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高阶矩湍流模型研究进展及挑战

王圣业 符翔 杨小亮 郑皓榜 邓小刚

王圣业, 符翔, 杨小亮, 郑皓榜, 邓小刚. 高阶矩湍流模型研究进展及挑战[J]. 力学进展, 2021, 51(1): 29-61. doi: 10.6052/1000-0992-20-029
引用本文: 王圣业, 符翔, 杨小亮, 郑皓榜, 邓小刚. 高阶矩湍流模型研究进展及挑战[J]. 力学进展, 2021, 51(1): 29-61. doi: 10.6052/1000-0992-20-029
WANG Shengye, FU Xiang, YANG Xiaoliang, ZHENG Haobang, DENG Xiaogang. Progresses and challenges of high-order-moment turbulence closure[J]. Advances in Mechanics, 2021, 51(1): 29-61. doi: 10.6052/1000-0992-20-029
Citation: WANG Shengye, FU Xiang, YANG Xiaoliang, ZHENG Haobang, DENG Xiaogang. Progresses and challenges of high-order-moment turbulence closure[J]. Advances in Mechanics, 2021, 51(1): 29-61. doi: 10.6052/1000-0992-20-029

高阶矩湍流模型研究进展及挑战

doi: 10.6052/1000-0992-20-029
基金项目: 

国家自然科学基金 (12002379)、湖南省自然科学基金 (2020JJ5648)、国防科技大学科研计 划 (ZK20-43) 和国家专项工程 (GJXM92579) 资助项目.

详细信息
    作者简介:

    *E-mail: xgdeng2000@vip.sina.com
    王圣业, 国防科技大学副教授, 主要从事计算流体力学方法研究与大型CAE软件平台开发工作. 2013年于北京航空航天大学获学士学位, 2018年于国防科技大学获博士学位, 师从邓小刚院士. 现为某国家重大工程副主任设计师、中国力学学会会员, 主持国家自然科学基金青年基金项目、湖南省自然科学基金青年基金项目、国防科技大学科研计划项目3项. 以第一通讯作者身份在国内外学术期刊上发表SCI论文9篇, EI论文3篇.

    通讯作者:

    邓小刚

  • 中图分类号: O357.5

Progresses and challenges of high-order-moment turbulence closure

More Information
    Corresponding author: DENG Xiaogang
  • 摘要: 高阶矩模型是湍流模式理论研究中的难点和前沿. 自周培源先生首次建立一般湍流的雷诺应力输运方程起, 为了更精确的预测复杂流动, 人们从未间断过对高阶矩模型的研究. 尤其进入新世纪以来, 随着计算机硬件水平的飞跃和高精度数值算法的突破, 湍流模拟方法正由RANS向LES转变. 而无论对于RANS框架、LES框架还是两者混合, 高阶矩模式都是其中先进封闭模式的代表. 基于此, 本文对高阶矩模型的发展情况进行了总结, 重点包括高阶矩模型中各项的建模方式、尺度提供方程的演化历程和数值求解技术的关键需求. 然后, 通过几类典型湍流问题展示了其相对于传统涡黏模型的优势, 并且给出了部分CFD软件对高阶矩模型的集成情况. 最后对高阶矩湍流模型未来面临的挑战和发展的方向进行了展望.

     

  • [1] 陈懋章. 2002. 黏性流体力学基础. 北京: 高等教育出版社

    (Chen M Z. 2002. Fundamentals of Viscous Fluid Dynamics. Beijing: Higher Education Press).
    [2] 董义道, 王东方, 王光学, 邓小刚. 2016. 雷诺应力模型的初步应用. 国防科技大学学报, 38(4):46-53

    (Dong Y D, Wang D F, Wang G X, Deng X G. 2016. Preliminary application of Reynolds stress model. Journal of National University of Defence Technology, 38(4):46-53).
    [3] 符松. 1994. 湍流模式--研究现状与发展趋势. 应用基础与工程科学学报, 2(1):1-15

    (Fu S. 1994. Turbulence models: Presnet status and future developement. Journal of Basic Science and Engineering, 2(1):1-15).
    [4] 傅德薰, 马延文, 李新亮, 王强. 2010. 可压缩湍流直接数值模拟. 北京: 科学出版社

