- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Improving the Spalart-Allmaras turbulence model for separated flows using field inversion and symbolic regression

Paul Bataillie1,2, Maxime Casanova1, and Pedro Stefanin Volpiani1,*

  • *Contact author: pedro.stefanin_volpiani@onera.fr
PDF

Phys. Rev. Fluids 11, 074604 – Published 16 July, 2026

DOI: https://doi.org/10.1103/dk9r-td14

Abstract

Reynolds-averaged Navier-Stokes (RANS) turbulence models rely on several simplifying assumptions that limit their accuracy, particularly in predicting separated flows. To address these limitations, significant efforts have focused on augmenting RANS models through field inversion and machine learning. In the present study, data assimilation and symbolic regression are used to formulate an analytical correction to the Spalart-Allmaras model, addressing local deficiencies in its production term. A key objective of this work is to develop a correction applicable to a wide range of flow configurations, improving predictions for separated flows without degrading the performance of the baseline model for wall-attached flows. To this end, symbolic regression is performed using assimilated data from two-dimensional separated-flow cases (converging-diverging channel, bump H42, and NASA wall-mounted hump) together with a reference case where the Spalart-Allmaras model remains accurate (flat plate). This methodology enforces the desired correction in separated flows while minimizing the impact on the model's accuracy for wall-attached flows. The symbolic correction accurately reproduces the assimilated quantities and its performance on unseen flows demonstrates that the resulting expression remains applicable across a broad range of flow configurations and Reynolds numbers (bump H38, square cylinder, and periodic hill). The correction is also assessed on three-dimensional cases, namely, the FAITH hill and the Ahmed body. While improvements are observed for some configurations, these tests also reveal limitations that warrant further investigation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. A. W. Cary, J. Chawner, E. P. Duque, W. Gropp, W. L. Kleb, R. M. Kolonay, E. Nielsen, and B. Smith, CFD vision 2030 road map: Progress and perspectives, in Proceedings of the AIAA Aviation Forum (American Institute of Aeronautics and Astronautics, Reston, VA, 2021).
  2. B. E. Launder and D. B. Spalding, The numerical computation of turbulent flows, Comput. Methods Appl. Mech. Eng. 3, 269 (1974).
  3. D. C. Wilcox, Formulation of the k-w turbulence model revisited, AIAA J. 46, 2823 (2008).
  4. P. Spalart and S. Allmaras, A one-equation turbulence model for aerodynamic flows, in Proceedings of the 30th Aerospace Sciences Meeting and Exhibit (American Institute of Aeronautics and Astronautics, Reston, VA, 1992).
  5. C. L. Rumsey, Exploring a method for improving turbulent separated-flow predictions with k-omega models, NASA Technical report, NASA, Washington, D.C., 2009.
  6. K. Duraisamy, Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence, Phys. Rev. Fluids 6, 050504 (2021).
  7. J. Ling, A. Kurzawski, and J. Templeton, Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, J. Fluid Mech. 807, 155 (2016).
  8. J.-X. Wang, J.-L. Wu, and H. Xiao, Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data, Phys. Rev. Fluids 2, 034603 (2017).
  9. P. S. Volpiani, R. F. Bernardini, and L. Franceschini, Neural network-based eddy-viscosity correction for RANS simulations of flows over bi-dimensional bumps, Int. J. Heat Fluid Flow 97, 109034 (2022).
  10. P. S. Volpiani, Are random forests better suited than neural networks to augment RANS turbulence models? Int. J. Heat Fluid Flow 107, 109348 (2024).
