Viscous potential free-surface flows in a fluid layer of finite depth

Denys Dutykh; Frédéric Dias

doi:10.1016/j.crma.2007.06.007

Mathematical Problems in Mechanics

Viscous potential free-surface flows in a fluid layer of finite depth

Presented by: Gérard Iooss

Denys Dutykh ¹; Frédéric Dias ¹

¹ CMLA, ENS Cachan, CNRS, PRES UniverSud, 61, avenue du Président Wilson, 94230 Cachan cedex, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 113-118.

Abstract
Résumé

It is shown how to model weakly dissipative free-surface flows using the classical potential flow approach. The Helmholtz–Leray decomposition is applied to the linearized 3D Navier–Stokes equations. The governing equations are treated using Fourier–Laplace transforms. We show how to express the vortical component of the velocity only in terms of the potential and free-surface elevation. A new predominant nonlocal viscous term is derived in the bottom kinematic boundary condition. The resulting formulation is simple and does not involve any correction procedure as in previous viscous potential flow theories (Joseph and Wang, 2004). Corresponding long wave model equations are derived.

Nous montrons comment modéliser les écoulements à surface libre faiblement dissipatifs en utilisant une approche potentielle. La décomposition de Helmholtz–Leray est appliquée aux équations de Navier–Stokes linéarisées. Le problème est étudié au moyen de la transformée de Fourier–Laplace. Nous montrons comment exprimer la partie rotationnelle de la vitesse en fonction du potentiel des vitesses et de l'élévation de la surface libre. Un nouveau terme nonlocal prépondérant apparaît dans la condition cinématique au fond. La formulation finale est simple et ne requiert pas de corrections ultérieures comme dans Joseph et Wang, 2004. Un modèle d'ondes longues est obtenu à partir de ces nouvelles équations.

Received: 2007-05-09
Accepted: 2007-06-05
Published online: 2007-07-13

DOI: 10.1016/j.crma.2007.06.007

Author's affiliations:

Denys Dutykh ¹; Frédéric Dias ¹

¹ CMLA, ENS Cachan, CNRS, PRES UniverSud, 61, avenue du Président Wilson, 94230 Cachan cedex, France

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     author = {Denys Dutykh and Fr\'ed\'eric Dias},
     title = {Viscous potential free-surface flows in a fluid layer of finite depth},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {113--118},
     publisher = {Elsevier},
     volume = {345},
     number = {2},
     year = {2007},
     doi = {10.1016/j.crma.2007.06.007},
     language = {en},
}

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Denys Dutykh; Frédéric Dias. Viscous potential free-surface flows in a fluid layer of finite depth. Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 113-118. doi : 10.1016/j.crma.2007.06.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.007/

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