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Exact coherent states in plane Couette flow under spanwise wall oscillation

Published online by Cambridge University Press:  15 August 2022

Yacine Bengana Open the ORCID record for Yacine Bengana [Opens in a new window] ,
Qiang Yang Open the ORCID record for Qiang Yang [Opens in a new window] ,
Guohua Tu  and
Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window]
Yacine Bengana*
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
Qiang Yang
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang 621000, PR China
Guohua Tu*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang 621000, PR China
Yongyun Hwang
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods $({A_w}, T)$. It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around ${T^{+}} \approx 100$. The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.

JFM classification

Flow Control: Flow Control Turbulent Flows: Turbulent Flows Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 947 , 25 September 2022 , A2
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Agostini, L. & Leschziner, M. 2021 Statistical analysis of outer large-scale/inner-layer interactions in channel flow subjected to oscillatory drag-reducing wall motion using a multiple-variable joint-probability-density function methodology. J. Fluid Mech. 923, A25.10.1017/jfm.2021.585CrossRefGoogle Scholar
Agostini, L., Touber, E. & Leschziner, M.A. 2014 Spanwise oscillatory wall motion in channel flow: drag-reduction mechanisms inferred from DNS-predicted phase-wise property variations at $Re_\tau =1000$. J. Fluid Mech. 743, 606–635.10.1017/jfm.2014.40CrossRefGoogle Scholar
del Álamo, J.C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.10.1017/S0022112006000607CrossRefGoogle Scholar
Auteri, F., Baron, A., Belan, M., Campanardi, G. & Quadrio, M. 2010 Experimental assessment of drag reduction by traveling waves in a turbulent pipe flow. Phys. Fluids 22 (11), 115103.10.1063/1.3491203CrossRefGoogle Scholar
Baron, A. & Quadrio, M. 1995 Turbulent drag reduction by spanwise wall oscillations. Appl. Sci. Res. 55 (4), 311–326.10.1007/BF00856638CrossRefGoogle Scholar
Bewley, T.R. 2014 Numerical Renaissance: Simulation, Optimisation and Control. Renaissance Press.Google Scholar
Bird, J., Santer, M. & Morrison, J.F. 2018 Experimental control of turbulent boundary layers with in-plane travelling waves. Flow Turbul. Combust. 100 (4), 1015–1035.CrossRefGoogle ScholarPubMed
Blesbois, O., Chernyshenko, S.I., Touber, E. & Leschziner, M.A. 2013 Pattern prediction by linear analysis of turbulent flow with drag reduction by wall oscillation. J. Fluid Mech. 724, 607–641.10.1017/jfm.2013.165CrossRefGoogle Scholar
Brandt, L. 2014 The lift-up effect: the linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. B/Fluids 47, 80–96.CrossRefGoogle Scholar
Budanur, N.B., Short, K.Y., Farazmand, M., Willis, A.P. & Cvitanovic, P. 2017 Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833, 274–301.10.1017/jfm.2017.699CrossRefGoogle Scholar
Butler, K.M. & Farrell, B.F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774–777.10.1063/1.858663CrossRefGoogle Scholar
Cassinelli, A., de Giovanetti, M. & Hwang, Y. 2017 Streak instability in near-wall turbulence revisited. J. Turbul. 18 (5), 443–464.10.1080/14685248.2017.1294757CrossRefGoogle Scholar
Chernyshenko, S.I. & Baig, M.F. 2005 The mechanism of streak formation in near-wall turbulence. J. Fluid Mech. 544, 99–131.10.1017/S0022112005006506CrossRefGoogle Scholar
Choi, H., Moin, P. & Kim, J. 1993 Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503–539.10.1017/S0022112093002575CrossRefGoogle Scholar
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech. 262, 75–110.CrossRefGoogle Scholar
Choi, J.-I., Xu, C.-X. & Sung, H.J. 2002 Drag reduction by spanwise wall oscillation in wall-bounded turbulent flows. AIAA J. 40 (5), 842–850.10.2514/2.1750CrossRefGoogle Scholar
