Numerical Modelling of Marine Hydrodynamics (eBook)

Front Cover 1
Numerical Modelling of Marine Hydrodynamics: Applications to Dynamic Physical Processes 4
Copyright Page 5
Contents 8
Preface 6
Chapter I. General Equations 14
1. The hydrodynamic differential equations of motion and continuity 15
2. The mass transport and free surface equations 22
3. One-dimensional motion 26
4. Frictional forces at the surface and the bottom 29
Chapter II. Steady motion - numerical methods 36
1. Numerical approximation of steady phenomena 36
2. Convergence and choice of grid step 42
3. Boundary conditions 44
4. Some general properties of the differential operators 46
5. Numerical methods of solution of partial differential equations of steady state 48
6. Simple iteration method 50
7. Gauss-Seidel method 51
8. Line inversion method 52
9. Application of line inversion in the alternating- direction method 55
Chapter III. Steady motion - oceanographical examples 58
1. Motion in channels - analytical solution 58
2. Wind-driven currents in a shallow sea 63
3. Ekman equations 64
4. The mass transport and stream function equations 65
5. Solution of the Dirichlet problem for elliptical equations in the multiply-connected domain 68
6. Determination of eddy viscosity 69
7. Some simplified models 70
8. Steady wind-driven circulation in the Baltic Sea 81
9 . The influence of density stratification on the vertical distribution of current 91
10. The wind- and density-driven currents 96
11. Different methods to estimate the exchange of momentum in the Ekman layer 100
12. Dimensional analysis 102
13. Constant eddy viscosity 105
14. A statistical method based on the equation of turbulent energy b2 and the scale length 1 109
15. Shallow, horizontally unbounded sea 111
16. Two-layer model 114
17. A statistical method based on the equations of energy and dissipation 118
18. Influence of the density stratification 122
Chapter IV. Unsteady motion - numerical methods 125
1. Principal equations and their difference form 126
2. Stability of the numerical solution 132
3. Stability of a system of equations 135
4. Wave deformation 140
5. Physical and numerical solutions 143
6. Nonlinear equations 145
7. Numerical filtering in time and space 150
8. Boundary and initial conditions 157
9. The explicit numerical scheme 161
10. An implicit numerical scheme 167
11. Computational example to compare explicit and implicit properties 174
12. A numerical system with mixed explicit-implicit properties 174
13. Steady state processes from a point of view of numerical methods used for solving an unsteady problem 178
Chapter V. Numerical treatment of tides 182
1. Introduction 182
2. A system of equations for the study of the tides 183
3. The boundary-value problem 188
4. The hydrodynamic-numerical method 191
5. Calculating the M2-tide in the Arctic Ocean - an example 195
6. The results of the computation of the tides in the Arctic Ocean 198
7. The energy balance equation 207
8. Numerical models to study the vertical distri-bution of velocity in the tide waves 210
9. A treatment of the boundary layer in the tide wave 214
Chapter VI. Models of shallow coastal areas and tidal rivers 219
1. Some remarks on one-dimensional models 220
2. River models 223
3. A one-dimensional treatment of river flow: the multi-channel system 227
4. A two-dimensional model of the River Elbe covering the area between Seemannshöft and Cuxhaven 234
5. Approximation of the coast-line 236
6. Modelling the flooding and uncovering of tidal flats 239
7. A special treatment of the bottom friction in extremely shallow waters 245
8. Residual currents 254
9. The application of the grid refinement and the irregular grid 256
10. Some results and conclusions derived from the nested model of the North Sea 262
11. Some examples of the application of hydrodynamic- numerical models on coastal engineering 263
Chapter VII. The application of the transport equations 300
1. Basic concepts 300
2. Two-dimensional turbulent diffusion 304
3. Numerical methods of solving the transport equation 310
4. The application of the transport equation in a multi-channel model 315
5. A comparison between an analytical and a numerical solution 321
6. The turbidity zone of the River Elbe 326
7. The transport of pollutants 331
Chapter VIII. Periodic motion 335
1. Introduction 335
2. The general properties of eigenvalues 339
3. Eigenvalues in two-dimensional oscillations: The construction of a symmetrical matrix 341
4. Galerkin's method and its application to the problem of eigenvalues and eigenvectors 344
5. A method of resonance iterations 353
6. A method of computing the longest period of free oscillation 358
7. A method of computing a set of eigenperiods 360
8. The influence of friction and small variations of frequency 364
9. Numerical schemes describing frictionless motion 366
List of Figures 373
Subject Index 377