Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-01T01:34:30.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonances and instabilities in a tilted rotating annulus

Published online by Cambridge University Press:  20 October 2023

S. Scollo Open the ORCID record for S. Scollo [Opens in a new window] ,
C. Nobili ,
E. Villermaux Open the ORCID record for E. Villermaux [Opens in a new window]  and
P. Meunier Open the ORCID record for P. Meunier [Opens in a new window]
S. Scollo*
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, 13384 Marseille, France
C. Nobili
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, 13384 Marseille, France
E. Villermaux
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, 13384 Marseille, France Institut Universitaire de France, 75005 Paris, France
P. Meunier
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, 13384 Marseille, France
*
†Email address for correspondence: scollo@irphe.univ-mrs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow inside a rotating annulus tilted with respect to gravity is characterized experimentally and theoretically. As in the case of a tilted rotating cylinder the flow is forced by the free surface, maintained flat by gravity. It leads to resonances of global inertial modes (Kelvin modes) when the height of fluid is a multiple of half the wavelength of the mode. The divergence of the mode is saturated by viscous effects at the resonance. The maximum amplitude scales as the Ekman number to the power $-1/2$ when surface Ekman pumping is dominant, and to the power $-1$ when volumic damping is dominant. An analytical prediction is given with no fitting parameter, in excellent agreement with experimental results. At lower Ekman numbers, the flow destabilizes with respect to a triadic resonance instability, as already observed by Xu & Harlander (Phys. Rev. Fluids, 2020). We provide here a linear stability analysis leading to the viscous threshold of the instability for small tilt angles. For large tilt angles, a centrifugal instability is observed due to the acceleration of the flow by the inner cylinder. Finally, the features of the turbulent flow and its mixing efficiency are characterized experimentally. We underline the potential interest of this configuration for bioreactors.

JFM classification

Geophysical and Geological Flows: Rotating flows Instability: Parametric instability Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 973 , 25 October 2023 , A34
Check for updates
Copyright
© The Author(s), 2023. 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

