Thermal boundary layer profiles in turbulent Rayleigh-Benard convection

We have studied the mean temperature boundary layer profile T(z) and root-mean-square (rms) temperature profile S(z) in turbulent Rayleigh-Benard convection along the central axis z of a convection cell, which has a thin vertical disk shape with an inner diameter D = 18 cm. The temperature measurements were made at fixed Prandtl numbers Pr = 4.3 and Pr = 7.6 and with the Rayleigh number Ra varied in the range between $1\times 10^9$ and $1\times 10^{10}$. The measured T(z) for different values of Pr and Ra can all be well described by the newly proposed boundary layer model [Shishkina et al., Phys. Rev. Lett. {\bf 114}, 114302 (2015)] with a parameter c varying from 1 to 2.1. The measured rms temperature profile S(z) is found to be a single-peaked function with the peak position located at $z\simeq 0.8 \delta$, where $\delta$ is the boundary layer thickness. The measured S(z) has two separate scaling lengths. Within the boundary layer, it scales with $\delta$ and can be fitted to a power law, $S(z)\sim (z/\delta)^{\alpha}$ with $\alpha\simeq 0.6$. Outside the boundary layer, it scales with the cell size D and follows a different power law, $S(z)\sim (z/D)^{\beta}$, with $\beta= -0.42$. *This work was supported by the Research Grants Council of Hong Kong SAR.