1932

Abstract

The dissolution of minute concentration of polymers in wall-bounded flows is well-known for its unparalleled ability to reduce turbulent friction drag. Another phenomenon, elasto-inertial turbulence (EIT), has been far less studied even though elastic instabilities have already been observed in dilute polymer solutions before the discovery of polymer drag reduction. EIT is a chaotic state driven by polymer dynamics that is observed across many orders of magnitude in Reynolds number. It involves energy transfer from small elastic scales to large flow scales. The investigation of the mechanisms of EIT offers the possibility to better understand other complex phenomena such as elastic turbulence and maximum drag reduction. In this review, we survey recent research efforts that are advancing the understanding of the dynamics of EIT. We highlight the fundamental differences between EIT and Newtonian/inertial turbulence from the perspective of experiments, numerical simulations, instabilities, and coherent structures. Finally, we discuss the possible links between EIT and elastic turbulence and polymer drag reduction, as well as the remaining challenges in unraveling the self-sustaining mechanism of EIT.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-fluid-032822-025933
2023-01-19
2024-03-29
Loading full text...

Full text loading...

/deliver/fulltext/fluid/55/1/annurev-fluid-032822-025933.html?itemId=/content/journals/10.1146/annurev-fluid-032822-025933&mimeType=html&fmt=ahah

Literature Cited

  1. Agarwal A, Brandt L, Zaki TA. 2014. Linear and nonlinear evolution of a localized disturbance in polymeric channel flow. J. Fluid Mech. 760:278–303
    [Google Scholar]
  2. Alves M, Oliveira P, Pinho F. 2021. Numerical methods for viscoelastic fluid flows. Annu. Rev. Fluid Mech. 53:509–41
    [Google Scholar]
  3. Avila K, Moxey D, de Lozar A, Avila M, Barkley D, Hof B. 2011. The onset of turbulence in pipe flow. Science 333:6039192–96
    [Google Scholar]
  4. Avila M, Barkley D, Hof B. 2023. Transition to turbulence in pipe flow. Annu. Rev. Fluid Mech. 55:575602
    [Google Scholar]
  5. Barnes HA, Hutton JF, Walters K. 1989. An Introduction to Rheology, Vol. 3 Amsterdam: Elsevier
  6. Batchelor GK. 1959. Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5:1113–33
    [Google Scholar]
  7. Batchelor GK, Howells ID, Townsend AA. 1959. Small-scale variation of convected quantities like temperature in turbulent fluid. Part 2. The case of large conductivity. J. Fluid Mech. 5:1134–39
    [Google Scholar]
  8. Berti S, Bistagnino A, Boffetta G, Celani A, Musacchio S. 2008. Two-dimensional elastic turbulence. Phys. Rev. E 77:5055306
    [Google Scholar]
  9. Bird R, Armstrong R, Hassager O 1987. Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory New York: Wiley-Intersci.
