The humble Table of Contents. With my holiday approaching, I thought it would be pleasant to talk about something less demanding, and which involved zero LaTeXing. So, I thought that I would speak up for the Table of Contents being included in journal articles. Of course its inclusion is commonplace in books, and one couldn’tContinue reading The humble Table of Contents.

An assessment of Onsager’s concept of scale invariance: 3 I will complete this sequence of posts by discussing the scale-invariance paradox and showing how the Onsager criterion for the inertial range should be modified in order to resolve it. We begin by noting that the first measurement of the transfer spectrum was made in 1963Continue reading An assessment of Onsager’s concept of scale invariance: 3

An assessment of Onsager’s concept of scale invariance: 2 For many years, arising from Onsager’s observation in 1945 [1], the condition for a range of wavenumbers has been seen as a criterion for the existence of an inertial range, and hence for the Kolmogorov spectrum holding over that range. It has been a cornerstone ofContinue reading An assessment of Onsager’s concept of scale invariance: 2

An assessment of Onsager’s concept of scale invariance: 1 We stated Onsager’s criterion [1] for an inertial range in terms of a scale-invariant flux as equation (4) in our post of 24 June 2024. In order to assess Onsager’s concept, we begin by considering the Lin equation in terms of the energy spectrum and theContinue reading An assessment of Onsager’s concept of scale invariance: 1

Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 4 In his 1964 paper, Corrsin [1] explained Onsager’s theory in a more up to date notation. We will build on that treatment here, but we shall use an even more modern notation. In particular, we will use for the energy spectrum and for the dissipation rate. InContinue reading Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 4

Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 3 Returning to this topic after my holiday, I will focus on Onsager’s 1945 abstract [1]. This is brief to the point of being cryptic and requires exegesis, but we shall defer that to the next post. For the moment we will concentrate on its relationship to KolmogorovContinue reading Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 3

Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 2 Before we discuss Onsager’s contribution, it will be helpful to first make some observations about Kolmogorov’s derivation of the law for the second-order structure function in the inertial range of scales [1]. This is sometimes referrred to as K41A and there are many treatments of it (e.g.Continue reading Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 2

Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 1 It is a well known fact that as one gets older time speeds up! This (apparent) reverse time-dilation is something to be reckoned with once one has passed `the big eight-oh’. So much so, that I was convinced that I had posted blogs quite recently only toContinue reading Onsager’s (1945) interpretation of Kolmogorov’s (1941a) theory: 1

Is it possible to achieve an infinite Reynolds number? There has been an increasing awareness in the turbulence community of the significance of finite-Reynolds-number (FRN) effects, corresponding to an impression that Kolmogorov’s theory requires an infinite Reynolds number. At the same time, there has been a recent growth of interest in Onsager’s Conjecture, which isContinue reading Is it possible to achieve an infinite Reynolds number?

Special Issue of the journal Atmosphere on Isotropic Turbulence. It has been some time since I last posted, and the title of this post suggests the main reason, as being a Guest Editor has proved to be quite time consuming. Nevertheless, I have agreed to be Guest Editor because, although I think it is aContinue reading Special Issue of the journal Atmosphere on Isotropic Turbulence.

Two-time correlations and temporal spectra: the analysis by Tennekes [1]. In this post we take a closer look at the analysis by Tennekes [1] in which he differed from the earlier analysis of Tennekes and Lumley [2] and concluded that large-scale sweeping is the determining factor in the decorrelation of the two-time correlation in theContinue reading Two-time correlations and temporal spectra: the analysis by Tennekes [1].

Two-time correlations and temporal spectra: the Lagrangian case. In my previous post on 27 April 2023, I promised to come back to the Lagrangian case. Over the years, I have taken the view that the discussion of the Lagrangian case along with the Eulerian case, which is the one that is of more practical importance,Continue reading Two-time correlations and temporal spectra: the Lagrangian case.

