Logarithmic temperature profiles in the bulk of turbulent Rayleigh–Bénard convection for a Prandtl number of 12.3

Abstract

AbstractWe report measurements of logarithmic temperature profiles $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\varTheta (z,r) = A(r)\times \ln (z/L) + B(r)$ in the bulk of turbulent Rayleigh–Bénard convection (here $\varTheta $ is a scaled and time-averaged local temperature in the fluid, $ z$ is the vertical and $r$ the radial position, and $L$ is the sample height). Two samples had aspect ratios $\varGamma \equiv D/L = 1.00$ and 0.50 (where $D=190\ \mathrm{mm}$ is the diameter). The fluid was a fluorocarbon with a Prandtl number of $\mathit{Pr} = 12.3$. The measurements covered the Rayleigh-number range $2\times 10^{10} \lesssim \mathit{Ra} \lesssim 2\times 10^{11}$ for $\varGamma = 1.00$ and $3\times 10^{11} \lesssim \mathit{Ra} \lesssim 2\times 10^{12}$ for $\varGamma = 0.50$. In contradistinction to what had been found for $\varGamma = 0.50$ and $\mathit{Pr} = 0.78$ by Ahlers et al. (Phys. Rev. Lett., vol. 109, 2012, art. 114501; J. Fluid Mech., 2014, in press), the measurements revealed no $\mathit{Ra}$ dependence of the amplitude $A(r)$ of the logarithmic term. Within the experimental resolution, the amplitude was also found to be independent of $\varGamma $. It varied with $r$ in a manner consistent with the function $A(\xi ) = A_1/\sqrt{2\xi - \xi ^2}$, where $\xi \equiv (R-r)/R$ with $R=D/2$ and $A_1 \simeq 0.0016$. The results for $A(r)$ are smaller than those obtained from experiments and direct numerical simulations (Ahlers et al., Phys. Rev. Lett., vol. 109, 2012, art. 114501) at similar values of $\mathit{Ra}$ for $\mathit{Pr} = 0.7$ and $\varGamma = \frac{1}{2}$ by a factor that depended slightly upon $\mathit{Ra}$ but was close to $2$.