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The evolution of the Kelvin-Helmholtz (KH) instability in the presence of classical hydrodynamics and magneto-hydro-dynamics is investigated numerically by using the magneto-hydro-dynamic (MHD) equations. The MHD equations are solved with the corner transport upwind plus constrained transport algorithm that guarantees the divergence-free constraint in the magnetic field. The numerical results are used to analyze the effects of magnetic field (
${M_{\rm{A}}} = 3.33$ ) on the vorticity and pressure evolution of mixing layer, and also compared with those in the hydrodynamics situation. Moreover, the mechanism of weakening the effect of magnetic field on the KH instability is revealed from the perspectives of the magnetic pressure and the magnetic tension. The results show that the external magnetic field has a great influence on the flow structure of the mixing layer. Specifically, the magnetic pressure has a major effect in the vorticity deposition on the interface, whereas the magnetic tension generates a torque to counter the scrolling of vortex. As a result, the large vortex structure is stretched and destroyed, and finally restrains the vortex rolling-up. In addition, with the development of mixing layer, the interface will separate at the points of maximum curvature under the joint effect of the magnetic pressure, the magnetic tension and the pressure field, and finally form a fishhook-like vortex structure.-
Keywords:
- Kelvin-Helmholtz instability /
- magnetic pressure /
- magnetic tension /
- magnetic fluid
[1] Rahmani M, Seymour B, Lawrence G 2014 Environ. Fluid Mech. 14 1275
Google Scholar
[2] Ryutova M, Berger T, Frank Z, Tarbell T, Title A 2010 Sol. Phys. 267 75
Google Scholar
[3] Zhelyazkov I, Zaqarashvili T V, Ofman L, Chandra R 2018 Adv. Space Res. 61 628
Google Scholar
[4] Ismayilli R F, Dzhalilov N S, Shergelashvili B M, Poedts S, Pirguliyev M S 2018 Phys. Plasmas 25 062903
Google Scholar
[5] Zhelyazkov I, Chandra R, Srivastava A K, Mishonov T 2015 Astrophys. Space Sci. 356 231
Google Scholar
[6] Wu C C 1986 J. Geophys. Res. Space Phys. 91 3042
Google Scholar
[7] Hasegawa H, Fujimoto M, Takagi K, Saito Y, Mukai T, Rème H 2006 J. Geophys. Res. Space Phys. 111 1
Google Scholar
[8] Leroy M H J, Keppens R 2016 Meeting of the French Society of Astronomy & Astrophysics Lyon, France, June 14–17, 2016 p107
[9] Ho C M, Huerre P 1984 Annu. Rev. Fluid Mech. 16 365
Google Scholar
[10] Gratton F T, Gnavi G, Farrugia C J, Bender L 2004 Braz. J. Phys. 34 1804
Google Scholar
[11] Zhao K G, Wang L F, Ye W H, Wu J F, Li Y J 2014 Chin. Phys. Lett. 31 030401
Google Scholar
[12] Leep L J, Button J C, Burr R F 1993 AIAA J. 31 2039
Google Scholar
[13] Brüggen M, Hillebrandt W 2001 Mon. Not. R. Astron. Soc. 323 56
Google Scholar
[14] Keppens R, Toth G, Westermann R H J, Goedbloed J P 1999 J. Plasma Phys. 61 1
Google Scholar
[15] Sharma R C, Srivastava K M 1970 Can. J. Phys. 48 2083
Google Scholar
[16] Sharma R C, Srivastava K M 1968 Aust. J. Phys. 21 917
Google Scholar
[17] Jeong H, Ryu D, Jones T W, Frank A 2000 Astrophys. J. 529 536
Google Scholar
[18] Tian C L, Chen Y 2016 Astrophys. J. 824 60
Google Scholar
[19] Liu Y, Chen Z H, Zhang H H, Lin Z Y 2018 Phys. Fluids 30 044102
Google Scholar
[20] Praturi D S, Girimaji S S 2019 Phys. Fluids 31 024108
Google Scholar
[21] Lin Z Y, Zhang H H, Chen Z H, Liu Y, Hong Y 2017 Int. J. Comut. Fluid Dyn. 31 21
Google Scholar
[22] 林震亚, 张焕好, 陈志华, 刘迎 2017 爆炸与冲击 37 748
Google Scholar
Lin Z Y, Zhang H H, Chen Z H, Liu Y 2017 Explosion and Shock Waves 37 748
Google Scholar
[23] Bogdanoff D W 1983 AIAA J. 21 926
Google Scholar
[24] 董国丹, 郭则庆, 秦建华, 张焕好, 姜孝海, 陈志华, 沙莎 2019 物理学报 68 165201
Google Scholar
Dong G D, Guo Z Q, Qin J H, Zhang H H, Jiang X H, Chen Z H, Sha S 2019 Acta Phys. Sin. 68 165201
Google Scholar
[25] 沙莎, 张焕好, 陈志华, 郑纯, 吴威涛, 石启陈 2020 物理学报 69 184701
Google Scholar
Sha S, Zhang H H, Chen Z H, Chun C, Wu W T, Shi Q C 2020 Acta Phys. Sin. 69 184701
Google Scholar
[26] Karimi M, Girimaji S S 2016 Phys. Rev. E 93 041102
Google Scholar
[27] Karimabadi H, Roytershteyn V, Wan M, Matthaeus W H, Daughton W, Wu P, Shay M, Loring B, Borovsky J, Leonardis E 2013 Phys. Plasmas 20 763
Google Scholar
[28] Patnaik P C, Sherman F S, Corcos G M 1976 J. Fluid Mech. 73 215
Google Scholar
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图 1 计算模型
Figure 1. Schematic of computational model.
