Fluid Mechanics Webinar Series – Barkley

[Music] so it's a great pleasure today to welcome Dwight Berkeley from Warwick Dwight obtained his PhD in physics from the University of Texas at Austin he worked the one year in Caltech with Phillips upman and then you spend three years in Princeton working with Janice Kepler Kiedis and Steve Orszag and in 1994 he joined the faculty at Warwick Dwight I'm sure he's known to many of you who's the recipient of many prizes and awards and there's done much seminal work his early work was into waves and excitable media he's done a lot of nonlinear dynamics he's also known according to Wikipedia for deriving an equation to estimate how long it will be until a child in the car ask the question are we there yet it sounds like good research he now works on transition into turbulence is the author of the first Jeff and perspective in fluid mechanics on this topic and is also part of an international team recently funded by the Simons Foundation to revisit the turbulence problem using statistical mechanics so welcome Dwight and so thank you and I want to thank the organizers not only for the kind invitation but for organizing this series so we've come to really look forward to these seminars on Friday afternoons and I want to thank Steve personally for his patience and help in getting set up for this talk so notwithstanding the title that I gave what I'm going to discuss today is the route to the turbulence in wall bounded shear flows and over the past proximately 20 years but particularly in the past ten years there's been a fundamental paradigm shift in our understanding of how these flows become turbulent as you increase my models number and we now know that what's important is it's a complicated spatial temporal process with deep connections to critical transitions in statistical physics and that's what I want to tell you about so let me start with something simple this is a subcritical transition also as a hardier first for a transition what you see here is a tape measure with a magnetic ball stuck on the end of it to add a little bit of weight and right now it's currently in the mechanical equilibrium there's a control parameter for the system which is the length of the tape and I'm going to increase that control parameter and I maintain a continuous variation of that first equilibrium however there's now a second equilibrium is available to the system there's a threshold between the two its reversible you can go back and forth a decrease control parameter a little bit that cyclic equilibrium so we do this I can transition between the two back and forth perfectly reversible as I continue to decrease that control parameter the length of the tape eventually I get to a point where I lose that second equilibrium here so I only have to one equilibrium that when I'll refer to as the sequel simple equilibrium so we're all familiar with this and we would typically plot a bifurcation diagram as shown here we plot the control parameter is a function of some amplitude think of it as a deviation from the simple state so we have one equilibrium which is the unamed tape measure it is this and is stable for a large range of control parameters but it's some finite value of the control parameter a second equilibrium is born through a saddle node bifurcation and we have two equilibria and a threshold separating the two so this idea is I think very familiar to everybody here now so critical tier flows have basically the same characteristic we have laminar state which is a mechanical equilibrium of the flow and it exists for all Reynolds numbers or all Reynolds numbers of concern and it's linearly stable for all Reynolds numbers of interest in this talk and then but it's some some finite Reynolds number typically on the order of a thousand it varies with the precise definition in the precise flow but anyway between a few hundred and a few thousand turbulence exists as a coexisting mechanical equilibrium of the systems so we and it appears at finite amplitude and it's separating from the Lamoreaux branch by some threshold which is not as simple as shown here but nevertheless it it exists okay and so this has led to the view historically that as one increases Reynolds number in these flows the reach to turbulence is discontinuous and one of the things that we've learned now and we understand quite well is that this is just an incorrect statement when viewed in the way that you would say properly or naturally view the system the route of turbulence is in fact continuous at least in many cases it follows a very specific statistic called phase transition with well-defined critical points and Universal scaling so that's what I want to tell you about now and wave a little bit more backgrounds let me turn to some observations by Reynolds from 1883 these are some sketches modeled after his paper so here we have pipe flow this is the entrance to the pipe and dye is injected upstream and Lovelock values of the flow rate or what we now call Reynolds number the flow is laminar and you get a straight streak line at high values of the Reynolds number the flow becomes turbulent within a short distance of the interest of the pipe and so these are two mechanical equilibrium now as was well understood by Reynolds in fact and what is not shown here is that at this higher Reynolds number the one shown here if you're very do this very experiment very very carefully in fact the laminar branch still exists it's still linearly stable it's still accessible it's just it's here it's masked by perturbations and noise now what Reynolds also noted was that there's a regime in which the flow is intermittent or transitional abuse of modern words in which turbulence is intermixed with in laminar flow alright and this is going to be our interest and I put this on here just to remind me because I'll do it without noticing these today we call these patches of turbulence within the laminar background we call these puffs that I will just I will tend to use that word so what we want to understand are two things in this talk we want to understand how it is that turbulent flow becomes intermittent and transitional and then we want to understand by what scenario is this transitional flow we prove revert stool and a flow that's what it is I want to discuss okay so let me show you some intermittent flows in a variety of cases I'm not going to discuss these in detail just with one exception with is this plane couette government here for preaching at hell from 2002 and I'm showing this for a couple reasons the first of which is is this is how I became interested in the problem what you're seeing here is plane couette flow now unfortunately I cannot see my my self so I hope to show it to you here this is a model of the experiment by preacher now so there's a plate moving in one direction on one side in a plate moving the other direction on the other side that's plain cloth and there's a fluid in between and this is approximately hopefully you can see that this is approximately the correct aspect ratio of this experiment and let me just say one more thing now while I'm thinking of it because I might forget later I'm going to refer to this as a planar situation but you should recall that it really is three-dimensional there's no fluid in there it's just that it's a very large aspect ratio in two dimensions okay so what they observed was that it's what you see here this these lighter regions are turbulent flow and the darker regions are laminar flow turbulent laminar and it organizes into this periodic banded structure tilted obliquely to the stream-wise direction so this is a nice problem in patter information and I got interested in this and distant work on it but I'm not going to describe that here but I just wanted to say how I got involved in these problems in the first place and let me just show you a movie by Mugen ladder and Henningsen of roughly the same circumstance and you can see the formation of these bands in this case you're seeing both orientations oblique this way and oblique that way but again to emphasize that this really is turbulence in there intermixed with laminar flow