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A Physics Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies Based on DNS Data
Jian-Xun Wang, Jin-Long Wu, Heng Xiao
TL;DR
The paper addresses the lack of universally predictive RANS models by learning Reynolds-stress discrepancies from mean-flow features with random forests. The approach is trained on benchmark data and applied to new periodic-hill flows, where corrected Reynolds stresses were significantly improved over baseline RANS predictions in both scenarios. However, improved Reynolds stresses do not necessarily guarantee improved propagated mean-flow fields.
Problem
Standard RANS models still lack universally applicable predictive capabilities, although improving them remains practically important for industrial CFD.
Method
The method uses random forests to learn Reynolds-stress discrepancy functions formulated from mean-flow features and trained with benchmark flow data.
Results
Corrected Reynolds stresses were significantly improved over baseline RANS predictions in both tested scenarios.
Takeaways & Limitations
Improving RANS-modeled Reynolds stresses is an important step toward predictive, data-driven turbulence modeling with standard RANS models.
Takeaways & Limitations
Improved Reynolds stresses from DNS-trained discrepancy functions cannot guarantee improved propagated mean-flow fields.
Abstract
from arXiv · showhide
Turbulence modeling is a critical component in numerical simulations of industrial flows based on Reynolds-averaged Navier-Stokes (RANS) equations. However, after decades of efforts in the turbulence modeling community, universally applicable RANS models with predictive capabilities are still lacking. Recently, data-driven methods have been proposed as a promising alternative to the traditional approaches of turbulence model development. In this work we propose a data-driven, physics-informed machine learning approach for predicting discrepancies in RANS modeled Reynolds stresses. The discrepancies are formulated as functions of the mean flow features. By using a modern machine learning technique based on random forests, the discrepancy functions are first trained with benchmark flow data and then used to predict Reynolds stresses discrepancies in new flows. The method is used to predict the Reynolds stresses in the flow over periodic hills by using two training flow scenarios of increasing difficulties: (1) the flow in the same periodic hills geometry yet at a lower Reynolds number, and (2) the flow in a different hill geometry with a similar recirculation zone. Excellent predictive performances were observed in both scenarios, demonstrating the merits of the proposed method. Improvement of RANS modeled Reynolds stresses enabled by the proposed method is an important step towards predictive turbulence modeling, where the ultimate goal is to predict the quantities of interest (e.g., velocity field, drag, lift) more accurately by solving RANS equations with the Reynolds stresses obtained therefrom.
1. Introduction
RANS remains practically important for industrial CFD, but improving its predictive capabilities is challenging. Data-driven discrepancy modeling aims to improve standard RANS predictions across similar yet different flows.
- RANS Models as Workhorse Tool in Industrial CFD: RANS remains a dominant and practically important tool for industrial CFD, despite the availability of more computationally expensive alternatives.Standard eddy-viscosity models remain important because advanced Reynolds-stress models are more expensive and less robust.
- Data-Driven Turbulence Modeling: Data-driven approaches use offline or online flow data to infer discrepancies in modeled turbulence quantities and improve predictive turbulence modeling.Prior work considered discrepancies in turbulent viscosity, transport-equation terms, or Reynolds-stress projections.
- Limitations of Prior Approaches: Earlier coordinate-based discrepancy functions were limited in extrapolating beyond the same geometry and location, with less success reported for different geometries.This limitation affected approaches indexed by physical coordinates rather than flow features.
- Proposed Approach: The proposed extension constructs Reynolds-stress discrepancy functions from well-chosen mean-flow features rather than physical coordinates.The formulation retains normalized Reynolds-stress projections, including anisotropy parameters and orientations, as discrepancy targets.
- Proposed Approach: Feature-based discrepancy functions are intended to support extrapolation across a wider range of flows sharing characteristics such as separation.Random forests are used as the function-construction method, with decision trees capturing feature interactions.
- Objective and Scope: The work targets improved RANS Reynolds-stress predictions as a step toward data-driven turbulence modeling using standard RANS models with offline databases.The stated longer-term objective is more accurate prediction of quantities of interest through RANS equations using improved Reynolds stresses.
2. Methodology
The paper develops a physics-informed machine-learning framework that learns Reynolds-stress discrepancies from high-fidelity training flows and predicts them in related test flows. The workflow uses RANS-derived mean-flow features, regression functions, and physical constraints to correct baseline RANS stresses.
- 2.1. Problem Statement: Physics-informed design incorporates domain knowledge through physically motivated inputs and realizability constraints during learning and prediction.The framework emphasizes accounting for physical domain knowledge at every machine-learning stage.
- 2.1. Problem Statement: The framework uses high-fidelity Reynolds-stress data from training flows to predict a new test flow lacking such data.Training and test flows are distinguished by the availability of high-fidelity simulation data.
- 2.1. Problem Statement: The study assumes training and test flows have similar complexities and dominant characteristics; broader test flows containing unseen characteristics remain outside its scope.The authors describe the broader scenario as more challenging and defer it to future work.
