The infinite Reynolds number limit: Onsager versus Batchelor: 3 In the preceding two posts, we have pointed out that the final statement by Onsager in his 1949 paper [1] is, in the absence of a proper limiting procedure, only a conjecture; and that the infinite Reynolds number limit, as introduced by Batchelor [2] and extendedContinue reading The infinite Reynolds number limit: Onsager versus Batchelor: 3

The infinite Reynolds number limit: Onsager versus Batchelor: 2 In the preceding post, we argued that the final statement by Onsager in his 1949 paper [1] is, in the absence of a proper limiting procedure, only a conjecture; and that the infinite Reynolds number limit, as introduced by Batchelor [2] and extended by Edwards [3],Continue reading The infinite Reynolds number limit: Onsager versus Batchelor: 2

The infinite Reynolds number limit: Onsager versus Batchelor: 1 A pioneering paper on turbulence by Onsager, which was published in 1949 [1], seems to have had a profound influence on some aspects of the subject in later years. In particular, he put forward the idea that as the turbulence was still dissipative in the limitContinue reading The infinite Reynolds number limit: Onsager versus Batchelor: 1

The role of Gaussians in turbulence studies. The Gaussian, or normal, distribution plays a key part in statistical field theory. This is partly because it is the only functional which can be integrated and partly because Gaussian distributions are frequently encountered in microscopic physics at, or near, thermal equilibrium. The latter is not the caseContinue reading The role of Gaussians in turbulence studies.

Is there actually a single ‘turbulence problem’? When I was preparing last week’s post, I consulted the Saffman lectures in order to find an example of the culture clash between theoretical physics and applied maths. In the process I noticed quite a few points that I felt tempted to write about and in particular thatContinue reading Is there actually a single ‘turbulence problem’?

Here’s to mathematics and may it never be of use to anyone! When I was a student, I read that mathematicians at conference dinners would drink a toast along the lines of the title of this piece. As an idealistic young man, I was quite shocked by this; and thought it very arrogant. Apart fromContinue reading Here’s to mathematics and may it never be of use to anyone!

Operational Large-Eddy Simulation. When I was visiting TU Delft in 1997, I stayed with my wife and daughter in the Hague, where we rented an apartment from one of the professors at Delft. He and his wife occupied the penthouse above us. They had originally bought the second apartment so that their teenage daughters couldContinue reading Operational Large-Eddy Simulation.

Peer Review in Wonderland In 1974 I completed a task which had begun during my PhD days, and found a way of rendering the Edwards statistical closure compatible with the Kolmogorov spectrum. The basic idea was that the entire transfer spectrum acted as a sink of energy at low wavenumbers and a source of energyContinue reading Peer Review in Wonderland

Formulation of Renormalization Group (RG) for turbulence: 2 In last week’s post, we recognised that the basic step of averaging over high-frequency modes was impossible in principle for a classical, deterministic problem such as turbulence. Curiously enough, for many years it has been recognized in the analogous subgrid modelling problem that a conditional average isContinue reading Formulation of Renormalization Group (RG) for turbulence: 2

Formulation of Renormalization Group (RG) for turbulence: 1 In my posts of 30 April and 7 May, I discussed the relevance of field-theoretic methods (and particularly RG) to the Navier-Stokes equation (NSE). Here I want to deal with some specific points and in the process highlight the snags involved in going from microscopic quantum randomnessContinue reading Formulation of Renormalization Group (RG) for turbulence: 1

Reynolds averaging re-formulated. At the beginning of the 1980s I was still involved in experimental work on drag reduction; while, on the theoretical side, I had begun numerical evaluation of the LET theory. One day I went into the lab to help a student who was having problems with his laser anemometer. In those daysContinue reading Reynolds averaging re-formulated.

Turbulent dissipation and other rates of change. When I was working for my PhD with Sam Edwards in the late 1960s, my second supervisor was David Leslie. We would meet up every so often to discuss progress, and I recall that David was invariably exasperated by our concentration on asymptotic behaviour at high wavenumbers. HeContinue reading Turbulent dissipation and other rates of change.

Heuristic is as heuristic does! In the early years of my career, I would sometimes encounter the word `heuristic’ in a mathematical theory. I understood that authors, when using this word, were in effect crossing their fingers behind their back and indicating that their work might not be entirely rigorous. But I found myself quiteContinue reading Heuristic is as heuristic does!

Peer review: some further thoughts. Vacation post No 4. I will be out of the virtual office until Monday 31 August. Peer review continues to cause concern, with widespread perceptions of unfairness. Although most of what I have noticed recently seems to be in the medical/public health communities, where one major gripe appears to beContinue reading Peer review: some further thoughts.

