Intermittency and Structure(s) of and/in Turbulence

Abstract

Intermittency specifically in genuine fluid turbulence is associated mostly with some aspects of its spatiotemporal structure. Hence, the close relation between the origin(s) and meaning of intermittency and structure of turbulence. Just like there is no general agreement on the origin and meaning of the former, there is no consensus regarding what are the origin(s) and what turbulence structure(s) really mean. At the present state of matters both issues are pretty speculative and an example of ‘ephemeral’ collection of such is given in this chapter. We have to admit at this stage that structure(s) is(are) just an inherent property of turbulence.

Structureless turbulence is meaningless. There is no turbulence without structure(s). Every part (just as the whole) of the turbulent field—including the so-called ‘structureless background’—possess structure. Structureless turbulence (or any of its part) contradicts both the experimental evidence and the Navier–Stokes equations. The claims for ‘structureless background’ is a reflection of our inability to ‘see’ more intricate aspects of turbulence structure: intricacy, complexity and ‘randomness’ are not synonymous for absence of structure.

What is definite is that turbulent flows have lots of structure(s). The term structure(s) is used here deliberately in order to emphasize the duality (or even multiplicity) of the meaning of the underlying problem. The first is about how turbulence ‘looks’. The second implies the existence of some entities. Objective treatment of both requires use of some statistical methods. It is thought that these methods alone may be insufficient to cope with the problem, but so far no satisfactory solution was found. One (but not the only) reason—as mentioned—is that it is not so clear what one is looking for: the objects seem to be still elusive. For example, there is still a non-negligible set of people in the community that are in a great doubt that the concept of coherent structure is much different from the Emperors’s new Clothes.

An example of acute difficulties described in this chapter is associated with high dimension of what is called structures so that simple single parameter thresholding is inappropriate to make on them statistics due to the painful question how really “similar” are all these if the individual members of such an ensemble are defined by one parameter only. The view that turbulence structure(s) is(are) simple in some sense and that turbulence can be represented as a collection of simple objects only seems to be a nice illusion which, unfortunately, has little to do with reality. It seems somewhat wishfully naive to expect that such a complicated phenomenon like turbulence can merely be described in terms of collections of only such ‘simple’ and weakly interacting object.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The nature and characterization of the structure(s) of turbulent flows are among the most controversial issues in turbulence research with extreme views on many aspects of the problem—in words of Richard Feynmann (1963, pp. 41–12), holding strong opinions either way. For example, it is common in the vast literature on turbulence to consider the terms statistical and structural (and also deterministic) as incompatible or even contradictory, see for example Dwoyer et al. (1985), Lumley (1989). However, there are common points as well. For instance, it is mostly agreed that turbulence definitely possesses structure(s) (whatever this means) and that intermittency, which is addressed in the following subsection, is intimately related to some aspects of the structure(s) of turbulence.

At present it became clear that it is a misconception to contrapose the statistical, the structural and the deterministic and that they represent different facets/aspects of the same problem, so that there is no real gap between structure(s), statistics (but not in the sense of absence of laws) and determinism. Just like it seems impossible to separate the structure(s) from the so called ‘random structureless background’ or the ‘random processes from the nonrandom processes’, Dryden (1948) due to strong interaction and nonlocality, both between individual structures, and between structures and the ‘background’. In other words, there is no turbulence without structure, every part of the turbulent field just like the whole possess structure.¹ Structureless turbulence or any its part contradicts both the experimental evidence and the Navier–Stokes equations. It is noteworthy that the statement that turbulence has structure is in a sense trivial: to say that turbulent flow is ‘completely random’ would define turbulence out of existence (Tritton 1988, p. 295).

One of the basic properties underlying both aspects, i.e. intermittency and structure(s) is the essentially non-Gaussian nature of turbulence, which follows from NSE, e.g. Novikov (1967), Lumley (1970), and there are numerous experimental observations on non-Gaussianity of turbulence, see e.g. Sect. 6.8 in Tsinober (2009) on non-Gaussian nature of turbulence, e.g. the outstanding non-zero odd moments: the strain and enstrophy production −〈s _ij s _jk s _ki 〉, 〈ω _i ω _j s _ij 〉. It has to be stressed that (i) non-Gaussianity and intermittency/structure are not synonymous as not any non-Gaussian field is turbulent and (ii) non-Gaussianity is only a statistical manifestation of intermittency/structure.

As a simple illustration one can see that even if the flow field is initially Gaussian, the dynamics of turbulence makes it non-Gaussian with finite rate. This is seen by looking 〈…〉 at the equation for 〈ω _i ω _j s _ij 〉 (dropping the viscous term).

$$D\langle \omega _{i}\omega _{j}s_{ij}\rangle /Dt=\langle \omega _{j}s_{ij}\omega _{k}s_{ik} \rangle -\langle \omega _{i}\omega _{j}\varPi _{ij} \rangle. $$

For a Gaussian velocity field 〈ω _i ω _j s _ij 〉 _G =0, 〈ω _i ω _j Π _ij 〉 _G =0 and $\langle \omega _{j}s_{ij}\omega _{k}s_{ik} \rangle _{G}=\frac{1}{6} \langle \omega ^{2}\rangle ^{2}>0$. Since the quantity ω _j s _ij ω _k s _ik ≡W ², W _i =ω _j s _ij , it is positive pointwise for any vector field. Hence for an initially Gaussian field

$$\bigl\{ D\langle \omega _{i}\omega _{j}s_{ij} \rangle /Dt \bigr\} _{t=0}= \bigl\{ \langle \omega _{j}s_{ij} \omega _{k}s_{ik} \rangle \bigr\} _{t=0}>0. $$

It follows that, at least for a short time interval t, the mean enstrophy production will become positive.

The above equations and similar ones for 〈s _ij s _jk s _ki 〉 can be seen as one of the manifestations of the statistical irreversibility of turbulent flows (Betchov 1974; Novikov 1974). The corresponding dynamical instantaneous (inviscid) equations are reversible. Hence, the term “statistical”. One would claim that the Kolmogorov 4/5 law in the conventionally defined inertial range belongs to the same category, but as described above it appears to be not a purely inertial relation. Another aspect of irreversibility is related to the dissipative nature of turbulence—its not “slightly” dissipative at whatever large Reynolds number.

Turbulence—being essentially non-Gaussian—is such a rich phenomenon that it can ‘afford’ a number of Gaussian-like manifestations, some of which are not obvious and even nontrivial, for examples see Sect. 6.8.2 in Tsinober (2009). Hence the importance of parameters related to the non-Gaussian nature of turbulence and not only in various contexts of intermittency and structure(s).

1 Intermittency

At any instant the production of small scales is… occurring vigorously in some places and only weakly in the others (Tritton 1988).

Intermittency is a phenomenon where Nature spends little time, but acts vigorously (Betchov 1993).

Typical distribution of scalar and vector fields is one in which there appear characteristic structures accompanied by high peaks or spikes with large intensity and small duration or spatial extent. The intervals between the spikes are characterized by small intensity and large extent (Zeldovich et al. 1990).

The term intermittency is used in two distinct (but not independent) aspects of turbulent flows. The first one is the so-called external intermittency. It is associated with what is called here partly turbulent flows, specifically with the strongly irregular and convoluted structure and random movement of the ‘boundary’ between the turbulent and nonturbulent fluid.

The second aspect is the so-called small scale, internal or intrinsic intermittency and is associate with spotty temporal and spatial patterns of the small scale structure(s) in the interior of turbulent flows.

1.1 The External Intermittency and Entrainment

This kind of intermittency was studied first by Townsend (1948) and involves the so called entrainment which is one of the most basic processes of transition from laminar to turbulent state with the coexistence of both in one flow. It is associated with the coexistence of laminar and turbulent flow regions—an observer located in the proximity of either side of the ‘mean’ boundary between these regions observes intermittently laminar and turbulent flow in the form of a signal similar to that as in Fig. 21.4 in Tritton (1988), clearly demonstrating the external intermittency of in the wake past a circular cylinder. Here we again encounter the question about what turbulent is. Looking at signals like the mentioned one clearly sees what is turbulent and what is laminar. But the question is how one can say whether a small part of flow is turbulent. In other words if turbulence is to be identified by statistical means, then what is the meaning of ‘turbulent’ locally? This involves taking decisions about what is turbulent using some conditional criterion, see discussion and references in Kuznetsov et al. (1992).

The main mechanism by which nonturbulent fluid becomes turbulent locally as it crosses the interface is due viscous diffusion of vorticity across the interface. As this process is associated with small scales it is thought to be the reason why the interface appears sharp compared to the scale of the whole flow.

