Skip to main content
    • Aa
    • Aa

Theoretical perspective on the route to turbulence in a pipe


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.

Corresponding author
Email address for correspondence:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

K. T. Allhoff  & B. Eckhardt 2012 Directed percolation model for turbulence transition in shear flows. Fluid Dyn. Res. 44 (3), 031201.

K. Avila , D. Moxey , A. de Lozar , M. Avila , D. Barkley  & B. Hof 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192196.

D. Barkley , B. Song , V. Mukund , G. Lemoult , M. Avila  & B. Hof 2015 The rise of fully turbulent flow. Nature 526 (7574), 550553.

H. Chaté  & P. Manneville 1987 Transition to turbulence via spatiotemporal intermittency. Phys. Rev. Lett. 58 (2), 112115.

J.-M. Chomaz 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.

P. Chossat  & G. Iooss 1985 Primary and secondary bifurcations in the Couette–Taylor problem. Japan J. Appl. Math. 2 (1), 3768.

C. R. Doering 1987 A stochastic partial differential equation with multiplicative noise. Phys. Lett. A 122 (3–4), 133139.

C. W. van Doorne  & J. Westerweel 2009 The flow structure of a puff. Phil. Trans. R. Soc. Lond. A 367 (1888), 489507.

Y. Duguet  & P. Schlatter 2013 Oblique laminar–turbulent interfaces in plane shear flows. Phys. Rev. Lett. 110 (3), 034502.

B. Eckhardt , T. M. Schneider , B. Hof  & J. Westerweel 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.

H. Faisst  & B. Eckhardt 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.

M. J. Feigenbaum 1978 Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19 (1), 2552.

G. Flores 1991 Stability analysis for the slow travelling pulse of the Fitzhugh–Nagumo system. SIAM J. Math. Anal. 22 (2), 392399.

J. P. Gollub  & H. L. Swinney 1975 Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35 (14), 927.

H. Hinrichsen 2000 Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49 (7), 815958.

A. L. Hodgkin  & A. F. Huxley 1952 A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. Lond. 117 (4), 500544.

B. Hof , A. Juel  & T. Mullin 2003 Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91 (24), 244502.

B. Hof , A. de Lozar , M. Avila , X. Tu  & T. M. Schneider 2010 Eliminating turbulence in spatially intermittent flows. Science 327 (5972), 14911494.

B. Hof , J. Westerweel , T. M. Schneider  & B. Eckhardt 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 5962.

E. Hopf 1948 A mathematical example displaying features of turbulence. Commun. Pure Appl. Maths 1 (4), 303322.

T. Itano  & S. Toh 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703716.

J. Jalife 2000 Ventricular fibrillation: mechanisms of initiation and maintenance. Annu. Rev. Phys. Chem. 62 (1), 2550.

D. D. Joseph 1976 Stability of Fluid Motions I, Springer Tracts in Natural Philosophy, vol. 27. Springer.

K. Kaneko 1985 Spatiotemporal intermittency in coupled map lattices. Prog. Theor. Phys. 74 (5), 10331044.

G. Kawahara , M. Uhlmann  & L. van Veen 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44 (1), 203225.

G. Lemoult , L. Shi , K. Avila , S. V. Jalikop , M. Avila  & B. Hof 2016 Directed percolation phase transition to sustained turbulence in Couette flow. Nat. Phys. 12 (3), 254258.

E. R. Lindgren 1969 Propagation velocity of turbulent slugs and streaks in transition pipe flow. Phys. Fluids 12 (2), 418425.

P. Manneville 2015 On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular. Eur. J. Mech. (B/Fluids) 49, 345362.

C. Marschler  & J. Vollmer 2014 Unidirectionally coupled map lattices with nonlinear coupling: Unbinding transitions and superlong transients. SIAM J. Appl. Dyn. Syst. 13 (3), 11371151.

F. Mellibovsky , A. Meseguer , T. M. Schneider  & B. Eckhardt 2009 Transition in localized pipe flow turbulence. Phys. Rev. Lett. 103 (5), 054502.

A. Meseguer  & L. N. Trefethen 2003 Linearized pipe flow to Reynolds number 107 . J. Comput. Phys. 186 (1), 178197.

D. Moxey  & D. Barkley 2010 Distinct large-scale turbulent-laminar states in transitional pipe flow. Proc. Natl Acad. Sci. USA 107 (18), 80918096.

R. Narasimha  & K. R. Sreenivasan 1979 Relaminarization of fluid flows. Adv. Appl. Mech. 19, 221309.

J. Peixinho  & T. Mullin 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96 (9), 094501.

Y. Pomeau 2015 The transition to turbulence in parallel flows: a personal view. C. R. Méc. 343 (3), 210218.

S. B Pope 2000 Turbulent Flows. Cambridge University Press.

O. Reynolds 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, 935982.

J. Rinzel  & D. Terman 1982 Propagation phenomena in a bistable reaction-diffusion system. SIAM J. Appl. Maths 42 (5), 11111137.

J. Rotta 1956 Experimenteller Beitrag zur Entstehung turbulenter Strömung im Rohr. Ing-Arch. 24 (4), 258281.

D. Ruelle  & F. Takens 1971 On the nature of turbulence. Commun. Math. Phys. 20 (3), 167192.

T. Schneider , B. Eckhardt  & J. Yorke 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.

H.-Y. Shih , T.-L. Hsieh  & N. Goldenfeld 2016 Ecological collapse and the emergence of travelling waves at the onset of shear turbulence. Nat. Phys. 12, 245248.

M. Shimizu  & S. Kida 2009 A driving mechanism of a turbulent puff in pipe flow. Fluid Dyn. Res. 41 (4), 045501.

M. Shimizu , P. Manneville , Y. Duguet  & G. Kawahara 2014 Splitting of a turbulent puff in pipe flow. Fluid Dyn. Res. 46 (6), 061403.

C. F. Starmer , V. N. Biktashev , D. N. Romashko , M. R. Stepanov , O. N. Makarova  & V. I. Krinsky 1993 Vulnerability in an excitable medium: analytical and numerical studies of initiating unidirectional propagation. Biophys. J. 65 (5), 1775.

H. L. Swinney  & J. P. Gollub 1985 Hydrodynamic Instabilities and the Transition to Turbulence, 2nd edn. Topics in Applied Physics, vol. 45. Springer.

K. A. Takeuchi , M. Kuroda , H. Chaté  & M. Sano 2007 Directed percolation criticality in turbulent liquid crystals. Phys. Rev. Lett. 99 (23), 234503.

G. I. Taylor 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289343.

J. Vollmer , T. M Schneider  & B. Eckhardt 2009 Basin boundary, edge of chaos and edge state in a two-dimensional model. New J. Phys. 11, 013040.

F. Waleffe 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883900.

A. P. Willis  & R. R. Kerswell 2007 Critical behavior in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett. 98 (1), 014501.

A. T. Winfree 1991 Varieties of spiral wave behavior: an experimentalist’s approach to the theory of excitable media. Chaos: An Interdisciplinary J. Nonlinear Sci. 1 (3), 303334.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Altmetric attention score

Full text views

Total number of HTML views: 127
Total number of PDF views: 1882 *
Loading metrics...

Abstract views

Total abstract views: 3199 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th July 2017. This data will be updated every 24 hours.