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

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

王圣业¹,
符翔¹,
杨小亮¹,
郑皓榜¹,
邓小刚^1,2,*, ,

¹ 国防科技大学空天科学学院, 长沙 410073
² 军事科学院, 北京 100190

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
计量
- 文章访问数: 2510
- HTML全文浏览量: 444
- PDF下载量: 374
- 被引次数: 0
出版历程
- 收稿日期: 2020-11-16
- 刊出日期: 2021-03-25

Progresses and challenges of high-order-moment turbulence closure

¹ College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China
² Academy of Military Sciences, Beijing 100190, China

More Information

Corresponding author: DENG Xiaogang

摘要

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.
- 湍流模型 /
- 计算流体力学 /
- 雷诺应力模型 /
- 雷诺平均模拟 /
- 大涡数值模拟
Abstract: High-order-moment model is one of the focuses and frontier topics in the research of turbulence closure theory. Since Mr. Chou first established the Reynolds-stress transport equation in general turbulence, scholars have never stopped to study the high-order-moment model in order to predict the complex flow more accurately. Especially in the new century, thanks to the rapid development of computer hardware and the breakthrough of high-order numerical methods, turbulence simulation is in the course of turning from RANS into LES. In either RANS framework, LES framework, or the hybrid one, high-order-moment models are representative of advanced closure models. According to this, the paper gives a general review of the high-order-moment models. The focus includes the modeling approach of each source item, the evolution process of scale providing equations, and the research demand of numerical solution technology. Through several typical turbulence problems, their advantages over the traditional eddy-viscosity models are shown. Moreover, the integration of high-order-moment models in some CFD software is given. Finally, the future challenges and development direction of the high-order-moment turbulence model are prospected.
- turbulence model /
- computational fluid dynamics /
- Reynolds-stress model /
- RANS /
- LES

HTML全文

参考文献(134)

[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.