7. The N’s of Turbulence

There is no consensus on the meaning of the term integrability, but it is mostly agreed that integrable systems behave nicely and are globally ‘regular’, whereas the nonintegrable systems are not ‘solvable exactly’ and exhibit chaotic behavior, see Zakharov (1990) and Kosmann-Schwarzbach et al. (2004) for more examples and discussion on what integrability is. The latter write It would fit for a course entitled “Integrability” to start with a definition of this notion. Alas, this is not possible. There exists a profusion of definitions and where you have two scientists you have (at least) three different definitions of integrability but mention the definition by Poincaré: to integrate a differential equation is to find for the general solution a finite expression, possibly multivalued, in a finite number of functions.

Otherwise, i.e. in the physical space there are no cascades and there is no necessity for such creatures. In other words it is a “technical issue” - turbulence “handles” the issue without any unnecessary mediators.

For example, see Tennekes and Lumley (1972): dissipation of energy at small scales occurs only if there exists a dynamical mechanism that transfers energy from large scales to small scales p. 60; the internal dynamics of turbulence must transfer energy from large scales to small scales p. 68; there exists a dynamical mechanism that transfers energy from large scales to small scales ...the energy transfer from large eddies to small eddies may be considered in terms of vortex stretching p. 75.

The latest example is by Jiménez et al. (2017), Jimenez (2018): “the three-dimensional energy cascade of the viscous Navier-Stokes equations is an attempt by the flow to fill the more numerous wavenumbers at small scales, frustrated by the vigorous viscous dissipation at those scales”

Again one encounters the question about the need for cascades!

This necessity as the only option is due to the view that small scales are a kind of passive sink of energy which is a major misconception: there is a rich direct and bidirectional coupling between large and small scales comprising an essential part of the complex interaction between the multitude of the degrees of freedom (which as mentioned is preferable to terms like eddies, scales and so forth) in turbulent flows. The small scales in turbulence (i) are very far from being simple as, e.g. objects used in statistical mechanics and (ii) interact non-trivially, bidirectionally and non-locally, with the rest of the flow. In other words, contrary to the common view, the small scales cannot be seen as a kind passive sink of energy and/or as ‘slaved’ to the large scales—the small scales react back in a nonlocal manner and (iii) they carry lots of the basic physics of turbulence, such as those associated with its fundamental properties as a rotational and dissipative phenomenon rather than just an essentially inviscid, inertial phenomenon, as these properties are mainly excited just by the nonlinearity in NSE, see next section.

The particular aspect of dissipation is because dissipation is strain, and the role of the field of strain is not limited just by dissipation - it has a number of other roles as it is an active field, see next section, and Tsinober (2009) Chap. 6, Sect. 6.2.2 Why strain too.

Among other things this results in an alternative to the conventional “cascade” (being inherently local) with an option of a nonlocal exchange between LS and SS of energy and not just energy as just mentioned above. This is due to the inherently non-local nature of turbulence - the just mentioned direct and bidirectional coupling between LS and SS being one of its important manifestations . For more on the issue of nonlocality, see Tsinober (2009), Sect. 6. For a discussion of a number of issues on “cascade” see Sect. 5 Cascade in Tsinober (2009) and the next section on nonlocality below.

The observation by Richardson was made by looking at the structure of clouds, i.e. condensed water vapour at the interface between laminar and turbulent flows in their bulk, which do not necessarily reflect the structure of the underlying velocity field and other dynamical variables.

Richardson (1922): Thus, because it is not possible to separate eddies into clearly defined classes according to the source of their energy; and as there is no object, for present purposes, in making a distinction based on size between cumulus eddies and eddies a few metres in diameter (since both are small compared with our coordinate chequer), therefore a single coefficient is used to represent the effect produced by eddies of all sizes and descriptions.

Kolmogorov (1941a) formulated a cascade picture supplemented by the assumption about the chaotic nature of cascade was used by Kolmogorov (1941a) in a 2/3 page footnote as a qualitative justification of his hypothesis on the local isotropy of turbulent flows for very large Reynolds numbers. The cascade picture is based on the intuitive notion that turbulent flows possess a hierarchical structure consisting of ‘eddies’ (Richardson’s ‘whirls’, Kolmogorov’s ‘pulsations’, etc.) as a result of successive instabilities. The essence of this picture is in its successive hierarchical process, and in this sense it is the same as the Landau–Hopf picture of transition to turbulence as a ‘cascade’ of successive instabilities. The difference is that the Richardson-Kolmogorov cascade refers to a process at some fixed Reynolds number, whereas the Landau–Hopf picture describes the process of changes occurring as the Reynolds number increases.

Kolmogorov (1941a): It is natural that in so general and somewhat indefinite a formulation the just proposition cannot be rigorously proved . We may indicate here only certain considerations speaking for the advanced hypothesis.

