Nonlinear spatial evolution of an externally excited instability wave in a free shear layer

Abstract

We consider a disturbance that evolves from a strictly linear finite-growth-rate instability wave, with nonlinear effects first becoming important in the critical layer. The local Reynolds number is assumed to be just small enough so that the spatial-evolution, nonlinear-convection, and viscous-diffusion terms are of the same order of magnitude in the interactive critical-layer vorticity equation. The numerical results show that viscous effects eventually become important even when the viscosity is very small due to continually decreasing scales generated by the nonlinear effects. The vorticity distribution diffuses into a more regular pattern vis-a-vis the inviscid case, and the instability-wave growth ultimately becomes algebraic. This leads to a new dominant balance between linear- and nonlinear-convection terms and an equilibrium critical layer of the Benney & Bergeron (1969) type begins to emerge, but the detailed flow field, which has variable vorticity within the cat's-eye boundary, turns out to be somewhat different from theirs. The solution to this rescaled problem is compared with the numerical results and is then used to infer the scaling for the next stage of evolution of the flow. The instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects in this latter stage of development and is eventually converted to decay. The neutral stability point is the same as in the corresponding linear case, however.