Nonlinear temporal-spatial modulation of near-planar Rayleigh waves in shear flows: formation of streamwise vortices

Abstract

The nonlinear temporal-spatial modulation of a near-planar Rayleigh instability wave is studied. The amplitude of the wave is allowed to be a slowly varying function of spanwise position as well as of time (or streamwise variable in the spatial evolution case). It is shown that the development of the disturbance is controlled by critical-layer nonlinear effects when the linear growth rate decreases to O(ε⅖), where ε is the magnitude of the disturbance. Nonlinear interactions influence the evolution by producing spanwise dependent mean-flow distortions. The evolution is governed by an integro-partial-differential equation containing history-dependent nonlinear terms of Hickernell (1984) type. A notable feature of the amplitude equation is that the highest derivative with respect to spanwise position appears in the nonlinear terms. These terms are associated with three-dimensionality. The possible properties of the amplitude equation are discussed. Numerical solutions show that a disturbance initially centred at a spanwise position can propagate laterally to form concentrated, quasi-periodic streamwise vortices. This qualitatively captures the phenomena observed in experiments. The focusing of vorticity may be associated with a localized singularity which can occur at a finite distance downstream or within a finite time. It is noted that the amplitude equation is rather generic and applies to a broad class of shear flows which is inviscidly unstable.