×

Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier-Stokes simulations: a data-driven, physics-informed Bayesian approach. (English) Zbl 1371.76082

Summary: Despite their well-known limitations, Reynolds-averaged Navier-Stokes (RANS) models are still the workhorse tools for turbulent flow simulations in today’s engineering analysis, design and optimization. While the predictive capability of RANS models depends on many factors, for many practical flows the turbulence models are by far the largest source of uncertainty. As RANS models are used in the design and safety evaluation of many mission-critical systems such as airplanes and nuclear power plants, quantifying their model-form uncertainties has significant implications in enabling risk-informed decision-making. In this work we develop a data-driven, physics-informed Bayesian framework for quantifying model-form uncertainties in RANS simulations. Uncertainties are introduced directly to the Reynolds stresses and are represented with compact parameterization accounting for empirical prior knowledge and physical constraints (e.g., realizability, smoothness, and symmetry). An iterative ensemble Kalman method is used to assimilate the prior knowledge and observation data in a Bayesian framework, and to propagate them to posterior distributions of velocities and other quantities of interest (QoIs). We use two representative cases, the flow over periodic hills and the flow in a square duct, to evaluate the performance of the proposed framework. Both cases are challenging for standard RANS turbulence models. Simulation results suggest that, even with very sparse observations, the obtained posterior mean velocities and other QoIs have significantly better agreement with the benchmark data compared to the baseline results. At most locations the posterior distribution adequately captures the true model error within the developed model form uncertainty bounds. The framework is a major improvement over existing black-box, physics-neutral methods for model-form uncertainty quantification, where prior knowledge and details of the models are not exploited. This approach has potential implications in many fields in which the governing equations are well understood but the model uncertainty comes from unresolved physical processes.

MSC:

76F55 Statistical turbulence modeling
62F15 Bayesian inference

Software:

