Advances in complex low speed flow around a prolate spheroid

CONG Chenghua; DENG Xiaogang; MAO Meiliang

doi:10.6052/1000-0992-20-036

Volume 51 Issue 3

Sep. 2021

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Cong C H, Deng X G, Mao M L. Advances in complex low speed flow around a prolate spheroid. Advances in Mechanics, 2021, 51(3): 467-619 doi: 10.6052/1000-0992-20-036

Citation:

Cong C H, Deng X G, Mao M L. Advances in complex low speed flow around a prolate spheroid. Advances in Mechanics, 2021, 51(3): 467-619 doi: 10.6052/1000-0992-20-036

Cong C H, Deng X G, Mao M L. Advances in complex low speed flow around a prolate spheroid. Advances in Mechanics, 2021, 51(3): 467-619 doi: 10.6052/1000-0992-20-036

Citation:

Cong C H, Deng X G, Mao M L. Advances in complex low speed flow around a prolate spheroid. Advances in Mechanics, 2021, 51(3): 467-619 doi: 10.6052/1000-0992-20-036

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Advances in complex low speed flow around a prolate spheroid

doi: 10.6052/1000-0992-20-036

1.
Key Laboratory of Unsteady Aerodynamics and Flow Control of Ministry of Industry and Information Technology, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2.
Facility Design and Instrumentation Institute; China Aerodynamic Research and Development Center, Mianyang 621000, Sichuan, China
3.
Military Academy of Sciences, Beijing 100010, China
4.
Institute of computational aerodynamics, China Aerodynamic Research and Development Center, Mianyang 621000, Sichuan, China

More Information

Corresponding author: cch_sd@163.com
Received Date: 2020-12-24
Accepted Date: 2021-04-08

Available Online: 2021-04-16

Publish Date: 2021-09-25

Abstract

Abstract

Understanding and predicting the flow around the prolate spheroid is of great engineering significance to guide the design of vehicles such as aircraft and submarines. In recent years, a lot of experimental and numerical studies have been carried out on the flow around the prolate spheroid. The qualitative description and quantitative research of flow separation around prolate spheroid at attack angle are presented, promoting the identification and topology research of three-dimensional separation. The experimental results of oil flow, smoke, dye, hydrogen bubble, and LDV are given. The flow field characteristics are analyzed, and the existing problems are pointed out. Based on the introduction of the above phenomena, the effects of separation on aerodynamic force, noise, and wake are introduced. The effects of test conditions such as transition zone, protrusion, depression, and tail support on flow are also discussed. There are obvious differences between the above steady flow and the unsteady maneuvering process. The unsteady maneuvering process can not be treated as a steady or quasi-steady problem. During the maneuvering process, the separation will be delayed obviously, and the aerodynamic force will also change obviously. The greater the angle of attack, the higher the maneuvering rate, the more noticeable this effect will be. At present, RANS turbulence model is still the primary engineering method to solve the large-scale separated flow around the prolate spheroid. However, LES, DES, and other methods have gradually been widely used. Due to the limitation of computer capability, DNS can only be used in the case of lower Reynolds number but not in high Reynolds number flow. The difference between the numerical simulation and the unsteady simulation is more significant. Finally, the research progress of prolate spheroid transition is introduced. The mechanism and identification of TS transition and cross-flow transition are more accurate. The numerical simulation results are basically consistent with the experimental results, but the understanding of reattachment transition is not clear enough, especially on the windward side. Therefore, the research of prolate spheroid transition still needs to rely on experiments.
- prolate spheroid,
- separation,
- vortex,
- wake,
- aerodynamic,
- sysnoise,
- unsteady maneuver,
- transition

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References(183)

