Nonlinear stage of instability development in a stratified shear flow with an inflection-free velocity profile

Abstract

A plane-parallel shear flow with an inflection-free velocity profile Vx=U(y), having an embedded thin layer of stable density stratification, is known to be unstable in ideal incompressible fluid, for an arbitrary bulk Richardson number J>0 [S. M. Churilov, J. Fluid Mech. 539, 25 (2005); S. M. Churilov, J. Fluid Mech.617, 301 (2008)], and it is the three- rather than two-dimensional (z-independent) disturbances that are most unstable within a wide range of parameters. We examine the weakly nonlinear evolution of a pair of unstable oblique waves in such a flow, in the unsteady critical layer regime. For this purpose, we derive the evolution equation which has the form of a nonlinear integral equation and is valid for both thin and thick critical layers, including the case where the critical layer width exceeds the stratification layer thickness. The solutions of this equation are inspected both analytically and numerically, and it is shown that during the nonlinear stage, the disturbance evolves, as a rule, explosively.