Two-phase flow analysis of unstable fluid mixing in one-dimensional geometry

Abstract

A two-phase flow model for an acceleration-driven compressible fluid mixing layer is applied to an initially planar/cylindrical/spherical fluid configuration. A conservative form of the one-dimensional compressible equations is derived under the assumption that the fluid concentration is continuous. With a hyperbolic conservation law for the concentration gradient, the model supports traveling discontinuities in this quantity. The primary examples of this wave type are the moving boundaries of a finite mixing layer, which determine the instability growth rate. Constitutive laws for interfacial averages, previously derived for planar incompressible mixing, are reinterpreted and shown to be applicable to other one-dimensional mixing problems of interest. The equations of motion for an incompressible mixing layer in planar, cylindrical, or spherical geometry are solved exactly, up to a history integral of a function of the edge trajectories, and without assuming incompressible flow outside the layer. Full solutions are obtained by numerically integrating a coupled system of ordinary differential equations for the volume fraction characteristics. Results for self-similar Rayleigh–Taylor mixing in planar geometry are compared to the work of others. This comparison suggests that the shape of the fluid concentration profile is primarily a consequence of mass conservation, parameterized by the expansion ratio of the mixing zone edges.