Self-similar collapse of a 3D straight vortex filament model

Abstract

As an example of 3D vortex collapse, the vortex dodecapole, the superposition of three equal-strength, orthogonal, vortex quadrupoles, was proposed by Rich Pelz and co-workers. In this article, we examine the simplest model of the vortex dodecapole in which the vortex tubes are replaced with straight vortex filaments of infinitesimal thickness and the entire motion is monitored by tracing the motion of a representative point on one vortex filament. It is demonstrated that this model permits a self-similar collapse solution which provides the time dependence of the length scale as (t _c − t)^1/2, (t < t _c ) where t _c , the collapse time, depends on the initial configuration. This time dependence implies that vorticity ω scales as (t _c − t)⁻¹, which agrees with the one observed in the direct numerical (pseudo spectral) simulations of the vortex dodecapole.

Keywords:

• Finite-time singularity
• Euler's equation
• Vortex filaments
• Self-similar collapse

Acknowledgments

This work was supported by Grant-in-Aid for Scientific Research (B) No. 18340025, Japan Society for the Promotion of Science. The author would like to thank Aimé Fournier for discussion.