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Small-Scale Sediment Transport Processes and Bedform Dynamics

Permanent Link: http://ufdc.ufl.edu/UFE0021220/00001

Material Information

Title: Small-Scale Sediment Transport Processes and Bedform Dynamics
Physical Description: 1 online resource (304 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bed, bedforms, live, modeling, morphology, sediment, transport
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The generation of small-scale sedimentary structures in the coastal environment is a complex process that occurs over a wide separation of scales in both time and space. These bedforms are ubiquitous features of the nearshore region, and yet specific information regarding their behavior and characteristics is still lacking. Specifically, it is unclear whether the bedload-dominated processes of the linear regime are as equally responsible for the generation of bedforms in the nonlinear regime, where flow separation, and subsequent vortex formation, tend to govern the dynamics of the bottom boundary layer. While a simple one-dimensional model is derived and used to explain incipient bedform growth in the linear regime, such an approach is not well-suited at addressing the complexities of the wave bottom boundary layer. Utilizing a new three-dimensional phase-resolving live-bed model, we simulate the dynamics of bedforms, such as sand ripples, in the nonlinear regime. Through forty-three independent simulations, the model has been found to reproduce oscillatory boundary layer flow, as well as provide accurate predictions of ripple geometry in both lab- and field-scale flows. Model results confirm that in the linear regime, bedform growth is promoted purely through bedload sediment transport, but inertial properties of the sediment are equally as important. In the nonlinear regime, bedform growth is also dominated by bedload transport; however, the entrainment and deposition of bed material plays an important role in maintaining ripple equilibrium, whereas it is mostly responsible for ripple decay.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Slinn, Donald N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021220:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021220/00001

Material Information

Title: Small-Scale Sediment Transport Processes and Bedform Dynamics
Physical Description: 1 online resource (304 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bed, bedforms, live, modeling, morphology, sediment, transport
Civil and Coastal Engineering -- Dissertations, Academic -- UF
Genre: Coastal and Oceanographic Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The generation of small-scale sedimentary structures in the coastal environment is a complex process that occurs over a wide separation of scales in both time and space. These bedforms are ubiquitous features of the nearshore region, and yet specific information regarding their behavior and characteristics is still lacking. Specifically, it is unclear whether the bedload-dominated processes of the linear regime are as equally responsible for the generation of bedforms in the nonlinear regime, where flow separation, and subsequent vortex formation, tend to govern the dynamics of the bottom boundary layer. While a simple one-dimensional model is derived and used to explain incipient bedform growth in the linear regime, such an approach is not well-suited at addressing the complexities of the wave bottom boundary layer. Utilizing a new three-dimensional phase-resolving live-bed model, we simulate the dynamics of bedforms, such as sand ripples, in the nonlinear regime. Through forty-three independent simulations, the model has been found to reproduce oscillatory boundary layer flow, as well as provide accurate predictions of ripple geometry in both lab- and field-scale flows. Model results confirm that in the linear regime, bedform growth is promoted purely through bedload sediment transport, but inertial properties of the sediment are equally as important. In the nonlinear regime, bedform growth is also dominated by bedload transport; however, the entrainment and deposition of bed material plays an important role in maintaining ripple equilibrium, whereas it is mostly responsible for ripple decay.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Slinn, Donald N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021220:00001


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SMALL-SCALE SEDIMENT TRANSPORT PROCESSES AND BEDFORM DYNAMICS


By

BRET MAXWELL WEBB



















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

2008

































Copyright 2008

by

Bret Maxwell Webb

































"It is unlikely that the natural process of sediment transport by flowing water

will be understood in precise I,;;,,;i,,,,/, terms in the foreseeable future."

R.A. Bagnold (1980)









ACKNOWLEDGMENTS

I would like to extend my gratitude first to my fellow office-mates for their support,

encouragement, assistance, and friendship. Robert, Jenn, Allison, Ty, and Jessica,

thanks for alv--, serving as a sounding board for my ideas, as well as my complaints.

Second, I am grateful for the constructive criticism of my supervisory committee

members, those that I have often sought out for advice, and particularly for the guidance

and encouragement of my advisor, Don Slinn. Mutlu Sumer is deserving of special

acknowledgment for the boundary l~- -r data used in the model-data comparison. Last, I

owe a great debt of gratitude to my family, especially my wife, Shannon, as none of this

would have been possible without their love and constant support.









TABLE OF CONTENTS

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ....................... ............. 8

LIST OF FIGURES .................................... 9

LIST OF SYMBOLS ....................... ............ 14

A BSTRA CT . . . . . . . . . . 19

CHAPTER

1 INTRODUCTION ...................... .......... 20

1.1 Background ...................... ........... 20
1.2 Motivation ...................... ........... 23
1.3 Approach ....................... ............ 26
1.4 O utline. . . . . . . ..... . 28

2 SEDIMENT TRANSPORT PROCESSES ........ .......... .... 30

2.1 Introduction ...................... ............ 30
2.2 Governing Hydrodynamics ................... ..... 30
2.2.1 W aves ...................... .......... 30
2.2.2 Currents ...................... .......... 33
2.2.3 Waves and Currents ................... ..... 34
2.2.4 Tides and Tidal Currents .................. ... 35
2.3 Transport .. .. .. ... .. .. .. .. ... .. .. .. .. ...... 36
2.3.1 M odes . . . . . . . .... 36
2.3.1.1 Bedload . . . . . . .. .. 37
2.3.1.2 Suspended Load .................. .. 40
2.3.2 Regim es . . . . . . . .... 43
2.4 Turbulence .. .. .. ... .. .. .. .. .. .. .. .. .. ...... 46
2.4.1 Dynam ics .................. ............. 47
2.4.2 Closure ................... . . ... 48
2.4.2.1 RANS M odels .................. .... 48
2.4.2.2 Large Eddy Simulations .................. 49
2.4.2.3 Direct Numerical Simulations ............... 49
2.5 M odels . . . . . . . . .. .. 50
2.5.1 Types. .................. ............... 50
2.5.1.1 Time-Averaged .................. ... 50
2.5.1.2 Quasi-Steady ............... .... .51
2.5.1.3 Semi-Unsteady ............... .. .. 52
2.5.1.4 Unsteady ............... ...... .. 53
2.5.2 Integrated Approaches ............... ... .54
2.5.3 Shortcom ings ... .. .. .. .. ... .. .. .. ...... 56









3 BEDFORM DYNAMICS ............


3.1 Introduction .............................. 59
3.2 Bedform Types .................. .............. .. 59
3.2.1 Rolling Grain Ripples .................. ..... .. 61
3.2.2 Vortex Ripples .................. . .... 63
3.2.3 ('!C i 'teristics . . . . .. . .. 66
3.3 Experiments .................. ................ .. 69
3.3.1 Laboratory .................. ............. .. 69
3.3.1.1 Oscillating Tray .................. .... .. 70
3.3.1.2 Oscillating Water Tunnel .................. .. 70
3.3.1.3 Flume .................. .......... .. 71
3.3.2 Field . . . . . . .. . ... 73
3.4 Ripple Predictors .................. ............. .. 74
3.4.1 Clifton (1976) .................. ........... .. 74
3.4.2 Nielsen (1981) .................. ........... .. 74
3.4.3 Grant and Madsen (1982) ................ ... .. 76
3.4.4 Wiberg and Harris (1994) .................. .... .. 77
3.4.5 Mogridge et al. (1994) .................. ..... .. 78
3.4.6 Khelifa and Ouellet (2000) ................ .... .. 79
3.4.7 Faraci and Foti (2002) ............ . ..... 80
3.4.8 W illiams et al. (2005) .................. ..... .. 81
3.4.9 Soulsby and Whitehouse (2006) ............. .. .. 81
3.5 Models ................. ................ ...... 82

4 MODEL DESCRIPTION ................... . 88

4.1 Introduction .................. . . .... 88
4.2 One-Dimensional Linear Model .................. .... .. 88
4.2.1 Hydrodynamics .................. .......... .. 89
4.2.1.1 Governing Equations ................... ... .. 89
4.2.1.2 Num erics .................. ...... .. .. 92
4.2.2 Sediment Transport ................... . ... 93
4.2.3 Morphology .............. . . . .... 93
4.3 Two- and Three-Dimensional Nonlinear Models ...... . . 93
4.3.1 Hydrodynamics .................. .......... .. 95
4.3.1.1 Governing Equations ............. .. .. .. 95
4.3.1.2 Turbulence Closure ................ .. 98
4.3.1.3 Numerics ... ............ ....... ..104
4.3.1.4 Boundary Conditions ............... 109
4.3.2 Sediment Transport .................. ........ .. 111
4.3.2.1 Bedload .................. ....... ..1 111
4.3.2.2 Suspended Load ................ .... .. 117
4.3.3 Morphology ............... . . ..... 123
4.3.3.1 Finite-Difference Methods ................. 125
4.3.3.2 Filtering Techniques ............... . 131









5 MODEL EXPERIMENTS ..................


5.1 Linear Model Experiments ........................... 146
5.2 Nonlinear Model Experiments .................. .... 147
5.2.1 Phase I: Model Validation ................. .... 147
5.2.2 Phase II: Model Capabilities ................. .... 149
5.2.3 Phase III: Sediment Transport Processes . . 150

6 RESULTS ................... ............ ...... 159

6.1 Linear Model Results .................. ........... 159
6.2 Nonlinear Model Results .................. ......... 160
6.2.1 Phase I: Model Validation ................. .... 160
6.2.1.1 Hydrodynamic Validation ................. .. 161
6.2.1.2 Computational Grid Tests ..... . . ..... 163
6.2.1.3 Model Domain Width Tests ................ 165
6.2.1.4 Sediment Transport Submodel Tests . . 170
6.2.1.5 Morphology Tests ................ .... .. 173
6.2.2 Phase II: Model Capabilities ................... .... .. 183
6.2.2.1 Steady Flow .................. .. ..... 184
6.2.2.2 Subcritical Flow ................. .... 185
6.2.2.3 Sheetflow Regime ................ .. 185
6.2.2.4 Ripple Growth, Equilibrium, and Decay . .... 187
6.2.2.5 Sediment Size Tests ................. . .. 193
6.2.3 Phase III: Sediment Transport Processes . . 197
6.2.3.1 Total Load Growth ............. .... . 198
6.2.3.2 Total Load Decay ................ .... .. 203
6.2.3.3 Bedload Only .................. ..... 205
6.2.3.4 Suspended Load Only ............... .. 206

7 CONCLUSIONS ................... . . ..... 274

7.1 Live-Bed Model Evaluation .................. ..... .. 274
7.2 Sediment Transport and Bedform Dynamics ................ ..278
7.3 Future Applications .................. .......... .. 281

APPENDIX

A EXPERIMENT MATRIX .................. ......... ..283

B FLUX TENDENCY PHASE DIAGRAM ................. .. ..285

REFERENCES .................. ................ .. .. 289

BIOGRAPHICAL SKETCH .................. ............. ..304










LIST OF TABLES


Tabl

21

22

31

32

33

3-4

51

52

53

54

55

56

57

58

61

6-2

63

A-


Power law formulations for bedload transport . .....

Empirical pick-up functions for suspended load transport . ..

List of common values for orbital ripple length.. . .

Summary of field and laboratory ripple experiments . .

Parameters related to bedform dynamics. . .....

Ripple predictor key ...... . .

Mesh parameters for grid comparisons..... . .....

Mesh parameters for domain width comparisons . .

Abbreviation key for transport equation tests . .....

Morphology test simulation matrix....... . .....

Ripple predictor citation key for the morphology test simulations .

Expected morphology results based on ripple predictors . .

Phase II experiment matrix ................. . .....

Simulation parameters for the Phase III experiments . .

Ripple predictor RMS error for morphology tests under lab-scale flows

Ripple predictor RMS error for morphology tests under field-scale flows

Ripple predictor model performance index. . .....

Complete experimental matrix....... . ......


e


Page

. 58

. 58

. 85

. 86

. 87

. .. 87

. 155

. 155

. 155

. 156

. 156

. 157

. 158

. 158

. 272

. 272

. 273

. 284









LIST OF FIGURES


Figure Page

1-1 Pictures of sand ripples in lab and field settings ................. .. 29

2-1 Velocity and stress time series for linear and nonlinear waves. . ... 57

2-2 Distribution of fluid and grain shear stress. .................. .... 57

3-1 Schematic of vortex shedding over a rippled bed ................. ..84

3-2 Effects of bedforms on boundary -1,- r thickness ................. ..85

4-1 Schematic of one-dimensional bedload model ........ . 135

4-2 Schematic of the three-dimensional modeling domain ............. 135

4-3 Model control volumes with variable locations .............. 136

4-4 Schematic of variable mesh scaling ................ ...... 136

4-5 Mesh clustering and variable scaling ................ ..... 137

4-6 A comparison of bed shear stress formulations .... . ... 138

4-7 Estimation of critical Shields' curve by Brownlie (1981) . . ..... 139

4-8 Estimation of critical Shields' curve by van Rijn (1993) ........... .139

4-9 Particle forces acting on longitudinal and transverse slopes . . ... 140

4-10 Gravitational forces on a sphere .................. ........ .. 141

4-11 Behavior of bedload transport equations .................. ..... 141

4-12 Behavior of sediment pick-up functions .................. ...... 142

4-13 Modification of relative settling velocity by concentration . . ..... 142

4-14 Sediment control volume and transport schematic ................ ..143

4-15 Amplification factors for the two-dimensional low-pass morphology filter . 144

4-16 Amplification factors for the hybrid morphology filter .............. ..145

5-1 Horizontal and vertical grid spacing for grid comparisons ............ ..152

5-2 Model domains used in the domain width comparison tests . . .... 153

5-3 Schematic of the model domain used for the Phase III simulations . ... 154

6-1 Timestacks of bed elevation for the 1DH linear model experiment . ... 208










6-2 Time evolution of ripple height, wavelength, and steepness for the 1DH linear


model experiment . .


. . . .. . 209


3 Model-data hydrodynamic comparison... . ......

4 Model-data comparison of boundary 1-,-r thickness . .

5 Assessment of hydrodynamic model error . .....

6 Computational budget for grid resolution tests.. . .

7 Velocity standard deviations for grid comparisons . .

8 Cumulative average turbulent kinetic energy (TKE) for grid comparisons

9 Phase- and volume-averaged TKE for grid comparisons . .

10 Phase- and y-averaged u-velocity profiles for grid comparisons . .

11 Phase- and y-averaged v-velocity profiles for grid comparisons . .

12 Phase- and y-averaged w-velocity profiles for grid comparisons . .

13 Computational budget for three-dimensional grid tests . .....


6-14 Cumulative average turbulent kinetic energy (TKE) for three-dimensional grid
com parsons . . . . . . . . . .

6-15 Phase- and volume-averaged TKE for three-dimensional grid comparisons .

6-16 Phase- and volume-averaged velocity components for grid comparisons .....

6-17 Average Fourier transforms of v-velocity for grid comparisons .. ........

6-18 Spatially-averaged standard deviation of bed elevation in the third dimension .

6-19 Comparisons of equilibrium ripple height for grid comparisons .. ........

6-20 Average Fourier transforms of bed elevation for grid comparisons ...

6-21 Effects of sediment transport submodels on equilibrium ripple characteristics .

6-22 Conservation of sediment mass .............

6-23 Bedform statistics in lab-scale flows .............

6-24 Fourier analysis timestack of bedform statistics for HL1v1 .. .........

6-25 Ripple predictor comparisons for lab-scale morphology tests .. .........

6-26 Bedform statistics in field-scale flows .. ....................

6-27 Fourier analysis timestack of bedform statistics for MF2v2 and HF1v1 ..


. 210

. 211

. 212

. 213

. 214

. 215

. 215

. 216

. 217

. 218

. 219









6-28 Ripple predictor comparisons for field-scale morphology tests . . ... 232

6-29 Comparison of model results to the ripple predictor equations of Faraci and Foti
(2002) . . . . . . .. . ... ... 233

6-30 Comparison of model results to the ripple predictor equations of Nielsen (1981),
Khelifa and Ouellet (2000), and Williams et al. (2005) ........... .234

6-31 Comparison of model results to the ripple predictor equations of Mogridge et al.
(1994) . . . . . . .. . ... ... 235

6-32 Bed elevation timestack for steady flow test ................ 236

6-33 Steady flow bed shear stress and sediment transport ... . . 237

6-34 Bed elevation timestack for subcritical flow test ................. ..238

6-35 Average bed shear stress and bedload transport in subcritical flow . ... 238

6-36 Bed elevation timestack for sheetflow sediment transport ............ ..239

6-37 Time-series of bedform statistics in the sheetflow regime ............ ..239

6-38 Energy density timestack for the sheetflow test .................. ..240

6-39 Isocontours of sediment concentration in the sheetflow regime . ... 240

6-40 Average sediment concentration profile in the sheetflow regime . ... 241

6-41 Bed elevation timestacks for ripple growth, equilibrium, and decay . ... 242

6-42 Timestack of ripple spectra during growth from flat bed ............ ..243

6-43 Ripple statistics for growth, equilibrium, and decay ............... ..244

6-44 Time evolution of ripple heights during growth, equilibrium, and decay . 245

6-45 Bed elevation timestacks for ripple coarsening and bifurcation . ... 246

6-46 Timestacks of ripple spectra during ripple wavelength saturation . ... 247

6-47 Ripple statistics during height and wavelength evolution ............ ..248

6-48 Effects of hindered settling on model predictions of ripple height and length 249

6-49 Modification of average concentration profile due to hindered settling . 250

6-50 Time-series of bedform statistics for fine gravel in subcritical flow . ... 250

6-51 Phase-averaged stress and transport for fine gravel in subcritical flow . 251

6-52 Timestacks of bed elevation for fine gravel in a strong flow . .... 251









6-53 Bedform statistics for fine gravel in a strong flow ................ ..252

6-54 Phase-averaged stress and transport for fine gravel in a strong flow ...... ..252

6-55 Timestacks of bed elevation for medium silt in a weak flow . ..... 253

6-56 Time evolution of ripple height and length for medium silt in a weak flow . 253

6-57 Phase-averaged stress, transport, and sediment concentration for medium silt in
a weak flow ............... ................ .. 254

6-58 Timestacks of bed elevation for medium silt in a strong, highly-concentrated flow 255

6-59 Time evolution of bedform height and length for medium silt in a strong flow 255

6-60 Phase-averaged stress, transport, and sediment concentration for medium silt in
a strong, highly-concentrated flow .................. ...... 256

6-61 Suspended sediment phase diagram .................. ..... 257

6-62 Time evolution of ripple height, wavelength, and steepness during ripple growth
and equilibration (TL2v2) .................. ........... .. 258

6-63 Timestacks of bed elevation during ripple growth (TL2v2) . ..... 259

6-64 Cumulative averaged and corrected transport fluxes for TL2v2 . ... 259

6-65 Phase dependence of bedload, entrainment, and deposition during ripple growth
and equilibrium .................. ............... .. .. 260

6-66 Phase dependence of corrected bedload and suspended load fluxes during ripple
growth and equilibrium .................. ............. .. 261

6-67 Transport flux tendency phase diagrams for ripple growth and equilibrium 262

6-68 Time evolution of ripple characteristics during ripple decay (TL2v2d) . 263

6-69 Cumulative averaged and corrected transport fluxes for TL2v2d . ... 264

6-70 Phase-averaged and corrected transport fluxes during ripple decay . ... 264

6-71 Phase dependence of transport constituents during ripple decay for TL2v2d. 265

6-72 Transport flux phase diagram for rapid ripple decay ............... ..266

6-73 Time evolution of ripple characteristics through bedload transport only . 267

6-74 Phase dependence of bedload transport during ripple growth for BL2v2 . 268

6-75 Time evolution of ripple characteristics through suspended load transport only 269

6-76 Phase dependence of vertical sediment fluxes during ripple decay for SL2v2 270









6-77 Time evolution of scaled ripple heights for Phase III simulations . ... 271

B-1 Derivation of the flux tendency phase diagram of bedform growth, equilibrium,
and decay . . . . . . . .. .. . . 288









LIST OF SYMBOLS

Greek symbols

a longitudinal bedslope.

/3 transverse bedslope.

X wave period parameter.

ft time step.

atf morphology hybrid filtering time step.

6xi x grid spacing.

6yj y grid spacing.

6Zk z grid spacing.

6 boundary lv. r thickness.

A filter width.

6d displacement boundary l1.-r thickness.

Ad relative orbital excursion.

6ij Kronecker delta.

rl free surface displacement.

rlr ripple height.

7Y smoothing coefficient.

K von Kdrmdn constant.

Amix mixed l-.- r thickness.

,r ripple wavelength.

p dynamic viscosity.

v kinematic viscosity.

Vt eddy viscosity.

w angular frequency.

Sediment angle of repose.

4b nondimensional bedload discharge.
















P
p

Ps

Pmix

-as



J-A

(r7







T,

iri
0




,Ow

02.5

Ocr







General



((90))


nondimensional entrainment rate.

velocity potential.

mobility number.

density of water.

density of sediment.

density of mixed IlV 1 r.

Schmidt number.

ripple height standard deviation.

ripple length standard deviation.

ripple steepness.

bed shear stress.

gravitational particle shear stress.

maximum bed shear stress.

residual-stress tensor.

anisotropic residual-stress tensor.

nondimensional (Shields) stress.

variable finite-difference coefficient.

nondimensional current (Shields) stress.

nondimensional wave (Shields) stress.

grain roughness Shields parameter.

critical Shields parameter.

critical Shields parameter for a horizontal bed.

wave-current angle.

primitive variable at time level n.

operators

filtered variable.

phase-averaged quantity.









(c) time-averaged quantity.

P spatially-averaged quantity.

cp! residual variable.

Roman symbols

fs Smagorinsky lengthscale.

Zb low-pass filtered bed elevation.

VQb sign-corrected bedload transport flux.

VQb sign-corrected suspended load transport flux.

C filtered volume concentration.

Cb filtered volume concentration above bed.

Sij filtered rate of strain.

S characteristic filtered rate of strain.

sR grain Reynolds number.

Rp particle Reynolds number.

R, wave orbital Reynolds number.

Zb hybrid filtered bed elevation.

k wavenumber vector.

Q+ right-biased sediment transport flux.

Q- left-biased sediment transport flux.

A orbital semi-excursion length.

A+ van Driest constant.

C volume concentration.

Cf coefficient of friction.

Cp bedform propagation phase speed.

Cs Smagorinsky coefficient.

Cf Courant-Freidrichs-Levy number.

C,ix mixed I,-r concentration.









d grain diameter.

d, dimensionless grain size.

d5o median grain diameter.

E(t) entrainment rate of bed material.

f coefficient of friction.

F(, ) Flux evaluated at time level n.

fb coefficient of resisting friction.

F, downslope component of particle weight.

fw wave friction factor.

F. buovi-nt weight of particle.

F, body force.

f2.5 grain roughness friction factor.

fmix coefficient of applied friction.

g gravitational acceleration.

h water depth.

H wave height.

k turbulent kinetic energy.

k, residual kinetic energy.

k, Nikuradse roughness length.

k, wave number.

kx wavenumber in x-direction.

ky wavenumber in y-direction.

n normal direction.

np porosity of bed material.

p pressure.

Prt turbulent Prandtl number.

Qb volumetric bedload transport rate.









Qs volumetric suspended load transport rate.

Qt total volumetric sediment transport rate.

s specific gravity of sediment.

S, nondimensional sediment parameter.

T wave period.

t time.

TKE turbulent kinetic energy.

U, friction velocity.

Us fluid velocity at edge of boundary l1--_r.

Uoo freestream velocity.

ub nearbed fluid velocity.

Uc current velocity.

ui velocity component.

Umax maximum velocity amplitude.

Umix mixed lv.--r velocity.

Us sediment velocity.

Ws hindered settling velocity.

W, constant settling velocity.

xi spatial direction component.

z+ normalized wall unit.

Zb vertical location of bed.

Zo reference height.









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

SMALL-SCALE SEDIMENT TRANSPORT PROCESSES AND BEDFORM DYNAMICS

By

Bret Maxwell Webb

M1 v 2008

C('! Ii: Donald N. Slinn
Major Department: Coastal and Oceanographic Engineering

The generation of small-scale sedimentary structures in the coastal environment

is a complex process that occurs over a wide separation of scales in both time and

space. These bedforms are ubiquitous features of the nearshore region, and yet specific

information regarding their behavior and characteristics is still lacking. Specifically, it

is unclear whether the bedload-dominated processes of the linear regime are as equally

responsible for the generation of bedforms in the nonlinear regime, where flow separation,

and subsequent vortex formation, tend to govern the dynamics of the bottom boundary

l~- -r. While a simple one-dimensional model is derived and used to explain incipient

bedform growth in the linear regime, such an approach is not well-suited at addressing

the complexities of the wave bottom boundary l1.--r. Utilizing a new three-dimensional

phase-resolving live-bed model, we simulate the dynamics of bedforms, such as sand

ripples, in the nonlinear regime. Through forty-three independent simulations, the

model has been found to reproduce oscillatory boundary l -r flow, as well as provide

accurate predictions of ripple geometry in both lab- and field-scale flows. Model results

confirm that in the linear regime, bedform growth is promoted purely through bedload

sediment transport, but inertial properties of the sediment are equally as important. In

the nonlinear regime, bedform growth is also dominated by bedload transport; however,

the entrainment and deposition of bed material p1l i an important role in maintaining

ripple equilibrium, whereas it is mostly responsible for ripple decay.









CHAPTER 1
INTRODUCTION

1.1 Background

Bedforms, such as sand ripples, are ubiquitous features of the coastal zone. And

although they have been the subject of numerous investigations-theoretical, experimental,

and numerical alike-dating back to the late nineteenth century (Hunt, 1882; Darwin,

1883), definitive information about their dynamics in the coastal environment remains

limited. This is not to ,i- that theory and understanding have not been markedly

advanced over the last century, but it underpins the necessity for continued research on

bedform dynamics at scales both large and small.

Morphological bedform features occurring in nature may range in scales from a few

centimeters in height and tens of centimeters in length for wave-generated sand ripples,

to larger sandwave features like dunes and mega-ripples that can have heights of a few

meters and lengths on the order of tens of meters. Regardless of their size, bedforms plih

an important role in both the energy and sediment budgets of the nearshore environment.

Bedforms have been found to strongly affect:

energy dissipation in the wave bottom boundary 1lV- r;

sediment transport characteristics; and

acoustic properties of the seafloor.

While momentum transfer above plane beds is due mostly to turbulent diffusion,

rippled beds induce flow separation resulting in organized vortices, or coherent motions,

that dominate momentum transfer in the wave bottom boundary 1l---r (WBBL) (\1 !,1 i

2004). These coherent motions are effective in dissipating wave energy in a nearbed

boundary 1i,-
2006). Above this nearbed 1l,- -r, coherent motions break down and are replaced by

random turbulence (Davies and Villaret, 1997). Tunstall (1973) and Tunstall and Inman









(1975) estimate that as much as 10' of wave energy may be dissipated through the

process of vortex formation.

Wave energy is also effectively dissipated through friction as bedforms affect the

hydraulic roughness of the bed for waves and currents (Soulsby and Whitehouse, 2006).

Bedform height controls the effective bottom roughness (Faraci and Foti, 2002), which is

of practical engineering importance. Parametrization of hydraulic roughness (Grant and

Madsen, 1982) are commonly applied to engineering wave and sediment transport models

in order to account for the additional energy dissipation that takes place in the WBBL.

Such practice permits the effects of bedforms on currents and waves to be treated in a

bulk manner, rather than having to account for them explicitly in the modeling approach.

The parametrization and application of roughness to models accounts for the thickening

of the WBBL in the presence of bedforms (Wiberg and Harris, 1994), whereby energy is

dissipated through form di I. skin friction, and turbulence damping due to an intense

1],-.-r of nearbed sediment transport (Grant and Madsen, 1982). The amount of dissipation

due to bedforms is not trivial; field observations by Ardhuin et al. (2002) indicate that

the form drag over large ripples is effective in wave attenuation across the continental

shelf. Accurate representations of equivalent bottom roughness, then, are of particular

importance for understanding the energy budget of the nearshore environment. However,

prior characterizations of roughness and their subsequent implementation in nearshore

models often neglect changes in bedform size, shape, and orientation, and do not account

for phase lag effects induced by flow separation in the boundary 1l,-.r.

Bedforms are manifestations of differential sediment transport near the seabed, and

can often be found in unique patterns along the seafloor as seen in Figures 1 -(a)-1-l(d).

Once large enough to induce flow separation, they have a profound impact on subsequent

sediment transport. Enhanced sediment suspension over rippled beds not only increases

potential sediment transport in the bottom boundary 1I,-.r, but affects the background

flow as well. Stratification of sediment in the water column results in a modification of the









velocity profile (Coleman, 1981), turbulence damping (\McLean, 1991), and an increase in

boundary roughness (Parker and Coleman, 1985). Therefore, sediment suspension over

rippled beds serves as an additional mechanism for the dissipation of energy in the bottom

boundary lv-r.

According to Nakato et al. (1977), suspension processes over rippled beds are

dominated by the formation, ejection, and motion of eddies. Organized vortices that

form in the lee-side of ripples entrain sediment from troughs or valleys in each successive

half-cycle of wave forcing. Immediately following flow reversal, the sediment-laden vortices

are ejected into the main flow above the bedforms where the sediment is subsequently

advected by the local fluid velocity field. Laboratory experiments by (van der Werf

et al., 2005) -,t-'-. -1 that such behavior results in three distinct peaks in the nearbed

concentration as a function of wave phase. Compared to suspension events over a flat

bed, field data -ti-'-. -1 that sediment-laden vortices in the wake of bedforms alters

sediment suspension (Gallagher et al., 1998), and may also enhance the phase lag between

suspension and transport (Inman and Bowen, 1963). Furthermore, van der Werf and

Ribberink (2004) propose that ripple-induced phase differences between peak suspended

sediment concentrations and peak fluid velocities result in net sediment transport rates

directed offshore under surface gravity waves. Such phase differences, however, are not

accounted for in common phase-averaging, coupled hydrodynamic and sediment transport

models. This may -it--.- -1 why some cross-shore sediment transport models fail when

oscillatory currents are larger than steady currents (Gallagher et al., 1998).

In recent years, the Navy has become increasingly interested in the acoustic

properties of the seafloor, specifically with respect to the ability of sonar devices to

detect both emergent and submerged munitions in the battle space environment (BSE).

Bedforms affect the acoustic response for sonar applications, either enhancing or inhibiting

the penetration of the sonar signal into the bed (Soulsby and Whitehouse, 2006). From

data collected during the SAX99 field experiment, Piper et al. (2002) show that both the









ripple height and wavelength strongly affect the level of sonar penetration for subcritical

grazing angles. Additionally, the orientation of the ripples relative to the incident sonar

field is of importance. The results of Piper et al. (2002) indicate that subsurface imaging

is enhanced when the sonar propagation direction is perpendicular to the mean direction

of ripple crests, whereby the amount of acoustic energy scattered by the bedforms into the

sediment is increased.

1.2 Motivation

In the linear regime of bedform growth-where rolling grain ripples persist-sediment

transport takes place purely through bedload processes. The bedload transport mode

consists of particles rolling, sliding, and saltating in small leaps on the order of a few grain

diameters above the bed. The linear regime is dominated by friction and inertial lags

between fluid forcing and particle transport. The absence of suspended sediment transport

in the linear regime-and at subcritical threshold values-have lead many to postulate

that bedforms are generated purely through bedload transport. It is unclear, however,

that this statement holds true in the nonlinear regime where coherent motions in the

boundary 1i-. r promote suspended load transport and induce phase differences between

fluid forcing and sediment entrainment.

The nonlinear regime contains both bedload and suspended load transport modes,

but their relative contributions to ripple growth, equilibration, and decay are unknown.

Additionally, the contributions of bedload and suspended load transport modes may vary

as a function of ripple position (profile and planform) and fluid forcing. The quantitative

roles of constructive (fluid forcing) and destructive (gravitational) forces in generating

sedimentary structures are not immediately evident. It would be beneficial, therefore, to

investigate the contributions of various transport modes to bedform dynamics, and to

better understand the roles of constructive and destructive forces during ripple, growth,

equilibration, and decay.









Field and laboratory measurement techniques are incapable of gathering in situ

data that clearly distinguish one transport mode from another. Such methods often

require the assumption of a threshold based on volumetric concentration to separate the

transport modes after the data have been collected. Additionally, measurement techniques

for bedload and suspended load transport are commonly invasive, thereby altering

the hydrodynamic and sediment transport fields that they are attempting to quantify.

Numerical simulations may provide useful insight into sediment transport processes in

small-scale bedform dynamics where physical experimentation is currently not possible.

Through the University of Florida, we are participating in the Office of N i,.

Research's (ONR) intensive study of Sand Ripples on the Inner Shelf (FY04-08), a

project involving fifteen principal investigators from across the nation (Ripples DRI).

This interdisciplinary project combines field observations with laboratory experiments and

numerical simulations from the biological, geological, and oceanographic sciences. The

primary goals of the project, as identified by ONR, are to

1. measure and model morphology;

2. investigate morphological response to forcing perturbations;

3. measure and model rates of bio-degradation;

4. measure and model the effects of grain-size distribution; and

5. understand the role in generating sedimentary structures.

Our participation in the project primarily involves the first two goals outlined above,

as well as the last. In order to address these objectives, we seek to develop a physics-based

model that couples hydrodynamics and morphology by updating the bed level at every,

or nearly every, hydrodynamic time-step. The review of noncohesive sediment transport

processes and bedform dynamics that follows in C'! lpters 2 and 3 should provide adequate

guidance for determining a suitable model framework.









Aside from the goals of the ONR Ripples DRI, the primary objective of this research

is to investigate sediment transport characteristics during ripple growth, equilibration, and

decay. In particular, we hope to provide answers to the following questions:

1. What are the relative contributions of bedload and suspended load to the generation

or obliteration of sedimentary structures?

2. Are there specific locations in the ripple profile/planform where one transport mode

dominates over the other?

3. Are there specific times-as a function of phase-when one mode dominates over the

other?

4. What are the dominant roles of: bedload, suspended load, and gravitational forces?

To address our objectives, we propose the development of an entirely new modeling

system capable of simulating phase-resolving small-scale sediment transport and

morphology. Capabilities of the modeling system will be assessed by evaluating the

hydrodynamics through model-data comparisons, and by also performing simulations

of bedform equilibration under a variety of scenarios (e.g. growth, equilibration, decay,

coarsening, bifurcation, steady flows, short- and long-period flows, highly-concentrated

flows, and extreme sediment sizes). In order to answer the questions posed above,

simulations of bedform growth, equilibration, and decay will be performed with

bedload and suspended load;

bedload only; and

suspended load only.

The design of the modeling system assumes that bedload transport is predicted using

Einstein's dimensionless bedload flux and common power-law formulations, and that

suspended load originates through an entrainment of sediment from the bed using

semi-empirical pick-up functions available in existing literature. Thus, our system allows

us to alternately turn bedload and suspended load on or off by simply setting bedload

transport rates, or entrainment/deposition, to zero, respectively. We further assume









that the bedload regime is comprised of two types of transport: one due to fluid forcing;

and another due to gravitational forces. Doing so permits us to determine the roles

of constructive and destructive forces independently, thereby allowing us to better

understand their roles in generating sedimentary structures. The development of this

modeling system is a secondary objective of this dissertation, and is necessary to address

our scientific questions posed above. Specific details about the system are outlined below,

as well as in the following chapters.

1.3 Approach

Modeling sediment transport continues to be a challenge for fluid dynamicists.

Indeed, much of the uncertainty in fluid-sediment models continues to lie mainly with the

particulate phase, whereas the fluid hydrodynamics are comparatively well understood.

Sediment transport models fall mainly into two broad categories:

1. time- (or phase-) averaged; and

2. unsteady (or phase-resolving).

Variations on phase-averaged and phase-resolving models include quasi-steady and

semi-unsteady models, respectively. Each of these models has its own benefits and

deficiencies, making some more useful in certain situations than others. Briefly, time- or

phase-averaged sediment transport models effectively integrate out intra-wave processes by

considering wave-averaged values of fluid velocity, sediment concentration, and sediment

transport (Bijker, 1971). So called quasi-steady models assume that the transport rate is

proportional to the instantaneous nearbed flow field raised to some power (Bailard, 1981),

but only provide an average transport rate over the wave period. Semi-unsteady models

incorporate additional complexity into the formulation of transport rates by considering

phase lags either through parametrizations (Nielsen, 1988; Dohmen-Janssen, 1999) or by

breaking a full wave period into wave half-cycles (Di i.iiiii and Watanabe, 1992). While

computationally intensive, the most robust approach is to describe the sediment transport

and hydrodynamics as coupled, time-dependent processes. Very few of these models









exist outside of the computational research arena, as their practicality for engineering

applications is still somewhat limited due to their complexity [see Drake and Calantoni

(2001); Gessler et al. (1999); and Lesser et al. (2004)].

To address a number of relevant engineering problems involving sediment transport

processes-from structure-induced scour to coastal erosion-it would be advantageous to

have a flexible, robust model capable of resolving time-dependent bed morphology under

various forcing conditions. Such a modeling system could also satisfy some of the primary

objectives of the ONR Ripples DRI project outlined above. Here we seek to develop and

evaluate the skill of a coupled fluid-sediment numerical model that resolves instantaneous

hydrodynamics, sediment transport, and time-dependent bed morphology. It is not evident

that such a model exists at the level of detail proposed.

The desired phase-resolving model would be capable of both two- and three-dimensional

simulations of (1) steady flow, (2) oscillatory flow, (3) combined oscillatory and steady

flow, and (4) surface gravity waves. While the time integration of the governing equations

will be limited to durations of 0(100 s), resolution of physical lengths will range from

0(10-4 m) to 0(1 m). Such detailed treatment of hydrodynamics near solid boundaries

should permit the resolution of high-intensity turbulent fluctuations that initiate sediment

transport. Integration of the bed level through time will be governed by the sediment

continuity (Exner) equation, where gradients in the bedload flux and the competition

between entrainment and deposition of suspended solids provide an estimation of the

instantaneous bed level. Sediment transport in the model will be estimated using

bulk, empirical formulations available in existing literature. Natural feedback between

morphology and the hydrodynamic flow field represents the one-way fluid-sediment

coupling in the model. Two-way coupling of the hydrodynamics and local suspended

sediment concentration may be considered in the future.









1.4 Outline

What follows is a general overview of knowledge pertaining to noncohesive sediment

transport and bedform dynamics, cast in a manner that, hopefully, underscores the need

for a hydrodynamic live-bed model having the general characteristics outlined previously.

A brief summary of noncohesive sediment transport processes is given in ('! Ilpter 2, with

a discussion on the governing hydrodynamics or forcing, modes of transport, the role

of turbulence, and a description of accepted models. C'!i lpter 3 provides information

about general bedform properties and dynamics, with particular attention given to

their classification and characteristics, mechanisms for growth, a summary of relevant

laboratory and field experiments, semi-empirical solutions for their geometrical properties,

and an overview of models ranging from simple to complex. A description of newly

developed models is provided in ('!i lpter 4 and outlines governing equations, as well as

methodologies for their implementation in the modeling system. The model experiments

are outlined in ('! Ilpter 5, and those simulation results are presented and discussed in

('!i lpter 6. Concluding statements are provided in Chapter 7, along with some words on

potential future applications of the newly-created modeling systems.
































(b) USA Wave Basin


(c) Yucatan Peninsula, Mexico


Figure 1 1.


(d) Yucatan Peninsula, Mexico


Pictures of sand ripples in lab [(a) and (b)] and field [(c) and (d)] settings.
The lab pictures were taken in the University of South Alabama Wave Basin
after draining, while the field pictures were taken in a depth of approximately
5 m of water off the East coast of the Yucatan Peninsula in Mexico.


(a) USA Wave Basin









CHAPTER 2
SEDIMENT TRANSPORT PROCESSES

2.1 Introduction

Sediment transport processes in the coastal environment are highly complex.

Transportation is both initiated and sustained by hydrodynamic forcing on the seabed and

water column. While the knowledge of sediment transport processes has been significantly

advanced in idealized, laboratory environments, detailed information about such processes

remains ambiguous under the stochastic forcing that is present in nature.

The following sections outline the governing hydrodynamics that drive sediment

transport in the coastal environment, provide a description of common transport modes

and regimes, discuss the mutual relationships between transport and turbulence, and

deliver an overview of various sediment transport models.

2.2 Governing Hydrodynamics

2.2.1 Waves

Surface gravity waves represent the most dominant mobilizing force for sediment in

coastal areas (van der Werf, 2004). In some cases the orbital velocities under these waves

can be quite large near the seabed, especially when waves steepen as they enter shallower

water. Furthermore, realistic surface gravity waves are not linear, typically having steeper

peaks and flatter troughs. A result of this nonlinear profile is an inherent .i-vmmetry in

the orbital velocity field with short-duration, high-intensity velocities directed shoreward

under wave crests and longer-duration, lower-intensity velocities directed seaward under

the broad, flat troughs (see Figure 2-la). This inequality in the orbital velocity time

series becomes very important for the determination of net sediment transport rates [see

Ribberink and Al-Salem (1990); Gallagher et al. (1998); and Elgar et al. (2001)].

As discussed earlier, the wave bottom boundary li-c-r (WBBL) is an important

mechanism for energy dissipation and influences the magnitude and direction of sediment

transport in the nearshore. This boundary lI--r develops near the seabed in response









to friction between it and the fluid. A general expression for boundary li-- r thickness is

given by

6 oc xVIT (2-1)

where v is the kinematic viscosity and T is the wave period. The thickness of the

boundary l~-.-r is also affected by the roughness of the boundary. Wave boundary

l.. --ir over smooth boundaries may be of 0(1 mm) while the thickness over rough

boundaries, like a real seabed, increases to 0(1 cm). Due to inertial effects in the

boundary 1l-v-r (slower moving fluid), the nearbed oscillatory flow is often out of phase

with the freestream forcing. This results in a nearbed phase-lead of approximately 450 for

laminar flow and 15 for turbulent boundary lV. -i with respect to the freestream forcing.

The phase lead of nearbed oscillatory flow has profound significance for the

estimation of sediment transport. Common sediment transport formulations incorporate

some threshold for incipient motion, such as Shields' parameter (Shields, 1936), that

requires the bed shear stress to exceed some critical value. A general expression of the bed

shear stress is

Tb =pCf U U (2-2)

where p is the fluid density, Cf is a coefficient of friction (drag), and U is representative of

the freestream forcing. Figure 2-lb shows a representation of bed shear stress under linear

and nonlinear waves using Equation 2-2 with p = 1025 kg/m3 and the canonical value of

0.0025 for Cf (Rivikre et al., 2004). This type of formulation for bed shear stress is typical

of very general models and does not include any information about phase differences

between flow at the bed and flow outside of the boundary lI- -r. It is evident, then, that

application of Equation 2-2 for unsteady flow in the boundary l-1i--r will incorrectly

predict the phase of sediment mobilization with respect to the applied forcing. Under

purely horizontal, regular oscillatory flow this is of less significance as the net sediment

transport from cycle-to-cycle will be zero. The phase difference for mobilization and

entrainment under nonlinear surface gravity waves, however, is significant as it can dictate









the direction of net transport [see Elgar et al. (2001) and van der Werf and Ribberink

(2004)].

An alternative to the general bed shear stress formula given by Equation 2-2,

commonly employ, l in phase-averaged or quasi-steady transport models, is to consider

the maximum bed shear stress for an individual wave based on a wave friction factor (f,).

Such a method was proposed by Jonsson (1966):


Tb,max = Pf.., (2-3)

where Ub is the nearbed orbital velocity. The wave friction factor of Swart (1974) is

applicable for fully-developed, rough turbulent flow and is given by


f, exp 5.213( 04) -5.997 (2-4)


where k, is the Nikuradse (1933) roughness lengthscale and A is the orbital semi-excursion

length (A = UbU). Jonsson's (1966) formulation of the maximum bed shear stress is more

complete than the formulation shown in Equation 2-2, as it accounts for the roughness of

the bottom and uses the nearbed orbital velocity as opposed to the freestream forcing, but

represents a bulk approximation of the stress during a wave period instead of treating it as

a time-dependent quantity.

A large portion of the total bed stress is due to wave pressure variations over the

seabed. According to McLean (1991), this part of the stress is not effective in mobilizing

sediment because the length-scale of the pressure variation is so much larger than the

particle diameter. Foda (2003) claims, however, that outside of the surf zone the wave

pressure pl .1 i an important role in sediment transport processes: low-energy waves

(mild pressure gradients) tend to drive sediment offshore, while high-energy waves (steep

pressure gradients) result in onshore transport. These two conflicting theories may

be harmonized by considering the context under which the statements were made. In

an attempt to determine the importance of wave pressure on bedload transport, Foda









(2003) treats the seabed as a viscoelastic fluid, not as individual particles; therefore, the

length-scales of the pressure variation and an "effectiu, length of the active viscoelastic

fluid may not be greatly different.

N. i-!1i.re waves, due in large part to their .i-viiili. l iry, produce secondary flows

outside of the WBBL that plan an important role in suspended sediment transport.

Considering the velocity time series for a nonlinear (Stokes) wave in Figure 2-la, one

can see that velocity under the wave crest is larger than that under the trough. This

inequality results in a net displacement of a parcel of fluid, or suspended sediment,

in the direction of wave propagation, often referred to as Stokes drift (van der Werf,

2004). In the nearshore region, this net displacement of shoreward-directed fluid is

balanced, over long periods of time, by a seaward-directed return flow near the bed often

called undertow. Elgar et al. (2001) --.-. -1 that offshore sandbar migration is linked to

cross-shore gradients of undertow, as these nearbed currents are often strong enough to

mobilize sediment. Furthermore, Gallagher et al. (1998) believe that the intensification

of undertow near a sandbar leads to cross-shore gradients in the suspended sediment flux

that further promote offshore bar migration.

In addition to the cross-shore directed secondary flows, surface gravity waves also

induce boundary l1-.-r, or steady, streaming. Vertical velocities generated in the WBBL

result in a diffusive vertical flux of momentum directed away from the horizontal boundary

liv.-r. Although weak compared to oscillatory flow outside of the boundary lV.- r, steady

streaming results in a non-zero time-averaged bed shear stress and has a significant effect

on suspended sediment transport (\ 1,iii, 2004). The steaming-induced flux away from the

boundary 1-.-r represents an additional mechanism for the entrainment of sediment into

the outer flow, where it can then be advected by the local flow field.

2.2.2 Currents

Steady currents, such as the flow found in rivers and hydraulic channels, are effective

in transporting sediment, once mobilized, through advection. Relative to the wave









boundary 1I-. r, the current boundary 1i-.r is often one or two orders of magnitude larger

as it develops over comparatively long durations of time. A number of scientists have

considered sediment transport in currents: A t. -l- r-Peter and Miiller (1948), Eintstein

(1950), Smith and McLean (1977), Bagnold (1980), and van Rijn (1984). Much of the

research indicates that the transporting capability of a steady flow is proportional to the

stream power, or velocity, raised to some power [Bagnold (1980); van Rijn (1993)] such

that

q = au

where q is the volumetric transport, and a and b are empirically determined constants

(Long et al., 2007).

2.2.3 Waves and Currents

The nearshore environment is often characterized as stochastic, having multiple

realizations of wave frequency, direction, and height, as well as current magnitude and

direction. Often times nearshore waves travel over shearing currents-a common example

being a shoaling wave propagating over an alongshore current. The combined effects

of wave- and current-induced velocities, however, cannot be found by a simple linear

superposition of the constituents (van der Werf, 2004). Rather, the presence of a current

modifies the wave-induced flow field in a nonlinear manner by alternatively adding and

subtracting from the orbital velocities during different phases of the wave. The wave

celerity and length are also affected by the current, where celerity increases (decreases)

as wavelength decreases (increases) in response to an opposing (following) current. This

effect is demonstrated for collinear waves and currents by considering the modified

dispersion relationship (Dean and Dalrymple, 1991):


L kw Uk, + gk tanh kh (2-5)


where w is the angular wave frequency, Uc is the current velocity, k" is the wave number

(k, = 27/L), g is gravitational acceleration, and h is water depth. It is also known that









the wave-induced orbital velocities reduce the nearbed current magnitude (van der Werf,

2004).

The nonlinear interactions between the current- and wave-induced boundary

1-v, -i s have a significant effect on sediment transport. The stirring effect of waves

coupled with a steady current has been found to increase the total sediment transport

significantly (Camenen and Larson, 2005). Wave orbital velocity amplitude (A), current

velocity (U,), and the angle of incidence between the wave and current (c,,) dictate the

resultant behavior of nearbed fluid motion. While a current flowing perpendicular to the

direction of wave propagation does not affect wave celerity or length, it does alter the

net sediment transport. In their laboratory experiments, Lacy et al. (2006) find that for

,, = 900 the maximum transport is symmetric about the current direction resulting in

wave-phase-averaged sediment transport in the downstream direction. As the wave-current

angle decreases from 0,, = 900, the maximum instantaneous bed shear stress increases

(similarly for O8, increasing from 900 to 1800). Lacy et al. (2006) found that the effect

of the current was to increase the nearbed velocity amplitude, thereby increasing the

relative importance of suspended sediment to bedload transport. Increased entrainment of

mobilized sediment into the current boundary 1v.,-r was also observed.

2.2.4 Tides and Tidal Currents

In general, tides do not have a substantial impact on sediment transport in the

nearshore region. Tidal-induced velocities are typically orders of magnitude smaller than

the instantaneous, orbital velocities produced under a surface gravity wave having a

frequency much higher than the tidal frequency. Therefore, tidal-induced velocities are

often not large enough to initiate motion at the seabed, nor do they p1 i, a significant

role in the advection of sediment to different locations. Of course there may be special

circumstances, such as a very shallow inlet or estuary, where tidal currents are sufficiently

strong to mobilize sediment in large quantities (i\ I!! r and Komar, 1980a).









Tidal currents generated by shoaling internal tidal-waves can increase the shear

stress sufficiently to develop nepheloid l1.1 r? or regions on high turbidity near the

seabed. These tidal currents are influential in suspending near-bottom sediments

in the absence of wind-generated waves and currents on the continental shelf and

slope (Cacchione et al., 1994). Once mobilized, the suspended sediment may then be

transported across the continental shelf by internal waves and tidal currents. Although

they found storm-generated waves to be the dominant forcing mechanism of transport

on the continental shelf, Puig et al. (2001) -i r--. -1. 1 near-inertial internal waves as a

mechanism for the maintenance of a nepheloid l1i.-r and suspended sediment during

milder wave climates.

2.3 Transport

Incipient sediment motion has been studied extensively in the laboratory since the

pioneering work of Shields (1936). The basic concept of incipient motion, offered by

Shields (1936), is that the sediment mobilizing forces of fluid lift and drag must exceed

the stabilizing force of gravity acting on the particle. The ratio of mobilizing to stabilizing

forces is referred to as the Shields parameter and is given by


0 b (26)
(s 1)G

where Tb is the bed shear stress, s is the specific gravity of the sediment relative to water,

g is gravity, and d is the particle diameter. The dimensionless Shields stress accounts for

skin friction, but not form drag which is the second component of total drag experienced

by the bed material. However, skin friction is the more effective component of drag on

mobilizing sediment (McLean, 1991).

2.3.1 Modes

Classical modes of sediment transport are divided into three categories:

1. wash load,

2. bedload, and









3. suspended load.

The wash load is characterized by very fine sediment particles, transported by

the fluid, that are not representative of the bed material (Fredsoe and Deigaard,

1992). Estimation of the wash load is difficult since it requires knowledge of sediment

characteristics from the point of origin... a location possibly far removed from the area of

interest. It is commonplace to discard wash load in the computation of the total sediment

load, which is then comprised of the bedload and suspended load material.

2.3.1.1 Bedload

Bedload is the part of the total sediment load that is in almost continuous contact

with the bed (Fredsoe and Deigaard, 1992). Under non-breaking waves, Dohmen-Janssen

and Hanes (2002) find that bedload transport accounts for nearly 9i '. of the total

sediment load. The bedload material tends to move along just above the static bed in

a thickness a few tens of grain diameters at most (Bagnold, 1980). The movement of

bedload particles is characterized by sliding, rolling, and/or saltation. Sliding occurs in

more loosely packed beds and for higher mobilizing stresses, where the particles slide

across one another in sheets. Bedload particles may also roll over top of their neighbors

if the moment of lift and drag forces are just large enough to counteract the moment of

stabilizing force of gravity acting on the particle (Luque and van Beek, 1976). Saltation

occurs when a particle is liberated from the bed material by an intense stress and, once

ejected into the flow, follows a more or less ballistic trajectory just a few grain diameters

above the bed (Bagnold, 1973).

Pioneering work in theoretical and empirical bedload transport was put forth by

Eintstein (1950). Einstein's (1950) empirical bedload function was the first of such

formulations to account for the randomness of flow and subsequent transport by equating

the number of particles deposited (eroded) per unit area to the number of particles in

motion (on the bed), and on the probability that the force balance on the particle is such

that it will be deposited (eroded). A central concept of the Einstein bedload function is









that bedload transport is proportional to the flow rate raised to some power. In contrast,

the experiments of Luque and van Beek (1976) show that the average length of individual

particle saltations is a constant, implying that the probability of deposition is independent

of the flow rate. Such a proportionality, however, has also been -ii--::. -1. by empirical

relationships "\!. I. r-Peter and Miiller (1948); Bagnold (1980); Ribberink and Al-Salem

(1990); Nielsen (1992); Ribberink (1998); Camenen and Larson (2005)], and by analytical

expressions derived from physical concepts as well [Bagnold (1966); Bowen (1980); Bailard

(1981); Kovacs and Parker (1994)].

Eintstein (1950) postulated that a functional relationship existed between the bedload

transport rate and the particle buov, i, v, such that a nondimensional bedload discharge

may defined as
Qb
b Qb (2-7)
V(s 1)gd3
where Qb is the volumetric rate of bedload transport per unit width, s is the specific

gravity of the particle, g is gravitational acceleration, and d is the particle diameter.

A number of physical laboratory experiments have been conducted to determine the

functional relationship between fluid forcing and resulting transport, represented in

Equation 2-7 by Kb. These functional relationships are termed I ..-- 1 1.--- since the

transport rate is proportional to some measure of the fluid forcing (velocity or stress)

raised to a power. A number of power laws, in various forms, have been -ir-i-- I. 1 for

transport by currents [e.g. A1t. r-Peter and Miller (1948); Nielsen (1992); Ribberink

(1998); Camenen and Larson (2005)] and by waves [e.g. Madsen and Grant (1976);

Bailard and Inman (1981); Dib l i and Watanabe (1992); Soulsby (1997); Ribberink

(1998); Camenen and Larson (2005)]. Most of these relationships are based on an

exceedance of the critical Shields stress and take the form:


b = a(O Oc)b (2-8)









where a is some constant of proportionality, b is an empirically-derived exponent providing

the best fit of Equation 2-8 to a set of data, and 0 and Oc, are the instantaneous and

critical Shields parameters, respectively. Various examples of bedload power laws are give

in Table 2-1.

Shields' description of incipient sediment motion is based on the principle that there

is absolutely no motion when the applied stress is below the critical threshold. Therefore,

when 0 < Ocr in Equation 2-8 the dimensionless bedload discharge 1b is actually equal to

zero. Camenen and Larson (2005) provide expressions for 4b that have the general form:


b =- Cod exp ( e ) (2-9)

where c, d, and e are empirically-derived constants. Equation 2-9 -,i--. -1- that transport

begins as soon as a stress is applied to the sediment (when 0 > 0), contradicting Shields'

concept of a critical threshold for sediment motion.

Initial treatment of bedload transport treated moving particles in a bulk fashion,

with the mobilizing stress being transmitted by the fluid to the bulk. By the definition

of bedload given above, however, it would seem that some stress is transmitted between

particles since they are in almost constant contact with one another. Bagnold (1954)

addressed this subject by considering the intergranular stresses transmitted between

particles under fluid shear and found a substantial radial dispersive pressure exerted

between the particles. This dispersive pressure is distributed in such a manner that

the moving grains are in equilibrium with their gravitational force. As shear stress is

applied to the particle matrix, dilation occurs and the dispersive pressure decreases as the

particles move further from each other. The total shear stress within the dispersed particle

matrix is then a combination of intergranular fluid and particle stresses such that


7 = f + 7 (2-10)









where Tf is the stress transmitted by the fluid within the pores, and 7T is the tangential

dispersive stress that represents momentum exchanged due to particle-particle interactions

(Fredsoe and Deigaard, 1992). The particle collisions that contribute to 7- subtract

momentum from moving sediment particles that must be replaced by the fluid forcing.

This momentum subtraction results in an apparent roughness greater than would be

expected for a static bed (McLean, 1991).

Through a series of laboratory experiments of particle transport in a closed

rectangular flow channel, Luque and van Beek (1976) found that bedload reduces the

maximum fluid shear stress at the bed level by exerting an average reaction force on the

surrounding fluid. According to Equation 2-10, a reduction in fluid stress -ii.;. -I that

the intergranular stresses must increase to maintain the total stress. At high bedload

transport rates, then, a in ii y of the total stress is exerted by particle collisions

while the fluid stresses remain small (Luque and van Beek, 1976). A schematic of the

distribution of fluid and granular stresses for an idealized open channel flow is shown in

Figure 2-2.

2.3.1.2 Suspended Load

Besides bedload, the other constituent of the total sediment load is the suspended

load. Suspended load may be defined as material advected by the fluid, maintained in

suspension by turbulence, and having very little contact with the bed. In this respect

the suspended load is transported by mechanisms similar to those responsible for the

transport of wash load, but its origins differ. While wash load consists of particles

not characteristic of bed material, suspended load sediment is entrained directly from

the bed material. Hence, some knowledge of the bed material may be utilized in the

characterization of suspended sediment properties.

Suspended load p1l 'i-, an important role in both sediment transport and hydrodynamic

processes. Once entrained from the bed material, these sediments can be advected

locally by nearbed wave orbital motion, and regionally by persistent cross-shore and









alongshore currents. As most natural sediment is multi-modal-having a naturally

occurring distribution of grain sizes-the entrainment of smaller particles near the surface

can leave behind a l-1v.r of larger particles. Bagnold (1980) s-l-::. -1 that the presence

of a larger grain size reduces the transport rate through a phenomenon called armoring.

Total load transport rates may also be reduced through the effects of the suspended

load on the nearbed flow field. McLean (1991) selI--: -1. I that density stratification by

suspended sediment damps turbulence, thereby limiting the ability of flow to transport

mass and momentum vertically. As a result, the upward dispersion of suspended sediment

maintained, against gravity, by random eddy currents is reduced (Bagnold, 1973), the

ability of the flow to keep sediment in suspension is impaired, and the entrainment

capacity of the flow decreases (M\cLean, 1991). Suspension and density stratification may

also modify the velocity profile, which indirectly alters the boundary roughness through

a subsequent change in bed morphology [Coleman (1981); Parker and Coleman (1985)].

Although suspended sediment may suppress turbulence, Nakato et al. (1977) found that

both the mean and fluctuating vertical fluid velocity were amplified by suspension.

Through a series of laboratory experiments, Coleman (1981) demonstrated the ability

of suspended sediment to reduce turbulence, as well as modify the shape of the velocity

profile. The distribution of velocity throughout the water column is of significance for

net sand transport, as the suspended particles are advected by the fluid velocity. This is

especially true for transport over a rippled bed where coherent motions, like organized

vortices, dictate entrainment, suspension, and advection of significant quantities of

sediment (van der Werf et al., 2006). Luque and van Beek (1976) found that the average

transport velocity of suspended particles just above the bed was approximately equal

to the turbulent fluid velocity minus a constant, which is speculated to be an inertial

effect. Knowledge of the velocity profile, then, could provide some indication of suspended

load concentrations. Such information is especially useful in the nearbed region, which

p1 i,-, an important role in the determination of net sand transport owing to the high









concentrations of sediment found there (van der Werf et al., 2006). A common approach

for estimating the nearbed velocity distribution, within the boundary l1- -r, utilizes some

functional form of the classical Prandtl-Kdrmin logarithmic law of the wall (von Kirmnn,

1930), or log law, given by

ln- + B (2-11)
Ut K Z0

where u is fluid velocity, u, is friction velocity, K is the von Krrmnn constant, z is the

vertical distance from the boundary (bed), and zo is a reference height. While various

estimates of K and B appear in literature pertaining to boundary 1-f-r flows, Pope (2000)

reports that generally all are within five percent of


S= 0.41, B 5.2. (2-12)


In the determination of velocity profiles containing suspended sediment, some have

-ii_, t-, -,.1 that K decreases with increasing suspended sediment concentration [Raudkivi

(1967); Graf (1971); Yalin (1977)]. A reduction of K in Equation 2-11 -"t-'.-, -1 that the

ratio of the fluid velocity to friction velocity increases. The experiments of Coleman

(1981), however, demonstrate that K is independent of sediment concentration. Values for

the reference height in Equation 2-11 (zo) have commonly been related to the sediment

grain diameter (Grant and Madsen, 1982). An estimate of Zo for intense sediment

transport over flat beds in oscillatory flows, determined from the lab data of Carstens

et al. (1969), is approximately 7 8 grain diameters. This concept of an equivalent sand

grain roughness height (Ks) evolved from the work of Nikuradse (1933).

Entrainment of bed particles into the flow may be computed in a manner similar

to that used for bedload discharge (Equation 2-7). Following Eintstein (1950), a

dimensionless entrainment parameter %p is given by

E
S- (2-13)
p S- l),.J,,









where E is the volumetric rate of entrainment of bed material, ps is the sediment density,

s is the sediment specific gravity, g is gravitational acceleration, and d50 is a median grain

diameter. Typically, one uses an available expression for 4~ and then computes the rate of

entrainment E, or pick-up rate. This methodology has been used for modeling suspended

load transport in particle trajectory models (Hansen et al., 1991) as well as in more

sophisticated hydrodynamic models Zedler and Street (2001). A number of relationships

for 4~ have been empirically derived from laboratory experiments, and a few common

relationships are given in Table 2-2.

2.3.2 Regimes

Sediment transport is often divided into the following three regimes (van der Werf,

2004):

1. bedload,

2. ripple, and

3. sheetflow.

Clear definitions of these regimes in the literature, and the constitutive relationships

that make them unique, are tenuous at best. The bedload regime is characterized by

bedload transport of sand particles in a l- -r not more than a few grain diameters thick

(Bagnold, 1980), and occurring over an otherwise horizontal bed. According to Bagnold

(1946) this transport persists from the onset of motion to a value about twice the critical

Shields parameter (Ocr < 0 < 20cr). For reference, a practical range of Ocr for the coastal

environment varies between 0.03 and 0.06 (van der Werf, 2004).

Although the distinction was not made at the time, early experiments on bedform

development by Darwin (1883) and Ayrton (1904) provided qualitative information on the

characteristics of incipient motion and transport in the bedload regime. Darwin (1883)

-~i--. -1. .1 that particles transported in the bedload regime, by oscillatory flow, would

., i-'regate in regions on increased friction. Similarly, the oscillating tank experiments of

Ayrton (1904) demonstrated that particles transported on a flat bed would congregate









in a specific location, that being the location of a standing wave node where the vertical

(horizontal) velocity is at a maximum (minimum). The oscillating tray experiments

of Bagnold (1946) further reinforce the bedload regime concept, where sand particles

transported in a thin, concentrated 1. -r over a flat bed would .i:- -regate in areas of higher

friction, and lower tangential forcing.

Once a sufficient number of particles have gathered to induce flow separation, the

transport behavior is modified due to the adverse pressure gradient formed in the lee

(sheltered side) of the perturbation. This behavior was consistent in the experiments

of Darwin (1883), Ayrton (1904), and Bagnold (1946), even though all three utilized

different testing devices. According to Bagnold (1946), the growth of substantial vortices

in the lee of a bedform occurs when the Shields parameter exceeds twice the critical

value, or 0 > 20cr. As opposed to the turbulent diffusion of momentum that takes place

above a horizontal bed, van der Werf et al. (2006) -ir--:. -i that momentum transfer and

sediment transport above rippled beds are dominated by organized vortex shedding in a

nearbed li--r approximately twice the ripple height. Coherent motions break down and

are replaced by random turbulence above this lIv-r (Davies and Villaret, 1997).

Transport modes in the ripple regime include both bedload and suspended load. The

particular mode of transport is determined by the ratio of particle settling velocity to

friction velocity:
wo (2-14)
t,
where w,8 is the particle settling (fall) velocity, and u, is the friction velocity (u,

b/p). The principle mode of sediment transport shifts from suspended load to bedload

when Equation 2-14 is equal to about 0.07 Nielsen (1979). The shedding of vortices from

ripple crests in oscillatory flow, as well as their subsequent ejection into the outer flow,

dictates the phase and quantity of sediment entrainment and deposition in the ripple

regime. Huang and Dong (2002) -ir---. -1. 1 that rippled beds induce a nearbed current

directed opposite from the wave propagation direction. Furthermore, van der Werf and









Ribberink (2004) found that ripple-induced phase differences between peak suspended

sediment concentrations and peak fluid velocities lead to net transport rates directed

seaward under surface gravity waves. This -'- .--- -I that the timing, or phase, of sediment

entrainment in the ripple regime is of particular importance for determining transport

direction under .i-vmmetric flow conditions.

The sheetflow regime is characterized by a relatively thin ( 10 100d) 1iv-.r of

particles transported in a highly concentrated suspension. This regime persists for values

of the Shields parameter much greater than the critical threshold (0 w 0.8 1.0) (van der

Werf, 2004). For very large values of the maximum Shields parameter (w 10 times as

large), sheetflow washes out ripples and planes off the bed (Li and Amos, 1999). Such

conditions exist under surfzone waves as transient pressure gradients result in fleeting

episodes of intense sheetflow transport (Drake and Calantoni, 2001). According to Dong

and Zhang (1999), the sheetflow regime is of particular importance due to the high

concentrations and large transport rates found within. Dominant forcing mechanisms

within the sheetflow 1.--r include intergranular and turbulent stresses, as well as the

interaction forces of fluid lift, drag, and inertia between the fluid and sediment particles

(Dong and Z!i ii:- 1999).

