基于WENO格式有限体积法的铁磁流体两相流相场方法
作者:
作者单位:

1.重庆大学 航空航天学院;2.沪渝人工智能研究院

中图分类号:

O359

基金项目:

国家自然科学基金(12102071,12172070);重庆市博士直通车项目(No.CSTB2022BSXM-JCX0086).


A finite volume-based phase field method for two-phase ferrofluid flows
Author:
Affiliation:

1.College of Aerospace Engineering,Chongqing University;2.Shanghai-Chongqing institute of Artificial Intelligence,Chongqing

Fund Project:

The National Natural Science Foundation of China (12102071,12172070), and Chongqing Doctoral Through Train Program (No. CSTB2022BSXM-JCX0086),

  • 摘要
  • | |
  • 访问统计
  • |
  • 参考文献 [33]
  • |
  • 相似文献
  • |
  • 引证文献
  • | |
  • 文章评论
    摘要:

    本文采用基于WENO格式的有限体积法,发展了包括铁磁流体的两相流相场方法。流体流动采用了不可压缩Navier-Stokes方程进行描述,利用Cahn-Hilliard方程来捕捉两相流界面运动,并采用Maxwell方程描述磁场分布。同时,在流体流动控制方程方程中加入了开尔文力和表面张力以实现磁场对界面动力学行为的描述。我们将四阶Cahn-Hilliard方程拆分为两个Helmholtz类型方程,从而克服四阶非线性项的离散和高精度计算带来的难题。我们采用五阶WENO格式对控制方程的对流项进行统一离散处理,从而提高数值计算的精度性,同时避免产生数值振荡。Zalesak’s圆盘问题数值模拟的结果表明,本文发展方法的相界面捕捉精度高于参考文献中所报道的离散方法,与高精度相场方法的精度相当;对剪切流中的液滴变形问题的数值模拟揭示我们数值方法可以更真实地捕捉到更多个卫星液滴。此外,针对磁场影响下较低毛细数条件的铁磁流体液滴剪切变形研究显示:当外加磁场方向与液滴水动力学变形方向一致时,磁场的作用会放大液滴变形,进一步增加磁场强度会诱发液滴分裂;而当外加磁场方向与液滴水动力学变形方向垂直时,较低强度的磁场作用能够改变液滴变形方向,而较高强度的磁场则会使液滴直接呈现出沿磁场方向变形。

    Abstract:

    In this paper, a finite volume method based on the WENO scheme is used to develop a two-phase flow phase field method including ferrofluids. The governing equations include the Navier-Stokes equations for incompressible flow, the Cahn-Hilliard equation for interfacial dynamics, and the Maxwell equation for the distribution of the external magnetic field. To overcome the challenges posed by the fourth-order nonlinear diffusion terms, we decompose the Cahn-Hilliard equation into two Helmholtz equations. The fifth-order WENO scheme is employed to handle the convection term of the governing equation, aiming to enhance accuracy and prevent numerical oscillations. The Zalesak"s disk problem shows that the proposed method has a higher phase interface capture accuracy compared to references, and is comparable to the high-precision phase field method. The proposed method is applied to the droplet shear deformation problem, and it is observed that the current method can capture more satellite droplets. Additionally, it is noted that the magnetic interfacial force favors droplet deformation when the direction of the external magnetic field aligns closely with the direction of droplet hydrodynamic deformation. Furthermore, an increase in the magnetic field intensity leads to droplet splitting. On the contrary, when the magnetic field is nearly perpendicular to the direction of droplet hydrodynamic deformation, a lower intensity magnetic field alters the direction of droplet deformation, while a higher intensity magnetic field directly deforms the droplet along the direction of the magnetic field.

