Sediment Transport Dynamics 1032380284, 9781032380285

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Sediment Transport Dynamics

This book focuses on the fundamentals of sediment transport in surface waters. It covers sediment properties, open channel flows, sediment particle settling, incipient motion, bed forms, bed load, suspended load, total load, cohesive sediments, water-sediment two-phase flows, hyperconcentrated flows, debris flows, wave-induced sediment transport, turbidity currents, and physical modeling. Besides the primary context of river sedimentation, this book extensively covers sediment transport under coexisting waves and currents in coasts and estuaries, hyperconcentrated and debris flows in rivers, as well as turbidity currents in lakes, reservoirs, channels, and the ocean. It includes a chapter on the water-sediment two-phase flow theory, which is considered the basis of many sediment transport models. It introduces some special topics that have emerged in recent years, such as the transport of mixed cohesive and noncohesive sediments, biofilm-coated sediments, and infiltrated sand within gravel and cobble beds. The text merges classical and new knowledge of sediment transport from various sources in English and non-English literature and includes important contributions made by many scientists and engineers from all over the world. It balances the breadth, depth, fundamental importance, practical applicability, and future advancement of the covered knowledge, and can be used as a text and reference book. The chapters are arranged in a useful sequence for teaching purposes. Certain homework problems are prepared, which also highlight the important topics for instructors to select. Solutions to homework problems are available from the author by request.

Sediment Transport Dynamics

Weiming Wu

A BALKEMA BOOK

Designed cover image: A braided reach of the Rakaia River, New Zealand (2009) Courtesy of Andrew Penny—CORE Education, New Zealand First published 2024 by CRC Press/Balkema Schipholweg 107C, 2316 XC Leiden, The Netherlands e-mail: [email protected] www.routledge.com—www.taylorandfrancis.com CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2024 Weiming Wu The right of Weiming Wu to be identified as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. ISBN: 978-1-032-38028-5 (hbk) ISBN: 978-1-032-38029-2 (pbk) ISBN: 978-1-003-34316-5 (ebk) DOI: 10.1201/9781003343165 Typeset in Times New Roman by Apex CoVantage, LLC

This book is dedicated to the memories of Prof. Jianheng Xie and Dr. Nick Kraus.

Contents

Preface List of Reviewers About the Author 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2

xviii xx xxii 1

Origins of Sediments  1 Classifications of Sediments  1 Sediment Transport Processes and Problems  2 Classifications of Water-Borne Sediment Loads  9 Sediment Yields in the World’s Rivers  10 Historical Development of Sediment Transport Dynamics  13 Coverage of This Book  16

Properties of Water and Sediment 2.1 Physical Properties of Water  18 2.1.1 Water Density and Specific Weight  18 2.1.2 Water Viscosity  18 2.2 Sediment Density and Specific Weight  19 2.3 Mineral Composition of Sediments  20 2.4 Electrochemical Properties of Sediment Particles  21 2.5 Geometric Properties of Sediment Particles  22 2.5.1 Particle Size  22 2.5.2 Particle Shape  24 2.6 Size Gradation of a Sediment Mixture  28 2.6.1 Size Distribution  28 2.6.2 Characteristic Diameters  30 2.6.3 Gradation and Nonuniformity  31 2.6.4 Measurements of Size Distribution  32 2.6.5 Bed Material Sampling  34 2.7 Porosity and Dry Density of Sediment Deposits  36 2.7.1 Initial Porosity and Dry Density  37 2.7.2 Variations in Porosity and Dry Density due to Consolidation  39

18

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2.7.3 Effects of Particle Filling and Packing on Porosity and Dry Density  40 2.7.4 Measurements of Porosity and Dry Density  45 2.8 Geotechnical Properties of Sediment Deposits  45 2.8.1 Angle of Repose  45 2.8.2 Shear Strength  48 2.9 Physical Properties of Sediment-Laden Water  49 2.9.1 Sediment Concentration and Transport Rate  49 2.9.2 Viscosity of Sediment-Laden Water  51 3

Open Channel Flows

53

3.1 Classifications of Open Channel Flows  53 3.2 Basic Hydrodynamic Equations  54 3.3 Turbulence Characteristics  55 3.3.1 Turbulence Cascade Process  55 3.3.2 Reynolds Shear Stress  57 3.3.3 Turbulence Intensity  58 3.3.4 Turbulence Dissipation Rate  59 3.3.5 Eddy Viscosity  60 3.4 Velocity Profiles of Uniform Turbulent Flows  61 3.4.1 Velocity Profile Near the Bed  61 3.4.2 Log Law of Clear Water Flows  63 3.4.3 Von Karman Coefficient in Sediment-Laden Flows  66 3.4.4 Log-Wake Law  69 3.4.5 Power Law  71 3.5 Average Velocity and Boundary Shear Stress of Uniform Flows  72 3.5.1 Uniform Flows in Simple Channels  72 3.5.2 Composite Roughness  74 3.5.3 Partition of Bed and Sidewall Shear Stresses  76 3.5.4 Measurements of Boundary Shear Stress  79 3.6 Three-Dimensional Flow Features  81 3.6.1 Flow Features in Straight Channels  81 3.6.2 Flow Features in Curved Channels  85 3.7 Coherent Structures in Turbulent Shear Flows  88 3.7.1 Inner-Scale Coherent Structures  88 3.7.2 Outer-Scale Coherent Structures  90 3.7.3 Effects of Coherent Structures on Sediment Transport  91 4

Settling of Sediment Particles 4.1 Settling Process of Sediment Particles  95 4.2 General Formula of Particle Settling Velocity  96 4.3 Settling Velocity of Spherical Particles  97

95

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4.4 Settling Velocity of Sediment Particles  99 4.4.1 Formulas Based on Particle Size  99 4.4.2 Formulas Based on Particle Size and Sphericity  103 4.4.3 Formulas Based on Particle Size, Sphericity, and Roundness  106 4.4.4 Comparison of Existing Formulas for Sediment Settling Velocity  109 4.5 Effects of Sediment Concentration on Settling Velocity  110 4.5.1 Settling Velocity of Uniform Sediments in Concentrated Water  110 4.5.2 Settling Velocity of Nonuniform Sediments in Concentrated Water  113 4.6 Effects of Turbulence on Sediment Settling Velocity  115 4.7 Measurements of Sediment Settling Velocity  117 4.8 Relationship Between Fall and Nominal Diameters of Sediment Particles  118 4.9 Time and Distance to Terminal Settling of Sediment Particles  119 Appendix 4.1  Settling Velocity of Common Natural Sediment Particles  122 5

Incipient Motion of Sediments 5.1 Drag and Lift Forces on Bed Sediment Particles  124 5.2 Incipient Motion Thresholds of Individual Sediment Particles  129 5.3 Approaches to Describing the Incipient Motion of Bed Particle Ensembles 131 5.3.1 Stochastic Approach for Entrainment Probability  131 5.3.2 Deterministic Approach to Incipient Motion Thresholds  133 5.4 Critical Average Velocity for the Incipient Motion of Uniform Sediments 136 5.5 Critical Shear Stress for the Incipient Motion of Uniform Sediments  138 5.5.1 Original Shields Diagram  138 5.5.2 Modified Shields Diagrams  140 5.5.3 Comparison of Original and Modified Shields Diagrams  142 5.6 Incipient Motion Thresholds of Nonuniform Sediments  144 5.7 Incipient Motion Thresholds of Sediment Particles on a Steep Slope  149 5.8 Other Factors Affecting the Incipient Motion of Sediment Particles  151 5.8.1 Effect of Particle Exposure on Incipient Motion  151 5.8.2 Effect of Particle Shape on Incipient Motion  152 5.8.3 Effect of Bed Roughness on Incipient Motion  153 5.8.4 Effect of Particle Submergence on Incipient Motion  154 5.8.5 Effect of Turbulence on Incipient Motion  156 5.8.6 Effect of Impulse Duration on Incipient Motion  157 5.9 Probabilities of Sediment Entrainment  158 5.9.1 Entrainment Probability Formulas Based on Lift Force  158 5.9.2 Entrainment Probability Formulas Based on Flow Velocity  159

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5.9.3 Entrainment Probability Formulas Based on Bed Shear Stress 161 5.9.4 Entrainment Probability Formulas Based on Bivariate Distributions 163 5.9.5 Particle Rolling and Lifting Probabilities  168 5.9.6 Entrainment Probability of Nonuniform Sediments  169 Appendix 5.1 Comparison of Incipient Rolling, Sliding, and Lifting Thresholds of Spheres  170 Appendix 5.2 Comparison of Reference Thresholds of Sediment Incipient Motion  172 6

Bed Forms

175

6.1 Classifications of Bed Forms  175 6.1.1 Classification of Small-Scale Bed Forms in Sand-Bed Rivers  175 6.1.2 Classification of Large-Scale Bed Forms in Alluvial Rivers  178 6.1.3 Classification of Bed Forms in Mountainous Rivers  180 6.2 Development Mechanisms and Regimes of Bed Forms  182 6.2.1 Formation Theories for Sand Ripples, Dunes, and Antidunes  182 6.2.2 Regime Diagrams of Small-Scale Sandy Bed Forms  185 6.2.3 Stability Models and Regime Diagrams of Bars in Alluvial Channels  188 6.2.4 Formations and Regime Diagrams of Coarse-Grained Bed Forms  190 6.3 Dimensions and Speeds of Equilibrium Bed Forms  191 6.3.1 Sand Ripples  191 6.3.2 Sand Dunes  192 6.3.3 Sand Antidunes  197 6.3.4 Alternate Bars  198 6.3.5 Transverse Ribs, Riffle-Pool Sequences, and Step-Pool Systems  199 6.4 Partition of Grain and Form Resistances  200 6.5 Resistance Formulas for Sand-Bed Rivers  203 6.6 Resistance Formulas for Gravel- and Cobble-Bed Rivers  210 6.7 Comments on Movable Bed Roughness Formulas  212 7

Bed-Load Transport 7.1 7.2 7.3 7.4

Bed-Load Regimes  215 Bed-Load Particle Dynamics  216 Average Characteristics of Bed-Load Particle Motion  218 Unisized Bed-load Transport Capacity  221 7.4.1 Bed-Load Formulas Based on Hydraulic Traction  222 7.4.2 Bed-Load Formulas Based on Mass Flux  226 7.4.3 Bed-Load Formulas Based on Stream Power  229

215

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7.5 7.6 7.7 7.8 7.9

8

7.4.4 Bed-Load Formulas Based on Bed-Form Migration  232 7.4.5 Bed-Load Formulas Based on Stochastic Theory  234 Multisized Bed-Load Transport Capacity  241 Comparison of Bed-Load Formulas  248 Effect of Steep Slope on Bed Load  250 Fluctuations of Bed Load  251 Measurements of Bed Load  254 7.9.1 Bed-Load Samplers  254 7.9.2 Bed-Load Discharge Measurements  258

Suspended-Load Transport

261

8.1 8.2 8.3 8.4

Suspended-Load Transport Processes  261 Criteria of Incipient Suspension  262 Time-Averaged Transport Equation for Suspended Load  265 Vertical Distribution of Suspended-Load Concentration  266 8.4.1 Theoretical Concentration Profiles  266 8.4.2 Correction Factor βs for Sediment Diffusivity  269 8.5 Depth-Averaged Concentration of Suspended Load  273 8.6 Near-Bed Concentration of Suspended Load  275 8.7 Transport Capacity of Suspended Load  278 8.8 Nonequilibrium Transport of Suspended Load  283 8.9 Measurements of Suspended Load  285 8.9.1 Suspended-Load Samplers  285 8.9.2 Measurements of Suspended-Load Discharge  288 8.9.3 Estimations of Unmeasured Load  289 8.9.4 Surrogate Measurements of Suspended Load  291 Appendix 8.1  Calculations of Einstein Integrals  292 9

Total-Load Transport 9.1 9.2 9.3 9.4 9.5

Total Transport Capacity of Bed-Material Load  296 Fractional Transport Capacity of Bed-Material Load  301 Comparison of Bed-Material Load Formulas  304 Sediment Rating Curve  307 Pickup Rate of Noncohesive Sediments  309 9.5.1 Sediment Pickup Rate under Uniform Flows  310 9.5.2 Sediment Pickup Rate under Nonuniform Flows  313 9.6 Wash Load  315 9.6.1 Division of Bed-Material Load and Wash Load  315 9.6.2 Estimation of Wash Load  317 9.7 Infiltrated Sand Transport in Immobile Gravel and Cobble Beds  319 9.7.1 Infiltration of Fine Sediment into a Coarse-Grained Bed  319 9.7.2 Hydrodynamics Near and in Gravel and Cobble Beds With Infiltrated Sand  319

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9.7.3 Transport Rate of Infiltrated Sand in Gravel and Cobble Beds  320 9.7.4  Cleanout Depth of Sand in Gravel Beds  321 10 Cohesive Sediment Transport 10.1  Distinctive Features of Cohesive Sediments  325 10.1.1  Cohesive Sediment Transport Processes  325 10.1.2 Vertically Layered Concentration Profile of Cohesive Sediments 326 10.1.3  Characteristics of Flocs  327 10.2  Settling and Deposition of Cohesive Sediments  330 10.2.1  Factors Affecting Floc Settling Velocity  330 10.2.2 Formulas for Floc Settling Velocity Considering Multiple Factors 334 10.2.3  Deposition Rate of Cohesive Sediments  335 10.3  Erosion of Cohesive Sediments  336 10.3.1  Erosion Modes of Cohesive Sediments  336 10.3.2  Cohesive Forces on Bed Sediment Particles  337 10.3.3  Threshold Conditions for Incipient Erosion  339 10.3.3.1 Thresholds for Surface Erosion of Cohesive Sediments  339 10.3.3.2 Unified Erosion Thresholds of Cohesive and Noncohesive Sediments  341 10.3.4  Surface Erosion Rate of Cohesive Sediments  342 10.3.4.1  Linear Functions of Erosion Rate  342 10.3.4.2  Nonlinear Functions of Erosion Rate  345 10.3.5  Mass Erosion Rate of Cohesive Sediments  345 10.3.6  Fluid Mud Entrainment  347 10.3.7  Measurements of Cohesive Sediment Erodibility  347 10.4  Exclusive Versus Continuous Erosion and Deposition  349 10.5  Transport of Mixed Cohesive and Noncohesive Sediments  350 10.5.1  Settling of Mixed Cohesive and Noncohesive Sediments  350 10.5.2 Erosion Threshold of Mixed Cohesive and Noncohesive Sediments 352 10.5.3 Erosion Rate of Mixed Cohesive and Noncohesive Sediments 354 10.5.4 Transport Rate and Bed Forms of Mixed Cohesive and Noncohesive Sediments  356 10.6  Biological Effects on Sediment Transport  357 10.6.1  Effects of Benthic Organisms on Sediment Stability  357 10.6.2  Flocculation and Settling of Biofilm-Coated Sediments  359 10.6.3 Erosion of Sediment Beds Affected by Biofilms  361 10.6.3.1  Erosion Modes of Sediments with Biofilms  361

325

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10.6.3.2 Incipient Erosion Thresholds of Sediments with Biofilms  361 10.6.4 Transport Rate and Bed Forms of Biofilm-Coated Sediments 363 10.7 Comments on Cohesive Sediment Transport  364 11 Sediment-Laden Two-Phase Flows

366

11.1  Two-Phase Perspective of Sediment-Laden Flows  366 11.1.1  Controlling Parameters for Fluid-Particle Interactions  366 11.1.2  Experimental Observations  367 11.1.3  Mathematic Descriptions  368 11.2  Two-Fluid Model of Sediment-Laden Flows  369 11.2.1  Governing Equations of the Two-Fluid Model  369 11.2.2  Pressure Actions in the Two-Fluid Model  370 11.2.3  Reynolds-Averaged Equations for the Two-Fluid Model  371 11.2.4  Favre-Averaged Equations for the Two-Fluid Model  372 11.2.5  Closures of the Two-Fluid Model  373 11.2.6  Interphase Forces Between Water and Sediment  375 11.3  Mixture Model of Sediment-Laden Flows  377 11.3.1  Governing Equations of the Mixture Flow Model  377 11.3.2  Constitutive Relationships of the Mixture Flow Model  378 11.3.3  Interphase Velocity Lag Equation  378 11.3.4 Reynolds-Averaged Equations for the Mixture Flow Model  379 11.3.5  Favre-Averaged Equations for the Mixture Flow Model  380 11.4  Applications to Steady Uniform Sediment-Laden Flows  381 11.4.1  Mean Velocities and Shear Stresses  381 11.4.2  Vertical Profile of Streamwise Velocity Lag  383 11.4.3  Vertical Profile of Sediment Concentration  386 12 Hyperconcentrated and Debris Flows 12.1  Classifications of Hyperconcentrated and Debris Flows  393 12.2  Rheology of Hyperconcentrated Sediment Slurries  395 12.2.1  Bingham Viscoplastic Fluid Model  395 12.2.2  Granular Flow Models  398 12.2.3  Quadratic Rheological Model  399 12.2.4  Herschel-Bulkley Rheological Model  400 12.3  Uniform Muddy Flows  401 12.3.1  State of Muddy Flows  402 12.3.2  Velocity and Resistance of Laminar Muddy Flows  403 12.3.3  Velocity and Resistance of Turbulent Muddy Flows  406 12.3.4  Suspended Load in Turbulent Muddy Flows  408 12.4  Uniform Granular Flows  410

393

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12.5 

12.6 

12.7 

12.8 

12.4.1  State of Granular Flows  410 12.4.2  Initiation of Granular Debris Flows  411 12.4.3 Velocity and Resistance of Laminated Granular Debris Flows 412 12.4.4 Velocity and Resistance of Turbulent Granular Debris Flows 413 12.4.5 Sediment Concentration Distribution of Granular Debris Flows 414 12.4.6  Sediment Transport Capacity of Granular Debris Flows  415 Uniform Mixed-Type Hyperconcentrated and Debris Flows  416 12.5.1  Settling Velocity of Coarse Particles in a Bingham Fluid  416 12.5.2  Division of Fine and Coarse Particles  418 12.5.3  Velocity and Resistance of Mixed-Type Flows  419 12.5.4  Incipient Motion of Coarse Sediments in Muddy Slurries  422 12.5.5 Transport Capacity of Coarse Sediments in Muddy Slurries 422 Gradually-Varied Hyperconcentrated and Debris Flows  423 12.6.1  General Model Formulations  423 12.6.2  Application to Runout Distance of Debris Flows  425 Rapidly-Varied Hyperconcentrated and Debris Flows  426 12.7.1  Bottom-Tearing Erosion by Hyperconcentrated Flows  426 12.7.2 Intermittent Motions of Hyperconcentrated and Debris Flows 427 12.7.3  Impact of Debris Flows on Structures  430 12.7.4 Superelevation of Hyperconcentrated and Debris Flows in Channel Bends  433 Morphological Features of Hyperconcentrated and Debris Flows  434 12.8.1  Morphological Features of Hyperconcentrated Flows  434 12.8.2  Morphological Features of Debris Flows  434

13 Coastal Sediment Transport 13.1  Wave Propagation  438 13.1.1  Types of Water Waves  438 13.1.2  Small Amplitude Waves  439 13.1.3  Finite Amplitude Waves  442 13.1.4  Characteristics of Irregular Waves  443 13.1.5  Wave Reflection and Diffraction  446 13.1.6  Wave Shoaling and Refraction  446 13.1.7  Wave Breaking  449 13.1.8  Wave Radiation Stress  451 13.2  Wave Boundary Layer and Bed Shear Stress  452 13.2.1  Near-Bed Orbital Velocity of Waves  452

438

Contents xv

13.2.2  Oscillatory Flow in the Wave Boundary Layer  454 13.2.3  Wave Boundary Layer Streaming  456 13.2.4  Profile of Turbulent Current Velocity With Wave Effects  458 13.2.5 Bed Shear Stress  460 13.2.5.1  Bed Shear Stress due to Waves  460 13.2.5.2  Bed Shear Stress due to Currents  462 13.2.5.3 Bed Shear Stress due to Combined Currents and Waves  462 13.3  Sediment Transport in General Wave Environments  465 13.3.1  Incipient Motion of Sediment Particles  465 13.3.2  Bed Forms and Roughness  467 13.3.3  General Description of Sediment Transport by Waves  473 13.3.4  Bed-Load Transport  474 13.3.5  Suspended-Load Transport  480 13.3.5.1  Concentration Profile of Suspended Load  480 13.3.5.2  Near-Bed Concentration of Suspended Load  482 13.3.5.3  Transport Rate of Suspended Load  483 13.3.6  Cohesive Sediment Transport  485 13.4  Wave-Induced Currents and Sediment Transport in the Surf Zone  486 13.4.1  Wave Setdown and Setup  486 13.4.2  Wave-Induced Nearshore Currents  488 13.4.3  Cross-Shore Sediment Transport  489 13.4.4  Longshore Sediment Transport  490 13.5  Wave Motions and Sediment Transport in the Swash Zone  492 13.5.1  Wave Runup  492 13.5.2  Sediment Transport by Wave Swash  493 14 Turbidity Currents 14.1  General Considerations  496 14.1.1  Causes of Turbidity Currents  496 14.1.2  Effective Gravity  498 14.1.3  Flow Regimes and Classifications  498 14.2  Uniform Turbidity Currents  500 14.2.1  Hydrostatic Pressure  500 14.2.2  Current Velocity Profile  501 14.2.3  Boundary Resistance  502 14.2.4  Partition of Bed Grain and Form Resistances  503 14.2.5  Sediment Transport  504 14.2.6  Interfacial Stability  507 14.2.7  Critical Bed Slope  507 14.3  Gradually-Varied Turbidity Currents  508 14.3.1  Governing Equations  508

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14.4 

14.5 

14.6 

14.7 

14.3.2  Ambient Fluid Entrainment  509 14.3.3  Sediment Erosion and Deposition  510 14.3.4  Reduction to Steady Turbidity Currents  511 14.3.5  Simplified Solutions of Depositional Turbidity Currents  512 Rapidly-Varied Turbidity Currents  513 14.4.1  Head Velocity  513 14.4.2  Lateral Spreading  515 14.4.3  Hydraulic Jump  517 14.4.4  Sudden Expansion  518 14.4.5  Retention by Obstacles  519 14.4.6  Local Head Loss  520 Turbidity Currents in Reservoirs  521 14.5.1  Plunge at Reservoir Entrances  521 14.5.2  Propagation and Deposition in Reservoirs  524 14.5.3  Venting Through Orifice Outlets  524 Turbidity Currents in Channels Open to Sediment-Laden Water  526 14.6.1  Formation at Channel Entrances  526 14.6.2  Propagation and Deposition in Channels  528 Turbidity Currents in Seas and Lakes  529 14.7.1  Formation at River Estuaries  529 14.7.2  Other Formation Mechanisms  531 14.7.3  Propagation and Sedimentation in Seas and Lakes  531

15 Physical Modeling and Similitude 15.1  General Similitude Theories and Analytical Methods  534 15.1.1  General Similitudes Between Model and Prototype  534 15.1.2  Buckingham Π Theorem and Dimensional Analysis  535 15.1.3  Deduction of Similitudes from Governing Equations  536 15.2  Scale Models of Open Channel Flows  537 15.2.1  Similitude Criteria Required by Governing Equations  537 15.2.2  Similitude Criteria Required by Boundary Conditions  541 15.2.3  Limitation of Vertically Distorted Models  542 15.2.4  Incomplete Reynolds Similitude  543 15.2.5  Adjustment of Bed Roughness in Fixed-Bed Models  544 15.2.6  Deviations From Froude and Friction Similitudes  546 15.2.7  Design Procedure of Flow Models  546 15.3  Scale Models of Sediment Transport Over Alluvial Beds  546 15.3.1  Similitude of Sediment Incipient Motion  546 15.3.1.1 Similitude Criterion Based on Critical Average Velocity  547 15.3.1.2 Similitude Criterion Based on Critical Shear Stress 547

534

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15.3.2  Similitude of Sediment Suspension  548 15.3.3  Similitude of Sediment Transport Capacity  550 15.3.3.1  General Considerations  550 15.3.3.2  Similitude of Bed-Load Transport Capacity  552 15.3.3.3 Similitude of Suspended-Load Transport Capacity 553 15.3.3.4 Similitude of Total-Load Transport Capacity  553 15.3.4  Similitude of Bed Changes  554 15.3.5  Modeling of Nonuniform Sediment Transport  555 15.3.6  Similarity of Bed Roughness in Movable-Bed Models  556 15.3.7  Selection of Model Sediments and Relevant Similitudes  557 15.4  Scale Models of Other Sediment Transport Processes  559 15.4.1  Similitude of Local Scour Near Hydraulic Structures  559 15.4.2  Similitude of Cohesive Sediment Erosion and Transport  560 15.4.3  Similitude of Bank Erosion  561 15.4.4  Similitude of Turbidity Currents  563 15.5  Comments on Scale Models of Flow and Sediment Transport  564 Notations Symbols  568 Subscripts 574 Units  574 References Index

568

575 638

Preface

Sediments widely exist in surface waters on Earth. They are eroded from uplands and transported through rivers down to lakes, reservoirs, estuaries, coastal zones, and the ocean. They affect channel conveyance, flood control, water supply, irrigation, navigation, power generation, environment quality, aquatic habitats, and more. Scientists and engineers have gradually elucidated the processes of sediment transport and established basic principles and analytical methods to solve sedimentation problems. These are described by the sediment transport dynamics introduced in this book. The present book is a companion text of my first book, “Computational River Dynamics”. It focuses on the fundamentals of sediment transport in surface waters, covering open channel flows, sediment properties, particle settling, incipient motion, bed forms, bed load, suspended load, total load, cohesive sediments, hyperconcentrated flows, debris flows, wave-induced sediment transport, turbidity currents, water-sediment two-phase flows, and physical modeling. Due to the length limit, it does not cover practical sedimentation problems, which are left for a possible future project. This book is written primarily as a text and reference book for students, scholars, and practitioners of sedimentation engineering. Certain contents are derived from existing books by Graf (1971), Yalin (1972), Vanoni (1975), Xie (1981), Chien and Wan (1983, 1999), Simons and Senturk (1992), van Rijn (1993), Yang (1996), Garcia (2008), Julien (2010), and Dey (2014), but I take a step further by merging classical and new knowledge from various sources in English and non-English literature. It has been a great challenge for me to comprehend and compile this vast literature into a book form. I intend to be inclusive, while balancing the breadth, depth, fundamental importance, practical applicability, and future advancement of the covered knowledge. For teaching purposes, I have prepared certain homework problems, which highlight the important topics for instructors to select. My fascination with sediment transport began with my MS and PhD supervisor, Prof. Jianheng Xie. Under his guidance, I had been very fortunate to participate in studies of sedimentation problems with several hydraulic projects on the Yangtze River, and had gained firsthand experience in sedimentation engineering. My writing of the present book was inspired by the Chinese textbook “River Sedimentation Engineering” edited by Prof. Xie and his colleagues in 1981. My later knowledge of sediment transport by waves was credited to Dr. Nick Kraus, a great scientist in coastal engineering. His immense support encouraged me to cross the boundary of river engineering to coastal engineering. Though this book mainly covers river sedimentation problems, it is complemented with an extensive chapter on coastal sediment transport. I would like to dedicate this book to the memories of Prof. Jianheng Xie and Dr. Nick Kraus.

Preface xix

I am greatly indebted to Prof. Wolfgang Rodi, Prof. Sam S.Y. Wang, and Prof. Hung Tao Shen for their guidance, collaboration, and encouragement throughout my career. I am very grateful to Prof. James K. Edzwald for his honorable endowment, entitling me Chair Professor for Water Engineering at Clarkson University. I sincerely thank my colleagues and friends all over the world for their care and encouragement. I am also very grateful to the many experts who reviewed this manuscript and provided important suggestions. They are specially acknowledged on the list of reviewers. I am solely responsible for any imperfections that may remain in the text. Many thanks go to Senior Publisher Mr. Janjaap Blom, Assistant Editors Mr. Kaustav Ghosh and Ms. Jahnavi Vaid, and the production team of Ms. Jennifer Hicks, Mr. Balaji Karuppanan, and others from Taylor & Francis/CRC Press for their guidance during the preparation and publication of this text. I would like to extend my sincere thanks to Mr. Barrie Matthews, who helped find the beautiful cover picture from the Rakaia River, New Zealand. Sincere thanks also to the many friends who provided images, data, permissions, and assistance with figures and tables cited in this book. These individuals include Goodarz Ahmadi, Abdul-Sahib T. Al-Madhhachi, Christophe Ancey, Paolo Billi, John M. Buffington, Hervé Capart, Yuehong Chen, Tjalling de Haas, David Dennis, Valentin Glosov, Pamela Green, Philippa Green, Robert Grabowski, Ernst Hauber, Guojian He, Bernd Hentschel, Nils Peter Huber, A. Paul Jenkin, Junhua Li, Zhiwei Li, Jing Luo, Chao Ma, Ashish Mehta, David M. Paterson, Octavio E. Sequeiros, Estibaliz Serrano, Xiaonan Tang, Moritz Thom, Dario Ventra, John W.F. Waldron, Desmond Walling, Mengzhen Xu, Hao Zhang, and Hongping Zhang, among many more. Others are acknowledged for their provision of figures and tables. Special thanks go to my wife Ling and daughter Siyuan, who gave me tremendous support during this endeavor. Weiming Wu Clarkson University, May 2023

Reviewers

Prof. Abul B. Baki, Clarkson University, USA Dr. Benoît Camenen, River Hydraulics—INRAE, France Prof. Dong Chen, Institute of Geographic Sciences and Natural Resources Research, CAS, China Prof. Niansheng Cheng, Zhejiang University, China Prof. Panayiotis Diplas, Lehigh University, USA Prof. Xiping Dou, Nanjing Institute of Hydraulic Research, China, Dr. Kamal El Kadi Abderrezzak, Électricité de France (EDF), France Prof. Robert Ettema, Colorado State University, USA Prof. Junke Guo, University of Nebraska—Omaha, USA Prof. Zhiguo He, Zhejiang University, China Prof. Peng Hu, Zhejiang University, China Dr. Jianchun Victor Huang, Bureau of Reclamation, USA Prof. Pierre Y. Julien, Colorado State University, USA Dr. Roger A. Kuhnle, USDA-ARS National Sedimentation Laboratory, USA Dr. Eddy J. Langendoen, USDA-ARS National Sedimentation Laboratory, USA Prof. Magnus Larson, Lund University, Sweden Prof. Wei Li, Zhejiang University, China Prof. Xiaofeng Liu, Penn State University, USA Prof. Liqun Lyu, Beijing Forestry University, China Prof. Reza Marsooli, Stevens Institute of Technology, USA Prof. Kim Dan Nguyen, Saint-Venant Laboratory for Hydraulics, Ecole des Ponts, EDF, France Prof. Prem Lal Patel, Sardar Vallabhbhai National Institute of Technology, Surat, India Dr. Chamil Perera, WSP Global, USA

Reviewers xxi

Prof. Damien Pham van Bang, Université du Québec, Canada Dr. Alejandro Sanchez, Hydrologic Engineering Center, US Army Corps of Engineers, USA Prof. Hung Tao Shen, Clarkson University, USA Prof. Anping Shu, Beijing Normal University, China Dr. Daniel G. Wren, USDA-ARS National Sedimentation Laboratory, USA Prof. Junqiang Xia, Wuhan University, China Prof. Abbas Yeganeh-Bakhtiary, Iran University of Science and Technology, Iran Prof. Deyu Zhong, Tsinghua University, China

About the Author

Dr. Weiming Wu is James K. Edzwald Professor of Water Engineering at Clarkson University, NY, USA. Dr. Wu earned his PhD from Wuhan University of Hydraulic and Electric Engineering, China in 1991. He was Lecturer/Associate Professor at his alma mater in 1991–1995; Research Fellow of the Alexander von Humboldt Foundation at the Institute for Hydromechanics, University of Karlsruhe, Germany in 1995–1997; and a faculty member at the National Center for Computational Hydroscience and Engineering of the University of Mississippi in 1997–2013. His research interests include fundamental sediment transport; hydro- and morphodynamics in rivers, estuaries, coastal waters and uplands; surge and wave attenuation by vegetation; interaction between surface and subsurface flows; free surface flow and sediment transport modeling; dam/levee breach and flood modeling; and water quality and aquatic ecosystem/ecotoxicology modeling. He has developed a suite of computational models for flow, sediment transport, pollutant transport, and aquatic ecology in riverine and coastal waters. He authored the book Computational River Dynamics, published through Taylor & Francis, UK in November 2007. In addition, he has published more than 150 articles in journals and conferences. He received a Best Paper Award in 2007 from the World Association for Sedimentation and Erosion Research (WASER). He is a fellow of American Society of Civil Engineers (ASCE) and a member of the International Association for Hydro-Environment Engineering and Research (IAHR). He served as Associate Editor for the International Journal of Sediment Research in 2008–2010 and for the ASCE Journal of Hydraulic Engineering in 2010–2019, and was Chair of the ASCE Computational Hydraulics Committee (2010–2012), the ASCE Task Committee on Dam/Levee Breaching (2009–2012), and the ASCE Sedimentation Committee (2016–2018). He currently serves as Vice President for WASER.

Chapter 1

Introduction

This chapter briefly introduces the origins and classifications of sediments, sediment transport processes and problems in surface waters, typical sediment transport modes in rivers, sediment loads in the world’s rivers, the developmental history of sediment transport dynamics, and the coverage of this book. 1.1  Origins of Sediments Sediments are solid particles that are broken down from rocks by processes of weathering and erosion. They may be transported to depositional sites by the actions of water, air, ice, or gravity. Sediments are often considered synonymous with soils, but the emphasis on “transport” distinguishes sediments from general soils. Weathering involves disintegration (physical weathering) and decomposition (chemical weathering). Disintegration describes all the physical processes by which rocks are broken into smaller pieces, including fracturing and fragmenting of rocks by the disruptive forces of temperature change, frost, pressure change, and diastrophism, as well as abrasion of rocks by moving water, ice, wind, and solid particles. Rocks can also be disintegrated by plant root growth, lichen pitting, and animal burrowing. On the other hand, decomposition produces secondary minerals, such as clay minerals, silica, calcite, and oxides, as rock minerals react with the weak carbonic acid, oxygen, and other chemical agents in water and air. In addition, biochemical processes produce organic matter in soils. Typically, sediments are eroded from uplands and transported by water flows through rivers to lakes, reservoirs, and seas, as well as by waves and currents in coastal zones. Dust storms and sand dunes in open lands (e.g., deserts) are examples of wind-transported sediments, and glacial deposits and tills are examples of ice-transported sediments. On a steep slope, sediments may fall or move down by the action of gravity. This phenomenon is called an avalanche or slide. Among these processes, transport by water contributes the majority of sediment fluxes on the Earth’s surface and has been the main focus of sedimentation research and practice. 1.2  Classifications of Sediments Sediments are classified as boulder (256–4,096 mm in size), cobble (64–256 mm), gravel (2–64 mm), sand (0.0625–2 mm), silt (0.004–0.0625 mm), and clay (0.00024–0.004 mm) based on the size grade scale in Table 1.1. Each class is further divided into several subclasses, such as coarse, medium, and fine. This grade scale is extended from that suggested by Wentworth (1922) and commonly used in sedimentation engineering (Vanoni, 1975). DOI: 10.1201/9781003343165-1

2 Introduction Table 1.1  Grade scale of sediments (data from Vanoni, 1975) Class

Size range (mm)

Class

Size range (mm)

Very large boulder Large boulder Medium boulder Small boulder

4,096 2,048 1,024 512 –

Large cobble Small cobble

256 – 128 128 – 64

Very coarse gravel Coarse gravel Medium gravel Fine gravel Very fine gravel

64 – 32 32 – 16 16 – 8 8–4 4–2

Very coarse sand Coarse sand Medium sand Fine sand Very fine sand Coarse silt Medium silt Fine silt Very fine silt Coarse clay Medium clay Fine clay Very fine clay

2–1 1 – 0.5 0.5 – 0.25 0.25 – 0.125 0.125 – 0.0625 0.0625 – 0.031 0.031 – 0.016 0.016 – 0.008 0.008 – 0.004 0.004 – 0.002 0.002 – 0.001 0.001 – 0.0005 0.0005 – 0.00024

– 2,048 – 1,024 – 512 256

In geosciences, sediment size is also defined by the φ scale (Krumbein and Aberdeen, 1937) as follows: Dmm  2 , or    log 2 Dmm (1.1) where Dmm is sediment size in mm. When Dmm is 1 mm, φ is 0, and when Dmm is 0.0625 mm, φ is 4. The grade scale shown in Table 1.1 can be well described with the φ scale. The sizes -8φ, -6φ, -1φ, 4φ, and 8φ approximately divide the boulder, cobble, gravel, sand, silt, and clay categories. Note that the division of silt and clay is 0.004 mm (4 μm) in Table 1.1, whereas it is 0.002 mm in the soil size scale suggested by the International Organization for Standardization (ISO, 2002). One should be aware of this difference when dealing with quantities of silt and clay. In addition, the International System of Units (SI) is used in this text. The related abbreviations are described in the section “Units”. Fine-grained sediments, such as silt and clay, exhibit cohesive properties and are called cohesive sediments. They tend to form flocs (aggregates) due to interparticle electrostatic forces greater than that of gravity (see Section 2.4). Coarse-grained sediments, such as boulder, cobble, gravel, and sand, are noncohesive. The threshold size that divides cohesive and noncohesive sediments is often set at about 0.03 (or 0.01–0.06) mm. These two types of sediment exhibit significantly different dynamic behaviors. Cohesive sediment transport is mainly discussed in Chapter 10, whereas noncohesive sediment transport is covered in other chapters of this text. 1.3  Sediment Transport Processes and Problems Sediment transport in surface waters generally comprises erosion, transportation, and deposition. Erosion is the removal or detachment of sediments from the bed, whereas deposition is the settling of sediments to the bed. Erosion occurs when the flow strength increases locally or when the upstream sediment loading decreases to a certain level. Deposition occurs conversely. Transportation (also termed “transport” for short) includes advection (or convection) with flow velocity, diffusion by turbulence, and collision between particles. Noncohesive sediments usually move in particles,

Introduction 3

whereas cohesive sediments move in flocs that aggregate and disaggregate continuously. Cohesive sediment deposits undergo consolidation due to the pressure of the overlying water and sediments. Note that the term “sedimentation” often includes all the aforementioned sediment transport processes. For example, sedimentation engineering is the engineering discipline that deals with all sediment-related processes and problems (e.g., Vanoni, 1975; Garcia, 2008). However, sometimes sedimentation refers only to the deposition of sediments as the opposite of erosion (e.g., Julien, 2010). The latter usage of sedimentation is avoided in the present book to reduce confusion. Because of different hydrodynamic and geomorphological settings, sediment transport in uplands, rivers, reservoirs, estuaries, and coastal zones exhibit distinct features and problems, as briefly described in the following text. Erosion in Watersheds In a rainfall event, unprotected upland soil is disturbed by raindrops and transported by a thin surface flow with erosive capacity due to turbulence enhanced by raindrop impacts. This process is called interrill or sheet erosion. Surface flows are concentrated in low areas, where erosion creates narrow and shallow channels called rills (Figure 1.1a). If uncontrolled, rills in regularly eroded areas may eventually develop into large, incised channels called gullies (Figure 1.1b). Interrill, rill, and gully erosion can result in significant soil loss on uplands. Soil erosion is often intensified due to poor agricultural practices and land vegetation removal. A part of the eroded soil may be deposited in lowland areas, depressions, and ponds, while the rest is transported as sediment load to downstream rivers. The sediment delivery ratio is defined as the sediment load divided by the total eroded soil. It is related to factors such as soil characteristics, surface slope, vegetation, and weather. Serious erosion on uplands can cause

Figure 1.1 (a) Rill and interrill erosion on a winter cereal field in Rottingdean, East Sussex, UK, October 1987 (from Boardman, 2013), and (b) gully erosion on a pasture field (photo courtesy of Natural Resources Conservation Service, USA)

4 Introduction

significant deposition in downstream river systems, and thus rational watershed management is essential for both uplands and river systems. Sediment Transport in Rivers Rivers are the passages that transport runoff and sediments from the land to the ocean. River channels are created by water flows over the land surface. A river channel tends to evolve toward an equilibrium state between sediment transport capacity and upstream sediment load, but this state continually changes because of the impacts of human activities and variations in natural environmental conditions. Sediments may be eroded from or deposited to the bed when the flow becomes stronger or weaker. Bank erosion occurs due to hydraulic shearing at the toe and slope failure driven by gravity and other forces (Figure 1.2). Therefore, channel planforms can migrate, meander, and bifurcate. These changes may cause land loss, flooding, levee breach, instream structure failure, aquatic habitat loss, etc. Once adverse impacts occur, training, mitigation, and restoration are needed to change river systems to more favorable, stable states. Figure 1.3 shows that wing dikes are used to help maintain the shipping channel in a bend near the Mississippi River mile 100. In such a bend, sediment transport may cause the channel to become too shallow for ship navigation. River training structures, including wing dikes and bendway weirs, are used to concentrate the flows and create deep channels. A combination of structures, revetments, and dredging keep the river navigable. Reservoir Sedimentation Sediments carried from upstream rivers are likely deposited in a reservoir due to the reduction in flow strength after the water level increases. Coarse sediments settle first and may form a delta in the upper part of the reservoir. Fine sediments are transported in suspension and deposited in the middle and lower parts. The sediment-laden water may plunge and form a density current called turbidity current, which can carry fine sediments a long distance to the deep area near the

Figure 1.2  Devastating impact of bank erosion (from Binzaid and Chowdhury, 2014)

Introduction 5

Figure 1.3 Wing dikes have been used since the 1830s to help maintain the shipping channel near Mississippi River mile 100 (photo courtesy of U.S. Army Corps of Engineers)

Figure 1.4  Reservoir sedimentation processes

dam (Figure 1.4). With time, the deposition wedge grows upstream, raising the water level in the tail region of the reservoir, and extends downstream toward the dam, reducing the reservoir storage capacity. In the meantime, reservoir detention causes sediment deficit and erosion in downstream channels. The reservoir deposition and downstream channel erosion processes, along with the resulting equilibrium bed profiles, are topics of concern. Measures such as sediment flushing, bypass, and dredging are often used to reduce reservoir deposition and prolong the reservoir lifespan. After a reservoir reaches the equilibrium state, the efficiencies of flood control, power generation, water supply, and sediment detention are significantly reduced, and handling the deposited sediments becomes an important issue for the decommission and rehabilitation of an aged dam (Figure 1.5).

6 Introduction

Erosion and Deposition Near Instream Structures Water intake structures and navigation facilities are susceptible to sediment deposition. Coarse sediments can damage power generators, spillways, and sluice gates. Appropriate designs and mitigation techniques are often needed to reduce sediment deposition around or transport through these structures. For example, a water intake structure should be placed in a location where less sediment deposition occurs, such as on the outer bank of a channel bend.

Figure 1.5 Sedimentation in the Matilija Reservoir, Ojai, Calif., USA. The reservoir, built in 1947, has lost its capacity for water storage and flood control and is awaiting removal (photo courtesy of A. Paul Jenkin, Matilija Coalition)

Figure 1.6 Scour near bridge piers

Introduction 7

Bridge piers, abutments, spur dikes, and weirs change water flows significantly and induce considerable local erosion (Figure 1.6). Erosion also occurs due to jet impingement downstream of sluice gates, spillways, and overfalls. Local erosion is the major reason for the failure of many instream structures. Because of the complexity of the physical processes involved, the prediction and prevention of local erosion around such structures are very challenging. Sediment Problems in Estuaries Morphodynamic processes under the actions of river and tidal flows in estuaries are very complex (Figure 1.7). In the upper reaches of an estuary, strong tides push salinity upriver beneath the outflowing river water and resuspend sediments and other particulate materials from the bed. The flocculation of cohesive sediments transported by the river flow is intensified when they come into contact with the salt wedge pushing its way upstream. These two processes result in elevated concentrations of suspended particulate materials, a phenomenon called estuarine turbidity maximum. A large amount of fine-grained sediments coming from the river tend to be deposited in the lower estuarine regions, forming river delta and mouth bars. Fine-grained sediments also enter harbors and settle there. Training and dredging works are necessary to maintain the navigation channels. The turbid river plume brings sediments into the ocean, and a portion of these sediments are transported along the coastal shores. Figure 1.8 shows an image of the Mississippi River Estuary and Delta on October 10, 2001. The Mississippi River has changed courses numerous times over the past 10,000 years, wandering across a roughly 320 km range along the Gulf of Mexico. It probably settled on its current course about six centuries ago (http://visibleearth.nasa.gov/view.php?id=8103). The delta was formed by excessive sediment deposition offsetting land subsidence. After permanent human settlement, the Mississippi River has been kept on one course by the construction of dikes, locks, and canals, and sediment deposition has been reduced by the construction of dams and the conservation of soil and water in the watershed. Many factors, including source sediment reduction, storms, waves, climate change, gas extraction, and land subsidence, have caused the loss of marshlands, threatening coastal ecosystems and infrastructure and increasing flood vulnerability along the northern coast of the Gulf of Mexico.

Figure 1.7  Sediment transport in an estuary

8 Introduction

Figure 1.8 Sedimentation in the Mississippi River Estuary and Delta (NASA image on October 10, 2001, created by Jesse Allen using data provided by the University of Maryland’s Global Land Cover Facility)

Coastal Sediment Problems Coastal environments are very dynamic due to the variable forces of waves, tides, storm surges, and winds. Longshore and cross-shore currents generated by waves are often responsible for sediment imbalance in the surf zone and in turn cause shoreline erosion. Natural and anthropogenic morphological changes can damage coastal structures, threaten flood control, and affect port and harbor operations. Potential sea level rise can intensify shore erosion, land loss, and flood threat. Figure 1.9 shows that significant erosion occurred the week before August 5, 2015, along the beach of Ocean City, Maryland, created a steep drop-off of as high as 1.2 or 1.5 m with a pronounced ledge at the water edge (https://mdcoastdispatch.com/2015/08/05/). This was caused by about 6 hr of steep waves with short periods, peaking right around high tides spurred by two full moons in July, as well as by the strong longshore currents generated by waves running south to north. Such phenomena often occur on many beaches. Sediment-Related Environmental Problems Waste from industry, agriculture, and residences impairs not only water quality, but also sediment quality in the receiving waters. Fine sediments, such as clay and silt, can absorb a variety of pollutants. The contaminated sediments carried by water may be deposited when the flow

Introduction 9

Figure 1.9 Significant erosion at Ocean City Beach in the week before August 5, 2015 (photo courtesy of Chris Parypa/Maryland Coast Dispatch)

becomes weak. Typical deposition sites include channel expansions, reservoirs, lakes, estuaries, wetlands, and floodplains. Contaminants accumulated in a sediment bed may later become a major source of pollution through resuspension, desorption, and diffusion to the water column, as well as by undergoing biochemical processes that may deplete the dissolved oxygen in the water column and release toxic gases to the environment. 1.4  Classifications of Water-Borne Sediment Loads Sediments on the bed or banks of a water body (e.g., river) are called bed or bank materials. They are stationary but susceptible to transport. The overall amount of sediment transported by a given body of water is called its total sediment load. The total load can be divided into bed-material load and wash load depending on the sediment source. The bed-material load can be further divided into bed load and suspended load depending on the sediment transport mode. Figure 1.10 depicts these classifications. Bed load consists of sediment particles that slide, roll, or saltate in the layer several particle sizes above the bed surface (Figure 1.11). Suspended load is composed of sediment particles that move in suspension in the water column above the bed-load layer. The weight of the suspended load is continuously supported by the turbulence of flow, whereas bed load is mainly dislodged by flow traction on the bed surface. Bed load and suspended load exhibit different transport behaviors. Bed-material load is made up of moving sediment particles that are found in appreciable quantity in the channel bed. Bed-material load constantly exchanges with bed material and contributes significantly to channel morphology. Wash load is comprised of moving sediment particles derived from upstream sources and infrequently exchanges with bed material. Wash load is finer than bed-material load. Bed-material load moves as both bed load and suspended load. Wash load consists of fine particles that move mainly in suspension. Because wash load

10 Introduction

Figure 1.10 Classifications of sediment loads

Figure 1.11 Sketch of bed load and suspended load

does not significantly contribute to local bed morphology, it is ignored in certain problems. Bedmaterial load is related to the local flow and bed sediment conditions, but wash load is not. Thus, excluding wash load allows us to establish reliable bed-material load formulas as functions of the local flow and sediment parameters. Silt and clay primarily move in suspension, and gravel usually moves as bed load. Sand may move as both suspended load and bed load depending on flow conditions and sediment properties. Based on measurements from 35 river stations in South Korea, Yang and Julien (2019) found that at least 90% of sand and finer fractions are suspended in gravel- and sand-bed rivers when the flow discharge is greater than the mean annual discharge. In addition, soluble sediment material, such as ions from chemical weathering, may be transported with water in solution. This portion of sediment is called the dissolved load. It is significantly different from the solid loads, and not covered in this book. 1.5  Sediment Yields in the World’s Rivers Lvovich et al. (1991) mapped the global suspended sediment yields from the world’s rivers, as shown in Figure 1.12, based on data from intermediate-sized drainage basins (1000–10,000 km2). A similar map was also developed by Walling and Webb (1983). The highest sediment yield was greater than 50,000 t/(km2∙yr), found at the Huangfuchuan River, a tributary of the

Introduction 11

Figure 1.12 Global pattern of suspended sediment yield (after Lvovich et al., 1991; image from Walling and Webb, 1996)

Yellow River in China. This was due to the presence of highly erodible loess soils, lack of vegetation cover, and semiarid climate with intense storm rainfalls. High sediment yields were also found in high-relief mountain areas, such as the Cenozoic mountain areas around the Pacific margin, as well as in regions with Mediterranean, semiarid, and humid tropical climates. Low sediment yields were associated with desert regions and the low-relief, glaciated regions of the Canadian Shield and northern Eurasia (Walling and Webb, 1996). Table 1.2 lists the annual mean runoff and sediment loads in selected rivers of the world. Among these, the Yellow River in China has the highest mean suspended sediment concentration of 25.6 kg/m3. This corresponds with the high sediment yield there, as shown in Figure 1.12. The Amazon River of Brazil has the largest basin area and runoff discharge, but low suspended sediment concentration. The St. Lawrence River in North America has very little sediment due to detention by the Great Lakes upstream. Milliman and Meade (1983) estimated that the total suspended sediment flux from the world’s rivers to the ocean was 13.5×109 t/yr. Their estimate excluded the sediment deposition in major reservoirs and extrapolated the gauged data to ungauged areas. Considering that ungauged areas are usually mountainous and have high erosion potential, Milliman and Syvitski (1992) suggested that the total sediment flux to the ocean was likely about 20×109 t/yr. Several other investigators suggested similar estimates of approximately (15–20)×109 t/yr (Walling and Webb, 1996). Note that the sediment yields in Figure 1.12 and Table 1.2 were determined decades ago. Recently, the world’s rivers have been experiencing significant anthropogenic and natural impacts,

12 Introduction Table 1.2  Mean annual runoff and sediment loads in selected world’s rivers River

Measurement station

Basin area Runoff (10 3 km 2) (10 9 m 3/yr)

Sediment load Sediment Reference (10 6 t/yr) concentration (kg/m 3)

Amazon Amur Congo Danube Ganges Indus Lena MacKenzie Mekong Mississippi Nile Ob Parana Rhine Rio Grande St. Lawrence Yangtze Yellow Yenisei

Mouth

6130.5 1800 4013 810 1059 958 2500 1800 790 3300 3000 2447 2600 160 69.3 1100 1705 752 2600

404.6 52 72.23 67 1637 481 12 42 160 400/210* 120** 17.13 79 2.78 9.43 4 468 1100 13

Mouth Delta Kotri

Mouth Cairo Salekhard Mouth Lobith San Acacia Datong Lijing

5714 324 1252 202.5 445 213.6 512.5 306 466 495 90 394 429 69 2.4 478.5 911 43.1 572

0.07 0.16 0.06 0.33 3.68 2.25 0.023 0.14 0.34 0.8/0.42* 1.33** 0.043 0.18 0.04 3.9 0.008 0.53 25.6 0.023

J-P M-S J-P M-S J-P J-P M-S M-S M-S M-S M-S J-P M-S J-P J-P M-S Z Z M-S

*: pre-dam/post-dam; **: derived from reservoir deposit; J-P: Jansen and Painter (1974); M-S: Milliman and Syvitski (1992); and Z: Zhang et al. (1998).

Figure 1.13 Recent trends in the annual runoff and suspended sediment load of the Yellow River at Tongguan Station, China (adapted from Hu and Zhang, 2018)

such as dam construction, land use changes, increasing water demand, sediment mining, and climate change. As an example, Figure 1.13 shows the recent trends in annual runoff and sediment load of the Yellow River (Hu and Zhang, 2018). Sediment load in this river has been reduced due to precipitation decrease, water abstraction, water and sediment conservation, and reservoir trapping. In the Rio Grande, dam construction and water abstraction have reduced post-dam annual water discharge and sediment load to approx. 4% of the pre-dam values (Vörösmarty et al.,

Introduction 13

2003). In the Yangtze, Mississippi, and Danube Rivers, sediment levels have decreased because of dam construction, watershed management, and climate change, while runoff levels have not exhibited significant reduction (Walling, 2008; Zhu et al., 2015; Mize et al., 2018). Differently, some rivers have seen increases in sediment load due to human activities such as deforestation and sediment mining in watersheds (Walling, 2008). 1.6  Historical Development of Sediment Transport Dynamics During the Renaissance, Italian artist and engineer Leonardo da Vinci (1452–1519) carried out an observational study of water flow and sediment transport. He traced the lateral distribution of stream velocity by releasing a rod at different places in the cross-section of a river. He described that rivers erode rocks and carry sediments to the sea in a grand, continuous cycle (http://www. waterencyclopedia.com/La-Mi/Leonardo-da-Vinci.html). In 1851, G. G. Stokes derived a formula, now known as Stokes’ law, for the drag force exerted on a spherical particle in a laminar flow by solving the Navier-Stokes equations for small Reynolds numbers. Rubey (1933) proposed the first semi-empirical formula for solid particle settling velocity covering all laminar, transitional, and turbulent flow regimes. The Rubey formula has large errors for natural sediment particles due to a limited quantity of data used in its development. Improvements have been made by the U.S. Interagency Committee (1957), Zhang (1961), Dietrich (1982), Cheng (1997a), Wu and Wang (2006), among others, using much more data. Du Boys (1879) established a bed-load formula, which was the first empirical formula for sediment transport. Although it has not been widely used, the formula of Du Boys relates bedload transport to (excess) bed shear stress, which has been recognized as one of the key parameters for sediment transport. Shields (1936) developed a diagram for the critical bed shear stress at which bed sediment particles start to move. The Shields diagram has been improved upon by many scholars (e.g., Yalin and Karahan, 1979b; Chien and Wan, 1983, 1999). Shamov (1959), Zhang (1961), and Neill (1968), among others, developed formulas for the critical flow velocity of sediment incipient motion. Both approaches became the bases of stable channel design. Rouse (1938a) derived an analytical function for the vertical distribution of suspended sediment concentration by using the turbulent diffusion theory, which is based on the law of mass conservation. The Rouse distribution represented a milestone after which sediment transport dynamics has emerged as a distinct discipline of science and technology. The turbulent diffusion theory has been widely used to explain the transport of suspended sediments (e.g., Lane and Kalinske, 1941; Vanoni, 1946; van Rijn, 1984b). Based on experimental data, Meyer-Peter and Mueller (1948) developed an empirical formula that relates bed-load transport to grain shear stress. This is an important concept advocated by many scientists (e.g., Einstein, 1942, 1950; van Rijn, 1984a; Wu et al., 2000). The formula of Meyer-Peter and Mueller (1948) is still widely used. Shamov (1959), Yalin (1972, 1977), and van Rijn (1984a), among others, established bedload transport formulas by multiplying the bed-load particle velocity, concentration, and layer thickness, each of which is related to particle size, density, and flow conditions (e.g., velocity, bed shear stress). The energy conservation law was applied by Velikanov (1954, 1955) to explain sediment transport, by Bagnold (1956, 1962, 1966) in his sediment transport formulas and auto suspension theory, by Zhang (1961) in his suspended-load transport capacity formula and turbulence

14 Introduction

attenuation concept, and by Yang (1973) for his minimum stream power concept. Stream power is one of the key factors affecting sediment transport and has been widely used in many formulas (e.g., Dou, 1964; Bagnold, 1966; Engelund and Hansen, 1967; Yang, 1973; Ackers and White, 1973; Wu et al., 2000). Sediment transport exhibits random characteristics due to fluctuating turbulent flows, heterogeneous sediments, and complex water-sediment-bed interactions. Einstein (1942, 1950) was the first investigator who applied stochastic theory to sediment research, followed by Han and He (1984), among others. Einstein (1942, 1950) derived the entrainment probability and bed-load transport rate using elegant mathematics and fluid dynamics. Han and He (1984) established a comprehensive stochastic theory of sediment transport, including the probability of incipient motion, the life distribution of resting bed particles, and the intensity of sediment transport. Another contribution of Einstein (1950) was his approach to calculating the suspended-load transport rate by integrating the sediment flux (flow velocity times sediment concentration) over the flow depth. Then, he defined the depth-averaged concentration of suspended load based on the vertically integrated sediment flux. This is well accepted by the sedimentation engineering community. Einstein (1950) recognized the hiding and exposure phenomenon among nonuniform bed material particles, in which fine particles may be sheltered by coarse particles and coarse particles have more exposure to the flow. He first introduced a correction method to account for the hiding-exposure effects on the fractional transport of nonuniform sediments. Since then, several such correction factors have been developed and used in a variety of formulas for incipient motion and transport capacity of nonuniform sediments (e.g., Egiazaroff, 1965; Parker et al., 1982; Patel and Ranga Raju, 1996; Wu et al., 2000). Bed forms such as ripples, dunes, and antidunes are generated due to the instability of a movable bed surface under the action of fluids. Bed stability has been analyzed by Kennedy (1963), Engelund (1970), Bose and Dey (2012), and others. Bed-form regimes have been studied by Garde and Albertson (1959), Simons and Richardson (1961), and Brownlie (1983), among others. Formulas have been developed to determine the bed-form heights and roughness coefficients (e.g., van Rijn, 1984c; Wu and Wang, 1999). Einstein (1942) and Engelund (1966) proposed methods to divide form and grain resistances. Fine-grained sediments (less than about 0.03 mm in diameter) exhibit cohesive properties and form flocs (aggregates) due to the action of electrostatic forces. Unlike noncohesive sediments, cohesive sediments move as flocs. Many formulas have been established to calculate the deposition, erosion, flocculation, and consolidation of cohesive sediments (e.g., Krone, 1962; Partheniades, 1965; Mehta and McAnally, 2008). The vertical profile of uniform shear flow velocity follows the log law in the inner region near the bed according to the classical theory of boundary layer flows (Prandtl, 1925), whereas the log law itself or in the form of a defect law can be extended to a significant portion of the outer region in river flows (e.g., von Karman, 1934; Satkevich, 1934; Keulegan, 1938; Einstein, 1950). Due to the effects of free surface and side walls, the velocity profile deviates from the log law and may exhibit a dip near the water surface under certain conditions (e.g., Nezu and Rodi, 1986). Several other profiles, such as the log-wake law, have been developed for general situations (e.g., Coleman, 1981; Guo and Julien, 2001). When sediment concentration is higher, the effects of sediment on flow become important. Fine sediment particles attenuate turbulence, whereas coarse particles create turbulence in the flow. The water-sediment mixture behaves as a non-Newtonian (e.g., Bingham

Introduction 15

viscoplastic) fluid when the sediment concentration is higher than a certain level, e.g., about 200 kg/m3. Hyperconcentrated and debris flows have been studied by Wang and Chien (1984) and Takahashi (1991, 2014), among many others. These flows are particularly emphasized in the Chinese sedimentation engineering community because the Yellow River exhibits high sediment concentrations and poses significant flood threats to adjacent areas (Zhang and Xie, 1993). Coastal sedimentation processes are quite different from those in rivers due to the combined effects of waves and currents. The periods of wind waves usually range from 1 to 30 seconds. An oscillatory wave boundary layer exhibits greater apparent resistance than a steady flow boundary layer (Grant and Madsen, 1979). Wave-generated longshore and crossshore currents are responsible for sediment transport and morphological changes along the shoreline. Formulas have been developed to calculate movable bed roughness, bed shear stress, sediment incipient motion, bed load, and suspended load under combined waves and currents (e.g., Jonsson, 1966; Bailard, 1981; Soulsby, 1997; Camenen and Larson, 2007; Wu and Lin, 2014). Turbidity currents represent a special mode of sediment transport driven by density differences due to suspended sediments in reservoirs, channels, and seas. Their formation, propagation, ambient fluid entrainment, and sedimentation mechanisms have been increasingly better elucidated through the works of Gariel (1949), Fan (1960), Parker (1982), and many others. Sediment-laden flows are a type of liquid-solid two-phase flow. Two-phase flow theories have been applied to study sediment-laden flows (e.g., Hetsroni, 1982; Liu, 1993). There have also been attempts to treat sediment particles individually and apply discrete element models for sediment transport. However, there is still a lot of work to do to fully take advantage of these theories and models. Thousands of sediment transport data have been reported in the literature, e.g., Gilbert (1914), Guy et al. (1966), Toffaleti (1968), Brownlie (1981b), Williams and Rosgen (1989), Camenen and Larson (2007), and Hotchkiss and Hinton (2016). These data are invaluable for the sedimentation engineering community and have been used to calibrate and validate sediment transport formulas and models (e.g., Wu et al., 2000; Wu and Lin, 2014). It is expected that increasingly more data, particularly those collected in the field, will be available for this purpose. Physical modeling has been an important tool for investigating flows and sediment transport processes in surface water bodies. Movable-bed scale models were designed using the Froude, Shields, and Rouse similitudes based on dimension analysis (Stevens et al., 1942; USBR, 1953). Later on, Xie (1981), among others, developed a complete list of similitude criteria for free surface flows, sediment incipient motion, suspension, transport capacity, and bed deformation by analyzing the governing equations, boundary conditions, and empirical formulas of the physical processes. Key problems, such as scale distortions and model sediments, have been better understood and tackled (Xie, 1981; Ettema et al., 2000). Many numerical models of sediment transport have been developed since the 1950s. Most of the early models assumed the instantaneous equilibrium of bed load or total load at each computational node (e.g., Thomas, 1982; Spasojevic and Holly, 1993). In recent decades, nonequilibrium sediment transport modeling has emerged to more realistically describe the temporal and spatial lags between flow and sediment transport (e.g., Han, 1980; Bell and Sutherland, 1983; Wu, 2004). Such model formulations were comprehensively introduced by Wu (2007). Computational modeling is cost-effective and has gradually become the primary tool for solving practical sedimentation problems.

16 Introduction

After a long history of development, the field of sediment transport dynamics has become increasingly more mature, though it still relies on many empirical approaches. It uses the knowledge of hydrodynamics regarding turbulent flows in alluvial channels, as well as currents and waves in coastal waters. It has a series of theories, formulas, and methods to understand and determine sediment settling, incipient motion, erosion, transportation, deposition, movable bed roughness, etc. It also incorporates the study of geomorphological processes, including evolution laws and regime theories, in fluvial, estuarine, and coastal systems. The study of sediment transport dynamics provides physical principles and analysis methods for sedimentation engineering. 1.7  Coverage of This Book Sediment transport dynamics has been reviewed and introduced in previous publications by Graf (1971), Yalin (1972), Vanoni (1975), Xie (1981), Chien and Wan (1983, 1999), Simons and Senturk (1992), van Rijn (1993), Yang (1996), Soulsby (1997), Zhang et al. (1998), Garcia (2008), Julien (2010), and Dey (2014). The present book complements these texts by including the early classical works and recent developments in sediment transport theories and methods from English and non-English literature, and by merging the knowledge and experience gained by sediment scholars all over the world. It serves as a reference book for sedimentation engineering professionals and a textbook for graduate and senior undergraduate students. The present book focuses on the processes of sediment transport in surface waters. The subjects are organized into fifteen chapters. This chapter presents a general introduction, including the origins and classifications of sediments, a brief overview of sediment transport processes in surface waters, sediment loads in the world’s rivers, and the development history of sediment transport dynamics. Chapter 2 introduces the properties of water and sediment. Chapter 3 provides an overview of open channel hydraulics, including turbulence characteristics, velocity distribution, channel resistance, secondary flows, and coherent structures. Chapter 4 describes the affecting factors and often-used formulas of sediment settling velocity. Chapter 5 presents the deterministic threshold criteria for incipient motion of uniform and nonuniform sediments, as well as the probability of incipient motion. Chapter 6 explains the characteristics, regimes, and dimensions of bed forms, as well as the empirical formulas of movable bed resistance. Chapters 7–9 introduce the transport of bed load, suspended load, and total load in rivers, respectively, as well as measurements of sediments. These nine chapters cover the properties of sediments and the fundamentals of river sedimentation. The remaining six chapters cover special topics on different aspects of sediment transport. Chapter 10 describes the physical processes and empirical formulas of cohesive sediment transport, which often occurs in estuaries and reservoirs. Chapter 11 applies two-phase flow theories to study sediment-laden flows. Chapter 12 outlines the classifications, rheological relationships, velocity distributions, resistances, sediment transport, and morphological features of hyperconcentrated and debris flows. Chapter 13 describes the movable bed roughness, bed load, suspended load, and nearshore transport induced by waves and currents in coastal waters. Chapter 14 introduces the basic characteristics of uniform, gradually-varied, and rapidly-varied turbidity currents, as well as the special features of turbidity currents in reservoirs, channels, and seas. Chapter 15 presents the similitude theories for designing scale models of flow and sediment transport in rivers.

Introduction 17

Homework Problems 1.1 What are sediments? Are sediments and soils the same? What agents are responsible for sediment transport? 1.2 What are the size ranges of gravel, sand, silt, and clay? 1.3 What are erosion and deposition? 1.4 Describe the sediment transport processes from uplands to the ocean. Where are the sources and sinks of sediments? 1.5 Describe the influences of hydraulic structures on sediment transport. 1.6 Describe bed load, suspended load, bed-material load, wash load, and total load. 1.7 How have human activities affected the sediment yields on uplands and sediment loads to rivers and the ocean? 1.8 Describe the sediment problems of a large river in your country or region. 1.9 What are sediment transport dynamics?

