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

Sub-grid models

dedaLES impements a variety of models that approximate the impact of unresolved, turbulent, ‘subgrid’ stress and tracer flux on the evolution of the resolved momentum and tracer fields. These models are often called ‘subgrid closures’.

The models that dedaLES implements are ‘eddy viscosity’ and ‘eddy diffusivity’ models, because the subgrid stress and tracer flux are assumed proportional to the resolved rate of strain and tracer gradients. The constants of proportionality are the eddy viscosity and diffusivity.

The subgrid stress tensor \(F^{\bu}_{ij}\) is thus written

\[F^{\bu}_{ij} = 2 \nu_e S_{ij} \c\]

where \(\nu_e\) is the eddy viscosity, and

\[S_{ij} = \tfrac{1}{2} \left ( \d_i u_j + \d_j u_i \right ) \c\]

is the rate of strain tensor, often abbreviated as the ‘strain tensor’. The subgrid flux of a tracer \(\theta\) is similarly

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

where \(\bnabla \theta\) is the resolved tracer gradient and \(\kappa_e\) is the eddy diffusivity. The differences between models lies entirely in how \(\nu_e\) is calculated.

Implemented closures

The following closure schemes for the subgrid-scale turbulent stress and tracer flux are implemented in dedaLES:

• Constant Smagorinsky
  • References
• Modified constant Smagorinsky
  • References
• Anisotropic minimum dissipation
  • Basic form
  • The eddy viscosity predictor
  • The eddy diffusivity predictor
  • References