Chapter
1.2.6 Gaussian and Mean Curvature
1.2.7 Principal Curvature Sections
1.2.9 Divergence Representations
1.3 Initial and Boundary-Value Problems
1.3.1 Types of Boundary Conditions
1.3.2 Initial-Value Problems
1.3.3 Two-Point Boundary-Value Problem
1.3.4 General Equilibrium Problems
1.4 Classification of Partial Differential Equations
1.4.1 Linear, First-Order Equation
1.4.2 Systems of First-Order PDE's
1.4.3 Classification of Quasi-linear Systems
1.4.4 Second Order Equations
1.4.5 Classification of Second-Order Equations
2.2.1 Young-Laplace Equation
2.2.2 Wettability and Contact Angle
2.2.4 Marangoni Forces and the Bond Number
2.2.5 Surface Free Energy
2.2.6 Minimum Surface Energy
2.2.8 Circular Hydraulic Jump
2.3 Free Surface Boundary Conditions
2.3.1 Dynamic Surface Condition
2.3.1.2 Tangential Forces
2.3.2 Scaling the Dynamic Surface Condition
2.3.3 Dynamic Condition for Potential Flow
2.3.4 Kinematic Surface Condition
2.3.5 Steady Flow in Two Dimensions
2.3.6 Kinematic Bottom Condition
2.3.7 Rigid Lid Approximation
2.3.8 Boundary Conditions at Contact Lines
2.3.9 Boundary Condition for Pressure Poisson Equation
2.4 Simple Viscous Flows With a Free Surface
2.4.1 Channel Flow Under Calm Wind
2.4.2 The Rate of Streamwise Energy Dissipation
2.4.3 Flow Driven by Wind Shear
2.4.4 Suddenly Accelerated Air-Water Interface
2.5 Transfer Processes at the Air-Water Interface
2.5.1 Drag Coefficient at Air-Water Interface
2.5.2 Significant Wave Height
2.5.3 Random Wave Analysis
2.5.4 Wave Frequency Spectrum
2.5.4.1 Pierson-Moskowitz Spectrum
2.6 Atmospheric Surface Layer
2.6.1 Wind and Wave Stresses
2.6.2 Constant Flux Layer
2.6.4 Monin-Obukhov Similarity Theory
2.8 Large Scale Interface Disturbances
2.8.2 Meteorological Tsunami
3.2 Small-Amplitude Gravity Waves
3.3 Two-Dimensional Oscillatory Waves
3.4 Airy's Theory for Gravity Waves
3.4.1 Boundary Conditions
3.4.2 Velocity Potential for Sinusoidal Waves
3.4.3 Dispersion Relation
3.4.4 Shallow-Water Limit
3.4.5 Pressure Distribution
3.5 Dispersion of Non-sinusoidal Waves
3.5.4 Dispersion of a Composite Wave
3.5.5 Dispersion of a Gaussian Wave Packet
3.6 Superposition of Linear Gravity Waves
3.6.1 Reflection on a Solid Boundary
3.7.1 Two-Dimensional Seiche
3.8 Mass Transport by Gravity Waves
3.9 Progressive Wave Energy
3.12.1 Diffraction Theory
3.12.2 Waves Incident Obliquely on the Breakwater
4 Shallow-Water Approximation
4.2 Shallow-Water Equations
4.2.1 Depth-Averaged Equations
4.2.1.1 Equation of Continuity
4.2.1.2 Equation of Streamwise Momentum
4.2.1.3 Equation of Transverse Momentum
4.2.1.4 Vector Form of Shallow-Water Equations
4.2.2 The Gas Dynamics Analogy
4.2.3 Vorticity Transport in Shallow Water
4.3 Waves in Shallow Water
4.3.2 Gravity Waves on a Rotating Earth
4.3.3 Gravity Waves Along the Coast
4.3.4 Barotropic Vorticity Waves
4.4 Dispersion Relations for Nonlinear Waves
4.5 Higher-Order Long-Wave Approximation
4.5.1 Zero-Order Approximation
