Transition Problem and Localized Turbulent Structures in Pipes

Abstract

In the classical experiments of O. Reynolds made in 1883 the critical value of a dimensionless parameter, named now the Reynolds number, \({\text{R}}{{{\text{e}}}_{c}} \approx 2000\), was determined. As this value is exceeded in a pipe of circular cross-section, a turbulent flow regime can occur. The attempts to define this value more exactly undertaken during the twentieth century have not met with success. In this study, we present a review of theoretical, experimental, and numerical investigations of flows in a round pipe at the stage of transition to turbulence performed in recent years, which make it possible to formulate a new view on the nature of laminar-turbulent transition in these flows.

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REFERENCES

  1. 1

    O. Reynolds, “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. Roy. Soc. London 174, 935–982 (1883).

    ADS  MATH  Article  Google Scholar 

  2. 2

    W. Pfenniger, “Transition in the inlet length of tubes at high Reynolds numbers,” in: Boundary Layer and Flow Control, Ed. by G. V. Lachman (Pergamon, 1961), p. 970.

    Google Scholar 

  3. 3

    H. Salwen, F. W. Cotton, and C. E. Grosch, “Linear stability of Poiseuille flow in a circular pipe,” J. Fluid Mech. 98, 273–284 (1980).

    ADS  MATH  Article  Google Scholar 

  4. 4

    A. Meseguer and L. N. Trefethen, “Linearized pipe flow at Reynolds numbers 10 000 000,” J. Comp. Phys. 186, 178–197 (2003).

    ADS  MATH  Article  Google Scholar 

  5. 5

    A. M. Binnie and J. S. Fowler, “A study by a double refraction method of the development of turbulence in a long cylindrical tube,” Proc. Roy. Soc. London. A 192, 32 (1947).

    ADS  Article  Google Scholar 

  6. 6

    E. R. Lindgren, The transition process and other phenomena in viscous flow,” Arkiv für Physik 12, 1–169 (1958).

    Google Scholar 

  7. 7

    R. J. Leite, “An experimental investigation of the stability of Poiseuille flow,” J. Fluid Mech. 5, 81–97 (1959).

    ADS  MATH  Article  Google Scholar 

  8. 8

    A. G. Darbyshire and T. Mullin, “Transition to turbulence in constant-mass-flux pipe flow,” J. Fluid Mech. 289, 83–114 (1995).

    ADS  Article  Google Scholar 

  9. 9

    A. A. Pavelyev, A. I. Reshmin, S. Kh. Teplovodskii, and S. G. Fedoseev, “On the lower critical Reynolds number for flow in a circular tube,” Fluid Dynamics 38(4), 545—551 (2003).

    Article  Google Scholar 

  10. 10

    A. A. Pavelyev, A. I. Reshmin, and V. V. Trifonov, “Effect of the pattern of initial perturbations on the steady pipe flow regime,” Fluid Dynamics 41(6), 916—922 (2006).

    ADS  Article  Google Scholar 

  11. 11

    B. L. Rozhdestvensky and I. N. Simakin, “Secondary flows in a plane channel: their relationship and comparison with turbulent flows,” J. Fluid Mech. 147, 261–289 (1984).

    ADS  MATH  Article  Google Scholar 

  12. 12

    J. Kim, P. Moin, and R. Moser, “Turbulence statistics in fully developed channel flow at low Reynolds number,” J. Fluid Mech. 177, 133–166 (1987).

    ADS  MATH  Article  Google Scholar 

  13. 13

    N. Itoh, “Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances,” J. Fluid Mech. 82, 469–479 (1977).

    ADS  MATH  Article  Google Scholar 

  14. 14

    A. T. Patera and S. A. Orszag, “Finite amplitude stability of axisymmetric pipe flow,” J. Fluid. Mech. 112, 467–474 (1981).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. 15

    N. V. Nikitin, “Hard excitation of auto-oscillations in a Hagen—Poiseuille flow,” Fluid Dynamics 19(5), 828—830 (1984).

    ADS  MATH  Article  Google Scholar 

  16. 16

    V. G. Priymak and B. L. Rozhdestvenskii, “Secondary flows of a viscous incompressible fluid in a round pipe and their statistical properties,” Dokl. Akad. Nauk SSSR 297(6), 1326–1330 (1987).

