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Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

Published online by Cambridge University Press:  15 February 2016

L. J. Dempsey ,
K. Deguchi ,
P. Hall  and
A. G. Walton
L. J. Dempsey
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
K. Deguchi
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
P. Hall
Affiliation:
School of Mathematical Sciences, Monash University, Victoria 3800, Australia
A. G. Walton*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
†Email address for correspondence: a.walton@imperial.ac.uk
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Abstract

Strongly nonlinear three-dimensional interactions between a roll–streak structure and a Tollmien–Schlichting wave in plane Poiseuille flow are considered in this study. Equations governing the interaction at high Reynolds number originally derived by Bennett et al. (J. Fluid Mech., vol. 223, 1991, pp. 475–495) are solved numerically. Travelling wave states bifurcating from the lower branch linear neutral point are tracked to finite amplitudes, where they are observed to localize in the spanwise direction. The nature of the localization is analysed in detail near the relevant spanwise locations, revealing the presence of a singularity which slowly develops in the governing interaction equations as the amplitude of the motion is increased. Comparisons with the full Navier–Stokes equations demonstrate that the finite Reynolds number solutions gradually approach the numerical asymptotic solutions with increasing Reynolds number.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Instability: Nonlinear instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 791 , 25 March 2016 , pp. 97 - 121
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Copyright
© 2016 Cambridge University Press 

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References

Alavyoon, F., Henningson, D. S. & Alfredsson, P. H. 1986 Turbulent spots in plane Poiseuille flow–flow visualization. Phys. Fluids 29, 1328–1331.Google Scholar
Bennett, J., Hall, P. & Smith, F. T. 1991 The strong nonlinear interaction of Tollmien–Schlichting waves and Taylor–Görtler vortices in curved channel flow. J. Fluid Mech. 223, 475–495.Google Scholar
Benney, D. J. 1984 The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Maths 70, 1–19.Google Scholar
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234, 511–527.Google Scholar
Deguchi, K. & Hall, P. 2014 The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99–112.Google Scholar
Deguchi, K., Hall, P. & Walton, A. G. 2013 The emergence of localized vortex–wave interaction states in plane Couette flow. J. Fluid Mech. 721, 58–85.Google Scholar
Ehrenstein, U. & Koch, W. 1991 Three-dimensional wavelike equilibrium states in plane Poiseuille flow. J. Fluid Mech. 228, 111–148.Google Scholar
Emmons, H. W. 1951 The laminar-turbulent transition in a boundary layer. Part 1. J. Aero. Sci. 18, 490–498.Google Scholar
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.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
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
Hall, P. & Smith, F. T. 1988 The nonlinear interaction of Tollmien–Schlichting waves and Taylor–Görtler vortices in curved channel flows. Proc. R. Soc. Lond. A 417, 255–282.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.Google Scholar
Henningson, D. S., Spalart, P. & Kim, J. 1987 Numerical simulations of turbulent spots in plane Poiseuille flow and boundary-layer flow. Phys. Fluids 30, 2914–2917.Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 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.CrossRefGoogle 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
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34.CrossRefGoogle Scholar
Lin, C. C. 1945 On the stability of two-dimensional parallel flows. Part III – Stability in a viscous fluid. Q. Appl. Maths 3, 277–301.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
Nishioka, M., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731–751.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Schmid, P. T. & Henningson, D. S. 1992 A new mechanism for rapid transition involving a pair of oblique waves. Phys. Fluids A4, 1986–1989.Google Scholar
Schneider, T. M., Gibson, J. F. & Burke, J. 2010 Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.Google Scholar
Smith, F. T. 1979 Instability of flow through pipes of general cross-section, Part 1. Mathematika 26, 187–210.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Walton, A. G. & Patel, R. A. 1999 Singularity formation in the strongly nonlinear wide-vortex/Tollmien–Schlichting-wave interaction equations. J. Fluid Mech. 400, 265–293.Google Scholar
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 204501.Google Scholar