide these two factors could not be correlated. He also concluded that wave force is most important for introducing sediment into the water column. 00 #. 0.48 g W 0 6 . " ' 0 CO 0.08 w o, zoo 0 2 4 6 8 TIME AFTER START OF EXPERIMENT(hrs) Fig. 21. A Plot of the Suspended Solids and Bed Shear Stress for a Wave Resuspension Test (After Alishahi and Krone, 1964). Jackson (1973) conducted laboratory experiments to study the effects of water waves on the resuspension of muds on a beach. Breaking waves were involved because of the sloping beach. He found that waves produced little erosion or resuspension when the bottom orbital velocity was less than 10 12 cm/s. For orbital velocities greater than 20 cm/s, particlebyparticle erosion behavior shifted to mass erosion. Thimakorn (1980) conducted nonbreaking wave resuspension tests of mud taken from the Sakohon River mouth, an estuary in central Thailand. His wave flume had dimensions of 45 cm width, 60 cm depth, and 45 m length. The grain size distribution and clay composition were not given but the investigator noted that all sediments were washed through a U.S. standard sieve no. 200. The mud was rinsed with fresh water and the experiments were conducted in fresh water. Mud slurry was uniformly distributed in the entire flume, and was allowed to settle for two weeks. The final bed thickness was 2.5 cm. He took water samples at nine elevations at only one location. Figure 22 shows examples of the concentration profile. The data show a strong vertical mixing mechanism illustrated by the uniform concentration profile for the top 25 cm and gradient over the bottom 5 cm. He integrated these profiles and obtained the time variation of total suspended sediment concentra tion. From these data, he concluded that the wave erosion process did reach a steady state after 3 or 4 hours. However, the possibility of a strong sediment concentration gradient in the longitudinal direc tion was not considered. This gradient, for example, could be induced by the longitudinal variation of bed shear stress. Therefore, his statement regarding the steady state condition may not be fully 20 L U. 30 w z t a~in 0 0 00 00 500100 t 00min min 0) 500 1ID C(mg/H) h= 29, 4 cm H = 5.4 cm T= 1.7 s Rw =4900 Fig. 22. Measured Suspended Sediment Concentration Profiles from a Wave Resuspension Test (after Thimakorn, 1980). 15 justified in case of a strong longitudinal gradient of sediment concen tration. Thimakorn (1984) also presented a mathematical model to simulate the suspended sediment concentration profile. This was composed of (1) a shear model for predicting the sediment mass eroded by wave shear stress (which had a similar function to that of the erosion rate) and (2) a diffusion model for distributing the eroded sediment into the water column. The diffusion model, based on a constant diffusivity, should have had the wellknown loglinear distribution curve for the sediment concentration profile at steady state. However, Thimakorn did not demonstrate this characteristic but introduced a depthvarying diffusivity profile and asserted that the results were good compared to measured data. Some relevant conclusions based on the aforementioned studies are as follows: 1. The critical shear stress concept for erosion by steady flow (Partheniades, 1965; Mehta et al., 1982) can also be applied for wave erosion. In other words, waveinduced bed shear stress must exceed a critical value to cause noticeable erosion. 2. The mud beds used in these studies were quite thin. They were on the order of 1 to 2.5 cm. Therefore, the interaction between the mud bed and water waves was limited. Water wave attenuation was not mentioned in these studies, possibly because it was not noticeable for such a small mud thickness. However, this is not the case in the prototype, where significant wave decay has been reported (Tubman and Suhayda, 1976; Wells, 1983). Therefore, a thicker mud bed should be employed for wave erosion/resuspension tests. 3. Wave erosion rates were not mentioned even though this information is most important for estimation of sediment transport. 4. Data on the vertical structure of suspended sediment profiles are rare. One practical reason is the huge amount of sediment samples that must be gathered for this purpose. An easy and reliable measurement technique for rapidly recording cohesive sediment con centration is not available yet. It is believed that light attenu ation caused by individual sediment particles or flocs can be utilized for this purpose, e.g., the Iowa Sediment Concentration Measuring System for suspended sand (Glover et al., 1969). However, unresolved problems still remain in measuring silt and clay suspen sions. 5. It remains unclear whether the wave erosion process reaches a steady state or not. Suppose a steady state can be reached, then it must reflect a downward increase of erosion resistance in the bed, provided no deposition is occurring. A similar behavior has been observed for steady flow erosion processes (Mehta et al., 1982). However, more data are required to support the premise of a steady state for wave erosion. 6. The sudden movement of bed material may be caused by the cyclic mobility or liquefaction of the bed by water waves. The former, for clays subjected to oscillatory shear loads, has been demon strated by Schuckman and Yamamoto (1982) for bentonite beds. The latter, for silt or fine sand under cyclic loads, also has been verified by Turcotte et al. (1984). Pamukcu et al. (1983) claimed that these two terminologies (cyclic mobility and liquefaction) actually describe the same phenomenon but for a different type of soil. The important results from this "loosening" are the changing flow structure (for both water and mud) as well as the erosion behavior. 7. In the previous studies noted, the bed layers were assumed to be rigid. Bed shear stresses were evaluated from linear wave theory and the wave friction factor concept. Such an assumption is valid for a hard soil or compacted sediment of low water content. In general, particularly for soft muds, motion within the bed can be significant, and the shear stress between the eroding fluid and the bed must be evaluated by considering the dynamics of mud motion. However, the question of which bed properties should be selected has not been resolved. 2.3 Constitutive Models for Soft Muds Elastic, poroelastic, viscous fluid, viscoplastic, and visco elastic behavior are the five behavioral models which have been proposed for simulating mud properties. They are discussed in this section. It may be unrealistic to say that any simple model, such as those described below, is good enough to represent the rather complex behavior of mud. However, in general, the goal is not to obtain an absolutely correct answer but to find a relatively reasonable predictive model. Before discussing any model, it may be helpful to examine the nature of a clayey soil under a dynamic shear load. Although the 18 characteristics of the clay bed discussed are for quite low water contents, it does give some insight into soft bed behavior. 2.3.1 Nature of Clayey Soil Hardin and Drnevich (1972) measured clayey soil response under cyclic load. They followed the definition of energy loss ratio given by Kolsky (1963) as the ratio of the energy dissipation in taking a spec imen through a stress cycle and the elastic energy stored in the spec imen when the strain is a maximum. See Fig. 23. For a stressstrain cycle not too far from the linear range, the energy dissipation is small (see the small hysteresis loop). However, when the strain is far off the linear range, the stressstrain path follows the large hysteresis loop and results in high energy dissipation. Kovacs et al. (1971) indicated that the ratio of energy dissipation, X, increases at large shear strains. See Fig. 24. This implies that, for a given shear load, soft muds have high energy dissipation rates due to the large shear strain. The more dense muds, however, have smaller strains and consequently small energy dissipation as well. This also implies that energy dissipation is velocitydependent (strickly speaking, shear displacementdependent) since high velocities usually mean large shear displacements. Thiers and Seed (1968) demonstrated that the shear strength of clays can be reduced when subjected to a sufficient number of cyclic shear loads. See Fig. 25. This phenomenon was also observed by Schuckman and Yamamoto (1982) for marine sediment from measurement of the undrained shear strength, Su, of a bentonite bed before and after Strain 7 /Hysteresis Loop Fig. 23. Shear StressStrain Loop for Clay under Cyclic Loading. 0.16 x0O08 0.04 . 0.1 0.4 1.0 40 10.0 SHEAR STRAIN (%) Fig. 24. Energy Dissipation Ratio A versus Shear Strain (after Kovacs et al., 1971). Fig. 25. Shear StressStrain Loops and Effects of Cyclic Loading (after Thiers and Seed, 1968). 