Home › Azimuth Project › › Azimuth Blog

It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.4K
- Chat 505
- Study Groups 21
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 2
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- Baez ACT 2019: Online Course 339
- Baez ACT 2019: Lectures 79
- Baez ACT 2019: Exercises 149
- Baez ACT 2019: Chat 50
- UCR ACT Seminar 4
- General 75
- Azimuth Code Project 111
- Statistical methods 4
- Drafts 10
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 148
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 719

Options

Stub for a blog post about turbulence

as a place to collect some material.

Turbulence is an important phenomenon in the oceans and the atmosphere, but I don't know enough about that to write about it. Instead this will become a simple introduction to some basic aspects, mostly links to interesting literature.

## Comments

Great!

Since turbulence is a mystery, I'm not sure

anyoneknows enough about it. But I would enjoy learning more about it and writing about it. For starters, it would be fun to explain the Navier-Stokes equations: what they say, and maybe a tiny bit about why they're hard.`Great! Since turbulence is a mystery, I'm not sure _anyone_ knows enough about it. But I would enjoy learning more about it and writing about it. For starters, it would be fun to explain the Navier-Stokes equations: what they say, and maybe a tiny bit about why they're hard.`

We can write a blog post as a group, of course. If you like to, free feel to chime in (everybody).

The storyline I had in mind for this post is:

Turbulence = Eddies,

Eddies = mass transfer -> important for atmosphere and oceans, the planet would be different without it,

eddies + viscosity = dissipation, with the nice formula from enstrophy.

`We can write a blog post as a group, of course. If you like to, free feel to chime in (everybody). The storyline I had in mind for this post is: Turbulence = Eddies, Eddies = mass transfer -> important for atmosphere and oceans, the planet would be different without it, eddies + viscosity = dissipation, with the nice formula from [[enstrophy]].`

I have finished a first draft, but I think I'll have to rework some of the pictures, because the labels are not readable in the standard format for the blog.

`I have finished a first draft, but I think I'll have to rework some of the pictures, because the labels are not readable in the standard format for the blog.`

Hi Tim,

some really minor comments

this is the first place where "eddy" appears. According to a dictionary it's a circular current, but is it a very well known word outside of physics? Perhaps you could add a sentence for those who don't know it.

Besides, this reminds me of:

Another comment:

Do you recognize

her? ;)Again besides, about:

in your formula it's $\mu \cdot \epsilon$ so for high viscosity eddies are suppressed. And I think (but I'm not completely sure) that for higher Reynolds numbers, the lower length scale down towards the flow still exhibits turbulence becomes smaller and smaller.

`Hi Tim, some really minor comments > As a first guess we could say that a characteristic property of turbulent flow is the presence of eddies. this is the first place where "eddy" appears. According to a dictionary it's a circular current, but is it a very well known word outside of physics? Perhaps you could add a sentence for those who don't know it. Besides, this reminds me of: > So I wrote stuff like, “Aeronautical science is important in the analysis of the eddies, vortices, and whirlpools formed in the atmosphere behind the aircraft...” — I knew that eddies, vortices, and whirlpools are the same thing, but mentioning them three different ways sounds better! That was the only thing I would not have ordinarily done on the test. The teacher who corrected my examination must have been impressed by eddies, vortices, and whirlpools, because I got a 91 on the exam — while my literary friends, who chose topics the English teachers could more easily take issue with, both got 88. Another comment: > Do you recognize it? Do you recognize **her**? ;) Again besides, about: > Since eddies maximize dissipation, natural fluid flows should somehow tend towards the production of eddies. in your formula it's $\mu \cdot \epsilon$ so for high viscosity eddies are suppressed. And I think (but I'm not completely sure) that for higher Reynolds numbers, the lower length scale down towards the flow still exhibits turbulence becomes smaller and smaller.`

Ah, yes, Mister Feynman and his disdain for the fine arts (for those who don't know: Frederik was citing from the autobiography of Richard Feynman, if memory serves: "Surely you're joking Mr. Feynman!").

I'm not sure if I can address a hurricane as "her" in English. I did in the first version and "corrected" it later.

Ugh, the decrease of kinetic energy is proportional to viscosity, so the higher the viscosity, the faster the decay of kinetic energy. If a fluid flow "wants" to dissipate kinetic energy fast, it needs to increase its viscosity or it needs to create eddies. Is some part of this message obscured in the text?

`Ah, yes, Mister Feynman and his disdain for the fine arts (for those who don't know: Frederik was citing from the autobiography of Richard Feynman, if memory serves: "Surely you're joking Mr. Feynman!"). I'm not sure if I can address a hurricane as "her" in English. I did in the first version and "corrected" it later. <blockquote> <p> in your formula it's μ⋅ϵ so for high viscosity eddies are suppressed </p> </blockquote> Ugh, the decrease of kinetic energy is proportional to viscosity, so the higher the viscosity, the faster the decay of kinetic energy. If a fluid flow "wants" to dissipate kinetic energy fast, it needs to increase its viscosity or it needs to create eddies. Is some part of this message obscured in the text?`

I thought Frederik's "her" referred to the English teacher, not the hurricane. Maybe not.

Anyway, the United States has ended the sexist practice of naming hurricanes only after women; there are now both male and female hurricanes. You can already see what the hurricanes in 2016 will be called.

I'll have to read Tim's new post!

