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Physics-Informed Machine Learning Approach for Augmenting Turbulence Models: A Comprehensive Framework
Jin-Long Wu, Heng Xiao, Eric Paterson
TL;DR
Data-driven turbulence models face ill-conditioning when improved Reynolds stresses are used to obtain mean velocities. The paper presents a physics-based implicit treatment that systematically selects mean-flow features and separately models linear and nonlinear Reynolds-stress components, enabling accurate mean-velocity prediction.
Problem
Ill-conditioning in the RANS equations remains a common challenge for data-driven turbulence models, even when Reynolds-stress discrepancies are modeled.
Method
The framework systematically chooses input features based on mean-flow tensors and uses machine learning to model Reynolds stress through separate linear and nonlinear components.
Results
The proposed treatment enabled accurate prediction of mean velocities with a data-driven Reynolds-stress model.
Takeaways & Limitations
The framework addresses ill-conditioning by using a physics-based implicit treatment of Reynolds stress rather than directly relying on improved Reynolds stresses alone.
Takeaways & Limitations
The discussion identifies missing physics in the assumption underlying the choice of mean-flow feature variables.
Abstract
from arXiv · showhide
Reynolds-averaged Navier-Stokes (RANS) equations are widely used in engineering turbulent flow simulations. However, RANS predictions may have large discrepancies due to the uncertainties in modeled Reynolds stresses. Recently, Wang et al. demonstrated that machine learning can be used to improve the RANS modeled Reynolds stresses by leveraging data from high fidelity simulations (Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Physical Review Fluids. 2, 034603, 2017). However, solving for mean flows from the improved Reynolds stresses still poses significant challenges due to potential ill-conditioning of RANS equations with Reynolds stress closures. Enabling improved predictions of mean velocities are of profound practical importance, because often the velocity and its derived quantities (QoIs, e.g., drag, lift, surface friction), and not the Reynolds stress itself, are of ultimate interest in RANS simulations. To this end, we present a comprehensive framework for augmenting turbulence models with physics-informed machine learning, illustrating a complete workflow from identification of input features to final prediction of mean velocities. This work has two innovations. First, we demonstrate a systematic procedure to generate mean flow features based on the integrity basis for mean flow tensors. Second, we propose using machine learning to predict linear and nonlinear parts of the Reynolds stress tensor separately. Inspired by the finite polynomial representation of tensors in classical turbulence modeling, such a decomposition is instrumental in overcoming the ill-conditioning of RANS equations. Numerical tests demonstrated merits of the proposed framework.
1. Introduction
RANS remains central to engineering turbulent-flow simulation, but modeled Reynolds stresses can produce large discrepancies and ill-conditioned mean-velocity solutions. This work develops a physics-informed machine-learning framework that systematically selects mean-flow features and separately learns linear and nonlinear Reynolds-stress components to improve mean-velocity prediction.
- RANS remains a work-horse tool for engineering turbulent-flow simulations, despite large discrepancies in flows with swirl, pressure gradients, or streamline curvature.
- Modeled Reynolds stresses are identified as the dominant cause of discrepancies in RANS predictions.
- Machine-learning approaches leverage high-fidelity data to predict Reynolds stresses or their discrepancies, with encouraging results in canonical separated and secondary-flow cases.
- Small Reynolds-stress errors can become large mean-velocity errors because RANS equations with Reynolds-stress closures may be ill-conditioned, especially as Reynolds number increases.
- The framework systematically constructs mean-flow features from tensorial invariance and avoids ad hoc scalar-feature selection.
- Reynolds stresses are learned as separate linear and nonlinear parts, enabling implicit treatment of the linear term and improving conditioning without ad hoc blending.
- Numerical examples show that the proposed treatment enables accurate prediction of mean velocities with a data-driven Reynolds-stress model.
2. Methodology
The framework augments RANS turbulence models with physics-informed machine learning while addressing feature selection, rotational invariance, and ill-conditioning in Reynolds-stress closures. It separates Reynolds-stress anisotropy into linear and nonlinear parts so the linear contribution can be treated implicitly when solving RANS equations.
- Motivation: RANS turbulence models map mean velocity fields to Reynolds stresses, whose inaccuracies can limit CFD prediction accuracy.Mean velocity and derived quantities such as drag and lift are often the engineering targets.
