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Comptes Rendus Mathématique
Volume 346, n° 15-16
pages 833-838 (août 2008)
Doi : 10.1016/j.crma.2008.06.003
Received : 26 October 2006 ;  accepted : 3 June 2008
Vers des lois de parois multi-échelle implicites
Towards implicit multi-scale wall-laws
 

Didier Bresch a , Vuk Milisic b
a LAMA UMR5127, Université de Savoie, 73000 Le Bourget du lac cedex, France 
b LMC-IMAG UMR5223, 51, rue des Mathématiques, B.P. 53, 38041 Grenoble cedex 9, France 

Résumé

Le but de cette Note est de présenter une approche unifiée des approximations de type couche limite pour lʼopérateur 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 sʼexpriment 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 lʼimpossibilité de construire une loi dʼordre 2 effectif et qui soit moyennée dans les variables rapides. Pour citer cet article : D. Bresch, V. Milisic, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

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Abstract

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. To cite this article: D. Bresch, V. Milisic, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.


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