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Data-enabled discovery of specific and generalisable turbulence closures

Published online by Cambridge University Press:  25 July 2025

Zhongxin Yang Open the ORCID record for Zhongxin Yang [Opens in a new window] ,
Xianglin Shan Open the ORCID record for Xianglin Shan [Opens in a new window] ,
Xiang I.A. Yang Open the ORCID record for Xiang I.A. Yang [Opens in a new window]  and
Weiwei Zhang Open the ORCID record for Weiwei Zhang [Opens in a new window]
Zhongxin Yang
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, PR China International Joint Institute of Artificial Intelligence on Fluid Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China National Key Laboratory of Aircraft Configuration Design, Xi’an 710072, PR China
Xianglin Shan
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, PR China International Joint Institute of Artificial Intelligence on Fluid Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China National Key Laboratory of Aircraft Configuration Design, Xi’an 710072, PR China
Xiang I.A. Yang
Affiliation:
Mechanical Engineering, Pennsylvania State University, State College 16802, USA
Weiwei Zhang*
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, PR China International Joint Institute of Artificial Intelligence on Fluid Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China National Key Laboratory of Aircraft Configuration Design, Xi’an 710072, PR China
*
Corresponding author: Weiwei Zhang, aeroelastic@nwpu.edu.cn
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Abstract

Turbulence closures are essential for predictive fluid flow simulations in both natural and engineering systems. While machine learning offers promising avenues, existing data-driven turbulence models often fail to generalise beyond their training datasets. This study identifies the root cause of this limitation as the conflation of generalisable flow physics and dataset-specific behaviours. We address this challenge using symbolic regression, which yields interpretable, white-box expressions. By decomposing the learned corrections into inner-layer, outer-layer and pressure-gradient components, we isolate universal physics from flow-specific features. The model is trained progressively using high-fidelity datasets for plane channel flows, zero-pressure-gradient turbulent boundary layers (ZPGTBLs), and adverse pressure-gradient turbulent boundary layers (PGTBLs). For example, direct application of a model trained on channel flow data to ZPGTBLs results in incorrect skin friction predictions. However, when only the generalisable inner-layer component is retained and combined with an outer-layer correction specific to ZPGTBLs, predictions improve significantly. Similarly, a pressure-gradient correction derived from PGTBL data enables accurate modelling of aerofoil flows with both favourable and adverse pressure gradients. The resulting symbolic corrections are compact, interpretable, and generalise across configurations – including unseen geometries such as aerofoils and Reynolds numbers outside the training set. The models outperform baseline Reynolds-averaged Navier–Stokes closures (e.g. the Spalart–Allmaras and shear stress transport models) in both a priori and a posteriori tests. These results demonstrate that explicit identification and retention of generalisable components is key to overcoming the generalisation challenge in machine-learned turbulence closures.

