HyperFLOW软件非结构网格亚跨声速湍流模拟的验证与确认
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摘要
计算流体力学(computational fluid dynamics, CFD)数值模拟在航空航天等领域发挥越来越重要的作用,然而CFD数值模拟结果的可信度仍然需要通过不断地验证与确认来提高.本文给出了从制造解精度测试、简单到复杂外形湍流模拟网格收敛性研究等三个方面开展CFD软件验证与确认的方法,并对自主研发的CFD软件平台HyperFLOW在非结构网格上模拟亚跨声速湍流问题的能力进行了验证与确认.首先通过基于Euler方程和标量扩散方程的制造解精度测试,分别验证了HyperFLOW在非结构网格上对Euler方程和黏性项的求解精度,结果表明其能够在任意非结构网格上达到设计的二阶精度.其次,通过NASA Turbulence Modeling Resource中的湍流平板、二维翼型近尾迹流动、二维Bump等几个典型的亚声速湍流算例的网格收敛性研究,量化考察了数值结果的观测精度阶和网格收敛性指数,并与国外知名CFD解算器CFL3D,FUN3D的计算结果进行了对比,验证了HyperFLOW对简单湍流问题的模拟能力,且具有良好的网格收敛性和计算精度(阶).最后,通过NASA Common Research Model标模定升力系数的网格收敛性研究和升阻极曲线预测,验证了软件在复杂外形亚跨声速湍流流动数值模拟中也具有良好的可信度.
Computational fluid dynamics(CFD) is playing a more and more important role in aerospace and relevant industries. While the credibility of CFD numerical simulation requires continuous verification and validation. This paper proposes for CFD solvers a complete verification and validation procedure including rigorous accuracy tests based on the method of manufactured solutions(MMS), simple and complex turbulent flow simulation and grid convergence study.The procedure is implemented on the in-house CFD solver HyperFLOW on verifying and validating its ability on subsonic and transonic turbulent flow simulation. We firstly verified that HyperFLOW can achieve designed second-order accuracy on arbitrary unstructured grids via numerical accuracy tests based on the MMS of the Euler equation and the diffusion equation. Then, simple subsonic turbulent flow cases, such as the turbulent flat plate, the 2D Airfoil Near-Wake and the 2D Bump cases from NASA Turbulence Modeling Resources are taken into consideration for grid convergence tests.Observed accuracy order and grid convergence index are calculated and compared with the results obtained by CFL3D and FUN3D, which verified and validated the accuracy and grid convergence performance of HyperFLOW in simple case simulation. Finally, the NASA Common Research Model inDrag Prediction Workshop VI is simulated with fixed lift coefficient, and grid convergence study is carried out on a series of refined grids.Drag polar is predicted by HyperFLOW and compared with other verified and validated results. It demonstrated that HyperFLOW has good accuracy and grid convergence performance in complex configuration high Reynolds subsonic and transonic turbulent flow simulation on unstructured/hybrid grids.
Computational fluid dynamics(CFD) is playing a more and more important role in aerospace and relevant industries. While the credibility of CFD numerical simulation requires continuous verification and validation. This paper proposes for CFD solvers a complete verification and validation procedure including rigorous accuracy tests based on the method of manufactured solutions(MMS), simple and complex turbulent flow simulation and grid convergence study.The procedure is implemented on the in-house CFD solver HyperFLOW on verifying and validating its ability on subsonic and transonic turbulent flow simulation. We firstly verified that HyperFLOW can achieve designed second-order accuracy on arbitrary unstructured grids via numerical accuracy tests based on the MMS of the Euler equation and the diffusion equation. Then, simple subsonic turbulent flow cases, such as the turbulent flat plate, the 2D Airfoil Near-Wake and the 2D Bump cases from NASA Turbulence Modeling Resources are taken into consideration for grid convergence tests.Observed accuracy order and grid convergence index are calculated and compared with the results obtained by CFL3D and FUN3D, which verified and validated the accuracy and grid convergence performance of HyperFLOW in simple case simulation. Finally, the NASA Common Research Model inDrag Prediction Workshop VI is simulated with fixed lift coefficient, and grid convergence study is carried out on a series of refined grids.Drag polar is predicted by HyperFLOW and compared with other verified and validated results. It demonstrated that HyperFLOW has good accuracy and grid convergence performance in complex configuration high Reynolds subsonic and transonic turbulent flow simulation on unstructured/hybrid grids.
