Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-12-01T02:30:56.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Rayleigh–Taylor instability in a finite cylinder: linear stability analysis and long-time fingering solutions

Published online by Cambridge University Press:  09 October 2013

H. Sweeney ,
R. R. Kerswell  and
T. Mullin
H. Sweeney
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
R. R. Kerswell*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
T. Mullin
Affiliation:
School of Physics and Astronomy, University of Manchester. Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the Rayleigh–Taylor instability problem of two initially stationary immiscible viscous fluids positioned with the denser above the less dense in a finite circular cylinder, such that their starting fluid–fluid interface is the horizontal midplane of the cylinder. The ensuing linear instability problem has a five-dimensional parameter space – defined by the density ratio, the viscosity ratio, the cylinder aspect ratio, the surface tension between the fluids and the ratio of viscous to gravitational time scales – of which we explore only part, motivated by recent experiments where viscous fluids exchange in vertical tubes (Beckett et al., J. Fluid Mech., 2011, vol. 682, pp. 652–670). We find that for these experiments, the instability is invariably ‘side-by-side’ (of azimuthal wavenumber 1 type) but we also uncover parameter regions where the preferred instability is axisymmetric. The fact that both ‘core-annular’ (axisymmetric) and ‘side-by-side’ (asymmetric) long-time flows are seen experimentally highlights the fact that the initial Rayleigh–Taylor instability of the interface does not determine the long-time flow configuration in these situations. Finally, long-time flow solutions are presented on the basis that they will be slowly varying fingering solutions.