However, at large Reynolds numbers, the entrainment rate and the propagation velocity of the interface relative to the fluid are known to be independent of viscosity. Therefore the slow process of diffusion into the ambient fluid must be accelerated by interaction with velocity fields of eddies of all sizes, from viscous eddies to the energy-containing eddies so that the overall rate of entrainment is set by large-scale parameters of the flow (Townsend 1976). That is although the spreading is brought about by small eddies [viscosity] its rate is governed by the larger eddies. The total area of the interface, over which the spreading is occurring at any instant, is determined by these larger eddies (Tritton 1988). This is analogous to independence of dissipation of viscosity in turbulent flows at large Reynolds numbers. In other words, small scales do the ‘work’, but the amount of work is fixed by the large scales in such a way that the outcome is independent of viscosity. This shows that independence of some parameter of viscosity at large Reynolds numbers does not mean that viscosity is unimportant. It means only that the (cumulative) effect of viscosity is Reynolds number independent, for more see Tsinober (2009), Wolf et al. (2012a, 2012b) and references therein.

1.2 The Small Scale, Internal or Intrinsic Intermittency

The second aspect is the so-called small scale, internal or intrinsic intermittency. It is usually associated with the tendency to spatial and temporal localization of the ‘fine’ or small scale structure(s) of turbulent flows with large probability of taking values of some variable both very large and very small compared to its standard deviation. An important feature is that regions of exceptionally high values make disproportionately large contribution (sometimes even dominant) to the integral properties even though occupying a small fraction of the flow. Consequently, regions with “voids” (low intensity) occupy a “disproportionately” large fraction of the flow and thus being statistically dominating, but contribute far less to the dynamics. For example, in field experiments with Re _λ ≈10⁴ the dissipative events with dissipation exceeding the mean 12 times (i.e. ϵ<12〈ϵ〉) contribute to 〈ϵ〉 about 20 % while taking about 1 % of the total volume, see Fig. 9.1. The instantaneous values of dissipation exceed 10⁴〈ϵ〉.

Fig. 9.1

Percent of the strong dissipative events as defined in the text (a) and their contribution to the total dissipation (b) as a function of the threshold q for various separations r. The value of separation is not relevant as concerns the contribution of the dissipative events to the total dissipation (Kholmyansky and Tsinober 2009)

Similar behavior is observed for other variables exhibiting intermittency, e.g. enstrophy and enstrophy and strain production, see Tsinober (2009).

Our concern here is with the intermittency of this second kind. It is noteworthy that in a broad sense intermittency is a ubiquitous phenomenon occurring in a great variety of qualitatively different systems, see Zeldovich et al. (1990) for a lively exposition of a wide number of different systems exhibiting intermittency, and also Vassilicos (2001). The main common features of all of them is (space/time) irregularity and localization (both spatial and temporal) of their ‘fine’ structure. However, this is not enough to define intermittency. For example, almost any nonlinear function or almost any nonlinear functional of a random Gaussian field is intermittent in the above sense, though random Gaussian fields by definition lack any intermittency.

We do not make a special distinction between the intermittency in the inertial and dissipative ranges as both are not well defined.

The intermittent nature of the small scale structure of turbulent flows was foreseen by Taylor (1938b): …the view frequently put forward by the author that the dissipation of energy is due chiefly to the formation of very small regions where vorticity is very high. ² Indeed, such behavior is not unexpected since the viscous term in the NSE contains the highest-order derivative while high Reynolds number turbulence involves the limit ν→0. This limit is a singular perturbation problem and localized regions in which gradients are large should be expected to form (Orszag 1977). Along with nonlocality this means that the consequence of intermittency is not just a small “correction” in the properties of turbulence with the possible exception of the quasi-Gaussian ones.

The phenomenon of small scale intermittency was observed by Batchelor and Townsend (1949) in experiments with turbulent grid flows and in a wake past a circular cylinder.

Batchelor and Townsend (1949) wrote: The basic observation which requires explanation is that activation of large wave-numbers is very unevenly distributed in space. These space variations in activation can be described as fluctuations in the spectrum at large wave-number… As the wave-number is increased the fluctuations seem to tend to an approximate on-off, or intermittent variation. Whatever the reason for the occurrence of these fluctuations, they appear to be intrinsic to the equilibrium range of wave-numbers. All the evidence is consistent with the inference that the fluctuations are small in the region of smallest wave-numbers of equilibrium range and become increasingly large at larger wave-numbers… the mean separation of the visible activated regions is comparable with the integral scale of the turbulence, i.e. with the size of the energy-containing eddies. Batchelor and Townsend (1949, pp. 252–253) obtained some evidence that the deviation from Gaussianity is stronger as the Reynolds number is increased. However, they did not appreciate this effect and claimed that the flatness factors seems to vary a little with Reynolds number, though this factor changed from 5 to 7 for the third order derivative; see their Fig. 5 for the flatness factor of velocity derivatives of different orders and different Reynolds numbers. This was confirmed by a number of subsequent experiments such as by Kuo and Corrsin (1971), for an updated overview of the subsequent results see, e.g. Sreenivasan and Antonia (1997). An example of time records of the streamwise velocity component, and their derivatives obtained in a field experiment at Re _λ =10⁴ is shown in Fig. 1.17 in Tsinober (2009). The increasingly intermittent behavior of the signal with the derivative order is seen quite clearly. Also shown are records for the enstrophy ω ², total strain 2s ²≡2s _ij s _ij and their surrogate (∂u ₁/∂x ₁)², and enstrophy production ω _i ω _j s _ij , s _ij s _jk s _ki and their surrogate (∂u ₁/∂x ₁)³. The experiment was performed in the atmospheric surface layer at a height 10 m in approximately neutral (slightly unstable) conditions.

A qualitative summary is that small scale intermittency of turbulence is associated with its spotty (spatio-temporal) structure which among other things is manifested as a particular kind of non-Gaussian behavior of turbulent flows. This deviation from Gaussianity, increases with both (1) increasing the Reynolds number and (2) decreasing the ‘scale’. In other words, intermittency involves two (not independent) aspects of turbulent flows—their structure/geometry and statistics. These two aspects are reflected in attempts to ‘define’ intermittency, see references in Tsinober (2009).

1.3 Measures/Manifestations of Intermittency

There are numerous quasi-Gaussian manifestations of turbulent flows, for some see Sect. 7.2.2 in Chap. 7, Tsinober (2009). Hence the importance of parameters related to the non-Gaussian nature of turbulence in various contexts of intermittency and structure(s). It is important that intermittency implies non-Gaussianity, but not necessarily vice versa—practically any parameter can at most indicate the degree of intermittency of a flow already known to be intermittent. A statistical measure such as flatness or some similar intermittency factors may deviate strongly from a Gaussian value without any intermittency in the flow field. The simplest example is the Gaussian field itself, which by definition lacks any intermittency. However, any nonlinear function (or functional) of a variable, which is Gaussian, is non-Gaussian. For instance 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 (ω ²)=〈ω ⁴〉/〈ω ²〉²=5/3 and F _G (s ²)=〈s ⁴〉/〈s ²〉²=7/5. But this by no means indicates that, for a Gaussian velocity field, these quantities are intermittent as claimed sometimes. Moreover, the flatness of enstrophy is larger than that of total strain, $F_{\omega ^{2}}-F_{s^{2}}=4/15$. Similarly, the Reynolds stress u _i u _j exhibits ‘intermittency’. The main contribution to this intermittency comes from the fact that u _i u _j is a product of two random variables both distributed close to Gaussian. For example, the PDF of the u ₁ u ₂ of the strongly intermittent signal obtained by Lu and Willmarth (1973) in a turbulent boundary layer is strongly non-Gaussian. However, the PDF of u ₁ u ₂ is approximated with high precision by assuming both u ₁ and u ₂ to be Gaussian with a correlation coefficient between them adjusted from the experiment −0.44.³

Passive objects (scalars like heat, vectors like magnetic field) in a random velocity field (real or artificially prescribed) are nonlinear functionals. of the velocity field and forcing/excitation. Therefore, even when both the velocity field and forcing are Gaussian the field of a passive object is expected to be strongly non-Gaussian as usually (but not always) is the case (Majda and Kramer 1999). Such kinematic intermittency is observed in a great number of theoretical and some experimental works. The term ‘kinematic’ is used here in the sense that there is no relation to the dynamics of fluid motion, which does not enter in the problems in question, and the velocity field is prescribed and often assumed to be Gaussian.

Odd Moments

Any odd moment of a Gaussian variable vanishes, for example skewness S _G (a)≡〈a ³〉/〈a ²〉^3/2=0. Therefore, odd moments are very sensitive to deviations from Gaussianity, so that non-zero odd moments may be especially good indicators of intermittency. Build-up of odd moments is a result of both the (kinematic) evolution of a passive field in any random velocity field and the dynamics of turbulence itself. In the latter case, non-vanishing odd moments are the most important, dynamically significant manifestations of non-Gaussianity, i.e. they reflect directly the dynamic aspects of intermittency. The most prominent odd moments are the third order structure function for longitudinal velocity increments $S_{3}^{\|}= \langle \{[ \mathbf{u}(\mathbf{x}+\mathbf{r})-\mathbf{u}(\mathbf{x})]\cdot (\mathbf{r}/r)\}^{3} \rangle$ entering the 4/5 law, the enstrophy production 〈ω _i ω _k s _ik 〉 and the third order moment of the strain tensor 〈s _ij s _jk s _ki 〉. Note that all these and other odd moments vanish in a Gaussian velocity field. In contrast similar “odd” moments for passive objects, 〈G _i G _k s _ik 〉, 〈B _i B _k s _ik 〉 do not vanish. Hence, the passive objects in some sense are “more intermittent”.