The citations at the beginning of this section belong to the minority. But even in the minority some authors change their minds and return to exploiting the cascade concept, see e.g. Falkovich (2009) and Goto and Kraichnan (2004).

This belief was massively adopted in the community along with some closely related. For example, Tennekes and Lumley (1972) believed that: the energy transfer from large eddies to small eddies may be considered in terms of vortex stretching p. 75; vortex stretching is the only known spectral energy-transfer mechanism p. 91; the existence of energy transfer from large eddies to small eddies, driven by vortex stretching and leading to viscous dissipation of energy near the Kolmogorov microscale p 256.

Nevertheless, Lumley wrote in (1992)

The dominance of the terms \(\omega _{i}\omega _{j}s_{ij}\) and \( -s_{ij}s_{jk}s_{ki}\) appears (empirically) very robust in a broad range of Reynolds numbers \(Re_{\lambda }\) from 60 to \(10^{4}\) almost everywhere in the flow field except in irrotational regions where the pressure-Hessian acts instead to produce the strain, e.g. on the almost irrotational side of the entrainment interface Holzner et al. (2009) and close to the grid element centerline in the turbulence producing grid Paul et al. (2017)

Quantities directly related to the mean flow MF, pressure Hessian pH and external forcing EF in other – not “energy” equations are not negligible almost in any sense in a multitude of other aspects...and even in the equations below at least in several contexts different from the one discussed here.

Generally, the “energy” balance equations alone and alike are not “exhaustive” and cannot be saved from this malady by any “cascade”. Also they require some closure, unless , e.g. using DNS of NSE and analyzing the data in terms of balance equations , e.g. scale by scale budgets etc , involving painful efforts in interpretation of the chosen statistical info. The terms \(( {\frac{1}{2}} )D\omega {{}^2} /Dt\), \(( {\frac{1}{2}} )Ds {{}^2} /Dt\) are not necessarily statistically dominant even in non-stationary turbulent flows, but play essential role and, obviously cannot be just neglected even if not statistically dominant.

Moreover, all terms are (potentially) important if conditioned on different ranges of the normalized dissipation and/or enstrophy Yeung et al. (2015), the latter being potentially true for multitude of other conditioning criteria.

The above is consistent with the results on a Lagrangian experiment using a 3D particle tracking velocimetry with access to velocity derivatives (Lüthi et al. 2005; Guala et al. 2006). Namely, it was found that the statistical evolution of strain and enstrophy can be interpreted as a kind of a life-cycle for strain and enstrophy and can be summarized in a sequence of processes starting with the strain self-amplification in low strain low enstrophy regions. This is followed by enstrophy production and growth, leading to the formation of high strain high enstrophy regions. The depletion of both strain and its production in parallel to the growth of enstrophy is related to the evolution of these regions into high enstrophy low strain regions, i.e. to the evolution of vortex sheets (shear layers) into vortex filaments. These regions evolve into weak enstrophy - strain regions since the enstrophy production, in presence of low strain and preferential alignment between \(\omega \) and \(\lambda _{2}\), cannot oppose the viscous destruction of enstrophy. This cyclic sequence consists of local and non-local processes of different Lagrangian time scales which governs the dynamics of small-scale turbulence.

This expectation takes its origin mainly from the incorrect analogy with material lines and other passive vectors, see Sect. 9.4 Vorticity versus passive vectors in Chap. 9 on analogies and misconceptions in Tsinober (2009). In particular, the nonlocality of the relation between vorticity and strain - which does not have an analogue with passive vectors - play an important role in the issue of \(\varvec{\omega ,\lambda }_{i}\) alignments. Hamlington et al. (2008).

An important general aspect is that the strongest interaction between vorticity and strain occurs in regions with \(\varvec{\omega },\mathbf { \lambda }_{1}\) alignments and large strain, see Chap.  6, pp. 150–153 in Tsinober (2009).

There is a kind of tautological aspect since “cascade” by definition is local, i.e. the consequence of assuming a cascade implies that the Navier–Stokes equations are local. The latter is generally not correct in any space - be it physical, Fourier or any other. This is one of the main themes of the present section.

This cannot be underestimated for several reasons. The main is that the Eqs. (7.2, 7.3) imply a nontrivial relation between the fields of velocity (energy) and that of strain (dissipation). One of the problems here is that in real fluids the role of the strain field is not limited just by dissipation with a number of other important functions. One of the consequences is that the nature of dissipation makes an essential qualitative difference including nonlocal effects so that different dissipative mechanisms result in different outcomes including the “inertial range” contrary to common belief. Hence problems in interpretation of results when replacing Newtonian viscous term by some “surrogate” such a the hyperviscous one. For example, this concerns studying properties of “inertial range” in fluids with hyperviscosity, Barjona and da Silva (2017), and the so called truncated Euler equations with the smaller scales considered as thermalized modes, Frisch et al. (2008).

