Intermittency and Structure(s) of and/in Turbulence

  • Arkady Tsinober

Abstract

Intermittency specifically in genuine fluid turbulence is associated mostly with some aspects of its spatiotemporal structure. Hence, the close relation between the origin(s) and meaning of intermittency and structure of turbulence. Just like there is no general agreement on the origin and meaning of the former, there is no consensus regarding what are the origin(s) and what turbulence structure(s) really mean. At the present state of matters both issues are pretty speculative and an example of ‘ephemeral’ collection of such is given in this chapter. We have to admit at this stage that structure(s) is(are) just an inherent property of turbulence.

Structureless turbulence is meaningless. There is no turbulence without structure(s). Every part (just as the whole) of the turbulent field—including the so-called ‘structureless background’—possess structure. Structureless turbulence (or any of its part) contradicts both the experimental evidence and the Navier–Stokes equations. The claims for ‘structureless background’ is a reflection of our inability to ‘see’ more intricate aspects of turbulence structure: intricacy, complexity and ‘randomness’ are not synonymous for absence of structure.

What is definite is that turbulent flows have lots of structure(s). The term structure(s) is used here deliberately in order to emphasize the duality (or even multiplicity) of the meaning of the underlying problem. The first is about how turbulence ‘looks’. The second implies the existence of some entities. Objective treatment of both requires use of some statistical methods. It is thought that these methods alone may be insufficient to cope with the problem, but so far no satisfactory solution was found. One (but not the only) reason—as mentioned—is that it is not so clear what one is looking for: the objects seem to be still elusive. For example, there is still a non-negligible set of people in the community that are in a great doubt that the concept of coherent structure is much different from the Emperors’s new Clothes.

An example of acute difficulties described in this chapter is associated with high dimension of what is called structures so that simple single parameter thresholding is inappropriate to make on them statistics due to the painful question how really “similar” are all these if the individual members of such an ensemble are defined by one parameter only. The view that turbulence structure(s) is(are) simple in some sense and that turbulence can be represented as a collection of simple objects only seems to be a nice illusion which, unfortunately, has little to do with reality. It seems somewhat wishfully naive to expect that such a complicated phenomenon like turbulence can merely be described in terms of collections of only such ‘simple’ and weakly interacting object.

Keywords

Coherent Structure Turbulence Structure Vortex Tube Inertial Range Large Reynolds Number 
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.

