Skip to main content
×
×
Home

Streaming patterns in Faraday waves

  • Nicolas Périnet (a1), Pablo Gutiérrez (a1) (a2), Héctor Urra (a3), Nicolás Mujica (a1) and Leonardo Gordillo (a4)...
Abstract

Wave patterns in the Faraday instability have been studied for decades. Besides the rich wave dynamics observed at the interface, Faraday waves hide elusive flow patterns in the bulk – streaming patterns – which have not been studied experimentally. The streaming patterns are responsible for a net circulation in the flow, which is reminiscent of the circulation in convection cells. In this article, we analyse these streaming flows by conducting experiments in a Faraday-wave set-up using particle image velocimetry. To visualise the flows, we perform stroboscopic measurements to both generate trajectory maps and probe the streaming velocity field. We identify three types of patterns and experimentally show that identical Faraday waves can mask streaming patterns that are qualitatively very different. Next, we consider a three-dimensional model for streaming flows in quasi-inviscid fluids, whose key is the complex coupling occurring at all of the viscous boundary layers. This coupling yields modified boundary conditions in a three-dimensional Navier–Stokes formulation of the streaming flow. Numerical simulations based on this framework show reasonably good agreement, both qualitative and quantitative, with the velocity fields of our experiments. The model highlights the relevance of three-dimensional effects in the streaming patterns. Our simulations also reveal that the variety of streaming patterns is deeply linked to the boundary condition at the top interface, which may be strongly affected by the presence of contaminants.

Copyright
Corresponding author
Email address for correspondence: leonardo.gordillo@usach.cl
References
Hide All
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.
Chen, P., Luo, Z., Güven, S., Tasoglu, S., Ganesan, A. V., Weng, A. & Demirci, U. 2014 Microscale assembly directed by liquid-based template. Adv. Mater. 26 (34), 59365941.
Chladni, E. F. F. 1787 Entdeckungen über die Theorie des Klangen. Wittenberg.
Chorin, A. J. 1968 Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745762.
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385425.
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.
Falkovich, G., Weinberg, A., Denissenko, P. & Lukaschuk, S. 2005 Surface tension: floater clustering in a standing wave. Nature 435 (7045), 10451046.
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. A 121, 299340.
van Gerner, H. J., van der Weele, K., van der Hoef, M. A. & van der Meer, D. 2011 Air-induced inverse Chladni patterns. J. Fluid Mech. 689, 203220.
Gordillo, L.2012 Non-propagating hydrodynamic solitons in a quasi-one-dimensional free surface subject to vertical vibrations. PhD thesis, Universidad de Chile, Santiago.
Gordillo, L. & Mujica, N. 2014 Measurement of the velocity field in parametrically excited solitary waves. J. Fluid Mech. 754, 590604.
Gutiérrez, P. & Aumaître, S. 2016 Clustering of floaters on the free surface of a turbulent flow: an experimental study. Eur. J. Mech. B 60, 2432.
Henderson, D. & Miles, J. W. 1994 Surface wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 275, 285299.
Higuera, M. & Knobloch, E. 2006 Nearly inviscid Faraday waves in slightly rectangular containers. Prog. Theor. Phys. Supp. 161, 5367.
Higuera, M., Vega, J. M. & Knobloch, E. 2002 Coupled amplitude–streaming flow equations for nearly inviscid Faraday waves in small aspect ratio containers. J. Nonlinear Sci. 12 (5), 505551.
Lamb, H. 2006 Hydrodynamics, 6th edn. Cambridge Mathematical Library. Cambridge University Press.
Lide, D. R. 2004 Handbook of Chemistry and Physics, 85th edn. Chemical Rubber Company Press.
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535.
Martin, E., Martel, C. & Vega, J. M. 2002 Drift instability of standing Faraday waves. J. Fluid Mech. 467, 5779.
Martin, E. & Vega, J. M. 2005 The effect of surface contamination on the drift instability of standing Faraday waves. J. Fluid Mech. 546, 203225.
Meinhart, C. D., Wereley, S. T. & Santiago, J. G. 2000 A PIV algorithm for estimating time-averaged velocity fields. Trans. ASME J. Fluids Engng 122 (2), 285289.
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459.
Miles, J. W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419448.
Miles, J. W. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143165.
Nicolás, J. A. & Vega, J. M. 2000 A note on the effect of surface contamination in water wave damping. J. Fluid Mech. 410, 367373.
Nicolás, J. A. & Vega, J. M. 2003 Three-dimensional streaming flows driven by oscillatory boundary layers. Fluid Dyn. Res. 32 (4), 119139.
Raffel, M., Willert, C. E., Wereley, S. T. & Kompenhans, J. 2007 Particle Image Velocimetry: A Practical Guide, 2nd edn. Springer.
Riley, N. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33, 4365.
Sanli, C., Lohse, D. & van der Meer, D. 2014 From antinode clusters to node clusters: the concentration-dependent transition of floaters on a standing Faraday wave. Phys. Rev. E 89 (5), 053011.
Schlichting, H. 1932 Berechnung ebener periodischer Grenzschichtströmungen. Phys. Z 33 (1932), 327335.
Schlichting, H. T. & Gersten, K. 1999 Boundary Layer Theory, 8th edn. Springer.
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441473.
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8.
Strickland, S. L., Shearer, M. & Daniels, K. E. 2015 Spatiotemporal measurement of surfactant distribution on gravity–capillary waves. J. Fluid Mech. 777, 523543.
Van Dyke, M. 1982 An Album of Fluid Motion. Parabolic.
Vega, J. M., Knobloch, E. & Martel, C. 2001 Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio. Physica D 154 (3), 313336.
Wallet, A. & Ruellan, F. 1950 Trajectoires Internes Dans un Clapotis Partiel, vol. 5. La Houille Blanche.
Wright, P. H. & Saylor, J. R. 2003 Patterning of particulate films using Faraday waves. Rev. Sci. Instrum. 74 (9), 4063.
Wu, J., Keolian, R. & Rudnick, I. 1984 Observation of a nonpropagating hydrodynamic soliton. Phys. Rev. Lett. 52 (1), 14211424.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 9
Total number of PDF views: 206 *
Loading metrics...

Abstract views

Total abstract views: 767 *
Loading metrics...

* Views captured on Cambridge Core between 21st April 2017 - 21st September 2018. This data will be updated every 24 hours.