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Aming at the applicability of low magnetic Reynolds number method, in this paper we analyze the differences in the application of low magnetic Reynolds number condition and the limitation of full MHD method when it is applied to hypersonic flow. According to the low magnetic Reynolds number magneto-hydrodynamic control numerical simulation method, computing magnetic vector potential through integrating induced current, and considering the reduction of computation domain caused by truncation factors, we propose a low magnetic Reynolds number MHD computation method which is adjusted by the induced magnetic field, and the validation of this method is also presented. Through the numerical simulation of RAM-C blunt cone in flight test condition, we analyze the discrepancy caused by “neglecting induced magnetic field”, and also discuss the principle of the application of low magnetic Reynolds number assumption of hypersonic flow. The obtained results are as follows. (1) The adjusted computation method developed in this paper breaks through the limit of low magnetic Reynolds number, and expands the application range of low magnetic Reynolds number method to hypersonic flow, the numerical simulation result is reliable; Compared with direct integration of Biot-Savart law, the computation efficiency is considerably improved. (2) In the hypersonic flow, the influence of induced magnetic field is presented, thus weakening and distorting the applied magnetic field macroscopically, as a result weakening the effect of magnetic control to some extent. Under the condition of this paper, the low magnetic Reynolds number condition “Rem < 0.1” is probably too conservative, and it is better to adopt Rem < 1.0, and the characteristic conductivity and characteristic length should be chosen according to the actual plasma distribution.
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Keywords:
- magnetohydrodynamic /
- low magnetic Reynolds number condition /
- plasma /
- numerical simulation
[1] 田正雨 2008 博士学位论文 (长沙: 国防科学技术大学)
Tian Z Y 2008 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)
[2] 潘勇 2007 博士学位论文(南京: 南京航空航天大学)
PanY 2007 Ph. D. Dissertation (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[3] 张向洪 2014 博士学位论文 (南京: 南京航空航天大学)
Zhang X H 2014 Ph. D. Dissertation (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[4] Palmer G 1993 J. Thermophys Heat Transfer 7 294
Google Scholar
[5] Barmin A A, Kulikovskiy A G 1996 J. Comput. Phys. 126 77
Google Scholar
[6] Nagata Y, Otsu H, Yamada K 2012 43rd AIAA Plasmadynamics and Lasers Conference NewOrleans, Louisiana, USA, June 25−8, 2012 p2734
[7] Otsu H 2005 36th AIAA Plasmadynamics and Lasers Conference Toronto, Ontario, Canada, June 6−9, 2005 p5049
[8] Fujino T, Ishikawa M 2013 44th AIAA Plasmadynamics and Lasers Conference SanDiego, California, USA, June 24−27, 2013 p3000
[9] Takahashi T, Shimosawa Y, Masuda K, Fujino T 2015 46th AIAA Plasma dynamics and Lasers Conference Dallas, Texas, USA, June 22−26, 2015 p3365
[10] Bisek N J, Poggie J 2011 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition Orlando, Florida, USA, January 4−7, 2011 p897
[11] Bisek N J, Boyd I D 2010 J. Spacecraft Rockets 47 816
