Heat transport by turbulent Rayleigh–Bénard convection for Pr ≃ 0.8 and 3 × 1012 ≲ Ra ≲ 1015: aspect ratio Γ = 0.50

Abstract

We report on the experimental results for heat-transport measurements, in the form of the Nusselt number Nu , by turbulent Rayleigh–Bénard convection (RBC) in a cylindrical sample of aspect ratio Γ ≡ D / L = 0.50 ( D = 1.12 m is the diameter and L = 2.24 m the height). The measurements were made using sulfur hexafluoride at pressures up to 19 bar as the fluid. They are for the Rayleigh-number range 3 × 10 ^12 ≲ Ra ≲ 10 ^15 and for Prandtl numbers Pr between 0.79 and 0.86. For Ra < Ra * _1 ≃ 1.4 × 10 ^13 we find Nu = N _0 Ra ^γ ^eff_eff with γ _eff = 0.312 ± 0.002, which is consistent with classical turbulent RBC in a system with laminar boundary layers below the top and above the bottom plate. For Ra * _1 < Ra < Ra * _2 (with Ra * _2 ≃ 5 × 10 ^14 ) γ _eff gradually increases up to 0.37 ± 0.01. We argue that above Ra * _2 the system is in the ultimate state of convection where the boundary layers, both thermal and kinetic, are also turbulent. Several previous measurements for Γ = 0.50 are re-examined and compared with our results. Some of them show a transition to a state with γ _eff in the range from 0.37 to 0.40, albeit at values of Ra in the range from 9 × 10 ^10 to 7 × 10 ^11 which is much lower than the present Ra * _1 or Ra * _2 . The nature of the transition found by them is relatively sharp and does not reveal the wide transition range observed in this work. In addition to the results for the genuine Rayleigh–Bénard system, we present measurements for a sample which was not completely sealed; the small openings permitted external currents, imposed by density differences and gravity, to pass through the sample. That system should no longer be regarded as genuine RBC because the externally imposed currents modified the heat transport in a major way. It showed a sudden decrease of γ _eff from 0.308 for Ra < Ra _t ≃ 4 × 10 ^13 to 0.25 for larger Ra . A number of possible experimental effects are examined in a sequence of appendices; none of these effects is found to have a significant influence on the measurements.