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.
Similar content being viewed by others
REFERENCES
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).
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.
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).
A. Meseguer and L. N. Trefethen, “Linearized pipe flow at Reynolds numbers 10 000 000,” J. Comp. Phys. 186, 178–197 (2003).
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).
E. R. Lindgren, The transition process and other phenomena in viscous flow,” Arkiv für Physik 12, 1–169 (1958).
R. J. Leite, “An experimental investigation of the stability of Poiseuille flow,” J. Fluid Mech. 5, 81–97 (1959).
A. G. Darbyshire and T. Mullin, “Transition to turbulence in constant-mass-flux pipe flow,” J. Fluid Mech. 289, 83–114 (1995).
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).
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).
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).
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).
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).
A. T. Patera and S. A. Orszag, “Finite amplitude stability of axisymmetric pipe flow,” J. Fluid. Mech. 112, 467–474 (1981).
N. V. Nikitin, “Hard excitation of auto-oscillations in a Hagen—Poiseuille flow,” Fluid Dynamics 19(5), 828—830 (1984).
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).
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).
N. V. Nikitin, “Direct numerical modeling of three-dimensional turbulent flows in pipes of circular cross-section,” Fluid Dynamics 29(6), 749—758 (1994).
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).
H. Faisst and B. Eckhardt, “Traveling waves in pipe flow,” Phys. Rev. Lett. 91(22), 224502(4) (2003).
H. Wedin and R. R. Kerswell, “Exact coherent structures in pipe flow: travelling wave solutions,” J. Fluid Mech. 508, 333–371 (2004).
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).
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).
R. R. Kerswell and O. R. Tutty, “Recurrence of travelling waves in transitional pipe flow,” J. Fluid Mech. 584, 69–102 (2007).
B. Eckhardt, T. M. Schneider, B. Hof, and J. Westerweel, “Turbulence transition in pipe flow,” Annu. Rev. Fluid Mech. 39, 447–468 (2007).
E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, 1993).
H. Faisst and B. Eckhardt, “Sensitive dependence on initial conditions in transition to turbulence in pipe flow,” J. Fluid Mech. 504, 343–352 (2004).
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).
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).
Y. Duguet, A. P. Willis, and R. R. Kerswell, “Slug genesis in cylindrical pipe flow,” J. Fluid Mech. 663, 180–208 (2010).
D. Barkley, B. Song, V. Mukund, G. Lemoult, M. Avila, and B. Hof, “The rise of fully turbulent flow,” Nature 526, 550–553 (2015).
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).
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).
V. G. Priymak and T. Miyazaki, “Direct numerical simulation of equilibrium spatially localized structures in pipe flow,” Phys. Fluids 16(12), 4221–4234 (2004).
N. V. Nikitin and V. O. Pimanov, “Numerical study of localized turbulent structures in pipes,” Fluid Dynamics 50(5), 655—664 (2015).
N. Nikitin, “Finite-difference method for incompressible Navier—Stokes equations in arbitrary orthogonal curvilinear coordinates,” J. Comput. Phys. 217(2), 759–781 (2006).
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).
D. Moxey and D. Barkley, “Distinct large-scale turbulent-laminar states in transitional pipe flow,” PNAS 107(18), 8091–8096 (2010).
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).
B. Hof, A. de Lozar, M. Avila, X. Tu, and T. M. Schneider, “Eliminating turbulence in spatially intermittent flows,” Science 327, 1491–1494 (2010).
M. Shimizu and S. Kida, “A driving mechanism of a turbulent puff in pipe flow,” Fluid Dyn. Res. 41, 045501(27) (2009).
C. W. H. van Doorne and J. Westerweel, “The flow structure of a puff,” Phil. Trans. Roy. Soc. A 367, 489–507 (2009).
W. Schoppa and F. Hussain, “Coherent structure generation in near-wall turbulence,” J. Fluid Mech. 453, 57–108 (2002).
J. D. Skufca, J. A. Yorke, and B. Eckhardt, “Edge of chaos in a parallel shear flow,” Phys. Rev. Lett. 96(17), 174101 (2006).
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).
N. V. Nikitin and V. O. Pimanov, “Sustainment of oscillations in localized turbulent structures in pipes,” Fluid Dynamics 53(1), 65—73 (2018).
J. Peixinho and T. Mullin, “Decay of turbulence in pipe flow,” Phys. Rev. Let. 96, 094501(4), (2006).
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).
B. Hof, J. Westerweel, T. M. Schneider, and B. Eckhardt, “Finite lifetime of turbulence in shear flows,” Nature 443, 59–62 (2006).
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).
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).
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).
D. Barkley, “Theoretical perspective on the route to turbulence in a pipe,” J. Fluid Mech. 803, P1 (2016).
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.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The Author declares no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Additional information
Translated by M. Lebedev
Rights and permissions
About this article
Cite this article
Nikitin, N.V. Transition Problem and Localized Turbulent Structures in Pipes. Fluid Dyn 56, 31–44 (2021). https://doi.org/10.1134/S0015462821010092
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0015462821010092