Choi, K.-S. 1989 Near-wall structure of a turbulent boundary layer with riblets. J. Fluid Mech. 208, 417–458.10.1017/S0022112089002892CrossRefGoogle Scholar
Choi, K.-S., Debisschop, J. & Clayton, B.R. 1998 Turbulent boundary-layer control by means of spanwise-wall oscillation. AIAA J. 36 (7), 1157–1163.10.2514/2.526CrossRefGoogle Scholar
Choi, K.-S. & Graham, M. 1998 Drag reduction of turbulent pipe flows by circular-wall oscillation. Phys. Fluids 10 (1), 7–9.10.1063/1.869538CrossRefGoogle Scholar
Choi, K.-S., Jukes, T. & Whalley, R. 2011 Turbulent boundary-layer control with plasma actuators. Phil. Trans. R. Soc. A 369, 1443–1458.10.1098/rsta.2010.0362CrossRefGoogle ScholarPubMed
Chung, Y.M., Hurst, E. & Yang, Q. 2013 DNS of turbulent drag reduction by spanwise wall forcing: the Reynolds number effect. In 8th International Symposium on Turbulence and Shear Flow Phenomena, article CON1C.10.1615/TSFP8.1330CrossRefGoogle Scholar
Davis, E.A. & Park, J.S. 2020 Dynamics of laminar and transitional flows over slip surfaces: effects on the laminar–turbulent separatrix. J. Fluid Mech. 894, A16.10.1017/jfm.2020.282CrossRefGoogle Scholar
Doohan, P., Bengana, Y., Yang, Q., Willis, A.P. & Hwang, Y. 2021 The state space and travelling-wave solutions in two-scale wall-bounded turbulence. J. Fluid Mech. (submitted).Google Scholar
Doohan, P., Willis, A.P. & Hwang, Y. 2019 Shear stress-driven flow: the state space of near-wall turbulence as $Re_\tau \rightarrow \infty$. J. Fluid Mech. 874, 606–638.10.1017/jfm.2019.472CrossRefGoogle Scholar
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, 255–274.10.1017/S0022112008003248CrossRefGoogle Scholar
Duque-Daza, C.A., Baig, M.F., Lockerby, D.A., Chernyshenko, S.I. & Davies, C. 2012 Modelling turbulent skin-friction control using linearized Navier–Stokes equations. J. Fluid Mech. 702, 403–414.10.1017/jfm.2012.189CrossRefGoogle Scholar
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Eckhardt, B. & Zammert, S. 2018 Small scale exact coherent structures at large Reynolds numbers in plane Couette flow. Nonlinearity 31 (2), R66–R77.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487–488.10.1063/1.861156CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle Scholar
Fukagata, K., Iwamoto, K. & Kasagi, N. 2002 Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14 (11), L73–L76.CrossRefGoogle Scholar
García-Mayoral, R. & Jiménez, J. 2011 Drag reduction by riblets. Phil. Trans. R. Soc. A 369 (1940), 1412–1427.10.1098/rsta.2010.0359CrossRefGoogle ScholarPubMed
Gatti, D., Güttler, A., Frohnapfel, B. & Tropea, C. 2015 Experimental assessment of spanwise-oscillating dielectric electroactive surfaces for turbulent drag reduction in an air channel flow. Exp. Fluids 56 (5), 110.CrossRefGoogle Scholar
Gatti, D. & Quadrio, M. 2013 Performance losses of drag-reducing spanwise forcing at moderate values of the Reynolds number. Phys. Fluids 25 (12), 125109.CrossRefGoogle Scholar
Gatti, D. & Quadrio, M. 2016 Reynolds-number dependence of turbulent skin-friction drag reduction induced by spanwise forcing. J. Fluid Mech. 802, 553–582.10.1017/jfm.2016.485CrossRefGoogle Scholar
Gibson, J.F., Halcrow, J. & Cvitanovic, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Gibson, J.F., Halcrow, J. & Cvitanovic, P. 2009 Equilibrium and traveling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.10.1017/S0022112009990863CrossRefGoogle Scholar
de Giovanetti, M., Hwang, Y. & Choi, H. 2016 Skin-friction generation by attached eddies in turbulent channel flow. J. Fluid Mech. 808, 511–538.10.1017/jfm.2016.665CrossRefGoogle Scholar
Goodman, J., Hou, T. & Tadmor, E. 1994 On the stability of the unsmoothed Fourier method for hyperbolic equations. Numer. Math. 67 (1), 93–129.10.1007/s002110050019CrossRefGoogle Scholar
Graham, M.D. & Floryan, D. 2021 Exact coherent states and the nonlinear dynamics of wall-bounded turbulent flows. Annu. Rev. Fluid Mech. 53, 227–253.CrossRefGoogle Scholar
Gustavsson, L.H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.10.1017/S002211209100174XCrossRefGoogle Scholar