Google Scholar
[12] Lee J, Huerta M, Zha G 2010 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition Orlando, Florida, USA, January 4−7, 2010 p229
[13] Cristofolini A, Borghi C A, Neretti G 2012 18th AIAA/3 AF International Space Planes and Hypersonic Systems and Technologies Conference Tours, France, September 24−28, 2012 p5804
[14] Cristofolini A, Borghi C A, Neretti G 2012 43rd AIAA Plasmadynamics and Lasers Conference New Orleans, Louisiana, USA June 25−28, 2012 p2730
[15] 李开, 柳军, 刘伟强 2017 物理学报 66 084702
Google Scholar
Li K, Liu J, Liu W Q 2017 Acta Phys. Sin. 66 084702
Google Scholar
[16] 李开, 刘伟强 2016 物理学报 65 064701
Google Scholar
Li K, Liu W Q 2016 Acta Phys. Sin. 65 064701
Google Scholar
[17] 姚霄, 刘伟强, 谭建国 2018 物理学报 67 174702
Google Scholar
Yao X, Liu W Q, Tan J G 2018 Acta Phys. Sin. 67 174702
Google Scholar
[18] 何淼生, 杨文将, 郑小梅, 刘宇 2013 航空动力学报 28 365
He M S, Yang W J, Zheng X M, Liu Y 2013 J. Aerosp. Power 28 365
[19] 陈刚, 张劲柏, 李椿萱 2010 北京航空航天大学学报 36 135
Chen G, Zhang J B, Li C X 2010 J. BUAA 36 135
[20] 孙晓辉 2013 博士学位论文(南京: 南京理工大学)
Sun X H 2008 Ph. D. Dissertation (Nanjing: Nanjing University of Science and Technology) (in Chinese)
[21] Poggie J, Gaitonde D 2002 Phys. Fluids 14 1720
Google Scholar
[22] Fujino T, Kondo S, Ishikawa M 2007 38th AIAA Plasmadynamics and Lasers Conference Miami, Florida, USA, June 25−28, 2007 p4248
[23] Khan O U, Hoffmann K A, Dietiker J F 2007 38th AIAA Plasmadynamics and Lasers Conference Miami, Florida, USA June 25−28, 2007 p4374
[24] MacCormack R W 2008 46th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, USA, January 7−10, 2008 p1070
[25] Kim M K 2009 Ph. D. Dissertation (Michigan: University of Michigan)
[26] Bocharov A N 2010 High Temp. 48 483
[27] Yoshino T, Fujino T, Ishikawa M 2007 38th AIAA Plasmadynamics and Lasers Conference Miami, Florida, USA, June 25−28, 2007 p4249
[28] 李开, 刘伟强 2016 国防科技大学学报 38 25
Google Scholar
Li K, Liu W Q 2016 J. NUDT 38 25
Google Scholar
[29] 郭硕鸿 1997 电动力学 (北京: 高等教育出版社) 第1−38页
Guo S H 1997 Electrodynamics (Beijing: Higher Education Press) pp1−38 (in Chinese)
[30] MacCormack R 2005 43rd AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, USA, January 10−13, 2005 p0559
[31] Khan O U, Hoffmann K A, Dietiker J F 2006 44th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, USA, January 9−12, 2006 p966
[32] 黄富来, 黄护林 2009 航空学报 30 1834
Google Scholar
Huang F L, Huang H L 2009 Acta Aeronaut. Astronaut. Sin. 30 1834
Google Scholar
[33] Maccormack R W 2005 36th AIAA Plasmadynamics and Lasers Conference Toronto, Ontario, Canada, June 6−9, 2005 p4780
[34] 丁明松, 江涛, 董维中, 高铁锁, 刘庆宗 2017 航空学报 38 121030
Ding M S, Jiang T, Dong W Z, Gao T S, Liu Q Z 2017 Acta Aeronaut. Astronaut. Sin. 38 121030
[35] 丁明松, 江涛, 刘庆宗, 董维中, 高铁锁 2019 航空学报 40 123009
Ding M S, Jiang T, Liu Q Z, Dong W Z, Gao T S 2019 Acta Aeronaut. Astronaut. Sin. 40 123009
[36] 丁明松, 江涛, 董维中, 高铁锁, 刘庆宗 2019 物理学报 68 174702
Google Scholar
Ding M S, Jiang T, Dong W Z, Gao T S, Liu Q Z 2019 Acta Phys. Sin. 68 174702
Google Scholar
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图 1 流场中电导率和环形感应电流分布[34] (a)电导率; (b)电流密度
Figure 1. Distribution of electronic conductivity and annular electric current: (a) Conductivity; (b) current.