21 wave loading. Su (N/m2) was empirically related to the shear modulus G (N/m2) according to G = a Sum, with a = 0.66 and m= 1.75, applicable to G values on the order of 102 to 104 N/m2. A maximum reduction of Su of about 55% was reported. The reduction was fully recovered 9 days after stopping the wave action. The reduction of shear modulus is typically due to dynamic softening experienced by the sediment when subjected to cyclic shear stresses. 2.3.2 Elastic Model For an isotropic and homogeneous linear elastic body, the consti tutive equation for an incompressible material can be represented by T'= 20 E' (21) with E. = ( ) (21a) E. 2 3x i ax where T and E are the stress and strain tensor, respectively, the prime indicates the deviator parts of those two tensors, i,j, are free in dices, and ri is the displacement in the xi direction. Mallard and Dalrymple (1977) used this model to study the stress and displacement of a soil bed. Dawson (1978) also used this model but included the inertia term to improve upon Mallard and Dalrymple's results. A serious drawback of this model is that no damping effects can be obtained because no viscous or friction terms are involved. There fore, this model can not justifiably represent the interaction of the wavemud system, where energy dissipation is important. 22 2.3.3 Poroelastic Material with Coulomb Friction Yamamoto and Takahashi (1985) suggested that the Internal loss of energy is due to Coulomb friction and is independent of the velocity of soil movement. Because of the nature of the energy loss mechanism they chose, a direct measure of the energy dissipation of mud was selected. This direct measure does not require any assumption concerning the nature of internal friction. However, the results are dependent on the strain amplitude as well as the previous stress experience. This approach complicates the modeling of soil behavior because more pa rameters are involved. Yamamoto (1982) incorporated the inviscid fluid flow assumption for the water waves (therefore, the momentum is transferred to the mud by normal pressure only). He then solved the equations of motion for a poroelastic soil skeletal frame and Darcy's equation for the pore water pressure. Energy dissipation was evaluated by using a complex shear modulus, in which the imaginary part stands for the friction damping coefficient. Because of their inviscid fluid assumption for water, the bed shear stress (an important parameter for erosion) was ignored. The applicability of "poroelastic material" to muds (which have low per meability and can be considered as a continuum) is not apparent. 2.3.4 Viscous Fluid Model In this model the sediment bed is treated as a fluid which has a greater viscosity and higher density than water. The constitutive equation is T' = 2p D' (22) with I. i + ._u. (22a) mij i x where D' is the deviator part of the rate of deformation tensor D which is close to E for small strains, and ui is the velocity in the xi direction. Gade(1958) may have been the first to adopt this model to simulate energy dissipation in a twolayered system. He assumed the overlying water to be an inviscid fluid and the driving force to be restricted to long waves only. In addition to theoretical derivation, he also con ducted experiments to verify his results. Unfortunately, his experi ments did not provide any information on real mud behavior because he used a sugar solution instead of mud. Since a general model based on the same viscous fluid assumption was developed later by Dalrymple and Liu (1978), Gade's work is not further discussed. Dalrymple and Liu (1978) published their."complete model," in which they used two layers of (laminar) viscous fluid to represent the overlying water and the soft mud bed. They then solved the linearized NavierStokes equations for wave propagation. The boundary conditions were specified as follows. First, the noslip condition was satisfied at the rigid bottom. Second, at the mudwater interface the horizontal velocity, vertical velocity, shear stress, and normal stress were matched. Third, the zero shear stress and zero normal stress conditions were specified at the free surface. Finally, the linearized kinematic boundary condition at the free surface and the watermud interface were invoked. This resulted in 10 equations and 10 unknowns. After some 24 manipulations, the 10 equations were reduced to a single one, and a numerical method was utilized to solve it. Velocities, pressure, and the interracial wave amplitude were typical outputs. Although the shear stress profile was not shown it can be worked out easily from the velocity gradients. Based on their model, Fig. 26 shows the horizontal velocity amplitude profiles for four mud viscosities, v2. All other parameters are the same: wave height, H = 0.026 m, period, T = 1.75 sec, water density, pl, and mud density, P2, equal to 1000 and 1260 kg/m3, respec tively, and water viscosity v, = 106 m2/sec. The interfacial shear stress, Tb, and the decay coefficient, kim, are also displayed. It is observed that because of the viscous fluid assumption, the interfacial shear stress turns out to be small. Figure 26 indicates that the horizontal velocity in the bottom layer is of the same order as that in the upper water layer provided the kinematic viscosity of mud is less than 0.01 m2/sec (1000 times higher than that for water). The dif ference in the velocity phase for these two layers is found to be about 45 degrees. This means that the velocity gradient at the watermud interface is small, which explains why the interface shear stress was small. Also notice in Fig. 26, that for small v2, e.g., less than 103 m2/sec, the viscous wave boundary layer 22a is limited to the mud layer. For high mud viscosity, the wave boundary layer thickness extends up to the interface and a second wave boundary layer in the water layer is developed. This second wave boundary is characterized by the velocity overshoot above the interface. This overshooting is due 0.0001 0.017 0.018 i I 0.001 0.057 0.019 1 f j 0.01 0.235 0.099 0.05 0.148 0.179 H / 010 00 0 41 6 81 2' 1'B0t HORIZONTAL VELOCITY AM=LIUD (01S/) Fig. 26. A Plot of the Horizontal Velocity Amplitude Profiles by Using the TwoLayered Viscous Fluid Model. 26 to the influence of the viscosity within the second boundary layer. This phenomenon has been explained by Dean and Dalrymple (1984) and observed by Sleath (1970) for a smooth bed, and Jonsson and Carlsen (1976) for a rough bed. The velocity in the lower layer is significantly reduced for high viscosity because the entire mud layer is inside the viscous wave bound ary layer. However, the fact that measured viscosity data (see Chapter 5) indicate typically low viscosities, e.g., 103 to 104 m2/sec, for the soft mud prohibits the use of high viscosity in the model. However, model results from a high viscosity are closer to what has been observed (Migniot, 1968; Nagai el al., 1982; Schuckman and Yamamoto, 1982). This hints at another velocity restraining mechanism, which can be in corporated, for instance, into the viscosity term to create a high apparent viscosity. This can then be used to simulate the mud motion. Either the Bingham model's yield strength or the viscoelastic model's elasticity could be used. These two models will be discussed next. Despite the drawback in the selection of a viscous fluid model to simulate wavemud interaction, Dalrymple and Liu's model is indeed the one with the most complete approach. Efforts to generalize their approach as well as to incorporate other restraining mechanisms have been made and are presented in Chapter 4. 2.3.5 Viscoplastic model Viscoplastic material, also called a Bingham plastic, is charac terized by a special feature recognized as the yield strength. Basi cally, a viscoplastic material can sustain a shear stress even when it 27 is at rest, but when the shear stress intensity reaches a critical value the material flows, with viscous stresses proportional to the excess stress intensity. The constitutive equation (Malvern, 1969) is + K(+ )' (23) with D' 2 i D! (23a) I' ij lj where IID, is the second invariant of the deviator part of the rate of deformation tensor, K and 4 are the yield strength and apparent viscosity. Notice that the extra term, i.e., the second invariant in the constitutive equation, results in a nonlinear equation of motion which is difficult to solve analytically except in a few simple steady flow cases. Unfortunately, the wave propagation problem is not in cluded. Although many researchers have studied the theological proper ties of mud by using this model, none has been able to solve the equa tions of motion of a Bingham material under an oscillatory force. Krone (1965) used a fairly extensive experimental procedure to measure the rheological properties, i.e., yield strength and apparent viscosity, of cohesive sediment. The maximum sediment concentration in his experiments was about 110 g/l. The period of consolidation was not mentioned. However, it can be inferred that he did not allow the sediment to settle down because he remolded the mud quite frequently to keep the sediment in suspension. He found that the yield strength for all the samples in his experiments was proportional to the 2.5th power of sediment concentration. See Fig. 27. The apparent visco sity for mud, pm, was 1 to 10 times higher than those for water, 1w. See Fig. 28. O San Francisco Bay Mud () Wilmington Mud B runswick Mud Grundite 10 Koolinite Suspension 10 10 0 4 2 16 _ SEDIMENT CONCENTRATIO(g/.t) Fig. 27. Relationship between Yield Strength and Sediment Concentration (Data from Krone, 1965; Gularte, 1978; Engelund and Wan, 1983). I I I I I I I @ Son Francisco Bay Mud (Z) Wilmington Mud () Brunswick Mud @ Grundite (5) Koolinite Suspension 0 ~1~I I I I I I I I I I '0 200 400 600 800 1000 SEDIMENT CONCENTRATION (g/) Fig. 28. Relationship between Apparent Viscosity and Sediment Concentration. (Data from Krone, 1965; Gularte, 1978; Engelund and Wan, 1983). 