In the lower part of the sheetflow Il.-1-r, sediment concentrations are very high

and the stresses generated by particle collisions significantly affect the velocity of fluid

and sediment (Dong and Zhang, 1999). Through a series of laboratory experiments on

sheetflow transport, Ribberink et al. (1994) -,-.-. -1 that a three- i-.i r system exists with

an active pick-up 1.v.-r near the bed, a sheetflow li-V-r, and a suspension 1iv.r in the

outer flow. Ribberink et al. (1994) found that a majority of the horizontal fluxes were

concentrated in the sheetflow and pick-up lv. -i and an increasing phase lag between

fluid velocity and sediment concentration with increasing elevation in the suspension l-cr.

Almost no phase lag between fluid velocity and concentration was found to exist in the

sheetflow and pick-up lI-.-i where net sediment transport rates were proportional to the









third-order velocity moment ((u3)) and the concentration profile was predicted well by a

power law formulation.

2.4 Turbulence

In the rippled bed regime, nearbed momentum transfer is dominated by vortex

shedding rather than random turbulence in oscillatory forcing conditions [Sleath (1991);

Nielsen (1992); Malarkey and Davies (2004)]. Flow separation at ripple crests represents

a significant mechanism for the production of turbulence in the boundary 1.T--r during

the phase of maximum flow (Barr et al., 2004). Turbulence p1l ,i-, an important role in the

hydrodynamics of the bottom boundary l~-.-r and affects the suspension and transport

of sediment. Turbulent eddies maintain an upward dispersion of suspended sediment

against the counteracting force of gravity (Bagnold, 1973), and turbulent stresses serve

as additional forcing in the sheetflow regime (Dong and Zhang, 1999). The subsequent

stratification of suspended sediment, in turn, affects the background flow by modifying

the velocity profile (Coleman, 1981), damping turbulence (M\cLean, 1991), and increasing

friction (Parker and Coleman, 1985).

Modeling the effects of the particle phase on fluid turbulence is difficult. Most

numerical models cannot accurately simulate sediment-fluid interactions and coherent

motions in the turbulent boundary l1-v-r (Zedler and Street, 2001). Closure of the

nonlinear momentum equations, as it pertains to turbulence, has been approached a

number of v--iv ranging from the more simple one-dimensional eddy viscosity approach

(Davies and Thorne, 2002) to the more complex large eddy simulation (Wang and Squires,

1996). One option for bypassing the explicit treatment of turbulence closure through

one- and two-equation models, or eddy viscosity models, is to perform direct numerical

simulations of the particle-fluid interactions [Elghobashi and Truesdell (1992); Drake and

Calantoni (2001)]. Direct numerical simulations of high Reynolds numbers flows, however,

are still computationally prohibitive for large spatial scales and long durations.









2.4.1 Dynamics

Turbulence and sediment transport are interconnected in the sense that they affect

one another. Perhaps the first to incorporate the role of turbulence in sediment transport

formulations, Eintstein (1950) considered the probability of particle deposition and erosion

based on the randomness of nearbed fluid velocity. Bagnold (1954) further considered

the role of turbulence in the fluid-particle matrix through laboratory experiments on

the dispersion of spheres in a shearing flow. In the clear fluid, Bagnold (1954) found the

in i i ily of bed shear stress to be due almost wholly to turbulence, whereas increasing

concentrations of sediment suppressed turbulence. These concepts were confirmed through

additional experiments by Luque and van Beek (1976). Bagnold (1973) later -ii--.- -1. .

that the dissipation of turbulence in fluid-particle flows occurred through the development

and maintenance of a mean velocity equal in magnitude, and opposite in direction, to

the particle settling velocity. The role of turbulence in concentrated particle flow, then,

is to maintain the sediment in suspension: work performed on the particles represents

the energy dissipated by the turbulence (Parker and Coleman, 1985). Subsequently, the

transport rates of solids in suspension is limited by the rate of generation of turbulent

kinetic energy (Bagnold, 1973).

In the rippled bed and sheetflow transport regimes, intense episodes of suspension

can lead to density stratification. Stratification serves as an additional mechanism for

the suppression of turbulence by limiting the ability of the flow to transport momentum

and particles vertically (\McLean, 1991). This often results in higher suspended sediment

concentrations in the nearbed region (Ribberink et al., 1994), which may limit the

entrainment capacity of the flow (McLean, 1991). As turbulence is damped, drag decreases

as does the relative velocity maintaining particles in suspension, thereby leading to

particle settling (McLean, 1991).









2.4.2 Closure

Turbulence closure typically refers to the solution of an additional equation that

provides an estimate of turbulent stress. Various closure methods have been presented in

the literature, where most seek to solve for the turbulent Reynolds stresses (uu'j) in the

Reynolds-averaged momentum equation(s). The most basic, and widely used, concept was

first introduced by Boussinesq who postulated that turbulent stresses were proportional to

mean turbulent velocity gradients such that


-u v( ui 2 k (215)


where vt is a turbulent eddy viscosity, and k is the turbulent kinetic energy which is

equal to one-half the trace of the turbulent Reynolds-stress tensor. Equation 2-15 is often

referred to as the generalized eddy viscosity concept (van der Werf, 2004). There are three

basic types of eddy viscosity models:

1. zero-equation models;

2. one-equation models; and

3. two-equation models.

Zero-equation models include both time-invariant and time-dependent formulations,

mixing length models, and von Kdrmdn momentum integral methods (van der Werf,

2004). One-equation models seek to solve for the turbulent kinetic energy k, such as in

Davies and Li (1997). Popular two-equation closure models include k c and k U, where

e and w are turbulent dissipation rates [see Wilcox (1998); Andersen (1999); Andersen and

Freds0e (1999)].

2.4.2.1 RANS Models

Reynolds-Averaged N .',1. i-Stokes (RANS) models solve the Reynolds-averaged

momentum equations, where dependent variables (like velocity and pressure) are

decomposed into mean and fluctuating components. Two-equation closure models,

such as k e and k w models, are often employ, ,1 in the RANS approach. With respect









to their two-dimensional sediment transport model, Andersen and Fredsoe (1999) found

the k u performed well in estimating turbulence above ripples in unsteady flow. C'!i ig

and Scotti (2004) compared simulations of turbulent flow over stationary ripples using the

k u closure model of Wilcox (1998) and large eddy simulations, and found deficiencies in

the RANS approach. RANS models require averaging processes that can bias the highly

unsteady, time-dependent flow that occurs in an oscillatory boundary l-.-r above a rippled

bed.

2.4.2.2 Large Eddy Simulations

While RANS models seek to separate dependent quantities into mean and fluctuating

components, the basic concept of large eddy simulation is that variables are decomposed

into resolved (large) and filtered (small) motions. The LES approach assumes that the

computational mesh is sufficiently fine enough to resolve the larger turbulent motions,

and yet acknowledges the fact that smaller eddies will be filtered by the grid near physical

boundaries, unless the resolution is near the viscous lengthscale (Pope, 2000). In high

Reynolds numbers flows, the viscous lengthscale is often prohibitively small to resolve

in the computational mesh so wall models are typically used to compensate for the lack

of resolution there. Particularly relevant large eddy simulations of turbulent flow, with

respect to sediment transport and bedforms, have been successfully performed by Wang

and Squires (1996), Calhoun and Street (2001), Zedler and Street (2001), Barr et al.

(2004), C'!I ,ig and Scotti (2004) and others.

2.4.2.3 Direct Numerical Simulations

It is theoretically possible to bypass the turbulence closure problem altogether by

performing direct numerical simulations (DNS) of turbulent flow. The N i.',1 -Stokes

equations fully describe all quantities of the time-dependent flow field, in all three

dimensions, if the computational mesh is capable of resolving all scales of motion. Until

recently, the application of such models was limited to low or moderate Reynolds number

flows over smooth boundaries (van der Werf, 2004). Technological advancements in









micro-processors have made it viable to perform DNS simulations at high Reynolds

numbers over complex physical boundaries. Recent examples of three-dimensional direct

numerical simulations of moderate and high Reynolds number flows over stationary ripples

are found in Scandura et al. (2000) and Barr et al. (2004). Detailed three-dimensional

DNS modeling of fluid-particle, and particle-particle, interactions has also been performed

by Elghobashi and Truesdell (1992) and Drake and Calantoni (2001).

2.5 Models

A number of sediment transport models exist in published literature, ranging

from simple empirical models to more complex three-dimensional unsteady models.

These models fall into one of four categories based on their treatment of transport and

hydrodynamics: time-averaged, quasi-steady, semi-unsteady, and fully unsteady. Each

have benefits and limitations, some of which will be discussed in the following sections.

Additionally, most coupled models of fluid hydrodynamics and sediment transport require

parametrization of either the fluid or particle phase, but some exceptions do exist. The

following sections provide a brief overview of typical model types, their assumptions, and

also their shortcomings.

2.5.1 Types

The most commonly used sediment transport models may be classified as either

time-averaged, quasi-steady, semi-unsteady, or fully unsteady.

2.5.1.1 Time-Averaged

While simple, time-averaged models are not necessarily considered robust. These

models often rely on flow statistics that have been averaged over durations much longer

than would be considered relevant in the coastal environment, such as an individual wave

period. Time-averaged suspended load transport may be computed using a C u integral

approach:

Q, = C(z) u(z)dz (2-16)
Jo 0h









where the dependent variables of sediment concentration (C) and fluid velocity (u) are

averaged over one or many wave periods (van der Werf, 2004). A popular time-averaged

model for current-related suspended load and bedload transport is given by Bijker (1971).

With respect to applications in the coastal environment, the 1i i ir, disadvantage of any

time-averaged sediment transport model is that the wave-related oscillatoryy) component

of transport is integrated out of the solution.

2.5.1.2 Quasi-Steady

Quasi-steady models seek to account for both the wave- and current-related sediment

transport components by equating the instantaneous transport to the instantaneous

nearbed flow field raised to some power. Implementation of quasi-steady transport models,

however, are not meant to provide intra-wave statistics of sediment transport. Rather, the

quasi-steady transport is computed as


Q J1 u3(t)dt (2-17)
T Jo

where Q8 is the wave-averaged sediment transport rate, T is the wave period, and u(t)

is an expression of the wave and current velocity field. Considering the example given

by van der Werf (2004), the velocity time-series (u(t)) for a second-order Stokes wave

superimposed on a steady current is given by


u(t) = u + u1 cos(wt) + u2 cos(2wt) (2-18)

where u is the current velocity and i1 and u2 are the wave velocity amplitudes. Substituting

Equation 2-18 into Equation 2-17, and integrating over the wave period, gives the

wave-averaged transport rate as a function of both wave and current velocities:

3 2 3 2 3 2, 2
Q, oc 3 1 u+2 4 + tu2). (2-19)

An example of a widely-used quasi-steady transport model is given by the total load

formulation of Bailard (1981), which is an extension of the earlier energetic model of









Bagnold (1966). Useful quasi-steady and unsteady bedload formulations are also provided

by Ribberink (1998). Due to the time-dependent process of vortex shedding, however,

quasi-steady models are not sufficiently robust to estimate the suspended sediment

transport above rippled beds (van der Werf et al., 2006).

2.5.1.3 Semi-Unsteady

An implicit assumption in time-averaged and quasi-steady transport models is that

sediment transport is ahv--, in phase with wave forcing. This is not ahv-- i the case,

and inertial effects of both the boundary 1 -, r flow and the weight of the sediment itself

result in phase differences between applied forcing and subsequent transport [Parker

(1975); Luque and van Beek (1976)]. Semi-unsteady models attempt to account for these

phase differences either through parametrizations, or by considering transport over two

successive wave half-cycles.

Nielsen (1988) developed a simple 5i ,1> and dump" model of wave-related sediment

transport by assuming that transport over ripples occurs through two distinct mechanisms:

first, sediment is transported along the ripple face (stoss) and carried over the crest while

a lee vortex entrains sediment from the trough and second, liberated sand is lifted by the

vortex and subsequently advected by the main flow. Since this process happens twice

during each wave cycle, Nielsen (1988) accounts for the forward and backward transport

separately through distinct entrainment coefficients based on the instantaneous velocity

amplitudes. In this manner, the ,i i1> and dump" model is capable of accounting for

wave .,-vmmetry. Nielsen (1988) reports that the model provides reasonable estimates

of transport for both fine and coarse sands, resolves the phase of maximum transport

events, and is extremely practical as it requires only an estimate of a nearbed reference

concentration.

The concept of dividing transport into successive wave half-cycles to account for

phase lags was popularized by the model of Dil iii and Watanabe (1992). Instead of

computing an average transport rate over the entire wave period, the half-cycle model









solves for the net transport rate over the entire wave period. This is done by taking the

linear sum of the transport rates in the positive (first half) and negative (second half)

phases of a wave. The half-cycle model consists of two elements of transport for each

half-cycle: sediment entrained and transported within the same half-cycle, and sediment

entrained during the previous half-cycle and transported during the following half-cycle

(van der Werf, 2004). The half-cycle model is considered to be semi-, il-/, ,.i/; because

while it is unsteady over the entire wave period, it is steady for each half-cycle.

2.5.1.4 Unsteady

Unsteady models provide estimates of time-dependent sediment transport, typically

through integration with an unsteady hydrodynamics model. Time-dependent values

of fluid velocity and sediment concentration may be substituted into a C u integral

approach, similar to Equation 2-16, in order to model the unsteady suspended sediment

transport rate:
oh
Qs(t)= C(z,t) u(z,t)dz. (2-20)

Most quasi-steady bedload transport models [e.g. At. i-, r-Peter and Miller (1948);

Ribberink (1998)] may also be adapted for unsteady use by considering a time-dependent

Shields parameter 0(t), based on the instantaneous bed shear stress, in the formulation

(\! ,, en and Grant, 1976). For example, the popular A\t, i, r-Peter and Miller (1948)

bedload transport equation may be cast in an unsteady form:


b(t = 8(0(t) 8,)3/2 (2-21)

where )b(t) is the time-dependent, dimensionless bedload function of Eintstein (1950).

Various references to the 1 iv, r-Peter and Miiller (1948) bedload equation have been

made in the literature and some have -ir-- -I 1..1 that the coefficient (8) and exponent (3/2)

vary somewhat depending on flow and sediment characteristics. Although the functional

form was derived from laboratory data on steady flow, Madsen (1991) -i r-.-. -Is that this

bedload formula performs well for coastal applications, too. Depending on the modeling









approach, these unsteady approximations of suspended load and bedload transport

rates may either be averaged over individual wave periods to provide estimates of net

transport (van der Werf, 2004), or integrated over fractional time steps to provide discrete

representations of instantaneous transport.

2.5.2 Integrated Approaches

For unsteady models in particular, there are a variety of integrated fluid-sediment

transport models described in existing literature. The coupling of hydrodynamics and

sediment transport often requires either special treatment of the fluid or parametrizations

of the sediment phase. Some models solve the detailed hydrodynamics explicitly, often

through DNS (Elghobashi and Truesdell, 1992) or using LES turbulence closure (Wang

and Squires, 1996), and then incorporate one-way coupling by modeling the particle

momentum equations (\! i::;-y and Riley, 1983). Others choose to model the particulate

phase by solving an advection-diffusion equation for the suspended sediment concentration

(Zedler and Street, 2001), along with a sediment pick-up function (van Rijn, 1984) and

appropriate boundary conditions (Nielsen et al., 1978).

A slightly more detailed approach for simulating fluid-particle flow involves modeling

the water and sediment as a continuum, such as in Phillips et al. (1992). An advantage of

the continuum approach is that two-way coupling of mass and momentum is implicit in

the formulation. The continuum model has three major components: mixture momentum

equations, an advection-diffusion equation for the mixture concentration, and often times

a sophisticated diffusivity model. The application of continuum models to large coastal

applications remains a relatively new area of research.

Explicit treatment of three-dimensional hydrodynamics, especially for DNS and

LES models, can be time-consuming. Unsteady hydrodynamic models, however, are

necessary for accurately resolving the time-dependent coherent motions that develop in

the boundary liv. r above a rippled bed. Since the vortex motions determine a iii i ilry

of entrainment and transport in the rippled bed regime (Sleath, 1991; Nielsen, 1992),









it is possible to parametrize the bulk hydrodynamics by considering the transport of

vorticity in the flow field rather than the momentum. Such an approach is described in

Malarkey and Davies (2002), where a discrete vortex model (DVM) is used to simulate

the time-dependent vorticity field above a rippled bed in oscillatory flow. Malarkey and

Davies (2002) present results of simulations using a simple inviscid DVM, and a slightly

more advanced "cloud-in-cell" (CIC) model that considers a distribution of vorticity

point values within the overall vortex. These models are capable of estimating reasonable

values of vorticity, but often fail to accurately capture the phase of vortex formation and

ejection. Compared to RANS models, the discrete vortex models are better-suited for

resolving sharp gradients in the vorticity field since they do not suffer from the numerical

diffusion related to advection (i\ I, Il:;.y and Davies, 2004).

An alternative method for simulating coherent motions in the boundary 1,-v.r was

presented by Davies and Villaret (1997), who sel---- -1. that vortex shedding could

be modeled as a "coi,. I I ,. stress represented by a time-dependent, height-invariant

eddy viscosity. The convectivee eddy vi -... ii relates the convective shear stress to the

velocity gradient. Examples of one-dimensional convective eddy viscosity models are given

by Davies and Thorne (2002) and Malarkey and Davies (2004). Model simulations by

Malarkey and Davies (2004) sirl--. -1 that the convective eddy viscosity approach is valid

in a thickness of about one or two ripple heights above the mean bed where the coherent

motions exist in a convective 1V.-,r.

More advanced modeling techniques exist for the simulation of multi-phase flows in

coastal applications. For example, Calantoni (2002) presents a discrete particle model for

bedload transport in the surf zone. The discrete particle model simulates the dynamics

of the fluid flow and accounts for the kinematics of each particle on an individual basis.

Therefore, terms that are typically parametrized in other models, such as granular

stresses and momentum transfer through elastic collisions, are accounted for explicitly

through first principles. Solving momentum equations for the fluid, as well as each









particle, is computationally expensive and applications of discrete particle models are

currently reserved for research problems. An alternative approach is to model all of

the sediment particles as a separate phase, much as the fluid phase is modeled as a

homogeneous medium. Following this methodology, Dong and Zhang (1999) developed

a two-phase model of fluid and sediment flow in oscillatory sheetflow. The model solves

the continuity and linearized momentum equations for fluid and sediment. Turbulent

and intergranular stresses are incorporated in the solution algorithm, thereby providing a

complete description of the interaction forces between the fluid and sediment phases.

2.5.3 Shortcomings

Analytical models of sediment transport are not readily available. Most sediment

transport relations may be characterized as either empirical or semi-empirical at best.

Indeed, much of the literature published on sediment transport has focused on providing

parametrizations of transport based on laboratory, and sometimes field, experiments. This

tends to be acceptable when developing a model for a specific application to a specific set

of criteria (e.g. forcing, sediment size and gradation, transport mode and regime), but

makes it difficult to develop universal sediment transport models. Regardless of the model

framework chosen, the limiting factor on accuracy and predictability seems to lie in the

treatment of sediment transport and not necessarily the hydrodynamics.


















(a) 1


-05


-1



(b) 3

2

_ 1


Linear
-----------. Stokes

1 2 3 4 5
Time (s)


Figure 2-1. Comparison of (a) velocity and (b) shear stress time series for linear and
nonlinear waves with period T = 6 s in water depth h = 1 m.


Figure 2-2. Approximate distribution of fluid and grain shear stress in steady open

channel flow. Adapted from Fredsoe and Deigaard (1992).











Table 2-1. Common power law formulations for bedload sediment transport by currents
and waves. Shields' parameter based on currents is Oc and for waves is O0.
Reference Dimensionless Bedload Discharge
Currents 4b


AI, iv r-Peter and Muller (1948)


8(0c Ocr)1.5

12v/O(Oc Ocr)

11(Oc _Ocr)1.65


Nielsen (1992)

Ribberink (1998)


Camenen and Larson (2005)


Waves


12015 exp(-4.50cr/Oc)


)1/2


Madsen and Grant (1976)


Soulsby (1997)


Waves


Bailard and Inman (1981)


CbfwU(t)3/[(S 1)22 tan d5o]

11(18,(t)l OCr)1.65- l (t)/l W(t)]


Ribberink (1998)


Waves


Dil, i i; I and Watanabe (1992)

Camenen and Larson (2005)


0.001W~For5(F/ F)/ (8s- 1),~.l

a Ocwon + cw,offOw,m exp(-bOcr/0w)


Table 2-2. Empirical entrainment functions for suspended load sediment transport.
Reference Dimensionless Pick-Up Rate, 4,
Fernandez-Luque (1974) 0.02(0 Ocr)1.5


N .1;: 1i. a and Tsujimoto (1980)


0.02(1 0.035/0)3o


van Rijn (1984)


S(t)


12.5W, 0 3 ( 1),/.,,


5.1(0w Ocr)1.5


0.00033[(s 1),l',,/'v2]]0.1/(/ cr )1.5









CHAPTER 3
BEDFORM DYNAMICS

3.1 Introduction

Differential sediment transport on the seabed creates morphological features that

range in size from small-scale sand ripples to much larger, shore-parallel sandbars.

Regardless of their size, the resulting morphological structure plh-,v an important role in

both sediment transport and hydrodynamic processes. Shore-parallel sandbars have been

shown to intensify the cross-shore currents that promote offshore bar migration through

gradients in the cross-shore sediment transport (Gallagher et al., 1998). Small-scale

bedforms, such as sand ripples, determine the generation of turbulence and entrainment

of sand near the bed (Hanes et al., 2001) largely through flow separation that leads to an

organized pattern of time-dependent vortex shedding in each half-cycle of a wave. These

vortices often determine the timing of sediment entrainment and deposition (Nielsen,

1988), the amount of sediment carried in suspension (van der Werf et al., 2005), and

are responsible for the dissipation of wave energy in the boundary 1-. r (Tunstall, 1973;

Tunstall and Inman, 1975). The presence of small-scale bedforms has also been found to

cause natural sorting of sediment, resulting in a redistribution of fine and coarse material

on the seabed (Faraci and Foti, 2002).

The focus of this chapter is on small-scale bedforms, such as sand ripples, so further

discussion of larger features like sandbars will be limited. The following sections provide

a brief overview of bedform types, their classification and characteristics, mechanisms for

their growth, their affect on the bottom boundary I .;r, and methods for predicting their

length and height. Additional information on both simple and complex morphological

models is also provided.

3.2 Bedform Types

Small-scale bedforms may be broadly categorized by the dominant flow conditions

under which they are formed: current-generated bedforms or wave-generated bedforms.









Early accounts of wave-generated ripples in the sea date back to the observations of

Hunt (1882), who found that "ripple n i Il:- were formed on a sandy seabed by a slight

oscillation of the water. Laboratory experiments by Darwin (1883), and later by Ayrton

(1904), provided mostly qualitative data about the formation of ripples in oscillating flow.

Wave-generated bedforms are further distinguished by Bagnold (1946) as either rolling

grain or vortex ripples. A distinction is made between these two types of wave-generated

ripples not only because they grow through different processes, but also because their

characteristic length, height, and steepness scale differently. Briefly, rolling grain ripples

grow through an .,.::-regation of sand particles in areas of high friction (Darwin, 1883)

but do not typically scale with wave-related quantities. Vortex ripples form once a

perturbation on the bed is large enough to induce flow separation (Bagnold, 1946), such

that an adverse pressure gradient is formed in the lee of the ripple crest (Ayrton, 1904).

This process may occur either through the growth of rolling grain ripples or due to

existing perturbations on the seafloor. Specific details about the mechanisms for growth

and evolution are given in the following sections.

Current-generated bedforms include those formed in rivers and open channel

hydraulic flows. The similarity between these two environments is that the predominant

forcing is, more or less, one-dimensional steady flow. While not ahv--, truly -I. ily,"

the term is used here to convey an absence of a persistent oscillatory, or repeating, flow.

Shallow tidally-dominated inlets and estuaries may even be a special case of -I. ,Ily"

flow if the ambient oscillatory motion is weak compared to the tidal-induced currents.

Current-generated ripples initially form in much the same way(s) as wave-generated

ripples, but once the bedform is large enough to induce flow separation the dynamics are

different. Since the steady flow has a constant phase, a recirculation zone is only formed

on one side-the lee side-of the bedform. The dynamics of the bedform are controlled

by a balance between sand transported along the stoss, sand transported over the crest

onto the lee side, gravitational forces pulling sediment down the lee side, and the transport









of sand up the lee side from the attached vortex. This commonly results in a bedform

with an .,i-vii,,. I ic profile where the stoss side has a milder slope than that of the lee

side (Darwin, 1883). It is common for the lee side of an .,i-vii,,. I ii. ripple to have a slope

very close to the natural angle of repose. Once in equilibrium with the surrounding flow,

steady currents are capable of maintaining the ripple through a dynamic equilibrium but

are often displaced downstream due to higher velocities at the crest than at the trough

(Darwin, 1883).

3.2.1 Rolling Grain Ripples

Bagnold (1946) was the first to classify the two types of wave-generated ripples as

either rolling grain or vortex, but a physical distinction between them was made earlier

by Darwin (1883). The laboratory experiments of Darwin (1883) demonstrated that

oscillatory motion first created what he called transient ripples as a result of particle

..-i-regation in areas of high friction. The process of ..-:'regation, he noted, increases the

friction further and traps even more particles in that area. Others have characterized

the process similarly, -i -.-. -1i i-; that each grain creates a region of weaker flow in its lee

ultimately causing grains to group into transverse zones that form ripples with larger

shadow zones (Bagnold, 1946; Blondeaux, 1990; Vittori and Blondeaux, 1990; Andersen,

1999). The particle model of Andersen (2001) demonstrates that rolling grain ripples form

and coarsen due to gradients in the transport velocity from one side of the .r.- 'egation to

the other.

Transient ripples were found to have a wavelength approximately one-half of the final

ripple wavelength in the experiments of Darwin (1883), which seems to be a consistent

observation in other laboratory experiments (Sleath, 1976). Rolling grain ripples continue

to grow in height through ..-. -regation and trapping until the lee slopes are steep enough

to allow a vortex to form in the sheltered space behind the crest (Bagnold, 1946),

eventually forming vortex ripples (Scherer et al., 1999; Faraci and Foti, 2001). During

the initial growth process, the steepness (height/length) of rolling grain ripples has been









found to be consistently near 0.1 by Sleath (1976) and others. Darwin (1883) qualitatively

-Ii.-.. -I. 1 that the initial growth process only occurred between an upper and lower limit

of fluid velocity, and that the limits were a function of average sediment properties like

size and density. These upper and lower limits were found by Bagnold (1946) to be 20cr

and 0r,, respectively.

The stability of rolling grain ripples formed during laboratory experiments is unclear.

Due to their low relief and transient nature, rolling grain ripples are rarely observed in

the field. Rolling grain ripples were found to be stable in the experiments of Bagnold

(1946) and Sleath (1976), but Miller and Komar (1980b) -i,-i-. -1 that these two instances

may be a result of the testing apparatus used during the experiments (oscillating tray).

Experiments in rotating annular cells (Scherer et al., 1999; Stegner and Wesfreid, 1999)

and oscillating water tunnels ii-.-. -1 that rolling grain ripples are transient, unstable

features (\ I!1. r and Komar, 1980b). The prevailing theory, now, is that rolling grain

ripples are transient bedforms that initiate the growth of larger, vortex ripples through an

inverse cascade mechanism: the dynamical evolution involves coarsening from an initially

short wavelength (rolling grain ripples) to a longer wavelength (vortex ripples), with

saturation at a final equilibrium condition (Stegner and Wesfreid, 1999; Andersen et al.,

2002).

While Darwin (1883) and Bagnold (1946) provided qualitative descriptions of

the growth of rolling grain ripples from an initially flat bed, Kennedy (1963, 1969)

demonstrated that if a lag is assumed to exist between velocity and transport, an

instability exists at the sand-water interface (Coleman and Melville, 1994). This lag

is often attributed to inertial effects of the sediment (Parker, 1975), and has been

identified as the ripple "i,'-i.'i:; by Smith (1970). Initiation of growth through this

ripple instability, however, should not be confused with the stability of rolling grain ripples

described above; the former refers to a mechanism that initiates growth while the latter









conveys an ability of the bedform to achieve an equilibrium condition with respect to its

flow conditions.

If a spatial lag is assumed to exist between velocity and transport, Parker (1975)

finds, through a linear stability analysis of derived relations for flow and transport, an

inherent instability of flat beds which leads to the formation of river antidunes. However,

if velocity (shear) and transport are assumed to be in phase with one another, flat sand

beds are unconditionally stable (Parker, 1975). Under pure bedload transport, a flat bed

is also unstable at all wavelengths to small perturbations in topography (Smith, 1970).

This sir.-.- -I that one of two necessary conditions must be met for the initiation of ripple

growth from a flat bed; either forcing and transport must be decoupled, or some initial

perturbation in bed topography must be present. Hayakawa (1970) was able to develop a

theory of dune formation in open channel flows by applying the ripple instability concept,

and others have had similar success (Coleman and Melville, 1994).

Some predictive models of rolling grain ripple geometry, based on the ripple

instability concept, have also been developed. Sleath (1976) created a stability analysis

model capable of predicting the wavelength of rolling grain ripples. More recently, Foti

and Blondeaux (1995a) developed a predictive theory of ripple formation under waves that

agrees well with experimental data. Assuming a fully turbulent boundary lv.-vr, Foti and

Blondeaux (1995a) obtained closed form solutions of turbulent oscillatory flow over a wavy

surface through a linear stability analysis of the governing equations. Detailed particle

models of rolling grain ripples also exist (Andersen, 2001).

3.2.2 Vortex Ripples

Bagnold (1946) -ir-.-.- -1. I that wave-generated ripples fall into one of two categories:

rolling grain or vortex. The deciding factor, Bagnold determined, was based on the range

of applied stress at the bed. Once the Shields parameter increases beyond 20cr, the lee

slopes of the ripple grow to a height large enough that a vortex is formed in the lee

(Bagnold, 1946). The process that Bagnold (1946) observed in the laboratory involved a









transition first from a flat bed to one with rolling grain ripples; and second, from rolling

grain ripples to vortex ripples. It is also possible for vortex ripples to form without rolling

grain ripples if there is some sort of natural perturbation on the bed large enough to

induce flow separation (Ayrton, 1904). So while rolling grain ripples are a sufficient

mechanism for the growth of vortex ripples, the induced flow separation and subsequent

vortex formation are necessary conditions. The ability of vortex ripples to grow from

natural perturbations may also explain why rolling grain ripples are rarely observed in

the field (Mi!!,. r and Komar, 1980b), whereas vortex ripples are often detected (M!1!!, r and

Komar, 1980a; Hanes et al., 2001; Ardhuin et al., 2002).

Vortex ripples pliv an integral role in both sediment transport and energy dissipation,

as discussed previously. Central to the effects of vortex ripples on boundary l-v. r

processes is the formation and ejection of vortices from the ripple profile into the main

flow during each successive half-cycle of oscillatory forcing. Vortices ejected over the

ripple crest can entrain significant amounts of sediment within the boundary 1~. -r that is

subsequently advected by the outer flow (Gallagher et al., 1998; van der Werf et al., 2005).

This is thought to increase the phase lag between suspension and transport relative to

transport over a flat bed (Inman and Bowen, 1963), leading to time-dependent suspended

sediment transport processes that cannot be modeled in a phase-averaged or quasi-steady

manner (van der Werf et al., 2006).

The formation of vortices in the lee of bedforms results from flow separation at the

crest, similar in many v--v to flow separation around bluff bodies, cylinders, and the

like. In her laboratory experiments, Ayrton (1904) identified two conditions necessary for

vortex formation in the lee of ripples:

1. a reduction of pressure in the lee; and

2. an upward-directed resultant gravity pressure along the lee ridge.

The first condition results from flow separating at the crest, creating a sheltered region

in the lee of the ripple. The second condition is due to an adverse pressure gradient that









develops due to phase differences between the boundary li.-r flow and the freestream

forcing-an inertial property of boundary l~-v-r flows. Adverse pressure gradients may also

be attributed to surface gravity waves, but are not present in steady flows over flat beds.

Ayrton (1904) speculated that vortex formation could not occur in steady currents and

that ripples could not be created or maintained by such flow conditions. Recent laboratory

experiments of bedform growth from a flat bed in steady currents have proved this to be

false (\I! ii. 1990; Coleman and Melville, 1994, 1996).

Under oscillatory forcing, vortices are formed and ejected in alternating half-cycles.

A schematic of vortex formation and ejection as a function of fluid phase and forcing is

shown in Figure 3-1. Vortices form at the beginning of each half-cycle (Figures 3-la, 3-1c, 3-le),

growing in strength and size to maximum values at a phase of approximately 90 degrees

(Earnshaw and Greated, 1998). During phases of maximum forcing, strong flow separation

and vortex shedding near the ripple crest serve as mechanisms for the production of

turbulence in the boundary 1-, -r (Barr et al., 2004). After contracting in size slightly

due to a reduction in the applied forcing, the ejected vortices in Figures 3-lc and 3-le

continue to increase in size as they move vertically into areas of decreasing pressure

(Ayrton, 1904). Neglecting small perturbations due to turbulence, these processes are

more or less symmetric from one half-cycle to the next. Under real waves, however, vortex

formation and ejection is .,i-viii,. iir i owing to Stokes' law (Earnshaw and Greated, 1998).

This can have a significant effect on the net sediment transport in the cross-shore as

suspended load may be advected shoreward by the Eulerian flow.

Flow separation and vortex development are imperative for the growth and stability

of vortex ripples under a variety of forcing conditions (Nielsen, 1981). During the growth

process, the nearbed hydrodynamics result in net sediment transport toward the ripple

crest (Andersen and Freds0e, 1999). The impinging flow accelerates as it travels along

the upwind side of the ripple (the stoss) creating strong shear stresses at the bed that

drive sediment transport from the trough toward the crest. In the lee of the ripple, the









counter-rotating vortex induces a bed shear stress that is directed up the lee-side of the

ripple resulting in transport toward the crest. This is the primary growth mechanism for

vortex ripples in near-equilibrium conditions. Experimental data of bedforms in a wave

flume i--.- -i; that the alternating vortices support the ripple in each successive half-cycle

when 0 < 0.2 Faraci and Foti (2002). For 0 > 0.2, the stress at the crest is so large that

the induced erosion cannot be overcome by the stabilizing effect of the vortices.

3.2.3 Characteristics

The growth and evolution of vortex ripples from a flat bed may occur through one of

two processes; one involves a transient state prior to equilibration and the other -i-.-i, -1-

relatively constant growth from a flat bed. For weak flows and under regular waves, vortex

ripples form through coarsening of transient rolling grain ripples (Stegner and Wesfreid,

1999; Faraci and Foti, 2002). The coarsening process is characterized by an increase

in wavelength, resulting in fewer ripples, with saturation to an equilibrium condition

(Andersen et al., 2002). The coarsening process does not occur for bedforms in stronger

flows or under irregular wave conditions where they reach equilibrium directly through

constant growth (Stegner and Wesfreid, 1999; Faraci and Foti, 2002). An overwhelming

amount of physical data demonstrates the ability of vortex ripples to transform from a

smaller to a larger wavelength, but rarely does this happen in reverse. Vortex ripples in

the field and lab have been found to exhibit strong hysteresis (Traykovski et al., 1999;

Stegner and Wesfreid, 1999) whereby changes in stable ripple patterns from larger to

shorter wavelengths are not observed. This -i.-..-- I that larger ripples must be completely

destroyed by an increase in the forcing conditions before a smaller stable ripple pattern

grows in response to new forcing.

Rolling grain ripples and vortex ripples each have unique characteristics due in

large part to the forcing conditions that create them. The rolling grain ripples observed

and measured in a number of lab experiments (Bagnold, 1946; Sleath, 1976; Faraci

and Foti, 2002) consistently have a steepness (T,]/Ar) of 0.1 regardless of the type of









flow and testing apparatus, where rl, and A, are the ripple height and wavelength,

respectively. Vortex ripples tend to have a maximum steepness almost twice that of

rolling grain ripples at approximately 0.18 (Nielsen, 1981; Faraci and Foti, 2002). For

0 < 0.2, Faraci and Foti (2002) find that the ripple steepness remains constant at its

maximum value that corresponds well with the value given by 0.32 tan Q. The wavelength

of vortex ripples commonly increase along with the height-keeping a more or less

constant steepness-until an equilibrium is reached. This equilibrium is a function of flow

and sediment characteristics (Lofquist, 1978) and the final steepness tends to maximize

resistance to the local flow conditions (Davies, 1980). Doucette and O'Donoghue (2006)

identify three distinct processes through which the ripple wavelength evolves: slide,

merge, and split. The slide process identifies a gradual increase (decrease) of wavelength

as .il.i i :ent ripples move away from (toward) each other. A merge occurs when two

larger flanking ripples increase in height and move toward each other, d. -1 vii._-; the

smaller ripple in between. A split refers to a situation when one large ripple decreases

in size through the crest splitting into what appear to be two or more ripples of shorter

wavelength. When bedload transport dominates over suspended load, vortex ripples

grow through the processes of merge and slide and decay through the slide and split

mechanisms (Doucette and O'Donoghue, 2006).

The characteristics of vortex ripples have been found in rn inr: cases to scale with

properties of the fluid forcing. Clifton (1976) first sl---,- -1. 1 that a particular ripple regime

existed whereby the ripple wavelengths were constant multiples of the nearbed orbital

diameter 2A:

A Ua (3-1)

where Umax is the maximum nearbed velocity amplitude and w is the angular frequency

of the forcing. Here, A represents the orbital semi-excursion length which is one-half

the orbital diameter (2A). A number of relationships between ripple wavelength A, and

orbital semi-excursion A have been presented in the literature, and range from Ar 0.78A









for suspended dominated ripples (Andersen and Fredsoe, 1999) to the upper estimate of

A, w 1.33A of Nielsen (1992) for bedload dominated conditions. Some common wavelength

scaling relationships for orbital ripples are given in Table 3-1.

An alternative theory of bedform scaling is offered by Wiberg and Harris (1994) who

ir-,-.. -1 that for field-scale flows, vortex ripples scale best with the grain size. This is

not ahv--, true, however, as Ardhuin et al. (2002) found that wave-formed vortex ripple

wavelengths on the continental shelf were proportional to the nearbed orbital excursions,

not the grain size. Data from lab and field experiments do consistently show discrepancies

between vortex ripple scaling [e.g. Miller and Komar (1980a); Nielsen (1981); Faraci and

Foti (2002)] and many attribute these differences to the disparity between flow conditions.

Whereas the in ii i ly of laboratory experiments are confined to regular oscillatory forcing

with short periods, conditions in the field are represented by a broad spectrum of wave

heights, frequencies, directions, and other flow properties. Attempts to unify appropriate

vortex ripple scales for field and lab data have received increasing attention recently [e.g.

Miller and Komar (1980a); O'Donoghue and Clubb (2001); Williams et al. (2005); Lacy

et al. (2006)] and will be discussed further in the following sections.

Wiberg and Harris (1994) expand on the earlier concepts of Clifton (1976) who

identified three categories of symmetric ripples that differ in their characteristics:

1. orbital ripples;

2. anorbital ripples; and

3. suborbital ripples.

Where orbital ripples scale with the orbital fluid motion and are commonly found in lab

experiments, Wiberg and Harris (1994) -,-.::. -1 that anorbital ripples scale with sediment

properties and are indicative of vortex ripples found in the field. Suborbital ripples are

classified as transitional bedforms lying somewhere between orbital and anorbital ripples.

Wiberg and Harris (1994) state that anorbital and orbital ripple height T,] scale with the

oscillatory boundary lv. -r thickness 6 such that orbital ripple heights are approximately









26 while anorbital ripple heights are about 6/4. This is demonstrated in Figure 3-2 and

shows that while anorbital ripples are completely immersed within the boundary l-r,

orbital ripples tend to protrude from it. The significance of this is made evident through

the choice of roughness parametrizations in coastal hydrodynamic models, where one is

forced to choose appropriate values of boundary roughness based on the interaction of

bedforms with the boundary 1-v'.

3.3 Experiments

A number of experiments on bedform dynamics have been performed in an effort to

provide better understanding of ripple growth and evolution, as well as sediment transport

characteristics in the rippled bed regime. These experiments range from small- to

large-scale laboratory experiments utilizing a number of different testing methods, to field

experiments and observations of bedform dynamics under the stochastic conditions found

in nature. The experimental data have been used to develop new empirically derived

ripple predictors (discussed in the next section) and have lead to the conclusion that

discrepancies exist between bedforms produced by lab- and field-scale flows. Additionally,

some have found that the type of laboratory apparatus used in experiments may affect

bedform characteristics and behavior. A brief overview of experiments on bedform

dynamics is provided in the following sections and a summary of historical experiments

and data sets is provided in Table 3-2, along with citation keys used in the following

sections.

3.3.1 Laboratory

Miller and Komar (1980b) provide a helpful review of commonly-used experimental

techniques used for simulating flows in the laboratory. Although some experimental data

from experiments in rotating annular cells exist in published bedform literature (Scherer

et al., 1999; Stegner and Wesfreid, 1999), by far the most widely used experimental setups

include:

oscillating tray;









oscillating water tunnel; and

flume.

Each of the devices listed, however, have unique characteristics that have some consequence

on the characteristics and dynamics of bedforms.

3.3.1.1 Oscillating Tray

The oscillating tray device has been used to study hydrodynamics in the vicinity of

stationary (simulated) ripples (Earnshaw and Greated, 1998) and also bedform evolution

in a sand bed (Bagnold, 1946; Manohar, 1955; Sleath, 1976, 1985; Scherer et al., 1999;

Lacy et al., 2006). In these experiments, a tray of sand or stationary bedforms is oscillated

at a particular frequency through a tank of still water thereby simulating the oscillatory

motion of water waves. Lacy et al. (2006) used a modified setup to simulate combined

wave and current forcing by passing a steady current over the tray as it oscillated in a

flume at various angles to the steady flow.

The flow induced by the oscillating tray apparatus, however, is not a direct analog

of oscillatory wave motion over a static bed. Steady streaming effects found in oscillatory

boundary l--r flows are not simulated by the apparatus, thereby excluding the additional

vertical flux of momentum directed away from the bed. The motion of the tray has also

been found to induce a mean flow directed opposite to the direction of travel (Scherer

et al., 1999). Observed bedform characteristics in oscillating tray experiments also appear

to be unique, where Scherer et al. (1999) note that the apparatus imparts an additional

inertial force to the bedforms. Furthermore, rolling grain ripples have only been found

to be stable in oscillating tray experiments where they are transient in oscillating water

tunnel and flume experiments, and rarely observed in the field (1i,!! r and Komar, 1980b).

3.3.1.2 Oscillating Water Tunnel

Among others, the oscillating water tunnel has been used to study bedform

characteristics by Foti and Blondeaux (1995a), V, o'i' v- et al. (1999), O'Donoghue

and Clubb (2001), and Doucette and O'Donoghue (2006). Similar to the oscillating









tray apparatus, oscillating water tunnels are designed to simulate the oscillatory motion

of waves. Most oscillating water tunnels consist of a long closed-conduit horizontal

test section and pistons located in vertical risers at each end that are used to drive

the oscillatory flow. The resulting oscillatory flow is purely horizontal and mostly

one-dimensional. Like the oscillating tray apparatus, the horizontal flow is of constant

phase and lacks any vertical component due to boundary l1.-r streaming. In some cases it

is possible to simulate a collinear steady current superimposed on the oscillatory motion.

A distinct advantage of the oscillating water tunnel is that the flow characteristics can

be precisely prescribed and controlled. While most flume and oscillating tray experiments

are limited to lab-scale flows of short-period waves, low to moderate current velocities,

and low to moderate mobility numbers, the oscillating water tunnel has been used to

simulate wave conditions more commonly found in nature. O'Donoghue and Clubb (2001)

and Doucette and O'Donoghue (2006) utilized the water tunnel apparatus to simulate

long-period waves and to analyze the differences between symmetric and .,i-i, i,,. 1 ic

forcing on bedform characteristics. While no significant difference in ripple height and

wavelength was found for symmetric and .i-vmmetric flows, O'Donoghue and Clubb

(2001) did find that previously sl.-.- -1. ,1 methods for predicting bedform geometry were

insufficient, especially for three-dimensional ripples.

3.3.1.3 Flume

Flumes have been used extensively to study bedform dynamics under steady currents

(Yalin, 1985; Mantz, 1990; Coleman and Melville, 1994, 1996), waves (1M'1!. r and Komar,

1980b; Marsh et al., 1999; Faraci and Foti, 2002), and combined wave and current forcing

(Khelifa and Ouellet, 2000; Lacy et al., 2006). An advantage of flume experiments over

oscillating tray and oscillating water tunnel experiments is that the true oscillatory

nearbed motion is simulated by the surface gravity waves. However, most laboratory

flumes are only capable of producing short-period, uni-directional waves not truly

representative of field conditions. Certain exceptions do exist, such as the large wave









flumes at Oregon State and Delft University of Technology, and experiments in these

facilities help bridge the information gap between lab-scale and field-scale bedform

characteristics (Mi !!. r and Komar, 1980b).

As mentioned earlier, the oscillatory motion produced by oscillating tray and

oscillating water tunnel experiments is not a true representation of wave-induced motion

in the boundary 1.-v-r. Progressive waves induce a steady streaming in the boundary l- v-r

that, while weak compared to the oscillatory flow, has a significant effect on sediment

transport (\1 iii, 2004). Boundary l-iv-r streaming is characterized by a vertical diffusive

flux of momentum directed away from the horizontal boundary 1 -V-r. Dohmen-Janssen

and Hanes (2002) -ir-:.- -1 that this streaming results in net transport rates under waves

that are 2.5 times larger than those in uniform horizontal oscillatory flow. Furthermore,

the wave bottom boundary 1-vi-r contains components of velocity in both horizontal and

vertical directions. These vertical velocities, which are not exactly out of phase with the

horizontal velocities in the bottom boundary 1-;V-r, lead to a non-zero time-averaged bed

shear stress (\I i ii, 2004).

Field conditions are rarely classified by one type and size of wave. Rather, field

data are characterized by a broad spectrum of wave frequencies, directions, heights, and

current velocities (\i!!,. r and Komar, 1980a). Even the few large wave flumes capable

of producing long-period, irregular waves are unable to simulate multi-directional waves

or waves superimposed on a steady current. Therefore, methods for predicting ripple

geometry that have been based on laboratory data tend to fail when applied to field data.

In an effort to overcome at least one of these shortcomings, Khelifa and Ouellet

(2000) performed laboratory experiments on bedform characteristics in combined wave

and current flows. They accomplished this by using two intersecting flumes at Laval

University and were able to simulate progressive waves combined with steady currents

at 600 and 900 degree angles to one another. The resulting data were used to formulate

new empirically derived expressions for bedform height and length in combined flows,









but unfortunately their utility is somewhat limited by the short (< 2 s) wave periods

generated in the flume. More recently, Lacy et al. (2006) attempted to overcome this scale

discrepancy by simulating field-scale wave periods (> 8 s) and current velocities with

an oscillating tray in a current flume. The fully-instrumented tray was filled with sand

and then oscillated at various angles-both oblique and orthogonal-to a steady current

generated in the flume. The data from their experiments are awaiting publication.

3.3.2 Field

Observations of ripple characteristics under field conditions have been found to

differ substantially (about 3i ii) from those found in laboratory experiments (Faraci

and Foti, 2002). In the laboratory, Marsh et al. (1999) -,i--.- -1 that bedform length

scales rapidly under the monochromatic forcing typical of most wave flumes. They

speculate, however, that ripple wavelength must be difficult to change in the field since the

bedforms are subjected to a broad spectrum of wave frequencies. Often times observed

ripple characteristics are not in direct equilibrium with the local forcing and experience

significant hysteresis (Traykovski et al., 1999). More specifically, bedforms in the field may

be a result of an antecedent wave climate that remain static as local forcing conditions

subside below the critical threshold required for sediment mobilization. This has made it

difficult to formulate unified expressions for bedform geometry under field-scale conditions.

Additional complications arise when trying to associate ripple processes with a

single, statistical representation of a forcing spectrum composed of many wave frequencies

and directions. Miller and Komar (1980a) -ir.-.-, I 1 that significant wave parameters

computed from field spectra should be used in the calculation of relevant bedform

parameters such as maximum orbital velocity amplitude, orbital excursion length, mobility

number, and Shields' parameter. Nevertheless, it remains difficult to formulate new

expressions based on field data and the predictive capability of methods derived from

field-scale lab experiments is still poor. Observations and measurements of bedform

characteristics in the field, however, continue to supplement existing theory and provide









more detailed information about their geometry and behavior under natural forcing

conditions (Li and Amos, 1999; Traykovski et al., 1999; Hanes et al., 2001; Ardhuin et al.,

2002).

3.4 Ripple Predictors

The following section presents some commonly used methodologies from existing

literature for predicting ripple height ,]r and wavelength A,. These ripple predictors are

based on extensive sets of data from lab and field experiments (see Table 3-2), and in

many cases are based on nondimensional groups that relate sediment characteristics

with flow properties. A number of these dimensionless parameters used to classify or

characterize bedforms are listed in Table 3-3.

3.4.1 Clifton (1976)

The semi-quantitative model of Clifton (1976) (C76) is based on the nearbed orbital

velocity Ub and nearbed orbital velocity .i-vimmetry Aub:


Ub = Au (3-2a)

H2
Aub 14.8- sinh4 kh (3-2b)
LT

where H is wave height, L is wavelength, k is the wavenumber, and h is water depth.

These parameters are then used to identify one of four bed states: (1) no sediment

movement; (2) symmetric ripples; (3) .,-vmmetric ripples; and (4) sheetflow. The

symmetric ripple bed state is further divided by Clifton (1976) into orbital, anorbital,

and suborbital ripples as discussed previously.

3.4.2 Nielsen (1981)

Nielsen (1981) (N81) offers semi-empirical formulations for the size and shape of

vortex ripples derived from an analysis of various sets of lab and field data. A strong

dependence on Shields' parameter 0 was found for ripple steepness while the ripple

wavelength A, data collapsed best when plotted against the mobility number T (Brebner,

1980), which is a ratio between sediment destabilizing and stabilizing forces. The









formulations offered by Nielsen (1981) are particularly useful since they differentiate

between lab and field conditions. When compared to the BASEX data of Vincent and

Osborne (1993), Marsh et al. (1999) find that the two sets of equations serve as bounding

equations for the data.

For relatively weak conditions (T < 20), the ripple wavelength and steepness are given

by

A, a 1.3A (3-3a)

(r,/A,)max 0.32 tan Q. (3-3b)

For conditions outside of this range, or for general use, Nielsen (1981) provides the

following relationships for regular laboratory waves;


A,/A = 2.2 0.3450.34 (2 < T < 230), (3-4a)


qr/Ar 0.182 0.240, (3-4b)

0.275 0.022V for ( < 156
r ,/A = (3-4c)
0 for T > 156

and for irregular field waves;


/A exp 693 0.371n8 (35a)
1000 + 0.75 In7 qj

or/Ar = 0.342 0.34 5, (3-5b)

r,/A = 214-1.85 (4 > 10). (3-5c)

Equations 3-4c and 3-5c correspond specifically to quartz sand with s = 2.65.

Parameters such as A, T, and 02.5 used in the ripple predictors for irregular waves

(Equations 3-5a 3-5c) should be based on significant wave parameters (Nielsen, 1981).









In the preceding expressions, the grain roughness Shields parameter 02.5 is computed

as
1/2 f2.5p(A)2 1
02.5 2p(.5 (3-6a)
p(s 1)gd 2
where the grain roughness friction factor f2.5 is based on the formulation of Swart (1974)

(Equation 2-4) with a roughness (k,) of 2.5d50,
S 2 5d 0.194 -
f2.5 exp 5.213 2.5 5.977 (3-6b)


3.4.3 Grant and Madsen (1982)

Based on the lab data of C69, the formulations of Grant and Madsen (1982) (GM82)

for bedform characteristics are based on the skin friction component of boundary shear

stress. Grant and Madsen (1982) identify two specific ranges for bedform behavior; an

equilibrium range where ripple steepness remains constant and the wavelength changes

with the shear stress; and a break-off range at higher shear stress where ripple steepness

decreases. For the equilibrium range (0/Ocr) < (O/Ocr)B, the ripple height and steepness

are given by Equations 3-7a and 3-7b, respectively. Ripple characteristics in the break-off

range (0/Oc) > (0/Ocr)B are computed using Equations 3-8a and 3-8b. The break-off

range is determined as (0/0cr)B 1.8S.6, where S, is a dimensionless sediment parameter

defined in Table 3-3.

lr 0.22A(0/0,)-0.16 (3 7a)

T,//A, = 0.16(0/0r)-0.04 (3-7b)



rr = 0.48ASo0(8/o0) -15 (3-8a)

r/Ar = 0.28S0.6(0/0)-1.0 (38b)

Unfortunately, accurate values of the boundary shear stress are not commonly known

a prior, nor are they characterized by one specific value. These two shortcomings make









the ripple predictor of Grant and Madsen (1982) a cumbersome choice for both estimating

and comparing ripple characteristics.

3.4.4 Wiberg and Harris (1994)

When plotted against the relative orbital diameter 2A/d, Wiberg and Harris (1994)

(WH94) find that nondimensional ripple properties from field and lab data fall into unique

groups. They propose that a lack of substantial overlap in forcing conditions between

lab (regular waves, short-period flows) and field (irregular waves, long-period flows) data

makes it difficult to effectively classify ripples or to characterize their height and length.

Indeed, there are few existing ripple predictors that perform equally as well at predicting

lab- and field-scale ripples.

Through an analysis of the laboratory data of C69, KF65, MK72, D74, and the field

data of 157 and D74, Wiberg and Harris (1994) find that anorbital ripple wavelength

Xno found in field data is independent of the nearbed orbital excursions. Instead, they

find that ,no scales proportionally with the grain diameter d ranging between 400d <

Ano < 600d, similar to Clifton and Dingier (1984). The methodology of Wiberg and

Harris (1994) involves a sequence of steps that first assumes that the ripples are classified

as anorbital ripples, where the wavelength and steepness are given by Equations 3-9a and

3-9b, respectively. Of course, the expression for the steepness has the added disadvantage

of requiring an iterative process to solve for /ano given Aano and A.


ano = 535d (3-9a)


(/A)ano exp 0.095 In + 0.442 In -- 2.28 (3-9b)
/ ano ) Tano
Once the anorbital ripple height has been calculated using Equation 3-9b, the

following ranges are used to determine whether the ripple should be classified as orbital









(Equation 3-10a), anorbital (Equation 310b), or suborbital (Equation 3-10c):


2A/ lno < 20 orbital ripples (3-10a)

2A/riano > 100 anorbital ripples (3O10b)

20 < 2A/riano < 100 suborbital ripples (3-10c)

If the computed value of 2A/,ano does not fall in the anorbital range, then the ripple

properties must then be computed using either orbital (Equations 3- 1a and 3-1 b) or

suborbital (Equations 3-12a and 3-12b) formulations. Wiberg and Harris (1994) -,i-.-. -1

that the wavelengths of suborbital ripples fall between those of orbital and anorbital

ripples; therefore, Equation 3-12a represents a weighted geometric average of the orbital

and anorbital ripple wavelengths given by Equations 3- 1a and 3-9a, respectively. The

suborbital ripple height is determined iteratively using Equation 3-12b, which is simply a

modification of Equation 3-9b previously defined.


Aorb = 1.24A (3- 1a)

(q/A),,b = 0.17 (3-11b)



Asub = exp L 1n2 lnlOO ) (ln orb In Aano) + In Aano (3-12a)

/)sub exp -0.095 In + 0.442 In 2.28 (312b)
L \ sub rlsub

3.4.5 Mogridge et al. (1994)

Mogridge et al. (1994) (\!')4) conducted a substantial analysis of existing lab (B46,

M55, YR62, KF65, HW67, C69, MK72, S76, W93) and field (157, DI76, K88) data, along

with a reduction of methodologies and data presented in Miller and Komar (1980a),

Nielsen (1981), and Vongvisessoini 1i (1984). The data are evaluated in terms of the

relative nearbed orbital diameter 2A/d and a wave period parameter X (see Table 3-3),









and a dependence between ripple properties and the wave period parameter is found. For

very small values of x (< 0.15 x 10-6)-representative of field data-Mogridge et al. (1994)

find that the ripple wavelength depends only on the grain size and scales according to

A = 1394d. Most of the laboratory data fall in the range X > 0.15 x 10-6 and expressions

for maximum ripple wavelength (Equation 3-13a) and height (Equation 3-13b) are found

through curve fitting.

loglo(A/d) = 13.373 13.7720.02054 (3 13a)

logjo(rl/d) = 8.542 10.8220.03967 (3 13b)

These equations have been found to perform well at predicting bedform geometry in

field-scale oscillatory flows (O'Donoghue and Clubb, 2001), yet it should be noted that

Equation 3-13b is based solely on laboratory data.

3.4.6 Khelifa and Ouellet (2000)

Khelifa and Ouellet (2000) (KO00) performed laboratory experiments of bedforms

in combined wave and current flows using an intersecting wave basin. The experiments

consisted of short-period waves (T < 2.0s) of low velocity amplitude (Umax < 30cm/s),

and relatively weak currents (Uc < 30cm/s). The new bedform data was added to existing

data sets from wave and combined wave-current experiments, and Khelifa and Ouellet

(2000) derived new empirical formulations for ripple wavelength (Equation 3-14a) and

height (Equation 3-14b) in combined wave-current flows. The new formulations were

compared to existing ripple predictors for wave forcing (1\ i!! r and Komar, 1980b; Nielsen,

1981; Mogridge et al., 1994; Tanaka and Shuto, 1984) and for combined wave-current flows

(Tanaka and Shuto, 1984; Tanaka and Dang, 1996).


A,,/A,, = 1.9 + 0.08 1n2(1 + ,) 0.74 ln(1 + T,,) (3-14a)

/wc/A,, = 0.32 + 0.0171n2(1 + c) 0.142 ln(1 + ,,) (3-14b)









Contrary to behavior in purely oscillatory motion, ripple height and wavelength

were found to increase along with a modified wave-current mobility number for the

combined wave-current flows. Khelifa and Ouellet (2000) attribute this incongruity to a

current-dominated effect, but note that the expressions given by Equations 3-14a and

3-14b reproduced the observed increase. The parameters A,, and w,,c used above are

modified versions of the orbital excursion amplitude and mobility number, respectively,

that account for the combined wave and current effects by utilizing a wave-current velocity

amplitude U,, [see Khelifa and Ouellet (2000)].

3.4.7 Faraci and Foti (2002)

Faraci and Foti (2002) (FF02) collected bedform data from laboratory experiments

of noncohesive, quartz sand under regular and irregular waves at low mobility numbers.

Discrepancies in ripple characteristics from the two forcing types were not evident, but the

bedforms were observed to reach equilibrium through differing mechanisms. Under regular

waves, Faraci and Foti (2002) find that the equilibrium ripples grow from a flat bed

through a transitional state where rTl/A, A 0.10 that -t-'-.- -I the presence of rolling grain

ripples. The data for irregular waves -ii-:.: -i that the ripples reached their equilibrium

height and wavelength through constant growth.

Bedform data were evaluated in terms of specific gravity s, the grain Reynolds

number Rg, the orbital Reynolds number R,, and the mobility number T. In the

equilibrium range, Faraci and Foti (2002) find that ripple wavelength and height are

given by Equation 3-15a and Equation 3-15b, respectively, and that steepness follows the

empirical relationship of Nielsen (1981) (Equation 3-15c).

0.08291
A/A, so09 (3-15a)
QRg

Tyr/A 1 -02 I exp-(0.0076R5 + 0.1681) (3-15b)

rT,/A, 0.32 tan (3-15c)








3.4.8 Williams et al. (2005)

The ripple predictor of Williams et al. (2005) (W05) is empirically derived from
sets of field and lab data on bedform growth. They distinguish between two types of
commonly observed ripples; small wave-generated ripples (SWRs) that may be classified
as orbital, anorbital, or suborbital having heights of centimeters and lengths of tens
of centimeters; and large wave-generated ripples (LWRs) that have long wavelengths

(O(m)) and comparatively low amplitudes (O(cm)). Williams et al. (2005) note that
the empirically derived formulations for SWRs prevalent in the literature perform badly
at predicting the characteristics of LWRs. Through an analysis of many data sets,
Williams et al. (2005) provide formulations for SWR and LWR characteristics based on
the mobility number T. The ripple height and wavelength of LWRs are best predicted by
Equation 3-16a and Equation 3-16b, respectively.

1LWR/A exp [ 0.2043 ln()2 + 1.2791n(0) 4.808] (3-16a)

ALWR/A exp 0.20071n(0)2 + 1.4671n(Q) 1.718] (3-16b)

For SWRs, the ripple height and wavelength are given by Equation 3-17a and Equation 3-17b,
respectively.

swR/A = exp 0.0282 n(Q-)2 1.418 In() + 1.249] (3-17a)

ASWR/A exp [0.0542 In(0)2 1.3071n(T ) + 2.843] (3-17b)

3.4.9 Soulsby and Whitehouse (2006)
By re-evaluating field and laboratory data of bedform characteristics Soulsby and
Whitehouse (2006) (SW06) provide empirically derived formulations for ripple wavelength
and height. They found that, when combined, the lab and field data collapsed best when









plotted against a relative orbital excursion length Ad = A/d. The nondimensional ripple

wavelength and height are given by Equation 3-18a and Equation 3-18b, respectively.


rA/A= [L + 1.87 x 10"oAd(1 exp (2.0 x o-4Ad)15}) (3 18a)

7qr/Ar = 0.15 I exp { (5000/Ad)3}] (3-18b)

3.5 Models

While simple analytical models of bedform characteristics exist-such as the ripple

predictors given in the previous section-there are few detailed models of hydrodynamics

and sediment transport that seek to simulate unsteady bedform characteristics and

time-dependent morphology. This is due in large part to the complexity of accurately

describing the time-dependent, nonlinear processes that occur above vortex ripples. Where

a linear stability win 1 ,i-;- was enough to effectively describe bedform growth in the linear

regime, such an approach is not applicable for modeling the evolution of vortex ripples

(Nielsen, 1981). Indeed, a linchpin of the stability approach of Foti and Blondeaux (1995a)

requires that the flow remain in the linear regime thereby precluding flow separation and

vortex shedding.

Some empirical and semi-empirical models of bedforms have been developed

through laboratory experiments. Baas (1994) created an empirical model to simulate

the development and equilibrium characteristics of small-scale bedforms in fine sand. The

model was based on steady flow experiments in a flume, and demonstrated that the time

required for equilibrium ripples to develop from a flat sand bed is related to an inverse

power of the flow velocity. The semi-empirical models of Andersen and Fredsoe (1999),

Andersen et al. (2001), and Andersen et al. (2002) incorporate additional complexity

through additional treatment of the hydrodynamics, providing more detailed information

about the nonlinear evolution of vortex ripples. None of these models, however, seek to









characterize the hydrodynamic, sediment transport, and morphological processes in great

detail.

A number of lab and field experiments, combined with advancements in computational

technology, have improved basic understanding of coupled hydrodynamic and sediment

transport processes. This has made it possible for scientists to develop more sophisticated

morphological models. While scarce, some detailed models of the time-dependent coupling

between hydrodynamics and morphology exist (Gessler et al., 1999; Lesser et al., 2004).

Gessler et al. (1999) developed a sediment transport and morphology component

for the three-dimensional hydrodynamic model CH3D, allowing for simulations of

time-dependent bed morphology. The model does not specifically address the generation

or dynamics of bedforms but shows some skill at predicting erosion and deposition in river

bends. While the Navier-Stokes equations are used in the horizontal plane, the model

invokes the Boussinesq equation in the vertical that assumes a hydrostatic pressure field.

The Boussinesq approximation is not expected to hold true above a rippled bed where

coherent motions create a spatially dependent pressure field. Gessler et al. (1999) do,

however, incorporate a number of useful components into CH3D-SED such as the ability

to model a number of different sediment sizes, sediment exchange between bedload and

suspended load regimes, and turbulent damping by suspended particles.

A morphological component integrated with the DELFT3D model was created by

Lesser et al. (2004), resulting in a fully coupled three-dimensional model of hydrodynamics

and morphology. The hydrodynamic model solves the three-dimensional shallow

water equations, incorporates a number of turbulence closure schemes, and uses an

advection-diffusion equation to model the suspended load. Lesser et al. (2004) apply the

modeling system to a number of coastal problems but tend to concentrate on large-scale

morphology.














(a) phase = 60 deg.








(b) phase = 120 deg.







(c) phase = 240 deg.







(d) phase = 300 deg.







(e) phase = 400 deg.


Figure 3-1.


An illustration of vortex shedding over a rippled bed as a function of fluid
phase. Vortex sizes, locations, and fluid phases are approximate and not to
scale. The magnitude of large arrows indicate forcing strength and direction.
As the flow accelerates from left to right, (a) a vortex forms in the lee of each
ripple and (b) increases in size with increasing fluid velocity. The original
vortices are ejected over the ripple crest upon flow reversal. New vortices form
(c) in the lee of each ripple and (d) grow as the ejected vortices are advected
by the main flow. The process repeats itself (e) during the next half-cycle.
Adapted from Earnshaw and Greated (1998).














Orbital Ripples


Anorbital Ripples





Figure 3-2. Effects of bedform type on boundary li. -r thickness.












