Order, disorder and entropy (Lecture - 01) by Daan Frenkel

very good afternoon to all the students present here a respected principal we have an eminent speaker today dr. Samriddhi Shankar Ray and the entire coordinating team the Faculty of the mathematics department I welcome you all to this fascinating talk on the fascinating world of turbulent flows see since it is related to mechanical aeronautical and to an extent civil Department I have requested your HODs to send you for to attend this talk today see it's my privilege to introduce today's speaker dr. Samriddhi Shankar Ray who is a reader at International Centre for Theoretical Sciences Tata Institute of fundamental research Bangalore he's a physics graduate from Presidency College Calcutta University Kolkata in 2003 and then he's did his MS in physics in 2006 from the Department of Physics Indian Institute of Science Bangalore he completed his PhD in physics in 2010 again from the Department of Physics Indian Institute of Science Bangalore and his research work focus is statistical studies of fluid passive scalar and Berger's turbulence he held many research positions starting from the postdoc fellow from April 2010 to December 2012 at the laboratory Lagrangian Observatory CNRS NICE France followed by a junior faculty position from January 2013 to 2015 at ICTS Tata Institute of fundamental research Bangalore and at present he is a reader in International Centre for Theoretical Sciences TIFR Bangalore he has been a Observatory or oh actually a visiting professor to many universities India and abroad to name a few it is CNRS nice France from May to June 2018 also Federation Dublin University of nice sophia antipolis France he's also a visitor to University of Rome May 2015 again at CNRS nice France May 2015 He was also a visitor to nordita at Stockholm Sweden again a visitor to Max Planck Institute for dynamics and self-organization göttingen Germany from October to November 2011 he's also a visitor to laboratory Poncelet Moscow Russia in September 2008 and their list is huge so I am just naming a few here he's also serving as the referees to the journals including Physical Review Letters euro physical journal physica D Journal of fluid mechanics Journal of statistical mechanics and non-linearity he was also a scientific secretary problems of turbulence 50 years after the turbulence colloquium Marseille 1961 France September 2011 his principal research interest lies in fluid magnetohydrodynamics passive scalar and Berger's turbulence inertial particles in turbulent flows truncated systems thermalization and statistical mechanics of turbulent flows singularities in the Euler equation multi-phase flows etcetera dr. Ray has also published around 36 publications in various peer-reviewed journals I will name a few here European physics journal Physical Review Letters Journal of fluid mechanics physics of fluid letters perspectives in nonlinear dynamics pramana journal of physics euro physics letter physics of fluids journal of fluid mechanics just to name a few at present he is guiding many PhD students and master students and he keeps organizing many conferences workshops and seminars he has given more than 15 invited talks at various places of repute like IISc Bangalore TIFR and JNC ASR bangalore iit mumbai Kanpur and Kharagpur University of Rome Italy Stockholm Sweden Vienna Austria Leiden Holland Saha Institute of nuclear physics Kolkata blacks Planck Institute for dynamics and self-organization göttingen Germany Marseille France and so on he is also the recipient of many awards and grants one of them is ECR grant from DST India from 2016 to 19 he's a principal investigator of Airbus group corporate foundation chair in mathematics of complex systems he's a CO PI and member of Indo french Center for Applied Mathematics he is got funding from European Research Council under the european community 7th Framework Programme he is also a member of European cooperation in science and technology on flowing matter cost action and he is also a member of European corporation in science and technology on particles in turbulence cost action I think if I go on reading the credentials that he has to his credits I only will take most of the time so I would like to leave the stage open for dr. Ray and I welcome him to the Dayananda Sagar College of Engineering I request our principal sir to welcome him with the bouquet stages yours while the lights are dimming let me thank dr. Gupta for than generous action and it's an absolute pleasure to be here it's the first time in this Institute and it's been wonderful the last hour and a bit that I've been to meet many of your teachers colleagues students etcetera it's really been a pleasure to be here so thank you once again I'm going to talk about is so I did interact with some of the students in the last few minutes or so and you know so most of you come from various engineering back grounds so what I will be doing is to give you a flavour of why some of us are interested in turbulent flows I will certainly sacrifice some rigor at the altar of being accessible to most of you so they will you know I won't indulge in a lot of complicated ideas and stuff but I hope