Large-amplitude interfacial waves on a linear shear flow in the presence of a current

Abstract

The properties of two-dimensional steady periodic interfacial gravity waves between two fluids in relative motion and of constant vorticities and finite depths are investigated analytically and numerically. Particular attention is given to the effect of uniform vorticity, in the presence of a current velocity, on the two factors (identified in the literature as dynamical and geometrical limits) which limit the existence of steady gravity wave Solutions. The dynamical limit to the existence of steady Solutions is found to be significantly influenced by the uniform vorticity of the lower fluid. In particular, the effect of non-zero vorticity is qualitatively different between a very shallow and a relatively deep lower fluid. Profiles and flow fields corresponding to very steep waves are calculated for a wide range of parameter values. The effect of uniform vorticity on the interfacial wave structure is demonstrated through a direct comparison with irrotational waves. For negative vorticity and high current velocity, a new flow structure is found, consisting of a closed eddy attached to the interface below the crest. Resemblance with shallow water waves breaking under strong air flow, (described in the experimental literature as roll waves) is noted.