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Representation of stress tensor perturbations with application in machine-learning-assisted turbulence modeling. (English) Zbl 1440.76042

Summary: Numerical simulations based on Reynolds-Averaged Navier-Stokes (RANS) equations are widely used in engineering design and analysis involving turbulent flows. However, RANS simulations are known to be unreliable in many flows of engineering relevance, which is largely caused by model-form uncertainties associated with the Reynolds stresses. Recently, a machine-learning approach has been proposed to quantify the discrepancies between RANS modeled Reynolds stress and the true Reynolds stress. However, it remains a challenge to represent discrepancies in the Reynolds stress eigenvectors in machine learning due to the requirements of spatial smoothness, frame-independence, and realizability. This challenge also exists in the data-driven computational mechanics in general where quantifying the perturbation of stress tensors is needed. In this work, we propose three schemes for representing perturbations to the eigenvectors of RANS modeled Reynolds stresses: (1) discrepancy-based Euler angles, (2) direct-rotation-based Euler angles, and (3) unit quaternions. We compare these metrics by performing a priori and a posteriori tests on two canonical flows: fully developed turbulent flows in a square duct and massively separated flows over periodic hills. The results demonstrate that the direct-rotation-based Euler angles representation lacks spatial smoothness while the discrepancy-based Euler angles representation lacks frame-independence, making them unsuitable for being used in machine-learning-assisted turbulence modeling. In contrast, the representation based on unit quaternion satisfies all the requirements stated above, and thus it is an ideal choice in representing the perturbations associated with the eigenvectors of Reynolds stress tensors. This finding has clear importance for uncertainty quantification and machine learning in turbulence modeling and for data-driven computational mechanics in general.

MSC:

76F65 Direct numerical and large eddy simulation of turbulence
65F35 Numerical computation of matrix norms, conditioning, scaling
68T05 Learning and adaptive systems in artificial intelligence
76M99 Basic methods in fluid mechanics

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References:

