The vortex method for two-dimensional ideal flows in exterior domains

Diogo Arsénio, Emmanuel Dormy, Christophe Lacave

Research output: Contribution to journalArticlepeer-review

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 and that the total vorticity around the obstacle is conserved. In this work, we provide a rigorous justification of this method for any smooth exterior domain, one of the main mathematical difficulties being that the Biot-Savart kernel defines a singular integral operator when restricted to a curve (here, the boundary of the domain). We also introduce an alternative method-the fluid charge method-which, as we argue, is better conditioned and therefore leads to significant numerical improvements.

Original languageEnglish (US)
Pages (from-to)3881-3961
JournalSIAM Journal on Mathematical Analysis
Volume52
Issue number4
DOIs
StatePublished - 2020

Keywords

  • Cauchy integrals
  • Discretization of singular integral operators
  • Double layer potential
  • Elliptic problems in exterior domains
  • Euler equations
  • Poincaré-Bertrand formula
  • Spectral analysis

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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