Hostname: page-component-7f64f4797f-8cpv7 Total loading time: 0 Render date: 2025-11-04T09:50:35.728Z Has data issue: false hasContentIssue false
  • English
  • Français

The onset of transient turbulence in minimal plane Couette flow

Published online by Cambridge University Press:  10 January 2019

Julius Rhoan T. Lustro Open the ORCID record for Julius Rhoan T. Lustro [Opens in a new window] ,
Genta Kawahara Open the ORCID record for Genta Kawahara [Opens in a new window] ,
Lennaert van Veen Open the ORCID record for Lennaert van Veen [Opens in a new window] ,
Masaki Shimizu Open the ORCID record for Masaki Shimizu [Opens in a new window]  and
Hiroshi Kokubu Open the ORCID record for Hiroshi Kokubu [Opens in a new window]
Julius Rhoan T. Lustro
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Genta Kawahara*
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Lennaert van Veen
Affiliation:
Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, Ontario L1H 7K4, Canada
Masaki Shimizu
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
Hiroshi Kokubu
Affiliation:
Graduate School of Science, Kyoto University, Kitashirakawa Oiwake, Sakyo, Kyoto 606-8502, Japan
*
†Email address for correspondence: kawahara@me.es.osaka-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

The onset of transient turbulence in minimal plane Couette flow has been identified theoretically as homoclinic tangency with respect to a simple edge state for the Navier–Stokes equation, i.e., the gentle periodic orbit (the lower branch of a saddle-node pair) found by Kawahara & Kida (J. Fluid Mech., vol. 449, 2001, pp. 291–300). The first tangency of a pair of distinct homoclinic orbits to this periodic edge state has been discovered at Reynolds number $Re\equiv Uh/\unicode[STIX]{x1D708}=Re_{T}\approx 240.88$ ($U$, $h$, and $\unicode[STIX]{x1D708}$ being half the difference of the two wall velocities, half the wall separation, and the kinematic viscosity of fluid, respectively). At $Re>Re_{T}$ a Smale horseshoe appears on the Poincaré section through transversal homoclinic points to generate a transient chaos that eventually relaminarises. In numerical experiments a sustaining chaos, which is a consequence of period-doubling cascade stemming from the upper branch of another saddle-node pair of periodic orbits, is observed in a narrow range of the Reynolds number, $Re\approx 240.40$–240.46. At the upper edge of this $Re$ range it is found that the chaotic set touches the lower branch of this pair, i.e., another edge state. The corresponding chaotic attractor is replaced by a chaotic saddle at $Re\approx 240.46$, and subsequently this saddle touches the gentle periodic edge state on the boundary of the laminar basin at the tangency Reynolds number $Re=Re_{T}$. After this crisis on the boundary of the laminar basin, for $Re>Re_{T}$, chaotic transients that eventually relaminarise can be observed.

JFM classification

Nonlinear Dynamical Systems: Chaos Instability: Transition to turbulence

Information

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , R2
Copyright
© 2019 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.)

Article purchase

Temporarily unavailable

References

Avila, M., Mellibovsky, F., Roland, N. & Hof, B. 2013 Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110, 224502.Google Scholar
Budanur, N. B. & Hof, B. 2017 Heteroclinic path to spatially localized chaos in pipe flow. J. Fluid Mech. 827, R1.Google Scholar
Chian, A. C.-L., Muñoz, P. R. & Rempel, E. L. 2013 Edge of chaos and genesis of turbulence. Phys. Rev. E 88, 052910.Google Scholar
Doedel, E. J., Kooi, B. W., van Voorn, G. A. K. & Kuznetsov, Y. A. 2009 Continuation of connecting orbits in 3D-ODEs (II): cycle-to-cycle connections. Intl J. Bifurcation Chaos Appl. Sci. Eng. 19, 159–169.Google Scholar
Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T. M. 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. Lond. A 366, 1297–1315.Google Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.10.1146/annurev.fluid.39.050905.110308Google Scholar
Grebogi, C., Ott, E. & Yorke, J. A. 1983 Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7, 181–200.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.10.1017/S0022112095000978Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716.10.1143/JPSJ.70.703Google 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, 015105.10.1063/1.1825451Google Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Kawahara, G. 2005 Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids 17, 041702.10.1063/1.1890428Google Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.10.1017/S0022112001006243Google Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.10.1146/annurev-fluid-120710-101228Google Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–44.10.1088/0951-7715/18/6/R01Google Scholar
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.10.1017/S0022112087000892Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.10.1063/1.4757227Google Scholar
McFadden, G. B., Murray, B. T. & Boisvert, R. F. 1990 Elimination of spurious eigenvalues in the Chebyshev tau spectral method. J. Comput. Phys. 91, 228–239.10.1016/0021-9991(90)90012-PGoogle Scholar
Muñoz, P. R., Barroso, J. J., Chian, A. C.-L. & Rempel, E. L. 2012 Edge state and crisis in the Pierce diode. Chaos 22, 033120.Google Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.10.1017/S0022112090000829Google Scholar
Ott, E. 2002 Chaos in Dynamical Systems, 2nd edn. Cambridge University Press.Google Scholar
Palis, J. & Takens, F. 1993 Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. 174, 935–982.Google Scholar
Riols, A., Rincon, F., Cossu, C., Lesur, G., Longaretti, P.-Y., Ogilvie, G. I. & Herault, J. 2013 Global bifurcations to subcritical magnetorotational dynamo action in Keplerian shear flow. J. Fluid Mech. 731, 1–45.10.1017/jfm.2013.317Google Scholar
Sánchez, J., Net, M., García-Archilla, B. & Simó, C. 2004 Newton–Krylov continuation of periodic orbits for Navier–Stokes flows. J. Comput. Phys. 201, 13–33.Google Scholar
Schneider, T. M., Gibson, J. F., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78, 037301.Google Scholar
Shimizu, M., Kawahara, G., Lustro, J. R. T. & van Veen, L. 2014 Route to chaos in minimal plane Couette flow. In ECCOMAS Congress, International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain.Google Scholar
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.Google Scholar
van Veen, L. & Kawahara, G. 2011 Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107, 114501.Google Scholar
van Veen, L., Kawahara, G. & Matsumura, A. 2011 On matrix-free computation of 2D unstable manifolds. SIAM J. Sci. Comput. 33, 25–44.10.1137/100789804Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.10.1017/S0022112007005459Google Scholar
Vollmer, J., Schneider, T. M. & Eckhardt, B. 2009 Basin boundary, edge of chaos, and edge state in a two-dimensional model. New J. Phys. 11, 1–23.Google Scholar