We remind that the non-Gaussian nature of genuine turbulent flows and of passive objects is qualitatively different just like is intermittency in a great variety of physically different systems.

Scaling Exponents and PDFs

It is commonly believed that among the manifestations of the small scale intermittency in the commonly defined inertial range⁴ is the experimentally observed deviation of the scaling exponents for structure functions $S_{p}^{\|}=\langle \{[\mathbf{u}(\mathbf{x}+\mathbf{r})- \mathbf{u}(\mathbf{x})]\cdot (r/r)\}^{p}\rangle$ for p>3 from the values implied by the Kolmogorov theory, i.e. anomalous scaling, which in turn is due to rare strong events. Namely, $S_{p}^{\parallel }(r)\propto r^{\zeta _{p}^{\parallel }}$, where $\zeta _{p}^{\parallel }=p/3-\mu _{p}<p/3$ is a convex nonlinear function of p.

However, there are major problems with scaling as follows.

First, there exists no one-to-one relation between simple statistical manifestations and the underlying structure(s) of turbulence. Moreover, qualitatively different phenomena can and do possess the same set of scaling exponents, so that one needs more subtle statistical characterizations of turbulence structure(s) and intermittency. For example, until recently one of the common beliefs was that the observed vortex filaments/worms are mainly responsible for the phenomenon of intermittency understood as anomalous scaling. However, it appears that this is not the case, see the evidence and references given in Tsinober (1998a, 2009). More specifically, the problem with the intermittency in the conventionally defined inertial range (CDIR) and the related anomalous scaling is in the object (CDIR) itself as it is ill defined and does not exist in reality. Hence the problem with theories attempting explanations of intermittency in this nonexistent phenomenon, e.g. breakdown coefficients/multipliers (Novikov 1974, 1990a), multi-fractals (Frisch 1995) and the so-called ‘hierarchical symmetry’ (She and Zhang 2009).

There are no ‘corrections’ to the scaling exponent in 4/5 law—it is an exact consequence of NSE. However, as (i) it manifests the non-Gaussian nature of turbulence and (ii) the PDFs of the longitudinal velocity increments especially at small r have flaring tails, i.e. hanging far above the Gaussian PDF, the 4/5 law should be considered as related to intermittency. This shows that ‘intermittency corrections’ are not that reliable as indicators of intermittency, if at all. Reminding that recent experiments at high Reynolds numbers showed that the 4/5 law is not a pure inertial relation (which is one of the manifestations of the ill posedness of the CDIR) since the PDFs of the velocity increments contain strong dissipative events with nonnegligible contributions to the structure functions $S_{p}^{\parallel }(r)$ increasing with the order and among them $S_{3}^{\parallel }(r)$.

The next example is represented by numerous models attempted to reproduce the anomalous scaling. A partial list of references is given in Sreenivasan and Antonia (1997) and Tsinober (1998b). These models followed the Kolmogorov (1962) refined similarity hypothesis (RSH) in which the mean dissipation 〈ϵ〉 was replaced by ‘local’ dissipation ϵ _r averaged over a region of size r. The scaling exponents obtained in all of these models are in good agreement with the experimental and numerical evidence, e.g. these models exhibit the same scaling properties (and some other such as PDFs) as in real turbulence. It is noteworthy that many of these models are based on qualitatively different premises/assumptions and with few exceptions have no direct bearing on the Navier–Stokes equations. Therefore the success of such models can hardly be evaluated on the basis of how well they agree with experiments. Phenomenology and models only will hardly be useful and convincing, since almost any dimensionally correct model, both right or wrong, will lead to correct scaling without appealing to NSE and/or elaborate physics. For example, there exist many theories which produce the k ^−5/3 energy spectrum for qualitatively and/or physically different reasons. A recent example is a suggestion that the spectrum of fully developed turbulence is determined by the equilibrium statistics of the Euler equations and that a full description of turbulence requires only a perturbation, small in some appropriate metric, of a Gibbsian equilibrium (Chorin 1996). The most common justification for the preoccupation with such models is that they (at least some of them) share the same basic symmetries, conservation laws and some other general properties, etc. as the Navier–Stokes equations. The general belief is that this—along with the diversity of such systems (there are many having nothing to do with fluid dynamics, e.g. granular systems, financial markets, brain activity)—is the reason for the above mentioned agreement. However, this is not really the case, e.g. in Kraichnan (1974) a counter example of a ‘dynamical equation is exhibited which has the same essential invariances, symmetries, dimensionality and equilibrium statistical ensembles as the Navier–Stokes equations but which has radically different inertial-range behavior’! The majority of models exhibit temporal chaos only. Therefore, such and most other models hardly can be associated with the intermittency of real fluid turbulence, which involves essentially spatial chaos as well. Again, for the above reasons the agreement between such models and experiments (both laboratory and numerical) cannot be used for evaluation of the success of such models. There are proposals to use two sets of independent exponents $\zeta _{p}^{\parallel }$ and $\zeta _{p}^{\perp }$ (Chen et al. 2003) and there exist other ‘universality’ proposals involving ‘much more’ scaling exponents, see e.g. Biferale and Procaccia (2005), Frisch (1995) and references therein.

Scaling laws alone are not necessarily theories. With all the attractiveness of scaling, turbulence phenomena are infinitely richer than their manifestation in scaling and related things. Most of these manifestations are beyond the reach of phenomenology. Phenomenology is inherently unable to handle the structure of turbulence in general, and phase and geometrical relations in particular, to say nothing of dynamical features such as build up of odd moments, interaction of vorticity and strain resulting in positive net strain and enstrophy production/predominant vortex stretching. It seems that there is little promise for progress in understanding the basic physics of turbulence in going on dealing with scaling and related matters only, without looking into the structure and, where possible, basic mechanisms which are specific to turbulent flows. In fact, the main question of principle which should have been asked long ago is: Why on earth should we perform so many elaborate measurements of various scaling exponents without looking into the possible concomitant physics and/or without asking why and how more precise knowledge of such exponents, even assuming their existence, can aid our understanding of turbulent flows? This is not to say that one has to abandon the issues of scaling. An example of affirmative answer is given in Sect. 7.1 concerning the ill-posedness of the concept of inertial range.

Second, as discussed in the previous section the very existence of scaling exponents in statistical sense (as, e.g. for various structure functions or corresponding PDFs, etc.) which is taken for granted, is a problem by itself.

A similar question arises in respect with multifractality which was designed to ‘explain’ the ‘anomalous’ scaling, since there is no direct experimental evidence on the multifractal structure of turbulent flows. So there is a possibility that multifractality in turbulence is an artifact (see Frisch 1995, p. 190). Moreover, in reality multifractality in fact is a kind of description of finite Reynolds number effect at whatever large Re due to ill posedness of the inertial range as mentioned several times above.

The PDFs of an intermittent variable are quite useful as, e.g. they carry the information showing that extremely small and extremely large values are much more likely than for a Gaussian variable. However, they contain no information on the structure of the underlying weak and strong events, nor on the structure of the background field. Hence, the same PDFs can have qualitatively different underlying structure(s) of the flow, i.e. ‘how the flow looks’. Similar PDFs of some quantities can correspond to qualitatively different structure(s) and quantitatively different values of Reynolds number, see references in Tsinober (1998a, 1998b, 2009). For example, the qualitative difference in the behavior and properties of regions dominated by strain and those with large enstrophy cannot be captured by such means and other conventional measures of intermittency. Also the PDFs, like scaling exponents, do not allow us to infer much about the underlying dynamics. This, however, is true of ‘conventional’ PDFs like those of velocity increments, but not of any PDFs such as those directly associated with geometrical flow properties.

Note that the largest deviation from Gaussianity occurs at small scales. In this sense, the field of velocity derivatives, ∂u _i /∂x _k , is more intermittent than the field of velocity, u _i , itself. One of the possible reasons for this is in the different nature of nonlinearity at the level of velocity field, i.e. in the Navier–Stokes equations and, for example, in the equation for vorticity.

1.4 On Possible Origins of Small Scale Intermittency

At the present state of matters the issue is pretty speculative, and an example of ‘ephemeral’ collection of such is given below.

As one of manifestations of turbulence structure(s), intermittency has its origins in the structure of turbulence, see next section. Therefore we briefly address here the issue on possible origins of intermittency. There are roughly two kinds of origins of intermittency: kinematic and dynamic.

