CR-07-1

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ABSTRACT: The Coastal Inlets Research Program (CIRP) is developing predictive numerical models for simulating the waves, currents, sediment transport, and morphology change at and around coastal inlets. Water motion at a coastal inlet is a combination of quasi-steady currents such as river flow, tidal current, wind-generated current, and seiching, and of oscillatory flows generated by surface waves. Waves can also create quasi-steady currents, and the waves can be breaking or non-breaking, greatly changing potential for sediment transport. These flows act in arbitrary combinations with different magnitudes and directions to mobilize and transport sediment. Reliable prediction of morphology change requires accurate predictive formulas for sediment transport rates that smoothly match in the various regimes of water motion. This report describes results of a research effort conducted to develop unified sediment transport rate predictive formulas for application in the coastal inlet environment. The formulas were calibrated with a wide range of available measurements compiled from the laboratory and field and then implemented in the CIRP's Coastal Modeling System.

Emphasis of the study was on reliable predictions over a wide range of input conditions. All relevant physical processes were incorporated to obtain greatest generality, including: (1) bed load and suspended load, (2) waves and currents, (3) breaking and non-breaking waves, (4) bottom slope, (5) initiation of motion, (6) asymmetric wave velocity, and (7) arbitrary angle between waves and current. A large database on sediment transport measurements made in the laboratory and the field was compiled to test different aspects of the formulation over the widest possible range of conditions. Other phenomena or mechanisms may also be of importance, such as the phase lag between water and sediment motion or the influence of bed forms. Modifications to the general formulation are derived to take these phenomena into account. The performance of the new transport formulation was compared to several popular existing predictive formulas, and the new formulation yielded the overall best predictions among the formulas investigated. Results of this report are thus considered to represent a significant and operational step toward a unified formulation for sediment transport at coastal inlets and the nearshore where transport of non-cohesive sediment is common.

Contents

Figures and Tables vi


Preface xii


1 Introduction 1
  Background 1
  Objectives 5
  Procedure 6


2 General Sediment Transport Properties 8
  Physical properties of particles 8
    Granulometry 8
    Porosity and friction angle 9
    Settling velocity 10
  Shear stresses and friction coefficients 13
    Bottom boundary layer flow 13
    Current-related shear stress 14
    Wave related shear stress 15
    Combined wave and current shear stress 17
  Bed forms effects and roughness computation 18
    Current ripples, dunes, and wave ripples 19
    Computation of various roughnesses 22
    Calculation of total roughness 24
  Shields parameter and sediment transport 24
    Threshold of motion and critical Shields parameter 24
    Mode of sediment transport 25
    Inception of sheet flow 28


3 Bed Load 34
  Introduction 34
  Previous studies on bed-load transport under wave and current interaction 35
    Bijker formula 36
    Bailard formula 37
    Van Rijn formula 38
    Dibajnia and Watanabe formula 38
    Ribberink formula 41
  Bed-load transport by currents 42
    Existing formulas 42
    Comparison with data 43
    New formula for bed-load transport 47
  Bed-load transport by waves 50
    Existing formulas 50
    Development of new formula 52
    Comparison with experimental data 54
  Bed-load transport by waves and currents 62
    Development of new formula 62
    Comparison with experimental data 65
    Comparison with existing formulas for waves and current 66
  Phase-lag effects on sediment transport in sheet flow 68
    Introduction 68
    Simple conceptual model 69
    Dibajnia and Watanabe formula 72
    Modification of Camenen and Larson formula for phase lag 73
    Experimental data 75
    Calibration of conceptual model 76
    Influence of median grain size 77
    Influence of wave orbital velocity 78
    Influence of wave period 81
    Comparison with all data 83
    Concluding remarks on phase-lag effects 85


