Hostname: page-component-5b777bbd6c-gcwzt Total loading time: 0 Render date: 2025-06-21T20:06:16.366Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent–laminar patterns in shear flows without walls

Published online by Cambridge University Press:  24 February 2016

Matthew Chantry ,
Laurette S. Tuckerman  and
Dwight Barkley
Matthew Chantry*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), UMR CNRS 7636; PSL – ESPCI, Sorbonne Université – UPMC, Univ. Paris 06; Sorbonne Paris Cité – UDD, Univ. Paris 07, – 10 rue Vauquelin, 75005 Paris, France
Laurette S. Tuckerman
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), UMR CNRS 7636; PSL – ESPCI, Sorbonne Université – UPMC, Univ. Paris 06; Sorbonne Paris Cité – UDD, Univ. Paris 07, – 10 rue Vauquelin, 75005 Paris, France
Dwight Barkley
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
*
†Email address for correspondence: matthew.chantry@espci.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Turbulent–laminar intermittency, typically in the form of bands and spots, is a ubiquitous feature of the route to turbulence in wall-bounded shear flows. Here we study the idealised shear between stress-free boundaries driven by a sinusoidal body force and demonstrate quantitative agreement between turbulence in this flow and that found in the interior of plane Couette flow – the region excluding the boundary layers. Exploiting the absence of boundary layers, we construct a model flow that uses only four Fourier modes in the shear direction and yet robustly captures the range of spatiotemporal phenomena observed in transition, from spot growth to turbulent bands and uniform turbulence. The model substantially reduces the cost of simulating intermittent turbulent structures while maintaining the essential physics and a direct connection to the Navier–Stokes equations. We demonstrate the generic nature of this process by introducing stress-free equivalent flows for plane Poiseuille and pipe flows that again capture the turbulent–laminar structures seen in transition.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Pattern formation Instability: Transition to turbulence
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 791 , 25 March 2016 , R8
Check for updates
Copyright
© 2016 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.)

