Langevin models of turbulence: Renormalization group, distant interaction algorithms or rapid distortion theory?

Abstract

A new dynamical turbulence model is validated by comparisons of its numerical simulations with fully resolved, direct numerical simulations (DNS) of the Navier–Stokes equations in three-dimensional, isotropic, homogeneous conditions. In this model the small-scale velocities are computed using a Langevin, linear, inhomogeneous, stochastic equation that is derived from a quasi-linear approximation of the Navier–Stokes equations, in the spirit of rapid distortion theory (RDT). The values of the turbulent viscosity involved in our Langevin model are compared with a theoretical prescription based on the renormalization group and the distant interaction algorithms (DSTA) model. We show that the empirical turbulent viscosities derived from simulations of the Langevin model are in good quantitative agreement with the DSTA predictions. Finally, Langevin simulations are compared with DNS and large eddy simulations based on the eddy-damped quasi-normal Markovian closure. The Langevin RDT model is able to reproduce the correct spectrum shape, intermittency statistics, and coherent flow structures for both the resolved and the largest sub-grid scales. It also predicts the evolution of the resolved scales better than the alternative models.