WALE SGS model in OpenFOAM


OpenFOAM Version: OpenFOAM-dev, OpenFOAM-1612+

Implementation in OpenFOAM

In the WALE (Wall-Adapting Local Eddy-viscosity) model, the subgrid scale viscosity is computed 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.

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.

The subgrid-scale kinetic energy 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 🙂

I will upload some animations next year.

Author: fumiya

CFD engineer in Japan

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