Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows

Abstract

Direct numerical simulations of the three-dimensional Euler equations at resolutions up to 2563 for general periodic flows and 8643 for the symmetric Taylor–Green vortex are presented. The spontaneous emergence of flat pancakelike structures that shrink exponentially in time is observed. A simple self-similar model that fits these observations is discussed. Focusing instabilities similar to those leading to streamwise vortices in the context of free shear layers [J. Fluid Mech. 143, 253 (1984)], are expected to subsequently concentrate the vorticity and produce isolated vortex filaments. A finite time singularity for the Euler equation is not excluded as the result of interactions among these filaments.