Nonlinear Instability of a Uni-directional Transversely Sheared Mean Flow

It is well known that the presence of a weak cross flow in an otherwise two-dimensional shear flow results in a spanwise variation in the mean streamwise velocity profile that can lead to an amplification of certain three-dimensional disturbances through a kind of resonant-interaction mechanism (Goldstein and Wundrow 1994). The spatial evolution of an initially linear, finite-growth-rate, instability wave in such a spanwise-varying shear flow is considered, The base flow, which is governed by the three-dimensional parabolized Navier-Stokes equations, is initiated by imposing a spanwise- periodic cross-flow velocity on an otherwise two-dimensional shear flow at some fixed streamwise location. The resulting mean-flow distortion initially grows with increasing streamwise distance, reaches a maximum and eventually decays through the action of viscosity. This decay, which coincides with the viscous spread of of the shear layer, means that the local growth rate of the instability wave will eventually decrease as the wave propagates downstream. Nonlinear effects can then become important within a thin spanwise-modulated critical layer once the local instability-wave amplitude and growth rate become sufficiently large and small, respectively. The amplitude equation that describes this stage of evolution is shown to be a generalization of the one obtained by Goldstein and Choi (1989) who considered the related problem of the interaction of two oblique modes in a two-dimensional shear layer.