Is there any place for personal taste in science? Vacation post No 3. I will be out of the virtual office until Monday 31 August. It has long been the case that physicists talk approvingly about a physical theory as being `elegant’ or even `beautiful’. Like so much else, this seems to have become commonplaceContinue reading Is there any place for personal taste in science?

My list of jobs to do from 17 November 2009. Vacation post No 2. I will be out of the virtual office until Monday 31 August. Recently I was tidying up some papers and I came across this list from 2009. At that time I had just entered my fourth year of retirement (now inContinue reading My list of jobs to do from 17 November 2009.

Can mathematicians solve problems in physics? Vacation post No 1. I will be out of the virtual office until Monday 31 August. When I used to lecture final-year undergraduates in mathematical physics, there were often quite a few mathematicians attending and I would sometimes tease them by pointing out that mathematicians try to prove theContinue reading Can mathematicians solve problems in physics?

Should turbulence researchers dare to be dull? I recently read a book review in The Times which was headed `Scientists must dare to be dull’. Well, that was attention grabbing, because most of the general population probably think that we already are. The author of the review then went further in a subheading: `We shouldContinue reading Should turbulence researchers dare to be dull?

The modified Lin equation. In my post of 27 February I discussed the importance of being aware of the full form of the Lin equation as this reveals the existence of a cascade in wavenumber space. In this post I want to take this a bit further, using my resolution of the scale-invariance paradox [1].Continue reading The modified Lin equation.

Local Energy Transfer (LET): a curate’s egg theory? The LET theory began well as a modification to the Edwards theory [1,2], which was a single-time theory, and then underwent a rather heuristic extension to two-time form to become in effect a modification of Kraichnan’s DIA theory [3]. It was successfully computed for freely decaying turbulenceContinue reading Local Energy Transfer (LET): a curate’s egg theory?

Peer review: the role of the editor. In 1985 I published a paper in JFM on laser-doppler measurements in drag-reducing fibre suspensions. This was the only paper on experimental work that I published in that journal and the refereeing process was not without interest. There was the usual iteration process and Referees A and BContinue reading Peer review: the role of the editor.

Further thoughts on free decay of isotropic turbulence. In the previous post I discussed the initial value problem posed by the free decay of the energy in isotropic turbulence, along with things that we ought to bear in mind when considering its experimental or DNS realisations. We should also mention the more general problem ofContinue reading Further thoughts on free decay of isotropic turbulence.

Free decay of isotropic turbulence as a test problem. When I began my postgraduate research in 1966, I quickly decided that there was one problem that I would never work on. That was the free decay of the kinetic energy of turbulence from some initial value. Although, as the subject of my postgraduate research wasContinue reading Free decay of isotropic turbulence as a test problem.

Stationary isotropic turbulence as a test problem. When I was first publishing, in the early 1970s, referees would often say something like `the author uses the turbulence in a box concept’ before going on to reveal a degree of incomprehension about what I might be doing, let alone what I actually was doing. A fewContinue reading Stationary isotropic turbulence as a test problem.

Asymptotic behaviour of the Direct Interaction Approximation. As mentioned previously, Kraichnan’s asymptotic solution of the DIA, for high Reynolds numbers and large wavenumbers, did not agree with the observed asymptotic behaviour of turbulence. His expression for the spectrum was , where is the root-mean-square velocity and is a constant. In 1964 (see [1] for theContinue reading Asymptotic behaviour of the Direct Interaction Approximation.

A brief summary of two-point renormalized perturbation theories. In the previous post we discussed the introduction of Kraichnan’s DIA, based on a combination of a mean-field assumption and a new kind of perturbation theory, and how it was supported by Wyld’s formalism, itself based on a conventional perturbation expansion of the NSE. This was notContinue reading A brief summary of two-point renormalized perturbation theories.

Theories versus formalisms. After the catastrophe of quasi-normality, the modern era of turbulence theory began in the late 1950s, with a series of papers by Kraichnan in the Physical Review, culminating in the formal presentation of his direct-interaction approximation (DIA) in JFM in 1959 [1]. The next step was the paper by Wyld [2], whichContinue reading Theories versus formalisms

Marseille (1961): a paradoxical outcome. When I was first at Edinburgh, in the early 1970s, a number of samizdat-like documents, of entirely mysterious provenance, were being passed around. One that came my way, was a paper by Lumley which contained some rather interesting ideas for treating the problem of turbulent diffusion. I expect that itContinue reading Marseille (1961): a paradoxical outcome.

Which Navier-Stokes equation do you use? In the first half of 1999, a major turbulence programme was held at the Isaac Newton Institute in Cambridge. On those days when there were no lectures or seminars during the morning, a large group of us used to meet for coffee and discussions. In my view these discussionsContinue reading Which Navier-Stokes equation do you use?

