Free-Surface Flow:

Free-Surface Flow:

Shallow Water Dynamics

1st Edition - August 30, 2018

Write a review

  • Author: Nikolaos Katopodes
  • eBook ISBN: 9780128154885
  • Paperback ISBN: 9780128154878

Purchase options

Purchase options
DRM-free (EPub, Mobi, PDF)
Available
Sales tax will be calculated at check-out

Institutional Subscription

Free Global Shipping
No minimum order

Description

Free-Surface Flow: Shallow-Water Dynamics presents a novel approach to this phenomenon. It bridges the gap between traditional books on open-channel flow and analytical fluid mechanics. Shallow-water theory is established by formal integration of the Navier-Stokes equations, and boundary resistance is developed by a rigorous construction of turbulent flow models for channel flow. In addition, the book presents a comprehensive description of shallow-water waves by mathematical analysis. These methods form the foundation for understanding flood routing, sudden water releases, dam and levee break, sluice gate dynamics and wave-current interaction.

Key Features

  • Bridges the gap between traditional books on open-channel flow and wave mechanics
  • Presents a comprehensive description of shallow-water waves by characteristic and bicharacteristic analysis
  • Presents techniques for wave control and active flood mitigation

Readership

Civil and Environmental Engineers

Table of Contents

  • Prologue xix

    References xxvii

    1. Basic Concepts

    1.1 Introduction 4

    1.1.1 Shallow-Water Models 4

    1.2 Surfaces in Three-Dimensional Space 9

    1.2.1 Analytic Representation of Surfaces 9

    1.2.2 Implicit Surfaces 9

    1.2.3 Parametric Representation 11

    1.2.4 Area of a Two-Dimensional Surface 12

    1.2.5 Oriented Surfaces and Fundamental Forms 14

    1.2.6 Gaussian and Mean Curvature 16

    1.2.7 Principal Curvature Sections 17

    1.2.8 Euler’s Theorem 18

    1.2.9 Divergence Representations 19

    1.3 Initial and Boundary-Value Problems 21

    1.3.1 Types of Boundary Conditions 21

    1.3.2 Initial-Value Problems 22

    1.3.3 Two-Point Boundary-Value Problem 23

    1.3.4 General Equilibrium Problems 23

    1.3.5 Evolution Problems 24

    1.4 Classification of Partial Differential Equations 27

    1.4.1 Linear, First-Order Equation 28

    1.4.2 Systems of First-Order PDE’s 31

    1.4.3 Classification of Quasi-linear Systems 33

    1.4.4 Second Order Equations 34

    1.4.5 Classification of Second-Order Equations 36

    Problems 38

    References 40

    Note 41

    2. Air-Water Interface

    2.1 Introduction 44

    2.2 Surface Tension 46

    2.2.1 Young-Laplace Equation 47

    2.2.2 Wettability and Contact Angle 49

    2.2.3 Meniscus Profile 50

    2.2.4 Marangoni Forces and the Bond Number 52

    2.2.5 Surface Free Energy 53

    2.2.6 Minimum Surface Energy 54

    2.2.7 Floating Bodies 55

    2.2.8 Circular Hydraulic Jump 56

    2.3 Free Surface Boundary Conditions 60

    2.3.1 Dynamic Surface Condition 60

    2.3.1.1 Normal Forces 62

    2.3.1.2 Tangential Forces 63

    2.3.2 Scaling the Dynamic Surface Condition 63

    2.3.3 Dynamic Condition for Potential Flow 64

    2.3.4 Kinematic Surface Condition 65

    2.3.5 Steady Flow in Two Dimensions 66

    2.3.6 Kinematic Bottom Condition 68

    2.3.7 Rigid Lid Approximation 69

    2.3.8 Boundary Conditions at Contact Lines 70

    2.3.9 Pressure Boundary Condition for Poisson Equation 73

    2.4 Simple Viscous Flows With a Free Surface 74

