Spatial Development of Viscous Flow Induced by Wave Propagation over Vortex Ripples

In the present study, numerical simulations of the free-surface flow, developing by the propagation of nonlinear water waves over a rippled bottom, are performed assuming that the corresponding flow is two-dimensional, incompressible and viscous. The simulations are based on the numerical solution of the unsteady, two-dimensional, Navier-Stokes equations subject to the fully-nonlinear free-surface boundary conditions and appropriate bottom, inflow and outflow boundary conditions. The equations are properly transformed so that the computational domain becomes time-independent. For the spatial discretization, a hybrid scheme is used, in which the finite-difference method, in the horizontal direction, and a pseudo-spectral approximation method with Chebyshev polynomials, in the vertical direction, are applied. A fractional time-step scheme, which consists of three different stages, is used for the temporal discretization. A damping zone is placed at the outflow region in order to efficiently minimize reflection of waves by the outflow boundary. Over the rippled bed, the wave boundary layer thickness increases significantly, while flow separation at the ripple crests generates a circulation region. The amplitude of the wall shear stress over the ripples increases with increasing ripple height, while the corresponding friction force is insensitive to this increase. The amplitude of the form drag forces due to dynamic and hydrostatic pressures also increase with increasing ripple height, therefore, the percentage of friction in the total drag force decreases with increasing ripple height.