Before proceeding we reiterate again that non-Gaussianity and intermittency are not synonymous just like the origins of non-Gaussian statistics in various systems and genuine turbulence are generally quite different even qualitatively. Therefore, it is misleading to ‘explain’ such properties of genuine turbulence by analogy with non-Gaussian behavior of, e.g. Burgers and/or restricted Euler equations. An important point is that these are integrable equations, and exhibit random behavior only under random forcing and or initial conditions, otherwise their solutions are not random and should be distinguished from problems involving genuine turbulence. Navier–Stokes equations at sufficiently large Reynolds number have the property of intrinsic mechanisms of becoming complex without any external aid including strain and vorticity production. There is no guarantee that the outcome, e.g. such as structure(s) is the same from, e.g. natural “self-randomization” and with random forcing and even with different kinds of forcing. Moreover there is evidence that the outcome is indeed different.

Direct and Bidirectional Interaction/Coupling Between Large and Small Scales

As discussed, direct and bidirectional interaction/coupling between large and small scales is one of the elements of the nonlocality of turbulence. It is both of kinematic and dynamic nature.

The first recognized manifestation of such interaction is that the small scales do not forget the anisotropy of the large ones. There is a variety of mechanisms producing and influencing the large scales: various external constraints like boundaries with different boundary conditions, including the periodic ones, initial conditions, forcing (as in DNS), mean shear/strain, centrifugal forces (rotation), buoyancy, magnetic field, external intermittency in partially turbulent flows, etc. Most of these factors usually act as organizing elements, favoring the formation of coherent structures of different kinds (quasi-two-dimensional, helical, hairpins, etc.). These, as a rule, large scale features depend on the particularities of a given flow that are not universal. Therefore the direct interaction between large and small scales leads to ‘contamination’ of small scales by the large ones, e.g. the edges of large scale structures are believed to be responsible for such ‘contamination’ in any kind of flow. This contamination is unavoidable even in homogeneous and isotropic turbulence, since there are many ways to produce such a flow, i.e. many ways to produce the large scales. It is the difference in the mechanisms of large scale production which ‘contaminates’ the small scales. Hence, non-universality.

The direct and bidirectional interaction/coupling of large and small scales, i.e. nonlocality, is a generic property of all turbulent flows and one of the main reasons for small scale intermittency, non universality, and quite modest manifestations of scaling. This dates back to the famous Landau remark stating that the important part will be played by the manner of variation of ε over times of the order of the periods of large eddies (of size ℓ) (Landau and Lifshits 1944, see 1987, p. 140).

“Near” Singularities

It is not known for sure whether Navier–Stokes equations at large Reynolds numbers develop a genuine singularity in finite time, though there no evidence in favor of this, so the term “near” singularities is just another term for strong events not necessarily just dissipative. Nevertheless, it seems a reasonable speculation that the ‘near’ singularities trigger topological change and large dissipation events; their presence is felt at the dissipation scales and is perhaps the source of small scale intermittency (Constantin 1996).

In any case, the ‘near’ singular objects may be among the origins of intermittency of a dynamical nature.⁵ However, there is a problem with two-dimensional ‘turbulence’. Namely, in this case everything is beautifully regular, but there is intermittency in the sense of the above definitions, with the exception of scaling exponents for velocity structure functions and corresponding quasi-Gaussian behavior. However, non-Gaussianity is strong at the level of velocity derivatives of a second order. Hence the possible formation of singularities in 3D is not necessarily the underlying reason for intermittency in 3D turbulence. Another example relates to modified Navier–Stokes equations such as those using hyperviscosity replacing the Laplacian by a higher order operator (−1)^h+1∇^2h with h>1 with the underlying assumption that this manipulation changes only the small scales. In this case too everything is beautifully regular too for h>5/4, i.e. the solution remains regular for all times and any Reynolds number (Ladyzhenskaya 1975; Lions 1969) and some features of turbulence are reproduced well (such k ^−5/3 spectrum) including intermittency, but its structure(s) appear quite different from those as for true NSE.

Multiplicative Noise, Intermittency of Passive Objects in Random Media

It has been known for about thirty years that passive objects (scalars, vectors) exhibit ‘anomalous scaling’ behavior and other strong manifestations of intermittency even in pure Gaussian random velocity field, see references in Tsinober (2009). These are dynamically linear systems, but they are of the kind which involve the so-called multiplicative ‘noise’, i.e. the coefficients in the equations that depend on the velocity field. Therefore, statistically they are ‘nonlinear’, since the field of passive objects is a nonlinear functional of the velocity field. Therefore, passive objects exhibit strong deviations from Gaussianity. In such systems, intermittency results either from external pumping (forcing term on RHS of the equations), or in systems without external forcing from instability (self-excitation) of a passive object in a random velocity field under certain conditions.

The velocity field does not ‘know’ about the passive objects. In this sense, problems involving passive objects are kinematic in respect with the velocity field in real fluid turbulence. They may reflect the contribution of kinematic nature in real turbulent flows. In view of the recent progress in this field it was claimed that investigation of the statistics of the passive scalar field advected by random flow is interesting for the insight it offers into the origin of intermittency and anomalous scaling of turbulent fluctuations (Pumir et al. 1997), for later references see Tsinober (2009). More precisely it offers an insight into the origin of intermittency and anomalous scaling of fluctuations in random media generally and independently of the nature of the random motion (Zeldovich et al. 1990), i.e. it gives some insight into the contributions of kinematic nature, but does not offer much regarding the specific dynamical aspects of strong turbulence in fluids. Moreover, anomalous diffusion and scaling of passive objects occurs in purely laminar flows in Eulerian sense (E-laminar flows) as a result of Lagrangian chaos (L-turbulent flows), i.e. intermittency of passive objects may even have nothing to do with the random nature of fluid motion in Eulerian setting.

Thus in real turbulent flows there are two contributions to the behavior of passive objects, kinematic and dynamic. It seems hopeless to separate them in any sense.

Summarizing, intermittency specifically in genuine fluid turbulence is associated mostly with some aspects of its spatiotemporal structure, especially the spatial one. Hence, the close relation between the origin(s) and meaning of intermittency and structure of turbulence. Just like there is no general agreement on the origin and meaning of the former, there is no consensus regarding what are the origin(s) and what turbulence structure(s) really mean. What is definite is that turbulent flows have lots of structure(s). The term structure(s) is used here deliberately in order to emphasize the duality (or even multiplicity) of the meaning of the underlying problem. The first is about how turbulence ‘looks’. The second implies the existence of some entities. Objective treatment of both requires use of some statistical methods. It is thought that these methods alone may be insufficient to cope with the problem, but so far no satisfactory solution was found. One (but not the only) reason—as mentioned—is that it is not so clear what one is looking for: the objects seem to be still elusive. For example, there is still a non-negligible set of people in the community that are in a great doubt that the concept of coherent structure is much different from the Emperors’s new Clothes.

2 What Is(Are) Structure(s) of Turbulent Flows? What We See Is Real. The Problem Is Interpretation

What we see is real. The problem is interpretation as there is even a problem of defining of “seeing”.

The issue is pretty speculative, and an example of ‘ephemeral’ collection of such is given below. We have to admit at this stage that structure(s) is(are) just an inherent property of turbulence. Structureless turbulence is meaningless.

The difficulties of definition what the structure(s) of turbulence are of the same nature as the question about what is turbulence itself. So before and in order to ‘see’ or ‘measure’ the structure(s) of turbulence one encounters the most difficult questions such as: what is (say, dynamically relevant) structure?, Structure of what? Which quantities possess structure in turbulence? What is the relation between structure(s) and ‘scales’—unfortunately both ill defined? Can structure exist in ‘structureless’ (artificial) pure random Gaussian fields? Which ones? All this—like many other issues—are intimately related to the skill/art to ask the right and correctly posed questions. These impossibly difficult questions are made not easier due to quite a bit of turbulence in terminological aspects and terminological abuse by use of a variety of ill defined terms (eddies, worms, sheets, tubes, pancakes, ribbons, vortex or vortical structures/filaments, vortons, ‘eigensolutions’, significant shear layers, etc. Some people would include in this list also coherent structures frequently used as synonyms of components of some decompositions or similar “executions” of the real flow field, which are usually followed by studies—sometimes pretty sophisticated—of their interaction not necessarily reflecting any physics, at least as concerns physical space. One of the popular games of this kind is looking for confirmation of the classical energy cascade picture, such as in the latest examples in Aluie (2012), Leung et al. (2012) and references therein.

The main starting point here is at the end of the previous section: just like there is no general agreement on the origin and meaning of intermittency, there is no consensus regarding what are the origin(s) and what turbulence structure(s) really mean. What is definite is that turbulent flows have lots of structure(s). The term structure(s) is used here deliberately in order to emphasize the duality (or even multiplicity) of the meaning of the underlying problem. The first is about how turbulence ‘looks’. The second implies the existence of some entities. This follows by more serious issues which unfortunately are mostly even not properly posed.