Table 3-1. List of common orbital ripple length predictors based on the orbital
semi-excursion. A differentiation is made for some of the values with SL
referring to suspended load dominated bedforms and BL denoting bedload
dominated bedforms.
Reference A,
Andersen and Fredsoe (1999) (SL) 0.78A
Andersen and Fredsoe (1999) (BL) 1.25A
Andersen et al. (2001) 1.28A
Mogridge and Kamphuis (1972) 1.3A
Miller and Komar (1980b) 1.3A
Nielsen (1981) 1.3A
Nielsen (1992) 1.33A
Wiberg and Harris (1994) 1.24A













Table 3-2. A summary of lab and field experiments, and ripple data sets.
keys F and L refer to field and lab experiments, respectively.
Reference Experiment
Bagnold (1946) L
Manohar (1955) L
Inman (1957) F
Yalin and Russell (1962) L
Kennedy and Falcon (1965) L
Horikawa and Watanabe (1967) L
Carstens et al. (1969) L
Mogridge and Kamphuis (1972) L
Dingier (1974) F
Tunstall and Inman (1975) L
Dingier and Inman (1976) F
Sleath (1976) L
Lofquist (1978) L
Nielsen (1979) L
Miller and Komar (1980b) L
Miller and Komar (1980a) F
Boyd et al. (1988) F
K.-'iii, (1988) F
Southard et al. (1990) L
Osborne and Vincent (1993) F
Vincent and Osborne (1993) F
Willis et al. (1993) L
Coleman and Melville (1994) L
Hay and Wilson (1994) F
Wheatcroft (1994) F
Thornton et al. (1998) F
Li and Amos (1999) F
Marsh et al. (1999) L
Traykovski et al. (1999) F
Khelifa and Ouellet (2000) L
Hanes et al. (2001) F
O'Donoghue and Clubb (2001) L
Ardhuin et al. (2002) F
Faraci and Foti (2002) L
Doucette and O'Donoghue (2006) L
Lacy et al. (2006) L


The experiment

Citation Key
B46
M55
157
YR62
KF65
HW67
C69
MK72
D74
TI75
DI76
S76
L78
N79


B88
K88
S90
OV93
V093
W93
C'\1)4
HW94
W94
T98
LA99
M99
T99
KO00
HO1
OC01
A02
FF02
D006
L06














Table 3-3. Common dimensionless parameters and variables used in the classification of
bedforms and computation of bedform characteristics.
Name Symbol Equation
Orbital Semi-Excursion A Umax /W


A/d


Relative Orbital Excursion

Displacement Thickness

Orbital Reynolds Number

Grain Reynolds Number

Sediment Parameter

Mobility Number

Wave Period Parameter


0.08A


Aud/v


(d/4v)[(s- 1)gd]0.5

(Aw)2/[(s 1)gd]

d/[(s )gT2]


Table 3-4. A list of common ripple predictors, the values they provide, and classifications
of their supporting data sets as either laboratory L, field F.


Reference
Nielsen (1981)
Grant and Madsen (1982)
Wiberg and Harris (1994)
Mogridge et al. (1994)
Khelifa and Ouellet (2000)
Faraci and Foti (2002)
Williams et al. (2005)
Soulsby and Whitehouse (2006)


Value Computed Data Sets


'Tr, Ar
T'r, Ar
Tlorb,ano,sub, Aorb,ano,sub
rImax, Amax
7wc, Ae
TrR, Ar
tILWR,SWR, ALWR,SWR
Trj, Ar


L, F
L
L, F
L, F
L
L
L, F
L, F









CHAPTER 4
MODEL DESCRIPTION

4.1 Introduction

This chapter provides a description of newly-developed models for simulating

unsteady fluid hydrodynamics, noncohesive sediment transport, and time-dependent

morphology in linear and nonlinear forcing regimes. The models range from a simple

one-dimensional bedload model based on a simplified momentum equation, to more

complex two- and three-dimensional models that solve the unsteady Navier-Stokes

equations for fluid flow with a set of equations for turbulence closure, and simulate

both suspended load and bedload sediment transport. Each of the new models may

be classified as unsteady, or phase-resolving, and do not employ process averaging

or parametrizations of flow or transport characteristics inherent in phase-averaged,

quasi-steady, and semi-unsteady models. Sediment transport is treated in a bulk fashion

in the models: no attempt is made at solving independent sets of kinematic equations for

each sand particle, although such models do exist (Calantoni, 2002).

The following models are particularly well-suited to handle a variety of forcing

conditions, but each was developed with a specific application in mind; the one-dimensional

model is used to simulate dynamics in the linear regime; and the two- and three-dimensional

models are applied to problems in the nonlinear regime where flow separation, coherent

motions, and turbulence p1 iv significant roles in boundary 1. -r sediment transport

processes. It should be noted that the two-dimensional model is simply a unique

application of the full, three-dimensional model, so they will be discussed simultaneously.

4.2 One-Dimensional Linear Model

The growth of bedforms from an initially flat bed occurs through two distinct

regimes; a linear regime where the freestream forcing and applied bed stress are in phase

with one another; and a nonlinear regime where phase lags exist between stress in the

boundary 1-v.-r and fluid forcing in the outer flow. In the linear regime, Bagnold (1946)









observed that rolling grain ripples formed through an ..- .- regation of sediment particles

in areas of increased friction. Such a process is challenging to simulate with numerical

models that do not incorporate parametrizations of spatially-dependent friction as there

is no inherent mechanism to initiate ..-.- regation. Linear stability models have proven to

be useful tools in simulating transport and bedform growth in the linear regime, but often

incorporate parametrizations specific to a set of fluid or sediment parameters (Parker,

1975; Sleath, 1976; Coleman and Melville, 1994; Foti and Blondeaux, 1995a).

The following sections introduce the methodology used to formulate a new one-dimensional

model of transport and bedform dynamics in the linear regime, and the constitutive

relationships for fluid flow and sediment transport are provided. A schematic of the

model is provided in Figure 4-1, and will serve as a useful reference for explanation of

methodologies and assumptions used to derive the governing equations employ, -1 in the

new model. The objectives of the one-dimensional model are to identify the instability

of the flat bed leading to bedform growth in the linear regime, and to simulate the

time-dependent morphology resulting from gradients in the bedload transport.

4.2.1 Hydrodynamics

4.2.1.1 Governing Equations

The one-dimensional horizontal (1DH) model is based on a simplified version of the

one-dimensional momentum equation for fluid flow in the x-direction (Equation 4-1).

A right-handed coordinate system is used where x and z are the horizontal and vertical

directions as shown if Figure 4-1. Although the z-direction is shown in Figure 4-1, the

computational domain is confined solely to the horizontal x-direction. Therefore, terms

of v and w and derivatives in y and z implicit in the full one-dimensional momentum

equation are excluded from the simplified 1DH momentum equation:

u+ au + i P X 0 (4-1)
at Ox p Ox









where u is the x-directed fluid velocity, p is pressure, p is fluid density, and 'T is a

horizontal shear stress.

For the purpose of simulating the single 1~-,--r mixture of sediment and water in the

linear regime, the pressure term in Equation 4-1 is assumed to be hydrostatic where

p = pg(qr z) (Dean and Dalrymple, 1991). Substituting the hydrostatic assumption for

the pressure variable in the pressure gradient term of Equation 4-1 yields:

Sa g9 (4-2)
p x Qx

where = (zb + Amix) is the displacement height of the surface mixture, and Zb is the

height of the bed level. The thickness of the mixed lI---r, Amix, is a free parameter that

depends on three parameters: the number of grain-thicknesses assumed in the 1I-,r (N),

the grain diameter (d), and the concentration in the mixed 1. -r (Cmix). The relationship

between the mixed l-1~--r thickness and the three parameters is given by


Amix = Nd( Cmix). (4-3)



The concentration of the mixed 1 i-.r, like the l V.-r thickness, is a free parameter.

Bearing in mind that the maximum packing concentration of spherical particles is nearly

0.60, the concentration in the mixed l1-,-r (Cmix) will be somewhat less due to the

presence of fluid in the 1l -r. A typical assumption for model applications is that the

mixed 1l---r contains 5(0' sediment by volume, or Cix = 0.30. The choice of the mixed

1 ,-,r concentration subsequently has an effect on the density of the mixed 1-~-,r:


Pmix = PsCmix + P(1 Cmix), (4-4)

where ps is the sediment density. Present simulations with the 1DH model assume that

the sediment is comprised of quartz material with a specific gravity of s = 2.65.









The horizontal stress gradient in Equation 4-1 is replaced by a dimensionally

equivalent expression representing the balance of stresses on the mixed li--r shown if

Figure 4-1:
1 aTr 1
07 --,--( rs (4-5)
p OX PmixAmix (
where Tb is the bed shear stress resisting motion of the mixed 1li.-r, and 'T is the surface

stress applied to the mixed l-?v-r from the freestream forcing. The parametrizations of

these two stress terms, Tb and r,, are given by Equations 4-7 and 4-8, respectively.

Substituting the previous simplifications into the original one-dimensional momentum

equation (Equation 4-1), and denoting the velocity u in the mixture l-,v-r as Umix, the

resulting 1DH general mixed-1 iv.r momentum equation for the linear regime is given by

auuix 9um ar 1 /
+ Umix +g + Ts + F, = 0 (4-6a)
dt OX OX PmixAmix



F, = Uuw cos(wt) (4-6b)

where F, is a body force, U, is the freestream forcing velocity, w is the angular wave

frequency, and t is time. The body force Fx represents the oscillatory nearbed forcing

induced by a horizontal pressure gradient, and is applied uniformly to the fluid.

Parametrizations for the stresses applied to (7-) and resisting (Tr) the mixed l-,v-r are

required to close the 1DH mixed l-iv-r momentum equation (Equation 4-6a). The resisting

stress of the bed acting on the mixed 1l-v-r is given by


b = Pmix fbUmix Umix (4 7)

where fb is a coefficient of friction between the bed and the mixed l1. -r. The constant

stress applied to the mixed li- -r from the simulated freestream forcing is given by


Ts= Pf. ,", "f (4-8)









where fmix is a constant representing friction between the fluid and mixed l1'- rs, and uf

is the fluid velocity above the mixed li r. At any time t the fluid velocity Uf is found by

integrating the body force F, (Equation 4-6b) with respect to time such that


f(t) = E(t) dt= JU,w cos(wt) dt (4-9)



4.2.1.2 Numerics

The 1DH mixed-li-, "r momentum equation (Equation 4-6a) is integrated forward in

time using an explicit third-order Adams-Bashforth time-marching scheme:

n+1 =n+ [+j23F() 16F(- 1) + 5F(9-2)] (4-10)


where jo" may be taken as the mixture velocity umz% at time level n and the terms F(p"),
F(p"-1), and F(pn-2) correspond to the evaluation of fluxes from Equation 4-6a at the n,

n-1, and n-2 time levels, respectively. In general, a flux term F(p" from Equation 4-6a

is evaluated as

Fu%") -u'. -- n Ts n x (4-11)
mx OX g ax Pmixnmix b

while the fluxes at time levels n 1 and n 2 are evaluated in a similar fashion. Since the

third-order Adams-Bashforth scheme (Equation 4-10) requires information at two previous

time levels (n 1 and n 2), a first-order upwinding time integration (Equation 4-12) is

used for the first two time steps.

Sn+l = p + 1tF(pn) (4-12)

The spatial discretization of the 1DH mixed-l iv-r momentum equation (Equation 4-6a)

is achieved using second-order central differences:

O i+l 26 (4-13)
Oxj 26x









where 90 may be taken as any primitive variable of the 1DH mixed-1 vivr momentum

equation evaluated at grid location i and time level n, and 6x is the grid spacing. The

horizontal grid spacing in the x-direction is assumed to be constant, where periodic

boundary conditions are employ, 1

4.2.2 Sediment Transport

The volumetric rate of bedload sediment transport (per unit width) in the 1DH model

is assumed to be a function of the mixed-li-v- r velocity, thickness, and concentration such

that

Qb UmixrmixCmix (4-14)

where the mixed-1 i-- r thickness Amix and concentration Cmix are assumed to remain

constant. The magnitude of bedload transport predicted by Equation 4-14 will vary

depending on values chosen for the number of particle thicknesses (N) and concentration

(Cmix) in the mixed-liv- r through Equation 4-3.
4.2.3 Morphology

The time-dependent bed morphology is predicted by the sediment continuity, or

Exner, equation:
a zb aQb
(- ) + 0 (415)
at ax

where Zb is the vertical location of the bed, Qb is the bedload transport rate, and n, is

the porosity of the bed material. Equation 4-15 is integrated forward in time using the

third-order Adams-Bashforth scheme (Equation 4-10), and discretized spatially using

second-order central differences (Equation 4-13).

4.3 Two- and Three-Dimensional Nonlinear Models

In comparison to the one-dimensional linear model presented previously, the two-

and three-dimensional nonlinear models are quite complex. The complexity is justifiable

since in the nonlinear regime of bedform growth it is no longer sufficient to assume that

the applied stress at the bed is uniform in space, and in phase with the fluid forcing.









Rather, phase differences in the nonlinear regime are manifest through inertial effects in

the boundary 1--, r where applied forcing on the sediment tends to lead the freestream

oscillatory forcing outside the boundary 1 .-r. Early investigations of bedform growth by

Ayrton (1904) and Bagnold (1946) gave particular attention to this matter, noting the

regime change from the frictionally-dominated case of rolling grain ripples to the orbital,

or vortex, ripples that form and grow due to flow separation.

A unique, coupled modeling system has been developed to capture all of the salient

mechanisms governing the behavior of orbital and anorbital ripples in the nonlinear

regime. A schematic of the three-dimensional modeling domain is provided in Figure 4-2.

The modeling system employs a three-dimensional, N ,i,. r-Stokes solver for the fluid

hydrodynamics and incorporates a Large Eddy Simulation to model the sub-grid scale

(SGS) turbulent stresses. Semi-empirical formulations, available in existing literature,

govern the bedload and suspended load noncohesive sediment transport. Coupling

between the fluid and sediment dynamics is achieved by solving a discretized version of

the sediment continuity equation, which provides time-dependent information about the

spatial location of the bed (morphology). Unlike the discrete particle model of Drake and

Calantoni (2001), or the two-phase model of Dong and Zhang (1999), sediment is not

inherent to the current modeling scheme. Instead, the sediment should be considered as a

distinct second phase acting only as a passive scalar with its own inertial properties. The

model does not yet consider the two-way coupling between fluid and sediment interactions

that may lead to significant alterations of the hydrodynamic field through turbulence

damping at high concentrations (McLean, 1991; Ribberink et al., 1994). Specific details

about the hydrodynamics, turbulence closure, sediment transport, and morphology

techniques employ, .1 in the new, nonlinear modeling system are provided in the following

sections.









4.3.1 Hydrodynamics

4.3.1.1 Governing Equations

In a right-handed coordinate system, where the x-, y-, and z-directions (xi, x2, x3)

are associated with the u-, v-, and w-velocity components (ul, u2, u3), fluid hydrodynamics

in the model are governed by the continuity (Equation 4-16a) and Navier-Stokes

(Equation 4-16b) equations.
u = 0 (4-16a)
axi


9Oui 9Oui 9 fi,6ijp 9OuiA
S+ U + + 6ijF = 0 (4-16b)
at 9xj 9xj p 9xj )


F,= [U,,w cos(wt), 0, g] (4-16c)

where ui is the fluid velocity, p is pressure, v is kinematic viscosity, 6ij is the Kronecker

delta, and Fi is an appropriate body force applied to the fluid. As shown in Equation 4-16c,

a constant horizontal pressure gradient is applied in the x-direction, and gravitational

acceleration (g) acts in the z-direction. The horizontal pressure gradient drives an

oscillatory flow having the form


U(x, t) U sin(t) (4-17)

where U, is the prescribed freestream forcing and w is the angular frequency of

oscillation. The applied forcing given in Equation 4-16c is derived by simply taking

the time-derivative of the velocity time-series given by Equation 4-17.

The time-dependent, finite-difference modeling system employs an Eulerian

reference frame, such that the fluid motions must be tracked as they progress through

the computational mesh, or grid. Primary dependent variables in the model consist

of the three components of velocity (u, v, w), and the total fluid pressure (p). The

velocity and pressure variables are arranged on a ,.:- r'ed, Cartesian grid, such that









numerical representations of pressure gradients are coincident with the locations of the

face-centered, velocity variables in the momentum equation (Equation 4-16b). The

-1 I,--.: red arrangement ensures numerical fidelity, and reduces the computational overhead

associated with variable memory storage. A representation of the -i .1:: red grid is

provided in Figures 4-3(a) and 4-3(b).

Other features of the computational grid include optional variable mesh clustering

and scaling, which can be utilized to enhance numerical resolution in desired regions while

permitting a more coarse resolution for far-field solutions. With the current memory

capacity of most computational platforms exceeding one-gigabyte (1GB), the additional

storage requirement of three vectors containing the variable grid spacing values is of minor

importance; however, the variable mesh scaling does add complexity to the derivation

of appropriate finite-difference schemes as 6x, 6y, and 6z are no longer constants.

Nevertheless, the benefits of increased resolution in numerically intensive areas such

as boundary 1iv.-r- and regions of organized, coherent motions far outweigh the negatives.

The variable mesh scaling is set using a simple geometric function:


6xi = A + Bxi + Cxf + Dx4 (4-18)


where the coefficients A, B, C, and D are a function of the number of grid points, as

well as the minimum specified grid -I' ii;- in each submesh. A schematic of the variable

mesh scaling is shown by Figures 4-4(a) and 4-4(b). In order to simplify numerical

algorithms used in the model, the variable mesh scaling is considered to be isotropic in

the two orthogonal directions: for instance, the grid spacing in the x-direction will be

a geometric function of the grid location i only, such that 6x(i) is constant in the y-

and z-directions. It is also possible to define submeshes in each of the three dimensions

that permit clustering of grid points in multiple areas where higher resolution is needed

(i.e., a boundary 1iv-r and an interface between two fluids). Such an example is shown in

Figures 4-5(a)-4-5(d) where the x- and y-directions use one clustered mesh, and the









z-direction includes two clustered submeshes. In Figure 4-5(a), the grid volume ratio is

defined as

6v Y (4 19a)
6max
where

JS^z = 6x(i) Sy(j) 6z(k) (4-19b)

and

6max = max{6x(i)}max{y(j)} max{6z(k)}. (4-19c)

For simulations incorporating free or rigid boundaries, care must be taken to

avoid smoothing of flow properties across the boundary. Finite-difference formulations

often require the averaging of flow properties on the computational grid to ensure the

co-location of variable quantities. This type of averaging is common in the evaluation of

convective flux terms in the momentum equation (Equation 4-16b), but contributes to

"Eulerian smoothiin if flow properties are averaged across a discontinuous boundary

such as a free surface (Nichols et al., 1980). The fractional Volume of Fluid (VOF)

method of Hirt and Nichols (1981) is used to avoid the numerical inaccuracies associated

with averaging flow properties across free boundaries. The volume of fluid is a time-

and spatially-dependent quantity denoted in the model as F(x, y, z, t). A step function

representation of F is used, where F = 1 in a cell whose volume is completely filled with

fluid, and F = 0 elsewhere. When averaged over a control volume, the fractional volume

of fluid may fall within the range given by Equation 4-20. If Equation 4-20 is true for

a particular control volume, then that control volume is assumed to contain an interface

between the fluid corresponding to F = 1 and the alternate media represented by values

of F = 0. The alternate media may either be void space, whose density and viscosity are

undefined, or an alternate fluid with its own unique density. In such a manner, free surface

or two-phase flows may be simulated.


0 < F(x,y,z,t) < 1 (4-20)









The fractional volume of fluid for any particular cell is advected through the

computational mesh corresponding to the local velocity field. At this particular time,

there is no diffusion of the VOF scalar. The governing equation for F(x, y, z, t) is

OF OF
F+ F 0 (421)


In the VOF method, boundary normals are identified by evaluating the location of highest

spatial gradients of F throughout the computational mesh. The value of F for a cell

containing an interface may be used in conjunction with the boundary normal direction to

reconstruct the slope of the interface in that cell. Since F is a step function, caution must

be exercised when computing the spatial derivatives of Equation 4-21 to avoid unwanted

smearing of the fractional volume between zero and one. A special flux approximation

for F, the Donor-Acceptor method, is used to retain the discontinuous nature of the

free surface where necessary (Johnson, 1970). The Donor-Acceptor method considers

maximum and minimum values of F available for fluxing between control volumes based

on the local advecting velocity field, and the fractional value of F contained in neighboring

cells. In this way, no control volume can ever donate more mass to its neighbor than it has

to give, and no volume can ever accept more mass from its downstream neighbor than it

can contain within its own control volume. One limitation of the current VOF scheme is

that it is only accurate to O[(6t)] in time and O[(6x)] in space.

4.3.1.2 Turbulence Closure

The foundation of turbulence closure starts with a well-chosen decomposition of the

instantaneous velocity field into mean (average) and fluctuating components. For instance,

consider the standard Reynolds decomposition of a velocity component into mean and

fluctuating components:

U(x, t) = U(x, t) + u(x, t). (4-22)

As demonstrated by Reynolds (1894), substituting the decomposition (Equation 4-22)

into the N ,li, r-Stokes equation(s) (Equation 4-16b) provides a mean momentum









equation. The resulting set of mean momentum equations is commonly referred to as the

Reynolds-Averaged Navier-Stokes (RANS) equations and form the basis of two-equation

turbulence models such as the commonplace k c and k w formulations.

An alternative to the Reynolds-stress approach for modeling turbulence, large-eddy

simulation (LES), has gained popularity in the disciplines of atmospheric and oceanographic

sciences in recent years. In level of complexity and computational cost, LES lies

somewhere between Reynolds-stress models and direct numerical simulations (DNS).

However, in highly unsteady three-dimensional flows, LES is found to be superior to

Reynolds-averaged models in predicting turbulent flow over bluff bodies (Pope, 2000). By

comparing the estimates of RANS and LES model results to DNS simulations of turbulent

flow over ripples, ('!i ing and Scotti (2004) demonstrate that a two-equation (k w) RANS

model underestimates both the magnitude of Reynolds stress, as well as the oscillatory

amplitude of vertical velocity. Large-eddy simulation has been applied to a number of

turbulent flows over simple and complex geometries with promising results [e.g. Zedler

and Street (2001); Calhoun and Street (2001); Mahesh et al. (2004); Tseng and Ferziger

(2004)].

The central concept of LES is that larger-scale turbulent motions are well resolved in

the computational mesh, and therefore directly represented, while the small-scale motions

are modeled. There are four conceptual steps involved in large-eddy simulation:

1. filtering of dependent variables,

2. derivation of the filtered governing equations,

3. modeling the subgrid scale residual-stress tensor (closure), and

4. solve the filtered momentum equations from step 2.

The filtered velocity field is given by the convolution


U(x) G(r)U(x r)dr (4-23)









where G(r) is a homogeneous filter. Common filter types include box, Gaussian, sharp

spectral, Cauchy, and Pao as given by Pope (2000). Regardless of the filter chosen, their

effect on the velocity field is to remove small wavenumber fluctuations while maintaining

the overall trends of the velocity field.

Prior to general filtering concepts, Deardorff (1970) -I.-.-. -1. 1 that mean quantities

could be represented by volume-averaging their values over a cell in a rectangular grid.

That is, for a cell centered at x, a mean quantity is given by

_1 r3+h 3/2 xa2+h2/2 xi+hi/2
U(x, t) U(x, t)dxidx2dx3 (4-24)
1Jh2h3 J3-h3/2 Ja2-h2/2 Jxi-hi/2

where hi, h2, andh3 are filter widths of a three-dimensional, inhomogeneous grid. The

equivalent homogeneous filter, for use in Equation 4-23, is
3
G(r) J H A() |r() (4-25)
i= 1

where A(i) is representative of the filter size. Applying this volume-averaging filter to the

dependent variables on a I:.-.-- red, rectangular grid results in a filter-to-grid ration of

two. According to C'! i. -- and Moin (2003), a filter-to-grid ratio of at least four is desired

for a second-order finite-difference scheme. A filter-to-grid ratio of four may be obtained

by passing an unfiltered variable through a succession of second-order filters:


U(x, t) = U(x, t) + 7 (U(x 6x, t) 2U(x, t) + U(x + 6x, t)) (4-26a)



U(x, t) = U(x, t) + U(0x 6x, t) 2U(x, t) + U(x + 6x, t)) (4-26b)

where 6x is the grid spacing and 7, is a smoothing coefficient, typically taken equal to

0.25. Current model simulations employ the more simple volume-averaging approach given

by Equation 4-24, as it is much easier to apply to complex geometries near the bed.

While the approach of Reynolds (1894) involves the decomposition of dependent

variables into average and fluctuating quantities, LES employs a decomposition that









separates variables into filtered (resolved) and residual (modeled) components. Consider

the following decompositions for the velocity and pressure fields:

ui = ui + u (4-27a)



p =p + p' (4-27b)

where ui and p are the filtered velocity and pressure and u' and p' are the residual

components of velocity and pressure, respectively. Derivation of the filtered momentum

equations is achieved by substituting the velocity (Equation 4-27a) and pressure

(Equation 4-27b) decompositions into the conservation of mass and momentum equations

given by Equation 4-16a and Equation 4-16b, respectively. After simplification, the

filtered continuity and momentum equations are

au= 0 (4 28a)
axi

Oui Oui a ( ijp Oui R\ -
-+ U + v + ) + VjF = 0 (4-28b)
at Ujaxj a xx

where P^ is the residual-stress tensor, which is equivalent to the Reynolds-stress tensor.

More specifically, Tr, represents the difference between the filtered product (T)uj) and the

product of the filtered velocities (uiuj), which arises through decomposition of the velocity

field in the N ',i. r-Stokes equation.

In order to close the system of filtered equations (Equations 4-28a and 4-28b), it

is necessary to introduce a third equation that relates the residual-stress tensor (r-,)

to the filtered velocity field. Turbulence closure is provided by a linear eddy-viscosity

model proposed by Smagorinsky (1963). The Smagorinsky (1963) model assumes that the

anisotropic residual-stress tensor is proportional to the filtered rate of strain (Sij) through

the following equation:

,j = -2vt Sj (4-29)









where vt is the so-called eddy viscosity and the filtered rate of strain is


42 xj 30x(
Si -i(aita i)+ (430)

The relationship between the anisotropic and isotropic residual-stress tensors is given by
Equation 4-31a, where k, (Equation 4-31b) is the residual kinetic energy.


Sr[T + 2 k6 (4-31a)

k, -t \ (4-31b)

The Smagorinsky (1963) model defines the eddy viscosity as

Vt f2

S (CsA)2 (4-32)

where is is the Smagorinsky lengthscale, Cs is the Smagorinsky coefficient, A is
representative of the filter width, and S is the characteristic filtered rate of strain,

S 2S JS (4-33)

In the current modeling approach, the Smagorinsky lengthscale is defined as Cs
0.17 (Pope, 2000) and the filter width is taken as A = ( ,5.Iz)1/3. However, using a
constant value of is in the viscous wall region is not necessarily correct as it leads to
a non-zero residual viscosity at the wall (Pope, 2000); therefore, the definition of is
requires additional treatment in the near-wall region, especially for turbulent shear flows.
According to Pope (2000), special consideration must be given to the viscous wall region
in LES turbulence closure models, and two methods are identified. Large-eddy simulation
with near-wall resolution (LES-NWR) is an appropriate choice when at least ',I '. of the
total energy is resolved by the filter and grid. If the filter and grid are only capable of
resolving 'II'. of the energy outside of the boundary lv-r (the near-wall eddy motions are
not resolved), then large-eddy simulation with near-wall modeling (LES-NWM) should









be considered. Even though the numerical model considered here employs variable mesh

i ,ii.- it is computationally prohibitive to resolve eddy motions at the viscous lengthscale

as the number of grid points necessary to do so is large in the high-Reynolds number flows

considered here. Therefore, LES-NWM is implemented in the turbulence closure scheme

by applying a van Driest damping function to fs similar to Moin and Kim (1982):


is = CsA [1 exp (z+/A+)] (4-34a)



z+= z (4-34b)

where z+ is the normalized distance from the wall, A+ = 26 is the van Driest constant

(Pope, 2000), and u, = I/bp is the friction velocity. Use of the van Driest damping

function is a rather simple wall model which allows the eddy viscosity to go to zero at the

bed. While more sophisticated wall models exist, such as those proposed by Piomelli and

Balaras (2002), the method of Moin and Kim (1982) provides the flexibility necessary to

adapt to the changing bed level during simulations. Moreover, the linear eddy-viscosity

concept employ, 1 in the Smagorinsky (1963) model itself has come under some scrutiny

for application to high-Reynolds number flows over complex geometries (Ferziger, 1996b).

One of the limitations of the Smagorinsky (1963) model is that it does not include

backscatter: the eddy viscosity t is alv--,i- greater than or equal to zero. Conceptually,

a negative eddy viscosity implies an inverse energy cascade, or one where energy is

transferred from small- to large-scale motions. Dynamic subgrid-scale eddy-viscosity

models, such as the one proposed by Germano et al. (1991), are more robust and allow for

negative values of eddy viscosity. However, such an approach requires additional filtering

of the eddy viscosity field (Zikanov et al., 2002) to prevent numerical instability when the

eddy viscosity is negative over large spatial extents (Ferziger, 1996b).









4.3.1.3 Numerics

Governing equations in the hydrodynamic modeling system are discretized on the

orthogonal Cartesian grid and solved using a variety of finite-difference techniques in space

and time. Discretization of the first derivatives implicit in the advective accelerations of

the Navier-Stokes equation (Equation 4-16b) are complicated by the non-constant grid

spacing. Since two .,.i ,i:ent control volumes may have different widths or heights, it is not

possible to solve the momentum equation in conservative form where

u O(UU) (4-35)
Oxj Oxj

For the computation of advective fluxes on computational grids with non-uniform spacing,

Nichols et al. (1980) report no reduction in accuracy if the non-conservative notation of

Equation 4-16b is employ, 1l Furthermore, Nichols et al. (1980) demonstrate that the

advective fluxes may be discretized using a theta-weighted variable differencing scheme

such that 0 = 1 corresponds to pure one-sided donor cell differencing and 0 = 0

represents a standard second-order central difference. Values of 0 between zero and

one may also be used to construct a mixed finite-difference scheme incorporating the

benefits of improved accuracy inherent in the second-order solution, while retaining

some diffusive properties of the first-order scheme. An example of the theta-weighted

discretization for an advective flux term at the grid location (i + ~ j, k) is given by

Equations 4-36a and 4-36b.

aui+ U+ [i+l (u'i .
ae J +2' UtL^ } +
:1-- 1 '2 {i --(Ui ) -U6+Ui+13-U
2 9i 6xH [ 6xi 2 2 2xi+l 2 (u
U U




e J + 6x + +l + 2 (6xi+l 6x ) (4-36b)
U +1









With the exception of the special theta-weighted discretization of the advective

fluxes, the remainder of first and second spatial derivatives are solved using modified,

second-order two- and three-point finite-difference stencils. The finite-difference schemes

are derived through application of a Taylor series expansion about a particular grid

location, such as (i + -,j, k), while accounting for the non-uniform grid spacing. The

resulting second-order two-point stencil is just a special case of Equation 4-36a with

0 0 and without the leading velocity variable ui+i. An example of the second-order

three-point stencil used for second derivatives for a point at (i + j, k) is given by

Equation 4-37.

02U,, r u. i Ui ui, 3
=2 2 2 + + iix+ (4-37)
X2 6xi(6xi + 6xi+) 6xi+(6xi6+

The hydrodynamic modeling system utilizes two different types of explicit time-marching,

or time advancement, finite-difference schemes; the first-order Euler upwinding scheme

and the third-order Adams-Bashforth scheme. As given in Duran (1999), the general

formulation of the Euler and Adams-Bashforth schemes are, respectively, given by

Equations 4-38a and 4-38b.

n+l =/ + 6tF(pn) (4-38a)


n+ n + 6 [+ 23F(qn) 16F(p- ) + 5F(9s-2)] (4 38b)


where po is a generic substitute for any primitive variable, the superscripts n denote the

time level, and F(yp) represents the evaluation of flux terms at the indicated time level.

For the case of the momentum equation (Equation 4 16b), the flux F(p") would represent

the evaluation of advective fluxes, viscous fluxes, and the pressure gradient term evaluated

at time level n. As evident in Equation 4-38b, the third-order Adams-Bashforth scheme

requires information at four different time levels, two of which are evaluations of fluxes in

previous time steps. Since in many cases the flow conditions are not known a priori, a new

simulation starting at t = 0 requires an explicit time-marching scheme that only considers









fluxes evaluated at the current time level. In the current modeling system, Equation 4-38a

is used for the first two time steps, and those fluxes are stored for use in Equation 4-38b

once the third time step has been reached.

Caution should be exercised when using Equation 4-38a with second-order spatial

differencing. Such a marriage is often referred to as the forward-time centered-space

scheme, or FTCS, and it has strict stability requirements. For convection dominated

problems, the FTCS scheme is unconditionally unstable (Fletcher, 2000). Typical

applications of the N ',i. i-Stokes equation (Equation 4-16b), such as those considered

in the current model, are more representative of transport equations with a balance

between advection (convection) and diffusion. Fletcher (2000) states that the FTCS

scheme is conditionally stable when applied to advection-diffusion transport equations, so

long as

Cf< (439)

where Cfl = u6t/6x is typically called the Courant number of the CFL (Courant-Freidrichs-Levy)

criterion. When starting simulations from equilibrium conditions, typical u velocities

are negligibly small, and in the current modeling system the time step 6t is also very

small. In general, the third-order Adams-Bashforth scheme (Equation 4-38b) has a

CFL limit similar to Equation 4-39, but the utility of the scheme is its reduction in

amplification and phase errors. The amplification and phase errors associated with

the first-order Euler scheme (Equation 4-38a) are of 0[(6t)2], while the errors of the

third-order Adams-Bashforth scheme (Equation 4-38b) are of 0[(6t)4]. In our explicit

time-marching model, the hydrodynamic time steps are much less than unity; therefore,

the errors associated with the third-order Adams-Bashforth scheme are often orders of

magnitude smaller than those of the first-order Euler scheme.

In all varieties of numerical modeling, the appropriate choice of a time step is

often precursory. The chosen time step may often be too large or too small, depending

on the current flow conditions and the resolution of the computational grid. It is









often convenient, then, to utilize a dynamic time step that adjusts accordingly to the

instantaneous flow conditions and grid resolution. This type of adjustable time step

is utilized in the current modeling system to optimize the time efficiency of the code.

The instantaneous model time step is selected by taking the minimum of time steps

computed using the CFL limit, as well as a viscous diffusivee) stability criterion, given

by the formulations of Equations 4-40a and 4-40b, respectively. A moderate amount of

conservatism is applied to the CFL limit in Equation 4-40a, such that the instantaneous

time step will alv-- produce a Cfl < 0.5, keeping well within the stability requirements

and reducing potential phase and amplification errors associated with the magnitude of 6t.


1 6xi 6yj 6zk
6t < -min i j' I (4-40a)
2 |Ui,j,k I I' I


1 6X26y26z2
v6t < (4 40b)
2 62 + 6y2 + z2
The explicit time-marching formulations given by Equations 4-38a and 4-38b are

strictly valid for a constant time step at. No formal changes are required when using a

dynamic time step in Equation 4-38a, as there is only a single time step being evaluated

between time levels n and n + 1. The derivation of a third-order Adams-Bashforth scheme

with a non-constant time step requires the application of a Taylor series expansion about

time levels n 2, n 1, n, and n + 1, and a number of substitutions. The result of such

a procedure produces roughly eight hand-written pages of numerical derivation, and the

final result is given by Equation 4-41a, in its most compact form possible.

at[ F(cp")

SF(t )t2 ) 6(&[) t2- + 3(t)(t_)+ + 2(


+ (-2) 3(t)( ) + 2(6t)2 (4
(6t2-)[62- U1t1











6t_ = t t"-1


6t2- t"-2 (4 41c)

The basic solution algorithm of the hydrodynamic code can be summarized in four

procedures:

1. Compute an initial first guess for the velocity field u4 using the discretized

momentum equation (Equation 4-16b) and variables at time level n.

2. Apply appropriate boundary conditions to the velocity field.

3. Iteratively solve for the pressure variable at the newest time level p"+l until the

continuity equation (Equation 4-16a) has been satisfied within some user-defined

threshold (typically e < 1 x 10-3).

4. Update the volume of fluid scalar F(x, y, z, t) to advect any material interfaces

through the computational mesh.

Direct solutions of the pressure field are not considered in the current hydrodynamic

model due to the additional complexity of multiple free boundaries. Instead, a two-step

projection method similar to that of Chorin (1968) is employ, -1 to iteratively adjust cell

pressures and velocities until continuity is obeyed in each control volume. The first step of

the projection involves computing the initial velocity field estimate u4 at an intermediate

time level between n and n + 1, by solving the discretized form of the Navier-Stokes

equation (Equation 4-16b). If the divergence of the velocity field (the left-hand side of

Equation 4-16a) is greater than zero, the cell pressure is reduced to encourage inflow to

the control volume. If the divergence of the velocity field is less than zero, the cell pressure

is increased to discourage inflow. These incremental cell pressures are used in the second

step of the projection to update the velocity estimates and pressure field. The general

formulations for the second step of the projection are given by Equations 4-42a 4-42c

(Nichols et al., 1980). A successive over-relaxation (SOR) method is used to enhance the


(4-41b)









convergence of the iterative solution, where a typical value of 2 = 1.8 is applied to the

incremental pressure adjustments shown in Equations 4-42b and 4-42c.


Pi,j,k + Pj,k + 6p (4-42a)




li+'Jk i k + 2t 6 (4-42b)



u- = j ut 6ot (4 42c)


4.3.1.4 Boundary Conditions

The imposition of suitable boundary conditions for numerical models is alv-7x-

challenging. To be as efficient as possible, computational grids are often selected to

be no larger than what is necessary to resolve the desired phenomenon. If the size of

the domain is not sufficiently large such that the simulated processes are far removed

from the numerical boundaries, inaccuracies in the modeled processes will arise. Various

combinations of six different types of boundary conditions are employ, l1 in the current

hydrodynamic model. Possible options include free-slip and no-slip walls, periodic

boundaries, inflow-outflow boundaries, time-dependent radiative boundaries, and sponge

I ir -. The free- and no-slip walls are often termed mixed boundary conditions because

they incorporate both Neumann and Dirichlet specifications. Periodic boundary conditions

are useful in modeling monochromatic processes where one dominant oscillatory frequency

is readily identifiable: they are intended to mimic an infinitely long domain in the

direction of periodicity. Inflow-outflow boundaries employ Dirichlet-type boundary

conditions where the flow or pressure is specified either upstream or downstream to

prohibit reflection from the boundary. The time-dependent radiative boundary condition

may be utilized to simulate a surface gravity wave for free surface simulations. When the

time-dependent boundary condition for wave forcing is used in free surface simulations,









it is often advantageous to employ a sponge l-1 vr at the opposite end of the modeling

domain. Such a l-v-r minimizes reflections from the far boundary, and prevents the need

for an outflow boundary condition.

As given in Boccotti (2000), the expressions for the free surface and three-dimensional

velocity potential are given by Equations 4-43a and 4-43b, respectively.

H
q(x, y, t) = cos(kmx + ky ut) (4-43a)



H g cosh k(h + z)
S(x, y, z, t) =- --- sin(kxx + kyy ut) (4-43b)
2 u cosh kh
Here, 0, is the velocity potential, H is the wave height, h is the water depth, kx and ky

are the wavenumbers in the x- and y-directions, respectively, and k is the wavenumber

magnitude such that k = / + k$. Assuming that the model describes irrotational

waves at the radiative boundary, a velocity potential for the fluid exists and the total fluid

velocity may be taken as i = Vo,. The three components of velocity used to force the

time-dependent wave signal may then be found using the following expressions:

0 0 '*.. '.*..
x' ay dz

These velocities constitute the Dirichlet boundary conditions applied along a vertical wall

in the modeling domain, such as at (1,j, k) for all j and k indices.

Current applications of the hydrodynamic modeling system to the simulation of

bedforms employ a free-slip boundary on the top (lid) of the domain, a no-slip condition

along the bed (sand-water interface), and periodic boundary conditions along the vertical

walls in the x- and y-directions. A summary of the Neumann and Dirichlet boundary










conditions used for the top and bottom of the modeling domain are provided below.

ap 0 aap 0
az az
au 0 u 0
Top : Bottom :
= 0 v 0

w 0 w 0

An example of the periodic, Dirichlet boundary condition used on the vertical domain

walls in the x-direction is provided below. The periodic boundary conditions result in

an overlapping of six computational grid points in order to define three common, periodic

points. The same methodology may be applied to the v- and w-velocity components,

as well as the pressure variable p. A similar procedure is used to define the periodic

boundary conditions in the y-direction.


Ul,j,k Uimax-2,j,k
Sides : Uimax-l,j,k 1
Uimnax,j,k ,k



4.3.2 Sediment Transport

4.3.2.1 Bedload

The transport of noncohesive sediment in the bedload regime occurs through a

destabilization of sediment particles due to the applied fluid stress. The total stress within

the dispersed particle matrix, however, is a function of both the applied fluid stress and a

resisting stress due to particle-particle interactions (Bagnold, 1954). Luque and van Beek

(1976) found that the particle-particle stresses dominated the total stress only for very

large bedload transport rates; therefore, only the applied fluid stress will be considered in

the current bedload formulations.

Most modern sediment transport models assume that bed particles remain static until

the ratio of destabilizing forces to stabilizing forces reaches some critical threshold. This

nondimensional shear stress considers the ratio of applied fluid stress to the buo-'in-_y









of a sediment particle resting in a fluid, and is often called the Shields stress after the

experiments of Shields (1936). The instantaneous Shields stress is given by Equation 4-44,

where rb(t) is the instantaneous applied fluid stress, s is the specific gravity of the

sediment, g is gravitational acceleration, and d is the particle diameter. While Shields'

stress is typically taken as a constant for use in steady flows, Madsen and Grant (1976)

found it suitable to apply it discretely in time by using the instantaneous bed shear stress.


0(t) = () (4-44)
(s l)gd

It is often common in both coastal and riverine applications to use a standard,

quadratic drag law (Equation 4-45) to estimate the bed shear stress. In Equation 4-45, f

is an empirical drag coefficient (f f 2.5 x 10-3), and Us is the fluid velocity at the edge of

the boundary 1liv r.

Tb pf U0 Us (4-45)

Common bed shear stress parametrizations, however, do not account for phase differences

that are often present in oscillatory boundary -1.- rs. Inertial effects in the boundary li. r

cause the fluid velocity at the bed to lead the freestream forcing, resulting in a phase lag

outside of the boundary 1-.-r with respect to the applied forcing on the bedload sediment.

In such cases, the bed shear stress parametrization given by Equation 4-45 leads to errors

in both magnitude and direction (see Figure 4-6). If instead we consider the Newtonian

definition of fluid stress, where stress is proportional to velocity shear, it is possible to

define the instantaneous bed shear stress as

au
Tb T/an (4-46)

where p is the fluid dynamic viscosity, u is the fluid velocity near the bed, and n is the

direction normal to the bed. Using this formulation, however, results in bed shear stress

values that are not in agreement with the order of magnitude of stress predicted by

Equation 4-45. To bring these formulations into agreement, the turbulent eddy viscosity









predicted by the Smagorinsky (1963) method is added to the absolute viscosity, resulting

in a more accurate representation of the total viscosity near the bed (Equation 4-47). The

relationship between Equations 4-45 4-47 is demonstrated in Figure 4-6 as a function of

fluid phase over a simulated ripple.


b ( + PVt) (4 47)

Through a series of laboratory experiments Shields (1936) determined the critical

threshold for incipient sediment motion on a horizontal bed. In terms of the nondimensional

Shields stress (Equation 4-44, this critical threshold is a function of the applied forcing,

as well as fluid and sediment properties. Values of the critical Shields stress may be read

from Shields' curve (Shields, 1936), but are also commonly found in textbooks (Julien,

1998). Shields' curve may be reproduced using the empirical approximation of Brownlie

(1981) for steady, open channel flow:


0* = 0.22 R 0.6 + 0.06 exp(-17.77R0.6) (4-48a)



u,d
Rip (4 1-1)

where 0* is the critical Shields stress for a horizontal bed, and Rp is a particle Reynolds

number based on the friction velocity u,. The predicted curve of Equation 4-48a is plotted

in Figure 4-7 as a function of the particle Reynolds number. For a given value of Rp,

sediment motion will occur for values of 0r above Shields' curve in Figure 4-7, while no

motion is predicted for values falling below the curve. There is some speculation that

Shields' method of extrapolation to the point of zero transport may overestimate the

critical stress as both grain stress and form drag due to bedforms are inherent in the

methodology (Chiew and Parker, 1194). This may partially explain the scattering of data

often seen when plotted against the original Shields curve (Brownlie, 1981; Chiew and









Parker, 1194). Theoretical derivations of Shields' curve have also been -ii-:.- -i .1 by Ikeda

(1982) and Wiberg and Smith (1985), but only considered a mostly horizontal bed.

For unsteady flow applications it is more convenient to use an empirically derived

expression for the critical Shields stress based on unchanging properties such as grain size.

The empirical expression of van Rijn (1993) is given by Equations 4-49a and 4-49b, where

d, is a dimensionless grain size parameter. Values of 0* predicted by Equation 4-49a are

plotted as a function of d, and d50 in Figure 4-8.


0.24d*1 1 < d, < 4

0.14d 0.64 4 < d* < 10

r = 0.04d;-01 10 < d < 20 (4-49a)

0.01:'./"-" 20 < d, < 150

0.055 d, > 150


d d50 2l)- 3 (4-49b)

To this point, we have only considered incipient motion on a horizontal bed. Chiew

and Parker (1194) evaluated Shields' (1936) curve for incipient motion through laboratory

experiments and found that the critical Shields stress was overestimated (underestimated)

for downward (upward) slopes. This -ii::- -I that the slope of the bed also pl .- a role in

determining the critical threshold of motion. Modification of the critical Shields parameter

for transverse and longitudinal bedslopes is achieved using

Sw r A tan2 ] tan1
Ocr O cos [ cosa 1 tan (4-50)
Stan2 tan1

where f is the angle of transverse slope, a is the angle of longitudinal slope, and Q is the

angle of sediment repose (Calantoni, 2002). Figures 4-9(a) and 4-9(b) demonstrate

the modification of the critical Shields parameter due to bedslope, as well as the

linear superposition of fluid and gravitational forcing. In Figure 4-9(b), for example,









Equation 4-50 acts to increase (decrease) the critical stress when the fluid forcing and

gravitational component of the sediment oppose (align with) one another.

The bedslope modification for the critical Shields parameter Ocr given by Equation 4-50

applies only to bedslope angles less than the angle of sediment repose, or a, 3 < Q.

Therefore, simply increasing or decreasing the critical threshold will not prevent bedforms

from becoming too steep, nor will it simulate gravity-induced transport. Watanabe (1988)

-,-.-. -i .,1 modifying the local transport rates using

8zb
Qb= QbO- O, I (4-51a)
Ozb
Qby Qbyo lQbol aO (4-51b)

where E (F 2) is a diffusivity parameter and Qb.o and Qbyo are the x- and y- components

of bedload transport on a flat bed, respectively. The effect of Equations. 4-51a and 4-51b

is to enhance downslope transport while diminishing upslope transport. Unfortunately,

the choice of an appropriate value for the diffusivity parameter E is somewhat vague as it

is often used as a tuning parameter (Watanabe, 1988; Johnson and Zyserman, 2002). A

poor choice of E leads to excessive smoothing (diffusion) of the transport and morphology

when it is too large, and is not effective at preventing unrealistically steep bedforms when

chosen too small.

An alternative method to simulating gravitational transport is to derive a particle

shear stress based on the buovint weight of the spherical particle submerged in the fluid,

and the local bedslope angle. A schematic of the particle and resultant static forces

is shown in Figure 4-10 where a and f are the longitudinal and transverse bedslopes,

respectively. The particle shear stress is derived by considering the buovint weight of the

sphere where

F =- 3 (ps p)g (4-52a)

F = Fsin a (4-52b)









Fgy = F, sin f


and the downslope gravitational components Fx and Fgy act over an area roughly equal to

the cross-sectional area of a sphere such that

2
mT = ,gd(ps p) sin a for a > Q (4-52d)

and
2
7~= 3gd(p,- p)sin 3 for 3 > (4-52e)

When a > (> Q), the particle shear stress of Equation 4-52d (Equation 4-52e) is

added to the bed shear stress Tb (rT) to simulate gravity-induced transport. Current

model simulations -,.., -1 that the addition of the gravitational stress component is

sufficient to prevent bedforms from growing steeper than the angle of repose.

The nondimensional bedload function of Eintstein (1950) (Equation 2-7) describes

the relationship between the volumetric bedload transport rate and the particle buov, i v.

By measuring the rate of bedload transport in a laboratory, it is then possible to derive a

relationship for f)b from Equation 2-7 that equates the amount of transport to the level

of applied forcing. A number of functional relationships for )b have been empirically

derived for currents (\ I. v- r-Peter and Miiller, 1948; Nielsen, 1992; Ribberink, 1998;

Camenen and Larson, 2005) and parametrizations have also been developed for waves

(\! k, .en and Grant, 1976; Bailard and Inman, 1981; D1i.b iiii and Watanabe, 1992;

Soulsby, 1997; Ribberink, 1998; Camenen and Larson, 2005). Most of these functional

equations (see Table 2-1) follow a 1"'".- i 1 .-- formulation as given by Equation 2-8,

where the exceedance of the critical Shields stress is raised to a specified power, and then

multiplied by a constant of proportionality. Examples of commonly used power laws for

bedload transport by currents (Equation 4-53) and by waves (Equation 4-54) are found in

At, v, r-Peter and Miller (1948) and Ribberink (1998), respectively.


b((t) 8 ( (t)- 0 (4-53)
10(01l


(4-52c)











4eb(t) t)-t 1.65 (4-54)

More recently, Camenen and Larson (2005) --.- -1I,1 that most power laws

underestimate bedload transport for Shields stress values just slightly larger than the

critical threshold; and of course they failed to predict any motion for Shields stress values

below the critical threshold. Camenen and Larson (2005) offered an alternative equation

(Equation 4-55) for bedload transport based on the concept of continuing transport for all

values of Shields stress greater than zero.
+bt(t) 21.t
b(t)= 12 0 (t) i. exp(-4.50cr/10(t)) (4-55)

The behavior of Equations 4-53 4-55, as well as the power law of Nielsen (1992)

(Table 2-1), is demonstrated in Figure 4-11 for a range of Shields stress values typically

found in the flatbed and rippled-bed transport regimes. The inset panel in Figure 4-11

shows that Equation 4-55 predicts larger values of Nb below and slightly above the

specified critical Shields stress of Ocr = 0.04, but predicts smaller values of )b as Shields'

stress increases, when compared to the power law formulations.

The choice of which bedload transport equation to use is somewhat arbitrary, but

Madsen (1991) advocates strongly for use of the M iv, r-Peter and Miller (1948) power

law in coastal applications. Substituting Equation 4-53 for Kb(t) in Equation 2-7 yields a

direct solution for the volumetric bedload transport rate:

0(t) l5
Qb(t) 8 0 (1 I (t)l 15 (- 1)gd3 (4 56)

where the term 0(t)/l0(t) is used to assign direction to the bedload transport rate in

keeping with the sign of the bed shear stress Tb.

4.3.2.2 Suspended Load

Parametrizations for the volumetric suspended sediment transport rate exist

(McLean, 1991; van der Werf et al., 2006), but their implementation in a time-dependent,









three-dimensional model would be unnecessarily ad hoc. Often times these parametrizations

require a priori knowledge of either the flow field or the sediment transport conditions,

such as a reference concentration above the bed. Others have -ii-:.- -I. 1 adapting power

laws similar to those used for bedload transport by increasing the critical threshold level

and the exponent (Yalin, 1977; Hogg et al., 1997). An alternative approach for computing

the suspended load transport is the Cu-integral approach discussed in Nielsen (1992). In

this approach, the instantaneous volumetric suspended load transport rate is defined as

fD
Qs Jt) Cz,t )z (4 57)

where Zb is the vertical location of the bed, D is the depth, C is the filtered volumetric

concentration, and us, is the sediment velocity. The suspended load transport rate

Qs, given in Equation 4-57 is computed for the horizontal x- and y-directions
by substituting the appropriate component of the sediment velocity u,, given by

Equation 4-59.

The Cu-integral approach requires an estimation of the local volumetric concentration

field, which may be determined from either a particle trajectory model (Nielsen, 1992),

or by modeling an advection-diffusion equation (Equation 4-58) for the scalar quantity

(Zedler and Street, 2001). While both approaches incorporate some order of empiricism

into the modeling approach-mainly through the selection of a pick-up function-the

particle trajectory approach also requires the solution of a momentum equation for

the sediment phase. The trajectory method introduces further uncertainty through

parametrizations of drag and inertial forces; therefore, the particle trajectory model is not

used to simulate suspended load. Instead, the volumetric concentration field is modeled

using

a + a (u w)C V + ai 0 (4-58)
at 9xj as ) 9xjI
where C is the filtered volume concentration, vt is the turbulent eddy viscosity predicted

by the Smagorinsky (1963) model, w, is the hindered settling velocity (Equation 4-61a),









and as is the Schmidt number. The Schmidt number as represents the balance between

momentum and mass diffusivity (viscosity and molecular diffusion), and is often used

as a scaling parameter in boundary l-1v-r models. We assume here that as = 1 for our

computations, as in Zedler and Street (2001).

In the model, it is assumed that the sediment velocity is defined by Equation 4-59.

ui [u, v, (w w,)] (4 59)

The settling velocity of a single particle in quiescent fluid is a function of sediment

characteristics and fluid properties. A general expression for this constant settling velocity

(wso), as proposed by van Rijn (1984), is

10vlj' 0.01(s 1)gd3 1
ws d =O 1 + 2O.(s -) 1 (4-60)


where d is the mean grain diameter, and s is the specific gravity of sediment (taken here

as s = 2.65). However, the settling velocity is affected by the local concentration of

particles in the fluid; regions of high concentration have a smaller settling velocity relative

to that of a single particle, as the fluid drag on the concentrated region is increased.

Richardson and Zaki (1954) provide a parametrization for the hindered settling velocity

based on the local concentration and sediment characteristics:

w, w,(1 C)q (4-61a)



4.35 P-0.03 0.2 < Rp < 1.0
q = 4.45 R- 010 1.0 < Rp < 500 (4-61b)

2.39 500 < ~R

dw(4
RP so (4 61c)
V









where C is the filtered volume concentration and Rp is the settling particle Reynolds

number. A graphical plot showing the reduction of settling velocity for increasing

concentrations is shown in Figure 4-13 for three different values of the settling particle

Reynolds number. For volumetric concentrations less than about 0.5' ,. Figure 4-13

demonstrates that there is very little reduction of the settling velocity wo. The effects of

hindered settling, as predicted by Equation 4-61a, are negligible for very dilute flows, but

are significant for highly concentrated flows (Figure 4-13). The affect of hindered settling

velocity on sediment transport and morphology is discussed in C'! lpter 6, and is found to

have a modest effect due to the relatively low suspended sediment concentrations modeled

in most simulations (10-6 < C < 10-3).

Speculation exists regarding the applicability of Equation 4-58 to simulating

suspended sediment transport, mostly due to the absence of particle-fluid or particle-particle

interactions (Zedler and Street, 2001). In the advection-diffusion equation, we assume

that the volumetric concentration is a conservative scalar deviating from the behavior

of a fluid particle only by the particle settling velocity. In this manner, only the

fluid-particle interaction is modeled, and other interactions such as turbulence damping

from particle-fluid interactions, and energy dissipation through particle-particle interactions,

are neglected. When working with sand particles whose density is much greater than that

of fluid, Elghobashi (1994) report that particle-fluid interactions are important when

10-6 < C < 10-3, and that particle-particle interactions become significant when

C > 10-3. However, Villaret and Davies (1995) -,i--.- -1 that Equation 4-58 is appropriate

for concentrations less than approximately C < 0.1 and has been successfully used for

concentrations up to C a 1 Current simulations of the model system have persistent

concentration values in the range 10-6 < C < 10-3 where Elghobashi (1994) states that

particle-fluid interactions should be considered; therefore, it may be necessary to consider

the particle-fluid turbulence damping in future versions of the modeling system.









There are inherent difficulties in modeling the concentration field with the advection-diffusion

equation (Equation 4-58). Similar to the volume of fluid function F, the concentration

values vary through the range 0 < C < 1. Without the inclusion of flux-limiting schemes,

traditional finite-difference formulations tend to result in values of concentration less

than zero and greater than one. These inaccuracies result in non-conservative values

of the concentration scalar, sI.-. -. ii-; that sediment is either gained or lost in the

computational grid. To overcome these difficulties, we adopt a two-step process for solving

Equation 4-58. First, the Donor-Acceptor method of Johnson (1970) is used to estimate

the proper amount of concentration (mass) to flux from one cell to its neighbor based on

the advecting velocity and the amount of mass available to either donate or accept. The

concentration values are updated in time to an intermediate value between time level n

and time level n + 1. The second step of the solution sweeps through the concentration

field and computes the diffusive fluxes using the three-point stencil for a second-order

central difference given by Equation 4-37. Once the diffusive fluxes have been calculated,

a final first-order time advancement from the intermediate time level to n + 1 is performed.

This two-step process is given by Equations 4-62a and 4-62b in abbreviated form.

Although the second step is accurate to O[(6x)2], the leading order of error for the entire

process is similar to the VOF method with 0[(6t), (6x)].


C* = C" 6t(uj wo)VC (4-62a)



C"' = C* + 6t V + vt VC (4-62b)

The boundary conditions on the concentration field are specified as free-slip along

the top of the rigid lid domain, and periodic in the x- and y-directions (along the

vertical walls of the model domain). As written in Equation 4-58, the advection-diffusion

equation does nothing more than transport sediment from one place to another based on

the local hydrodynamics. Hence, a special boundary condition must be used along the









bed to entrain the sediment into the flow (Nielsen et al., 1978). The Neumann boundary

conditions for the top and bottom of the modeling domain are given by


Top : C 0 Bottom : ac E(t)
5z an v

where E(t) is a time-dependent sediment pick-up function. Similar to the nondimensional

bedload function of Eintstein (1950), a nondimensional entrainment function may be

expresses as
E
4 (4-63)
Ps (s 1)gd
which relates the volumetric rate of bed material exchange to the buoi i, i- of the

sediment particle. Various forms of 4, have been -ii -.-. -I1. in the literature. Similar to

the power law bedload relations for I)b, the nondimensional pick-up functions often are

formulated in terms of some exceedance of a critical Shields stress raised to an empirically

determined exponent. Three examples of nondimensional entrainment functions are found

in Fernandez-Luque (1974), N I, iI: i.- and Tsujimoto (1980), and van Rijn (1984), and

are given by Equations 4-64a 4-64c.


Ip = 0.02(|10 0cr)15 (4-64a)



4, 0.020(- 0.0) (4-64b)



(i 0.00033 [(s )gd 0"1 0- 0, (4-64c)

The behavior of Equations 4-64a 4-64c with respect to 0 are demonstrated in

Figure 4-12. Although Equations 4-64a 4-64c were empirically determined for steady

flow, Nielsen (1992) '--- -1. .1 that the entrainment function of van Rijn (1984) could be

adapted to unsteady flow by considering the time-dependent Shields parameter 0(t). The









adapted form of the entrainment function given by Nielsen (1992) is

E(t) = 0.00033 S )0. dj ( 0 -W ) 1.5 (4-65)
00.2 C T

when 0(t) > Oc0, and E(t) = 0 if the time-dependent Shields parameter (Equation 4-44)
is below the critical value. The pick-up rate E(t) has units of volume rate of exchange of
bed material per unit area per unit time. Also, when substituted for 4p in Equation 4-63,
Equation 4-64c is equivalent to Equation 4-65 with the exception of the time-dependent
Shields parameter.
4.3.3 Morphology

The two-way coupling between fluid hydrodynamics and sediment transport is
produced through differential changes in 11llr:metry with respect to time. These
time-dependent, morphological changes produce positive feedback loops between the
hydrodynamics and resulting sediment transport. In this case, fluid forcing applied to
the sediment produces differential transport in the horizontal (bedload) and vertical
(deposition and entrainment) directions leading to changes in bed elevation, which further
act to alter the nearbed hydrodynamics. In the most simple form, the time rate of change
of bed elevation is predicted by the sediment continuity, or Exner, equation:


(- n) + VQt 0 (4-66)

where n, is the porosity of the bed material, zb is the vertical location of the bed, and Qt
is the total sediment transport (Qt = Qb + Q), or the sum of the bedload and suspended
load components. In Equation 4-66, V is the gradient operator acting on Qt in the x-
and y-directions such that
VQ (Qt + Qt
Q Ox a y
differentiates the total sediment transport Qt with respect to x and y, individually.
The simple sediment continuity equation (Equation 4-66) has been applied to a
number of general riverine and coastal morphology problems (Foti and Blondeaux, 1995b;









Gessler et al., 1999; Parker et al., 2000; Johnson and Zyserman, 2002; Long et al., 2007).

In the current modeling system, the suspended load sediment transport rate is predicted

by the Cu-integral approach given by Equation 4-57. Instead of lumping the bedload Qb

and suspended load Q8 transport rates together as Qt in Equation 4-66, it is advantageous

to separate them into horizontal and vertical flux components for application of the

control volume approach employ, ,1 in the modeling system. If the divergence of suspended

load transport is instead replaced by the balance between the rates of entrainment and

deposition, the resulting two-dimensional sediment continuity equation may be written as

aZb
(1 np) + VQb + E- woCb 0 (4-67)

where E is the entrainment rate, wo is the constant settling velocity, and Cb is the

average volume concentration immediately above the bed (Nielsen, 1992). When taken

as a whole, the term (E W8oCb) in Equation 4-67 is equivalent to the divergence of

the suspended load transport rate VQs, as it represents the differential flux of sediment

through the top of the control volume. The physical interpretation of the differential

vertical flux term is that when the rate of entrainment is greater than the rate of

deposition (E WsoCb) > 0, the time rate of change of zb is negative, corresponding

to a decrease in bed elevation. Conversely, when the rate of deposition is larger than

the rate of entrainment (E WsoCb) < 0, the time rate of change of Zb is positive,

corresponding to an increase in bed elevation. A schematic of a two-dimensional (x z)

sediment control volume is provided in Figure 4-14, along with a representation of the

terminology used in Equation 4-67.

Numerous variations of the Exner equation (Equation 4-67) are found in existing

literature. For instance, Parker et al. (2000) include the effects of sediment mixtures,

and Gessler et al. (1999) incorporate multiple sediment size classes and simulate the

exchange of particles between the bedload and suspended load lV. -r. Such intricacies

are reserved for employment in this modeling system at a later date since merely solving









Equation 4-67 numerically presents special challenges. Details regarding the numerical

solution of Equation 4-67 and filtering requirements follow.

4.3.3.1 Finite-Difference Methods

The Exner equation (Equation 4-67) may be generalized as a one- or two-dimensional

advection/convection equation. Such equations are difficult to solve numerically, as there

is no physical mechanism (diffusion) for balancing the advection component. Additionally,

the simulation of bedforms require special "shock capturing techniques so as not to

artificially smooth steep-sided morphology. The following finite-difference formulations

were tested and found to be insufficient for solving Equation 4-67:

Forward Time, Backward Space (FTBS);

Forward Time, Centered Space (FTCS); and

FTCS-Lax-Wendroff (FTCS-LW).

The FTBS upwinding scheme is first order accurate in time and space, and has

a complex amplification factor resulting in strong dissipation of zb (Fletcher, 2000).

The dissipation of the upwinding scheme is sufficiently large to control the advection,

but artificially damps perturbations in the bed level. When applied to the simple

Exner equation (Equation 4-66), the FTBS upwinding scheme results in the following

discretization for the x-direction:


zbi Zb 26x(1t- np) 1 ) + Q )] (4 68)

where 6x and Rt are taken as constants for the sake of brevity, and Cp, is a bedform

propagation phase speed given by (Long et al., 2007) as
aQt
C,(zb) < ax (4-69)

The FTBS upwind scheme is conditionally stable for |Cp, < 1. Estimation of Cp has

undesirable effects when applied to morphology with little to no change in slope ( = 0),

as the value of Cp, approaches infinity (Long et al., 2007); therefore, discretization









requiring the estimation of Cp should not be used when simulating small-scale bedforms

or mostly flat beds.

The FTCS scheme is unconditionally unstable for all applications to Equation 4-66,

but is conditionally stable for advection-diffusion transport equations, as discussed

earlier. The leading order truncation error of the FTCS scheme for the Exner equation is

second order in space and first order time O[(6x2, 6t)], and when applied to the advection

equation results in an imaginary amplification factor (Fletcher, 2000). According to Long

et al. (2007), a typical method for counteracting the instability of the FTCS scheme is

to add artificial dissipation, or viscous effects, to the finite difference formulation. The

Lax and Wendroff (1960) scheme addresses this issue by incorporating diffusion of the

bed elevation (zb). As presented in Johnson and Zyserman (2002) and also in Long et al.

(2007), the discretization of Equation 4-66 for the FTCS-Lax-Wendroff scheme yields:

+l at 6 t2C
zbn = z1 (Q Q ) + ( -1}t1 2 2z + z b ) (4 70)
bi bi 26x(1 np) "+I ti-1 26X2 bi-

where again, 6x and 6t are assumed to be constant in space and time, respectively. The

addition of the diffusive term in Equation 4-70 produces a scheme that is second order in

both space and time O[(6x2, 6t2)], and is conditionally stable for |Cp, t < 1. As noted by

Long et al. (2007), an overestimation of C,, since it is taken to the second power in front

of the diffusive term in Equation 4-70, produces excessive smoothing of the morphology;

however, if Cpi is underestimated the diffusive term is no longer large enough to balance

the advection, and Equation 4-70 reduces to a standard FTCS scheme.

