The vortex method for 2D ideal flows in the exterior of a disk

TL;DR

This paper rigorously justifies the vortex method for simulating 2D ideal flows around a disk, modeling boundary effects with point vortices and addressing mathematical challenges of singular integral operators.

Contribution

It provides a rigorous mathematical foundation for the vortex method in exterior domains, specifically for flow around a disk, including boundary condition enforcement.

Findings

Justification of vortex method in exterior domain

Handling of singular integral operators on boundary curves

Application to flow around a disk with uniform vortex distribution

Abstract

The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation around the obstacle. The density of these point vortices is chosen in order that the flow remains tangent at midpoints between adjacent vortices. In this work, we provide a rigorous justification for this method in exterior domains. One of the main mathematical difficulties being that the Biot-Savart kernel defines a singular integral operator when restricted to a curve. For simplicity and clarity, we only treat the case of the unit disk in the plane approximated by a uniformly distributed mesh of point vortices. The complete and general version of our work is available in [arXiv:1707.01458].