The nonlinear stability of a free shear layer in the viscous critical layer régime

Abstract
The nonlinear evolution of weakly amplified waves in a hyperbolic tangent free shear layer is described for spatially and temporally growing waves when the shear layer Reynolds number is large and the critical layer viscous. An artificial body force is introduced in order to keep the mean flow parallel. Jump conditions on the perturbation velocity and mean vorticity are derived across the critical layer by applying the method of matched asymptotic expansions and it is shown that viscous effects outside the critical layer have to be taken into account in order to obtain a uniformly valid solution. Consequently the true neutral wavenumber and frequency are lower than their inviscid counterparts. When only the harmonic fluctuations are considered, it is known that the Landau constant is negative so that linearly amplified disturbances reach an equilibrium amplitude. It is shown that when the mean flow distortion generated by Reynolds stresses is also included, the Landau constant becomes positive. Thus, in both the spatial and temporal case, linearly amplified waves are further destabilized and damped waves are unstable above a threshold amplitude.

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