R. Kerswell - Extracting Order from Disorder : Periodic Orbits Buried in Fluid Turbulence

thank you very much for that very generous introduction it's a pleasure to be here what I wanted to talk about today is something I've been thinking about recently and I think I think is sufficiently accessible at least at some level to be interesting to a broad cross-section of people so hopefully you'll get something out even if you're not an expert in turbulence so since this is a colloquium I quickly downloaded some images from Google just to sort of set the scene and these really are perhaps from the front page of Google this is a this is a very popular picture that helps me detect if the projectors working because it's got nice red this is an aircraft coming in to land and these are the trailing vortices breaking up and there's small scales so this is a you know typical engineering picture this is something where you can clearly see this in in in the domestic setting you know a nice laminar flow of fluid hitting the base and I'm not quite sure they've got her cup overflowing and producing complicated motions this is flow out of a pipe this is ice a contours of or tissa t amplitude in three dimensional decaying turbulence this is flow past sphere I think and this I like this picture because this is this is a generic scientist I put his name down there but pointing to the non-linearity in the navier-stokes equations which is the cause of all the trouble and I guess what I'm trying to show in this overhead is that turbulence is is a lot of things to a lot of different people but I guess the best way to describe it is is a flow is a property of the flow such that there are lots of time scales and length scales and we basically don't really understand it and that's that's a very good easy definition of turbulence now I'm not going to consider these complicated flows I'm going to consider something very manageable this is one of these canonical flow I've stolen this from John Gibson has done a lot of work on this flow so it's one of the simplest things that you can generate it's playing couette flow so that the top of this box is flowing in this direction as indicated by the arrow the bottom of the box is flowing in that direction so there's a simple shear and in between these two upper and lower surfaces is a fluid and what's shown here is the flow not being a nice simple unidirectional flow in other words sharing the same symmetries as the forcing so if if the flow was just the simple shear which you get at very small values of this shearing then the colors would just be uniform but gradually going from below to red up here but as you can see they vary in the span-wise direction indicating that the flow you know the broken symmetry this is time dependent and this is the you know the start the long journey towards the turbulent state where this is a this is a nice cartoon that I've taken from one of Poorman little papers he describes it as featureless turbulence so in other words you know that the wide variety of length scales and time scales and the flow is very complicated so this particular flow here is probably about here what's different or what I should point out with this picture down here this sort of regime diagram is this is for plane couette flow extending extended in all the direction so very large plane couette flow whereas typically when you do computational studies you impose a certain periodicity in the stream-wise direction here and in this bound wise direction in fact I'm afraid to say I'm going to be looking at those sorts of flows in this talk so I'm going to be talking about very sort of clean model systems where we can study what's going on turbulently so now I'm going to go back to my title so I sort of try to indicate what turbulence could be or a rough definition of it and I want to spend first half of this talk talking about periodic orbits the evidence so far for why they're interesting why they're part of turbulence the current state of the art for finding them and then I want to talk about a new method for finding them which hopefully will replace the current state of the art to become the future state of the art just to mention for some postdocs that have helped me with this work over the years Gary Chandler Dan Lucas who's now electric eel and Jacob Paige who's still with me so this this is the oh we had really that got me started thinking about periodic orbits nearly twenty years ago a very famous paper now by Kawahara and kida in kyoto i think they were at the time so this is the plane couette flow situation here so I've sketched it and this is a small domain so the normal way you talk about this flow is to take your length scale as twice sorry is as your length scale is being twice the separation between the plates so that's why L Zed is 2h and then these are the the periodicity is imposed on the floor I'm not quite sure why they do chose these but they did because this is 2001 they were only using about 15,000 degrees of freedom and a Reynolds number of 400 so this is you could imagine it's but fairly weakly turbulent but it is turbulent now this plot I should explain it because it's it's going to come up a lot here's the energy input this is I energy input and D that's energy output that's always known as the dissipation rate these are rates and so on average you know statistically steady stay the flow should be sort of centered around this line where they're equal okay so that's why you see this roughly centered on this line of equal input and output what's plotted here is a long dns signal so in every time that you choose some time increment you plot the input and the dissipation the green dots are supposed to indicate equal time intervals so you get an idea of how quickly the flow is transiting through a part of this plane so for example here the orbit is moving quickly whereas its spending a lot of time down here with a very densely packed and of course Uriah is drawn to this red closed loop and that is a periodic orbit that they extracted from the DNS and they extracted it from the yellow loop which is actually part of the trajectory and is unclosed okay so really the underlying question in this talk is how do you detect those yellow loops to then find the red loop