DNS of thermal channel flow for $Re_\tau=5000$

A new DNS of a thermal channel flow has been conducted at $Re_\tau=5000$. The thermal field has been considered as a passive scalar and the Mixed Boundary Condition is employed. The Prandtl number of air, $0.71$, has been used. A large enough computational box of dimensions $2\pi h$ x $2 h$ x $\pi h$ has been set to obtain accurate statistics. The mesh used had a total of $6144$ x $1285$ x $6144$ $\approx$ 5e10 points. The simulation has run on 2048 cores of the supercomputer MareNostrum. The logarithmic region of the mean temperature profile is starting to be properly developed with a von Karman constant of $\kappa_t = 0.444$. Maxima of the temperature intensities increase and move towards the wall with respect to other cases with lower Reynolds numbers. A power function has been obtained to calculate the Nusselt number as a function of Reynolds, for $Pr = 0.71$. Turbulent Prandtl number does not show remarkable differences with the ones obtained for lower Reynolds and it keeps being close to $1$ near the wall. Finally, turbulent budgets for heat fluxes, temperature variance and its dissipation rate have been calculated. Scaling of all terms is analysed and discussed.