Chapter 2

Properties of Water and Sediment

Introduced in this chapter are the properties of water and sediment, including the viscosities of pure and sediment-laden waters, the mineral and electrochemical properties of sediments, the size and shape of sediment particles, the size gradation of sediment mixtures, as well as the porosity, repose angle, and shear strength of sediment deposits. 2.1  Physical Properties of Water 2.1.1  Water Density and Specific Weight The density and specific weight of a matter are defined as the mass and weight per unit volume, often in kg/m3 and N/m3, respectively. Water is an incompressible fluid. The density of pure water, ρ, is 1,000 kg/m3 at 4°C and varies slightly with temperature (Table 2.1). Water density may change due to dissolved or suspended materials. The density of sediment-laden water increases as sediment concentration increases; this is discussed in Section 2.9.1. The density of saline water increases with increasing salinity and is about 1,025 kg/m3 in open seas. The specific weight of water, γ, is related to ρ by γ = ρg

(2.1)

where g is the gravitational acceleration, which is about 9.80665 m/s2 at sea level on the Earth’s surface. 2.1.2  Water Viscosity Water deforms under the action of shearing. The dynamic viscosity of water, μ, is the coefficient of proportionality relating the shear stress, τ, to the deformation rate, du/dy, as follows: τ = μ du/dy

(2.2)

where u is the flow velocity and y is the coordinate normal to the flow direction. The kinematic viscosity of water, v, is the ratio of the dynamic viscosity to the density of water: v = μ /ρ

DOI: 10.1201/9781003343165-2

(2.3)

6

Properties of Water and Sediment  19 Table 2.1  Density and viscosity of pure water (data from Cengel and Cimbala, 2014) Temperature (°C)

Density (kg/m3)

Dynamic viscosity (x10⁻ 3 N·s/m2)

Kinematic viscosity (x10⁻ 6 m2/s)

0.01 5 10 15 20 25 30 35 40 50 60 70 80 90 100

999.8 999.9 999.7 999.1 998.0 997.0 996.0 994.0 992.1 988.1 983.3 977.5 971.8 965.3 957.9

1.792 1.519 1.307 1.138 1.002 0.891 0.798 0.720 0.653 0.547 0.467 0.404 0.355 0.315 0.282

1.792 1.519 1.307 1.139 1.004 0.894 0.801 0.724 0.658 0.554 0.475 0.413 0.365 0.326 0.294

The units often used for viscosities μ and v are N·s/m2 and m2/s, respectively. Water viscosity is directly related to molecular interactions. It decreases as water temperature increases, as shown in Table 2.1. Kinematic viscosity can be approximated with the following formula (R2 = 0.998):

  1.781  5.548  102 T  1.002  103 T 2  9.341 106 T 3  3.385  108 T 4   106



T 3  3.385  108 T 4  106 m2/s

(2.4)

where T is temperature in °C. 2.2  Sediment Density and Specific Weight The sediment specific weight, γs, is related to density, ρs, as follows: γs = ρs g

(2.5)

Sediment particles submerged in water experience gravity and buoyancy in the vertical direction. According to Archimedes’ principle, the specific weight of submerged sediment is the difference between the specific weights of sediment and water, γs – γ. It is an important parameter for sediment transport in water. The specific gravity of sediment, G, is the ratio of the specific weight of sediment to the specific weight of water at a temperature of 4°C: G = γs / γ = ρs / ρ

(2.6)

Specific gravity depends on the mineral composition and is about 2.65 for natural sediments, as explained in the next section. It does not vary significantly with temperature.

20

2.3

Properties of Water and Sediment

Mineral Composition of Sediments

The mineral composition of sediments is highly related to the production processes described in Section 1.1. Since disintegration processes do not change chemical compositions, sediments contain primary rock-forming minerals, such as feldspars, hornblende, pyroxene, quartz, and micas. However, decomposition processes produce many secondary minerals in sediments. For example, clay minerals, silica, and calcite are produced when feldspars react with the weak carbonic acid (H2CO3) formed by carbon dioxide (CO2) and water. Many rock minerals are transformed to clay minerals by hydration. Oxides, such as those of aluminum and iron, are formed when rock minerals react with oxygen in air and water. Some minerals are removed from sediments by dissolution into water. Typically, coarse-grained sediments are produced by disintegration, so they contain the primary rock minerals. Finer sediments are more likely produced or altered by decomposition, so they have more secondary minerals. USDA-SCS (1983) summarized the occurrence frequency of common minerals in average igneous rocks and sediments in Table 2.2 based on the early analyses of Clarke (1924) and Leith and Mead (1915). The occurrence frequency of a mineral in sediments depends on its abundance in the original rocks and its chemical stability. Quartz is abundant in igneous rocks and relatively resistant to chemical decomposition, so it is the most common mineral in sediments. Feldspars are the most common minerals in igneous rocks but much less chemically stable than quartz, so they are the second most common minerals in sediments. Micas have greater chemical stability than feldspars; thus, although micas are much less abundant than feldspars in igneous rocks, both are almost equally abundant in sediments. Because hornblende and pyroxene are less stable, they are abundant in igneous rocks but not in sediments. Clay minerals are the most abundant secondary minerals in sediments. Several minerals, such as dolomite, calcite, and limonite, occur in sediments but are not abundant in igneous rocks. They are either secondary minerals or primary minerals in other types of rocks (e.g., sedimentary and metamorphic rocks). According to Tian et al. (2021), the sediments sampled from shallow bank-attached and midchannel bars in the middle and lower Yellow River contain about 82.9% non-clay minerals and Table 2.2 Occurrence frequency, specific gravity, and hardness of common minerals in average igneous rocks and sediments (USDA-SCS, 1983) Mineral

Feldspars Hornblende, pyroxene Quartz Micas Titanium minerals Clay minerals Dolomite Calcite Limonite Apatite Gypsum Others Total

Mohs scale of hardness

Specific gravity

6 5–6 7 2–4 5–6

2.6–2.8 2.9–3.3 2.65 2.7–3.1 3.4–5.5 2.0–3.0 2.8–2.9 2.7 3.4–4.0 3.2 2.2–2.4

3.5–4 3 1–5.5 4.5–5 1.5–2

Frequency of occurrence (%) Average igneous rocks

Sediments

59.5 16.8 12.0 3.8 1.5 — — — — 0.6 — 5.8 100.0

15.6 — 34.8 15.1 trace 14.5 9.1 4.2 4.0 0.4 1.0 1.3 100.0

Properties of Water and Sediment  21

about 17.1% clay minerals. The main non-clay minerals are quartz (about 45.3% of the total soil sample), feldspars (27.8%), calcite (8.1%), and hornblende (1.7%). The main clay minerals are montmorillonite (6.6% of the total soil sample), illite (6.4%), and kaolinite/chlorite (4.1%). As sediment size increases, the clay mineral content decreases and the non-clay mineral content increases. The Loess Plateau soils in the Yellow River watershed have a similar mineral composition. Table 2.2 shows that the specific gravity is 2.65 for quartz, 2.6–2.8 for feldspars, and 2.7–3.1 for micas. Since quartz is the most common mineral in natural sediments, the specific gravity of natural sediments is often estimated as about 2.65 before direct measurement is conducted. The Mohs scale of hardness is a qualitative measure (1-softest to 10-hardest) of scratch resistance of minerals as gauged by the ability of a harder mineral to scratch a softer mineral. As shown in Table 2.2, quartz and feldspars have high hardness of about 7 and 6, respectively. They may abrade spillway surfaces, turbine blades, etc. 2.4  Electrochemical Properties of Sediment Particles According to the global database of Müller et al. (2021), the main chemical compound of suspended sediments in the world’s major rivers is silica (SiO2), corresponding to an average of about 65.3% of the total weight. Next, alumina (Al2O3) and iron oxide (Fe2O3 total) occupy about 16.1% and 7.4%, respectively. Other minor compounds include calcium oxide (CaO, 3.6%), magnesium oxide (MgO, 2.2%), potassium oxide (K2O, 2.2%), sodium oxide (Na2O, 1.9%), and titanium dioxide (TiO2, 0.9%). Similar estimates were reported previously by Martin and Meybeck (1979) and Xie (1981). The electrochemical properties of sediment particles are governed by the dominant compounds, particularly silica. An electric double-layer (DL) structure exists around a sediment particle exposed to water (Figure 2.1a). The chemical interactions between silica and water molecules create negative charges and firmly bind cations on the particle surface. The bound cations form the first layer, called the stern layer, of the DL structure. The second layer, called the diffuse layer, is loosely composed of free cations attracted by the negative surface charge via electrostatic force and thermal motion. Correspondingly, the water in the DL structure forms a film attached to the particle.

Figure 2.1 Cohesion of fine-grained sediments: (a) electrical double layer around a particle and (b) role of cations in flocculation

22  Properties of Water and Sediment

An electrokinetic (or ζ  ) potential is generated at the slipping plane that separates the attached and mobile fluids. The ζ potential indicates the degree of electrostatic repulsion between similarly charged adjacent sediment particles. It increases as the diffuse layer thickness increases. Positively charged cations, such as calcium (Ca2+), magnesium (Mg2+), potassium (K+), and sodium (Na+), dissolved in water can neutralize the particle surface charges and reduce the diffuse layer, thus allowing the particles to attract each other by van der Waals forces (Figure 2.1b). The absorption capacity of a sediment particle is proportional to its specific surface area, which is defined as the ratio of the surface area to volume of the particle and is equal to 6/D for a sphere. Therefore, fine-grained sediments, particularly clay and fine silt, have a strong capacity for absorbing cations in water and experience high interparticle attractive forces (i.e., cohesion) that are comparable to or stronger than gravity. Such particles tend to stick together and form flocs or aggregates. This is called “flocculation”. Flocculation greatly increases as the cation valence or concentration increases. Thus, flocculation is intensified in saline water due to the increased concentrations of Ca2+, Mg2+, K+, and Na+. Iron (Fe3+) and aluminum (Al3+) sulfates are often added to water to enhance the flocculation and settling of fine sediments. In addition, sediment particles in aquatic environments are often coated with organic materials, such as polysaccharides, proteins, and colloids. These organic materials, if positively charged, can enlarge the binding forces between cohesive particles. More importantly, many of these materials are extracellular polymeric substances (EPS) produced by microalgae and bacteria, which form biofilms on the interfaces of sediment and water. They can adhere sediment particles together. Note that the term “cohesion” indicates herein the van der Waals attractions between sediment particles of the same medium, whereas “adhesion” is any additional binding action due to the presence of a second interparticle medium, such as a biofilm. These two terms are also defined as physical and biological cohesions. The combined effect of cohesion and adhesion can create large, strong flocs. More discussions on them are given in Chapter 10. 2.5  Geometric Properties of Sediment Particles 2.5.1  Particle Size Particle size is one of the most important sediment properties because it is closely associated with the weight of the particle. It is often given in meters (m) and millimeters (mm). It is represented by the nominal, sieve, and fall diameters obtained with different measurement techniques, as described herein. Nominal diameter, D, is defined as the diameter of a sphere that has the same volume as the given particle, i.e., D  3 6V  

(2.7)

where V is the volume of the particle and π is the circumference-diameter ratio (≈ 3.14159…). V can be obtained by submerging the particle in a liquid and measuring the volume of liquid displaced by the particle. A sediment particle may be considered an ellipsoid. The diameters along its longest, intermediate, and shortest mutually perpendicular axes are denoted as a, b, and c, respectively. The diameters a, b,

Properties of Water and Sediment  23

Figure 2.2  Measurement of sediment diameters a, b, and c with calipers (from Krumbein, 1941)

Figure 2.3  Sieves with square opening

and c can be measured using calipers (Figure 2.2). The particle volume may be approximated as V ≈ πabc/6. Substituting this formula into Equation 2.7 yields D  3 abc



(2.8)

Note that nominal diameter is usually used only for coarse particles, such as boulders, cobbles, and coarse gravels, whose volume or diameters a, b, and c can be easily measured. Sieve diameter is the width of the square sieve opening (Figure 2.3) through which the given particle just passes. It is approximately equal to the intermediate diameter b. The sieve diameter is slightly smaller than the nominal diameter. For naturally worn sediment particles in the range of about 0.2 to 20 mm, the sieve diameter is approximately 0.9 times the nominal diameter on average (U.S. Interagency Committee, 1957; Raudkivi, 1998). Sieve analysis is introduced in Section 2.6.4. Sieving is not convenient for very coarse and fine particles, so sieve diameter is usually used for only fine gravel and sand. Fall diameter is defined as the diameter of a sphere that has the same density and the same terminal settling velocity as the particle in the same fluid. The standard fall diameter is the diameter of a sphere that has a specific gravity of 2.65 and the same settling velocity as the given particle in quiescent, distilled water at a temperature of 24°C. Fall diameter can be obtained from settling velocity for all sediments from boulder to clay, but in practice, it is used only for silt and clay that are too fine to be sieved conveniently.

24  Properties of Water and Sediment

Based on the data collected by Schulz et al. (1954) and the U.S. Interagency Committee (1957) for a variety of sediment particles coarser than 0.1 mm, the ratio of fall diameter to nominal diameter increases from about 0.2 to 1.0 with increasing particle sphericity, and the effect of particle shape on the ratio decreases as sediment size decreases. The fall diameter is approximately 0.85 times the nominal diameter for naturally worn sediment particles finer than 0.1 mm (see Section 4.8). Differences likely exist between the measured nominal, sieve, and fall diameters, each of which is applied to the specific size range described previously. To be consistent, sieve and fall diameters are preferably converted to nominal diameter. However, this is not a universally accepted rule, so attention should be paid to which diameter is specifically used in a formula or model. In the present text, sediment size is set as the nominal diameter by default, but exceptions are noticed when sieve and fall diameters are used. 2.5.2  Particle Shape The shape of natural sediment particles is quite irregular and significantly affects their hydrodynamic behaviors (Schulz et al., 1954; Define, 2002; Wu and Wang, 2006). This irregularity results from the original sediment production processes and evolves due to the abrasion by particle-particle and particle-bed collisions and the actions of the carrier media during transport. Therefore, particle shape is related to mineral composition, cleavage, hardness, transport distance, etc. Many particle shape factors have been proposed in the literature (e.g., Wadell, 1932; Krumbein, 1941; Corey, 1949; Define, 2002). The commonly used ones are introduced in the following text, while a more detailed review can be found in Smart (2013). Particle Sphericity Wadell (1932) defined particle sphericity as the ratio of the surface area of a sphere with the same volume as the particle to the actual surface area of the particle. However, because the surface area of irregular sediment particles is not convenient to measure, this sphericity factor is rarely used. Riley (1941) proposed a more practical sphericity factor as the diameter ratio of the largest inscribed circle (sphere) and the smallest circumscribed circle (sphere). Zingg (1935) used the ratios b/a and c/b to classify particles as discoidal (b/a > 2/3, c/b < 2/3), spherical (b/a > 2/3, c/b > 2/3), blade-like (b/a < 2/3, c/b < 2/3), and rod-like (b/a < 2/3, c/b > 2/3) (Figure 2.4). Krumbein (1941) combined these two ratios into a single sphericity index: 2

 b  c 3 bc  p  3   a2 a b



(2.9)

which is the same as the operational sphericity proposed by Wadell (1932) based on the ratio   p 3 abc a 3 3 bc a 2 . of the particle volume to the volume of the smallest circumsphere, Thus, it is called the Wadell-Krumbein sphericity index. The contours of this index are plotted in the Zingg diagram in Figure 2.4. Corey (1949) proposed the following shape factor: Sp  c

ab



(2.10)

Properties of Water and Sediment  25

Figure 2.4 Zingg diagram showing contour lines of the Wadell-Krumbein sphericity index (after Krumbein, 1941)

The Corey shape factor is equal to 1 for a perfectly spherical particle, and is smaller for a less spherical sediment particle. Therefore, it can be used as a measure of particle sphericity. It is similar to the maximum projection sphericity factor 3 c 2 ab proposed by Sneed and Folk (1958) and the shape factor c/[(a2 + b2 + c2)/3]1/2 proposed by Janke (1966). It is inversely proportional to the flatness factor (a + b)/(2c) proposed by Wentworth (1923). The study of Sneed and Folk (1958) showed that particle sphericity depends most importantly on the inherent abrasive properties of different rock types, is a strong function of particle size as well as transport distance, and is only weakly affected by selective sorting. Sphericity tends to decrease with the decreasing size of particles (Chien and Wan, 1983; Simons and Senturk, 1992). Fine fractions of clay consist principally of flake-shaped particles (Terzaghi et al., 1996). The Corey shape factor of naturally worn particles is about 0.7 on average (U.S. Interagency Committee, 1957). The Corey shape factor has been widely used in sedimentation engineering (e.g., Schulz et al., 1954; Dietrich, 1982; Wu and Wang, 2006). It is believed to represent the hydrodynamic behaviors of particles in a fluid (Sneed and Folk, 1958), since a particle usually settles along its shortest axis in a large range of particle Reynolds numbers (see Section 4.1). McNown and Malaika (1950) proposed the following shape factor considering the direction of particle motion: S m  l1

l2 l3



(2.11)

where l1 is the diameter of the particle along the moving direction, and l2 and l3 are diameters on the other two axes normal to the moving direction. Equation 2.11 is the same as Equation 2.10 when the particle moves along its shortest axis. A more complicated shape factor was proposed by Define (2002) using the streamwise, spanwise, and vertical diameters of a particle on a stream bed. This type of shape factor encounters uncertainties since the particle orientation may change during motion.

26  Properties of Water and Sediment

Particle Roundness Wadell (1932) quantitatively defined roundness, Rp, as the ratio of the average radius of curvature of the corners or edges on the particle surface to the radius of the maximum inscribed sphere (Figure 2.5). It is expressed as 1 Rp   N



N

 r  i 1

i

rins 

(2.12)

where ri is the radius of curvature of the ith corner, N is the total number of corners, and rins is the radius of the maximum inscribed sphere. Note that rins is the maximum value of ri among the corners. Rp = 1 for a sphere. Powers (1953) suggested the following verbal roundness classes: Very angular: corners sharp and jagged Angular Subangular Subrounded Rounded Well-rounded: corners completely rounded The corresponding Wadell roundness limits of each Powers verbal class are given in Table 2.3. The ratio of the upper and lower roundness limits of each class interval is approximately equal to 2 . Folk (1955) developed a logarithmic transformation of this scale to a PR scale, which varies from 0.0 to 6.0 for very angular to well-rounded particles (Table 2.3).

Figure 2.5  Measurement of particle roundness Table 2.3  Particle roundness classes (Powers, 1953; Folk, 1955) Powers verbal class

Wadell roundness R P values

Folk’s scale (P R )

Very angular Angular Subangular Subrounded Rounded Well-rounded

0.12–0.17 0.17–0.25 0.25–0.35 0.35–0.49 0.49–0.70 0.70–1.0

0.0–1.0 1.0–2.0 2.0–3.0 3.0–4.0 4.0–5.0 5.0–6.0

Properties of Water and Sediment  27

Sediment particles from freshly crushed rocks are usually angular and have small PR values, whereas those on riverbeds and coastal beaches are usually rounded and have high PR values. Roundness tends to decrease with decreasing particle size (Chien and Wan, 1983; Simons and Senturk, 1992). For naturally-worn sediment particles, the average value of PR is about 3.5. It is tedious to measure the roundness of a natural sediment particle using Equation 2.12. Krumbein (1941) and Powers (1953) provided visual charts for sediment particles with known roundness, as shown in Figures 2.6 and 2.7. The roundness of a particle can be quickly estimated by visually comparing its largest projection with the images. However, such visual estimation is subjective. Operator errors can be considerable, and the reproducibility of results even by the same operator is low (Folk, 1955). Operators tend to underestimate particle roundness, particularly in the intermediate range (Hryciw et al., 2016). Visual estimation methods yield only rough approximations.

Figure 2.6  Krumbein’s (1941) chart of particle roundness

Figure 2.7  Chart of the verbal roundness classes from Powers (1953)

28  Properties of Water and Sediment

In recent years, measurements of particle shape factors, including sphericity and roundness, using computer image analysis have become popular and efficient (e.g., Kuo and Freeman, 2000; Altuhafi et al., 2013; Hryciw et al., 2016). For example, Zheng and Hryciw (2015) proposed a computational geometry algorithm to automate the measurement procedure of Wadell (1932), and Zheng and Hryciw (2016) extended it to the three-dimensional (3-D) assembly of particle images. The algorithm includes the following steps: (1) digitize the outline of the particle and obtain the average perimeter by removing surface roughness; (2) find the maximum inscribed circle whose center has the maximum distance among all the interior points to the nearest particle perimeter; (3) find corners, each of which is a convex outward section of the perimeter with three or more consecutive points, and then fit the identified corners with appropriate circles. The algorithm can accurately reproduce the roundness values for the particles shown in Figure 2.6. Note that this particle roundness definition excludes the small-scale roughness elements on the particle surface; however, these may significantly affect the dynamic behaviors (e.g., friction) of sediment particles. The roughness height can be represented with the root-mean-square of the variations from the average perimeter in the aforementioned image analysis of Zheng and Hryciw (2015, 2016). In addition, Equation 2.12 equally weighs the contributions of large and small corners, and the derived Rp value may vary with the threshold scale below which roughness elements are excluded. These need to be clarified in the future. 2.6  Size Gradation of a Sediment Mixture Sediments in natural waters often comprise particles of multiple sizes, as shown in Figure 2.8. The fine and coarse particles in a sediment mixture behave differently under the action of water flow. How to quantify and measure the size distribution of sediment mixture is introduced in this section. 2.6.1  Size Distribution A mixture of sediment particles with nonuniform sizes can be represented by a suitable number of size classes. Each size class is defined by its lower and upper bound diameters, denoted as Dlk and Duk, respectively. Here, subscript k is the index of size class. The representative diameter, Dk, of size class k can be determined using one of the following equations: Dk  Dlk Duk 

(2.13)

Dk = (Dlk + Duk)/2

(2.14)

Equation 2.13 calculates the geometric mean of Dlk and Duk, and Equation 2.14 calculates the arithmetic mean. The geometric mean is lower than the arithmetic mean. Another equation is Dk  ( Dlk  Duk  Dlk Duk ) 3, which gives a value between those of Equations 2.13 and 2.14 (Xie, 1981). Equation 2.13 is convenient when the logarithmic scale of sediment size is used to handle a widely graded mixture. This is further discussed in the next section. For consistency, it is necessary to report the specific method used to calculate Dk. The fraction of the kth size class, pk, is the ratio of its weight (volume or number) to the total weight (volume or number) of the mixture, ranging from 0 to 1. Note that pk is also often defined by percent, ranging from 0 to 100%. They have different usages. For example, fraction is often used in calculations, and percent is often used in graphical presentations.

Properties of Water and Sediment  29

Figure 2.8 Sediment mixtures: (a) well sorted, (b) poorly sorted (photos by the author), and (c) layered structure of gravel bed, River Wharfe, UK (from Powell, 1998; photo courtesy of Ian Reid)

Table 2.4 presents how to calculate the size distribution of a sediment mixture in a sieve analysis. The second column gives the US sieve number. The third column gives the sieve size, which is set as the lower bound diameter of the size class on the sieve. Note that the lower bound diameter of size class k is the upper bound diameter of size class k + 1. The fourth and fifth columns are the representative diameters of each size class calculated from the bound diameters in the third column using Equations 2.13 and 2.14, respectively. The sixth column shows the percentage by weight of each size class in the sediment mixture. The seventh column specifies the cumulative percentage of sediment finer than the given particle, which is obtained by adding the percentages of size classes finer than the size given in the third column. The size distribution of a sediment mixture is often presented using a size-frequency histogram and cumulative size-frequency curve. The histogram is constructed by plotting the size class intervals on the abscissa and the actual percent by weight (volume or number) of each size class in the total sample on the ordinate, as shown in Figure 2.9a. The cumulative size-frequency curve indicates the percent of material finer than a given sediment size in the total sample, as shown in Figure 2.9b. Because the sediment size range is usually wide in field conditions, the logarithmic scale of sediment size is often used on the abscissa. If there is only one peak in the size distribution histogram, the sediment mixture is unimodal (Figure 2.9a); otherwise, the sediment mixture is multimodal. A bimodal mixture is comprised of two dominant size classes of particles. Each mode of the sediment mixture likely comes from a different source in the watershed.

30

Properties of Water and Sediment

Table 2.4 Example of the size distribution of sediment mixture Size class no.

US sieve size

D lk (mm)

Geometric D k (mm)

Arithmetic D k (mm)

pk (%)

% finer than D lk

1 2

10" 5/2" 5/8" No. 5 No. 18 No. 60 No. 230

256 64 16 4 1 0.25 0.0625

128 32

160 40

4 11

8

10

25

100 96 85 60 30 8 0

3 4 5 6

2 0.5

2.5 0.625

30 22

0.125

0.156

8

Figure 2.9 Size distribution: (a) histogram and (b) cumulative frequency curve

For a unimodal natural sediment mixture, the logarithmic value of particle size often approximately exhibits a normal distribution, which has the following probability density function:  ( x  x )2  exp    2 2  2  1

 f ( x)

(2.15)

and the cumulative probability of x: x

Pcf    f  x  dx 

 ( x  x )2  exp   dx  2 2  2   1

x

(2.16)

where x = ln D, x is the mean value, and σ is the standard deviation of x. This distribution is also called the lognormal distribution of sediment size D. The geometric standard deviation of D is defined as σg = exp(σ), and the geometric mean diameter is Dg  exp(x ). 2.6.2

Characteristic Diameters

The median diameter, D50, is the particle size for which 50% by weight of the sample is finer. Likewise, D10 and D90 are the particle sizes for which 10% and 90% by weight of the sample

Properties of Water and Sediment  31

are finer, respectively. They can be read from the cumulative size-frequency curve, as shown in Figure 2.9b. For the lognormal size distribution, the inversion of Equation 2.16 gives D p  Dg 

g

2erf 1 ( Pcf 50 1)



(2.17)

where DP is the particle size for which Pcf percent by weight of the sample is finer, and erf −1 is z 2  the inverse error function defined by erf ( z ) (2  ) 0 exp  t  dt . By using Equation 2.17, 2.326 one can derive D10  Dg  g1.282 , D50 = Dg, D90  Dg  g1.282, and D99  Dg  g . For a general sediment mixture, the geometric mean diameter is calculated by D D 1p1 100 D 2p2 100  D NpN g

100



(2.18)

and the arithmetic mean diameter is Dm 

N

p k 1

k

Dk 100



(2.19)

where pk is given by percent and N is the total number of size classes. To be consistent, Dg and Dm are calculated by using Dk determined with Equations 2.13 and 2.14, respectively. Generally, Dg is close to D50, and Dm is larger than D50. These three diameters are often used to represent the average size of the sediment mixture. Likewise, D10 and D90 represent the particle sizes of the fine and coarse tail portions. 2.6.3  Gradation and Nonuniformity In the case of lognormal size distribution, D15.9 and D84.1 are one standard deviation away from the median or geometric mean diameter, i.e., ln D15.9 = ln D50 – σ,  ln D84.1 = ln D50 + σ

(2.20)

Therefore, D15.9 and D84.1 are used to define the geometric standard deviation σg for a general sediment mixture: σg = (D84.1/D15.9)1/2

(2.21)

and similarly the gradation coefficient Gr:  Gr

1  D84.1 D50     2  D50 D15.9 



For the lognormal distribution, Gr = σg since σg = D84.1/D50 = D50 /D15.9.

(2.22)

32  Properties of Water and Sediment

Kramer (1935) defined a uniformity index as the ratio of the mean sizes of the two portions in the cumulative size-frequency curve separated by D50: M 

50



pk Dk

100



Pcf ,k 0  Pcf ,k 50

pk Dk 

(2.23)

where Pcf,k is the cumulative percentage of sediment finer than size Dk. The indices σg, Gr, and M characterize the size gradation or nonuniformity of a sediment mixture. For a uniform sediment mixture, σg = 1, Gr = 1, and M = 1. As σg and Gr increase or M decreases, the mixture becomes more graded or less sorted. According to Dyer (1986), if σg  4, the mixture is poorly sorted. Examples of well and poorly sorted sediment mixtures are shown in Figures 2.8a and b. 2.6.4  Measurements of Size Distribution Traditionally, the size distribution of sediment mixtures is analyzed by sieving and settling. Settling analyses include pipette, hydrometer, visual accumulation tube, and bottom withdrawal tube methods. More recently, non-intrusive technologies, such as X-ray and laser diffraction analysis, have been developed. The procedures, principles, and limitations of these methods are described in this section. Sieve Analysis Sieving includes wet and dry sieving. For dry sieving, the sample is oven-dried at a temperature of about 80°C and carefully inspected to make sure that no aggregates exist. Then, sieving is carried out in the following steps: (1) Weigh a beaker on a scale. Pour the sediment sample into the beaker and determine the combined weight of the sample and beaker. Calculate the weight of the sample and record it on a datasheet. (2) Select sieves according to the size classes required. Nest the sieves in proper order, with the coarsest on the top and pan at the bottom (Figure 2.3). Divide the sieves into two stacks, if they are too many to be handled by the sieve shaker at one time. Pour the sample onto the top sieve of the first stack and put the lid on. Place and secure the stack of sieves onto the shaker. Set the timer to a standard time (10–15 minutes) and turn on the shaker. When the shaker turns off, take the stack of sieves out. Pour the sediment in the pan of the first stack onto the top sieve of the second stack, and again, shake the second stack of sieves. Care should be taken to ensure that all the particles finer than the sieve size pass through the mesh. (3) Empty the sediment in each sieve, one at a time, onto a piece of glazed white paper. Invert the sieve, and gently wipe the bottom of the screen with a soft brush so that all the lodged particles fall back through the mesh and onto the paper. Carefully pour the sediment of each size class into a beaker. (4) Weigh the sediment in each of the sieves and the lower pan. Record the weights on the datasheet.

Properties of Water and Sediment  33

(5) Measure the size of the largest particle on the coarsest sieve. If needed, store the fine sediment (D < about 0.063 mm) on the lower pan in a separate beaker to be measured using a settling method or laser diffraction analysis. (6) Determine the weight of sediment in each size class and sum the weights of all size classes. Determine the portion of the sample lost during sieving by comparing the total summed weight of all size classes and the total initial weight before sieving. The lost portion should be very small, e.g., < 2% of the total sample. Calculate the percentage by weight for each class and the cumulative percentage of the sediment finer than that size class (see Table 2.4). Wet sieving uses a similar procedure, but with water poured from the top of the shaker and the water and fine sediment collected through a hose at the bottom. If a sample includes a considerable amount of silt and clay, it should normally be wet-sieved. Both dry and wet sieve analyses are applicable only for sediments coarser than about 0.063 mm. Sediments finer than this size are usually measured by settling analysis or laser diffraction analysis, as described in the following text. Settling Methods The settling methods measure the size distribution of sediment mixture by sorting out the size classes of the sample through differential settling in a water tube or column. To apply a settling method, the sample is first treated with sodium hexametaphosphate or another dispersion agent to disaggregate clay and silt thoroughly into individual particles. The sample is then diluted with distilled water and stirred to form a homogeneous suspension in a vertical tube. The sediment particles in the suspension settle due to the action of gravity. Coarse particles settle faster than fine particles; thus, over a certain elapsed time, coarse size classes settle to the bottom of the tube and fine size classes remain in suspension. The time needed for each sediment size class to settle through a certain depth is determined using the Stokes law (described in Section 4.3) assuming spherical geometry of the particles. The size composition of the sample is determined by measuring either the residue remaining in the suspension or the sediment settled to the bottom of the tube at successive time intervals. Pipette, hydrometer, and X-ray methods can be used to measure the residual size fraction in the suspension. With the pipette method, small volumes of the suspension are extracted from the tube at predefined, successive times through a pipette inserted into the tube at a given depth. The sediment in the extracted subsamples is obtained by filtering (very fine particles may be lost) or by evaporation (the dissolved salts need to be excluded). With the hydrometer method, the density of the suspension is measured with a hydrometer at the beginning, after the sand settles, and again after the silt settles out from the tube; and the density is converted to sediment concentration in the suspension (Ashworth et al., 2001; Burt et al., 2014). As a third, non-intrusive alternative, the concentration of the residue remaining in suspension can be inferred by measuring the attenuation of X-rays passing through the suspension at a given depth and time (Coakley and Syvitski, 1991). The sediment settled to the bottom of the tube can be measured using visual accumulation and bottom withdrawal tube methods. The visual accumulation tube method involves the direct visual tracing of the height of sediment accumulation in a contracted section at the bottom of the tube (Colby and Christensen, 1956). The bottom withdrawal tube method involves removing the lowest portion of the settling suspension at successive, logarithmically spaced time intervals through a tap at the bottom of the tube (Vanoni, 1975).

34  Properties of Water and Sediment

Each of the aforementioned settling methods has certain variations depending on the type of dispersion solution, the volume of the suspension, the time of settling before taking readings or extracting subsamples, or the handling of the raw readings. Each method has its application range. For example, the visual accumulation tube was developed primarily for sand (Colby and Christensen, 1956), the X-ray method applies only to sediment up to 0.3 mm in size (Bianchi et al., 1999; Cheetham et al., 2008), and the pipette and hydrometer methods are applicable for sediments finer than about 0.063 mm in size (Burt et al., 2014). The major source of error in this group of methods is incomplete dispersion of the clay fraction, which is cemented by various chemical agents and organic materials into aggregates of larger sizes in the original sample and even during measurement. Incomplete dispersion results in low percentages for clay and high percentages for silt and sand (Cheetham et al., 2008). Additional error is caused by Stokes’ law, which assumes spherical particles and is invalid for medium and coarse sands. This is discussed in Chapter 4. Laser Diffraction Analysis Laser diffraction technology has been developed since the 1970s to analyze particle size distribution. The technology is based on the principle that smaller particles scatter a parallel beam of monochromatic light at higher angles than larger particles. Like the settling methods, laser diffraction analysis requires that the sediment sample is thoroughly dispersed and homogeneously suspended in a water column. When the sample in suspension is passed through a parallel beam of monochromatic light, the angle of the diffracted light is measured by a multi-element detector. By using the Fraunhofer-Mie theory for the scattering of light from spherical particles, the output of the analysis is calculated as the percentage by volume of size ranges in the sediment sample (Xu, 2000). The laser diffraction analysis is faster than the traditional sieving-pipette method. However, the algorithms are developed for spherical particles and exhibit bias for platy clays and micas. Incomplete dispersion of clay aggregates in the suspension also causes errors. Many tests have shown that laser diffraction analysis works well for sand and silt but underestimates the clay fraction (Agrawal et al., 1991; Konert and Vandenberghe, 1997; Cheetham et al., 2008). 2.6.5  Bed Material Sampling The size distribution of bed materials needs to be measured because it affects the hydraulic, geomorphic, and ecologic features of the channel. Bed materials often exhibit vertical stratification and horizontal segregation. A gravel bed in particular often has a coarse surface pavement (or armoring) layer, a finer subpavement layer, and different substrata in the bottom, as shown in Figure 2.8c (Church et al., 1987; Diplas and Sutherland, 1988). Segregation of bed materials is often observed in different areas of a depositional bar (Bluck, 1982; Diplas, 1994) and between riffles and channels (Sear, 1996), and fining is perceived along the downstream direction (Church and Kellerhals, 1978; Parker, 1991). A suitable number of samples should be collected using appropriate methods to describe the vertical and horizontal inhomogeneity of bed materials. Bed material sampling techniques are divided into volumetric, grid, and areal methods. Volumetric or bulk sampling is the most desirable and commonly used method. The sample is extracted from the bed by grabbing, coring (Figure 2.10a), or excavating. For volumetric

Properties of Water and Sediment  35

Figure 2.10 (a) Core sample (photo by the author), and (b) areal sampling tool with a thin layer of clay applied on a flat plate inside the piston (from Diplas, 2008)

methods, the sample size should be large enough that its dimensions are independent of the dimensions of individual particles (Kellerhals and Bray, 1971). It is recommended that the minimum sampling depth be about twice the size of the largest particle present in the sampled deposit (Diplas and Fripp, 1992). The size distribution of the sample is then analyzed by sieving or using another method described earlier, and the results are plotted in terms of particle size versus percentage by weight of particles that are finer than the size in the sample. This is called volume-by-weight sampling. Volumetric sampling is appropriate for sediment deposits in sandy streams and the bottom layers of gravel streams where no size stratification occurs. It can provide unbiased estimates of the size distribution of bed sediments (Diplas, 2008). If vertical size stratification occurs in a gravel bed, each of the pavement and subpavement layers is roughly as thick as the diameter of the coarsest particle presented (Figure 2.8c). A volumetric sample would include the pavement and subpavement layers and maybe some of the bottom layers (Kellerhals and Bray, 1971). The resulting size distribution cannot accurately describe the layered structure. Therefore, surface-oriented methods should be used to collect sediment from each stratum separately. The surface pavement layer can be sampled using a grid or areal method. The grid method uses particles that lie directly below an established grid covering the area of interest. A wire mesh or frame may be used as the grid. For large areas, Wolman (1954) suggested a walking method, in which an operator paces off at regular intervals and picks up the particle below his/her toe. The Wolman method may yield coarser results than the pebble counts obtained with the gridded sampling frame, especially for smaller percentiles (Strom et al., 2010). Sampling on a regular grid gives the highest accuracy for a given number of collected particles, whereas random sampling is not as efficient (Underwood, 1970; Diplas, 2008). The size distribution is obtained by plotting particle size versus percentage by number of particles finer in the sample. This method is called grid-by-number sampling. To avoid the effects of structural features (e.g., particle cluster) on the bed surface, a grid spacing of at least twice the largest particle size present is recommended (Rice and Church, 1998).

36  Properties of Water and Sediment

Areal sampling uses all the particles that are exposed on the surface of the sediment deposit. An areal sample can be collected using paint, wax, clay, or other adhesives (Kellerhals and Bray, 1971; Adams, 1979; Diplas and Sutherland, 1988; McEwan et al., 2000; Diplas, 2008). After paint is sprayed on the bed surface, the painted particles can be collected later by hand. Wax poured onto the bed surface hardens and collects all of the surface particles and possibly some below that. The wax among the collected particles is melted and poured away. Moist pottery clay may be used to retrieve a surface sample coarser than the clay (Figure 2.10b). It works underwater as well as on dry surfaces, and thus is more suitable than wax for field sampling (Diplas and Fripp, 1992). The size distribution is obtained by plotting particle size versus percentage by area of particles finer in the sample. This is called area-by-area sampling. To reduce the effort spent in the field, grid and areal sampling can be conducted using a photograph of bed surface material. In this method, one counts the particles below a predefined grid or from the entire image. However, users of this method encounter difficulty in determining the actual dimensions of the particles from the image. The results are typically smaller than the real particle sizes measured in the field (Kellerhals and Bray, 1971; Church et al., 1987; Diplas, 2008; Strom et al., 2010; Stähly et al., 2017). This bias is attributed to irregular particle shape, overlapping angle, particle packing, shadow, vegetation residue, and scale distortion. Nevertheless, the photographic method can estimate with moderate accuracy the median size of a bed material consisting of gravel and larger particles (Church et al., 1987). Improvements have been made by using enlarged images (Ibbeken and Schleyer, 1986; Russ and Dehoff, 2000). Stähly et al. (2017) found that the grain sizes obtained by imaging processing can be directly compared with those obtained by square-hole sieving, as they are of the same order of magnitude. A comparison conducted by Cislaghi et al. (2016) showed that manual imaging processing is more reliable than automatic processing, particularly in field conditions. If the samples do not have any bias, the size distributions obtained by volume-by-weight, areaby-area, and grid-by-number sampling are equivalent (Kellerhals and Bray, 1971; Diplas, 2008). Kellerhals and Bray (1971) developed models to convert between the particle size distributions obtained by different analysis methods, such as area-by-number and volume-by-number. Diplas (2008) and Graham et al. (2012) validated and improved these models; interested readers are encouraged to review these articles. Such conversions can be avoided if the size distributions of volumetric, area, and grid samples are given with percentages by weight, area, and number, respectively, as suggested previously. Nevertheless, besides errors from operator handling, devices, and environmental conditions, certain sampling methods inherent possible bias. For example, areal sampling using wax may include particles below the bed surface, areal sampling using clay may omit coarse particles if the clay adhesion is not strong enough, and volumetric sampling by grabbing underwater may omit fine particles. These possible biases should be minimized during sampling and analysis, and caution should be taken in data reporting and application. 2.7  Porosity and Dry Density of Sediment Deposits A sediment deposit is a porous material that has voids among solid particles. Porosity, φv, is defined as the volume of voids per unit volume of the deposit:

v 

Vv  Vv  Vs

(2.24)

Properties of Water and Sediment  37

where Vv and Vs are the volume of voids and solids, respectively. Porosity is required to convert between the mass and volume of sediment deposit. Dry density, ρd, and dry specific weight, γd, are the mass and weight of the solids per unit volume of the sediment deposit, respectively. They are related with porosity by ρd = ρs(1 − φv),  γd = γs (1 − φv)

(2.25)

Dry density is also called dry bulk density. Wet bulk density is used if the mass includes pore water and solids in the deposit. Bulk density generally indicates the wet bulk density. Thus, (wet) bulk density varies with water content in the deposit. Porosity and dry density depend on the sediment size gradation, mineral composition, particle shape, particle packing, deposition environment, etc. They vary with time and depth due to consolidation in fine-grained sediment deposits. Their initial values, variations, and measurements are discussed in this section. 2.7.1  Initial Porosity and Dry Density Initial porosity and dry density apply to freshly deposited sediments. Han et al. (1981) proposed the following semi-empirical formula to calculate the initial porosity of a uniform sediment deposit: 3    Dmm  1  0.525     Dmm  4 f  v      0.095 Dmm  D0  D0 0.3  0.175e

Dmm  1 mm



(2.26)

Dmm  1 mm

where Dmm is the sediment size in mm; D0 is a reference size, set to 1 mm; and δf is the thickness of the water film attaching to sediment particles, given a value of about 0.0004 mm. Komura (1963) proposed an empirical formula for the initial porosity of a sediment deposit based on measurement data:

v 0.245  

0.0864 (0.1D50, mm )0.21



(2.27)

where D50,mm is the median diameter of the sediment mixture in mm. Wu and Wang (2006) tested the formulas of Komura (1963) and Han et al. (1981) using extensive laboratory and field data, as shown in Figure 2.11. The Komura formula slightly underestimates the dry density for sand and gravel and overestimates for silt and coarse clay. The semi-empirical formula of Han et al. (1981) has more errors, and its two segmented functions have different trends for fine- and coarse-grained deposits. A more accurate curve was obtained in Figure 2.11 and expressed as

v 0.13  

0.21 ( D50, mm  0.002)0.21



(2.28)

38  Properties of Water and Sediment

Figure 2.11 Initial dry density of a deposit as a function of median diameter (from Wu and Wang, 2006)

Figure 2.12 Porosity of unimodal sediment deposit as a function of geometric standard deviation (from Wooster et al., 2008)

Equations 2.26–2.28 are limited to uniform sediments or nonuniform sediment mixtures with narrow size ranges. Size gradation affects the particle packing configuration and, in turn, the porosity of the sediment mixture. Wooster et al. (2008) measured the porosity of 32 unimodal sediment mixtures with geometric standard deviations from 1.2 to 3.06 and size classes of 0.075 to 22 mm. The obtained relationship between porosity and geometric standard deviation is shown in Figure 2.12. The porosity decreases with increasing geometric standard deviation. This relationship agrees with the observations of Allen (1985) using several types of granular materials. Additional confidence in this relationship could be gained if further tested using more data with wider standard deviations in different types of sediment mixtures. Besides particle median size and standard deviation, porosity is also related to particle shape. An irregular shape hinders the ability of the particles to attain a dense packing configuration. Platy particles can bridge gaps and create large, open voids (Guimaraes, 2002). Porosity increases as the particle roundness and sphericity decrease (Fraser, 1935; Jia and Williams, 2001; Cho et al., 2006).

Properties of Water and Sediment  39

2.7.2  Variations in Porosity and Dry Density due to Consolidation Consolidation is a compaction process of deposited material under the influence of gravity and water pressure, with a simultaneous expulsion of pore water and an increase in strength of the material. It depends on sediment size, mineralogical composition, deposit layer thickness, etc. Consolidation usually occurs for fine-grained sediment or mud deposits. The term “mud” here includes clay and silt. Consolidation of coarse-grained sediments, such as gravel and sand, is usually negligible. According to NEDECO (1965) and Hamm and Migniot (1994), consolidation can be described as a three-stage process. The first stage is the settlement of flocs to form a particlesupported matrix or fluid mud, which happens within several hours of deposition. The second stage is the elimination of interstitial water, which occurs in one to two days. The third stage is the gelling of clay, which is a very slow process. The mud surface sinks linearly with time t in the initial stage, with t 0.5 in the second stage, and with log(t) in the third stage. According to the degree of consolidation, the deposited muds are classified as fluid, plastic, and solid muds. The fluid mud is unconsolidated, the plastic mud is undergoing consolidation, and the solid mud is consolidated. The porosity and dry density of mud deposits vary with elapsed time and deposit depth due to consolidation. Porosity decreases and dry density increases as consolidation progresses. Figure 2.13 shows the mean dry bed density varying with consolidation time for Avonmouth mud (Owen, 1975), commercial grade kaolinite in saltwater (salinity = 35 ppt) (Parchure, 1980), and kaolinite in tap water (no salinity) (Dixit, 1982). Noteworthy is the very rapid increase in d over the first 48 hours, after which the porosity increases much less rapidly and asymptotically approaches to the final value, d. The temporal variation of the mean dry density for the three muds can be approximated by (Hayter, 1983)

 d  d   1  a e

 b t



(2.29)

where a  1   d 0  d , with  d 0 representing the dry density at t = 0. The coefficient bρ varies with many factors, including the deposit thickness. A thicker layer consolidates slower because the pore water has a larger travel distance to the mud surface (van Rijn, 1993).

Figure 2.13 Variation of mean bed density with consolidation time (from Dixit, 1982)

40

Properties of Water and Sediment

Lane and Koelzer (1953) proposed the following formula to determine the dry density of a reservoir deposit undergoing the consolidation process: ρd = ρd0 + βρ log t

(2.30)

where ρd is the dry density (kg/m3) at time t, ρd0 is the dry density after one year of consolidation, t is the consolidation time (years), and βρ is a coefficient. Table 2.5 gives ρd0 and βρ for sand, silt, and clay under different reservoir operation conditions. Equation 2.30 corresponds with the third stage of consolidation. As experimentally observed by Owen (1975), Dixit (1982), and Hayter (1983), the dry bed density varies along the depth below the bed surface. For a homogeneous deposit, the bottom layer usually has lower porosity and higher dry density since it deposits earlier and experiences higher loading from the upper layer and water column. 2.7.3

Effects of Particle Filling and Packing on Porosity and Dry Density

The porosity of a nonuniform sediment deposit is highly related to its particle packing configuration. As shown in Figure 2.14, there are three packing modes between fine and coarse

Table 2.5 ρ d0 and β ρ in Equation 2.30 (from Lane and Koelzer, 1953) Reservoir operation

Sediment always or nearly submerged Normally a moderate reservoir drawdown Normally a considerable reservoir drawdown Reservoir normally empty

Sand

Silt

Clay

ρ d0

βρ

ρ d0

βρ

ρ d0

βρ

1489

0

1041

91

481

256

1489

0

1185

43

737

171

1489

0

1265

16

961

96

1489

0

1313

0

1249

0

Figure 2.14 Packing of fine and coarse particles: (a) no filling, (b) coarse packing, and (c) fine packing (from Wu and Li, 2017)

Properties of Water and Sediment  41

particles in a bimodal sediment deposit. The first mode indicates no mixing between the two size classes. The second mode is called coarse packing, whereby fine particles fill in the voids of coarse particles. The third mode is called fine packing, whereby coarse particles are dispersed by fine particles. In both fine and coarse packing, the voids of coarse particles are completely or partially lost (Han et al., 1981; Marion et al., 1992; Wu and Li, 2017). Because of imperfect mixing, all the three packing modes may coexist in reality. In addition, particles may bridge gaps, especially if the particle shape is irregular. Particle filling occurs when fine particles are finer than the voids of coarse particles. Therefore, if a sediment mixture is composed of only fine particles (D < 0.05 mm) or if its size range is narrow, particle filling is negligible (Han et al., 1981). In this case, the overall porosity of the deposit can be calculated using the Colby (1963) method: 1  1  v

N

pk

 1 k 1



(2.31)

vk

where pk is the fraction by weight and φvk is the porosity of size class k in the sediment mixture. The fraction-wise porosity φvk can be calculated using one of the Equations 2.26–2.28 or similar formulas developed for uniform or well-sorted sediment mixtures. If the size range is wide, particle filling is important, and Equation 2.31 significantly overestimates the mixture porosity. Many particle filling and packing models have been developed in the literature (e.g., Clarke, 1979; Han et al., 1981; Marion et al., 1992; Kwan et al., 2013; Wu and Li, 2017; Perera et al., 2022). Some of these porosity models are introduced in the following text. Filling and Packing Models of Bimodal Mixtures Consider a deposit comprising a bimodal sediment mixture, with a coarse component of size Dc and fraction pc, and a fine component of size Df and fraction pf. The portion of the coarse size class participating in filling is denoted as Bf. If the fine and coarse particles have the same density, the volume budget for the sediment mixture of a unit mass can be expressed as (Han et al., 1981; Wu and Li, 2017) pf pc 1  1  B f   pc B f   1  v 1  vc 1  vf



(2.32)

The first, second, and third terms on the right-hand side of Equation 2.32 represent the volumes of the coarse size class without filling, the coarse size class with filling, and the fine size class, respectively. When Bf = 0, Equation 2.32 is reduced to Equation 2.31 with subscript k = c and f, corresponding to the case of no particle filling shown in Figure 2.14a. Bf = 1 indicates the ideal fine packing shown in Figure 2.14c, where the voids between coarse particles are lost completely. For the ideal coarse packing shown in Figure 2.14b, all the fine particles stay inside the voids of coarse particles, so Equation 2.32 is reduced to 1/(1 − φv) = pc  /(1 − φvc) or Bf = pf(1 − φvc)/[pcφvc

42  Properties of Water and Sediment

(1 − φvf)] (Clarke, 1979; Marion et al., 1992; Wu and Li, 2017). In reality, Bf is more complex, since all the three packing modes exist in a given sediment mixture. Han et al. (1981) proposed a random filling theory to determine Bf  . The theory assumes that the contact probability of sediment particles is proportional to their specific surface area. If the par2 3 ticles are spherical, the specific surface area of each size class is Dk pk  Dk 6   6 pk Dk . Thus, the probability of a coarse particle contacting a fine particle is Pf 

p f Df pc Dc  p f D f



(2.33)

The probability for the voids of a coarse particle to be filled by fine particles is PfnP , where nP is the number of layers of fine particles required. Therefore, the portion of coarse particles losing voids due to filling by fine particles is B f Pf  PfnP PfnP 1



(2.34)

Han et al. (1981) derived a formula for nP based on a simplified two-dimensional (2-D) particle packing configuration. Wu and Li (2017) improved the formula by considering 3-D packing, as shown in Figure 2.15a, in which four coarse particles are in tight contact with one another. The tetrahedron formed by the centers of the four coarse particles has a circumradius of R  ( 6 4) Dc , and the radius of the largest sphere that the void of the four coarse particles can hold is Δh = R – Dc/2 (Figure 2.15b). Thus, the number of layers of fine particles needed to fill the void of each coarse particle is D  D 1  6 h  nP  Dc  c   0.1124 c   Df 2  Df Df  4

(2.35)

Figure 2.15 Three-dimensional coarse packing used by Wu and Li (2017): (a) four coarse particles packing tightly, and (b) tetrahedron formed by the centers of the four coarse particles

Properties of Water and Sediment  43

For filling to occur, at least one fine particle fills in the void of the coarse particles, i.e., Δh > Df /2. By using Δh given by Equation 2.35, this filling condition is expressed as (Wu and Li, 2017) Dc > 4.45Df

(2.36)

The random filling model is valid when Dc /Df > 4.45. In the range of 1 ≤ Dc /Df ≤ 4.45, although particle filling does not occur, the fine particles may disturb the coarse particles by loosening and wedging (Kwan et al., 2013), and the packing between fine and coarse particles still affects the deposit porosity. Therefore, the filling coefficient in Equation 2.34 is modified into a packing coefficient by multiplying it with a function of the coarse particle fraction ( pc) in the mixture as follows (Perera et al., 2022): B f   B exp  B pc  PfnP 1



(2.37)

where nP is determined by extending Equation 2.35 to the range of 1 ≤ Dc /Df ≤ 4.45:



0.1124 Dc D f nP   0.5 



Dc D f  4.45 1  Dc D f  4.45



(2.38)

The parameters ξB and ςB show strong relationships with Dc/Df as ξB = 1 − aξ exp( − bξ Dc/Df)

(2.39)

ςB = aς exp( − bς Dc/Df)

(2.40)

where aξ, bξ, aς, and bς are empirical coefficients. For spherical glass beads, aξ = 1.72, bξ = 0.39, aς = 3.95, and bς = 0.55. For natural sediments, aξ = 1.47, bξ = 0.445, aς = 3.54, and bς = 0.615. When Dc/Df is larger than about 10, Equations 2.39 and 2.40 give ξB ≈ 1 and ςB ≈ 0, and Equation 2.37 is reduced to Equation 2.34. The threshold Dc /Df ratio of 10 is different from the theoretical value of 4.45, indicating that a transition range exists for particle packing to become filling-dominant. Wu and Li (2017) developed a formula for the filling coefficient Bf based on how many fine particles are available and how many are needed to cover the surface of a coarse particle. The number of fine particles available to cover a coarse sediment particle is equal to the ratio of the numbers of fine and coarse particles, RN, defined as  RN

6 p f D3f )   pc Dc3 )

p f Dc3 pc D3f



(2.41)

The number of fine particles needed to cover the surface of a coarse particle, Nc, is given as Nc 



arcsin  D f 







Dc  D f  

2

 cos 30

(2.42)

44  Properties of Water and Sediment

Figure 2.16  M easured and calculated porosities of the sand-pebble mixtures in the investigation of Phillips (2007) (from Wu and Li, 2017)

Because nP layers of fine particles are needed to fill the void of a coarse particle, the total number of fine particles needed is approximately nP Nc. The filling coefficient Bf is set to the ratio of Pf RN and nP Nc:   Pf RN , B fmax   B f  min  n N  P c 

(2.43)

where Pf is determined using Equation 2.33, and Bfmax is the maximum value of Bf. Bfmax depends on the preparation process and the properties of the sediment mixture. Bfmax was determined to be about 0.85 for natural pebble-sand mixtures and about 0.65 for natural sand-clay mixtures in shallow water bodies such as rivers (Wu and Li, 2017). The values of Bfmax need to be further validated using more measured data. The models introduced in this section were applied to mixtures of sand (Df = 0.385 mm) and pebbles (Dc = 11.11 mm) experimentally studied by Phillips (2007). The measured and calculated porosities are compared in Figure 2.16. The measured data show that the porosity decreases first and then increases as the fine fraction increases. The particle filling models developed by Han et al. (1981) and Wu and Li (2017) can capture this variation trend with reasonable accuracy. The ideal packing model systematically underestimates porosity. The nofilling Colby model gives a monotonic variation trend, which is qualitatively wrong. Therefore, particle filling needs to be considered when calculating the porosity of sediment mixtures with wide size ranges. Extension to Trimodal Mixtures A trimodal sediment mixture is composed of three dominant size classes. According to Han et al. (1981), the porosity of a trimodal mixture deposit can be determined by applying the aforementioned bimodal-mixture random filling model to the sub-mixture of two of the three size classes first, and then to the mixture generated by adding the third size class into the submixture. Perera et al. (2022) found that the results depend on the mixing sequence of the three size classes, and a model using fine-medium-coarse serial mixing (i.e., mixing the fine and medium size classes first) performs relatively well.

Properties of Water and Sediment  45

Alternatively, Perera et al. (2022) extended the random filling-packing model of bimodal sediment mixtures to trimodal mixtures, considering all three size classes to interact randomly with one another. This model performs as well as the fine-medium-coarse serial mixing model. Details of these two models can be found in Perera et al. (2022). 2.7.4  Measurements of Porosity and Dry Density Direct coring and excavation are traditionally used to measure the porosity and dry density of sediment beds. Dry density is obtained by measuring the volume of the undisturbed deposit sample and the mass of the sediment after pore water is removed by oven drying (at 105°C for about 3 hr). Note that when the pore water is evaporated, the originally dissolved chemicals may be precipitated and need to be excluded from the solids. In addition, organic matter may be burned during the drying process. Direct coring and excavation are relatively time- and laborintensive and may alter the bed configuration. The attenuation of γ- and X-rays through a material is highly related to the bulk density of the material. Therefore, γ- and X-rays have been used as non-intrusive tools for measuring sediment bulk density by recording the intensities of the incident and receiving rays and calibrating the attenuation coefficient (Jackson and Hawkes, 1981; Orsi, 1994). Comparison and evaluation of γ- and X-ray devices can be found in Fortin et al. (2013). The big concern in using γ- and X-rays is operational safety and potential contamination. Similarly, acoustic echo strength is proportional to the acoustic impedance (i.e., the product of the speed of sound and the bulk density of the material). Acoustic profilers have been used for decades to measure the geo-acoustic properties of subsurface sediments, but such instruments generally have low vertical resolutions for identifying the changes in bulk density within the near-surface sediment layer. Nevertheless, promising advances have been made in the development of parametric echo-sounders (Schrottke et al., 2006) and the use of high-frequency (300–700 kHz) chirp acoustic waves (Ha et al., 2010). 2.8  Geotechnical Properties of Sediment Deposits 2.8.1  Angle of Repose The angle of repose is the angle of the slope, with respect to the horizontal, formed by a deposit of loose sediment particles under incipient sliding conditions (Figure 2.17a). It is an important parameter for the slope stability of noncohesive sediment banks and shores.

Figure 2.17  Angle of repose: (a) definition, and (b) slope sliding stability

46  Properties of Water and Sediment

Figure 2.18  Angle of repose for tightly packed spheres: (a) top view and (b) side view at θ = 0º

Consider a sphere lying on three tightly packed spheres on a slope (Figure 2.18). The top particle is assumed to roll along the slope direction, which has an angle θ with respect to the centerline of the top and frontal particles. The angle θ is between 0° and 60°. At θ = 0°, the top particle rolls over the gap in the middle of the two downstream particles. At θ = 60°, the top particle rolls over the apex of the immediate downstream particle. The following threshold condition is obtained when the particle mass center is vertically above the pivot point: tan  r 

1 cos  3(1  D2 D1 ) 2  4



(2.44)

where ϕr is the angle of repose, D1 is the diameter of base particles, and D2 is the diameter of the top particle. If the spheres are unisized (i.e., D1 = D2), Equation 2.44 results in 19.5° ≤ ϕr ≤ 35.3°. Because of the randomness of particle packing, the observed angles of repose of spherical particles may roughly fall into this range. In reality, the angle of repose of granular materials depends on many factors. Figure 2.19 shows the relationship proposed by Simons (1957) for the angle of repose of noncohesive sediments submerged in water. For naturally worn sediments, the angle of repose increases from 30° to 41° as the sediment size increases from 0.2 to 600 mm. Rounded particles have lower angles of repose than crushed and angular particles. Similarly, Figure 2.20 shows that the angle of repose decreases with increasing particle roundness (Cho et al., 2006). In addition, surface roughness increases the friction between particles and, in turn, the angle of repose. According to Yang et al. (2009), the angle of repose of a nonuniform sediment deposit generally increases as the mean particle size increases. However, the data exhibit noticeable scatter because the particles of different sizes interact in complex mechanisms. Coarse particles may act as keystones and form clusters with fine particles on the slope. Fine particles may fill in the voids of coarse particles and increase the slope stability. Thus, a more nonuniform sediment mixture tends to have a higher angle of repose. Other affecting factors include the random packing and possible segregation of particles, as well as the particle shape described previously. Because of the buoyancy effect, a noncohesive sediment deposit fully submerged in water has a higher angle of repose than that dried in air (Yang et al., 2009). However, a sand deposit with unsaturated moisture in air exhibits apparent cohesion, in which negative capillary pressure

Properties of Water and Sediment  47

Figure 2.19 Angle of repose for granular materials (from Simons, 1957)

Figure 2.20 Effect of roundness (R P) on the angle of repose (S R is Riley’s sphericity) (from Cho et al., 2006)

tends to hold particles together and increases the angle of repose. This effect occurs in a certain range of moisture contents and disappears when the moisture content is close to saturation (e.g., submerged in water) or is fully dried in air (Chou et al., 2010). Because finer particles usually tend to be more angular, rough, and even cohesive, there is a threshold particle size at which the angle of repose changes from a decreasing trend to an increasing trend as the particle size decreases. The threshold particle size increases as the specific weight of sediment decreases (Simons and Senturk, 1992). Cohesive sediments can form very steep slopes and even overhanging banks. Generally, the angle of repose is not used for cohesive sediments, as explained in the next section. The angle of repose can be measured by piling the sediment deposit as shown in Figure 2.17a. Other measurement methods include the use of a tilting box, tilting cylinder, hollow cylinder, fixed funnel, or revolving drum. Each of these methods is designed for specific materials and applications, and measurements are affected by the device size, confinement, boundary friction, particle segregation, jamming, etc. The measured values can be significantly different among these methods. Currently, there is no consistent guidance for converting between outputs (Al-Hashemi and Al-Amoudi, 2018). Caution should be exercised when applying the measured data.

48  Properties of Water and Sediment

2.8.2  Shear Strength Shear strength is the magnitude of shear stress that a soil can sustain in a shearing process. The shear strength, τs , consists of cohesion and internal friction, as expressed in the following MohrCoulomb equation: τs = σs tan ϕi + Ccoh 

(2.45)

where ϕi is the friction angle; σs is the applied stress normal to the shearing plane; and Ccoh is the cohesion, defined as the cohesion force per unit area, often in Pa. Cohesion is caused by the electrostatic interparticle forces mainly existing in clay and fine silt, as explained in Section 2.4. The clay fraction and minerals in the soil significantly affect cohesion. Cohesion usually increases as the clay fraction increases. Soil cohesion also depends on water content. Moist, sandy soils exhibit apparent cohesion, as described in the previous section. The cohesion of silty soils decreases when dried, whereas the cohesion of soils with large amounts of clay generally increases when they are dried (Kemper and Rosenau, 1984; Zhang et al., 1998). In addition, Ccoh can be extended to include adhesion caused by the cementing effect of chemical and organic materials and the bonding effect of plant roots grown in the soil. Because shear strain involves particle rotation and contact slippage, the friction angle is related to the particle surface texture and packing configuration. The friction angle depends on soil type. For noncohesive soils, the friction angle is approximately equal to the angle of repose. This can be explained by considering the sliding of particles along an infinitely long slope with the angle of repose shown in Figure 2.17b. The resistant force, Fres, and driving force, Fdri, along the sliding plane are Fres = W cos ϕr tan ϕi

(2.46)

Fdri = W sin ϕr

(2.47)

where W is the weight of the failure material. Equating Fres and Fdri results in ϕi = ϕr. However, this equality is limited to the sliding plane where the angle of repose is defined. Because of the heterogeneous microscopic structures of soil, the friction angle may be anisotropic and depend on the direction of shearing. The friction angle measured in a shear test may be different from the angle of repose. Nevertheless, such anisotropy is usually ignored. Thus, according to Figures 2.19 and 2.20, the friction angle of coarse-grained sediments decreases as the particle size and angularity decrease. However, for fine-grained sediments, the friction angle deviates from the angle of repose due to the existence of cohesion. Generally, the friction angle decreases as the particle size decreases or the clay fraction increases. Cohesion and friction angle highly depend on compaction, drainage, deformation, and loading conditions. Compaction increases both cohesion and friction angle and, in turn, effective shear strength. This is particularly significant in cohesive soils under consolidation. In the analysis of effective shear strength, the pore water pressure, pw, can be considered by replacing σs with σs − pw in Equation 2.45. In unsaturated soils, negative pw increases the effective strength. Saturated soils have significantly lower effective strength due to positive pw under hydrostatic conditions. According to Terzaghi et al. (1996), the friction angle for the effective stress of normally consolidated cohesive soils decreases as the plasticity index increases (Figure 2.21). The

Properties of Water and Sediment  49

Figure 2.21  Friction angle as a function of plasticity index (from Terzaghi et al., 1996)

plasticity index (PI) is the range of water contents over which the soil exhibits plastic properties. It is the difference between the liquid and plastic limits. Clay tends to have a high PI, and silt tends to have a low PI. The effective friction angle is higher than 30° when PI < 20% and lower than 25° when PI > 50%. Because many factors are involved, the cohesion and friction angle of cohesive sediments are usually measured directly in real-life applications. The drained shear test follows the ASTM D3080-98 standard (ASTM, 1998) or similar guidelines. 2.9  Physical Properties of Sediment-Laden Water 2.9.1  Sediment Concentration and Transport Rate Figure 2.22 shows a sketch of a mixture comprising a volume of water, Vf , and a volume of sediment, Vf . Sediment concentration is defined by volume as c

Vs  V f  Vs

(2.48)

or by weight as  c

 sVs   V f   sVs

(2.49)

Both sediment concentrations defined here are unitless. They are correlated by  c

 Gc c   or  c   1  (G  1)c G  (G  1)c



(2.50)

Alternatively, sediment concentration is defined as sediment mass per unit volume of the water-sediment mixture, which is equal to ρsc. The corresponding units are kg/m3, g/l, mg/l,

50  Properties of Water and Sediment

Figure 2.22  Sketch of a water-sediment mixture

etc. The sediment concentration is also given in parts per million (ppm) by weight, which is  equivalent to 106 c . The density of the water-sediment mixture, ρm, is determined by ρm = ρ(1 − c) + ρsc

(2.51)

The specific weight of the water-sediment mixture is γm = ρmg. In a dilute or low-concentration condition, c ≪ 1, then ρm ≈ ρ. Suspended load is often quantified by using either concentration or transport rate. The transport rate is also called flux or discharge. The transport rate, Qs, through a cross-section of the channel is determined with Qs = QC

(2.52)

where Q is the flow discharge and C is the average concentration of suspended sediment on the cross-section. The specific (or unit) transport rate, qs, per unit channel width on a vertical is qs = qC

(2.53)

where q is the specific (or unit) discharge of flow and C is the average concentration of suspended sediment on the vertical. Because bed load moves in a thin layer near the bed, it is usually quantified using transport rate. This is either the total transport rate, Qb, through a cross-section or the specific transport rate, qb, per unit channel width. They have the following relationship: B

Qb   qb dz 0



(2.54)

where B is the channel width at the water surface. If qb exhibits a uniform distribution or represents the average specific transport rate over the cross-section, Equation 2.54 becomes Qb = qbB. A similar relationship exists between Qs and qs. The units of Qs and Qb are typically m3/s, kg/s, or N/s, when the volume, mass, or weight of sediment per unit time is used. Correspondingly, the units of qs and qb are m2/s, kg/(s·m), or N/(s·m).

Properties of Water and Sediment  51

2.9.2  Viscosity of Sediment-Laden Water The sediment-laden water behaves differently from clear water. Sediment particles in water change the flow streamlines and in turn the fluid behavior. When sediment concentration is low, the water-sediment mixture is still a Newtonian fluid and exhibits a linear relationship between the shear stress and deformation rate shown in Equation 2.2. However, the viscosity becomes a function of sediment concentration, which can be generally expressed as follows (Einstein, 1906; Chien and Wan, 1983, 1999): μm = μ(1 + k1c + k2c2)

(2.55)

where μm is the viscosity of sediment-laden water, μ is the viscosity of clear water, and c is the volumetric concentration of sediment. The coefficients are k1 = 2.5 and k2 = 0 for very low concentrations (Einstein, 1906). For slightly higher concentrations, k2 varies from 2.5 to 14.1 (Chien and Wan, 1983, 1999). The viscosity increases as c increases. Similarly, Maron and Pierce (1956), Krieger and Dougherty (1959), and Quemada (1977) proposed the following formula:

  1  c cmax  m

 nc



(2.56)

where cmax is the maximum concentration of sediment suspension, with about 0.6 for sand particles. Krieger and Dougherty (1959) used nc = 2.5cmax. Maron and Pierce (1956) and Quemada (1977) suggested nc = 2. As sediment concentration increases to a certain level, the water-sediment mixture becomes a non-Newtonian fluid, and the flows are called hyperconcentrated and debris flows. The threshold concentration for this transition depends on the size and mineral compositions of sediment. The features of hyperconcentrated and debris flows are introduced in Chapter 12. Homework Problems 2.1 How do you measure the mass density of sediment, ρs? 2.2 Determine the mass density, specific gravity, specific weight, and submerged specific weight of quartz gravel. 2.3 Why is quartz the most abundant mineral in riverine sediments? 2.4 What are the average Corey shape and particle roundness factors for naturally worn sediment particles? 2.5 A sediment particle has diameter a = 10 mm, b = 3 mm, and c = 1 mm in the longest, intermediate, and shortest axes. Calculate the Corey shape factor and find the particle type using the Zingg diagram. 2.6 How do you measure particle sizes of cobble, gravel, sand, silt, and clay? 2.7 For a bed material that consists of cobble, gravel, sand, silt, and clay, what methods should you use to measure the size distribution? 2.8 The sediment concentration by volume is measured as 0.1. The specific gravity of sediment is 2.65. (a) Determine the sediment concentration by weight in mass per unit volume and in ppm by weight. (b) Calculate the density of the water-sediment mixture.

