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Free-Surface Flow:
Shallow Water Dynamics
- 1st Edition - August 30, 2018
- Author: Nikolaos D. Katopodes
- Language: English
- Paperback ISBN:9 7 8 - 0 - 1 2 - 8 1 5 4 8 7 - 8
- eBook ISBN:9 7 8 - 0 - 1 2 - 8 1 5 4 8 8 - 5
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 mec… Read more
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Request a sales quoteFree-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.
- 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
Civil and Environmental Engineers
Prologue xix
References
xxvii1. Basic Concepts
1.1 Introduction
41.1.1 Shallow-Water Models 4
1.2 Surfaces in Three-Dimensional Space
91.2.1 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
211.3.1 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
271.4.1 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
38References
40Note
412. Air-Water Interface
2.1 Introduction
442.2 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
602.3.1 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
742.4.1 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
862.5.1 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
982.6.1 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
1062.7.1 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
1102.8.1 Cyclogenesis 110
2.8.2 Meteorological Tsunami 111
Problems 113
References
1143. Gravity Waves
3.1 Introduction
1183.2 Small-Amplitude Gravity Waves
1213.3 Two-Dimensional Oscillatory Waves
1243.4 Airy’s Theory for Gravity Waves
1263.4.1 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
1343.5.1 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
1413.6.1 Reflection on a Solid Boundary 141
3.6.2 Standing Waves 142
3.7 Seiches
1453.7.1 Two-Dimensional Seiche 146
3.8 Mass Transport by Gravity Waves
1493.9 Progressive Wave Energy
1523.9.1 Wave Energy Flux 153
3.10 Group Velocity
1553.11 Wave Refraction
1583.12 Wave Diffraction
1623.12.1 Diffraction Theory 162
3.12.2 Waves Incident Obliquely on the Breakwater 166
3.13 Wave Breaking
1693.13.1 Radiation Stress 171
3.13.2 Wave Set-Up 172
Problems
176References
1774. Shallow-Water Approximation
4.1 Introduction
1824.2 Shallow-Water Equations
1834.2.1 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
1944.3.1 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
1994.5 Higher-Order Long-Wave Approximation
2024.5.1 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
2114.7 Long Waves in Trapezoidal Channels
2154.7.1 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
2204.9 The Korteweg-De Vries Equation
2254.9.1 Solitary Wave 226
4.10 Hamiltonian Approach to Water Waves
2294.10.1 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
244References
2455. Tidal Forcing
5.1 Introduction
2505.2 Equilibrium Theory of Tides
2515.2.1 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
2615.3.1 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
268Problems
271References
2726. Long Waves
6.1 Introduction
2766.2 Flow in One-Dimensional Channels
2786.3 Integral Relations
2856.4 The Saint-Venant Equations
2876.5 Energy Considerations in an Open Channel
2906.5.1 The Choice Between Momentum and Energy 294
6.6 Vector Representation
2986.6.1 Broad-Channel Representation 298
6.6.2 Saint-Venant Equations 299
6.7 Further Simplifications
3006.8 Linearized Equations
3046.9 Symmetric Equations
3066.10 Steady, Non-uniform Flow
3076.11 Shallow-Water Flow in Two Space Dimensions
308Problems
311References
3127. Channel Transitions
7.1 Introduction
3167.2 Regimes of Steady Flow
3197.3 Nearly-Horizontal Flow
3217.3.1 Steep Channels 323
7.3.2 Kinetic Energy Correction Factor 325
7.4 Transitions in Geometry and Bathymetry
3267.5 Flow Under a Vertical Sluice Gate
3297.5.1 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
3337.7 The Specific Energy
3347.7.1 Dimensionless E-h Diagram 336
7.8 Critical Velocity and Gravity Wave Speed
3397.9 The Froude Number
3427.9.1 Alternative Scaling Approaches 343
7.10 Critical Flow in Channels of Arbitrary Cross-Sectional Shape
3467.10.1 Channels With a Floodplain 347
7.10.2 Channel Shape for Unconditional Critical Flow 349
7.11 Subcritical Flow Over a Smooth Ridge
3517.11.1 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
3587.12.1 Occurrence of Critical Flow 359
7.13 Downstream Control – Formation of a Hydraulic Jump
3627.13.1 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
3697.14.1 Dimensionless F-h Diagram 371
7.14.2 Flow Under a Submerged Sluice Gate 372
7.15 Fluid Force on Transition Structures
