Camassa-Holm type equations for axisymmetric Poiseuille pipe flows

TL;DR

This paper derives Camassa-Holm type equations to model nonlinear disturbances in axisymmetric pipe flows, revealing singular wave solutions called peakons that relate to vortex structures and transition phenomena.

Contribution

It introduces a novel reduction of Navier-Stokes equations to Camassa-Holm type equations for axisymmetric flows, highlighting the existence of peakon solutions and their physical interpretation.

Findings

Discovery of singular inviscid traveling wave solutions called peakons.

Peakons correspond to localized vortex structures in pipe flows.

Potential link between vortexons and flow transition to turbulence.

Abstract

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.