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

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ 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 Γ follows Rac,Γ∼Rac,∞(1+CΓ-2)2, with C ≲ 1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32)≤Γ≤32, the rescaling Ra→RaℓRa[Γ2/(C+Γ2)]3/2 collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Γ). Our findings predict the Γ dependence of the onset of the ultimate regime Rau,Γ∼[Γ2/(C+Γ2)]-3/2 in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ=1, 1/2, and 1/3) for the transition in OB RB convection and explains why, in small-Γ cells, much larger Ra (namely, by a factor Γ-3) must be achieved to observe the ultimate regime.