Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-04T07:44:21.100Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic descriptions of oblique coherent structures in shear flows

Published online by Cambridge University Press:  08 October 2015

Kengo Deguchi  and
Philip Hall
Kengo Deguchi*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Philip Hall*
Affiliation:
School of Mathematics, Monash University, Melbourne, VIC 3800, Australia
*
†Email addresses for correspondence: k.deguchi@imperial.ac.uk, phil.hall@monash.edu
†Email addresses for correspondence: k.deguchi@imperial.ac.uk, phil.hall@monash.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Exact coherent states in plane Couette flow are extended to an oblique parallelogram computational domain and the large-Reynolds-number asymptotic descriptions of the states are derived. When the tilt angle of the domain is increased, the states first obey vortex–wave interaction theory and then develop a new asymptotic structure. The latter asymptotic structure is characterized by a self-similar nonlinear interaction localized in the fluid layer. For the largest scale of the self-similar nonlinear structure, the theory predicts an inverse Reynolds number dependence for the tilt angle of the oblique pattern. That result is consistent with the numerical and experimental observations of the laminar–turbulent banded pattern in shear flows.

JFM classification

Instability: Instability Instability: Nonlinear instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 782 , 10 November 2015 , pp. 356 - 367
Check for updates
Copyright
© 2015 Cambridge University Press 

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

Barkley, D. & Tuckerman, L. S. 2005 Computational study of turbulent–laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502.Google Scholar
Barkley, D. & Tuckerman, L. S. 2007 Mean flow of turbulent–laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014a The high Reynolds number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99–112.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014b Canonical exact coherent structures embedded in high Reynolds number flows. Phil. Trans. R. Soc. Lond. A 372, 20130352, 1–19.Google ScholarPubMed
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
Deguchi, K. & Altmeyer, S. 2013 Fully nonlinear mode competitions of nearly bicritical spiral and Taylor vortices in Taylor–Couette flow. Phys. Rev. E 87 (4), 043017.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.CrossRefGoogle ScholarPubMed
Hall, P. 1985 The Görtler vortex mechanism in three-dimensional boundary layers. Proc. R. Soc. Lond. A 399, 135–152.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.CrossRefGoogle 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
Blackburn, H. M., Hall, P. & Sherwin, S. 2013 Lower branch equilibria in Couette flow: the emergence of canonical stats for arbitrary shear flows. J. Fluid Mech. 726, R2.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Itano, T. & Generalis, S. C. 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102, 114501.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. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 1–24.Google Scholar
Prigent, A., Greǵoire, G., Chaté, H., Dauchot, O. & van Saarloos, W. 2002 Large-scale finite-wavelength modulation within turbulent shear flows. Phys. Rev. Lett. 89, 014501.CrossRefGoogle ScholarPubMed
Taylor, G. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A A223, 289–343.Google Scholar
Tsukahara, T., Seki, Y. & Tochio, D. 2005 DNS of turbulent channel flow at very low Reynolds numbers. In Proceedings of the 4th International Symposium on Turbulence and Shear Flow Phenomena, pp. 935–940.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.Google Scholar