Hostname: page-component-54dcc4c588-trf7k Total loading time: 0 Render date: 2025-09-14T03:27:42.028Z Has data issue: false hasContentIssue false
  • English
  • Français

Faraday instability on a sphere: numerical simulation

Published online by Cambridge University Press:  10 May 2019

A. Ebo-Adou Open the ORCID record for A. Ebo-Adou [Opens in a new window] ,
L. S. Tuckerman Open the ORCID record for L. S. Tuckerman [Opens in a new window] ,
S. Shin ,
J. Chergui  and
D. Juric
A. Ebo-Adou
Affiliation:
Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, 75005, France Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur (LIMSI), Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay, Bât. 507, Rue du Belvédère, Campus Universitaire, 91405 Orsay, France Institut des Sciences de la Terre, Centre d’Études et de Recherche de Djibouti, Route de l’aéroport B.P 486 Djibouti-ville, République de Djibouti
L. S. Tuckerman*
Affiliation:
Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, 75005, France
S. Shin
Affiliation:
Department of Mechanical and System Design Engineering, Hongik University, Seoul 121-791, Republic of Korea
J. Chergui
Affiliation:
Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur (LIMSI), Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay, Bât. 507, Rue du Belvédère, Campus Universitaire, 91405 Orsay, France
D. Juric
Affiliation:
Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur (LIMSI), Centre National de la Recherche Scientifique (CNRS), Université Paris Saclay, Bât. 507, Rue du Belvédère, Campus Universitaire, 91405 Orsay, France
*
†Email address for correspondence: laurette@pmmh.espci.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate–prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves Nonlinear Dynamical Systems: Pattern formation

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 433 - 459
Copyright
© 2019 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.)

Article purchase

Temporarily unavailable

References

Arbell, H. & Fineberg, J. 2002 Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65, 036224.Google Scholar
Basaran, O. A. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169–198.Google Scholar
Busse, F. H. 1975 Patterns of convection in spherical shells. J. Fluid Mech. 72, 67–85.Google Scholar
Busse, F. H. & Riahi, N. 1982 Patterns of convection in spherical shells. Part 2. J. Fluid Mech. 123, 283–301.Google Scholar
Chorin, A. J. 1968 Numerical simulation of the Navier–Stokes equations. Math. Comput. 22, 745–762.Google Scholar
Chossat, P., Lauterbach, R. & Melbourne, I. 1991 Steady-state bifurcation with O (3) symmetry. Arch. Rat. Mech. Anal. 113, 313–376.Google Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.Google Scholar
Ebo-Adou, A. & Tuckerman, L. S. 2016 Faraday instability on a sphere: Floquet analysis. J. Fluid Mech. 805, 591–610.Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. A 121, 299–340.Google Scholar
Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and Groups in Bifurcation Theory: Vol. II. Springer.Google Scholar
Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182.Google Scholar
Ihrig, E. & Golubitsky, M. 1984 Pattern selection with O (3) symmetry. Physica D 13, 1–33.Google Scholar
Kahouadji, L., Périnet, N., Tuckerman, L. S., Shin, S., Chergui, J. & Juric, D. 2015 Numerical simulation of supersquare patterns in Faraday waves. J. Fluid Mech. 772, R2.Google Scholar
Kudrolli, A., Pier, B. & Gollub, J. P. 1998 Superlattice patterns in surface waves. Physica D 123, 99–111.Google Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. Lond. 452, 1113–1126.Google Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.Google Scholar
Kwak, D. Y. & Lee, J. S. 2004 Multigrid algorithm for the cell-centered finite-difference method II: discontinuous coefficient case. Numer. Methods Partial Differential Equations 20, 723–741.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lundgren, T. S. & Mansour, N. N. 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510.Google Scholar
Matthews, P. C. 2003 Pattern formation on a sphere. Phys. Rev. E 67, 036206.Google Scholar
Meradji, S., Lyubimova, T. P., Lyubimov, D. V. & Roux, B. 2001 Numerical simulation of a liquid drop freely oscillating. Cryst. Res. Technol. 36, 729–744.Google Scholar
Patzek, T., Benner, R., Basaran, O. & Scriven, L. 1991 Nonlinear oscillations of inviscid free drops. J. Comput. Phys. 97, 489–515.Google Scholar
Périnet, N., Juric, D. & Tuckerman, L. S. 2012 Alternating hexagonal and striped patterns in Faraday surface waves. Phys. Rev. Lett. 109, 164501.Google Scholar
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252.Google Scholar
Politis, A.2013 Real/complex spherical harmonic transform, Gaunt coefficients and rotations. http://www.mathworks.com/matlabcentral/fileexchange/43856-real-complex-spherical-harmonic-transform-gaunt-coefficients-and-rotations/.Google Scholar
Politis, A.2016 Microphone array processing for parametric spatial audio techniques. PhD thesis, Aalto University, Finland.Google Scholar
Popinet, S. 1993 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.Google Scholar
Rajchenbach, J., Leroux, A. & Clamond, D. 2011 New standing solitary waves in water. Phys. Rev. Lett. 107, 024502.Google Scholar
Rayleigh, L. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Riahi, N. 1984 Nonlinear convection in a spherical shell. J. Phys. Soc. Japan 53, 2506–2512.Google Scholar
Rucklidge, A. M. & Skeldon, A. C. 2015 Can weakly nonlinear theory explain Faraday wave patterns near onset? J. Fluid Mech. 777, 604–632.Google Scholar
Shin, S. 2007 Computation of the curvature field in numerical simulation of multiphase flow. J. Comput. Phys. 222, 872–878.Google Scholar
Shin, S., Chergui, J. & Juric, D. 2017 A solver for massively parallel direct numerical simulation of three-dimensional multiphase flows. J. Mech. Sci. Technol. 31, 1739–1751.Google Scholar
Shin, S. & Juric, D. 2007 High order level contour reconstruction method. J. Mech. Sci. Technol. 21, 311–326.Google Scholar
Shin, S. & Juric, D. 2009 A hybrid interface method for three-dimensional multiphase flows based on front-tracking and level set techniques. Intl J. Numer. Meth. Fluids 60, 753–778.Google Scholar
Shu, C. W. & Osher, S. 1989 Efficient implementation of essentially non-oscillatory shock capturing schemes, II. J. Comput. Phys. 83, 32–78.Google Scholar
Silber, M. & Proctor, M. R. E. 1998 Nonlinear competition between small and large hexagonal patterns. Phys. Rev. Lett. 81, 2450–2453.Google Scholar
Trinh, E. & Wang, T. G. 1982 Large-amplitude free and driven drop-shape oscillations: experimental observations. J. Fluid Mech. 122, 315–338.Google Scholar
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulation of Gas–Liquid Multiphase Flows. Cambridge University Press.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 514–537.Google Scholar

