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

Arsénio, Diogo; Dormy, Emmanuel; Lacave, Christophe

Journées équations aux dérivées partielles (2014), article no. 5, 22 p.

La méthode des vortex est une approche théorique et numérique couramment utilisée afin d’implémenter le mouvement d’un fluide parfait, dans laquelle le tourbillon est approché par une somme de points vortex, de sorte que les équations d’Euler se réécrivent comme un système d’équations différentielles ordinaires. Une telle méthode est rigoureusement justifiée dans le plan complet, grâce aux formules explicites de Biot et Savart. Dans un domaine extérieur, nous remplaçons également le bord imperméable par une collection de points vortex, générant une circulation autour de l’obstacle. La densité de ces points est choisie de sorte que le flot demeure tangent au bord sur certains points intermédiaires aux paires de tourbillons adjacents sur le bord. Dans ce travail, nous proposons une justification rigoureuse de cette méthode dans des domaines extérieurs. L’une des principales difficultés mathématiques étant que le noyau de Biot-Savart définit un opérateur intégral singulier lorsqu’il est restreint à une courbe. Par souci de simplicité et de clarté, nous traitons seulement le cas du disque unité dans le plan, approché par un maillage de points uniformément répartis. La version complète et générale de notre travail est disponible en [1].

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].

Classification : 35Q31, 35Q35, 76B47
Keywords: Euler equations, elliptic problem in exterior domains, Hilbert transform and discrete Hilbert transform
Mots clés : équations d’Euler, problème elliptique dans des domaines extérieurs, transformée de Hilbert et transformée de Hilbert discrète

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