Hostname: page-component-cd4964975-96cn4 Total loading time: 0 Render date: 2023-03-28T15:30:41.434Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Linear and nonlinear optimal growth mechanisms for generating turbulent bands

Published online by Cambridge University Press:  17 March 2022

E. Parente*
DynFluid – Arts et Métiers Paris, 151 Bd de l'Hôpital, 75013 Paris, France Dipartimento di Meccanica, Matematica e Management (DMMM), Politecnico di Bari, Via Re David 200, 70126 Bari, Italy
J.-Ch. Robinet
DynFluid – Arts et Métiers Paris, 151 Bd de l'Hôpital, 75013 Paris, France
P. De Palma
Dipartimento di Meccanica, Matematica e Management (DMMM), Politecnico di Bari, Via Re David 200, 70126 Bari, Italy
S. Cherubini
Dipartimento di Meccanica, Matematica e Management (DMMM), Politecnico di Bari, Via Re David 200, 70126 Bari, Italy
Email address for correspondence:


Recently, many authors have investigated the origin and growth of turbulent bands in shear flows, highlighting the role of streaks and their inflectional instability in the process of band generation and sustainment. Recalling that streaks are created by an optimal transient growth mechanism, and motivated by the observation of a strong increase of the disturbance kinetic energy corresponding to the creation of turbulent bands, we use linear and nonlinear energy optimisations in a tilted domain to unveil the main mechanisms allowing the creation of a turbulent band in a channel flow. Linear transient growth analysis shows an optimal growth for wavenumbers having an angle of approximately $35^\circ$, close to the peak values of the premultiplied energy spectra of direct numerical simulations. This linear optimal perturbation generates oblique streaks, which, for a sufficiently large initial energy, induce turbulence in the whole domain, due to the lack of spatial localisation. However, spatially localised perturbations obtained by adding nonlinear effects to the optimisation or by artificially confining the linear optimal to a localised region in the transverse direction are characterised by a large-scale flow and lead to the generation of a localised turbulent band. These results suggest that two main elements are needed for inducing turbulent bands in a tilted domain: (i) a linear energy growth mechanism, such as the lift-up, for generating large-amplitude flow structures, which produce inflection points; (ii) spatial localisation, linked to the presence or generation of large-scale vortices. We show that these elements alone generate isolated turbulent bands also in large non-tilted domains.

JFM Papers
© The Author(s), 2022. Published by 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.)



