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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: [email protected]
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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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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