  11. E. J. Parish and K. Duraisamy, A paradigm for data-driven predictive modeling using field inversion and machine learning, J. Comput. Phys. 305, 758 (2016).
  12. A. P. Singh, S. Medida, and K. Duraisamy, Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils, AIAA J. 55, 2215 (2017).
  13. A. Ferrero, A. Iollo, and F. Larocca, Field inversion for data-augmented RANS modelling in turbomachinery flows, Comput. Fluids 201, 104474 (2020).
  14. C. Yan, H. Li, Y. Zhang, and H. Chen, Data-driven turbulence modeling in separated flows considering physical mechanism analysis, Int. J. Heat Fluid Flow 96, 109004 (2022).
  15. A. S. Cato, P. S. Volpiani, V. Mons, O. Marquet, and D. Sipp, Comparison of different data-assimilation approaches to augment RANS turbulence models, Comput. Fluids 266, 106054 (2023).
  16. P. S. Volpiani, M. Meyer, L. Franceschini, J. Dandois, F. Renac, E. Martin, O. Marquet, and D. Sipp, Machine learning-augmented turbulence modeling for RANS simulations of massively separated flows, Phys. Rev. Fluids 6, 064607 (2021).
  17. F. Jäckel, A closed-form correction for the Spalart-Allmaras turbulence model for separated flows, AIAA J. 61, 2319 (2023).
  18. Z. He, C. Wu, and Y. Zhang, A field inversion and symbolic regression enhanced Spalart-Allmaras model for airfoil stall prediction, Phys. Fluids 36, 065111 (2024).
  19. R. McConkey, E. Yee, and F. S. Lien, Deep structured neural networks for turbulence closure modeling, Phys. Fluids 34, 035110 (2022).
  20. M. A. Cruz, R. L. Thompson, L. E. Sampaio, and R. D. Bacchi, The use of the Reynolds force vector in a physics informed machine learning approach for predictive turbulence modeling, Comput. Fluids 192, 104258 (2019).
  21. C. L. Rumsey, G. N. Coleman, and L. Wang, In search of data-driven improvements to RANS models applied to separated flows, in Proceedings of the AIAA SCITECH Forum (American Institute of Aeronautics and Astronautics, Reston, VA, 2021).
  22. M. Cranmer, Interpretable machine learning for science with PySR and SymbolicRegression.jl, arXiv:2305.01582v3.
  23. W. Tenachi, R. Ibata, and F. I. Diakogiannis, Deep symbolic regression for physics guided by units constraints: Toward the automated discovery of physical laws, Astrophys. J. 959, 99 (2023).
  24. P. Spalart, An old-fashioned framework for machine learning in turbulence modeling, arXiv:2308.00837.
  25. J. Weatheritt and R. Sandberg, The development of algebraic stress models using a novel evolutionary algorithm, Int. J. Heat Fluid Flow 68, 298 (2017).
  26. M. Schmelzer, R. P. Dwight, and P. Cinnella, Discovery of algebraic Reynolds-stress models using sparse symbolic regression, Flow Turbul. Combust. 104, 579 (2020).
  27. T. Buchanan, M. Lăcătuş, A. West, and R. P. Dwight, Data-driven RANS closures using a relative importance term analysis based classifier for 2D and 3D separated flows, Comput. Fluids 305, 106899 (2026).
  28. C. Wu and Y. Zhang, Enhancing the shear-stress-transport turbulence model with symbolic regression: A generalizable and interpretable data-driven approach, Phys. Rev. Fluids 8, 084604 (2023).
  29. C. Wu, S. Zhang, and Y. Zhang, Development of a generalizable data-driven turbulence model: Conditioned field inversion and symbolic regression, AIAA J. 63, 687 (2025).
  30. B. An, Z. Jiang, C. Yi, M. Han, and C. Yan, Improvement of the SST model for streamline curvature effect based on field inversion and symbolic regression, Aerosp. Sci. Technol. 157, 109828 (2025).
  31. H. Liao, X. Sun, Y. Liu, and W. Zhang, Data-driven turbulence modeling: A mutually coupled framework for symbolic regression and data assimilation, Phys. Fluids 37, 075211 (2025).
  32. A. Crivellini, V. D'Alessandro, and F. Bassi, A Spalart-Allmaras turbulence model implementation in a discontinuous Galerkin solver for incompressible flows, J. Comput. Phys. 241, 388 (2013).
