Nonlinear evolution of spiral density waves generated by the instability of the shear layer in a rotating compressible fluid

Abstract

In a previous paper we considered the nonlinear stability of a cylindrical mixing layer in an incompressible fluid at large Reynolds numbers. Nonlinear evolution results in the formation of vortex structures in the vicinity of the corotation radius rc. This paper considers the same model but in a compressible fluid. A fundamental difference implied by the presence of compressibility is the possibility of the generation of disturbances which are no longer localized near the shear layer but embrace the entire region. These are acoustic waves generated in the region of corotation resonance and emitted into the periphery. In the r > rc region lines of equal density are trailing spirals. The nonlinear evolution of such disturbances is determined by redistribution of the mean flow inside the critical layer (CL). It is shown that only two possible types of CL, viscous and unsteady, can be realized here. For both types of these regimes, evolution equations describing the dynamics of a spiral density wave amplitude are obtained and their solutions analysed. It appears that at any values (provided that they are small enough) of initial supercriticality of the flow, an explosive growth of amplitude occurs which continues as long as values comparable with background ones are reached.