The route to turbulence by Dwight Barkley

thank you and thank you for coming it's a pleasure to be here so I want to talk about the route to turbulence in fluid flow so I think it's generally known turbulence is the great unsolved problem of classical physics and there are many there are many ideas about what the solution to the turbulence problem would be I want to talk about something very specific which is how do flows become turbulent so you you have a flow that under some conditions some low Reynolds sum or some low stresses the flow is smooth and laminar and then as you increase some control parameter some flow rate of the fluid becomes turbulent and that's one one important aspect of the turbulence problem to understand how you make a transition from laminar flow to turbulence and the thing is for when I'm going to focus on in this talk are this class of problems wall bounded shear flows such as and I've drawn the same ones here because I want to point to them from time to time particularly the flow through a pipe but also flow through a channel or couette flow where your have disappear of fluid between parallel plates these kind of flows there's been a fundamental change in the way in which we understand the onset of turbulence and these flows has taken place in the past 10 to 15 years and and that's what I wanted to tell you about and to to kind of motivate it I want to say a few things I was under the impression from earlier today many people know these kind of flows better than I do and I realized maybe not everybody and if you don't know about them it doesn't really matter but there are flows that are flow between concentric cylinders the Taylor couette problem so I have an inner cylinder here I'll just consider the case where I'm going to spend another cylinder I have an hour cylinder and there's fluid between those two and that's a fluid dynamical situation or you can have rainy bernard convection where i have heated from below cooled from above a fluid layer in there and i have some control which is the the temperature difference and in these cases starting in the early twentieth century there was huge success in understanding how these flows become unstable and then eventually through the later parts of the 20th century how they become turbulent all right and in the simplest case and again i don't want to spend too much time on this but the the the reason why at least in most parameter regimes not in all régimes this works is because as you adjust the parameter you have some amplitude adjustment I'm just trying to distinguish different states here it doesn't really matter I have some control parameter temperature difference or speed of the inner cylinder I can find a place where I can actually do an expansion a linear expansion and I can find actually an instability I can find a transition point and I can often do it with paper and pencil and then I can do some weakly nonlinear theory and understand what comes out of that bifurcation and then if I'm really good probably turning to a computer I can continue on and I can find in fact that this becomes unstable again to yet another state etc and then apply some dynamical systems theories some symmetries ideas like this you can eventually understand that it will become turbulent sorry ran out of room at least in a weakly nonlinear sense of chaotic dynamics okay and this problem is this is very well understood now I mean many open questions but at general program is is as well in hand and it's been extremely successful and extremely influential and really outside of fluid dynamics as well as I think people know and let me just point out something so I don't fail to to say it's not always this is what would be called the super critical case whether the new branch that appears is stable from the stable to the stable but it can't it can't be the case that they then it turns backwards I'm trying to draw it this way where this actually turns backwards and then eventually stabilizes this would be a subcritical case or weakly subcritical but that still falls into this approach to understanding the onset of turbulence okay now that the success of that approach has actually been somewhat detrimental I believe to understanding this that I think a lot of time was wasted by people including myself trying to take that picture and contort it to what happens in these flows and and that just is a mistake it's not the way to think about it and so one way to get to kind of motivate this is what's a question is what is the critical Reynolds number for the onset of turbulence in a pipe see over here I told you in both of these cases we know what it is we know how to compute it with basically a paper-and-pencil GI taylor did this 1923 he found this first critical point and and as well and you can reproduce in an experiment people can reproduce in all kinds of calculations and everybody agrees what it means and how to find it so Reynolds knew that this is pipe flow he knew that the critical Reynolds number for pipe flow I'm not telling you quite what it means but he knew it's about 2,000 in the way you would define Reynolds number for a pipe flow anyway he knew what it was and then in the 20th century there were some results Arata approximated is 2300 and then people largely because of Tom Mullins work began to be really interested in the pipe problem and these and these wall bounded to your problems and to look at them you know very very carefully with a kind of rigor and that we that people had looked at these problems and so then in the 21st century people began to look at this more carefully and these are the values that were coming out of studies and and just to say I mean they look funny but these are serious studies these are these are serious scientists these are published in high procedure NAL's and people simply could not agree what the critical reynolds number was for this flow okay and what it tells you is there's a lack of understanding of what we're doing okay it's not that people are stupid I mean these people are really bright and just to emphasize something that it really the fact that you could have looked in Wikipedia and this is something that went around at the time in this community is Wikipedia was full of all kinds of different answers depending on what a particular article nobody knew what their critical Reynolds number for pipe flow was and it's surprising how many places at Wikipedia this comes up because people say as in pipe flow the critical Reynolds number is and then they could just quote whatever they remembered and it didn't seem to have a well-defined value okay alright so I want to discuss this and I would it liked it presented in a way look you can read the papers I mean we've written papers on this and they're accessible and you can just read them I'd like to present it a little bit different than what you would just read in a paper because it is a talk format so I'd like to do that so let me just start with right now this is this is not mr. Reynolds that's a mr.foster but that's the pipe and just that's why I draw it that way that's why most people that's where the pipe is this experiment still exists in Manchester you can go and you get your picture taken on it there it is that's me I don't know you can't quite tell all right but anyway so this is this is what Reynolds told us that um at low Reynolds number this is a streak line that's tie at low Reynolds number the flow is laminar you get a straight streak line at high Reynolds numbers high flow rates let me just say I haven't defined Reynolds remember I should the flow is very quickly after the inlet it becomes fully turbulent you have a turbulent pipe flow but in between you have this you have an intermittent regime where you see these what he called flashes of turbulence and let me say now because I forgot to write it on those slides I'm gonna refer to these it's puffs and I'll just I can't stop myself because that's what we say so just if you see these if I use the word puffs that's what I mean by these little patches of turbulence in a pipe okay so that's what we know from for metals and the thing is what differs between these flows and this flows is that in these flows we have a finite amplitude instability alright without laminar flow first becoming unstable and in fact let me let me just illustrate that here this is not a perfect plot I'm sorry I did this is Reynolds number and as you can see it's on a logarithmic scale you just have to remember that okay and this is a friction factor which you think of as the pressure drop you need to drive a given plus okay it's a measure of the the resistance of the flow to forcing alright and that's also in a logarithmic scale you can't tell that and then there's a laminar branch and a turbulent branch okay and these are disconnected okay all right and this branch as far as we know never goes undergoes a linear instability that laminar flow of the laminar solution to pipe flow exists for arbitrarily high Reynolds number and is stable okay it never has a bifurcation so it is not at all in this class of problems okay so but that's it this has been known for a long time so it's so it's a it's a finite amplitude transition okay it's a finite amplitude transition and this is my favorite example of a finite amplitude transition that we can all do you know you could say it's a coke can it's under pressure all right but it's stable it you know there's fluctuations going around there's trucks on the road and so forth it's and it will just stay in this state indefinitely right so you're here and you'll just stay there indefinitely this is laminar flow you introduce