Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T16:58:58.878Z Has data issue: false hasContentIssue false
  • English
  • Français

Axisymmetric travelling waves in annular sliding Couette flow at finite and asymptotically large Reynolds number

Published online by Cambridge University Press:  27 February 2013

K. Deguchi  and
A. G. Walton
K. Deguchi*
Affiliation:
Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto, 606-8501, Japan
A. G. Walton*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The relationship between numerical finite-amplitude equilibrium solutions of the full Navier–Stokes equations and nonlinear solutions arising from a high-Reynolds-number asymptotic analysis is discussed for a Tollmien–Schlichting wave-type two-dimensional vortical flow structure. The specific flow chosen for this purpose is that which arises from the mutual axial sliding of co-axial cylinders for which nonlinear axisymmetric travelling-wave solutions have been discovered recently by Deguchi & Nagata (J. Fluid Mech., vol. 678, 2011, pp. 156–178). We continue this solution branch to a Reynolds number $R= 1{0}^{8} $ and confirm that the behaviour of its so-called lower branch solutions, which typically produce a smaller modification to the laminar state than the other solution branches, quantitatively agrees with the axisymmetric asymptotic theory developed in this paper. We further find that this asymptotic structure breaks down when the disturbance wavelength is comparable with $R$. The new structure which replaces it is investigated and the governing equations are derived and solved. The flow visualization of the resultant solutions reveals that they possess a streamwise localized structure, with the trend agreeing qualitatively with full Navier–Stokes solutions for relatively long-wavelength disturbances.

JFM classification

Instability: Nonlinear instability Nonlinear Dynamical Systems: Bifurcation Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 720 , 10 April 2013 , pp. 582 - 617
Copyright
©2013 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

Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181–204.Google Scholar
Bodonyi, R. J., Smith, F. T. & Gajjar, J. 1983 Amplitude-dependent stability of boundary layer flow with a strongly nonlinear critical layer. IMA J. Appl. Maths 30, 1–19.Google Scholar
Brown, S. N. & Stewartson, K. 1978 The evolution of a small inviscid disturbance to a marginally unstable stratified shear flow; stage two. Proc. R. Soc. Lond. A 363, 174–194.Google Scholar
Cherhabili, A. & Ehrenstein, U. 1995 Spatially localized two-dimensional equilibrium states in plane Couette flow. Eur. J. Mech. (B/Fluids) 14, 677–696.Google Scholar
Cipolla, K. M. & Keith, W. L. 2003 High Reynolds number thick axisymmetric turbulent boundary layer measurements. Exp. Fluids 35, 477–485.Google Scholar
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.Google Scholar
Cowley, S. J. & Smith, F. T. 1985 On the stability of Poiseuille–Couette flow: a bifurcation of infinity. J. Fluid Mech. 156, 83–100.Google Scholar
Davis, R. E. 1969 On the high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36, 337–346.CrossRefGoogle Scholar
Davies, S. J. & White, C. M. 1928 An experimental study of the flow of water in pipes of rectangular section. Proc. R. Soc. Lond. A 119, 92–107.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. (in press).Google Scholar
Deguchi, K. & Nagata, M. 2011 Bifurcations and instabilities in sliding Couette flow. J. Fluid Mech. 678, 156–178.Google Scholar
Ehrenstein, U., Nagata, M. & Rincon, F. 2008 Two-dimensional nonlinear plane Poiseuille–Couette flow homotopy revisited. Phys. Fluids 20, 064103.Google Scholar
Frei, Ch., Lüscher, P & Wintermantel, E. 2000 Thread-annular flow in vertical pipes. J. Fluid Mech. 410, 185–210.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Gittler, Ph. 1993 Stability of Poiseuille–Couette flow between concentric cylinders. Acta Mechanica 101, 1–13.Google Scholar
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139–161.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
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.Google Scholar
Healey, J. J. 1995 On the neutral curve of the flat-plate boundary layer: comparison between experiment, Orr–Sommerfeld theory and asymptotic theory. J. Fluid Mech. 288, 59–73.CrossRefGoogle Scholar
Henningson, D., Spalart, P. & Kim, J. 1987 Numerical simulations of turbulent spots in plane Poiseuille and boundary-layer flow. Phys. Fluids 30 (10), 2914–2917.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716.Google Scholar
Joseph, D. D. & Carmi, S. 1969 Stability of Poiseuille flow in pipes, annuli, and channels. Q. Appl. Maths 26, 575–599.CrossRefGoogle Scholar
Lemoult, G., Aider, J. & Wesfreid, 2012 Experimental scaling law for the subcritical transition to turbulence in plane Poiseuille flow. Phys. Rev. E 85, 025303(R).Google Scholar
Mehta, P. G. 2004 A unified well-posed computational approach for 2D Orr–Sommerfeld problem. J. Comput. Phys. 199 (2), 541–557.CrossRefGoogle Scholar
Mehta, P. G. & Healey, T. J. 2005 On steady solutions of symmetry-preserving perturbations of the two-dimensional Couette flow problem. Phys. Fluids 17, 094108.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 Couette flow. J. Fluid Mech. 72 (4), 731–751.Google Scholar
Orszag, S. A. & Kells, L. C. 1980 Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech. 72 (4), 731–751.Google Scholar
Rincon, F. 2007 On the existence of two-dimensional nonlinear steady states in plane Couette flow. Phys. Fluids 19, 074105.CrossRefGoogle Scholar
Schlichting, H. 1933 Laminare Strahlenausbreitung. Z. Angew. Math. Mech. 13, 260–263.Google Scholar
Shands, J., Alfredsson, H. & Lindgren, E. R. 1980 Annular pipe flow subject to axial motion of the inner boundary. Phys. Fluids 23 (10), 2144–2145.CrossRefGoogle Scholar
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101–174104.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1982a Amplitude-dependent neutral modes in the Hagen–Poiseuille flow through a circular pipe. Proc. R. Soc. Lond. A 384, 463–489.Google Scholar
Smith, F. T. & Bodonyi, R. J. 1982b Nonlinear critical layers and their development in streaming-flow instability. J. Fluid Mech. 118, 165–185.Google Scholar
Tollmien, W. 1929 Uber die Entstehung der Turbulenz. Nachr. Ges. Wiss. Gottingen Math-Phys. Kl. II 21–44.Google Scholar
Tsukahara, T., Seki, S., Kawamura, H. & Tochio, D. 2005 DNS of turbulent channel flow at very low Reynolds numbers. In Proceedings of the Fourth International Symposium on Turbulence and Shear Flow Phenomena (ed. J. A. C. Humphrey et al.), pp. 935–940.Google Scholar
Tutty, O. R. 2008 Flow along a long thin cylinder. J. Fluid Mech. 602, 1–37.Google Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81 (19), 4140–4143.Google Scholar
Walton, A. G. 2002 The temporal evolution of neutral modes in the impulsively started flow through a pipe of circular cross-section. J. Fluid Mech. 457, 339–376.CrossRefGoogle Scholar
Walton, A. G. 2003 The nonlinear instability of thread-annular flow at high Reynolds number. J. Fluid Mech. 477, 227–257.Google Scholar
Walton, A. G. 2004 Stability of circular Poiseuille–Couette flow to axisymmetric disturbances. J. Fluid Mech. 500, 169–210.Google Scholar
Walton, A. G. 2005 The linear and nonlinear stability of thread-annular flow. Phil. Trans. R. Soc. A 363, 1223–1233.Google Scholar
Walton, A. G. 2011 The stability of developing pipe flow at high Reynolds number and the existence of nonlinear neutral centre modes. J. Fluid Mech. 684, 284–315.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Wong, A. W. H. & Walton, A. G. 2012 Axisymmetric travelling waves in annular Couette–Poiseuille flow. Q. J. Mech. Appl. Maths 65, 293–311.Google Scholar