引文
1 Park MA,Lee-Rausch EM,Rumsey CL.FUN3D and CFL3D computations for the first high lift prediction workshop.AIAA Paper2011-936,2011
2 Park MA,Laflin KR,Chaffin MS,et al.CFL3D,FUN3D,NSU3Dcontributions to the fifth drag prediction workshop.AIAA Paper2013-0050,2013
3 Rumsey CL,Thomas JL.Application of FUN3D and CFL3D to the 3rd workshop on CFD uncertainty analysis.NASA/TM-2008-215537,2008
4 Cambier L,Veuillot JP.Status of the elsA CFD software for flow simulation and multidisciplinary applications.AIAA Paper 2008-664,2008
5 Cambier L,Gazaix M.elsA:An efficient object-oriented solution to CFD complexity.AIAA Paper 02-13597,2002
6 Langer S,Schwoppe A,Kroll N.The DLR Flow Solver TAU-Status and Recent Algorithmic Developments.AIAA Paper 2014-0080,2014
7 ANSYS Fluent14.0 Help.ANSYS Inc,2011
8 ANSYS CFX14.0 Help.ANSYS Inc,2011
9王运涛,王光学,张玉伦.TRIP2.0软件的确认:DPW II复杂组合体的数值模拟.航空学报,2008,29(1):34-40(Wang Yuntao,Wang Guangxue,Zhang Yulun.Validation of TRIP2.0:Numerical simulation of DPW II complex configuration.Acta Aeronautica et Astronautica Sinica,2008,29(1):34-40(in Chinese))
10 Chen JT,Zhang YB,Zhou NC,et al.Numerical investigation of the high-lift configuration with MFlow solver.AIAA Journal,2015,52(4):1051-1061
11 He X,Zhao Z,Ma R,et al.Validation of HyperFLOW in subsonic and transonic flow.Acta Aerodynamica Sinica,2016,34(2):267-275
12 He X,He XY,He L,et al.HyperFLOW:A structured/unstructured hybrid integrated computational environment for multi-purpose fluid simulation.Procedia Engineering,2015,126:645-649
13邓小刚,宗文刚,张来平等.计算流体力学中的验证与确认.力学进展,2007,37(2):279-288(Deng Xiaogang,Zong Wengang,Zhang Laiping,et al.Verification and validation in computational fluid dynamics.Advances in Mechanics,2007,37(2):279-288(in Chinese))
14 Porter JL,Agarwal R,Azad RS,et al.Guide for the verification and validation of computational fluid dynamics Simulation.AIAAG-077-1998,1998
15 Roy CJ,Tinoco EN.Summary of data from the Sixth AIAA CFDDrag Prediction Workshop:Case 1 Code Verification.AIAA Paper2017-1206,2017
16 Tinoco EN,Brodersen OP,Keye S,et al.Summary of data from the Sixth AIAA CFD Drag Prediction Workshop:CRM Cases 2 to 5.AIAA Paper 2017-1208,2017
17 Rumsey CL,Slotnick JP.Overview and summary of the Second AIAA High-Lift Prediction Workshop.AIAA Paper 2014-0747,2014
18 Rumsey CL,Slotnick JP,Sclafani AJ.Overview and summary of the Third AIAA High-Lift Prediction Workshop.AIAA Paper 2018-1258,2018
19 Eca L,Hoekstra M,Roache P,et al.Code verification,solution verification and validation:An overview of the 3rd Lisbon Workshop.AIAA Paper 2009-3647,2009
20白文,李立,李周明等.CFD验证、确认和共享基准数据库.中国航空学报(英文版),2006,19(2):160-167(Bai Wen,Li Li,Li Zhouming,et al.CFD V&V and Open Benchmark Database.Chinese Journal of Aeronautics,2006,19(2):160-167(in Chinese))
21王年华,常兴华,赵钟等.基于HyperFLOW平台的客机标模CHN-T1气动性能预测及可信度研究.空气动力学学报,2019,37(2):-(Wang Nianhua,Chang Xinghua,Zhao Zhong,et al.Aerodynamic performance prediction and credibility study of the transport model CHN-T1 based on HyperFLOW solver.Acta Aerodynamica Sinica,2019,37(2):-(in Chinese))(in press)
22 Iannelli J.An exact non-linear Navier-Stokes compressible-flow solution for CFD code verification.International Journal for Numerical Methods in Fluids,2013,72:157-176
23 Ghia U,Bayyuk S,Roy C,et al.The AIAA code verification projecttest cases for CFD code verification.AIAA Paper 2010-125,2010