a perturbation and you transition to turbulence okay finite amplitude perturbation is necessary and then you transition and it's an abrupt transition and it's dramatic okay so this is this is the way you should think about it I have a control parameter which is pressure and I have a state which is the inverse height of the can okay so of my laminar State which is the uncrushed cocaine and I have the turbulent state and there's a threshold depending on pressure and if I cross that threshold with a finite perturbation I jump to the crush state but you notice that these two are very widely separated this is nothing at all like this and it's not a weakly nonlinear theory you're not going to understand the crushing of a coke can well through this method now that's not entirely true people do oh no let me don't get into that all right let me just say that from our purposes that's not a good approach you're right I ice that's why I only rotating the inner cylinder that's right if you start because then if you start rotating both you can definitely put it in into that so that's that is exactly right I'm saying that if you only rotate the inner cylinder it does this I said if you start counter rotate look if you're doing exact counter rotation and working it and narrow gap then you're in this you're gonna be in this limit and it's gonna be subcritical and there in this case by the way we know it's provable that there is no linear instability this case we don't know but it almost surely is the case okay so what what we know from this then is it's the subcritical it's nice with they're called subcritical shear flows these wall bound of clothes that are in this case that there's as I tried to draw here there's a laminar branch and a turbulent branch here I'm drawing with friction factor on here I'm just going to drawing is a kind of generic bifurcation diagram and that there's a finite threshold that you cross and then you get a very very different state okay so that one is not a small perturbation of the other and there doesn't seem to be much in between and that transition where you begin to see turbulence and I didn't label it on here it's typically of order a thousand in the any usual definitions of rent it varies with between the particular flow and again depending on how you define Reynolds number but it's of the order of a thousand just to say I don't want to okay so what we've what we've known has been known Landau was the first person I know that said this he said that that the route to turbulence in pipe flow and in particular he said it's a hard transition it's a subcritical or hard transition do I use that word hard I used it someplace okay land I'll use this and he understood very well the thing is it turns out that that's incorrect and this is what took so long for people to understand and understand the right way to look at the problem so despite the fact that this diagram is correct and what I've drawn here is correct you have laminar flow that never becomes unstable you have a discontinued disconnected turbulent branch which is in no way close to this branch there's no measure in which these two are close no meaningful measure nevertheless the transition between laminar flow and turbulence is a continuous transition okay it has well defined a very specific Universal exponents with well-defined critical points and this is what we now understand okay so that's what I'm gonna tell you about okay I have to tell you a little bit about intermittency about this state all right so this is just a picture of I'm not going to go through all these in detail these are just a variety of different shear flows it just doesn't happen in these three or a variety of cases and shear flows in which when you look between the fully turbulent state and the laminar state you see these states here exists as long lives turbulent intermittent turbulent States oh by the way just to say you're looking at an orientation generally perpendicular to the to the walls and the length scales are large just add these or details but that's about 20 times the gap between the cylinders and these are dynamic this is a these are so called turbulent bands it's like this there's a patch of turbulent laminar turbulent laminar and these things have they just replay this you know this this is really turbulence there's really dynamics here this is not a steady state but it forms these kind of striped patterns this intermittent pattern that's a whole nother talk to talk about these patterns I'm not going to do that here if look this is really much sped up movie this is an experimental movie this is turbulence and that's laminar flow and it's intermittent and it'll just stay like this for as long as you watch it okay it's an intermittent mixture of turbulent and laminar flow and that is pretty much in these flows almost always observed between a fully turbulent case when you're up on this branch and the laminar branch in between there are these intermittent this intermittent flow but where you have a mixture of turbulent and laminar flow okay okay well that too is a long story so yeah as far as I know nobody ever won you probably know the Don calls cube of the oven where you measure the different states and you win the cube if you explained it anyway so as far as I know it's going on claim that price yes that's a complication I would still put it in this category of things it's things we can certainly now understand pretty well with a computer and that there are kind of low dimensional phenomena and they're not kind of the statistical phenomena that I'm going to be describing later but yes there's even within this there are a lot of complications with multiple solutions etc yes yeah what I mean is that if experimentalists comes along and measures friction factor they will lie and does it with good statistics over you know a sufficient long periods of time etc they will lie on a curb that people now everybody's data agrees that's what I mean and yeah and things are known about that although their arguments still about that it represents a yeah that's right but to come when we get to the end if I get to the end which I probably maybe it'll be that state I consider that's I consider this from my point of view simple because you can read it in Steve Pope's book anything you can open up a textbook and read in about five pages I'll call it simple so in that sense it's I I believe it's it's it's Lord all right so here's what we have what I'm telling you if we have laminar flow we have some fully turbulent flow which you could read in Steve Pope's book if you haven't done it yet you can read that in Steve Pope's book and in between we have this which I will claim for a long time it was thought just to be due to dynamical noise but we know it's fundamentally really a dynamical state of the system just I want to just organize it this way so control parameter which is Reynolds number I'm just going to use turbulence fraction which is a percentage of the flow that is turbulent after initial conditions have past efforts reach some equilibrium right so laminar flow has no turbulence so it's serving up rocks into zero if you have fully turbulent flow kind of by definition it's turbulence fraction is one that flows everywhere turbulent but these intermittent states they have a non-trivial turbines friction some of the flow is turbulent and some of its laminar again you have to run this for a long time run your experiment for a long time run your simulation for a long time and get an equilibrium value and then you're allowed to put a point there okay so it's an equilibrium value it does have an equilibrium and you can find it and it grows and so the thing is below here turbulence could exist but only transiently asymptotically there will be no turbulence above some critical values there will be turbulence and associated with this is critical phenomena haven't explained any of that yet that's what I'm going to tell you okay that because of the complexity of this spatial temporal complexity you go from something that is simple to something as spatio-temporal complex right from the onset which is very different from that picture and by the way it's also you have to work in a thermodynamic where the system size is infinite and the timescales are infinite and so that's you know makes difficulties [Music] whether it's obviously new silk number scaling is a function of rayleigh which is what every probably one of the big important problems so there are things like that but it's not exactly turbulent fraction I'm thinking about something that really you can dissociate distinguish turbulent and laminar but yeah let me think about that oh you mean it does everybody agree what that should be you're saying does everybody agree what the Devon yeah it's just the volume practice what you're defining in turbulence and then there's a question of whether you're precisely you have everybody agrees on that same definition of turbulence I'll just say for those if you're doing critical phenomena it that doesn't matter as long as you don't have something that's non smooth you'll get the same critical exponents for any but definition of turbulent fraction that's the beauty of that kind of critical phenomena okay so here's the way you should think about so let's ask you a question so is turbulence in a pipe sustained at Reynolds number 1900 so who thinks yes and who thinks no oh okay so let me give you a hint cuz you probably know much about this so let's show you an experiment so this is the experiments Reynolds number 1900 this is by Tom Mullen and George Pacino and there's a pipe there and they introduced a