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Annual Review of Fluid Mechanics

Volume 53, 2021

Exact Coherent States and the Nonlinear Dynamics of Wall-Bounded Turbulent Flows

Abstract

Wall-bounded turbulence exhibits patterns that persist in time and space: coherent structures. These are important for transport processes and form a conceptual framework for important theoretical approaches. Key observed structures include quasi-streamwise and hairpin vortices, as well as the localized spots and puffs of turbulence observed during transition. This review describes recent research on so-called exact coherent states (ECS) in wall-bounded parallel flows at Reynolds numbers Re 104; these are nonturbulent, nonlinear solutions to the Navier–Stokes equations that in many cases resemble coherent structures in turbulence. That is, idealized versions of many of these structures exist as distinct, self-sustaining entities. ECS are saddle points in state space and form, at least in part, the state space skeleton of the turbulent dynamics. While most work on ECS focuses on Newtonian flow, some advances have been made on the role of ECS in turbulent drag reduction in polymer solutions. Emerging directions include applications to control and connections to large-scale structures and the attached eddy model.

Keyword(s): coherent structures, dynamical systems theory, turbulence
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2021-01-05
2024-04-16
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Literature Cited

  1. Adrian RJ. 1994. Stochastic estimation of conditional structure: a review. Appl. Sci. Res. 53:291–303
     [Google Scholar]
  2. Adrian RJ, Meinhart CD, Tomkins CD 2000. Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422:1–54
     [Google Scholar]
  3. Arnold VI. 1992. Ordinary Differential Equations Berlin: Springer-Verlag. , 3rd ed..
  4. Avila M, Mellibovsky F, Roland N, Hof B 2013. Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110:224502
     [Google Scholar]
  5. Azimi S, Schneider TM. 2020. Self-similar invariant solution in the near-wall region of a turbulent boundary layer at asymptotically high Reynolds numbers. J. Fluid Mech. 888:A15–13
     [Google Scholar]
  6. Baltzer JR, Adrian RJ, Wu X 2013. Structural organization of large and very large scales in turbulent pipe flow simulation. J. Fluid Mech. 720:236–79
     [Google Scholar]
  7. Barkley D. 2016. Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803:15–80
     [Google Scholar]
  8. Barkley D, Song B, Mukund V, Lemoult G, Avila M, Hof B 2015. The rise of fully turbulent flow. Nature 526:550–53
     [Google Scholar]
  9. Barkley D, Tuckerman LS. 2005. Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94:014502
     [Google Scholar]
  10. Beaume C, Chini GP, Julien K, Knobloch E 2015. Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91:043010
     [Google Scholar]
  11. Bird RB, Curtiss CF, Armstrong RC, Hassager O 1987. Dynamics of Polymeric Liquids 2 New York: Wiley-Intersci. , 2nd ed..
  12. Blackburn HM, Hall P, Sherwin SJ 2013. Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows. J. Fluid Mech. 726:R2
     [Google Scholar]
  13. Brand E, Gibson JF. 2014. A doubly-localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750:R3
     [Google Scholar]
  14. Budanur NB, Short KY, Farazmand M, Willis AP, Cvitanović P 2017. Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833:274–301
     [Google Scholar]
  15. Burger E, Munk W, Wahl H 1982. Flow increase in the trans Alaska pipeline through use of a polymeric drag-reducing additive. J. Petrol. Technol. 34:377–86
     [Google Scholar]
  16. Carlson DR, Widnall SE, Peeters MF 1982. A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech. 121:487–505
     [Google Scholar]
  17. Chantry M, Willis AP, Kerswell RR 2014. Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow. Phys. Rev. Lett. 112:164501
     [Google Scholar]
  18. Chaudhary I, Garg P, Shankar V, Subramanian G 2019. Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow. J. Fluid Mech. 881:119–63
     [Google Scholar]
  19. Cherubini S, De Palma P 2013. Nonlinear optimal perturbations in a Couette flow: bursting and transition. J. Fluid Mech. 716:251–79
     [Google Scholar]
  20. Chini GP, Montemuro B, White CM, Klewicki J 2017. A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows. Philos. Trans. R. Soc. A 375:2016009018
     [Google Scholar]
  21. Choueiri GH, Lopez JM, Hof B 2018. Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett. 120:124501
     [Google Scholar]
  22. Clever R, Busse F. 1992. Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234:511–27
     [Google Scholar]
  23. Clever R, Busse FH. 1997. Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344:137–53
     [Google Scholar]
  24. Cossu C, Hwang Y. 2017. Self-sustaining processes at all scales in wall-bounded turbulent shear flows. Philos. Trans. R. Soc. A 375:20160088
     [Google Scholar]
  25. Deguchi K, Hall P. 2014a. Canonical exact coherent structures embedded in high Reynolds number flows. Philos. Trans. R. Soc. A 372:20130352
     [Google Scholar]
  26. Deguchi K, Hall P. 2014b. Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech. 752:602–25
     [Google Scholar]
  27. Deguchi K, Hall P. 2014c. The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750:99–112
     [Google Scholar]
  28. 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
     [Google Scholar]
  29. Doering CR, Gibbon JD. 1995. Applied Analysis of the Navier-Stokes Equations Cambridge, UK: Cambridge Univ. Press. , 1st ed..
