跨流域高超声速绕流环境Boltzmann模型方程统一算法研究

李志辉; 彭傲平; 方方; 李四新; 张顺玉

doi:10.7498/aps.64.224703

摘要

如何准确可靠地模拟从外层空间高稀薄流到近地面连续流的航天器高超声速绕流环境与复杂流动变化机理是流体物理的前沿基础科学问题. 基于对Boltzmann方程碰撞积分的物理分析与可计算建模, 确立了可描述自由分子流到连续流区各流域不同马赫数复杂流动输运现象统一的Boltzmann模型速度分布函数方程, 发展了适于高、低不同马赫数绕流问题的离散速度坐标法和直接求解分子速度分布函数演化更新的气体动理论数值格式, 建立了模拟复杂飞行器跨流域高超声速飞行热环境绕流问题的气体动理论统一算法. 对稀薄流到连续流不同Knudsen数0.002 Kn 1.618、不同马赫数下可重复使用卫星体再入过程(11070 km)中高超声速绕流问题进行算法验证分析, 计算结果与典型文献的Monte Carlo直接模拟值及相关理论分析符合得较好. 研究揭示了飞行器跨流域不同高度高超声速复杂流动机理、绕流现象与气动力/热变化规律, 提出了一个通过数值求解介观Boltzmann模型方程, 可靠模拟高稀薄自由分子流到连续流跨流域高超声速气动力/热绕流特性统一算法.

关键词:

Abstract

How to solve hypersonic aerothermodynamics and complex flow mechanism covering various flow regimes from high rarefied free-molecular flow of outer-layer space to continuum flow of near-ground is one of the frontier basic problems in the field of fluid physics. In this work, the unified Boltzmann model equation based on the molecular velocity distribution function is presented for describing complex hypersonic flow transport phenomena covering all flow regimes by physics analysis and model processing of the collision integral to the Boltzmann equation. The discrete velocity ordinate method is developed to simulate complex flows from low Mach numbers to hypersonic flight, and the gas-kinetic coupling-iteration numerical scheme is constructed directly to solve the evolution and updating of the molecular velocity distribution function by employing the unsteady time-splitting method and the NND finite-difference technique. Then, the gas-kinetic unified algorithm (GKUA) is presented to~simulate the three-dimensional hypersonic aerothermodynamics and flow problems around space vehicles covering various flow regimes from free-molecule to continuum. To verify the accuracy and reliability of the present GKUA and simulate gas thermodynamic transport phenomena covering various flow regimes, firstly, the two-dimensional supersonic flows around a circular cylinder are simulated in the continuum regime of Kn= 0.0001 and in the high rarefied regime of Kn= 0.3 through the comparison between the Navier-Stokes (N-S) solution and the direct simulation Monte Carlo (DSMC) result, respectively. It is indicated that the GKUA can exactly converge to the N-S solution in the continuum flow regime, and the computed results of the GKUA are consistent with the DSMC simulation with a small deviation of 0.45% in the high rarefied flow regime. Then, the three-dimensional complex hypersonic flows around reusable satellite shape are studied as one of the engineering applications of the GKUA with a wide range 0.002 Kn 1.618 of the free-stream Knudsen numbers and different Mach numbers during re-entering Earth atmosphere with the flying altitudes of 110-70~km. The computed results are found to be in high resolution of the flow fields and in good agreement in a deviation range of 0.27%-8.56% by comparison among the relevant reference data, DSMC and theoretical predictions. The complex flow mechanism, flow phenomena and changing laws of hypersonic aerothermodynamics are revealed for spacecraft re-entry into the atmosphere, and the effects of rarefied gas and wall temperature on the aerothermodynamics characteristics of re-entry satellite shape are compared and analysed with different Knudsen numbers and wall temperature ratios of Tw/T = 1.6, 10 and 15.6. It is validated that the non-dimensional heat flux coefficient in the rarefied transitional flow regime is higher than that of the continuum and near-continuum flow regimes, the high wall temperature results in the enlarging amplitude of temperature variation on the stagnation line and the serious effect on the heat flux of the stagnation point, and wall temperature becomes lower, the heat flux coefficient of wall surface becomes bigger, and the friction force and pressure coefficients decrease. The non-equilibrium level of flow velocity slip and temperature jump on the surface of space vehicle becomes severer, and the stronger heat transfer effect between the space vehicle and the gas flow is produced as the Mach number or Knudsen number of the free-stream flow increases. It can be realized from this study that the gas-kinetic unified algorithm directly solving the Boltzmann model velocity distribution function equation may provide an important and feasible way that complex hypersonic aerothermodynamic problems and flow mechanisms from high rarefied free-molecule to continuum flow regimes can be solved effectively and reliably.

