Axisymmetric convection in a cylinder

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 R_c 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* = 1.32R_c, 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 ≤ 100R_c and p ≥ 0.01. The results are similar to those for two-dimensional rolls and for R ≥ 5R_c the Nusselt number shows only a weak dependence on p. For p > 1 there is a viscous regime with N ≈ 2(R/R_c)^1/3; when R/R_c [gsim ] p^3/2, N increases more rapidly, approximately as R^0.4. At high Rayleigh numbers a large isothermal region develops, in which the ratio of vorticity to distance from the axis is nearly constant.