Nonlinear interactions between oblique instability waves on nearly parallel shear flows

Abstract

Asymptotic methods are used to describe the nonlinear self-interaction between a pair of oblique instability modes that eventually develops when initially linear, spatially growing instability waves evolve downstream in nominally two-dimensional, unbounded or semibounded, laminar shear flows. The first nonlinear reaction takes place locally within a so-called ‘‘critical layer’’ with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves together with an associated plane wave. The instability wave amplitudes, which are completely determined by nonlinear effects within the critical layer, satisfy a pair of integral differential equations with quadratic to quartic-type nonlinearities. The most important feature of these equations is the oblique mode, self-interaction term that usually leads to a singularity at a finite downstream position. It is shown that this type of interaction is quite ubiquitous and is the dominant nonlinear interaction in many apparently unrelated shear flows—even when the oblique modes do not exhibit the most rapid growth in the initial linear stage.