The Essence of Turbulence as a Physical Phenomenon

The issue is about turbulence as a natural physical phenomenon as related to observations as distinct from the problem of turbulence. The dichotomic distinction between laminar and turbulent flows is problematic in several respects. First, most flows termed “turbulent” are in reality partly turbulent: some portions of the flow as turbulent and some as laminar—the coexistence of the two regimes, partly turbulent flows in one flow is a common feature with continuous transition of laminar into turbulent state via the entrainment process through the boundary between the two. Moreover, the reality is not that simple as the laminar/turbulent dichotomy as, e.g. the behavior of passive objects in flows with small Reynolds number looks as perfectly turbulent reflecting the qualitative difference between the chaotic flow properties in Eulerian and Lagrangian settings. These examples illustrate the enormous difficulties in defining what is both (i) turbulence and (ii) the turbulence problem. As concerns (i) one can only provide a description of major qualitative universal (sic) features of turbulent flows as obtained almost exclusively from observations (rather than by any theoretical deliberations) which form most important part of the “essence” of turbulence. This is because these mostly widely known qualitative features of all turbulent flows are essentially the same, i.e., it is meaningful to speak about qualitative universality of turbulent flows. It has to be stressed that the term “phenomenon of turbulence” as used above is mostly associated with the observational aspects, which in turbulence play far more important role due the unsatisfactory state of “theory”: there seems to exist no such a thing based on first principles. Hence it is vital to put the emphasis on the physical aspects based in the first place on observations as a basis aiding to form the notion of what turbulence and its essince are.

The issue involves a set of questions concerning theory(ies) of turbulence and questions such as what is physics and what is mathematics of turbulence and what are the physical/mathematical problems of turbulence. Though there is no acceptable definition of (what is) turbulence as a physical phenomenon, the relative “easiness” of observing its diverse manifestations leaves less problems than in the above. There is no consensus on what is the problem(s) of turbulence are and what would constitute their solution. There are even doubts about the very existence of the problem or any essence in it, so that the failure so far of theoretical efforts may be blamed on the problem itself. Neither is there agreement on what constitutes understanding. There is also no consensus on what the main difficulties are and why turbulence is so impossibly difficult: almost every aspect of turbulence is controversial, which by itself is one of the greatest difficulties. As concerns the basic aspects of the problem there is far more to say about the difficulties rather than the achievements. The problem was recognized by Neumann and Kolmogorov among others. The most acute difficulty is of basic and conceptual nature and concerns the physics of turbulence and the lack of knowledge about the basic physical processes of turbulence, its generation and origin, and poor understanding of the very few processes which are already known. One of the key physical processes is the predominant production of the velocity derivatives, to stress both strain/dissipation and vorticity as almost equal partners. The “almost” is important because it is the strain production is responsible for production of both contrary to the common view about amplification of vorticity. Moreover these are not just production - they are self-amplification. These attributes are, probably, the most specific and important as concerns fluid dynamic turbulence. Another set of much neglected issues is about the nonlocal properties of turbulence and related questions such as the ill-posedness of the concepts/pseudo-paradigms of inertial range and cascade, and the role of large scales and viscosity/dissipation; scale invariance, symmetries and universality in turbulence; origins of intermittency and the so-called ‘anomalous’ scaling in the inertial range just to mention a few.

As for today the standpoint of continuum mechanics reflected by the Navier–Stokes equations as a coarse graining over the molecular effects is considered as adequate and Perhaps the biggest fallacy about turbulence is that it can be reliably described (statistically) by a system of equations which is far easier to solve than the full time-dependent three-dimensional Navier–Stokes equations (Bradshaw, Exp Fluids 16:203–216, 1994).

