The question of finite time singularities in the Euler and Navier-Stokes equation is one of the outstanding open problems in fluid dynamics with strong implications on the intermittency properties of turbulent flows. The mathematical question whether smooth initial condition stay smooth for all times or develop singularities in finite time was recognized as one of the Millenium problems by the Clay Mathematics Institute. We try to gain further insight into this problem by performing high resolution adaptive mesh refinement simulations using our new developed framework racoon for massive parallel computations. Recent mathematical theorems highlight the importance of the geometry (curvature) of vortex lines and the divergence of the direction of vortex lines in a vortex tube near the possible singularity. In order to testify the mathematical assumptions of theorems which provide necessary conditions for the existence of global solutions, we advect passive tracer particles with the flow to follow the dynamics of the vortex lines and their geometric properties.

Principal Investigators:

Prof. Dr. R. Grauer

Institute for Theoretical Physics I
Ruhr-University Bochum

Dr. Jürgen Dreher

Institute for Theoretical Physics I
Ruhr-University Bochum

PhD Students

Tobias Grafke

Institute for Theoretical Physics I
Ruhr-University Bochum

Thorsten Hater

Institute for Theoretical Physics I
Ruhr-University Bochum)