Camassa-Holm type equations for axisymmetric Poiseuille pipe flows

Francesco Fedele (1) , Denys Dutykh (2, 3)

1 ECE GeorgiaTech - School of Electrical and Computer Engineering - Georgia Insitute of Technology
2 LAMA - Laboratoire de Mathématiques
3 School of Mathematical Sciences [Dublin]

We present a study on the nonlinear dynamics of a disturbance to the laminar state in non-rotating axisymmetric Poiseuille pipe flows. The associated Navier-Stokes equations are reduced to a set of coupled generalized Camassa-Holm type equations. These support singular inviscid travelling waves with wedge-type singularities, the so called peakons, which bifurcate from smooth solitary waves as their celerity increase. In physical space they correspond to localized toroidal vortices or vortexons. The inviscid vortexon is similar to the nonlinear neutral structures found by Walton (2011) and it may be a precursor to puffs and slugs observed at transition, since most likely it is unstable to non-axisymmetric disturbances.

Mots clés: pipe flow; Poiseuille flow; Camassa-Holm equation; peakons

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Mathématiques générales [math.GM]; Modélisation et simulation; Dynamique des Fluides [physics.flu-dyn]; Physique Numérique [physics.comp-ph]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Systèmes Solubles et Intégrables [nlin.SI]; Formation de Structures et Solitons [nlin.PS]; Physique Classique [physics.class-ph]; Dynamique Chaotique [nlin.CD]

Francesco Fedele, Denys Dutykh. Camassa-Holm type equations for axisymmetric Poiseuille pipe flows. 2012. ⟨hal-00752999v2⟩