In fluid dynamics, fluid motions can be separated into two very distinct types of flows. In Laminar flows, the fluid particles move in ordered Layers with little to no mixing between different Layers. A simple example of a laminar flow would be a viscous fluid flowing through a pipe. In contrast, turbulent flows are chaotic in nature. They can be observed in many aspects of everyday life, but are also of particular importance for many aspects of modern physics. Examples of this would be oceanic mixed layers, or chemical mixing in giant molecular clouds. While the statistical properties of turbulence are relatively well understood, mixing processes are still a field of active research, especially in astrophysical conditions.
There are two major simulation methods to model hydrodynamic systems. Many commonly used codes like AREPO use a mesh-based approach, where the simulation domain is subdivided into many smaller cells. Alternatively, smoothed particle hydrodynamics (SPH) based codes introduce particles that can move through the simulation domain and carry all necessary quantities like pressure and density without requiring any grid. Such SPH-based codes are particularly well suited to study mixing problems. As the simulation method in Lagrangian in nature, mixing can simply be studied by tracing the SPH-particles without requiring any additional tracer particles.
For this reason, our group is interested in using the new SPH-EXA simulation code to study mixing problems related to giant molecular clouds and star formation. SPH-EXA is a new, GPU-accelerated simulation code that is developed by research teams at the universities in Basel and Zürich. It is specifically designed for high-resolution simulations and can handle billions of SPH-particles with relative ease. To achieve the necessary computational power, we use the Alps supercomputer, one of the largest supercomputers in the world.
So far, our work has mainly focused on validating and improving this new simulation code. This is especially important for subsonic turbulence, a regime with which the SPH method has historically struggled. Through several improvements in the simulation method, SPH-EXA can now produce accurate probability density functions, power spectra and structure functions in the regime of subsonic turbulence using the SPH method. Our results match those from established mesh-based codes like AREPO (see Cabézon et al (2025) for more details). This is a very important milestone, as we are now confident that our code accurately models turbulent flows in both the sub- and supersonic regime.
The SPH method is not only beneficial for studying mixing coefficients. It also gives easy access to diagnostics like the forward and backward in time finite-time Lyapunov exponent (FTLE). The FTLE is, broadly speaking, a measurment of chaos. Here, we use this quantity to identify the most attracting and repelling surfaces in a flow and is useful to visualize coherent structures in complex flows. The Figure above shows a slice through a simulation of isothermal, supersonic turbulence with periodic boundary conditions. This simulation was carried out with a Mach number of 4 and a resolution of SPH-particles. One can see that both the forward and backward in time the high-FTLE regions correspond to the shock fronts in the fluid. This is expected, as the shock fronts are regions where the gas is compressed and then decompressed very rapidly.
With all this in mind, we can now move forward to studying mixing processes under astrophysical conditions. An specific application is mixing of hot and cold gas phases in the interstellar medium where cold gas is able to form stars but stars then heat gas due to their radiation and, finally, during their supernova explosion. These idealised turbulence simulations help to better understand how the different gas phases mix and interact to regulate star formation.
This work has been done during the Master Thesis of Oliver Avril.