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From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow

Published online by Cambridge University Press:  29 August 2012

F. Mellibovsky  and
B. Eckhardt
F. Mellibovsky*
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, 08034, Barcelona, Spain
B. Eckhardt
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany J.M. Burgerscentrum, Delft University of Technology, 2638 CD Delft, The Netherlands
*
†Email address for correspondence: [email protected]
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Abstract

We study numerically a succession of transitions in pipe Poiseuille flow that lead from simple travelling waves to waves with chaotic time-dependence. The waves at the origin of the bifurcation cascade are twofold azimuthally periodic, shift–reflect symmetric, and have a non-dimensional axial wavelength of diameters. As the Reynolds number is increased, successive transitions result in a wide range of time-dependent solutions that include spiralling, modulated travelling, modulated spiralling, doubly modulated spiralling and mildly chaotic waves. Numerical evidence suggests that the latter spring from heteroclinic tangles of the stable and unstable invariant manifolds of two shift–reflect symmetric, modulated travelling waves. The chaotic set thus produced is confined to a limited range of Reynolds numbers, bounded by the occurrence of manifold tangencies. The subspace of discrete symmetry to which the states studied here belong makes many of the bifurcation and path-following investigations presented readily accessible. However, we expect that most of the phenomenology carries over to the full state space, thus suggesting a mechanism for the formation and break-up of invariant states that can give rise to chaotic dynamics.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Chaos Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 709 , 25 October 2012 , pp. 149 - 190
Copyright
Copyright © Cambridge University Press 2012

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F. Mellibovsky and B. Eckhardt supplementary movies

Modulated travelling wave (mtw) at Re=2335, κ=1.63. Top left: Phase map projection. Top right: axial phase speed (cz, red) and mean axial pressure gradient ((∇p)z, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ 〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle-(tw) and upper-middle-branch (tw2) travelling.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 3.2 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Modulated travelling wave (mtw) at Re=2335, κ=1.63. Top left: Phase map projection. Top right: axial phase speed (cz, red) and mean axial pressure gradient ((∇p)z, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ 〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle-(tw) and upper-middle-branch (tw2) travelling.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 1.7 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Modulated spiralling wave (msw) at Re=2185, κ=1.63. Top left: Phase map projection. The projection on the horizontal plane corresponds to a projection on the shift-reflect subspace. Top right: axial (cz, red) and azimuthal phase speed (c&theta, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated alongside lower-branch spiralling-wave (sw) values for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 3.4 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Modulated spiralling wave (msw) at Re=2185, κ=1.63. Top left: Phase map projection. The projection on the horizontal plane corresponds to a projection on the shift-reflect subspace. Top right: axial (cz, red) and azimuthal phase speed (c&theta, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated alongside lower-branch spiralling-wave (sw) values for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 1.6 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Doubly-modulated spiralling wave (m2sw) at Re=2205, κ=1.63. Top left: Phase map projection. The projection on the horizontal plane corresponds to a projection on the shift-reflect subspace. Top right: axial (cz, red) and azimuthal phase speed (c&theta, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated alongside lower-branch spiralling-wave (sw) values for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 7.9 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Doubly-modulated spiralling wave (m2sw) at Re=2205, κ=1.63. Top left: Phase map projection. The projection on the horizontal plane corresponds to a projection on the shift-reflect subspace. Top right: axial (cz, red) and azimuthal phase speed (c&theta, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated alongside lower-branch spiralling-wave (sw) values for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 3 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Mildly chaotic spiralling wave (cw) at Re=2205, κ=1.63. Top left: Phase map projection. The projection on the horizontal plane corresponds to a projection on the shift-reflect subspace. Top right: axial (cz, red) and azimuthal phase speed (c&theta, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated alongside lower-branch spiralling-wave (sw) values for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 19.3 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Mildly chaotic spiralling wave (cw) at Re=2205, κ=1.63. Top left: Phase map projection. The projection on the horizontal plane corresponds to a projection on the shift-reflect subspace. Top right: axial (cz, red) and azimuthal phase speed (c&theta, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated alongside lower-branch spiralling-wave (sw) values for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 7.3 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Unstable modulated travelling wave (umtw) at Re=2209.67, κ=1.63. Top left: Phase map projection. Top right: axial phase speed (cz, red) and mean axial pressure gradient ((∇p)z, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 3.3 MB

F. Mellibovsky and B. Eckhardt supplementary movies

Unstable modulated travelling wave (umtw) at Re=2209.67, κ=1.63. Top left: Phase map projection. Top right: axial phase speed (cz, red) and mean axial pressure gradient ((∇p)z, blue) along a full period. Bottom left: z-averaged axial velocity contours relative to the parabolic profile (contour spacing: Δ〈uz〉z=0.1 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Lower-middle- (tw) and upper-middle-branch (tw2) travelling-wave values are indicated for reference.

Download F. Mellibovsky and B. Eckhardt supplementary movies(Video)
Video 1.5 MB