图 3 不同磁场强度混合层失稳过程的涡量分布 (a)
$\tau = $ $ 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ Figure 3. Snapshots of the vorticity field at different instants: (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ .
图 4 经典流体(HD)混合层失稳过程的涡量分布 (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 276$ Figure 4. Vorticity distribution during the instability process of the classical fluid mixing layer: (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 276$ .
图 5 经典流体(HD)混合层失稳过程的压力分布 (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 276$ Figure 5. Pressure distribution during the instability process of the classical fluid mixing layer: (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 276$ .
图 6 磁流体(MHD)混合层失稳过程的涡量分布 (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 197$ Figure 6. Vorticity distribution of the MHD mixing layer: (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 197$ .
图 7 磁流体(MHD)混合层失稳过程的压力分布 (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 197$ Figure 7. Pressure distribution of the MHD mixing layer: (a)
$\tau = 30$ ; (b)$\tau = 51$ ; (c)$\tau = 73$ ; (d)$\tau = 119$ ; (e)$\tau = 141$ ; (f)$\tau = 197$ .
图 8 HD和MHD情况下, 纵向总动能随时间变化
Figure 8. The y-component of the total kinetic energy for the HD and MHD situations.
图 9 HD和MHD情况下涡量厚度(a)和环量(b)随时间的变化
Figure 9. Temporal evolutions of the vorticity thickness (a) and circulation (b) in the case of HD and MHD.
图 10 混合层发展过程中涡量及磁感应线分布 (a)
$\tau = 73$ ; (b)$\tau = 97$ ; (c)$\tau = 119$ ; (d)$\tau = 141$ Figure 10. Distribution of the vorticity and magnetic induction line at different times: (a)
$\tau = 73$ ; (b)$\tau = 97$ ; (c)$\tau = 119$ ; (d)$\tau = 141$ .
图 11 MHD情况, 平均磁场强度随时间的变化
Figure 11. Change of average magnetic field intensity with time for the MHD situation.
图 12 磁压力分布(上图: 纵向磁压力; 下图: 横向磁压力) (a)
$\tau = 51$ ; (b)$\tau = 119$ Figure 12. Magnetic pressure distribution (upper: longitudinal magnetic pressure; lower: transverse magnetic pressure): (a)
$\tau = 51$ ; (b)$\tau = 119$ .
图 13 界面上磁压力的方向 (a)
$\tau = 51$ ; (b)$\tau = 119$ Figure 13. Directions of magnetic pressure on the interfaces: (a)
$\tau = 51$ ; (b)$\tau = 119$ .
图 14 平均涡量随时间的变化
Figure 14. Variation of the mean vorticity over time.
图 15 磁张力分布(上图: 纵向磁张力; 下图: 横向磁张力) (a)
$\tau = 51$ ; (b)$\tau = 119$ Figure 15. Magnetic tension distribution (upper: longitudinal magnetic tension; lower: transverse magnetic tension): (a)
$\tau = 51$ ; (b)$\tau = 119$ .