Hack, M.J.P. & Zaki, T.A. 2015 Modal and non-modal stability of boundary layers forced by spanwise wall oscillations. J. Fluid Mech. 778, 389–427.10.1017/jfm.2015.387CrossRefGoogle Scholar
Hamilton, J.M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Hicks, P.D. & Ricco, P. 2015 Laminar streak growth above a spanwise oscillating wall. J. Fluid Mech. 768, 348–374.CrossRefGoogle 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 travelling waves in turbulent pipe flow. Science 1305, 1594.10.1126/science.1100393CrossRefGoogle Scholar
Hof, B., de Lozar, A., Avila, M., Tu, X. & Schneider, T.M. 2010 Eliminating turbulence in spatially intermittent flows. Science 327, 1491–1494.CrossRefGoogle ScholarPubMed
Hurst, E., Yang, Q. & Chung, Y.M. 2014 The effect of Reynolds number on turbulent drag reduction by streamwise travelling waves. J. Fluid Mech. 759, 28–55.10.1017/jfm.2014.524CrossRefGoogle Scholar
Hwang, Y. 2013 Near-wall turbulent fluctuations in the absence of wide outer motions. J. Fluid Mech. 723, 264–288.10.1017/jfm.2013.133CrossRefGoogle Scholar
Hwang, Y. & Bengana, Y. 2016 Self-sustaining process of minimal attached eddies in turbulent channel flow. J. Fluid Mech. 795, 708–738.10.1017/jfm.2016.226CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
Hwang, Y., Willis, A.P. & Cossu, C. 2016 Invariant solutions of minimal large-scale structures in turbulent channel flow for $Re_\tau$ up to 1000. J. Fluid Mech. 802, R1.CrossRefGoogle Scholar
Ibrahim, J.I., Yang, Q., Doohan, P. & Hwang, Y. 2019 Phase-space dynamics of opposition control in wall-bounded turbulent flows. J. Fluid Mech. 861, 29–54.10.1017/jfm.2018.905CrossRefGoogle Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.10.1143/JPSJ.70.703CrossRefGoogle Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.10.1017/S0022112091002033CrossRefGoogle Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.10.1017/S0022112099005066CrossRefGoogle Scholar
Jozsa, T.I. 2019 Analytical solutions of incompressible laminar channel and pipe flows driven by in-plane wall oscillations. Phys. Fluids 31 (8), 083605.CrossRefGoogle Scholar
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), 1605–1607.10.1063/1.858381CrossRefGoogle Scholar
Karniadakis, G.E. & Choi, K. -S. 2003 Mechanisms on transverse motions in turbulent wall flows. Annu. Rev. Fluid Mech. 35, 45–62.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
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.10.1017/S0022112001006243CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.10.1146/annurev-fluid-120710-101228CrossRefGoogle Scholar
Kerswell, R.R. 2005 Recent progress in understanding the transition to turbulence. Nonlinearity 18, R17–R44.10.1088/0951-7715/18/6/R01CrossRefGoogle Scholar
Kerswell, R.R. & Tutty, O.R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.10.1017/S0022112007006301CrossRefGoogle 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
Kravchenko, A.G., Choi, H. & Moin, P. 1993 On the relation of near-wall streamwise vortices to wall skin friction in turbulent boundary layers. Phys. Fluids A 5, 3307.CrossRefGoogle Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22 (4), 047505.CrossRefGoogle ScholarPubMed
Kreilos, T., Gibson, J.F. & Schneider, T.M. 2016 a Localized travelling waves in the asymptotic suction boundary layer. J. Fluid Mech. 795, R3.CrossRefGoogle Scholar
Kreilos, T., Khapko, T., Schlatter, P., Duguet, Y., Henningson, D.S. & Eckhardt, B. 2016 b Bypass transition and spot nucleation in boundary layers. Phys. Rev. Fluids 1 (4), 043602.10.1103/PhysRevFluids.1.043602CrossRefGoogle Scholar
Kühnen, J., Song, B., Scarselli, D., Budanur, N.B., Riedl, M., Willis, A.P., Avila, M. & Hof, B. 2018 Destabilizing turbulence in pipe flow. Nat. Phys. 14 (4), 386–390.CrossRefGoogle Scholar
Kushwaha, A. 2017 Near-wall turbulence and utilisation of the nonlinear dynamics towards control of turbulent flow. PhD thesis, University of Wisconsin-Madison.Google Scholar
