Sigma Model (LES Subgrid-Scale Model)
Theory: eddy viscosity from singular values
Overview
Using the Smagorinsky model in LES, viscosity grew even in regions that should be laminar, and transition was delayed.
Smagorinsky sets eddy viscosity from strain-rate magnitude alone, so it adds viscosity wherever there's a velocity gradient, turbulent or not — even in simple shear. The sigma model, created by Nicoud and co-workers in 2011 to address this, uses a combination of the three singular values of the velocity gradient tensor. It is designed to vanish automatically for gradient shapes that can't be turbulence: simple shear, solid rotation and two-dimensional flow.
Model equations
$\sigma_1 \ge \sigma_2 \ge \sigma_3 \ge 0$ are the singular values of the velocity gradient tensor and $\Delta$ the filter width. In 2D flow $\sigma_3 = 0$; in solid rotation or isotropic expansion all three are equal; in axisymmetric strain $\sigma_2 = \sigma_3$ — so the numerator vanishes in each case.
The Smagorinsky form used for comparison, set by strain-rate magnitude $S_{ij}$ alone.
Sixty years of trimming spurious viscosity
The first subgrid model was built in 1963 by meteorologist Smagorinsky for general circulation calculations of the atmosphere. Simple and robust, it gave too much viscosity near walls and in laminar regions. In the 1990s Germano and colleagues' dynamic procedure set the coefficient locally; WALE (1999) made viscosity decay correctly near walls; Vreman (2004) kept computation simple while vanishing in shear. The 2011 sigma model expresses a form satisfying these 'should vanish here' conditions together, using the clear quantity of singular values.
Worked example: comparison on basic flows
For basic flows with 1 mm filter width and velocity gradient magnitude 100 1/s, eddy viscosity (m²/s) from four models was computed. Coefficients: Smagorinsky 0.17, WALE 0.325, Vreman 2.5×0.17², sigma 1.35.
| Flow | Smagorinsky | WALE | Vreman | Sigma |
|---|---|---|---|---|
| Pure shear | 2.9×10⁻⁶ | 0 | 0 | 0 |
| Solid rotation | 0 | 4.8×10⁻⁶ | 2.6×10⁻⁶ | 0 |
| Isotropic expansion | 2.4×10⁻⁶ | 0 | 2.4×10⁻⁶ | 0 |
| Axisymmetric strain | 5.0×10⁻⁶ | 0.8×10⁻⁶ | 4.4×10⁻⁶ | 0 |
| Plane strain (2D) | 5.8×10⁻⁶ | 0.9×10⁻⁶ | 5.1×10⁻⁶ | 0 |
| General 3D gradient | 4.8×10⁻⁶ | 1.1×10⁻⁶ | 4.5×10⁻⁶ | 2.5×10⁻⁶ |
Is Smagorinsky's 2.9×10⁻⁶ in pure shear large?
Air's kinematic viscosity is 1.5×10⁻⁵ m²/s, so that's an extra 19%. A laminar boundary layer that much more viscous won't let small disturbances grow, delaying transition. In plane strain it's 39%. Sigma is zero for every 2D-type gradient and adds viscosity only when three-dimensional fluctuations appear. WALE and Vreman don't vanish in solid rotation, so they add viscosity in vortex cores. The table shows clearly which model adds spurious viscosity where.
Example 2: near-wall behaviour
Near a no-slip wall, the power of wall distance y with which eddy viscosity decays was computed from the velocity gradient form. Turbulent eddy viscosity correctly decays as y cubed.
| Model | Near-wall decay | Verdict |
|---|---|---|
| Smagorinsky | y⁰ | No decay (needs damping) |
| Vreman | y¹ | Decays, but slowly |
| WALE | y³ | Correct |
| Sigma | y³ | Correct |
Practical guidance
- For transition or flows with laminar regions, prefer sigma or WALE.
- For persistent vortex cores (tip vortices, wakes), sigma has the edge.
- The 1.35 coefficient is a guideline; the optimum depends on grid and numerics. Check on a reference flow such as channel flow.
- Near walls, refine to y⁺ ≈ 1 or combine with a wall model.
- Comparing results across models is a good way to gauge SGS model influence.
“The tip vortex vanishes too soon”
An LES tracking the tip vortex behind a wing saw it weaken far sooner than in experiments. The vortex core is nearly solid rotation with little turbulence, but the model used added eddy viscosity even in solid rotation, gradually eroding the core's spin. Switching to sigma made eddy viscosity nearly zero at the core, and the vortex persisted downstream. When vortices are the main subject, whether the model stays silent in 'pure rotation' decides the result.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Smagorinsky for transition | Delayed transition | Sigma or WALE |
| Trusting the default coefficient | Too much or too little dissipation | Check on reference flows |
| Coarse near-wall grid | Wall friction off | y⁺ ≈ 1 or wall model |
| Ignoring numerical dissipation | Model effect masked | Low-dissipation schemes |
| Concluding from model changes alone | Confused with grid effects | Vary the grid too |
I'd like to learn related topics.
The turbulence model simulator and turbulent boundary layer calculator cover turbulence basics. For other models see Smagorinsky model, dynamic Smagorinsky, WALE model and Vreman model; for the big picture, LES fundamentals and wall-modelled LES.
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