Self-similar supersonic variable-density shear layers in binary systems

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Abstract
Some characteristic features of supersonic/supersonic, laminar, variable-density shear layers are studied by examining the self-similar behavior of five nitrogen/hydrogen streams. With the Levy-Lees transformation, the flow-field variables, which include the transverse velocity and dilatation, are obtained through the solution of the coupled set of nonlinear conservation equations. The issue of the appropriate ''third boundary condition,'' first given for the supersonic/supersonic case by Ting [J. Math. Phys. 28, 153 (1959)], is addressed and implemented in the present formulation. Expressions for the thermal conductivity, viscosity, specific heat, and binary diffusion coefficients of an arbitrary mixture are utilized so that the Prandtl and Lewis numbers and the Chapman-Rubesin parameter can vary freely across the shear layer. In the particular cases considered, these three quantities varied by factors of approximately 3, 7, and 22, respectively. The region of high vorticity moves toward the less-dense hydrogen stream for large density ratios (approximately 9:1), and becomes nearly decoupled from the density profile. Because the vorticity is responsible for the kinematic mixing of the two streams, this mixing of the two laminar streams is likely to be inhibited. Even though laminar flows are considered here, this effect is consistent with the experimental observation that as density ratios become very large, further increases in the density ratio have no effect on the turbulent shear layer growth rate. The density and hydrogen mass-fraction profiles are quite elongated in the transverse direction. An increase in the velocity ratio exaggerates both of these effects. Results obtained in this work are compatible with earlier work on incompressible, variable-density flows; more importantly, these results qualitatively resemble those from experiments of compressible and incompressible turbulent flows.