perturbation and running down the pipe and here's a simulation by Kabila also at Reynolds number 1900 and you see there's this patch of turbulence again it's it's localized it's this puff so up here is laminar flow and there's laminar flow and this thing is moving downstream it's chaotic is it it's um but it's really a fixed intent you can see it's really a fixed intensity the cameras moving downstream it's not speeding up or slowing down much you know so it looks really it's pretty stable alright so so that's a Reynolds number 1900 so we say yes and then so what we do next is we're gonna lower the Reynolds number and see if it's still stable there right no okay I tricked you so the thing is if you wait longer let's wait a little bit longer here so it looked fine it looks stable oops it wasn't okay so all right so the answer was no okay so what we're gonna do is increase the Reynolds number and do the next case they're both ideas are bad ideas the point is it's statistical and what looking at one realization is not going to tell you the answer all right you have to do multiple realizations okay it's a statistical phase transition so this brings me to the world of statistical phrase transitions and this yan sand and grass burger published papers on this in the 1980s and made conjectures about about systems and this would apply to heterogeneity heterogeneous chemical reactions to forest fires to flow through porous media and flu epidemics okay and many other things and just a quote from the paper it suggests another type of universality comprising all critical points with an absorbing state and a single order parameter so I have to explain all those words but that's the idea is that there's a kind of universality associated with these systems and you'll notice you could just see well this look at the forest fire example it's not like that okay it's not like that it's a different it's a different kind of example okay and Aoife mow realized in the 1980s not long after yawns and in graz burger published their work that these sub critical shear flows would also be of this type and they ought to also exhibit that kind of universality and this was taken up by the group in sac clay unfortunately for them I mean they had really the right ideas but they were too early and they didn't they didn't their system sizes would turned out to be just very very much too small so the answers turned out to be incorrect although the ideas were certainly there okay so I want to discuss this point I want to suggest what's going on there that's gonna probably take up most the rest of the talk I will see how things go so let me give you a mini tutorial on absorbing state transition so um the standard example and people you will hear directed percolation directed percolation that's the canonical example that is used in the statistical physics world and people refer to this is the director percolation problem and the universality class is director percolation I don't find that so useful to explaining it to people who don't already know it I find a much more natural example to use flu epidemics because they are spatial-temporal process as is this is a spatial temporal process and it just the connection is much much better in my opinion so what I want you to do is I want you to think about turbulence as a disease and so so here's this turbulent patch a puff and I want you to think of it as an infection and see there that's me so I am infected it's why I have the disease okay so that's why I want you to think about it all right so what we know of what I've already showed you is that if you wait awhile even though you're infected I stay in bed I drink the soup and then I get better and I go to the office and I'm working it and these equations well they would appear later but I'm not going to get to that part but anyway that's that's what I'm working on trying to solve this turbulence problem okay all right now the thing is so this is a space-time diagram of what I just showed you in this movie it's not exactly the same state but morally it is so this is space-time so that puff that turbulent patch exists robust again it's fluctuating but its intensity is more or less constant its speed isn't changing much it's moot and then a bit of abruptly it decays reverts to laminar flow okay it recovered and the thing is it's it's statistical if you run a bunch of different realizations you'll find that when this occurs it occurs at different times all right and I want to show you this because I think there's nothing like seeing it so this is I'm going to show you a simulation here I'll just show you and then I'll explain so this is running and this is this is pipe flow just take it to be pipe flow and this this top curve here which you can see that's red that is the turbulence intensity and at zero right now because I'm in laminar flow okay but if I I have to you know it's unless I kick it nothing will happen but I gave it a little kick there and I generated one of these turbulent puffs it's periodic so it goes out bad and it comes back in this end but it goes for a while the bottom curve by the way is the let's just talk about so you see it went for a while and then it reverted back to laminar flow I'm back to laminar flow now right okay and I can't get again and it goes to run and you can time this with a stopwatch sometimes they last a long time sometimes not so long well wow that's doing that oh it looked about the same here sometimes they're fast sometimes they're short anyway this other curve by the way is is the mean shear and that's gonna come up again later you see at UK anyway so if you do a bunch of these let me just pause it so I'm not using up up my CPU time on coming back over here again you'll get a you'll get what collect statistics and you can get a survival probability as a function of time so this is the the survival function how long you know what's the probability that it survives up to a given time 2000 time units and if you plot that from from either experiments or numerical simulations you find that's a logarithmic scale you find their exponential okay so that they occur just randomly but along this exponential so what you can measure from this is a mean lifetime they and that is the sensible thing to measure the the life of this particular realization really doesn't mean anything is indicative of something but it is not the thing to measure the thing to measure is a bunch of them get the distribution and find the half-life basically the mean lifetime okay how long on average did they last before they decay alright so that is the meaningful quantity and that's one of the first things that it took a while to understand is that you need to look at multiple realizations to get something meaningful oh and let me just say something about that little simulation I was showing that simulation I was showing was completely deterministic it's exactly reproducible okay I was changing the initial condition in the in the six-digit and running different cases okay that's otherwise they were exactly the same and just if this means something you I didn't point to this before but just in case it means something to you if you have a billiard bouncing around in and stay yet a chaotic stadium and then you just put a little hole here and look at the and and look at the escape rate from this this will also be memoryless applause own process and anyway if that means something to you that's that's the way to think about it so it's deterministic but it acts statistically alright well tonight this is 1900 here it's gonna be actually below the critical value wait wait you have to wait okay I don't need I don't need that I don't need this for the minute but I'm gonna need it later this is the time to tell you that as you would have guessed that the mean lifetimes depend on Reynolds number and they grow with Reynolds number it's a Reynolds number increases they're more they're less and less likely to decay they'll live longer they still live they still live exactly as these puffs but they live longer at high Reynolds number just put that aside and I'll come back to it now we're talking about disease remember now so disease so this is again infection but the disease can spread so here I am you think this is me as the disease I mean as you know there's a disease this is me I'm sick and in bed all right but what can happen is that these puffs can often split off a daughter puff okay it's known as puff splitting okay so a daughter puff has come off and so what you should think about that is this is my daughter and now she's sick too okay so we're all familiar with this idea right that's how diseases work you can I'm not going to do this for you it would take too much time and it's kind of too obvious at this point but you can measure it's also statistical you can measure lifetimes you get district you get distributions that are exponential it's a memoryless process now as you increase Reynolds number they become they the timescales become shorter they're more likely to split at higher Reynolds number I'll show you this again later just to tell you we can measure those things okay so here's the message so we have turbulence and with some degree of randomness it could either revert to laminar if that's just meant to represent them a bit of turbulence with some degree of randomness it can go to turbulent flow okay the laminar flow or it can take it can resize some bit of a laminar flow to multiply in it now I have two pieces of turbulence okay and then laminar flow on the other hand it's linearly stable and the absence of noise and everything it cannot continuously become turbulent if