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

TL;DR

This study numerically investigates turbulent convection with internal heat sources and sinks, revealing how the Prandtl number influences heat transfer scaling laws and identifying different regimes based on boundary conditions and Prandtl number values.

Contribution

It provides new insights into the Prandtl number's role in convection driven by localized internal heating, confirming experimental scaling laws and characterizing different flow regimes.

Findings

Heat transfer scales as Nu ~ Ra^{1/2} with Pr dependence varying by boundary conditions.

For stress-free surfaces, Prandtl number affects heat transfer with an exponent ~1/2.

For no-slip surfaces, a transition occurs from Pr^{1/2} to Pr^{1/6} scaling at Pr ≈ 0.04.

Abstract

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 $Nu \sim Ra^\gamma Pr^\chi$ 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 $\gamma\approx 1/2$ for the dependence on $Ra$, we identify several regimes of dependence on $Pr$. For a stress-free bottom surface and within a range as broad as $Pr \in [0.003, 10]$, we observe the exponent $\chi\approx 1/2$, in agreement with Spiegel's mixing length theory. For a no-slip bottom surface we observe a transition from $\chi\approx 1/2$ for $Pr \leq 0.04$ to $\chi\approx 1/6$ for $Pr \geq 0.04$, 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.