Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion

Abstract

In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier-Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute the maximum vorticity ${\omega }_{max}$ as a function of vortex Reynolds number $R_{\Gamma }$ in the range $2000\le R_{\Gamma }\le 3400$, and deduce a compatible behaviour ${\omega }_{max}\sim {\omega }_0\exp \left[1+220{\left(\log \left[R_{\Gamma }/2000\right]\right)}^2\right]$ as $R_{\Gamma }\to \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_{\Gamma }\gtrsim 4000$.

10 pages, 8 figures, accepted for JFM Rapids

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