Large eddy simulation (LES)

Different models for the subgrid stresses are available. Due to the spatial filtering, the models below all use the cell size

\begin{equation} \Delta = V^{\frac 1 3} \end{equation}

determined by the cell volume \(V\).

Smagorinsky model

The Smagorinsky model is implemented according to [51] . It relates the subgrid eddy viscosity to the norm of the strain tensor via

\begin{equation} \mu_T = \rhobar (C_s \Delta)^2 \sqrt{2S_{ij} S_{ij}}. \end{equation}

The constant is given by

\begin{equation} C_s = 0.17. \end{equation}

Wall adaptive local eddy viscosity (WALE) model

The WALE model is implemented according to [39] . Like the Smagorinsky model it determines an eddy viscosity locally from resolved quantities.

\begin{equation} \mu_T = \rhobar (C_w \Delta)^2 \frac{(S_{ij}^d S_{ij}^d)^{\frac 3 2}}{(S_{ij} S_{ij})^{\frac 5 2} + (S_{ij}^d S_{ij}^d)^{\frac 5 4}} \end{equation}

The WALE tensor is given by

\begin{equation} S_{ij}^d = S_{ik} S_{kj} + W_{ik} W_{kj} - \frac 1 3 \delta_{ij} (S_{nm} S_{nm} - W_{nm} W_{nm}) \end{equation}

and the constant is

\begin{equation} C_w = 0.5587. \end{equation}

This constant is computed from the relation

\begin{equation} C_w^2 = \alpha C_s^2 \end{equation}

with \(C_s = 0.17\) as in the Smagorinsky model. The factor \(\alpha\) is the average of Table I in [39] .