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The state space and travelling-wave solutions in two-scale wall-bounded turbulence

Published online by Cambridge University Press:  30 August 2022

Patrick Doohan Open the ORCID record for Patrick Doohan [Opens in a new window] ,
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] ,
Ashley P. Willis Open the ORCID record for Ashley P. Willis [Opens in a new window]  and
Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window]
Patrick Doohan
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Yacine Bengana
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Qiang Yang
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang 621000, PR China
Ashley P. Willis
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
Yongyun Hwang*
Affiliation:
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: y.hwang@imperial.ac.uk
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Abstract

The computation of invariant solutions and the visualisation of the associated state space have played a key role in the understanding of transition and the self-sustaining process in wall-bounded shear flows. In this study, an extension of this approach is sought for a turbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multi-scale near-wall turbulence, which resolves two adjacent spanwise integral length scales of motion, is considered using a shear stress-driven flow model (Doohan, Willis & Hwang J. Fluid Mech., vol. 913, 2021, A8). The edge state, 26 travelling waves and two periodic orbits are computed, which represent either the large- or small-scale self-sustaining processes. Given that the spanwise length scales are not widely separated here, it could be envisaged that turbulent trajectories visit these solutions in the state space. Considering the intra- and inter-scale dynamics of the flow, numerous phase portraits are examined, but the turbulent state is not found to approach any of these solutions. A detailed analysis reveals that this is due to the lack of scale interaction processes captured by the invariant solutions, including the mean–fluctuation interaction, the energy cascade in the streamwise wavenumber space and the cascade-driven energy production discovered recently. There is a single solution that resembles turbulence much more than the others, which captures two-scale energetics and a scale interaction process involving energy feeding from small to large spanwise scales through the subharmonic sinuous streak instability mode. Based on these observations, it is conjectured that the state space view of turbulent trajectories wandering between solutions would need suitable refinement to model multi-scale turbulence, when each solution does not represent multi-scale processes of turbulence. In particular, invariant solutions that are inherently multi-scale would be required.

JFM classification

Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
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Journal of Fluid Mechanics , Volume 947 , 25 September 2022 , A41
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© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Adrian, R.J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301.CrossRefGoogle Scholar
Agostini, L. & Leschziner, M. 2016 Predicting the response of small-scale near-wall turbulence to large-scale outer motions. Phys. Fluids 28 (1), 015107.CrossRefGoogle Scholar
del Álamo, J.C. & Jimenez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
del Álamo, J.C. & Jiménez, J. 2009 Estimation of turbulent convection velocities and corrections to Taylor's approximation. J. Fluid Mech. 640, 5–26.CrossRefGoogle Scholar
del Álamo, J.C., Jimenez, J., Zandonade, P. & Moser, R.D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J.L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.CrossRefGoogle Scholar
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.CrossRefGoogle ScholarPubMed
Azimi, S. & Schneider, T.M. 2020 Self-similar invariant solution in the near-wall region of a turbulent boundary layer at asymptotically high Reynolds numbers. J. Fluid Mech. 888, A15.CrossRefGoogle Scholar
Baars, W.J., Hutchins, N. & Marusic, I. 2017 Reynolds number trend of hierarchies and scale interactions in turbulent boundary layers. Phil. Trans. R. Soc. Lond. A 375, 20160077.Google ScholarPubMed