and then be able to say something about the turbulent flow okay now I'm going to touch on all of those issues during this talk but the very interesting thing of this piece of work was not necessarily this diagram nor this picture but what they did with this periodic orbit so then they took this periodic orbit and compared it with a statistics of the turbulence so here their periodic orbit that they found had a period of sixty four point seven that little H over u the velocity imposed on the boundaries of your plane couette flow system and they compared it with us with the low order statistics over the DNS signal six times ten to the four so they did a maybe they took the average over the whole of this trajectory and compared it with the periodic orbit just over its period and of course what what one of the first things you want to look at when you're trying to predict turbulence is what I mean profile is so this diagram isn't is it really great in terms of orientating it with this little cartoon here so I sort of rotated it and inverted it and this is what the mean profile looks like compared to this cartoon here if it was just the basic uni-directional 1d state it would be a straight line shear between these two so that you get this characteristic s with the mean flow gradually losing its shear in the interior and this is the prediction from the periodic orbit the periodic orbit is this is the solid line and then the turbulence are there discrete symbols and then this is the rms values of the fluctuations and you can see the correspondence is not so great there but it's still not bad and this is all using one periodic orbit so you know did they get lucky it's this some great new way of analyzing turbulence you just find one periodic orbit and then you're done well unfortunately not but you know this is this is the sort of piece of work that gets you interested in intrigued and you want to work on it so in fact there are more periodic orbits so this is some follow-up work in 2007 this one ass was involved in this Gibson and Frederick's at an average they looked at plane couette flow same Reynolds number up the degrees of freedom so now we're doing a hundred thousand and this is some funky representation that they used based on one of the equilibrium states they found in the flow and then the unstable directions of that state that's not really important it's just some motivated way to try and project the dynamics down this is the plot not to focus on this is the plot I want you to look at there are five unstable periodic orbits on here and this dotted line is the DNS in this projection so you can see or there are lots of periodic orbits buried inside this turbulence and then the question is well which one is the right one which one is the coherent key to one that I want to then take the statistics off well in fact if you do that you'll find that none of these really work that well and so then you come to think all okay maybe I need some weighted mean of them and and that's you know that's that's an issue I'll come back to but anyway so here were five and here's a different flow where we found 50 and we were still counting so then we just explained this flow a little bit it's 2d you know if you if you're trying to really push something you've got a 2d very famous flow where you force it at a large wavelength and you can do the same thing you can find then this is now PDF of the energy input versus the dissipation so dark colors are where the trajectory spends most of its time lighter colors where it goes through sort of bursting episodes and the all these clotheslines these different colors represent periodic orbits and I think I've shown about 20 here so you can see they're really pile in to cover where the turbine spends most of its time and of course it suggests you a sort of a dynamical systems perspective where you have a turbulent attractor and you've got periodic orbits which is densely packed inside so you can really hope that maybe finding these finding enough of them you might be able to make some progress understanding what's going on I should also say what's happening over here because it's 2d I've just drawn the component the vorticity out of the flow domain and red is positive and blue is negative so you can see it goes through episodes where it sort of pseudo settles down like there and then it goes through sort of bursting situations where it really shears and becomes very energetic and that of course corresponds to being up here but really most of the time it's down here this turns out to be very interesting flow actually because what what it turns out to be is two structures which have frustrated in this domain if you open up this domain you get all sorts of interesting dynamics because then they can coexist but that's that's by the by here the point of this overhead is to show you now we have 50 in this situation and to show you that what sort of things that you can do so this is a very good example where this is the flow this is all slowed down now this is the DNS you can see there's a coherent structure in the center of the domain and then it breaks down but there's certainly coherence there and this is what you can extract out as that DNS is a periodic orbit and that that's a solution of the navier stokes equations it just happens to be unstable but only in a very small way now what do I mean by small I mean I mean that it only has a very small number of unstable directions so let's say this computation had a hundred thousand degrees of freedom there might only be ten on state or directions so there's you know nine hundred and ninety you know ninety nine thousand nine hundred and ninety Direction's so you can imagine that the flow wants to come in you know be attracted to this date and then it ultimately it gets flung out the other thing to take out from this comparison is that your eyes naturally dawn to their dynamics in the center of the domain whereas if you look at the corner there's a very large strong vortex just sitting there that's not really replicated here if you look at that picture again the vortex is much weaker that there so really you know we were lucky to converge this based on what was happening here really this is you know it's got the same sort of