A new algebraic RANS model for the wall sublayer of the hybrid RANS/LES method based on machine learning methods
DOI:
https://doi.org/10.7242/1999-6691/2026.19.3.20Keywords:
hybrid RANS/LES methods, wall-bounded flows, algebraic RANS models, machine learningAbstract
Today, there is a special focus on the development of Hybrid RANS/LES methods, in which the RANS model, describing the near-wall region of the flow, is one of their basic components. Since this region accounts for approximately 10% of the boundary layer thickness, one would expect that relatively simple algebraic closures can perform well in it. However, they are either non-local or implicit. The goal of this paper is to create a new explicit local algebraic RANS model for the near-wall sublayer, which will be used as a part of a hybrid RANS/LES method. The development of the model would involve the machine learning methods, which have found wide application in turbulence modeling. This procedure consists of two stages. First, by performing a priori testing based on the DNS data on the developed turbulent flow in a plane channel at the friction Reynolds numbers from 550 to 5200, the functional form of the new model is constructed using genetic programming. Then, the optimization of coefficients is carried out by applying a model-consistent learning approach that is based on the solution of a turbulent flow in a plane channel. The results calculated by the new model are compared with those obtained with the use of the classical Prandtl and Spalart-Allmaras models. It is shown that the new model outperforms them in accuracy. The new model was tested at a friction Reynolds number of 20000. It has been found that with this model it is possible to construct mean velocity profiles that agree well with the Nickels approximation and are close to those calculated by other models. Thus, this paper demonstrates that a combination of a priori and model-consistent machine learning approaches has been rather successful in creating an original algebraic RANS turbulence model.
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References
Garbaruk A.V., Strelets M.K., Travin A.K., Shur M.L. Modern approaches to the modeling of turbulence. Saint Petersburg: Izd-vo Politekhnicheskogo un-ta, 2016. 233 p.
White F.M., Christoph G.H. A Simple Theory for the Two-Dimensional Compressible Turbulent Boundary Layer. Journal of Basic Engineering. 1972. Vol. 94, no. 3. P. 636–642. DOI: 10.1115/1.3425519
Peng S.-H. Algebraic Hybrid RANS-LES Modelling Applied to Incompressible and Compressible Turbulent Flows. 36th AIAA Fluid Dynamics Conference and Exhibit. 2006. DOI: 10.2514/6.2006-3910.3910.
Peng S.-H. Hybrid RANS-LES modeling based on zero- and one-equation models for turbulent flow simulation. Proceeding of Fourth International Symposium on Turbulence and Shear Flow Phenomena. Begel House Inc., 2005. DOI: 10.1615/TSFP4.1930
Menter F.R., Kolmogorov D.K., Garbaruk A.V., Stabnikov A.S. Direct- and Large Eddy Simulations of Turbulent Flow in CS0 Diffuser on Resolved and Under-resolved Meshes. Flow, Turbulence and Combustion. 2023. Vol. 110, no. 3. P. 515–546. DOI: 10.1007/s10494-023-00399-1
Nebenführ B., Yao H., Peng S.-H., Davidson L. Hybrid RANS/LES Simulations for Aerodynamic and Aeroacoustic Analysis of a Multi-Element Airfoil. 19th AIAA/CEAS Aeroacoustics Conference. 2013. DOI: 10.2514/6.2013-2066.2066.
Nebenführ B., Peng S.-H., Davidson L. Hybrid RANS-LES Simulation of Turbulent High-Lift Flow in Relation to Noise Generation. Progress in Hybrid RANS-LES Modelling. Springer Berlin Heidelberg, 2012. P. 303–314. DOI: 10.1007/978-3-642-31818-4_26
Li H., Yakovenko S.N. Application of machine learning methods to develop algebraic Reynolds-stress models for flows in channels. Proceeding of 10th International Symposium on Turbulence, Heat and Mass Transfer, THMT-23, Rome, Italy, 11-15 September 2023. Begel House Inc., 2023. DOI: 10.1615/ichmt.thmt-23.710
Smith A.M., Cebeci T. Numerical solution of the turbulent-boundary-layer equations: Technical report / Douglas Aircraft Company. 1967. DAC33735. DOI: 10.21236/ad0656430
Baldwin B., Lomax H. Thin-layer approximation and algebraic model for separated turbulent flows. 16th Aerospace Sciences Meeting. 1978. DOI: 10.2514/6.1978-257.257.
