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

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Abstract

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.

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References

  1. Ansys fluent user guide, release 15.0. ANSYS Inc (2013)

  2. Banerjee, S., Krahl, R., Durst, F., Zenger, C.: Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches. J. Turbul. 8, N32 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheung, S.H., Oliver, T.A., Prudencio, E.E., Prudhomme, S., Moser, R.D.: Bayesian uncertainty analysis with applications to turbulence modeling. Reliab. Eng. Syst. Saf. 96(9), 1137–1149 (2011)

    Article  Google Scholar 

  4. Craft, T., Launder, B., Suga, K.: Development and application of a cubic eddy-viscosity model of turbulence. Int. J. Heat Fluid Flow 17(2), 108–115 (1996)

    Article  Google Scholar 

  5. Dow, E., Wang, Q.: Quantification of structural uncertainties in the \(k\)–\(\omega \) turbulence model. In: 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA, Denver, Colorado, pp. 2011–1762 (2011)

  6. Duan, L., Choudhari, M.M.: Analysis of numerical simulation database for pressure fluctuations induced by high-speed turbulent boundary layers. AIAA Paper 2014-2912 (2014)

  7. Duan, L., Beekman, I., Martin, M.: Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of mach number. J. Fluid Mech. 672, 245–267 (2011)

    Article  MATH  Google Scholar 

  8. Duan, L., Choudhari, M.M., Wu, M.: Numerical study of acoustic radiation due to a supersonic turbulent boundary layer. J. Fluid Mech. 746, 165–192 (2014)

    Article  Google Scholar 

  9. Duan, L., Choudhari, M.M., Zhang, C.: Pressure fluctuations induced by a hypersonic turbulent boundary layer. J. Fluid Mech. 804, 578–607 (2016)

    Article  MathSciNet  Google Scholar 

  10. Duraisamy, K., Iaccarino, G., Xiao, H.: Turbulence modeling in the age of data. (2018). arXiv preprint arXiV:1804.00183

  11. Durbin, P.A., Reif, B.P.: Statistical Theory and Modeling for Turbulent Flows. Wiley, Hoboken (2011)

    MATH  Google Scholar 

  12. Edeling, W., Cinnella, P., Dwight, R.P.: Predictive RANS simulations via Bayesian model-scenario averaging. J. Comput. Phys. 275, 65–91 (2014a)

    Article  MathSciNet  MATH  Google Scholar 

  13. Edeling, W., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the \(k\)-\(\varepsilon \) turbulence model. J. Comput. Phys. 258, 73–94 (2014b)

    Article  MathSciNet  MATH  Google Scholar 

  14. Emory, M., Larsson, J., Iaccarino, G.: Modeling of structural uncertainties in Reynolds-averaged Navier–Stokes closures. Phys. Fluids 25(11), 110,822 (2013)

    Article  Google Scholar 

  15. Friedman, J., Hastie, T., Tibshirani, R.: The elements of statistical learning, vol 1. Springer series in statistics New York, NY, USA (2001)

  16. Gatski, T., Jongen, T.: Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Prog. Aerospace Sci. 36(8), 655–682 (2000)

    Article  Google Scholar 

  17. Goldberg, U., Batten, P., Palaniswamy, S., Chakravarthy, S., Peroomian, O.: Hypersonic flow predictions using linear and nonlinear turbulence closures. J. Aircr. 37(4), 671–675 (2000)

    Article  Google Scholar 

  18. Huang, P., Coleman, G., Bradshaw, P.: Compressible turbulent channel flows: DNS results and modelling. J. Fluid Mech. 305, 185–218 (1995)

