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Prog. Theor. Phys. Vol. 78 No. 2 (1987) pp. 282-304

[ Full Text PDF : FREE ACCESS (1405K) ]

Competition of the Two Unstable Modes in the Rayleigh-Benard Convection

Hideo Yahata

Department of Materials Science, Faculty of Science, Hiroshima University, Hirosima 730

(Received March 26, 1987)

Abstract:

A model dynamical system which governs time evolution of the convection rolls in a rectangular box is investigated. With increase of the vertical temperature gradient R, either the real mode or the complex mode instability triggers the transition of the steady convection state depending on whether the side length of the box perpendicular to the roll axis Γy is less or greater than a certain critical value. In the neighborhood of this critical value a variety of dynamical behavior originating from competition of these real and complex modes are observed. With further increase of R it is found by numerical time integration of the system that the bifurcation sequences leading to chaos take various forms depending on the value of Γy. In addition, temporal evolution of the spatial patterns of convection is presented using the isopleths of the temperature and the velocity fields for several different time-dependent states.


URL : http://ptp.ipap.jp/link?PTP/78/282/
DOI : 10.1143/PTP.78.282

[ Full Text PDF : FREE ACCESS (1405K) ] Citation:

References:

  1. H. Yahata, Prog. Theor. Phys. 75 (1986), 790[PTP].
  2. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, 1983).
  3. V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, 1983).
  4. J. Guckenheimer and E. Knobloch, Geophys. Astrophys. Fluid Dynamics 23 (1983), 247.
  5. E. Knobloch and M. R. E. Proctor, J. Fluid Mech. 108 (1981), 291[CrossRef].
  6. H. R. Brand, P. C. Hohenberg and V. Steinberg, Phys. Rev. A30 (1984), 2548[APS].
  7. I. Rehberg and G. Ahlers, Phys. Rev. Lett. 55 (1985), 500[APS].
  8. A. Libchaber, S. Fauve and C. Laroche, Physica 7D (1983), 73[Elsevier].
  9. J. P. Gollub and S. V. Benson, J. Fluid Mech. 100 (1980), 449[CrossRef].
  10. M. Sano and Y. Sawada, in Turbulence and Chaotic Phenomena in Fluids, ed. T. Tatsumi (North-Holland, 1984), p. 167.
  11. H. Yahata, Prog. Theor. Phys. Suppl. No. 79 (1984), 26[PTP].
  12. G. E. Willis and J. W. Deardorff, J. Fluid Mech. 44 (1970), 661[CrossRef].
  13. R. M. Clever and F. H. Busse, J. Fluid Mech. 75 (1974), 625[CrossRef].
  14. F. B. Lipps, J. Fluid Mech. 75 (1976), 113[CrossRef].
  15. R. Kessler, U. Dallmann and H. Oertel, in Turbulence and Chaotic Phenomena in Fluids, ed. T. Tatsumi (North-Holland, 1984), p. 173.
    R. Kessler, J. Fluid Mech. 174 (1987), 357[CrossRef].
  16. F. H. Busse and R. M. Clever, J. Fluid Mech. 91 (1979), 319[CrossRef]; J. Appl. Math. Phys. (ZAMP) 29 (1978), 711.
  17. J. P. Gollub, A. R. McCarriar and J. F. Steinman, J. Fluid Mech. 125 (1982), 259[CrossRef].
  18. S. Nasuno, M. Sano and Y. Sawada, a talk presented at the meeting of the physical Society of Japan in September, 1986.
  19. J. Maurer and A. Libchaber, J. de Phys. 40 (1979), L419; ibid. 41 (1980), L515.
  20. A. Libchaber and J. Maurer, in Nonlinear Phenomena at Phase Transitions and Instabilities, ed. T. Riste (Plenum, 1982), p.259.
  21. W. F. Langford, SIAM J. Appl. Math. 37 (1979), 22.
  22. L. P. Silinikov, Sov. Math. Dokl. 6 (1965), 163.
  23. S. Sato, M. Sano and Y. Sawada, Preprint.
    R. W. Walden, P. Kolodner, A. Passner and C. M. Surko, Phys. Rev. Lett. 53 (1984), 242[APS].
  24. R. P. Behringer, H. Gao and J. N. Shaumeyer, Phys. Rev. Lett. 50 (1983), 1199[APS].
  25. R. P. Behringer, Rev. Mod. Phys. 57 (1985), 657[APS].
  26. M. Nakayama, Senior thesis at Hiroshima University (1987).

Citing Article(s) :

  1. Journal of the Physical Society of Japan 57 (1988) pp. 3357-3364 :
    Spatial Instabilities and Onset of Chaos in Rayleigh-Benard System with Intermediate Aspect-Ratio
    Satoru Nasuno, Masaki Sano and Yasuji Sawada
  2. Journal of the Physical Society of Japan 66 (1997) pp. 79-90 :
    Sequential Transitions of the Thermal Convection in a Square Cavity
    Jiro Mizushima and Takahiro Adachi
  3. Journal of the Physical Society of Japan 66 (1997) pp. 2337-2341 :
    Onset of 3D Thermal Convection in a Cubic Cavity
    Jiro Mizushima and Osamu Matsuda
  4. Journal of the Physical Society of Japan 74 (2005) pp. 1750-1761 :
    Construction of a Dynamical System Governing the Rayleigh–Bénard Convection in a Rectangular Box
    Hideo Yahata
  5. Progress of Theoretical Physics Vol. 81 No. 3 (1989) pp. 592-602 :
    Statics and Dynamics of the Benard Convection Rolls in an Intermediate-Aspect-Ratio Vessel
    Hideo Yahata

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