Beyond Chaos: The Continuing Enigma of Turbulence - Nigel Goldenfeld (UIUC) [2017]

so I'd like to thank Lars for keeping the introduction mercifully short and I want to say it's a great pleasure to be speaking to you with some of the friends in the audience and my former postdoc advisor I was a postdoc here in the years 1982 to 1985 and I got a huge amount of value out of ki TP it was just a life-changing experience and this is my chance to give something back to the community and so I'm happy to do that so what I want to talk about is is turbulence and the title of my talk is beyond chaos the continuing enigma of turbulence I will be mostly talking about the subject itself and the intellectual debate that goes on in the subject but at the end I'll talk a somewhat personalised view about about some of the things that we're debating here at the ke ITP so there are things I'm going to talk about it's my own personal views and if somebody else equally qualified were to give this talk they would choose a different set of topics in a different way to present it but hopefully you'll find this interesting I will talk about some original work and that was done with students hung and she and an undergraduates song Lindsay who is now in the graduate program at Princeton alright so we're at the Kaveri Institute for Theoretical Physics and so the question that I really want to ask is well you know why are we as theoretical physicists talking about turbulence what does theoretical physics itself have to say about turbulence now one of the great highlights of my time as a postdoc in in ki TP in Santa Barbara was when Richard Fineman the very famous Nobel laureate who got the Nobel Prize for his work on the quantum theory of electricity and magnetism so he came and visited and gave a wonderful talk about negative probabilities and I got to fulfill my my sort of lifetime ambition I got to talk with him I'd followed his his lectures when I was at high school and so that was a great a great moment so when I was thinking about this this this talk and trying to prepare it I remembered passages that he'd written in famous lectures on physics and so I thought I would share those with you so what can theoretical physics tell us about turbulence well basically nothing according to turbulence so he says there's a physical problem common to many field that is very old it has not been solved it is not the problem of finding new elementary particles new fundamental particles at all but something left over from a long time ago over a hundred years nobody in physics has really been able to analyze it mathematically it's as satisfactorily in spite of its importance to the sister Sciences to the analysis of circulating or turbulent fluids and he goes on to say what the problem is all you have to do is take a pipe that is very long and push water for it at high speed and the question is this to push a given amount of water through that pipe how much pressure do you need okay it's a very very simple problem and we still don't know how to calculate it we still don't know the answer and and if the water flows very slowly or if we use a thick goo like honey then we can do fine job that's easy okay but this problem when it goes at high speed we still don't know and that's why this problem is so hard in fact people say that this is the greatest unsolved problem of classical physics so what we should find man said was this is the central problem which we ought to solve some day and we have not and we still have and even though he wrote these words in 1963 but I think I wanted I'll show you tonight that we've made some progress in understanding some aspects of this problem okay so if I only asked answer the question what can theoretical physics tells are turbulent let's ask the other question bit more fundamental written theoretical physics tells about anything at all okay no obvious okay so and actually while we're at it let's let's you know really you know roll up our sleeves you know what what is clear article physics anyway so let me see if I can try to try to try to answer these questions for you so let's start off by talking about these people these are the winners of the Nobel Prize in Physics for last year 2016 they run it for a field that's my intellectual home condensed matter theory David Thewlis mic oscillates and Duncan holding all of these by the way are British theoretical physicists who made their careers in the United States so in some sense Margaret Thatcher owes a lot of credit for this for this for this prize as well so I want to tell you a little bit about what they got the Nobel Prize for at least cost alerts and customers and Fowler's because it's germane to this subject so let's start off by talking not about regular fluid like the ones that you're used to but let's talk about super fluids so here's the sort of fluids that you're used to this is honey it's a regular normal fluid with a flow is very slowly it's very viscous its sticky its gooey so we all know what honey is this is superfluid helium and this is a photograph taken at one degree or so above absolute zero and it shows an effect called the fountain effect where if you heat the fluid up it actually causes it to move in a way that's very hard to explain unless you use quantum mechanics so this is superfluid helium but what's remarkable about superfluid helium is that in some sense or other it flows without any viscosity at all so if you you know try to stir it up it will just keep on going it's basically the newest thing we have in nature - perpetual motion and and so this it clearly is the opposite of perpetual motion you walk through honey you get stuck very quickly this stuff has no viscosity at all now what's special about about this fluid is the following if I put my fingers into a classical fluid like water doesn't work so well in honey and stir it up I can make it swirl around an arbitrarily an arbitrary amount I'm using swirl here quotation marks to let you know that I'm making some kind of simplification that the experts and the audience will understand if you take a quantum fluid like the superfluid helium and try to stir it up and make it rotate what you find is that it can't swirl slower than a certain amount the angular momentum the the rotational properties of the fluid come in lumps just like energy does as I was one of the first to show so this is the this is what makes it quantum this this fact that it can swirl as slow as you like is what makes it classical now what cost is a thousand sidered a model which is depicted here in this picture this is a lattice it's a it's a lattice in space it has little arrows on it those arrows you can think of as representing the flow of a quantum fluid of the super fluid they just represent the direction in which the fluid is flowing what they discovered was something amazing just by looking at the properties of that very simple model and really all they had in the model was that there exists on ours there's nothing quantum mechanical in the model there's nothing like that at all there's nothing even very super fluid about it nothing even fluid they discovered that they could predict how the how the fluid underwent this transition as you cooled it to low to low temperatures how it went from being a normal fluid normal helium to being a superfluid and I'm not going to explain this graph but it shows the experimental data and their prediction and all of that works works very well so so the the remarkable thing is that as you can imagine super fluids and these are super fluids in two dimensions not three dimensions are very complicated you have to worry about if you started to think about this from scratch you'd think you'd have to include everything you need to know about helium and its atomic structure and it's it's it's chemistry and so on but none of that seems to matter at all they showed how it goes from helium to being a superfluid just by looking at the simplest way that you can arrange arrows on a plane in in in terms of their energetics it's a very very simple thing and the surprise is that this very simple model which doesn't look anything like the original physical system that you're