Keywords:

Boltzmann model equation /
discrete velocity ordinate method /
gas-kinetic unified algorithm covering various flow regimes /
hypersonic flows

作者及机构信息

1.
中国空气动力研究与发展中心超高速空气动力研究所, 绵阳 621000;

2.
中国空气动力研究与发展中心, 空气动力学国家重点实验室, 绵阳 621000;

3.
中国空间技术研究院, 载人航天总体部, 北京 100094

通信作者: 李志辉, zhli0097@x263.net

基金项目: 国家重点基础研究发展计划(批准号: 2014CB744100)、国家自然科学基金(批准号: 11325212, 91016027)和国防基础科研基金(批准号: 51313030104)资助的课题.

Authors and contacts

1.
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China;

2.
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China;

3.
Institute of Manned Space System Engineering, China Academy of Space Technology, Beijing 100094, China

Corresponding author: Li Zhi-Hui, zhli0097@x263.net

Funds: Project supported by the National Basic Research Program of China (Grant No. 2014CB744100), the National Natural Science Foundation of China (Grant Nos. 11325212, 91016027), and the National Defense Basic Scientific Research Program of China (Grant No. 51313030104).

参考文献

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[51]	Li Z H, Zhang H X 2010 Acta Aerodynam. Sin. 28 7 (in Chinese) [李志辉, 张涵信 2010 空气动力学学报 28 7]
[52]	Li Z H, Zhang H X, Fu S 2005 Sci. China: Phys. Mech. Astron. 48 496
[53]	Li Z H, Zhang H X 2009 J. Comput. Phys. 228 1116
[54]	Li Z H, Zhang H X 2007 Acta Mechan. Sin. 23 121
[55]	Li Z H, Peng A P, Zhang H X, Yang J Y 2015 Prog. Aerospace Sci. 74 81
[56]	Xu K, Huang J C 2010 J. Comput. Phys. 229 7747
[57]	Chen S Z, Xu K, Lee C B, Cai Q D 2012 J. Comput. Phys. 231 6643
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[61]	Wu L, White C, Scanlon T J, Reese J M, Zhang Y H 2013 J. Comput. Phys. 250 27
[62]	Jin S, Li Q 2013 Numerical Methods for Partial Differential Equations 29 1056
[63]	Wu L, Reese J M, Zhang Y H 2014 J. Fluid Mech. 746 53
[64]	Li Z H, Zhang H X 2008 Int. J. Comput. Fluid Dynam. 22 623
[65]	Zhang H X, Zhuang F G 1992 Adv. Appl. Mech. 29 193
[66]	Sharipov F 2003 Brazilian J. Phys. 33 398