1200 102 ,o3 30 Migniot (1968) also made a series of laboratory experiments to study the physical properties of cohesive sediments and their behavior under hydrodynamic forces. He observed that muds have an "initial rigi dity" and that it is proportional to the 5th power of sediment concen tration, ranging from 120 to 800 g/l. The initial rigidity was defined as the yield strength extrapolated from a curve (not a straight line) which fitted the measured data on shear stress versus rate of shear strain. The apparent viscosities were small for low sediment concen tration but increased in significance when the sediment concentration became greater than 200 g/l. For example, m 20 pw when the sediment concentration was around 200 g/l. However, pm 150 Vw when the concen tration went up to 500 g/l. Gularte (1978) used a vane viscometer to measure the viscosity and yield strength of grundite, which is an equal mixture of illite and silt. The sediment concentration ranged from 600 g/l to 1000 g/l. He found that the Bingham yield strength was proportional to the 10th power of sediment concentration. See Fig. 27. The apparent viscosity, pm, was 1000 times higher than pw. See Fig. 28. Englund and Wan (1984) studied the instability of the hyper concentrated flow. They found that the Bingham model is quite satis factory for the description of the instability of the fluid surface elevation. They found the yield strength to be proportional to the 3rd power of sediment volume concentration. The correlation between Um and sediment concentration is also shown in Fig. 28. Parker and Kirby (1982) pointed out that, although mud suspensions have been characterized as Newtonian fluid at low solid concentration, to Bingham plastic at high concentration, the true behavior should follow a pseudoplastic model, and that the yield strengths obtained by many researchers are really extrapolations (to zero shear strain rate) based on measurements at higher shear strain rates. They also presented data from Williams and James (1978) to support their view point. To summarize the aforedescribed studies on the viscoplastic model, some conclusions may be stated as follows: 1. The yield strengths and apparent viscosities observed vary signi ficantly. No unique, correlative relationships have so far been found. 2. Mud suspensions at high sediment concentration do not follow Newtonian fluid behavior. Although a Bingham model is commonly suggested, a pseudoplastic model may be more appropriate. 3. In previous studies, attention does not seem to have been paid to the effects of selfweight consolidation. This may be important, how ever, because muds possess a shear strength and a behavior which approaches that of an elastic solid. Fieldmeasured acceleration data (in two horizontal and one vertical direction) of mud do not seem to show the quiescencemovingquiescence behavior predicted by the Bingham fluid model (Tubman and Suhayda, 1976). Instead, a continuously varied acceleration, which implies an elastic behavior, is observed. 4. The Bingham model predicts no motion in the mud whenever the shear stress is less than the yield strength. Therefore, no energy loss can be predicted under this condition. However, unless the equation 32 of motion for a Bingham fluid can be solved, it would be difficult to say whether there is a movement or not. 5. To further examine the suitability of the Bingham model for the wavemud system, solving the velocity field for both water and mud, even onedimensionally, is important. The velocity and associated energy dissipation data can provide information for the justifica tion of using this model. Efforts to evaluate the onedimensional watermud system have been made and the results are discussed in Chapter 3. 2.3.6 Viscoelastic Model Muds can have elastic properties when the sediment concentration exceeds a specific value. Golden et al. (1982) reported that kaolinite suspensions can have a shear strength when the solid fraction exceeds 0.04 (sediment concentration over 107 g/l) and it increases markedly with higher sediment concentration. Mud beds usually have a sediment concentration greater than 200 g/l. Therefore, elasticity can not be neglected. Furthermore, in order to account for the energy loss due to mud motion, a dissipation function must be included. Therefore, visco elastic models may be used. Viscoelastic models account for the energy loss in an elastic solid in an indirect manner. It is assumed that the restoring forces are proportional to the amplitude of vibration, while the dissipative forces are proportional to the velocity. Two simple linear models, namely those based on the VoIgt and Maxwell elements, are available. These are based on the analogy of a 2G 2,u a) Voigt Element 2G 2. b) Maxwell Eement Fig. 29. System Diagram for Simple Viscoelastic Models and Response under Constant Loading. 8 0 z 0D4 e (rad,/,ec) o .206 2 o .0265 a .0122 F .00135 0I I I 0 .02 .O1 ,rJ .08 SEAR STRAIN (rad) Fig. 210. Viscoelastic Behavior of Clay (after Stevenson, 1973). The rate of angular displacement is denoted by . 34 springdashpot system. They are plotted in Fig. 29 and the constitu tive equations are presented as Voigt element: T' = 20 E + 2P E' (24a) Maxwell element: T' + p/G T' = 2p E' (2'4b) where T and E are the stress and strain tensor, respectively, the prime indicates the deviator part of those two tenors, the dot indicates the derivative with respect to time, and G and p are constants. Figure 29 shows the shear strain as function of time under a constant shear load. For the Voigt element, the response of elastic shear strain is delayed, but it finally reaches the value that the spring alone would reach. However, the Maxwell element will keep creeping because of the nature of the dashpot. The Maxwell element is often used to describe the relaxation of shear stresses under a constant strain. In the Voigt element, the spring and dashpot are subject to the same displacement but different stresses. It is easily shifted between a fluid and an elastic solid and the behavior is frequencyindependent. The Maxwell element, however, behaves as an elastic solid at a high frequency load and as a fluid at a low frequency load. Hsiao and Shemdin (1980) studied the wave decay problem by assuming that soft muds have Volgt properties. They solved the linearized two dimensional equation of motion for the Voigt element given by Kolsky (1963) and assumed the overlying water to be an inviscid fluid. Kolsky gave 2 2 a 2 . p ax 3 G + ( P t  9t x x 2 ( at (25 J J where i is the displacement in the xi direction, t is the time, p is the pressure, and p is the material density. Hsiao and Shemdin (1980) gave results showing a rapid wave energy dissipation. Macpherson (1980) also studied the wave attenuation problem by assuming that muds have the properties of a Voigt element. He had a different treatment of Eq. 25 and arrived at a linearized Navier Stokes equation but with a complex viscosity. His apparent viscosity for the mud is given in Eq. 26a. Here he represented the shear modulus as the imaginary part of the apparent viscosity term and kept the kinematic viscosity as the real part. Although his model could not predict a bed shear stress because he neglected the viscosity of the water layer, his approach greatly simplified Eq. 25. Voigt element: ve V + jG(26a) (1 0 Maxwell element: ve ( + j P (26b) 1+(a/G)2 Following Macpherson's approach for an oscillatory load, the apparent viscosity for a Maxwell element is also given in Eq. 26b. However, unlike the Voigt element, the apparent viscosity is frequency dependent. For an infinite G, the Maxwell element behaves as a viscous fluid but the Voigt element becomes a rigid body. For zero V, the Voigt element is a perfect elastic material while the Maxwell element is not defined. Also the Maxwell element is not defined for zero G; however, the Voigt element is a viscous fluid at this condition. 