24 Maruli VK,Burg COE.Verification of 2D Navier-Stokes codes by the method of manufactured solutions.AIAA Paper 2002-3109,2002
25 Eca L.Klaij CM,Vaz G,et al.On code verification of RANSsolvers.Journal of Computational Physics,2016,310:418-439
26 Petri L,Sartori P,Rogenski J,et al.Verification and validation of a direct numerical simulation code.Computer Methods in Applied Mechanics and Engineering,2015,291:266-279
27 Veluri SP,Roy CJ,Hebert S,et al.Verification of the Loci-CHEMCFD code using the method of manufactured solutions.AIAA Paper2008-661,2008
28 Veluri SP,Roy CJ,Luke EA.Comprehensive code verification for an unstructured finite volume CFD code.AIAA Paper 2010-127,2010
29余云龙,林忠,王瑞利等.辐射流体力学Lagrange方程组一类人为解构造方法.应用数学和力学,2015,36(1):110-118(Yu Yunlong,Lin Zhong,Wang Ruili,et al.A method of manufacturing solutions for verification of Lagrangian radiation hydrodynamics codes.Applied Mathematics and Mechanics,2015,36(1):110-118(in Chinese))
30王瑞利,林忠,袁国兴.科学计算程序的验证和确认.北京理工大学学报,2010,30(3):353-360(Wang Ruili,Lin Zhong,Yuan Guoxing.Verification and validation in scientific computing code.Transactions of Beijing Institute of Technology,2010,30(3):353-360(in Chinese))
31王年华,张来平,赵钟等.基于制造解的非结构二阶有限体积离散格式的精度测试与验证.力学学报,2017,49(3):627-637(Wang Nianhua,Zhang Laiping,Zhao Zhong,et al.Accuracy verification of unstructured second-order finite volume discretization schemes based on the method of manufactured solutions.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):627-637(in Chinese))
32 Richardson LF.The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a Masonry Dam.Transactions of the Royal Society of London,Series A,1908,210:307-357
33 Roache PJ.Verification and Validation in Computational Science and Engineering.Hermosa,Albuquerque,NM,1998
34 Roy CJ.Grid convergence error analysis for mixed order numerical schemes.AIAA Journal,2003,41(4):595-604
35 Roache PJ.Perspective:A method for uniform reporting of grid refinement studies.Journal of Fluids Engineering,1994,116(3):405-413
36赵训友,林景松,童晓艳.基于Richardson外推法的CFD中离散不确定度估计.系统仿真学报,2014,26(10):2315-2320(Zhao Xunyou,Lin Jinsong,Tong Xiaoyan.Discretization uncertainty estimation in CFD based on Richardson extrapolation method.Journal of System Simulation,2014,26(10):2315-2320(in Chinese))
37陈坚强,张益荣.基于Richardson插值法的CFD验证和确认方法的研究.空气动力学学报,2012,30(2):176-183(Chen Jianqiang,Zhang Yirong.Verification and validation in CFD based on the Richardson extrapolation method.Acta Aerodynamica Sinica,2012,30(2):176-183(in Chinese))
38李钊,陈海昕,张宇飞.基于广义Richardson外插方法的阻力预测精度分析.航空学报,2015,36(7):2105-2114(Li Zhao,Chen Haixin,Zhang Yufei.Accuracy analysis of drag prediction based on generalized Richardson extrapolation.Acta Aerodynamica et Astronautica Sinica,2015,36(7):2105-2114(in Chinese))
39赵钟,赫新,张来平等.HyperFLOW软件数值模拟TrapWing高升力外形.空气动力学学报,2015,33(5):594-602(Zhao Zhong,He Xin,Zhang Laiping,et al.Numerical research of NASA high-lift trap wing model based on HyperFLOW.Acta Aerodynamica Sinica,2015,33(5):594-602(in Chinese))
40 NASA Turbulence Modeling Resources,http://turbmodels.larc.nasa.gov/
41 Childs ML,Pulliam TH,Jespersen DC.OVERFLOW,Turbulence Modeling Resource verification results.NAS Technical Report:NAS-2014-03,2014
42 Rumsey CL.Recent developments on the turbulence modeling resource website.AIAA Paper 2015-2927,2015
43 Celik IB,Ghia U,Roache PJ,et al.Procedure for estimation and reporting of uncertainty due to discretization in CFD applications.Journal of Fluids Engineering,2008,130:078001
