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

Journal of Fluid Mechanics

Volumn 670, Issue , 2011, Pages 96-129

  Mellibovsky, F ^a   Eckhardt, B ^b,c  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^b PHILIPPS UNIVERSITY MARBURG   (Germany)

^c DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

bifurcation; transition to turbulence

Indexed keywords

BIFURCATION (MATHEMATICS); NEWTON-RAPHSON METHOD; PHASE SPACE METHODS; REYNOLDS EQUATION; REYNOLDS NUMBER; STABILITY; WAVE TRANSMISSION;

BIFURCATION SCENARIO; PHASE SPACE STRUCTURES; RELATIVE EQUILIBRIA; SADDLE LOOP BIFURCATIONS; STABILITY PROPERTIES; TAKENS BOGDANOV BIFURCATION; TRANSITION TO TURBULENCE; TRAVELLING WAVE SOLUTION;

HOPF BIFURCATION;

BIFURCATION; FLOW STABILITY; FLUID DYNAMICS; PIPE FLOW; REYNOLDS NUMBER; TURBULENT FLOW;

EID: 79952743511     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S0022112010005239     Document Type: Article

References (64)

1. Barkley, D. 1990 Theory and predictions for finite-amplitude waves in two-dimensional plane Poiseuille flow. Phys. Fluids A 2 (6), 955-970.
2. Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. A: Phys. Sci. 43, 697-726.
3. Bogdanov, R. 1975 Versal deformations of a singular point on the plane in the case of zero eigenvalues. Funct. Anal. Appl. 9, 144-145.
4. Brosa, U. & Grossmann, S. 1999 Minimum description of the onset of pipe turbulence. Eur. Phys. J. B 9 (2), 343-354. (Pubitemid 129558560)
5. Chossat, P. & Lauterbach, R. 2000 Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific.
6. Darbyshire, A. G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech. 289, 83-114.
7. Dennis, J. E. & Schnabel, R. B. 1996 Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM.
8. Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008a Relative periodic orbits in transitional pipe flow. Phys. Fluids 20 (11), 114102.
9. Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008b Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255-274.
10. Dumortier, F., Roussarie, R., Sotomayor, J. & Zoladek, H. 1991 Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals. Springer.
11. Eckhardt, B. 2009 Introduction. Turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds' paper. Phil. Trans. R. Soc. Lond. A 367, 449-455.
12. 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, 1297-1315. (Pubitemid 351301968)
13. Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447-468. (Pubitemid 46210837)
14. Ehrenstein, U. & Koch, W. 1991 Three-dimensional wave-like equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111-148. (Pubitemid 21681626)
15. Ehrenstein, U. & Koch, W. 1995 Homoclinic bifurcation in Blasius boundary-layer flow. Phys. Fluids 7, 1282-1291. (Pubitemid 26440471)
16. Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.
17. Frayssé, V., Giraud, L., Gratton, S. & Langou, J. 2003 A set of GMRES routines for real and complex arithmetics on high performance computers. Tech. Rep. TR/PA/03/3. CERFACS. Available at: http://www.cerfacs/ algor/Softs.
18. Freitag, M. A. 2007 Inner-outer iterative methods for eigenvalue problems: convergence and preconditioning. PhD thesis, University of Bath, Bath, UK.
19. 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), 1-3.
20. Golubitsky, M., LeBlanc, V. G. & Melbourne, I. 2000 Hopf bifurcation from rotating waves and patterns in physical space. J. Nonlinear Sci. 10 (1), 69-101.
21. Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72 (2), 603-618.
22. Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
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), 1594-1598. (Pubitemid 39296349)
24. Hof, B., Schneider, T. M., Westerweel, J. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59-62. (Pubitemid 44344046)
25. Iooss, G. & Adelmeyer, M. 1998 Topics in Bifurcation Theory and Applications, 2nd edn. World Scientific.
26. Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69-102. (Pubitemid 47215387)
27. Knobloch, E. 1986 Normal forms for bifurcations at a double zero eigenvalue. Phys. Lett. A 115 (5), 199-201.