36 Questions concerning which element simulates mud behavior more closely are not discussed by Macpherson (1980) or by Hsiao and Shemdin (1980) and still remain unclear. Stevenson's (1973) data, see Fig. 210, showed that the Voigt element was applicable for his soil sam ples. However, his samples had a high shear strength and were not like a soft mud. Therefore, experiments to observe the response of mud under constant shear loads were conducted in the present study. The objective was to find whether the Voigt model or the Maxwell model was better in modeling the stressstrain relation, and also to find the material constants G and p. Details are given in Chapter 5. 2.4 Bed Shear Stress The bed shear stress is well known as the most important factor in the erosion process in steady flow as well as in an oscillatory wave flow. Unlike steady flow, for a smooth bed (ratio of the bed roughness r to the semiexcursion distance just outside the boundary layer ab, r/ab < 0.01), the wave bed shear stress has a period which is basically equal to the wave period. For a rough bed, r/ab > 0.01, higher order harmonics may be induced by the rhythmic formation and release of vor tices (Lofquist, 1980). Most papers dealing with bed shear stress in the oscillatory flow assume that the bed is rigid and that the water is free of sediment. This is a reasonable assumption for a sandy bed or a clayey bed with low water content since there is no appreciable motion in the bed. However, for a cohesive sediment bed with high water content, this assumption is not always applicable because the bed may also move (Tubman and Suhayda, 1976; Nagai et al., 1982). This would 37 mean that the bed shear stress would be different because the velocity gradient is altered at the interface. Therefore, to accurately estimate the bed shear stress one must consider the dynamic response of mud as well. In this section, only the wave bed shear stress for a rigid bed is briefly summarized. Jonsson (1966) proposed the following equation to evaluate the maximum bed shear stress (which is one of the most popular equations still used): 1 2 = 2 p fw Ub (27) where ub=abu is the maximum particle velocity just outside the bottom wave boundary layer which can be predicted by classical wave theory, and fw is the wave friction factor. This is a simple form and the only unknown is fw. The wave friction factor is a function of the surface roughness and wave Reynolds number Rw, defined by Jonsson (1966): Rw= u2/ va (28) Here v is the kinematic viscosity of the water. For the case when Rw < 1.25 x 104, the viscous wave boundary condition, Jonsson (1966) gave fw= 21 Aww (29) w w At higher wave Reynolds numbers, the bottom roughness is involved. This is the transition zone between viscous and full turbulent flow. Kamphuis (1975) presented a chart (Fig. 211) to evaluate fw. For . Rough Turtbient 0"  80 r  40 N. 80 Z Ierpoloted 20 o ldExtrpolated 400 i Smootho_ r Lowen Limit of Rough %be ** l lt I I ,itt*l g *pttl llllplI a i flit .21 I0 1 1 0,1 1 1 111 1 1 0111 C REYNOLDS NUMBER, Rw Fig. 211. Wave Friction Factor (after Kamphuis, 1975). The grain size, which is larger than 90% of the total mass of soil sample, is denoted as D90. 39 higher Rw, only the roughness is important. Jonsson (1966, 1975, 1980) also gave a formula to evaluate the wave friction factor under this condition. 2.5 Wave Diffusion Coefficient Fisher (1968) pointed out that the concept of the apparent diffu sion coefficient Dm (turbulent diffusion and dispersion) has been widely employed to predict the transport of pollutants. This concept has been used for sediment transport as well. However, the problem of evaluation of the diffusion coefficient for a nonbreaking wave environ ment has not been fully clarified yet. Previous studies relative to this coefficient, although having no consistent results, are briefly summarized here. Kennedy and Locher (1972) gave a good summary of the early studies. Key points are given here. The pioneering experimental work done by Shinohara et al. (1958) showed that a constant diffusivity is good for sand with a mean diameter of 0.2 mm. However, specifying only one constant diffusivity for pulverized coal with a mean diameter of 0.3 mm was not sufficient. This was because of the discontinuity in the concentration profile near the bed. Hattori (1969) also employed the constant diffusion concept and showed good agreement with his experiments for the top 80% of the water column. Das (1971) concluded, from his oscillating bed experiment, that the diffusion coefficient varies linearly with distance above the bed. Homma and Horikawa (1963) gave Eq. 210 for the diffusion coef ficient under a wave field. D 1 Ho sinh3kz" m 2B sinh kb cosh 2 kz (210) where is a constant related to the ripple geometry, H is wave height, h is water depth, k is wave number, z" is elevation above bed, and c is wave celerity. This formula uses the velocity profile from the small amplitude wave theory and, therefore, the diffusion coefficient in creases slowly upward since the velocity profile does not vary sharply with elevation. However, their sediment concentration data (for the upper 70% of water layer) showed quite good linearity on a loglinear plot which implies a constant Dm. Horikawa and Watanabe (1970) measured the turbulent velocity fluctuations, u' and w', rather than deduce the variation of diffusion coefficient from the vertical distribution of the mean sediment con centration. They concluded that the root mean square (RMS) value of w' is almost independent of z, whereas the RMS value of u', which is greatest near the bed, decreases first just above the bed, and then becomes nearly constant for the most part. Jonsson and Carlsen (1976) measured wave velocity profiles and then calculated the profile of eddy viscosity, c. Their flow had a high Rw and a rough bed. Their results showed that the maximum E is located within the turbulent wave boundary layer. Outside the boundary layer, c decreases drastically and approaches zero for the upper water column. Here c = 0 reflects the fact that turbulence is not important in the calculation of velocity for wave flows. However, turbulence is indeed important in sediment transport. For cohesive sediment, momentum transfer coefficients are almost the same as mass transfer coefficient, diffusion coefficient, (Jobson and Sayre, 1970). Therefore, it can be inferred that the turbulent diffusion coefficient should be also the largest near a rough bed. Wells et al. (1978) presented Eq. 211 as "the most justifiable estimate for suspendedsediment concentration" along the Louisiana coast as a result of wave resuspension: C(z)= 5.31 exp(1.8 z") (211) where C is the sediment concentration in kg/m3 and applies to a water depth h < 5 m. Their result implies a constant diffusion coefficient since there is a loglinear relationship between C and the vertical coordinate, z". Hwang and Wang (1982) summarized the models available for the turbulent diffusion coefficient outside the wave boundary and suggested that a modified version of their model, Eq. 212, would be the most plausible one: D =oH2a sinh2kz" (212) m 2 sinh2kh where a is a constant. Thimakorn (1984) also gave an equation for the diffusion coef ficient profile, which was close to that given by Hwang and Wang (1982), based on the concept of turbulent energy. The above information is summarized in Table 2P1. The wave Reynolds number, Rw, which indicates the flow regime near the mud sur face, is also included. Notice that some of the laboratory studies 42 have had a wide range of Rw which indicates that both the yiscous and turbulent wave boundary layers were included. Two types of diffusion coefficient profiles have been introduced. Data from Shinohara et al. (1958), Hattori (1969), Horikawa and Watanbe (1970), and Wells et al. (1978) suggest a constant coefficient for the top 80% of the water layer, the remainder apparently having another constant value. However, to the contrary, the others (Homma and Horikawa, 1963; Das, 1971; Hwang and Wang, 1982; Thimakorn, 1984) suggest a minimum diffusivity at the bottom and increasing upward. Table 21. Diffusion Coefficients in Wave Flow Researcher Sediment Rw Suggested Dm Shinohara et al. Sand, < 6700 Constant Coal Hattori Sand < 24000 Constant Homma & Horikawa Sand < 26400 Eq. 210 Horikawa & > 12000 Constant Watanabe Wells et al. Mud > 12000 Constant Hwang & Wang Sand Eq. 212 Thimakorn Mud < 10000 similar to Eq. 212 It is interesting to note that none of these studies has mentioned a possible longitudinal dispersion phenomenon. In actual shear flow, fluid layers near the bottom tend to exhibit a sharp velocity gradient. This results in a much greater longitudinal dispersion. Taylor (1954) 43 gave the example of steady uniform pipe flow and showed that the disper sion coefficient can be about 200 times larger than the mean value of the turbulent diffusion coefficient. However, Awaya (1969) pointed out that in an oscillatory flow the dispersion coefficient can not be that high, because of the limited timescale associated with this type of flow, tidal flow in his case. It can still be inferred, however, for waveinduced flow, that there must be a strong longitudinal mixing layer above the bed surface because of sharp velocity gradients in the bottom boundary layer. It is further noted that the bed surface is the main source of turbulence (Nielsen, 1984). This may explain what was observed by Shinohara et al. and others mentioned. To further under stand the wave diffusion behavior, dye diffusion tests were conducted in the present study. Suspended sediment concentration profiles were also collected. Details are discussed in Chapter 6 and Chapter 7. CHAPTER 3 ONEDIMENSIONAL BINGHAM FLUID MODEL 3.1 Introduction In this chapter a numerical solution of the velocity field and energy dissipation in a vertical onedimensional watermud system is presented. The mud is assumed to have Bingham plastic properties with constant density but variable viscosity and yield strength. The over lying water is assumed to have constant density and viscosity. The objectives are to examine the velocity profile and the associated energy dissipation in this system, and to justify this 1D model for simulating the watermud system. 