you're laminar unless you kick it you're never gonna become turbulent at these Runnels subs I mean eventually you know practically any any noise will drive you but this flow is not cannot spontaneously become target and this generically is referred to as the absorbing state because if you get in that state if the entire system is in this state you will never leave it it's absorbing if once you're in it you never leave and this turbulence is generically referred to as the excited state okay and these are the exactly the conditions that Yance and in grass burger we're discussing when they discuss things like forest fires and epidemics and so forth it also applies to turbulence as I've just shown you so again let me just show you some generic behavior they'll be immediately clear what it is because we know about epidemics so the generic terms are in the excited state I mean the active state or the excited state in the absorbing state which are is turbulence or infection laminar or healthy I think that those ideas are clear right and the control parameter the in I'm gonna show you some little trivial model simulations in a model I'm not really going to explain but just to communicate the ideas the relevant control parameter is the rate of spreading divided by the rate of decay okay so if you're so far as big it means you're spreading faster than you're decaying if I were small yet the way around okay so here are some simulations and just so you know it's a coupled map lattice that's what the onsite dynamics is it's a it's a tap map with a laminar State anyway it has all these correct properties and I don't want to discuss it further so I'm just going to show you two simulations both starting with initial seed so white here so this is space-time is going to always be going up here black is the active or turbulent state white is the laminar or absorbing state so this is an infection so you infect some individuals here I've infected a few and they spread they infect their neighbors and they recover and they affect their neighbors but if the if the spreading rate is small relative to the to the recovery rate then eventually the epidemic dies out and then everybody's healthy and once everybody's healthy that disease is never coming back it's like smallpox right no more smallpox and then here we have it we here we have the infection has become in to the population it just persist indefinitely in all time now notice it's not that one part of the not it's not that one individual stays in fact you know it's not that the population is I'm infected you're infected you're infected and that's the way it goes for all time no it's spreading around in this complex spatial temporal way and that's how it persists okay so we're familiar with this idea and I'm going to just show you this and it doesn't matter about the initial conditions it doesn't matter that it was a point you can do lots of you can infect everybody and then just see what happens okay and so there's let me just show you here you can very are a little bit more finally anyway what you'll learn from this is that there's a there's a well-defined critical point above which you have in the thermodynamic limit of an infinitely large system infinite time you will have the disease the infection list the persisting to infinite time below that with probability one the disease will not go to infinite time okay and there's a well-defined point that separates these two here the disease will necessarily died out before infinite time and here some will persist for infinite time okay all right yes if there's a question back there No all right and so the shear flows actually behaved this way well I'm gonna just give you some hints about this so we did this this ice this is the data that I showed you already about the decay and the split I told you we could measure those timescales so this is the mean lifetime as a function of Reynolds number and this is the decay events and these are the splitting events and these are the timescales and you'll see that they cross here at this point and that point is 2040 plus or minus 10 and so the way you should think about that is at this time if you're to the left of here the decay time scale is shorter than the splitting timescale so you so if you start off with turbulence you're decaying more your decay of 2 to the laminar flow is more likely than your spreading which means asymptotically you expect the flow to be just laminar above this point this this timescale is shorter than the decay time scale so you're spreading more rapidly than you're decaying so asymptotically you expect the turbulence to persist here right let me just tell for the experts what I'm not obviously not taking into account in this picture is that there's correlations they were going to begin to happen which is gonna change things a little bit and let me just continue with that soft-foot again for the experts but these these curves are super exponential and the rate at which these saints cross each other is so fast that within this plus or minus ten that the difference of the decay and splitting time is enormous that's just these little details but just those who worry about it anyway so if you look a little bit above that critical point you see exactly in these face line diagrams the kind of thing I was showing you with the disease so you have the patch it's splitting and there's some decay events and again a correlations begin to matter here but the turbulence persists and as far as we know it persists indefinitely through this combination of process and this is a way in which we can get at this critical phenomena by these measurements and let me just say that we know that the 2040 plus or minus 10 is correct because that's what it says it everywhere at Wikipedia now okay so it must be correct all right okay so let me just say a little bit more so we can do more I I mean it's gonna flash this I'll talk a little bit about this tomorrow so I don't want to really do it now but there's it there's there's a lot more associated with this is just a basic idea it was the first direct measurement of a critical point that really eliminated meant we correctly understood things we took we were working in a huge hugely long systems of our very very long times things that people want to ask me about later I can talk about but the thing is there's other critical phenomena associated with these and let me just give you a quick sense of that so I already showed you this picture so this is you know the control parameter this is a little toy model this is not actual turbulence and what you can see from this picture is that after I cross this this critical reynolds number in which the turbulence persists or the epidemic persists you can see that if you take an average a mean up here some you know large time and again you have to be careful about make sure you're in a statistical equilibrium etc etc but you could do that you could see the turbulence fraction is increasing you're going to see with your eyes and it's intuitively obvious and so the turbulence fraction is the right parameter for this system it is the right order parameter for this system and what you expect and what this little model will certainly show you is that it grows and it grows continuously and it grows continuously whether very well defined exponent beta okay I'll come back to the universe out in a minute there are two other exponents if you if you look now going to a decreasing or what you can see again with your eye is this patterns becoming sparser okay it's becoming and and because space and time are not equivalent they're kind of two characteristic scales one is associated with the time intervals and the other is associated with the space intervals okay those scale differently but you can see that they diverge as you as you come closer to this critical point so you can define and I'm not gonna worry about here you can define characteristic scales we have a way to measure them and you'll find that they diverge hence the negative exponents and they're a pair of those one for the space and went to the time all right so there are three exponents one scaling for the order parameter and two associated with the the characteristic space and time scales and so these are the three scalings you have and then you have many many other scalings which are all derived from these by manipulations but these would I'll take them to be the three fundamental ones and the thing is and this really is our conjecture at this point is that what yan said in brass Berger said is that these are to be universal and they don't depend in any way on the details of the system other than its spatial dimension is at one dimensional two dimensional three dimensional but otherwise it doesn't care about the details as with many critical phenomena okay and this has been observed over and over and over again in models yes or questions yes yeah no I'm sorry I probably I gave a misleading impression all I'm saying is that I'd like to think of it one is going up and when it's going down but no all of these things you should think of as equilibrium that doesn't say I'm going to take a parameter value I'm gonna run till I get statistical equilibrium make my measurements and then I don't care that that will now be it doesn't care what I increase right on sum or decrease Reynolds number etc all it cares about is that I started with some black that's the only requirement I'll come back to see I had to have put some black in here if I started with all white there wouldn't be anything to measure but once I put black and wait long enough and get equilibrium there's no hysteresis there's