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Smagorinsky-Lilly model

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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}}


\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
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