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On singularity formation via viscous vortex reconnection
Published online by Cambridge University Press: 06 February 2020
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.
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- JFM Rapids
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- © The Author(s), 2020. Published by Cambridge University Press
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