Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations
We study the interplay between the local geometric properties
and the non-blowup of the 3D incompressible Euler equations. We consider
the interaction of two perturbed antiparallel vortex tubes [Phys. Fluids 5 (1993), 1725].
We use a pseudo-spectral method with very high
resolution to resolve the nearly singular behavior of the Euler equations.
Our numerical results demonstrate that the maximum vorticity does not grow
faster than double exponential in time until the solution is numerically
resolved. The velocity, the enstrophy and enstrophy production rate remain
bounded throughout the computations. As the flow evolves, the vortex tubes
are flattened severely and turned into thin vortex sheets, which roll up
subsequently. The vortex lines near the region of the maximum vorticity
are relatively straight. This local geometric regularity of vortex lines
seems to be responsible for the dynamic depletion of vortex stretching.