A new scaling law for temperature variance profile in the mixing zone of turbulent Rayleigh-B\'{e}nard convection

We report a combined experimental and numerical study of the scaling properties of the temperature variance profile $\eta $(z) along the central z axis of turbulent Rayleigh-B\'{e}nard convection in a thin disk cell and an upright cylinder of aspect ratio unity. In the mixing zone outside the thermal boundary layer region, the measured $\eta $(z) is found to scale with the cell height H in both cells and obey a power law, $\eta $(z)\textasciitilde (z/H)$^{\mathrm{\varepsilon }}$, with the obtained values of $\varepsilon $ being very close to -1. Based on the experimental and numerical findings, we derive a new equation for $\eta $(z) in the mixing zone, which has a power-law solution in good agreement with the experimental and numerical results. Our work thus provides a common framework for understanding the effect of boundary layer fluctuations on the scaling properties of the temperature variance profile in turbulent Rayleigh- B\'{e}nard convection.