  33. D. A. Ham, P. H. Kelly, L. Mitchell, C. Cotter, R. C. Kirby, K. Sagiyama, N. Bouziani, S. Vorderwuelbecke, T. Gregory, J. Betteridge, D. R. Shapero, R. Nixon-Hill, C. Ward, P. E. Farrell, P. D. Brubeck, I. Marsden, T. H. Gibson, M. Homolya, T. Sun, A. T. McRae et al., Firedrake User Manual, Imperial College London and University of Oxford and Baylor University and University of Washington, 1st ed., 2023.
  34. M. S. Alnæs, A. Logg, K. B. Ølgaard, M. E. Rognes, and G. N. Wells, Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Trans. Math. Softw. 40, 1 (2014).
  35. Y. Bao, D. Zhou, C. Huang, Q. Wu, and X. Chen, Numerical prediction of aerodynamic characteristics of prismatic cylinder by finite element method with Spalart-Allmaras turbulence model, Comput. Struct. 89, 325 (2011).
  36. M. Olshanskii, G. Lube, T. Heister, and J. Löwe, GRAD-DIV stabilization and subgrid pressure models for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng. 198, 3975 (2009).
  37. P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent, and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl. 23, 15 (2001).
  38. P. R. Amestoy, A. Buttari, J.-Y. L'Excellent, and T. Mary, Performance and scalability of the block low-rank multifrontal factorization on multicore architectures, ACM Trans. Math. Softw. 45, 1 (2019).
  39. M. L. Shur, M. K. Strelets, A. K. Travin, and P. R. Spalart, Turbulence modeling in rotating and curved channels: Assessing the Spalart-Shur correction, AIAA J. 38, 784 (2000).
  40. B. Fanizza, P. Stefanin Volpiani, F. Renac, E. Martin, and D. Sipp, Adjoint-based optimization for non-linear inverse problems with high-order discretization of the compressible RANS equations, Appl. Math. Modell. 142, 115984 (2025).
  41. G. K. Kenway, C. A. Mader, P. He, and J. R. Martins, Effective adjoint approaches for computational fluid dynamics, Prog. Aerosp. Sci. 110, 100542 (2019).
  42. S. K. Mitusch, S. W. Funke, and J. S. Dokken, dolfin-adjoint 2018.1: Automated adjoints for FEniCS and Firedrake, J. Open Source Software 4, 1292 (2019).
  43. C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization, ACM Trans. Math. Softw. 23, 550 (1997).
  44. R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput. 16, 1190 (1995).
  45. L. Franceschini, D. Sipp, and O. Marquet, Mean-flow data assimilation based on minimal correction of turbulence models: Application to turbulent high Reynolds number backward-facing step, Phys. Rev. Fluids 5, 094603 (2020).
  46. M. Marquillie, U. Ehrenstein, and J.-P Laval, Instability of streaks in wall turbulence with adverse pressure gradient, J. Fluid Mech. 681, 205 (2011).
  47. R. Matai and P. Durbin, Large-eddy simulation of turbulent flow over a parametric set of bumps, J. Fluid Mech. 866, 503 (2019).
  48. A. Uzun and M. R. Malik, Large-eddy simulation of flow over a wall-mounted hump with separation and reattachment, AIAA J. 56, 715 (2018).
  49. M. Cranmer, A. Sanchez-Gonzalez, P. Battaglia, R. Xu, K. Cranmer, D. Spergel, and S. Ho, Discovering symbolic models from deep learning with inductive biases, arXiv:2006.11287.
  50. P. R. Spalart, Philosophies and fallacies in turbulence modeling, Prog. Aerosp. Sci. 74, 1 (2015).
  51. B. D. Tracey, K. Duraisamy, and J. J. Alonso, A machine learning strategy to assist turbulence model development, in Proceedings of the 53rd AIAA Aerospace Sciences Meeting (American Institute of Aeronautics and Astronautics, Kissimmee, FL, 2015).
  52. H. J. Herring and G. L. Mellor, Two methods of calculating turbulent boundary layer behavior based on numerical solutions of the equations of motion (Stanford, CA 1969), https://ntrs.nasa.gov/citations/19700036233.
