Simulation of water waves by Boussinesq models

A new set of time-dependent Boussinesq equations is derived to simulate nonlinear long wave propagation in coastal regions. Following the approaches by Nwogu (1993) and later by Chen and Liu (1995), the velocity (or velocity potential) at a certain water depth corresponding to the optimum linear dispersion property is used as a dependent variable. Further, no assumption for small non-linearity is made throughout the derivation. Therefore, the resulting equations are valid in intermediate water depth as well as for highly nonlinear waves. Co-efficients for second order bound waves and the third order Schodinger equation are derived and compared with exact solutions.; A numerical model using a combination of second and fourth order schemes to discretize equation terms is developed for obtaining solutions to the equations. A fourth order predictor-corrector scheme is employed for time stepping and the first order derivative terms are finite differenced to fourth order accuracy, making the truncation errors smaller than the dispersive terms in the equations. Linear stability analysis is performed to determine the corresponding numerical stability range for the model. To avoid the problem of wave reflection from the conventional incident boundary condition, internal wave generation by source function is employed for the present model. The linear relation between the source function and the property of the desirable wave is derived. Numerical filtering is applied at specified time steps in the model to eliminate short waves (about 2 to 5 times of the grid size) which are generated by the nonlinear interaction of long waves.; To simulate the wave breaking process, additional terms for artificial eddy viscosity are included in the model equations to dissipate wave energy. The dissipation terms are activated when the horizontal gradient of the horizontal velocity exceeds the specified breaking criteria. Some of the existing models for simulating the process of wave runup are reviewed and we attempt to incorporate the present model to simulate the process by maintaining a thin layer of water over the physically dry grids.; Extensive tests are made to examine the validity of the present model for simulating wave propagation under various conditions. For the one dimensional case, the present model is applied to study the evolution of solitary waves in constant depth, the permanent solution of high nonlinear solitary waves, the shoaling of solitary waves over constant slopes, the propagation of undular bores, and the shoaling and breaking of random waves over a beach. For the two dimensional case, the present model is applied to study the evolution of waves (whose initial surface elevation is a Gaussian distribution) in a closed basin, the propagation of monochromatic waves over submerged shoals of Berkhoff et al. (1982) and of Chawla (1995). Results from the present model are compared in detail with available analytical solutions, experimental data, and other model results.