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Mathematical Foundations: Machine learning Turbulent Flows: Turbulence modelling

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Journal of Fluid Mechanics , Volume 1016 , 10 August 2025 , R1
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© The Author(s), 2025. Published by Cambridge University Press

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References

Arranz, G., Ling, Y., Costa, S., Goc, K. & Lozano-Durán, A. 2024 Building-block-flow computational model for large-eddy simulation of external aerodynamic applications. Commun. Engng 3 (1), 127.CrossRefGoogle ScholarPubMed
Baldwin, B. & Lomax, H. 1978 Thin-layer approximation and algebraic model for separated turbulent flows. In 16th Aerospace Sciences Meeting. American Institute of Aeronautics and Astronautics.Google Scholar
Bin, Y., Huang, G., Kunz, R. & Yang, X.I.A. 2024 Constrained recalibration of Reynolds-averaged Navier–Stokes models. AIAA J. 62 (4), 1434–1446.CrossRefGoogle Scholar
Bobke, A., Vinuesa, R., Örlü, R. & Schlatter, P. 2017 History effects and near equilibrium in adverse-pressure-gradient turbulent boundary layers. J. Fluid Mech. 820, 667–692.CrossRefGoogle Scholar
Bose, S.T. & Park, G.I. 2018 Wall-modeled large-eddy simulation for complex turbulent flows. Annu. Rev. Fluid. Mech. 50 (1), 535–561.CrossRefGoogle ScholarPubMed
Cao, W., Song, J. & Zhang, W. 2024 A solver for subsonic flow around airfoils based on physics-informed neural networks and mesh transformation. Phys. Fluids 36 (2), 027134.CrossRefGoogle Scholar
Cécora, R.-D., Radespiel, R., Eisfeld, B. & Probst, A. 2015 Differential Reynolds-stress modeling for aeronautics. AIAA J. 53 (3), 739–755.CrossRefGoogle Scholar
Chen, P.E.S., Bin, Y., Yang, X.I.A., Shi, Y., Abkar, M. & Park, G.I. 2023 a A priori screening of data-enabled turbulence models. Phys. Rev. Fluids 8 (12), 124606.CrossRefGoogle Scholar
Chen, P.E.S., Lv, Y., Xu, H.H.A., Shi, Y., Yang, X.I.A. 2022 a LES wall modeling for heat transfer at high speeds. Phys. Fluids 7 (1), 014608.CrossRefGoogle Scholar
Chen, P.E.S., Wu, W., Griffin, K.P., Shi, Y. & Yang, X.I.A. 2023 b A universal velocity transformation for boundary layers with pressure gradients. J. Fluid Mech. 970, A3.CrossRefGoogle Scholar
Chen, Y., Wang, S. & Liu, W. 2022 b Data-driven transition models for aeronautical flows with a high-order numerical method. Aerospace 9 (10), 578.CrossRefGoogle Scholar
Choi, H. & Moin, P. 2012 Grid-point requirements for large eddy simulation: Chapman’s estimates revisited. Phys. Fluids 24 (1), 011702.CrossRefGoogle Scholar
Cranmer, M. 2023 Interpretable machine learning for science with PySR and SymbolicRegression.jl, arXiv preprint arXiv: 2305.01582.Google Scholar
Cranmer, M. 2024 PySR: high-performance symbolic regression in Python. Available at: https://astroautomata.com/PySR/api-advanced/ (Accessed: 2024-07-12).Google Scholar
Durbin, P.A. 2018 Some recent developments in turbulence closure modeling. Annu. Rev. Fluid. Mech. 50 (1), 77–103.CrossRefGoogle Scholar
Eitel-Amor, G., Örlü, R. & Schlatter, P. 2014 Simulation and validation of a spatially evolving turbulent boundary layer up to Re $_\theta$ = 8300. Intl J. Heat Fluid Flow 47, 57–69.10.1016/j.ijheatfluidflow.2014.02.006CrossRefGoogle Scholar
Gao, C., Zhang, W., Li, X., Liu, Y., Quan, J., Ye, Z. & Jiang, Y. 2017 Mechanism of frequency lock-in in transonic buffeting flow. J. Fluid Mech. 818, 528–561.CrossRefGoogle Scholar