4 Suspended Load 86
  Introduction 86
  Equilibrium profile for suspended sediment 89
    Mass conservation equation 89
    Schmidt number 90
    Sediment diffusivity and concentration profiles 91
  Sediment diffusivity due to steady current 94
    Experimental data 94
    Shape of concentration profile 97
    Estimation of Schmidt number 102
  Sediment diffusivity due to nonbreaking waves 107
    Theoretical profiles 107
    Estimation of sediment diffusivity profiles for oscillatory flows 107
    Starting point for suspension load 113
    Shape of concentration profile 117
    Relationships for mean sediment diffusivity due to waves 123
    New formula for mean sediment diffusivity due to waves 126
    Interaction between waves and current 130
  Effect of breaking waves on sediment diffusivity 132
    Extension of sediment diffusion expression 133
    Experimental data with breaking waves 136
    Energy dissipation due to breaking waves 136
    Influence of Irribaren parameter and u*w/Ws on sediment diffusivity 137
  Reference concentration 141
    Effect of current 143
    Effect of waves 147
    Wave-current interaction 155
    Cases with breaking waves 159
  Suspended load transport 163
    Existing formulas for suspended load under wave-current interaction 163
    A simple formula 165
    Experimental data 167
    Validation of hypothesis 167
    Comparison with experimental data in case of current only 174
    Comparison with experimental data for waves-current interaction 175
  Suspended sediment transport for rippled beds 178
    Effects of ripples on suspended load 178
    Simple conceptual model for phase-lag effects on suspended load 180
    Modification of formula for asymmetric waves 183
    Observations of phase-lag effects on suspended load over ripples 184
    Empirical formulas for  and αpl,s 186
    Sensitivity analysis for different formulas 189
    Concluding remarks on phase-lag effects 194


5 Unified Sediment Transport Formula for Coastal Inlet Application 195
  Summary of total load formula 195
    Bed-load transport 195
    Suspended load transport 197
    Velocity profiles for varying slope 200
  Application to coastal inlet studies 202
    Validation for longshore sediment transport 202
    Validation of cross-shore sediment transport 210
    Comments on morphological evolution using total load formulas 215


6 Conclusions 216


References 219


Appendix A: Notation 231


Appendix B: Computation of Mean Values for Onshore and Offshore Shields Parameter 239
  Sinusoidal wave without current 239
  2nd-order stokes wave without current 239
  Sinusoidal wave with current 240
  2nd-order stokes wave with current 240

