3D Modeling of Nonlinear Wave Phenomena on Shallow Water Surfaces

Preface vii

Introduction ix

1 Equations of Hydrodynamics 1

1.1 Features of the Problems in the Formulation of Mathematical Physics 1

1.2 Classification of Linear Differential Equations with Partial Derivatives of the Second Order 3

1.3 Nonlinear Equations of Fluid Dynamics 5

1.4 Methods for Solving Nonlinear Equations 9

1.5 The Basic Laws of Hydrodynamics of an Ideal Fluid 11

1.6 Linear Equations of Hydrodynamic Waves 16

Conclusions 20

2 Modeling of Wave Phenomena on the Shallow Water Surface 23

2.1 Waves on the Sea Surface 23

2.2 Review of Research on Surface Gravity Waves 26

2.3 Investigation of Surface Gravity Waves 39

2.4 Spatial modeling of Wave Phenomena on Shallow Water Surface 50

2.5 Actual Observations of Wave Phenomena on the Surface of Shallow Water 57

2.6 Ship Waves. “Reactive” Ducks of the Alexander Garden 58

Conclusions 64

3 Modeling of Nonlinear Surface Gravity Waves in Shallow Water 65

3.1 Overview of Studies on Nonlinear Surface Gravity Waves in Shallow Water 65

3.2 Nonlinear Models of Surface Gravity Waves in Shallow Water 74

3.3 Solution of the Nonlinear Shallow Water Equation by the Method of Successive Approximations 81

3.4 Modeling the Propagation of Nonlinear Surface Gravity Waves in Shallow Water 86

3.5 Modeling the Refraction of Nonlinear Surface Gravity Waves 99

3.6 Modeling of Propagation and Refraction of Nonlinear Surface Gravity Waves Under Shallow Water Conditions with Account of Dispersion 109

Conclusions 117

4 Numerical Simulation of Nonlinear Surface Gravity Waves in Shallow Water 119

4.1 Review of Studies on Computational Modeling of Surface Waves 119

4.2 Statement of the Problem 132

4.3 The Research of a Discrete Model 138

4.4 Results of Numerical Modeling based on Shallow Water Equations 139

4.5 Discussion and Comparison of Results 148

Conclusions 152

5 Two-Dimensional Numerical Simulation of the Run-Up of Nonlinear Surface Gravity Waves 155

5.1 Statement of the Problem 155

5.2 Construction of a Discrete Finite-Volume Model 160

5.3 Discrete Model Research 169

5.4 Results of Two-Dimensional Numerical Modeling and Their Analysis 171

5.5 Discussion and Comparison of Results 185

Conclusions 188

6 Three-Dimensional Numerical Modeling of the Runup of Nonlinear Surface Gravity Waves 191

6.1 Statement of the Problem. Boundary and Initial Conditions 191

6.2 Construction of a Discrete Model 195

6.3 The Construction of a Discrete Finite-Volume Model 206

6.4 Discrete Model Research 229

6.5 Results of Three-Dimensional Numerical Modeling and Their Analysis 230

6.6 Discussion and Comparison of Results 241

Conclusions 242

Conclusion 245

References 24