Transition in pipe flow: The saddle structure on the boundary of turbulence

Journal of Fluid Mechanics

Volumn 613, Issue , 2008, Pages 255-274

  Duguet, Y ^a   Willis, A P ^b   Kerswell, R R ^c  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

^b ECOLE POLYTECHNIQUE   (France)

^c UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

PHASE SPACE METHODS; PIPE FLOW; SPACE FLIGHT;

COHERENT SOLUTIONS; COHERENT STRUCTURE; HETEROCLINIC CONNECTION; INITIAL CONDITIONS; NUMERICAL TECHNIQUES; PHASE SPACE TRAJECTORY; ROTATIONAL SYMMETRIES; TURBULENT BOUNDARY;

WAVE TRANSMISSION;

FLOW STRUCTURE; LAMINAR FLOW; NUMERICAL METHOD; PIPE FLOW; ROTATING FLOW; TURBULENT BOUNDARY LAYER; TURBULENT FLOW; WAVE;

EID: 54049091638     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S0022112008003248     Document Type: Article

References (46)

1. BROWN, P. N. & SAAD, Y. 1990 Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statist. Comput. 11, 450-481.
2. DENNIS, J. E., JR & SCHNABEL, R. E. 1995 Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM.
3. EISENSTAT, C. & WALKER, H. 1996 Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput. 17, 16-32.
4. FAISST, H. & ECKHARDT, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
5. FAISST, H. & ECKHARDT, B. 2004 Sensitive dependence on initial conditions in transition to turbulence. J. Fluid Mech. 504, 343-352.
6. GIBSON, J. F., HALCROW, J. & CVITANOVIĆ, P. 2008 The geometry of state-space in plane Couette flow. J. Fluid Mech. (submitted).
7. HAGEN, G. H. L. 1839 Über die Bewegung des Wassers in engen zylindrischen Rohren. Poggendorfs Annal. Physik Chemie 16, 423.
8. HAMILTON, J. M., KIM, J. & WALEFFE, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317-348.
9. 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 non-linear traveling waves in turbulent pipe flow. Science 305, 1594-1597.
10. HOF, B., VAN DOORNE, C. W. H., WESTERWEEL, J. & NIEUWSTADT, F. T. M. 2005. Transition in pipe flow: The saddle structure on the boundary of turbulence. Phys. Rev. Lett. 95, 214502.
11. ITANO, T. & TOH, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703-716.
12. JIMÉNEZ, J. & MOIN P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213-240.
13. JOSEPH, D. D. & CARMI, S. 1969 Stability of Poiseuille flow in pipes, annuli, and channels. Quart. Appl. Math. 26, 575-599.
14. KAWAHARA, G. 2005 Laminarization of minimal plane Couette flow: Going beyond the basin of attraction of turbulence. Phys. Fluids 17, 041702.
15. KERSWELL, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17-R44.
16. KERSWELL, R. R. & TUTTY, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69-102.
17. MARQUÉS, F. 1990 On boundary conditions for velocity potentials in confined flows: Application to Couette flow. Phys. Fluids A 2, 729-737.
18. MELLIBOVSKY, F. & MESEGUER, A. 2007 Pipe flow dynamics on the critical threshold. In Proc. 15th Intl Couette-Taylor Workshop (ed. I. Mutabazi), Le Havre, France.
19. NAGATA, M. 1990 Three dimensional finite amplitude solutions in plane Couette flow: Bifurcation from infinity. J. Fluid Mech. 217, 519-527.
20. NIKITIN N. 2007 Spatial periodicity of spatially evolving flows caused by inflow boundary condition. Phys. Fluids 19, 091703.
21. PEIXINHO, J. & MULLIN, T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96, 094501.
22. PEIXINHO, J. & MULLIN, T. 2007 Finite-amplitude thresholds for transition in pipe flow. J. Fluid Mech. 582, 169-178.
23. 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. Cambridge University Press.
24. POISEUILLE, J. L. M. 1840 Recherches experimentales sur le mouvement des liquides dans les tubes de trés petits diamètres. CR Acad. Sci. 11, 961.
25. PRINGLE, C. C. T. & KERSWELL, R. R. 2007 Asymmetric, helical and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.
26. PRINGLE, C. C. T., DUGUET, Y. & KERSWELL, R. R. 2008 Highly-symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. (submitted).
27. PRIYMAK, V. G. MIYAZAKI, T 2004 Direct numerical simulation of equilibrium spatiallly-localized structures in pipe flow. Phys. Fluids, 16, 4221-4234.
28. 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. 174, 935-982.
29. SAAD, Y. & SCHULTZ, M. H. 1986 GMRES: A generalized minimal residual method for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856-869.
30. SCHNEIDER, T. M., ECKHARDT, B. & VOLLMER, J. A. 2007a Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75, 066313.
31. SCHNEIDER, T. M., ECKHARDT, B. & YORKE, J. A. 2007b Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.
32. SCHNEIDER, T. M., GIBSON J. F., LAGHA, M., DE LILLO, F. & ECKHARDT, B. 2008 Laminar-turbulent boundary in plane Couette flow. Preprint (http://arxiv.org/abs/0805.1015).
33. SKUFCA, J. D., YORKE, J. A. & ECKHARDT, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.
34. TOH, S. & ITANO, T. 1999 Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence: Traveling-wave solution is a saddle point? In Proc. IUTAM Symp. on Geometry and Statistics of Turbulence (ed. T. Kambe), Kluwer.
35. VISWANATH, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339-358.
36. VISWANATH, D. 2008 The dynamics of transition to turbulence in plane Couette. In Mathematics and Computation: A Contemporary View Springer.
37. WALEFFE, F. 1997 On the self-sustaining process in shear flows. Phys. Fluids 9, 883-900.
38. WALEFFE, F. 1998 Three dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140-4143.
39. WALEFFE, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93-102.
40. WALEFFE, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 1517-1534.
41. WANG, J., GIBSON, J. & WALEFFE, F. 2007 Lower branch coherent states in shear flows: Transition and control. Phys. Rev. Lett. 98, 204501.
42. WEDIN, H. & KERSWELL, R. R. 2004 Exact coherent structures in pipe flow: Travelling wave solutions. J. Fluid Mech. 508, 333-371.
43. WILLIS, A. P. & KERSWELL, R. R. 2007 Critical behaviour in the relaminarisation of localised turbulence in pipe flow. Phys. Rev. Lett. 98, 14501.
44. WILLIS, A. P. & KERSWELL, R. R. 2008a Coherent structures in local and global pipe turbulence. Phys. Rev. Lett. 100, 124501.
45. WILLIS, A. P. & KERSWELL, R. R. 2008b Turbulent dynamics of pipe flow captured in a reduced model: Puff relaminarisation and localised 'edge' states. J. Fluid Mech. (submitted).
46. 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.