    (Fu D X, Ma Y W, Li X L, Wang Q. 2010. Direct Numerical Simulation of Compressible Turbulence. Beijing: Science press).
    [5] 聂胜阳, 王垠, 刘志强, 金朋, 焦瑾. 2019. 基于S-A与SSG/LRR-$omega$两种湍流模型的CHN-T1标模计算与分析. 空气动力学学报, 37(2):310-319

    (Nie S Y, Wang Y, Liu Z Q, Jin P, Jiao J. 2019. Numerical inverstigation and discussion on CHN-T1 benchmark model using Spalart-Allmaras model and SSG/LRR-$omega $ model. Acta Aerodynamica Sinica, 37(2):310-319).
    [6] 是勋刚. 1992. 湍流. 天津: 天津工业大学出版社

    (Shi X G. 1992. Turbulent Flow. Tianjin: Tianjin Polytechnic University Press).
    [7] 王圣业. 2018. 高精度WCNS 格式在亚/跨声速分离流动中的应用研究. [博士论文]. 长沙: 国防科技大学

    (Wang S Y. 2018. Application research of high-order weighted compact nonlinear schemes in subsonic/transonic separated flows. [PhD Thesis]. Changsha: National University of Defense Technology).
    [8] 王圣业, 王光学, 董义道, 邓小刚. 2017. 基于雷诺应力模型的高精度分离涡模拟方法. 物理学报, 66:184701

    (Wang S Y, Wang G X, Dong Y D, Deng X G. 2017. High-order detached-eddy simulation method based on a Reynolds-stress background model. Acta Physica Sinica, 66:184701).
    [9] 王运涛, 刘刚, 陈作斌. 2019. 第一届航空CFD可信度研讨会总结. 空气动力学学报. 37(2):247-261

    (Wang Y T, Liu G, Chen Z B. 2019. Summary of the first aeronautical computational fluid dynamics credibility workshop. Acta Aerodynamica Sinica, 37(2):247-261).
    [10] 张伟伟, 朱林阳, 刘溢浪, 寇家庆. 机器学习在湍流模型构建中的应用进展. 空气动力学报, 37(3):444-454

    (Zhang W W, Zhu L Y, Liu Y L, Kou J Q. Progresses in the application of mechine learning in turbulence modeling. Acta Aerodynamica Sinica, 37(3):444-454).
    [11] 郑晓静, 王国华. 2020. 高雷诺数壁湍流的研究进展及挑战. 力学进展, 42:522-537

    (Zheng X J, Wang G H. 2020. Progresses and challenges of high Reynolds number wall-bounded turbulence. Advances in Mechanics, 42:522-537).
    [12] 周培源. 1940. 关于Reynolds求似应力方法的推广和湍流的性质. 中国物理学报, 4:1.

    (Chou P Y. 1940. Chin. Journ. of Phys, 4:1).
    [13] 周铸, 黄江涛, 黄勇, 等. 2017. CFD 技术在航空工程领域的应用、挑战与发展. 航空学报, 38(3):020891