The meaning of structure(s) depends largely on what is meant by turbulence itself, and especially structure(s) of the particular field one is looking. For example, the velocity field may have no structure, but the passive tracer may well have a pretty nontrivial one, simple laminar Eulerian velocity field (E-laminar) creates complicated Lagrangian field (L-turbulent). Purely Gaussian, i.e. ‘structureless’ velocity field creates structure in the field of passive objects. The structure(s) seen in the velocity field depend on the motion of the observer. Finally, what is called “coherent structures” or “organized motion”, which has been rediscovered many times, may be not directly related to the turbulent nature of the flow (such as mixing layer), but are rather a result of large scale instability of the flow as ‘whole’ (zooming out).

2.1 On the Origins of Structure(s) of/in Turbulence

This question—in some sense—is a ‘philosophical’ one. But its importance is in direct relation to even more important questions about the origin of turbulence itself. Hence again an ‘ephemeral’ collection of such possible reasons/causes of structure(s) in turbulence flows keeping in mind that structure(s) is (are) just an inherent property of turbulence. There is no turbulence without structure(s).

Instability

As mentioned in Chap. 2, the most commonly accepted view on the origin of turbulence is flow instability. An additional factor is that instability is considered as one of the origins of structure(s) in/of turbulence. However, this latter view requires to assume that turbulence has a pretty long ‘memory’ of or, alternatively, that the ‘purely’ turbulent flow regime, i.e. at large enough Reynolds numbers, has instability mechanisms similar to those existing in the process of transition from laminar to turbulent flow state. The problem is that speaking about (in)stability requires one to define the state of flow (in)stability of which is considered, which is not a simple matter in the case of a turbulent flow.

Note the observation made by Goldshtik and Shtern (1981): The fact that the phenomenon of intermittency and structures are observed in the proximity of the outer boundary of turbulent flow or in close to the wall and in the small scale “tail” of turbulent flow flows, i.e. when the characteristical Reynolds numbers are relatively not large, prompts an assumption that “structureness” is associated with mechanisms of turbulence origins. This may be an underlying reason of some similarity between some flow patterns (“structures”) in transitional and developed turbulent flows. This idea appeared in a number of subsequent publications, e.g. Blackwelder (1983), Pullin et al. (2013) and references therein. It should be stressed that even if the above hypothesis is true there in no escape from nonlocal effects!

Emergence

Another less known view holds that structure(s) emerge in large Reynolds number turbulence out of ‘purely random structureless’ background, e.g. via the so-called inverse cascades or negative eddy viscosity. Among the spectacular examples, are the ‘geophysical vortices’ in the atmosphere, and ocean, as well as astrophysical objects. Another example is the emergence of coherent entities, such as vortex filaments/worms and other structure(s), out of an initially random Gaussian velocity field via the NSE dynamics, for examples see references in Tsinober (2009).

Anderson (1972) emphasizes the concept of ‘broken symmetry’, the ability of a large collection of simple objects to abandon its own symmetry as well as the symmetries of the forces governing it and to exhibit the ‘emergent property’ of a new symmetry. One of the difficulties in turbulence research is that no objects simple enough have been found so far such that a collection of these objects would adequately represent turbulent flows. It is not clear how meaningful is the very question on the existence of such objects.

It ‘Just Exists’, or Do Flows Become or Are They Are ‘Just’ Possessing Structure?

To the flows observed in the long run after the influence of the initial conditions has died down there correspond certain solutions of the Navier–Stokes equations. These solutions constitute a certain manifold $\mathcal{M}=\mathcal{M}(\mu)$ (or $\mathcal{M}=\mathcal{M}(\mathit{Re})$ ) in phase space invariant under phase flow (Hopf 1948). Kolmogorov’s scenario was based on the complexity of the dynamics along the attractor rather than its stability (Arnold 1991), see also Keefe (1990a), Keefe et al. (1992).

This view is a reflection of one of the modern beliefs that the structure(s) of turbulence—as we observe in physical space—is (are) the manifestation of the generic structural properties of mathematical objects in phase space, which are called attractors and which are invariant in some sense. In other words, here the structure(s) are assumed to be ‘built in’ the turbulence independently of its origin, hence the tendency to universality.⁶ It is noteworthy that the assumed attractor existence makes sense for statistically stationary turbulent flows. However, for flows which are not such, e.g. decaying turbulent flows past a grid or a DNS simulated flow in box the attractor is trivial. Nevertheless, these flows possess many properties which are essentially the same as their statistically stationary counterparts provided that their Reynolds numbers are not too small (Re _λ ≥10²).

The above refers to the dynamical aspects of real turbulent flows. We mention again here also the

Emergence of Structures in Passive Objects in Random Media

In which the velocity field and the external forcing are prescribed. Whatever their nature—even Gaussian—structure is emerging in the field of passive objects (Zeldovich et al. 1990; Ott 1999 and references therein).

2.2 How Does the Structure of Turbulence ‘Look’?

For long time the first and the only impression/answer to this question was obtained by employing visualization techniques. First in experiments using mainly passive markers and later using the DNS simulations looking mainly at objects bounded by isosurfaces of some quantity, such as enstrophy, ω ².

The first important result was that even turbulence which is ‘homogeneous’ and ‘isotropic’ has structure(s), i.e. contains a variety of strongly localized events. The primary evidence is related to spatial localization of subregions with large enstrophy, i.e. intense vorticity, which are organized in long, thin tubes-filaments-worms. Some evidence was obtained about regions with large strain, s _ij s _ij , i.e. dissipation, being sheet-like objects with very sharp edges (razors/flakes), see references in Tsinober (2009) for both.

The relatively simple appearance of the observed ‘structures’ is due simple techniques as looking at isosurfaces of some quantity, e.g. enstrophy ω ², with thresholding using conditional sampling techniques. This is how the first evidence of concentrated vorticity/filaments/worms was obtained.

This prompted a rather popular view that turbulence structure(s) is (are) simple in some sense and that essential aspects of turbulence structure and its dynamics may be adequately represented by a random distribution of simple (weakly interacting) objects.

In particular, it is commonly believed that most of the structure of turbulence is associated with and is due to various strongly localized intense events/structures, e.g. mostly regions of concentrated vorticity so that turbulent flow is dominated by vortex tubes of small cross-section and bounded eccentricity (Chorin 1994, p. 95), for other quotes and references see Tsinober (1998a, 2009), and that these events are mainly responsible for the phenomenon of intermittency. It is demonstrated in Tsinober (1998a, 2009) that such views are inadequate. It appears that—though important—these structures are not the most dynamically important ones and are the consequence of the dynamics of turbulence rather than its dominating factor. Namely, regions other than those involving concentrated vorticity such as: (i) ‘structureless’ background, (ii) regions of strong vorticity/strain (self) interaction and largest enstrophy and strain production dominated by large strain rather than large enstrophy, and (iii) regions with negative enstrophy production are all dynamically significant and in some important respects more significant than those with concentrated vorticity, strongly non-Gaussian, and possess structure. Due to the strong nonlocality of turbulence in physical space all the regions are in continuous interaction and are strongly coupled. A similar statement can be made regarding the so called streamwise vortices observed in many turbulent flow. Moreover, as described in Chap. 7 the anomalous scaling is due to strong dissipative events, i.e. large strain s ², so that the conventionally defined inertial range is an ill defined concept and turning it out of existence.

To emphasize, the above conclusions are the outcome of the use of quantitative manifestations of turbulence structure, which just like intermittency are in the first place of statistical nature independently of how specifically the individual structures look and whether they do exist at all.

Though the isosurfacing/thresholding approach is useful and ‘easy’, it is inherently limited and reflects at best the simplest aspects of the problem. Even for characterization of some aspects of the local (i.e. in a sense ‘point’-wise) structure of the flow field in the frame following a fluid particle requires at least two parameters.⁷ Therefore attempts to adequately characterize finite scale structure(s) by one parameter only are unlikely to be successful. The one parameter approaches are not made any better (but rather more misleading) by adding to thresholding and isosurfacing some decomposition, e.g. Leung et al. (2012) and references therein. One can study some geometrical issues of the isosurfaces in the filtered fields and even the “interaction” between such “structures” belonging to the fields corresponding to different filter bands with the remaining acute problem as to how all this is related to the whole flow field in the real physical space. More generally, the problem is related to pattern recognition and requires defining a conditional sampling scheme involving more than two parameters. This scheme is in turn based on what a particular investigator thinks are the most important physical processes, features, etc. This in turn opens a Pandora box of possibilities and contains an inherent element of subjectivity and arbitrariness, since the physics of turbulence is not well understood. In this sense, the circle is closed: in order to objectively define and educe some structure, one needs clear understanding of the physics of turbulence, which, it is in turn believed, can be achieved via study of turbulence structure(s).

There are other serious problems in observations of individual structures obtained via isosurfacing and thresholding or similar and alike.