4.5.2 First-Order Approximation
4.5.3 Second-Order Approximation
4.5.4 Second-Order Oscillatory Wave
4.6 The Boussinesq Equations
4.7 Long Waves in Trapezoidal Channels
4.7.1 Boussinesq Equations for Trapezoidal Channel
4.7.1.1 First Approximation
4.7.1.2 Second Approximation
4.9 The Korteweg-De Vries Equation
4.10 Hamiltonian Approach to Water Waves
4.10.1 Approximation of the Kinetic Energy
4.10.1.1 Kinetic Energy Below the Mean Water Level
4.10.1.2 Kinetic Energy Above the Mean Water Level
4.10.1.3 Hamiltonian for Fairly Low Long Waves
4.10.1.4 Canonical Equations
4.10.2 Horizontal Channel
4.10.3 Approximate Hamiltonian
4.10.4 The Free-Surface Approximation
4.10.5 Extension to Uneven Bottom
4.10.6 Canonical Equations for the Average Velocity
5.2 Equilibrium Theory of Tides
5.2.2 Equilibrium Tidal Surface
5.2.3 Planetary Complications
5.3 Dynamic Theory of Tides
5.3.1 Standing Tidal Wave
5.3.3 Co-tidal Lines and Amphidromic Points
5.4 Harmonic Analysis and Tide Prediction
6.2 Flow in One-Dimensional Channels
6.4 The Saint-Venant Equations
6.5 Energy Considerations in an Open Channel
6.5.1 The Choice Between Momentum and Energy
6.6 Vector Representation
6.6.1 Broad-Channel Representation
6.6.2 Saint-Venant Equations
6.7 Further Simplifications
6.10 Steady, Non-uniform Flow
6.11 Shallow-Water Flow in Two Space Dimensions
7.2 Regimes of Steady Flow
7.3 Nearly-Horizontal Flow
7.3.2 Kinetic Energy Correction Factor
7.4 Transitions in Geometry and Bathymetry
7.5 Flow Under a Vertical Sluice Gate
7.5.1 The Contraction Coefficient
7.5.2 Discharge Through a Free-Flowing Gate
7.5.3 Fluid Force on Sluice Gate
7.6 Flow Over a Smooth Bottom Ridge
7.7.1 Dimensionless E-h Diagram
7.8 Critical Velocity and Gravity Wave Speed
7.9.1 Alternative Scaling Approaches
7.10 Critical Flow in Channels of Arbitrary Cross-Sectional Shape
7.10.1 Channels With a Floodplain
7.10.2 Channel Shape for Unconditional Critical Flow
7.11 Subcritical Flow Over a Smooth Ridge
7.11.1 Occurrence of Critical Flow
7.11.2 Supercritical Flow Over a Smooth Ridge
7.11.3 Experimental Validation
7.11.4 Force Exerted on Bottom Ridge
7.12 Flow Through a Smooth Transition in Width
7.12.1 Occurrence of Critical Flow
7.13 Downstream Control - Formation of a Hydraulic Jump
7.13.1 Conservation of Momentum Across a Hydraulic Jump
7.13.2 Hydraulic Jump in a Rectangular channel
7.13.3 Dissipation of Energy
7.14.1 Dimensionless F-h Diagram
7.14.2 Flow Under a Submerged Sluice Gate
7.15 Fluid Force on Transition Structures
7.15.1 Blocks Assisting the Formation of a Jump
7.15.2 Control of Hydraulic Jump by Abrupt Drop
7.15.3 Control of Hydraulic Jump by Abrupt Rise
7.15.4 Choking Mechanisms
7.16 Other Rapidly-Varied Flow Transitions
7.16.1 Outflow From a Reservoir
7.16.3 Lateral Outflow Through a Smooth Downspout
7.16.4 Flow Around a Bend in Subcritical Flow
7.16.4.1 Channel Bed Adjustment
8.2 Uniform Flow in a Sloping Channel
8.2.1 Reynolds Numbers Limits for Open-Channel Flow
8.3 Logarithmic Velocity Profiles
8.3.1 Smooth Wall Boundary
8.3.2 Rough Wall Boundary
8.3.3 The Velocity Intercept
8.3.4 Classification of "Smooth" and "Rough" Walls
8.4 Depth-Averaged Velocities
8.5 Bed Shear in Shallow-Water Flow
8.5.1 Newton's Law of Flow Resistance
8.6.1 Computation by Velocity Measurements
8.7 Flow Resistance in Open Channels
8.7.2 Chézy Equation for General Cross Sections
8.7.3 The Gauckler-Kutter Equation
8.9 Optimal Cross-Sectional Shape
8.9.1 Rectangular Channel
8.9.2 Trapezoidal Channel
8.10 Classification of Uniform Flow Regimes
9.2.1 Other Forms of the GVF Equation
9.2.1.1 Section Factor Form
9.2.1.2 Critical Discharge Form
9.2.1.3 Hydraulic Exponent Form
9.2.1.4 Bresse's Wide Channel Approximation
9.3 Classification of Gradually-Varied Flow Profiles
9.3.2 Steep Slope Profiles
9.3.4 Adverse Slope Profiles
9.3.5 Critical Slope Profiles
9.3.6 Frictionless Channel Profiles
9.3.7 Zero-Inertia Profiles
9.4 Direct Integration of the GVF Equation
9.4.1 Frictionless Channel
9.4.2 Wide Horizontal Channel
9.4.3 Sloping Wide Channel - Bresse Solution
9.4.4 General Channel - Ven Te Chow Solution
9.4.4.1 Horizontal Bottom
9.4.5 Singular Perturbation Solution
9.5 Numerical Solution of the GVF Equation
9.6 Dimensionless GVF Profiles
9.7 Lake Outflow Into Channel With Mild Slope
9.7.2.1 Dimensionless Lake to M2 Profile
9.7.2.2 Dimensionless Lake to H2 Profile
9.8 Spatially-Varied Flow
10 Characteristic Analysis
10.2 Discontinuities of the Free-Surface Profile
10.2.1 Waves and Wave Fronts
10.3 Classification of Shallow-Water Equations
10.3.1 de Saint Venant Equations
10.3.2 Zero-Inertia Equations
10.3.3 Kinematic-Wave Equation
10.5 Transport of Wave Fronts
10.6 Identification of Characteristic Directions
10.6.1 Characteristic Form of Scalar Wave Equation
10.6.2 Characteristic Form of Kinematic Wave Equation
10.6.3 Kinematic Shock Wave
10.6.4 Impact of Lateral Inflow
10.7 Characteristics of St. Venant Equations
10.7.1 Characteristic Equations
10.7.2 Universal Celerity Variable
10.7.3 Compatibility Equations
10.7.4 Riemann Invariants
10.7.5 Canonical Equations
10.7.5.1 Gravity Waves in a Frictionless Horizontal Channel
10.7.7 Compatibility Equations
10.7.8 Contact Discontinuities
10.8 Specification of Initial and Boundary Conditions
10.8.1 The Characteristic Network
10.8.2 Interference of Boundaries
10.8.3 Non-reflecting Boundaries
10.9 Steady Flow in Two Dimensions
10.9.1 Impact of Froude Number
10.9.2 Compatibility Equations
10.10 The Hodograph Plane
10.10.1 Characteristics on the Hodograph Plane
10.10.2 Polar Form of Hodograph Equations
10.11 Change of Depth Across a Characteristic
11.1.1 Propagation of Initial Data
11.1.1.1 Eigenvalues as Characteristic Surface Normals
11.2 Characteristic Surfaces and Bicharacteristics
11.2.1 Construction of Interior Operators
11.3 Characteristic Surface Families
11.3.1 Characteristic Flow Surfaces
11.3.2 Characteristic Wave Surfaces