Albrecht, T., Blackburn, H., Lopez, J., Manasseh, R. & Meunier, P. 2015 Triadic resonances in precessing rapidly rotating cylinder flows. J. Fluid Mech. 778, R11–R112.CrossRefGoogle Scholar
Albrecht, T., Blackburn, H., Lopez, J., Manasseh, R. & Meunier, P. 2018 On triadic resonance as an instability mechanism in precessing cylinder flow. J. Fluid Mech. 841, R3.CrossRefGoogle Scholar
Albrecht, T., Blackburn, H., Lopez, J., Manasseh, R. & Meunier, P. 2021 On the origins of steady streaming in precessing fluids. J. Fluid Mech. 910, A51.CrossRefGoogle Scholar
Batchelor, G.K. 1959 Small-scale variation of convected quantities like temperature in a turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5 (1), 113–133.CrossRefGoogle Scholar
Cherry, R.S. & Papoutsakis, E.T. 1986 Hydrodynamic effects on cells in agitated tissue culture reactors. Bioprocess Engng 1 (1), 29–41.CrossRefGoogle Scholar
Doran, P.M. 1999 Design of mixing systems for plant cell suspensions in stirred reactors. Biotechnol. Prog. 15 (3), 319–335.10.1021/bp990042vCrossRefGoogle ScholarPubMed
Ekman, V.W. 1905 On the influence of the Earth's rotation on ocean-currents. Arkiv for Matematik, Astronomi och Fysik 2 (11), 1–52.Google Scholar
Gans, R.F. 1970 On the precession of a resonant cylinder. J. Fluid Mech 476, 865–872.CrossRefGoogle Scholar
Gao, D., Meunier, P., Le Dizès, S. & Eloy, C. 2021 Zonal flow in a resonant precessing cylinder. J. Fluid Mech. 923, A29.CrossRefGoogle Scholar
Garcia-Ochoa, F. & Gomez, E. 2009 Bioreactor scale-up and oxygen transfer rate in microbial processes: an overview. Biotechnol. Adv. 27 (2), 153–176.CrossRefGoogle ScholarPubMed
Gauthier, G., Gondret, P. & Rabaud, M. 1998 Motions of anisotropic particles: application to visualization of three-dimensional flows. Phys. Fluids 10 (9), 2147–2154.10.1063/1.869736CrossRefGoogle Scholar
Giesecke, A., Albrecht, T., Gundrum, T., Herault, J. & Stefani, F. 2015 Triadic resonances in nonlinear simulations of a fluid flow in a precessing cylinder. New J. Phys. 17 (11), 113044.CrossRefGoogle Scholar
Giesecke, A., Vogt, T., Gundrum, T. & Stefani, F. 2018 Nonlinear large scale flow in a precessing cylinder and its ability to drive dynamo action. Phys. Rev. Lett. 120 (2), 024502.CrossRefGoogle Scholar
Goto, S., Shimizu, M. & Kawahara, G. 2014 Turbulent mixing in a precessing sphere. Phys. Fluids 26 (11), 115106.CrossRefGoogle Scholar
Greenspan, H.P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H.P. 1969 On the lnviscid theory of rotating fluids. Stud. Appl. Maths 48, 19–28.CrossRefGoogle Scholar
Heraulta, J., Gundrum, T., Giesecke, A. & Stefani, F. 2015 Subcritical transition to turbulence of a precessing flow in a cylindrical vessel. Phys. Fluids 27 (12), 124102.CrossRefGoogle Scholar
Kelvin, L. 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Kerswell, R. & Barenghi, C. 1995 On the viscous decay rates of inertial waves in a rotating circular cylinder. J. Fluid Mech. 285, 203–214.CrossRefGoogle Scholar
Kerswell, R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.CrossRefGoogle Scholar
Lagrange, R., Eloy, C., Nadal, F. & Meunier, P. 2008 Instability of a fluid inside a precessing cylinder. Phys. Fluids 20 (8), 081701.CrossRefGoogle Scholar
Lagrange, R., Meunier, P., Nadal, F. & Eloy, C. 2011 Precessional instability of a fluid cylinder. Exp. Fluids 666, 104–145.Google Scholar
Le Bars, M., Cébron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193.10.1146/annurev-fluid-010814-014556CrossRefGoogle Scholar
Lin, Y., Noir, J. & Jackson, A. 2014 Experimental study of fluid flows in a precessing cylindrical annulus. Phys. Fluids 26 (4), 046604.10.1063/1.4871026CrossRefGoogle Scholar
Lopez, J. & Marques, F. 2016 Nonlinear and detuning effects of the nutation angle in precessionally forced rotating cylinder flow. Phys. Rev. Fluids 1, 023602.CrossRefGoogle Scholar
Lopez, J. & Marques, F. 2018 Rapidly rotating precessing cylinder flows: forced triadic resonances. J. Fluid Mech. 839, 239–270.CrossRefGoogle Scholar
Manasseh, R. 1992 Breakdown regimes of inertia waves in a precessing cylinder. J. Fluid Mech. 243, 261–296.CrossRefGoogle Scholar
Manasseh, R. 1994 Distorsions of inertia waves in a precessing cylinder forced near its fundamental mode resonance. J. Fluid Mech. 265, 345–370.CrossRefGoogle Scholar
Marques, F. & Lopez, J. 2015 Precession of a rapidly rotating cylinder flow: traverse through resonance. J. Fluid Mech. 782, 63–98.CrossRefGoogle Scholar
McEwan, A. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40 (3), 603–640.10.1017/S0022112070000344CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2003 Analysis and minimization of errors due to high gradients in particule image velocimetry. Exp. Fluids 35 (5), 408–421.CrossRefGoogle Scholar
Meunier, P., Eloy, C., Lagrange, R. & Nadal, F. 2008 A rotating fluid cylinder subject to weak precession. J. Fluid Mech. 599, 405–440.CrossRefGoogle Scholar
Meunier, P. 2020 Geoinspired soft mixers. J. Fluid Mech. 903, A15.CrossRefGoogle Scholar
Nagata, S. 1975 Mixing: Principles and Applications. Halsted.Google Scholar
Pizzi, F., Giesecke, A., Šimkanin, J. & Stefani, F. 2021 Prograde and retrograde precession of a fluid-filled cylinder. New J. Phys. 23 (12), 123016.CrossRefGoogle Scholar
Rayleigh, Lord 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Taylor, G.I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289.Google Scholar
Thompson, R. 1970 Diurnal tides and shear instabilities in a rotating cylinder. J. Fluid Mech. 40, 737–751.10.1017/S0022112070000411CrossRefGoogle Scholar
Villermaux, E. 2019 Mixing versus stirring. Annu. Rev. Fluid Mech. 51, 245–273.CrossRefGoogle Scholar
Watanabe, D. & Goto, S. 2022 Simple bladeless mixer with liquid–gas interface. Flow 2, E28.CrossRefGoogle Scholar
Xu, W. & Harlander, U. 2020 Inertial mode interactions in a rotating tilted cylindrical annulus with free surface. Phys. Rev. Fluids 5, 9.CrossRefGoogle Scholar