52  Properties of Water and Sediment

2.9 A sediment mixture has the following size distribution: D (mm)

0.001 0.004 0.01 0.062 0.25 1

Percent finer (%) 0

5

15

43

25 100

62.5 80 90 100

(a) Determine the arithmetic and geometric mean diameters. (b) Determine the geometric standard deviation and gradation coefficient. (c) Calculate the porosity using Colby’s, Komura’s, and Wu and Wang’s formulas. 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19

How do you measure the dry density of a sediment deposit? What is the relationship between the porosity and dry density of a sediment deposit? How do you measure the repose angle of a sediment deposit? For sediments with sizes 1 and 20 mm, determine their angles of repose using the Simons diagram. Assume average particle roundness. Consider a mixture of sand and gravel. Why and how does the porosity of the mixture vary with the fraction of sand? How do you sample the bed material in a sand-bed stream? How do you sample the bed material in a gravel-bed stream? Why and how does the dry density of a cohesive sediment deposit change during consolidation? Describe how the viscosity of turbid water changes with sediment concentration. Derive Equation 2.44 for the angle of repose for a sphere lying on three tightly packed spheres on a slope.

Chapter 3

Open Channel Flows

This chapter briefly reviews basic knowledge of open channel flows, including governing equations, turbulence characteristics, velocity distribution, channel resistance, boundary shear stress, secondary flows, and coherent structures. 3.1  Classifications of Open Channel Flows A flow is steady if its velocity at any given point does not vary with time; otherwise, it is unsteady. A flow is uniform if its velocity does not change in either magnitude or direction along any streamline; otherwise, it is nonuniform. Unsteady or nonuniform flows can be further divided into gradually and rapidly varied flows. In a rapidly varied flow, the velocity changes abruptly in a short time period or in a short channel reach. In a gradually varied flow, the velocity changes gently. Flows are laminar or turbulent, depending on the Reynolds number Re. For open channel flows, Re is defined as Re = Uh/v

(3.1)

where U is the depth-averaged flow velocity, h is the flow depth (or replaced by the hydraulic radius), and ν is the kinematic viscosity of water. When Re < about 500, the flow is laminar, exhibiting smooth streamlines. When Re > about 1,000, the flow becomes turbulent, featuring velocity fluctuations and highly disordered motions. The threshold Reynolds number for the transition from laminar to turbulent flow depends on channel geometry, boundary surface roughness, inflow conditions, etc. Turbulent flows are further classified as hydraulically smooth, transitional, or rough using the roughness shear Reynolds number: k s  u* k s  

(3.2)

where u* is the shear or friction velocity and ks is the bed roughness height. The turbulent flow is hydraulically smooth if ks+ < 5, transitional if 5  k s  70 , and rough if k s  70. Open channel flows are subcritical or supercritical, depending on the Froude number Fr. For a rectangular channel, Fr is defined as Fr  U

gh 

(3.3)

where g is the gravitational acceleration. For an irregular channel, h is replaced by the hydraulic depth A/B, in which A is the cross-sectional area of the flow and B is the channel width at the water surface. The flow is critical at Fr = 1, subcritical if Fr < 1, and supercritical if Fr > 1. DOI: 10.1201/9781003343165-3

54  Open Channel Flows

3.2  Basic Hydrodynamic Equations Flows in natural rivers usually are turbulent due to irregular and rough boundaries. Thus, the flow velocity exhibits fluctuations as shown in Figure 3.1. However, engineers are usually not interested in the details of these turbulent fluctuations. In practice, it is important to describe and solve for mean motions. As suggested by Osborne Reynolds, the instantaneous quantity  is divided into two parts as

    

(3.4)

where  is the mean quantity and φ' is the fluctuating quantity. The mean quantity is determined as

 

1 T



t T 2 t T 2

 (t ) dt 

(3.5)

where t is the time and T is the period considered. T should be much longer than the fluctuating periods of turbulence, as illustrated in Figure 3.1. The fluctuating quantity satisfies    0 . Applying the time-averaging (Reynolds-averaging) procedure to the continuity and momentum (Navier-Stokes) equations of flow yields ui  0 xi

(3.6)

ui  (ui u j ) 1  2 ui 1 p 1  tij   Fi     t   xi x j x j x j  x j

(3.7)

where i and j are coordinate indices (= 1, 2, 3); xi is the ith coordinate; ui is the component of flow velocity in the xi direction; Fi is the external force, such as gravity, in the xi direction; ρ is the density of water; p is the pressure; and  tij    uiu j is the turbulent or Reynolds stress tensor. Note that the overbar on ui, p, and Fi is omitted for brevity (except in Chapter 11). The subscripts i and j are subject to the Einstein summation convention: any product or quotient of terms with subscript (or superscript) repeated twice is summed over the entire value range of that subscript (or superscript). Boussinesq’s eddy viscosity concept is widely used to model the turbulent stresses in Equation 3.7 as follows:  u u j  tij   t  i   x j xi 

 2    k ij   3 

Figure 3.1 Reynolds’ time-averaging procedure

(3.8)

Open Channel Flows  55

where νt is the turbulent or eddy viscosity; k = uiui / 2 is the turbulent kinetic energy (TKE); and δij is the Kronecker delta, with a value of 1 when i = j and of 0 when i ≠ j. Unlike the molecular viscosity ν, the eddy viscosity νt is not a fluid property but strongly depends on the state of turbulence. The eddy viscosity is usually much greater than the molecular viscosity and varies widely in time and space. The TKE is governed by the following equation (Rodi, 1993): k  (ui k )    t k        Pk   xi t xi   k xi 

(3.9)

where σk is an empirical coefficient of about 1.0. The second term on the left-hand side and the first term on the right-hand side represent the changes in TKE due to advection and diffusion, respectively. Pk and ε are the production and dissipation rates of TKE, defined as Pk  uiu j

 

 u u j ui t  i   x j xi x j 

 ui    x j 

ui ui  x j x j

(3.10)

(3.11)

3.3  Turbulence Characteristics 3.3.1  Turbulence Cascade Process A turbulent flow consists of eddies on various scales. An eddy is a turbulent motion that is at least moderately coherent over a local region. Larger eddies continuously break up into smaller eddies in a cascade process through which the turbulent energy is transferred from a larger to smaller scale until dissipated into heat by molecular viscosity (Figure 3.2). In contrast, the energy is transferred from smaller to larger scales by a process called backscatter. The backscatter process usually happens at a much lower rate than the cascade process. In a shear flow, large eddies are anisotropic since they are confined by asymmetric flow boundaries. Smaller eddies tend to be more isotropic.

Figure 3.2 Cascade process of turbulent energy (after Nezu, 2005)

56  Open Channel Flows

The largest eddies have a characteristic velocity in the order of k1/2. Their characteristic time is commonly referred to as turnover time, and their length is characterized with the integral scale of turbulence: Li ∝ k3/2 /ε

(3.12)

The smallest eddies are characterized by the Kolmogorov microscale: lK = (ν3/ε)1/4

(3.13)

Equation 3.13 is constructed by dimensional analysis using ε and ν as the independent variables since the smallest eddies are constrained by the action of viscosity. Correspondingly, the velocity and time scales of the smallest eddies are described as uK = (νε)1/4

(3.14)

tK = lK/uK = (ν/ε)1/2

(3.15)

Kolmogorov proposed a model of locally isotropic turbulence in which an intermediate inertial subrange exists between the largest and smallest eddies. In this subrange, the process of energy transfer is independent of ν and is governed solely by ε, and the eddy sizes are given by the Taylor microscale: lT ≈ (10νk/ε)1/2

(3.16)

Figure 3.3 shows the general energy spectral distribution of turbulent eddies. The wave number is defined as Kw = 2π/le, with le being the eddy size ranging from the integral scale to the Kolmogorov scale. E(Kw) is the energy spectrum density of eddies with wave number Kw, satis fying k  E  K w  dK w. The most energetic eddies are large eddies, which contain the major



0

kinetic energy of turbulence and are effective in the transfer and mixing processes (Rodi, 2017). The energy spectrum in the inertial subrange can be described with the well-known − 5/3 power law derived by Kolmogorov: E  K w   CK  2 / 3 K w5 / 3  2 / Li  K w  2 / lK  

(3.17)

where CK is the Kolmogorov constant. Note that the turbulent energy spectrum shown in Figure 3.3 is observed for 3-D turbulent eddies that are generated locally by bottom friction and restricted by water depth. Large, 2-D horizontal eddies can exist in shallow water bodies, such as coastal waters, lakes, reservoirs, and rivers. The large, 2-D horizontal eddies and the small, 3-D bed-generated eddies are associated with ranges of low and high wave numbers, respectively, yielding a two-range spectrum (Nadaoka and Yagi, 1998). The 2-D horizontal eddies are responsible for horizontal mixing in shallow flows and the 3-D eddies enable vertical mixing (Rodi, 2017). In a sediment-laden flow, particles undergoing weak transport (e.g., bed load) near the bed may increase turbulence by inducing flow instability, increasing bed roughness, generating shedding eddies, etc. (Elata and Ippen, 1961; Best et al., 1997; Dey et al., 2012). In the case of suspended load, energy is extracted from large eddies to suspend sediment particles,

Open Channel Flows  57

Figure 3.3 Energy spectrum of locally generated turbulence, including the integral scale, Kolmogorov scale, and intermediate inertial subrange with the −5/3 power law

whereas small eddies may be generated by sediment particles during the motions relative to the fluid, e.g., flow-particle and particle-particle interactions. In experiments using large, nearly neutrally buoyant particles, Bellani et al. (2012) observed that turbulent energy was removed from large eddies and reinserted into small eddies. However, Wang and Qian (1989) and Lyn (1992) observed that large scales of turbulence slightly gained importance in the energy spectrum relative to small scales in experiments using natural and artificial sediments. These inconsistent results are likely associated with different flow and sediment conditions or due to the difficulty of performing accurate measurements. The overall turbulent energy is attenuated by suspended sediments due to sediment-induced stratification and increased fluid viscosity (e.g., Monin, 1959; Zhang, 1961; Hino, 1963; Yalin, 1972; Chien and Wan, 1983; Bellani et al., 2012). The degree of turbulence attenuation increases with increasing sediment concentration. Therefore, turbulence-sediment interactions involve very complex processes depending on sediment concentration, relative size (le /D), relative density (ρs/ρ), particle shape, etc. More discussion is presented in Section 11.1 from the perspective of twophase flows. 3.3.2  Reynolds Shear Stress For a steady, uniform flow in a wide channel, as shown in Figure 3.4, the shear stress (τ = τxy) is governed by the following equation:

    u v  

du y    b 1    dy  h

(3.18)

where τb is the shear stress at the bed; uʹ and vʹ are the fluctuating velocities in x and y directions, respectively; x is the streamwise coordinate; and y is the coordinate normal to the bed. The total shear stress τ has a linear profile, with a value of zero at the water surface and τb at the bed surface. The Reynolds and viscous components are represented with the first and second terms in the middle section of this equation, respectively. Figure 3.5 shows the vertical profiles of the Reynolds shear stress in clear water flows over smooth beds as measured by Nezu and Rodi (1986), Nezu and Azuma (2004), and Muste et al.

58  Open Channel Flows

Figure 3.4 Uniform flow along an inclined channel slope

Figure 3.5 Distribution of the Reynolds shear stress in open channel flows (symbols: measured data; solid lines: Reynolds shear stress; dashed lines: total shear stress; cases P2 and P4: Nezu and Rodi, 1986; NA: Nezu and Azuma, 2004; M: Muste et al., 2005)

(2005). In the main water column away from the bed, the Reynolds shear stress is approximately equal to the total shear stress and follows the linear profile defined by Equation 3.18 irrespective of the flow conditions. Near the bed, the Reynolds shear stress decreases greatly due to the wall effect, and the viscous shear stress becomes dominant. 3.3.3  Turbulence Intensity The components of turbulence intensity in the streamwise (x), vertical (y), and spanwise  , and wrms  ( u 2 ) , vrms  , respectively. Figure 3.6 (z) directions can be represented by urms shows the streamwise and vertical components for a uniform flow in an open channel (Grass, 1971). The turbulence intensity increases rapidly and then decreases gradually with increasing height (y) above the bed. The maximum value occurs at about (0.04–0.1)h for the streamwise component and at about (0.1–0.2)h for the vertical component. The near-bed turbulence is suppressed by the wall effect and related to the bed roughness. As the roughness shear Reynolds number increases, the streamwise turbulence intensity near the bed decreases, but the vertical turbulence intensity increases. The turbulence intensities in the main water column over smooth and rough beds collapse into single curves by scaling with the shear velocity. The streamwise intensity is greater than the vertical intensity, which is about the shear velocity.

Open Channel Flows  59

Figure 3.6 Turbulence intensity in streamwise and vertical directions (from Grass, 1971)

Nezu (1977) proposed the following exponential functions for the streamwise, vertical, and spanwise components of turbulence intensity in the main water column (0.1 < y/h < 0.9):    y urms  y  vrms y  wrms   ,  w exp   w   u exp   u  v exp   v , h u* h  u* h  u*   

(3.19)

Nezu (1977) obtained ξu = 2.3, ξv = 1.27, ξw = 1.63, and ζu = ζv = ζw = 1.0 using a hot-film anemometer, whereas Nezu and Rodi (1986) obtained ξu = 2.26, ξv = 1.23, ζu = 0.88, and ζv = 0.67 using a laser Doppler anemometer. Kironoto and Graf (1994) reported ξu = 2.04, ξv = 1.14, ζu = 0.97, and ζv = 0.76 for gravel and artificial rough beds. Dey and Raikar (2007) reported ξu = 2.07, ξv = 1.17, ζu = 0.95, and ζv = 0.69 for gravel beds at incipient motion thresholds. Similar values were obtained by Song et al. (1994) for movable gravel beds and by Muste and Patel (1997) for flows with suspended sediment of low concentration. Differences in ξu, ξv, and ξw indicate that the turbulence in open channel flows is anisotropic. By combining the three components in Equation 3.19, the following distribution was derived for the TKE k  (u 2  v2  w2 ) / 2 (Nezu and Nakagawa, 1993): k y   4.78 exp  2   2 u*  h

(3.20)

Similarly, Nikora and Goring (2000) proposed logarithmic functions to approximate the turbulence intensity profiles in the main water column. 3.3.4  Turbulence Dissipation Rate It is difficult to evaluate the dissipation rate of TKE using Equation 3.11 because instantaneous velocity gradients are involved. Therefore, ε is often estimated using a result applicable to isotropic turbulence (Tennekes and Lumley, 1972): 2

u 2  u     15   15 2   lT  x 

(3.21)

where the Taylor microscale lT and u 2 can be estimated using single-point measurements.

60  Open Channel Flows

Nezu (1977) proposed the following distribution for ε:

h  y  E1   3 u* h

1/ 2

 y exp  3    h

(3.22)

where E1 is a coefficient related to the Reynolds number. At moderate Reynolds numbers of 104 to 105, E1 is approximately equal to 9.8. The turbulence dissipation rate decreases with increasing y in the main water column. The turbulent flow in an open channel is usually divided into inner (y/h < ~0.2) and outer (y/h > ~0.2) regions. In the inner region, turbulence is intensively produced due to bed shear. TKE production and dissipation are approximately equal in the lower part of the outer region, whereas in the upper part (y/h > ~0.6), the flow is significantly affected by the free surface and TKE dissipation exceeds production. 3.3.5  Eddy Viscosity Cases of Clear Water One of the turbulence models widely used for the eddy viscosity is the mixing length model of Prandtl (1925):

 t  lm2

u  y

(3.23)

where lm is the mixing length. For general boundary layer flows, lm = κy (Prandtl, 1925). For open channel flows, lm is often given as follows (Satkevich, 1934): y   lm  y  1    h  

(3.24)

where κ is the von Karman coefficient. The mixing length model is suitable for flows where turbulence is in local equilibrium (i.e., TKE production = dissipation) rather than where the advective and/or diffusive transport of turbulence is important. The mixing length model is often used for simple shear flows where lm can be specified empirically. It is rarely used for rapidly varied flows, such as recirculating flows, in channels with complex geometry. For the uniform flow in an open channel, the eddy viscosity yields a parabolic profile: y    t  u* y 1    h  

(3.25)

Many advanced turbulence models, such as k-ε models, Reynolds stress models, and large eddy simulations (LES), have been applied to simulate turbulent flows (Rodi, 1993, 2017). They are usually incorporated with multidimensional numerical modeling (e.g., Wu, 2007; Stoesser et al., 2015).

Open Channel Flows  61

Effects of Stratification and Buoyancy Eddy viscosity is strongly influenced by buoyancy effects and particularly reduced by stable stratification (Rodi, 1993). In a stably stratified flow where the bottom fluid is denser than the upper fluid, turbulence is attenuated because the bottom fluid lifted by turbulent mixing is moved back by gravity. Conversely, turbulence is enhanced in an unstably stratified flow. To account for these effects, the following function is usually applied:

  t 1   Ri  tm

 m



(3.26)

where νtm is the eddy viscosity of the stratified flow; νt is the eddy viscosity of the flow without stratification; αv and mv are empirical coefficients, set as 10 and 0.5, respectively (Munk and Anderson, 1948); and Ri is the gradient Richardson number: Ri  

g  y    u y 2

(3.27)

The effects of buoyancy on mixing length are taken into account with the Monin-Obukhov relation, which adjusts the mixing length with a damping function similar to that in Equation 3.26 (Busch, 1972). It should be pointed out that buoyancy effects have been studied mainly in thermally induced stratified flows. In the case of sediment-laden flows, not only buoyancy effects but also particle inertia and interparticle collision play important roles. 3.4  Velocity Profiles of Uniform Turbulent Flows For a uniform turbulent flow in a wide channel, the velocity profile exhibits layered structures along the depth (Figure 3.7). Immediately above a smooth bed is a viscous sublayer, in which the viscous force is dominant and the flow remains relatively smooth. The viscous sublayer is altered over a rough bed. The flow velocity obeys the log law, defect law, or log-wake law in the main water column over both smooth and rough beds. These velocity profiles are introduced in this section. 3.4.1  Velocity Profile Near the Bed In the viscous sublayer near a smooth bed, Newton’s viscosity law prevails, and the shear stress is virtually constant. Thus



du  b  dy

(3.28)

Integrating Equation 3.28 and using the definition of shear velocity u*   b  yield the following linear velocity profile: u u* y   u*  where the origin of the y axis is set at the bed surface.

(3.29)

62  Open Channel Flows

Figure 3.7 Velocity distribution of turbulent flow over a smooth bed, plotted in the log(y) scale with layered structures; shaded area represents the range of measured data; a buffer zone exists between the viscous sublayer and the log-law layer; the linear and log laws intercept at y+ = 11.6, which is set as the nominal division of the two layers (after Crowe et al., 2009)

The thickness of the viscous sublayer, δ', is approximated by δ' = 5ν/u*

(3.30)

The viscous sublayer is very thin. It is about 0.5 mm thick when u* is 0.01 m/s, and decreases as u* increases. In the case of a rough bed, comparing the roughness height ks with the viscous sublayer   k s u thickness δ' leads to the roughness shear Reynolds number, * k s v 5k s  , which is used to classify turbulent flows, as described in Section 3.1. In a hydraulically smooth turbulent flow (k s  5 or ks< δ'), the roughness elements are sheltered within the viscous sublayer, so the primary flow does not experience the effect of bed roughness. In a fully rough turbulent flow (k s  70), the viscous sublayer disappears and bed roughness plays a dominant role. In the hydraulically transitional regime, both the viscous sublayer and bed roughness affect the turbulent flow characteristics. The flow near a sediment bed is complex at the scale of grain roughness (Figure 3.8a). The protruding particles deflect the flow, form separation zones with wake eddies, and create uneven pressure fields. Nikora et al. (2001, 2004) defined two near-bed sublayers: the forminduced sublayer located immediately above the crests of the surface particles, and the interfacial sublayer located between the crests and troughs of the surface particles. A subsurface flow may exist in the pores of bed particles. In the interfacial sublayer, the time-averaged flow is highly 3-D and heterogeneous in space. A more suitable approach is to apply the Navier-Stokes equations averaged in both time and space. This is called double-averaging. The double-averaged flow velocity in the interfacial sublayer has a linear, exponential, or polynomial profile, depending on the roughness features and flow conditions (e.g., Nikora et al., 2001, 2004).

Open Channel Flows  63

Figure 3.8 Turbulent flow over a rough bed: (a) layer structure and (b) theoretical bed

3.4.2  Log Law of Clear Water Flows In the main water column, the turbulent shear stress is approximately equal to the total shear stress: 2

 du  y du      b 1     t  lm2    h dy    dy 

(3.31)

Substituting the mixing length Equation 3.24 or the parabolic eddy viscosity Equation 3.25 into Equation 3.31 yields

b /  du   y dy

(3.32)

Integrating this equation yields u 1  ln y  C1 u 

(3.33)

Equation 3.33 is the well-known log law of velocity. It was first derived by Prandtl (1925) for a turbulent boundary layer flow over a flat plate; however, Prandtl’s original derivation assumes constant shear stress and uses the mixing length lm = κy. The assumption of constant shear stress is limited to the near-bed region. The derivation described here avoids this limitation by using the mixing length Equation 3.24 or the parabolic eddy viscosity Equation 3.25. The integral constant C1 needs to be determined using measurement data. Based on the experiments of Nikuradse (1932) in smooth pipes, κ = 0.4 and C1 = 5.5 + (1/κ) ln(u*/ν), which leads to the following velocity distribution (Keulegan, 1938): u 1 u* y  ln  5.5  u*  

(3.34)

Detailed laser-Doppler measurements by Nezu and Rodi (1986) showed that κ = 0.412 and C1 = 5.29+(1/κ) ln(u*/ν) for clear water flows in smooth channels. These values agree well between pipe and open channel flows. The lower validity limit of Equation 3.34 is y+ = u*y/ν ≈ 30 (Figure 3.7). The combination of the linear and log laws is called the law of the wall. A buffer zone exists between the viscous

64  Open Channel Flows

sublayer and the log-law layer, but it is usually ignored. The division of the viscous sublayer and the log-law layer is nominally set at u*  y/v = 11.6, where the linear and logarithmic velocity distributions intercept. Therefore, the viscous sublayer has a nominal thickness: δ = 11.6ν/u*

(3.35)

According to the experiments of Nikuradse (1932) in rough pipes, the integral C1 is related to the bed roughness height ks, and the velocity distribution is written as (Keulegan, 1938) u 1 y  ln  8.5 u*  k s

(3.36)

For a bed with static sediment particles, the theoretical bed (y = 0) was set at about 0.2ks (Einstein and El-Samni, 1949; Kironoto and Graf, 1994), 0.25ks (van Rijn, 1984a; Afzalimehr and Rennie, 2009), and 0.2D–0.3D (Bridge and Bennett, 1992) below the crests of the surface particles (Figure 3.8b). Here, D is the diameter of sediment particles on the bed surface. The roughness height ks was set as D67 by Einstein and El-Samni (1949), D50 by Kironoto and Graf (1994), and D84 by Afzalimehr and Rennie (2009), with more options presented in Section 3.5.1 about flow resistance. Furthermore, Plott et al. (2013) found that this concept is also valid for large-scale roughness elements, such as rigid vegetation and mobile bed forms, for which the theoretical bed is 0.2–0.4 times the roughness height below the crests of the roughness elements. The lower end of this range is suitable for smaller, sediment-scale roughness types, and the upper end is for larger, channel-spanning roughness elements. The log law is not strictly valid in the form-induced sublayer over a rough bed (Figure 3.8a). However, for convenience in practice, the lower limit of the log-law layer is assumed to be at y0, where the log law gives a zero velocity (Figure 3.8b). By using Equation 3.36, y0 = ks/30 is derived. Considering the theoretical bed, this value of y0 is below the crests of the surface particles. Thus, the log-law layer nominally includes the form-induced sublayer and the upper part of the interfacial sublayer. Equations 3.34 and 3.36 can be written in the following form covering hydraulically smooth to rough turbulent flows (Einstein, 1950):  30  s y  u  5.75 log   u*  ks 

(3.37)

where χs is a coefficient that varies with ks/δ, as shown in Figure 3.9. The relationship in Figure 3.9 is reduced to χs = 1 for a hydraulically rough bed andχss  0.3k s for a hydraulically smooth bed. Higher values of χs appear in the hydraulically transitional regime. These trends result from the different turbulent flow structures in the three regimes described previously. Note that κ = 0.4 is used for deriving Equation 3.37. Equation 3.37 can also be written as (von Karman, 1934; Cebeci and Bradshaw, 1977) u 1  y  1 y  ln ln  Bs   u*   y0   k s

(3.38)

where y0 = ks/(30χs), and Bs = 5.75log(30χs).  Bs 1 ln k s  5.5 for a hydrauliThe parameter Bs = 8.5 for a hydraulically rough flow, and cally smooth flow. Several functions of Bs or y0 for the entire turbulent flow regime were

Open Channel Flows  65

Figure 3.9 Coefficient χ s as a function of k s/δ, showing variations in characteristics of turbulent flow through the hydraulically smooth, transitional, and rough regimes (from Einstein, 1950)

proposed by Cebeci and Bradshaw (1977), Christoffersen and Jonsson (1985), Yalin (1992), and da Silva and Bolisetti (2000), among others. The function proposed by Christoffersen and Jonsson (1985) is y0 

ks   u* k s 1  exp   30   27

     9u   *

(3.39)

Equation 3.39 is reduced to y0 = ν/(9u*) for a hydraulically smooth flow, and y0 = ks/30 for a hydraulically rough flow. The function of da Silva and Bolisetti (2000) is  2.55

 Bs (2.5 ln k s  5.5)e 0.0705(ln ks )

 2.55

 8.5[1  e 0.0594(ln ks )

 ] (0.2  log k s  3.2) 

(3.40)

The upper limit of the log-law layer has been debated for a long time. It is suggested by the classical boundary layer theory as y+ ≈ 500 or y ≈ 0.2h, i.e., within the inner region (e.g., Graf and Altinakar, 1998; Crowe et al., 2009). Nevertheless, the log law can be extended to a significant portion of the outer region in an open channel flow, except the region near the water surface (e.g., Keulegan, 1938; Einstein, 1950; Vanoni, 1975). An alternative is to rewrite Equation 3.33 in the following defect law (von Karman, 1934): umax  u y 1   ln  u  m

(3.41)

where umax is the maximum velocity and δm is the height at which the maximum velocity occurs. The maximum velocity is usually located at the water surface (δm = h) in a wide channel but may be lowered to a certain depth below the water surface in a narrow channel (see Section 3.6.1). Equation 3.41 is derived by subtracting Equation 3.33 from umax/u* = (1/κ) lnδm + C1 to eliminate the integral constant C1. Equations 3.33 and 3.41 are equivalent if κ and C1 are constant along the flow depth. Equation 3.41 is applied in the outer region below δm. It works for both smooth and rough beds.

66  Open Channel Flows

Integrating Equation 3.41 over the flow depth (δm = h) yields umax  U 1   u 

(3.42)

By eliminating umax from Equations 3.41 and 3.42 and using the relationship u U  g Ch described in Equation 3.65, the defect or log law can be written in the following form: g  u  y   1 1  ln       Ch  U  h 

(3.43)

where Ch is the Chezy coefficient. This equation clearly describes how the Chezy resistance coefficient affects the vertical profile of flow velocity. In addition, the velocity profile of a flow over a rough bed is essentially affected by the relative submergence of roughness elements, h/ks (Nikora et al., 2001). For high relative submergence  h / k s  1, the flow has the log-law layer and near-bed sublayers described earlier. For small relative submergence (1 ≤ h/ks < 2 to 5), the log-law layer diminishes and the forminduced sublayer dominates the water column. For partial submergence (h/ks105. According to Kironoto and Graf (1994), κ = 0.4 and Π is about 0.09 and −0.03 for uniform flows over rough plates and gravel beds, respectively. Holmes and Garcia (2008) tested the log-wake law in a large dune-bed river and found that Π varies from 0.25 at the crest to −0.25 at the lee of the dune. For the velocity profile spatially averaged over the dune bed, the log or defect law without the wake function is accurate enough, except in the near-bed zone. Notwithstanding these findings, applications of the log-wake law are still limited because Π varies case by case and needs a reliable formula to cover the variety in flow conditions (Wang et al., 2001). Note that Equation 3.48 is invalid at the water surface, where it yields a finite gradient of velocity, du/dy = u*/(κh). The velocity gradient at the free surface should be zero if there is no wind driving. To consider this, Guo and Julien (2001) suggested the following modified log-wake law:   y  1 y / m umax  u y 1   ln  0 cos 2    u  m   2 m 

(3.50)

Figure 3.13 Relationship between the wake strength coefficient and global Richardson number (from Coleman, 1981)

Open Channel Flows  71

where Ω0 = 2Π/κ. Based on the experimental data of Wang and Qian (1992) and Coleman (1981), κ varies with sediment concentration and global Richardson number, and Ω0 is related to the aspect ratio B/h and slightly affected by the global Richardson number and sediment concentration. Equation 3.50 needs to be tested more, particularly in field conditions. 3.4.5  Power Law Another often-used velocity profile of uniform turbulent flows is the power law: u umax

1/ m

 y    h

(3.51)

where m is an empirical coefficient that ranges from 4 to 12, with a typical value of 6 or 7 for flows in alluvial rivers and estuaries. Integrating Equation 3.51 over the flow depth leads to U m   umax m 1

(3.52)

and thus, Equation 3.51 can be rewritten as 1/ m

u m 1  y    U m  h 

(3.53)

Eliminating umax from Equations 3.42 and 3.52 results in (Chen, 1991) m = κU/u*

(3.54)

The ratio U/u* is related to the channel resistance coefficient. For example, the Darcy-Weisbach coefficient f gives 2

f  u    8  U 

(3.55)

Using Equations 3.54 and 3.55 leads to the following formula of Hinze (1975): m = 1.2f −0.5

(3.56)

Karim and Kennedy (1987) extended Equation 3.56 to sediment-laden flows and related the coefficient to the increased rate of fluid-shear energy dissipation produced by the moving sediment. Based on laboratory data, Cheng (2007) found that the one-sixth power (i.e., m = 6) is acceptable in general, while higher indexes would be required for flows over very rough boundaries. Cheng (2007) proposed the following empirical formula: m = 1.37f −0.43

(3.57)

72  Open Channel Flows

Through Equation 3.54, m can also be related to the Manning and Chezy coefficients (defined in Section 3.5.1). The International Organization for Standardization (ISO, 1997) suggested the following relationship between m and the Chezy coefficient Ch:  m

Ch  2  0.3  g  1  Ch

  g 

(3.58)

Note that the aforementioned log, defect, log-wake, and power laws are valid in the central region of a channel. As explained in Section 3.6.1, the flow near sidewalls exhibits strong 3-D features. 3.5 Average Velocity and Boundary Shear Stress of Uniform Flows 3.5.1  Uniform Flows in Simple Channels Consider a uniform flow in a prismatic open channel, as shown in Figure 3.14a. For the general cross-section shown in Figure 3.14b, the flow area A, flow width B, and wetted perimeter P are functions of the flow depth h. For the segment of length L shown in Figure 3.14a, the acting forces include the gravity (W ) on the water body, the friction force (F ) over the wetted boundary, and the pressure forces on the upstream and downstream faces. The pressure forces cancel each other, so the force equilibrium in the streamwise direction is expressed as F = W sin β

(3.59)

where β is the bed slope angle. The gravity is W = γAL, and the friction force is F = τ0PL. Substituting these terms into Equation 3.59 leads to A   0  sin   R sin   P

(3.60)

where γ is the specific weight of water, R = A/P is the hydraulic radius of the channel, and τ0 is the average boundary shear stress over the wetted perimeter. For a wide channel, τ0 is equal to the bed shear stress τb. For a gentle bed slope, sin β ≈ tan β = S. Thus, Equation 3.60 is rewritten as τ0 = γRS

(3.61)

Figure 3.14 Uniform flow in a prismatic open channel: (a) forces on a segment and (b) crosssection geometric parameters

Open Channel Flows  73

Using Equations 3.55 and 3.61 leads to the Darcy-Weisbach equation: S f

1 U2 or U  4R 2 g

8 gRS  f

(3.62)

Thus, the energy or friction slope, i.e., the right-hand term of the first equation here, is equal to the bed slope S in the case of uniform flows. For a laminar flow, the Darcy-Weisbach coefficient is f = 24/Re

(3.63)

For a turbulent flow, integrating Equation 3.37 over the flow depth yields the log law for the channel resistance (Einstein, 1950; Keulegan, 1938): U  u

 Rs 8  5.75 log 12.27 f ks 

  

(3.64)

Note that the factor “12.27” in Equation 3.64 is different from the theoretical value of “11”. This difference is minor because ks is usually calibrated. Equation 3.64 works well for open channel flows. For immobile beds, the roughness height ks is often set to a substantial multiple of sediment diameter, such as 2D50 (Yalin, 1972), 2D65 (Engelund, 1966), 2D90 (Kamphuis, 1974; Parker and Peterson, 1980), 3D90 (van Rijn, 1984a), 3.5D84 (Hey, 1979), and about 4D84 (Ferguson, 2007). Hey (1979) found that ks ≈ D99, which can be approximated with  g2.33 D50 for unimodal sediment mixtures (Cheng, 2016a; see Section 2.6.2). The roughness height ks in Equations 3.37 and 3.64 may have different values because ks in Equation 3.64 additionally accounts for the effects of sediment transport and cross-section shape and compensates for the errors of Equation 3.37 in the upper water column. Chezy suggested the following equation for the average flow velocity: U  Ch RS 

(3.65)

The Chezy coefficient C h is dimensional. It is given in m 1/2/s when U is in m/s and R is in m. It is sometimes replaced with the dimensionless Chezy coefficient ch  Ch g . The bottom friction coefficient cf is often used to relate the bed shear stress and average flow velocity: τ0 = cf   ρU 2

(3.66)

Comparing Equations 3.62, 3.65, and 3.66 leads to the following relationships among the Chezy, Darcy-Weisbach, and bottom friction coefficients: c f

f g 1    2 8 Ch ch2

(3.67)

Thus, the Darcy-Weisbach coefficient increases as the Chezy coefficient decreases. For open channel flows, Manning suggested Ch 

1 1/ 6 R  n

(3.68)

74  Open Channel Flows

where n is the Manning roughness coefficient, in s/m1/3. This equation is primarily applied to hydraulically rough turbulent flows. Substituting it into Equation 3.65 leads to the Manning equation: U

1 2 / 3 1/ 2 R S  n

(3.69)

Now U can be calculated using one of Equations 3.64, 3.65, and 3.69. Then, the flow discharge is calculated as Q = AU

(3.70)

Combining Equations 3.69 and 3.70 yields  Q

1  AR 2 / 3 S 1/ 2 KS 1/ 2  n

(3.71)

where K = AR2/3/n = A5/3/(nP2/3) and is the channel conveyance. The Manning n accounts for the effects of bed roughness on the flow field, and its determination is essential to the accuracy of the calculated flow and sediment transport. For a movable bed with sediment grains and bed forms, n can be determined by using one of the empirical formulas introduced in Chapter 6. In general, n depends on many factors, including cross-section shape, channel alignment, channel curvature, surface roughness, bed forms, obstructions, vegetation, sediment transport, and temperature. Therefore, n should be calibrated if gauged water surface profiles and high-water marks are available, or estimated by referring to the n values in similar streams. The often-used references include Chow (1959), Fasken (1963), Barnes (1967), and Hicks and Mason (1991). 3.5.2  Composite Roughness If hydraulic properties, such as roughness and conveyance, are nonuniform across the channel, their composite values need to be computed. The often-used methods include hydraulic radius division, energy slope division, and conveyance methods. Einstein (1942) and Horton (1933) proposed a method to compute the composite hydraulic properties for the cross-section with rough vertical sidewalls or steep bank slopes, based on the division of hydraulic radius. The wetted perimeter is divided into several segments according to the surface roughness distribution, as shown in Figure 3.15. The total shear force on the wetted perimeter is calculated as P 0 

M

P j 1



j 0j



(3.72)

M

where P  P , Pj is the wetted perimeter in segment j,  0 j is the shear stress in segment j, j 1 j and M is the number of segments. In the hydraulic radius division method, the boundary shear stress is determined by using Equation 3.61 on the entire cross-section and the following formula on segment j:

 0 j   Rj S 

(3.73)

Open Channel Flows  75

Figure 3.15 Channel with different roughness on cross-sectional segments

where Rj is the hydraulic radius of segment j. Assuming equal velocity in all the segments and applying the Manning equation in the entire cross-section and segment j yield R = (nU/S1/2)3/2, Rj = (njU/S1/2)3/2

(3.74)

Substituting Equations 3.61, 3.73, and 3.74 into Equation 3.72 leads to the following equation for the composite Manning n:  n  

M

 j 1

Pj n3j / 2

 P  

2/3



(3.75)

Based on the division of energy slope, Engelund (1966) proposed an alternative option to determine the composite hydraulic properties. This method replaces Equation 3.73 with

 0 j   RS j 

(3.76)

and applies the equal velocity assumption and the Manning equation in the entire cross-section and segment j: S = (nU/R2/3)2, Sj = (njU/R2/3) 2

(3.77)

where Sj is the energy slope on segment j. Substituting Equations 3.61, 3.76, and 3.77 into Equation 3.72 yields  n  

M

 j 1

1/ 2

Pj n 2j

 P   

(3.78)

The divisions of hydraulic radius and energy slope lead to different exponents in Equations 3.75 and 3.78. Equation 3.78 gives a larger value than Equation 3.75, but the difference is small. The equal velocity assumption used in the aforementioned methods is only applicable in simple channels. Because the flow velocities in the main channel and floodplains may be significantly different (Figure 3.16), a more adequate method for determining the composite hydraulic properties in a compound channel is the conveyance method. In the conveyance method, the cross-section is divided into subsections in such a way that the equal velocity assumption can be approximately valid in each subsection. Each subsection can

76  Open Channel Flows

Figure 3.16 Composite roughness of compound channel

be further divided into segments according to the surface roughness. The flow area, wetted perimeter, and conveyance of each subsection can be calculated normally. The conveyances of all subsections are then summed to obtain the total conveyance of the cross-section. For example, the compound cross-section shown in Figure 3.16 is divided into three subsections: main channel, left floodplain, and right floodplain, and the total conveyance is determined as K

5/ 3 ALF

2/3 nLF PLF



5/ 3 AMC

2/3 nMC PMC



5/ 3 ARF

2/3 nRF PRF



(3.79)

where the subscripts LF, MC, and RF denote the left floodplain, main channel, and right floodplain, respectively. The Manning n in each subsection can be determined using the hydraulic radius or energy slope division method. The conveyance method is simple but does not consider the interactions between the main channel and floodplains that have different velocities. Methods considering the momentum transfer and apparent shear stress at the interfaces of the subsections have been developed by Knight et al. (2010), Moreta and Martín-Vide (2010), and Yang et al. (2012, 2014b), among others. Interested readers can find details in the references. 3.5.3  Partition of Bed and Sidewall Shear Stresses Consider a steady, uniform flow in a trapezoidal channel, as shown in Figure 3.15. The sidewalls have a uniform roughness, and the bed may have a different roughness. The channel bed slope is S, the flow depth is h, and the average flow velocity is U. These conditions allow one to determine only the average boundary shear stress using the formulas presented in Section 3.5.1. The average shear stress needs to be separated into parts that are responsible for bed and bank sediment movements. Particularly, the sidewall effect in a narrow channel needs to be corrected because only the bed shear stress is usually used to calculate sediment transport over the bed. The cross-sectional hydraulic radius is R 

(b0  ms h)h A   P b  2 1  m2 h 0 s

where b0 is the channel width at the bottom and ms is the side slope (horizontal/vertical).

(3.80)

Open Channel Flows  77

The cross-sectional area is divided into Ab and Aw in Figure 3.15. Ab is the area of part 2, and Aw is the combination of parts 1 and 3 that are assumed to have the same sidewall roughness. Applying Equation 3.72 yields P  0 Pb b  Pw w 

(3.81)

where τb is the average bed shear stress, τw is the average sidewall shear stress, Pb = b0,

 Pw 2 1  ms2 h, and P = Pb+ Pw. Equation 3.81 alone is not enough to determine both bed and sidewall shear stresses, so an additional equation is needed. The hydraulic radius or energy slope division method described in the previous section can be used. For example, Einstein (1942) determined the sidewall hydraulic radius Rw using Equation 3.74 with an estimated Manning coefficient nw for the sidewalls. Because the Manning equation is primarily used for hydraulically rough turbulent flows, the Einstein method was modified by Johnson (1942) and Vanoni and Brooks (1957) for smooth sidewalls as follows. Under the assumption of equal velocity, the Darcy-Weisbach formula gives U2 8   gR S f

8  gRb fb

8 gRw  fw

(3.82)

which is rewritten as R Rb Rw    f fb fw

(3.83)

Substituting τ0 = γRS, τb = γRbS, and τw = γRwS into Equation 3.81 and then using Equation 3.83 yield Pf = Pb fb + Pw fw

(3.84)

Equation 3.84 can also be derived using Equation 3.83 with A = Aw + Ab, A = PR, Ab = PbRb, and Aw = Pw Rw. In the methods modified by Johnson (1942) and Vanoni and Brooks (1957), the sidewall friction factor fw is calculated by using the following von Karman-Prandtl formula for smooth sidewalls: 1 fw

 4URw  2 log   

 f w   0.8  

(3.85)

Then, the bed friction factor fb is obtained using Equation 3.84 from the total friction factor f, which is calculated with the first part of Equation 3.82. The calculation procedure is given as follows: (1) Calculate R using Equation 3.80 and f using the first part of Equation 3.82 with the experimental data (i.e., U, h, S). (2) Assume a trial value for Rw and calculate fw using Equation 3.85.

78  Open Channel Flows

(3) Compute Rw using Equation 3.83 and compare it with the assumed value. Return to Step 2 if the new Rw value is different from the assumed value beyond a certain tolerance. (4) Obtain fb from Equation 3.84, and Rb from Equation 3.83. Then the bed shear stress is calculated with τb = γRbS. The aforementioned method has been widely used in laboratory experiments. However, it adopts the equal velocity assumption, which may not be strictly valid. It is inconvenient because of the iterations involved. Cheng (2011) proposed a direct approximate solution method, and Guo (2017) proposed an explicit and exact solution method. Moreover, Equation 3.85 is limited to smooth sidewalls. For more general applications, Equation 3.85 can be replaced with Equation 3.64, the Colebrook-White formula, or a similar formula considering both smooth and rough sidewalls. More discussion can be found in Guo (2017). Williams (1970) proposed the following factor to adjust the slope, shear stress, and unit stream power: Y

1 1  0.055 h B 2



(3.86)

where B is the channel width. The units of h and B are meters. The experimental data used to develop Equation 3.86 were measured in rectangular flumes with flow widths of 0.076–0.6 m, flow depths of 0.03–0.21 m, and a sediment size of 1.35 mm. Multiplying the laboratory values of slope, unit stream power, and shear stress by the correction factor Y gives the corresponding values in a wide channel for the same water depth and unit sediment transport rate (Williams, 1970). For example, applying the correction factor to the bed shear stress yields τb = γhSY. Knight et al. (1994) derived the following relationship for the integrated sidewall shear force versus the wetted perimeter ratio Pb/Pw using data obtained from smooth and uniformly roughened rectangular and trapezoidal channels, as well as ducts:   P  % SFw  Cef exp  4.605  3.23 log  b  1     C2 Pw  

(3.87)

where %SFw = 100τwPw/(τ0P) is the percentage of the sidewall shear force over the total shear force carried by the channel, and empirical coefficients Cef = 1.0 and C2 = 1.5. Knight and Sterling (2000) examined Equation 3.87 in smooth circular conduits and found that Cef and C2 slightly varied with the Froude number and the wetted perimeter ratio. Equation 3.87 can be used with Equation 3.81 to calculate the mean bed and sidewall shear stresses. Using the bisector as the division between sidewall and bed zones and the equal velocity assumption, Ramana Prasad and Russell Manson (2002) proposed the following formula for the %SFw in a prismatic rectangular channel with homogeneous roughness: 25  4  b0 h  b0 h  2 %SFw    b0 h  2  100 h b0

(3.88)

As described in Section 3.6.1, the distribution of the boundary shear stress is affected by the secondary flows that appear due to anisotropic features of turbulence, even in a straight prismatic

Open Channel Flows  79

channel. Guo and Julien (2005) determined the average bed and sidewall shear stresses in rectangular channels with smooth boundaries by approximately solving the continuity and momentum equations in the cross section and introducing correction factors to consider the effects of the secondary flows. The derived formula for the average bed shear stress is   h  h    h b 4 arctan exp      exp       hS   b0   b0   4 b0 

(3.89)

and the average sidewall shear stress is determined by substituting the calculated τb into Equation 3.81. Javid and Mohammadi (2012) applied the approach of Guo and Julien (2005) to trapezoidal channels. Guo (2015) developed an alternative method to determine the average bed and sidewall shear stresses on a smooth, rectangular channel based on a cross-sectional distribution of the streamwise velocity that takes into account the effects of secondary flows. The details can be found in the cited articles. Khodashenas et al. (2008) and Guo (2015) compared some of the aforementioned formulas against measurement data. Khodashenas et al. (2008) found that the method of Guo and Julien (2005) works well for smooth rectangular channels, the method of Ramana Prasad and Russell Manson (2002) works well for rough rectangular channels, and the method of Knight et al. (1994) modified by Knight and Sterling (2000) performs well for circular channels with flat beds. Guo (2015) found that the Einstein method agrees with the measured data for rectangular channels with rough beds. 3.5.4  Measurements of Boundary Shear Stress For a smooth bed, Preston (1954) proposed a method to estimate the skin friction using the dynamic and static pressures measured with a Pitot-static tube. However, when applied to a rough bed, the Preston method encounters challenges due to the involvement of roughness height and zero-plane displacement. Therefore, shear stress on sediment beds is usually determined from other directly measurable flow parameters, such as water surface slope, velocity, Reynolds stress, and TKE. For a steady, uniform flow, the reach-averaged boundary shear stress is related to the hydraulic radius R and the friction slope S by Equation 3.61. The hydraulic radius can be readily calculated using the channel cross-section geometry for a given water level. The friction slope is approximately equal to the water surface slope, which can be calculated using the water level readings at two stations along the channel reach of study. Since the water surface slope is usually a small value, the distance between the two stations should be long enough to have a meaningful reading of the elevation difference. The estimated average shear stress includes the components on the bed and banks, as well as form drag and skin friction. If the bed roughness height is known, the bed shear stress can be estimated using Equation 3.64 with the measured average velocity or using Equation 3.37 with a single-point streamwise velocity measured in the log-law layer. The estimate obtained using the depth-averaged velocity usually has much less variability than using a single-point velocity (Wilcock, 1996). If the bed roughness height is not known, one can use the pair values of u and log(y) in the log-law layer to derive the regression of u on log(y) as shown in Equation 3.44. Then, the regression slope au is used to calculate the shear velocity u* =   κau/2.303 and the bed shear stress  b   u*2. This method has been widely used (e.g.,

80  Open Channel Flows

Bergeron and Abrahams, 1992; Kim et al., 2000), but it has several limitations. Because κ varies with sediment concentration, this method is used only for clear water flows, in which κ is about 0.4. In addition, the y coordinate origin needs to be estimated by trial and error since the bed roughness height is not known. Moreover, the log law is inadequate when the flow is complex. The Reynolds shear stress is approximately equal to the total shear stress in a large portion of the flow depth away from the bed, i.e., 

 u v y  1  b h

(3.90)

The Reynolds shear stress can be determined by measuring the streamwise and vertical velocity fluctuations. Then, the bed shear stress can be estimated using Equation 3.90 with a single-point value of the Reynolds shear stress or by extrapolating the shear stress profile to the bed surface. The 2-D turbulence fluctuations can be measured using laser Doppler velocimetry (LDV), acoustic Doppler velocimetry (ADV), particle image velocimetry (PIV), etc. This method encounters errors when the sensor is misaligned with local streamlines and the shear stress profile is affected by secondary flows (Stapleton and Huntley, 1995; Kim et al., 2000; Biron et al., 2004; Pu, 2021). Equation 3.20 shows that the TKE k is related to shear velocity. Thus, bed shear stress can be determined using the following relationship:

 b =cc12ρk  v 2

(3.91)

where c1 is a calibration coefficient. The values of c1 have been determined as 0.19 by Stapleton and Huntley (1995), 0.2 by Soulsby and Dyer (1981), and 0.21 by Kim et al. (2000), primarily for tidal environments. The TKE method requires 3-D measurements of velocity fluctuations in a single near-bed point. Equation 3.19 shows that the Reynolds normal stresses, i.e., individual components of TKE, can also be used to determine the bed shear stress. The vertical normal stress is usually less affected by waves and other factors, and thus used as

 b  c2  v2 

(3.92)

where c2 is a calibration coefficient given as 0.9 by Kim et al. (2000) in an estuarine environment. This method may have errors when the sensor is misaligned with local streamlines (Pu, 2021). Biron et al. (2004) compared the bed shear stresses estimated from the log law, Reynolds shear stress, and TKE approaches. The results show that in a simple boundary layer in a laboratory flume with Plexiglas and sand beds, the log law and Reynolds shear stress methods are more appropriate. In a complex flow field around deflectors, the TKE method works best as it is not affected by local streamline variations or increased turbulent fluctuations. It is suggested that when single-point measurements are used, the instrument should be positioned at ~0.1h, where the peak Reynolds shear stress and TKE occur (see Figure 3.6). Though the TKE method works well in the studies conducted by Kim et al. (2000) and Biron et al. (2004), the coefficient c1 depends on the location of the measurement point above the bed and may vary in different environments. According to Wren et al. (2017), c1 is affected by bed

Open Channel Flows  81

roughness, with a value of about 0.24 for a gravel bed and 0.48 for a cobble bed. It is reduced when sand fills the voids among the gravel and cobble particles. For shear stress on stream banks with and without vegetation, c1 ranges from 0.11 to 0.53 due to the effects of vegetation, secondary flows, and instrumentation (Hopkinson and Wynn-Thompson, 2012). Thus, the TKE method requires more tests in stream environments. 3.6  Three-Dimensional Flow Features Flows in natural rivers are truly 3-D and very complex. Secondary flows are generated due to anisotropic turbulence structures, nonuniform boundary roughness, varying channel geometries (curvature, expansion, contraction, branch, confluence), etc. Secondary flows and their effects on the primary flows in straight and curved channels are briefly introduced in this section. 3.6.1  Flow Features in Straight Channels Generation of Secondary Flows As shown in Figure 3.17a, a pair of secondary flow cells are generated due to anisotropic turbulence near a sidewall in a prismatic rectangular channel. The sidewall affects a region of about 2.5h in the lateral distance (Xie, 1981; Nezu and Rodi, 1985; Kironoto and Graf, 1994). Thus, when the aspect ratio B/h < 5, the flow in the entire cross-section is 3-D. When B/h > 5, the flow is 2-D in the central region about 2.5h away from each sidewall. Secondary flows are also generated due to anisotropic turbulence in a prismatic trapezoidal channel, as shown in Figure 3.18 (Knight et al., 2010). Likewise, the aspect ratio plays an important role in the flow structure. Additional complexity arises from the side slope angle. If the side slope is steep, a pair of secondary flow cells exist at each side of the channel, with one over the bed and the other one along the side slope. If the side slope is gentle, multiple cells may appear along the side slope (Kamran, 2011). Similar secondary flows also appear in natural rivers (Nezu, 2005). Figure 3.19a illustrates the multicellular secondary flows created by alternate smooth and rough bed strips in a straight rectangular channel, as observed by McLean (1981), Nakagawa et al. (1981), Wang and Cheng (2005, 2006), and others. Near the sidewall, two triangular secondary flow cells are separated roughly by the corner bisector. The central region is characterized by pairs of counter-rotating

Figure 3.17 Flows in a straight rectangular channel: (a) secondary flows near a sidewall and (b) streamwise velocity profile at the channel centerline (after Nezu and Rodi, 1985)

82  Open Channel Flows

Figure 3.18 Secondary flow structures, primary flow velocity, and boundary shear stress measured in a prismatic trapezoidal channel (from Knight et al., 2010)

secondary flow cells approximately on the scale of h, with each cell approximately aligning with the edge of the smooth and rough strips. The secondary flows point from a rough strip to the neighboring smooth strip near the bed and oppositely near the water surface (Figure 3.19b). Similar secondary flow patterns were observed in straight open channels with static gravel beds (Albayrak and Lemmin, 2011) and irregular beds with ridges and troughs (Vachtman and Laronne, 2011). The generation of the aforementioned secondary flows can be explained by using the following vorticity equation of steady, uniform flow along the cross-section of a straight open channel (Einstein and Li, 1958; Culbertson, 1967): v





 2  2  2 v2  w2   2  2 w   y y z yz z 

 2  vw      

(3.93)

where v and w are velocities in the vertical (y) and lateral (z) directions, respectively; and Ω = ∂w/∂y − ∂v/∂z is vorticity. Equation 3.19 shows that v2 and w2 are anisotropic, even for a uniform turbulent flow. Figure 3.19c presents the distribution of normalized turbulence anisotropy on the cross-section of the channel simulated by Stoesser et al. (2015) using a large eddy simulation (LES) model. The results support the origin of the secondary currents at the channel corners, where w2  v2 is of the opposite sign. Substantial gradients of w2  v2 are also found over the rough strips and near the bed between the rough and smooth strips, where secondary flows are observed. Variation of Streamwise Velocity Secondary flows induce additional momentum transfer and thus modify the primary flow velocity structures. As illustrated in Figure 3.17, an upper secondary flow near the water surface transfers the lower momentum from the sidewall to the centerline, resulting in a shift of the maximum streamwise velocity from the water surface to a lower location. This phenomenon is called a velocity dip. A detailed explanation is given herein. The streamwise momentum equation on the cross-section of a straight channel is written as   uv  y



  uw  z



   u   u   u w   g siin    u v     z z   y  y   

(3.94)

Open Channel Flows  83

Figure 3.19 Secondary flows over alternate smooth and rough bed strips: (a) schematic pattern in half of the channel cross-section, (b) measured secondary flow vectors between two rough strips (from Wang and Cheng, 2005, 2006), and (c) contours of the normalized normal stress anisotropy simulated using an LES model (from Stoesser et al., 2015)

where β is the channel bed slope angle. In the centerline of the channel, the spanwise velocity w and gradient ∂u/∂z are set to zero, so Equation 3.94 is reduced to  uv   u    u v   g sin   y y  y 

(3.95)

Integrating Equation 3.95, applying a shear stress of zero at the free surface, and treating the Reynolds shear stress with the Boussinesq eddy viscosity concept lead to uv 



 t 

u  g  h  y  sin   y

(3.96)

84  Open Channel Flows

In the central region of a wide channel (B/h > 5), the vertical velocity v is zero, and Equation 3.96 leads to the log law, as explained in Section 3.4.2. In a narrow channel, v is not zero due to the secondary flows, as shown in Figure 3.17a. Tominaga et al. (1989) experimentally demonstrated that a negative v appears near the water surface in the channel centerline, which leads to a negative uv. If the magnitude of uv is larger than the last term of Equation 3.96, a negative gradient of u appears near the water surface. Thus, the maximum streamwise velocity shifts down, yielding the velocity dip as shown in Figure 3.17b. The location of δm depends on the aspect ratio of the channel (Wang et al., 2001). If the profile of vertical velocity v is known, the vertical profile of u can be derived by using Equation 3.96. Such profiles have been proposed by Guo and Julien (2001), Absi (2011), and Pal and Ghoshal (2016), among others, based on a variety of approximations. Distribution of Boundary Shear Stress Secondary flows significantly affect the distribution of the boundary shear stress in a prismatic channel. This has been intensively investigated by Olsen and Florey (1952), Khodashenas and Paquier (1999), Yang and Lim (2005), and Knight et al. (2010), among many others. When the aspect ratio b0/h is large, the secondary flows do not significantly affect the central region of the channel, so that the bed shear stress in the central region is uniform and the maximum bed shear stress occurs at the center of the channel. When the aspect ratio is small, the effects of the secondary flows become important, leading to multiple maxima of the bed and sidewall shear stresses and the shift of the maximum bed shear stress toward the sides (Figure 3.18). Figure 3.20 shows the maximum bed and sidewall shear stresses as functions of the side slope ms (horizontal/vertical) and the aspect ratio b0/h in straight trapezoidal channels. These curves were based on the work of Olsen and Florey (1952) and expanded by the U.S. Federal Highway Research Board (1970). They are often used for stable channel design in practice. In more recent studies, Khodashenas and Paquier (1999) and Yang and Lim (2005) developed empirical formulas for the local boundary shear stress in prismatic channels, without considering the effects of secondary flows. To accurately calculate secondary flows and in turn boundary shear stress, high-order turbulence models, such as nonlinear k-ε turbulence models, Reynolds stress models, and LESs, are required with the incorporation of advanced numerical algorithms (Rodi, 1993; Stoesser et al., 2015).

Figure 3.20 Maximum boundary shear stress in a straight trapezoidal channel: (a) on the bed and (b) on side slopes (after Olsen and Florey, 1952; U.S. Federal Highway Research Board, 1970)

Open Channel Flows  85

3.6.2  Flow Features in Curved Channels Generation of the Helical Flow Consider the flow in a curved, open channel, as shown in Figure 3.21. The curved primary flow experiences a centrifugal force, which points toward the outer bank and raises the water surface elevation there. Figure 3.22 illustrates the force diagram for a vertical water column that extends from the bed to the water surface and has a base with side widths of ds and dn in the streamwise (s) and lateral (n) directions, respectively. The water column experiences a net lateral pressure force, which is depth-invariant and points to the inner bank because of the tilting water surface (Figure 3.22a). The centrifugal force is proportional to the streamwise velocity squared and thus increases from the bottom to the top layer (Figure 3.22b). The depth-varying centrifugal force acts against the depthinvariant net pressure force, resulting in a secondary flow called helical flow. The helical flow points to the outer bank in the top layer and to the inner bank in the bottom layer (Figures 3.21b and 3.22c). For an infinitesimal element of height dy in the water column, the force equilibrium in the lateral direction for the fully developed helical flow is expressed as      u2 p    0 dn   dsdy   n   n  n dy   dsdn   s dsdydn  p  p  y r n        

(3.97)

where τn is the shear stress due to the helical flow, us is the streamwise velocity, and r is the radius of curvature. By assuming the hydrostatic pressure p = γ(ys − y), one obtains y p  s J  n n

(3.98)

where ys is the water surface elevation and J is the water surface slope along the lateral direction. Inserting Equation 3.98 into Equation 3.97 leads to  n u2  J  s  y r

(3.99)

Integrating Equation 3.99 over the flow depth and assuming τn at the bed and water surfaces to be negligible, one obtains

 Jh 

 r



h

0

us2 dy 

Figure 3.21 Flow in a curved channel: (a) horizontal plane and (b) cross-section

(3.100)

86  Open Channel Flows

Figure 3.22 Force diagram and helical flow velocity for a water column in the curved channel: (a) pressure force, (b) centrifugal force, and (c) helical flow velocity

Substituting Equation 3.100 into Equation 3.99 and treating the shear stress τn with the Boussinesq eddy viscosity concept lead to the following equation for the helical flow:   un  1   t  y  y  rh



h

0

us2 dy 

us2  r

(3.101)

where vt is the eddy viscosity and un is the lateral velocity of the helical flow. Rozovskii (1957) derived the following distribution of helical flow velocity by using Equation 3.101 with the parabolic profile of eddy viscosity and the log law Equation 3.43 of the streamwise velocity:  un

hU s    2   2 r  





0

g  ln  d  1  1   Ch 

  





0

 ln 2  d  2    1  

(3.102)

where Us is the depth-averaged velocity of the primary flow and η = y/h. Equation 3.102 is complex and inconvenient to use. If the Chezy coefficient Ch is larger than 50, the Rozovskii distribution can be simplified to a linear distribution (Odgaard, 1986):  un bs

hU s r

 y   2 h  1   

(3.103)

where bs is a coefficient with a value of about 6.0. The following analytical solution of Equation 3.101 is derived by the author using the power distribution Equation 3.53 for the streamwise flow velocity and the depth-invariant eddy viscosity vt = αu*h: un 

2 2 2/ m  hU s (m  1) 2  y  m  y 5m  2        3(m  1)(3m  2)  f 2 r m(m  2)  h  (m  1)  h 

1



(3.104)

where f is the Darcy-Weisbach friction coefficient; m is the reciprocal of the exponent in Equation 3.53, set here as 6; and α = κ/6 = 0.0667. In the derivation, ∂un/∂y = 0 is set at the free h surface, and  un dy  0 is applied for mass conservation. 0

Open Channel Flows  87

Figure 3.23 Comparison of helical flow velocities calculated with Equation 3.104 and measured by de Vriend and Koch (1977) in a 90° bend

Equation 3.104 was tested using the helical flow velocity measured by de Vriend and Koch (1977) in a 90° bend with a rectangular cross-section. The bend had a radius of curvature of 50 m and a channel width of 6 m. The flow had a discharge of 0.305 m3/s and a depth of 0.25 m. The friction coefficient f was given as 0.021 for the concrete bed. The measured helical flow velocity profile was adjusted slightly by subtracting the depth-averaged transverse velocity for mass conservation. The analytical model agrees well with the measured data, as shown in Figure 3.23. It needs to be tested in different conditions, such as uneven and movable beds in natural rivers. Note that Equations 3.102–3.104 are valid only for the developed helical flow in the downstream portion of the bend without sidewall effects. The evolution of the helical flow from the crossover to the bend apex is complex and requires a 3-D or special depth-averaged 2-D model, as described by Wu (2007). Distributions of Primary Flow Velocity and Boundary Shear Stress The helical flow transfers the momentum from the inner bank toward the outer bank and sediment from the outer bank toward the inner bank, thus creating a shift in the primary flow toward the outer bank and forming a deep channel on the outer side and a point bar on the inner side. This eventually leads to channel meandering in alluvial plains. A comprehensive description of the flows, sediment transport, and morphology in meandering channels with various curvatures can be found in da Silva and Yalin (2017). The primary flow shift leads to greater shear stress on the outer bank. Figure 3.24 shows the distribution of boundary shear stress in a 60° trapezoidal channel bend with rough boundaries measured by Ippen and Drinker (1962). The maximum shear stress appears on the outer bank near the bend exit. The ratio of the maximum boundary shear stress to the average shear stress of the approach channel, Kb, is a function of the relative curvature of the channel (Ippen and Drinker, 1962; U.S. Army Corps of Engineers, 1994; Federal Highway Administration, 2005).

88  Open Channel Flows

Figure 3.24 Contour lines of  0  0,in in a 60° curved trapezoidal channel (τ 0 is the boundary shear stress and  0,in is the average boundary shear stress in the approach channel) (after Ippen and Drinker, 1962)

The following regression formula was proposed by Sin et al. (2012) to determine Kb for a meandering channel: Kb = 2.5(r/B) −0.32

(3.105)

where B is the top width of the channel. An upper envelope formula was also proposed for design purposes. Sin et al. (2012) measured the boundary shear stress using the Preston method and the Reynolds shear stress extrapolation. The measurement data varied significantly with measurement methods and had large uncertainties. Thus, Equation 3.105 needs to be further tested using more consistent measurement data. 3.7  Coherent Structures in Turbulent Shear Flows Turbulence is traditionally treated as a random phenomenon of eddies with various scales and characterized by statistical properties. In recent decades, researchers have revealed the existence of coherent structures in turbulent shear flows, which are irregular but repetitive eddy structures in time and space, with distinctive shapes and histories of formation, evolution, and dissipation. The coherent structures form a part of large-scale eddies in the turbulence cascade process described in Figure 3.2 and play important roles in sediment transport and morphology evolution. 3.7.1  Inner-Scale Coherent Structures Sequential burst-sweep events occur in a turbulent shear flow over a bed (or solid wall in general), due to the inherent instability of the flow at certain high Reynolds numbers. The key feature in each burst-sweep cycle is the so-called hairpin or horseshoe vortex shown in Figure 3.25 (Hinze, 1975; Smith, 1996; Southard, 2006; Adrian and Marusic, 2012). Such a vortex originates from the near-bed sublayers. It is accelerated and then ejected outward as a violent burst into a region of flow farther from the bed. The vortex collapses into eddies of various sizes and generates

Open Channel Flows  89

Figure 3.25 Evolution of a hairpin vortex, showing its role in the burst-sweep cycle (after Hinze, 1975; Southard, 2006)

pressure waves. The high-speed inrush fluid sweeps back at a shallow angle toward the bed, interacts with the low-speed fluid near the bed, and creates acceleration and small-scale turbulence. The period TB and longitudinal spacing λL of bursts over smooth beds depend on the viscous length scale ν/u*. The reported dimensionless longitudinal burst spacing u*λL/ν is about 500 (Raudkivi, 1998). Values of the nondimensional burst period TB  TB u*2  have been reported as about 100 (Willmarth and Sharma, 1984), 480 (Smith and Metzler, 1983), and 500 (Niño and Garcia, 1996). In some cases, TB can be as large as 2,500 (Smith and Metzler, 1983; Nezu and Nakagawa, 1993). The bursts near a rough bed are related to the roughness height ks because the instability originates from the deflected and separation flows and uneven pressure fields around the protruding roughness elements (Figure 3.8). As u*ks/ν increases, the ejection intensity increases and sweeping decreases; multiple ejections can occur at the same time and impede the sweeping jets. When the bed roughness is large enough, the multi-ejection phenomenon becomes dominant (Mao, 2003). On the other hand, moving coarse particles (i.e., bed load) tend to break eddy coherence, thus reducing the frequency of large events and increasing the frequency of small events. Ejection events suffer particularly great reduction (Santos et al., 2014). Other effects of moving sediment on turbulence are described in Sections 3.3.1, 3.4.3, and 11.1. Burst-sweep events are associated with the existence of streamwise-oriented, low-speed streaks immediately above the bed (Kline et al., 1967; Smith, 1996; Nezu, 2005). The streaks are low-speed zones that lie between intervening high-speed zones, as portrayed in Figure 3.26. In the low-speed zones, slow-moving fluid has an upwelling component of motion as a result of downwelling in the high-speed zones. The streaks greatly extend along the flow direction, waver and shift irregularly from side to side, and appear intermittently. The lateral spacing of the streaks, λ, weakly depends on the mean-flow Reynolds number, with u*λ/ν = about 100, for a hydraulically smooth flow; and λ is 3 to 4 times the size of the close-packed roughness elements for a fully rough flow. The situation is less straightforward for a transitionally rough flow (Southard, 2006).