References

[1]	卞于中, 张孝棣, Iuso G, 等. 1989. 绕旋转体复杂三维流动的实验研究. 空气动力学学报, 7: 435-441 (Bian Y Z, Zhang X D, Iuso G, et al. 1989. Experimental investigation of the complex 3D flow around an revolutinoary body. Acta Aerodynamica Sinica, 7: 435-441).
[2]	陈亮中. 2010. 双三角翼非定常分离流动的数值模拟研究 [博士论文]. 中国空气动力研究与发展中心 Chen L Z. 2010. Numerical investigation on the unsteady separaton flows around double delta wings [PhD Thesis]. China Aerodynamic Research and Development Center
[3]	丛成华, 邓小刚, 毛枚良. 2011. 微可压缩模型预处理技术研究. 力学学报, 43: 775-779 (Cong C H, Deng X G, Mao M L. 2011. Preconditioning technique in slightly compressible model. Chinese Journal of Theoretical and Applied Mechanics, 43: 775-779). doi: 10.6052/0459-1879-2011-4-lxxb2010-530
[4]	符松, 王亮湍. 2007. 流转捩模式研究进展. 力学进展, 37: 409-416 (Fu S, Wang L T. 2007. Progress in turbulent transiton modeling. Advances in Mechanics, 37: 409-416).
[5]	胡偶, 赵宁, 沈志伟. 2017. SST-DDES模型在大分离流动问题中的应用. 南京航空航天大学学报, 49: 206-211 (Hu O, Zhao N, Shen Z W. 2017. Simulation of large separated flows with SST-DDES model. Journal of Nanjing University of Aeronautics and Astronautics, 49: 206-211).
[6]	鞠胜军, 阎超, 叶志飞. 2017. r-Re_θt-CF转捩模型在Spalart-Allmaras湍流模型中的推广及验证. 航空学报, 38: 120383 (Ju S J, Yan C, Ye Z F. 2017. Genevalization and validation of r-Re_θt-CF transiton modeling in combination with Spalart-Allmaras turbulence model. Acta Aeronautica et Astronautica Sinica, 38: 120383).
[7]	李存标. 1998. 转捩和湍流研究的最新进展. 流体力学实验与测量, 12: 8-28 (Li C B. 1998. Recent development in the study of transition and turbulence. Experiments and Measurements in Fluid Mechanics, 12: 8-28).
[8]	刘沛清, 邓学蓥, 孔繁美. 2002. 绕细长旋成体非对称涡非定常性的实验研究. 流体力学实验与测量, 16: 39-46 (Liu P Q, Deng X Y, Kong F M. 2002. Experimental investigation of asymmetry vortex unsteadiness over slender cylinders. Experiments and Measurements in Fluid Mechanics, 16: 39-46).
[9]	刘伟, 杨小亮, 张涵信, 等. 2008. 大攻角运动时的机翼摇滚问题研究综述. 力学进展, 38: 214-228 (Liu W, Yang X L, Zhang H X, et al. 2008. A review on investigations of wing rock problems under high angles of attack. Advances in Mechanics, 38: 214-228).
[10]	潘翀, 王晋军. 2011. 自由来流扰动引起的旁路转捩研究进展. 力学进展, 41: 668-685 (Pan C, Wang J J. 2011. Progress in bypass transition induced by free-stream disturbance. Advances in Mechanics, 41: 668-685).
[11]	邱磊. 2004. 船舶操纵相关黏性流及水动力计算[博士论文]. 武汉理工大学 Qiu L. 2004. Computation of ship manoeuvring related viscous flow and hydrodynamics forces [PhD Thesis]. Wuhan University of Technology
[12]	邱磊, 邹早建. 2005. 船舶操纵黏性流求解器的开发与应用. 华中科技大学学报, 33: 27-29 (Qiu L, Zou Z J. 2005. Development of a RANS solver for viscous flows from ship maneuver and its application. Journal of Huazhong University of Science and Technology, 33: 27-29).
[13]	温功碧, 陈作斌. 2004. 三维非定常/定常不可压缩流动N-S方程基于人工压缩性方法的数值模拟. 应用数学和力学, 25: 53-66 (Wen G B, Chen Z B. 2004. Unsteady/steady numerical simulation of three dimensional incompressible Navier-Stokes equations on artificial compressibility. Applied Mathematics and Mechanics, 25: 53-66). doi: 10.3321/j.issn:1000-0887.2004.01.007
[14]	吴天佐. 2014. 超声速来流下支撑干扰数值模拟研究[硕士论文]. 国防科学技术大学 Wu T Z. 2014. Numerical simulation of sting interference in supersonic flow [Master Thesis]. National of defense technology
[15]	向大平, 邓小刚, 毛枚良. 2005. 微可压缩模型与可压缩N-S方程数值计算对比研究. 空气动力学学报, 23: 195-199 (Xiang D P, Deng X G, Mao M L. 2005. Study of slightly compressible model (SCM) and compressible N-S equations on low Mach number flow computation. Acta Aerodynamica Sinica, 23: 195-199). doi: 10.3969/j.issn.0258-1825.2005.02.012
[16]	肖昌润, 刘巨斌, 毕毅, 等. 2007. 椭球体水动力数值计算. 武汉理工大学学报, 31: 842-845 (Xiao C R, Liu J B, Bi Y, et al. 2007. Numerical computation of flow and hydrodynamic force of prolate ellipsoid at high incidence. Journal of Wuhan University of Technology, 31: 842-845).
[17]	肖志祥, 陈海昕, 李启兵. 2006. 采用RANS/LES混合方法研究分离流动. 空气动力学学报, 24: 218-222 (Xiao Z X, Chen H X, Li Q B. 2006. Simulation of separation flows with RANS/LES hybrid methods. Acta Aerodynamica Sinica, 24: 218-222). doi: 10.3969/j.issn.0258-1825.2006.02.015
[18]	熊英. 2019. 密度分层流中长椭球俯仰振荡和自主运动的数值模拟研究[博士论文]. 大连理工大学 Xiong Y. 2019. Numerical simulation of the pitching oscillation and free motion of a long prolate spheroid in a stratified flow [PhD Thesis]. Dalian University of Technology
[19]	徐家宽. 2019. 基于RANS方程的多速域边界层转捩模式构造方法及应用研究[博士论文]. 西北工业大学 Xu J K. 2019. Modelling methods and application research of boundary layer transition in multi-speed range based on RANS equations [PhD Thesis]. Northwestern Polytechnical University
[20]	徐晶磊, 周禹, 乔磊, 等. 2019. 基于湍动能输运的一方程转捩模型. 推进技术, 40: 741-749 (Xu J L, Zhou Y, Qiao L, et al. 2019. One-equation transition model based on turbulent kinetic energy transport. Journal of Propulsion Technology, 40: 741-749).
