Skip to main content
SpringerLink
Log in

Laminar-Turbulent Transition: The Change of the Flow State Temperature with the Reynolds Number

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Representing the fluid flow as a collection of coherent structures of various size, the statistical temperature of the flow state is determined as a function of the Reynolds number. It is shown that at small Reynolds numbers, associated with laminar states, the temperature is positive, while at large Reynolds numbers, associated with turbulent states, it is negative. At intermediate Reynolds numbers, the temperature changes from positive to negative as the size of the coherent structures increases, similar to what was predicted by Onsager for a system of parallel point-vortices in an inviscid fluid. It is also shown that in the range of intermediate Reynolds numbers the temperature exhibits a critical divergence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Onsager, L.: Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279–287 (1949)

    Article  MathSciNet  Google Scholar 

  2. Kraichnan, R.H., Montgomery, D.: Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  3. Joyce, G., Montgomery, D.: Negative temperature states for the two-dimensional guiding-centre plasma. J. Plasma Phys. 10, 107–121 (1973)

    Article  ADS  Google Scholar 

  4. Edwards, S.F., Taylor, J.B.: Negative temperature states of two-dimensional plasmas and vortex fluids. Proc. R. Soc. Lond. A 336, 257–271 (1974)

    Article  ADS  Google Scholar 

  5. Montgomery, D., Joyce, G.: Statistical mechanics of negative temperature states. Phys. Fluids 17, 1139–1145 (1974)

  6. Pointin, Y.B., Lundgren, T.S.: Statistical mechanics of two-dimensional vortices in a bounded container. Phys. Fluids 19, 1459–1470 (1976)

    Article  ADS  MATH  Google Scholar 

  7. Lundgren, T.S., Pointin, Y.B.: Statistical mechanics of two-dimensional vortices. J. Stat. Phys. 17, 323–355 (1977)

  8. Fröhlich, J., Ruelle, D.: Statistical mechanics of vortices in an inviscid two-dimensional fluid. Commun. Math. Phys. 87, 1–36 (1982)

    Article  ADS  MATH  Google Scholar 

  9. Miller, J.: Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett. 65, 2137–2140 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Chorin, A.J.: Equilibrium statistics of a vortex filament with applications. Commun. Math. Phys. 141, 619–631 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Eyink, G.L., Spohn, H.: Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence. J. Stat. Phys. 70, 833–886 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Tabeling, P.: Two-dimensional turbulence: a physicist approach. Phys. Rep. 362, 1–62 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Jung, S., Morrison, P.J., Swinney, H.L.: Statistical mechanics of two-dimensional turbulence. J. Fluid Mech. 554, 433–456 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Chavanis, P.-H.: Virial theorem for Onsager vortices in two-dimensional hydrodynamics. Eur. Phys. J. Plus 127, 159 (2012)

    Article  Google Scholar 

  15. Eyink, G.L., Sreenivasan, K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87–135 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Campa, A., Dauxois, T., Ruffo, S.: Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480, 57–159 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  17. Chekmarev, S.F.: Tendency to occupy a statistically dominant spatial state of the flow as a driving force for turbulent transition. Chaos 23, 013144 (2013)

    Article  ADS  Google Scholar 

  18. Kusukawa, K.: The foundation of the quantisation of turbulence. J. Phys. Soc. Jpn. 6, 86–89 (1951)

    Article  ADS  Google Scholar 

  19. Stanley, H.E.: Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, Oxford (1971)

    Google Scholar 

  20. Sornette, D.: Critical Phenomena in Natural Sciences. Spinger, Berlin (2006)

    MATH  Google Scholar 

  21. Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North-Holland, Amsterdam (1972)

  22. Kolmogorov, A. N.: The local structure of turbulence in an incompressible viscous fluid at very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303 (1941). (Reprinted in Proc. R. Soc. Lond. A 434, 9–13, 1991).

  23. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Pergamon, New York (1987)

    MATH  Google Scholar 

  24. Landau, L.D., Lifshitz, E.M.: Statistical Physics. Pergamon, New York (1987)

    MATH  Google Scholar 

  25. Ruelle, D.P.: Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics. Proc. Natl. Acad. Sci. USA 109, 20344–20346 (2012)

    Article  ADS  Google Scholar 

  26. Moisy, F., Jiménez, J.: Geometry and clustering of intense structures in isotropic turbulence. J. Fluid Mech. 513, 111–133 (2004)

    Article  ADS  MATH  Google Scholar 

  27. Jiménez, J., Wray, A.A., Saffman, P.G., Rogallo, R.S.: The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65–90 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Lesieur, M.: Turbulence in Fluids. Springer, Dordrecht (2008)

    Book  MATH  Google Scholar 

  29. Reynolds, O.: 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. Phil. Trans. R. Soc. Lond. 174, 935–982 (1883)

    Article  MATH  Google Scholar 

  30. Pfenninger, W.: Transition in the inlet length of tubes at high Reynolds numbers. In: Boundary Layer and Flow Control, edited by G. V. Lachman. Pergamon, Oxford (1961) pp. 970–980.

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  34. Chabaud, B., Naert, A., Peinke, J., Chillà, F., Castaing, B., Hébral, B.: Transition toward developed turbulence. Phys. Rev. Lett. 73, 3227–3230 (1994)

    Article  ADS  Google Scholar 

  35. Tabeling, P., Willaime, H.: Transition at dissipative scales in large-Reynolds-number turbulence. Phys. Rev. E 65, 066301 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  36. Cortet, P.-P., Chiffaudel, A., Daviaud, F., Dubrulle, B.: Experimental evidence of a phase transition in a closed turbulent flow. Phys. Rev. Lett. 105, 214501 (2010)

    Article  ADS  Google Scholar 

  37. Castaing, B.: The temperature of turbulent flows. J. Phys. II (France) 6, 105–114 (1996)

    Article  Google Scholar 

  38. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)

Download references

Acknowledgments

I thank R. Khairulin for useful discussions of the critical phenomena.

Author information

Authors and Affiliations

  1. Institute of Thermophysics, SB RAS, 630090, Novosibirsk, Russia

    Sergei F. Chekmarev

  2. Department of Physics, Novosibirsk State University, 630090, Novosibirsk, Russia

    Sergei F. Chekmarev

Authors
  1. Sergei F. Chekmarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergei F. Chekmarev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chekmarev, S.F. Laminar-Turbulent Transition: The Change of the Flow State Temperature with the Reynolds Number. J Stat Phys 157, 1019–1030 (2014). https://doi.org/10.1007/s10955-014-1112-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1112-x

Keywords

Search

Navigation