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Equilibrium and travelling-wave solutions of plane Couette flow

  • J. F. GIBSON (a1), J. HALCROW (a1) and P. CVITANOVIĆ (a1)

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.

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Corresponding author

Email address for correspondence: gibson@cns.physics.gatech.edu

References

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Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Cherhabili, A. & Ehrenstein, U. 1997 Finite-amplitude equilibrium states in plane Couette flow. J. Fluid Mech. 342, 159177.
Christiansen, F., Cvitanović, P. & Putkaradze, V. 1997 Spatio-temporal chaos in terms of unstable recurrent patterns. Nonlinearity 10, 5570.
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal layer subjected to constant shear. J. Fluid Mech. 234, 511527.
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137153.
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.
Dennis, J. E. Jr., & Schnabel, R. B. 1996 Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM.
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102, arXiv:0807.2580.
Ehrenstein, U., Nagata, M. & Rincon, F. 2008 Two-dimensional nonlinear plane Poiseuille-Couette flow homotopy revisited. Phys. Fluids 20, 064103-1–064103-4.
Faisst, H. & Eckhardt, B. 2003 travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
Frisch, U. 1996 Turbulence. Cambridge University Press.
Gibson, J. F. 2008 a Channelflow: a spectral Navier–Stokes simulator in C++. Tech Rep. Georgia Institute of Technology. http://www.Channelflow.org.
Gibson, J. F. 2008 b Movies of plane Couette. Tech Rep. Georgia Institute of Technology. http://www.ChaosBook.org/tutorials.
Gibson, J. F. & Cvitanović, P. 2009 Periodic orbits of plane Couette flow. http://www.channelflow.org/database.
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state-space in plane Couette flow. J. Fluid Mech. 611, 107130. arXiv:0705.3957.
Gilmore, R. & Letellier, C. 2007 The Symmetry of Chaos. Oxford University Press.
Golubitsky, M. & Stewart, I. 2002 The Symmetry Perspective. Birkhäuser.
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.
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365376. arXiv:0808.1865.
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.
Harter, W. G. 1993 Principles of Symmetry, Dynamics, and Spectroscopy. Wiley.
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), 15941598. http://www.sciencemag.org/cgi/reprint/305/5690/1594.pdf.
Hoyle, R. 2006 Pattern Formation: An Introduction to Methods. Cambridge University Press.
Itano, T. & Generalis, S. C. 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102, 114501.
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Jpn 70, 701714.
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.
Kim, H., Kline, S. & Reynolds, W. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133160.
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.
Lan, Y. & Cvitanović, P. 2008 Unstable recurrent patterns in Kuramoto–Sivashinsky dynamics. Phys. Rev. E 78, 026208. arXiv.org:0804.2474.
Marsden, J. E. & Ratiu, T. S. 1999 Introduction to Mechanics and Symmetry. Springer.
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.
Nagata, M. 1997 Three-dimensional travelling-wave solutions in plane Couette flow. Phys. Rev. E 55, 20232025.
Peyret, R. 2002 Spectral Methods for Incompressible Flows. Springer.
Pringle, C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.
Rincon, F. 2007 On the existence of two-dimensional nonlinear steady states in plane Couette flow. Phys. Fluids 19, 4105. arXiv:0706.1165.
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.
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.
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.
Tuckerman, L. S. & Barkley, D. 2002 Symmetry breaking and turbulence in perturbed plane Couette flow. Theoret. Comput. Fluid Dyn. 16, 9197. arXiv:physics/0312051.
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339358. arXiv:physics/0604062.
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.
Waleffe, F. 1990 Proposal for a self-sustaining mechanism in shear flows. Unpublished Preprint. Center for Turbulence Research, Stanford University/NASA Ames.
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Math. 95, 319343.
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883900.
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 41404143.
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93102.
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.
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 15171543.
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.
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Equilibrium and travelling-wave solutions of plane Couette flow

  • J. F. GIBSON (a1), J. HALCROW (a1) and P. CVITANOVIĆ (a1)

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