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Numerical analysis of the Rayleigh–Taylor instability in an electric field

Published online by Cambridge University Press:  03 March 2016

Qingzhen Yang ,
Ben Q. Li ,
Zhengtuo Zhao ,
Jinyou Shao  and
Feng Xu
Qingzhen Yang
Affiliation:
The Key Laboratory of Biomedical Information Engineering of Ministry of Education, School of Life Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China Bioinspired Engineering and Biomechanics Center (BEBC), Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China
Ben Q. Li*
Affiliation:
Department of Mechanical Engineering, University of Michigan, Dearborn, MI 48128, USA
Zhengtuo Zhao
Affiliation:
Department of Mechanical Engineering, University of Michigan, Dearborn, MI 48128, USA
Jinyou Shao
Affiliation:
State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China
Feng Xu
Affiliation:
The Key Laboratory of Biomedical Information Engineering of Ministry of Education, School of Life Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China Bioinspired Engineering and Biomechanics Center (BEBC), Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China
*
†Email address for correspondence: benqli@umich.edu
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Abstract

A numerical analysis is presented of the Rayleigh–Taylor instability (RTI) in the presence of an external electric field, with an emphasis on nonlinear phenomena associated with the evolution of complex interfacial morphology. The Poisson equation for the electric field and the Navier–Stokes equation for fluid flow field are solved simultaneously along with the Cahn–Hilliard phase field equation for interface deformation and morphology development. Numerical model is validated against the existing data and the results of linear analysis. Extensive numerical simulations are carried out for a wide range of fluid flow and electric field conditions. Computed results show that, in both linear and nonlinear regimes, a horizontal field suppresses the RTI, while a vertical electric field aggravates it. However, the vertical field does not affect the secondary instability; specifically, it does not contribute to the baroclinical generation of vorticity and consequently does not affect the roll-up formation. Linear analysis predicts that the RTI remains the same with the interchange of the dielectric constants of the two fluids, which is also confirmed by the numerical model for small interface deformations. This prediction, however, does not hold true in the nonlinear regimes in that complex interfacial morphology may evolve quite differently if the dielectric constants of two fluids are interchanged.

JFM classification

Flow Control: Instability control MHD and Electrohydrodynamics: MHD and Electrohydrodynamics Multiphase and Particle-laden Flows: Multiphase flow

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Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 397 - 434
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© 2016 Cambridge University Press 

