Filtering, averaging and scale dependency in homogeneous variable density turbulence

• Published in: arXiv.org
• Main Authors: Saenz, J A, Aslangil, D, Livescu, D
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 12.12.2020
• Subjects: Balances (scales); Computational fluid dynamics; Covariance; Density; Destruction; Direct numerical simulation; Filtration; Kinetic energy; Mathematical models; Physics - Fluid Dynamics; Realizability; Specific volume; Statistics; Turbulence; Turbulent flow
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2012.06851

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.