Eulerian mass density variations in a flow relate to compressibility and material inhomogeneities in the fluid. These variations can be caused due to high flow speeds, heat transfer, thermo-chemical reactions and/or phase change. From a local perspective, density gradient in space affects the velocity gradient dynamics due to variable inertia, in the presence of pressure-gradient driven acceleration, and therefore indirectly, the dissipation rate of kinetic energy and enstrophy. In turbulent flows, density variations and their effects on the velocity field influences the interscale interactions. Of particular interest is the turbulent dynamics in the presence of large vorticity generation by baroclinic torque. Although these effects are usually transient (in space or time) as turbulent mixing homogenizes the density field, the deviation from constant-density dynamical evolution can be statistically significant, particularly in instability-dominated flows with high sensitivity to initial/boundary conditions. In unsteady reacting flows, sustained chemi-acoustic interactions result in turbulent vorticity dynamics that is markedly different from the well-studied incompressible constant-density turbulence. Large-eddy simulations of high Reynolds number variable-density flows require adequate representation of unresolved small-scale variable-density effects. The present work is an effort to understand subgrid-scale (SGS) variable-density effects to improve the fidelity and accuracy of our simulations in these regimes. The thesis focuses on Reynolds-filtered governing equations to compute the large-scale vorticity dynamics more precisely. A novel equation set for coarse-grained mass, momentum and energy is derived that employs only second order moment based closures, and allows explicit representation of subgrid-scale compressibility and inertial effects. The new form of the filtered equations has terms that represent the SGS mass flux, pressure-gradient acceleration, and velocity-dilatation correlation. We attempt to quantify the dynamical significance of these terms with direct numerical and large eddy simulations.