Landahl, M.T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.10.1017/S0022112080000122CrossRefGoogle Scholar
Langham, J., Eaves, T.S. & Kerswell, R.R. 2020 Stably stratified exact coherent structures in shear flow: the effect of Prandtl number. J. Fluid Mech. 882, A10.CrossRefGoogle Scholar
Lardeau, S. & Leschziner, M.A. 2013 The streamwise drag-reduction response of a boundary layer subjected to a sudden imposition of transverse oscillatory wall motion. Phys. Fluids 25 (7), 075109.CrossRefGoogle Scholar
Lee, J., Jelly, T.O. & Zaki, T.A. 2015 Effect of Reynolds number on turbulent drag reduction by superhydrophobic surface textures. Flow Turbul. Combust. 95, 277–300.CrossRefGoogle Scholar
Leschziner, M.A. 2020 Friction-drag reduction by transverse wall motion – a review. J. Mech. 36 (5), 649–663.10.1017/jmech.2020.31CrossRefGoogle Scholar
Leschziner, M.A., Choi, H. & Choi, K. -S. 2011 Flow-control approaches to drag reduction in aerodynamics: progress and prospects. Phil. Trans. R. Soc. A 369 (1940), 1349–1351.CrossRefGoogle ScholarPubMed
Lessen, M., Sadler, S.G. & Liu, T.-Y. 1968 Stability of pipe Poiseuille flow. Phys. Fluids 11 (7), 1404–1409.10.1063/1.1692122CrossRefGoogle Scholar
Li, W. & Graham, M.D. 2007 Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19 (8), 083101.CrossRefGoogle Scholar
Li, W., Xi, L. & Graham, M.D. 2006 Nonlinear travelling waves as a framework for understanding turbulent drag reduction. J. Fluid Mech. 565, 353–362.CrossRefGoogle Scholar
Lustro, J.R.T., Kawahara, G., Van Veen, L., Shimizu, M. & Kokubu, H. 2019 The onset of transient turbulence in minimal plane Couette flow. J. Fluid Mech. 862, R2.CrossRefGoogle Scholar
Marensi, E., Ding, Z., Willis, A.P. & Kerswell, R.R. 2020 Designing a minimal baffle to destabilise turbulence in pipe flows. J. Fluid Mech. 900, A31.10.1017/jfm.2020.518CrossRefGoogle 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, 850–875.10.1017/jfm.2018.1012CrossRefGoogle Scholar
Marusic, I., Chandran, D., Rouhi, A., Fu, M.K., Wine, D., Holloway, B., Chung, D. & Smits, A.J. 2021 An energy-efficient pathway to turbulent drag reduction. Nat. Commun. 12 (1), 5805.CrossRefGoogle ScholarPubMed
McKeon, B.J. & Sharma, A.S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.10.1017/S002211201000176XCrossRefGoogle Scholar
Moarref, R. & Jovanović, M.R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205–240.10.1017/jfm.2012.272CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Nagata, M. & Deguchi, K. 2013 Mirror-symmetric exact coherent states in plane Poiseuille flow. J. Fluid Mech. 735, R4.10.1017/jfm.2013.515CrossRefGoogle Scholar
Park, J.S. & Graham, M.D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430–454.CrossRefGoogle Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21, 015109.CrossRefGoogle Scholar
Quadrio, M. 2011 Drag reduction in turbulent boundary layers by in-plane wall motion. Phil. Trans. R. Soc. A 369 (1940), 1428–1442.10.1098/rsta.2010.0366CrossRefGoogle ScholarPubMed
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise oscillations. J. Fluid Mech. 521, 251–271.10.1017/S0022112004001855CrossRefGoogle Scholar
Quadrio, M., Ricco, P. & Viotti, C. 2009 Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction. J. Fluid Mech. 627, 161–178.10.1017/S0022112009006077CrossRefGoogle Scholar
Quadrio, M. & Sibilla, S. 2000 Numerical simulation of turbulent flow in a pipe oscillating around its axis. J. Fluid Mech. 424, 217–241.10.1017/S0022112000001889CrossRefGoogle 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
Reetz, F., Kreilos, T. & Schneider, T.M. 2019 Exact invariant solution reveals the origin of self-organized oblique turbulent–laminar stripes. Nat. Commun. 10 (1), 2277.10.1038/s41467-019-10208-xCrossRefGoogle ScholarPubMed
Ricco, P. 2004 Modification of near-wall turbulence due to spanwise wall oscillations. J. Turbul. 5 (1), 024.CrossRefGoogle Scholar
Ricco, P., Ottonelli, C., Hasegawa, Y. & Quadrio, M. 2012 Changes in turbulent dissipation in a channel flow with oscillating walls. J. Fluid Mech. 700, 77–104.10.1017/jfm.2012.97CrossRefGoogle Scholar