    ADS  Google Scholar 

  17. 17

    V. G. Priymak, “Results and possibilities of direct numerical simulation of viscous turbulent flows in a round pipe,” Dokl. Akad. Nauk SSSR 316(1), 71–76 (1991).

    Google Scholar 

  18. 18

    N. V. Nikitin, “Direct numerical modeling of three-dimensional turbulent flows in pipes of circular cross-section,” Fluid Dynamics 29(6), 749—758 (1994).

    ADS  Article  Google Scholar 

  19. 19

    J. G. M. Eggels, F. Unger, M. H. Weiss, J. Westerweel, R. J. Adrian, R. Friedrich, and F. T. M. Nieuwstadt, “Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment,” J. Fluid Mech. 268, 175–209 (1994).

    ADS  Article  Google Scholar 

  20. 20

    H. Faisst and B. Eckhardt, “Traveling waves in pipe flow,” Phys. Rev. Lett. 91(22), 224502(4) (2003).

  21. 21

    H. Wedin and R. R. Kerswell, “Exact coherent structures in pipe flow: travelling wave solutions,” J. Fluid Mech. 508, 333–371 (2004).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. 22

    C. C. T. Pringle and R. R. Kerswell, “Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow,” Phys. Rev. Lett. 99(7), 074502(4) (2007).

  23. 23

    B. Hof, C. W. H. van Doorne, J. Westerweel, F. T. M. Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R. R. Kerswell, and F. Waleffe, “Experimental observation of nonlinear traveling waves in turbulent pipe flow,” Science 305, 1594–1598 (2004).

    ADS  Article  Google Scholar 

  24. 24

    R. R. Kerswell and O. R. Tutty, “Recurrence of travelling waves in transitional pipe flow,” J. Fluid Mech. 584, 69–102 (2007).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. 25

    B. Eckhardt, T. M. Schneider, B. Hof, and J. Westerweel, “Turbulence transition in pipe flow,” Annu. Rev. Fluid Mech. 39, 447–468 (2007).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. 26

    E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, 1993).

    MATH  Google Scholar 

  27. 27

    H. Faisst and B. Eckhardt, “Sensitive dependence on initial conditions in transition to turbulence in pipe flow,” J. Fluid Mech. 504, 343–352 (2004).

    ADS  MATH  Article  Google Scholar 

  28. 28

    I. J. Wygnanski and F. H. Champagne, “On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug,” J. Fluid Mech. 59, 281–335 (1973).

    ADS  Article  Google Scholar 

  29. 29

    M. Nishi, B. Ünsal, F. Durst, and G. Biswas, “Laminar-to-turbulent transition of pipe flows through puffs and slugs,” J. Fluid Mech. 614, 425–446 (2008).

    ADS  MATH  Article  Google Scholar 

  30. 30

    Y. Duguet, A. P. Willis, and R. R. Kerswell, “Slug genesis in cylindrical pipe flow,” J. Fluid Mech. 663, 180–208 (2010).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. 31

    D. Barkley, B. Song, V. Mukund, G. Lemoult, M. Avila, and B. Hof, “The rise of fully turbulent flow,” Nature 526, 550–553 (2015).

    ADS  Article  Google Scholar 

  32. 32

    I. Wygnanski, M. Sokolov, and D. Friedman, “On transition in a pipe. Part 2. The equilibrium puff,” J. Fluid Mech. 69(2), 283–304 (1975).

    ADS  Article  Google Scholar 

  33. 33

    H. Shan, B. Ma, Z. Zhang, and F. T. M. Nieuwstadt, “Direct numerical simulation of a puff and a slug in transitional cylindrical pipe flow,” J. Fluid Mech. 387, 39–60 (1999).

    ADS  MATH  Article  Google Scholar 

  34. 34

    V. G. Priymak and T. Miyazaki, “Direct numerical simulation of equilibrium spatially localized structures in pipe flow,” Phys. Fluids 16(12), 4221–4234 (2004).

    ADS  MATH  Article  Google Scholar 

  35. 35

    N. V. Nikitin and V. O. Pimanov, “Numerical study of localized turbulent structures in pipes,” Fluid Dynamics 50(5), 655—664 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  36. 36

    N. Nikitin, “Finite-difference method for incompressible Navier—Stokes equations in arbitrary orthogonal curvilinear coordinates,” J. Comput. Phys. 217(2), 759–781 (2006).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  37. 37

    N. Nikitin, “Third-order-accurate semi-implicit Runge-Kutta scheme for incompressible Navier—Stokes equations,” Int. J. Numer. Meth. Fluids 51(2), 221–233 (2006).

