The Influence of the Prandtl Number on the Style of Vigorous Thermal Convection

Abstract

The effect of the Prandtl number on convection in a planar three-dimensional geometry is investigated in this study. We have employed a numerical scheme to integrate the governing equations. Differently from previous studies we have chosen stress-free boundaries. Experiments have been performed at a Rayleigh number of Ra ¼ 10 6 for Prandtl numbers (Pr) ranging from 0.025 to 100. We have further conducted one experiment in the limiting case of infinite Prandtl number. Despite the differences in the geometry and the boundary conditions, as compared to other studies, we find a similar transition in the dynamics of the flow when the Prandtl number is increased. While the velocity and the temperature structure show diffusive character at low Pr, sharp thermal boundary layers form at high Pr. The heat transport efficiency increases with Pr until a transition value is reached, from there on Nu behaves almost asymptotically. The transition can not be caused by a change in hierarchies between velocity and thermal boundary layers, as suggested in other studies. Due to the stress-free boundaries, a velocity boundary layer does not exist. We observe that the toroidal part of the flow is strong at low Pr and looses its strength with increasing Pr, thus it is likely to be responsible for the transition. In a further chapter we demonstrate that due to the neglect of the toroidal part in two-dimensional calculations at low Pr results are obtained which are misleading, even in a qualitative sense. Infinite Pr results from 2D calculations closely resemble the dynamics of fully 3D flows.

Canonical Models of Geophysical and Astrophysical Flows: Turbulent Convection Experiments in Liquid Metals

“…However, these moderate P r LSC flows generate relatively large P e c values and, thus, tend to thermally mix and isothermalize the bulk fluid. These P r 1 LSC flows, then, do not generate such strong mid plane temperature anomalies, as are found here as well as in other comparable low P r convection studies [40,50].…”

“…Instead, large-scale, large-amplitude thermal anomalies are generated in the bulk (cf. [40,50]). Figure 6b shows the time-averaged, maximum horizontal thermal anomaly on the mid-plane, |δζ|,…”

“…However, such high Re c , moderate P e c flows are unable to isothermalize the fluid bulk, as in moderate P r fluids. Instead, these high velocity, large scale flows are able to generate container-scale gradients in the perturbation temperature field [40,50].…”

“…Instead, such flows will tend then to create large amplitude, large scale temperature perturbations in the fluid bulk (e.g., see Figure 1c in [40]). Vigorous small-scale flows in liquid metals will be largely unable to perturb the thermal field.…”

Numerical Simulations of Thermal Convection at High Prandtl Numbers

“…Some numerical studies deal with varying P r numbers ( [17], [12], [65], [41], [77]). Only a few of them ( [12], [77]) mimic a set-up similar to experiments, thus allowing a proper comparison of the results.…”