Boundary layer fluctuations and their effects on mean and variance temperature profiles in turbulent Rayleigh-B{\'e}nard convection

We report simultaneous measurements of the mean temperature profile $\theta(z)$ and temperature variance profile $\eta(z)$ near the lower conducting plate of a specially designed quasi-two-dimensional cell for turbulent Rayleigh-B{\'e}nard convection. The measured $\theta(z)$ is found to have a universal scaling form $\theta(z/\delta)$ with varying thermal boundary layer (BL) thickness $\delta$, and its functional form agrees well with the recently derived BL equation by Shishkina et al. The measured $\eta(z)$, on the other hand, is found to have a scaling form $\eta(z/\delta)$ only in the near-wall region with $z/\delta < 2$. Based on the experimental findings, we derive a new BL equation for $\eta(z/\delta)$, which is in good agreement with the experimental results. The new BL equations thus provide a common framework for understanding the effect of BL fluctuations. *This work was supported by the Research Grants Council of Hong Kong SAR and by the China Thousand Young Talents Program.