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Mixing pp 361-383 | Cite as

Fluctuations and Mixing of a Passive Scalar in Turbulent Flow

  • Boris I. Shraiman
  • Eric D. Siggia
Part of the NATO ASI Series book series (NSSB, volume 373)

Abstract

The article reviews the aspects of the intermittency phenomena exhibited by the advected passive scalar and summarizes the recent theoretical ideas concerning the statistics of large fluctuations and the origin of anomalous scaling of multipoint correlators.

Keywords

Probability Distribution Function Zero Mode Passive Scalar Inertial Range Exponential Tail 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    K.R. Sreenivasan, Proc. R. Soc. London A 434, 165 (1991).ADSGoogle Scholar
  2. [2]
    H. J. Catrakis and P. Dimotakis, J Fluid Mech 317, 369 (1996); P.L. Miller and P. Dimotakis, J Fluid Mech 308 129 (1996).ADSCrossRefGoogle Scholar
  3. [3]
    R. A. Antonia and C. W.Van Atta, J. Fluid Mech. 84, 561 (1980).ADSCrossRefGoogle Scholar
  4. [4]
    F. Anselmet, Y. Gagne, E.J. Hopfinger and R.A. Antonia, J. Fluid Mech. 140, 63 (1984).ADSCrossRefGoogle Scholar
  5. [5]
    A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 301 (1941).ADSGoogle Scholar
  6. [6]
    A. M. Obukhov: Structure of the Temperature Field in a Turbulent Flow. Izv. Akad. Nauk SSSR, Geogr.Google Scholar
  7. [7]
    S. Corrsin, NACA R & M 58B1 (1958).Google Scholar
  8. [8]
    K. R. Sreenivasan and R. Antonia, Annual Review of Fluid Mechanics, 29, 435, (1997).MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    U. Frisch, Turbulence: the Legacy of A.N. Kolmogorov, (Cambridge University Press, Cambridge, 1995).zbMATHGoogle Scholar
  10. [10]
    M. Holzer and E.D. Siggia, Phys. Fluids 6, 1820 (1994).MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. [11]
    A. Pumir, Phys. Fluids 6, 2118 (1994).MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. [12]
    R. H. Kraichnan, V. Yakhot, and S. Chen, Phys. Rev. Lett. 75, 240, (1995).ADSCrossRefGoogle Scholar
  13. [13]
    R. H. Kraichnan, Phys. Fluids 11, 945 (1968).MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. [14]
    R. H. Kraichnan, J. Fluid Mech. 64, 737 (1974).MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. [15]
    R. H. Kraichnan, Phys. Rev. Lett. 72, 1016 (1994).ADSCrossRefGoogle Scholar
  16. [16]
    B. I. Shraiman and E. D. Siggia, Phys. Rev. E49, 2912 (1994).MathSciNetADSGoogle Scholar
  17. [17]
    B.I. Shraiman and E.D. Siggia, C. R. Acad. sci. Paris, 321, Sér. IIb, 279 (1995).zbMATHGoogle Scholar
  18. [18]
    M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev Phys. Rev. E 52, 4924 (1995).MathSciNetADSGoogle Scholar
  19. [19]
    K. Gawçdzki and A. Kupiainen, Phys. Rev. Lett. 75, 3834 (1995).ADSCrossRefGoogle Scholar
  20. [20]
    B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett, 77, 2463, (1996).ADSCrossRefGoogle Scholar
  21. [21]
    P.G. Mestayer, J. Fluid Mech. 125, 475, (1982).ADSCrossRefGoogle Scholar
  22. [22]
    J. Gollub et al, Phys. Rev. Lett. 67, 3507, (1991).ADSCrossRefGoogle Scholar
  23. [23]
    Jayesh and Z. Warhaft, Phys. Rev. Lett. 67, 3503, (1991).ADSCrossRefGoogle Scholar
  24. [24]
    G. K. Batchelor, J. Fluid Mech. 5, 113 (1959).MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. [25]
    J.L. Lumley, Phys. Fluids 10, 855 (1967).MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    S. Tavoulares and S. Corrsin, J. Fluid Mech. 104, 311, (1981).ADSCrossRefGoogle Scholar
  27. [27]
    S. Saddoughi and S.V. Veeravalli, J. Fluid Mech. 268, 333, (1994).ADSCrossRefGoogle Scholar
  28. [28]
    The Batchelor k--1 spectrum is even more controversial (see [1, 2]_and B.Williams and J. Gollub, to be published) although the numerical simulations [10, 11]_suggest that it only appears at very high Pe, the existence of this long crossover covering most experimentally relevant situations is interesting and requires an explanation in itself.Google Scholar