JFM classification

Convection: Buoyancy-driven instability Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , pp. 338 - 362
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abarzhi, S. I. 2010 Review of theoretical modelling approaches of Rayleigh–Taylor instabilities and turbulent mixing. Phil. Trans. R. Soc. 368, 1809–1828.CrossRefGoogle ScholarPubMed
Arakeri, J. H., Avila, F. E., Dada, J. M. & Tovar, R. O. 2000 Convection in a long vertical tube due to unstable stratification: a new type of turbulent flow? Curr. Sci. 79, 859–866.Google Scholar
Arnett, W. D., Bahcall, J. N., Kirshner, R. P. & Woosley, S. E. 1989 Supernova 1987A. Annu. Rev. Astron. Astrophys. 27, 629–700.CrossRefGoogle Scholar
Batchelor, G. K. & Nitsche, J. M. 1993 Instability of stratified fluid in a vertical cylinder. J. Fluid Mech. 252, 419–448.CrossRefGoogle Scholar
Beckett, F., Mader, H. M., Phillips, J. C., Rust, A. & Witham, F. 2011 An experimental study of low-Reynolds-number exchange flow of two Newtonian fluids in a vertical pipe. J. Fluid Mech 682, 652–670.CrossRefGoogle Scholar
Bellman, R. & Pennington, R. H. 1954 Effects of surface tension and viscosity on Taylor instability. Q. Appl. Maths 12, 151–162.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chen, C. Y. & Meiburg, E. 1996 Miscible displacements in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech. 326, 57–90.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duprez, M. (1851 & 1854) Sur un cas particulier de l’équilibre des liquides. Nouv. Mém. Acad. Belg..CrossRefGoogle Scholar
Hinds, W., Ashley, A., Kennedy, N. & Bucknam, P. 2002 Conditions for cloud settling and Rayleigh–Taylor instability. Aerosol Sci. Technol. 36, 1128–1138.CrossRefGoogle Scholar
Huppert, H. E. & Hallworth, M. A. 2007 Bi-directional flows in constrained systems. J. Fluid Mech. 578, 95–112.CrossRefGoogle Scholar
Jacobs, J. W, Bunster, A., Catton, I. & Plesset, M. S. 1985 Experimental Rayleigh–Taylor instability in a circular tube. Trans. ASME I: J. Fluids Engng 107, 460–466.Google Scholar
Jacobs, J. W. & Catton, I. 1988 Three-dimensional Rayleigh–Taylor instability. Part 2. Experiment. J. Fluid Mech. 187, 353–371.CrossRefGoogle Scholar
Kerswell, R. R. 2011 Exchange flow of two immiscible fluids and the principle of maximum flux. J. Fluid Mech. 682, 132–159 (referred to as K11 in the text).CrossRefGoogle Scholar
Kerswell, R. R. & Davey, 1996 A. On the linear instability of elliptic pipe flow. J. Fluid Mech. 316, 307–324.CrossRefGoogle Scholar
Kuang, J., Maxworthy, T. & Petitjeans, P. 2003 Miscible displacements between silicone oils in capillary tubes. Eur. J. Mech. B 22, 271–277.CrossRefGoogle Scholar
Lindl, J. D., McCrory, R. L. & Campbell, E. M. 1992 Progress toward ignition and burn propagation in inertial confinement fusion. Phys. Today 45, 32–40.CrossRefGoogle Scholar
Malekmohammadi, S., Carrasco-Teja, M., Storey, S., Frigaard, I. A. & Martinez, D. M. 2010 An experimental study of laminar displacement flows in narrow vertical eccentric annuli. J. Fluid Mech. 649, 371–398.CrossRefGoogle Scholar
Matson, G. P. & Hogg, A. J. 2012 Viscous exchange flows. Phys. Fluids 24, 023102.CrossRefGoogle Scholar
Maxwell, J. C.  (1890) Scientific Papers, vol. II (ed. W. D. Niven), p. 587. Cambridge University Press.Google Scholar
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37–56.CrossRefGoogle Scholar
Rayleigh, J. W. S. 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
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 3–18.CrossRefGoogle Scholar
Stevenson, D. S. & Blake, S. 1998 Modelling the dynamics and thermodynamics of volcanic degassing. Bull. Volcanol. 60, 307–317.CrossRefGoogle Scholar
Sweeney, H. The Rayleigh–Taylor instability in a cylinder: linear instability analysis and long-time solutions. MSc thesis, School of Mathematics, University of Bristol, May 2011.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. A 201, 192–196.Google Scholar
Vanaparthy, S. H., Meiburg, E. & Wilhelm, D. 2003 Density-driven instabilities of miscible fluids in a capillary tube: linear stability analysis. J. Fluid Mech. 497, 99–121.CrossRefGoogle Scholar
Vanaparthy, S. H. & Meiburg, E. 2008 Variable density and viscosity, miscible displacements in capillary tubes. Eur. J. Mech. B 27, 268–289.CrossRefGoogle Scholar
Whitehead, J. A. & Luther, D. S. 1975 Dynamics of laboratory diaper and plume models. J. Geophys. Res. 80, 705–717.CrossRefGoogle Scholar
Wilkinson, J. P. & Jacobs, J. W. 2007 Experimental study of the single-mode three-dimensional Rayleigh–Taylor instability. Phys. Fluids 19, 124102.CrossRefGoogle Scholar
Wooding, R. A. 1959 The stability of a viscous liquid in a vertical tube containing porous material. Proc. R. Soc. Lond. 252, 120–134.Google Scholar
Yih, C.-S. 1980 Stratified Flows. Academic.Google Scholar
Yu, H. & Livescu, D. 2008 Rayleigh–Taylor instability in cylindrical geometry with compressible fluids. Phys. Fluids 20, 104103.CrossRefGoogle Scholar

Sweeney et al. supplemenatry material

This video shows the motion induced by overturning a 15 cm long, 6.3 cm diameter glass cylinder which is filled 50:50 with golden syrup (ρ1 = 1.1 g/cm3, ν1 = 1200cm2/s) and silicone fluid (ρ2 = 0.98g/cm3, ν2 = 600cm2/s). The video starts just as the motion begins which is ≈ 2 secs. after overturning. Note the creation of a pair of cusps in the middle of the cylinder.

Download Sweeney et al. supplemenatry material(Video)
Video 5.4 MB