Gortler vortices in compressible mixing layers

In experiments, Plesniak, Mehta & Johnson (1994) have noted that curved two-stream mixing layers are susceptible to centrifugal instabilities under the condition that the slower of the streams curves towards the faster one; this condition is analogous to the concave curvature condition for the stability of the flow over a plate. The modes which arise manifest themselves as vortices aligned with the dominant flow direction. Previous numerical and analytical work has elucidated the structure of these Vortices within incompressible mixing layers; Otto, Jackson & Hu (1996). In this paper we go on to investigate the roles of compressibility and heating in determining the streamwise fate of Gortler vortices within these situations. The development of the disturbances is monitored downstream and curves of neutral stability are plotted. The effect of changing the Mach number and free-stream temperatures is studied in detail. It is found that for certain parameter regimes modes can occur within convexly curved, or 'stable' mixing layers; these 'thermal modes' have no counterpart within incompressible mixing layers. By making use of a large Gortler number analysis we are able to verify our numerical results, and derive a very simple condition which yields information about the parameter ranges for which certain modes are likely to occur. As an aside this method can be used to show that no degree of wall cooling will allow sustained growth of Gortler vortices within boundary layers over convex plates.