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

  • F. Mellibovsky (a1) and B. Eckhardt (a2) (a3)

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.

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Corresponding author

Email address for correspondence: fmellibovsky@fa.upc.edu

References

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1. Anischenko, V. S., Safonova, M. A. & Chua, L. O. 1993 Confirmation of the Afraimovich–Shilnikov torus-breakdown theorem via a torus circuit. IEEE Trans. Cir. Sys. 40 (11), 792800.
2. Avila, K., Moxey, D., de Lozar, A., Avila, M. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192196.
3. Avila, M., Willis, A. P. & Hof, B. 2010 On the transient nature of localized pipe flow turbulence. J. Fluid Mech. 646, 127136.
4. Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. A 43, 697726.
5. Brosa, U. & Grossmann, S. 1999 Minimum description of the onset of pipe turbulence. Eur. Phys. J. B 9 (2), 343354.
6. Chenciner, A. & Iooss, G. 1979 Bifurcation of invariant torus. Arch. Rat. Mech. Anal. 69 (2), 109198.
7. Chossat, P. & Lauterbach, R. 2000 Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific.
8. Crawford, J. D. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu. Rev. Fluid Mech. 23, 341387.
9. Darbyshire, A. G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech. 289, 83114.
10. Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20 (11), 114102.
11. Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255274.
12. Eckhardt, B. 2009 Introduction. Turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds’ paper. Phil. Trans. R. Soc. Lond. A 367 (1888), 449455.
13. Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T. M. 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. Lond. A 366, 12971315.
14. Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.
15. Ehrenstein, U. & Koch, W. 1991 Three-dimensional wave-like equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111148.
16. Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.
17. Gaspard, P. 1990 Measurement of the instability rate of a far-from-equilibrium steady state at an infinite period bifurcation. J. Phys. Chem. 94 (1), 13.
18. Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.
19. Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243266.
20. Golubitsky, M., LeBlanc, V. G. & Melbourne, I. 2000 Hopf bifurcation from rotating waves and patterns in physical space. J. Nonlinear Sci. 10 (1), 69101.
21. Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and groups in bifurcation theory, vol. 2. In Applied Mathematical Sciences, vol. 69. Springer.
22. Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72 (2), 603618.
23. Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear travelling waves in turbulent pipe flow. Science 305 (5690), 15941598.
24. Hof, B., Schneider, T. M., Westerweel, J. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 5962.
25. Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69102.
26. Krupa, M. 1990 Bifurcations of relative equilibria. SIAM J. Math. Anal. 21 (6), 14531486.
27. Kuznetsov, Y. A. 1995 Elements of Applied Bifurcation Theory, third edition. Springer.
28. Kuznetsov, Y. A., Meijer, H. G. E. & Van Veen, L. 2004 The fold-flip bifurcation. Intl J. Bifurcation Chaos 14 (7), 22532282.
29. Landau, L. 1944 On the problem of turbulence. Dokl. Akad. Nauk SSSR 44, 339342.
30. Marques, F., Lopez, J. M. & Shen, J. 2001 A periodically forced flow displaying symmetry breaking via a three-tori gluing bifurcation and two-tori resonances. Physica D 156, 8197.
31. Meca, E., Mercader, I., Batiste, O. & Ramírez-Piscina, L. 2004 Blue sky catastrophe in double-diffusive convection. Phys. Rev. Lett. 92 (23), 234501.
32. Mellibovsky, F. & Eckhardt, B. 2011 Takens–Bogdanov bifurcation of travelling wave solutions in pipe flow. J. Fluid Mech. 670, 96129.
33. Mellibovsky, F. & Meseguer, A. 2009 Critical threshold in pipe flow transition. Phil. Trans. R. Soc. Lond. A 367 (1888), 545560.
34. Meseguer, A. & Mellibovsky, F. 2007 On a solenoidal Fourier–Chebyshev spectral method for stability analysis of the Hagen–Poiseuille flow. Appl. Numer. Math. 57, 920938.
35. Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number . J. Comput. Phys. 186, 178197.
36. Nagata, M. 1997 Three-dimensional travelling-wave solutions in plane Couette flow. Phys. Rev. E 55 (2), 20232025.
37. Newhouse, S., Ruelle, D. & Takens, F. 1978 Occurrence of strange axiom attractors near quasi-periodic flows on . Commun. Math. Phys. 64, 3540.
38. Pfenniger, W. 1961 Boundary Layer and Flow Control. (ed. Lachman, G. V. ). pp. 970980. Pergamon.