structure but it's much weaker than the actual state here the courses this essentially highlights there's the struggle or the challenge when you have a large domain and you're trying to extract coherency you know the coherency might only be in a certain part of your domain and you know if you try and converge the whole domain is a whole dynamical structure you might miss it well of course you know our eyes are very good at picking up that this is obviously related to that and we were lucky here that we must get the whole whole situation but you know in terms of is the trajectory and face base getting close to this at this point here well it is in the center but what about at the sides and how do you sort of develop a proper metric for that okay so one let's say you've got one of these pure log orbits remember this is a sort of motivation part of the talk where I say the periodic orbits are interesting well at the very least you can take one of these and this is one of the longer period ones so this is 37 in those units of H over you and you can see what happens as you go around the periodic orbit you can sort of diagnose the dynamics and I have to apologize the color schemes change slightly here so positive vorticity is white and negative autistic red it just so happens that this reproduces better and black why so journals like prefer this or they did in the old days and you can see that you know this is the nice quiet region here where these two structures just sort sit next to each other and then you go through this shearing episode and then you settle down and you come back again and there's a very good example of exactly this approach being used very profitably in shear flows and it's essentially through this approach that the self-sustaining process was discovered this is the key Peck the Hamilton Kim and more laughs and I was looking at the citations recently and this now has five hundred and three citations probably many more now and these are the subsequent papers build built on this there's the coherent key to one and this paper was one of the first to actually find the sorts of structures with in turbulent flow experimentally and of course what's happening here is that this is a very generic mechanism now we now appreciate where so-called streaks interact with waves and vortices to produce solutions which can sustain themselves against viscosity so I won't really I don't really want to spend time going into this but this has been a very important realization in shear flows over the last 20 years and that came about by doing this sort of analysis where you try and see where the flow becomes nearly periodic and he did it by shrinking the domain looking for a minimal flow unit recognizing these structures and then ultimately converging exact solutions of the equations in terms of traveling waves and then if you're really lucky you've got enough of these periodic orbits you can start thinking about making some predictions so here let's say gamma is a key quantity you're interested in let's say dissipation or shear on the surface then you can take a weighted average of that property associated with a periodic orbit where the waiting here is WI for the periodical which is labeled I of course the big question is how to choose these w's there is some theory for low dimensional dynamical systems called periodic orbits Theory goes back to prodrug certain of which in the late 90s and you know you can try and do this so we did this with the 50 new POS we found in Comodoro flow so this is the PDF of the energy this is the PDF of the dissipation so let's maybe just focus on this one this is probably the most informative with a periodic orbit Theory you can take your periodic orbits and you can produce this purple curve you can see it's very lumpy so that indicates that we haven't got enough orbits but also what's interesting is just taking equal weighting for taking W itv1 just produces just as good a prediction really if not better than the DNS which is the blue and you can also say that the the tail of the distribution is not being picked up at all because we haven't got enough statistics I'm going to appear on orbits that reach that far so this is the hope but we're a long way away from realizing it then of course the central issue here is we don't have enough of them so we need to improve we need to improve our capability of finding these periodic orbits so that's what the rest of the talk isn't it so this is the current state of the art it's very naive but it's been what's been used since Kawahara heat of 2001 and that is to do a very simple search so you do your DNS for whatever flow you're interested in so this is a velocity field you get out at certain time at some point in space and you compare it with the history of the flow you choose some capital T to go back over time so here I've chosen a history of 50 and I basically look for where the flow starts to look like it was some time previously normalize appropriately and it's just an l2 norm and you can see this is the simplest thing you write down there's nothing optimized about it but it sort of worked up till now but obviously it can be improved so let me explain this plot here so I've drawn a line at this point I think it's like 158 so this is my simulation I run it to about T equals 158 then I look back through the history so this is fixed at 158 and now I'm just going through history with capital T going back in time and I look for a minimum and you can see there's a minimum there it's blue and it's not it's not an eye-catching minimum right these these values don't really change by very much it's not as if I'm getting orders of magnitude improvement but it is a minimum all the same and then you can try and try and converge this as an initial guess with a period appropriate to this gap and all these black dots converged to give periodic orbits so this one in particular converges I won't go into into the how you converge them it's basically a glorified newton-raphson in multi-dimensions but the point is that you identify them like this and you can potentially converge them now there's one very important fact from this diagram that is you require the flow to stay close to the periodic orbit for almost a whole of the period otherwise you