    2.4.1 Channel Flow Under Calm Wind 74

    2.4.2 The Rate of Streamwise Energy Dissipation 76

    2.4.3 Flow Driven by Wind Shear 78

    2.4.3.1 Wind Set-Up 81

    2.4.4 Suddenly Accelerated Air-Water Interface 82

    2.5 Transfer Processes at the Air-Water Interface 86

    2.5.1 Drag Coefficient at Air-Water Interface 87

    2.5.2 Significant Wave Height 89

    2.5.3 Random Wave Analysis 91

    2.5.4 Wave Frequency Spectrum 93

    2.5.4.1 Pierson-Moskowitz Spectrum 95

    2.5.4.2 JONSWAP Spectrum 96

    2.6 Atmospheric Surface Layer 98

    2.6.1 Wind and Wave Stresses 99

    2.6.2 Constant Flux Layer 99

    2.6.3 Obukhov Length 100

    2.6.4 Monin-Obukhov Similarity Theory 101

    2.7 Storm Surge 106

    2.7.1 Barometric Tide 106

    2.7.2 Wind Shear Tide 107

    2.7.3 Bathystrophic Tide 108

    2.7.4 Wave Set-Up 109

    2.8 Large Scale Interface Disturbances 110

    2.8.1 Cyclogenesis 110

    2.8.2 Meteorological Tsunami 111

    Problems 113

    References 114

    3. Gravity Waves

    3.1 Introduction 118

    3.2 Small-Amplitude Gravity Waves 121

    3.3 Two-Dimensional Oscillatory Waves 124

    3.4 Airy’s Theory for Gravity Waves 126

    3.4.1 Boundary Conditions 127

    3.4.2 Velocity Potential for Sinusoidal Waves 128

    3.4.3 Dispersion Relation 129

    3.4.4 Shallow-Water Limit 130

    3.4.5 Pressure Distribution 132

    3.5 Dispersion of Non-sinusoidal Waves 134

    3.5.1 Fourier Series 134

    3.5.2 Fourier Integral 135

    3.5.3 Fourier Transform 136

    3.5.4 Dispersion of a Composite Wave 136

    3.5.4.1 Gaussian Packet 137

    3.5.5 Dispersion of a Gaussian Wave Packet 139

    3.6 Superposition of Linear Gravity Waves 141

    3.6.1 Reflection on a Solid Boundary 141

    3.6.2 Standing Waves 142

    3.7 Seiches 145

    3.7.1 Two-Dimensional Seiche 146

    3.8 Mass Transport by Gravity Waves 149

    3.9 Progressive Wave Energy 152

    3.9.1 Wave Energy Flux 153

    3.10 Group Velocity 155

    3.11 Wave Refraction 158

    3.12 Wave Diffraction 162

    3.12.1 Diffraction Theory 162

    3.12.2 Waves Incident Obliquely on the Breakwater 166

    3.13 Wave Breaking 169

    3.13.1 Radiation Stress 171

    3.13.2 Wave Set-Up 172

    Problems 176

    References 177

    4. Shallow-Water Approximation

    4.1 Introduction 182

    4.2 Shallow-Water Equations 183

    4.2.1 Depth-Averaged Equations 185

    4.2.1.1 Equation of Continuity 185

    4.2.1.2 Equation of Streamwise Momentum 186

    4.2.1.3 Equation of Transverse Momentum 187

    4.2.1.4 Vector Form of Shallow-Water Equations 187

    4.2.2 The Gas Dynamics Analogy 189

    4.2.3 Vorticity Transport in Shallow Water 191

    4.3 Waves in Shallow Water 194

    4.3.1 Gravity Waves 194

    4.3.2 Gravity Waves on a Rotating Earth 194

    4.3.3 Gravity Waves Along the Coast 195

    4.3.4 Barotropic Vorticity Waves 196

    4.4 Dispersion Relations for Nonlinear Waves 199

    4.5 Higher-Order Long-Wave Approximation 202

    4.5.1 Zero-Order Approximation 204

    4.5.2 First-Order Approximation 205

    4.5.3 Second-Order Approximation 207

    4.5.4 Second-Order Oscillatory Wave 209

    4.6 The Boussinesq Equations 211

    4.7 Long Waves in Trapezoidal Channels 215

    4.7.1 Boussinesq Equations for Trapezoidal Channel 215

    4.7.1.1 First Approximation 216

    4.7.1.2 Second Approximation 217

    4.8 The Serre Equations 220

    4.9 The Korteweg-De Vries Equation 225

