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    Fukazawa, Yoshinari and Funakoshi, Mitsuaki 2015. Onset of thermal convection and its flow patterns in a rectangular cavity. Fluid Dynamics Research, Vol. 47, Issue. 6, p. 065505.

    Lappa, Marcello 2007. Secondary and oscillatory gravitational instabilities in canonical three-dimensional models of crystal growth from the melt. Part 1: Rayleigh–Bénard systems. Comptes Rendus Mécanique, Vol. 335, Issue. 5-6, p. 253.

    Pylaev, A. M. 2005. Problem of Critical Convective Flows in Horizontal-Cylindrical Cavities. Fluid Dynamics, Vol. 40, Issue. 3, p. 349.

    Lir, J.T. and Lin, T.F. 2001. Visualization of roll patterns in Rayleigh–Bénard convection of air in a rectangular shallow cavity. International Journal of Heat and Mass Transfer, Vol. 44, Issue. 15, p. 2889.

    Gelfgat, Alexander Yu. 1999. Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures. Journal of Computational Physics, Vol. 156, Issue. 2, p. 300.

    Daniels, P. G. and Weinstein, M. 1996. On finite-amplitude patterns of convection in a rectangular-planform container. Journal of Fluid Mechanics, Vol. 317, Issue. -1, p. 111.

    Chen, Yih-Yuh 1992. Boundary conditions and linear analysis of finite-cell Rayleigh–Bénard convection. Journal of Fluid Mechanics, Vol. 241, Issue. -1, p. 549.

    Edwards, Boyd F. Wilder, Joseph W. and Showalter, Kenneth 1991. Onset of convection for autocatalytic reaction fronts: Laterally unbounded system. Physical Review A, Vol. 43, Issue. 2, p. 749.

  • Journal of Fluid Mechanics, Volume 191
  • June 1988, pp. 583-597

Crossed rolls at onset of convection in a rigid box

  • Boyd F. Edwards (a1)
  • DOI:
  • Published online: 01 April 2006

Critical Rayleigh numbers, roll configurations, and growth-rate derivative are calculated at onset of convection for a rigid box with conducting upper and lower plates and insulating sidewalls. When the sidewalls form a square or a near square, the linearized Oberbeck–Boussinesq equations favour crossed rolls, a superposition of three-dimensional rolls in the x- and y-directions, over unidirectional rolls. These crossed rolls preserve the four-fold rotation symmetry about the vertical axis of a square box only when the aspect ratio (ratio of width to depth of the box) demands an even number of rolls in each direction. The analysis explains patterns observed by Stork & Müller.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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