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A mechanism for streamwise localisation of nonlinear waves in shear flows

Published online by Cambridge University Press:  17 August 2015

Fernando Mellibovsky  and
Alvaro Meseguer
Fernando Mellibovsky*
Affiliation:
Castelldefels School of Telecom and Aerospace Engineering, Universitat Politècnica de Catalunya, 08034, Barcelona, Spain
Alvaro Meseguer
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, 08034, Barcelona, Spain
*
†Email address for correspondence: fernando.mellibovsky@upc.edu
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Abstract

We present the complete unfolding of streamwise localisation in a paradigm of extended shear flows, namely two-dimensional plane Poiseuille flow. Exact solutions of the Navier–Stokes equations are computed numerically and tracked in the streamwise wavenumber–Reynolds number parameter space to identify and describe the fundamental mechanism behind streamwise localisation, a ubiquitous feature of shear flow turbulence. Unlike shear flow spanwise localisation, streamwise localisation does not follow the snaking mechanism demonstrated for plane Couette flow.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Instability: Instability Turbulent Flows: Turbulent transition
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 779 , 25 September 2015 , R1
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Copyright
© 2015 Cambridge University Press 

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References

Avila, K., Moxey, D., de Lozar, A., Avila, M. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192–196.Google Scholar
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
Barkley, D. 1990 Theory and predictions for finite-amplitude waves in 2-dimensional plane Poiseuille flow. Phys. Fluids A 2 (6), 955–970.Google Scholar
Beaume, C., Bergeon, A., Kao, H. S. & Knobloch, E. 2013a Convectons in a rotating fluid layer. J. Fluid Mech. 717, 417–448.Google Scholar
Beaume, C., Knobloch, E. & Bergeon, A. 2013b Nonsnaking doubly diffusive convectons and the twist instability. Phys. Fluids 25 (11), 114102.Google Scholar
Bergeon, A., Burke, J., Knobloch, E. & Mercader, I. 2008 Eckhaus instability and homoclinic snaking. Phys. Rev. E 78 (4), 046201.Google Scholar
Brand, E. & Gibson, J. F. 2014 A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3,1–12.Google Scholar
Budd, C. J. & Kuske, R. 2005 Localized periodic patterns for the generalized Swift–Hohenberg equation. Physica D 208, 73–95.Google Scholar
Burke, J., Houghton, S. M. & Knobloch, E. 2009 Swift–Hohenberg equation with broken reflection symmetry. Phys. Rev. E 80 (3), 036202.CrossRefGoogle ScholarPubMed
Burke, J. & Knobloch, E. 2006 Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73 (5), 056211.Google Scholar
Burke, J. & Knobloch, E. 2007a Homoclinic snaking: structure and stability. Chaos 17 (3), 037102.CrossRefGoogle ScholarPubMed
Burke, J. & Knobloch, E. 2007b Snakes and ladders: localized states in the Swift–Hohenberg equation. Phys. Lett. A 360 (6), 681–688.CrossRefGoogle Scholar
Chantry, M. & Kerswell, R. R. 2015 Localization in a spanwise-extended model of plane Couette flow. Phys. Rev. E 91 (4), 043005.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
Chen, T. S. & Joseph, D. D. 1973 Subcritical bifurcation of plane Poiseuille flow. J. Fluid Mech. 58, 337–351.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851–1112.Google Scholar
Dawes, J. H. P. 2008 Localized pattern formation with large-scale mode: slanted snaking. SIAM J. Appl. Dyn. Syst. 7 (1), 186–206.Google Scholar
Drissi, A., Net, M. & Mercader, I. 1999 Subharmonic instabilities of Tollmien–Schlichting waves in two-dimensional Poiseuille flow. Phys. Rev. E 60 (2), 1781–1791.Google Scholar
Duguet, Y., Schlatter, P. & Henningson, D. S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21 (11), 111701.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., 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. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.CrossRefGoogle ScholarPubMed
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.CrossRefGoogle Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.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 (5690), 1594–1598.Google Scholar
