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Laminar–turbulent transition in channel flow with superhydrophobic surfaces modelled as a partial slip wall

Published online by Cambridge University Press:  24 October 2019

Francesco Picella Open the ORCID record for Francesco Picella [Opens in a new window] ,
J.-Ch. Robinet Open the ORCID record for J.-Ch. Robinet [Opens in a new window]  and
S. Cherubini Open the ORCID record for S. Cherubini [Opens in a new window]
Francesco Picella
Affiliation:
DynFluid – Arts et Métiers Paris, 151 Bd de l’Hôpital, 75013 Paris, France
J.-Ch. Robinet*
Affiliation:
DynFluid – Arts et Métiers Paris, 151 Bd de l’Hôpital, 75013 Paris, France
S. Cherubini
Affiliation:
Dipartimento di Meccanica, Matematica e Management (DMMM), Politecnico di Bari, Via Re David 200, 70126 Bari, Italy
*
†Email address for correspondence: jean-christophe.robinet@ensam.eu
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Abstract

Superhydrophobic surfaces are capable of trapping gas pockets within the micro-roughnesses on their surfaces when submerged in a liquid, with the overall effect of lubricating the flow on top of them. These bio-inspired surfaces have proven to be capable of dramatically reducing skin friction of the overlying flow in both laminar and turbulent regimes. However, their effect in transitional conditions, in which the flow evolution strongly depends on the initial conditions, has still not been deeply investigated. In this work the influence of superhydrophobic surfaces on several scenarios of laminar–turbulent transition in channel flow is studied by means of direct numerical simulations. A single phase incompressible flow has been considered and the effect of the micro-structured superhydrophobic surfaces has been modelled imposing a slip condition with given slip length at both walls. The evolution from laminar, to transitional, to fully developed turbulent flow has been followed starting from several different initial conditions. When modal disturbances issued from linear stability analyses are used for perturbing the laminar flow, as in supercritical conditions or in the classical K-type transition scenario, superhydrophobic surfaces are able to delay or even avoid the onset of turbulence, leading to a considerable drag reduction. Whereas, when transition is triggered by non-modal mechanisms, as in the optimal or uncontrolled transition scenarios, which are currently observed in noisy environments, these surfaces are totally ineffective for controlling transition. Superhydrophobic surfaces can thus be considered effective for delaying transition only in low-noise environments, where transition is triggered mostly by modal mechanisms.

JFM classification

Instability: Transition to turbulence Flow Control: Drag reduction
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 462 - 497
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© 2019 Cambridge University Press 

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References

Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301.Google Scholar
Aghdam, S. K. & Ricco, P. 2016 Laminar and turbulent flows over hydrophobic surfaces with shear-dependent slip length. Phys. Fluids 28 (3), 035109.Google Scholar
Alinovi, E.2018 Modelling the flow over superhydrophobic and liquid-impregnated surfaces. PhD dissertation, DICCA, University of Genova.Google Scholar
Alinovi, E. & Bottaro, A. 2018 Apparent slip and drag reduction for the flow over superhydrophobic and lubricant-impregnated surfaces. Phys. Rev. Fluids 3 (12), 124002.Google Scholar
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150.Google Scholar
Arenas, I., García, E., Fu, M. K., Orlandi, P., Hultmark, M. & Leonardi, S. 2019 Comparison between super-hydrophobic, liquid infused and rough surfaces: a direct numerical simulation study. J. Fluid Mech. 869, 500–525.Google Scholar
Auteri, F., Carini, M., Fournié, M., Fratantonio, D. & Giannetti, F. 2016 Global linear stability analysis of the flow around a superhydrophobic circular cylinder. In Springer Proceedings in Physics, pp. 165–170. Springer.Google Scholar
