Smagorinsky Subgrid Scale Model

The Smagorinsky Subgrid Scale model is the original and most basic subgrid scale model, upon which many more advanced models are based. This model uses a mixing length hypothesis to model the subgrid scale stresses.

One shortcoming of the Smagorinsky Subgrid Scale model is that it contains a model coefficient $C_{s}$ that is not universal and depends on the local flow conditions. Additionally, a damping function is needed for proper results in wall-bounded flows. The Van Driest damping function is a non-local operation, requiring the use of a KD-tree data structure to communicate wall information to the interior. In simulations with very complex geometries, this KD-tree must be broadcast to each parallel node, resulting in a memory and communication overhead. An update policy can be used to improve the communication overhead, but not the memory overhead. Therefore, this model should be avoided for large-scale computations on complex geometries. It is, however, useful for reference simulations, since it is the original algebraic subgrid scale model formulation.

Smagorinsky Subgrid Scale Viscosity

The Smagorinsky Subgrid Scale model [355] provides the following mixing-length type formula for the subgrid scale viscosity:

Figure 1. EQUATION_DISPLAY

μ_{t} = ρ Δ^{2} S

(1387)

where:

$ρ$ is the density.
$Δ$ is the length scale or grid filter width.
$S$ is given by Eqn. (1129) and computed from the resolved velocity field $\tilde{v}$ .

The length scale $Δ$ is directly related to the cell volume $V$ and the wall distance $d$ as follows:

Figure 2. EQUATION_DISPLAY

Δ = {\begin{array}{l} {{f_{v} C}_{s} V}^{1 / 3} & if length scale limit not applied \\ f_{v} min (κ d, {C_{s} V}^{1 / 3}) & if  length scale limit is applied \end{array}

(1388)

where:

$f_{v}$ is the Van Driest damping function.
$C_{s}$ is a Model Coefficient.
$κ$ is the von Karman constant (see Model Coefficients).

The Van Driest damping function $f_{v}$ computed as follows:

Figure 3. EQUATION_DISPLAY

f_{v} = {\begin{array}{l} \begin{matrix} 1 & no damping \\ 1 - exp (- \frac{y^{+}}{A}) & standard \\ \sqrt{1 - exp {(- \frac{y^{+}}{A})}^{3}} & modified \end{matrix} \end{array}

(1389)

where A is a Model Coefficient.

$y^{+}$ is the dimensionless wall distance defined as:

y^{+} = \frac{u^{*} d}{ν}

(1390)

where:

$u^{*}$ is the friction velocity computed from the instantaneous wall shear stress at the nearest wall face.
$d$ is the distance to the wall.
$ν$ is the kinematic viscosity.

The modified version of the Van Driest damping function [357] is attributed to Piomelli and co-workers [351].

Model Coefficients


$C_{s}$	$κ$	$A$
0.1	0.41	25

The model coefficient

C_{s}

is not universal. Typical values of

C_{s}

reported in the literature range from 0.165 for homogeneous isotropic decaying turbulence to 0.07 for channel flows. The default value of

C_{s} = 0.1

is a compromise between these two extremes.