won't recognize that it's nearly come back to itself okay so here's my periodic orbit I need my trajectory to come in go around and then with off again and I need this to be sufficiently close such that I can recognize it's almost completed a loop and that's a serious issue as a Reynolds number increases as your domain increases and that's really been a bit of a bottleneck for this approach that's why we started looking for different approaches and it's the different approach that I think might be quite interesting to people interested in fluid mechanics or people not interested in fluid mechanics yeah this big traditional stable none of these orbits are stable well this picture um well look so first of all this is this is a chaotic system right so if I if I change my dynamical system slightly and I start with the same initial condition I've got no hope of finding this one precisely at this point sometime later but what I do have a hope of is finding some of these other ones but at different times so it sort of structurally robust but not point wise as it were I wouldn't expect that okay so okay so this is a just a brief summary of where we are problem finding periodic orbits embedded in turbulent flows just introduce you to recurrent flow analysis a very naive dumb approach really needs some sort of shadow a whole periodic orbit less likely surrounds number goes to infinity can we recognize D recurrent behavior using a more sophisticated approach well ok so this is what I'm now going to talk about you know well there's two heavy overheads and then I'll talk about it in a very light and breezy way so what I'm going to do is I'm going to use Koopman analysis no physicists usually a very comfortable with Koopman analysis they know it's very strong but useless in the sense that it's very hard to find the modes and it's a very nice idea but impractical but anyway I'll come on to that let's just first introduce what it is so let's say I've got a nonlinear dynamical system just written very simply here f is nonlinear you know no restrictions on F except you know smooth differentiable and this is my time stepper little F so the Koopman operator is a linear infinite dimensional operator I'll come back to the infinite dimensional later suggested by Copeman in 1931 which acts on observables of unless let's just cut to the chase let's say you is it a velocity field doesn't have to be but it's it's my dependent variable so basically this this observable side takes you which is in a and produces a number and so the Koopman operator essentially moves this observable forward in time okay so this looks a like a really good way of stepping forward in time something I'm interested in the flow let's say I'm interested in the dissipation well the dissipation is an observable because it takes the U field all over the whole domain calculates the dissipation and produces a number so anyway what's interesting about this is it's linear or in a very trivial way that I tried to outline here and I'm very glad that we've got read on the overhead now so this just takes two observables when I when I do the Koopman operator just distributes and then I can reconstitute it here so it's linear so I can I've now got all the linear analysis tricks that work we're all told at graduate school so I can expect eigenfunctions eigenvalues of this linear operator and I can in particular look at I can functions which this time step in a very nice way okay so if I just Koopman operate on this eigenfunction it only involves in an exponential way like this just rescales lambda could be complex okay so as soon as i got i can functions and eigenvalues i can then say well okay what if I take any observable can I just have an expansion like this of eigenfunctions and then I can do all the usual linear things I could say well okay so if I want this dissipation times depth then I I bring in the time operator on you that changes the argument but if I got this expansion I know how to deal with all these things because all it does is just bring these exponential terms to so that really looks great because you know even though this kappa this Koopman operator is linear all the complication is in this what's happening to the argument okay this looks like magic I'll tell you it's not magic in them and I'll show you why it's not magic now what you can do is stack lock so these are the observables into a vector and I think the easiest way of absorbing that sentence is to say well what happens if the observables are values of U at grid points that's certainly an observable right so I can I can say at this point in my spatial domain what is the velocity field that is a scalar or it's a three vector if I want to take account of the three directions in three space but you know I can write it as this okay so this is U at those grid points these are the Koopman eigenfunctions this is now called the Koopman mode so this is like the big vector which has the same form as this observable here so these are just basically the coefficients which map the eigenfunction and eigen values over here so this is the expansion you're looking at okay so let's just do a simple example before I actually do something a little bit more important so here's a very simple system u dot time derivative of U equals minus au solution that Koopman eigen functions look like this so they're just U to the power N and I've made an assumption here that I'm only interested in analytic functions Phi right now I'll come down to that in a moment and then lambda n is minus na here where n is a positive number including zero and you can see what happens when I just apply the Koopman operator on here U to the N if I and then I use that and you can see this exponential coming down here and then is clearly the eigenfunction here - na so important little observation here is the eigenfunctions can be nonlinear functions of you despite the linear equation ok and these eigenfunctions really can span the space of real analytic functions because you I don't think too many people would balk at seeing this expression here where these are the coefficients in front of the Koopman eigen functions and of course I can build this up into vectors like this if I've got more than one observable these theses tell me which which of these eigen functions I need another example pitchfork bifurcation so this is a very simple example where you're going from a repeller at R equals 0 to an attractor at R equals 1 right the very simple equation can i express this using this approach as a function of time well it's very easy to write down the solution to this equation it's here ok and this is the dynamics again and here are the two expansions it turns out you need two expansions one around the origin and one around the destination point these are very familiar easily calculate it but these are the Koopman eigenfunctions these are the cooling modes and these are the coolant eigenvalues exponentiated so indeed you have this representation now what we're trying to do is saying well actually can we use this to find points in the flow which are nearly recurrent that's where we're heading and just to get just to give you a feel for why this is a well essentially why this operator is linear and it's infinite dimensional let me go back to this system here and do something called common linearize a so this is a one dimensional nonlinear equation because of the R cubed but I can turn it into an infinite set of linear equations just by defining new dependent variables so here for example if I say well why don't I just make R cubed a new dependent variable x2 if I do that I need a new equation for x2 here's the new equation but what I do is I generate a new non-linearity which is R to the 5 I'll just make that X 3 I need an equation for x 3 and I can just keep doing this but the system will never close and what a large fine is an infinite linear system here in terms of my new variables and it turns out that the eigenfunctions of this are the eigen of the Koopman eigen functions and the eigenvalues of this of the Koopman eigen values and the left eigen vectors define the eigen modes the Koopman modes so this is where this infinite dimensionalities come from this is how I can turn a nonlinear system into a linear system but the payback is it's an infinite dimensional system that's rather interesting that Carmen came up with it a year after Koopman but you can't really see any connection in the literature which is sort of fascinating ok so why now well it turns out that the numerical procedure invented recently or 2010 is recent by Peter Schmidt and quickly developed by others can take data and generate Koopman eigen values and eigen functions in certain limits so let me just briefly discuss this and the reason this is interesting is you can do this on data either numerical or experimental you can say well ok I'm going to develop a vector observables at a certain time which is a function of my velocity field I'm going to measure it again at certain time later and keep on measuring it and then build up this data set and then I'm going to measure that precisely delta T later from each of these observation times and then I'm going to take the best fit linear operator between these two okay so this is the best fit linear operator and essentially what I'm going to do then is look at the eigenvalues and eigenfunctions of this operator k and it turns out to be intimately linked to the Koopman operator you know in a sort of a fairly deep way now what's been interesting is that this is revolutionized how people are now starting to use Koopman and isis they're basically doing d-md using the ideas and a framework of Koopman analysis to then try and make deductions about dynamical systems I'm interested in fluid flows but this is very general you can do to any experiment any data so this is the point here connection to the Koopman operator at the MD gives you this sort of representation so now the blue indicates the eigen function for the DMD and the eigenvalue here exponentiated in a direct connection down here between the eigen function Koopman mode and eigenvalue so this blue here corresponds to exactly that point there a n is this value here when it and the eigen values are the same all right so let me just briefly illustrate this for the van der Pol oscillator I wrote I will come back to flow mechanics at the moment so this is a two-dimensional OD e I can obviously write it like this to first order ode E's very famous picture in any mechanical engineering course you have this limit cycle in red this is X and X dot and anything any initial condition placed in the vicinity of this limit cycle in fact anywhere in the whole domain will quickly converge you in and then follow this limit cycle now what do you do you can basically do exactly I said take data along this loop and I took two thousand to thousands of snapshots of what's happening and then found the eigenvalues of the operator and you find exactly the frequencies associated with the periodic orbit in red and you also find the least decay rate of the end of the stable direction it all comes out without doing any analysis I don't have to find this exact orbit here and then do find the linear subspace around it to find the eigenvalues I can just do this DMD where I take data from a simulation in here just eject direct numerical solution of this and I get the structure coming out so this level here is negative it's a decay and it's a stable direction of the flow coming in and here by the way is the vector of data I took so I take X X dot x squared X X dot this is the data I take along this trajectory as it comes in this blue one here is the one I talk and you can do it with an unstable so somebody when I gave this talk somewhere else said are you cheated because that's a stable stable structure so I said okay either under stable direction so here's the unstable Direction Z dot equals Sigma 0 as Sigma is a half I think and so it goes round the orbit then spirals out the where I start and what happens then is I get the same eigen values out but I also pick up the why not unstable direction too so that Sigma is 1/2 coming out nice and cleanly all I'm doing is taking data along this DNS processes processing it using DMD and I'm picking up all the important things about the dynamics we're doing minimal work really back to navier-stokes so the idea is I'm now going to do DMD on short windows of the turbulent DNS can I find near a current phenomena well what's the idea here well what I'm going to do