Scale analysis and wall-layer model for the temperature profile in a turbulent thermal convection

Three sets of characteristic scales for the conduction layer, the transition layer and the convection layer are proposed to analyze the mean thermal structure in a turbulent thermal convection without mean motion. These scales are formulated based on molecular or turbulent eddy contribution to the momentum and heat transports in each layer. Using the proposed scales and a gradient matching technique at the interface between two adjacent layers, Kraichnan's (Physics Fluids 5, 1374 (1962)) multi-layered structure of the mean temperature gradient profile is re-established. If the conduction scales are used to nondimensionalize mean temperature gradient data near the wall, they form a plausible correlation curve that is nearly independent of the Prandtl number and the Rayleigh number for the range of experiments. From the correlation curve, it is found that the convection layer or the similarity layer with the slope of - 4 3 begins to appear after about z₊ ~ 15 and the proportionality constant of the - 4 3 power law of the mean temperature gradient is found to be about 0.6 or dΘ₊/dz₊ = 0.6z₊^{- 4 3}, where Θ₊ and z₊ are nondimensional temperature and distance scaled by the respective conduction scales. Further, a wall-layer model for the mean temperature gradient profile is formulated in accordance with the power law, dΘ₊/dz₊ ~ z^-α₊, across the layers, which is in good agreement with the data.