3.2 Problem Formulation Figure 31 shows the system diagram. A variable grid system, with grid size Azj, was chosen to give better resolution at the bottom, the interface, and the free surface. The grid number starts from I at the rigid bottom, and ends at the free surface with number J. The total depth is h. The fluid properties, e.g., density p, kinematic viscosity v, and yield strength K, as well as the horizontal velocity u, are specified at the center of each grid element. Shear stresses, T, are specified at the node points. The onedimensional linearized equation of motion is 1 J+I SWL J ._Azj I kJi T_  Fig. 31. System Diagram for the 1D Numerical Bingham Fluid Model. du dI 1 dT (3I) with + K du (31a) where dn/dx is the slope of the water surface profile, t is time, g is gravity, and z is the vertical direction. The boundary conditions are as follows: 1. noslip at the rigid bottom, u=0 at z=O. 2. velocity is maximum at the free surface, du/dz=0 at z=h, or, uj~uj+I 3. The initial condition says that the fluid is at rest at the beginning, u=0 at t=O. To ensure that the flow velocity is maximum at the free surface, a fictitious layer (from J to J+1) is added at the top of the water column. Equation 3"1, with the boundary conditions, can be solved by the double sweep method (Abbott, 1980). The finite difference formula tion is n+1 n n+1 n+1 n n uj Uuj = dn )n+ /2+ 1 (P+ lA 1= ) + .L ( J+ I J) At dx 2pj Az. 2p. z (32) where k n 1 un+1 n+1 j1(Az++ Az+1) 2 uj+1 i (32a) 2 luj+1 u I CAZ + Azj+I) n+1 n+1 k. (Azj+ Az _) 2 (u. un+I In =[P + n j1 u j1 (32b) 2 jun ju11 (AZj + AZjI) 47 Notice that the velocities at time n are also used to evaluate the velocity gradient in the denominator for the shear stress term at time n+1. The superscript n+1/2 implies the center of each time step. Equation 32 can be rearranged in terms of un+1 and rewritten as n+1 un+1 n+1 Au .B uj. = D (33) J1+ .j j1 where K (Azi + Az +I) Au+1 _un 1 I (Az ) 7Z. (33a) 2 + AZ i Z K 1(Azj + Azj I) B = 1 + ( j + I + ) j A t K. ( Azj +(.j 3 Azj1) At 2 n n p (Az + Az 1 ) Azj (33b) 2 1 uj un_1P1 (zi+A J1A K.(Azj + Azj1 ) At 2 (uj n _u n P (Azi + Az j) AZ (33c) 2lu n I jAj z D= gAt (dnfn1/ + dx Kj+ (Azj + Azj+1 At Un (Pj+I 2 Jun1 n pj(Azj + Az ) Az. j+1 j j j j Kj (AZj ) Az +U 1 uj Aj+ +Kj(Az + AZjI At U + 31(1+1 + 2 Jun+ unI P .(Az. + Azj~ ) AZ (Ij + nuKj(Azj + AzA] 2 u n u11 PJ(AZ + AZ1) A1 j Kj (Azj +z 2 n U (33d) 2 lu' ujI pj(Azj + Azj I) Azj j 3d Now introduce the auxiliary equation, Eq. 34, and substitute into Eq. 33 to obtain Eq. 35 n+1 n+1 u+1 =E i+ F (34) uj+I = j 3 3 n+1 C n+1 (D A F.) j A E.+ B uj1 + (A E j+ B) n+1I =Ej_ ujI + IF (35) i J1 Ji Comparing Eq. 34 and Eq. 35 reveals that Ej and Fj, from j=J1 to 1, can be obtained because Ej=1 and Fj=O are specified by the zero velocity gradient condition. Therefore, the first sweep yields all the Ej and Fj. The second sweep, from Eq. 34 and the noslip boundary condition, u1=O, solves for the velocity for all the grid elements. Because there is a velocity gradient term in the denominator (Eq. 31a) for evaluating the shear stress of the Bingham fluid, a problem of calculating A, B, C, and D was encountered because of the zero velocity gradient at the onset. However, letting the first sweep stop at the uppermost element, S, where the yield strength is zero, and moving the noslip condition to the element one cell below element S, allows calculation of the velocity profile above this element. By comparing the shear stress acting at the bottom of the moving fluid and the yield strength one layer below, a decision can be made as to whether the next layer will move or not and therefore to stop the first sweep 49 for the next time step. If this is done, the first sweep can go down stepbystep to a cell where the shear stress and shear strength are equal, or to the rigid bottom provided the driving force is large enough. Wave energy components are evaluated separately as kinetic energy and potential energy. The timeaveraged total energy per unit surface area is E= 4 pgdz dt + 1 T hpu2dz dt 2 =aa a a exp( 2k.X) (36) 0 m where a is a constant coefficient, kim is the wave decay coefficient, and ao is the wave amplitude at x = 0. The timeaveraged energy dis sipation per unit surface area is calculated as follows: I T h du zd Ed 1 0 hd dz N J T 1 n n n n = I I (T j1 u + /2 u I/2)]At (37) n1 j=1 J where T and u are given by the model and N is the total number of time steps for a complete wave cycle. The energy dissipation can also be written as Sc DE = 2 cga kima exp(2k X) d ax g 0in im = 2 cgkimEn (38) where Cg is the wave group velocity and can be replaced by vg because of the long wave assumption. Therefore, the decay coefficient can be obtained as k im E / 2V9 En) (39) 3.3 Results The numerical scheme was tested for the steadystate solutions first, simply because analytical solutions are available. The water surface gradient was arbitrarily selected to be 105. The time incre ment was also arbitrarily chosen as 50 seconds since the implicit scheme is unconditionally stable. Figure 32a shows the velocity profile for a twolayered viscous fluid at selected times. Squares show the analytical solution. Figure 32b shows the velocity profile for an assumed fluid which has exponentially distributed yield stress and viscosity. The results are satisfactory in both cases. The pressure forcing term (dn/dx) in Eq. 31 was then changed to a cosine function to simulate a periodic wave load. The time increment was also reduced to 0.05 second because the wave period was relatively short and also to minimize possible error introduced by the stepbystep downward computational procedure. The scheme was compared with the results from Dalrymple and Liu's twolayered viscous fluid model and showed reasonable agreement. See Fig. 33. The observed discrepancy was due to the long wave assumption in the numerical model. In this example, the long wave assumption could not account for the viscosity variation at the interface. This may be interpreted as follows. Long waves are characterized by a constant dynamic pressure in the entire water depth (i.e., pressure is hydrostatic) and by the absence of 30 t=166 min t 7 mi 66 mi 2S n t=332 min t=75 mi 20 t=500 min tal50 mi 15 t=666 min LwA 10 a: Top Bottom K=0 I N/m2 v=3.5xl06 S p(kg/m3) 1000 1160 ixlO' m2/s ci pl60 kg/n3 (a) v(m2/s) x106 2xO4 (b) 0 s0 oo SO 200 250 0 20 40 so 80 100 UELOCITY U (cm/s) UELOCITY U (cm/s) Fig. 32. Horizontal Velocity Profiles for Two Steady Flows. (a) Two;Layerd Viscous Fluid; (b) Bingham Fluid with Exponentially Increasing K and U. W 2 I I '15 iInterface Elevation_ W 10 () I z I 05 I H T 7 Legend Model (cmXsec ID 2.5 3 DaL Z53  D L 5 40 5 10 15 20 25 VELOCITY AMPLITUDE G and Q (cm/sec) Fig. 33. Comparison of Wave Velocity Amplitude Profiles from 1D Model and Dalrymple and Liu's Model with P=11000 kg/m3, P2=1160 kg/m3, \i=IxIO6 and v2=2x104 rm2/s. 53 vertical velocity. For cases where the viscosity of the bottom layer is not too large, the viscous boundary layer, of thickness 6L, is far below the interface. Then the equation of motion, for the part outside 6L, can be simplified by neglecting the shear stress term, the last term in Eq. 31. Although the shear stress in the bottom layer is larger than in water, because v2 > v1, it is still small when compared with the pressure and inertia terms. Therefore, a single equation can be used from the water surface down to a particular elevation which is below the interface but above the viscous boundary layer. This phe nomenon was demonstrated by another run of Dalrymple and Liu's model (see Fig. 33) but with a long wave period, T = 40 sec. A simulation of waterBingham fluid system was then performed. Input data were specified as follows. The water layer had a constant density (1000 kg/m3) and viscosity (1 x 106 m2/sec). The bottom Bingham layer also had a constant density, 1160 kg/m3, viscosity, 0.0002 m2/sec, and yield strength, 0.1 N/m2. Fig. 34a,b shows the velocity profiles under two wave conditions. It was expected that the low layer would not move for a small wave height (Fig. 