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The non-Markovian nature of turbulence 8: Almost-Markovian models and 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.

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.

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

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.

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.

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 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.

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.

Turbulence renormalization and the Euler equation: 5

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

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

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

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

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.

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.

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 minus five thirds in wavenumber to plus two-thirds in real space.

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

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 last post of the weekly Blogs … but intermittently hereafter!

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?

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?

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?

Summary of the Kolmogorov-Obukhov (1941) theory: overview.

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.

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.

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 x-space and k-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?

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?

Large-scale resolution and finite-size effects.

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.

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.

Viscous and forcing corrections to Kolmogorov’s ‘4/5’ law.

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.

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?

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?

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!

Superstitions in turbulence theory 2: that intermittency destroys scale-invariance! At the moment I am busy revising a paper (see [1] 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!

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 [1]; 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.

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.

The exactness of mathematics and the inexactness of physics.

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.

Why am I so concerned about Onsager’s so-called conjecture?

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?

How do we identify the presence of turbulence?

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.

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.

Are Kraichnan’s papers difficult to read? Part 1: Galilean Invariance

Are Kraichnan’s papers difficult to read? Part 1: Galilean Invariance. When I was first at Edinburgh, in the early 1970s, I gave some informal talks on turbulence theory. One of my colleagues became sufficiently interested to start doing some reading on the subject. Shortly afterwards he came up to me at coffee time and said.Continue reading Are Kraichnan’s papers difficult to read? Part 1: Galilean Invariance

The Kolmogorov (1962) theory: a critical review Part 2

The Kolmogorov (1962) theory: a critical review Part 2 Following on to last week’s post, I would like to make a point that, so far as I know, has not previously been made in the literature of the subject. This is, that the energy spectrum is (in the sense of thermodynamics) an intensive quantity. ThereforeContinue reading The Kolmogorov (1962) theory: a critical review Part 2

The Kolmogorov (1962) theory: a critical review Part 1

The Kolmogorov (1962) theory: a critical review Part 1 As is well known, Kolmogorov interpreted Landau’s criticism as referring to the small-scale intermittency of the instantaneous dissipation rate. His response was to adopt Obukhov’s proposal to introduce a new dissipation rate which had been averaged over a sphere of radius , and which may beContinue reading The Kolmogorov (1962) theory: a critical review Part 1

The Landau criticism of K41 and problems with averages

The Landau criticism of K41 and problems with averages The idea that K41 had some problem with the way that averages were taken has its origins in the famous footnote on page 126 of the book by Landau and Lifshitz [1]. This footnote is notoriously difficult to understand; not least because it is meaningless unlessContinue reading The Landau criticism of K41 and problems with averages