Gimenez, J.M. & Bre, F. 2019 Optimization of RANS turbulence models using genetic algorithms to improve the prediction of wind pressure coefficients on low-rise buildings. J. Wind Engng Ind. Aerodyn. 193, 103978.CrossRefGoogle Scholar
Gungor, A.G., Maciel, Y., Simens, M.P. & Soria, J. 2016 Scaling and statistics of large-defect adverse pressure gradient turbulent boundary layers. Intl J. Heat Fluid Flow 59, 109–124.CrossRefGoogle Scholar
Guo, X., Xia, Z. & Chen, S. 2021 Computing mean fields with known Reynolds stresses at steady state. Theor. Appl. Mech. Lett. 11 (3), 100244.CrossRefGoogle Scholar
He, Z., Wu, C. & Zhang, Y. 2024 A field inversion and symbolic regression enhanced Spalart–Allmaras model for airfoil stall prediction. Phys. Fluids 36 (6), 065111.CrossRefGoogle Scholar
Hoyas, S., Oberlack, M., Alcántara-Ávila, F., Kraheberger, S.V. & Laux, J. 2022 Wall turbulence at high friction Reynolds numbers. Phys. Rev. Fluids 7 (1), 014602.CrossRefGoogle Scholar
Huang, X.L.D., Yang, X.I.A. & Kunz, R.F. 2019 Wall-modeled large-eddy simulations of spanwise rotating turbulent channels – comparing a physics-based approach and a data-based approach. Phys. Fluids 31 (12), 125105.10.1063/1.5129178CrossRefGoogle Scholar
Kays, W.M., Crawford, M.E. & Weigand, B. 1980 Convective Heat and Mass Transfer. McGraw-Hill.Google Scholar
Klewicki, J.C. 2010 Reynolds number dependence, scaling, and dynamics of turbulent boundary layers. ASME J. Fluids Engng 132 (9), 094001.CrossRefGoogle Scholar
Kou, J. & Zhang, W. 2021 Data-driven modeling for unsteady aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 125, 100725.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_\tau \approx 5200$ . J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Li, J., Bin, Y., Huang, G. & Yang, X.I.A. 2024 Enhancing generalizability of machine-learning turbulence models. In AIAA SCITECH 2024 Forum, p. 1574. American Institute of Aeronautics and Astronautics.Google Scholar
Li, J.-Q.J.L., Yang, X.I.A. & Kunz, R.F. 2022 Grid-point and time-step requirements for large-eddy simulation and Reynolds-averaged Navier–Stokes of stratified wakes. Phys. Fluids 34 (11), 115125.CrossRefGoogle Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.CrossRefGoogle Scholar
Menter, F.R. 1994 Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 1598–1605.CrossRefGoogle Scholar
Nickels, T.B. 2004 Inner scaling for wall-bounded flows subject to large pressure gradients. J. Fluid Mech. 521, 217–239.CrossRefGoogle Scholar
Parish, E.J. & Duraisamy, K. 2016 A paradigm for data-driven predictive modeling using field inversion and machine learning. J. Comput. Phys. 305, 758–774.CrossRefGoogle Scholar
Piomelli, U. 2008 Wall-layer models for large-eddy simulations. Prog. Aerosp. Sci. 44 (6), 437–446.CrossRefGoogle Scholar
Pope, S.B. 2001 Turbulent flows. Meas. Sci. Technol. 12 (11), 2020–2021.CrossRefGoogle Scholar
Prandtl, L. 1925 Uber die ausgebildete Turbulenz. Z. Angew. Math. Mech. 5, 136–139.CrossRefGoogle Scholar
Romero, S., Zimmerman, S., Philip, J., White, C. & Klewicki, J. 2022 Properties of the inertial sublayer in adverse pressure-gradient turbulent boundary layers. J. Fluid Mech. 937, A30.CrossRefGoogle Scholar
Rumsey, C. 2019 Langley Research Center turbulence modeling resource. Available at: https://turbmodels.larc.nasa.gov/ (Accessed: 2024-07-12).Google Scholar