    MathSciNet  MATH  Article  Google Scholar 

  38. 38

    D. Moxey and D. Barkley, “Distinct large-scale turbulent-laminar states in transitional pipe flow,” PNAS 107(18), 8091–8096 (2010).

    ADS  Article  Google Scholar 

  39. 39

    N. V. Nikitin and V. O. Pimanov, “Localized turbulent structures in round pipes,” Uch. Zap. Kazan Univ. Ser. Fiz.-Mat. Nauki 157(3), 111–116 (2015).

    Google Scholar 

  40. 40

    B. Hof, A. de Lozar, M. Avila, X. Tu, and T. M. Schneider, “Eliminating turbulence in spatially intermittent flows,” Science 327, 1491–1494 (2010).

    ADS  Article  Google Scholar 

  41. 41

    M. Shimizu and S. Kida, “A driving mechanism of a turbulent puff in pipe flow,” Fluid Dyn. Res. 41, 045501(27) (2009).

  42. 42

    C. W. H. van Doorne and J. Westerweel, “The flow structure of a puff,” Phil. Trans. Roy. Soc. A 367, 489–507 (2009).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. 43

    W. Schoppa and F. Hussain, “Coherent structure generation in near-wall turbulence,” J. Fluid Mech. 453, 57–108 (2002).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  44. 44

    J. D. Skufca, J. A. Yorke, and B. Eckhardt, “Edge of chaos in a parallel shear flow,” Phys. Rev. Lett. 96(17), 174101 (2006).

    ADS  Article  Google Scholar 

  45. 45

    M. Avila, F. Mellibovsky, N. Roland, and B. Hof, “Streamwise-localized solutions at the onset of turbulence in pipe flow,” Phys. Rev. Lett. 110(22), 224502 (2013).

    ADS  Article  Google Scholar 

  46. 46

    N. V. Nikitin and V. O. Pimanov, “Sustainment of oscillations in localized turbulent structures in pipes,” Fluid Dynamics 53(1), 65—73 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  47. 47

    J. Peixinho and T. Mullin, “Decay of turbulence in pipe flow,” Phys. Rev. Let. 96, 094501(4), (2006).

  48. 48

    A. P. Willis and R. R. Kerswell, “Critical behavior in the relaminarization of localized turbulence in pipe flow,” Phys. Rev. Lett. 98, 014501(4) (2007).

  49. 49

    B. Hof, J. Westerweel, T. M. Schneider, and B. Eckhardt, “Finite lifetime of turbulence in shear flows,” Nature 443, 59–62 (2006).

    ADS  Article  Google Scholar 

  50. 50

    B. Hof, A. de Lozar, D. J. Kuik, and J. Westerweel, “Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow,” Phys. Rev. Lett. 101, 214501(4) (2008).

  51. 51

    D. J. Kuik, C. Poelma, and J. Westerweel, “Quantitative measurement of the lifetime of localized turbulence in pipe flow,” J. Fluid Mech. 645, 529–539 (2010).

    ADS  MATH  Article  Google Scholar 

  52. 52

    K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, “The onset of turbulence in pipe flow,” Science 333, 192–196 (2011).

    ADS  MATH  Article  Google Scholar 

  53. 53

    D. Barkley, “Theoretical perspective on the route to turbulence in a pipe,” J. Fluid Mech. 803, P1 (2016).

    ADS  MathSciNet  MATH  Article  Google Scholar 

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Funding

The study was carried out with the support of the Russian Foundation for Basic Research (project 19-11-50023) using the facilities of the Supercomputer Center of Moscow State University.

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Correspondence to N. V. Nikitin.

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The Author declares no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Translated by M. Lebedev

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Nikitin, N.V. Transition Problem and Localized Turbulent Structures in Pipes. Fluid Dyn 56, 31–44 (2021). https://doi.org/10.1134/S0015462821010092

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Keywords:

  • flows in round pipes, transition to turbulence, Reynolds’ experiments, critical Reynolds number, localized structures, turbulent puff, Navier
  • Stokes equations, direct numerical simulation