`> I'm not sure if I can address a hurricane as "her" in English. I thought Frederik's "her" referred to the English teacher, not the hurricane. Maybe not. Anyway, the United States has ended the sexist practice of naming hurricanes only after women; there are now both male and female hurricanes. You can [already see what the hurricanes in 2016 will be called](http://www.nhc.noaa.gov/aboutnames.shtml). I'll have to read Tim's new post!`

When you read it you'll see that both the post and Frederik are referring to the hurricane "Katrina", and to the question if it should be "it, the hurricane" or "her, the hurricane Katrina".

All hurricanes are named after women? Then the movie "Abyss" by James Cameron is not realistic. The hurricane that causes the deep core desaster was named Bob or something, with people joking that hurricanes

shouldalwyays be named after women.`When you read it you'll see that both the post and Frederik are referring to the hurricane "Katrina", and to the question if it should be "it, the hurricane" or "her, the hurricane Katrina". All hurricanes are named after women? Then the movie "Abyss" by James Cameron is not realistic. The hurricane that causes the deep core desaster was named Bob or something, with people joking that hurricanes <i>should</i> alwyays be named after women.`

What I meant was that for high viscosity, the tendency to create eddies would be less strong ("suppressed") than for low viscosity. Actually, other things being equal, I believe this is the case: for high viscosity, there is not more decay of kinetic energy, there are less eddies (I think).

I am not an expert on this but I thought that the onset of turbulence is relatively well captured by the Reynolds number, which, other things being equal (length scales and velocities), is governed by the viscosity. If not, it's not really your message that's obscure, but rather the introductory textbook from which I orginally learned the basics of fluid mechanics.

Btw, is there always the tendency to "want to" decay kinetic energy into heat, or is it rather connected to differences, e.g. turbulence smoothens the kinetic energy profile (among others)? E.g. for the Kelvin-Helmholtz instability, I thought the eddies are there because there's a difference in kinetic energy between the two fluid flows and the turbulence "helps" to mix the velocity profile.

Minor comment: I was citing from "What do you care other people think". Anyway, this was the first place where I read that eddies, whirlpools and vortices are really words for the same phenomenon. Before that, for me eddy vaguely meant "something related to turbulence" as I was too lazy to look it up in a dictionary (I also thought it was a word like "action" which has a physics meaning different from the everyday sense). So this is why I think one line that explains the word eddy wouldn't be redundant for some readers. [later edit: I've seen you've clarified it]

I was referring to the hurricane (Katrina), as Tim deduced. Perhaps it's possible to use "her" in Dutch and German, but not in English. Actually I'm not sure about either supposition.

`> Is some part of this message obscured in the text? What I meant was that for high viscosity, the tendency to create eddies would be less strong ("suppressed") than for low viscosity. Actually, other things being equal, I believe this is the case: for high viscosity, there is not more decay of kinetic energy, there are less eddies (I think). I am not an expert on this but I thought that the onset of turbulence is relatively well captured by the Reynolds number, which, other things being equal (length scales and velocities), is governed by the viscosity. If not, it's not really your message that's obscure, but rather the introductory textbook from which I orginally learned the basics of fluid mechanics. Btw, is there always the tendency to "want to" decay kinetic energy into heat, or is it rather connected to differences, e.g. turbulence smoothens the kinetic energy profile (among others)? E.g. for the Kelvin-Helmholtz instability, I thought the eddies are there because there's a difference in kinetic energy between the two fluid flows and the turbulence "helps" to mix the velocity profile. Minor comment: I was citing from "What do you care other people think". Anyway, this was the first place where I read that eddies, whirlpools and vortices are really words for the same phenomenon. Before that, for me eddy vaguely meant "something related to turbulence" as I was too lazy to look it up in a dictionary (I also thought it was a word like "action" which has a physics meaning different from the everyday sense). So this is why I think one line that explains the word eddy wouldn't be redundant for some readers. [later edit: I've seen you've clarified it] > I thought Frederik's "her" referred to I was referring to the hurricane (Katrina), as Tim deduced. Perhaps it's possible to use "her" in Dutch and German, but not in English. Actually I'm not sure about either supposition.`

John wrote:

There are two posts that are in "first draft" status, Eddy Who? and A Quantum of Warmth.

`John wrote: <blockquote> <p> I'll have to read Tim's new post! </p> </blockquote> There are two posts that are in "first draft" status, [[Eddy Who?]] and [[A Quantum of Warmth]].`

Frederik wrote:

You are right, of course. I was thinking about a fluid with fixed viscosity, that is driven by heat transfer, like the atmosphere. Atmosphere and oceans transport heat from the equator to the poles. If such a flow fulfills some maximum entropy production, it should tend to produce eddies, or at least I think so. But I haven't found any results into that direction.

`Frederik wrote: <blockquote> <p> What I meant was that for high viscosity, the tendency to create eddies would be less strong ("suppressed") than for low viscosity. </p> </blockquote> You are right, of course. I was thinking about a fluid with fixed viscosity, that is driven by heat transfer, like the atmosphere. Atmosphere and oceans transport heat from the equator to the poles. If such a flow fulfills some maximum entropy production, it should tend to produce eddies, or at least I think so. But I haven't found any results into that direction.`

Do I understand correctly that the eddies you're referring to here are e.g. cyclones? It sounds reasonable that these redistribute energy (well, at least they seem to collect some of it at the tropics and deposit it at higher latitudes)

I am asking this because it seems we have eddies in different spatial directions. Can the Hadley cells be considered as eddies on a very large scale (for a fixed longitude), which distribute the heat from the tropics towards the poles?

So superposed on that there are other large (but smaller) scale eddies like cyclones, which seem to have (spherical) r fixed instead of longitude. But maybe I'm mistaken here, because there are updrafts and downdrafts, so the eddy spiralling anticlockwise is accompanied by an eddy which is reminiscent of a Hadley cell, only on a smaller scale. Is it correct that the westerlies and trade winds are also connected to the Hadley cells? In that case there would be a lot of similarity.

