The last post … of the first year!
A year ago, when I began this blog, few of us can have had any idea of what the year had in store from the coronavirus, now known to us as covid-19. Over the years, I have sometimes reflected on the very fortunate lives of my generation. I was born at the beginning of World War 2 and it impinged very little on my life or consciousness. In contrast, my grandparents all were adults during WW1, and would have suffered from that; while my parents must have endured fear and anxiety during WW2, but did not pass on any of that to me or my siblings. Basically all that I can remember was the occasional comment about the wonderful things (e.g. unlimited cream or butter) that one could get before the war!

So perhaps the pandemic is our war? Well, for many people it must seem like it; but, for those of us who are retired and have not been touched personally by the fatal consequences of the virus, it really only amounts to a degree of anxiety and some disruption of our lives. In my own case, I have not been able to go to my university office since last February. But this lack of access to my papers and books has merely been an inconvenience. Although, I do have plans to write a couple of review articles in the coming year; and, if I don’t have access to my office, only certain preliminaries will be possible.

In my first post, I referred to a paper of mine which I speculated might be my last as it had bounced from four different journals. I mentioned that I had let my guard down and made some sweeping statements without justifying them in detail. At the time I hadn’t mastered the art or science of incorporating references in my blogs, so I can now remedy the omission and this paper can be found as reference [1] below. So you can judge for yourself. Comments would be welcome. Just as a foretaste of something that I shall return to, is that in my view such a paper should have been unnecessary. The point it makes is that K41 scaling is observed for spectra and K62 scaling is not.

Incidentally, my speculation about publishing no more papers turned out to be overly pessimistic: see reference [2] below. There is rather a nice story attached to this, but I won’t go into that at the moment. Suffice it to say that it quite encouraged me and I have to confess that I now have a number of papers at various stages of preparation. At worst their fate when submitted to journals should make interesting anecdotes under the generic title of `peer review’.

To close on an upbeat note, I intend to integrate some of my blogs with the preparation of the two review articles that I have in mind. First, I intend to review the general topic of energy transfer and dissipation. In particular, the existing literature on the subject is unhelpful to the point of being quite bizarre. For instance, I recently read a discussion of the paper known as K41 (see reference [3] below) in which the author purports to quote this paper and in the process uses the word `wavenumber’, when in fact K41 derives the two-thirds law for the second-order structure function (i.e. $S_2(r) \sim r^{2/3}$), and the word wavenumber does not appear in the paper! Moreover, there is not a single exegesis (so far as I know) of K41 in the literature. Given its seminal nature, this is absolutely astonishing. It needs to be put right.

Secondly, I intend to write an article on statistical theories of turbulence, which will be much more accessible to those who are not theoretical physicists, and who balk at the word renormalization. In deciding which words not to use, I shall be guided by the acerbic remarks of the late Philip Saffman, which are to be found in his published lecture notes. Basically, I remain optimistic about this activity.

[1] W. D. McComb and M. Q. May. The effect of Kolmogorov (1962) scaling on the universality of turbulence energy spectra. arXiv:1812.09174[physics.flu-dyn], 2018.
[2] W. D. McComb. A modified Lin equation for the energy balance in isotropic turbulence. Theoretical & Applied Mechanics Letters, 10:377-381, 2020.
[3] A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS, 30:301,1941.

How important are the higher-order moments of the velocity field?

Up until about 1970, fundamental work on turbulence was dominated by the study of the energy spectrum, and most work was carried out in wavenumber space. In 1963 Uberoi measured the time-derivative of the energy spectrum and also the dissipation spectrum, in grid turbulence; and used the Lin equation to obtain the form of the transfer spectrum $T(k)$ [1]. Later on, this work was extended and refined by van Atta and Chen, who obtained the transfer spectrum more directly from the third-order correlation [2]. This seems to have been the peak of experimental interest in spectra, and from then on there was a growing concentration on the behaviour of the moments (strictly speaking, in the form of structure functions) in real space [3], [4].

Introducing the structure function of order $n$ by \[S_n(r) = \langle \delta u_L^n(r) \rangle,\] where $\delta u_L^n(r)$ is the longitudinal velocity difference, taken over a distance $r$, it is well known that, on dimensional grounds, they are expected to take the form \[S_n=C_n \,(\varepsilon r)^{n/3},\] whereas investigations like [3] and [4] (and many following them over the years) found deviations from this that increased with order $n$. Such results gave increased traction to belief in intermittency corrections and anomalous exponents.