  30. Doi M, Edwards SF. 1986. The Theory of Polymer Dynamics New York: Oxford Univ. Press
  31. Drazin PG, Reid WH. 1981. Hydrodynamic Stability Cambridge, UK: Cambridge Univ. Press
  32. Dubief Y, Terrapon VE, Soria J 2013. On the mechanism of elasto-inertial turbulence. Phys. Fluids 25:110817
     [Google Scholar]
  33. Duguet Y, Pringle CCT, Kerswell RR 2008. Relative periodic orbits in transitional pipe flow. Phys. Fluids 20:114102
     [Google Scholar]
  34. Duguet Y, Schlatter P, Henningson DS 2009. Localized edge states in plane Couette flow. Phys. Fluids 21:111701
     [Google Scholar]
  35. Eckhardt B, Zammert S. 2018. Small scale exact coherent structures at large Reynolds numbers in plane Couette flow. Nonlinearity 31:R66–77
     [Google Scholar]
  36. Faisst H, Eckhardt B. 2003. Traveling waves in pipe flow. Phys. Rev. Lett. 91:224502
     [Google Scholar]
  37. Fink JK. 2012. Petroleum Engineer's Guide to Oil Field Chemicals and Fluids Waltham, MA: Gulf Prof. Publ.
  38. Garg P, Chaudhary I, Khalid M, Shankar V, Subramanian G 2018. Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121:024502
     [Google Scholar]
  39. Gibson JF, Brand E. 2014. Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745:25–61
     [Google Scholar]
  40. Gibson JF, Halcrow J, Cvitanović P 2008. Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611:107–30
     [Google Scholar]
  41. Girimaji SS, Pope SB. 1990. Material-element deformation in isotropic turbulence. J. Fluid Mech. 220:427–58
     [Google Scholar]
  42. Golubitsky M, Stewart I, Schaeffer DG 1988. Singularities and Groups in Bifurcation Theory 2 New York: Springer-Verlag
  43. Graham MD. 2004. Drag reduction in turbulent flow of polymer solutions. Rheol. Rev. 2:143–70
     [Google Scholar]
  44. Graham MD. 2014. Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26:101301
     [Google Scholar]
  45. Graham MD. 2015. Turbulence spreads like wildfire. Nature 526:508–9
     [Google Scholar]
  46. Graham MD. 2018. Microhydrodynamics, Brownian Motion, and Complex Fluids Cambridge, UK: Cambridge Univ. Press
  47. Guckenheimer J, Holmes P. 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields New York: Springer-Verlag. , 1st ed..
