Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T05:49:50.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Studying edge geometry in transiently turbulent shear flows

Published online by Cambridge University Press:  23 April 2014

Matthew Chantry  and
Tobias M. Schneider
Matthew Chantry*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
Tobias M. Schneider
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany Institute of Mechanical Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In linearly stable shear flows at moderate Reynolds number, turbulence spontaneously decays despite the existence of a codimension-one manifold, termed the edge, which separates decaying perturbations from those triggering turbulence. We statistically analyse the decay in plane Couette flow, quantify the breaking of self-sustaining feedback loops and demonstrate the existence of a whole continuum of possible decay paths. Drawing parallels with low-dimensional models and monitoring the location of the edge relative to decaying trajectories, we provide evidence that the edge of chaos does not separate state space globally. It is instead wrapped around the turbulence generating structures and not an independent dynamical structure but part of the chaotic saddle. Thereby, decaying trajectories need not cross the edge, but circumnavigate it while unwrapping from the turbulent saddle.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 747 , 25 May 2014 , pp. 506 - 517
Copyright
© 2014 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

Avila, M., Mellibovsky, F., Roland, N. & Hof, B. 2013 Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110 (22), 224502.Google Scholar
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192–196.Google Scholar
De Lozar, A., Mellibovsky, F., Avila, M. & Hof, B. 2012 Edge state in pipe flow experiments. Phys. Rev. Lett. 108 (21), 214502.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 (1), 255–274.Google Scholar
Faisst, H. & Eckhardt, B. 2004 Sensitive dependence on initial conditions in transition to turbulence in pipe flow. J. Fluid Mech. 504, 343–352.Google Scholar
Gibson, J. F.2012 Channelflow: a spectral Navier–Stokes simulator in C $++$ . Tech. Rep. University of New Hampshire. Channelflow.org.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.Google Scholar
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59–62.Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.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
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.CrossRefGoogle 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.Google Scholar
Kim, L. & Moehlis, J. 2008 Characterizing the edge of chaos for a shear flow model. Phys. Rev. E 78 (3), 036315.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane couette flow. Chaos 22 (4), 047505.Google Scholar
Lebovitz, N. R. 2009 Shear-flow transition: the basin boundary. Nonlinearity 22 (11), 2645–2655.Google Scholar
Lebovitz, N. R. 2012 Boundary collapse in models of shear-flow transition. Commun. Nonlinear Sci. Numer. Simul. 17 (5), 2095–2100.Google Scholar
Lebovitz, N. & Mariotti, G. 2013 Edges in models of shear flow. J. Fluid Mech. 721, 386–402.Google Scholar
Moxey, D. & Barkley, D. 2010 Distinct large-scale turbulent-laminar states in transitional pipe flow. Proc. Natl Acad. Sci. USA 107 (18), 8091–8096.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
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.Google Scholar
Schneider, T. M., Gibson, J. F. & Burke, J. 2010a Snakes and ladders: localized solutions of plane couette flow. Phys. Rev. Lett. 104 (10), 104501.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 (3), 037301.Google Scholar
Schneider, T. M., Lillo, F. D., Buehrle, J., Eckhardt, B., Dörnemann, T., Dörnemann, K. & Freisleben, B. 2010b Transient turbulence in plane Couette flow. Phys. Rev. E 81, 015301(R).CrossRefGoogle ScholarPubMed
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.Google Scholar
Viswanath, D. 2008 The dynamics of transition to turbulence in plane couette flow. In Mathematics and Computation, A Contemporary View, pp. 109–127. Springer.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar