Hostname: page-component-f554764f5-nwwvg Total loading time: 0 Render date: 2025-04-20T16:16:13.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Theoretical perspective on the route to turbulence in a pipe

Published online by Cambridge University Press:  30 August 2016

D. Barkley
D. Barkley*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The route to turbulence in pipe flow is a complex, nonlinear, spatiotemporal process for which an increasingly clear understanding has emerged in recent years. This paper presents a theoretical perspective on the problem, focusing on what can be understood from relatively few physical features and models that encompass these features. The paper proceeds step-by-step with increasing detail about the transition process, first discussing the relationship to phase transitions and then exploiting an even deeper connection between pipe flow and excitable and bistable media. In the end a picture emerges for all stages of the transition process, from transient turbulence, to the onset of sustained turbulence in a percolation transition, to the modest and then rapid expansion of turbulence, ultimately leading to fully turbulent pipe flow.

JFM classification

Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent Flows
Type
JFM Perspectives
Information
Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , P1
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Allhoff, K. T. & Eckhardt, B. 2012 Directed percolation model for turbulence transition in shear flows. Fluid Dyn. Res. 44 (3), 031201.CrossRefGoogle Scholar
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192–196.Google Scholar
Avila, M., Willis, A. P. & Hof, B. 2010 On the transient nature of localized pipe flow turbulence. J. Fluid Mech. 646, 127–136.Google Scholar
Bandyopadhyay, P. R. 1986 Aspects of the equilibrium puff in transitional pipe flow. J. Fluid Mech. 163, 439–458.Google Scholar
Barkley, D. 2011a Simplifying the complexity of pipe flow. Phys. Rev. E 84 (1), 016309.Google Scholar
Barkley, D. 2011b Modeling the transition to turbulence in shear flows. J. Phys.: Conf. Ser. 318 (3), 032001.Google Scholar
Barkley, D. 2012 Pipe flow as an excitable medium. Rev. Cub. Fis. 29, 1E27.Google Scholar
Barkley, D., Song, B., Mukund, V., Lemoult, G., Avila, M. & Hof, B. 2015 The rise of fully turbulent flow. Nature 526 (7574), 550–553.CrossRefGoogle ScholarPubMed
Barkley, D. & Tuckerman, L. S. 2007 Mean flow of turbulent–laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137.Google Scholar
Chantry, M., Tuckerman, L. S. & Barkley, D. 2016 Turbulent–laminar patterns in shear flows without walls. J. Fluid Mech. 791, R8.Google Scholar
Chaté, H. & Manneville, P. 1987 Transition to turbulence via spatiotemporal intermittency. Phys. Rev. Lett. 58 (2), 112–115.Google Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Chossat, P. & Iooss, G. 1985 Primary and secondary bifurcations in the Couette–Taylor problem. Japan J. Appl. Math. 2 (1), 37–68.Google Scholar
Coles, D. 1962 Interfaces and intermittency in turbulent shear flow. Méc. Turbul. 108 (108), 229–250.Google Scholar
Darbyshire, A. G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe-flow. J. Fluid Mech. 289, 83–114.CrossRefGoogle Scholar
Doering, C. R. 1987 A stochastic partial differential equation with multiplicative noise. Phys. Lett. A 122 (3–4), 133–139.CrossRefGoogle Scholar
van Doorne, C. W. & Westerweel, J. 2009 The flow structure of a puff. Phil. Trans. R. Soc. Lond. A 367 (1888), 489–507.Google Scholar
Duguet, Y. & Schlatter, P. 2013 Oblique laminar–turbulent interfaces in plane shear flows. Phys. Rev. Lett. 110 (3), 034502.Google Scholar
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, 255–274.Google Scholar
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2010 Slug genesis in cylindrical pipe flow. J. Fluid Mech. 663, 180–208.Google Scholar
Eckert, M. 2010 The troublesome birth of hydrodynamic stability theory: Sommerfeld and the turbulence problem. Eur. Phys. J. H 35 (1), 29–51.Google Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.CrossRefGoogle ScholarPubMed
Faisst, H. & Eckhardt, B. 2004 Sensitive dependence on initial conditions in transition to turbulence in pipe flow. J. Fluid Mech. 504, 343–352.Google Scholar
