A review of uncertainty quantification methods for Computational Fluid Dynamics

CHEN Xin; WANG Gang; YE Zheng Yin; WU Xiaojun

doi:10.7638/kqdlxxb-2021.0012

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A review of uncertainty quantification methods for Computational Fluid Dynamics

更新时间：2026-05-28

- A review of uncertainty quantification methods for Computational Fluid Dynamics
- Vol. 39, Issue 4, Pages: 1-13(2021)
- 作者机构：
  
  1. 西北工业大学航空学院,西安,710072
  2. 中国空气动力研究与发展中心计算空气动力研究所,绵阳,621000
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2021.0012
  CLC： V211.3
- Published：2021
- 稿件说明：
移动端阅览
陈鑫, 王刚, 叶正寅, 等. CFD不确定度量化方法研究综述[J]. 空气动力学学报, 2021, 39(4): 1−13.

CHEN Xin, WANG Gang, YE Zheng Yin, et al. A review of uncertainty quantification methods for Computational Fluid Dynamics[J]. 2021, 39(4): 1-13.
陈鑫, 王刚, 叶正寅, 等. CFD不确定度量化方法研究综述[J]. 空气动力学学报, 2021, 39(4): 1−13. DOI： 10.7638/kqdlxxb-2021.0012

CHEN Xin, WANG Gang, YE Zheng Yin, et al. A review of uncertainty quantification methods for Computational Fluid Dynamics[J]. 2021, 39(4): 1-13. DOI： 10.7638/kqdlxxb-2021.0012.

摘要

随着计算流体力学（CFD）算法和软件的不断发展和完善，CFD数值模拟已经在涉及流体力学的各个领域发挥着日益重要的作用。不确定性因素在CFD计算过程中普遍存在，并且会对数值模拟结果造成影响。发展CFD不确定度量化方法，既能满足工程实践中对CFD可信度评估的需求，同时也能够支撑飞行器的精细化设计。本文旨在总结不确定度量化方法及其在CFD领域中的发展与应用。首先介绍CFD计算中的不确定性来源，以及不确定性的表现形式—随机不确定性和认知不确定性。然后按照不确定性的表现形式介绍对应的不确定度量化方法。最后介绍不确定度量化方法在CFD计算中的发展与应用，并且给出进一步开展不确定度量化工作的建议。

Abstract

With the rapid development of Computational Fluid Dynamics (CFD) algorithms and software

CFD has been playing an increasingly important role in various fields involving fluid mechanics. However

unavoidable uncertainties in CFD simulations have great influence on the numerical results. Therefore

developing CFD uncertainty quantification methods which characterizes all major uncertainty sources during the simulation and quantify their effects on the quantities of interest can not only satisfy the demands of CFD credibility evaluation in engineering practices

but also support the precise design of aircrafts. The development and application of uncertainty quantification methods in CFD are summarized in this paper. Firstly

the premise of CFD uncertainty quantification

i.e.

uncertainty sources are introduced. Generally

uncertainties can be categorized into stochastic and epistemic ones. Their corresponding uncertainty quantification methods have been proposed. Then the development and application of uncertainty quantification methods are surveyed in typical CFD processes including the geometry generation

meshing

turbulence modeling

and inflow conditions. In the end

sensitivity analysis methods are introduced

which are used to make out the influence of different uncertainty factors on numerical results. Summarizing the previous work on uncertainty

suggestions for the future development of uncertainty quantification methods in CFD are given.

关键词

Keywords

references

JAMESON A. Computational aerodynamics for aircraft design[J]. Science, 1989, 245(4916): 361-371. DOI: 10.1126/science.245.4916.361.

AGARWAL R. Computational fluid dynamics of whole-body aircraft[J]. Annual Review of Fluid Mechanics, 1999, 31(1): 125-169. DOI: 10.1146/annurev.fluid.31.1.125.

OBERKAMPF W L, SINDIR M N, CONLISK A T. Guide: Guide for the verification and validation of computational fluid dynamics simulations(AIAA G-077-1998(2002))[M]. American Institute of Aeronautics and Astronautics, 1998.

OBERKAMPF W L, DELAND S M, RUTHERFORD B M, et al. Estimation of total uncertainty in modeling and simulation[R]. Sandia National Laboratories, SAND 2000-0824, 2000.