90  Open Channel Flows

Figure 3.26 Low-speed wall streaks and high-speed zones near a bed: (a) top view, revealed by the cross-stream lines of hydrogen bubbles generated just above the bed; (b) upstream view of coherent structures on the cross-section (from Smith, 1996)

Burst-sweep cycles and the associated streak structures are features of the near-boundary inner layer, where the majority of turbulence energy is produced and dissipated. In open channel flows, the near-wall region in which ejections and sweeps occur violently is approximately within y+( = u*y/ν) < 100 (Nezu, 2005). 3.7.2  Outer-Scale Coherent Structures In the outer region away from the bed, fewer coherent structures occur, but their scales are much larger than those of the inner-region structures. The outer-region coherent structures consist of sweeps/bursts and low-/high-speed zones in a width scale of about 1h (or 1δm) in the cross section (Kinoshita, 1967; Nezu, 2005; Zhong et al., 2016). The value of the dimensionless burst period TB  TBU 0 h is about 5 (Raudkivi, 1998). Here, U0 is the free stream velocity. In the streamwise vertical plane, the coherent structures have a length scale of about (2–10)h (Adrian and Marusic, 2012 for a summary). Balakumar and Adrian (2007) divided these structures into large-scale (< 3h) motions (LSM) and very large scale (> 3h) motions (VLSM) that exhibit different characteristics. They showed that LSM wavelengths persist out to about 0.5h, whereas VLSM wavelengths do not extend beyond the inner region (< 0.2h). In the horizontal plane, long coherent structures composed of meandering, elongated low- and high-speed regions were observed by Hutchins and Marusic (2007) and Dennis and Nickels (2011) (Figure 3.27). The streamwise lengths are well over 10h. Some coherent structures are several times the channel width in length and likely associated with sidewall boundaries (Yalin, 1992). Inner- and outer-region coherent structures are interconnected. The large coherent structures in the outer region may agglomerate and modulate the small ones in the inner region (Nezu, 2005; Adrian and Marusic, 2012). Tomkins and Adrian (2003) observed the growth of a spanwise eddy scale with the distance above the bed by merging small coherent structures into large ones in the inner region (< 0.2h). Moreover, Wang et al. (2017a) observed that the scales of coherent structures are continuous and grow linearly from the inner to the outer region (< 0.6h) in an open channel flow. Similar observations were made by Bagherimiyab and Lemmin (2018). This can be explained using the attached-eddy hypothesis, which addresses the scale evolution of the dominant eddies along the direction normal to the bed (Townsend, 1961; Woodcock and Marusic, 2015). In the upper part of the outer region (> 0.6h), the scales of the coherent structures decrease with increasing distance above the bed due to the effect of the free surface as a moving boundary (Wang et al., 2017a).

Open Channel Flows  91

Figure 3.27 Large coherent structures in a turbulent boundary layer at R δ = 4,700 observed using high-frame-rate stereo-PIV measurements. The black isocontours show swirl strength, indicating the corresponding location of vortical structures with lowspeed (blue) and high-speed (red) zones (from Dennis and Nickels, 2011)

Laboratory and field measurements have shown that large-scale coherent structures over rough beds are similar to those observed in laboratory experiments over smooth beds (Roy et al., 2004; Detert et al., 2010a; Cameron et al., 2017; Bagherimiyab and Lemmin, 2018). The largescale structures grow upward from the bed and reach the surface (Roy et al., 2004; Hurther et al., 2007; Nikora et al., 2007; Bagherimiyab and Lemmin, 2018). Cameron et al. (2017) reported that the length of the LSMs over a rough bed is scaled well with depth, whereas that of the VLSMs is rather related to the channel aspect ratio (B/h) or relative submergence. The transverse scale of the LSMs grows with elevation, whereas the VLSMs are fixed with a width of about 2h. Roy et al. (2004) observed that the large coherent structures in a gravel-bed river have a length scale of (3–5)h and a width scale of (0.5–1)h, which are smaller than those over smooth beds. This is due to the deflection by bed roughness elements, which generates structures that grow more rapidly (with a larger ejection angle, such as 45°, compared to the ~15° angle of the hairpin vortex). Bed roughness reduces the streamwise correlation length (Adrian and Marusic, 2012). The similarity of large-scale coherent structures and the validity of the log law in flows over smooth and rough beds indicate that the main flow in the upper layers tends to be self-organized rather than dictated by the bed surface conditions. 3.7.3  Effects of Coherent Structures on Sediment Transport Coherent structures significantly affect sediment transport and morphology evolution in rivers. Sweeps increase the bottom pressure and skin friction and create a Bernoulli-type vertical lift force. Sweep events play a predominant role in dislodging bed particles (Drake et al., 1988; Dey et al., 2011; Bagherimiyab and Lemmin, 2018; Schobesberger et al., 2020). On the other hand, upward ejection events generate forces by vertical and horizontal accelerations and may entrain the less stable particles from the bed (Cecchetto et al., 2017). Ejections are the prevailing mechanism for sediment suspension (Sumer and Deigaard, 1981; Mao, 2003; Salim et al., 2017). Vertical

92  Open Channel Flows

turbulent momentum flux is dominated by ejections and sweeps over most of the water column, with sweeps contributing more near the bed and ejections contributing more at the mid-depth (Bagherimiyab and Lemmin, 2018). In addition, particle-particle interactions, such as direct impact and indirect interference through induced fluid motion, can also trigger sediment entrainment and affect sediment movement (Cecchetto et al., 2017). Figure 3.28 shows the multicellular flow structures and the associated streamwise sand ridges on a movable bed in a rectangular channel (Culbertson, 1967; Karcz, 1973; Nezu, 2005). Elongated low-/high-speed zones of about 1h in width appear periodically in the cross section. Long streets of boils (bursts) appear in the low-speed zones due to fluid upwelling, while fluid downwelling occurs in the high-speed zones (Kinoshita, 1967; Nezu, 2005; Zhong et al., 2016). Because the fluid in the high-speed zones moves slightly obliquely toward the low-speed zones, sediment particles tend to be swept into the low-speed zones, where some particles move with the bursts and the others form sand ridges on the bed. In the case of nonuniform sediment, fine particles appear at the ridges and coarse particles stay at the troughs. With these uneven bed features, the coherent structures evolve to the multicellular secondary mean flows described previously in Section 3.6.1, which are more persistent and further stabilize the sand ridges. Note that secondary flows are mean motions, whereas coherent structures are turbulent motions. They are interconnected but exhibit different behaviors. Likewise, the generation of sand ripples in alluvial rivers is highly related to inner-region coherent structures. Sand dunes in alluvial rivers are likely caused by large-scale vertical coherent structures, and thus exhibit a length of several times the flow depth, e.g., (5–10)h. Alternate bars in relatively narrow straight channels are associated with horizontal coherent structures, having a length scale of several times the channel width B, e.g., about 6B (Yalin, 1992; da Silva and Yalin, 2017). These bed forms are described in detail in Chapter 6 of this book. Meanwhile, bed forms create complex coherent structures in the flow (Mao, 2003; Best, 2005; Nezu, 2005; Kwoll et al., 2017). Figure 3.29 sketches the flow structures developed over a bed covered with a series of asymmetric sand dunes (Best, 2005). As the flow passes over the crest of each dune, a separation flow zone is formed on the lee side, and a shear layer is created outside the separation zone. Large-scale turbulence is generated due to the Kelvin-Helmholtz instabilities along this shear layer. A wake zone is created by the flow expanding from the dune crest, and a shear layer is formed outside the wake zone. Downstream of the reattachment point, an internal boundary layer grows as the flow reestablishes itself over the stoss slope of the next dune. The flow reaches its maximum horizontal velocity over the dune crest and then expands to the lee side

Figure 3.28 Boil streets, multicellular secondary flows, and longitudinal sand ridges along the cross section of a rectangular channel (after Nezu, 2005)

Open Channel Flows  93

Figure 3.29 Schematic diagram of the principal regions of flow over sand dunes (after Best, 2005)

and repeats the aforementioned features (Best, 2005). Sediment is picked up by strong turbulence eddies, such as the Kolk vortices and boils reported by Nezu and Nakagawa (1993) and Mao (2003), associated with the shear layer outside the separation flow zone, as well as the internal boundary layer flow on the stoss slope of the dune. A portion of the sediment is suspended into the expanding flow, and another portion is dropped into the separation zone on the lee side. Turbulent flows, sediment transport, and bed forms constitute a complex, interactive system. Understanding of coherent structures and their effects on sediment transport and bed forms in open channel flows has been significantly improved over the last decades through advanced measurement techniques. However, the knowledge of coherent structures is far from complete, and much needs to be done to incorporate the knowledge into sediment transport theory and practice. It is quite a challenging task in this field. Homework Problems 3.1 How do you classify laminar and turbulent flows in open channels? How do you classify subcritical and supercritical flows in open channels? 3.2 A rectangular laboratory channel has a width of B = 4 m. The Manning coefficient n is 0.025 m-1/3/s. Determine the bed slope to achieve a critical uniform flow for a discharge of Q = 10 m3/s. 3.3 Describe the cascade process of turbulence energy. What are the integral, Kolmogorov, and Taylor scales of turbulence? 3.4 How do sediments affect the intensity of turbulence? 3.5 Consider a uniform turbulent flow in a wide, straight channel with a bed slope of 0.0005 and a flow depth of 1.4 m. Draw the vertical distributions of turbulence intensity, TKE, and TKE dissipation rate. 3.6 Consider a uniform turbulent flow in a wide, straight channel with a bed slope of 0.0005 and a flow depth of 1.4 m. Compare the production and dissipation rates of TKE and analyze the kinetic energy budget. (Hint: use Equations 3.10, 3.22, and 3.34) 3.7 Can you derive the parabolic eddy viscosity profile from the mixing length theory? 3.8 Describe the vertical distribution of streamwise velocity for a steady, uniform flow in the central region of a wide, straight channel with a smooth bed. 3.9 Consider a steady uniform flow in a wide, straight channel with a constant bed slope. Derive the vertical distribution of streamwise velocity in cases of laminar and turbulent flows.

94  Open Channel Flows

Figure 3.30 Sketch of a compound channel

3.10 Consider a uniform turbulent flow in a wide, straight open channel. If the velocities at 0.2h and 0.8h are measured, can you estimate the unit discharge? h is the flow depth. (Hint: use the log law or power law) 3.11 Describe how the von Karman coefficient varies with sediment concentration. 3.12 Consider a steady, uniform flow in a straight prismatic channel with a compound crosssection, as shown in Figure 3.30. The sidewalls in the main channel and floodplains are vertical. The Manning n values in the left floodplain, main channel, and right floodplain are 0.035, 0.025, and 0.045, respectively. The bed slope is 0.001. (a) Determine the conveyance of the channel as flow depth varies from 1 to 10 m with an interval of 1 m. (b) If the water depth in the main channel is 2.5 m, determine the flow discharge. (c) If the flow discharge is 400 m3/s, determine the water depth. 3.13 Explain the hydraulic radius and energy slope division methods for the composite roughness in a trapezoidal channel. 3.14 Consider a straight rectangular channel. The bed slope is 0.001, the average flow velocity is 1 m/s, and the flow depth is 0.5 m. Determine the bed shear stress when the channel width is 0.5 and 5 m using the following methods: (a) Vanoni and Brooks (1957) (b) Williams (1970) (c) Knight et al. (1994) (d) Guo and Julien (2005) 3.15 How do you measure or estimate bed shear stress? Outline the strength and weakness of each method. 3.16 Describe the velocity dip phenomenon in a narrow, straight channel. 3.17 Describe the helical flow mechanism and velocity profile in a curved channel. How does the helical flow affect the primary flow? 3.18 Derive the vertical distribution of the helical flow velocity from Equation 3.101 by assuming a constant viscosity νn and using the power law for streamwise velocity. 3.19 What are coherent structures in a turbulent shear flow? Describe the differences between coherent structures in the inner and outer regions.

Chapter 4

Settling of Sediment Particles

This chapter introduces the settling process of sediment particles in water; the formulas of settling velocity; the effects of particle shape, sediment concentration, and turbulence on settling; the measurement of settling velocity; the time and distance to reach terminal settling; and the relationship between fall and nominal diameters. 4.1  Settling Process of Sediment Particles When a sediment particle is placed in quiescent, distilled water, it settles due to the downward action of the submerged weight, i.e., gravity minus buoyancy force. As the particle moves through the water, a drag force is generated due to surface friction and pressure difference on the particle, an inertia force (including the added mass force) is produced due to the acceleration of the particle relative to the fluid, and the Basset (1961) force is created by the unsteadiness of the boundary layer flow around the particle (see Section 7.2). The drag force increases as the particle velocity increases, whereas the inertia and Basset forces decrease as the particle acceleration decreases. Particle settling reaches a terminal stage when the inertia and Basset forces become negligible and the drag force is equal to the submerged weight. The acceleration period from rest to the terminal stage is very short; e.g., about 0.1 s for a particle of 1 mm in diameter. The average velocity of the particle in the terminal stage is called settling or fall velocity, denoted as ωs . Figure 4.1 illustrates the flow around a settling sphere. The flow may be laminar or turbulent, depending on the particle Reynolds number, defined as Red = ωs D/v, where D is the sediment diameter and ν is the kinematic viscosity of water. When Red is smaller than about 1.0 (i.e., D < about 0.1 mm), the flow is usually laminar and the particle settles vertically in a straight line (Figure 4.1a). This regime is called laminar or viscous settling. At an intermediate Red between about 1 and 1,000, the flow is transitional (Figure 4.1b and c). A stable and symmetric recirculation zone exists behind the sphere when Red is about 10–200, and hairpin-shaped vortex shedding occurs when Red is about 300–800 (Sakamoto and Haniu, 1990; Wu and Faeth, 1993). When Red is larger than about 1,000 (D > about 4 mm), the flow is turbulent, vortex shedding and turbulent wake eddies coexist (Figure 4.1d), and the particle may not settle in a straight line. This regime is called turbulent or inertial settling. The settling pattern of a non-spherical particle highly depends on the particle Reynolds number and asymmetric shape (McNown et al., 1951; Stringham et al., 1969; Chien and Wan, 1983, 1999). An elliptical particle usually settles stably along its initial orientation at low Red values (< 0.1). However, for a large range of Red (about 10–105), an elliptical particle adjusts its orientation and ultimately settles along the shortest axis c. The shortest axis allows the particle to have DOI: 10.1201/9781003343165-4

96  Settling of Sediment Particles

Figure 4.1  Flow around a settling sphere: (a) laminar, (b)-(c) transitional, and (d) turbulent

a large frontal area and, in turn, a high drag force. A disc-shaped particle stably settles along the c axis at Red < 100. As Red increases, the disc particle oscillates, then glides and tumbles (Stringham et al., 1969). The settling of a naturally-worn sediment particle is significantly affected by its sphericity, roundness, and surface roughness. A sediment particle with high sphericity and high roundness may settle like an ellipse, whereas a platy sediment particle may settle like a disc. When a mixture of sediment particles settles in a water column, each particle may be affected by the upward compensating flows generated by other settling particles, as well as by particleparticle collisions. Therefore, the settling velocity is influenced by sediment concentration and size gradation. Moreover, cohesive sediment particles may form flocs when they collide. The settling process of cohesive sediments is much more complex than that of noncohesive sediments, as described in Section 10.2. 4.2  General Formula of Particle Settling Velocity A sediment particle experiences mainly the submerged weight and drag force in the terminal settling stage, as described previously and shown in Figure 4.1a. The submerged weight, Ws, is expressed as W  s

 s    g

D 3 6



(4.1)

where D is the nominal particle diameter, ρs is the sediment density, and g is the gravitational acceleration. Note that ρ is given as the density of pure water because a single particle or a dilute case is considered here. The drag force, FD, has the following general form: FD  CD 

D 2 s2 4 2



(4.2)

Settling of Sediment Particles  97

where CD is the drag coefficient. Note that πD2/4 is the projection area of a sphere on the plane normal to the direction of settling. For a non-spherical particle, the effect of particle shape is considered in CD. In the terminal stage of settling, the drag force in Equation 4.2 is equal to the submerged weight in Equation 4.1, yielding 1/ 2

  4 s   s   gD   3CD  

(4.3)

Equation 4.3 is a general formula for particle settling velocity. However, it is not explicit because CD may be related to ωs. How to determine the drag coefficient is discussed in the next sections. 4.3  Settling Velocity of Spherical Particles Stokes (1851) derived the drag force induced by the flow around a sphere in the laminar settling regime (Red < 1) by solving the Navier-Stokes equations without inertia terms. The derived drag coefficient is CD = 24/Red(4.4) Inserting Equation 4.4 into Equation 4.3 leads to the Stokes law, which explicitly determines the settling velocity of a sphere in the laminar regime as:

s 

1 s   D 2 g 18  



(4.5)

Oseen (1927) solved the approximate Navier-Stokes equations including some inertia terms. Furthermore, Goldstein (1929) derived a relatively complete solution of Oseen’s approximation as follows: CD 

24 Red

3 19 2 71   3 1  16 Red  1280 Red  20480 Red  ...   



(4.6)

However, Equation 4.6 is valid only for Red values up to 2. Beyond this range, the drag coefficient has to be determined by experiments rather than theoretical solutions. Rouse (1938b) compiled the available experimental data and obtained the relationship between CD and Red as shown in Figure 4.2, which can be used to determine CD and, in turn, the settling velocity of spherical particles. Figure 4.2 shows that when Red > 1,000, i.e., in the turbulent settling regime, the drag coefficient is independent of Red and has a value of about 0.45. Substituting this CD value into Equation 4.3 yields 1/ 2

     s 1.72  s  1 gD     



(4.7)

98  Settling of Sediment Particles

Figure 4.2  Relationship between C D and R ed for spheres (after Rouse, 1938b)

The drag coefficient data in Figure 4.2 can be fitted to the following formula when Red < 800 (Schiller and Naumann, 1933; Rowe, 1961):  CD

24 0.687 1  0.15 Red Red







(4.8)

or more generally (White, 1991): C  D

24 6   0.4 Red 1  Red



(4.9)

The accuracy of Equation 4.9 is ±10% for Red values up to 2×105, above which the turbulent boundary layer markedly thins the wake and reduces the drag on the particle. The settling velocity can be calculated by inserting Equation 4.8 or 4.9 into Equation 4.3 and iteratively solving the derived equation. Brown and Lawler (2003) examined the existing data on sphere settling velocities and drag coefficients published throughout the twentieth century. They applied the correction method proposed by Fidleris and Whitmore (1961; see Section 4.7) to exclude the wall effect for 178 of the 480 collected data sets. Then, Brown and Lawler (2003) developed the following explicit formula for the settling velocity of spheres for Red up to 2×105:

s

1/ 3

 g   s     

 18 0.8981 0.936 D*  1 D*   0.317 0.449     2    D*    D*    

1.114



(4.10)

where D * = D[(ρ s/ρ‒1)g/v 2] 1/3, often called the nondimensional diameter. D * is used in many sediment formulas. Values of D * for common sediment sizes are given in Appendix 4.1.

Settling of Sediment Particles  99

Based on the data collected by Brown and Lawler (2003), Barati et al. (2014) developed a more accurate but more complicated formula for CD by applying multi-gene genetic programming. The developed formula, along with a dozen other existing formulas for the drag coefficient of spherical particles, can be found in Barati et al. (2014). 4.4  Settling Velocity of Sediment Particles Formulas for the settling velocity of sediment particles are grouped by their ability to consider the effects of particle size and shape. The first group takes into account only particle size and is applicable for naturally worn sediment particles with general sphericity and roundness. The second group considers particle size and sphericity, whereas the third group adds particle roundness. These three groups of formulas are introduced in the following sections. 4.4.1  Formulas Based on Particle Size Rubey’s (1933) Formula Rubey (1933) approximated the drag coefficient of sediment particles as C  D

24 2 Red



(4.11)

and substituted it into Equation 4.3, yielding 1/ 2

     s  R  s  1 gD     



(4.12)

where ξR = 0.79 for particles larger than 1 mm at water temperatures between 10 and 25°C. For smaller particles, ξR is determined with 1/ 2

2  36 2     R 3 3 gD (  1 )   s  

1/ 2

  36 2  3  gD (  1 )   s  



(4.13)

The Rubey formula is the first to cover all the laminar, transitional, and turbulent settling regimes. However, it was based on very limited data. It is reduced to the Stokes law Equation 4.5 of sphere settling, which overestimates the settling velocity of fine sediments if D is the nominal diameter (McNown et al., 1951). For naturally worn coarse sediment particles, the Rubey formula significantly underestimates the settling velocity because Equation 4.11 gives CD = 2, which is too large. Zhang’s (1961) Formula Zhang (1961; see Zhang and Xie, 1993) assumed the drag force in the transitional settling regime to be a linear combination of those in the laminar and turbulent regimes:  FD C1  Ds  C2  D 2s2



(4.14)

100  Settling of Sediment Particles

where C1 and C2 are coefficients. By equating Equations 4.1 and 4.14 and calibrating the coefficients C1 and C2 using measurement data, Zhang (1961) obtained the following formula for the settling velocity of naturally worn sediment particles: 2

      s  13.95   1.09  s  1 gD  13.95  D D    

(4.15)

The Zhang formula can be used for a wide range of sediment sizes. In the laminar settling regime, it is simplified to

ss 

11 ss  D D22 gg 25  25..66 



(4.16)

Equation 4.16 is similar to the Stokes law Equation 4.5, with the coefficient of 1/18 being changed to 1/25.6. Goncharov (1954; see Cheng, 1997a) and Sha (1956) also found that the laminar settling of natural sediment particles deviates from Stokes’ law, with the coefficient 1/18 in Equation 4.5 being changed to 1/24. In the turbulent regime, the Zhang formula is reduced to 1/ 2

     s 1.044  s  1 gD     



(4.17)

Equation 4.17 is similar to Equation 4.7 with a different coefficient. The coefficient value of 1.044 in Equation 4.17 is equivalent to a drag coefficient of 1.2, which is larger than the spherical drag coefficient of 0.45. Similar findings were reported by Goncharov (1954; see Cheng, 1997a) and Sha (1956), among others. Zanke’s (1977) and Soulsby’s (1997) Formulas Zanke (1977) developed the following formula for sediment settling velocity:  s  gD 3      1 2   1  0.01 D       

1/ 2

s  10

  1 



(4.18)

Soulsby (1997) replaced the coefficients 10 and 0.01 with 10.36 and 0.00977, respectively. Van Rijn (1984b) adopted the Stokes law Equation 4.5 to compute the settling velocity for sediment particles smaller than 0.1 mm, the Zanke formula for particles from 0.1 to 1 mm, and the following formula for particles larger than 1 mm: 1/ 2

     s 1.1  s  1 gD     



(4.19)

The formulas of Zanke (1977) and Soulsby (1997) have small errors, which the segmented method of van Rijn (1984b) somewhat improves, as explained in the following sections.

Settling of Sediment Particles  101

Cheng’s (1997a) Formula The formulas of Rubey (1933), Zhang (1961), Zanke (1977), and Soulsby (1997) described earlier have the same formulation with different coefficients. Similar formulas were also proposed by Goncharov (1954; see Cheng, 1997a), Sha (1956, 1965), Raudkivi (1998), and Julien (2010). In general, the drag coefficient can be approximated with (Cheng, 1997a; Wu and Wang, 2006; Camenen, 2007)  M 1/ ns  CD   s   N s1/ ns   Red  

ns



(4.20)

where Ms, Ns, and ns are coefficients. Equation 4.20 is an extension of Rubey’s Equation 4.11. It can be reduced to CD = Ms/Red whereby Ms replaces the coefficient value of 24 in Equation 4.4 for the laminar settling regime. Equation 4.20 is reduced to CD = Ns such that Ns represents CD in the turbulent regime. The exponent ns determines the variation trend of CD in the transitional regime. Using Ms = 32, Ns = 1, and ns = 1.5, Cheng (1997a) derived the following formula for the settling velocity of naturally worn sediment particles:

s 

 D



25  1.2 D2  5



1.5



(4.21)

For calcareous sand particles sampled from the coast of the Yucatan Peninsula, Mexico, Alcerreca et al. (2013) developed a formula similar to Equation 4.21 and equivalent to Equation 4.20 with slightly different coefficients: Ms = 31.7, Ns = 1.1, and ns = 1.5. However, caution needs to be exercised because calcareous sand particles usually exhibit a wide variety of shapes associated with their biological origins (Smith and Cheung, 2003). Table 4.1 lists the values of Ms, Ns, and ns given by different authors for naturally worn sediment particles. Most of the authors suggested values for Ns in a narrow range of 1.0–1.2, which Table 4.1  Values of M s, N s, and n s for naturally worn sediment particles Author(s)

Ms

Ns

ns

Comments

Rubey (1933)

24

2

1

Goncharov (1954) Sha (1956) Zhang (1961) Zanke (1977) Soulsby (1997) Van Rijn (1984b) Raudkivi (1998) Julien (2010)

32 32 34 26.7 26.3 24 32 24

1.2 1.0 1.2 1.33 1.27 1.1 1.2 1.5

1 1 1 1 1

Cheng (1997a) Alcerreca et al. (2013) Wu and Wang (2006) Camenen (2007)

32 31.7 33.9 24.6

1 1.1 0.98 0.97

1.5 1.5 1.33 1.53

Fall diameter in laminar turbulent regime Nominal diameter Nominal diameter Nominal diameter Small errors Small errors Fall diameter in laminar Nominal diameter Fall diameter in laminar turbulent regime Nominal diameter Nominal diameter Nominal diameter Fall diameter in laminar

regime, large errors in

regime regime, deviations in

regime

102  Settling of Sediment Particles

agree well with the measurement data of coarse sediment particles, such as gravels, cobbles, and boulders. Rubey (1933) set Ns as 2, yielding significant underestimation of the settling velocity. Soulsby (1997), Zanke (1977), and Julien (2010) used Ns = 1.27, 1.33, and 1.5, respectively, which have certain deviations likely due to different data used, such as the nominal diameter estimated from the a-, b- and c-diameters rather than the volume of the particles (see Section 2.5.1). The coefficient Ms was set at a value of 24 by Rubey (1933), van Rijn (1984b), and Julien (2010); 26.3–26.7 by Soulsby (1997) and Zanke (1977); and 32–34 by Goncharov (1954, see Cheng, 1997a), Sha (1956), Zhang (1961), Raudkivi (1998), Cheng (1997a), and Alcerreca et al. (2013). When the nominal diameter is used, Ms = 32–34 agrees well with the measurement data in the laminar settling regime. The formulas using Ms = 24 are reduced to the Stokes law of spheres, which is not accurate for non-spherical particles. To minimize the errors, the formulas with Ms = 24 should use the fall diameter in the laminar regime. The intermediate values of Ms = 26.3–26.7 used by Soulsby (1997) and Zanke (1977) yield small errors. The rightmost column of Table 4.1 gives comments and suggestions for the listed formulas. Note that the segmented method proposed by van Rijn (1984b) reduces the errors of the Zanke formula by using the fall diameter in the laminar regime and the nominal diameter in the turbulent regime. In the transitional regime, a compromise is made in choosing the sieve diameter, with possible inconsistency existing between the regimes. Formulas of Ahrens (2000) and Chang and Liou (2001) According to Yalin (1972) and Hallermeier (1981), the particle Reynolds number Red is proportional to Ar and Ar in the laminar and turbulent settling regimes, respectively. Here, Ar  (  s   1) gD 3  2  D3 is the Archimedes buoyancy number. Thus, the following relationship is a logical approach for the transitional regime (Ahrens, 2000):  Red C3 Ar  C4 Ar 

(4.22)

where C3 and C4 are coefficients. Solving this equation for the settling velocity yields

s  C3

( s   1) gD 2  C4 ( s   1) gD  

(4.23)

0.50  where C3 0.055 tanh 12 Ar 0.59 exp(  0.0004 Ar ) , and C4 1.06 tanh 0.016 Ar exp (120 Ar ) , as determined by Ahrens (2000) using the data of Hallermeier (1981) based on the sieve diameter in the quartz density range (2.58 < ρs/ρ < 2.67). Chang and Liou (2001) proposed an alternative relationship between ωs and Ar:

s 

 Arn0  D 18 1   Arn0 1







where α = 30.22, and n0 = 0.463, as calibrated with the same data used by Ahrens (2000).

(4.24)

Settling of Sediment Particles  103

Formula of Ferguson and Church (2004) Ferguson and Church (2004) proposed the following formula of sediment settling velocity, which is applicable for the laminar, transitional, and turbulent regimes: ( s   1) gD

s

 C5



 0.75C6

( s   1) gD3



(4.25)

Equation 4.25 is an explicit function of ωs. It is equivalent to the following drag coefficient formula:   2C5 CD    C6   3(    1) gD3  s  

2



(4.26)

Ar  D1.5 , Equation 4.25 is similar to Equation 4.23, and Considering ( s   1) gD3   Equation 4.26 is similar to Equation 4.20. For smooth spheres, C5 = 18 and C6 = 0.4. For naturally worn sediment particles, C5 = 18 and C6 = 1.0 when D is the sieve diameter, or C5 = 20 and C6 = 1.1 when D is the nominal diameter. Equation 4.25 can be approximately reduced to the Stokes law Equation 4.5 for small particles, and to Equation 4.19, as used by van Rijn (1984b), for coarse particles. 4.4.2  Formulas Based on Particle Size and Sphericity Consider the settling of an elliptical particle along one of its longest, intermediate, and shortest axes. Denote the diameter along the settling axis as l1 and the diameters on the other two axes as l2 and l3. The particle has a nominal diameter of D  3 l1l2 l3 and a frontal area of πl2l3/4. Substituting these expressions into Equations 4.1 and 4.2 yields 1/ 2

2/3   l   4 s   1    s  gD   l2 l3    3CD     



(4.27)

where CD is the drag coefficient when the projection area is set as πl2l3/4 in Equation 4.2. Equation 4.27 indicates that the settling velocity is affected by the particle shape factor l1 l2 l3 of McNown and Malaika (1950), and perhaps additional factors involved in the drag coefficient. McNown et al. (1951) studied the effect of particle shape on the settling velocity of machined particles, such as ellipsoids, prisms, double pyramids, cylinders, and double cones. They introduced the correction factor Ks as the ratio of the settling velocity of a sphere with the same volume and weight as the particle to the settling velocity of the particle. Figure 4.3 shows the experimental data for these particles at Red < 0.1, and the theoretical curves obtained by McNown and Malaika (1950) solving the basic equations for ellipsoids in the laminar regime. The results indicate that Ks varies with l1 l2 l3 and l2/l3. The non-spherical particles experience higher drag and settle more slowly than the equivalent spheres, except for ellipsoids with a value of l1 l2 l3 around 2 and a value of l2/l3 close to 1.

104  Settling of Sediment Particles

Figure 4.3 Correction factor for the settling velocity of non-spherical particles at R ed < 0.1 (numbers beside symbols indicate values of l 2 / l 3) (from McNown et al., 1951)

As described in Section 4.1, in a large range of particle Reynolds numbers, a sediment particle settles along the shortest axis c, so l1 l2 l3 in Equation 4.27 becomes the Corey shape factor, S p  c ab . This was supported by Romanovskii (1972) in the following formula for settling velocity in the turbulent regime:  c  s  1.72    ab 

2/3

s   gD  

(4.28)

For spheres, the Corey shape factor Sp is 1, and Equation 4.28 is reduced to Equation 4.7. Substituting Sp = 0.7 into Equation 4.28 yields a coefficient of 1.36, which is larger than the value of 1.044 used in Equation 4.17 and of 1.1 in Equation 4.19. Therefore, Equation 4.28 significantly overestimates the settling velocity of naturally worn sediment particles. Based on the experiments of Krumbein (1942), Corey (1949), McNown et al. (1951), Wilde (1952), and Schulz et al. (1954), the U.S. Interagency Committee (1957) developed a series of curves shown in Figure 4.4 for the settling velocity of sediment particles as functions of particle size, Corey shape factor, and water temperature. Because several interpolations are involved, the curves in Figure 4.4 are inconvenient to use. In addition, all of the data used in the calibration were determined when Red > 3. The relationship was extended to the range of Red < 3 by assuming that it approaches the Stokes law Equation 4.5 as Red decreases. The curves yield errors in the laminar settling regime if the nominal diameter is used, as illustrated by McNown et al. (1951) in Figure 4.3. Therefore, Wu and Wang (2006) developed a formula to replace and improve the curves in Figure 4.4, as described herein. Wu and Wang (2006) related the coefficients Ms, Ns, and ns in Equation 4.20 to the Corey shape factor Sp as follows using the natural sediment settling data of Krumbein (1942), Corey (1949), Wilde (1952), Schulz et al. (1954), and Romanovskii (1972): M s  53.5e 0.65 SP , N s  5.65e 2.5 SP,  ns = 0.7 + 0.9Sp

(4.29)

Settling of Sediment Particles  105

Figure 4.4 Fall velocity of sediment particles as functions of particle size, shape factor, and water temperature (from the U.S. Interagency Committee, 1957)

Figure 4.5 Drag coefficient as a function of particle Reynolds number and shape factor (from Wu and Wang, 2006)

Figure 4.5 compares the measured drag coefficients with those calculated using Equation 4.20 with the Ms, Ns, and ns values determined using Equation 4.29. The agreement is generally good. Because the data in Figure 4.5 were for Red > 3, the trend of the CD − Red relationship for Red < 3 was determined using the data from Zegzhda (1934), Arkhangel’skii (1935), and Sarkisyan (1958), as compiled by Cheng (1997a). The Corey shape factor was assumed to be 0.7 since naturally worn sediment particles were used. Figure 4.6 shows that these data match well with the relationship of CD and Red given by Equation 4.20 with Equation 4.29, rather than the Stokes law of spheres.

106  Settling of Sediment Particles

Figure 4.6 Drag coefficient as a function of particle Reynolds number for naturally worn sediment particles (S p = 0.7) (from Wu and Wang, 2006)

It should be noted that when Sp = 1.0, Equation 4.20 with coefficients determined from Equation 4.29 deviates from the curve of spheres in Figure 4.2 obtained by Rouse (1938b). This is because the naturally worn sediment particles with Sp = 1.0 may not be exactly spherical, and other factors such as particle roundness and roughness also affect the settling process. Inserting Equation 4.20 into Equation 4.3 yields the general formula of settling velocity (Wu and Wang, 2006; Camenen, 2007): 1/ ns   M s  1  4 N s 3  1 D  s     N s D  4  3M s2 2   

ns

(4.30)

Note that D in Equation 4.30 is the nominal diameter (in meters), on which the drag coefficient CD in Figure 4.5 is based. Equation 4.30 is applied with Ms, Ns, and ns determined using Equation 4.29. It is explicit and thus convenient to use. If the particle shape is not measured, Equation 4.30 can still be used for naturally worn sediment particles by assuming Sp = 0.7 and thus setting Ms = 33.9, Ns = 0.98, and ns = 1.33. Figure 4.7 compares the settling velocities calculated using Equation 4.30 and the U.S. Interagency Committee’s 1957 curves at a water temperature of 24°C. These two methods give close predictions for coarse particles (D > 0.2 mm) with an average deviation of 2.75%. However, larger deviations are expected for fine sediments because the U.S. Interagency Committee’s curves approach the Stokes law Equation 4.5, which has large errors for sediment particles if the nominal diameter is used. 4.4.3  Formulas Based on Particle Size, Sphericity, and Roundness Wilde (1952) and Schulz et al. (1954) experimentally demonstrated that the settling velocity of freshly crushed rocks is significantly affected by particle roundness. According to Gogus

p

Settling of Sediment Particles  107

Figure 4.7 Comparison of Equation 4.30 and the method of the U.S. Interagency Committee (from Wu and Wang, 2006)

et al. (2001), machined particles, such as cylindrical, cubic, wedge-shaped, and box-shaped prisms, experience higher drag and slower settling velocity than sediment particles with the same Corey shape factor. Therefore, the Corey shape factor alone cannot represent the complex geometry of these particles. Gogus et al. (2001) proposed a more complex shape factor based on their data. Dietrich (1982), Jimenez and Madsen (2003), and Camenen (2007) developed more general methods to consider the effects of particle size, sphericity, and roundness on the settling velocity. The formula of Dietrich (1982) for all three settling regimes is written as

s3  10 R1  R2 R3     g  s 



(4.31)

where R1, R2, and R3 account for the effects of particle size, sphericity, and roundness, respectively, as follows: R1  3.76715  1.92944  log Ar   0.09815  log Ar   0.00575  log Ar   0.00056  log Ar  2



2

 0.00575  log Ar   0.00056  log Ar  3



2.3





4

4

R log 1  (1  S p ) 0.85  1  S p 2



3

tanh  log Ar  4.6   0.3 0.5  S p 1  S p



2.3

tanh  log Ar  4.6  

  log A  4.6  2

r

1  3.5  PR  / 2.5

0.65  ( S p 2.83) tanh  log Ar  4.6   R3 

where PR is the particle roundness factor defined in Section 2.5.2.





 Sp 1 Sp

  log A  4.6  2

r

108  Settling of Sediment Particles

The Dietrich formula is relatively complicated. Jimenez and Madsen (2003) simplified it to the following formula:  B    A0  0  S D  ( s   1) gD 

s

1



(4.32)

( s   1) gD 3 ( 4 )  D1.5 4 . The coefficients A0 and B0 are related to particle where S D  sphericity and roundness, as shown in Figure 4.8. For naturally worn sediments, A0 = 0.954 and B0 = 5.121. For spherical particles, A0 = 0.794 and B0 = 4.606. Like Wu and Wang (2006), Camenen (2007) used Equations 4.20 and 4.30 to calculate the settling velocity of sediment particles. The effect of particle roundness is considered in addition to the particle size and sphericity through the coefficients Ms, Ns, and ns as follows: a3

b3

        Ms  a1  a2 1  sin  S p   , N s  b1  b2 1  sin  S p   ,  2     2       (4.33) ns  n1 sin n2  S p  2  where a1 = 24, a2 = 100, a3 = 2.1 + 0.06PR, b1 = 0.39 + 0.22(6 − PR), b2 = 20, b3 = 1.75 + 0.35PR, n1 = 1.2 + 0.12PR, and n2 = 0.47. Camenen (2007) calibrated Equation 4.33 using the results of Dietrich (1982) and additional measurement data. For natural sediment particles with Sp = 0.7 and PR = 3.5, Equation 4.33 yields Ms = 24.6, Ns = 0.97, and ns = 1.53. The value of Ms = 24.6 is close to that in Stokes law, so in this case, the fall diameter should be used in the laminar settling regime. Ns = 0.97 works well in the turbulent regime where the nominal diameter is used. In the transitional regime, the sieve diameter is likely a compromise choice (see Section 4.4.1).

Figure 4.8 Coefficients A 0 and B 0 in Equation 4.32 as functions of particle sphericity and roundness (from Jimenez and Madsen, 2003)

Settling of Sediment Particles

4.4.4

109

Comparison of Existing Formulas for Sediment Settling Velocity

Jimenez and Madsen (2003) compared the formulas of Zanke (1977), Julien (2010), Soulsby (1997), Cheng (1997a), Ahrens (2000), and Jimenez and Madsen (2003). The used data were collected from Cheng (1997a), Engelund and Hansen (1972), Hallermeier (1981), and Raudkivi (1998). Table 4.2 shows the prediction scores measured as percentages of cases exhibiting specific errors of prediction (calculated/measured) for each formula. The formulas of Ahrens (2000) and Jimenez and Madsen (2003) perform somewhat better than the other four formulas. Wu and Wang (2006) compared the formulas of Dietrich (1982), Jimenez and Madsen (2003), and Wu and Wang (2006), which all consider the effect of particle shape on sediment settling. A total of 571 measurement data sets, including those in Figure 4.5 and some reported by Briggs et al. (1962), were used to test the three formulas. Because the particle roundness was unknown, PR = 3.5 was assumed for the Dietrich and Jimenez-Madsen formulas. Table 4.3 shows that the mean relative errors of the three formulas are very close. Wu and Wang (2006) compared several existing formulas against the measurements of naturally worn sediment particles, including the Russian data shown in Figure 4.6 and data reported by Hallermeier (1981) and Raudkivi (1998). They analyzed 100 data sets in total. A Corey shape factor of 0.7 and a roundness index PR of 3.5 were assumed. Table 4.4 shows the mean relative errors of the tested formulas. Rubey’s (1933) formula has large errors. The formulas of Zhang (1961), Hallermeier (1981), Dietrich (1982), Cheng (1997a), Ahrens (2000), Jimenez and Madsen (2003), and Wu and Wang (2006) have comparable reliabilities, with average errors less than 9%.

Table 4.2 Prediction scores of settling velocity in terms of percentages of cases exhibiting specific errors of prediction (calculated/measured) (from Jimenez and Madsen, 2003) Error percentage

Zanke (1977)

Julien (2010)

Soulsby (1997)

Cheng (1997a)

Ahrens (2000)

Jimenez and Madsen (2003)

10% 20%

56 91

46 89

62 91

62 91

77 91

72 94

Table 4.3 Comparison of formulas against data with the Corey shape factor (from Wu and Wang, 2006) Data range

Fine particles (D * < 30) Coarse particles (D * ≥ 30) Total

Number of data

Mean relative errors (%) Dietrich (1982)

Jimenez and Madsen (2003)

Wu and Wang (2006)

289 282 571

7.7 14.2 10.9

8.1 13.3 10.7

8.1 10.1 9.1

110  Settling of Sediment Particles Table 4.4 Comparison of formulas against data without particle shape information (from Wu and Wang, 2006) Formula Rubey (1933) Mean relative error (%) 20.5

Zhang (1961) 8.5

Hallermeier (1981) Dietrich (1982) 8.7 8.2

Formula

Cheng (1997a) Ahrens (2000) Jimenez and Madsen (2003) Mean relative error (%) 7.4 7.6 7.9

Wu and Wang (2006) 6.8

4.5  Effects of Sediment Concentration on Settling Velocity 4.5.1  Settling Velocity of Uniform Sediments in Concentrated Water General Formulation Consider a group of uniform noncohesive sediment particles settling in a finite water column. The volumetric concentration of sediment is c and the settling velocity of each particle in clear water is ωs. The settling of the particle group induces a return flow of water, which has an upward velocity ωsc/(1 − c) according to the balance of displaced water and sediment volumes. Therefore, the particles have an apparent settling velocity ωs[1 − c/(1 − c)] (with respect to a fixed reference frame), which is approximately equal to ωs(1 − c) when c is small. The density of the water-sediment mixture is given with Equation 2.51. Due to the increase in buoyant force, the submerged specific weight of the sediment in concentrated water is reduced to 1 − c times that in clear water: γs – γm = (γs − γ)(1 − c)(4.34) where γ and γm are the specific weights of clear and sediment-laden waters, respectively. By using the Stokes law Equation 4.5 and considering the aforementioned effects of return flow and buoyancy, the following formula can be derived (Chien and Wan, 1983, 1999):

sm  2  1  c  s m



(4.35)

where μ and μm are the dynamic viscosities of clear and sediment-laden waters, respectively; and ωsm is the sediment settling velocity in the sediment-laden water. By determining the viscosity ratio μm/ μ with Equation 2.55, Equation 4.35 can be used to calculate ωsm. Because μm > μ and 0 ≤ c < 1, Equation 4.35 indicates that the settling velocity is reduced in the concentrated water. This phenomenon is known as hindered settling. Equation 4.35 is limited to fine sediments since it is based on the Stokes law. Cheng (1997b) suggested an alternative approach for a wider range of particles using the settling velocity Equation 4.21 with similar corrections for the effects of return flow, buoyancy, and viscosity.

Settling of Sediment Particles  111

Formula of Richardson and Zaki (1954) Numerous formulas based on Equation 4.35 have been developed to quantify the hindered settling of solid particles in fluids (Richardson and Zaki, 1954; Sha, 1965; Garside and Al-Dibouni, 1977; Chien and Wan, 1983; Baldock et al., 2004; Kramer et al., 2019). The most widely used is the following formula proposed by Richardson and Zaki (1954):

  s 1  c  h  sm n

(4.36)

where nh is an empirical exponent, related to the particle Reynolds number Red = ωs D/v as Red  0.2  4.65  4.4 R 0.034 0.2  R  1  ed ed nh   0.097 4 . 4 R 1  R  500 ed ed   2.4 Red  500



(4.37)

Later studies have shown that Equation 4.37 has significant bias for natural sands at small Red values (Figure 4.9). Chien and Wan (1983) prepared an earlier version of Figure 4.9 that included the data of Liacenko (1957, see Chien and Wan, 1983) and Zhao (1964), as well as the empirical relationships of nh and Red developed by Chien and Wan (1983), Guo and Zhuang (1963), and Zaminiyan (1957, see Chien and Wan, 1983). The figure is updated here by including the data measured by Wilhelm and Kwauk (1948), Cleasby and Woods (1975), Cleasby and Fan (1981), and Baldock et al. (2004) primarily for natural and filter sands. The Chien-Wan, Guo-Zhuang, and Zaminiyan relationships of nh and Red show improvements over Equation 4.37. Garside and Al-Dibouni (1977) proposed an alternative function for nh, which is written in the following general form: nhL  nh   Red v nh  nhT

(4.38)

where αω and βω are empirical coefficients, nhL is the upper limit of nh in the laminar settling regime (low Red), and nhT is the lower limit of nh in the turbulent regime (high Red). Table 4.5

Figure 4.9  Exponent n h in Equation 4.36 as a function of particle Reynolds number

112

Settling of Sediment Particles

Table 4.5 Parameters in Equation 4.38 given by different authors Author(s)

n hL

n hT

Richardson and Zaki (1954) Garside and Al-Dibouni (1977) Chien and Wan (1983, 1999) Rowe (1987) Kramer et al. (2019) Present fitting

4.65 5.1 4.91 4.7 4.8 4.9

2.4 2.7 2.25 2.35 2.4 2.1

αω

βω

0.1

0.9

0.175 0.043 0.053

0.75 0.75 0.75

Particle type Spheres Spheres Various types Spheres Garnet grains, calcite pellets Natural and filter sands

summarizes the values of these coefficients suggested for various granular materials (Garside and Al-Dibouni, 1977; Rowe, 1987; Kramer et al., 2019). These coefficients are recalibrated here as nhL = 4.9, nhT = 2.1, αω = 0.053, and βω = 0.75 by fitting Equation 4.38 with the sand data in Figure 4.9, as denoted with “present fitting”. Equation 4.38 with the newly calibrated coefficients performs overall better than the other models. Baldock et al. (2004) established a relationship between nh and sediment diameter, which is approximated as (Te Slaa et al., 2015) nh = 4.4(Dref / D)0.2

(4.39)

where Dref is a reference diameter set as 0.0002 m. Equation 4.39 is plotted in Figure 4.9 by converting D to Red with the help of the settling velocity formula of Wu and Wang (2006) at a water temperature of 20°C. Equation 4.39 shows a generally good agreement with the measured data, except for overshooting at the lower and upper ends. Kumbhakar et al. (2017) developed the following formula for nh by applying an entropy concept: 0      Red  nh 2.41 2 ln 1  0.83exp a0      c(  s   1)    

(4.40)

where a0 = 0.061 and η0 = 0.45. Note that a0 had an error in the original paper of Kumbhakar et al. (2017) (personal communication with the authors). Equation 4.40 was tested with the fixed concentration c = 0.2 and specific gravity ρs/ρ = 2.65. The effects of c and ρs/ρ on nh need to be verified by using more data. Figure 4.9 indicates that the exponent nh varies from about 2 to 5 for sands. However, higher nh values, such as 7, were reported for fine sediments (Xia and Wang, 1982; Wang, 1984b). The deviation might be partly due to the effect of silt and clay in the sediment mixtures. Further investigation is needed to clarify this effect. Formulas of Camenen (2008), Te Slaa et al. (2015), and Spearman and Manning (2017) The correction factor 1 c  h in Equation 4.36 combines the effects of return flow, buoyancy, and viscosity, as illustrated in Equation 4.35. Camenen (2008) and Spearman and Manning (2017) reconstructed it by considering the viscosity effect in analogy to Equation 2.56: n

  s 1  c  sm

mh

1  c cmax 

nh

(4.41)

Settling of Sediment Particles  113

where cmax is the maximum concentration of sediment suspension. Camenen (2008) set mh = nh − 1  and nh  cmax. Spearman and Manning (2017) set mh = 2.7 − 0.15nh; and nh 0.62nh  1.46, with nh  0 in the turbulent settling regime and nh  1.5 in the laminar regime. Similarly, Te Slaa et al. (2015) proposed the following formula to consider the three effects in Equation 4.35 separately by using the viscosity Equation 2.56 of Krieger and Dougherty (1959) and reformulating the factor of return flow:

  s 1  c cstruct  sm

m1

1  c  1  c cmax 

2.5 cmax



(4.42)

where cstruct is the structural concentration of sediment at which the apparent settling velocity is zero due to the return flow. The exponent m1 is likely between 1 and 2. The concentrations cmax and cstruct depend on the particle size gradation. Normally cstruct ≤ cmax. Equations 4.41 and 4.42 eliminate the problem of Equation 4.36 that gives a non-zero settling velocity as c approaches cmax or min(cstruct, cmax). Formulas of Sha (1965) and Zhang and Zhang (1992) Sha (1965) and Zhang and Zhang (1992) introduced sediment size into the correction factors for settling velocity in concentrated water. The formula of Sha (1965) is  c sm s 1    2 D50, mm 

   

3

(4.43)

and the formula of Zhang and Zhang (1992) is  c sm s 1    2.25 D50, mm 

   

3.5

1  1.25c 

(4.44)

where D50,mm = D50 in mm. The factors 2 D50, mm and 2.25 D50, mm act as the concentration cmax or cstruct. Note that Equations 4.36 and 4.41–4.44 are primarily used for noncohesive sediments. Cohesive sediments exhibit significantly different settling behaviors. Moreover, at high concentrations, sediment-laden water exhibits non-Newtonian fluid features. These cases are introduced in Chapters 10 and 12. 4.5.2 Settling Velocity of Nonuniform Sediments in Concentrated Water When sediment concentration is very low, the interactions between size classes in a nonuniform sediment mixture can be ignored, and thus the settling velocity of each size class can be calculated separately using the settling velocity formulas of single particles described in the previous sections. As the concentration increases, the effects of return flow, buoyancy, and viscosity increase, and the interactions between size classes become important. In particular, differential settling promotes collision between coarse and fine particles, which in turn modifies the settling velocity of all the particles.

114  Settling of Sediment Particles

Masliyah (1979) and Lockett and Bassoon (1979) generalized the equation of Richardson and Zaki (1954) for the settling velocity of the ith size class of a sediment mixture:

sm , i

1  c 

nh  2

 s ,i  susp N s , j  susp   s , j c j  s ,i  s ,i   s , j   j 1 

    

(4.45)

where susp 1  c     j 1 c j s , j , c is the volumetric concentration of total sediment, cj is the volumetric concentration of the jth size class of sediment, ρs,j is the density of the jth size class of sediment, nh is the exponent of the Richardson-Zaki equation, and N is the total number of size classes. By setting N = 1, Equation 4.45 can be reduced to Equation 4.36. Thus, Equation 4.45 takes into consideration the effects of return flow, buoyancy, and viscosity. To account for the effects of relative size and density, Batchelor (1982) proposed the following formula for the fraction-wise settling velocity of a suspended polydisperse mixture to the first order with respect to the fractional volumetric concentration under the assumption of small Peclet numbers: N

N    sm ,i s ,i 1   Sij c j   j 1  

(4.46)

where Sij is an empirical parameter related to the particle diameter ratio λij = Dj/Di, the submerged density ratio αij = (ρs,j – ρ)/(ρs,i – ρ) and the Peclet number Pec = uD/ԑs (Batchelor and Wen, 1982). The Peclet number represents the relative importance of particle advection (represented by particle velocity u) and diffusion (Brownian motion, represented by diffusion coefficient εs). Pec ≫1 if advection is dominant, and Pec ≪1 if diffusion is dominant. By using the Richardson-Zaki formula, Davis and Gecol (1994) extended the work of Batchelor (1982) and Batchelor and Wen (1982) to

  s ,i 1  c  sm , i

 Sii

N   1   Sij  Sii c j   j i  





(4.47)

where Sii is −5.6 for non-colloidal particles (D > about 1 μm) and −6.5 for colloidal particles (D < about 1 μm), and Sij is given as 2 3  3.5  1.1ij  1.02ij  0.002ij Sij   2 3 3.42  1.96ij  1.21ij  0.013ij

for non-colloidal particless for colloidal particles



(4.48)

Equation 4.48 does not take into account differences in density. Ha and Liu (2002) proposed the following formula for Sij that considers both particle size and density differences at large Peclet numbers:  1.87ij Sij 2.5   ij  ij2  3ij  1   1  0.0024ij2 

    

(4.49)

Settling of Sediment Particles  115

The term (1 c)  Sii in Equation 4.47 considers the effect of hindered settling, which slows down all size classes. The term inside the last pair of parentheses in Equation 4.47 can consider the effect of differential settling, which slows down coarse particles but speeds up fine particles. Overall, the settling of coarse particles is slowed down, whereas that of fine particles depends on the dominant factor between the hindered and differential settling. Equation 4.47 was first developed to determine the hindered settling velocity of polydisperse solid particles in chemical engineering. It was validated in the Stokes regime and can be applied to determine the fraction-wise settling velocity of noncohesive sediment mixtures in concentrated water. Modifications are introduced in Section 10.5.1 to adapt Equation 4.47 to scenarios involving mixed cohesive and noncohesive sediments. 4.6  Effects of Turbulence on Sediment Settling Velocity Turbulence consists of eddies that exist on a variety of scales (Figures 3.2 and 4.10). There are several mechanisms by which these eddies affect the settling of sediment particles. A particle may be trapped within the center of a turbulence vortex or delayed in the up-flow side of a vortex, which reduces the average settling velocity (Tooby et al., 1977). On the other hand, the settling velocity is increased when a heavy particle is preferentially swept into the down-flow border areas between vortices (Wang and Maxey, 1993). Nielsen (1993) identified these areas as “fast tracks” (Figure 4.10). In a mix of these two modes, a particle may randomly enter vortex center, up-flow, and down-flow zones on its settling path, so the average settling velocity may be reduced or increased. Lhermitte (1962), Fan et al. (1964), Field (1968), Murray (1970), Nielsen (1993), and Jacobs et al. (2016) examined the settling velocities of solid particles in turbulence fields generated by the oscillation of grids in still water. Fan et al. (1964) and Jobson and Sayre (1970) investigated the settling velocity of sediment particles released from point sources near the water surface in open-flume shear flows. Field (1968) showed that the average settling velocity of a particle in turbulent water decreased relative to that observed in still water, whereas Jobson and Sayre’s (1970) data indicated an increase. Lhermitte (1962) and Fan et al. (1964) did not find any significant reduction or increase. Murray’s (1970) data demonstrated that average settling velocity was reduced by turbulence for heavy particles but increased for quasineutral particles. Nielsen (1993) found that changes in settling velocity were related to the relative turbulence intensity, i.e., the ratio of turbulence intensity to settling velocity. The settling velocity is slowed by 20%–40% in

Figure 4.10  Sketch of fast settling tracks in a turbulence field

116  Settling of Sediment Particles

relatively weak grid turbulence fields, but could be increased considerably up to a factor of 4 in strong turbulence fields. The existence of “fast tracks” was observed by Jacobs et al. (2016) in experiments involving natural, synthetic, and industrial particles (Red = 0.4–123) settling in a grid turbulence facility. Using PIV and 2-D particle tracking, they found that the trajectories of small particles followed the turbulent flow patterns well, whereas those of large particles did not correspond to any particular flow pattern. The obtained relationship between the relative settling velocity ωst/ωs and the Stokes number StK is shown in Figure 4.11. Here, ωst and ωs are the settling velocities in turbulent and still waters, and StK is defined as the ratio of the particle relaxation (response) time tp and the Kolmogorov turbulence time scale tK. With tK given in Equation 3.15 and tp 

s D2 18 



(4.50)

such that StK is expressed as S tK

t p  s D 2   1/ 2  D  s     3  tK 18     18   lK 

2



(4.51)

where lK is the Kolmogorov length scale of turbulence defined in Equation 3.13. Figure 4.11 indicates that the settling velocity of fine particles is enhanced significantly by turbulence when StK < 0.23. According to Equation 4.51, StK = 0.23 when D/lK = 1.25 for natural sediment particles. Thus, fine particles with D < 1.25lK are likely affected by turbulence, whereas coarse particles are largely unaffected. Kawanisi and Shiozaki (2008) measured the settling velocities of sediment particles in openflume turbulent flows via ADV. The water depth was about 0.3 m, and the depth-averaged velocity varied from 0.05 to 0.3 m/s. Seven types of particles were used, with sizes ranging from 0.038 to 0.5 mm and specific gravities ranging from 1.06 to 3.13. The ADV sampling volume was a cylinder of 9 mm in length and 6 mm in diameter, located 0.06 m above the bed. Figure 4.12 shows the relative settling velocity as a function of the relative turbulence intensity and the Stokes number, St = tp/tT. Here, tT  15  is the Taylor time scale derived from Equation 3.21.

Figure 4.11 Relationship between relative settling velocity and Stokes number (from Jacobs et al., 2016)

Settling of Sediment Particles  117

Figure 4.12 Relative settling velocity (ω st/ω s) of a sediment particle as a function of the relative turbulence intensity and Stokes number in turbulent shear flows (after Kawanisi and Shiozaki, 2008)

The data of Murray (1970) and Nielsen (1993) are also included in Figure 4.12, denoted as “M” and “N”, respectively. The data in Figure 4.12 follow the same trend in shear flows and grid turbulence fields. The particle settling velocity is unchanged or slightly reduced in relatively weak turbulence fields. In relatively strong turbulence fields, the settling velocity increases considerably up to a factor of about 2–6 at σp/ωs = 15. Here, σp is the particle turbulence intensity, taken as the standard deviation of the vertical velocity of a particle. At intermediate turbulence intensities, the settling velocity decreases with large Stokes numbers but increases with small Stokes numbers. 4.7  Measurements of Sediment Settling Velocity Settling velocity is usually measured in a settling column or tube, as shown in Figure 14.13a. The column is divided into the accelerating and terminal settling regions. The column should be long enough to ensure terminal settling in the bottom region, where measurement takes place. The accelerating region is usually short, as shown in Section 4.9. A stopwatch is often used to record the time taken for a particle to fall through the measurement region. The settling velocity is obtained by dividing the settling length by the settling time. In recent years, photography and imaging technologies have been used to measure particle settling velocity. Because it affects viscosity, the water temperature should be measured. Fine particles may form aggregates, which can be disaggregated by adding a dispersion agent, such as sodium hexametaphosphate or sodium carbonate. Particle settling in a tube of finite size may be slowed by the return flow of the surrounding fluid. This is called the wall effect. The wall effect for spherical particles was thoroughly investigated by Fidleris and Whitmore (1961), among others. Based on over 3,000 velocity measurements, Fidleris and Whitmore related the wall effect to two variables: the ratio of the particle diameter D to the tube diameter dtube, and the particle Reynolds number Red,∞ = ωs,∞D/v, as shown in Figure 4.13b. Here, ωs is the terminal velocity in the tube, and ωs,∞ is the terminal velocity in an infinite fluid. When D/dtube is less than 0.1, the wall effect is very small, with Red,∞ > 100. The wall effect becomes significant at low Red,∞ and high D/dtube values.

118  Settling of Sediment Particles

Figure 4.13  (a) Sketch of a settling tube, and (b) wall effect on the settling velocity of sphere in a tube of finite size (from Fidleris and Whitmore, 1961, © IOP Publishing, reproduced with permission)

Figure 4.13b can be used to correct ωs to ωs,∞ when the wall effect is not negligible. Di Felice (1996) proposed the following formula to represent the graphical relationship shown in Figure 4.13b: na

 1  D dtube  s    s ,  1  0.33D dtube 

(4.52)

where na is related to Red,∞ as 3.3  na  0.1Red ,   na  0.85

(4.53)

4.8 Relationship Between Fall and Nominal Diameters of Sediment Particles The U.S. Interagency Committee (1957) established the relationship between the standard fall diameter (at a water temperature of 24°C) and nominal diameter of naturally worn quartz particles with a range of Corey shape factors, as shown in Figure 4.14. The relationship was based on the measurement data of particles larger than about 0.2 mm and assumed to approach Stokes’ law in the laminar settling regime. The difference between the fall and nominal diameters increases as the Corey shape factor decreases. Fall and nominal diameters can be correlated by equating the settling velocities of the equivalent sphere and sediment particle calculated with Equations 4.10 and 4.30, respectively. With a given Corey shape factor, the fall diameter of the sediment particle can be converted to the nominal diameter in two steps. First, use Equation 4.10 to calculate the settling velocity of the equivalent sphere having the same diameter as the given fall diameter. Then, substitute the calculated settling velocity and the given Corey shape factor into Equation 4.30 to calculate the nominal diameter of the sediment particle. The calculations are straightforward, since Equation 4.30 has the following inverse equation:

Settling of Sediment Particles  119

Figure 4.14  Relationship between the nominal and standard fall diameters of naturally worn sand particles (from U.S. Interagency Committee, 1957)

1/ ns   3 N s s 2   1 1  4 M s ( s   1) g    D    4  3 N s2s3   4( s   1) g   2 

ns

    

(4.54)

Conversely, the nominal diameter can be converted to the fall diameter by employing Equation 4.30 first to calculate the settling velocity with the given nominal diameter and Corey shape factor, and then substituting the calculated settling velocity into Equation 4.10 to determine the diameter of the equivalent sphere. Because Equation 4.10 cannot be reversed explicitly, iteration is needed to solve Equation 4.10 for the sphere diameter. In the laminar settling regime, Equation 4.30 is reduced to

s 

4 s   D 2 g 3M s  



(4.55)

Table 4.1 shows that when the nominal diameter is used, Ms is between 32 and 34 for naturally worn sediment particles. Comparing the Stokes law Equation 4.5 and the natural sediment settling velocity Equation 4.55 leads to the conclusion that the fall diameter is about 0.85 times the nominal diameter. Therefore, the nominal diameter can be obtained by dividing the fall diameter by 0.85 for fine sediments (D < 0.1 mm). 4.9 Time and Distance to Terminal Settling of Sediment Particles After a sediment particle is dropped into a settling column, it accelerates because the gravity is greater than the buoyant and drag forces. In the accelerating or initial period, the particle

120  Settling of Sediment Particles

additionally experiences the inertia force and the Basset (1961) force. These two forces are introduced in detail in Section 7.2. The inertia force is given as FM  CM 

D 3 d sa  6 dt

(4.56)

where ωsa is the settling velocity at time t and CM is the inertia coefficient. CM is about 1.5–2.0 for a sphere and may be set as 0.5 to consider only the added mass force in certain studies. According to the second law of Newton, the particle motion is governed by mp

d sa  Ws  FD  FM  dt

(4.57)

where mP = ρs πD3/6 is the particle mass. The Basset force is ignored here since it is usually much smaller than the drag force. A complicated model including the Basset force can be found in Guo (2011). Substituting Equations 4.1, 4.2, and 4.56 into Equation 4.57 yields

  s  CM  

D 3 d sa  6 dt

 s    g

D 3 D 2 sa2  CD  6 4 2



(4.58)

The drag coefficient CD is determined using Equation 4.20 with ns = 1 assumed for convenience. Thus, Equation 4.58 is rewritten as  M  d sa s    3 2  g   s  Ns  sa  dt (  s  CM  )  sa D  (  s  CM  ) 4 D

(4.59)

Integrating Equation 4.59 and using the initial condition ωsa = 0 at t = 0 yields N 1  exp   2 ( s3CMs  ) D  0 t  sa   s 1  0 exp   2 (  3CN s  ) D  0 t  s M  

(4.60)

2

where  0  4 D(  s   ) g (3 N s )  [ M s (2 DN s )] ; ωs = α0 – Msv/(2DNs) is the terminal settling velocity, derived from Equation 4.30 with ns = 1; and Ω0 = [α0 – Msv/(2DNs)]/[α0+Msv/ (2DNs)]. By using Equation 4.60, the time required for a sediment particle to attain 99% of its terminal settling velocity can be derived as follows: t99% 

2(  s  CM  ) D ln 100  990   3  N s 0

(4.61)

The settling distance sa is governed by dsa /dt = ωsa 

(4.62)

Settling of Sediment Particles  121

Figure 4.15  Variations of velocity with distance in the accelerating stage of particle settling Table 4.6  Time and distance to 99% of the terminal settling velocity D (mm)

t 99% (s)

s 99% (cm)

0.0625 0.125 0.25 0.5 1 2 4 8 16 32 64 128

0.00325 0.0122 0.0353 0.0658 0.100 0.143 0.203 0.288 0.408 0.577 0.816 1.154

  0.00063   0.0091   0.085   0.35   0.90   1.94   3.99   8.05  16.2  32.4  64.7 129.5

Integrating Equation 4.62 and using the initial condition sa = 0 at t = 0 yield  1  2(  s  CM  ) D   sa s t  1   N  3  0 s 0  



3 Ns   1  0 exp  2 ( s  CM  ) D  0 t  ln  1  0  

     

(4.63)

Figure 4.15 shows the validation of the initial settling model derived here against the measurement data of Allen (1900) for a steel sphere settling in water at a temperature of 17.8°C. The particle has a diameter of 3.18 mm, a relative density ρs/ρ of 7.82, and a terminal settling velocity of about 0.83 m/s. Ms = 24 and Ns = 0.4 are fixed, while CM is given as 2.0 (i.e., inertia coefficient) and 0.5 (i.e., added mass coefficient only) for a sensitivity analysis. Because the particle is coarse, the terminal settling velocity is mainly controlled by Ns. The variation of particle velocity with distance is significantly affected by CM and well predicted when CM = 2.0. This supports the use of the inertia coefficient CM in Equation 4.56 rather than the added mass coefficient only. The time and distance needed to reach 99% of the terminal settling velocity for natural sediment particles with different sizes in water are presented in Table 4.6. Time and distance are calculated by using Equations 4.61 and 4.63 with ν = 10-6 m 2/s, ρs/ρ = 2.65, Ms = 34,

122  Settling of Sediment Particles Table 4.7  Sediment size D and corresponding D *, R ed, and ω s values D (mm)

D*

R ed

ω s (cm/s)

0.004 0.008 0.016 0.031 0.0625 0.125 0.25 0.5 1 2 4 8 16 32 64 128

0.101 0.202 0.405 0.784 1.58 3.16 6.32 12.6 25.3 50.6 101.2 202.4 404.7 809.5 1618.9 3237.9

0.000041 0.000326 0.00260 0.0189 0.152 1.13 7.01 32.6 118.9 379.1 1133.2 3287.1 9405.4 26743.5 75825.5 214704.8

0.0010 0.0041 0.0163 0.0609 0.243 0.902 2.80 6.52 11.9 19.0 28.3 41.1 58.8 83.6 118.5 167.7

Ns = 1.2, and CM = 2.0. For a sediment particle of 1 mm in diameter, the acceleration period is 0.1 s, and the distance is 0.9 cm. The time and distance increase as the sediment size increases. For sediment particles smaller than 128 mm, the acceleration region is shorter than 130 cm. This can be used as a reference for determining the acceleration region in a settling tube. Appendix 4.1 Settling Velocity of Common Natural Sediment Particles Assume kinematic viscosity ν = 10-6 m2/s and specific gravity G = 2.65 at a water temperature of 20°C. The corresponding D*, Red, and ωs values for natural sediment particles with D varying from 0.004 to 128 mm are given in Table 4.7. The terminal settling velocity is calculated using Wu and Wang’s (2006) formula. Homework Problems 4.1 Describe the characteristics of laminar and turbulent settling of sediment particles. 4.2 Describe the relationship between the drag coefficient and sediment size. How do particle shape and angularity affect the drag coefficient? 4.3 Consider a spherical solid particle and a natural sediment particle with the same density and nominal size in the same liquid at the same temperature. Which one settles faster? 4.4 Calculate the settling velocities of natural sediment particles with nominal sizes of 0.01 and 1 mm at a water temperature of 20°C using the formulas of Zhang (1961), Wu and Wang (2006), Ahrens (2000), and Jimenez and Madsen (2003). Note that the formula of Ahrens (2000) uses the sieve diameter, whereas the other three formulas use the nominal diameter. 4.5 How does sediment concentration affect settling velocity? 4.6 How does water temperature affect the settling velocity of sediment? 4.7 How does turbulence affect the settling velocity of sediment?

