THE LINEAR STABILITY ANALYSIS OF FLOW PAST A SPHERICAL BUBBLE UNDER THE EFFECT OF STREAMWISE MAGNETIC FIELD
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摘要: 电磁冶金中氩气通常作为动力和载体将脱硫剂、脱氧剂吹入到液态金属中, 因此存在磁场环境下气泡在液态金属中自由运动的问题, 而绕流作为自由运动的特殊形态, 是研究自由运动问题的第一步. 本文通过有限元方法研究流向磁场作用下球形气泡绕流的全局线性稳定性, 讨论
$\mathit{Re}\leqslant 1000,N\leqslant 60$ 参数范围内稳态轴对称基本流对时间−方位角模态小扰动的响应, 发现了8个不稳定的驻定模态, 并绘制它们在$ \mathit{Re}-N $ 参数平面与$ \mathit{Re}-Ha $ 参数平面的中性曲线. 结果显示方位角波数$ m = 1 $ 的驻定模态首先导致第一次常规分岔, 该模态已被广泛证实为轴对称绕流中最不稳定的模态, 使轴对称尾迹转变为由一对反向旋转涡组成的单平面对称尾迹. 同时第一次分岔的中性曲线展示磁场对球形气泡绕流先失稳后致稳的作用. 后续分岔依次由$m = 2, 3, \cdots, 8$ 的不稳定模态导致. 这些分岔为认识磁场环境下气泡绕流的尾迹结构提供重要的参考价值.Abstract: In electromagnetic metallurgy, argon is usually used as a power and carrier to blow desulfurizer and deoxidizer into liquid metal, so there is a problem of free movement of bubbles in liquid metal under a magnetic field environment. Flow past a fixed bubble as a special form of free movement, is the first step to study the problem of free movement. In this paper, the global linear stability analysis of the flow past a spherical bubble under the effect of a streamwise magnetic field is simulated by the finite element method. The response of the steady axisymmetric basic flow to the small perturbation of the independent time-azimuthal mode in the range of$\mathit{Re}\leqslant 1000,N\leqslant 60$ is discussed. Eight unstable stationary modes are found, and their neutral curves in the$ \mathit{Re}-N $ parameter plane or$ \mathit{Re}-Ha $ parameter plane are displayed. The results show that the stationary mode with azimuthal wave number m = 1 leads to the first regular bifurcation, this mode has been widely confirmed as the most unstable mode in the flow past axisymmetric objects, which transforms the axisymmetric wake into a plane symmetric wake composed of a pair of opposite vortices. In addition, the results of the neutral curve show the effect of the magnetic field on the instability of the flow past the spherical bubble. The subsequent bifurcations are successively caused by the unstable modes of m = 2, 3,..., 8, these bifurcations provide an important reference value for understanding the wake structure of the flow past a bubble in the magnetic field environment.-
Key words:
- flow past a spherical bubble /
- linear stability analysis /
- magnetic field
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图 2 无磁场环境下不同纵横比
$ \chi $ 的气泡第一次常规分岔和第二次Hopf分岔对应的临界$ Re $ 与文献结果的对比Figure 2. The comparison of the critical Reynolds number at the first regular bifurcation and the second Hopf bifurcations with results from literatures at different aspect ratios
$ \chi $ without magnetic fields
图 3 定常轴对称基本流的流线和涡量分布
Figure 3. Streamlines and vorticity of steady axisymmetric base flow
图 4
$ \mathit{Re} = 600 $ 时回流区长度和分离角随相互作用数$ N $ 的变化Figure 4. The variations of recirculation length and separation angle with
$ N $ at$ \mathit{Re} = 600 $ 
图 5 Re=600时最大涡量随相互作用数
$ N $ 的变化Figure 5. The variations of the maximum vorticity with
$ N $ at$ \mathit{Re} = 600 $ 
图 6
$ \mathit{Re} = 600,N = 20 $ 时特征值谱在$m = 0, 1,2, \cdots,7$ 的分布Figure 6. The eigenvalue spectra for
$m = 0, 1,2, \cdots,7$ at$ \mathit{Re} = 600,N = 20 $ 
图 7 不稳定模态的流向扰动速度
$ {u}_{x} $ 的空间分布Figure 7. The spatial distributions of streamwise perturbation velocity
$ {u}_{x} $ for unstable modes
图 8
