Skip to main content
    • Aa
    • Aa

Higher harmonic resonance of two-dimensional disturbances in Rayleigh-Bénard convection

  • Jiro Mizushima (a1) and Kaoru Fujimura (a2)

A higher harmonic resonance with wavenumber ratio of 1:3 is found to take place in Rayleigh-Bénard convection under rigid-rigid boundary conditions. Bifurcation diagrams for two-dimensional motion are obtained for various values of the Prandtl number P. It is found that a pure mode and mixed mode solutions exist as nonlinear equilibrium states of primary roll solutions for relatively high-Prandtl-number fluids (P ≥ 0.13) while the pure mode, mixed modes, travelling wave and modulated wave solutions exist for relatively low-Prandtl-number fluids (P ≤ 0.12).

Hide All
Armbruster, D. 1987 O(2)-symmetric bifurcation theory for convection rolls. Physica 27D, 433439.
Armbruster, D., Guckenheimer, J. & Holmes, P. 1989 Kuramoto—Sivashinsky dynamics on the centre-unstable manifold. SIAM J. Appl. Maths 49, 676691.
Busse, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below. J. Maths and Phys. 46, 140150.
Busse, F. H. 1987 Transition to asymmetric convection rolls. In Bifurcation: Analysis, Algorithms, Applications (ed. T. Küpper, R. Seydel & H. Troger), pp. 2626. Birkhäuser.
Busse, F. H. & Clever, R. M. 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319335.
Busse, F. H. & Or, A. C. 1986 Subharmonic and asymmetric convection rolls. Z. Angew. Math. Phys. 37, 608623.
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305320.
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65, 625645.
Dangelmayr, G. 1986 Steady-state mode interactions in the presence of O(2)-symmetry. Dyn. Stab. Syst. 1, 159185.
Dangelmayr, G. & Armbruster, D. 1986 Steady-state mode interactions in the presence of O(2)-symmetry and in non-flux boundary value problems. Contemp. Maths 56, 5367.
Fujimura, K. 1989 The equivalence between two perturbation methods in weakly non-linear stability theory for parallel shear flows. Proc. R. Soc. Lond. A 424, 373392.
Fujimura, K. & Mizushima, J. 1987 Nonlinear interaction of disturbances in free convection between vertical parallel plates. In Nonlinear Wave Interactions in Fluids (ed. R. W. Miksad, T. R. Akylas & T. Herbert), pp. 130130. ASME.
Jeffreys, H. 1928 Some cases of instability in fluid motion. Proc. R. Soc. Lond. A 118, 195208.
Kidachi, H. 1982 Side wall effect on the pattern formation of the Rayleigh—Bénard convection. Prog. Theor. Phys. 68, 4963.
Knobloch, E. & Guckenheimer, J. 1983 Convective transitions induced by a varying aspect ratio. Phys. Rev. A 27, 408417.
Knobloch, E. & Proctor, M. R. E. 1988 The double Hopf bifurcation with 2:1 resonance. Proc. R. Soc. Lond. A 415, 6190.
Li, R. 1986 Analysis for Taylor vortex flow. Ph.D. thesis, Virginia Polytechnic Institute.
Meyer-Spasche, R. & Keller, H. B. 1985 Some bifurcation diagrams for Taylor vortex flows. Phys. Fluids 28, 12481252.
Mizushima, J. & Saito, Y. 1988 Equilibrium characteristics of the secondary convection in a vertical fluid layer between two flat plates. Fluid Dyn. Res. 2, 183191.
Nagata, M. & Busse, F. H. 1983 Three-dimensional tertiary motions in a plane shear layer. J. Fluid Mech. 135, 126.
Pellew, A. & Southwell, R. V. 1940 On maintained convective motion in a fluid heated from below. Proc. R. Soc. Lond. A 176, 312343.
Proctor, M. R. E. & Hughes, D. W. 1990 Chaos and the effect of noise for the double Hopf bifurcation with 2:1 resonance. In Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems (ed. F. H. Busse & L. Kramer), pp. 384384. Plenum.
Reid, W. H. & Harris, D. L. 1958 Some further results on the Bénard problem. Phys. Fluids 1, 102110.
Schlüter, A., Lortz, D. & Busse, F. H. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129144.
Specht, H., Wagner, M. & Meyer-Spasche, R. 1989 Interactions of secondary branches of Taylor vortex solutions. Z. Angew. Math. Mech. 69, 339352.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 11 *
Loading metrics...

Abstract views

Total abstract views: 46 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.