Bae, H.J., Lozano-Duran, A. & McKeon, B.J. 2021 Nonlinear mechanism of the self-sustaining process in the buffer and logarithmic layer of wall-bounded flows. J. Fluid Mech. 914, A3.CrossRefGoogle Scholar
Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.CrossRefGoogle Scholar
Benny, D.J. 1984 The evolution of disturbances in shear flows at high Reynolds numbers. Stud. Appl. Maths 70, 1–19.CrossRefGoogle Scholar
Bewley, T.R. 2014 Numerical Renaissance: Simulation, Optimization, & Control. Renaissance.Google Scholar
Blackburn, H., Deguchi, K. & Hall, P. 2021 Distributed vortex–wave interactions: the relation of self-similarity to the attached eddy hypothesis. J. Fluid Mech. 924, A8.CrossRefGoogle Scholar
Brand, E. & Gibson, J.F 2014 A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R3.CrossRefGoogle Scholar
Budanur, N.B., Short, K.Y., Farazmand, M., Willis, A.P. & Cvitanović, P. 2017 Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833, 274–301.CrossRefGoogle Scholar
Butler, K.M. & Farrell, B.F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A: Fluid 5 (3), 774–777.CrossRefGoogle Scholar
Cassinelli, A., de Giovanetti, M. & Hwang, Y. 2017 Streak instability in near-wall turbulence revisited. J. Turbul. 18 (5), 443–464.CrossRefGoogle Scholar
Chandler, G.J. & Kerswell, R.R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554–595.CrossRefGoogle Scholar
Cho, M., Hwang, Y. & Choi, H. 2018 Scale interactions and spectral energy transfer in turbulent channel flow. J. Fluid Mech. 854, 474–504.CrossRefGoogle Scholar
Cimarelli, A., De Angelis, E., Jimenez, J. & Casciola, C.M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417–436.CrossRefGoogle Scholar
Clever, R.M. & Busse, F.H. 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Cvitanovic, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G., Whelan, N. & Wirzba, A. 2005 Chaos: Classical and Quantum. Niels Bohr Institute.Google Scholar
Cvitanovic, P. & Gibson, J.F. 2010 Geometry of the turbulence in wall-bounded shear flows: periodic orbits. Phys. Scr. T142, 014007.CrossRefGoogle Scholar
Deguchi, K. 2015 Self-sustained states at Kolmogorov microscale. J. Fluid Mech. 781, R6.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014 a Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech. 752, 602–625.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014 a The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99–112.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2017 The relationship between free-stream coherent structures and near-wall streaks at high Reynolds numbers. Phil. Trans. R. Soc. Lond. A 375 (2089), 20160078.Google ScholarPubMed
Deguchi, K., Hall, P. & Walton, A. 2013 The emergence of localized vortex–wave interaction states in plane Couette flow. J. Fluid Mech. 721, 58–85.CrossRefGoogle 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.CrossRefGoogle Scholar
Doohan, P., Willis, A.P. & Hwang, Y. 2021 Minimal multi-scale dynamics of near-wall turbulence. J. Fluid Mech. 913, A38.CrossRefGoogle Scholar
Duguet, Y., Schlatter, P. & Henningson, D.S. 2009 Localized edge states in plane Couette flow. Phys. Fluids 21 (11), 111701.CrossRefGoogle 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.CrossRefGoogle Scholar
Duvvuri, S. & McKeon, B.J. 2015 Triadic scale interactions in a turbulent boundary layer. J. Fluid Mech. 767, R4.CrossRefGoogle 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.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91 (22), 224502.CrossRefGoogle ScholarPubMed
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2017 Optimal bursts in turbulent channel flow. J. Fluid Mech. 20, 35–60.CrossRefGoogle Scholar
Farrell, B.F., Ioannou, P.J., Jiménez, J., Constantinou, N.C., Lozano-Durán, A. & Nikolaidis, M.-A. 2016 A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. J. Fluid Mech. 809, 290–315.CrossRefGoogle Scholar
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22 (7), 071704.CrossRefGoogle Scholar
Gibson, J.F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.CrossRefGoogle 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.CrossRefGoogle Scholar
de Giovanetti, M., Sung, H.J. & Hwang, Y. 2017 Streak instability in turbulent channel flow: the seeding mechanism of large-scale motions. J. Fluid Mech. 832, 483–513.CrossRefGoogle Scholar
Goto, S. & Vassilicos, J.C. 2015 Energy dissipation and flux laws for unsteady turbulence. Phys. Lett. A 379, 1144–1148.CrossRefGoogle Scholar
