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Numerical simulation of supersquare patterns in Faraday waves

Published online by Cambridge University Press:  08 May 2015

L. Kahouadji
Affiliation:
PMMH (UMR 7636 CNRS - ESPCI - UPMC Paris 6 - UPD Paris 7 - PSL), 10 rue Vauquelin, 75005 Paris, France Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
N. Périnet
Affiliation:
Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile
L. S. Tuckerman*
Affiliation:
PMMH (UMR 7636 CNRS - ESPCI - UPMC Paris 6 - UPD Paris 7 - PSL), 10 rue Vauquelin, 75005 Paris, France
S. Shin
Affiliation:
Department of Mechanical and System Design Engineering, Hongik University, Seoul 121-791, Republic of Korea
J. Chergui
Affiliation:
LIMSI-CNRS, Bât 508, rue John von Neumann - 91405 Orsay, France
D. Juric
Affiliation:
LIMSI-CNRS, Bât 508, rue John von Neumann - 91405 Orsay, France
*
Email address for correspondence: laurette@pmmh.espci.fr

Abstract

We report the first simulations of the Faraday instability using the full three-dimensional Navier–Stokes equations in domains much larger than the characteristic wavelength of the pattern. We use a massively parallel code based on a hybrid front-tracking/level-set algorithm for Lagrangian tracking of arbitrarily deformable phase interfaces. Simulations performed in square and cylindrical domains yield complex patterns. In particular, a superlattice-like pattern similar to those of Douady & Fauve (Europhys. Lett., vol. 6, 1988, pp. 221–226) and Douady (J. Fluid Mech., vol. 221, 1990, pp. 383–409) is observed. The pattern consists of the superposition of two square superlattices. We conjecture that such patterns are widespread if the square container is large compared with the critical wavelength. In the cylinder, pentagonal cells near the outer wall allow a square-wave pattern to be accommodated in the centre.

Type
Rapids
Copyright
© 2015 Cambridge University Press 

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References

Arbell, H. & Fineberg, J. 2002 Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65, 036224.CrossRefGoogle ScholarPubMed
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013 The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496523.CrossRefGoogle Scholar
Chorin, A. J. 1968 Numerical simulation of the Navier–Stokes equations. Maths Comput. 22, 745762.CrossRefGoogle Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381398.CrossRefGoogle Scholar
Crawford, J. D. 1991 Normal forms for driven surface waves: boundary conditions, symmetry, and genericity. Physica D 52, 429457.CrossRefGoogle Scholar
Crawford, J. D., Knobloch, E. & Riecke, H. 1990 Period-doubling mode interactions with circular symmetry. Physica D 44, 340396.CrossRefGoogle Scholar
Das, S. P. & Hopfinger, E. J. 2008 Parametrically forced gravity waves in a circular cylinder and finite-time singularity. J. Fluid Mech. 599, 205228.CrossRefGoogle Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.CrossRefGoogle Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Europhys. Lett. 6, 221226.CrossRefGoogle 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. 121, 299340.CrossRefGoogle Scholar
Gomes, M. G. M. & Stewart, I. 1994 Steady PDEs on generalized rectangles: a change of genericity in mode interactions. Nonlinearity 7, 253272.CrossRefGoogle 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.CrossRefGoogle Scholar
Kudrolli, A., Pier, B. & Gollub, J. P. 1998 Superlattice patterns in surface waves. Physica D 123, 99111; See arXiv:chao-dyn/9803016 for more legible versions of figures.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4968.CrossRefGoogle Scholar
Kwak, D. Y. & Lee, J. S. 2004 Multigrid algorithm for the cell-centered finite-difference method II: discontinuous coefficient case. Numer. Meth. Part. Differ. Equ. 20, 723741.CrossRefGoogle Scholar
Meron, E. 1987 Parametric excitation of multimode dissipative systems. Phys. Rev. A 35, 48924895.CrossRefGoogle ScholarPubMed
Périnet, N., Juric, D. & Tuckerman, L. S. 2009 Numerical simulation of Faraday waves. J. Fluid Mech. 635, 126.CrossRefGoogle Scholar
Périnet, N., Juric, D. & Tuckerman, L. S. 2012 Alternating hexagonal and striped patterns in Faraday waves. Phys. Rev. Lett. 109, 164501.CrossRefGoogle Scholar
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220252.CrossRefGoogle Scholar
Rajchenbach, J., Clamond, D. & Leroux, A. 2013 Observation of star-shaped surface gravity waves. Phys. Rev. Lett. 110, 094502.Google ScholarPubMed
Shin, S. 2007 Computation of the curvature field in numerical simulation of multiphase flow. J. Comput. Phys. 222, 872878.CrossRefGoogle Scholar
Shin, S., Chergui, J. & Juric, D. 2014 A solver for massively parallel direct numerical simulation of three-dimensional multiphase flows. Comput. Fluids (submitted) arXiv:1410.8568 [physics.flu-dyn].Google Scholar
Shin, S. & Juric, D. 2009a A hybrid interface method for three-dimensional multiphase flows based on front-tracking and level set techniques. Intl J. Numer. Meth. Fluids 60, 753778.CrossRefGoogle Scholar
Shin, S. & Juric, D. 2009b Simulation of droplet impact on a solid surface using the level contour reconstruction method. J. Mech. Sci. Technol. 23, 24342443.CrossRefGoogle Scholar
Shu, C. W. & Osher, S. 1989 Efficient implementation of essentially non-oscillatory shock capturing schemes, II. J. Comput. Phys. 83, 3278.CrossRefGoogle Scholar
Silber, M. & Knobloch, E. 1989 Parametrically excited surface waves in square geometry. Phys. Lett. A 137, 349354.CrossRefGoogle Scholar
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471494.CrossRefGoogle Scholar

Kahouadji et al. supplementary movie

Evolution of interface height (left) and vertical velocity (right) over one oscillation period in a square container.

Download Kahouadji et al. supplementary movie(Video)
Video 13 MB

Kahouadji et al. supplementary movie

Evolution of interface height (left) and vertical velocity (right) over one oscillation period in a cylindrical container.

Download Kahouadji et al. supplementary movie(Video)
Video 14 MB
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