Settling of Sediment Particles  123

4.8 Describe the wall effect on the settling of sediment particles in a finite water column. 4.9 A naturally-worn sediment particle is measured to have a settling velocity of 2 cm/s in a settling tube with a diameter of 0.05 m at a water temperature of 20°C. Determine its settling velocity in a large, still water body. 4.10 Derive Equation 4.63. Draw a figure to illustrate the relationship between the accelerating region length and the sediment size. 4.11 Following the calculation procedure described in Section 4.8, draw a graph similar to Figure 4.14 to illustrate the relationship of the standard fall diameter with the nominal diameter and shape factor for naturally worn quartz particles. 4.12 What is the relationship between the fall and nominal diameters in the laminar settling regime? If a particle has a fall diameter of 0.01 mm, what is the nominal diameter approximately?

Chapter 5

Incipient Motion of Sediments

Bed sediment particles start moving when the acting flow strength exceeds a certain threshold. The incipient motion of noncohesive sediments is introduced in this chapter. Topics include drag and lift forces on bed particles, general threshold conditions for individual particle entrainment, critical bed shear stress and critical depth-averaged velocity of uniform and nonuniform sediments, and probabilities of particle entrainment, as well as effects of channel slope, bed roughness, particle shape, exposure height, shallow submergence, turbulence, and impulse duration on incipient motion. 5.1  Drag and Lift Forces on Bed Sediment Particles A sediment particle on a bed under flowing water experiences drag and lift forces (Figure 5.1), which are the streamwise and vertical components of the hydrodynamic force created by surface friction and pressure difference. The drag force on a settling particle, as described in Section 4.2, can be extended to a bed particle. Complexities arise from the asymmetric shear flow near a rough bed, which essentially induces lift force in addition to drag force. Though analytical formulas are available for a sphere at low particle Reynolds numbers (Saffman, 1965), the drag and lift forces on a natural sediment particle are usually calculated using the following formulas with experimentally determined coefficients: FD  CD  Ap

ub2  2

(5.1)

FL  CL  Ap

ub2  2

(5.2)

where FD is the drag force, FL is the lift force, CD is the drag coefficient, CL is the lift coefficient, ρ is the water density, AP is the projection area of the particle, and ub is the near-bed flow velocity. Ideally, ub in Equations 5.1 and 5.2 is the approaching flow velocity ahead of the particle, but this is applicable only when the particle is directly exposed to the flow. When the particle is blocked by upstream particles, ub is treated as a reference velocity near the bed, e.g., at 0.15D above the particle apex (Einstein and El-Samni, 1949). The projection area AP is related to the particle exposure (protrusion) height ∆e, which is defined as the elevation difference between the particle apex and the bed formed by the apexes of the surrounding particles, especially the nearest upstream particle (Figure 5.1) (Fenton and Abbott, 1977; Wu et al., 2000). Determination of AP is very complicated DOI: 10.1201/9781003343165-5

Incipient Motion of Sediments  125

Figure 5.1 Forces on a sediment particle on the bed surface (O is the pivot point; k WD, k DD, k LD are the moment arms; α sl is the slide angle; and β is the bed slope angle)

Figure 5.2 Distributions of pressure and the streamwise and vertical components of the pressure-difference force on a gravel particle at four positions on the bed surface (adapted from Zhang et al., 1965)

when the particle is naturally worn and partially blocked. For simplicity, AP is set as πD2/4 of a fully-exposed sphere, and the effects of particle shape and frontal blockage are considered in the drag and lift coefficients. Here, D is the nominal particle diameter unless stated otherwise. Drag and lift forces are highly related to particle position on the bed surface. Zhang et al. (1965) measured the pressure-difference forces on a gravel particle placed at four positions on a bed under flowing water (Figure 5.2). The test particle was 6.2, 5.0, and 3.9 cm long on the longest, intermediate, and shortest axes, respectively. It was drilled with 12 holes for

126  Incipient Motion of Sediments

Figure 5.3 Drag coefficient (based on shear velocity) on spheres at the four positions shown in Figure 5.2 under flowing oil, reported by Watters and Rao (1971) (from Chien and Wan, 1983)

measuring the pressure distribution on the particle surface along the longitudinal-vertical section. At position 1, the particle was located in a co-planar bed. At position 2, the particle sat on a triangular interstice formed by particles on a planar bed. At positions 3 and 4, the particle was placed in front of and behind a group of particles, respectively. The measured streamwise and vertical pressure-difference forces were of the same magnitude, except at position 1. The forces at positions 2 and 3 were significantly larger than those at positions 1 and 4. The streamwise force at position 1 was negative but became positive when the particle slightly protruded into the flow. Note that the measured streamwise force was not exactly the drag force because the surface friction was not included. Nevertheless, the measured vertical force was likely close to the lift force. Watters and Rao (1971) measured the drag forces on spheres at the same four positions shown in Figure 5.2 under flowing oil. The results, as presented by Chien and Wan (1983) in Figure 5.3, showed that the drag coefficient (based on shear velocity) varied with the particle shear Reynolds number Re*, defined as u*D/v, where u* is the shear velocity and ν is the kinematic viscosity. Position 1 yielded the lowest drag among the four positions. Position 4 had the next lowest drag. Position 2 had higher drag than position 3 at high Re* values, but the opposite was true for low Re* values. The results of Watters and Rao (1971) agree qualitatively with those of Zhang et al. (1965). Chien and Wan (1983) created Figure 5.4 to compare the drag coefficient of a sphere rolling down an inclined slope or lying on a bed (Garde and Sethuraman, 1969; Watters and Rao, 1971; Coleman, 1972) against that of a sphere settling in a water column. The drag coefficient of an isolated sphere lying on a bed (position 2 in Figure 5.2) in flowing water is similar to that of a free settling sphere and is smaller than that of a sphere rolling down a slope in still water. The drag coefficient decreases as the particle Reynolds number increases. Li et al. (1983) reported CD = 0.7 for a sphere on a flat bed at ubD/ν > 2000. Aksoy (1973) obtained CD = 1.04. Schmeeckle et al. (2007) measured the drag force on an isolated sphere (19 mm diameter), cube (25.4 mm), and natural gravel (22 mm average diameter) on smooth beds, with mean drag coefficients of 0.76, 1.36, and 0.91, respectively. The results of Aksoy (1973), Li et al. (1983), and Schmeeckle et al. (2007) agree well with those of Coleman (1972) and Watters and Rao (1971) in the turbulent regime (Table 5.1).

Incipient Motion of Sediments  127

Figure 5.4 Drag coefficients of spheres lying on a bed in flowing water, rolling down a slope, and free settling (from Chien and Wan, 1983)

Dwivedi et al. (2010, 2011) measured the drag and lift forces on a sphere of D = 38.3 mm placed with other particles, similar to position 1 in Figure 5.2, with exposure heights of 0–10 mm. The drag coefficient was 0.11–0.14 (using πD2/4 as the projection area). For natural stones packed at the same bed level, Wang and Fontijn (1993) measured CD ≈ 0.36. These low values for the drag coefficient are due to the blockage by upstream particles and the fact that the reference velocity is set at 0.15D above the particle apex, rather than as the approaching velocity. Einstein and El-Samni (1949) measured pressure differences as the lift forces on densely packed spherical plastic balls (D = 68.6 mm) and natural gravels (D = 17 – 76.6 mm) (i.e., position 1 in Figure 5.2). The theoretical bed was assumed at 0.2D below the particle apex, and the reference flow velocity ub was measured at 0.15D above the particle apex. The mean lift coefficient was about 0.178. The lift force followed a normal error distribution, with a standard deviation/mean value ratio of  L FL  0.364 . Similarly, Dwivedi et al. (2011) obtained CL = 0.13 – 0.30 for spheres and confirmed that the normal error distribution matched the measured data in the range of lift force within ±3σL. Wang and Fontijn (1993) obtained CL = 0.15 for stones placed at the same bed level. Li et al. (1983) measured CL = 0.18 for a sphere on a flat bed, and Aksoy (1973) obtained CL = 0.14. These CL values are very close, even for particles at positions 1 and 2. However, negative lift forces on spheres were observed at Re* < 100 by Watters and Rao (1971), at Re* < 15 by Coleman (1972), and at Re* < 5 by Davies and Samad (1978). Davies and Samad (1978) reported that negative lift occurred due to significant underflows beneath the spheres, but the mechanism of negative lift has not been understood completely. Lamb et al. (2017) measured the drag and lift forces acting on spherical particles of 75 – 218 mm in diameter sitting on planar fixed gravel-cobble beds with slopes of 0.004 – 0.3 under shallow flows. The local flow velocity in front of the test particles was used as ub. CD was about 0.7 for submerged particles but increased significantly for partially submerged particles. In contrast, CL decreased from near unity to zero as the flow became shallow and was strongly negative for partially submerged particles.

128

Incipient Motion of Sediments

Table 5.1 Experimental results of drag and lift forces on spherical and natural bed particles Investigator(s) Zhang et al. (1965)

Experiment setup

Gravel at four positions (Figure 5.2) Watters and Sphere in Rao (1971) oil at four positions (Figure 5.2) Einstein and Spheres and El-Samni gravels (1949) densely packed Wang and Stones placed Fontijn at same bed (1993) level Dwivedi et al. Spheres (2010, 2011) densely packed Apperley Sphere laid (1968) on gravels Coleman Isolated sphere (1972) on densely packed spheres Aksoy (1973) Sphere

Particle size (mm)

R e*

62 × 50 × 39

–

95.3

–

Spheres: 68.6 Gravels: 17 – 76.6 33

3,300 – 5,600 –

Velocity u b –

0.5D above bed 0.15D above particle apex

38.3

1,983 – 3,240

6.4

70

0.15D above particle apex 0.15D above particle apex –

13; 0.6 – 20

10 – 1,500; 6.5 – 1,500

0.5D above bed

20

UR/ν = 2,700 – 6,000 u bD/ν > 2,000

– Li et al. (1983) Isolated sphere on flat bed Schmeeckle Sphere, cube, 19, 25.4, 22, respectively et al. (2007) natural gravel on flat bed 75 – 218 Lamb et al. Isolated Shallow (2017) sphere on flows gravelcobble bed 16 Bagnold (1974) Sphere 800 – Garde and Sphere – Sethuraman rolling on (1969) the bed in still water

–

Results Drag and lift at the same order of magnitude, Figure 5.2 C D in Figure 5.3; negative lift for R e* ~0.7 C L < ~1 F L/F D ≈ 0.5 C D larger than free settling, Figure 5.4

Apperley (1968) found that the lift to drag ratio increased from 0.5 to 0.78 when a sphere was lifted by 0.25 times its diameter, but decreased as it was lifted further. Dwivedi et al. (2011) found that the lift to drag ratio increased from 1.22 to 2.40 as particle exposure was changed from 0.0 to a quarter of the diameter. Bagnold (1974) observed that the lift force

Incipient Motion of Sediments  129 Table 5.2  Experimental results of drag and lift forces on hemispherical bed particles Investigator(s)

Experiment setup

Particle size (mm)

R e*

Chepil (1961)

3 – 102

16 – 13,680

Chen and Clyde (1972)

Hemispheres arranged with 3D distance in air flows Densely packed hemispheres

305

35,800 –  63,200

Brayshaw et al. (1983)

Hemisphere in wind tunnel

115

52,000

Velocity u b –

Average velocity –

Results FL/FD = 0.53 –  1.32, with average 0.83 Drag and lift— normal error distributions F L/F D ≈ 1.8

decreased with increasing clearance between the particle and the boundary. The results of Bagnold (1974) and Dwivedi et al. (2011) agree qualitatively with those of Apperley (1968). Chepil (1961) reported a lift to drag ratio in the range of 0.53 – 1.32 with an average of 0.83 for hemispheres in air flows. A ratio of about 1.8 was measured by Brayshaw et al. (1983). Chen and Clyde (1972) found that the lift and drag forces on densely packed hemispheres followed normal error distributions, with normalized standard deviations  L FL  0.18 and  D FD  0.4 to 0.8 depending on flow depth (Table 5.2).  5.2  Incipient Motion Thresholds of Individual Sediment Particles Consider a noncohesive sediment particle on an alluvial bed under a steady, uniform flow, as shown in Figure 5.1. Besides the drag and lift forces discussed previously, the particle experiences the submerged weight Ws. The particle starts moving when the drag and lift forces are strong enough to overcome the submerged weight. This is called incipient motion or entrainment. The modes of incipient motion include rolling, sliding, and lifting (saltating). In the case of incipient rolling, the moment balance for the particle is expressed as kD  D·FD + kL D·FL − kW  D·Ws = 0

(5.3)

where kD D, kL D, and kW D are the moment arm distances of forces FD, FL, and Ws to the point of pivot, respectively. These arm distances depend on the size, shape, position, and packing of bed particles. The submerged weight is given in Equation 4.1. Inserting Equations 4.1, 5.1, and 5.2 into Equation 5.3 yields 1/ 2

ucr , r

  4kW s    gD   3  k D CD  k L CL   



(5.4)

where ucr,r is the critical near-bed velocity for incipient motion by rolling, ρs is the particle density, and g is the gravitational acceleration.

130  Incipient Motion of Sediments

On the sediment bed, a particle slides on the curved surface of other particles below it, rather than on the average bed slope. Therefore, for the particle shown in Figure 5.1, the balance of the resistance and driving forces along the sliding direction reads Ws cos   FL  cos  sl  Ws sin   FD  sin  sl  tan i   Ws sin   FD  cos  sl  Ws cos   FL  sin  sl

(5.5)

where ϕi is the friction angle, β is the bed slope angle, and αsl is the slide angle with respect to the bed slope. Substituting Equations 4.1, 5.1, and 5.2 into Equation 5.5 leads to the critical near-bed velocity for incipient sliding: 1/ 2

 4  cos  tan( sl  i )  sin    s    ucr , s   gD     3 CD  CL tan( sl  i ) 



(5.6)

When the particle starts lifting from a gentle bed slope, the lift force equals the submerged weight: FL = Ws

(5.7)

Substituting Equations 4.1 and 5.2 into Equation 5.7 yields the critical near-bed velocity for incipient lifting: 1/ 2

ucr ,l

 4 s     gD   3CL  



(5.8)

The lifted particle may start to saltate or fall back to the bed. Han and He (1984) defined the critical saltation condition as when the particle is lifted a distance of one diameter. This implies that the saltation threshold is higher than the lifting threshold. This difference is not quantified in the present text, but the incipient saltation concept along with Han and He’s probability theory of sediment transport is introduced in Section 7.4.5. Particle saltation dynamics are described in Sections 7.2 and 7.3. For general applications, Equations 5.4, 5.6, and 5.8 are collectively written as 1/ 2

   ucr   r  s gD    



(5.9)

where ucr is the overall critical near-bed velocity of incipient motion, taking the minimum of ucr,r, ucr,s, and ucr,l. The coefficient ξr takes the formulation from Equation 5.4, 5.6, or 5.8, depending on the primary mode of incipient motion. As analyzed in Appendix 5.1, for spherical particles, rolling requires the smallest critical flow velocity among the three modes and thus is the primary mode of incipient motion. According to Carling et al. (1992), elliptic particles are usually entrained by rolling, but often after adjustment

Incipient Motion of Sediments  131

of local position by sliding in the bed pocket. This is likely also true for naturally worn sediment particles that have relatively high sphericity and roundness. Thus, ξr likely has the following formulation based on the rolling mode: 1/ 2

  4kW r     3  k D CD  k L CL  



(5.10)

In practice, ξr needs to be measured. This is due to the complexities of particle shape and packing of natural sediments on the bed surface as well as the uncertainties of the involved parameters. Because the near-bed flow velocity is not often measured, ucr in Equation 5.9 is converted to the depth-averaged velocity or the bed shear stress. For example, using the log law Equation 3.37 for ucr in Equation 5.9 yields 2

   cr r    s    D  5.75 log  30 yd  s ks  

(5.11)

where τcr is the critical shear stress of incipient motion, γs is the specific weight of sediment, γ is the specific weight of water, yd is the height at which the near-bed velocity is located, and χs is the correction factor defined in Figure 3.9. Equations 5.9 and 5.11 can be applied to determine the critical near-bed velocity and shear stress for a specific sediment particle to start motion at a specific position on the bed surface. The determined values are instantaneous. These two equations form a basis for the development of statistical mean thresholds for the incipient motion of sediment mixtures, as presented in the next sections. 5.3 Approaches to Describing the Incipient Motion of Bed Particle Ensembles The incipient motion of bed sediment particles as an ensemble is usually handled with both deterministic and stochastic approaches since it obeys the principles of dynamics but exhibits randomness. 5.3.1  Stochastic Approach for Entrainment Probability Einstein (1942, 1950) pioneered the application of the stochastic approach to sediment transport. He proposed an entrainment probability model considering the random fluctuation of lift force. Since then, entrainment probability has been studied by Dou (1962), Paintal (1971b), He and Han (1982), Wu and Yang (2004a), and Tregnaghi et al. (2012), among others. Its general concept is briefly described here. As introduced in the previous section, a sediment particle starts to move when the applied near-bed flow velocity is greater than the critical velocity. Instantaneous values of these two velocities are denoted as ũb and ũcr. In a turbulent flow, ũb is typically a random variable due to the intermittent and stochastic behaviors of turbulent fluctuations. For the ensemble of sediment particles on the bed, ũcr is also a random variable due to the randomness of particle position and

132  Incipient Motion of Sediments

Figure 5.5  Probability of incipient motion based on ũ b and ũ cr distributions

packing as well as the heterogeneity of particle size, shape, and density. When the probability distributions of ũb and ũcr overlap, the particles have a probability of beginning to move (Figure 5.5). In general, ũb and ũcr are assumed to be statistically independent. Thus, the entrainment probability pE is defined as (He and Han, 1982; Zanke, 1990; Wu and Yang, 2004a) pE  p ( ub  ucr )  1 

ucr ,max

ucr ,min

 ucr f (u )du  Y (u ) du  b b cr cr   ucr 

(5.12)

where ũcr,max and ũcr,min are the maximum and minimum values of ũcr, respectively; Y(ũcr) is the probability density function of ũcr; and f(ũb) is the probability density function of ũb. In homogenous turbulence fields, velocity fluctuations obey the normal or Gaussian probability distribution (Townsend, 1947; Batchelor, 1953). The normal distribution is approximately applicable for turbulent shear flows, except near rigid walls where large velocity gradients and roughness elements create anisotropic features of turbulence. Thus, the normal distribution has been widely used to describe the probability of ũb (Kalinske, 1943; Dou, 1962; He and Han, 1982; Cheng and Chiew, 1998; Hofland and Battjes, 2006):  f  ub 

  ub  ub 2    exp   2 u2  2 u   1

(5.13)

where ūb and σu are the mean value and standard deviation of ũb, respectively. The probability distribution of ũcr is little understood and thus avoided in many studies. If ũcr varies in a narrow range, it can be represented by a single value denoted as ucrm, and the entrainment probability Equation 5.12 can be simplified as ucrm 1 ucrm ub   u  x2 2 pE  1  f (ub )dub  1 e dx   ucrm 2 ucrm  ub   u



(5.14)

where x = (ũb − ūb)/σu. Alternatively, the entrainment probability can be defined as follows by considering the instantaneous bed shear stress b and critical shear stress cr as independent random variables (Grass, 1970; Lopez and Garcia, 2001; Tregnaghi et al., 2012):  pE p b  cr   

cr ,max

cr ,min

  ( )d  X ( )d b b cr cr  cr 



(5.15)

Incipient Motion of Sediments  133

where cr , min and cr , max are the minimum and maximum values of cr , respectively; X (cr ) is the probability density function of cr ; and  (b ) is the probability density function of b . A number of entrainment probability models have been developed based on the general concept described here. Details of these models are introduced in Section 5.9. 5.3.2  Deterministic Approach to Incipient Motion Thresholds The deterministic approach adopts statistically mean threshold flow conditions for the incipient motion of a sediment mixture. As described in Section 5.2 and Appendix 5.1, the thresholds of incipient motion for particles at specific positions on the bed surface can be theoretically derived using a force or moment balance analysis. However, the theoretical analyses have limitations due to the assumptions of spherical particles and certain packing configurations. The derived thresholds vary widely with particle positions and do not represent the mean characteristics in reality. Therefore, the mean thresholds adopted in practice are usually obtained through laboratory and field measurements using methods such as flow competence, visual observation, and reference transport. Flow Competence Gilbert (1914) defined “flow competence” as the value of a controlling factor (e.g., bed shear stress, flow velocity) that causes the onset of sediment transport. For a flow able to move a subset of the particle sizes in a mixed sediment bed, he suggested using the largest mobile particle as a measure of the flow competence. With samples from a range of flows, a relationship between the flow factor and the largest mobile particle can be derived, and then used to estimate the critical flow condition for individual particle sizes in the bed sediment. The flow competence method has been adopted by Andrew (1983) and Komar (1987), among others. However, it uses an extreme value of the transport particle size distribution, which may be subject to large errors and sensitive to the effect of widely varying sample volume (Wilcock, 1992). Observations of incipient motion in coarse sediment mixtures suggest that all sizes may begin moving in a narrow range of flow conditions, within which the flow competence method may not give a meaningful estimate on a size-by-size basis (Parker et al., 1982; Andrews, 1983; Wilcock, 1992). Visual Observation This approach uses visual techniques, including manual counting, photography, and video recording, to observe the status of bed particles as the flow strength is sequentially increased. The critical flow condition is obtained when the stationary bed particles start to move. Kramer (1935) defined three intensities of sediment movement near the critical condition: weak, medium, and general. Weak movement indicates that a few of the finest sand particles move at isolated spots, and those moving over a small bed area (e.g., one square centimeter) can be counted. Medium movement indicates that an uncountable number of sand particles of mean diameter are in motion. Such movement is no longer local, but it is not yet strong enough to affect bed configuration and does not result in appreciable sediment discharge. The general movement indicates that all the particles, including the coarsest, are in motion over all parts of the bed at all times. The criteria of Kramer are easy to use but only qualitative and somewhat subjective.

134  Incipient Motion of Sediments

For fine sediments, a turbulent flow moves bed particles intermittently in gusts or bursts. The frequency of these bursts is correlated with the number of particles per burst moving in a small area. Thus, Vanoni (1964) used the frequency of 1/3 to 1 burst per second over an area of 8–18 mm in diameter as the critical incipient condition. This criterion appears to agree with Kramer’s weak movement. However, it is not appropriate for coarse sediments, such as gravel. Consider a flat bed with sediment of size D. The number of particles per unit bed area is about 1/D2. The time for a particle to be lifted from the bed is assumed proportional to D/u*, where the shear velocity u* is used to characterize the velocity of the lifted particle. The number of particles displaced from unit bed area per unit time is denoted as n p . Thus, based on the kinematic similarity of sediment particles with different sizes, Neill and Yalin (1969) derived the following nondimensional entrainment intensity of sediment particles: n 

n p

1 D  u 2



D



n p D 3 u



(5.16)

Shvidchenko and Pender (2000) set the incipient motion by using a specified value (10-4 s−1) of the sediment movement intensity n p n p. Here, np is the number of particles per unit bed area. This intensity can account for the bed material packing density through ma = np πD2/4, but it is dimensional. It can be nondimensionalized as follows by applying Equation 5.16 and considering the relationship np∝1/D2: d 

n p D n p u



(5.17)

The criteria in Equations 5.16 and 5.17 are correlated by Гd = Гnπ /(4ma). For generally packed sediment beds, Гd ≈ Гn. Reference Transport In the reference transport approach, the bed-load transport or bed erosion rate is measured under a series of flow conditions, and a curve of bed-load transport (or bed erosion) rate versus bed shear stress (or flow velocity) is established. The critical flow condition is then obtained by extrapolating the established curve to a reference value of bed-load transport or bed erosion rate. Shields (1936) used a zero transport rate of bed load as the reference threshold. Numerous experiments (e.g., Paintal, 1971a; Salim et al., 2017) have shown that even when the flow strength is much weaker than the threshold condition proposed by Shields (1936), there are still particles moving on the bed. One reason is the random behavior of sediment transport, as described in Section 5.3.1. Another reason is that the extrapolated curve used by Shields (1936) may overshoot the zero transport rate under a higher bed shear stress. This indicates that a zero transport rate of bed load is just a nominal reference. Shields’ threshold condition is similar to the weak movement criterion of Kramer (Vanoni, 1964). Therefore, several low values of bed-load transport or bed erosion rate have been used as the reference. For example, qb = 0.00041 kg/(s·m) was suggested by the U.S. Army Corps of Engineers’ Waterways Experiment Station (USWES, 1936), and an erosion rate of 10-4 cm/s was adopted by McNeil et al. (1996). The former method was proposed for sand, whereas the

Incipient Motion of Sediments  135

latter was meant for cohesive sediments. As shown in Appendix 5.2, these two methods specify relatively strong transport for fine sediments. Neill and Yalin (1969) defined the bed-load transport rate qb  n p lb m p , in which lb is the average saltation step length and mp is the particle mass. The saltation step length is proportional to the sediment size D, and the particle mass is equal to (π/6)ρs D3. Thus, by substituting this rate into Equation 5.16, Neill and Yalin (1969) derived the following dimensionless bedload transport rate:  qb 

qb   s Du

(5.18)

where qb is the bed-load transport rate by mass per unit time per unit width. Comparing Equations 5.16 and 5.18 yields Гqb/Гn = (π/6)lb /D. As shown in Appendix 5.2, Гqb has a value of about 0.0001 as the lower limit of incipient motion threshold. Since lb is about 100D (Einstein, 1950), Гn ≈ 2×10 − 6. Equations 5.16 and 5.18 work well for gravel and sand. Based on a stochastic theory, Han and He (1984) suggested the following nondimensional bed-load transport threshold for incipient motion:

 qbk 

qbk  pbk  s Dk sk

(5.19)

where qbk is a stochastically mean transport rate for the kth size class of bed load by mass per unit time per unit width, Dk is the diameter, ωsk is the settling velocity, and pbk is the fraction by weight of the kth size class in bed material. The value of ηqbk was calibrated as about 0.000317 for uniform sediments (pbk = 1.0) but found to be about 0.000113 for the fraction-wise incipient motion of nonuniform sediments. Possible reasons for the different ηqbk values include interactions among different size classes in nonuniform sediments and different measurement data used to calibrate the thresholds. Parker et al. (1982) suggested the following dimensionless bed-load transport as the reference for the incipient motion of nonuniform sediments: Wk 

qbk (  s /  1)  0.002 pbk  s ( ghS )1/ 2 hS

(5.20)

where qbk is the transport rate for the kth size class of bed load, h is the flow depth, and S is the energy or bed slope. Chien and Wan (1983) proposed the following reference transport threshold based on the Einstein transport number: b 

s

 s

qb

  1 gD 3

 0.0001



(5.21)

This criterion was also used by Shvidchenko and Pender (2000). It corresponds to an incipient motion probability of 0.433%, according to Einstein’s bed-load transport formula described in Section 7.4.5.

136  Incipient Motion of Sediments Table 5.3  Dou’s (1962) probability criteria of particle incipient motion Threshold

Near-bed flow velocity

Probability p E

Impending motion Little motion General motion

ũ b > u crm = ū b +3σ u ũ b > u crm = ū b +2σ u ũ b > u crm = ū b +σ u

0.135% 2.28% 15.9%

As explained in Section 5.5, the bed shear stress at incipient motion is proportional to sediment diameter D; thus, Equations 5.18 and 5.20 are similar to Equation 5.21. Such similarity also exists between Equations 5.19 and 5.21, since the settling velocity is related to sediment diameter, with the square root function for coarse particles (D > 4 mm). Dou (1962) defined three states of incipient motion: impending, little, and general motion (Table 5.3), which correspond to values of critical near-bed velocity ucrm at ūb + 3σu, ūb + 2σu, and ūb + σu, and have incipient motion probabilities of 0.135%, 2.28%, and 15.9%, respectively, as calculated with Equation 5.14. The 0.135% and 2.28% probabilities of Dou (1962) are around the threshold proposed by Chien and Wan (1983). Dou (1962) suggested the 2.28% probability as the incipient motion threshold. This is close to the threshold of 2% probability used by Papanicolaou et al. (2002). However, the 0.135% probability is at the lower limit of Kramer’s weak movement, and thus may be more adequately used as the incipient motion threshold than the 2.28% probability. The aforementioned transport thresholds are compared in Appendix 5.2. It can be concluded that the criteria of Parker et al. (1982), Neill and Yalin (1969), Dou (1962) with pE = 0.135%, Chien and Wan (1983), and Han and He (1984) are generally applicable for sand and gravel. The first three of these five methods usually specify weaker transport and thus are safer in terms of channel bed stability than the last two methods. The criteria of USWES (1936) and McNeil et al. (1996) are less adequate since they give larger quantities for fine sediments in motion. 5.4 Critical Average Velocity for the Incipient Motion of Uniform Sediments The critical near-bed velocity ucr is set at yd = ςdD above the bed. Here, ςd is an empirical coefficient. Using the power law Equation 3.53, Equation 5.9 is converted to the following formula for the critical average velocity Ucr in m/s (Shamov, 1959; Zhang, 1961; Neill, 1968): 1/ 2

   U cr   c  s gD    

1/ m

h D  



(5.22)

where m is the reciprocal of the exponent in Equation 3.53 and  c  r m [(m  1) 1d/ m ] . These two coefficients are usually determined with laboratory and field measurements. Values reported in the literature include m = 6 and ζc = 1.14 (Shamov, 1959), m = 7 and ζc = 1.34 (Zhang, 1961), and m = 6 and ζc = 1.414 (Neill, 1968). For riprap design, Garcia (2008) suggested m = 6 and ζc = 1.204. Equation 5.22 can be applied for coarse particles, e.g., D > 1 mm. Its extension to fine particles is described in Section 10.3.3.2.

Incipient Motion of Sediments  137

Using the log law Equations 3.37 and 3.64 to determine ucr in Equation 5.9, Goncharov (1954) and Dou (1960) derived 1/ 2

   U cr   d  s gD    

log

a0 h  ks

(5.23)

where ks is the bed roughness height. Goncharov (1954) used ζd = 1.07, a0 = 8.8, and ks = D95, whereas Dou (1960) used ζd = 0.74, a0 = 11.0, and ks = D with a lower limit of 0.0005 m. Van Rijn (1993) proposed the following formula for natural sediments (ρs = 2,650 kg/m3) in fresh water at 15°C: 0.1 0.19 D50 log  4h D90  for 0.00005  D50  0.0005 m U cr    0.6 for 0.0005  D50  0.002 m  8.5 D50 log  4h D90 

(5.24)

where Ucr and D50 are given in m/s and m, respectively. Considering the similarity of the drag coefficients for incipient motion and free settling in Figure 5.4, as observed by Coleman (1972), and assuming that the lift coefficient is proportional to the drag coefficient, Yang (1973) established U cr 0.66  2.5  log Re*  0.06  1.2  Re*  70  s  Re*  70 2.05



(5.25)

where Re* = u*D/v is the shear Reynolds number. For coarse sediments, the critical average velocity is about twice the settling velocity. Equation 5.25 was calibrated using the data from only laboratory experiments. It does not consider the effect of flow depth. Its applicability in large rivers needs to be investigated (Chien and Wan, 1983). Note that the aforementioned formulas for the critical average velocity of incipient motion were calibrated against data measured at the deterministic reference thresholds described in Section 5.3.2, so they should be used as a kind of statistically mean criteria. Similarly, threshold flow conditions are defined for particle motion cessation, bed erosion, deposition, and nonerosion. The critical velocity of particle motion cessation, U cr , indicates the threshold condition at which sediment particles stop motion and settle to the bed. It is slightly   U cr 1.2, smaller than the critical velocity of incipient motion. Shamov (1959) proposed U cr with Ucr given by Equation 5.22. Hjulström (1935) prepared Figure 5.6 for the critical average velocities for erosion and deposition on a movable bed with quartz sediments in water. The critical velocity of erosion, UE, decreases first and then increases as the mean diameter of sediment increases. Higher critical velocities for fine sediments are observed due to the effects of cohesion and viscous sublayer. The cohesion effect is discussed in more detail in Section 10.3.3. The critical velocity of deposition, UD, increases as sediment size increases. Erosion and incipient motion have approximately the same thresholds, as do deposition and motion cessation. The critical average velocity of nonerosion indicates the maximum permissible flow velocity, Ump, below which a sediment bed does not erode. Ump is approximately equal to or slightly smaller

138  Incipient Motion of Sediments

Figure 5.6 Critical average velocities of erosion and deposition as functions of mean diameter (from Hjulström, 1935)

than UE. It is often used in channel design. Lischtvan and Lebediev (1959) developed a relationship between the nondimensionalized Ump and the relative flow depth h/D using data observed in Russian channels for wide ranges of quartz sediment sizes (0.005 mm < D < 500 mm) and flow depths (0.4 m < h < 10 m). This relationship was separately fitted with the following equations for gravel-/cobble-bed streams and sand-bed streams by García-Flores and Maza (1996): 1.630  h D 0.1283    s   1 gD 0.453  h D 0.3221 U mp

h D  744.2 h D  744.2



(5.26)

5.5 Critical Shear Stress for the Incipient Motion of Uniform Sediments 5.5.1  Original Shields Diagram For practical applications, a variety of empirical formulas have been developed for critical shear stress by inferring from Equation 5.11 and using measurement data (e.g., Shields, 1936; White, 1940; Wiberg and Smith, 1987). Among them, a widely accepted relationship was derived by Shields (1936) as follows: cr  f  Recr  

(5.27)

where Θcr = τcr/[(γs – γ)D] is the critical Shields number; and Re*cr = u*cr D/v is the particle shear Reynolds number, with ucr   cr  being the critical shear velocity. Shields (1936) established the relationship between Θcr and Re*cr, as shown in Figure 5.7, using experimental data available at that time. The sediments used in the experiments had median sizes of 0.4 – 3.4 mm, geometric standard deviations of 1.3 – 2.3, and specific weights of 1.06 – 4.25 g/cm3. The sediment size D was represented by the median sieve diameter. Shields’s criterion corresponds closely to Kramer’s weak movement criterion (Vanoni, 1964),

Incipient Motion of Sediments  139

and to an entrainment probability of about 1% (Cheng and Chiew, 1998; Choi and Kwak, 2001; see Sections 5.9.2 and 5.9.3). In the original Shields (1936) diagram, Θcr has a value of 0.06 when Re*cr > 600. Θcr decreases as Re*cr decreases from 600 to 10. Θcr reaches a minimum value of 0.032 near Re*cr = 10. Then, Θcr increases as Re*cr decreases further. The dip in Θcr is mainly due to the characteristics of turbulent flows indicated by the χs – ks/δ curve in Figure 3.9. As the flow shifts through hydraulically rough, transitional, and smooth regimes, χs increases and then decreases, and in turn Θcr decreases and then increases according to Equation 5.11. In addition, sand (at intermediate Re*cr) has a lower friction angle and in turn a lower critical Shields number than gravel (at high Re*cr). Moreover, sand particles may experience more exposure on the bed surface since they are too small to allow for individual leveling in experiments, whereas gravel particles may have less exposure since they are large enough that careful leveling is possible (Fenton and Abbott, 1977). More exposed particles have smaller Θcr values. The effects of particle exposure and bed friction on incipient motion are discussed in detail in Sections 5.8.1 and 5.8.3, respectively. In the range of Re*cr < 2, Shields (1936) did not have any data but suggested cr  0.1Re*1cr by extending the curve trend of Re*cr > 2 and considering the effect of the viscous sublayer. This function is only qualitatively correct and has large errors. More accurate functions were obtained later using measurement data, as introduced in the next section. Because bed shear stress is correlated with shear velocity, the Shields diagram in Figure 5.7 is actually an implicit relationship. Iteration is required to obtain the critical shear stress for a given sediment size. To avoid iteration, Vanoni (1964) added to Figure 5.7 a series of inclined lines, which can be expressed as

 cr u D  K  cr     D  s    

2



Figure 5.7  Shields diagram of sediment incipient motion (from Vanoni, 1964)

(5.28)

140  Incipient Motion of Sediments

where K is a constant for each line. Eliminating u*cr from this equation yields D 

s   gD  

1 K



(5.29)

D which implies that   s   1 gD is constant at each inclined line. The inclined lines intercept with the gridline of Θcr =  0.1 at Re*cr D 0.1 s   1 gD . For convenience, the values of D 0.1 s   1 gD are marked for the inclined lines in Figure 5.7. Intercepts of the inclined  lines with the Shields curve explicitly give values of Θcr.

D The parameter  0.1 s   1 gD is associated with the nondimensional particle size D* = D[(ρs/ρ − 1)g/v2]1/3. Moreover, Θcr, Re*cr, and D* are correlated by Re2*cr  cr D*3 . Thus, the relationship between Θcr and Re*cr in Figure 5.7 can be converted to explicit relationships between Θcr and D*, such as the following ones suggested by van Rijn (1984a):

 0.24 D1 , D  4  0.64 4  D  10  0.14 D ,  cr   0.04 D0.1 , 10  D  20 0.013D 0.29 , 20  D  150    D  150  0.055,



(5.30)

and by Soulsby (1997):





cr  0.24 D1  0.055 1  e 0.02 D 

(5.31)

A similar explicit formula was proposed by Brownlie (1981a):





cr  0.22 Rep0.6  0.06 exp 17.73Rep0.6 

(5.32)

where  Rep   s   1 gD3   D*3/ 2 . Though the original Shields diagram in Figure 5.7 has drawbacks at low and high Re*cr values, it has been one of the greatest contributions in the vast literature of sedimentation engineering. 5.5.2  Modified Shields Diagrams The value of Θcr = 0.06 at high Re*cr in the original Shields diagram is the upper bound of Θcr measured in experiments. In the laminar and hydraulically smooth regimes, the experiments of Yalin and Karahan (1979b) yielded cr  0.1Re*0cr.3, rather than the original cr  0.1Re*1cr (Figure 5.8). Therefore, Mantz (1977), Yalin and Karahan (1979b), Chien and Wan (1983), and Paphitis (2001), among others, modified the original Shields diagram using more measurement data. Figure 5.8 shows the Shields diagrams modified by Yalin and Karahan (1979b) and Chien and Wan (1983). The data used by Chien and Wan included measurements from White (1940), Tison

Incipient Motion of Sediments  141

Figure 5.8 Shields diagrams modified by Chien and Wan (1983) and Yalin and Karahan (1979b) (after Chien and Wan, 1983)

(1948), Li and Sun (1964), White (1970), Mantz (1977), and Yalin and Karahan (1979b). Some data at Re*cr > 1000 used by Yalin and Karahan were excluded, likely because of high D/h ratios. Both modifications improve the original Shields diagram under hydraulically smooth and rough regimes. In the rough regime, mean Θcr values of 0.045 and 0.052 are used in the diagrams of Yalin and Karahan (1979b) and Chien and Wan (1983), respectively. In the smooth regime, the diagram of Yalin and Karahan aligns more closely with the data measured in laminar flows, whereas the diagram of Chien and Wan ignores those data. The Shields diagram modified by Yalin and Karahan (1979b) can be approximated with the following formulas proposed by García-Flores and Maza (1996):  0.137 D0.377 , 0.1074  D  2.084  cr  0.178 D0.73  0.0437 exp{[31.954 ( D  10)]2.453 }, 2.084  D  47.75   0.045, D  47.75 

(5.33)

da Silva and Bolisetti (2000): 2



 cr 0.13D0.392 e 0.015 D*  0.045 1  e 0.068 D





(5.34)

and Cao et al. (2006):  0.1414 Rep0.231 , Rep  6.6  0.677 2.836 0.354 cr  [1  (0.0223Rep ) ] (3.095 Rep ) , 6.6  Rep  282.8  0 . 045 , Rep  282.8 



(5.35)

The first expressions in Equations 5.33 and 5.35 are close to the function cr  0.135 D*0.39 , which is converted from cr  0.1Re*0cr.3, as indicated by the data of Yalin and Karahan in the laminar and hydraulically smooth regimes. The formula of Cao et al. (2006) aligns more toward the data points measured in laminar flows than the formulas suggested by García-Flores and Maza (1996) and da Silva and Bolisetti (2000).

142  Incipient Motion of Sediments

The threshold diagram of Chien and Wan (1983) is given by the upper and lower bound curves in Figure 5.8. The general trend of the mean Θcr can be approximated with (Wu and Wang, 1999)  0.126 D0.44 , D  1.5  0.55 0 . 131 D , 1 . 5  D  10   0 . 27  0.0685 D , 10  D  20 cr   0.19  0.0173D , 20  D  40  0.0115 D 0.30 , 40  D  150    0 . 052 , D  150  



(5.36)

Paphitis (2001) re-examined the Shields diagram using a large quantity of data reported by many investigators in the literature. The data covered a variety of uniform and well-sorted mixtures of natural and artificial particles with densities of 2.49 – 2.71 on flat beds in flowing water, oil, and glycerol. The threshold conditions were determined through visual definition or extrapolation of sediment transport rates to either zero or a low reference value. Paphitis (2001) derived the following threshold formula: cr 





0.188  0.0475 1  0.699e 0.015 Recr  1  Recr

(5.37)

Equation 5.37 gives a value of 0.0475 in the hydraulically rough regime. Equation 5.37 needs to be solved iteratively. Paphitis (2001) also gave an explicit formula of Θcr versus D*, which, however, somewhat deviates from Equation 5.37. Guo (2020) proposed the following function to represent the Shields diagram: cr 

1 1 1   (14 3) Recr  4 (2 3) Recr  18 18



(5.38)

This function has asymptotes at Θcr = 1/4 and 1/18 (≈ 0.0556) for lower and higher grain Reynolds numbers, respectively, and a minimum value of Θcr = 0.0333 in the hydraulically transitional regime. By applying cr  D*3 Re2*cr ,v Equation 5.38 is converted to a quartic equation that can be solved explicitly, but not as straightforwardly as other explicit relations introduced earlier. 5.5.3  Comparison of Original and Modified Shields Diagrams The original and modified Shields diagrams are compared in Figure 5.9 against the data compiled by Buffington and Montgomery (1997) from published incipient motion studies spanning eight decades. The data were limited to uniform and well-sorted sediment mixtures with D50/h ≤ 0.2 to minimize the effects of sediment nonuniformity and shallow submergence (see Sections 5.6 and 5.8.4). The bed-material size characteristics were derived from either the initial mixture or the bed surface layer. The critical shear stress values were determined using reference transport, visual observation, and flow competence approaches. These data are used here because they were carefully selected by an independent group rather than the

Incipient Motion of Sediments  143

Figure 5.9  Comparison of the Shields diagrams

developers of the compared Shields diagrams. Some data for platy particles and laminar flows are included for reference only. The threshold diagrams are plotted as Θcr–D* curves in Figure 5.9. The limitation of the original Shields diagram at low D* is presented by the curve of Soulsby (1997). The explicit formulas proposed by Brownlie (1981a) and van Rijn (1984a) provide similar curves and thus are not included. Equation 5.38, proposed by Guo (2020), is close to the original Shields curve approximated by Soulsby (1997), except for D* < 1. Equation 5.37 proposed by Paphitis (2001) works well in the rough turbulent regime but exhibits biases in smooth and transitional turbulent regimes. The biases are likely due to different data used, particularly those data collected in laminar flows but overlapping with the hydraulically smooth regime. The Shields diagrams modified by Yalin and Karahan (1979b) and Chien and Wan (1983) are represented in Figure 5.9 by Equations 5.34 and 5.36, suggested by da Silva and Bolisetti (2000) and Wu and Wang (1999), respectively. They agree well with the data, except in the range of about 20 < D* < 200. The following equation is derived by revising Equation 5.31 for better fitting with the data in Figure 5.9: cr 

0.19  0.05 1  e 0.0105 D 0.45  D0.75







(5.39)

Equation 5.39 is represented by the curve denoted as “present” in Figure 5.9. If the exponent −0.0105D* in the last term is replaced with −0.015D*, Equation 5.39 can substitute the segmented Equation 5.36 of Wu and Wang (1999) to represent the Shields diagram modified by Chien and Wan (1983). In general, Equation 5.39 can be used for the incipient motion of uniform and well-sorted bed sediments in typical turbulent river flows. Figure 5.9 incudes a theoretical Θcr – D* curve generated using Equation 5.11 with  yd 0.32k s (1  0.95 k s ), ξr = 1.26, ks = D, and χs given by Equation 3.40. Here, k s  u k s  . The function  yd 0.32k s (1  0.95 k s ) is reduced to yd = 0.32ks when k s  1, and to   yd u 0.3 when k s  1. Due to the lack of sufficient data, the values of yd and ξr given here * yd 

144  Incipient Motion of Sediments

are just one set of possible solutions. The values of yd = 0.32ks and ξr = 1.26 are between the limiting values corresponding to particles placed at positions 1 and 2 in Figure 5.2 (see Appendix 5.1). The theoretical curve gives Θcr = 0.05 at high D* values and approximately predicts the Θcr dip around D* = 15. It works well in hydraulically smooth and rough regimes, but has bias in the hydraulically transitional regime. The bias is likely caused by different exposure heights between sand and gravel particles, as perceived by Fenton and Abbott (1977), and can be corrected by adjusting yd and ξr. Buffington and Montgomery (1997) examined the data scatter in Figure 5.9. The data are stratified with different threshold definition approaches. The values determined using reference transport rates are generally higher than those obtained using visual observation and flow competence approaches. Other contributing factors include particle properties (particle shape, size distribution, cohesion, friction angle), bed configurations (particle packing, bed roughness, bed slope), flow conditions (turbulence intensity, sidewall effect, shallow water, antecedent flow history), and measurement methods (shear stress measurement, observation duration). Perret et al. (2023) obtained similar findings. Uncertainties resulting from these factors should be considered when Equation 5.39 is applied to a stable channel design. Equation 5.39 divided by a safety factor of 1.9 (or 1.6) contains 98% (or 90%) of the data points (including the platy particles) in Figure 5.9. Likewise, a safety factor of 2 was suggested by Parker (2005). Zanke (2003), Vollmer and Kleinhans (2007), and Ali and Dey (2016) developed hydrodynamic models to describe how some of the aforementioned factors affect the incipient motion of sediment. They reconstructed the Shields diagram as a series of Θcr – Re*cr curves, each of which corresponds to a constant value given to one of the affecting factors, such as relative bed roughness (Zanke, 2003; Vollmer and Kleinhans, 2007; Ali and Dey, 2016), turbulence intensity, friction angle (Zanke, 2003), particle exposure, and bed slope (Vollmer and Kleinhans, 2007). Details of these models are not introduced here. Instead, the general effects of several relatively well-understood factors are discussed in the next three sections. 5.6  Incipient Motion Thresholds of Nonuniform Sediments Interactions exist among the size classes of nonuniform bed sediments. Typically, coarse particles have a higher chance of exposure to the flow, and fine particles are more likely hidden by coarse particles, as illustrated in Figure 5.10. Hiding and exposure mechanisms significantly affect the entrainment and transport of nonuniform sediments. These effects are usually considered by introducing correction factors into the existing formulas of uniform sediments, as presented in this section.

Figure 5.10  Hiding and exposure of nonuniform sediment particles on the bed surface

Incipient Motion of Sediments  145

Formula of Egiazaroff (1965) and its Revisions Egiazaroff (1965) set ks = Dm, χs = 1, and yd = 0.63Dk in Equation 5.11 to determine the critical shear stress, τcrk, for the kth size class in the bed sediment mixture, yielding 2

   crk r   x ( s   ) Dk  5.75 log(19 Dk Dm ) 

(5.40)

where Dm is the arithmetic mean diameter of the sediment mixture and Dk is the diameter of the kth size class. By comparing Equation 5.40 with its reduced form for size Dm (i.e., setting Dk = Dm), Egiazaroff (1965) derived the following formula for the incipient motion of nonuniform bed material:  crk  log 19   crR  log(19 Dk Dm ) 

2



(5.41)

where Θcrk = τcrk/[(γs − γ)Dk], and ΘcrR = τcrR/[(γs − γ)Dm] with τcrR representing the critical shear stress corresponding to Dm. The term on the right-hand side of Equation 5.41 is a correction factor, which leads to Θcrk < ΘcrR for Dk /Dm > 1, and Θcrk > ΘcrR for Dk /Dm < 1. Thus, it takes into account the exposure effect on coarse particles and the hiding effect on fine particles. Considering that the fine particles sheltered by coarse particles start to move when the coarse particles move, Ashida and Michiue (1971, 1972) modified the Egiazaroff formula as 2 crk  log 19 log(19 Dk Dm )   crR  0.85 Dm Dk

Dk Dm  0.4 Dk Dm  0.4



(5.42)

and Hayashi et al. (1980) proposed a similar modification: 2 crk  log 8 log(8 Dk Dm )   crR  Dm Dk

Dk Dm  1 Dk Dm  1



(5.43)

The critical shear stress calculated with Equations 5.42 and 5.43 does not change with sediment size for particles finer than 0.4Dm and Dm, respectively. Egiazaroff (1965) used ΘcrR = 0.06, which is relatively large. Ashida and Michiue (1971, 1972) suggested ΘcrR = 0.05. Misri et al. (1984) found ΘcrR to fall in the range of 0.023 – 0.0303. Formula of Parker et al. (1982) and Others Parker et al. (1982), Andrew (1983), and Buscombe and Conley (2012), among others, proposed the following correction factor based on nondimensional particle size: crk  Dk    crR  DR 

 m



(5.44)

146  Incipient Motion of Sediments

where DR is a reference particle diameter, ΘcrR is the critical or reference Shields number corresponding to DR, and mτ is an empirical exponent. Parker et al. (1982) and Andrew (1983) used the subsurface bed material D50 as the reference diameter DR in Equation 5.44. Surface and subsurface bed materials may have significantly different size distributions, particularly when armoring exists on the bed surface. Because the surface bed material directly affects flow and sediment transport, it is likely more reasonable to use the surface bed material D50 as DR. Moreover, Parker et al. (1982) and Andrew (1983) set ΘcrR at large values, such as 0.0875 and 0.083, which represent the upper limit of the critical Shields number shown in Figures 5.7 – 5.9, and thus interpreted ΘcrR as a reference Shields number. For relatively well-graded gravel mixtures, Petit (1994) found ΘcrR ≈ 0.05, which agrees better with the Shields diagram. The exponent mτ characterizes the size-selective entrainment of a nonuniform sediment mixture. A value of mτ = 1 indicates that all particles start to move at the same threshold, and mτ = 0 indicates that all particles start to move independently without interactions. Petit (1994) suggested mτ ≈ 0.7. Buffington and Montgomery (1997) reviewed the literature data and found mτ to fall in the range of 0.29 – 1.0. According to Kuhnle et al. (1996), mτ depends on the size distribution of the sediment mixture (see Section 9.2). Wilcock and Crowe (2003) and Buscombe and Conley (2012) used Dm as the reference diameter DR in Equation 5.44. Wilcock and Crowe (2003) proposed mτ as a function of Dk /Dm, which is described in Section 7.5. The following function was proposed by Buscombe and Conley (2012):

 u    m  1  1.04  *   m   u*crR   Dm 

(5.45)

where u*crR is the critical shear velocity for DR; and σm is the arithmetic standard deviation of 2

particle sizes, defined  by  m

N

 (D k 1

k

 Dm ) 2 pk , in which pk is the fraction of the kth size class

and N is the total number of size classes in the sediment mixture. Equation 5.45 is applied with the constraint of mτ ≥ 0 for Dk > Dm. Day (1980), Proffitt and Sutherland (1983), and Patel et al. (2013) defined several reference diameters different from D50 and Dm. The correction factors of Day (1980) and Proffitt and Sutherland (1983) are directly used for the total-load transport rate introduced in Section 9.2. Patel et al. (2013) used D R = D gσg, where D g is the geometric mean diameter and σg is the geometric standard deviation. They set ΘcrR as a function of Kramer’s uniformity coefficient M: ΘcrR = 0.0456M 0.55

(5.46)

and mτ as  0.96 m   0.37 g 2.67e

 g  2.85  g  2.85



(5.47)

Incipient Motion of Sediments  147

Formula of Qin (1980) and Others Qin (1980) modified Equation 5.22 as follows for the critical average velocity Ucrk (m/s) of the kth size class of the bed sediment mixture: 1/ 6

 h   D    U crk 0.786 1  2.5 d m  s gDk   Dk     D90 



(5.48)

where ψd is related to ηd = D60/D10 as 0.6 d  2  d   0.7606  0.6801  d  2.235   d  2



(5.49)

When Dk = Dm, 1 + 2.5ψd Dm / Dk = 1 + 2.5ψd and Equation 5.48 is reduced to Equation 5.22 for uniform sediment, with ζc = 1.24. For Dk < Dm, 1 + 2.5ψd Dm / Dk > 1 + 2.5ψd, and for Dk > Dm, 1 + 2.5ψd Dm /Dk < 1 + 2.5ψd. Thus, Equation 5.48 can consider the hiding-exposure effects. Similar formulas were developed by Xu et al. (2008): 1/ 2

  0.431 U crk   15 / 7 1/ 7  . ( ) . ( ) 0 02 D D  0 22 D D k m k m  

1/ 6

 h  s   gDk     Dk 



(5.50)



(5.51)

and Chen and Xie (1988): 1/ 8

U crk

1/ 7

 D   D75   0.56  k      Dm   D25 

log[11.1h ( x Dm )] s   gDk  log[15.1Dk ( x Dm )]

where ϕx = 2. Formula of Wu et al. (2000) Consider a mixture of bed particles with nonuniform diameters, as shown in Figure 5.10. Particle position on a bed surface can be represented by the exposure height Δe. The particle is in an exposed state if Δe > 0 and in a hidden state if Δe < 0. For a particle with diameter Dk in the bed surface layer, Δe is in the range between −Dj and Dk. Here, Dj is the diameter of the upstream particle. Because the particles are randomly distributed on the bed, Δe is a random variable. It is assumed to have a uniform probability density function (Wu et al., 2000): 1 ( Dk  D j )  D j   e  Dk f ( e )   0 otherwise 



(5.52)

The probability of particles of Dj staying in front of particles of Dk is assumed to be the fraction of particles of Dj in the bed material, denoted as pbj. Therefore, the probabilities

148  Incipient Motion of Sediments

of particles of Dk hidden and exposed due to particles of Dj are obtained from Equation 5.52 as phk , j  pbj

Dj Dk  D j

pek , j  pbj

,

Dk  Dk  D j

(5.53)

The total hiding and exposure probabilities, phk and pek, of particles of Dk are then obtained by summing Equation 5.53 over all the N size classes: N

Dj

j 1

Dk  D j

phk   pbj

,

N

pek   pbj j 1

Dk  Dk  D j

(5.54)

A relationship of phk + pek = 1 exists. For uniform sediment particles, phk = pek = 0.5, indicating equal hiding and exposure probabilities. In a nonuniform mixture, phk ≤ pek for coarse particles, and phk ≥ pek for fine particles. This can be demonstrated with a simple example. For a mixture of two size classes with D 1 = 1 mm, p b1 = 0.4, D 2 = 5 mm, and p b2 = 0.6, one can obtain p h1 = 0.7 > pe1 = 0.3 and ph2 = 0.3667 < pe2 = 0.63333. It is shown that coarse particles are more likely exposed and fine particles are more likely hidden. Based on the hiding and exposure probabilities, the following correction factor is introduced for the critical Shields number (Wu et al., 2000): crk  pek    crR  phk 

 mw



(5.55)

where ΘcrR is the reference Shields number and mw is an empirical exponent. Their values were calibrated as ΘcrR = 0.03 and mw = 0.6 using laboratory and field measurement data of sand and gravel mixtures. In each test case, the measured bed shear stress was set as the critical shear stress of the size class whose bed-load transport rate met the reference threshold defined in Equation 5.20 by Parker et al. (1982). The agreement between measurements and predictions is generally good, as shown in Figure 5.11a.

Figure 5.11 Comparison of measured and calculated critical shear stresses (from Wu et al., 2000)

Incipient Motion of Sediments  149

Figure 5.11b shows that the formula of Egiazaroff (1965) significantly over-predicts the same measurements, particularly at lower critical shear stresses. The value 0.06 of ΘcrR used by Egiazaroff (1965) is relatively large. The formula of Hayashi et al. (1980) gives somehow improved results, but still has large errors at lower critical shear stresses (Wu et al., 2000). The advantage of Equation 5.55 over Equations 5.41–5.44 is its probability concept considering the particle exposure and size distribution of bed material. 5.7 Incipient Motion Thresholds of Sediment Particles on a Steep Slope For a sediment particle on a sloped bed or bank, incipient motion is driven not only by the flow, but also by the component of gravity along the slope. In the case of incipient sliding on the slope shown in Figure 5.12, the particle experiences the flow tractive force F, the submerged weight Ws, and the friction force FR. The driving and resistance forces can be expressed as  FT

F 2  Ws sin    2 FWs sin  cos(90o   ) 2



(5.56)

FR = Ws cos β tan ϕi

(5.57)

where β is the slope angle with positive values for downslope, α is the angle between the flow direction and the horizontal line of the slope, and ϕi is the friction angle. The force equilibrium at incipient sliding, i.e., FT = FR, yields F Ws sin  sin  

Ws cos  tan i   Ws sin  cos   2

2



(5.58)

In the case of a particle sliding on a horizontal bed (α and β = 0°), the tractive force, denoted as Fˊ, has the following relationship: Fˊ = Ws tan ϕi

Figure 5.12  Forces on a particle over a slope

(5.59)

150  Incipient Motion of Sediments

Comparing Equations 5.58 and 5.59 leads to F sin  sin     cos 2  F tan i

 sin  cos      tan i 

2



(5.60)

The tractive forces on the particle are associated with the bed shear stresses by F = a3D2τcrβ and Fˊ = a3 D2τcr. Here, a3 D2 is the bed area occupied by the particle, τcrβ is the critical shear stress on the slope, and τcr is the critical shear stress on the horizontal bed. Therefore, Equation 5.60 leads to (Brooks, 1963)

 cr   cr

sin  sin  sin 2  cos 2      cos 2   tan i tan 2 i

(5.61)

By setting F  U cr2  and F   U cr2 , Equation 5.60 leads to an equivalent formula for Ucrβ /Ucr proposed by B.A. Pyshkin (see Xie, 1981). Here, Ucrβ is the critical average velocity for the incipient motion of sediment particles on the slope. When α = 0°, i.e., for a particle on a sloped bank with a horizontal channel slope, Equation 5.61 is reduced to Lane’s (1955) relationship:

 cr   cr

 cos  1 

tan 2  2

tan i

 1

sin 2  sin 2 i



(5.62)

When α = 90°, i.e., for a particle on a channel slope with an angle of β, Equation 5.61 is reduced to

 cr   cr



sin i    sin i



(5.63)

Equation 5.63 was first introduced by Armin Schoklitsch in the early 1900s and has been used by many investigators, including Stevens et al. (1976), Smart (1984), Chiew and Parker (1994), Damgaard et al. (1997), and Lau and Engel (1999). Note that for a particle on a slope of repose β = ϕi, Equation 5.63 gives τcrβ=0. This implies that any downslope flow can cause sediment to move on the slope of repose. By combining Equations 5.62 and 5.63, van Rijn (1993) suggested the following approximate method to determine τcrβ on a general slope: τcrβ = k1k2τcr

(5.64)

where k1 = sin(ϕi – βL) / sin ϕi describes the correction factor in the flow direction and  k2 cos T 1  tan 2 T tan 2 i describes the correction factor in the direction normal to the flow. Here, βL and βT are the slope angles in the streamwise and sideward directions, respectively.

Incipient Motion of Sediments  151

5.8 Other Factors Affecting the Incipient Motion of Sediment Particles 5.8.1  Effect of Particle Exposure on Incipient Motion Fenton and Abbott (1977) experimentally demonstrated how particle exposure (protrusion) affects the incipient motion of individual particles. They used spheres and rounded gravels. In several experimental runs, a test particle was placed on a hexagonal array of particles (highest   0.01 was observed. exposure, position 2 in Figure 5.2), and a critical Shields number of  cr In most of the other runs, the particle was placed in a hole and pushed up by a rod to various exposure heights on the bed (Figure 5.13). For a particle on a co-planar bed (zero exposure,  was in the range of 0.08 – 0.24. In general,   position 1 in Figure 5.2), the measured  cr cr decreased as the exposure height Δe increased. The measured data in Figure 5.13 can be fitted with (R2 = 0.82)   0.146e 3.4 e  cr

D



(5.65)

 denotes the instantaneous critical Shields number of specific particles on the Note that  cr bed surface and is different from the statistical mean Θcr defined by the Shields diagram for a sediment mixture on the bed. Equation 5.65 may have errors for normally packed bed particles since the particle placed in the hole has different pivot angles with the neighboring particles. Specifically, a particle with negative exposure on a normally packed bed is usually sheltered by the overlying particles and thus is more difficult to thrust into motion than the particle placed in the hole. Nevertheless, Equation 5.65 reveals the significant effect of particle exposure on the incipient motion. According to Perret et al. (2018), a packed gravel bed has lower particle exposure on average and, in turn, higher critical shear stress than a loose gravel bed. The particle exposure height is related to the initial particle packing on a flume bed and changed by flow work during the

Figure 5.13 Critical Shields number as a function of particle exposure (data from Fenton and Abbott, 1977)

152  Incipient Motion of Sediments

experiment. Thus, it is more adequate to measure the incipient motion threshold after the sediment bed adjusts its configuration for some time, e.g., 1 hr (Vanoni, 1964). The effect of particle position on the critical shear stress can be considered using Equation 5.11 with the parameters yd, CD, CL, kW, kD, and kL varying with the particle exposure height.  is about 0.18 and This is described in detail in Appendix 5.1. The results there show that  cr 0.01 for particles to start rolling at positions 1 and 2 in Figure 5.2, respectively. These values  agree well with the data measured in the aforementioned experiments of Fenton and Abof  cr bott (1977). Furthermore, Vollmer and Kleinhans (2007) developed a sophisticated model to  and Δ shown quantify the variations of these parameters and reproduce the relationship of  e cr in Figure 5.13. In an alternative approach, Dwivedi et al. (2012) proposed the following Froude-type parameter to consider the effect of particle exposure on the incipient motion: FmS 

| u   u u |2yd

( s   1) gD



(5.66)

where u is the time-averaged velocity, σu is the standard deviation of velocity, αu is an empirical coefficient, and yd is the height at which the near-bed velocity is measured. The theoretical bed is set at 0.2D below the apexes of the surface particles, and the reference flow velocity is located at 0.15D above the apex of the test particle, yielding yd = 0.15D + 0.2D + Δe = 0.35D + Δe. In addition, the effect of large-scale turbulence events, such as bursts and sweeps, on incipient motion can be considered in Equation 5.66 by setting u + αuσu as the reference flow velocity. The coefficient αu is related to the particle exposure via αu = max[( −16.5Δe /D + 3.5), 0]. The velocity u is determined with the log law Equation 3.37, and the standard deviation σu is determined with Equation 3.19 or a similar method. Note that FmS here is not comparable with the critical Shields number Θcr but about 100 times Θcr in the order of magnitude. More work is needed to make Equation 5.66 applicable in practice. 5.8.2  Effect of Particle Shape on Incipient Motion The incipient motion of a non-spherical bed particle is affected by its shape orientation along the flow direction. This was experimentally investigated by Carling et al. (1992) and Gogus and Defne (2005). The studied particles included cubes, rectangular prisms, spheres, ellipses, discs, and irregular particles lying on flume beds. The results show that if the particle is longer in the streamwise direction, it is more difficult to start moving due to low drag and high resistance. On a sediment bed, the particles with the shortest (c) axis aligned with the flow likely start moving at lower thresholds. The particles with the longest (a) or intermediate (b) axis aligning with flow may adjust their local positions first by vibrating, swiveling, and sliding to a transverse attitude, and then roll out (Carling et al., 1992). Sediment particles on a flow-worked bed tend to adjust their orientations such that the a-axis is along the streamwise direction (Penna et al., 2021), and a relatively stable surface layer is formed against disturbance, yielding higher inception thresholds. Unlike round particles, individual platy or flat particles (e.g., slates, flakes) may have relatively low thresholds of incipient motion (see Figure 5.9), according to Mantz (1977). This is likely due to the elevated lift force. However, platy particles tend to form imbricate cluster

Incipient Motion of Sediments  153

Figure 5.14 Imbricate cluster structure of flat particles on a bed surface (from Wang and Dittrich, 1999)

structures on the bed surface, whereby one particle rests against another with one end tilting up in the streamwise direction (Figure 5.14). The imbrication offers an armoring effect, so the particles exhibit high resistance to incipient motion (Lane and Carlson, 1954; Brayshaw et al., 1983; Gomez, 1994; Wang and Dittrich, 1999). Wang and Dittrich (1999) reported that when the imbricate cluster structure was developed, the critical Shields parameter for the incipient motion of flat particles was 65% larger than that for round particles with the same nominal diameter. Likewise, particle angularity affects incipient motion via two opposite mechanisms. More angular particles experience higher drag but greater friction and have a higher chance of interlocking. Joshi et al. (2017) measured the critical shear stress of maerl biogenic gravel particles collected from beach, intertidal, and open marine environments in Galway Bay, west of Ireland. The maerl particles were convex, branched, and rough. The highly irregular shapes allowed the particles to have higher drag and start moving more easily than quartz grains with the same nominal diameter. However, when the maerl particles were interlocked on the bed, they exhibited higher resistance to movement. This interlocking effect was also observed by Li and Komar (1986), among others. Smith (2003) experimentally observed that sand particles with an average Corey shape factor of 0.56 collected from Oahu, Hawaii, had higher critical shear stress in the hydraulically smooth regime and lower critical shear stress in the hydraulically rough regime than common, naturallyworn sediment particles. This might be because the hydrodynamic force on non-spherical particles behaved differently in the hydraulically smooth and rough regimes, or because the fine and coarse particles had different shapes. In summary, incipient motion thresholds exhibit significant uncertainty and even inconsistency due to the randomness of particle shape and orientation, the formation of imbrication and interlocking on sediment beds, and the different behaviors of hydrodynamic forces in different flow regimes. The effect of particle shape on incipient motion has not been fully understood, and most of the knowledge is just descriptive. Caution should be exercised when dealing with this in practice. 5.8.3  Effect of Bed Roughness on Incipient Motion Preexisting bed features, such as large immobile particles, clusters, and sand ripples, significantly affect particle incipient motion. The incipient motion Equation 5.11 indicates that the critical shear stress τcr increases as the bed roughness height ks increases. This is qualitatively correct, although the parameters yd and ξr may vary as the pivoting point and particle exposure vary with ks. This is also confirmed by the hydrodynamic models of Zanke (2003), Vollmer and Kleinhans (2007), and Ali and Dey (2016), as well as by two simpler approaches introduced herein.

