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Axisymmetric rotating flow with free surface in a cylindrical tank

Published online by Cambridge University Press:  28 December 2018

Wen Yang Open the ORCID record for Wen Yang [Opens in a new window] ,
Ivan Delbende Open the ORCID record for Ivan Delbende [Opens in a new window] ,
Yann Fraigneau Open the ORCID record for Yann Fraigneau [Opens in a new window]  and
Laurent Martin Witkowski Open the ORCID record for Laurent Martin Witkowski [Opens in a new window]
Wen Yang
Affiliation:
Sorbonne Université, Collège Doctoral, F-75005 Paris, France LIMSI-CNRS, Bât 507, rue du Belvédère, F-91405 Orsay CEDEX, France FAST UMR 7608, Parc-Club Orsay Université, F-91405 Orsay CEDEX, France
Ivan Delbende
Affiliation:
LIMSI-CNRS, Bât 507, rue du Belvédère, F-91405 Orsay CEDEX, France Sorbonne Université, Faculté des Sciences et Ingénierie, UFR d’Ingénierie, F-75005 Paris, France
Yann Fraigneau
Affiliation:
LIMSI-CNRS, Bât 507, rue du Belvédère, F-91405 Orsay CEDEX, France
Laurent Martin Witkowski*
Affiliation:
LIMSI-CNRS, Bât 507, rue du Belvédère, F-91405 Orsay CEDEX, France Sorbonne Université, Faculté des Sciences et Ingénierie, UFR d’Ingénierie, F-75005 Paris, France
*
†Email address for correspondence: laurent.martin_witkowski@sorbonne-universite.fr
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Abstract

The flow induced by a disk rotating at the bottom of a cylindrical tank is characterised using numerical techniques – computation of steady solutions or time-averaged two-dimensional and three-dimensional direct simulations – as well as laser-Doppler velocimetry measurements. Axisymmetric steady solutions reveal the structure of the toroidal flow located at the periphery of the central solid body rotation region. When viewed in a meridional plane, this flow cell is found to be bordered by four layers, two at the solid boundaries, one at the free surface and one located at the edge of the central region, which possesses a sinuous shape. The cell intensity and geometry are determined for several fluid-layer aspect ratios; the flow is shown to depend very weakly on Froude number (associated with surface deformation) or on Reynolds number if sufficiently large. The paper then focuses on the high Reynolds number regime for which the flow has become unsteady and three-dimensional while the surface is still almost flat. Direct numerical simulations show that the averaged flow shares many similarities with the above steady axisymmetric solutions. Experimental measurements corroborate most of the numerical results and also allow for the spatio-temporal characterisation of the fluctuations, in particular the azimuthal structure and frequency spectrum. Mean azimuthal velocity profiles obtained in this transitional regime are eventually compared to existing theoretical models.

JFM classification

Boundary Layers: Boundary Layers Interfacial Flows (free surface): Interfacial Flows (free surface) Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 861 , 25 February 2019 , pp. 796 - 814
Copyright
© 2018 Cambridge University Press 

