Dynamic Model for LES Without Test Filtering: Quantifying the Accuracy of Taylor Series Approximations

Abstract

The dynamic model for large-eddy simulation (LES) of turbulent flows requires test filtering the resolved velocity fields in order to determine model coefficients. However, test filtering is costly to perform in LES of complex geometry flows, especially on unstructured grids. The objective of this work is to develop and test an approximate but less costly dynamic procedure which does not require test filtering. The proposed method is based on Taylor series expansions of the resolved velocity fields. Accuracy is governed by the derivative schemes used in the calculation and the number of terms considered in the approximation to the test filtering operator. The expansion is developed up to fourth order, and results are tested a priori based on direct numerical simulation data of forced isotropic turbulence in the context of the dynamic Smagorinsky model. The tests compare the dynamic Smagorinsky coefficient obtained from filtering with those obtained from application of the Taylor series expansion. They show that the expansion up to second order provides a reasonable approximation to the true dynamic coefficient (with errors on the order of about 5% for c 2 s ), but that including higher-order terms does not necessarily lead to improvements in the results due to inherent limitations in accurately evaluating high-order derivatives. A posteriori tests using the Taylor series approximation in LES of forced isotropic turbulence and channel flow confirm that the Taylor series approximation yields accurate results for the dynamic coefficient. Moreover, the simulations are stable and yield accurate resolved velocity statistics.

A dynamic Smagorinsky model with dynamic determination of the filter width ratio

“…Furthermore, although results may be independent of the test filter shape, they may not be optimal. 12 Additionally, results can be sensitive to the filter width ratio as will be shown later and as is shown by Chester et al, 13 thus motivating the need for its accurate determination. This sensitivity issue deserves special attention specifically for cases when LES grids are coarsened, leading to stronger subgrid-scale model dissipations.…”

Subgrid Modeling Approaches for Scalar Transport and Mixing in LES of Turbulent Shear Flows

“…One possible solution is to approximate the test filter with a derivative-based method, thus eliminating the need for explicit test filtering. This approach, developed by Chester et al [12], approximates test filtering with Taylor series expansions of resolved-scale quantities. Retaining terms up to second order in the expansion, a test-filtered quantity, f, is expressed in terms of the resolved-scale (grid-scale) quantity, f , as:…”

Large-Eddy Simulation of a Particle-Laden Turbulent Channel Flow

“…Next we simplify the dynamic model in line with the approaches described by Pope (2000, p. 623) and Chester et al (2001). Thus, in the following Taylor expansions are used to approximate the tensors L ij and M ij .…”