Direct Numerical Simulations (DNS) of a spatially developing supersonic turbulent shear layer are conducted for a range of convective Mach numbers (Mc), velocity parameters, and density Atwood numbers (A) to examine the effects of compressibility, advection, and multi-fluid global density variation on the growth rate, self-similarity, flow statistics, asymmetry, and entrainment of the layer. At distant downstream locations, self-similarity is attained for all the examined cases. The self-similar region is identified by the collapse of normalized mean streamwise velocity, the constant peak of normalized Reynolds stresses, and the linear growth rate of the shear layer thickness and momentum thickness. Despite significant variations in the lower-order and higher-order statistics across different convective Mach numbers, velocity parameters, and density Atwood numbers, the profiles collapse within the self-similar region using our proposed self-similar scalings. It is demonstrated that the observed numerical trends and profiles are consistent with the literature and can be explained via compressible self-similar equations and models. The self-similar forms of continuity, streamwise momentum, transverse momentum, and energy equations have been formulated, incorporating both compressibility and centerline shifts. The self-similar normalized density distribution inside the layer is used to explain the effects of compressibility on various flow statistics including the far-field cross-stream velocity. The density variation is linked to dissipation effects as revealed by our analysis of the self-similar energy equation. An approximate equation for the cross-stream velocity is developed and the profiles of cross-stream velocity obtained from this equation are compared with the DNS results. A geometric interpretation of the entrainment ratio is presented and the approximate equation for the cross-stream velocity is used to provide the general expression of the entrainment ratio. The entrainment ratio increases with convective Mach numbers and velocity parameters, favoring excess entrainment on the high-speed side. Introducing global density variation in the flow enhances the layer asymmetry compared to the single-fluid shear layer, meaning that the shear layer centerline and the peak of Reynolds stresses shift more towards the lower momentum side. Apart from enhanced asymmetry, the increase in global density variation causes a further reduction in the shear layer growth rate compared to compressibility alone. A comparative study of the effects of compressibility and global density change on flow variables like mean density or cross-stream velocity reveals some of the interesting features of the simulated compressible multi-fluid shear layer. Despite significant differences in the lower and higher order statistics at different density Atwood numbers, the mean flow profiles collapse within the self-similar zone using our suggested self-similar scaling. A geometric interpretation of the entrainment ratio is presented, which helps to explain the decrease in the entrainment ratio with increasing Atwood numbers.
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