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Numerical simulation of Faraday waves

Published online by Cambridge University Press:  10 September 2009

NICOLAS PÉRINET ,
DAMIR JURIC  and
LAURETTE S. TUCKERMAN
NICOLAS PÉRINET
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI), Centre National de la Recherche Scientifique (CNRS), UMR 7636, Université Paris 6 et Paris 7, 10 rue Vauquelin, 75231 Paris Cedex 5, France
DAMIR JURIC
Affiliation:
Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur (LIMSI), Centre National de la Recherche Scientifique (CNRS), UPR 3251, BP133, 91403 Orsay Cedex, France
LAURETTE S. TUCKERMAN*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI), Centre National de la Recherche Scientifique (CNRS), UMR 7636, Université Paris 6 et Paris 7, 10 rue Vauquelin, 75231 Paris Cedex 5, France
*
†Email address for correspondence: laurette@pmmh.espci.fr
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Abstract

We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier–Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, p. 49). The finite-amplitude results are compared with the experiments of Kityk et al (Phys. Rev. E, vol. 72, 2005, p. 036209). In particular, we reproduce the detailed spatio-temporal spectrum of both square and hexagonal patterns within experimental uncertainty. We present the first calculations of a three-dimensional velocity field arising from the Faraday instability for a hexagonal pattern as it varies over its oscillation period.

JFM classification

Waves/Free-surface Flows: Faraday waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 635 , 25 September 2009 , pp. 1 - 26
Copyright
Copyright © Cambridge University Press 2009

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References

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Perinet et al. supplementary movie

Movie 1. Three dimensional numerical simulation of the hexagonal Faraday wave pattern during one subharmonic period which is the fundamental period of patterns occuring within the Faraday experiment. The simulation parameters match those in the experiment of Kityk et al. (2005) where two superposed fluids are shaken vertically with a forcing acceleration a=38.0 m/s2 and frequency ω/2π=12 Hz (period T=0.0833 s). For the lower fluid ρ1 =1346 kg/m3, μ1=7.2 mPa s. For the upper fluid ρ2=949 kg/m3, μ2=20 mPa s. The surface tension at the interface is σ=35 mN/m, the total height of the container in z is 1.0 cm, and the mean height of the interface is 1.6 mm. The Floquet analysis for these parameters yields a critical wavelength of λc =2π/kc=13.2 mm. The Bond number Bo=|ρ1-ρ2| g /(σ kc2)=0.49 and the Reynolds numbers for the lower and upper fluids are Re1=ρ1/(μ1kc2 T)=9.9 and Re2=ρ2/(μ2kc2 T)=2.52 respectively. The horizontal dimensions of the calculation domain 2λc/√3 in x and 2λc in y support one spatial hexagonal period and the numerical grid resolution is 58×100×180 in the x,y,z directions respectively. For clarity each horizontal direction shown is twice that of the calculation domain. The movie begins when the interface has reached its maximum height exhibiting large peaks (up hexagon structure seen in figure 10c in Kityk et al. 2005) which are beginning to collapse. Although the interface never becomes entirely flat, its amplitude goes through a minimum; during which its structure becomes more complex. At this time, the critical modes kc compete with modes of higher wavenumber which are temporarily of similar magnitude. The initial peaks continue their descent to form wide flat craters (down hexagon structure in figure 10a in Kityk et al.) whose floors are only 0.1mm deep. The rims of these craters now collapse inwards, forming circular waves which invade the craters, whose remnants are visible as dimples. At this point the interface amplitude again goes through a minimum. The competition between the critical modes and the higher wavenumbers gives rise to a small scale six-fold flower petal pattern surrounding the dimples (figure 10b in Kityk et al.) Finally the dimples erupt upward at high velocity to reconstitute the large peaks at the beginning of the cycle. Hexagonal symmetry (reflexion and π/3 rotation invariance) is retained throughout.

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