Standard k-omega model (Wilcox)
Standard k-omega model (Wilcox): Theoretical Foundations
Overview
Teacher! What's the difference between the standard k-ฯ model and SST k-ฯ?
The standard k-ฯ model was developed by Wilcox (1988, 2006) and served as the base for SST. It uses the specific dissipation rate $\omega = \varepsilon/(C_\mu k)$ as a transport variable, offering the advantage of more natural handling near walls compared to the k-ฮต model.
You mean it's strong at walls?
Yes. The wall boundary condition for ฯ can be defined analytically (Dirichlet condition), and unlike ฮต, it does not have a singularity at the wall. It can directly resolve down to the viscous sublayer without wall functions. However, it has a significant weakness of high sensitivity to free-stream boundary conditions.
Governing Equations
Please teach me the equations.
The revised k-ฯ equations by Wilcox (2006):
k equation:
ฯ equation:
Eddy viscosity: $\mu_t = \rho k / \omega$
Model constants: $\alpha = 13/25$, $\beta = \beta_0 f_\beta$, $\beta^* = 9/100$, $\sigma = 1/2$, $\sigma^* = 3/5$, $\sigma_d = 1/8$ (only when $\nabla k \cdot \nabla \omega > 0$).
It's interesting that the $\sigma_d$ cross-diffusion term is conditional.
That's an important improvement in the Wilcox 2006 version. It significantly mitigated the free-stream sensitivity problem. The 1988 version lacked this term, causing the solution to vary greatly depending on the free-stream $\omega$ value.
Free-Stream Sensitivity Problem
What exactly is the free-stream sensitivity problem?
Changing the boundary value of $\omega$ in the free-stream (region far from the wall) also changes the solution near the wall. This is physically unreasonable. Menter's primary motivation for developing the blend with k-ฮต (SST) was this problem.
It was greatly improved by the cross-diffusion term in the 2006 version, but not completely resolved. This is why SST k-ฯ is overwhelmingly more used than standard k-ฯ in industrial CFD.
The Model Wilcox Continuously Updated for 50 Years
The standard k-ฯ model has been continuously revised since its proposal by David C. Wilcox in 1988, with versions in 1998, 2006, and 2008. Free-stream sensitivity (the problem where results depend sensitively on the inlet boundary ฯ value) has long been known as a weakness, and Wilcox himself recognized that "improvement is needed" and continued revisions. It is rare in the engineering world for a researcher to publicly acknowledge their model's shortcomings and keep improving it. When using it, it's important to check which version is implemented, as the equations differ by era even when simply referred to as "standard k-ฯ".
Computational Methods for Standard k-omega model (Wilcox)
Numerical Implementation
Are there any specific points to note when implementing standard k-ฯ?
The wall boundary condition for ฯ is the most important point.
Wall Boundary Conditions
How is the wall value for ฯ determined?
In the limit $y^+ \to 0$:
This is a Dirichlet condition, and numerically there are several implementation methods:
1. Set as the value at the first cell center from the wall: $\omega_P = \frac{6\nu}{\beta_1 (\Delta y_1)^2}$
2. Set as the value at a wall ghost cell: Recommended by Wilcox
3. Switch with wall functions: For $y^+ > 2.5$, use $\omega = u_\tau / (\sqrt{\beta^*} \kappa y)$
With method 1, ฯ gets larger as the mesh gets finer, right?
Yes. At $y^+ = 1$, $\omega \sim O(10^6)$, at $y^+ = 0.1$, $\omega \sim O(10^8)$. Care is needed to maintain matrix diagonal dominance.
Discretization Schemes
What schemes are recommended for discretizing k-ฯ?
| Variable | Recommended Scheme | Remarks |
|---|---|---|
| k | Second Order Upwind | With TVD limiter function |
| ฯ | First/Second Order Upwind | First order acceptable for ฯ as it changes rapidly near walls |
| Momentum | Second Order Upwind or higher | |
| Cross-diffusion term | Central Difference | Note the dot product of gradients and conditional branching |
OpenFOAM Settings
Please teach me how to use it in OpenFOAM.
OpenFOAM has the Wilcox 2006 version implemented as kOmega.
```
RAS
{
RASModel kOmega;
turbulence on;
printCoeffs on;
}
```
However, in practice, kOmegaSST is overwhelmingly more common. kOmega is used only for specific benchmark validation or when you want to exclude the influence of SST blending.
Convergence Improvement
Any techniques for when convergence is poor?
The Compatibility of k-ฯ and y+=1โUnexpected Benefits of Wall Resolution
The k-ฯ model has the property that ฯโโ near the wall due to its boundary condition, and since an analytical solution for ฯ on the wall exists, it pairs very well with fine meshes around y+=1 that directly resolve the viscous sublayer. Thanks to this characteristic, it is strong for boundary layer calculations of transitional flows and low Reynolds number regions, which is why it has long been used for aircraft wing boundary layer analysis. Conversely, the practical guideline "if you choose k-ฯ, aim for y+ โค 1" naturally arises from this model's physical characteristics. That "k-ฯ's true potential is unleashed with finer meshes" is important knowledge for practicing engineers.
Standard k-omega model (Wilcox) in Practice
Practical Guide
Are there any situations where standard k-ฯ should be used? If SST exists, isn't it un...
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