SCHWER L E. Guide for verification and validation in computational solid mechanics: an overview of the PTC 60/VV 10[M ] . The American Society of Mechanical Engineers, 2006. https://cstools.asme.org/csconnect/FileUpload.cfm?View=yesID=24816

ASME VV 20-2009. Standard for verification and validation in computational fluid dynamics and heat transfer[S ] . The American Society of Mechanical Engineers, 2009. https://files.asme.org/Catalog/Codes/PrintBook/21356.pdf

ASME VV 10.1-2012. An illustration of the concepts of verification and validation in computational solid mechanics[S ] . The American Society of Mechanical Engineers, 2012. https://www.asme.org/getmedia/ae188d7f-e6ad-483f-bf6f-194d1049d17a/31917.pdf

HIRSCH C. NODESIM-CFD: Non-deterministic simulation for CFD based design methodologies[R ] . AST5-CT-2006-030959, 2006. https://trimis.ec.europa.eu/sites/default/files/project/documents/20121026_100225_36867_Aerodays-2011.pdf

SLOTNICK J, KHODADOUST A, ALONSO J, et al. CFD vision 2030 study: a path to revolutionary computational aerosciences[R ] . NASA Langley Research Center, Hampton, Virginia, NASA CR-2014-218178. https://core.ac.uk/download/pdf/42732819.pdf

ZHANG H X. On the uncertainty about CFD results[J]. Acta Aerodynamica Sinica, 2008, 26(1): 47-49, 90. (in Chinese)

WANG R L, JIANG S. Mathematical methods for uncertainty quantification in nonlinear multi-physics systems and their numerical simulations[J]. SCIENTIA SINICA Mathematica, 2015, 45(6): 723-738. (in Chinese)

LIANG X, WANG R L. Quantification of aleatory uncertainty in blast wave problem[J]. Chinese Journal of High Pressure Physics, 2016, 30(6): 531-536. (in Chinese)

LIANG X, WANG R L. Mixed uncertainty quantification of blast wave problem[J]. Chinese Journal of Computational Physics, 2017, 34(5): 574-582. (in Chinese)

WANG Y T, LIU G, CHEN Z B. Summary of the first aeronautical computational fluid dynamics credibility workshop[J]. Acta Aerodynamica Sinica, 2019, 37(2): 247-261, 246. (in Chinese)

CHEN J Q. Advances in the key technologies of Chinese national numerical windtunnel project[J ] . SCIENTIA SINICA Technologica, 2020(online). (in Chinese) doi: 10.1360/SST-2020-0334

LEE H B, GHIA U, BAYYUK S, et al. Development and use of engineering standards for computational fluid dynamics for complex aerospace systems[C ] //46th AIAA Fluid Dynamics Conference, Washington D C, Reston, Virginia: AIAA, 2016. doi: 10.2514/6.2016-3811

NASA-STD-7009. Standard for Models and Simulations[S ] . National Aeronautics and Space Administration, 2008. https://standards.nasa.gov/sites/default/files/nasa-std-7009.pdf

SCHAEFER J A, ROMERO V J, SCHAFER S R, et al. Approaches for quantifying uncertainties in computational modeling for aerospace applications[C ] //AIAA Scitech 2020 Forum, Orlando, FL. Reston, Virginia: AIAA, 2020. doi: 10.2514/6.2020-1520

CULLEN A, FREY H C. Probabilistic techniques in exposure assessment: A handbook for dealing with variability and uncertainty in models and inputs[M]. Plenum Press, New York, 1999.

SÁNDOR Z, ANDRÁS P. Alternative sampling methods for estimating multivariate normal probabilities[J]. Journal of Econometrics, 2004, 120(2): 207-234. DOI: 10.1016/S0304-4076(03)00212-4.

LE MAÎTRE O P, KNIO O M. Spectral methods for uncertainty quantification: with applications to computational fluid dynamics[M ] . Springer, 2010. doi: 10.1007/978-90-481-3520-2

MCKAY M D, BECKMAN R J, CONOVER W J. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[J]. Technometrics, 1979, 21(2): 239. DOI: 10.2307/1268522.

SANTNER T J, WILLIAMS B J, NOTZ W I. The design and analysis of computer experiments[M ] . New York, NY: Springer New York, 2003. doi: 10.1007/978-1-4757-3799-8

HELTON J C, DAVIS F J. Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems[J]. Reliability Engineering System Safety, 2003, 81(1): 23-69. DOI: 10.1016/S0951-8320(03)00058-9.

