1932

Abstract

Data from experiments and direct simulations of turbulence have historically been used to calibrate simple engineering models such as those based on the Reynolds-averaged Navier–Stokes (RANS) equations. In the past few years, with the availability of large and diverse data sets, researchers have begun to explore methods to systematically inform turbulence models with data, with the goal of quantifying and reducing model uncertainties. This review surveys recent developments in bounding uncertainties in RANS models via physical constraints, in adopting statistical inference to characterize model coefficients and estimate discrepancy, and in using machine learning to improve turbulence models. Key principles, achievements, and challenges are discussed. A central perspective advocated in this review is that by exploiting foundational knowledge in turbulence modeling and physical constraints, researchers can use data-driven approaches to yield useful predictive models.

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2019-01-05
2024-03-29
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Literature Cited

  1. Aster RC, Borchers B, Thurber CH 2012. Parameter Estimation and Inverse Problems Waltham, MA: Academic. 2nd ed.
  2. Banerjee S, Krahl R, Durst F, Zenger C 2007. Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches. J. Turbul. 8:N32
    [Google Scholar]
  3. Brunton SL, Proctor JL, Kutz JN 2016. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. PNAS 113:3932–37
    [Google Scholar]
  4. Busse F 1970. Bounds for turbulent shear flow. J. Fluid Mech. 41:219–40
    [Google Scholar]
  5. Cheung SH, Oliver TA, Prudencio EE, Prudhomme S, Moser RD 2011. Bayesian uncertainty analysis with applications to turbulence modeling. Reliab. Eng. Syst. Saf. 96:1137–49
    [Google Scholar]
  6. Comte-Bellot G, Corrsin S 1966. The use of a contraction to improve the isotropy of grid-generated turbulence. J. Fluid Mech. 25:657–82
    [Google Scholar]
  7. Coupland J 1993. ERCOFTAC (European Research Community on Flow, Turbulence and Combustion) classic collection database Test Case Database, Comput. Fluid Dyn. Turbul. Mech. Res. Group, Univ. Manchester, Manchester, UK. http://cfd.mace.manchester.ac.uk/ercoftac
  8. Darrigol O 2005. Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl Oxford: Oxford Univ. Press
  9. Doering CR, Constantin P 1994. Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49:4087
    [Google Scholar]
  10. Dow E, Wang Q 2011. Quantification of structural uncertainties in the k−w turbulence model Paper presented at AIAA Structures, Structural Dynamics and Materials Conference, 52nd, Denver, CO, AIAA Pap. 2011-1762
    [Google Scholar]
  11. Duraisamy K, Zhang ZJ, Singh AP 2015. New approaches in turbulence and transition modeling using data-driven techniques Paper presented at AIAA Aerospace Sciences Meeting, 53rd, Kissimmee, FL, AIAA Pap. 2015-1284
  12. Durbin PA 2018. Some recent developments in turbulence closure modeling. Annu. Rev. Fluid Mech. 50:77–103
    [Google Scholar]
  13. Durbin PA, Speziale CG 1994. Realizability of second-moment closure via stochastic analysis. J. Fluid Mech. 280:395–407
    [Google Scholar]
  14. Edeling WN, Cinnella P, Dwight RP 2014.a Predictive RANS simulations via Bayesian model-scenario averaging. J. Comput. Phys. 275:65–91
    [Google Scholar]
  15. Edeling WN, Cinnella P, Dwight RP, Bijl H 2014.b Bayesian estimates of parameter variability in the k−ε turbulence model. J. Comput. Phys. 258:73–94
    [Google Scholar]
  16. Edeling WN, Iaccarino G, Cinnella P 2017. Data-free and data-driven RANS predictions with quantified uncertainty. Flow Turbul. Combust. 100:593–616
    [Google Scholar]
  17. Efron B, Hastie T 2016. Computer Age Statistical Inference: Algorithms, Evidence and Data Science Cambridge, UK: Cambridge Univ. Press
  18. Eisfeld B 2017. Reynolds stress anisotropy in self-preserving turbulent shear flows Tech. Rep. DLR-IB-AS-BS-2017-106, Dtsch. Zent. Luft Raumfahrt (DLR), Braunschweig, Ger.
