Dynamic Smagorinsky Subgrid Scale Model

The Dynamic Smagorinsky Subgrid Scale model has the same basic form as the Smagorinsky model, however instead of using a single user-defined $C_{s}$ coefficient, the model computes a local time-varying coefficient by test-filtering the flow field at a length scale greater than the grid length scale. This dynamic variation of the constant gives the model its name, and allows it to compute the correct result for wall-bounded flows without the use of damping functions.

Dynamic Smagorinsky Subgrid Scale Viscosity

The calculation of the turbulent viscosity in the Dynamic Smagorinsky SGS model ([352], [353]) follows the same equation as for the standard Smagorinsky SGS model:

Figure 1. EQUATION_DISPLAY

μ_{t} = ρ Δ^{2} S

(1391)

where:

$ρ$ is the density.
$S$ is given by Eqn. (1129) and computed from the resolved velocity field $\tilde{v}$ .

However, the parameter $Δ^{2}$ is computed dynamically as a function of a number of test-filtered variables:

Figure 2. EQUATION_DISPLAY

Δ^{2} = {C_{s}^{2} V}^{2 / 3}

(1392)

where $V$ is the cell volume.

If $\tilde{ϕ}$ is the grid-filtered LES variable in a cell, then the corresponding test-filtered value $\hat{\tilde{ϕ}}$ is given as:

Figure 3. EQUATION_DISPLAY

\hat{\tilde{ϕ}} = \frac{1}{\sum_{n = 0}^{N} V_{n}} \sum_{n = 1}^{N} {\tilde{ϕ}}_{n} V_{n}

(1393)

where the subscripts denote cell number, cell 0 is the current cell, and cells 1 to N are the face neighbors of cell 0.

Continuing with the superscript “~” notation to denote test-filtered variables, the dynamic parameter $C_{s}^{2}$ is computed as follows:

Figure 4. EQUATION_DISPLAY

L_{i j} = \tilde{u_{i}} \hat{\tilde{u_{j}}} - \hat{\tilde{u_{i}}} \hat{\tilde{u_{j}}}

(1394)

Figure 5. EQUATION_DISPLAY

M_{i j} = 2 {\tilde{L}}^{2} (| \tilde{S} | {\hat{\tilde{S}}}_{i j} - \frac{{\hat{L}}^{2}}{{\tilde{L}}^{2}} | \hat{\tilde{S}} | {\hat{\tilde{S}}}_{i j})

(1395)

where:

$\tilde{L}$ is the grid filter length.
$\hat{L} / \tilde{L}$ is the filter width ratio and a Model Coefficients.

The tensor $M_{i j}$ arises during the derivation of the model by subtracting the Smagorinsky subgrid stress tensor at the grid filter scale from the same tensor at the test filter scale. Finally, $C_{s}^{2}$ is computed as:

Figure 6. EQUATION_DISPLAY

C_{s}^{2} = \frac{〈 L_{i j} M_{i j} 〉}{〈 M_{i j} M_{i j} 〉}

(1396)

where the brackets <> denote averaging. Since, in most simulations, there is no homogeneous direction over which to average, the same approach is used as for the filtering operation. This averaging is optional; you can activate or deactivate it in the Average Parameter expert property of the model in the GUI.

To increase stability in the calculation, the values of $C_{s}^{2}$ are clipped using minimum and maximum values. See Model Coefficients.

Model Coefficients


$\hat{L} / \tilde{L}$	${C_{s}^{2}}_{\min}$	${C_{s}^{2}}_{\max}$
2	0	1000000