HELTON J C, JOHNSON J D, SALLABERRY C J P, et al. Survey of sampling-based methods for uncertainty and sensitivity analysis.[R ] . Sandia National Laboratories, SAND 2006-2901, 2006. https://digital.library.unt.edu/ark:/67531/metadc891681/m2/1/high_res_d/886897.pdf

GHANEM R G, SPANOS P D. Stochastic finite elements: A spectral approach[M ] . New York, NY: Springer New York, 1991. doi: 10.1007/978-1-4612-3094-6

DINESCU C, SMIRNOV S, HIRSCH C, et al. Assessment of intrusive and non-intrusive non-deterministic CFD methodologies based on polynomial chaos expansions[J]. International Journal of Engineering Systems Modelling and Simulation, 2010, 2(1/2): 87-98.. DOI: 10.1504/ijesms.2010.031874.

MYERS R H, MONTGOMERY D C. Response surface methodology[J]. IIE Transactions, 1996, 28(12): 1031-1032. DOI: 10.1080/15458830.1996.11770760.

HOSDER S, WALTERS R W, BALCH M. Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics[J]. AIAA Journal, 2010, 48(12): 2721-2730. DOI: 10.2514/1.39389.

ELDRED M. Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design[C ] //50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California. Reston, Virginia: AIAA, 2009. doi: 10.2514/6.2009-2274

XIU D B, LUCOR D, SU C H, et al. Stochastic modeling of flow-structure interactions using generalized polynomial chaos[J]. Journal of Fluids Engineering, 2002, 124(1): 51-59. DOI: 10.1115/1.1436089.

WITTEVEEN J A S, BIJL H. Modeling arbitrary uncertainties using gram-Schmidt polynomial chaos[C ] //44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. Reston, Virginia: AIAA, 2006. doi: 10.2514/6.2006-896

RESMINI A, PETER J, LUCOR D. Sparse grids-based stochastic approximations with applications to aerodynamics sensitivity analysis[J]. International Journal for Numerical Methods in Engineering, 2016, 106(1): 32-57. DOI: 10.1002/nme.5005.

BOMPARD M, PETER J, DÉSIDÉRI J S. Surrogate models based on function and derivative values for aerodynamic global optimization[C ] //V European Conference on Computational, 2010, Lisbonne, Portugal. INRIA-00537120. https://hal.inria.fr/inria-00537120/document

HANSEN E, WALSTER G W. Global optimization using interval analysis[M]. New York: Marcel Dekker, 1992.

TABER R. The fuzzy systems handbook: a practitioner's guide to building, using, and maintaining fuzzy systems (earl cox)[J]. SIAM Review, 1995, 37(2): 281-282. DOI: 10.1137/1037078.

TUCKER W T, FERSON S. Sensitivity in risk analyses with uncertain numbers[R ] . Sandia Report, SAND 2006-2801, 2006. https://digital.library.unt.edu/ark:/67531/metadc874250/m2/1/high_res_d/886899.pdf doi:10.2172/886899

FERSON S, TUCKER W T. Sensitivity analysis using probability bounding[J]. Reliability Engineering System Safety, 2006, 91(10-11): 1435-1442. DOI: 10.1016/j.ress.2005.11.052.

GOODMAN I R, NGUYEN H T. Probability updating using second order probabilities and conditional event algebra[J]. Information Sciences, 1999, 121(3-4): 295-347. DOI: 10.1016/S0020-0255(99)00089-4.

SWILER L P, PAEZ T L, MAYES R L. Epistemic uncer tainty quantification tutorial[C ] //IMAC XXVII conference and exposition on structural dynamics, Orlando, FL, 2009. https://cfwebprod.sandia.gov/cfdocs/CompResearch/docs/294_swi.pdf

ELDRED M S, SWILER L P, TANG G. Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation[J]. Reliability Engineering System Safety, 2011, 96(9): 1092-1113. DOI: 10.1016/j.ress.2010.11.010.

WILLIAMSON R C, DOWNS T. Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds[J]. International Journal of Approximate Reasoning, 1990, 4(2): 89-158. DOI: 10.1016/0888-613X(90)90022-T.

