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On the transient nature of localized pipe flow turbulence

Published online by Cambridge University Press:  08 March 2010

MARC AVILA ,
ASHLEY P. WILLIS  and
BJÖRN HOF
MARC AVILA*
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, 37073 Göttingen, Germany
ASHLEY P. WILLIS
Affiliation:
Laboratoire d'Hydrodynamique, École Polytechnique, 91128 Palaiseau, France
BJÖRN HOF
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, 37073 Göttingen, Germany
*
†Email address for correspondence: mavila@ds.mpg.de
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Abstract

The onset of shear flow turbulence is characterized by turbulent patches bounded by regions of laminar flow. At low Reynolds numbers localized turbulence relaminarizes, raising the question of whether it is transient in nature or becomes sustained at a critical threshold. We present extensive numerical simulations and a detailed statistical analysis of the lifetime data, in order to shed light on the sources of the discrepancies present in the literature. The results are in excellent quantitative agreement with recent experiments and show that turbulent lifetimes increase super-exponentially with Reynolds number. In addition, we provide evidence for a lower bound below which there are no meta-stable characteristics of the transients, i.e. the relaminarization process is no longer memoryless.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 646 , 10 March 2010 , pp. 127 - 136
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Borrero-Echeverry, D., Tagg, R. & Schatz, M. F. 2009 Transient turbulence in Taylor–Couette flow. arXiv:0905.0147v3.CrossRefGoogle Scholar
Bottin, S. & Chaté, H. 1998 Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6 (1), 143–155.CrossRefGoogle Scholar
Bottin, S., Daviaud, F., Manneville, P. & Dauchot, O. 1998 Discontinuous transition to spatiotemporal intermittency in plane Couette flow. Europhys. Lett. 43 (2), 171–176.CrossRefGoogle Scholar
Brosa, U. 1989 Turbulence without strange attractor. J. Stat. Phys. 55 (5), 1303–1312.CrossRefGoogle Scholar
Darbyshire, A. G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech. 289, 83–114.CrossRefGoogle Scholar
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102.CrossRefGoogle Scholar
Eckhardt, B. 2008 Turbulence transition in pipe flow: some open questions. Nonlinearity (London) 21 (1), T1–T11.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle 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. J. Fluid Mech. 504, 343–352.CrossRefGoogle Scholar
Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72 (2), 603–618.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Hof, B., Juel, A. & Mullin, T. 2003 Scaling of the turbulence transition threshold in a pipe. Phys. Rev. Lett. 91 (24), 244502.CrossRefGoogle Scholar
Hof, B., de Lozar, A., Kuik, D. J. & Westerweel, J. 2008 Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett. 101 (21), 214501.CrossRefGoogle ScholarPubMed
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature (London) 443 (7), 59–62.CrossRefGoogle ScholarPubMed
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity (London) 18 (6), R17–R44.CrossRefGoogle Scholar
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.CrossRefGoogle Scholar
Lawless, J. F. 2003 Statistical Models and Methods for Lifetime Data, 2nd edn. Wiley.Google Scholar
de Lozar, A. & Hof, B. 2009 An experimental study of the decay of turbulent puffs in pipe flow. Phil. Trans. R. Soc. A 367 (1888), 589–599.CrossRefGoogle ScholarPubMed
Marques, F. 1990 On boundary conditions for velocity potentials in confined flows: application to Couette flow. Phys. Fluids A 2, 729–737.CrossRefGoogle Scholar
Mellibovsky, F. & Meseguer, A. 2009 Critical threshold in pipe flow transition. Phil. Trans. R. Soc. A 367 (1888), 545–560.CrossRefGoogle ScholarPubMed
Peixinho, J. & Mullin, T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96 (9), 094501.CrossRefGoogle ScholarPubMed
Peixinho, J. & Mullin, T. 2007 Finite-amplitude thresholds for transition in pipe flow. J. Fluid Mech. 582, 169–178.CrossRefGoogle Scholar
Pfenniger, W. 1961 Transition in the inlet length of tubes at high Reynolds numbers. In Boundary Layer and Flow Control (ed. Lachman, G. V.), pp. 970–980. Pergamon.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.CrossRefGoogle ScholarPubMed
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. Proc. R. Soc. Lond. 35, 84–99.Google Scholar
Schneider, T. M. & Eckhardt, B. 2008 Lifetime statistics in transitional pipe flow. Phys. Rev. E 78 (4), 046310.CrossRefGoogle ScholarPubMed
Schoepe, W. 2004 Fluctuations and stability of superfluid turbulence at mK temperatures. Phys. Rev. Lett. 92 (9), 95301.CrossRefGoogle ScholarPubMed
Tél, T. & Lai, Y. C. 2008 Chaotic transients in spatially extended systems. Phys. Rep. 460 (6), 245–275.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Willis, A. P. & Kerswell, R. R. 2007 Critical behaviour in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett. 98 (1), 014501.CrossRefGoogle ScholarPubMed
Willis, A. P. & Kerswell, R. R. 2008 Reply to comment on ‘Critical behaviour in the relaminarization of localized turbulence in pipe flow’. arXiv:0707.2684.CrossRefGoogle Scholar
Willis, A. P. & Kerswell, R. R. 2009 Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarization and localized ‘edge’ states. J. Fluid Mech. 619, 213–233.CrossRefGoogle Scholar
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 (2), 281–335.CrossRefGoogle Scholar