Exploring finite time singularity via collision of inviscid vortex rings

Motivated by the recent model on the possible formation of finite-time singularity (FTS) by Moffatt & Kimura [J. Fluid Mech. 861, 930 (2019); J. Fluid Mech. 870, R1 (2019)], we conducted a numerical study of two colliding slender vortex rings with the same condition as those used in Yao & Hussain (J. Fluid Mech., vol. 888, 2020a, R2) for three-dimensional incompressible Euler equations. While most metrics – such as the separation s and curvature κ at the two tipping points – exhibit similar trends as predicted by the Moffatt-Kimura model, significant differences are indeed found between them. Specifically, computed κ and the maximum vorticity ∥ω(x, t)∥∞ grow much slower than the model’s prediction. Several phenomena that may affect the formation of a finite-time singularity are observed. First, when the two vortices come close (i.e., the separation distance is comparable to the core size), significant core flattening occurs – deforming the initially circular vortex cross-section into elliptical and then into the typical head-tail structures. In addition, near the end of the simulation, the curvature κ at the tipping points eventually saturates. Finally, the strong core dynamics along the vortex centerline can further prevent core size from decreasing and the maximum vorticity from increasing considerably. Whether a finite-time singularity can develop for this configuration still remains an unresolved issue; if it does at all, the singularity time should be much later than the predicted value (i.e., t_c ≈ 0.243).