  53. P. R. Spalart, S. Deck, M. L. Shur, K. D. Squires, M. K. Strelets, and A. Travin, A new version of detached-eddy simulation, resistant to ambiguous grid densities, Theor. Comput. Fluid Dyn. 20, 181 (2006).
  54. M. Mani, D. Babcock, C. Winkler, and P. Spalart, Predictions of a supersonic turbulent flow in a square duct, in Proceedings of the 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition (AIAA, Reston, VA, 2013).
  55. V. Srivastava, C. L. Rumsey, G. N. Coleman, and L. Wang, On generalizably improving RANS predictions of flow separation and reattachment, in Proceedings of the AIAA SCITECH Forum (American Institute of Aeronautics and Astronautics, Orlando, FL, 2024).
  56. Y. Fang, Y. Zhao, F. Waschkowski, A. S. H. Ooi, and R. D. Sandberg, Toward more general turbulence models via multicase computational-fluid-dynamics-driven training, AIAA J. 61, 2100 (2023).
  57. R. B. Langtry and F. R. Menter, Correlation-based transition modeling for unstructured parallelized computational fluid dynamics codes, AIAA J. 47, 2894 (2009).
  58. K. Wieghardt and W. Tillmann, On the turbulent friction layer for rising pressure, Technical Memorandum No. 1314 (National Advisory Committee for Aeronautics, Washington, 1951).
  59. A. M. Savill, in Further Progress in the Turbulence Modelling of By-Pass Transition, Engineering Turbulence Modelling and Experiments, edited by W. Rodi and F. Martelli (Elsevier, Oxford, UK, 1993), p. 583.
  60. C. Wu and Y. Zhang, Field inversion and machine learning based on the rubber-band Spalart-Allmaras model, Theor. Appl. Mech. Lett. 15, 100564 (2025).
  61. L. Franceschini, D. Sipp, O. Marquet, J. Moulin, and J. Dandois, Identification and reconstruction of high-frequency fluctuations evolving on a low-frequency periodic limit cycle: Application to turbulent cylinder flow, J. Fluid Mech. 942, A28 (2022).
  62. X. Gloerfelt and P. Cinnella, Large eddy simulation requirements for the flow over periodic hills, Flow Turbul. Combust. 103, 55 (2019).
  63. D. M. Driver and H. L. Seegmiller, Features of a reattaching turbulent shear layer in divergent channel flow, AIAA J. 23, 163 (1985).
  64. Y. Bentaleb, S. Lardeau, and M. A. Leschziner, Large-eddy simulation of turbulent boundary layer separation from a rounded step, J. Turbul. 13, N4 (2012).
  65. J. Bell, J. Heineck, G. Zilliac, R. Mehta, and K. Long, Surface and flow field measurements on the FAITH hill model, in Proceedings of the 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition (American Institute of Aeronautics and Astronautics, Reston, VA, 2012).
  66. H. Lienhart, C. Stoots, and S. Becker, in Flow and Turbulence Structures in the Wake of a Simplified Car Model (Ahmed Modell), edited by S. Wagner, U. Rist, H.-J. Heinemann, and R. Hilbig, New Results in Numerical and Experimental Fluid Mechanics III Vol. 77 (Springer, Berlin, Heidelberg, 2002), p. 323.
  67. J. P. Huijing, R. P. Dwight, and M. Schmelzer, Data-driven RANS closures for three-dimensional flows around bluff bodies, Comput. Fluids 225, 104997 (2021).
  68. R. McConkey, T. Buchanan, T. Smidt, A. Bodner, R. Dwight, and P. Cinnella, The closure challenge: A benchmark task for machine learning in turbulence modelling, arXiv:2603.28884.
  69. E. Guilmineau, Computational study of flow around a simplified car body, J. Wind Eng. Ind. Aerodyn. 96, 1207 (2008).
  70. M. Oulghelou, S. Cherroud, X. Merle, and P. Cinnella, Machine-learning-assisted blending of data-driven turbulence models, Flow Turbul. Combust. 115, 1095 (2025).
  71. https://github.com/p-bataillie/2026-PRF-SA-FISR.

Outline

Information

Phys. Rev. Fluids 11, 074604– Published 16 July, 2026
  • Received 27 October 2025
  • Accepted 10 June 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation