RNG k-epsilon model
Theory and Physics
Overview
Professor! How is the RNG k-ε model different from the standard k-ε model?
The RNG k-ε model is a model derived statistically and mechanically by Yakhot & Orszag (1986) using Renormalization Group theory. A major difference is that while the model constants in the standard k-ε model are determined empirically, in the RNG version they are derived theoretically.
Does being theoretically derived automatically mean higher accuracy?
The values of the constants themselves don't change that much; what's important is the R-term (Additional Rate-of-Strain Term) added to the ε equation. This improves accuracy in rapidly straining flows and swirling flows.
Governing Equations
Please show me the specific equations.
The k equation is almost the same as the standard k-ε.
The ε equation has the RNG-specific R-term added.
Here, the modified dissipation coefficient is:
RNG constants: $C_{\varepsilon 1}=1.42$, $C_{\varepsilon 2}=1.68$, $C_\mu=0.0845$, $\eta_0=4.38$, $\beta=0.012$.
So this $\eta$ parameter is the key, right?
Yes. When $\eta > \eta_0 \approx 4.38$ (rapidly straining flow), the numerator $(1-\eta/\eta_0)$ becomes negative, increasing $C_{\varepsilon 2}^*$. This increases the dissipation of $\varepsilon$, resulting in a decrease in turbulent viscosity $\mu_t = \rho C_\mu k^2/\varepsilon$. In other words, it has the effect of suppressing excessive turbulent viscosity in rapidly straining flows.
So it mitigates the problem of excessive diffusion of swirling flows in the standard k-ε model.
Correct. However, since the R-term only becomes significant where $\eta$ is large, it doesn't dramatically improve all cases.
What is "Renormalization Group"? — The Day Physics Techniques Came to CFD
It's natural to hear the name RNG (Renormalization Group) and think, "What's that difficult-sounding name?" Originally, renormalization group is a mathematical technique developed in particle physics, a tool for systematically handling phenomena at different scales. It began when Yakhot and Orszag realized in 1986 that "this might be applicable to energy transfer between scales in turbulence" and brought it into CFD. The modified ε in high-strain regions is naturally derived from this group-theoretic operation. The interesting aspect of RNG k-ε is the boldness of the idea to "apply physics tools to engineering."
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 "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? Looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, solving first with steady-state is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? 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 air, as a "carrier," 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'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, after a while it naturally mixes. 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 becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly 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 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 applied to molten metal by a factory electromagnetic pump... These are all actions that "inject energy or force into the fluid from outside," expressed by source terms. What happens if you forget the 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 heated winter room.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular 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): Consider density changes 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
What are the differences from standard k-ε when implementing RNG k-ε?
From a solver perspective, the discretization of the k equation is identical, and only the R-term is added to the source term of the ε equation. However, since the R-term depends on $\eta = Sk/\varepsilon$ and also on $\varepsilon$ itself, implicit treatment is required.
Linearization of the R-term
What specifically do you do for implicit treatment?
Linearize the $R$ term with respect to $\varepsilon$. Since $R = \frac{C_\mu \rho \eta^3(1-\eta/\eta_0)}{(1+\beta\eta^3)} \frac{\varepsilon^2}{k}$, when $\eta < \eta_0$, $R > 0$ (source term), and when $\eta > \eta_0$, $R < 0$ (sink term).
For sink terms, add to the diagonal term and treat implicitly; for source terms, add to the source vector. This separation improves numerical stability.
Wall Treatment
How do you handle the near-wall region?
The RNG k-ε model is inherently a high-Re model, so wall functions are used. However, in Fluent or CFX, there is an "Enhanced Wall Treatment" option that can handle low-Re regions with a two-layer model.
| Wall Treatment | Required $y^+$ | Accuracy | Application |
|---|---|---|---|
| Standard Wall Function | $30 < y^+ < 300$ | Medium | General industrial use |
| Non-equilibrium Wall Function | $30 < y^+ < 300$ | Medium-High | Flows with separation and reattachment |
| Enhanced Wall Treatment | $y^+ \approx 1$ | High | Heat transfer, separation prediction |
OpenFOAM Settings
How do you use RNG k-ε in OpenFOAM?
Set it in constant/turbulenceProperties as follows.
```
RAS
{
RASModel RNGkEpsilon;
turbulence on;
printCoeffs on;
}
```
Wall functions are specified in the variable files in the 0/ directory. For example, use nutkWallFunction for nut and epsilonWallFunction for epsilon.
Fluent Settings
What about in Fluent?
Select Models → Viscous → k-epsilon → RNG. Options include:
- Differential Viscosity Model: Uses an effective viscosity formula including low-Re number effects
- Swirl Dominated Flow: Adds swirl correction (effective when swirl number is large)
These options are RNG-specific features not available in standard k-ε.
Why RNG k-ε Shines in Swirl Combustors
In gas turbine combustor design, swirling (swirl) flow is used to mix fuel and air. It is known that standard k-ε overestimates eddy viscosity for this strong swirl, causing swirl intensity to decay faster than in experiments. The high-strain correction term in RNG k-ε mitigates this problem, so gas turbine manufacturers have established a procedure of using RNG k-ε for initial combustor design and transitioning to RSM for detailed design. It's a practical choice balancing computational cost and model accuracy.
Upwind Differencing (Upwind)
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 shocks 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 judged when residuals for continuity, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Typical initial values: Pressure: 0.2-0.3, Velocity: 0.5-0.7. If diverging, lower the relaxation factor. After convergence, increase 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 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 catchball is repeated to approach the correct answer. 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 the source of the water by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Accuracy is first-order, but it's highly stable because it correctly captures flow direction.
Practical Guide
Practical Guide
Please tell me about practical situations where RNG k-ε is effective.
There are cases where RNG k-ε is clearly superior to standard k-ε.
Recommended Application Cases
| Flow Type | RNG Superiority | Reason |
|---|---|---|
| CFD for Indoor Air Conditioning | High |