154  Incipient Motion of Sediments

Wiberg and Smith (1987) proposed the following formula for the critical shear stress of particle incipient sliding using Equation 5.6 with the slide angle αsl = 0:

 cr 4  cos  tan i  sin   1  ( s   ) D 3 (CD  CL tan i ) f ub2



(5.67)

where fub is a function relating the near-bed velocity with the shear velocity, such as the log law used in Equation 5.11. Then, Wiberg and Smith (1987) considered the effect of heterogeneous bed roughness on the critical shear stress using the following formula for friction angle:  D k s  y  i  cos 1    D ks  1 



(5.68)

where y* is the average level of the entraining particle bottom, depending on the particle sphericity and roundness, set as y* = −0.02. Because the slide angle αsl is set as zero, its effect is considered through ϕi in Equation 5.68. Equation 5.68 was tested by using the data of Miller and Byrne (1966) with naturally sorted sediments. The τcr values calculated with Equations 5.67 and 5.68 for uniformly sized sediments (D/ks = 1) correspond closely to the Shields diagram. The critical Shields number of the moving particles decreases when D/ks > 1 and increases when D/ks < 1. When large form features exist on the bed, only the bed shear stress acting on the grains is effective to incipient motion. Using the hydraulic radius partition method described in Section 6.4, the critical Shields number based on the total bed shear stress, ΘcrT, can be calculated with n crT     n 

3/ 2

cr 

(5.69)

where Θcr is the critical Shields number on a flat bed, determined with the Shields diagram, as in Equation 5.39; n is the Manning coefficient for the total bed roughness; and n′ is the Manning coefficient for the grain roughness, determined with n′ = D1/6/21.5 (Strickler, 1923) or a similar formula introduced in Section 6.4. Both Equations 5.67 and 5.69 suggest that critical shear stress increases with bed roughness. However, exceptions may exist around large-scale roughness elements (e.g., boulders, vegetation stems), which create local flow features (e.g., deflected flows, vortices) inducing the nearby bed particles to start moving earlier. 5.8.4  Effect of Particle Submergence on Incipient Motion Based on their compiled literature data of uniform and well-sorted sediment mixtures, Buffington and Montgomery (1997) illustrated in Figure 5.15 the variation of the critical Shields number Θcr50 with the relative particle size D50 / h. At D50 / h < 0.01, Θcr50 increases with decreasing D50 / h due to the effect of the viscous sublayer in the hydraulically smooth regime. At D50 / h > 0.01, Θcr50 increases with increasing D50 / h, indicating a more stable bed at a lower submergence of bed particles or in shallower water. Shvidchenko and Pender (2000) reported a similar finding.

Incipient Motion of Sediments  155

Figure 5.15 Variation of Θ cr50 with relative size D 50 /  h (from Buffington and Montgomery, 1997)

According to Mueller et al. (2005), Lamb et al. (2008), and Perret et al. (2023), the critical Shields number for sediment incipient motion increases with bed slope S, particularly in steep streams. The empirical formula of Lamb et al. (2008) is written as Θcr = 0.15S

0.25



(5.70)

This slope dependence of Θcr is contradictory to the standard models described in Section 5.7 that predict reduced stability with increasing slope due to the streamwise component of gravity. This is likely due to an increase in drag coefficient alongside a decrease in lift coefficient caused by the coincident increase in the relative roughness D50 / h (i.e., decrease in h) as the bed slope increases (Lamb et al., 2017). Another reason is that immobile boulders, particle clusters, and bed forms in steep streams increase the overall resistance to the flow, but the increased form resistance is not directly associated with particle entrainment (Ferguson, 2012). Equation 5.69 can help to explain this. Armanini and Gregoretti (2005) and Gregoretti (2008) proposed theoretical formulas to determine the incipient motion of sediment particle sliding on steep slopes at low and partial submergences. They considered the low-submergence flow velocity profile, the seepage flow in the coarse sediment bed, and the projection area blocked by upstream particles. Interested readers can find further details in these references. By substituting the Manning-Strickler friction factor (Equation 6.87) modified by Cheng (2017) for large bed roughness into Equation 5.67, Cheng et al. (2018) proposed the following critical Shields number for sediment incipient sliding on a steep slope under shallow submergence: 2

1/ 3

ks   ks   cr  0.013 d 1  0.75 h   D  2   50  tan i 1  tan   tan i  tan 



(5.71)

where ζd = 4 tan ϕi/[3(CD + CL tan ϕi)] calibrated as about 2.1. By using k s   g2.33 D50 and S = tan β = Θcr(ρs /ρ − 1)D50 / h, this equation can be transformed to cr  0.013 d

tan i  cr (  s   1) D50 h tan i 1  [cr (  s   1) D50

2

 2.33 D50  0.78 1  0.75 g  g 2  h  h]



(5.72)

156  Incipient Motion of Sediments

or an alternative formula in terms of bed slope S. Here, σg is the geometric standard deviation of bed sediment. The tanβ term in Equation 5.71 is similar to that in Equation 5.63 for the effect of steep slope, showing that Θcr decreases as the bed slope tanβ increases. The ks / h term considers the effect of large bed roughness or shallow submergence, yielding that Θcr increases as ks / h increases. The overall variation depends on which of these two terms is dominant. According to Cheng et al. (2018), Equation 5.72 can reproduce the increasing trend of Θcr at high D50 / h shown in Figure 5.15. 5.8.5  Effect of Turbulence on Incipient Motion The critical average velocity and shear stress presented in Sections 5.4 and 5.5 are the timeaveraged flow conditions at which the bed particles start to move. As described in Section 3.7.3, among turbulence eddies, only the large-scale sweep-ejection events induce particle entrainment. The mean flow condition is represented by ub, and the sweep-ejection events are represented by ub + αuσu in analogy to Equation 5.66. Thus, the mean shear stress τcr and the effective shear stress τcr,eff created by the sweep-ejection events at incipient motion are correlated as

 cr ,eff  cr

 u   u u  b ub 

2

  u    1   u  ub   

2



(5.73)

Equation 5.73 indicates that τcr,eff is greater than the critical shear stress τcr given by the Shields diagram. The coefficient αu is similar to that in Equation 5.66, but further investigation is needed because Equation 5.73 has a different application. Zanke (2003) used Equation 5.73 to account for the effect of turbulence on incipient motion by defining τcr,eff = 0.7 tan ϕi as the critical shear stress without turbulence and τcr as the critical shear stress with turbulence. Furthermore, using Equation 5.67 and FL / FD = CL / CD, Zanke (2003) obtained 2

     F  cr 0.7(tan i ) 1   u u  1  L tan i   u F b   D  

1



(5.74)

In addition, Zanke (2003) considered the effect of cohesion through the bulk friction angle ϕi of fine-grained particles. Zanke (2003) reconstructed the Shields diagram by using Equation 5.74 and relating σu / ub and FL / FD to the turbulence intensity. For uniform flows, σu is closely related to ub, and in turn the ratio τcr,eff /τcr in Equation 5.73 is well defined; thus, τcr is used as a nominal critical shear stress for sediment incipient motion. In nonuniform flows, such as the rapidly-varied flows around bridge piers and abutments, σu is not necessarily related to ub, and thus ub cannot represent the effect of the locally-induced turbulence on sediment entrainment. Instead, ub + αuσu or a similar composite parameter is often used to account for the contributions of both the mean and fluctuating motions. This is discussed in Section 9.5.2 for sediment entrainment under nonuniform flows. In stochastic approaches, the effect of turbulence on sediment entrainment is considered through the entrainment probability concept, which treats the acting bed shear stress or flow velocity as a random variable, as described in Section 5.9.

Incipient Motion of Sediments  157

5.8.6  Effect of Impulse Duration on Incipient Motion The incipient motion thresholds introduced in Section 5.2 consider only the magnitude of the hydrodynamic force acting on individual particles. In reality, a short-lived, instantaneous force may not result in complete particle entrainment because the particle needs enough time to move out of its resting position. Diplas et al. (2008) proposed an approach to consider both the magnitude and duration of the force, by defining “impulse” as the product of the hydrodynamic force F(t) and the duration T in which the critical resisting force Fcr is exceeded (Figure 5.16):  I



ti  T

ti

F (t )dt , F (t )  Fcr 

(5.75)

The critical force Fcr can be determined using the force or moment analyses described in Section 5.2 for particle rolling, sliding, and lifting. Potential entrainment events can be identified by comparing the measured instantaneous force F and the critical force Fcr. However, complete removal of the exposed particle from its local pocket occurs only when an impulse exceeds a critical level Icr. For particle rolling, Valyrakis et al. (2010) proposed Icr = FDTroll, with Troll 

 FD cos( 0   )  Ws sin  0  Larm ( 75  s   K m )V  arcsinh  2W    FD cos( 0   )  Ws sin  0 FD sin( 0   )  Ws cos  0  

(5.76)

where ξθ = cos θ0 sin β + (1 – sin θ0)(cos β – ρ / ρs); FD and Troll are the average drag force and duration of the impulse event for particle rolling, respectively; Larm is the lever arm, defined as the distance between the gravity center and the rotation point of the particle; θ0 is the pivoting angle formed between the horizontal and the lever arm; β is the bed slope angle; V, W, and Ws are the volume, weight, and submerged weight of the rolling particle, respectively; and Km is the added mass coefficient. The lift force is ignored in Equation 5.76 for simplicity. Valyrakis et al. (2010) also proposed a critical impulse level for particle lifting. The impulse concept provides physical insight into particle incipient motion. However, it is difficult to use in practice compared with the conventional approaches based on bed shear stress and average flow velocity.

Figure 5.16 Sketch of a temporal history of hydrodynamic force F(t) acting on a particle (Fcr is the critical force level; impulse event i—incomplete, i+1—complete entrainment) (after Valyrakis et al., 2010)

158  Incipient Motion of Sediments

5.9  Probabilities of Sediment Entrainment The entrainment probability is also called pickup probability. Following the general concept briefly introduced in Section 5.3.1, available models based on lift force, flow velocity, and bed shear stress to calculate the entrainment probability of uniform sediments are presented in this section. In addition, the probabilities of particle rolling and lifting, as well as the entrainment probability of nonuniform sediments, are discussed briefly. 5.9.1  Entrainment Probability Formulas Based on Lift Force Einstein (1942, 1950) assumed lift force as the main driving force for sediment entrainment and defined the entrainment probability pE as the probability of the lift force being larger than the submerged weight, p(FL > Ws). The submerged weight and the lift force are expressed in Equations 4.1 and 5.2, respectively. By using the log law of flow velocity, the lift force can be determined as C D 2  FL  L 8

   s yd 5.75u log  30 ks  

2

   1   L    

(5.77)

where ηL is the relative fluctuation of lift force with respect to the mean value. Entrainment indicates Ws 1  B   1 FL  L  1  L 0



(5.78)

where B* = 4/{3CL[5.75log(30χsyd / ks)]2 σL0},   ( s   ) D  b is the reciprocal of the Shields number, ηL* = ηL / σL0, and σL0 is the standard deviation of ηL. Considering that the fluctuation of lift force is attributed to the fluctuation of flow velocity that can be positive or negative, Einstein (1942, 1950) adjusted Equation 5.78 as |ηL* + 1/σL0| > B*Ψ

(5.79)

and then derived the following entrainment probability by using the normal error distribution for the fluctuation of lift force observed by Einstein and El-Samni (1949): pE  1 

1

 

B  1/  L 0

 B  1/  L 0

2

e  t dt



(5.80)

Einstein (1950) calibrated σL0 = 0.5 and B* = 0.143 using the measurements of Einstein and El-Samni (1949) and the bed-load data described in Section 7.4.5. However, with these coefficients, Equation 5.80 performs poorly against the data of Fernandez Luque (1974) and the Fort Collins data cited by Guy et al. (1966), as shown in Figure 5.17. This may be due to different data used but is more likely because the model has defects.

Incipient Motion of Sediments  159

Figure 5.17 Comparison of measured and predicted entrainment probabilities for sediment particles(Lift force based model — Equation 5.80; velocity based model — Equation 5.83; velocity based (revised) model — Equation 5.84; bivariate velocity based model — Equation 5.111; bed shear stress based model — Equations 5.95 and 5.107, which give the same curve)

The adjustment from Equation 5.78 to 5.79 is incorrect since a particle cannot be entrained by only a downward lift force (Wang et al., 2008b). In reality, this adjustment does not cause a significant error because the entrainment due to negative fluctuating velocity is very small. A rather important defect is that the model ignores the effect of drag force and assumes lifting as the entrainment mode. In addition, although the normal error distribution for the lift force was positively confirmed by the experiments of Chen and Clyde (1972) and Dwivedi et al. (2011), it is questionable because FL  ub2 vand ũb usually follows the normal distribution (He and Han, 1982; Papanicolaou et al., 2002). 5.9.2  Entrainment Probability Formulas Based on Flow Velocity The entrainment probability model based on the normal distribution of the near-bed flow velocity is expressed with Equation 5.14. The integral in Equation 5.14 can be treated using the following approximation suggested by Guo (1990): 1

 2

a

0

2

e  x 2 dx  0.5

 2a 2  a 1  exp    a   

for a ≠ 0

(5.81)

The error associated with this approximation is less than 0.7%. Thus, Equation 5.14 is converted to  2 2  2  u u    2 u u   u u b (5.82) pE  1  0.5 crm b 1  exp    crm b    0.5 1  exp    crm   ucrm  ub     u       u  

160  Incipient Motion of Sediments

Cheng and Chiew (1998) considered the incipient lifting of a particle lying on closely packed particles (i.e., at position 2 in Figure 5.2). They obtained ūb = 5.52u* using Equation 3.37 with ks = 2D, χs = 1.0, and yd = 0.6D. They set σu ≈ 2u* as suggested by Kironoto and Graf (1994), and determined ucrm using Equation 5.8. By inserting these parameters into Equation 5.82, Cheng and Chiew (1998) derived  pE  1  0.5

2 2         0.46 0.46     2.2    2.2   0.5 1  exp   1  exp         CL   CL         (5.83)

0.21  CL 0.21  CL

where    u*2 [( s   ) D] . The lift coefficient CL in Equation 5.83 is treated as a calibrated parameter. It varies between 0.1 and 0.4 and has a default value of 0.25, according to Cheng and Chiew (1998). On average, these calibrated values are above the measured values of CL shown in Table 5.1, even though the model is based on the particle at position 2 in Figure 5.2, which likely has a high entrainment probability. This defect is because lifting is considered the entrainment mode. The entrainment probability based on the rolling mode is herein derived as follows by substituting Equation 3.37, Equation 5.4, and σu ≈ 2u* into Equation 5.82:

p E

 0.46  Y 0.5  0.5

R

K R



0.46  YR K R

2     0.46   YR    1  exp       K R    

(5.84)

where KR = (kDCD + kLCL)/kW, and YR = 2.3log(30yd χs/ks). Note that the portion of the entrainment probability due to negative velocity fluctuation is herein neglected, since it is about 1% of the total entrainment probability and in the same order of magnitude as the error caused by the approximation in Equation 5.81. Equation 5.84 has only two controlling parameters: KR and YR. They are calibrated as KR = 0.38 and YR = 1.5 by trial and error using the data in Figure 5.17. YR = 1.5 corresponds to yd = 0.15ks /χs, and KR = 0.38 is a combination of the drag and lift coefficients given by Einstein and El-Samni (1949), Wang and Fontijn (1993), and Dwivedi et al. (2010) listed in Table 5.1 for particles in a co-planar bed, i.e., position 1 in Figure 5.2. The curve generated by Equation 5.84 is denoted with “velocity based (revised)” in Figure 5.17. The comparison shows that Equation 5.84 fits better with the measured data than Equation 5.83. This is partly because Equation 5.84 has one more parameter to tune. Overall, Equation 5.84 is more general than Equation 5.83 since it is based on the rolling mode of particles driven by both drag and lift forces. Equation 5.84 gives an entrainment probability of 1.1% at Θ = 0.052. Similarly, Sun and Donahue (2000) defined the entrainment probability pE as the probability of force moments that cause particles to roll exceeding those that keep the particles at rest, and proposed the following formula: pE  1 

1

 2

2.7 ( 0.0822  1)

2.7 ( 0.0822  1)

2

e t 2 dt



(5.85)

Incipient Motion of Sediments  161

Equation 5.85 can be approximated as Equation 5.83 with CL = 0.55. This CL value is significantly above the range of 0.1 – 0.4 given by Cheng and Chiew (1998). Thus, Equation 5.85 likely over-predicts the entrainment probability. This may be offset by using the grain shear stress in the calculation of the reciprocal Shields number (see Section 7.5). Equations 5.83 – 5.85 are simple, but mixed opinions exist in the literature regarding the normal distribution of the streamwise flow velocity ũb. Bose and Dey (2013) proposed a lift-based entrainment probability model using the two-sided exponential distribution of ũb, but no improvement is gained in comparison with the model of Cheng and Chiew (1998). Wu and Lin (2002) used a lognormal distribution, and furthermore, Wu and Yang (2004a) used the following fourth-order Gram-Charlier-type probability density function of ũb to account for the higherorder correlations associated with turbulent bursting: f GC 4  uˆ  

exp( uˆ2 2)  Su 3 Ku  3 4  2 1  3!  uˆ  3uˆ   4 !  uˆ  6uˆ  3   2  

(5.86)

 ub  ub , Su  ub3  u3 describes the skewness of ub , K u  ub4  u4 where uˆ  ub  u , u b describes the kurtosis of ub , and 〈 〉 denotes ensemble mean. Equation 5.86 is reduced to the normal distribution if Su = 0 and Ku = 3. Wu and Yang (2004a) related the high-order moments to the roughness shear Reynolds number k s ( k s u  ) as follows:

 u u*  max[0.187 ln(k s )  2.93, 2.14] Su  min[0.102 ln( k s ), 0.43]





 K u min[0.136 ln(k s )  2.3, 2.88] 

(5.87) (5.88) (5.89)

Equations 5.87 – 5.89 indicate that the high-order moments vary with k s in the hydraulically smooth and transitional regimes (k s  70 ) but are constant in the hydraulically rough regime ( k s  70). According to Wu and Yang (2004a), Equation 5.86 is more adequate than the normal and lognormal distributions for ũb. However, this is still under debate. The measurements of Tregnaghi et al. (2012) indicate that the normal distribution is adequate for the streamwise velocity fluctuation, though not for the vertical velocity fluctuation. Note that the contribution of vertical flow fluctuation is ignored in all the aforementioned velocity-based entrainment probability models. Further investigations are needed for a better understanding of the effects of both the streamwise and vertical flow fluctuations, as well as their correlations (i.e., Reynolds shear stress), on sediment entrainment probability. 5.9.3  Entrainment Probability Formulas Based on Bed Shear Stress Paintal (1971b), Gessler (1970), and van Rijn (1993) applied a normal probability distribution for the instantaneous bed shear stress b . This was supported by the experiments of Chen and Clyde (1972). However, many experiments have revealed that b tends to be positively skewed (e.g., Grass, 1970; Blinco and Simons, 1974; Lopez and Garcia, 2001). Papanicolaou et al.

162  Incipient Motion of Sediments

(2002) suggested the χ2 distribution for b by considering b  ub2 and the normal distribution of ũb. Similarly, Hofland and Battjes (2006) developed the following probability distribution based on b   t ub ub to consider the direction of velocity:

  b 

1

 1 exp     2 2 2 t  b



b  t  sgn (b ) t

  2



(5.90)

where δt = ūb /σu is the reciprocal of turbulence intensity, and αt is assumed to be constant for convenience. The χ2 distribution and Equation 5.90 have complex formulations for mean value and standard deviation. Thus, a simpler approach is preferred. For example, Sarmiento and Falcon (2006) used a skewed normal distribution for b with empirically calibrated parameters. A more widely accepted simple approach is the lognormal distribution. By defining x  ln b , the normal distribution of x is   x  x 2    exp   2 x2  2 x   1

 f  x

(5.91)

where x is the mean value and σx is the standard deviation of x. Note that Equation 5.91 is applied in the range of b  0; otherwise, f = 0. Cheng and Law (2003) validated the lognormal distribution of b using the measurements reported in the literature, including the study of Blinco and Simons (1974), as shown in Figure 5.18. The theoretical curves agree well with the measurement data. The probability of bed shear stress approximately follows the lognormal distribution. The mean value and standard deviation of x are related to those of b by



 x ln  b

1  I2





(5.92)





 s b   b   (histogram: Figure 5.18 Probability density distribution of bed shear stress  measured by Blinco and Simons (1974), and lines: lognormal distribution) (from Cheng and Law, 2003)

Incipient Motion of Sediments  163

 x





ln 1  I2 

(5.93)

where I     b is the intensity of bed shear stress fluctuation. For flows over smooth flat beds where Re > 1000, I  150 Re0.671, as determined by Cheng and Law (2003). Here, Re is the Reynolds number based on the free-stream velocity and the flow depth. Grass (1970) measured Iτ = 0.4 for bed sediments with D = 0.09 – 0.195 mm. Gessler (1970) reported Iτ = 0.57 for coarse sediments. More investigations are needed to validate the lognormal probability distribution of bed shear stress and quantify Iτ in different flow regimes over sediment beds. If the randomness of cr is ignored, cr is represented with a single value, τcrm. The entrainment probability Equation 5.15 is simplified and then formulated with Equation 5.91 as  pE



f ( x)dx  ln  crm

1



2 

ln  crm  x   x

e t

2

2

dt



(5.94)

Using Equation 5.81, Equation 5.94 is approximated as p 0.5  0.5 E

  ln  

ln crm  ln 

1  I2

ln crm

1  I2

 

 2 1  exp  2   x



ln   ln  crm 





2  1  I2    

(5.95)

where Θcrm = τcrm /[(γs − γ)D] and    b [( s   ) D] . The entrainment probability in Equation 5.95 depends on Θcrm and Iτ. They are calibrated as Θcrm = 0.18 and Iτ = 0.75 by comparing the calculated entrainment probabilities with the  range of 0.08 – 0.24 observed by measured data in Figure 5.17. This Θcrm value is within the  cr Fenton and Abbott (1977) for particles in a co-planar bed. It is close to the value of 0.16 used by Sarmiento and Falcon (2006). The value of Iτ = 0.75 is significantly higher than those of 0.4 and 0.57 measured by Grass (1970) and Gessler (1970). This is because Iτ lumps the intensities of b and cr , as explained in the next section. In addition, the sensitivity of the entrainment probability with respect to Θcrm and Iτ is shown in Figure 5.19. The entrainment probability decreases as Θcrm increases. As Iτ increases, the curve rotates clockwise, with the probability increasing at low Shields numbers and deceasing at high Shields numbers. Equation 5.95 with Θcrm = 0.18 and Iτ = 0.75 gives pE = 36.9% at Θ = Θcrm and pE = 1.2% at Θ = 0.052 on hydraulically rough beds. Thus, τcrm corresponds to a much higher entrainment probability than the Shields critical shear stress τcr does. 5.9.4 Entrainment Probability Formulas Based on Bivariate Distributions Probability Distributions of Critical Velocity and Shear Stress The entrainment probability models introduced in the previous sections assign single values to the critical velocity ũcr and the critical shear stress cr . In reality, ũcr and cr are random variables due to the stochastic characteristics of the shape, position, and packing of bed particles. The particle projection area, exposure height, and friction angle vary widely and

164  Incipient Motion of Sediments

Figure 5.19  Sensitivity of entrainment probability with respect to (a) Θ crm and (b) I τ

Figure 5.20  Definition sketch of s, forces, and moment arms

randomly on flow-worked beds (Kirchner et al., 1990). Grass (1970) measured the probability distribution of cr , with the ratio of standard deviation to mean value,   cr  cr , as about 0.3. However, the measured cr probability distributions varied case by case and did not support a conclusive model, likely due to insufficient sample numbers. Gessler (1973) suggested a triangular probability distribution, whereas Lopez and Garcia (2001) used a lognormal distribution for cr . He and Han (1982) developed a theoretical probability distribution model of ũcr for unisized spherical particles to start rolling on a horizontal bed, considering the randomness of particle position on the bed surface. They used a simple 2-D particle packing configuration shown in Figure 5.20. The particle position is represented by the particle bottom-pivot elevation difference, s, which is related to the exposure height Δe by s = (D − Δe) / 2 and assumed to obey the following uniform probability distribution: 1 ( s  s ) smin  s  smax  ( s )   max min 0 otherwise 



(5.96)

Incipient Motion of Sediments  165

where smin = 0.5D(1 – cos30°) = 0.067D and smax = 0.5D as the minimum and maximum values of s, respectively. The critical near-bed velocity Equation 5.4 was rewritten as 1/ 2

 4 s    ucr    gD   3CD  

 0 

(5.97)

where  0 [ 3C4D (  s   1) gD]1/ 2 is a characteristic velocity determined additionally (Han and He, 1984); and ς = [kW / (kD + kL CL / CD)]1/2, in which kW, kD, and kL vary with the particle position, i.e., s as follows: kW D 

0.25 D 2  (0.5 D  s ) 2 



Ds  s 2

(5.98)

kD D = bD D + (0.5D − s)

(5.99)

kLD = bLD + kWD

(5.100)

where bD D and bLD are the moment arm distances for the drag and lift forces about the particle center, respectively. He and Han (1982) assumed bD = 1/6, bL = 1/6, and CL / CD = 1/4, and then obtained: 1/ 2

  s D  ( s D)2      2  2 3  s D  [1 6  s D  ( s D) ] 4 



(5.101)

As s varies from smim to smax, ũcr varies from 0.59ω0 to 1.22ω0. Equation 5.101 is a monotonous function, so the probability density function of ũcr is





Y  ucr    11 (ucr )  11 (ucr )  

(5.102)

where ς1 = ςω0, and 11 is the inverse function of ũcr = ς1(s). The probability density function of ũcr is shown graphically in Figure 5.21. It increases first and then decreases as ũcr / ω0 increases. The mean value of ũcr is 0.85ω0. The calculated probability distribution of ũcr is plausible. However, possible errors exist since the lift-to-drag ratio is fixed at 1/4 and the drag coefficient is assumed constant. Using Equation 5.96 for s and the normal distribution for the near-bed velocity, the entrainment probability Equation 5.12 is rewritten as (He and Han, 1982) pE  1  

0.5 D

smin

 1

 ucr  s  f (u )du  ( s )ds b b   ucr  s   1

0.866 2



1

0.134



ucr  sˆ   ub   u

 ucr  sˆ   ub   u

2



e t 2 dt dsˆ



(5.103)

where sˆ  2 s D and the critical velocity is calculated with Equation 5.97 for a given s. He and Han's model is an important contribution, but its applicability is limited due to the assumptions used.

166  Incipient Motion of Sediments

Figure 5.21  Probability density function of critical near-bed velocity (from He and Han, 1982)

Wu and Yang (2004a) developed an entrainment probability model similar to Equation 5.103 but with several improvements. Not only the moment arms but also the projection area and the acting flow velocity are associated with the particle position, which is represented with the uniformly-distributed exposure heights on the front and back sides. The projection area excludes the portion blocked by the frontal adjacent particle. The acting flow velocity is the area-weighted average velocity on the projection area. The variable projection area and acting flow velocity allow nearly constant drag and lift coefficients. The fourth-order Gram-Charliertype probability distribution is used for ũb. The model does not give an explicit probability distribution of ũcr but directly calculates the entrainment probability instead. The model details are not given here because of the complexity. According to Tregnaghi et al. (2012), both b and cr are related to bed configurations, so b and cr are conditionally dependent. This likely improves the bivariate entrainment probability Equation 5.15, in which b and cr are assumed to be independent. The measurements of Tregnaghi et al. (2012) indicate that the particle elevation relative to the bed approximately follows a normal probability distribution, rather than the uniform distributions used by Han and He (1984) and Wu and Yang (2004a). Tregnaghi et al. (2012) conducted a numerical modeling for the rolling-based entrainment probability of sediment particles with elevation measured on a flat bed. The results show that the probability of cr can be well described with the log-logistic distribution and approximately with the lognormal distribution. Note that the probability distributions of ũcr and cr adopted or developed by Gessler (1973), Lopez and Garcia (2001), He and Han (1982), and Tregnaghi et al. (2012) have not been validated directly against measurements. High-quality measurements of ũcr and cr are required for better understanding and quantification of the entrainment probability of sediment particles. Approximations of Bivariate Entrainment Probability Models In general, Equation 5.15 needs to be calculated using a numerical method. For convenience of application, Lopez and Garcia (2001) approximated Equation 5.15 with a polynomial function by assuming b  cr to obey the normal distribution. Two similar but improved approximations are presented here. Because b and cr likely follow lognormal distributions, it is adequate to assume that  x ln b  ln cr follows the normal distribution expressed in Equation 5.91 with the mean value and standard deviation:

Incipient Motion of Sediments  167

 



x   ln b mean   ln cr mean ln  b  x







ln 1  I2b  ln 1  I2cr



1  I2b  ln  cr

1  I2cr







(5.104) (5.105)

where I b    b  b , I cr    cr  cr ,  b is the mean value of b ,   b is the standard deviation of b ,  cr is the mean value of cr , and   cr is the standard deviation of cr . Then, pE is determined as  pE  p  x  0



f ( x)dx  0

1

 2



x x

2

e t 2 dt



(5.106)

By using Equation 5.81, Equation 5.106 is approximated as

p E

 0.5  0.5 ln  

ln cr cr

  2  1  exp  2   x 

    ln 

1  I2cr  ln 

1  I2b

1  I2cr

1  I2b

   ln  cr   2   1  I cr

 

    ln       1  I2 b  

    

2



(5.107)

   

where  cr  cr [( s   ) D] . It is interesting that Equations 5.95 and 5.107 are equivalent if I  (1  I2b )(1  I2cr )  1 and  cr crm (1  I2cr ) . Thus, Iτ lumps the contributions of I b and I cr , and Θcrm is proportional to cr . However, these two relationships do not give definite solutions for converting Iτ and Θcrm in Equation 5.95 to I b , I cr , and cr in Equation 5.107. Using the ratio I b I cr  0.4 0.3 measured by Grass (1970) as the additional constraint, I b  0.57 , I cr  0.43 , and cr  0.21 are obtained. The same “bed shear stress based” curve shown in Figure 5.17 can also be generated using Equation 5.107 with these values of I b , I cr , and cr . The value of I b  0.57 is comparable with those of 0.4 and 0.57 measured by Grass (1970) and Gessler (1970), respectively, and I cr  0.43 is larger than the value of 0.3 measured by Grass (1970). The differences are likely due to the different sediments used. The value of cr  0.21 is in the range of 0.08 – 0.24 measured by Fenton and Abbott (1977) for particles in a co-planar bed. Similarly, the bivariate velocity-based entrainment probability Equation 5.12 can be simplified by assuming that ũb and ũcr are independent and normally distributed, so their difference x = ũb − ũcr obeys the normal distribution Equation 5.91 with x ub  ucr   x

 u2b   u2cr 

(5.108) (5.109)

168  Incipient Motion of Sediments

where ūcr is the mean critical velocity,  ucr is the standard deviation of ũcr, and  ub is the standard deviation of ũb. The entrainment probability pE is defined in Equation 5.106 and approximated as 

1.0  0.5 p E

 2 (ucr  ub ) 2   2 (ucr  ub ) 2  1  exp      0.5 1  exp   2 2 2 2    ub   ucr )     ub   ucr ) 

ucr  ub ucr  ub

(5.110)

In analogy to Equations 5.83 and 5.84, Equation 5.110 is converted as follows by considering  ub  2u and  u  ucr and by neglecting the portion due to negative velocity fluctuation: cr

p E

 0.46  Y 0.5  0.5

R

K R

0.46  YR K R



2     0.46 1     YR  1  exp    1   R (K R ) 1    K R    

(5.111)

where αR is an empirical coefficient introduced so that 1  ( ucr  ub ) 2  1   R (K R ) 1 . Figure 5.17 includes a curve generated using Equation 5.111 with KR = 0.3, YR = 1.68, and αR = 0.005. The values of KR and YR are slightly adjusted from those used in Equation 5.84. Among the models tested in Figure 5.17, Equations 5.84, 5.95, and 5.111 perform relatively well, particularly in the intermediate range of entrainments. Equations 5.84 and 5.111 yield shorter tails at weak entrainments and longer tails at strong entrainments than Equation 5.95. This is likely due to differences in the normal and lognormal distributions. Compared with Equation 5.84, Equation 5.111 includes a different term, 1/[1 + αR(ΘKR) − 1], which can help to improve the performance in the ranges of weak and strong entrainments. However, the limited data on the two tailing ranges do not conclusively support which model among the three is more adequate. As discussed for the Einstein model, all the models included in Figure 5.17 need to be further tested because the quantity of supporting data is limited. More data are needed to clarify whether the entrainment probabilities in Figure 5.17 represent average situations in general or only in those experimental cases. 5.9.5  Particle Rolling and Lifting Probabilities Among the three incipient motion modes, rolling is normally the primary mode that occurs at a low flow velocity. If the flow velocity increases, a rolling particle may be lifted into saltation or jump mode. Sliding is an alternative to rolling, but it is less likely to occur except for platy or streamwise-elongated particles. Therefore, rolling and lifting are usually considered two consecutive modes of incipient motion and have separately defined probabilities (Wu and Chou, 2003; Li et al., 2018b): pR = p(ucr,r < ũb ≤ ucr,l)

(5.112)

pL = p(ũb > ucr,l)

(5.113)

where ucr,r and ucr,l are the critical near-bed velocities for rolling and lifting, respectively, as described in Section 5.2.

Incipient Motion of Sediments  169

Figure 5.22  L ifting, rolling, and total entrainment probabilities calculated with Equation 5.95

When rolling is the primary mode, ucr,r = ucr. Rolling and lifting are treated as independent events so that the rolling and lifting probabilities sum to the overall entrainment probability pE: pR + pL = pE

(5.114)

The lifting-based entrainment probability Equation 5.80 from Einstein (1942, 1950) and Equation 5.83 from Cheng and Chiew (1998) were calibrated using entrainment probability data. After recalibration, they can be used to determine the lifting probability pL only. For example, Li et al. (2018b) calculated the lifting probability pL by applying Equation 5.83 with CL = 0.178. The rolling-based Equations 5.84 and 5.95 can be directly used to determine the overall entrainment probability pE. Then, the rolling probability pR is determined by using Equation 5.114, i.e., pR = pE − pL. Figure 5.22 compares the lifting and rolling probabilities against the entrainment probability pE calculated by using Equation 5.95 with Θcrm = 0.18, as discussed in Section 5.9.3. The thresh for particle lifting in a co-planar bed (i.e., at position 1 in Figure 5.2) is old Shields number  cr , l  = 0.52 into Equation 5.95 determined as 0.52, as explained in Appendix 5.1. Substituting  cr , l yields the lifting probability pL shown in Figure 5.22. Then, the rolling probability is obtained with pR = pE − pL. Due to the lack of measurement data, the calculated pL and pR values shown in Figure 5.22 are only qualitative. As the Shields number increases, the lifting probability pL monotonically increases, but the rolling probability pR increases first and then decreases. Rolling is dominant in weak flows, whereas lifting is dominant in strong flows. Similar trends were reported by Wu and Chou (2003) and Li et al. (2018b), but these authors questionably used the data in Figure 5.22 as the lifting probability only. 5.9.6  Entrainment Probability of Nonuniform Sediments For a nonuniform sediment mixture, the particle availability is represented by the size frequency fpd (D) in the bed surface layer. The entrainment probability of particles with size D is denoted

170  Incipient Motion of Sediments

as pE (D). Because fpd and pE are independent, the total entrainment probability of the mixture is determined as (Gessler, 1970; Wu and Yang, 2004a) pE ,tot 



Dmax

Dmin

f pd ( D) pE ( D)dD 

(5.115)

where Dmin and Dmax are the minimum and maximum particle diameters in the sediment mixture, respectively; and



Dmax

Dmin

f pd ( D)dD  1. Then, the cumulative size frequency of the moving

sediment particles is

Pcf

  

D

Dmin Dmax

Dmin

f pd ( D) pE ( D)dD f pd ( D) pE ( D)dD



(5.116)

Gessler (1970) determined pE(D) using an entrainment probability model based on the normal distribution of bed shear stress. In general, the aforementioned entrainment probability models of uniform sediment mixtures can be used for pE(D). Importantly, the hiding-exposure effects in nonuniform bed materials need to be considered (Sun and Donahue, 2000; Wu and Yang, 2004a). This is usually achieved by applying the correction factors described in Section 5.6 to the critical shear stress or velocity. Appendix 5.1 Comparison of Incipient Rolling, Sliding, and Lifting Thresholds of Spheres The critical near-bed velocities of incipient rolling, sliding, and lifting introduced in Section 5.2 are compared here regarding positions 1 and 2 shown in Figure 5.2 on a horizontal bed of uniform spherical particles. In the case of position 2, Figure 5.23a and b illustrate the 3-D configuration for the particle to roll about the pivot points on the two immediately downstream particles that form a line normal to the flow. This configuration allows the least flow strength for the particle to start into motion. The particle exposure Δe = 0.817D, and the bottom-pivot elevation difference s = 0.092D. The involved parameters are listed in Table 5.4. The moment arms of the drag and lift forces are calculated with Equations 5.99 and 5.100, and the moment arm of the submerged weight is

Figure 5.23 (a) Top view and (b) side view of an entrainment particle on top of three particles; and (c) top view of an entrainment particle in front of two co-planar particles

Incipient Motion of Sediments  171

 kW D

2 0.25 D 2 cos 2 30   0.5 D  s  

(5.117)

The drag force is assumed to act at the particle center elevation, i.e., bD = 0, whereas bL = 0.167 was used by He and Han (1982). CD = 0.8 and CL = 0.18 are estimated by referring to the experimental results of Apperley (1968), Li et al. (1983), and Schmeeckle et al. (2007) shown in Table 5.1. The sliding angle αsl is about 16.8°. The friction angle ϕi is set as 32°. Then, Equations 5.4, 5.6, and 5.8 with these parameters lead to ucr,l = 2.21ucr,s = 3.8ucr,r

(5.118)

Equation 5.118 indicates that among the three entrainment modes, rolling requires the lowest critical velocity, so the particle at position 2 starts motion primarily by rolling. This agrees with the analyses of Ling (1995) and Choi and Kwak (2001), showing that at position 2, rolling requires a weaker flow and has a higher probability than lifting, with that of sliding in between. In the case of position 1, the entrainment particle lies in a co-planar bed, where Δe = 0 and s = 0.5D. Because the sliding plane angle αsl is 90° and Equation 5.6 gives a large critical velocity, sliding is unlikely to occur. Figure 5.23c shows the 3-D configuration for the particle to roll about the pivot points on the two downstream particles. The drag force is assumed to act at about 0.15D above the theoretical bed that is set at 0.25D below the apexes of the surface particles, so bD = 0.4, whereas bL = 0.167 is unchanged. The moment arms kW, kD, and kL are calculated using Equations 5.99, 5.100, and 5.117. CD = 0.29 and CL = 0.18 are estimated using the measurements of Einstein and El-Samni (1949), Wang and Fontijn (1993), and Dwivedi et al. (2010, 2011) shown in Table 5.1. Comparing Equations 5.4 and 5.8 leads to ucr,l = 1.7ucr,r

(5.119)

Thus, the particle at position 1 is entrained primarily by rolling. Since positions 1 and 2 are the two position bounds of the particles on the bed surface, it can be concluded that rolling is the primary mode of incipient motion for spherical particles. This agrees with the experimental observations of Carling et al. (1992). Likewise, the critical shear stresses are compared. For the particles at positions 1 and 2 in Figure 5.2, the reference flow velocity is set at yd = 0.15D and 0.567D above the theoretical bed, respectively. With the values of kW, kD, kL, CD, and CL listed in Table 5.4 and ks = D, Equa = 0.18 and 0.01 for particles to tion 5.10 gives ξr = 1.60 and 0.71, and Equation 5.11 yields  cr

Table 5.4  Parameters in Equations 5.4 and 5.11 for example particles Parameter

Particle at position 1

Particle at position 2

Parameter

Particle at position 1

Particle at position 2

Δe bD bL kW kD kL

0 0.4 0.167 0.433 0.4 0.6

0.817D 0 0.167 0.144 0.408 0.311

CD CL yd χs ξr

0.29 0.18 0.15D 1 1.60 0.18

0.8 0.18 0.567D 1 0.71 0.01

  cr

172

Incipient Motion of Sediments

 start rolling at position 1 and 2, respectively. Correspondingly, the critical Shields number  cr ,l for particle lifting is determined as 0.52 and 0.15 from positions 1 and 2, respectively, by using  > Equation 5.11 with ξr = [4/(3CL)]1/2. This ξr formula is derived from Equation 5.8. Thus,  cr,l  , indicating that rolling is the primary entrainment mode. Then, the overall critical Shields  cr numbers at positions 1 and 2 are 0.18 and 0.01, respectively, which are within the ranges of about 0.08 – 0.24 and 0.0094 – 0.012, respectively, measured by Fenton and Abbott (1977). Appendix 5.2

Compa rison of Reference Thresholds of Sediment Incipient Motion

Table 5.5 compares the numbers of sediment particles moving on the bed calculated using the reference transport thresholds described in Section 5.3.2. Three uniform particles sized 10, 1, and 0.1 mm are used. The specific gravity is set as 2.65. For the methods of Parker et al. (1982) and Neill and Yalin (1969), the bed shear stress or shear velocity at incipient motion is determined with Equation 5.39. For the method of Han and He (1984), the settling velocity is calculated with Equation 4.15. The 0.135% and 2.28% probabilities of Dou (1962) are equivalent to Φb = 0.000031 and 0.000536, respectively, according to Einstein’s bed-load transport equation described in Section 7.4.5. Of all the reference transport thresholds compared in Table 5.5, the methods of Parker et al. (1982), Neill and Yalin (1969) with Γqb = 0.0001, and Dou (1962) with pE = 0.135% give the weakest transports and thus define safer inception thresholds from the design point of view. The methods of Chien and Wan (1983) and Han and He (1984) define somewhat higher thresholds. The method of USWES (1936) specifies a very large number of particles in motion and is inadequate for the incipient motion of sand. Table 5.6 compares the numbers of sediment particles dislodged from a 1 m2 bed area per second calculated using the methods of Vanoni (1964), Neill and Yalin (1969), McNeil et al. (1996), and Shvidchenko and Pender (2000). For the method of McNeil et al. (1996), the bed porosity is set as 0.4. For the method of Shvidchenko and Pender (2000), the areal packing density ma is set as 0.7. In Vanoni’s (1964) experiments with a sediment of D = 0.102 mm, in one second, roughly 10/3 to 40 particles were observed to start moving on a bed area of 8 mm in diameter, i.e., 66,314 – 795,775 particles/m2. Vanoni’s criterion is significantly higher than Neill and Yalin’s Γn and Shvidchenko and Pender’s criteria, and is lower than the criterion of

Table 5.5 Comparison of reference transport thresholds Author(s)

USWES (1936) Neill and Yalin (1969) Parker et al. (1982) Chien and Wan (1983) Han and He (1984) Dou (1962) Dou (1962)

Reference threshold

q b = 0.00041 kg/(m·s) Γ qb = 0.0001 q b in Equation 5.20 q b in Equation 5.21 q b/(ρ sDω s) = 0.000317 p E = 0.135% p E = 2.28%

Number of moving particles per meter of channel width per second D = 10 mm

D = 1 mm

D = 0.1 mm

0.3 0.17 0.17 0.77 2.5 0.24 4.1

295 4.1 2.3 24.3 72.4 7.5 130.2

295,392 216 339 768 375 238 4,119

Incipient Motion of Sediments

173

Table 5.6 Comparison of reference entrainment thresholds Author(s)

Vanoni (1964) Neill and Yalin (1969) Shvidchenko and Pender (2000) McNeil et al. (1996)

Reference threshold

Number of particles dislodged per m 2 of bed area per second D = 10 mm

D = 1 mm

D = 0.1 mm

Burst frequency Γ n = 2×10 -6 in Equation 5.16 n p np = 10 -4 s -1

– 0.18

– 42.5

66,314 – 795,775 22,570

0.89

89

8913

Erosion rate of 10 -4 cm/s

1.15

1,146

1,145,916

McNeil et al. (1996). Shvidchenko and Pender’s criterion gives higher numbers of entrainment particles for gravel and coarse sand but a lower number for fine sand than Neill and Yalin’s Γn criterion. The criterion of McNeil et al. is less adequate since it gives relatively large numbers for fine sediments in motion. The criteria in Tables 5.5 and 5.6 can be roughly compared by applying qb  n p lb m p and assuming lb = 100D. Vanoni’s criterion is equivalent to about 676 – 8,117 bed-load particles per meter of channel width per second. It is significantly higher than the criteria listed in Table 5.5, except Dou’s pE = 2.28% and the criterion of USWES (1936). Neill and Yalin’s Γn criterion is approximately equivalent to their Γqb criterion, comparable to those of Parker et al. (1982) and Dou (1962) with pE = 0.135%, and less but close enough to those of Chien and Wan (1983) and Han and He (1984). The criterion of Shvidchenko and Pender (2000) is close to those of Chien and Wan (1983) and Han and He (1984) for gravel, but gives the lowest number for fine sand among all the criteria. This inconsistency is due to its dimensional form. Though the aforementioned reference thresholds exhibit large differences, much lower differences exist in the derived bed shear stresses. This is because the bed-load transport rate is approximately an 8th – 16th power function of bed shear stress in the proximity of incipient motion (Paintal, 1971a; Wilcock and Crowe, 2003), as explained in Chapter 7. A ratio of 100 between two reference transport rates corresponds to a ratio of 1.3 – 1.8 between the derived values of critical shear stress. Homework Problems 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Describe the incipient motion modes of bed sediment particles. Which is usually the primary mode? Why is the incipient motion of sediment often treated as a random process? Suggest one method to calculate the probability of incipient motion. Describe the reference thresholds for determining the critical flow condition of incipient motion. Explain how and why the critical Shields number varies as the particle shear Reynolds number increases. Explain the reasons for the data scatter in the Shields diagram. Can you convert the Shields diagram of Θcr and Re*cr to a relationship between Θcr and D*? How does particle shape affect the incipient motion of sediment?

174  Incipient Motion of Sediments

5.8 How does particle exposure affect the incipient motion of sediment? 5.9 Derive the velocity-based entrainment probability by assuming that ũb and ũcr are independent and ũb − ũcr follows a normal distribution. 5.10 Describe the hiding and exposure mechanisms in nonuniform bed material. How do these mechanisms affect the incipient motion of fine and coarse particles on the bed surface? How does one account for the hiding-exposure effects? 5.11 Explain how the longitudinal slope affects the incipient motion of sediment. 5.12 Given sediment diameter D = 1 mm and a flow depth of h = 2 m, calculate the critical average velocity using the Shamov formula. 5.13 Given sediment diameters D = 0.01 and 1 mm, calculate the critical shear stress using one of the modified Shields diagrams. Assume a water temperature of 20°C. 5.14 Given a shear velocity of 0.05 m/s, determine the sediment size at incipient motion using one of the modified Shields diagrams. 5.15 Given a channel slope of 0.002, determine the flow depth at which sediment with a 10 mm diameter starts to move. Use one of the modified Shields diagrams. 5.16 Consider a nonuniform sediment mixture consisting of three size classes with representative diameters 0.05, 0.5, and 5 mm and fractions 0.3, 0.4, and 0.3, respectively. Determine the critical shear stress using the formulas of Egiazaroff (1965) and Wu et al. (2000). 5.17 Compare the original and modified Shields diagrams by graphically showing these relationships in terms of Θcr and D*. 5.18 Determine the stability of riprap of D = 12 cm placed on a 1V:2H bank slope if the shear stress is 18 Pa. 5.19 Design a stable, straight, rectangular channel with a bed made of noncohesive sediment with D50 = 16 mm under clear-water flow conditions. The channel has a slope of 0.001 and conveys a flow discharge of Q = 15 m3/s. The Manning n is 0.025 s/m1/3. Assume that banks are nonerodible. 5.20 Design a stable, straight, trapezoidal channel made of noncohesive sediment with D50 = 16 mm under clear-water flow conditions. The channel has a slope of 0.001 and conveys a flow discharge of Q = 15 m3/s. The Manning n is 0.025 s/m1/3. The same sediment is used on banks and bed.

Chapter 6

Bed Forms

Bed forms are created due to interactions of flow, sediment transport, and bed. They are important components of bed roughness and significantly affect channel hydro-morphodynamics. Introduced in this chapter are the characteristics, regimes, and dimensions of bed forms, as well as movable bed resistance in rivers. 6.1 Classifications of Bed Forms Bed forms are categorized as small- or large-scale. Small bed forms scale to sediment size or flow depth, whereas large ones scale to channel width. These two groups are associated separately with vertical and horizontal turbulence structures. In addition, different bed-form features exist in alluvial and mountainous rivers. 6.1.1 Classification of Small-Scale Bed Forms in Sand-Bed Rivers Consider a straight open channel with a movable sand bed and sufficient sediment supply under the action of a uniform flow. As illustrated in Figure 6.1, the bed may evolve through the following stages as the flow strength increases: (1) In the initial stage when the flow is weak, only a small number of sediment particles start to move on the stationary flat bed. (2) When the flow is stronger, more sediment particles participate in motion, mainly as bed load; small wavelets form and evolve into ripples on the bed surface. (3) Then, a significant number of particles move as bed load or suspended load, and dunes appear on the bed. (4) As the flow strength continually increases, sediment particles are largely suspended and transported further downstream, dunes are washed out, and in turn, the bed becomes a plane with moving sediment particles. (5) Next, antidunes appear, accompanying intensive sediment transport and strong interactions among flow, moving sediment, and bed. (6) Finally, chutes and pools form when water flow and sediment transport are very strong. The stationary flat bed, ripples, and dunes comprise the lower flow regime, while the moving plane bed, antidunes, and chute-pools are parts of the upper flow regime. The moving plane bed is also considered a transition between the lower and upper flow regimes. The general characteristics of these bed forms are briefly described in the following text. DOI: 10.1201/9781003343165-6

176  Bed Forms

Sand ripples are asymmetric with a gentle stoss side and a steep lee side (Figures 6.1b and 6.2). Ripples are characteristic of low transport rates associated with fine to medium sand smaller than about 0.6 mm (Zhang et al., 1998; Raudkivi, 1998; Parker, 2005). They appear when the particle shear Reynolds number Re* = u*D/ν is less than 10–26 (Sumer and Bakioglu, 1984; Raudkivi, 1998). Sand ripples are related to the viscous sublayer and near-bed coherent structures, so they scale well with sediment size. They migrate downstream. Dunes are the most common bed forms in sand-bed rivers. Figure 6.3 shows sand dunes on a sand bar in the Shishou Reach of the middle Yangtze River, China. Figure 6.4 shows the migration of sand dunes observed on May 26–27, 1958, in a section of the Yangtze River, China (Zhang, 1961) and on May 6–7, 1980, in a section of the Ijssel River in the Netherlands

Figure 6.1 Evolution of bed forms as the flow strength increases (after Xie, 1981; Richardson and Simons, 1967)

Figure 6.2 Sand ripples in the Hunter River, New South Wales, Australia (image courtesy of Michael C. Rygel, via CC-BY-SA-3.0 license)

Bed Forms  177

Figure 6.3 Sand dunes superimposed on a sand bar in the middle Yangtze River, China (photo courtesy of Zhiwei Li)

Figure 6.4 Migration of sand dunes: (a) in the Yangtze River, China (from Zhang, 1961), and (b) in the Ijssel River, the Netherlands (after Havinga, 1982)

(Havinga, 1982). Dunes are characteristic of subcritical flow (with a Froude number Fr sufficiently below 1). Fluvial dunes are usually asymmetric, with a gentle stoss slope and a steep lee slope. They are out of phase and interact weakly with the water surface, so the flow accelerates over the crests (Figure 6.1c). Flow acceleration causes sediment erosion on the stoss slope, while flow deceleration and separation cause sediment deposition on the lee slope. Thus, dunes migrate downstream in certain shapes (Figure 6.4). Dunes are associated with the depth-scaled vertical eddies of free-surface flow, so they scale well with flow depth. According to van Rijn (2007a), dunes are generally absent when the bed sediment is finer than about 0.1 mm. Under this condition, the bed generally consists of a flat mobile surface and/ or of large-scale sand waves with or without small-scale ripples superimposed. Antidunes usually appear in rivers with Fr values larger than 1. While water and sediment move downstream, bed and water surface waves likely propagate upstream in phase, showing little

178  Bed Forms

Figure 6.5 Antidunes beneath a fast-flowing stream in the Kennetcook River (antidunes migrated slowly from right to left while the flow moved from left to right) (© John W.F. Waldron)

Figure 6.6 Cyclic steps in a coastal outflow channel on a beach at Calais, France (photo courtesy of Hervé Capart)

asymmetry (Figures 6.1e and 6.5). They may break as surf or subside as standing waves. The sediment movement is strongly influenced by free surface flow. Antidunes scale well with flow depth. Chutes and pools appear in very steep streams (Richardson and Simons, 1967). They are also called cyclic steps (Taki and Parker, 2005). Each step has a subcritical upstream portion followed by a supercritical chute portion and ends with a hydraulic jump in a pool (Figures 6.1f and 6.6). Erosion occurs in the chute portion, and deposition occurs in the pool portion downstream of the jump. The steps migrate upstream while preserving their form. 6.1.2 Classification of Large-Scale Bed Forms in Alluvial Rivers Large-scale bed forms, such as alternate, point, and mid-channel bars, appear in alluvial rivers. They may be submerged at high stages but are emergent and shape the river flows at low stages. They may be superimposed with small-scale bed forms (see Figure 6.3). The development of these bars is associated with the planform and evolution of river channels. Figure 6.7 shows the conceptual framework for classifying alluvial rivers proposed by Schumm (1981), revised by Church (2006), and further revised here. As the channel width/depth ratio and the sediment supply increase, the channel planform changes from straight to meandering, braided, anastomosed, and wandering; meanwhile, the number of bars increases and the channel stability decreases. The

Bed Forms  179

Figure 6.7 Classification of alluvial rivers (modified from Church, 2006 and Schumm, 1981)

wandering channel type is added here for cases, such as that in the lower Yellow River, where sediment concentration is considerably high, bed materials mainly consist of fine sand and silt, the channel is wide (e.g., ~10 km), and the main flow course shifts frequently (Zhang and Xie, 1993). Alternate bars often exist in nearly straight channels (Figure 6.8a) with sufficiently large (> ~12) but not too large width-depth ratios B/h (Parker, 2005). When the channel widthdepth ratio is even larger, free multi-row bars (e.g., linguoid bars) occur, and the channel becomes braided (Figure 6.8b). These alternate and multi-row bars result from the morphodynamic instability of the riverbed, often appearing in rhythmic sequences and migrating through the river. Thus, they are called free bars (Crosato and Mosselman, 2020). They scale well to channel width and usually migrate downstream. However, Palucis et al. (2018) experimentally observed upstream-migrating alternate gravel bars in high-gradient streams. Point bars typically form along the inner bank and deep channels form along the outer bank of curved or meandering channels (Figure 6.9). They are created by the helical flow that carries sediments toward the inner bank and shifts the main flow toward the outer bank, as described in Section 3.6.2. Shallow crossovers exist in the straight reaches between bends. The point bars are crescent-shaped. Their size is highly related to channel width and curvature. They prograde outward, causing channel meandering if the outer bank retreats. In addition, certain mid-channel bars and islands are created due to sediment deposition as a river enters expanded channels and river mouths, and side bars are generated in the separation flow zones downstream of tributary confluences and bank protrusions. The positions and sizes of these bars are controlled by factors such as channel geometry, sediment size, bank erodibility, and channel slope. The mid-channel bars and islands may migrate downstream and laterally to a certain extent as the branch channels evolve. These mid-channel and side bars, as well as the aforementioned point bars, are called forced bars. A review of fluvial bars can be found in Crosato and Mosselman (2020).

180  Bed Forms

Figure 6.8 (a) Alternate bars in the Rhine River, Switzerland (from Jaeggi, 1984), and (b) multirow bars in the Waimakariri River, New Zealand (photo courtesy of B. Federici) (Seminara, 2006)

Figure 6.9 Point bar in a meandering river through the Innoko National Wildlife Refuge (photo courtesy of U.S. Fish and Wildlife Service)

6.1.3 Classification of Bed Forms in Mountainous Rivers Small-scale bed forms in mountainous rivers with gravel to boulder beds are likely different from those in sandy rivers since strong flows are required to achieve similar sediment transport intensities. Dune-shaped bed forms on gravel beds appear only during floods (Dinehart, 1989). Typical small bed features in gravel- and cobble-bed streams include clusters and transverse

Bed Forms  181

ribs. Clusters are formed by discrete, organized groups of coarse particles that sit above the average elevation of the surrounding bed (Figure 6.10a). Transverse ribs are arrays of accumulated coarse particles oriented perpendicularly to the flow direction (Figure 6.10b). Large-scale bed forms in mountainous rivers include bar, riffle-pool, step-pool, cascade, and colluvial systems. Some of these bed forms are illustrated in the left column of the river classification diagram in Figure 6.7. Sequential riffles and pools often appear as the flow alternates in gravel-bed streams with slopes less than about 0.02 (Figure 6.11). Riffles are formed in shallow areas by coarser bed materials, and pools are formed in deep areas with finer bed materials. The riffle acts as a hydraulic control on flows through the upstream pool at low flows and as a submerged large-scale roughness element at high flows. If the stream is curved or meandering, pools appear on the outer sides of bends, point bars on the inner sides, and riffles at the crossovers between bends. The riffle-pool undulations scale well to channel width.

Figure 6.10 (a) Clusters formed on a gravel-cobble bed (photo by the author); and (b) transverse ribs (TR) of gravels on the dry bed of the Dandero River, Danakil, Eritrea (from Billi et al., 2014)

Figure 6.11 Riffle-pool system in a gravel-bed stream (from Buffington and Montgomery, 2013)

182  Bed Forms

Figure 6.12 Step-pool system in a boulder-bed stream (from Billi et al., 2014)

Step-pool, cascade, and colluvial systems occur in the uppermost steep streams. In these locations, beds are composed of rocks and shallow colluvial/alluvial materials, and channels are typically confined by valley walls, directly coupled to hillslopes, and often affected by debris flows. In a step-pool system, the steps are made of transverse arrays of boulders, woody debris, or resistant bedrock, and pools are formed with cobble- to sand-sized bed materials, as shown in Figure 6.12. The step-pool systems effectively dissipate flow energy, enhance bed stability, and increase biodiversity (Wang et al., 2009). In a cascade stream, boulders are chaotically distributed on the bed and the flow exhibits continuous macroscale turbulence. The colluvial regime occurs in first-order channels incised into colluvial valleys by overland flow and seepage erosion. The streamflow may be perennial or ephemeral, and the bed morphology is strongly controlled by randomly occurring obstructions, such as boulders, woody debris, and in-channel vegetation (Buffington and Montgomery, 2013). 6.2 Development Mechanisms and Regimes of Bed Forms 6.2.1 Formation Theories for Sand Ripples, Dunes, and Antidunes The initiation and evolution of bed forms have been extensively studied over the past century. According to Exner (1925), the evolution of bed forms can be described with the sediment continuity equation:

1  v 

yb qb   0 t x

(6.1)

where t is the time, x is the streamwise coordinate, qb is the volumetric transport rate of sediment, yb is the bed elevation, and φv is the porosity of bed sediment. This equation is primarily applied to bed-load transport since it ignores the turbulent diffusion of suspended load and the temporospatial lags of sediment transport relative to fluid flow (Wu, 2007). By assuming qb to be a power function of the flow velocity under a steady flow with a constant water level, Equation 6.1 is rewritten as yb  b mv q mv yb   0 t 1  v  ( ys  yb ) mv 1 x

(6.2)

Bed Forms  183 mv where ζb and mv are coefficients such that  qb   b q mv ( ys  yb ) mv, U is the average flow bU velocity, q is the unit flow discharge, and ys is the water surface elevation. Thus, the bed forms migrate with a celerity of cd  b mv q mv [(1  v )( ys  yb ) mv 1 ] . Figure 6.13 shows the evolution of an initially sine-shaped sand ripple described by the analytical solution of Equation 6.2 derived by Exner (1925). Because the ripple crest has a higher celerity than the trough, the sand ripple becomes more and more asymmetric, and eventually the crest overhangs downstream. The evolution process in the initial period is qualitatively reasonable, but the overhanging ripple crest is physically inaccurate because the sediment particles should slide into the trough when the lee slope is steeper than the repose angle. Dashed lines are added in Figure 6.13 to show the realistic pattern of sand ripples on the lee side. The Exner model can illustrate the evolution of sand ripples but not the initiation. It has been recognized that bed forms such as ripples, dunes, and antidunes are created due to the inherent instability of movable granular beds under the action of shear flows. The instability originates from the uneven bed surface and the large variety of turbulent eddies, particularly the coherent structures. In addition, the Exner model assumes a constant water level, which is not valid for sand dunes and antidunes. Therefore, improved models have been developed by incorporating advanced calculation methods for turbulent flow and sediment transport (e.g., Zhang et al., 2018). Meanwhile, many sediment bed stability models have been proposed to explain the initiation and evolution of bed forms, as introduced herein. Kennedy (1963) proposed a linear stability model of bed forms by solving for the vertical two-dimensional potential flow over a wavy bed. The model assumes the sediment transport rate to be a power function of the bottom flow velocity at an upstream cross-section with a phase lag due to fluid inertia, and then solves Equation 6.1 to derive the sand-wave celerity and amplitude. Reynolds (1965) improved the analysis of Kennedy (1963) and extended it to 3-D sand waves. The linear stability model leads to the phase diagram shown in Figure 6.14. The wave numberdependent criterion for critical flow over a bed form is expressed as (Kennedy, 1963)

Fr2 

tanh(kh)  kh

(6.3)

where Fr  U gh is the Froude number, h is the flow depth, and k = 2π/Ld is the wave number with Ld being the length of bed forms. The occurrence condition is Fr2  [tanh(kh)] (kh) for dunes and Fr2  [tanh(kh)] (kh) for antidunes. Moreover, antidunes may migrate either

Figure 6.13 Evolution of a sand ripple: solid lines indicate the analytical solution derived by Exner (1925) and dashed lines indicate the slope of repose

184  Bed Forms

upstream or downstream. The potential-flow criterion dividing the upstream and downstream migrating antidunes is (Reynolds, 1965) Fr2 

1  kh tanh(kh)

(6.4)

Antidunes migrate downstream when Fr2  [kh tanh(kh)]1 and upstream when [tanh( kh)] (kh)  Fr2  [ kh tan anh( kh)] (kh)  Fr2  [ kh tanh( kh)]1. Downstream-migrating antidunes are rare, so antidunes are usually observed to migrate upstream. The phase diagram in Figure 6.14 can approximately predict the existence regimes of dunes and antidunes. However, it has limitations because the effects of viscosity, turbulence, bed slope, and sediment properties are ignored. Many scholars have examined and enriched the linear stability theory. Engelund (1970) used a simple concept of eddy viscosity to calculate the perturbed flow, Fredsøe (1974) considered the contribution of suspended load, Hayashi (1970) took into account the effect of gravity on bed-load transport over a local bed slope, and Parker (1975) considered the effect of bed-load inertia. According to Engelund (1970), the bed is unstable in the region of [tanh( kh)] ( kh)  Fr2  [ kh tanh( kh)]1 (i.e., the upstream-migrating antidune regime in Figure 6.14) and stable outside of this region. Richards (1980) followed the analyses of Engelund (1970) and Fredsøe (1974) but used a one-equation turbulence model to describe the near-bed turbulent flow. His analysis yields two maximum sand-wave growth rates: one depends strongly on the flow depth and the other essentially does not. The former growth rate is attributed to the formation of sand dunes, and the latter is associated with the formation of sand ripples. Sumer and Bakioglu (1984) extended the analysis of Richards (1980) by including the effect of bed slope on bed load and found that sand ripples occur only when the particle shear Reynolds number is less than about 10–26. In this regime, the viscous sublayer is effective. The aforementioned linear stability theories are based on the assumption of small bed perturbations, which is valid only during the initial stage of sand wave evolution (Coleman and Melville, 1994). The ripples and dunes observed in nature are usually in nonlinear regimes (e.g., their aspect ratios are not much smaller than 1). Moreover, the experiments of Coleman

Figure 6.14 Phase diagram of dunes and antidunes based on the linear stability model (after Engelund, 1970 and Parker, 2005)

Bed Forms  185

and Melville (1994, 1996) and Raudkivi (2006) and the numerical modeling of Zhang et al. (2018) have shown that sand-wave growth is essentially a process of smaller sand waves merging into larger ones. In the beginning, wavelets result from the interactions of turbulent bursts and sediment particles at the bed surface. Sand waves with different sizes propagate at different speeds. A faster sand wave overtakes a slower one, and then two waves merge and form a new larger one. Through this merging process, wavelets grow into ripples and then into dunes. The sand-wave sizes increase during the merging process and eventually reach a stable state under specific flow and sediment conditions. Ji and Mendoza (1997) conducted a weakly nonlinear stability analysis of dune formation by using the steady vorticity transport equation and Bagnold’s bed-load transport formula that considers the effect of the local dune-surface slope. Zhou and Mendoza (2005) applied a nonlinear stability theory to study the growth of sand waves from an initial flat bed. Their results agree with the observations of Coleman and Melville (1994, 1996). Bose and Dey (2009) developed a stability theory for a turbulent shear flow over an undulating sand bed based on the Reynolds-averaged Navier-Stokes equations and suspended-load transport equation over the entire flow depth. The results show that bed instability leads to the formation of dunes at lower Froude numbers (less than 0.8) and the formation of standing waves and antidunes at higher Froude numbers. Bose and Dey (2012) extended their theory to sand ripples associated with bed-load transport and found that the ratio of ripple length to sand size increases with an increasing Shields number. Based on the energy balance over a symmetrical antidune, Núñez-González and Martín-Vide (2011) proposed Fr = (kh) −1 to replace the criterion in Equation 6.4 for the migration direction of antidunes. Fourrière et al. (2010) performed a linear stability analysis of sand waves considering the spatial lag from a nonequilibrium state of sediment transport to the equilibrium state. Their results show that ripples are created by linear instabilities, whereas dunes are formed by the coalescence of ripples. The initial wavelength of ripples essentially scales to the sediment adaptation length. The growth rate of ripples is four to six orders of magnitude greater than that of dunes. Following Fourrière et al. (2010), Franklin (2012) performed a nonlinear stability analysis considering freesurface effects and obtained a simple model for the length and celerity of dunes. His model can explain the growth and saturation of sand ripples and the evolution from sand ripples to dunes. Due to the complexity of the stability of the movable bed under turbulent shear flows, the aforementioned theoretical models have to adopt many assumptions and approximations. They provide valuable physical insights but are not yet general enough. Engineering practice still relies on empirical diagrams and formulas developed using laboratory and field data, as introduced in the following section. 6.2.2 Regime Diagrams of Small-Scale Sandy Bed Forms Predication of bed-form regimes under given flow and sediment conditions is an important task in sedimentation engineering. Choices for sand-bed streams include the diagrams and methods developed by Garde and Albertson (1959), Simons and Richardson (1961), Brownlie (1983), van Rijn (1984c), and Karim (1995). Garde and Albertson’s (1959) and Karim’s (1995) Methods Garde and Albertson (1959) developed a diagram based on the Shields number Θ = τb/[(γs   −   γ)D] and Froude number using laboratory and field data, as shown in Figure 6.15. Here, τb is the bed shear stress, γs is the specific weight of sediment, γ is the specific weight of water, and D is the

186  Bed Forms

Figure 6.15 Bed-form regimes in sand-bed streams in terms of Shields number and Froude number (from Garde and Albertson, 1959)

sediment diameter. For a given Shields number, the bed develops from ripples to dunes to antidunes as the Froude number increases. The upper flow regime occurs in supercritical flows. Karim (1995) proposed a similar diagram by using the Froude number and h/D50 as the independent variables. Here, D50 is the median diameter of bed material. The lower and upper flow regimes are divided using the following limit Froude numbers: Frl = 2.716(h/D50) −0.25, Fru = 4.785(h/D50) −0.27

(6.5)

such that the lower flow regime occurs if Fr ≤ Frl, the upper flow regime occurs if Fr ≥ Fru, and a transitional regime exists in the range of Frl 25. Note that van Rijn (1993) introduced mega-ripples, which have a wavelength of about the flow depth. The mega-ripples can be treated as small dunes or a transitional type of bed form between ripples and dunes. Brownlie’s (1983) Diagram Brownlie (1983) delineated the lower and upper flow regimes in sand-bed streams using the particle densimetric Froude number  Frd U ( s   1) gD50 and the channel slope S, as shown in Figure 6.17. For slopes greater than 0.006, all bed forms are in the upper flow regime. For slopes less than 0.006, the upper and lower flow regimes are roughly divided by the line of

188  Bed Forms

Figure 6.17 Delineation of the lower and upper flow regimes in sand-bed streams (after Brownlie, 1983)

F  F 1.74 S 1/ 3 . A transition zone is defined along this line by using the following relationrd rd * ships for the lower limit of the upper flow regime: log

Frd 0.02469  0.1517 log( D50  )  0.8381[log( D50  )]2  Frd *  log 1.25

D50   2  D50   2

(6.7)

and the upper limit of the lower flow regime: log

Frd 0.2026  0.07026 log( D50  )  0.9330[log( D50  )]2  Frd *  log 0.8

D50   2  D50   2

(6.8)

where δ = 11.6ν/u* is the nominal thickness of the viscous sublayer. 6.2.3 Stability Models and Regime Diagrams of Bars in Alluvial Channels Since the late 1960s, many channel stability models have been proposed to predict the development of alternate and multi-row bars in straight channels, as well as the pattern and evolution of meandering and braided channels (Callander, 1969; Struiksma et al., 1985). These models are divided into bar (e.g., Schielen et al., 1993; Tubino et al., 1999) and bend models (e.g., Ikeda et al., 1981; Odgaard, 1989). They are typically based on analytical or semi-analytical solutions of the simplified two-dimensional St. Venant-Exner shallow water model with a roughness formula and a bed load predictor. The bend models are additionally implemented with bank erosion calculations. Crosato and Mosselman (2009) proposed the following bar mode predictor based on the simplified version of the linear morphodynamic stability model of Struiksma et al. (1985): 2 M bar  0.17 g

(mv  3) (  s   1) D50

B3 S  Ch Q

(6.9)

Bed Forms  189

where Mbar is the bar mode; B is the channel width (m); Ch is the Chezy coefficient (m1/2/s); Q is the bankfull flow discharge (m3/s); and mv is an exponent in the sediment transport law expressed as qb  U mv, with a value of mv = 4 for sand-bed rivers and mv = 10 for gravel-bed rivers. Equation 6.9 gives a real number that is converted to the closest integer number, M bar. If M bar = 0 (Mbar < 0.5), the river is morphodynamically stable and does not have any alternate bars. If M bar  1 (0.5 ≤ Mbar < 1.5), alternate bars occur. If M bar  2 (1.5 ≤ Mbar < 2.5), central or mid-channel bars appear. If M bar  2 (Mbar ≥ 2.5), multi-row bars appear in the cross-section. Correspondingly, the river is single-threaded (meandering or straight) if M bar  1 and braided if M bar  2. However, Equation 6.9 does not apply to forced bars, such as point bars, which are independent of morphodynamic instability and form close to the sources of force. Due to the limitation of the linear approach, Equation 6.9 overestimates the bar mode for width-to-depth ratios larger than 100, but correctly predicts the bar pattern (Crosato and Mosselman, 2009). Ahmari and da Silva (2011) developed a regime diagram of bar formation, channel meandering, and braiding on the B/h and h/D plane shown in Figure 6.18. The regions of alternate and multi-row bars are separated. The multi-row bars are further divided into 2- and 2+-row bars. The data of alternate bars and meandering channels overlap approximately in the same region with the same upper boundary but different lower boundaries. Some authors consider the early occurrence of alternate bars as the cause of channel meandering, but this has not been completely understood. The region of alternate bars in Figure 6.18 has the lower limit (line L0,1):  25(h D) 0.55 h D  25  B    (2 13) (h D) 25  h D  130   h 0,1  h D  130 20 

(6.10)

and the upper limit (line L1, 2): 25(h D)1/ 3 B     h 1, 2  146

h D  200  h D  200

(6.11)

Figure 6.18  Regime diagram of bars and channel planforms (after Ahmari and da Silva, 2011)

190  Bed Forms

The analysis of Lyster et al. (2022), based on a large database of river planforms, indicates that the depth/width ratio alone is sufficient to discriminate between single-thread (h/B > 0.02) and multithread (h/B < 0.02) rivers. Thus, bank cohesion is a critical determinant of channel planform. Furthermore, multithread rivers are likely to be anastomosing (avulsion-dominated) when S/Fr < 0.003 and braided (bifurcation-dominated) when S/Fr > 0.003. 6.2.4 Formations and Regime Diagrams of Coarse-Grained Bed Forms The step-pool systems in coarse-grained streams share similar ranges of kh and Fr with antidunes in sand-bed streams (Figure 6.14), so the antidune theory of Kennedy (1963) has been applied to describe the formation and development of step-pool systems (Chin, 1999; Chartrand and Whiting, 2000). However, these two types of bed forms are different in certain aspects. The water and bed surfaces are strongly coupled in sandy antidunes, but such coupling is less likely to occur in step-pool systems due to the much lower mobility of poorly-sorted gravel, cobble, and boulder mixtures. The stability of a step-pool system is controlled by the large stones comprising the steps. When the steps are entrained by a high or extreme flood, the step-pool system is completely mobilized. The antidune theory may be applicable in this stage. The formation of a new step-pool system likely starts when large stones first fall and become keystones on the bed during the receding flood. Then, steps form as more particles accumulate around the keystones. The flow over each step plunges and creates a pool downstream, at the end of which a hydraulic jump may form. Therefore, the keystone hypothesis likely holds in the receding flood period (Zimmerman and Church, 2001). Similarly, transverse ribs and clusters are created by the deposition of coarse particles during receding floods. Riffle-pool systems are highly related to the heterogeneity of bed materials and the feedback effects of flow structures. In straight and weakly sinuous streams, erosion creates pools in the areas with finer bed particles, whereas riffles form in the areas with coarser particles. In curved or meandering streams, pools and point bars are created and maintained by helical flow, which tends to sweep sediments toward the inner bank and shift the main flow toward the outer bank. Sediment transport in a riffle-pool system is size-selective, temporally sporadic, and patchy. At low flows, only sand particles are mobilized; they move over the gravel-bed riffles and settle onto the coarser beds in the pools. At high flows, the sand particles deposited in pools are eroded and transported throughout the system, while gravel particles are partially mobilized along preferential pathways, e.g., with high mobility observed in the centers and downstream ends of pools. Transverse ribs, step-pools, and riffle-pools are considered parts of the continuum of coarsegrained bed forms in mountain streams (Billi et al., 2014). The main fields of existence of these bed forms can be distinguished using the bed slope S and the ratio of bed material D84−100 to channel width B, as shown in Figure 6.19. D84−100 is either D84 or D100 indistinctly. Step-pool systems occur in steeper streams with higher D84−100/B ratios than riffle-pool systems do. The step-pool and riffle-pool regimes are divided roughly at a channel slope of 0.02. Buffington and Montgomery (2013) proposed a conceptual framework for the continuum of bed forms in steep streams. The bed-form regimes are a function of the imposed basin conditions, including topography (valley gradient, confinement), streamflow, sediment supply (caliber, volume), and valley substrate type (alluvial, bedrock, colluvial). Colluvial, cascade, and step-pool regimes occur in the uppermost steep streams prone to debris flow passages, whereas plane-bed, riffle-pool, and dune-ripple regimes occur in lower steep streams with the possible influence of riparian vegetation and large woody debris. More discussion on this can be found in Buffington and Montgomery (2013).