    参考文献
    [1] ZHANG S-T, NIU X-D, LI Q-P, et al. A numerical investigation on the deformation of ferrofluid droplets [J]. Physics of Fluids, 2023, 35(1).
    [2] KHAN A, NIU X D, LI Y, et al. Motion, deformation, and coalescence of ferrofluid droplets subjected to a uniform magnetic field [J]. International Journal for Numerical Methods in Fluids, 2020, 92(11): 1584-603.
    [3] WU Z, TROLL J, H-HJEONG, et al. A swarm of slippery micropropellers penetrates the vitreous body of the eye [J]. Science Advances, 2018, 4(11).
    [4] FAN X, SUN M, SUN L, et al. Ferrofluid Droplets as Liquid Microrobots with Multiple Deformabilities [J]. Advanced Functional Materials, 2020, 30(24).
    [5] MEDINA‐SáNCHEZ M, MAGDANZ V, GUIX M, et al. Swimming Microrobots: Soft, Reconfigurable, and Smart [J]. Advanced Functional Materials, 2018, 28(25).
    [6] SUN M, HAO B, YANG S, et al. Exploiting ferrofluidic wetting for miniature soft machines [J]. Nature Communications, 2022, 13(1).
    [7] GaoDonghong, MorleyNeil B.,DhirVijay.Understanding Magnetic Field Gradient Effect From a Liquid Metal Droplet Movement.J. Fluids Eng. 2004, 126(1): 120-124.
    [8] KorlieMark S , MukherjeeArup, NitaBogdan G, StevensJohn G, DavidTrubatchA, YeckoPhilip. Modeling bubbles and droplets in magnetic fluids[J]. J Phys Condens Matter 2008 21;20(20):204143.
    [9] CUNHA L H P, SIQUEIRA I R, OLIVEIRA T F, et al. Field-induced control of ferrofluid emulsion rheology and droplet break-up in shear flows [J]. Physics of Fluids, 2018, 30(12).
    [10] NOCHETTO R H, SALGADO A J, TOMAS I. A diffuse interface model for two-phase ferrofluid flows [J]. Computer Methods in Applied Mechanics and Engineering, 2016, 309: 497-531.
    [11] HU Y, LI D, NIU X. Phase-field-based lattice Boltzmann model for multiphase ferrofluid flows [J]. Physical Review E, 2018, 98(3).
    [12] KHAN A, NIU X-D, LI Q-Z, et al. Dynamic study of ferrodroplet and bubbles merging in ferrofluid by a simplified multiphase lattice Boltzmann method [J]. Journal of Magnetism and Magnetic Materials, 2020, 495.
    [13] TAO T, ZHONGHUA Q. Efficient numerical methods for phase-field equations [J]. SCIENTIA SINICA Mathematica, 2020, 50(6).
    [14] YUE P, FENG J J, LIU C, et al. A diffuse-interface method for simulating two-phase flows of complex fluids [J]. Journal of Fluid Mechanics, 2004, 515: 293-317.
    [15] DONG S, SHEN J. A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios [J]. Journal of Computational Physics, 2012, 231(17): 5788-804.
    [16] JiangGuang-Shan, ShuChi-Wang,Efficient Implementation of Weighted ENO Schemes [J].Journal of Computational Physics,1996, 202-228.
    [17] Huang Z, Lin G, Ardekani A M. Consistent, essentially conservative and balanced-force PhaseField method to model incompressible two-phase flows[J]. Journal of Computational Physics,Academic Press Inc., 2020, 406.
    [18] Guermond J L, Minev P, Shen J. An overview of projection methods for incompressible flows[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195(44–47): 6011–6045.
    [19] Rhie C M, Chow W L. Numerical study of the turbulent flow past an airfoil with trailing edgeseparation[J]. AIAA Journal, 1983, 21(11): 1525–1532.
    [20] HUANG Z, LIN G, ARDEKANI A M. A mixed upwind/central WENO scheme for incompressible two-phase flows [J]. Journal of Computational Physics, 2019, 387: 455-80.
    [21] Kim J. A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198(37–40): 3105–3112.
    [22] Liu H, Valocchi A J, Zhang Y, et al. Lattice Boltzmann phase-field modeling of thermocapillary flows in a confined microchannel[J]. Journal of Computational Physics, Academic Press Inc,2014, 256: 334–356.
    [23] WANG H L, CHAI Z H, SHI B C, et al. Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations [J]. Phys Rev E, 2016, 94(3-1): 033304.
    [24] LIANG H, SHI B C, GUO Z L, et al. Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows [J]. Phys Rev E Stat Nonlin Soft Matter Phys, 2014, 89(5): 053320.
    [25] XIAO Y, ZENG Z, ZHANG L, et al. A highly accurate bound-preserving phase field method for incompressible two-phase flows [J]. Physics of Fluids, 2022, 34(9).
    [26] Taylor Geoffrey Ingram .The viscosity of a fluid containing small drops of another fluid Proc[J] ,1932, 41–48.
    [27] GUIDO S, VILLONE M. Three-dimensional shape of a drop under simple shear flow [J]. Journal of Rheology, 1998, 42(2): 395-415.
    [28] M. Kennedy, C. Pozrikidis, and R. Skalak, “Motion and deformation of liquid drops, and the rheology ofdilute emulsions in simple shear flow,” Comput. Fluids 23, 251–278 (1994).
    [29] Cox RG. The deformation of a drop in a general time-dependent fluid flow [J]. Journal of Fluid Mechanics. 1969;37(3):601-623.
    [30] HASSAN M R, WANG C. Magnetic field induced ferrofluid droplet breakup in a simple shear flow at a low Reynolds number [J]. Physics of Fluids, 2019, 31(12).
    [31] HU Y, LI D, JIN L, et al. Hybrid Allen-Cahn-based lattice Boltzmann model for incompressible two-phase flows: The reduction of numerical dispersion [J]. Phys Rev E, 2019, 99(2-1): 023302.
    [32] LI Q-Z, LU Z-L, CHEN Z, et al. An efficient simplified phase-field lattice Boltzmann method for super-large-density-ratio multiphase flow [J]. International Journal of Multiphase Flow, 2023, 160.
    [33] FLAMENT, LACIS, BACRI, et al. Measurements of ferrofluid surface tension in confined geometry [J]. 1996, 53 5: 4801-6.
    相似文献
    引证文献
    网友评论
    网友评论
    分享到微博
    发 布
引用本文
分享
文章指标
  • 点击次数:220
  • 下载次数: 0
  • HTML阅读次数: 0
  • 引用次数: 0
历史
  • 收稿日期:2024-01-31
  • 最后修改日期:2024-04-18
  • 录用日期:2024-05-07
文章二维码