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Smagorinsky
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In the Smagorinsky model, :math:`k_{I}` is taken to be of order :math:`(I|S|)^{2}`. 
The subgrid viscosity is then

.. math::
    \nu_{\mathrm{SGS}}&=C_{\mathrm{S}}^{2}\left(\Delta x_{n}\right)^{2}\left(2S_{\alpha\beta}S_{\alpha\beta}\right)^{1/2}\\
    S_{\alpha\beta}S_{\alpha\beta}&=S_{xx}^{2}+S_{yy}^{2}+S_{zz}^{2}+2\left(S_{xy}^{2}+S_{xz}^{2}+S_{yz}^{2}\right)\\
    :label: smagorinsky

where :math:`C_{\mathrm{S}}` is the Smagorinsky constant, usually given a value of 0.17. 
Note that the bar symbol over the filtered rate-of-strain was dropped. 
This model has a number of shortcomings, the most important of which is that :math:`\nu_{SGS}` doesn't reduce to zero near a wall. 
Damping functions might be used to correct that issue.

.. However, the greatest advance from this model was the derivation of the dynamic Smagorinsky model (ref), because it adjusts the characteristic scale according to flow conditions. 
.. This method isn't going to be pursued since it requires an explicit filter operation (non-local) over the resolved variables.

By separating :math:`S_{\alpha\beta}` as

.. math::
    S_{\alpha\beta}=\frac{1}{2\rho c_{s}^{2}\Delta t \tau}Q_{\alpha}{\beta}
    :label: shearseparated

where :math:`Q_{\alpha\beta}` is

.. math::
    Q_{\alpha\beta}=\Pi_{\alpha\beta}^{\mathrm{neq}}+\frac{1}{2}\left(F_{\alpha}u_{\beta}+F_{\beta}u_{\alpha}\right)
    :label: qalphabeta

An explicit expression for the effective relaxation time :math:`\tau^{*}=\tau + \tau_{\mathrm{SGS}}` can be found as:

.. math::
    \tau^{*}=\frac{1}{2}\left(\tau+\sqrt{\tau^{2}+\frac{2(C_{\mathrm{S}}^{2}\sqrt{2Q_{\alpha\beta}Q_{\alpha\beta}})}{\Delta t \rho c_{s}^{4}}}\right)\\
    :label: explicit_tau

The time-updated value of :math:`\tau` is used to find the filtered macroscopic variables and to perform the collision operation.

.. footbibliography::