\[ \begin{align}\begin{aligned}\newcommand{\b}[1]{\boldsymbol{#1}} \newcommand{\r}[1]{\mathrm{#1}} \newcommand{\bz}{\b{z}} \newcommand{\bu}{\b{u}} \newcommand{\bcdot}{\b{\cdot}} \newcommand{\d}{\partial}\\\newcommand{\p}{\, .} \newcommand{\c}{\, ,}\end{aligned}\end{align} \]

Constant Smagorinsky

In the first-order ‘constant Smagorinsky’ turbulence closure, the subgrid stress \(F^\bu_{ij}\) defined in terms of the resolved rate of strain tensor \(S_{ij} = \tfrac{1}{2} \left ( \d_i u_j + \d_j u_i \right )\) and an eddy viscosity \(\nu_e\):

\[F^\bu_{ij} = 2 \nu_e S_{ij} \p\]

In dedaLES, the eddy viscosity \(\nu_e\) is defined via a slight generalization of traditional constant Smagorinsky,

\[\nu_e = \left [ \Delta_{\r{const}}^2 + \left ( C \Delta \right )^2 \right ] | \b{S} | \, ,\]

where \(\Delta_{\r{const}}\) is a constant ‘filter width’, \(C\) is the Poincaré constant, and \(\Delta\) is a filter width defined by some multiple of the grid resolution, and thus dependent on position within the chosen grid in general. The invariant of the resolved strain tensor \(|\b{S}|\) is

\[| \b{S} | \equiv \sqrt{ 2 S_{ij} S_{ji} } \, .\]

Note that \(S_{ij}\) is symmetric, so that \(S_{ij} = S_{ji}\). The subgrid buoyancy flux is

\[\b{F}^b = -\kappa_e \nabla b \c\]

with \(\kappa_e = \nu_e / Pr_e\) for effective turbulent Prandtl number \(Pr_e\).