and I think that's all I really want to say about this slide right so here's what we have we have laminar flow we have fully developed turbulent flow and in between is this interesting complicated spatial temporal complexity flow and to organize things I want to plot as a function of turbulent fraction so that's the fraction of the flow that's turbulent so in the language state there's no turbulence it's zero and the fully turbulent state it's one by definition so it's one and then in these this intermittent regime is going to be someplace between zero and one depending on precisely how you find it so that's this is the scenario that I'm going to be dealing with I'm going to describe two things as I've already alluded to I want to describe tell you why it is fully developed turbulence we've become intermittent in the first place and this is going to be the organizing Center for transition I would say that's what I'm going to argue I'm going to focus on what happens in this intermittent regime but I just want to say here now that in fact what I'm going to describe in terms of this process this organizing Center also describes what you see at high Reynolds number for several thousand and Reynolds number depending on the flow okay so that's thing one the other thing I want to discuss is what has attracted a great deal of attention in in recent years which is a critical point for the onset of turbulence and the universality associated with that so that's the second thing I'll discuss okay before beginning let me just say that this is a rich field there's a lot of work on this problem and related problems and I won't be able to discuss all that understandably I do just want to acknowledge colleagues and collaborators and I list many people here from whom I've learned a lot about this problem and I'm sorry I can't cite all of your work okay so let me the first thing I then I want to discuss is the mechanism underlying intermittency here's a list of references you can find this in my jfm perspective and these are I think are the good references which give you the basic mechanics and mechanisms involved here and then I also want to acknowledge particularly discussions with both I'm song Martha billet and Bjorn Hoff from whom I've learned a great deal about this all right so here's intermittency I'm going to show you in pipe flow I'm going to mostly focus on pipe flow at least for this first half of the talk um what you're seeing just take it to be turbulent kinetic energy in the comoving reference frame and this is a nice movie by Poe fine song which are saying here is initially the flow in a turbulent state just a little bit above where there's intermittency and what both hang is going to do is he's going to drop the Reynolds number instantaneously to a number of probably Reynolds number mm that would be the normal choice that what would make it's very much in the intermittent regime it's a nice round number and you see the formation of these puffs as I've already described and what I want to explain is why does this happen right that's my goal and to do that the what I found is the fastest clearest way to explain this is to understand what happens to the flow following localized perturbation in the pipe so I'm imagining I have laminar flow and I'm going to inject a perturbation this is what experimentalists do all the time and I'm going to ask what happens well what happens depends on Reynolds number at transitional Reynolds numbers if these intermittent Reynolds numbers again typically 2000 if you look in a space-time diagram you see one of these localized pups it travels down the pike and canes localized structure as I've discussed if you go up to high Reynolds number 5000 the turbulence expands and the word it's used is slug and I will use that word so you have it a downstream and upstream front in between those the turbulence is expanding a lot of the focus here is going to be on what happens at this upstream front I'll just alert you to that in advance okay just to show some experimental work though so those were just my little sketches here's some actual experimental measurements from the paper by Nishi at Allen and jfm I invite you to go look at that again the Puffin son these are actually time-space diagrams but they can fade the same information I will I think it's pretty clear that you see the localized Puffin expanding slug I do want to emphasize that what you're seeing here is measurements of centerline velocity that's quite an important quantity it's going to come up later so I'll just alert you to that now in particular I want you to notice with this puff when you measure the centerline velocity it is a symmetric about the front and back right so anyway we have puffs here we have slugs depending on the reynolds one final thing I want to show you is I want to show you puffs in experiment and simulation and this gives me the opportunity to highlight a work that comes out of Tom Mullins lab so what you're going to see here at the top is a movie by George Pacino and Tom Mullen and it's what I just described is laminar flow in a pipe at Reynolds number 1900 and what they're going to do is it's going to give it a kick it's going to form a puff and then I want to start this movie down below by marca bila which is showing streamwise four tissa T and right in the pipe it's also a puff and this is moving downstream you can see the cameras moving with a puff it's moving in roughly constant speed as const intensity constant streamlines lengths it's pretty much a solitary structure as it moves down the pike okay so this is the key so if you'd stop paying attention momentarily this is the time to pay attention I am now going to explain in this one slide why intermittency forms at least that's my intention so I want to focus first on the slow so what I show here is the slug again from mouthing song it's I want to imagine we're looking at it in the comoving a frame in which this this front this upstream front is stationary and what we have here is upstream we have this laminar fully developed laminar flow and I want you to think of that as fuel there's a lot of kinetic energy in the Sun or flow remember kinetic energy goes as the square the velocity the centerline velocity is very large there's a lot of kinetic energy there it flows into this front and production turbulent production exceeds dissipation there's a lot of burn there you should think of it as burning not laminar kinetic energy and turning into turbulent kinetic energy when you do that necessarily you blood the shear profile I hope this shows up on over the Internet anyway you'll see more this in just a second you blood you've taken energy out of the the the mean shear so you've blunted it and that's with it all right and then you go into this equilibrium where production is equal to dissipation and then you just remain in that state for arbitrarily as long as your pipe is you will just remain in that state all right so um let me just say one way I think it's very well I think is very clear way to think about this um if I look at an upstream cross-section there is no kinetic turbulent kinetic energy crossing that upstream section if I look at a downstream section there's definitely turbulent kinetic energy passing out of that downstream section everywhere in this flow that there's a balance between products will be between production and dissipation everywhere except here this is the source of the kinetic energy leaving the pipe all right there has to be source away that's it then the other thing I want you to again pay attention to is that we have these two equilibria that I've stressed and I'm going to continue to stress throughout this talk we have the laminar state and we have the and we have the turbulent state the other mechanical equilibrium of the system okay now what I want you to think about is what happens when I decrease the Reynolds number when I decrease the Reynolds number I changed my control parameter I already told you what happened in slide two what I'm going to do is I'm going to lose this secondary state through some sort of bifurcation okay so I'm going to do that and that state