ASSESSMENT OF LARGE-EDDY SIMULATION BASED ON RELAXATION FILTERING: APPLICATION TO THE TAYLOR-GREEN VORTEX

Abstract

Results obtained for a Taylor-Green vortex at a Reynolds number of 3000, using Large-Eddy Simulation (LES) based on Relaxation Filtering (RF), are presented in order to assess the quality of the RF-LES methodology. The RF is applied every time step to the velocity components, using a standard filters of orders k ≥ 4 at a fixed strength σ , to relax subgrid energy from scales at wave numbers close to the grid cut-off wave number. Various combinations of k and σ are considered, for k ranging from 4 to 14 and σ from 0.15 to 1. Error landscapes are obtained by comparing the 643 LES results, filtered in post-processing to an effective resolution of four points per wavelength, to 3843 Direct Numerical Simulation data, filtered at identical resolution. For filters of order k ≤ 6, the LES accuracy is found to be rather poor and varies significantly with the filtering strength σ . However, for higher order filters, i.e. for k > 6, the accuracy is good and nearly independent of the strength σ . INTRODUCTION In Large-Eddy Simulation (LES) of a turbulent flow, the most significant scales of motion, i.e. the largest and most significant scales of motion in the energy-containing range and inertial range, are resolved in order to obtain a statistically sufficiently accurate prediction of the flow. Since the small scales in the dissipation range are not resolved, their effects must be accounted for by an artificial dissipation mechanism, in order to avoid a pile-up of energy at the cut-off wavenumber imposed by the computational grid. This is usually done by replacing the residual stress tensor in the filtered Navier-Stokes equations with an eddy-viscosity model, or by applying dissipative numerical discretization schemes for the convective terms as in Implicit LES methods. We refer to the reviews by Lesieur and Metais (1996), Grinstein and Fureby (2002), and Domaradzki (2010), and to the books by Geurts (2004) and Sagaut (2005). The amount of dissipation, as well as its spectral distribution, may, however, be difficult to control in these methods as pointed out in Domaradzki et al. (2000, 2002, 2003) and Bogey and Bailly (2005, 2006b). This has led to the development of alternative LES methodologies relying on high-order dissipation mechanisms, such as hyper-viscosity models (Passot and Pouquet 1988, Dantinne et al. 1998) or the relaxation term in the Approximate Deconvolution Model (Stolz et al. 2001). In recent years, an LES approach based on a Relaxation Filtering (RF) to account for the subgrid dissipation, has been proposed, and applied successfully to various flow configurations by Visbal and Rizzetta (2002), Rizzetta et al. (2003), Mathew et al. (2003) and Bogey et al. (2006a, 2009, 2011), among others. In order to relax the turbulent energy from the small scales at wave numbers close to the grid cut-off wavenumber, a low-pass filter is applied to the components of the velocity field, every nth time step in each Cartesian direction, as follows ũ(x, t) = u(x, t)−σ n ∑ j=−n d ju(x j, t) (1) where ũ and u denote respectively the filtered and unfiltered variables, and d j represents the weighting coefficients that determine the dissipative contribution of the (2n+1)-point symmetric filter. The filtering strength σ is between 0 and 1. To obtain the necessary energy dissipation, criteria could be developed to adjust dynamically the filtering frequency and strength to the flow features, e.g. in Tantikul and Domaradzki (2010). For practical reasons, however, the filtering is usually applied every time step at a constant strength σ (typically σ ≃ 1). The results of the RF procedure depend in this case on the shape of the filter and the filtering strength. Since the selected filter must provide sufficient dissipation to the smallest resolved scales while leaving the largest scales mostly unaffected, the influence of the filter-shape, determined by the