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Conditioning and accurate solutions of Reynolds average Navier–Stokes equations with data-driven turbulence closures

Published online by Cambridge University Press:  29 March 2021

Bernardo P. Brener ,
Matheus A. Cruz ,
Roney L. Thompson Open the ORCID record for Roney L. Thompson [Opens in a new window]  and
Rodrigo P. Anjos
Bernardo P. Brener
Affiliation:
COPPE, Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, 21941-909, Rio de Janeiro, RJ, Brazil
Matheus A. Cruz
Affiliation:
COPPE, Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, 21941-909, Rio de Janeiro, RJ, Brazil
Roney L. Thompson*
Affiliation:
COPPE, Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Ilha do Fundão, 21941-909, Rio de Janeiro, RJ, Brazil
Rodrigo P. Anjos
Affiliation:
Chemical and Biochemical Process Engineering (EPQB), School of Chemistry (EQ), Federal University of Rio de Janeiro (UFRJ), 21941-909, Rio de Janeiro, RJ, Brazil
*
†Email address for correspondence: rthompson@mecanica.coppe.ufrj.br
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Abstract

The possible ill conditioning of the Reynolds average Navier–Stokes (RANS) equations when an explicit data-driven Reynolds stress tensor closure is employed is a discussion of paramount importance. This matter has far-reaching consequences on the emerging field of data-driven turbulence modelling, as well as in Reynolds stress models and in epistemic uncertainty quantification. In the present work, we explore fundamental aspects of this problem with the aid of direct numerical simulation (DNS) databases of the turbulent flows in a plane channel, in a square duct and in periodic hill geometries. We show that the RANS equations are ill conditioned in the whole range of cases analysed, even when the linear term of the Reynolds stress tensor is treated implicitly, when no information with respect to the DNS mean velocity field is provided. That is, in more than ten different simulations varying Reynolds number and geometry, using the DNS Reynolds stress tensor solely, which carries a small error, the error propagation to the mean velocity field is significantly amplified. Accurate solutions can be obtained with implicit or explicit procedures that include information from the DNS mean velocity field. We propose a new strategy along these lines for solving the RANS equations that combines ideas put forward in the literature and show that this new procedure outperforms the ones previously adopted to mitigate the error propagation to the mean velocity field. We have shown advantages of adopting the Reynolds force vector, the divergence of the Reynolds stress tensor, as a quantity with its own identity, which can impact on the choice of the target quantity to be modelled in RANS equations.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A110
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Andrade, J.R., Martins, R.S., Thompson, R.L., Mompean, G. & Neto, A.S. 2018 Analysis of uncertainties and convergence of turbulent wall-bounded flows by means of a physically-based criterion. Phys. Fluids 30, 045106.CrossRefGoogle Scholar
Bae, H.J., Lozano-Durán, A., Bose, S.T. & Moin, P. 2018 Dynamic slip wall model for large–eddy simulation. J. Fluid Mech. 859, 400–432.CrossRefGoogle Scholar
Bauer, C., Feldmann, D. & Wagner, C. 2017 On the convergence and scaling of high-order statistical moments in turbulent pipe flow using direct numerical simulations. Phys. Fluids 29, 125105.CrossRefGoogle Scholar
Brunton, S.T., Noack, B.R. & Koumoutsakos, P. 2020 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508.CrossRefGoogle Scholar
Chaouat, B. & Schiestel, R. 2009 Progress in subgrid-scale transport modelling for continuous hybrid non-zonal RANS/LES simulations. Intl J. Heat Fluid Flow 30, 602–616.CrossRefGoogle Scholar
Cruz, M.A., Thompson, R.L., Sampaio, L.E.B. & Bacchi, R.D.A. 2019 The use of the Reynolds force vector in a physics informed machine learning approach for predictive turbulence modeling. Comput. Fluids 192, 1–14.CrossRefGoogle Scholar
Davidson, L. & Peng, S. 2003 Hybrid LES-RANS modelling: a one-equation SGS model combined with a k-epsilon model for predicting recirculating flows. Intl J. Numer. Meth. Fluids 43, 1003–1018.CrossRefGoogle Scholar
