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Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned

Published online by Cambridge University Press:  29 April 2019

Jinlong Wu ,
Heng Xiao Open the ORCID record for Heng Xiao [Opens in a new window] ,
Rui Sun  and
Qiqi Wang
Jinlong Wu
Affiliation:
Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24060, USA
Heng Xiao*
Affiliation:
Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24060, USA
Rui Sun
Affiliation:
Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24060, USA
Qiqi Wang
Affiliation:
Department of Aeronautics and Astronautics, MIT, Cambridge, MA 02139, USA
*
†Email address for correspondence: hengxiao@vt.edu
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Abstract

Reynolds-averaged Navier–Stokes (RANS) simulations with turbulence closure models continue to play important roles in industrial flow simulations. However, the commonly used linear eddy-viscosity models are intrinsically unable to handle flows with non-equilibrium turbulence (e.g. flows with massive separation). Reynolds stress models, on the other hand, are plagued by their lack of robustness. Recent studies in plane channel flows found that even substituting Reynolds stresses with errors below 0.5 % from direct numerical simulation databases into RANS equations leads to velocities with large errors (up to 35 %). While such an observation may have only marginal relevance to traditional Reynolds stress models, it is disturbing for the recently emerging data-driven models that treat the Reynolds stress as an explicit source term in the RANS equations, as it suggests that the RANS equations with such models can be ill-conditioned. So far, a rigorous analysis of the condition of such models is still lacking. As such, in this work we propose a metric based on local condition number function for a priori evaluation of the conditioning of the RANS equations. We further show that the ill-conditioning cannot be explained by the global matrix condition number of the discretized RANS equations. Comprehensive numerical tests are performed on turbulent channel flows at various Reynolds numbers and additionally on two complex flows, i.e. flow over periodic hills, and flow in a square duct. Results suggest that the proposed metric can adequately explain observations in previous studies, i.e. deteriorated model conditioning with increasing Reynolds number and better conditioning of the implicit treatment of the Reynolds stress compared to the explicit treatment. This metric can play critical roles in the future development of data-driven turbulence models by enforcing the conditioning as a requirement on these models.

JFM classification

Mathematical Foundations: Computational methods Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 869 , 25 June 2019 , pp. 553 - 586
Copyright
© 2019 Cambridge University Press 

