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Doubly localized states in plane Couette flow

Published online by Cambridge University Press:  07 October 2014

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]
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Abstract

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Much of our understanding of the transition to turbulence in flows without a linear instability came with the discovery and characterization of fully three-dimensional solutions to the Navier–Stokes equation. The first examples in plane Couette flow were periodic in both spanwise and streamwise directions, and could explain the transitions in small domains only. The presence of localized turbulent spots in larger domains, the spatiotemporal decoherence on larger scales and the ability to trigger turbulence with pointwise perturbations require solutions that are localized in both directions, like the one presented by Brand & Gibson (J. Fluid Mech., vol. 750, 2014, R3). They describe a steady solution of the Navier–Stokes equations and characterize in unprecedented detail, including an analytic computation of its localization properties. The study opens up new ways to describe localized turbulent patches.

Type
Focus on Fluids
Copyright
© 2014 Cambridge University Press 

References

Barkley, D. & Tuckerman, L. 2005 Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502.Google Scholar
Brand, E. & Gibson, J. F. 2014 A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3.Google Scholar
Chantry, M., Willis, A. P. & Kerswell, R. R. 2014 Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow. Phys. Rev. Lett. 112, 164501.Google Scholar
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137153.CrossRefGoogle Scholar
Cvitanović, P. 2013 Recurrent flows: the clockwork behind turbulence. J. Fluid Mech. 726, 14.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, 119129.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.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 2561.CrossRefGoogle Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2010 Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717, R6.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.Google Scholar
Lundbladh, A. & Johansson, A. V. 1991 Direct simulation of turbulent spots in plane Couette flow. J. Fluid Mech. 229, 499516.CrossRefGoogle Scholar
Melnikov, K., Kreilos, T. & Eckhardt, B. 2014 Long-wavelength instability of coherent structures in plane Couette flow. Phys. Rev. E 89, 043008.Google Scholar
Mullin, T. 2011 Experimental studies of transition to turbulence in a pipe. Annu. Rev. Fluid Mech. 43, 124.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.Google Scholar
Schneider, T. M., Gibson, J. F. & Burke, J. 2010a Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.Google Scholar
Schneider, T. M., Marinc, D. & Eckhardt, B. 2010b Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441451.Google Scholar
Teramura, T. & Toh, S. 2014 Damping filter method for obtaining spatially localized solutions. Phys. Rev. E 89, 052910.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 15171534.Google Scholar