- Motivation: Machine learning is used to predict Reynolds-stress discrepancies from mean-flow features and DNS data.The discrepancy is defined as ∆τ ≡ τ_DNS − τ_RANS.
- Reynolds-stress representation: The anisotropy Reynolds-stress tensor is decomposed into linear and nonlinear parts, with the linear term treated implicitly to improve RANS conditioning.The nonlinear component remains important near walls and in swirling or impinging flows, where linear models can miss relevant physics.
- Reynolds-stress representation: Machine learning regression functions predict the optimal eddy-viscosity coefficient and the nonlinear anisotropy component from the selected features.The optimal eddy viscosity minimizes the discrepancy between the anisotropy stress tensor and its linear part.
- Feature construction: Input features are normalized and designed to support physical meaning, extrapolation, and invariance under coordinate and reference-frame transformations.Raw features are normalized by local quantities consistent with traditional turbulence modeling practice.
- Feature construction: The single-point constitutive assumption is extended with pressure-gradient and TKE-gradient inputs because turbulence can exhibit nonequilibrium effects and convection or diffusion.The framework includes low-Reynolds-number information for near-wall modeling and supplements tensor features with turbulence scales.
- Feature construction: The framework constructs mean-flow features from physically motivated variables, including strain rate, rotation rate, pressure gradient, and TKE gradient.Supplementary features provide wall-distance, turbulence length-scale, and time-scale information.
- Invariance considerations: Rotational invariance requires the learned tensor mapping to transform consistently under any proper orthogonal rotation.This property is critical when augmenting traditional tensor models.
3. Numerical results
Numerical tests on square-duct and periodic-hill flows show that implicit Reynolds-stress treatment improves mean-velocity predictions and addresses RANS ill-conditioning. Machine-learning predictions reproduce key Reynolds-stress features and yield mean flows closer to DNS or experimental data.
- Two canonical tests target stress-induced secondary flow in a square duct and massive separation over periodic hills, both challenging for traditional RANS modeling.
- Propagation of DNS Reynolds stresses: DNS Reynolds-stress propagation shows explicit Reynolds-stress treatment can produce large mean-velocity errors, whereas implicit treatment yields well-conditioned momentum equations.
- Flow over periodic hills: The implicit model produces mean velocities that agree much better with DNS than explicit Reynolds-stress or baseline RANS treatments in the periodic-hill test.
- Flow in a square duct: In the square duct, machine learning improves normal-stress imbalance and predicts secondary-flow magnitude, vortex shape, corner behavior, and a small inverse-flow vortex more accurately.
- Flow in a square duct: The predicted linear and nonlinear Reynolds-stress components better match DNS patterns, while downstream improvement in secondary-flow prediction becomes more prominent.
- Flow over periodic hills: For periodic hills, the framework improves shear-stress and turbulent-kinetic-energy agreement with DNS and predicts reverse flow more accurately from x/H = 1 to x/H = 3.
4. Discussion: Potentials and Limitations of Data-Driven Turbulence Models
The discussion frames data-driven turbulence models as flow-specific augmentations whose predictive reach, universality, and interpretability remain limited. The paper argues that invariant features and broader training databases can improve extrapolation, while practical use involves trade-offs between predictive capability and interpretability.
- The framework addresses ill-conditioning in a class of data-driven turbulence models by using implicit treatment of Reynolds-stress components.
- Universality: Existing data-driven turbulence models generally show limited predictive capability outside flows close to their training flows, especially across different flow configurations.
- Universality: The model’s rotationally and Galilean-invariant inputs and outputs support invariance under coordinate transformations, but extrapolation still depends on training-data diversity.
- Universality: Training on flows with specific characteristics has demonstrated prediction of unknown flows with similar characteristics, while more diversified high-fidelity databases may extend extrapolation.
- Practical augmentation: Data-driven models can be treated as flow-specific corrections that turn off and revert to a traditional baseline when a flow is too far from the training data.
- Interpretability: Neural networks and random forests are not analytical forms amenable to direct interpretation, although regularization and feature-importance analysis may improve interpretability.
- Interpretability: The approaches reflect a trade-off between predictive capability and interpretability, while a universal turbulent constitutive relation remains an unresolved question.