Feigenbaum, M. J. 1978 Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19 (1), 25–52.Google Scholar
Flores, G. 1991 Stability analysis for the slow travelling pulse of the Fitzhugh–Nagumo system. SIAM J. Math. Anal. 22 (2), 392–399.CrossRefGoogle Scholar
Goldenfeld, N., Guttenberg, N. & Gioia, G. 2010 Extreme fluctuations and the finite lifetime of the turbulent state. Phys. Rev. E 81 (3), 035304.Google Scholar
Gollub, J. P. & Swinney, H. L. 1975 Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35 (14), 927.Google Scholar
Hinrichsen, H. 2000 Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49 (7), 815–958.CrossRefGoogle Scholar
Hodgkin, A. L. & Huxley, A. F. 1952 A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. Lond. 117 (4), 500–544.Google Scholar
Hof, B., Juel, A. & Mullin, T. 2003 Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91 (24), 244502.Google Scholar
Hof, B., de Lozar, A., Avila, M., Tu, X. & Schneider, T. M. 2010 Eliminating turbulence in spatially intermittent flows. Science 327 (5972), 1491–1494.Google Scholar
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59–62.CrossRefGoogle ScholarPubMed
Holzner, M., Song, B., Avila, M. & Hof, B. 2013 Lagrangian approach to laminar–turbulent interfaces in transitional pipe flow. J. Fluid Mech. 723, 140–162.Google Scholar
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Pure Appl. Maths 1 (4), 303–322.Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.Google Scholar
Jalife, J. 2000 Ventricular fibrillation: mechanisms of initiation and maintenance. Annu. Rev. Phys. Chem. 62 (1), 25–50.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions I, Springer Tracts in Natural Philosophy, vol. 27. Springer.Google Scholar
Kaneko, K. 1985 Spatiotemporal intermittency in coupled map lattices. Prog. Theor. Phys. 74 (5), 1033–1044.Google Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44 (1), 203–225.Google Scholar
Keener, J. & Sneyd, J. 2008 Mathematical Physiology I: Cellular Physiology, 2nd edn. Springer.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Volume 6 of A Course of Theoretical Physics. Pergamon Press.Google Scholar
Landau, L. D. 1944 On the problem of turbulence. Dokl. Akad. Nauk SSSR 44 (8), 339–349.Google Scholar
Lemoult, G., Gumowski, K., Aider, J.-L. & Wesfreid, J. E. 2014 Turbulent spots in channel flow: an experimental study. Eur. Phys. J. E 37, 25.Google Scholar
Lemoult, G., Shi, L., Avila, K., Jalikop, S. V., Avila, M. & Hof, B. 2016 Directed percolation phase transition to sustained turbulence in Couette flow. Nat. Phys. 12 (3), 254–258.Google Scholar
Lindgren, E. R. 1957 The transition process and other phenomena in viscous flow. Ark. Fys. 12, 1–169.Google Scholar
Lindgren, E. R. 1969 Propagation velocity of turbulent slugs and streaks in transition pipe flow. Phys. Fluids 12 (2), 418–425.Google Scholar
Manneville, P. 2015 On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular. Eur. J. Mech. (B/Fluids) 49, 345–362.Google Scholar
Manneville, P. 2016 Transition to turbulence in wall-bounded flows: Where do we stand? Bull. JSME 3 (2), 15–00684.Google Scholar
Marschler, C. & Vollmer, J. 2014 Unidirectionally coupled map lattices with nonlinear coupling: Unbinding transitions and superlong transients. SIAM J. Appl. Dyn. Syst. 13 (3), 1137–1151.Google Scholar
McKeon, B. J., Swanson, C J, Zagarola, M. V., Donnelly, R. J. & Smits, A. J. 2004 Friction factors for smooth pipe flow. J. Fluid Mech. 511, 41–44.Google Scholar
Mellibovsky, F., Meseguer, A., Schneider, T. M. & Eckhardt, B. 2009 Transition in localized pipe flow turbulence. Phys. Rev. Lett. 103 (5), 054502.Google Scholar
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number 107 . J. Comput. Phys. 186 (1), 178–197.Google Scholar
Moxey, D. & Barkley, D. 2010 Distinct large-scale turbulent-laminar states in transitional pipe flow. Proc. Natl Acad. Sci. USA 107 (18), 8091–8096.Google Scholar
Narasimha, R. & Sreenivasan, K. R. 1979 Relaminarization of fluid flows. Adv. Appl. Mech. 19, 221–309.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38 (02), 279–303.CrossRefGoogle Scholar
