Viscous and Inviscid Reconnection of Vortex Rings on Logarithmic Lattices

TL;DR

This study uses a novel logarithmic lattice numerical method to analyze vortex ring reconnection, revealing finite vorticity peaks at high Reynolds numbers and suggesting no finite-time singularity in viscous cases.

Contribution

The paper demonstrates that the log-lattice technique captures key physical processes and extends analysis to very high Reynolds numbers, providing new insights into vortex reconnection dynamics.

Findings

Vorticity peak remains finite at high Re_gamma in viscous flows.

The log-lattice method efficiently captures vortex dynamics with reduced computational cost.

A finite-time singularity occurs only in the inviscid case, not in viscous flows.

Abstract

To address the possible occurrence of a Finite-Time Singularity (FTS) during the oblique reconnection of two vortex rings, Moffatt-Kimura (MK) (J. Fluid Mech., 2019a; J. Fluid Mech., 2019b) developed a simplified model based on the Biot-Savart law and claimed that the vorticity amplification $\omega_{\text{max}}/\omega_0$ becomes very large for vortex Reynolds number $Re_{\Gamma} \geq 4000$. However, with Direct Numerical Simulation (DNS), Yao and Hussain (J. Fluid Mech., 2020) were able to show that the vorticity amplification is in fact much smaller and increases slowly with $Re_{\Gamma}$. The suppression of vorticity was linked to two key factors - deformation of the vortex core during approach and formation of hairpin-like bridge structures. In this work, a recently developed numerical technique called log-lattice (Campolina and Mailybaev, Nonlinearity, 2021), where interacting Fourier modes are logarithmically sampled, is applied to the same oblique vortex ring interaction problem. It is shown that this technique is not only capable of capturing the two key physical processes overlooked by the MK model but also other quantitative and qualitative attributes generally seen with DNS, at a fraction of the computational cost. Furthermore, the sparsity of the Fourier modes allows us to probe very large $Re_{\Gamma} = 10^8$ until which the peak of the maximum norm of vorticity, while increasing with $Re_{\Gamma}$, remains finite, and a blowup is observed only for the inviscid case.