studying works and predict very complex phenomena now in fact everything we know about the physical world comes from looking at simple mathematical models that have the right property of symmetry and topology would be examples of those but this the the beauty is you can sometimes construct surprising models that seem to have no relation at all to the problem you're trying to understand and yet they turn out to be identical and we're going to see an example of that later on in this talk so here's another example this is not superfluous in two dimensions this is a simulation of superfluid turbulence in three dimensions and this here shows you a very complicated pattern of what are called vortex lines quantum vortex lines in the super fluid these are mathematical singularities inside the fluid they're kind of like cosmic strings which some of you may have heard about they have the same sort of mathematical properties as those and as they flow they go through each other they cut each other they form this incredibly complicated space-time pattern and from this you can construct all sorts of you can make various predictions about what such a fluid would behave like and those predictions agree with experiment what's surprising is that this code the simulation that did this was a 5-line computer program okay it used the same kind of simple mathematical structure that concerts and thalis used in their work but to study something as complicated as this quantum turbulence now that's an introduction to you know some recent theoretical physics but you know what is theoretical physics and how do you recognize the article physicists when you see them well these are some of the theoretical physicists that you might run into at the University of Illinois they look more or less normal at least they do to me I hope they they do to you so you can't tell much from that how about this how about a the lab so here's the question I want you to think about this is a blackboard okay and the question we want to talk about now is this is this theoretical physics thing how many people think it is yeah I see a lot of people nodding their heads well the thing is that it's my blackboard and so it probably is right because I'm a theoretical physicist however what you didn't hear in the introduction is that I work half my time on condensed matter physics and turbulence and things like and the other half of my time I work on biology on genomic biology microbiology ecology and evolution this is my blackboard in the Institute for genomic biology at my university where i work so who knows maybe this isn't a play medical physics well let's have a look at this part so you might say aha well let's take a look at that is that theoretical physics nope it's not theoretical physics this is mathematical physics it's not theoretical physics okay what's the difference well this what's what's sketched over here is just some complicated calculation it's basically like a math homework problem except on steroids okay how about this part anybody want to guess theoretical physics well this one is yeah this this one is theoretical physics has it happened why what this is this is a guess about how something behaves it's actually a guess about the topic of this conference it's a guess about how fluids become turbulent so here you've got mathematical physics just some hard calculation that's been done here you've got theoretical physics and basically theoretical physics is basically maths plus a simple gas it's some gas that starts you off in doing a calculation and trying to understand what's going on in the physical world all right so now you know the difference between mathematics and theoretical physics if you see theoretical physicists they're usually doing complicated mathematics but it doesn't mean they're doing theoretical physics they may just be trying to get through the day okay so now we want to talk about turbulence so we need to do theoretical physics and turbulence and the question I want to ask is why do we need to guess in turbulence because turbulence is the flow of a fluid and now we're not going to worry about quantum mechanics we're not going to worry about any of that we're just going to worry about regular classical physics the solar physics that you're familiar with it you see every day so why do we need to guess well because think about it you know we shouldn't need to guess because we know what governs the motion of all matter in outside the quantum regime and that is F equals MA okay so it's basically the only equation that's going to be in this talk F equals MA that goes back 400 years everybody here knows that and what this stands for is force the mass of whatever the object is and its acceleration now this isn't actually a very useful way to write down this equation the best way to write it down is like this AE equal Zeff and for M we're just going to divide both sides why well we write it like this because this tells you what's happening to the stuff you're looking at this tells you what force you are applying and then this tells you how the thing you're looking at starts to move as a result of that and that's what you want to know you want to know if I push this thing how fast is it going to go what's it going to do and so on so this is the way this this way is how we should really write down Newton's equation so what about fluid mechanics well in fluid mechanics the force is usually gravity I mean it rains the water goes down to the ground we can be pressure it can be the viscosity that we talked about before it could also be magnetic fields and stars or if you're trying to do a plasma fusion experiment to try to make a tokamak to solve the energy crisis it would be a magnetic field in that case - or it could even come from quarks and gluons in in a particle physics experiment like the sort of experiment that's done at the relativistic heavy ion collider in Brookhaven so basically we know how to write down what the what the force is the mass is just the density of the fluid and then we the question we need to understand them is how do we calculate and what do we do with the acceleration now if we know acceleration we can calculate speed and so we're going to do a little a little test calculation just to see if everyone is awake and is following so what does acceleration means well you know what acceleration means because you here and if you drove here your speed changed as you accelerate it and slow down as you've got to traffic lights and stoplights and so on so that means your speed changes every second so let's do let's do a midterm exam okay I know this is a public lecture and you are here as the guests of the ki TP and and and so on and so forth but you're not going to get off that easily so we're going to do a have a midterm so here's the question a car accelerates from the west and nine meters per second per second for three seconds and the question is this what is the brand of the car okay so let's let's work this problem okay so the answer is this after one second it's going nine meters per second after two seconds it's going 18 meters per second after three seconds it's going 27 meters per second 27 meters per second is 60 miles an hour 0 to 60 and three seconds it has to be a Porsche okay so now you know so a tester would do it as well we don't okay well you know I got my data in 2014 so it's probably you know a bit out of date yet and so anyway so so okay so this tells you basically the moral of this story is this if I know the acceleration I can then calculate how fast it's going to go and if I know how fast it's going to go I can tell where it's going to go okay so let's pause to that people take photographs I also keep keep going yes this this is the easiest midterm ever right right so now there's asked us ourselves the question if we want to apply this formula a equals F over m not to one particle but to every point in the fluid so it's not just a single particle the fluid there's many many points that's when things start to get messy so let's have a look and see why so let's suppose I start off by taking a particle and I accelerate it so this this little red arrow is meant to represent its speed in say