Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation

TL;DR

This study uses a Fourier-Chebyshev pseudospectral DNS to investigate potential singularities in 3D axisymmetric Euler equations, revealing the formation of tygers and estimating singularity time through an extended analyticity-strip method.

Contribution

It introduces a pseudospectral DNS approach for 3D Euler equations and extends the analyticity-strip method to estimate singularity formation time.

Findings

Tygers form before singularity in spectrally truncated DNS.

Analyticity-strip method can estimate potential singularity time.

Tygers are similar to structures in Burgers and 2D Euler equations.

Abstract

We develop a Fourier-Chebyshev pseudospectral direct numerical simulation (DNS) to examine a potentially singular solution of the radially bounded, three-dimensional (3D), axisymmetric Euler equations [G. Luo and T.Y. Hou, Proc. Natl. Acad. Sci. USA, 111.36 (2014)]. We demonstrate that: (a) the time of singularity is preceded, in any spectrally truncated DNS, by the formation of oscillatory structures called tygers, first investigated in the one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) the analyticity-strip method can be generalized to obtain an estimate for the (potential) singularity time.