Bake, S., Meyer, D. G. W. & Rist, U. 2002 Turbulence mechanism in Klebanoff transition: a quantitative comparison of experiment and direct numerical simulation. J. Fluid Mech. 459, 17–243.Google Scholar
Barthlott, W. & Neinhuis, C. 1997 Purity of the sacred lotus, or escape from contamination in biological surfaces. Planta 202 (1), 1–8.Google Scholar
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105–129.Google Scholar
Bidkar, R. A., Leblanc, L., Kulkarni, A. J., Bahadur, V., Ceccio, S. L. & Perlin, M. 2014 Skin-friction drag reduction in the turbulent regime using random-textured hydrophobic surfaces. Phys. Fluids 26 (8), 085108.Google Scholar
Brandt, L., Cossu, C., Chomaz, J.-M., Huerre, P. & Henningson, D. S. 2003 On the convectively unstable nature of optimal streaks in boundary layers. J. Fluid Mech. 485, 221–242.Google Scholar
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.Google Scholar
Busse, A. & Sandham, N. D. 2012 Influence of an anisotropic slip-length boundary condition on turbulent channel flow. Phys. Fluids 24 (5), 055111.Google 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
Byun, D., Kim, J., Ko, H. S. & Park, H. C. 2008 Direct measurement of slip flows in superhydrophobic microchannels with transverse grooves. Phys. Fluids 20 (11), 113601.Google Scholar
Cassie, A. B. D. & Baxter, S. 1944 Wettability of porous surfaces. Trans. Faraday Soc. 40, 546.Google Scholar
Choi, C.-H., Westin, K. J. A. & Breuer, K. S. 2003 Apparent slip flows in hydrophilic and hydrophobic microchannels. Phys. Fluids 15 (10), 2897.Google Scholar
Daniello, R. J., Waterhouse, N. E. & Rothstein, J. P. 2009 Drag reduction in turbulent flows over superhydrophobic surfaces. Phys. Fluids 21 (8), 085103.Google Scholar
Davies, J., Maynes, D., Webb, B. W. & Woolford, B. 2006 Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs. Phys. Fluids 18 (8), 087110.Google Scholar
De Donati, E.2015 Stability of the flow in a channel with grooved wall. Master’s dissertation, AERO, Politecnico di Milano.Google Scholar
Duan, H. 2017 Underwater superhydrophobiciy: fundamentals and applications. Procedia IUTAM 20, 128–135.Google Scholar
Emami, B., Hemeda, A. A., Amrei, M. M., Luzar, A., el Hak, M. G. & Tafreshi, H. V. 2013 Predicting longevity of submerged superhydrophobic surfaces with parallel grooves. Phys. Fluids 25 (6), 062108.Google Scholar
Fairhall, C. T., Abderrahaman-Elena, N. & García-Mayoral, R. 2018 The effect of slip and surface texture on turbulence over superhydrophobic surfaces. J. Fluid Mech. 861, 88–118.Google Scholar
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2015 Hairpin-like optimal perturbations in plane Poiseuille flow. J. Fluid Mech. 775, R2.Google Scholar
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2017 Optimal bursts in turbulent channel flow. J. Fluid Mech. 817, 35–60.Google Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31 (8), 2093.Google Scholar
Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G.2008 nek5000. Available at: http://nek5000.mcs.anl.gov.Google Scholar
Fukagata, K., Kasagi, N. & Koumoutsakos, P. 2006 A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces. Phys. Fluids 18 (5), 051703.Google Scholar
Gilbert, N. & Kleiser, L. 1990 Near-wall phenomena in transition to turbulence. Near-Wall Turbulence (ed. Kline, S. J. & Afgan, N. H.), pp. 7–27.Google Scholar
Gogte, S., Vorobieff, P., Truesdell, R., Mammoli, A., van Swol, F., Shah, P. & Brinker, C. J. 2005 Effective slip on textured superhydrophobic surfaces. Phys. Fluids 17 (5), 051701.Google Scholar
Gose, J. W., Golovin, K., Boban, M., Mabry, J. M., Tuteja, A., Perlin, M. & Ceccio, S. L. 2018 Characterization of superhydrophobic surfaces for drag reduction in turbulent flow. J. Fluid Mech. 845, 560–580.Google Scholar
Guo, H., Borodulin, V. I., Kachanov, Y. S., Pan, C., Wang, J. J., Lian, Q. X. & Wang, S. F. 2010 Nature of sweep and ejection events in transitional and turbulent boundary layers. J. Turbul. 11, N34.Google Scholar
