Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-04-30T14:45:26.250Z Has data issue: false hasContentIssue false
  • English
  • Français

The subcritical transition to turbulence of Faraday waves in miscible fluids

Published online by Cambridge University Press:  19 January 2022

M. Cavelier ,
B.-J. Gréa Open the ORCID record for B.-J. Gréa [Opens in a new window] ,
A. Briard Open the ORCID record for A. Briard [Opens in a new window]  and
L. Gostiaux Open the ORCID record for L. Gostiaux [Opens in a new window]
M. Cavelier
Affiliation:
CEA, DAM, DIF, F-91297Arpajon, France Univ Lyon, CNRS, École Centrale de Lyon, INSA Lyon, Univ Claude Bernard Lyon 1, LMFA, UMR5509, F-69134Écully, France
B.-J. Gréa*
Affiliation:
CEA, DAM, DIF, F-91297Arpajon, France
A. Briard
Affiliation:
CEA, DAM, DIF, F-91297Arpajon, France
L. Gostiaux
Affiliation:
Univ Lyon, CNRS, École Centrale de Lyon, INSA Lyon, Univ Claude Bernard Lyon 1, LMFA, UMR5509, F-69134Écully, France
*
†Email address for correspondence: benoit-joseph.grea@cea.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the development and the breaking process of standing waves at the interface between two miscible fluids of small density contrast. In our experiment, a subharmonic wave is generated by a time-periodic vertical acceleration via the Faraday instability. It is shown that its wavelength may be selected not only by the linear process predicted by the Floquet theory and favouring the most unstable modes allowed by the tank geometry, but also by a nonlinear mode competition mechanism giving the preference to subcritical modes. Subsequently, as the standing wave amplitude grows, a secondary destabilization process occurs at smaller scales and produces turbulent mixing at the nodes. We explain this phenomenon as a subcritical parametric resonance instability. Different approaches derived from local and global stability analysis are proposed to predict the critical wave steepness. These theories are then assessed against various numerical and experimental data varying the frequencies and amplitudes of the forcing acceleration.

JFM classification

Instability: Parametric instability Instability: Transition to turbulence Waves/Free-surface Flows: Faraday waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 934 , 10 March 2022 , A34
Copyright
© The Author(s), 2022. Published by 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

