Scaling transition of thermal dissipation in turbulent convection

Direct numerical simulation (DNS) of non-slip two dimensional (2-D) Rayleigh-Benard convection (RBC) is conducted for a wide range of Rayleigh number (Ra up to 1013) at Prandtl number Pr = 0.7 and aspect ratio Γ =1. The thermal dissipation rate is shown to display an evident scaling transition through the compensated plot, i.e. ⟨εθ⟩ ∝ Raγ with γ ≈ −0.17 for 106 {less than or equal to} Ra {less than or equal to} 109, whilst γ ≈ −0.19 for Ra = 109 {less than or equal to} Ra {less than or equal to} 1013. To track the transition, separate contributions from the thermal boundary layer (BL) and the bulk flow region are examined, incorporated also with the mean and fluctuation decomposition. It turns out that the mean temperature gradient in the BL is the dominant contribution, and together with other parts (i.e. fluctuations in the BL and bulk, and the mean gradient in the bulk), they all exhibit an obvious transition at Ra ≈ 109. We have further checked the Nusselt number (Nu) which also shows the transition at Ra ≈ 109, and interestingly, Nu ∝ Ra0.33 is observed for small Ra whilst Nu ∝ Ra2/7 is absent in 2-D RBC cases. To understand the physical origin of transition, spatial distributions and probability density functions of thermal dissipation rate are finally discussed, with notable statistical features changed at Ra ≈ 109.