34a). This is because the bed shear stress for this wave was less than the yield strength. For larger waves, the lower layer is observed to move with a zero velocity gradient in much of the lower layer, indicating a plastic sliding. Nevertheless, the results show that the velocities in the mud bed were either zero or close to that in the water layer. Energy dissipation and total energy at each time step were evalu ated. Timeaveraged values over one wave period were calculated next. tsw LWL    30 0tS aat Q t6 a6a c~20 1! IT C=)Interface o Interfacei 7T T Cr 2 (a) (b) 5 0 10 is 5 0 5 10 is 20 UELOCITY (cm/9) UELOCITY (cm/s) Fig. 314. Wave Velocity Profiles for the WaterBingham Fluid System with T =3 see. (a) H 2.5 cm and ki,= 0.0011 m'1; (b) H = 5 cm and kim=0.016 m'1. The decay coefficient was then obtained based on Eq. 39 and is given in Fig. 34. Yield strength and dynamic viscosity data were obtained from viscometer measurement (see Chapter 5 for detail) for a kaolinite bed with a twoday consolidation period, which was the softest mud bed in the wave erosion tests (described later in Chapter 6 and Chapter 7). Results are given in Fig. 35. The yield strength and dynamic viscosity varied from 2.5 to 4.4 N/m2 and 0.1 to 0.2 Ns/m2, respectively. The corresponding kinematic viscosity was on the order of 1 x 104 m2/s. Because of the relatively high yield strength, the modelsimulated mud bed showed no motion at all. See Fig. 36. The energy dissipation was significantly low because there was no motion in the bottom layer. 3.4 Conclusion It must be realized that muds can dissipate energy only if they are moving. The 1D numerical model presented here shows no motion in the mud layer, therefore no energy dissipation, when the yield shear strength is higher than the bed shear stress. However, such a condition of no motion no dissipation was not observed in experiments described later, and thus indicated that the 1D Bingham model is not generally applicable for the simulation of watermud interaction in erosion studies. IC I I I Leptd Elev. Below Mud Surface (cm) o 1.5 6.3 w + 9.3 t;6 1 K=4.4N/n,2 x 2 I I I I I I 2 4 6 SHEAR STRAIN RATE (sec1) Fig. 35. Shear Stress versus Shear Strain Rate for Kaolinite Bed with TwoDay Consolidation Period. 10 1 LSWL' 6tO 61 i . 2 Interfacei' 5 0 5 10 15 20 UELOCITY U (cm/s) Fig. 36. Wave Velocity Profiles for WaterBingham Fluid System with Measured Yield Strength and Viscosity. 3 CHAPTER 4 TWODIMENSIONAL MULTILAYERED HYDRODYNAMIC MODEL 4.1 Introduction A multilayered hydrodynamic model to simulate the two important parameters, wave damping coefficient and interracial shear stress, is presented in this chapter. The water layer was assumed to be charac terized by a constant kinematic and eddy viscosity. A simple linear viscoelastic behavior, the Voigt element, and the linearized equations of motion were selected for modeling mud response under wave loading. This selection was based on the discussion presented in Chapter 2, Chapter 3 and the results from Chapter 5. Depth variation of mud properties can be simulated by further dividing the mud bed into layers. The equations of motion for the water layer are also linearized to simplify the problem. The water and mud layers are considered incompressible and immiscible such that density, viscosity, and shear modulus are constant in each layer. 4.2 Formulation Figure 41 is a definition sketch of the watermud system. Water waves propagate in the xdirection. The water layer has a thickness, dl, and the underlying mud bed can be considered as composed of several layers of linear viscoelastic material with thicknesses di. The free 58 Z Pi ,vi ,Gi diI Fig. 41. Schematic Figure for the MultiLayered Model. surface displacement is n. Here i 2, 3, ..., n. The displacement at each interface is i. The displacements n and i can be expressed as n = ao exp[ J(kxot)] (41a) i= boi exp[ j(kxot)] (41b) 59 where ao is the given water wave amplitude at x = 0, boi a e unknown complex variables for interface wave amplitudes and phases at x = 0, J /CT, o=2ir/T, T is the wave period, t is time, and k is a complex wave number. The real part, kr=27/L, is related to the wave length, L. The imaginary part, kim, represents the decay coefficient defined as a(x) = aoexp(kimx) (42) This expression correctly represents the wave decay in a flume (Schuckman and Yamamoto, 1982; Nagai et al., 1982). The linearized equations of motion for the incompressible upper fluid and incompressible, viscoelastic Voigt element layers under oscillatory load are u a t a 2 U 2u __. = 1 + Vei( ) (43a) at P 3x ei x2 z 2 t 2i 2i Dw 1 1 Pi + e ( i + i )(43b) at P z ei x2 3z2 where u and w are velocities in x and z direction, respectively. The apparent viscosity, vei, actually contains two terms. For the water layer, it contains the kinematic viscosity, vI, and a constant eddy viscosity, e. The purpose of involving e in the model is for the bed shear stress calculation. The approach presented later demonstrates that this choice of e is not connected with the turbulent diffusion coefficient, Dm. Each mud layer is characterized by a kinematic viscosity, vi, and a shear modulus, Gi, as discussed in Chapter 2. The 60 subscript i1, 2 ..., N indicate the top, second, and subsequent layers, respectively, pt is the total pressure defined as t 0 Pi = Pi Pigz Pi and 0, if i = 1 o ii iil (44) Pi I(mP)g E },d] if i > 1 m=1r= where m and r are dummy indices. The continuity equation is Sui w i ax + = 0 (45) The solutions of these variables are assumed separable according to ui(x,z,t) = u1 (z) exp( j(kxat)) wi(x,z,t) = wi(z) exp( j(kxat)) (46) Pi(xz,t) = pi(z) exp( j(kxat)) Substititing u i wi into the continuity equation one obtains u = J ~ (47) i k i where the prime indicates differentiation with respect to z. Intro ducing this equation for u, into the horizontal momentum equation (43a) yields an expression for pi, i.e., 'ei w 2 w'A ) (48) ;i k2 1 ii with 2 2 1 Ai k jvei Substituting P into Eq. 43b yields 61 2 2 22 k X + k A wi 0 (49) The solutions for the water layer and the subsequent mud layers are in the form: w1(z)=A sinh Z, + B10osh Z+ C1exp(A1z) +D1exp(A1(z+dl)) (410a) wi(z)=A sinh Zi + Bicosh Zi + Cisinh Ni + Dicosh Ni (4l0b) with i=2,3 ,...,N, Zik(d1+d2+...+di+z), and Ni=Ai(d1+d2+...+di+z). For only one mud layer, there are eight unknown complex coefficients (Al, ...,D2) for the vertical velocities and the two unknown complex varia bles, i.e., wave number, k, and interfacial wave amplitude bo. There fore, 10 boundary conditions are required for solving for the 10 un known coefficients. The 10 boundary conditions will be summarized here to show how the coefficient matrix is constructed. Details are shown in Appendix A. At the free surface there is no external driving force, i.e., 2w (2u 2w normal stress, an + 2.e a, and shear stress, T = p L + L). are zero. Because the actual free surface is unknown a prior, these two boundary conditions are applied at the mean or still water surface. Taylor expansion technique is applied as shown in Appendix A to obtain an approximate solution. Only the basic harmonic term is considered. All high harmonic terms are neglected. The zero normal stress condi tion at the free surface gives M ( A cosh kd1+ B1sinh kd ) 2veIX1C = plgao (411) The linearized kinematic free surface boundary condition yields A 1sinh kd1+ B 1cosh kd1+ CI joa0 (412) Equations 411 and 412 are combined to give Eq. 413 c1,1 A,+ c1,2 B, + c1,3 C = 0 (413) The coefficients cij are given in Appendix A. The zero shear stress condition at the mean free surface gives c2,1 A, + c2,2 B, ++c2,3 CI = 0 (414) At the watermud interface E, (a prior unknown elevation), the hori zontal and vertical velocities for the water and mud layer are matched. See Eq. 415 and 416. The linearized kinematic boundary condition at the interface gives Eq. 417. The matched normal stresses and shear stresses conditions between the water layer and the mud layer give Eqs. 418 and 4119. For details, see Appendix A. c Ac Dc A2+c_ .c Cc 0 (415) c3,1A c 3,4 D1+ 3,5 2 3,6B c3,7 C2+ c3,8D 2 c 4,2Bi+ c 4,4 c 4,5A2+ 4,6B2 c 4,7C+ c 0 4,8D2 0 (416) c5,2 B1 + c5,4 D1+ c5,9 b 10 0 (417) c6,1A, + c6,4D+ c 6,5 A2 + c6,6 B2+ c6,7C2 + c6,8 D2 + c6,9 b 10 0 (418) c7,2B+ c7,4D+ c 7,5 A2+ c7,6 B2+ c7,7 C2+ c7,8D2 0 (419) At the rigid bottom (z=dld2), the noslip condition (uw=O0) must be satisfied, i.e., 08,5 A2 + c8,7 C2. 