no history in this whatsoever I'm at oh yeah yeah okay let me get this is this is this good so this is this is a real simulation of pipe flow maybe I don't understand the question so what you're saying here is I believe stream wise for tissa t that would be the normal thing to plot in the pipe okay so now maybe I can find a dresser so what you're seeing is black and the other things what this thing correspond into one black dot in those little model pictures I was showing you but yeah okay yeah right you're asking very good questions and I should have addressed this would I the scales that were interested in here are are very long on the scale of this phenomena okay so this by the way this is blowing up by a factor too but let's not worry about that so this is roughly the turbulence that you see in a pipe is roughly on the scale of ten pipe diameters so if you have a diameter of a pipe the turbulence inside its stream-wise length is about ten pipe diameters okay now what I want to imagine doing is taking my pipe and what you have to imagine is that what the pipe really looks like is more like this and probably sooner than that piece of chalk it's four thousand I'm not sure if I'm answering your quick it's four thousand times the diameter is the length of the pipe so one of these little patches of thing here and this scale looks like a little dot there and that's where it's turbulent so I've sort of kind of collapsed mentally all of this to kind of just one discrete state either it's a turbulent puff or it's not but I still don't think I've answered your question because of the way your butt yeah yes that's right yeah you've got it okay yeah so space here this is the axis of the pipe it's along the axis of the pipe sorry and it's good you asses but and I should have made that clear but the pipes are going to be very very very long and so I've kind of worrying about the cross like what I see through the cross section I'm just collapsing all that to a single okay but I think I'm a bit better go on but also alright so and I showed these X yeah yeah okay so we those in this community have spent I'm not talking about the the flat pipe boundary layer we would love to understand the flat plate boundary layer and understand how much it has to do with this and how much it doesn't have to do with this it's a hard problem for the it's a developing flow it's a hard problem and it's still under work that's are you yeah and when we talking about the successes not the failures okay so let me just I'm just gonna flash this so because it's not our work so this is a work that came out of yarn Hoffs lab you know really a tremendous set of experiments going on there the first author is Gregoire lulu and what they built you can't really see this it's a taylor couette device but the fluid is essentially it has very long circumference relative to either the height there's a little tightening the height is 8 times the gap the gap is the gap and the fluid is basically three thousand times that so it's very again as I was trying to communicate here with the pipe the idea is and I didn't really says we're treating this is one-dimensional obviously it's a three dimensional flow there's there's turbulence in here but it's got one really long direction in which the fluid can explore this intermittency okay and that's what this is this is essentially a one dimensional fluid it's really turbulent in there this really but but it can only explore this this one large dimension okay and if this won't convince you this is real experimental data I mean it this is what the turbulence is you have these patches of turbulence within the flow and they're and they're going along in this way they're just like these little mount toy models that I was showing you so it kind of percolates through the through the system so this is again the disease so the disease almost died out if it had died out it wouldn't you know it wouldn't over with but it continued on this is very very near critical and I'm not gonna flash a bunch of log-log plots you do a bunch of log-log plots you do a lot of hard work it's really a lot of hard work and they find the critical exponents and they find that they agree extremely well with the known theoretical values okay and just to say I don't want to spend too much time but this is the first direct confirmation that the sub critical root to turbulence really is continuous via this very specific phase transition the way we know that in particular is that there's these Universal exponents which this look at that flow exhibits okay there really is this this this way in what you understand it is being a continuous transition and let me just say if you don't try if you don't do it that way and maybe I come back a little later if something you're just gonna be in trouble this is what caused all this trouble at the beginning I said about trying to define the critical Reynolds number if you try and give some other definition it just seems to be the wrong thing to do you know and I'm gonna say some more things about that when I conclude okay yeah yeah yes that's right it is it is okay we have a long discussion about that puffs actually are moving upstream the book there's a mean flow they're moving downstream on average they're moving about the speed of the mean flow they're not moving at exactly the speed the moon flow that's kind of important that they're not moving like but there's definitely slower and in the other direction I would be interested to why you're asking that that's a very insightful question okay so I yeah this cap you're taking this off my time right okay go ahead sorry I'm talking about one hour one dimensional DP bottles what we know from one dimensional DP week we cannot I'm kind of unclear on that because I think they're only known numerically I don't think anyway I mean we they're irrational those exponents are irrational and I think they're only known to find a precision from that's my understanding you can read that look you all right so I'll talk about this tomorrow I'm not going to say it now but with Matt Chantry and he really did it all we needed so that was a very nice work on a one-dimensional problem but I told you it depends on the number of space dimensions and so we wanted to look at a planar shear flow don't think consider a two-dimensional problem let me just show you these are this kind of intermittency in one of these planar flows again you're looking at it on there's a wall normal direction which you don't see here this was a little the largest numerical experiment that had been done to date and this is the largest a physical experiment that had been done to date and that's perfectly those are perfectly fine for understanding what you see in this plan this figure but the thing is Matt had to work out a way to go to much much larger domains and even this is kind of marginal so that's fine for this for this year but the thing is this turbulence becomes very very sparse as you go close to criticality and you have to be able to resolve things like this and I'll tell you about that tomorrow okay okay so let me so I have still a good 10 minutes right all right okay all right so yeah all right I'm trying to think of how I want to do this this is this was a blank slide in there to make me think all right so I don't always explain things the same way so let me let me say this what I've been describing is you know how you think about going from laminar flow to turbulence you know that well again let me just summarize a few ideas that we've had one is that for a long time it was thought that these things were just noise driven in perfection and experiment we know that that's not true that they are very very much they act statistically but they are definitely deterministic phenomena and again think about billiard bouncing around this is a completely deterministic phenomena but it acts like a memoryless process so that's the way these things are and and this really provided a lot of clarity in trying to understand things like what is a critical reynolds number and so forth it really until this occurred there was huge fights about this and now all that is gone away we at least understand the context in which we should be asking the questions but from a fluid mechanics point of view that's not the important quote I mean that's solved that you know that stopped a lot of fighting but it it doesn't really explain why there's intermittency in the first place which is really the more important question because I told you that these phenomena were universal they happen in in flow through porous media they happen in disease epidemics they happen in for as far as it well if they happen in all those systems well how could it not happen in I mean there's pressures there's a global problem I mean there's recent stuff but the thing is why did it become intermittent in the first place because once it's intermittent then probably it had to obey the universality so the question is why is it intermittent in the first place I believe that that's the more fundamental question and if you understand that then everything else will follow now yeah why there isn't maybe yeah you guys ask a lot of good questions yeah all right all right yeah put that aside all right all right let me show you let me go back to this let's let's play with this a little bit so this is my simulation and again so let me just remind you what I was doing if I give it a little kick there I can generate one of these puffs I'll show you so so this is this is a simulation of these puffs and then as I as I as I showed you if you I didn't I didn't actually run this in the in the model let