  19. Emory M, Larsson J, Iaccarino G 2013. Modeling of structural uncertainties in Reynolds-averaged Navier-Stokes closures. Phys. Fluids 25:110822
    [Google Scholar]
  20. Emory M, Pecnik R, Iaccarino G 2011. Modeling structural uncertainties in Reynolds-averaged computations of shock/boundary layer interactions Paper presented at AIAA Aerospace Sciences Meeting, 49th, Orlando, FL, AIAA Pap. 2011-479
  21. Gamahara M, Hattori Y 2017. Searching for turbulence models by artificial neural network. Phys. Rev. Fluids 2:054604
    [Google Scholar]
  22. Gatski T, Jongen T 2000. Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Prog. Aerosp. Sci. 36:655–82
    [Google Scholar]
  23. Giles MB, Duta MC, Müller J-D, Pierce NA 2003. Algorithm developments for discrete adjoint methods. AIAA J. 41:198–205
    [Google Scholar]
  24. Girimaji SS 2006. Partially-averaged Navier-Stokes model for turbulence: a Reynolds-averaged Navier-Stokes to direct numerical simulation bridging method. J. Appl. Mech. 73:413–21
    [Google Scholar]
  25. Gorlé C, Larsson J, Emory M, Iaccarino G 2014. The deviation from parallel shear flow as an indicator of linear eddy-viscosity model inaccuracy. Phys. Fluids 26:051702
    [Google Scholar]
  26. Hamlington PE, Dahm WJ 2008. Reynolds stress closure for nonequilibrium effects in turbulent flows. Phys. Fluids 20:115101
    [Google Scholar]
  27. Howard LN 1972. Bounds on flow quantities. Annu. Rev. Fluid Mech. 4:473–94
    [Google Scholar]
  28. Iaccarino G, Mishra AA, Ghili S 2017. Eigenspace perturbations for uncertainty estimation of single-point turbulence closures. Phys. Rev. Fluids 2:024605
    [Google Scholar]
  29. Iglesias MA, Law KJ, Stuart AM 2013. Ensemble Kalman methods for inverse problems. Inverse Probl. 29:045001
    [Google Scholar]
  30. Jofre L, Domino SP, Iaccarino G 2018. A framework for characterizing structural uncertainty in large-eddy simulation closures. Flow Turbul. Combust. 100:341–63
    [Google Scholar]
  31. Kennedy MC, O'Hagan A 2001. Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B 63:425–64
    [Google Scholar]
  32. Kim J, Moin P, Moser R 1987. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177:133–66
    [Google Scholar]
  33. Launder B, Reece GJ, Rodi W 1975. Progress in the development of a Reynolds-stress turbulence closure. J. Mech. 68:3537–66
    [Google Scholar]
  34. Lefantzi S, Ray J, Arunajatesan S, Dechant L 2015. Estimation of k − ε parameters using surrogate models and jet-in-crossflow data Tech. Rep., Sandia Natl. Lab., Livermore, CA
  35. Li Y, Perlman E, Wan M, Yang Y, Meneveau C et al. 2008. A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9:N31
    [Google Scholar]
  36. Ling J, Jones R, Templeton J 2016.a Machine learning strategies for systems with invariance properties. J. Comput. Phys. 318:22–35
    [Google Scholar]
  37. Ling J, Kurzawski A, Templeton J 2016.b Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807:155–66
    [Google Scholar]
  38. Ling J, Templeton J 2015. Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty. Phys. Fluids 27:085103
    [Google Scholar]
  39. Lumley JL 1979. Computational modeling of turbulent flows. Adv. Appl. Mech. 18:123–76
    [Google Scholar]
  40. Ma M, Lu J, Tryggvason G 2015. Using statistical learning to close two-fluid multiphase flow equations for a simple bubbly system. Phys. Fluids 27:092101
    [Google Scholar]
  41. Ma M, Lu J, Tryggvason G 2016. Using statistical learning to close two-fluid multiphase flow equations for bubbly flows in vertical channels. Int. J. Multiphase Flow 85:336–47
