Smagorinsky#
In the Smagorinsky model, \(k_{I}\) is taken to be of order \((I|S|)^{2}\). The subgrid viscosity is then
(1)#\[\begin{split}\nu_{\mathrm{SGS}}&=C_{\mathrm{S}}^{2}\left(\Delta x_{n}\right)^{2}\left(2S_{\alpha\beta}S_{\alpha\beta}\right)^{1/2}\\
S_{\alpha\beta}S_{\alpha\beta}&=S_{xx}^{2}+S_{yy}^{2}+S_{zz}^{2}+2\left(S_{xy}^{2}+S_{xz}^{2}+S_{yz}^{2}\right)\\\end{split}\]
where \(C_{\mathrm{S}}\) 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 \(\nu_{SGS}\) doesn’t reduce to zero near a wall. Damping functions might be used to correct that issue.
By separating \(S_{\alpha\beta}\) as
(2)#\[S_{\alpha\beta}=\frac{1}{2\rho c_{s}^{2}\Delta t \tau}Q_{\alpha}{\beta}\]
where \(Q_{\alpha\beta}\) is
(3)#\[Q_{\alpha\beta}=\Pi_{\alpha\beta}^{\mathrm{neq}}+\frac{1}{2}\left(F_{\alpha}u_{\beta}+F_{\beta}u_{\alpha}\right)\]
An explicit expression for the effective relaxation time \(\tau^{*}=\tau + \tau_{\mathrm{SGS}}\) can be found as:
(4)#\[\begin{split}\tau^{*}=\frac{1}{2}\left(\tau+\sqrt{\tau^{2}+\frac{2(C_{\mathrm{S}}^{2}\sqrt{2Q_{\alpha\beta}Q_{\alpha\beta}})}{\Delta t \rho c_{s}^{4}}}\right)\\\end{split}\]
The time-updated value of \(\tau\) is used to find the filtered macroscopic variables and to perform the collision operation.