is gone however here at this front production exceeded dissipation and that is going to survive that collapse of this downstream state okay and that is what a pup is right so we look at it here's incoming fuel he gets burnt you convert laminar kinetic energy to turbulent kinetic energy you blood the shear profile once you've blended that shear profile you can no longer in the presence of this deform shear profile you can no longer extract enough kinetic energy from this your profile and the turbulence decays once that turbulence the case the the laminar profile is free to re to re-accelerate and reenergize nap in a nutshell is the process so what a pup is it's a it's just a front it's the upstream front of a slug it's just continually burning left laminar flow but leaving behind no down stream Lee and I want to say that a puff necessarily includes this refractory region in which the flow rate accelerates and this is an observation first made by hospital science 2010 it's really quite crucial to the full story here and one final thing I invite you to think about it this way this is a Bunsen burner I think you realize that the air fuel mixture is moving upwards the flame front is moving downwards and that's your equilibrium and it will really really help you to think about pulse if you view them this way and with that in mind let's go back to both lying simulation and I'll just show it to you again and now maybe you can see that what this is is this is a this is like a plane front so there's incoming laminar flow on fuel it gets burnt the the profile gets modified the turbulence is no longer sustained you've run the fuel however unlike a Bunsen burner you can we accelerate and you can have plenty fuel left for the next puff etc etc and while I'm thinking of it because I didn't say this before these look relatively periodic they are not in general pretty to be spaced more or less arbitrarily except for the fact that there's a minimum distance based on this recovery period okay so the turbulence can be stained in this intermittent state precisely because it's intermittent precisely because you can pick up kinetic energy in these laminar gaps there is no equilibrium you've lost the the uniform equilibrium it's gone to the system it doesn't exist anymore but this spatially intermittent if equilibrium does okay hopefully that's clear I don't want to spend too much time on this just to say that this is from the paper song in our 2017 in which this is all things thesis work in which he analyzed these things very carefully on this cover quickly this is a puff and these are two different kinds of slugs in space-time visualizations and I think you can see that this upstream front of the slug continuously conforms and becomes the puff likewise in terms of these turbulent kinetic energy budgets you can see this is the kinetic energy budget of the upstream front of production and dissipation this is the core of the slug and you can see that once you lose that that equilibrium of the core of the slug you remain with this upstream front which is the puff okay just saying the same thing over and over so let me come back to this so we have these two equilibrium this one in this one again the two the two states of the tape measure what we're interested now in this is fronts between these two equilibria and well what I would like to present is a spatial temporal theory for transition based on these ideas these ingredients now this idea these ideas go back a long ways probably even further back than what I'm showing here but I want to mention three things in particular Donna Kohl's who in this marseilles proceedings described these issues very very clearly Aoife Moe who also described these issues very clearly and then in Landau and licious and I meant to look up the page number uh sorry I forgot to do that I have the second edition if you open it up to the to the theory on to the section pipe flow transition you'll find exactly this kind of description and in fact what it is described in Landau Lipchitz in just a couple paragraphs is a very clear explanation of what transition looks like where you have a patch of turbulence surrounded by laminar flow and that you have fronts between turbulent and laminar flow and then they're either expanding or correct contracting depending on the Reynolds number so these ideas are not new however what's missing from this theory and when I showed you about intermittency and this is the thing which in certain out with intermittency intermittency is missing from from all this work and it's missing because of what I told you that the shear profile and excuse me the mean profile is so important to this problem it's that interplay the fact that the PUF modifies this your profile and what's up your profile is modified you can no longer sustain turbulence and you get that those localized States that is missing from this work and we have to add it right and as you can see from here what I need to somehow is incorporate the the the main profile and I'm going to do it a very nice proxy for the state of this year profile is the centerline velocity you can see it's large for laminar flow and it's shorter for a blunted turbulent profile and so that provides me with a nice scalar quantity which I can use to distinguish these two states okay so I've written this up together with collaborators in a number of papers some of which are and particularly this jmf perspective which is available to you and I'll just jump ahead now the thing is I don't know if you can hear this in that so the reason I knew how to do this is because I also study these kinds of problems and this is oh that's been around again this is an electrocardiogram for a beating heart and what it's an electrical measurement the electrical activity in the heart is what controls the heart activity and leads to pumping of blood organizations organize pumping of blood and what these these pulses are effectively puffs the mathematical description of these is nearly identical to what happens in puff in pipe smoke there's an excitation with a nearly a constant amplitude and where followed by a refractory period before the next one can occur okay and this exists throughout biology is how your brain works that's how your your excellent your your nerves fire it's an extremely robust mechanism it has to be your main biology wouldn't pick something that was not robust for these quick mission-critical tasks it's a highly nonlinear state there is no small puff there's big pops or no puffs okay and it's spatial temporal in nature there is no there's no act the potential that sits there it fires okay and and these things is what's known in the mathematical biology literature is that these are generic features and there will captured and well understood using simple polynomial nonlinearities and for those of you who know this is in fact dates back to van der Pol van der Pol if you think about van der Pol oscillator this is a large amplitude oscillator with simple polynomial equations and it's really kind of part of what's going on here okay so here's the simple of PDE that I could write down that describes what it is I want to describe and I'm not so I want to be so concerned with the equations but the pictures and hopefully you can get this so there are two variables I already told you one is the turbulence intensity and the other is the centerline velocity but I want you to think of it as a proxy for the state of the mean shear profile all right so I'm plotting these two in a phase plane and just to orient you this is laminar fluid about laminar flow the turbulence is zero and the laminar and the centerline velocity is at its maximum value so I now want to describe the dynamics of the two variables in the absence of any spatial effects so the new dynamics the the the the centerline velocity or the mean profile is described this way this is the curve on which u dot is equal to zero and it divides u dot negative from u dot positive and the way you think about the only important thing you need to know is that when you have turbulence this the shear profile Bloods okay the centerline velocity decreases when you