Le but de cette Note est de présenter une approche unifiée des approximations de type couche limite pour l'opérateur de Laplace dans un domaine à bord rugueux périodique. On montre un résultat négatif pour une loi de paroi moyennée du second ordre. Pour contourner la difficulté, on propose de nouvelles lois de parois multi-échelles incluant les oscillations microscopiques sur la frontière fictive. Dans un premier temps, elles sont explicites et s'expriment comme des conditions de Dirichlet non-homogènes, ensuite on dérive une loi multi-échelle implicite de type Saffman–Joseph mais à coefficient variable. On établit des ordres de convergence et on montre leur validité numérique. On montre également sur un contre-exemple l'impossibilité de construire une loi d'ordre 2 effectif et qui soit moyennée dans les variables rapides.
The purpose of this Note is to present a unifying approach of boundary layer approximations for the Laplace operator in domains with periodic rugous boundaries. We show a negative result for an averaged second-order like wall-law. To circumvent this difficulty, we propose new multi-scale wall-laws that include microscopic oscillations on the fictitious boundary. In a first step they are explicit non-homogeneous Dirichlet conditions, afterwards an implicit multi-scale Saffman–Joseph-like wall-law is derived. We establish theoretical orders of convergence and provide their numerical assessment, as well as a counter-example that demonstrates the impossibility of a real averaged second order wall-law.
Accepté le :
Publié le :
Didier Bresch 1 ; Vuk Milisic 2
@article{CRMATH_2008__346_15-16_833_0, author = {Didier Bresch and Vuk Milisic}, title = {Vers des lois de parois multi-\'echelle implicites}, journal = {Comptes Rendus. Math\'ematique}, pages = {833--838}, publisher = {Elsevier}, volume = {346}, number = {15-16}, year = {2008}, doi = {10.1016/j.crma.2008.06.003}, language = {fr}, }
Didier Bresch; Vuk Milisic. Vers des lois de parois multi-échelle implicites. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 833-838. doi : 10.1016/j.crma.2008.06.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.06.003/
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AS. Mathematical Models Methods in Applied Sciences, Volume 25 (2015) no. 7, pp. 1257-1297 | DOI:10.1142/s0218202515500323 | Zbl:1321.74059 - High order multi-scale wall-laws. I: The periodic case, Quarterly of Applied Mathematics, Volume 68 (2010) no. 2, pp. 229-253 | DOI:10.1090/s0033-569x-10-01135-0 | Zbl:1426.76226
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