    [Google Scholar]
  10. Buza G, Page J, Kerswell RR. 2022. Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers. J. Fluid Mech. 940:A11
    [Google Scholar]
  11. Chandra B, Shankar V, Das D. 2018. Onset of transition in the flow of polymer solutions through microtubes. J. Fluid Mech. 844:1052–83
    [Google Scholar]
  12. Chandra B, Shankar V, Das D. 2020. Early transition, relaminarization and drag reduction in the flow of polymer solutions through microtubes. J. Fluid Mech. 885:A47
    [Google Scholar]
  13. 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]
  14. Chaudhary I, Garg P, Subramanian G, Shankar V. 2021. Linear instability of viscoelastic pipe flow. J. Fluid Mech. 908:A11
    [Google Scholar]
  15. Choueiri GH, Lopez JM, Hof B. 2018. Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett. 120:12124501
    [Google Scholar]
  16. Choueiri GH, Lopez JM, Varshney A, Sankar S, Hof B. 2021. Experimental observation of the origin and structure of elastoinertial turbulence. PNAS 118:45e2102350118
    [Google Scholar]
  17. Doering CR, Eckhardt B, Schumacher J. 2006. Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers. J. Non-Newton. Fluid Mech. 135:2–392–96
    [Google Scholar]
  18. Dong M, Zhang M. 2022. Asymptotic study of linear instability in a viscoelastic pipe flow. J. Fluid Mech. 935:A28
    [Google Scholar]
  19. Dubief Y, Delcayre F. 2000. On coherent-vortex identification in turbulence. J. Turbul. 1:N11
    [Google Scholar]
  20. Dubief Y, Page J, Kerswell RR, Terrapon VE, Steinberg V. 2022. A first coherent structure in elasto-inertial turbulence. Phys. Rev. Fluids 7:073301
    [Google Scholar]
  21. Dubief Y, Terrapon V, White C, Shaqfeh E, Moin P, Lele S. 2005. New answers on the interaction between polymers and vortices in turbulent flows. Flow Turbul. Combust. 74:4311–29
    [Google Scholar]
  22. Dubief Y, Terrapon VE, Soria J. 2013. On the mechanism of elasto-inertial turbulence. Phys. Fluids 25:11110817
    [Google Scholar]
  23. Dubief Y, White C. 2011. Elastic turbulence in high Reynolds number polymer drag reduced flows Abstract presented at APS Division of Fluid Dynamics Meeting Baltimore, MD: Nov. 20–22, Abstr. M8–002
  24. Dubief Y, White CM, Shaqfeh ESG, Terrapon VE. 2010. Polymer maximum drag reduction: a unique transitional state. Annual Research Briefs395–404 Stanford, CA: Cent. Turbul. Res.
    [Google Scholar]
  25. Eckhardt B, Schneider TM, Hof B, Westerweel J. 2007. Turbulence transition in pipe flow.. Annu. Rev. Fluid Mech. 39:447–68
    [Google Scholar]
  26. Fattal R, Kupferman R. 2005. Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newton. Fluid Mech. 126:23–37
    [Google Scholar]
  27. Forame PC, Hansen RJ, Little RC. 1972. Observations of early turbulence in the pipe flow of drag reducing polymer solutions. AIChE J. 18:213–17
    [Google Scholar]
  28. Garg P, Chaudhary I, Khalid M, Shankar V, Subramanian G. 2018. Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett. 121:2024502
    [Google Scholar]
  29. Goldstein M, Hultgren LS. 1989. Boundary-layer receptivity to long-wave free-stream disturbances. Annu. Rev. Fluid Mech. 21:137–66
    [Google Scholar]
  30. Gorodtsov V, Leonov A. 1967. On a linear instability of a plane parallel Couette flow of viscoelastic fluid. J. Appl. Math. Mech. 31:2310–19
    [Google Scholar]
  31. Graham MD. 2014. Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids 26:10101301
    [Google Scholar]
  32. Graham MD, Floryan D. 2021. Exact coherent states and the nonlinear dynamics of wall-bounded turbulent flows. Annu. Rev. Fluid Mech. 53:227–53
    [Google Scholar]
