Three-Dimensional Model of Navier-Stokes Equations for Water Waves

Abstract

A three-dimensional numerical model based on the complete Navier-Stokes equations is developed and presented in this paper. The model can be used for the problem of propagation of fully nonlinear water waves. The Navier-Stokes equations are first transformed from an irregular calculation domain to a regular one using sigma coordinates. The projection method is used to separate advection and diffusion terms from the pressure terms in Navier-Stokes equations. MacCormack's explicit scheme is used for the advection and diffusion terms, and it has second-order accuracy in both space and time. The pressure variable is further separated into hydrostatic and hydrodynamic pressures so that the computer rounding errors can be largely avoided. The resulting hydrodynamic pressure equation is solved by a multigrid method. A staggered mesh and central spatial finite-difference scheme are used. The model is tested against the experimental data of Luth et al., and the comparison shows that higher harmonics can be modeled well. Comparison of the model solutions with the elliptic shoal case confirms that the present model works well for wave refraction and diffraction with strong wave focusing.