ENGINEERING AND CFD

Introduction

Turbulent convection flows, which evolve in horizontally extended domains, are often organized in prominent and regular patterns on scales that exceed the characteristic layer height. Furthermore these patterns, which we term turbulent superstructures of convection, evolve gradually with respect to time [1,2]. This large-scale organization challenges the classical picture of turbulence in which a turbulent flow is considered as a tangle of irregular and chaotically moving vortices and swirls. Examples for superstructures in nature are cloud streets in the atmosphere of our planet or the granulation at the surface of the Sun. In the latter astrophysical case, this structure formation is additionally affected by magnetic fields, which are generated inside the Sun. Our understanding of the origin of turbulent superstructures, their mechanics, their role for the turbulent transport of heat and momentum as well as the influence of magnetic fields on their structure is presently still incomplete. High-resolution direct numerical simulations of the equations of turbulent fluid motion in the simplest setting of a turbulent convection flow, Rayleigh-Bénard convection in a layer that is uniformly heated from below and cooled from above, aim at a detailed study of these superstructure pattern formation processes in several working fluids with very different kinematic viscosities and thermal conductivities. In order to fully resolve the flows in horizontally extended domains, we have to rely on massively parallel super-computers for our numerical investigations [2, 3].

Results and Methods

We solve the equations of motions that couple the dynamics of the velocity and temperature fields numerically. These are the three-dimensional Boussinesq equations of thermal convection. The external magnetic field is assumed to be strong such that we can apply the quasi-static limit of magnetohydrodynamics [4]. The thermal driving by the applied outer temperature difference is quantified by the Rayleigh number Ra, the properties of the working fluid by the Prandtl number Pr, and the strength of the applied magnetic field by the Hartmann number Ha. Astrophysical flows are mostly found at very low Prandtl numbers, which cause a very vigorous fluid turbulence in the convection system. This is one point that makes our numerical simulations very challenging since all vortices have to be resolved. Two numerical methods are applied, a spectral element method [3] and a second-order finite difference method [4]. The latter is used when convection with magnetic fields is considered. The simulation domains are closed square cells with no-slip boundary conditions at all walls. The sidewalls are thermally insulated. Typical production runs in domains with an aspect ratio of 25:25:1 required 16384 SuperMUC cores for the non-magnetic cases. The simulations with magnetic field require 4096 cores for a box of aspect ratio 4:4:1. All simulations are long-term runs that involved sequences of several 48-hour runs in a row. In the course of the two project years, this sums up to 80 million consumed core hours.

On-going Research / Outlook

The presented numerical investigations would not have been possible without the use of the most powerful supercomputers. In both discussed examples, we studied turbulent convection in horizontally extended domains. The numerical effort for such runs grows typically with the square of the aspect ratio at a given Rayleigh and Prandtl number. The very large aspect ratio of 25 in the first part was necessary to minimize sidewall effects in the pattern formation processes and to reliably extract the typical superstructure pattern scales. The SuperMUC computer made furthermore long-term simulations possible that resolved the very slow dynamics of the turbulent super-structures for the first time. An important question of the future work will be how these superstructures vary once the Rayleigh number is further increased. In view to astrophysical convection phenomena a further decrease of the Prandtl number would be a second challenge that want to address in the near future. Along these lines machine learning methods, such as deep convolutional neural networks [7], are currently applied to reduce the very large amount of data and to extract essential physical information on the turbulent transport properties in these turbulent flows.