Validation and analysis of gas-kinetic unified algorithm for solving Boltzmann model equation with vibrational energy excitation

Peng Ao-Ping; Li Zhi-Hui; Wu Jun-Lin; Jiang Xin-Yu

doi:10.7498/aps.66.204703

Abstract

With the increase of temperature in flow field,gas molecules possess not only rotational degree of freedom,but also vibrational energy excitation.In order to simulate and study the influence of internal energy excitation on polyatomic gas flow with high temperature and high Mach number,according to the general Boltzmann equation,we consider the rotational and vibrational energy modes as the independent variables of gas molecular velocity distribution function.It is assumed that the rotational and vibrational energy modes are described by continuous distribution with degree of freedom and temperature.Based on the Borgnakke-Larsen collision model used in direct simulation Monte Carlo (DSMC) method, the collision term of Boltzmann equation with internal energy excitation is divided into elastic and inelastic collision terms.The inelastic collision is decomposed into translational-rotational energy relaxation and translational-rotationalvibrational energy relaxation according to a certain relaxation rate obtained from the reciprocalities of rotational and vibrational collisions numbers per one elastic collision.Then a kind of Boltzmann model equation considering the excitation of vibrational energy is constructed.For showing the consistency between the present model equation and Boltzmann equation,the conservation of summational invariants and the H-theorem of this model are proved.When solving the present model equation with numerical methods,because of the continuous energy modes,it is difficult to simulate this model equation directly.In this paper,three control equations are derived and solved by the LU-SGS (lower-upper symmetric Gauss-Seidel) method,and the cell-centered finite volume method with multi-block patched grid technique in physical space.As a result,these gas-kinetic unified algorithm (GKUA) with vibrational energy excitation has been developed.Results are presented for N2 with different Knudsen numbers around cylinder from continuum to rarefied gas flow by using the present Boltzmann model equation,GKUA with simple gas model,and DSMC method. Very good agreement between the present model and DSMC results is obtained,which shows that the accuracy and reliability of the present model.Comparing the translational,rotational,vibrational,and total temperatures computed by different methods,the effects of the rotational and vibrational degrees of freedom are demonstrated.For the simple gas model,the translational temperature is much higher than those for the other two models with internal energy excitation. At the same time,the distance from shock wave to wall for the simple gas model is about twice those for the other two models.On the other hand,the obtained aerodynamic force coefficients of the cylinder are increasing according to the sequence from the simple gas model to the rotational energy excitation model to the vibrational energy excitation model, but the variation range is very small.By reducing the gas characteristic vibrational temperature,the temperature after the shock wave is much lower,and the heat flux declines evidently at the stagnation point with the same temperature as the wall temperature.This implies that with the wall temperature increasing the heat flux declines.

Keywords:

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.
National Laboratory for Computational Fluid Dynamics, Beijing 100191, China

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

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

References

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[33]	Li Z, Zhu T, Levin D A 2013 AIAA Paper AIAA 2013-1201
[34]	Wang C S, Uhlenbeck G E, Boer J D 1964 Studies in Statistical Mechanics (Amsterdam:North-Holland Publishing Company) p2
[35]	Wang C S (translated by Ying C T, Zhang C Z) 1994 The Kinetic Theory of a Gas (Beijing:Atom Energy Press) pp71-75(in Chinese)[王承书著(应纯同, 张存镇译) 1994气体分子运动论(北京:原子出版社)第7175页]
[36]	Morse T F 1964 Phys. Fluids 7 2012
[37]	Andries P, Le Tallec P, Perlat J P, Perthame B 2000 Eur. J. Mech. B:Fluid 19 813
[38]	Brull S, Schneider J 2009 Continuum Mech. Thermodyn. 20 489
[39]	Rykov V A 1975 Fluid Dynam.+ 10 959
[40]	Rykov V A, Titarev V A, Shakhov E M 2008 Fluid Dynam.+ 43 316
[41]	Rykov V A, Titarev V A, Shakhov E M 2007 Comp. Math. Math. Phys.+ 47 136
[42]	Wu L, White C, Thomas J S, Reese J M, Zhang Y H 2015 J. Fluid Mech. 763 24
[43]	Tantos C, Valougeorgis D, Frezzotti A 2015 Int. J. Heat Mass Trans. 88 636
[44]	Tantos C, Ghiroldi G P, Valougeorgis D, Frezzotti A 2016 Int. J. Heat Mass Trans. 102 162
[45]	Allu P, Mazumder S 2016 Int. J. Heat Mass Tran. 100 165
[46]	Li Z H, Jiang X Y, Wu J L, Peng A P 2014 Chin. J. Theor. Appl. Mech. 46 336 (in Chinese)[李志辉, 蒋新宇, 吴俊林, 彭傲平2014力学学报46 336]
[47]	Wu J L, Peng A P, Li Z H, Fang M 2015 Acta Aerodynam. Sin. 33 5(in Chinese)[吴俊林, 彭傲平, 李志辉, 方明2015空气动力学学报33 5]
[48]	Jiang X Y, Li Z H, Wu J L 2014 Chin. J. Comput. Phys. 31 403(in Chinese)[蒋新宇, 李志辉, 吴俊林2014计算物理31 403]
[49]	Xu A G, Zhang G C, Li Y J, Li H 2014 Prog. Phys. Sin. 34 136(in Chinese)[许爱国, 张广财, 李英骏, 李华2014物理学进展34 136]
[50]	Ying C T 1990 Theory and Application of Gases Transport (Beijing:Tsinghua University Press) p62(in Chinese)[应纯同1990气体输运理论及应用(北京:清华大学出版社)第62页]
[51]	Laurent B, Brnice G, Milana P C, Francesco S 2013 Proc. Appl. Math. Mech. 13 353
[52]	Bird G A 2005 Proceedings of the 24th International Symposium on Rarefied Gas Dynamics Melville, Canada 2005 p541