Currently, the two-dimensional Exner equation (Equation 4-67) is solved using an

Euler-WENO (Weighted Essentially Non-Oscillatory) scheme after Long et al. (2007).

The "Euler" part refers to the first order time marching scheme, while \V\NO" describes

the discretization of the horizontal sediment transport fluxes. WENO methods are based

on the ENO (Essentially Non-Oscillatory) method of Harten (1983), which employs

a smoothed stencil to evaluate sediment fluxes over a number of .,Ii i,:ent grid points.









By taking a weighted average of the ENO stencils, Long et al. (2007) demonstrate the

ability to avoid oscillations, and excessive smoothing, near sharp discontinuities in the bed

elevation. The WENO scheme has been successfully applied to cross-shore morphology

models by Long et al. (2007), but there are no references to its utility in small-scale

modeling.

For a simple, one-dimensional Exner equation (Equation 4-66) where Q = Qt for

simplification, the flux of sediment is approximated by the WENO method as

dQ Qi+ Qi-1/2
2 (4-71)
dx 6x

where Qi 1 and Qj_1 are approximations of the sediment transport rate at grid points

S+ and i respectively. Each approximation of the sediment transport rate may be

further divided into left- and right-biased fluxes such that


Q+ = Q + Q+, (4-72a)
2 2 +2

and

Qi Q + (4-72b)
2 ?- 2 2

where Q- and Q+ correspond to bedform propagation in the negative and positive

x-directions, respectively. For brevity, the remainder of the formulations will only focus

on the approximation of the transport component Qi+ given by Equation 4-72a, but

a similar methodology may be applied to Qi_ by shifting indices back one full step.

Moreover, the following formulations are easily applied for transport in two directions

(x, y) by replacing all i indices with the corresponding j index.
The left- and right-biased fluxes in Equation 4-72a each have their own unique

five-point finite-difference stencil and weighting coefficients. The left-biased flux is given

by
= 70 + 72Q + 3+ for C +> 0 (4-73a)











Q_-= 0 for Cp 1<0

1 7 11

S 3 Qi-2 Qi-1 + Qi


1 5 1
Q Qi-1 + Qi + Qi+
2 6 6 3


(4-73b)


(4-73c)



(4-73d)


1 5 1
Q3 = 1 Q + Qi+ tQi+2 (4 73e)

are three finite-difference stencils used to compute the value of Qi+. The weighting

coefficients 71, 72, and 73 in Equation 4-73a, as determined by Jiang and Shu (1996),

produce a fifth-order accurate estimation of the transport component at i + I and are

computed as follows:


71- = 1
I + 2 + 3

72 +2

73 3
7 + 2 + 3


(4-74a)

(4-74b)

(4-74c)


0.1
1 0. (4 75a)
(S1 + C)2
0.6
S- (4-75b)
(S2& )2
0.3
1 0. (4-75c)
(S3+ )2
with c is taken as a very small number (t 10-20) and the smoothness measurements Si,

S2, and S3 are computed as:


13 1
S1 = (vi 2v2 + V3)2 + (1 4V2 + 33)2


S2 (V= 2 2, + 4)2 + (V2 V4)2
12 4


and


where


(4 76a)


(4 76b)









13
S3 = NV3
12


1
2v4 + .)2 + (3V3
4


1 = Qi-2

v2 Qi-1

V3 Qi

V4 = Qi+

' = Qi+2.


Following a similar procedure, the right-biased flux Q+ of Equation 4-72a is defined
'+2


2 -1Ii^ 1 )2 3
1Qi+ _, + 3Qi+l


for Cp


0 for C, >0
+2


1 5 1



1 5 1
Qi- + Qi 6Qi+l2
+2 3 6 6


11
Q3 Q
B2 6


(4-78c)



(4-78d)


(4 78e)


7 1
7Qi+2 + Qi+3
6 3


are three finite-difference stencils used to compute the value of Qi+ The weighting

coefficients 71, 72, and 73 in Equation 4-78a are computed as follows:


1 + 2 + 3


1I + 2 + 63


(4-79a)


(4-79b)


with


4v4 + .)2


(4 76c)


(4 77a)

(4-77b)

(4-77c)

(4-77d)

(4 77e)


and


where


2i


(4 78a)


(4-78b)










73- 3 (4-79c)

where
S 0.3
0 .= (4-80a)
(S1+e)2
0.6
(2 (2 (4-80b)
(S2 +)2
0.1
(1 (4-80c)
(S3 +)2
where S1, S2, and S3 are computed as

13 1
S1 = (v2 2 + U4)2 + (2 4V3 + 3U4)2 (4 81a)
12 4
13 1
2 -= ( 24 +' )2 + ( .)2 (4 81b)
12 4
13 1
3 -(4 2, .. + + -(4 '4 .. + 2 (4 81c)
12 4
where ',. =Qi+3 is added to the previous definitions given by Equations 4-77a 4-77e.

Unlike the FTBS (Equation 4-68) and FTCS-Lax-Wendroff (Equation 4-70)

schemes that required an estimation of the bedform propagation phase speed Cp, only

the sign, or direction of bedform propagation, is needed in the WENO scheme (see

Equations 4-73a 4-73b and Equations 4-78a 4-78b). One simple method of computing

the propagation phase speed is given by the simple upwinding scheme of Roe (1987) where

CP 1 Q- I Q for z~ bi+ z. (4-82)
2 (1 np) zbi+ bi

When Cp > 0 the bedform propagates from left to right, or in the positive x-direction.
2+
Alternatively, the bedform propagation direction is from right to left when Cp,+ < 0. The

WENO scheme, however, only uses the sign of Cp, to select the left- or right-biased fluxes,

so a much less restrictive method is given by

(Qi+l Q)(bi ) (483a)
+ I(Qi+l Qi)(zbi+l bi)l









which produces values of Cp {- 1,0, 1} corresponding to bedform propagation in

the {-x, +x, +x} directions. A similar methodology is applied to the y-direction for

two-dimensional transport calculations where

r+ (QJ+I Qi)(zbj+ Zbj) (483b)
C (4-83b)
I+ |(Qj+l Qj)(zbj+, -bj )

produces values of C, = {-1, 0, 1} corresponding to bedform propagation in the

{-y, +y,+y} directions.
Combining the WENO discretization of the horizontal sediment transport flux

terms with the first-order Euler time-marching scheme, and applying them to our

two-dimensional Exner equation (Equation 4-67) results in the following finite-difference

formulation:
n+1, ,) [(Q/+. 1 .- Qb. i, + Qb.. Qb ,+ j-w (484)
(1-r,-- n,) [K 2 2 ) K 2 + -WCbij (484)
( np) _6x 6y

which is first-order accurate in time and fifth-order accurate in space O[(6x5,6t)].

4.3.3.2 Filtering Techniques

The fifth-order WENO finite-difference scheme presented in the previous section

is often not ..- --ressive enough to damp high-frequency oscillations of the bed level by

merely smoothing the transport gradients. Johnson and Zyserman (2002) demonstrate

that high-frequency spatial perturbations in simulated morphology are produced

through a dependence of the bed celerity with bed elevation. As a consequence, these

high-frequency signals can bread even higher harmonics until the entire bed elevation

signal is truncated by numerical inaccuracies. It is often necessary, therefore, to apply

some artificial smoothing to the morphology in order to damp the high wavenumber

errors while allowing the low wavenumber signal to dominate. This type of smoothing

may be accomplished by invoking a low-pass filter of the type given by Equation 4-26a

for a one-dimensional field. For morphology that varies in both the x- and y-directions,









however, the two-dimensional filter of Johnson and Zyserman (2002) is emplovc.,


5ij 45i,j +'s(QPi+l,j 2i,j + i--l,j) + 's(Vi,j+l 2pi,j + 4ij-l) +
1
+ 2 s(i+1,j+l i+lj-1 + 9i- -l1 i-lj+l) (4-85)

where p is any primitive variable, ( is the low-pass filtered result, and 7s is a smoothing

coefficient. Unlike the LES filtering process where Equation 4-26a is applied independently

in the x-, y-, and z-directions, smoothing the morphology requires use of the

two-dimensional filter as it includes cross-slope components represented by the last

group of terms on the right-hand side of Equation 4-85.

Johnson and Zyserman (2002) show that, when performed, a linear stability analysis

of Equation 4-85 results in the following amplification factor:


Gi -p 1 47t + 27,(cos (x + cos (y + sin ( sin (y) (4-86a)


( 2 cos (4-86b)
a
S2 sin Ob
( (4 86c)
a

where Ob is the direction of bedform propagation, and a is the number of grid points

per wavelength. The 2D low-pass filter given by Equation 4-85 is stable (unstable)

when 7s < 0.25 (> 0.25): values of 7y > 0.25 result in amplification factors GI-p

greater than unity. The amplification factors of the 2D low-pass filter are shown in

Figures 4-15(a) and 4-15(b) for smoothing coefficient values of s = 0.125 and 7, = 0.25,

respectively. While application of Equation 4-85 to the time-dependent morphology

successfully damps the high-frequency disturbances, it also tends to damp low-frequency

(or large-scale) features over a given amount of time, many of which may be physically

realistic. This behavior is demonstrated in Figures 4-15(a) and 4-15(b), where even for

, = 0.125 up to 5'. of the signal is damped for bed features resolved by ten or more grid

points (Ar > 10 6x). When applied on morphological time-scales, subsequent damping of









5'. during each application of Equation 4-85 may completely obscure physically realistic

bedforms, especially for the small-scale features considered in this study.

The shortcomings of the single-application 2D low-pass filter (Equation 4-85) may

be overcome by considering the hybrid filtering algorithm -i--., -1. I by Jensen et al.

(1999). As outlined by Jensen et al. (1999), the filtering algorithm consists of the following

four-steps:

1. Filtering Zb to obtain Zb;

2. Computing 6zb Zb b;

3. Filtering 6zb to obtain 6zb; and

4. Updating the new bed elevation using Zb =b + 6b.

Jensen et al. (1999) report that this hybrid filtering algorithm effectively damps signals on

the order of the grid size O[(6x, 6y)]. The -i-i.: 1.1 hybrid filtering algorithm is applied

to new estimates of the bed elevation z+1, which are provided by the 2D Exner equation

(Equation 4-67), in the following manner:

S+l '+1 (z+1 2+ + z ) + 7 1 +l 2 "+1 + ,+ +

2-n+1 -z z" zb ) (4 87a)
Sbi+ j ,j +1 i
-I "+ 1 'nr+l I Z+l + +, (4 87a)
+ 28b +1,^ _i+ijj-i + '-l i-ij+i

n+l = -n+l_ n+1 (4-87b)

Sn+1
+1 l (Z7s(z -+l 26-n+1 + zn+l) +

+7s(6n+l1- 267n+1 + 6n+l )+

1 0+1 b -
+ 1(6z S i+l,j+l b bb+1j- i 6 ,j-1 b +1,
n+1
n +l= n+l + 6 (4-87d)

where ?n+l is the final smoothed estimate of the bed elevation at the newest time level.

In the current modeling scheme, one must be cautious as to how often the hybrid filtering

algorithm given by Equations 4-87a 4-87d are applied to the morphology as it is still









possible to over-filter the bed elevation if used too frequently (i.e., at every hydrodynamic

time-step). An optimum frequency interval (6tf) for the hybrid smoothing applied to the

current modeling system corresponds to a range of roughly 0.05 s < 6tf < 0.1 s.

Applying the 2D low-pass filter (Equation 4-85) in the manner -ii-.-. -I. 1 by

Jensen et al. (1999) (Equations 4-87a 4-87d), Johnson and Zyserman (2002) find

the amplification factor of the total algorithm to be


Gf = G1_p(2 GIp) (488)


where Gl-p is the amplification factor of the single-application 2D low-pass filter given

by Equation 4-86a. Amplification factors computed with Equation 4-88 for values of

s = 0.125 and 7s = 0.25 are plotted as a function of wavelength resolution (number of

grid points per unit disturbance wavelength) in Figures 4-16(a) and 4-16(b), respectively.

When compared to the behavior of the single-application low-pass filter (Equation 4-85)

for = 0.125, Figure 4-16(a) demonstrates that the hybrid filtering algorithm damps

< 1 of the morphology signal for features resolved by six grid points (6 6x) or more,

while leaving features resolved by ten grid points or more unbiased. Although the behavior

of the hybrid filtering algorithm for >7 = 0.25 completely damps the 2 6x wave, as shown

in Figure 4-16(b), it continues to have a diminutive effect on features resolved at the

10 6x wavelength. Model simulations of small-scale bedforms in the current study have

confirmed the assertion of Johnson and Zyserman (2002) that using s = 0.125 sufficiently

damps high-frequency oscillations in the bed elevation.































Figure 4-1. A schematic of the one-dimensional linear model of bedload transport.

z




L












0

Figure 4-2. A representation of a generic three-dimensional model domain with domain
length labels Lx, L,, and L, the bed elevation zb, and velocity vectors. The
blue color represents fluid and the brown color represents sediment.














z

1


p-l U i+ ,j,k


(k )


x(i) z l )

(a) x z plane


5(k) --* -


-i


y(j- )


-*t---V,j +J,


YJb ) Y( + 1)

(b) y z plane


Figure 4-3. Model control volumes showing the location of primitive model variables and
definition of indices for a I.:--- red grid in the x z and y z planes.


Z





z


II


6xi


ti1'


J SZk


(a) x z plane


(b) y z plane


Figure 4-4. A representation of variable mesh scaling in the x z and y z planes. For
each plane, the horizontal grid spacing is uniform in the orthogonal direction.






136


z


----,X w,j,c+
7- 1


kh + 2


i i -


z(k) --- -


(k


Fi,3,k
-0-----
mink
^JL


2







Pi,3,


'( )
2


- --------- -- ------ ---


- - - - - - -


_/i





















































(a) Grid volume ratio


y(cm)

(c) Grid spacing in y


3xyz 3. max
6 1
09
08
07 '
06
05 C
04 -
02 0
01


x(cm)

(b) Grid spacing in x


50


40


30


20


10


0 02 04 06 08 1 12 14 16 18
Grid Spacing, 6z (cm)

(d) Grid spacing in z


representation of (a) variable mesh clustering and mesh scaling in the (b)

-, (c) y-, and (d) z-directions.


18
16

S14
1 12


S108
U

, 06
06
04
02


Figure 4-5. A

X-





























_ 5-- ----------------------------------------
I 1/10 z_ _




gp 5
---- 1 ----- ----------------



x (cm)
150
o 3U/10
15











531/10

5 x (cm) 10

S15












S10 7 10


10



0 E-----------
5 ---------------------------------












1,
15













5 (cm)


0 10




0 x (c 10







x(cm)
15
0 10 /10







S5
----------------- -

















1o
x (cm)


8 6 /10









10 /10



5
0 /10--------



S5 X(cm) 10
10 8 10
5







...5r--....-7777757 --'7-- 1



E 5


1 -


Predictions of bed shear stress with three different formulations. The

red, green, and blue lines correspond to the stress formulations give in

Equations 4-45, 4-46, and 4-47, respectively. The solid black line denotes

the location of the bed, in the horizontal, relative to the stress predictions.


Figure 4-6.























0,

10'








102
10


Figure 4-7.


10' 102 103
9 (U Ud/v)


Estimation of Shields' curve for incipient sediment motion based on the
equation of Brownlie (1981).


0 0.2 d50 (cm)


dso (cm)
0 0.2 d ( ) 0.4 0.



















0 50 100 15


Figure 4-8. Critical Shields stress curve based on the dimensionless grain size method of
van Rijn (1993).








































Flow Direction

W Wsin a a
31 W sin,,3 > Flow Direction W ia
(a) Transverse slope (b) Longitudinal slope


Figure 4-9. Particle forces acting on longitudinal and transverse slopes.



























F9


Figure 4-10. A schematic of gravitational forces acting on a sphere resting on a sloping
bed with angle a or 3 to the horizontal.





0.12 1 1

MPM48
0.1 N92
------ R98 /
-..-.... CL05 / /



0.0 / / / /


0.04 / /


0.02 /
/ o I,




0 0.02 0.04 0.06 0.08 0.1
00



Figure 4-11. Behavior of bedload transport equations at low Shields numbers. The critical
threshold for this example is taken arbitrarily as cr = 0.04. Legend names
correspond to the transport relations of M1, i, r-Peter and Miller (1948)
(\!PM48), Nielsen (1992) (N92), Ribberink (1998) (R98), and Camenen and
Larson (2005) (CL05). The limits of the inset panel are represented by the
dark black box on the larger figure.















0.002


0.0015




0.001





0.0005




0


FL74
NT80
VR84










004 0045 005 .





0.04 0.06 0.08 C
0


Figure 4-12. Behavior of sediment pick-up functions at low Shields numbers. The critical
threshold for this example is taken arbitrarily as r, = 0.04. Legend names
correspond to the transport relations of Fernandez-Luque (1974) (FL74),
N ,; .1: .vt and Tsujimoto (1980) (NT80), and van Rijn (1984) (VR84). The
limits of the inset panel are represented by the dark black box on the larger
figure.









0.8
=9p, 0.5


0.6- 9I = 40


p \9= 9 7000
0.4



0.2



0 0.2 0.4 0.6 0.8 1
C (cm3 / cm3)


Figure 4-13. The modification of the relative settling velocity as a function of
concentration using the equations of Richardson and Zaki (1954).


0 0001


6E-05k


4E-05

2E-05

(903 0030


I I I I *


i.




























WsoCb E






C(x,y,z,t)


Qb Qb + dQ4 dx
dx




Figure 4-14. A two-dimensional sediment control volume showing relevant transport
terminology and a schematic of sediment distribution. In the model, the true
control volume extends into the y-direction, too.

























0.8


0.6


0.4


0.2


---- e=o
- Ob = 22.5
........... = 45

I I I I I I I I
2 4 6 8 10 12 14 16 18
Grid Points/Wavelength


(a) 7, 0.125


6 8 10 12 14
Grid Points/Wavelength


(b) y, 0.25


Figure 4-15. Amplification factors for the 2D low-pass morphology filter (after Johnson
and Zyserman (2002)).






































2 4 6 8 10 12 14
Grid Points/Wavelength


16 18 20


(a) 7 = 0.125


Grid Points/Wavelength


(b) y, 0.25


Figure 4-16. Amplification factors for the four-step hybrid morphology filter of Jensen
et al. (1999) (after Johnson and Zyserman (2002)).


0.8


0.6


0.4


0.2


/i








Ob = 0
- eb = 22.5
........... b = 45












CHAPTER 5
MODEL EXPERIMENTS

A total of forty simulations are performed to evaluate the capabilities of our 1DH

linear, and 3D nonlinear, phase-resolving live-bed modeling systems. An additional four

simulations are conducted to address our specific scientific questions regarding the roles of

bedload and suspended load sediment transport, as a function of both space and time, in

generating and d, -lr ,ii- ;. bedforms. Additional details regarding simulation parameters

for all forty-four experiments can be found in Table A-1 of Appendix A. Information

regarding the experiments follows, and is first segregated by model type, and then into

three distinct phases for the nonlinear model experiments.

5.1 Linear Model Experiments

A proof-of-concept simulation is proposed with the 1DH model of bedform growth

in the linear regime. The simulation will be used to illustrate the validity of our derived

model dynamics, and similarly prove that bedforms in the linear regime may be formed

completely through bedload sediment transport, as si- -l. -1. I in the literature. Other

than the basic domain and flow parameters, there are two free parameters that must be

chosen: N and Cmix. For the simulation here, we assign a mixed-li i-r thickness of N = 2

grain-diameters, and a mixed-1li vr concentration of Cix = 0.3, which is one-half of

the commonly accepted close-packing concentration of 0.6. A 300 pm grain diameter is

selected for this experiment. In reality, the region of active bedload sediment transport

may be as much as 10 20 grain-diameters in thickness (Bagnold, 1980), but the model

results seem relatively insensitive to this parameter. For the coefficient of resisting friction

in Equation 4-7, the canonical value of fb = 2.5 x 10-3 is selected. The periodic length of

the x-domain is set at Lx = 60 cm, with a horizontal grid spacing of 6x = 2.0 cm. The

freestream velocity and period are chosen as U, = 30 cm/s and T = 4 s, respectively.









5.2 Nonlinear Model Experiments

5.2.1 Phase I: Model Validation

Evaluation of the hydrodynamic modeling system is carried out through model-data

comparisons using the oscillatory boundary l-1-.-r data of Jensen et al. (1989). Specifically,

simulations of Test 10 from Jensen et al. (1989) are performed with the following flow

characteristics: Uo = 200 cm/s, and T = 9.72 s. The flow parameters tested produce

an orbital Reynolds number of R = 6 x 106. In the horizontal periodic x y plane, the

model domain has lengths Lx = Ly = 3.2 cm, whereas the entire 30 cm water column

is modeled in the vertical. Model predictions of the horizontally- and phase-averaged (10

wave periods) streamwise boundary l-1-.-r velocity, as well as the boundary l-1-.-r thickness,

are compared to the Test 10 experimental data reported in Jensen et al. (1989), and

obtained from Mutlu Sumer.

Beyond the model-data comparisons of boundary lv-.-r flow, four simulations with

domain sizes, flow parameters, and boundary conditions similar to those that will be

used extensively in the following investigation are performed to determine appropriate

grid metrics. Additionally, these tests are performed to ensure convergence of normalized

flow metrics for a variety of horizontal and vertical grid resolutions. The variable mesh

resolution for the computational grid tests is shown in Figures 5-1(a)-5-1(d), and the grid

parameters are provided in Table 5-1. These resolution tests are performed over a single,

stationary, sinusoidal ripple having ,r = 1.0 cm and A = 13.6 cm, in a flow with Uo = 40

cm/s and T = 2 s.

Prior to performing the majority of model simulations, seven simulations with

increasingly large domain widths in the spanwise (y) direction are performed with the

coupled, live-bed model. These domain width tests are conducted to determine the effect

of the third dimension on both hydrodynamics and predicted morphology, the results of

which will be used to determine appropriate domain sizes for the Phase III simulations.

Model domain parameters for the seven domain width tests are given in Table 5-2,









and the undistorted model domains are shown graphically in Figures 5-2(a)-5-2(f).

Simulations are performed with a freestream velocity of U" = 40 cm/s, wave period of

T = 2 s, and grain size of d = 0.3 mm, and the bottom boundary is initialized with a 2D

sinusoidal ripple having r] = 1.0 cm and A, = 13.6 cm. The domain width tests are run

for a total of 40 wave periods, or t = 80 s. These flow parameters will be used extensively

throughout the following investigation, and correspond to what we are deeming our

baseline experimental parameters.

In order to evaluate the sensitivity of morphology predictions to the choice of

sediment transport submodels, four simulations are proposed. The sediment transport

submodel tests employ the bedload transport equations of AT. -, r-Peter and Miller

(1948) (Equation 4-53) and Camenen and Larson (2005) (Equation 4-55), and the

pick-up functions of van Rijn (1984) (Equation 4-64c) and Fernandez-Luque (1974)

(Equation 4-64a), in the varying combinations listed in Table 5-3. These particular

transport submodel combinations are chosen for their uniqueness, or popularity, in

available literature; for example, the power-law bedload formulations are conceptually

the same, differing only through the choice of a constant and an exponent; however, the

formulation of Camenen and Larson (2005) is unique in the literature, as it is one of

very few that do not incorporate the commonly accepted critical threshold concept. For

the sake of brevity, a quasi-three-dimensional domain is used, having only two physical,

non-periodic grid points in the spanwise direction. The domain and initial morphology

are similar to that shown in Figure 5-2(a). The model simulations are initialized with

a single, sinusoidal ripple having rr = 1.0 cm and A, = 13.6 cm. Flow parameters

correspond to the baseline case where U, = 40 cm/s and T = 2 s, yielding a mobility

number =- 32.95.

The final suite of simulations performed in Phase I are aimed at assessing the

ability of the 3D live-bed modeling system at predicting morphology over a range of

flow conditions. With respect to the mobility number T, a nondimensional parameter









commonly used in semi-empirical ripple predictors for determining equilibrium height

and length, the experiments range by one order of magnitude from low (T w 10) to

high (T w 100). A total of nine morphology test simulations are conducted utilizing the

quasi-three-dimensional configuration, where only two non-periodic points are used in the

spanwise direction. The simulations are performed with 200 pm, 300 pm, and 400 pm

grain diameters, flow velocities ranging from 30 cm/s < U o < 80 cm/s, for lab-scale (T = 2

s) and field-scale (T = 8 s) flows. Where possible, the morphology is initialized with two

parabolic-shaped ripples having T1r = re/2 and Ar = A,, where r1e and A, correspond to

the equilibrium ripple height and length, respectively, computed using the semi-empirical

ripple predictor of Nielsen (1981) for both field-scale (regular) and lab-scale (irregular)

flows. Forcing and sediment characteristics of the nine morphology test simulations are

given in Table 5-4. The expected ripple heights and lengths, as predicted by the equations

of Nielsen (1981), Grant and Madsen (1982), Mogridge et al. (1994), Wiberg and Harris

(1994), Khelifa and Ouellet (2000), Faraci and Foti (2002), Williams et al. (2005), and

Soulsby and Whitehouse (2006), are provided in Table 5-6.

5.2.2 Phase II: Model Capabilities

A total of fourteen simulations are performed in Phase II to determine additional

model capabilities, as well as evaluate the sensitivity of the coupled modeling system to

initial morphology, periodic domain length, and sediment size. Simulation parameters,

such as model domain lengths, sediment size, and flow characteristics, are provided in

Table 5-7. These test cases are also performed with the quasi-three-dimensional model

configuration, which only employs two non-periodic grid points in the spanwise direction.

We feel that the limited domain width will be mostly insignificant for the purposes of

exploring the model sensitivity and parameter space, as mostly qualitative evaluations of

system capabilities will be pursued. In general, the model experiments conducted in Phase

II are illustrative of typical results.









One simulation of bedform propagation under a steady current is performed, as

well as an experiment yielding subcritical flow conditions to ensure that the morphology

and transport submodels do not produce spurious results. A high mobility number

simulation (T = 206) of morphology and transport in the sheetflow regime is also

proposed, and will assess the capability of the suspended load transport submodel at

predicting typical concentrations in sheetflow lI. -i, as well as the morphological response

to the highly-concentrated and energetic flow conditions. Half of the model experiments

conducted in Phase III are aimed at evaluating the sensitivity of morphology submodel

predictions to initial conditions, periodic domain lengths, and the effects of hindered

settling; these simulations address ripple growth from 1/2 the expected height, as well as

growth from a nearly flat bed; morphological response under equilibrium conditions; ripple

decay from initialized morphology that is out of equilibrium with the flow conditions; as

well as ripple wavelength evolution through the processes of ripple coarsening (slide) and

bifurcation (split).

While the majority of simulations conducted with the live-bed modeling system use

fine to medium cohesionless sand, four experiments are performed with extreme grain

sizes ranging from silt, d = 25 pm, to gravel, d = 6000 pm. Although most of the

semi-empirical ripple predictors discussed previously are not well-suited for predicting

morphology under such extreme conditions, the predicted bed states are compared to

a recent bed state phase diagram provided by Kleinhans (2005) for morphology under

wave-induced flows. A total of four experiments are conducted with these extreme

sediment sizes, with one simulation expected to develop a consistent morphology, and one

to remain below the critical threshold of motion, for both grain sizes.

5.2.3 Phase III: Sediment Transport Processes

The third and final phase of model experiments are formulated, and conducted, to

determine the relative contributions of bedload and suspended load sediment transport,

as a function of both space and time, during ripple growth, equilibrium, and decay. Four









simulations are proposed to address these specific scientific objectives: one employing total

load sediment transport for ripple growth; another simulating ripple decay under total

sediment load transport; a third incorporating only bedload sediment transport; and the

fourth employing only suspended load transport in the bed-level updating scheme.

Using the results of many of the simulations performed in Phases I and II, specifically

those pertaining to acceptable grid resolution (GRID1) and model domain width

(Ly > 2.0 cm), we select the most desirable parameters for computational mesh and

domain size. The four Phase III production simulations are all performed with the full

three-dimensional live-bed modeling system to ensure accurate representation of turbulent

structures that form in the spanwise direction. A schematic of the model domain showing

the initial morphology-two parabolic-shaped ripples having r1, = 1.0 cm and A, = 13.6

cm is provided in Figure 5-3. Model domain lengths, as well as flow and sediment

characteristics, of the final simulations are provided in Table 5-8. The flow parameters for

the Phase III simulations correspond to our baseline test case where U" = 40 cm/s and

T = 2 s. These four simulations are initially run for 3 wave periods to allow equilibration

of model hydrodynamics prior to engaging the transport and morphology submodels, after

which time they are run out for an additional 72 wave periods for a total simulation time

of t = 150 s.
























S4


2


0-

10:





10


8


6


N 4


2


0

10:






Figure 5-1.


10'
Grid Spacing (cm)

(a) GRID1


10
Grid Spacing (cm)

(c) GRID3


S4


2


0 -

10 10:





10r


8


6


S 4


2


0

10 10:


10
Grid Spacing (cm)

(b) GRID2


10
Grid Spacing (cm)

(d) GRID4


The horizontal (6x, by) and vertical (6z) grid spacing for (a) GRID1, (b)

GRID2, (c) GRID3, and (d) GRID4 as a function of the vertical z-dimension.

The values of grid spacing 6x, by, and 6z are plotted on a logarithmic scale.
























k 2 0 2 4 1 o
( 01


(a) WIDE3D_5


r j- 12 1
4" 68
0 0


(c) WIDE3D_2)


(e) WIDE3D_40


(b) WIDE3D_10


(d) WIDE3D_30


10





0
8 1
IzIR


(f) WIDE3D_50


Figure 5-2. The model domains for the domain width comparison tests. The domain for
WIDE3D_2 was omitted since it is only two grid points wide. Color contours
represent the volumetric sediment concentration C.






153


10


N 5



0

- 1
019


6 788
4 6 "
klb,4













z





20 C


0 95
09
15 0 85
08
0 75
07
N 10 065
06
"C 055
05
0 45
04
0 35
03
0 25
02
25 015
15 005






Figure 5 3. A schematic of the model domain and initial morphology used in the Phase

III simulations: TL2v2, BL2v2, and SL2v2. The morphology is initialized

with two parabolic-shaped ripples having = 1.0 cm and A, = 13.6 cm.















































154










Table 5-1. Numerical mesh parameters for the grid comparison tests.
Grid L,, L,, Lz 6x, 6y, 6z nx, ny, nz Tot


tal Number of Points


cm
13.6, 0.4, 11.0
13.6, 0.8, 11.0
13.6, 0.2, 11.0
13.6, 0.2, 11.0


cm
0.2, 0.2, 0.05
0.4, 0.4, 0.1
0.1, 0.1, 0.025
0.1, 0.1, 0.05


71, 5, 92
37, 5, 47
139, 5, 182
139, 5, 92


Table 5-2. Numerical mesh parameters for the domain width comparison tests.


Lx, Ly,, Lz
cm
13.6, 0.4, 11.0
13.6, 1.0, 11.0
13.6, 2.0, 11.0
13.6, 4.0, 11.0
13.6, 6.0, 11.0
13.6, 8.0, 11.0
13.6, 10.0, 11.0


Table 5-3. Abbreviations used in
Reference
M. i,. r-Peter and Muller (1948)
Camenen and Larson (2005)
Fernandez-Luque (1974)
van Rijn (1984)


6x, 6y, 6z
cm
0.2, 0.2, 0.05
0.2, 0.2, 0.05
0.2, 0.2, 0.05
0.2, 0.2, 0.05
0.2, 0.2, 0.05
0.2, 0.2, 0.05
0.2, 0.2, 0.05


nx, ny, nz

71, 5, 92
71, 8, 92
71, 13, 92
71, 23, 92
71, 33, 92
71, 43, 92
71, 53, 92


the sediment transport equation tests.
Equation Transport Mode
4 53 Bedload
4-55 Bedload
4-64a Suspension
4-64c Suspension


Total Number of Points

32660
52256
84916
150236
215556
280876
346196


Abbreviation
MPM
CL
FL
VR


GRID1
GRID2
GRID3
GRID4


32660
8695
126490
63940


Grid


WIDE3D_
WIDE3D_
WIDE3D_
WIDE3D_
WIDE3D_
WIDE3D_
WIDE3D_















Table 5-4.


Case Name

LL1v3
LF1v1
ML1v2
ML2v2
ML3v1
MF1v2
MF2v2
HL1v1
HF1vI


The morphology test simulation matrix showing alpha-numeric case names,
fluid forcing and sediment parameters, and values of nondimensional
parameters commonly used in semi-empirical ripple predictors.


Uo0
(cm/s)
30
30
30
50
40
30
50
80
60


A
(cm)
9.
38.
9.
15. f
12.
38.
63.
25.
76.


d
Ipm
400
400
200
300
300
200
300
300
200


13.
13.
27.
51.
33.
27.
51.
132.
111.


X
(x10-6)
6. 18
0. 386
3. 089
4. 63
4. 63
0. 193
0. 290
4. 63
0. 193


(x104)
2. 86
11. 5
2. 86
7. 96
5. 09
11. 5
31. 8
20. 4
45. 8


Table 5-5. Citation keys of ripple predictor equations used in the morphology test
simulations.


Reference
Nielsen (1981)
Grant and Madsen (1982)
Mogridge et al. (1994)
Wiberg and Harris (1994)
Khelifa and Ouellet (2000)
Faraci and Foti (2002)
Williams et al. (2005)
Soulsby and Whitehouse (2006)


Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations
Equations


3a-3-5c
-7a-38b
13a-3-13b
9a-3-12b
14a-3-14b
15a-3-15c
16a-3-17b
18a-3-18b


Citation Key
N81
GM82
M94
WH94
KO00
FF02
W05
SW06










_^ 03 c 0 Cl 0 C O -
"o t oc



0 0





0 0 0 0
-; V ^ ,- *-*- oi TO O ( t
CC CtL- 'I t L'




' C o0 CD






coc
O


^ s~ 3- o~ooo ooo
C.) c> CI I03- 7t
0*







0 cc I- 03 I -
^E~)r n ~~ io-Loo oo




0;
a D 1 0c 3 0^




0 r Cl^ ClOO^
ci o 03 oc oo6
So --










'. 1 t-
0 B 030 030303 00 -








S s~ ^-oc^ o oc- Lr!o c
0;
0





-S o -, 03 0l 0


- 0 Cl CO Cl 03 Cl 0 3 Cl 0n

bic




CTJCD
o i-i3~o dC~ jC j h V. .




C'A OC C'









C. C I- h V 0 0 -



0 ~ ~ o cCl C lC l O V h
0 CI

0 c CO Vc CO 1i CC V
"3 ,_,0 0 1 0 10 0












-4 oi Cl ho i Cco 0




CIA st' -' -y. CI
bC









0&O0-hO&O&L-C~ OCl
0S 0






H_ O



C0 V CO1 C1 C








t5
0S 0 C C<^--1 ( 00 30
H






t C1^-t 5
w~~. E01'0 O

L c ?^^_-C ^ ^0 ^-^- ;- 0 0



S C ^ o o CM^- o o

Z < ^- 0 -- -











Table 5-7. A list of experiments and relevant simulation parameters for simulations
performed in Phase II.


Case Name

STEADY
SUBCR
SHEET
GROW1
GROW2
EQUILIB
DECAY
HS1v2
SLIDE
SPLIT
GRAVLOW
GRAVHI
SILTLOW
SILTHI


Lx
(cm)
27. 2
27. 2
13. 6
13. 6
13. 6
13. 6
13. 6
27. 2
27. 2
27. 2
30. 0
30. 0
30. 0
30. 0


Ly
(cm)
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4
0. 4


Lz
(cm)
12. 0
11. 0
11. 0
11. 0
11. 0
11. 0
11. 0
11. 5
11. 5
11. 5
11. 5
11. 5
11. 5
11. 5


nx ny nz U.
(cm/s)
136 2 110 30
136 2 90 5
68 2 90 100
68 2 90 40
68 2 90 40
68 2 90 40
68 2 90 40
136 2 100 40
136 2 100 40
136 2 100 40
150 2 90 30
150 2 90 80
150 2 90 10
150 2 90 40


Table 5-8. A full list of experiments and relevant simulation parameters for the Phase III


Case Name

TL2v2
TL2v2d
BL2v2
SL2v2


model experiments of sediment transport processes and bedform dynamics.
Lx Ly Lz nx ny nz U. T R d
(cm) (cm) (cm) (cm/s) (s) (x104) (pm)
27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300
27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300
27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300
27. 2 3. 2 21. 5 136 16 110 40 2. 0 5. 0929 300


(x104)
0. 00
0. 07958
31. 83
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
2. '.*,,
20. 372
0. 3183
5. 0929


d
(3m)
300
300
300
300
300
300
300
300
300
300
6000
6000
25
25









CHAPTER 6
RESULTS

Simulation results for the linear and nonlinear phase-resolving live-bed models of

hydrodynamics and sediment transport are provided in the following sections. A total of

forty-four (44) simulations, outlined in Chapter 5 and listed in Table A 1, were performed,

and provide insight about model skill and capabilities. The model results reinforce two

important concepts: first, bedform growth in the linear regime is a process dominated

by sediment inertia, and second, growth in the nonlinear regime is dominated by the

formation of vortices through flow separation at the crest of bedforms.

6.1 Linear Model Results

The results of the 1DH conceptual model experiments indicate that in order to create

bedforms in the linear regime, it is necessary to incorporate a spatial lag between the

mixture velocity and transport in Equation 4-14. Without such a lag, the 1DH model

predicts a flattening of the initialized morphology, with no recovery or generation of

bedforms. This behavior has both physical and numerical significance, and is discussed

in terms of linear stability in Parker (1975). The linear stability analysis of Parker (1975)

demonstrates that flat sand beds are unconditionally stable in a linear regime where

velocity and transport are assumed to be in phase with one another. Physically, this

spatial lag may be interpreted as an inertial effect of the sediment, whereby the weight

of the sediment causes transport to lag behind the applied forcing. The spatial lag 6s is

implemented in the 1DH transport equation (Equation 4-14) according to Equation 6-1.

For the conceptual 1DH experiment discussed here, the lag was found to be 6s = 6 cm,

which yields a lag distance-wavelength ratio, ks/27 =0.1, similar to the results of Parker

(1975).

Qb(X) = Umix(X 6s)AmixCmix (6-1)

Timestacks of bed elevation from the 1DH linear model experiment are plotted

in Figure 6-1. After a flattening of the initial morphology, having an amplitude equal









to three grain diameters and a wavelength equal to the periodic domain, the bedform

instability begins to develop, and grows to a point where the linear model approximations

are no longer valid. The time evolution of bedform height and length is demonstrated

in Figure 6 2(a), and shows the rapid scaling of wavelength coincident with bedform

development at approximately t = 19 s. Rolling grain ripples, characteristic of the linear

regime, typically have steepness values of r = 0.1 (Sleath, 1976). A time-series of bedform

steepness (or = -qr/Ar) is given in Figure 6-2(b), and shows an increase in steepness

to about one-half the expected value for rolling grain ripples before nonlinearity in the

mixed velocity causes the model to fail. More specifically, this model failure is a result of a

nonlinearity between the mixed velocity directed up the bedform slopes and a competing

gravitational stress directed in the opposite sense. The results of the experiment using

our conceptual 1DH bedload transport model, however, successfully point out that in the

linear regime, where applied stress and transport are in phase with one another, inertial

characteristics of the sediment are responsible for bedform development.

6.2 Nonlinear Model Results

Results of the nonlinear model simulations are divided into three phases corresponding

to: model validation of hydrodynamics and morphology; an exploration of model

capabilities, sensitivity, and parameter space for sediment transport and morphology;

and information about the roles of bedload and suspended load transport in generating

sedimentary structures.

6.2.1 Phase I: Model Validation

A total of twenty-five (25) simulations were performed to evaluate the predictive skill

of the hydrodynamic and live-bed submodels, as well as their sensitivity to numerical

resolution and domain size. The results of these simulations are provided in the following

sections.









6.2.1.1 Hydrodynamic Validation

Assessment of the LES hydrodynamic model was accomplished by comparing model

predictions of the phase-averaged streamwise velocity to the oscillatory boundary l1 v.-

data of Jensen et al. (1989) (Test 10). The tests simulate an oscillatory boundary l1'v-r

over a smooth boundary having U, = 200 cm/s and T = 9.72 s, and an orbital Reynolds

number of R, = 6 x 106. The results of the hydrodynamic validation are shown in

Figures 6-3(a)-6-3(1). Streamwise velocity profiles predicted by the hydrodynamic model

were phase-averaged over nine wave periods, with the first cycle being ignored due to

model spinup .,-vi-iii:, I ry. Experimental data for the second half of the phase-averaged

wave cycle were not available, but the results are simply a mirror image of the first

half-cycle. Laboratory velocity profile data were only collected over the bottom 15 cm of

the water column, so only that portion of the model profile is shown; however, the model

domain extended to L, = 30 cm to simulated the full water column. For additional clarity,

only every other data point is plotted in Figures 6-3(a)-6-3(1).

Shortcomings in the hydrodynamic model are evident in the comparison of predicted

and observed velocity profile data shown by Figures 6-3(a)-6-3(1). Although the shape

of the predicted velocity profile tends to agree with the results, the model velocity tends

to lag slightly behind the laboratory data outside of the boundary 1l- .-r. On further

inspection, the lag appears to be somewhat consistent having a phase lag of approximately

3, or about t = 0.08 s when considering the simulated wave period of T = 9.72 s. This

disagreement may be the result of systemic time-keeping errors, either in the laboratory

data, or manifest through a summation of residual computational errors in the model.

Differences are also seen in the modeled and observed boundary 11-rvr thickness, as

demonstrated in Figure 6-4. With exception to a few phases, the LES model consistently

overestimates the thickness of the boundary 1-v -r (6) by approximately 2 5 mm. The

marked overestimation of 6 at t = 107r/12 is a result of the predicted flow beginning

to reverse phase ahead of the observed data, whereby the boundary l-?v1-r is released









prior to reforming near the wall in the succeeding phase interval of t = ll1r/12. This

premature flow reversal, as seen in Figures 6-3(j) and 6-3(k), is an inertial property

of the boundary 1l -.-r, where the slower flow (thicker boundary 1i-~v-r) predicted by the

model possesses less momentum than the faster flow (thinner boundary 1.-~-r ) observed

in the experiment. The overestimation of the boundary 1- v-r thickness, and the resulting

differences between modeled and observed data in boundary 1-v.-r phase, are most likely

due to an overestimation of the turbulent eddy viscosity Vt. Such overestimations are

characteristic of the Smagorinsky (1963) closure method in high Reynolds number flows,

but have been shown to perform well in limited cases (Ferziger, 1996a; Meneveau, 1994).

Additional inaccuracies in the boundary l- v-r predictions may be the result of the

near-wall model, which uses the van Driest damping function of Moin and Kim (1982)

to ensure that Vt goes to zero at the wall. Ferziger (1996b) argues that the application of

van Driest damping as a near-wall model is difficult to justify in LES models, and more

sophisticated models are offered by Piomelli and Balaras (2002).

An evaluation of average model error, with respect to the experimental data, is shown

in Figures 6 5(a) and 6 5(b). Model predictions of velocity were obtained at twice the

resolution of the available lab data in the lower 15 cm of the water column; therefore,

for the purpose of determining model error through direct comparison, it was necessary

to interpolate the model predictions to the distribution of experimental data over the

vertical. The interpolation was performed using an inverse distance weighted (IDW)

procedure, but no attempt was made to ensure conservation over the profile. The phase-

and horizontally-averaged absolute percent difference between model predictions and

experimental data is plotted in Figure 6 5(a) as a function of vertical location and wave

phase. As noted earlier, the poorest agreement in Figure 6 5(a) tends to be during periods

of flow reversal, and within the boundary 11-~v-r. This is more clearly demonstrated by

the vertically-averaged values of absolute difference plotted in Figure 6-5(b). With the









exception of the large differences found at times of flow reversal, the model predictions are

generally within i l'. of the expected values.

The relatively poor performance of the LES model, especially within the boundary

l~v-.r, does have implications for the performance of the sediment transport and

morphology submodels. An overestimation of the boundary 1l- -r thickness, due mostly

to a larger vt, yields lower vertical velocity gradients, and, therefore, lower values of

bed shear stress (Equation 4-46). This difficulty may be overcome by accounting for

the effects of the eddy viscosity in the shear stress definition, as previously discussed in

C'! lpter 4, and demonstrated in Equation 4-47. A thicker boundary l1-, -r may also lead to

higher concentrations of suspended sediment over a greater distance above the bed. The

phase lead in the boundary l~- -r (ahead of the data), and corresponding lag in the outer

flow, may also affect the timing of entrainment and deposition of suspended sediments,

although we would expect the presence of vortices over rippled beds to p1 i, a much more

dominant role in this regard. It should be noted that while the LES model yielded, in

some cases significant, differences between expected and observed results, the 1D! iiP ily of

simulations used for assessing characteristics of sediment transport and morphology were

performed at Reynolds numbers two orders of magnitude smaller than that used in the

model-data comparison.

6.2.1.2 Computational Grid Tests

A number of quasi-three-dimensional (having only a few grid points in the y or

spanwise direction) simulations were performed in order to determine the effects of

grid resolution on normalized flow metrics, as well as computational expense. Four

grids with varying degrees of resolution (see Table 5-1) were tested for an oscillatory

flow with U, = 40 cm/s and T = 2 s over a fixed, sinusoidal bottom boundary. The

overall computational expense, with respect to computational resolution, is demonstrated

in Figure 6-6, and the relationship between the two is found to follow a logarithmic

relationship. The total number of grid points within the tested domains ranged from









8695 at the lowest resolution, to 126490 for the highest resolution, and required 1.2 and

401.5 hours of wall-clock time to simulate 30 s of model time, respectively. The most

time-consuming component of the modeling system is by far the iterative pressure solver,

requiring increasingly larger number of iterations as the number of grid points increases.

As exhibited by the overall computational performance plotted in Figure 6-6, a roughly

ten-fold increase in the number of grid points results in a four-hundred-fold increase in

required wall-clock time.

While the computational expense, with respect to time, is sensitive to the number

of grid points used, certain normalized flow metrics should not vary significantly with

numerical resolution. The time-dependent, volume-averaged u-, v-, and w-velocity

standard deviations are plotted in Figures 6-7(a)-6-7(c). The standard deviations were

filtered (50 passes with 7s = 0.25) using a diffusive smoothing filter (Equation 4-26a) to

enhance the low-frequency detail of the signals. Convergence of the average streamwise

velocity standard deviation signal is observed in Figure 6-7(a) for all four grids. Similar

convergence is found in the vertical velocity standard deviation signal for GRID1, GRID3,

and GRID4 in Figure 6-7(c), with the lowest resolution grid (GRID2) predicting slightly

lower values. The spanwise velocity standard deviation signals shown in Figure 6-7(b) are

nearly dubious since the domains only have two non-periodic points in the y-direction;

however, GRID2 demonstrates an inability of the numerics to reach a steady state in the

spanwise direction within the simulated model time.

The turbulent kinetic energy [k = (u'2 + v'2 + w'2)/2] is also used as a normalized

metric for comparing the effects of grid resolution on model predictions. Figure 6-8 shows

the cumulative volume-averaged turbulent kinetic energy (TKE) as a function of time for

the grid comparison tests. Similar to the velocity standard deviations, the TKE signals

were filtered (50 passes with 7s = 0.25) using a diffusive smoothing filter (Equation 4-26a)

to enhance the signal-to-noise ratio. The maximum absolute difference between the four

signals in Figure 6-8 is approximately 10C' and is reduced by a factor of one-half when









the lower resolution GRID2 is removed. The phase- and volume-averaged TKE (((TKE)))

is plotted in Figure 6-9 as a function of forcing phase (wt), and demonstrates similarly

good agreement between GRID1, GRID3, and GRID4, indicating numerical convergence of

the grid parametrics.

Phase- and spanwise-averaged velocity profiles for the grid comparison tests are

plotted as a function of ripple location in Figures 6-10(a) 6-10(f), Figures 6-11(a) 6-11(f),

and Figures 6-12(a) 6-12(f), for the ((u)), ((v)), and ((w)) velocity components,

respectively. Since the applied horizontal pressure gradient produces a symmetric

oscillatory flow, only the first one-half of the forcing cycle is shown in each of the figures.

Although the average velocity profiles are not necessarily considered a normalized metric,

there appears to be generally good agreement between the predictions of the four grids

tested. The most significant differences in profile shape and magnitude are found for the

((w)) velocity component plotted in Figures 6-12(a) 6-12(f), and may be attributed to

the difference in resolution of the sinusoidal bottom boundary between the four grids.

While the velocity profile and normalized metric comparisons tended to yield favorable

results for GRID1, GRID3, and GRID4, the former (GRID1) was selected for use in the

remainder of the simulations as it provided the most advantageous balance of resolution

and expense.

6.2.1.3 Model Domain Width Tests

Simulations with increasingly larger spanwise (y) domains were performed in order

to evaluate the effects of domain width, and therefore three-dimensionality, on both

normalized flow metrics and the equilibrium bed morphology. These tests are useful for

determining the extent of the third dimension required for resolving pertinent details

in flow and morphology, as it is computationally beneficial to use the smallest domain

possible. The simulations were performed with a two-dimensional (x z) oscillatory

flow having U, = 40 cm/s and T = 2 s, over a solitary sinusoidal ripple initialized

at lr = 1.2 cm for a total duration of 40 wave periods, or t = 80 s. Mesh parameters









for the seven three-dimensional (3D) tests are provided in Table 5-2, and range from

domains having two to fifty grid points in the y-direction. The overall computational

budget for the seven domains tested is plotted in Figure 6-13, and shows a logarithmic

relationship between the number of grid points and wall-clock time similar to that found

in the resolution tests (Figure 6-6). A ten-fold increase in the number of grid points used

in the three-dimensional tests results in simulation times that are nearly four-times longer.

Unfortunately, simulations were run on different platforms with some of the larger tests

executed on more recent CPU models; therefore, a direct comparison or analysis between

the simulations is not appropriate, but is nevertheless insightful. It should also be noted

here that the 3D tests utilized the sediment transport and morphology submodels to

update the bed elevation at every timestep, thereby requiring an increasing number of

iterations by the pressure solver to ensure flow continuity within the fluid domain. A more

extensive evaluation of the sediment transport and morphology submodels is provided in

the following sections.

Similar to the numerical resolution tests discussed previously, normalized metrics

of flow and morphology are used to compare the effects of the domain width on model

predictions. The filtered (Equation 4-26a) cumulative volume-averaged TKE (TKE)

is plotted in Figure 6-14 as a function of time for the seven 3D grid comparisons. The

cumulative TKE for all seven grids is found to grow immediately during model spinup,

and then achieves an approximately average value of TKE = 200 cm2/s2. As opposed to

the fixed bed resolution tests, there is more high frequency oscillation in the cumulative

signal due to the change in morphology at every hydrodynamic timestep; however, they

do tend to attain a mean value over the duration of the simulation. Figure 6-15 shows the

phase- and volume-averaged TKE (((TKE))) as a function of forcing phase (wt) for the

seven 3D domains. Phase-averaged quantities are computed using ensemble averages over

39 wave cycles, with the first cycle being excluded due to model spinup .,-i-iliii I ry. With

the exception of the 40- (WIDE3D_40) and 50-point domains (WIDE3D_50), the phase-









and volume-averaged TKE appears to be similar between the grids. The slightly lower

values of average TKE in the two largest domains, as indicated by Figures 6-14 and 6-15,

may indicate higher levels of either physical or numerical dissipation in the grid.

The general effect of the third dimension (y) should be to increase dissipation due to

the formation of turbulence. Without a sufficient width in the spanwise direction, rotation

generated in the x z plane is prohibited from breaking down into turbulent structures,

thereby limiting the dissipation of energy through turbulence production. In theory, one

would seek to increase the spanwise dimension until full development of the spanwise

velocity is achieved. The phase- and volume-averaged u, v, and w velocity components are

plotted in Figures 6-16(a)-6-16(c) as a function of forcing phase for the three-dimensional

grid comparisons. The oscillatory u-velocity signal is clearly defined in Figure 6-16(a),

and no discernible differences are evident between the seven different grids. The width

of the third dimension (Ly) does appear to impact the development and behavior of

the spanwise (v) velocity, with the three largest domains reaching general agreement.

Interestingly, the four largest domains ranging in width from 4 cm to 10 cm show a mean

spanwise velocity oriented in the positive y-direction; however, even the largest values

(0.3 cm/s) are two orders of magnitude smaller than the streamwise forcing velocity. The

magnitudes of the phase- and volume-averaged vertical (w) velocity are fleetingly small,

having values less than 1 x 10-3 cm/s, and appear to have a zero mean. This -I.-.-. -1-

that the development of vortices on either side of the ripple crest are well-balanced during

ripple development.

A Fourier ain i1, -i- using a Fast Fourier Transform (FFT) package, was performed

to determine the dominant energy-containing lengthscales for flow in the spanwise

direction of the seven 3D domain tests. Figures 6-17(a) and 6-17(b) show the time- and

spatially-averaged direct Fourier transforms of the v velocity as a function of wavenumber

(ky). Applying direct Fourier transforms to the v velocity provides information regarding

the relationship between energy density and dominant lengthscales in the velocity field. As









shown in Figure 6-17(a), the largest values of energy density for each domain correspond

to the lowest wavenumbers (k = 1/Ly). The lowest wavenumber for each domain

corresponds to the full periodic length of that domain. If the peak energy density were

to occur at a higher wavenumber, then the domains would contain significant amounts of

energy at smaller integral lengthscales, or wavenumbers, in the y-direction. Furthermore,

the spectral density plots in Figure 6-17(a) tend to saturate for the four largest domains

having widths Ly > 2.0 cm. Figure 6-17(b) indicates that the largest velocity amplitudes

also occur at the lowest wavenumbers, but are still one order of magnitude smaller than

the streamwise forcing amplitude (U, = 40 cm/s). The absence of significant amounts

of energy at integral lengthscales smaller than the periodic length of the y-direction for

the three-dimensional domains, as well as the small values of velocity amplitude in the

y-direction, reinforce the two-dimensional nature of the applied oscillatory forcing.

In addition to the effects of the domain width on hydrodynamics, particularly as

they relate to the generation of turbulence, it is beneficial to evaluate those effects on

equilibrium ripple morphology. Even for the strictly two-dimensional horizontal pressure

gradient applied to generate the oscillatory boundary 1-, r forcing, some modification

of the flow, and therefore morphology, is anticipated in the spanwise dimension. The

standard deviation of the bed elevation a,, in the spanwise dimension, averaged over the

x-direction, is plotted as a function of time in Figure 6-18. The standard deviation of

the bed elevation in the third (y) dimension provides an indication of ripple variability,

about the mean, in the third dimension for the different three-dimensional grids. The

smallest domain, containing only two physical grid points (WIDE3D_2), has essentially

zero variability in the y-direction as shown in Figure 6-18. The lack of variability in the

third dimension for WIDE3D_2 indicates that it is behaving more two-dimensionally, with

little to no transport in the y-direction. For domains having greater than five physical

grid points in the third dimension, Figure 6-18 demonstrates that variability of the ripple

profile in the y-direction saturates at about a, = 0.75 cm, and remains relatively









constant after the first few wave periods. Therefore, the width of the spanwise dimension

does appear to influence properties of the morphology in the third dimension, but the

variability for domains larger than or equal to Ly = 2.0 cm appears to be relatively

similar, and remains constant in time.

The most significant effect of the y-domain width (Ly) on the morphology would

manifest itself through discrepancies in the equilibrium ripple characteristics. Equilibrium

ripple heights for the seven domains are shown in Figure 6-19, and have been averaged

over thirty wave periods (20 s < t < 80 s). The standard deviation of ,]r about the

time-averaged value (Tr,) is also given in Figure 6-19 for each domain test. There

appears to be good agreement for lr not only amongst the seven domains, but also to

the equilibrium ripple height (Tr = 1.89 cm) computed using the ripple predictor of

Nielsen (1981) for regular flows. There is approximately a twelve-percent difference

between the highest and lowest predictions of l, for the seven domains, with the smallest

]r = 1.80 cm occurring for the quasi-three-dimensional domain (WIDE3D_2). For domains

having widths greater than or equal to Ly = 2.0 cm, there is less than I!'. difference in

the average equilibrium ripple heights. Since all seven domain tests were run with one full

period of a ripple, the equilibrium ripple length remains locked at A = 13.6 cm for all

simulations.

Fourier analysis was also used to evaluate the impact of the domain width on the

development of significant bed features in the third dimension. Figures 6-20(a) and

6-20(b) show the time- and spatially-averaged direct Fourier transforms of the bed

elevation zb in the y-direction. The transforms were first applied to the morphology in

the y-direction, averaged spatially in the x- and z-directions above the ripple crests,

and then time-averaged over the last thirty wave periods of the simulations. Similar to

the results of the Fourier analysis applied to the v velocity field, the maximum spectral

density of zb occurs at the lowest wavenumber (k, = 1/L,) for each domain, corresponding

to the full periodic length of each domain in the y-direction (Figure 6-20(a)). The energy









contained in perturbations of the ripple profile in the y-direction, even at the lowest

wavenumbers (largest lengthscales), is weak with most peaks less than 0.1 cm3. The

maximum perturbation amplitudes, shown in Figure 6-20(b), are also small and tend to

be less than 0.1 cm with peaks occurring at the lowest wavenumber. Once again, there

appears to be saturation in both the spectral density and perturbation amplitude at the

lowest wavenumbers for the four largest three-dimensional domains (Ly > 2.0 cm), with

very little energy contained in the higher wavenumbers. Taking into account the impact of

the third dimension on both hydrodynamics and morphology, it appears advantageous to

perform the final production simulations with domain widths of Ly > 2.0 cm. This length

is in direct proportion to the expected equilibrium ripple height, as predicted by Nielsen

(1981), sl-.-,- -I ii.-; a dependence of domain width on ripple height. Similar observations

were made by Barr et al. (2004) for oscillatory flow over fixed, rippled boundaries.

6.2.1.4 Sediment Transport Submodel Tests

Four combinations of various bedload transport and sediment pick-up equations,

found in available literature, were used to determine the sensitivity of morphology to

transport submodels. As outlined in Table 5-3, the following combinations of transport

submodels are shown in Figures 6-21(a) and 6-21(b): MPMVR, AT i r-Peter and

Miller (1948) and van Rijn (1984); MPMFL, At, i* r-Peter and Miller (1948) and

Fernandez-Luque (1974); CLVR, Camenen and Larson (2005) and van Rijn (1984);

CLFL, Camenen and Larson (2005) and Fernandez-Luque (1974). The instantaneous

ripple height, shown in Figure 6-21(a), was calculated using Equation 6-2, where a ,(t)

is the standard deviation of the bed elevation zb in the streamwise direction at each time

t. The ripple height formulation given in Equation 6-2 was -I ---- -I. '1 by Hanes et al.

(2001), and has more recently been used in laboratory analysis of transient ripples by

Doucette and O'Donoghue (2006). In this current study, the method of Hanes et al. (2001)

for calculating ripple heights is favored over the more traditional method of identifying

maximums and minimums in the bed elevation signal-typically through application of









a zero-crossing technique-for two reasons: first, the zero-crossing techniques are not

applicable when simulating only one ripple wavelength; and second, Equation 6-2 is much

less sensitive to the rapid changes in bed elevation experienced during the simulations as it

relies on average statistics of the morphology instead of single-point measurements.



r (t) 2 V2a(t) (6-2)

Time-averaged statistics of the ripple heights for each combination of sediment

transport submodels are shown in the inset plot of Figure 6-21(a), where the error bars

represent the standard deviation of the instantaneous ripple height about the mean (lr).

The time-averaged ripple heights (Qlr) were determined by averaging ripple height statistics

over the last twenty wave periods of the simulation, The data indicate that there is some

dependence of the predicted morphology on the choice of sediment transport submodels,

with a difference of roughly ten percent between the highest and lowest average ripple

height predictions, and a factor of two difference between the ripple height standard

deviations. The maximum departure from the expected ripple height (rr = 1.89 cm) is

just over seven percent, and corresponds to the sediment transport submodel (I PMFL)

employing the bedload transport equation (Equation 4-53) of At. i, r-Peter and Miller

(1948) and sediment pick-up function (Equation 4-64a) of Fernandez-Luque (1974).

The MPMVR and CLFL transport submodels yielded the minimum departure from the

expected ripple height at +1.5'. and -1.5'. respectively; however, the CLFL submodel

yielded the largest standard deviation of all the transport submodels at a = 0.11 cm, while

the MPMVR submodel produced the smallest at a = 0.057 cm.

The data in Figure 6-21(a) may be used to assess the predictive capabilities

of submodel combinations, as well as their constituent components. For example,

the MPMFL and CLFL submodels both employ the sediment pick-up function of

Fernandez-Luque (1974), which yielded the largest standard deviations in the bed

elevation time-series. In contrast, the MPMVR and CLVR submodels produced the









smallest standard deviations and ripple height predictions both within 2.,' of the

expected value. These results sl-.-,- -I the the sediment pick-up function of van Rijn (1984)

may perform better at predicting suspension events, with respect to time-dependent

bed morphology, to that of Fernandez-Luque (1974). Isolating the performance of the

bedload transport equations from the data in Figure 6-21(a) is more difficult, but the

combination (\PIMVR) of the AM. i'- r-Peter and Miller (1948) bedload transport equation

and sediment pick-up function of van Rijn (1984) appears to produce the best overall

results. This conclusion agrees well with those of Madsen (1991) and Garcia and Parker

(1991), who recommended using the A1. i*, r-Peter and Miiller (1948) bedload transport

equation and van Rijn (1984) pick-up function, respectively.

The ripple steepness, defined as r/A,, for the sediment transport submodel

comparison tests is shown in Figure 6-21(b). Since the ripple wavelength A, for these

simulations was fixed at one periodic length in the streamwise direction (A, = 13.6 cm),

the ripple steepness time-series mimics that of the ripple height shown in Figure 6-21(a).

Classical orbital ripple predictors si---.- -1 that there is only one value of ripple steepness,

or ratio of ripple height to wavelength, for a particular combination of fluid forcing and

sediment characteristics [e.g. Nielsen (1981)]; therefore, the ripple steepness should remain

constant during ripple growth as the height and wavelength grow large in proportion to

one another. This constant ripple steepness was observed in the laboratory experiments

of O'Donoghue and Clubb (2001), but is not reflected in the results of Figure 6-21(b)

since Ar remains fixed at the periodic length of the streamwise domain. Instead, the ripple

steepness grows quickly along with the ripple height until the morphology is in equilibrium

with the hydrodynamics, after which time it remains relatively constant at roughly

r/Ar = 0.14. It is encouraging that the ripple steepness remains both relatively constant

and below the maximum (gravitational) limit as indicated in Figure 6-21(b), as it sir-l-- -1

that the morphology reaches a dynamic equilibrium with the fluid forcing characteristics

instead of gravitational limits imposed by the sediment's avalanche angle, or static angle









of sediment repose. The maximum steepness limit shown in Figure 6-21(b) corresponds

to 0.32 tan 4, a practical limit of steepness for vortex ripples indicated by laboratory data

(Nielsen, 1992; Faraci and Foti, 2002). In the transport submodel tests, Q = 300 is the

static angle of sediment repose for the cohesionless sediment grain size (ds0 0.3 mm)

assumed in the simulations (Julien, 1998).

Anecdotal evidence of stable rolling grain ripples has appeared in the literature,

but typically only for laboratory experiments utilizing oscillating trays (Bagnold, 1946)

or oscillating annular cells (Stegner and Wesfreid, 1999). Analytical models of rolling

grain ripples, such as those of Sleath (1976), Andersen (2001), and Andersen et al.

(2001), have also -i---. -1. the presence of stable rolling grain ripples, typically having a

characteristic steepness of rr/Ar = 0.10 similar to that -it-i-. -1i 1 by early experiments of

Bagnold (1946). Furthermore, the experiments of Bagnold (1946) -,;;-i -1. .1 that ripple

development involved a primary formation of stable rolling grain ripples, with a transition

to vortex ripples at higher Shields' stresses. Laboratory experiments in oscillating water

tunnels, however, -,'-1.: -1 that rolling grain ripples are transient at best; and Miller and

Komar (1980b) state that they are rarely observed in the field. The ripple steepness

results in Figure 6-21(b) show no clear evidence of a transient state at the rolling grain

ripple steepness of 0.10, and instead -ii.:: -1 that there is continuous transformation of the

steepness as the ripple grows in height.

6.2.1.5 Morphology Tests

A total of nine simulations were conducted over a range of fluid forcing and sediment

characteristics to:

1. Assess the predictive capabilities of the sediment transport and morphology

submodels; and

2. Evaluate the performance of the coupled modeling system over a range of mobility

numbers (T) (Brebner, 1980).









The mobility number T is a metric that describes the ratio of mobilizing (destabilizing) to

gravitational (stabilizing) forces for a sand bed, and is commonly used in semi-empirical

ripple predictors (Nielsen, 1981; Khelifa and Ouellet, 2000; Faraci and Foti, 2002;

Williams et al., 2005) to determine equilibrium ripple height and length. As there are

many references to the role of the wave period-and hence the nearbed wave orbital

amplitude-on determining ripple characteristics (\il r and Komar, 1980a; Mogridge

et al., 1994; Wiberg and Harris, 1994), simulations were conducted at what are considered

lab-scale flows having wave periods of T = 2 s, and field-scale flows having periods of

T = 8 s. The simulation parameters for the morphology tests, along with the expected

ripple heights and lengths determined by the ripple predictors discussed in ('!i Ilter 3, are

provided in Tables 5-4 and 5-6, respectively.

