Viscous attenuation of mean drift in water waves

Abstract

The initial-value problem for slightly viscous, two-dimensional, spatially periodic waves is examined. Matched asymptotic expansions in space for small wave amplitude a and multiple scales in time allow the boundary layers and viscous attenuation to be described. The bottom and surface boundary layers of thickness δ are equivalent to those of Longuet-Higgins except that wave attenuation is included. For progressive waves one solution for the interior motion independent of the magnitude of δ/a is an attenuating version of the conduction solution of Longuet-Higgins, but with modified structure, the O(a2) vorticity at the boundaries ultimately diffusing into the entire field. There are certain critical depths for which there is secular behaviour and these do not correspond to quasi-steady flows. Other solutions may be possible. For standing waves the interior flow depends on the magnitude of the steady-drift Reynolds number Rs∝ (a/δ)2 introduced by Stuart. When Rs [Lt ] 1, the interior is viscous with an O(a2) vorticity ultimately diffusing into the entire field. When Rs [Gt ] 1 there is a doubleboundary-layer structure on the bottom and on the surface. Within the outer layers, the O(a2) steady drift decays to the potential flow interior. A direct analogy with the flow structure on a circular cylinder oscillating along its diameter is introduced and pursued. Finally, all of the above fields are converted to Lagrangian fields so that masstransport profiles can be obtained. Comparisons are made with previous theoretical and experimental work.