Numerical study on the dissipation of water waves over a viscous fluid-mud layer

Direct simulations of the Navier-Stokes equations are performed to investigate the interaction between a nonlinear wave at the water surface and an interfacial wave at the fluid-mud layer below. A level-set method is employed to capture the air-water and water-mud interfaces. Despite the nonlinearity of the governing equations, the direct numerical simulation shows that the total wave-damping rate exhibits a remarkable consistency with the prediction of the linear theory. To reveal the underlying mechanism, we analyze the velocity and vortcity fields, energy transfer from the water to the mud, and energy dissipation. The detailed analysis of velocity and vorticity fields shows an appreciable nonlinear effect in the water but a relatively weak nonlinear effect in the fluid mud. The major pathway of transferring energy from the water to the mud is the pressure acting on the water-mud interface. The viscous dissipation of the energy also exhibits a local, significant nonlinear effect in the water. However, the excess and deficiency in the dissipation rate at different wave phases compared with the linear theory largely cancel each other, resulting in an overall wave-damping rate close to the prediction of the linear theory. Furthermore, the analysis of energy budgets elucidates a comprehensive picture of energy transport and dissipation in the wave-mud system. In the water, the horizontal motion first loses energy to the vertical motion through the pressure-strain correlation. The energy of the vertical motion is then transported downwards by the pressure and vertical advection. Across the water-mud interface, the vertical motion of the water transmits energy to the mud through the pressure work. In the mud, the energy of the vertical motion is transported downwards by the pressure and then redistributed to the horizontal motion through the pressure-strain correlation again. The energy of the horizontal motion is transported towards the mud bottom through the viscous diffusion and is finally dissipated.