Hence the problematic character of the claims like “physically justified approximation that velocity \(\mathbf {u}\) and strain \(\mathbf {S}\) are uncorrelated” because “In a turbulent flow, the velocity field \(\mathbf {u}\) and the rate of strain \(\mathbf {S}\) are expected to be independent of each other” Pumir et al. (2016).

In non-newtonian fluids the issue is more complicated, e.g. in hyperviscous case the viscous dissipation \(\epsilon =\nu _{h}(\nabla ^{h}\mathbf {u})\).

Kolmogorov did these modifications following the Landau objection to universality in the first Russian edition of Fluid Mechanics by Landau and Lifshits (1944) about the role of large-scale fluctuations of energy dissipation rate, i.e., non-universality of both the scaling exponents and the prefactors: important part will be played by the manner of variation of \(\epsilon \) over times of the order of the periods of large eddies (of size \(\ell \)), see Landau and Lifshits (1987), p. 140.

This assumption is due to Obukhov (1962) because as he wrote it is not very restrictive as an approximate hypothesis since the distribution of any essentially positive characteristic can be represented by a logarithmically Gaussian distribution with correct values of the first two moments.

This is correct for empirical purposes, but when it goes about the right results for the right reasons it is not sufficient.

We point out that SAS does not require a large scale flow and there is no need for a “cascade” to ‘deliver’ the energy from LS to SS. It is independent of the existence (or not) of such a cascade, but depends on specific factors, such as the nature of turbulence excitation (LS, SS, broadband, etc - which may exclude “cascades”) and on BC’c and IC’c. Neither is there no need for two-point statistics.

SAS is a specific (!!) feature of genuine turbulence having no counterpart in e.g. the behavior of passive objects.

Among the main reasons that true quilty party was missed by the community is that because their attention was concentrated elsewhere. The search was focused on the two-point statistical equations such as the Karman–Howarth equation, the “energy balance” equation (scale-by-scale) for the second structure function and the like, see e.g. Chap. 7 in McComb (2014), Vassilicos (2015), Antonia et al. (2017) and references therein. This was motivated by the success of Kolmogorov (1941b) with his 4/5 law. This allowed an explicit relation to be obtained between the third order longitudinal velocity structure function and the mean over the scale r dissipation \(\epsilon ,\) \(\left\langle (\Delta u)^{3}\right\rangle \) \( =-4/5\epsilon r\) for large Reynolds numbers for globally homogeneous and isotropic flows or assuming local homogeneity and isotropy. The above shows that one cannot handle the velocity field and/or its increments without the field of velocity derivatives when dealing with issues like those disscussed here.

The features concerning the TL balance like the (7.1a, 7.2a) appear to be true for temporally modulated turbulent flows and for flows with hyperviscosity of different orders, \(h=2,4,8,\) \(h=1\) corresponds to Newtonian fluid. This means that the self amplification of velocity derivatives will depend on index h.

It would be interesting to see these “dissipation images” of the specific whirls, individual intense structures identified in real space as claimed by Cardese et al. (2017) and Jiménez et al. (2017).

Thus the Eq. 7.3, 7.4 alone prevent the pretty popular kind of interpretation of Kolmogorov as e.g.given by Sawford and Yeung (2015):

Kolmogorov similarity theory is the starting point for our discussion of Lagrangian intermittency. According to this theory, the small scales of turbulence have a universal structure depending only on the viscosity and the mean rate at which energy is transferred down the spectrum from the largest scales where turbulence is created to the smallest scales where viscosity dissipates this energy. Yeung et al. (2015).

Simply otherwise the flow does not obey the NSE!

However, any nonlinear function, functional and alike of a variable, which is Gaussian, is non-Gaussian. For instance the enstrophy, dissipation, pressure, etc. of a Gaussian velocity field possess exponential tails and their flatness is quite different from 3. For example, for a Gaussian velocity field \(F_{G}(\omega ^{2})=\langle \omega ^{4}\rangle /\langle \omega ^{2}\rangle ^{2}=5/3\) and \(F_{G}(s^{2})=\langle s^{4}\rangle /\langle s^{2}\rangle ^{2}=7/10\). But this by no means indicates that, for a Gaussian velocity field, these quantities are intermittent and possess structure as sometimes claimed. These and similar “non-Gaussian features” resulting by using Gaussian objects do not cure the problem from the original impotences of the Gaussian fields even if producing results in agreement with, say, DNS. In basic research the right results should be for the right reasons. Among the reasons is that multiplicative models are able to produce intermittency for a purely nonintermittent field as is the Gaussian velocity field. See YaB et al. (1990) on interesting observations on this and related matters. We also to point out a similar concern regrading “Lagrangian” intermittency, e.g. “intermittency” of a purely Eulerian Gaussian field in its Lagrangian representation, see Sect. 3.​4 above and Sect. 9.​1.​3 below.

The above is not a serious concern for engineering applications as even wrong theories may help in designing machines Feynmann (1996).

For recent examples see Wilczek and Meneveau (2014) Wilczek (2016) and references therein.