    Article  MATH  Google Scholar 

  19. James, G., Witten, D., Hastie, T., Tibshirani, R.: An Introduction to Statistical Learning, vol. 112. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  20. Jiang, G., Shu, C.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Johnson, R.W.: Handbook of Fluid Dynamics. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  22. Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 63(3), 425–464 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Keyes, F.G.: A summary of viscosity and heat-conduction data for He, A, \(H_2\), \(O_2\), \(CO\), \(CO_2\), \(H_2 O\), and air. Trans. Am. Soc. Mech. Eng. 73, 589–596 (1951)

    Google Scholar 

  24. Ling, J., Templeton, J.: Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty. Phys. Fluids (1994-present) 27(8), 085,103 (2015)

    Article  Google Scholar 

  25. Ling, J., Jones, R., Templeton, J.: Machine learning strategies for systems with invariance properties. J. Comput. Phys. 318, 22–35 (2016a)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ling, J., Kurzawski, A., Templeton, J.: Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166 (2016b)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ling, J., Ruiz, A., Lacaze, G., Oefelein, J.: Uncertainty analysis and data-driven model advances for a jet-in-crossflow. J. Turbomach. 139(2), 021,008 (2017)

    Article  Google Scholar 

  28. Lvd, Maaten, Hinton, G.: Visualizing data using t-SNE. J. Mach. Learn. Res. 9(Nov), 2579–2605 (2008)

    MATH  Google Scholar 

  29. Menter, F.R.: Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32(8), 1598–1605 (1994)

    Article  Google Scholar 

  30. Nieckele, A., Thompson, R., Mompean, G.: Anisotropic Reynolds stress tensor representation in shear flows using DNS and experimental data. J. Turbul. 17(6), 602–632 (2016)

    Article  MathSciNet  Google Scholar 

  31. Oliver, T.A., Moser, R.D.: Bayesian uncertainty quantification applied to RANS turbulence models. In: Journal of Physics: Conference Series, IOP Publishing vol 318, p. 042032 (2011)

  32. Parish, E.J., Duraisamy, K.: A paradigm for data-driven predictive modeling using field inversion and machine learning. J. Comput. Phys. 305, 758–774 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Ray, J., Lefantzi, S., Arunajatesan, S., Dechant, L.: Bayesian parameter estimation of a \(k\)-\(\varepsilon \) model for accurate jet-in-crossflow simulations. AIAA J. 54(8), 1–17 (2016)

    Article  Google Scholar 

  35. Rumsey, C.L.: Compressibility considerations for \(k\)-\(\omega \) turbulence models in hypersonic boundary-layer applications. J. Spacecr. Rockets 47(1), 11–20 (2010)

    Article  Google Scholar 

  36. Sebastian, J.J., James, S.E., Suryan, A.: Computational study of hypersonic flow past spiked blunt body using RANS and DSMC method. Procedia Technol. 25, 892–899 (2016)

    Article  Google Scholar 

  37. Singh, A.P., Duraisamy, K.: Using field inversion to quantify functional errors in turbulence closures. Phys. Fluids 28(045), 110 (2016)

    Google Scholar 

  38. Smits, A.J., Dussauge, J.P.: Turbulent Shear Layers in Supersonic Flow, 2nd edn. American Institute of Physics, College Park (2006)

    Google Scholar 

  39. So, R., Jin, L., Gatski, T.: An explicit algebraic reynolds stress and heat flux model for incompressible turbulence: part i non-isothermal flow. Theor. Comput. Fluid Dyn. 17(5), 351–376 (2004)

    Article  MATH  Google Scholar 

  40. Taylor, E.M., Wu, M., Martín, M.P.: Optimization of nonlinear error sources for weighted non-oscillatory methods in direct numerical simulations of compressible turbulence. J. Comput. Phys. 223, 384–397 (2006)

    Article  MATH  Google Scholar 

  41. Thompson, K.W.: Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68(1), 1–24 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  42. Touber, E., Sandham, N.D.: Oblique shock impinging on a turbulent boundary layer: low-frequency mechanisms. AIAA Paper 2008-4170 (2008)

