Axisymmetric convection in a cylinder

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Abstract
In three-dimensional Bénard convection regions of rising and sinking fluid are dissimilar. This geometrical effect is studied for axisymmetric convection in a Boussinesq fluid contained in a cylindrical cell with free boundaries. Near the critical Rayleigh number Rc the solution is obtained from a perturbation expansion, valid only if both the Reynolds number and the Péclet number are small. For values of the Nusselt number N ≤ 2 accurate solutions are provided by an expansion in a finite number of vertical modes. For Prandtl numbers p < 1 the form of the solution changes at large Reynolds number and becomes independent of p; in the limit p → 0 there is an effective critical Rayleigh number R* = 1.32Rc, which can also be derived by a perturbation procedure, and the Nusselt number is a function of the Rayleigh number only. Numerical experiments yield solutions for Rayleigh numbers R ≤ 100Rc and p ≥ 0.01. The results are similar to those for two-dimensional rolls and for R ≥ 5Rc the Nusselt number shows only a weak dependence on p. For p > 1 there is a viscous regime with N ≈ 2(R/Rc)1/3; when R/Rc [gsim ] p3/2, N increases more rapidly, approximately as R0.4. At high Rayleigh numbers a large isothermal region develops, in which the ratio of vorticity to distance from the axis is nearly constant.

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