don't have turbulence the centerline velocity increases back towards laminar flow I believe that's the only important thing you need to know there as far as the turbulence goes on turbulence so this is my by stability that I've been talking about since slide - I have a laminar state I'm a turbulent state and a special dividing the two now we're here I'm sewing as a function of the it now depends on the state of this year profile if I blunt to share a profile I can lose that equilibrium all right so now it also depends on this year profile that's the only thing I've added compared to what you'd find in Landau and Lipschitz and another people so now the other thing is that we need Reynolds number and Reynolds number will if I vary Reynolds number the the the null clients that the zeros will do this and you'll understand as I put it all together so now I plot the blue and the red together wherever the blue curve intersects the red curve I have a fixed point of the full system okay and this fixed point here is my fully developed laminar flow it exists and is linearly stable for all values of the Reynolds number as I vary the Reynolds number what happens is the system changes I get another intersection and that is my stable fully about turbulent flow that's my second equilibrium that has now appeared okay it's what I've been discussing since the beginning right the difference between this and the tape measure and what you read and land on Lipschitz now oh I'm sorry let me just say you have to add downstream infection and some spatial coupling to eat the full model so now let's go so let's discuss them what is different between this and what is in these classical works is that I now have a puff now on the slug and a puff and you can understand that source fellow so let me give the slug first I always do this slug first as I look in space I have laminar flow upstream I then generate turbulence the shear profile blunts and then I reach an equilibrium that is what you see here I'm in the laminar State the turbulence increases as a result the shear profile blunts and I reach an equilibrium the puffs on the other hand starts the same laminar flow turbulence increases blending of the shear profile but now there's no equilibrium and the turbulence decays and then I recover laminar flow which again you can see here there's no equilibrium and I return ok so this so that's the picture and in the again the mathematical biology literature and many other letters you infer to this case is Phi stable again the word I've been saying since the beginning you have to use stable fixed points and this is the case that you refer to it's excitable so you excite the system but there is no other fixed point available so these are two systems to have a complete theory you're going to have to introduce fluctuations there are two ways to do that either you can do it with deterministic chaos or you can do it with multiplicative noise or you could perhaps do it with something else we know that the the the never stokes equations are deterministic so there's that in some ways is an ideal thing to do but the pluses and minuses is another case and I'm not going to say how I done this you can read the papers so let me just make sure a lot of time here so I'm now going to go fairly quickly through just some examples of things again I've already mentioned this this worked by Bell fine song in which he did direct numerical simulations of puffs and slugs actually measuring real quantities of the turbulence intensity in the center line velocity and shows that you get the transition as expected between excitable case of a puff and then the slug there was in fact different kinds of slogans I've already highlighted and this was all studied in this paper I want to mention it Rinaldi it out who recently extended this work and considering what happens when you bend a pipe and what the effect is on the fronts and and so forth in those cases and I'll just again I'll let you read that there's also a I wrote a focus on fluids about this work and you can read that as well I don't really want to go through into a long explanation of this again there are these different kinds of slugs which i've alluded to I don't want to explain we've analysed this you're able to understand certain things about the speed and the onset of that the only thing I really want to focus to mention about this is that again these ideas well I'm specifically interested in the intermittent flows they really apply too much high Reynolds numbers than just the intermittent urging so I believe it's much broader than that and then the final thing I'll say about part one is this that's a bit much to take in what you're seeing is three panels and in each panel it's divided into two on the left is reality and by reality I mean something that comes from a direct numerical simulation of the navvy Stokes equations or experiments on experiments or both and then in the right column is what comes out of the model one of the various models that have been proposed to understand these things and I am NOT going to go through all these and I hope you don't find this too weird but the reason for presenting this is is twofold one is to just say that there are a lot of there's a rich variety of phenomena that people in the field are interested in have studied continue to study associated with this that again I'm not going to talk about each one of these things would take five minutes to discuss but also to say that these that this idea of excitability Debye stability with with with fluctuations captures pretty well of the majority if not all of the phenomena I don't want to oversell it I mean it's its qualitative at best semi quantitative there's a lot of work that can be done to improve it but the essential mechanism unconvinced is correct this essential mechanism a bi-stability excitability is what's driving the phenomena okay so I seem to be good on times so let me go on now to the to the second part so I've discussed now this what I in simple terms what I believe is the the origin of intermittency in these flows what is the basic cause why it exists and now I wanted to turn to this this critical point and the onset of turbulence and this brings me to the issue of statistical phase transition so this was originated in the early 1980s with papers by Hansen and grass Berger and they described statistical phase transition or theory of statistical phase transitions to apply to a lot of systems such as heterogeneous chemical reactions or forest fires or some idealized forest fire perhaps or flow through porous media or flu epidemics and and let me just give you a quote from grass burgers paper it says it suggests another type of universality comprising all critical points with an absorbing state and a single order parameter okay so there's a lot to take in there some of which will explain some of which I won't but you'll notice that there's universality in here that should apply to all of these systems but there are critical points well define critical points for these systems right and let me just let me say this was almost surely going to make this mistake in the statistical physics literature the class of problems see the set of exponents and and and the universality class of this set of problems is referred to as directed percolation so you think of percolation through a porous media I am NOT going to use that as the example but I probably can't help but say directed percolation directed percolation if I do that I just want you to know why I think a much better example in terms of what we're going to see is flu epidemics and of course the the current situation lends itself to that naturally anyway because in most both space and time in a natural way so it was Eva mo who actually also in the 1980s not very long after grass burger and Prakash I mean you know Anson grass burger realized that the subcritical shear flows had the characteristics of what gansan and grass burger described and therefore you might be able to deserve that kind of universality those critical points in these systems and that was taken up early on by the group in Saclay and they did a lot I