Characteristics of temperature fluctuation in two-dimensional turbulent Rayleigh-Bénard convection

We study the characteristics of temperature fluctuation in two-dimensional turbulent Rayleigh-Bénard convection in a square cavity by direct numerical simulations. The Rayleigh number range is 1\times 10^8\le Ra\le 1\times 10^13, and the Prandtl number is selected as Pr=0.7 and Pr=4.3. It is found that the temperature fluctuation profiles with respect to Ra exhibit two different distribution patterns. In the thermal boundary layer, the normalized fluctuation \theta_\rm rms/\theta_\rm rms,max is independent of Ra and a power law relation is identified, i.e., \theta _\rm rms/\theta_\rm rms,max\sim \left( z / \delta \right)^0.99\pm 0.01, where z / \delta is a dimensionless distance to the boundary (\delta is the thickness of thermal boundary layer). Out of the boundary layer, when Ra\le 5\times 10^9, the profiles of \theta _\rm rms/\theta_\rm rms,max descend, then ascend, and finally drop dramatically as z / \delta increases. While for Ra\ge 1\times 10^10, the profiles continuously decrease and finally overlap with each other. The two different characteristics of temperature fluctuations are closely related to the formation of stable large-scale circulations and corner rolls. Besides, there is a critical value of Ra indicating the transition, beyond which the fluctuation \langle \theta_\rm rms\rangle _V has a power law dependence on Ra, given by \langle \theta_\rm rms\rangle _V\sim Ra^-0.14\pm 0.01.