Camassa-Holm equations and vortexons for axisymmetric pipe flows

In this paper, we study the nonlinear dynamics of an axisymmetric disturbance to the laminar state in non-rotating Poiseuille pipe flows. In particular, we show that the associated Navier-Stokes equations can be reduced to a set of coupled Camassa-Holm type equations. These support inviscid and smooth localized travelling waves, which are numerically computed using the Petviashvili method. In physical space they correspond to localized toroidal vortices that concentrate near the pipe boundaries (wall vortexons) or wrap around the pipe axis (centre vortexons) in agreement with the analytical soliton solutions derived by Fedele (2012) for small and long-wave disturbances. Inviscid singular vortexons with discontinuous radial velocities are also numerically discovered as associated to special traveling waves with a wedge-type singularity, viz. peakons. Their existence is confirmed by an analytical solution of exponentially-shaped peakons that is obtained for the particular case of the uncoupled Camassa-Holm equations. The evolution of a perturbation is also investigated using an accurate Fourier-type spectral scheme. We observe that an initial vortical patch splits into a centre vortexon radiating vorticity in the form of wall vortexons. These can under go further splitting before viscosity dissipate them, leading to a slug of centre vortexons. The splitting process originates from a radial flux of azimuthal vorticity from the wall to the pipe axis in agreement with Eyink (2008). The inviscid and smooth vortexon is similar to the nonlinear neutral structures derived 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.

💡 Research Summary

The paper investigates the nonlinear evolution of axisymmetric disturbances superimposed on the laminar Poiseuille flow in a straight, non‑rotating pipe. By expanding the Navier‑Stokes equations in a basis of radial eigenfunctions and performing a Galerkin projection, the authors reduce the governing dynamics to a set of coupled Camassa‑Holm (CH) type equations. These equations retain the essential nonlinear advection and dispersion terms while incorporating viscous diffusion in a dimensionless form.

The authors first demonstrate analytically that, in the long‑wave, small‑amplitude limit, the CH system admits smooth solitary‑wave solutions that correspond to localized toroidal vortices. Two families of such solutions are identified: (i) “wall vortexons,” which are concentrated near the pipe wall, and (ii) “centre vortexons,” which wrap around the pipe axis. Both families are neutral, inviscid structures that travel downstream without changing shape.

To compute fully nonlinear travelling waves, the Petviashvili iteration scheme is employed. This method, originally devised for finding solitary‑wave solutions of nonlinear wave equations, converges rapidly for the CH system. Numerical results confirm the existence of the wall and centre vortexons and provide detailed profiles of the radial and azimuthal velocity components. In addition, the authors discover a second class of solutions—peakons—characterised by a wedge‑type singularity in the radial velocity. An explicit exponential peakon solution is derived for the uncoupled CH equation, establishing that such singular vortexons are admissible within the same mathematical framework.

Time‑dependent dynamics are explored using a high‑order Fourier‑type spectral code. Starting from a localized vortical patch, the simulation shows a cascade of events: the patch first collapses into a centre vortexon, which then radiates azimuthal vorticity outward. This outward flux generates a series of wall vortexons that can themselves split further, creating a “slug” of multiple centre vortexons separated by wall‑bound structures. The observed splitting process is interpreted as a radial flux of azimuthal vorticity from the wall toward the axis, in agreement with Eyink’s (2008) theoretical description of vortex‑flux dynamics in pipe flow.

The paper situates its findings within the broader literature. The smooth vortexons resemble the nonlinear neutral structures reported by Walton (2011), while the analytical soliton solutions echo Fedele’s (2012) small‑amplitude, long‑wave analysis. The peakon vortexons extend the catalogue of admissible solutions, highlighting the richness of the CH system beyond smooth solitons. Importantly, the authors argue that the axisymmetric vortexons are likely unstable to non‑axisymmetric perturbations, suggesting that they may act as precursors to the experimentally observed puffs and slugs that mark the transition to turbulence.

In conclusion, the study provides a rigorous derivation of a reduced CH model for axisymmetric pipe flow, validates the existence of both smooth and singular travelling‑wave vortexons through analytical and numerical means, and elucidates a plausible dynamical pathway from an initial disturbance to a complex array of vortex structures. These results deepen our theoretical understanding of the early stages of pipe‑flow transition and open avenues for future work on three‑dimensional stability, experimental verification, and potential flow‑control strategies.