Complex-space singularities of 2D Euler flow in Lagrangian coordinates

TL;DR

This paper demonstrates that for 2D incompressible flows, Lagrangian coordinates enable highly accurate numerical evaluation and reveal complex singularities closer to the real domain than Eulerian ones, providing insights into fluid particle behavior.

Contribution

It introduces a numerical method in Lagrangian coordinates for 2D Euler flow that accurately identifies complex singularities and compares them to Eulerian singularities.

Findings

Lagrangian singularities are closer to the real domain than Eulerian ones

Lagrangian singularities correspond to particles escaping to complex infinity

The method achieves spectral accuracy comparable to Eulerian spectral methods

Abstract

We show that, for two-dimensional space-periodic incompressible flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy achieved in standard Eulerian spectral methods. This allows the determination of complex-space Lagrangian singularities. Lagrangian singularities are found to be closer to the real domain than Eulerian singularities and seem to correspond to fluid particles which escape to (complex) infinity by the current time. Various mathematical conjectures regarding Eulerian/Lagrangian singularities are presented.