Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Benard (RB) cell are rather independent of the aspect ratio Gamma (diameter/height) for large Gamma, a small-Gamma cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Gamma follows Ra-c,Ra-Gamma similar to Ra-c,Ra-infinity(1 + C Gamma(-2))(2), with C less than or similar to 1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32) <= Gamma <= 32, the rescaling Ra -> Ra-l Ra[Gamma(2)/(C+Gamma(2))](3/2) collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Gamma). Our findings predict the Gamma dependence of the onset of the ultimate regime Ra-u,Ra-Gamma similar to [Gamma(2)/(C+Gamma(2))](-3/2) in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Gamma = 1, 1/2, and 1/3) for the transition in OB RB convection and explains why, in small-Gamma cells, much larger Ra (namely, by a factor Gamma(-3)) must be achieved to observe the ultimate regime.