Prior to discussing any results from the morphology test cases, it is imperative that

we demonstrate conservation of mass within the coupled modeling system. Conservation

of fluid mass in the hydrodynamic model is ensured through the SOR iterative pressure

solver, which iterates until continuity is satisfied within IV ui < 1 x 10-3. As outlined in

C'! lpter 4, the morphology submodel utilizes gradients in bedload transport flux, as well

as the entrainment and deposition modeled through the sediment concentration field, to

update the bed elevation at every hydrodynamic timestep. There are then three possible

areas where conservation errors may be introduced: (1) the finite-difference formulations

used to estimate the bedload transport fluxes, (2) incorrect implementation of the

sediment pick-up function as a bottom boundary condition on the scalar concentration

field, and (3) numerical errors associated with the first-order donor-acceptor method used

for tracking the sediment concentration field. Figure 6-22 demonstrates the conservative

nature of the sediment transport and morphology component of the modeling system,

and is representative of typical model performance. At every time step (6t = 0.0001 s),

the percent change in sediment volume, from the initial conditions, was computed and

recorded; the resulting time-series, after filtering at 1 Hz to remove noise, is shown in









Figure 6-22. The signal mean, regardless of the filtering process, remains well below 0.1.

- -.-, I ii-; that conservation of sediment mass is obtained.

In general, most of the available literature on bedforms tends to agree that orbital,

vortex ripples scale in some proportion to the nearbed wave orbital semi-amplitude

A = Umax/. A selection of such relationships between ripple wavelength A, and

A are provided in Table 3-1, and show general agreement for bedload-dominated

transport regimes. Commonly, ripple predictor equations for height and length are

nondimensionalized by the semi-amplitude A and cast in terms of the mobility number T.

The most commonly observed behavior of orbital ripples in fine to coarse sand, in terms

of T, is that for an increase in destabilizing forces (larger T), the ripple height tends to

decrease while the ripple length increases. The laboratory data of YR62, KF65, C69, N79

(refer to Table 3-2 for citation keys) tend to confirm this behavior for T < 150, above

which r1, -- 0 and A, -- oc, and the bed becomes mostly flat.

In order to determine the skill of the coupled modeling system at predicting ripple

geometry in lab-scale flows, a total of five simulations were performed having mobility

numbers ranging by an order of magnitude from 13 to 130. Using the classification system

of Wiberg and Harris (1994), the bedforms in the five lab-scale flow morphology tests fall

into the orbital range; therefore, the bedforms are expected to scale with hydrodynamic

properties within the wave bottom boundary l-iv-r (WBBL). Each simulation was allowed

to run for three wave periods (t = 6 s) before the morphology subroutine was engaged to

allow the hydrodynamic .,-i-iii.ii I ry associated with model spinup to subside, Simulations

were initialized with two parabolic-shaped ripples having heights approximately one-half

of the expected equilibrium and wavelengths corresponding to the expected value As

a baseline predictor for all simulations, the regular and irregular ripple predictors of

Nielsen (1981) were used to determine the expected geometry of lab-scale and field-scale

bedforms, respectively. Time-series of bedform statistics from the lab-scale morphology

tests are show in Figures 6-23(a)-6-23(e). The bed elevation signal was recorded at a









frequency of 10 Hz during model simulations, and then post-processed to determine both

time-dependent and time-averaged ripple height and length. The time-dependent ripple

length in Figures 6-23(a)-6-23(d) were computed using a zero downcrossing technique

after averaging the signal in the y-direction. We found the zero downcrossing technique

to routinely overestimate the ripple height by a factor of 1.25, so the statistical method

of Hanes et al. (2001) (Equation 6-2) was used to determine the instantaneous ripple

height. The method of Hanes et al. (2001) is also particularly helpful when analyzing a

signal that contains only one bedform, as there are not successive downcrossings in such

a signal. For the more intense flow of simulation HL1v1, a direct Fourier transform was

applied to the spanwise-averaged bed elevation to determine the time-dependent ripple

height and wavelength (Figure 6-24). The ripple height and length were determined

from the Fourier analysis by first selecting the wavelength A, with the largest energy

density, and then by multiplying the real Fourier coefficient, which describes the signal

amplitude at the selected wavelength, by two. Often, if the morphology is highly irregular

it is impossible to use the zero downcrossing technique to select wavelengths once the

mean of the bed elevation signal is removed, as it cannot identify successive crossings of

zero. The ripple heights computed using the Fourier analysis generally agree well with the

equation of Hanes et al. (2001), but the analysis is particularly helpful at selecting the

energy-containing wavelengths in the elevation signal. Average ripple heights and lengths,

along with the associated signal standard deviations, are denoted by the symbols and error

bars to the right of the time-series in Figures 6-23(a)-6-23(e). The time-averaged ripple

characteristics are obtained by averaging the quantities over the last twenty wave-periods

(t = 20 s) of the signals.

The time-averaged ripple heights ((Tlr)) and lengths ((A,)) obtained from the lab-scale

morphology simulation are compared to several semi-empirical ripple predictors in

Figures 6-25(a) and 6-25(b), respectively. The citation keys listed in the legends of

Figures 6-25(a) and 6-25(b) correspond to the ripple predictors defined in Table 5-5,









and discussed in C'i ipter 3. The expected ripple heights and lengths for the lab- and

field-scale flow simulations, determined using the ripple predictors, are provided in

Table 5-6. A one-to-one comparison of modeled abscissaa) and expected ordinatee) ripple

height r1, is shown in Figure 6-25(a), where the dark solid line -ii--.- -;perfect agreement,

and the dashed (--) and dotted (...) lines represent deviations of 10'. and :IG'. from

the expected values, respectively. The symbols are colored according to their numeric

value of the mobility number T, and correspond to the included color scale. There appears

to be much better agreement in the low to medium mobility numbers in Figure 6-25(a),

whereas only one comparison of ripple height (N81) falls within the :l' I. lines for the

high mobility number test. The model predictions of equilibrium ripple height appear to

agree best with the N81 and SW06 ripple predictors, with all of the comparisons falling

within the 3:l'. range. With the exception of the high mobility number case, all of the

N81 comparisons fall within 1(0' of the expected value.

As demonstrated in Figure 6-25(a), the model tends to follow the expected trend

of ripple height variation (decreasing) with mobility number (increasing). A similar

observation is made regarding the trend of ripple length (increasing) with mobility

number (increasing) when considering the one-to-one comparison of modeled abscissaa)

and expected ordinatee) ripple lengths in Figure 6-25(b). Admittedly, the comparison

of predicted and expected ripple wavelengths is less definitive, since the periodic length

of the streamwise domain is thought to p1 i, a role in wavelength selection. Since the

simulations were initialized at the expected wavelength, as determined by N81, the low to

medium mobility number cases show perfect agreement when model results are compared

to the N81 values in Figure 6-25(b). Comparisons using SW06 and M94 all fall within

:I '. of the expected values, while nearly all of the comparisons using FF02 fall within

10''. of the expected values.

The root-mean-square error (RMSE) between modeled and expected values for

ripple height and length were computed for each ripple predictor used in the one-to-one









comparisons of Figures 6-25(a) and 6-25(b), and are provided in Table 6-1. The
RMSE for ripple height and length were computed using Equations 6-3a and 6-3b,
respectively, where N is the number of tests, rli and Ai are the average ripple height
and length obtained by the model, and qi and Ai are the expected ripple height and
length, respectively. A total of eight ripple predictors were used in direct comparisons

to model results, for five different lab-scale flows, resulting in a total of 40 data points
for use in model-predictor evaluation. Each ripple predictor was assigned a score of 1
through 8 based on the computed value of TrRMSE and ARMSE, with 1 representing the
model-predictor comparison with the lowest RMS error, and 8 signifying the largest
RMS error between model and expected results. The individual scores for TIRMSE and

ARMSE were then combined, yielding an overall score for each model-predictor comparison.
The eight ripple predictors listed in Table 6-1 are ranked from lowest to highest score,
representing the smallest and largest combined RMSE, respectively. The analysis of
RMSE -,i-.-. -I that while the model results for r, agree best with those of Nielsen (1981),
the average ripple wavelengths A tend to be closer to the predictor of Faraci and Foti
(2002). This is an interesting result since morphology simulations were initialized using
expected ripple wavelengths predicted by Nielsen (1981); therefore, some modification of
the ripple wavelength is found to occur during the morphology simulations. Modification

of the ripple wavelength during the simulations is encouraging, as it -i .-.-. -1 that the
periodic streamwise domain is not prohibiting changes in the ripple geometry.


T1RMSE N i)2 (6-3a)
(1 i=l )

ARMSE = A i)2 (6-3b)
i=
There is much debate in the literature on bedforms over what parameters ripples

measured in field conditions actually scale best with. One reason for the debate stems
from the vast discrepancies found in collected ripple data from the field. Traykovski et al.









(1999) attributes some of these differences to hysteresis, whereby ripples in the field often

do not relax from larger to smaller wavelengths. Furthermore, Miller and Komar (1980a)

-,i--.- -1 that the spread of field data on ripple geometry may suffer from data collected

during times when bedforms are not in true equilibrium with the observed hydrodynamic

conditions. This lack of harmony in the expected ripple geometry for field-scale flows has

created an opportunity for a number of investigators to relate expected characteristics

to a variety of different parameters. While Wiberg and Harris (1994) argue that ripples

commonly found in the field can be characterized as anorbital, scaling with sediment

size as opposed to hydrodynamic properties. Conversely, Nielsen (1981) and Mogridge

et al. (1994) s-l--.- -1 a relationship between expected ripple geometry and hydrodynamic

properties, but do not agree on the nondimensional scaling groups used, with the former

using the mobility number T and the latter using the wave period parameter X. According

to the classification system of Wiberg and Harris (1994), the expected bedforms in the

field-scale morphology tests are expected to be suborbital, with the exception of the high

mobility number case which is anticipated to fall in the anorbital range. Suborbital ripples

represent a distinct subset of data containing ripples that do not tend to scale well with

either hydrodynamic or sediment properties.

A total of four morphology simulations were conducted under field-scale, or longer

wave period, flows where T = 8 s. Similar to the lab-scale simulations, the bedforms were

initialized at one-half the expected height, and at the expected equilibrium wavelength,

using the predictor of Nielsen (1981) for irregular, or field-type flows. Model simulations

were run for two wave periods (t = 16 s) prior to engaging the morphology submodel.

Figures 6-26(a)-6-26(d) show the time-series of bedform statistics for the lab-scale flow

simulations. The symbols and error bars to the left of the time-series correspond to

the time-averaged ripple heights and lengths, and their respective standard deviations,

respectively. The time-averaged ripple height (Trr) and length (Ar) were computed by

averaging the instantaneous results over the last five wave periods of the simulation









corresponding to a time of t = 40 s. Again, the bed elevation was recorded at a frequency

of 10 Hz throughout the simulation, and then post-processed to determine the average

height and wavelength of the bedforms. A zero downcrossing technique was used to

compute the average bedform wavelength, and Equation 6-2 was used to compute the

average ripple height as a function of time, in Figures 6-26(a) and 6-26(b). Fourier

a' -1 i was used to determine the ripple height and wavelength for the more energetic

simulations MF_2v2 and HF_1v1, and the timestacks are provided in Figures 6-27(a) and

6-26(d).

Direct comparisons of model results to expected ripple height and length for the

field-scale flow tests are shown in Figures 6-28(a) and 6-28(b), respectively. Expected

ripple heights and lengths for the field-scale morphology tests are also provided in

Table 5-6 for each of the eight ripple predictors used in the direct comparisons. With

respect to the lab-scale flows, the direct comparisons of model results to predicted ripple

height and length show a much larger spread in the expected results. The solid dark line

in the one-to-one plots shown in Figures 6-28(a) and 6-28(b) represents perfect agreement

between model predictions and expected results, while the dashed (--) and dotted ( ..)

lines represent deviations of 10'.' and nl' from perfect agreement. For the one-to-one

comparison of average ripple heights (9Tr) plotted in Figure 6-28(a), only the W05 and

FF02 ripple predictors consistently fall within n:' of perfect agreement. The symbols

in Figures 6-28(a) and 6-28(b) are colored according to the corresponding value of T for

that test case. Similar to the behavior found in the lab-scale flows, the model consistently

predicts a reduction in r1, for an increase in the mobility number, which may also be

taken as a proxy for the strength of the flow. The one-to-one comparisons of modeled and

expected ripple wavelengths are demonstrated in Figure 6-28(b), and reveal a similarly

large amount of disagreement between the eight ripple predictors used. The N81, W05,

and WH94 predictors appear to perform similarly with regard to predicting equilibrium









ripple wavelength, with most of the direct comparisons falling within ~3' of perfect

agreement.

The RMS errors between model predictions and expected ripple height and

wavelength for the field-scale flow tests were computed using Equations 6-3a and 6-3b.

Each ripple predictor was assigned a score of 1 through 8 based on the computed RMS

error for both ripple height and wavelength, with 1 corresponding to the smallest error,

and 8 representing the largest error. The scores for T/RMSE and ARMSE were then added,

yielding a total combined score for each ripple predictor. These scores are listed in

Table 6-2, and -r.-.;. -i that the model predictions of equilibrium ripple geometry in

field-scale flows agree best with the ripple predictors of Williams et al. (2005), Faraci

and Foti (2002), and Nielsen (1981). While exhibiting a larger degree of spread between

the predictions of ripple geometry in field-scale flows, the RMS errors listed in Table 6-2

are consistently of the same order of magnitude as those for the lab-scale flows provided

in Table 6-1. The similarity of these errors --. --; that the model is equally adept at

predicting ripple geometry in both lab- and field-scale flow regimes.

The combined scores for ripple predictors in lab-scale (Table 6-1) and field-scale

(Table 6-2) are further combined to produce a model performance index (\! PI) score in

Table 6-3. The MPI scores in Table 6-3 are reflective of the capability of the modeling

system to predict morphology over a large range of flow conditions. The rankings show

that for all of the morphology simulations, the best agreement between model predictions

and expected ripple height and length is obtained when using the ripple predictors of

Faraci and Foti (2002), Nielsen (1981), and Williams et al. (2005). The predictors of

Mogridge et al. (1994) and Khelifa and Ouellet (2000) complete the top five, with the

equations of Wiberg and Harris (1994), Soulsby and Whitehouse (2006), and Grant and

Madsen (1982) following in that order.

Further analysis of model predictions from the morphology tests is performed

with respect to the five ripple predictors having the lowest MPI scores in Table 6-3









(FF02, N81, W05, M94, and KO00). The analysis is conducted by plotting the model

predictions against the respective ripple predictor equations given in ('!i Ilter 3. The

ripple predictor equations (Equations 3-15b and 3-15a) of Faraci and Foti (2002) are

plotted in Figures 6-29(a) and 6-29(b) as a function of nondimensional ripple height and

length, respectively. Equation 3-15b is a function of both mobility number (') and orbital

Reynolds number ( f), so the results are plotted as a series of curves in Figure 6-29(a).

The family of curves for R, correspond to the median values of the ranges indicated

in the figure legend, where open and filled symbols signify lab-scale and field-scale flow

simulations, respectively. The model predictions appear to agree well with the family of

curves, and capture the general trends of bedform behavior where ripple height decreases

as the mobility number and orbital Reynolds numbers increase. The nondimensional ripple

wavelength, computed using Equation 3-15a, is plotted in Figure 6-29(b) as a function

of the orbital Reynolds number R, for a family of curves corresponding to the grain

Reynolds numbers ?Rg shown in the figure legend. Again, the open and closed symbols

in Figure 6-29(b) denote the lab-scale and field-scale flow simulations, respectively, and

the symbol shapes correspond to the range of grain Reynolds number provided in the

legend. The model predictions of ripple wavelength, when nondimensionalized by the

wave orbital semi-amplitude, agree well with the family of curves given by Faraci and

Foti (2002). Figure 6-29(b) demonstrates that the model predictions are following two

expected relationships, where the wavelength increases with increasing values of Rg, and

decreases with larger values of R,.

Model predictions of equilibrium ripple height and length, for all nine morphology

simulations, are plotted in Figures 6-30(a) and 6 30(b), respectively, along with the

ripple predictors of Nielsen (1981), Khelifa and Ouellet (2000), and Williams et al.

(2005). The expressions of N81, KO00, and W05 are functions of T exclusively, so each

curve in Figures 6-30(a) and 6-30(b) represents the ripple predictor corresponding to

the key given in the figure legend. Furthermore, the open and closed symbols represent









the lab-scale and field-scale morphology simulations, respectively. The references to

N81r and N81i in the figures correspond to the ripple predictors of Nielsen (1981) for

regular (lab) and irregular (field) flows. The model predictions of nondimensional

ripple height for lab-scale flows appear to agree very well with the rippled predictor of

Nielsen (1981) for regular flows. For the field-scale results, better agreement between

model predictions and expected results are found for the Williams et al. (2005) ripple

predictor, specifically for the short wave ripple (SWR) equations. Similarly, the model

predictions of ripple wavelength in Figure 6-30(b), when nondimensionalized by the wave

orbital semi-amplitude A, for lab-scale flows agree best with the curve of Nielsen (1981)

for regular flows, while the field-scale morphology results are better predicted by the

equations of Williams et al. (2005) for short wave ripples (SWR).

Figures 6-31(a) and 6-31(b) demonstrate the agreement between model predictions of

ripple height and length, respectively, and the ripple predictors of Mogridge et al. (1994)

for lab- and field-scale morphology results. The nondimensional ripple characteristics

in Figures 6-31(a) and 6-31(b) are plotted as a function of the wave period parameter

X. Although the formulation of X is similar to that of T, the wave period parameter

depends explicitly on the wave period squared, which is found in the denominator of the

nondimensional relationship. Therefore, larger wave periods yield lower values of X while

shorter, lab-scale flows yield values of X that can be one or two orders of magnitude larger.

The equations of Mogridge et al. (1994) provide estimates of the maximum expected

ripple height and length as a function of X, with shorter period flows (larger X values)

having smaller ripple heights and lengths. Figures 6-31(a) and 6-31(b) demonstrate fair

agreement between model predictions and expected values of ripple height and length,

respectively, particularly for the lab-scale flows (open symbols).

6.2.2 Phase II: Model Capabilities

Fourteen (14) simulations were performed to evaluate the skill and capability of the

coupled modeling system at predicting morphology over a range of flow and sediment









parameters, as well as its sensitivity to initial conditions. The results of these simulations

are provided in the sections that follow.

6.2.2.1 Steady Flow

A timestack of the spanwise-averaged bed elevation is shown in Figure 6-32. The

bed state diagrams of Southard and Boguchwal (1990) and Kleinhans (2005) SI---:. -1

that ripples should form for the given flow (U, = 30 cm/s) and sediment (d = 0.3 mm)

characteristics, while stronger forcing would tend to produce two-dimensional dunes. If

the applied bed stress in the steady flow case was overestimated, owing in part to the

larger eddy viscosity predicted by the LES submodel, then the formation of the dune in

the model simulation would be in agreement with the bed state diagrams of Southard and

Boguchwal (1990) and Kleinhans (2005). Although ripples did not form, the translation

of the hump is similar to the results of numerical simulations performed by Long et al.

(2007), which may be used to verify that the Euler-WENO morphology subroutine is

working properly.

As seen in Figure 6-32, the morphology reaches a steady state shortly after t = 100

s. This steady state is the result of a thickening, or diffusion, of the boundary 11- r to the

point where the applied bed shear stress drops below the critical threshold for motion,

and sediment transport goes to zero, as shown in Figure 6-33(a). As the boundary 1-iv r

diffuses vertically through the water column, the velocity shear near the bed tends to

decrease. This reduction in velocity shear is typically balanced by an increase in the

turbulent eddy viscosity in order to maintain a relatively constant bed shear stress

(Equation 4-47). Time-series of spatially-averaged nearbed velocity shear and eddy

viscosity are plotted in Figure 6-33(b) and demonstrate such behavior; however, the

increase in eddy viscosity does not appear to be substantial enough to overcome the

reduction of velocity shear in maintaining a higher bed shear stress.









6.2.2.2 Subcritical Flow

The flow and sediment characteristics of the subcritical flow simulation are such that

no sediment motion is anticipated. As found in Julien (1998), the threshold for sediment

motion of a sediment with a particle size of d = 0.3 mm corresponds to a critical shear

stress of Tr, = 0.21 Pa, or 2.1 dyne/cm2. The phase- and spatially-averaged bed shear

stress and bedload transport rate are plotted in Figure 6-35 as a function of fluid phase.

The yellow region in Figure 6-35 denotes the range of fluid phase over which the average

bed shear stress falls below the threshold for motion. For the subcritical flow simulation,

the average stress clearly stays well below the critical stress, resulting in zero bedload

transport. However, the timestack of bed elevations plotted in Figure 6-34 -i-.;. -I that

there is a subtle change from the initial to final ripple shapes. The smoothing of the bed

profile evident in Figure 6-34 is a result of the mildly diffusive morphology filter used

to prevent spurious oscillations in the bed elevation signal. Although the filter tends to

smooth the ripple profile over time, it does not appear to have an effect on the ripple

height, as demonstrated by the color contours in Figure 6-34.

6.2.2.3 Sheetflow Regime

As discussed in C'! Ilpter 2, the sheetflow regime of suspended sediment transport

is characterized by persistent, high volumes of sediment contained in a nearbed 1iv.r

approximately 10 to 100 grain-diameters thick. The suspended sediment transport regime

shifts from the vortex-dominated rippled bed regime to the shear-dominated sheetflow

regime for values of 0 w 0.8 1.0 (van der Werf, 2004). For values of 0 > 100oc, Li and

Amos (1999) find that intense sheetflow transport produces a flat bed. Dingier (1974)

-~i -I- -i that, in terms of the mobility number, the planar bed state is achieved for

T > 240.

Although the suspended sediment transport submodel is not anticipated to perform

well at such high sediment concentrations (see C'! Ilpter 4), a simulation was performed

to evaluate model behavior in the sheetflow regime. A timestack of the bed elevation









made from successive vectors of Zb(x,t), recorded at a frequency of 10 Hz, is shown in

Figure 6-36. The morphology was initialized with a parabolic-shaped ripple having

lr = 2.0 cm and A, = 13.6 cm. The bed elevation timestack in Figure 6-36 clearly

demonstrates the effects of the intense sheetflow transport on the morphology signal. As

expected, no persistent bed state is reached during 40 wave periods of the simulation. The

mobility number corresponding to the flow and sediment characteristics of the sheetflow

simulation (U, = 100 cm/s, T = 2 s, and d = 0.3 mm) yield a value of T = 206, which

is less than the flat bed limit (T > 240) of Dingier (1974) mentioned previously. Time-

and horizontally-averaged values of the Shields parameter are found to be (0) = 0.825,

which is within the range of values anticipated in the sheetflow regime (van der Werf,

2004). Time-series of ripple height and length are provided in Figure 6-37, and the

symbols and error bars at the end of the time-series denote average values and standard

deviations computed over the last twenty wave periods of the simulation. The time-series

of ripple characteristics was constructed from samplings of zb(t) taken at a frequency of

10 Hz, and then an !.i-. .1 using direct Fourier transforms to determine the instantaneous

ripple heights and lengths containing the largest energy density. A plot of spectral energy

density as a function of both ripple length A, and time, is given in Figure 6-38, and

demonstrates a lack of coherency in the bed state during the intense sheetflow transport.

The advection-diffusion equation (Equation 4-58) used to model the volumetric

concentration field in the coupled modeling system is not expected to produce reliable

answers for concentrations C > 1 (Villaret and Davies, 1995). Additionally, the

modeling system does not account for particle-particle interactions that are -i-i-:: -I '1

to pl i, an important role for concentrations C > 10-3 (Elghobashi, 1994). The

time- and volume-averaged suspended sediment concentration during the sheetflow

simulation was computed as (C) = 2.15 x 10-2, or (C) a 60 g/l. Isocontours of the

time-averaged volumetric concentration field (C(x, y, z)) are shown in Figure 6-39, and

are used to determine the average nearbed concentrations predicted by the suspended









sediment submodel. Values of (C(x, y, z)) isocontours correspond to the provided legend,

and -i-i.:: -i average concentrations as high as 265 g/1 within 1 cm of the average bed

elevation ((C(x, y, z)) = 0.6). The time- and horizontally-averaged concentration profile

(C(z)) is also provided in Figure 6-40 and shows similar behavior due to the horizontal

stratification of suspended sediment above the mostly flat bed. Ribberink and Al-Salem

(1995) report suspended concentrations as high as 500 g/1 in the upper sheetflow 1-lv.r,

where peaks in suspended sediment concentration are approximately in phase with the

freestream velocity. This agreement in phase is atypical of suspended sediment behavior

in the rippled-bed regime, where peak concentrations tend to lag behind the oscillatory

forcing outside of the boundary 1 .Ir. In volumetric units, sheetflow concentrations of 500

g/1 equate to C w 1'-. for quartz sand having a specific gravity of S = 2.65.

6.2.2.4 Ripple Growth, Equilibrium, and Decay

A total of seven unique simulations were performed in order to evaluate the

sensitivity of ripple height and wavelength to:

Initial morphology characteristics;

Periodic domain length; and

Hindered settling.

These experiments were conducted using the baseline simulation flow and sediment

characteristics of U, = 40 cm/s, T = 2 s, and d = 300 pm. Additional simulation

parameters for GROW1, GROW2, EQUILIB, DECAY, SLIDE, SPLIT, and HS1v2, such

as domain lengths and grid resolution, are listed in Table A-1. The purpose of these

simulations is to determine how different aspects of the coupled modeling system affect

the expected morphology. Eight predictions (N81, GM82, M94, WH94, KO00, FF02,

W05, and SW06) of equilibrium ripple height re1 and wavelength A, for our baseline test

are given in Table 5-6, and correspond to the simulation case name ML3v1. Where

applicable, the ripple predictor of Nielsen (1981) (N81) was used to determine expected

ripple heights rT1 and lengths A, a prior.









The first set of four experiments, namely GROW1, GROW2, EQUILIB, and DECAY,

were performed to evaluate the sensitivity of ripple height and length to the initial

morphology used in the model simulations. Timestacks of the spanwise-averaged bed

elevation zb, recorded at a frequency of 10 Hz during the simulations, are plotted in

Figures 6-41(a)-6-41(d) for the four simulations. The morphology for GROW1 was

initialized at e/2 and A, (based on N81) using a parabolic ripple shape. In GROW2, a

small Gaussian hump 5 grain-diameters in height (1.5 mm) and approximately 1 cm in

length was used to break the model symmetry experienced when using a perfectly smooth

bed. The initial morphology used in the EQUILIB simulation corresponds to the expected

ripple height and length, r and A,, as given by Nielsen (1981). In DECAY, the ripple

wavelength was initialized at A, while the initial ripple height was set at 150'. of the

expected value re.

The primary mechanism for growth from e/2, as demonstrated in Figure 6-41(a),

appears to be through a deepening of the ripple trough, and a subsequent increase in the

height of the ripple crest. This is similar to the expected behavior of ripple growth in

the orbital range, which was discussed in C'! lpter 3, where shear at the bed is directed

up the lee and stoss ripple slopes during both phases of flow due to flow separation in

the lee of the ripple crest. The crest-directed stress results in net transport of sediment

from the troughs to the crests during ripple growth, until it reaches an equilibrium with

entrainment of sediment from the crest and deposition in the trough. The behavior

exhibited by GROW2 (Figure 6-41(b)), where the simulation was started from a mostly

flat bed, is more indicative of growth in the linear regime, where groups of sand grains

tend to cluster together to form rolling grain ripples until the crests are high enough

to induce flow separation and vortex formation, finally saturating at the equilibrium

wavelength. A timestack of energy density is plotted in Figure 6-42 as a function of

A, and time for the GROW2 simulation, and demonstrates that there is no coherency

between energy and A, until saturation is reached at approximately t = 60 s, or after 30









wave cycles. Once flow separation begins, the time evolution of the ripple is similar to

that of GROW1 where sediment is taken from the trough and moved toward the crest,

thereby increasing the height. As anticipated, the ripple geometry remains relatively

constant throughout the EQUILIB simulation shown in Figure 6-41(c), where the

morphology was initialized at r]e and A,. When initialized at 150'. of the expected

height, the results of the DECAY simulation shown in Figure 6-41(d) demonstrate a rapid

decrease in height due to gravitational bedload transport, after which the morphology

remains relatively constant.

The time evolution of ripple height and wavelength for the GROW1, GROW2,

EQUILIB, and DECAY simulations are shown in Figures 6-43(a)-6-43(d), respectively.

These time series were reconstructed from the bed elevation Zb(X, y, t) sampled at 10

Hz, averaged in the spanwise direction, and then analyzed using Equation 6-2 and a

zero downcrossing technique to determine the ripple height qr,(t) and wavelength A,(t)

statistics, respectively, as a function of simulation time. Whereas growth from r1e/2 to

a relatively constant value occurred in about 10 wave cycles in Figure 6-43(a), growth

from a flat bed took nearly 50 wave cycles to reach a constant value (Figure 6-43(b).

After an initial rapid reduction in height due to gravitation transport, continued ripple

decay shown in Figure 6-43(d) occurred over approximately 15 wave cycles until a value

consistent with the final average was obtained. The time evolution of ripple height

for all four simulations is plotted in nondimensional form in Figure 6-44. Here, the

time-dependent ripple height rl,(t) is scaled by the expected value r1e as predicted by the

ripple predictor of Nielsen (1981), while the simulation time is scaled by the total duration

of each experiment, tf, provided in the figure. Figure 6-44 demonstrates that equilibration

of ripple height from an initially flat bed took approximately 150'. longer than required

when starting from one-half re. With exception of the flat bed case, GROW1, EQUILIB,

and DECAY all reach and/or remain at the expected ripple height 10' .









The preceding figures of ripple height evolution demonstrate that the final ripple

height predicted by the coupled modeling system is irrespective of the initial morphology

for a given flow condition (U, and T) and sediment characteristics (d). It would also

be beneficial to ensure that the choice of a periodic domain in the streamwise direction

is not forcing the selection of a particular wavelength, like L,, for the ripple geometry.

Two simulations, SLIDE and SPLIT, were performed to determine the effects of periodic

domain length L, on Ar, and demonstrate ripple coarsening and bifurcation.

Timestacks of spanwise-averaged bed elevation zb(t) from the SLIDE and SPLIT

simulations are plotted in Figures 6-45(a) and 6-45(b), respectively. In SLIDE, the

morphology was initialized with e/2 and A,/2, while the periodic domain was set at 2 A,.

The process of ripple coarsening, whereby ripples with wavelengths out of equilibrium with

the given flow condition increase in length until saturation at the expected wavelength A,

is achieved. This coarsening process is clearly evident in the timestacks of Figure 6-45(a),

as well as in the timestacks of ripple spectra plotted in Figure 6-46(a). The ripple spectra

demonstrate the energy density as a function of ripple wavelength A,, and show that the

initial energy confined to the smaller ripples having A, = 6.8 cm decays over the first

14 s until saturation at A, = 13.6 cm, after which time the equilibrium ripples continue

to grow in height. The immediate shift from one wavelength to another is characteristic

of the laboratory data reported by Doucette and O'Donoghue (2006), who classify such

behavior as ripple slide (see C'!i pter 3). A time-series of ripple statistics for the SLIDE

simulation, shown in Figure 6-47(a), demonstrates that the saturation of A, at t = 14 s is

accompanied by a rapid increase in ripple height until a dynamic equilibrium is attained.

The symbol and error bars to the right of the time-series represent the time-averaged

ripple height ((Tr) = 1.98 cm) and standard deviation (" .), respectively, computed

using the last 20 wave periods of the simulation. According to the ripple predictor of

Nielsen (1981) for regular waves, the SLIDE simulation over predicts the expected ripple









height of r- = 1.89 cm by roughly 5'. while underestimating the expected ripple length of

A = 13.6 cm by nearly the same amount.

The inverse of the coarsening experiment is represented by the SPLIT simulation,

whereby one ripple initialized with a Gaussian shape, and having a continuous length

equal to L,, splits to form two ripples having A, = L,/2. Timestacks of spanwise-averaged

bed elevation zb, sampled from the simulation at a frequency of 10 Hz, are plotted

in Figure 6-45(b) and show the progression from initial to final morphology. The

time evolution of ripple height is more clearly shown by the time-series plotted in

Figure 6-47(b), where the ripple height ,] corresponds to the left ordinate. These

time-series were produced by analyzing the bed elevation field zb(x, y, t) at a frequency

of 10 Hz, making use of Equation 6-2 and a zero downcrossing technique to determine

the spatially averaged ripple height and length, respectively. The time-averaged ripple

height ((rr) = 2.39 cm) and standard deviation (5'.), computed over the last 20 wave

periods of the simulation, are given by the square symbol and error bars. With respect

to the equilibrium ripple height predicted by Nielsen (1981), the SPLIT simulation over

predicts the average ripple height by 12.' ; when combined with the rather large value of

r = 5' over the last twenty wave periods, it is possible that the morphology had not

yet reached an equilibrium with the flow conditions. As is often the case with incoherent

bed elevation signals, the zero downcrossing technique failed to consistently identify an

appropriate ripple wavelength as demonstrated in the time-series of Figure 6-47(b);

therefore, Fourier analysis was employ, 1 to identify ripple wavelengths containing

the most energy in the signal zb(x, t). Timestacks of ripple spectra are provided in

Figure 6-46(b); initially, the energy in the signal is contained at A, = Lx, but shifts to

Ar = Lx/2 after approximately 7 wave periods. According to Doucette and O'Donoghue

(2006), this type of wavelength evolution is most similar to the process of ripple split,

where one ripple with a large wavelength decreases through a bifurcation into two or more

ripples having shorter wavelengths.









As demonstrated in Figure 4-13, the modification of the constant settling velocity

due to hindered settling, in suspensions having C < 5 x 10-3, is minimal. A simulation

(HS1v2) having the flow and sediment characteristics of our baseline test case (U, = 40

cm/s and d = 0.3 mm) was performed and incorporated the hindered settling velocity

in the advection-diffusion equation (Equation 4-58) for sediment concentration. While

time- and volume-averaged values of sediment concentration are small for the baseline

simulations, (C) < 10-3, instantaneous values can often be two or three times as large;

therefore, it would be beneficial to evaluate modifications of stratification and ripple

morphology from hindered settling.

The time- and horizontally-averaged concentration profiles for simulations HS1v2

and ML3v1 (baseline) are plotted in Figure 6-49. The reduced gradients of OC/az in

HS1v2 demonstrate an increase in vertical stratification within the boundary 1 ,-.r due

to hindered settling, as anticipated (Baldock et al., 2004). Time- and volume-averaged

concentrations ((C)) for HS1v2 and ML3v1 were found to be 5.7 x 10-3 and 3.3 x

10-3, respectively. For the typical values of (C) observed in the baseline simulations,

modification of the constant settling velocity due to hindered settling is only 0."-'

Time-series of ripple height and length for the comparative simulations are given in

Figures 6-48(a) and 6-48(b), respectively. For the simulations considered here, there

does not appear to be a significant effect on the ripple characteristics. The time-averaged

ripple height (over 20 wave periods) (plr) for the two simulations differ by only 2.5'. while

the average wavelengths (A,) differ by 2.>'. One noticeable difference between the time

evolution of ripple height in Figure 6-48(a) is the rate of growth, which appears to be

higher for the simulation with hindered settling. This increase in rate may be a result of

suppressed entrainment of sediment from the bed due to the lower vertical gradients in the

nearbed concentration field.

Recently, Baldock et al. (2004) published modifications to the exponent used in the

hindered setting velocity formulation of Richardson and Zaki (1954). These modifications









were a result of settling velocity experiments using real sand, and demonstrated an

increase in typical exponents by a factor of approximately 1.5. For the flow and sediment

characteristics used in the baseline simulations, estimates of w8 by Richardson and Zaki

(1954) and Baldock et al. (2004) differ by only 0.'"-., therefore, employing the more recent

results of Baldock et al. (2004) for real sand would not have a significant effect on the

stratification or the morphology.

6.2.2.5 Sediment Size Tests

With respect to the coupled modeling system, the previous simulations have

addressed model sensitivity to flow characteristics (mobility number, '), transport

regime, initial morphology bias, and periodic domain length. These simulations, however,

have only explored the parameter space of grain size in a very narrow range: 0.2 < d < 0.4

mm. It is anticipated that the grain size will have an effect on morphology and sediment

transport predictions through the Shields parameter (Equation 4-44), the fall velocity

(Equation 4-60), and the rate of entrainment of bed material (Equation 4-65).

Four simulations were conducted to test model sensitivity to large and small sediment

sizes, each for weak and strong flows. The bed state diagram of Kleinhans (2005) was

used to determine extreme combinations of flow and sediment characteristics that would

yield bedforms in one simulation and not the other, for each sediment size. The large

sediment size tested, d = 6 mm, is classified as pebble on the Wentworth scale, and fine

gravel in the Unified Soil Classification (USC) system (Dean and Dalrymple, 2004). The

smaller sediment size, d = 0.025 mm, corresponds to silt in both the Wentworth and

USC systems, or a medium silt in Julien (1998). There are some computational reasons

why these two sediment sizes represent extremes for the current modeling system: first, it

doesn't make physical sense to have a sediment diameter that is smaller than even your

highest numerical resolution through the bed, and second, using a grain size as large as

d = 6 mm would typically warrant a parametrization of bottom roughness to account for

the additional friction at the bed.









The fine gravel, having d = 6 mm, was tested in a weak flow having the characteristics

U, = 30 cm/s and T = 2 s, yielding a mobility number of =- 0.93 (GRAVLOW). Due

to the dependence of T on the grain size, the same flow characteristics yielded a value of

T = 28 for a fine sand having d = 0.2 mm in the morphology test case ML1v2 in which

ripples formed. According to the bed state diagram of Kleinhans (2005), no motion is

anticipated for the combination of large grain size and comparatively weak fluid forcing

used in the GRAVLOW simulation. Indeed, the ratio of destabilizing to stabilizing forces

represented by the Shields parameter 0 is difficult to elevate above the critical threshold

of Ocr = 0.055 due to the weight of the particle. The phase- and horizontally-averaged

bed shear stress and sediment transport rates are plotted in Figure 6-51, and demonstrate

that the the shear stress applied to the bed is never large enough to surpass the critical

threshold for motion, which is represented by the shaded yellow area in the figure. The

simulation is initialized with a flat bed, and the time-series of bedform height and length

shown in Figure 6-50 indicate that no morphology develops during the simulation.

The second simulation performed for the fine gravel employ .1 stronger fluid forcing

with U, = 80 cm/s and T = 2 s yielding a mobility number of T = 6.59 for the GRAVHI

test. Comparatively, the HL1v1 test case used the same fluid forcing, but yielded a

mobility number of T = 132 due to the medium sand size of 0.3 mm. The bed state

diagram of Kleinhans (2005), for oscillatory flows, -,ii--.- -I that the flow parameters of

the GRAVHI simulation are such that gravel ripples may be expected. Timestacks of

bed elevation (Figure 6-52), and time-series of ripple height evolution (Figure 6-53),

demonstrate that some morphology develops from the initially flat bed; however, the

computed average bedform height in Figure 6-53 is less than 1 mm, and our grain size is

d = 6 mm. Hence, it is difficult to ?-v whether these results are actually meaningful. It

is encouraging, however, that the model predicts both bedload and suspended sediment

transport during periods of strong flow, as demonstrated in Figure 6-54, as transport

would be necessary to generate the gravel ripples.









Simulations of the silt-sized sediment with d = 0.025 mm in weak and strong flows

are represented by the simulations SILTLOW and SILTHI, respectively. In SILTLOW,

the fluid forcing characteristics of U, = 10 cm/s and T = 2 s combine with the small

sediment size to yield a mobility number T = 24.7. Timestacks of the spanwise-averaged

bed elevation zb, sampled at a frequency of 10 Hz during the simulation, are plotted

in Figure 6-55 and demonstrate that the bed remained flat through approximately 30

wave cycles, after which time a low-crested feature began to form. The time evolution of

bedform height and length, determined through Fourier analysis, is shown in Figure 6-56

and reveals three wavelengths in the morphology: A, = 4.7 cm, A, = 7.6 cm, and the

largest, A, = 30 cm. The average height of the feature having the wavelength A, = 30 cm

is rr = 0.03 cm, which is on the order of ten grain sizes. It is unclear whether this feature

has physical relevance, or if it is a result of positive feedback between the suspended

load transport, shown in Figure 6-57(a), and bed-level updating submodel. The bed

state diagram of Kleinhans (2005) predicts no formation of persistent morphology for

the combination of weak flow and small sediment used in the SILTLOW simulation.

Figures 6-57(a) and 6-57(b) show the phase-averaged relationships between bed

shear stress and sediment transport; and bed shear stress and sediment concentration,

respectively. Average values of bed shear stress predicted by the model eclipse the critical

threshold required for sediment motion, represented by the shaded yellow areas in these

figures (Julien, 1998), but only yield significant values of suspended load transport and not

bedload transport. The phase- and volume-averaged suspended sediment concentration,

plotted in Figure 6-57(b), -,-.; --; an average value of (C) a 4 x 10-3 during the

simulation.

Fluid forcing characteristics for the simulation of silt in strong flows were chosen

such that we would expect highly-concentrated flows, as well as the development of

morphology. When combined with the small sediment size d = 0.025 mm, the forcing

characteristics U, = 40 cm/s and T = 2 s yield a very large mobility number:









T = 395. The bed state diagram of Kleinhans (2005) predicts the development of

hummocks or other complicated, three-dimensional bedforms for the parameters used in

the SILTHI simulation. Timestacks of bed elevation for the SILTHI simulation are shown

in Figure 6-58 and demonstrate a very complicated time evolution of bedform heights and

lengths through the processes of slide, split, and merge (Doucette and O'Donoghue, 2006).

Time-series of spatially-averaged ripple height and length are plotted in Figure 6-59, along

with time-averaged values and standard deviations obtained over the last 20 wave periods

of the simulation. The predicted average height (Tr) = 2.074 cm and length (A,) = 18.07

cm of the final morphology agree reasonably well with the predicted values of Grant

and Madsen (1982) (e = 2.17 cm, Ae 14.5 cm) and Soulsby and Whitehouse (2006)

(re = 1.80 cm, Ae 12.0 cm).

The phase- and spatially-averaged bed shear stress and sediment transport rates

are plotted in Figure 6-60(a), and sel---- -1 a dominance of suspended load transport

over bedload transport for the SILTHI simulation. With respect to the phase-averaged

suspended load transport ((Q,)) for the SILTLOW experiment (Figure 6-57(a)), the

phase lag of ((Q,)) for the SILTHI test is enhanced by approximately 200. These phase

lags are characteristic of the upper sheetflow li.--r (Ribberink et al., 1994), but are

typically not found within the active pick-up l. I-i~ near the bed. The phase lag between

average shear stress and average sediment concentration is approximately 450, as shown

in Figure 6-60(b). Time- and volume-averaging the sediment concentration field during

the SILTHI simulation yields a value of (C) = 0.025, or (C) = 66.25 g/1. According

to Elghobashi (1994), such large values are indicative of a range where particle-particle

interactions become important; therefore, it is unclear whether the current suspended

sediment submodel (Equation 4-58) is applicable here, although Villaret and Davies

(1995) cite successful applications of similar models for concentrations as high as C 1

A phase diagram of model applicability, with respect to the suspended sediment transport

submodel, is provided in Figure 6-61. The phase diagram shows the predicted time- and









volume-averaged suspended sediment concentrations from 15 different model simulations

conducted over large ranges of fluid forcing and sediment size. The color of each data

point in Figure 6-61 represents the predicted value of (C) for the specific combination

of orbital Reynolds number R, and sediment size. Orbital Reynolds numbers are plotted

on a logarithmic (base ten) scale in Figure 6-61, while the sediment sizes correspond

to the Phi scale of Krumbein (1936), where grain size is distributed from large to small

progressing from left to right along the abscissa. For reference, typical results of a baseline

simulation, such as ML3v1, are indicated on the phase diagram.

6.2.3 Phase III: Sediment Transport Processes

In order to address our specific scientific questions posed in ('! Ilpter 1, four

independent simulations were performed to evaluate the roles of bedload and suspended

load sediment transport during ripple growth, equilibrium, and decay. The simulation

results also provide previously undocumented information regarding the roles of bedload

and suspended load transport as a function of the ripple profile, as well as phase of

fluid forcing. These comparative simulations were conducted by first obtaining the

morphological response to bedload and suspended load for our baseline test case having

U, = 40 cm/s, T =2 s, and d = 0.3 mm (TL2v2). The morphology in TL2v2 was

initialized with r, = 1.0 cm and A, = 13.6 cm, and model hydrodynamics were simulated

for three wave periods (t = 6 s) prior to engaging the sediment transport and morphology

submodels. While the TL2v2 simulation simulates ripple growth in response to the

baseline simulation conditions, an additional experiment (TL2v2d) was performed

to evaluate the behavior of ripple decay under total load conditions. In TL2v2d, the

morphology was initialized with Tr, = 3.0 cm and A, = 13.6 cm. Two additional

simulations were then performed incorporating only bedload (BL2v2) or suspended

load (SL2v2) transport modes.

As a result of the computational grid and model domain width test simulations

performed in Phase I, these final production runs were performed using the resolution of









GRID1, and conducted with the three-dimensional model having domain widths of 3.2 cm.

The resolution throughout the ripple profile (6x = 6y = 0.2 cm, 6z = 0.05 cm) is such that

each control volume would contain approximately 85 spherical particles having d = 0.3

mm and a porosity np = 0.4. These final production simulations, then, are the equivalent

of modeling 9.2 x 106 sand particles.

6.2.3.1 Total Load Growth

The total load simulation TL2v2 was initialized with two parabolic-shaped ripples

having 1, = 1.0 cm and A, = 13.6 cm. The time evolution of ripple height, shown in

Figure 6-62(a), indicates that after an initially rapid increase in height during the first

20 wave periods (O9]r/t = 3.5 x 10-2 cm/s), the rate of growth slows considerably

(Or]r/t = 4.7 x 10-4 cm/s) until t U 110 s when a stable ripple height is reached. A

time-average of the ripple height taken over the last 20 wave periods of the simulation

yields a value of (Tr) = 2.43 cm having a standard deviation of 0.;:' The average ripple

height predicted by the TL2v2 simulation is within 5'. of the prediction of Nielsen (1981)

using Equation 3-3b. As demonstrated in Figure 6-62(b), there is very little modification

of the ripple wavelength, from the initialized length, during the course of the simulation.

The time-average ripple wavelength, taken over the last 20 wave periods, was found to be

(Ar) = 13.68 cm and had a standard deviation of 0.7'-. Time evolution of the ripple

steepness, computed as ,/Ar,, is provided in Figure 6-62(c), and generally follows the

evolution of height since the length remains nearly constant.

The predicted shape of the bedforms, clearly demonstrated by the bed elevation

timestacks in Figure 6-63, is mostly triangular with slightly smoothed crests. The

smoothing at the crests is most likely an artifact of the bed level filtering process, the

effects of which were demonstrated by the SUBCRIT simulation. However, the mostly

triangular shape of the ripple flanks is characteristic of shapes often observed in the

field (e.g. Figures 1 l(c) and 1 l(d)) and also found in the laboratory experiments of

Stegner and Wesfreid (1999). For their lower-frequency experiments (w = 0.5 Hz),









which corresponds to the angular frequency used in the baseline simulations, Stegner and

Wesfreid (1999) observed triangular-shaped ripples with peaked crests and linear flanks

while the higher-frequency tests ( = 1 Hz) produced a more sinusoidal ripple having

smooth crests and troughs.

The TL2v2 simulation results -i-i-:: -1 an initially linear growth rate for the

morphology, as demonstrated in Figure 6-62(a), followed by a gradual approach to the

final value of rr = 2.43 cm. The red boxes in Figure 6-62(a) highlight two specific areas of

interest: first, the period of growth that occurs during the first 20 wave cycles, and second,

a period of equilibrium during the last 20 wave cycles of the simulation. For the period

of growth, the rate of increase is found to be r9r/0t = 3.5 x 10-2 cm/s, and is followed

by a marked decrease over the next 33 wave periods to 9Or/at = 4.7 x 10-4 cm/s. These

same areas of growth and equilibration are identified in Figure 6-64, where the cumulative

spatially-averaged and corrected transport fluxes are plotted as a function of time. We

call these fluxes "corrected," because we have altered the sign of the transport fluxes to

produce a more intuitive convention where positive (+) fluxes correspond to bedform

growth, and negative (-) fluxes represent bedform decay (Penko, 2007). The alteration

of sign must be performed due to fluxes either above or below the bed elevation zb = 0

representing different growth tendencies. The cumulative, spatially-averaged and corrected

bedload (Qt) and suspended load (Qf) fluxes are computed using Equations 6-4a and

6-4b, respectively. In Figure 6-64, the corrected fluxes s-l--:- -1 that the bedload fluxes

are responsible for the i_, .ii i ly of the ripple growth throughout the simulation, and,

therefore, represents a constructive force with respect to ripple growth. Although the

corrected suspended flux is found to be negative initially, thereby acting to decrease

the ripple height (a destructive force), it too becomes increasingly positive during the

simulation. Once in the equilibrium phase of ripple evolution, corresponding to the

right-most box in Figure 6-64, we see that the cumulative average bedload and suspended









load fluxes are of the same magnitude. This s-l--:- -1- a balance between bedload and

suspended load during times of constant ripple geometry.

nx,ny


Vb (6 4a)
nx, n n

S(wC E)t
i, 1,1
VQ (6-4b)
nx ny

The qualitative relationship between bedload transport, entrainment, and deposition

to the phase of hydrodynamic forcing appear to vary little during times of ripple

growth and equilibration. The phase-averaged bed shear stress, gravitational stress,

bedload transport rate, and vertical sediment fluxes are provided in Figures 6-65(a) and

6-65(b) during ripple growth, and Figures 6-65(c) and 6-65(d) during ripple equilibrium.

Although the bedload transport rate and vertical sediment fluxes decrease in magnitude

from growth to equilibration, their behavior with respect to the phase of applied shear

remains similar. The bedload transport rate simply mimics the phase-dependence of the

applied shear, with both leading the freestream forcing by approximately 600. Phase

leads such as these are common in both laminar and turbulent oscillatory boundary

1~.v. r, but this value is higher than values typically found in the literature (Nielsen and

Callaghan, 2003). It is unclear whether this more pronounced phase lead has physical

significance, owing to the evolution of the bedforms, or is an artifact of deficiencies in the

LES closure model discussed previously. The vertical sediment fluxes -,-II-- -1 a slightly

different behavior, with deposition 1 S-:-ii_-; slightly behind periods of maximum flow, and

entrainment leading the maximum flow (Figures 6-65(b) and 6-65(d)). The model clearly

demonstrates the inverse relationship between the entrainment and deposition of solids

at the bed; when entrainment is large, deposition is small; and when deposition is large,

entrainment is low.









While Figures 6-65(a)-6-65(d) do show the phase-dependence of bedload transport

and vertical sediment fluxes, they do not provide information as to whether those

sediment transport processes are producing net growth or decay. If we instead take

the horizontally-averaged and corrected bedload and suspended load fluxes given by

Equations 6-4a and 6-4b, and then phase-average them over 20 wave periods during

times of growth and equilibrium, we clearly see the effects of fluid forcing on building and

decaying fluxes in Figures 6-66(a) and 6-66(b). During ripple growth, the phase-averaged

and corrected fluxes indicate that bedload and suspended load fluxes are of the same

magnitude, and behave inversely; when building (positive) bedload fluxes are low (1 ,.--iir-;

behind maximum freestream forcing), the growth of the bedform is dominated by vertical

fluxes; when building (positive) bedload fluxes are large (1 .-.-i i-; slightly behind flow

reversals), the bedforms experience decay through vertical sediment fluxes. When the

curves in Figure 6-66(a) are integrated and the areas summed, the result is a net positive

in sediment fluxes corresponding to ripple growth; furthermore, we find that i.' of the

transport fluxes associated with growth may be attributed to bedload processes, while the

remaining ;:".. is due to the vertical sediment fluxes of entrainment and deposition. These

results are, of course, consistent with our observations of ripple growth through the time

evolution of ripple height shown in Figure 6-62(a).

The phase-dependence of building and decaying sediment fluxes is similarly intriguing

during ripple equilibrium, as plotted in Figure 6-66(b), -, --.- -Iii':-; an unexpected

compensation between the two transport modes to maintain equilibrium. More specifically,

when the vertical sediment fluxes are building (decaying) the ripple following the

maximum freestream fc.n ivr. the bedload fluxes are compensating by decaying (building).

Integrating the two curves in Figure 6-66(b), we find the net action of bedload and

vertical sediment fluxes to be nearly zero, which is consistent with our observations of

ripple equilibrium. Additionally, comparing their integrated magnitudes si -.-. -i- that only









-:'. of the transport fluxes are due to bedload, while i'-, of the fluxes may be attributed

to suspension and deposition events.

Thus far, we have only discussed the temporal and phase dependence of the sediment

transport fluxes with respect to ripple growth and equilibrium. Using the sign convention

for building (+) and decaying (-) fluxes as described earlier, but dropping the horizontal

averaging, we define the time-averaged and corrected bedload and suspended load

transport fluxes in Equations 6-5a and 6-5b, respectively. Such a convention is useful

for determining the distribution and tendency of transport fluxes throughout the ripple

profile. Time-averaging over 20 wave periods during ripple growth and equilibrium,

these corrected fluxes are plotted in Figures 6-67(a) and 6-67(b), respectively. Each

symbol represents a point on the horizontal bed plane, and their colors correspond

to the time-averaged bed elevation (zb) during the respective period. Plotting the

bedload fluxes against the suspended load fluxes in this manner yields quadrants of

growth and decay for each transport constituent, as well as a line of equilibrium. A

more thorough explanation of the transport flux tendency phase diagram is provided

in Appendix B. During times of ripple growth, the data in Figure 6-67(a) indicate a

bedload-dominated (closer to the horizontal axis) growth at the ripple crest (warmer

colors), and a suspension-dominated (closer to the vertical axis) growth in the ripple

troughs (cooler colors). This flux tendency phase diagram confirms the observations of

ripple growth from the time-series of ripple heights (Figure 6-62(a)), as the concentration

of data points is highest above the equilibrium line (the solid diagonal line), in the growth

tendency region. The locus of transport flux data during ripple equilibrium is found to fall

almost entirely along the equilibrium line in Figure 6-67(b), which also agrees with the

observation of ripple equilibrium during the last 20 wave periods of the TL2v2 simulation.

The distribution of the data points in Figure 6-67(b) -ir-.-. -I that in order to maintain

equilibrium, the ripple crest experiences an almost perfect balance between bedload flux









growth and vertical flux decay; the opposite is observed in the troughs, where suspended

fluxes promote growth and balance the bedload fluxes that cause decay.

nt fX nxly .

Q 1 ij 1,1 (6 5a)
(V b,) fllk^l -l (6-5a)


SZ (wsCb E)
(VQ-) -1 ij 11 (6-5b)

6.2.3.2 Total Load Decay

The results of the total load simulation TL2v2 provide information about the

spatial and phase dependence of the various transport components with respect to ripple

growth and equilibrium. A similar test was performed to evaluate the phase and spatial

dependence of the transport components, as well as the bedform behavior, during times

of decay (TL2v2d). This was achieved by repeating the simulation parameters of TL2v2,

with the exception of initializing the morphology with a ripple height that was three-times

larger, or rl, = 3.0 cm. The initial ripple wavelength was maintained at A, = 13.6 cm,

thus creating a bedform that would be out of equilibrium in both height and steepness

(or = 0.22 cm/cm). Initially, it was unclear whether the ripple wavelength would also

have to adjust to compensate for the higher than normal steepness; maximum steepness

values are found in vanishing flow conditions and have typical values of cr = 0.32 tan Q

or about a, = 0.18 for a sediment having an angle of repose of = 300 (Nielsen, 1992).

As demonstrated by the time evolution of ripple wavelength shown in Figure 6-68(b),

A, tended to remain constant throughout the simulation. In order to then reach the

anticipated equilibrium steepness ,r = 0.18, plotted in Figure 6-68(c), the ripple height

d.'- i, d rapidly over the first 10 wave periods as shown in Figure 6-68(a). Following

the initial rapid decay in height, the ripple experienced a much more gradual decay over

the next 40 wave periods until it finally reached an equilibrium value near t = 106 s,









remaining constant for the remainder of the simulation. The time-averaged ripple height

and standard deviation, taken over the last 20 wave periods of the simulation, are plotted

at the end of the time-series in Figure 6-68(a).

Applying the segregation of signs to the transport fluxes to signify their tendency

to promote bedform growth (+) or decay (-), and then using Equations 6-4a and 6-4b,

we find that the initial rapid decay of ripple height is due to the vertical transport fluxes

of entrainment and deposition. These results are plotted in Figure 6-69, and indicate

that the bedload transport flux is perfectly balanced with equal magnitudes promoting

both growth and decay, while the suspended load fluxes are clearly negative, indicating

their tendency to promote ripple height decay. By extracting the first 20 s of data shown

in Figure 6-69 and phase-averaging over those 10 wave periods corresponding to the

time of rapid ripple height decay, we are able to plot the phase-averaged and corrected

transport fluxes, as shown in Figure 6-70. During rapid decay, the phase-averaged

bedload transport fluxes are perfectly balanced in magnitude between growth and decay

(((VQ )) = 0), while the vertical sediment fluxes are strongly negative throughout most

of the wave cycle ((((wsCb E) ) < 0). When integrated over the wave cycle, the

phase- and spatially-averaged and corrected vertical sediment flux is negative, verifying

its tendency to promote ripple decay. As a function of wave phase, the vertical flux term

tends to promote weak building during flow deceleration, and stronger decay during flow

acceleration. The phase dependence of the bedload, entrainment, and deposition transport

components is demonstrated in Figures 6-71(a) and 6-71(b), and shows a behavior similar

to that observed during the simulation of growth and equilibrium. The bedload transport

tends to mimic the phase behavior of the motivating bed stress; peaks in the entrainment

signal lead the maximum freestream forcing; and deposition events lag slightly behind the

maximum flow. One clear difference between the growth and decay simulations, however,

is also evident in Figures 6-71(a) and 6-71(b); the magnitude of the gravitational stresses









is much larger in the decaying phase, -ii-:: -1ii-:; a strong gravitational component in the

bedload sediment transport signal.

Similar to the simulation of bedform growth and equilibrium discussed earlier, the

transport fluxes also exhibit a spatial dependence with respect to their location along

the ripple profile. The spatial dependence of the bedload and suspended load transport

fluxes is demonstrated in the flux tendency phase diagram shown in Figure 6-72. Here,

the data points represent the time-averaged and corrected flux tendencies computed using

Equations 6-5a and 6-5b and are colored according to their vertical position along the

time-averaged ripple profile ((zb)); warmer colors represent the time-averaged ripple crest

location, while cooler colors indicate the average ripple trough elevation. Using the flux

tendency phase diagram maps provided in Appendix B, we can infer from Figure 6-72

that most of the ripple decay is experienced through suspension events at the crest, and

deposition in the trough. The upper ripple flanks experience a weak tendency to build

through bedload transport, while the lower flanks experience weak building through

deposition events.

6.2.3.3 Bedload Only

Having already determined the behavior and tendencies of bedload transport

fluxes to promote ripple growth in the total sediment load simulations, performing a

simulation incorporating only the bedload transport term in our bed-level updating

scheme (Equation 4-67) is done for the sake of completeness. In the BL2v2 simulation,

the vertical sediment fluxes corresponding to sediment entrainment (E) and deposition

(w8Cb) are set to zero in the transport and morphology submodels. If our assessment of

bedload flux tendency in the total load simulations holds true, then the modeling system

should still predict bedform growth and equilibration at or near the expected conditions.

The time evolution of ripple height, length, and steepness is provided in Figures 6-73(a),

6-73(b), and 6-73(c), respectively, and confirms our previous evaluation. Figure 6-73(a)

-Ii--.- -I that while the ripple height does grow to within 12''. of the value observed in the









total load simulation TL2v2, the growth period occurs over approximately 30 wave periods

instead of 20 as before. The slower observed growth rate (Olr/a/t = 2.1 x 10-2 cm/s) is due

to the absence of suspended load transport in the BL2v2 simulation, which, according to

Figure 6-64, also contributes to bedform growth under certain conditions.

The phase-averaged shear stress, gravitational stress, and bedload transport

rates during ripple growth and equilibration in the BL2v2 simulation are provided

in Figures 6-74(a) and 6-74(b), respectively. There are no substantial differences in

the phase dependence of bedload transport during ripple growth or equilibrium, nor

does the behavior differ significantly from the total load simulation results provided

in Figures 6-65(a) and 6-65(c). However, Figures 6-74(a) and 6-74(b) do -ii--. -1 an

enhanced phase lead of the bed shear stress and bedload transport, ahead of the maximum

freestream velocity, as ripple height increases. This result is consistent with the enhanced

phase lead found for the total load simulation (TL2v2).

6.2.3.4 Suspended Load Only

Simulations of suspended load only were achieved by setting the bedload transport

rate to zero in the bed-level updating scheme given by Equation 4-67. Such an experiment

holds little physical relevance, as it is unlikely to ever occur in nature; however, the results

are illustrative.

Previously, we observed that the suspended load transport fluxes contributed to

positive bedform growth under certain circumstances during the total load simulation

TL2v2 (see Figures 6-64 and 6-66(a)). If this were generally true, then would a simulation

incorporating only the entrainment and deposition of bed material in the morphology

submodel yield bedform growth? The answer, interestingly, is no. The time evolution of

ripple height, length, and steepness for the SL2v2 simulation is shown in Figures 6-75(a),

6-75(b), and 6-75(c), respectively. After an initially rapid decay in ripple height over

the first 10 wave periods, Figure 6-75(a) demonstrates a more gradual decay during the

subsequent 10 wave periods, followed then by a generally static time where the height









does not vary more than :'-. After 72 wave periods, the final ripple height and length are

(Tr) = 8.5 x 10-2 cm and (A) = 14.4 cm, respectively; the time-averaged height over the

last 20 wave periods of the simulation is on the order of 3 grain-diameters, -~i--.- -1 iir-; that

the bed is mostly flat.

The phase dependence of suspension and deposition events for the rapidly decaying

ripple height observed in simulation SL2v2 shows significantly different behavior from

that observed in previous simulations. Phase-averaged bed shear stress, entrainment,

and deposition are plotted in Figure 6-76, and are i-.-- -I i,.- of turbulent flow and

transport over a rough bed, where both lead the freestream forcing by approximately 18.

In the total load and bedload simulations, periods of entrainment tended to lag slightly

behind the maximum freestream forcing, while depositional events occurred during times

of flow reversal. The depositional behavior, with respect to wave phase, is unchanged

for the simulation incorporating only suspended load transport; however, peaks in the

entrainment signal tend to closely follow the phase signal of the applied fluid forcing and

bed stress. This result is encouraging, as it agrees well with the observations of Ribberink

et al. (1994) who found almost no phase difference between flow and suspension in the

sheetflow and pick-up l V.-ir for transport in turbulent flows over rough beds.

The results of the total load, total load decaying, bedload, and suspended load

simulations provide interesting detail about the phase and spatial tendencies of bedload

and suspended load transport fluxes to promote ripple growth, equilibration, and decay.

While bedload transport is completely capable of generating and sustaining orbital ripples

in the simulated flow regime (moderate mobility number, T = 32.95), suspended load

transport, in the absence of bedload sediment transport, tends to promote ripple decay

through strong suspension events at the ripple crests, subsequently leading to deposition

in the troughs. The time evolution of ripple heights for all four simulations are plotted in

Figure 6-77. Here, ripple heights are scaled by the final equilibrium height of the total

load simulation ((Tr) = 2.43 cm), and time is nondimensionalized by the wave period.
















Z, (cm) k

03 x
024
0 18
012
0 06
0
-0 06
-0 12
-0 18
S-0 24
-0 32


15


10

0
10 5
20
30
40
xCn 50 60 0






Figure 6-1. The timestacks of bed elevation for the 1DH linear model experiment.




















100


80

E

60 <


a-
40


20
20


Time (s)


(a) Height and Length


4 8 12 16 20
Time (s)


(b) Steepness


Figure 6-2. Time-series of ripple (a) height (left ordinate) and length (right ordinate), and

(b) steepness from the 1DH linear model experiment.


S0.06
C

CL

) 0.04

0.
0.

0.02



0
0
0.2-


,r














































0o


C.)
Cor
E 2

M
^ a


I I I I I I I I
In o ICn o

(wU3) z




C,


I I I I
In o In o
(wo) z


















So I0 o

(wo) z


2,
Co)
E -










C.)
4
I3:
















2 -


vv






E -i
4
V



v~-





u ~~
- t
SCo1
13 ,-

^^ -


D- O
c"
U 0














S o in oi
(uwo) z








c

c 00


c























) I) C
(uwo) z


In o n o In o In
(wU3) Z (wUo3) Z


0









O
v .
C) 'S









0
O 0









z
m -
0b






a r


















2-d






O























0I
1m r-
E

.2 b


'-5;D






























Phase (radians)


Figure 6-4. A comparison of the modeled (
1-vr thickness (6).


) and observed (o) phase-averaged boundary
















Difference (%)
0 15 30 45 60 75 90 105 120 135 150


On 0.257c 0.57 0.757c
Phase (radians)


(a) Phase-Averaged Difference


0.257c 0.57 0.757c
Phase (radians)

(b) Phase- and Vertically-Averaged Difference


Figure 6-5. The (a) phase-averaged and (b) phase- and vertically-averaged absolute
percent difference between model and experimental data.


150


S125

C
2 100

5
Z 75

0


so
a 50


: 25





















10'
I-


103 104 105 106
Number of Grid Points

Figure 6-6. The computational budget for the grid comparison tests (GRID1, GRID2,
GRID3, GRID4). The equation and statistics of the least-squares fit to the
four data points are provided in the figure.



























