Numerical study of pattern formation in weakly damped parametric surface waves

Wenbin Zhang, Jorge Viñals

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.

Original languageEnglish (US)
Pages (from-to)225-243
Number of pages19
JournalPhysica D: Nonlinear Phenomena
Volume116
Issue number1-2
DOIs
StatePublished - Jan 1 1998
Externally publishedYes

Fingerprint

surface waves
aspect ratio
fluids
symmetry
gravity waves
predictions
numerical analysis
viscosity
boundary conditions
expansion
wavelengths

Keywords

  • Faraday waves
  • Parametric instability
  • Quasipatterns

Cite this

Numerical study of pattern formation in weakly damped parametric surface waves. / Zhang, Wenbin; Viñals, Jorge.

In: Physica D: Nonlinear Phenomena, Vol. 116, No. 1-2, 01.01.1998, p. 225-243.

Research output: Contribution to journalArticle

@article{5196e49fe68449a0955cc85fa33663c5,
title = "Numerical study of pattern formation in weakly damped parametric surface waves",
abstract = "We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.",
keywords = "Faraday waves, Parametric instability, Quasipatterns",
author = "Wenbin Zhang and Jorge Vi{\~n}als",
year = "1998",
month = "1",
day = "1",
doi = "10.1016/S0167-2789(97)00172-3",
language = "English (US)",
volume = "116",
pages = "225--243",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "1-2",

}

TY - JOUR

T1 - Numerical study of pattern formation in weakly damped parametric surface waves

AU - Zhang, Wenbin

AU - Viñals, Jorge

PY - 1998/1/1

Y1 - 1998/1/1

N2 - We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.

AB - We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.

KW - Faraday waves

KW - Parametric instability

KW - Quasipatterns

UR - http://www.scopus.com/inward/record.url?scp=0000355977&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000355977&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(97)00172-3

DO - 10.1016/S0167-2789(97)00172-3

M3 - Article

AN - SCOPUS:0000355977

VL - 116

SP - 225

EP - 243

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -