Advances in Turbulence Modelling with Machine Learning

Turbulence is a complex and chaotic motion that occurs in fluids, like air and water. Understanding this phenomenon is essential for many fields, including engineering, meteorology, and oceanography. In many cases, it is not feasible to capture all the details of turbulent flows due to the immense computing power required. Therefore, scientists and engineers use specific methods, known as Turbulence Models, to simplify these flows and make predictions.

#Types of Turbulence Models

There are several techniques for modelling turbulence, each with its advantages and limitations. The most commonly used methods include:

1. Reynolds-Averaged Navier-Stokes (RANS)
2. Large Eddy Simulation (LES)
3. Detached Eddy Simulation (DES)

Each of these methods serves different purposes based on the available computing resources and the nature of the flow being studied.

#Reynolds-Averaged Navier-Stokes (RANS)

RANS is the most popular method in industry because it is simpler and requires less computational power. This approach averages the effects of turbulence to create a single set of equations. However, it often leads to less accurate predictions, especially in complex flows.

#Large Eddy Simulation (LES)

LES captures larger turbulent structures while modelling smaller ones. It offers better accuracy than RANS but is more demanding in terms of computational power. Consequently, it is typically used in research rather than in everyday industrial applications.

#Detached Eddy Simulation (DES)

DES is a hybrid approach that combines RANS and LES, allowing for efficient computations while still capturing essential details of turbulent flows. It is useful for flows with significant separation and reattachment.

#The Role of Machine Learning in Turbulence Modelling

Recently, there has been growing interest in applying machine learning techniques to enhance turbulence modelling. By leveraging large datasets, machine learning can help create more accurate turbulence models that adapt to specific flow conditions.

Machine learning methods can:

• Speed up simulations
• Improve model calibration
• Optimize coefficients for different flows
• Enhance the accuracy of turbulence predictions

One crucial aspect of this approach is the training of specialized turbulence models using existing datasets. This helps to refine the predictions for specific flows, increasing the reliability of the models.

#Challenges in Turbulence Modelling

While machine learning presents new opportunities, several challenges remain in developing and implementing these models:

#Data Quality and Quantity

Machine learning models require high-quality data to be effective. However, gathering sufficient data for complex turbulent flows can be difficult. Often, available datasets do not cover the entire range of possible flow conditions, limiting the model's generalization ability.

#Generalization to New Flows

Machine learning models often struggle to adapt to new flow configurations. This challenge is known as the "no-free-lunch theorem," which states that no single model works best for every situation. Therefore, while a model might excel in one scenario, it may not perform well in another, especially if the training data does not represent that flow type.

#Realizability

Realizability refers to the physical feasibility of the model predictions. Many traditional turbulence models do not guarantee that the predicted values are physically realistic, leading to results that might not accurately represent real-world behaviour.

#Proposed Approach to Address the Challenges

To tackle the issues mentioned above, a new framework has been proposed that integrates machine learning with traditional turbulence modelling techniques. This framework aims to maintain the benefits of established models while embedding machine-learning capabilities to enhance accuracy and stability.

#Realizability-Informed Loss Function

A key aspect of this new approach is the implementation of a loss function that penalizes non-realizable predictions during the training process. By incorporating this penalty, the model learns a bias towards producing more physically valid predictions.

#Input Features Selection

Choosing the right input features is essential for the performance of machine learning models. The proposed method focuses on using a comprehensive set of features that provide valuable information about the flow conditions. This selection is done to ensure the model can effectively learn the relationships between the input features and the outcomes.

#Modified Neural Network Architecture

Improvements have been made to the structure of the neural networks used in the model. These modifications enhance the training and testing stability, ensuring better performance when applied to different flow cases.

#Applications of the New Approach

The new framework has been applied to various flow scenarios, demonstrating its effectiveness in enhancing turbulence modelling. These scenarios include:

1. Flow over a flat plate
2. Flow through a square duct
3. Flow over periodic hills

#Flow Over a Flat Plate

This scenario involves a developing turbulent boundary layer, which is a common condition in fluid dynamics. The model was tested against reference data and showed significant improvements in predicting the anisotropy tensor, a key component in turbulence modelling.

#Flow Through a Square Duct

Flows through square ducts present unique challenges due to the occurrence of secondary flows. The proposed framework successfully captured these secondary flows, highlighting its ability to handle complex flow patterns effectively.

#Flow Over Periodic Hills

This case involves turbulent flow in areas with changing geometry. The model accurately predicted flow behaviour, including reattachment and separation, essential for understanding adverse pressure gradients.

#Results and Findings

The implementation of the new approach yielded several key findings:

#Enhanced Generalization

The modified model demonstrated improved generalization capabilities across different flow configurations. This shows that with sufficient training data and the right architecture, machine learning can effectively augment conventional turbulence models.

#Improved Stability

By embedding machine learning into traditional models, the overall stability of the predictions during the injection process was enhanced. This stability is crucial for practical applications in engineering and industry.

#Realizability Improvements

The use of a realizability-informed loss function successfully reduced the number of non-realizable predictions, resulting in outputs that align more closely with physical expectations.

#Future Directions

While significant progress has been made, there are still areas to explore in turbulence modelling with machine learning. Future research should focus on:

• Generating larger and more diverse datasets to improve model training.
• Investigating further enhancements in model architectures and training processes.
• Exploring the application of these techniques to more complex and varied flow scenarios.

#Conclusion

The integration of machine learning into turbulence modelling presents a promising avenue for improving predictions in fluid dynamics. By addressing challenges related to data quality, generalization, and realizability, the proposed framework demonstrates how traditional methods can evolve with new technologies.

As research continues in this field, we can expect to see advances that will bridge the gap between classical turbulence modelling and modern computational capabilities, ultimately leading to more accurate and reliable simulations in various applications.