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References

Abarzhi, S. I. 2010 Review of theoretical modelling approaches of Rayleigh–Taylor instabilities and turbulent mixing. Phil. Trans. R. Soc. Lond. A 368 (1916), 1809–1828.Google Scholar
Barannyk, L. L., Papageorgiou, D. T. & Petropoulos, P. G. 2012 Suppression of Rayleigh–Taylor instability using electric fields. Math. Comput. Simul. 82 (6), 1008–1016.Google Scholar
Boffetta, G., Mazzino, A., Musacchio, S. & Vozella, L. 2009 Kolmogorov scaling and intermittency in Rayleigh–Taylor turbulence. Phys. Rev. E 79, 065301.Google ScholarPubMed
Boomkamp, P. A. M. & Miesen, R. H. M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22 (6), 67–88.Google Scholar
Celani, A., Mazzino, A., Muratore-Ginanneschi, P. & Vozella, L. 2009a Phase-field model for the Rayleigh–Taylor instability of immiscible fluids. J. Fluid Mech. 622, 115–134.Google Scholar
Celani, A., Mazzino, A., Muratore-Ginnaneschi, P. & Vozella, L. 2009b Rayleigh–Taylor instability in two dimensions and phase-field method. In Advances in Turbulence XII, pp. 169–172. Springer.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, pp. 448–453. Clarendon.Google Scholar
Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30 (1), 329–364.CrossRefGoogle Scholar
Chertkov, M. 2003 Phenomenology of Rayleigh–Taylor turbulence. Phys. Rev. Lett. 91 (11), 115001.Google Scholar
Cimpeanu, R., Papageorgiou, D. T. & Petropoulos, P. G. 2014 On the control and suppression of the Rayleigh–Taylor instability using electric fields. Phys. Fluids 26 (2), 022105.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.Google Scholar
Daly, B. J. 1967 Numerical study of two fluid Rayleigh–Taylor instability. Phys. Fluids 10 (2), 297–307.Google Scholar
Dalziel, S. B. 1993 Rayleigh–Taylor instability: experiments with image analysis. Dyn. Atmos. Oceans 20 (1), 127–153.CrossRefGoogle Scholar
Dimonte, G. 2004 Dependence of turbulent Rayleigh–Taylor instability on initial perturbations. Phys. Rev. E 69 (5), 056305.Google Scholar
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12, 304–321.CrossRefGoogle Scholar
Ding, H., Spelt, P. D. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226 (2), 2078–2095.Google Scholar
Grandison, S., Papageorgiou, D. T. & Vanden-Broeck, J. M. 2007 Interfacial capillary waves in the presence of electric fields. Eur. J. Mech. (B/Fluids) 26 (3), 404–421.Google Scholar
Grandison, S., Papageorgiou, D. T. & Vanden-Broeck, J. M. 2012 The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets. Z. Angew. Math. Phys. 63 (1), 125–144.CrossRefGoogle Scholar
Guermond, J. L. & Quartapelle, L. 2000 A projection FEM for variable density incompressible flows. J. Comput. Phys. 165 (1), 167–188.Google Scholar
He, X., Chen, S. & Zhang, R. 1999 A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability. J. Comput. Phys. 152 (2), 642–663.CrossRefGoogle Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1), 201–225.CrossRefGoogle Scholar
Hua, J., Lim, L. K. & Wang, C.-H. 2008 Numerical simulation of deformation/motion of a drop suspended in viscous liquids under influence of steady electric fields. Phys. Fluids 20 (11), 113302.Google Scholar
Jackson, J. D. 1999 Classical Electrodynamics. Wiley.Google Scholar
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1), 96–127.CrossRefGoogle Scholar
Joshi, A., Radhakrishna, M. C. & Rudraiah, N. 2010 Rayleigh–Taylor instability in dielectric fluids. Phys. Fluids 22 (6), 064102.Google Scholar
Kilkenny, J. D., Glendinning, S. G., Haan, S. W., Hammel, B. A., Lindl, J. D., Munro, D., Remington, B. A., Weber, S. V., Knauer, J. P. & Verdon, C. P. 1994 A review of the ablative stabilization of the Rayleigh–Taylor instability in regimes relevant to inertial confinement fusion. Phys. Plasmas 1 (5), 1379–1389.CrossRefGoogle Scholar
Korovin, V. M. 2011 Effect of tangential electric field on the evolution of the Rayleigh–Taylor instability of a dielectric liquid film. Tech. Phys. 56 (10), 1390–1397.Google Scholar
Kull, H. J. 1991 Theory of the Rayleigh–Taylor instability. Phys. Rep. 206 (5), 197–325.Google Scholar
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 1.Google Scholar