the horizontal direction so that's predictable if I take a fluid and the speed isn't too high I can think of the fluid as basically you know a little I could have fluid over here a little packet of fluid next to it a little packet of fluid next to that and so on and so on many times filling the whole room that's I would try to describe the air in this room if I was trying to do a calculation or the simulation so that's what these boxes are meant to represent little packets of fluid and they're moving in some uniform way and so again the flow is perfectly predictable and we can calculate everything they're just as fine them set so this is called laminar this is just flowing steadily deterministically predictably no surprises now is asked what happens if the flow has been perturbed a little bit or if we've the flow is going so large is going fast enough then what happens is this you get some random directions for these fluids these fluid particles start to go off in slightly different directions but the point is that how far they get depends on their acceleration and that and the and it depends on their velocity and and that depends on the acceleration so the place that they get to you can't predict in advance you have to know how they accelerated to get there so this regular pattern of the fluid parcels now get disrupted in you know a second later or something like this and now this this process keeps on going so if you started off with just a regular pattern of these fluid parcels after some period of time they're all going to be higgledy-piggledy all over the place now the other thing about it is these fluid parcels were initially next to each other as these things move around they're going to become further and further apart because this one has a high acceleration so it moves fast so the next time you work out how fast it's going to go it's moved much further than this one and that's the point that how far they move depends on what their acceleration is so they don't all feel the same the same kind of force so after a time if this these packets are scattered higgledy-piggledy and that happens to every packet that you start with your fluid instead of being nice regular and uniform has got contorted up into this messed-up thing like a like a piece of bread that you're cooking that you've swirled up like this and it's become very very disordered and so it's unpredictable and that's what we call turbulence now what about chaos well if you think about this behavior as being bad in some sense then what chaos is is one particle behaving badly in time so think about it there may be a system let's say some kind of pendulum that if in normally it would swing like this but you can construct pendulums which are coupled to each other in some way and their motion can be chaotic and unpredictable and and and that unpredictability is part of the definition of what chaos is but it really only applies to one particle behaving in a kind of unpredictable way but really only in time now when you have turbulence you have many of these fluid packets all crashing into each other and affecting each other's motion and so on they are behaving badly not just in time but also in space so turbulence is chaos but an infinite amount of chaos chaos at every single point in space simultaneously and coupled together that's why the problem of chaos has sort of more or less understood and we understand many aspects of it but turbulence is much more complex because it's chaos in space as well as in time so in a sense then chaos is unpredictable but with some kind of simple structure whereas turbulence is unpredictable but with very complex structure so to summarize the velocity of a fluid blob depends on where it was where it is going to and how fast it is going and if you write that down mathematically what you see is that the velocity of the fluid governed its own behavior in some kind of self referential way and that that self referential way is the thing that when you write down the equation it makes it nonlinear it makes it incredibly hard to compute the consequences and so on but that's really where it comes from in at least in cartoon okay so what is turbulence then well turbulence is is a flow that is stochastic and wildly fluctuating so here's an experiment that was done by our water Goldberg this is who I collaborated with this is a flowing soap fill and you can visualize the the flow structure very easily and you can see it has this very complicated structure of swirling motions all intertwined with each other just as you can imagine those cartoon blobs that I showed you before after they become separated and start moving around in some random way they start screwing the fluid up in some really crazy way like this and this is what you end up with so it's stochastic and wildly fluctuating the other thing you can see about this is that you have structure but the structure is on many scales so there's structural on this large scale and then if you look inside these swirly bits you see other swirly bits and if you look inside those you see other swirly bits as well and so on it's like a Russian doll and that fact is what makes turbulence interesting but in some sense also makes it susceptible to to some kind of analysis so now I could tell you about what we have to guess in turbulence so what is the big gest that starts everything off we do we can write down the equations for fluid flow and they describe fluid turbulence we can write down Newton's equation and we believe that those equations would correctly describe turbulence so we can write down those equations but they're useless it's with few exceptions it's basically true that almost everything we know about turbines did not come from solving the equations but came from gases and then doing calculations and the biggest guess of all probably the biggest progress in understanding turbos was made by comma graph and also other people particularly a Richardson and his idea was the following this is the big guess when you look that these swirly swirly motions like this you might say what would happen if I animated that so what you might imagine is I start off with some swirly motion of the fluid and then it spins off some other swirly motion some other wor vortex whatever you want to call it and then that one breaks up into smaller ones and so on and you can sort of see that that could have happened here now usually when things break up and make smaller bits come out of them it means that they're falling apart okay if you were riding your car and first they first the tires came off and then the axle came off and then the ball bearings came off and then the engine fell apart and then you crashed into a tree or something well you know the the failure mode is all the bits coming off the bigger bits first and then the smaller bits so you might think that these swirling motions coming off and making smaller and smaller swirling motions would be a hallmark of turbulence sort of running down but that's not true at all these Eddie's is what what people like to call them they spin off other edges but they do it without losing any energy at all it's not like when you get a swirling motion and Eddy and it breaks up into two it's not like it's kind of run out of steam and so it's exhausted itself and just makes two little ones and hopes that they'll continue not at all the actual dynamics of that process conserves energy it's just what they do that's what makes it so complicated it's they don't lose energy in doing it so there's no friction involved this isn't due to friction there's breaking up of the eddies it's just what these flows do and so as a result what they guessed Richardson and collagraph they guessed that turbulence is a scale-invariant cascade of energy and I'm going to show you exactly what that means okay so we're going to see that this idea this very simple guess turns out to have remarkable explanatory power all right so what do I mean by yourself as a scale-invariant cascade so I'm sure you're very literate so you know Jonathan Swift and you may know another he wrote a poem and that was modified by orgasm de Morgan and the poem goes like this great fleas have little