References

  1. Aluie H (2012) Scale locality and the inertial range in compressible turbulence. J Fluid Mech (submitted). arXiv:1101.0150
  2. Anderson PW (1972) More is different. Science 177:393–396 CrossRefGoogle Scholar
  3. Armi L, Flament P (1987) Cautionary remarks on the spectral interpretation of turbulent flows. J Geophys Res 90:11,779–11,782 Google Scholar
  4. Arnold VI (1991) Kolmogorov’s hydrodynamics attractors. Proc R Soc Lond A 434:19–22 CrossRefGoogle Scholar
  5. Batchelor GK, Townsend AA (1949) The nature of turbulent motion at large wave-numbers. Proc R Soc Lond A 199:238–255 MATHCrossRefGoogle Scholar
  6. Betchov R (1974) Non-Gaussian and irreversible events in isotropic turbulence. Phys Fluids 17:1509–1512 CrossRefGoogle Scholar
  7. Betchov R (1976) On the non-Gaussian aspects of turbulence. Arch Mech 28(5–6):837–845 Google Scholar
  8. Betchov R (1993) In: Dracos T, Tsinober A (eds) New approaches and turbulence. Birkhäuser, Basel, p 155 Google Scholar
  9. Biferale L, Procaccia I (2005) Anisotropy in turbulent flows and in turbulent transport. Phys Rep 414:43–164 MathSciNetCrossRefGoogle Scholar
  10. Blackwelder RF (1983) Analogies between transitional and turbulent boundary layers. Phys Fluids 26:2807–2815 CrossRefGoogle Scholar
  11. Bonnet JP (ed) (1996) Eddy structure identification. Springer, Berlin MATHGoogle Scholar
  12. Chen Q, Chen S, Eyink GL, Holm DD (2003) Intermittency in the joint cascade of energy and helicity. Phys Rev Lett 90:214503 CrossRefGoogle Scholar
  13. Chorin AJ (1994) Vorticity and turbulence. Springer, Berlin MATHGoogle Scholar
  14. Chorin AJ (1996) Turbulence cascades across equilibrium spectra. Phys Rev 54:2616–2619 Google Scholar
  15. Constantin P (1996) Navier–Stokes equations and incompressible fluid turbulence. Lect Appl Math 31:219–234 MathSciNetGoogle Scholar
  16. Cvitanović P, Gibson P (2010) Phys Scr T 142:014007 CrossRefGoogle Scholar
  17. Dowker M, Ohkitani K (2012) Intermittency and local Reynolds number in Navier–Stokes turbulence: a cross-over scale in the Caffarelli-Kohn-Nirenberg integral. Phys Fluids 24:115112 CrossRefGoogle Scholar
  18. Dryden H (1948) Recent advances in boundary layer flow. Adv Appl Mech 1:1–40 MathSciNetCrossRefGoogle Scholar
  19. Dwoyer DL, Hussaini MY, Voigt RG (eds) (1985) Theoretical approaches to turbulence. Springer, Berlin MATHGoogle Scholar
  20. Elliott FW, Majda AJ (1995) A new algorithm with plane waves and wavelets for random velocity fields with many spatial scales. J Comput Phys 117:146–162 MathSciNetMATHCrossRefGoogle Scholar
  21. Elsinga GE, Marusic I (2010) Universal aspects of small-scale motions in turbulence. J Fluid Mech 662:514–539 MATHCrossRefGoogle Scholar
  22. Ferchichi M, Tavoularis S (2000) Reynolds number dependence of the fine structure of uniformly sheared turbulence. Phys Fluids 12:2942–2953 CrossRefGoogle Scholar
  23. Feynmann R (1963) Lect Phys 2:41–42 Google Scholar
  24. Frenkiel FN, Klebanoff PS, Huang TT (1979) Grid turbulence in air and water. Phys Fluids 22:1606–1617 CrossRefGoogle Scholar
  25. Frisch U (1995) Turbulence: the legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge MATHGoogle Scholar
  26. Gibson CH, Stegen GS, Williams RB (1970) Statistics of the fine structure of turbulent velocity and temperature fields measured at high Reynolds numbers. J Fluid Mech 41:153–167 CrossRefGoogle Scholar
  27. Gibson CH, Friehe CA, McConnell SO (1977) Structure of sheared turbulent fields. Phys Fluids 20(II):S156–S167 CrossRefGoogle Scholar
