Spatial distribution of heat flux and fluctuations in turbulent Rayleigh-Bénard convection

Abstract

We numerically investigate the radial dependence of the velocity and temperature fluctuations and of the time-averaged heat flux $\overline{j}(r)$ in a cylindrical Rayleigh-B\'enard cell with aspect ratio $\ensuremath{\Gamma}=1$ for Rayleigh numbers Ra between $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ and $2\ifmmode\times\else\texttimes\fi{}{10}^{9}$ at a fixed Prandtl number $\text{Pr}=5.2$. The numerical results reveal that the heat flux close to the sidewall is larger than in the center and that, just as the global heat transport, it has an effective power law dependence on the Rayleigh number, $\overline{j}(r)\ensuremath{\propto}{\text{Ra}}^{{\ensuremath{\gamma}}_{j}(r)}$. The scaling exponent ${\ensuremath{\gamma}}_{j}(r)$ decreases monotonically from $0.43$ near the axis ($r\ensuremath{\approx}0$) to $0.29$ close to the sidewalls ($r\ensuremath{\approx}D/2$). The effective exponents near the axis and the sidewall agree well with the measurements of Shang et al. [Phys. Rev. Lett. 100, 244503 (2008)] and the predictions of Grossmann and Lohse [Phys. Fluids 16, 1070 (2004)]. Extrapolating our results to large Rayleigh number would imply a crossover at $\text{Ra}\ensuremath{\approx}{10}^{15}$, where the heat flux near the axis would begin to dominate. In addition, we find that the local heat flux is more than twice as high at the location where warm or cold plumes go up or down than in plume depleted regions.