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 $ω_{max}$ as a function of vortex Reynolds number $R_Γ$ in the range $2000\le R_Γ\le 3400$, and deduce a compatible behaviour $ω_{max}\sim ω_{0}\exp{\left[1 + 220 \left(\log\left[R_Γ/2000\right]\right)^{2}\right]}$ as $R_Γ\rightarrow \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_Γ\gtrsim 4000$.