5. Conclusion
The framework addresses the challenge of obtaining accurate mean velocities from improved Reynolds stresses by implicitly treating linear and nonlinear stress components. Tests across square-duct and periodic-hill flows demonstrated satisfactory mean-velocity prediction, including an unseen flow pattern.
- Improved mean velocity remains challenging even when Reynolds stresses are accurately predicted because mean velocity is sensitive to Reynolds-stress errors.The conclusion identifies this sensitivity as the main motivation for the proposed implicit treatment.
- The framework uses machine learning to predict the optimal eddy viscosity and nonlinear part of the Reynolds stress separately.This decomposition provides the physics-based implicit treatment used to solve for mean velocity.
- Three a posteriori tests evaluate training on a square duct at Re = 2200 and periodic hills at Re = 5600, with predictions at different duct Reynolds numbers and a steeper hill profile.The square-duct predictions use Re = 3500 and Re = 1.25 × 10^5; the periodic-hill test uses a steeper profile.
- The proposed framework achieves satisfactory prediction performance for the mean velocity field across the tested cases.The reported tests demonstrate the predictive capability of the machine-learning-assisted framework.
- The trained model successfully predicts the mean flow pattern in the second case, although that pattern is not shown in the training flow.This result is presented as evidence of predictive capability beyond direct reproduction of the training case.
- The authors state that machine-learning-assisted turbulence modeling can reveal flow physics from existing data rather than merely fit that data.The practical framework targets applications requiring mean velocity and other quantities of interest.
Appendix A. Data-Driven Reynolds Stress Model: Detailed Algorithms
The detailed algorithm combines baseline RANS simulations, discrepancy regression, separate treatment of linear and nonlinear Reynolds-stress components, and an implicit RANS solve for mean velocity.
- Baseline RANS simulations provide mean-flow features q and the modeled Reynolds stress tensor τ_RANS for training and test flows.
- Regression functions f1 and f2 predict the Reynolds-stress discrepancies Δτ and Δτ_L from mean-flow features.For a test flow, the predicted discrepancies are added to τ_RANS to obtain τ and τ_L.
- The predicted ν_L^t, τ, and τ_L are inserted into the RANS equations to solve for the corresponding mean velocity field.Equation A.1 expresses the anisotropy stress using the implicitly treated linear contribution and the nonlinear remainder.
- The linear and nonlinear Reynolds-stress parts are predicted separately, with the linear part represented through the optimal eddy viscosity ν_L^t.The nonlinear part is calculated by subtracting the linear stress from the Reynolds stress itself.
- A modified OpenFOAM simpleFoam solver enables implicit treatment of the machine-learning-predicted Reynolds stresses.The solver uses SIMPLE, second-order central differences except for second-order upwind convection discretization.
- The implementation applies fixed zero values for predicted ν_L^t at walls, periodic streamwise conditions, and standard mean-velocity and pressure boundary conditions.
Appendix B. Integrity Basis of Mean Flow Features
The appendix constructs rotationally invariant mean-flow features using a minimal integrity basis generated from symmetric and antisymmetric tensors.
- The minimal integrity basis is defined for a symmetric strain-rate tensor bS and antisymmetric tensors bΩ, bA_p, and bA_k.The antisymmetric tensors are associated with rotation rate, pressure-gradient, and turbulent-kinetic-energy-gradient information.
- Spencer’s procedure systematically generates minimal invariant bases from sets of symmetric and antisymmetric tensors.
- The pressure and turbulent-kinetic-energy gradient vectors are transformed into corresponding antisymmetric tensors before generating the basis.
Appendix C. Galilean Invariance
The appendix shows that the material derivative of mean velocity is Galilean invariant, supporting feature normalization and model validity across inertial frames.
- The material derivative of velocity remains invariant under a Galilean transformation between stationary and uniformly translating reference frames.The transformed velocity is U*(x*, t) = U(x − Ct, t) + C, with x* = x − Ct.
- The resulting normalization factors are Galilean invariant, addressing the concern that ρ|Du/Dt| was not evidently invariant.
- Galilean invariance makes the trained model valid in all inertial frames, expressed as g(U) = g(U + C).The appendix presents this as enhancing the generality of the trained model.
- Several earlier turbulence-modeling features were not Galilean invariant, including raw pressure-gradient, turbulent-kinetic-energy, and turbulence-intensity terms.
- The proposed invariant inputs and outputs are expected to improve extrapolation capability.