Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-04T19:46:21.035Z Has data issue: false hasContentIssue false
  • English
  • Français

Equilibrium and travelling-wave solutions of plane Couette flow

Published online by Cambridge University Press:  29 September 2009

J. F. GIBSON ,
J. HALCROW  and
P. CVITANOVIĆ
J. F. GIBSON*
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
J. HALCROW
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
P. CVITANOVIĆ
Affiliation:
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
†Email address for correspondence: gibson@cns.physics.gatech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new travelling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 638 , 10 November 2009 , pp. 243 - 266
Copyright
Copyright © Cambridge University Press 2009

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

Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Cherhabili, A. & Ehrenstein, U. 1997 Finite-amplitude equilibrium states in plane Couette flow. J. Fluid Mech. 342, 159–177.CrossRefGoogle Scholar
Christiansen, F., Cvitanović, P. & Putkaradze, V. 1997 Spatio-temporal chaos in terms of unstable recurrent patterns. Nonlinearity 10, 55–70.CrossRefGoogle Scholar
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal layer subjected to constant shear. J. Fluid Mech. 234, 511–527.CrossRefGoogle Scholar
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.CrossRefGoogle Scholar
Cvitanović, P., Davidchack, R. L. & Siminos, E. 2009 On state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain. SIAM J. Appl. Dynam. Systems. To appear. arXiv:0709.2944.CrossRefGoogle Scholar
Dennis, J. E. Jr., & Schnabel, R. B. 1996 Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM.CrossRefGoogle Scholar
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102, arXiv:0807.2580.CrossRefGoogle Scholar
Ehrenstein, U., Nagata, M. & Rincon, F. 2008 Two-dimensional nonlinear plane Poiseuille-Couette flow homotopy revisited. Phys. Fluids 20, 064103-1–064103-4.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Frisch, U. 1996 Turbulence. Cambridge University Press.Google Scholar
Gibson, J. F. 2008 a Channelflow: a spectral Navier–Stokes simulator in C++. Tech Rep. Georgia Institute of Technology. http://www.Channelflow.org.Google Scholar
Gibson, J. F. 2008 b Movies of plane Couette. Tech Rep. Georgia Institute of Technology. http://www.ChaosBook.org/tutorials.Google Scholar
Gibson, J. F. & Cvitanović, P. 2009 Periodic orbits of plane Couette flow. http://www.channelflow.org/database.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state-space in plane Couette flow. J. Fluid Mech. 611, 107–130. arXiv:0705.3957.CrossRefGoogle Scholar
Gilmore, R. & Letellier, C. 2007 The Symmetry of Chaos. Oxford University Press.CrossRefGoogle Scholar
Golubitsky, M. & Stewart, I. 2002 The Symmetry Perspective. Birkhäuser.CrossRefGoogle Scholar
Halcrow, J. 2008 Geometry of turbulence: An exploration of the state-space of plane Couette flow. PhD thesis, School of Physics, Georgia Institute of Technology, Atlanta, GA. http://www.ChaosBook.org/projects/theses.html.Google Scholar
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365–376. arXiv:0808.1865.CrossRefGoogle Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Harter, W. G. 1993 Principles of Symmetry, Dynamics, and Spectroscopy. Wiley.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. http://www.sciencemag.org/cgi/reprint/305/5690/1594.pdf.CrossRefGoogle ScholarPubMed
Hoyle, R. 2006 Pattern Formation: An Introduction to Methods. Cambridge University Press.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.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Jpn 70, 701–714.CrossRefGoogle Scholar
Jiménez, J., Kawahara, G., Simens, M. P., Nagata, M. & Shiba, M. 2005 Characterization of near-wall turbulence in terms of equilibrium and bursting solutions. Phys. Fluids 17, 015105.CrossRefGoogle Scholar
Kim, H., Kline, S. & Reynolds, W. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133–160.CrossRefGoogle Scholar
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows. In Proceedings of the Third GAMM Conference on Numerical Methods in Fluid Mechanics (ed. E. Hirschel), pp. 165–173. GAMM, Vieweg.CrossRefGoogle Scholar
Lan, Y. & Cvitanović, P. 2008 Unstable recurrent patterns in Kuramoto–Sivashinsky dynamics. Phys. Rev. E 78, 026208. arXiv.org:0804.2474.CrossRefGoogle ScholarPubMed
Marsden, J. E. & Ratiu, T. S. 1999 Introduction to Mechanics and Symmetry. Springer.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
Nagata, M. 1997 Three-dimensional travelling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023–2025.CrossRefGoogle Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Flows. Springer.CrossRefGoogle Scholar
Pringle, C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.CrossRefGoogle ScholarPubMed
Rincon, F. 2007 On the existence of two-dimensional nonlinear steady states in plane Couette flow. Phys. Fluids 19, 4105. arXiv:0706.1165.CrossRefGoogle Scholar
Schmiegel, A. 1999 Transition to turbulence in linearly stable shear flows. PhD thesis, Philipps-Universität Marburg, Maburg, Germany. archiv.ub.uni-marburg.de/diss/z2000/0062.Google Scholar
Schneider, T., Gibson, J., Lagha, M., Lillo, F. D. & Eckhardt, B. 2008 Laminar–turbulent boundary in plane Couette flow. Phys. Rev. E. 78, 037301. arXiv:0805.1015.CrossRefGoogle ScholarPubMed
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.CrossRefGoogle Scholar
Tuckerman, L. S. & Barkley, D. 2002 Symmetry breaking and turbulence in perturbed plane Couette flow. Theoret. Comput. Fluid Dyn. 16, 91–97. arXiv:physics/0312051.CrossRefGoogle Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358. arXiv:physics/0604062.CrossRefGoogle Scholar
Viswanath, D. 2008 The dynamics of transition to turbulence in plane Couette flow. In Mathematics and Computation, a Contemporary View. The Abel Symposium 2006, Abel Symposia, vol. 3. (ed Munthe-Kaas, H. and Owren, B.) Springer. arXiv:physics/0701337.Google Scholar
Waleffe, F. 1990 Proposal for a self-sustaining mechanism in shear flows. Unpublished Preprint. Center for Turbulence Research, Stanford University/NASA Ames.Google Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Math. 95, 319–343.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.CrossRefGoogle Scholar
Waleffe, F. 2002 Exact coherent structures and their instabilities: Toward a dynamical-system theory of shear turbulence. In Proceedings of the International Symposium on ‘Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices’ (ed. S. Kida), pp. 115–128. National Center of Sciences.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1543.CrossRefGoogle Scholar
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.CrossRefGoogle ScholarPubMed
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar