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Magnetohydrodynamic (MHD) heat shield system is a novel-concept thermal protection technique for hypersonic vehicles, which has been proved by lots of researchers with both numerical and experimental methods. Most of researchers neglect the Hall effect in their researches. However, in the hypersonic reentry process, the Hall effect is sometimes so significant that the electric current distribution in the shock layer can be changed by the induced electric field. Consequently, the Lorentz force as well as the Joule heat is varied, and thus the efficiency of the MHD heat shield system is affected.In order to analyze the influence of Hall effect, the induced electric field must be taken into consideration. According to the weakly-ionized characteristics of hypersonic flow post bow shock, the magneto-Reynolds number is assumed to be small. Therefore, the Maxwell equations are simplified with the generalized Ohm's law, and the induced electric field is governed by the potential Possion equation. Numerical methods are hence established to solve the Hall electric field equations in the thermochemical nonequilibrium flow field. The electric potential Poisson equation is of significant rigidity and difficult to solve for two reasons. One is that the coefficient matrix may not be diagonally dominant when the Hall parameter is large in the shock layer, and the other is that this matrix including the electric conductivity is discontinuous across the shock. In this paper, a virtual stepping factor is included to strengthen the diagonal dominance and improve the computational stability. Moreover, approximate factor and alternating direction implicit method are employed for further improving the stability. With these methods, a FORTRAN code is written and validated by comparing the numerical results with the analytical ones as well as results available from previous references. After that, relation between the convergence property and the virtual stepping factor is revealed by theoretical analysis and numerical simulations. Based on these work, a local variable stepping factor method is proposed to accelerate the iterating process. Results show that the convergence property is closely related to the mesh density and Hall parameter, and there exists a best stepping factor for a particular mesh as well as a particular Hall parameter. Since the best stepping factor varies a lot for different meshes and different Hall parameter, its appropriate value is hard to choose. The best value of stepping factor coefficient still exists in the local step factor method, but its value range is relatively smaller. More importantly, the local stepping factor method yields better convergence property than the regular constant one when employing a locally refined mesh.
[1] Zhu Y J, Jiang Y S, Hua H Q, Zhang C H, Xin C W 2014 Acta Phys. Sin. 63 244101 (in Chinese) [朱艳菊, 江月松, 华厚强, 张崇辉, 辛灿伟 2014 物理学报 63 244101]
[2] Yin J F, You Y X, Li W, Hu T Q 2014 Acta Phys. Sin. 63 044701 (in Chinese) [尹纪富, 尤云祥, 李巍, 胡天群 2014 物理学报 63 044701]
[3] Zhao G Y, Li Y H, Liang H, Hua W Z, Han M H 2015 Acta Phys. Sin. 64 015101 (in Chinese) [赵光银, 李应红, 梁华, 化为卓, 韩孟虎 2015 物理学报 64 015101]
[4] Yu H Y 2014 Acta Phys. Sin. 63 047502 (in Chinese) [于红云 2014 物理学报 63 047502]
[5] Bityurin V A, Bocharov A N 2014 52nd Aerospace Sciences Meeting National Harbor, Maryland, January 13-17, 2014
[6] Bisek N J, Gosse R, Poggie J 2013 J. Spacecraft Rockets 50 927
