Abstract
One of the concerns is that statistics only can be misused and misleading, so there are difficult issues of interpretation, validation and related, especially if the information is not in physical space, e.g. Fourier or any other decomposition.
One of the concerns is the issue of statistical dominance versus dynamical relevance. The statistical predominance not necessarily corresponds to the dynamical relevance as, e.g. in the case of sweeping decorrelation hypothesis or “exotic” averaging of turbulent flow fields such as represented in the local coordinate system defined by the eigenvectors of the strain rate tensor at each point.
Similar concern is arising in issues handling turbulence structure(s). These are finite objects which nevertheless are mostly hunted by isosurfacing and thresholding and are defined at some time moment only and, moreover, they cannot be followed in time. Therefore, producing statistics is performed out of collections of “similar objects” obtained from snapshots at the same and different time moments. But the painful question is how really “similar” are all these if they are typically defined by one parameter only? It is almost obvious that such kind of “statistical” processing, i.e. another kind of “exotic” averaging is killing most of essential features of the real “structure” and leaves the question of relevance, say the dynamical one, of these “structures” at best open.
The bottom line is that with all the respect, relying on statistics only such as, e.g. very strong and very weak correlations, ‘exotic averaging’, etc., may bring one to nowhere by missing essential dynamical effects.
Among the consequences is a serious misuse and interpretational (and terminological either) abuse of observations, which is essentially “aided” by the absence of genuine theory.
It is not unusual that when it goes about validation that the hard data (both from physical observations and numerical) is considered as kind of inferior as compared to ‘models’ so that the former are tested against the latter and not vice a versa as it is done even in fields being in possession of rigorous theories. Not many care about the observational evidence. The approach in reality is in some sense reverse. The widespread view of both mathematicians and theoretical physicists is that the main function of all experiments/observations both physical and numerical is to “validate theories”—paradoxically nonexistent so far. In such a situation the issue of interpretation and validation as concerns the right results for the right reasons or “theories” versus hard evidence becomes of more than of utmost importance. A related issue of importance is on ergodicity and similar.
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Notes
- 1.
For example, it is clear that if a result can be derived by dimensional analysis alone… then it can be derived by almost any theory, right or wrong, which is dimensionally-correct and uses the right variables (Bradshaw 1994).
- 2.
It is noteworthy that this spectrum is not precisely the “right” one. Indeed, if one looks at the data by Grant et al. (1962), especially unpublished, but see Long (2003), the error bar is not that small as to exclude the k −6/3 spectrum which correspond just to a single sharp change in velocity, see also Tsinober (2009, p. 334) and references therein for recent results on the “approximately” k −5/3. Moreover, the “small” differences are essential and increase as concerns higher order quantities, derivatives and “strong events”.
- 3.
In this context it is of interest to quote Goto and Kraichnan himself (2004): Multifractal models of turbulence have not been derived from the NS equation but they are supported by theoretical arguments and their parameters can be tuned to agree well with a variety of experimental measurements… Multifractal cascade models raise the general issue of distinction between what is descriptive of physical behavior and what can be used for analysis of data… Multifractal models may or may not express well the cascade physics at large but finite Reynolds numbers.
- 4.
Frisch (1995) presents this in the form of his hypothesis H1 (p. 74), but omits to mention that it is due to Kolmogorov: there is no presentation of the hypothesis of local isotropy in his book.
It is noteworthy that Kolmogorov theory is in reality based on similarity and dimensionality and has no connection to NSE, see e.g. Monin and Yaglom (1971, p. 21): The great attention paid in this book to, similarity and dimensionality is also conditioned by the fact that Kolmogorov’s theory of locally isotropic turbulence (which is based entirely on these methods) is given a great deal of space here. In other words, experimental validation of Kolmogorov (1941a) theory, as all theories of this kind, has a limited value. Again, it is clear that if a result can be derived by dimensional analysis alone… then it can be derived by almost any theory, right or wrong, which is dimensionally-correct and uses the right variables (Bradshaw 1994).
- 5.
In the language of mathematicians invariant probability measures, and there is a question which one is selected in experiments (Ruelle 1983).
- 6.
The problem with this ergodicity assumption is that nobody has ever even come close to proving it for the Navier–Stokes equation (Foiaş 1997), though some mathematical results, which are claimed to be relevant to turbulence are given in Foiaş et al. (2001). Namely, they have shown that there are measures—in the language of physics ensembles—on a function space that are time-invariant. However, invariance under time evolution is not enough to specify a unique measure which would describe turbulence. Another problem is that it is not clear how the objects that the authors have constructed and used in their proofs are relevant/related or even have anything to do with turbulence.
- 7.
Turbulent flows possess (empirically) stable statistical properties (SSP), not just averages but almost all statistical properties. In case of statistically stationary flows the existence of SSP seems to be an indication of the existence of what mathematicians call attractors. But matters are more complicated as many statistical properties of time-dependent in the statistical sense turbulent flows (possessing no attractor, but stable SSP) are quite similar at least qualitatively to those of statistically stationary ones as long as the Reynolds number of the former is not too small at any particular time moment of interest. This can be qualified as some manifestation of qualitative temporal universality/memory.
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Tsinober, A. (2014). Additional Issues of Importance Related to the Use of Statistical Methods. In: The Essence of Turbulence as a Physical Phenomenon. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7180-2_6
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