It is a common (but not the only) view that the origin of turbulence is in the instabilities of some basic laminar flow. As the Reynolds number increases, some instability sets in, which is followed by further instabilities/bifurcations, transition and a fully developed turbulent state and the processes by which flows become turbulent are quite diverse. One process deserves special attention as a specific universal phenomenon. It is the continuous transition of laminar flow into turbulent flow via the entrainment process through the boundary between laminar/turbulent regions in the partly turbulent flows. In all these there is a distinct Lagrangian aspect: the abrupt transition of fluid particles (i.e. Lagrangian objects) from the laminar to turbulent state when passing across the laminar/turbulent ‘interface ’. Abrupt is a key word here, i.e. without any cascade whatsoever. Another issue of special interest is that there is a conceptual difference between the two kinds of flows: those arising ‘naturally’, e.g. by a simple time independent and smooth in space deterministic forcing, i.e. governed by a purely deterministic system and those produced by some external random source/excitation. The point is illustrated by the examples of stochastic forcing of integrable nonlinear equations (Burgers, Korteveg de Vries, etc. or just NSE at low Reynolds numbers) which without such forcing do not exhibit any “stochastic” behavior whatsoever. In other words, the properties of a turbulent flow are not neutral to the nature of excitation. Likewise boundary, initial and inflow conditions may cause qualitative differences especially due to essential role of nonlocal properties of the system. There is a qualitative difference between transition to turbulence as a phenomenon characterized by a large number of strongly interacting degrees of freedom and transition to chaotic behavior, in general, and between such notions as degrees of freedom and the dimension of attractor (assumed to exist), in particular. Any fluid flow which is adequately represented by a low-dimensional system is not turbulent—a kind of definition of ‘non-turbulence’. The immediate examples are laminar in Euler setting (E-laminar) which are low-dimensional chaotic fluid flows (L-turbulent).

The main dispute about the origins and nature of turbulence involves a number of aspects and issues in the frame of the dichotomy of deterministic versus random. In science this dispute covers an enormous spectrum of themes such as philosophy of science, mathematics, physics and the other natural sciences. Fortunately, we do not have to venture into this ocean of debate and opposing and intermediate opinions. This is mainly because (as it now stands) turbulence is described by the NSE which are purely deterministic equations with extremely complex behavior enforcing use of statistical methods, but this does not mean that the nature of such systems is statistical in any/some sense as frequently claimed. The bottom line is that turbulence is only apparently random: the apparently random behavior of turbulence is a manifestation of properties of a purely deterministic law of nature in our case adequately described by NSE. An important point is that this complex behavior does not make this law either probabilistic or indeterminate.

One of the most serious concerns is that statistics alone can be misused and misleading due the absence of theory based on first principles and inadequate tools for handling both the problem and the phenomenon and difficult issues of (mis)interpretation, validation, oversimplification and related, especially if the information is not in physical space, e.g. Fourier or any other decomposition. Among the concerns is the issue of statistical (pre)dominance versus dynamical relevance. The statistical predominance does not necessarily corresponds to the dynamical relevance as, e.g. in the case of sweeping decorrelation hypothesis or “exotic” averaging of turbulent flow fields such as represented in the local coordinate system defined by the eigenvectors of the strain rate tensor at each point.

We start with the N’s of turbulence. These comprise most of the reasons why turbulence is so impossibly difficult along with the essential constructive aspects facilitating all that is found in this book, i.e to a large extent the “essence” of turbulence. Whatever the approach there are important common issues, difficulties, features. Most of theses belong to the following categories: nonlinearity, nonlocality (and consequently “nondecomposabilty”) and non-integrability, non-Gaussianity and non-Markovianity, non-equilibrium and (time) irreversible, no scale invariance and no other symmetries, no small parameters and no low-dimensional description. As a consequence there no theory based on first principles as such NSE equations – a real frustration for a theoretician. In other words, the terms without the “non”s (e.g., non-linearity, non-locality, etc.) do belong to the category that theory can handle, but this seems unfortunately to exclude turbulence. There are also other closely related issues as, e.g. uncertainty and unpredictability.