  43. Trettel, A., Larsson, J.: Mean velocity scaling for compressible wall turbulence with heat transfer. Phys. Fluids 28(2), 026,102 (2016)

    Article  Google Scholar 

  44. Wang, J., Wu, J., Ling, J., Iaccarino, G., Xiao, H.: Physics-informed machine learning for predictive turbulence modeling: toward a complete framework. In: 2016 Proceedings of the CTR Summer Program, Stanford University, CA, p. 1 (2016a)

  45. Wang, J.X., Sun, R., Xiao, H.: Quantification of uncertainties in turbulence modeling: a comparison of physics-based and random matrix theoretic approaches. Int. J. Heat Fluid Flow 62(B), 577–592 (2016b)

    Article  Google Scholar 

  46. Wang, J.X., Wu, J.L., Xiao, H.: Incorporating prior knowledge for quantifying and reducing model-form uncertainty in RANS simulations. Int. J. Uncertain. Quantif. 6(2), 109–126 (2016c)

    Article  MathSciNet  Google Scholar 

  47. Wang, J.X., Wu, J., Ling, J., Iaccarino, G., Xiao, H.: A comprehensive physics-informed machine learning framework for predictive turbulence modeling, submitted, (2017a). arXiv:1701.07102

  48. Wang, J.X., Wu, J., Xiao, H.: Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2(3), 1–22 (2017b)

    Google Scholar 

  49. Williamson, J.: Low-storage Runge–Kutta schemes. J. Comput. Phys. 35(1), 48–56 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wu, J., Wang, J., Xiao, H., Ling, J.: Visualization of high dimensional turbulence simulation data using t-SNE. In: 19th AIAA Non-Deterministic Approaches Conference, p. 1770 (2017a)

  51. Wu, J.L., Wang, J.X., Xiao, H.: A Bayesian calibration–prediction method for reducing model-form uncertainties with application in RANS simulations. Flow, Turbulence and Combustion, pp. 1–26 (2015)

  52. Wu, J.L., Wang, J.X., Xiao, H., Ling, J.: A priori assessment of prediction confidence for data-driven turbulence modeling. Flow, Turbulence and Combustion, pp. 1–22 (2017b)

  53. Wu, J.L., Xiao, H., Paterson, E.: Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3(074), 602 (2018). https://doi.org/10.1103/PhysRevFluids.3.074602

    Google Scholar 

  54. Wu, M., Martín, M.P.: Direct numerical simulation of supersonic boundary layer over a compression ramp. AIAA J. 45(4), 879–889 (2007)

    Article  Google Scholar 

  55. Wu, X.: Inflow turbulence generation methods. Annu. Rev. Fluid Mech. 49, 23–49 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xiao, H., Wu, J.L., Wang, J.X., Sun, R., Roy, C.: Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: a data-driven, physics-informed Bayesian approach. J. Comput. Phys. 324, 115–136 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  57. Xu, S., Martín, M.P.: Assessment of inflow boundary conditions for compressible turbulent boundary layers. Phys. Fluids 16(7), 2623–2639 (2004)

    Article  MATH  Google Scholar 

  58. Zhang, C., Duan, L., Choudhari, M.M.: Effect of wall cooling on boundary-layer-induced pressure fluctuations at Mach 6. J. Fluid Mech. 822, 5–30 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, Center of Informatics and Computational Science, University of Notre Dame, Notre Dame, IN, 46556, USA

    Jian-Xun Wang

  2. Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Junji Huang & Lian Duan

  3. Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA, 24061, USA

    Heng Xiao

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  1. Jian-Xun Wang
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Correspondence to Jian-Xun Wang.

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Wang, JX., Huang, J., Duan, L. et al. Prediction of Reynolds stresses in high-Mach-number turbulent boundary layers using physics-informed machine learning. Theor. Comput. Fluid Dyn. 33, 1–19 (2019). https://doi.org/10.1007/s00162-018-0480-2

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