| The mass transport set up by various kinds of two and three-dimensional water-waves is studied numerically. The characteristics of the equations governing the mass transport depend on the ratio of viscous length scale to the amplitude of the free surface displacement. When this ratio is small, the nonlinearities are important and the mass transport flow acquires a boundary layer character. For two-dimensional problems the governing equations are cast into the streamfunction-vorticity formulation and two numerical solutions are presented. The first, a spectral method that relies on a Fourier-Chebyshev expansion, is adopted when the mass transport is periodic in the horizontal direction. It is used to investigate the mass transport in a partially reflected wave. The second, a finite element solution, is adopted when the problem has an irregular geometry. It is developed to investigate mass transport in a flow oscillating between two concentric cylinders, and in a wave traveling above a hump in the seabed. Another spectral scheme is developed to study mass transport in three dimensional water-waves where the steady flow is assumed to be periodic in two horizontal directions. The velocity-vorticity formulation is adopted, and boundary conditions for the vorticity are derived analytically to enforce the no-slip conditions. The numerical scheme is used to calculate the mass transport under two intersecting wave trains. The resulting flow is reminiscent of the Langmuir circulation patterns. The scheme is then applied to study the steady flow in a three-dimensional standing wave. |
