Realizable k-epsilon model
Realizable k-epsilon: Theoretical Foundations
Overview
The Realizable k-ε model means "realizable" k-ε, right? What exactly is "realizable"?
It's a model proposed by Shih et al. (1995). "Realizable" means it satisfies the physical realizability constraints of the Reynolds stress tensor. Specifically, it ensures that the normal Reynolds stresses are non-negative ($\overline{u_\alpha'^2} \geq 0$) and that Schwarz's inequality ($\overline{u_\alpha' u_\beta'}^2 \leq \overline{u_\alpha'^2}\cdot\overline{u_\beta'^2}$) is satisfied.
Can the standard k-ε model violate this condition?
Yes. In the standard k-ε model, $C_\mu = 0.09$ is a constant, so in regions with very high strain rates (e.g., the center of swirling flows), normal stresses can become negative. This is physically impossible.
Transport Equations
The $k$ equation is the same as in the standard k-ε, but the $\varepsilon$ equation is significantly different.
The key point here is the definition of $C_1$.
Variable $C_\mu$
The fact that $C_\mu$ becomes a variable is the biggest feature, right?
Exactly. In the Realizable model, the eddy viscosity is defined as follows.
Here, $U^* = \sqrt{S_{ij}S_{ij} + \tilde{\Omega}_{ij}\tilde{\Omega}_{ij}}$, $A_0 = 4.04$, and $A_s = \sqrt{6}\cos\phi$ (where $\phi$ is calculated from the third invariant of the strain rate tensor).
So, does that mean $C_\mu$ automatically decreases when the strain rate becomes large?
Precisely. In the center of swirling flows, $S$ is large, so $C_\mu$ decreases, suppressing the overprediction of turbulent kinetic energy. In calm flows ($S k/\varepsilon \to 0$), $C_\mu \to 1/A_0 \approx 0.25$, but in typical shear flows, it takes values close to about 0.09, aligning with the standard k-ε model.
Model Constants
| Constant | Value |
|---|---|
| $C_2$ | 1.9 |
| $\sigma_k$ | 1.0 |
| $\sigma_\varepsilon$ | 1.2 |
| $A_0$ | 4.04 |
Does the Realizable k-ε model have any weaknesses?
The source term in the $\varepsilon$ equation contains $\sqrt{\nu\varepsilon}$, which avoids singularity even when $\varepsilon \to 0$. However, in multi-zone calculations using multiple rotating reference frames, non-physical turbulent viscosity can sometimes arise. In Fluent, this is addressed with a modified version (correction for sliding mesh).
The Meaning of "Realizable" – The Effort to Uphold the Obvious
The name "Realizable" in Realizable k-ε means that it satisfies the mathematical constraints of turbulence—non-negativity of normal stresses and the Schwarz inequality. In fact, the standard k-ε model does not always satisfy these constraints and can predict unphysical negative normal stresses in strong strain flows. Shih et al. (1994) demonstrated that "simply adhering to these obvious physical constraints significantly improves the model." The reason Realizable k-ε is preferred as the standard choice in the k-ε family for compression, expansion, and separated flows lies in this "guarantee of the obvious."
Computational Methods for Realizable k-epsilon
Discretization Notes
Are there any differences in numerical handling between Realizable k-ε and standard k-ε?
The form of the $\varepsilon$ equation is different, so care is needed in linearizing the source term. The denominator $k + \sqrt{\nu\varepsilon}$ is particularly important; it's designed to prevent division by zero even in regions where $k \to 0$ (e.g., very near walls or non-turbulent regions).
The recommended discretization schemes are the same as for standard k-ε.
| Term | Recommended Scheme |
|---|---|
| Convection Term | Second Order Upwind |
| Diffusion Term | Central Differencing |
| Temporal Term | Second Order Implicit |
Numerical Calculation of $C_\mu$
Since $C_\mu$ depends on the flow field, does it need to be recalculated every iteration?
Yes. For each cell, $S_{ij}$ and $\Omega_{ij}$ are calculated, then $U^*$ and $\phi$ are computed to update $C_\mu$. The computational cost increases slightly compared to standard k-ε, but since no additional transport equations are solved, the difference is not significant.
Ansys Fluent
```
Models → Viscous → k-epsilon → Realizable
Near-Wall Treatment → Enhanced Wall Treatment (Recommended)
```
In Fluent, Realizable is the default recommended k-ε model. When combined with Enhanced Wall Treatment, the $y^+$ constraint is relaxed.
OpenFOAM
Specify the following in constant/turbulenceProperties.
```
RAS
{
RASModel realizableKE;
turbulence on;
printCoeffs on;
}
```
STAR-CCM+
Select K-Epsilon Turbulence → Realizable K-Epsilon Two-Layer. The Two-Layer model is typically used in combination with the All y+ Wall Treatment.
Convergence Comparison
Is it harder to converge compared to standard k-ε?
Generally, Realizable k-ε has better convergence. This is because the variable $C_\mu$ suppresses non-physical explosions of $\mu_t$. However, if $C_\mu$ fluctuates significantly in the first few tens of iterations, it can become unstable. In such cases, reducing the URF for turbulent viscosity ratio to around 0.8 is advisable.
How Realizable k-ε Became the Star in Automotive External Aerodynamics Analysis
Since the 2000s, Realizable k-ε has spread as the standard model for external aerodynamics analysis in automotive CFD departments. One reason is its higher accuracy in reproducing separation/reattachment near the A-pillar and side mirror wake flows compared to standard k-ε. There are multiple reports of OEM design teams comparing four models with the same mesh and settings, finding that the drag coefficient difference from wind tunnel test values was smallest with Realizable k-ε, leading to its adoption in subsequent design cycles. Even today, it's often taught in ANSYS Fluent introductory seminars: "For external aerodynamics, start with Realizable k-ε."
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