Lee, H. G., Kim, K. & Kim, J. 2011 On the long time simulation of the Rayleigh–Taylor instability. Intl J. Numer. Meth. Engng 85 (13), 1633–1647.Google Scholar
Lin, Y., Skjetne, P. & Carlson, A. 2012 A phase field model for multiphase electro-hydrodynamic flow. Intl J. Multiphase Flow 45, 1–11.Google Scholar
Melcher, J. R. & Schwarz, W. J. Jr 1968 Interfacial relaxation overstability in a tangential electric field. Phys. Fluids 11 (12), 2604–2616.Google Scholar
Michioka, H. & Sumita, I. 2005 Rayleigh–Taylor instability of a particle packed viscous fluid: implications for a solidifying magma. Geophys. Res. Lett. 32 (3), L03309.Google Scholar
Mohamed, A. & El Shehawey, E. S. F. 1983a Nonlinear electrohydrodynamic Rayleigh–Taylor instability. II: a perpendicular field producing surface charge. Phys. Fluids 26 (7), 1724–1730.Google Scholar
Mohamed, A. E. M. A. & Shehawey, E. S. F. 1983b Nonlinear electrohydrodynamic Rayleigh–Taylor instability. Part 1. A perpendicular field in the absence of surface charges. J. Fluid Mech. 129, 473–494.Google Scholar
Olson, D. H. & Jacobs, J. W. 2009 Experimental study of Rayleigh–Taylor instability with a complex initial perturbation. Phys. Fluids 21 (3), 034103.Google Scholar
Rahmat, A., Tofighi, N., Shadloo, M. S. & Yildiz, M. 2014 Numerical simulation of wall bounded and electrically excited Rayleigh–Taylor instability using incompressible smoothed particle hydrodynamics. Colloid Surf. A 460, 60–70.Google Scholar
Ramaprabhu, P., Dimonte, G., Woodward, P., Fryer, C., Rockefeller, G., Muthuraman, K., Lin, P.-H. & Jayaraj, J. 2012 The late-time dynamics of the single-mode Rayleigh–Taylor instability. Phys. Fluids 24 (7), 074107.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177.Google Scholar
Rayleigh, Lord 1900 Scientific Papers. vol. II. Cambridge University Press.Google Scholar
Renoult, M. C., Petschek, R. G., Rosenblatt, C. & Carlès, P. 2011 Deforming static fluid interfaces with magnetic fields: application to the Rayleigh–Taylor instability. Exp. Fluids 51 (4), 1073–1083.Google Scholar
Ribeyre, X., Tikhonchuk, V. T. & Bouquet, S. 2004 Compressible Rayleigh–Taylor instabilities in supernova remnants. Phys. Fluids 16 (12), 4661–4670.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64.CrossRefGoogle Scholar
Schneider, M. B., Dimonte, G. & Remington, B. 1998 Large and small scale structure in Rayleigh–Taylor mixing. Phys. Rev. Lett. 80 (16), 3507.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1), 3–18.Google Scholar
Sohn, S. I. 2009 Effects of surface tension and viscosity on the growth rates of Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. E 80 (5), 055302.Google ScholarPubMed
Taylor, G. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192–196.Google Scholar
Taylor, G. I. & Mcewan, A. D. 1965 The stability of a horizontal fluid interface in a vertical electric field. J. Fluid Mech. 22 (01), 1–15.Google Scholar
Thorpe, A. J., Volkert, H. & Ziemianski, M. J. 2003 The Bjerknes’ circulation theorem: a historical perspective. Bull. Am. Meteorol. Soc. 4, 471–480.Google Scholar
Tomar, G., Gerlach, D., Biswas, G., Alleborn, N., Sharma, A., Durst, F., Welch, S. W. J. & Delgado, A. 2007 Two-phase electrohydrodynamic simulations using a volume-of-fluid approach. J. Comput. Phys. 227 (2), 1267–1285.Google Scholar
Tryggvason, G. 1988 Numerical simulations of the Rayleigh–Taylor instability. J. Comput. Phys. 75 (2), 253–282.CrossRefGoogle Scholar
White, J., Oakley, J., Anderson, M. & Bonazza, R. 2010 Experimental measurements of the nonlinear Rayleigh–Taylor instability using a magnetorheological fluid. Phys. Rev. E 81 (2), 026303.Google Scholar
Wu, N. & Russel, W. B. 2009 Micro-and nano-patterns created via electrohydrodynamic instabilities. Nano Today 4 (2), 180–192.Google Scholar
Yang, Q., Li, B. Q. & Ding, Y. 2013 3D phase field modeling of electrohydrodynamic multiphase flows. Intl J. Multiphase Flow 57, 1–9.Google Scholar
Yang, Q., Li, B. Q., Shao, J. & Ding, Y. 2014 A phase field numerical study of 3D bubble rising in viscous fluids under an electric field. Intl J. Heat Mass Transfer 78, 820–829.Google Scholar
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27 (02), 337–352.Google Scholar
Youngs, D. L. 1984 Numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12 (1), 32–44.Google Scholar