that I will be able to tread the tight rope of not being inaccurate scientifically that at the same time trying to explain certain features to all of you but do stop me especially the students anytime you feel that I'm getting ahead of myself or things are just too vague for us to continue alright so so with that sort of preamble let me begin my talk but before that let me sort of ask you how many of you heard the word turbulence in any context it doesn't matter so there is show of hands amongst the students all right and when you don't what we usually do is we Google all right and when we google the word turbulance this it's not a selection that I've made to to make a talk more dramatic when you actually Google turbulence these are the sort of hits that you get nothing really to do with mechanical engineering there is for people who like football it has to do with Liverpool football club it has to do with hampshire cultural trust and things like this it's not surprising that this is what you get when you go because if you look at the history of the subject then you realize that the word turbulence in whatever sort of you know idea that you have in your mind you know how you associate turbulence with whatever phenomena that you have in your mind you realize that historically the word comes from the Latin word which means the disorderly motion of a crowd and later in the Middle Ages the word turbulence but you know variants were used essentially to mean trouble so I'm going to cause some trouble in the next 45 minutes or so and and try to sort of understand and appreciate while some of us are fascinated with this rather troubling phenomena right so of course you know before I begin let's have a sort of reasonable starting point and this we will modify as we go along so a reasonable starting talking point for turbulance as a scientific term of political or economic term has to do with the fact that it's essentially describes complex unpredictable and seemingly random motions of fluids all right and again just for historical perspective it's interesting to note that a brief timeline and here I should be careful this brief timeline is just a personal preference if you have someone else who works in the field you'll find different names different time stamps alright so but this is just my personal feeling it sort of begins with Euler and I'll talk a little bit about Euler note of the date seventeen hundred and fifty seven it's about two hundred and sixty-one years down the road and then it sort of continues with navier-stokes with Reynolds Prandtl the two of them were fantastic engineers as was Navier a French engineer Von Karman Richardson and then you know this is sort of Kolmogorov who was part of the great Russian school of mathematics and he sort of launched what one might call the modern ideas of turbulence and I'm going to certainly talk a little bit about the work of people who have been highlighted in red and then we come to really modern era Robert Kraichnan and these dates are you know their respective sort of you know they're they're sort of seminal work which changed our understanding of the field and it's interesting that here we have an Einstein connection Robert Kraichnan was a student or in those days what was known as an assistant to Albert Einstein who sort of set him off on this rather strange path and then more recently you know we have Uriel frisch and Giorgio Parisi in spite of this long history as I will argue or was in the abstract which we all had to read the problem is and I'll sort of try to just little what we mean by that half way through the stock and you see the day it stops at 1985 so we are really looking towards young people like you whose names and years will get added when similar talks are given in the future all right okay again before we proceed it is always a good idea to have a sort of a feeling or a rough idea of what sort of phenomena we are talking about interestingly turbulence shows up in a whole bunch of objects you could see things in the motion of these little small organisms especially people who come from close to the sea you see these sort of algae bloom and patchy ness in the water that's really small little organs which float around turbulence effects that certainly on Astrophysical scales where you have you know swirls of dust and heat and mass which is transported those flows are turbulent in geophysics you know this is a famous sort of volcanic eruption in the nineteen eighty so you have these plumes of particles which are being ejected and that flow is also turbulent again with that sort of very weak definition of irregular unpredictable so if you keep that in mind you sort of tend to appreciate for us living in in most parts of the world actually now the issue of pollutant dispersion smoke coming out of chimneys that seems to be turbulent then closer home in mechanical engineering if you push a flow past an obstacle then the flows have other weird structures actually called Karman streets then in aerospace engineering one of the problems that all of you have to deal with at some point in your curriculum has to do with boundary layers and what happenes when there is flow going past a hot plate