Itano, T. & Toh, S. 2000 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.Google Scholar
Jimenez, J. 1990 Transition to turbulence in 2-dimensional Poiseuille flow. J. Fluid Mech. 218, 265–297.CrossRefGoogle Scholar
Kelley, C. T. 1995 Iterative Methods for Linear and Nonlinear Equations. SIAM.Google Scholar
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.Google Scholar
Kirchgässner, K. 1982 Wave solutions of reversible problems and solutions. J. Differ. Equ. 45 (1), 113–127.Google Scholar
Kreilos, T., Veble, G., Schneider, T. M. & Eckhardt, B. 2013 Edge states for the turbulence transition in the asymptotic suction boundary layer. J. Fluid Mech. 726, 100–122.Google Scholar
Kuznetsov, Y. A. 2004 Elements of Applied Bifurcation Theory, 3rd edn. Springer.Google Scholar
Mellibovsky, F. & Eckhardt, B. 2012 From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow. J. Fluid Mech. 709, 149–190.Google Scholar
Mellibovsky, F. & Meseguer, A. 2007 Pipe flow transition threshold following localized impulsive perturbations. Phys. Fluids 19 (4), 044102.Google Scholar
Mellibovsky, F., Meseguer, A., Schneider, T. S. & Eckhardt, B. 2009 Transition in localized pipe flow turbulence. Phys. Rev. Lett. 103 (5), 054502.Google Scholar
Melnikov, K., Kreilos, T. & Eckhardt, B. 2014 Long-wavelength instability of coherent structures in plane Couette flow. Phys. Rev. E 89 (4), 043008.Google Scholar
Mercader, I., Batiste, O., Alonso, A. & Knobloch, E. 2009 Localized pinning states in closed containers: homoclinic snaking without bistability. Phys. Rev. E 80 (2), 025201.Google Scholar
Nagata, M. 1990 3-dimensional finite-amplitude solutions in plane Couette-flow – bifurcation from infinity. J. Fluid Mech. 217, 519–527.Google Scholar
Orszag, S. A. 1971 Numerical simulation of incompressible flows with simple boundaries – accuracy. J. Fluid Mech. 49, 75–112.Google Scholar
Prat, J., Mercader, I. & Knobloch, E. 1998 Resonant mode interactions in Rayleigh–Bénard convection. Phys. Rev. E 58 (3), 3145–3156.Google Scholar
Pugh, J. D. & Saffman, P. G. 1988 Two-dimensional superharmonic stability of finite-amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295–307.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935–982.Google Scholar
Schneider, T. M., Gibson, J. F. & Burke, J. 2010 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
Soibelman, I.1989 A study of finite amplitude bifurcations in plane Poiseuille flow. PhD thesis, California Institute of Technology, Pasadena, CA, USA.Google Scholar
Soibelman, I. & Meiron, D. I. 1991 Finite-amplitude bifurcations in plane Poiseuille flow – 2-dimensional Hopf-bifurcation. J. Fluid Mech. 229, 389–416.Google Scholar
Teramura, T. & Toh, S.2015 Chaotic self-sustaining structure embedded in turbulent–laminar interface. arXiv:1503.04551.Google Scholar
Trefethen, L. N. & Bau, D. 1997 Numerical Linear Algebra. SIAM.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
Wygnanski, I. J. & Champagne, F. H. 1973 On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech. 59, 281–335.Google Scholar
Zahn, J. P., Toomre, J., Spiegel, E. A. & Gough, D. O. 1974 Nonlinear cellular motions in Poiseuille channel flow. J. Fluid Mech. 64, 319–345.Google Scholar
Zammert, S. & Eckhardt, B. 2014 Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348–359.CrossRefGoogle Scholar

Mellibovsky et al. supplementary movie

Tollmien Schlichting wave at Re=2800.

Download Mellibovsky et al. supplementary movie(Video)
Video 16.2 MB

Mellibovsky et al. supplementary movie

Tollmien Schlichting wave at Re=2800.

Download Mellibovsky et al. supplementary movie(Video)
Video 1.2 MB

Mellibovsky et al. supplementary movie

Modulated Tollmien Schlichting wave at Re=2800

Download Mellibovsky et al. supplementary movie(Video)
Video 17.7 MB

Mellibovsky et al. supplementary movie

Modulated Tollmien Schlichting wave at Re=2800

Download Mellibovsky et al. supplementary movie(Video)
Video 1.3 MB

Mellibovsky et al. supplementary movie

Localised modulated Tollmien Schlichting wave at Re=2800.

Download Mellibovsky et al. supplementary movie(Video)
Video 11.4 MB

Mellibovsky et al. supplementary movie

Localised modulated Tollmien Schlichting wave at Re=2800.

Download Mellibovsky et al. supplementary movie(Video)
Video 932.1 KB