Introduction to RANS Turbulence Models

What are RANS Turbulence Models?

Turbulence is a fundamental phenomenon in fluid mechanics that occurs when fluid flows become highly complex and unpredictable. While the Navier-Stokes equations can describe the behavior of turbulent flows theoretically, they are extremely challenging to solve in practice due to their nonlinearity. To mitigate this difficulty, engineers have developed various turbulence models that offer simplified representations of turbulent flows. One popular category of turbulence models is known as two-equation turbulence models.

The Reynolds' Averaged Navier-Stokes Equations

At the core of two-equation turbulence models lies the Reynolds-averaged Navier-Stokes (RANS) equations. These equations are fundamental in understanding and predicting the behavior of turbulent flows in various engineering applications. Turbulence, characterized by chaotic and unpredictable fluid motion, poses significant challenges in fluid dynamics.

The RANS equations are obtained by averaging the governing Navier-Stokes equations over time. This averaging process effectively separates the flow variables into mean (time-averaged) and fluctuating components. By doing so, the RANS equations provide a simplified representation of turbulence, where the time-dependent fluctuations are parameterized by additional transport equations.

By employing the RANS equations, two-equation turbulence models offer a practical means to predict the mean flow characteristics of turbulent flows. These models strike a balance between accuracy and computational efficiency, making them widely used in engineering applications. Compared to more computationally intensive methods such as large eddy simulation, two-equation models provide a cost-effective approach to studying turbulent flows.

One of the key advantages of two-equation turbulence models is their ability to capture the essential features of turbulence without explicitly resolving all the small-scale turbulent structures. This is achieved through the introduction of additional transport equations for the turbulent kinetic energy and its dissipation rate. These equations account for the energy transfer between the mean flow and the turbulent fluctuations, allowing for a more comprehensive understanding of the flow behavior.

Furthermore, two-equation models provide insights into the turbulence closure problem, which involves determining the appropriate closure coefficients for the transport equations. These coefficients are typically calibrated using experimental data or high-fidelity simulations, ensuring that the model accurately represents the specific flow conditions under investigation.

Despite their simplifications, two-equation turbulence models have been extensively validated and proven to yield reliable predictions for a wide range of turbulent flows. They have been successfully applied in various fields, including aerospace engineering, automotive design, and environmental fluid dynamics.

In conclusion, the Reynolds-averaged Navier-Stokes equations form the foundation of two-equation turbulence models, enabling engineers and researchers to study and predict the behavior of turbulent flows. These models strike a balance between accuracy and computational efficiency, making them invaluable tools in engineering applications. By parameterizing the time-dependent fluctuations, two-equation models provide a practical approach to understanding turbulence and its impact on fluid dynamics.

The Closure Problem and the Reynolds' Stress Tensor

One of the essential assumptions underlying two-equation turbulence models is the closure problem. The closure problem arises from the need to parameterize the effects of fluctuating turbulent stresses, known as Reynolds stresses, in terms of the mean flow variables. In other words, it involves finding an appropriate relationship between the Reynolds stresses and the mean flow velocity gradients.

To address this challenge, two-equation turbulence models propose simplifying assumptions and closure relations. These models use transport equations for two turbulence variables, typically turbulence kinetic energy (k) and its dissipation rate (ε). Through these transport equations and additional assumptions, the models aim to capture the effects of turbulence on the mean flow.

The Reynolds' stress tensor is a key quantity in two-equation turbulence models as it represents the anisotropy of turbulence. While the RANS equations provide equations for the mean flow variables, they do not explicitly account for the Reynolds stresses. Two-equation turbulence models introduce additional equations for the Reynolds stress tensor components to approximate its behavior based on the turbulence variables.

Let's delve deeper into the closure problem and its significance in turbulence modeling. The closure problem arises due to the inherent complexity of turbulent flows. Turbulence is characterized by chaotic and random fluctuations in velocity and pressure, making it challenging to accurately predict its behavior. In order to simulate turbulent flows numerically, simplifications and assumptions are necessary.

Two-equation turbulence models attempt to solve the closure problem by introducing transport equations for turbulence variables. These equations describe the evolution of turbulence kinetic energy and its dissipation rate throughout the flow domain. By solving these equations along with the governing equations for mean flow variables, a more comprehensive understanding of the flow behavior can be obtained.

However, the closure problem still persists as the transport equations for turbulence variables rely on closure relations. These relations provide a link between the Reynolds stresses and the mean flow velocity gradients. Various closure relations have been proposed over the years, each with its own set of assumptions and limitations.

The Reynolds' stress tensor plays a crucial role in two-equation turbulence models. It represents the anisotropy of turbulence, capturing the directional dependence of turbulent fluctuations. The Reynolds stress tensor consists of six components, three normal stresses and three shear stresses, which describe the distribution of turbulent stresses in different directions.

In order to approximate the behavior of the Reynolds stress tensor, two-equation turbulence models introduce additional equations for its components. These equations are derived based on statistical assumptions and empirical data. By solving these equations along with the transport equations for turbulence variables, the Reynolds stresses can be estimated and incorporated into the mean flow equations.

It is important to note that while two-equation turbulence models provide a practical approach to tackle the closure problem, they are not without limitations. The simplifications and assumptions inherent in these models may lead to inaccuracies in certain flow regimes or geometries. Researchers continue to develop and refine turbulence models to improve their predictive capabilities and address the challenges posed by the closure problem.

The Boussinesq Eddy Viscosity Assumption

A fundamental assumption made by two-equation turbulence models is the Boussinesq eddy viscosity assumption. This assumption posits that the Reynolds stresses can be related to the mean velocity gradients through an eddy viscosity coefficient. The eddy viscosity coefficient represents the turbulent momentum transfer and effectively replaces the molecular viscosity in the Navier-Stokes equations.

The Boussinesq eddy viscosity assumption simplifies the modeling of turbulence by reducing the number of unknowns in the system of equations. However, this assumption also introduces limitations, as it assumes that the turbulent flows are isotropic and in equilibrium. Consequently, two-equation turbulence models may struggle to accurately capture highly anisotropic or rapidly changing turbulence phenomena.

Spalart–Allmaras, k–epsilon, and k–omega Models

There are several well-known eddy viscosity turbulence models that have been developed and widely adopted in engineering practice. Three notable examples are the Spalart–Allmaras model, the k–epsilon model, and the k–omega model.

The Spalart–Allmaras model is a one-equation model that assumes a turbulent viscosity based on a single transport equation for the turbulence variable. It offers computational efficiency and has been particularly successful in simulating transitional flows.

The k–epsilon model is a two-equation model that accounts for both turbulence kinetic energy (k) and the rate of dissipation (epsilon). It offers good performance for a wide range of turbulent flows, but may struggle in the presence of adverse pressure gradients and swirling flows.

The k–omega model is another two-equation model that is designed to address some of the limitations of the k–epsilon model. It employs a transport equation for turbulence kinetic energy (k) and another for the specific dissipation rate (omega). The k–omega model performs well in simulating boundary layer flows but may require additional modifications for scenarios involving complex flow separation.

Conclusion

In conclusion, two-equation turbulence models represent a significant advancement in the simulation of turbulent flows. By utilizing the RANS equations, closure relations, and the Boussinesq eddy viscosity assumption, these models offer a balance between accuracy and computational efficiency. While limitations exist, the Spalart–Allmaras, k–epsilon, and k–omega models have proven to be valuable tools in engineering analyses where a simplified description of turbulence is desired.