11.3.3 Characteristic Cone
11.3.4 Characteristic Conoid
11.3.5 Existence and Uniqueness of Solution
11.3.7 Parametric Representation of Bicharacteristics
11.3.8 Bicharacteristic Tangency Condition
11.4 Compatibility Relations
11.4.1.1 Propagation of Scalar Properties
11.4.1.2 Propagation of Shear Waves
11.4.3 Interior Differential Equations
11.4.4 Interdependence of Compatibility Conditions
11.4.5 Canonical Equations
11.5 Bicharacteristics of Turbid Underflows
11.5.1 Canonical Equations
12 Simple Waves, Surges, and Shocks
Dynamic Equation of Motion
12.2 Properties of Simple Waves
12.2.1 Profile Deformation in Simple Wave Region
12.2.2 Regressive Depression Wave
12.3 Progressive Depression Wave
12.3.1 Supercritical Initial Flow
12.3.2 Centered Depression Waves
12.4 Progressive Elevation Wave
12.4.1 Occurrence of First Discontinuity
12.4.2 Surge Formation by Flowrate Control
12.5 Regressive Elevation Wave
12.6 Interaction of Simple Waves
12.7.1 Conservation of Mass
12.7.2 Conservation of Momentum
12.7.3 Conservation of Energy
12.7.4 Choice of Jump Conditions
12.8 Weak Solutions of Conservation Laws
12.8.1 Properties of Weak Solutions
12.9 Algebraic Jump Conditions
12.10 Instantaneous Jump Formation
12.10.1 Surge Resulting From Upstream Gate Opening
12.10.2 Shock Resulting From Downstream Gate Closing
12.11 Compatibility Conditions at a Discontinuity
12.11.1 High Side on the Right of Jump (r>1)
12.11.2 High Side on the Left of Jump (r<1)
12.12 Energy Loss Across a Jump
12.13 Interaction of Shock Waves
12.14 Interaction of Shocks and Simple Waves
13.2.1 Dimensionless Depth Profile
13.2.2 Characteristics of Ritter Solution
13.2.3 Conservation Properties of Ritter Solution
13.2.4 Evolution of the Ritter Dam-Break Wave
13.3 Dam-Break on Still Water of Constant Depth
13.3.1 Evolution of Dam-Break Wave in Wet Channel
13.3.2 Dam-Break in a Channel With Base Flow
13.3.2.1 Dimensionless Solution
13.3.2.2 Limiting Depth Ratio
13.4.1 Free Flowing Breach
13.4.2 Hydraulic Jump Within Breach
13.5 Effects of Bed Slope and Resistance
13.5.1 Dam-Break in Frictionless, Sloping Channel
13.5.2 Wave Front on Rough, Dry Bed
13.5.3 Whitham's Approximation of the Wave Tip
13.5.3.1 Conservation of Wave Tip Volume
13.5.3.2 Conservation of Wave Tip Momentum
13.5.3.3 Wave Front Advance
13.5.3.4 Wave Front Profile
13.5.3.5 Matched Asymptotic Expansions
13.7 Sluice Gate Operation
13.7.1 Sudden Complete Opening
13.7.2 Sudden Complete Closing
13.7.3 Sudden Partial Opening
13.7.4 Sudden Partial Closing
14.2 Adjoint Equations for Open-Channel Flow
14.2.1 Characteristic Analysis
14.2.2 Sensitivity Equations
14.2.3 Alternative Formulation of the Adjoint Problem
14.2.4 Physical Meaning of Adjoint Variables
14.3 Levee Breach Control
14.4 Control of Plane Waves
14.4.1 Characteristic Form of Adjoint Equations
14.4.2 Evaluation of Sensitivities
Free-Surface Flow:
：Shallow Water Dynamics
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