0 (420) c9,6 B2 + c9,8 D2= 0 (421) These nine equations, Eq. 413 through Eq. 421, can be written as 63 a homogeneous matrix equation c X = 0, where c is a 9 by 9 matrix with nonzero terms specified from Eqs. 413 through 421. Here X is a column matrix which contains coefficients A,, BI, C1, D1, A2, B2, C2, D2, and bol. The unknown complex wave number k is involved in the coefficient matrix c. This is an Elgenvalue problem since the system has solutions only for special k values such that the matrix c is singular, i.e., the determinant of matrix c is zero. Adding one more mud layer in the bottom gives five more unknowns: four coefficients (A3, B3, C3, D3) in the added vertical velocity equa tion, and one (bo2) for the unknown new interracial amplitude and phase. Five more boundary conditions must also be specified similar to Eq. 415 through 419. The newly added layer should be specified as the lowest one, directly above the rigid bottom. Then, the added equations can be listed as c8,5 A2+ c8,7C 2+ c8,10A3 c 8,11 B3 + c8,12 C3+ c8,14D3' 0 (422) c9,6 B 2+ c09,8D C9,10 A3+ c9,11B 3 c9,12 C + c9.3 D 3 0 (423) c0,6 B2 + c10,8D 2 c10,14 b2o' 0 (424) c11,5A2 + c11,7 C2+ c11,10 A3 + c11,11B 3 + c11,12C3 + c11,13 D + c11,14 b2o 0 (425) c12,6B2 + c12,8D 2+ c12,10A 3 + c12,11B 3+ c12,12C3 + c 12,13D3 0 (426) The noslip condition, now at z=dld2d3, is replaced by c13,10 A3 + c13,12 C3 0 (427) c14,11 B3 + c14,13 D3' 0 (428) 64 The size of the new coefficient matrix is then expanded to 14 by 14, and five more coefficients (A3 D3, and b,2) are appended to the column matrix X. The homogeneous matrix equation, however, remains in a similar form, and the same technique may be utilized to solve it. When adding one more layer, matching conditions at the new interface and the noslip condition at the rigid bottom are similar to Eqs. 422 through 4,28 because of the same equation for w. However, the subscripts are increased by 5. The matrix c expands to 19 by 19. Figure 42 shows the layout of matrix equation, c X = 0, for N layers. The calculation of matrix c may be performed in an iterative manner (a DOloop in the computer program). This approach allows greater flexibility to add layers in meeting the depth variation of mud proper ties. Any number of mud layers may be assigned provided this is ne cessary. Basically, the fewer the layers, the faster the rate of con vergence. 4.3 Solution Technique To find the complex wave number k for a zero determinant, the NewtonRaphson method was used. The principle of this method for a complex function is the same as that for a real function. The first guess of k value used is the wave number obtained from Airy's (linear) wave theory for a water depth dl. The second guess is arbitrarily set to be 1% off from the first one. These two values of k are real numbers, but then k shifts to a complex number because the coefficient matrix is a complex function. Examining the trajectory of the complex wave number and the value of the determinant suggests that the function xI I I I x x x Ilx iX I x x xix xix i i x x X x x  x x xx xl x x KXX X X X X___ _ x Xl X I L L I I I i~~ ~ I [ I lEI x le Loyer 2ndLayer 3,d Layer NI I Fig. 42. Layout of the Coefficient Matrix for the MultiLayered Model. X I IX I IIXIXIXIXI X X X X _Vj 11 thLayer Nth Layer 66 has a tornadolike shape. To hit the target (exact k value for zero determinant), high precision in the computer program is required. Sometimes, an absolute zero determinant is difficult to obtain due to the large matrix and limited accuracy of the computer. Therefore, another convergence criterion was desirable. Comparing the results, e.g., velocity and pressure, obtained from a finite determinant value but with Ak < 1014 and that from a zero determinant showed no account able difference. Therefore, Ak < 1014 was selected as the new con vergence criterion. This criterion requires a reasonable number of iterations, i.e., less than 20, to obtain the results. The elements of the column matrix X, after solving the matrix equation, do not represent the true answer but are proportional to the true value. Equation 411 (or Eq. 412) is invoked to obtain the pro portionality constant and then to obtain the true Ai,Bi,..., and bi1. Pressure and velocity profiles are then determined easily from Eqs. 47, 48, and 410. Shear stress profile is obtained as = (u + 9w) Pe~z ax SPe(W" + k2 w) exp[ j(kx ot)] (429) 4.4 Input data The model was run through all the experimental cases (see Chapter 6 for detail). Wave amplitude, period, and the thickness, bulk densi ty, viscosity, and shear modulus for each layer are the input data needed. The number of layers and thicknesses di were selected to meet the variation of measured density profile. For the water layer, the 67 density was selected to be 1000 kg/m3. The kinematic viscosity, vI, was 1 x 106 m2/sec. The eddy viscosity, e, is dependent on the wave Reynolds number and is discussed later. Details for measuring Gi and vi for the selected muds are given in Chapter 5. In this study, G and p for the six experimental runs were correlated with the dry density. The correlative equations are presented in Chapter 5. See Table 52. Bed density information follows in Chapter 7, see Eq. 71 through 73. Wave conditions for the six experimental runs are presented in Chapter 6 (Table 64). The thickness of each layer for the six experi mental runs are listed in Table 41. Each run had two wave loadings except Run 6 which had three. See Chapter 6 for detail. An assumption concerning the eddy viscosity was that E (water layer) is only a function of the waterwave parameters and is indepen dent of mud bed rigidity. In other words, the eddy viscosity in the water column for a rigid bed was assumed to be the same as for a soft movable bed, given the same wave conditions. This assumption allows the determination and use of eddy viscosity in the model. Details are described next. Given a high shear modulus for the mud layer, e.g., G > 10000 N/m2, the model may be used to simulate the wave bed shear stress for a rigid bed. By comparing the modelpredicted bed shear stress and that calculated from Jonsson's friction factor concept (using the friction factor diagram presented by Kamphuis, 1975: see Fig. 211 in Chapter 2), a value of the apparent eddy viscosity, Vel = "I + E, can be esti mated. From this, c can be obtained given vI. Figure 43 shows an example with d, 3 m, ao = 0.25 m, a = 0.79 rad/sec, Pi = 1035 kg/m3, Table 41. Input Thickness for Each Layer (cm) Run dI d2 d3 d4 d 11 21.65 2.5 2.5 4.0 5.0 2 24.15 1.5 3.0 3.0 4.0 21 19.15 1.5 3.0 3.0 4.0 2 19.65 1.0 3.0 3.0 4.0 31 16.15 2.5 3.0 4.0 5.0 2 18.15 1.5 2.0 4.0 5.0 41 26.35 1.3 2.0 3.0 3.0 2 28.65 1.0 1.0 2.0 3.0 51 19.65 2.0 3.0 5.0 6.0 2 21.05 1.6 2.0 5.0 6.0 61 24.65 1.0 2.0 3.0 5.0 2 25.15 0.5 2.0 3.0 5.0 3 25.15 0.5 2.0 3.0 5.0 Fig. 43. Comparison of Computed Bed Shear Stress (Model and Kamphuis) at the Mud Surface to Determine the Eddy Viscosity of Water. 69 vi = 1x106 m2/s, bed surface roughness r = 20 Om and Rw = 3.0 x 105. Values of d2, P2, and P2 were immaterial because of the high G2 value. With a fw from Kamphuis' diagram, Eq. 27 gives a maximum bed shear stress of 0.6 N/m2 which is equivalent to the model having a total viscosity of 2.4 x 106 m2/sec. Therefore, the corresponding eddy viscosity would be 1.4 x 106 m2/sec and may be used to calculate the interfacial shear stress for a soft movable bed under the same wave load. This procedure illustrates a practical method for estimating the watermud interfacial shear stress for any wave flow condition. For the six experimental runs, however, the value of c was set to be equal to zero because all Rw < 104 (presented later in Table 42). This is because Rw < 104 indicates a viscous sublayer just above the mud surface and consequently zero eddy viscosity for the purpose of calcu lating the bed shear stress. In oscillatory flows, the pressure force is balanced mainly by the inertia force. The shear force plays little role when outside the bottom boundary layer, 6, even though turbulence is involved. This can be understood by noting that a high value of Reynolds number, R=udl/vel 300 2000, implies that the shear stress term can be neglected outside the boundary layer. This explains why the potential flow ap proach works quite well for water wave problems. For wave flows in volving viscosity, an example of the horizontal velocity profile given by Dean and Dalrymple (1984, p. 264) indicates that the only influence of increased vel is an increase in 6. The velocity profile above 6 does not change at all. Therefore, the value of c for the water layer is primarily necessary for the determination of the bed shear stress. 