First, the “boundaries” of flow regions isolated by such methods cannot be qualified as “natural” in any sense and serve as a technical means only. One cannot take such an approach for granted as reliable for getting the “natural” boundaries of these structures and serving simultaneously as definitions of those “structures”. Both can be qualified as wishful thinking at best. On the other hand, there exists a number of attempts to define what is, e.g. a coherent structure, a vortex, etc., see references in Malm et al. (2012) and also Monin (1991), Townsend (1987), Bonnet (1996) and Holmes et al. (1996).

Second, these structures are just single time snapshots in space having no identity beyond the particular time moments of their (infinitesimal) life time, so that one cannot observe their time evolution. The latter difficulty is nontrivial, because one ventures to deal with finite objects. Namely, even having defined such a finite relatively simple object at some time moment one is loosing it in a pretty short time even if there is a possibility to follow this object as in case of purely Lagrangian objects. This is illustrated in Fig. 9.2. It is for these reasons people produce statistics out of collections of “similar objects” obtained from snapshots at the same and different time moments via isosurfacing and thresholding and other tricks to justify these “surgeries”, etc. 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 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.

Fig. 9.2

Evolution of a tetrahedron with edge of ≈4η at t=0 using the data base of Johns Hopkins University (Li et al. 2008), for a 1024⁴ space-time history of a direct numerical simulation of isotropic turbulent flow in incompressible fluid in 3D. Courtesy Beat Luethi. It seen that a simple Lagrangian tetrahedron, i.e. consisting of fluid particles, becomes not so simple after just one Kolmogorov time scale τ _η and turns into a non-trivial object in time of few Kolmogorov time scales

The above points to acute problems in defining instantaneous structures not to mention studying them, though it is commonly assumed that there exist instantaneous “structures” which are in some sense “key” objects from some point of view and that these unknown objects even govern the dynamics of the flow. Claims of this kind are pretty frequent, but without much—if any—explanation/justification or whatever.

As concerns the individual structure it was already mentioned that the regions of concentrated vorticity are of limited dynamical relevance. First, these regions are characterized by approximate balance between enstrophy production ω _i ω _k s _ik and its viscous destruction in a way similar to that of Tennekes and Lumley balance, see Tsinober (2009). Second, they belong to the category of flow patterns with predominant alignment of vorticity and the strain eigenvector λ ₂ corresponding to the intermediate strain eigenvalue, Λ₂. However, the major contribution to the enstrophy production comes from the regions with the ω,λ ₁ alignment (corresponding to the largest strain eigenvalue, Λ₁) and in which there is no approximate balance between enstrophy production ω _i ω _k s _ik and its viscous destruction with strong dominance of ω _i ω _k s _ik . Moreover, the regions with the ω,λ ₁ alignment comprise a large part of those where the vorticity/strain interaction is strongest, see Chap. 7 above and Chap. 6 in Tsinober (2009).

Recently there is some trend of reviving and ascribing some excessive importance to thin shear layers (Hunt et al. 2010; Elsinga and Marusic 2010; Worth and Nickels 2011; Ishihara et al. 2011 and references therein). However, they are not more relevant than the “worms” as they belong to the same category with predominant alignment of vorticity and the strain eigenvector corresponding to Λ₂ the intermediate strain eigenvalue Λ₂,⁸ whereas as mentioned the most dynamically active are flow patterns with predominant ω,λ ₁ alignment, e.g. as concerns enstrophy production and other essential nonlinear processes, see Chap. 7 above (Tsinober 2009 and references therein).

A final note is that though the patterns with predominant alignment of vorticity and the strain eigenvector corresponding to the intermediate strain eigenvalue, Λ₂ (worms, shear layers and more involved patterns) are statistically dominant they are not the most dynamically relevant. In other words, statistical dominance is not synonymous to dynamical relevance. The qualification of “shear layer” as (several previous “key” objects, e.g. worms) belongs to the category of oversimplified concepts and vague terminology—as mentioned more preferable are well defined strain and vorticity. The oversimplification (thin layers!) is seen clearly from the above as neglecting important issues of geometrical nature among others, e.g. the aspects of alignments of vorticity and the eigenframe of the rate of strain tensor.

2.3 Structure Versus Statistics

The ‘not objective enough’ nature of a variety of conditional sampling procedures resulted in a whole ‘zoo’ of ‘structures’ in different turbulent flows, which some people believe to be significant in some sense, but many do not. The zoo seems to have a tendency to grow at least exponentially with the introduction of multiscale approaches, but one cannot help reminding the question by Kadanoff (1986) when the mulifractal “formalism” was just born: Where is the physics? Among the reasons for such skepticism is some evidence that the attempts at adequate representation of such a complicated phenomenon like turbulence as a collection of simple objects/structures only are unlikely to succeed. As mentioned, until recently it was believed that concentrated vorticity/filaments is the dominating structure in turbulent flows in the sense that most of the structure of turbulence is associated with and is due to regions of concentrated vorticity. It appears that—though important—these structures are not the most dynamically important ones and are the consequence of the dynamics of turbulence rather than being its dominating factor. A similar statement is true in respect of recently revived sheer layer.

Nevertheless, some ‘objectiveness’ can be achieved using quantities appearing in the NSE and/or the equations which are exact consequences of NSE.

The question about what structure(s) of turbulence mean(s) can be answered via a statement of impotence: speaking about ‘structure(s)’ in turbulence the implication is that there exist something ‘structureless’, e.g. Gaussian random field as a representative of full/complete disorder. Gaussian field is appropriate/natural to represent the absence of structure in the statistical sense. Hence all non-Gaussian manifestations of turbulent flows can be seen as some statistical signature of turbulence structure(s). This does not imply that an exactly Gaussian field does not necessarily possess any spatial or temporal structures, see, e.g. Fig. 3 in She et al. (1990)—any individual realization of a Gaussian field does have structures. However, an exactly Gaussian field does not possess dynamically relevant structure(s), it is dynamically impotent.

So the next most difficult question is about the relevance/significance of some particular aspect of non-Gaussianity for a specific problem in question. It seems that here one enters the subjective realm: the criteria of significance (which is the matter of physics!) are decided by the researchers. However, the following examples show that objective choice of the structure sensitive statistics is dictated by general dynamical aspects of the problem.

For instance, the build up of odd moments is an important specific manifestation of structure of turbulence along with being the manifestation of its nonlinearity. The two most important examples are the third order velocity structure function S ₃(r)=〈{[u(x+r)−u(x)]⋅r/r}³〉 and the mean enstrophy production 〈ω _i ω _k s _ik 〉. The first one is associated with the −4/5 Kolmogorov law S ₃(r)=−4/5〈ϵ〉r (Kolmogorov 1941b), which is the first strong indication of the presence of structure in the inertial range showing that both non-Gaussianity and the structure of turbulence are directly related to it’s dissipative nature. It is remarkable that the title of this paper by Kolmogorov is Dissipation of energy in the locally isotropic turbulence. The −4/5 Kolmogorov law clearly overrules the claims that ‘Kolmogorov’s work on the fine-scale properties ignores any structure which may be present in the flow’ (Frisch 1995, p. 182) and that it is associated with near-Gaussian statistics, see references in Tsinober (2009) among multitude of others. As concerns the near Gaussian statistics it is correct that single point statistics is known to be quite close to the Gaussian one. However, the conclusion that velocity fluctuations are really almost Gaussian is a misconception, not to mention the field of velocity derivatives. This is already seen when one looks at two-point velocity statistics. For instance, in such a case the odd moments are significantly different from zero, e.g. Frenkiel et al. (1979).

The essentially positive value of the mean enstrophy production 〈ω _i ω _k s _ik 〉, discovered by Taylor (1938a) is the first indication of the presence of structure in the small scales, where turbulence is particularly strongly non-Gaussian and intermittent. The above two examples show that both the essential turbulence dynamics and its structure are associated with those aspects of it’s non-Gaussianity exhibited in the build up of odd moments, which among other things means phase and geometrical coherency, i.e. structure. Hence, the importance of odd moments as indicators of intermittency. It is to be noted that the non-Gaussianity found experimentally both in large and small scales is exhibited not only in the nonzero odd moments, but also in strong deviations of even moments from their Gaussian values. Thus both the large and small scales differ essentially from Gaussian indicating that both possess structure.

However, an important point is that probability criteria are insufficient, since one can find in statistical data irrelevant structures with high probability (Lumley 1981). In other words the structure(s) should be relevant/significant in some sense. For example, it should be dynamically relevant for velocity field, and related quantities such as vorticity and strain. This does not mean that kinematical aspects of turbulence structure(s) are of no importance. For example, anisotropy is a typical kinematic statistical characteristic of turbulent flows of utmost dynamical significance/impact which hardly can be applied to individual structures, e.g. a turbulent flow consisting mostly of ‘anisotropic’ individual structures can be statistically isotropic. Among the first statistical treatments of turbulence structure is, of course, the first paper by Kolmogorov (1941a), the very title of which is The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.

