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

A two-dimensional disturbance evolving from a strictly linear, finite-growth-rate instability wave, with nonlinear effects first becoming important in the critical layer is considered. The analysis is carried out for a general weakly nonparallel mean flow using matched asymptotic expansions. The flow in the critical layer is governed by a nonlinear vorticity equation which includes a spatial-evolution term. As in Goldstein and Hultgren (1988), the critical layer ages into a quasi-equilibrium one and the initial exponential growth of the instability wave is converted into a weak algebraic growth during the roll-up process. This leads to a next stage of evolution where the instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects and is eventually converted to decay. Expansions for the various streamwise regions of the flow are combined into a single composite formula accounting for both shear-layer spreading and nonlinear critical-layer effects and good agreement with the experimental results of Thomas and Chu (1989) and Freymuth (1966) is demonstrated.