Free-Surface Flow:

Prologue xix

References xxvii

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

2.1 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

3.1 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

4.1 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

5.1 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

6.1 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

7.1 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

8.1 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

9.1 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

10.1 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

12.1 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

13.1 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

14.1 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