Logarithmic temperature and velocity profiles in Rayleigh-B\'enard turbulence

In this study, we present results from two-dimensional DNS for the temperature and velocity profiles in Rayleigh-B\’enard convection. The mean temperature profiles, averaged both vertically and horizontally, are logarithmic in the boundary layers. The logarithmic region extends wider with increasing Rayleigh number and reaches one and half decades at $Ra = 10^{14}$. Similarly, there is also a logarithmic dependence of the mean velocity profile when the shear Reynolds number $Re_s \approx 420$ , as suggested by Grossmann-Lohse theory. We further separate the boundary into plume emitting and impacting regions and show that the heat transport has different scalings in the two regions. From low to high Rayleigh number, the system undergoes a transition from the impacting dominant regime to the emitting dominant regime with respect to the heat transport.