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Comptes Rendus Physique
Volume 13, n° 9-10
pages 878-888 (novembre 2012)
Doi : 10.1016/j.crhy.2012.09.003
Geometrical statistics of fluid deformation: Restricted Euler approximation and the effects of pressure
Statistiques de la géométrie de la déformation dʼun fluide : Lʼapproximation dʼEuler restreint et les effets de pression

Yi Li
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK 


The geometrical statistics of fluid deformation are analyzed theoretically within the framework of the restricted Euler approximation, and numerically using direct numerical simulations. The restricted Euler analysis predicts that asymptotically a material line element becomes an eigenvector of the velocity gradient regardless its initial orientation. The asymptotic stretching rate equals the intermediate eigenvalue of the strain rate tensor. Analyses of numerical data show that the pressure Hessian is the leading cause to destroy the alignment between the longest axis of the material element and the strongest stretching eigendirection of the strain rate. It also facilitates the alignment between the longest axis of the element and the intermediate eigendirection of the strain rate during initial evolution, but tends to oppose the alignment later.

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Les statistiques de la géométrie de la déformation sont analysées dans le cadre théorique de lʼapproximation dʼEuler restreint, et numériquement en utilisant une simulation numérique directe des équations de Navier–Stokes. Sous lʼapproximation dʼEuler restreint, il est prédit quʼune ligne matérielle devienne asymptotiquement un vecteur propre du tenseur des gradients, quʼelle que soit son orientation initiale. De plus, le taux dʼétirement devient égal à la valeur propre intermédiaire du tenseur de déformation. Lʼanalyse des simulations numériques montre que le hessien de pression est la cause principale de la détérioration de lʼalignement du plus grand axe de lʼélément matériel avec la direction principale de plus grande déformation. Le hessien favorise aussi lʼalignement de ce plus grand axe avec la direction principale intermédiaire de la déformation lors de lʼévolution aux temps courts, mais a tendance à sʼy opposer par la suite.

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Keywords : Turbulence, Geometrical statistics, Restricted Euler approximation, Modeling

Mots-clés : Turbulence, Statistiques de la géométrie, Approximation dʼEuler restreint, Modélisation

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