[Une méthode d'éléments finis adaptative pour la simulation de la sédimentation d'un corps solide dans un fluide Newtonien]
La simulation numérique de la sédimentation d'un corps solide dans un fluide visqueux est un problème difficile car il exige, entre autres, l'emploi de domaines de calcul de plusieurs ordres de grandeur plus grands que le corps solide, ceci afin d'éviter l'influence des frontières artificielles. Dans le but de construire un maillage de calcul optimal, dans ce contexte, nous proposons un méthode basée sur des estimations d'erreur a posteriori avec poids pour l'approximation par éléments finis utilisée pour simuler le couplage fluide/solide. Un élément clé de l'approche proposée dans cet article est la reformulation des équations de l'écoulement, et du mouvement du corps solide, dans un repère mobile, rigidement attaché au solide ; par ailleurs, via une formulation variationelle bien choisie, nous évitons d'avoir à calculer, explicitement, la résultante et le moment des forces hydrodynamiques que le fluide exerce sur le solide. Les informations fournies par la solution d'un problème dual bien choisi, permettent de contrôler l'erreur de discrétisation pour des fonctionnelles données de la solution (traînée, par exemple). Notre analyse couvre le calcul de la vitesse de sédimentation, l'orientation du corps solide, la résultante et le moment des forces que le fluide exerce sur le solide. Des essais numériques, concernant la résolution d'un problème de sédimentation bi-dimensionnel, valident la méthode proposée.
The numerical simulation of the free fall of a solid body in a viscous fluid is a challenging task since it requires computational domains which usually need to be several order of magnitude larger than the solid body in order to avoid the influence of artificial boundaries. Toward an optimal mesh design in that context, we propose a method based on the weighted a posteriori error estimation of the finite element approximation of the fluid/body motion. A key ingredient for the proposed approach is the reformulation of the conservation and kinetic equations in the solid frame as well as the implicit treatment of the hydrodynamic forces and torque acting on the solid body in the weak formulation. Information given by the solution of an adequate dual problem allows one to control the discretization error of given functionals. The analysis encompasses the control of the free fall velocity, the orientation of the body, the hydrodynamic force and torque on the body. Numerical experiments for the two dimensional sedimentation problem validate the method.
Mot clés : Mécanaique des fluides numérique, Méthode des élements finis, Estimation d'erreur a posteriori, Écoulements particulaires, Couplage fluide–structure
@article{CRMECA_2005__333_12_896_0,
author = {Vincent Heuveline},
title = {Adaptive finite elements for the steady free fall of a body in a {Newtonian} fluid},
journal = {Comptes Rendus. M\'ecanique},
pages = {896--909},
publisher = {Elsevier},
volume = {333},
number = {12},
year = {2005},
doi = {10.1016/j.crme.2005.10.004},
language = {en},
}
Vincent Heuveline. Adaptive finite elements for the steady free fall of a body in a Newtonian fluid. Comptes Rendus. Mécanique, Volume 333 (2005) no. 12, pp. 896-909. doi : 10.1016/j.crme.2005.10.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2005.10.004/
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