EnKF; UQTk
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Wilcox, D. C., Turbulence Modeling for CFD (2006), DCW Industries
[2] Pope, S. B., Turbulent Flows (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0802.76033
[3] Roy, C. J.; Oberkampf, W. L., A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing, Comput. Methods Appl. Mech. Eng., 200, 25, 2131-2144 (2011) · Zbl 1230.76049
[5] Kennedy, M. C.; O’Hagan, A., Bayesian calibration of computer models, J. R. Stat. Soc., Ser. B, Stat. Methodol., 63, 3, 425-464 (2001) · Zbl 1007.62021
[6] Xiong, Y.; Chen, W.; Apley, D.; Ding, X., A non-stationary covariance-based Kriging method for metamodelling in engineering design, Int. J. Numer. Methods Eng., 71, 6, 733-756 (2007) · Zbl 1194.74553
[7] Huang, D.; Allen, T.; Notz, W.; Miller, R., Sequential Kriging optimization using multiple-fidelity evaluations, Struct. Multidiscip. Optim., 32, 5, 369-382 (2006)
[8] Conti, S.; Gosling, J. P.; Oakley, J. E.; O’Hagan, A., Gaussian process emulation of dynamic computer codes, Biometrika, 96, 3, 663-676 (2009) · Zbl 1437.62015
[9] Higdon, D.; Kennedy, M.; Cavendish, J. C.; Cafeo, J. A.; Ryne, R. D., Combining field data and computer simulations for calibration and prediction, SIAM J. Sci. Comput., 26, 2, 448-466 (2004) · Zbl 1072.62018
[10] Brynjarsdóttir, J.; O’Hagan, A., Learning about physical parameters: the importance of model discrepancy, Inverse Probl., 30, 114007 (2014) · Zbl 1307.60042
[11] Oliver, T.; Moser, R., Uncertainty quantification for RANS turbulence model predictions, (APS Division of Fluid Dynamics Meeting Abstracts, vol. 1 (2009))
[12] Oliver, T. A.; Moser, R. D., Bayesian uncertainty quantification applied to RANS turbulence models, J. Phys. Conf. Ser., 318, Article 042032 pp. (2011), IOP Publishing
[13] 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)
[14] Gorlé, C.; Iaccarino, G., A framework for epistemic uncertainty quantification of turbulent scalar flux models for Reynolds-averaged Navier-Stokes simulations, Phys. Fluids, 25, 5, Article 055105 pp. (2013)
[15] Gorlé, C.; Larsson, J.; Emory, M.; Iaccarino, G., The deviation from parallel shear flow as an indicator of linear eddy-viscosity model inaccuracy, Phys. Fluids, 26, 5, Article 051702 pp. (2014)
[16] Emory, M.; Larsson, J.; Iaccarino, G., Modeling of structural uncertainties in Reynolds-averaged Navier-Stokes closures, Phys. Fluids, 25, 11, 110822 (2013)
[17] Emory, M.; Pecnik, R.; Iaccarino, G., Modeling structural uncertainties in Reynolds-averaged computations of shock/boundary layer interactions, (AIAA Paper, vol. 479 (2011)), 1-16
[18] Emory, M. A., Estimating model-form uncertainty in Reynolds-averaged Navier-Stokes closures (2014), Stanford University, Ph.D. thesis
[19] Dow, E.; Wang, Q., Quantification of structural uncertainties in the \(k-ω\) turbulence model, (52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA Paper (2011), AIAA: AIAA Denver, Colorado), 2011-1762
[20] Wikle, C. K.; Milliff, R. F.; Nychka, D.; Berliner, L. M., Spatiotemporal hierarchical bayesian modeling tropical ocean surface winds, J. Am. Stat. Assoc., 96, 454, 382-397 (2001) · Zbl 1022.62117
[21] Berliner, L. M., Physical-statistical modeling in geophysics, J. Geophys. Res., Atmos., 108, D24 (2003)
[22] Launder, B.; Reece, G.; Rodi, W., Progress in development of a Reynolds-stress turbulence closure, J. Fluid Mech., 68, 537-566 (1975) · Zbl 0301.76030
[23] Tracey, B.; Duraisamy, K.; Alonso, J. J., A machine learning strategy to assist turbulence model development, (AIAA Paper, vol. 1287 (2015)), 2015
[24] 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) · Zbl 1349.76006
[25] Singh, A. P.; Duraisamy, K., Using field inversion to quantify functional errors in turbulence closures, Phys. Fluids, 28, Article 045110 pp. (2016)
[26] Wu, J.-L.; Wang, J.-X.; Xiao, H., Quantifying model form uncertainty in rans simulation of wing-body junction flow (2016), available at
[27] Tennekes, H.; Lumley, J. L., A First Course in Turbulence (1972), MIT Press · Zbl 0285.76018
[28] 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
[29] Xiao, H.; Wang, J.-X.; Ghanem, R. G., A random matrix approach for quantifying model-form uncertainties in turbulence modeling (2016), submitted, available at
[30] 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 (2016)
[31] Le Maître, O. P.; Knio, O. M., Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics (2010), Springer · Zbl 1193.76003
[32] Daubechies, I., Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41, 7, 909-996 (1988) · Zbl 0644.42026
[33] Buhmann, M. D., Radial Basis Functions: Theory and Implementations (2003), Cambridge University Press · Zbl 1038.41001
[34] Iglesias, M. A.; Law, K. J.; Stuart, A. M., Ensemble Kalman methods for inverse problems, Inverse Probl., 29, 4, Article 045001 pp. (2013) · Zbl 1311.65064
[35] Evensen, G., Data Assimilation: The Ensemble Kalman Filter (2009), Springer · Zbl 1157.86001
[36] Dennis, B.; Ponciano, J. M.; Lele, S. R.; Taper, M. L.; Staples, D. F., Estimating density dependence, process noise, and observation error, Ecol. Monogr., 76, 3, 323-341 (2006)
[37] Law, K.; Stuart, A., Evaluating data assimilation algorithms, Mon. Weather Rev., 140, 3757-3782 (2012)
[38] Debusschere, B.; Sargsyan, K.; Safta, C., UQTk user manual (June 2014), Sandia National Laboratories: Sandia National Laboratories Albuquerque, NM 87185 and Livermore, CA 94550, version 2.1 Edition
[39] Patankar, S. V.; Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transf., 15, 10, 1787-1806 (1972) · Zbl 0246.76080
[40] Rhie, C. M.; Chow, W. L., A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation, AIAA J., 21, 11, 1525-1532 (1983) · Zbl 0528.76044
[41] Launder, B. E.; Sharma, B. I., Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc, Lett. Heat Mass Transf., 1, 131-138 (1974)
[42] 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
[43] Houtekamer, P.; Mitchell, H. L.; Deng, X., Model error representation in an operational ensemble Kalman filter, Mon. Weather Rev., 137, 7, 2126-2143 (2009)
[44] Wang, J.-X.; Wu, J.-L.; Xiao, H., Incorporating prior knowledge for quantifying and reducing model-form uncertainty in RANS simulations (2015), submitted, available at
[45] Schillings, C.; Stuart, A. M., Analysis of the ensemble Kalman filter for inverse problems (2016), submitted, available at
[46] Neuman, S. P., Maximum likelihood Bayesian averaging of uncertain model predictions, Stoch. Environ. Res. Risk Assess., 17, 5, 291-305 (2003) · Zbl 1036.62113
[47] Huser, A.; Biringen, S., Direct numerical simulation of turbulent flow in a square duct, J. Fluid Mech., 257, 65-95 (1993) · Zbl 0800.76189
[48] AGARD, A selection of test cases for the validation of large-eddy simulations of turbulent flows (1998), Tech. Rep. 345, AGARD advisory report
[49] Demuren, A.; Rodi, W., Calculation of turbulence-driven secondary motion in non-circular ducts, J. Fluid Mech., 140, 189-222 (1984) · Zbl 0563.76056
[50] Perkins, H., The formation of streamwise vorticity in turbulent flow, J. Fluid Mech., 44, 04, 721-740 (1970) · Zbl 0205.57201
[51] Constantine, P. G.; Dow, E.; Wang, Q., Active subspace methods in theory and practice: applications to Kriging surfaces, SIAM J. Sci. Comput., 36, 4, A1500-A1524 (2014) · Zbl 1311.65008
[52] Cui, T.; Martin, J.; Marzouk, Y. M.; Solonen, A.; Spantini, A., Likelihood-informed dimension reduction for nonlinear inverse problems, Inverse Probl., 30, 11, 114015 (2014) · Zbl 1310.62030
[53] Haario, H.; Laine, M.; Mira, A.; Saksman, E., DRAM: efficient adaptive MCMC, Stat. Comput., 16, 4, 339-354 (2006)
[54] Wu, J.-L.; Wang, J.-X.; Xiao, H., A Bayesian calibration-prediction method for reducing model-form uncertainties with application in RANS simulations, Flow Turbul. Combust. (2016)
[55] Hua, C., An inverse transformation for quadrilateral isoparametric elements: analysis and application, Finite Elem. Anal. Des., 7, 2, 159-166 (1990) · Zbl 0718.73081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.