[21]	严崇禄, 曹露洁, 林发布, 等. 2002. 4: 1椭球体分离流动湍流特性的试验研究. 力学学报, 34: 501-507 (Yan C L, Cao L J, Lin F B, et al. 2002. Experimental investigation on turbulent behavior in a separated flow around a 4: 1 prolate spheroid at incidence. Acta Mechanic Sinica, 34: 501-507). doi: 10.3321/j.issn:0459-1879.2002.04.003
[22]	严家祥, 席德科. 1989. 利用流线法决定绕扁长椭球流动分离的位置. 空气动力学学报, 7: 474-451 (Yan J X, Xi D K. 1989. Determiniation of prolate spheroid flow separation by using streamline method. Acta Aerodynamica Sinica, 7: 474-451).
[23]	严家祥, 席德科. 1991. 绕扁长椭球体三维分离的积分法. 空气动力学学报, 9: 428-434 (Yan J X, Xi D K. 1991. An integral method for a prolate spheroid 3D flow separation. Acta Aerodynamica Sinica, 9: 428-434).
[24]	于向阳, 刘巨斌, 肖昌润. 2011. 采用DES模型对6: 1椭球体进行水动力数值计算. 海军工程大学学报, 23: 100-103 (Yu X Y, Liu J B, Xiao C R. 2011. Numerical computation of flow and hydrodynamic force of ellipsoid based on DES hybrid methods. Journal of Naval University of Engineering, 23: 100-103). doi: 10.3969/j.issn.1009-3486.2011.01.020
[25]	袁礼. 2002. 三维不可压N-S方程的多重网格求解. 计算物理, 39: 23-29 (Yuan L. 2002. Multigrid solutions for the three dimensional incompressible Navier-Stokes equations in artificial compressibility formulation. Chinese Journal of Computational Physics, 39: 23-29). doi: 10.3969/j.issn.1001-246X.2002.01.005
[26]	张丹, 郭雪岩. 2008. 平流层双轴椭球体飞艇绕流场的数值分析. 力学季刊, 29: 556-564 (Zhang D, Guo X Y. 2008. Numerical analysis on ambient flow of a double axis ellipsoidal stratospheric airship. Chinese Quarterly of Mechanics, 29: 556-564).
[27]	张涵信. 1997. 分离流和涡运动横截面流态的拓扑. 空气动力学学报, 15: 1-12 (Zhang H X. 1997. Crossflow topology of three dimensional separated flows and vortex motion. Acta Aerodynamica Sinica, 15: 1-12).
[28]	张涵信. 1998. 我国计算空气动力学发展中存在的问题及其对策. 世界科技研究与发展, 20: 26-28 (Zhang H X. 1998. Issues concerning development of computing aerodynamics in China. World Sci-tech Reasearch and Development, 20: 26-28).
[29]	张涵信, 周恒. 2001. 流体力学的基础研究. 世界科技研究与发展, 23: 15-18 (Zhang H X, Zhou H. 2001. On basic research of fluid mechanics. World Sci-tech Reasearch and Development, 23: 15-18). doi: 10.3969/j.issn.1006-6055.2001.01.004
[30]	张涵信. 2002. 分离流与旋涡运动的结构分析. 北京: 国防工业出版社 Zhang H X. 2002. Structure Analysis of Separated Flow and Vortex Motion. Beijing: National Defense Industry Press
[31]	张涵信, 张树海, 田浩, 等. 2012. 三维可压缩非定常流的壁面分离判据及其分离线附近的流动形态. 空气动力学学报, 30: 421-430 (Zhang H X, Zhang S H, Tian H, et al. 2012. Separation on fixed surface for three dimensional compressible unsteady flows. Acta Aerodynamica Sinica, 30: 421-430). doi: 10.3969/j.issn.0258-1825.2012.04.001
[32]	张涵信. 2016. 关于CFD高精度保真的数值模拟研究. 空气动力学学报, 34: 1-4 (Zhang H X. 2016. Investigation on fidelity of high order accurate numerical simulation for computational fluid dynamics. Acta Aerodynamica Sinica, 34: 1-4).
[33]	张来平, 邓小刚, 张涵信. 2010. 动网格生成技术及非定常计算方法进展综述. 力学进展, 40: 424-447 (Zhang L P, Deng X G, Zhang H X. 2010. Reviews of moving grid generation techniques and numerical methods for unteady flow. Advances in Mechanics, 40: 424-447).
[34]	张毅锋. 2010. 高阶精度格式(WCNS)加速收敛和复杂流动数值模拟的应用研究 [博士论文]. 中国空气动力研究与发展中心 Zhang Y F. 2010. Investigation of convergence acceleration and complex flow numerical simulation for high order accurate scheme WCNS [PhD Thesis]. China Aerodynamic Research and Development Center
[35]	郑世超. 2019. 高精度格式在高超声速流动数值模拟中的验证与确认研究[博士论文]. 国防科学技术大学 Zheng S C. Verification and validation study of high order schemes in numerical simulations of hypersonic flows [PhD Thesis]. National University of Defense Technology
[36]	祝成民, 忻鼎定, 庄逢甘. 2002. 改进油流显示法研究层流和湍流下绕椭球分离流. 空气动力学学报, 20: 312-319 (Zhu C M, Xin D D, Zhuang F G. 2002. Investigation of separate flow around a prolate spheroid in laminar and turbulent flow by an improved oil-flow visualization technique. Acta Aerodynamica Sinica, 20: 312-319). doi: 10.3969/j.issn.0258-1825.2002.03.008
[37]	祝成民, 忻鼎定, 庄逢甘. 2003a. 两种湍流模式对绕椭球体三维分离流计算的比较. 航空学报, 24: 491-494 (Zhu C M, Xin D D, Zhuang F G. 2003a. A comparison of two turbulence models on computation of three dimensional separated flow around a prolate spheroid. Acta Aeronautica et Astronautica Sinica, 24: 491-494).
[38]	祝成民, 忻鼎定, 庄逢甘. 2003b. 绕椭球三维流动的分离结构随雷诺数的变化. 空气动力学学报, 21: 75-81 (Zhu C M, Xin D D, Zhuang F G. 2003b. The variation of the threr dimensional structures of flow separation with Reynolds number for a prolate spheroid. Acta Aerodynamica Sinica, 21: 75-81).
[39]	Ahn S, Simpson R L. 1992. Cross-flow separation on a prolate spheroid at angles of attack. AIAA Paper 92-0428
[40]	Alin N, Fureby C, Svennberg S U, et al. 2005. 3D unsteady computations for submarine-like bodies. AIAA Paper 2005-1104