  48. Halcrow J, Gibson JF, Cvitanović P, Viswanath D 2009. Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621:365–76
     [Google Scholar]
  49. Hall P. 2018. Vortex–wave interaction arrays: a sustaining mechanism for the log layer. ? J. Fluid Mech. 850:46–82
     [Google Scholar]
  50. 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
     [Google Scholar]
  51. Hall P, Smith FT. 1991. On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227:641–66
     [Google Scholar]
  52. Hamilton JM, Kim J, Waleffe F 1995. Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287:317–48
     [Google Scholar]
  53. Hirsch MW, Smale S, Devaney RL 2013. Differential Equations, Dynamical Systems, and an Introduction to Chaos Waltham, MA: Academic. , 3rd ed..
  54. Holmes P. 2015. Dynamical systems. Princeton Companion to Applied Mathematics NJ Higham 383–93 Princeton, NJ: Princeton Univ. Press
     [Google Scholar]
  55. Holmes P, Lumley JL, Berkooz G, Rowley CW 2012. Turbulence, Coherent Structures, Dynamical Systems and Symmetry Cambridge, UK: Cambridge Univ. Press. , 2nd ed..
  56. Hopf E. 1948. A mathematical example displaying features of turbulence. Commun. Pure Appl. Math. 1:303–22
     [Google Scholar]
  57. Hwang Y, Willis AP, Cossu C 2016. Invariant solutions of minimal large-scale structures in turbulent channel flow for Reτ up to 1000. J. Fluid Mech. 802:R1
     [Google Scholar]
  58. Ibrahim JI, Yang Q, Doohan P, Hwang Y 2018. Phase-space dynamics of opposition control in wall-bounded turbulent flows. J. Fluid Mech. 861:29–54
     [Google Scholar]
  59. Itano T, Toh S. 2001. The dynamics of bursting process in wall turbulence. J. Phys. Soc. Jpn. 70:703–16
     [Google Scholar]
  60. Jiménez J. 1990. Transition to turbulence in two-dimensional Poiseuille flow. J. Fluid Mech. 218:265–97
     [Google Scholar]
  61. Jiménez J. 2012. Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44:27–45
     [Google Scholar]
  62. Jiménez J, Kawahara G, Simens MP, Nagata M, Shiba M 2005. Characterization of near-wall turbulence in terms of equilibrium and bursting solutions. Phys. Fluids 17:015105
     [Google Scholar]
  63. Jiménez J, Moin P. 1991. The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225:213–40
     [Google Scholar]
  64. Jiménez J, Pinelli A. 1999. The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389:335–59
     [Google Scholar]
  65. Kawahara G, Kida S. 2001. Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449:291–300
     [Google Scholar]
  66. Kerswell RR. 2018. Nonlinear nonmodal stability theory. Annu. Rev. Fluid Mech. 50:319–45
     [Google Scholar]
  67. Khapko T, Duguet Y, Kreilos T, Schlatter P, Eckhardt B, Henningson DS 2014. Complexity of localised coherent structures in a boundary-layer flow. Eur. Phys. J. E 37:32
     [Google Scholar]
  68. Khapko T, Kreilos T, Schlatter P, Duguet Y, Eckhardt B, Henningson DS 2013. Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717:R6
     [Google Scholar]
  69. Khapko T, Kreilos T, Schlatter P, Duguet Y, Eckhardt B, Henningson DS 2016. Edge states as mediators of bypass transition in boundary-layer flows. J. Fluid Mech. 801:R2
     [Google Scholar]
  70. Kida S, Goto S. 2002. Line statistics: stretching rate of passive lines in turbulence. Phys. Fluids 14:352–61
     [Google Scholar]
  71. Kim K, Li CF, Sureshkumar R, Balachandar S, Adrian RJ 2007. Effects of polymer stresses on eddy structures in drag-reduced turbulent channel flow. J. Fluid Mech. 584:281–99
     [Google Scholar]
  72. King GE. 2012. Hydraulic fracturing 101: what every representative, environmentalist, regulator, reporter, investor, university researcher, neighbor and engineer should know about hydraulic fracturing risk. J. Petrol. Technol. 64:434–42
     [Google Scholar]
  73. Knobloch E. 2015. Spatial localization in dissipative systems. Annu. Rev. Condens. Matter Phys. 6:325–59
     [Google Scholar]
  74. Kushwaha A. 2017. Near-wall turbulence and utilisation of the nonlinear dynamics towards control of turbulent flows PhD Thesis, Univ. Wis Madison:
  75. Kushwaha A, Park JS, Graham MD 2017. Temporal and spatial intermittencies within channel flow turbulence near transition. Phys. Rev. Fluids 2:024603
     [Google Scholar]
