Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-08T06:06:53.282Z Has data issue: false hasContentIssue false
  • English
  • Français

Radiative instability of the flow around a rotating cylinder in a stratified fluid

Published online by Cambridge University Press:  07 February 2011

XAVIER RIEDINGER ,
STÉPHANE LE DIZÈS  and
PATRICE MEUNIER
XAVIER RIEDINGER
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS/Universités Aix-Marseille I and II, 49, rue F. Joliot-Curie, BP 146, F-13384 Marseille CEDEX 13, France
STÉPHANE LE DIZÈS*
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS/Universités Aix-Marseille I and II, 49, rue F. Joliot-Curie, BP 146, F-13384 Marseille CEDEX 13, France
PATRICE MEUNIER
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS/Universités Aix-Marseille I and II, 49, rue F. Joliot-Curie, BP 146, F-13384 Marseille CEDEX 13, France
*
†Email address for correspondence: ledizes@irphe.univ-mrs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of the flow around a rotating cylinder in a fluid linearly stratified along the cylinder axis is studied numerically and experimentally for moderate Reynolds numbers. The flow is assumed potential and axisymmetric with an angular velocity profile Ω = 1/r2, where r is the radial coordinate. Neglecting density diffusion and non-Boussinesq effects, the properties of the linear normal modes are first provided. A comprehensive stability diagram is then obtained for Froude numbers between 0 and 3 and Reynolds numbers below 1000. The main result is that the potential flow, which is stable for a homogeneous fluid, becomes unstable for Froude number close to one and for Reynolds numbers larger than 360. The numerical results are then compared with experimental results obtained using shadowgraph and synthetic Schlieren techniques. Two symmetrical helical modes are found to be simultaneously unstable. We show that these modes exhibit an internal gravity wave structure extending far from the cylinder in agreement with the theory. Their wavelength and frequency are shown to be in good agreement with the numerical predictions for a large range of Froude and Reynolds numbers. These experimental results are the first indisputable evidence of the radiative instability.

