Assessing late-time singular behaviour in symmetry-plane models of 3D Euler flow

etc15 Tracking Number 84

Motivated by work on stagnation-point type exact solutions of the 3D Euler fluid equations by Gibbon [Gibbon et. al. Phys. D, 132, 497, (1999)] and the subsequent demonstration of finite-time blowup by Constantin [Constantin, Math. Res. Notices, 9, 455, (2000)] we introduce a one-parameter family of models of the 3D Euler equations on a 2D symmetry plane. These models provide a collection of blow-up scenarios which admit analytical solutions and are computationally inexpensive in comparison to the full 3D Euler equations. We take advantage of these features to examine the efficacy of novel methods which aid the assessment of finite-time blow-up in numerical simulations. The principal of these is the mapping to regular systems [Bustamante, Phys. D, 240, 1092, (2011)]; a bijective nonlinear mapping of time and the prognostic variables based on a Beale-Kato-Majda (BKM) type supremum norm regularity condition [Beale et. al. Commun. Math. Phys. 94, 61, (1984)]. We show a 3 order of magnitude increase of accuracy of the singularity time when employing the mapping with negligible additional computational expense. An investigation of the spectra of the primary field (vortex stretching rate) allows us to confirm a power law decrement of the analyticity-strip width with time in agreement with rigorous bounds bridging between the global spatial behaviour and BKM theorems [Bustamante & Brachet, Phys. Rev. E. 86, (2012)].