The large Reynolds number behavior is of special importance from several points of fundamental nature which include such issues as restoring (or not) in somesensethe symmetries of Navier–Stokes equations (e.g. locally in Kolmogorov, Dokl Akad Nauk SSSR 30:299–303, (1941a)), universality, the role of viscosity/dissipation and the concept of inertial range, the role of the nature of forcing/excitation, inflow, initial and boundary conditions. In view of the arguments and experimental results on nonlocality and the direct and bidirectional coupling between large and small scales in the previous section a natural question arises what is the impact of nonlocality on all the above and whether there are enough reasons and evidence for a discussion and reexamination of the above issues, generally, and in relation to the nonlocal properties of turbulent flows among others, especially. Navier–Stokes equations at sufficiently large Reynolds number have the property of intrinsic mechanisms of becoming complex without any external aid such as strain and vorticity self-amplification. There is no guarantee that the outcome is the same from, e.g., natural “self-randomization” and with random forcing, on one hand, and different kinds of forcing, boundary and initial conditions, on the other hand. Moreover, there is serious evidence that the outcome may be and indeed is different.

Intermittency specifically in genuine fluid turbulence is associated mostly with some aspects of its spatiotemporal structure and its relation to the essential dynamical aspects. Hence, the close relation between the origin(s) and meaning of intermittency and structure of turbulence on one hand and of intermittency, structure and the dynamics, on the other hand. Just like there is no general agreement on the origin and meaning of the former, there is no consensus regarding the origin(s) and or meaning of turbulence structure(s). The present state of matters is that both issues are pretty speculative, an example of an ‘ephemeral’ collection of such is given in this chapter. We must 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’—possesses structure. Structureless turbulence (or any of its part) contradicts both the experimental evidence and the Navier–Stokes equations. The claims for ‘structureless background’ are a reflection of our inability to ‘see’ the more intricate aspects of turbulence structure: intricacy, complexity and ‘randomness’ are not synonymous with 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 no satisfactory solution has been found. One (but not the only) reason—as mentioned—is that it is not so clear what one is looking for: the objects still seem to be elusive. For example, a non-negligible set of people in the community doubt that the concept of coherent structure is much different from the Emperors’s New Clothes. An example of the acute difficulties described in this chapter is associated with the high dimensionality of what is called structures. Simple single parameter thresholding is inappropriate in drawing statistics due to the painful question of how really “similar” all these structure are if the individual members of such an ensemble are defined by one parameter only. An acute issue is about the spatiotemporal identity of what are called “structures”, which is not the same what is denoted by the term “structure” of turbulence. 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 high-dimensional phenomenon like turbulence can merely be described in terms of collections of only such ‘simple’ and weakly interacting objects. On the contrary, intermittency is an important feature of the common (!) nonlocal life of large and small scales and small scales both dynamically and kinematically.

The cental issue here is the long-lasting and continuing paradigmatic crisis in fundamental turbulent research especially in theoretical/mathematical but also experimental (both observational and numerical) aspects. This is the general reason of special importance for discussion of the conceptual and problematic aspects, physical phenomena, observations, misconceptions and unresolved issues rather than conventional formalistic aspects, models and similar. We start from the question on why the turbulence and the turbulence problem is so impossibly difficult with a number of specific aspects as seen from the subtitle below. These involve the major qualitative properties of turbulent flows and reiteration of problematic main/selected issues of a general nature, on one hand and theory versus experiments and/or theoreticians versus experimentalists and vice versa with some emphasis on what comes next. The following chapter is a colection of Essential Quotations in view of the importance of the outstanding theoretical thinking and efforts of great people in the field.

Along with a relatively limited number of citations in the text we bring a small collection of citations in this Appendix, more can be found in Tsinober (2009) especially in the text and also Appendices A and B. In this reference one of the aims was an extensive treatment of the dialogue within the turbulence community with an emphasis on problems of a conceptual nature.