Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T05:21:55.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Bénard convection in a slowly rotating penny-shaped cylinder subject to constant heat flux boundary conditions

Published online by Cambridge University Press:  02 November 2022

A.M. Soward Open the ORCID record for A.M. Soward [Opens in a new window] ,
L. Oruba Open the ORCID record for L. Oruba [Opens in a new window]  and
E. Dormy Open the ORCID record for E. Dormy [Opens in a new window]
A.M. Soward*
Affiliation:
School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK
L. Oruba*
Affiliation:
Laboratoire Atmosphères Milieux Observations Spatiales (LATMOS/IPSL), Sorbonne Université, UVSQ, CNRS, Paris, France
E. Dormy*
Affiliation:
Département de Mathématiques et Applications, UMR-8553, École Normale Supérieure, CNRS, PSL University, 75005 Paris, France
*
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider axisymmetric Boussinesq convection in a shallow cylinder of radius $L$ and depth $H (\ll L)$, which rotates with angular velocity $\varOmega$ about its axis of symmetry aligned to the vertical. Constant heat flux boundary conditions, top and bottom, are adopted, for which the onset of instability occurs on a long horizontal length scale provided that $\varOmega$ is sufficiently small. We investigate the nonlinear development by well-established two-scale asymptotic expansion methods. Comparisons of the results with the direct numerical simulations (DNS) of the primitive governing equations are good at sufficiently large Prandtl number $\sigma$. As $\sigma$ is reduced, the finite amplitude range of applicability of the asymptotics reduces in concert. Though the large meridional convective cell, predicted by the DNS, is approximated adequately by the asymptotics, the azimuthal flow fails almost catastrophically, because of significant angular momentum transport at small $\sigma$, exacerbated by the cylindrical geometry. To appraise the situation, we propose hybrid methods that build on the meridional streamfunction $\psi$ derived from the asymptotics. With $\psi$ given, we solve the now linear azimuthal equation of motion for the azimuthal velocity $v$ by DNS. Our ‘hybrid’ methods enable us to explain features of the flow at large Rayleigh number, found previously by Oruba et al. (J. Fluid Mech., vol. 812, 2017, pp. 890–904).

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Rotating flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 951 , 25 November 2022 , A5
Check for updates
Copyright
© The Author(s), 2022. 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

REFERENCES

Calkins, M.A., Hale, K., Julien, K., Nieves, D., Driggs, D. & Marti, P. 2015 The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection. J. Fluid Mech. 784, R2.CrossRefGoogle Scholar
Cessi, P. & Young, W.R. 1992 Fixed-flux convection in a tilted slot. J. Fluid Mech. 237, 57–71.CrossRefGoogle Scholar
Chapman, C.J., Childress, S. & Proctor, M.R.E. 1980 Long wavelength thermal convection between non-conducting boundaries. Earth Planet. Sci. Lett. 54, 362–369.CrossRefGoogle Scholar
Chapman, C.J. & Proctor, M.R.E. 1980 Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101 (4), 759–782.CrossRefGoogle Scholar
Cox, S.M. 1998 Long-wavelength rotating convection between poorly conducting boundaries. SIAM J. Appl. Maths 58 (4), 1338–1364.CrossRefGoogle Scholar
Cross, M.C. & Hohenberg, P.C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–273.CrossRefGoogle Scholar
Depassier, M.C. & Spiegel, E.A. 1981 The large-scale structure of compressible convection. Astron. J. 86 (3), 496–512.CrossRefGoogle Scholar
Dowling, T.E. 1988 Rotating Rayleigh–Bénard convection with fixed flux boundaries. Woods Hole Oceanog. Inst. Tech. Rep. WHOI-89-26, pp. 230–247.Google Scholar
Fiedler, B.H. 1999 Thermal convection in a layer bounded by uniform heat flux: application of a strongly nonlinear analytic solution. Geophys. Astrophys. Fluid Dyn. 91, 223–250.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag.CrossRefGoogle Scholar
Guervilly, C., Hughes, D. & Jones, C.A. 2014 Large-scale vortices in rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech. 758, 407–435.CrossRefGoogle Scholar
Küppers, G. & Lortz, D. 1969 Transition from laminar convection to thermal turbulence in a rotating fluid layer. J. Fluid Mech. 35, 609–620.CrossRefGoogle Scholar
Matthews, P.C. & Cox, S.M. 2000 Pattern formation with a conservation law. Nonlinearity 13, 1293–1320.CrossRefGoogle Scholar
Oruba, L., Davidson, P.A. & Dormy, E. 2017 Eye formation in rotating convection. J. Fluid Mech. 812, 890–904.CrossRefGoogle Scholar
Oruba, L., Davidson, P.A. & Dormy, E. 2018 Formation of eyes in large-scale cyclonic vortices. Phys. Rev. Fluids 3, 013502.CrossRefGoogle Scholar
Pons, A.J., Sagués, F. & Bees, M.A. 2004 Chemoconvection patterns in the methylene-blue–glucose system: weakly nonlinear analysis. Phys. Rev. E 70, 066304.CrossRefGoogle ScholarPubMed
Sivashinsky, G.I. 1982 Large cells in nonlinear Marangoni convection. Physica D 4 (2), 227–235.CrossRefGoogle Scholar
Soward, A.M. 1985 Bifurcation and stability of finite amplitude convection in a rotating layer. Physica D 14 (2), 227–241.CrossRefGoogle Scholar
Takehiro, S.-I., Masaki, I., Nakajima, K. & Hayashi, Y.-Y. 2002 Linear instability of thermal convection in rotating systems with fixed heat flux boundaries. Geophys. Astrophys. Fluid Dyn. 96 (6), 439–459.CrossRefGoogle Scholar