  76. Lasota A, Mackey MC. 1994. Chaos, Fractals and Noise New York: Springer. , 2nd ed..
  77. Lebovitz NR. 2012. Boundary collapse in models of shear-flow transition. Commun. Nonlinear Sci. Numer. Simul. 17:2095–100
     [Google Scholar]
  78. Lebovitz NR, Mariotti G. 2013. Edges in models of shear flow. J. Fluid Mech. 721:386–402
     [Google Scholar]
  79. Li W, Graham MD. 2007. Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19:083101
     [Google Scholar]
  80. Li W, Stone PA, Graham MD 2005. Viscoelastic nonlinear traveling waves and drag reduction in plane Poiseuille flow. IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions T Mullin, R Kerswell 285–308 Dordrecht, Neth: Springer
     [Google Scholar]
  81. Li W, Xi L, Graham MD 2006. Nonlinear travelling waves as a framework for understanding turbulent drag reduction. J. Fluid Mech. 565:353–62
     [Google Scholar]
  82. Lumley JL. 1969. Drag reduction by additives. Annu. Rev. Fluid Mech. 1:367–84
     [Google Scholar]
  83. Lumley JL. 1973. Drag reduction in turbulent flow by polymer additives. J. Polymer Sci. Macromol. Rev. 7:263–90
     [Google Scholar]
  84. Lundbladh A, Johansson AV. 1991. Direct simulation of turbulent spots in plane Couette flow. J. Fluid Mech. 229:499–516
     [Google Scholar]
  85. Lustro JRT, 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
     [Google Scholar]
  86. Marusic I, Monty JP. 2019. Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51:49–74
     [Google Scholar]
  87. Maslowe SA. 1986. Critical layers in shear flows. Annu. Rev. Fluid Mech. 18:405–32
     [Google Scholar]
  88. McComb WD. 1990. The Physics of Fluid Turbulence Oxford, UK: Oxford Univ. Press
  89. McKeon BJ. 2017. The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. 817:P1
     [Google Scholar]
  90. McKeon BJ, Sharma AS. 2010. A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658:336–82
     [Google Scholar]
  91. Nagata M. 1990. Three-dimensional finite-amplitude solutions in plane Couette-flow: bifurcation from infinity. J. Fluid Mech. 217:519–27
     [Google Scholar]
  92. Park JS, Graham MD. 2015. Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782:430–54
     [Google Scholar]
  93. Park JS, Shekar A, Graham MD 2018. Bursting and critical layer frequencies in minimal turbulent dynamics and connections to exact coherent states. Phys. Rev. Fluids 3:014611
     [Google Scholar]
  94. Perry AE, Chong MS. 1982. On the mechanism of wall turbulence. J. Fluid Mech. 119:106–21
     [Google Scholar]
  95. Pringle CCT, Duguet Y, Kerswell RR 2009. Highly symmetric travelling waves in pipe flow. Philos. Trans. R. Soc. A 367:457–72
     [Google Scholar]
  96. 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. 782:515–40
     [Google Scholar]
  97. Rawat S, Cossu C, Rincon F 2016. Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow. C. R. Méc 344:448–55
     [Google Scholar]
  98. Reetz F, Kreilos T, Schneider TM 2019. Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes. Nat. Commun. 10:499
     [Google Scholar]
  99. Robinson SK. 1991. Coherent motions in the turbulent boundary-layer. Annu. Rev. Fluid Mech. 23:601–39
     [Google Scholar]
  100. Rosenberg K, McKeon BJ. 2019. Computing exact coherent states in channels starting from the laminar profile: a resolvent-based approach. Phys. Rev. E 100:021101
     [Google Scholar]
  101. Samanta D, Dubief Y, Holzner M, Schäfer C, Morozov AN et al. 2013. Elasto-inertial turbulence. PNAS 110:10557–62
     [Google Scholar]
  102. Schneider TM, Eckhardt B. 2009. Edge states intermediate between laminar and turbulent dynamics in pipe flow. Philos. Trans. R. Soc. A 367:577–87
     [Google Scholar]