If necessary, please correct.

`> Atmosphere and oceans transport heat from the equator to the poles. If such a flow fulfills some maximum entropy production, it should tend to produce eddies, or at least I think so Do I understand correctly that the eddies you're referring to here are e.g. cyclones? It sounds reasonable that these redistribute energy (well, at least they seem to collect some of it at the tropics and deposit it at higher latitudes) I am asking this because it seems we have eddies in different spatial directions. Can the [Hadley cells](http://en.wikipedia.org/wiki/Hadley_cell) be considered as eddies on a very large scale (for a fixed longitude), which distribute the heat from the tropics towards the poles? So superposed on that there are other large (but smaller) scale eddies like cyclones, which seem to have (spherical) r fixed instead of longitude. But maybe I'm mistaken here, because there are updrafts and downdrafts, so the eddy spiralling anticlockwise is accompanied by an eddy which is reminiscent of a Hadley cell, only on a smaller scale. Is it correct that the westerlies and trade winds are also connected to the Hadley cells? In that case there would be a lot of similarity. If necessary, please correct.`

Yes, but I think we should try to classify them a little bit:

convection in vertical direction, this is a major cause of "weather"

convection on a synoptic scale with mass and heat transport from the equator to the poles (e.g. Hadley cells), this is also a cause of significant weather events on a synoptic scale,

eddies on a microscale that contribute to dissipation, that's what I was talking about.

When people talk about turbulence in the oceans and atmosphere they are talking about eddies and their contribution to heat and mass transfer and mixing, not about dissipation. I'd like to write about that aspect, too, but will have to read a little bit more about it first.

I think that a principle of maximum entropy production could be applicable to the microscale phenomenon of dissipation, but also to the synoptic scale phenomenon of heat transfer by convection from the equator to the poles.

`<blockquote> <p> Can the Hadley cells be considered as eddies on a very large scale (for a fixed longitude), which distribute the heat from the tropics towards the poles? </p> </blockquote> Yes, but I think we should try to classify them a little bit: - convection in vertical direction, this is a major cause of "weather" - convection on a synoptic scale with mass and heat transport from the equator to the poles (e.g. Hadley cells), this is also a cause of significant weather events on a synoptic scale, - eddies on a microscale that contribute to dissipation, that's what I was talking about. When people talk about turbulence in the oceans and atmosphere they are talking about eddies and their contribution to heat and mass transfer and mixing, not about dissipation. I'd like to write about that aspect, too, but will have to read a little bit more about it first. I think that a principle of maximum entropy production could be applicable to the microscale phenomenon of dissipation, but also to the synoptic scale phenomenon of heat transfer by convection from the equator to the poles.`

You're talking about interesting things. On a much less interesting note:

All hurricanes

werenamed after women, starting in 1953 but ending in 1979 when Hurricane Bob came along. I remember thinking it was very strange to have hurricanes named after men! I'm pretty sure it was some feminists who realized it was strange to only name hurricanes after women.Anyway, by now it seems strange to refer to hurricane Katrina as 'her' rather than 'it'. I don't know if people ever did this.`You're talking about interesting things. On a much less interesting note: > All hurricanes are named after women? All hurricanes _were_ named after women, [starting in 1953](http://en.wikipedia.org/wiki/List_of_previous_tropical_cyclone_names#Names_used_between_1950.E2.80.931959) but [ending in 1979](http://en.wikipedia.org/wiki/List_of_previous_tropical_cyclone_names#Names_used_between_1970.E2.80.931979) when Hurricane Bob came along. I remember thinking it was very strange to have hurricanes named after men! I'm pretty sure it was some feminists who realized it was strange to only name hurricanes after women. _Anyway_, by now it seems strange to refer to hurricane Katrina as 'her' rather than 'it'. I don't know if people ever did this.`

Ok, maybe I should explain my personal background:

Both my parents worked as teachers in groundschool. I have a sister that is 3 years older. The first fact has led me to believe that men and women should be equal in the workplace, with equal career choices and possibilities, the second one that women aren't better beings in any sense. I consider myself to be a mild feminist.

I'll leave it as it is now, then.

`<blockquote> <p> I'm pretty sure it was some feminists who realized it was strange to only name hurricanes after women. </p> </blockquote> Ok, maybe I should explain my personal background: Both my parents worked as teachers in groundschool. I have a sister that is 3 years older. The first fact has led me to believe that men and women should be equal in the workplace, with equal career choices and possibilities, the second one that women aren't better beings in any sense. I consider myself to be a mild feminist. <blockquote> <p> Anyway, by now it seems strange to refer to hurricane Katrina as 'her' rather than 'it'. I don't know if people ever did this. </p> </blockquote> I'll leave it as it is now, then.`

By the way, Tim's remark makes me want to emphasize: I wasn't trying to make a political statement by noting that:

I think it's just a fact. Yes, here's some evidence:

