  This work extends the framework of the partially-averaged Navier-Stokes
(PANS) equations to variable-density flow, \text{i.e.}, multi-material and/or
compressible mixing problems with density variations and production of
turbulence kinetic energy by both shear and buoyancy mechanisms. The proposed
methodology is utilized to derive the PANS BHR-LEVM closure. This includes
\textit{a-priori} testing to analyze and develop guidelines toward the
efficient selection of the parameters controlling the physical resolution and,
consequently, the range of resolved scales of PANS. Two archetypal test-cases
involving transient turbulence, hydrodynamic instabilities, and coherent
structures are used to illustrate the accuracy and potential of the method: 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 processes, constitute the initial
validation space of the new model, and their results are comprehensively
discussed in two subsequent studies. The computations indicate that PANS can
accurately predict the selected 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.
