Penetrative internally heated convection in two and three dimensions

TL;DR

This study uses direct numerical simulations to analyze how internally heated convection behaves in two and three dimensions, revealing differences in heat transfer and temperature scaling in turbulent regimes.

Contribution

It provides new insights into the dimensional dependence of heat transfer and temperature scaling in internally heated convection at high Rayleigh numbers.

Findings

In 2D, bottom heat escape fraction initially decreases then increases in turbulence.

In 3D, bottom heat escape fraction decreases monotonically with convection strength.

Mean fluid temperature scales as $H^{4/5}$ in turbulent regimes.

Abstract

Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows proportionally to the heating rate ($H$) without convection, grows proportionally to $H^{4/5}$ when convection is strong in both two and three dimensions. The ratio of the heating rate to the fluid temperature is likened to the Nusselt number of Rayleigh-B\'enard convection. Simulations are reported for Prandtl numbers between 0.1 and 10 and for Rayleigh numbers (defined in terms of the heating rate) up to $5\times10^{10}$.