Turbulent Convection at Very Low Prandtl Numbers

**Principal Investigator:**

Jörg Schumacher

**Affiliation:**

Technische Universität Ilmenau (Germany)

**Local Project ID:**

hil09

**HPC Platform used:**

JUQUEEN (JSC)

**Date published:**

*In many turbulent convection flows in nature and technology the thermal diffusivity is much higher than the kinematic viscosity which means that the Prandtl number is very low. Applications of this regime reach from deep solar convection, via convection in the liquid metal core of the Earth to liquid metal batteries for grid energy storage and nuclear engineering technology. Laboratory experiments in low-Prandtl-number convection for Pr < 0.1 have to be conducted in liquid metals which are inaccessible for laser imaging techniques and require analysis by ultrasound or X-rays. Direct numerical simulations of this regime of turbulent convection at high Rayleigh numbers are the only way to reveal the full three-dimensional structure of temperature and velocity fields.*

In many turbulent convection flows in nature and technology the thermal diffusivity is much higher than the kinematic viscosity which means that the Prandtl number is very low. Applications of this regime reach from deep solar convection, via convection in the liquid metal core of the Earth to liquid metal batteries for grid energy storage and nuclear engineering technology. Laboratory experiments in low-Prandtl-number convection for Pr < 0.1 have to be conducted in liquid metals which are inaccessible for laser imaging techniques and require analysis by ultrasound or X-rays. Direct numerical simulations of this regime of turbulent convection at high Rayleigh numbers are the only way to reveal the full three-dimensional structure of temperature and velocity fields [1,2].

In Rayleigh-Bénard convection, a fluid cell is kept at a constant temperature difference between top and bottom plates. If the temperature difference is big enough, the fluid motion becomes turbulent. One of the fundamental questions in the field of turbulent convection is that of the turbulent transport mechanisms of heat and momentum. The thin boundary layers of the temperature and velocity fields, which form in the vicinity of the heating and cooling plates, are the key for a deeper understanding of the global transport mechanisms. For low Prandtl number flows the velocity boundary layer is much thinner than the one of the temperature – both boundary layers are to some degree decoupled of each other. Our numerical studies show highly fluctuating velocity boundary layers as visible for example in the skin friction field in the bottom row of figure 1. Based on these simulation runs, we can also predict a Rayleigh number range at which a transition to a fully turbulent velocity boundary layer and thus a crossover into the ultimate regime of turbulent convection is possible. We found that the Rayleigh numbers for such a transition to the ultimate regime decrease as the Prandtl number decreases [3,4].

In the framework of GCS Large Scale Projects, we conducted high-precision spectral element simulations which resolved the fine-scale structure of turbulent RBC, in particular the statistical fluctuations of the temperature and velocity gradients in the bulk as well as close to the cooling and heating plates. The massively parallel simulations required up to 262144 MPI tasks on the JUQUEEN supercomputer. Snapshots of three simulations in liquid sodium, liquid mercury and air are shown in Figure 1 at Rayleigh numbers of Ra=1e7, 1e8 and 1e10, respectively. The highly diffusive temperature field with the coarse thermal plumes (see Fig. 1 left) turns out to be an efficient driver of fluid turbulence in the interior of the convection cell [1]. The global transport of heat becomes much less efficient in liquid metals than in air or water at the same Rayleigh number. In turn, the global momentum transport is found to be increased significantly [2]. This generates the strong fluctuations in the velocity boundary layer.

Figure 2 shows a snapshot of the velocity boundary layer in a mercury cell. We plot streamlines of the near-wall velocity field in combination with high-amplitude vortex filaments. The complex flow structure, which can be seen in the plot, underlines the highly transient character of the velocity boundary layer. Such vortical structures are typical in transient and fully turbulent boundary layers in other flows, such as pipe flows. The quantitative measures for the prediction of the transition range to the ultimate regime are based on velocity derivatives that can be calculated close to and at the walls by accurate spectral methods in our simulations [3,4].

This research work is supported by the Deutsche Forschungsgemeinschaft within the Research Training Group GRK 1567 and the Priority Programme SPP 1881.

**Publications**

[1] Jörg Schumacher, Paul Götzfried and Janet D. Scheel, Enhanced enstrophy generation for turbulent convection in low-Prandtl number fluids, Proc. Natl. Acad. Sci. USA 112, 9530-9535 (2015).

[2] Janet D. Scheel and Jörg Schumacher, Global and local statistics in turbulent convection at low Prandtl numbers, J. Fluid Mech. 802, 147-173 (2016).

[3] Jörg Schumacher, Vinodh Bandaru, Ambrish Pandey and Janet D. Scheel, Transitional boundary layers in low-Prandtl-number convection, Phys. Rev. Fluids 1, 084402 (2016).

[4] Janet D. Scheel and Jörg Schumacher, Predicting transition ranges to fully turbulent viscous boundary layers in low Prandtl number convection flows, Phys. Rev. Fluids 2, 123501 (2017).

Prof. Dr. Jörg Schumacher

Institut für Thermo- und Fluiddynamik

Technische Universität Ilmenau

Postfach 100565, D-98684 Ilmenau

e-mail: joerg.schumacher[at]tu-ilmenau.de