    4.9.1 Solitary Wave 226

    4.10 Hamiltonian Approach to Water Waves 229

    4.10.1 Approximation of the Kinetic Energy 232

    4.10.1.1 Kinetic Energy Below the Mean Water Level 233

    4.10.1.2 Kinetic Energy Above the Mean Water Level 235

    4.10.1.3 Hamiltonian for Fairly Low Long Waves 235

    4.10.1.4 Canonical Equations 236

    4.10.2 Horizontal Channel 237

    4.10.3 Approximate Hamiltonian 239

    4.10.4 The Free-Surface Approximation 241

    4.10.5 Extension to Uneven Bottom 242

    4.10.6 Canonical Equations for the Average Velocity 243

    Problems 244

    References 245

    5. Tidal Forcing

    5.1 Introduction 250

    5.2 Equilibrium Theory of Tides 251

    5.2.1 Tidal Forces 252

    5.2.2 Equilibrium Tidal Surface 257

    5.2.3 Planetary Complications 257

    5.2.4 Solar Tide 259

    5.3 Dynamic Theory of Tides 261

    5.3.1 Standing Tidal Wave 262

    5.3.2 Kelvin Tidal Wave 263

    5.3.3 Co-tidal Lines and Amphidromic Points 265

    5.4 Harmonic Analysis and Tide Prediction 268

    Problems 271

    References 272

    6. Long Waves

    6.1 Introduction 276

    6.2 Flow in One-Dimensional Channels 278

    6.3 Integral Relations 285

    6.4 The Saint-Venant Equations 287

    6.5 Energy Considerations in an Open Channel 290

    6.5.1 The Choice Between Momentum and Energy 294

    6.6 Vector Representation 298

    6.6.1 Broad-Channel Representation 298

    6.6.2 Saint-Venant Equations 299

    6.7 Further Simplifications 300

    6.8 Linearized Equations 304

    6.9 Symmetric Equations 306

    6.10 Steady, Non-uniform Flow 307

    6.11 Shallow-Water Flow in Two Space Dimensions 308

    Problems 311

    References 312

    7. Channel Transitions

    7.1 Introduction 316

    7.2 Regimes of Steady Flow 319

    7.3 Nearly-Horizontal Flow 321

    7.3.1 Steep Channels 323

    7.3.2 Kinetic Energy Correction Factor 325

    7.4 Transitions in Geometry and Bathymetry 326

    7.5 Flow Under a Vertical Sluice Gate 329

    7.5.1 The Contraction Coefficient 330

    7.5.2 Discharge Through a Free-Flowing Gate 330

    7.5.3 Fluid Force on Sluice Gate 331

    7.6 Flow Over a Smooth Bottom Ridge 333

    7.7 The Specific Energy 334

    7.7.1 Dimensionless E-h Diagram 336

    7.8 Critical Velocity and Gravity Wave Speed 339

    7.9 The Froude Number 342

    7.9.1 Alternative Scaling Approaches 343

    7.10 Critical Flow in Channels of Arbitrary Cross-Sectional Shape 346

    7.10.1 Channels With a Floodplain 347

    7.10.2 Channel Shape for Unconditional Critical Flow 349

    7.11 Subcritical Flow Over a Smooth Ridge 351

    7.11.1 Occurrence of Critical Flow 352

    7.11.2 Supercritical Flow Over a Smooth Ridge 354

    7.11.3 Experimental Validation 355

    7.11.4 Force Exerted on Bottom Ridge 356

    7.12 Flow Through a Smooth Transition in Width 358

    7.12.1 Occurrence of Critical Flow 359

    7.13 Downstream Control – Formation of a Hydraulic Jump 362

    7.13.1 Conservation of Momentum Across a Hydraulic Jump 365

    7.13.2 Hydraulic Jump in a Rectangular channel 366

    7.13.3 Dissipation of Energy 368

    7.14 The Specific Force 369

    7.14.1 Dimensionless F-h Diagram 371

    7.14.2 Flow Under a Submerged Sluice Gate 372

    7.15 Fluid Force on Transition Structures 376

    7.15.1 Blocks Assisting the Formation of a Jump 377

    7.15.2 Control of Hydraulic Jump by Abrupt Drop 379

    7.15.3 Control of Hydraulic Jump by Abrupt Rise 381