[1] Wilcox, D. C., Turbulence Modeling for CFD (2006), DCW Industries
[2] Wang, J.-X.; Wu, J.-L.; Xiao, H., Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data, Phys. Rev. Fluids, 2, 3, Article 034603 pp. (2017)
[3] Ling, J.; Kurzawski, A.; Templeton, J., Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, J. Fluid Mech., 807, 155-166 (2016) · Zbl 1383.76175
[4] A.P. Singh, S. Medida, K. Duraisamy, Machine learning-augmented predictive modeling of turbulent separated flows over airfoils. arXiv preprint arXiv:1608.03990; A.P. Singh, S. Medida, K. Duraisamy, Machine learning-augmented predictive modeling of turbulent separated flows over airfoils. arXiv preprint arXiv:1608.03990
[5] Tracey, B.; Duraisamy, K.; Alonso, J. J., A machine learning strategy to assist turbulence model development, AIAA Paper, 1287, 2015 (2015)
[6] Singh, A. P.; Duraisamy, K., Using field inversion to quantify functional errors in turbulence closures, Phys. Fluids, 28, Article 045110 pp. (2016)
[7] Pope, S. B., Turbulent Flows (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0966.76002
[8] H. Xiao, J.-L. Wu, J.-X. Wang, Physics-informed machine learning for predictive turbulence modeling: Progress and perspectives, in: 2017 AIAA SciTech Forum and Exposition, 2017, Grapevine, TX, 2017, AIAA, Reston, VA, 2017, paper 2017-1712.; H. Xiao, J.-L. Wu, J.-X. Wang, Physics-informed machine learning for predictive turbulence modeling: Progress and perspectives, in: 2017 AIAA SciTech Forum and Exposition, 2017, Grapevine, TX, 2017, AIAA, Reston, VA, 2017, paper 2017-1712.
[9] King, R. N.; Hamlington, P. E.; Dahm, W. J., Autonomic closure for turbulence simulations, Phys. Rev. E, 93, 3, Article 031301 pp. (2016)
[10] Vollant, A.; Balarac, G.; Corre, C., Subgrid-scale scalar flux modelling based on optimal estimation theory and machine-learning procedures, J. Turbul., 1-25 (2017)
[11] Emory, M.; Larsson, J.; Iaccarino, G., Modeling of structural uncertainties in Reynolds-averaged Navier-Stokes closures, Phys. Fluids, 25, 11, Article 110822 pp. (2013)
[12] Emory, M.; Pecnik, R.; Iaccarino, G., Modeling structural uncertainties in Reynolds-averaged computations of shock/boundary layer interactions, AIAA Paper, 479, 1-16 (2011)
[13] Emory, M. A., Estimating Model-Form Uncertainty in Reynolds-Averaged navier-Stokes Closures (2014), Stanford University, (Ph.D. thesis)
[14] Banerjee, S.; Krahl, R.; Durst, F.; Zenger, C., Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches, J. Turbul., 8, 32, 1-27 (2007) · Zbl 1273.76244
[15] Lumley, J. L.; Newman, G. R., The return to isotropy of homogeneous turbulence, J. Fluid Mech., 82, 01, 161-178 (1977) · Zbl 0368.76055
[16] Thompson, R.; Sampaio, L.; Edeling, W.; Mishra, A. A.; Iaccarino, G., A strategy for the eigenvector perturbations of the Reynolds stress tensor in the context of uncertainty quantification, (Proceedings of the Summer Program (2016), Center for Turbulence Research), 10
[17] Iaccarino, G.; Mishra, A. A.; Ghili, S., Eigenspace perturbations for uncertainty estimation of single-point turbulence closures, Phys. Rev. Fluids, 2, 2, Article 024605 pp. (2017)
[18] 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 Flows, 62, 577-592 (2016)
[19] Wang, J.-X.; Wu, J.-L.; Ling, J.; Iaccarino, G.; Xiao, H., (Towards a Complete Framework of Physics-Informed Machine Learning for Predictive Turbulence Modeling, Tech. Rep.. Towards a Complete Framework of Physics-Informed Machine Learning for Predictive Turbulence Modeling, Tech. Rep., Proceedings of Summer Research Program (2016), Center of Turbulence Research, Stanford University: Center of Turbulence Research, Stanford University Stanford, CA, USA)
[20] Duraisamy, K.; Iaccarino, G.; Xiao, H., Turbulence modeling in the age of data, Annu. Rev. Fluid Mech., 51, 357-377 (2019) · Zbl 1412.76040
[21] Wu, J.-L.; Xiao, H.; Paterson, E., Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework, Phys. Rev. Fluids, 3, 7, Article 074602 pp. (2018)
[22] Wu, J.-L.; Wang, J.-X.; Xiao, H.; Ling, J., A priori assessment of prediction confidence for data-driven turbulence modeling, Flow Turbul. Combust., 99, 25-46 (2017)
[23] Huynh, D. Q., Metrics for 3d rotations: Comparison and analysis, J. Math. Imaging Vision, 35, 2, 155-164 (2009) · Zbl 1490.68249