  33. Grillet AM, Bogaerds AC, Peters GW, Baaijens FP. 2002. Stability analysis of constitutive equations for polymer melts in viscometric flows. J. Non-Newton. Fluid Mech. 103:2–3221–50
    [Google Scholar]
  34. Groisman A, Steinberg V. 2000. Elastic turbulence in a polymer solution flow. Nature 405:678253–55
    [Google Scholar]
  35. Hameduddin I, Gayme DF, Zaki TA. 2019. Perturbative expansions of the conformation tensor in viscoelastic flows. J. Fluid Mech. 858:377–406
    [Google Scholar]
  36. Hameduddin I, Meneveau C, Zaki TA, Gayme DF. 2018. Geometric decomposition of the conformation tensor in viscoelastic turbulence. J. Fluid Mech. 842:395–427
    [Google Scholar]
  37. Hariharan G, Jovanović MR, Kumar S. 2018. Amplification of localized body forces in channel flows of viscoelastic fluids. J. Non-Newton. Fluid Mech. 260:40–53
    [Google Scholar]
  38. Ho TC, Denn MM. 1977. Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newton. Fluid Mech. 3:2179–95
    [Google Scholar]
  39. Hoda N, Jovanović MR, Kumar S. 2008. Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech. 601:407–24
    [Google Scholar]
  40. Hoda N, Jovanović MR, Kumar S. 2009. Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. Fluid Mech. 625:411–34
    [Google Scholar]
  41. Hof B, De Lozar A, Kuik DJ, Westerweel J. 2008. Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett. 101:21214501
    [Google Scholar]
  42. Hof B, Samanta D, Wagner C. 2011. The maximum drag reduction asymptote Abstract presented at APS Division of Fluid Dynamics Meeting Baltimore, MD: Nov. 20–22, Abstr. M8-005
  43. Hof B, Westerweel J, Schneider TM, Eckhardt B. 2006. Finite lifetime of turbulence in shear flows. Nature 443:710759–62
    [Google Scholar]
  44. Hoyt J. 1977. Laminar-turbulent transition in polymer solutions. Nature 270:508–9
    [Google Scholar]
  45. Hunt JC, Wray AA, Moin P. 1988. Eddies, streams, and convergence zones in turbulent flows. Studying Turbulence Using Numerical Simulation Databases, II: Proceedings of the 1988 Summer Program193–208 Stanford, CA: Cent. Turbul. Res.
    [Google Scholar]
  46. Jha NK, Steinberg V. 2021. Elastically driven Kelvin–Helmholtz-like instability in straight channel flow. PNAS 118:34e2105211118
    [Google Scholar]
  47. Jiménez J. 1994. On the structure and control of near wall turbulence. Phys. Fluids 6:2944–53
    [Google Scholar]
  48. Jiménez J, Flores O, Garca-Villalba M. 2001. The large-scale organization of autonomous turbulent wall regions. Annual Research Briefs 2001317–27 Stanford, CA: Cent. Turbul. Res.
    [Google Scholar]
  49. Jiménez J, Moin P. 1991. The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225:213–40
    [Google Scholar]
  50. Jiménez J, Pinelli A. 1999. The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389:335–59
    [Google Scholar]
  51. Jovanović MR, Kumar S. 2009. Variance amplification in channel flows of strongly elastic polymer solutions. 2009 American Control Conference842–47 New York: IEEE
    [Google Scholar]
  52. Jovanović MR, Kumar S. 2010. Transient growth without inertia. Phys. Fluids 22:2023101
    [Google Scholar]
  53. Jovanović MR, Kumar S. 2011. Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newton. Fluid Mech. 166:14–15755–78
    [Google Scholar]
  54. Kerswell RR. 2005. Recent progress in understanding the transition to turbulence in a pipe.. Nonlinearity 18:6R17–44
    [Google Scholar]
  55. Khalid M, Chaudhary I, Garg P, Shankar V, Subramanian G 2021a. The centre-mode instability of viscoelastic plane Poiseuille flow. J. Fluid Mech. 915:A43
    [Google Scholar]
  56. Khalid M, Shankar V, Subramanian G 2021b. Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow. Phys. Rev. Lett. 127:13134502
