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Centrifugal effects in rotating convection: axisymmetric states and three-dimensional instabilities

Published online by Cambridge University Press:  21 May 2007

F. MARQUES ,
I MERCADER ,
O. BATISTE  and
J. M. LOPEZ
F. MARQUES
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
I MERCADER
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
O. BATISTE
Affiliation:
Departament de Física Aplicada, Univ. Politècnica de Catalunya, Barcelona 08034, Spain
J. M. LOPEZ
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
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Abstract

Rotating convection is analysed numerically in a cylinder of aspect ratio one, for Prandtl number about 7. Traditionally, the problem has been studied within the Boussinesq approximation with density variation only incorporated in the gravitational buoyancy term and not in the centrifugal buoyancy term. In that limit, the governing equations admit a trivial conduction solution. However, the centrifugal buoyancy changes the problem in a fundamental manner, driving a large-scale circulation in which cool denser fluid is centrifuged radially outward and warm less-dense fluid is centrifuged radially inward, and so there is no trivial conduction state. For small Froude numbers, the transition to three-dimensional flow occurs for Rayleigh number R ≈ 7.5 × 103. For Froude numbers larger than 0.4, the centrifugal buoyancy stabilizes the axisymmetric large-scale circulation flow in the parameter range explored (up to R = 3.5 × 104). At intermediate Froude numbers, the transition to three-dimensional flow is via four different Hopf bifurcations, resulting in different coexisting branches of three-dimensional solutions. How the centrifugal and the gravitational buoyancies interact and compete, and the manner in which the flow becomes three-dimensional is different along each branch. The centrifugal buoyancy, even for relatively small Froude numbers, leads to quantitative and qualitative changes in the flow dynamics.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 580 , 10 June 2007 , pp. 303 - 318
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Barcilon, V. & Pedlosky, J. 1967 On the steady motions produced by a stable stratification in a rapidly rotating fluid. J. Fluid Mech. 29, 673–690.Google Scholar
Barkley, D. & Tuckerman, L. S. 1988 Global bifurcation to traveling waves in axisymmetric convection. Phys. Rev. Lett. 61, 408–411.Google Scholar
Becker, N., Scheel, J. D., Cross, M. C. & Ahlers, G. 2006 Effect of the centrifugal force on domain chaos in Rayleigh-Bénard convection. Phys. Rev. E 73, 066309.Google ScholarPubMed
Bergeon, A., Henry, D., Benhadid, H. & Tuckerman, L. S. 1998 Marangoni convection in binary mixtures with Soret effect. J. Fluid Mech. 375, 143–177.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.Google Scholar
Buell, J. C. & Catton, I. 1983 Effect of rotation on the stability of a bounded cylindrical layer of fluid heated from below. Phys. Fluids 26, 892–896.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Fornberg, B. 1998 A Practical Guide to Pseudospectral Methods. Cambridge University Press.Google Scholar
Frayssé, V., Giraud, L. & Langou, J. 2003 A set of GMRES routines for real and complex arithmetics on high performance computers. Tech. Rep. TR/PA/03/3. CERFACS.Google Scholar
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1993 Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers. J. Fluid Mech. 248, 583–604.Google Scholar
Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1994 Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers. J. Fluid Mech. 262, 293–324.Google Scholar
Hart, J. E. 2000 On the influence of centrifugal buoyancy on rotating convection. J. Fluid Mech. 403, 133–151.Google Scholar
Herrmann, J. & Busse, F. H. 1993 Asymptotic theory of wall-attached convection in a rotating fluid layer. J. Fluid Mech. 255, 183–194.Google Scholar
Homsy, G. M. & Hudson, J. L. 1969 Centrifugally driven thermal convection in a rotating cylinder. J. Fluid Mech. 35, 33–52.Google Scholar
Homsy, G. M. & Hudson, J. L. 1971 Centrifugal convection and its effect on the asymptotic stability of a bounded rotating fluid heated from below. J. Fluid Mech. 48, 605–624.CrossRefGoogle Scholar
Hughes, S. & Randriamampianina, A. 1998 An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations. Intl. J. Numer. Meth. Fluids 28, 501–521.Google Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.Google Scholar
Kuo, E. Y. & Cross, M. C. 1993 Traveling-wave wall states in rotating Rayleigh-Bénard convection. Phys. Rev. E 47, R2245–R2248.Google Scholar
Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory, 2nd edn. Springer.Google Scholar
Lopez, J. M., Rubio, A. & Marques, F. 2006 Traveling circular waves in axisymmetric rotating convection. J. Fluid Mech. 569, 331–348.Google Scholar
Mamun, L. S. & Tuckerman, L. S. 1995 Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91.Google Scholar
Mercader, I., Batiste, O. & Alonso, A. 2006 Continuation of travelling-wave solutions of the Navier-Stokes equations. Intl J. Numer. Meth. Fluids 52, 707–721.Google Scholar
Mercader, I., Net, M. & Falqués, A. 1991 Spectral methods for high order equations. Comput. Meth. Appl. Mech. Engng 91, 1245–1251.Google Scholar
Ning, L. & Ecke, R. E. 1993 Rotating Rayleigh-Bénard convection: Aspect-ratio dependence of the initial bifurcations. Phys. Rev. E 47, 3326–3333.Google ScholarPubMed
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Pfotenhauer, J. M., Niemela, J. J. & Donnelly, R. J. 1987 Stability and heat transfer of rotating cryogens. Part 3. Effects of finite cylindrical geometry and rotation on the onset of convection. J. Fluid Mech. 175, 85–96.Google Scholar
Rossby, H. T. 1969 A study of Benard convection with and without rotation. J. Fluid Mech. 36, 309–335.Google Scholar
Sánchez-Álvarez, J. J., Serre, E., Crespo, del Arco, E. & Busse, F. H. 2005 Square patterns in rotating Rayleigh-Bénard convection. Phys. Rev. E 72, 036307.Google Scholar
Veronis, G. 1959 Cellular convection with finite amplitude in a rotating fluid. J. Fluid Mech. 5, 401–435.Google Scholar
Zhong, F., Ecke, R. & Steinberg, V. 1991 Asymmetric modes and the transition to vortex structures in rotating Rayleigh-Bénard convection. Phys. Rev. Lett. 67, 2473–2476.CrossRefGoogle ScholarPubMed
Zhong, F., Ecke, R. & Steinberg, V. 1993 Rotating Rayleigh-Bénard convection: asymmetric modes and vortex states. J. Fluid Mech. 249, 135–159.CrossRefGoogle Scholar