  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.
