Filtering, averaging and scale dependency in homogeneous variable density turbulence

TL;DR

This paper develops a scale-dependent framework for analyzing turbulence statistics, bridging DNS and Reynolds-averaged descriptions, and investigates the scale interactions in variable density turbulence through numerical simulations.

Contribution

It introduces a generalized filtering approach for turbulence statistics that smoothly transitions between DNS and Reynolds-averaged limits, providing a scale-resolving framework.

Findings

Scale-resolving statistics vary smoothly between DNS and RANS limits.

Intermediate scales show interactions not present in pure RANS.

Density spectrum development influences turbulence variable behaviors.

Abstract

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.