A priori and a posteriori analysis in Large eddy simulation of the two-dimensional decaying turbulence using Explicit filtering approach

In this work, we consider turbulence closures of LES (Large Eddy Simulation) type for classical decaying 2D turbulence in a priori and a posteriori experiments using explicit filtering approach. According to Germano 1986 decomposition, full subfilter stress for Gaussian filter is decomposed into three Galilean-invariant parts: Leonard, Cross and Reynolds stresses. By analysing spectral transfer of energy and enstrophy, we show that Leonard stress redistributes resolved energy toward large scales and dissipates substantial part of enstrophy, while Cross stress provides an additional enstrophy dissipation at subfilter scales and Reynolds stress predominantly injects energy into middle scales (i.e., Kinetic Energy Backscatter). Substantial part of enstrophy dissipation is located on subfilter scales, and it should be accounted for by choosing base filter wide enough compared to mesh step of LES model. Otherwise, significant fraction of enstrophy dissipation will correspond to subgrid scale stress, which is less universal and harder to approximate. As a result of a priori analysis, we propose LES closure consisting of three parts: SSM (Scale Similarity Model), which is equivalent to Leonard stress, biharmonic Smagorinsky damping as a Cross stress counterpart and ADM (Approximate Deconvolution Model) approximation for Reynolds stress ("backscatter"). The proposed model have two free parameters: Smagorinsky constant and amplitude of the backscatter. These parameters are estimated in a posteriori experiments utilizing dynamic approach and energy-enstrophy balance equation, correspondingly. The proposed model have the following distinctive features: it reproduces energy and enstrophy transfer spectra in accordance to the individual components of the subfilter forces, "reproduces" base filter and reproduces energy growth in accordance to the filtered DNS (Direct Numerical Simulation) solution.

The work was supported by the Russian Foundation for Basic Research (projects 19-35-90023, 18-05-60184) and Moscow Center for Fundamental and Applied Mathematics (agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2019-1624).