Viscous evolution of point vortex equilibria: The collinear state

TL;DR

This paper investigates how a collinear three-vortex equilibrium in 2D Euler flows evolves under viscosity, revealing initial unsteady rotation, instability, and a transition to a long-term Lamb-Oseen vortex with topological bifurcations.

Contribution

It introduces a multi-Gaussian core-growth model to analyze viscous evolution of vortex equilibria, highlighting instability mechanisms and bifurcation sequences.

Findings

Immediate unsteady rotation due to viscous effects

Sequence of topological bifurcations during evolution

Passive particle trajectories reflect flow pattern changes

Abstract

When point vortex equilibria of the 2D Euler equations are used as initial conditions for the corre- sponding Navier-Stokes equations (viscous), typically an interesting dynamical process unfolds at short and intermediate time scales, before the long time single peaked, self-similar Oseen vortex state dom- inates. In this paper, we describe the viscous evolution of a collinear three vortex structure that cor- responds to an inviscid point vortex fixed equilibrium. Using a multi-Gaussian 'core-growth' type of model, we show that the system immediately begins to rotate unsteadily, a mechanism we attribute to a 'viscously induced' instability. We then examine in detail the qualitative and quantitative evolution of the system as it evolves toward the long-time asymptotic Lamb-Oseen state, showing the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. The evolution of passive particles in this viscously evolving flow is shown and interpreted in relation to these evolving streamline patterns.