WALE Model
WALE: Theoretical Foundations
Overview
Professor, I heard the WALE model is the most popular in industrial LES. What kind of model is it?
The WALE (Wall-Adapting Local Eddy-viscosity) model is an LES SGS model proposed by Nicoud-Ducros (1999). By using the traceless symmetric part $S_{ij}^d$ of the squared velocity gradient tensor $g_{ij}^2 = \bar{g}_{ik}\bar{g}_{kj}$, it automatically achieves $\nu_{\text{sgs}} \to 0$ near walls. It requires neither a Van Driest damping function nor wall distance.
Governing Equations
Could you please show me the specific equations?
The SGS eddy viscosity is calculated by the following equation.
Here, $S_{ij}^d$ is the symmetric traceless part of the squared velocity gradient tensor.
The model constant is $C_w = 0.325$ (generally corresponds to $C_s = 0.1$ for Smagorinsky).
WALE Wall Behavior
Why does it automatically become zero at the wall?
Near the wall, the velocity field becomes $u \sim y$ (linear profile). At this point, $\bar{S}_{ij} \sim O(1)$ but $S_{ij}^d \sim O(y)$. Therefore, the numerator of WALE becomes $O(y^3)$ and the denominator $O(1)$, causing $\nu_{\text{sgs}} \sim y^3$ to decay. This matches the theoretical requirement $\nu_t \sim y^3$ near the wall.
For comparison, in the Smagorinsky model, $\nu_{\text{sgs}} \sim |\bar{S}| \sim O(1)$ (does not decay at the wall), so the Van Driest damping $f = 1 - \exp(-y^+/A^+)$ was necessary.
| Model | $\nu_{\text{sgs}}$ at Wall | Additional Treatment |
|---|---|---|
| Smagorinsky | $O(1)$ (No decay) | Van Driest damping required |
| Dynamic Smagorinsky | $O(y^3)$ (Automatic) | Not required |
| WALE | $O(y^3)$ (Automatic) | Not required |
So WALE achieves the same wall behavior as dynamic Smagorinsky without the need for test filter calculations. That's why it's popular for industrial applications.
How the WALE Model Achieved "No van Driest Needed at Walls Either"
In the conventional Smagorinsky model, a van Driest damping function was necessary to prevent excessive dissipation near walls. The WALE (Wall-Adapting Local Eddy-viscosity) model proposed by Nicoud & Ducros in 1999 cleverly combines invariants of the velocity gradient tensor to achieve the property that "the eddy viscosity naturally approaches zero with $y^3$ order near walls." The fact that the van Driest function is unnecessary is a bigger advantage than it appears, as it eliminates the need to calculate wall distances even for complex geometries.
Computational Methods for WALE
Implementation Details
Are there any points to be careful about when implementing the WALE model?
Calculating $S_{ij}^d$ requires the 9 components of the velocity gradient tensor $\bar{g}_{ij} = \partial \bar{u}_i / \partial x_j$ and its square $\bar{g}_{ij}^2 = \bar{g}_{ik}\bar{g}_{kj}$. The computational cost is not much different from the Smagorinsky model.
| Calculation Step | Required Operations |
|---|---|
| 1. Calculate $\bar{g}_{ij}$ | Velocity gradient tensor (9 components) |
| 2. Calculate $\bar{g}_{ij}^2$ | Tensor product (9 components) |
| 3. Calculate $S_{ij}^d$ | Symmetrization + trace removal (6 components) |
| 4. Calculate $\nu_{\text{sgs}}$ | Scalar operations |
Solver-Specific Settings
Could you tell me how to set it up in each solver?
WALE is standard-supported in all major CFD solvers. Its simple setup and lack of need for special parameter tuning are highly valued in industrial applications.
Grid Width $\Delta$ Definition
How is the grid width defined in the WALE model?
Generally, the volume-equivalent width $\Delta = V^{1/3}$ is used. For structured grids, $\Delta = (\Delta_x \Delta_y \Delta_z)^{1/3}$. The WALE model is highly robust to the definition of $\Delta$ because it does not depend on wall distance.
WALE has correct wall behavior despite its simple implementation and requires no constant tuning. It's an optimal choice as the default SGS model for industrial LES.
The $S_{ij}^d$ Tensor in WALEโWhy Use the "Square of the Velocity Gradient"?
The key to the WALE model is the symmetric deviatoric component $S_{ij}^d$ of $\mathbf{g}^2 = \mathbf{g} \cdot \mathbf{g}$ (the square of the velocity gradient tensor). Why go to the trouble of squaring it? If you use the velocity gradient tensor to the first power, the eddy viscosity vanishes proportional to $y$ near the wall, but the correct physics at a solid wall requires it to vanish with $y^3$ order. By using the square of the velocity gradient, this correct scaling is automatically obtained. It's a good example often featured in turbulence modeling textbooks where a simple modification yields a major improvement.
WALE in Practice
Applicable Scenarios
In what situations should the WALE model be used?
| Application Scenario | Reason |
|---|---|
| Automotive Aerodynamics LES | Correct behavior near walls, simple setup |
| Wind Environment Analysis Around Buildings | Robust to mesh non-uniformity |
| Mixing/Stirring LES | No wall distance needed, handles complex geometries |
| SGS Model for LES region in DES/DDES | Good compatibility due to lack of wall dependency |
Mesh Requirements
What mesh resolution is required for the WALE model?
It's the same as the standard requirements for Wall-Resolved LES.
| Parameter | Recommended Value (wall units) |
|---|---|
| $y^+$ (First layer) | < 1 |
| $\Delta x^+$ (Streamwise) | 20~50 |
| $\Delta z^+$ (Spanwise) | 10~20 |
| Number of layers within boundary layer | 15~25 |
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