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A probabilistic protocol for the assessment of transition and control

Published online by Cambridge University Press:  18 May 2020

Anton Pershin Open the ORCID record for Anton Pershin [Opens in a new window] ,
Cédric Beaume Open the ORCID record for Cédric Beaume [Opens in a new window]  and
Steven M. Tobias Open the ORCID record for Steven M. Tobias [Opens in a new window]
Anton Pershin*
Affiliation:
School of Mathematics, University of Leeds, LeedsLS2 9JT, UK
Cédric Beaume
Affiliation:
School of Mathematics, University of Leeds, LeedsLS2 9JT, UK
Steven M. Tobias
Affiliation:
School of Mathematics, University of Leeds, LeedsLS2 9JT, UK
*
†Email address for correspondence: [email protected]
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Abstract

Transition to turbulence dramatically alters the properties of fluid flows. In most canonical shear flows, the laminar flow is linearly stable and a finite-amplitude perturbation is necessary to trigger transition. Controlling transition to turbulence is achieved via the broadening or narrowing of the basin of attraction of the laminar flow. In this paper, a novel methodology to assess the robustness of the laminar flow and the efficiency of control strategies is introduced. It relies on the statistical sampling of the phase-space neighbourhood around the laminar flow in order to assess the transition probability of perturbations as a function of their energy. This approach is applied to a canonical flow (plane Couette flow) and provides invaluable insight: in the presence of the chosen control, transition is significantly suppressed whereas plausible scalar indicators of the nonlinear stability of the flow, such as the edge-state energy, do not provide conclusive predictions. The methodology presented here in the context of transition to turbulence is applicable to any nonlinear system displaying finite-amplitude instability.

