A Comprehensive Physics-Informed Machine Learning Framework for Predictive Turbulence Modeling

Abstract: Although an increased availability of computational resources has enabled high-fidelity simulations of turbulent flows, the RANS models are still the dominant tools for industrial applications. However, the predictive capabilities of RANS models are limited by potential inaccuracy driven by hypotheses in the Reynolds stress closure. Recently, a Physics-Informed Machine Learning (PIML) approach has been proposed to learn the functional form of Reynolds stress discrepancy in RANS simulations based on available data. It has been demonstrated that the learned discrepancy function can be used to improve Reynolds stresses in different flows where data are not available. However, owing to a number of challenges, the improvements have been demonstrated only in the Reynolds stress prediction but not in the corresponding propagated quantities of interest. In this work, we introduce the procedures toward a complete PIML framework for predictive turbulence modeling, including learning Reynolds stress discrepancy function, predicting Reynolds stresses in different flows, and propagating to mean flow fields. The process of Reynolds stress propagation and predictive accuracy of the propagated velocity field are investigated. To improve the learning-prediction performance, the input features are enriched based on an integrity basis of invariants. The fully developed turbulent flow in a square duct is used as the test case. The discrepancy model is trained on flow fields obtained from several Reynolds numbers and evaluated on a duct flow at a Reynolds number higher than any of the training cases. The predicted Reynolds stresses are propagated to velocity field through RANS equations. Numerical results show excellent predictive performances in both Reynolds stresses and their propagated velocities, demonstrating the merits of the PIML approach in predictive turbulence modeling.

• The paper introduces a physics-informed ML workflow that corrects Reynolds stress discrepancies in RANS models.
• It leverages an invariant input feature set and random forest regression to achieve predictions closely aligned with DNS benchmarks.
• Propagation of corrected stresses significantly improves mean velocity predictions, advancing turbulence modeling in complex flows.

A Comprehensive Physics-Informed Machine Learning Framework for Predictive Turbulence Modeling

The study presents a refined, comprehensive framework for leveraging Physics-Informed Machine Learning (PIML) to enhance predictive capabilities in Reynolds-Averaged Navier-Stokes (RANS) turbulence modeling. Despite the capability of high-fidelity simulations like LES and DNS, RANS models remain widely utilized for industrial turbulent flow analysis due to computational expense constraints. However, RANS models suffer from inaccuracies mainly due to limitations in Reynolds stress closures.

Methodological Approach and Contributions

The paper introduces a noteworthy advancement, presenting a workflow that utilizes PIML to specifically address discrepancies within RANS-modeled Reynolds stresses. This approach is designed to improve RANS predictions by training on high-fidelity data from simpler flows and applying this knowledge to flows lacking such data, thus effectively enhancing the accuracy of mean velocity and other propagated quantities of interest.

Key components of the methodology include:

1. Invariant Input Feature Set Construction: An expanded feature set is constructed, drawing on invariance properties and high-fidelity mean flow characteristics to form a more robust basis for regression models. This systematic inclusion of invariants helps optimize the predictive capability of machine learning models for turbulence.
2. Reynolds Stress Discrepancy Representation: Discrepancies in Reynolds stresses are represented across multiple dimensions—magnitude, shape, and orientation. By incorporating the eigen-decomposition technique, the approach accounts for more precise tensor component predictions.
3. Machine Learning Model Deployment: A Random Forest algorithm is selected to predict Reynolds stress discrepancies based on the comprehensive feature set due to its strength in handling high-dimensional regression issues effectively.
4. Propagation of Corrected Predictions: Lastly, the corrected Reynolds stresses are reintegrated into RANS models to yield better predictions of the mean velocity fields, evaluated through comparison with DNS benchmarks.

Results and Insights

The fully developed turbulent duct flow scenario serves to demonstrate the framework, with training conducted on lower Reynolds number cases and predictions made on a significantly higher Reynolds flow. The findings reveal:

• Reynolds Stress Predictions: Leveraging invariant features substantially enhances prediction accuracy for discrepancies in both the magnitude of turbulent kinetic energy (TKE) and anisotropy of the Reynolds tensor, particularly in complex near-wall regions.
• Velocity Field Propagation: The propagation of corrected Reynolds stresses yields velocity fields that align closely with DNS results, surpassing RSTM baselines, especially in secondary flow scenarios governed by normal-stress imbalances.
• Input Feature Space Augmentation: The paper conclusively shows that enriching the invariant input feature space contributes significantly to optimized predictions, mitigating non-physical resultant artifacts in the velocity field.

Implications and Future Directions

This framework enhances the viability of PIML approaches for practical turbulence modeling by addressing inherent biases in RANS models, suggesting broader applications where high-fidelity data are scarce. These results suggest future investigations may extend to include broader flow regimes and further refine physical closure models, fostering developments across domains requiring empirical sub-model corrections.

Despite the promising performance in the examined case of a square duct, generalization across various turbulent configurations remains a challenge and area for further exploration. Moreover, addressing the prediction smoothness and derivative accuracy in propagated flow fields represents an ongoing opportunity for enhancing the robustness of velocity predictions.

Ultimately, while the paper refrains from sensational claims, it underscores significant potential in synergizing machine learning with physical models to expedite advancements in simulation fidelity, paving the way for innovative turbulence modeling methodologies.