On the role of the Prandtl number in convection driven by heat sources and sinks

We report on a numerical study of turbulent convection driven by a combination of internal heat sources and sinks. Motivated by a recent experimental realisation (Lepot et al. 2018), we focus on the situation where the cooling is uniform, while the internal heating is localised near the bottom boundary, over approximately one tenth of the domain height. We obtain scaling laws $Nu\sim R{a}^{\gamma }P{r}^{\chi }$ for the heat transfer as measured by the Nusselt number $Nu$ expressed as a function of the Rayleigh number $Ra$ and the Prandtl number $Pr$. After confirming the experimental value $\gamma\approx 1/2$forthedependenceon$Ra$,weidentifyseveralregimesofdependenceon$Pr$\mathrm{.}Forastress-freebottomsurfaceandwithinarangeasbroadas$Pr \in [0.003, 10]$,weobservetheexponent$\chi\approx 1/2$, in agreement with Spiegel’s mixing length theory. For a no-slip bottom surface we observe a transition from $\chi \approx 1\mathrm{/}2$ for $Pr\le 0.04$ to $\chi \approx 1\mathrm{/}6$ for $Pr\ge 0.04$, in agreement with scaling predictions by Bouillaut et al. The latter scaling regime stems from heat accumulation in the stagnant layer adjacent to a no-slip bottom boundary, which we characterise by comparing the local contributions of diffusive and convective thermal fluxes.