Duraisamy, K., Iaccarino, G. & Xiao, H. 2019 Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377.CrossRefGoogle Scholar
Edeling, W.N., Cinnella, P. & Dwight, R.P. 2014 a Predictive RANS simulations via Bayesian model-scenario averaging. J. Comput. Phys. 275, 65–91.CrossRefGoogle Scholar
Edeling, W.N., Cinnella, P., Dwight, R.P. & Bijl, H. 2014 b Bayesian estimates of parameter variability in the k-epsilon turbulence model. J. Comput. Phys. 258, 73–94.CrossRefGoogle Scholar
Edeling, W.N., Schmelzer, M., Dwight, R.P. & Cinnella, P. 2018 Bayesian predictions of Reynolds-averaged Navier–Stokes uncertainties using maximum a posteriori estimates. AIAA Comput. Fluid Dyn. 56 (5), 2018–2029.Google Scholar
Emory, M., Larsson, J. & Iaccarino, G. 2013 Modeling of structural uncertainties in Reynolds-averaged Navier–Stokes closures. Phys. Fluids 25, 110822.CrossRefGoogle Scholar
Flageul, C. & Tiselj, I. 2017 Impact of unresolved smaller scales on the scalar dissipation rate in direct numerical simulations of wall bounded flows. Intl J. Heat Fluid Flow 68, 173–179.CrossRefGoogle Scholar
Flageul, C. & Tiselj, I. 2018 Convergence rate of individual and global quantities in direct numerical simulations. Phys. Fluids 30, 111704.CrossRefGoogle Scholar
Friess, C., Manceau, R. & Gatski, T.B. 2015 Toward an equivalence criterion for Hybrid RANS/LES methods. Comput. Fluids 20, 233–246.CrossRefGoogle Scholar
Gorlé, C. & Iaccarino, G. 2013 A framework for epistemic uncertainty quantification of turbulent scalar flux models for Reynolds-averaged Navier–Stokes simulations. Phys. Fluids 25, 055105.CrossRefGoogle Scholar
Hamlington, P.E., Krasnov, D., Boeck, T. & Schumacher, J. 2012 Local dissipation scales and energy dissipation-rate moments in channel flow. J. Fluid Mech. 701, 419–429.CrossRefGoogle Scholar
Iaccarino, G., Mishra, A.A. & Ghili, S. 2017 Eigenspace perturbations for uncertainty estimation of single-point turbulence closures. Phys. Rev. Fluids 2, 024605.CrossRefGoogle Scholar
Kaandorp, M.L.A. & Dwight, R.P. 2020 Data-driven modelling of the Reynolds stress tensor using random forests with invariance. Comput. Fluids 202, 104497.CrossRefGoogle Scholar
Kozuca, M., Seki, Y. & Kawamura, H. 2009 DNS of turbulent heat transfer in a channel flow with a high spatial resolution. Intl J. Heat Fluid Flow 30, 514–524.CrossRefGoogle Scholar
Kutz, J.N. 2017 Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4.CrossRefGoogle Scholar
Launder, B.E. & Spalding, D.B. 1974 The numerical computation of turbulent flows. Comput. Meth. Appl. Mech. Engng 3, 269–289.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_{\tau }=5200$. J. Fluid Mech. 774, 395–415.CrossRefGoogle Scholar
Ling, J., Jones, R. & Templeton, J. 2016 a Machine learning strategies for systems with invariance properties. J. Comput. Phys. 318, 22–35.CrossRefGoogle Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 b Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.CrossRefGoogle Scholar
Ling, J., Ruiz, A., Lacanze, G. & Oefelein, J. 2016 c Uncertainty analysis and data-driven model advances for a jet-in-crossflow. Trans. ASME: J. Turbomach. 139, 021008.Google Scholar
Margheri, L., Meldi, M., Salvetti, M.V. & Sagaut, P. 2014 Epistemic uncertainties in RANS model free coefficients. Comput. Fluids 102, 315–335.CrossRefGoogle Scholar
Meyer, C. 2000 Matrix analysis and applied linear algebra. In Other Titles in Applied Mathematics. SIAM. ISBN: 9780898719512.CrossRefGoogle Scholar
Oliver, T.A., Malaya, N., Ulerich, R. & Moser, R.D. 2014 Estimating uncertainties in statistics computed from direct numerical simulation. Phys. Fluids 26, 035101.CrossRefGoogle Scholar
Parish, E. & Duraisamy, K. 2016 A paradigm for data-driven predictive modeling using field inversion and machine learning. J. Comput. Phys. 305, 758–774.CrossRefGoogle Scholar
Passalacqua, A. & Fox, R.O. 2011 Implementation of an iterative solution procedure for multi-fluid gas–particle flow models on unstructured grids. Powder Technol. 213, 174–187.CrossRefGoogle Scholar
Patankar, S.V. 1980 Numerical Heat Transfer and Fluid Flow. Taylor & Francis.Google Scholar
Perot, B. 1999 Turbulence modeling using body force potentials. Phys. Fluids 11 (9), 2645–2656.CrossRefGoogle Scholar
