Second order nonlinear spatial stability analysis of compressible mixing layers

Abstract

Second order nonlinear spatial stability to three-dimensional perturbation waves is analyzed for compressible mixing layers by expanding the perturbations into amplitude-dependent harmonic waves and truncating the Landau equation to the second term. This leads to a system of nonlinear ordinary differential equations for the harmonics. The two constants in Landau equation are calculated, wherein the independent variable, time t, is replaced by the streamwise coordinate direction x. The basic procedure in this paper is similar to that by Liu for compressible laminar wakes [Phys. Fluids 12, 1763 (1969)]. However, unlike this reference, which does not provide any results for their analysis, the present paper obtained many interesting results. The linear results from the present work compare very favorably with those reported by Day, Reynolds, and Mansour [Phys. Fluids 10, 993 (1998)], who employed a different procedure and limited their analysis to the linear regime. In the present studies, both the linear and nonlinear problems were analyzed in exactly the same manner, with the implication that the nonlinear results are probably accurate. These results include the convergence of the amplitude to an equilibrium value that depends on the two constants in the amplitude equation from Landau’s procedure. The present analysis is restricted to exponentially decaying linear solutions at the boundaries and hence to region one in the phase speed-Mach number diagram. However, we have observed that nonlinear effects could introduce constant, decaying, or outgoing wave solutions at the boundaries, depending on the velocity and density ratios and the Mach number of the fast stream. Other effects of these parameters are reported.