When Does Eddy Viscosity Damp Subfilter Scales Sufficiently?

Large eddy simulation (LES) seeks to predict the dynamics of spatially filtered turbulent flows. The very essence is that the LES-solution contains only scales of size ≥Δ, where Δ denotes some user-chosen length scale. This property enables us to perform a LES when it is not feasible to compute the full, turbulent solution of the Navier-Stokes equations. Therefore, in case the large eddy simulation is based on an eddy viscosity model we determine the eddy viscosity such that any scales of size <Δ are dynamically insignificant. In this paper, we address the following two questions: how much eddy diffusion is needed to (a) balance the production of scales of size smaller than Δ; and (b) damp any disturbances having a scale of size smaller than Δ initially. From this we deduce that the eddy viscosity νe has to depend on the invariants $q = \frac{1}{2}\mathrm{tr}(S^{2})$ and $r= -\frac{1}{3}\mathrm{tr}(S^{3})$ of the (filtered) strain rate tensor S. The simplest model is then given by $\nu_{e} = \frac{3}{2}(\Delta/\pi)^{2} |r|/q$. This model is successfully tested for a turbulent channel flow (Re τ=590).

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