Yilmaz I. An Algebraic Hybrid RANS-LES Model with Application to Turbulent Heat Transfer. Progress in Hybrid RANS-LES Modelling. Springer International Publishing, 2018. P. 503–512. DOI: 10.1007/978-3-319-70031-1_42
Zhu H., Fu S., Shi L., Wang Z.J. Implicit Large-Eddy Simulation for the High-Order Flux Reconstruction Method. AIAA Journal. 2016. Vol. 54, no. 9. P. 2721–2733. DOI: 10.2514/1.j054826
Duraisamy K., Iaccarino G., Xiao H. Turbulence Modeling in the Age of Data. Annual Review of Fluid Mechanics. 2019. Vol. 51, no. 1. P. 357–377. DOI: 10.1146/annurev-fluid-010518-040547
Data Driven Analysis and Modeling of Turbulent Flows / ed. by K. Duraisamy. Elsevier, 2025. P. 401. DOI: 10.1016/C2021-0-02739-5
Poroseva S.V., Colmenares F. J.D., Murman S.M. On the accuracy of RANS simulations with DNS data. Physics of Fluids. 2016. Vol. 28, no. 11. 115102. DOI: 10.1063/1.4966639
Thompson R.L., Sampaio L.E.B., Bragança Alves F.A.V. de, Thais L., Mompean G. A methodology to evaluate statistical errors in DNS data of plane channel flows. Computers & Fluids. 2016. Vol. 130. P. 1–7. DOI: 10.1016/j.compfluid.2016.01.014
Reissmann M., Hasslberger J., Sandberg R.D., Klein M. Application of Gene Expression Programming to a-posteriori LES modeling of a Taylor Green Vortex. Journal of Computational Physics. 2021. Vol. 424. 109859. DOI: 10.1016/j.jcp.2020.109859
Ben Hassan Saïdi I., Schmelzer M., Cinnella P., Grasso F. CFD-driven symbolic identification of algebraic Reynolds-stress models. Journal of Computational Physics. 2022. Vol. 457. 111037. DOI: 10.1016/j.jcp.2022.111037
Duraisamy K. Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence. Physical Review Fluids. 2021. Vol. 6, no. 5. 050504. DOI: 10.1103/PhysRevFluids.6.050504
Holland J.R., Baeder J.D., Duraisamy K. Field Inversion and Machine Learning With Embedded Neural Networks: Physics-Consistent Neural Network Training. AIAA Aviation 2019 Forum. 2019. DOI: 10.2514/6.2019-3200.3200.
Jiang C., Vinuesa R., Chen R., Mi J., Laima S., Li H. An interpretable framework of data-driven turbulence modeling using deep neural networks. Physics of Fluids. 2021. Vol. 33, no. 5. 055133. DOI: 10.1063/5.0048909
Ling J., Kurzawski A., Templeton J. Reynolds-averaged turbulence modelling using deep neural networks with embedded invariance. Journal of Fluid Mechanics. 2016. Vol. 807. P. 155–166. DOI: 10.1017/jfm.2016.615
Kaandorp M.L.A., Dwight R.P. Data-driven modelling of the Reynolds stress tensor using random forests with invariance. Computers & Fluids. 2020. Vol. 202. 104497. DOI: 10.1016/j.compfluid.2020.104497
Bernard A., Yakovenko S.N. Enhancement of RANS models by means of the tensor basis random forest for turbulent flows in two-dimensional channels with bumps. Journal of Applied Mechanics and Technical Physics. 2023. Vol. 64, no. 3. P. 437–441. DOI: 10.1134/S0021894423030094
Wu J.-L., Xiao H., Paterson E. Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework. Physical Review Fluids. 2018. Vol. 3, no. 7. 074602. DOI: 10.1103/physrevfluids.3.074602
Cramer N.L. A Representation for the Adaptive Generation of Simple Sequential Programs. Proceedings of an International Conference on Genetic Algorithms and their Applications. Pittsburgh, USA, 24-26 July 1985. 1985. P. 183–187.
Ferreira C. Gene Expression Programming: a New Adaptive Algorithm for Solving Problems. Complex Systems. 2001. Vol. 13. P. 87–129. DOI: 10.48550/arXiv.cs/0102027
Tibshirani R. Regression Shrinkage and Selection Via the Lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology. 1996. Vol. 58, no. 1. P. 267–288. DOI: 10.1111/j.2517-6161.1996.tb02080.x
Weatheritt J., Sandberg R. A novel evolutionary algorithm applied to algebraic modifications of the RANS stress-strain relationship. Journal of Computational Physics. 2016. Vol. 325. P. 22–37. DOI: 10.1016/j.jcp.2016.08.015
Weatheritt J., Sandberg R.D. The development of algebraic stress models using a novel evolutionary algorithm. International Journal of Heat and Fluid Flow. 2017. Vol. 68. P. 298–318. DOI: 10.1016/j.ijheatfluidflow.2017.09.017
Menter F. Zonal two-equation k-ω turbulence models for aerodynamic flows. 23rd Fluid Dynamics, Plasmadynamics, and Lasers Conference. 1993. DOI: 10.2514/6.1993-2906.2906.
