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Rapid Growth Events of Enstrophy in Fluid Turbulence

Prof. Dr. Jörg Schumacher, TU Ilmenau

Although the Navier-Stokes equations have been known for the last 170 years, the uniqueness and regularity of their solutions for the three-dimensional case is still an open issue and one of the millennium problems in mathematics. The uniqueness of their solutions is tightly connected with the enstrophy which is nothing else but the square of the magnitude of the fluid vorticity collected over some volume. Turbulence researchers are puzzled if this quantity can blow up in time in a viscous turbulent fluid. If this is not the case, it is still interesting to study how rapidly with respect to time enstrophy can grow.

The present supercomputer simulations are studying exactly this question and will be compared with recently found analytical results. We therefore generate turbulence in a periodic box which is resolved with 2048 grid points in each direction. The local growth of the enstrophy is monitored in the quasi-Lagrangian frame. A little observational cube is comoving therefore with a tracer particle. Think of riding on a roller coaster through the flow and monitoring your local environment. Such analysis removes the so-called sweeping, i.e., that smaller vortices are swimming on bigger ones. The animation illustrates first the tube-like filaments (see figure) in which the strongest vortices are found. Exactly these filaments will be monitored with the cubes. The second part of the animation demonstrates that interacting vortex tubes can cause a very rapid local growth of the enstrophy. The simulations were conducted on the JUMP cluster at the NIC in Jülich and took 45 days on 512 CPUs. It was a project of the Deep Computing Initiative of the DEISA.