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

We investigate relationships between statistics obtained from filtering and from ensemble or Reynolds-averaging turbulence flow fields as a function of length scale. Generalized central moments in the filtering approach are expressed as inner products of generalized fluctuating quantities, $q'(\xi,x)=q(\xi)-\overline q(x)$, representing fluctuations of a field $q(\xi)$, at any point $\xi$, with respect to its filtered value at $x$. For positive-definite filter kernels, these expressions provide a scale-resolving framework, with statistics and realizability conditions at any length scale. In the small-scale limit, scale-resolving statistics become zero. In the large-scale limit, scale-resolving statistics and realizability conditions are the same as in the Reynolds-averaged description. Using direct numerical simulations (DNS) of homogeneous variable density turbulence, we diagnose Reynolds stresses, $\mathcal{T}_{ij}$, resolved kinetic energy, $k_r$, turbulent mass-flux velocity, $a_i$, and density-specific volume covariance, $b$, defined in the scale-resolving framework. These variables, and terms in their governing equations, vary smoothly between zero and their Reynolds-averaged definitions at the small and large scale limits, respectively. At intermediate scales, the governing equations exhibit interactions between terms that are not active in the Reynolds-averaged limit. For example, in the Reynolds-averaged limit, $b$ follows a decaying process driven by a destruction term; at intermediate length scales it is a balance between production, redistribution, destruction, and transport, where $b$ grows as the density spectrum develops, and then decays when mixing becomes strong enough. This work supports the notion of a generalized, length-scale adaptive model that converges to DNS at high resolutions, and to Reynolds-averaged statistics at coarse resolutions.