Rumsey, C.L., Coleman, G.N. & Wang, Li 2022 In search of data-driven improvements to RANS models applied to separated flows. In AIAA Scitech 2022 Forum, p. 0937. American Institute of Aeronautics and Astronautics.Google Scholar
Sarghini, F., de Felice, G. & Santini, S. 2003 Neural networks based subgrid scale modeling in large eddy simulations. Comput. Fluids 32 (1), 97–108.CrossRefGoogle Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.CrossRefGoogle Scholar
Shan, X., Liu, Y., Cao, W., Sun, X. & Zhang, W. 2023 Turbulence modeling via data assimilation and machine learning for separated flows over airfoils. AIAA J. 61 (9), 3883–3899.10.2514/1.J062711CrossRefGoogle Scholar
Shan, X., Sun, X., Cao, W., Zhang, W. & Xia, Z. 2024 Modeling Reynolds stress anisotropy invariants via machine learning. Acta Mechanica Sin. 40 (6), 323629.CrossRefGoogle Scholar
Shan, X. & Zhang, W. 2025 Data-driven adverse pressure gradient correction for turbulence model. AIAA J. 63, 1–17.CrossRefGoogle Scholar
Slotnick, J.P., Khodadoust, A., Alonso, J., Darmofal, D., Gropp, W., Lurie, E. & Mavriplis, D. 2014 CFD Vision 2030 study: a path to revolutionary computational aerosciences. Tech. Rep. NASA/CR-2014-218178. NASA Langley Research Center.Google Scholar
Spalart, P.R. 2000 Strategies for turbulence modelling and simulations. Intl J. Heat Fluid Flow 21 (3), 252–263.CrossRefGoogle Scholar
Spalart, P.R. & Allmaras, S.R. 1992 A one-equation turbulence model for aerodynamic flows. In 30th Aerospace Sciences Meeting and Exhibit, p. 439. American Institute of Aeronautics and Astronautics.Google Scholar
Srivastava, V., Rumsey, C.L., Coleman, G.N. & Wang, L. 2024 On generalizably improving RANS predictions of flow separation and reattachment. In AIAA SCITECH 2024 Forum, p. 2520. American Institute of Aeronautics and Astronautics.Google Scholar
Taghizadeh, S., Witherden, F.D., Hassan, Y.A. & Girimaji, S.S. 2021 Turbulence closure modeling with data-driven techniques: investigation of generalizable deep neural networks. Phys. Fluids 33 (11), 115132.CrossRefGoogle Scholar
Tanarro, Á, Vinuesa, R. & Schlatter, P. 2020 Effect of adverse pressure gradients on turbulent wing boundary layers. J. Fluid Mech. 883, A8.CrossRefGoogle Scholar
Tang, H., Wang, Y., Wang, T. & Tian, L. 2023 Discovering explicit Reynolds-averaged turbulence closures for turbulent separated flows through deep learning-based symbolic regression with non-linear corrections. Phys. Fluids 35 (2), 025118.10.1063/5.0135638CrossRefGoogle Scholar
Tracey, B.D., Duraisamy, K. & Alonso, J.J. 2015 A machine learning strategy to assist turbulence model development. In 53rd AIAA Aerospace Sciences Meeting, p. 1287. American Institute of Aeronautics and Astronautics.Google Scholar
Vaddireddy, H., Rasheed, A., Staples, A.E. & San, O. 2020 Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data. Phys. Fluids 32 (1), 015113.CrossRefGoogle Scholar
Vadrot, A., Yang, X.I.A. & Abkar, M. 2023 Survey of machine-learning wall models for large-eddy simulation. Phys. Rev. Fluids 8 (6), 064603.10.1103/PhysRevFluids.8.064603CrossRefGoogle Scholar
van Driest, E.R. 1956 On turbulent flow near a wall. J. Aeronaut. Sci. 23 (11), 1007–1011.10.2514/8.3713CrossRefGoogle Scholar
Wang, Z. & Zhang, W. 2023 A unified method of data assimilation and turbulence modeling for separated flows at high Reynolds numbers. Phys. Fluids 35 (2), 025124.CrossRefGoogle Scholar
Weatheritt, J. & Sandberg, R. 2016 A novel evolutionary algorithm applied to algebraic modifications of the RANS stress–strain relationship. J. Comput. Phys. 325, 22–37.CrossRefGoogle Scholar
