Non-dimensional meshing criterion of mean flow field discretization for RANS and LES

Abstract

When turbulent flows occur, Reynolds Average Navier–Stokes (RANS) and Large-Eddy Simulation (LES) approaches are now valuable strategies to numerically study complex systems. An open question is still to be able to define an adequate mesh, i.e. guaranteeing accuracy of the numerical simulations but limiting the number of mesh elements to limit computational cost. RANS and LES approaches differ in term of level of description of the turbulent fields, but these approaches share the same objective to obtain mean fields independent of the mesh. Based on the Reynolds equation, a new mesh size based Reynolds number, $R{e}_{\Delta }$, is derived. This new criterion is the upper bound of a non-dimensional error estimation of the mean velocity field. This new criterion can also be interpreted by analogy with the Kolmogorov scale, $\eta$. Indeed, $\eta$ can be interpreted as the scale where the instantaneous dynamic is dominated by (molecular) diffusive effects, leading to the Kolmogorov Reynolds number, $R{e}_{\eta }\sim 1$. Similarly, $R{e}_{\Delta }$ will be close to 1 at scale $\Delta$ where molecular and turbulent diffusive effects dominate the mean field dynamic. This allows to define the local mesh size to guarantee a correct discretization of the mean field. This criterion is applied in various flow configuration for LES, with and without law of the wall, as well as RANS simulations with great accuracy. In practice, it is found that the value $R{e}_{\Delta }\sim 1$ appears indeed as a good compromise in terms of number of elements and precision. This allows to easily obtain an adequate mesh for the mean flow velocity field, without a priori knowledge of the flow dynamic.