[24] Kuffner, J. J., Effective sampling and distance metrics for 3d rigid body path planning, (2004 IEEE International Conference on Robotics and Automation. 2004 IEEE International Conference on Robotics and Automation, ICRA’04. 2004 IEEE International Conference on Robotics and Automation. 2004 IEEE International Conference on Robotics and Automation, ICRA’04, 2004 Proceedings, vol. 4 (2004), IEEE), 3993-3998
[25] Horn, B. K., Closed-form solution of absolute orientation using unit quaternions, J. Opt. Soc. Amer. A, 4, 4, 629-642 (1987)
[26] Heeger, D. J.; Jepson, A., Simple method for computing 3d motion and depth, (Third International Conference on Computer Vision. Third International Conference on Computer Vision, 1990 Proceedings (1990), IEEE), 96-100 · Zbl 0825.68384
[27] Domingos, P., A few useful things to know about machine learning, Commun. ACM, 55, 10, 78-87 (2012)
[28] Oliver, T. A.; Moser, R. D., Bayesian uncertainty quantification applied to RANS turbulence models, (Journal of Physics: Conference Series, 318 (2011), IOP Publishing), Article 042032 pp.
[29] Wang, J.-X.; Wu, J.-L.; Ling, J.; Iaccarino, G.; Xiao, H., (Physics-informed Machine Learning for Predictive Turbulence Modeling: Towards a Complete Framework, Tech. Rep.. Physics-informed Machine Learning for Predictive Turbulence Modeling: Towards a Complete Framework, Tech. Rep., Proceedings of the Summer Program (2016), Center of Turbulence Research, Stanford University)
[30] Simonsen, A.; Krogstad, P.-Å., Turbulent stress invariant analysis: Clarification of existing terminology, Phys. Fluids, 17, 8, Article 088103 pp. (2005) · Zbl 1187.76490
[31] Pope, S., A more general effective-viscosity hypothesis, J. Fluid Mech., 72, 02, 331-340 (1975) · Zbl 0315.76024
[32] Ling, J.; Jones, R.; Templeton, J., Machine learning strategies for systems with invariance properties, J. Comput. Phys., 318, 22-35 (2016) · Zbl 1349.76124
[33] Ling, J.; Jones, R.; Templeton, J., Machine learning strategies for systems with invariance properties, J. Comput. Phys., 318, 22-35 (2016) · Zbl 1349.76124
[34] Goldstein, H., The Euler angles, (Classical Mechanics (1980)), 143-148
[35] Kuipers, J. B., Quaternions and Rotation Sequences, Vol. 66 (1999), Princeton University Press: Princeton University Press Princeton · Zbl 1053.70001
[36] Breiman, L., Random forests, Mach. Learn., 45, 1, 5-32 (2001) · Zbl 1007.68152
[37] Data-driven turbulence modeling with physics-informed machine learning, 2018. https://github.com/xiaoh/turbulence-modeling-PIML; Data-driven turbulence modeling with physics-informed machine learning, 2018. https://github.com/xiaoh/turbulence-modeling-PIML
[38] Huser, A.; Biringen, S., Direct numerical simulation of turbulent flow in a square duct, J. Fluid Mech., 257, 65-95 (1993) · Zbl 0800.76189
[39] Weller, H. G.; Tabor, G.; Jasak, H.; Fureby, C., A tensorial approach to computational continuum mechanics using object-oriented techniques, Comput. Phys., 12, 6, 620-631 (1998)
[40] Patankar, S., Numerical Heat Transfer and Fluid Flow (1980), CRC Press · Zbl 0521.76003
[41] Gibson, M.; Launder, B., Ground effects on pressure fluctuations in the atmospheric boundary layer, J. Fluid Mech., 86, 03, 491-511 (1978) · Zbl 0377.76062
[42] Pinelli, A.; Uhlmann, M.; Sekimoto, A.; Kawahara, G., Reynolds number dependence of mean flow structure in square duct turbulence, J. Fluid Mech., 644, 107-122 (2010) · Zbl 1189.76265
[43] Laizet, S.; Li, N., Incompact3d: A powerful tool to tackle turbulence problems with up to \(O(1 0^5)\) computational cores, Internat. J. Numer. Methods Fluids, 67, 11, 1735-1757 (2011) · Zbl 1419.76481
[44] Laizet, S.; Lamballais, E., High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy, J. Comput. Phys., 228, 16, 5989-6015 (2009) · Zbl 1185.76823
[45] Breuer, M.; Peller, N.; Rapp, C.; Manhart, M., Flow over periodic hills: Numerical and experimental study in a wide range of Reynolds numbers, Comput. & Fluids, 38, 2, 433-457 (2009) · Zbl 1237.76026
[46] Launder, B.; Sharma, B., 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 (1974)
[47] Breuer, M.; Peller, N.; Rapp, C.; Manhart, M., Flow over periodic hills-numerical and experimental study in a wide range of Reynolds numbers, Comput. & Fluids, 38, 2, 433-457 (2009) · Zbl 1237.76026
[48] 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, Article 085103 pp. (2015)
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