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.
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Notes
- 1.
Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the “old one.” I, at any rate, am convinced that He does not throw dice (Einstein 1926).
Not only does God definitely play dice, but He sometimes confuses us by throwing them where they can’t be seen (Hawking and Penrose 1996).
Today “chaotic” and “determinstic” are not considered as counterparts of a false dichotomy. This takes it origin to Poincare (1952b, pp. xxiii–iv).
- 2.
However, we repeat that since Leray (1934) until recently one was not sure about the theoretical, but not observational, possibility that turbulence is a manifestation of breakdown of the Navier–Stokes equations. Also note the statement by Ladyzhenskaya (1969): …it is hardly possible to explain the transition from laminar to turbulent flows within the framework of the classical Navier–Stokes theory.
- 3.
The en masse comes from the analogy with statistical physics. But there one has literally many similar objects—molecules. So one realization there may well suffice either, see below.
- 4.
The basic question (which usually is not asked) concerning statistical description is whether such complex behavior permits a closed representation that is simple enough to be tractable and insightful but powerful enough to be faithful to the essential dynamics (Kraichnan and Chen 1989).
The problem is that in such an approach the rotational and dissipative aspects are not considered as belonging to the essential dynamics.
- 5.
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Tsinober, A. (2014). Nature of Turbulence. In: The Essence of Turbulence as a Physical Phenomenon. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7180-2_5
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