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

Published online by Cambridge University Press:  07 May 2019

H. K. Moffatt Open the ORCID record for H. K. Moffatt [Opens in a new window]  and
Yoshifumi Kimura
H. K. Moffatt*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Yoshifumi Kimura
Affiliation:
Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
*
†Email address for correspondence: [email protected]
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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 $R_{\unicode[STIX]{x1D6E4}}\gtrsim 4000$.

JFM classification

Vortex Flows: Vortex dynamics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , R1
Copyright
© 2019 Cambridge University Press 

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References

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