Graham, M.D. & Floryan, D. 2020 Exact coherent states and the nonlinear dynamics of wall-bounded turbulent flows. Annu. Rev. Fluid. Mech. 53, 227–253.CrossRefGoogle Scholar
Hall, P. 2018 Vortex–wave interaction arrays: a sustaining mechanism for the log layer? J. Fluid Mech. 850, 46–82.CrossRefGoogle Scholar
Hall, P. & Horseman, N.J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary-layers. J. Fluid Mech. 232, 357–375.CrossRefGoogle Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.CrossRefGoogle Scholar
Hall, P. & Smith, F.T. 1991 On strongly nonlinear vortex–wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.CrossRefGoogle Scholar
Hamilton, J.M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
Hernandez, C.G. & Hwang, Y. 2021 Spectral energetics of quasilinear approximation in uniform shear turbulence. J. Fluid Mech. 904, A11.CrossRefGoogle Scholar
Herring, J.R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20 (4), 325–338.2.0.CO;2>CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. Lond. A 365 (1852), 647–664.Google ScholarPubMed
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254–289.CrossRefGoogle Scholar
Hwang, Y. & Bengana, Y. 2016 Self-sustaining process of minimal attached eddies in turbulent channel flow. J. Fluid Mech. 795, 708–738.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 a 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. & Cossu, C. 2010 b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 c Self-sustained process at large scales in turbulent channel flow. Phys. Rev. Lett. 105 (4), 044505.CrossRefGoogle ScholarPubMed
Hwang, Y. & Cossu, C. 2011 Self-sustained processes in the logarithmic layer of turbulent channel flows. Phys. Fluids 23 (6), 061702.CrossRefGoogle Scholar
Hwang, Y. & Eckhardt, B.E. 2020 Attached eddy model revisited using a minimal quasilinear approximation. J. Fluid Mech. 894, A23.CrossRefGoogle Scholar
Hwang, Y. & Lee, M. 2020 The mean logarithm emerges with self-similar energy balance. J. Fluid Mech. 903, R6.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
Itano, T. & Generalis, S.C. 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102 (11), 114501.CrossRefGoogle ScholarPubMed
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70 (3), 703–716.CrossRefGoogle Scholar
Jimenez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215–236.CrossRefGoogle Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.CrossRefGoogle Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.CrossRefGoogle Scholar
Jiménez, J. & Simens, M.P. 2001 Low-dimensional dynamics of a turbulent wall flow. J. Fluid Mech. 435, 81–91.CrossRefGoogle Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle 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.CrossRefGoogle Scholar
Kawata, T. & Alfredsson, P.H. 2018 Inverse interscale transport of the Reynolds shear stress in plane Couette turbulence. Phys. Rev. Lett. 120 (24), 244501.CrossRefGoogle ScholarPubMed
Kerswell, R.R. 2005 Recent progress in understanding the transition to turbulence. Nonlinearity 18, R17–R44.CrossRefGoogle Scholar
Kerswell, R.R. & Tutty, O.R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102.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
Khoo, Z.C., Chan, C.H. & Hwang, Y. 2021 A sparse optimal closure for a reduced-order model of wall-bounded turbulence. J. Fluid Mech. 939, A11.CrossRefGoogle Scholar
Kim, J. 1989 On the structure of pressure fluctuations in simulated turbulent channel flow. J. Fluid Mech. 205, 421–451.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59 (2), 308–323.CrossRefGoogle Scholar
Kline, S.J., Reynolds, W.C., Schraub, F.A. & Runstadler, P.W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS 30, 301–305.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.CrossRefGoogle ScholarPubMed
Kreilos, T., Veble, G., Schneider, T.M. & Eckhardt, B. 2013 Edge states for the turbulence transition in the asymptotic suction boundary layer. J. Fluid Mech. 726, 100–122.CrossRefGoogle Scholar
Lee, M. & Moser, R.D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. J. Fluid Mech. 860, 886–938.CrossRefGoogle Scholar
Long, R.R. & Chen, T. 1981 Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432–471.CrossRefGoogle Scholar
