A 3-D model based on Navier–Stokes equations for regular and irregular water wave propagation

A spatial fixed σ-coordinate is used to transform the Navier–Stokes equations from the sea bed to the still water level. In the fixed σ-coordinate system only a very small number of vertical grid points are required for the numerical model. The time step for using the spatial fixed σ-coordinate is efficiently larger than that of using a time dependent σ-coordinate, as there is substantial truncation error involved in the time dependent σ-coordinate transformation. There is no need to carry out the σ-coordinate transformation at each time step, which can reduce computational times. It is important that wave breaking can be potentially modeled in the fixed σ-coordinate system, but in a time-dependent σ-coordinate system the wave breaking cannot be modeled. A projection method is used to separate advection and diffusion terms from the pressure terms in Navier–Stokes equations. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. In order to reduce computational time of solving the hydrodynamic pressure equation, at every time step the initial pressure is extrapolated in time domain using computed pressures from previous time steps, and then corrected in spatial domain using a multigrid method. For each time step, only a few of iterations (typically six iterations) are required for solving the pressure equation. The model is tested against available experimental data for regular and irregular waves and good agreement between calculation results and the measured data has been achieved.