    (Zhou Z, Huang J T, Huang Y, et al. 2017. CFD technology in aeronautic engineering field: Applications, challenges and development. Acta Aeronautica et Astronautica Sinica, 38(3):020891).
    [14] Al-Sharif S F. 2011. Reynolds stress transport modelling. Computational Simulations and Applications, InTech, 3-26.
    [15] Bassi F, Crivellini A, Rebay S, Savini M. 2011. Disontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and $k$-$omega $ turbulence model equations. Comput. Fluids, 34:507-540.
    [16] Boussinesq J. 1877. Theorie de l' Ecoulement Tourbillant. Mem. Presents par Divers Savants Acad. Sci. Inst. Fr, 23:46-50.
    [17] Brandt A. 2005. Multiscale solvers and systematic upscaling in computational physics. Computer Physics Communications, 169(1-3):438-441.
    [18] Brun G, Herard J M, Jeandel D, Uhlmann M. 2000. An approximate Roe-type Riemann solver for a class of realizable second order closure. Int. J. Comput. Fluid Dyn., 13:223-249.
    [19] Cecora R D, Radespiel R, Eisfeld B, et al. 2015. Differential Reynolds-stress modeling for aeronautics. AIAA Journal, 53(3):739-755.
    [20] Chaouat B. 2006. Reynolds stress transport modeling for high-lift airfoil flows. AIAA J. 44(10):2390-2403.
    [21] Chaouat B. 2011. An efficient numerical method for RANS/LES turbulent simulations using subfilter scale stress transport equations. Int. J. Numer. Methods Fluids, 67:1207-1233.
    [22] Chaouat B. 2017. The State of the art of hybrid RANS/LES modeling for the simulation of turbulent flows. Flow Turbulence Combust, 99:279-327
    [23] Chassaing J C, Gerolymos G A, Vallet I. 2003. Efficient and robust Reynolds-stress model computation of three-dimensional compressible flows. AIAA J., 41(5):763-773.
    [24] Cheng Y, Canuto V M, Howard A M. 2005. Nonlocal convective PBL model based on new third- and fourth-order moments. Journal of Atmospheric Science, 62:2189-2204.
    [25] Chien K Y. 1982. Predictions of channel and boundary-layer flows with a low-Reynolds-Number turbulence model. AIAA J., 20(1):33-38.
    [26] Chou P Y. 1945. On the velocity correlations and the solution of the equations of turbulent fluctuation. Quart. Appl. Math., 3:38.
    [27] Chow J S, Zilliac G G, Bradshaw P. 1993. Measurements in the near-field of a turbulent wingtip vortex//31st Aerospace Sciences Meeting, AIAA Paper 1993-0551.
    [28] Chow J S, Zilliac G G, Bradshaw P. 1997. Mean and turbulence measurements in the near field of a wingtip vortex. AIAA Journal, 35(10):1561-1567.
    [29] Chu J, Luckring J. 1996. Experimental surface pressure data obtained on $65^circ$ delta wing across Reynolds number and Mach number ranges. NASA TM 4645.
    [30] Craft T J. 1998. Developments in a low-Reynolds-number second-moment closure and its application to separating and reattaching flows. Int. J. Heat Fluid Flow, 19(5):541-548.
    [31] Craft T J, Launder B E. 1992. New wall-reflection model applied to the turbulent impinging jet. AlAA Journal, 30(12):2970-2972.
    [32] Craft T J, Launder B E. 1996. A Reynolds stress closure designed for complex geometries. Int. J. Heat Fluid Flow, 17(3):245-254.
    [33] Crow S C. 1968. Viscoelastic properties of fine-grained incompressible turbulence. Journal of Fluid Mechanics, 33(1):1-20.
    [34] Daly B J. 1970. Transport equations in turbulence. Phys. Fluids, 13(11):2634-2649.
    [35] Deardorff J. 1973. The use of subgrid transport equations in a three-dimensional model of atmospheric turbulence. J. Fluids Eng., ASME, 95:429-438.
    [36] Deardorff J. 1974. Three-dimensional numerical study of the height and mean structure of heated planetary boundary layer. Bound.-Layer Meteorol, 7:81-106.
    [37] Dekeyser I, Launder B E. 1985. A comparison of triple-moment temperature-velocity correlations in the asymmetric heated jet with alternative closure models//Turbulent Shear Flows 4, Springer Berlin Heidelberg, 102-117.