Figures and Tables

Figures

Figure 1. Hydrodynamic processes controlling sediment transport in an inlet environment 2
Figure 2. Natural processes around an inlet for which predictions of sediment transport and morphological evolution are of importance 3
Figure 3. Hydrodynamic forcing determining conditions for longshore sediment
transport 		4
Figure 4. Sediment transport formulation 5
Figure 5. Example of grain-size distribution 9
Figure 6. Settling velocity for sediment with comparison of several formulas against experimental data 12
Figure 7. Turbulent boundary layer structure with mean velocity profile 13
Figure 8. Wave friction coefficients for plane bed using different formulas 17
Figure 9. Schematic diagram for nonlinear interaction between wave and current bed shear stress 18
Figure 10. Schematic of ripples due to current or waves 19
Figure 11. Influence of grain size diameter, water depth, and wave orbital velocity; on bed-form predictions for current ripples, dunes, and wave ripples; and on roughness prediction according to different formulas 20
Figure 12. Equivalent roughness ratio '''ks'/'''d50 'versus total Shields parameter θ for compiled data set together with predictions by studied formulas 23
Figure 13. Critical Shields parameter plotted against dimensionless grain size 26
Figure 14. Modes of sediment transport 26
Figure 15. Schematic representation of different bed forms and sediment transport regimes for increasing current or wave orbital velocity 27
Figure 16. Classification of different types of sediment transport with respect to Shields parameter '''θ 'and ratio '''Uc/'''Ws '28
Figure 17. Comparison between observed critical wave orbital velocity '''Uw,crsf,meas 'for inception of sheet flow and predicted value '''Uw,crsf,pred 'using Equation 57 33
Figure 18. Definition of wave and current directions, and horizontal time-dependent velocity variation at bottom in direction of wave propagation 39
Figure 19. Time variation of bottom velocity in wave direction, and induced shear stress for waves and current combined 41
Figure 20. Distribution of median grain size and mean current speed for database compiled on sediment transport under steady current 44
Figure 21. Comparison between Meyer-Peter and Müller formula and compiled database on sediment transport rates 45
Figure 22. Effect of critical Shields parameter on bed-load transport rate: comparison between data and studied formulas 47
Figure 23. Influence of critical Shields parameter on bed-load transport rate illustrated through data and Equation 81 49
Figure 24. Comparison between bed-load transport for current only predicted by new formula and measurements 50
Figure 25. Typical wave velocity variation, and instantaneous Shields parameter variation over wave period in direction of waves 52
Figure 26. Calibration of coefficient '''aw 'using all experimental data with waves only 57
Figure 27. Comparison between bed-load transport predicted by new formula and experimental data with waves 58
Figure 28. Comparison between bed-load transport predicted by Equation 90 and experimental data over full wave cycle 61
Figure 29. Definition sketches for wave and current interaction, and typical velocity variation over wave period in direction of waves including effect of steady current 63
Figure 30. Comparison between bed-load transport predicted by Equation 99 and experimental data with current 67
Figure 31. Phase-lag effect on sediment transport for sinusoidal wave with superimposed current when phase lag φ is introduced for concentration at bottom 71
Figure 32. Phase-lag effects on sediment transport for second-order Stokes wave with positive or negative, and adding current introducing phase lag  for concentration at bottom and with '''rw '= 0.20 71
Figure 33. Notations for colinear wave and current interaction 72
Figure 34. Calibration of conceptual model against data 77
Figure 35. Influence of grain size on bedload sediment transport 78
Figure 36. Influence of wave orbital velocity on sediment transport 80
Figure 37. Influence of wave period on sediment transport 82
Figure 38. Comparison between predicted and measured sediment transport rate 84
Figure 39. Computation of suspended load over depth 87
Figure 40. Concentration profile for steady conditions 90
Figure 41. Three analytical relationships for vertical sediment diffusivity versus '''z'''/'''h 'with σE '= 1 = 1/2σP '= 1/6σB '94
Figure 42. Comparison between "energy slope" method and "velocity profile" method to estimate Nikuradse roughness '''ks 'and shear velocity '''u*c '96
Figure 43. Vertical profile of sediment diffusivity obtained from Equation 129 using measured concentration profiles 98
Figure 44. Examples of comparisons between predicted concentration profiles, using fitted exponential or power-law profiles, and measured concentration 100
Figure 45. Comparison between predicted concentration using fitted exponential profile, "linear" power-law profile, or "parabolic" power-law profile and measured concentration using all data 101
Figure 46. Estimation of Schmidt number as function of ratio '''Ws/'''u*c '104
Figure 47. Estimated coefficient '''σE 'compared to coefficients σP 'and σB '105
Figure 48. Estimated values of Schmidt number as function of ratio '''Ws/'''u*c, together with predictive equations 106
Figure 49. Vertical profiles of eddy diffusivity obtained from Equation 130 using measured concentration profiles with interaction between nonbreaking waves and current 111
Figure 50. Vertical profile of eddy diffusivity obtained from Equation 130 using measured concentration profiles with interaction between nonbreaking waves and current 112
Figure 51. High sediment concentration close to bottom using data from Dohmen Janssen 114
Figure 52. Characteristic sediment diffusion profile (Equation 142) and induced sediment concentration profile; division of induced concentration profile to different layers and application of an exponential and parabolic logarithmic profile to estimate suspended load 116
Figure 53. Schematic representation of sediment concentration within moving mixing layer for rippled bed 117
Figure 54. Examples of comparison between predicted concentration using fitted exponential profile and power-law profiles and measured concentration for interaction between nonbreaking waves and current 118
Figure 55. Examples of comparison between predicted concentration using fitted exponential profile and power-law profile and measured concentration for interaction between nonbreaking waves and current 119
Figure 56. Examples of concentration profiles and corresponding sediment diffusivity profiles using data from Steetzel and Van der Velden 120
Figure 57. Dimensionless sediment diffusivity w,E/(κ'''hu*w) versus wave period
and ratio '''Uw/'''Ws 		'126