  28. Goldshtik MA, Shtern VN (1981) Structural turbulence theory. Dokl Akad Nauk SSSR 257(6):1319–1322 (in Russian) Google Scholar
  29. Guckenheimer J (1986) Strange attractors in fluids: another view. Annu Rev Fluid Mech 18:15–31 MathSciNetCrossRefGoogle Scholar
  30. Hill RJ (1997) Applicability of Kolmogorov’s and Monin’s equations to turbulence. J Fluid Mech 353:67–81 MathSciNetMATHCrossRefGoogle Scholar
  31. Holmes PJ, Berkooz G, Lumley JL (1996) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge MATHCrossRefGoogle Scholar
  32. Holmes PJ, Lumley JL, Berkooz G, Mattingly JC, Wittenberg RW (1997) Low-dimensional models of coherent structures in turbulence. Phys Rep 287:337–384 MathSciNetCrossRefGoogle Scholar
  33. Hopf E (1948) A mathematical example displaying features of turbulence. Commun Pure Appl Math 1:303–322 MathSciNetMATHCrossRefGoogle Scholar
  34. Hunt JCR, Eames I, Westerweel J, Davidson PA, Voropayev SI, Fernando J, Braza M (2010) Thin shear layers—the key to turbulence structure? J Hydro-Environ Res 4:75–82 CrossRefGoogle Scholar
  35. Ishihara T, Hunt JCR, Kaneda Y (2011) Conditional analysis near strong shear layers in DNS of isotropic turbulence at high reynolds number. J Phys Conf Ser 318(4):042004 CrossRefGoogle Scholar
  36. Kadanoff LP (1986) Fractals: where is the physics? Phys Today 39:3–7 Google Scholar
  37. Kawahara G, Uhlmann M, van Veen L (2012) The significance of simple invariant solutions in turbulent flows. Annu Rev Fluid Mech 44:203–225 CrossRefGoogle Scholar
  38. Keefe L (1990a) Connecting coherent structures and strange attractors. In: Kline SJ, Afgan HN (eds) Near wall turbulence—1988 Zaric memorial conference. Hemisphere, Washington, pp 63–80 Google Scholar
  39. Keefe L (1990b) In: Lumley JL (ed) Whither turbulence? Springer, Berlin, p 189 Google Scholar
  40. Keefe L, Moin P, Kim J (1992) The dimension of attractors underlying periodic turbulent Poiseulle flow. J Fluid Mech 242:1–29 MathSciNetMATHCrossRefGoogle Scholar
  41. Kholmyansky M, Tsinober A (2009) On an alternative explanation of anomalous scaling and how well-defined is the concept of inertial range. Phys Lett A 273:2364–2367 CrossRefGoogle Scholar
  42. Kolmogorov AN (1941a) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl Akad Nauk SSSR 30:299–303. For English translation see Tikhomirov VM (ed) (1991) Selected works of AN Kolmogorov, vol I, Kluwer, pp 318–321 Google Scholar
  43. Kolmogorov AN (1941b) Dissipation of energy in locally isotropic turbulence. Dokl Akad Nauk SSSR 32:19–21. For English translation see Tikhomirov VM (ed) (1991) Selected works of AN Kolmogorov, vol I, Kluwer, pp 324–327 Google Scholar
  44. Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence is a viscous incompressible fluid at high Reynolds number. J Fluid Mech 13:82–85 MathSciNetMATHCrossRefGoogle Scholar
  45. Kraichnan RH (1974) On Kolmogorov’s inertial-range theories. J Fluid Mech 62:305–330 MathSciNetMATHCrossRefGoogle Scholar
  46. Kuo AY-S, Corrsin S (1971) Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid. J Fluid Mech 50:285–319 CrossRefGoogle Scholar
  47. Kuznetsov VR, Praskovsky AA, Sabelnikov VA (1992) Finescale turbulence structure of intermittent shear flows. J Fluid Mech 243:595–622 CrossRefGoogle Scholar
  48. Ladyzhenskaya OA (1975) Mathematical analysis of NSE for incompressible liquids. Annu Rev Fluid Mech 7:249–272 CrossRefGoogle Scholar
  49. Landau LD, Lifshits EM (1944) Fluid mechanics, 1st Russian edn Google Scholar