`By the way, Tim's remark makes me want to emphasize: I wasn't trying to make a political statement by noting that: > I'm pretty sure it was some feminists who realized it was strange to only name hurricanes after women. I think it's just a fact. Yes, [here's](http://www.msnbc.msn.com/id/9400889/ns/weather-weather_news/t/hurricane-center-may-run-out-names/) some evidence: > The naming of hurricanes has a long and interesting history. For many centuries, hurricanes in the West Indies were named after particular Catholic saint's days on which they occurred. Hurricane "San Felipe" struck Puerto Rico on September 13, 1876. When another hurricane struck Puerto Rico on the same day more than fifty years later, it was christianed "San Felipe the second." > Later, latitude-longitude positions were used, but this method quickly proved cumbersome. > Military weather forecasters began giving women's names to significant storms during WWII, then in 1950 the WMO agreed to an alphabetical naming system, using the military's radio code. The first named Atlantic hurricane was Able in 1950. > Officials soon realized the naming convention would cause problems in the history books if more than one powerful Hurricane Able made landfall. So, in 1953 the organization adopted a rotating series of women's names, planning to retire names of significant storms. > Feminists urged the WMO to add men's names, which was done in 1979. The boy-girl-boy-girl naming convention evolved to include French and Spanish names in the Atlantic system, reflecting the languages of the nations affected by Carribean hurricanes. > The twenty-one names reserved each year (the letters q, u, x, y and z are not used) are recycled every six years, minus those retired (such as Hugo and Andrew and, you can bet, Katrina). When a name is retired, the WMO chooses a new name to replace it.`

On a very slightly more substantial topic: many Americans may be puzzled by the use of $rot \vec{u}$ for what they usually call $curl \vec{u}$. Luckily however you say right away that $rot$ means 'curl'.

Personally, when I teach kids vector calculus, I always write $\nabla \cdot \vec{u}$ and $\nabla \times \vec{u}$ for the divergence and curl of a vector field. Somehow I think it's better for people to start wondering about what $\nabla$ is, rather than using two different strings of letters to denote two different things you do with it.

`On a very slightly more substantial topic: many Americans may be puzzled by the use of $rot \vec{u}$ for what they usually call $curl \vec{u}$. Luckily however you say right away that $rot$ means 'curl'. Personally, when I teach kids vector calculus, I always write $\nabla \cdot \vec{u}$ and $\nabla \times \vec{u}$ for the divergence and curl of a vector field. Somehow I think it's better for people to start wondering about what $\nabla$ is, rather than using two different strings of letters to denote two different things you do with it.`

What do we need to do, to finish this blog post? We shouldn't be so ambitious that it becomes hard to write.

`What do we need to do, to finish this blog post? We shouldn't be so ambitious that it becomes hard to write.`

John asked:

I incorporated your feedback (nabla instead of "rot" and missing information for the references) and put in another graphic to show laminar and turbulent flow with their Reynolds numbers. For people who haven't heard much about turbulence and the Navier-Stokes equations yet, the post in its current form could even turn out to be interesting, so feel free to post it if you agree.

I don't like the nabla notation, but it is international. I thought that "rot" refers to "rotation" in English, but it would seem that it refers to the German word "Rotation" instead :-)

(That's why people from continental Europe should spend some years of their Academic career in one of the English speaking countries, which I haven't.)

`John asked: <blockquote> <p> What do we need to do, to finish this blog post? We shouldn't be so ambitious that it becomes hard to write. </p> </blockquote> I incorporated your feedback (nabla instead of "rot" and missing information for the references) and put in another graphic to show laminar and turbulent flow with their Reynolds numbers. For people who haven't heard much about turbulence and the Navier-Stokes equations yet, the post in its current form could even turn out to be interesting, so feel free to post it if you agree. I don't like the nabla notation, but it is international. I thought that "rot" refers to "rotation" in English, but it would seem that it refers to the German word "Rotation" instead :-) (That's why people from continental Europe should spend some years of their Academic career in one of the English speaking countries, which I haven't.)`

I'm trying to catch up the discussion.

Do you actually mean "some maximum entropy production principle"'? It is quite different from "maximum entropy principle" and indeed might be rather controversial.

Just a remark, you are all seeking for some deep physical reason for turbulence (and I'm fully agree with such an approach). However, as far as I understand, the common approach is to say that the reason for turbulence is that laminar solution for NS becomes unstable for some conditions. And as such conditions occur the laminar flow is switched to turbulent one. Hence there are several mechanisms for the development of instability.

For example (as far as I remember) Kelvin-Helmholtz instability is inviscid mechanism of turbulence development, that is it is modeled with Euler equations rather than NS. There are also mechanisms for the development of the instability that are viscid (that are require NS or Euler equations to be studied). The mechanisms are also classified as 2D or 3D (2D require only 2D NS equations, rather than 3D).

And by the way, I think it is worth mentioning the energy cascade, just one-two sentences to have a better picture of turbulence.

`I'm trying to catch up the discussion. <blockquote> <p> This connection of eddies and dissipation could indicate that there is also a connection of eddies and some maximum entropy principle. </p> </blockquote> Do you actually mean "some maximum entropy production principle"'? It is quite different from "maximum entropy principle" and indeed might be rather controversial. <blockquote> <p> Btw, is there always the tendency to "want to" decay kinetic energy into heat, or is it rather connected to differences, e.g. turbulence smoothens the kinetic energy profile (among others)? E.g. for the Kelvin-Helmholtz instability, I thought the eddies are there because there's a difference in kinetic energy between the two fluid flows and the turbulence "helps" to mix the velocity profile. </p> </blockquote> Just a remark, you are all seeking for some deep physical reason for turbulence (and I'm fully agree with such an approach). However, as far as I understand, the common approach is to say that the reason for turbulence is that laminar solution for NS becomes unstable for some conditions. And as such conditions occur the laminar flow is switched to turbulent one. Hence there are several mechanisms for the development of instability. For example (as far as I remember) Kelvin-Helmholtz instability is inviscid mechanism of turbulence development, that is it is modeled with Euler equations rather than NS. There are also mechanisms for the development of the instability that are viscid (that are require NS or Euler equations to be studied). The mechanisms are also classified as 2D or 3D (2D require only 2D NS equations, rather than 3D). And by the way, I think it is worth mentioning the energy cascade, just one-two sentences to have a better picture of turbulence.`

Big whirls have little whirls that feed on their velocity,

and little whirls have lesser whirls and so on to viscosity

Maybe it will come in hand for the blog.

