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Additional Issues of Importance Related to the Use of Statistical Methods

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The Essence of Turbulence as a Physical Phenomenon

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Abstract

One of the most serious concerns is that statistics alone can be misused and misleading due the absence of theory based on first principles and inadequate tools for handling both the problem and the phenomenon and difficult issues of (mis)interpretation, validation, oversimplification and related, especially if the information is not in physical space, e.g. Fourier or any other decomposition. Among the concerns is the issue of statistical (pre)dominance versus dynamical relevance. The statistical predominance does 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.

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Notes

  1. 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. 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 (1962), especially unpublished, but see Long (2003), the error bar is not that small as to exclude the \(k^{-6/3}\) spectrum which correspond just to a single sharp change in velocity, see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 extreme/strong events.

  3. 3.

    In this context it is of interest to quote Goto and Kraichnan (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. 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 in reality is 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. 5.

    This seems to be the reason why Batchelor (1953) called this statistical regime as “universal equilibrium” - a somewhat misleading term.Kolmogorov would never use the term “equilibrium” especially as in a variety of “explanations” followed his publications. He would definitely not use the “truncated” Fourier version of the so called Lin equation as did Batchelor (1953) and some later authors for the high wave numbers \( T(k)=2\nu k^{2}E(k)\) - ‘equilibrium range’, Eq. 6.6.5, p. 126. A “small” \(\partial E/\partial t\) can (and mosly will) make an essential difference turning the ‘parabolic’ equation into an ‘elliptical’ one. In employing statistics a “small” difference between approximately and exactly stationary is not necessarily synonymous to ‘unimportant’ especially in the context of statistical predominance versus dynamical significance. Kolmogorov would also never use the Fourier transform either.

  6. 6.

    One cannot be sure even whether one can always consider the statistically stationary turbulent flows as “equilibrium” except those with large scale time independent excitation.

  7. 7.

    In the language of mathematicians invariant probability measures, and there is an unsolved question/problem which one is selected in experiments. Ruelle (1983).

  8. 8.

    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). Nameley, they have shown that there are measures – in the language of physicists 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.

  9. 9.

    Turbulent flows possess (empirically) stable statistical properties (SSP), not just averages but almost all statistical properties. In case if 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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  1. School of Mechanical Engineering, Tel Aviv University, Tel Aviv, Israel

    Arkady Tsinober

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Tsinober, A. (2019). Additional Issues of Importance Related to the Use of Statistical Methods. In: The Essence of Turbulence as a Physical Phenomenon. Springer, Cham. https://doi.org/10.1007/978-3-319-99531-1_6

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