- 2.1. Problem Statement: The method represents Reynolds-stress discrepancies as regression functions mapping RANS-derived mean-flow features q to discrepancy responses Δτα.The functions are trained from discrepancies computed using high-fidelity data and later queried for test-flow predictions.
- 2.1. Problem Statement: The five-step procedure performs baseline RANS simulations, computes feature fields and training discrepancies, constructs regression functions, and corrects test-flow stresses.The final correction combines baseline RANS predictions with evaluated discrepancies.
- 2.3. Choice of Mean Flow Features as Regression Input: Random forests are selected to construct the discrepancy regression functions, using mean-flow features such as pressure-gradient measures, curvature, Q-criterion, and wall-distance Reynolds number.The features are designed to be Galilean-invariant, primarily local, and computable from RANS-predicted fields.
3. Numerical Results
The PIML framework improves Reynolds-stress predictions for square-duct and periodic-hill flows by learning discrepancies from related training flows. Performance is strongest in supported flow regions and weakens where test-flow features are absent from training data.
- Square-duct flow: Square-duct DNS anisotropy approaches the one-component state away from the wall, unlike the baseline RANS trajectory toward three-component anisotropy.Near-wall RANS anisotropy also differs substantially from DNS, affecting secondary-flow prediction.
- Square-duct flow: PIML-corrected square-duct anisotropy agrees well with DNS, with near-identical predictions on the line y/H = 0.75.Improvements extend across barycentric, TKE, and orientation projections.
- Square-duct flow: The baseline RSTM overestimates both τyy and τzz, producing an inaccurate normal-stress imbalance and unreliable secondary mean flow motion.τyy errors are largest near the wall, whereas τzz errors are largest farther from it.
- Square-duct flow: PIML predictions nearly overlap DNS for τyy and τzz, with improvements reported for all Reynolds-stress tensor components.These results demonstrate strong performance of the proposed framework with RSTM as baseline.
4. Discussion
The framework uses random-forest feature importance to interpret how mean-flow features relate to Reynolds-stress discrepancies. It shows strong predictive performance across stress projections while retaining important limitations in interpretability and spatial smoothness.
- Feature importance: Feature importance identifies q3, the wall-distance-based Reynolds number Red, as the most important predictor of anisotropy discrepancy Δη.This agrees with predictions showing that Δη depends notably on distance from the wall.
- Feature importance: Feature q2, turbulence intensity, is the most important predictor of the turbulence-kinetic-energy discrepancy Δlog k.
- Interpretability: Random forests can interpret feature–response relationships to a certain extent, although feature-importance measures may be biased under some conditions.Feature interactions may also be examined because decision trees can capture interactions among mean-flow variables.
- Framework contribution: The framework aims to improve baseline RANS-predicted Reynolds stresses by learning discrepancy functions from mean-flow features in closely related flows.Its novelty lies in using machine learning to identify functional forms for Reynolds-stress discrepancies.
- Predictive performance: Excellent PIML performance was obtained for anisotropy, turbulent kinetic energy, and turbulent shear stress, indicating extrapolation to complex flows with similar characteristics.The improvement targets Reynolds stresses as a step toward better predictions of velocities and other quantities of interest.
- Limitations: Corrected Reynolds stresses do not guarantee improved velocities because unsupported feature regions, pointwise regression, and insufficient features can produce nonsmooth or abnormal corrections.Small-amplitude stress fluctuations can create abnormal behavior in the divergence term and predicted velocities.
5. Conclusion
The proposed physics-informed machine learning approach reconstructs Reynolds stress discrepancies from mean-flow features and improves corrected Reynolds stresses across two increasingly challenging flow scenarios. Its broader value is supporting predictive RANS modeling while exposing which mean-flow features contribute to modeling discrepancies.
- Contribution: The approach formulates Reynolds stress discrepancies as functions of mean-flow features and learns them with random forest regression for prediction in new flows.The target can represent Reynolds stress magnitude, anisotropy shape, and anisotropy orientation rather than individual stress components.
- Evaluation: The evaluation considers lower-Reynolds-number training flows in the same periodic-hill geometry and separated training flows in different geometries.The two scenarios increase difficulty from shared geometry to geometric extrapolation.
- Results: Corrected Reynolds stresses are significantly improved over baseline RANS predictions in all test cases.The reported improvement is observed across both training scenarios.
- Results: The improvement is less drastic for different training and prediction geometries because that scenario requires more extrapolation in feature space.The prediction flow is less similar to the training flows than in the identical-geometry scenario.
- Implications: Improving modeled Reynolds stresses is an important step toward predictive RANS capabilities, but further challenges remain before accurate engineering quantities of interest can be predicted.Future work is needed to connect corrected stresses with quantities such as drag and lift coefficients.
- Implications: Random forests provide insights into the relative importance of mean-flow features contributing to Reynolds stress discrepancies.These insights can assist future turbulence-model development by identifying responses models should represent.