S10 -- -

8Y
6
S-- GRID1
4 ---- GRID2
4 ---- GRID3
> --- GRID4
> 2

I I I I I I I I I I
0 4 8 12 16 20 24 28
Time (s)


(a) (u,





04


S02


0 4 8 12 16 20 24 28
Time (s)


(b) a,


6


S4






0 4 8 12 16 20 24 28
Time (s)

(c) u,


Figure 6-7. The (a) u- and (b) w-velocity standard deviations of volume-averaged

velocities as a function of time for GRID1 ( ), GRID2 (-), GRID3 (
D-









and GRID4 (-).















214



















80
I^
E
S60

I-
" 40

E
C.


Figure 6-8.


Time (s)


The cumulative, volume-averaged turbulent kinetic energy (TKE) as a
function of time for GRID1 ( ), GRID2 (-), GRID3 (-), and GRID4 (-).


100



80



- 60
E


S40



20



0


Figure 6-9.


17C
Phase (radians)


The phase- and volume-averaged turbulent kinetic energy (((TKE))) as a
function of forcing phase for GRID1 ( ), GRID2 (-), GRID3 (-), and
GRID4 ().














0 O

















-4 0
biCk


oc
E "

u Q~







HCCl


0





cn
13 --












b-














bci
z 0





^ E -s .





X9 ~
0 a



























oc



















(w3) z
2 0
-&^


E^-C 0









Hm
i-i 11























(w3o)z b-* O 0
2m
c,^

0 ^ "
o
= r c
^ ^ -^ -
-1 -, o s i
^^ ^ i< ^ G

m~ \N ^' ^
\ ^ 2 --- '
(Ur3) Z
m bO--
__________ ^-' i





~i -- i -- -- J: -- ^ ^



_______________ d00 a




I Cb ~^ fi n
E o" 8



I ^-, r^ CC
\ ^ ^ ^
a- ^_^ 3 S
ri --

r r
c -' -' -











216


(w3) z


(wU3)z













0 O







cb
0


E |
----- == ^-I I ^ '*









0 a0


(w13)Z z
,-Q ,- '

or m





=0
? ^ m



E -- ^ r










00^
cb O
k 0

:i O
0 -- n0
E !
















-m o

0'' ^ '







(w3)z b S^
217
o ^ ^ || '
,- QU /\ C
^ g S^
----- --- ^ ^ S 3
fOs
cba



-~ ~~ ~ ~ \i : ^ I
\ ^-, r m
\ ~ r ^ ^^
= a_





^00

(luo)d z~ i~

o bf i

217O


(w3) z


(wo3)z















o 0


oo


430
0










0















-4-a
06 m












0-
-H I































oc' C
(w23) z I
00
0









--- -- ^ ^ ^ --- ^ ^ ^ 2

____ ^ T









00 *^ a
\ ^ ^ ^ *
^_ -- ^ ^ ^ ^ ^ ^ 1 ^ o ^
I ^ x co ^ 3 I
\ ^ ^' ^ ^ ^






(iuo)z i kj


218


(w3) z


(wU3)z





















103
E


104 105 106
Number of Grid Points

Figure 6-13. The computational budget for the three-dimensional grid comparison
tests (WIDE3D_2, WIDE3D_5, WIDE3D_10, WIDE3D_20, WIDE3D_30,
WIDE3D_40, and WIDE3D_50). The equation and statistics of the
least-squares fit to the seven data points are provided in the figure.




















102



SI-

WIDE3D_2
S10 -- WIDE3D_5
E
S-- WIDE3D_10
0 WIDE3D 20
WIDE3D_30
WIDE3D 40
WIDE3D_50

100 I I
0 20 40 60 80
Time (s)


Figure 6-14. The cumulative, volume-averaged turbulent kinetic energy (TKE) as a
function of time for domains having increasingly large widths in the third (y)
dimension.


103








E 102

-)





101


Phase (radians)


Figure 6-15. The phase- and volume-averaged turbulent kinetic energy (((TKE))) as a
function of forcing phase for domains having increasingly larger widths in the
third (y) dimension.
















60 I I


40


20 \


0

WIDE3D_2
-20 -- WIDE3D_5
--- WIDE3D_10
-- WIDE3D_20
-40 -- WIDE3D 30
-- WIDE3D 40
WIDE3D 50

-60 0 5x 1M 1 5n' 2x
Phase (radians)


(a) u-velocity




1 i i




05 -



05




-05 -




005 0 5 17 1 51 2.
Phase (radians)


(b) v-velocity




001




0 005 -








-0 005 -




-00 0 5x 17 1 5x 2x
Phase (radians)


(c) w-velocity



Figure 6-16. The phase- and volume-averaged (a) ((u)), (b) ((v)), and (c) ((w)) velocity

components for seven different three-dimensional grids.



221





















102


S101

E
100


a 10 -
a

S102-
:


c) 10 -

10-4


10 -
10-2


ky (cm-1)


(a) Spectral Density


S100





0
>. 10-
E


o 2







10-3
10-2


10-1 100
ky (cm-)


(b) Velocity Amplitude


Figure 6-17. The time- and spatially-averaged (x and z) (a) spectral density, and

(b) amplitude of the v-velocity as a function of wavenumber (ky) for

three-dimensional grid comparisons.


WIDE3D_2
WIDE3D 5
WIDE3D_10
WIDE3D_20
WIDE3D_30
WIDE3D_40
WIDE3D_50


WIDE3D_2
- WIDE3D_5
- WIDE3D_10
- WIDE3D_20
- WIDE3D_30
- WIDE3D_40
WIDE3D 50
























10


10-1



S
e 10.2
._


O
m 10-4 WIDE3D 2
C -o WIDE3D-5
.- WIDE3D_10
U) 10 -- WIDE3D_20
-- WIDE3D_30
106 -- WIDE3D_40
WIDE3D_50

10
0 20 40 60 80
Time (s)


Figure 6-18. The x-averaged standard deviation of the bed elevation (ab) in the
y-direction as a function of time for three-dimensional grid comparisons.


2.5


0.5


2 4 6
k (cm)


8 10


Figure 6-19. Time-averaged equilibrium ripple heights ((tr,)) as a function of domain
width (Ly) in the third dimension. The error bars correspond to the
standard deviation of the ripple height r1, about (Tr,). The dashed line
represents the equilibrium ripple height as predicted by the equations of
Nielsen (1981).


WIDE3D_2
A WIDE3D_5
y WIDE3D_10
I WIDE3D_20
4 WIDE3D_30
S WIDE3D_40
WIDE3D_50
- Nielsen(1981)

I I I I I l





















10 -


10-2
E
U 10-4

C)

S10o-4

) 10 -,
a


10-8




10-7


10-8
102








10 -





10-1



10 -2





10-





10-4
10-2


ky (cm-1)


(a) Spectral Density


10-1 100
ky (cm-)


(b) Perturbation Amplitude


Figure 6-20. The time- and spatially-averaged (x and z) (a) spectral density, and (b)

amplitude of bed elevation perturbations in the y-direction as a function of

wavenumber (ky) for three-dimensional grid comparisons.


WIDE3D_2
WIDE3D_5
WIDE3D_10
WIDE3D_20
WIDE3D_30
WIDE3D_40
WIDE3D 50


WIDE3D 2
WIDE3D 5
WIDE3D_10
WIDE3D_20
WIDE3D_30
WIDE3D_40
WIDE3D_50
























2.5


2
E

S1.5


1


0.5


0.


Time (s)


(a) Ripple Height


Maximum






Rolling Grain Ripples






20 40 60 8
Time (s)


(b) Ripple Steepness


Figure 6-21. The equilibrium ripple (a) height, and (b) steepness as a function of time for
various combinations of bedload transport equations and sediment pick-up
functions (see Table 5-3 for abbreviation keys). The average ripple heights
(rTr) and corresponding standard deviations are shown in the inset of panel
(a), and represent statistics averaged over 20 wave periods (40 s < t < 80 s).





































100
E



10 -






a) 10-2
C
0)






103
20 40 60 80 100 120 140 160
Time (s)


Figure 6-22. The percent change in sediment volume as a function of time during a
morphology test (LL1v3). The signal has been filtered at 1 Hz to remove
some of the noise.


















25

2 -8

15




05


0 20 40 60 80 100 120 140 160
Time (s)


(a) LL1v3


Time (s)


- Height
- -- Length







L'0 .. L .. .. . . "





20 40 60 80 100 120 140
Time (s)


(b) ML1v2


0 20 40
Time (s)


(c) ML2v2


60 80


(d) ML3vl


Time (s)


(e) HLlvl


Figure 6-23. Time-series of bedform height and length for morphology tests in lab-scale

flows (T = 2 s). The average height ((Tlr)) and length ((A,)) are given by the

symbols at the end of the time-series, and the error bars denote the standard

deviation about the mean. Average bedform statistics are computed using

the last twenty wave-periods in the signal.


227


. ---- Height
-*-- Length
15
4 -


-10





























ri (cm)

0 010203040506070809 1


...... 1 E

-05

25

0 0 5



Figure 6-24. A timestack of bedform statistics using direct Fourier analysis for simulation
HL1v1, where the ripple heights are inferred from the real Fourier
coefficients.





































2 3 4
Modeled Ripple Height (cm)


130
118
106
94
82
70
58
46
34
0 22
10


(a) Ripple Heights


40 I




/30 130
1 18
/ / 106
C X 94
S- 82
S- 70
20 / 58
S 46
C, N81 34
S// WH94 221
A GM82
10 v M94
X1 x 4 KOOO
L 0* FF02
S. W05
0 x SW06

0 I I
0 10 20 30 40
Modeled Ripple Length (cm)


(b) Ripple Lengths


Figure 6-25. One-to-one comparisons of modeled and expected ripple (a) heights and
(b) wavelengths for the lab-scale morphology tests. The dark solid line
represents perfect agreement, while the dashed (--) and dotted ( ... )
lines represent deviations of 1(0' and :i ', respectively. The coloring of
the symbols indicates the numeric value of the mobility number T for that
case.















- ---- Height
-0--- Length











40 80 120 160 200 240 280 320
Time (s)


(a) LFlv1


SI I I 'I I


I I 'I


--a-- Height
-- Length







~""ir


0 20 40 60 80 100 120 140 160
Time (s)


(c) MF2v2


---- Height
- Length











20 40 60 80 100 120 140 160
Time (s)


(b) MF1v2


(d) HFlvi


Figure 6-26. Time-series of bedform height and length for morphology tests in field-scale
flows (T = 8 s). The average height ((Q,)) and length ((A,)) are given by the
symbols at the end of the time-series, and the error bars denote the standard
deviation about the mean. Average bedform statistics are computed using
the last twenty wave-periods in the signal.























230


Im












ri (cm)

0 010203040506070809 1


160 ;1,
.......ilE~ir.... 5




20

0 l~


(a) MF2v2


rnr (cm)

0 002 004 006 008 01 012 014 0 16018 02











r~j0 6 ,

0 4








(b) HFlv1


Figure 6-27. A timestack of bedform statistics using direct Fourier analysis for simulations
(a) MF2v2 and (b) HF1v1, where the ripple heights are inferred from the
real Fourier coefficients.


















14

E








6
01
2 -
0



U
U :






00


110
100
90
80
70
60
50
40
30
S20
10


(a) Ripple Heights


W
110
100
90
80
70
60
50
40
30
E 20
10


Modeled Ripple Length (cm)


(b) Ripple Lengths


Figure 6-28. One-to-one comparisons of modeled and expected ripple (a) heights and
(b) wavelengths for the field-scale morphology tests. The dark solid line
represents perfect agreement, while the dashed (--) and dotted ( ... )
lines represent deviations of 1(0'. and 3:Ii '. respectively. The coloring of

the symbols indicates the numeric value of the mobility number I for that

case.


2 4
Modeled Ripple Height (cm)


100



S80
E



o 60
-J


40


S 20
. 20
40r



w 20

















Re = 25x103

Re =65x103


R =---150-
Re = 150x10/ "*'. \3


Re= 400x103/



E Re= (20-30)x103
A Re= (50-80)x103
v Re= (100-200)x103
0 Re = (300-500)x103


" 10-1



10-2




a 10-3
ci




C
(.







0

z

10-1




101
C


10-

















100

C 10
-J




0
' 10-
C
a)





E
a,





c

z


105
Orbital Reynolds Number, Re,


(b) Nondimensional Ripple Length


Figure 6-29. A comparison of model results to the ripple predictor equations of Faraci and
Foti (2002) for nondimensional ripple (a) height and (b) length. The open
and filled symbols denote the lab-scale and field-scale morphology results,
respectively.


I1
ii


101 102
Mobility Number, y


(a) Nondimensional Ripple Height


10-21
104


Re, = 60
- Re, =120
.................. Re = 240
a 30 < Re < 90
S 90 0 Re = 240


*\


I































Mobility Number, y


(a) Nondimensional Ripple Height


10-1- '" ..




10-2 I I- I
101 102
Mobility Number, y

(b) Nondimensional Ripple Length

Figure 6-30. A comparison of model results to the ripple predictor equations of
Nielsen (1981), Khelifa and Ouellet (2000), and Williams et al. (2005) for
nondimensional ripple (a) height and (b) length. The open and filled symbols
denote the lab-scale and field-scale morphology results, respectively.



















103

.S
1-r

l 102
Q.

E
0.
E
E' 101
xI1




100
1c


104


104



101

CJ



0o
E





101
1(


10-1 100 101 10
Wave Period Parameter, xx 106

(a) Nondimensional Ripple Height


(b) Nondimensional Ripple Length


Figure 6-31. A comparison of model results to the ripple predictor equations of Mogridge
et al. (1994) for nondimensional ripple (a) height and (b) length. The open
and filled symbols denote the lab-scale and field-scale morphology results,
respectively.













z

Zb (cm)
04 X
0 32
024
0 16
008
-008 200
-0 16
-0 24
-0 32 160
-0 4 140
120



60
40
15 20 20

x (Cm) 30






Figure 6-32. A timestack of bed elevation for the steady flow simulation.





















o10-2
10.
E





10-5
10-



1 0
1I-
10-6 ,



108 1

10-1I


Time (s)


(a) Stress and Transport


140 0.04

u120 laz E-
120 C
vn V,
N0 0.03
3 100

80
80-
0.02

C

40
0.01
>- I-
4 20


0 20 40 60 80 100 120 140 160 180 20
Time (s)

(b) li,, ,i and Eddy Viscosity


Figure 6-33. Time-series of spatially-averaged (a) bed shear stress and bedload transport,
and (b) velocity shear and eddy viscosity for the steady flow simulation.
















Zb (cm)
055
0 495
044
0 385
033
0 275
022
0165
011
0 055
0


(cm) 10


Figure 6-34. A timestack of y-averaged bed elevation profiles for the subcritical flow
simulation.


-4
0:


7C


17l
Phase (radians)


1.57


0.1





0.05

E


-0.0 E





-0.05


Figure 6-35. The phase- and spatially-averaged bed shear stress and bedload transport as

a function of phase. The yellow area shows the positive and negative limits
for the threshold of motion.


H b


I I I I I I I I I I I I I I I


Z













Zb (cm)
2
18
16
1 1




*


Figure 6-36. A timestack of y-averaged bed elevation profiles during sheetflow sediment
transport.


6


5


4

E
C 3


2


1


-


18

16

14

12

10

8 4


20 40 60 80
Time (s)


Figure 6-37. A time-series of average ripple height (r,) and length (A,) for the sheetflow
simulation. The symbols .,.1i ,:ent to the time-series denote the ripple height
and length averaged over twenty wave periods, and the error bars represent
the signal standard deviation from the mean.


z


_I























E (cm C. E


I,

le





Figure 6-38. A timestack of the energy density contained in the morphology profile for the
sheetflow simulation.


S(C)=0.001

S(C)=0.1
S(C = 0.5


Figure 6-39. Time-averaged isocontours of sediment concentration in the sheetflow regime.


4--x






































3



2 -

N

Sc ) (cmi/cm3)




0 -


0 0.2 0.4 0.6 0.8 1
( ) (cm3/cm3)


Figure 6-40. The time- and horizontally-averaged sediment concentration profile in the
sheetflow regime. The inset plot shows the full extent of the profile to the
top of the modeling domain.




























z, (cm)
I2


"icm)


(a) Growth from r,/2


z, (cm)

4


(c) Equilibrium Conditions (ne, A,)


(b) Growth from flat bed


z (cm)
2 7
24


( CM)


(d) Decay from 3r,/2


Figure 6-41. Timestacks of the spanwise averaged bed elevation zb for live-bed

simulations initialized with (a) r7 /2 and Ae, (b) a small Gaussian hump

five grain-diameters tall (1.5 mm), (c) r, and A,, and (d) 3,e/2 and Ae.


z, (cm)


160





0
10
X (Crn 150










z

Ytx


E (cm' )




-I:-
1 5 J,\e e
05
0


Figure 6-42. Timestacks of spectral energy density as a function of ripple wavelength and
time for the GROW2 simulation.






























5 -

2 10







0 0
1 5

5 Height
Length

S20 40 60 80
Time (s)

(a) Growth from ,e/2



4 .20

5 ~Height
Length
3 15

5 -
E
2 10 U

5 -

1 5

5 -


0 20 40
Time (s)


60 80


(c) Equilibrium Conditions (ne, Ae)


Time (s)


(b) Growth from flat bed


0 20 40
Time (s)


60 80


(d) Decay from 3 e/2


Figure 6-43. Time-series of ripple height T, and length A, for live-bed simulations

initialized with (a) re/2 and Ae, (b) a small Gaussian hump five

grain-diameters tall (1.5 mm), (c) qe and A,, and (d) 3 q,/2 and A,. The

time-dependent average ripple height (rl) corresponds to the left ordinate,

while the average ripple wavelength (A,) corresponds to the right ordinate.

The symbols and error bars denote the averaged and standard deviations of

ripple height ((fq)) and length ((A,)) computed using the last twenty wave

periods of the simulation.












244


- --- Height
-0-- Length
















--- Growth
t,=80: Decay
-- Equilibrium
S1.5 -t,=160s: FlatBed


.S
I
C.

0.





0
0 0.2 0.4 0.6 0.8
Nondimensional Time, t t,


Figure 6-44. The time evolution of ripple height, scaled by the expected height from
Nielsen (1981) rT, = 1.89 cm, versus the nondimensional simulation time. The
simulation time is scaled by the total time duration of the simulations, tf,
which are given in the figure.
































x (cm)


(a) Ripple Coarsening


Zb (cm)
2
17
14


I I

*7


x (cm)


(b) Ripple Bifurcation


Figure 6-45. Timestacks of the spanwise averaged bed elevation zb for live-bed simulations
initialized with (a) qe/2 and A,/2, and (b) qe/2 and 2 A, demonstrating
ripple evolution through slide and split, respectively .

































1 IpP 8


1
0l
0


(a) Ripple Coarsening


E (cm)' E

III ,

.0
i. _l.I. 21




0

(b) Ripple Bifurcation


Figure 6-46. The timestacks of spectral density as a function of ripple wavelength and
time during wavelength saturation through (a) coarsening (slide), and (b)
bifurcation (split).


















15




10 _




5




0


Time (s)


(a) Ripple Coarsening


40

35

30

25

E
20 )

15

10

5

0


Time (s)


(b) Ripple Bifurcation


Figure 6-47. Time-series of ripple height tr, and length A, for live-bed simulations
initialized with (a) q7,/2 and A,/2, and (b) q7,/2 and 2 A,. The
time-dependent average ripple height (r,) corresponds to the left ordinate,
while the average ripple wavelength (A,) corresponds to the right ordinate.
The symbols and error bars denote the averaged and standard deviations of
ripple height ((rf)) and length ((A,)) computed using the last twenty wave
periods of the simulation.





























0.5 -


20 40 60 8(
Nondimensional Time, t / T

(a) Scaled Ripple Heights


20 40 60
Nondimensional Time, t/ T


(b) Scaled Ripple Lengths


Figure 6-48. Time-series of (a) ripple height, and (b) ripple length scaled by their
expected values (rle = 1.89 cm, A, = 13.6 cm) from Nielsen (1981).
Simulation time t is scaled by the wave period T = 2 s, resulting in a
measure of the number of wave cycles completed.


HS1v2
ML3v1





.....-...... ....... ._". .**.*.. .** *
I 0- *O. >D o0icP -z rcc ox
-0


S HS1v2
S ML3v





























-1 Cc ) (cfm'cm-)



0


-1
0 0.2 0.4 0.6 0.8 1
( c ) (cm/cm3)


Figure 6-49. The time- and horizontally-averaged concentration profiles (C(z)) from
simulations HS1v2 and ML3v1. The inset figure shows the full vertical extent
of the profiles, and the dashed black line (--) shows the upper extent of the
profiles plotted in the larger figure.


4

3.5

3

2.5

^ 2

F 1.5

1

0.5

0

-0.5


I I I 1 I I I I


20 40 60 80 100
Time (s)


120 140 160


30


25


20


15 E


10


5


0


Figure 6-50. A time-series of bedform height and length for the GRAVLOW simulation.






250


--- Height
0- Length -






































Phase (radians)


Figure 6-51. The phase- and horizontally-averaged bed shear stress (('b)) (left ordinate)
and bedload ((Qb)) and suspended load ((Q,)) sediment transport (right
ordinate). The yellow shaded areas indicate regions of subcritical flow as
predicted by Julien (1998).


z


Zb (cm)
02
016
S012
008
004
0
- -004
- -008
-0 12
01C


x (cm)


Figure 6-52. Timestacks of the spanwise-averaged bed elevation zb for the GRAVHI
simulation.


U 1


-- Qb






0 -I
0 0







0 --


-5


1 .57


U7C
















3.5 -- Height .
Length

3

2.5

2

1.5

1

0.5
o-


35

30

25

20 E
20

15 <

10

5

0


0 20 40 60 80 100 120 140 160
Time (s)


Figure 6-53. A time-series of bedform height (left ordinate) and length (right ordinate)
for the GRAVHI simulation. The symbols and error bars to the right of each
signal represent the time-average value and standard deviation computed
over the last 20 wave periods of the simulation.




200 .15

150 / b
--0- Q 10
100 -

c1 -5 E
E 50 U
4-
C C .................................... ... i.1... .. ... E









A 0.57 A
S-Pho (r
--5
-100 -
-10
-150 -

200 215
0U 0.5c 1c 1.5c C
Phase (radians)


Figure 6-54. The phase- and horizontally-averaged bed shear stress ((Tb)) (left ordinate)
and bedload ((Qb)) and suspended load ((Q,)) sediment transport (right
ordinate) for the GRAVHI simulations. The yellow shaded areas indicate
regions of subcritical flow as predicted by Julien (1998).


















Zb (cm)
003
0 024
0018
S0012
0006
0
- -0 006
--0012
-0018
-0 024
-003


Figure 6-55. Timestacks of spanwise-averaged bed elevation zb for the SILTLOW

simulation.


I. (cm)





0uu
S0008
0 006
0 004
0 002


Figure 6-56. Timestacks of bedform height T1, and length A, computed using direct

Fourier transforms on the spanwise-averaged bed elevation signal Zb(, t)

sampled at 10 Hz.


z


I.,































Phase (radians)


(a) Stress and Sediment Transport


1 a
Phase (radians)


0.005


(b) Stress and Sediment Concentration

Figure 6-57. The phase- and horizontally-averaged (a) bed stress ((rb)) (left ordinate)
and bedload ((Qb)) and suspended load ((Q,)) sediment transport
(right ordinate), and (b) bed stress ((rb)) (left ordinate) and sediment
concentration ((C)) for the SILTLOW simulation. The yellow shaded areas
indicate regions of subcritical flow as predicted by Julien (1998).














Zb (cm)

*2


Z


X~


-
2 30
X(cm)1~~~i


Figure 6-58. The timestacks of spanwise-averaged bed elevation zb sampled at 10 Hz
during the SILTHI simulation.


4

3.5

3

2.5

E
2

1.5

1

0.5


40

35

30

25

20
U
15

10

5

0


Time (s)


Figure 6-59. A time-series of bedform height r, (left ordinate) and length A, (right
ordinate) for the I i:. o- highly-concentrated SILTHI simulation.






















E

0(
I"

a



-5




-10








10







E

0




-5




-10


Phase (radians)


(a) Stress and Sediment Transport


chase (radians)
Phase (radians)


(b) Stress and Sediment Concentration


Figure 6-60. The phase- and horizontally-averaged (a) bed stress ((Tb)) (left ordinate)
and bedload ((Qb)) and suspended load ((Q,)) sediment transport
(right ordinate), and (b) bed stress ((rb)) (left ordinate) and sediment
concentration ((C)) for the SILTHI simulation. The yellow shaded areas
indicate regions of subcritical flow as predicted by Julien (1998).



















6 (C)
S1 OE-02
4 0E-03
1 6E-03
) Baseline- 6 3E-04
Baseline 2 5E04
t 2 5E-04
S- 1 OE-04
4 Particle-Fluid 4 0E-05
0 1 6E-05
0 6 3E-06
E 2 5E-06
1 OE-06
2


Fluid-Particle

-4 -2 0 2 4 6
Diameter, Phi Units


Figure 6-61. A phase diagram for sediment suspension showing the three ranges of
interactions as -i-.- -1. .1 by Elghobashi (1994): fluid-particle interactions,
C < 10-6; particle-fluid interactions, 10-6 < C < 10-3; and particle-particle
interactions, C > 10- The data points are taken from a selection of
simulations having ranges of orbital Reynolds number and sediment diameter
of 700 < sR, < 460, 000 and 0.025 < d < 6 mm, respectively.
















U
3

2. 2- Equilibrium

*H. Growth
0.-

0 20 40 60 80 100 120 140 160
Time (s)

(a) Ripple Height



U 1-

15
10
t-

a. 5- -
-
O I I I I I I, I ,I, I ,I, I I
0 20 40 60 80 100 120 140 160
Time (s)

(b) Ripple Length


E

0 0.2



I ~ I I I I I I I ,
0.1 -




0 20 40 60 80 100 120 140 160
Time (s)

(c) Ripple Steepness


Figure 6-62. Time-series of ripple (a) height, (b) length, and (c) steepness for the
total load simulation, TL2v2. The shaded yellow areas correspond to
upper and lower estimates or ripple geometry using Grant and Madsen
(1982) and Nielsen (1981), respectively. The red boxes in (a) represent the
20-wave-periods corresponding to the growth and equilibrium phases of
height evolution. The symbols and error bars represent the time-averaged
ripple (a) height ((rlr)) and standard deviation (o,,), and (b) length ((A,))
and standard deviation (a\) computed using the last 20 wave periods of the
simulation.






258













Z


X


Zb (cm)
I 2
17
14


140








40

20

x (cm) '




Figure 6-63. The timestacks of spanwise-averaged bed elevation zb during ripple growth
and equilibrium for the total load simulation TL2v2.


20 40 60 80
Time (s)


100 120 140 160


Figure 6-64. The cumulative, spatially-averaged and corrected bedload and suspended
load transport fluxes as a function of simulation time for TL2v2.


2

E
S1.5
X


U-



I- 0.5
0


0


5 -0.5
E

-1


I I I I I I I


I-


S Suspended Flux
Bedload Flux
, I I I I I I I ,

























































(a) Ripple Growth


1P r
Phase (radians)


(c) Ripple Equilibrium


Phase (radians)


(b) Ripple Growth


Phase (radians)


(d) Ripple Equilibrium


Figure 6-65.


The phase dependence of (a) bedload sediment transport (Qb), and

(b) entrainment (E) and deposition (wsCb) during ripple growth, and

equilibrium in (c) and (d) for the TL2v2 simulation.


005



0 W
o

0-o


E



0 05
_=


I- E
8- --

6 E


2


S-

-4

-6

-8

10


1 5. 2.




















x
LL
S0.01











0 -0.01
U-


0-
S-0.02
o -0.02


Phase (radians)


(a) Ripple Growth


-- VQb
-- w. E


OTC 0.57c 17 1.5(7 2
Phase (radians)


(b) Ripple Equilibrium



Figure 6-66. The phase- and horizontally-averaged, corrected bedload (VQ ) and

suspended load (VQ ) transport fluxes during ripple (a) growth and (b)

equilibrium for the TL2v2 simulation.


0
U_
U3











x
LU-





02
U_
ci












Q




















(zb) (cm)
15
13
1 1
09
07
05
03
01
-01
-03
-05


- Bedload Flux < + Bedload Flux

(a) Ripple Growth


S002 (cm)
0.02 / 0

(/ 17
+ 14
11
S 0* 05
S- -' 04

X .* -04
LLV 7-
-0.02
0.



).04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

Bedload Flux I + Bedload Flux

(b) Ripple Equilibrium


Figure 6-67. The time-averaged and corrected bedload ((VQ~ )) and suspended load
((VQ )) transport fluxes during ripple (a) growth and (b) equilibrium
for the TL2v2 simulation. The colored symbols represent points in the
horizontal x y bed plane, and are contoured with respect to their location
in the ripple profile with the warmer colors representing the ripple crest, and
cooler colors denoting the ripple troughs. The dashed (--) lines signify the
one-to-one agreement in each quadrant.














3-

._ 2
SRapidGradual Equilibrium

a 1

O I I I I I I I I I
0 20 40 60 80 100 120 140 160
Time (s)

(a) Ripple Height



U
15


10

,.- 5

O I I I I I ,I I I I
0 20 40 60 80 100 120 140 160
Time (s)

(b) Ripple Length

I I I I ~ I I I I I I


S0.2-


0.1 -
0.1
0.
SI I I I I I I ,
0 20 40 60 80 100 120 140 160
Time (s)

(c) Ripple Steepness


Figure 6-68. Time-series of ripple (a) height, (b) length, and (c) steepness for the
decaying total load simulation, TL2v2d. The shaded yellow areas correspond
to upper and lower estimates or ripple geometry using Grant and Madsen
(1982) and Nielsen (1981), respectively. The red boxes in (a) represent the
rapid, gradual, and equilibrium phases of height evolution. The symbols and
error bars represent the time-averaged ripple (a) height ((ir,)) and standard
deviation (o,,), and (b) length ((A,)) and standard deviation (ax) computed
using the last 20 wave periods of the simulation.
















I I I I I I I


20 40 60 80
Time (s)


100 120 140 160


Figure 6-69. The cumulative, spatially-averaged and corrected bedload and suspended

load transport fluxes as a function of simulation time during ripple decay

(TL2v2d).


0.02


E
x

U-c

i0







x
L-
'0

0D
D -


0.01


0.01


0.02 '
0


7n


l1
Phase (radians)


Figure 6-70. The phase-averaged and corrected bedload and vertical sediment fluxes

during rapid ripple decay in TL2v2d.


-- Suspended Flux
Bedload Flux


I I I I I I I I I


-- VQb
w c E

I II I I







































Phase (radians)


E
0.05


0 Q.

o
-0

-0.05 -
m




71


(a) Stress and Bedload Transport


Phase (radians)


0.05 E

x
LL
C
0 a)
E




7 o
-0.05 '=


(b) Stress and Vertical Sediment Fluxes


Figure 6-71. The phase-averaged (a) bed shear stress, gravitational stress, and bedload
transport rate, and (b) bed shear stress, gravitational stress, and vertical
sediment fluxes for the TL2v2d simulation.































a) /
(Zb) (cm)
0.01 / ( M)

0/) -t / 18
12
0 1 2
-* / 14

J -*08
S* 106
S* 04
LL lLl 02
[i ./ 0 2
s*/ e e
-0.01
/


-0. 02
-0.02 -0.01 0 0.01 0.02
Bedload Flux <= + Bedload Flux


Figure 6-72. The time-averaged and corrected bedload ((VQf)) and suspended load
((VQ')) transport fluxes during rapid ripple decay for the TL2v2d
simulation. The colored symbols represent points in the horizontal x y
bed plane, and are contoured with respect to their location in the ripple
profile with the warmer colors representing the ripple crest, and cooler colors
denoting the ripple troughs. The dashed (--) lines signify the one-to-one
agreement in each quadrant.


















3-
S~Equilibrium
*u 2
2 i-----i--- --------------


0.-
H. 1 Growth

0 I I I ,I ,I, I ,II I
0 20 40 60 80 100 120 140 160
Time (s)

(a) Ripple Height




15

10

0-
a. 5- -
-
O I I I IIII I I
0 20 40 60 80 100 120 140 160
Time (s)

(b) Ripple Length



A 0.3 2 -
E

S0.2


0.

.2 -
0 I I I I I I I I I ,
0 20 40 60 80 100 120 140 160
Time (s)

(c) Ripple Steepness


Figure 6-73. Time-series of ripple (a) height, (b) length, and (c) steepness produced solely
through bedload transport. The shaded yellow areas correspond to upper
and lower estimates or ripple geometry using Grant and Madsen (1982) and
Nielsen (1981), respectively. The red boxes in (a) represent the growth and
equilibrium phases of height evolution. The symbols and error bars represent
the time-averaged ripple (a) height ((Tr,)) and standard deviation (o,,), and
(b) length ((A,)) and standard deviation (a\) computed using the last 20
wave periods of the simulation.







267



















6 I. I -2I. I
6- 005 6- -005

0- 4 ---------- --- 0 -?--- 4 --^ O
4 -4
-
0 0


-0 05 -0 0 5
-6 -- -8 -

-10 015 1 115 2 U100K 05. 1I 150 2 0
Phase (radians) Phase (radians)

(a) Ripple Growth (b) Ripple Equilibrium


Figure 6-74. The phase dependence of bedload sediment transport (Qb) during ripple (a)

growth, and (b) equilibrium for the BL2v2 simulation.

























































268













S0.8 Decay
F
S0.6-

S0.4-

S0.2 -

0 20 40 60 80 100 120 140 160
Time (s)

(a) Ripple Height

40 I I I I I

30

.2 -
S20
-1
L 10


0 20 40 60 80 100 120 140 160
Time (s)

(b) Ripple Length


E

e0.15 -

0.1


O 0.05


0 20 40 60 80 100 120 140 160
Time (s)

(c) Ripple Steepness


Figure 6-75. Time-series of ripple (a) height, (b) length, and (c) steepness produced solely
through suspended load transport (SL2v2). The red box in (a) represents
the time of rapid height decay. The symbols and error bars represent the
time-averaged ripple (a) height ((Tr,)) and standard deviation (o,,), and (b)
length ((A,)) and standard deviation (a\) computed using the last 20 wave
periods of the simulation.

















15 0.15


S10 -0.1 E
o X
S5 0.05 .

S- ----- -0--- 0

-5 -0.05 U

C -10 b -0.1 .

E
-15 c -0.15
Ws Cb

-20 -0.2
020 0.57c 17 1.57c 27C
Phase (radians)


Figure 6-76. The phase dependence of entrainment (E) and deposition (w2Cb) fluxes for
rapid ripple decay during the suspension-only simulation SL2v2.








































C-
1

S05 -- TL2v2
-- TL2v2d
BL2v2
O ___SL2v2
U)



0 20 40 60
Nondimensional Time, t/ T


Figure 6-77. The evolution of ripple height for the total load (TL2v2), decaying total
load (TL2v2d), bedload only (BL2v2), and suspended load only (SL2v2)
simulations as a function of the number of wave cycles (t/T). The ripple
heights are nondimensionalized by the time-averaged ripple height ((7,)) of
the TL2v2 simulation.










Table 6-1.



Ripple Pre

Faraci and
Mogridge e
Nielsen (19
Soulsby an
Khelifa and
Williams et
Grant and
Wiberg an


The RMS error between model and expected ripple height and length for
morphology tests under lab-scale flows, with associated numerical scores and
combined rankings. Numerical scores are assigned values ranging from 1 to 8
corresponding to the lowest and highest RMS errors, respectively.
doctor r]RMSE Score ARMSE Score Combined Score
(cm) (cm)
Foti (2002) 0. 652 2 2. 25 1 3
t al. (1994) 1. 052 3 2. 77 2 5
81) 0. 114 1 4. 22 4 5
d Whitehouse (2006) 1. 57 6 3. 53 3 9
i Ouellet (2000) 1. 19 4 10. 2 8 12
Sal. (2005) 1. 30 5 10. 2 7 12
Madsen (1982) 2. 23 7 7. 055 6 13
i Harris (1994) 2. 38 8 6. 42 5 13


Table 6-2. The RMS error between model and expected ripple height and length for
morphology tests under field-scale flows, with associated numerical scores and
combined rankings. Numerical scores are assigned values ranging from 1 to 8
corresponding to the lowest and highest RMS errors, respectively.
Ripple Predictor r1RMSE Score ARMSE Score Combined Score
(cm) (cm)
Williams et al. (2005) 0. 435 1 5. 47 1 2
Faraci and Foti (2002) 0. 914 2 7. 66 4 6
Nielsen (1981) 1. 56 4 6. 42 2 6
Khelifa and Ouellet (2000) 1. 17 3 8. 76 5 8
Wiberg and Harris (1994) 2. 84 5 7. 34 3 8
Mogridge et al. (1994) 9. 69 7 30. 8 6 13
Soulsbv and Whitehouse (2006) 7. 40 6 37. 5 7 13


Grant and Madsen (1982)


10. 8


8 55. 9


8 16




























Table 6-3.


The model performance index (l\!PI) associated with the ripple predictors used
to assess morphology test results. The MPI score is found by summing the
combined scores from the lab-scale and field-scale flow rankings in Tables 6-1
and 6-2, respectively.


MPI Score


Ripple Predictor
Faraci and Foti (2002)
Nielsen (1981)
Williams et al. (2005)
Mogridge et al. (1994)
Khelifa and Ouellet (2000)
Wiberg and Harris (1994)
Soulsby and Whitehouse (2006)
Grant and Madsen (1982)









CHAPTER 7
CONCLUSIONS

Through forty-four simulations, we have not only demonstrated the capabilities of

new phase-resolving live-bed modeling systems in the linear and nonlinear regime, but

have enhanced our understanding of small-scale sediment transport processes, and their

role in generating bedforms. In some cases, we have reinforced widely-accepted theories

about the roles of sediment transport modes during ripple growth, while in other areas we

have advanced knowledge on the subject of ripple equilibrium and decay. The following

sections provide a brief summation of pertinent model results as they pertain to the

validity and predictive capabilities of our newly developed live-bed models at simulating

morphology under a wide variety of flow regimes. Additionally, conclusions are drawn

from the results of Phase III simulations, and shed light on the roles of bedload and

suspended load transport during ripple growth, equilibrium, and decay in the nonlinear

regime as a function of both ripple location and wave phase.

7.1 Live-Bed Model Evaluation

The simple one-dimensional-horizontal (1DH) model of bedload transport in the

linear regime, derived from first principles, reinforces a concept widely held in literature

on bedform growth for the previous century; incipient bedform growth from a flat sand

bed may occur entirely through bedload sediment transport. An interesting result of our

conceptual 1DH model simulation was discovered, and appears to emphasize the role of

particle inertia in promoting an initial instability of the flat bed. A central theme, or

assumption, of the simple 1DH model is that the motivating stress applied to the bed

by the mixed 1lvr is constant across the bed, and in phase (space) with the sediment

transport. The latter part of this assumption was found to be incorrect; in fact, a spatial

lag between applied stress and particle transport is necessary for the linear instability

of the sand-water interface to develop, as found in the analysis of Parker (1975). This

phase lag is explained as an inertial effect of the sediment, whereby its weight creates a









tendency for the particle motion to lag slightly behind the mobilizing forces. Without

such a lag, the sand bed remains unconditionally stable at all wavelengths, and bedload

transport within the mixed 1. -r tends to promote a stable, flat bed. While our simple

1DH model is of little use for practical applications, it is nonetheless illustrative of the

dominant sediment transport and particle processes that lead to bedform growth in the

linear regime. The generation of rolling grain ripples on a flat bed may serve as a useful

physical analog to the characteristics of the 1DH model.

Through model-data comparisons of phase- and horizontally-averaged velocity profiles

in oscillatory boundary 1- -r flow (Jensen et al., 1989), we demonstrate the ability of

the LES hydrodynamic model to reproduce realistic flow characteristics in the boundary

1 -,- r. Although the model predictions of boundary 1-- r flow exhibited a consistent lag

of 30 with respect to the experimental data, and tended to overestimate the boundary

1 -,-r thickness by 2 5 mm, simulation results are generally within Si"' of the observed

values. The overestimation of boundary 1l-.-r thickness tends to promote early flow

reversal near the bed, leading to undesirable phase discrepancies in the model-data

comparisons, particularly near flow reversal. These shortcomings may be harmonized by

employing a more advanced closure model, as the Smagorinsky (1963) method is known to

perform poorly in high Reynolds number flows, tending to overestimate the turbulent eddy

viscosity; model predictions in the boundary l1..-r, particularly near the wall, may further

benefit from a more sophisticated near-wall model. Generally speaking, though, the LES

model numerics tended to show positive performance in the numerical convergence tests,

demonstrating an independence of normalized flow metrics to the choice of grid resolution.

Aside from evaluating the hydrodynamic component of the coupled modeling system,

the predictive capability of the sediment transport and morphology submodels was

also investigated through the simulation of nine distinct live-bed model experiments of

morphology in a wide range of flow conditions. The distribution of flow and sediment

characteristics used in these simulations yielded mobility numbers that ranged by one









order of magnitude. Due to the overwhelming number of references to discrepancies in

the characteristics of morphology created under lab-scale and field-scale flow conditions,

flow periods were chosen to mimic oscillatory flows likely to be found in both settings. In

the absence of a unified set of laboratory and field data regarding bedform characteristics,

model predictions of ripple height and length were compared to expected values computed

using eight different semi-empirical ripple predictors prevalent in existing literature on the

subject [i.e. Nielsen (1981); Grant and Madsen (1982); Mogridge et al. (1994); Wiberg and

Harris (1994); Khelifa and Ouellet (2000); Faraci and Foti (2002); Williams et al. (2005);

Soulsby and Whitehouse (2006)]. As the 1 i in i ly of these expressions were developed

using multiple sets of data on bedform and flow characteristics, the end result is equivalent

to comparing our model predictions of morphology to over 30 different sets of laboratory

and field data (see Table 3-2). Model predictions of ripple height and length tend to

agree well with the ripple predictors of Nielsen (1981) for both field- and lab-scale flows,

similar to the observations of Marsh et al. (1999); however, the best overall fit to all of

the morphology test predictions by the new live-bed model was found using the ripple

predictor of Faraci and Foti (2002). Owing to the large degree of scatter in the expected

values of ripple height and length computed using the ripple predictors (see Table 5-6)

for the field-scale flow simulations, it is unclear whether the model is particularly adept

at predicting morphology under such conditions. A lack of harmony in the analysis of

field ripple data, therefore, precludes us from drawing any definite conclusions about

the applicability of the live-bed modeling system to field-scale flows; however, the model

predictions of ripple height and length for lab-scale flows, especially those at low to

moderate mobility numbers, are encouraging.

Numerous other simulations were performed in Phase II, the results of which highlight

additional capabilities of the live-bed modeling system, as well as address sensitivity

issues regarding initial conditions, periodic domain length, and the effects of hindered

settling on morphology. While it would seem that the methodology chosen for modeling









the suspended load transport through an advection-diffusion equation would not lend itself

well in simulating suspension and transport in the sheetflow regime, model simulations

of highly-concentrated flow at high mobility numbers -i .-.-. -I reasonable performance.

Without a detailed model-data comparison to sheetflow experiments, the only conclusion

that may be drawn from the sheetflow simulation is that the suspended load transport

submodel predicts concentrations in the lower and upper sheetflow lV. -i typical of those

found in laboratory experiments (Ribberink and Al-Salem, 1995). Furthermore, the

morphology model correctly predicted a flattening of the initial bedform, and the lack of a

coherent signal in the bed elevation for such energetic flow and transport conditions.

Simulations of our baseline flow and sediment parameters were conducted to evaluate

the sensitivity of the morphology submodel to initial conditions. One experiment

simulated the growth of a solitary ripple from one-half its expected equilibrium height,

while another simulated growth from a nearly flat bed (a small Gaussian hump having

a height equivalent to 5 grain-diameters). Still others simulated equilibrium ripple

conditions, as well as decay from morphology initialized with a ripple height that was

out of equilibrium with the flow conditions. An additional experiment of growth from

one-half the expected height was performed, and incorporated the effects of hindered

settling (Richardson and Zaki, 1954) in the suspended sediment transport submodel.

The final predictions of ripple height for all five cases varied by less than 1('. ,-ii-. -li:-.;

that the initialized morphology p1. i'- only a minor role in the final model predictions.

Although the in, i.i iy of simulations were performed with morphology initialized at the

equilibrium or expected ripple wavelength, in order to satisfy the periodic streamwise

boundary conditions, two model experiments were conducted to determine the domain

effect on wavelength selection. These simulations successfully reproduced both the

processes of ripple coarsening and bifurcation, referred to in the literature as ripple slide

and split (Doucette and O'Donoghue, 2006), whereby ripple evolution occurs through a

change in wavelength first, and height second. Whereas the coarsening process involves









the saturation of ripple wavelength at a value larger than what it started with, ripple

bifurcation involves a splitting of the initial bedform into two distinct ripples having

wavelengths smaller than the initial morphology. Both experiments yielded ripple heights

and lengths at or very near the expected values.

7.2 Sediment Transport and Bedform Dynamics

The final set of four experiments conducted for the Phase III experiments were

formulated with our specific scientific objectives, outlined in C'! lpter 1, in mind. Each

simulation was conducted using the full three-dimensional live-bed modeling system,

where information regarding acceptable grid resolutions and spanwise domain widths

were obtained through the Phase I model experiments. In order to determine the relative

contributions of bedload and suspended load transport during ripple growth, equilibrium,

and decay, and also to determine their dependence on both wave phase and location

throughout the ripple profile, two total load transport simulations were conducted, each

focusing on either ripple growth and equilibration, or decay and equilibration. Taking

advantage of our unique model capabilities, two additional simulations were performed

incorporating either bedload transport only, or suspended load transport only, in order to

determine their respective roles in bedform dynamics in the absence of the other.

Utilizing the total load simulation results of bedform growth, equilibrium, and decay,

along with the newly developed flux tendency phase diagrams discussed in Chapter 6,

and more thoroughly in Appendix B, we present previously unpublished information

regarding the phase dependence of transport fluxes with respect to bedform dynamics.

With respect to wave phase, the dominant transport fluxes leading to ripple growth

occur during times of maximum acceleration and deceleration of the freestream flow.

During maximum acceleration, the model predictions -,ii--. -i bedload-dominated growth,

with decay occurring through suspension events. The opposite behavior is observed

during times of maximum flow deceleration, with deposition-dominated growth and a

comparatively weak decay through bedload transport. These results also appear to hold









true during times of ripple equilibrium, yet the suspension and bedload fluxes are balanced

in a manner that yields no net growth or decay. The simulations of ripple decay under the

influence of total sediment transport load -ii.:: -1 drastically different behavior; here, the

bedload transport fluxes contain equal magnitudes tending to promote both decay and

growth, resulting in no net contribution throughout the entire wave phase. Conversely,

the vertical sediment fluxes exhibit a dependence on wave phase similar to that observed

during growth and equilibration, although the signal is displaced almost entirely to the

decaying (-) tendency. In summary, the model simulations r--.- -I that during ripple

growth, i.'. (35'.) of the total fluxes may be attributed to bedload (suspended load)

sediment transport; and during ripple equilibrium, I.'. (-: '.) of the total fluxes are

related to suspended load (bedload) sediment transport; and during rapid ripple decay,

10' of the net sediment fluxes are due to suspension and deposition events, because the

bedload transport fluxes are equally balanced between promoting growth and decay.

In order to explain the spatial dependence of the transport fluxes, as well as their

tendency to either promote ripple growth or decay, we make use of the newly developed

flux tendency phase diagram. For the purposes of this discussion, the spatial dependence

is related to the ripple profile through three distinct regions:

1. the ripple crest;

2. the ripple flanks; and

3. the ripple troughs.

The simulation results of the total load simulations -ii-.- -I that during times of growth,

the ripple experiences bedload-dominated growth at the crest and upper flanks, while

suspension-dominated growth occurs in the ripple troughs; a balance of fluxes is found to

exist through the lower and middle ripple flanks. During times of ripple equilibrium, we

find the locus of data points falling along the equilibrium (solid diagonal line) line in the

flux tendency phase diagram, where ripple crests experience a balance between bedload

growth and suspension decay, while ripple troughs tend to exhibit decay through bedload









fluxes and growth through suspension, causing them to scour deeper. Not surprisingly,

a i: i i i ly of data points are found in the lower left triangle of the flux tendency phase

diagram during ripple decay. The highest density of data points tends to fall around the

vertical axis, indicating very strong decaying fluxes produced through suspension at the

ripple crests, and deposition in the ripple troughs. A comparatively weak growth tendency

due to depositional fluxes is observed along the ripple flanks.

These data reinforce the common interpretation of processes leading to bedform

growth, whereby vortices in the lee of ripple crests tend to liberate particles from

the ripple troughs, which subsequently experience crest-directed movement through

bedload sediment transport. An unexpected result, however, is the lack of significant

gravitationally-induced bedload transport during ripple equilibrium. Although the

phase-averaged gravitational stresses where considerably larger for the total load decaying

simulation, their role in substantial ripple decay must be minor. This lack of a significant

gravitational component to the bed stress and transport is an encouraging result of the

growth and equilibrium simulations, and '--. -; that the live-bed modeling system

predicts morphology in equilibrium with the hydrodynamic flow conditions rather than

steepness-limited bedforms.

The bedload and suspended load simulations verify the information extracted from

the flux tendency phase diagrams, where we find that bedload is the dominant transport

mechanism promoting bedform growth while the entrainment and deposition of bed

material tend to promote decay. These simulations are unique as they are not something

that could be easily achieved in a laboratory experiment; it would be unlikely that

such experiments could be derived in the nonlinear regime where one could alternately

turn bedload and suspended load on or off. One attainable analog would be to perform

experiments with extremes in the grain size spectrum, whereby gravel and cobbles

would move only through bedload transport, and fine silts and (1 li--i.. 1 particles would

mostly be transported through suspension. Using such extremes, however, presents









additional difficulties due to their inertial properties and tendencies to create bedforms

other than orbital ripples. In retrospect, the independent simulations using only bedload

sediment transport or suspended load transport were frivolous, owing to the utility of

the flux tendency phase diagrams derived from model simulations of total sediment

transport load during periods of ripple growth, equilibrium, and decay. However, aspects

of the simulations that could not have been directly inferred from the phase diagrams

is the remarkable tendency of bedload transport to produce orbital-type ripples in the

nonlinear regime, and that of suspension events to rapidly decay bedforms in the absence

of bedload sediment transport. The bedload simulations also demonstrated that when

acting without suspended load sediment transport, the ripple growth rate is much slower

when compared to that observed in the total load simulations, with the time to reach

equilibrium increasing by 150'. In this regard, the additional bedload and suspended load

transport simulations were particularly illustrative.

7.3 Future Applications

An extensive list of "to-do" items, with respect to the current modeling system,

has been compiled over the course of this investigation. It would appear that the

hydrodynamic modeling system could benefit from a more robust closure scheme,

potentially a dynamic closure method that simulates backscatter of energy. Also, it

would be advantageous to introduce spectral methods for the solution of the pressure field,

thereby promoting future optimization of the entire live-bed modeling system through

domain decomposition and parallelization. With respect to the transport and morphology

submodels, the Euler-WENO scheme should be adapted to permit variable mesh scaling

in the horizontal plane if simulations of large-scale morphology are to be performed in

the future. Additionally, an alternate method of simulating the suspended sediment

concentration field may be considered for simulating more typical coastal-type flows

having suspended sediment concentrations, by mass, on the order of 100 g/1. Of course,

the 3D live-bed modeling system could benefit from additional validation and calibration









to recent laboratory data, such as those collected during the CROSSTEX experiments at

the O.H. Hinsdale Wave Research Laboratory at Oregon State University, or the USGS

oscillating tray experiments under combined wave-current flow reported in Lacy et al.

(2006).









APPENDIX A
EXPERIMENT MATRIX

The complete simulation matrix for test cases discussed in the text is given in

Table A-l. The experimental matrix lists the case name, domain sizes and resolutions,

relevant flow characteristics, and sediment sizes.











Table A-1. A full list of experiments and relevant simulation parameters for test cases


discussed in C'! plters 5


Case Name

1DH
JSF-4v8
GRID1
GRID2
GRID3
GRID4
WIDE3D_2
WIDE3D_5
WIDE3D_10
WIDE3D_20
WIDE3D_30
WIDE3D_40
WIDE3D_50
MPMVR
MPMFL
CLVR
CLFL
LL1v3
LFlvO
ML1v2
ML2v2
ML3v1
MF1v2
MF2v2
HL1v1
HFlv1
STEADY
SUBCR
SHEET
GROW
GROW2
EQUILIB
DECAY
HSlv2
SLIDE
SPLIT
GRAVLOW
GRAVHI
SILTLOW
SILTHI
TL2v2
TL2v2d
BL2v2
SL2v2


Lx
(cm)
60.0
3.2
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
13.6
25.6
34.8
21.6
28.0
13.6
25.2
22.8
29.6
24.4
27.2
27.2
13.6
13.6
13.6
13.6
13.6
27.2
27.2
27.2
30.0
30.0
30.0
30.0
27.2
27.2
27.2
27.2


Ly
(cm)
0.0
3.2
0.4
0.8
0.2
0.2
0.4
1.0
2.0
4.0
6.0
8.0
10.0
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
3.2
3.2
3.2
3.2


Lz
(cm)
0.0
30.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.0
11.5
32.5
11.5
11.5
11.0
11.5
11.5
11.5
11.5
12.0
11.0
11.0
11.0
11.0
11.0
11.0
11.5
11.5
11.5
11.5
11.5
11.5
11.5
21.5
21.5
21.5
21.5


and 6.
nx

30
32
68
34
136
136
68
68
68
68
68
68
68
68
68
68
68
128
174
108
140
68
126
114
148
122
136
136
68
68
68
68
68
136
136
136
150
150
150
150
136
136
136
136


ny nz Uo
(cm/s)
N/A N/A 30
32 100 200
3 90 40
3 45 40
3 180 40
3 90 40
2 90 40
5 90 40
10 90 40
20 90 40
30 90 40
40 90 40
50 90 40
2 90 40
2 90 40
2 90 40
2 90 40
2 100 30
2 160 30
2 100 30
2 100 50
2 90 40
2 100 30
2 100 50
2 100 80
2 100 60
2 110 30
2 90 5
2 90 100
2 90 40
2 90 40
2 90 40
2 90 40
2 100 40
2 100 40
2 100 40
2 90 30
2 90 80
2 90 10
2 90 40
16 110 40
16 110 40
16 110 40
16 110 40


T
(s)
4.0
9.72
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
8.0
2.0
2.0
2.0
8.0
8.0
2.0
8.0
0.00
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0


(x104)
5. 730
618.8
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
2. .i ,',
11.46
2. .i ,',
7.958
5.0929
11.46
31.83
20.37
45.84
0.00
0.07958
31.83
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929
2. .i ,',
20.372
0.3183
5. 0929
5. 0929
5. 0929
5. 0929
5. 0929


d
(pm)
300
N/A
N/A
N/A
N/A
N/A
300
300
300
300
300
300
300
300
300
300
300
400
400
200
300
300
200
300
300
200
300
300
300
300
300
300
300
300
300
300
6000
6000
25
25
300
300
300
300









APPENDIX B
FLUX TENDENCY PHASE DIAGRAM

An explanation regarding the use of the phase diagram, as well as a description of its

derivation, is provided. The flux tendency phase diagram is a useful tool for determining

the relationship between sediment transport fluxes and their tendency to promote either

growth, equilibrium, or decay. While this investigation made particular use of the phase

diagram while explaining the spatial dependence of transport fluxes throughout the ripple

profile, the same methodology could be applied to determining phase dependence, as

well, by coloring the symbols with respect to wave phase. Observations made regarding

the behavior of transport fluxes during the Phase III simulations fully support the flux

tendency phase diagram concept, as well as the inferences that can be drawn from it

regarding the predilection of bedload and suspended load transport flux components to

either promote growth or decay. The flux tendency diagram is elegant in its simplicity,

and quite powerful in its utility.

In order to utilize the flux tendency phase diagram shown in Figures B-l(a)-B-l(d),

it is first necessary to segregate transport flux components by their tendency to either

promote bedform growth or decay. This process is similar to that used in Penko (2007),

and is found to be quite illustrative. Transport fluxes producing bedform growth are

assigned a positive (+) value, while those tending to yield bedform decay are given a

negative (-) value. These assignments are given in Equations B-la and B-lb. Special

care must be taken to ensure sign fidelity when segregating transport fluxes above and

below Zb = 0. Once the flux components have been successfully sorted by sign, which

indicates its tendency for growth or decay, the data can then be plotted as scatter in the

flux tendency phase diagram.


7Q+ VQb,s Qb,s 01b41
= |vQbs for > 0 (B-la)

VQ = -|VQb, for < 0 (B-b)









The concept of the flux tendency phase diagram began as a simple one-to-one plot

of bedload transport flux versus suspended load fluxes with the raw, absolute flux data

plotted as scatter. Utilizing the segregation principle outlined in Equations B-la and

B-lb required further division of the simple one-to-one plot, as it initially only applied to

the positive-positive quadrant since only the absolute flux magnitudes were being plotted.

This primary division is shown in Figure B-l(a), and represents the first of four steps

in the derivation of the flux tendency phase diagram. Here, the phase diagram is clearly

divided into four distinct quadrants, labeled I -IV using a convention similar to that

found in Euclidean geometry.

In Figure B-l(a), we see that the overall plot is bisected by the bedload and

suspended load transport flux axes, indicating an origin in the center of the diagram.

In this manner, quadrants to the right of the vertical bisector (I and II) are -i-i -1 .-

of bedload flux growth, while those to the left (III and IV) represent regions of bedform

flux decay. Alternatively, quadrants above the horizontal bisector (I and IV) indicate

bedform growth through suspended fluxes, with those below the horizontal bisector (II

and III) denoting suspended flux decay. Now, each quadrant can be interpreted as its

own distinct one-to-one comparison of VQt and VQ', as demonstrated in Figure B-l(b),

with the lines of perfect agreement represented by the dashed (--) lines. Data falling

along the lines of perfect agreement indicate an equality in bedload and suspended load

flux magnitudes for that quadrant. Points falling closer to the horizontal bisectors signify

bedload-dominated tendencies, while those falling closer to the vertical bisector represent

suspension- or deposition-dominated tendencies.

In an attempt to generalize the flux tendency diagram further, the diagram

described by Figure B-l(b) is simplified to define regions of growth (+) and decay

(-) in Figure B-l(c). Moreover, the tendency of the one-to-one lines in each quadrant is

rationalized and labeled in Figure B-l(c), where the solid diagonal line represents a line of

equilibrium; in quadrant II, bedload flux growth is counteracted by suspended flux decay









along the one-to-one line; the opposite is true in quadrant IV. In quadrant I, the line

of perfect agreement produces growth regardless of the flux mechanism, while quadrant

III represents total decay through contributions from both flux mechanisms. Regions of

growth (+) and decay (-) are combined in Figure B-l(d), yielding the final flux tendency

phase diagram. Here, the phase diagram is subdivided into only two distinct regions for

simplicity: growth and decay. Each region may be further subdivided by the respective

action on either side of the one-to-one line in order to generalize the dominant flux

behavior in that region of the phase diagram. Data points falling along the equilibrium

line are indicative of morphology that is experiencing dynamic equilibrium through an

equality in magnitudes of both bedload and suspended load transport fluxes. Simulation

results of bedform growth, equilibrium, and decay confirm the utility of the flux tendency

phase diagram.




























- Bedload Flux <


Figure B-1.


>+ Bedload Flux


(a) Ordered Quadrants


(c) Reduction


Bedload Flux + Bedload Flux
(b) Dominant Fluxes and Tendency
Bedload Flux =! i=> + Bedload Flux
(b) Dominant Fluxes and Tendency


0
- Bedload Flux =z> + Bedload Flux
(d) Simplified Map


An explanation of the transport flux tendency phase diagram of bedform
growth, equilibrium, and decay. The quadrants are (a) ordered one
through four using a common convention, (b) labeled corresponding to
the dominant flux and its tendency, (c) divided in half by the one-to-one
(perfect agreement) line and labeled according to the growth (+) or decay
(-) tendency, and (d) simplified to two triangular regions associated with
either bedform growth (upper right green triangle) or decay (lower left red
triangle). The solid diagonal line (-) represents ripple equilibrium, while the
dashed line (--) and dashed-dotted line (- -) represent balanced growth
and decay, respectively.


IV I




III II









REFERENCES


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BIOGRAPHICAL SKETCH

Bret Maxwell Webb was born in Fort Myers, Florida, on the 8th of May, 1979. The

son of Ross and Stephanie Webb, and the younger of two children, Bret was raised in

the City of Palms by his parents, grand parents, and extended family. Spending most

of his time on the waters of San Carlos Bay and Pine Island Sound either sailing or

fishing for Redfish, Bret did manage to schedule in some academics, graduating from

Bishop Verot Catholic High School in 1997. After receiving a B.S. in Civil Engineering

from the University of Florida in 2001, Bret pursued a graduate degree in Coastal and

Oceanographic Engineering, obtaining a Master of Science degree in 2004, also from the

University of Florida.

Upon completion of the master's program, Bret was awarded an Alumni Fellowship

from the Department of Civil and Coastal Engineering at the University of Florida,

providing him funding to seek a doctorate in his chosen field. Bret has been studying sand

ripples for the past four years, specifically developing numerical models to explain their

behavior and characteristics. In August 2007, Bret was offered a position in academia at

the University of South Alabama in the Department of Civil Engineering. Since that time,

he has served as an instructor, teaching undergraduate courses in the civil engineering

curriculum. Upon successful fulfillment of the requirements necessary to obtain his Doctor

of Philosophy, Bret will return to the University of South Alabama to accept a tenure

track faculty position. Bret currently resides in Mobile, Alabama, with his wife of five

years, Shannon Leigh.