`For those who love rhymes, the one about energy cascade, by Lewis Fry Richardson himself: Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity Maybe it will come in hand for the blog.`

Hi Grigory, are you new to this forum?

You can cite by simply entering the HTML node "blockquote".

I'm not sure if I use the correct terminology, but I'm looking for some law that applies to non-equillibrium situations. I looked for references for some hours and found two that I wrote down on the page entropy.

Yes, but on the other hand I have read claims that the heat and mass transfer observed on a planetary scale can best be described by some maximum entropy (production?) principle, so if we take as granted that the fluid flow on planets have passed the laminar-turbulent threshhold the question is what can we say about the kind of turbulence with respect to entropy production?

Good point, but I don't want that blog post to become too long, and since it is aimed mostly at mathematicians that don't know anything about the topic at all, I think there is enough material in it. Besides, I don't have a good intuitive understanding of Kolmogorov's concept of energy scales, so I don't feel like explaining it :-)

`Hi Grigory, are you new to this forum? You can cite by simply entering the HTML node "blockquote". <blockquote> <p> Do you actually mean ``some maximum entropy production principle''? It is quite different from ``maximum entropy principle'' and indeed might be rather controversial. </p> </blockquote> I'm not sure if I use the correct terminology, but I'm looking for some law that applies to non-equillibrium situations. I looked for references for some hours and found two that I wrote down on the page [[entropy]]. <blockquote> <p> However, as far as I understand, the common approach is to say that the reason for turbulence is that laminar solution for NS becomes unstable for some conditions. And as such conditions occur the laminar flow is switched to turbulent one. Hence there are several mechanisms for the development of instability. </p> </blockquote> Yes, but on the other hand I have read claims that the heat and mass transfer observed on a planetary scale can best be described by some maximum entropy (production?) principle, so if we take as granted that the fluid flow on planets have passed the laminar-turbulent threshhold the question is what can we say about the kind of turbulence with respect to entropy production? <blockquote> <p> And by the way, I think it is worth mentioning energy cascade, just one-two sentences to have a better picture of turbulence. </p> </blockquote> Good point, but I don't want that blog post to become too long, and since it is aimed mostly at mathematicians that don't know anything about the topic at all, I think there is enough material in it. Besides, I don't have a good intuitive understanding of Kolmogorov's concept of energy scales, so I don't feel like explaining it :-)`

Yes, I am :-)

We are all looking for it, I guess the search started around 70 years ago. But as you know there is no such universally recognized all-embracing theory of non-equilibrium/irreversible processes.

For maximum entropy principle I refer to the wikipedia article. I haven't read it, but it seems to be exactly about it. To be brief maximum entropy principle, I'll state my understanding of it. To apply it, you write down the expression for the entropy of the system as functional of a distribution function (or anything that is believed to describe the system, but to my current knowledge systems are described by some distribution function/density matrix). Then you maximize this functional under certain constraints. That's why it is maximum entropy principle --- the distribution function of the system is the one that gives the maximum for entropy under the constraints.

This principle works great for equilibrium, indeed if you maximize the entropy functional, constraining the average energy, you'll get the Gibbs distribution. However it is not so easy in the non-equilibrium case. Note that those mentioned constraints should be know in advance, actually they are in some sense dual to macroscopic description of the system. Just like the average energy constraint is connected to the temperature or the average number of particles constraint is connected to the chemical potential of the system. So to apply the principle to non-equilibrium states one should actually know how the system is described macroscopically (by what entities). Anyway this principle is almost 60 years old and it didn't help to construct any good turbulence model (or even maybe a bad one).

As for the the maximum entropy production principle --- I've just noticed my huge mistake. Actually I meant the minimum entropy production theorem of Prigogine. However, there is also a maximum entropy production principle, I've heard a little about it before --- two times actually from lectures in my institute (one is studying turbulence, the other is studying stability of flows). The first one was very skeptical about it and said that the principle was wrong in general case, the second one told that it was once used successfully to interpret some results. Unfortunately that's all I know about it. I've searched wikipedia and it gave me the section about both maximum and minimum entropy production principles. Both seem to be controversial.

I would like to write more, but I'm too sleepy, hope I'll continue tomorrow, had to study that maximum production principle controversy. And to speak about the energy cascade. And about the maximum entropy principle I should have also mentioned the book "Statistical Mechanics of Nonequilibrium Processes" by Zubarev, Morozov, Röpke, but enough for now.

`<blockquote><p>Hi Grigory, are you new to this forum?</p></blockquote> Yes, I am :-) <blockquote><p>I'm not sure if I use the correct terminology, but I'm looking for some law that applies to non-equillibrium situations.</p></blockquote> We are all looking for it, I guess the search started around 70 years ago. But as you know there is no such universally recognized all-embracing theory of non-equilibrium/irreversible processes. For maximum entropy principle I refer to the [wikipedia article](http://en.wikipedia.org/wiki/Principle_of_maximum_entropy). I haven't read it, but it seems to be exactly about it. To be brief maximum entropy principle, I'll state my understanding of it. To apply it, you write down the expression for the entropy of the system as functional of a distribution function (or anything that is believed to describe the system, but to my current knowledge systems are described by some distribution function/density matrix). Then you maximize this functional under certain constraints. That's why it is maximum entropy principle --- the distribution function of the system is the one that gives the maximum for entropy under the constraints. This principle works great for equilibrium, indeed if you maximize the entropy functional, constraining the average energy, you'll get the Gibbs distribution. However it is not so easy in the non-equilibrium case. Note that those mentioned constraints should be know in advance, actually they are in some sense dual to macroscopic description of the system. Just like the average energy constraint is connected to the temperature or the average number of particles constraint is connected to the chemical potential of the system. So to apply the principle to non-equilibrium states one should actually know how the system is described macroscopically (by what entities). Anyway this principle is almost 60 years old and it didn't help to construct any good turbulence model (or even maybe a bad one). As for the the maximum entropy production principle --- I've just noticed my huge mistake. Actually I meant the minimum entropy production theorem of Prigogine. However, there is also a maximum entropy production principle, I've heard a little about it before --- two times actually from lectures in my institute (one is studying turbulence, the other is studying stability of flows). The first one was very skeptical about it and said that the principle was wrong in general case, the second one told that it was once used successfully to interpret some results. Unfortunately that's all I know about it. I've searched wikipedia and it gave me [the section](http://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics#Speculated_thermodynamic_extremum_principles_for_energy_dissipation_and_entropy_production) about both maximum and minimum entropy production principles. Both seem to be controversial. I would like to write more, but I'm too sleepy, hope I'll continue tomorrow, had to study that maximum production principle controversy. And to speak about the energy cascade. And about the maximum entropy principle I should have also mentioned the book "Statistical Mechanics of Nonequilibrium Processes" by Zubarev, Morozov, Röpke, but enough for now.`

Li and Vitanyi's Kolmogorov complexity and applications is parly in google books.

hth

`Ed Jaynes' Probability the logic of science is my favourite bayes book, online at bayes.wustl.edu Li and Vitanyi's Kolmogorov complexity and applications is parly in google books. hth`

How do you use TeX in forums?

`How do you use TeX in forums?`

What do you think about splitting the blog post into two parts --- one about some basics of hydrodynamics (Navier-Stokes equations) and the second part solely about turbulence? Just to make a better structured, slightly deeper introduction?

`What do you think about splitting the blog post into two parts --- one about some basics of hydrodynamics (Navier-Stokes equations) and the second part solely about turbulence? Just to make a better structured, slightly deeper introduction?`

The Forum and Wiki use something called iTeX, which isn't the same as TeX (there's a thread where Andrew Stacey explains the differences) but uses rather similar syntax.

If a certain TeX command (keep it simple) doesn't work, you can post a specific question in the category "Technical" where someone may give the answer.

(searching for TeX or iTeX discussions may also help, there are already a few)

`> How do you use TeX in forums? The Forum and Wiki use something called iTeX, which isn't the same as TeX (there's a thread where Andrew Stacey explains the differences) but uses rather similar syntax. If a certain TeX command (keep it simple) doesn't work, you can post a specific question in the category "Technical" where someone may give the answer. (searching for TeX or iTeX discussions may also help, there are already a few)`

Hi, Grigory! The really short answer to

is: you write something like this:

`$\nabla \times (\nabla f) = 0$`

and you get something like this

$\nabla \times (\nabla f) = 0$

iffyou click "Format comments as: Markdown+Itex" below your comment. (If you enable cookies, the Forum should remember your choice here, and you'll never need to click "Format comments as: Markdown+Itex" again.)If you do this a while you'll note some differences between Itex and standard LaTeX. It's probably easiest to ask questions when this causes trouble.

I'm eager for Tim—and everyone else here!—to write blog posts on Azimuth. One key principle is

don't worry too much. It's much better to write an imperfect article than tonotwrite aperfectone. It's quite possible that the idea of splitting the post into two parts and making the article deeper will prevent Tim from writing it at all. If so, that's a bad idea.On the other hand, if Tim wants to do that, it would be great!

And on the third hand, if

youthink you can write a longer and deeper introduction to hydrodynamics and then turbulence, then it's aidea forgreatto write it!you(This is of course an example of the Fundamental Law of Internet Cooperation:

If you want something to be done better, just go ahead and do it better.)We have a standard method of writing blog articles; you can see many examples of how it's done here:

You can copy this method (or ask questions about it).

I will be happy to post Tim's short introduction

anda more detailed 2-part introduction. The more the better!`Hi, Grigory! The really short answer to > How do you use TeX in forums? is: you write something like this: ` $\nabla \times (\nabla f) = 0$ ` and you get something like this $\nabla \times (\nabla f) = 0$ **iff** you click "Format comments as: Markdown+Itex" below your comment. (If you enable cookies, the Forum should remember your choice here, and you'll never need to click "Format comments as: Markdown+Itex" again.) If you do this a while you'll note some differences between Itex and standard LaTeX. It's probably easiest to ask questions when this causes trouble. > What do you think about splitting the blog post into two parts --- one about some basics of hydrodynamics (Navier-Stokes equations) and the second part solely about turbulence? Just to make a better structured, slightly deeper introduction? I'm eager for Tim—and everyone else here!—to write blog posts on Azimuth. One key principle is _don't worry too much_. It's much better to write an imperfect article than to _not_ write a _perfect_ one. It's quite possible that the idea of splitting the post into two parts and making the article deeper will prevent Tim from writing it at all. If so, that's a bad idea. On the other hand, if Tim wants to do that, it would be great! And on the third hand, if _you_ think you can write a longer and deeper introduction to hydrodynamics and then turbulence, then it's a _**great**_ idea for _**you**_ to write it! (This is of course an example of the Fundamental Law of Internet Cooperation: _If you want something to be done better, just go ahead and do it better_.) We have a standard method of writing blog articles; you can see many examples of how it's done here: * [[Blog articles in progress]] You can copy this method (or ask questions about it). I will be happy to post Tim's short introduction _and_ a more detailed 2-part introduction. The more the better!`

By the way, Grigory, it would be great if you could introduce yourself by posting a comment here in the category Chat. We'd be very interested to hear bit about what you've done and what you're interested in doing.

