Turbulent Convection in Slender Cells up to Ra=10^{15}

**Principal Investigator:**

Jörg Schumacher

**Affiliation:**

Institute of Thermodynamics and Fluid Mechanics, TU Ilmenau (Germany)

**Local Project ID:**

chil12

**HPC Platform used:**

JUWELS of JSC

**Date published:**

**Project name: Eulerian and Lagrangian analysis of transport in thermal convection in thermal convection flows in large aspect ratio cells**

The law of heat transfer remains one of the central questions in the research field of turbulent thermal convection with its numerous applications in atmospheric and astrophysical phenomena and technology. The Rayleigh-Bénard setup includes a fluid layer that is uniformly heated from below and cooled from above. It is the simplest turbulent thermal convection flow and serves as a paradigm for many real-life flows. Our present knowledge of the heat transport law in this setting, which connects the Nusselt number *Nu*, a dimensionless measure of turbulent heat transfer across the convection layer, with the Rayleigh number *Ra*, a dimensionless number that probes the vigor of convective turbulence, is still inconclusive for very high Rayleigh numbers. Laboratory experiments in closed cylindrical cells with diameters similar to the cell height for *Ra ≥ **10*^{12} have reported different outcomes. Typical working fluids to obtain these large Rayleigh numbers are cryogenic helium or compressed sulfur hexafluoride. The resulting Prandtl numbers *Pr* are close to those of convection in air.

Our direct numerical simulations of turbulent Rayleigh-Bénard convection in closed cylindrical cells advanced to Rayleigh numbers as high as *Ra =** 10*^{15} when the aspect ratio (which relates the diameter *d *to the height *H*) of the cell is chosen sufficiently small, *d**/H**=**1**/10* for *Pr = 1. *Without a decrease of the aspect ratio to such a small value, direct numerical simulations which resolve all relevant scales in the flow cannot access such large Rayleigh numbers. We found that the classical scaling law of turbulent heat transfer, *Nu~**Ra*^{1/3}*,* is satisfied. This scaling assumes that the thin boundary layers of the temperature and velocity fields at the top and bottom plates are marginally stable, a model that goes back to the seminal theoretical works by Malkus and Spiegel in the 50s and 60s of the past century. Even the prefactor of the detected transport law was very close to the theoretical predictions. For the highest Rayleigh number in our simulations and the chosen slender cell geometry, the ratio of the boundary layer thickness to the cell diameter is still about 1 to 1000. Figure 1 displays the boundary layer structure for three different Rayleigh numbers. It is seen how the turbulence fields become more and more filamented, developing ever finer thermal plumes structures and vortices. We also showed that the fluctuations, e.g. those of the wall shear stress components, grow steadily as *Ra* grows without an abrupt transition into a new state. Therefore, the Rayleigh numbers were varied over a range of 8 orders of magnitude.

The present direct numerical simulations applied a spectral element method (nek5000) which guarantees an exponentially fast decrease of numerical approximation errors. The simulation runs were carried out on both sides of the Atlantic Ocean, the runs up to *Ra=**10*^{12} on JUWELS at the JSC and the remaining ones for larger Rayleigh numbers on BG/Q Mira at the Argonne National Laboratory in the United States within a parallel INCITE project (PI J. D. Scheel). For the highest Rayleigh numbers the spectral element grid comprised more than 17 million spectral elements. On each of these spectral elements the fields are expanded in interpolation polynomials of degree 11 in each spatial direction which required massively parallel simulations with more than half a million processing cores on BG/Q.

In the future, we plan to extend these studies to even higher Rayleigh numbers which would then come already close to typical values of convection in an atmospheric boundary layer.

**Project Team Members**

- Kartik P. Iyer, New York University, New York, USA
- Janet D. Scheel, Occidental College, Los Angeles, USA
- Katepalli R. Sreenivasan, New York University, USA

**References**

[1] Kartik P. Iyer, Janet D. Scheel, Jörg Schumacher, and Katepalli R. Sreenivasan, Classical 1/3 scaling of convection holds up to Ra=1015, PNAS 117, 7594-7598 (2020). https://www.pnas.org/cgi/doi/10.1073/pnas.1922794117

[2] https://www.tu-ilmenau.de/tsm

**Scientific Contact**

Prof. Dr. Jörg Schumacher

Technische Universität Ilmenau

Department of Mechanical Engineering

Institute of Thermodynamics and Fluid Mechanics

P.O.Box 100 565, D-98684 Ilmenau (Germany)

e-mail: Joerg.Schumacher [@] tu-ilmenau.de

*JSC project ID: chil12*

*April 2020*