Nonlinear stability of a stratified shear flow in the regime with an unsteady critical layer. Part 2. Arbitrary stratification of asymmetric flow

Abstract

A weakly nonlinear analysis of the downstream evolution of weakly unstable disturbances in a stably stratified mixing layer with a large Reynolds number is carried out. No other requirements are imposed upon velocity and density profiles, thus making it possible to overcome the restrictions placed in earlier studies (Brown & Stewartson 1978; Brown et al. 1981; Churilov & Shukhman 1987, 1988) by a particular choice of weakly supercritical flow models assuming symmetry. For each of the two critical layer regimes possible here, viscous and unsteady, evolution equations are obtained, their solutions and competition between nonlinearities in the course of instability development are analysed, and evolution scenarios for unstable disturbances are constructed for different levels of their supercriticality. It is established that the regime with a nonlinear critical layer does not arise in an evolutionary manner, except for the previously studied case of a weak stratification (Shukhman & Churilov 1997). It is shown that while in the viscous critical layer regime the relaxation of assumptions of the symmetry and weak supercriticality of the flow produces no fundamental changes in the theory, in the unsteady critical layer regime a new (non-dissipative cubic) nonlinearity appears which governs the instability development on equal terms with two already known nonlinearities. Results are illustrated by calculations for two families of flow models with a controlled degree of asymmetry.