mean they clearly had exactly the right ideas they were following on along the correct path their system sizes turned out to be what we now understand to be too small I may come back to say that but I really want to highlight this work because because the ideas were spot on as to what was going on okay so I need to describe now a little bit more about the phenomenology in the transitional regime I'm not going to describe everything but I'm going to describe a couple of key ingredients and what I want to do is I wanted to use the model to do this so I've described to you so I'm showing here again the comparison of DNS and model story I'm not saying this for us the best way and I've described to you that there are puffs and I've described to you that there are slugs but I haven't really described to you what goes on in detail and rather than showing you images from DNS and experiments and so forth I want to use the model which captures these features from the purposes of this talk perfectly well and it just allows me a lot of freedom in discussion so that's what I'm going to do so let me turn over here to this and hopefully this if it does not work I'm going to jump to a movie but hopefully this will work clear pipe top pipe okay here we go so what I'm showing you is another me not touched anything what I'm showing you is a simulation of the model equations and what's shown here in red is the turbulent intensity cue they're two variables turbulent intensity cue and what's shown here in blue is the centerline velocity U and this is the their state for laminar flow and I'm running the model it's a simulating but as we know laminar flow is linearly stable it's gonna do this all day long as long as I sit there but if I give it a kick as I as I described you kick the flow someplace upstream and you'll generate one of these puffs and the people in this field will know this really looks a whole lot like a puff in pipe flow you see this this decreasing centerline velocity with the recovery this refractory region and so this will go up the decay oh that's too bad here let's start it again so it's going to go down it goes out the downstream end of the pipe and it comes back in the upstream I just have periodic boundary conditions so it moves as more or less a solitary structure there's a little bit of weight limits here I'll hold off CTU's time is being spent you're on the zoom in my talk but it's going to go around and eventually what's going to happen is this one it's going to decay to in fact you if you keep doing this all day long if you got out of stopwatch and recorded at the time and wrote it down every time you did this and you did this a few thousand times and pause this you would realize that the statistics of this or poissonian this acts like a memoryless process it's a completely deterministic system by the way it's completely reproducible the only thing I'm changing with each realization is I'm mounting a very very small random number to each initial condition right so what we see is that these turbulent puffs and this is well-documented in lots and lots of flows and lots of lots of cases these face are in fact metastable they look as though they're stable but if you look at them and analyze them carefully and analyze them statistically you find out in fact that they're not and this is why you know again statistical phase transitions so now let me increase the Reynolds number in the model and and start an app up again so I'm now going to a higher Reynolds number and again I have a puff and it's going to go along is again more or less you can see it's wiggling about a bit more but it's going to go along and more or less the solid story structure and again I have no control about how long this is gonna last I'll just keep talking till something happens yeah okay it's gonna come around again so now I have two puffs all right this is a process known as puff splitting all right let me do another one and hopefully and here too you could you can measure statistics you can you can start the simulation you can run and you can wait and oh now I got three now I got more okay and you can wait and you can measure the time until the very first time it splits and if you do that you'll find that that's also obeys porcelain statistics it's a memoryless process and the relevant the relevant quantity that you can draw from that is the mean lifetime right and so there's one more thing I'm gonna really just this is not part of what I need to discuss now but just to show you the model this is what a slug looks like by the way if you run the model it spreads out and if I know just decrease the Reynolds number I can recover one of these intermittent states okay so this is what intermittency looks like in a point and I will just show you one more thing if I can do it and if I can't home give up if you look this is extremely long long spatial scales and I'm not going to run the simulation that would take up all my CPU power but if you look at this they're not equally spaced they may come in clumps like this and I'm gonna discuss those statistics in just a moment all right let me come back to my talk okay so hopefully we're clear on that so that's the movie I already showed you if the if the simulations didn't work with the simulation at Stanford so again back to that we have two states we have turbulent and laminar flow so in the statistical physics analogy one would refer to the turbulent state as the excited state which aligns nicely with the idea of excitable media etc and the laminar flow as the absorbing state and what I just showed you is that in the intermittent regime it was some degree of randomness turbulence will either revert to laminar flow or it will excite neighboring diameter flow and basically proliferate through one of these tough splitting processes laminar flow on the other hand is pretty boring it will not spontaneously become turbulent it's a linearly stable State and if you start with the system in a linear and laminar flow it will never ever ever leave it you have to take it somehow in that state so these features are precisely those that dancing in grass burger describe the precisely the features that they were interested in so turbulent intermittent other sites should say transitional turbulence has these has these features and this was the observation of p'mote in the 1980s so let me show you it a little bit more detail so I'm going to show you now plot I'm going to have an active state in an absorbing state the active state you should think of a turbulence and black and the absorbing state will you just think of a laminar flow or wait and the controlling parameter is the rate of spreading over the rate of decay let me just take the opportunity to say you can also think of as I said flu epidemics you could also think of the turbulent state as infected individuals and even think of the laminar state is unaffected but susceptible individuals so I'm gonna run a generic model it's a coupled map lattice who's on site dynamics as this I don't want to say anything more about it it's just a typical model I just want to illustrate illustrate the generic behavior which will all recognize immediately so I'm going to start both of these with an initial C so I'm an initially susceptible population I'm an initial laminar state all-white I introduced the seed of turbulence or seed of infection if the value of R is small as we know for epidemics then that then the infection will die out it will not persist if in a time and once you're clear of the infection once you're all white you will never ever come back if the value of our sufficiently large then the infection will spread become endemic to the population okay I think that's clear enough so if you look on now in finer steps in the control parameter R and this time is just variety I started from kind of uniformly turbulent or infected state it doesn't really matter how you start you this is what you see so what you see is that at low values again asymptotically you you're in this laminar State this is a absorbing state and you'll never come out however beyond on some value you do asymptotically for a long time you