    [38] Deng G B, Visonneau M. 1997. Near-wall modelization for dissipation in second-moment closures//11th Symposium on Turbulent Shear Flows, 2: P2-101-P2-106.
    [39] Deng X G, Mao M L, Tu G H, Liu H Y, Zhang H X. 2011. Geometric conservation law and application to high-order finite difference schemes with stationary grids. Journal of Computational Physics, 230:1100-1115.
    [40] Deng X G, Min Y B, Mao M L, Liu H Y, Tu G H, Zhang H X. 2013. Further study on geometric conservation law and application to high-order finite difference schemes with stationary grids. Journal of Computational Physics, 239:90-111.
    [41] Donaldson C duP, Rosenbaum H. 1968. Calculation of the turbulent shear flows through closure of the reynolds equations by invariant modeling. ARAP Report 127, Aeronautical Research Associates of Princeton, Princeton, NJ.
    [42] Eisfeld B, Brodersen O. 2005. Advanced turbulence modelling and stress analysis for the DLR-F6 configuration//23rd AIAA Applied Aerodynamics Conference, AIAA Paper 2005-4727.
    [43] Eisfeld B, Rumsey C L, Togiti V. 2016. Verification and validation of a second-moment-closure model. AIAA Journal, 54(5):1524-1541.
    [44] Eisfeld B, Rumsey C L. 2020. Length-scale correction for Reynolds-stress modeling. AIAA Journal, 58(4):1518-1528.
    [45] Frohlich J, von Terzi D. 2008. Hybrid LES/RANS methods for the simulation of turbulent flows. Progress in Aerospace Sciences, 44:349-377.
    [46] Fu S. 1998. Modelling of pressure-strain correlations for Taylor-Proudman turbulence. Science in China ( Series A), 41(6):638-646.
    [47] Fu S, Launder B E, Leschziner M A. 1987. Modeling strongly swirling recirculating jet flow with Reynolds-stress transport closures//Sixth Symposium on Turbulent Shear Flows, Toulouse, France.
    [48] Fu S, Launder B E, Tselepidakis D P. 1987. Accommodating the effects of high strain rates in modelling the pressure-strain correlation. Technical Report TFD/87/5.
    [49] Georgiadis N, Rizzetta D P, Fureby C. 2010. Large-eddy simulation: Current capabilities, recommended practices, and future research. AIAA Journal, 48(8):1772-1784.
    [50] Gerolymos G A, Vallet I. 2007. Low-diffusion approximate Riemann Solvers for Reynolds-stress transport//18th AIAA Computational Fluid Dynamics Conference, Miami, FL, AIAA paper 2007-4467.
    [51] Gibson M M, Launder B E. 1978. Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics, 86(3):491-511.
    [52] Gilbert N, Kleiser L. 1991. Turbulence model testing with the aid of direct numerical simulation results//8th Symp. on Turbulent Shear Flows, TU Munchen, p. 29. 1.
    [53] Gordeyev S, Post M, McLaughlin T, et al. 2007. Aero-optical environment around a conformal-window turret. AIAA Journal, 45(7):1514-1524.
    [54] Greschner B, Thiele F, Jacob M, et al. 2008. Prediction of sound generated by a rod--airfoil configuration using EASM DES and the generalised Lighthill/FW-H analogy. Computers & Fluids, 37:402-413.
    [55] Grossman S A, Narayan R. 1993. A Theory of nonlocal mixing-length convection. 2: Generalized smoothed particle hydrodynamics simulations. Astrophysical Journal Supplement Series, 89:361-394.
    [56] Gryanik V M, Hartmann J, Raasch S, Schroter M. 2005. A renement of the Millionshchikov quasi-normality hypothesis for convective boundary layer turbulence. Journal of Atmospheric Sciences, 62:2632-2638.
    [57] Hanjalic K, Jakirlic S. 1993. A model of stress dissipation in second-moment closures. Appl. Sci. Res, 51:513-518.
    [58] Hanjalié K, Launder B E. 1976. Contribution towards a Reynolds stress closure for low-Reynolds-number turbulence. Journal of Fluid Mechanics, 74(4):593-610.
    [59] Hanjali? K, Launder B. 2011. Modelling Turbulence in Engineering and the Environment, Second-Moment Routes to Closure. Cambridge: Cambridge University Press.
    [60] Hanjali? K, Jakirli? S, Had?i? I. 1997. Expanding the limits of "equilibrium" second-moment turbulence closures. Fluid Dyn. Res., 20(1-6):25-41.