Lozano-Durán, A., Nikolaidis, M.-A., Constantinou, N.C. & Karp, M. 2021 Cause-and-effect of linear mechanisms sustaining wall turbulence. J. Fluid Mech. 914, A8.CrossRefGoogle Scholar
Malkus, W.V.R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539.CrossRefGoogle Scholar
Marusic, I., Monty, J.P., Hultmark, M. & Smits, A.J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.CrossRefGoogle Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.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
Motoki, S., Kawahara, G. & Shimizu, M. 2021 Multi-scale steady solution for Rayleigh–Bénard convection. J. Fluid Mech. 914, A14.CrossRefGoogle 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.CrossRefGoogle Scholar
Park, J., Hwang, Y. & Cossu, C. 2011 On the stability of large-scale streaks in the turbulent Couette and Poiseuille flows. C. R. Méc 339 (1), 1–5.CrossRefGoogle 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
Pausch, M., Yang, Q., Hwang, Y. & Eckhardt, B. 2019 Quasi-linear approximation of exact coherent states in parallel shear flows. Fluid Dyn. Res. 51, 011402.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 (1), 015109.CrossRefGoogle Scholar
Rawat, S., Cossu, C., Hwang, Y. & Rincon, F. 2015 On the self-sustained nature of large-scale motions in turbulent Couette flow. J. Fluid Mech. 782, 515–540.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schneider, T.M., Eckhardt, B. & Yorke, J.A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.CrossRefGoogle ScholarPubMed
Schneider, T.M., Gibson, J.F. & Burke, J. 2010 a Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104 (10), 104501.CrossRefGoogle ScholarPubMed
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
Schneider, T.M., Marinc, D. & Eckhardt, B. 2010 b Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646, 441–451.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Shekar, A. & Graham, M.D. 2018 Exact coherent states with hairpin-like vortex structure in channel flow. J. Fluid Mech. 849, 76–89.CrossRefGoogle Scholar
Skouloudis, N. & Hwang, Y. 2021 Scaling of turbulence intensities up to $Re_\tau =10^{6}$ with a resolvent-based quasilinear approximation. Phys. Rev. Fluids 6 (3), 034602.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
Talluru, K.M., Baidya, R., Hutchins, N. & Marusic, I. 2014 Amplitude modulation of all three velocity components in turbulent boundary layers. J. Fluid Mech. 746, R1.CrossRefGoogle Scholar
Thomas, V.L., Lieu, B.K., Jovanovic, M.R., Farrell, B.F., Ioannou, P.J. & Gayme, D.F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26 (10), 105112.CrossRefGoogle Scholar
Tomkins, C.D. & Adrian, R.J. 2003 Spanwise structure and scale growth in turbulent boundary layers. J. Fluid Mech. 490, 37–74.CrossRefGoogle Scholar
Townsend, A.A. 1956 The Structure of Turbulent Shear Flow, 1st edn. Cambridge University Press.Google Scholar
Townsend, A.A. 1980 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Vassilicos, J.C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid. Mech. 47, 95–114.CrossRefGoogle Scholar
van Veen, L., Vela-Martín, A. & Kawahara, G. 2019 Time-periodic inertial range dynamics. Phys. Rev. Lett. 123 (13), 134502.CrossRefGoogle ScholarPubMed
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.CrossRefGoogle Scholar
Viswanath, D. 2009 The dynamics of transition to turbulence in plane Couette flow. Phil. Trans. R. Soc. 367, 561–576.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81 (19), 4140.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 1517–1534.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
Wedin, H. & Kerswell, R.R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Wei, T., Fife, P., Klewicki, J.C. & McMurtry, P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303–327.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 (3), 036321.CrossRefGoogle ScholarPubMed
Willis, A.P., Short, K.Y. & Cvitanović, P. 2016 Symmetry reduction in high dimensions, illustrated in a turbulent pipe. Phys. Rev. E 93 (2), 022204.CrossRefGoogle 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
Zammert, S. & Eckhardt, B. 2015 Crisis bifurcations in plane Poiseuille flow. Phys. Rev. E 91 (4), 041003.CrossRefGoogle ScholarPubMed
Zhang, C. & Chernyshenko, S.I. 2016 Quasisteady quasihomogeneous description of the scale interactions in near-wall turbulence. Phys. Rev. Fluids 1 (1), 014401.CrossRefGoogle Scholar
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