# Nature of Turbulence

## Abstract

The main dispute about the origins and nature of turbulence involves a number of aspects and issues in the frame of the dichotomy of deterministic versus random. In science this dispute covers an enormous spectrum of themes such as philosophy of science, mathematics, physics and other natural sciences. Fortunately, we do not have to dwell into this ocean of debate and opposing and intermediate opinions. This is mainly because (as it stands now) turbulence is described by the NSE which are purely deterministic equations with extremely complex behavior enforcing use of statistical methods, but this does not mean that the nature of such systems is statistical in any/some sense as frequently claimed. The bottom line is that turbulence is only apparently random: the apparently random behavior of turbulence is a manifestation of properties of a purely deterministic law of nature in our case adequately described by NSE. An important point is that this complex behavior does make this law neither probabilistic nor indeterminate.

One of the problems of turbulent research is that we are forced to use statistical methods in one sense/way or another. All statistical methods have inherent limitations the most acute reflected in the inability of all theoretical attempts (both physical and mathematical) to create a rigorous theory along with other inherent limitations of handling data such as description and interpretation of observations. However, the technical necessity of using statistical methods is commonly stated as the only possibility in the theory of turbulence. The consequence of this leads to the necessity of low-dimensional description with the removal of small scale and high-frequency components of the dynamics of a flow including quantities containing a great deal of fundamental physics of the whole flow field such as rotational and dissipative nature of turbulence among others. Thus, relying on statistical methods only (again with all the respect) one is inevitably loosing/missing essential aspects of basic physics of turbulence. So one stays with the troublesome question whether it is possible to penetrate into the fundamental physics of turbulence via statistics only. In other words, there is an essential difference between the enforced necessity to employ statistical methods in view absence of other methods so far and the impossibility in principle to study turbulence via other approaches. This is especially discouraging all attempts to get into more than just “en masse”. Also such a standpoint means that there is not much to be expected as concerns the essence of turbulence using exclusively statistical methods.

## Keywords

Similar Object Hydrogen Fluoride Deterministic Equation Random Behavior Individual Realization## References

