On Taylor-series expansions of residual stress

Abstract

Although subgrid-scale models of similarity type are insufficiently dissipative for practical applications to large-eddy simulation, in recently published a priori analyses, they perform remarkably well in the sense of correlating highly against exact residual stresses. Here, Taylor-series expansions of residual stress are exploited to explain the observed behavior and “success” of similarity models. Specifically, the first few terms of the exact residual stress τkl are obtained in (general) terms of the Taylor coefficients of the grid filter. Also, by expansion of the test filter, a similar expression results for the resolved turbulent stress tensor Lkl in terms of the Taylor coefficients of both the grid and test filters. Comparison of the expansions for τkl and Lkl yields the grid- and test-filter dependent value of the constant cL in the scale-similarity model of Liu et al. [J. Fluid Mech. 275, 83 (1994)]. Until recently, little attention has been given to issues related to the convergence of such expansions. To this end, we re-express the convergence criterion of Vasilyev et al. [J. Comput. Phys. 146, 82 (1998)] in terms of the transfer function and the cutoff wave number of the filter. As a rule of thumb, the less dissipative the filter (e.g., the higher the cutoff), the faster the rate of convergence.