    [Google Scholar]
  57. Kim J, Moin P, Moser R. 1987. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177:133–66
    [Google Scholar]
  58. Larson RG. 1992. Instabilities in viscoelastic flows. Rheol. Acta 31:3213–63
    [Google Scholar]
  59. Larson RG, Shaqfeh E, Muller S. 1990. A purely elastic instability in Taylor–Couette flow. J. Fluid Mech. 218:1573–600
    [Google Scholar]
  60. Layec Y, Layec-Raphalen MN. 1983. Instability of dilute poly (ethylene-oxide) solutions. J. Phys. Lett. 44:3121–28
    [Google Scholar]
  61. Li CF, Sureshkumar R, Khomami B. 2006. Influence of rheological parameters on polymer induced turbulent drag reduction. J. Non-Newton. Fluid Mech. 140:1–323–40
    [Google Scholar]
  62. Li W, Graham M. 2007. Polymer induced drag reduction in exact coherent structures of plane Poiseuille flow. Phys. Fluids 19:083101
    [Google Scholar]
  63. Little RC, Hansen R, Hunston D, Kim O, Patterson R, Ting R. 1975. The drag reduction phenomenon. Observed characteristics, improved agents, and proposed mechanisms. Ind. Eng. Chem. Fundamen. 14:4283–96
    [Google Scholar]
  64. Little RC, Wiegard M. 1970. Drag reduction and structural turbulence in flowing polyox solutions. J. Appl. Polymer Sci. 14:2409–19
    [Google Scholar]
  65. Lopez JM, Choueiri GH, Hof B. 2019. Dynamics of viscoelastic pipe flow at low Reynolds numbers in the maximum drag reduction limit. J. Fluid Mech. 874:699–719
    [Google Scholar]
  66. Lumley J. 1969. Drag reduction by additives. Ann. Rev. Fluid Mech. 1:367–84
    [Google Scholar]
  67. McKinley GH, Byars JA, Brown RA, Armstrong RC. 1991. Observations on the elastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid. J. Non-Newton. Fluid Mech. 40:2201–29
    [Google Scholar]
  68. Meulenbroek B, Storm C, Morozov AN, van Saarloos W. 2004. Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow. J. Non-Newton. Fluid Mech. 116:2–3235–68
    [Google Scholar]
  69. Min T, Yoo JY, Choi H. 2001. Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows. J. Non-Newton. Fluid Mech. 100:1–327–47
    [Google Scholar]
  70. Morozov AN, van Saarloos W. 2005. Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. Phys. Rev. Lett. 95:2024501
    [Google Scholar]
  71. Morozov AN, van Saarloos W. 2007. An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows. Phys. Rep. 447:3–6112–43
    [Google Scholar]
  72. Morozov AN, van Saarloos W. 2019. Subcritical instabilities in plane Poiseuille flow of an Oldroyd-B fluid. J. Stat. Phys. 175:3–4554–77
    [Google Scholar]
  73. Mukund V, Hof B. 2018. The critical point of the transition to turbulence in pipe flow. J. Fluid Mech. 839:76–94
    [Google Scholar]
  74. Orlandi P, Jiménez J. 1994. On the generation of turbulent wall friction. Phys. Fluids 6:2634–41
    [Google Scholar]
  75. Ostwald W, Auerbach R. 1926. Ueber die Viskosität kolloider Lösungen im Struktur-, Laminar- und Turbulenzgebiet. Kolloid-Zeitschrift 38:3261–280
    [Google Scholar]
  76. Page J, Dubief Y, Kerswell RR. 2020. Exact traveling wave solutions in viscoelastic channel flow. Phys. Rev. Lett. 125:15154501
    [Google Scholar]
  77. Page J, Zaki TA. 2014. Streak evolution in viscoelastic Couette flow. J. Fluid Mech. 742:520–51
    [Google Scholar]
  78. Page J, Zaki TA. 2015. The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow. J. Fluid Mech. 777:327–63
    [Google Scholar]
  79. Páll S, Zhmurov A, Bauer P, Abraham M, Lundborg M et al. 2020. Heterogeneous parallelization and acceleration of molecular dynamics simulations in GROMACS. J. Chem. Phys. 153:13134110