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References

Bach, B., Linnartz, E. C., Vested, M. H., Andersen, A. & Bohr, T. 2014 From Newton’s bucket to rotating polygons: experiments on surface instabilities in swirling flows. J. Fluid Mech. 759, 386–403.Google Scholar
Bergmann, R., Tophøj, L., Homan, T. A. M., Hersen, P., Andersen, A. & Bohr, T. 2011 Polygon formation and surface flow on a rotating fluid surface. J. Fluid Mech. 679, 415–431.Google Scholar
Cogan, S. J., Ryan, K. & Sheard, G. J. 2011 Symmetry breaking and instability mechanisms in medium depth torsionally open cylinder flows. J. Fluid Mech. 672, 521–544.Google Scholar
Escudier, M. P. 1984 Observations of the flow produced in a cylindrical container by a rotating end wall. Exp. Fluids 2, 189–196.Google Scholar
Fabre, D. & Mougel, J. 2014 Generation of three-dimensional patterns through wave interaction in a model of free surface swirling flow. Fluid Dyn. Res. 46 (6), 061415.Google Scholar
Herrada, M. A., Shtern, V. N. & Lopez-Herrera, J. M. 2013 Off-axis vortex breakdown in a shallow whirlpool. Phys. Rev. E 87, 063016.Google Scholar
Iga, K. 2017 Axisymmetric flow in a cylindrical tank over a rotating bottom. Part I. Analysis of boundary layers and vertical circulation. Fluid Dyn. Res. 49 (6), 065502.Google Scholar
Iga, K., Yokota, S., Watanabe, S., Ikeda, T., Niino, H. & Misawa, N. 2014 Various phenomena on a water vortex in a cylindrical tank over a rotating bottom. Fluid Dyn. Res. 46 (3), 031409.Google Scholar
Iga, K., Yokota, S., Watanabe, S., Ikeda, T., Niino, H. & Misawa, N. 2017 Axisymmetric flow in a cylindrical tank over a rotating bottom. Part II. Deformation of the water surface and experimental verification of the theory. Fluid Dyn. Res. 49 (6), 065501.Google Scholar
Iima, M. & Tasaka, Y. 2016 Dynamics of flow structures and surface shapes in the surface switching of rotating fluid. J. Fluid Mech. 789, 402–424.Google Scholar
Iwatsu, R. 2004 Analysis of flows in a cylindrical container with rotating bottom and top underformable free surface. JSME Intl J. 47 (3), 549–556.Google Scholar
Jansson, T. R. N., Haspang, M. P., Jensen, K. H., Hersen, P. & Bohr, T. 2006 Polygons on a rotating fluid surface. Phys. Rev. Lett. 96, 174502.Google Scholar
Kahouadji, L.2011 Analyse de stabilité linéaire d’écoulements tournants en présence de surface libre. PhD thesis, Université Pierre et Marie Curie, Paris, France.Google Scholar
Kahouadji, L. & Martin Witkowski, L. 2014 Free surface due to a flow driven by a rotating disk inside a vertical cylindrical tank: axisymmetric configuration. Phys. Fluids 26, 072105.Google Scholar
Kahouadji, L., Martin Witkowski, L. & Le Quéré, P. 2010 Seuils de stabilité pour un écoulement à surface libre engendré dans une cavité cylindrique tournante à petit rapport de forme. Mécanique et Industries 11, 339–344.Google Scholar
von Kármán, T. 1921 Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1, 233–252.Google Scholar
Lopez, J. M., Marques, F., Hirsa, A. H. & Miraghaie, R. 2004 Symmetry breaking in free-surface cylinder flows. J. Fluid Mech. 502, 99–126.Google Scholar
Moisy, F., Bouvard, J. & Herreman, W. 2018 Counter-rotation in an orbitally shaken glass of beer. Europhys. Lett. 122 (3), 34002.Google Scholar
Moisy, F., Doaré, O., Pasutto, T., Daube, O. & Rabaud, M. 2004 Experimental and numerical study of the shear layer instability between two counter-rotating disks. J. Fluid Mech. 507, 175–202.Google Scholar
Mougel, J.2014 Ondes et instabilités dans les écoulements tournants à surface libre. PhD thesis, Université Toulouse 3 Paul Sabatier, Toulouse, France.Google Scholar
Mougel, J., Fabre, D., Lacaze, L. & Bohr, T. 2017 On the instabilities of a potential vortex with a free surface. J. Fluid Mech. 824, 230–264.Google Scholar
Peaudecerf, F. J., Landel, J. R., Goldstein, R. E. & Luzzatto-Fegiz, P. 2017 Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces. Proc. Natl Acad. Sci. USA 114 (28), 7254–7259.Google Scholar
Piva, M. & Meiburg, E. 2005 Steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall. Phys. Fluids 17 (6), 063603(12).Google Scholar
Poncet, S. & Chauve, M. P. 2007 Shear-layer instability in a rotating system. J. Flow Visual. Image Process. 14 (1), 85–105.Google Scholar
Serre, E. & Bontoux, P. 2007 Vortex breakdown in a cylinder with a rotating bottom and a flat stress-free surface. Intl J. Heat Fluid Flow 28, 229–248.Google Scholar
Suzuki, T., Iima, M. & Hayase, Y. 2006 Surface switching of rotating fluid in a cylinder. Phys. Fluids 18, 101701.Google Scholar
Tasaka, Y. & Iima, M. 2009 Flow transitions in the surface switching of rotating fluid. J. Fluid Mech. 636, 475–484.Google Scholar
Tasaka, Y. & Iima, M. 2017 Surface switching statistics of rotating fluid: disk-rim gap effects. Phys. Rev. E 95, 043113.Google Scholar
Tophøj, L., Mougel, J., Bohr, T. & Fabre, D. 2013 Rotating polygon instability of a swirling free surface flow. Phys. Rev. Lett. 110, 194502.Google Scholar
Tuerke, F., Pastur, L., Fraigneau, Y., Sciamarella, D., Lusseyran, F. & Artana, G. 2017 Nonlinear dynamics and hydrodynamic feedback in two-dimensional double cavity flow. J. Fluid Mech. 813, 1–22.Google Scholar
Vatistas, G. H. 1990 A note on liquid vortex sloshing and Kelvin’s equilibria. J. Fluid Mech. 217, 241–248.Google Scholar
Young, D. L., Sheen, H. J. & Hwu, T. Y. 1995 Period-doubling route to chaos for a swirling flow in an open cylindrical container with a rotating disk. Exp. Fluids 18, 389–396.Google Scholar

Yang et al. supplementary movie 1

Movie of the experiment at G = 0.1856, Re = 30000 and Fr = 0.0335: top view of the water layer seeded with Kalliroscope flakes. The red point has been artificially inserted and rotates at the disk angular speed. A m = 3 pattern is clearly identified.

Download Yang et al. supplementary movie 1(Video)
Video 8.8 MB

Yang et al. supplementary movie 2

Same parameters as movie 1, viewed in a meridional plane. One can observe that the liquid surface is horizontal and remains flat. The turbulent recirculation region can be easily seen in the radial range r=0.6 to r=1, in fair agreement with the numerical results shown in movie3. The movie has been slowed down by a factor 1.5 compared to the real time experiment.

Download Yang et al. supplementary movie 2(Video)
Video 38.9 MB

Yang et al. supplementary movie 3

Temporal evolution of the azimuthal vorticity in a meridional plane, taken from the 3D numerical simulation at G=0.1856, Re=30000 and Fr=0. Time is scaled back to dimensional value and slowed down so that the movie is running at the same speed as in Movie 2.

Download Yang et al. supplementary movie 3(Video)
Video 2.7 MB