SST k-omega model (Menter)
Theory and Physics
Overview
Teacher! I get the impression that the SST k-ω model is the most commonly used one in CFD, but why is it so popular?
The SST (Shear Stress Transport) k-ω model developed by Menter (1994) is a hybrid model that combines the superiority of the k-ω model near walls with the stability of the k-ε model in free stream flow. Furthermore, the addition of a limiter that considers the transport of turbulent shear stress has significantly improved the prediction of separation under adverse pressure gradients.
It seems like a model that kills three birds with one stone.
Exactly. It is so widely used that it can be called the "default model" for industrial CFD.
Governing Equations
Please show me the equations.
k equation:
ω equation:
The last term, the Cross-Diffusion Term, arises from the transformation from k-ε and is controlled by the $F_1$ blending function.
Blending function:
Near the wall, $F_1 \to 1$ (k-ω behavior), and far away, $F_1 \to 0$ (k-ε behavior).
Eddy viscosity definition (SST limiter):
$a_1 = 0.31$, $F_2$ is the second blending function. This limiter approximates Bradshaw's hypothesis ($-\overline{u'v'} = a_1 k$) and improves separation prediction under adverse pressure gradients.
So $F_1$ switches based on the distance from the wall.
More precisely, it switches adaptively using not only the wall distance $y$ but also local flow field quantities ($k$, $\omega$, $\nu$). This ensures a smooth connection between the k-ω region near the wall and the k-ε region far away.
Taking the "Best of Both Worlds" – Menter's Idea
The SST model, whose paper was published by Florian Menter in 1994, was a novel hybrid technique at the time, blending the free-stream stability of k-ε with the near-wall accuracy of k-ω using a blending function. This switching mechanism, where it behaves as k-ω in boundary layers around wings and as something close to k-ε in free stream, is the reason it became the standard model for aerospace, automotive, and turbomachinery CFD. It is now so widespread that it is the default setting in CFD solvers worldwide.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of 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 "period of change" is described by the temporal term. The pulsation of blood flow due to a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since this drastically reduces computational cost, starting with a steady-state solution 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. The 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 things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is 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 it naturally mixes. That's molecular diffusion. Now, next question—honey or 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 flow (slow, viscous), diffusion is dominant. Conversely, in high Re number flow, 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, and the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are densely packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—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, it might be due to confusion between absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed upward 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 an electromagnetic pump in a factory... These are all actions that "inject energy or force into the fluid from the outside" and are expressed by source terms. What happens if you forget a source term? In natural convection analysis, if you forget to include buoyancy, 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 (mean free path of molecules ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (viscosity models are needed for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms.
- Non-applicable Cases: Rarefied gases (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 | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition. |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability. |
Numerical Methods and Implementation
Key Points in Numerical Implementation
What points require special attention when implementing SST?
There are three important points.
1. Calculation of Wall Distance
The blending function requires the wall distance $y$, right?
The wall distance is sometimes obtained by solving a Poisson equation before the calculation starts, and sometimes by geometrically calculating the distance to the nearest wall cell. For complex geometries, the Poisson equation-based method (e.g., OpenFOAM's wallDist) is more robust.
Fluent automatically calculates wall distance. In OpenFOAM, it is specified via wallDist in fvOptions. The accuracy of the wall distance affects the results through the blending function, so mesh orthogonality to the wall is important.
2. Production Term Limiter
Does $\tilde{P}_k$ have a limiter applied?
Yes. In Menter's original paper, $\tilde{P}_k = \min(P_k,\; 10 \beta^* \rho k \omega)$, limiting the production term so it does not exceed 10 times the dissipation term. This prevents excessive accumulation of turbulent kinetic energy at stagnation points (Stagnation Point Anomaly).
Some solvers have an option to use a vorticity-based production term, which further mitigates the stagnation point problem.
3. Handling the Cross-Diffusion Term
Is the cross-diffusion term in the $\omega$ equation numerically stable?
$\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}$ can be positive or negative. To prevent division by zero, $CD_{k\omega} = \max\left(2\rho\sigma_{\omega 2}\frac{1}{\omega}\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j},\; 10^{-10}\right)$ is used.
OpenFOAM Settings
Please show me typical settings in OpenFOAM.
constant/turbulenceProperties:
```
RAS
{
RASModel kOmegaSST;
turbulence on;
printCoeffs on;
}
```
Wall boundary conditions:
k:kqRWallFunction(usefixedValue 1e-10for $y^+ < 1$)omega:omegaWallFunctionnut:nutUSpaldingWallFunction(compatible with all $y^+$)
Recommended Settings in Fluent
| Parameter | Recommended Value | Remarks |
|---|---|---|
| Model | k-omega SST | Production Limiter ON by default |
| Wall Treatment | Resolve to $y^+ \approx 1$ | Enhanced Wall Treatment is unnecessary (SST handles low-Re) |
| URF (k, omega) | 0.7-0.8 | Lower to 0.5 if convergence is difficult |
| Discretization | Second Order Upwind | For both k and ω |
The True Nature of Blending Function F1 – The Boundary Between "k-ε from Here, k-ω from Here"
The blending function F1, the core of the SST model, is a dimensionless quantity calculated from wall distance, turbulent kinetic energy, eddy viscosity, etc., and varies smoothly between 0 and 1. In regions where F1=1, k-ω dominates, and where F1=0, equations close to k-ε are active. Implementation-wise, wall distance calculation is necessary; OpenFOAM uses the wallDist library, but for complex shapes, if this wall distance calculation becomes inaccurate, blending can go wrong. If you feel "SST results look strange," I recommend starting by visualizing wallDist.
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 > 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 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 methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is typically judged when residuals for continuity, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2-0.3, Velocity: 0.5-0.7 are typical initial values. Lower the factors if divergence occurs. Increase them after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution is reached. 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 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—this method reflects the physics that upstream information determines downstream conditions. Although it is first-order accurate, it is highly stable because it correctly captures the flow direction.
Practical Guide
Practical Guide
Please tell me the best practices for using the SST model in practical work.
SST is highly versatile, but there are several points to note.
Mesh Requirements
What should I do about the wall $y^+$?
For SST, it is recommended to resolve the viscous sublayer. Aim for $y^+ \approx 1$ for the first cell. If you cannot achieve this due to mesh constraints, using wall functions is possible, but the accuracy of separation prediction may degrade.
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