图 2 不同位置的磁感应强度 (a)
$r = 1.0\; {\rm{m}}$ ; (b)$x = 50.0 \;{\rm{m}}$ Figure 2. Magnetic induction intensity in different locations: (a)
$r = 1.0 \;{\rm{m}}$ ; (b)$x = 50.0 \;{\rm{m}}$ 
图 6 采用电导率模型M6计算的流场电导率分布 (a)全场云图; (b)驻点线参数分布
Figure 6. Distribution of electronic conductivity using M6: (a) Full contour map; (b) parameters along stagnation line.
图 7 修正方法计算的钝锥RAM-C感应磁场 (a)
${B_x}$ 分量; (b)${B_y}$ 分量Figure 7. Induced magnetic field of RAM-C using improved method: (a) Component
${B_x}$ ; (b) component${B_y}$ .
图 8 钝锥RAM-C外加磁场和修正方法计算的全磁场分布 (a)全磁场
${B_x}$ 分量; (b)全磁场${B_y}$ 分量; (c)外加磁场${B_x}$ 分量; (b)外加磁场${B_y}$ 分量Figure 8. Total magnetic field computed using improved method and externally applied magnetic field of RAM-C: (a) Total
${B_x}$ ; (b) total${B_y}$ ; (c) externally${B_x}$ ; (c) externally${B_y}$ .
图 9 修正方法和一般低磁雷诺数MHD方法计算得到的热流分布比较
Figure 9. Heat flux computed using Low Rem method or improvbed method.
图 10 采用不同修正方法计算得到的热流和残差收敛曲线 (a)热流; (b)残差
Figure 10. Heat flux and residual error computed using different modified method: (a) Heat flux; (b) residual error.
表 1 钝锥RAM-C阻力系数
Table 1. Drag coefficient of RAM-C.
计算方法或条件 总阻力系数 增大比例 No Mag. 0.292 — 一般低$R{e_{\rm{m}}}$方法 0.991 239% 修正方法 0.980 234% 
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[1] 田正雨 2008 博士学位论文 (长沙: 国防科学技术大学)
Tian Z Y 2008 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese)
[2] 潘勇 2007 博士学位论文(南京: 南京航空航天大学)
PanY 2007 Ph. D. Dissertation (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[3] 张向洪 2014 博士学位论文 (南京: 南京航空航天大学)
Zhang X H 2014 Ph. D. Dissertation (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)
[4] Palmer G 1993 J. Thermophys Heat Transfer 7 294
Google Scholar
[5] Barmin A A, Kulikovskiy A G 1996 J. Comput. Phys. 126 77
Google Scholar
[6] Nagata Y, Otsu H, Yamada K 2012 43rd AIAA Plasmadynamics and Lasers Conference NewOrleans, Louisiana, USA, June 25−8, 2012 p2734
[7] Otsu H 2005 36th AIAA Plasmadynamics and Lasers Conference Toronto, Ontario, Canada, June 6−9, 2005 p5049
[8] Fujino T, Ishikawa M 2013 44th AIAA Plasmadynamics and Lasers Conference SanDiego, California, USA, June 24−27, 2013 p3000
[9] Takahashi T, Shimosawa Y, Masuda K, Fujino T 2015 46th AIAA Plasma dynamics and Lasers Conference Dallas, Texas, USA, June 22−26, 2015 p3365
[10] Bisek N J, Poggie J 2011 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition Orlando, Florida, USA, January 4−7, 2011 p897
[11] Bisek N J, Boyd I D 2010 J. Spacecraft Rockets 47 816
Google Scholar
[12] Lee J, Huerta M, Zha G 2010 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition Orlando, Florida, USA, January 4−7, 2010 p229
[13] Cristofolini A, Borghi C A, Neretti G 2012 18th AIAA/3 AF International Space Planes and Hypersonic Systems and Technologies Conference Tours, France, September 24−28, 2012 p5804