施引文献

[1]	Tsien H S 1946 J. Aeronaut. Sci. 13 653
[2]	Chapmann S, Cowling T G 1970 The Mathematical Theory of Non-uniform Gases (3rd Ed.) (Cambridge: Cambridge University Press) p62
[3]	Bertin J J, Cummings R M 2003 Prog. Aerospace Sci. 39 511
[4]	Frantziskonis G, Muralidharan K 2009 J. Comput. Phys. 228 8085
[5]	D'Souza S N, Sarigul-Klijn N 2014 Prog. Aerospace Sci. 68 64
[6]	Bird G A 1963 Phys. Fluids 6 1518
[7]	Pham-Van Diep G, Erwin D, Muntz E P 1989 Science 245 624
[8]	Haas B L, Boyd L D 1993 Phys. Fluids A 5 478
[9]	Bird G A 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows (London: Oxford University Press) p195
[10]	Koppenwallner G, Legge H 1986 Progress in Astronautics and Aeronautics: Thermophysical Aspects of Reentry Flows. AIAA Paper 85-0998
[11]	Li Z H, Wu Z Y 1996 Acta Aerodynam. Sin. 14 230 (in Chinese) [李志辉, 吴振宇 1996 空气动力学学报 14 230]
[12]	Ivanov M S, Vashchenkov P, Kashkovsky A 2007 Numerical Investigation of the EXPERT Reentry Vehicle Aerothermodynamics along the Descent Trajectory AIAA 2007-4145
[13]	Li Z H, Fang M, Jiang X Y, Wu J L 2013 Sci. China: Phys. Mech. Astron. 56 404
[14]	Kostoff R N, Cummings R M 2013 Aerospace Sci. Technol. 26 216
[15]	Cercignani C 1988 The Boltzmann Equation and its Applications (Berlin: Springer Verlag) p192
[16]	Whitehead Jr A 1989 NASP Aerodynamics AIAA Paper 89-5013
[17]	Kirk B S, Stogner R H, Bauman P T, Oliver T A 2014 Computers Fluids 92 281
[18]	Wang C S (translated by Ying C T, Zhang C Z) 1994 The Kinetic Theory of a Gas (Beijing: Atom Energy Press) pp71-222 (in Chinese) [王承书著 (应纯同, 张存镇译) 1994气体运动论论文选集 (北京: 原子能出版社) 第71222页]
[19]	Peng H W, Xu X S 1998 The Fundamentals of Theoretical Physics (The Series of Advanced Physics of Peking University) (Beijing: Peking University Press) pp143-255 (in Chinese) [彭恒武, 徐锡申 1998 理论物理基础, 北京大学物理学丛书(教材) (北京: 北京大学出版社) 第143255页]
[20]	Bhatnagar P L, Gross E P, Krook M 1954 Phys. Rev. 94 511
[21]	Holway Jr. L H 1966 Phys. Fluids 9 1658
[22]	Shakhov E M 1968 Fluid Dyn. 3 158
[23]	Abe T, Oguchi H 1977 Progress in Astronautics and Aeronautics (Vol. 51) (NewYork: AIAA) pp781-793
[24]	Pullin D I 1980 J. Comput. Phys. 34 231
[25]	Macrossan M N 1989 J. Comput. Phys. 80 204
[26]	Prendergast K H, Xu K 1993 J. Comput. Phys. 109 53
[27]	Xu K 2001 J. Comput. Phys. 171 289
[28]	Xu K, Li Z H 2004 J. Fluid Mech. 513 87
[29]	Frisch U, Hasslacher B, Pomeau Y 1986 Phys. Rev. Lett. 56 1505
[30]	Qian Y H, Succi S, Orszag S 1995 Annu. Rev. Compt. Phys. 3 195
[31]	Chen S, Doolen G D 1998 Annu. Rev. Fluid Mech. 30 329
[32]	Ran Z 2009 Chin. Phys. B 18 2159
[33]	Zhong C W, Xie J F, Zhuo C S, Xiong S W, Yin D C 2009 Chin. Phys. B 18 4083
[34]	Chen F, Xu A G, Zhang G C 2011 Commun. Theor. Phys. 55 325