Two-time correlations and temporal spectra I previously discussed this topic in my posts of 25 February 2021 and 10 March 2022. In the succeeding months I have become increasingly aware that there is dissension in the literature, with people citing the temporal spectrum as , if the arguments of Kolmogorov apply; and , if convectiveContinue reading Two-time correlations and temporal spectra

Mode elimination: taking the phases into account: 5 When I began this series of posts on the effects of phase, I had quite forgotten that I had once looked into the effects of phase in quite a specific way. This only came back to me when I was using my own book [1] to remindContinue reading Mode elimination: taking the phases into account: 5

Mode elimination: taking the phases into account: 4 In the previous post we came to the unsurprising conclusion that as a matter of rigorous mathematics, we cannot average out the high-wavenumber modes while leaving the low-wavenumber modes unaffected. However, turbulence is a matter of physics rather than pure mathematics and the initial conditions are notContinue reading Mode elimination: taking the phases into account: 4

Mode elimination: taking the phases into account: 3 In this post we look at some of the fundamental problems involved in taking a conditional average over the high-wavenumber modes, while leaving the low-wavenumber modes unaffected. Let us consider isotropic, stationary turbulence, with a velocity field in wavenumber space which is defined on . Note thatContinue reading Mode elimination: taking the phases into account: 3

Mode elimination: taking the phases into account: 2 In last week’s post, we mentioned Saffman’s criticism of models like Heisenberg’s theory of the energy spectrum in terms of their failure to take the phases into account. In this post we explore this idea and try to elucidate this criticism a little further. We can takeContinue reading Mode elimination: taking the phases into account: 2

Mode elimination: taking the phases into account: 1 This is my first blog this year, very largely because I have been working on a review article as my contribution to a journal issue commemorating Jack Herring, who died last year. We were asked to include some personal recollections of Jack, in addition to the physics,Continue reading Mode elimination: taking the phases into account: 1

What are the first and second laws of turbulence? Occasionally I still see references in the literature to the Zeroth Law of Turbulence. The existence of a zeroth law would seem to imply that there is at least a first law as well. But, so far as I know, there are no other laws ofContinue reading What are the first and second laws of turbulence?

The non-Markovian nature of turbulence 9: large-eddy simulation (LES) using closure theories. In this series of posts we have argued that the three pioneering theories of turbulence (due to Kraichnan, Edwards and Herring, respectively) are all Markovian with respect to wavenumber interactions. Thus, despite their many successful features, the ultimate failure of these theories toContinue reading The non-Markovian nature of turbulence 9: large-eddy simulation (LES) using closure theories.

The non-Markovian nature of turbulence 8: Almost-Markovian models and theories Previously, in my post of 10 November 2022, I mentioned, purely for completeness, the work of Phythian [1] who presented a self-consistent theory that led to the DIA. The importance of this for Kraichnan was that it also led to a model representation of theContinue reading The non-Markovian nature of turbulence 8: Almost-Markovian models and theories

The non-Markovian nature of turbulence 7: non-Markovian closures and the LET theory in particular. We can sum up the situation regarding the failure of the pioneering closures as follows. Their form of the transfer spectrum , with its division into input and output parts, with the latter being proportional to the amount of energy inContinue reading The non-Markovian nature of turbulence 7: non-Markovian closures and the LET theory in particular.

The non-Markovian nature of turbulence 6: the assumptions that led to the Edwards theory being Markovian. Turbulence theories are usually referred to by acronyms e.g. DIA, SCF, ALHDIA, \dots, and so on. Here SCF is Herring’s theory and, to avoid confusion, Herring and Kraichnan referred to the Edwards SCF as EDW [1]. Later on, whenContinue reading The non-Markovian nature of turbulence 6: the assumptions that led to the Edwards theory being Markovian.

The non-Markovian nature of turbulence 5: implications for Kraichnan’s DIA and Herring’s SCF Having shown that the Edwards theory is Markovian, our present task is to show that Kraichnan’s DIA and Herring’s SCF are closely related to the Edwards theory. However, we should first note that, in the case of the DIA, one can seeContinue reading The non-Markovian nature of turbulence 5: implications for Kraichnan’s DIA and Herring’s SCF

The non-Markovian nature of turbulence 4: the Edwards energy balance as a Master Equation. In this post we will rely on the book [1] for background material and further details. We begin with the well-known Lin equation for the energy spectrum in freely decaying turbulence, thus: (1)   where is the transfer spectum: see [1]Continue reading The non-Markovian nature of turbulence 4: the Edwards energy balance as a Master Equation.

The non-Markovian nature of turbulence 3: the Master Equation. In the previous post we established that the ‘loss’ term in the transport equation depends on the number of particles in the state currently being studied. This followed straightforwardly from our consideration of hard-sphere collisions. Now we want to establish that this is a general consequenceContinue reading The non-Markovian nature of turbulence 3: the Master Equation.

