Effect of Density Gradients in Confined Compressible Shear Layers.

The effects of basic flow density gradients on confined supersonic shear layers are investigated on an inviscid basis using both linear analysis and numerical simulations. The compressible Rayleigh equation for a two species system is solved using the solution of the steady compressible boundary layer equations for the basic profiles. First, a test case corresponding to the parameters used by Tam and Hu (1989) who used a hyperbolic tangent velocity profile, were investigated. Three generalized inflection points were found for the basic profiles. Due to the confinement of the shear layer, all of these inflection points are related to critical neutral solutions. Three sets of instability modes were found, two of which start from the familiar non-inflectional neutral solutions and the third mode is a new mode which starts from the middle inflection point. As the density ratio was increased from the test case, the basic profiles changed in such a way that eventually only the lower inflection point remained. This change in the profiles leads to a linear resonance between the instability modes which results in a mode with multiple growth rate peaks, a much larger unstable bandwidth and much larger growth rates than those of the test case modes. The numerical portion encompasses two types of study. The first is a study of the feasibility of a cell averaged based multi-dimensional ENO scheme for the numerical simulation of shear layers, and the second involves numerical spatial simulations at two density ratios. The initial attempt in obtaining solutions of the test case using the multi-dimensional ENO scheme failed due to occurrence of negative pressure within the flowfield. Improvements were made on the ENO recipe such as resorting to a coefficient by coefficient limiting of polynomials and applying ENO directly on pressure which allowed solutions to be obtained. Finally a TVD scheme is used for the spatial simulations of the instabilities for two density ratios. For the test case, formation of strong shock-expansion structures are observed within the flowfield. The acceleration and rotation of the lower part of these structures due to the generation of vorticity near the walls increase the magnitude of the pressure change near the lower wall, causing a blowing effect from the wall which improves the entrainment process. For the higher density ratio, some interesting phenomena such as generation of Kelvin-Helmholtz type structures in the subsonic portion and compression-expansion waves in the supersonic portion of the shear layer were observed. Also, the fundamental mode seemed to saturate quickly and then, after a long plateau, seemed to grow again.