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What is the final size of turbulent mixing zones driven by the Faraday instability?

Published online by Cambridge University Press:  21 December 2017

B.-J. Gréa Open the ORCID record for B.-J. Gréa [Opens in a new window]  and
A. Ebo Adou
B.-J. Gréa*
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France
A. Ebo Adou
Affiliation:
CMLA, ENS Cachan, France
*
†Email address for correspondence: benoit-joseph.grea@cea.fr
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Abstract

Miscible fluids of different densities subjected to strong time-periodic accelerations normal to their interface can mix due to Faraday instability effects. Turbulent fluctuations generated by this mechanism lead to the emergence and the growth of a mixing layer. Its enlargement is gradually slowed down as the resonance conditions driving the instability cease to be fulfilled. The final state corresponds to a saturated mixing zone in which the turbulence intensity progressively decays. A new formalism based on second-order correlation spectra for the turbulent quantities is introduced for this problem. This method allows for the prediction of the final mixing zone size and extends results from classical stability analysis limited to weakly nonlinear regimes. We perform at various forcing frequencies and amplitudes a large set of homogeneous and inhomogeneous numerical simulations, extensively exploring the influence of initial conditions. The mixing zone widths, measured at the end of the simulations, are satisfactorily compared to the predictions, and bring a strong support to the proposed theory. The flow dynamics is also studied and reveals the presence of sub-harmonic as well as harmonic modes depending on the initial parameters in the Mathieu phase diagram. Important changes in the flow anisotropy, corresponding to the large scale structures of turbulence, occur. This phenomenon appears directly related to the orientation of the most amplified gravity waves excited in the system, evolving due to the enlargement of the mixing zone.

JFM classification

Convection: Buoyancy-driven instability Turbulent Flows: Stratified turbulence Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 293 - 319
Copyright
© 2017 Cambridge University Press 

