Filtering, averaging, and scale dependency in homogeneous variable density turbulence*

We investigate relationships between filtering and Reynolds-averaging statistics as a function of length scale. Generalized central moments are expressed as inner products of generalized fluctuating quantities q′(ξ, x) = q(ξ) − q(x), representing fluctuations of a field q(ξ) with respect to its filtered values at x. These expressions provide a scale-resolving (SR) framework, with statistics, governing equations and realizability conditions at any length scale. In the small-scale limit, SR statistics become zero. In the large-scale limit, SR statistics and realizability conditions are the same as in the Reynolds-averaged description. Using DNS of homogeneous variable density turbulence, we diagnose Reynolds stresses T_ij, resolved kinetic energy k_r, turbulent mass-flux velocity a_i, and density-specific volume covariance b, defined in the SR framework. At intermediate scales, the governing equations for these variables exhibit interactions between terms that are not active in the Reynolds-averaged limit: in the Reynolds-averaged limit, b follows a decaying process; b peaks at intermediate length scales, where it is a balance between production, redistribution, destruction, and transport. This work supports the notion of a hybrid, length-scale adaptive model.