WU Jun-lin, LI Zhi-hui, PENG Ao-ping, JIANG Xin-yu. Calculation of Boltzmann-Rykov Model Equation by Finite Volume Method[J]. Applied Mathematics and Mechanics, 2014, 35(2): 121-129. doi: 10.3879/j.issn.1000-0887.2014.02.002
 Citation: WU Jun-lin, LI Zhi-hui, PENG Ao-ping, JIANG Xin-yu. Calculation of Boltzmann-Rykov Model Equation by Finite Volume Method[J]. Applied Mathematics and Mechanics, 2014, 35(2): 121-129.

# Calculation of Boltzmann-Rykov Model Equation by Finite Volume Method

##### doi: 10.3879/j.issn.1000-0887.2014.02.002
Funds:  The National Natural Science Foundation of China(91016027); The National Basic Research Program of China (973 Program)（2014CB744100）
• Rev Recd Date: 2013-12-09
• Publish Date: 2014-02-15
• A three order precision finite volume scheme was formulated to numerically solve the Boltzmann-Rykov model equation in which rotational energy was considered. This model equation was discretized into a series of equations at each discrete velocity point, and then a high order half-discretization finite volume scheme was used to compute these equations. Three order Runge-Kutta method was introduced for time marching, and central value in each cell was taken to approximate the average collision term. This finite volume scheme was of three order precision in convection term, while positive definiteness of the distribution functions and flux conservation were ensured. Results were compared with those of finite difference method and Riemann exact solution in continuum regime. The good coincidence shows validity of the solving process for the model equation by finite volume method.
•  [1] Bird G A. Approach to translational equilibrium in a rigid sphere gas[J]. Phys Fluids,1963,6: 1518-1519. [2] Filbet F, Russo G. High order numerical methods for the space non-homogeneous Boltzmann equation[J]. Journal of Computational Physics,2003,186(2): 457-480. [3] 李志辉, 吴振宇. 阿波罗指令舱稀薄气体动力学特征的蒙特卡罗数值模拟[J]. 空气动力学学报, 1996,14(2): 230-233.(LI Zhi-hui, WU Zhen-yu. Monte-Carlo numerical simulation of rarefied aerodynamic characters for Apollo spacecraft[J]. Acta Aerodynamica Sinica,1996,14(2): 230-233.（in Chinese）) [4] Mieussens L. Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics[J]. Mathematical Models and Methods in Applied Sciences,2000,10(8): 1121-1150. [5] Bird G A. Molecular Gas Dynamics [M]. Oxford: Clarendon Press, 1976. [6] 沈青. DSMC方法与稀薄气流计算的发展[J]. 力学进展, 1996,26(1): 1-13.（SHEN Qing. DSMC method and the calculation of rarefied gas flow[J]. Advances in Mechanics,1996,26(1): 1-13.（in Chinese）） [7] Bobylev A V, Cergignani C. Exact eternal solutions of the Boltzmann equation[J]. Journal of Statistical Physics,2002,106(5/6): 1019-1038. [8] Kolobov V I, Bayyuk S A. Construction of a unified continuum/kinetic solver for aerodynamic problems[J]. Journal of Spacecraft and Rockets,2005,42(4): 598-606. [9] Cheremisin F G, Agarwal R K. Computation of hypersonic shock structure in diatomic gases with rotational and vibrational relaxation using the generalized Boltzmann equation[C]// 46th AIAA Aerospace Sciences Meeting and Exhibit,2008. doi: 10.2514/6.2008-1269. [10] LI Zhi-hui, ZHANG Han-xin. Study on gas kinetic unified algorithm for flows from rarefied transition to continuum[J]. Journal of Computational Physics,2004,193(2): 708-738. [11] LI Zhi-hui, ZHANG Han-xin. Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation[J]. International Journal for Numerical Methods in Fluids,2003,42(4): 361-382. [12] ZHENG Ying-song, Struchtrup H. Ellipsoidal statistical Bhatnagar-Gross-Krook model with velocity-dependent collision frequency[J]. Physics of Fluids,2005,17(12): 103-127. [13] 李志辉, 张涵信. 稀薄流到连续流的气体运动论统一数值算法初步研究[J]. 空气动力学学报, 2000,18(3): 255-263.（LI Zhi-hui, ZHANG Han-xin. Study on gas kinetic algorithm for flows from rarefied transition to continuum[J]. Acta Aerodynamica Sinica,2000,18(3): 255-263.（in Chinese）） [14] Rykov V A. Model kinetic equation of a gas with rotational degrees of freedom[J]. Fluid Dynamics,1975,10(6): 959-966. [15] Rykov V A, Titarev V A, Shakhov E M. Shockwave structure in a diatomic gas based on a kinetic model[J]. Fluid Dynamics,2008,43(2): 316-326. [16] 张涵信, 沈孟育. 计算流体力学——差分方法的原理和应用[M]. 北京: 国防工业出版社, 2003.(ZHANG Han-xin, SHEN Meng-yu. Computational Fluid Dynamics: Principles of Differential Methods and Applications [M]. Beijing: National Defence Industry Press, 2003.(in Chinese))

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142