me just show you this you can this will split see oh there see you pop splitting I told you you know I've increased the Reynolds number now it now it went from one to two does everybody get that and now it would is a three right so this is this splitting phenomena so I'm in this intermittent regime so this eventually will kind of fill up this this geometry with these with these patches and they'll kind of decay and spread and kind of keep it in this intermittent thing and you can compute the statistics of this and you find that it a base of percolation transition is directed percolation transition with their critical exponents okay so that's the thing but let me just say that we I wanted to show you what this if I increase the Reynolds number further and I hope I did it about the right amount and I give it a perturbation what happens is that perturbation just spreads and it just takes over the flow just becomes turbulent everywhere okay and this is what most experimentalists in pipe flow do I mean they probably don't even bother with the perturbation because if you work up at high enough Reynolds number the perturbation that you need is practically the scale of your you know the defects in your experiment and you just simply jump up to the turbulent branch okay the thing is then if you bring down the rent on someone just say what you know you'll get back to this intermittent state okay and the question is why did it do that and that's one way of looking at it you know forget that I came up from below why does it do this why did it become intermittent like this and I'll you be hit so there's two things here that's the turbulence and here's the mean shear so let's talk about that again I'm gonna pause it so I don't use up on my computer CPU time so let's see if we can explain why there's intermittency so for that I need to show you two things so this is what we've been discussing up to now this is Reynolds number 2000 this is this intermittent state just one patch of turbulence a puff again this is space and this is time in a code moving frame these things move downstream but I'm looking at a code moving frame and and just to answer your question this speed code moving frame and Reynolds number 2000 this is this is in the frame of reference of the mean flow and it hardly moves it all in that frame of reference okay so asymptotically I have one of these or multiple of these and that's the intermittent state if I go to high Reynolds number then turbulence what I just showed you the turbulence will just spread and fill out and again this is all moving downstream but we're Co moving with it so if we see it's spreading both ways and then you get to this and this is what you just called turbulent pipe flow that's just what you know as in Steve Pope's book okay so I want to understand these two and but I really want to understand it in this spatial temporal way I don't want to just say this and this I really want to talk about this okay that's why I'm showing you these all right so it's easy to understand oh I didn't you say the words let me say the words this is called a slug this is called a puff as I've already said this is called a slug that's just the word that's used I can't help myself I will use that word okay so let's we can understand this slug very easily all right so upstream there's laminar flow that looks like this let's see what's gonna and you should think of this this has this has a high centerline velocity there's a lot of kinetic energy in this laminar flow okay there's a lot of kinetic energy in this laminar flow it's gonna flow into it's moving relative to this interface you saw that before that interface in the frame of reference of this mean flow this interface is moving that way this front is moving into it so anyway though these two are moving together and what happens is right at the beginning and you could the intensity of the turbulent what you're seeing here in colors is the turbulent kinetic energy I should have said that that's the turbulent you can see it's really intense their production exceeds dissipation there because you have all this basically a source of fuel which is this upstream laminar flow there's a huge amount of kinetic energy in there it comes in there is you extract kinetic energy from the mean flow and put it into the turbulence okay and then what happens is that it's because you've extracted kinetic energy from the from the profile you've blunted the profile to a typical turbulent profile this is just a sketch okay you blunt the profile and then you reach some some this equilibrium downstair downstream where the production is equal to the dissipation okay and I really really want you understand that this is an equilibrium and you can write down Reynolds average navier-stokes equations okay write down the Reynolds average so I'm not gonna do it you'll find the production term is just this it's a product between the the mean shear and there and this component of the Reynolds stress and this is the only term that enters into the production and again in case people are in experts the word is used as production means you're producing turbulent kinetic energy but really it is a transfer but this is doing it's taking energy out of this mean shear and putting it into the turbulence so from the point of view of the mean shear this is this is your losing energy by this but that's what it's that's what this is so there's an equilibrium setup between this shear profile and the turbulent fluctuations that just exists for forever and again this is extremely well understood it's been known for a long time just open up Steve Pope's book you'll have a complete description of this problem okay you can't analytically solve for these things but numerically we know what they all are okay so again that's the clue now what happens when we turned down the Reynolds number when we turn down the Reynolds number that means we increase the dissipation the production doesn't care about Reynolds number not directly the dissipation does there's a factor of viscosity in front of it well what happens is this equilibrium is lost and it's gone however you can see that this where production exceeded dissipation appear that does not die at the same place that this does so this equilibrium is lost but you can still maintain this upstream in ten front okay and so that's what a puff is it's just what remains of this turbulent slug after that downstream core is lost and you get an you can see it very clearly there's lots and lots of way you can see it this upstream front is just what survives and becomes this that's all it is so you lose this this downstream equilibrium it's a car yeah it's a computational result and that's a problem under consideration is to do a much better job with that okay yeah you're right okay so what happens there's a whole long story here too there are the this this undergoes a couple transitions in the process asymptotically at large Reynolds number these two burns do become the same because what happens here is you're over running downstream laminar flow and the exactly a symmetric thing happens to upstream but in between this front speed the the difference in the local advection speed changes that picture the fact is that this is what my arrows aren't on here the fact that there is a difference I have flow coming in here like this so they're not symmetric so a transitional Reynolds numbers right when this first forms it is not symmetric front to back but asymptotically you know large Reynolds numbers it becomes again that's probably good way to say it certainly say whether it's yes that's right that's right exactly that's right yeah you guys are right good yeah so here so here's the thing so I think you got this at what a puff is it's just this it's just the upstream front that remains after you lose the thing well what is key and what I told you essentially was that the mean shear profile provides a negative feedback mechanism providing the localization it's the fact that here I have this when this goes through that when the kinetic energy is extracted that blunts this profile and it blunts it to the point that that the disturb you lence you can this turbulence could no longer extract enough kinetic energy out of this mean to sustain yourself so it decays then once the turbulence is decayed then that then that Lambert profile can recover okay and I'm going to show you a movie and I hope it's clear enough here to me this makes it just so clear but it's whether you can see it or not so this is again a movie by both Lang songs we did a much of this work and that's a high Reynolds number turbulence say you see it's not completely uniform but and then what he did is you drop the Reynolds number and let's just watch it and then I'll point to some things and you'll see that you could have imagined the whole thing just collapsed you know although all the turbulence went away to zero but in fact it doesn't it forms these localized patches now they seem to have come out regular rather evenly spaced that's not always the case but let's just think about what what's going on here the thing is at this Reynolds number if you tried to put turbulence everywhere in this pipe there would be too much dissipation there would be too much dissipation relative to the pressure drop and it could not sustain itself energetically it would collapse however it can sustain itself in this state and the reason it can is because there's turbulence oh look whatever here the flow there's there's there's no turbulence here so there's less dissipation the flow can reaccelerating you can re-energize the flow through that acceleration