Singularity formation of vortex sheets in 2D Euler equations using the characteristic mapping method

Abstract

The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for non-smooth initial data, in particular for vortex sheets. To this end high resolution computations of vortex layers in 2D incompressible Euler flows are performed using the characteristic mapping method (CMM). This semi-Lagrangian method evolves the flow map using the gradient-augmented level set method (GALS). The semi-group structure of the flow map allows its decomposition into sub-maps (each over a finite time interval), and thus the precision can be controlled by choosing appropriate remapping times. Composing the flow map yields exponential resolution in linear time, a unique feature of CMM, and thus fine scale flow structures can be resolved in great detail. Here the roll-up process of vortex layers is studied varying the thickness of the layer showing its impact on the growth of palinstrophy and possible blow up of absolute vorticity. The curvature of the vortex sheet shows a singular-like behavior. The self-similar structure of the vortex core is investigated in the vanishing thickness limit. Conclusions on the non-uniqueness of weak solutions of 2D Euler for non-smooth initial data are drawn and the presence of flow singularities is revealed tracking them in the complex plane.