    [Google Scholar]
  80. Pan L, Morozov A, Wagner C, Arratia P. 2013. Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett. 110:17174502
    [Google Scholar]
  81. Porteous KC, Denn MM. 1972. Linear stability of plane Poiseuille flow of viscoelastic liquids. Trans. Soc. Rheol. 16:2295–308
    [Google Scholar]
  82. Ptasinski P, Boersma B, Nieuwstadt F, Hulsen M, Van den Brule B, Hunt J. 2003. Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms. J. Fluid Mech. 490:251–91
    [Google Scholar]
  83. Qin B, Arratia PE. 2017. Characterizing elastic turbulence in channel flows at low Reynolds number. Phys. Rev. Fluids 2:8083302
    [Google Scholar]
  84. Ram A, Tamir A. 1964. Structural turbulence in polymer solutions. J. Appl. Polymer Sci. 8:62751–62
    [Google Scholar]
  85. Reiner M. 1926a. Ueber die Strömung einer elastischen Flüssigkeit durch eine Kapillare. Kolloid-Zeitschrift 39:180–87
    [Google Scholar]
  86. Reiner M. 1926b. Zur Theorie der “Strukturturbulenz. .” Kolloid-Zeitschrift 39:4314–15
    [Google Scholar]
  87. Reynolds O. 1883. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philos. Trans. R. Soc. Lond. 174:935–82
    [Google Scholar]
  88. Samanta D, Dubief Y, Holzner M, Schäfer C, Morozov AN et al. 2013. Elasto-inertial turbulence. PNAS 110:2610557–62
    [Google Scholar]
  89. Sánchez HAC, Jovanović MR, Kumar S, Morozov A, Shankar V et al. 2022. Understanding viscoelastic flow instabilities: Oldroyd-B and beyond. J. Non-Newton. Fluid Mech. 302:104472
    [Google Scholar]
  90. Schmid PJ. 2007. Nonmodal stability theory. Annu. Rev. Fluid Mech. 39:129–62
    [Google Scholar]
  91. Schmid PJ, Henningson D. 2000. Stability and Transition in Shear Flows New York: Springer
  92. Shaqfeh E. 1996. Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28:129–85
    [Google Scholar]
  93. Shekar A, McMullen RM, McKeon BJ, Graham MD. 2020. Self-sustained elastoinertial Tollmien–Schlichting waves. J. Fluid Mech. 897:A3
    [Google Scholar]
  94. Shekar A, McMullen RM, McKeon BJ, Graham MD. 2021. Tollmien-Schlichting route to elastoinertial turbulence in channel flow. Phys. Rev. Fluids 6:9093301
    [Google Scholar]
  95. Shekar A, McMullen RM, Wang SN, McKeon BJ, Graham MD. 2019. Critical-layer structures and mechanisms in elastoinertial turbulence. Phys. Rev. Lett. 122:12124503
    [Google Scholar]
  96. Sid S, Terrapon VE, Dubief Y. 2018. Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Phys. Rev. Fluids 3:1011301
    [Google Scholar]
  97. Srinivas SS, Kumaran V. 2017. Effect of viscoelasticity on the soft-wall transition and turbulence in a microchannel. J. Fluid Mech. 812:1076–118
    [Google Scholar]
  98. Steinberg V. 2021. Elastic turbulence: an experimental view on inertialess random flow. Annu. Rev. Fluid Mech. 53:27–58
    [Google Scholar]
  99. Stone P, Roy A, Larson R, Waleffe F, Graham M. 2004. Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids 16:3470
    [Google Scholar]
  100. Sureshkumar R, Beris AN. 1995a. Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows. J. Non-Newton. Fluid Mech. 60:153–80
    [Google Scholar]
  101. Sureshkumar R, Beris AN. 1995b. Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newton. Fluid Mech. 56:2151–82
    [Google Scholar]
  102. Sureshkumar R, Beris AN, Handler R. 1997. Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids 9:743
    [Google Scholar]
  103. Tabor M, De Gennes P. 1986. A cascade theory of drag reduction. Europhys. Lett. 2:519–22
    [Google Scholar]
  104. Terrapon VE, Dubief Y, Soria J. 2015. On the role of pressure in elasto-inertial turbulence. J. Turbul. 16:126–43
    [Google Scholar]
