NONLINEAR-INTERACTIONS BETWEEN OBLIQUE INSTABILITY WAVES ON NEARLY PARALLEL SHEAR FLOWS

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 down stream 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 consis ting of a locally parallel mean flow plus a pair of oblique instabilit y waves together with an associated plane wave. The instability wave a mplitudes, 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 fea ture of these equations is the oblique mode, self-interaction term tha t usually leads to a singularity at a finite downstream position. It i s shown that this type of interaction is quite ubiquitous and is the d ominant nonlinear interaction in many apparently unrelated shear flows -even when the oblique modes do not exhibit the most rapid growth in t he initial linear stage.