Direct Numerical Simulations of turbulent RayleighBénard Convection
The large structures occurring in the fluid flow on the Sun's surface, in the atmosphere and oceans of planets, including our Earth, are primarily driven by convection. Their actual shape but also the efficiency of the heat transport is however significantly influenced by the Coriolis force due to rotation. Crystal growth and the ventilation of buildings and aircrafts originate within the same physical framework. Understanding these fundamental processes is thus not only utterly important for geoand astrophysics, but also in industry.
That is, where the idealization, the socalled RayleighBénard convection comes into play: The above mentioned highly complex phenomena are simplified to a fluid heated from below and cooled from above, with solely gravity, and hence buoyancy, acting on it.

Figure 1: Scaling behaviour of the FLOWSI code (pink diamonds) on HLRB II. The perfect scalability of an ideal code is shown as well (black dashed line).

In most of the theoretical and numerical investigations of natural convection the OberbeckBoussinesq (OB) approximation is considered. This means that all physical properties of the fluid are assumed to be independent of temperature and pressure, except the density in the buoyancy term which is considered linearly dependent on the temperature. These assumptions are evidently never fulfilled in reality and the deviations of the flow characteristics due to their violation are called nonOberbeckBoussinesq (NOB) effects.
Thus the objective of our studies is to investigate the influence of rotation and NOB effects on turbulent thermal convection, also, but not exclusively, at very high Rayleigh numbers. We address these issues by means of Direct Numerical Simulations (DNS).
Numerical Method
We performed highresolved direct numerical simulations of RayleighBénard convection making use of the welltested fourth order finite volume code FLOWSI [2]. The code solves the incompressible NavierStokes equations and has shown an almost perfect scalability on the HLRB II cluster (Fig. 1).

Figure 2: Validity range of the OB approximation according to Gray & Giorgini [2] for water (top) and glycerol (bottom). The pink stars mark the parameters for our NOB simulations.

For the purpose of investigating NOB effects, the code has been advanced by taking temperaturedependent material properties into account. Furthermore, we incorporated a module to model the effects of rotation.
The mesh size was chosen in a way to fulfil the criterion by Shishkina et al. [5], which guarantees the resolution of the smallest relevant scales, i.e. the Kolmogorov and the Batchelor scale.
So far, our simulations focussed on two fluids, water (Pr = 4.38) and glycerol (Pr = 2547.9). Their material properties were adopted from experimental data by Ahlers et al. [1]. The parameter ranges for our performed NOB simulations are given in Fig. 2, together with the estimation of the validity range of the OB approximation according to Gray & Giorgini [2]. In sum, we covered the range of Rayleigh numbers
105≤ Ra ≤ 109 and NOB conditions up to a temperature difference ∆ between the top and bottom plate of 80 K.
Selfevidently, we performed the corresponding OB simulations for the purpose of comparison as well. Furthermore, we also conducted DNS for different rotation rates under OB and NOB conditions.
However, not only the required resolution but also the necessary computation time to obtain reliable statistics, that is
at least several months, pushes us to the limits of nowadays supercomputers.
Results
Our NOB simulations revealed that the temperature dependencies of the material properties are able to significantly influence the global flow structures. In general they lead to a breakdown of the topbottom symmetry typical for OB simulations. The NOB effects that we observe, include, but are not limited to, different thermal and viscous boundary layers, asymmetric plume dynamics and an increase of the bulk temperature Tc (Fig. 3, 4). In glycerol for the NOB caseΔ = 80 K we obtain
a Tc that is 15 K higher than the arithmetic mean temperature between the plates, whereas in contrast, in the case of water the observed increase of Tc is only about 5 K.
Nevertheless, the Nusselt number Nu and the Reynolds number Re and their scaling with Ra show only a slight deviation from the OB case.

Figure 3: Instantaneous temperature isosurfaces of water (Pr = 4.38, Ra = 108, magenta indicates the warm, cyan the cold fluid), shown are the nonrotating OB (upper left) and NOB case (upper right), as well as a moderately rotating OB (lower left) and NOB case (lower right).

In rotating convection another physical phenomenon plays an important role: the formation of columnar vortices, as seen in Fig. 3. These socalled Ekman vortices are able to extract hot and cold fluid from the bottom and top boundary layers, respectively, and thereby increase the heat transport. They are also responsible for a persisting mean temperature gradient in the bulk. Their size, and related to that their efficiency, is determined by the heat diffusivity and viscosity, thus they are sensitive to NOB effects. Indeed, while Tc is the same as in the nonrotating case, the absolute value of the temperature gradient is diminished. In line with this, the enhancement of Nuunder rotation is stronger under NOB than under OB conditions.
Ongoing Research/Outlook
A next step on studying NOB effects will be the inclusion of compressible convective flows, i.e. almost all gases, where the Low Mach Number (LMN) approximation will be used, which generally admits the working fluid to be a nonperfect gas. And since in an astronomical and geophysical context Ra is typically rather in the order of 1020, we plan to go to higher Ra, which are of course computational more expensive, because of the mesh size and the required time for sufficient statistics. Additionally, we plan to test existing and as appropriate develop subgrid scale models and perform largeeddy simulations (LES).

Figure 4: Instantaneous temperature isosurfaces of glycerol (Pr = 2547.9, Ra = 109) under OB (left) and the NOB conditions (right).

Acknowledgments
All simulations were performed at the Leibniz Supercomputing Centre on the national supercomputer HLRB II and on the HPC Cluster SCART (DLR Göttingen). The authors acknowledge support by the Deutsche Forschungsgemeinschaft under grant SH405/21 and would also like to thank the staff of the LRZ for their continuous support.
References
[1] Ahlers, G., Brown, E., Fontenele Araujo, F., Funfschilling, D., Grossmann, S., Lohse, D. J. Fluid Mech., 2006
[2] Gray, D.D., Giorgini, A. Int. J. Heat Mass Transfer 19, 1996
[3] Horn, S., Shishkina, O., Wagner, C. J. Phys.: Conf. Ser. 318, 082005., 2011
[4] Shishkina, O., Wagner, C. C. R. Mecanique 333, 2005
[5] Shishkina, O., Stevens, R.J.A.M., Grossmann, S., Lohse, D. New J. Physics 12, 2010
• Susanne Horn
• Matthias Kaczorowski
• Olga Shishkina
Institute of Aerodynamics and Flow Technology German Aerospace Center (DLR), Göttingen
top

