Comptes Rendus
Van Hove singularities in Probability Density Functions of scalars
Comptes Rendus. Mécanique, Volume 335 (2007) no. 3, pp. 162-167.

A general theory for the Probability Density Function (PDF) of a scalar stirred in an axisymmetric time-dependent flow is derived. This theory reveals singularities, discontinuities and cusps occurring as soon as the spatial gradient of the scalar concentration vanishes somewhere in the field. These singularities are similar to the Van Hove singularities obtained in the density of vibration modes of a crystal. This feature, ubiquitous in convection–diffusion problems, is documented experimentally for the mixing of a dye in a Lamb–Oseen vortex.

Nous dérivons une théorie générale pour la Densité de Probabilité (PDF) d'un scalaire étiré dans un champ de vitesse axisymmétrique et dépendant du temps. Cette théorie révèle des singularités (discontinuités et cusps), qui apparaissent dès lors que le gradient spatial du scalaire s'annule en un endroit du champ. Ces singularités sont similaires aux singularités de Van Hove obtenue pour la densité des modes de vibration d'un cristal. Ce phénomène, omniprésent dans les problèmes de diffusion–convection, est documenté expérimentalement pour le mélange d'un colorant dans un vortex de Lamb–Oseen.

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DOI: 10.1016/j.crme.2007.02.001
Keywords: Computational fluid mechanics, Mixing, Singularities, PDF
Mot clés : Mécanique des fluides numérique, Mélange, Singularités, PDF

Patrice Meunier 1; Emmanuel Villermaux 1, 2

1 Université de Provence and CNRS, institut de recherche sur les phénomènes hors équilibre, 49, rue Frédéric-Joliot-Curie, 13384 Marseille cedex 13, France
2 Institut Universitaire de France
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Patrice Meunier; Emmanuel Villermaux. Van Hove singularities in Probability Density Functions of scalars. Comptes Rendus. Mécanique, Volume 335 (2007) no. 3, pp. 162-167. doi : 10.1016/j.crme.2007.02.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.02.001/

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