Article purchase

Temporarily unavailable

References

Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.CrossRefGoogle ScholarPubMed
Barkley, D. 2011 Simplifying the complexity of pipe flow. Phys. Rev. E 84, 016309.Google Scholar
Barkley, D., Song, B., Mukund, V., Lemoult, G., Avila, M. & Hof, B. 2015 The rise of fully turbulent flow. Nature 526, 550–553.Google Scholar
Barkley, D. & Tuckerman, L. S. 2005 Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502.Google Scholar
Barkley, D. & Tuckerman, L. S. 2007 Mean flow of turbulent–laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137.Google Scholar
Beaume, C., Chini, G. P., Julien, K. & Knobloch, E. 2015 Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91, 043010.Google Scholar
Bottin, S. & Chaté, H. 1998 Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6, 143–155.Google Scholar
Bottin, S., Daviaud, F., Manneville, P. & Dauchot, O. 1998 Discontinuous transition to spatiotemporal intermittency in plane Couette flow. Europhys. Lett. 43, 171–176.Google Scholar
Chantry, M. & Kerswell, R. R. 2015 Localization in a spanwise-extended model of plane Couette flow. Phys. Rev. E 91, 043005.Google Scholar
Coles, D. 1962 Interfaces and intermittency in turbulent shear flow. Mécanique de la Turbulence 108, 229–248.Google Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Couliou, M. & Monchaux, R. 2015 Large-scale flows in transitional plane Couette flow: a key ingredient of the spot growth mechanism. Phys. Fluids 27, 034101.Google Scholar
Darbyshire, A. G. & Mullin, T. 1995 Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech. 289, 83–114.Google Scholar
Dawes, J. H. P. & Giles, W. J. 2011 Turbulent transition in a truncated one-dimensional model for shear flow. Proc. R. Soc. Lond. A 467, 3066–3087.Google Scholar
Doering, C. R., Eckhardt, B. & Schumacher, J. 2003 Energy dissipation in body-forced plane shear flow. J. Fluid Mech. 494, 275–284.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duguet, Y. & Schlatter, P. 2013 Oblique laminar–turbulent interfaces in plane shear flows. Phys. Rev. Lett. 110, 034502.CrossRefGoogle ScholarPubMed
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
Gibson, J. F.2014 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep., University of New Hampshire. Channelflow.org.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
Lagha, M. 2007 Turbulent spots and waves in a model for plane Poiseuille flow. Phys. Fluids 19, 124103.Google Scholar
Lagha, M. & Manneville, P. 2007a Modeling of plane Couette flow. I. Large scale flow around turbulent spots. Phys. Fluids 19, 094105.Google Scholar
Lagha, M. & Manneville, P. 2007b Modeling transitional plane Couette flow. Eur. Phys. J. B 58, 433–447.Google Scholar
Lemoult, G., Gumowski, K., Aider, J.-L. & Wesfreid, J. E. 2014 Turbulent spots in channel flow: an experimental study. Eur. Phys. J. E 37, 1–11.Google ScholarPubMed
Manneville, P. 2004 Spots and turbulent domains in a model of transitional plane Couette flow. Theoret. Comput. Fluid Dyn. 18, 169–181.CrossRefGoogle Scholar
Manneville, P. 2009 Spatiotemporal perspective on the decay of turbulence in wall-bounded flows. Phys. Rev. E 79, 025301.Google Scholar
Manneville, P. 2015 On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular. Eur. J. Mech. (B/Fluids) 49, 345–362.Google Scholar
Meseguer, A., Mellibovsky, F., Avila, M. & Marques, F. 2009 Instability mechanisms and transition scenarios of spiral turbulence in Taylor–Couette flow. Phys. Rev. E 80, 046315.Google Scholar
Mizuno, Y. & Jiménez, J. 2013 Wall turbulence without walls. J. Fluid Mech. 723, 429–455.CrossRefGoogle Scholar
Moehlis, J., Faisst, H. & Eckhardt, B. 2004 A low-dimensional model for turbulent shear flows. New J. Phys. 6, 56.Google Scholar
Moehlis, J., Faisst, H. & Eckhardt, B. 2005 Periodic orbits and chaotic sets in a low-dimensional model for shear flows. SIAM J. Appl. Dyn. Syst. 4, 352–376.Google Scholar
Moxey, D. & Barkley, D. 2010 Distinct large-scale turbulent–laminar states in transitional pipe flow. Proc. Natl Acad. Sci. USA 107, 8091–8096.Google Scholar
Podvin, B. & Fraigneau, Y. 2011 Synthetic wall boundary conditions for the direct numerical simulation of wall-bounded turbulence. J. Turbul. 12, 1–26.Google Scholar
Pomeau, Y. 1986 Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23, 3–11.Google Scholar
Prigent, A., Grégoire, G., Chaté, H. & Dauchot, O. 2003 Long-wavelength modulation of turbulent shear flows. Physica D 174, 100–113.Google Scholar
Prigent, A., Grégoire, G., Chaté, H., Dauchot, O. & van Saarloos, W. 2002 Large-scale finite-wavelength modulation within turbulent shear flows. Phys. Rev. Lett. 89, 014501.Google Scholar
Schumacher, J. & Eckhardt, B. 2001 Evolution of turbulent spots in a parallel shear flow. Phys. Rev. E 63, 046307.Google Scholar
Seshasayanan, K. & Manneville, P. 2015 Laminar–turbulent patterning in wall-bounded shear flows: a Galerkin model. Fluid Dyn. Res. 47, 035512.Google Scholar
Shi, L., Avila, M. & Hof, B. 2013 Scale invariance at the onset of turbulence in Couette flow. Phys. Rev. Lett. 110, 204502.Google Scholar
Tsukahara, T., Iwamoto, K., Kawamura, H. & Takeda, T.2014 DNS of heat transfer in transitional channel flow accompanied by turbulent puff-like structure. arXiv:1406.0586.Google Scholar
Tuckerman, L. S. & Barkley, D. 2011 Patterns and dynamics in transitional plane Couette flow. Phys. Fluids 23, 041301.CrossRefGoogle Scholar
Tuckerman, L. S., Kreilos, T., Schrobsdorff, H., Schneider, T. M. & Gibson, J. F. 2014 Turbulent–laminar patterns in plane Poiseuille flow. Phys. Fluids 26, 114103.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google 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.Google Scholar