$Re = 600$ 时$m = 0, 1, 2,\cdots , 8 $ 的驻定模态的增长率随$ N $ 的连续性变化Figure 8. The variations of growth rate with
$ N $ for$m = 0, 1, 2,\cdots , 8$ stationary modes at$ Re = 600 $ 
图 9
$m = 1, 2, \cdots, 8$ 的定常不稳定模态的中性曲线,$ \mathit{Re} = 600 $ 不同颜色的实线表示不同模态占优对应的$ N $ 和$ Ha $ 的范围Figure 9. The neutral curves of
$m = 1, 2, \cdots, 8$ stationary unstable modes, solid lines of different colors indicate the range of$ N $ and$ Ha $ corresponding to different dominant modes at$ \mathit{Re} = 600 $ -
[1] 高翱. 新型电磁出钢系统的研发及其效果分析. [博士论文]. 沈阳: 东北大学, 2010Gao Xiang. Research and development of new electromagnetic tapping system and its effect analysis. [PhD Thesis]. Shenyang: Northeast University, 2010 (in Chinese)) [2] 成钢. 氩气在炼钢生产中的应用. 深冷技术, 1981, 4: 78-80 (Cheng Gang. Application of argon in steelmaking production. Cryogenic Technology, 1981, 4: 78-80 (in Chinese) [3] 丰文祥, 陈伟庆, 赵继增. 中间包吹氩去除钢水夹杂物. 北京科技大学报, 2010, 32(4): 425-431 (Feng Wenxiang, Chen Weiqing, Zhao Jizeng. Removal of inclusions in molten steel by argon blowing in tundish. Beijing University of Science and Technology, 2010, 32(4): 425-431 (in Chinese) [4] Pan JH, Zhang NM, Ni MJ. The wake structure and transition process of a flow past a sphere affected by a streamwise magnetic field. Journal of Fluid Mechanics, 2018, 842: 248-272 doi: 10.1017/jfm.2018.133 [5] Mougin G, Magnaudet J. Path instability of a rising bubble. Physical Review Letters, 2001, 88(1): 014502 doi: 10.1103/PhysRevLett.88.014502 [6] Magnaudet J, Mougin G. Wake instability of a fixed spheroidal bubble. Journal of Fluid Mechanics, 2007, 572: 311-337 doi: 10.1017/S0022112006003442 [7] Leal LG. Vorticity transport and wake structure for bluff bodies at finite Reynolds number. Physics of Fluids A: Fluid Dynamics, 1989, 1(1): 124-131 doi: 10.1063/1.857540 [8] Blanco A, Magnaudet J. The structure of the axisymmetric high‐Reynolds number flow around an ellipsoidal bubble of fixed shape. Physics of Fluids, 1995, 7(6): 1265-1274 doi: 10.1063/1.868515 [9] Yang B, Prosperetti A. Linear stability of the flow past a spheroidal bubble. Journal of Fluid Mechanics, 2007, 582: 53-78 doi: 10.1017/S0022112007005691 [10] Tchoufag J, Magnaudet J, Fabre D. Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble. Physics of Fluids, 2013, 25(5): 054108 doi: 10.1063/1.4804552 [11] Lahjomri J, Capéran P, Alemany A. The cylinder wake in a magnetic field aligned with the velocity. Journal of Fluid Mechanics, 1993, 253: 421-448 doi: 10.1017/S0022112093001855 [12] Josserand J, Marty P, Alemany A. Pressure and drag measurements on a cylinder in a liquid metal flow with an aligned magnetic field. Fluid Dynamics Research, 1993, 11(3): 107 doi: 10.1016/0169-5983(93)90010-8 [13] Mutschke G, Gerbeth G, Shatrov V, et al. Two-and three-dimensional instabilities of the cylinder wake in an aligned magnetic field. Physics of Fluids, 1997, 9(11): 3114-3116 doi: 10.1063/1.869429 [14] Mutschke G, Shatrov V, Gerbeth G. Cylinder wake control by magnetic fields in liquid metal flows. Experimental Thermal and Fluid Science, 1998, 16(1-2): 92-99 doi: 10.1016/S0894-1777(97)10007-3 [15] Yonas G. Measurements of drag in a conducting fluid with an aligned field and large interaction parameter. Journal of Fluid Mechanics, 1967, 30(4): 813-821 doi: 10.1017/S002211206700179X [16] Zhang J, Ni MJ. Direct simulation of single bubble motion under vertical magnetic field: Paths and wakes. Physics of Fluids, 2014, 26(10): 102102 doi: 10.1063/1.4896775 [17] Richter T, Keplinger O, Shevchenko N, et al. Single bubble rise in GaInSn in a horizontal magnetic field. International Journal of Multiphase Flow, 2018, 104: 32-41 doi: 10.1016/j.ijmultiphaseflow.2018.03.012 [18] 刘德华, 黎一锴. 