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

TL;DR

This paper introduces a novel fractional subgrid-scale model for turbulent flows based on nonlocal, non-Gaussian statistics, derived from the filtered Boltzmann equation, and evaluates its predictive capabilities through a priori analysis.

Contribution

The paper develops a new nonlocal fractional-order model for turbulence that captures non-Gaussian statistics and spatial nonlocality, starting from the filtered Boltzmann equation.

Findings

The fractional model effectively captures non-Gaussian turbulence features.

Model parameter depends on filter width and Reynolds number.

A priori tests show promising predictive performance.

Abstract

Coherent structures/motions in turbulence inherently give rise to intermittent signals with sharp peaks, heavy-skirt, and skewed distributions of velocity increments, highlighting the non-Gaussian nature of turbulence. That suggests that the spatial nonlocal interactions cannot be ruled out of the turbulence physics. Furthermore, filtering the Navier-Stokes equations in the large eddy simulation of turbulent flows would further enhance the existing nonlocality, emerging in the corresponding subgrid scale fluid motions. That urges the development of new nonlocal closure models, which respect the corresponding non-Gaussian statistics of the subgrid stochastic motions. To this end and starting from the filtered Boltzmann equation, we model the corresponding equilibrium distribution function with a \textit{L\'evy}-stable distribution, leading to the proposed fractional-order modeling of subgrid-scale stresses. We approximate the filtered equilibrium distribution function with a power-law term, and derive the corresponding filtered Navier-Stokes equations. Subsequently in our functional modeling, the divergence of subgrid-scale stresses emerges as a single-parameter fractional Laplacian, $(-\Delta)^{\alpha}(\cdot)$, $\alpha \in (0,1]$, of the filtered velocity field. The only model parameter, i.e., the fractional exponent, appears to be strictly depending on the filter-width and the flow Reynolds number. We furthermore explore the main physical and mathematical properties of the proposed model under a set of mild conditions. Finally, the introduced model undergoes \textit{a priori} evaluations based on the direct numerical simulation database of forced and decaying homogeneous isotropic turbulent flows at relatively high and moderate Reynolds numbers, respectively. Such analysis provides a comparative study of predictability and performance of the proposed fractional model.