We analyse and improve a recently-proposed two-phase flow model
for the statistical
evolution of two-fluid mixing. A hyperbolic equation for the volume fraction,
whose
characteristic speed is the average interface velocity v*,
plays a central role. We
propose a new model for v* in terms of the volume fraction
and fluid velocities, which
can be interpreted as a constitutive law for two-fluid mixing. In the incompressible
limit, the two-phase equations admit a self-similar solution for an arbitrary
scaling of
lengths. We show that the constitutive law for v* can be expressed
directly in terms
of the volume fraction, and thus it is an experimentally measurable quantity.
For
incompressible Rayleigh–Taylor mixing, we examine the self-similar
solution based
on a simple zero-parameter model for v*. It is shown that
the present approach gives
improved agreement with experimental data for the growth rate of a Rayleigh–Taylor
mixing layer.Closure of the two-phase flow model requires boundary conditions for
the surfaces
that separate the two-phase and single-phase regions, i.e. the edges of
the mixing layer.
We propose boundary conditions for Rayleigh–Taylor mixing based on
the inertial,
drag, and buoyant forces on the furthest penetrating structures which define
these
edges. Our analysis indicates that the compatibility of the boundary conditions
with
the two-phase flow model is an important consideration. The closure assumptions
introduced here and their consequences in relation to experimental data
are compared
to the work of others.