Cited By

[1]	Votta R, Schettino A, Bonfiglioli A 2013 Aerosp. Sci. Technol. 25 253
[2]	Shevyrin A A, Vashchenkov P V, Bondar Y A, Ivanov M S 2014 Proceedings of the 29th International Symposium on Rarefied Gas Dynamics Xi' an, China, July 13-18, 2014 p155
[3]	Bird G A 1994 Molecular Gas Dynamics and The Direct Simulation of Gas Flows (Oxford:Oxford University Press) pp50-54
[4]	Shen Q 2003 Rarefied Gas Dynamics (Beijing:National Defense Industry Press) pp38, 83-88(in Chinese)[沈青2003稀薄气体动力学(北京:国防工业出版社)第38, 8388页]
[5]	Struchtrup H 2005 Macroscopic Transport Equations for Rarefied Gas Flows (Berlin:Springer) p27
[6]	Chapman S, Cowling T G 1970 The Mathematical Theory of Non-uniform Gases (Cambridge:Cambridge University Press) pp46-48
[7]	Cercignani C 1988 The Boltzmann Equation and Its Applications (New York:Springer Science Business Media) pp64-66
[8]	Kremer G M 2010 An Introduction to the Boltzmann Equation and Transport Processes in Gases (Berlin:Springer) p37
[9]	Bhatnagar P L, Gross E P, Krook M 1954 Phys. Rev. 94 511
[10]	Holway L H 1966 Phys. Fluids 9 1658
[11]	Shakhov E M 1968 Fluid Dynam. 3 95
[12]	Yang J Y, Huang J C 1995 J. Comput. Phys. 120 232
[13]	Li Z H, Zhang H X 2002 Acta Mech. Sin. 34 145 (in Chinese)[李志辉, 张涵信2002力学学报34 145]
[14]	Olga I R, Alexey P P, Irina A G 2013 Comput. Fluids 80 71
[15]	Titarev V, Dumbser M, Utyuzhnikov S 2014 J. Comput. Phys. 256 17
[16]	Li Z H, Peng A P, Fang F, Li S X, Zhang S Y 2015 Acta Phys. Sin. 64 224703 (in Chinese)[李志辉, 彭傲平, 方方, 李四新, 张顺玉2015 64 224703]
[17]	Xu K, Huang J C 2010 J. Comput. Phys. 229 7747
[18]	Cai Z N, Li R 2014 J. Comput. Phys. 267 63
[19]	Li Z H, Peng A P, Zhang H X, Yang J Y 2015 Prog. Aerosp. Sci. 74 81
[20]	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)]
[21]	Li Z H, Zhang H X 2004 J. Comput. Phys. 193 708
[22]	Li Z H, Zhang H X 2005 Adv. Mech. Sin. 35 559 (in Chinese)[李志辉, 张涵信2005力学进展35 559]
[23]	Li Z H, Zhang H X 2007 Acta Mech. Sin.:PRC 23 121
[24]	Li Z H, Zhang H X 2009 J. Comput. Phys. 228 1116
[25]	Li Z H, Peng A P, Zhang H X, Deng X G 2011 Sci. Sin.:Phys. Mech. Astron. 54 1687
[26]	Peng A P, Li Z H, Wu J L, Jiang X Y 2016 Chin. J. Theor. Appl. Mech. 48 95 (in Chinese)[彭傲平, 李志辉, 吴俊林, 蒋新宇2016力学学报48 95]
[27]	Peng A P, Li Z H, Wu J L, Jiang X Y 2016 J. Comput. Phys. 327 919
[28]	Li Z H, Wu J L, Jiang X Y, Ma Q 2015 Acta Aeronaut. Astron. Sin. 36 201 (in Chinese)[李志辉, 吴俊林, 蒋新宇, 马强2015航空学报36 201]