[41]	Alin N, Fureby C, Svennberg S U, et al. 2007. Large eddy simulation of the transient flow around a submarine during maneuver. AIAA Paper 2007-1454
[42]	Alpman E, Long L N. 2005. Separated turbulent flow simulations using a Reynolds stress model and unstructured meshes. AIAA Paper 2005-1094
[43]	Ambo T, Otsuki T, Taniguchi S, et al. 2016. Investigation rear-end of flow structures on a 6: 1 prolate spheroid by using the magnetic suspension and balance system. AIAA Paper 2016-4158
[44]	Ashok A, Buren T V, Smits A J. 2015. The structure of the wake generated by a submarine model in yaw. Experiments in Fluids, 56: 123-131. doi: 10.1007/s00348-015-1997-4
[45]	Baldwin B S, Lomax H. 1978. Thin layer approximation and algebraic model for separated turbulent flows. AIAA Paper 78-0257
[46]	Barber K M, Simpson R L. 1991. Mean velocity and turbulence measurements of flow around a 6: 1 prolate spheroid. AIAA Paper 91-0255
[47]	Barberis D, Molton P. 1995. Experimental study of three-dimensional separation on a large-scale model. AIAA Journal, 33: 2107-2113. doi: 10.2514/3.12954
[48]	Boltz F W, Kenyon G C. 1956. Measurements of boundary-layer transition at low speed on two bodies of revolution in a low-turbulence wind tunnel. NACA RM-R56G17
[49]	Carmody T. 1964. Establishment of the wake behind a disk. Journal of Basic Engineering, 87: 869-882.
[50]	Cebeci T, Khattab A K, Stewartson K. 1980. On nose separation. Journal of Fluid Mechanics, 97: 435-454. doi: 10.1017/S0022112080002649
[51]	Cebeci T, Khattab A K, Stewartson K. 1981. Three-dimensional laminar boundary layers and the OK of accessibility. Journal of Fluid Mechanics, 107: 57-87.
[52]	Cebeci T, Meier H U. 1987. Turbulent boundary layers on a prolate spheroid. AIAA Paper 87-1299
[53]	Cebeci T, Stewartson K. 1980. On stability and transition in three-dimensional flows. AIAA Journal, 18: 398-405. doi: 10.2514/3.50772
[54]	Cebeci T, Stewartson K, Schimke S M. 1984. Some important problems in unsteady boundary layers including separation II unsteady boundary layers close to the stagnation region of slender bodies. AFOSR TR-0861
[55]	Cebeci T, Su W H. 1988a. Separation of three-dimensional laminar boundary layers on a prolate spheroid. Journal of Fluid Mechanics, 191: 47-77. doi: 10.1017/S002211208800151X
[56]	Cebeci T, Su W H. 1988b. The birth of open separation on a prolate spheroid. International Journal of Computational Fluid Dynamics, 2: 283-307.
[57]	Chang P K. 1961. Separation of flow. Journal of the Franklin Institute, 272: 433-448. doi: 10.1016/0016-0032(61)90875-4
[58]	Chapman D R. 1979. Computatisnal aerodynamics development and outlook. AIAA Journal, 17: 1293-1313. doi: 10.2514/3.61311
[59]	Chapman G T. 1986. Topological classification of flow separation on three-dimensional bodies. AIAA Paper 86-0485
[60]	Chesnakas C J, Simpson R L. 1996. Measurements of the turbulence structure in the vicinity of a 3-d separation. Journal of Fluids Engineering, 118: 268-275. doi: 10.1115/1.2817373
[61]	Chesnakas C J, Simpson R L. 1997. Detailed investigation of the three-dimensional separation about a 6: 1 prolate spheroid. AIAA Journal, 35: 990-999. doi: 10.2514/2.208
[62]	Chevray R. 1968. The turbulent wake of a body of revolution. Journal of Basic Engineering, 91: 275-284.
[63]	Cianferra M, Armenio V, Ianniello S. 2018. Hydroacoustic noise from different geometries. International Journal of Heat and Fluid Flow, 70: 348-362. doi: 10.1016/j.ijheatfluidflow.2017.12.005
[64]	Constantinescu G S, Pasinato H, WangY Q, et al. 2002. Numerical investigation of flow past a prolate spheroid. Journal of Fluids Engineering, 124: 904-910. doi: 10.1115/1.1517571
[65]	Constantinescu G, Squires K. 2004. Numerical investigations of flow over a sphere in the subcritical and supercritical regimes. Physics of Fluids, 16: 1449-1466. doi: 10.1063/1.1688325
[66]	Costis C E, Hoang N T, Telionis D P. 1989. Laminar separating flow over a prolate spheroid. Journal of Aircraft, 26: 810-816. doi: 10.2514/3.45845
[67]	Costis C E, Polen D M, Hoang N T. 1987. Laminar separating flow over a prolate spheroid. AIAA Paper 1212
[68]	Costis C E, Telionis D P. 1984. Unsteady vortical wakes over a prolate spheroid. AIAA Paper 84-0419
[69]	Costis C E, Telionis D P. 1988. Vortical wakes over a prolate spheroid. AIAA Journal, 26: 1189-1193. doi: 10.2514/3.10027
[70]	Dallmann U, Herberg Th, Gebing H, et al. 1995. Flow field diagnostics topological flow changes and spatio-temporal flow structure. AIAA Paper 95-0791
[71]	Delery J M. 1992. Physics of vortical flows. Journal of Aircraft, 29: 856-876. doi: 10.2514/3.46256
