Numerical Study of Compressibility and Velocity Parameter Effects on Spatially Evolving Supersonic Turbulent Shear Layers

Abstract

Direct Numerical Simulations (DNS) of a spatially developing supersonic turbulent shear layer are conducted for a range of convective Mach numbers ($M_c$) and velocity parameters ($λ$) to examine the effects of compressibility and advection on the growth rate, self-similarity, flow statistics, asymmetry, and entrainment of the layer. At distant downstream locations, self-similarity is attained for all 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 lower-order and higher-order statistics across different $M_c$ and $λ$ values, profiles of all turbulence quantities examined collapse within the self-similar region using our proposed self-similar scalings. The self-similar forms of continuity, momentum, and energy equations have been formulated, incorporating 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 show good agreement 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 a general closed-form expression of the entrainment ratio. The entrainment ratio increases with $M_c$ and $λ$, favoring excess entrainment on the high-speed side.