The Smagorinsky Model -- Theory and Governing Equations

That's correct. It was proposed by Joseph Smagorinsky in 1963 for atmospheric circulation models and can be considered the origin of SGS models in LES. The concept is simple: it calculates the SGS eddy viscosity from the magnitude of the local strain rate and the filter width.

In the Smagorinsky model, the SGS eddy viscosity $\nu_{sgs}$ is defined as follows.

Here, $C_s$ is the Smagorinsky constant, $\Delta$ is the filter width, and $|\bar{S}|$ is the magnitude of the filtered strain rate tensor.

Good observation. In fact, this model can also be interpreted as the LES version of the mixing length theory. It has the same structure as Prandtl's mixing length theory in RANS, $\nu_t = l_m^2 |S|$, where the mixing length $l_m$ is replaced by the filter width $C_s \Delta$.

According to Lilly's theoretical analysis (1967), assuming isotropic turbulence and aligning it with Kolmogorov's energy spectrum $E(k) = C_K \varepsilon^{2/3} k^{-5/3}$, we get:

(where $C_K \approx 1.6$). However, in practice, values in the range $C_s = 0.1$ to $0.2$ are often used.

Exactly. While $C_s \approx 0.17$ is appropriate for isotropic turbulence, in shear flows it needs to be around $C_s \approx 0.1$ to avoid excessive dissipation. Near walls, an even smaller value is needed, and it's common to use the Van Driest damping function to make $C_s \to 0$ at the wall.

Here, $A^+ \approx 25$. This drawback of "needing to adjust $C_s$ according to the flow" was the motivation for developing the dynamic Smagorinsky model.

For each timestep and each cell, calculate the following.

1. Calculate the filtered velocity gradient $\partial \bar{u}_i / \partial x_j$

2. Calculate the strain rate tensor $\bar{S}_{ij}$

3. Calculate $|\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}}$

4. Calculate the filter width $\Delta$ (e.g., cube root of cell volume)

5. Compute the SGS eddy viscosity $\nu_{sgs} = (C_s \Delta)^2 |\bar{S}|$

6. Reflect it in the diffusion term of the momentum equation as the effective viscosity $\nu_{eff} = \nu + \nu_{sgs}$

No additional transport equations for the SGS model are required, so the implementation is very simple.

Correct. Representative filter width definitions are as follows.

For cells with high aspect ratios (e.g., prism layers near walls), the filter width definition significantly affects the results.

Write the following in constant/turbulenceProperties.

```

simulationType LES;

LES

{

LESModel Smagorinsky;

turbulence on;

printCoeffs on;

delta cubeRootVol;

SmagorinskyCoeffs

{

Ck 0.094;

Ce 1.048;

}

}

```

In OpenFOAM, it's defined with $C_k$ and $C_e$ instead of $C_s$, with the relation $C_s = C_k^{3/4}/\pi$.

In Fluent, select LES in the Viscous Model dialog and choose the Smagorinsky-Lilly Model as the SGS model. The Smagorinsky constant is set to $C_s = 0.1$ by default and can be changed as needed. Van Driest damping near walls is applied automatically, so no special settings are required.

The Smagorinsky model guarantees $\nu_{sgs} \geq 0$ at all times, making it very stable numerically. This is a major advantage, especially for beginners starting with LES, as it can be used with confidence. However, conversely, this also means it cannot represent inverse energy cascade (energy transfer from small to large scales) at all, which is a physical limitation.

Exactly. When using the Smagorinsky model in practice, if you carefully choose the value of $C_s$ and properly set wall damping, you can obtain sufficient accuracy for many industrial problems.