Wu, C. & Zhang, Y. 2023 a Data-driven turbulence modeling using field inversion and deep symbolic regression. In Aerospace Europe Conference 2023. EUCASS.Google Scholar
Wu, C. & Zhang, Y. 2023 b Enhancing the shear-stress-transport turbulence model with symbolic regression: a generalizable and interpretable data-driven approach. Phys. Rev. Fluids 8 (8), 084604.CrossRefGoogle Scholar
Wu, J.-L., Xiao, H. & Paterson, E. 2018 Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3 (7), 074602.10.1103/PhysRevFluids.3.074602CrossRefGoogle Scholar
Wu, X., Zuo, Z., Ma, L. & Zhang, W. 2024 Multi-fidelity neural network-based aerodynamic optimization framework for propeller design in electric aircraft. Aerosp. Sci. Technol. 146, 108963.10.1016/j.ast.2024.108963CrossRefGoogle Scholar
Yamamoto, Y. & Tsuji, Y. 2018 Numerical evidence of logarithmic regions in channel flow at $Re_\tau$ = 8000. Phys. Rev. Fluids 3 (1), 012602.10.1103/PhysRevFluids.3.012602CrossRefGoogle Scholar
Yan, C., Zhang, Y. & Chen, H. 2022 Data augmented turbulence modeling for three-dimensional separation flows. Phys. Fluids 34 (7), 075101.10.1063/5.0097438CrossRefGoogle Scholar
Yang, X., Zhang, W., Abkar, M. & Anderson, W. 2024 a Computational fluid dynamics: its carbon footprint and role in carbon reduction. J. Renew. Sustain. Energy 16 (5), 055906.10.1063/5.0217320CrossRefGoogle Scholar
Yang, X.I.A., Chen, P.E.S., Zhang, W. & Kunz, R. 2024 b Predictive near-wall modelling for turbulent boundary layers with arbitrary pressure gradients. J. Fluid Mech. 993, A1.CrossRefGoogle Scholar
Yang, X.I.A. & Griffin, K.P. 2021 Grid-point and time-step requirements for direct numerical simulation and large-eddy simulation. Phys. Fluids 33 (1), 015108.10.1063/5.0036515CrossRefGoogle Scholar
Yang, X.I.A., Zafar, S., Wang, J.-X. & Xiao, H. 2019 Predictive large-eddy-simulation wall modeling via physics-informed neural networks. Phys. Rev. Fluids 4 (3), 034602.10.1103/PhysRevFluids.4.034602CrossRefGoogle Scholar
Zhang, W., Li, X., Ye, Z. & Jiang, Y. 2015 Mechanism of frequency lock-in in vortex-induced vibrations at low Reynolds numbers. J. Fluid Mech. 783, 72–102.CrossRefGoogle Scholar
Zhang, W., Xuhao, P., Jiaqing, K. & Xu, W. 2024 Heterogeneous data-driven aerodynamic modeling based on physical feature embedding. Chin. J. Aeronaut. 37 (3), 1–6.10.1016/j.cja.2023.11.010CrossRefGoogle Scholar
Zhang, Z.J. & Duraisamy, K. 2015 Machine learning methods for data-driven turbulence modeling. In 22nd AIAA Computational Fluid Dynamics Conference, p. 2460. American Institute of Aeronautics and Astronautics.Google Scholar
Zhou, D. & Bae, H.J. 2024 Sensitivity analysis of wall-modeled large-eddy simulation for separated turbulent flow. J. Comput. Phys. 506, 112948.CrossRefGoogle Scholar
Zhou, D., Whitmore, M.P., Griffin, K.P. & Bae, H.J. 2023 Large-eddy simulation of flow over Boeing Gaussian bump using multi-agent reinforcement learning wall model. In AIAA Aviation 2023 Forum, p. 3985. American Institute of Aeronautics and Astronautics.Google Scholar
Zhu, L., Zhang, W., Kou, J. & Liu, Y. 2019 Machine learning methods for turbulence modeling in subsonic flows around airfoils. Phys. Fluids 31 (1), 015105.10.1063/1.5061693CrossRefGoogle Scholar
Zhu, L., Zhang, W., Sun, X., Liu, Y. & Yuan, X. 2021 Turbulence closure for high Reynolds number airfoil flows by deep neural networks. Aerosp. Sci. Technol. 110, 106452.10.1016/j.ast.2020.106452CrossRefGoogle Scholar