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Triggering turbulence efficiently in plane Couette flow

Published online by Cambridge University Press:  27 September 2012

S. M. E. Rabin ,
C. P. Caulfield  and
R. R. Kerswell
S. M. E. Rabin*
Affiliation:
Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield
Affiliation:
Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK
R. R. Kerswell
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
*
†Email address for correspondence: S.Rabin@damtp.cam.ac.uk
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Abstract

We use a variational formulation incorporating the full Navier–Stokes equations to identify initial perturbations with finite kinetic energy ${E}_{0} $ which generate the largest gain in perturbation kinetic energy at some time $T$ later for plane Couette flow. Using the flow geometry originally used by Butler & Farrell (Phys. Fluids A, vol. 4, 1992, pp. 1637–1650) to identify the linear transient optimal perturbations for ${E}_{0} \ensuremath{\rightarrow} 0$ and incorporating $T$ as part of the optimization procedure, we show how the addition of nonlinearity smoothly changes the result as ${E}_{0} $ increases from zero until a small but finite ${E}_{c} $ is reached. At this point, the variational algorithm is able to identify an initial condition of completely different form which triggers turbulence – called the minimal seed for turbulence. If instead $T$ is fixed at some asymptotically large value, as suggested by Pringle, Willis & Kerswell (J. Fluid Mech., vol. 703, 2012, pp. 415–443), a fundamentally different ‘final’ optimal perturbation emerges from our algorithm above some threshold initial energy ${E}_{f} \in (0, {E}_{c} )$ which shows signs of localization. This nonlinear optimal perturbation clearly approaches the structure of the minimal seed as ${E}_{0} \ensuremath{\rightarrow} { E}_{c}^{\ensuremath{-} } $, although for ${E}_{0} \lt {E}_{c} $, its maximum gain over all time intervals is always less than the equivalent maximum gain for the ‘quasi-linear optimal perturbation’, i.e. the finite-amplitude manifestation of the underlying linear optimal perturbation. We also consider a wider flow geometry recently studied by Monokrousos et al. (Phys. Rev. Lett., vol. 106, 2011, 134502) and present evidence that the critical energy for transition ${E}_{c} $ they found by using total dissipation over a time interval as the optimizing functional is recovered using energy gain at a fixed target time as the optimizing functional, with the same associated minimal seed emerging. This emphasizes that the precise form of the functional does not appear to be important for identifying ${E}_{c} $ provided it takes heightened values for turbulent flows, as postulated by Pringle, Willis & Kerswell (J. Fluid Mech., vol. 703, 2012, pp. 415–443). All our results highlight the irrelevance of the linear energy gain optimal perturbation for predicting or describing the lowest-energy flow structure which triggers turbulence.

JFM classification

Instability: Nonlinear instability Instability: Transition to turbulence Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 244 - 272
Copyright
©2012 Cambridge University Press

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References

Bottin, S. & Chate, H. 1998 Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6 (1), 143–155.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.Google Scholar
Cherubini, S., De Palma, P., Robinet, J. C. & Bottaro, A. 2010 Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow. Phys. Rev. E 82, 066302.Google Scholar
Cherubini, S., De Palma, P., Robinet, J. C. & Bottaro, A. 2011a Edge states in a boundary layer. Phys. Fluids 23, 051705.Google Scholar
Cherubini, S., De Palma, P., Robinet, J. C. & Bottaro, A. 2011b The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689, 221–253.Google Scholar
Duguet, Y., Brandt, L. & Larsson, B. R. J. 2010 Towards minimal perturbations in transitional Couette flow. Phys. Rev. E 82, 026316.CrossRefGoogle ScholarPubMed
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255–274.CrossRefGoogle Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716.CrossRefGoogle Scholar
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106 (13), 134502.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 (15), 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. 703, 415–443.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S. & Henningson, D. S. 1998 On the stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.CrossRefGoogle Scholar
Romanov, V. A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Appl. 7, 137–146.Google Scholar
Schmid, P. J. 2007 Nonmodel stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.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.CrossRefGoogle Scholar
Taylor, J. R. 2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego.Google Scholar