Skip to main content Accessibility help
×
Home

Visualizing the geometry of state space in plane Couette flow

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

Abstract

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier–Stokes equations. We construct a dynamical 105-dimensional state-space representation of plane Couette flow at Reynolds number Re = 400 in a small periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds partially tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of symmetry-induced heteroclinic connections.

Copyright

References

Hide All
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of turbulent boundary layer. J. Fluid Mech. 192, 115173.
Barenghi, C. 2004 Turbulent transition for fluids. Phys. World 17 (12), 1718.
Busse, F. H. 2004 Visualizing the dynamics of the onset of turbulence. Science 305, 15741575.
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.
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G., Whelan, N. & Wirzba, A. 2007 Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen. ChaosBook.org.
Cvitanović, P., Davidchack, R. L. & Siminos, E. 2008 State space geometry of a spatio-temporally chaotic Kuramoto-Sivashinsky flow. Available at arxiv.org:0709.2944.
Dauchot, O. & Vioujard, N. 2000 Phase space analysis of a dynamical model for the subcritical transition to turbulence in plane Couette flow. Eur. Phys. J. B 14, 377381.
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
Foias, C., Nicolaenko, B., Sell, G. R. & Temam, R. 1985 Inertial manifold for the Kuramoto–Sivashinsky equation. C. R. Acad. Sci. I-Math 301, 285288.
Gibson, J. F. 2002 Dynamical systems models of wall-bounded, shear-flow turbulence. PhD thesis, Cornell University.
Gibson, J. F. 2007 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep. Georgia Institute of Technology.
Golubitsky, M. & Stewart, I. 2002 The Symmetry Perspective. Birkhäuser, Boston.
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. ChaosBook.org/projects/theses.html.
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.
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 traveling waves in turbulent pipe flow. Science 305 (5690), 15941598.
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Appl. Maths 1, 303322.
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703716.
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.
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.
Kawahara, G., Kida, S. & Nagata, M. 2005 Unstable periodic motion in plane Couette system: the skeleton of turbulence. In One Hundred Years of Boundary Layer Research. Kluwer.
Kerswell, R. R. & Tutty, O. 2007 Recurrence of travelling wave solutions in transitional pipe flow. J. Fluid Mech. 584, 69102.
Kevrekidis, I. G., Nicolaenko, B. & Scovel, J. C. 1990 Back in the saddle again: a computer assisted study of the Kuramoto–Sivashinsky equation. SIAM J. Appl. Maths. 50, 760790.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows. In Proc. 3rd GAMM Conf. Numerical Methods in Fluid Mechanics (ed. Hirschel, E.), pp. 165–173. GAMM, Viewweg, Braunschweig.
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Rundstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741773.
Li, W. & Graham, M. 2007 Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19 (083101), 115.
López, V., Boyland, P., Heath, M. T. & Moser, R. D. 2006 Relative periodic solutions of the complex Ginzburg–Landau equation. SIAM J. Appl. Dyn. Systems 4, 10421075.
Manneville, P. 2004 Spots and turbulent domains in a model of transitional plane Couette flow. Theoret. Comput. Fluid Dyn. 18, 169181.
Moehlis, J., Faisst, H. & Eckhardt, B. 2004 A low-dimensional model for turbulent shear flows. New J. Phys. 6, 56.
Moehlis, J., Faisst, H. & Eckhardt, B. 2005 Periodic orbits and chaotic sets in a low-dimensional model for shear flows. SIAM J. Appl. Dyn. Systems 4, 352376.
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 traveling-wave solutions in plane Couette flow. Phys. Rev. E 55, 20232025.
Panton, R. L. (ed.) 1997 Self-Sustaining Mechanisms of Wall Turbulence. Computational Mechanics, Southhampton.
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601639.
Schmiegel, A. 1999 Transition to turbulence in linearly stable shear flows. PhD thesis, Philipps-Universität Marburg.
Schmiegel, A. & Eckhardt, B. 1997 Fractal stability border in plane Couette flow. Phys. Rev. Lett. 79, 5250.
Schneider, T. M., Eckhardt, B. & Yorke, J. 2007 Turbulence, transition, and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.
Sirovich, L. & Zhou, X. 1994 Reply to ‘Observations regarding “Coherence and chaos in a model of turbulent boundary layer” by X. Zhou and L. Sirovich’. Phys. Fluids 6, 15791582.
Skufca, J. D. 2005 Understanding the chaotic saddle with focus on a 9-variable model of planar Couette flow. PhD thesis, University of Maryland.
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.
Smith, T. R., Moehlis, J. & Holmes, P. 2005 Low-dimensional models for turbulent plane Couette flow in a minimal flow unit. J. Fluid Mech. 538, 71110.
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339358.
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. Springer.
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 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 Proc. Int. Symp. on Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices (ed. Kida, S.), pp. 115128. National Center of Sciences, Tokyo, Japan.
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 15171543.
Waleffe, F. & Wang, J. 2005 Transition threshold and the self-sustaining process. In IUTAM Symp. on Laminar–Turbulent Transition and Finite Amplitude Solutions (ed. Mullin, T. & Kerswell, R. R.), pp. 85106. Kluwer.
Wang, J., Gibson, J. F. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 14.
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.
Zhou, X. & Sirovich, L. 1992 Coherence and chaos in a model of turbulent boundary layer. Phys. Fluids A 4, 28552874.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Visualizing the geometry of state space in plane Couette flow

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

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed