The dynamics of nonlinear instability waves in laminar heated and unheated compressible mixing layers

Abstract

Previous studies have shown that when using instability wave theories to model compressible laminar mixing layers, explicitly accounting for nonlinear processes such as the mean flow correction and modal interactions is necessary in order to properly capture vortex roll-up and pairing phenomena. In this study, we examine the effects of heating when using the nonlinear instability wave formulation on two-dimensional compressible mixing layers. We compare the results of both heated and unheated, supersonic and subsonic laminar mixing layer calculations using both the nonlinear parabolized stability equations (PSE) and Navier–Stokes equations. For all supersonic mixing layers, we find that the nonlinear stability method adequately captures the roll-up process and agrees well with direct calculations. While the same is true for the unheated subsonic mixing layer, in the heated subsonic mixing layer, we find that baroclinic vorticity generation, in addition to nonlinear interactions, plays a significant role in governing the vorticity dynamics. We observe that in the presence of large density stratifications in subsonic flows, approximations made in the streamwise pressure gradient of the PSE formulation can lead to discrepancies in the vortex dynamics. Error estimates for the approximations to the baroclinic term show that discrepancies are directly proportional to the temperature difference and inversely proportional to the Reynolds number.