44 Oberkampf WL,Roy CJ.Verification and Validation in Scientific Computing.Cambridge University Press,2010
45王年华,张来平,马戎等.非结构网格质量对梯度重构及无黏流动模拟精度的影响.计算力学学报,2017,34(5):555-563(Wang Nianhua,Zhang Laiping,Ma Rong,et al.Mesh quality effects on the accuracy of gradient reconstruction and inviscid flow simulation on isotropic unstructured grids.Chinese Journal of Computational Mechanics,2017,34(5):555-563(in Chinese))
46王年华,李明,张来平.非结构网格二阶有限体积法中黏性通量离散格式精度分析与改进.力学学报,2018,50(3):527-537(Wang Nianhua,Li Ming,Zhang Laiping.Accuracy analysis and improvement of viscous flux schemes in unstructured second-order finitevolume discretization.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):527-537(in Chinese))
47 Roache PJ.Code verification by method of manufactured solutions.Transactions of ASME,2002,124:4-10
48 Mani M,Babcock DA,Winkler CM,et al.Prediction of supersonic turbulent flow in a square duct.AIAA Paper 2013-0860,2013
49王运涛,孙岩,孟德虹等.包含支撑装置和机翼变形的CRM-WB构型气动特性数值模拟.航空学报,2017,38(10):121202(Wang Yuntao,Sun Yan,Meng Dehong,et al.Numerical simulation of aerodynamic characteristics of CRM WB configuration with support system and wing deformation.Acta Aeronautica et Astronautica Sinica,2017,38(10):121202(in Chinese))
50陈树生,刘丽媛,阎超等.CFD软件自动化验证确认云平台设计与实现.航空学报,2017,38(3):120209(Chen Shusheng,Liu Liyuan,Yan Chao,et al.Design and realization of automated testing cloud platform for CFD verification and validation.Acta Aeronautica et Astronautica Sinica,2017,38(3):120209(in Chinese))
51何磊,赫新,马戎等.大型CFD软件自动化测试平台的初步设计与实现.空气动力学学报,2016,34(4):418-425(He Lei,He Xin,Ma Rong,et al.Preliminary design and application of CFD software automatic testing platform.Acta Aerodynamica Sinica,2016,34(4):418-425(in Chinese))
52张涵信,查俊.关于CFD验证确认中的不确定度和真值估算.空气动力学学报,2010,28(1):39-45(Zhang Hanxin,Zha Jun.The uncertainty and truth-value assessment in the verification and validation of CFD.Acta Aerodynamica Sinica,2010,28(1):39-45(in Chinese))
53赵钟,张来平,何磊等.适用任意网格的大规模并行CFD计算框架PHengLEI.计算机学报,2018,41,网络出版(Zhao Zhong,Zhang Laiping,He Lei,et al.PHengLEI:A large scale parallel CFDframework for arbitrary grids.Chinese Journal of Computers,2018,41,Published online(in Chinese))
2 Park MA,Laflin KR,Chaffin MS,et al.CFL3D,FUN3D,NSU3Dcontributions to the fifth drag prediction workshop.AIAA Paper2013-0050,2013
3 Rumsey CL,Thomas JL.Application of FUN3D and CFL3D to the 3rd workshop on CFD uncertainty analysis.NASA/TM-2008-215537,2008
4 Cambier L,Veuillot JP.Status of the elsA CFD software for flow simulation and multidisciplinary applications.AIAA Paper 2008-664,2008
5 Cambier L,Gazaix M.elsA:An efficient object-oriented solution to CFD complexity.AIAA Paper 02-13597,2002
6 Langer S,Schwoppe A,Kroll N.The DLR Flow Solver TAU-Status and Recent Algorithmic Developments.AIAA Paper 2014-0080,2014
7 ANSYS Fluent14.0 Help.ANSYS Inc,2011
8 ANSYS CFX14.0 Help.ANSYS Inc,2011
9王运涛,王光学,张玉伦.TRIP2.0软件的确认:DPW II复杂组合体的数值模拟.航空学报,2008,29(1):34-40(Wang Yuntao,Wang Guangxue,Zhang Yulun.Validation of TRIP2.0:Numerical simulation of DPW II complex configuration.Acta Aeronautica et Astronautica Sinica,2008,29(1):34-40(in Chinese))
10 Chen JT,Zhang YB,Zhou NC,et al.Numerical investigation of the high-lift configuration with MFlow solver.AIAA Journal,2015,52(4):1051-1061
11 He X,Zhao Z,Ma R,et al.Validation of HyperFLOW in subsonic and transonic flow.Acta Aerodynamica Sinica,2016,34(2):267-275
