Asymptotic properties of wall-induced chaotic mixing in point vortex pairs

TL;DR

This paper analyzes how a wall influences chaotic mixing around a vortex pair, revealing that wall effects induce bifurcations and stochastic layers that enhance fluid mixing, with layer thickness depending on wall proximity.

Contribution

It provides an asymptotic analysis of wall effects on vortex pair flow, demonstrating the formation of stochastic layers and their scaling with wall distance, supported by Melnikov and separatrix map methods.

Findings

Wall induces homoclinic bifurcations in vortex flow.

Inner stochastic layer thickness scales inversely with wall distance squared.

Outer stochastic layer exists but is significantly thinner than the inner layer.

Abstract

The purpose of this work is to analyze the flow due to a potential point vortex pair in the vicinity of a symmetry line (or "wall"), in order to understand why the presence of the wall, even far from the vortices, accelerates fluid mixing around the vortex pair. An asymptotic analysis, in the limit of large distances to the wall, allows to approximate the wall effect as a constant translation of the vortex pair parallel to the wall, plus a straining flow which induces a natural blinking vortex mechanism with period half the rotation period. A Melnikov analysis of lagrangian particles, in the frame translating and rotating with the vortices, shows that a homoclinic bifurcation indeed occurs, so that the various separatrices located near the vortex pair (and rotating with it) do not survive when a wall is present. The thickness of the resulting inner stochastic layer is estimated by using the separatrix map method, and is shown to scale like the inverse of the squared distance to the wall. This estimation provides a lower-bound to the numerical thickness measured from either Poincar\'e sections or simulations of lagrangian particles transported by the exact potential velocity field in the laboratory frame. In addition, it is shown that the outer homoclinic cycle, separating the vortices from the external (open) flow, is also perturbed from inside by the rotation of the vortex pair. As a consequence, a stochastic layer is shown to exist also in the vicinity of this cycle, allowing fluid exchange between the vortices and the outer flow. However, the thickness of this outer stochastic zone is observed to be much smaller than the one of the inner stochastic zone near vortices, as soon as the distance to the wall is large enough.