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 $ #1{} #1{ {#1}} ≤ = ≤ = ≥ = ≥ = {Pr} {Fr} {Re}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 = [ T(z) - Tₘ]/Δ T (with Tₘ the mean temperature, Δ T 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-η , with η 0.12 in the classical and