We're getting a lot of new members! Here are some recent examples of self-introductions:

Didier Paillard

Jim Stuttard

Marcel Bokstedt

`By the way, Grigory, it would be great if you could introduce yourself by posting a comment here in the category [Chat](http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/?CategoryID=11). We'd be very interested to hear bit about what you've done and what you're interested in doing. We're getting a lot of new members! Here are some recent examples of self-introductions: [Didier Paillard](http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=765&Focus=5251#Comment_5251) [Jim Stuttard](http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=802&Focus=5265#Comment_5265) [Marcel Bokstedt](http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=765&Focus=4942#Comment_4942)`

I like to have one central and simple message per post. For the one at hand that is "turbulence = eddies". I don't have two messages right now in my head, so there is no natural way to split the post into two parts for me.

Of course there is enough material that one could mention, but if I write about that now, it would simply deteriorate into a mediocre introductory lecture about hydrodynamics.

On the other hand, I don't claim any copyright to the text I wrote, so if you, Grigory, would like to take the text as a basis for a two post series, that's fine with me.

`I like to have one central and simple message per post. For the one at hand that is "turbulence = eddies". I don't have two messages right now in my head, so there is no natural way to split the post into two parts for me. Of course there is enough material that one could mention, but if I write about that now, it would simply deteriorate into a mediocre introductory lecture about hydrodynamics. On the other hand, I don't claim any copyright to the text I wrote, so if you, Grigory, would like to take the text as a basis for a two post series, that's fine with me.`

I believe I should post Tim's article now and let Grigory and others (including Tim) write more posts later. The number of posts people dream of writing, or even start writing, significantly exceeds the number that are ever actually finished.

It's fine if there's some overlap between posts... so if Grigory wants to use Tim's post as the basis of some more detailed discussion, that's fine.

I have to take my sister to Arab Street for some touristic activities today. Tonight, if nobody convinces me otherwise in the meantime, I'll try to post Tim's article.

`I believe I should post Tim's article now and let Grigory and others (including Tim) write more posts later. The number of posts people dream of writing, or even start writing, significantly exceeds the number that are ever actually finished. It's fine if there's some overlap between posts... so if Grigory wants to use Tim's post as the basis of some more detailed discussion, that's fine. I have to take my sister to Arab Street for some touristic activities today. Tonight, if nobody convinces me otherwise in the meantime, I'll try to post Tim's article.`

I'll do that right now.

Since the energy cascade is actually the idea of "turbulence = eddies", it highlights the structure of relations between eddies, I'll try to explain the energy cascade. Thus I'll find out whether I can explain something before trying to write for the blog.

Indeed the idea is extremely simple. First of all, we could think of a turbulent flow as a collection of eddies of different sizes. Then we say, that large eddies generate the smaller ones, that is large eddies break into smaller eddies. If the eddies are large enough the transfer of energy from larger to smaller scale occurs without dissipation. This last fact can be easily seen if you recall that Reynolds number represents the strength of inviscid terms compared to the viscid ones in the Navier-Stokes equations. Thus given the eddy of the scale $\lambda$ and of characteristic velocity of $u_\lambda$ (the velocity relative to the average velocity of flow) the corresponding Reynolds number would be

$Re_\lambda = \frac{u_\lambda \lambda}{\nu}$

So large-scaled processes are governed by dominating non-viscid terms of Navier-Stokes equations and dissipation does not occur.

We actually could easily find the energy transfered from larger scale to the smaller ones. There is an assumption, that for a certain scales the energy transfer does not depend on the flow as a whole and this transfer is governed by some universal law. Although these scales are smaller than the overall size of the flow, these eddies are still big enough to neglect viscosity. Hence we have only $\varepsilon$ [J/ (kg * s)], $\lambda$ [m] and $u_\lambda$ [m / s] (but not $\nu$) to form the physical relation. With the help of the pi theorem:

$\varepsilon \sim \frac{u_{\lambda}^3}{\lambda}$

However, as the eddies become smaller and smaller the role of viscosity increases. For some scale it is so significant that the energy dissipates into heat instead of transferring to new smaller eddies.

Actually the process of generation of smaller eddies from larger ones can be seen on the video from the Wikipedia article about Kelvin-Helmholtz instability you've been referring to. It is clearly seen how new smaller eddies are generated on the exterior of larger ones. If there was no viscosity you would have seen the increasingly smaller eddies, however indeed there is no eddies under certain scale $\lambda_0$ that is called Kolmogorov scale.

To find out the size of the smallest eddies we resort to another hypothesis (by Kolmogorov). According to it the rate of the dissipation is close to the energy flux $\varepsilon$ that small eddies receive from their elder brothers. Moreover, it is assumed, that processes at the dissipation scale depend only on $nu$ and $\varepsilon$. Hence $\lambda_0$, $\nu$ and $\varepsilon$ are to from the physical relation:

$\lambda_0 \sim \left(\frac{\nu^3}{\varepsilon} \right)^{1/4}$

This estimation is used to demonstrate why Navier-Stokes equations are not used to calculate turbulent flows around planes. Let $L$ and $U$ be the parameters for the whole flow, thus from (1) and (2) approximately

$\frac{\lambda}{L} = \left(\frac{\nu^3}{L \varepsilon} \right) \sim \left(\frac{\nu^3}{L} \frac{L}{U^3} \right) = \frac{1}{Re_L^{3/4}}$

Even if $Re$ is as small as 10^4 it is needed at least

$\frac{L}{\lambda} \sim 10^9$

cells in the simulation area to resolve the flow. That's why Navier-Stokes equations cannot be used, for example, to find the flow around a plane ($Re$ is roughly 10^7).

