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.

Numerically induced high-pass dynamics in large-eddy simulation

“…We notice that the filter corresponding to a higher-order spatial discretization is itself a higher-order filter; with increasing order, the Fourier transform of the kernel is seen to stay closer to the identity operator for a wider range of wave numbers. 21,22,24 Each discretization method gives rise to a particular damping of the amplitude of individual modes, which induces a specific dynamic contribution in an actual large-eddy simulation based on this method. The role of the total filter is discussed further in, e.g., Refs.…”

“…3 A filter kernel is characterized by an externally specified length scale parameter , which defines the effective filter width ⌬; e.g., as 22 1…”

Lagrangian dynamics of commutator errors in large-eddy simulation

“…Most studies of the commutator error that were reported in literature relied on an analytical evaluation of the commutator error 5,6,9,12 or on a priori investigations using DNS data. 7 In this paper an explicit Lagrangian interpretation of the dynamics of commutator errors is provided which directly suggests a corresponding SFS model. We examine the effect of commutator errors on the transport of resolved kinetic energy.…”

“…We will formulate the explicit Lagrangian commutator-error model and test it against other commutator-error models ͑CE models͒ that are based on similarity assumptions instead. [6][7][8] The type of nonuniform filter in the definition of the commutator error explicitly affects the type of dynamic contributions of these terms. We will investigate both the commonly used symmetric filters as well as skewed filters.…”

“…In fact, a nonuniform filter width induces a number of additional closure terms, generally referred to as commutator errors. [5][6][7][8][9] These commutator errors express the noncommutation between filtering and differentiation. For a field f we may define the commutator error as…”