The non-Markovian nature of turbulence 2: The influence of the kinetic equation of statistical physics. The pioneering theories of turbulence which we discussed in the previous post were formulated by theoretical physicists who were undoubtedly influenced by their background in statistical physics. In this post we will look at one particular aspect of this, theContinue reading The non-Markovian nature of turbulence 2: The influence of the kinetic equation of statistical physics.

The non-Markovian nature of turbulence 1: A puzzling aspect of the pioneering two-point closures. When I began my postgraduate research on turbulence in 1966, the field had just gone through a very exciting phase of new developments. But there was a snag. These exciting new theories which seemed so promising were not quite correct. TheyContinue reading The non-Markovian nature of turbulence 1: A puzzling aspect of the pioneering two-point closures.

Work in progress. In my blog of 13 August 2020 I posted a `to-do list’ that dated from November 2009. None of these jobs ever got done, because other jobs cropped up which had greater priority. I’m fairly confident that this won’t happen with my current `to-do list’ as I see all these jobs asContinue reading Work in progress.

Turbulence renormalization and the Euler equation: 5 In the preceding posts we have discussed the fact that the Euler equation can behave like the Navier-Stokes equation (NSE) as a transient for sufficiently short times [1], [2]. It has been found that spectra at lower wavenumbers are very similar to those of turbulence, and there appearsContinue reading Turbulence renormalization and the Euler equation: 5

Turbulence renormalization and the Euler equation: 4 In the previous post we mentioned that Kraichnan’s DIA theory [1] and Wyld’s [2] diagrammatic formalism both depended on the use of an externally applied stirring force to define the response function. This is also true of the later functional formalism of Martin, Siggia and Rose [3]. BothContinue reading Turbulence renormalization and the Euler equation: 4

Turbulence renormalization and the Euler equation: 3 In the previous post we saw that the mean-field and self-consistent assumptions/approximations are separate operations, although often referred to in the literature as if they could be used interchangeably. We also saw that the screened potential in a cloud of electrons could be interpreted as a Coulomb potentialContinue reading Turbulence renormalization and the Euler equation: 3

Turbulence renormalization and the Euler equation: 2 In the early 1970s, my former PhD supervisor Sam Edwards asked me to be the external examiner for one of his current students. It was only a few years since I had been on the receiving end of this process so naturally I approached the task in aContinue reading Turbulence renormalization and the Euler equation: 2

Turbulence renormalization and the Euler equation: 1 The term renormalization comes from particle physics but the concept originated in condensed matter physics and indeed could be said to originate in the study of turbulence in the late 19th and early 20th centuries. It has become a dominant theme in statistical theories of turbulence since theContinue reading Turbulence renormalization and the Euler equation: 1

Alternative formulations for statistical theories: 2. Carrying on from my previous post, I thought it would be interesting to look at the effect of the different formulations on statistical closure theories. In order to keep matters as simple as possible, I am restricting my attention to single-time theories and their forms for the transfer spectrumContinue reading Alternative formulations for statistical theories: 2.

Alternative formulations for statistical theories: 1. In the spectral representation of turbulence it is well known that interactions in wavenumber space involve triads of wave vectors, with the members of each triad combining to form a triangle. It is perhaps less well known that the way in which this constraint is handled can have practicalContinue reading Alternative formulations for statistical theories: 1.

From to . From time to time, I have remarked that all the controversy about Kolmogorov’s (1941) theory arises because his real-space derivation is rather imprecise. A rigorous derivation relies on a wavenumber-space treatment; and then, in principle, one could derive the two-thirds law for the second-order structure function from Fourier transformation of the minusContinue reading From minus five thirds in wavenumber to plus two-thirds in real space.

Compatibility of temporal spectra with Kolmogorov (1941) and with random sweeping. I previously wrote about temporal frequency spectra, in the context of the Taylor hypothesis and a uniform convection velocity of , in my post of 25 February 2021. At the time, I said that I would return to the more difficult question of whatContinue reading Compatibility of temporal spectra with Kolmogorov (1941) and with random sweeping

The question of notation. In recent years, when I specify the velocity field for turbulence, I invariably add a word of explanation about my use of Greek letters for Cartesian tensor indices. I point out that these Greek indices should not be confused with those used in four-space for four-dimensional tensors, as encountered in Einstein’sContinue reading The question of notation.

The last post of the weekly Blogs … but intermittently hereafter! I posted the first of these blogs on 6 February 2020, just as the pandemic was getting under way. Since then (and slightly to my surprise) I have managed to post a blog every week. In the case of holidays, I wrote an appropriateContinue reading The last post of the weekly Blogs … but intermittently hereafter!