`<blockquote> <p> By the way, Grigory, it would be great if you could introduce yourself by posting a comment here in the category Chat. We'd be very interested to hear bit about what you've done and what you're interested in doing. </p> </blockquote> I'll do that right now. <blockquote> <p> I like to have one central and simple message per post. For the one at hand that is "turbulence = eddies". </p> </blockquote> Since the energy cascade is actually the idea of "turbulence = eddies", it highlights the structure of relations between eddies, I'll try to explain the energy cascade. Thus I'll find out whether I can explain something before trying to write for the blog. Indeed the idea is extremely simple. First of all, we could think of a turbulent flow as a collection of eddies of different sizes. Then we say, that large eddies generate the smaller ones, that is large eddies break into smaller eddies. If the eddies are large enough the transfer of energy from larger to smaller scale occurs without dissipation. This last fact can be easily seen if you recall that Reynolds number represents the strength of inviscid terms compared to the viscid ones in the Navier-Stokes equations. Thus given the eddy of the scale $\lambda$ and of characteristic velocity of $u_\lambda$ (the velocity relative to the average velocity of flow) the corresponding Reynolds number would be $Re_\lambda = \frac{u_\lambda \lambda}{\nu}$ So large-scaled processes are governed by dominating non-viscid terms of Navier-Stokes equations and dissipation does not occur. We actually could easily find the energy transfered from larger scale to the smaller ones. There is an assumption, that for a certain scales the energy transfer does not depend on the flow as a whole and this transfer is governed by some universal law. Although these scales are smaller than the overall size of the flow, these eddies are still big enough to neglect viscosity. Hence we have only $\varepsilon$ [J/ (kg * s)], $\lambda$ [m] and $u_\lambda$ [m / s] (but not $\nu$) to form the physical relation. With the help of the pi theorem: $\varepsilon \sim \frac{u_{\lambda}^3}{\lambda}$ However, as the eddies become smaller and smaller the role of viscosity increases. For some scale it is so significant that the energy dissipates into heat instead of transferring to new smaller eddies. Actually the process of generation of smaller eddies from larger ones can be seen on the video from the Wikipedia article about Kelvin-Helmholtz instability you've been referring to. It is clearly seen how new smaller eddies are generated on the exterior of larger ones. If there was no viscosity you would have seen the increasingly smaller eddies, however indeed there is no eddies under certain scale $\lambda_0$ that is called Kolmogorov scale. To find out the size of the smallest eddies we resort to another hypothesis (by Kolmogorov). According to it the rate of the dissipation is close to the energy flux $\varepsilon$ that small eddies receive from their elder brothers. Moreover, it is assumed, that processes at the dissipation scale depend only on $nu$ and $\varepsilon$. Hence $\lambda_0$, $\nu$ and $\varepsilon$ are to from the physical relation: $\lambda_0 \sim \left(\frac{\nu^3}{\varepsilon} \right)^{1/4}$ This estimation is used to demonstrate why Navier-Stokes equations are not used to calculate turbulent flows around planes. Let $L$ and $U$ be the parameters for the whole flow, thus from (1) and (2) approximately $\frac{\lambda}{L} = \left(\frac{\nu^3}{L \varepsilon} \right) \sim \left(\frac{\nu^3}{L} \frac{L}{U^3} \right) = \frac{1}{Re_L^{3/4}}$ Even if $Re$ is as small as 10^4 it is needed at least $\frac{L}{\lambda} \sim 10^9$ cells in the simulation area to resolve the flow. That's why Navier-Stokes equations cannot be used, for example, to find the flow around a plane ($Re$ is roughly 10^7).`

What you just wrote could easily be expanded into a nice blog entry, Grigory! Or if you like, I could try it.

I published Tim van Beek's blog post here, after making a few small edits:

If Tim or anyone else spots typos or other errors, let me know on the blog and I'll fix them.

`What you just wrote could easily be expanded into a nice blog entry, Grigory! Or if you like, I could try it. I published Tim van Beek's blog post here, after making a few small edits: * [Eddy Who?](http://johncarlosbaez.wordpress.com/2011/08/25/eddy-who/), Azimuth Blog. If Tim or anyone else spots typos or other errors, let me know on the blog and I'll fix them.`

I could not find your "thumbs up emoticon" here:

http://math.ucr.edu/home/baez/emoticons

so, well, thumbs up!

`I could not find your "thumbs up emoticon" here: http://math.ucr.edu/home/baez/emoticons so, well, thumbs up!`

Thanks. And to you, too!

(For people who are new here, that image was created by typing:

`<img src = "http://math.ucr.edu/home/baez/emoticons/thumbsup.gif" alt = ""/>`

and clicking 'Markdown+Itex' when entering this comment.)

`Thanks. And <img src = "http://math.ucr.edu/home/baez/emoticons/thumbsup.gif" alt = ""/> to you, too! (For people who are new here, that image was created by typing: `<img src = "http://math.ucr.edu/home/baez/emoticons/thumbsup.gif" alt = ""/>` and clicking 'Markdown+Itex' when entering this comment.)`