Analysis of the gradient-diffusion hypothesis in large-eddy simulations based on transport equations

Abstract

The gradient-diffusion hypothesis is frequently used in numerical simulations of turbulent flows involving transport equations. In the context of large-eddy simulations (LES) of turbulent flows, one modeling trend involves the use of transport equations for the subgrid-scale (SGS) kinetic energy and SGS scalar variance. In virtually all models using these equations, the diffusion terms are lumped together, and their joint effect is modeled using a “gradient-diffusion” model. In this work, direct numerical simulations of homogeneous isotropic turbulence are used to analyze the local dynamics of these terms and to assess the performance of the “gradient-diffusion” hypothesis used in their modeling. For this purpose a priori tests are used to assess the influence of the Reynolds and Schmidt numbers and the size of the implicit grid filter in this modeling assumption. The analysis uses correlations, variances, skewnesses, flatnesses, probability density functions, and joint probability density functions. The correlations and joint probability density functions show that provided the filter width is within or close to the dissipative range the diffusion terms pertaining to the SGS kinetic energy and SGS scalar variance transport equations are well represented by a gradient-diffusion model. However, this situation changes dramatically for both equations when considering inertial range filter sizes and high Reynolds numbers. The reason for this lies in part in a loss of local balance between the SGS turbulent diffusion and diffusion caused by grid/subgrid-scale (GS/SGS) interactions, which arises at inertial range filter sizes. Moreover, due to the deficient modeling of the diffusion by SGS pressure-velocity interactions, the diffusion terms in the SGS kinetic energy equation are particularly difficult to reconcile with the gradient-diffusion assumption. In order to improve this situation, a new model, inspired by Clark’s SGS model, is developed for this term. The new model shows very good agreement with the exact SGS pressure-velocity term in a priori tests and better results than the classical model in a posteriori (LES) tests.