Vreman Model
Theory: eddy viscosity that vanishes in laminar flow
Overview
I've heard Smagorinsky in LES makes near-wall flow reluctant to transition.
Smagorinsky sets eddy viscosity from strain magnitude alone. Near-wall laminar flow and pre-transition boundary layers still have shear, so it adds viscosity where there's no turbulence. Seeds of turbulence get damped, delaying transition and overpredicting wall friction. Damping functions that reduce the coefficient near walls and dynamic models that infer it from the flow were introduced to avoid this. Vreman combines velocity gradients so that eddy viscosity automatically vanishes in simple shear, at about the same cost as Smagorinsky.
The Vreman model
Δ is the grid width. B_β is the second invariant of β and is zero when the velocity gradient α varies in only one direction (rank one). The Smagorinsky formula is shown for comparison.
A model built to vanish in laminar flow
In his 2004 paper Vreman classified flows by their velocity-gradient components and showed eddy viscosity vanishes in many non-turbulent flows such as shear layers. He stressed that the model is local and needs no averaging like dynamic procedures. It doesn't vanish for every laminar flow, though; the sigma model proposed by Nicoud and colleagues in 2011 was refined to vanish also for solid-body rotation and axisymmetric expansion.
Worked examples
Example 1: eddy viscosity in canonical flows (Δ = 1 mm, gradient 100 1/s, Cs = 0.17, c = 0.072)
| Flow | Smagorinsky | Vreman | Ratio to air ν (S / V) |
|---|---|---|---|
| Simple shear | 2.9×10⁻⁶ | 0 | 0.19 / 0 |
| Shear + 10% cross component | 2.9×10⁻⁶ | 7.2×10⁻⁷ | 0.19 / 0.05 |
| Plane strain | 5.8×10⁻⁶ | 5.1×10⁻⁶ | 0.39 / 0.34 |
| Axisymmetric strain | 5.0×10⁻⁶ | 4.4×10⁻⁶ | 0.33 / 0.29 |
| Solid rotation | 0 | 5.1×10⁻⁶ | 0 / 0.34 |
| Isotropic random (mean of 20,000) | 3.2×10⁻⁶ | 3.1×10⁻⁶ | 0.21 / 0.21 |
Units are m²/s; air's kinematic viscosity is taken as 1.5×10⁻⁵ m²/s. In isotropic turbulence the ratio is 0.97, showing c = 2.5·Cs² is chosen to give about the same strength.
Example 2: near-wall laminar shear
| Wall shear rate | Smagorinsky | Vreman |
|---|---|---|
| 100 1/s | 2.9×10⁻⁶ (19% of air) | 0 |
| 1,000 1/s | 2.9×10⁻⁵ (193% of air) | 0 |
At 1,000 1/s shear, Smagorinsky adds nearly twice air's own viscosity.
Yes. With a 1 mm grid, near-wall flow that should be laminar is computed as a fluid two to three times as viscous as reality. Turbulent seeds die, the boundary layer fails to transition and separation moves. Real meshes refine toward the wall, reducing the value, but without damping functions wall friction still tends to come out high. Vreman gives zero here and needs no damping function. As the table's solid rotation shows, though, it adds viscosity in non-turbulent rotation; take care in rotating containers and vortex cores, where comparing with sigma or WALE models helps.
How to use it
- Start with c = 2.5·Cs² (about 0.07 for Cs = 0.17).
- Don't add wall damping functions; the model shrinks near walls by itself.
- Compare with Smagorinsky in flows mixing laminar and transitional regions.
- In rotating flows and vortex cores, compare eddy-viscosity fields with the sigma model etc.
- Check grid convergence.
“An LES with no transition over the wing”
A low-Reynolds-number wing LES with Smagorinsky failed to reproduce the leading-edge separation bubble and reattachment seen in experiments; flow stayed laminar, separated massively and lift came out low. Spurious eddy viscosity in the laminar shear near the leading edge delayed transition of the separated shear layer. Switching to Vreman produced the bubble and reattachment, bringing lift close to experiment at almost the same computational cost.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Using Cs directly as c | Viscosity off by 2×+ | c = 2.5·Cs² |
| Adding damping functions | Too little near-wall viscosity | Leave them off |
| Overtrusting in rotation | Spurious viscosity | Compare with sigma |
| Grid too coarse | Strong model dependence | Refine grid |
| Using negative B_β | Square root fails | Clip at zero |
I'd like to learn related topics.
Related pages include the Smagorinsky model, the dynamic Smagorinsky model, the WALE model, the sigma model and LES fundamentals.
Related Topics
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