Waves in a shear flow: transition between the KH, Holmboe and Miles instability

We study the generation of waves at the interface of two immiscible fluids, arising due to shear-driven, inviscid instabilities. The background profile is chosen to be the classic exponential velocity profile (Morland transitioning from having a sharp jump at the critical location for 1 (Miles) to smooth variation through this location for 0. 5 (H). A theoretical explanation for this is presented. In the higher density ratio regime (0. 5 0. 9), by comparing against stability results from the corresponding piecewise linear (PL) velocity profile, we show the possibility of the H and the KH instability in the exponential model. Simulations, solving the incompressible Euler’s equation with gravity and surface tension are reported. In the Miles regime (=0. 01) excellent agreement with linear theory upto five wave periods is observed. As these waves saturate, tiny surface ripples appear. Increasing to 0. 1, finite-amplitude waves resulting from the instability display larger-amplitude ripples on their surface; these are reminiscent of Stokes waves with capillary effects. With further increase (=0. 5), exponentially growing waves display a sheared cusp at the crest in the nonlinear regime. They emit spume droplets and resemble the asymmetric H waves in the simulations and experiments of Lawrence et al. (1998). At =0. 9, growing interfacial waves rapidly distort into classic KH spirals within five wave periods. Several comparisons reported between the PL and the exponential profile clarify the role of background profile curvature and the critical location. To our knowledge, this is the first demonstration of three canonical instabilities, within these background states and without usage of the Boussinesq approximation.