CFD Online Discussion Forums - Relashionship between subgrid-scale kinetic energy and subgrid-scale iscosity

Hi everybody.

Inside a paper I found this equation:
$k_{sgs}=\left( \frac{\nu_{sgs}}{C_K \Delta} \right)^2$
This was applied to a LES case using the static Smagorinsky turbulence model.

I tried to understand where this come from, but with no success.
I found THIS wiki page, but honestly it didn't help me too much. I even started to fear that this applies to a completely different turbulence model, rather than to Smagorinsky one.

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Here below I report some though of mine.

I think that it's correct to write that the $k_{sgs}$ is
$k_{sgs}=k-k_R$
where

is the resolved kinetic energy. Now
$k=\frac{1}{2} \overline{{u_i^{'}}^2}$
where

is the time averaging operator, and
$k_{sgs}=\frac{1}{2} \overline{\widetilde{{u_i^{'}}}^2}$
where $\widetilde{}$ is the LES space-filtering operator.

In this case the wiki equation seams wrong to me, since in LES should be
$\overline{\widetilde{{u_i^{'}}}^2} \neq \overline{{u_i^{'}}}^2$
as time-averaging and space-filtering are different operations.

One more thing that I cannot explain myself is that in the Boussinesque hypothesis that relates $\tau_{ij}$ to $\nu_T$ the rate-of-strain tensor appears and I cannot figure how to remove it or relate it to the $k_{sgs}$ .

Any hint would be welcome. Thank you very much.

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EDIT:
I found this equation inside a chapter on the Fluent guide http://www.arc.vt.edu/ansys_help/flu...gs_models.html (last sub-chapter in the link). This refers to a paper by Kim & Menon (http://arc.aiaa.org/doi/pdf/10.2514/6.1997-210). This however seams to point in the direction that this equation applies to a different turbulence model, rather than to the Smagorinsky one.