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.

Smagorinsky SGS model in OpenFOAM

In this blog post, I give a description of the implementation of the Smagorinsky subgrid scale model in OpenFOAM.

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:

Tensor Operations in OpenFOAM

In this blog post, I will pick out some typical tensor operations and give brief explanations of them with some usage examples in OpenFOAM.

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 [1].

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

[1] F. Nicoud and F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62(3), 183-200, 1999.

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 [1] is not easy to understand. I want to develop my SGS model someday 🙂

Author: fumiya

CFD engineer in Japan View all posts by fumiya