图 16 涡量和磁张力矢量分布图 (a)
$\tau = 51$ ; (b)$\tau = $ $ 119$ Figure 16. Vorticity and magnetic tension vector distribution: (a)
$\tau = 51$ ; (b)$\tau = 119$ . -
[1] Rahmani M, Seymour B, Lawrence G 2014 Environ. Fluid Mech. 14 1275
Google Scholar
[2] Ryutova M, Berger T, Frank Z, Tarbell T, Title A 2010 Sol. Phys. 267 75
Google Scholar
[3] Zhelyazkov I, Zaqarashvili T V, Ofman L, Chandra R 2018 Adv. Space Res. 61 628
Google Scholar
[4] Ismayilli R F, Dzhalilov N S, Shergelashvili B M, Poedts S, Pirguliyev M S 2018 Phys. Plasmas 25 062903
Google Scholar
[5] Zhelyazkov I, Chandra R, Srivastava A K, Mishonov T 2015 Astrophys. Space Sci. 356 231
Google Scholar
[6] Wu C C 1986 J. Geophys. Res. Space Phys. 91 3042
Google Scholar
[7] Hasegawa H, Fujimoto M, Takagi K, Saito Y, Mukai T, Rème H 2006 J. Geophys. Res. Space Phys. 111 1
Google Scholar
[8] Leroy M H J, Keppens R 2016 Meeting of the French Society of Astronomy & Astrophysics Lyon, France, June 14–17, 2016 p107
[9] Ho C M, Huerre P 1984 Annu. Rev. Fluid Mech. 16 365
Google Scholar
[10] Gratton F T, Gnavi G, Farrugia C J, Bender L 2004 Braz. J. Phys. 34 1804
Google Scholar
[11] Zhao K G, Wang L F, Ye W H, Wu J F, Li Y J 2014 Chin. Phys. Lett. 31 030401
Google Scholar
[12] Leep L J, Button J C, Burr R F 1993 AIAA J. 31 2039
Google Scholar
[13] Brüggen M, Hillebrandt W 2001 Mon. Not. R. Astron. Soc. 323 56
Google Scholar
[14] Keppens R, Toth G, Westermann R H J, Goedbloed J P 1999 J. Plasma Phys. 61 1
Google Scholar
[15] Sharma R C, Srivastava K M 1970 Can. J. Phys. 48 2083
Google Scholar
[16] Sharma R C, Srivastava K M 1968 Aust. J. Phys. 21 917
Google Scholar
[17] Jeong H, Ryu D, Jones T W, Frank A 2000 Astrophys. J. 529 536
Google Scholar
[18] Tian C L, Chen Y 2016 Astrophys. J. 824 60
Google Scholar
[19] Liu Y, Chen Z H, Zhang H H, Lin Z Y 2018 Phys. Fluids 30 044102
Google Scholar
[20] Praturi D S, Girimaji S S 2019 Phys. Fluids 31 024108
Google Scholar
[21] Lin Z Y, Zhang H H, Chen Z H, Liu Y, Hong Y 2017 Int. J. Comut. Fluid Dyn. 31 21
Google Scholar
[22] 林震亚, 张焕好, 陈志华, 刘迎 2017 爆炸与冲击 37 748
Google Scholar
Lin Z Y, Zhang H H, Chen Z H, Liu Y 2017 Explosion and Shock Waves 37 748
Google Scholar
[23] Bogdanoff D W 1983 AIAA J. 21 926
Google Scholar
[24] 董国丹, 郭则庆, 秦建华, 张焕好, 姜孝海, 陈志华, 沙莎 2019 物理学报 68 165201
Google Scholar
Dong G D, Guo Z Q, Qin J H, Zhang H H, Jiang X H, Chen Z H, Sha S 2019 Acta Phys. Sin. 68 165201
Google Scholar
[25] 沙莎, 张焕好, 陈志华, 郑纯, 吴威涛, 石启陈 2020 物理学报 69 184701
Google Scholar
Sha S, Zhang H H, Chen Z H, Chun C, Wu W T, Shi Q C 2020 Acta Phys. Sin. 69 184701
Google Scholar
[26] Karimi M, Girimaji S S 2016 Phys. Rev. E 93 041102
Google Scholar
[27] Karimabadi H, Roytershteyn V, Wan M, Matthaeus W H, Daughton W, Wu P, Shay M, Loring B, Borovsky J, Leonardis E 2013 Phys. Plasmas 20 763
Google Scholar
[28] Patnaik P C, Sherman F S, Corcos G M 1976 J. Fluid Mech. 73 215
Google Scholar
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