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

  • F. MELLIBOVSKY (a1) and B. ECKHARDT (a2) (a3)
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 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.

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Corresponding author
Email address for correspondence: fmellibovsky@fa.upc.edu
References
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Type Description Title
VIDEO
Movies

Mellibovsky et al. supplementary material
Modulated travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: axial phase-speed (cz) time-series and three-dimensional energy (ε3D) vs mean axial pressure gradient ((∇p)z) phase map. Middle right: Axial velocity contours relative to the parabolic profile at the z=0 and z=Λ/4 cross-sections (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Green/Blue dashed line and square refer to the upper/lower branch travelling wave (twub/twlb). The red dot following the solid line and loop represents the modulated wave (mtw). The phase map dashed loop is an unstable modulated wave at the same parameter values. Axial cross-sections shown are indicated with black lines/rings.

 Video (3.8 MB)
3.8 MB
VIDEO
Movies

Mellibovsky et al. supplementary movie
Upper-branch travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: Axial velocity contours relative to the parabolic profile at the marked z-cross-section (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Current axial position indicated with a black line/ring.

 Video (1.5 MB)
1.5 MB
VIDEO
Movies

Mellibovsky et al. supplementary movie
Lower-branch travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: Axial velocity contours relative to the parabolic profile at the marked z-cross-section (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Current axial position indicated with a black line/ring.

 Video (3.7 MB)
3.7 MB
VIDEO
Movies

Mellibovsky et al. supplementary movie
Upper-branch travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: Axial velocity contours relative to the parabolic profile at the marked z-cross-section (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Current axial position indicated with a black line/ring.

 Video (3.8 MB)
3.8 MB
VIDEO
Movies

Mellibovsky et al. supplementary movie
Unstable modulated travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: axial phase-speed (cz) time-series and three-dimensional energy (ε3D) vs mean axial pressure gradient ((∇p)z) phase map. Middle right: Axial velocity contours relative to the parabolic profile at the z=0 and z=Λ/4 cross-sections (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Green/Blue dashed line and square refer to the upper/lower branch travelling wave (twub/twlb). The red dot following the solid line and loop represents the modulated wave (mtw). The phase map dashed loop is the stable modulated wave coexisting at the same parameter values. Axial cross-sections shown are indicated with black lines/rings.

 Video (6.8 MB)
6.8 MB
VIDEO
Movies

Mellibovsky et al. supplementary material
Modulated travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: axial phase-speed (cz) time-series and three-dimensional energy (ε3D) vs mean axial pressure gradient ((∇p)z) phase map. Middle right: Axial velocity contours relative to the parabolic profile at the z=0 and z=Λ/4 cross-sections (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Green/Blue dashed line and square refer to the upper/lower branch travelling wave (twub/twlb). The red dot following the solid line and loop represents the modulated wave (mtw). The phase map dashed loop is an unstable modulated wave at the same parameter values. Axial cross-sections shown are indicated with black lines/rings.

 Video (7.7 MB)
7.7 MB
VIDEO
Movies

Mellibovsky et al. supplementary movie
Unstable modulated travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: axial phase-speed (cz) time-series and three-dimensional energy (ε3D) vs mean axial pressure gradient ((∇p)z) phase map. Middle right: Axial velocity contours relative to the parabolic profile at the z=0 and z=Λ/4 cross-sections (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Green/Blue dashed line and square refer to the upper/lower branch travelling wave (twub/twlb). The red dot following the solid line and loop represents the modulated wave (mtw). The phase map dashed loop is the stable modulated wave coexisting at the same parameter values. Axial cross-sections shown are indicated with black lines/rings.

 Video (12.1 MB)
12.1 MB
VIDEO
Movies

Mellibovsky et al. supplementary movie
Lower-branch travelling wave at Re=1600, κ=1.52. Left: Radial velocity contours at r=0.65 (contour spacing: Δur=0.008 U). Top right: Axial velocity contours relative to the parabolic profile at the marked z-cross-section (contour spacing: Δuz=0.12 U) with in-plane velocity vectors. Bottom right: Axial vorticity iso-surfaces at ωz= ±1.0 U/D. Blue for negative, red for positive. Current axial position indicated with a black line/ring.

 Video (1.4 MB)
1.4 MB

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