Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-19T11:30:18.463Z Has data issue: false hasContentIssue false
  • English
  • Français

Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures

Published online by Cambridge University Press:  16 August 2010

PHILIP HALL  and
SPENCER SHERWIN
PHILIP HALL*
Affiliation:
Institute of Mathematical Sciences, Imperial College London, London SW7 2AZ, UK
SPENCER SHERWIN
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: philhall@ic.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The relationship between asymptotic descriptions of vortex–wave interactions and more recent work on ‘exact coherent structures’ is investigated. In recent years immense interest has been focused on so-called self-sustained processes in turbulent shear flows where the importance of waves interacting with streamwise vortex flows has been elucidated in a number of papers. In this paper, it is shown that the so-called ‘lower branch’ state which has been shown to play a crucial role in these self-sustained processes is a finite Reynolds number analogue of a Rayleigh vortex–wave interaction with scales appropriately modified from those for external flows to Couette flow, the flow of interest here. Remarkable agreement between the asymptotic theory and numerical solutions of the Navier–Stokes equations is found even down to relatively small Reynolds numbers, thereby suggesting the possible importance of vortex–wave interaction theory in turbulent shear flows. The relevance of the work to more general shear flows is also discussed.

JFM classification

Flow Control: Control theory Instability: Nonlinear instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 661 , 25 October 2010 , pp. 178 - 205
Copyright
Copyright © Cambridge University Press 2010

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.)

References

REFERENCES

Ainsworth, M. & Sherwin, S. 1999 Domain decomposition preconditions for p and hp finite element approximation of Stokes equations. Comp. Meth. Appl. Mech. Engng 175, 243–266.Google Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 1435–1450.Google Scholar
Bassom, A. & Hall, P. 1989 On the generation of mean flows by the interaction of Görtler vortices and Tollmien–Schlichting waves in curved channel flows. Stud. Appl. Math. 81, 185–219.Google Scholar
Bennett, J., Hall, P. & Smith, F. 1991 The strong nonlinear interaction of Tollmien–Schlichting waves and Taylor–Görtler vortices in curved channel flow. J. Fluid Mech. 223, 475–495.CrossRefGoogle Scholar
Benney, D. 1984 The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Math. 70, 1–19.CrossRefGoogle Scholar
Blackaby, N. 1991 On viscous, inviscid and centrifugal instability mechanisms in compressible boundary layers, including non-linear vortex/wave interactions and the effects of large Mach number on transition. PhD thesis, University of London.Google Scholar
Blackaby, N. & Hall, P. 1994 On the nonlinear secondary instability of a streamwise vortex flow. Phil. Trans. Soc. A 352, 1700.Google Scholar
Brown, P., Brown, S., Smith, F. & Timoshin, S. 1993 On the starting process of strongly nonlinear vortex/Rayleigh-wave interactions. Mathematika 40 (1), 7–29.Google Scholar
Chapman, S. J. 2002 On the subcritical instability of channel flows. J. Fluid Mech. 451, 35–97.Google Scholar
Clever, R. & Busse, F. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.CrossRefGoogle Scholar
Davey, A., DiPrima, R. & Stuart, J. 1968 On the instability of Taylor vortices. J. Fluid Mech. 31 (01), 17–52.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.CrossRefGoogle ScholarPubMed
Fitzgerald, R. 2004 New experiments set the scale for the onset of turbulence in pipe flow. Phys. Today 57 (2), 21–24.Google Scholar
Guermond, J. & Shen, J. 2003 Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41, 112–134.Google Scholar
Hall, P. 1994 On the initial stages of vortex wave interactions in highly curved boundary layer flows. Mathematika 41 (1), 40–67.Google Scholar
Hall, P. & Horseman, N. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.CrossRefGoogle Scholar
Hall, P. & Morris, H. 1992 On the instability of boundary layers on heated flat plates. J. Fluid Mech. 245, 367–400.Google Scholar
Hall, P. & Smith, F. 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. 1989 Nonlinear a Tollmien–Schlichting/vortex interaction in boundary layers. Eur. J. Mech. B/Fluids 8 (3), 179–205.Google Scholar
Hall, P. & Smith, F. 1990 Near Planar TS Waves and Longitudinal Vortices in Channel Flow: Nonlinear Interaction and Focussing, pp. 5–39. Springer.Google Scholar
Hall, P. & Smith, F. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.Google Scholar
Hof, B., van Doorne, C., Westerweel, J., Nieuwstadt, F., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305 (5690), 1594–1598.Google Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414–443.Google Scholar
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn.Oxford University Press.Google Scholar
Kerswell, R. & Tutty, O. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.Google Scholar
LeTallec, P. & Patra, A. 1997 Non-overlapping domain decomposition methods for adaptive hp approximation of the Stokes problem with discontinuous pressure fields. Comp. Meth. Appl. Mech. Engng 145, 361–379.Google Scholar
Li, F. & Malik, M. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.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
Riley, N. 1967 Oscillatory viscous flows: Review and extension. J. Inst. Math. Appl. 3, 419–434.Google Scholar
Schneider, T., Gibson, J., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar–turbulent boundary in plane Couette flow. Phys. Rev. E, 037301.Google Scholar
Stuart, J. T. 1966 Double boundary layers in oscillatory flow. J. Fluid Mech. 24, 673–687.Google Scholar
Swearingen, J. & Blackwelder, R. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290.Google Scholar
Taylor, G. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A A223, 289–343.Google Scholar
Timoshin, S. & Smith, F. 1997 Vortex/inflectional-wave interactions with weakly three-dimensional input. J. Fluid Mech. 348, 247–294.Google Scholar
Tuckerman, L. S. & Barkley, D. 2000 Bifurcation analysis for timesteppers. In Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (ed. Doedel, E. & Tuckerman, L. S.), pp. 453–566. Springer.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Math. 95 (3), 319–343.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.Google Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81 (19), 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 (6), 1517–1534.Google Scholar
Walton, A. G. & Smith, F. 1992 Properties of strongly nonlinear vortex/Tollmien–Schlichting-wave interactions. J. Fluid Mech. 244, 649–676.CrossRefGoogle Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.Google Scholar
Wedin, H. & Kerswell, R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Yu, X. & Liu, J. 1991 The secondary instability in Goertler flow. Phys. Fluids 3 (8), 1845–1847.CrossRefGoogle Scholar