    [61] Hartmann R, Held J, Leicht T. 2011. Adjoint-based error estimation and adaptive mesh refinement for the RANS and $k$-$omega $ turbulence model equations. J. Comput. Phys, 230(11):4268-4284.
    [62] Jakirli? S, Hanjali? K. 2002. A new approach to modelling near-wall turbulence energy and stress dissipation. J. Fluid Mech., 459:139-166.
    [63] Jeyapaul E, Coleman G N, Rumsey C L. 2014. Assessment of higher-order rans closures in a decelerated planar wall-bounded turbulent flow. Int. J. Heat and Fluid Flow, 10(4):282-300.
    [64] Jovanovi'c J, Durst F, Johansson T G. 1993. Statistical analysis of the dynamic equations for higher-order moments in turbulent wall bounded flows. Physics of Fluids A: Fluid Dynamics, 5:2886-2900.
    [65] Kalitzin G, Gould A, Benton J. 1996. Application of two-equation turbulence models in aircraft design//34th Aerospace Sciences Meeting and Exhibit, AIAA Paper 1996-0327.
    [66] Kawamura H, Sasaki J, Kobayashi K. 1995. Budget and modelling of triple-moment velocity correlations in a turbulent channel flow based on DNS//10th Symposium on Turbulent Shear Flows, August 14-16, 1995, pp. 13-18.
    [67] Kebede W, Launder B E, Younis B A. 1985. Large-amplitude periodic pipe flow: A Second-Moment Closure study//5th Symp. on Turbulent Shear Flows, Cornell University, Ithaca, New York, p. 16. 23. 1.
    [68] Kok J C, Spekreijse S P. 2000. Efficient and accurate implementation of the $k$-$omega $ turbulence model in the NLR multi-block Navier-Stokes system. NAL NLR-TP-2000-144.
    [69] Kolmogorov A N. 1941. Local Structure of turbulence in incompressible viscous fluid for very large reynolds number. Dokl. Akad. Nauk SSSR, 30:299-303.
    [70] Kurbatskii A F, Poroseva S V. 1997. A model for calculating the three components of the excess for the turbulent field of flow velocity in a round pipe rotating about its longitudinal axis. High Temperature, 35(3):432-440.
    [71] Langtry R B, Menter F R. Correlation-based transition modeling for unstructured parallelized computational fluid dynamics codes. AIAA Journal, 47(12):2894-2906.
    [72] Launder B E. 1996. An introduction to single-point closure methodology. Simulation and Modeling of Turbulent Flows, Oxford University Press, 243-310.
    [73] Launder B E, Reece G L, Rodi W. 1975. Progress in the development of a Reynolds-stress turbulence closure. Journal of Fluid Mechanics, 68(3):537-566.
    [74] Launder B E, Tselepidakis D P. 1993. Progress and paradoxes in modelling near-wall turbulence. Turbulent Shear Flows, 8:81-96.
    [75] Lav C, Sandberg R D, Philip J. 2019. A framework to develop data-driven turbulence models for flows with organised unsteadiness. Journal of Computational Physics, 383:148-165.
    [76] Lee-Rausch E M, Rumsey C L, Eisfeld B. 2016. Application of a full reynolds stress model to high lift flows. AIAA 2016-3944.
    [77] Levy D W, Laflin K R, Tinoco E N, et al. 2013. Summary of data from the fifth AIAA CFD Drag Prediction Workshop. AIAA Paper 2013-0046.
    [78] Liu C B, Nithiarasu P T P T. 2010. Wall distance calculation using the Eikonal/Hamilton-Jacobi equations on unstructured meshes. Eng. Comput., 27:645-657.
    [79] Lumley J L. 1978. Computational modeling of turbulent flows. Adv. Appl. Mech. 18(4b):123-176.
    [80] Lumley J L. 1983. Turbulence modeling. J. Appl. Mech., 50:1097-1103.
    [81] Lumley J L, Khajeh-Nouri B. 1974. Computational modeling of turbulent transport. Advances in Geophysics, 18A:169-192.
    [82] Malik M R, Bushnell D, eds. 2012. Role of Computational fluid dynamics and wind tunnels in aeronautics R&D. NASA TP 2012-217602.
    [83] Mellor G L, Herring H J. 1973. A survey of mean turbu1ent fie1d closure mode1s. AlAA Journal, 11(5):590-599.
    [84] Menter F R. 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J., 32(8):1598-1605.
    [85] Menter F, Kuntz M, Bender R. 2003. A scale-adaptive simulation model for turbulent flow predictions. AIAA Paper 2003-0767.