Barkley, D. & Tuckerman, L.S. 2005 Computational study of turbulent laminar patterns in couette flow. Phys. Rev. Lett. 94 (1), 014502.10.1103/PhysRevLett.94.014502CrossRefGoogle ScholarPubMed
Barkley, D. & Tuckerman, L.S. 2007 Mean flow of turbulent–laminar patterns in plane couette flow. J. Fluid Mech. 576, 109137.10.1017/S002211200600454XCrossRefGoogle Scholar
Butler, K.M. & Farrell, B.F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5 (3), 774777.10.1063/1.858663CrossRefGoogle Scholar
Carlson, D.R., Widnall, S.E. & Peeters, M.F. 1982 A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech. 121, 487505.CrossRefGoogle Scholar
Chantry, M., Tuckerman, L.S & Barkley, D. 2017 Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1.10.1017/jfm.2017.405CrossRefGoogle Scholar
Cherubini, S. & De Palma, P. 2013 Nonlinear optimal perturbations in a couette flow: bursting and transition. J. Fluid Mech. 716, 251279.10.1017/jfm.2012.544CrossRefGoogle Scholar
Cherubini, S., De Palma, P. & Robinet, J.-C. 2015 Nonlinear optimals in the asymptotic suction boundary layer: transition thresholds and symmetry breaking. Phys. Fluids 27 (3), 034108.10.1063/1.4916017CrossRefGoogle 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 (6), 066302.10.1103/PhysRevE.82.066302CrossRefGoogle Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011 The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689, 221253.10.1017/jfm.2011.412CrossRefGoogle Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2012 A purely nonlinear route to transition approaching the edge of chaos in a boundary layer. Fluid Dyn. Res. 44, 031404.10.1088/0169-5983/44/3/031404CrossRefGoogle Scholar
Cherubini, S., De Tullio, M.D., De Palma, P. & Pascazio, G. 2013 Transient growth in the flow past a three-dimensional smooth roughness element. J. Fluid Mech. 724, 642670.10.1017/jfm.2013.177CrossRefGoogle Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D.S. 2013 Minimal transition thresholds in plane couette flow. Phys. Fluids 25 (8), 084103.10.1063/1.4817328CrossRefGoogle Scholar
Duguet, Y. & Schlatter, P. 2013 Oblique laminar–turbulent interfaces in plane shear flows. Phys. Rev. Lett. 110, 034502.10.1103/PhysRevLett.110.034502CrossRefGoogle ScholarPubMed
Duguet, Y., Schlatter, P. & Henningson, D.S. 2010 Formation of turbulent patterns near the onset of transition in plane couette flow. J. Fluid Mech. 650, 119.10.1017/S0022112010000297CrossRefGoogle Scholar
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2015 Hairpin-like optimal perturbations in plane Poiseuille flow. J. Fluid Mech. 775, R1.10.1017/jfm.2015.320CrossRefGoogle Scholar
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2017 Optimal bursts in turbulent channel flow. J. Fluid Mech. 817, 3560.10.1017/jfm.2017.107CrossRefGoogle Scholar
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2016 Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow. Fluid Dyn. Res. 48 (6), 061409.10.1088/0169-5983/48/6/061409CrossRefGoogle Scholar
Gibson, J.F., et al. 2021 Channelflow 2.0. in preparation.Google Scholar
Gomé, S., Tuckerman, L.S. & Barkley, D. 2020 Statistical transition to turbulence in plane channel flow. Phys. Rev. Fluids 5 (8), 083905.10.1103/PhysRevFluids.5.083905CrossRefGoogle Scholar
Hamilton, J.M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.10.1017/S0022112095000978CrossRefGoogle Scholar
Kashyap, P.V., Duguet, Y. & Chantry, M. 2020 a Far field of turbulent spots. Phys. Rev. Fluids 5, 103902.10.1103/PhysRevFluids.5.103902CrossRefGoogle Scholar
Kashyap, P.V., Duguet, Y. & Dauchot, O. 2020 b Flow statistics in the transitional regime of plane channel flow. Entropy 22 (9), 1001.10.3390/e22091001CrossRefGoogle ScholarPubMed
Kerswell, R.R. 2018 Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50 (1), 319345.10.1146/annurev-fluid-122316-045042CrossRefGoogle Scholar
Landahl, M.T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98 (2), 243251.10.1017/S0022112080000122CrossRefGoogle Scholar
Liu, J., Xiao, Y., Zhang, L., Li, M., Tao, J. & Xu, S. 2020 Extension at the downstream end of turbulent band in channel flow. Phys. Fluids 32 (12), 121703.10.1063/5.0032272CrossRefGoogle Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289309.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), 134502.10.1103/PhysRevLett.106.134502CrossRefGoogle ScholarPubMed
Paranjape, C.S., Duguet, Y. & Hof, B. 2020 Oblique stripe solutions of channel flow. J. Fluid Mech. 897, A7.10.1017/jfm.2020.322CrossRefGoogle Scholar
Parente, E., Robinet, J.C., De Palma, P. & Cherubini, S. 2021 Minimal seeds for turbulent bands. J. Fluid Mech. arXiv:2107.10157.Google Scholar
Pralits, J.O., Bottaro, A. & Cherubini, S. 2015 Weakly nonlinear optimal perturbations. J. Fluid Mech. 785, 135151.10.1017/jfm.2015.622CrossRefGoogle Scholar
Prigent, A., Grégoire, G., Chaté, H., Dauchot, O. & van Saarloos, W. 2002 Large-scale finite-wavelength modulation within turbulent shear flows. Phys. Rev. Lett. 89 (1), 014501.CrossRefGoogle ScholarPubMed
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), 154502.10.1103/PhysRevLett.105.154502CrossRefGoogle ScholarPubMed
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, 415443.10.1017/jfm.2012.192CrossRefGoogle Scholar
Rabin, S.M.E., Caulfield, C.P. & Kerswell, R.R. 2012 Triggering turbulence efficiently in plane couette flow. J. Fluid Mech. 712, 244272.CrossRefGoogle Scholar
Reddy, S.C. & Henningson, D.S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209238.CrossRefGoogle Scholar
Shimizu, M. & Manneville, P. 2019 Bifurcations to turbulence in transitional channel flow. Phys. Rev. Fluids 4 (11), 113903.10.1103/PhysRevFluids.4.113903CrossRefGoogle Scholar
Skufca, J.D., Yorke, J.A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.10.1103/PhysRevLett.96.174101CrossRefGoogle Scholar
Song, B. & Xiao, X. 2020 Trigger turbulent bands directly at low Reynolds numbers in channel flow using a moving-force technique. J. Fluid Mech. 903, A43.CrossRefGoogle Scholar
Tao, J.J., Eckhardt, B. & Xiong, X.M. 2018 Extended localized structures and the onset of turbulence in channel flow. Phys. Rev. Fluids 3 (1), 011902.10.1103/PhysRevFluids.3.011902CrossRefGoogle Scholar
Tsukahara, T., Kawaguchi, Y. & Kawamura, H. 2014 An experimental study on turbulent-stripe structure in transitional channel flow. arXiv:1406.1378.Google Scholar
Tsukahara, T., Seki, Y., Kawamura, H. & Tochio, D. 2005 DNS of turbulent channel flow at very low Reynolds numbers. In Fourth International Symposium on Turbulence and Shear Flow Phenomena. Begel House.Google Scholar
Tuckerman, L.S. & Barkley, D. 2011 Patterns and dynamics in transitional plane Couette flow. Phys. Fluids 23 (4), 041301.10.1063/1.3580263CrossRefGoogle Scholar
Tuckerman, L.S., Chantry, M. & Barkley, D. 2020 Patterns in wall-bounded shear flows. Annu. Rev. Fluid Mech. 52, 343367.CrossRefGoogle Scholar
Tuckerman, L.S., Kreilos, T., Schrobsdorff, H., Schneider, T.M. & Gibson, J.F. 2014 Turbulent-laminar patterns in plane Poiseuille flow. Phys. Fluids 26 (11), 114103.10.1063/1.4900874CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows . Phys. Fluids 9, 883901.CrossRefGoogle Scholar
Wang, Z., Guet, C., Monchaux, R., Duguet, Y. & Eckhardt, B. 2020 Quadrupolar flows around spots in internal shear flows. J. Fluid Mech. 892, A27.CrossRefGoogle Scholar
Xiao, X. & Song, B. 2020 The growth mechanism of turbulent bands in channel flow at low Reynolds numbers. J. Fluid Mech. 883, R1.10.1017/jfm.2019.899CrossRefGoogle Scholar
Xiong, X., Tao, J., Chen, S. & Brandt, L. 2015 Turbulent bands in plane-Poiseuille flow at moderate Reynolds numbers. Phys. Fluids 27 (4), 041702.10.1063/1.4917173CrossRefGoogle Scholar