Figure 58. Vertical sediment diffusivity w,E 'estimated from data compiled versus w,E 'calculated with Equation 152 127
Figure 59. Estimated value of coefficient '''σw 'using Equation 154 as function of ratio '''Ws/'''u*w 'with roughness ratio '''ks/'''d50 'indicated 129
Figure 60. Estimation of sediment diffusivity '''cw 'by adding current- and wave-related sediment diffusivity as function of parameter '''Ws'/'''u*w 'with ratio '''Uc/Uw 'emphasized 131
Figure 61. Vertical sediment diffusivity cw''','''E 'estimated from compiled data versus cw,E 'calculated with Equation 156 for wave and current interaction 133
Figure 62. Comparison between estimated energy dissipation from measured wave height variation and calculated energy dissipation from bore analogy using data from Peters 138
Figure 63. Ratio b,meas/b''','''pred 'versus Irribaren parameter ξ∞ or ratio '''u*w/'''Ws 'using data from Table 24 139
Figure 64. Vertical sediment diffusivity v,E 'estimated from compiled data versus v''','''E 'calculated with Equations 157 and 175 for breaking waves 141
Figure 65. Comparison between observed reference concentrations assuming an exponential profile or power-law profile at reference level '''za '= '''ks '= 2 '''d50 '142
Figure 66. Predicted reference concentration '''ca 'and '''cR 'versus Shields parameter from various formulas 144
Figure 67. Predicted reference concentration '''cR 'using Equations 185 and 186 versus experimental reference concentration assuming an exponential profile for concentration 147
Figure 68. Bottom concentration '''c0 'versus modified skin Shields parameter θr 'using data collected by Nielsen; new equation is based on Equation 190 with calibrated coefficient value from Equation 191 149
Figure 69. Histograms of grain-size distribution for current data set and wave
data set 		152
Figure 70. Reference concentration '''cR 'estimated from compiled data versus '''cR 'calculated with Equation 190 and 186 with roughness ratio emphasized 153
Figure 71. Estimated roughness ratio '''ks'/'''d50 'versus total Shields parameter with ripple height emphasized 154
Figure 72. Reference concentration '''cR 'estimated from data compiled for waves only versus '''cR 'calculated with Equations 190 and 186 156
Figure 73. Reference concentration '''cR 'estimated from compiled data set with wave-current interaction versus '''cR 'calculated with Equations 192, 193, and 186 with absolute mean current '''Uc or roughness ratio '''ks/'''d50 'emphasized 158
Figure 74. Reference concentration '''cR 'estimated from data compiled with wave-current interaction versus '''cR 'calculated with Equations 192, 193, and 186 159
Figure 75. Reference concentration '''cR 'estimated from data compiled with breaking waves excluding data from Peters and using data set from Peters only versus '''cR 'calculated with Equations 192, 193, and 186 161
Figure 76. Ratio between estimated reference concentration and predicted reference concentration from Equations 192 and 186 as function of Irribaren parameter ξ'∞ 'using data sets from Table 24 162
Figure 77. Reference concentration '''cR 'estimated from data compiled with breaking waves versus '''cR 'calculated with Equations 192, 186, and 196 162
Figure 78. Comparison between observed and calculated suspended sediment load for steady current using Equation 210 with experimental values on '''cR 'and  169
Figure 79. Comparison between observed and calculated suspended sediment load for wave-current interaction using Equation 210 with experimental values on '''cR 'and  171
Figure 80. Vertical velocity profile and sediment concentration profile inside surf zone, and close to breaker line 173
Figure 81. Comparison between observed suspended sediment load and calculated load using Equation 210 and predicted values for '''cR 'and 'for current only 176
Figure 82. Comparison between measured and calculated values of wave and current interaction for reference concentration '''cR 'and sediment diffusivity , and for resulting suspended sediment load using Equation 210 177
Figure 83. Schematic of transport processes in asymmetric wave motion over
rippled bed 		179
Figure 84. Phase-lag effects on sediment transport for second-order Stokes wave with positive or negative current introducing phase lag 'pl '''''''for concentration at bottom and with asymmetry of '''rw '= 0.20 182
Figure 85. Coefficient αpl,s 'as function of sediment phase lag pl 'with varying values on '''Uc/'''Uw 'and an asymmetry of '''rw '= 0.20 182
Figure 86. Notation for colinear wave and current interaction 183
Figure 87. Comparison between measured and calculated net sediment transport rate using Equations 210 and 218 with α'pl''','''s '= 0 185
Figure 88. Dimensionless suspended sediment transport rate as function of phase lag parameter '''pWR '186
Figure 89. Comparison between measured and estimated net sediment transport rate using Van der Werf and Ribberink formula; Equations 210 and 218 with '''rpl''','''s 'from Equation 213; Equations 210 and 218 with αpl''','''s 'from Equation 217; or Equation 221 189
Figure 90. Influence of wave orbital velocity on sediment transport 191
Figure 91. Influence of wave period on sediment transport 192
Figure 92. Influence of wave asymmetry on sediment transport 193
Figure 93. Definition of current and wave direction and velocity variation at bed in direction of wave propagation 197
Figure 94. Velocity profiles according to Equations 244, 245, and 246 201
Figure 95. Cross-shore variations in hydrodynamic parameters and beach profile for an LSTF experimental case together with measured longshore suspended sediment transport and calculated transport using six studied formulas 206
Figure 96. Cross-shore variations in hydrodynamic parameters and beach profile for an LSTF experimental case together with measured longshore suspended sediment transport and calculated transport using six studied formulas 207
Figure 97. Cross-shore variations in hydrodynamic parameters and beach profile for Sandy Duck experiment together with measured longshore suspended sediment transport, and calculated transport using six studied formulas 208
Figure 98. Cross-shore variations in hydrodynamic parameters and beach profile for Sandy Duck experiment together with measured longshore suspended sediment transport and calculated transport using six studied formulas 209
Figure 99. Predictive results for longshore sediment transport rate across beach profile using present formula for LSTF data, and Sandy Duck data 211
Figure 100. Cross-shore variations in hydrodynamic parameters and beach profile for Sandy Duck experimental case together with measured cross-shore suspended sediment transport, and calculated transport using six studied formulas 212
Figure 101. Prediction of cross-shore suspended load across profile line using new formula with Sandy Duck data 214

