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/periodic 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-00752999v1⟩