REFERENCES

Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Amiroudine, S., Zoueshtiagh, F. & Narayanan, R. 2012 Mixing generated by Faraday instability between miscible liquids. Phys. Rev. E 85, 016326.CrossRefGoogle ScholarPubMed
Andrews, M.J. & Spalding, D.B. 1990 A simple experiment to investigate two-dimensional mixing by Rayleigh–Taylor instability. Phys. Fluids A 2 (6), 922–927.CrossRefGoogle Scholar
Baker, G., Caflisch, R.E. & Siegel, M. 1993 Singularity formation during Rayleigh–Taylor instability. J. Fluid Mech. 252, 51–78.CrossRefGoogle Scholar
Banner, M.L. & Peregrine, D.H. 1993 Wave breaking in deep water. Annu. Rev. Fluid. Mech. 25, 373–397.CrossRefGoogle Scholar
Bechhoefer, J., Ego, V., Manneville, S. & Johnson, B. 1995 An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech. 288, 325–350.CrossRefGoogle Scholar
Benielli, D. & Sommeria, J. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.CrossRefGoogle Scholar
Benjamin, T.B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225 (1163), 505–515.Google Scholar
Birkhoff, G. 1962 Helmholtz and Taylor Instabilities. AMS.CrossRefGoogle Scholar
Bouruet-Aubertot, P., Sommeria, J. & Staquet, C. 1995 Breaking of standing internal gravity waves through two-dimensional instabilities. J. Fluid Mech. 285, 265–301.CrossRefGoogle Scholar
Briard, A., Gostiaux, L. & Gréa, B.-J. 2020 The turbulent Faraday instability in miscible fluids. J. Fluid Mech. 883, A57.CrossRefGoogle Scholar
Briard, A., Gréa, B.-J. & Gostiaux, L. 2019 Harmonic to subharmonic transition of the Faraday instability in miscible fluids. Phys. Rev. Fluids 4, 044502.CrossRefGoogle Scholar
Caulfield, C.P. 2021 Layering, instabilities, and mixing in turbulent stratified flows. Annu. Rev. Fluid Mech. 53 (1), 113–145.CrossRefGoogle Scholar
Caulfield, C.-C.P. 1994 Multiple linear instability of layered stratified shear flow. J. Fluid Mech. 258, 255–285.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chen, P. & Viñals, J. 1999 Amplitude equation and pattern selection in Faraday waves. Phys. Rev. E 60, 559–570.CrossRefGoogle ScholarPubMed
Ciliberto, S. & Gollub, J.P. 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922–925.CrossRefGoogle Scholar
Ciliberto, S. & Gollub, J.P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.CrossRefGoogle Scholar
Daly, B.J. 1967 Numerical study of two fluid Rayleigh–Taylor instability. Phys. Fluids 10 (2), 297–307.CrossRefGoogle Scholar
Davies Wykes, M.S. & Dalziel, S.B. 2014 Efficient mixing in stratified flows: exerimental study of a Rayleigh–Taylor unstable interface within an otherwise stable stratification. J. Fluid Mech. 756, 1027–1057.CrossRefGoogle Scholar
Dimotakis, P.E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37 (1), 329–356.CrossRefGoogle Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.CrossRefGoogle Scholar
Edwards, W.S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.CrossRefGoogle Scholar
Faraday, M. 1831 On the forms and states of fluids on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–340.Google Scholar
Gaponenko, Y.A., Torregrosa, M., Yasnou, V., Mialdun, A. & Shevtsova, V. 2015 Dynamics of the interface between miscible liquids subjected to horizontal vibration. J. Fluid Mech. 784, 342–372.CrossRefGoogle Scholar
Godrèche, C. & Manneville, P. (Ed.) 2005 Hydrodynamics and Nonlinear Instabilities. Cambridge University Press.Google Scholar
Gollub, J.P. & Ramshankar, R. 1991 Spatiotemporal Chaos in Interfacial Waves, pp. 165–194. Springer.Google Scholar
Gréa, B.-J. 2013 The rapid acceleration model and the growth rate of a turbulent mixing zone induced by Rayleigh–Taylor instability. Phys. Fluids 25 (1), 015118.CrossRefGoogle Scholar
Gréa, B.-J. & Ebo Adou, A. 2018 What is the final size of turbulent mixing zones driven by the Faraday instability?. J. Fluid Mech. 837, 293–319.CrossRefGoogle Scholar
Gréa, B.-J. & Briard, A. 2019 Frozen waves in turbulent mixing layers. Phys. Rev. Fluids 4, 064608.CrossRefGoogle Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51 (1), 39–61.CrossRefGoogle Scholar
Hunt, J.C.R. & Carruthers, D.J. 1990 Rapid distortion theory and the problems of turbulence. J. Fluid Mech. 212, 497–532.CrossRefGoogle Scholar
Jiang, L., Perlin, M. & Schultz, W.W. 1998 Period tripling and energy dissipation of breaking standing waves. J. Fluid Mech. 369, 273–299.CrossRefGoogle 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.CrossRefGoogle Scholar
Kalinichenko, V.A. 2005 Development of a shear instability in nodal zones of a standing internal wave. Fluid Dyn. 40, 956–964.CrossRefGoogle Scholar
Kalinichenko, V.A. 2009 Breaking of Faraday waves and jet launch formation. Fluid Dyn. 44 (4), 577–586.CrossRefGoogle Scholar
Kelly, R.E. 1965 The stability of an unsteady Kelvin–Helmholtz flow. J. Fluid Mech. 22 (3), 547–560.CrossRefGoogle Scholar
Keulegan, G.H. 1959 Energy dissipation in standing waves in rectangular basins. J. Fluid Mech. 6 (1), 33–50.CrossRefGoogle Scholar
Khenner, M.V., Lyubimov, D.V., Belozerova, T.S. & Roux, B. 1999 Stability of plane-parallel vibrational flow in two-layer system. Eur. J. Mech. (B/Fluids) 18, 1085–1101.CrossRefGoogle Scholar
Kiger, T.K. & Duncan, J.H. 2012 Air-entrainment mechanisms in plunging jets and breaking waves. Annu. Rev. Fluid. Mech. 44, 563–596.CrossRefGoogle Scholar
Kudrolli, A. & Gollub, J.P. 1996 Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97 (1), 133–154.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L.S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Landau, L.D. & Lifshitz, E.M. 2013 Fluid Mechanics: Volume 6. Elsevier Science.Google Scholar