is I'm going to do data just like I did on the van der Pol offer later of a DNS signal I'm going to take the eigenvalues of the operator K which is my best fit linear operation between all these snapshots and then I'm going to plot them on this lambda R lambda I diagram and of course if I find like this that they're essentially equally spaced along the imaginary axis so in other words they're just oscillations this starts to look like the flows nearly periodic this is an actual picture here so this looks that's precisely on zero that's the first harmonic that's the second harmonic that's the thirds not quite there it's just off that lying guideline and that one's clearly off being the fourth harmonic but this is a pretty good example where the flow is looking like it's recurrent now anything like this this is indicator all right it's got an unstable direction but I don't care about that I just want the structure and I can based on these eigenvalues you know this is this is Omega 1 I'm sorry I'm good 0 and go 1 I'm gonna go to I'm go 3 I can make a best guess of what that fundamental frequency is and then I can I can develop some metric of how close it is to being recurrent and you'll note here that when I do this I take a fairly small value of n n is 2 or 3 I'm not taking a lot I'm just looking for dominant periodicity okay so you can see this is a very general generic approach and then once I've once I decided I've got a nearly periodic episode let's say with N equals 2 or 3 here then I need to build a representation of what the velocity field is just using those modes alright so this is a very low dimensional representation of the flow this is the corresponding Koopman mode for these these values and I use this as an initial gas for a periodic orbit within the turbulence so this is just some nitty-gritty about how you choose the AJ it's not particularly important not particularly unique either but but there seems to work quite well and so this is what you find so let's say I take a signal and it's this blue line juggling around this is time this is the predicted period of the periodicity so let me just go back up here or if I got it up there yes I've got it here so once I make a prediction about what the basic frequency is I can then define a period just simply like that I plot my data against this this guess of what the period is and underneath I also plot this measure of how close I am to being periodic okay and yeah that's that's epsilon Omega looks like Omega epsilon but anyway it's it's the epsilon Armiger and my threshold I just set as five times ten to the minus three I think well no five times to the minus four must be and these green dots indicate where I think it's nearly periodic behavior right there's a lot more episodes here than in the previous picture I showed you right at the beginning of the talk and indeed this new approach gives me these purple boxes as well as these orange boxes so the previous state-of-the-art there are current flow analysis only picked up these orange e boxes managed to converge some periodic behavior in there whereas this new approach were able to pick up here a periodicity in here too and the key difference is that this approach doesn't require us to collect data over the period of the periodic orbit we don't need to shadow the orbit over its entirety and I'll give you an explicit example of that so here is a convergence on an energy input dissipation plot so the turbulent trajectory is this red dotted line it starts there that's what that helpful s is supposed to indicate to me it goes out here round around here over here comes in and I think finishes somewhere I can't see the finish now o finishes Danna and what we managed to find using this DMD approach is that this this flow looks periodic basically from here all right doesn't look at just plotting on this graph but anyway we then take our low dimensional approximation of what the flow is doing plug it into the navier-stokes equation it carries through this blue line now the reason it stops being solid here is because this is the data window and then we continue it to reach the actual predicted period of the recurrent orbit so the data window is 50 and our predicted upo is 70 now what happens next is you put this guess into a glorified solver and this blue line gradually converges through these green curves you can just about see the first iteration here to this purple curve which is a periodic orbit okay and this this is an example where we've taken data from a shorter data set converge the periodic of much longer period or much longer at least 40% and we've used this now to find a lot more periodic orbits so we're quite hopeful this is useful so in terms of what I've tried to say Koopman and ice is an interesting new perspective may practical recently by the discoverer the DMD algorithm I think that's that's a general message whether you're interested in turbulence or not could this help you in your your your research DMD can identify ups in turbulent DNS without the need for a near occurrence so this is a major step forward from the current state of the art and we're very helpful this is going to help us find a lot more pure orbits and the other thing we're we're very interested in doing is actually studying transient dynamics transient flow dynamics where you don't have simpler variant says everything's decaying down to zero but can you still extract the slow dynamic slow important dynamics and this is very important in in mixing events in stratified flows for example where you start with an initial condition the flow mixes and then just herbs away to give you a 0 velocity State at the moment there aren't many good techniques for actually trying to extract what's going on because it's not a steady dynamical system and here are just two two little pictures here this is an ohm upo and then this is one of the one I just showed you this is the vorticity stream wise velocity and you can just see they're not particularly rigorous vigorous but you know we're making progress in that ok that's a good place for me to stop so any floor statistically steady yes so you there must be some concept I mean if you have to if you have these states you there must be some concept of the at least the the the outside