JFM classification

Instability: Transition to turbulence Instability: Nonlinear instability Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 895 , 25 July 2020 , A16
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333 (6039), 192–196.CrossRefGoogle ScholarPubMed
Baron, A. & Quadrio, M. 1995 Turbulent drag reduction by spanwise wall oscillations. Appl. Sci. Res. 55 (4), 311–326.CrossRefGoogle Scholar
Brunton, S. L. & Noack, B. R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67 (5), 050801.Google Scholar
Budanur, N. B., Marensi, E., Willis, A. P. & Hof, B. 2020 Upper edge of chaos and the energetics of transition in pipe flow. Phys. Rev. F 5 (2), 023903.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Chantry, M. & Schneider, T. M. 2014 Studying edge geometry in transiently turbulent shear flows. J. Fluid Mech. 747, 506–517.CrossRefGoogle Scholar
Chantry, M., Tuckerman, L. S. & Barkley, D. 2017 Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1.CrossRefGoogle Scholar
Dauchot, O. & Daviaud, F. 1995 Finite amplitude perturbation and spots growth mechanism in plane Couette flow. Phys. Fluids 7 (2), 335–343.CrossRefGoogle Scholar
Duguet, Y., Brandt, L. & Larsson, B. R. J. 2010 Towards minimal perturbations in transitional plane Couette flow. Phys. Rev. E 82 (2), 026316.Google ScholarPubMed
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25 (8), 084103.CrossRefGoogle Scholar
Faranda, D., Lucarini, V., Manneville, P. & Wouters, J. 2014 On using extreme values to detect global stability thresholds in multi-stable systems: the case of transitional plane Couette flow. Chaos, Solitons Fractals 64, 26–35.CrossRefGoogle Scholar
Gibson, J. F.2014 Channelflow: A spectral Navier–Stokes simulator in $\text{C}++$. Tech. Rep. U. New Hampshire, Channelflow.org.Google Scholar
Hof, B., De Lozar, A., Avila, M., Tu, X. & Schneider, T. M. 2010 Eliminating turbulence in spatially intermittent flows. Science 327 (5972), 1491–1494.CrossRefGoogle ScholarPubMed
Hof, B., Westerweel, J., Schneider, T. M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59–62.CrossRefGoogle ScholarPubMed
Jung, W.-J., Mangiavacchi, N. & Akhavan, R. 1992 Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations. Phys. Fluids A 4 (8), 1605–1607.CrossRefGoogle Scholar
Kasagi, N., Suzuki, Y. & Fukagata, K. 2009 Microelectromechanical systems–based feedback control of turbulence for skin friction reduction. Annu. Rev. Fluid Mech. 41, 231–251.CrossRefGoogle Scholar
Kawahara, G. 2005 Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids 17 (4), 041702.CrossRefGoogle Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2013 Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717, R6.CrossRefGoogle Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2016 Edge states as mediators of bypass transition in boundary-layer flows. J. Fluid Mech. 801, R2.CrossRefGoogle Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
Marensi, E., Willis, A. P. & Kerswell, R. R. 2019 Stabilisation and drag reduction of pipe flows by flattening the base profile. J. Fluid Mech. 863, 850–875.CrossRefGoogle Scholar
Mellibovsky, F., Meseguer, A., Schneider, T. M. & Eckhardt, B. 2009 Transition in localized pipe flow turbulence. Phys. Rev. Lett. 103 (5), 054502.CrossRefGoogle ScholarPubMed
Menck, P. J., Heitzig, J., Marwan, N. & Kurths, J. 2013 How basin stability complements the linear-stability paradigm. Nat. Phys. 9 (2), 89–92.CrossRefGoogle Scholar
Moehlis, J., Faisst, H. & Eckhardt, B. 2004 A low-dimensional model for turbulent shear flows. New J. Phys. 6 (1), 56.CrossRefGoogle Scholar
Olvera, D. & Kerswell, R. R. 2017 Optimizing energy growth as a tool for finding exact coherent structures. Phys. Rev. F 2, 083902.Google Scholar
Pershin, A., Beaume, C. & Tobias, S. M. 2019 Dynamics of spatially localized states in transitional plane Couette flow. J. Fluid Mech. 867, 414–437.CrossRefGoogle 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.CrossRefGoogle Scholar
Quadrio, M. 2011 Drag reduction in turbulent boundary layers by in-plane wall motion. Phil. Trans. R. Soc. Lond. A 369 (1940), 1428–1442.CrossRefGoogle ScholarPubMed
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise wall oscillations. J. Fluid Mech. 521, 251–271.CrossRefGoogle Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2014 Designing a more nonlinearly stable laminar flow via boundary manipulation. J. Fluid Mech. 738, R1.CrossRefGoogle Scholar
Schmiegel, A. & Eckhardt, B. 1997 Fractal stability border in plane Couette flow. Phys. Rev. Lett. 79 (26), 5250.CrossRefGoogle 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 (3), 037301.Google ScholarPubMed
Shi, L., Avila, M. & Hof, B. 2013 Scale invariance at the onset of turbulence in Couette flow. Phys. Rev. Lett. 110 (20), 204502.CrossRefGoogle ScholarPubMed
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.CrossRefGoogle Scholar
Spalart, P. R., Moser, R. D. & Rogers, M. M. 1991 Spectral methods for the Navier–Stokes equations with one infinite and two periodic directions. J. Comp. Phys. 96 (2), 297–324.CrossRefGoogle Scholar
Sreenivasan, K. R. 1982 Laminarescent, relaminarizing and retransitional flows. Acta Mechanica 44, 1–48.CrossRefGoogle Scholar
Stone, P. A., Roy, A., Larson, R. G., Waleffe, F. & Graham, M. D. 2004 Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids 16 (9), 3470–3482.CrossRefGoogle Scholar
Stone, P. A., Waleffe, F. & Graham, M. D. 2002 Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett. 89 (20), 208301.CrossRefGoogle Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (20), 204501.CrossRefGoogle ScholarPubMed
Watanabe, T., Iima, M. & Nishiura, Y. 2016 A skeleton of collision dynamics: hierarchical network structure among even-symmetric steady pulses in binary fluid convection. SIAM J. Appl. Dyn. Sys. 15 (2), 789–806.CrossRefGoogle Scholar
Zammert, S. & Eckhardt, B. 2015 Crisis bifurcations in plane Poiseuille flow. Phys. Rev. E 91, 041003(R).Google ScholarPubMed