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

  • H. K. Moffatt (a1) and Yoshifumi Kimura (a2)

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 $\unicode[STIX]{x1D714}_{max}$ as a function of vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}$ in the range $2000\leqslant R_{\unicode[STIX]{x1D6E4}}\leqslant 3400$ , and deduce a compatible behaviour $\unicode[STIX]{x1D714}_{max}\sim \unicode[STIX]{x1D714}_{0}\exp [1+220(\log [R_{\unicode[STIX]{x1D6E4}}/2000])^{2}]$ as $R_{\unicode[STIX]{x1D6E4}}\rightarrow \infty$ . This may be described as a physical (although not strictly mathematical) singularity, for all $R_{\unicode[STIX]{x1D6E4}}\gtrsim 4000$ .

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Corresponding author

Email address for correspondence: hkm2@damtp.cam.ac.uk

References

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Hussain, F. & Duraisami, K. 2011 Mechanics of viscous vortex reconnection. Phys. Fluids 23, 021701.
Jeong, J. T. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds-number. J. Fluid Mech. 839, R2.
Kerr, R. M. 2018 Enstrophy and circulation scaling for Navier–Stokes reconnection. J. Fluid Mech. 241, 122.
McKeown, R., Ostilla-Monico, R., Pumir, A., Brenner, M. P. & Rubinstein, S. M. 2018 A cascade leading to the emergence of small structures in vortex ring collisions. Phys. Rev. Fluids 3, 124702.
Melander, M. V. & Hussain, F. 1989 Cross-linking of two antiparallel vortex tubes. Phys. Fluids A 1, 633636.10.1063/1.857437
Moffatt, H. K. & Kimura, Y. 2019 Towards a finite-time singularity of the Navier–Stokes equations. Part 1. J. Fluid Mech. 861, 950967.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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