[72]	Deng X G, Zhuang F G. 2002. A novel slightly compressible model for low Mach number perfect gas flow calculation. Acta Mechanica Sinica, 18: 193-208. doi: 10.1007/BF02487948
[73]	Dorrington G E. 2006. Drag of spheroid-cone shaped airship. Journal of Aircraft, 43: 363-371. doi: 10.2514/1.14796
[74]	Dress D A. 1990. Drag measurements on a modified prolate spheroid using a magnetic suspension and balance system. Journal of Aircraft, 27: 523-528. doi: 10.2514/3.25314
[75]	Farhat C, Rajasekharan A, Koobus B. 2006. A dynamic variational multiscale method for large eddy simulations on unstructured meshes. Computer Methods in Applied Mechanics and Engineering, 195: 1667-1691. doi: 10.1016/j.cma.2005.05.045
[76]	Gee K, Cummings R M. 1992. Turbulence model effects on separated flow about a prolate spheroid. AIAA Journal, 30: 655-664. doi: 10.2514/3.10969
[77]	Geissler W. 1974. Three-dimensional laminar boundary layer over a body of revolution at incidence and with separation. AIAA Journal, 12: 1743-1745. doi: 10.2514/3.49593
[78]	Goman M, Khrabrov A. 1994. State-space representation of aerodynamic characteristics of an aircraft at high angles of attack. Journal of Aircraft, 31: 1109-1115. doi: 10.2514/3.46618
[79]	Goody M C, Simpson R L, Chesnakas C J. 2000a. Separated flow surface pressure fluctuations and pressure–velocity correlations on prolate spheroid. AIAA Journal, 38: 266-274. doi: 10.2514/2.953
[80]	Goody M C, Simpson R L. 2000b. Surface pressure fluctuations beneath two- and three-dimensional turbulent boundary layers. AIAA Journal, 38: 1822-1931. doi: 10.2514/2.863
[81]	Goody M C. 2004. Empirical spectral model of surface pressure fluctuations. AIAA Journal, 42: 1788-1794. doi: 10.2514/1.9433
[82]	Grabe C, Krumbein A. 2013. Correlation-based transition transport modeling for three-dimensional aerodynamic configurations. Journal of Aircraft, 50: 1533-1539. doi: 10.2514/1.C032063
[83]	Granlund K, Simpson R L. 2009. Experimentally obtained forces and moments on slender bodies during steady and unsteady maneuvers. AIAA Paper 2009-1292
[84]	Han T, Patel V C. 1979. Flow separation on a spheroid at incidence. Journal of Fluid Mechanics, 92: 643-657. doi: 10.1017/S002211207900080X
[85]	Hartwich P M, Hall R M. 1990. Navier-Stokes solutions for vortical flows over a tangent-ogive cylinder. AIAA Journal, 28: 1171-1179. doi: 10.2514/3.25188
[86]	Hedin P O, Berglund M, Alin N, et al. 2001. Large eddy simulation of the flow around an inclined prolate spheroid. AIAA Paper 2001-1035
[87]	Hirsh R S, Cebeci T. 1977. Calculation of three-dimensional boundary layers with negative cross flow on bodies of revolutions. AIAA Paper 77-0683
[88]	Hoang N T, Wetzel T G, Simpson R L. 1994a. Unsteady measurements over a 6: 1 prolate spheroid undergoing a pitch-up maneuver. AIAA Paper 94-0197
[89]	Hoang N T, Wetzel T G, Simpson R L. 1994b. Surface pressure measurements over a 6: 1 prolate spheroid undergoing time-dependent maneuvers. AIAA Paper 94-1908
[90]	Hoang N T, Telionis D P. 1991. The dynamic character of the wake of an axisymmetric body at an angle of attack. AIAA Paper 91-3268
[91]	Holt J, Garry K, Smith T. 2016. Investigation of the aerodynamic characteristics of a lifting body in ground proximity. AIAA Paper 2016-3881
[92]	Hosder S, Simpson R L. 2001. Unsteady turbulent skin friction and separation location measurements on a maneu-vering undersea vehicle. AIAA Paper 2001-1000
[93]	Huang J C, Lin H, Yang J Y. 2009. Implicit preconditioned WENO scheme for steady viscous flow computation. Journal of Computational Physics, 228: 420-438. doi: 10.1016/j.jcp.2008.09.017
[94]	Jiang F J, Gallardo J P, Andersson H I. 2014. The laminar wake behind a 6: 1 prolate spheroid at 45 incidence angle. Physics of Fluids, 26: 113602. doi: 10.1063/1.4902015
[95]	Jiang F J, Gallardo J P, Andersson H I, et al. 2015. The transitional wake behind an inclined prolate spheroid. Physics of Fluids, 27: 093602. doi: 10.1063/1.4929764
[96]	Johnson D A, King L S. 1985. A mathematically simple turbulence closure model for attached and separated turbulent boundary layers. AIAA Journal, 23: 1684-1692. doi: 10.2514/3.9152
[97]	Johnson T A, Patel V C. 1999. Flow past a sphere up to a Reynolds number of 300. Journal Fluid Mechanics, 378: 19-70. doi: 10.1017/S0022112098003206
[98]	Karlsson A, Fureby C. 2009. LES of the flow past a 6: 1 prolate spheroid. AIAA Paper 2009-1616
[99]	Khoury G K EL, Andersson H I, Pettersen B. 2010. Crossflow past a prolate spheroid at Reynolds number of 10000. Journal of Fluid Mechanics, 659: 365-374. doi: 10.1017/S0022112010003216
