# Intermittency and Structure(s) of and/in Turbulence

## 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## References

- Aluie H (2012) Scale locality and the inertial range in compressible turbulence. J Fluid Mech (submitted). arXiv:1101.0150
- Anderson PW (1972) More is different. Science 177:393–396 CrossRefGoogle Scholar
- Armi L, Flament P (1987) Cautionary remarks on the spectral interpretation of turbulent flows. J Geophys Res 90:11,779–11,782 Google Scholar
- Arnold VI (1991) Kolmogorov’s hydrodynamics attractors. Proc R Soc Lond A 434:19–22 CrossRefGoogle Scholar
- Batchelor GK, Townsend AA (1949) The nature of turbulent motion at large wave-numbers. Proc R Soc Lond A 199:238–255 MATHCrossRefGoogle Scholar
- Betchov R (1974) Non-Gaussian and irreversible events in isotropic turbulence. Phys Fluids 17:1509–1512 CrossRefGoogle Scholar
- Betchov R (1976) On the non-Gaussian aspects of turbulence. Arch Mech 28(5–6):837–845 Google Scholar
- Betchov R (1993) In: Dracos T, Tsinober A (eds) New approaches and turbulence. Birkhäuser, Basel, p 155 Google Scholar
- Biferale L, Procaccia I (2005) Anisotropy in turbulent flows and in turbulent transport. Phys Rep 414:43–164 MathSciNetCrossRefGoogle Scholar
- Blackwelder RF (1983) Analogies between transitional and turbulent boundary layers. Phys Fluids 26:2807–2815 CrossRefGoogle Scholar
- Bonnet JP (ed) (1996) Eddy structure identification. Springer, Berlin MATHGoogle Scholar
- 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
- Chorin AJ (1994) Vorticity and turbulence. Springer, Berlin MATHGoogle Scholar
- Chorin AJ (1996) Turbulence cascades across equilibrium spectra. Phys Rev 54:2616–2619 Google Scholar
- Constantin P (1996) Navier–Stokes equations and incompressible fluid turbulence. Lect Appl Math 31:219–234 MathSciNetGoogle Scholar
- Cvitanović P, Gibson P (2010) Phys Scr T 142:014007 CrossRefGoogle Scholar
- 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
- Dryden H (1948) Recent advances in boundary layer flow. Adv Appl Mech 1:1–40 MathSciNetCrossRefGoogle Scholar
- Dwoyer DL, Hussaini MY, Voigt RG (eds) (1985) Theoretical approaches to turbulence. Springer, Berlin MATHGoogle Scholar
- 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
- Elsinga GE, Marusic I (2010) Universal aspects of small-scale motions in turbulence. J Fluid Mech 662:514–539 MATHCrossRefGoogle Scholar
- Ferchichi M, Tavoularis S (2000) Reynolds number dependence of the fine structure of uniformly sheared turbulence. Phys Fluids 12:2942–2953 CrossRefGoogle Scholar
- Feynmann R (1963) Lect Phys 2:41–42 Google Scholar
- Frenkiel FN, Klebanoff PS, Huang TT (1979) Grid turbulence in air and water. Phys Fluids 22:1606–1617 CrossRefGoogle Scholar
- Frisch U (1995) Turbulence: the legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge MATHGoogle Scholar
- 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
- Gibson CH, Friehe CA, McConnell SO (1977) Structure of sheared turbulent fields. Phys Fluids 20(II):S156–S167 CrossRefGoogle Scholar
- Goldshtik MA, Shtern VN (1981) Structural turbulence theory. Dokl Akad Nauk SSSR 257(6):1319–1322 (in Russian) Google Scholar
- Guckenheimer J (1986) Strange attractors in fluids: another view. Annu Rev Fluid Mech 18:15–31 MathSciNetCrossRefGoogle Scholar
- Hill RJ (1997) Applicability of Kolmogorov’s and Monin’s equations to turbulence. J Fluid Mech 353:67–81 MathSciNetMATHCrossRefGoogle Scholar
- Holmes PJ, Berkooz G, Lumley JL (1996) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge MATHCrossRefGoogle Scholar
- 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
- Hopf E (1948) A mathematical example displaying features of turbulence. Commun Pure Appl Math 1:303–322 MathSciNetMATHCrossRefGoogle Scholar
- 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
- 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
- Kadanoff LP (1986) Fractals: where is the physics? Phys Today 39:3–7 Google Scholar
- 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
- 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
- Keefe L (1990b) In: Lumley JL (ed) Whither turbulence? Springer, Berlin, p 189 Google Scholar
- Keefe L, Moin P, Kim J (1992) The dimension of attractors underlying periodic turbulent Poiseulle flow. J Fluid Mech 242:1–29 MathSciNetMATHCrossRefGoogle Scholar
- 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
- 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
- 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
- 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
- Kraichnan RH (1974) On Kolmogorov’s inertial-range theories. J Fluid Mech 62:305–330 MathSciNetMATHCrossRefGoogle Scholar
- 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
