Heat transport in turbulent Rayleigh-Bénard convection: Effect of finite top- and bottom-plate conductivities

Abstract

We describe three apparatus, known as the large, medium, and small apparatus, used for high-precision measurements of the Nusselt number N as a function of the Rayleigh number R for cylindrical samples of fluid and present results illustrating the influence of the finite conductivity of the top and bottom plates on the heat transport in the fluid. We used water samples at a mean temperature of 40°C (Prandtl number σ=4.4). The samples in the large apparatus had a diameter D of 49.69cm and heights L≃116.33, 74.42, 50.61, and 16.52cm. For the medium apparatus we had D=24.81cm, and L=90.20 and 24.76cm. The small apparatus contained a sample with D=9.21cm and L=9.52cm. For each aspect ratio Γ≡D∕L the data covered a range of a little over a decade of R. The maximum R≃1×1012 with Nusselt number N≃600 was reached for Γ=0.43. Measurements were made with both aluminum (conductivity λp=161W∕mK) and copper (λp=391W∕mK) top and bottom plates of nominally identical size and shape. For the large and medium apparatus the results with aluminum plates fall below those obtained with copper plates, thus confirming qualitatively the prediction by [Verzicco, “Effects of nonperfect thermal sources in turbulent thermal convection,” Phys. Fluids 16, 1965 (2004)] that plates of finite conductivity diminish the heat transport in the fluid. The Nusselt number N∞ for plates with infinite conductivity was estimated by fitting simultaneously aluminum- and copper-plate data sets to an effective power law for N∞ multiplied by a correction factor f(X)=1−exp[−(aX)b] that depends on the ratio X of the thermal resistance of the fluid to that of the plates, as suggested by Verzicco. Within their uncertainties the parameters a and b were independent of Γ for the large apparatus and showed a small Γ dependence for the medium apparatus. The correction was larger for the large, smaller for the medium, and negligible for the small apparatus.