
Free-Surface Flow:
Shallow Water Dynamics
Description
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
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
Affiliations and Expertise
Ratings and Reviews
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.