WALE Model
Theory and Physics
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.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out spluttering and unstable, but after a while, the flow becomes steady, right? This "period of change" is described by the temporal term. The pulsation of blood flow due to heartbeats, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled"—in other words, setting this term to zero. This significantly reduces computational cost, so trying a steady-state solution first is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the air, the "carrier," transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while they naturally mix. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (viscosity model needed for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. Consider compressibility effects for Mach number 0.3 and above
- Boussinesq Approximation (Natural Convection): Consider density changes only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³@20°C, Water: approx. 998 kg/m³@20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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 WALE—Why 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.
Upwind Differencing (Upwind)
1st-order Upwind: Large numerical diffusion but stable. 2nd-order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
2nd-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Scheme (MUSCL, QUICK, etc.)
Maintains high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex geometries and multi-physics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for continuity, momentum, and energy each drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Pressure: 0.2~0.3, Velocity: 0.5~0.7 are typical initial values. Reduce the factor if diverging. Increase after convergence to accelerate.
Unsteady Calculation Inner Iterations
Iterate within each time step until a steady solution converges. Inner iteration count: 5~20 times is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Although it's first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
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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