Nishi, M., Ünsal, B., Durst, F. & Biswas, G. 2008 Laminar-to-turbulent transition of pipe flows through puffs and slugs. J. Fluid Mech. 614, 425.Google Scholar
Orr, W. M.’F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: a viscous liquid. Proc. R. Irish Acad. A 27, 69–138.Google Scholar
Peixinho, J. & Mullin, T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96 (9), 094501.CrossRefGoogle ScholarPubMed
Pomeau, Y. 1986 Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23 (1–3), 3–11.Google Scholar
Pomeau, Y. 2015 The transition to turbulence in parallel flows: a personal view. C. R. Méc. 343 (3), 210–218.CrossRefGoogle Scholar
Pope, S. B 2000 Turbulent Flows. Cambridge University Press.Google Scholar
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. A 174, 935–982.Google Scholar
Rinzel, J. & Terman, D. 1982 Propagation phenomena in a bistable reaction-diffusion system. SIAM J. Appl. Maths 42 (5), 1111–1137.Google Scholar
Rotta, J. 1956 Experimenteller Beitrag zur Entstehung turbulenter Strömung im Rohr. Ing-Arch. 24 (4), 258–281.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20 (3), 167–192.Google Scholar
Salwen, H., Cotton, F. W. & Grosch, C. E. 1980 Linear stability of poiseuille flow in a circular pipe. J. Fluid Mech. 98 (02), 273–284.Google Scholar
Samanta, D., De Lozar, A. & Hof, B. 2011 Experimental investigation of laminar turbulent intermittency in pipe flow. J. Fluid Mech. 681, 193–204.CrossRefGoogle Scholar
Schlichting, H. 1968 Boundary-Layer Theory. McGraw-Hill.Google Scholar
Schneider, T., Eckhardt, B. & Yorke, J. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.Google Scholar
Segel, L. A. 1969 Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38 (01), 203–224.Google Scholar
Shih, H.-Y., Hsieh, T.-L. & Goldenfeld, N. 2016 Ecological collapse and the emergence of travelling waves at the onset of shear turbulence. Nat. Phys. 12, 245–248.Google Scholar
Shimizu, M. & Kida, S. 2009 A driving mechanism of a turbulent puff in pipe flow. Fluid Dyn. Res. 41 (4), 045501.Google Scholar
Shimizu, M., Manneville, P., Duguet, Y. & Kawahara, G. 2014 Splitting of a turbulent puff in pipe flow. Fluid Dyn. Res. 46 (6), 061403.Google Scholar
Sipos, M. & Goldenfeld, N. 2011 Directed percolation describes lifetime and growth of turbulent puffs and slugs. Phys. Rev. E 84 (3), 035304.Google Scholar
Song, B., Barkley, D., Avila, M. & Hof, B.2016 Speed and structure of turbulent fronts in pipe flow. arXiv:1603.04077.Google Scholar
Starmer, C. F., Biktashev, V. N., Romashko, D. N., Stepanov, M. R., Makarova, O. N. & Krinsky, V. I. 1993 Vulnerability in an excitable medium: analytical and numerical studies of initiating unidirectional propagation. Biophys. J. 65 (5), 1775.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4 (01), 1–21.Google Scholar
Swinney, H. L. & Gollub, J. P. 1985 Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn. Topics in Applied Physics, vol. 45. Springer.Google Scholar
Takeuchi, K. A., Kuroda, M., Chaté, H. & Sano, M. 2007 Directed percolation criticality in turbulent liquid crystals. Phys. Rev. Lett. 99 (23), 234503.Google Scholar
Takeuchi, K. A., Kuroda, M., Chaté, H. & Sano, M. 2009 Experimental realization of directed percolation criticality in turbulent liquid crystals. Phys. Rev. E 80 (5), 051116.Google ScholarPubMed
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Tyson, J. J. & Keener, J. P. 1988 Singular perturbation-theory of traveling waves in excitable media. Physica D 32 (3), 327–361.Google Scholar
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.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.Google Scholar
Willis, A. P. & Kerswell, R. R. 2007 Critical behavior in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett. 98 (1), 014501.Google Scholar
Winfree, A. T. 1991 Varieties of spiral wave behavior: an experimentalist’s approach to the theory of excitable media. Chaos: An Interdisciplinary J. Nonlinear Sci. 1 (3), 303–334.Google Scholar
Wygnanski, I. & Champagne, H. 1973 Transition in a pipe. Part 1. The origin of puffs and slugs and flow in a turbulent slug. J. Fluid Mech. 59, 281–335.Google Scholar
Wygnanski, I., Sokolov, M. & Friedman, D. 1975 Transition in a pipe. Part 2. The equilibrium puff. J. Fluid Mech. 69, 283–304.Google Scholar