JFM classification

Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex instability Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 130 - 146
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Balmforth, N. J. 1999 Shear instability in shallow water. J. Fluid Mech. 387, 97–127.Google Scholar
Billant, P. & Chomaz, J.-M. 2000 Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167–188.Google Scholar
Billant, P. & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.CrossRefGoogle Scholar
Billant, P. & Le Dizès, S. 2009 Waves on a columnar vortex in a strongly stratified fluid. Phys. Fluids 21, 106602.Google Scholar
Boubnov, B. M., Gledzer, E. B. & Hopfinger, E. J. 1995 Stratified circular Couette flow: instability and flow regimes. J. Fluid Mech. 292, 333–358.Google Scholar
Boulanger, N., Meunier, P. & Le Dizès, S. 2007 Structure of a tilted stratified vortex. J. Fluid Mech. 583, 443–458.CrossRefGoogle Scholar
Boulanger, N., Meunier, P. & Le Dizès, S. 2008 Instability of a tilted vortex in stratified fluid. J. Fluid Mech. 596, 1–20.Google Scholar
Broadbent, E. G. & Moore, D. W. 1979 Acoustic destabilization of vortices. Phil. Trans. R. Soc. Lond. A 290, 353–371.Google Scholar
Caton, F., Janiaud, B. & Hopfinger, E. J. 2000 Stability and bifurcations in stratified Taylor–Couette flows. J. Fluid Mech. 419, 93–124.Google Scholar
Dubrulle, B., Marie, L., Normand, C., Richard, D., Hersant, F. & Zahn, J.-P. 2005 A hydrodynamic shear instability in stratified disks. Astron. Astrophys. 429, 1–13.CrossRefGoogle Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 The Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235–274.CrossRefGoogle Scholar
Ford, R. 1994 The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water. J. Fluid Mech. 280, 303–334.Google Scholar
Guimbard, D., Le Dizès, S., Le Bars, M., Le Gal, P. & Leblanc, S. 2010 Elliptic instability of stratified fluid in a rotating cylinder. J. Fluid Mech. 660, 240–257.Google Scholar
Hayashi, Y.-Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flows in shallow waters. J. Fluid Mech. 184, 477–504.Google Scholar
Knessl, C. & Keller, J. B. 1995 Stability of linear shear flows in shallow water. J. Fluid Mech. 303, 203–214.CrossRefGoogle Scholar
Le Bars, M. & Le Gal, P. 2007 Experimental analysis of the stratorotational instability in a cylindrical Couette flow. Phys. Rev. Lett. 99, 064502.Google Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602.Google Scholar
Le Dizès, S. & Riedinger, X. 2010 The strato-rotational instability of Taylor–Couette and Keplerian flows. J. Fluid Mech. 660, 147–161.CrossRefGoogle Scholar
Lindzen, R. S. & Barker, J. W. 1985 Instability and wave over-reflection in stably stratified shear flow. J. Fluid Mech. 151, 189–217.Google Scholar
Luo, K. H. & Sandham, N. D. 1997 Instability of vortical and acoustic modes in supersonic round jets. Phys. Fluids 9, 1003–1013.Google Scholar
Meunier, P. & Leweke, T. 2003 Analysis and optimization of the error caused by high velocity gradients in particle image velocimetry. Exp. Fluids 35 (5), 408–421.Google Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2001 Instability and equilibration of centrifugally stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 5270–5273.Google Scholar
Narayan, R., Goldreich, P. & Goodman, J. 1987 Physics of modes in a differentially rotating system: analysis of the shearing sheet. Mon. Not. R. Astron. Soc. 228, 1–41.Google Scholar
Ooyama, K. 1966 On the stability of baroclinic circular vortex: a sufficient criterion for instability. J. Atmos. Sci. 23, 43–53.Google Scholar
Parras, L. & Le Dizès, S. 2010 Temporal instability modes of supersonic round jets. J. Fluid Mech. 662, 173–196.CrossRefGoogle Scholar
Riedinger, X., Le Dizès, S. & Meunier, P. 2010 a Viscous stability properties of a Lamb–Oseen vortex in a stratified fluid. J. Fluid Mech. 645, 255–278.Google Scholar
Riedinger, X., Meunier, P. & Le Dizès, S. 2010 b Instability of a vertical columnar vortex in a stratified fluid. Exp. Fluids 49, 673–681.Google Scholar
Roberts, P. H. 2003 On vortex waves in compressible fluids. I. The hollow-core vortex. Phil. Trans. R. Soc. Lond. A 459, 331–352.Google Scholar
Satomura, T. 1981 An investigation of shear instability in a shallow water. J. Met. Soc. Japan 59, 148–167.Google Scholar
Schecter, D. A. 2008 The spontaneous imbalance of an atmospheric vortex at high Rossby number. J. Atmos. Sci. 65, 2498–2521.Google Scholar
Schecter, D. A. & Montgomery, M. T. 2004 Damping and pumping of a vortex Rossby wave in a monotonic cyclone: critical layer stirring versus inertia–buoyancy wave emission. Phys. Fluids 16, 1334–1348.Google Scholar
Shalybkov, D. & Rüdiger, G. 2005 Stability of density-stratified viscous Taylor–Couette flows. Astron. Astrophys. 438, 411–417.Google Scholar
Thorpe, S. A. 1966 Notes on 1966 summer geophysical fluid dynamics, p. 80. Woods Hole Oceanographic Institute.Google Scholar
Withjack, E. M. & Chen, C. F. 1974 An experimental study of Couette instability of stratified fluids. J. Fluid Mech. 66, 725–737.Google Scholar
Withjack, E. M. & Chen, C. F. 1975 Stability analysis of rotational Couette flow of stratified fluids. J. Fluid Mech. 68, 157–175.Google Scholar
Yavneh, I., McWilliams, J. C. & Molemaker, M. J. 2001 Non-axisymmetric instability of centrifugally stable stratified Taylor–Couette flow. J. Fluid Mech. 448, 1–21.Google Scholar

Riedinger et al. supplementary movie

Synthetic schlieren visualisation. Vertical density gradient pattern associated with the radiative instability for Re=343 and F=1.01. Speed x2. The size of the image is $40*27$ cm$^2$.

Download Riedinger et al. supplementary movie(Video)
Video 7.9 MB

Riedinger et al. supplementary movie

Synthetic schlieren visualisation. Vertical density gradient pattern associated with the radiative instability for Re=343 and F=1.01. Speed x2. The size of the image is $40*27$ cm$^2$.

Download Riedinger et al. supplementary movie(Video)
Video 8.1 MB