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Dynamics of spatially localized states in transitional plane Couette flow

  • Anton Pershin (a1), Cédric Beaume (a1) and Steven M. Tobias (a1)


Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider et al., Phys. Rev. Lett., vol. 104, 2010, 104501). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed ( $Re<175$ ) to transitional ones ( $Re\approx 325$ ), these spatially localized states traverse various regimes where their relaminarization time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterized in this paper for a $4\unicode[STIX]{x03C0}$ -periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarizing. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarizing. Very long chaotic transients ( $t>10^{4}$ ) can be observed without difficulty for relatively low values of the Reynolds number ( $Re\approx 250$ ).


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Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192196.
Avila, M., Willis, A. P. & Hof, B. 2010 On the transient nature of localized pipe flow turbulence. J. Fluid Mech. 646, 127136.
Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.
Barkley, D. & Tuckerman, L. S. 2005 Turbulent-laminar patterns in plane Couette flow. In IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions, pp. 107127. Springer.
Beaume, C., Bergeon, A. & Knobloch, E. 2013 Convectons and secondary snaking in three-dimensional natural doubly diffusive convection. Phys. Fluids 25 (2), 024105.
Beaume, C., Bergeon, A. & Knobloch, E. 2018 Three-dimensional doubly diffusive convectons: instability and transition to complex dynamics. J. Fluid Mech. 840, 74105.
Bergeon, A. & Knobloch, E. 2008 Spatially localized states in natural doubly diffusive convection. Phys. Fluids 20, 034102.
Burke, J. & Dawes, J. H. P. 2012 Localized states in an extended Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 11 (1), 261284.
Burke, J. & Knobloch, E. 2006 Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73 (5), 056211.
Chantry, M., Tuckerman, L. S. & Barkley, D. 2017 Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1.
Coullet, P., Riera, C. & Tresser, C. 2000 Stable static localized structures in one dimension. Phys. Rev. Lett. 84, 30693072.
Dauchot, O. & Daviaud, F. 1995 Finite amplitude perturbation and spots growth mechanism in plane Couette flow. Phys. Fluids 7 (2), 335343.
Duguet, Y., Le Maître, O. & Schlatter, P. 2011 Stochastic and deterministic motion of a laminar-turbulent front in a spanwisely extended Couette flow. Phys. Rev. E 84 (6), 066315.
Duguet, Y. & Schlatter, P. 2013 Oblique laminar-turbulent interfaces in plane shear flows. Phys. Rev. Lett. 110, 034502.
Duguet, Y., Schlatter, P. & Henningson, D. S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21 (11), 111701.
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447468.
Gibson, J. F.2014 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep., University of New Hampshire,
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243266.
Gibson, J. F. & Schneider, T. M. 2016 Homoclinic snaking in plane Couette flow: bending, skewing and finite-size effects. J. Fluid Mech. 794, 530551.
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365376.
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.
Knobloch, E. 2015 Spatial localization in dissipative systems. Annu. Rev. Condens. Matter Phys. 6 (1), 325359.
Lemoult, G., Shi, L., Avila, K., Jalikop, S. V., Avila, M. & Hof, B. 2016 Directed percolation phase transition to sustained turbulence in Couette flow. Nat. Phys. 12 (3), 254258.
Lloyd, D. J. B., Gollwitzer, C., Rehberg, C. & Richter, R. 2015 Homoclinic snaking near the surface instability of a polarizable fluid. J. Fluid Mech. 783, 283305.
Mercader, I., Batiste, O., Alonso, A. & Knobloch, E. 2011 Convectons, anticonvectons and multiconvectons in binary fluid convection. J. Fluid Mech. 667, 586606.
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number 107 . J. Comput. Phys. 186, 178197.
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.
Olvera, D. & Kerswell, R. R. 2017 Optimizing energy growth as a tool for finding exact coherent structures. Phys. Rev. Fluids 2, 083902.
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689703.
Pomeau, Y. 1986 Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23 (1–3), 311.
Romanov, V. A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Appl. 7 (2), 137146.
Saarloos, W. V. 2003 Front propagation into unstable states. Phys. Rep. 386, 29222.
Sano, M. & Tamai, K. 2016 A universal transition to turbulence in channel flow. Nat. Phys. 12 (3), 249253.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, vol. 142. Springer.
Schmiegel, A. & Eckhardt, B. 2000 Persistent turbulence in annealed plane Couette flow. Europhys. Lett. 51 (4), 395400.
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.
Schneider, T. M., Gibson, J. F. & Burke, J. 2010a Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104 (10), 104501.
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 (3), 037301.
Schneider, T. M., Marinc, D. & Eckhardt, B. 2010b Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441451.
Shi, L., Avila, M. & Hof, B. 2013 Scale invariance at the onset of turbulence in Couette flow. Phys. Rev. Lett. 110 (20), 204502.
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339358.
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883900.
Waleffe, F. 2009 Turbulence and Interactions: Exact Coherent Structures in Turbulent Shear Flows. Springer.
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.
Willis, A. P. & Kerswell, R. R. 2007 Critical behavior in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett. 98 (1), 014501.
Woods, P. D. & Champneys, A. R. 1999 Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian–Hopf bifurcation. Physica D 129, 147170.
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Dynamics of spatially localized states in transitional plane Couette flow

  • Anton Pershin (a1), Cédric Beaume (a1) and Steven M. Tobias (a1)


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