Turbulence as a quantum field theory: 2 In the previous post, we specified the problem of stationary, isotropic turbulence, and discussed the nature of turbulence phenomenology, insofar as it is relevant to taking our first steps in a field-theoretic approach. Now we will extend that specification in order to allow us to concentrate on renormalizationContinue reading Turbulence as a quantum field theory: 2

Turbulence as a quantum field theory: 1 In the late 1940s, the remarkable success of arbitrary renormalization procedures in quantum electrodynamics in giving an accurate picture of the interaction between matter and the electromagnetic field, led on to the development of quantum field theory. The basis of the method was perturbation theory, which is essentiallyContinue reading Turbulence as a quantum field theory: 1

Is there an alternative infinite Reynolds number limit? I first became conscious of the term dissipation anomaly in January 2006, at a summer school, where the lecturer preceding me laid heavy emphasis on the term, drawing an analogy with the concept of anomaly in quantum field theory, as he did so. It seemed that thisContinue reading Is there an alternative infinite Reynolds number limit?

What relevance has theoretical physics to turbulence theory? The question is of course rhetorical, as I intend to answer it. But I have to pause on the thought that it is also unsatisfactory in some respects. So why ask it then? Well my reply to that is that various turbulence researchers have over the yearsContinue reading What relevance has theoretical physics to turbulence theory?

The Kolmogorov `5/3’ spectrum and why it is important An intriguing aspect of the Kolmogorov inertial range spectrum is that it was not actually derived by Kolmogorov. This fact was unknown to me when, as a new postgraduate student, I first encountered the `5/3’ spectrum in 1966. At that time, all work on the statisticalContinue reading The Kolmogorov `5/3’ spectrum and why it is important

Scientific discussion in the turbulence community. Shortly after I retired, I began a two-year travel fellowship, with the hope of having interesting discussions on various aspects of turbulence. I’m sure that I had many interesting discussions, particularly in trying out some new and half-baked ideas that I had about that time, but what really sticksContinue reading Scientific discussion in the turbulence community.

Intermittency corrections (sic) and the perversity of group think. In The Times of 11 January this year, there was a report by their Science Editor which had the title Expert’s lonely 30-year quest for Alzheimer’s cure offers new hope. Senile dementia is the curse of the age (even if temporarily eclipsed by the Corona virus)Continue reading Intermittency corrections (sic) and the perversity of group think

Bad proofs and `curate’s egg’ theories. At about the time I took up my appointment at Edinburgh, I heard about a pure mathematician who wanted to be remembered for his bad proofs. Some years later I read his obituary in The Times and this fact was mentioned again. I had thought that I had keptContinue reading Bad proofs and `curate’s egg’ theories

The infinite-Reynolds number limit: a first look. I notice that MSRI at Berkeley have a programme next year on math problems in fluid dynamics. The primary component seems to be an examination of the relationship between the Euler and Navier-Stokes equations, `in the zero-viscosity limit’. The latter is, of course, the same as the limitContinue reading The infinite-Reynolds number limit: a first look

A first look at Kolmogorov (1941) Around the turn of the new millennium, I attended the PhD oral of one of my own students for the last time as Internal Examiner. After that the regulations were changed; or perhaps it was frowned on for the supervisor to also be the Internal. Later still I stoppedContinue reading A first look at Kolmogorov (1941)

The energy balance equations: or what’s in a name?  Over the last few years I have noticed that the Karman-Howarth equation is sometimes referred to nowadays as the `scale-by-scale energy budget equation’. Having thought about it carefully, I have concluded that I understand that description; but I think the mere fact that one has toContinue reading The energy balance equation: or what’s in a name?

Wavenumber Murder and other grisly tales. When I was first at Edinburgh, I worked on developing a theory of turbulent drag reduction by additives. But, instead of considering polymers, I studied the much less well-known phenomenon involving macroscopic fibres. This was because it seemed to me that the fibres were probably of a length whichContinue reading Wavenumber Murder and other grisly tales

HIT: Do three-letter acronyms always win out? In 1997, I visited Delft Technical University and while I was there gave a course of lectures on turbulence theory. During these lectures, I mentioned that nowadays people seemed to refer to homogeneous, isotropic turbulence; whereas, when I started out, it was commonplace to simply say isotropic turbulence.Continue reading HIT: Do three-letter acronyms always win out?

The First Post Many years ago, early in my career, I learned the hard way that every paper submitted for publication should be ruthlessly pared down to consist solely of factual material and fully justified statements. Any personal opinions, speculations, whimsical thoughts, comments or suchlike, should be eliminated; as, in the words of the poetContinue reading The First Post