A LES-Langevin model for turbulence

We propose a new model of turbulence for use in large-eddy simulations (LES). The turbulent force, represented here by the turbulent Lamb vector, is divided in two contributions. The contribution including only subfilter fields is deterministically modeled through a classical eddy-viscosity. The other contribution including both filtered and subfilter scales is dynamically computed as solution of a generalized (stochastic) Langevin equation. This equation is derived using Rapid Distortion Theory (RDT) applied to the subfilter scales. The general friction operator therefore includes both advection and stretching by the resolved scale. The stochastic noise is derived as the sum of a contribution from the energy cascade and a contribution from the pressure. The LES model is thus made of an equation for the resolved scale, including the turbulent force, and a generalized Langevin equation integrated on a twice-finer grid. We compare the full model with several approximations. In the first one, the friction operator of the Langevin equation is simply replaced by an empirical constant, of the order of the resolved scale correlation time. In the second approximation, the integration is replaced by a condition of instantaneous adjustment to the stochastic force. In this approximation, our model becomes equivalent to the velocity-estimation model of Domaradzki et al. [7, 5, 8]. In the isotropic, homogeneous situations we study, both approximations provide satisfactory results, at a reduced computational cost. The model is finally validated by comparison to DNS and is tested against classical LES models for isotropic homogeneous turbulence, based on eddy viscosity. We show that even in this situation, where no walls are present, our inclusion of backscatter through the Langevin equation results in a better description of the flow.