Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow

TL;DR

This paper investigates the Takens-Bogdanov bifurcation in pipe flow solutions, revealing how traveling waves emerge and evolve with Reynolds number, contributing to understanding turbulence onset.

Contribution

It provides a detailed numerical analysis of bifurcation scenarios in pipe flow, including the unfolding of Takens-Bogdanov bifurcations and the emergence of modulated waves.

Findings

Identification of Takens-Bogdanov bifurcation in pipe flow solutions

Extension of normal form to include higher order terms

Prediction of stable upper-branch solutions influencing turbulence

Abstract

The appearance of travelling-wave-type solutions in pipe Poiseuille flow that are disconnected from the basic parabolic profile is numerically studied in detail. We focus on solutions in the 2-fold azimuthally-periodic subspace because of their special stability properties, but relate our findings to other solutions as well. Using time-stepping, an adapted Krylov-Newton method and Arnoldi iteration for the computation and stability analysis of relative equilibria, and a robust pseudo-arclength continuation scheme we unfold a double-zero (Takens-Bogdanov) bifurcating scenario as a function of Reynolds number (Re) and wavenumber (k). This scenario is extended, by the inclusion of higher order terms in the normal form, to account for the appearance of supercritical modulated waves emanating from the upper branch of solutions at a degenerate Hopf bifurcation. These waves are expected to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, thereby leaving stable upper-branch solutions whose subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.