Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells with aspect ratio one and less

Abstract

We present high-precision measurements of the Nusselt number $\cal N$ as a function of the Rayleigh number $R$ for cylindrical samples of water (Prandtl number $\sigma \,{=}\, 4.4$) with a diameter $D$ of 49.7 cm and heights $L \,{=}\, 116.3, 74.6$, and 50.6 cm, as well as for $D \,{=}\, 24.8$ cm and $L \,{=}\, 90.2$ cm. For each aspect ratio $\Gamma \,{\equiv}\, D/L \,{=}\, 0.28, 0.43, 0.67$, and 0.98 the data cover a range of a little over a decade of $R$. The maximum $R \,{\simeq}\, 10^{12}$ and Nusselt number ${\cal N} \,{\simeq}\, 600$ were reached for $\Gamma \,{=}\, 0.43$ and $D \,{=}\, 49.7$. The data were corrected for the influence of the finite conductivity of the top and bottom plates on the heat transport in the fluid to obtain estimates of $\cal N_{\infty}$ for plates with infinite conductivity. The results for ${\cal N}_{\infty}$ and $\Gamma \,{\geq}\, 0.43$ are nearly independent of $\Gamma$. For $\Gamma \,{=}\, 0.275$${\cal N}_{\infty}$ falls about 2.5% below the other data. For $R \,{\lesssim}\,10^{11}$, the effective exponent $\gamma_{\hbox{\scriptsize\it eff}}$ of ${\cal N}_{\infty} \,{=}\, N_0 R^{\gamma_{\hbox{\scriptsize\it eff}}}$ is about 0.32, larger than those of the Grossmann–Lohse model with its current parameters by about 0.01. For the largest Rayleigh numbers covered for $\Gamma \,{=}\, $0.98, 0.67, and 0.43, $\gamma_{\hbox{\scriptsize\it eff}}$ saturates at the asymptotic value $\gamma \,{=}\, 1/3$ of the Grossmann–Lohse model. The data do not reveal any crossover to a Kraichnan regime with $\gamma_{\hbox{\scriptsize\it eff}} \,{>}\, 1/3$.