12 He X,He XY,He L,et al.HyperFLOW:A structured/unstructured hybrid integrated computational environment for multi-purpose fluid simulation.Procedia Engineering,2015,126:645-649
13邓小刚,宗文刚,张来平等.计算流体力学中的验证与确认.力学进展,2007,37(2):279-288(Deng Xiaogang,Zong Wengang,Zhang Laiping,et al.Verification and validation in computational fluid dynamics.Advances in Mechanics,2007,37(2):279-288(in Chinese))
14 Porter JL,Agarwal R,Azad RS,et al.Guide for the verification and validation of computational fluid dynamics Simulation.AIAAG-077-1998,1998
15 Roy CJ,Tinoco EN.Summary of data from the Sixth AIAA CFDDrag Prediction Workshop:Case 1 Code Verification.AIAA Paper2017-1206,2017
16 Tinoco EN,Brodersen OP,Keye S,et al.Summary of data from the Sixth AIAA CFD Drag Prediction Workshop:CRM Cases 2 to 5.AIAA Paper 2017-1208,2017
17 Rumsey CL,Slotnick JP.Overview and summary of the Second AIAA High-Lift Prediction Workshop.AIAA Paper 2014-0747,2014
18 Rumsey CL,Slotnick JP,Sclafani AJ.Overview and summary of the Third AIAA High-Lift Prediction Workshop.AIAA Paper 2018-1258,2018
19 Eca L,Hoekstra M,Roache P,et al.Code verification,solution verification and validation:An overview of the 3rd Lisbon Workshop.AIAA Paper 2009-3647,2009
20白文,李立,李周明等.CFD验证、确认和共享基准数据库.中国航空学报(英文版),2006,19(2):160-167(Bai Wen,Li Li,Li Zhouming,et al.CFD V&V and Open Benchmark Database.Chinese Journal of Aeronautics,2006,19(2):160-167(in Chinese))
21王年华,常兴华,赵钟等.基于HyperFLOW平台的客机标模CHN-T1气动性能预测及可信度研究.空气动力学学报,2019,37(2):-(Wang Nianhua,Chang Xinghua,Zhao Zhong,et al.Aerodynamic performance prediction and credibility study of the transport model CHN-T1 based on HyperFLOW solver.Acta Aerodynamica Sinica,2019,37(2):-(in Chinese))(in press)
22 Iannelli J.An exact non-linear Navier-Stokes compressible-flow solution for CFD code verification.International Journal for Numerical Methods in Fluids,2013,72:157-176
23 Ghia U,Bayyuk S,Roy C,et al.The AIAA code verification projecttest cases for CFD code verification.AIAA Paper 2010-125,2010
24 Maruli VK,Burg COE.Verification of 2D Navier-Stokes codes by the method of manufactured solutions.AIAA Paper 2002-3109,2002
25 Eca L.Klaij CM,Vaz G,et al.On code verification of RANSsolvers.Journal of Computational Physics,2016,310:418-439
26 Petri L,Sartori P,Rogenski J,et al.Verification and validation of a direct numerical simulation code.Computer Methods in Applied Mechanics and Engineering,2015,291:266-279
27 Veluri SP,Roy CJ,Hebert S,et al.Verification of the Loci-CHEMCFD code using the method of manufactured solutions.AIAA Paper2008-661,2008
28 Veluri SP,Roy CJ,Luke EA.Comprehensive code verification for an unstructured finite volume CFD code.AIAA Paper 2010-127,2010
29余云龙,林忠,王瑞利等.辐射流体力学Lagrange方程组一类人为解构造方法.应用数学和力学,2015,36(1):110-118(Yu Yunlong,Lin Zhong,Wang Ruili,et al.A method of manufacturing solutions for verification of Lagrangian radiation hydrodynamics codes.Applied Mathematics and Mechanics,2015,36(1):110-118(in Chinese))
30王瑞利,林忠,袁国兴.科学计算程序的验证和确认.北京理工大学学报,2010,30(3):353-360(Wang Ruili,Lin Zhong,Yuan Guoxing.Verification and validation in scientific computing code.Transactions of Beijing Institute of Technology,2010,30(3):353-360(in Chinese))
31王年华,张来平,赵钟等.基于制造解的非结构二阶有限体积离散格式的精度测试与验证.力学学报,2017,49(3):627-637(Wang Nianhua,Zhang Laiping,Zhao Zhong,et al.Accuracy verification of unstructured second-order finite volume discretization schemes based on the method of manufactured solutions.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):627-637(in Chinese))