[35]	Xie H Q, Zeng Z, Zhang L Q, Liang G Y, Hiroshi M, Yoshiyuki K 2012 Chin. Phys. B 21 124703
[36]	Chen L, He Y L, Kang Q J, Tao W Q 2013 J. Comput. Phys. 255 83
[37]	Xie W J, Teng P F 2014 Acta Phys. Sin. 63 164301 (in Chinese) [解文军, 腾鹏飞 2014 63 164301]
[38]	Liu F F, Wei S S, Wei C Z, Ren X F 2014 Acta Phys. Sin. 63 194704 (in Chinese) [刘飞飞, 魏守水, 魏长智, 任晓飞 2014 63 194704]
[39]	Vahid E, Esmaeil D, Amir M D-S 2014 Chin. Phys. B 23 084702
[40]	Yang J Y, Huang J C 1995 J. Comput. Phys. 120 323
[41]	Shakhov E M 1984 Proceedings of 14th International Symposium on Rarefied Gas Dynamics Tsukuba, Japan, 1984 pp137-148
[42]	Aoki K, Kanba K, Takata S 1997 Phys. Fluids. 9 1144
[43]	Mieussens L 2000 J. Comput. Phys. 162 429
[44]	Li Z H, Zhang H X 2000 Proc. of 22nd International Symposium on Rarefied Gas Dynamics Sydney, Australia, July 9-14, 2000 pp628-636
[45]	Li Z H 2001 Ph. D. Dissertation (Mianyang: China Aerodynamics Research and Development Center) (in Chinese) [李志辉 2001 博士学位论文(绵阳: 中国空气动力研究与发展中心)]
[46]	Li Z H, Zhang H X 2003 Int. J. Numer. Meth. Fluids 42 361
[47]	Li Z H, Zhang H X 2004 J. Comput. Phys. 193 708
[48]	Li Z H, Zhang H X 2005 Adv. Mech. 35 557 (in Chinese) [李志辉, 张涵信 2005力学进展 35 557]
[49]	Zhang H X, Shen M Y 2003 Computational Fluid Dynamics-Fundamentals and Applications of Finite Difference Methods (Beijing: National Defence Industry Press) p240 (in Chinese) [张涵信, 沈孟育 2003 计算流体力学-差分方法的原理和应用 (北京: 国防工业出版社) 第240页]
[50]	Li Z H, Zhang H X 2008 Chin. J. Comput. Phys. 25 65 (in Chinese) [李志辉, 张涵信 2008 计算物理 25 65]
[51]	Li Z H, Zhang H X 2010 Acta Aerodynam. Sin. 28 7 (in Chinese) [李志辉, 张涵信 2010 空气动力学学报 28 7]
[52]	Li Z H, Zhang H X, Fu S 2005 Sci. China: Phys. Mech. Astron. 48 496
[53]	Li Z H, Zhang H X 2009 J. Comput. Phys. 228 1116
[54]	Li Z H, Zhang H X 2007 Acta Mechan. Sin. 23 121
[55]	Li Z H, Peng A P, Zhang H X, Yang J Y 2015 Prog. Aerospace Sci. 74 81
[56]	Xu K, Huang J C 2010 J. Comput. Phys. 229 7747
[57]	Chen S Z, Xu K, Lee C B, Cai Q D 2012 J. Comput. Phys. 231 6643
[58]	Guo Z L, Xu K, Wang R J 2013 Phys. Rev. E 88 033305
[59]	Bobylev A V, Rjasanow S 1999 Eur. J. Mech. B 18 869
[60]	Pareschi L, Russo G 2000 SIAM J. Numer. Anal. 37 1217
[61]	Wu L, White C, Scanlon T J, Reese J M, Zhang Y H 2013 J. Comput. Phys. 250 27
[62]	Jin S, Li Q 2013 Numerical Methods for Partial Differential Equations 29 1056
[63]	Wu L, Reese J M, Zhang Y H 2014 J. Fluid Mech. 746 53
[64]	Li Z H, Zhang H X 2008 Int. J. Comput. Fluid Dynam. 22 623
[65]	Zhang H X, Zhuang F G 1992 Adv. Appl. Mech. 29 193
[66]	Sharipov F 2003 Brazilian J. Phys. 33 398

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[20]	周义昌, 余超凡. 一维线性链的亚声速和超声速孤立子. , 1992, 41(12): 2016-2023. doi: 10.7498/aps.41.2016

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