Zhao Y., Akolekar H.D., Weatheritt J., Michelassi V., Sandberg R.D. RANS turbulence model development using CFD-driven machine learning. Journal of Computational Physics. 2020. Vol. 411. 109413. DOI: 10.1016/j.jcp.2020.109413
Di Fabbio T., Fang Y., Tangermann E., Sandberg R.D., Klein M. Strategies for Enhancing One-Equation Turbulence Model Predictions Using Gene-Expression Programming. Fluids. 2024. Vol. 9, no. 8. 191. DOI: 10.3390/fluids9080191
Spalart P., Allmaras S. A one-equation turbulence model for aerodynamic flows. 30th Aerospace Sciences Meeting and Exhibit. 1992. DOI: 10.2514/6.1992-439.439.
Li H., Zhao Y., Wang J., Sandberg R.D. Data-driven model development for large-eddy simulation of turbulence using gene-expression programing. Physics of Fluids. 2021. Vol. 33, no. 12. 125127. DOI: 10.1063/5.0076693
Moin P., Squires K., Cabot W., Lee S. A dynamic subgrid-scale model for compressible turbulence and scalar transport. Physics of Fluids A: Fluid Dynamics. 1991. Vol. 3, no. 11. P. 2746–2757. DOI: 10.1063/1.858164
Liu S., Meneveau C., Katz J. On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. Journal of Fluid Mechanics. 1994. Vol. 275. P. 83–119. DOI: 10.1017/S0022112094002296
Weatheritt J., Sandberg R.D. Hybrid Reynolds-Averaged/Large-Eddy Simulation Methodology from Symbolic Regression: Formulation and Application. AIAA Journal. 2017. Vol. 55, no. 11. P. 3734–3746. DOI: 10.2514/1.j055378
Li H., Waschkowski F., Zhao Y., Sandberg R.D. Turbulence Model Development based on a Novel Method Combining Gene Expression Programming with an Artificial Neural Network. 2023. DOI: 10.48550/arXiv.2301.07293
Li H., Zhao Y., Waschkowski F., Sandberg R.D. Evolutionary neural networks for learning turbulence closure models with explicit expressions. Physics of Fluids. 2024. Vol. 36, no. 5. 055126. DOI: 10.1063/5.0203975
Prandtl L. Bericht über Untersuchungen zur ausgebildeten Turbulenz. Zeitschrift für Angewandte Mathematik und Mechanik. 1925. Vol. 5. P. 136–139.
Van Driest E.R. On Turbulent Flow Near a Wall. Journal of the Aeronautical Sciences. 1956. Vol. 23, no. 11. P. 1007–1011. DOI: 10.2514/8.3713
Lee M., Moser R.D. Direct numerical simulation of turbulent channel flow up to Reτ ≈ 5200. Journal of Fluid Mechanics. 2015. Vol. 774. P. 395–415. DOI: 10.1017/jfm.2015.268
The Julia Programming Language. URL: https://julialang.org/ (accessed: 30.10.2026)
SymbolicRegression.jl. URL: https://juliapackages.com/p/symbolicregression (accessed: 30.10.2026)
GeneExpressionProgramming.jl. URL: https : / / github . com / maxreiss123 / GeneExpressionProgramming . jl (accessed: 30.10.2026)
DifferentialEquations.jl: Efficient Differential Equation Solving in Julia. URL: https://docs.sciml.ai/DiffEqDocs/ stable/ (accessed: 30.10.2026)
Sundials.jl. URL: https://docs.sciml.ai/DiffEqDocs/dev/api/sundials/ (accessed: 30.10.2026)
Hindmarsh A.C., Taylor A.G. User documentation for IDA, a differential-algebraic equation solver for sequential and parallel computers: Technical report / Lawrence Livermore National Laboratory. 1999. UCRL-MA–136910
Optim.jl. URL: https://juliapackages.com/p/optim (accessed: 30.10.2026)
BoundaryValueDiffEq.jl. URL: https://docs.sciml.ai/BoundaryValueDiffEq/dev/ (accessed: 30.10.2026). Julia package.
Verner J.H. Numerically optimal Runge–Kutta pairs with interpolants. Numerical Algorithms. 2010. Vol. 53, no. 2/3. P. 383–396. DOI: 10.1007/s11075-009-9290-3
Nickels T.B. Inner scaling for wall-bounded flows subject to large pressure gradients. Journal of Fluid Mechanics. 2004. Vol. 521. P. 217–239. DOI: 10.1017/s0022112004001788
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