No abstract

We have investigated the influence of the Prandtl number on the dynamics of high Rayleigh number thermal convection. A numerical parameter study in a three-dimensional Rayleigh-Bénard configuration was carried out, where we varied the Prandtl number between 10(-3)< or =Pr< or =10(2). The Rayleigh number was fixed at a value of Ra=10(6). Our main focus lay on the question how the value of the Prandtl number affects the spatial structure of the flow. We investigated the functional dependence of the Nusselt number and the Reynolds number and compared our results with a recent theoretical approach of Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000); Phys. Rev. Lett. 86, 3316 (2001)].

High-resolution computer simulations of two-dimensional (2D) convection are often used to investigate turbulent flows. In this paper we compare numerical simulations in 2D with three-dimensional (3D) simulations. We investigate flows at a fixed Rayleigh number of Ra = 10 6 . The velocity boundary conditions are rigid for the upper and lower boundary and stress-free for the side walls. For high values of the Prandtl number the flow structure and global quantities such as the Nusselt number (Nu) and the Reynolds number (Re) show similar behaviour in 3D and 2D simulations. For values of the Prandtl number smaller than unity, however, the 2D results diverge strongly form the 3D findings. This is true not only for global properties (Nu, Re) but also for local properties such as the structure of the boundary layer and the shapes of the up-and down-wellings. We relate this behaviour to the different large-scale structure of the flow and the toroidal component of the kinetic energy.

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.

[1] In a terrestrial magma ocean, the metal-silicate separation involved small metal droplets. Our goal is to better understand the dynamics of the metal droplet scenario. The mechanism of sedimentation in a vigorously convecting and strongly rotating magma ocean may differ significantly from settling droplets in a still fluid. In order to systematically study the parameter dependence on the style of motion we utilize two-dimensional and three-dimensional numerical convection models combined with a tracer-based sedimentation method. We investigate the characteristic flow patterns resulting from the competing effects of convection and droplet settling and find three styles of motion: a temperature-dominated style where most droplets remain suspended, a droplet-dominated style where the droplets separate from the fluid, and a style of repetitive motion. We find that the droplet-dominated style is relevant to the magma ocean. In this scenario the droplets settle and form a dense bottom layer. One of the key findings of this work is that the formation of the dense bottom layer, and therefore the separation of metal droplets from the liquid silicate, occurs on a characteristic timescale, which is identical with the Stokes' settling time. Finally, we study the chemical interaction of settling metal droplets and liquid silicate and discuss how accurately a simple parameterized model can describe the chemical equilibration in the metal-rain scenario of a terrestrial magma ocean. We find agreement between the parameterized model and our two-dimensional model. Our simulations, which include thermal convection into the metal-rain model, confirm the fundamental observation that metal droplets are not permanently suspended in the convecting silicate melt, but that they sink and rapidly form a dense layer at the base of the magma ocean.