Ebo-Adou et al. supplementary movie 1

Visualisation of $\ell=1$ mode for spherical drop. The drop is displaced alternately to the left and the right. Length and colors of arrows indicate the velocity of the surrounding air.

Download Ebo-Adou et al. supplementary movie 1(Video)
Video 6.2 MB

Ebo-Adou et al. supplementary movie 2

Visualisation of $\ell=2$ prolate-oblate pattern of gravitational harmonic waves. Drop interface and velocity field on and outside drop are shown.

Download Ebo-Adou et al. supplementary movie 2(Video)
Video 49.5 MB

Ebo-Adou et al. supplementary movie 3

Visualisation of $\ell=3$ tetrahedral pattern of capillary subharmonic waves.

Download Ebo-Adou et al. supplementary movie 3(Video)
Video 2.3 MB

Ebo-Adou et al. supplementary movie 4a

Visualisation of $\ell=4$ cubic-octahedral pattern of capillary subharmonic waves.

Download Ebo-Adou et al. supplementary movie 4a(Video)
Video 1.4 MB

Ebo-Adou et al. supplementary movie 4b

Visualisation of $\ell=4$ axisymmetric pattern for subharmonic capillary waves. In this stroboscopic film, only snapshots at a single temporal phase are included, emphasizing the overall drift of the pattern

Download Ebo-Adou et al. supplementary movie 4b(Video)
Video 4 MB

Ebo-Adou et al. supplementary movie 5

Visualisation of $\ell=5$ mode for subharmonic capillary waves, showing evolution from axisymmetric to $D_4$ pattern.

Download Ebo-Adou et al. supplementary movie 5(Video)
Video 14.8 MB

Ebo-Adou et al. supplementary movie 6

Visualisation of $\ell=6$ icosahedral-dodecahedral pattern for subharmonic capillary waves.

Download Ebo-Adou et al. supplementary movie 6(Video)
Video 2.6 MB