    7.15.4 Choking Mechanisms 383

    7.16 Other Rapidly-Varied Flow Transitions 386

    7.16.1 Outflow From a Reservoir 386

    7.16.2 Free Overfall 387

    7.16.3 Lateral Outflow Through a Smooth Downspout 389

    7.16.4 Flow Around a Bend in Subcritical Flow 390

    7.16.4.1 Channel Bed Adjustment 392

    Problems 394

    References 399

    8. Channel Bed Resistance

    8.1 Introduction 402

    8.2 Uniform Flow in a Sloping Channel 405

    8.2.1 Reynolds Numbers Limits for Open-Channel Flow 406

    8.3 Logarithmic Velocity Profiles 408

    8.3.1 Smooth Wall Boundary 409

    8.3.2 Rough Wall Boundary 411

    8.3.3 The Velocity Intercept 412

    8.3.4 Classification of “Smooth” and “Rough” Walls 415

    8.4 Depth-Averaged Velocities 417

    8.5 Bed Shear in Shallow-Water Flow 420

    8.5.1 Newton’s Law of Flow Resistance 422

    8.6 The Friction Factor 423

    8.6.1 Computation by Velocity Measurements 424

    8.7 Flow Resistance in Open Channels 427

    8.7.1 The Chézy Equation 427

    8.7.2 Chézy Equation for General Cross Sections 430

    8.7.3 The Gauckler-Kutter Equation 430

    8.7.4 The Manning n 432

    8.8 Uniform Flow 436

    8.9 Optimal Cross-Sectional Shape 439

    8.9.1 Rectangular Channel 439

    8.9.2 Trapezoidal Channel 439

    8.10 Classification of Uniform Flow Regimes 441

    Problems 443

    References 444

    9. Gradually-Varied-Flow

    9.1 Introduction 448

    9.2 Non-uniform Flow 449

    9.2.1 Other Forms of the GVF Equation 451

    9.2.1.1 Section Factor Form 451

    9.2.1.2 Critical Discharge Form 451

    9.2.1.3 Hydraulic Exponent Form 452

    9.2.1.4 Bresse’s Wide Channel Approximation 454

    9.3 Classification of Gradually-Varied Flow Profiles 456

    9.3.1 Mild Slope Profiles 456

    9.3.1.1 Backwater 456

    9.3.1.2 Drawdown 459

    9.3.1.3 Tailwater 461

    9.3.2 Steep Slope Profiles 462

    9.3.2.1 Backwater 462

    9.3.2.2 Drawdown 463

    9.3.2.3 Tailwater 464

    9.3.3 Zero Slope Profiles 464

    9.3.4 Adverse Slope Profiles 465

    9.3.5 Critical Slope Profiles 466

    9.3.6 Frictionless Channel Profiles 467

    9.3.7 Zero-Inertia Profiles 468

    9.4 Direct Integration of the GVF Equation 470

    9.4.1 Frictionless Channel 470

    9.4.2 Wide Horizontal Channel 471

    9.4.3 Sloping Wide Channel – Bresse Solution 473

    9.4.4 General Channel – Ven Te Chow Solution 474

    9.4.4.1 Horizontal Bottom 474

    9.4.5 Singular Perturbation Solution 475

    9.5 Numerical Solution of the GVF Equation 478

    9.6 Dimensionless GVF Profiles 482

    9.7 Lake Outflow Into Channel With Mild Slope 484

    9.7.1 Long Channel 484

    9.7.2 Short Channel 485

    9.7.2.1 Dimensionless Lake to M2 Profile 486

    9.7.2.2 Dimensionless Lake to H2 Profile 488

    9.8 Spatially-Varied Flow 490

    9.8.1 Lateral Inflow 490

    9.8.2 Lateral Outflow 493

    Problems 495

    References 497

    10. Characteristic Analysis

    10.1 Introduction 500

    10.2 Discontinuities of the Free-Surface Profile 501

    10.2.1 Waves and Wave Fronts 501

    10.3 Classification of Shallow-Water Equations 505

    10.3.1 de Saint Venant Equations 505

    10.3.2 Zero-Inertia Equations 505

    10.3.3 Kinematic-Wave Equation 506

    10.4 The x t Plane 507

    10.5 Transport of Wave Fronts 510

    10.6 Identification of Characteristic Directions 513

    10.6.1 Characteristic Form of Scalar Wave Equation 513

    10.6.2 Characteristic Form of Kinematic Wave Equation 516