  103. Schneider TM, Gibson JF, Burke J 2010a. Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104:104501
     [Google Scholar]
  104. Schneider TM, Gibson JF, Lagha M, de Lillo F, Eckhardt B 2008. Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78:037301
     [Google Scholar]
  105. Schneider TM, Marinc D, Eckhardt B 2010b. Localized edge states nucleate turbulence in extended plane Couette cells. J. Fluid Mech. 646:441–51
     [Google Scholar]
  106. Schoppa W, Hussain F. 2002. Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453:57–108
     [Google Scholar]
  107. Sekimoto A, Jiménez J. 2017. Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow. J. Fluid Mech. 827:225–49
     [Google Scholar]
  108. Sharma AS, Mezić I, McKeon BJ 2016a. Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations. Phys. Rev. Fluids 1:032402
     [Google Scholar]
  109. Sharma AS, Moarref R, McKeon BJ, Park JS, Graham MD, Willis AP 2016b. Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence. Phys. Rev. E 93:021102
     [Google Scholar]
  110. Shekar A, Graham MD. 2018. Exact coherent states with hairpin-like vortex structure in channel flow. J. Fluid Mech. 849:76–89
     [Google Scholar]
  111. Shekar A, McMullen RM, McKeon BJ, Graham MD 2019a. Elastoinertial turbulence and connections to linear mechanisms Paper presented at the 11th International Symposium on Turbulence and Shear Flow Phenomena Southampton, UK: July 30
  112. Shekar A, McMullen RM, Wang SN, McKeon BJ, Graham MD 2019b. Critical-layer structures and mechanisms in elastoinertial turbulence. Phys. Rev. Lett. 122:124503
     [Google Scholar]
  113. Sid S, Terrapon VE, Dubief Y 2018. Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Phys. Rev. Fluids 3:011301
     [Google Scholar]
  114. Skufca JD, Yorke JA, Eckhardt B 2006. Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96:174101
     [Google Scholar]
  115. Smits AJ, McKeon BJ, Marusic I 2011. High–Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43:353–75
     [Google Scholar]
  116. Stone PA, Graham MD. 2003. Polymer dynamics in a model of the turbulent buffer layer. Phys. Fluids 15:1247–56
     [Google Scholar]
  117. Stone PA, Roy A, Larson RG, Waleffe F, Graham MD 2004. Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids 16:3470–82
     [Google Scholar]
  118. Stone PA, Waleffe F, Graham MD 2002. Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett. 89:208301
     [Google Scholar]
  119. Tillmark N, Alfredsson PH. 1992. Experiments on transition in plane Couette flow. J. Fluid Mech. 235:89–102
     [Google Scholar]
  120. Toh S, Itano T. 2003. A periodic-like solution in channel flow. J. Fluid Mech. 481:67–76
     [Google Scholar]
  121. Toms BA. 1949. Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. Proceedings of the First International Congress on Rheology135–41 Amsterdam: North-Holland
     [Google Scholar]
  122. Toms BA. 1977. On the early experiments on drag reduction by polymers. Phys. Fluids 20:S3–8
     [Google Scholar]
  123. Townsend AA. 1961. Equilibrium layers and wall turbulence. J. Fluid Mech. 11:97–120
     [Google Scholar]
  124. Tuckerman LS, Chantry M, Barkley D 2020. Patterns in wall-bounded shear flows. Annu. Rev. Fluid Mech. 52:343–67
     [Google Scholar]
  125. Uhlmann M, Kawahara G, Pinelli A 2010. Travelling-waves consistent with turbulence-driven secondary flow in a square duct. Phys. Fluids 22:084102
     [Google Scholar]
  126. Vadarevu S, Sharma A, Ganapathisubramani B 2016. Exact coherent states for grooved Couette flows. arXiv:1612.07723 [physics.flu-dyn]
  127. van Veen L, Kawahara G 2011. Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107:114501
     [Google Scholar]
  128. van Veen L, Uhlmann M, Kawahara G 2012. The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44:203–25
