Regularization of the problem of vortex reconnection

As a model for studying the evolution towards finite-time singularity of the Navier-Stokes equation, a dynamical system was proposed for describing the behavior of vortex reconnection of two vortex rings placed symmetrically on two tilted planes [1][2].

In this paper, we investigate this problem using the Biot-Savart model with the Rosenhead regularization numerically and theoretically. The Rosenhead regularization introduces a positive parameter in the denominator of the Biot-Savart integral which averts the logarithmic singularity in the limit when the denominator vanishes. It is demonstrated that with this regularization, the development of curvature at the tipping points, the closest approach on the vortex rings, is moderated. The effect of moderation extends in a small circular region around a tipping points the size of which is a function of the value of the parameter. It is shown that at large values of the parameter, the development of curvature deviates significantly to produce a spurious bubble. We analize this phenomenon theoretically.

[1] Moffatt, H.K. & Kimura, Y., Towards a finite-time singularity of the Navier-Stokes equations. Part 1. Derivation and analysis of dynamical system. J. Fluid Mech. (2019) 861 930-967.

[2] Moffatt, H.K. & Kimura, Y., Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion. J. Fluid Mech. (2019) 870 R1.