Standard k-omega model (Wilcox)
Theory and Physics
Overview
Teacher! What's the difference between the standard k-ω model and SST k-ω?
The standard k-ω 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-ε model.
You mean it's strong at walls?
Yes. The wall boundary condition for ω can be defined analytically (Dirichlet condition), and unlike ε, 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-ω equations by Wilcox (2006):
k equation:
ω 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-ε (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-ω is overwhelmingly more used than standard k-ω in industrial CFD.
The Model Wilcox Continuously Updated for 50 Years
The standard k-ω 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 ω 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-ω".
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, it becomes a steady flow, right? This "during the change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens/closes, are all unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. This significantly reduces computational cost, so solving steady-state 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 "carrier," air, 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" → They're 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, it naturally mixes after a while. That's molecular diffusion. Now a question—honey and water, which flows easier? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re number flows, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure term $-\nabla p$: When you push a syringe plunger, liquid shoots out forcefully from the needle tip, right? Why? The plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force pushing the fluid. Dam water release works on the same principle. On a weather map, where isobars are densely packed? That's right, strong winds blow. "Flow occurs 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 go wrong immediately after switching to compressible analysis, confusion between 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 applied to molten metal by a factory electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget a source term? In natural convection analysis, forgetting 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: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility assumption (for Ma < 0.3): Treat density as constant. For Mach number ≥ 0.3, consider compressibility effects
- Boussinesq approximation (Natural convection): Consider density variation 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
Numerical Implementation
Are there any specific points to note when implementing standard k-ω?
The wall boundary condition for ω is the most important point.
Wall Boundary Conditions
How is the wall value for ω 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, ω 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-ω?
| Variable | Recommended Scheme | Remarks |
|---|---|---|
| k | Second Order Upwind | With TVD limiter function |
| ω | First/Second Order Upwind | First order acceptable for ω 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-ω and y+=1—Unexpected Benefits of Wall Resolution
The k-ω model has the property that ω→∞ near the wall due to its boundary condition, and since an analytical solution for ω 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-ω, aim for y+ ≤ 1" naturally arises from this model's physical characteristics. That "k-ω's true potential is unleashed with finer meshes" is important knowledge for practicing engineers.
Upwind Scheme
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number, diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multi-physics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: 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 typically judged when residuals for the continuity equation, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2-0.3, Velocity: 0.5-0.7. Reduce the factor if diverging. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5-20 iterations 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 obtained (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised with 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 the Upwind Scheme
The upwind scheme 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. It's first-order accurate but highly stable because it correctly captures flow direction.
Practical Guide
Practical Guide
Are there any situations where standard k-ω should be used? If SST exists, isn't it un...
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