conditions of the the boundary conditions having some steady state do you think that any steady state flow has a periodic opt is there some kind of existence theorem or force or existence belief of these periodic orbits whenever you have a steady state or what are the conditions for having such a steady state that's a very good question so I think I don't I don't think you you need to convince many people that as soon as you have a complicated flow say turbulent or even just chaotic you've got periodic orbits you know anybody with any background in dynamical systems would be quite happy with that now can I say for any in any fluid flow with steady initial conditions there we'll be periodic snow I can't the periodic orbits no I can't because I know situations where you can have steady conditions and you have no turbulence in fact there's only one state which is a global attractor for all Reynolds numbers but those are rare and I don't you know what they're hard for to try and prove that they are so that so there's gonna be no existence you know if you have chaos chaos or turbulent then yes but then that's the harder question finding chaos in terms how far can we go with a method like that I mean it seems that as you increase Reynolds number yes will become much much much more complicated and many more so it seems that we were very limited in that's wrong ago that's right and I'm not so hopefully in my lifetime of being able to use is practically but I was thinking about this other day I was thinking well okay so all this sort of stuff community looking at exact structures of the equations you know simple various solutions of the turbulent attractor whereas can you go with this approach certainly you can you can pick up simple solutions solutions which you think are particularly relevant to your parameter regime but ultimately if you're going to try and validate the dynamical systems approach one has to go down this route now whether you can actually do it is another question now I would say I would say 50 years ago I bet you nobody had would have any idea that I would be standing here talking about this right and you know go back to the literature and you read papers fifty years ago you know the world has really changed just look at computing I mean like you know the the very fact that this work is now possible there's only due to computing I mean you know I could do these computations on that laptop it's embarrassing really so I don't know 50 years time will would be doing this automatically I don't know or will just be doing DNS and nobody will care about this or will we would be all doing machine learning I don't know well so this this is you know something I didn't mention in my talk but this is eminently convertible into machine learning in fact that's that's one thing we're doing now so we're mushin learning down to an optimal basis and then looking for periodic orbits in that that's very successful well I'm not sure that you yeah I don't know I haven't tried it yet but maybe that's that's yeah maybe that's something but you know it's a philosopher called question right you know so okay so let's say we don't do this what else we going to do are we just gonna say we'll just do DNS or we're going to piece together sort of little cartoons of what's happening in the flow you know you could we could do that but you know there's a long term picture here that you can see working it's just a question of how hard it is yeah I have a more basic question because I'm not a fluid dynamics guy so first of all how do you know that these cockman operator is going to be diagonalizable how do I know if in some basis but how do you know that this will work well you know I mean like are you done not for certainly for the navier-stokes equations okay okay so but this is the classic case you suck it and see all right you know DNS is very cheap and most the time people are looking for what to do with their DNS and so you just try it no and then I have more questions so like why would you expect that to be periodic orbits well unstable but with very few unstable directions I mean right so that's a discovery okay so we did so one of the things you know when Christophe was introducing me and he mentioned troubling waves and these structures in pipe flow it was a surprise when we found that these structures although unstable have very small dimensional unstable manifold it doesn't have to be that way I don't like you know I can't see any reason why it should be like that maybe 50 percent were unstable and fifty percent more stable we'd find a very different flow but that's just what it is it seems to be so it might be as a function of Reynolds number actually does get a 50% rule like feature of dynamical systems or is it like - you're kind of fluid dynamics situation I don't know it's just what we were so far in a number of different canonical flows and all these flows are typically linked by a unidirectional driving so that might be key I don't know whether if you were to drive it in various different directions that would disrupt that I don't know but that's the observation so far working basis that's what we find very nice thank you very much but actually there seems to be a remarkable lack of communication with the rich literature along the same lines in you know you you are obviously you have colleagues that endear yourself to gfd and and Astrophysical you employ dynamics we knew this stuff for 30 years that typically you know there is a dissipation you are you know there's a nonlinear laplacian there is no negative eigenvalues and then there are just a few directions that associated with instabilities yes and they're specific calculations which are done for you know full general circulation models with many many maybe at the time when we first looked at these things it was only a few ten to the four or five now it's ten to the six or seven or more variables and there are maybe a hundred directions which aren't stable which are realized from other instabilities than necessarily the Jacobian you know the thing that your Bhattacharya is pointing to in in the first slide and well of course coupe and and DMD has been very fashionable in non gfd and JFD recently you know we've had various kinds of so-called empirical