## WALE SGS model in OpenFOAM

Last Updated: May 6, 2019

The WALE (Wall-Adapting Local Eddy-viscosity) SGS model is one of the major SGS models. It is an algebraic eddy viscosity model (0-equation model) as with the Smagorinsky SGS model, but it has some excellent features that the Smagorinsky model does not have.

##### Implementation in OpenFOAM

In the WALE model, the subgrid scale eddy viscosity $$\nu_{sgs}$$ is evaluated as
\begin{equation}
\nu_{sgs} = C_{k} \Delta \sqrt{k_{sgs}}, \tag{1} \label{eq:nusgs}
\end{equation}
where $$C_{k}$$ is a model constant and $$k_{sgs}$$ is the subgrid scale kinetic energy. You can find its definition in this post.

The traceless symmetric part of the square of the velocity gradient tensor $$S^d$$ is calculated in the following function.
\begin{eqnarray}
S_{ij}^d = \frac{1}{2} \left( \frac{\partial \overline{u}_k}{\partial x_i}\frac{\partial \overline{u}_j}{\partial x_k} + \frac{\partial \overline{u}_k}{\partial x_j}\frac{\partial \overline{u}_i}{\partial x_k} \right) – \frac{1}{3} \delta_{ij} \frac{\partial \overline{u}_k}{\partial x_l}\frac{\partial \overline{u}_l}{\partial x_k}, \tag{2} \label{eq:sd}
\end{eqnarray}
where $$\delta_{ij}$$ is the Kronecker delta.

These tensor operations used above are summarized in the following post:

The subgrid scale kinetic energy $$k_{sgs}$$ is
\begin{equation}
k_{sgs} = \left( \frac{C_w^2 \Delta}{C_k} \right)^2 \frac{\left( S_{ij}^d S_{ij}^d \right)^3}{\left( \left( \overline{S}_{ij} \overline{S}_{ij} \right)^{5/2} + \left( S_{ij}^d S_{ij}^d \right)^{5/4} \right)^2}, \tag{3} \label{eq:ksgs}
\end{equation}
where
\begin{equation}
\overline{S}_{ij} = \frac{1}{2} \left( \frac{\partial \overline{u}_{i}}{\partial x_{j}} + \frac{\partial \overline{u}_{j}}{\partial x_{i}}\right), \tag{4} \label{eq:s}
\end{equation}
is the resolved-scale strain rate tensor.

Finally, we can get the following expression by substituting Eq. \eqref{eq:ksgs} into Eq. \eqref{eq:nusgs}:
\begin{eqnarray}
\nu_{sgs} = \left( C_w \Delta \right)^2 \frac{\left( S_{ij}^d S_{ij}^d \right)^{3/2}}{\left( \overline{S}_{ij} \overline{S}_{ij} \right)^{5/2} + \left( S_{ij}^d S_{ij}^d \right)^{5/4}}, \tag{5} \label{eq:nusgs2}
\end{eqnarray}
It is the same as Eq. (13) in .

##### Features
• Algebraic eddy viscosity model (0-equation model)
• The rotation rate is taken into account in the calculation of $$\nu_{sgs}$$
• To be able to handle transition
• Damping is Not necessary for $$\nu_{sgs}$$ in the near-wall region
##### References

Final expression of the SGS eddy viscosity is relatively simple and the implementation into OpenFOAM is not so complicated but the process of deriving the expression described in  is not easy to understand. I want to develop my SGS model someday 🙂