Commutator errors in the filtering approach to large-eddy simulation

Abstract

We analyze the large-eddy equations that are obtained from the application of a spatially nonuniform filter to the Navier–Stokes equations. Next to the well-known turbulent stress tensor a second group of subgrid terms arises; the so-called commutator errors. These additional subgrid terms emerge in the large-eddy equations solely as a consequence of the nonuniformity of the filter. We compare the magnitude of the divergence of the turbulent stress tensor with that of the commutator errors and pay attention to the role of the explicit filter that is adopted. It is shown that the turbulent stress contributions and the commutator errors display the same scaling behavior on the filter width and its derivative. Correspondingly, the use of higher-order filters is shown not only to decrease the commutator errors but likewise the turbulent stresses are uniformly reduced with increasing filter order. In addition, we establish that skewness of the spatial filter has a strong influence on the magnitude of the commutator errors while leaving the turbulent stress contributions roughly unaltered. The analysis of the order of magnitude of the various terms provides an impression of the flow conditions and filter width nonuniformities which necessitate explicit commutator-error modeling next to the more traditional turbulent stress subgrid modeling. Some explicit models for the commutator errors are put forward and an a priori assessment of these commutator-error models in turbulent mixing layers is obtained. Generalized similarity models appear promising in this respect and display high correlation with the exact commutator errors.