Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow

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.

26 pages, 15 figures (pdf and png). Submitted to J. Fluid Mech

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• Exact coherent states and connections to turbulent dynamics in minimal channel flow
• From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow
• Transition to turbulence in shear flows
• Small scale exact coherent structures at large Reynolds numbers in plane Couette flow
• Long wavelength instability of coherent structures in plane Couette flow
• Self-sustainment of coherent structures in counter-rotating Taylor-Couette flow
• Cartography of high-dimensional flows: A visual guide to sections and slices
• Spatial localisation beyond steady states in the neighbourhood of the Takens--Bogdanov bifurcation
• The role of periodic orbits and bubbles of chaos during the transition to turbulence