From ‘wavenumber murder’ to wavenumber muddle? In my post of 20 February 2020, I told of the referee who described my use of Fourier transformation as ‘the usual wavenumber murder’. I speculated that the situation had improved over the years due to the use of pseudo-spectral methods in direct numerical simulation, although I was ableContinue reading From ‘wavenumber murder’ to wavenumber muddle?

Is the concept of an energy cascade a help or a hindrance? In his 1947 exegesis of Kolmogorov’s theory, Batchelor [1] explained the underlying idea of a transfer of energy from large eddies to progressively smaller eddies, until the (local) Reynolds number becomes too small for new eddies to form. He pointed out that theContinue reading Is the concept of an energy cascade a help or a hindrance?

Chaos and Complexity. In the previous blog we discussed the growth of interest in deterministic chaos in low-dimensional dynamical systems, and the way in which it impinged on turbulence theory. Altogether, it seemed like a paradigm shift; in that we learned that only quantum effects were truly random, and that all classical effects were deterministic.Continue reading Chaos and Complexity.

Fashions in turbulence theory. Back in the 1980s, fractals were all the rage. They were going to solve everything, and turbulence was no exception. The only thing that I can remember from their use in microscopic physics was that the idea was applied to the problem of diffusion-limited aggregation, and I’ve no memory of howContinue reading Fashions in turbulence theory.

Summary of the Kolmogorov-Obukhov (1941) theory: overview. In the last three posts we have summarised various aspects of the Kolmogorov-Obukhov (1941) theory. When considering this theory, the following things need to be borne in mind. [a] Whether we are working in -space or -space matters. See my posts of 8 April and 15 April 2021Continue reading Summary of the Kolmogorov-Obukhov (1941) theory: overview.

Summary of the Kolmogorov-Obukhov (1941) theory. Part 3: Obukhov’s theory in k-space. Obukhov is regarded as having begun the treatment of the problem in wavenumber space. In [1] he referred to an earlier paper by Kolmogorov for the spectral decomposition of the velocity field in one dimension and pointed out that the three-dimensional case isContinue reading Summary of the Kolmogorov-Obukhov (1941) theory. Part 3: Obukhov’s theory in k-space.

Summary of the Kolmogorov-Obukhov (1941) theory. Part 2: Kolmogorov’s theory in x-space. Kolmogorov worked in -space and his two relevant papers are cited below as [1] (often referred to as K41A) and [2] (K41B). We may make a pointwise summary of this work, along with more recent developments as follows. [a] In K41A, Kolmogorov introducedContinue reading Summary of the Kolmogorov-Obukhov (1941) theory. Part 2: Kolmogorov’s theory in x-space.

Summary of Kolmogorov-Obukhov (1941) theory. Part 1: some preliminaries in -space and -space. Discussions of the Kolmogorov-Obukhov theory often touch on the question: can the two-thirds law; or, alternatively, the minus five-thirds law, be derived from the equations of motion (NSE)? And the answer is almost always: ‘no, they can’t’! Yet virtually every aspect ofContinue reading Summary of Kolmogorov-Obukhov (1941) theory. Part 1: some preliminaries in x-space and k-space.

The importance of terminology: stationarity or equilibrium? When I began my post-graduate research in 1966, I found that I immediately had to get used to a new terminology. For instance, concepts like homogeneity and isotropy were a definite novelty. In physics one takes these for granted and they are never mentioned. Indeed the opposite isContinue reading The importance of terminology: stationarity or equilibrium?

Turbulence in a box. When the turbulence theories of Kraichnan, Edwards, Herring, and so on, began attracting attention in the 1960s, they also attracted attention to the underlying ideas of homogeneity, isotropy, and Fourier analysis of the equations of motion. These must have seemed very exotic notions to the fluid dynamicists and engineers who workedContinue reading Turbulence in a box.

Large-scale resolution and finite-size effects. This post arises out of the one on local isotropy posted on 21 October 2021; and in particular relates to the comment posted by Alex Liberzon on the need to choose the size of volume within which Kolmogorov’s assumptions of localness may hold. In fact, as is so often theContinue reading Large-scale resolution and finite-size effects.

The second-order structure function corrected for systematic error. In last week’s post, we discussed the corrections to the third-order structure function arising from forcing and viscous effects, as established by McComb et al [1]. This week we return to that reference in order to consider the effect of systematic error on the second-order structure function,Continue reading The second-order structure function corrected for systematic error.