39. Pringle, C. C. T., Duguet, Y. & Kerswell, R. R. 2009 Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. Lond. A 367 (1888), 457472.
40. Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99 (7), 074502.
41. Pugh, J. D. & Saffman, P. G. 1988 Two-dimensional superharmonic stability of finite-amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295307.
42. Rand, D. 1982 Dynamics and symmetry: predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79 (1), 137.
43. Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935982.
44. Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167192.
45. Sanchez, J., Net, M. & Simo, C. 2010 Computation of invariant tori by Newton–Krylov methods in large-scale dissipative systems. Physica D 239, 123133.
46. Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory, eighth edition. Springer.
47. Schmid, P. J. & Henningson, D. S. 1994 Optimal energy growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197225.
48. Schneider, T. M. & Eckhardt, B. 2009 Edge states intermediate between laminar and turbulent dynamics in pipe flow. Phil. Trans. R. Soc. Lond. A 367 (1888), 577587.
49. Schneider, T. M., Eckhardt, B. & Vollmer, J. 2007 Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75 (6), 066313.
50. Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.
51. Shan, H., Ma, B., Zhang, Z. & Nieuwstadt, F. T. M. 1999 On the spatial evolution of a wall-imposed periodic disturbance in pipe Poiseuille flow at . Part 1. Subcritical disturbance. J. Fluid Mech. 398, 181224.
52. Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.
53. Soibelman, I. & Meiron, D. I. 1991 Finite-amplitude bifurcations in plane Poiseuille flow: two-dimensional Hopf bifurcation. J. Fluid Mech. 229, 389416.
54. Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Westview.
55. van Veen, L. & Kawahara, G. 2011 Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107 (11), 114501.
56. Viswanath, D. 2006 Recurrent motions within plane-Couette turbulence. J. Fluid Mech. 580, 339358.
57. Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. Lond. A 367 (1888), 561576.
58. Vollmer, J., Schneider, T. M. & Eckhardt, B. 2009 Basin boundary, edge of chaos and edge state in a two-dimensional model. New J. Phys. 11, 013040.
59. Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.
60. Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.
61. Willis, A. P. & Kerswell, R. R. 2008 Coherent structures in localized and global pipe turbulence. Phys. Rev. Lett. 100 (12).
62. Wygnanski, I. J. & Champagne, F. H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59, 281335.
63. Wygnanski, I. J., Sokolov, M. & Friedman, D. 1975 On transition in a pipe. Part 2. The equilibrium puff. J. Fluid Mech. 69, 283304.
64. Zikanov, O. Y. 1996 On the instability of pipe Poiseuille flow. Phys. Fluids 8 (11), 29232932.
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JFM classification

Type Description Title
VIDEO
Movies

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 〈uzz=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.

 Video (3.2 MB)
3.2 MB
VIDEO
Movies

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 〈uzz=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.

 Video (1.7 MB)
1.7 MB
VIDEO
Movies

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〈uzz=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.

 Video (3.4 MB)
3.4 MB
VIDEO
Movies

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〈uzz=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.

 Video (1.6 MB)
1.6 MB
VIDEO
Movies

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〈uzz=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.

 Video (7.9 MB)
7.9 MB
VIDEO
Movies

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〈uzz=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.

 Video (3.0 MB)
3.0 MB
VIDEO
Movies

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〈uzz=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.

 Video (19.3 MB)
19.3 MB
VIDEO
Movies

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〈uzz=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.

 Video (7.3 MB)
7.3 MB
VIDEO
Movies

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〈uzz=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.

 Video (3.3 MB)
3.3 MB
VIDEO
Movies

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〈uzz=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.

 Video (1.5 MB)
1.5 MB

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

  • F. Mellibovsky (a1) and B. Eckhardt (a2) (a3)

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