FERSON S, KREINOVICK V, GINZBURG L, et al. Constructing probability boxes and dempster-shafer structures[R ] . Sandia Report, SAND 2002-4015, 2003. https://www.researchgate.net/publication/2898381_Constructing_Probability_Boxes_and_Dempster-Shafer_Structures doi: 10.2172/809606

ROMERO V. Approximate probability boxes and other shortcuts in a broad-before-deep approach to balanced UQ[C ] //ASME 2015 VV Symposium, Las Vegas, NV. SAND 2015-3605C. https://www.osti.gov/servlets/purl/1252925

RICHARDSON L F. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam[J]. Philosophical Transactions of the Royal Society A, 1911, 210(1): 459-470. DOI: 10.1098/rsta.1911.0009 https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1911.0009.

CELIK I, KARATEKIN O. Numerical experiments on application of Richardson extrapolation with nonuniform grids[J]. Journal of Fluids Engineering, 1997, 119(3): 584-590. DOI: 10.1115/1.2819284.

CELIK I B, GHIA U,. ROACHE P J, et al.. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications[J]. Journal of Fluids Engineering, 2008, 130(7): 078001. DOI: 10.1115/1.2960953.

ZHAO X Y, LIN J S, TONG X Y. Discretization uncertainty estimation in CFD based on Richardson extrapolation method[J]. Journal of System Simulation, 2014, 26(10): 2315-2320. (in Chinese)

SCHAEFER J A, HOSDER S, MANI M, et al. The effect of grid topology and flow solver on turbulence model closure coefficient uncertainties for a transonic airfoil[C ] //46th AIAA Fluid Dynamics Conference, Washington D C. Reston, Virginia: AIAA, 2016. doi: 10.2514/6.2016-4400

LOEVEN A, BIJL H. Airfoil analysis with uncertain geometry using the probabilistic collocation method[C ] // 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL, 2008. AIAA 2008-2070. doi: 10.2514/6.2008-2070

PARUSSINI L, PEDIRODA V, POLONI C. Prediction of geometric uncertainty effects on Fluid Dynamics by Polynomial Chaos and Fictitious Domain method[J]. Computers Fluids, 2010, 39(1): 137-151. DOI: 10.1016/j.compfluid.2009.07.008.

LIU D S, LITVINENKO A, SCHILLINGS C, et al. Quantification of airfoil geometry-induced aerodynamic uncertainties---comparison of approaches[J]. SIAM/ASA Journal on Uncertainty Quantification, 2017, 5(1): 334-352. DOI: 10.1137/15m1050239.

LIU Z Y, WANG X D, KANG S. Stochastic performance evaluation of horizontal axis wind turbine blades using non-deterministic CFD simulations[J]. Energy, 2014, 73: 126-136. DOI: 10.1016/j.energy.2014.05.107.

TROJAK W, WATSON R, SCILLITOE A, et al. Effect of mesh quality on flux reconstruction in multi-dimensions[J]. Journal of Scientific Computing, 2020, 82(3): 1-36. DOI: 10.1007/s10915-020-01184-2.

Liu S Y, Wang Y B, Qin N, Zhao N. Quantification of Airfoil Aerodynamic Uncertainty due to Pressure-Sensitive Paint Thickness[J]. AIAA JOURNAL, 2020, 58(4): 1432-1440. DOI:10.2514/1.J058801.

XIU D B, KARNIADAKIS G E. The Wiener: askey polynomial chaos for stochastic differential equations[J]. SIAM Journal on Scientific Computing, 2002, 24(2): 619-644. DOI: 10.1137/s1064827501387826.

LOEVEN G J A, BIJL H. Probabilistic Collocation used in a Two-Step approach for efficient uncertainty quantification in computational fluid dynamics[J]. Computer Modeling in Engineering and Sciences, 2008, 36(3): 193-212. DOI: 10.3970/cmes.2008.036.193.

MARIOTTI A, SALVETTI M V, SHOEIBI OMRANI P, et al. Stochastic analysis of the impact of freestream conditions on the aerodynamics of a rectangular 5: 1 cylinder[J]. Computers Fluids, 2016, 136: 170-192. DOI: 10.1016/j.compfluid.2016.06.008.