竖直振动无黏液滴的法拉第不稳定性分析. 力学学报, 2022, 54(2): 369-378 (Liu Dehua, Li Yikai. Instability analysis of inviscid droplet subject to vertical vibration. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 369-378 (in Chinese) doi: 10.6052/0459-1879-21-483 [19] Stuart JT, 周恒. 流动稳定性及转捩(特约稿). 力学进展, 1984, 14(2): 21-33Stuart JT, Zhou Heng. flow stability and transition (contributed). Advances in Mechanics, 1984, 14(2): 21-33 (in Chinese)) [20] Wan Z, Yang H, Zhou L, et al. Linear stability analysis of supersonic axisymmetric jets. Theoretical and Applied Mechanics Letters, 2014, 4(6): 062005 doi: 10.1063/2.1406205 [21] Davidson PA. An Introduction to Magnetohydrodynamics, 2002 [22] Moreau RJ. Magnetohydrodynamics. Springer Science & Business Media, 1990 [23] Ghidersa B, Dušek JAN. Breaking of axisymmetry and onset of unsteadiness in the wake of a sphere. Journal of Fluid Mechanics, 2000, 423: 33-69 doi: 10.1017/S0022112000001701 [24] Pier B. Local and global instabilities in the wake of a sphere. Journal of Fluid Mechanics, 2008, 603: 39-61 doi: 10.1017/S0022112008000736 [25] 胡军, 李杰权. 相邻双侧边盖驱动方腔流动的三维线性整体稳定性. 气体物理, 2018, 3(3): 1-8 (Hu Jun, Li Jiequan. Three-dimensional linear global stability of flow in a square cavity driven by two adjacent side covers. Gas Physics, 2018, 3(3): 1-8 (in Chinese) [26] Mück B, Günther C, Müller U, et al. Three-dimensional MHD flows in rectangular ducts with internal obstacles. Journal of Fluid Mechanics, 2000, 418: 265-295 doi: 10.1017/S0022112000001300 [27] Li L, Ni M, Zheng W. A charge-conservative finite element method for inductionless MHD equations. Part I: Convergence. SIAM Journal on Scientific Computing, 2019, 41(4): B796-B815 doi: 10.1137/17M1160768 [28] Li L, Ni M, Zheng W. A charge-conservative finite element method for inductionless MHD equations. Part II: A robust solver. SIAM Journal on Scientific Computing, 2019, 41(4): B816-B842 [29] Verfürth R. Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition. Numerische Mathematik, 1986, 50(6): 697-721 [30] Verfürth R. Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition II. Numerische Mathematik, 1991, 59(1): 615-636 doi: 10.1007/BF01385799 [31] Natarajan R, Acrivos A. The instability of the steady flow past spheres and disks. Journal of Fluid Mechanics, 1993, 254: 323-344 doi: 10.1017/S0022112093002150 [32] Meliga P, Chomaz JM, Sipp D. Unsteadiness in the wake of disks and spheres: instability, receptivity and control using direct and adjoint global stability analyses. Journal of Fluids and Structures, 2009, 25(4): 601-616 doi: 10.1016/j.jfluidstructs.2009.04.004 [33] Tchoufag J, Fabre D, Magnaudet J. Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders. Journal of Fluid Mechanics, 2014, 740: 278-311 doi: 10.1017/jfm.2013.642 [34] Szaltys P, Chrust M, Przadka A, et al. Nonlinear evolution of instabilities behind spheres and disks. Journal of Fluids and Structures, 2012, 28: 483-487 doi: 10.1016/j.jfluidstructs.2011.10.004 -
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