Wall-Modeled LES (WMLES)
Theory: leaving the near-wall region to a model
Overview
I want to run automotive aerodynamics with LES, but the near-wall grid is so fine it never finishes.
Near-wall eddies have a very small, viscosity-set size that shrinks further as Reynolds number rises. LES that resolves them directly (wall-resolved LES) sees cell counts explode at high Reynolds number. Wall-modelled LES instead leaves the thin near-wall layer unresolved, represents its velocity profile with a wall model such as the log law, and returns wall shear stress to the LES. Cells can be sized by boundary-layer thickness, cutting counts by orders of magnitude.
Log-law wall model
Take the LES velocity $u$ at height $y$ from the wall, solve for friction velocity $u_\tau$, and return wall shear $\tau_w$. $y$ is often about a tenth of the boundary-layer thickness, tens to hundreds in $y^+$.
Boundary-layer estimates
The wall problem from LES's beginnings
In 1970 Deardorff already treated the near-wall region with the log law in LES of atmospheric boundary layers and channel flow, since computers couldn't resolve the wall. LES and DNS resolving the wall advanced later, but in 1979 Chapman estimated that resolving wing flows to the wall makes cell counts climb steeply with Reynolds number, arguing wall models are indispensable at high Reynolds numbers. In the 2010s Choi, Moin and others revisited these estimates, and research on using wall-modelled LES in aircraft design advanced greatly.
Worked examples
Example 1: estimating cell counts
For an air flat-plate boundary layer 0.5 m from the leading edge, cells needed inside the boundary layer per m² of wall were estimated. Wall-resolved LES: 50 streamwise and 15 spanwise wall units, first cell y+ = 1 stretched by 1.1 to the boundary-layer edge. Wall-modelled LES: cells a tenth of boundary-layer thickness, 20 across it:
| Speed | Re_x | y+ = 1 height | BL thickness | Wall-resolved | Wall-modelled | Ratio |
|---|---|---|---|---|---|---|
| 10 m/s | 3.3×10⁵ | 31 µm | 14.5 mm | 5.7×10⁷ | 9.5×10⁶ | 6 |
| 50 m/s | 1.7×10⁶ | 7.3 µm | 10.5 mm | 1.3×10⁹ | 1.8×10⁷ | 74 |
| 250 m/s | 8.3×10⁶ | 1.7 µm | 7.6 mm | 2.9×10¹⁰ | 3.4×10⁷ | 858 |
Units are cells per m² of wall (250 m/s is indicative, ignoring compressibility).
Each fivefold increase in speed widens the ratio from 6 to 74 to 858.
Resolving the wall requires sizing cells to the viscous length (the y+ = 1 height), which shrinks from 31 µm to 1.7 µm as speed rises. Wall-modelled cells are set by boundary-layer thickness, which if anything thins only slightly. So the gap grows with Reynolds number. At full-scale automotive or aircraft Reynolds numbers, wall-resolved LES becomes impractical, leaving wall-modelled LES or the DES family that solves the near-wall region with RANS.
Example 2: wall-model sensitivity
With LES velocity 40 m/s at 1 mm from the wall, the log law gives friction velocity 2.31 m/s and wall shear 6.42 Pa (matching point y+ 154). LES velocity 10% low gives −17% wall shear; 10% high gives +18%. In wall-modelled LES, velocity accuracy at the matching point is roughly doubled in its effect on wall friction.
Analysis workflow
- Estimate wall-resolved cell counts from Reynolds number to judge whether a wall model is needed.
- Size cells at roughly 1/10 to 1/20 of boundary-layer thickness.
- Place the matching point a few cells off the wall to avoid numerical error from the first cells.
- Where separation or strong adverse pressure gradients occur, consider non-equilibrium wall models.
- Compare skin friction and surface pressure with experiment or DNS data.
“Skin friction 20% high”
Verifying flat-plate skin friction with wall-modelled LES gave about 20% above experiment. The matching point was the first wall cell, whose velocity carried the LES grid's numerical error (the first cell is under-resolved). Given the log-law sensitivity, a 10% velocity error becomes 20% in friction. Moving the matching point three or four cells off the wall brought close agreement. In wall-modelled LES, which velocity you pass decides the accuracy.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Matching at the first cell | Friction off | Move a few cells out |
| Log law in separation | Separation misplaced | Non-equilibrium models |
| Grid coarse versus BL thickness | Outer-layer eddies unresolved | Below 1/10 of thickness |
| WMLES at low Re | Little benefit | Consider wall-resolved |
| No inflow turbulence | Delayed transition | Use turbulence generation |
I'd like to learn related topics.
Check boundary layers with the turbulent boundary layer calculator and turbulence model simulator. Related methods include LES fundamentals, DES, DDES, DNS fundamentals and the sigma model.
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