Boundary layer fluctuations in turbulent Rayleigh–Bénard convection

Abstract

We report a combined experimental and numerical study of the effect of boundary layer (BL) fluctuations on the scaling properties of the mean temperature profile$\unicode[STIX]{x1D703}(z)$and temperature variance profile$\unicode[STIX]{x1D702}(z)$in turbulent Rayleigh–Bénard convection in a thin disk cell and an upright cylinder of aspect ratio unity. Two scaling regions are found with increasing distance$z$away from the bottom conducting plate. In the BL region, the measured$\unicode[STIX]{x1D703}(z)$and$\unicode[STIX]{x1D702}(z)$are found to have the scaling forms$\unicode[STIX]{x1D703}(z/\unicode[STIX]{x1D6FF})$and$\unicode[STIX]{x1D702}(z/\unicode[STIX]{x1D6FF})$, respectively, with varying thermal BL thickness$\unicode[STIX]{x1D6FF}$. The functional forms of the measured$\unicode[STIX]{x1D703}(z/\unicode[STIX]{x1D6FF})$and$\unicode[STIX]{x1D702}(z/\unicode[STIX]{x1D6FF})$in the two convection cells agree well with the recently derived BL equations by Shishkinaet al.(Phys. Rev. Lett., vol. 114, 2015, 114302) and by Wanget al.(Phys. Rev. Fluids, vol. 1, 2016, 082301). In the mixing zone outside the BL region, the measured$\unicode[STIX]{x1D703}(z)$remains approximately constant, whereas the measured$\unicode[STIX]{x1D702}(z)$is found to scale with the cell height$H$in the two convection cells and follows a power law,$\unicode[STIX]{x1D702}(z)\sim (z/H)^{\unicode[STIX]{x1D716}}$, with the obtained values of$\unicode[STIX]{x1D716}$being close to$-1$. Based on the experimental and numerical findings, we derive a new equation for$\unicode[STIX]{x1D702}(z)$in the mixing zone, which has a power-law solution in good agreement with the experimental and numerical results. Our work demonstrates that the effect of BL fluctuations can be adequately described by the velocity–temperature correlation functions and the new BL equations capture the essential physics.