Turbulent Convection at Very Small Prandtl Numbers

**Principal Investigator:**

Jörg Schumacher

**Affiliation:**

Technische Universität Ilmenau

**Local Project ID:**

pr62se, pn68ni

**HPC Platform used:**

SuperMUC and SuperMUC-NG of LRZ

**Date published:**

**Introduction**

Turbulent convection is one essential process to transport heat across a fluid layer or closed domain. In many of the astrophysical or technological applications of convection the working fluid is characterized by a very low dimensionless Prandtl number *Pr=**ν**/κ* which relates the kinematic viscosity to temperature diffusivity. Two important cases are (i) turbulent convection in the outer shell of the Sun at *Pr~**10*^{-6} in the presence of rotation, radiation, magnetic fields, and even changes of the chemical composition close to the surface [1,2] and (ii) turbulent heat transfer processes in the cooling blankets of nuclear fusion reactors at *Pr~**10*^{-2}[3]. These are rectangular ducts which are exposed to very strong magnetic fields that keep the 100 million Kelvin hot plasma confined. Our understanding of the complex interaction of turbulence with the other physical processes in these two examples is still incomplete. High-resolution direct numerical simulations of the equations of turbulent fluid motion in the simplest setting of a turbulent convection flow, Rayleigh-Bénard convection in a layer or a straight duct that is uniformly heated from below and cooled from above, help to reveal some of these aspects at a reduced physical complexity and to discuss the basic heat transfer mechanisms that have many of these applications in common. Such studies have to rely on massively parallel supercomputers.

**Results and Methods**

We solve the three-dimensional Boussinesq equations of thermal convection. They couple the turbulent velocity and temperature fields. The external magnetic field in the nuclear fusion application (ii) is typically very strong such that we can apply the quasi-static limit of magnetohydrodynamics [3]. The vigor of convective turbulence is quantified by a further dimensionless parameter, the Rayleigh number *Ra*, and the strength of the applied external magnetic field by the Hartmann number *Ha*. Turbulent flows at very low Prandtl numbers are known to cause highly inertial fluid turbulence which makes our numerical simulations very challenging since all turbulent vortices down to the smallest ones have to be resolved. We apply a second-order finite difference method [3]. The simulation domains are cuboid cells or ducts with no-slip boundary conditions at all walls. The sidewalls are thermally insulated. The numerical simulations at *Pr=**10*^{-3} in domains of aspect ratio 25:25:1 require 38400 SuperMUC-NG cores for a grid with *12800×12800×800* points in the non-magnetic case (i). The simulations at a Hartmann number *Ha=**10*^{3} require up to 7680 cores for a long duct with *15360×1280×384* grid points. All simulations are long-term runs that involved sequences of several 48-hour runs in a row. In the course of two project years, this sums up to 80 million core hours which will eventually be consumed for the Large Scale Projects pr62se and pn68ni.

Figure 1 illustrates the impact of a strong magnetic field on the turbulent mixing and thus the heat transfer properties in a rectangular duct flow (case ii). The flow enters the duct as a planar jet which is immediately rotated into the direction of the external magnetic field (along blue axis) and develops quasi-2d vortical structures downstream (along the red axis) that can mix the liquid metal coolant effectively. Also visible are so-called Shercliff layers at the front and back face that become unstable and contribute additionally to the turbulent transport.

Figure 2 (left) is a view from the top onto the streamlines in the whole turbulent Rayleigh-Bénard convection layer for *Ra=**10*^{6} and *Pr=**10*^{-3}, one of our biggest simulation runs so far. Clearly, these parameters are by many orders of magnitude away from any realistic dynamics of solar convection. Nevertheless, they provide us already new and interesting insights into the low-Prandtl-number turbulence.

We observe how the convection is organized in large patterns of circulation rolls that fill the whole layer and extend from the top to the bottom. Further turbulence fields are replotted in the right column of this figure. Characteristic for low-Prandtl-number convection is a very coarse-grained temperature field which is obvious from the blurred contours that are visible. Gradients of temperature will be washed out quickly due to the large diffusivity which implies that heat transport is very inefficient and close to the diffusive lower bound. In contrast, we observe many fine filaments for quantities that probe the small-scale structure of the corresponding fluid turbulence, such as the kinetic energy dissipation rate field or the turbulent kinetic energy. Both fields suggest a large Reynolds number flow and thus a strong momentum transfer. As the fine-scale features of the convective turbulence cannot be resolved in more complex numerical simulations of bigger domains, our present data records provide an ideal testing bed to calculate turbulent eddy viscosities and diffusivities which can be used to close the underlying equations of motion in reduced models that describe the large-scale features of turbulent convection only. These investigations are currently underway.

**On-going Research / Outlook**

As demonstrated, our numerical investigations require access to the most powerful supercomputers. In both discussed examples, we were thus able to study turbulent convection in parameter ranges that are not accessible in laboratory experiments. For example, the smallest possible Prandtl number in a laboratory experiment is *Pr≈0.005* for liquid sodium.

The SuperMUC-NG computer made furthermore long-term simulations possible that resolved the evolution of the large-scale patterns. An important point of the future work will be to extend the complexity of the studies in both cases to more realistic temperature-dependent material parameters *ν(T)* and *κ**(T)**, * a step which has been started within the present Large Scale Project pn68ni.

**References and Links**

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

[2] J. Schumacher and K. R. Sreenivasan, Rev. Mod. Phys. **92**, 041001 (2020).

[3] I. Belyaev, D. Krasnov et al. Phys. Fluids** 32**, 094106 (2020).

**Research Team**

Dmitry Krasnov^{1}, Ambrish Pandey^{2, }Jörg Schumacher (PI)^{1}

^{1}Technische Universität Ilmenau^{2}New York University Abu Dhabi

**Scientific Contact**

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

http://www.tu-ilmenau.de/tsm

*LRZ project IDs: pr62se, pn68ni*

*November 2020*