  105. Thais L, Tejada-Mart AE, Gatski TB, Mompean G et al. 2011. A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow. Comput. Fluids 43:1134–42
    [Google Scholar]
  106. Toms BA. 1948. 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: N. Holland
    [Google Scholar]
  107. Vaithianathan T, Collins L. 2003. Numerical approach to simulating turbulent flow of a viscoelastic polymer solution. J. Comput. Phys. 187:1–21
    [Google Scholar]
  108. Vaithianathan T, Robert A, Brasseur JG, Collins LR. 2006. An improved algorithm for simulating three-dimensional, viscoelastic turbulence. J. Non-Newton. Fluid Mech. 140:1–33–22
    [Google Scholar]
  109. Vinogradov GV, Manin VN. 1965. An experimental study of elastic turbulence. Kolloid-Z. Z. Polym. 201:293–98
    [Google Scholar]
  110. Virk P, Mickley H, Smith K. 1970. The ultimate asymptote and mean flow structure in Toms' phenomenon. J. Appl. Mech 37:488–93
    [Google Scholar]
  111. Waleffe F. 1997. On a self-sustaining process in shear flows. Phys. Fluids 9:4883–900
    [Google Scholar]
  112. Waleffe F. 2001. Exact coherent structures in channel flow. J. Fluid Mech. 435:93–102
    [Google Scholar]
  113. Wan D, Sun G, Zhang M. 2021. Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows. J. Fluid Mech. 929:A16
    [Google Scholar]
  114. Warholic M, Massah H, Hanratty T. 1999. Influence of drag-reducing polymers on turbulence: effects of Reynolds number concentration and mixing. Exp. Fluids 27:5461–72
    [Google Scholar]
  115. Warner HR. 1972. Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind. Eng. Chem. Fundam. 11:3379–87
    [Google Scholar]
  116. Wen C, Poole RJ, Willis AP, Dennis DJC. 2017. Experimental evidence of symmetry-breaking supercritical transition in pipe flow of shear-thinning fluids. Phys. Rev. Fluids 2:3031901
    [Google Scholar]
  117. White C, Mungal M. 2008. Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40:235–56
    [Google Scholar]
  118. Wilson HJ, Renardy M, Renardy Y. 1999. Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids. J. Non-Newton. Fluid Mech. 80:2–3251–68
    [Google Scholar]
  119. Xi L. 2019. Turbulent drag reduction by polymer additives: fundamentals and recent advances. Phys. Fluids 31:121302
    [Google Scholar]
  120. Xi L, Graham M 2010. Active and hibernating turbulence in minimal channel flow of Newtonian and polymeric fluids. Phys. Rev. Lett. 104:21218301
    [Google Scholar]
  121. Zhang M. 2021. Energy growth in subcritical viscoelastic pipe flows. J. Non-Newton. Fluid Mech. 294:104581
    [Google Scholar]
  122. Zhang M, Lashgari I, Zaki TA, Brandt L. 2013. Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. Fluid Mech. 737:249–79
    [Google Scholar]
  123. Zhang WH, Shao QQ, Li YK, Ma Y, Zhang HN, Li FC. 2021a. On the mechanisms of sheet-like extension structures formation and self-sustaining process in elasto-inertial turbulence. Phys. Fluids 33:8085107
    [Google Scholar]
  124. Zhang WH, Zhang HN, Li YK, Yu B, Li FC. 2021b. Role of elasto-inertial turbulence in viscoelastic drag-reducing turbulence. Phys. Fluids 33:8081706
    [Google Scholar]
  125. Zhang WH, Zhang HN, Wang ZM, Li YK, Yu B, Li FC 2022. Repicturing viscoelastic drag-reducing turbulence by introducing dynamics of elasto-inertial turbulence. J. Fluid Mech. 940:A31
    [Google Scholar]
  126. Zhu L, Xi L. 2021. Nonasymptotic elastoinertial turbulence for asymptotic drag reduction. Phys. Rev. Fluids 6:1014601
    [Google Scholar]
/content/journals/10.1146/annurev-fluid-032822-025933
Loading
/content/journals/10.1146/annurev-fluid-032822-025933
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error