Max Planck Institute for Dynamics and Self-Organization, Göttingen (Germany)
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Turbulent thermal convection is ubiquitous in nature and technical applications. Inclined convection, where a fluid is confined between two differently heated parallel surfaces, which are inclined with respect to gravity, is one of the main model systems to study the physics of turbulent thermal convection. In this project, we focus on the investigation of the interaction between shear and buoyancy and want to know, how they influence the development of the flow superstructures and contribute to the mean heat transport enhancement in the system.
Being one of the three ways to transport heat, thermal convection is ubiquitous in nature and technical applications. It is studied in atmospheric physics, astro and geophysics. Additionally, thermal convection is also studied in engineering as it is very important for technical applications as well, e.g. in cooling processes. The utilized cooling fluids range from water (cooling CPUs) to liquid metals, which are used for cooling nuclear power plants. Thus it becomes necessary to understand and predict convective heat transport in fluids with different properties.
Rayleigh-Bénard convection (RBC), where a fluid is confined between a hot bottom plate and a cold top plate, is one of the main model systems to investigate the physics of turbulent thermal convection.
In our project, we focus on the investigation of the interaction between the shear and buoyancy force. In paticular we would like to find out how these forces can enhance the mean heat transport across a cylindrical domain (represented by the dimensionless Nusselt number Nu) and which kind of superstructures emerge inside the flow, and how they contribute to the global heat transport .
How does our model system work? We consider convection in fluids, whose density decreases with increasing temperature. Thus, here buoyancy drives convection. The strength of the driving is determined by the Rayleigh number Ra, which is proportional to the temperature difference between the plates. Additionally, shear is induced in our system by inclining the cylindrical cell with respect to gravity. This splits the buoyancy force into two components, and the additional component is directed parallel to the hot/cold plates, creating a shear flow along the hot and cold thermal boundary layers.
Previous studies for cylinders of the diameter-to-height aspect ratio one (Γ=1) evince complex behavior of the heat transport for various fluids . In our current studies we include an additional aspect which is geometrical confinement of the convection cell. The aspect ratio is reduced to Γ=1/5 (i.e., the height H of the cylinder equals five times the diameter D), and our setup becomes a slender cylinder .
Recent experiments conducted for a slender cylinder of even stronger confinement (H = 20D) showed that the heat transport in such configurations can significantly increase (~10x) compared to that in RBC . These experiments used liquid sodium as a working fluid, which has a very small Prandtl number Pr, i.e. the ratio of the viscosity to the thermal diffusivity is very small.
The main goal of our study is to provide insight into the flow fields in this kind of convective setup, because it is very difficult to do this experimentally. Furthermore, we want to analyze the heat transport scaling relations with Rayleigh number and Prandtl number.
Results and Methods
Our model system is described by the momentum (incompressible Navier-Stokes) and energy equations. The Boussinesq approximation is used, which means that the change of the temperature only affects the buoyancy term, all other fluid properties are independent of the temperature. These equations are solved by the Goldfish code, which implements a highly advanced high-order finite-volume method. We perform direct numerical simulations (DNS), which do not need any closure models, and our results are free from additional assumptions. We use cylindrical coordinates and non-equidistant, staggered meshes, to adapt the geometry of the cylinder.
In Table 1 we list typical numbers about the mesh, CPU time and memory requirements. The mesh is denser in proximity to the boundaries of the cylinder in order to properly resolve the boundary layers (BLs). The necessary resolution of the mesh is determined by the smallest turbulent scale. For small Prandtl numbers this is the Kolmogorov scale. The boundary conditions for the velocity are no-slip at all walls and for the temperature adiabatic, side walls are used and the temperatures at the cold top and hot bottom plates are kept constant.
A non-dimensional form of the equation has four input parameters: the Rayleigh number (strength of the thermal driving), the Prandtl number (fluid property), the diameter-to-height aspect ratio and the inclination angle of the cell. The inclination angle is varied from 0, which corresponds to Rayleigh-Bénard convection (RBC), to π/2, which is so-called vertical convection (VC). Currently the Rayleigh number reaches up to 10⁹ for Prandtl numbers 1 and 0.1.
In this report we focus on one of our most interesting findings. These results illustrate how geometrical confinement and inclination of the cylinder can alter the flow structure in such a way, that the mean heat transport significantly increases. In RBC, that means without inclination, the heat transport is similar for both aspect ratios (less than 2% difference). Fig. 1 shows that in the Γ=1 case there is a large-scale circulation (LSC) but in the slender cylinder there is not. Thus, the LSC does not play an important roll in RBC. The similarity between the flows can be seen in Fig 2. It shows that the sheet-like thermal plumes have similar dimensions in the case without inclination for both aspect ratios. Nevertheless, other results show that in the inclined slender cylinder the LSC is present, its direction is fixed by the splitting of the buoyancy force and its strength plays an important role in the heat transport.
When the inclination angle reaches π/4, the mean heat transport is optimal (maximal) for both aspect ratios. However, in the slender cylinder the heat transport increases by about 40%, while in the Γ=1 cylinder the increase amounts only 6% compared to the no inclination case. This significant difference is observed also in Figs. 1 and 2. The sheet-like thermal plumes do not change much in the aspect ratio one case. Inside the slender cylinder, however, we do not observe any sheet-like plumes. Instead there is one large zone of an impinging cold plume and one large zone of a rising hot plume.
Our simulations show also for other Rayleigh numbers and Prandtl numbers that this kind of appearance of the thermal BLs seems to be favorable for the heat transport enhancement in the system. Apparently the interaction of the LSC and the BLs is stronger in the sense that a more rapid LSC squeezes the thermal BL together, thus enhancing the heat transport.
On-going Research and Outlook
Going to lower Prandtl numbers requires particularly fine meshes to resolve the relevant turbulent scales and therefore these simulations consume much more CPU hours. In our future work we plan to directly compare the results of our simulations with measurements from our collaborators. This will help us gain more detailed insight into the heat and momentum transport in turbulent thermal convection of liquid metals.
The simulations in the desired parameter range have already started.
Olga Shishkina (PI), Lukas Zwirner
References and Links
 Shishkina, O., Horn, S. 2016. J. Fluid Mech. 790, R3
 Zwirner, L., Shishkina, O., under revision
 Vasilev, A. Yu., et al. 2015. Technical Physics 60, 1063–7842
PD Dr. Olga Shishkina
Max Planck Institute for Dynamics and Self-Organization
Am Fassberg 17, D37077 Göttingen (Germany)
NOTE: This report was first published in the book "High Performance Computing in Science and Engineering – Garching/Munich 2018".
LRZ project IDs: pr84pu, pr92jo