fleas upon their backs to bitin and little fleas have lesser fleas and so I had infinitum okay so that's the idea you've got a flea and then on its back it's got a flea and then you flee and another flea and so on there's this this there's this Russian doll structure this continuum of all these different creatures all which live on something and have something living on that okay it's like turtles all the way down except this is fleas all the way down so that's that's that's a classic appellant but Richardson made one up for turbulence and his version is this big worlds have little worlds that feed on their velocity and little worlds have lesser walls and so on to viscosity so what happens is this spinning off goes on and on it keeps on going eventually what happens is this eventually you have to pay the price because you're putting energy into a fluid you're sitting down your bathtub with your fingers in it sloshing it around like this and you're putting energy in what this picture says is those swirls go down to the little whirls and little ones and little ones and so on but you're putting energy in maybe the only thing that can happen is this either the whole thing will blow up because you've put so much energy in all that energy will go somewhere and it'll stop it blowing up and that's what happens it doesn't blow up when you stir your the water in your bath it does actually heat up but it doesn't explode what happens is the energy goes down to smaller and smaller world until eventually it gets down to a scale where the fluid viscosity does matter and then it stops making that structure that I showed you it does something actually rather boring and and then you generate heaters as Joule heating and as the the water slightly heats up but along the way the energy the procced the energy went from scale to scale without losing energy just trampling down to smaller and smaller world and so that's what a scale invariant cascade is it's a cascade because you go from big to small to small as the smaller still and small and so on and it's it's called scale invariant because wherever you look you can't tell where you are in the system if you're one of those fleas you look around you say okay am I the middle flea well I've got a flea above me and a flea below me so I could be anywhere okay go down ten fleas hopefully also has a flea above it and a flea below and you can't tell so you can't tell where you are and so that's why we that's what we mean by scale-invariant now I'm going to show you a graph I don't I want you to basically ignore it but but what did what this shows is is energy in fact it tells me well it's basically the kinetic energy of the fluid and in some sense what it shows is this this is the length scales this goes from large to small and the picture of turbines that we generally have is you put energy in here the energy flows through a range which is scale invariant and then when you get to small length scales it dissipates as heat and this scale invariant cascade the mathematical way that we see this it's scale invariant is that when we look at space and this K variable means the inverse of space okay so just think of that as 1 over space what it means is the the energy that's the kinetic energy is basically going as this parallel don't worry about the functional form don't worry where that comes from but that's the mathematical expression of scale invariance and so knowing this function knowing the form is what enables us to make someone just some headway now artists have been really really good at depicting turbulence here we have a beautiful picture from a Leonardo da Vinci showing the big world and the smaller walls and smaller ones and smaller and smaller ones here we have a van Gogh famous starry nights picture and I want to just tell you a little bit about the different kinds of cascades that we know of in in in in physics so the regular one is called a 3d Ford cascade it basically says energy flows to smaller scales so you stir the fluid up the energy goes down to smaller scales and make some nice picture like this is called Ford cascade and it behaves in some way which you don't really need to worry about but the main thing is that energy flows to small scales if you live in two dimensions okay like the superfluid that I showed you before in two dimensions it turns out that when you stir fluid the energy and the angular momentum go to large scales not to small scales you've all seen the big red spot of Jupiter you all seen pictures of that okay a planetary atmosphere is roughly two dimensional you've got a huge big planet and you've got a thin skin of atmosphere around it as as it flows and does whatever whatever it does it's basically behaving like a two-dimensional fluid and what happens in two dimensional fluid is you get these huge big tornadoes or hurricanes or vortices like the Big Red Spot that's something that happens generically in two-dimensional fluids and it's called the inverse cascade because energy flows to large scales now there's another kind of thing that can happen if you live in two dimensions this was the inverse cascade that I told you about before where energy flows not from large to small but from small to large and makes something like the big red spot of Jupiter the other thing you can have though in two dimensions is that swirl enos because angular momentum if you like this this kind of rotational motion that can be transferred to smallest scales that behaves like the energy does and why I'm not going to I can't go into but the thing about it is and I do want you to pay attention to this when you write down the mathematical structure of that it behaves as the inverse of space so K is distance but the inverse existence raised to power minus three whereas when you have energy being transferred it goes as as wave number that's the inverse of space to the power 5/3 so they are different mathematical Cascades and that happens because you are in two dimensions and because you are transferring a different physical quantity in one case swirl enos and rotational motion in the other case energy so you have two separate cascades in two dimensions so I told you about planetary atmospheres so let's see does it work let me show you some real data so here is data from the Earth's atmosphere these are very famous data and a strong engage this is the the e of K the spectral density I'm not going to go and explain exactly what that is but it's roughly the kinetic energy of the of the of the fluid motion as a function of wavelength or the inverse wavelength is wave number so this is basically space on top here is the inverse of space the distances the way this graph is plotted this is large length scales over here this is small length scales over here and one you can see they this is a graph which is a log-log graph it's not a regular graph you're not plotting the regular things you're plotting the logarithms ten to the one 10 to the 2 10 to the 3 up to 10 to the Aged over here and the same thing let's bless you the same thing along the horizontal axis so a straight line on this graph means a power law and that means it's scale invariant so the fact that these data fit these power laws shows you proves to you that we do have scale invariant Cascades in real fluids and what you can see here is you go from this five third power law to this power law minus three okay so we can see all of these different Cascades in real experimental data there's a very strange puzzle about this picture some of the experts and the audience will know what it is anybody know what it is maybe some of the experts don't know there's something very perplexing about this that's not that's that's not that's not the problem the problem is much much worse than than is the is the is the layer thin or thick it's one much much worse than this the problem is this if you think about these Cascades the the energy is flowing in the wrong direction what people expected was that over here you would have the five third cascade and over here at small length scales you'd have the K to the minus you'd have the the swirling this cascade why because you expect the energy flowing from large to small all the