Haase, A. S., Wood, J. A., Lammertink, R. G. H. & Snoeijer, J. H. 2016 Why bumpy is better: the role of the dissipation distribution in slip flow over a bubble mattress. Phys. Rev. Fluids 1 (5), 054101.Google Scholar
Henoch, C., Krupenkin, T., Kolodner, P., Taylor, J., Hodes, M., Lyons, A., Peguero, C. & Breuer, K. 2006 Turbulent drag reduction using superhydrophobic surfaces. In 3rd AIAA Flow Control Conference. American Institute of Aeronautics and Astronautics.Google Scholar
Huang, S., Lv, P. & Duan, H. 2019 Morphology evolution of liquid–gas interface on submerged solid structured surfaces. Extreme Mech. Lett. 27, 34–51.Google Scholar
van Ingen, J. 2008 The eN method for transition prediction. Historical review of work at TU Delft. In 38th Fluid Dynamics Conference and Exhibit. American Institute of Aeronautics and Astronautics.Google Scholar
Jacobs, R. G. & Durbin, P. A. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.Google Scholar
Jelly, T. O., Jung, S. Y. & Zaki, T. A. 2014 Turbulence and skin friction modification in channel flow with streamwise-aligned superhydrophobic surface texture. Phys. Fluids 26 (9), 095102.Google Scholar
Jiménez, J. 2013 Near-wall turbulence. Phys. Fluids 25 (10), 101302.Google Scholar
Jung, Y. C. & Bhushan, B. 2009 Biomimetic structures for fluid drag reduction in laminar and turbulent flows. J. Phys.: Condens. Matter 22 (3), 035104.Google Scholar
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26 (1), 411–482.Google Scholar
Kendall, J. 1998 Experiments on boundary-layer receptivity to freestream turbulence. In 36th AIAA Aerospace Sciences Meeting and Exhibit. American Institute of Aeronautics and Astronautics.Google Scholar
Kleiser, L. & Zang, T. A. 1991 Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech. 23 (1), 495–537.Google Scholar
Lee, C., Choi, C.-H. & Kim, C.-J. 2016 Superhydrophobic drag reduction in laminar flows: a critical review. Exp. Fluids 57 (12), 176.Google Scholar
Lee, C. & Kim, C.-J. 2009 Maximizing the giant liquid slip on superhydrophobic microstructures by nanostructuring their sidewalls. Langmuir 25 (21), 12812–12818.Google Scholar
Lee, C. & Kim, C.-J. 2011 Underwater restoration and retention of gases on superhydrophobic surfaces for drag reduction. Phys. Rev. Lett. 106 (1), 014502.Google Scholar
Lee, M. K., Eckelman, L. D. & Hanratty, T. J. 1974 Identification of turbulent wall eddies through the phase relation of the components of the fluctuating velocity gradient. J. Fluid Mech. 66 (1), 17–33.Google Scholar
Li, Y., Alame, K. & Mahesh, K. 2017 Feature-resolved computational and analytical study of laminar drag reduction by superhydrophobic surfaces. Phys. Rev. Fluids 2 (5), 054002.Google Scholar
Ling, H., Srinivasan, S., Golovin, K., McKinley, G. H., Tuteja, A. & Katz, J. 2016 High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces. J. Fluid Mech. 801, 670–703.Google Scholar
Liu, Y., Wexler, J. S., Schönecker, C. & Stone, H. A. 2016 Effect of viscosity ratio on the shear-driven failure of liquid-infused surfaces. Phys. Rev. Fluids 1 (7), 074003.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.Google Scholar
Maali, A. & Bhushan, B. 2012 Measurement of slip length on superhydrophobic surfaces. Phil. Trans. R. Soc. A 370 (1967), 2304–2320.Google Scholar
Malm, J., Schlatter, P. & Sandham, N. D. 2011 A vorticity stretching diagnostic for turbulent and transitional flows. Theor. Comput. Fluid Dyn. 26 (6), 485–499.Google Scholar
Manneville, P. 2015 On the transition to turbulence of wall-bounded flows in general, and plane couette flow in particular. Eur. J. Mech. (B/Fluids) 49, 345–362.Google Scholar
Mao, X., Zaki, T. A., Sherwin, S. J. & Blackburn, H. M. 2017 Transition induced by linear and nonlinear perturbation growth in flow past a compressor blade. J. Fluid Mech. 820, 604–632.Google Scholar
Martell, M. B., Perot, J. B. & Rothstein, J. P. 2009 Direct numerical simulations of turbulent flows over superhydrophobic surfaces. J. Fluid Mech. 620, 31–41.Google Scholar
