Towards systematic grid selection in LES: iterative identification of the coarse-graining length scale by minimizing the solution sensitivity

Abstract

The accuracy of a large eddy simulation (LES) is determined by the accuracy of the model used to describe the effect of unresolved scales, the numerical errors of the resolved scales, and the optimality of the length scale that separates resolved from unresolved scales (the filter-width, or the coarse-graining length scale). This paper is focused entirely on the last of these, proposing a systematic algorithm for identifying the "optimal" spatial distribution of the coarse-graining length scale and its aspect ratio. The core idea is that the "optimal" coarse-graining length scale for LES is the largest length scale for which the LES solution is minimally sensitive to it. This idea is formulated based on an error indicator that measures the sensitivity of the solution and a criterion that determines how that error indicator should vary in space and direction to minimize the overall sensitivity of the solution. The solution to this optimization problem is that the cell-integrated error indicator should be equi-distributed; a corollary is that one cannot link the accuracy in LES to quantities that are not cell-integrated, including the common belief that LES is accurate whenever 80-90\% of the energy is resolved. The full method is tested on the wall-resolved LES of turbulent channel flow and the flow over a backward-facing step, with final length scale fields (or filter-width fields, or grids) that are close to what is considered "best practice" in the LES literature. Finally, the derivation of the error indicator offers an alternative explanation for the success of the dynamic procedure.