Logarithmic temperature profiles of turbulent Rayleigh-Bénard convection in the classical and ultimate state for a Prandtl number of 0.8

We report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh-Bénard convection for a cylindrical sample with an aspect ratio (diameter D over height L) equal to 0.50, in both the classical and the ultimate state. The measurements are for Rayleigh numbers Ra from 6 × 10¹¹ to 10¹³ in the classical and 7 × 10¹⁴ to 1.1 × 10¹⁵ (our maximum accessible Ra) in the ultimate state. The Prandtl number was close to 0.8. Although to lowest order the bulk is often assumed to be isothermal in the time average, we found a 'logarithmic layer' (as reported briefly by Ahlers et al., Phys. Rev. Lett., vol. 109, 2012, 114501) in which the reduced temperature Θ = [〈T(z)〉 - T_m]/ΔT (with T_m the mean temperature, ΔT the applied temperature difference and 〈⋯〉 a time average) varies as A ln(z/L) + B or A′ ln(1 - z/L) + B′ with the distance z from the bottom plate of the sample. In the classical state, the amplitudes -A and A′ are equal within our resolution, while in the ultimate state there is a small difference, with -A/A′ ≃ 0.95. For the classical state, the width of the log layer is approximately 0.1L, the same near the top and the bottom plate as expected for a system with reflection symmetry about its horizontal midplane. For the ultimate state, the log-layer width is larger, extending through most of the sample, and slightly asymmetric about the midplane. Both amplitudes A and A′ vary with radial position r, and this variation can be described well by A = A₀[(R - r)/R]^-0.65, where R is the radius of the sample. In the classical state, these results are in good agreement with direct numerical simulations (DNS) for Ra = 2 × 10¹²; in the ultimate state there are as yet no DNS. The amplitudes -A and A′ varied as Ra^-n, with n ≃ 0.12 in the classical and n ≃ 0.18 in the ultimate state. A close analogy between the temperature field in the classical state and the 'law of the wall' for the time-averaged downstream velocity in shear flow is discussed. A two-sublayer mean-field model of the temperature profile in the classical state was analysed and yielded a logarithmic z dependence of Θ. The Ra dependence of the amplitude A given by the model corresponds to an exponent n_th = 0.106, in good agreement with the experiment. In the ultimate state the experimental result n ≃ 0.18 differs from the prediction n_th ≃ 0.043 by Grossmann & Lohse.