Bed Forms  191

Figure 6.19 Diagram of bed-form regimes in coarse-grained streams in terms of S and D 84−100 (from Billi et al., 2014)

6.3 Dimensions and Speeds of Equilibrium Bed Forms 6.3.1 Sand Ripples Ripples are about (100–1000)D in length (Nordin, 1976; Yalin, 1972) and (50–100)D in height (Nordin, 1976). Baas (1993) developed the following formulas for the length (Lr) and height (Δr) of sand ripples: Lr = 75.4logD50 + 197, Δr = 3.4logD50+18

(6.12)

Raudkivi (1997) developed similar formulas: Lr  245 D500.35,  r  18.16 D500.097

(6.13)

Note that Lr, Δr, and D50 in Equations 6.12 and 6.13 are in mm. These two sets of formulas have limitations since only the effects of sediment size are considered. Because the generation of sand ripples is affected by the viscous sublayer and the turbulence bursts near the bed, Lr and Δr also depend on the flow conditions. Yalin (1985) obtained the graphical relationship in Figure 6.20 for the relative ripple length Lr /D. The particle shear Reynolds number, Re* ( u D/ ), is a key parameter. In the range of Re* < about 3.5, Lr /D decreases as Re* increases. When Re* is beyond 3.5, Lr /D increases with increasing Re* in a sediment-size-dependent manner. The sand ripples diminish as Re* increases beyond about 11. Note that Re* is proportional to the ratio of the sediment size D and the viscous sublayer thickness δ ′(=5ν/u*). Thus, the decrease and subsequent increase in Lr /D in Figure 6.20 coincide with the ranges of D/δ ′ smaller and larger than about 1, respectively. Da Silva and Yalin (2017) presented the following formula to approximate the graphical relationship in Figure 6.20: Lr 3000   0 . 88 D D*  1  0.22 





(6.14)

192  Bed Forms

Figure 6.20 Length of sand ripples (from Yalin, 1985)

where η* = (u*/u*cr)2. The ripple steepness is given as (da Silva and Yalin, 2017)  r Lr  0.14r r e1 r r 

(6.15)

where ζr = 0.1(η* − 1); r = 1 if ζr ≤ 1, and r = ζr(2 − ζr) if 1 < ζr ≤ 2; ψr = 1 if Re* ≤ 2.5, and ψr = exp{ − [(Re* − 2.5)/14]2} if Re* > 2.5. 6.3.2 Sand Dunes Dune Length and Height Using a large quantity of data from various sources, Yalin (1972) established the relationship of dune length Ld with the particle shear Reynolds number and the flow depth/sediment size ratio shown in Figure 6.21a. The curve family can be represented with the following equation:   Z  40   Z  400  0.055 Ld  6 Z 1  0.01 e D Z 

Z  0.04 Re*

  

(6.16)

where Z = h/D. If the turbulent flow over the initially flat bed is hydraulically rough, the effect of viscous sublayer is negligible and Equation 6.16 is reduced to Ld = 6h ≈ 2πh, as proposed by Yalin (1964). Van Rijn (1984c) suggested the dune length Ld = 7.3h. The results of Yalin (1964) and van Rijn (1984c) agree well with the observations of Nordin (1976) that dunes are usually about (5–10)h long. The experiments of Bradley and Venditti (2019) show that dune length is nearly constant at low transport stages but increases with flow strength at high transport stages. Zhang (1961) reported that sand dunes were up to several meters high and hundreds of meters long in the Yangtze River (see Figure 6.4a). He derived the following formula for the dune height Δd:

Bed Forms  193

Figure 6.21 Sand dunes: (a) relative dune length as a function of h/D and R e* (from Yalin, 1972), and (b) dune steepness as a function of Θ/Θ cr and h/D (from Yalin and Karahan, 1979a)

1/ 4

d U h  0.086   h gh  D 



(6.17)

Barekyan (1962) developed a similar formula: d g  U U0  Kd 2  h Ch  U 0

  

(6.18)

where U0 is the noneroding mean velocity and Kd is a coefficient equal to about 20.6 (Simons et al., 1965). Ranga Raju and Soni (1976) related the height and length of sand ripples and dunes to the grain-related Shields number Θ′, the Froude number Fr, the particle densimetric Froude number Frd, and the submergence ratio Rb/D, as shown in Figure 6.22. They derived the nondimensional parameters Fr3 Frd  d D and  Rb D  Fr3 Frd Ld D based on the concept that bed-form migration results from bed load. Here,    b [( s   ) D], Fr  U gRb , and  b is the grain-related bed shear stress, as discussed in the next section. Van Rijn (1984c) established a relationship of dune height, as shown in Figure 6.23 and expressed as d D   0.11 50  h  h 

0.3

1  e

0.5T

  25  T   

(6.19)

where T* is defined in Equation 6.6. Karim (1995) developed an empirical formula for dune height using experimental data reported by Guy et al. (1966) and field data measured in the Missouri River. The graphical relationship between Δd and u*/ωs is shown in Figure 6.24. The data show that the dune height

194  Bed Forms

Figure 6.22 Relationships for the geometry of sand ripples and dunes: (a) height and (b) length (from Ranga Raju and Soni, 1976)

Figure 6.23 Relationship of sand dune height (from van Rijn, 1984c)

is less than 0.5h. This agrees well with Nordin’s (1976) observations that dunes are usually (0.1–0.5)h high. In the range of 0.15 < u*/ωs< 3.64, Δd is determined by 2

3

u  u  u  u  d  0.04  0.294     0.00316     0.0319     0.00272    h  s   s   s   s 

4



(6.20)

Bed Forms  195

Figure 6.24 Relative dune height as a function of u */ω s (from Karim, 1995)

where ωs is the settling velocity of sediment particles with size D50. Equations 6.17 and 6.18 do not describe the complete evolutionary picture of sand dunes because they indicate only a monotonous increase in dune height with increasing flow strength. The later-developed Equations 6.19 and 6.20 show that the dune height increases first and then decreases as the flow strength increases. Equations 6.19 and 6.20 reasonably represent the general process of dune formation and wash-out (e.g., Bradley and Venditti, 2019). However, Julien and Klaassen (1995) reported that van Rijn’s Equation 6.19 has significant errors for sand dunes in large rivers. In addition, Figures 6.23 and 6.24 do not have enough data to support the relationships in the regime from dunes washed-out to moving plane bed. Dune Steepness The dune steepness Δd/Ld can be calculated by using Equation 6.19 and Ld = 7.3h from van Rijn (1984c). Fredsøe (1975) developed the following formula for dune steepness using the data reported by Guy et al. (1966): 2

d   0.06  0.119 1   0.4    Ld  

(6.21)

Equation 6.21 gives zero values for Δd/Ld at Θ = 0.0615 and 2.4385 and a maximum value of Δd/Ld = 0.0567 at Θ ≈ 0.4. It covers ripples and dunes. Ripples occur at Θ < 0.3 and correspond to the rising part of the function (Raudkivi, 1998, p. 78). Yalin and Karahan (1979a) obtained the relationship of dune steepness with relative shear stress Θ/Θcr and relative depth h/D, as shown in Figure 6.21b. As Θ/Θcr increases, dune height increases first and then decreases, representing the processes of dune formation and wash-out. In addition, dune steepness increases as h/D increases.

196  Bed Forms

Using more data from various sources, da Silva and Yalin (2017) consolidated the graphical relationship in Figure 6.21b and presented the following formula: d  d    d e1 d  Ld  Ld  max





md

d 

(6.22) 0.4

135 ) (1  e 0.074 Z )  5 , md  1  0.6e 0.1(5 log Z ) ,  d  1  e  ( Re* ( 1d)  ((ˆd  1), (˘ˆd  where  d  0.47  ( Re* 10 )2, and   d  1 e   d Ld max 0.00047 Z 1.2 e0.17 Z  0.04 1  e0.002 Z . 3.6





Flemming (2000) collected 1,491 sets of data for a variety of bed forms from ripples (a few centimeters long) to mega-dunes (several hundred meters long) and obtained the following regression relationship between height and length (R2 = 0.98):  d  0.0677 L0d.81 

(6.23)

where Δd and Ld are both in meters. Equation 6.23 supports the self-similarity hypothesis of sand waves. The steepness of sand waves decreases as the wavelength increases. This indicates that smaller bed forms have a steeper face slope. Dune Speed Zhang (1961) and Barekyan (1962) derived the following formula for the migration speed of sand dunes, cd: cd U   c Fr2

(6.24)

where ζc is a coefficient, calibrated as 0.0144 by Zhang (1961) using the measurement data mostly from the Yangtze River, 0.92 by Barekyan (1962), and about 0.5 by Simons et al. (1965) using data from laboratory experiments. The large discrepancies between the ζc values need to be clarified by further measurements. Kondap and Garde (1973) proposed a similar formula: cd U  0.021Fr3

(6.25)

The weakness of Equations 6.24 and 6.25 is the absence of sediment properties. The following formula derived by Shinohara and Tsubaki (1959) considers more factors: m0

   b  0      s   ) D   s   1 gD3 cd h

(6.26)

where  b is the portion of bed shear stress related to sand dunes. Coefficient values are ζ0 = 48.6 and m0 = 1.5 for sediments 0.69–1.46 mm in size, and ζ0 = 76.1 and m0 = 2.5 for sediments 0.1–0.21 mm in size. Through a space-time Fourier decomposition of bed elevations, Guala et al. (2014) proposed the following function for the length and time scales of migrating bed forms of different types and sizes (e.g., ripples, dunes):

10 )2

Bed Forms  197

Ld/D = 2(Tdu*/D)1/2

(6.27)

where Ld and Td are the wavelength and period of bed forms, respectively. Considering cd = Ld /Td, Equation 6.27 gives a relationship of cd  Td1/ 2  Ld1. It indicates that larger bed forms migrate slower than smaller bed forms. The data used to validate Equation 6.27 cover the range of 0.2 < Fr < 0.5. 6.3.3 Sand Antidunes The linear stability analysis of Kennedy (1963) results in the following constraint for the antidune wavelength: Ld = 2πU 2/g, or Fr2  1 (kh)

(6.28)

Recking et al. (2009) presented Figure 6.25 to compare Equation 6.28 with the data obtained on both gentle and steep slopes. Equation 6.28 agrees relatively well with the data but tends to under-predict as slope increases. Equations 6.3 and 6.4 give a good estimate of the flow domain permitting antidunes. Based on dimensional analysis and the collected data set, Recking et al. (2009) developed the following formula for the wavelength of antidunes: Ld   0.093 3 Fr  D cr

(6.29)

where Θcr = 0.15S0.275 (Recking et al., 2008). Equation 6.29 gives results very similar to Equation 6.28 for gentle slopes but performs better on steep slopes. In addition, Recking et al. (2009) found that the ratio of antidune amplitude and wavelength is about 0.033, but the used data are quite scattered. The surface waves over antidunes break at high velocities. According to Kennedy (1963), the wave steepness at incipient breaking is about 0.14. The analysis of Núñez-González and Martín-Vide (2011) shows that the steepness of upstream-migrating antidunes has an upper limit of roughly 0.15, and the downstream-migrating antidunes can have a higher steepness than the upstream-migrating antidunes.

Figure 6.25 Comparison of Equation 6.28 and antidune data (from Recking et al., 2009)

198  Bed Forms

6.3.4 Alternate Bars Many channel regime studies have suggested that the alternate-bar and meander wavelengths La are roughly proportional to the channel width B as follows: La   a B ma 

(6.30)

where αa and ma are empirical coefficients. Leopold and Wolman (1957) reported that αa = 6.5 and ma = 1.1 for the wavelengths of meanders and riffles. Yalin (1972) found αa to be about 2π and ma to be about 1 for alternate bars. Ikeda (1984) suggested that αa varies between 4 and 17, with an average value of 9 for alternate sand bars. Palucis et al. (2018) found that αa is about 8 for alternate bars in a gravel-bed flume at slopes of 10% and 20%. Based on a linear stability analysis, Parker and Anderson (1975) suggested the following relationship for bar wavelength: La  2 1ch Bh 

(6.31)

where α1 is a function of Fr and is about 1.41 when Fr ≪1, h is the mean depth of flow, and ch  U ghS is the dimensionless Chezy resistance coefficient. According to Ikeda (1984), Equation 6.31 is valid at Fr 0.74 and 6 < B/h < 40: 1.45

a B  0.0442   h h

h   D

0.45



(6.35)

Jaeggi (1984): a B  0.219   B D

0.15



(6.36)

Bed Forms  199

and Yalin (1992): a h  0.18   B D

0.45



(6.37)

Cheng and da Silva (2019) compared Equations 6.35–6.37 against 192 sets of measurement data. Ikeda’s Equation 6.35 and Yalin’s Equation 6.37 have similar performance, whereas Jaeggi’s Equation 6.36 has a relatively large bias. Based on these data, Cheng and da Silva (2019) developed a more comprehensive formula: a3 a  0.22 a1e a2 [(1a ) 1]  h

(6.38) 2

where a1 = 0 if Re* 12; e* ≤ 12, and a1 2 a2 [0.68e 7.2 ( 0.05 Re* 1)  1]1[2.32(h D) 0.5  0.0038(h D)0.5  3.2] ; a3 = 2.5+3200(h/D)−3; and ςa = [(B/h) − (B/h)0,1]/[(B/h)1,2 − (B/h)0,1], with (B/h)0,1 and (B/h)1,2 determined using Equations 6.10 and 6.11. Equations 6.32–6.38 can be used to determine the alternate-bar steepness Δa/La. Palucis et al. (2018) found that Δa/La decreases as the Shields number increases. 6.3.5 Transverse Ribs, Riffle-Pool Sequences, and Step-Pool Systems Leopold et al. (1964) pointed out that the riffle-pool sequences in meandering streams commonly occur at an interval ranging from 5 to 7 times stream width. This is similar to the spacing of alternate bars and meanders in sandy streams. The data collected by Billi et al. (2014) from literature and field measurements show that for low-sinuosity, single-thread streams, the ratio of riffle spacing and channel width, LRP/B, ranges from 1.2 to 6.7 with a mean value of 3.9, whereas for braided streams, both the mean and the range are much less, i.e., 0.5 and 0.2–1.6, respectively. The significant reduction of this ratio in braided channels might be because the riffle spacing was measured on the main channel but the bankfull channel width was used as the reference width. Sear et al. (2003) proposed the following relationship for riffle spacing: LRP = 7.36B0.896S −0.03

(6.39)

Billi et al. (2014) suggested that Equation 6.39 is only significant for low-sinuosity singlethread streams. They found that for both low-sinuosity and braided streams the riffle spacing LRP (in m) is related to the bed slope S and particle size D84 (in m) as follows (R2 = 0.72): LRP = 0.0007D84 /S+115

(6.40)

Carling and Orr (2000) derived the following relationship between the riffle height ΔRP and length LRP:  RP  0.06 L0RP.666 

(6.41)

Equation 6.41 is similar to Equation 6.23 for sand waves, although the mechanical laws for sand ripples-dunes and gravel riffle-pool systems are significantly different.

200  Bed Forms

In step-pool systems, the steps are made of large particles that are imbricated and/or rooted in the underlain sediments. Thus, the step height ΔSP is closely related to the size of the particles comprising the steps. Egashira and Ashida (1991) reported that ΔSP is approximately equal to the mean particle diameter of armor coats. Similarly, Chin (1999) found that ΔSP is about 1.2 times the particle diameters, and Chartrand and Whiting (2000) reported that ΔSP ranges from about 1 to 1.5 median diameters. Billi et al. (2014) found that ΔSP is less than D84 and typically about 0.5–0.6 times D84. According to Chin (1999), the pool length is approximately 10 times the step height for rivers with slopes of 4%–12%. Considering that step-pools mostly occur in very steep streams (bed slopes > 2%), the stream bed slope is likely one of the controlling parameters for the step spacing. Many authors have suggested that the step steepness (height/spacing) ΔSP /LSP is a function of channel slope S (Judd, 1964; Abrahams et al., 1995; Chartrand and Whiting, 2000):  SP LSP  a1 S b1

(6.42)

where a1 and b1 are empirical coefficients. The value of b1 ranges between 0.68 and 1.19 (Chartrand and Whiting, 2000). Abrahams et al. (1995) experimentally derived that b1 is 1 and a1 is between 1 and 2. The values of a1 fall within the 1–2 range for more than 60% of the field data compiled by Billi et al. (2014). Transverse ribs usually are as high as the size of the coarsest particles. According to Billi et al. (2014), the transverse rib spacing/channel width ratio LTR /B lies between 1.5 and 3.0 for 94% of their data. The rib spacing is about 7 times the maximum particle size Dmax and about 4.5 times the flow depth h. 6.4 Partition of Grain and Form Resistances For a channel bed with sediment particles and bed forms, the bed resistance is composed of grain-related friction and form-related drag, as shown in Figure 6.26. The grain-related friction is proportional to the velocity squared, and the form-related drag varies with the evolution of bed forms. Thus, the bed shear stress is partitioned as

 b   b   b

Figure 6.26 Partition of movable bed resistance

(6.43)

Bed Forms  201

where  b and  b are the grain- and form-related shear stresses on the bed, respectively. This partition is theoretically valid, due to the fact that bed shear force is additive. Note that there is a third component of resistance corresponding to the portion of flow energy spent on carrying sediment particles. Its importance increases as the intensity of sediment transport increases. It is often considered in the coastal context (see Section 13.3.2) but is omitted or combined with the grain-related component in the riverine context, as it is usually small. The bed shear stress is usually calculated by τb = γRbS

(6.44)

If the bed shear stress on the left-hand side of Equation 6.44 is partitioned according to Equation 6.43, the term on the right-hand side can be handled by either partitioning the hydraulic radius Rb or the energy slope S since the specific weight of water is considered constant. Einstein (1942) suggested the partition of the hydraulic radius Rb into two parts: Rb and Rb , corresponding to grain and form roughness, respectively: Rb  Rb  Rb 

(6.45)

and determined the grain- and form-related shear stresses as

 b   Rb S,  b   RbS

(6.46)

The Manning equation is applied to the total, grain, and form resistances on the channel bed: U  Rb2 / 3 S 1/ 2 n , U  Rb2 / 3 S 1/ 2 n , U  Rb2 / 3 S 1/ 2 n 

(6.47)

where n, n′, and nʹʹ are the Manning coefficients corresponding to the total, grain, and form roughness, respectively. Note that equal velocity is assumed among the total, grain, and form resistances. This assumption is valid since all these resistances are applied on the bed. This is different from the partition of bed and bank resistances described in Section 3.5.2. Using Equations 6.44, 6.46, and 6.47 yields

 b   n n   b 

(6.48)

 b   n n   b c

(6.49)

3/ 2

3/ 2

Inserting Equations 6.48 and 6.49 into Equation 6.43 leads to 3/ 2

n3/ 2   n 

3/ 2

  n 



(6.50)

Unlike Einstein’s method, Engelund (1966) suggested a partition of bed shear stress according to the energy slope: S = S′ + Sʹʹ

(6.51)

and determined the grain- and form-related shear stresses as

 b   Rb S ,  b   Rb S  

(6.52)

202  Bed Forms

where S′ and Sʹʹ represent the portions of the energy slope corresponding to the grain and form roughness, respectively. The Manning equation is applied to the total, grain, and form resistances under the same flow velocity as follows: U  Rb2 / 3 S 1/ 2 n, U  Rb2 / 3 S 1/ 2 n , U  Rb2 / 3 S 1/ 2 n 

(6.53)

Using Equations 6.44, 6.52, and 6.53 yields

 b   n n   b 

(6.54)

 b   n n   b 

(6.55)

2

2

Inserting Equations 6.54 and 6.55 into Equation 6.43 leads to 2

2

n 2   n    n  

(6.56)

The grain roughness coefficient n′ is often calculated using Strickler-type formulas, 1/ 6 such as n′ = D1/6/21.5 (Strickler, 1923), n  D50 20 (Li and Liu, 1963; Wu and Wang, 1999), 1/ 6 1/ 6 n  D65 24 (Patel and Ranga Raju, 1996), n′ = (2D65)1/6/25 (Wilcock et al., 2009), n  D84 20.4 1/ 6 (Rickenmann and Recking, 2011), and n  D90 26 (Meyer-Peter and Mueller, 1948). Here, the units of sediment sizes and n′ are m and s/m1/3, respectively. Substituting one of these n′ formulas and the Manning Equation 3.69 into Equation 6.54 or 6.48 yields a Manning-Strickler type equation for  b (Laursen, 1958; see Section 9.1). Note that the exponents are 3/2 in Equation 6.50 but 2 in Equation 6.56. A linear partition of Manning n is sometimes used in the literature, which is not so theoretically sound as Equations 6.50 and 6.56. Despite of the difference in these exponents, both Einstein’s and Engelund’s methods give the following relationship: 1 Ch2



1 1  2  2 Ch Ch

(6.57)

where Ch, Ch , and Ch are the Chezy coefficients corresponding to the total, grain, and form roughness, respectively. Equation 6.57 can also be derived by applying Equation 3.65 to the total, grain, and form roughness, i.e.,  b   gU 2 Ch2,  b   gU 2 Ch2 and  b   gU 2 Ch2 , and then substituting these relationships into Equation 6.43. Similarly, the following formula can be derived by applying Equation 3.55 as τb = ρfU2/8,  b   f U 2 8 and  b   f U 2 8 and then substituting them into Equation 6.43: f = f ′ + f ʹʹ

(6.58)

where f, f ′, and fʹʹ are the Darcy-Weisbach resistance coefficients for the total, grain, and form roughness, respectively. Correspondingly, the portion of the grain shear stress over the total bed shear stress is given as

 b  b   Ch Ch   f  f  2

(6.59)

Bed Forms  203

Equation 6.59 is the same for both the Einstein and Engelund approaches. The following similar formula is derived by defining f   8 gn2 Rb1/ 3 and substituting it into Equation 6.48 (Schneider et al., 2015; Wu and Lin, 2019):

 b  b   f  f   3/ 4

(6.60)

Different exponents appear in Equations 6.59 and 6.60 because f   8 gn2 Rb1/ 3 is based on the Engelund approach but substituted into Equation 6.48, which belongs to the Einstein approach. Equation 6.60 is treated as an empirical approach, and the difference may be compromised by recalibrating the roughness coefficients. The bed roughness height ks is often partitioned into grain and form roughness heights, k s and k s , as follows: k s  k s  k s 

(6.61)

Unlike Equations 6.57 and 6.58, Equation 6.61 is not strictly derived from Equation 6.43. The grain roughness height k s is given a variety of values in the literature, such as D65 (Einstein and Barbarossa, 1952), 2D65 (Engelund, 1966), 2D90 (Kamphuis, 1974), (1−3)D90 (van Rijn, 1984c, 2007a), (1.5−3)D90 (Wu and Lin, 2014), 3.5D84 (Hey, 1979), and about 4D84 (Ferguson, 2007). A typical example of this approach is given in Equation 6.75. Bed forms of different scales, such as ripples, dunes, and bars, may coexist in natural rivers. In such cases, the form roughness is further partitioned into subcomponents, and the form roughness terms in Equations 6.43, 6.50, 6.56–6.58, and 6.61 are replaced with ∑ b, ∑(nʹʹ)3/2, ∑(nʹʹ)2, ∑(1 Ch2 ), ∑  f ʹʹ, and ∑ k s, respectively. An alternative approach is given in Equation 6.78 proposed by van Rijn (2007a).



6.5 Resistance Formulas for Sand-Bed Rivers Einstein and Barbarossa (1952), Engelund (1966), and Alam and Kennedy (1969) proposed empirical methods for separately calculating the grain and form resistances to flow. Li and Liu (1963) and Wu and Wang (1999) suggested direct calculations of the total roughness coefficient. Van Rijn (1984c), Yalin (1992), and Karim (1995) established empirical relationships of the resistance coefficient as functions of the height and length of bed forms. Brownlie (1983) proposed a formula for directly calculating the flow depth. Alam and Kennedy’s (1969) method determines f ′ and fʹʹ in Equation 6.58 using two experimentally obtained graphical relationships, which are not described here. The other methods in this list are introduced in the following text. Method of Einstein and Barbarossa (1952) In Einstein and Barbarossa’s (1952) method, the movable bed resistance is divided into grain and form components according to Equation 6.45. The grain-related resistance is determined by using the formula of Keulegan (1938) with k s = D65: U u  5.75 log 12.27 Rb  s k s   where u is the grain shear velocity and χs is the coefficient described in Figure 3.9.

(6.62)

204  Bed Forms

Figure 6.27 Bed-form resistance relationship of Einstein and Barbarossa (1952)

To determine the form resistance, Einstein and Barbarossa (1952) established the relation ( s   ) D35 ( Rb S ) shown in Figure 6.27. Here, Rb is used in the ship between U u and  formula for Ψ based on the concept that only the grain shear stress contributes to bed-load transport. When Rb and S are given, the following calculation procedure is required to obtain the flow velocity U: (1) (2) (3) (4) (5)

Guess a value of Rb. Calculate u  gRb S and U using Equation 6.62. Calculate Ψ and find the value of U u from Figure 6.27. Determine u from U u , and then Rb from u  gRbS . Check whether Rb  Rb  Rb is satisfied. If yes, the calculation stops; otherwise, guess a new value of Rb and repeat from step 2.

Einstein and Barbarossa’s method is only applied in the lower flow regime. It performed well with field data in the Rio Grande (Nordin, 1964) and the Missouri and Mississippi Rivers (Harrison and Mellema, 1967). However, poor performance was obtained with flume and field data from Garde and Ranga Raju (1966). A modification of Einstein and Barbarossa’s method was proposed by Shen (1962) by adding the particle Reynolds number ωsD/ν. Method of Engelund (1966) In the method of Engelund (1966), the total bed shear stress is determined using the partition of energy slope according to Equation 6.51. However, the grain shear stress is calculated using the hydraulic radius partition approach by iteratively solving u  gRb S and the following logarithmic formula:

Bed Forms  205

Figure 6.28 Relationship of bed-form resistance (from Engelund, 1967)

U u  5.75 log  Rb k s   6 

(6.63)

with k s  2 D65 . Based on the flume data of Guy et al. (1966) and the similarity theory of alluvial rivers, Engelund (1966) developed the relationship between Θ and Θ′ shown in Figure 6.28. The relationship is discontinuous at about Θ′ = 0.55. It can be expressed as follows in the lower flow regime: Θ′ = 0.06 + 0.4Θ2 for Θ′ < 0.55

(6.64)

and in the upper flow regime:      1.8 1/1.8 (0.3  0.7 )

0.55    1    1

(6.65)

The calculation procedure of Engelund’s (1966) method is similar to that of Einstein and Barbarossa’s (1952) method and also needs iteration. Wright and Parker (2004) found that Equation 6.64 performs poorly for large, low-slope sand-bed rivers. They suggested the following modification:



  0.05  0.7 Fr0.7



0.8

for Θ′ < 0.55

(6.66)

Method of Yalin (1964, 1992) Yalin’s (1964, 1992) method is based on the partition of energy slope. Equation 3.65 is applied to determine the grain-related energy slope: S 

U2  ch2 gh

(6.67)

206  Bed Forms

where ch is the dimensionless Chezy coefficient related to grain roughness, determined using the following equation derived by depth-averaging Equation 3.38: ch 

1  h   Bs  ln 0.368   2 D 

(6.68)

where Bs is given in Equation 3.40. As shown in Figure 6.1c, the flow over the crest of a bed form may detach and form a recirculation zone at the lee side. This results in a local head loss, which can be approximated as hf 

U1  U 2  2g

2



(6.69)

where U1 is the average flow velocity above the crest and U2 is the average velocity over the trough. Equation 6.69 can be derived by applying the continuity and momentum equations. Equation 6.69 is manipulated as 2

2

 U 2  d  q q 1  hf       2 g  h  0.5 d h  0.5 d  2 g  h 

(6.70)

where Δd is the bed-form height and h is the average flow depth. The last term of this equation is derived by assuming a small Δd /h ratio and applying the first-order approximation to the two quotients in the parentheses in the middle section. The form-related energy slope Sʹʹ is the head loss hf divided by the bed-form wavelength L d, i.e., 2

S  

U 2  d   2 gLd  h 

(6.71)

Equation 6.71 was derived by Yalin (1964) and Engelund (1966). However, Engelund (1966) did not directly apply it. Substituting Equations 3.65, 6.67, and 6.71 into Equation 6.51 yields (Yalin, 1964) 2

1 1 1  L  2  d d  2 2  Ld  h ch ch

(6.72)

where ch is the dimensionless Chezy coefficient due to the total roughness. The values of Ld and Δd are calculated using Equations 6.16 and 6.22, respectively. Then, the total Chezy coefficient is used in Equation 3.65 to compute the flow velocity. Iteration is needed in this procedure. Since energy losses are additive, Equation 6.72 can be generalized as follows for the case of superimposed sand ripples, dunes, and bars (Yalin, 1992): 2 2 2   d   a  1 1 1   r        L L L       r d a ch2 ch2 2h  Lr    Ld   La  



(6.73)

Bed Forms  207

where Δa and La are the height and length, respectively, of alternate bars or other types of bars. According to da Silva and Yalin (2017), the contribution of alternate bars is approximately one order of magnitude smaller than that of sand dunes and ripples. The importance of other sand bars needs to be investigated. Method of Brownlie (1983) Using a large database, Brownlie (1983) developed the following regression equations for the lower and upper flow regimes separately:  0.3724q0.6539 S 0.2542 g0.1050 R  D50 0.2836q0.6248 S 0.2877 g0.08013 

lower regime upper regime



(6.74)

where R is the hydraulic radius, σg is the geometric standard deviation of bed sediment, and q  q ( gD503 )1/ 2 . The used data cover D50 from 0.088 to 2.8 mm, q from 0.012 to 40 m2/s, S from 3.0×10 −6 to 3.7×10 −2, R from 0.025 to 17 m, and water temperatures from 0 to 63°C. Equation 6.74 can be used as a hydraulic geometry equation for an alluvial channel under given flow discharge, sediment size, and channel slope. For a given set of independent variables, Equation 6.74 gives two possible depths or hydraulic radii. The correct solution needs to be determined using the flow regime diagram of Brownlie (1983) shown in Figure 6.17. Van Rijn’s (1984c, 2007a) Formulas In van Rijn’s (1984c) method, the grain roughness height is determined as k s  3D90 and the form roughness height is k s 1.1 d (1  e 25d Ld ) . Therefore, the total bed roughness is given according to Equation 6.61 as



ks  3D90  1.1 d 1  e 25d

Ld



(6.75)

The Chezy coefficient is then calculated as Ch = 18log(12Rb/ks)

(6.76)

where Rb is determined by using Vanoni and Brooks’ (1957) method to exclude the sidewall effect. In Equation 6.75, Ld = 7.3h, and Δd is determined using Equation 6.19. The Chezy coefficient is used in Equation 3.65 to compute the flow velocity. All these equations are solved together iteratively. The grain-related Chezy coefficient Ch is calculated using Equation 6.76 with k s  3D90 , i.e., Ch  18 log 12 Rb k s   18 log  4 Rb D90  

(6.77)

Van Rijn (2007a) proposed a different bed-form roughness formula to consider the contributions of superimposed ripples, mega-ripples, and dunes:



k s  k s, 2r  k s, 2mr  ks, 2d



1/ 2



(6.78)

208  Bed Forms

In this formula, k s, r , k s, mr , and k s, d are the roughness heights due to ripples, mega-ripples, and dunes, respectively: k s, r  f cs D50 {85  65 tanh[0.015(  150]}



(6.79)

 k s, mr 0.00002 f fs h[1  exp( 0.05 )](550  )  k s, d 0.00008 f fs h[1  exp( 0.02 )](600  )

 

(6.80) (6.81)

where ψ = U2/[(ρs/ρ − 1)gD50] is the mobility parameter, fcs = min[1, (0.25Dgravel/D50)1.5] with Dgravel = 0.002 m, and ffs = min[1, D50/(1.5Dsand)] with Dsand = 0.000062 m. The correction factors fcs and ffs account for the effects of gradually decreasing ripple height in coarse sediment beds and decreasing mega-ripple and dune heights in fine sediment beds, respectively. The expressions of fcs and ffs were guessed by van Rijn (2007a). Equation 6.78 is not theoretically derived and thus is treated as an empirical approach, like Equation 6.61. Karim’s (1995) Formula Karim (1995) proposed the following formula to determine the Manning roughness coefficient on a movable bed after the dune height Δd is determined with Equation 6.20:  n 0.037 D500.126 1.2  8.92  d h 

0.465



(6.82)

where h is the hydraulic depth, i.e., the flow area divided by the water surface width. The units of n and D50 are s/m1/3 and m, respectively. Formulas of Li and Liu (1963) and Wu and Wang (1999) Following the Strickler formula, the Manning n on an alluvial bed is related to the bed sediment size D by

Figure 6.29 Relationship between A n and U/U cr (from Li and Liu, 1963)

Bed Forms  209

n = D1/6/An

(6.83)

where An is a parameter related to bed-material size composition, particle shape, bed forms, flow conditions, etc. For a stationary flat bed with nonuniform sediment particles, D is often set as the median size D50, and An is about 20 (Li and Liu, 1963; Wu and Wang, 1999). Here, the units of n and D are the same as those used for Equation 6.82. If sediment particles with slightly irregular shapes are tightly placed on the bed, An may have a larger value of up to 24 (i.e., lower resistance to flow). If sediment particles with rather irregular shapes are loosely placed on the bed, An has a smaller value between 17 and 20. Other values of D and An can be found in Section 6.4. For a movable bed, An should consider the effects of bed forms. Li and Liu (1963) proposed a relationship between An and U/Ucr for natural rivers as shown in Figure 6.29. Here, Ucr is the critical average velocity of incipient motion. The relationship reasonably represents the formation and wash-out of ripples and dunes in the lower flow regime. However, Li and Liu (1963) set the exponent of D in Equation 6.83 to 1/6 for the Yangtze River and 1/5 for the Yellow and Gan Rivers. This inconsistency limits the applicability. The relationship does not agree well with most of the flume and field data used in the testing performed by Wu and Wang (1999). To improve this, Wu and Wang (1999) established a relationship between An ( g 1/ 2 Fr1/ 3 ) and  b  cr 50 , as shown in Figure 6.30. For convenience of use, the relationship in the range of 1   b  cr 50  55 is approximated with (Wu and Wang, 2001) An 8[1  0.0235( b  cr 50 )1.25 ]   g 1/ 2 Fr1/ 3 ( b  cr 50 )1/ 3

(6.84)

The critical shear stress τcr50 in Equation 6.84 is calculated using the Shields curve modified by Chien and Wan (1983). The grain shear stress  b is calculated using Equation 6.48, with n′ 1/ 6 calculated by n  D50 20 and τb by Equation 6.44. The bed hydraulic radius Rb is determined using Williams’ (1970) Equation 3.86. The Manning n calculated with Equations 6.83 and 6.84 is used in Equation 3.69 to calculate the flow velocity. Iteration is needed in this calculation procedure.

Figure 6.30 Relationship between An ( g1/ 2 Fr1/ 3 ) and  b  cr 50 as sand ripples and dunes are generated first and then washed out in the lower flow regime (from Wu and Wang, 1999)

210  Bed Forms

6.6 Resistance Formulas for Gravel- and Cobble-Bed Rivers In gravel- and cobble-bed streams, besides the friction of basal grain roughness, form drag exists due to large roughness elements such as bars, dunes, ribs, clusters, riffle-pools, steppools, boulders, and woody debris. The form drag becomes increasingly important at low flow stages. Traditionally, the Manning equation and log law have been applied to coarse-bed streams with calibrated roughness parameters. Normally, the Manning n increases as the flow depth decreases or the channel slope increases (Jarrett, 1984). Hey (1979) adopted Equation 3.64 with ks = 3.5D84 for the friction coefficient in gravel-bed streams:  h 8  6.25  5.75 log  f  3.5 D84

 (6.85)   where h is the average flow depth. Similarly, López and Barragán (2008) set ks = 2.4D90, 2.8D84, and 6.1D50. Ferguson (2007) developed the following variable power law of flow resistance to consider the effect of shallow submergence: 8  f

a1a2  R D84  2 1

2 2

a a

R

D84 



(6.86)

5/ 3

with a1 = 6.5 and a2 = 2.5. Equation 6.86 is reduced to the Manning-Strickler type equation 8 f  a1 ( R D84 )1/ 6 at high R/D84 values. This equation is derived by substituting the Strickler 1/ 6 type equation n  D84 into the Manning Equation 3.69. Cheng (2017) modified the Manning-Strickler equation as follows to account for the flow constriction induced by large-scale roughness elements: 2

1/ 3

k  k    f 0.115 1  0.75 s   s   h h 

(6.87)

where k s   g2.33 D50 for unimodal sediment mixtures (Cheng, 2016a). Equation 6.87 may have limitations because it ignores the seepage flow effects, which can be important under shallow submergence (Cheng, 2017). Using a large set of data, Rickenmann and Recking (2011) proposed a formula similar to Equation 6.86 written as a nondimensional hydraulic geometry relationship: U gSD84

  q   1.5471  gSD843   

0.7062

  q 1     10.31 gSD 3 84  

   

0.6 6317

   

0.493



(6.88)

Furthermore, Nitsche et al. (2012) introduced the concept of boulder concentration to represent the macro-roughness in steep streams for this type of relationship: U gSD84

0.6

  q   (1.4  1.73)    gSD843   

(6.89)

Bed Forms  211

where Γ is the boulder concentration, defined as  N b Db2 (4WR LR ), in which Nb is the number of boulders, Db is the boulder diameter, WR is the channel width, and LR is the channel length. However, Equation 6.89 was tested with limited available data compared to those used for Equation 6.88. To consider the effects of sediment transport and shallow submergence, Recking et al. (2008) modified Equation 6.85 into the following form:   8 R  6.25  5.75 log   f   RL BR D 

(6.90)

where αRL = 4(R/D)−0.43 with 1 ≤ αRL ≤ 4, and αBR = 7S 0.85(R/D) with 1< αBR ≤ 2.6. These two parameters consider the increasing influence of the roughness layer at small relative depths and the flow resistance caused by bed load. Note that Equation 6.90 was developed for flat beds, and R was corrected to exclude the sidewall effect. Alternatively, Mendicino and Colosimo (2019) modified several existing resistance formulas by adding correction terms formulated as functions of the Froude number and the Shields parameter to consider the effects of sediment transport and flow regime. For example, the Limerinos (1970) equation is modified as  0.041   0.3 F 0.47  0.05 1.2  R D  520.65 0.1129 R1/ 6 cr r 84 n   1.16  2.0 log( R D84 ) 0.026   cr 0.85 Fr1.47  0.05 0.14  R D84  1.2 

(6.91)

According to Palucis et al. (2018), Equations 6.86 and 6.90 significantly underpredict the resistance coefficient when alternate bars appear. Equations 6.85 and 6.88 likely have similar behaviors. Parker and Peterson (1980) proposed an empirical formula for bar resistance and used the slope partition method to determine the reach-averaged bed shear stress. Some of the partition methods described in Section 6.5 can be applied to coarse-bed streams, with modifications for different roughness elements, such as bars, dunes, clusters, steps, riffles, boulders, and woody debris. Typically, the logarithmic or Strickler-type friction equation is used to determine grain shear stress. For example, Parker and Peterson (1980) used Equation 6.63 with k s  2 D90. Ricken1/ 6 20.4 , which is rewritten as follows: mann and Recking (2011) used n  D84 1/ 6

 R  8  6.5    f  D84 

(6.92)

David et al. (2011) applied an additive partition method to determine the flow resistance in step-pool and cascade streams. The drag force on a submerged object (e.g., step, boulder, woody debris) is calculated by using the following quadratic law: FD = CD ρAFU2/2

(6.93)

where CD is the drag coefficient and AF is the frontal area of the object. The drag force is converted to the bed shear stress by dividing FD by the channel surface area over which FD is

212  Bed Forms

applied, and then converted to the energy slope corresponding to the object. The total energy slope is obtained by summing all the individual components. David et al. (2011) compared a variety of methods for the individual components and found that uncertainties and limitations exist in this partition approach. Even for the grain roughness component, it is difficult to determine which method is most accurate. The drag coefficients for steps and wood debris vary with flow conditions and require further investigation. In addition, energy dissipations due to other unconsidered mechanisms, such as submerged nappes, plunging pools, and skimming flows, may be important, particularly at low flows. 6.7 Comments on Movable Bed Roughness Formulas Though the prediction of flow resistance in movable-bed rivers has been improved in recent decades, the available formulas and methods for bed-form regimes, dimensions, and resistance effects are empirical or semi-empirical. The formulas introduced in Sections 6.5 and 6.6 consider only certain types of bed forms, such as ripples and dunes. Applying them in complex field cases is restricted due to the presence of other roughness elements, such as bars, vegetation, and structures. These formulas are calibrated and validated using only certain measurement data, so applications to different situations should be made cautiously and further validation with site-specific data is necessary. The Manning n values calibrated for a specific site normally change with the flow depth or discharge and need to be recalibrated once the flow regimes and channel configurations have changed. Uncertainties can be reduced by incorporating more physical processes through the formulas introduced in the previous sections. For example, Ferguson (2021) suggested replacing D84 in Equations 6.85 and 6.86 with an effective roughness height that can be calibrated by using measurement data. This approach enhances predictability when the n values vary considerably with the flow depth. This is because these two equations can consider such variations to some extent. The present formulas of bed forms and resistance primarily focus on either sand- or gravel-bed streams under transport-limited conditions. It is recognized that the nonuniformity of sand-gravel mixtures affects bed surface features, such as sand ribbons, barchans, dunes, and bars (Kleinhans et al., 2002). In addition, the supply-limited streams exhibit different bed features. Much work is required to examine bed forms in mixed sand-gravel-bed streams and under supply-limited conditions and develop analytical tools to tackle these cases in practice. The effects of bed forms can be considered through either bed roughness or channel topography in hydrodynamic analyses. In simple analytical models and one-dimensional numerical models, bed forms are usually treated as roughness elements. In two- and three-dimensional numerical models, fine meshes can be used to describe the large-scale bed forms, such as bars and islands, on the channel topography, so only the bed forms somewhat smaller than the mesh resolution need to be treated as roughness elements (Wu, 2007). Homework Problems 6.1 In a straight flume with a sandy bed, how do the bed forms evolve as the flow strength increases?

Bed Forms  213

6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

What are the typical large-scale bed forms in alluvial rivers? What are the typical bed forms in mountainous rivers? Describe the phase diagram in Figure 6.14 derived from the linear stability theory. What is grain roughness? What is form roughness? Describe a method to determine the Manning coefficients corresponding to grain and form roughness. Figure 6.26 portrays a relationship between the Darcy-Weisbach friction factor f and the flow velocity in an alluvial river. Explain the characteristics of bed roughness in different stages of bed forms. Derive Equation 6.57 using Einstein’s and Engelund’s partition methods. Derive Equations 6.59 and 6.60, and then explain the reasons for the different exponents. Consider a uniform flow in a sand-bed channel. The mean velocity is 0.7 m/s, the flow depth is 0.6 m, the bed slope is 0.0005, the water temperature is 20°C, the bed sediment size D50 = 0.35 mm, and D90 = 1 mm. Predict the regime of bed forms using the following methods: (a) Garde and Albertson (1959) (b) Simons and Richardson (1961) (c) van Rijn (1984c)

6.10 Given a mean velocity U = 0.8 m/s, flow depth h = 0.6 m, bed slope S = 0.00075, and median diameter D50 = 0.25 mm, determine the regime of bed forms using the following methods: (a) Brownlie (1983) (b) Karim (1995) 6.11 Use the regime diagram of Ahmari and da Silva (2011) shown in Figure 6.18 to determine the channel type and bar characteristics for a given channel with width B = 100 m, flow depth h = 1.2 m, and sediment median diameter D50 = 0.25 mm. 6.12 Given a unit flow discharge of 1 m2/s in a wide channel with a bed slope of 0.0007, sediment size D50 = 0.45 mm, and D90 = 1.5 mm, calculate the dune height using the following methods: (a) Karim (1995) (b) van Rijn (1984c) 6.13 Consider a wide channel with a bed slope of 0.001, sediment size D50 = 0.5 mm, and geometric standarad deviation σg = 2. If the unit flow discharge is 0.5 and 10 m2/s, determine the uniform flow depth using the following methods: (a) Brownlie (1983) (b) Engelund (1966) 6.14 Given a unit flow discharge of 2 m2/s in a wide channel with a bed slope of 0.0005 and a bed-material median size of 0.4 mm, determine the Manning coefficient using the following methods: (a) Yalin (1964, 1992) (b) Wu and Wang (1999)

214  Bed Forms

6.15 Given a unit flow discharge of 5 m2/s in a wide channel with a bed slope of 0.005, a median bed-material size of 120 mm, and a standard deviation σg = 1.5, determine the Darcy-Weisbach resistance coefficient using the following methods: (a) Cheng (2017) (b) Ferguson (2007) (c) Hey (1979) (d) Recking et al. (2008)

Chapter 7

Bed-Load Transport

Bed load consists of the sediment particles that move in proximity to the bed. The regimes, particle dynamics, mean motion characteristics, unisized and multisized transport capacities, fluctuations, and measurements of bed load are introduced in this chapter. 7.1  Bed-Load Regimes At a low flow, bed-load particles usually move by rolling, sliding, and saltation on a flat bed (Figure 1.11). Saltation is the dominant mode, while rolling and to a lesser extent sliding occur only near the threshold of incipient motion and between individual saltations (Bagnold, 1956; Bridge and Dominic, 1984). Thus, this bed-load regime is called the saltation regime. If the flow strength increases, sand ripples and dunes are generated and bed-load transport is controlled by the migration of these bed forms. This is illustrated in Figure 7.1 with a contour map of the mean sediment flux cu over mobile dunes. The sediment concentration c and velocity u were measured by Naqshband et al. (2014) using an acoustic system. A significant part of the total sediment flux is bed load, which is entrained from and moves along the stoss side of the dunes and is entirely captured on the lee side with zero-transport rates at the flow reattachment points. This regime is called the ripple-dune regime. When the flow strength reaches a certain stage, dunes disappear and bed load moves in a laminated layer over a plane bed. This regime is called sheet flow or laminated load (Bagnold, 1956; Wilson, 1987; Sumer et al., 1996; Gao, 2008). The bed-load regimes described here coincide approximately with the lower and transitional flow regimes in sand-bed streams shown in Figure 6.1. In the upper flow regime, antidunes

Figure 7.1  M ean streamwise sediment flux cu along a dune (Δ d and L d are the dune height and length, respectively; open circles indicate the flow reattachment points) (from Naqshband et al., 2014) DOI: 10.1201/9781003343165-7

216  Bed-Load Transport

and chute-pools likely play important roles for bed-load transport. In contrast, the ripple-dune regime may not exist in gravel-bed streams. Thus, saltation and sheet flow are the main regimes of gravel transport. In addition, migration of bars (e.g. alternate bars, middle-channel bars) and erosion of bed features (e.g., steps, riffles) also induce bed-load transport. 7.2  Bed-Load Particle Dynamics The motion of a sediment particle with the flow of water (Figure 7.2) can be described with Newton’s second law in vector form:

s

D 3 dus  F 6 dt

(7.1)



where t is the time, D is the nominal particle diameter, ρs is the sediment density, us is the vector of particle velocity, and F is the vector for forces acting on the particle. Equation 7.1 is valid not only for bed load, but also for suspended load. Besides gravity and buoyancy, the particle experiences drag force, inertia force, Basset (1961) force, Magnus force, Saffman (1965) force, lift force due to asymmetric particle shape, and collision force. The gravity and buoyancy forces result in the submerged weight of the particle:  W s

 s   

D 3 g 6 

(7.2)

where ρ is the water density and g is the vector of gravitational acceleration. The drag force on the particle is  FD

1 CD  Ap u f  us u f  us 2





(7.3) 

where CD is the drag coefficient; uf is the flow velocity; and Ap is the projection area of the particle, given as πD2/4. Inertia force includes the pressure-gradient force and the added mass force due to the relative acceleration of the surrounding fluid around the particle. By simplifying the momentum equation, the acceleration-induced pressure gradient can be approximated as − ∇p ≈ ρd(uf − us)/dt.

Figure 7.2  Motion of a sediment particle in water

Bed-Load Transport  217

Integrating this equation around the particle and considering the surrounding fluid moving together yield (Morison et al., 1950; Sorensen, 2006)  FM CM 

D 3 d u f  us 6 dt







(7.4)

where CM is the inertia coefficient, expressed as 1 + Km, in which 1 represents the acceleration-induced pressure-gradient force and Km is the added mass coefficient accounting for the surrounding fluid moving with the particle. Theoretically, Km = 1/2 for a sphere in an inviscid, incompress1.05  0.066 Ac2  0.12 , ible fluid. Odar and Hamilton (1964) experimentally obtained K m  in which Ac = ǀuf – usǀ2/[Dd(uf – us)/dt] is the acceleration modulus. Note that there has been a debate in the literature on whether CM is 1 + Km or Km. In the present text, CM is treated as a calibrated parameter, as shown in Section 4.9. Generally, CM varies with particle shape. It is about 1.5–2.0 for a sphere and about 2.0 for a vertical cylinder. When a particle is impulsively set into motion from rest, the unsteadiness of the boundary layer exerts a force, called the Basset (1961) force, on the particle. This force is related to the viscosity of water and the course of acceleration as follows:



Fba  K ba 

t

D 2  1 d  u f  us  dt  0 t  t  dt 4 









(7.5)

where v is the kinematic viscosity of water and t is an integration variable in the range of 0  t  t . For a spherical particle, Basset (1961) analytically derived Kba = 6, and Odar and Hamilton (1964) experimentally obtained Kba = 2.88 + 3.12/(Ac + 1)3. When a rotating particle moves in water, it experiences a force perpendicular to the rotation axis and the relative velocity uf − us. This force is called the Magnus force and expressed as follows (Rubinow and Keller, 1961):  F  ml

D 3   u f  us 8







(7.6)

where ω is the angular velocity of rotation. If a particle moves in the flow field with a transverse velocity gradient, the particle experiences a transverse lift force, called the Saffman (1965) force. If the particle Reynolds number is very small, the Saffman force on a sphere is Fsa

1.62  D

2



du f dy

1/ 2

u f  us nL 

(7.7)

where y is the transverse direction and nL is the unit vector normal to the relative velocity uf − us. The sediment particle may experience another lift force if its shape is asymmetric. If the orientation of particles is random, the lift forces due to asymmetric particle shape on a group of

218  Bed-Load Transport

solid particles may offset one another. For the sake of simplicity, this lift force is often combined with the Magnus and Saffman forces into a total lift force expressed as  FL

2 1 CL  Ap u f  us nL  2

(7.8)

where CL is the lift coefficient. Compared with the drag force, the Basset force is important only at the initial stage of acceleration, the Magnus force needs to be considered only for particles rotating at high speeds, and the Saffman force is strong only in the region of large velocity gradients (Liu, 1993). According to Saffman (1965), the Magnus force is an order of magnitude smaller than the Saffman force for a viscous flow. As observed by Lee and Hsu (1994), the Magnus effect increases the saltation height and length by up to 12%. According to Chepil (1961), once a particle moves, the lift force tends to decrease while the drag force increases. As indicated by Hu and Hui (1996) and Nasrollahi et al. (2008), the drag and lift coefficients vary in the rising and descending stages of saltation, as well as with the distance above the bed due to the effect of the boundary. In addition, when a moving sediment particle collides with or contacts another moving particle or the bed surface, it experiences a collision or friction force. The collision and friction forces increase as sediment concentration increases (Bialik, 2011). Equation 7.1 has been solved to obtain the saltation trajectories of individual sediment particles in shear flows over smooth and rough beds and then derive the characteristics of bed-load transport (e.g., Yalin, 1972; van Rijn, 1984a; Hu and Hui, 1996; Lee et al., 2000; Nasrollahi et  al., 2008; Maldonado and Borthwick, 2015). However, wide applications of Equation 7.1 encounter difficulties due to the tremendously large number of heterogeneous sediment particles with complex shapes and random packing configureurations in field conditions. 7.3  Average Characteristics of Bed-Load Particle Motion For bed-load transport over a flat bed, Einstein (1942, 1950) set the bed-load layer thickness δb as 2D, the particle saltation step length lb as about 100D, and the bed-load velocity usb as 11.6 times the grain-related shear velocity. Later studies have demonstrated that these estimates are oversimplified. Poreh et al. (1970) observed step lengths in the range of (5–40)D for bed-load particles of D = 1.9 mm. Fernandez Luque (1974) observed step lengths of about (16 ± 5)D in experiments of bed-load transport using sediment, magnetite, and walnut particles (D = 0.9–3.3 mm). Van Rijn (1984a) solved Equation 7.1 for a sediment particle in saltation considering the submerged weight, drag force, added mass force, Saffman force, and Magnus force, and using the log law for the flow velocity. He obtained the following relationships to approximate the solutions for saltation height δb and step length lb:

b lb  0.3D*0.7T0.5 ,  3D*0.6T0.9 D D   



(7.9)

where D* is the nondimensional particle diameter D[(ρs/ρ − 1)g/v2]1/3 and T* is the transport stage number defined in Equation 6.6. The particle saltation height is often treated as the bedload layer thickness.

Bed-Load Transport  219

Lee and Hsu (1994) measured the instantaneous saltation trajectories of sediment particles with sizes of 1.36 and 2.47 mm in a slope-adjustable recirculating flume using a real-time flow visualization technique. They proposed the following formulas based on the measured data:

b lb 0.575 0.788 ,  14.27    cr   196.3    cr  D   D



(7.10)

where Θ is the Shields number τb/[(γs − γ)D], Θcr is the critical Shields number τcr /[(γs − γ)D], τb is the bed shear stress, τcr is the critical shear stress of incipient motion, γs is the specific weight of sediment, and γ is the specific weight of water. By conducting more experiments and solving Equation 7.1 for bed-load particle motion, Lee et al. (2000) established a set of formulas similar to Equation 7.9. As found by Hu and Hui (1996), the bed-load saltation height, length, and velocity exhibit different characteristics over smooth and rough beds. The obtained formulas for saltation height and length over rough beds are b    1.78  s  D  

0.86

  l 0.69 ,  b  27.54  s  D  

0.94

0.9 

(7.11)

Using data in the literature, Sklar and Dietrich (2004) obtained b lb  1.44T*0.5,  8.0T*0.88  D   D

(7.12)

with T* determined using the critical shear stress τcr = 0.03(γs − γ)D. Yalin (1972) made several assumptions for analytically solving Equation 7.1 with the drag force and submerged weight only, and derived the following formula for the average bed-load velocity:   1 usb  C1u 1  ln 1  aY Tt     aY Tt 

(7.13)

where u* is the shear velocity, Tt = (Θ − Θcr)/Θcr, aY  2.45 cr    s  , and C1 is an empirical coefficient. Considering the equilibrium between the drag and friction forces on a particle moving on the bed, Ashida and Michiue (1972) obtained 0.4

 s

usb

 a0   1 gD



  cr





(7.14)

The basic form of Equation 7.14 was also derived by Kalinske (1947), Goncharov (1954), Bagnold (1956, 1973), and Bridge and Dominic (1984), among others. The coefficient a0 was

220  Bed-Load Transport

set as 8.5 by Ashida and Michiue (1972) and as 8–12 by Bridge and Dominic (1984). The following alternative form of Equation 7.14 was used by Engelund and Fredsøe (1976) and several others:

 s

usb

 ae   1 gD



  be cr





(7.15)

where ae is a coefficient similar to a0, and be is the ratio of the critical shear velocities for particle motion cessation and initiation. Fernandez Luque and van Beek (1976) obtained ae = 11.5 and be = 0.7 by using experiments involving the bed-load transport of coarse sand and fine gravel in an 8 m long and 0.1 m wide closed rectangular channel. Fredsøe and Deigaard (1992) used ae = 10 and be = 0.7 based on data for particle diameters between 1.8 and 7.8 mm. Hu and Hui (1996) found ae = 11.9 and be = 0.44 through experiments using spherical and irregular particles with diameters of 1.34–7.6 mm and specific gravities of 1.043–2.65 in a 16 m long and 0.5 m wide flume. Van Rijn (1984a) derived the following formula for the average velocity of saltating particles:

 s

usb

  1 gD

 1.5T0.6 

(7.16)

This formula was verified by Wu et al. (2006) using the experimental data measured by Francis (1973), Fernandez Luque and van Beek (1976), and Lee and Hsu (1994), as shown in Figure 7.3. The right-hand side of Equation 7.16 was revised as 1.64T0.5 , with T* = τb/τcr − 1, where τb is the bed shear stress measured in the experiments without bed forms. Likewise, Sklar and Dietrich (2004) revised the right-hand side of Equation 7.16 as 1.56T0.56 . Using data from experiments in a 3 m long and 0.1 m wide flume with flow depths of 12.5– 22 mm and a particle size of 11 mm and from a variety of literature, Cheng and Emadzadeh (2014) established the following formula for the average velocity of particles over smooth and rough beds: usb k   13.00.5  1.5  s  D   s   1 gD

0.6



(7.17)

where ks is the bed roughness height. The coefficients vary slightly for different particle shapes and bed conditions. Equation 7.17 indicates that usb decreases as ks increases. However, an opposite trend was observed by Demiral et al. (2022) in the experiments of the roughness effect on particle saltation in supercritical flows. This needs to be studied further. Julien and Bounvilay (2013) developed the following formula for the average velocity of bedload particles rolling over rough beds:

 s

usb

  1 gD

0.95

0.21

 11.5 D

D    ks 

0.36

 s

  1

0.28



(7.18)

Bed-Load Transport  221

Figure 7.3  Bed-load velocity as a function of transport stage parameter (from Wu et al., 2006)

Shim and Duan (2019) measured the instantaneous velocity of particles in a 2.4 m long and 0.15 m wide flume using images captured with a high-speed camera. The time- and spaceaveraged particle velocity is related to the Shields number as usb 

 s

  1 gD     34.6     26.3 s   Km  cr 

(7.19)

where Km is the added mass coefficient set as 0.5 and Θcr = 0.031. The exponents of Θ in Equations 7.18 and 7.19 are significantly higher than those in other equations described here, likely due to different flow and sediment conditions used in the experiments. Equation 7.18 was developed for rolling particles. In addition, some of the aforementioned experiments used small flumes with shallow submergence (i.e., high D/h), the scale effects of which need to be studied. All the formulas introduced thus far are limited to the saltation regime of bed load on flat beds. When ripples and dunes appear, the bed-load particle motion is likely to be interrupted at the lee side of bed forms. According to Bagnold (1971), the particle step length is the same as the ripple length for wind-induced ripples. As observed by Grigg (1970) and Lee and Jobson (1977), the particle step length has a mean value of 0.5–0.6 times the dune length. This factor of 0.5–0.6 agrees well with the shape factor of sand dunes. More discussion on the contribution of bed forms to bed load is given in Section 7.4.4. 7.4  Unisized Bed-load Transport Capacity The transport rate or concentration of noncohesive sediments in the modes of bed load and suspended load or jointly as bed-material load at the equilibrium state in a steady, uniform flow is a function of flow conditions and sediment properties. This is often called sediment transport capacity or capacity of flow carrying sediment. Since Du Boys (1879) proposed his hydraulic traction theory of bed load, sediment transport capacity has been extensively studied and many

222  Bed-Load Transport

formulas have been developed by using fluid dynamics, stochastic theory, dimensional analysis, and laboratory/field measurements. The bed-load formulas are introduced in this chapter, while the formulas of suspended load and bed-material load are presented in Chapters 8 and 9, respectively. A uniform or well-sorted nonuniform sediment mixture is often represented with a single size class, whereas a poorly sorted sediment mixture needs to be divided into multiple size classes since interactions exist among the size classes. Unisized and multisized bed-load formulas are introduced in this and next sections, respectively. Because the number of unisized bed-load formulas is large, they are arranged into five groups based on hydraulic traction, stream power, mass flux, bed-form migration, and stochastic theory. 7.4.1  Bed-Load Formulas Based on Hydraulic Traction Bed-load transport is directly caused by the traction of a shear flow over the bed, so the bed-load transport rate is related to the bed shear stress. The pioneers of this approach include Du Boys (1879), Meyer-Peter and Mueller (1948), and Paintal (1971a), followed by many others. Formula of Du Boys (1879) According to the concept of Du Boys (1879), bed load moves in layers by flow shear, as shown in Figure 7.4. Layer thickness δ0 is on the same order of magnitude as the sediment diameter. If nd layers are activated, the bottom layer starts moving with a zero velocity and the top nd – 1 layers exhibit a velocity increment of Δu successively. Then, the bed-load average velocity is (nd – 1)Δu / 2, and the bed-load transport rate per unit channel width is  qb

1 nd  0  nd  1 u 2



(7.20)

The bed shear stress τb is balanced by the frictional force between the successive layers, i.e.,   b s   s    nd  0  where φs is the friction coefficient.