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SMALL-SCALESEDIMENTTRANSPORTPROCESSESANDBEDFORMDYNAMICSByBRETMAXWELLWEBBADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2008 1

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Copyright2008byBretMaxwellWebb 2

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Itisunlikelythatthenaturalprocessofsedimenttransportbyowingwaterwillbeunderstoodinprecisedynamicaltermsintheforeseeablefuture." -R.A.Bagnold 3

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ACKNOWLEDGMENTSIwouldliketoextendmygratitudersttomyfellowoce-matesfortheirsupport,encouragement,assistance,andfriendship.Robert,Jenn,Allison,Ty,andJessica,thanksforalwaysservingasasoundingboardformyideas,aswellasmycomplaints.Second,Iamgratefulfortheconstructivecriticismofmysupervisorycommitteemembers,thosethatIhaveoftensoughtoutforadvice,andparticularlyfortheguidanceandencouragementofmyadvisor,DonSlinn.MutluSumerisdeservingofspecialacknowledgmentfortheboundarylayerdatausedinthemodel-datacomparison.Last,Ioweagreatdebtofgratitudetomyfamily,especiallymywife,Shannon,asnoneofthiswouldhavebeenpossiblewithouttheirloveandconstantsupport. 4

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TABLEOFCONTENTS ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 LISTOFSYMBOLS .................................... 14 ABSTRACT ........................................ 19 CHAPTER 1INTRODUCTION .................................. 20 1.1Background ................................... 20 1.2Motivation .................................... 23 1.3Approach .................................... 26 1.4Outline ...................................... 28 2SEDIMENTTRANSPORTPROCESSES ..................... 30 2.1Introduction ................................... 30 2.2GoverningHydrodynamics ........................... 30 2.2.1Waves .................................. 30 2.2.2Currents ................................. 33 2.2.3WavesandCurrents ........................... 34 2.2.4TidesandTidalCurrents ........................ 35 2.3Transport .................................... 36 2.3.1Modes .................................. 36 2.3.1.1Bedload ............................ 37 2.3.1.2SuspendedLoad ....................... 40 2.3.2Regimes ................................. 43 2.4Turbulence ................................... 46 2.4.1Dynamics ................................ 47 2.4.2Closure .................................. 48 2.4.2.1RANSModels ......................... 48 2.4.2.2LargeEddySimulations ................... 49 2.4.2.3DirectNumericalSimulations ................ 49 2.5Models ...................................... 50 2.5.1Types ................................... 50 2.5.1.1Time-Averaged ........................ 50 2.5.1.2Quasi-Steady ......................... 51 2.5.1.3Semi-Unsteady ........................ 52 2.5.1.4Unsteady ........................... 53 2.5.2IntegratedApproaches ......................... 54 2.5.3Shortcomings .............................. 56 5

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3BEDFORMDYNAMICS ............................... 59 3.1Introduction ................................... 59 3.2BedformTypes ................................. 59 3.2.1RollingGrainRipples .......................... 61 3.2.2VortexRipples .............................. 63 3.2.3Characteristics .............................. 66 3.3Experiments ................................... 69 3.3.1Laboratory ................................ 69 3.3.1.1OscillatingTray ........................ 70 3.3.1.2OscillatingWaterTunnel ................... 70 3.3.1.3Flume ............................. 71 3.3.2Field ................................... 73 3.4RipplePredictors ................................ 74 3.4.1Clifton1976 .............................. 74 3.4.2Nielsen81 .............................. 74 3.4.3GrantandMadsen82 ....................... 76 3.4.4WibergandHarris94 ........................ 77 3.4.5Mogridgeetal.994 ......................... 78 3.4.6KhelifaandOuellet000 ....................... 79 3.4.7FaraciandFoti02 ......................... 80 3.4.8Williamsetal.2005 .......................... 81 3.4.9SoulsbyandWhitehouse06 .................... 81 3.5Models ...................................... 82 4MODELDESCRIPTION .............................. 88 4.1Introduction ................................... 88 4.2One-DimensionalLinearModel ........................ 88 4.2.1Hydrodynamics ............................. 89 4.2.1.1GoverningEquations ..................... 89 4.2.1.2Numerics ........................... 92 4.2.2SedimentTransport ........................... 93 4.2.3Morphology ............................... 93 4.3Two-andThree-DimensionalNonlinearModels ............... 93 4.3.1Hydrodynamics ............................. 95 4.3.1.1GoverningEquations ..................... 95 4.3.1.2TurbulenceClosure ...................... 98 4.3.1.3Numerics ........................... 104 4.3.1.4BoundaryConditions ..................... 109 4.3.2SedimentTransport ........................... 111 4.3.2.1Bedload ............................ 111 4.3.2.2SuspendedLoad ....................... 117 4.3.3Morphology ............................... 123 4.3.3.1Finite-DierenceMethods .................. 125 4.3.3.2FilteringTechniques ..................... 131 6

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5MODELEXPERIMENTS .............................. 146 5.1LinearModelExperiments ........................... 146 5.2NonlinearModelExperiments ......................... 147 5.2.1PhaseI:ModelValidation ....................... 147 5.2.2PhaseII:ModelCapabilities ...................... 149 5.2.3PhaseIII:SedimentTransportProcesses ............... 150 6RESULTS ....................................... 159 6.1LinearModelResults .............................. 159 6.2NonlinearModelResults ............................ 160 6.2.1PhaseI:ModelValidation ....................... 160 6.2.1.1HydrodynamicValidation .................. 161 6.2.1.2ComputationalGridTests .................. 163 6.2.1.3ModelDomainWidthTests ................. 165 6.2.1.4SedimentTransportSubmodelTests ............ 170 6.2.1.5MorphologyTests ....................... 173 6.2.2PhaseII:ModelCapabilities ...................... 183 6.2.2.1SteadyFlow .......................... 184 6.2.2.2SubcriticalFlow ....................... 185 6.2.2.3SheetowRegime ....................... 185 6.2.2.4RippleGrowth,Equilibrium,andDecay .......... 187 6.2.2.5SedimentSizeTests ...................... 193 6.2.3PhaseIII:SedimentTransportProcesses ............... 197 6.2.3.1TotalLoadGrowth ...................... 198 6.2.3.2TotalLoadDecay ....................... 203 6.2.3.3BedloadOnly ......................... 205 6.2.3.4SuspendedLoadOnly .................... 206 7CONCLUSIONS ................................... 274 7.1Live-BedModelEvaluation .......................... 274 7.2SedimentTransportandBedformDynamics ................. 278 7.3FutureApplications ............................... 281 APPENDIX AEXPERIMENTMATRIX .............................. 283 BFLUXTENDENCYPHASEDIAGRAM ...................... 285 REFERENCES ....................................... 289 BIOGRAPHICALSKETCH ................................ 304 7

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LISTOFTABLES Table Page 2{1Powerlawformulationsforbedloadtransport ................... 58 2{2Empiricalpick-upfunctionsforsuspendedloadtransport ............. 58 3{1Listofcommonvaluesfororbitalripplelength ................... 85 3{2Summaryofeldandlaboratoryrippleexperiments ................ 86 3{3Parametersrelatedtobedformdynamics ...................... 87 3{4Ripplepredictorkey ................................. 87 5{1Meshparametersforgridcomparisons ....................... 155 5{2Meshparametersfordomainwidthcomparisons .................. 155 5{3Abbreviationkeyfortransportequationtests ................... 155 5{4Morphologytestsimulationmatrix ......................... 156 5{5Ripplepredictorcitationkeyforthemorphologytestsimulations ........ 156 5{6Expectedmorphologyresultsbasedonripplepredictors .............. 157 5{7PhaseIIexperimentmatrix ............................. 158 5{8SimulationparametersforthePhaseIIIexperiments ............... 158 6{1RipplepredictorRMSerrorformorphologytestsunderlab-scaleows ..... 272 6{2RipplepredictorRMSerrorformorphologytestsundereld-scaleows ..... 272 6{3Ripplepredictormodelperformanceindex ..................... 273 A{1Completeexperimentalmatrix ............................ 284 8

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LISTOFFIGURES Figure Page 1{1Picturesofsandripplesinlabandeldsettings .................. 29 2{1Velocityandstresstimeseriesforlinearandnonlinearwaves. .......... 57 2{2Distributionofuidandgrainshearstress. ..................... 57 3{1Schematicofvortexsheddingoverarippledbed .................. 84 3{2Eectsofbedformsonboundarylayerthickness .................. 85 4{1Schematicofone-dimensionalbedloadmodel .................... 135 4{2Schematicofthethree-dimensionalmodelingdomain ............... 135 4{3Modelcontrolvolumeswithvariablelocations ................... 136 4{4Schematicofvariablemeshscaling ......................... 136 4{5Meshclusteringandvariablescaling ........................ 137 4{6Acomparisonofbedshearstressformulations ................... 138 4{7EstimationofcriticalShields'curveby Brownlie 1981 .............. 139 4{8EstimationofcriticalShields'curveby vanRijn 1993 .............. 139 4{9Particleforcesactingonlongitudinalandtransverseslopes ............ 140 4{10Gravitationalforcesonasphere ........................... 141 4{11Behaviorofbedloadtransportequations ...................... 141 4{12Behaviorofsedimentpick-upfunctions ....................... 142 4{13Modicationofrelativesettlingvelocitybyconcentration ............. 142 4{14Sedimentcontrolvolumeandtransportschematic ................. 143 4{15Amplicationfactorsforthetwo-dimensionallow-passmorphologylter .... 144 4{16Amplicationfactorsforthehybridmorphologylter ............... 145 5{1Horizontalandverticalgridspacingforgridcomparisons ............. 152 5{2Modeldomainsusedinthedomainwidthcomparisontests ............ 153 5{3SchematicofthemodeldomainusedforthePhaseIIIsimulations ........ 154 6{1Timestacksofbedelevationforthe1DHlinearmodelexperiment ........ 208 9

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6{2Timeevolutionofrippleheight,wavelength,andsteepnessforthe1DHlinearmodelexperiment ................................... 209 6{3Model-datahydrodynamiccomparison ....................... 210 6{4Model-datacomparisonofboundarylayerthickness ................ 211 6{5Assessmentofhydrodynamicmodelerror ...................... 212 6{6Computationalbudgetforgridresolutiontests ................... 213 6{7Velocitystandarddeviationsforgridcomparisons ................. 214 6{8CumulativeaverageturbulentkineticenergyTKEforgridcomparisons .... 215 6{9Phase-andvolume-averagedTKEforgridcomparisons .............. 215 6{10Phase-andy)]TJ/F15 11.9552 Tf 9.2985 0 Td[(averagedu)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityprolesforgridcomparisons ......... 216 6{11Phase-andy)]TJ/F15 11.9552 Tf 9.2985 0 Td[(averagedv)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityprolesforgridcomparisons ......... 217 6{12Phase-andy)]TJ/F15 11.9552 Tf 9.2985 0 Td[(averagedw)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityprolesforgridcomparisons ......... 218 6{13Computationalbudgetforthree-dimensionalgridtests .............. 219 6{14CumulativeaverageturbulentkineticenergyTKEforthree-dimensionalgridcomparisons ...................................... 220 6{15Phase-andvolume-averagedTKEforthree-dimensionalgridcomparisons .... 220 6{16Phase-andvolume-averagedvelocitycomponentsforgridcomparisons ..... 221 6{17AverageFouriertransformsofv)]TJ/F15 11.9552 Tf 9.2985 0 Td[(velocityforgridcomparisons .......... 222 6{18Spatially-averagedstandarddeviationofbedelevationinthethirddimension .. 223 6{19Comparisonsofequilibriumrippleheightforgridcomparisons .......... 223 6{20AverageFouriertransformsofbedelevationforgridcomparisons ........ 224 6{21Eectsofsedimenttransportsubmodelsonequilibriumripplecharacteristics .. 225 6{22Conservationofsedimentmass ........................... 226 6{23Bedformstatisticsinlab-scaleows ......................... 227 6{24FourieranalysistimestackofbedformstatisticsforHL1v1 ............ 228 6{25Ripplepredictorcomparisonsforlab-scalemorphologytests ........... 229 6{26Bedformstatisticsineld-scaleows ........................ 230 6{27FourieranalysistimestackofbedformstatisticsforMF2v2andHF1v1 ..... 231 10

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6{28Ripplepredictorcomparisonsforeld-scalemorphologytests ........... 232 6{29Comparisonofmodelresultstotheripplepredictorequationsof FaraciandFoti 2002 ......................................... 233 6{30Comparisonofmodelresultstotheripplepredictorequationsof Nielsen 1981 KhelifaandOuellet 2000 ,and Williamsetal. 2005 .............. 234 6{31Comparisonofmodelresultstotheripplepredictorequationsof Mogridgeetal. 1994 ......................................... 235 6{32Bedelevationtimestackforsteadyowtest .................... 236 6{33Steadyowbedshearstressandsedimenttransport ................ 237 6{34Bedelevationtimestackforsubcriticalowtest .................. 238 6{35Averagebedshearstressandbedloadtransportinsubcriticalow ........ 238 6{36Bedelevationtimestackforsheetowsedimenttransport ............. 239 6{37Time-seriesofbedformstatisticsinthesheetowregime ............. 239 6{38Energydensitytimestackforthesheetowtest ................... 240 6{39Isocontoursofsedimentconcentrationinthesheetowregime .......... 240 6{40Averagesedimentconcentrationproleinthesheetowregime .......... 241 6{41Bedelevationtimestacksforripplegrowth,equilibrium,anddecay ........ 242 6{42Timestackofripplespectraduringgrowthfromatbed ............. 243 6{43Ripplestatisticsforgrowth,equilibrium,anddecay ................ 244 6{44Timeevolutionofrippleheightsduringgrowth,equilibrium,anddecay ..... 245 6{45Bedelevationtimestacksforripplecoarseningandbifurcation .......... 246 6{46Timestacksofripplespectraduringripplewavelengthsaturation ......... 247 6{47Ripplestatisticsduringheightandwavelengthevolution ............. 248 6{48Eectsofhinderedsettlingonmodelpredictionsofrippleheightandlength .. 249 6{49Modicationofaverageconcentrationproleduetohinderedsettling ...... 250 6{50Time-seriesofbedformstatisticsfornegravelinsubcriticalow ........ 250 6{51Phase-averagedstressandtransportfornegravelinsubcriticalow ...... 251 6{52Timestacksofbedelevationfornegravelinastrongow ............ 251 11

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6{53Bedformstatisticsfornegravelinastrongow ................. 252 6{54Phase-averagedstressandtransportfornegravelinastrongow ....... 252 6{55Timestacksofbedelevationformediumsiltinaweakow ............ 253 6{56Timeevolutionofrippleheightandlengthformediumsiltinaweakow .... 253 6{57Phase-averagedstress,transport,andsedimentconcentrationformediumsiltinaweakow ...................................... 254 6{58Timestacksofbedelevationformediumsiltinastrong,highly-concentratedow 255 6{59Timeevolutionofbedformheightandlengthformediumsiltinastrongow .. 255 6{60Phase-averagedstress,transport,andsedimentconcentrationformediumsiltinastrong,highly-concentratedow .......................... 256 6{61Suspendedsedimentphasediagram ......................... 257 6{62Timeevolutionofrippleheight,wavelength,andsteepnessduringripplegrowthandequilibrationTL2v2 .............................. 258 6{63TimestacksofbedelevationduringripplegrowthTL2v2 ............ 259 6{64CumulativeaveragedandcorrectedtransportuxesforTL2v2 .......... 259 6{65Phasedependenceofbedload,entrainment,anddepositionduringripplegrowthandequilibrium .................................... 260 6{66Phasedependenceofcorrectedbedloadandsuspendedloaduxesduringripplegrowthandequilibrium ................................ 261 6{67Transportuxtendencyphasediagramsforripplegrowthandequilibrium ... 262 6{68TimeevolutionofripplecharacteristicsduringrippledecayTL2v2d ...... 263 6{69CumulativeaveragedandcorrectedtransportuxesforTL2v2d ......... 264 6{70Phase-averagedandcorrectedtransportuxesduringrippledecay ........ 264 6{71PhasedependenceoftransportconstituentsduringrippledecayforTL2v2d ... 265 6{72Transportuxphasediagramforrapidrippledecay ................ 266 6{73Timeevolutionofripplecharacteristicsthroughbedloadtransportonly ..... 267 6{74PhasedependenceofbedloadtransportduringripplegrowthforBL2v2 ..... 268 6{75Timeevolutionofripplecharacteristicsthroughsuspendedloadtransportonly 269 6{76PhasedependenceofverticalsedimentuxesduringrippledecayforSL2v2 ... 270 12

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6{77TimeevolutionofscaledrippleheightsforPhaseIIIsimulations ......... 271 B{1Derivationoftheuxtendencyphasediagramofbedformgrowth,equilibrium,anddecay ....................................... 288 13

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LISTOFSYMBOLSGreeksymbols longitudinalbedslope. transversebedslope. waveperiodparameter. t timestep. tf morphologyhybridlteringtimestep. xi xgridspacing. yj ygridspacing. zk zgridspacing. boundarylayerthickness. lterwidth. d displacementboundarylayerthickness. d relativeorbitalexcursion. ij Kroneckerdelta. freesurfacedisplacement. r rippleheight. s smoothingcoecient. vonKarmanconstant. mix mixedlayerthickness. r ripplewavelength. dynamicviscosity. kinematicviscosity. t eddyviscosity. angularfrequency. sedimentangleofrepose. b nondimensionalbedloaddischarge. 14

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p nondimensionalentrainmentrate. v velocitypotential. mobilitynumber. densityofwater. s densityofsediment. mix densityofmixedlayer. S Schmidtnumber. rippleheightstandarddeviation. ripplelengthstandarddeviation. r ripplesteepness. b bedshearstress. g gravitationalparticleshearstress. b;max maximumbedshearstress. Rij residual-stresstensor. rij anisotropicresidual-stresstensor. nondimensionalShieldsstress. variablenite-dierencecoecient. c nondimensionalcurrentShieldsstress. w nondimensionalwaveShieldsstress. 2:5 grainroughnessShieldsparameter. cr criticalShieldsparameter. cr criticalShieldsparameterforahorizontalbed. cw wave-currentangle. 'n primitivevariableattimeleveln.Genearaloperators lteredvariable. hh'ii phase-averagedquantity. 15

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h'i time-averagedquantity. spatially-averagedquantity. '0 residualvariable.Romansymbols `S Smagorinskylengthscale. ^zb low-passlteredbedelevation. rQb sign-correctedbedloadtransportux. rQs sign-correctedsuspendedloadtransportux. C lteredvolumeconcentration. Cb lteredvolumeconcentrationabovebed. Sij lteredrateofstrain. S characteristiclteredrateofstrain.
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d graindiameter. d dimensionlessgrainsize. d50 mediangraindiameter. Et entrainmentrateofbedmaterial. f coecientoffriction. Fn Fluxevaluatedattimeleveln. fb coecientofresistingfriction. Fg downslopecomponentofparticleweight. fw wavefrictionfactor. Fw buoyantweightofparticle. Fx bodyforce. f2:5 grainroughnessfrictionfactor. fmix coecientofappliedfriction. g gravitationalacceleration. h waterdepth. H waveheight. k turbulentkineticenergy. kr residualkineticenergy. ks Nikuradseroughnesslength. kw wavenumber. kx wavenumberinx)]TJ/F15 11.9552 Tf 9.2985 0 Td[(direction. ky wavenumberiny)]TJ/F15 11.9552 Tf 9.2985 0 Td[(direction. n normaldirection. np porosityofbedmaterial. p pressure. Prt turbulentPrandtlnumber. Qb volumetricbedloadtransportrate. 17

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Qs volumetricsuspendedloadtransportrate. Qt totalvolumetricsedimenttransportrate. s specicgravityofsediment. S nondimensionalsedimentparameter. T waveperiod. t time. TKE turbulentkineticenergy. u frictionvelocity. U uidvelocityatedgeofboundarylayer. U1 freestreamvelocity. ub nearbeduidvelocity. Uc currentvelocity. ui velocitycomponent. Umax maximumvelocityamplitude. umix mixedlayervelocity. us sedimentvelocity. ws hinderedsettlingvelocity. wso constantsettlingvelocity. xi spatialdirectioncomponent. z+ normalizedwallunit. zb verticallocationofbed. zo referenceheight. 18

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySMALL-SCALESEDIMENTTRANSPORTPROCESSESANDBEDFORMDYNAMICSByBretMaxwellWebbMay2008Chair:DonaldN.SlinnMajorDepartment:CoastalandOceanographicEngineeringThegenerationofsmall-scalesedimentarystructuresinthecoastalenvironmentisacomplexprocessthatoccursoverawideseparationofscalesinbothtimeandspace.Thesebedformsareubiquitousfeaturesofthenearshoreregion,andyetspecicinformationregardingtheirbehaviorandcharacteristicsisstilllacking.Specically,itisunclearwhetherthebedload-dominatedprocessesofthelinearregimeareasequallyresponsibleforthegenerationofbedformsinthenonlinearregime,whereowseparation,andsubsequentvortexformation,tendtogovernthedynamicsofthebottomboundarylayer.Whileasimpleone-dimensionalmodelisderivedandusedtoexplainincipientbedformgrowthinthelinearregime,suchanapproachisnotwell-suitedataddressingthecomplexitiesofthewavebottomboundarylayer.Utilizinganewthree-dimensionalphase-resolvinglive-bedmodel,wesimulatethedynamicsofbedforms,suchassandripples,inthenonlinearregime.Throughforty-threeindependentsimulations,themodelhasbeenfoundtoreproduceoscillatoryboundarylayerow,aswellasprovideaccuratepredictionsofripplegeometryinbothlab-andeld-scaleows.Modelresultsconrmthatinthelinearregime,bedformgrowthispromotedpurelythroughbedloadsedimenttransport,butinertialpropertiesofthesedimentareequallyasimportant.Inthenonlinearregime,bedformgrowthisalsodominatedbybedloadtransport;however,theentrainmentanddepositionofbedmaterialplaysanimportantroleinmaintainingrippleequilibrium,whereasitismostlyresponsibleforrippledecay. 19

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CHAPTER1INTRODUCTION1.1BackgroundBedforms,suchassandripples,areubiquitousfeaturesofthecoastalzone.Andalthoughtheyhavebeenthesubjectofnumerousinvestigations|theoretical,experimental,andnumericalalike|datingbacktothelatenineteenthcentury Hunt 1882 ; Darwin 1883 ,denitiveinformationabouttheirdynamicsinthecoastalenvironmentremainslimited.Thisisnottosaythattheoryandunderstandinghavenotbeenmarkedlyadvancedoverthelastcentury,butitunderpinsthenecessityforcontinuedresearchonbedformdynamicsatscalesbothlargeandsmall.Morphologicalbedformfeaturesoccurringinnaturemayrangeinscalesfromafewcentimetersinheightandtensofcentimetersinlengthforwave-generatedsandripples,tolargersandwavefeatureslikedunesandmega-ripplesthatcanhaveheightsofafewmetersandlengthsontheorderoftensofmeters.Regardlessoftheirsize,bedformsplayanimportantroleinboththeenergyandsedimentbudgetsofthenearshoreenvironment.Bedformshavebeenfoundtostronglyaect: energydissipationinthewavebottomboundarylayer; sedimenttransportcharacteristics;and acousticpropertiesoftheseaoor.Whilemomentumtransferaboveplanebedsisduemostlytoturbulentdiusion,rippledbedsinduceowseparationresultinginorganizedvortices,orcoherentmotions,thatdominatemomentumtransferinthewavebottomboundarylayerWBBL Marin 2004 .Thesecoherentmotionsareeectiveindissipatingwaveenergyinanearbedboundarylayerthatisapproximatelytwotimestherippleheight vanderWerfetal. 2006 .Abovethisnearbedlayer,coherentmotionsbreakdownandarereplacedbyrandomturbulence DaviesandVillaret 1997 Tunstall 1973 and TunstallandInman 20

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1975 estimatethatasmuchas10%ofwaveenergymaybedissipatedthroughtheprocessofvortexformation.Waveenergyisalsoeectivelydissipatedthroughfrictionasbedformsaectthehydraulicroughnessofthebedforwavesandcurrents SoulsbyandWhitehouse 2006 .Bedformheightcontrolstheeectivebottomroughness FaraciandFoti 2002 ,whichisofpracticalengineeringimportance.Parametrizationofhydraulicroughness GrantandMadsen 1982 arecommonlyappliedtoengineeringwaveandsedimenttransportmodelsinordertoaccountfortheadditionalenergydissipationthattakesplaceintheWBBL.Suchpracticepermitstheeectsofbedformsoncurrentsandwavestobetreatedinabulkmanner,ratherthanhavingtoaccountforthemexplicitlyinthemodelingapproach.TheparametrizationandapplicationofroughnesstomodelsaccountsforthethickeningoftheWBBLinthepresenceofbedforms WibergandHarris 1994 ,wherebyenergyisdissipatedthroughformdrag,skinfriction,andturbulencedampingduetoanintenselayerofnearbedsedimenttransport GrantandMadsen 1982 .Theamountofdissipationduetobedformsisnottrivial;eldobservationsby Ardhuinetal. 2002 indicatethattheformdragoverlargeripplesiseectiveinwaveattenuationacrossthecontinentalshelf.Accuraterepresentationsofequivalentbottomroughness,then,areofparticularimportanceforunderstandingtheenergybudgetofthenearshoreenvironment.However,priorcharacterizationsofroughnessandtheirsubsequentimplementationinnearshoremodelsoftenneglectchangesinbedformsize,shape,andorientation,anddonotaccountforphaselageectsinducedbyowseparationintheboundarylayer.Bedformsaremanifestationsofdierentialsedimenttransportneartheseabed,andcanoftenbefoundinuniquepatternsalongtheseaoorasseeninFigures 1{1a { 1{1d .Oncelargeenoughtoinduceowseparation,theyhaveaprofoundimpactonsubsequentsedimenttransport.Enhancedsedimentsuspensionoverrippledbedsnotonlyincreasespotentialsedimenttransportinthebottomboundarylayer,butaectsthebackgroundowaswell.Straticationofsedimentinthewatercolumnresultsinamodicationofthe 21

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velocityprole Coleman 1981 ,turbulencedamping McLean 1991 ,andanincreaseinboundaryroughness ParkerandColeman 1985 .Therefore,sedimentsuspensionoverrippledbedsservesasanadditionalmechanismforthedissipationofenergyinthebottomboundarylayer.Accordingto Nakatoetal. 1977 ,suspensionprocessesoverrippledbedsaredominatedbytheformation,ejection,andmotionofeddies.Organizedvorticesthatforminthelee-sideofripplesentrainsedimentfromtroughsorvalleysineachsuccessivehalf-cycleofwaveforcing.Immediatelyfollowingowreversal,thesediment-ladenvorticesareejectedintothemainowabovethebedformswherethesedimentissubsequentlyadvectedbythelocaluidvelocityeld.Laboratoryexperimentsby vanderWerfetal. 2005 suggestthatsuchbehaviorresultsinthreedistinctpeaksinthenearbedconcentrationasafunctionofwavephase.Comparedtosuspensioneventsoveraatbed,elddatasuggestthatsediment-ladenvorticesinthewakeofbedformsalterssedimentsuspension Gallagheretal. 1998 ,andmayalsoenhancethephaselagbetweensuspensionandtransport InmanandBowen 1963 .Furthermore, vanderWerfandRibberink 2004 proposethatripple-inducedphasedierencesbetweenpeaksuspendedsedimentconcentrationsandpeakuidvelocitiesresultinnetsedimenttransportratesdirectedoshoreundersurfacegravitywaves.Suchphasedierences,however,arenotaccountedforincommonphase-averaging,coupledhydrodynamicandsedimenttransportmodels.Thismaysuggestwhysomecross-shoresedimenttransportmodelsfailwhenoscillatorycurrentsarelargerthansteadycurrents Gallagheretal. 1998 .Inrecentyears,theNavyhasbecomeincreasinglyinterestedintheacousticpropertiesoftheseaoor,specicallywithrespecttotheabilityofsonardevicestodetectbothemergentandsubmergedmunitionsinthebattlespaceenvironmentBSE.Bedformsaecttheacousticresponseforsonarapplications,eitherenhancingorinhibitingthepenetrationofthesonarsignalintothebed SoulsbyandWhitehouse 2006 .FromdatacollectedduringtheSAX99eldexperiment, Piperetal. 2002 showthatboththe 22

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rippleheightandwavelengthstronglyaectthelevelofsonarpenetrationforsubcriticalgrazingangles.Additionally,theorientationoftheripplesrelativetotheincidentsonareldisofimportance.Theresultsof Piperetal. 2002 indicatethatsubsurfaceimagingisenhancedwhenthesonarpropagationdirectionisperpendiculartothemeandirectionofripplecrests,wherebytheamountofacousticenergyscatteredbythebedformsintothesedimentisincreased.1.2MotivationInthelinearregimeofbedformgrowth|whererollinggrainripplespersist|sedimenttransporttakesplacepurelythroughbedloadprocesses.Thebedloadtransportmodeconsistsofparticlesrolling,sliding,andsaltatinginsmallleapsontheorderofafewgraindiametersabovethebed.Thelinearregimeisdominatedbyfrictionandinertiallagsbetweenuidforcingandparticletransport.Theabsenceofsuspendedsedimenttransportinthelinearregime|andatsubcriticalthresholdvalues|haveleadmanytopostulatethatbedformsaregeneratedpurelythroughbedloadtransport.Itisunclear,however,thatthisstatementholdstrueinthenonlinearregimewherecoherentmotionsintheboundarylayerpromotesuspendedloadtransportandinducephasedierencesbetweenuidforcingandsedimententrainment.Thenonlinearregimecontainsbothbedloadandsuspendedloadtransportmodes,buttheirrelativecontributionstoripplegrowth,equilibration,anddecayareunknown.Additionally,thecontributionsofbedloadandsuspendedloadtransportmodesmayvaryasafunctionofripplepositionproleandplanformanduidforcing.Thequantitativerolesofconstructiveuidforcinganddestructivegravitationalforcesingeneratingsedimentarystructuresarenotimmediatelyevident.Itwouldbebenecial,therefore,toinvestigatethecontributionsofvarioustransportmodestobedformdynamics,andtobetterunderstandtherolesofconstructiveanddestructiveforcesduringripple,growth,equilibration,anddecay. 23

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Fieldandlaboratorymeasurementtechniquesareincapableofgatheringinsitudatathatclearlydistinguishonetransportmodefromanother.Suchmethodsoftenrequiretheassumptionofathresholdbasedonvolumetricconcentrationtoseparatethetransportmodesafterthedatahavebeencollected.Additionally,measurementtechniquesforbedloadandsuspendedloadtransportarecommonlyinvasive,therebyalteringthehydrodynamicandsedimenttransporteldsthattheyareattemptingtoquantify.Numericalsimulationsmayprovideusefulinsightintosedimenttransportprocessesinsmall-scalebedformdynamicswherephysicalexperimentationiscurrentlynotpossible.ThroughtheUniversityofFlorida,weareparticipatingintheOceofNavalResearch'sONRintensivestudyofSandRipplesontheInnerShelfFY04-08,aprojectinvolvingfteenprincipalinvestigatorsfromacrossthenationRipplesDRI.Thisinterdisciplinaryprojectcombineseldobservationswithlaboratoryexperimentsandnumericalsimulationsfromthebiological,geological,andoceanographicsciences.Theprimarygoalsoftheproject,asidentiedbyONR,areto 1. measureandmodelmorphology; 2. investigatemorphologicalresponsetoforcingperturbations; 3. measureandmodelratesofbio-degradation; 4. measureandmodeltheeectsofgrain-sizedistribution;and 5. understandtheroleingeneratingsedimentarystructures.Ourparticipationintheprojectprimarilyinvolvesthersttwogoalsoutlinedabove,aswellasthelast.Inordertoaddresstheseobjectives,weseektodevelopaphysics-basedmodelthatcoupleshydrodynamicsandmorphologybyupdatingthebedlevelatevery,ornearlyevery,hydrodynamictime-step.ThereviewofnoncohesivesedimenttransportprocessesandbedformdynamicsthatfollowsinChapters 2 and 3 shouldprovideadequateguidancefordeterminingasuitablemodelframework. 24

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AsidefromthegoalsoftheONRRipplesDRI,theprimaryobjectiveofthisresearchistoinvestigatesedimenttransportcharacteristicsduringripplegrowth,equilibration,anddecay.Inparticular,wehopetoprovideanswerstothefollowingquestions: 1. Whataretherelativecontributionsofbedloadandsuspendedloadtothegenerationorobliterationofsedimentarystructures? 2. Aretherespeciclocationsintherippleprole/planformwhereonetransportmodedominatesovertheother? 3. Aretherespecictimes|asafunctionofphase|whenonemodedominatesovertheother? 4. Whatarethedominantrolesof:bedload,suspendedload,andgravitationalforces?Toaddressourobjectives,weproposethedevelopmentofanentirelynewmodelingsystemcapableofsimulatingphase-resolvingsmall-scalesedimenttransportandmorphology.Capabilitiesofthemodelingsystemwillbeassessedbyevaluatingthehydrodynamicsthroughmodel-datacomparisons,andbyalsoperformingsimulationsofbedformequilibrationunderavarietyofscenariose.g.growth,equilibration,decay,coarsening,bifurcation,steadyows,short-andlong-periodows,highly-concentratedows,andextremesedimentsizes.Inordertoanswerthequestionsposedabove,simulationsofbedformgrowth,equilibration,anddecaywillbeperformedwith bedloadandsuspendedload; bedloadonly;and suspendedloadonly.ThedesignofthemodelingsystemassumesthatbedloadtransportispredictedusingEinstein'sdimensionlessbedloaduxandcommonpower-lawformulations,andthatsuspendedloadoriginatesthroughanentrainmentofsedimentfromthebedusingsemi-empiricalpick-upfunctionsavailableinexistingliterature.Thus,oursystemallowsustoalternatelyturnbedloadandsuspendedloadonorobysimplysettingbedloadtransportrates,orentrainment/deposition,tozero,respectively.Wefurtherassume 25

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thatthebedloadregimeiscomprisedoftwotypesoftransport:oneduetouidforcing;andanotherduetogravitationalforces.Doingsopermitsustodeterminetherolesofconstructiveanddestructiveforcesindependently,therebyallowingustobetterunderstandtheirrolesingeneratingsedimentarystructures.Thedevelopmentofthismodelingsystemisasecondaryobjectiveofthisdissertation,andisnecessarytoaddressourscienticquestionsposedabove.Specicdetailsaboutthesystemareoutlinedbelow,aswellasinthefollowingchapters.1.3ApproachModelingsedimenttransportcontinuestobeachallengeforuiddynamicists.Indeed,muchoftheuncertaintyinuid-sedimentmodelscontinuestoliemainlywiththeparticulatephase,whereastheuidhydrodynamicsarecomparativelywellunderstood.Sedimenttransportmodelsfallmainlyintotwobroadcategories: 1. time-orphase-averaged;and 2. unsteadyorphase-resolving.Variationsonphase-averagedandphase-resolvingmodelsincludequasi-steadyandsemi-unsteadymodels,respectively.Eachofthesemodelshasitsownbenetsanddeciencies,makingsomemoreusefulincertainsituationsthanothers.Briey,time-orphase-averagedsedimenttransportmodelseectivelyintegrateoutintra-waveprocessesbyconsideringwave-averagedvaluesofuidvelocity,sedimentconcentration,andsedimenttransport Bijker 1971 .Socalledquasi-steadymodelsassumethatthetransportrateisproportionaltotheinstantaneousnearbedoweldraisedtosomepower Bailard 1981 ,butonlyprovideanaveragetransportrateoverthewaveperiod.Semi-unsteadymodelsincorporateadditionalcomplexityintotheformulationoftransportratesbyconsideringphaselagseitherthroughparametrizations Nielsen 1988 ; Dohmen-Janssen 1999 orbybreakingafullwaveperiodintowavehalf-cycles DibajniaandWatanabe 1992 .Whilecomputationallyintensive,themostrobustapproachistodescribethesedimenttransportandhydrodynamicsascoupled,time-dependentprocesses.Veryfewofthesemodels 26

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existoutsideofthecomputationalresearcharena,astheirpracticalityforengineeringapplicationsisstillsomewhatlimitedduetotheircomplexity[see DrakeandCalantoni 2001 ; Gessleretal. 1999 ;and Lesseretal. 2004 ].Toaddressanumberofrelevantengineeringproblemsinvolvingsedimenttransportprocesses|fromstructure-inducedscourtocoastalerosion|itwouldbeadvantageoustohaveaexible,robustmodelcapableofresolvingtime-dependentbedmorphologyundervariousforcingconditions.SuchamodelingsystemcouldalsosatisfysomeoftheprimaryobjectivesoftheONRRipplesDRIprojectoutlinedabove.Hereweseektodevelopandevaluatetheskillofacoupleduid-sedimentnumericalmodelthatresolvesinstantaneoushydrodynamics,sedimenttransport,andtime-dependentbedmorphology.Itisnotevidentthatsuchamodelexistsatthelevelofdetailproposed.Thedesiredphase-resolvingmodelwouldbecapableofbothtwo-andthree-dimensionalsimulationsof1steadyow,oscillatoryow,combinedoscillatoryandsteadyow,andsurfacegravitywaves.WhilethetimeintegrationofthegoverningequationswillbelimitedtodurationsofO00s,resolutionofphysicallengthswillrangefromO0)]TJ/F23 7.9701 Tf 6.5865 0 Td[(4mtoOm.Suchdetailedtreatmentofhydrodynamicsnearsolidboundariesshouldpermittheresolutionofhigh-intensityturbulentuctuationsthatinitiatesedimenttransport.IntegrationofthebedlevelthroughtimewillbegovernedbythesedimentcontinuityExnerequation,wheregradientsinthebedloaduxandthecompetitionbetweenentrainmentanddepositionofsuspendedsolidsprovideanestimationoftheinstantaneousbedlevel.Sedimenttransportinthemodelwillbeestimatedusingbulk,empiricalformulationsavailableinexistingliterature.Naturalfeedbackbetweenmorphologyandthehydrodynamicoweldrepresentstheone-wayuid-sedimentcouplinginthemodel.Two-waycouplingofthehydrodynamicsandlocalsuspendedsedimentconcentrationmaybeconsideredinthefuture. 27

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1.4OutlineWhatfollowsisageneraloverviewofknowledgepertainingtononcohesivesedimenttransportandbedformdynamics,castinamannerthat,hopefully,underscorestheneedforahydrodynamiclive-bedmodelhavingthegeneralcharacteristicsoutlinedpreviously.AbriefsummaryofnoncohesivesedimenttransportprocessesisgiveninChapter 2 ,withadiscussiononthegoverninghydrodynamicsorforcing,modesoftransport,theroleofturbulence,andadescriptionofacceptedmodels.Chapter 3 providesinformationaboutgeneralbedformpropertiesanddynamics,withparticularattentiongiventotheirclassicationandcharacteristics,mechanismsforgrowth,asummaryofrelevantlaboratoryandeldexperiments,semi-empiricalsolutionsfortheirgeometricalproperties,andanoverviewofmodelsrangingfromsimpletocomplex.AdescriptionofnewlydevelopedmodelsisprovidedinChapter 4 andoutlinesgoverningequations,aswellasmethodologiesfortheirimplementationinthemodelingsystem.ThemodelexperimentsareoutlinedinChapter 5 ,andthosesimulationresultsarepresentedanddiscussedinChapter 6 .ConcludingstatementsareprovidedinChapter 7 ,alongwithsomewordsonpotentialfutureapplicationsofthenewly-createdmodelingsystems. 28

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aUSAWaveBasin bUSAWaveBasin cYucatanPeninsula,Mexico dYucatanPeninsula,MexicoFigure1{1.Picturesofsandripplesinlab[aandb]andeld[candd]settings.ThelabpicturesweretakenintheUniversityofSouthAlabamaWaveBasinafterdraining,whiletheeldpicturesweretakeninadepthofapproximately5mofwaterotheEastcoastoftheYucatanPeninsulainMexico. 29

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CHAPTER2SEDIMENTTRANSPORTPROCESSES2.1IntroductionSedimenttransportprocessesinthecoastalenvironmentarehighlycomplex.Transportationisbothinitiatedandsustainedbyhydrodynamicforcingontheseabedandwatercolumn.Whiletheknowledgeofsedimenttransportprocesseshasbeensignicantlyadvancedinidealized,laboratoryenvironments,detailedinformationaboutsuchprocessesremainsambiguousunderthestochasticforcingthatispresentinnature.Thefollowingsectionsoutlinethegoverninghydrodynamicsthatdrivesedimenttransportinthecoastalenvironment,provideadescriptionofcommontransportmodesandregimes,discussthemutualrelationshipsbetweentransportandturbulence,anddeliveranoverviewofvarioussedimenttransportmodels.2.2GoverningHydrodynamics2.2.1WavesSurfacegravitywavesrepresentthemostdominantmobilizingforceforsedimentincoastalareas vanderWerf 2004 .Insomecasestheorbitalvelocitiesunderthesewavescanbequitelargeneartheseabed,especiallywhenwavessteepenastheyentershallowerwater.Furthermore,realisticsurfacegravitywavesarenotlinear,typicallyhavingsteeperpeaksandattertroughs.Aresultofthisnonlinearproleisaninherentasymmetryintheorbitalvelocityeldwithshort-duration,high-intensityvelocitiesdirectedshorewardunderwavecrestsandlonger-duration,lower-intensityvelocitiesdirectedseawardunderthebroad,attroughsseeFigure 2{1 a.Thisinequalityintheorbitalvelocitytimeseriesbecomesveryimportantforthedeterminationofnetsedimenttransportrates[see RibberinkandAl-Salem 1990 ; Gallagheretal. 1998 ;and Elgaretal. 2001 ].Asdiscussedearlier,thewavebottomboundarylayerWBBLisanimportantmechanismforenergydissipationandinuencesthemagnitudeanddirectionofsedimenttransportinthenearshore.Thisboundarylayerdevelopsneartheseabedinresponse 30

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tofrictionbetweenitandtheuid.Ageneralexpressionforboundarylayerthicknessisgivenby/p T{1whereisthekinematicviscosityandTisthewaveperiod.Thethicknessoftheboundarylayerisalsoaectedbytheroughnessoftheboundary.WaveboundarylayersoversmoothboundariesmaybeofOmmwhilethethicknessoverroughboundaries,likearealseabed,increasestoOcm.Duetoinertialeectsintheboundarylayerslowermovinguid,thenearbedoscillatoryowisoftenoutofphasewiththefreestreamforcing.Thisresultsinanearbedphase-leadofapproximately45forlaminarowand15forturbulentboundarylayers,withrespecttothefreestreamforcing.Thephaseleadofnearbedoscillatoryowhasprofoundsignicancefortheestimationofsedimenttransport.Commonsedimenttransportformulationsincorporatesomethresholdforincipientmotion,suchasShields'parameter Shields 1936 ,thatrequiresthebedshearstresstoexceedsomecriticalvalue.Ageneralexpressionofthebedshearstressisb=CfUjUj{2whereistheuiddensity,Cfisacoecientoffrictiondrag,andUisrepresentativeofthefreestreamforcing.Figure 2{1 bshowsarepresentationofbedshearstressunderlinearandnonlinearwavesusingEquation 2{2 with=1025kg/m3andthecanonicalvalueof0.0025forCf Riviereetal. 2004 .Thistypeofformulationforbedshearstressistypicalofverygeneralmodelsanddoesnotincludeanyinformationaboutphasedierencesbetweenowatthebedandowoutsideoftheboundarylayer.Itisevident,then,thatapplicationofEquation 2{2 forunsteadyowintheboundarylayerwillincorrectlypredictthephaseofsedimentmobilizationwithrespecttotheappliedforcing.Underpurelyhorizontal,regularoscillatoryowthisisoflesssignicanceasthenetsedimenttransportfromcycle-to-cyclewillbezero.Thephasedierenceformobilizationandentrainmentundernonlinearsurfacegravitywaves,however,issignicantasitcandictate 31

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thedirectionofnettransport[see Elgaretal. 2001 and vanderWerfandRibberink 2004 ].AnalternativetothegeneralbedshearstressformulagivenbyEquation 2{2 ,commonlyemployedinphase-averagedorquasi-steadytransportmodels,istoconsiderthemaximumbedshearstressforanindividualwavebasedonawavefrictionfactorfw.Suchamethodwasproposedby Jonsson 1966 :b;max=1 2fwu2b{3whereubisthenearbedorbitalvelocity.Thewavefrictionfactorof Swart 1974 isapplicableforfully-developed,roughturbulentowandisgivenbyfw=exp"5:213ks A0:194)]TJ/F15 11.9552 Tf 11.9552 0 Td[(5:997#{4whereksisthe Nikuradse 1933 roughnesslengthscaleandAistheorbitalsemi-excursionlengthA=ub!.Jonsson's966formulationofthemaximumbedshearstressismorecompletethantheformulationshowninEquation 2{2 ,asitaccountsfortheroughnessofthebottomandusesthenearbedorbitalvelocityasopposedtothefreestreamforcing,butrepresentsabulkapproximationofthestressduringawaveperiodinsteadoftreatingitasatime-dependentquantity.Alargeportionofthetotalbedstressisduetowavepressurevariationsovertheseabed.Accordingto McLean 1991 ,thispartofthestressisnoteectiveinmobilizingsedimentbecausethelength-scaleofthepressurevariationissomuchlargerthantheparticlediameter. Foda 2003 claims,however,thatoutsideofthesurfzonethewavepressureplaysanimportantroleinsedimenttransportprocesses:low-energywavesmildpressuregradientstendtodrivesedimentoshore,whilehigh-energywavessteeppressuregradientsresultinonshoretransport.Thesetwoconictingtheoriesmaybeharmonizedbyconsideringthecontextunderwhichthestatementsweremade.Inanattempttodeterminetheimportanceofwavepressureonbedloadtransport, Foda 32

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2003 treatstheseabedasaviscoelasticuid,notasindividualparticles;therefore,thelength-scalesofthepressurevariationandaneective"lengthoftheactiveviscoelasticuidmaynotbegreatlydierent.Nearshorewaves,dueinlargeparttotheirasymmetry,producesecondaryowsoutsideoftheWBBLthatplananimportantroleinsuspendedsedimenttransport.ConsideringthevelocitytimeseriesforanonlinearStokeswaveinFigure 2{1 a,onecanseethatvelocityunderthewavecrestislargerthanthatunderthetrough.Thisinequalityresultsinanetdisplacementofaparcelofuid,orsuspendedsediment,inthedirectionofwavepropagation,oftenreferredtoasStokesdrift vanderWerf 2004 .Inthenearshoreregion,thisnetdisplacementofshoreward-directeduidisbalanced,overlongperiodsoftime,byaseaward-directedreturnownearthebedoftencalledundertow. Elgaretal. 2001 suggestthatoshoresandbarmigrationislinkedtocross-shoregradientsofundertow,asthesenearbedcurrentsareoftenstrongenoughtomobilizesediment.Furthermore, Gallagheretal. 1998 believethattheintensicationofundertownearasandbarleadstocross-shoregradientsinthesuspendedsedimentuxthatfurtherpromoteoshorebarmigration.Inadditiontothecross-shoredirectedsecondaryows,surfacegravitywavesalsoinduceboundarylayer,orsteady,streaming.VerticalvelocitiesgeneratedintheWBBLresultinadiusiveverticaluxofmomentumdirectedawayfromthehorizontalboundarylayer.Althoughweakcomparedtooscillatoryowoutsideoftheboundarylayer,steadystreamingresultsinanon-zerotime-averagedbedshearstressandhasasignicanteectonsuspendedsedimenttransport Marin 2004 .Thesteaming-induceduxawayfromtheboundarylayerrepresentsanadditionalmechanismfortheentrainmentofsedimentintotheouterow,whereitcanthenbeadvectedbythelocaloweld.2.2.2CurrentsSteadycurrents,suchastheowfoundinriversandhydraulicchannels,areeectiveintransportingsediment,oncemobilized,throughadvection.Relativetothewave 33

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boundarylayer,thecurrentboundarylayerisoftenoneortwoordersofmagnitudelargerasitdevelopsovercomparativelylongdurationsoftime.Anumberofscientistshaveconsideredsedimenttransportincurrents: Meyer-PeterandMuller 1948 Eintstein 1950 SmithandMcLean 1977 Bagnold 1980 ,and vanRijn 1984 .Muchoftheresearchindicatesthatthetransportingcapabilityofasteadyowisproportionaltothestreampower,orvelocity,raisedtosomepower[ Bagnold 1980 ; vanRijn 1993 ]suchthatq=aubwhereqisthevolumetrictransport,andaandbareempiricallydeterminedconstants Longetal. 2007 .2.2.3WavesandCurrentsThenearshoreenvironmentisoftencharacterizedasstochastic,havingmultiplerealizationsofwavefrequency,direction,andheight,aswellascurrentmagnitudeanddirection.Oftentimesnearshorewavestravelovershearingcurrents|acommonexamplebeingashoalingwavepropagatingoveranalongshorecurrent.Thecombinedeectsofwave-andcurrent-inducedvelocities,however,cannotbefoundbyasimplelinearsuperpositionoftheconstituents vanderWerf 2004 .Rather,thepresenceofacurrentmodiesthewave-inducedoweldinanonlinearmannerbyalternativelyaddingandsubtractingfromtheorbitalvelocitiesduringdierentphasesofthewave.Thewavecelerityandlengtharealsoaectedbythecurrent,wherecelerityincreasesdecreasesaswavelengthdecreasesincreasesinresponsetoanopposingfollowingcurrent.Thiseectisdemonstratedforcollinearwavesandcurrentsbyconsideringthemodieddispersionrelationship DeanandDalrymple 1991 :!=Uckw+p gktanhkh{5where!istheangularwavefrequency,Ucisthecurrentvelocity,kwisthewavenumberkw=2=L,gisgravitationalacceleration,andhiswaterdepth.Itisalsoknownthat 34

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thewave-inducedorbitalvelocitiesreducethenearbedcurrentmagnitude vanderWerf 2004 .Thenonlinearinteractionsbetweenthecurrent-andwave-inducedboundarylayershaveasignicanteectonsedimenttransport.Thestirringeectofwavescoupledwithasteadycurrenthasbeenfoundtoincreasethetotalsedimenttransportsignicantly CamenenandLarson 2005 .WaveorbitalvelocityamplitudeA,currentvelocityUc,andtheangleofincidencebetweenthewaveandcurrentcwdictatetheresultantbehaviorofnearbeduidmotion.Whileacurrentowingperpendiculartothedirectionofwavepropagationdoesnotaectwavecelerityorlength,itdoesalterthenetsedimenttransport.Intheirlaboratoryexperiments, Lacyetal. 2006 ndthatforcw=90themaximumtransportissymmetricaboutthecurrentdirectionresultinginwave-phase-averagedsedimenttransportinthedownstreamdirection.Asthewave-currentangledecreasesfromcw=90,themaximuminstantaneousbedshearstressincreasessimilarlyforcwincreasingfrom90to180. Lacyetal. 2006 foundthattheeectofthecurrentwastoincreasethenearbedvelocityamplitude,therebyincreasingtherelativeimportanceofsuspendedsedimenttobedloadtransport.Increasedentrainmentofmobilizedsedimentintothecurrentboundarylayerwasalsoobserved.2.2.4TidesandTidalCurrentsIngeneral,tidesdonothaveasubstantialimpactonsedimenttransportinthenearshoreregion.Tidal-inducedvelocitiesaretypicallyordersofmagnitudesmallerthantheinstantaneous,orbitalvelocitiesproducedunderasurfacegravitywavehavingafrequencymuchhigherthanthetidalfrequency.Therefore,tidal-inducedvelocitiesareoftennotlargeenoughtoinitiatemotionattheseabed,nordotheyplayasignicantroleintheadvectionofsedimenttodierentlocations.Ofcoursetheremaybespecialcircumstances,suchasaveryshallowinletorestuary,wheretidalcurrentsaresucientlystrongtomobilizesedimentinlargequantities MillerandKomar 1980a 35

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Tidalcurrentsgeneratedbyshoalinginternaltidal-wavescanincreasetheshearstresssucientlytodevelopnepheloidlayers,orregionsonhighturbidityneartheseabed.Thesetidalcurrentsareinuentialinsuspendingnear-bottomsedimentsintheabsenceofwind-generatedwavesandcurrentsonthecontinentalshelfandslope Cacchioneetal. 1994 .Oncemobilized,thesuspendedsedimentmaythenbetransportedacrossthecontinentalshelfbyinternalwavesandtidalcurrents.Althoughtheyfoundstorm-generatedwavestobethedominantforcingmechanismoftransportonthecontinentalshelf, Puigetal. 2001 suggestednear-inertialinternalwavesasamechanismforthemaintenanceofanepheloidlayerandsuspendedsedimentduringmilderwaveclimates.2.3TransportIncipientsedimentmotionhasbeenstudiedextensivelyinthelaboratorysincethepioneeringworkof Shields 1936 .Thebasicconceptofincipientmotion,oeredby Shields 1936 ,isthatthesedimentmobilizingforcesofuidliftanddragmustexceedthestabilizingforceofgravityactingontheparticle.TheratioofmobilizingtostabilizingforcesisreferredtoastheShieldsparameterandisgivenby=b s)]TJ/F15 11.9552 Tf 11.9551 0 Td[(1G{6wherebisthebedshearstress,sisthespecicgravityofthesedimentrelativetowater,gisgravity,anddistheparticlediameter.ThedimensionlessShieldsstressaccountsforskinfriction,butnotformdragwhichisthesecondcomponentoftotaldragexperiencedbythebedmaterial.However,skinfrictionisthemoreeectivecomponentofdragonmobilizingsediment McLean 1991 .2.3.1ModesClassicalmodesofsedimenttransportaredividedintothreecategories: 1. washload, 2. bedload,and 36

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3. suspendedload.Thewashloadischaracterizedbyverynesedimentparticles,transportedbytheuid,thatarenotrepresentativeofthebedmaterial FredseandDeigaard 1992 .Estimationofthewashloadisdicultsinceitrequiresknowledgeofsedimentcharacteristicsfromthepointoforigin...alocationpossiblyfarremovedfromtheareaofinterest.Itiscommonplacetodiscardwashloadinthecomputationofthetotalsedimentload,whichisthencomprisedofthebedloadandsuspendedloadmaterial.2.3.1.1BedloadBedloadisthepartofthetotalsedimentloadthatisinalmostcontinuouscontactwiththebed FredseandDeigaard 1992 .Undernon-breakingwaves, Dohmen-JanssenandHanes 2002 ndthatbedloadtransportaccountsfornearly90%ofthetotalsedimentload.Thebedloadmaterialtendstomovealongjustabovethestaticbedinathicknessafewtensofgraindiametersatmost Bagnold 1980 .Themovementofbedloadparticlesischaracterizedbysliding,rolling,and/orsaltation.Slidingoccursinmorelooselypackedbedsandforhighermobilizingstresses,wheretheparticlesslideacrossoneanotherinsheets.Bedloadparticlesmayalsorollovertopoftheirneighborsifthemomentofliftanddragforcesarejustlargeenoughtocounteractthemomentofstabilizingforceofgravityactingontheparticle LuqueandvanBeek 1976 .Saltationoccurswhenaparticleisliberatedfromthebedmaterialbyanintensestressand,onceejectedintotheow,followsamoreorlessballistictrajectoryjustafewgraindiametersabovethebed Bagnold 1973 .Pioneeringworkintheoreticalandempiricalbedloadtransportwasputforthby Eintstein 1950 .Einstein's950empiricalbedloadfunctionwastherstofsuchformulationstoaccountfortherandomnessofowandsubsequenttransportbyequatingthenumberofparticlesdepositederodedperunitareatothenumberofparticlesinmotiononthebed,andontheprobabilitythattheforcebalanceontheparticleissuchthatitwillbedepositederoded.AcentralconceptoftheEinsteinbedloadfunctionis 37

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thatbedloadtransportisproportionaltotheowrateraisedtosomepower.Incontrast,theexperimentsof LuqueandvanBeek 1976 showthattheaveragelengthofindividualparticlesaltationsisaconstant,implyingthattheprobabilityofdepositionisindependentoftheowrate.Suchaproportionality,however,hasalsobeensuggestedbyempiricalrelationships[ Meyer-PeterandMuller 1948 ; Bagnold 1980 ; RibberinkandAl-Salem 1990 ; Nielsen 1992 ; Ribberink 1998 ; CamenenandLarson 2005 ],andbyanalyticalexpressionsderivedfromphysicalconceptsaswell[ Bagnold 1966 ; Bowen 1980 ; Bailard 1981 ; KovacsandParker 1994 ]. Eintstein 1950 postulatedthatafunctionalrelationshipexistedbetweenthebedloadtransportrateandtheparticlebuoyancy,suchthatanondimensionalbedloaddischargemaydenedasb=Qb p s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd3{7whereQbisthevolumetricrateofbedloadtransportperunitwidth,sisthespecicgravityoftheparticle,gisgravitationalacceleration,anddistheparticlediameter.Anumberofphysicallaboratoryexperimentshavebeenconductedtodeterminethefunctionalrelationshipbetweenuidforcingandresultingtransport,representedinEquation 2{7 byb.Thesefunctionalrelationshipsaretermedpowerlaws"sincethetransportrateisproportionaltosomemeasureoftheuidforcingvelocityorstressraisedtoapower.Anumberofpowerlaws,invariousforms,havebeensuggestedfortransportbycurrents[e.g. Meyer-PeterandMuller 1948 ; Nielsen 1992 ; Ribberink 1998 ; CamenenandLarson 2005 ]andbywaves[e.g. MadsenandGrant 1976 ; BailardandInman 1981 ; DibajniaandWatanabe 1992 ; Soulsby 1997 ; Ribberink 1998 ; CamenenandLarson 2005 ].MostoftheserelationshipsarebasedonanexceedanceofthecriticalShieldsstressandtaketheform:b=a)]TJ/F24 11.9552 Tf 11.9552 0 Td[(crb{8 38

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whereaissomeconstantofproportionality,bisanempirically-derivedexponentprovidingthebesttofEquation 2{8 toasetofdata,andandcraretheinstantaneousandcriticalShieldsparameters,respectively.VariousexamplesofbedloadpowerlawsaregiveinTable 2{1 .Shields'descriptionofincipientsedimentmotionisbasedontheprinciplethatthereisabsolutelynomotionwhentheappliedstressisbelowthecriticalthreshold.Therefore,when0,contradictingShields'conceptofacriticalthresholdforsedimentmotion.Initialtreatmentofbedloadtransporttreatedmovingparticlesinabulkfashion,withthemobilizingstressbeingtransmittedbytheuidtothebulk.Bythedenitionofbedloadgivenabove,however,itwouldseemthatsomestressistransmittedbetweenparticlessincetheyareinalmostconstantcontactwithoneanother. Bagnold 1954 addressedthissubjectbyconsideringtheintergranularstressestransmittedbetweenparticlesunderuidshearandfoundasubstantialradialdispersivepressureexertedbetweentheparticles.Thisdispersivepressureisdistributedinsuchamannerthatthemovinggrainsareinequilibriumwiththeirgravitationalforce.Asshearstressisappliedtotheparticlematrix,dilationoccursandthedispersivepressuredecreasesastheparticlesmovefurtherfromeachother.Thetotalshearstresswithinthedispersedparticlematrixisthenacombinationofintergranularuidandparticlestressessuchthat=f+g{10 39

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wherefisthestresstransmittedbytheuidwithinthepores,andgisthetangentialdispersivestressthatrepresentsmomentumexchangedduetoparticle-particleinteractions FredseandDeigaard 1992 .Theparticlecollisionsthatcontributetogsubtractmomentumfrommovingsedimentparticlesthatmustbereplacedbytheuidforcing.Thismomentumsubtractionresultsinanapparentroughnessgreaterthanwouldbeexpectedforastaticbed McLean 1991 .Throughaseriesoflaboratoryexperimentsofparticletransportinaclosedrectangularowchannel, LuqueandvanBeek 1976 foundthatbedloadreducesthemaximumuidshearstressatthebedlevelbyexertinganaveragereactionforceonthesurroundinguid.AccordingtoEquation 2{10 ,areductioninuidstresssuggeststhattheintergranularstressesmustincreasetomaintainthetotalstress.Athighbedloadtransportrates,then,amajorityofthetotalstressisexertedbyparticlecollisionswhiletheuidstressesremainsmall LuqueandvanBeek 1976 .AschematicofthedistributionofuidandgranularstressesforanidealizedopenchannelowisshowninFigure 2{2 .2.3.1.2SuspendedLoadBesidesbedload,theotherconstituentofthetotalsedimentloadisthesuspendedload.Suspendedloadmaybedenedasmaterialadvectedbytheuid,maintainedinsuspensionbyturbulence,andhavingverylittlecontactwiththebed.Inthisrespectthesuspendedloadistransportedbymechanismssimilartothoseresponsibleforthetransportofwashload,butitsoriginsdier.Whilewashloadconsistsofparticlesnotcharacteristicofbedmaterial,suspendedloadsedimentisentraineddirectlyfromthebedmaterial.Hence,someknowledgeofthebedmaterialmaybeutilizedinthecharacterizationofsuspendedsedimentproperties.Suspendedloadplaysanimportantroleinbothsedimenttransportandhydrodynamicprocesses.Onceentrainedfromthebedmaterial,thesesedimentscanbeadvectedlocallybynearbedwaveorbitalmotion,andregionallybypersistentcross-shoreand 40

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alongshorecurrents.Asmostnaturalsedimentismulti-modal|havinganaturallyoccurringdistributionofgrainsizes|theentrainmentofsmallerparticlesnearthesurfacecanleavebehindalayeroflargerparticles. Bagnold 1980 suggeststhatthepresenceofalargergrainsizereducesthetransportratethroughaphenomenoncalledarmoring.Totalloadtransportratesmayalsobereducedthroughtheeectsofthesuspendedloadonthenearbedoweld. McLean 1991 suggestedthatdensitystraticationbysuspendedsedimentdampsturbulence,therebylimitingtheabilityofowtotransportmassandmomentumvertically.Asaresult,theupwarddispersionofsuspendedsedimentmaintained,againstgravity,byrandomeddycurrentsisreduced Bagnold 1973 ,theabilityoftheowtokeepsedimentinsuspensionisimpaired,andtheentrainmentcapacityoftheowdecreases McLean 1991 .Suspensionanddensitystraticationmayalsomodifythevelocityprole,whichindirectlyalterstheboundaryroughnessthroughasubsequentchangeinbedmorphology[ Coleman 1981 ; ParkerandColeman 1985 ].Althoughsuspendedsedimentmaysuppressturbulence, Nakatoetal. 1977 foundthatboththemeananductuatingverticaluidvelocitywereampliedbysuspension.Throughaseriesoflaboratoryexperiments, Coleman 1981 demonstratedtheabilityofsuspendedsedimenttoreduceturbulence,aswellasmodifytheshapeofthevelocityprole.Thedistributionofvelocitythroughoutthewatercolumnisofsignicancefornetsandtransport,asthesuspendedparticlesareadvectedbytheuidvelocity.Thisisespeciallytruefortransportoverarippledbedwherecoherentmotions,likeorganizedvortices,dictateentrainment,suspension,andadvectionofsignicantquantitiesofsediment vanderWerfetal. 2006 LuqueandvanBeek 1976 foundthattheaveragetransportvelocityofsuspendedparticlesjustabovethebedwasapproximatelyequaltotheturbulentuidvelocityminusaconstant,whichisspeculatedtobeaninertialeect.Knowledgeofthevelocityprole,then,couldprovidesomeindicationofsuspendedloadconcentrations.Suchinformationisespeciallyusefulinthenearbedregion,whichplaysanimportantroleinthedeterminationofnetsandtransportowingtothehigh 41

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concentrationsofsedimentfoundthere vanderWerfetal. 2006 .Acommonapproachforestimatingthenearbedvelocitydistribution,withintheboundarylayer,utilizessomefunctionalformoftheclassicalPrandtl-Karmanlogarithmiclawofthewall vonKarman 1930 ,orloglaw,givenbyu u=1 lnz zo+B{11whereuisuidvelocity,uisfrictionvelocity,isthevonKarmanconstant,zistheverticaldistancefromtheboundarybed,andzoisareferenceheight.WhilevariousestimatesofandBappearinliteraturepertainingtoboundarylayerows, Pope 2000 reportsthatgenerallyallarewithinvepercentof=0:41;B=5:2:{12Inthedeterminationofvelocityprolescontainingsuspendedsediment,somehavesuggestedthatdecreaseswithincreasingsuspendedsedimentconcentration[ Raudkivi 1967 ; Graf 1971 ; Yalin 1977 ].AreductionofinEquation 2{11 suggeststhattheratiooftheuidvelocitytofrictionvelocityincreases.Theexperimentsof Coleman 1981 ,however,demonstratethatisindependentofsedimentconcentration.ValuesforthereferenceheightinEquation 2{11 zohavecommonlybeenrelatedtothesedimentgraindiameter GrantandMadsen 1982 .Anestimateofzoforintensesedimenttransportoveratbedsinoscillatoryows,determinedfromthelabdataof Carstensetal. 1969 ,isapproximately7)]TJ/F15 11.9552 Tf 12.2887 0 Td[(8graindiameters.Thisconceptofanequivalentsandgrainroughnessheightsevolvedfromtheworkof Nikuradse 1933 .EntrainmentofbedparticlesintotheowmaybecomputedinamannersimilartothatusedforbedloaddischargeEquation 2{7 .Following Eintstein 1950 ,adimensionlessentrainmentparameterpisgivenbyp=E sp s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd50{13 42

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whereEisthevolumetricrateofentrainmentofbedmaterial,sisthesedimentdensity,sisthesedimentspecicgravity,gisgravitationalacceleration,andd50isamediangraindiameter.Typically,oneusesanavailableexpressionforpandthencomputestherateofentrainmentE,orpick-uprate.Thismethodologyhasbeenusedformodelingsuspendedloadtransportinparticletrajectorymodels Hansenetal. 1991 aswellasinmoresophisticatedhydrodynamicmodels ZedlerandStreet 2001 .Anumberofrelationshipsforphavebeenempiricallyderivedfromlaboratoryexperiments,andafewcommonrelationshipsaregiveninTable 2{2 .2.3.2RegimesSedimenttransportisoftendividedintothefollowingthreeregimes vanderWerf 2004 : 1. bedload, 2. ripple,and 3. sheetow.Cleardenitionsoftheseregimesintheliterature,andtheconstitutiverelationshipsthatmakethemunique,aretenuousatbest.Thebedloadregimeischaracterizedbybedloadtransportofsandparticlesinalayernotmorethanafewgraindiametersthick Bagnold 1980 ,andoccurringoveranotherwisehorizontalbed.Accordingto Bagnold 1946 thistransportpersistsfromtheonsetofmotiontoavalueabouttwicethecriticalShieldsparametercr<<2cr.Forreference,apracticalrangeofcrforthecoastalenvironmentvariesbetween0:03and0:06 vanderWerf 2004 .Althoughthedistinctionwasnotmadeatthetime,earlyexperimentsonbedformdevelopmentby Darwin 1883 and Ayrton 1904 providedqualitativeinformationonthecharacteristicsofincipientmotionandtransportinthebedloadregime. Darwin 1883 suggestedthatparticlestransportedinthebedloadregime,byoscillatoryow,wouldaggregateinregionsonincreasedfriction.Similarly,theoscillatingtankexperimentsof Ayrton 1904 demonstratedthatparticlestransportedonaatbedwouldcongregate 43

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inaspeciclocation,thatbeingthelocationofastandingwavenodewheretheverticalhorizontalvelocityisatamaximumminimum.Theoscillatingtrayexperimentsof Bagnold 1946 furtherreinforcethebedloadregimeconcept,wheresandparticlestransportedinathin,concentratedlayeroveraatbedwouldaggregateinareasofhigherfriction,andlowertangentialforcing.Onceasucientnumberofparticleshavegatheredtoinduceowseparation,thetransportbehaviorismodiedduetotheadversepressuregradientformedintheleeshelteredsideoftheperturbation.Thisbehaviorwasconsistentintheexperimentsof Darwin 1883 Ayrton 1904 ,and Bagnold 1946 ,eventhoughallthreeutilizeddierenttestingdevices.Accordingto Bagnold 1946 ,thegrowthofsubstantialvorticesintheleeofabedformoccurswhentheShieldsparameterexceedstwicethecriticalvalue,or>2cr.Asopposedtotheturbulentdiusionofmomentumthattakesplaceaboveahorizontalbed, vanderWerfetal. 2006 suggestthatmomentumtransferandsedimenttransportaboverippledbedsaredominatedbyorganizedvortexsheddinginanearbedlayerapproximatelytwicetherippleheight.Coherentmotionsbreakdownandarereplacedbyrandomturbulenceabovethislayer DaviesandVillaret 1997 .Transportmodesintherippleregimeincludebothbedloadandsuspendedload.Theparticularmodeoftransportisdeterminedbytheratioofparticlesettlingvelocitytofrictionvelocity:wso u{14wherewsoistheparticlesettlingfallvelocity,anduisthefrictionvelocityu=p b=.TheprinciplemodeofsedimenttransportshiftsfromsuspendedloadtobedloadwhenEquation 2{14 isequaltoabout0.07 Nielsen 1979 .Thesheddingofvorticesfromripplecrestsinoscillatoryow,aswellastheirsubsequentejectionintotheouterow,dictatesthephaseandquantityofsedimententrainmentanddepositionintherippleregime. HuangandDong 2002 suggestedthatrippledbedsinduceanearbedcurrentdirectedoppositefromthewavepropagationdirection.Furthermore, vanderWerfand 44

