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  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 . 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  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 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 , . 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  and Wyld’s  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 . 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  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  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  (often referred to as K41A) and  (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 . 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.
Viscous and forcing corrections to Kolmogorov’s ‘4/5’ law. The Kolmogorov `4/5′ law for the third-order structure function is widely regarded as the one exact result in turbulence theory. And so it should be: it has a straightforward derivation from the Karman-Howarth equation (KHE), which is an exact energy balance derived from the Navier-Stokes equation. Nevertheless,Continue reading Viscous and forcing corrections to Kolmogorov’s ‘4/5’ law.
Local isotropy, local homogeneity and local stationarity. In last week’s post I reiterated the argument that the existence of isotropy implies homogeneity. However, Alex Liberzon commented that there could be inhomogeneous flows that exhibited isotropy on scales that were small compared to the overall size of the flow. This comment has the great merit ofContinue reading Local isotropy, local homogeneity and local stationarity.
Is isotropy the same as spherical symmetry? To which you might be tempted to reply: ‘Who ever thought it was?’ Well, I don’t know for sure, but I’ve developed a suspicion that such a misconception may underpin the belief that it is necessary to specify that turbulence is homogeneous as well as isotropic. When IContinue reading Is isotropy the same as spherical symmetry?
Various kinds of turbulent dissipation? The current interest in Onsager’s conjecture (see my blog of 23 September 2021) has sparked my interest in the nature of turbulent dissipation. Essentially a fluid only moves because a force acts on it and does work to maintain it in motion. The effect of viscosity is to convert thisContinue reading Various kinds of turbulent dissipation?
Superstitions in turbulence theory 2: that intermittency destroys scale-invariance! At the moment I am busy revising a paper (see  below) in order to meet the comments of the referees. As is so often the case, Referee 1 is supportive and Referee 2 is hostile. Naturally, Referee 2 writes at great length, so it isContinue reading Superstitions in turbulence theory 2: that intermittency destroys scale-invariance!
Superstitions in turbulence theory 1: the infinite Re limit of the Navier-Stokes equation is the Euler equation! I recently posted blogs about the Onsager conjecture ; the need to take limits properly (Onsager didn’t!); and the programme at MSRI Berkeley, which referred to the Euler equation as the infinite Reynolds number limit, in a seriesContinue reading Superstitions in turbulence theory 1: the infinite Re limit of the Navier-Stokes equation is the Euler equation!
Peer review: the role of the referee. In earlier years I used to get the occasional phone call from George Batchelor, at that time the editor of Journal of Fluid Mechanics, asking for suggestions of new referees on the statistical theory of turbulence. To avoid confusion I should point out that by this I meanContinue reading Peer review: the role of the referee.
Peer review: The role of the author. I have previously posted on the role of the editor (see my blog on 09/07/2020) and had intended to go on to discuss the role of the referee. However, before doing that it occurred to me that it might be helpful to first discuss the role of theContinue reading Peer review: The role of the author.
The exactness of mathematics and the inexactness of physics. This post was prompted by something that came up in a previous one (i.e. see my blog on 12 August 2021), where I commented on the fact that an anonymous referee did not know what to make of an asymptotic curve. The obvious conclusion from thisContinue reading The exactness of mathematics and the inexactness of physics.
Nightmare on Buccleuch Street. Staycation post No 4. I will be out of the virtual office until 30 August. I haven’t been into the university since the pandemic began but recently I dreamt that I was in the university library, in the section where magazines and journals are kept. In this dream, I was sittingContinue reading Nightmare on Buccleuch Street.
Why am I so concerned about Onsager’s so-called conjecture? Staycation post No 3. I will be out of the virtual office until 30 August. In recent years, Onsager’s (1949) paper on turbulence has been rediscovered and its eccentricities promoted enthusiastically, despite the fact that they are at odds with much well-established research in turbulence, beginningContinue reading Why am I so concerned about Onsager’s so-called conjecture?
That’s the giddy limit! Staycation post No 2. I will be out of the virtual office until 30 August. The expression above was still in use when I was young, and vestiges of its use linger on even today. It referred, often jocularly, to any behaviour which was deemed unacceptable. Why giddy? I’m afraid thatContinue reading That’s the giddy limit!
When is a conjecture not a conjecture? Staycation post No 1. I will be out of the virtual office until 30 August. That sounds like the sort of riddle I used to hear in childhood. For instance, when is a door not a door? The answer was: when it’s ajar!  Well, at least weContinue reading When is a conjecture not a conjecture?
How do we identify the presence of turbulence? In 1971, when I began as a lecturer in Engineering Science at Edinburgh, my degree in physics provided me with no basis for teaching fluid dynamics. I had met the concept of the convective derivative in statistical mechanics, as part of the derivation of the Liouville equation,Continue reading How do we identify the presence of turbulence?
Are Kraichnan’s papers difficult to read? Part 2: The DIA. In 2008, or thereabouts, I took part in a small conference at the Isaac Newton Institute and gave a talk on the LET theory, its relationship to DIA, and how both theories could be understood in terms of their relationship to Quasi-normality. During my talk,Continue reading Are Kraichnan’s papers difficult to read? Part 2: The DIA.