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Homoclinic snaking in plane Couette flow: bending, skewing and finite-size effects

Published online by Cambridge University Press:  06 April 2016

J. F. Gibson  and
T. M. Schneider
J. F. Gibson*
Affiliation:
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA
T. M. Schneider
Affiliation:
Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
*
†Email address for correspondence: john.gibson@unh.edu
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Abstract

Invariant solutions of shear flows have recently been extended from spatially periodic solutions in minimal flow units to spatially localized solutions on extended domains. One set of spanwise-localized solutions of plane Couette flow exhibits homoclinic snaking, a process by which steady-state solutions grow additional structure smoothly at their fronts when continued parametrically. Homoclinic snaking is well understood mathematically in the context of the one-dimensional Swift–Hohenberg equation. Consequently, the snaking solutions of plane Couette flow form a promising connection between the largely phenomenological study of laminar–turbulent patterns in viscous shear flows and the mathematically well-developed field of pattern-formation theory. In this paper we present a numerical study of the snaking solutions of plane Couette flow, generalizing beyond the fixed streamwise wavelength of previous studies. We find a number of new solution features, including bending, skewing and finite-size effects. We establish the parameter regions over which snaking occurs and show that the finite-size effects of the travelling wave solution are due to a coupling between its fronts and interior that results from its shift-reflect symmetry. A new winding solution of plane Couette flow is derived from a strongly skewed localized equilibrium.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Nonlinear Dynamical Systems: Pattern formation Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 794 , 10 May 2016 , pp. 530 - 551
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Copyright
© 2016 Cambridge University Press 

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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, 224502.Google Scholar
Barkley, D. & Tuckerman, L. S. 2005 Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502.CrossRefGoogle ScholarPubMed
Barkley, D. & Tuckerman, L. S. 2007 Mean flow of turbulent-laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137.Google Scholar
Batiste, O. & Knobloch, E. 2005 Simulations of localized states of stationary convection in He-3-He-4 mixtures. Phys. Rev. Lett. 95 (24), 244501.CrossRefGoogle ScholarPubMed
Batiste, O., Knobloch, E., Alonso, A. & Mercader, I. 2006 Spatially localized binary-fluid convection. J. Fluid Mech. 560, 149–158.CrossRefGoogle 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 (4), 043010.Google Scholar
Beck, M., Knobloch, J., Lloyd, D. J. B., Sandstede, B. & Wagenknecht, T. 2009 Snakes, ladders, and isolas of localized patterns. SIAM J. Math. Anal. 41, 936–972.Google Scholar
Brand, E. & Gibson, J. F. 2014 A doubly-localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3.Google Scholar
Burke, J. & Knobloch, E. 2006 Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73, 056211.Google ScholarPubMed
Burke, J. & Knobloch, E. 2007a Homoclinic snaking: structure and stability. Chaos 17 (3), 037102.Google Scholar
Burke, J. & Knobloch, E. 2007b Snakes and ladders: localized states in the Swift–Hohenberg equation. Phys. Lett. A 360, 681–688.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 (16), 164501.CrossRefGoogle ScholarPubMed
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.Google Scholar
Deguchi, K., Hall, P. & Walton, A. 2013 The emergence of localized vortex-wave interaction states in plane Couette flow. J. Fluid Mech. 721, 58–85.Google Scholar
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102.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
Dummit, D. S. & Foote, R. M. 2004 Abstract Algebra. Wiley.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.Google Scholar
Gibson, J. F. & Brand, E. 2014 Spatially localized solutions of planar shear flow. J. Fluid Mech. 745, 25–61.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.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and traveling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 1–24.Google Scholar
Hall, P. 2012 Vortex–wave interactions: long-wavelength streaks and spatial localization in natural convection. J. Fluid Mech. 703, 99–110.Google Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.Google Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Kao, H. C., Beaume, C. & Knobloch, E. 2014 Spatial localization in heterogeneous systems. Phys. Rev. E 89, 012903.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, 203–225.Google Scholar
Knobloch, E. 2015 Spatial localization in dissipative systems. Annu. Rev. Condens. Matter Phys. 6, 325–359.CrossRefGoogle Scholar
Mellibovsky, F. & Meseguer, A. 2015 A mechanism for localization of nonlinear waves in shear flows. J. Fluid Mech. 779, R3.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. 2006 Edge of chaos in pipe flow. Chaos 16 (4), 041103.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 Localised 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 The edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.Google 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
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.Google Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: traveling wave solutions. J. Fluid Mech. 508, 333–371.Google Scholar
Woods, P. D. & Champneys, A. R. 1999 Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Physica D 129, 147–170.Google Scholar
Zammert, S. & Eckhardt, B. 2014a Periodically bursting edge states in plane Poiseuille flow. Fluid Dyn. Res. 46 (4), 041419.Google Scholar
Zammert, S. & Eckhardt, B. 2014b Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348–359.Google Scholar