[100]	Khoury G K EL, Andersson H I, Pettersen B. 2012. Wakes behind a prolate spheroid in crossflow. Journal of Fluid Mechanics, 701: 98-136. doi: 10.1017/jfm.2012.135
[101]	Kim S E, Patel V C. 1991. Laminar flow separation on a spheroid at incidence. AIAA Paper 91-1803
[102]	Kim S E, Rhee S H. 2003. Application of modern turbulence models to vortical flow around a prolate spheroid. AIAA Paper 2003-0429
[103]	Kotapati-Apparao R B, Squires K D. 2003. Prediction of a prolate spheroid undergoing a pitchup maneuver. AIAA Paper 2003-0269
[104]	Kreplin H P, Vollmers H, Meier H V. 1980. Experimental determination of wall shear-stress vector on an inclined prolate spheroid. Air Force Flight Dynamics Laboratory, TR-80-3088
[105]	Kreplin H U, Vollmers H, Meier H U. 1982. Measurements of the wall shear stress on an inclined prolate spheroid. Zeitschriftfur Flugwissenschaft Wetraumforschung, 6: 248-252.
[106]	Kreplin H P, Vollmers H, Meier H U. 1985. Wall shear stress measurements on an inclined prolate spheroid in the DFVLR 3 m×3 m low speed wind tunnel. Gottingen DFVLR AVA Report IB22 84A33
[107]	Krimmelbein N, Krumbein A. 2011. Automatic transition prediction for three-dimensional configurations with focus on industrial application. Journal of Aircraft, 48: 1878-1887. doi: 10.2514/1.C031230
[108]	Krimmelbein N, Radespiel R. 2009. Transition prediction for three-dimensional flows using parallel computation. Computers and Fluids, 38: 121-136. doi: 10.1016/j.compfluid.2008.01.004
[109]	Krimmelbein N, Radespiel R, Nebel C. 2005. Numerical aspects of transition prediction for three-dimensional configurations. AIAA Paper 2005-4764
[110]	Kubota H, Arai I, Matsuzakat M. 1983. Flat spin of slender bodies at high angles of attack. Journal of Spacecraft and Rockets, 20: 108-114. doi: 10.2514/3.28365
[111]	Liu C, Zheng X, Sung C H. 1998. Preconditioned multigrid methods for unsteady incompressible flows. Journal of Computational Physics, 139: 35-57. doi: 10.1006/jcph.1997.5859
[112]	Madden M M, Simpson R L. 1997. Octant analysis of the Reynolds stresses in the three-dimensional turbulent boundary layer of a prolate spheroid. VPI AOE-252
[113]	Manhart M. 2004. A zonal grid algorithm for DNS of turbulent boundary layers. Computers and Fluids, 33: 435-461. doi: 10.1016/S0045-7930(03)00061-6
[114]	Matthews C W. 1952. A comparison of the experimental subsonic pressure distributions about several bodies of revolution with pressure distributions computed by means of the linearized theory. NACA Report 1155
[115]	Morgan K, Hughes T G, Taylor C. 1978. A numerical model of turbulent shear flow behind a prolate spheroid. Applied Mathematical Modelling, 2: 271-274. doi: 10.1016/0307-904X(78)90020-3
[116]	Morrison J H, Panaras A G, Gatski T B, et al. 2003. Analysis of extensive cross-flow separation using higher-order RANS closure models. AIAA Paper 2003-3532
[117]	Newcomb A W. 1988. Effect of sting interference at low speeds on the drag coefficient of an ellipsoidal body using a magnetic suspension and balance system. NASA CR-181611
[118]	Newsome R W, Adams M S. 1988. Vortical flow over an elliptical-body missile at high angles of attack. Journal of Spacecraft and Rockets, 25: 24-30. doi: 10.2514/3.25983
[119]	Nie S Y, Krimmelbein N, Krumbein A, et al. 2018a. Coupling of a Reynolds stress model with the γ-Re_θ-t transition model. AIAA Journal, 56: 146-157. doi: 10.2514/1.J056167
[120]	Nie S Y, Krimmelbein N, Krumbein A, et al. 2018b. Extension of a Reynolds-stress-based transition transport model for crossflow transition. Journal of Aircraft, 55: 1641-1654. doi: 10.2514/1.C034586
[121]	Panzer E C, Simpson R L. 1995. Simulation of high Reynolds number flows over a prolate spheroid model using turbulence grid. AIAA Paper 95-0784
[122]	Patel V C, Baek J H. 1985. Boundary layers and separation on a spheroid at incidence. AIAA Journal, 23: 55-63. doi: 10.2514/3.8871
[123]	Radwan S F. 1988. Higher-order accurate calculations of the compressible boundary layers on a prolate spheroid. AIAA Paper 88-3585
[124]	Radwan S F, Lekoudis S G. 1986. Inverse mode calculations of the incompressible turbulent boundary layer on an ellipsoid. AIAA Journal, 24: 1628-1635. doi: 10.2514/3.9493
[125]	Ragab S A. 1982. A method for the calculation of three dimensional boundary layers with circumferential reversed flow on bodies. AIAA Paper 82-1023
[126]	Ragab S A. 1985. Steady and unsteady boundary layers on prolate spheroids at high incidence. AIAA Paper 85-1708
[127]	Ragab S A. 1986. The laminar boundary layer on a prolate spheroid started impulsivelyfrom rest at high incidence. AIAA Paper 86-1109
