Realizable k-epsilon model
Theory and Physics
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."
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, the 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 from 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 setting this term to zero. Since computational cost is significantly reduced, 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. 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 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 it naturally mixes, right? 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 high, 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 overwhelms, 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 provides the force that pushes 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 common point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
- 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 source terms. What happens if you forget a 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 heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (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): Density variation considered only in the buoyancy term; constant density used in other terms
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flows (requires shock capturing), free surface flows (requires VOF/Level Set, etc.)
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
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."
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.)
Suppress numerical oscillations while maintaining high accuracy 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 multiphysics. 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 timestep.
Residual Monitoring
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