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Gas-kinetic unified algorithm for hypersonic aerothermodynamics covering various flow regimes solving Boltzmann model equation

Li Zhi-Hui Peng Ao-Ping Fang Fang Li Si-Xin Zhang Shun-Yu

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Gas-kinetic unified algorithm for hypersonic aerothermodynamics covering various flow regimes solving Boltzmann model equation

Li Zhi-Hui, Peng Ao-Ping, Fang Fang, Li Si-Xin, Zhang Shun-Yu
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  • 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.
      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).
    [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 物理学报 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 物理学报 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

  • [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 物理学报 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 物理学报 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

Metrics
  • Abstract views:  5145
  • PDF Downloads:  299
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Publishing process
  • Received Date:  11 May 2015
  • Accepted Date:  23 June 2015
  • Published Online:  05 November 2015

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