k-epsilon Models: Standard, Realizable, and RNG

The k-epsilon model is widely used to simulate turbulent flows in CFD. It estimates turbulence by solving for two variables: turbulent kinetic energy (k), which represents the energy contained in the turbulent eddies, and dissipation rate (epsilon), which describes how quickly this turbulent kinetic energy is converted. These models fall under the RANS (Reynolds-Averaged Navier-Stokes) approach, where the effects of turbulence are averaged over time, allowing the prediction of mean flow characteristics. In this article, we will explore the features, applications, and differences between three main variants of the k-epsilon model: Standard, RNG, and Realizable.

1. The Standard k-epsilon model: A good starting point

The Standard k-epsilon model is a fundamental turbulence model developed by Launder and Spalding. It is widely used due to its simplicity and applicability in fully turbulent flows. This model is a reliable starting point. It is best suited for cases where the turbulence is well-developed and the flow does not involve complex factors like strong swirling or flow separation.

The Standard k-epsilon model solves two key transport equations:

• Turbulent kinetic energy (k): Represents the energy contained in the turbulent eddies, governed by the equation:

• Dissipation rate (epsilon): Describes how quickly this turbulent kinetic energy is converted, represented by the equation:

The Standard model is ideal for general-purpose simulations. It performs well in cases where turbulence is well-established and the flow does not involve excessive complexity. It’s a great starting point for:

• Homogeneous turbulent flows where the behavior is uniform.
• Well-developed turbulent conditions that do not involve significant complexities like swirling or strong pressure gradients.

However, this model can be limiting when dealing with more complex flows, such as strong rotation or flow separation, which leads to more specialized approaches.

2. The RNG k-epsilon model: Enhanced for complex flows

The RNG (Renormalization Group) k-epsilon model builds on the standard model by incorporating additional features that improve its accuracy for complex flows. This model is better adapted to problems involving rapid strain or swirling motion.

The RNG model modifies the dissipation rate equation by adding an extra term R_epsilon, which helps capture the effects of rapid strain and swirling. This enhancement allows for more precise predictions in scenarios with more chaotic turbulence.

• Turbulent kinetic energy (k) equation:

• Dissipation rate (epsilon) equation:

The RNG k-epsilon model is particularly used for:

• Flows with high strain rates: This is adapted to flows with sudden changes in velocity.
• Swirling and rotational flows: It effectively handles scenarios where swirling motion plays a significant role, such as cyclones or vortex flows.

The RNG model is more accurate than the Standard model in handling complex turbulence, particularly in cases with significant rotational or rapid changes in flow velocity.

3. The Realizable k-epsilon Model: The most versatile model

The Realizable k-epsilon model represents the most advanced variant in the k-epsilon family. It is called "realizable" because it adheres to certain mathematical constraints that the other models do not, allowing for more accurate predictions in complex flows.

The Realizable model modifies the formulation for turbulent viscosity, enabling it to adjust dynamically based on local flow conditions. This adaptability enhances the model's performance in predicting flow separation, recirculation, and swirling flows.

• Turbulent kinetic energy (k) equation:

• Dissipation rate (epsilon) equation:

The Realizable model excels in:

• Flows with significant flow separation: It’s particularly effective at predicting flow patterns around surfaces where separation is critical.
• Recirculating flows: This model is designed to handle complex flow patterns effectively.
• Jet flows: It is excellent for capturing the behavior of jets and their spreading rates.

A key feature of the Realizable model is the dynamic formulation of Cμ. Unlike the constant Cμ used in the Standard model, the Realizable model computes Cμ as a function of the local flow field, which provides:

• Adaptability: The dynamic definition of Cμ allows the turbulent viscosity to respond to the local flow environment, particularly in regions of strong shear or complex geometries. This is crucial for accurately modeling turbulence in flows where conditions change rapidly or flow structures are intricate.
• Enhanced Predictions: By incorporating a variable Cμ, the Realizable k-epsilon model can more effectively simulate the effects of turbulence on momentum transfer, resulting in better predictions for flows with significant non-linear characteristics, such as swirling or recirculating flows.

4. Which k-epsilon model should you choose?

Choosing the right k-epsilon model depends on the flow nature:

• The Standard k-epsilon model is ideal for simple, homogeneous turbulent flows, such as pipe flows or boundary layers, where turbulence is well-established and steady.
• The RNG k-epsilon model excels in flows with high strain rates and swirling, such as combustion chambers or cyclones, where turbulence evolves rapidly.
• The Realizable k-epsilon model is the most versatile. It performs well in complex flows with flow separation, recirculation, or adverse pressure gradients, making it suitable for applications like aerospace, automotive aerodynamics, and turbomachinery.

Understanding each model's strengths, allows adapting turbulence simulations to fit your flow's specific behavior.

Conclusion

The Standard, RNG, and Realizable k-epsilon models each bring unique strengths to turbulence modeling. Understanding their differences will allow you to choose the appropriate model for your specific flow conditions and achieve more accurate simulation results. As always, validating your results with experimental or benchmark data and fine-tuning model parameters like mesh size, boundary conditions, and turbulence intensity are crucial to ensure accuracy in complex flows.

References

For more details, please check this Fluent reference: https://courses.washington.edu/mengr544/handouts-10/Fluent-k-epsilon.pdf