Skip to main content Accessibility help

Nonlinear optimal perturbations in a Couette flow: bursting and transition

  • S. Cherubini (a1) (a2) and P. De Palma (a2)

This paper provides the analysis of bursting and transition to turbulence in a Couette flow, based on the growth of nonlinear optimal disturbances. We use a global variational procedure to identify such optimal disturbances, defined as those initial perturbations yielding the largest energy growth at a given target time, for given Reynolds number and initial energy. The nonlinear optimal disturbances are found to be characterized by a basic structure, composed of inclined streamwise vortices along localized regions of low and high momentum. This basic structure closely recalls that found in boundary-layer flow (Cherubini et al., J. Fluid Mech., vol. 689, 2011, pp. 221–253), indicating that this structure may be considered the most ‘energetic’ one at short target times. However, small differences in the shape of these optimal perturbations, due to different levels of the initial energy or target time assigned in the optimization process, may produce remarkable differences in their evolution towards turbulence. In particular, direct numerical simulations have shown that optimal disturbances obtained for large initial energies and target times induce bursting events, whereas for lower values of these parameters the flow is directly attracted towards the turbulent state. For this reason, the optimal disturbances have been classified into two classes, the highly dissipative and the short-path perturbations. Both classes lead the flow to turbulence, skipping the phases of streak formation and secondary instability which are typical of the classical transition scenario for shear flows. The dynamics of this transition scenario exploits three main features of the nonlinear optimal disturbances: (i) the large initial value of the streamwise velocity component; (ii) the streamwise dependence of the disturbance; (iii) the presence of initial inclined streamwise vortices. The short-path perturbations are found to spend a considerable amount of time in the vicinity of the edge state (Schneider et al., Phys. Rev. E, vol. 78, 2008, 037301), whereas the highly dissipative optimal disturbances pass closer to the edge, but they are rapidly repelled away from it, leading the flow to high values of the dissipation rate. After this dissipation peak, the trajectories do not lead towards the turbulent attractor, but they spend some time in the vicinity of an unstable periodic orbit (UPO). This behaviour led us to conjecture that bursting events can be obtained not only as homoclinic orbits approaching the UPO, as recently found by van Veen & Kawahara (Phys. Rev. Lett., vol. 107, 2011, p. 114501), but also as heteroclinic orbits between the equilibrium solution on the edge and the UPO.

Corresponding author
Email address for correspondence:
Hide All
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 16371650.
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2010a Rapid path to transition via nonlinear localized optimal perturbations. Phys. Rev. E 82, 066302.
Cherubini, S., De Palma, P., Robinet, J. C. & Bottaro, A. 2011 The minimal seed of turbulent transition in a boundary layer. J. Fluid Mech 689, 221253.
Cherubini, S., Robinet, J.-C., Bottaro, A. & De Palma, P. 2010b Optimal wave packets in a boundary layer and initial phases of a turbulent spot. J. Fluid Mech. 656, 231259.
Cossu, C., Brandt, L., Bagheri, S. & Henningson, D. S. 2011 Secondary threshold amplitudes for sinuous streak breakdown. Phys. Fluids 23, 074103.
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition of pipe flow. Annu. Rev. Fluid Mech. 39, 447468.
Ehrenstein, U. & Gallaire, F. 2008 Global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech. 614, 315327.
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and traveling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243.
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, 15941598.
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703716.
Jimenez, J., Kawahara, G., Simens, M. P. & Nagata, 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.
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69102.
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Rundstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741773.
Monokrousos, A., Akervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181214.
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Non-equilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106, 134502.
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow. J. Fluid Mech. 217, 519527.
Nagata, M. 1997 Three-dimensional traveling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023.
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.
Pringle, C. C. T., Willis, A. P. & Kerswell, R. R. 2012 Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid. Mech. 702, 415443.
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601.
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Flows. Springer.
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.
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.
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for the three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123 (2), 402414.
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883901.
Waleffe, F. 1998 Three-dimensional states in plane shear flow. Phys. Rev. Lett. 81, 41404143.
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 15171534.
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 20450.
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.
Zuccher, S., Luchini, P. & Bottaro, A. 2004 Algebraic growth in a Blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime. Eur. J. Mech. B (Fluids) 513, 135160.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


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