Evidence for a singularity of the three-dimensional, incompressible Euler equations

Abstract

Three-dimensional, incompressible Euler calculations of the interaction of perturbed antiparallel vortex tubes using smooth initial profiles in a bounded domain with bounded initial vorticity are discussed. A numerical method that uses symmetries and additional resolution in the direction and location of maximum compression is used to simulate periodic boundary conditions in all directions. For an initial condition that yields singular behavior, the growth of the peak vorticity, the peak axial strain, and the enstrophy production obey (tc−t)−1, and the enstrophy grows logarithmically. The enstrophy growth is associated with the energy spectrum approaching k−3. Self-similar development and equal rates of collapse in all three directions are shown.