28. Krupa, M. 1990 Bifurcations of relative equilibria. SIAM J. Math. Anal. 21 (6), 1453-1486.
29. Kuznetsov, Y. A. 1995 Elements of Applied Bifurcation Theory, 3rd edn. Springer.
30. Lehoucq, R. & Scott, J. A. 1996 An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices. Tech. Rep. MCS-P547-1195.
31. Argonne National Laboratory. Available at: http://www.caam.rice.edu/ software/ARPACK.
32. Mamun, C. K. & Tuckerman, L. S. 1995 Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 7 (1), 80-91.
33. Meerbergen, K., Spence, A. & Roose, D. 1994 Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric. BIT 34 (3), 409-423.
34. Mellibovsky, F. & Meseguer, A. 2009 Cri tical threshold in pipe flow transition. Phil. Trans. R. Soc. Lond. A 367, 545-560.
35. Meseguer, A., Avila, M., Mellibovsky, F. & Marques, P. 2007 Solenoidal spectral formulations for the computation of secondary flows in cylindrical and annular geometries. Eur. Phys. J. Special Topics 146, 249-259.
36. Meseguer, A. & Mellibovsky, F. 2007 On a solenoidal Fourier-Chebyshev spectral method for stability analysis of the Hagen-Poiseuille flow. Appl. Numer. Maths 57, 920-938. (Pubitemid 46754512)
37. 7. J. Comput. Phys. 186, 178-197. (Pubitemid 36374157)
38. Nagata, M. 1997 Three-dimensional traveling-wave solutions in plane Couette flow. Phys. Rev. E 55 (2), 2023-2025. (Pubitemid 127607562)
39. Pfenniger, W. 1961 Transition in the inlet length of tubes at high Reynolds numbers. In Boundary Layer and Flow Control (ed. G. V. Lachman), pp. 970-980.
40. Pergamon Pringle, C. C. T., Duguet, Y. & Kerswell, R. R. 2009 Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. Lond. A 367, 457-472.
41. Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow. Phys. Rev. Lett. 99 (7), 074502.
42. Pugh, J. D. & Saffman, P. G. 1988 Two-dimensional superharmonic stability of finite-amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295-307. (Pubitemid 18664141)
43. Quarteroni, A., Sacco, R. & Saleri, F. 2007 Numerical Mathematics, 2nd edn. Springer.
44. Rand, D. 1982 Dynamics and symmetry: predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79 (1), 1-37.
45. 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, 935-982.
46. Sanchez, J., Marques, F. & Lopez, J. M. 2002 A continuation and bifurcation technique for Navier-Stokes flows. J. Comput. Phys. 180 (1), 78-98.
47. Schmid, P. J. & Henningson, D. S. 1994 Optimal energy growth in Hagen-Poiseuille flow. J. Fluid Mech. 277, 197-225. (Pubitemid 283959)
48. Schneider, T. M., Eckhardt, B. & Vollmer, J. 2007a Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75 (6), 066313.
49. Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007b Turbulence transition and edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.
50. 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 Re = 3000. Part 1. Subcritical disturbance. J. Fluid Mech. 398, 181-224. (Pubitemid 129657153)
51. Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.
52. Smale, S. 1967 Differentiable dynamical systems. I. Diffeomorphisms. Bull. Am. Math. Soc. 73 (6), 747-817.
53. Soibelman, I. & Meiron, D. I. 1991 Finite-amplitude bifurcations in plane Poiseuille flow: two-dimensional Hopf-bifurcation. J. Fluid Mech. 229, 389-416. (Pubitemid 21693995)
54. Takens, F. 1974 Singularities of vector fields. Publ. Math. IHES 43, 47-100.
55. Viswanath, D . 2007 Recurring motions within plane Couette turbulence. J. Fluid Mech. 580, 339-358.
56. Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 95 (3), 319-343.
57. Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883-900.
58. Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.
59. Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333-371.
60. Wiggins, S. 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer.
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, 281-335.
63. Wygnanski, I. J., Sokolov, M. & Frie dman, D. 1975 On transition in a pipe. Part 2. The equilibrium puff. J. Fluid Mech. 69, 283-304.
64. Zikanov, O. Y. 1996 On the instability of pipe Poiseuille flow. Phys. Fluids 8 (11), 2923-2932. (Pubitemid 126616793)