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On singularity formation via viscous vortex reconnection

Published online by Cambridge University Press:  06 February 2020

Jie Yao Open the ORCID record for Jie Yao [Opens in a new window]  and
Fazle Hussain Open the ORCID record for Fazle Hussain [Opens in a new window]
Jie Yao
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79409, USA
Fazle Hussain*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX79409, USA
*
†Email address for correspondence: fazle.hussain@ttu.edu
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Abstract

Recognizing the fact that the finite-time singularity of the Navier–Stokes equations is widely accepted as a key issue in fundamental fluid mechanics, and motivated by the recent model of Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019a, pp. 930–967; J. Fluid Mech., vol. 870, 2019b, R1) on this issue, we have performed direct numerical simulation (DNS) for two colliding slender vortex rings of radius $R$. The separation between the two tipping points $2s_{0}$ and the scale of the core cross-section $\unicode[STIX]{x1D6FF}_{0}$ are chosen as $\unicode[STIX]{x1D6FF}_{0}=0.1s_{0}=0.01R$; the vortex Reynolds number ($Re=\text{circulation/viscosity}$) ranges from 1000 to 4000. In contrast to the claim that the core remains compact and circular, there is notable core flattening and stripping, which further increases with $Re$ – akin to our previous finding in the standard anti-parallel vortex reconnection. Furthermore, the induced motion of bridges arrests the curvature growth and vortex stretching at the tipping points; consequently, the maximum vorticity grows with $Re$ substantially slower than the exponential scaling predicted by the model – implying that, for this configuration, even physical singularity is unlikely. Our simulations not only shed light on the longstanding question of finite-time singularities, but also further delineate the detailed mechanisms of reconnection. In particular, we show for the first time that the separation distance $s(\unicode[STIX]{x1D70F})$ before reconnection follows 1/2 scaling exactly – a significant DNS result.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex interactions Turbulent Flows: Turbulence theory
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , R2
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Yao et al. supplementary movie

Simulation of two slender inclined vortex rings for Reynolds number up to 4000.

Download Yao et al. supplementary movie(Video)
Video 15.1 MB