Longuet-Higgins, M.S. 2001 Vertical jets from standing waves; the Bazooka effect. In IUTAM Symposium on Free Surface Flows (ed. A.C. King & Y.D. Shikhmurzaev), vol. 62. Kluwer.CrossRefGoogle Scholar
Lyubimov, D.V. & Cherepanov, A. 1987 Development of a steady relief at the interface of fluids in a vibrational field. Fluid Dyn. 86, 849–854.Google Scholar
Lyubimov, D.V., Khilko, G.L., Ivantsov, A.O. & Lyubimova, T.P. 2017 Viscosity effect on the longwave instability of a fluid interface subjected to horizontal vibrations. J. Fluid Mech. 814, 24–41.CrossRefGoogle Scholar
McEwan, A.D. & Robinson, R.M. 1975 Parametric instability of internal gravity waves. J. Fluid Mech. 67 (4), 667–687.CrossRefGoogle Scholar
Meron, E. 1987 Parametric excitation of multimode dissipative systems. Phys. Rev. A 35, 4892–4895.CrossRefGoogle ScholarPubMed
Meron, E. & Procaccia, I. 1986 Low-dimensional chaos in surface waves: theoretical analysis of an experiment. Phys. Rev. A 34 (4), 3221.CrossRefGoogle ScholarPubMed
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22 (1), 143–165.CrossRefGoogle Scholar
Miles, J.W. & Benjamin, T.B. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297 (1451), 459–475.Google Scholar
Peltier, W.R. & Caulfield, C.P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
Périnet, N., Falcón, C., Chergui, J., Juric, D. & Shin, S. 2016 Hysteretic Faraday waves. Phys. Rev. E 93, 063114.CrossRefGoogle ScholarPubMed
Poulin, F.J., Flierl, G.R. & Pedlosky, J. 2003 Parametric instability in oscillatory shear flows. J. Fluid Mech. 481, 329–353.CrossRefGoogle Scholar
Rajchenbach, J. & Clamond, D. 2015 Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited. J. Fluid Mech. 777, R2.CrossRefGoogle Scholar
Salehipour, H., Peltier, W.R. & Mashayek, A. 2015 Turbulent diapycnal mixing in stratified shear flows: the influence of Prandtl number on mixing efficiency and transition at high Reynolds number. J. Fluid Mech. 773, 178–223.CrossRefGoogle Scholar
Skeldon, A.C. & Rucklidge, A.M. 2015 Can weakly nonlinear theory explain Faraday wave patterns near onset? J. Fluid Mech. 777, 604–632.CrossRefGoogle Scholar
Soliman, M.S. & Thompson, J.M.T. 1992 Indeterminate sub-critical bifurcations in parametric resonance. Proc. R. Soc. Lond. A 438 (1904), 511–518.Google Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (1), 559–593.CrossRefGoogle Scholar
Sutherland, B.R. 2010 Internal Gravity Waves. Cambridge University Press.CrossRefGoogle Scholar
Taylor, G.I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132 (820), 499–523.Google Scholar
Thorpe, S.A. 1968 On standing internal gravity waves of finite amplitude. J. Fluid Mech. 32 (3), 489–528.CrossRefGoogle Scholar
Thorpe, S.A. 1977 Turbulence and mixing in a Scottish loch. Phil. Trans. R. Soc. Lond. A 286 (1334), 125–181.Google Scholar
Winters, K., Lombard, P.N., Riley, J.J. & D'Asaro, A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–128.CrossRefGoogle Scholar
Wolf, G.H. 1970 Dynamic stabilization of the interchange instability of a liquid-gas interface. Phys. Rev. Lett. 24, 444–446.CrossRefGoogle Scholar
Wright, J., Yon, S. & Pozrikidis, C. 2000 Numerical studies of two-dimensional Faraday oscillations of inviscid fluids. J. Fluid Mech. 402, 1–32.CrossRefGoogle Scholar
Wunenburger, R., Evesque, P., Chabot, C., Garrabos, Y., Fauve, S. & Beysens, D. 1999 Frozen wave induced by high frequency horizontal vibrations on a $\mathrm {CO}_2$ liquid-gas interface near the critical point. Phys. Rev. E 59, 5440–5445.CrossRefGoogle Scholar
Yalim, J., Lopez, J.M. & Welfert, B.D. 2020 Parametrically forced stably stratified flow in a three-dimensional rectangular container. J. Fluid Mech. 900, R3.CrossRefGoogle Scholar
Yalim, J., Welfert, B.D. & Lopez, J.M. 2019 Parametrically forced stably stratified cavity flow: complicated nonlinear dynamics near the onset of instability. J. Fluid Mech. 871, 1067–1096.CrossRefGoogle Scholar
Yoshikawa, H.N. & Wesfreid, J.E. 2011 Oscillatory Kelvin–Helmholtz instability. Part 2. An experiment in fluids with a large viscosity contrast. J. Fluid Mech. 675, 249–267.CrossRefGoogle Scholar
Zhang, W. & Viñals, J. 1997 Pattern formation in weakly damped parametric surface waves driven by two frequency components. J. Fluid Mech. 341, 225–244.CrossRefGoogle Scholar
Zoueshtiagh, F., Amiroudine, S. & Narayanan, R. 2009 Experimental and numerical study of miscible Faraday instability. J. Fluid Mech. 628, 43–55.CrossRefGoogle Scholar

Cavelier et al. supplementary movie 1

Movie of experiment EXPa1 (see parameters in table 1).

Download Cavelier et al. supplementary movie 1(Video)
Video 2.3 MB

Cavelier et al. supplementary movie 2

Movie of experiment EXPb1 (see parameters in table 1).

Download Cavelier et al. supplementary movie 2(Video)
Video 2.3 MB

Cavelier et al. supplementary movie 3

Movie of a vertical slice of the concentration eld extracted from simulation DNSa2 (see parameters in table 2).

Download Cavelier et al. supplementary movie 3(Video)
Video 1 MB

Cavelier et al. supplementary movie 4

Movie of a vertical slice of the concentration eld extracted from simulation DNSa3 (see parameters in table 2).

Download Cavelier et al. supplementary movie 4(Video)
Video 1.5 MB

Cavelier et al. supplementary movie 5

Movie of a vertical slice of the concentration eld extracted from simulation DNSa8 (see parameters in table 2).

Download Cavelier et al. supplementary movie 5(Video)
Video 5.1 MB

Cavelier et al. supplementary movie 6

Movie of the interface extracted from simulation DNSf (see pa- rameters in table 2).

Download Cavelier et al. supplementary movie 6(Video)
Video 12.3 MB