always remain there's always some amount of the turbulent state there's always some amount of media infection now is you kind of go to a thermodynamic limit where you have to keep making your system sizes larger and larger and wait longer and longer in time you find that there's actually a precise short critical point which distinguishes these cases below that point asymptotically you will be laminar with probability one above that critical point with probability one you will you will have some turbulence some infection remaining for all time again you have to take a thermodynamic limit this to be precise sorry my computer is going right I think I'll just keep going but I may have to switch no it's not okay sorry I thought it was still running my model so I thought I let's stop it all right so we're now going to get to critical exponents in what you would measure and how it precisely characterizes and the first thing it is turbulent fraction you can see here if you just look at asymptotically the turbulent fraction the percentage of black in these diagrams you can see that it increases as you increase the distance from the critical value so that's our order parameter if you think back to the yangdding grass burger statement of what's going on there is critical in the reserves in a scalar order parameter that scalar order parameter for us is the turbulent fraction below the critical value it's zero and above it it grows continuously it grows continuously with a universal exponent which is independent of the system being investigated okay so if you wait long enough your system science is big enough and you collect it the statistics of the turbulent fraction you will get a curve that goes like this universally right and you can also see if you look at these diagrams you can see an increased sparseness of the pattern as you decrease okay so the scales are diverging and there's both a length scale and a time scale those are not equivalent time is this way spaces this way those scales are not equivalent but both scales diverged and I'm not going to define them precisely for you but you can easily see that with your eyes and so we have negative exponents indicating the divergence of both length and time scales so we have then the this critical point is then characterized by three the divergent or the by three critical scalings when for the order parameter and two for the characteristic scales any other scalings associated with this transition would be determined from these three these are as I said Universal they only depend on the physical dimensions of the system and otherwise independent of details let me just take the moment to emphasize something so that I don't make sure it's really clear this this kind of case here this is a three dimensional flow okay there's turbulence your real terms of three-dimensional flow I'm going to call this two-dimensional okay just to be clear we call this two-dimensional in this directed percolation world so there have been some a number of papers on this I want to just mostly discuss in detail to the person is experimental it's the work of the MOOC out out out of your own half slab it's a beautiful beautiful experiment they built a very large aspect ratio Taylor couette system in which the circumference of the flow is about three thousand times the fluid gap so again in this kind of percolation transition sophistical transition world we're calling this a one-dimensional system even though it's a full three-dimensional flow and what you're seeing here are then experimental measurements of space circumference of the apparatus time now going downward sometimes down sometimes up here it goes down for three different values of the Reynolds number and we see just what I showed you in those previous images if you're below critical the blue is turbulence and the yellow is laminar flow you you have turbulence it persists for a while but eventually dies out and once you're in laminar flow you'll never recover above you remain intermittent for forever if you're right near critical you're kind of marginally going to sustain or not again as you make the system size increasingly large that will become really a sharp point then comes the hard part which is also kind of uninteresting to present and these talks are over there quickly is that you need to then measure things very very carefully what they were able to show is that there really is a continuous transition of the turbulence fraction from zero it has the correct scaling you need to get two orders of magnitude that's the gold standard to order two decades of scaling and they did this for the different critical exponents and they found that they agree with those predicted by by directed percolation by this University class described by Hansen in grass burgers so this is the first direct confirmation that a subcritical group to turbulence really is a continuous transition when viewed in this way it occurs if you precisely define critical points with Universal scaling exponents we know exactly then how to do this exactly how to put this in a class of problems that the people study in other fields now as I told so that's the frankly a one-dimensional system I told you that that the exponents depend on dimensionality so we and we means Matt Chantry decided want to do this in a two dimensional system and so what Matt considered was stress a stress free flow driven by a body force what we've referred to as wall F flow and it mimics plane couette flow but avoids the boundary layers which which caused you a lot of computational cost so it's an idealized planar shear flow but by going to this Matt was able to simulate domains that look like this now you can die again I can't see was using so this was the what was considered this is a very large aspect ratio domain and let me just say this unfortunately is not nearly large enough it turns out you get bad news if you public so this is Matt's domain I just I don't know if you can see it so this is this is Matt's on the same you know the same scale this way so this is what you have to go to and this is what max was able to achieve and I have to say it's marginal it's good enough but you would like to have gone to four times that size anyway so what Matt did also let me just show you this probably more interesting than the exponents so I'm going to plot here now turbulence fraction is a function of time on a log-log scale and I'm going to show you again the typical thing of three cases when it's below critical when it's above critical and one is pretty near critical again you see these these banded patterns disbanded turbulence it's typical of these planar shear flows well just in case you're worried here this for the for the one that's going to decay math only simulated in a smaller domain and then just tiled it it wasn't worth the expense to simulate decaying turbulence in such a large domain so that one's going to decay this one above the Reynolds number is above it's going to saturate and reach a statistical equilibrium that the movie is frozen but that will continue to evolve and you reach out an equilibrium turbulent fraction this one is this one very near critical is going to continue to decay what I didn't tell you is that - lying there is the exponent associated with directed percolation okay and so map was able to to verify all the scalings and so this is the first convincing evidence let me say it this way of of this process in a planar shear flow we have this continuous transition with these precisely can precisely define a critical point with the universal scales so that's the only the one I show you in detail let me just say there's ongoing work I just want to flash it and give a shout out to these various people who are working on this plain channel flow has a lot of interesting tricks about it and it's very quite an interesting thing it's a hot topic currently pipes also has there certain interesting things having to do with universality there I'll highlight my my colleagues who are working on this it's unpublished work so obviously not prepared to talk about but I just want to give a shout out to those things okay so is the exciting feel so let me let me turn to concluding remarks um so I discuss these two things I've discussed the organizing Center which gives rise to this intermittency in the first place why we have it why it exists and then the other thing I focuses on are this critical phenomena associated with statistical phase transitions so I want to discuss this a little bit for the first thing I'm just going to just review what I've already told you not so much to discuss but when I told you is that we you have two equilibria available to the system laminar flow and turbulent flow but which you should really see in terms of the intermittency is that when you lose this equilibrium this second equilibrium that doesn't mean the loss of turbulence it means it goes into this excitable regime an excitable regime is the exciting regime where all this fun stuff happens you have to add fluctuations to it but once you do add fluctuations to it you really do have I will say everything you have a lot of phenomena so it really provides my view is that this transition between by stability and excitability provides the organizing Center for these wall bounded chute flows and I really want to emphasize that this is a strongly nonlinear highly robust mechanism and it dictates the spatial temporal dynamics in toluca turbulence so that's that's my deal on that so Oh in particular it gives us the thing that we're going to care about then is the spatial temporal complex phenomena then I want to discuss this a little bit in more depth so coming back to this this quote of a grass burger and the fact that we have this universality in these systems and I want to discuss that so and his connection to the route to turbulence I originally wrote this that classical route to turbulence but I could get into a lot of trouble for that certainly I mean I gave route to turbulences props itself a very rich area but certainly one idea of the route to turbulence it dates back to Landau and Hoth the idea that you have a sequence a cascade of instabilities each entry increasing the temporal complexity of the flow that was then later modified by ideas of Llewellyn Hawkins and universality was brought in by eigen bomb here I show experiments like Elton swinney illustrating this this idea of this classical group to turbulence we're here again this is Tanika wet flow where only the inner cylinder is rotating and I increase the Reynolds number hi they increase the Reynolds number by increasing the rotation rate of the inner cylinder and what seize it and create a sequence of instabilities eventually gives giving rise to turbulence we see a an increase in temporal temporal of complexity giving rise to turbulent flow and so there's a lot of stability analysis bifurcation dynamical systems goes into this view and kind of through Feigenbaum and I'll give a shout-out to my colleague Padraig Sultana we get a lot of understanding of the universality in how this happens this is very very distinct from what I've described here this these take place were spatial the spatial temporal character of it is fundamental it doesn't exist without the spatial temporal characters and you'll come back to laminar flow you have laminar flow and we know it's a linearly stable State if you want to understand the transition to turbulence in pipe flow we know you don't do it by studying the linear stability of pipe of the laminar flow that's for sure but you don't even do it by weakly nonlinear theory that will get you nowhere it's that's not the way the flow becomes turbulent and I like the thing if you don't look here you can look at trees you go look at trees and you can look at trees and you can look at trees and you will never understand anything about a forest fire if you want to understand a forest or you have to light the trees on fire that's what you have to do and you have to it's a statistical thing even like one tree on fire might not tell you because you might one tree on fire and it might not persist what you have to do is you have to be able to you have to initiate the things and then look over long space timescales and then ask the statistical question is what persists and that's the way we now understand the way to look at and to organize these systems and not understand the root of turbulence so we inject we wait and then we collect statistics the hard part is the size of the systems I'll refer you back to Matt's numerical computations that's what's the hard part and that's why this is challenging however let me just say it is brought to mendes this view has brought tremendous clarity to the to this field and when people study these systems which seem big and I again I'm not criticizing him you had to you you couldn't have known how big you had to make your system when people study systems of this size or or other sizes this was the original Cyclades from us and again I'm not criticizing you didn't know where to start that the thing is you get different answers depending on on what the system was and you didn't know whether it was continuous or discontinuous you didn't know whether there's any universality right and we now this is now we've gotten over this by at least looking at some systems in which we've been fully able to characterize them and fully been able to understand them and the final thing is that that this has also provided a beautiful confirmation of these Universal scalings predicted by Hanson and grass burger many years ago and so for that I'll end and thank you and take your questions right that's that's a good clarifying question I'm I'm using there is certainly intermittent seed within turbulence even when I'm calling the fully turbulent state I'm viewing that as an equilibrium as a simple I'm basically collapsing it to a fixed point but of course that's not true there's intermittency with a slightly different meaning of the word within that turbulence itself so I'm not sure if I'm answering your question correctly but it's certainly good for me to clarify to the audience that when I say intermittency I mean a very large scale intermittency between terminal laminar flow and not the kind of interment see but I think you're referring to and then to maybe perhaps trying to answer your question is to say I don't know I mean that's to say I don't know whether the kind of intermittency did I think you're referring to plays any role I've kind of washed it out let me see that oh well most definitely I mean the Reynolds number I mean the viscosity appears in the disc in the dissipation interim of course not in the production but it appears in the and the answer patient that's expects precisely that's precisely how it appears in fact I didn't get into that detail but yeah it appears what is changing as you change the Reynolds number is the the that dissipation well a lot of things change but the thing in terms of the equations that you see is the viscosity in the dissipation well it's going to be around to 2040 2,400 I mean 2040 we know that because we can measure the individual decay and spreading processes it's going to be slightly different from that because of correlation and I'll defer that that issue to Bjorn Hoffman and his collaborators who will be telling you in publication hopefully very soon but it'll be I think below to 2060 between 2016 and 2015 2040 in 2060 Shirley no I don't think I need that I mean I don't like what I need what I need are two equilibria I need a spatial temporal system with the two equilibria and just to say I mean I'm gonna get answer I can see the questions ago so everything I know I came from this book and I don't know anything that isn't in this book you combine from Cambridge University Press and and and in this book and this EC I have it all marked up here is this discussion of basically what people just refer to as turbulent pipe loan and the balance between the means your profiles and and I'm kind of ignoring everything else about that state so there's lots of interesting questions about how you would say to geophysical phenomenon and in fact it's one of the things we're kind of discussing but currently I don't know but I don't think I need any of that phenomena for these kind of