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Slug genesis in cylindrical pipe flow

Published online by Cambridge University Press:  05 October 2010

Y. DUGUET*
Affiliation:
School of Mathematics, University of Bristol, BS8 1TW Bristol, UK Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden LIMSI-CNRS, UPR 3251, 91403 Orsay, France
A. P. WILLIS
Affiliation:
School of Mathematics, University of Bristol, BS8 1TW Bristol, UK Laboratoire d'Hydrodynamique, Ecole Polytechnique, 91128 Palaiseau, France
R. R. KERSWELL
Affiliation:
School of Mathematics, University of Bristol, BS8 1TW Bristol, UK
*
Email address for correspondence: duguet@mech.kth.se

Abstract

Transition to uniform turbulence in cylindrical pipe flow occurs experimentally via the spatial expansion of isolated coherent structures called ‘slugs’, triggered by localized finite-amplitude disturbances. We study this process numerically by examining the preferred route in phase space through which a critical disturbance initiates a ‘slug’. This entails first identifying the relative attractor – ‘edge state’ – on the laminar–turbulent boundary in a long pipe and then studying the dynamics along its low-dimensional unstable manifold, leading to the turbulent state. Even though the fully turbulent state delocalizes at Re ≈ 2300, the edge state is found to be localized over the range Re = 2000–6000, and progressively reduces in both energy and spatial extent as Re is increased. A key process in the genesis of a slug is found to be vortex shedding via a Kelvin–Helmholtz mechanism from wall-attached shear layers quickly formed at the edge state's upstream boundary. Whether these shedded vortices travel on average faster or slower downstream than the developing turbulence determines whether a puff or a slug (respectively) is formed. This observation suggests that slugs are out-of-equilibrium puffs which therefore do not co-exist with stable puffs.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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References

Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301.CrossRefGoogle Scholar
Bandyopadhyay, P. R. 1986 Aspects of the equilibrium puff in transitional pipe flow. J. Fluid Mech. 163, 439458.CrossRefGoogle Scholar
Binnie, A. M. & Fowler, J. S. 1947 A study of a double refraction method of the development of turbulence in a long cylindrical tube. Proc. R. Soc. Lond. A 192, 32.CrossRefGoogle Scholar
Darbyshire, A. G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech. 289, 83.CrossRefGoogle Scholar
Duguet, Y., Pringle, C. & Kerswell, R. R. 2008 b Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102.CrossRefGoogle Scholar
Duguet, Y., Schlatter, P. & Henningson, D. S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21, 111701.CrossRefGoogle Scholar
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008 a Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255274.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2009 Equilibrium and travelling wave solutions of plane Couette flow. J. Fluid Mech. 638, 124.CrossRefGoogle Scholar
Hagen, G. H. L. 1839 Über die Bewegung des Wassers in engen zylindrischen Röhren. Poggendorfs Annal. Physik Chemie 16, 423.CrossRefGoogle Scholar
Halcrow, J., Gibson, J. F., Cvitanovic, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365376.CrossRefGoogle Scholar
Herron, I. H. 1991 Observations on the role of vorticity on the stability of wall bounded flows. Stud. Appl. Math. 85, 269286.CrossRefGoogle Scholar
Hof, B., DeLozar, A., Avila, M., Tu, X. & Schneider, T. M. 2010 Eliminating turbulence in spatially intermittent flows. Science 327, 14911494.CrossRefGoogle ScholarPubMed
Hof, B., van Doorne, C. W. H., Westerweel, J. & Nieuwstadt, F. T. M. 2005 Turbulence regeneration in pipe flow at moderate Reynolds numbers. Phys. Rev. Lett. 95, 214502.CrossRefGoogle ScholarPubMed
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 traveling waves in turbulent pipe flow. Science 305, 15941597.CrossRefGoogle ScholarPubMed
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703.CrossRefGoogle Scholar
Joseph, D. D. & Carmi, S. 1969 Stability of Poiseuille flow in pipes, annuli, and channels. Q. Appl. Math. 26, 575599.CrossRefGoogle Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17R44.CrossRefGoogle Scholar
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69102.CrossRefGoogle Scholar
Leite, R. J. 1959 An experimental investigation of the stability of Poiseuille flow. J. Fluid Mech. 5, 81.CrossRefGoogle Scholar
Leonard, A. & Reynolds, W. C. 1985 Turbulence research by numerical simulation. In Perspectives in Fluid Mechanics: Proceedings of a Symposium Held on the Occasion of the 70th Birthday of Hans Wolfgang Liepmann (ed. Coles, D.), Lecture Notes in Physics, vol. 320, p. 113. Springer.Google Scholar
Lindgren, E. R. 1958 The transition process and other phenomena in viscous flow. Ark. Phys. 12, 1.Google Scholar
Lindgren, E. R. 1969 Propagation velocity of turbulent slugs and streaks in transition pipe flow. Phys. Fluids 12, 418.CrossRefGoogle Scholar
Mellibovsky, F. & Meseguer, A. 2009 Critical threshold in pipe flow transition. Phil. Trans. R. Soc. A 367, 545560.CrossRefGoogle Scholar
Mellibovsky, F., Meseguer, A., Schneider, T. M. & Eckhardt, B. 2009 Transition in localized pipe flow turbulence. Phys. Rev. Lett. 103, 054502.CrossRefGoogle Scholar
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number 107. J. Comp. Phys. 186, 178197.CrossRefGoogle Scholar
Moxey, D. & Barkley, D. 2010 Distinct large-scale turbulent–laminar states in transitional pipe flow. PNAS 107, 80918096.CrossRefGoogle ScholarPubMed
Mullin, T. 2011 Transition to turbulence in a pipe: a historical perspective. Annu. Rev. Fluid Mech. 43.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.CrossRefGoogle Scholar
Nishi, M., Unsal, B., Durst, F. & Biswas, G. 2008 Laminar-to-turbulent transition of pipe flows through puffs and slugs. J. Fluid Mech. 614, 425446.CrossRefGoogle Scholar
Peixinho, J. & Mullin, T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96, 094501.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.). NEW YORK: Pergamon.Google Scholar
Poiseuille, J. L. M. 1840 Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres. C. R. Acad. Sci. 11, 961.Google Scholar
Pringle, C. C. T., Duguet, Y. & Kerswell, R. R. 2009 Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. A 367, 457472.CrossRefGoogle 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 Scholar
Priymak, V. G. & Miyazaki, T. 2004 Direct numerical simulation of equilibrium spatially localized structures in pipe flow. Phys. Fluids 16, 42214234.CrossRefGoogle Scholar
Reuter, J. & Dempfer, D. 2004 Analysis of pipe flow transition. Part 1. Direct numerical simulation. Theor. Comput. Fluid Dyn. 17, 273.CrossRefGoogle 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. 174, 935982.CrossRefGoogle Scholar
Schneider, T. M. & Eckhardt, B. 2009 Edge states intermediate between laminar and turbulent dynamics in pipe flow. Phil. Trans. R. Soc. A 367, 577587.CrossRefGoogle ScholarPubMed
Schneider, T. M., Eckhardt, B. & Vollmer, J. A. 2007 a Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75, 066313.Google ScholarPubMed
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 b Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.CrossRefGoogle ScholarPubMed
Schneider, T. M., Gibson, J. F., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar–turbulent boundary in plane Couette flow. Phys. Rev. E 78, 037301.Google ScholarPubMed
Schneider, T. M., Marinc, D. & Eckhardt, B. 2010 Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441451.CrossRefGoogle Scholar
Shan, H., Ma, B., Zhang, Z. & Nieuwstadt, F. T. M. 1999 Direct numerical simulation of a puff and a slug in transitional cylindrical pipe flow. J. Fluid Mech. 387, 3960.CrossRefGoogle Scholar
Shimizu, M. & Kida, S. 2008 Structure of a turbulent puff in pipe flow. J. Phys. Soc. Japan 77, 114401.CrossRefGoogle Scholar
Shimizu, M. & Kida, S. 2009 A driving mechanism of a turbulent puff in pipe flow. Fluid Dyn. Res. 41, 045501.CrossRefGoogle Scholar
Toh, S. & Itano, T. 1999 Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence: traveling-wave solution is a saddle point? In Proceedings of IUTAM Symposium on Geometry and Statistics of Turbulence (ed. Kambe, T.). Kluwer.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech., 580, 339358.CrossRefGoogle Scholar
Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. 367, 561576.CrossRefGoogle ScholarPubMed
Waleffe, F. 1997 On the self-sustaining process in shear flows. Phys. Fluids 9, 883900.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 41404143.CrossRefGoogle Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 204501.CrossRefGoogle ScholarPubMed
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.CrossRefGoogle Scholar
Willis, A. P. & Kerswell, R. R. 2007 Critical behavior in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett. 98, 14501.CrossRefGoogle ScholarPubMed
Willis, A. P. & Kerswell, R. R. 2008 Coherent structures in local and global pipe turbulence. Phys. Rev. Lett. 100, 124501.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, 213233.CrossRefGoogle Scholar
Willis, A. P., Peixinho, J., Kerswell, R. R. & Mullin, T. 2009 Experimental and theoretical progress in pipe flow transition. Phil. Trans. R. Soc. A 366, 26712684.CrossRefGoogle ScholarPubMed
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, 281351.CrossRefGoogle Scholar

Duguet et al. supplementary movie

Movie 1: Genesis of a turbulent puff starting from the edge state at Re=2000. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black). The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 7 MB

Duguet et al. supplementary movie

Movie 1: Genesis of a turbulent puff starting from the edge state at Re=2000. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black). The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 6 MB

Duguet et al. supplementary movie

Movie 2: Genesis of a turbulent slug starting from the edge state at Re=3000. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black).The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 11 MB

Duguet et al. supplementary movie

Movie 2: Genesis of a turbulent slug starting from the edge state at Re=3000. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black).The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 6 MB

Duguet et al. supplementary movie

Movie 3: Genesis of a turbulent slug starting from the edge state at Re=4500. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black).The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 8 MB

Duguet et al. supplementary movie

Movie 3: Genesis of a turbulent slug starting from the edge state at Re=4500. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black).The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 5 MB

Duguet et al. supplementary movie

Movie 4: Genesis of a turbulent slug starting from the edge state at Re=6000. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black).The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 16 MB

Duguet et al. supplementary movie

Movie 4: Genesis of a turbulent slug starting from the edge state at Re=6000. Azimuthal vorticity perturbation in a plane containing the axis. Scale is logarithmic, from -5 (white) to -1 (black).The length of the pipe is 33.51 diameters, with periodic boundary conditions.

Download Duguet et al. supplementary movie(Video)
Video 6 MB
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