Martell, M. B., Rothstein, J. P. & Perot, J. B. 2010 An analysis of superhydrophobic turbulent drag reduction mechanisms using direct numerical simulation. Phys. Fluids 22 (6), 065102.Google Scholar
Min, T. & Kim, J. 2004 Effects of hydrophobic surface on skin-friction drag. Phys. Fluids 16 (7), L55–L58.Google Scholar
Min, T. & Kim, J. 2005 Effects of hydrophobic surface on stability and transition. Phys. Fluids 17 (10), 108106.Google Scholar
Navier, C. L. M. H.1823 Mémoire sur les lois du mouvement des fluides Mémoires de l’Académie Royale des Sciences de l’Institut de France.Google Scholar
Nishioka, M., Iid, A. S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72 (04), 731.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (04), 689.Google Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16 (12), 4635–4643.Google Scholar
Park, H., Park, H. & Kim, J. 2013 A numerical study of the effects of superhydrophobic surface on skin-friction drag in turbulent channel flow. Phys. Fluids 25 (11), 110815.Google Scholar
Park, H., Sun, G. & Chang-Jin Kim, C.-H. 2014 Superhydrophobic turbulent drag reduction as a function of surface grating parameters. J. Fluid Mech. 747, 722–734.Google Scholar
Philip, J. R. 1972 Flows satisfying mixed no-slip and no-shear conditions. Z. Angew. Math. Phys. ZAMP 23 (3), 353–372.Google Scholar
Picella, F., Bucci, M. A., Cherubini, S. & Robinet, J.-C. 2019 A synthetic forcing to trigger laminar–turbulent transition in parallel wall bounded flows via receptivity. J. Comput. Phys. 92–116.Google Scholar
Pralits, J. O., Alinovi, E. & Bottaro, A. 2017 Stability of the flow in a plane microchannel with one or two superhydrophobic walls. Phys. Rev. Fluids 2 (1), 013901.Google Scholar
Rastegari, A. & Akhavan, R. 2015 On the mechanism of turbulent drag reduction with super-hydrophobic surfaces. J. Fluid Mech. 773, R4.Google Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S. & Henningson, D. S. 1998 On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.Google Scholar
Reshotko, E. 2001 Transient growth: a factor in bypass transition. Phys. Fluids 13 (5), 1067–1075.Google Scholar
Rosenberg, B. J., Van Buren, T., Fu, M. K. & Smits, A. J. 2016 Turbulent drag reduction over air- and liquid-impregnated surfaces. Phys. Fluids 28 (1), 015103.Google Scholar
Rowin, W. A., Hou, J. & Ghaemi, S. 2017 Inner and outer layer turbulence over a superhydrophobic surface with low roughness level at low Reynolds number. Phys. Fluids 29 (9), 095106.Google Scholar
Rowin, W. A., Hou, J. & Ghaemi, S. 2018 Turbulent channel flow over riblets with superhydrophobic coating. Exp. Therm. Fluid Sci. 94, 192–204.Google Scholar
Sandham, N. D. & Kleiser, L. 1992 The late stages of transition to turbulence in channel flow. J. Fluid Mech. 245, 319.Google Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34 (1), 291–319.Google Scholar
Sayadi, T., Hamman, C. W. & Moin, P. 2013 Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480–509.Google Scholar
Schäffel, D., Koynov, K., Vollmer, D., Butt, H.-J. & Sönecker, C. 2016 Local flow field and slip length of superhydrophobic surfaces. Phys. Rev. Lett. 116 (13), 134501.Google Scholar
Schellenberger, F., Encinas, N., Vollmer, D. & Butt, H.-J. 2016 How water advances on superhydrophobic surfaces. Phys. Rev. Lett. 116 (9), 096101.Google Scholar
Schlatter, P., Stolz, S. & Kleiser, L. 2006 Large-eddy simulation of spatial transition in plane channel flow. J. Turbul. 7, N33.Google Scholar
Schlatter, P. C.2005 Large-eddy simulation of transition and turbulence in wall-bounded shear flow. PhD thesis, ETH Zurich.Google Scholar
Schmid, P. J. & Brandt, L. 2014 Analysis of fluid systems: stability, receptivity, sensitivity: lecture notes from the FLOW-NORDITA summer school on advanced instability methods for complex flows, Stockholm, Sweden, 2013. Appl. Mech. Rev. 66 (2), 024803.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schnitzer, O. & Yariv, E. 2018 Small-solid-fraction approximations for the slip-length tensor of micropillared superhydrophobic surfaces. J. Fluid Mech. 843, 637–652.Google Scholar