The advantage of such an approach is that it allows one to get insights into the structure of turbulence without the necessity of knowing much (if anything) about the actual appearance of it’s structures, since the very question of this kind may well be just meaningless. This is especially important in view of numerous problems/ambiguities in definitions of individual structures in turbulent flows, their identification and statistical characterization as well as their incorporation in ‘theories’. The main reason is that there exist an intrinsic problem of both defining what the relevant structures are, see Bonnet (1996) for references and a review of existing techniques which all are based on statistics anyhow, e.g. of defining extracting/educing and characterizing the so-called coherent structures. For a number of reasons, it is very difficult, if not impossible, to quantify the information on the instantaneous structures of turbulent flows into dynamically relevant/significant form. The observed individual structures strongly depend on the method observation/extracting, but more importantly none of them are simple, neither are they weakly interacting between themselves or with the background which in fact is not structureless as assumed by many. Indeed, you can find structures, essentially arbitrary, which have equal probability to the ones we have latched onto over the years: bursts, streaks, etc… If structures are defined as those objects which can be extracted by conditional sampling criteria, then they are everywhere one looks in turbulence (Keefe 1990b). For instance, looking at a snapshot of the enstrophy levels of a purely Gaussian velocity field in She et al. (1990) one can see a number of filaments—the irrelevant ones—like those observed in real turbulent flows, i.e. pure Gaussian velocity field has some structure(s) too.

2.4 What Kinds of Statistics Are Most Appropriate to Characterize at Least Some Aspects for Turbulence Structure

Returning to isosurfacing and thresholding it should be mentioned that these allow to handle regions (rather than “structures”) with some properties of interest, for latest example see Malm et al. (2012). These authors used a measure of vorticity stretching related to the magnitude of the vortex stretching vector $W_{i}\equiv \omega _{is_{ij}}$, which is exemplifying that the main criteria should be of dynamical relevance, and sensitiveness to the non-Gaussian properties of turbulence, so that one can speak about statistics weakly sensitive to structure and structure sensitive statistics.

Examples of Statistics Weakly Sensitive to Structure(s)

The first examples of this kind are energy spectra in which the phase (and geometric) information is lost. Hence their weak sensitivity to the structure of turbulence. This insensitivity, in particular, is exhibited in the scaling exponents when/if such exist. For example, the famous −5/3 exponent can be obtained for a great variety of qualitatively different real systems—not necessarily fluid dynamical—and theoretical models, for a partial list of references see Tsinober (2009). One can also construct a set of purely Gaussian velocity fields, i.e. lacking any dynamically relevant structure(s), with any desired length of the −5/3 ‘inertial’ range (Elliott and Majda 1995). An extreme example is a single sharp change in velocity represented in Fourier space has an energy spectrum ∝k ^−6/3 which is not so easy to distinguish from k ^−5/3! Vice versa the spectral slope can change, but the structure remains essentially the same ‘yet retaining all the phase information’ (Armi and Flament 1987). Moreover, not only ‘the spectral slope alone is inadequate to differentiate between theories’, alone it does not correspond to any particular structure(s) in turbulence or it’s absence: there is no one-to-one relation between scaling exponents and structure(s) of turbulence. This is true not only of exponents related to Fourier decomposition with its ambiguity (Tennekes 1976), but of many other scaling exponents including those obtained in some wavelet space, SO(3) decomposition and in the physical space—a much overstressed aspect of turbulent flows. Likewise, similar PDFs of some quantities can correspond to qualitatively different structure(s) and quantitatively different values of Reynolds number. The emphasis is on some quantities like pressure or some other usually (but not necessarily) even order quantities in velocities or their derivatives, since the PDFs of other appropriately chosen quantities are sensitive to structure (see below).

As mentioned turbulence possess a number of quasi-Gaussian manifestations. The corresponding statistics belongs to the weakly sensitive to structure(s).

Structure Sensitive Statistics

It is noteworthy that—as shown by Hill (1997)—the −4/5 Kolmogorov law is more sensitive to the anisotropy, i.e. the third-order statistics (again odd moments), than the second-order statistics. Likewise the structure functions of higher odd orders $S_{p}^{\parallel }(r)=\langle (\Delta u_{\parallel })^{p}\rangle$ are essentially different from zero, see references in Betchov (1976), Sreenivasan and Antonia (1997), Tsinober (1998b).

Odd Moments and Related PDFs

This is an example how structure sensitive statistics can help in looking for the right reasons of measured spectra in the lower mesoscale range (Lindborg 1999). The procedure involves using the third order structure functions which are generally positive in the two dimensional case contrary to the three-dimensional case. Calculations based on wind data from airplane flights, reported in the MOZAIC data set. It is argued that the k ⁻³-range is due to two-dimensional turbulence and can be interpreted as an enstrophy inertial range, while the k ^−5/3-range is probably not due to two-dimensional turbulence and should not be interpreted as a two-dimensional energy inertial range. There is a competing hypothesis that the large scale −5/3 range is the spectrum of weakly non-linear internal gravity waves with a forward energy cascade (Van Zandt 1982). A third claim is that the spectral slope in the enstrophy range is more shallow than −3 and is close to −7/3 (Tsinober 1995). This range and related anomalous diffusion is explained in terms of the phenomenon of spontaneous breaking of statistical isotropy (rotational and/or reflectional) symmetry—locally and/or globally.

Another example is the demonstration that the small scale structure of a homogeneous turbulent shear flow is essentially anisotropic at Reynolds number up to Re _λ ≈1000 (Shen and Warhaft 2000); see also Ferchichi and Tavoularis (2000). In order to detect this anisotropy the authors measured the velocity structure functions of third and higher odd orders of both longitudinal and transverse velocity components and corresponding moments of velocity derivatives. In particular, they found a skewness of order 1 of the derivative of the longitudinal velocity in the direction of the mean gradient, which should be very small (or ideally vanish) for a locally isotropic flow. Similar results were obtained in DNS, see references in Tsinober (2009). We should recall that analogous ‘misbehavior’ of large Reynolds-number turbulence regarding the skewness of temperature fluctuations in the atmospheric boundary layer is known since late sixties (Stewart 1969; Gibson et al. 1970, 1977).

Odd moments such as strain and enstrophy production are obviously of primary importance.

Geometrical Statistics

This example shows how conditional sampling based on geometrical statistics can help to get insight into the nature of various regions of turbulent flow, e.g. those associated with strong/weak vorticity, strain, various alignments, and other aspects of dynamical importance. The first general aspect is the qualitative difference in the behavior and properties of regions with large enstrophy from strain dominated regions, which is also one of the manifestations of intermittency. Various alignments comprise important simple geometrical characteristics and manifestation of the dynamics and structure of turbulence. For example, there is a distinct qualitative difference between the PDFs of cos(ω,λ _i ) for a real turbulent flow and a random Gaussian velocity field. In the last case, all these PDFs are precisely flat. An example of special dynamical importance is the strict alignment between vorticity, ω, and the vortex stretching vector W _i ≡ω _j s _ij , since the enstrophy production is just their scalar product, ω _i ω _j s _ij =ω⋅W. In real turbulent flows, the PDF of cos(ω,W) is strongly asymmetric whereas it is symmetric for a random Gaussian field. It remains essentially positively skewed for any part of the turbulent field, e.g. in the ‘weak background’ involving whatever definition based on enstrophy, strain, both and/or any other relevant quantity. Thus, contrary to common beliefs, the so called ‘background’ is not structureless, dynamically not inactive and essentially non-Gaussian, just like the whole flow field or any part of it. The structure of the apparently random ‘background’ seems to be rather complicated. The previous qualitative observations (mostly from DNS) about the ‘little apparent structure in the low intensity component’ or the ‘bulk of the volume’ with ‘no particular visible structure’ should be interpreted as meaning that no simple visible structure has been observed so far in the bulk of the volume in the flow. It is a reflection of our inability to ‘see’ more intricate aspects of turbulence structure: intricacy and ‘randomness’ are not synonyms for absence of structure.

Pressure Hessian

Some quantities like pressure or other usually (but not necessarily) even order quantities in velocity or their derivatives are less sensitive to structure. The example below present an opposite case

Of special interest is the pressure Hessian $\varPi _{ij}\equiv \frac{\partial ^{2}p}{\partial x_{i}\partial x_{j}}$. Among the general reasons for such an interest is that the pressure Hessian is intimately related to the nonlocality of turbulence in physical space, see references in Tsinober (1998a, 1998b, 2009).

One of the quantities in the present context directly associated with the pressure Hessian is the scalar invariant quantity ω _i ω _j Π _ij . It is responsible for the nonlocal effects in the rate of change of enstrophy production ω _i ω _k s _ik . What is special about this quantity, which is of even order in velocity, is that for a Gaussian velocity field 〈ω _i ω _k Π _ij 〉 _G ≡0, whereas in a real flow it is essentially positive and $\langle \omega _{i}\omega _{k}\varPi _{ij} \rangle \sim \frac{1}{3}\langle W^{2} \rangle$, where W _i ≡ω _k s _ik is the vortex stretching vector. Thus interaction between the pressure Hessian and the vorticity is one of the essential features of turbulence structure associated with its nonlocality. It is noteworthy that a similar useful quantity involving strain is non-vanishing for a Gaussian velocity field, $\langle s_{ik}s_{kj}\varPi _{ij}\rangle _{G}=-\frac{1}{20}\langle \omega ^{2}\rangle _{G}^{2}$.