Tables

Table 1. Classification of different types of sands 8
Table 2. Internal friction coefficient 9
Table 3. Summary of data sets on inception of sheet flow under oscillatory flow 29
Table 4. Prediction of critical wave orbital velocity for inception of sheet flow within factor of 1.25 together with mean value and standard deviation of Δ'''Uw '32
Table 5. Data base compiled to study bed-load sediment transport in steady current 44
Table 6. Prediction of bed-load transport rate within factor of 2 and 5 of measured values and root-mean-square errors using current 'only' data 46
Table 7. Data summary for bed-load sediment transport experiments carried out in oscillatory flow with and without current 55
Table 8. Predictive capability of bed-load transport rate within factor of 2 and 5 of measured values and root-mean-square errors, data from waves only 59
Table 9. Prediction of bed-load transport rate within factor of 2 and 5 of measured values and root-mean-square errors using data on waves and current combined 68
Table 10. Summary of data on bed-load sediment transport in full-cycle oscillatory flow 75
Table 11. Experimental conditions for studied cases on median grain-size effect 78
Table 12. Experimental conditions for studied cases on wave orbital velocity effects 80
Table 13. Experimental conditions for studied cases on wave period effects 82
Table 14. Prediction of bed-load transport rate within factor of 2 or 5 of measured values, together with mean value and standard deviation of Δ'''qs '84
Table 15. Data summary for suspended sediment experiments under steady currents 95
Table 16. Percentage of sediment concentration within +/- 20 percent of measured values obtained using an exponential law or power law for 'c 'in fitting against data 99
Table 17. Data summary for analysis on Schmidt number 103
Table 18. Prediction of Schmidt number using parabolic profile and exponential profile for steady current 103
Table 19. Data summary for suspended sediment experiments under oscillatory flows 109
Table 20. Percentage of predicted sediment concentrations within +/- 20 percent of measured values using exponential-law or power-law profiles for studied data sets 121
Table 21. Input parameters for four study cases in Figure 56 122
Table 22. Predictive skill of different formulas for sediment diffusivity for
waves only 		126
Table 23. Prediction of sediment diffusivity for wave and current interaction 131
Table 24. Data summary for suspended sediment experiments along beach profiles employed for investigating breaking wave effects 136
Table 25. Prediction of sediment diffusivity for transport under breaking waves 140
Table 26. Selected relationships for reference concentration as they chronologically appeared in literature 143
Table 27. Prediction of reference concentration assuming parabolic power-law or an exponential sediment concentration profile 146
Table 28. Experimental data used by Nielsen 148
Table 29. Prediction of bottom concentration using data from Nielsen 150
Table 30. Prediction of reference concentration using studied data set encompassing waves only 151
Table 31. Prediction of reference concentration using compiled data set with waves and current interaction 157
Table 32. Prediction of reference concentration using compiled data set
with breaking waves 		160
Table 33. Data summary for suspended sediment experiments under oscillatory flows 168
Table 34. Prediction of suspended load transport for steady current 170
Table 35. Prediction of suspended load transport for interaction between current and waves 172
Table 36. Summary of data on suspended load sediment transport over ripples in
full-cycle oscillatory flow 		184
Table 37. Prediction of suspended sediment transport rate within factor of 2 or 5 of measured values, together with mean value and standard deviation on '''f'''('''qss) 187
Table 38. Experiment conditions for studied cases on wave orbital velocity effects 190
Table 39. Experiment conditions for studied cases on wave period effects 192
Table 40. Experiment conditions for studied cases on wave asymmetry effects 193
Table 41. Experiment conditions for studied cases on longshore sediment transport 203
Table 42. Predictive capability of different transport formulas for longshore suspended load transport for LSTF and Sandy Duck experiments 210
Table 43. Predictive capability of different transport formulas regarding suspended load transport in cross-shore direction for Sandy Duck experiments 213
Table 44. Predictive capability of total load sediment transport in cross-shore direction for sheet flow experiments by Dohmen-Janssen and Hanes (2002) 214

