\[ \begin{align}\begin{aligned}\newcommand{\b}[1]{\boldsymbol{#1}}
\newcommand{\r}[1]{\mathrm{#1}}
\newcommand{\bz}{\b{z}}
\newcommand{\bu}{\b{u}}
\newcommand{\bcdot}{\b{\cdot}}
\newcommand{\d}{\partial}\\\newcommand{\p}{\, .}
\newcommand{\c}{\, ,}\end{aligned}\end{align} \]
Modified constant Smagorinsky¶
The modified constant Smagorinsky closure attempts to improve the Constant Smagorinsky closure in the presence of buoyancy gradients by multiplying the Smagorinsky subgrid stress by a ‘buoyancy factor’ \(\lambda\) such that
\[F^\bu_{ij} = \lambda \, \nu_e S_{ij} \p\]
The eddy viscosity \(\nu_e\) and strain tensor \(S_{ij}\) are defined as in Constant Smagorinsky. The buoyancy factor is
\[\begin{split}\lambda = \left \{ \begin{matrix}
1 & \quad \r{for} \quad N^2 \le 0 \\
\max \left [ 0, \sqrt{ 1 - N^2 / Pr | S|^2 } \right ] & \quad \r{for} \quad N^2 > 0
\end{matrix} \right . \c\end{split}\]
where \(Pr\) is the turbulent Prandtl number. The subgrid tracer flux in Constant Smagorinsky is not modified.
When a flow is affected by stratification such that the Richardson-like number \(N^2/|S|^2\) is greater than zero, the buoyancy factor \(\lambda\) acts to reduce the subgrid stress.