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Finite lifetime of turbulence in shear flows


Generally, the motion of fluids is smooth and laminar at low speeds but becomes highly disordered and turbulent as the velocity increases. The transition from laminar to turbulent flow can involve a sequence of instabilities in which the system realizes progressively more complicated states1, or it can occur suddenly2,3. Once the transition has taken place, it is generally assumed that, under steady conditions, the turbulent state will persist indefinitely. The flow of a fluid down a straight pipe provides a ubiquitous example of a shear flow undergoing a sudden transition from laminar to turbulent motion4,5,6. Extensive calculations7,8 and experimental studies9 have shown that, at relatively low flow rates, turbulence in pipes is transient, and is characterized by an exponential distribution of lifetimes. They8,9 also suggest that for Reynolds numbers exceeding a critical value the lifetime diverges (that is, becomes infinitely large), marking a change from transient to persistent turbulence. Here we present experimental data and numerical calculations covering more than two decades of lifetimes, showing that the lifetime does not in fact diverge but rather increases exponentially with the Reynolds number. This implies that turbulence in pipes is only a transient event (contrary to the commonly accepted view), and that the turbulent and laminar states remain dynamically connected, suggesting avenues for turbulence control10.

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Figure 1: Sketch of the experimental apparatus.
Figure 2: Lifetime distributions.
Figure 3: Variation of characteristic times with Reynolds number.


  1. Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840 (2000)

    Article  ADS  CAS  PubMed  Google Scholar 

  2. Grossmann, S. The onset of shear flow turbulence. Rev. Mod. Phys. 72, 603–618 (2000)

    Article  ADS  Google Scholar 

  3. Hof, B. et al. Experimental observation of nonlinear travelling waves in turbulent pipe flow. Science 305, 1594–1598 (2004)

    Article  ADS  CAS  PubMed  Google Scholar 

  4. Reynolds, O. 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. Proc. R. Soc. Lond. 35, 84–99 (1883)

    Article  Google Scholar 

  5. Kerswell, R. R. Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–R44 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  6. Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. (submitted)

  7. Brosa, U. Turbulence without strange attractor. J. Stat. Phys. 55, 1303–1312 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  8. Faisst, H. & Eckhardt, B. Sensitive dependence on initial conditions in transition to turbulence in pipe flow. J. Fluid Mech. 504, 343–352 (2004)

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  10. Shinbrot, T,, Grebogi, C., Ott, E. & Yorke, J. A. Using small perturbations to control chaos. Nature 363, 411–417 (1993)

    Article  ADS  Google Scholar 

  11. Drazin, P. G. & Reid, W. H. Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, UK, 1981)

    MATH  Google Scholar 

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

    Article  ADS  PubMed  Google Scholar 

  13. Eckhardt, B. & Mersmann, A. Transition to turbulence in a shear flow. Phys. Rev. E 60, 509–517 (1999)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  14. Wedin, H. & Kerswell, R. R. Exact coherent solutions in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  15. Kadanoff, L. P. & Tang, C. Escape from strange repellers. Proc. Natl Acad. Sci. USA 81, 1276–1279 (1984)

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  16. Kantz, H. & Grassberger, P. Repellers, semi-attractors, and long-lived chaotic transients. Physica D 17, 75–86 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  17. Hof, B. in Proc. IUTAM Symp. on Laminar-Turbulent Transition and Finite Amplitude Solutions (eds Mullin, T. & Kerswell, R. R.) 221–231 (Springer, Dordrecht, 2005)

    Book  Google Scholar 

  18. Bottin, S. & Chaté, H. Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6, 143–155 (1998)

    Article  ADS  CAS  Google Scholar 

  19. Schmiegel, A. & Eckhardt, B. Fractal stability border in plane Couette flow. Phys. Rev. Lett. 79, 5250–5253 (1997)

    Article  ADS  CAS  Google Scholar 

  20. Wygnanski, I. J. & Champagne, F. H. 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 (1973)

    Article  ADS  Google Scholar 

  21. Wygnanski, I. J., Sokolov, M. & Friedman, D. On transition in a pipe. Part 5. The equilibrium puff. J. Fluid Mech. 69, 283–304 (1975)

    Article  ADS  Google Scholar 

  22. Hof, B., vanDoorne, C. W. H., Westerweel, J. & Nieuwstadt, F. T. M. Turbulence regeneration at moderate Reynolds numbers. Phys. Rev. Lett. 95, 214502 (2005)

    Article  ADS  PubMed  Google Scholar 

  23. Rotta, J. C. Experimenteller Beitrag zur Entstehung turbulenter Strömungen im Rohr. Ing. Archiv 24, 258–281 (1956)

    Article  Google Scholar 

  24. Schmiegel, A. Transition to Turbulence in Linearly Stable Shear Flows. PhD thesis, Philipps-Univ. Marburg (1999)

    Google Scholar 

  25. Moehlis, J., Faisst, H. & Eckhardt, B. A low-dimensional model for turbulent shear flows. N. J. Phys. 6, 56 (2004)

    Article  MathSciNet  Google Scholar 

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This work was funded through a RCUK fellowship (B.H.), the Foundation of Fundamental Research on Matter (J.W.) and Deutsche Forschungsgemeinschaft (T.M.S. and B.E.). B.H. and J.W. thank R. Delfos for discussions and W. Tax and P. Tipler for technical assistance. We thank the Dorset Tube Company for supplying the precision pipe.

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Correspondence to Björn Hof.

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Supplementary information

Supplementary Data

DIn this supplement we provide further information on various aspects of the experiment, of the numerical simulations and an account of the re-analysis of the data for plane Couette flow that could not be included in the main text. Also included are supplementary figures 1–4. (PDF 221 kb)

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Hof, B., Westerweel, J., Schneider, T. et al. Finite lifetime of turbulence in shear flows. Nature 443, 59–62 (2006).

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