Figure 7.4  Bed-load movement model assumed by Du Boys (1879)

(7.21)

Bed-Load Transport  223

Upon incipient motion, only the surface layer starts moving. Thus, by setting nd = 1, Equation 7.21 is reduced to τcr = φs(γs − γ )δ0

(7.22)

Comparing Equations 7.21 and 7.22 gives nd = τb/τcr

(7.23)

Substituting Equation 7.23 into Equation 7.20 leads to the following bed-load formula, proposed by Du Boys (1879): qb = ψτb (τb – τcr)

(7.24)

where    0 u  2 cr  is a characteristic sediment coefficient. Based on experiments in small 3 / 4 laboratory flumes with uniform sand, Straub (1935) obtained   0.17 Dmm in m3/(kg·s) and 2 τcr = 0.061 + 0.093Dnm in kg/m , with Dnm being the sediment diameter in mm. The bed-load movement mechanism assumed by Du Boys is unrealistic and the data used to test the formula are very limited, so the Du Boys formula is rarely used in practice. However, the formulation of bed-load transport rate as a function of the (excess) bed shear stress has been widely adopted in later studies. 2

Formula of Meyer-Peter and Mueller (1948) Meyer-Peter and Mueller (1948) developed a bed-load formula using measurement data and the similarity principle. First, using a limited number of experimental data, Meyer-Peter et al. (1934) developed a simple empirical formula for the transport rate of uniform bed load with a specific gravity of 2.68:

 q 

2/3

S q 2 / 3  am  bm b D D



(7.25)

where qb is the discharge of bed load in submerged weight per unit time per unit channel width, S is the energy slope, and q is the unit flow discharge. By excluding the bank effect, q = URb, in which Rb is the hydraulic radius of the channel bed and U is the average flow velocity. The coefficients am = 17.0 and bm = 0.547 if the basic units are kg, m, and s. By conducting more experiments with wider ranges of energy slope and specific weight and correcting the bed shear stress for dune beds, Meyer-Peter and Mueller (1948) improved Equation 7.25 to yield the relationship shown in Figure 7.5 and expressed as Q Q

3/ 2

 Kb   RS 0.047   s    Dm  0.25  1/ 3 qb2 / 3    K  b



(7.26)

where R is the hydraulic radius of the channel (m); Dm is the arithmetic mean diameter of the bed sediment mixture (m); Kb = 1/n, with n being the Manning coefficient of the channel bed;

224  Bed-Load Transport

Figure 7.5 Bed-load transport as a function of grain shear stress (from Meyer-Peter and Mueller, 1948) 1/ 6 , with n′ representing the Manning coefficient due to grain roughness and K b  1 n  26 D90 D90 representing the particle size for which 90% by weight of the sediment mixture are finer; Q is the flow discharge over the cross-section; and Q* = BRbU, with B being the channel width at the water surface. In a wide channel, Q* = Q, and Equation 7.26 can be rewritten in the nondimensional form:

  K b K b 3 / 2  RS   8  0.047  3    s    Dm  s  s   1 gDm qb

3/ 2



(7.27)

where qb is the bed-load transport rate by mass per unit time per unit channel width, given in kg/ (m·s) if SI units based on kg, m, and s are used. The data used to calibrate Equation 7.27 covered flow depths of 0.01–1.2 m, specific discharges of 0.002–2 m2/s, energy slopes of 0.04%–2%, specific gravities of 1.25–4.2, and sediment sizes of 0.40–30 mm. Equation 7.27 is reliable for coarse sand and gravel. Its use of the critical Shields number of 0.047 is adequate for coarse sediments, but is too large and results in poor performance for coarse silt and fine sand. It gives zero calculated transport rates in cases near the incipient motion thresholds, as described in the last paragraph of Section 7.6. Meyer-Peter and Mueller’s (1948) formula is one of the earliest developed bed-load formulas that are still widely used without significant modification. Its significant contribution is the use of grain shear stress. As described in Section 6.4, the grain shear stress is the tangential or skin friction stress on the particles lying on the bed, whereas the form shear stress represents mainly the pressure difference and local head loss on bed forms. Bed load is affected directly by the grain shear stress rather than the form shear stress. This concept was also presented by Einstein (1942, 1950) and adopted by van Rijn (1984a), Wu et al. (2000), and Schneider et al. (2015), among many others. Formula of Paintal (1971a) Equations 7.24 and 7.27 use the excess shear stress (τb − τcr) and assume a zero bed-load transport rate when τb < τcr. However, according to the experiments of Paintal (1971a), near the

Bed-Load Transport  225

Figure 7.6  Variations of bed-load transport rate at low and high shear stresses (after Paintal, 1971a)

threshold of incipient motion, there is not a distinct condition for the initiation of sediment movement, as shown in Figure 7.6. Even at a very low shear stress, sediment particles can still be entrained and transported. Paintal (1971a) proposed a 16th power function for the bed-load transport at low shear stresses: Φb = 6.56 × 1018Θ16 for Θ < 0.06

(7.28)

3 where Φb is the Einstein number qb [  s (  s  1) gD ]. Other power index values of Θ, such as 7.5 and 8.345, were reported by Wilcock and Crowe (2003) and Khullar et al. (2007). These high-order power functions indicate that the bed-load transport rate is highly sensitive to any small change in the bed shear stress. Paintal (1971a) obtained the following 2.5th power function of bed-load transport at high shear stresses by applying the experimental data of Gilbert (1914), USWES (1935), and Casey (see Johnson, 1943):

Φb = 13Θ2.5 for Θ > 0.06

(7.29)

Guo (2021) adopted the following simple equation to fit the data of Paintal (1971a) across low and high shear stresses:   16   b 105 / 3 exp   3  1  (30) 



(7.30)

As suggested by Shih and Diplas (2018), the variation in the exponents from low to high shear stress in Equations 7.28 and 7.29 is due to the bed-load motion intermittency, which increases as the flow strength decreases. By taking a conditional average excluding the intermittency in the time series of hydrodynamic forces acting on a bed particle, Shih and Diplas (2018) derived a 1.5th power relationship between Φb and Θ across low to high transport regimes. If converted to the conventional time-averaging, this relationship is the same for Θ > 0.06, whereas for Θ  0.2 mm. Note that in Equations 7.37–7.39, qb is given in kg/(m·s), h and D are in m, and velocities are in m/s. Yalin’s (1972) Formula Using an alternative form of Equation 7.32, Yalin (1972) set the mass transport rate of bed load as qb   s

Ws usb  s 

(7.40)

where Ws is the submerged weight of bed load. It is related to γs − γ, ρ, D, τb, ν, etc. Yalin (1972) derived the following relationship for Ws by using dimensional analysis:   cr Ws   f  , cr   C2 cr  s    D

(7.41)

228  Bed-Load Transport

where C2 is a coefficient. As described in Section 7.2, Yalin (1972) derived Equation 7.13 for usb. By substituting Equations 7.13 and 7.41 into Equation 7.40 and calibrating the coefficients using measurement data, Yalin (1972) established qb  0.635Tt s u D

  1 ln 1  aY Tt    1   aY Tt 

(7.42)

Yalin’s formula was tested using measurement data of sediment sizes ranging from 0.315 to 28.65 mm. Formula of Engelund and Fredsøe (1976) Engelund and Fredsøe (1976) determined bed load using the following variant of Equation 7.32 as the mass of possibly-moving particles per unit bed area times the average particle velocity: qb  s

 3 pE D 2 usb  6 D

(7.43)

where pE is the probability for bed particles to move and 1/D2 is the number of particles on a unit bed. The bed-load velocity usb is determined with Equation 7.15. Engelund and Fredsøe (1976) set the total bed shear stress to be the sum of the critical shear stress and the friction force acting on the bed-load particles:

b   cr  nm   s   

D3 d 6



(7.44)

where μd is the friction coefficient; and nm is the number of bed-load particles at motion, related to pE by nm = pE /D2. Substituting this relationship into Equation 7.44 yields  pE

6    cr    d

(7.45)

Substituting Equations 7.15 and 7.45 into Equation 7.43, Engelund and Fredsøe (1976) obtained  b

30    cr   d



  0.7 cr





(7.46)

where Θcr = 0.05. Engelund and Fredsøe (1976) used μd = tan 27̊ and 30/(πμd) = 18.74, whereas Fredsøe and Deigaard (1992) used μd = 1 and 30/(πμd) = 9.55. Lajeunesse et al. (2010) developed a similar formula. They recalibrated the coefficient 30/ (πμd) in Equation 7.46 to be about 10.6 using uniform sediment data in the weak transport range of Θ < 0.26.

Bed-Load Transport  229

Van Rijn’s (1984a) Formula Van Rijn (1984a) considered saltation as the dominant motion mode of bed-load particles and derived Equations 7.9 and 7.16 for the saltation height δb and the average velocity usb. Then, he determined the bed-load concentration by using cb = qb/(δbusb) from bed-load transport rates measured in 130 sets of flume experiments selected from literature with median particle diameters of 0.2–2 mm, water depths larger than 0.1 m, and Froude numbers smaller than 0.9. By analyzing those data, van Rijn (1984a) obtained the following relationship for cb: cb T  0.18   c0 D

(7.47)

where c0 is the maximum volumetric concentration, set as 0.65. Substituting Equations 7.9, 7.16, and 7.47 into Equation 7.32 leads to the following equation (van Rijn, 1984a):  b  0.053D0.3T2.1



(7.48)

where T* is determined by using Equation 6.6 with the grain shear stress. The sediment size is represented by the median diameter D50 of the sediment mixture. The formula of van Rijn (1984a) was tested by using sand transport data. It was modified by van Rijn (2007a) for bedload transport in both riverine and coastal waters, as described in Section 13.3.4. 7.4.3  Bed-Load Formulas Based on Stream Power The stream power per unit channel width is defined as P = γqS = τbU

(7.49)

The energy of sediment-laden flow is divided into three parts to overcome bed friction, support suspended load through turbulence, and carry bed load. Thus, the sediment transport rate is related to stream power. This concept has been adopted to develop sediment transport formulas, including the bed-load formulas of Schoklitsch (1926, 1962), Bagnold (1956, 1973), Dou (1964), and Abrahams and Gao (2006) described herein, as well as several formulas of suspended load and bed-material load introduced in Chapters 8 and 9. Schoklitsch’s (1926, 1962) Formula Schoklitsch (1926) proposed the following formula by arguing that sediment transport rate is proportional to excess stream power:  qb  S ns  q  qcr  

(7.50)

where χ is a characteristic sediment coefficient and qcr is the unit flow discharge at which bed particles begin to move. The exponent ns is equal to 1 if the stream power is strictly considered. It is treated as a calibrated parameter and given a value of 3/2, which is used in the formula of Meyer-Peter and Mueller (1948).

230  Bed-Load Transport

Schoklitsch (1962) calibrated the coefficients of Equation 7.50 as follows using laboratory experiments performed by Gilbert (1914) and himself, as well as field measurements from the Danube and Aare Rivers in central Europe: qb = 2500S 3/2 (q − qcr)

(7.51)

where qb is given in kg/(m·s), and qcr is given in m2/s as     qcr  0.26  s    

5/ 3

D3/ 2 S 7/6



(7.52)

where D is given as D40 of the sediment mixture, in m. Bathurst (2007) recalibrated the coefficients of Equation 7.51 for the transport of coarser material as the armor layer breaks up in mountainous rivers. Bagnold’s (1956, 1966, 1973) Formula According to Bagnold (1956, 1966, 1973), the potential energy of water flow is transmitted to bed load by shear force, and the bed-load particles are supported by the dispersion (normal) force generated by particle collisions. The power spent to overcome the friction force induced by bed-load transport is  Ebl W qb s u sb tan i

( s   ) g tan i  s

(7.53)

where Ws is the submerged weight of bed load, qb is the mass transport rate of bed load, and tan φi is the particle friction coefficient. The portion of stream power transmitted to bed load is Pbl = τbUeb

(7.54)

where eb is the efficiency coefficient. Equating Ebl and Pbl with Equations 7.53 and 7.54 leads to the following formula for bed-load transport rate: qb 

s  bU eb  ( s   ) g tan i

(7.55)

In the early work of Bagnold (1966), eb/tan ϕi is set as 0.17 for D < 0.5 mm and Θ < 1; otherwise eb and tan ϕi are determined using empirical curves. The stream power concept of Bagnold (1966) is well accepted and adopted by many scholars in sedimentation engineering. However, the graphical relationship is inconvenient and limits its application. Bagnold (1973) analyzed the dynamics of particle saltation near the bed and reformulated the power on bed load as Ebl = Fx(und − ur). Here, Fx is the streamwise force, und is the flow velocity acting on bed-load particles, and ur is the relative velocity between water and sediment. By

Bed-Load Transport  231

assuming Fx = τb(u* − u*cr)/u* and ur = ωs, and using the log law for und, Bagnold (1973) derived a formula for eb and, in turn, the following formula for bed-load transport: qb 

s  bU u  ucr u ( s   ) g tan i

 5.75u log  0.37h (nB D)   s    1  U  

(7.56)

where tan ϕi ≈ 0.63, nBD represents the average height of the force acting on saltating particles, nB = 1.4(u*/u*cr)0.6, and ωs is the particle settling velocity. Equation 7.56 was developed for the saltation regime over flat beds. When it is applied to the ripple-dune regime, the term in the curly brackets may give a negative value. In such a case, the shear velocity in the curly brackets can be replaced with the grain shear velocity. This treatment needs to be tested. Formula of Abrahams and Gao (2006) In the concept of Bagnold (1956, 1966), bed load is supported entirely by the dispersion stress arising from particle collisions. As argued by Abrahams and Gao (2006), this may be true at a high transport stage, but at a low transport stage bed load is largely supported by fluid drag and lift. Therefore, Abrahams and Gao replaced the friction coefficient tan ϕi in Equation 7.55 with a stress coefficient: sb = 0.6(1 – Θcr/Θ)−2

(7.57)

and determined the power efficiency coefficient as eb = 0.6(1 – Θcr/Θ)1.4

(7.58)

By substituting Equations 7.57 and 7.58 into Equation 7.55, Abrahams and Gao (2006) derived the following formula for bed-load transport over plane beds:

s bU  cr  qb  1   ( s   ) g 

3.4



(7.59)

Equation 7.59 was tested using laboratory data for sand transport with ωs/u* > 0.8 under saltation and sheet-flow regimes. Its application to the ripple-dune regime needs to be validated. Gao (2011) modified Equation 7.59 for bed-load transport in gravel-bed rivers. One can find additional details in his article. Dou’s (1964) Formula Dou (1964) established a formula for bed-load transport based on the stream power and mass flux concepts. The flow energy used to carry bed load is expressed as  b U  U cr  eb . If n1 particles are entrained from a unit bed at a speed of vs, the following relation exists:

 b U  U cr  eb n1

D 3  s    vs  6

(7.60)

232  Bed-Load Transport

At the equilibrium state, the numbers of entrained and deposited sediment particles should be equal, i.e., n1vs = n2ωs

(7.61)

where n2 is the number of deposited particles. The bed-load velocity is assumed to be proportional to the flow velocity as K1U. Thus, the bed-load transport rate is formulated by using the mass flux concept as  D 3  qb  s K1U  n2   6  

(7.62)

Substituting Equations 7.60 and 7.61 into Equation 7.62 yields (Dou, 1964) qb  K 0

s U  b U  U cr   s   g s

(7.63)

where qb is given in kg/(m·s), and K0 = K1eb is calibrated as about 0.01 for sandy bed load. Equation 7.63 can also be used for bed-material load transport, with K0 = 0.1 calibrated using Gilbert’s data. 7.4.4  Bed-Load Formulas Based on Bed-Form Migration As shown in Figure 7.1, a significant portion of the bed load moves through bed forms when ripples and dunes appear (Exner, 1925; Zhang, 1961; Simons et al., 1965; Engel and Lau, 1980, 1981; van Rijn, 1986). The bed-load transport rate is related to the height Δd and celerity cd of bed forms as follows (Zhang, 1961; Simons et al., 1965):  qb  d s 1  v   d cd  qb 0



(7.64)

where φv is the porosity of bed sediment; αd is the shape factor of bed forms; and qb0 is the portion of bed load that does not enter the migration of bed forms, usually set as zero. Engel and Lau (1980) suggested Δd and cd as the representative height and celerity of the upper portion of the bed forms above the flow reattachment points (Figure 7.1). Moreover, Aberle et al. (2012) proposed a general formula to consider the vertical variations of porosity and celerity in each bed form due to the heterogeneity of bed material. However, the most widely used approach sets Δd as the crest-to-trough height and φv and cd as the bulk porosity and celerity of the bed forms, respectively. The shape factor αd is related to the bed-form steepness and, to a lesser extent, to the sediment size (Engel and Lau, 1981). It has a value of 0.5 for a triangle-shaped bed form. The calibrated αd values for naturally shaped bed forms range from 0.3 to 0.8, mostly falling between 0.5 and 0.6 (Aberle et al., 2012). According to Le Coz et al. (2022), αd is 0.47 obtained by using the simple spatial average over the measured bed elevations and 0.63 by using the celerity-weighted average, between which the value of 0.63 is more meaningful for the calculation of bed load in Equation 7.64.

Bed-Load Transport  233

In an alternative approach, Equation 7.64 is rewritten as follows by assuming the mean height of bed forms to be proportional to the standard deviation of bed surface elevation, σzb (Crickmore, 1970; Willis and Kennedy, 1977): qb = αd0 ρs(1 − φv)σzbcd

(7.65)

where αd0 is a coefficient. Crickmore (1970) theoretically determined that αd0 = 1.06 by assuming a Gaussian distribution of bed elevation. For sinusoidal propagating sand waves, Willis and Kennedy (1977) theoretically obtained  d 0  2 . For naturally shaped bed forms, αd0 needs to be calibrated. Equations 7.64 and 7.65 provide methods to measure bed-load transport by tracking bed forms (Zhang, 1961; Simons et al., 1965; Havinga, 1982) or bathymetric changes (Abraham et al., 2011). This is done by successively measuring the bed topography with a three-dimensional position system, which is usually complicated and labor-intensive. A simpler approach is to track the bed profiles along a prefixed streamwise section (Figure 6.4), but this may result in an overall inaccuracy of about 40%–50% under flume conditions and as large as 100% under field conditions (Engel and Wiebe, 1979; van Rijn, 1986). Equations 7.64 and 7.65 were confirmed for downstream-migrating dunes and ripples. Through a laboratory study, Pascal et al. (2021) proposed the following relationship between bed-load transport rate and celerity for upstream-migrating antidunes: qb 

s (1  v )cd D50 ( Ld Ld , R  0.57) 2 / 3



(7.66)

where Ld is the wavelength of antidunes measured; and Ld,R is the wavelength of dominant antidunes on steep slopes, predicted by Equation 6.29 (Recking et al., 2009). Substituting Equations 6.17 and 6.24 for the dune height and speed in Equation 7.64, Zhang (1961) developed the following formula for the bed-load transport rate in dune-bed streams: qb  0.00124

 d s 1  v U 4 g 3/ 2 h1/ 4 D1/ 4



(7.67)

where qb is given in kg/(m∙s). Similarly, by applying Equations 6.18 and 6.24 from Barekyan (1962) for the dune height and migration speed, Simons et al. (1965) derived  U U0  qb  K b bU     U0 

(7.68)

where U0 is the noneroding flow velocity, as defined in Section 5.4. The coefficient Kb is about 0.3 when qb and τbU are given in pounds per second per foot of width. The bed-form based Equations 7.67 and 7.68 are not widely used in practice, perhaps due to the limitations of the original formulas for dune height and celerity. Nevertheless, considering

234  Bed-Load Transport

its similarity to the stream-power based Equation 7.63 of Dou (1964), Equation 7.68 is likely also applicable to plane beds. Note that the bed-form migration concept can be unified with the mass flux concept described in Section 7.4.2 for plane beds. The bed-load flux on dune beds can be determined by applying Equation 7.32 for the stoss slope of the dunes, where the bed-load velocity, layer thickness, and concentration are related to the traction force on the slope surface. Therefore, bed-load transport on plane and dune beds can be collectively described by using grain shear stress as the controlling variable. 7.4.5  Bed-Load Formulas Based on Stochastic Theory As described in Section 5.3.1, sediment transport exhibits stochastic characteristics due to the fluctuation of turbulent flow, the heterogeneity of bed material, and the randomness of particle packing. Einstein (1942, 1950) was the pioneer who applied stochastic theory to sediment research, followed by Paintal (1971b), Han and He (1984), Sun and Donahue (2000), and Wu and Yang (2004b), among others. The bed-load theory of Einstein (1942, 1950), the stochastic theory of Han and He (1984), and a few modifications are presented in the following text. Einstein’s (1942, 1950) Formula Einstein (1942, 1950) defined bed load and bed material as two states of a closely related system, in which sediment particles change their states by entrainment, deposition, and re-entrainment in steps. The particle travel distance in one step is called step length, lb. The step length depends on the flow and sediment conditions, as described in Section 7.2. Einstein (1942) set lb = λbD, in which λb is assumed as a constant. The entrainment probability is denoted as pE, and correspondingly, the deposition probability is 1 − pE. After a particle finishes a moving step and falls to the bed, it has the probability 1 − pE of resting there and the probability pE of continuing to the next step. Thus, among N particles in motion, N(1 − pE) particles are deposited after the first step and the remaining NpE particles continue to move, as shown in Figure 7.7. Among the NpE moving particles, NpE(1 − pE) particles are deposited after the second step and the remaining NpE2 particles continue to move. As this continues, the average step distance is 1 L0  N



 N (1  p n 1

E

l ) pEn1nlb  b  1  pE

Figure 7.7  Movement step pattern of bed-load particles (Einstein, 1942)

(7.69)

Bed-Load Transport  235

Because the sediment particles passing through a cross-section are deposited at an average distance of L0, the deposition rate per unit bed area, Db, is related to the bed-load transport rate as qb qb 1  pE    L0 lb

D b

(7.70)

If the particle packing density is ma, the number of particles resting on a unit area of bed surface is 4ma/(πD2). Each particle has a mass of ρsπD3/6. Thus, the sediment entrainment rate, Eb, is defined as Eb 

p 2 ma D s E  3 ts

(7.71)

where ts is the exchange time. At the equilibrium state, the amount of sediment entrained from the bed is equal to that deposited to the bed. Equating Db and Eb in Equations 7.70 and 7.71 yields the following probabilitybased formula for bed-load transport rate: qb 

pE lb 2 ma D s   3 1  pE  t s

(7.72)

Einstein (1942, 1950) assumed the exchange time to be proportional to D/ωs, which is the time for the particles to settle at a distance of one sediment size. The settling velocity ωs is related to sediment size via Equation 4.3, yielding

 D s  g

ts  a3



(7.73)

where a3 is a coefficient. Thus, using lb = λbD and Equation 7.73, Equation 7.72 is rewritten as pE  A  b  1 pE

(7.74)

where A* = 3a3 /(2maλb). Einstein (1942, 1950) defined pE as the probability that the lift force is larger than the submerged weight, p(FL > Ws), as introduced in Section 5.9.1. Combining Equations 5.80 and 7.74 leads to the following bed-load formula: 1

1

 

B  1  L 0

 B  1  L 0

2 A  b e  t dt   1  A  b

(7.75)

236  Bed-Load Transport

Figure 7.8  Einstein’s (1950) bed-load function

where 1/σL0 = 2.0, A* = 43.5, and B* = 1/7, as calibrated by comparing the calculated and measured bed-load transport rates shown in Figure 7.8. Considering that only the grain shear stress affects bed-load transport, the reciprocal Shields number Ψ is redefined as 

 s    D  Rb S



(7.76)

where Rb is the hydraulic radius due to grain roughness, determined by using Einstein’s movable bed roughness method described in Section 6.5. The bed-load transport theory of Einstein (1942, 1950) is well recognized due to its significant contribution to sedimentation engineering. However, this theory has been criticized in several aspects. It ignores the effect of drag force on the entrainment probability and oversimplifies several parameters, such as step length. The assumption of ts ∝ D/ωs is not reasonable (Vanoni, 1975; Wang et al., 2008b; Zee and Zee, 2017). The formula calibration is very limited. In addition, Equation 7.76 is coupled with the movable roughness formula, which involves tedious iterations. The Einstein model has been modified by Sun and Donahue (2000) and Wang et al. (2008b), among others. Han and He’s (1984) Theory Han and He (1984) proposed a comprehensive stochastic theory for sediment transport considering the exchange among bed material, bed load, and suspended load. In this theory, sediments are divided into four basic states: rest, rolling, saltation, and suspension. Rest indicates the stationary sediment at the bed surface layer. Rolling and saltation are the major modes of bed-load movement, whereas sliding is assumed to be negligible. Four basic probabilities are defined for incipient motion (ε1), incipient saltation (ε2), incipient suspension (εsus), and motion cessation (1 − ε0): ε1 = p(|ub| > ubc,1)

(7.77)

Bed-Load Transport  237

ε2 = p(|ub| > ubc,2)

(7.78)

εsus = p(v'b > ωs)

(7.79)

1 − ε0 = p(|ub| ≤ ubc,0)

(7.80)

where ub is the near-bed velocity of flow acting on sediment particles; ubc,1, ubc,2, and ubc,0 are the critical near-bed flow velocities for incipient rolling, incipient saltation, and motion cessation, respectively; and vb is the fluctuation of the vertical velocity of flow. The transfer probabilities pij among the four states can be derived by using the basic probabilities and assuming that the streamwise and vertical velocity fluctuations are two independent variables. The transfer probability from a moving state (i = 2, 3, 4) to the rest state ( j = 1) is the joint probability of motion cessation and no suspension: pi1  p  vb  s    ub  ubc , 0    p  vb  s  p  ub  ubc , 0    1   sus  1   0 

(7.81)

The transfer probability from a non-saltation state (i = 1, 2, 4) to the saltation state (j = 3) is the joint probability of incipient saltation and no suspension: pi 3 p  ub  ubc , 2    vb  s   p  ub  ubc , 2  p  vb  s      2 1   sus 

(7.82)

The transfer probability from a non-suspension state (i = 1, 2, 3) to the suspension state ( j = 4) is the probability of incipient suspension:

pi 4  p  vb  s    sus 

(7.83)

The transfer probability from a non-rolling state (i = 1, 3, 4) to the rolling state ( j = 2) depends on the previous state of the particle. For a particle to transfer from rest (i = 1) to rolling ( j = 2), the probability is p12  p  vb  s    ub  ubc ,1    ub  ubc , 2     1   sus   1   2 

(7.84)

and for a particle to transfer from saltation or suspension (i = 3, 4) to rolling (j = 2), it is affected by the criterion of motion cessation rather than incipient motion, so the probability is pi 2  p  vb  s    ub  ubc , 0    ub  ubc , 2     1   sus    0   2 

(7.85)

238  Bed-Load Transport

The probability of staying in the same state, pii, can be obtained using the constraint condition 4 p  1. The long-run limiting probability of a sediment particle at one of the four states is j 1 ij 4 denoted as Pj ( j = 1, 2, 3, 4) and obtained by using the Markov chain Pj   i 1 Pi pij . The number of particles resting on the bed surface is denoted as n0. Thus, the transfer rates from rest to rolling, saltation, and suspension are given as



 1i 

p1i n0 ti , 0

(i = 2, 3, 4)

(7.86)

where ti,0 is the single-step time for particles to transfer from rest to rolling, saltation, and suspension when i = 2, 3, and 4, respectively. If the areal packing density is ma, then n0 = 4ma/(πD2). The number, mean step length, and mean velocity of sediment particles in a moving state i are denoted as ni, li, and usi (i = 2, 3, 4), respectively. The transfer rate from one of these three states to rest is determined as ψi1 = pi1niusi /li (i = 2, 3, 4)

(7.87)

The equilibrium between states 1 and i ( = 2, 3, or 4) indicates ψ1i = ψi1, which leads to the transport rate by mass of sediment in state i: p l D3 2  qmi  ni usi ma  s D 1i i s pi1ti ,0 6 3

(i = 2, 3, 4)

(7.88)

When i = 4, Equation 7.88 gives the suspended-load transport rate, which is not presented here. When i = 2 and 3, Equation 7.88 gives the transport rates of bed load by rolling and saltation, respectively. The sum of rolling and saltation rates results in the total bed-load transport rate: qb 

   1   2  l2   2 l3 2 ma  s D     3  1   0  t2, 0 1   0  t3, 0 

(7.89)

If rolling and saltation are combined into a single transport mode of bed load, then l2 = l3, t2,0 = t3,0, and Equation 7.89 is simplified as qb 

1l2 2 ma  s D 3 1   0  t2,0



(7.90)

Equation 7.90 is essentially the same as Equation 7.72 used by Einstein (1950), by setting ε0 = ε1 = pE, l2 = lb and t2,0 = ts. The difference is that t2,0 is the single-step time, but ts is the exchange time. Han and He (1984) studied the dynamics and stochastic characteristics of individual sediment particle motions and determined the parameters of Equation 7.89 for bed-load transport. They found that the particle step distance and travel time obey exponential distributions. The details

Bed-Load Transport  239

are not given here because of the complexity. The probabilities of incipient rolling, incipient saltation, and incipient suspension are introduced in Sections 5.9 and 8.2. Sun and Donahue’s (2000) Formula Sun and Donahue (2000) modified the bed-load models of Einstein (1942) and Han and He (1984) by using a Markov process for the exchange between bed material (static state denoted as 1) and bed load (moving state denoted as 2). They derived the following equation for the mean bed-load transport rate: qb 

pE t 2 ma s D usb 2  3 1  pE  t1

(7.91)

where ti is the mean single-step holding time of a particle at state i ( = 1, 2), and usb is the average velocity of single-step bed-load particle movement. By setting usbt2 = lb, Equation 7.91 becomes Equation 7.72, derived by Einstein (1950), but t1 and ts have different meanings. Sun and Donahue (2000) defined the entrainment probability pE as the probability of force moments causing particle rolling to exceed those keeping the particles at rest, as expressed in Equation 5.85. They related the average bed-load velocity and the ratio of averaged single-step motion and static holding times to the reciprocal Shields number Ψ as usb

t2  C1  B1    s   1 gD t1

(7.92)

where Ψ is calculated with Equation 7.76, and Rb or  b is determined using the ManningStickler equation. Substituting Equation 7.92 into Equation 7.91 yields b 

pE 1  B1 A1 1  pE

(7.93)

where A1 = 3/(2m0C1). The coefficients are calibrated as A1 = 10/3 and B1 = 3/4 using measurement data. Compared with Einstein’s model, the model of Sun and Donahue (2000) avoids the constant step length and the questionable exchange time. The possible overestimation of entrainment probability in Equation 5.85 is compensated by applying the grain shear stress and calibrating the coefficients A1 and B1. Formula of Wu and Yang (2004b) Wu and Yang (2004b) further modified the formula developed by Sun and Donahue (2000). They treated the movement of bed-load particles as a random combination of single-step motions described by a pseudo-four-state, continuous-time Markov process. The pseudo four states are the two states used by Sun and Donahue (2000) plus two virtual states added to consider transitions from state 1 to 1 and from state 2 to 2 in continuous time. The long-run moving

240  Bed-Load Transport

probability pM is evaluated with the instantaneous entrainment probability pE and the ratio of mean single-step holding times in the static and motion states, RT, as pM 

pE pE  RT (1  pE )



(7.94)

The mean bed-load transport rate is given with qb 

l 2 ma s DpM b  tM 3

(7.95)

where lb is the mean single-step length and tM is the mean single-step holding time in the motion state. Equation 7.95 avoids the weakness of Equation 7.91, which becomes indefinite as pE→1. The ratio lb /tM is set as the probability-weighted average velocity of rolling and saltating particles: lb  tM

pR p usR  L usL  pE pE

(7.96)

where pR and pL are the rolling and lifting probabilities, respectively; and usR and usL are the average rolling and saltating velocities of bed-load particles, respectively. The probabilities pR and pL are defined in Equations 5.112 and 5.113 using the lognormal distribution of the bottom flow velocity (Wu and Chou, 2003) and are functions of the grain Shields number Θ′. The particle velocities and the time ratio are related to Θ′ as usL (  s   1) gD usR (  s   1) gD

 7.5()0.5  0.5

 0.305 ln   1.4

RT = 5.45(Θ′)−0.6



(7.97)



(7.98) (7.99)

Equations 7.98 and 7.99 were obtained by Wu and Yang (2004b), and Equation 7.97 by Sun and Donahue (2000), using laboratory data. The grain shear stress in Θ′ is determined using the log law with a grain roughness height of D65 (Wilcock and McArdell, 1993). Formula of Li et al. (2018b) In analogy to Han and He’s Equation 7.89, Li et al. (2018b) considered different step lengths for the rolling and lifted particles and formulated the mean bed-load transport rate as qb 

( p l  pL l L ) 2 ma s D R R 3 tE



(7.100)

Bed-Load Transport  241

where lR and lL are the step lengths of the rolling and lifted particles, respectively, and tE is the time scale of entrainment. By assuming lR ∝ Θ, lL ∝ Θ, and tE ∝ D/u*, the final formula is written as follows: Φb = 20(pL + 0.15pR)Θ1.5

(7.101)

where pR and pL are determined using the formulas of Li et al. (2018b). The calibrated factor of 0.15 accounts for the ratio of the step length of rolling particles to that of lifted particles, and pL + 0.15pR acts as an effective entrainment probability. Equation 7.101 is similar to Equation 7.31, with pL + 0.15pR considering the intermittency of bed-load particles. Furthermore, Tsai and Lai (2014) developed a stochastic model for bed-load and suspendedload transport. The concept is similar to that of Han and He (1984) described previously, but is enhanced with a three-state, continuous-time Markov chain model for the exchanges among bed material, bed load, and suspended load. 7.5  Multisized Bed-Load Transport Capacity As described in Section 5.6, the hiding and exposure phenomena significantly affect the fraction-wise incipient motion of nonuniform sediment mixtures on the bed surface. These effects propagate to fractional transport rates. Moreover, fractional transport rates are modulated by interactions, such as collisions, among the moving sediment particles when the sediment concentration reaches a certain level. Einstein (1950) conducted pioneering research on the fractional transport rates of nonuniform sediments. Then, Ashida and Michiue (1972), Parker et al. (1982), Misri et al. (1984), Bridge and Bennett (1992), Hsu and Holly (1992), Patel and Ranga Raju (1996), Wu et al. (2000), Wilcock and Crowe (2003), and Schneider et al. (2015), among others, proposed a variety of methods to calculate nonuniform bed load. Some of these methods are introduced in this section. Einstein’s (1950) Formula Einstein (1950) and Einstein and Chien (1953a) extended the uniform bed-load Equation 7.75 to the fractional transport rates of nonuniform bed load: 1

1

 

B  k 1  L 0

 B  k 1  L 0

2 A  bk  e  t dt  1  A  bk

(7.102)

where  bk qbk  pbk  s (  s   1) gDk3  , in which qbk is the bed-load transport rate of size    class k by mass per unit time per unit channel width, pbk is the fraction of the kth size class of sediment in the bed surface layer, and Dk is the diameter of the kth size class of sediment. To consider the hiding-exposure effects of nonuniform bed material, the reciprocal Shields number Ψk is formulated as k 

 f YL   log 10.6    L  log(10.6 X d  s ) 

2

 s    Dk  Rb S



(7.103)

242  Bed-Load Transport

where ξf, YL, and θL are correction factors; Δs is the apparent bed roughness ks/χs, in which ks = D65 and χs is defined in Figure 3.9; and Xd is the characteristic size of bed material at which finer particles experience the hiding effect due to the viscous sublayer: Xd = 0.77Δs if Δs/δ > 1.8 and Xd = 1.39δ if Δs/δ ≤ 1.8, with δ = 11.6 /u representing the viscous sublayer thickness. The correction factor ξf considers the hiding effects on fine particles due to both the viscous sublayer and coarse particles on the bed surface. It is a function of Dk/Xd  and the following parameter, as shown in Figure 7.9a:  90 S0 

D75 ( s   ) D90 YL D25  Rb S



(7.104)

The factor YL corrects the lift coefficient to 0.178/YL in mixtures with various roughness conditions. YL is a function of ks/δ as shown in Figure 7.9b. The factor θL corrects the lift coefficient when u*Dk/v is less than 3.5, as shown in Figure 7.9c. The correction factors ξf, YL, and θL are much more complicated than those described in Section 5.6. The physical insights are thoughtful; however, Einstein’s bed-load formula has large errors for widely graded sediments (Misri et al., 1984; Samaga et al., 1986a). The reasons for this include the limited availability of high-quality data at that time and the complexity of sediment transport in nature. Sun and Donahue (2000) extended Equation 7.93 to nonuniform bed load by applying a much simpler correction factor to Ψk as follows: k 

 s    Dk 0.5  b g0.25  Dk Dm 



(7.105)

Figure 7.9 Correction factors: (a) ξ f as a function of D k/X d and Ψ *90S 0, (b) Y L as a function of k s/δ, and (c) θ L as a function of u *D k/v (after Einstein, 1950; Einstein and Chien, 1953a)

Bed-Load Transport  243

Wu and Yang (2004b) and Tsai and Lai (2014) also adopted this correction factor to the grain-related Shields number when they applied their uniform sediment transport formulas to nonuniform sediments. Formula of Ashida and Michiue (1972) By analyzing the forces acting on particles sliding on the bed, Ashida and Michiue (1972) derived Equation 7.14 for bed-load velocity and the following equation for the sediment mass (i.e., concentration × thickness) of the bed-load layer: cb b   cr   D d

(7.106)

where μd is the dynamic friction coefficient with an assumed value of 0.5. Substituting Equations 7.14 and 7.106 into Equation 7.32 for each size class leads to the following formula (Ashida and Michiue, 1972):    bk  17k3/ 2 1  crk k 

 crk  1  k 

   

(7.107)

where Θk = τb/[(γs − γ)Dk], k   b [( s   ) Dk ] , and Θcrk = τcrk/[(γs − γ)Dk]. Θcrk is determined using Equation 5.42 to consider the hiding-exposure effects. The grain shear stress is determined with   U R  5.75 log  6  u  Dm (1  2) 

(7.108)

Equation 7.108 uses the grain roughness height k s Dm (1  2) . For uniform sediments on flat beds, Equation 7.107 is similar to Equation 7.46 of Engelund and Fredsøe (1976) with different coefficient values. Formula of Parker et al. (1982) The bed-load transport formula developed by Parker et al. (1982) is mainly for gravel-bed streams. In such streams, a surface layer, called pavement or armor, is markedly coarser than the underlying substrate, and the moving sediment is different from the immobile armor. Parker et al. (1982) plotted the relationship between the dimensionless fractional bedload transport rate Wk and the dimensionless shear stress θk in Figure 7.10 using the field data for sediment sizes of 18–28 mm in Oak Creek, Oregon. Wk is given in Equation 5.20, i.e.,  Wk qbk ( s   1) [ pbk s ( ghS )1/ 2 hS ], and θk is defined as

k 

hS   1 Dk rk*    s

(7.109)

244  Bed-Load Transport

Figure 7.10  Gravel transport function of Parker et al. (1982)    k0,.0875 Dk where  rk with Ds 50 representing subpavement sediment size. The parameter  0.0875 Ds 50rk D *  rk can be written as  rk [( s   ) Dk ] , in which  rk is the reference shear stress of size class k. Note that the reference shear stress is not associated with sediment incipient motion, unlike the critical shear stress defined by Shields (1936). According to the equal mobility concept, all the grain sizes tend to have approximately equal mobility. Thus, only the subpavement size Ds50 is used to characterize the bed-load transport rate as follows:

Wk

0.0025 exp 14.2   1  9.28   12  50 50     4.5 11.2 1  0.822 /  50  

0.95   50  1.65

 50  1.65



(7.110)

where θ50 is the dimensionless shear stress, as defined in Equation 7.109, corresponding to Ds50. Considering that bed-load transport is accomplished through the mobilization of exposed particles on the bed surface rather than the substrate particles, Parker (1990) transformed Equation 7.110 into a surface-based relationship. Details can be found in the referenced study. Note that the equal mobility concept has not been fully validated. It may exist only at high flows when all the sediment particles are in motion. Thus, the data scatter significantly in Figure 7.11, and Equation 7.110 likely has errors at low flows. Hsu and Holly’s (1992) Method The method proposed by Hsu and Holly (1992) determines the size distribution of the transported sediment and the total transport rate. The fraction of each size class in the transported material is assumed to be proportional to the joint probability of two factors: mobility under the prevailing hydraulic conditions and availability on the bed surface. By applying the Gaussian probability distribution for the flow velocity fluctuation, the mobility of the kth size class of sediment is derived as  Pmo , k

1

 U 2



 U crk

 x2  exp   2  dx U 1  2 U 



(7.111)

Bed-Load Transport  245

where Ū is the mean velocity of flow; Ucrk is the incipient velocity of size class k, determined using Qin’s (1980) Equation 5.48 modified by recalibrating the constant 0.786 as 1.5; and σU is the standard deviation of the normalized fluctuating velocity U′/Ū, with a value of about 0.2. The availability of size class k is its fractional representation within the active bed surface layer, pbk. Thus, the fraction of size class k in the transported material is pk 

P



mo , k Dmax Dmin

pbk

Pmo , k pbk



(7.112)

where Dmin and Dmax are the minimum and maximum sizes of the sediment mixture, respectively. This transport probability model is similar to the entrainment probability model of nonuniform sediments proposed by Gessler (1970), which is described in Section 5.9.6. After the size distribution is obtained, the mean size Dmt and the mean critical velocity Ucrt are calculated. Then, the total bed-load transport rate, qb in kg/(m·s), is calculated using the following modified Shamov formula: 3

1/ 4

 U   Dmt  1/ 2  qb 12.5 Dmt U  U crmin      U crt   h 







(7.113)

where Ucrmin is the critical velocity for the incipient motion of the smallest size class, in m/s. Finally, the fractional transport rate is calculated with qbk = pkqb. Methods of Ranga Raju, Patel, and Colleagues Misri et al. (1984) extended Paintal’s (1971a) uniform bed-load relationship shown in Figure 7.6 to nonuniform bed load by introducing a hiding-exposure correction factor based on the assumption that the motions of fine and coarse particles are dominated by the lift and drag forces, respectively. Subsequently, Samaga et al. (1986a), Patel and Ranga Raju (1996), and Patel et al. (2015) revised the correction factor. In the version of Patel and Ranga Raju (1996), the effective shear stress is determined as

 eff  b b



(7.114)



where  b   Rb S, Rb  Un S 1/ 2 tor given by X mb  0.0713  X s k 

0.75144



3/ 2

1/ 6 24 , and ξb is a hiding-exposure correction fac, n  D65



(7.115) 2

3

0.1957  0.9571 log  b  cr   0.1949 log  b  cr    0.0644 log  b  cr   ; with log X s  Xm = 0.7092logM + 1.293 for 0.05 < M ≤ 0.38, and Xm = 1.0 for M > 0.38; M is the Kramer uniformity coefficient; and τcr is the critical shear stress for the arithmetic mean size Dm.

246  Bed-Load Transport

The relationship between Φbk and Θek = τeff /[(γs − γ)Dk] is expressed as (Khullar et al., 2007)

 bk

 108 8ek.345 0.02  ek  0.062  3 2 5 4  2545.5ek  412.23ek  518.81ek  81.01ek  6.19ek  0.178 0.062  ek  0.175  13.8951ek.9356 0.175  ek  1.83 



(7.116)

Note that the Paintal (1971a) formula uses the total bed shear stress, whereas Equation 7.114 uses the grain shear stress. The latter is likely more efficient for handling the effects of bed forms. Formula of Wu et al. (2000) Following the unisized bed-load formula of Meyer-Peter and Mueller (1948), Wu et al. (2000) related the fractional transport rate of nonuniform bed load to the nondimensional excess grain shear stress T*k   b  crk  1 . The grain shear stress  b is calculated using Equation 6.48. The established relationship is shown in Figure 7.11 and expressed as  bk

 n  3 / 2  b   0.0053    1  n   crk 

2.2



(7.117)

where n and n′ are Manning’s coefficients corresponding to the total and grain roughness of the channel bed, respectively. Note that pbk used in Φbk is the fraction of size class k in the surface layer of bed sediment. The critical shear stress τcrk is determined with Equation 5.55, which

Figure 7.11  Relationship of fractional bed-load transport rate (from Wu et al., 2000)

Bed-Load Transport  247

considers the hiding-exposure effects in nonuniform bed material. Equation 5.55 uses a critical Shields number of 0.03 for the reference sediment size, which allows Equation 7.117 to be applied in a wider size range than the formula of Meyer-Peter and Mueller (1948). Equation 7.117 was validated by using laboratory data for nonuniform bed load measured by Samaga et al. (1986a), Liu (1986), Kuhnle (1993), and Wilcock and McArdell (1993), as well as field data from the Susitna, Chulitna, Black, Toutle, and Yampa Rivers compiled by Williams and Rosgen (1989). In each set of the selected field data, the flow and sediment parameters were measured at the same time, and the bed-load transport rate and bed-material size composition were averaged from multiple samples on the cross-section. These data sets covered flow discharges up to 2,800 m3/s and sediment sizes from 0.062 to 128 mm. The calibrated exponent of 2.2 is close to the value of 2.1 used in van Rijn’s (1984a) formula and significantly different from the value of 1.5 in Meyer-Peter and Mueller’s (1948) formula. 1/ 6 An , with An = 20. As discussed in Section In Equation 7.117, Wu et al. (2000) used n  D50 6.5, the value of An may vary from 17 to 24 for various rough beds. Moreover, Hassan (2015) found that higher values of An could be used to account for the effect of “memory stress” on bedload transport. Memory stress is the time-dependency of processes specific to sediment particle arrangements and structures that enhance the bed stability and thus alter sediment entrainment and transport under the action of flow stress. As the time duration of the applied flow stress increases, An increases from about 18 to 25, and in turn, the bed-load transport rate decreases. Formulas of Wilcock and Crowe (2003) and Schneider et al. (2015) To consider the interactions between different size classes in a sediment mixture, Wilcock and Crowe (2003) proposed a surface-based method for nonuniform bed-load transport: Wk

0.002 k7.5   0.5 14 1  0.894 /  k





 k  1.35 4.5

 k  1.35



(7.118)

where θk = τb/τrk, with τrk determined by

 rk  Dk     rm  Dm 

mb



(7.119)

where Dm is the mean diameter of sediment in the bed surface layer, τrm is the reference shear stress for Dm, and mb is an empirical coefficient, given with

 rm  0.021  0.015 exp  20 ps   ( s   ) Dm mb 

0.67 1  exp(1.5  Dk Dm )



where ps is the fraction of sand in the surface-layer sediment mixture.

(7.120) (7.121)

248  Bed-Load Transport

Wilcock and Crowe (2003) tested Equation 7.118 using experimental data. Schneider et al. (2015) found that the use of the total shear stress in Equation 7.118 led to the overestimation by many orders of magnitude of the bed-load transport rate in steep mountain streams. Thus, Schneider et al. (2015) excluded the energy losses due to macro-roughness and revised Equation 7.118 as follows for predicting bed-load transport in streams with a wide range of slopes: 0.002 k6.82  Wk   14 1  0.894 /  k0.5  





 k  1.33 4.5

 k  1.33



(7.122)

where  Wk qbk ( s   1) g ( pbk s u*3 ) ; and  k   b  rk , with τrk using Equation 7.119 and τrm  = 0.03(γs − γ)Dm. The grain shear stress  b is determined using Equation 6.48 with n′  =  (2D65)1/6/25, as suggested by Wilcock et al. (2009), or using Equation 6.60 with f and f ′calculated according to Equations 6.86 and 6.92, as suggested by Ferguson (2007) and Rickenmann and Recking (2011). The former approach of  b and the method of τrm = 0.03(γs − γ)Dm are similar to those used by Wu et al. (2000) in Equation 7.117. 7.6  Comparison of Bed-Load Formulas Comparison of Bed-Load Formulas Using Unisized Transport Data Chien and Wan (1983, 1999) compared the formulas of Einstein (1942), Meyer-Peter and Mueller (1948), Bagnold (1966), and Yalin (1972) against measurement data, as shown in Figure 7.12. For weak sediment transport (Ψ > 2), the Yalin formula underpredicts the bed-load transport rates, but the other three formulas provide reasonably good predictions. For strong sediment transport (Ψ < 2), the predictions of the Einstein and Meyer-Peter-Mueller formulas are significantly different from measured values, whereas the Yalin and Bagnold formulas perform relatively well.

Figure 7.12  Comparison of bed-load formulas (from Chien and Wan, 1983)

Bed-Load Transport

249

Wu et al. (2000) compared the formulas of Meyer-Peter and Mueller (1948), Bagnold (1973), Engelund and Fredsøe (1976), and Wu et al. (2000) against 1,345 sets of uniform bed-load data. These data were selected from Brownlie’s (1981b) compilation by limiting the geometric standard deviation σg < 1.2, the Shields number Θ > 0.055, and the Rouse number ωs /(κu*) > 2.5. They covered flow discharges of 0.00094–297 m3/s, flow depths of 0.01–2.56 m, flow velocities of 0.086–2.88 m/s, surface slopes of 0.0000735–0.0367, and sediment sizes of 0.088–28.7 mm. None of these data were used to calibrate the formula of Wu et al. As shown in Table 7.1, the formula of Wu et al. (2000) provides the best results, likely because it was validated by more data than the other three formulas. Comparison of Bed-Load Formulas Using Multisized Transport Data Ribberink et al. (2002) compared the multisized bed-load transport formulas of Parker (1990), Wu et al. (2000), Ackers and White (1973), and Meyer-Peter and Mueller (1948). The formula of Ackers and White (1973) was used with the hiding-exposure correction factors of Day (1980) and Proffitt and Sutherland (1983), which are introduced in Section 9.2. The formula of Meyer-Peter and Mueller (1948) was used with the correction factors of Egiazaroff (1965) and Ashida and Michiue (1972). The unisized formulas of Engelund and Hansen (1967) and van Rijn (1984a) without any hiding and exposure correction were added as references. The used data covered the bed-load transport of widely graded sediment mixtures in the lower Shields regime. The results are expressed in mean under- or overestimation scores (factor n over/underestimation gives a score of 1/n) for the predicted total transport rate and mean transport diameter, as well as an average score between them (Table 7.2). Table 7.1 Calculated versus measured transport rates of uniform bed load (from Wu et al., 2000) Error Range

0.8≤r≤1.25 0.667≤r≤1.5 0.5≤r≤2

% Calculated transport rates in the error range Engelund-Fredsøe

Bagnold

Meyer-Peter-Mueller

Wu et al.

21.4 37.4 54.1

21.4 38.9 57.2

21.3 39.4 66.2

38.7 59.3 80.1

Note: r is the ratio of calculated and measured transport rates.

Table 7.2 Verification scores of multisized bed-load formulas (from Ribberink et al., 2002) Formula

Score for transport rate

Score for mean diameter

Average score

Wu et al. Engelund-Hansen A&W + Day Parker (surface) A&W + P&S Van Rijn MP&M + Egiazaroff MP&M + A&M

0.43 0.34 0.37 0.23 0.34 0.18 0.26 0.29

0.86 0.63 0.59 0.73 0.49 0.54 0.34 0.29

0.64 0.49 0.48 0.48 0.42 0.36 0.30 0.29

Note: A&W = Ackers and White; A&M = Ashida and Michiue; MP&M = Meyer-Peter and Mueller; P&S = Proffitt and Sutherland.

250  Bed-Load Transport

Of all the compared multisized transport formulas, the formula of Wu et al. (2000) gives the best scores, followed by the Ackers-White formula with the hiding-exposure correction factor of Day (1980). Surprisingly, the Engelund-Hansen formula, which was not developed for multifraction use, performs well. The formula of Meyer-Peter and Mueller with both the Egiazaroff and Ashida-Michiue correction factors gives poor scores, mainly due to many cases with zero predicted transport rates. This indicates that the critical Shields number of 0.047 used by MeyerPeter and Mueller (1948) is too large, particularly for sand. 7.7  Effect of Steep Slope on Bed Load Most of the bed-load formulas described in the previous sections are valid for gently sloped channels. Bed-load transport on a steep slope is affected by changes in force configuration, bed particle packing, flow regime, etc. One approach to considering this effect is to correct the critical shear stress τcr (Smart, 1984; Damgaard et al., 1997). For example, Damgaard et al. (1997) modified the bed-load Equation 7.27 of Meyer-Peter and Mueller (1948) as Φb = 8(Θ − Θcr β)3/2 fslope

(7.123)

where Θ = τb/[(γs − γ)D50]; Θcr β = τcr β/[(γs − γ)D50] is the critical Shields number on the sloped bed, determined using τcr β from Equation 5.63; and fslope is a correction factor applied to the transport rate: 1  f slope   1.5   cr 0.2 1  0.8(cr ) (1  cr  cr )

i    0 0    i



(7.124)

where ϕi is the friction or repose angle; and β is the bed slope angle, with a positive value for a downslope bed. A disadvantage of this approach is that when β = ϕi, the corrected critical shear stress goes to zero, so the calculated bed-load transport rate tends to be infinite for formulas that express qb as functions of  b  cr or τb/ τcr. The second approach adds the streamwise component of gravity to the grain shear stress  b or the bed shear stress τb without modifying τcr. The effective tractive force τbe is thus determined by (Wu, 2004)

 be   b   s

as   s    gD sin  6



(7.125)

where as is a coefficient related to the shape and packing of sediment particles and ξs is a coefficient for converting the streamwise gravity to shear stress on the bed slope. When the bed slope angle β is equal to the repose angle ϕi, sediment particles start moving (τbe = τcr) without any hydraulic force ( b  0 ). Under this condition, Equation 7.125 yields ξs = τcr/[(asπ/6)(ρs − ρ) Dsin ϕi]. Inserting this relationship into Equation 7.125 yields

 be   b  0 cr sin  sin i



(7.126)

Bed-Load Transport  251

where λ0 is introduced to improve the performance for downslopes. When Equation 7.126 is applied to the sediment transport formulas of Wu et al. (2000), λ0 has the following relationship (Wu, 2004): 1  0   0.15 2 sin  /sin i 1  0.22( b  cr ) e

 0  0



(7.127)

Though the two factors in Equations 7.124 and 7.127 are applied differently to the transport rate and the effective shear stress, they serve the same purpose of accounting for additional changes in bed-load transport over downslope beds. 7.8  Fluctuations of Bed Load The bed-load transport rates in natural rivers vary dramatically over a wide range of tempospatial scales. The instantaneous transport rates can be divided into mean and fluctuation rates. The mean transport rates are associated with the reach-averaged flow motions in storm events. The fluctuations can be roughly classified into two groups. The first group is caused by bed-form migration, macro-roughness (e.g., boulders, gravel clusters) variation, grain sorting (e.g., fine and coarse segregation), and bed armoring (formation, removal) (Iseya and Ikeda, 1987; Frey et al., 2003; Ghilardi et al., 2014; Dhont and Ancey, 2018; Recking et al., 2023). The second group is called diffusion at grain scale due to intermittent particle motion and turbulent flow fluctuations (Nikora et al., 2002; Cecchetto et al., 2018). The intrinsic fluctuations of bed-load transport cause significant uncertainties in measurement data and hinder the predictive capability of bed-load transport formulas. Figure 7.13a shows the temporal variations of bed-load transport rate, as measured by Kuhnle (2008) using a trap sampler in the Goodwin Creek during the falling stage of a flood event. The bed-load transport rate fluctuated in a cyclical pattern with periods of a few to tens of minutes, which did not appear in the flow hydrograph. Figure 7.13b shows similar bed-load fluctuations measured by Dhont and Ancey (2018) at the outlet of a laboratory flume under steady inflow conditions using six accelerometers in a time interval of 1 minute. This type of cyclical fluctuation captured with relatively long intervals is mainly in the scale of bed forms, i.e., the first group described herein, rather than the grain scale.

Figure 7.13 Temporal variations of bed-load transport rates: (a) measured at Goodwin Creek, Mississippi (from Kuhnle et al., 1989), and (b) measured in a flume (data from Dhont and Ancey, 2018)

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The physical mechanisms governing bed-load fluctuations at grain scale have been investigated in recent years by applying image-based measurement techniques. Nikora et al. (2002), Martin et al. (2012), Campagnol et al. (2015), and Cecchetto et al. (2018) analyzed the diffusive characteristics of bed-load motion using particle displacement data. Particle trajectory fluctuations behave differently on three scales: the local scale between two successive significant collisions with the bed, the intermediate scale between two particle-rests, and the global scale of multiple intermediate trajectories. Particle motion is superdiffusive at the local scale and subdiffusive at the global scale. Bed-particle-fluid interactions play important roles in these processes. Local-scale fluctuations are related to a balance between particle inertia and fluid drag. The fluctuations in the intermediate range are a result of the balance between intermittent transport and near-bed turbulence. In the global range, extreme values exist in the heavy-tailed distribution of particle resting times, as related to surface motion, vertical mixing, and bed scour. Hamamori (1962) proposed a theoretical probability distribution function for the relative bed-load transport rates over stream beds with secondary ripples on top of primary dunes under constant flow conditions. The distribution is shown in Figure 7.14 and expressed as  4qb  qb  qb  F 1  ln     qb  qb  4qb 

    

(7.128)

where F is the cumulative distribution function, i.e., the probability that an observed relative transport rate is less than or equal to the indicated value; qb is the individual transport rate; and qb is the long-term mean transport rate of all individuals. The Hamamori distribution has the following probability density function: pd ( x) 

1 4 In 4  x 



(7.129)

where x is the sample value of qb qb .

Figure 7.14  Probability distribution of relative bed-load transport rate (from Carey, 1985)

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In the Hamamori distribution, individual bed-load transport rates vary from zero to 4qb , and 60% of all rates are less than qb (Carey, 1985). The fluctuation amplitude of 4qb is likely due to the migration of dune-type bed forms. Later experiments have shown that the fluctuation amplitude can be as large as 12qb , likely caused by alternate bars (Dhont and Ancey, 2018). Birnbaum and Saunders (1968) proposed a two-parameter exponential distribution to model the times to failure due to fatigue of material under cyclic stresses. The Birnbaum-Sanders distribution was applied by Turowski (2010) to describe the distribution of bed-load fluctuations. It is expressed as follows: pd  x  

 ( x  b )2  exp   2  2 bs x 2 b x  2 x  b bs  x  b



(7.130)

where βb is the scale parameter and γbs is the shape parameter. The following gamma distribution is often used to model the probability density of the relative bed-load transport rate: pd  x  



g g ( g )

 g 1   g x

x

e

for x >0

(7.131)

where αg is the shape parameter, βg is the rate parameter, and Γ(αg) is the gamma function of αg. Both αg and βg are larger than 0. If αg and βg are set as fg/2, the gamma distribution is reduced to the F(fg,∞) distribution, in which fg is the degree of freedom. Tan (1982) tested the F(fg,∞) distribution using data from a sand-bed reach on the Yangtze River and a gravel-bed reach on the Min River (Figure 7.15). The data show lower probabilities at low and high transport rates and a peak in the lower middle range for sand beds, but a monotonously decreasing trend with increasing transport rate for gravel beds. To fit with the measured data, fg is set as 4 (i.e., αg = βg = 2) for a sand bed and 1 (αg = βg = 1/2) for a gravel bed. Kuhnle and Willis (1998) tested the Hamamori, gamma, exponential, and normal distributions against the bed-load transport rates measured in Goodwin Creek, Mississippi, USA. The measured data show a distribution similar to Figure 7.15a. The gamma distribution can capture the trend of the entire range of data. The Hamamori and exponential distributions agree closely

Figure 7.15 Probability density of relative bed-load transport rate: (a) sand bed, and (b) gravel bed (from Tan, 1982)

254  Bed-Load Transport

with the data at high transport rates but deviate significantly at low transport rates below the mean. Similar findings were reported by Saletti et al. (2015) in an experimental study of nonuniform bed-load transport in a step-pool system. Different distributions for sand and gravel beds in Figure 7.15 might be due to different bed features at those sites. Another possible reason is sampling time. According to Gomez et al. (1990), increasing sampling time causes the distribution to change from a Hamamori-type distribution (Figure 7.15b) to a skewed-normal-type distribution (Figure 7.15a). This is due to the assumption used by the Hamamori distribution that samples are collected at a point in an instantaneous time. The Hamamori distribution is adequate if the sampling time is short. The chance of yielding low transport is likely less when the sampling time is longer. Thus, the gamma or F(fg, ∞) distribution is more suitable to model bed-load fluctuations than the Hamamori distribution. The scale-dependent fluctuations in bed load are usually treated as random processes and described by using probabilistic and random walk models (e.g., Ancey et al., 2008; Furbish et al., 2012; Zhang et al., 2012; Fan et al., 2014). These models are not yet applicable in practice but have shown promising results. As derived by Ma et al. (2014) using the probability model of Ancey et al. (2008), the variance of bed-load flux decreases as the sampling time increases through three regimes: intermittent, invariant, and memoryless stages. The three temporal regimes may be associated with the three spatial scales observed by Nikora et al. (2002). Ancey and Pascal (2020) applied the probabilistic model to estimate mean bed-load transport rates and uncertainties in measurements. Details of these stochastic models can be found in the references. 7.9  Measurements of Bed Load 7.9.1  Bed-Load Samplers Bed-load samplers can be grouped into two types: pit-trough samplers installed in channel beds and manually operated portable samplers (Hubbell, 1964; Kuhnle, 2008). In addition, there are noninvasive measurement methods, such as bed-form tracking (Havinga, 1982), particle imaging (e.g., Lajeunesse et al., 2010; Shim and Duan, 2019), acoustic techniques (e.g., Conevski et al., 2019), geophonic/hydrophonic impact plates (e.g., Rickenmann, 2017), and magnetic tracking. These noninvasive methods have shown promise but have not yet been widely used in the field. Pit-trough and portable samplers are described in the following text. Pit-Trough Samplers Pit-trough samplers are installed in the channel bed by burying the sampler or digging a trap so that the sampler top remains level with the bed surface (Figure 7.16). These samplers are

Figure 7.16  Sketch of pit-trough sampler of bed load: (a) plan view, and (b) side view

Bed-Load Transport  255

designed to capture all the bed-load particles transported to the trap slot. They have been used in streams that can be easily accessed. The measurements yielded usually describe the total amount and size composition of bed load in a flood event or measurement interval. Pit-trough samplers are relatively accurate if designed and installed properly. However, they may not be useful if filled up at an unknown time before the flood event is finished or they are checked. To obtain the temporal variation of bed-load transport rate, instruments are needed to remove sediment from the sampler, weigh the sediment, and record the time and quantity. Such instruments are costly to install and operate and thus have been used only on a few rivers, e.g., the East Fork River, Wyoming (Leopold and Emmett, 1976; Emmett, 1980). For sand-bed channels, Einstein (1944) experimentally demonstrated that samplers with streamwise slot lengths of 100–200 grain diameters could collect nearly 100% of the bed load. For grain sizes between 1.88 and 4.5 mm, Poreh et al. (1970) showed in a laboratory flume that a channel-wide pit sampler with a streamwise slot length of about 35 grain diameters would have an efficiency close to 100%. Streamwise slot lengths much larger than necessary are not recommended because secondary flows in the trap increase with slot length and may exclude smaller grains moving as bed load (Ethembabaogla, 1978; Wilcock et al., 1996; Kuhnle, 2008). In addition, for pit traps with widths narrower than the channel width, sediment may enter the slot laterally from the sides and cause oversampling (Emmett, 1980). This can be minimized by installing fences along the sides of the sampler (Lewis, 1991). Portable Samplers Portable bed-load samplers include basket, pan (or tray), and pressure-difference samplers. A basket sampler has screens on all sides except for the front and possibly the bottom (Figure 7.17). It operates by screening sediment from the flow. A pan or tray sampler retains the

Figure 7.17 A basket bed-load sampler similar to the Ehrenberger (1931) model modified by the Slovak Water Research Institute, with a mesh size of 3 mm and a trapping efficiency of 0.7 (from Camenen et al., 2011)

256  Bed-Load Transport

sediment that drops into a slot or slots. Basket and pan samplers impose resistance, so the flow velocity inside is lower than that of the free stream. This causes deposition in the front and reduces sediment transport into the samplers. A pressure-difference sampler is designed to avoid reductions in flow velocity and bedload transport into the sampler by gradually increasing the cross-sectional area and in turn decreasing the pressure at the exit of the sampler nozzle. The Helley-Smith and Y-78 bed-load samplers are shown in Figure 7.18 as examples. The Helley-Smith sampler has a square, 7.6 cm (or 15.2 cm) wide entrance and an entrance/exit area ratio of 3.22 (Helley and Smith, 1971; Emmett, 1980). The Y-78 sampler has a square entrance of 10 cm in width. The Y-78 sampler has a smaller entrance/exit ratio and thus a lower sampling efficiency than the Helley-Smith sampler. Other pressure-difference samplers include the Federal Interagency Sedimentation Project FISP BL-84 (Davis, 2005), Toutle River-2 (Childers, 1992), Elwha River (Childers et al., 2000), Delft-Nile (van Rijn and Gaweesh, 1992), BM-2 (CAHE, 1992), and BTMA-2 (Duizendstra, 2001). A bed-load sampler should be used within the range of conditions for which it is designed. The limiting conditions include particle size, bed-load transport rate, water depth, and flow velocity. The sediment size range that a sampler can measure is restricted by the entrance size of the sampler and the mesh opening of the sample bag. The bed-load transport rate is limited by the catchment volume and sampling time. The applicable water depth depends on whether the sampler is designed for wading or cable suspension. The flow velocity is limited by the resistance of the sampler in the flow and the velocity range used in calibration. Table 7.3 lists the application-limiting conditions of selected portable bed-load samplers. Portable samplers are easy to set up and operate, but their efficiency is still of concern. The efficiency of a bed-load sampler is defined as the ratio of the sampled transport rate to the actual

Figure 7.18 Pressure-difference samplers: (a) Helley and Smith (1971) (photo courtesy of USGS), and (b) Y-78 (CAHE, 1992) (photo by the author)

Bed-Load Transport

257

rate. The efficiency is usually calibrated by comparing the amount of sediment collected by the sampler to the undisturbed bed-load movement without the sampler in place. Reported calibrations suggest efficiencies of 25%–70% for the basket type, 40%–75% for the pan or tray type, and 60%–180% for the pressure-difference type (Table 7.3). These efficiencies may vary with sediment transport rate, size gradation, bed configuration, and flow conditions. During a bed-load sampling event, the sampler may stir up the bed material, leave a gap above the bed surface due to uneven bed forms and erosion, scoop bed sediment when it is pulled up, misalign with the flow direction, or be clogged by clay aggregates and organic material. All these factors affect the integrity of the measurement data. To resolve these potential problems, stay lines, flexible bottoms, guide fins, larger collection bags, bottom sensors, and underwater video cameras have been designed for certain samplers (Childers, 1992; Duizendstra, 2001; Bunte et al., 2001; Dixon and Ryan, 2001). In addition, sampling and data analysis procedures have also been improved (Davis, 2005). Gaweesh and van Rijn (1994) recommended removing the highest and lowest 10% of the collected samples to reduce possible errors during field sampling. This may be a good approach when handling the measured raw data. Note that the portable samplers listed in Table 7.3 have inlet heights of 5.5–30 cm. These values may be higher than the bed-load layer thickness. Thus, the collected sediment may not include only bed load, but also some near-bed suspended load. This was demonstrated by Emmett (1980) in the calibration of the Helley-Smith sampler in the East Fork River. For sediment particle sizes between 0.5 and 16 mm, the Helley-Smith sampler has a near-perfect sampling efficiency. For particle sizes smaller than 0.5 mm, the Helley-Smith sampler has a high bed-load Table 7.3 Portable bed-load samplers Sampler name

Inlet width/height Sampler type (cm × cm)

YZ-80

50 (width)

Sediment Sampling Flow Sampler Reference size (mm) efficiency velocity capacity (%) (m/s) (kg)