Pinelli, A., Uhlmann, M., Sekimoto, A. & Kawahara, G. 2010 Reynolds number dependence of mean flow structure in square duct turbulence. J. Fluid Mech. 644, 107–122.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows, 10th edn. Cambridge University Press.CrossRefGoogle Scholar
Poroseva, S.V., Colmenares, J.D. & Murman, S.M. 2016 On the accuracy of RANS simulations with DNS data. Phys. Fluids 28, 115102.CrossRefGoogle ScholarPubMed
Shih, T., Liou, W.W., Shabbir, A., Yang, Z. & Zhu, J. 1995 A new k-epsilon eddy viscosity model for high Reynolds number turbulent flows. Comput. Fluids 24, 227–238.CrossRefGoogle Scholar
Strang, G. 2006 Linear Algebra and Its Applications, 4th edn. Cengage Learning.Google Scholar
Thompson, R.L., Mompean, G. & Thais, L. 2010 A methodology to quantify the non-linearity of the Reynolds stress tensor. J. Turbul. 11, 1–27.CrossRefGoogle Scholar
Thompson, R.L., Mishra, A.A., Iaccarino, G., Edeling, W.N. & Sampaio, L.E.B. 2019 Eigenvector perturbation methodology for uncertainty quantification of turbulence models. Phys. Rev. Fluids 4, 044603.CrossRefGoogle Scholar
Thompson, R.L., Sampaio, L.E.B., Alves, F.A.V.B., Thais, L. & Mompean, G. 2016 A methodology to evaluate statistical errors in DNS data of plane channel flows. Comput. Fluids 130, 1–7.CrossRefGoogle Scholar
Tracey, B., Duraisamy, K. & Alonso, J.J. 2013 Application of supervised learning to quantify uncertainties in turbulence and combustion modeling. In AIAA Aerospace Sciences Meeting. AIAA Paper 2013-0259.CrossRefGoogle Scholar
Tracey, B., Duraisamy, K. & Alonso, J.J. 2015 A machine learning strategy to assist turbulence model development. In AIAA Aerospace Sciences Meeting. AIAA Paper 2015-1287.CrossRefGoogle Scholar
Vreman, A.W. & Kuerten, J.G.M. 2014 a Comparison of direct numerical simulation databases of turbulent channel flow at $Re_{\tau } = 180$. Phys. Fluids 26, 015102.CrossRefGoogle Scholar
Vreman, A.W. & Kuerten, J.G.M. 2014 b Statistics of spatial derivatives of velocity and pressure in turbulent channel flow. Phys. Fluids 26, 085103.CrossRefGoogle Scholar
Wang, J.-X., Xiao, H. & Wu, J.-L. 2017 Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2, 034603.CrossRefGoogle Scholar
Wu, J.-L., Wang, J.-X. & Xiao, H. 2016 A Bayesian calibration–prediction method for reducing model–form uncertainties with application in RANS simulations. Flow Turbul. Combust. 97, 761–786.CrossRefGoogle Scholar
Wu, J.-L., Wang, J.-X., Xiao, H. & Ling, J. 2017 A priori assessment of prediction confidence for data-driven turbulence modeling. Flow Turbul. Combust. 99, 25–46.CrossRefGoogle Scholar
Wu, J.-L., Xiao, H. & Paterson, E. 2018 Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3, 074602.CrossRefGoogle Scholar
Wu, J.-L., Xiao, H., Sun, R. & Wang, Q. 2019 Reynolds-average Navier-Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned. J. Fluid Mech. 869, 553–586.CrossRefGoogle Scholar
Xiao, H. & Cinnella, P. 2019 Quantification of model uncertainty in RANS simulations: a review. Prog. Aerosp. Sci. 108, 1–31.CrossRefGoogle Scholar
Xiao, H. & Jenny, P. 2012 A consistent dual–mesh framework for hybrid LES/RANS modeling. J. Comput. Phys. 231, 1848–1865.CrossRefGoogle Scholar
Xiao, H., Wu, J.-L., Laizet, S. & Duan, L. 2020 Flows over periodic hills of parameterized geometries: a dataset for data-driven turbulence modeling from direct simulations. Comput. Fluids 200, 104431.CrossRefGoogle Scholar
Xiao, H., Wu, J.-L., Wang, J.-X., Sun, R. & Roy, C. 2016 Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: a data-driven, physics-informed bayesian approach. J. Comput. Phys. 324, 115–136.CrossRefGoogle Scholar
Yakhot, V., Thangam, S., Gatski, T.B., Orszag, S.A. & Speziale, C.G. 1991 Development of turbulence models for shear flows by a double expansion technique. In ICASE, pp. 1–24. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center.Google Scholar
Yang, X.I.A. & Lozano-Durán, A. 2017 A multifractal model for the momentum transfer process in wall–bounded flows. J. Fluid Mech. 824, R2.CrossRefGoogle ScholarPubMed
Zhao, Y., Akolekar, H.D., Weatheritt, J., Michelassi, V. & Sandberg, R.D. 2020 RANS turbulence model development using CFD-driven machine learning. J. Comput. Phys. 411, 109413.CrossRefGoogle Scholar
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