[29]	Li H Y 2007 Ph. D. Dissertation (Mianyang:China Aerodynamics Research and Development Center) (in Chinese)[李海燕2007博士学位论文(绵阳:中国空气动力研究与发展中心)]
[30]	Li H Y, Li Z H, Luo W Q, Li M 2014 Sci. Sin.:Phys. Mech. Astron. 44 194(in Chinese)[李海燕, 李志辉, 罗万清, 李明2014中国科学:物理学力学天文学44 194]
[31]	Boyd I D, Josyula E 2011 Phys. Fluids 23 057101
[32]	Yang H S 2013 M. S. Thesis (Shanghai:Shanghai Jiaotong University) (in Chinese)[杨浩森2013硕士学位论文(上海:上海交通大学)]
[33]	Li Z, Zhu T, Levin D A 2013 AIAA Paper AIAA 2013-1201
[34]	Wang C S, Uhlenbeck G E, Boer J D 1964 Studies in Statistical Mechanics (Amsterdam:North-Holland Publishing Company) p2
[35]	Wang C S (translated by Ying C T, Zhang C Z) 1994 The Kinetic Theory of a Gas (Beijing:Atom Energy Press) pp71-75(in Chinese)[王承书著(应纯同, 张存镇译) 1994气体分子运动论(北京:原子出版社)第7175页]
[36]	Morse T F 1964 Phys. Fluids 7 2012
[37]	Andries P, Le Tallec P, Perlat J P, Perthame B 2000 Eur. J. Mech. B:Fluid 19 813
[38]	Brull S, Schneider J 2009 Continuum Mech. Thermodyn. 20 489
[39]	Rykov V A 1975 Fluid Dynam.+ 10 959
[40]	Rykov V A, Titarev V A, Shakhov E M 2008 Fluid Dynam.+ 43 316
[41]	Rykov V A, Titarev V A, Shakhov E M 2007 Comp. Math. Math. Phys.+ 47 136
[42]	Wu L, White C, Thomas J S, Reese J M, Zhang Y H 2015 J. Fluid Mech. 763 24
[43]	Tantos C, Valougeorgis D, Frezzotti A 2015 Int. J. Heat Mass Trans. 88 636
[44]	Tantos C, Ghiroldi G P, Valougeorgis D, Frezzotti A 2016 Int. J. Heat Mass Trans. 102 162
[45]	Allu P, Mazumder S 2016 Int. J. Heat Mass Tran. 100 165
[46]	Li Z H, Jiang X Y, Wu J L, Peng A P 2014 Chin. J. Theor. Appl. Mech. 46 336 (in Chinese)[李志辉, 蒋新宇, 吴俊林, 彭傲平2014力学学报46 336]
[47]	Wu J L, Peng A P, Li Z H, Fang M 2015 Acta Aerodynam. Sin. 33 5(in Chinese)[吴俊林, 彭傲平, 李志辉, 方明2015空气动力学学报33 5]
[48]	Jiang X Y, Li Z H, Wu J L 2014 Chin. J. Comput. Phys. 31 403(in Chinese)[蒋新宇, 李志辉, 吴俊林2014计算物理31 403]
[49]	Xu A G, Zhang G C, Li Y J, Li H 2014 Prog. Phys. Sin. 34 136(in Chinese)[许爱国, 张广财, 李英骏, 李华2014物理学进展34 136]
[50]	Ying C T 1990 Theory and Application of Gases Transport (Beijing:Tsinghua University Press) p62(in Chinese)[应纯同1990气体输运理论及应用(北京:清华大学出版社)第62页]
[51]	Laurent B, Brnice G, Milana P C, Francesco S 2013 Proc. Appl. Math. Mech. 13 353
[52]	Bird G A 2005 Proceedings of the 24th International Symposium on Rarefied Gas Dynamics Melville, Canada 2005 p541

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