  50. Leung T, Swaminathan N, Davidson PA (2012) Geometry and interaction of structures in homogeneous isotropic turbulence. J Fluid Mech 710:453–481 MathSciNetCrossRefGoogle Scholar
  51. Li Y, Perlman E, Wan M, Yang Y, Meneveau C, Burns R, Chen S, Szalay A, Eyink G (2008) A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J Turbul 9(31):1–29 Google Scholar
  52. Liepmann HW (1979) The rise and fall of ideas in turbulence. Am Sci 67:221–228 MathSciNetGoogle Scholar
  53. Lindborg E (1999) Can atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J Fluid Mech 388:259–288 MATHCrossRefGoogle Scholar
  54. Lions JL (1969) Quelques méthodes de résolution des problèmes uax limites non linéaires. Dunod Gauthier-Villars, Paris Google Scholar
  55. Lu SS, Willmarth WW (1973) Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J Fluid Mech 60:481–511 CrossRefGoogle Scholar
  56. Lumley JL (1970) Stochastic tools in turbulence. Academic Press, New York MATHGoogle Scholar
  57. Lumley JL (1981) Coherent structures in turbulence. In: Meyer R (ed) Transition and turbulence. Academic Press, New York, pp 215–242 Google Scholar
  58. Lumley JL (1989) The state of turbulence research. In: George WK, Arndt R (eds) Advances in turbulence. Hemisphere/Springer, Washington, pp 1–10 Google Scholar
  59. Majda AJ, Kramer PR (1999) Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys Rep 314:237–574 MathSciNetCrossRefGoogle Scholar
  60. Malm J, Schlatter P, Sandham ND (2012) A vorticity stretching diagnostic for turbulent and transitional flows. Theor Comput Fluid Dyn 26:485–499 CrossRefGoogle Scholar
  61. Monin AS (1991) On definition of coherent structures. Sov Phys Dokl 36(6):424–426 MathSciNetMATHGoogle Scholar
  62. Newton KA, Aref H (2003) Chaos vs turbulence. In: Scott A (ed) Encyclopedia of nonlinear science, pp 114–116 Google Scholar
  63. Novikov EA (1967) Kinetic equations for a vortex field. Dokl Akad Nauk SSSR 177(2):299–301. English translation: Sov Phys Dokl 12(11), 1006–1008 (1968) Google Scholar
  64. Novikov EA (1974) Statistical irreversibility of turbulence. Arch Mech 4:741–745 Google Scholar
  65. Novikov EA (1990a) The effects of intermittency on statistical characteristics of turbulence and scale similarity of breakdown coefficients. Phys Fluids A 2:814–820 MathSciNetCrossRefGoogle Scholar
  66. Orszag SA (1977) Lectures on the statistical theory of turbulence. In: Balian R, Peube J-L (eds) Fluid dynamics. Gordon and Breach, New York, pp 235–374 Google Scholar
  67. Ott E (1999) The role of Lagrangian chaos in the creation of multifractal measures. In: Gyr A, Kinzelbach W, Tsinober A (eds) Fundamental problematic issues in turbulence. Birkhäuser, Basel, pp 381–403 CrossRefGoogle Scholar
  68. Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge MATHCrossRefGoogle Scholar
  69. Pouransari Z, Speetjens MFM, Clercx HJH (2010) Formation of coherent structures by fluid inertia in three-dimensional laminar flows. J Fluid Mech 654:5–34 MathSciNetMATHCrossRefGoogle Scholar
  70. Pullin DI, Inoue M, Saito N (2013) On the asymptotic state of high Reynolds number, smooth-wall turbulent flows. Phys Fluids 25:015116 CrossRefGoogle Scholar
  71. Pumir A, Shraiman BI, Siggia ED (1997) Perturbation theory for the δ-correlated model of passive scalar advection near the Batchelor limit. Phys Rev E 55:R1263 CrossRefGoogle Scholar
  72. Seiwert J, Morize C, Moisy F (2008) On the decrease of intermittency in decaying rotating turbulence. Phys Fluids 20:071702 CrossRefGoogle Scholar
  73. She Z-S, Zhang Z-X (2009) Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems. Acta Mech Sin 25:279–294 MATHCrossRefGoogle Scholar
