OpenFOAM SGS model

SGS kinetic energy $k=\frac{1}{2}(\overline{{\widetilde{v}}^2}-{\widetilde{v}}^2)$ , where $\widetilde{v}$ is the filtered density weighted velocity.
The following assumptions for the SGS density weighted stress tensor $\mathrm{B}$ and the filtered deviatoric part of the rate of strain tensor $\widetilde{D}_D$ are used:
$\mathrm{B}=\frac{2}{3}\overline{\rho}k\mathrm{I}-2\mu_B\widetilde{D}_D \tag{1}$ $\widetilde{D}_D=\widetilde{D}-\frac{1}{3}(tr\widetilde{D}\mathrm{I}) \tag{2}$ $\widetilde{D}=\frac{1}{2}(grad\widetilde{v}+grad\widetilde{v}^T) \tag{3}$ $\mu_B=c_k\overline{\rho}\sqrt{k}\vartriangle \tag{4}$ where $\mathrm{I}$ is the unit tensor, $\overline{\rho}$ is the filtered density, $\mu_B$ is the SGS viscosity and $\vartriangle$ represents the top-hat filter with a characteristic filter width estimated as the cubic root of the CV volume. An exact balance equation for $k$ can be derived, but some terms must be modeled. Instead, following Fureby this balance equation can be replaced by the a posteriori modeled equation:
$\partial_t(\overline{\rho}k)+div(\overline{\rho}\widetilde{v}k)=F_p+F_d-F_\varepsilon \tag{5}$ $F_p=-B\cdot D,\ F_d=div((\mu_B+\mu)grad\ k),\ F_\varepsilon=c_\varepsilon\overline{\rho}k^{\frac{3}{2}}/\vartriangle$ where $F_p$ is the production, $F_d$ diffusion and $F_\varepsilon$ dissipation terms, respectively.
The conventional Smagorinsky SGS model can be recovered from Eq. 5 by assuming local equilibrium, $F_p=F_\varepsilon$ . Thus, the SGS kinetic energy can be computed from the following relation: $B\cdot \widetilde{D}+c_\varepsilon\overline{\rho}k^{3/2}/\vartriangle=0 \tag{6}$
Using Eq. 1 and introducing the coefficients $a=\frac{c_\varepsilon}{\vartriangle},\ b=\frac{2}{3}tr\widetilde{D},\ c=-2c_k\vartriangle\widetilde{D}_D:\widetilde{D} \tag{7}$ The relation in Eq. 6 can be reformulated by the quadratic equation to the relation for $k$ $k=(\frac{-b+\sqrt{b^2-4ac}}{2a})^2\tag{8}$ The models constant are: $c_k=0.02$ and $c_\varepsilon=1.048$ . The relationship between the classical $C_s$ constant and the default constants $C_k$ and $C_\varepsilon$ from the Smagorinsky model implementation in OpenFOAM is $C_s=(c^3_k/c_\varepsilon)^{1/4}$ , which leads to a value $C_s=0.053$ . This value is slightly lower than the minimum conventional limit $C_s=0.065$ . It is worth noting that this relation can be recovered from the classical assumption for the eddy-viscosity $\mu_B=(C_s\vartriangle)^2\overline{\rho}\|\widetilde{D}\|$ and Eq. 4.
The dynamic model for the $k$ equation can be derived using the Germano identity $\mathrm{L}$ with another filter kernel of width $\overline{\vartriangle}=2\vartriangle$ . Again, one can find the theoretical background in the studies performed by Fureby. Here, we will concentrate on the model implementation only.
The model coefficient $c_k$ can not be removed from filtering and hence, a variational formulation is used to evaluate this: $c_k=\frac{\langle L_D\cdot M\rangle}{\langle M\cdot M\rangle}\tag{9}$ where $M=\vartriangle(\overline{k^{1/2}\widetilde{D}}-2(K+\overline{k})^{1/2}\overline{\widetilde{D}})$ , $L_D=(\overline{\widetilde{v}^2}-\overline{\widetilde{v}}^2)_D$ and $K=\frac{1}{2}(\overline{\|\widetilde{v}\|^2}-\|\overline{\widetilde{v}}\|^2)$ .
The second coefficient $c_\varepsilon$ is defined as $c_\varepsilon=\frac{\langle\zeta\cdot MM\rangle}{\langle MM\cdot MM\rangle}\tag{10}$ where $\zeta=2\vartriangle c_k(\overline{k^{1/2}\|\widetilde{D}\|^2}-2(K+\overline{k})^{1/2}\|\overline{\widetilde{D}}\|^2)$ and $MM=(K+\overline{k})^{3/2}/2\vartriangle-\overline{k^{3/2}}/\vartriangle$ .

Reference:Lysenko, D.A., I.S., E., K. E., R., 2012. Large-Eddy Simulation of the Flow Over a Circular Cylinder at Reynolds Number 3900 Using the OpenFOAM Toolbox. Flow Turbulence & Combustion 89, 491–518.