4.5 Model Results Model results in terms of velocities, pressure, and shear stress as well as the corresponding phase angles for the six experimental runs are summarized as follows. Details are presented in Appendix B. Five layers were chosen in all runs. 4.5.1 Velocity Two types of vertical water velocity profiles are obtained by model simulation. In the first example (see Fig. 44a), the vertical velocity decreased to a minimum a few centimeters above the mud surface and then increased to a maximum at the mud surface. The vertical velocity profile in the mud started from maximum at the mud surface and decreased to zero at the rigid bottom. This profile tends to show that there is a "false bottom" above the mud surface. The tendency to have a false bottom is due to the vertical motion of mud being about 90 degrees out of phase with the surface wave. Mallard and Dalrymple (1977) have shown mathematically that a false bottom can exist for a perfectly elastic bed which has a vertical motion 180 degrees out of phase with the surface wave. In the present example the viscosity damps out the response and therefore only shows a tendency. In the second type (see Fig. 45a for Run 51) the vertical velocity does not have a phase lag between the water and mud layers. Horizontal velocities were reduced significantly in the mud layers and also shifted 90 to 120 degrees. This implies that the velocity PHASE LAG (degree) 0 4 8 12 UELOCITY U 16 20 24 28 & U (cm/s) PHASE LAG (degree) 180 120 60 0 60 120 180 (b) SUL v Phase 13 *12 014 15 (9 measured 0.60 0.70 0.80 030 1.00 1.1,0 P Pig Fig. 44. Comparison of the Model Prediction and Measurement for Run 12. (a) Velocity; (b) Pressure. PHASE LAG (degree) PHASE LAG (degree) (a) Run S2 (b) SUL gW __R_ __ _ 3S : ^ 30.6 t ' "Ph a L~J 25 4, : Phase / dt  20 ^U "+ ,I/ Phase ;' 100 I" i t d ,4 d+I 12 15 (0 measured u) ds (emesured 0 5 to is 20 2S 30 3S 0.50 0.60 0.2o 0.80 0.0 1.00, 1.10 UELOCITY U & U (cm/9) 0/Plga Fig. 45. Comparison of the Model Prediction and Measurement for Run 52. (a) Velocity; (b) Pressure. 73 gradient at the watermud interface is larger than what meets the eye in Figs. 44a and 45a. Notice that the boundary layer thickness for the mud layers, Vve/2o 10 cm, was greater than the thickness of each layer and resulted in the buildup of a horizontal velocity overshoot above the watermud interface. The measured horizontal velocities (see Chapter 7 for details) in the water layer are also displayed. Measured values in these two examples show a reasonable agreement with model simulated values. Although the model gave information on the motion in the mud layers, experimentally measuring the motion was difficult. Here the horizontal mud displacements were measured by visual observation from the side wall for experimental runs 5 and 6. The amplitudes of move ment of sediment particles near the side wall were converted to the corresponding velocity amplitudes (by using the formula u = ao) and compared with model results. The observed mud velocities, however, were reduced, due to the side boundary effects; therefore, they were smaller than predicted (by a factor of about 1:3.7, see Appendix B). In justifying the measurements, a relationship for the lateral distri bution of the horizontal velocity, u(y), where y is the lateral coordinate, must be worked out. Efforts to interpret the results by considering the side wall influence are given in Appendix B. The resulting value of the ratio of u in the center of the flume and u near the side wall ranged from about 4 to 7, which is not far from the average value of 3.7 obtained by a direct comparison of prediction and measurement. 4.5.2 Pressure The pressure is always in phase both in the water and mud layers since it is the driving force. This phase relationship was also veri fied from experiments, see Section 7.2.2 in Chapter 7. The amplitude decreased in the water column, dropped at each interface between the layers, and increased within the mud layers. Figures 44b and 45b show two examples. Note that the measured pressure amplitudes (see Chapter 6 for details) were normalized by p1ga. At each interface, the normal stresses were matched from the two adjoining layers. Therefore, the pressure, P on 2p Dw/3z dropped between layers because of a stepincrease in p and almost the same 3w/az. Within a mud layer, the shear stress term becomes important because of the high ve value. The behavior in a vertical pressure profile can be depicted easily by substituting Eq. 46 and 47 into the vertical momentum equation (Eq. 43b). The vertical pressure gradient can be obtained as shown in Eq. 430. For low viscosity, e.g., water with v = 1 x 106 m2/s, the first term of the right hand side dominates, and therefore yields a positive pressure gradient. For large ve the second term is also important and may result in a negative pressure gradient as shown in Figs. 44b and 45b. T p [(jak v ) w w" ] (430) As already noted, the model uses layers of mud which have constant properties to simulate continuously varying properties. Therefore, the true pressure profile may be estimated by a smooth curve, e.g., the broken line in Fig. 45b. 75 Run 5 shows a reasonable agreement; however, Run 3 indicated that the measured pressure was larger (about 30%) than predicted. For other data, see Appendix B. Generally, the measured dynamic pressure was higher than that predicted by the model for kaolinite beds. The same was true for the Cedar Key mud bed, but only during the initial period of the wave loading. One possible reason is that the pressure trans ducer measured the total dynamic pressure, which also included the effect of timevarying mud structure and generation of pore water pres sure. The model, however, does not account for these effects. After the completion of these processes, the measured dynamic pressure principally reflected the waveinduced pressure. 4.5.3 Shear Stress The shear stress amplitude profile in this system showed a linear increase with depth characteristic of the mud bed over a rigid bottom, and negligible shear stresses in the water column except near the watermud interface. See Fig. 46 for an example. The small total mud thickness, less than half of the wave length, is the likely reason for a linearly increasing shear stress profile. For an infinite mud thick ness, the shear stress would eventually reach a maximum and then de crease (Mallard and Dalrymple, 1977). Shear stresses in the water layer are negligibly small because of the small velocity gradient except near the watermud interface. Notice the nearly 90 degree shift of shear stress in the mud layers. This implies that the maximum shear stress occurred between the wave crest and trough. It may further indicate that the shear stress in the mud was induced mainly by the 76 PHASE LAG (degree) 48 10 60 0 60 .120 180 240 Run 11 SWL T_ 35 30 LU 2s " 20 T .b=025N/r' U 4 Cr d3 d4 0 0 20 40 so 80 t0 120 140 SHEAR STRESS (N/m2) Fig. 46. An Example of the ModelPredicted Shear Stress Amplitude Profile. 77 longitudinal pressure variation at the mud surface rather than the watermud interracial shear stress. The maximum longitudinal pressure gradient occurs near the mean water elevation (90 degrees after wave crest), while the interfacial shear stress, being proportional to the velocity, is minimum. The figure shows oscillations of the shear stress just above mud surface. This would not be expected in reality and may be due to the calculation of phase lag by using small shear stress resulting in a numerical oscillation. Finally, it should be noted that the large shear stresses in the mud layer are due to the large viscosity, ve, and finite shear strain. The modelpredicted bed shear stress amplitude at x = 0, Tbo, is b e (w" + k2 w) (431) bo el k 1 1 The values of Tbo for the six runs are summarized in Table 42. Bed shear stress amplitudes based on the rigid mud bed assumption and Eq. 27 are also listed in the parentheses for comparison. It is observed that the model predicted generally larger Tbo (up to 30%) than those from a rigid bed. This is because of the opposite (100 120 degree phase lag) motion between the mud and the water. However, for a very soft mud, e.g., Run 5, the model predicted a smaller value. It should also be noted that the longitudinal shear stress profile also exhibits an exponential decay law for Rw < 104. The bed shear stress amplitude at any location can therefore be evaluated as Tb(x) = Tbo exp(kimx). The spatiallyaveraged bed shear stress amplitude is b 1 L Tbo <'b> T W bX dx [1 exp(kimL)] (4 32) b L b im Li 78 where L = 8 m is the length of mud bed section and kim is the measured wave decay coefficient (see Chapter 6 for details). The corresponding Rw are also included in Table 42. 4.5.4 Water Wave Decay A comparison of predicted and measured wave decay coefficient, kim, is an easy way to check the validity of the model. A viscous damping mechanism implies that the greater the mud velocity, the higher the rate of energy dissipation. Therefore, a highly consolidated (rigid) bed can not be expected to have a high energy dissipation. Both the predicted and measured kim show this behavior. See Fig. 47 and also Table 65, Chapter 6. Except for Run 6I, the predicted and measured wave decay coefficient show satisfactory agreement. Table 42. Interfacial Shear Stresses and Mud Wave Amplitudes Run T bo N/m2 N/m2 cm 1I 0.247 (0.187) 0.170 3600 0.24 2 0.386 (0.304) 0.242 4040 0.23 21 0.166 (0.163) 0.146 2360 0.05 2 0.291 (0.290) 0.245 3070 0.09 3I 0.210 (0.178) 0.124 2830 0.23 2 0.293 (0.258) 0.158 2720 0.27 4i 0.196 (0.192) 0.178 2510 0.06 2 0.297 (0.294) 0.251 2660 0.05 5I 0.272 (0.274) 0.191 5130 0.14 2 0.421 (0.425) 0.314 6220 0.15 61 0.226 (0.200) 0.149 3450 0.07 2 0.352 (0.306) 0.248 3590 0.13 3 0.431 (0.376) 0.323 3810 0.21 I I I I Digits Indicate Run Numbers 6 1 6 5 . 5r6 4 2 / I I I / aO 0.2 03 MODEL PREDICTED Kim(m"l) Fig. 417. Comparison of the Measured and Predicted Wave Decay Coefficient. Q3 E E 0.2 Ui 0.1 U1J L t .... 