Smagorinsky-Lilly model

The Smagorinsky model could be summarised as:

$\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\left( {C_s \Delta } \right)^2 \left| {\bar S} \right|S_{ij}$

In the Smagorinsky-Lilly model, the eddy viscosity is modeled by

$\mu _{sgs} = \rho \left( {C_s \Delta } \right)^2 \left| {\bar S} \right|$

Where the filter width is usually taken to be

$\Delta = \left( \mbox{Volume} \right)^{\frac{1}{3}}$

and

$\bar S = \sqrt {2S_{ij} S_{ij} }$

The effective viscosity is calculated from

$\mu _{eff} = \mu _{mol} + \mu _{sgs}$

The Smagorinsky constant usually has the value:

$C_s = 0.1 - 0.2$