Theoretical and Computational Fluid Dynamics

, Volume 33, Issue 1, pp 1–19 | Cite as

Prediction of Reynolds stresses in high-Mach-number turbulent boundary layers using physics-informed machine learning

  • Jian-Xun WangEmail author
  • Junji Huang
  • Lian Duan
  • Heng Xiao
Original Article


Modeled Reynolds stress is a major source of model-form uncertainties in Reynolds-averaged Navier–Stokes (RANS) simulations. Recently, a physics-informed machine learning (PIML) approach has been proposed for reconstructing the discrepancies in RANS-modeled Reynolds stresses. The merits of the PIML framework have been demonstrated in several canonical incompressible flows. However, its performance on high-Mach-number flows is still not clear. In this work, we use the PIML approach to predict the discrepancies in RANS-modeled Reynolds stresses in high-Mach-number flat-plate turbulent boundary layers by using an existing DNS database. Specifically, the discrepancy function is first constructed using a DNS training flow and then used to correct RANS-predicted Reynolds stresses under flow conditions different from the DNS. The machine learning technique is shown to significantly improve RANS-modeled turbulent normal stresses, the turbulent kinetic energy, and the Reynolds stress anisotropy. Improvements are consistently observed when different training datasets are used. Moreover, a high-dimensional visualization technique and a distance metrics are used to provide a priori assessment of prediction confidence based only on RANS simulations. This study demonstrates that the PIML approach is a computationally affordable technique for improving the accuracy of RANS-modeled Reynolds stresses for high-Mach-number turbulent flows when there is a lack of experiments and high-fidelity simulations.


Data-driven Reynolds-averaged Navier–Stokes High-speed flow Direct numerical simulation 


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The DNS database was produced based upon the work supported by AFOSR under Grant FA9550-14-1-0170 (Program Manager I. Leyva) and NASA Langley Research Center under Grant NNL09AA00A (through the National Institute of Aerospace). Computational resources for the DNS were provided by the NASA Advanced Supercomputing Division, the DoD High-Performance Computing Modernization Program, and the NSF’s Petascale Computing Resource Allocations Program (NSF ACI-1640865). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force. We also thank the anonymous reviewers for their comments, which helped improving the quality and clarity of the manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Aerospace and Mechanical Engineering, Center of Informatics and Computational ScienceUniversity of Notre DameNotre DameUSA
  2. 2.Department of Mechanical and Aerospace EngineeringMissouri University of Science and TechnologyRollaUSA
  3. 3.Kevin T. Crofton Department of Aerospace and Ocean EngineeringVirginia TechBlacksburgUSA

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