[14] Cristofolini A, Borghi C A, Neretti G 2012 43rd AIAA Plasmadynamics and Lasers Conference New Orleans, Louisiana, USA June 25−28, 2012 p2730
[15] 李开, 柳军, 刘伟强 2017 物理学报 66 084702
Google Scholar
Li K, Liu J, Liu W Q 2017 Acta Phys. Sin. 66 084702
Google Scholar
[16] 李开, 刘伟强 2016 物理学报 65 064701
Google Scholar
Li K, Liu W Q 2016 Acta Phys. Sin. 65 064701
Google Scholar
[17] 姚霄, 刘伟强, 谭建国 2018 物理学报 67 174702
Google Scholar
Yao X, Liu W Q, Tan J G 2018 Acta Phys. Sin. 67 174702
Google Scholar
[18] 何淼生, 杨文将, 郑小梅, 刘宇 2013 航空动力学报 28 365
He M S, Yang W J, Zheng X M, Liu Y 2013 J. Aerosp. Power 28 365
[19] 陈刚, 张劲柏, 李椿萱 2010 北京航空航天大学学报 36 135
Chen G, Zhang J B, Li C X 2010 J. BUAA 36 135
[20] 孙晓辉 2013 博士学位论文(南京: 南京理工大学)
Sun X H 2008 Ph. D. Dissertation (Nanjing: Nanjing University of Science and Technology) (in Chinese)
[21] Poggie J, Gaitonde D 2002 Phys. Fluids 14 1720
Google Scholar
[22] Fujino T, Kondo S, Ishikawa M 2007 38th AIAA Plasmadynamics and Lasers Conference Miami, Florida, USA, June 25−28, 2007 p4248
[23] Khan O U, Hoffmann K A, Dietiker J F 2007 38th AIAA Plasmadynamics and Lasers Conference Miami, Florida, USA June 25−28, 2007 p4374
[24] MacCormack R W 2008 46th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, USA, January 7−10, 2008 p1070
[25] Kim M K 2009 Ph. D. Dissertation (Michigan: University of Michigan)
[26] Bocharov A N 2010 High Temp. 48 483
[27] Yoshino T, Fujino T, Ishikawa M 2007 38th AIAA Plasmadynamics and Lasers Conference Miami, Florida, USA, June 25−28, 2007 p4249
[28] 李开, 刘伟强 2016 国防科技大学学报 38 25
Google Scholar
Li K, Liu W Q 2016 J. NUDT 38 25
Google Scholar
[29] 郭硕鸿 1997 电动力学 (北京: 高等教育出版社) 第1−38页
Guo S H 1997 Electrodynamics (Beijing: Higher Education Press) pp1−38 (in Chinese)
[30] MacCormack R 2005 43rd AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, USA, January 10−13, 2005 p0559
[31] Khan O U, Hoffmann K A, Dietiker J F 2006 44th AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, USA, January 9−12, 2006 p966
[32] 黄富来, 黄护林 2009 航空学报 30 1834
Google Scholar
Huang F L, Huang H L 2009 Acta Aeronaut. Astronaut. Sin. 30 1834
Google Scholar
[33] Maccormack R W 2005 36th AIAA Plasmadynamics and Lasers Conference Toronto, Ontario, Canada, June 6−9, 2005 p4780
[34] 丁明松, 江涛, 董维中, 高铁锁, 刘庆宗 2017 航空学报 38 121030
Ding M S, Jiang T, Dong W Z, Gao T S, Liu Q Z 2017 Acta Aeronaut. Astronaut. Sin. 38 121030
[35] 丁明松, 江涛, 刘庆宗, 董维中, 高铁锁 2019 航空学报 40 123009
Ding M S, Jiang T, Liu Q Z, Dong W Z, Gao T S 2019 Acta Aeronaut. Astronaut. Sin. 40 123009
[36] 丁明松, 江涛, 董维中, 高铁锁, 刘庆宗 2019 物理学报 68 174702
Google Scholar
Ding M S, Jiang T, Dong W Z, Gao T S, Liu Q Z 2019 Acta Phys. Sin. 68 174702
Google Scholar
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