Professor, how does the RNG k-ε model differ from the standard k-ε model?

The RNG k-ε model is a model derived statistically using Renormalization Group theory by Yakhot & Orszag (1986). A key difference is that while the model constants in the standard k-ε model are determined empirically, they are derived theoretically in the RNG version.

Does being theoretically derived automatically lead to higher accuracy?

The values of the constants themselves do not change drastically. The crucial difference is the additional rate-of-strain term (the R-term) introduced into the ε equation. This improves accuracy in flows with rapid strain and swirling flows.

So the parameter η is the key here.

Correct. When $\eta > \eta_0 \approx 4.38$ (in rapidly strained flow), the numerator $(1-\eta/\eta_0)$ becomes negative, increasing $C_{\varepsilon 2}^*$. This enhances the dissipation of ε, which in turn reduces the turbulent viscosity $\mu_t = \rho C_\mu k^2/\varepsilon$. In effect, this suppresses excessive turbulent viscosity in rapidly straining flows.

RNG k-epsilon model

Q: Could you provide the specific equations?

The k-equation is almost identical to the standard k-ε model. $$ \frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho u_j k)}{\partial x_j} = P_k - \rho\varepsilon + \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_j}\right] $$ The ε-equation includes the RNG-specific R-term. $$ \frac{\partial(\rho\varepsilon)}{\partial t} + \frac{\partial(\rho u_j \varepsilon)}{\partial x_j} = C_{\varepsilon 1}\frac{\varepsilon}{k}P_k - C_{\varepsilon 2}^{*}\rho\frac{\varepsilon^2}{k} + \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_\varepsilon}\right)\frac{\partial \varepsilon}{\partial x_j}\right] $$ Here, the modified dissipation coefficient is: $$ C_{\varepsilon 2}^{*} = C_{\varepsilon 2} + \frac{C_\mu \eta^3 (1 - \eta/\eta_0)}{1 + \beta \eta^3} $$ $$ \eta = S k / \varepsilon, \quad S = \sqrt{2S_{ij}S_{ij}} $$ RNG constants: $C_{\varepsilon 1}=1.42$, $C_{\varepsilon 2}=1.68$, $C_\mu=0.0845$, $\eta_0=4.38$, $\beta=0.012$.

Category: Fluid Analysis (CFD) | Integrated 2026-04-06

CAE visualization for k epsilon rng theory - technical simulation diagram

RNG k-ε Model

RNG k-epsilon: Theoretical Foundations

Overview

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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

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Please show me the specific equations.

🎓

The k equation is almost the same as the standard k-ε.

$$ \frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho u_j k)}{\partial x_j} = P_k - \rho\varepsilon + \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_j}\right] $$

The ε equation has the RNG-specific R-term added.

$$ \frac{\partial(\rho\varepsilon)}{\partial t} + \frac{\partial(\rho u_j \varepsilon)}{\partial x_j} = C_{\varepsilon 1}\frac{\varepsilon}{k}P_k - C_{\varepsilon 2}^{*}\rho\frac{\varepsilon^2}{k} + \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_\varepsilon}\right)\frac{\partial \varepsilon}{\partial x_j}\right] $$

Here, the modified dissipation coefficient is:

$$ C_{\varepsilon 2}^{*} = C_{\varepsilon 2} + \frac{C_\mu \eta^3 (1 - \eta/\eta_0)}{1 + \beta \eta^3} $$

$$ \eta = S k / \varepsilon, \quad S = \sqrt{2S_{ij}S_{ij}} $$

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.

Coffee Break Yomoyama Talk

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."

Computational Methods for RNG k-epsilon

Numerical Implementation

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What are the differences from standard k-ε when implementing RNG k-ε?

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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

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What specifically do you do for implicit treatment?

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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

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How do you handle the near-wall region?

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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

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How do you use RNG k-ε in OpenFOAM?

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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

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What about in Fluent?

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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.

Coffee Break Yomoyama Talk

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.

RNG k-epsilon in Practice

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Please tell me about practical situations where RNG k-ε is effective.

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There are cases where RNG k-ε is clearly superior to standard k-ε.

Recommended Application Cases

Practical CAE quality notes for RNG k-epsilon model

RNG k-epsilon model should be treated as an engineering model, not as an isolated formula. In fluid simulation, reliable results come from a clear chain of assumptions: governing physics, material data, boundary conditions, numerical discretization, solver settings, and post-processing criteria. Before using this note in a design review, identify which quantities are prescribed, which are solved, and which are only diagnostic indicators.

Model setup checklist

Define the scope: decide whether RNG k-epsilon model is being used for screening, detailed design, failure investigation, or verification of another simulation.
Check dimensions and units: keep SI units internally and document every conversion applied to loads, geometry, material constants, and time or frequency scales.
State assumptions explicitly: record linearity, steady-state or transient behavior, small-deformation limits, continuum assumptions, and any symmetry or ideal boundary conditions.
Compare with a baseline: use a hand calculation, limiting case, mesh refinement trend, or independent solver result before accepting the final value.

Validation signals

Flow Type	RNG Superiority	Reason
CFD for Indoor Air Conditioning	High

Review item	What to verify	Typical warning sign
Inputs	Geometry, material data, loads, and constraints match the intended fluid simulation problem.	Correct-looking plots with unrealistic magnitudes or units.
Numerics	Mesh, time step, convergence tolerance, and solver options are adequate for K Epsilon Rng.	Large changes after small mesh or tolerance adjustments.
Physics	The selected theory remains valid in the expected stress, temperature, velocity, or frequency range.	Results are used outside the assumptions stated in the model.

For production use, keep the model file, input table, result plots, and review comments together. This makes RNG k-epsilon model traceable and prevents the page from being used as a black-box answer without engineering judgment.

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About the Authors

RNG k-epsilon model

RNG k-epsilon: Theoretical Foundations

Overview

Governing Equations

What is "Renormalization Group"? — The Day Physics Techniques Came to CFD

Computational Methods for RNG k-epsilon

Numerical Implementation

Linearization of the R-term

Wall Treatment

OpenFOAM Settings

Fluent Settings

Why RNG k-ε Shines in Swirl Combustors

RNG k-epsilon in Practice

RNG k-epsilon in Practice

Recommended Application Cases

Practical CAE quality notes for RNG k-epsilon model

Model setup checklist

Validation signals

Related Topics

Related fields