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Coherent structures/motions in turbulence inherently give rise tointermittent signals with sharp peaks, heavy-skirt, and skewed distributions ofvelocity increments, highlighting the non-Gaussian nature of turbulence. Thatsuggests that the spatial nonlocal interactions cannot be ruled out of theturbulence physics. Furthermore, filtering the Navier-Stokes equations in thelarge eddy simulation of turbulent flows would further enhance the existingnonlocality, emerging in the corresponding subgrid scale fluid motions. Thaturges the development of new nonlocal closure models, which respect thecorresponding non-Gaussian statistics of the subgrid stochastic motions. Tothis end and starting from the filtered Boltzmann equation, we model thecorresponding equilibrium distribution function with a \textit{L\'evy}-stabledistribution, leading to the proposed fractional-order modeling ofsubgrid-scale stresses. We approximate the filtered equilibrium distributionfunction with a power-law term, and derive the corresponding filteredNavier-Stokes equations. Subsequently in our functional modeling, thedivergence of subgrid-scale stresses emerges as a single-parameter fractionalLaplacian, $(-\Delta)^{\alpha}(\cdot)$, $\alpha \in (0,1]$, of the filteredvelocity field. The only model parameter, i.e., the fractional exponent,appears to be strictly depending on the filter-width and the flow Reynoldsnumber. We furthermore explore the main physical and mathematical properties ofthe proposed model under a set of mild conditions. Finally, the introducedmodel undergoes \textit{a priori} evaluations based on the direct numericalsimulation database of forced and decaying homogeneous isotropic turbulentflows at relatively high and moderate Reynolds numbers, respectively. Suchanalysis provides a comparative study of predictability and performance of theproposed fractional model.

A Fractional Subgrid-scale Model for Turbulent Flows: Theoretical Formulation and a Priori Study