Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T22:35:55.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Edge states for the turbulence transition in the asymptotic suction boundary layer

Published online by Cambridge University Press:  30 May 2013

Tobias Kreilos ,
Gregor Veble ,
Tobias M. Schneider  and
Bruno Eckhardt
Tobias Kreilos*
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, Germany Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
Gregor Veble
Affiliation:
Pipistrel d.o.o. Ajdovščina, Goriška c. 50a, SI-5270 Ajdovščina, Slovenia University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia
Tobias M. Schneider
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA
Bruno Eckhardt
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, Renthof 6, D-35032 Marburg, Germany J. M. Burgerscentrum, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We demonstrate the existence of an exact invariant solution to the Navier–Stokes equations for the asymptotic suction boundary layer. The identified periodic orbit with a very long period of several thousand advective time units is found as a local dynamical attractor embedded in the stability boundary between laminar and turbulent dynamics. Its dynamics captures both the interplay of downstream-oriented vortex pairs and streaks observed in numerous shear flows as well as the energetic bursting that is characteristic for boundary layers. By embedding the flow into a family of flows that interpolates between plane Couette flow and the boundary layer, we demonstrate that the periodic orbit emerges in a saddle–node infinite-period (SNIPER) bifurcation of two symmetry-related travelling-wave solutions of plane Couette flow. Physically, the long period is due to a slow streak instability, which leads to a violent breakup of a streak associated with the bursting and the reformation of the streak at a different spanwise location. We show that the orbit is structurally stable when varying both the Reynolds number and the domain size.

JFM classification

Boundary Layers: Boundary Layers Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 100 - 122
Copyright
©2013 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.)

References

Abshagen, J., Lopez, J., Marques, F. & Pfister, G. 2005 Symmetry breaking via global bifurcations of modulated rotating waves in hydrodynamics. Phys. Rev. Lett. 94 (7), 10–13.Google Scholar
Alfredsson, P. H., Brandt, L., Bottaro, A., Henningson, D. S. & Sabatier, P. 2001 On the breakdown of boundary layer streaks. J. Fluid Mech. 428, 29–60.Google Scholar
Armbruster, D., Guckenheimer, J. & Holmes, P. 1988 Heteroclinic cycles and modulated travelling waves in systems with $O(2)$ symmetry. Physica D 29, 257–282.Google Scholar
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011 Edge states in a boundary layer. Phys. Fluids 23, 051705.Google Scholar
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.Google Scholar
Duguet, Y., Schlatter, P. & Henningson, D. S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21, 111701.Google Scholar
Duguet, Y., Schlatter, P. & Henningson, D. S. 2010 Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech. 650, 119–129.Google Scholar
Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. 2012 Self-sustained localized structures in a boundary-layer flow. Phys. Rev. Lett. 108, 044501.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
Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T. M. 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. A 366, 1297–1315.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. 2000 Transition from the Couette–Taylor system to the plane Couette system. Phys. Rev. E 61, 7227–7230.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Fransson, J. H. M. 2001 Investigations of the asymptotic suction boundary layer. PhD thesis, KTH, Stockholm.Google Scholar
Gibson, J. F. 2012 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep., University of New Hampshire.Google Scholar
Gibson, J. F., Halcrow, J & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.CrossRefGoogle Scholar
Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72, 603–618.Google Scholar
Halcrow, J. 2008 Charting the state space of plane Couette flow: equilibria, relative equilibria, and heteroclinic connections. PhD thesis, Georgia Institute of Technology.Google Scholar
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365–376.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Hocking, L. M. 1975 Non-linear instability of the asymptotic suction velocity profile. Q. J. Mech. Appl. Maths 28, 341–353.Google 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, 1594–1598.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Jeong, J., Hussain, F., Schoppa, W. & Kim, J. 1997 Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech. 332, 185–214.Google Scholar
Joslin, R. D. 1998 Aircraft laminar flow control. Annu. Rev. Fluid Mech. 30, 1–29.Google Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.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
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2013 Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717, R6 (11pp.).Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22 (4), 047505.Google Scholar
Levin, O., Davidsson, E. N. & Henningson, D. S. 2005 Transition thresholds in the asymptotic suction boundary layer. Phys. Fluids 17, 114104.Google Scholar
Mellibovsky, F., Meseguer, A., Schneider, T. M. & Eckhardt, B. 2009 Transition in localized pipe flow turbulence. Phys. Rev. Lett. 103 (5), 1–4.Google Scholar
Moehlis, J., Eckhardt, B. & Faisst, H. 2004 Fractal lifetimes in the transition to turbulence. Chaos 14, S11.Google Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.Google Scholar
Nagata, M. 1997 Three-dimensional travelling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023–2025.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23 (1), 601–639.Google Scholar
Schlichting, H. 2004 Boundary-Layer Theory. Springer.Google Scholar
Schmiegel, A. & Eckhardt, B. 1997 Fractal stability border in plane Couette flow. Phys. Rev. Lett. 79, 5250–5253.Google Scholar
Schneider, T. M. 2007 State space properties of transitional pipe flow. PhD thesis, University of Marburg.Google Scholar
Schneider, T. M. & Eckhardt, B. 2009 Edge states intermediate between laminar and turbulent dynamics in pipe flow. Phil. Trans. R. Soc. A 367 (1888), 577–587.Google Scholar
Schneider, T. M., Eckhardt, B. & Vollmer, J. 2007a Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75, 066313.Google Scholar
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007b Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.Google Scholar
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 Scholar
Schneider, T. M., Marinc, D. & Eckhardt, B. 2010 Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441–451.Google Scholar
Skufca, J.D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.Google Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering. Perseus.Google Scholar
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 67–76.Google Scholar
Tuckerman, L. S. & Barkley, D. 1988 Global bifurcations to traveling waves in axisymmetric convection. Phys. Rev. Lett. 61 (4), 408–411.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.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, 1–23.Google Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 95, 319–343.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 6–8.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., Cvitanović, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514–540.Google Scholar