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References

Amiroudine, S., Zoueshtiagh, F. & Narayanan, R. 2012 Mixing generated by Faraday instability between miscible liquids. Phys. Rev. E 85, 016326.Google Scholar
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.Google Scholar
Arbell, H. & Fineberg, J. 2002 Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65, 036224.Google Scholar
Batchelor, G. K., Canuto, V. M. & Chasnov, J. R. 1992 Homogeneous buoyancy-generated turbulence. J. Fluid Mech. 235, 349–378.Google Scholar
Batchelor, G. K. & Nitsche, J. M. 1991 Instability of stationary unbounded stratified fluid. J. Fluid Mech. 227, 357–391.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, reprinted by Springer (1999).Google Scholar
Benielli, D. & Sommeria, J. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.10.1017/S0022112098002602Google 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
Binks, D. & van de Water, W. 1997 Nonlinear pattern formation of Faraday waves. Phys. Rev. Lett. 78, 4043–4046.Google Scholar
Bosch, E. & van de Water, W. 1993 Spatiotemporal intermittency in the Faraday experiment. Phys. Rev. Lett. 70, 3420–3423.10.1103/PhysRevLett.70.3420Google Scholar
Burlot, A., Gréa, B.-J., Godeferd, F. S., Cambon, C. & Griffond, J. 2015 Spectral modelling of high Reynolds number unstably stratified homogeneous turbulence. J. Fluid Mech. 765, 17–44.10.1017/jfm.2014.726Google Scholar
Chen, P. & Viñals, J. 1999 Amplitude equation and pattern selection in Faraday waves. Phys. Rev. E 60, 559–570.Google Scholar
Chung, D. & Pullin, D. I. 2010 Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence. J. Fluid Mech. 643, 279–308.10.1017/S0022112009992801Google Scholar
Cook, A. W. & Dimotakis, P. E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 69–99.Google Scholar
Dimonte, G., Youngs, D. L., Dimits, A., Weber, S., Marinak, M., Wunsch, S., Garasi, C., Robinson, A., Andrews, M. J., Ramaprabhu, P. et al. 2004 A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: the Alpha-Group collaboration. Phys. Fluids 16 (5), 1668–1693.Google Scholar
Diwakar, S. V., Zoueshtiagh, F., Amiroudine, S. & Narayanan, R. 2015 The Faraday instability in miscible fluid systems. Phys. Fluids 27 (8), 084111.10.1063/1.4929401Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.Google Scholar
Falcón, C. & Fauve, S. 2009 Wave–vortex interaction. Phys. Rev. E 80, 056213.Google 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.Google Scholar
Godrèche, C. & Manneville, P.(Eds) 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.Google Scholar
Griffond, J., Gréa, B.-J. & Soulard, O. 2014 Unstably stratified homogeneous turbulence as a tool for turbulent mixing modeling. Trans. ASME J. Fluids Engng 136, 091201.10.1115/1.4025675Google Scholar
Hanazaki, H. & Hunt, J. C. R. 1996 Linear processes in unsteady stably stratified turbulence. J. Fluid Mech. 318, 303–337.10.1017/S0022112096007136Google Scholar
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and the problems of turbulence. J. Fluid Mech. 212, 497–532.10.1017/S0022112090002075Google Scholar
Jacobs, J. W., Krivets, V. V., Tsiklashvili, V. & Likhachev, O. A. 2013 Experiments on the Richtmyer–Meshkov instability with an imposed, random initial perturbation. Shock Waves 23, 407–413.10.1007/s00193-013-0436-9Google 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), 133–154.10.1016/0167-2789(96)00099-1Google Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.10.1017/S0022112094003812Google Scholar
Livescu, D. & Ristorcelli, J. R. 2007 Buoyancy-driven variable-density turbulence. J. Fluid Mech. 591, 43–71.Google Scholar
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22 (1), 143–165.Google Scholar
Morgan, B. E., Olson, B. J., White, J. E. & McFarland, J. A. 2017 Self-similarity of a Rayleigh–Taylor mixing layer at low Atwood number with a multimode initial perturbation. J. Turbul. 18 (10), 973–999.Google Scholar
Müller, H. W. 1994 Model equations for two-dimensional quasipatterns. Phys. Rev. E 49, 1273–1277.Google Scholar
Nazarenko, S., Kevlahan, N. K.-R. & Dubrulle, B. 1999 WKB theory for rapid distortion of inhomogeneous turbulence. J. Fluid Mech. 390, 325–348.10.1017/S0022112099005340Google Scholar
Poujade, O. & Peybernes, M. 2010 Growth rate of Rayleigh–Taylor turbulent mixing layers with the foliation approach. Phys. Rev. E 81, 016316.Google Scholar
Simonelli, F. & Gollub, J. P. 1989 Surface wave mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471–494.Google Scholar
Skeldon, A. C. & Guidoboni, G. 2007 Pattern selection for Faraday waves in an incompressible viscous fluids. SIAM J. Appl. Maths 67, 1064–1100.10.1137/050639223Google Scholar
Skeldon, A. C. & Rucklidge, A. M. 2015 Can weakly nonlinear theory explain Faraday wave patterns near onset? J. Fluid Mech. 777, 604–632.10.1017/jfm.2015.388Google Scholar
Soulard, O., Griffond, J. & Gréa, B.-J. 2015 Large-scale analysis of unconfined self-similar Rayleigh–Taylor turbulence. Phys. Fluids 27 (9), 095103.Google Scholar
Thornber, B. 2016 Impact of domain size and statistical errors in simulations of homogeneous decaying turbulence and the Richtmyer–Meshkov instability. Phys. Fluids 28 (4), 045106.Google Scholar
Watteaux, R.2011 Détection des grandes structures turbulentes dans les couches de mélange de type Rayleigh–Taylor en vue de la validation de modèles statistiques turbulents bi-structure. PhD thesis, ENS Cachan.Google 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.Google Scholar
Zoueshtiagh, F., Amiroudine, S. & Narayanan, R. 2009 Experimental and numerical study of miscible Faraday instability. J. Fluid Mech. 628, 43–55.Google Scholar

Gréa et al. supplementary movie 1

SHT 512^3 simulation with F=1: (Top) Time evolution of the mixing layer. (Bottom) Turbulent fluctuations of concentration in the periodic domain.

Download Gréa et al. supplementary movie 1(Video)
Video 36.6 MB

Gréa et al. supplementary movie 2

SHT 1024^3 simulation with F=1: (Top) Time evolution of the mixing layer. (Bottom left) Turbulent fluctuations of concentration in the periodic domain. (Bottom right) Spectra of kinetic energy and concentration.

Download Gréa et al. supplementary movie 2(Video)
Video 26.4 MB

Gréa et al. supplementary movie 3

MZ 512^3 simulation: (Top) time evolution of the mixing layer. (Bottom) Density field.

Download Gréa et al. supplementary movie 3(Video)
Video 4.8 MB