    10.6.3 Kinematic Shock Wave 520

    10.6.4 Impact of Lateral Inflow 521

    10.6.5 Overland Flow 522

    10.6.6 Recession 525

    10.7 Characteristics of St. Venant Equations 530

    10.7.1 Characteristic Equations 533

    10.7.2 Universal Celerity Variable 535

    10.7.3 Compatibility Equations 536

    10.7.4 Riemann Invariants 538

    10.7.5 Canonical Equations 538

    10.7.5.1 Gravity Waves in a Frictionless Horizontal

    Channel 539

    10.7.6 Turbid Underflows 540

    10.7.7 Compatibility Equations 542

    10.7.8 Contact Discontinuities 543

    10.8 Specification of Initial and Boundary Conditions 544

    10.8.1 The Characteristic Network 547

    10.8.2 Interference of Boundaries 548

    10.8.3 Non-reflecting Boundaries 550

    10.9 Steady Flow in Two Dimensions 553

    10.9.1 Impact of Froude Number 555

    10.9.2 Compatibility Equations 557

    10.10 The Hodograph Plane 559

    10.10.1 Characteristics on the Hodograph Plane 561

    10.10.2 Polar Form of Hodograph Equations 563

    10.11 Change of Depth Across a Characteristic 566

    Problems 569

    References 570

    11. Bicharacteristics

    11.1 Introduction 574

    11.1.1 Propagation of Initial Data 575

    11.1.1.1 Eigenvalues as Characteristic Surface Normals 576

    11.2 Characteristic Surfaces and Bicharacteristics 578

    11.2.1 Construction of Interior Operators 578

    11.3 Characteristic Surface Families 580

    11.3.1 Characteristic Flow Surfaces 580

    11.3.2 Characteristic Wave Surfaces 583

    11.3.3 Characteristic Cone 583

    11.3.4 Characteristic Conoid 585

    11.3.5 Existence and Uniqueness of Solution 588

    11.3.6 Bicharacteristics 589

    11.3.7 Parametric Representation of Bicharacteristics 590

    11.3.8 Bicharacteristic Tangency Condition 592

    11.4 Compatibility Relations 596

    11.4.1 Flow Surfaces 596

    11.4.1.1 Propagation of Scalar Properties 596

    11.4.1.2 Propagation of Shear Waves 597

    11.4.2 Wave Surfaces 598

    11.4.3 Interior Differential Equations 599

    11.4.4 Interdependence of Compatibility Conditions 601

    11.4.5 Canonical Equations 603

    11.5 Bicharacteristics of Turbid Underflows 605

    11.5.1 Canonical Equations 607

    Problems 612

    References 613

    12. Simple Waves, Surges, and Shocks

    12.1 Introduction 618

    12.2 Properties of Simple Waves 621

    12.2.1 Profile Deformation in Simple Wave Region 623

    12.2.2 Regressive Depression Wave 624

    12.3 Progressive Depression Wave 628

    12.3.1 Supercritical Initial Flow 629

    12.3.2 Centered Depression Waves 630

    12.3.2.1 Critical Outflow 631

    12.4 Progressive Elevation Wave 633

    12.4.1 Occurrence of First Discontinuity 635

    12.4.2 Surge Formation by Flowrate Control 637

    12.5 Regressive Elevation Wave 638

    12.6 Interaction of Simple Waves 640

    12.7 Surges and Shocks 646

    12.7.1 Conservation of Mass 647

    12.7.2 Conservation of Momentum 648

    12.7.3 Conservation of Energy 649

    12.7.4 Choice of Jump Conditions 650

    12.8 Weak Solutions of Conservation Laws 653

    12.8.1 Properties of Weak Solutions 655

    12.9 Algebraic Jump Conditions 658

    12.10 Instantaneous Jump Formation 661

    12.10.1 Surge Resulting From Upstream Gate Opening 661

    12.10.2 Shock Resulting From Downstream Gate Closing 664

    12.11 Compatibility Conditions at a Discontinuity 666

    12.11.1 High Side on the Right of Jump (r >1) 668

