Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T00:20:40.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear optimal perturbations in a Couette flow: bursting and transition

Published online by Cambridge University Press:  25 January 2013

S. Cherubini  and
P. De Palma
S. Cherubini*
Affiliation:
DynFluid, Arts et Metiers ParisTech, 151, boulevard de l’Hopital, 75013 Paris, France DIMeG, CEMeC, Politecnico di Bari, via Re David 200, 70125 Bari, Italy
P. De Palma
Affiliation:
DIMeG, CEMeC, Politecnico di Bari, via Re David 200, 70125 Bari, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Nonlinear instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 716 , 10 February 2013 , pp. 251 - 279
Copyright
©2013 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.)

References

Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
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.Google Scholar
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, 221–253.Google Scholar
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, 231–259.Google Scholar
Cossu, C., Brandt, L., Bagheri, S. & Henningson, D. S. 2011 Secondary threshold amplitudes for sinuous streak breakdown. Phys. Fluids 23, 074103.Google Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition of pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.Google Scholar
Ehrenstein, U. & Gallaire, F. 2008 Global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech. 614, 315–327.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and traveling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243.Google 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, 1594–1598.Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716.Google Scholar
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.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
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.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
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, 181–214.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow. J. Fluid Mech. 217, 519–527.Google Scholar
Nagata, M. 1997 Three-dimensional traveling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023.Google Scholar
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.Google Scholar
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, 415–443.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601.Google Scholar
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Flows. Springer.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
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
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for the three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123 (2), 402–414.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–901.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional states in plane shear flow. Phys. Rev. Lett. 81, 4140–4143.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 20450.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.Google Scholar
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, 135–160.Google Scholar