Ricco, P. & Quadrio, M. 2008 Wall-oscillation conditions for drag reduction in turbulent channel flow. Intl J. Heat Fluid Flow 29 (4), 891–902.CrossRefGoogle Scholar
Ricco, P., Skote, M. & Leschziner, M.A. 2021 A review of turbulent skin-friction drag reduction by near-wall transverse forcing. Prog. Aerosp. Sci. 123, 100713.CrossRefGoogle Scholar
Romanov, V.A. 1973 Stability of plane-parallel Couette flow. Funct. Anal. Applics. 7 (2), 137–146.10.1007/BF01078886CrossRefGoogle Scholar
Schlichting, H. 1979 Boundary-Layer Theory. Mc Graw-Hill.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 (3), 037301.CrossRefGoogle ScholarPubMed
Schoppa, S. & Hussain, F. 1998 A large-scale control strategy for drag reduction in turbulent boundary layers. Phys. Fluids 10, 1049.10.1063/1.869789CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Sendstad, O. 1992 The near-wall mechanics of three-dimensional turbulent boundary layers. PhD thesis, Stanford University.Google Scholar
Skote, M. 2013 Comparison between spatial and temporal wall oscillations in turbulent boundary layer flows. J. Fluid Mech. 730, 273–294.CrossRefGoogle Scholar
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
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), 208301.10.1103/PhysRevLett.89.208301CrossRefGoogle Scholar
Touber, E. & Leschziner, M.A. 2012 Near-wall streak modification by spanwise oscillatory wall motion and drag-reduction mechanisms. J. Fluid Mech. 693, 150–200.CrossRefGoogle Scholar
Viotti, C., Quadrio, M. & Luchini, P. 2009 Streamwise oscillation of spanwise velocity at the wall of a channel for turbulent drag reduction. Phys. Fluids 21, 115109.CrossRefGoogle Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.10.1063/1.869185CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.10.1103/PhysRevLett.81.4140CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.10.1017/S0022112001004189CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.CrossRefGoogle Scholar
Walsh, M.J. 1983 Riblets as a viscous drag reduction technique. AIAA J. 21 (4), 485–486.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R.R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Willis, A.P., Cvitanović, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514–540.CrossRefGoogle Scholar
Willis, A.P., Hwang, Y. & Cossu, C. 2010 Optimally amplified large-scale streaks and drag reduction in the turbulent pipe flow. Phys. Rev. E 82, 036321.10.1103/PhysRevE.82.036321CrossRefGoogle ScholarPubMed
Xu, C.-X. & Huang, W.-X. 2005 Transient response of Reynolds stress transport to spanwise wall oscillation in a turbulent channel flow. Phys. Fluids 17 (1), 018101.CrossRefGoogle Scholar
Xu, J., Dong, S., Maxey, M.R. & Karniadakis, G.E. 2007 Turbulent drag reduction by constant near-wall forcing. J. Fluid Mech. 582, 79–101.CrossRefGoogle Scholar
Yakeno, A., Hasegawa, Y. & Kasagi, N. 2014 Modification of quasi-streamwise vortical structure in a drag-reduced turbulent channel flow with spanwise wall oscillation. Phys. Fluids 26 (8), 085109.10.1063/1.4893903CrossRefGoogle Scholar
Yang, Q. 2016 Turbulent skin-friction drag reduction control by spanwise motion. PhD thesis, University of Warwick.Google Scholar
Yang, Q. & Hwang, Y. 2019 Modulation of attached exact coherent states under spanwise wall oscillation. In 11th International Symposium on Turbulence and Shear Flow Phenomena, article 257.Google Scholar
Yang, Q., Willis, A.P. & Hwang, Y. 2019 Exact coherent states of attached eddies in channel flow. J. Fluid Mech. 862, 1029–1059.CrossRefGoogle Scholar
Yao, J., Chen, X. & Hussain, F. 2018 Drag control in wall-bounded turbulent flows via spanwise opposed wall-jet forcing. J. Fluid Mech. 852, 678–709.CrossRefGoogle Scholar
Yao, J., Chen, X. & Hussain, F. 2019 Reynolds number effect on drag control via spanwise wall oscillation in turbulent channel flows. Phys. Fluids 31 (8), 085108.10.1063/1.5111651CrossRefGoogle Scholar