[128]	Ramamurti R, Sandberg W. 1994. Evaluation of a three dimensional finite element incompressible flow solver. AIAA Paper 94-0756
[129]	Ramaprian B R, Patel V C, Choi D H. 1981. Mean-flow measurements in the three-dimensional boundary layer over a body of revolution at incidence. Journal of Fluid Mechanics, 103: 479-504. doi: 10.1017/S0022112081001432
[130]	Ranjan R, Menon S. 2015. Hybrid two-level large-eddy simulation of turbulent flow in a channel, past a bump and around an inclined prolate spheroid. AIAA Paper 2015-1526
[131]	Rhee S H, HinoT. 2002. Numerical simulation of unsteady turbulent flow around maneuvering prolate spheroid. AIAA Journal, 40: 2017-2026. doi: 10.2514/2.1534
[132]	Rogers S E, Kwak D. 1990. Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA Journal, 28: 253-262. doi: 10.2514/3.10382
[133]	Rosemann H, Staeger R, Kreplin H P. 1996. Development and application of a quadruple hot-wire technique for turbulent flows. Measurement Science and Technology, 7: 1477-1491. doi: 10.1088/0957-0233/7/10/018
[134]	Rosenfeld M, Israeli M, Wolfshtein M. 1988. Numerical study of the skin friction on a spheroid at incidence. AIAA Journal, 26: 129-136. doi: 10.2514/3.9863
[135]	Rosenfeld M, Israeli M. 1985. Numerical solution of incompressible flows by a marching multigrid nonlinear method. AIAA Paper 85-1500
[136]	Rosenfeld M, Wolfshtein M, Israeli M. 1992. A numerical study of the laminar incompressible flow over a 6: 1 prolate spheroid at 10° incidence. International Journal Numerical Methmatics in Fluids, 15: 147-172. doi: 10.1002/fld.1650150203
[137]	Sanjeevi S K P, Padding J T. 2017. On the orientational dependence of drag experienced by spheroids. Journal of Fluid Mechanics, 820R1: 1-13.
[138]	Schonauer W, Straub D. 1975. Numerical solution of the steady laminar hypersonic boundary layer equations at chemical nonequilibrium. FTD HC-23-1164-75
[139]	Scott N W, Duque E P N. 2004. Unsteady Reynolds-averaged Navier-Stokes predictions of the flow around a prolate spheroid. AIAA Paper 2004-0055
[140]	Scott N W, Duque E P N. 2005. Using detached eddy simulation and overset grids to predict flow around a 6: 1 prolate spheroid. AIAA Paper 2005-1362
[141]	Sheng C, Taylor L K, Whitfield D L. 1995. Multigrid algorithm for three-dimensional incompressible high-Reynolds number turbulent flows. AIAA Journal, 33: 2073-2079. doi: 10.2514/3.12949
[142]	Shirayama S, Kuwahara K. 1987. Patterns of three- dimensional boundary layer separation. AIAA Paper 87-0461
[143]	Siclari M J, Ende R, Carpenter G. 1995. The application of Navier-Stokes computations to the design of high-speed, low-drag magnetically levitated (Maglev) vehicle shapes. AIAA Paper 95-1908
[144]	Simpson R L. 1996. Aspects of turbulent boundary-layer separation. Progress in Aerospace Sciences, 32: 457-521. doi: 10.1016/0376-0421(95)00012-7
[145]	Slotnick J, Khodadoust A, Alonso J, et al. 2014. CFD vision 2030 study a path to revolutionary computational aerosciences. NASA CR-218178
[146]	Spalart P R. 2009. Detached-eddy simulation. Annual Review of Fluid Mechanics, 41: 181-202. doi: 10.1146/annurev.fluid.010908.165130
[147]	Spalart P R, Deck S, Shur M L, et al. 2006. A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theoretical and Computational Fluid Dynamics, 20: 181-195. doi: 10.1007/s00162-006-0015-0
[148]	Strandenes H, Jiang F J, Pettersen B, et al. 2019. Near-wake of an inclined 6: 1 spheroid at Reynolds number 4000. AIAA Journal, 57: 1364-1372. doi: 10.2514/1.J057615
[149]	Stock H W. 2006. eN transition prediction in three-dimensional boundary layers on inclined prolate spheroids. AIAA Journal, 44: 108-118. doi: 10.2514/1.16026
[150]	Su W H, Tao B, Xu L. 1996. Three-dimensional separated flow over a prolate spheroid. AIAA Journal, 31: 2175-2176.
[151]	Sung C H, Griffin M J, Tsai J F, et al. 1993. Incompressible flow computation of forces and moments on bodies of revolution at incidence. AIAA Paper 93-0787
[152]	Tai T C. 1984. Integral prediction method for three-dimensional flow separation. AIAA Paper 84-0014
[153]	Taylor L K, Arabshahi A, Whitfield D L. 1995. Unsteady three-dimensional incompressible Navier-Stokes computations for a prolate spheroid undergoing time-dependent maneuvers. AIAA Paper 95-0313
[154]	Telionis D P, Costis C E. 1983. Three dimensional laminar separation. VPI E-48
[155]	Tezuka A, Suzuki K. 2006. Three-dimensional global linear stability analysis of flow around a spheroid. AIAA Journal, 44: 1697-1708. doi: 10.2514/1.16632