    12.11.2 High Side on the Left of Jump (r <1) 670

    12.12 Energy Loss Across a Jump 672

    12.13 Interaction of Shock Waves 674

    12.13.1 Shock Reflection 674

    12.13.2 Shock Collision 675

    12.14 Interaction of Shocks and Simple Waves 677

    Problems 679

    References 681

    13. Sudden Water Release

    13.1 Introduction 684

    13.2 Dam-Break Wave 685

    13.2.1 Dimensionless Depth Profile 690

    13.2.2 Characteristics of Ritter Solution 691

    13.2.3 Conservation Properties of Ritter Solution 692

    13.2.4 Evolution of the Ritter Dam-Break Wave 694

    13.3 Dam-Break on Still Water of Constant Depth 696

    13.3.1 Evolution of Dam-Break Wave in Wet Channel 698

    13.3.2 Dam-Break in a Channel With Base Flow 699

    13.3.2.1 Dimensionless Solution 701

    13.3.2.2 Limiting Depth Ratio 703

    13.4 Partial Dam Breach 706

    13.4.1 Free Flowing Breach 707

    13.4.2 Hydraulic Jump Within Breach 710

    13.4.3 Submerged Breach 711

    13.5 Effects of Bed Slope and Resistance 713

    13.5.1 Dam-Break in Frictionless, Sloping Channel 713

    13.5.2 Wave Front on Rough, Dry Bed 717

    13.5.3 Whitham’s Approximation of the Wave Tip 720

    13.5.3.1 Conservation of Wave Tip Volume 721

    13.5.3.2 Conservation of Wave Tip Momentum 721

    13.5.3.3 Wave Front Advance 722

    13.5.3.4 Wave Front Profile 725

    13.5.3.5 Matched Asymptotic Expansions 726

    13.6 Gradual Dam Breach 727

    13.7 Sluice Gate Operation 729

    13.7.1 Sudden Complete Opening 729

    13.7.2 Sudden Complete Closing 731

    13.7.3 Sudden Partial Opening 734

    13.7.4 Sudden Partial Closing 738

    Problems 741

    References 742

    14. Active Flood Control

    14.1 Introduction 745

    14.2 Adjoint Equations for Open-Channel Flow 747

    14.2.1 Characteristic Analysis 749

    14.2.2 Sensitivity Equations 751

    14.2.3 Alternative Formulation of the Adjoint Problem 754

    14.2.4 Physical Meaning of Adjoint Variables 755

    14.2.5 Gate Stroking 757

    14.2.6 Reservoir Control 759

    14.3 Levee Breach Control 762

    14.4 Control of Plane Waves 766

    14.4.1 Characteristic Form of Adjoint Equations 769

    14.4.2 Evaluation of Sensitivities 770

    Problems 774

    References 775

    Epilogue 777

    Note 778

    Bibliography 779

    Index 783

Product details

  • No. of pages: 848
  • Language: English
  • Copyright: © Butterworth-Heinemann 2018
  • Published: August 30, 2018
  • Imprint: Butterworth-Heinemann
  • eBook ISBN: 9780128154885
  • Paperback ISBN: 9780128154878

About the Author

Nikolaos Katopodes

Nikolaos D. Katopodes, University Michigan Ann Arbor, Department of Civil & Environmental Engineering, Ann Arbor, United States. Dr. Katopodes has chaired or co-chaired 28 PhD student theses. His research has resulted in over 200 publications, and several software packages that are used worldwide for the analysis and control of free-surface flows.

Affiliations and Expertise

Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, USA

Ratings and Reviews

Write a review

Latest reviews

(Total rating for all reviews)

  • CraigTacey Wed Nov 07 2018

    Very Enlightening!

    Professor Katopodes' book on Free-Surface Flow is carefully and beautifully crafted with ingenious proofs and wonderful graphics! I will certainly be continuing to review this book even after I graduate! I highly recommend this books to all students and professors studying higher level fluid mechanics.