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Ribberink 2004 foundthatripple-inducedphasedierencesbetweenpeaksuspendedsedimentconcentrationsandpeakuidvelocitiesleadtonettransportratesdirectedseawardundersurfacegravitywaves.Thissuggeststhatthetiming,orphase,ofsedimententrainmentintherippleregimeisofparticularimportancefordeterminingtransportdirectionunderasymmetricowconditions.Thesheetowregimeischaracterizedbyarelativelythin10)]TJ/F15 11.9552 Tf 12.7985 0 Td[(100dlayerofparticlestransportedinahighlyconcentratedsuspension.ThisregimepersistsforvaluesoftheShieldsparametermuchgreaterthanthecriticalthreshold0:8)]TJ/F15 11.9552 Tf 12.1308 0 Td[(1:0 vanderWerf 2004 .ForverylargevaluesofthemaximumShieldsparameter10timesaslarge,sheetowwashesoutripplesandplanesothebed LiandAmos 1999 .Suchconditionsexistundersurfzonewavesastransientpressuregradientsresultineetingepisodesofintensesheetowtransport DrakeandCalantoni 2001 .Accordingto DongandZhang 1999 ,thesheetowregimeisofparticularimportanceduetothehighconcentrationsandlargetransportratesfoundwithin.Dominantforcingmechanismswithinthesheetowlayerincludeintergranularandturbulentstresses,aswellastheinteractionforcesofuidlift,drag,andinertiabetweentheuidandsedimentparticles DongandZhang 1999 .Inthelowerpartofthesheetowlayer,sedimentconcentrationsareveryhighandthestressesgeneratedbyparticlecollisionssignicantlyaectthevelocityofuidandsediment DongandZhang 1999 .Throughaseriesoflaboratoryexperimentsonsheetowtransport, Ribberinketal. 1994 suggestthatathree-layersystemexistswithanactivepick-uplayernearthebed,asheetowlayer,andasuspensionlayerintheouterow. Ribberinketal. 1994 foundthatamajorityofthehorizontaluxeswereconcentratedinthesheetowandpick-uplayers,andanincreasingphaselagbetweenuidvelocityandsedimentconcentrationwithincreasingelevationinthesuspensionlayer.Almostnophaselagbetweenuidvelocityandconcentrationwasfoundtoexistinthesheetowandpick-uplayers,wherenetsedimenttransportrateswereproportionaltothe 45

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third-ordervelocitymomenthu3iandtheconcentrationprolewaspredictedwellbyapowerlawformulation.2.4TurbulenceIntherippledbedregime,nearbedmomentumtransferisdominatedbyvortexsheddingratherthanrandomturbulenceinoscillatoryforcingconditions[ Sleath 1991 ; Nielsen 1992 ; MalarkeyandDavies 2004 ].Flowseparationatripplecrestsrepresentsasignicantmechanismfortheproductionofturbulenceintheboundarylayerduringthephaseofmaximumow Barretal. 2004 .Turbulenceplaysanimportantroleinthehydrodynamicsofthebottomboundarylayerandaectsthesuspensionandtransportofsediment.Turbulenteddiesmaintainanupwarddispersionofsuspendedsedimentagainstthecounteractingforceofgravity Bagnold 1973 ,andturbulentstressesserveasadditionalforcinginthesheetowregime DongandZhang 1999 .Thesubsequentstraticationofsuspendedsediment,inturn,aectsthebackgroundowbymodifyingthevelocityprole Coleman 1981 ,dampingturbulence McLean 1991 ,andincreasingfriction ParkerandColeman 1985 .Modelingtheeectsoftheparticlephaseonuidturbulenceisdicult.Mostnumericalmodelscannotaccuratelysimulatesediment-uidinteractionsandcoherentmotionsintheturbulentboundarylayer ZedlerandStreet 2001 .Closureofthenonlinearmomentumequations,asitpertainstoturbulence,hasbeenapproachedanumberofwaysrangingfromthemoresimpleone-dimensionaleddyviscosityapproach DaviesandThorne 2002 tothemorecomplexlargeeddysimulation WangandSquires 1996 .Oneoptionforbypassingtheexplicittreatmentofturbulenceclosurethroughone-andtwo-equationmodels,oreddyviscositymodels,istoperformdirectnumericalsimulationsoftheparticle-uidinteractions[ ElghobashiandTruesdell 1992 ; DrakeandCalantoni 2001 ].DirectnumericalsimulationsofhighReynoldsnumbersows,however,arestillcomputationallyprohibitiveforlargespatialscalesandlongdurations. 46

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2.4.1DynamicsTurbulenceandsedimenttransportareinterconnectedinthesensethattheyaectoneanother.Perhapsthersttoincorporatetheroleofturbulenceinsedimenttransportformulations, Eintstein 1950 consideredtheprobabilityofparticledepositionanderosionbasedontherandomnessofnearbeduidvelocity. Bagnold 1954 furtherconsideredtheroleofturbulenceintheuid-particlematrixthroughlaboratoryexperimentsonthedispersionofspheresinashearingow.Intheclearuid, Bagnold 1954 foundthemajorityofbedshearstresstobeduealmostwhollytoturbulence,whereasincreasingconcentrationsofsedimentsuppressedturbulence.Theseconceptswereconrmedthroughadditionalexperimentsby LuqueandvanBeek 1976 Bagnold 1973 latersuggestedthatthedissipationofturbulenceinuid-particleowsoccurredthroughthedevelopmentandmaintenanceofameanvelocityequalinmagnitude,andoppositeindirection,totheparticlesettlingvelocity.Theroleofturbulenceinconcentratedparticleow,then,istomaintainthesedimentinsuspension:workperformedontheparticlesrepresentstheenergydissipatedbytheturbulence ParkerandColeman 1985 .Subsequently,thetransportratesofsolidsinsuspensionislimitedbytherateofgenerationofturbulentkineticenergy Bagnold 1973 .Intherippledbedandsheetowtransportregimes,intenseepisodesofsuspensioncanleadtodensitystratication.Straticationservesasanadditionalmechanismforthesuppressionofturbulencebylimitingtheabilityoftheowtotransportmomentumandparticlesvertically McLean 1991 .Thisoftenresultsinhighersuspendedsedimentconcentrationsinthenearbedregion Ribberinketal. 1994 ,whichmaylimittheentrainmentcapacityoftheow McLean 1991 .Asturbulenceisdamped,dragdecreasesasdoestherelativevelocitymaintainingparticlesinsuspension,therebyleadingtoparticlesettling McLean 1991 47

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2.4.2ClosureTurbulenceclosuretypicallyreferstothesolutionofanadditionalequationthatprovidesanestimateofturbulentstress.Variousclosuremethodshavebeenpresentedintheliterature,wheremostseektosolvefortheturbulentReynoldsstresses u0iu0jintheReynolds-averagedmomentumequations.Themostbasic,andwidelyused,conceptwasrstintroducedbyBoussinesqwhopostulatedthatturbulentstresseswereproportionaltomeanturbulentvelocitygradientssuchthat)]TJET1 0 0 1 238.9002 538.2004 cmq[]0 d0 J0.4782 w0 0.2391 m21.0883 0.2391 lSQ1 0 0 1 -238.9002 -538.2004 cmBT/F24 11.9552 Tf 238.9002 528.2201 Td[(u0iu0j=t@ui @xj+@uj @xi)]TJ/F15 11.9552 Tf 13.1506 8.0877 Td[(2 3k{15wheretisaturbulenteddyviscosity,andkistheturbulentkineticenergywhichisequaltoone-halfthetraceoftheturbulentReynolds-stresstensor.Equation 2{15 isoftenreferredtoasthegeneralizededdyviscosityconcept vanderWerf 2004 .Therearethreebasictypesofeddyviscositymodels: 1. zero-equationmodels; 2. one-equationmodels;and 3. two-equationmodels.Zero-equationmodelsincludebothtime-invariantandtime-dependentformulations,mixinglengthmodels,andvonKarmanmomentumintegralmethods vanderWerf 2004 .One-equationmodelsseektosolvefortheturbulentkineticenergyk,suchasin DaviesandLi 1997 .Populartwo-equationclosuremodelsincludek)]TJ/F24 11.9552 Tf 11.9619 0 Td[(andk)]TJ/F24 11.9552 Tf 11.9619 0 Td[(!,whereand!areturbulentdissipationrates[see Wilcox 1998 ; Andersen 1999 ; AndersenandFredse 1999 ].2.4.2.1RANSModelsReynolds-AveragedNavier-StokesRANSmodelssolvetheReynolds-averagedmomentumequations,wheredependentvariableslikevelocityandpressurearedecomposedintomeananductuatingcomponents.Two-equationclosuremodels,suchask)]TJ/F24 11.9552 Tf 12.1095 0 Td[(andk)]TJ/F24 11.9552 Tf 12.1095 0 Td[(!models,areoftenemployedintheRANSapproach.Withrespect 48

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totheirtwo-dimensionalsedimenttransportmodel, AndersenandFredse 1999 foundthek)]TJ/F24 11.9552 Tf 12.2059 0 Td[(!performedwellinestimatingturbulenceaboveripplesinunsteadyow. ChangandScotti 2004 comparedsimulationsofturbulentowoverstationaryripplesusingthek)]TJ/F24 11.9552 Tf 11.986 0 Td[(!closuremodelof Wilcox 1998 andlargeeddysimulations,andfounddecienciesintheRANSapproach.RANSmodelsrequireaveragingprocessesthatcanbiasthehighlyunsteady,time-dependentowthatoccursinanoscillatoryboundarylayerabovearippledbed.2.4.2.2LargeEddySimulationsWhileRANSmodelsseektoseparatedependentquantitiesintomeananductuatingcomponents,thebasicconceptoflargeeddysimulationisthatvariablesaredecomposedintoresolvedlargeandlteredsmallmotions.TheLESapproachassumesthatthecomputationalmeshissucientlyneenoughtoresolvethelargerturbulentmotions,andyetacknowledgesthefactthatsmallereddieswillbelteredbythegridnearphysicalboundaries,unlesstheresolutionisneartheviscouslengthscale Pope 2000 .InhighReynoldsnumbersows,theviscouslengthscaleisoftenprohibitivelysmalltoresolveinthecomputationalmeshsowallmodelsaretypicallyusedtocompensateforthelackofresolutionthere.Particularlyrelevantlargeeddysimulationsofturbulentow,withrespecttosedimenttransportandbedforms,havebeensuccessfullyperformedby WangandSquires 1996 CalhounandStreet 2001 ZedlerandStreet 2001 Barretal. 2004 ChangandScotti 2004 andothers.2.4.2.3DirectNumericalSimulationsItistheoreticallypossibletobypasstheturbulenceclosureproblemaltogetherbyperformingdirectnumericalsimulationsDNSofturbulentow.TheNavier-Stokesequationsfullydescribeallquantitiesofthetime-dependentoweld,inallthreedimensions,ifthecomputationalmeshiscapableofresolvingallscalesofmotion.Untilrecently,theapplicationofsuchmodelswaslimitedtolowormoderateReynoldsnumberowsoversmoothboundaries vanderWerf 2004 .Technologicaladvancementsin 49

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micro-processorshavemadeitviabletoperformDNSsimulationsathighReynoldsnumbersovercomplexphysicalboundaries.Recentexamplesofthree-dimensionaldirectnumericalsimulationsofmoderateandhighReynoldsnumberowsoverstationaryripplesarefoundin Scanduraetal. 2000 and Barretal. 2004 .Detailedthree-dimensionalDNSmodelingofuid-particle,andparticle-particle,interactionshasalsobeenperformedby ElghobashiandTruesdell 1992 and DrakeandCalantoni 2001 .2.5ModelsAnumberofsedimenttransportmodelsexistinpublishedliterature,rangingfromsimpleempiricalmodelstomorecomplexthree-dimensionalunsteadymodels.Thesemodelsfallintooneoffourcategoriesbasedontheirtreatmentoftransportandhydrodynamics:time-averaged,quasi-steady,semi-unsteady,andfullyunsteady.Eachhavebenetsandlimitations,someofwhichwillbediscussedinthefollowingsections.Additionally,mostcoupledmodelsofuidhydrodynamicsandsedimenttransportrequireparametrizationofeithertheuidorparticlephase,butsomeexceptionsdoexist.Thefollowingsectionsprovideabriefoverviewoftypicalmodeltypes,theirassumptions,andalsotheirshortcomings.2.5.1TypesThemostcommonlyusedsedimenttransportmodelsmaybeclassiedaseithertime-averaged,quasi-steady,semi-unsteady,orfullyunsteady.2.5.1.1Time-AveragedWhilesimple,time-averagedmodelsarenotnecessarilyconsideredrobust.Thesemodelsoftenrelyonowstatisticsthathavebeenaveragedoverdurationsmuchlongerthanwouldbeconsideredrelevantinthecoastalenvironment,suchasanindividualwaveperiod.Time-averagedsuspendedloadtransportmaybecomputedusingaC)]TJ/F24 11.9552 Tf 12.089 0 Td[(uintegralapproach: Qs=Zh0 Czuzdz{16 50

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wherethedependentvariablesofsedimentconcentration Canduidvelocityuareaveragedoveroneormanywaveperiods vanderWerf 2004 .Apopulartime-averagedmodelforcurrent-relatedsuspendedloadandbedloadtransportisgivenby Bijker 1971 .Withrespecttoapplicationsinthecoastalenvironment,themajordisadvantageofanytime-averagedsedimenttransportmodelisthatthewave-relatedoscillatorycomponentoftransportisintegratedoutofthesolution.2.5.1.2Quasi-SteadyQuasi-steadymodelsseektoaccountforboththewave-andcurrent-relatedsedimenttransportcomponentsbyequatingtheinstantaneoustransporttotheinstantaneousnearbedoweldraisedtosomepower.Implementationofquasi-steadytransportmodels,however,arenotmeanttoprovideintra-wavestatisticsofsedimenttransport.Rather,thequasi-steadytransportiscomputedas Qs/1 TZT0u3tdt{17where Qsisthewave-averagedsedimenttransportrate,Tisthewaveperiod,andutisanexpressionofthewaveandcurrentvelocityeld.Consideringtheexamplegivenby vanderWerf 2004 ,thevelocitytime-seriesutforasecond-orderStokeswavesuperimposedonasteadycurrentisgivenbyut=u+^u1cos!t+^u2cos!t{18whereuisthecurrentvelocityand^u1and^u2arethewavevelocityamplitudes.SubstitutingEquation 2{18 intoEquation 2{17 ,andintegratingoverthewaveperiod,givesthewave-averagedtransportrateasafunctionofbothwaveandcurrentvelocities: Qs/u3+3 2u^u21+3 2u^u22+3 4^u21^u2:{19Anexampleofawidely-usedquasi-steadytransportmodelisgivenbythetotalloadformulationof Bailard 1981 ,whichisanextensionoftheearlierenergeticsmodelof 51

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Bagnold 1966 .Usefulquasi-steadyandunsteadybedloadformulationsarealsoprovidedby Ribberink 1998 .Duetothetime-dependentprocessofvortexshedding,however,quasi-steadymodelsarenotsucientlyrobusttoestimatethesuspendedsedimenttransportaboverippledbeds vanderWerfetal. 2006 .2.5.1.3Semi-UnsteadyAnimplicitassumptionintime-averagedandquasi-steadytransportmodelsisthatsedimenttransportisalwaysinphasewithwaveforcing.Thisisnotalwaysthecase,andinertialeectsofboththeboundarylayerowandtheweightofthesedimentitselfresultinphasedierencesbetweenappliedforcingandsubsequenttransport[ Parker 1975 ; LuqueandvanBeek 1976 ].Semi-unsteadymodelsattempttoaccountforthesephasedierenceseitherthroughparametrizations,orbyconsideringtransportovertwosuccessivewavehalf-cycles. Nielsen 1988 developedasimplegrabanddump"modelofwave-relatedsedimenttransportbyassumingthattransportoverripplesoccursthroughtwodistinctmechanisms:rst,sedimentistransportedalongtheripplefacestossandcarriedoverthecrestwhilealeevortexentrainssedimentfromthetroughandsecond,liberatedsandisliftedbythevortexandsubsequentlyadvectedbythemainow.Sincethisprocesshappenstwiceduringeachwavecycle, Nielsen 1988 accountsfortheforwardandbackwardtransportseparatelythroughdistinctentrainmentcoecientsbasedontheinstantaneousvelocityamplitudes.Inthismanner,thegrabanddump"modeliscapableofaccountingforwaveasymmetry. Nielsen 1988 reportsthatthemodelprovidesreasonableestimatesoftransportforbothneandcoarsesands,resolvesthephaseofmaximumtransportevents,andisextremelypracticalasitrequiresonlyanestimateofanearbedreferenceconcentration.Theconceptofdividingtransportintosuccessivewavehalf-cyclestoaccountforphaselagswaspopularizedbythemodelof DibajniaandWatanabe 1992 .Insteadofcomputinganaveragetransportrateovertheentirewaveperiod,thehalf-cyclemodel 52

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solvesforthenettransportrateovertheentirewaveperiod.Thisisdonebytakingthelinearsumofthetransportratesinthepositiversthalfandnegativesecondhalfphasesofawave.Thehalf-cyclemodelconsistsoftwoelementsoftransportforeachhalf-cycle:sedimententrainedandtransportedwithinthesamehalf-cycle,andsedimententrainedduringtheprevioushalf-cycleandtransportedduringthefollowinghalf-cycle vanderWerf 2004 .Thehalf-cyclemodelisconsideredtobesemi-unsteady,becausewhileitisunsteadyovertheentirewaveperiod,itissteadyforeachhalf-cycle.2.5.1.4UnsteadyUnsteadymodelsprovideestimatesoftime-dependentsedimenttransport,typicallythroughintegrationwithanunsteadyhydrodynamicsmodel.Time-dependentvaluesofuidvelocityandsedimentconcentrationmaybesubstitutedintoaC)]TJ/F24 11.9552 Tf 12.7541 0 Td[(uintegralapproach,similartoEquation 2{16 ,inordertomodeltheunsteadysuspendedsedimenttransportrate:Qst=Zh0Cz;tuz;tdz:{20Mostquasi-steadybedloadtransportmodels[e.g. Meyer-PeterandMuller 1948 ; Ribberink 1998 ]mayalsobeadaptedforunsteadyusebyconsideringatime-dependentShieldsparametert,basedontheinstantaneousbedshearstress,intheformulation MadsenandGrant 1976 .Forexample,thepopular Meyer-PeterandMuller 1948 bedloadtransportequationmaybecastinanunsteadyform:bt=8t)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr3=2{21wherebtisthetime-dependent,dimensionlessbedloadfunctionof Eintstein 1950 .Variousreferencestothe Meyer-PeterandMuller 1948 bedloadequationhavebeenmadeintheliteratureandsomehavesuggestedthatthecoecientandexponent=2varysomewhatdependingonowandsedimentcharacteristics.Althoughthefunctionalformwasderivedfromlaboratorydataonsteadyow, Madsen 1991 suggeststhatthisbedloadformulaperformswellforcoastalapplications,too.Dependingonthemodeling 53

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approach,theseunsteadyapproximationsofsuspendedloadandbedloadtransportratesmayeitherbeaveragedoverindividualwaveperiodstoprovideestimatesofnettransport vanderWerf 2004 ,orintegratedoverfractionaltimestepstoprovidediscreterepresentationsofinstantaneoustransport.2.5.2IntegratedApproachesForunsteadymodelsinparticular,thereareavarietyofintegrateduid-sedimenttransportmodelsdescribedinexistingliterature.Thecouplingofhydrodynamicsandsedimenttransportoftenrequireseitherspecialtreatmentoftheuidorparametrizationsofthesedimentphase.Somemodelssolvethedetailedhydrodynamicsexplicitly,oftenthroughDNS ElghobashiandTruesdell 1992 orusingLESturbulenceclosure WangandSquires 1996 ,andthenincorporateone-waycouplingbymodelingtheparticlemomentumequations MaxeyandRiley 1983 .Otherschoosetomodeltheparticulatephasebysolvinganadvection-diusionequationforthesuspendedsedimentconcentration ZedlerandStreet 2001 ,alongwithasedimentpick-upfunction vanRijn 1984 andappropriateboundaryconditions Nielsenetal. 1978 .Aslightlymoredetailedapproachforsimulatinguid-particleowinvolvesmodelingthewaterandsedimentasacontinuum,suchasin Phillipsetal. 1992 .Anadvantageofthecontinuumapproachisthattwo-waycouplingofmassandmomentumisimplicitintheformulation.Thecontinuummodelhasthreemajorcomponents:mixturemomentumequations,anadvection-diusionequationforthemixtureconcentration,andoftentimesasophisticateddiusivitymodel.Theapplicationofcontinuummodelstolargecoastalapplicationsremainsarelativelynewareaofresearch.Explicittreatmentofthree-dimensionalhydrodynamics,especiallyforDNSandLESmodels,canbetime-consuming.Unsteadyhydrodynamicmodels,however,arenecessaryforaccuratelyresolvingthetime-dependentcoherentmotionsthatdevelopintheboundarylayerabovearippledbed.Sincethevortexmotionsdetermineamajorityofentrainmentandtransportintherippledbedregime Sleath 1991 ; Nielsen 1992 54

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itispossibletoparametrizethebulkhydrodynamicsbyconsideringthetransportofvorticityintheoweldratherthanthemomentum.Suchanapproachisdescribedin MalarkeyandDavies 2002 ,whereadiscretevortexmodelDVMisusedtosimulatethetime-dependentvorticityeldabovearippledbedinoscillatoryow. MalarkeyandDavies 2002 presentresultsofsimulationsusingasimpleinviscidDVM,andaslightlymoreadvancedcloud-in-cell"CICmodelthatconsidersadistributionofvorticitypointvalueswithintheoverallvortex.Thesemodelsarecapableofestimatingreasonablevaluesofvorticity,butoftenfailtoaccuratelycapturethephaseofvortexformationandejection.ComparedtoRANSmodels,thediscretevortexmodelsarebetter-suitedforresolvingsharpgradientsinthevorticityeldsincetheydonotsuerfromthenumericaldiusionrelatedtoadvection MalarkeyandDavies 2004 .Analternativemethodforsimulatingcoherentmotionsintheboundarylayerwaspresentedby DaviesandVillaret 1997 ,whosuggestedthatvortexsheddingcouldbemodeledasaconvective"stressrepresentedbyatime-dependent,height-invarianteddyviscosity.Theconvectiveeddyviscosity"relatestheconvectiveshearstresstothevelocitygradient.Examplesofone-dimensionalconvectiveeddyviscositymodelsaregivenby DaviesandThorne 2002 and MalarkeyandDavies 2004 .Modelsimulationsby MalarkeyandDavies 2004 suggestthattheconvectiveeddyviscosityapproachisvalidinathicknessofaboutoneortworippleheightsabovethemeanbedwherethecoherentmotionsexistinaconvectivelayer.Moreadvancedmodelingtechniquesexistforthesimulationofmulti-phaseowsincoastalapplications.Forexample, Calantoni 2002 presentsadiscreteparticlemodelforbedloadtransportinthesurfzone.Thediscreteparticlemodelsimulatesthedynamicsoftheuidowandaccountsforthekinematicsofeachparticleonanindividualbasis.Therefore,termsthataretypicallyparametrizedinothermodels,suchasgranularstressesandmomentumtransferthroughelasticcollisions,areaccountedforexplicitlythroughrstprinciples.Solvingmomentumequationsfortheuid,aswellaseach 55

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particle,iscomputationallyexpensiveandapplicationsofdiscreteparticlemodelsarecurrentlyreservedforresearchproblems.Analternativeapproachistomodelallofthesedimentparticlesasaseparatephase,muchastheuidphaseismodeledasahomogeneousmedium.Followingthismethodology, DongandZhang 1999 developedatwo-phasemodelofuidandsedimentowinoscillatorysheetow.Themodelsolvesthecontinuityandlinearizedmomentumequationsforuidandsediment.Turbulentandintergranularstressesareincorporatedinthesolutionalgorithm,therebyprovidingacompletedescriptionoftheinteractionforcesbetweentheuidandsedimentphases.2.5.3ShortcomingsAnalyticalmodelsofsedimenttransportarenotreadilyavailable.Mostsedimenttransportrelationsmaybecharacterizedaseitherempiricalorsemi-empiricalatbest.Indeed,muchoftheliteraturepublishedonsedimenttransporthasfocusedonprovidingparametrizationsoftransportbasedonlaboratory,andsometimeseld,experiments.Thistendstobeacceptablewhendevelopingamodelforaspecicapplicationtoaspecicsetofcriteriae.g.forcing,sedimentsizeandgradation,transportmodeandregime,butmakesitdiculttodevelopuniversalsedimenttransportmodels.Regardlessofthemodelframeworkchosen,thelimitingfactoronaccuracyandpredictabilityseemstolieinthetreatmentofsedimenttransportandnotnecessarilythehydrodynamics. 56

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Figure2{1.ComparisonofavelocityandbshearstresstimeseriesforlinearandnonlinearwaveswithperiodT=6sinwaterdepthh=1m. Figure2{2.Approximatedistributionofuidandgrainshearstressinsteadyopenchannelow.Adaptedfrom FredseandDeigaard 1992 57

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Table2{1.Commonpowerlawformulationsforbedloadsedimenttransportbycurrentsandwaves.Shields'parameterbasedoncurrentsiscandforwavesisw. ReferenceDimensionlessBedloadDischarge Currentsb Meyer-PeterandMuller 1948 8c)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr1:5 Nielsen 1992 12p cc)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr Ribberink 1998 11c)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr1:65 CamenenandLarson 2005 121:5cexp)]TJ/F15 11.9552 Tf 9.2985 0 Td[(4:5cr=cWaves1=2 MadsenandGrant 1976 12:5Ws3w=p s)]TJ/F15 11.9552 Tf 11.9551 0 Td[(1gd50 Soulsby 1997 5:1w)]TJ/F24 11.9552 Tf 11.9551 0 Td[(cr1:5Wavest BailardandInman 1981 bfwuwt3=[s)]TJ/F15 11.9552 Tf 11.9551 0 Td[(12g2tand50] Ribberink 1998 11jwtj)]TJ/F24 11.9552 Tf 17.9327 0 Td[(cr1:65wt=jwtjWaves DibajniaandWatanabe 1992 0:001Ws)]TJ/F23 7.9701 Tf 7.3141 4.3385 Td[(0:55)]TJ/F24 11.9552 Tf 11.8665 0 Td[(=j)]TJ/F27 11.9552 Tf 7.3142 0 Td[(j=p s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd50 CamenenandLarson 2005 ap cw;on+cw;offw;mexp)]TJ/F24 11.9552 Tf 9.2985 0 Td[(bcr=w Table2{2.Empiricalentrainmentfunctionsforsuspendedloadsedimenttransport. ReferenceDimensionlessPick-UpRate,p Fernandez-Luque 1974 0:02)]TJ/F24 11.9552 Tf 11.9552 0 Td[(cr1:5 NagakawaandTsujimoto 1980 0:02)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:035=3 vanRijn 1984 0:00033[s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd50=2]0:1=cr)]TJ/F15 11.9552 Tf 11.9552 0 Td[(11:5 58

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CHAPTER3BEDFORMDYNAMICS3.1IntroductionDierentialsedimenttransportontheseabedcreatesmorphologicalfeaturesthatrangeinsizefromsmall-scalesandripplestomuchlarger,shore-parallelsandbars.Regardlessoftheirsize,theresultingmorphologicalstructureplaysanimportantroleinbothsedimenttransportandhydrodynamicprocesses.Shore-parallelsandbarshavebeenshowntointensifythecross-shorecurrentsthatpromoteoshorebarmigrationthroughgradientsinthecross-shoresedimenttransport Gallagheretal. 1998 .Small-scalebedforms,suchassandripples,determinethegenerationofturbulenceandentrainmentofsandnearthebed Hanesetal. 2001 largelythroughowseparationthatleadstoanorganizedpatternoftime-dependentvortexsheddingineachhalf-cycleofawave.Thesevorticesoftendeterminethetimingofsedimententrainmentanddeposition Nielsen 1988 ,theamountofsedimentcarriedinsuspension vanderWerfetal. 2005 ,andareresponsibleforthedissipationofwaveenergyintheboundarylayer Tunstall 1973 ; TunstallandInman 1975 .Thepresenceofsmall-scalebedformshasalsobeenfoundtocausenaturalsortingofsediment,resultinginaredistributionofneandcoarsematerialontheseabed FaraciandFoti 2002 .Thefocusofthischapterisonsmall-scalebedforms,suchassandripples,sofurtherdiscussionoflargerfeatureslikesandbarswillbelimited.Thefollowingsectionsprovideabriefoverviewofbedformtypes,theirclassicationandcharacteristics,mechanismsfortheirgrowth,theiraectonthebottomboundarylayer,andmethodsforpredictingtheirlengthandheight.Additionalinformationonbothsimpleandcomplexmorphologicalmodelsisalsoprovided.3.2BedformTypesSmall-scalebedformsmaybebroadlycategorizedbythedominantowconditionsunderwhichtheyareformed:current-generatedbedformsorwave-generatedbedforms. 59

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Earlyaccountsofwave-generatedripplesintheseadatebacktotheobservationsof Hunt 1882 ,whofoundthatripplemarks"wereformedonasandyseabedbyaslightoscillationofthewater.Laboratoryexperimentsby Darwin 1883 ,andlaterby Ayrton 1904 ,providedmostlyqualitativedataabouttheformationofripplesinoscillatingow.Wave-generatedbedformsarefurtherdistinguishedby Bagnold 1946 aseitherrollinggrainorvortexripples.Adistinctionismadebetweenthesetwotypesofwave-generatedripplesnotonlybecausetheygrowthroughdierentprocesses,butalsobecausetheircharacteristiclength,height,andsteepnessscaledierently.Briey,rollinggrainripplesgrowthroughanaggregationofsandparticlesinareasofhighfriction Darwin 1883 butdonottypicallyscalewithwave-relatedquantities.Vortexripplesformonceaperturbationonthebedislargeenoughtoinduceowseparation Bagnold 1946 ,suchthatanadversepressuregradientisformedintheleeoftheripplecrest Ayrton 1904 .Thisprocessmayoccureitherthroughthegrowthofrollinggrainripplesorduetoexistingperturbationsontheseaoor.Specicdetailsaboutthemechanismsforgrowthandevolutionaregiveninthefollowingsections.Current-generatedbedformsincludethoseformedinriversandopenchannelhydraulicows.Thesimilaritybetweenthesetwoenvironmentsisthatthepredominantforcingis,moreorless,one-dimensionalsteadyow.Whilenotalwaystrulysteady,"thetermisusedheretoconveyanabsenceofapersistentoscillatory,orrepeating,ow.Shallowtidally-dominatedinletsandestuariesmayevenbeaspecialcaseofsteady"owiftheambientoscillatorymotionisweakcomparedtothetidal-inducedcurrents.Current-generatedripplesinitiallyforminmuchthesamewaysaswave-generatedripples,butoncethebedformislargeenoughtoinduceowseparationthedynamicsaredierent.Sincethesteadyowhasaconstantphase,arecirculationzoneisonlyformedononeside|theleeside|ofthebedform.Thedynamicsofthebedformarecontrolledbyabalancebetweensandtransportedalongthestoss,sandtransportedoverthecrestontotheleeside,gravitationalforcespullingsedimentdowntheleeside,andthetransport 60

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ofsanduptheleesidefromtheattachedvortex.Thiscommonlyresultsinabedformwithanasymmetricprolewherethestosssidehasamilderslopethanthatoftheleeside Darwin 1883 .Itiscommonfortheleesideofanasymmetricrippletohaveaslopeveryclosetothenaturalangleofrepose.Onceinequilibriumwiththesurroundingow,steadycurrentsarecapableofmaintainingtheripplethroughadynamicequilibriumbutareoftendisplaceddownstreamduetohighervelocitiesatthecrestthanatthetrough Darwin 1883 .3.2.1RollingGrainRipples Bagnold 1946 wasthersttoclassifythetwotypesofwave-generatedripplesaseitherrollinggrainorvortex,butaphysicaldistinctionbetweenthemwasmadeearlierby Darwin 1883 .Thelaboratoryexperimentsof Darwin 1883 demonstratedthatoscillatorymotionrstcreatedwhathecalledtransientripplesasaresultofparticleaggregationinareasofhighfriction.Theprocessofaggregation,henoted,increasesthefrictionfurtherandtrapsevenmoreparticlesinthatarea.Othershavecharacterizedtheprocesssimilarly,suggestingthateachgraincreatesaregionofweakerowinitsleeultimatelycausinggrainstogroupintotransversezonesthatformrippleswithlargershadowzones Bagnold 1946 ; Blondeaux 1990 ; VittoriandBlondeaux 1990 ; Andersen 1999 .Theparticlemodelof Andersen 2001 demonstratesthatrollinggrainripplesformandcoarsenduetogradientsinthetransportvelocityfromonesideoftheaggregationtotheother.Transientrippleswerefoundtohaveawavelengthapproximatelyone-halfofthenalripplewavelengthintheexperimentsof Darwin 1883 ,whichseemstobeaconsistentobservationinotherlaboratoryexperiments Sleath 1976 .Rollinggrainripplescontinuetogrowinheightthroughaggregationandtrappinguntiltheleeslopesaresteepenoughtoallowavortextoformintheshelteredspacebehindthecrest Bagnold 1946 ,eventuallyformingvortexripples Schereretal. 1999 ; FaraciandFoti 2001 .Duringtheinitialgrowthprocess,thesteepnessheight/lengthofrollinggrainrippleshasbeen 61

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foundtobeconsistentlynear0.1by Sleath 1976 andothers. Darwin 1883 qualitativelysuggestedthattheinitialgrowthprocessonlyoccurredbetweenanupperandlowerlimitofuidvelocity,andthatthelimitswereafunctionofaveragesedimentpropertieslikesizeanddensity.Theseupperandlowerlimitswerefoundby Bagnold 1946 tobe2crandcr,respectively.Thestabilityofrollinggrainripplesformedduringlaboratoryexperimentsisunclear.Duetotheirlowreliefandtransientnature,rollinggrainripplesarerarelyobservedintheeld.Rollinggrainrippleswerefoundtobestableintheexperimentsof Bagnold 1946 and Sleath 1976 ,but MillerandKomar 1980b suggestthatthesetwoinstancesmaybearesultofthetestingapparatususedduringtheexperimentsoscillatingtray.Experimentsinrotatingannularcells Schereretal. 1999 ; StegnerandWesfreid 1999 andoscillatingwatertunnelssuggestthatrollinggrainripplesaretransient,unstablefeatures MillerandKomar 1980b .Theprevailingtheory,now,isthatrollinggrainripplesaretransientbedformsthatinitiatethegrowthoflarger,vortexripplesthroughaninversecascademechanism:thedynamicalevolutioninvolvescoarseningfromaninitiallyshortwavelengthrollinggrainripplestoalongerwavelengthvortexripples,withsaturationatanalequilibriumcondition StegnerandWesfreid 1999 ; Andersenetal. 2002 .While Darwin 1883 and Bagnold 1946 providedqualitativedescriptionsofthegrowthofrollinggrainripplesfromaninitiallyatbed, Kennedy 1963 1969 demonstratedthatifalagisassumedtoexistbetweenvelocityandtransport,aninstabilityexistsatthesand-waterinterface ColemanandMelville 1994 .Thislagisoftenattributedtoinertialeectsofthesediment Parker 1975 ,andhasbeenidentiedastherippleinstabilityby Smith 1970 .Initiationofgrowththroughthisrippleinstability,however,shouldnotbeconfusedwiththestabilityofrollinggrainripplesdescribedabove;theformerreferstoamechanismthatinitiatesgrowthwhilethelatter 62

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conveysanabilityofthebedformtoachieveanequilibriumconditionwithrespecttoitsowconditions.Ifaspatiallagisassumedtoexistbetweenvelocityandtransport, Parker 1975 nds,throughalinearstabilityanalysisofderivedrelationsforowandtransport,aninherentinstabilityofatbedswhichleadstotheformationofriverantidunes.However,ifvelocityshearandtransportareassumedtobeinphasewithoneanother,atsandbedsareunconditionallystable Parker 1975 .Underpurebedloadtransport,aatbedisalsounstableatallwavelengthstosmallperturbationsintopography Smith 1970 .Thissuggeststhatoneoftwonecessaryconditionsmustbemetfortheinitiationofripplegrowthfromaatbed;eitherforcingandtransportmustbedecoupled,orsomeinitialperturbationinbedtopographymustbepresent. Hayakawa 1970 wasabletodevelopatheoryofduneformationinopenchannelowsbyapplyingtherippleinstabilityconcept,andothershavehadsimilarsuccess ColemanandMelville 1994 .Somepredictivemodelsofrollinggrainripplegeometry,basedontherippleinstabilityconcept,havealsobeendeveloped. Sleath 1976 createdastabilityanalysismodelcapableofpredictingthewavelengthofrollinggrainripples.Morerecently, FotiandBlondeaux 1995a developedapredictivetheoryofrippleformationunderwavesthatagreeswellwithexperimentaldata.Assumingafullyturbulentboundarylayer, FotiandBlondeaux 1995a obtainedclosedformsolutionsofturbulentoscillatoryowoverawavysurfacethroughalinearstabilityanalysisofthegoverningequations.Detailedparticlemodelsofrollinggrainripplesalsoexist Andersen 2001 .3.2.2VortexRipples Bagnold 1946 suggestedthatwave-generatedripplesfallintooneoftwocategories:rollinggrainorvortex.Thedecidingfactor,Bagnolddetermined,wasbasedontherangeofappliedstressatthebed.OncetheShieldsparameterincreasesbeyond2cr,theleeslopesoftheripplegrowtoaheightlargeenoughthatavortexisformedinthelee Bagnold 1946 .Theprocessthat Bagnold 1946 observedinthelaboratoryinvolveda 63

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transitionrstfromaatbedtoonewithrollinggrainripples;andsecond,fromrollinggrainripplestovortexripples.Itisalsopossibleforvortexripplestoformwithoutrollinggrainripplesifthereissomesortofnaturalperturbationonthebedlargeenoughtoinduceowseparation Ayrton 1904 .Sowhilerollinggrainripplesareasucientmechanismforthegrowthofvortexripples,theinducedowseparationandsubsequentvortexformationarenecessaryconditions.Theabilityofvortexripplestogrowfromnaturalperturbationsmayalsoexplainwhyrollinggrainripplesarerarelyobservedintheeld MillerandKomar 1980b ,whereasvortexripplesareoftendetected MillerandKomar 1980a ; Hanesetal. 2001 ; Ardhuinetal. 2002 .Vortexripplesplayanintegralroleinbothsedimenttransportandenergydissipation,asdiscussedpreviously.Centraltotheeectsofvortexripplesonboundarylayerprocessesistheformationandejectionofvorticesfromtherippleproleintothemainowduringeachsuccessivehalf-cycleofoscillatoryforcing.Vorticesejectedovertheripplecrestcanentrainsignicantamountsofsedimentwithintheboundarylayerthatissubsequentlyadvectedbytheouterow Gallagheretal. 1998 ; vanderWerfetal. 2005 .Thisisthoughttoincreasethephaselagbetweensuspensionandtransportrelativetotransportoveraatbed InmanandBowen 1963 ,leadingtotime-dependentsuspendedsedimenttransportprocessesthatcannotbemodeledinaphase-averagedorquasi-steadymanner vanderWerfetal. 2006 .Theformationofvorticesintheleeofbedformsresultsfromowseparationatthecrest,similarinmanywaystoowseparationaroundblubodies,cylinders,andthelike.Inherlaboratoryexperiments, Ayrton 1904 identiedtwoconditionsnecessaryforvortexformationintheleeofripples: 1. areductionofpressureinthelee;and 2. anupward-directedresultantgravitypressurealongtheleeridge.Therstconditionresultsfromowseparatingatthecrest,creatingashelteredregionintheleeoftheripple.Thesecondconditionisduetoanadversepressuregradientthat 64

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developsduetophasedierencesbetweentheboundarylayerowandthefreestreamforcing|aninertialpropertyofboundarylayerows.Adversepressuregradientsmayalsobeattributedtosurfacegravitywaves,butarenotpresentinsteadyowsoveratbeds. Ayrton 1904 speculatedthatvortexformationcouldnotoccurinsteadycurrentsandthatripplescouldnotbecreatedormaintainedbysuchowconditions.Recentlaboratoryexperimentsofbedformgrowthfromaatbedinsteadycurrentshaveprovedthistobefalse Mantz 1990 ; ColemanandMelville 1994 1996 .Underoscillatoryforcing,vorticesareformedandejectedinalternatinghalf-cycles.AschematicofvortexformationandejectionasafunctionofuidphaseandforcingisshowninFigure 3{1 .Vorticesformatthebeginningofeachhalf-cycleFigures 3{1 a, 3{1 c, 3{1 e,growinginstrengthandsizetomaximumvaluesataphaseofapproximately90degrees EarnshawandGreated 1998 .Duringphasesofmaximumforcing,strongowseparationandvortexsheddingneartheripplecrestserveasmechanismsfortheproductionofturbulenceintheboundarylayer Barretal. 2004 .Aftercontractinginsizeslightlyduetoareductionintheappliedforcing,theejectedvorticesinFigures 3{1 cand 3{1 econtinuetoincreaseinsizeastheymoveverticallyintoareasofdecreasingpressure Ayrton 1904 .Neglectingsmallperturbationsduetoturbulence,theseprocessesaremoreorlesssymmetricfromonehalf-cycletothenext.Underrealwaves,however,vortexformationandejectionisasymmetricowingtoStokes'law EarnshawandGreated 1998 .Thiscanhaveasignicanteectonthenetsedimenttransportinthecross-shoreassuspendedloadmaybeadvectedshorewardbytheEulerianow.Flowseparationandvortexdevelopmentareimperativeforthegrowthandstabilityofvortexripplesunderavarietyofforcingconditions Nielsen 1981 .Duringthegrowthprocess,thenearbedhydrodynamicsresultinnetsedimenttransporttowardtheripplecrest AndersenandFredse 1999 .Theimpingingowacceleratesasittravelsalongtheupwindsideoftheripplethestosscreatingstrongshearstressesatthebedthatdrivesedimenttransportfromthetroughtowardthecrest.Intheleeoftheripple,the 65

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counter-rotatingvortexinducesabedshearstressthatisdirectedupthelee-sideoftherippleresultingintransporttowardthecrest.Thisistheprimarygrowthmechanismforvortexripplesinnear-equilibriumconditions.Experimentaldataofbedformsinawaveumesuggeststhatthealternatingvorticessupporttherippleineachsuccessivehalf-cyclewhen<0:2 FaraciandFoti 2002 .For>0:2,thestressatthecrestissolargethattheinducederosioncannotbeovercomebythestabilizingeectofthevortices.3.2.3CharacteristicsThegrowthandevolutionofvortexripplesfromaatbedmayoccurthroughoneoftwoprocesses;oneinvolvesatransientstatepriortoequilibrationandtheothersuggestsrelativelyconstantgrowthfromaatbed.Forweakowsandunderregularwaves,vortexripplesformthroughcoarseningoftransientrollinggrainripples StegnerandWesfreid 1999 ; FaraciandFoti 2002 .Thecoarseningprocessischaracterizedbyanincreaseinwavelength,resultinginfewerripples,withsaturationtoanequilibriumcondition Andersenetal. 2002 .Thecoarseningprocessdoesnotoccurforbedformsinstrongerowsorunderirregularwaveconditionswheretheyreachequilibriumdirectlythroughconstantgrowth StegnerandWesfreid 1999 ; FaraciandFoti 2002 .Anoverwhelmingamountofphysicaldatademonstratestheabilityofvortexripplestotransformfromasmallertoalargerwavelength,butrarelydoesthishappeninreverse.Vortexripplesintheeldandlabhavebeenfoundtoexhibitstronghysteresis Traykovskietal. 1999 ; StegnerandWesfreid 1999 wherebychangesinstableripplepatternsfromlargertoshorterwavelengthsarenotobserved.Thissuggeststhatlargerripplesmustbecompletelydestroyedbyanincreaseintheforcingconditionsbeforeasmallerstableripplepatterngrowsinresponsetonewforcing.Rollinggrainripplesandvortexrippleseachhaveuniquecharacteristicsdueinlargeparttotheforcingconditionsthatcreatethem.Therollinggrainripplesobservedandmeasuredinanumberoflabexperiments Bagnold 1946 ; Sleath 1976 ; FaraciandFoti 2002 consistentlyhaveasteepnessr=rof0.1regardlessofthetypeof 66

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owandtestingapparatus,whererandraretherippleheightandwavelength,respectively.Vortexripplestendtohaveamaximumsteepnessalmosttwicethatofrollinggrainripplesatapproximately0:18 Nielsen 1981 ; FaraciandFoti 2002 .For<0:2, FaraciandFoti 2002 ndthattheripplesteepnessremainsconstantatitsmaximumvaluethatcorrespondswellwiththevaluegivenby0:32tan.Thewavelengthofvortexripplescommonlyincreasealongwiththeheight|keepingamoreorlessconstantsteepness|untilanequilibriumisreached.Thisequilibriumisafunctionofowandsedimentcharacteristics Lofquist 1978 andthenalsteepnesstendstomaximizeresistancetothelocalowconditions Davies 1980 DoucetteandO'Donoghue 2006 identifythreedistinctprocessesthroughwhichtheripplewavelengthevolves:slide,merge,andsplit.Theslideprocessidentiesagradualincreasedecreaseofwavelengthasadjacentripplesmoveawayfromtowardeachother.Amergeoccurswhentwolargerankingripplesincreaseinheightandmovetowardeachother,destroyingthesmallerrippleinbetween.Asplitreferstoasituationwhenonelargerippledecreasesinsizethroughthecrestsplittingintowhatappeartobetwoormoreripplesofshorterwavelength.Whenbedloadtransportdominatesoversuspendedload,vortexripplesgrowthroughtheprocessesofmergeandslideanddecaythroughtheslideandsplitmechanisms DoucetteandO'Donoghue 2006 .Thecharacteristicsofvortexrippleshavebeenfoundinmanycasestoscalewithpropertiesoftheuidforcing. Clifton 1976 rstsuggestedthataparticularrippleregimeexistedwherebytheripplewavelengthswereconstantmultiplesofthenearbedorbitaldiameter2A:A=Umax !{1whereUmaxisthemaximumnearbedvelocityamplitudeand!istheangularfrequencyoftheforcing.Here,Arepresentstheorbitalsemi-excursionlengthwhichisone-halftheorbitaldiameterA.Anumberofrelationshipsbetweenripplewavelengthrandorbitalsemi-excursionAhavebeenpresentedintheliterature,andrangefromr0:78A 67

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forsuspendeddominatedripples AndersenandFredse 1999 totheupperestimateofr1:33Aof Nielsen 1992 forbedloaddominatedconditions.SomecommonwavelengthscalingrelationshipsfororbitalripplesaregiveninTable 3{1 .Analternativetheoryofbedformscalingisoeredby WibergandHarris 1994 whosuggestthatforeld-scaleows,vortexripplesscalebestwiththegrainsize.Thisisnotalwaystrue,however,as Ardhuinetal. 2002 foundthatwave-formedvortexripplewavelengthsonthecontinentalshelfwereproportionaltothenearbedorbitalexcursions,notthegrainsize.Datafromlabandeldexperimentsdoconsistentlyshowdiscrepanciesbetweenvortexripplescaling[e.g. MillerandKomar 1980a ; Nielsen 1981 ; FaraciandFoti 2002 ]andmanyattributethesedierencestothedisparitybetweenowconditions.Whereasthemajorityoflaboratoryexperimentsareconnedtoregularoscillatoryforcingwithshortperiods,conditionsintheeldarerepresentedbyabroadspectrumofwaveheights,frequencies,directions,andotherowproperties.Attemptstounifyappropriatevortexripplescalesforeldandlabdatahavereceivedincreasingattentionrecently[e.g. MillerandKomar 1980a ; O'DonoghueandClubb 2001 ; Williamsetal. 2005 ; Lacyetal. 2006 ]andwillbediscussedfurtherinthefollowingsections. WibergandHarris 1994 expandontheearlierconceptsof Clifton 1976 whoidentiedthreecategoriesofsymmetricripplesthatdierintheircharacteristics: 1. orbitalripples; 2. anorbitalripples;and 3. suborbitalripples.Whereorbitalripplesscalewiththeorbitaluidmotionandarecommonlyfoundinlabexperiments, WibergandHarris 1994 suggestthatanorbitalripplesscalewithsedimentpropertiesandareindicativeofvortexripplesfoundintheeld.Suborbitalripplesareclassiedastransitionalbedformslyingsomewherebetweenorbitalandanorbitalripples. WibergandHarris 1994 statethatanorbitalandorbitalrippleheightrscalewiththeoscillatoryboundarylayerthicknesssuchthatorbitalrippleheightsareapproximately 68

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2whileanorbitalrippleheightsareabout=4.ThisisdemonstratedinFigure 3{2 andshowsthatwhileanorbitalripplesarecompletelyimmersedwithintheboundarylayer,orbitalripplestendtoprotrudefromit.Thesignicanceofthisismadeevidentthroughthechoiceofroughnessparametrizationsincoastalhydrodynamicmodels,whereoneisforcedtochooseappropriatevaluesofboundaryroughnessbasedontheinteractionofbedformswiththeboundarylayer.3.3ExperimentsAnumberofexperimentsonbedformdynamicshavebeenperformedinaneorttoprovidebetterunderstandingofripplegrowthandevolution,aswellassedimenttransportcharacteristicsintherippledbedregime.Theseexperimentsrangefromsmall-tolarge-scalelaboratoryexperimentsutilizinganumberofdierenttestingmethods,toeldexperimentsandobservationsofbedformdynamicsunderthestochasticconditionsfoundinnature.Theexperimentaldatahavebeenusedtodevelopnewempiricallyderivedripplepredictorsdiscussedinthenextsectionandhaveleadtotheconclusionthatdiscrepanciesexistbetweenbedformsproducedbylab-andeld-scaleows.Additionally,somehavefoundthatthetypeoflaboratoryapparatususedinexperimentsmayaectbedformcharacteristicsandbehavior.AbriefoverviewofexperimentsonbedformdynamicsisprovidedinthefollowingsectionsandasummaryofhistoricalexperimentsanddatasetsisprovidedinTable 3{2 ,alongwithcitationkeysusedinthefollowingsections.3.3.1Laboratory MillerandKomar 1980b provideahelpfulreviewofcommonly-usedexperimentaltechniquesusedforsimulatingowsinthelaboratory.Althoughsomeexperimentaldatafromexperimentsinrotatingannularcellsexistinpublishedbedformliterature Schereretal. 1999 ; StegnerandWesfreid 1999 ,byfarthemostwidelyusedexperimentalsetupsinclude: oscillatingtray; 69

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oscillatingwatertunnel;and ume.Eachofthedeviceslisted,however,haveuniquecharacteristicsthathavesomeconsequenceonthecharacteristicsanddynamicsofbedforms.3.3.1.1OscillatingTrayTheoscillatingtraydevicehasbeenusedtostudyhydrodynamicsinthevicinityofstationarysimulatedripples EarnshawandGreated 1998 andalsobedformevolutioninasandbed Bagnold 1946 ; Manohar 1955 ; Sleath 1976 1985 ; Schereretal. 1999 ; Lacyetal. 2006 .Intheseexperiments,atrayofsandorstationarybedformsisoscillatedataparticularfrequencythroughatankofstillwatertherebysimulatingtheoscillatorymotionofwaterwaves. Lacyetal. 2006 usedamodiedsetuptosimulatecombinedwaveandcurrentforcingbypassingasteadycurrentoverthetrayasitoscillatedinaumeatvariousanglestothesteadyow.Theowinducedbytheoscillatingtrayapparatus,however,isnotadirectanalogofoscillatorywavemotionoverastaticbed.Steadystreamingeectsfoundinoscillatoryboundarylayerowsarenotsimulatedbytheapparatus,therebyexcludingtheadditionalverticaluxofmomentumdirectedawayfromthebed.Themotionofthetrayhasalsobeenfoundtoinduceameanowdirectedoppositetothedirectionoftravel Schereretal. 1999 .Observedbedformcharacteristicsinoscillatingtrayexperimentsalsoappeartobeunique,where Schereretal. 1999 notethattheapparatusimpartsanadditionalinertialforcetothebedforms.Furthermore,rollinggrainrippleshaveonlybeenfoundtobestableinoscillatingtrayexperimentswheretheyaretransientinoscillatingwatertunnelandumeexperiments,andrarelyobservedintheeld MillerandKomar 1980b .3.3.1.2OscillatingWaterTunnelAmongothers,theoscillatingwatertunnelhasbeenusedtostudybedformcharacteristicsby FotiandBlondeaux 1995a Voropayevetal. 1999 O'DonoghueandClubb 2001 ,and DoucetteandO'Donoghue 2006 .Similartotheoscillating 70

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trayapparatus,oscillatingwatertunnelsaredesignedtosimulatetheoscillatorymotionofwaves.Mostoscillatingwatertunnelsconsistofalongclosed-conduithorizontaltestsectionandpistonslocatedinverticalrisersateachendthatareusedtodrivetheoscillatoryow.Theresultingoscillatoryowispurelyhorizontalandmostlyone-dimensional.Liketheoscillatingtrayapparatus,thehorizontalowisofconstantphaseandlacksanyverticalcomponentduetoboundarylayerstreaming.Insomecasesitispossibletosimulateacollinearsteadycurrentsuperimposedontheoscillatorymotion.Adistinctadvantageoftheoscillatingwatertunnelisthattheowcharacteristicscanbepreciselyprescribedandcontrolled.Whilemostumeandoscillatingtrayexperimentsarelimitedtolab-scaleowsofshort-periodwaves,lowtomoderatecurrentvelocities,andlowtomoderatemobilitynumbers,theoscillatingwatertunnelhasbeenusedtosimulatewaveconditionsmorecommonlyfoundinnature. O'DonoghueandClubb 2001 and DoucetteandO'Donoghue 2006 utilizedthewatertunnelapparatustosimulatelong-periodwavesandtoanalyzethedierencesbetweensymmetricandasymmetricforcingonbedformcharacteristics.Whilenosignicantdierenceinrippleheightandwavelengthwasfoundforsymmetricandasymmetricows, O'DonoghueandClubb 2001 didndthatpreviouslysuggestedmethodsforpredictingbedformgeometrywereinsucient,especiallyforthree-dimensionalripples.3.3.1.3FlumeFlumeshavebeenusedextensivelytostudybedformdynamicsundersteadycurrents Yalin 1985 ; Mantz 1990 ; ColemanandMelville 1994 1996 ,waves MillerandKomar 1980b ; Marshetal. 1999 ; FaraciandFoti 2002 ,andcombinedwaveandcurrentforcing KhelifaandOuellet 2000 ; Lacyetal. 2006 .Anadvantageofumeexperimentsoveroscillatingtrayandoscillatingwatertunnelexperimentsisthatthetrueoscillatorynearbedmotionissimulatedbythesurfacegravitywaves.However,mostlaboratoryumesareonlycapableofproducingshort-period,uni-directionalwavesnottrulyrepresentativeofeldconditions.Certainexceptionsdoexist,suchasthelargewave 71

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umesatOregonStateandDelftUniversityofTechnology,andexperimentsinthesefacilitieshelpbridgetheinformationgapbetweenlab-scaleandeld-scalebedformcharacteristics MillerandKomar 1980b .Asmentionedearlier,theoscillatorymotionproducedbyoscillatingtrayandoscillatingwatertunnelexperimentsisnotatruerepresentationofwave-inducedmotionintheboundarylayer.Progressivewavesinduceasteadystreamingintheboundarylayerthat,whileweakcomparedtotheoscillatoryow,hasasignicanteectonsedimenttransport Marin 2004 .Boundarylayerstreamingischaracterizedbyaverticaldiusiveuxofmomentumdirectedawayfromthehorizontalboundarylayer. Dohmen-JanssenandHanes 2002 suggestthatthisstreamingresultsinnettransportratesunderwavesthatare2:5timeslargerthanthoseinuniformhorizontaloscillatoryow.Furthermore,thewavebottomboundarylayercontainscomponentsofvelocityinbothhorizontalandverticaldirections.Theseverticalvelocities,whicharenotexactlyoutofphasewiththehorizontalvelocitiesinthebottomboundarylayer,leadtoanon-zerotime-averagedbedshearstress Marin 2004 .Fieldconditionsarerarelyclassiedbyonetypeandsizeofwave.Rather,elddataarecharacterizedbyabroadspectrumofwavefrequencies,directions,heights,andcurrentvelocities MillerandKomar 1980a .Eventhefewlargewaveumescapableofproducinglong-period,irregularwavesareunabletosimulatemulti-directionalwavesorwavessuperimposedonasteadycurrent.Therefore,methodsforpredictingripplegeometrythathavebeenbasedonlaboratorydatatendtofailwhenappliedtoelddata.Inaneorttoovercomeatleastoneoftheseshortcomings, KhelifaandOuellet 2000 performedlaboratoryexperimentsonbedformcharacteristicsincombinedwaveandcurrentows.TheyaccomplishedthisbyusingtwointersectingumesatLavalUniversityandwereabletosimulateprogressivewavescombinedwithsteadycurrentsat60and90degreeanglestooneanother.Theresultingdatawereusedtoformulatenewempiricallyderivedexpressionsforbedformheightandlengthincombinedows, 72

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butunfortunatelytheirutilityissomewhatlimitedbytheshort<2swaveperiodsgeneratedintheume.Morerecently, Lacyetal. 2006 attemptedtoovercomethisscalediscrepancybysimulatingeld-scalewaveperiods>8sandcurrentvelocitieswithanoscillatingtrayinacurrentume.Thefully-instrumentedtraywaslledwithsandandthenoscillatedatvariousangles|bothobliqueandorthogonal|toasteadycurrentgeneratedintheume.Thedatafromtheirexperimentsareawaitingpublication.3.3.2FieldObservationsofripplecharacteristicsundereldconditionshavebeenfoundtodiersubstantiallyabout300%fromthosefoundinlaboratoryexperiments FaraciandFoti 2002 .Inthelaboratory, Marshetal. 1999 suggestthatbedformlengthscalesrapidlyunderthemonochromaticforcingtypicalofmostwaveumes.Theyspeculate,however,thatripplewavelengthmustbediculttochangeintheeldsincethebedformsaresubjectedtoabroadspectrumofwavefrequencies.Oftentimesobservedripplecharacteristicsarenotindirectequilibriumwiththelocalforcingandexperiencesignicanthysteresis Traykovskietal. 1999 .Morespecically,bedformsintheeldmaybearesultofanantecedentwaveclimatethatremainstaticaslocalforcingconditionssubsidebelowthecriticalthresholdrequiredforsedimentmobilization.Thishasmadeitdiculttoformulateuniedexpressionsforbedformgeometryundereld-scaleconditions.Additionalcomplicationsarisewhentryingtoassociaterippleprocesseswithasingle,statisticalrepresentationofaforcingspectrumcomposedofmanywavefrequenciesanddirections. MillerandKomar 1980a suggestedthatsignicantwaveparameterscomputedfromeldspectrashouldbeusedinthecalculationofrelevantbedformparameterssuchasmaximumorbitalvelocityamplitude,orbitalexcursionlength,mobilitynumber,andShields'parameter.Nevertheless,itremainsdiculttoformulatenewexpressionsbasedonelddataandthepredictivecapabilityofmethodsderivedfromeld-scalelabexperimentsisstillpoor.Observationsandmeasurementsofbedformcharacteristicsintheeld,however,continuetosupplementexistingtheoryandprovide 73

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moredetailedinformationabouttheirgeometryandbehaviorundernaturalforcingconditions LiandAmos 1999 ; Traykovskietal. 1999 ; Hanesetal. 2001 ; Ardhuinetal. 2002 .3.4RipplePredictorsThefollowingsectionpresentssomecommonlyusedmethodologiesfromexistingliteratureforpredictingrippleheightrandwavelengthr.TheseripplepredictorsarebasedonextensivesetsofdatafromlabandeldexperimentsseeTable 3{2 ,andinmanycasesarebasedonnondimensionalgroupsthatrelatesedimentcharacteristicswithowproperties.AnumberofthesedimensionlessparametersusedtoclassifyorcharacterizebedformsarelistedinTable 3{3 .3.4.1Clifton976Thesemi-quantitativemodelof Clifton 1976 C76isbasedonthenearbedorbitalvelocityubandnearbedorbitalvelocityasymmetryub:ub=A!{2aub=14:8H2 LTsinh4kh{2bwhereHiswaveheight,Liswavelength,kisthewavenumber,andhiswaterdepth.Theseparametersarethenusedtoidentifyoneoffourbedstates:nosedimentmovement;symmetricripples;asymmetricripples;andsheetow.Thesymmetricripplebedstateisfurtherdividedby Clifton 1976 intoorbital,anorbital,andsuborbitalripplesasdiscussedpreviously.3.4.2Nielsen1981 Nielsen 1981 N81oerssemi-empiricalformulationsforthesizeandshapeofvortexripplesderivedfromananalysisofvarioussetsoflabandelddata.AstrongdependenceonShields'parameterwasfoundforripplesteepnesswhiletheripplewavelengthrdatacollapsedbestwhenplottedagainstthemobilitynumber Brebner 1980 ,whichisaratiobetweensedimentdestabilizingandstabilizingforces.The 74

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formulationsoeredby Nielsen 1981 areparticularlyusefulsincetheydierentiatebetweenlabandeldconditions.WhencomparedtotheBASEXdataof VincentandOsborne 1993 Marshetal. 1999 ndthatthetwosetsofequationsserveasboundingequationsforthedata.Forrelativelyweakconditions<20,theripplewavelengthandsteepnessaregivenbyr1:3A{3ar=rmax0:32tan:{3bForconditionsoutsideofthisrange,orforgeneraluse, Nielsen 1981 providesthefollowingrelationshipsforregularlaboratorywaves;r=A=2:2)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:3450:34<<230;{4ar=r=0:182)]TJ/F15 11.9552 Tf 11.9551 0 Td[(0:241:52:5;{4br=A=8><>:0:275)]TJ/F15 11.9552 Tf 11.9551 0 Td[(0:022p for<1560for>156{4candforirregulareldwaves;r=A=exp693)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:37ln8 1000+0:75ln7!;{5ar=r=0:342)]TJ/F15 11.9552 Tf 11.9552 0 Td[(0:344p 2:5;{5br=A=21)]TJ/F23 7.9701 Tf 6.5865 0 Td[(1:85>10:{5cEquations 3{4c and 3{5c correspondspecicallytoquartzsandwiths=2:65.ParameterssuchasA,,and2:5usedintheripplepredictorsforirregularwavesEquations 3{5a { 3{5c shouldbebasedonsignicantwaveparameters Nielsen 1981 75

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Intheprecedingexpressions,thegrainroughnessShieldsparameter2:5iscomputedas2:5=1=2f2:5A!2 s)]TJ/F15 11.9552 Tf 11.9552 0 Td[(1gd=1 2f2:5{6awherethegrainroughnessfrictionfactorf2:5isbasedontheformulationof Swart 1974 Equation 2{4 witharoughnessksof2:5d50,f2:5=exp"5:2132:5d50 A!0:194)]TJ/F15 11.9552 Tf 11.9551 0 Td[(5:977#:{6b3.4.3GrantandMadsen1982BasedonthelabdataofC69,theformulationsof GrantandMadsen 1982 GM82forbedformcharacteristicsarebasedontheskinfrictioncomponentofboundaryshearstress. GrantandMadsen 1982 identifytwospecicrangesforbedformbehavior;anequilibriumrangewhereripplesteepnessremainsconstantandthewavelengthchangeswiththeshearstress;andabreak-orangeathighershearstresswhereripplesteepnessdecreases.Fortheequilibriumrange=cr<=crB,therippleheightandsteepnessaregivenbyEquations 3{7a and 3{7b ,respectively.Ripplecharacteristicsinthebreak-orange=cr>=crBarecomputedusingEquations 3{8a and 3{8b .Thebreak-orangeisdeterminedas=crB=1:8S0:6,whereSisadimensionlesssedimentparameterdenedinTable 3{3 .r=0:22A=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(0:16{7ar=r=0:16=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(0:04{7br=0:48AS0:8=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(1:5{8ar=r=0:28S0:6=cr)]TJ/F23 7.9701 Tf 6.5865 0 Td[(1:0{8bUnfortunately,accuratevaluesoftheboundaryshearstressarenotcommonlyknownapriori,noraretheycharacterizedbyonespecicvalue.Thesetwoshortcomingsmake 76

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theripplepredictorof GrantandMadsen 1982 acumbersomechoiceforbothestimatingandcomparingripplecharacteristics.3.4.4WibergandHarris1994Whenplottedagainsttherelativeorbitaldiameter2A=d, WibergandHarris 1994 WH94ndthatnondimensionalripplepropertiesfromeldandlabdatafallintouniquegroups.Theyproposethatalackofsubstantialoverlapinforcingconditionsbetweenlabregularwaves,short-periodowsandeldirregularwaves,long-periodowsdatamakesitdiculttoeectivelyclassifyripplesortocharacterizetheirheightandlength.Indeed,therearefewexistingripplepredictorsthatperformequallyaswellatpredictinglab-andeld-scaleripples.ThroughananalysisofthelaboratorydataofC69,KF65,MK72,D74,andtheelddataofI57andD74, WibergandHarris 1994 ndthatanorbitalripplewavelengthanofoundinelddataisindependentofthenearbedorbitalexcursions.Instead,theyndthatanoscalesproportionallywiththegraindiameterdrangingbetween400d
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Equation 3{10a ,anorbitalEquation 3{10b ,orsuborbitalEquation 3{10c :2A=ano<20orbitalripples {10a 2A=ano>100anorbitalripples {10b 20<2A=ano<100suborbitalripples {10c Ifthecomputedvalueof2A=anodoesnotfallintheanorbitalrange,thentheripplepropertiesmustthenbecomputedusingeitherorbitalEquations 3{11a and 3{11b orsuborbitalEquations 3{12a and 3{12b formulations. WibergandHarris 1994 suggestthatthewavelengthsofsuborbita