you're extracting energy from the pressure gradient you're not spending any energy you know and dissipation into turbulence so you could re-energize the flow here that provides a nice source of energy and then you can burn some more okay so you can that is how this works and it's why it works is because of these two variables there's you know and that with a negative feedback okay now we'll see let's take a look at this does this remind anybody of anything yell out if you don't do does anybody here work with say the brain or the heart or anything does that remind you of anything it's their action potentials these are action potentials this is a model I've been not so clear about that I was gonna make its a 1d model okay so I find this remarkable that that this has not been known for a hundred years or 50 years but it doesn't seem to be the case that if you look this is actual data from a pipe this is not a model this is a cartoon representation but I have one of these puffs and there's a sharp peak in the in the kinetic energy and the flow and then the mean the the profile gets blunted and you have this localized patch here again the two variable thing and what does a neuron do again if you don't know about it well I think everybody knows the basic thing they have action potentials a signal electrical signal propagates down the down the neuron it's a depolarization between the inside and the outside of the axon so you measure it as a voltage and that's what it looks like and it's flow all right and so let me just go on so look we really have the situation many not all nerve cells are like this but that they will sit there in this unexcited state for as long as they're kept alive they'll just sit there like this just like pipe flow laminar pipe flow so you a Reynolds number right about here you know turbulence is possible for but it will sit there and if you start in the laminar State and don't kick it in any way it'll just stay like this right but if you do kick it either one of them if you kick the cell by having your signal coming in from a neighboring cell you'll excite this action potential and in this case in the case of pipe flow likewise if you kick it you looks like one of these and the really strong analogies between these two and in these kind of problems and I'll we're a point to it to an abstract that was just a talk that was held yesterday over there just work on this an excitable media again if you're not familiar with this just try and take it in a little bit we know that either you can go to the full physiological models this is for cardiac to your hearts also an excitable medium I'll get to that in a minute the waves propagating through it that once per second wave that goes through your heart is basically like a puff right anyway so there's a detailed physiological models which is you're thinking it was kinda like than every Stokes but you can capture the essence of these things were two simple two variable models they're two scalar variables here depending on space one is the voltage and the second is a gate variable and what I want you to think about that is we can do the same thing for for this pipe model where we have one variable is the the turbulence and the other is the mean shear now I know means here you have to say what you mean by that is a scalar representation but that's the idea that we're just gonna map all this down and I'm not gonna say anything more about I've written a lot of papers about it so if you're interested you can read those but this this model which is essentially just an excitability model is what I've been simulating up till now and here's again the basic idea and I'm not gonna explain in full details I don't want to but in poor people or whatever it just try and take it in if you're if he's not yet familiar with these ideas so this is my slug alright and this is um this is a model and this is the mean shear or you know and I have just a scale of representation that I'm like of way too much and this is the turbulence and this this state here is a fixed point corresponding to laminar flow the turbulence is zero and the means the shear profile is fully recovered just think of it that way and that corresponds to be up here in physical space that dot there then as I come through this this slug what do what happens what happens is I the the the turbulence increases like this the turbulent intensity increases as a result of that the shear profile blunts and I end up there so that the turbulence increases the shear profile blunts and then I reach this nice equilibrium downstream and that is that process and I reached this nice stable equilibrium in the case of a puff what happens is starts off the same I have this laminar fixed point and then I hit the turbulence in turbulent intensity goes up this year profile blunts but I can't I no longer have an equilibrium accessible to my system so what happens is the turbulence drops back down and then eventually the main she recovers and that by the way is the way in action potential works okay and so there's my arrows so I have this this excitable case which is the the localized state the puff and not the bi-stable case which is the slug the thing that fills the pipe and yeah yeah okay so let's see if I can do it with just this picture yeah yeah you could see it here so there is when this you're pro to see that this one has and you can work it out oh but now I've forgotten so let me get now I give you a number because I don't tell you the wrong number but this has quite a bit more kinetic energy than this you have to realize kinetic energy is the square of the velocity and of course I know that this is radial and so you have to take and took out the relative volume so that's all you have to be careful the answer but this actually has quite a bit more I believe it's about a factor of two more kinetic energy in it than this it's just simple acceleration of laminar flow down a pressure gradient oh you're talking about in this in this or in real pipe flow in real pipe flow there's a pressure gradient here and the pressure gradient is accelerating the flow the whole thing is driven by a pressure grade what what is going on in a pipe is I have a pressure drop here from here to there I have a pressure drop okay and everything that's going through that takes place in between is to is getting energy out of that pressure drop and it's either coming out the end with kinetic energy or it's dissipating it inside but one of those two things has happened to the kinetic energy you put in potential energy there's a certain flow rate so there's a certain power going into it and that power goes someplace either comes out the other end or it's lost to heat because what I'm saying is a transition the whole is captured by excitable device abilities together with fluctuations that you have to put in fluctuations on the upper branch and then this is no hold another talk about how do you put in the fact that this is not fluctuating but this is and you could either do it with multiplicative noise which is kind of formally the easiest thing to do or you could do it deterministically which is has a lot of advantages but you've got to put in the fluctuations but you have a nice thing you can look at it with or without fluctuations and I want to end cuz I don't want to keep people here too but the thing is with these ideas you can reproduce essentially every known phenomena in the in the pipe flow transition problem this is this is reality meaning either experiments or direct numerical simulations as in every Stokes this is a simple model this again is reality the crossing point things I did not talk about and then this is the model and I'll just tell you it gets the speeds almost exactly correctly there's a lot of details in those speeds that you were asking about okay I'm just gonna say just a word about this because Nigel is a colleague of mine we talk about this all the time I'm not talking behind his back I just I want to say this because I know he's gonna come here and sometime in the future we disagree on this I told you what I think I think I have tremendous amount of evidence mahai on my side Nigel is claiming this these phenomena are due to zonal flows I completely disagree and I think there are many problems with the zonal flow point of view and I'll be happy to discuss that privately again I'm not talking behind I chose back he knows I disagree with him and I just don't think it's right I think it's not right so let me not let me end on something lighter so again I will just show you something here this is so I told you the heart is also use the grey's Anatomy's Meredith spray saves the patient boom okay so now I don't know oh this is gonna be really hard for you to see can you see that or not see it do we throw the lights on no anyway yeah say this I know this is a TV show I I know that but but but but nevertheless it's for you so that's you know the save the page the heart went back those localized those localized things you see on the East ECG EKG or those are really these excitable the fact that you're the way it's going through your heart are localized is for exactly the same I mean not not the same physics going on when is electrophysiological one is fluid dynamics but the same negative feedback between an excitation variable and something that cuts off that expect that excitation burn it's really the same and they there's you could just property after property you can you can map one onto the other you get two speeds everybody in in in in neurophysiology knows exactly how speeds behaved and what the dispersion restitution and and these behaved I mean and and