Mass transport in viscous flow under a progressive water wave

Abstract

We consider two-dimensional free surface flow caused by a pressure wavemaker in a viscous incompressible fluid of finite depth and infinite horizontal extent. The governing equations are expressed in dimensionless form, and attention is restricted to the case δ[Lt ]ε[Lt ]1, where δ is the characteristic dimensionless thickness of a Stokes boundary layer and ε 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: bottom and surface Stokes layers of dimensionless thickness O(δ), bottom and surface Stuart layers of dimensionless thickness O(δ/ε) lying outside the Stokes layers, and an irrotational outer region of dimensionless thickness O(1). Equations describing the flow in all regions are derived, and the lowest-order steady streaming velocity in the near-field outer region is computed analytically.

It is shown that the flow far from the wavemaker is affected by thickening of the Stuart layers on the horizontal length scale O[(ε/δ)²], by viscous wave decay on the scale O(1/δ), and by nonlinear interactions on the scale O(1/ε²). The analysis of the flow in this region is simplified by imposing the restriction δ=O(ε²), so that all three processes take place on the same scale. The far-field flow structure is found to consist of a viscous outer core bounded by Stokes layers at the bottom boundary and water surface. An evolution equation governing the wave amplitude is derived and solved analytically. This solution and near-field matching conditions are employed to calculate the steady flow in the core numerically, and the results are compared with other theories and with observations.