Seo, J., García-Mayoral, R. & Mani, A. 2015 Pressure fluctuations and interfacial robustness in turbulent flows over superhydrophobic surfaces. J. Fluid Mech. 783, 448–473.Google Scholar
Seo, J., García-Mayoral, R. & Mani, A. 2017 Turbulent flows over superhydrophobic surfaces: flow-induced capillary waves, and robustness of air–water interfaces. J. Fluid Mech. 835, 45–85.Google Scholar
Seo, J. & Mani, A. 2016 On the scaling of the slip velocity in turbulent flows over superhydrophobic surfaces. Phys. Fluids 28 (2), 025110.Google Scholar
Seo, J. & Mani, A. 2018 Effect of texture randomization on the slip and interfacial robustness in turbulent flows over superhydrophobic surfaces. Phys. Rev. Fluids 3 (4), 044601.Google Scholar
Srinivasan, S., Kleingartner, J. A., Gilbert, J. B., Cohen, R. E., Milne, A. J. B. & McKinley, G. H. 2015 Sustainable drag reduction in turbulent Taylor–Couette flows by depositing sprayable superhydrophobic surfaces. Phys. Rev. Lett. 114 (1), 014501.Google Scholar
Tretheway, D. C. & Meinhart, C. D. 2002 Apparent fluid slip at hydrophobic microchannel walls. Phys. Fluids 14 (3), L9–L12.Google Scholar
Truesdell, R., Mammoli, A., Vorobieff, P., van Swol, F. & Brinker, C. J. 2006 Drag reduction on a patterned superhydrophobic surface. Phys. Rev. Lett. 97 (4), 044504.Google Scholar
Tsai, P., Peters, A. M., Pirat, C., Wessling, M., Lammertink, R. G. H. & Lohse, D. 2009 Quantifying effective slip length over micropatterned hydrophobic surfaces. Phys. Fluids 21 (11), 112002.Google Scholar
Watanabe, K., Yanuar & Udagawa, H. 1999 Drag reduction of Newtonian fluid in a circular pipe with a highly water-repellent wall. J. Fluid Mech. 381, 225–238.Google Scholar
Wenzel, R. N. 1936 Resistance of solid surfaces to wetting by water. Ind. Engng Chem. 28 (8), 988–994.Google Scholar
Wexler, J. S., Jacobi, I. & Stone, H. A. 2015 Shear-driven failure of liquid-infused surfaces. Phys. Rev. Lett. 114, 168301.Google Scholar
Woolford, B., Prince, J., Maynes, D. & Webb, B. W. 2009 Particle image velocimetry characterization of turbulent channel flow with rib patterned superhydrophobic walls. Phys. Fluids 21 (8), 085106.Google Scholar
Xiang, Y., Huang, S., Lv, P., Xue, Y., Su, Q. & Duan, H. 2017 Ultimate stable underwater superhydrophobic state. Phys. Rev. Lett. 119 (13), 134501.Google Scholar
Xu, H., Crick, C. R. & Poole, R. J. 2018 Evaluating the resilience of superhydrophobic materials using the slip-length concept. J. Mater. Chem. A 6 (10), 4458–4465.Google Scholar
Xu, M., Sun, G. & Kim, C.-J. 2014 Infinite lifetime of underwater superhydrophobic states. Phys. Rev. Lett. 113 (13), 136103.Google Scholar
Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P. & Bocquet, L. 2007 Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids 19 (12), 123601.Google Scholar
Ye, Q., Schrijer, F. F. J. & Scarano, F. 2018 On Reynolds number dependence of micro-ramp-induced transition. J. Fluid Mech. 837, 597–626.Google Scholar
Yu, K. H., Teo, C. J. & Khoo, B. C. 2016 Linear stability of pressure-driven flow over longitudinal superhydrophobic grooves. Phys. Fluids 28 (2), 022001.Google Scholar
Zampogna, G. A., Magnaudet, J. & Bottaro, A. 2019 Generalized slip condition over rough surfaces. J. Fluid Mech. 858, 407–436.Google Scholar
Zang, T. A. & Krist, S. E. 1989 Numerical experiments on stability and transition in plane channel flow. Theor. Comput. Fluid Dyn. 1 (1), 41–64.Google Scholar
Zhang, J., Tian, H., Yao, Z., Hao, P. & Jiang, N. 2015 Mechanisms of drag reduction of superhydrophobic surfaces in a turbulent boundary layer flow. Exp. Fluids 56 (9), 179.Google Scholar
Zhang, J., Yao, Z. & Hao, P. 2016 Drag reductions and the air–water interface stability of superhydrophobic surfaces in rectangular channel flow. Phys. Rev. E 94 (5), 053117.Google Scholar
Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396.Google Scholar
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