    [86] Menter F R, Langtry R B, Likki S R, et al. 2006. A correlation based transition model using local variables part 1: Model formulation. Journal of Turbomachinery, 128(3):413-422.
    [87] Menter F, Kuntz M, Langtry R. 2003. Ten years of industrial experience with the SST turbulence model. Begell House, 2003: 625-632.
    [88] Millionshtchikov M D. 1941. On the theory of homogeneous isotropic turbulence. C. R. Acad. Sci. SSSR, 32:615-619.
    [89] Mor-Yossef Y. 2014. Unconditionally stable time marching scheme for Reynolds stress models. Journal of Computational Physics, 276:635-664.
    [90] Mor-Yossef Y. 2016. Robust turbulent flow simulations using a Reynolds-stress-transport model on unstructured grids. Computers and Fluids, 129:111-133.
    [91] Nasr N B, Gerolymos G A, Vallet I. 2014. Low-diffusion approximate Riemann solvers for Reynolds-stress transport. J. Comput. Phys., 268(1):186-235.
    [92] Nie S Y, Krimmelbein N, Krumbein A, Grabe C. 2018. Coupling of a Reynolds stress model with the $gamma $-$Re_{ heta t}$ transition model. AIAA Journal, 56(1):146-157.
    [93] Pope S B. 2000. Turbulent Flows. Cambridge: Cambridge University Press.
    [94] Probst A, Radespiel R, Knopp T. 2011. Detached-eddy simulation of aerodynamic flows using a Reynolds-stress background model and algebraic RANS-LES sensors//20th AIAA Computational Fluid Dynamics Conference, 27-30 June 2011, Honolulu, Hawaii, AIAA 2011-3206.
    [95] Reynolds W C. 1970. Computation of turbulent flows-state of the art. Report No. MD-27, Dept. Mech. Eng, Stanford University,CA.
    [96] Rotta J C. 1951. Statistische theorie nichthomogener turbulenz. Zeitschrift fur Physik, 129:547-572.
    [97] Rubinstein R, Barton J M. 1990. Nonlinear Reynolds stress models and the renormalization group. Phys.Fluids A, 8:1472-1476
    [98] Rumsey C L. 2015. Application of Reynolds stress models to separated aerodynamic flows//Differential Reynolds Stress Modeling for Separating Flows in Industrial Aerodynamics, Eisfeld B. eds. Springer Tracts in Mechanical Engineering, Springer, New York, 19-37.
    [99] Rumsey C L. 2020. Turbulence Modeling Resource. NASA Langley Research Center, Hampton, VA, http://turbmodels.larc.nasa.gov.
    [100] Rumsey C L, Neuhart D H, Kegerise M A. 2016. The NASA juncture flow experiment: Goals, progress, preliminary testing//54th AIAA Aerospace Sciences Meeting, AIAA Paper 2016-1557.
    [101] Sanchez-Rocha M, Menon S. 2009. The compressible hybrid RANS/LES formulation using an additive operator. Journal of Computational Physics, 228(6):2037-2062.
    [102] Schoenawa S, Hartmann R. 2014. Discontinuous Galerkin discretization of the Reynolds-averaged Navier-Stokes equations with the shear-stress transport model. Journal of Computational Physics, 262:194-216.
    [103] Schumann U. 1977. Realizability of Reynolds-stress turbulence models. Phys. Fluids, 20:721-725.
    [104] Shih T H, Mansour N, Chen J Y. 1987. Reyno1ds stress models of homogeneous turbulence. studying turbulence using numerical simulation databases, NASA Ames/Stanford CTR-S87, pp 9.
    [105] Shima N. 1998. Low-Reynolds-number second-moment closure without wall-reflection redistribution terms. Int. J. Heat Fluid Flow, 19(5), 549-555.
    [106] Shur M L, Strelets M K, Travin A K, Spalart P R. 2000. Turbulence modeling in rotating and curved channels: assessing the spalart-shur correction. AIAA Journal, 38(5):784-792.
    [107] 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. NASA/CR-2014-218178.
    [108] Smagorinsky J. 1963. General circulation experiments with the primitive equations. I. the basic experiment. Mon. Weather Rev., 91:99-164.
    [109] Smits A J, Young S. Bradshaw P. 1979. The Effect of short regions of high surface curvature on turbulent boundary layers. J. Fluid Mech., 94(2):209-242.
    [110] Spalart P, Allmaras S. 1994. A one-equation turbulence model for aerodynamic flows. Recherche Aerospatiale, 1:5-21.
    [111] Spalart P, Jou W H, Strelets M, et al. 1997. Comments on the feas1ibility of LES for wings, on a hybrid RANS/LES approach//Proceedings of first AFOSR international conference on DNS/LES.
    [112] Spalart P R. 2000. Strategies for turbulence modelling and simulation. International Journal of Heat and Fluid Flow, 21:252-263.
    [113] Spalart P, Rumsey C. 2007. Effective inflow conditions for the turbulence models in aerodynamic calculations. AIAA Journal, 45(10):2544-2553.
    [114] Speziale C, Abid R, Durbin P. 1994. On the realizability of Reynolds stress turbulence closures. Journal of Scientific Computing, 9:369-403.
    [115] Speziale C G, Abid R, Anderson E C. 1992. Critical evaluation of two-equation models for near-wall turbulence. AIAA J., 30(2), 324-331.
    [116] Speziale C G, Sarkar S, Gatski T B. 1991. Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach. J. Fluid Mech., 227:245-272.
    [117] Stoellinger M, Heinz S, Saha P. 2015. Reynolds stress closure in hybrid RANS-LES methods//S. Girimaji et al. (eds.), Progress in Hybrid RANS-LES Modelling, 319-328.
    [118] Thompson K B, Hassan H A. 2015. Simulation of a variety of wings using a Reynolds stress model. Journal of Aircraft, 52(5):1668-1680.
    [119] Togiti V, Eisfeld B, Brodersen O. 2014. Turbulence model study for the flow around the NASA Common Research Model. Journal of Aircraft, 51(4):1331-1343.
    [120] Togiti V K, Eisfeld B. 2015. Assessment of $g$-equation formulation for a second-moment reynolds stress turbulence model//22nd AIAA Computational Fluid Dynamics Conference, AIAA 2015-2925.
    [121] Tracey B D, Duraisamy K, Alonso J J. 2015. A machine learning Strategy to assist turbulence model development//53rd AIAA Aerospace Science Meeting, AIAA Paper 2015-1287.
    [122] Tucker H J, Reynolds A J. 1968. The Distortion of turbulence by irrotational plane strain. Journal of Fluid Mechanics, 32(4):657-673.
    [123] Vassberg J C, DeHaan M A, Rivers S M, Wahls R A. 2008. Development of a common research model for applied CFD validation studies. AIAA Paper 2008-6919source.
    [124] Visbal R M, Gaitonde D V. 2002. On the Use of higher-order finite-difference schemes on curvilinear and deforming meshes. Journal of Computational Physics, 181:155-185.
    [125] Wang S Y, Deng X G, Wang G X, Yang X L. 2020. Blending the eddy-viscosity and Reynolds-stress models using uniform high-order discretization. AIAA Journal, Article in Advance.
    [126] Wang S Y, Dong Y D, Deng X G, et al. 2018. High-order simulation of aeronautical separated flows with a Reynolds stress model. Journal of Aircraft, 55(3):1177-1190.
    [127] Wilcox D C. 1988. Reassessment of the scale-determining equation for advanced turbulence models. AIAA J., 26(11),1299-1310.
    [128] Wilcox D C. 2006. Turbulence modeling for CFD. Third edition, DCW Industies, Inc.
    [129] Wilcox D C, Chambers T L. 1977. Streamline curvature effects on turbulent boundary layers. AIAA Journal, 15(4):574-580.
    [130] Wilcox D C, Rubesin M W. 1980. Progress in turbulence modeling for complex flow fie1ds including effects of compressibility. NASA TP-1517.
    [131] Yakhot V, Orszag S A. 1986. Renormalization group analysis of turbulence: I. Basic theory. Journal of Scientific Computing, 1(1):1-51.
    [132] Yap J C. 1987. Turbulent heat and momentum transfer in recirculating and impinging flows. [PhD Thesis]. Manchester: University of Manchester, Faculty of Technology.
    [133] Zhang Z J, Duraisamy K. 2015. Machine learning methods for data-driven turbulence modeling//22nd AIAA Computational Fluid Dynamics Conference, AIAA Paper 2015-2406.
    [134] Zhou Y. 2010. Renormalization group theory for fluid and plasma turbulence. Physics Reports, 488:1-49.
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  • 收稿日期:  2020-11-16
  • 刊出日期:  2021-03-25

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