3767.15.1 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
3867.16.1 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
394References
3998. Channel Bed Resistance
8.1 Introduction
4028.2 Uniform Flow in a Sloping Channel
4058.2.1 Reynolds Numbers Limits for Open-Channel Flow 406
8.3 Logarithmic Velocity Profiles
4088.3.1 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
4178.5 Bed Shear in Shallow-Water Flow
4208.5.1 Newton’s Law of Flow Resistance 422
8.6 The Friction Factor
4238.6.1 Computation by Velocity Measurements 424
8.7 Flow Resistance in Open Channels
4278.7.1 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
4398.9.1 Rectangular Channel 439
8.9.2 Trapezoidal Channel 439
8.10 Classification of Uniform Flow Regimes
441Problems
443References
4449. Gradually-Varied-Flow
9.1 Introduction
4489.2 Non-uniform Flow
4499.2.1 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
4569.3.1 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
4709.4.1 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
4789.6 Dimensionless GVF Profiles
4829.7 Lake Outflow Into Channel With Mild Slope
4849.7.1 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
4909.8.1 Lateral Inflow 490
9.8.2 Lateral Outflow 493
Problems
495References
49710. Characteristic Analysis
10.1 Introduction
50010.2 Discontinuities of the Free-Surface Profile
50110.2.1 Waves and Wave Fronts 501
10.3 Classification of Shallow-Water Equations
50510.3.1 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 50710.5 Transport of Wave Fronts
51010.6 Identification of Characteristic Directions
51310.6.1 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
53010.7.1 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
54410.8.1 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
55310.9.1 Impact of Froude Number 555
10.9.2 Compatibility Equations 557
10.10 The Hodograph Plane
55910.10.1 Characteristics on the Hodograph Plane 561
10.10.2 Polar Form of Hodograph Equations 563
10.11 Change of Depth Across a Characteristic
566Problems
569References
57011. 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
57811.2.1 Construction of Interior Operators 578
11.3 Characteristic Surface Families
58011.3.1 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
59611.4.1 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
60511.5.1 Canonical Equations 607
Problems
612References
61312. Simple Waves, Surges, and Shocks
12.1 Introduction
61812.2 Properties of Simple Waves
62112.2.1 Profile Deformation in Simple Wave Region 623
12.2.2 Regressive Depression Wave 624
12.3 Progressive Depression Wave
62812.3.1 Supercritical Initial Flow 629
12.3.2 Centered Depression Waves 630
12.3.2.1 Critical Outflow 631
12.4 Progressive Elevation Wave
63312.4.1 Occurrence of First Discontinuity 635
12.4.2 Surge Formation by Flowrate Control 637
12.5 Regressive Elevation Wave
63812.6 Interaction of Simple Waves
64012.7 Surges and Shocks
64612.7.1 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
65312.8.1 Properties of Weak Solutions 655
12.9 Algebraic Jump Conditions
65812.10 Instantaneous Jump Formation
66112.10.1 Surge Resulting From Upstream Gate Opening 661
12.10.2 Shock Resulting From Downstream Gate Closing 664
12.11 Compatibility Conditions at a Discontinuity
66612.11.1 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
67212.13 Interaction of Shock Waves
67412.13.1 Shock Reflection 674
12.13.2 Shock Collision 675
12.14 Interaction of Shocks and Simple Waves
677Problems
679References
68113. Sudden Water Release
13.1 Introduction
68413.2 Dam-Break Wave
68513.2.1 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
69613.3.1 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
70613.4.1 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
71313.5.1 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
72713.7 Sluice Gate Operation
72913.7.1 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
741References
74214. Active Flood Control
14.1 Introduction
74514.2 Adjoint Equations for Open-Channel Flow
74714.2.1 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
76214.4 Control of Plane Waves
76614.4.1 Characteristic Form of Adjoint Equations 769
14.4.2 Evaluation of Sensitivities 770
Problems
774References
775Epilogue 777
Note
778Bibliography 779
Index 783
- No. of pages: 848
- Language: English
- Edition: 1
- Published: August 30, 2018
- Imprint: Butterworth-Heinemann
- Paperback ISBN: 9780128154878
- eBook ISBN: 9780128154885
NK
Nikolaos D. 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, USARead Free-Surface Flow: on ScienceDirect