Yet, when one considers it, the moments of a distribution are equivalent to the distribution itself. It is well known that the moments are related to the distribution through its characteristic function which is its Fourier transform. From the simple example on page 529 of reference [5], we see that the characteristic function can be expanded out in terms of the moments. Hence the distribution can be recovered to any desired order from the infinite set of its moments. Therefore, when one measures moments to some order, one is merely assessing the accuracy with which one has measured the distribution itself. A plot of the measured exponent $\zeta_n$ against order $n$ is no more or less than a plot of systematic experimental error. A glance at the plots of measured distributions in both [3] and [4] will make this point with compelling force, especially when one considers the wings of the distribution.

A brief overview of this topic and a number of more recent references may be found in [6]. Note that in that reference, a standard laboratory method of reducing systematic error was used to measure $\zeta_2$ and showed that it tended towards the canonical value of $2/3$ as the Reynolds number was increased. As a matter of some slight interest, I learnt that method when I was about sixteen years old at school.

[1] M. S. Uberoi. Energy transfer in isotropic turbulence. Phys. Fluids, 6:1048, 1963.
[2] C. W. van Atta and W. Y. Chen. Measurements of spectral energy transfer in grid turbulence. J. Fluid Mech., 38:743-763, 1969.
[3] C. W. van Atta and W. Y. Chen. Structure functions of turbulence in the atmospheric boundary layer over the ocean. J. Fluid Mech., 44:145, 1970.
[4] F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Antonia. High-order velocity structure functions in turbulent shear flows. J. Fluid Mech., 140:63, 1984.
[5] W. D. McComb. The Physics of Fluid Turbulence. Oxford University Press, 1990.
[6] W. D. McComb, S. R. Yoffe, M. F. Linkmann, and A. Berera. Spectral analysis of structure functions and their scaling exponents in forced isotropic turbulence. Phys. Rev. E, 90:053010, 2014.

How big is infinity?
In physics it is usual to derive theories of macroscopic systems by taking an infinite limit. This could be the continuum limit or the thermodynamic limit. Or, in the theory of critical phenomena, the signal of a nontrivial fixed point is that the correlation length becomes infinite. Of course, what we mean by `infinity’ is actually just a very large number. But the mathematicians do not like this. In reference [1] below, the author states: ‘… statistical-mechanical theories of phase transitions tell us that phase transitions only occur in infinite systems’. She sees this as paradoxical because, as we all know, in everyday life we are surrounded by finite systems undergoing phase transitions. She further believes that the paradox can be resolved by working with constructive mathematics, rather than classical mathematics, which is what we all normally use.

My quotation from [1] is certainly open to deconstruction, and I doubt if many physicists would agree with it. What originally drew my attention to this particular problem is the situation in turbulence theory. As the Reynolds number is increased (or, the viscosity is decreased), the dissipation rate becomes independent of the viscosity. Physicists attribute this to the energy transfer by the nonlinear term in the equation of motion becoming scale-invariant. As the Reynolds number is increased even more, this scale-invariance extends further through wavenumber space, and nothing thereafter changes, either qualitatively or quantitatively. This in practical terms is the infinite Reynolds number limit, and it occurs at quite modest, finite values of the Reynolds number.

However, many mathematicians, harking back to a paper by Onsager [2] in 1949, believe that the infinite Reynolds number limit corresponds to zero viscosity; and, even more bizarrely, that the continuum properties of the fluid break down in this limit. Accordingly, they are driven to finding ways of making the Fourier representation of the inviscid Euler equation dissipative, by destroying its symmetry-based conservation properties. I have discussed this topic in three previous posts on 12, 19 and 26 November; and a paper, at that time in preparation, is now available on the arXiv as [3].

[1] Pauline van Wierst. The paradox of phase transitions in the light of constructive mathematics. Synthese, 196:1863, 2019.
[2] L. Onsager. Statistical Hydrodynamics. Nuovo Cim. Suppl., 6:279, 1949.
[3] W. D. McComb and S. R. Yoffe. The infinite Reynolds number limit and the quasi-dissipative anomaly. arXiv:2012.05614[physics.flu-dyn], 2020.

My life in wavenumber space
In September 1966, when I began work on my PhD, I almost immediately began to dwell in wavenumber space. After a brief nod to the real-space equations, I had to learn about Fourier transformation of the velocity field, with the wave-vector $\mathbf{k}$ replacing the position vector $\mathbf{x}$, and the Navier-Stokes equations being changed from real space to wavenumber space. In addition, it was usual in those days to begin with the velocity field in a cubic box and use Fourier series. Then at some stage one would let the box size tend to infinity, and replace summations by integrals. At the same time, the periodic boundary conditions would be replaced by good behaviour at infinity. So far as theoretical work was concerned, I was not to emerge from wavenumber space until around 2006, when I began to take an interest in the phenomenology of turbulence.

This narrowness was not unusual and indeed did not seem particularly narrow at the time. There had been an incursion of theoretical physicists into turbulence from the 1950s onwards; and, for theorists of the time, wavenumber space was just momentum space with Planck’s constant set equal to unity. So everyone working on the statistical theory of turbulence was quite at home in wavenumber space, and it fitted in with what was almost a tradition in turbulence theory, which had begun with Taylor’s introduction of spectral methods in the 1930s and had been carried on in the 1950s by Batchelor’s book in particular. Problems only arose when one’s papers were refereed by those who were not part of this grouping, and who were hostile to spectral methods. But I have written about that in other blogs and it is not what concerns me here, which is something rather more subtle.

The other day I was trying to work something out and was sure that I had done it previously. I’m not keen on doing anything that I, or indeed anyone else, has already done. Hence I was checking back in my notebooks and found what I was looking for dated May 1993. So, that was satisfactory, but it reminded me of why I had done the work originally. During the 1970s/80s, I became increasingly aware of referees who felt that theories predicting the Kolmogorov $-5/3$ law should not be published, because ‘intermittency corrections meant that it wasn’t correct’. It seemed to me that the very structure of renormalization theories was evidence for the correctness of the $-5/3$ law. But as such theories were very largely inaccessible to fluid dynamicists (especially, of course, when they were refereeing them!) I had wondered how one could extract the basic ideas without the full level of complication.

The essential feature, it seemed to me, was the occurrence of scale invariance, in which the inertial flux through wavenumber became constant independent of wavenumber. Beginning with the velocity field in $k$-space, one could exploit its complex nature to separate out amplitude and phase effects. Then, in the context of the energy balance equation (nowadays increasingly referred to as the Lin equation), one could determine the energy spectrum by power counting; with its prefactor being determined by an average over the phases.

I wrote this up and submitted it to PRL sometime in 1993. The response was interesting. It was rejected with a report that spoke approvingly of how it was written and presented but regretted that the energy-balance equation had already been used to derive the so-called ‘$4/5$’ law for the third-order structure function by Kolmogorov. I of course was happily ignorant of this. It was something done in real space. Which demonstrates the disadvantages of taking too limited or narrow an approach.

In 2006 I retired and began to take an interest in various phenomenological questions. This meant that at last I crossed over into real space and worked with the Kármán-Howarth equation as well as with the Lin equation. When working on the scale-invariance paradox, I decided to revisit my 1993 theory and this was published as [1] below. I was now able to point out that it answered the Landau criticism of Kolmogorov’s theory (as reinterpreted by Kraichnan [2]), in that its prefactor also depended on an average to the two-thirds power. If the original referee had been more familiar with spectral methods, he might have realised that my paper was a derivation of the inertial-range energy spectrum from the equations of motion, not the Fourier transform of the third-order structure function. So it was very much a different result from the Kolmogorov ‘$4/5$’ law. It also occurs to me as I write this, that the relationship between prefactors in the real-space and wavenumber-space formulations might be worth looking at.

Is there a moral in all this? I think there is. Basing my opinion on long experience of papers, discussions and referee reports, I believe that those fluid dynamicists who are uncomfortable with spectral methods understand less about the basic physics of turbulence than they otherwise might… and the New Year is a time for resolutions!

[1] David McComb. Scale-invariance and the inertial-range spectrum in three-dimensional stationary, isotropic turbulence. J. Phys. A: Math. Theor., 42:125501, 2009.
[2] R. H. Kraichnan. On Kolmogorov’s inertial-range theories. J. Fluid Mech., 62:305, 1974.