Van Hove singularities in Probability Density Functions of scalars

Patrice Meunier (1) , Emmanuel Villermaux (1)

1 IRPHE - Institut de Recherche sur les Phénomènes Hors Equilibre

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.

Mots clés: singularities; passive scalar; mixing

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]

Patrice Meunier, Emmanuel Villermaux. Van Hove singularities in Probability Density Functions of scalars. Comptes Rendus Mécanique, 2007, 335 (3), pp.162. ⟨10.1016/j.crme.2007.02.001⟩. ⟨hal-00141770⟩