way and the only way you can do it in two dimensions is to have it small scales but this this one the swirling this cascade and at large scales this cascade which is more less three-dimensional because the because of these are the things that Lars said and so that's what people expected and he and it goes the other way round and to this day nobody fully understands why that is the case okay so there's a mystery of terpenes one mystery so the story is so far the flow tells the flow how to change the flow this is one of these problems like general relativity where matter tells space how to move and study space tells matter how to move and matter tells space how to curve that's Einstein's equations in word so in a poem if you like ok that's that is that's basically the the general theory of relativity here the flow tells the flow how to change the flow this self-referential non-linearity is what makes turbulence hard and we can guess the outcome out of complexity comes a simple picture of scale-invariant cascades which isn't the whole story and it isn't the end of all of our questions but it's a lot of progress so it all sounds very nice so is it solved and what's new so what I'd like to do now is I'd like to talk about what's new and so I want to we'll start off by going back to 1883 to the first scientific studies of off turbulence made by Osborne Reynolds in in the 19th century so here you see his apparatus it's a pipe and he was interested in the flow of fluids in pipes he was interested in everything but this is where he did his controlled experiments so he sent him he had a reservoir here he sent in water through here and then he observed what happened as the water flowed through this pipe and he watched it very carefully what he realized was that what happens what you see depends on how fast the fluid is going but he also realized that it doesn't just depend on that it also depends on the visco because if you have a fluid that is not very viscous see like superfluid helium you only have to push it a little bit and it'll immediately become turbulent if on the other hand you have a fluid like honey you have to push it make it go really really fast before it will become turbulent so he realized that there's a number that you can define which is now called the Reynolds number which is defined in this way it's the flow speed times the system size would say the diameter of the pipe divided by the viscosity and also divided by the density okay so that's that number is it has no dimensions it's just a pure number and it tells us every fluid you can assign it that's in motion you can assign this number to it and what that number is tells you whether the fluid is going to be behaving well or whether it's going to be behaving badly so what he saw was when he changed the Reynolds number somewhere around 2000 he saw these things called flashes of turbulence and I want to tell you a little bit about those flashes of turbulence we call them puffs today but what they are basically is here the fluid is perfectly laminar everything is predictable and then here's a bit where it's completely chaotic and turbulent and you don't know what's happening then lamina again then turbulent then lamina so the way that turbulence arises is this alternating pattern of laminar and turbulent flow so let's just go back and think about what that means so here's our classical fluid this was our honey the velocity can swirl as slow as you like here's our quantum fluid superfluid helium where you can't swirl it slow within a certain amount and this is laminar this flow is typically much less than one whereas it doesn't take very much effort to make superfluid helium have a Reynolds number of about a million very turbulent okay so here is a map of fluid flow everything every phenomena you you you encounter in the flow of fluids you can put somewhere on this map and the nice thing about this map is it's only one dimensional it's only alive so it's much simpler than the two Google Maps the Google map of turbulence is one dimensional okay so Reynolds number being very large let's say close to infinity is pretty much to what describes a turbulent fluid like turbulent superfluid helium so this is where you would get that scale-invariant cascade in fully developed turbulent somewhere down here where Reynolds saw his his flashes that somewhere around Reynolds number 1800 to 2400 if you work in in experiments or you do observations of the Earth's atmosphere or something like that of course the Reynolds number can never be infinity but it turns out that this picture of the skin varying cascade still works even down to fairly low values of the Reynolds number it doesn't have to be just as infinity you can be at some medium medium value so most of our understanding of turbulence comes in this kind of crossover region here this is where you are okay just about here round about 2000 or so that's the Reynolds number for the flow of blood in your heart okay now it's rather unfortunate that you're here because right at that point that is where Reynolds saw his flashes of turbulence and when you have those flashes alternating turbulent slamming its heaviness laminar its shaking things up a lot and it can shake things up so much as the fluid goes through your blood vessels that it can burst them then the the fact that you are in this transitional range of turbulence where this transition is happening this the shocking behavior between turbulence and laminar turbulent and laminar that can disrupt your your arteries and give you an eel ism's when people die from this so yes turbulence can kill you down here down here in this range 1800 to 2400 that's where all the people who are at the ITP K ITP workshop line now that's where they were all working okay usually in a ITP when you have a workshop on turbulence people are more or less arguing about what happens up here up here but this time we're arguing about what happens down here so there's two real questions that one can ask about turbulence as of as a physicist as physicists I told you we like to ask questions that have simple answers okay I'd like to like coast alerts and fowlers did in their Nobel prize-winning work he took a very complicated problem and then they asked does this very simple model capture all the features that we want to see well we can't expect that to work for everything every single phenomenon but if it doesn't work well that's too bad but you should always look for that kind of simplicity when you work on a new phenomenon because that's where you're going to be able to make progress so you expect to be able to make progress up here and you expect to make progress when there's transitions and and and that's why people now work are studying this thing here so the two main questions in turbulence I would say is is there universal scaling behavior in fully developed turbulence in various varieties some people study just pipe flow some people study fluids heated from below so you get convection some people study fluids going round in in between cylinders some people study channels some people study jets some people push airplanes through fluids there's many many different ways you can ask these questions the second question and the one that we're going to talk about now for the next 11 minutes and 34 seconds is how do fluids become turbulent as you increase their flow speed that's what we really want to understand how does this thing actually actually happen and the answer I'm not going to be able to tell you the whole answer because it's highly technical and also we are still debating it but we've made a lot of progress and we've made a lot of progress because a lot of theoretical physics but of experimental physics okay so we have to remember this okay theoretical physics is wonderful but the primary way that we discover things about the natural world is by doing the experiments so what happened was in 1883 Reynolds did his measurements of turbulence in a pipe by 2008 a Burin Hoff and and collaborators did a highly precise measurement the laminate turbulence transition an incredibly beautiful experiment what they did was they took a nap thatís basically like Reynolds's but much much much longer and they put the fluid going in the fluid comes out the other end and the fluid just comes out with the following the yellow trajectory like so they asked a question that's a new question they said what is the probability that if I make a little bit of disturbance over here create artificially a little bit of turbulence what is the probability that it will survive until you get to the end of the pipe it's a different kind of question okay the sort of question that people had only just started to ask around about this time and what they realized was this they make a disturbance perturb the turbulence goes through the pipe but maybe the turbulence will die maybe or you're below the transition to turbulence so if the fluid wants to be laminar and you can try to make it turbulent the turbulence will eventually die away if the top fluid wants to be turbulent then when you kick it to be turbulent it'll just stay troubled so what they wanted to ask is this as I as I have a laminar fluid a fluid it doesn't want to be turbulent how long does it take for the turbulence to die and does that time get bigger and bigger as I get closer and closer to the actual transition to turbulence because when the fluid is turbulence you make a bit of turbulence it will live forever when the fluid excuse me when the fluid wants to be laminar below the critical Reynolds number you make some turbulence and it will eventually die away so the question is how long does it take and so what they did was they measured the fluid as it came out the end here and if the turbulence was still there it would follow the red trajectory it would be slightly there'll be more drag because turbulence causes drag turbines makes things go slower turbulence is why when you can if you want to save energy if you drive your car at 50 miles an hour rather than 70 miles an hour you actually save double the amount of energy the frictional drag is it goes down by a half so what happens then is is then is then they can measure they asked a simple question did the puffs survive or not yes or no yes or no many many times repeat this many times and then you can work out the probability there's a little bit of turbulence and will survive now when they did that they found an amazing result this is the the logarithm of the lifetime of the turbulent path and there there is a picture from a computer simulation that Marco Vela did and the the lifetime of this path is plotted here as a function of this number that Reynolds came up with the Reynolds number so as I told you the transition is somewhere around 2000 2100 and what you see is that the data show that this lifetime does indeed increase from a hundred in weather units these are up to ten to the eight by the time they got to somewhere over here the life time goes up enormously fast as you change this number and in fact it goes up in it with a functional form which is really unprecedented it goes as e to the e to the Reynolds number for those of you who who care very very rapid rising lee in the lifetime of turbulence when you go through the transition to turbulence what do you find is that if you make a little disturbance it doesn't just die away what happens is it kind of splits into two and eventually makes two puffs and those two puffs go along and then they'll either decay or maybe they'll splits as well and so you'll get some very complicated pattern of the behavior of the flow in this regime and the and you can measure the lifetime between puffs splitting and you can plot that and you get a graph like that and that's what beyond Hoffmann company discovered in 2008 now I want you to forget everything I've told you for the last 45 minutes okay I want to tell you about something completely different nothing to do with it remember the beginning I told you that I spent half my time studying biology and half my time studying condensed matter physics and I want to tell you about the biology okay you didn't sign up for this did you you can leave if you want but don't because there's a surprise coming I want I want to talk about and prey so I'm a theoretical physicist so basically I work in the spherical cow proximation so this could be a wall this this could be a shark or something like this doesn't really matter what they're called but this is the predator and this is and this is the prey I want you to ask what happens in an ecosystem where you have prey like this in creatures like this which are the Predators you have an ecosystem composed of all of these things and then you ask what is the fate of that ecosystem what can happen well let's think about it let's suppose that the prey birth rate is high so some up here so that means that these rabbits I guess they are rabbits right these rabbits are breeding like crazy if they breed like crazy it means that these this thing wherever it is what is it actually Lynx or something yeah there's links we'll have enough food to eat and so the Lynx population will also grow and what can happen is that they can come into some kind of balance the more prey there are the more predators are but then the Predators eat the prey and so the prey population goes down and so what actually happens is the population actually oscillates in some way like that it's not actually trivial to prove that it oscillates because the well anyway that's another story something that we work on but you can show that as high enough birth rate this population will be stable and so you get coexistence of both the predator and the prey in some kind of happy equilibrium or happy if you have a predator unhappy if you were prey on the other hand let's suppose that the birth weight of the prey is too small they don't make enough babies so what happens is the predator eats all the prey and then there aren't anymore foods for them to eat so they starve and they go extinct and then the prey eat all their food and they go extinct in the whole ecosystem collapses okay so that's what happens that can happen and there's a particular point where the birthrate is just large enough that the ecosystem suddenly starts to become stable and for the prey and the predator to coexist so hung and she my student did a simulation of a predator prey ecosystem it wasn't out of the blue we knew what ecosystem to study for because we did some fluid mechanics first but let me just tell you this surprising thing you'd start off with predators and prey and you vary the birthrate and you ask the following question as I change the birthrate how does the lifetime of the ecosystem change as I get close to that coexistence point when you do that calculation you get a graph that looks like this and then when you go above a certain amount you find that if you start off with a localized region of prey it splits into two just like the term impacted so what we've discovered then is that over here I have a calculation motivated purely by ecology and over here we have an experiment done purely from fluid mechanics it seems that ecology is the same as turbulence now this ecosystem that we simulated was not one that we had chosen at random this was one that we had discovered as being the appropriate modes the appropriate behavior of fluid turbulence very close to the transition we did numerical simulations of turbulence and we looked because I work in phase transition so I knew what kind of thing to look for and we looked for certain types of mode which I'm not going to go into and we discovered that they behaved we found those modes and we discovered that they behaved like a predator prey ecosystem strange but true and when we we measured this and we quantified it and so on and that picture I showed you at the beginning when you came in was a picture of that predator prey ecosystem though the white swirly pattern was the what's called a zonal flow into the it's the it's the predator and the red spaghetti was the the turbulence and that's the prey so ecology is equal to turbulence turbulence looks just like ecology in fact it's much much worse than that or much much better depending on your point of view the turbulence transition is connected to all sorts of things it turns out now what we did was we wanted to understand