     [Google Scholar]
  129. Virk PS. 1975. Drag reduction fundamentals. AIChE J 21:625–56
     [Google Scholar]
  130. Viswanath D. 2007. Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580:339–58
     [Google Scholar]
  131. Viswanath D. 2009. The critical layer in pipe flow at high Reynolds number. Philos. Trans. R. Soc. A 367:561–76
     [Google Scholar]
  132. Vollmer J, Schneider TM, Eckhardt B 2009. Basin boundary, edge of chaos and edge state in a two-dimensional model. New J. Phys. 11:013040
     [Google Scholar]
  133. Waleffe F. 1997. On a self-sustaining process in shear flows. Phys. Fluids 9:883–900
     [Google Scholar]
  134. Waleffe F. 1998. Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81:4140–43
     [Google Scholar]
  135. Waleffe F. 2001. Exact coherent structures in channel flow. J. Fluid Mech. 435:93–102
     [Google Scholar]
  136. Waleffe F. 2003. Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15:1517–34
     [Google Scholar]
  137. Wang J, Gibson J, Waleffe F 2007. Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98:204501
     [Google Scholar]
  138. Wang SN, Shekar A, Graham MD 2017. Spatiotemporal dynamics of viscoelastic turbulence in transitional channel flow. J. Non-Newton. Fluid Mech. 244:104–22
     [Google Scholar]
  139. Wedin H, Kerswell RR. 2004. Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508:333–71
     [Google Scholar]
  140. Whalley RD, Dennis DJC, Graham MD, Poole RJ 2019. An experimental investigation into spatiotemporal intermittencies in turbulent channel flow close to transition. Exp. Fluids 60:102
     [Google Scholar]
  141. Whalley RD, Park JS, Kushwaha A, Dennis DJC, Graham MD, Poole RJ 2017. Low-drag events in transitional wall-bounded turbulence. Phys. Rev. Fluids 2:034602
     [Google Scholar]
  142. White CM, Mungal MG. 2008. Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40:235–56
     [Google Scholar]
  143. Willis AP. 2019. Equilibria, periodic orbits and computing them. arXiv:1908.06730 [physics.flu-dyn]
  144. Willis AP, Kerswell RR. 2009. Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarization and localized ‘edge’ states. J. Fluid Mech. 619:213–33
     [Google Scholar]
  145. Willis AP, Short KY, Cvitanović P 2016. Symmetry reduction in high dimensions, illustrated in a turbulent pipe. Phys. Rev. E 93:022204
     [Google Scholar]
  146. Xi L. 2019. Turbulent drag reduction by polymer additives: fundamentals and recent advances. Phys. Fluids 31:121302
     [Google Scholar]
  147. Xi L, Graham MD. 2010. Active and hibernating turbulence in minimal channel flow of Newtonian and polymeric fluids. Phys. Rev. Lett. 104:218301
     [Google Scholar]
  148. Xi L, Graham MD. 2012a. Dynamics on the laminar-turbulent boundary and the origin of the maximum drag reduction asymptote. Phys. Rev. Lett. 108:028301
     [Google Scholar]
  149. Xi L, Graham MD. 2012b. Intermittent dynamics of turbulence hibernation in Newtonian and viscoelastic minimal channel flows. J. Fluid Mech. 693:433–72
     [Google Scholar]
  150. Yang Q, Hwang Y. 2019. Modulation of attached exact coherent states under spanwise wall oscillation Paper presented at the 11th International Symposium on Turbulence and Shear Flow Phenomena Southampton, UK: July 30
  151. Yang Q, Willis AP, Hwang Y 2019. Exact coherent states of attached eddies in channel flow. J. Fluid Mech. 862:1029–59
     [Google Scholar]
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Exact Coherent States and the Nonlinear Dynamics of Wall-Bounded Turbulent Flows
Annual Review of Fluid Mechanics 53, 227 (2021); https://doi.org/10.1146/annurev-fluid-051820-020223
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