  74. She Z-S, Jackson E, Orszag SA (1990) Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344:226–229 CrossRefGoogle Scholar
  75. Shen X, Warhaft Z (2000) The anisotropy of the small-scale structure in high Reynolds number, Re λ=1,000, turbulent shear flow. Phys Fluids 12:2976–2989 CrossRefGoogle Scholar
  76. Shlesinger MS (2000) Exploring phase space. Nature 405:135–137 CrossRefGoogle Scholar
  77. Sreenivasan KR, Antonia R (1997) The phenomenology of small-scale turbulence. Annu Rev Fluid Mech 29:435–472 MathSciNetCrossRefGoogle Scholar
  78. Stewart RW (1969) Turbulence and waves in stratified atmosphere. Radio Sci 4:1269–1278 CrossRefGoogle Scholar
  79. Taylor GI (1938a) Production and dissipation of vorticity in a turbulent fluid. Proc R Soc Lond A 164:15–23 MATHCrossRefGoogle Scholar
  80. Taylor GI (1938b) The spectrum of turbulence. Proc R Soc Lond A 164:476–490 CrossRefGoogle Scholar
  81. Tennekes H (1976) Fourier-transform ambiguity in turbulence dynamics. J Atmos Sci 33:1660–1663 CrossRefGoogle Scholar
  82. Townsend AA (1948) Local isotropy in the turbulent wake of cylinder. Aust J Sci Res 1:161–174 Google Scholar
  83. Townsend AA (1976) The structure of turbulent shear flow. Cambridge University Press, Cambridge MATHGoogle Scholar
  84. Townsend AA (1987) Organized eddy structures in turbulent flows. Physicochem Hydrodyn 8(1):23–30 Google Scholar
  85. Tritton DJ (1988) Physical fluid dynamics, 2nd edn. Clarendon, Oxford Google Scholar
  86. Tsinober A (1995) Variability of anomalous transport exponents versus different physical situations in geophysical and laboratory turbulence. In: Schlesinger M, Zaslavsky G, Frisch U (eds) Levy flights and related topics in physics. Lecture notes in physics, vol 450. Springer, Berlin, pp 3–33 Google Scholar
  87. Tsinober A (1998a) Is concentrated vorticity that important? Eur J Mech B, Fluids 17:421–449 MATHCrossRefGoogle Scholar
  88. Tsinober A (1998b) Turbulence—beyond phenomenology. In: Benkadda S, Zaslavsky GM (eds) Chaos, kinetics and nonlinear dynamics in fluids and plasmas. Lecture notes in physics, vol 511. Springer, Berlin, pp 85–143 Google Scholar
  89. Tsinober A (2009) An informal conceptual introduction to turbulence. Springer, Berlin MATHCrossRefGoogle Scholar
  90. Van Zandt TE (1982) A universal spectrum of buoyancy waves in the atmosphere. Geophys Res Lett 9:575–578 CrossRefGoogle Scholar
  91. Vassilicos JC (ed) (2001) Intermittency in turbulent flows. Cambridge University Press, Cambridge MATHGoogle Scholar
  92. Wolf M, Lüthi B, Holzner M, Krug D, Kinzelbach W, Tsinober A (2012a) Investigations on the local entrainment velocity in a turbulent jet. Phys Fluids 24:105110 CrossRefGoogle Scholar
  93. Wolf M, Lüthi B, Holzner M, Krug D, Kinzelbach W, Tsinober A (2012b) Effects of mean shear on the local turbulent entrainment process. J Fluid Mech (in press) Google Scholar
  94. Worth NA, Nickels TB (2011) Some characteristics of thin shear layers in homogeneous turbulent flow. Philos Trans R Soc Lond A 2011(369):709–722 MathSciNetCrossRefGoogle Scholar
  95. Zaslavsky GM (1999) Chaotic dynamics and the origin of statistical laws. Phys Today 51:39–45 CrossRefGoogle Scholar
  96. Zeldovich YaB, Ruzmaikin AA, Sokoloff DD (1990) The almighty chance. World Scientific, Singapore Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Arkady Tsinober
    • 1
  1. 1.School of Mechanical EngineeringTel Aviv UniversityTel AvivIsrael

Personalised recommendations