- Kuznetsov VR, Praskovsky AA, Sabelnikov VA (1992) Finescale turbulence structure of intermittent shear flows. J Fluid Mech 243:595–622 CrossRefGoogle Scholar
- Ladyzhenskaya OA (1975) Mathematical analysis of NSE for incompressible liquids. Annu Rev Fluid Mech 7:249–272 CrossRefGoogle Scholar
- Landau LD, Lifshits EM (1944) Fluid mechanics, 1st Russian edn Google Scholar
- Leung T, Swaminathan N, Davidson PA (2012) Geometry and interaction of structures in homogeneous isotropic turbulence. J Fluid Mech 710:453–481 MathSciNetCrossRefGoogle Scholar
- 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
- Liepmann HW (1979) The rise and fall of ideas in turbulence. Am Sci 67:221–228 MathSciNetGoogle Scholar
- Lindborg E (1999) Can atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J Fluid Mech 388:259–288 MATHCrossRefGoogle Scholar
- Lions JL (1969) Quelques méthodes de résolution des problèmes uax limites non linéaires. Dunod Gauthier-Villars, Paris Google Scholar
- 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
- Lumley JL (1970) Stochastic tools in turbulence. Academic Press, New York MATHGoogle Scholar
- Lumley JL (1981) Coherent structures in turbulence. In: Meyer R (ed) Transition and turbulence. Academic Press, New York, pp 215–242 Google Scholar
- 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
- Majda AJ, Kramer PR (1999) Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys Rep 314:237–574 MathSciNetCrossRefGoogle Scholar
- 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
- Monin AS (1991) On definition of coherent structures. Sov Phys Dokl 36(6):424–426 MathSciNetMATHGoogle Scholar
- Newton KA, Aref H (2003) Chaos vs turbulence. In: Scott A (ed) Encyclopedia of nonlinear science, pp 114–116 Google Scholar
- 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
- Novikov EA (1974) Statistical irreversibility of turbulence. Arch Mech 4:741–745 Google Scholar
- 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
- 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
- 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
- Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge MATHCrossRefGoogle Scholar
- 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
- 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
- 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 - Seiwert J, Morize C, Moisy F (2008) On the decrease of intermittency in decaying rotating turbulence. Phys Fluids 20:071702 CrossRefGoogle Scholar
- 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
- She Z-S, Jackson E, Orszag SA (1990) Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344:226–229 CrossRefGoogle Scholar
- 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 - Shlesinger MS (2000) Exploring phase space. Nature 405:135–137 CrossRefGoogle Scholar
- Sreenivasan KR, Antonia R (1997) The phenomenology of small-scale turbulence. Annu Rev Fluid Mech 29:435–472 MathSciNetCrossRefGoogle Scholar
- Stewart RW (1969) Turbulence and waves in stratified atmosphere. Radio Sci 4:1269–1278 CrossRefGoogle Scholar
- Taylor GI (1938a) Production and dissipation of vorticity in a turbulent fluid. Proc R Soc Lond A 164:15–23 MATHCrossRefGoogle Scholar
- Taylor GI (1938b) The spectrum of turbulence. Proc R Soc Lond A 164:476–490 CrossRefGoogle Scholar
- Tennekes H (1976) Fourier-transform ambiguity in turbulence dynamics. J Atmos Sci 33:1660–1663 CrossRefGoogle Scholar
- Townsend AA (1948) Local isotropy in the turbulent wake of cylinder. Aust J Sci Res 1:161–174 Google Scholar
- Townsend AA (1976) The structure of turbulent shear flow. Cambridge University Press, Cambridge MATHGoogle Scholar
- Townsend AA (1987) Organized eddy structures in turbulent flows. Physicochem Hydrodyn 8(1):23–30 Google Scholar
- Tritton DJ (1988) Physical fluid dynamics, 2nd edn. Clarendon, Oxford Google Scholar
- 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
- Tsinober A (1998a) Is concentrated vorticity that important? Eur J Mech B, Fluids 17:421–449 MATHCrossRefGoogle Scholar
- 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
- Tsinober A (2009) An informal conceptual introduction to turbulence. Springer, Berlin MATHCrossRefGoogle Scholar
- Van Zandt TE (1982) A universal spectrum of buoyancy waves in the atmosphere. Geophys Res Lett 9:575–578 CrossRefGoogle Scholar
- Vassilicos JC (ed) (2001) Intermittency in turbulent flows. Cambridge University Press, Cambridge MATHGoogle Scholar
- 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
- 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
- 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
- Zaslavsky GM (1999) Chaotic dynamics and the origin of statistical laws. Phys Today 51:39–45 CrossRefGoogle Scholar
- Zeldovich YaB, Ruzmaikin AA, Sokoloff DD (1990) The almighty chance. World Scientific, Singapore Google Scholar