these again one of the things I tried to emphasize let me take the opportunity to say both when I'm going to discussing why there's intermittency but then in terms of the statistical phase transitions everything I discuss is extremely robust I don't you have to have certain ingredients have to be present but the details of those ingredients I don't think matters very much that's why we see them and see them in lots of cases so when you you contrasted your view with the the more traditional route of turbulence to a sequence of bifurcations are you suggesting that that different system display one route over the other or that they're two times writing the same thing oh no sorry good that's good to clarify question no I'm saying that there are different there are different circumstances and these highly subcritical we're the only state you there is no wheat I mean some people would call me turbulence but there is no you have a puff or you don't have a puff there is no kind of half pump there's no weak pump so it's a strongly highly nonlinear mechanism from the beginning there is nothing else and in those systems and I'm not saying necessarily you have this but those are the classes you'd apply as opposed to the classic Taylor correct or you increase the temporal component so I'm saying and I'm not saying that these are the only two by the way either I'm saying I'm just distinguishing those two and there can be potentially other quite different ones I'm not don't have an opinion about that or not thank you thank you very much do you have any any insight into where the boundaries between those kinds of systems lie and if I can do it well yeah okay so unfortunately Taylor quick provides us a nice table because we know in the strongly counter-rotating case it's this sub critical scenario I just described the co-rotating it's the other scenario so there's a nice knob there you can play with I'll let you do that it's I don't know I would say it's not it's not a question of amplitude it's only a question about it's only a question of Reynolds number it's it's no I don't say it's not a question of amplitude I mean you might be able to get a puff split by playing with the amplitude but you will not get again the oh if you're in the proper game the only way you can sustain turbulence the only way it can exist is if there are laminar gaps in between if there are no more gaps in between either it will just fall down you cannot sustain so no it's either pop or not I mean there's there's this I didn't show you I mean there's there's more details than I'm telling you it if you get right to the the transition between puffs and slugs and you do get these kind of really kind of puffs like slug like things I'm not saying that there but but if you have if you're clearly in the puff regime then you're in the puff regime and you don't have slugs and then what amplitude you did okay and also the structure with a pop it doesn't play any role and all I said for them yeah well you know I'm skipping over a whole lot of things and just providing you know the kind of the skeleton the structure of a puff is really very important the whole lot is known about that both in terms of periodic orbit theory the structure of the fundamental role stry wave structure which maintains while bounded turbulence there's a whole lot of important phenomena there that I'm not describing which is important on so I'm not saying that the structure of the puffs isn't important to a puff I'm saying that the structure of a puff is I can reduce it to its essence which is that it's being energized on one side and leaving behind a wake on the other side that's the question I think I'll say that those are potentially those are differ I mean there's no reason they would be the same I would say that no in fact I will go I will say that that the lowest Reynolds number for which you can have an edge state will be below the critical Reynolds numbers I've defined because you can have puffs long-lived transients below their critical Reynolds number they have edge States the point is they're not stable the metastable I mean in fact that gets all back to the edge in fact the puffs I mean there is no edge in the sense that that pop eventually will cross that edge you go back to laminar flow well yeah that's actually very good question I mean I think I can't I prefer to have to answer that I mean I don't I I think I have to think about that because in some sense I say that it has to in the sense that that I'm putting that in and so but on the other hand it doesn't really matter too much what statistics I put into that determinacy yeah that means I say that's a good question I'll think about it the number of people tried that I would just say and I was going to put in a slide because I thought I would get that question at the end of this I did write down a model in 2011 and I to this day believe that model is essentially correct in in producing the periodic structure that you see in speaking element so you'll know these details that you see in couette flow if you work in a tilted domain and in fact there's a lot of things that you could there's some paper by shitao from two thousand thirteen or fourteen of your house she'll be a hawk and with that model you can reproduce that phenomenology almost exactly however what you don't get is the Tilted structure and that's still an ongoing kind of open question you can see that the percolation has become you know also you know more common knowledge than it used to be because people can understand what how things work you later don't percolate through a society you know I you know it'd be great if I actually had a real application I don't so presumably transition and control okay I do have an application that's not my own but it comes again from Bjorn Hawks laughs but it is captured again it's one of the things not on my you know comparison chart but it could be on a comparison chart which is it contains in there because you have the means to your profile in there you can one of the ideas of control is if you can somehow block the mean share profile of the shear profile such that turbulence can't sustain then you can kill turbulence while maintaining laminar flow so that's an idea and that is captured in the model well finding if I put it in its thesis where you can the model is this is one of the ways in which the model is least faithful that has a phenomena it has the effect but it's not perhaps is quite sufficiently quantitatively correct but the idea that and there's a there's a number of people let me just say actually Willis sorry I don't blank you but but and maybe then rich I've been looking at these ideas so there is an I'm sorry if I'm messing everybody's name but there is this idea of control methods some of that is picked up by the model known as I mean the Molly think just do DNS etcetera but there are ideas of control that closely tied with what I tell you because when I told you about the importance of this year profile and the idea of modifying this year profile so as to kill turbulence is certainly something that's a very active area of research yeah I mean you have channels and your pipes and then you have various kind of rotations and the things so quiet and so forth and you can drive them with walls or you can drive them with pressure gradients and you can put those together and then if you're you know ambitious you can start adding other effects this is also seen in MHD by the way that the idea of puffs Thomas Bach has has seen these things I I don't know how to answer the question so I mean frequency will have to have been so I assume that's kind of like an issue of turbine attraction because the you know statistical you've been there's no there's no periodicity here well it decreases I mean it did it the frequency increases with Reynolds number it decreases with decreasing Reynolds number mister it's it's scales with Reynolds number you

Channel: Cambridge University Press - Academic

Views: 704

Rating: 5 out of 5

Keywords: Turbulence, fluid mechanics, shear flows, directed percolation

Id: mkvUraQL4N0

Length: 65min 45sec (3945 seconds)

Published: Wed Jul 15 2020