Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-25T06:24:52.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Visualizing the geometry of state space in plane Couette flow

Published online by Cambridge University Press:  25 September 2008

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
Get access Rights & Permissions [Opens in a new window]

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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 611 , 25 September 2008 , pp. 107 - 130
Copyright
Copyright © Cambridge University Press 2008

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

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, 115–173.CrossRefGoogle Scholar
Barenghi, C. 2004 Turbulent transition for fluids. Phys. World 17 (12), 17–18.CrossRefGoogle Scholar
Busse, F. H. 2004 Visualizing the dynamics of the onset of turbulence. Science 305, 1574–1575.CrossRefGoogle ScholarPubMed
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
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.Google Scholar
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.Google Scholar
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, 377–381.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle ScholarPubMed
Foias, C., Nicolaenko, B., Sell, G. R. & Temam, R. 1985 Inertial manifold for the Kuramoto–Sivashinsky equation. C. R. Acad. Sci. I-Math 301, 285–288.Google Scholar
Gibson, J. F. 2002 Dynamical systems models of wall-bounded, shear-flow turbulence. PhD thesis, Cornell University.Google Scholar
Gibson, J. F. 2007 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep. Georgia Institute of Technology.Google Scholar
Golubitsky, M. & Stewart, I. 2002 The Symmetry Perspective. Birkhäuser, Boston.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. ChaosBook.org/projects/theses.html.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle 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 traveling waves in turbulent pipe flow. Science 305 (5690), 1594–1598.CrossRefGoogle ScholarPubMed
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.CrossRefGoogle Scholar
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Appl. Maths 1, 303–322.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716.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
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Kerswell, R. R. & Tutty, O. 2007 Recurrence of travelling wave solutions in transitional pipe flow. J. Fluid Mech. 584, 69–102.CrossRefGoogle Scholar
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, 760–790.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Rundstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.CrossRefGoogle Scholar
Li, W. & Graham, M. 2007 Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19 (083101), 1–15.CrossRefGoogle Scholar
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, 1042–1075.CrossRefGoogle Scholar
Manneville, P. 2004 Spots and turbulent domains in a model of transitional plane Couette flow. Theoret. Comput. Fluid Dyn. 18, 169–181.CrossRefGoogle Scholar
Moehlis, J., Faisst, H. & Eckhardt, B. 2004 A low-dimensional model for turbulent shear flows. New J. Phys. 6, 56.CrossRefGoogle Scholar
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, 352–376.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 traveling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023–2025.Google Scholar
Panton, R. L. (ed.) 1997 Self-Sustaining Mechanisms of Wall Turbulence. Computational Mechanics, Southhampton.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.CrossRefGoogle Scholar
Schmiegel, A. 1999 Transition to turbulence in linearly stable shear flows. PhD thesis, Philipps-Universität Marburg.Google Scholar
Schmiegel, A. & Eckhardt, B. 1997 Fractal stability border in plane Couette flow. Phys. Rev. Lett. 79, 5250.CrossRefGoogle Scholar
Schneider, T. M., Eckhardt, B. & Yorke, J. 2007 Turbulence, transition, and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.CrossRefGoogle ScholarPubMed
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, 1579–1582.CrossRefGoogle Scholar
Skufca, J. D. 2005 Understanding the chaotic saddle with focus on a 9-variable model of planar Couette flow. PhD thesis, University of Maryland.Google Scholar
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.CrossRefGoogle Scholar
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, 71–110.CrossRefGoogle Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.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. Springer.Google Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 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 Proc. Int. Symp. on Dynamics and Statistics of Coherent Structures in Turbulence: Roles of Elementary Vortices (ed. Kida, S.), pp. 115–128. National Center of Sciences, Tokyo, Japan.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1543.CrossRefGoogle Scholar
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. 85–106. Kluwer.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), 1–4.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
Zhou, X. & Sirovich, L. 1992 Coherence and chaos in a model of turbulent boundary layer. Phys. Fluids A 4, 2855–2874.CrossRefGoogle Scholar