pups behave the same way I just want to say that and there's a little fun okay I'm gonna conclude now because I don't want to keep you player too long so I've talked about you know about this and particularly about the idea that in these spatio-temporal complex systems you know we really do know how to now understand them through this critical phenomena I also claim that we understand a lot of the physics which gives you this in the first place and we understand that really it is a very good way to look at it as opposed to the way that those systems you know I'm not trying to put one over the other I'm just saying that this approach is really kind of necessary when you have these kind of phenomenon these strongly sub critical systems in which the turbulence behaves the way the way it does here and you know I think it's useful to think about this look if you want to say everybody can understand this if you want to understand forest fires okay you know um you're not likely to sit there and just watch it's not a good way to understand forest fires you know what you you know it's you have to light some trees on fire if you want to understand the way forest fires work you know and if you have a strongly sub critical system like these you know you have to introduce perturbations and then see what happens and this is what we learned is that you inject you weight the things you have to wait a really long time in a really long pipe and then till you reach a thermodynamic equilibrium I mean a statistical equilibrium and then you measure the statistical properties that is the way in which you can understand these these systems in a way that you don't cause all these arguments okay and it's really brought tremendous clarity to the our understanding of these problems and the other thing is again there's a there's a whole class you know the people over here I mean maybe some of you are those people - I mean this statistical physicists have known this for a long time there are many many many examples of this so why wouldn't you just put it in that natural class with all that understanding that already comes with it and it's a beautiful confirmation of these scaling laws that were predicted long ago and I'll just end with this very last thing is that you know this must be the right way to look at this and I'm gonna give you a quote from Osborne Reynolds from 1883 and it's really remarkable that Osborne Reynolds said this I'll just I'm gonna read it word for word I'll read from here it because word his language is not the best but so I'll have to translate a little bit even though it's in English okay it became clear to me that if in a tube of sufficient length again it had to be very long much longer than he thought if water were first admitted in a high state of disturbance are you gonna have to disturb the flow that's the right way to do it then as the water proceeded along the tube wait a long time that the disturbance was settled down into a steady steady condition which would be either one of Eddie's which we would now just refer to as turbulence or one of steady motion laminar flow according to whether the velocity was above or below what might be called a real critical value so that's exactly the description of directed percolation and everything I've told you that that you you have to you have to introduce the perturbation you have to wait wait wait go to some thermodynamic time and then see whether you have turbulence or not and measure those things and when you do that you get a real critical value without all thank you [Applause] the exact coherent solutions for the navier-stokes equations okay so that is an ingredient thing that I didn't discuss because they're just there's so many things yeah those there's certainly a lot of overlap I would say that that those ideas are trying to understand if you just kind of pick a random figure in here let me pick this one no yeah this so in other words this is quite a complex object even though it's relatively simple compared to a lot of things and you would like to understand why you know why does the turbulence have certain you knows length scales and things like this you one way you can understand it is by seeing that embedded in here are some simpler structures that we can get at exactly in these methods or even in the inter of this pipe and so they provided a lot of insight for this they provided a lot of insight into the boundary between turbulent and laminar flow they don't seem to at this point and they do begin to say something about the localization although I would criticize they reproduce the localization let me say it that way but what they the question is that they don't see anything about if I have two puffs all right buy one puff going to two puffs or if I have one puff dying they don't really address that problem and that's really what you kind of have to so there there are certain things they do tell us things about and other things they don't tell us about and most of what I talked about today they don't have much to say about which is why they didn't come up in my talk I think they're very valid they're very interesting but they just don't I don't think contribute much to our understanding of the percolation phenomena I think we should oh yes okay do you remember Fabian czar answer to non-normality he says yeah it's now the point is that the thing is you non-normality is throughout all of this because the operators are highly non normal it's not really it's about hold oh yeah okay break I I would say the distinction is non-linearity I say it's that's the non-linearity it's a it's by far the the important thing it's not I would say in fact I think you can probably find two systems which have whose operators have a you know the same pseudo spectrum I mean you know same degree of non-normality and one is it behaves this one won't but that's a conjecture I don't know that that's true but I almost it's not the non-normality I mean it's gonna be not normal but that is not the key distinction yeah but this gets back to the thing I mean it just to say the exact coherent structures tell us a lot about free Moi's for tissa tea and how that plays a role and i agree with all that by the way but the thing this is a what I'm claiming is really crucial here is the spatial when I shouldn't take it man but I can show you plots of the the all we have all the energy balances and everything going through these puffs and through these interfaces and the thing is what is going on here and along with here is very there from what is going on here it's a very spatially very inhomogeneous situation and not only is it spatially in homogeneous in the stream-wise direction the way that that shear profile bends is going through there is really quite complex so the thing is we don't those I'm sure you could I mean that's a it's a legitimate question it's just not known you know how do the stream-wise vortices and how does that picture you know what's I have to do here because people talk about the sheer layers you could see these things are slightly distorted in the shear layer so there's a lot of things there I mean if you look at one of these just after that you're gonna see a lot of stream-wise vorticity and you're gonna see a lot of these rolls in those structures that will always exist in a wall bound and shear flow but I don't think it's I think it's so spatially and homogeneous that there's a lot more going on talking about plane couette flow yeah yeah yeah okay yeah okay yes but from what I understand I'm not an expert on this but what from what I understand from pictures and Lirette may have something to say about this too is that you can see I believe if you're talking about say circular couette flow I believe that it really quite high right and this goes back a long time I believe even and Wrekin swinney noted this is that you can see some kind of large scale pattern you know when this flow has become very very highly turbulent there's still some structure in that turbulence it seems I see you nodding your head yes yeah so I'm it certainly it ceases to be intermittent and and it's fully turbulent but I believe that it yeah that I believe that there's still one sees some structure in that turbulence that is not just now I don't okay let me say I don't know it's easy is to say I don't know but yes I believe you know you turn up the Reynolds number at large enough and yes it becomes fully turbulent let me say this I think that if I understand your question correctly you know I showed you some model simulations based on a very simple some very simple ideas of in fact I'll just go to here and I was asked about this I showed you some very simple ideas with these pictures and then what I said because I did is it you need to add some sort of fluctuation I believe what you're denis those fluctuations in order to have the decay events the spreading events all those things those fluctuations have to be there and to this point the way we have added those fluctuation is either with multiplicative noise or with a certain kind of map models that are reminiscent of that and I think there's a lot of fluid mechanics that I'm ignoring that moment surely has to do with vorticity and other things that would explain what the nature of those fluctuations is I've just treated is kind of like a noise problem but it's not many people want to go though that we should end [Laughter] thank you [Applause]

Channel: International Centre for Theoretical Sciences

Views: 1,130

Rating: 5 out of 5

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Id: 197brZdt2q4

Length: 80min 54sec (4854 seconds)

Published: Thu Feb 20 2020