A pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation: tygers and thermalization

Sai Swetha Venkata Kolluru, Puneet Sharma, Rahul Pandit

Published 2020-12-28Version 1

G.Luo and T.Y.Hou (Proceedings of the National Academy of Sciences 111.36 (2014): 12968-12973) have investigated a potentially singular solution of the radially bounded, 3D, axisymmetric Euler equations via a hybrid, 6th-order Galerkin and 6th-order finite-difference method on an adaptive (moving) mesh. Given the importance of this problem, we have developed a pseudospectral, Fourier-Chebyshev scheme to study this solution by an independent numerical technique, which leads to new insights into this (potentially) singular solution: (a) the time of singularity is preceded, in any, practical, spectrally truncated direct numerical simulation (DNS), by the formation of structures called tygers, first investigated in the one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) these tygers herald the thermalization of our spectrally truncated system of equations. We demonstrate how we can analyse this (potential) singularity by using singularity-detection criteria based on (a) the Beale-Kato-Majda theorem and (b) a generalization of the analyticity-strip method. We find that the width of the analyticity strip, in the axial direction, decays faster than an exponential and, in the radial direction, it decays linearly; this linear decay can be extrapolated to obtain an estimate for the time at which the (potential) singularity occurs; our estimate is consistent with that of G.Luo and T.Y.Hou. We examine the DNS-resolution dependence of (a) the tyger-formation time, (b) thermalization, (c) energy and helicity conservation, and (d) our estimate for the singularity time.