AVDONIN A, POLIFKE W. Quantification of the impact of uncertainties in operating conditions on the flame transfer function with nonintrusive polynomial chaos expansion[J]. Journal of Engineering for Gas Turbines and Power, 2019, 141(1): 011020. DOI: 10.1115/1.4040745.

ZHU H Y, WANG G, LIU Y, et al. Numerical investigation of transonic buffet on supercritical airfoil considering uncertainties in wind tunnel testing[J]. International Journal of Modern Physics B, 2020, 34(14n16): 2040083. DOI: 10.1142/s0217979220400834.

LIU Z Y, WANG X D, KANG S. CFD simulations of uncertain tip clearance effect on compressor performance[J]. Journal of Engineering Thermophysics, 2013, 34(4): 628-631. (in Chinese)

WU X J, ZHANG W W, SONG S F, et al. Uncertainty quantification and global sensitivity analysis of transonic aerodynamics about airfoil[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(4): 587-595. (in Chinese)

WANG Y J, ZHANG S D. Uncertainty quantification of numerical simulation of flows around a cylinder using non-intrusive polynomial chaos[J]. Chinese Physics Letters, 2016, 33(9): 090501. DOI: 10.1088/0256-307x/33/9/090501.

DENG X B, CHEN Q, YUAN X X, et al. Study of aerodynamic uncertainty on the complex slender vehicle at high angle of attack[J]. SCIENTIA SINICA Technologica, 2016, 46(5): 493-499. (in Chinese)

XU L C, WANG G, WU J, et al. Uncertainty analysis of airfoil wind tunnel tests with automatic differentiation[J]. Acta Aeronautica et Astronautica Sinica, 2014, 35(8): 2102-2111. (in Chinese)

DURAISAMY K, IACCARINO G, XIAO H. Turbulence modeling in the age of data[J]. Annual Review of Fluid Mechanics, 2019, 51(1): 357-377. DOI: 10.1146/annurev-fluid-010518-040547.

XIAO H, CINNELLA P. Quantification of model uncertainty in RANS simulations: a review[J]. Progress in Aerospace Sciences, 2019, 108: 1-31. DOI: 10.1016/j.paerosci.2018.10.001.

LUMLEY J L, NEWMAN G R. The return to isotropy of homogeneous turbulence[J]. Journal of Fluid Mechanics, 1977, 82(1): 161-178. DOI: 10.1017/s0022112077000585.

BANERJEE S, KRAHL R, DURST F, et al. Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches[J]. Journal of Turbulence, 2007, 8: N32. DOI: 10.1080/14685240701506896.

EMORY M, PECNIK R, IACCARINO G. Modeling structural uncertainties in Reynolds-averaged computations of shock/boundary layer interactions[C ] //49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida. Reston, Virginia: AIAA, 2011. doi: 10.2514/6.2011-479

EMORY M, LARSSON J, IACCARINO G. Modeling of structural uncertainties in Reynolds-averaged Navier-Stokes closures[J]. Physics of Fluids, 2013, 25(11): 110822. DOI: 10.1063/1.4824659.

IACCARINO G, MISHRA A A, GHILI S, Eigenspace perturbations for uncertainty estimation of single-point turbulence closures[J]. Physical Review Fluids, 2017, 2(2): 024605.Doi: 10.1103/PhysRevFluids.2.024605.

MISHRA A A, MUKHOPADHAYA J, IACCARINO G, et al. Uncertainty estimation module for turbulence model predictions in SU 2 [J ] . AIAA Journal, 2018, 57(3): 1066-1077. DOI: 10.2514/1.J057187.

XIAO H, WANG J X, GHANEM R G. A random matrix approach for quantifying model-form uncertainties in turbulence modeling[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 941-965. DOI: 10.1016/j.cma.2016.10.025.

WANG J X, SUN R, XIAO H. Quantification of uncertainties in turbulence modeling: a comparison of physics-based and random matrix theoretic approaches[J]. International Journal of Heat and Fluid Flow, 2016, 62: 577-592. DOI: 10.1016/j.ijheatfluidflow.2016.07.005.

EDELING W N, IACCARINO G, CINNELLA P. Data-free and data-driven RANS predictions with quantified uncertainty[J]. Flow, Turbulence and Combustion, 2018, 100(3): 593-616. DOI: 10.1007/s10494-017-9870-6.

