MASS-TRANSPORT IN VISCOUS-FLOW UNDER A PROGRESSIVE WATER-WAVE

We consider two-dimensional free surface how caused by a pressure wave maker in a viscous incompressible fluid of finite depth and infinite h orizontal extent. The governing equations are expressed in dimensionle ss form, and attention is restricted to the case delta much less than epsilon much less than 1, where delta is the characteristic dimensionl ess thickness of a Stokes boundary layer and a is the Strouhal number. Our aim is to provide a global picture of the flow, with emphasis on the steady streaming velocity. The asymptotic flow structure near the wavenumber is found to consist of five distinct vertical regions: bott om and surface Stokes layers of dimensionless thickness O(delta), bott om and surface Stuart layers of dimensionless thickness O(delta/epsilo n lying outside the Stokes layers, and an irrotational outer region of dimensionless thickness O(1). Equations describing the flow in all re gions are derived, and the lowest-order steady streaming velocity in t he near-field outer region is computed analytically. It is shown that the flow far from the wavemaker is affected by thickening of the Stuar t layers on the horizontal length scale O[(epsilon/delta)(2)], by visc ous wave decay on the scale O(1/delta), and by nonlinear interactions on the scale O(1/epsilon(2)). The analysis of the flow in this region is simplified by imposing the restriction delta = O(epsilon(2)), so th at all three processes take place on the same scale. The far-field flo w structure is found to consist of a viscous outer core bounded by Sto kes layers at the bottom boundary and water surface. An evolution equa tion governing the wave amplitude is derived and solved analytically. This solution and near-field matching conditions are employed to calcu late the steady flow in the core numerically, and the results are comp ared with other theories and with observations.