32 Richardson LF.The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a Masonry Dam.Transactions of the Royal Society of London,Series A,1908,210:307-357
33 Roache PJ.Verification and Validation in Computational Science and Engineering.Hermosa,Albuquerque,NM,1998
34 Roy CJ.Grid convergence error analysis for mixed order numerical schemes.AIAA Journal,2003,41(4):595-604
35 Roache PJ.Perspective:A method for uniform reporting of grid refinement studies.Journal of Fluids Engineering,1994,116(3):405-413
36赵训友,林景松,童晓艳.基于Richardson外推法的CFD中离散不确定度估计.系统仿真学报,2014,26(10):2315-2320(Zhao Xunyou,Lin Jinsong,Tong Xiaoyan.Discretization uncertainty estimation in CFD based on Richardson extrapolation method.Journal of System Simulation,2014,26(10):2315-2320(in Chinese))
37陈坚强,张益荣.基于Richardson插值法的CFD验证和确认方法的研究.空气动力学学报,2012,30(2):176-183(Chen Jianqiang,Zhang Yirong.Verification and validation in CFD based on the Richardson extrapolation method.Acta Aerodynamica Sinica,2012,30(2):176-183(in Chinese))
38李钊,陈海昕,张宇飞.基于广义Richardson外插方法的阻力预测精度分析.航空学报,2015,36(7):2105-2114(Li Zhao,Chen Haixin,Zhang Yufei.Accuracy analysis of drag prediction based on generalized Richardson extrapolation.Acta Aerodynamica et Astronautica Sinica,2015,36(7):2105-2114(in Chinese))
39赵钟,赫新,张来平等.HyperFLOW软件数值模拟TrapWing高升力外形.空气动力学学报,2015,33(5):594-602(Zhao Zhong,He Xin,Zhang Laiping,et al.Numerical research of NASA high-lift trap wing model based on HyperFLOW.Acta Aerodynamica Sinica,2015,33(5):594-602(in Chinese))
40 NASA Turbulence Modeling Resources,http://turbmodels.larc.nasa.gov/
41 Childs ML,Pulliam TH,Jespersen DC.OVERFLOW,Turbulence Modeling Resource verification results.NAS Technical Report:NAS-2014-03,2014
42 Rumsey CL.Recent developments on the turbulence modeling resource website.AIAA Paper 2015-2927,2015
43 Celik IB,Ghia U,Roache PJ,et al.Procedure for estimation and reporting of uncertainty due to discretization in CFD applications.Journal of Fluids Engineering,2008,130:078001
44 Oberkampf WL,Roy CJ.Verification and Validation in Scientific Computing.Cambridge University Press,2010
45王年华,张来平,马戎等.非结构网格质量对梯度重构及无黏流动模拟精度的影响.计算力学学报,2017,34(5):555-563(Wang Nianhua,Zhang Laiping,Ma Rong,et al.Mesh quality effects on the accuracy of gradient reconstruction and inviscid flow simulation on isotropic unstructured grids.Chinese Journal of Computational Mechanics,2017,34(5):555-563(in Chinese))
46王年华,李明,张来平.非结构网格二阶有限体积法中黏性通量离散格式精度分析与改进.力学学报,2018,50(3):527-537(Wang Nianhua,Li Ming,Zhang Laiping.Accuracy analysis and improvement of viscous flux schemes in unstructured second-order finitevolume discretization.Chinese Journal of Theoretical and Applied Mechanics,2018,50(3):527-537(in Chinese))
47 Roache PJ.Code verification by method of manufactured solutions.Transactions of ASME,2002,124:4-10
48 Mani M,Babcock DA,Winkler CM,et al.Prediction of supersonic turbulent flow in a square duct.AIAA Paper 2013-0860,2013
49王运涛,孙岩,孟德虹等.包含支撑装置和机翼变形的CRM-WB构型气动特性数值模拟.航空学报,2017,38(10):121202(Wang Yuntao,Sun Yan,Meng Dehong,et al.Numerical simulation of aerodynamic characteristics of CRM WB configuration with support system and wing deformation.Acta Aeronautica et Astronautica Sinica,2017,38(10):121202(in Chinese))
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