  We extend the framework of the Partially-Averaged Navier-Stokes (PANS)
equations to variable-density flow, i.e., multi-material and/or compressible
mixing problems with density variations and production of turbulence kinetic
energy by both shear and buoyancy mechanisms. This includes \textit{a-priori}
testing to develop guidelines toward the efficient selection of the parameters
controlling the physical resolution (range of resolved scales) of PANS. The
method is then evaluated predicting two archetypal test-cases involving
transient turbulence, hydrodynamic instabilities, and coherent structures: the
Taylor-Green vortex (TGV) at Reynolds number $\mathrm{Re}=3000$, and the
Rayleigh-Taylor (RT) flow at Atwood number $0.5$ and
$(\mathrm{Re})_{\max}\approx 500$. These representative problems, for which
turbulence is generated by shear and buoyancy, constitute the initial
validation space of the new model. The results indicate that PANS can
accurately predict the test flow problems, resolving only a fraction of the
scales of large eddy simulation and direct numerical simulation strategies. The
results also reiterate that the physical resolution of the PANS model must
guarantee that the key instabilities and coherent structures of the flow are
resolved. The remaining scales can be modeled through an adequate turbulence
scale-dependent closure.