- Arnold VI (1991) Kolmogorov’s hydrodynamics attractors. Proc R Soc Lond A 434:19–22 CrossRefGoogle Scholar
- Biferale L, Lanotte AS, Federico Toschi F (2004) Effects of forcing in three-dimensional turbulent flows. Phys Rev Lett 92:094503 CrossRefGoogle Scholar
- Bonnet JP (ed) (1996) Eddy structure identification. Springer, Berlin zbMATHGoogle Scholar
- Borel E (1909) Sur les probabilites denombrables et leurs applications arithmetiques. Rend Circ Mat Palermo 41:247–271 CrossRefGoogle Scholar
- Bradshaw P (1994) Turbulence: the chief outstanding difficulty of our subject. Exp Fluids 16:203–216 Google Scholar
- Einstein A (1926) Letter to Max Born (4 December 1926); the Born-Einstein letters (translated by Irene Born). Walker and Company, New York. ISBN 0-8027-0326-7 Google Scholar
- Elsinga GE, Marusic I (2010) Universal aspects of small-scale motions in turbulence. J Fluid Mech 662:514–539 zbMATHCrossRefGoogle Scholar
- Foiaş C, Manley O, Rosa R, Temam R (2001) Navier–Stokes equations and turbulence. Cambridge University Press, Cambridge zbMATHGoogle Scholar
- Frenkiel FN, Klebanoff PS, Huang TT (1979) Grid turbulence in air and water. Phys Fluids 22:1606–1617 CrossRefGoogle Scholar
- Galanti B, Tsinober A (2006) Physical space helicity properties in quasi-homogeneous forced turbulence. Phys Lett A 352:141–149 zbMATHCrossRefGoogle Scholar
- Gkioulekas E (2007) On the elimination of the sweeping interactions from theories of hydrodynamic turbulence. Physica D 226:151–172 MathSciNetzbMATHCrossRefGoogle Scholar
- Guckenheimer J (1986) Strange attractors in fluids: another view. Annu Rev Fluid Mech 18:15–31 MathSciNetCrossRefGoogle Scholar
- Gulitskii G, Kholmyansky M, Kinzlebach W, Lüthi B, Tsinober A, Yorish S (2007a) Velocity and temperature derivatives in high Reynolds number turbulent flows in the atmospheric surface layer. Facilities, methods and some general results. J Fluid Mech 589:57–81 Google Scholar
- Gulitskii G, Kholmyansky M, Kinzlebach W, Lüthi B, Tsinober A, Yorish S (2007b) Velocity and temperature derivatives in high Reynolds number turbulent flows in the atmospheric surface layer. Part 2. Accelerations and related matters. J Fluid Mech 589:83–102 Google Scholar
- Gulitskii G, Kholmyansky M, Kinzlebach W, Lüthi B, Tsinober A, Yorish S (2007c) Velocity and temperature derivatives in high Reynolds number turbulent flows in the atmospheric surface layer. Part 3. Temperature and joint statistics of temperature and velocity derivatives. J Fluid Mech 589:103–123 Google Scholar
- Hawking S, Penrose R (1996) The nature and time. Princeton University Press, Princeton, p 26 zbMATHGoogle Scholar
- Holmes PJ, Berkooz G, Lumley JL (1996) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge zbMATHCrossRefGoogle Scholar
- Hoyle F (1957) The black cloud. Harper, New York Google Scholar
- Kolmogorov AN (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin. English translation: Kolmogorov AN (1956) Foundations of the theory of probability, Chelsea 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 (1956) The theory of probability. In: Aleksandrov AD et al. (eds) Mathematics, its content, methods and meaning. AN SSSR, Moscow. English translation: Am Math Soc, pp 229–264 (1963) Google Scholar
- Kolmogorov AN (1985) In: Notes preceding the papers on turbulence in the first volume of his selected papers, vol I. Kluwer, Dordrecht, pp 487–488. English translation: Tikhomirov VM (ed) (1991) Selected works of AN Kolmogorov Google Scholar
- Kraichnan RH (1959) The structure of isotropic turbulence at very high Reynolds numbers. J Fluid Mech 5:497–543 MathSciNetzbMATHCrossRefGoogle Scholar
- Kraichnan RH (1964) Kolmogorov’s hypotheses and Eulerian turbulence theory. Phys Fluids 7:1723–1734 MathSciNetzbMATHCrossRefGoogle Scholar
- Kraichnan RH, Chen S (1989) Is there a statistical mechanics of turbulence? Physica D 37:160–172 MathSciNetzbMATHCrossRefGoogle Scholar
- Ladyzhenskaya OA (1969) Mathematical problems of the dynamics of viscous incompressible fluids. Gordon and Breach, New York Google Scholar
- Landau LD, Lifshits EM (1959) Fluid mechanics. Pergamon, New York Google Scholar
- Laplace PS (1951) A philosophical essay on probabilities. Dover, New York. Translated by Truscott FW, Emory FL (Essai philosophique sur les probabilités. Rééd., Bourgeois, Paris, 1986. Texte de la 5éme éd., 1825) zbMATHGoogle Scholar
- Leray J (1934) Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math 63:193–248 MathSciNetzbMATHCrossRefGoogle Scholar
- Lorenz EN (1972) Investigating the predictability of turbulent motion. In: Rosenblatt M, van Atta CC (eds) Statistical models and turbulence. Lecture notes in physics, vol 12, pp 195–204 CrossRefGoogle Scholar
- Loskutov A (2010) Fascination of chaos. Phys Usp 53(12):1257–1280 MathSciNetCrossRefGoogle Scholar
- Lumley JL (1970) Stochastic tools in turbulence. Academic Press, New York zbMATHGoogle Scholar
- Lüthi B, Tsinober A, Kinzelbach W (2005) Lagrangian measurement of vorticity dynamics in turbulent flow. J Fluid Mech 528:87–118 zbMATHCrossRefGoogle Scholar
- Mollo-Christensen E (1973) Intermittency in large-scale turbulent flows. Annu Rev Fluid Mech 5:101–118 CrossRefGoogle Scholar
- Monin AS, Yaglom AM (1971) Statistical fluid mechanics, vol 1. MIT Press, Cambridge Google Scholar
- Ornstein S, Weiss B (1991) Statistical properties of chaotic systems. Bull Am Math Soc 24:11–116 MathSciNetzbMATHCrossRefGoogle 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
- Orszag SA, Staroselsky I, Yakhot V (1993) Some basic challenges for large eddy simulation research. In: Orszag SA, Galperin B (eds) Large eddy simulation of complex engineering and geophysical flows. Cambridge University Press, Cambridge, pp 55–78 Google Scholar
- Palmer TN, Hardaker PJ (2011) Introduction: handling uncertainty in science. Philos Trans R Soc Lond A 369:4681–4684 CrossRefGoogle Scholar
- Poincare H (1952b) Science and hypothesis. Dover, New York, pp xxiii–xiv zbMATHGoogle Scholar
- Ruelle D (1979) Microscopic fluctuations and turbulence. Phys Lett 72A(2):81–82 MathSciNetGoogle Scholar
- Tennekes H (1975) Eulerian and Lagrangian time microscales in isotropic turbulence. J Fluid Mech 67:561–567 zbMATHCrossRefGoogle Scholar
- Tikhomirov VM (ed) (1991) Selected works of AN Kolmogorov, vol I. Kluwer, Dordrecht Google Scholar
- Tritton DJ (1988) Physical fluid dynamics, 2nd edn. Clarendon, Oxford Google Scholar
- Tsinober A (2001) An informal introduction to turbulence. Kluwer, Dordrecht zbMATHGoogle Scholar
- Tsinober A (2009) An informal conceptual introduction to turbulence. Springer, Berlin zbMATHCrossRefGoogle Scholar
- Tsinober A, Vedula P, Yeung PK (2001) Random Taylor hypothesis and the behaviour of local and convective accelerations in isotropic turbulence. Phys Fluids 13:1974–1984 CrossRefGoogle Scholar
- Vishik MJ, Fursikov AV (1988) Mathematical problems of statistical hydromechanics. Kluwer, Dordrecht zbMATHCrossRefGoogle Scholar
- von Neumann J (1949) Recent theories of turbulence. In: Taub AH (ed) A report to the office of naval research. Collected works, vol 6. Pergamon, New York, pp 437–472 Google Scholar
- Wiener N (1938) Homogeneous chaos. Am J Math 60:897–936 MathSciNetCrossRefGoogle Scholar