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References

Basara, B. & Jakirlic, S. 2003 A new hybrid turbulence modelling strategy for industrial CFD. Intl J. Numer. Meth. Fluids 42 (1), 89–116.Google Scholar
Bernardini, M., Pirozzoli, S. & Orlandi, P. 2014 Velocity statistics in turbulent channel flow up to Re = 4000. J. Fluid Mech. 742, 171–191.Google Scholar
Breuer, M., Peller, N., Rapp, C. & Manhart, M. 2009 Flow over periodic hills: numerical and experimental study in a wide range of Reynolds numbers. Comput. Fluids 38 (2), 433–457.Google Scholar
Chandrasekaran, S. & Ipsen, I. C. F. 1995 On the sensitivity of solution components in linear systems of equations. SIAM J. Matrix Anal. Applics. 16 (1), 93–112.Google Scholar
Debnath, L. & Mikusiński, P. 2005 Hilbert Spaces with Applications. Academic Press.Google Scholar
Del Alamo, J. C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41–L44.Google Scholar
Gamahara, M. & Hattori, Y. 2017 Searching for turbulence models by artificial neural network. Phys. Rev. Fluids 2 (5), 054604.Google Scholar
Geneva, N. & Zabaras, N. 2019 Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks. J. Comput. Phys. 383, 125–147.Google Scholar
Hamlington, P. E. & Dahm, W. J. A. 2008 Reynolds stress closure for nonequilibrium effects in turbulent flows. Phys. Fluids 20 (11), 115101.Google Scholar
Hamlington, P. E. & Ihme, M. 2014 Modeling of non-equilibrium homogeneous turbulence in rapidly compressed flows. Flow Turbul. Combust. 93 (1), 93–124.Google Scholar
King, R. N., Hamlington, P. E. & Dahm, W. J. A. 2016 Autonomic closure for turbulence simulations. Phys. Rev. E 93 (3), 031301.Google Scholar
Laizet, S. & Lamballais, E. 2009 High-order compact schemes for incompressible flows: a simple and efficient method with quasi-spectral accuracy. J. Comput. Phys. 228 (16), 5989–6015.Google Scholar
Laizet, S. & Li, N. 2011 Incompact3d: a powerful tool to tackle turbulence problems with up to O (105) computational cores. Intl J. Numer. Meth. Fluids 67 (11), 1735–1757.Google Scholar
Lanczos, C. 1996 Linear Differential Operators. SIAM.Google Scholar
Launder, B. E. & Sharma, B. I. 1974 Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1 (2), 131–137.Google Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to Re = 5200. J. Fluid Mech. 774, 395–415.Google Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.Google Scholar
Maduta, R. & Jakirlic, S. 2017 Improved RANS computations of flow over the 25° -slant-angle Ahmed body. SAE Intl J. Passenger Cars – Mech. Syst. 10 (2), 649–661.Google Scholar
Maulik, R. & San, O. 2017 A neural network approach for the blind deconvolution of turbulent flows. J. Fluid Mech. 831, 151–181.Google Scholar
Maulik, R., San, O., Rasheed, A. & Vedula, P. 2019 Sub-grid modelling for two-dimensional turbulence using neural networks. J. Fluid Mech. 858, 122–144.Google Scholar
Menter, F. R. 1994 Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 1598–1605.Google Scholar
Parish, E. J. & Duraisamy, K. 2016 A paradigm for data-driven predictive modeling using field inversion and machine learning. J. Comput. Phys. 305, 758–774.Google 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.Google Scholar
Pope, S. B. 1975 A more general effective-viscosity hypothesis. J. Fluid Mech. 72 (2), 331–340.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Poroseva, S. V., Colmenares, F., Juan, D. & Murman, S. M. 2016 On the accuracy of RANS simulations with DNS data. Phys. Fluids 28 (11), 115102.Google Scholar
Singh, A. P. & Duraisamy, K. 2016 Using field inversion to quantify functional errors in turbulence closures. Phys. Fluids 28, 045110.Google Scholar
Singh, A. P., Medida, S. & Duraisamy, K. 2017 Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils. AIAA J. 55 (7), 2215–2227.Google Scholar
Slotnick, J., Khodadoust, A., Alonso, J., Darmofal, D., Gropp, W., Lurie, E. & Mavriplis, D.2014 CFD Vision 2030 Study: a path to revolutionary computational aerosciences. Tech. Rep. National Aeronautics and Space Administration, Langley Research Center, Hampton, VA.Google Scholar
Spalart, P. R. & Allmaras, S. R. 1994 A one-equation turbulence model for aerodynamic flows. Rech. Aerosp, 1, 5–21.Google Scholar
Speziale, C. G. & Xu, X.-H. 1996 Towards the development of second-order closure models for nonequilibrium turbulent flows. Intl J. Heat Fluid Flow 17 (3), 238–244.Google Scholar
Steele, J. M. 2004 The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press.Google Scholar
Strang, G. 2016 Introduction to Linear Algebra, 5th edn. Wellesley–Cambridge Press.Google Scholar
Thompson, R. L., Sampaio, L. E. B., de Bragança Alves, F. A. V., 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
Vollant, A., Balarac, G. & Corre, C. 2017 Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures. J. Turbul. 18 (9), 854–878.Google Scholar
Wang, J.-X., Wu, J.-L. & Xiao, H. 2017 Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2 (3), 034603.Google Scholar
Weatheritt, J. & Sandberg, R. 2016 A novel evolutionary algorithm applied to algebraic modifications of the RANS stress–strain relationship. J. Comput. Phys. 325, 22–37.Google Scholar
Weller, H. G., Tabor, G., Jasak, H. & Fureby, C. 1998 A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12 (6), 620–631.Google Scholar
Wilcox, D. C. 1988 Reassessment of the scale-determining equation for advanced turbulence models. AIAA J. 26 (11), 1299–1310.Google 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 (7), 074602.Google Scholar
Zhu, L., Zhang, W., Kou, J. & Liu, Y. 2019 Machine learning methods for turbulence modeling in subsonic flows around airfoils. Phys. Fluids 31 (1), 015105.Google Scholar