Wave Breaking: A Numerical Study

1: Introduction.- 1.1 Nature and scope of the work.- 1.2 Methodology.- 1.3 Innovations and conclusions.- 2: General aspects of incompressible flow. Theoretical review.- 2.1 Introduction.- 2.2 The Navier-Stokes equations for uniform, incompressible fluids.- 2.3 Initial and boundary conditions.- 2.4 The energy equation.- 2.5 The vorticity equation.- 2.6 The pressure Poisson equation for incompressible flows.- 2.7 General aspects of turbulent flows. Averaging methods and Reynolds equations.- 2.8 Turbulence transport equations.- 2.8.1 The exact K equation.- 2.8.2 The exact ? equation.- 2.8.3 The exact $$\overline {{u_i}{u_j}} $$ equation.- 2.9 Turbulence models.- 2.9.1 Definition and classification.- 2.9.2 Algebraic models.- 2.9.3 The K (one-equation) model.- 2.9.4 The two-equation K-? model.- 2.9.5 Reynolds-stress transport models.- 2.9.6 The K-?-A model.- 2.9.7 Practical considerations.- 2.10 Boundary conditions for K and ?.- 2.10.1 Wall boundary conditions.- 2.10.2 Planes and axes of symmetry.- 2.10.3 Turbulent non-turbulent interfaces.- 2.10.4 Free-surface boundary conditions.- 3: Mathematical modeling of breaking shallow water waves. Proposed methodology.- 3.1 Introduction.- 3.2 Physical processes.- 3.2.1 Wave breaking criteria.- 3.2.2 Shallow water steepening vs. dispersion.- 3.2.3 Wave overturning.- 3.2.4 Breaking wave propagation and decay.- 3.2.5 Other physical effects.- 3.3 Mathematical descriptions.- 3.3.1 Wave theories. Range of validity.- 3.3.2 Shallow water equations. Characteristics and discontinuities.- 3.3.3 Boussinesq-type shallow water equations.- 3.3.4 Overturning wave models.- 3.4 Wave theories for very shallow water.- 3.4.1 Solitary waves.- 3.4.2 Hydraulic jumps. Discrete forms of the conservation laws.- 3.5 Summary of experimental investigations.- 3.5.1 Wave-wave interactions.- 3.5.2 Hydraulic jumps.- 3.5.3 Waves breaking on a slope.- 3.6 Description of the proposed methodology.- 4: MAC-type methods for incompressible free-surface flows.- 4.1 Introduction.- 4.2 The choice of the mesh.- 4.3 The MAC (Marker-And-Cell) method.- 4.4 The projection method.- 4.5 The SMAC (Simplified-Marker-And-Cell) method.- 4.6 The pressure-velocity iteration method.- 4.7 Numerical treatment of free-surfaces.- 4.7.1 Free-surface representation methods.- 4.7.2 Methods for updating the free-surface.- 4.7.3 Discretization of the free-surface boundary conditions.- 4.8 Stability considerations.- 4.8.1 The convection-diffusion equation.- 4.8.2 Stability of the momentum equations.- 4.9 Conclusions.- 5: Description of the numerical model.- 5.1 Introduction.- 5.2 Momentum equation approximations.- 5.3 Continuity equation approximation.- 5.4 Approximations for the K and ? equations.- 5.5 Updating the fluid configuration.- 5.5.1 Algorithm for the convection of F.- 5.5.2 Determining interfaces within a cell.- 5.6 Velocity boundary conditions.- 5.6.1 Mesh boundaries.- 5.6.2 Free-surface boundaries.- 5.6.3 Internal-obstacle boundaries.- 5.7 Boundary conditions for the K and ? equations.- 5.8 Initial conditions for the K and ? equations.- 5.9 Stability considerations.- 5.10 Programming considerations.- 5.11 Selected test problems.- 5.11.1 Laminar cavity flow.- 5.11.2 Grid turbulence.- 5.11.3 Logarithmic boundary layer.- 5.11.4 Turbulent cavity flow.- 5.11.5 Dam-break problem.- 6: Numerical simulation of shallow water waves.- 6.1 Introduction.- 6.2 Propagation of a solitary wave over a horizontal bottom.- 6.3 Collision between solitary waves.- 6.3.1 Waves of equal amplitude.- 6.3.2 Waves of different amplitude.- 6.4 Simulation of undular, transitional and turbulent hydraulic jumps.- 6.4.1 Undular jump.- 6.4.2 Transitional jump.- 6.4.3 Turbulent hydraulic jump.- 6.5 Breaking of a solitary wave over a slope.- 6.6 Breaking of a train of solitary waves over a slope.- 7: Conclusions. Future research and development.- 7.1 Summary and conclusions.- 7.2 Future research and development.- References.