Commutator errors in large-eddy simulation

Commutator errors arise in large-eddy simulation of incompressible turbulent flow from the application of non-uniform filters to the continuity -- and Navier-Stokes equations. For non-uniform, high order filters with bounded moments the magnitude of the commutator errors is shown to be of the same order as that of the turbulent stress fluxes. Consequently, one cannot reduce the size of the commutator errors independently of the turbulent stress terms by any judicious construction of such filter operators. Independent control over the commutator errors compared to the turbulent stress fluxes can, instead, be obtained by appropriately restricting the spatial variations of the filter-width and filter-skewness. For situations in which the dynamical consequences of the commutator errors are significant, e.g., near solid boundaries, explicit similarity modelling for the commutator errors is proposed, including the application of Leray regularization. The performance of this commutator error parametrization is illustrated for the one-dimensional Burgers equation. The Leray approach is found to capture the filtered flow with higher accuracy than conventional similarity modelling, which is particularly relevant for large filter-width variations.