An eddy-viscosity model based on the invariants of the rate-of-strain tensor

Abstract

Turbulence can generally not be computed directly from the Navier-Stokes equations, because the flow possess far too many scales of motion. Therefore numerical simulations of turbulence have to resort to models of the small scales of motion for which numerical resolution is not available. Large-eddy simulation (LES) seeks to predict the dynamics of spatially filtered flows. If a spatial filter $u \mapsto \overline{u}$ is applied to the (incompressible) Navier-Stokes equations an expression depending on both full velocity field u and the filtered field $\overline{u}$ results, due to the nonlinearity. The dependence on u can be removed by adopting an eddy-viscosity model, for instance (Sagaut, 2001). The governing equation is then given by

$$\partial_t v + (v\cdot\nabla)v +\nabla p = 2\nabla \cdot \left( (\nu+\nu_e)\,S(v)\right)$$

where ν and ν _e denote the fluid viscosity and the eddy viscosity, respectively; S(v) is the symmetric part of the velocity gradient (the rate-of-strain tensor). The solution v is supposed to approximate the filtered Navier-Stokes solution $\overline{u}$.