how fluids become turbulence and what we do government was that they behave like a model which I'll show you called percolation but what we discovered along the way was that fluid turbulence behaves like ecology this behavioral ecology also behaves like a model in high energy physics or particles scattering against each other so it has exactly the same behavior as that and it has the same behavior as percolation fluid percolating like water percolating through the coffee granules in your coffee percolator in the morning all of these things turn out to be mathematically related last year Bureau often an amazing experiment he took two concentric cylinders so this you can see them over here and he had he spun just the outer cylinder around and when you looked at that aejong what you saw was a pattern like this gray is laminar black is turbulent laminar turbulent laminar turbulent laminar turbulent laminar turbulent now the fluid is going round and round like this so you know you take a snapshot one rotation two rotations three rotations you take those snapshots and you put them under each other you look at where those turbulent patches go so in the first time you do it these turbulent patches are like this the second time you do it they've moved slightly to the right or to the left in some way maybe a little bit later they've run into each other and they've merged maybe a little bit later they split they form some complicated pattern shown over here so here's his here's Buren's data turbulence laminar turbulent laminar turbulent and this is the the position of these spots in time as they do their experiment this is the ecosystem that we simulated here is the prey we're just showing you the prey here there's prey and there's predators elsewhere and you look at the patterns of the prey as the system evolves in time and it also makes the same structure there's pattern here when you analyze this pattern it looks kind of like a fractal I'm sure many of you have heard of a fractal this is this pattern has very interesting geometrical properties and they show the same kind of scale-invariant cascade behavior that I talked about at the beginning mathematically they look just like that so these so why the turbans looks like the ecology is because they both when when examined in the right way make this pattern this geometrical pattern in space and in time and that's called directed percolation all right so what did you learn today well they told you is that turbulence is a unpredictable complex flow we've structured a wide range of length scales I told you that in your body the flow of blood is at the edge of turbulence and crossing that transition can be fatal I told you that through great ingenuity many people saw them here at this workshop can now measure the transition very accurately and quantify its behavior and there's mathematical analogies between what happens in turbulence to other fields in in science like the extinction transition in ecosystem percolation for scattering of high-energy elementary particles and also in fact it turns out that the behavior of signals in nerves mathematically also that turns out to be related to what happens in turbulence so here's our the people who did the work I showed you this is hung in and me standing next to Reynolds's original experimental apparatus it's still on display at the University of Manchester in England and you can go there and be a scientific tourist and stand next to this apparatus this is a from Lynn in the sunnier place the Manchester England what is the take-home message I want you to know theoretical physics starts with a simple guess because the exact equations are too hard to solve and with the scientific method is we compare the consequences I guess with experiment and the experiment suggests new guesses and so we have this virtuous cycle of exponent and theory simple models but not too simple but still very very simple can describe very complicated phenomena why is physics so successful it's not because physicists are smarter than everybody else okay it's because we work on simple only that's the reason okay we choose the problems to be the simple easy ones okay okay biology much harder than physics economics much harder okay so we don't really know to what extent the physics approach can be used in more complex Sciences like biology we're discovering that it can be used in some weight cases maybe not as much as we would like but perhaps more than you might have expected and the problem of turbulence is not solved but we certainly understand a lot more than we did in fine Wednesday thanks to brains and controlled experiments and healthy interactions between experimenters theorists and computational physicists I want to end with a comment this is a fortune cookie that I got but think about this I want there's something serious here okay so usually it's the case in science particularly in physics that physics and physicists is applied to some other problem like biology for example the Nobel Prize year or two ago was given to the invention of super resolution microscopy the ability to be able to look on small scales inside living cells using very clever and microscopic methods these were physics problems physics techniques that were applied and totally revolutionized the biological sciences okay so that's usually the direction physics make some new microscope physics make some new computer physics has some other thing like some other technique like optical tweezers and then you can do manipulations and experiments you couldn't do otherwise particularly in biology there's a huge flowering of biology right now becoming quantitative becoming more mathematical becoming more physical based because of these techniques but this problem of turbulence that I showed you and this is just my own work they're not not the other people's work which is which I didn't have time to talk about obviously but what we did there was something rather interesting because we did the opposite thing we used biology to solve a problem in physics because I happen to know about stochastic ecosystems and that's what we do in our in our biology work we realized that we discovered just by luck it happened to be there the problem of the transition to turbulence turns out to be mathematically similar to that and we knew how to understand her because we'd already done that so we actually used our knowledge of biology to solve a very physics problem so turbulence is a life force it is opportunity let's love turbulence and use it for change thank you very much questions two quick questions for it's pretty stupid it's a Ponzi scheme yes yes yes that's right exactly so the question was are examples of these simple models cellular automata yes you're absolutely right the say automata produce very nice patterns that look like things you see in the real world but it's even more than that well I showed you the superfluid hydrodynamics picture the three the the the strings moving around in three dimensions that computer simulation that was a kind of cellular automata very very very simple rules producing behavior that I really mimic what you see in the real world so yes that's what that's one of the examples of that so that suggests that you can use an impeller to whip water and produce heat you can use a propeller to whip an impeller like like the Vitamix mixer for instance because when I asked the salesman about that he said well you're just whipping air into and compressing the air so which I wondered which is it if because sometimes I think he told you an alternative fact uh-huh yeah I was I was thinking that so it seemed funny at the time so you can you can basically propel water to a certain swirl velocity to Purdue C yes or you will produce we're okay no thank you practical application yes hi so you showed how when you change like the birth rate and the predator-prey rate rate you had like a different the stable populations changed when you did that yeah and how like when you change it there could be like multiple like stable populations given that rate as you changed it what happens is if you start off with us a small localized population prey they will start to spread out