Laminar Boundary Layer under a Wave

Abstract

The laminar boundary-layer flow resulting from a wave potential flow of form U = αc sin(kx — ωt), with α a parameter and c the wave velocity, is considered. It is shown, after suitable transformation, that there is an exact solution of the unsteady boundary-layer equations which is of the form of a power series in the phase kx — ωt. The coefficients φn are functions of a similarity variable, and are the solutions of an infinite set of linear third-order differential equations with nonlinear forcing terms. The forcing term in the equation for φn is a function of φ0, φ1, …, φn−1 and their derivatives. Solutions for φ0, φ1, and φ2 have been computed and are presented. The theory is applied to the laminar boundary layer under a progressive shallow-water wave, where α = a/h, and compared to a linearized theory. It is concluded that if α ≪ 1, or for any α in a sufficiently small region near kx — ωt = 0, the linearized theory is valid. Otherwise, the linear theory does not provide an adequate description of the flow.