Heat Transport Enhancement in Rotating Rayleigh-Bénard Convection

**Principal Investigator:**

Detlef Lohse

**Affiliation:**

Max Planck Institute for Dynamics and Self-Organization, Göttingen

**Local Project ID:**

pr74sa

**HPC Platform used:**

SuperMUC-NG at LRZ

**Date published:**

A tremendous variety of physical phenomena involve turbulence, such as the dynamics of the atmosphere or the oceans, avian and airplane flight, fish and boats, sailing, heating and ventilation, and even galaxy formation. Turbulent flow is characterized by chaotic swirling movements that vary widely in size, from sub-millimeter, over the extent of storm clouds, to galactic scales. The interaction of the chaotic movements on different scales makes it challenging to simulate and understand turbulent flows. Turbulent thermal convection plays an important role in a wide range of natural and industrial settings, from astrophysical and geophysical flows to process engineering. The paradigmatic representation of thermal convection is Rayleigh-Bénard flow [2] in which a layer of fluid is heated from below and cooled from above. The system is used to test new concepts in fluid dynamics, such as instabilities, non-linear dynamics, chaos, pattern formation, or turbulence. Rayleigh-Bénard convection is a relevant model for countless phenomena ranging from thermal convection in the atmosphere, oceans, and the outer layer of the Sun, to heating and ventilation of buildings and convection in various industrial applications. The Rayleigh-Bénard system is ideal for studying the interaction between the boundary layer and bulk dynamics, which will also shed more light on general wall-bounded turbulent flows.

One of the most striking features of Rayleigh-Bénard flow is the emergence of a large-scale convection roll. This roll is driven by small-scale thermal plumes detaching from the boundary layers at the top and bottom. However, under the influence of rotation [4,5] the flow reorganizes itself to a flow structure with vertically aligned vortices with increasing rotation rate, see figure 2a. Through the experiments by Rossby in 1969 and others, it is known that rotation can enhance heat transport, see figure 1. For water rotation can increase the heat transfer in the system with up to 25%. The heat transport enhancement is caused by Ekman pumping due to which hot or cold fluid near the bottom and top plates is drawn into the vertically aligned vortices and efficiently transported in vertical direction. The efficiency of Ekman pumping, and the rotation rate for which the highest heat transport enhancement is found, strongly depends on the non-dimensional control parameters Rayleigh and Prandtl. The Rayleigh (Ra) number indicates the thermal driving of the system, and the Prandtl (Pr) number the ratio of momentum diffusivity to thermal diffusivity of the fluid.

For lower Rayleigh the heat transfer at a given Rayleigh and Prandtl is highest at an optimal rotation rate, at which the thickness of the viscous and thermal boundary layer is about equal. From the scaling relations of the thermal and viscous boundary layer thicknesses, we derive that the optimal rotation rate scales as 1/Ro_{opt}≈0.12Pr^{1/2}Ra^{1/6}, see figure 3. In the low Rayleigh regime the heat transfer is similar in a periodic domain and cylindrical cells with different aspect ratios, i.e. the ratio of diameter to height. This is consistent with the view that the vertically aligned vortices are the dominant flow structure, see figure 1a and figure 2a. For higher Rayleigh the above scaling for the optimal rotation rate does not hold anymore. It turns out that in the high Rayleigh regime, the flow structures at the optimal rotation rate are very different than for lower Rayleigh, see figure 2b. Surprisingly, the heat transfer in the high Rayleigh regime differs significantly for a periodic domain and cylindrical cells with different aspect ratios, which originates from the sidewall boundary layer dynamics and the corresponding secondary circulation, see figure 1b.

We close the discussion by noting that the investigation of the phenomena discussed above may find wider applications. It has namely been shown that for relatively low Rayleigh different stabilizing forces, such as horizontal confinement and a second stabilizing scalar field (also known as double diffusive convection), can generate a surprisingly large heat transport enhancement. The heat flux enhancement due to a stabilizing force is observed in an intermediate regime in which the stabilization is enough to re-organize the flow, but not so strong that it severely suppresses the flow motions [3]. The observation that this process may be different in high Rayleigh number rotating convection may indicate that the effect of other stabilizing forces may be different in other high Rayleigh number settings.

In this project, we perform novel direct numerical simulations to perform novel computer simulations of highly turbulent flows. In these simulations we are considering highly turbulent flows dominated by rotation, which is relevant for better understanding of geophysical flow phenomena. To ensure that all turbulent length and time scales are adequately resolved very large computational grids are required, which requires tens of millions of CPU hours.

Simulations using up to 37 thousand computational cores are performed on SuperMUC-NG and generating overall database of hundreds of terabyte. The benefit of simulations is that they allow one to adjust control parameters arbitrarily and isolate physical effects in an attempt to identify the physical mechanisms that control heat transport in highly turbulent flow. The challenge of these simulations is that they are performed on huge computational domains, making the simulations very time-consuming. To perform these large-scale simulations, we developed an in-house second-order finite-difference flow solver specially developed for cylindrical geometries and periodic domains. To perform landmark simulations, we have completely rewritten our code to optimize its performance on large-scale computing platforms like SuperMUC-NG. Our code is written in Fortran 90. Large-scale parallelization is obtained using MPI to divide the computational domain over the outer computational loops, while OpenMP is used to obtain two-dimensional parallelization throughout. This approach is favored since it limits network communication between computational nodes by performing computations within a shared memory environment. To ensure computational efficiency, great care has been taken to vectorize computationally intensive parts of the code, limit memory movement by handwriting and in-lining computational operations and data movements, efficient use of different cache levels, and replace library calls with handwritten routines to perform case-specific optimizations.

SuperMUC-NG allowed us to perform unprecedented simulations, i.e., the largest turbulence simulations in a fully closed domain, which will be compared to the Göttingen Rayleigh-Bénard convection experiments, which have revealed the transition to the ultimate regime in which the boundary layers along the plates become turbulent and the heat transport increases faster. Computer simulations of such turbulent flows are notoriously computationally demanding due to the extensive range of length and time scales that needs to be resolved. Therefore, such groundbreaking simulations can only be performed on the largest supercomputers in the world, such as SuperMUC-NG. Our simulations were only possible due to algorithmic developments that limited the communication between different computational tasks. This improved our code's parallel efficiency and parallel efficiency on a vast number of processors. Long-term storage and data accessibility are assured using the open-source HDF5 data format. However, even with the massive computational and storage facilities offered by SuperMUC-NG, it is still an enormous challenge to simulate such highly turbulent flows, and more powerful supercomputers are required to study more challenging cases.

[1] Group website: pof.tnw.utwente.nl; Project details:

https://stevensrjam.github.io/Website/

[2] Y. Yang, R. Verzicco, D. Lohse, R.J.A.M. Stevens, Phys. Rev. Fluids 5, 053501 (2020).

[3] K.L. Chong, Y. Yang, S.-D. Huang, J.-Q. Zhong, R.J.A.M. Stevens, R. Verzicco, D. Lohse, K.-Q. Xia, Phys. Rev. Lett., 119, 064501 (2017).

[4] R.J.A.M. Stevens, H.J.H. Clercx, D. Lohse, European Journal of Mechanics B/Fluids 40, 41-49 (2013).

[5] J.Q. Zhong, R.J.A.M. Stevens, H.J.H. Clercx, R. Verzicco, D. Lohse, G. Ahlers, Phys. Rev. Lett. 102, 044502 (2009).