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The onset of transient turbulence in minimal plane Couette flow

  • Julius Rhoan T. Lustro (a1), Genta Kawahara (a1), Lennaert van Veen (a2), Masaki Shimizu (a1) and Hiroshi Kokubu (a3)...


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.


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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.
Budanur, N. B. & Hof, B. 2017 Heteroclinic path to spatially localized chaos in pipe flow. J. Fluid Mech. 827, R1.
Chian, A. C.-L., Muñoz, P. R. & Rempel, E. L. 2013 Edge of chaos and genesis of turbulence. Phys. Rev. E 88, 052910.
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, 159169.
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, 12971315.
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.
Grebogi, C., Ott, E. & Yorke, J. A. 1983 Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7, 181200.
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.
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.
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213240.
Kawahara, G. 2005 Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids 17, 041702.
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203225.
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R1744.
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.
McFadden, G. B., Murray, B. T. & Boisvert, R. F. 1990 Elimination of spurious eigenvalues in the Chebyshev tau spectral method. J. Comput. Phys. 91, 228239.
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.
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.
Ott, E. 2002 Chaos in Dynamical Systems, 2nd edn. Cambridge University Press.
Palis, J. & Takens, F. 1993 Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press.
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, 935982.
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, 145.
Sánchez, J., Net, M., García-Archilla, B. & Simó, C. 2004 Newton–Krylov continuation of periodic orbits for Navier–Stokes flows. J. Comput. Phys. 201, 1333.
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.
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.
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.
van Veen, L. & Kawahara, G. 2011 Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107, 114501.
van Veen, L., Kawahara, G. & Matsumura, A. 2011 On matrix-free computation of 2D unstable manifolds. SIAM J. Sci. Comput. 33, 2544.
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339358.
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, 123.
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