On Passive Objects and Lagrangian Coherent Structures

Above we discussed the dynamical aspects of the problem. The issues of structure(s) in various ‘kinematic’ issues, like the transport of passive objects (scalars, vectors, etc.), in which Gaussian or other prescribed velocity fields are used rather successfully, can be treated in a similar way as the one described in this section.

As mentioned in Lagrangian setting the dissipative effects are more “influencing” due to strong removal of sweeping effects. Hence stronger deviations from Gaussian statistics in the Lagrangian setting as compared to the Eulerian one just because “inbetween” there is a relation turning even a pure Gaussian velocity field in the Eulerian setting into strongly non-Gaussian one. Hence the so called Lagrangian coherent structures (LCS’s) even in pure laminar in the Eulerian setting and pure Gaussian Eulerian velocity field. In this sense the LCS are purely kinematic objects just like the structures in the passive objects evolving in a purely Gaussian velocity field due to the non-linear and non-integrable relation between the Eulerian and Lagrangian fields, for more see Tsinober (2009, p. 300) and Pouransari et al. (2010).

2.5 Structure(s) Versus Scales and Decompositions

It is natural to ask how meaningful is it to speak about different scales in the context of ‘structure(s)’ and in what sense, especially when looking at the ‘instantaneous’ structure(s) of/in turbulence. The known structures indeed possess quite different scales. Vortex filaments/worms—have at least two essentially different scales, their length can be of the order of the integral scale, whereas their cross-section is of the order of Kolmogorov scale. Similarly, the ramp-cliff fronts in the passive scalar fields have a thickness much smaller than the two other scales. This fact is consistent with the observation by Batchelor and Townsend (1949), that the mean separation of the visible activated regions is comparable with the integral scale of the turbulence, i.e. with the size of the energy-containing eddies.

It is believed that appropriately chosen decompositions may represent structure(s) of turbulence, e.g. Holmes et al. (1996, 1997). Here again several notes are in order. First, this position depends strongly on what is meant by structure(s). Second, such a possibility is realistic when the flow is dominated by (usually large scale) structures, when many, or practically any reasonable decompositions will do anyhow. And third, structure(s) (and related issues such as geometry) emerging in the ‘simplest’ case of turbulent flows, in a box with periodic boundary conditions, is(are) are inaccessible via Fourier decomposition, the most natural one in this case.

One of the popular ‘decompositions’ is into ‘coherent structures’ and random/dissipative ‘background’.⁹ This latter is generally considered as structureless and as a kind of passive sink of energy. None is true: the background is not passive at all, it is strongly coupled with the ‘coherent structures’, and possess lots of it ‘own’ structure(s).

There is no turbulence without structure(s). Every part (just as the whole) of the turbulent field—including the so-called ‘structureless background’—possess structure. Structureless turbulence (or any of its part) contradicts both the experimental evidence and the Navier–Stokes equations. The qualitative observations on the little apparent structure in the low intensity component or the bulk of the volume with no particular visible structure should be interpreted as indicating that no simple visible structure has been observed so far in the bulk of the volume in the flow. It is a reflection of our inability to ‘see’ more intricate aspects of turbulence structure: intricacy and ‘randomness’ are not synonyms for absence of structure.

Another kind of decomposition is represented by a latest example attempting to take into account the undeniable structure of the above mentioned “structureless background” by dividing the flow in two characteristic regions: the mentioned above “thin shear layers” occupying a small part of the volume of and the quasi-homogeneous rest (Hunt et al. 2010; Ishihara et al. 2011 and references therein). The assumption of “thin” is necessary in order to employ a kind of RDT approach. Another recent example is the mentioned above by Leung et al. (2012). Apart of problematic nature of such decompositions from the fundamental point, e.g. the claims that the components of some decomposition represent physically meaningful “structures”, there are many problems of conceptual and technical nature with what is called ‘coherent structures’, “thin shear layers”, vortex structures, filaments etc., starting from the very beginning of their definition (in fact non-existent or at best vague) and ending with their role in fluid flows both in Eulerian and Lagrangian setting. It is for this reason that At this stage, this alternative approach (i.e. the ‘structural’) has not led to a generally applicable quantitative model, neither—for better or worse—has it a major impact on the statistical approaches. Consequently the deterministic viewpoint is neither emphasized nor systematically presented (Pope 2000). This does not mean that there exists “generally applicable quantitative model” based on statistical approaches. It looks that so far Liepmann was correct (but a bit over-optimistic) in his prediction: Clearly, the exploration of the concept of coherent structure is still on the rise. Turbulence is and will remain the most difficult problem of fluid mechanics, and the past experience suggests that the subsequent fall of interest in the coherent structures is more than likely. The resulting net gain in understanding of turbulence may be less than our expectations of today but will certainly be positive (Liepmann 1979). Unfortunately, (so far) the resulting net gain in understanding of turbulence is far less than was expected in 1979 and on. Nevertheless, though essentially there is no acceptable definition of “coherent structure” the boldest part of the community wonders about “quantification” and even “the dynamical equations for coherent structures to predict their evolution”, see e.g. Holmes et al. (1997). Apart of sensible definition of this finite object one needs also a definition of the “incoherent components” not to mention the tools to handle their interaction. On top of this there is a not just technical question on coherent structures of what? It is the right place to remind that the objects termed “coherent structures” as other terms just structures or alike are still elusive, and may appear to be not much different from the Emperors’s new Clothes.

The reason for the above statement is as follows.

In dynamical systems, one looks for structure in the phase space (Shlesinger 2000; Zaslavsky 1999), since it is relatively ‘easy’ due to low dimensional nature of the problems involved. In turbulence nothing is known about its properties in the corresponding very high dimensional phase space.¹⁰ Therefore, it is common to look for structure in the physical space with the hope that the structure(s) of turbulence—as we observe it in physical space—is (are) the manifestation of the generic structural properties of mathematical objects in phase space, which are called attractors and which are invariant in some sense. In other words, the structure(s) is (are) assumed to be ‘built in’ in the turbulence independently of its (their) origin. The problem is that due to very high dimension and complex behavior of turbulent flows and structure of the underlying attractors one may never be able to realistically determine the fine-scale structure and dynamic details of attractors of even moderate dimension… The theoretical tools that characterize attractors of moderate or large dimensions in terms of the modest amounts of information gleaned from trajectories [i.e., particular solutions] …do not exist… they are more likely to be probabilistic than geometric in nature (Guckenheimer 1986). Therefore, it is indeed unlikely that one can succeed in hunting individual structures of finite dimensions using low-dimensional tools not to mention isosurfacing based on one parameter since evolution of finite objects is not low dimensional. The remaining question is what does one see in reality named as “structure”, “coherent structure” and so on. This question deserves far more serious attention beyond weakly founded speculations.

At present the dynamical systems community advocates an alternative approach to turbulence, based on recently found simple invariant solutions and connecting orbits in Navier–Stokes flows (Cvitanović and Gibson 2010; Kawahara et al. 2012). This requires handling of ODEs systems with a large number degrees of freedom for pretty moderate Reynolds numbers, typically larger 10⁵ for Re∼10², which hardly can be qualified as low-dimensional. This means that even here one cannot avoid statistics. However, though the system of these equations seems to be not far easier to solve than the full time-dependent three-dimensional Navier–Stokes equations there is an important hope and even promise to “get into” what von Neumann wrote about in 1949: nothing less than a thorough understanding of the [global behavior of the] system of all their solutions would seem to be adequate to elucidate the phenomenon of turbulence… There is probably no such thing as a most favored or most relevant, turbulent solution. Instead, the turbulent solutions represent an ensemble of statistical properties, which they share, and which alone constitute the essential and physically reproducible traits of turbulence. The results obtained so far are significant for a theoretical description of transition to turbulence. However, the claim that the same is true of “also fully turbulent flow” seems to be a bit premature even for low Reynolds numbers ≤10³ as long as one talks about things like “resembling the spatially coherent objects found in the near-wall region of true turbulent flows”, having “the potential to represent coherent structures”, “simple invariant solutions could represent turbulence dynamics, whereas the simple solutions themselves would represent coherent structures embedded in a turbulent state”, reminiscence of things observed in DNS as “regeneration cycle in the buffer layer”, etc. All this for low Reynolds numbers with considerably lower capability than DNS of NSE. The bottom line is that one is tempted to ask the question by Cvitanović and Gibson (2010) Should this be called ‘turbulence?’. Resemblance is far less than necessary and definitely not sufficient.