====Preface The Coastal Inlets Research Program (CIRP) is developing predictive numerical models for simulating the waves, currents, sediment transport, and morphology change at coastal inlets. Water motion at a coastal inlet can synoptically range through quasi-steady currents as in river flow, tide, wind, and seiching; oscillatory flow as under surface waves, which can create quasi-steady wave-induced currents; breaking and nonbreaking waves; and arbitrary combinations of these flows acting with different magnitudes and at different directions. Reliable prediction of morphology change requires accurate predictive formulas for sediment transport rates that will smoothly match in the aforementioned regimes of water motion and change according to the driving forces and water depth. This report describes a research effort conducted with the aim of developing unified sediment transport rate formulas for application in the coastal inlet environment. These formulas, calibrated with a wide range of available measurements compiled from the laboratory and field, have been implemented in CIRP's Coastal Modeling System.

CIRP is administered at the U.S. Army Engineer Research and Development Center (ERDC), Coastal and Hydraulics Laboratory (CHL) under the Navigation Systems Program for Headquarters, U.S. Army Corps of Engineers (HQUSACE). James E. Walker is HQUSACE Navigation Business Line Manager overseeing CIRP. James E. Clausner, CHL, is the Technical Director for the Navigation Systems Program. Dr. Nicholas C. Kraus, Senior Scientists Group (SSG), CHL, is the CIRP Program Manager.

The mission of CIRP is to conduct applied research to improve USACE capability to manage federally maintained inlets, which are present on all coasts of the United States, including the Atlantic Ocean, Gulf of Mexico, Pacific Ocean, Great Lakes, and U.S. territories. CIRP objectives are to advance knowledge and provide quantitative predictive tools to (a) make management of Federal coastal inlet navigation projects, principally the design, maintenance, and operation of channels and jetties, more effective and reduce the cost of dredging, and (b) preserve the adjacent beaches and estuary in a systems approach that treats the inlet, beaches, and estuary as sediment-sharing components. To achieve these objectives, CIRP is organized in work units conducting research and development in hydrodynamic, sediment transport and morphology change modeling; navigation channels and adjacent beaches; navigation channels and estuaries; inlet structures and scour; laboratory and field investigations; and technology transfer.

This report was prepared under contract with CIRP by Dr. Magnus Larson, Department of Water Resources Engineering, Lund University, Sweden, and by Dr. Benoît Camenen, presently at Cemagref Lyon, France, and formerly a post-doctoral researcher at Lund University, Sweden, and at the Disaster Prevention Research Institute, Kyoto University, Japan. J. Holley Messing, Coastal Engineering Branch, Navigation Division, CHL, typed the equations and format-edited this report. Dr. Kraus oversaw technical elements of this project during the 3 years of required research and development. Thomas W. Richardson was Director, CHL, and Dr. William D. Martin, Deputy Director, CHL, during the study and preparation of this report.

COL Richard B. Jenkins was Commander and Executive Director. Dr. James R. Houston was Director of ERDC.

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