Propagation of solitary waves over permeable rippled beds

The unsteady two-dimensional Navier-Stokes equations and Navier-Stokes type model equations for porous flow were solved numerically to simulate the propagation of solitary waves over a permeable rippled bed. The free surface boundary conditions and the interfacial boundary conditions between the water and the porous bed are in complete form. A boundary-fitted coordinate system was used in this model. The accuracy of the numerical scheme was verified by comparing the numerical results for the spatial distribution of wave amplitudes on the impermeable and permeable rippled bed at resonant conditions with the analytical solutions. Our numerical results showed that when the crest of a solitary wave propagates into the ripple section, flow separation with reattachment was formed at the lee side of each ripple crest. The flow separation develops gradually into a clockwise vortex with a dimension that covers the whole region between two successive ripple crests. The trajectories of fluid particles above the permeable rippled bed are similar to those on the impermeable rippled bed. Although all of the fluid particles on the impermeable rippled bed move eventually in the opposite direction of wave, some particles on the porous rippled bed do shift in the wave direction.