    [Google Scholar]
  42. Maulik R, San O 2017. A neural network approach for the blind deconvolution of turbulent flows. J. Fluid Mech. 831:151–81
    [Google Scholar]
  43. Mishra AA, Iaccarino G 2017. Uncertainty estimation for Reynolds-averaged Navier–Stokes predictions of high-speed aircraft nozzle jets. AIAA J. 55:3999–4004
    [Google Scholar]
  44. Oliver T, Moser R 2009. Uncertainty quantification for RANS turbulence model predictions Abstract presented at 62nd Annual Meeting of the APS Division of Fluid Dynamics, November 22–24, Minneapolis, MN, Abstr. LC.004
  45. Oliver TA, Moser RD 2011. Bayesian uncertainty quantification applied to RANS turbulence models. J. Phys. Conf. Ser. 318:042032
    [Google Scholar]
  46. Pan S, Duraisamy K 2018. Data-driven discovery of closure models. SIAM J. Appl. Dyn. Syst. 17:2381–413
    [Google Scholar]
  47. Parish EJ, Duraisamy K 2016. A paradigm for data-driven predictive modeling using field inversion and machine learning. J. Comput. Phys. 305:758–74
    [Google Scholar]
  48. Pope SB 1975. A more general effective-viscosity hypothesis. J. Fluid Mech. 72:331–40
    [Google Scholar]
  49. Pope SB 1985. PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11:119–92
    [Google Scholar]
  50. Pope SB 2000. Turbulent Flows Cambridge, UK: Cambridge Univ. Press
  51. Poroseva S, Colmenares FJD, Murman S 2016. On the accuracy of RANS simulations with DNS data. Phys. Fluids 28:115102
    [Google Scholar]
  52. Raissi M, Perdikaris P, Karniadakis GE 2017. Physics informed deep learning (part II): data-driven discovery of nonlinear partial differential equations. arXiv:1711.10566 [cs.AI]
  53. Rastegari A, Akhavan R 2018. The common mechanism of turbulent skin-friction drag reduction with superhydrophobic longitudinal microgrooves and riblets. J. Fluid Mech. 838:68–104
    [Google Scholar]
  54. Ray J, Lefantzi S, Arunajatesan S, Dechant L 2016. Bayesian parameter estimation of a k−ε model for accurate jet-in-crossflow simulations. AIAA J. 54:82432–48
    [Google Scholar]
  55. Ray J, Lefantzi S, Arunajatesan S, Dechant L 2018. Learning an eddy viscosity model using shrinkage and Bayesian calibration: a jet-in-crossflow case study. ASCE-ASME J. Risk Uncertain. Eng. Syst. B 4:011001
    [Google Scholar]
  56. Rudd RE, Broughton JQ 1998. Coarse-grained molecular dynamics and the atomic limit of finite elements. Phys. Rev. B 58:R5893
    [Google Scholar]
  57. Schmidt M, Lipson H 2009. Distilling free-form natural laws from experimental data. Science 324:81–85
    [Google Scholar]
  58. Schumann U 1977. Realizability of Reynolds-stress turbulence models. Phys. Fluids 20:721–25
    [Google Scholar]
  59. Seis C 2015. Scaling bounds on dissipation in turbulent flows. J. Fluid Mech. 777:591–603
    [Google Scholar]
  60. Shur ML, Strelets MK, Travin AK, Spalart PR 2000. Turbulence modeling in rotating and curved channels: assessing the Spalart–Shur correction. AIAA J. 38:784–92
    [Google Scholar]
  61. Singh AP, Duraisamy K 2016. Using field inversion to quantify functional errors in turbulence closures. Phys. Fluids 28:045110
    [Google Scholar]
  62. Singh AP, Duraisamy K, Zhang ZJ 2017.a Augmentation of turbulence models using field inversion and machine learning Paper presented at AIAA Aerospace Sciences Meeting, 55th, Grapevine, TX, AIAA Pap. 2017-0993
  63. Singh AP, Medida S, Duraisamy K 2017.b Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils. AIAA J. 55:2215–27
    [Google Scholar]