orthogonal functions and other basis functions yes in which we've been doing these things and looking at weakly unstable periodic orbits in the context of predictability at very long times compared to the characteristic time of the eddies that are the weather systems in the flow so you know this is this is really great but frankly there's a lot of other stuff well there's loads of stuff periodic orbits are everywhere it's gone that you know unstable periodic orbits and I said you know just very weakly unstable ones so so you're you mentioned many things in there so for example the dimension of the attractor right so that that that's obviously related to our unstable the structures are I mean what I've tried to talk about here is actually a systematic way of finding the orbits and then trying to go further than just saying okay there's one orbit here that seems to be interesting maybe we can compare it with some observations we notice so that's the challenge okay what's our question I don't know whether you work this no this work from hacker and crouppen in essence of what they did they took a classical Kotik system like a billiard and then they look for the periodic orbits and then well the periodic orbits when they come closer you can the idea is you can switch between the orbits yeah and putting these orbits together you can reconstruct the whole spectrum of eigenvalues so somehow you I mean using the good feeler trace formula so some other you try to do the same thing for childrens which i find the worst idea that's periodic orbit theory and a crucial thing with periodic orbit series you have some sort of symbolic dynamics so you can label the periodic orbits order them and then you can build an expansion based on where they are in that alphabet we don't have anything like that for terms that's the that's the problem of trying to go from a very low dimensional approach where you can as I say build periodic symbols and then order them and then in fact you know of a certain length that you've got all of them at that moment we're at the stage of just finding random ones and then hoping we've got enough of them as a function of the period and then build an expansion but the trouble is we don't have any symbol symbolic dynamics of the flow so we're at this stage now of finding lots of them and but we don't really know how they're ordered which ones are really important which ones are not so you can do you can do a few ad hoc things based on periodic orbits theory you can you can build the weights on how unstable they are in terms of you know functions of all they're unstable eigenvalues or functions of the UH number of unstable directions but these are all guesses at this stage because we just don't have a for the derivation of how to do it if you are the stochastic forcing to the system is there something and I look so good you can do I mean you don't have three other culprits there is there's something you can do that if you add well what would you do I've already thought about that so much I mean I guess here would increase my dissipation right and then I'd have to start thinking about periodic orbits in some sort of statistical way maybe I could do the same thing but after I've done that statistical averaging I don't know how anybody thought about it so I just want to understand so your periodic orbit is an exact solution but what you think it you think it is an exact solution of the navier stokes equations of under quit flow conditions for example yeah now you change the conditions of your quit flow of your system yeah I asked the question before but maybe I wasn't clear to you there should be some kind of response function right I mean there should be the the orbit is not going to go away immediately so it's a stable object in the sense that now you you know it has response functions yeah you can work on that you can you can you can work on it you can not make all of second you can then once you have some of them and there must be stable I mean they test that they cannot go away in something really amazing robust there must be robust yes what I'm saying yeah so that could be sometimes just asking so are you people working on to statistical mechanics I mean once you have response function then you have statistical mechanics yeah well I mean it relies on the fact that you've got enough of them let's say let's say a 1 Reynolds number so you know you basically use your DNS at 1 Reynolds number to find all these parallel orbits and then you use those periodic orbits to make a prediction at a different Reynolds number yes that's that's the hope but we already know there's an issue with that and and the issue is that some of the you know if you if you track these periodic orbits some of them go through bifurcations so they might not reach that point they might bend back in parameter space but then you'd hope that you have enough of them that when you average through your expansion that gets washed away but yeah those are all those are all the ideas that you're after yeah more questions I might have one additional question maybe yeah sorry so you showed 2d column ago of aspiration so in this case we expect so you will have inverse cascade and condensate with large-scale mode which I would expect will show low dimensional behavior more easily than the small scale so is it true to expect this type of beta to be more efficient for 2d turbulent flows and freely in we have large scale mode with possibly low dimensional be a girl I don't know I don't know I mean are you peeing is the fact that the dynamics aren't so so violent possibly yeah I mean it all comes down to how unstable each of these individual structures are so maybe you could argue that you know there's stable quite a few stable large wavelengths then they'd be more stable but there is no Chiara I don't think we know if we don't know enough about it I mean you know it's just it's not enough develop theory we still thought poking around trying to learn okay more questions good so thank you again okay [Applause]

Channel: Département de Physique de l'ENS

Views: 838

Rating: 5 out of 5

Keywords: Rich Kerswell, fluid turbulence, periodic orbits, colloquium ENS

Id: xh-beI4SMEg

Length: 61min 27sec (3687 seconds)

Published: Thu Jul 18 2019