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

Diogo Arsénio[1]; Emmanuel Dormy[2]; Christophe Lacave[3]

  • [1] CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586 Université Paris Diderot Sorbonne Paris Cité Sorbonne Universités UPMC Université Paris 06 F-75013 Paris, France
  • [2] CNRS & MAG (ENS/IPGP) Département de Physique 24 rue Lhomond 75005 Paris, France
  • [3] Université Paris Diderot Sorbonne Paris Cité Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR 7586, CNRS Sorbonne Universités UPMC Université Paris 06 F-75013, Paris, France

Journées Équations aux dérivées partielles (2014)

  • page 1-22
  • ISSN: 0752-0360

Abstract

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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 [1].

How to cite

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Arsénio, Diogo, Dormy, Emmanuel, and Lacave, Christophe. "The vortex method for 2D ideal flows in the exterior of a disk." Journées Équations aux dérivées partielles (2014): 1-22. <http://eudml.org/doc/275587>.

@article{Arsénio2014,
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 [1].},
affiliation = {CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586 Université Paris Diderot Sorbonne Paris Cité Sorbonne Universités UPMC Université Paris 06 F-75013 Paris, France; CNRS & MAG (ENS/IPGP) Département de Physique 24 rue Lhomond 75005 Paris, France; Université Paris Diderot Sorbonne Paris Cité Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR 7586, CNRS Sorbonne Universités UPMC Université Paris 06 F-75013, Paris, France},
author = {Arsénio, Diogo, Dormy, Emmanuel, Lacave, Christophe},
journal = {Journées Équations aux dérivées partielles},
keywords = {Euler equations; elliptic problem in exterior domains; Hilbert transform and discrete Hilbert transform},
language = {eng},
pages = {1-22},
publisher = {Groupement de recherche 2434 du CNRS},
title = {The vortex method for 2D ideal flows in the exterior of a disk},
url = {http://eudml.org/doc/275587},
year = {2014},
}

TY - JOUR
AU - Arsénio, Diogo
AU - Dormy, Emmanuel
AU - Lacave, Christophe
TI - The vortex method for 2D ideal flows in the exterior of a disk
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 22
AB - 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 [1].
LA - eng
KW - Euler equations; elliptic problem in exterior domains; Hilbert transform and discrete Hilbert transform
UR - http://eudml.org/doc/275587
ER -

References

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  12. D. Iftimie, M. C. Lopes Filho, H. J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Comm. Partial Differential Equations 28 (2003), 349-379 Zbl1094.76007MR1974460
  13. Andrew J. Majda, Andrea L. Bertozzi, Vorticity and incompressible flow, 27 (2002), Cambridge University Press, Cambridge Zbl0983.76001MR1867882
  14. Carlo Marchioro, Mario Pulvirenti, On the vortex-wave system, Mechanics, analysis and geometry: 200 years after Lagrange (1991), 79-95, North-Holland, Amsterdam Zbl0733.76015MR1098512
  15. N. I. Muskhelishvili, Singular integral equations, (1972), Wolters-Noordhoff Publishing, Groningen Zbl0174.16201MR355494
  16. Steven Schochet, The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math. 49 (1996), 911-965 Zbl0862.35092MR1399201
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