[156]	Tobak M, Peake D J. 1982. Topology of three-dimensional separated flows. Annual Review of Fluid Mechanics, 14: 61-85. doi: 10.1146/annurev.fl.14.010182.000425
[157]	Tsai C Y, Whitney A K. 1999. Numerical study of three-dimensional flow separation from a 6: 1 ellipsoid. AIAA Paper 99-0172
[158]	Tyll J S, Liu D, Schetz J A, et al. 1996. Experimental studies of magnetic levitation train aerodynamics. AIAA Journal, 34: 2465-2470. doi: 10.2514/3.13425
[159]	VanDalsem W R, Steger J L. 1987. Efficient simulation of separated three-dimensional viscous flows using the boundary-layer equations. AIAA Journal, 25: 395-340. doi: 10.2514/3.9636
[160]	Van Dommelen L L, Cowley S J. 1990. On the lagrangian description of unsteady boundary layer separation Part I: general theory. Journal of Fluid Mechanics, 210: 593-626. doi: 10.1017/S0022112090001410
[161]	Vatsa V N, Thomas J L, Wedan B W. 1987. Navier-Stokes computations of a prolate spheroid at angle of attack. Journal of Aircraft, 26: 986-993.
[162]	Vollmers H. 1982. Integration of streamlines from measured static pressure fields on a surface. AIAA Journal, 20: 1459-1460. doi: 10.2514/3.7985
[163]	Wang K C. 1972. Separation patterns of boundary layer over an inclined body of revolution. AIAA Journal, 10: 1044-1050. doi: 10.2514/3.50292
[164]	Wang K C. 1974a. Laminar boundary layer over a body of revolution at extremely high incidence. Physics of Fluids, 17: 1381-1385. doi: 10.1063/1.1694900
[165]	Wang K C. 1974b. Laminar boundary layer near the symmetry plane of a prolate spheroid. AIAA Journal, 12: 949-958. doi: 10.2514/3.49386
[166]	Wang K C. 1975. Boundary layer over a blunt body at low incidence with circumferential reversed flow. Journal of Fluid Mechanics, 72: 49-65. doi: 10.1017/S0022112075002935
[167]	Wang K C. 1976. Separation of three-dimensional flow. MML TR-54C
[168]	Wetzel T G, Simpson R L. 1996. Unsteady flow over a 6: 1 prolate spheroid. VPI AOE-232
[169]	Wetzel T G, Simpson R L, Chesnakas C J. 1998a. Measurement of three-dimensional crossflow separation. AIAA Journal, 36: 557-564. doi: 10.2514/2.429
[170]	Wetzel T G, Simpson R L. 1998b. Unsteady crossflow separation location measurements on a maneuvering 6: 1 prolate spheroid. AIAA Journal, 36: 2063-2071. doi: 10.2514/2.307
[171]	Wikstrom N, Svennberg S U, Alin N, et al. 2004. Large eddy simulation of the flow around an inclined prolate spheroid. Journal of Turbulence, 5: 37-41.
[172]	Wilson G R. 1971. Experimental study of a laminar boundary layer on a body of revolution [Master Thesis]. Air Force Institute of Technology
[173]	Wong T C, Kandil O A. 1989. Navier-Stokes computations of separated vortical flows past prolate spheroid at incidence. AIAA Paper 89-0553
[174]	Wu T, Shen S F. 1991. A multizone time-marching technique for unsteady separating three dimensional boundary layers and its application to the symmetry-plane solution of an impulsively started prolate spheroid. Journal of Fluids Engineering, 113: 228-239. doi: 10.1115/1.2909485
[175]	Wu T, Shen S F. 1992. Emergence of three-dimensional separation over a suddenly started prolate spheroid at incidence. AIAA Journal, 30: 2707-2715. doi: 10.2514/3.11288
[176]	Xiao Z X, Zhang Y F, Huang J B, et al. 2007. Prediction of separation flows around a 6: 1 prolate spheroid using RANS/LES hybrid approaches. Acta Mechanica Sinica, 23: 369-382. doi: 10.1007/s10409-007-0073-6
[177]	Xin D D. 2000. Algebraic turbulence model with memory for computation of 3-d turbulent boundary layers with validation. Chinese Journal of Aeronautics, 13: 65-74.
[178]	Yates L A, Chapman G T. 1988. Numerical investigation of crossflow separation on a three caliber tangent ogive cylinder. AIAA Journal, 26: 1223-1230. doi: 10.2514/3.10032
[179]	Yates L A, Chapman G T. 1992. Streamlines, vorticity lines, and vortices around three dimensional bodies. AIAA Journal, 30: 1819-1826. doi: 10.2514/3.11142
[180]	Yuan L. 2002. Comparison of implicit multigrid schemes for three-dimensional incompressible flows. Journal of Computational Physics, 177: 134-155. doi: 10.1006/jcph.2002.7007
[181]	Zamyshlyaev A A, Shrager G R. 2004. Fluid flows past spheroids at moderate reynolds numbers. Fluid Dynamics, 39: 376-383. doi: 10.1023/B:FLUI.0000038556.08179.ea
[182]	Zeiger M D, Telionis D P, Vlachos P P. 2004. Unsteady separated flows over three-dimensional slender bodies. Progress in Aerospace Sciences, 40: 291-320. doi: 10.1016/j.paerosci.2004.06.002
[183]	Zilliac G G. 1989. Computational and experimental study of the flow around a body of revolution at angle of attack. AIAA Journal, 27: 1008-1016. doi: 10.2514/3.10212