Interfacial progressive gravity waves in a two-layer shear flow

Abstract

Nonlinear interfacial gravity waves in a two-layer Boussinesq fluid are studied. In a previous paper, nonlinear waves on a vortex sheet separating two layers, each of constant density and velocity were considered. In the present paper the basic flow model consists of a constant vorticity upper layer bounded by a rigid surface and an irrotational lower layer of infinite depth with continuity of the unperturbed velocity at the density interface. Numerical solutions obtained from an exact formulation in terms of a complex-valued integral equation for the shape and local vortex-sheet strength of the wave profile are compared with results from a second-order Stokes expansion. It is found that the wave of maximum amplitude displays different geometrical features depending on the unperturbed flow parameters. These include waves containing an S-shaped section, waves with cusped crests and surface-constrained waves with long flat crests. Wave integral properties calculated, including the flux of momentum and energy in the wave propagation direction, showed monotonic variation with increasing wave amplitude.