Journal of Fluid Mechanics


Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow


a1 Departament de Física Aplicada, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain

a2 Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany

a3 Laboratory for Aero and Hydrodynamics, Delft University of Technology, 2628 CA Delft, The Netherlands


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 twofold 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 (κ). 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. We provide evidence that these modulated waves undergo a fold-of-cycles and compute some solutions on the unstable branch. These waves are shown to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, in accordance with the bifurcation scenario proposed. The travelling-wave upper-branch solutions are stable within the subspace of twofold periodic flows, and their subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.

(Received February 01 2010)

(Revised September 29 2010)

(Accepted October 01 2010)

(Online publication January 25 2011)

Key words:

  • bifurcation;
  • transition to turbulence


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