[7] Cristofolini A, Borghi C A, Neretti G, Battista F, Schettino A, Trifoni E, Filippis F D, Passaro A, Baccarella D 2012 18th AIAA/3AF International Space Planes and Hypersonic Systems and Technologies Conference Tours, France, September 24-28, 2012
[8] Lv H Y, Lee C H 2010 Chin. Sci. Bull. 55 1182 (in Chinese) [吕浩宇, 李椿萱 2010 科学通报 55 1182]
[9] Li K, Liu W Q 2016 Acta Phys. Sin. 65 064701 (in Chinese) [李开, 刘伟强 2016 物理学报 65 064701]
[10] Hu H Y, Yang Y J, Zhou W J 2011 Chin. J. Theo. App. Mechan. 43 453 (in Chinese) [胡海洋, 杨云军, 周伟江 2011 力学学报 43 453]
[11] Gaitonde D V, Poggie J 2002 40th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 14-17, 2002
[12] Wan T, Candler G V, Macheret S O, Shneider M N 2009 AIAA J. 47 1327
[13] Bisek N J 2010 Ph. D. Dissertation (Michigan: University of Michigan)
[14] Lv H Y, Lee C H, Dong H T 2009 Sci. Sin. Phys. Mechan. Astron. 39 435 (in Chinese) [吕浩宇, 李椿萱, 董海涛 2009 中国科学 G辑 39 435]
[15] Peng W, He Y G, Fang G F, Fan X T 2013 Acta Phys. Sin. 62 020301 (in Chinese) [彭武, 何怡刚, 方葛丰, 樊晓腾 2013 物理学报 62 020301]
[16] Fujino T, Matsumoto Y, Kasahara J, Ishikawa M 2000 Progress in Aerospace Sci. 36 1
[17] Zhang K P, Ding G H, Tian Z Y, Pan S, Li H 2009 J. National Univ. Defense Tech. 31 39 (in Chinese) [张康平, 丁国昊, 田正雨, 潘沙, 李桦 2009 国防科技大学学报 31 39]
[18] Tian Z Y, Zhang K P, Pan S, Li H 2008 Chin. Quar. Mechan. 29 72 (in Chinese) [田正雨, 张康平, 潘沙, 李桦 2008 力学季刊 29 72]
[19] Gnoffo P A, Gupta R N, Shinn J L 1989 NASA TP2867
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[1] Zhu Y J, Jiang Y S, Hua H Q, Zhang C H, Xin C W 2014 Acta Phys. Sin. 63 244101 (in Chinese) [朱艳菊, 江月松, 华厚强, 张崇辉, 辛灿伟 2014 物理学报 63 244101]
[2] Yin J F, You Y X, Li W, Hu T Q 2014 Acta Phys. Sin. 63 044701 (in Chinese) [尹纪富, 尤云祥, 李巍, 胡天群 2014 物理学报 63 044701]
[3] Zhao G Y, Li Y H, Liang H, Hua W Z, Han M H 2015 Acta Phys. Sin. 64 015101 (in Chinese) [赵光银, 李应红, 梁华, 化为卓, 韩孟虎 2015 物理学报 64 015101]
[4] Yu H Y 2014 Acta Phys. Sin. 63 047502 (in Chinese) [于红云 2014 物理学报 63 047502]
[5] Bityurin V A, Bocharov A N 2014 52nd Aerospace Sciences Meeting National Harbor, Maryland, January 13-17, 2014
[6] Bisek N J, Gosse R, Poggie J 2013 J. Spacecraft Rockets 50 927
[7] Cristofolini A, Borghi C A, Neretti G, Battista F, Schettino A, Trifoni E, Filippis F D, Passaro A, Baccarella D 2012 18th AIAA/3AF International Space Planes and Hypersonic Systems and Technologies Conference Tours, France, September 24-28, 2012
[8] Lv H Y, Lee C H 2010 Chin. Sci. Bull. 55 1182 (in Chinese) [吕浩宇, 李椿萱 2010 科学通报 55 1182]
[9] Li K, Liu W Q 2016 Acta Phys. Sin. 65 064701 (in Chinese) [李开, 刘伟强 2016 物理学报 65 064701]
[10] Hu H Y, Yang Y J, Zhou W J 2011 Chin. J. Theo. App. Mechan. 43 453 (in Chinese) [胡海洋, 杨云军, 周伟江 2011 力学学报 43 453]
[11] Gaitonde D V, Poggie J 2002 40th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 14-17, 2002
[12] Wan T, Candler G V, Macheret S O, Shneider M N 2009 AIAA J. 47 1327
[13] Bisek N J 2010 Ph. D. Dissertation (Michigan: University of Michigan)
[14] Lv H Y, Lee C H, Dong H T 2009 Sci. Sin. Phys. Mechan. Astron. 39 435 (in Chinese) [吕浩宇, 李椿萱, 董海涛 2009 中国科学 G辑 39 435]
[15] Peng W, He Y G, Fang G F, Fan X T 2013 Acta Phys. Sin. 62 020301 (in Chinese) [彭武, 何怡刚, 方葛丰, 樊晓腾 2013 物理学报 62 020301]
[16] Fujino T, Matsumoto Y, Kasahara J, Ishikawa M 2000 Progress in Aerospace Sci. 36 1
[17] Zhang K P, Ding G H, Tian Z Y, Pan S, Li H 2009 J. National Univ. Defense Tech. 31 39 (in Chinese) [张康平, 丁国昊, 田正雨, 潘沙, 李桦 2009 国防科技大学学报 31 39]
[18] Tian Z Y, Zhang K P, Pan S, Li H 2008 Chin. Quar. Mechan. 29 72 (in Chinese) [田正雨, 张康平, 潘沙, 李桦 2008 力学季刊 29 72]
[19] Gnoffo P A, Gupta R N, Shinn J L 1989 NASA TP2867
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