but they won't just decay away uniformly the system will actually bifurcate and form to little communities of prey that will wander around and do their anything and I was wondering if related to like climate and like climate forcing and that kind of stuff if you see given a set of like specific like Sun and wind energy like incident on the earth if there's multiple like like any by FERC ation or like stability in that system different types I don't know right I think I think what you're asking is is if I force the Earth's biosphere levy the the climate are there alternative states that it can occupy and does it make transitions between them for sure yes and and and and in fact one of the big questions that we have is is about tipping points so in other words you can have one of those states and then as you keep heating the earth up are we going to irreversibly go into the other one and then once we're there it's too late there's nothing you can do to stop it and that then that's the wall-e that people have that that could happen that if we don't start now to avert climate change that will happen the big problem is the got you in this is we don't know whether there's a tipping point or not but maybe people suspect videos and we certainly have absolutely no idea of where it is we don't know whether we should be comfortable with a three and a half degree fahrenheit change in in the earth or one degree or seven degrees we don't we simply don't know and but the whole point is absolutely like yeah I'm yeah I was wondering if you know anything about airflow in the lung and I'm thinking there's something like a cough as I understand that there's narrowing within the lung and if there is in fact turbulence it helps you move mucus in the lung so I I don't know if I do I don't know I haven't done the calculation but I suspect I suspect so yes oh yeah go ahead what they measure is the the measure the the velocity that you measure at that point in space and then you plot it as a function of time and so you measure how it's fluctuating and then from the way it is fluctuating then you compute mathematically what was known as its power spectrum and and so that's the quantity that's plotted and then it's converted into into wavelength but by assuming that the the time and space of related in some simple way so so basically you most the time you measure the speed at won't you put your finger up to the wind measure the fluctuations and then you know because the fluid is flowing if I'm measuring a fluctuation here you know a little bit later it moved over to here and so that's how you can work out the spatial part of it what the PSD yeah yes yes that's right exactly exactly that's right yep so on the Reynolds experiment there were little puffs yeah are those standing puffs so they're individual particles that are moving through an area of turbulence and then they become stable and then they become turbulent in the same particle become stable and if so can those areas of turbulence be predicted that's a really really good question so so you might think looking at those puffs that they're just moving along with the fluid okay and and and they're not okay you're asking how the stuff is the fluid moving through the puffs as it were might like when you look at a wave coming into the shore the fluid particles and actually going around in little circles like this they're not only traveling like this but the net effect is to cause a wave it's kind of like that it's a very good question those puffs move slightly slower than the rest of the fluid and actually we don't know why we don't really understand how to predict that I I think it's a very important clue as to trying to understand the thing is that we things about this transition determines that we haven't fully grasped yet and actually the people who work in this have not really focused on that aspect but it's definitely it definitely is like that we when we do the computer simulations you move at the same speed as the fluid and you see these puffs going the backwards a little bit we don't we don't really understand exactly why so thank you very good good question I'm not sure I got this right but I thought I saw a graph where you had turbulence cascading and terminating in viscous flow yes yes exactly that's the standard standard picture okay good so I'm trying to picture how does entropy change in that is it discontinuous at that termination well okay there's a long answer to that you know you know every speakers nightmare is that somebody will say entropy and then because we don't know cuz we don't know what it is and and you know the thing about turbulence is that you're putting energy in and then energy is being dissipated somehow so it's not equilibrium so it's not like doing thermodynamics where you can say here's a system it's closed I can work out what the entropy is when you do entropy in school physics clinically whatever you're really looking at closed systems now when you have a system that's out of equilibrium entropy isn't really defined you can ask how is entropy being generated okay maybe that's what you mean and that's a very complex question where is it being generated well this picture that I showed you if it's true and by the way it's not clear that it's true but if this cascade picture was true the entropy is only generated at the small scales where the viscous where the viscosity starts to come into play and there's heating so the whole process of the energy flowing down through this cascade would not generate any entropy if it's true now there's some people think that this cascade may be very close to being correct but maybe not quite and maybe there's ways that the energy can go from the large scale directly to the small scales without having to go through the intermediate scales and if that happens that would be another mechanism for increasing the entropy of the system so very subtle and and it's to answer it properly it requires a you know a lot of very careful definition and what we mean by entropy and and so on but yeah you're right it's and makes you think about that okay so maybe the only chemical engineer in the audience but the Reynolds number is near and dear to our heart we teach it immediately and the predictive power and the practical usefulness of that is great so my question for you is since you found another system that's just described by some other mathematics is there another system for which you can be a dimensionless number that you can call the golden phone number well I wouldn't I wouldn't I wouldn't I wouldn't do that but yes all all of these systems you can non-dimensionalize the units in in some way so in the ecology problem for example we know how to do that and so so almost always you know you have these dimensionless numbers as you know so so in the ecosystem one for sure yeah but it's nothing new I mean this particular ecosystem is interesting because it's you know derived from the from the turbulent model from the numerical simulations but once you've we know how to analyze this those equations using standard methods yeah yeah yeah yes but particularly I think in understanding plasma fusion because the big problem in plasma fusion is how do you maintain the plasma and the thing that that the problem is that you get turbulence and then the plasma sort of fluctuates out hits the walls of the tokamak and then it's all over you lose the energy and the plasma doesn't work anymore and so you can't use it to MIT to make energy and what we've learnt is that the same flow configuration that we see in in pipe flow as I showed you at the beginning in my little picture this one that that special flow configuration actually is a predator on turbulence and sustains it and and then so the interaction of this mode with turbines can keep the plasma fusion devices functioning longer than the otherwise would and so a lot of research is [Music] and as usual we do have cookies and some refreshments in the lounge you

Channel: Graduate Mathematics

Views: 1,863

Rating: 4.8596492 out of 5

Keywords: Nigel Goldenfeld, Chaos, Turbulence, classical physics

Id: Dw6WwEaPr54

Length: 73min 55sec (4435 seconds)

Published: Wed Aug 29 2018