  64. Spalart PR 2009. Detached-eddy simulation. Annu. Rev. Fluid Mech. 41:181–202
    [Google Scholar]
  65. Spalart PR, Allmaras S 1992. A one-equation turbulence model for aerodynamic flows Paper presented at AIAA Aerospace Sciences Meeting, 30th, Reno, NV, AIAA Pap. 1992-439
  66. Thompson RL, Sampaio LEB, de Bragança Alves FAV, Thais L, Mompean G 2016. A methodology to evaluate statistical errors in DNS data of plane channel flows. Comput. Fluids 130:1–7
    [Google Scholar]
  67. Tracey B, Duraisamy K, Alonso JJ 2013. Application of supervised learning to quantify uncertainties in turbulence and combustion modeling Paper presented at AIAA Aerospace Sciences Meeting, 51st, Dallas, TX, AIAA Pap. 2013-0259
  68. Tracey B, Duraisamy K, Alonso JJ 2015. A machine learning strategy to assist turbulence model development Paper presented at AIAA Aerospace Sciences Meeting, 53rd, Kissimmee, FL, AIAA Pap. 2015-1287
  69. Vollant A, Balarac G, Corre C 2017. Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures. J. Turbul. 18:854–78
    [Google Scholar]
  70. Vollant A, Balarac G, Geraci G, Corre C 2014. Optimal estimator and artificial neural network as efficient tools for the subgrid-scale scalar flux modeling Tech. Rep., Proceedings of the Summer Research Program, Center of Turbulence Research, Stanford University, Stanford, CA
  71. Wang JX, Sun R, Xiao H 2016.a Quantification of uncertainties in turbulence modeling: a comparison of physics-based and random matrix theoretic approaches. Int. J. Heat Fluid Flow 62:577–92
    [Google Scholar]
  72. Wang JX, Wu JL, Xiao H 2016.b Incorporating prior knowledge for quantifying and reducing model-form uncertainty in RANS simulations. Int. J. Uncertain. Quantif. 6:6109–26
    [Google Scholar]
  73. Wang JX, Wu JL, Xiao H 2017. Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2:034603
    [Google Scholar]
  74. Weatheritt J, Sandberg RD 2016. A novel evolutionary algorithm applied to algebraic modifications of the RANS stress–strain relationship. J. Comput. Phys. 325:22–37
    [Google Scholar]
  75. Weatheritt J, Sandberg RD 2017. The development of algebraic stress models using a novel evolutionary algorithm. Int. J. Heat Fluid Flow 68:298–318
    [Google Scholar]
  76. Wilcox DC 2006. Turbulence Modeling for CFD La Canada, CA: DCW Indust.
  77. Wu JL, Sun R, Laizet S, Xiao H 2018.a Representation of Reynolds stress perturbations with application in machine-learning-assisted turbulence modeling. Comput. Methods Appl. Mech. Eng. In press
  78. Wu JL, Wang JX, Xiao H 2016. A Bayesian calibration–prediction method for reducing model-form uncertainties with application in RANS simulations. Flow Turbul. Combust. 97:761–86
    [Google Scholar]
  79. Wu JL, Xiao H, Paterson EG 2018.b Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3:074602
    [Google Scholar]
  80. Wu JL, Xiao H, Sun R, Wang Q 2018.c RANS equations with Reynolds stress closure can be ill-conditioned. arXiv:1803.05581 [physics.flu-dyn]
  81. Xiao H, Cinnella P 2018. Quantification of model uncertainty in RANS simulations: a review. arXiv:1806.10434 [physics.flu-dyn]
  82. Xiao H, Wang JX, Ghanem RG 2017. A random matrix approach for quantifying model-form uncertainties in turbulence modeling. Comput. Methods Appl. Mech. Eng. 313:941–65
    [Google Scholar]
  83. Xiao H, Wu JL, Wang JX, Sun R, Roy CJ 2016. Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: a data-driven, physics-informed Bayesian approach. J. Comput. Phys. 324:115–36
    [Google Scholar]
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