POPE S B. Turbulent flows[M]. Cambridge, UK: Cambridge Univ. Press, 2000.

EISFELD B. Reynolds stress anisotropy in self-preserving turbulent shear flows[R ] . DLR-Interner Bericht. DLR-IB-AS-BS-2017-106, 158 S. https://elib.dlr.de/113887/

DUNN M C, SHOTORBAN B, FRENDI A. Uncertainty quantification of turbulence model coefficients via Latin hypercube sampling method[J]. Journal of Fluids Engineering, 2011, 133(4): 041402. DOI: 10.1115/1.4003762.

PLATTEEUW P D A, LOEVEN G J A, BIJL H. Uncertainty quantification applied to the k-epsilon model of turbulence using the probabilistic collocation method[C ] //49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL. Reston, Virginia: AIAA, 2008. doi: 10.2514/6.2008-2150

SHAH H R, HOSDER S, WINTER T. A mixed uncertainty quantification approach with evidence theory and stochastic expansions[C ] //16th AIAA Non-Deterministic Ap proaches Conference, National Harbor, Maryland. Reston, Virginia: AIAA, 2014. doi: 10.2514/6.2014-0298

WEST T K, HOSDER S, JOHNSTON C O. Multistep uncertainty quantification approach applied to hypersonic reentry flows[J]. Journal of Spacecraft and Rockets, 2013, 51(1): 296-310. DOI: 10.2514/1.A32592.

QUAGLIARELLA D, SERANI A, DIEZ M, et al. Benchmarking uncertainty quantification methods using the NACA2412 airfoil with geometrical and operational uncertainties[C ] //AIAA Aviation 2019 Forum, Dallas, Texas. Reston, Virginia: AIAA, 2019. doi: 10.2514/6.2019-3555

DUQUE E P, LAWRENCE S. Spectre: a computational environment for managing total uncertainty quantification of CFD studies[C ] //AIAA Scitech 2019 Forum, San Diego, California. Reston, Virginia: AIAA, 2019. doi: 10.2514/6.2019-2221

SCHAEFER J A, CARY A W, DUQUE E P, et al. Application of a CFD uncertainty quantification framework for industrial-scale aerodynamic analysis[C ] //AIAA Scitech 2019 Forum, San Diego, California. Reston, Virginia: AIAA, 2019. doi: 10.2514/6.2019-1492

WIGNALL T J, HOULDEN H. Uncertainty quantification for launch vehicle aerodynamic lineloads[C ] //AIAA Scitech 2020 Forum, Orlando, FL. Reston, Virginia: AIAA, 2020. doi: 10.2514/6.2020-1521

SCHAEFER J A, CARY A W, MANI M, et al. Uncertainty quantification and sensitivity analysis of SA turbulence model coefficients in two and three dimensions[C ] //55th AIAA Aerospace Sciences Meeting, Grapevine, Texas. Reston, Virginia: AIAA, 2017. doi: 10.2514/6.2017-1710

XIAO S N, LV Z Z, WANG W. A review of global sensitivity analysis for uncertainty structure[J]. Scientia Sinica (Physica, Mechanica Astronomica), 2018, 48(1): 8-25. (in Chinese)

STORLIE C B, SWILER L P, HELTON J C, et al. Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models[J]. Reliability Engineering System Safety, 2009, 94(11): 1735-1763. DOI: 10.1016/j.ress.2009.05.007.

SALTELLI A, ANNONI P, AZZINI I, et al. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index[J]. Computer Physics Communications, 2010, 181(2): 259-270. DOI: 10.1016/j.cpc.2009.09.018.

SUDRET B. Global sensitivity analysis using polynomial chaos expansions[J]. Reliability Engineering System Safety, 2008, 93(7): 964-979. DOI: 10.1016/j.ress.2007.04.002.

DUVIGNEAU R, PELLETIER D. A sensitivity equation method for fast evaluation of nearby flows and uncertainty analysis for shape parameters[J]. International Journal of Computational Fluid Dynamics, 2006, 20(7): 497-512. DOI: 10.1080/10618560600910059.

FIORINI C, DESPRÉS B, PUSCAS M A. Sensitivity equation method for the Navier-Stokes equations applied to uncertainty propagation[J]. International Journal for Numerical Methods in Fluids, 2021, 93(1): 71-92. DOI: 10.1002/fld.4875.

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