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Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container

  • Chuanshi Sun
  • Shuang Liu
  • Qi Wang
  • Zhenhua Wan
  • Dejun SunEmail author
Article
  • 8 Downloads

Abstract

The bifurcations of penetrative Rayleigh-Bénard convection in cylindrical containers are studied by the linear stability analysis (LSA) combined with the direct numerical simulation (DNS) method. The working fluid is cold water near 4°C, where the Prandtl number Pr is 11.57, and the aspect ratio (radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter θm. The relationship between the normalized critical Rayleigh number (Rac(θm)/Rac(0)) and θm is formulated, which is in good agreement with the stability results within a large range of θm. The aspect ratio has a minor effect on Rac(θm)/Rac(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system. Moreover, two kinds of qualitatively different steady axisymmetric solutions are identified.

Key words

bifurcation convection linear stability analysis (LSA) 

Nomenclature

θm

density inversion parameter

Ra

Rayleigh number

Pr

Prandtl number

T

temperature

m

wave number

a

aspect ratio

θ

dimensionless temperature

t

dimensionless time

u

dimensionless velocity

Chinese Library Classification

O357.1 

2010 Mathematics Subject Classification

76E20 34C23 

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References

  1. [1]
    VERONIS, G. Penetrative convection. Astrophysical Journal, 137, 641–663 (1963)CrossRefzbMATHGoogle Scholar
  2. [2]
    ZHANG, K. and SCHUBERT, G. Penetrative convection and zonal flow on Jupiter. Science, 273, 941–943 (1996)CrossRefGoogle Scholar
  3. [3]
    OLSON, P. and AURNOU, J. A polar vortex in the Earth’s core. nature, 402, 170–173 (1999)CrossRefGoogle Scholar
  4. [4]
    GUBBINS, D., THOMSON, C. J., and WHALER, K. A. Stable regions in the Earth’s liquid core. Geophysical Journal International, 68, 241–251 (1982)CrossRefGoogle Scholar
  5. [5]
    AHLERS, G., GROSSMANN, S., and LOHSE, D. Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Reviews of Modern Physics, 81, 503–537 (2009)CrossRefGoogle Scholar
  6. [6]
    LOHSE, D. and XIA, K. Q. Small-scale properties of turbulent Rayleigh-Bénard convection. Annual Review of Fluid Mechanics, 42, 335–364 (2010)CrossRefzbMATHGoogle Scholar
  7. [7]
    MOORE, D. R. and WEISS, N. O. Nonlinear penetrative convection. Journal of Fluid Mechanics, 61, 553–581 (1973)CrossRefGoogle Scholar
  8. [8]
    LARGE, E. and ANDERECK, C. D. Penetrative Rayleigh-Bénard convection in water near its maximum density point. Physics of Fluids, 26, 094101 (2014)CrossRefGoogle Scholar
  9. [9]
    HU, Y. P., LI, Y. R., and WU, C. M. Rayleigh-Bénard convection of cold water near its density maximum in a cubical cavity. Physics of Fluids, 27, 034102 (2015)CrossRefGoogle Scholar
  10. [10]
    LI, Y. R., OUYANG, Y. Q., and HU, Y. P. Pattern formation of Rayleigh-Bénard convection of cold water near its density maximum in a vertical cylindrical container. Physical Review E, 86, 046323 (2012)CrossRefGoogle Scholar
  11. [11]
    LI, Y. R., HU, Y. P., OUYANG, Y. Q., and WU, C. M. Flow state multiplicity in Rayleigh-Bénard convection of cold water with density maximum in a cylinder of aspect ratio 2. International Journal of Heat and Mass Transfer, 86, 244–257 (2015)CrossRefGoogle Scholar
  12. [12]
    HU, Y. P., LI, Y. R., and WU, C. M. Aspect ratio dependence of Rayleigh-Bénard convection of cold water near its density maximum in vertical cylindrical containers. International Journal of Heat and Mass Transfer, 97, 932–942 (2016)CrossRefGoogle Scholar
  13. [13]
    HUANG, X. J., LI, Y. R., ZHANG, L., and WU, C. M. Turbulent Rayleigh-Bénard convection of cold water near its maximum density in a vertical cylindrical container. International Journal of Heat and Mass Transfer, 116, 185–193 (2018)CrossRefGoogle Scholar
  14. [14]
    MA, D. J., SUN, D. J., and YIN, X. Y. Multiplicity of steady states in cylindrical Rayleigh-Bénard convection. Physical Review E, 74, 037302 (2006)CrossRefGoogle Scholar
  15. [15]
    ALONSO, A., MERCADER, I., and BATISTE, O. Pattern selection near the onset of convection in binary mixtures in cylindrical cells. Fluid Dynamics Research, 46, 041418 (2014)MathSciNetCrossRefGoogle Scholar
  16. [16]
    WANG, B. F., MA, D. J., CHEN, C., and SUN, D. J. Linear stability analysis of cylindrical Rayleigh-Bénard convection. Journal of Fluid Mechanics, 711, 27–39 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    GEBHART, B. and MOLLENDORF, J. C. A new density relation for pure and saline water. Deep Sea Research, 24, 831–848 (1977)CrossRefGoogle Scholar
  18. [18]
    VERZICCO, R. and ORLANDI, P. A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. Journal of Computational Physics, 123, 402–414 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    HUGUES, S. and RANDRIAMAMPIANINA, A. An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 28, 501–521 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    KNOLL, D. A. and KEYES, D. E. Jacobian-free Newton-Krylov methods: a survey of approaches and applications. Journal of Computational Physics, 193, 357–397 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    DOEDEL, E. and TUCKERMAN, L. S. Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, Springer Science and Business Media, New York, 453–466 (2012)Google Scholar
  22. [22]
    VAN DER VORST, H. A. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 13, 631–644 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    LEHOUCQ, R. B., SORENSEN, D. C., and YANG, C. ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Siam, Philadelphia (1998)CrossRefzbMATHGoogle Scholar
  24. [24]
    WANG, B. F., WAN, Z. H., GUO, Z. W., MA, D. J., and SUN, D. J. Linear instability analysis of convection in a laterally heated cylinder. Journal of Fluid Mechanics, 747, 447–459 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Chuanshi Sun
    • 1
  • Shuang Liu
    • 1
  • Qi Wang
    • 1
  • Zhenhua Wan
    • 1
  • Dejun Sun
    • 1
    Email author
  1. 1.Department of Modern MechanicsUniversity of Science and Technology of ChinaHefeiChina

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