Evolution of Finite Amplitude Kelvin–Helmholtz Billows in Two Spatial Dimensions

Abstract

A two-dimensional numerical model is used to calculate the nonlinear evolution of Kelvin-Helmholtz (KH) billows for various Reynolds numbers in the range where the turbulent collapse of the waves is expected. The onset of disordered motions is not observed in these numerical experiments, presumably because the transition requires the third spatial degree of freedom. Although we have shown elsewhere that these two-dimensional KH wave states are unstable with respect to three-dimensional perturbations, the spanwise coherent large-scale structure is observed to persist in presence of the small-scale fluctuations. Thus the two-dimensional wave structure is of importance in itself and the present paper is devoted to a detailed study of the laminar evolution of the dominant large-scale vortices. An analysis of the transfer of energy between the wave and the mean flow firmly establishes that the wave does not enter a steady state upon achieving maximum amplitude. Rather, it begins an almost periodic exchange of energy with the mean flow, and its amplitude begins to oscillate. This oscillation is associated with the nutation of the nonlinear vortex about a state for which the net Reynolds stress vanishes. We also demonstrate that diffusion of the mean flow can play an important role in the evolution of Kelvin?Helmholtz waves when the Reynolds number associated with the initial parallel flow is significantly lower than 500.