On singularity formation via viscous vortex reconnection

Abstract

Recognizing the fact that the finite-time singularity of the Navier–Stokes equations is widely accepted as a key issue in fundamental fluid mechanics, and motivated by the recent model of Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019a, pp. 930–967; J. Fluid Mech., vol. 870, 2019b, R1) on this issue, we have performed direct numerical simulation (DNS) for two colliding slender vortex rings of radius . The separation between the two tipping points and the scale of the core cross-section are chosen as ; the vortex Reynolds number ( ) ranges from 1000 to 4000. In contrast to the claim that the core remains compact and circular, there is notable core flattening and stripping, which further increases with – akin to our previous finding in the standard anti-parallel vortex reconnection. Furthermore, the induced motion of bridges arrests the curvature growth and vortex stretching at the tipping points; consequently, the maximum vorticity grows with substantially slower than the exponential scaling predicted by the model – implying that, for this configuration, even physical singularity is unlikely. Our simulations not only shed light on the longstanding question of finite-time singularities, but also further delineate the detailed mechanisms of reconnection. In particular, we show for the first time that the separation distance before reconnection follows 1/2 scaling exactly – a significant DNS result.