  29. [29]
    E.D. Siggia, J. Fluid Mech. 107, 375, (1981).ADSzbMATHCrossRefGoogle Scholar
  30. [30]
    S. Douady, Y. Couder, and M.E. Brachet, Phys. Rev. Lett. 67, 983, (1991).ADSCrossRefGoogle Scholar
  31. [31]
    G.I. Taylor, Proc. London Math. Soc., ser 2, 20, 196,(1921).zbMATHCrossRefGoogle Scholar
  32. [32]
    The appearance of a cusp in the center is ubiquitous and is observed experimentally both in the PDFs of ΔrΘ and ΔrU. The stretch exponentials employed in fitting these PDFs (see P. Kailasnath, K. Sreenivasan, and G. Stolovitzky, Phys. Rev. Lett. 68, 2766, (1992); P. Tabeling, G. Zocci, F. Belin, J. Maurer, and H. Willaime, Phys. Rev. E, 53, 1613, (1996) and E. Ching, Phys. Rev A44, 3622 (1991)) are to a large degree controlled by these cusps rather than the tails of the distribution. This interesting phenomenon deserves further investigation.Google Scholar
  33. [33]
    M. Chertkov, G. Falkovich, I. Kolokolov, and V. Lebedev, Phys. Rev. E 51, 3974, (1995).MathSciNetCrossRefGoogle Scholar
  34. [34]
    G. Falkovich, I. Kolokolov, V. Lebedev, and A. Migdal, Phys. Rev. E 54,4896,(1996).ADSCrossRefGoogle Scholar
  35. [35]
    M. Chertkov, Phys. Rev. E 55, (1997).Google Scholar
  36. [36]
    A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, (MIT Press, Cambridge, 1971).Google Scholar
  37. [37]
    Y. Sinai and V. Yakhot, Phys Rev Lett, 63, 1962, (1989).ADSCrossRefGoogle Scholar
  38. [38]
    E.S.C. Ching, Phys. Rev. Lett. 70, 283 (1993); S.B. Pope and E.S.C.Ching, Phys. Fluids A 5, 1529 (1993).ADSCrossRefGoogle Scholar
  39. [39]
    A. Fairhall, O. Gat, V. Lvov, and I. Procaccia, Phys Rev. E, 53, 3518, (1996).MathSciNetADSCrossRefGoogle Scholar
  40. [40]
    E. Ching, V. L’vov, A. Podivilov, and I. Procaccia, chao-dyn/9608008.Google Scholar
  41. [41]
    The irrelevance of zero modes and non-anomalous behavior should not be taken for granted: e.g. M. Vergassola have demonstrated the anomalous scaling of the 2-point function in the passive vector model (see Phys.Rev.E 53, R3021 (1996).Google Scholar
  42. [42]
    R. Courant and D. Hilbert, Mathematical Methods of Physics, (Wiley Inter-science, New York, 1953).Google Scholar
  43. [43]
    D. Bernard, K. Gawedski, and A. Kupiainen, Phys. Rev. E 54, 2564,(1996).MathSciNetADSCrossRefGoogle Scholar
  44. [44]
    A. Pumir, Europh. Lett. 34, 25 (1996).ADSCrossRefGoogle Scholar
  45. [45]
    M. Chertkov and G. Falkovich, PRL 76, 2706, (1996).ADSCrossRefGoogle Scholar
  46. [46]
    J-D. Fournier, U. Frisch, and H. A. Rose, J Phys A 11, 187, (1978).MathSciNetADSzbMATHCrossRefGoogle Scholar
  47. [47]
    R.H. Kraichnan, preprint (1996).Google Scholar
  48. [48]
    A. Pumir, B.I. Shraiman, and E.D. Siggia, PRE, Rapid Comm., to appear (1996).Google Scholar
  49. [49]
    E. Balkovsky, M. Chertkov, I. Kolokolov, and V. Lebedev, JETP Lett. 61, 1012, (1995).Google Scholar
  50. [50]
    L.P. Kadanoff, “Scaling, Universality and Operator Algebras,” in Phase transitions and Critical Phenomena, Vol. 5A, Domb and Green eds., (Academic Press, Boston, 1976).Google Scholar
  51. [51]
    G. Vilenkin, in Representations of groups and special functions Kluwer Acad. Publ., 1991.Google Scholar
  52. [52]
    B.I. Shraiman and E.D. Siggia, to be published.Google Scholar
  53. [53]
    M. Chertkov, G. Falkovich, and V. Lebedev, Phys. Rev. Lett., 76, 3707, (1996).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Boris I. Shraiman
    • 1
  • Eric D. Siggia
    • 2
  1. 1.Bell Laboratories, Lucent TechnologiesMurray HillUSA
  2. 2.Laboratory of Atomic & Solid State PhysicsCornell UniversityIthacaUSA

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