Bengana et al. supplementary movie 1

\item \textbf{Caption for Movie 1} Dynamical variation of the streamwise-averaged low-speed (blue contour) and high-speed (red contour) streaks and in-plan vortex structure (pink arrows) during one spanwise wall oscillation period for case A ($Re=400$, $T^+=166.4$, $A_w^+=2.6$). The streaks are visualised by the contour of $\langle u' \rangle _x$, and the vortex structure is visualised by the vector of $\left(\langle v' \rangle _x, \langle w' \rangle _x - \langle w' \rangle _{x,z}\right)$.

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Video 3.8 MB

Bengana et al. supplementary movie 2

\item \textbf{Caption for Movie 2} Dynamical variation of the streamwise-averaged low-speed (blue contour) and high-speed (red contour) streaks and in-plan vortex structure (pink arrows) during one spanwise wall oscillation period for case B ($Re=400$, $T^+=166.4$, $A_w^+=4.4$). The streaks are visualised by the contour of $\langle u' \rangle _x$, and the vortex structure is visualised by the vector of $\left(\langle v' \rangle _x, \langle w' \rangle _x - \langle w' \rangle _{x,z}\right)$.

Download Bengana et al. supplementary movie 2(Video)
Video 3.3 MB

Bengana et al. supplementary movie 3

\item \textbf{Caption for Movie 3} Exact coherent state under spanwise wall oscillation for case A ($Re=400$, $T^+=166.4$, $A_w^+=2.6$). The time variation during one spanwise wall oscillation period is shown. Yellow iso-surface ($u'^+=2.0$) indicates the high-speed streak, and green iso-surface ($v'^+=-0.2$) represents the vortical structure. Viewed from the wall into the flow.

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Video 1.4 MB

Bengana et al. supplementary movie 4

\item \textbf{Caption for Movie 4} Exact coherent state under spanwise wall oscillation for case B ($Re=400$, $T^+=166.4$, $A_w^+=4.4$). The time variation during one spanwise wall oscillation period is shown. Yellow iso-surface ($u'^+=2.0$) indicates the high-speed streak, and green iso-surface ($v'^+=-0.1$) represents the vortical structure. Viewed from the wall into the flow.

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Video 871 KB
Supplementary material: PDF

Bengana et al. supplementary material

Captions for movies 1-4

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