or the aerofoil of a plane the one that sort of interests me a lot these days has to do with cloud turbulence and if we have time I'll sort of elaborate a little on this later oceanography so it's essentially everywhere including your everyday use of stirring sugar in a cup of tea all right one of the things that you could or should do it's but sugar people who take sugar in a cup of tea or coffee and don't stir and then wait wait and wait and see how long it takes for the you know for your cup of tea or coffee to be uniformly sweet and then you stir ofcourse all of you know thats rather quick that all right I've cheated here a little it's actually a coffee cup but never mind all right so what are the ingredients that go in to making a fluid move so this is sort of a very rough summary of things one is of course viscosity are all of you familiar should have viscosity in some way yeah is that a yes because nodding heads have different sign conventions depending on where you're from so some nods might mean yes and some might mean no deppending on your yeah okay so of course we all flows viscous we all know that honey flows less easily than water less easily than air and that sort of acts at small scales where essentially viscosity is like friction so if there is some energy then what viscosity tries to do is to get rid of that energy typically got rid of as heat molecules or little constituents of the fluid rubbing against each other to make something flow you of course have to stir the bottom water at rest is not moving and I stir it and it starts to move and that scientifically is one would call injecting energy into the system and that is often done at large scale so I saw there's a nice wind tunnel in your Institute and so those of you certainly hopefully the guys in mechanical engineering and aerospace would have looked at it and you'd see that there are grids at the front of the wind tunnel which sort of pushes the flow through the wind tunnel and because these grids as little flaps which it sort of drives the flow and hence sort of stirs it up I mean roughly exactly what you do when you stir your coffee all right and that's sort of important at large scales and the third very important thing and for engineering students this is your life essentially is to understand what boundaries do to flow the bane of all engineering students often is how to deal with boundaries as you will see people like me cheat massively and get rid of those boundary you come to it in a bit all right so for example here's a simulation which is it's this is a golf ball and this is an ordinary smooth ball cricket tennis what-have-you and if you can look carefully you will see that given that the golf ball surface is more puckered the sort of flow past a golf ball is very different from the flow past you know a ball which has a smooth surface so that's the role of boundary it's sort of generating the peculiarities of a flow which makes things complicated so where do we go from here and as I was telling most of you when interacting with you before the talk began this is the only slide where I'll have some equations ok so bear with me but I think for the sake of Completeness it's important that I show this so of course that's the phenomena we like there scientists often very often the starting point is to actually write down the description of the flow so for example when you were you know in high school you were taught Newton's laws of motion right you knew that if you hit ball the ball would move but then you actually wrote down an equation for that and often it was Newton's laws of motion you know an Apple falls you write down the equation of motion of a falling object under gravity so similarly we can apply the very same Newton's laws of motion on a flow and say that in a region somewhere here I assigned a certain velocity I'll call that 'u' of the fluid at that point and now I just need to write an equation which will tell me that if I had the velocity 'u' at some point X and time T then what would be that velocity at some other time T or at some other point X or at some other point X comma T that is roughly equation of motion is right so that's where Euler comes in who's you know one of the great heroes in in mathematics his contribution in across several fields in mathematics in 1757 and that's Euler in 1757 he wrote down the equation of motion for a fluid don't worry about the precise details equation the only if you want to worry about is you should keep in mind that it has terms which are U squared so when terms are you know of the form of squares which are called nonlinear terms it doesn't make us happy that's because we really don't have a way to solve such equations and typically and this remains a unsolved equation so I'll come to that in a bit however in real fluids as we discussed there is viscosity enter Navier and Stokes about 70 years after Euler Euler wrote down his equations without it it's viscosity what's known as an ideal fluid now here who's an engineer so he knows that the real world are complicated and Stokes was a mathematician said well real fluids have viscosity so what they did was they added this viscous term alright and that's all all the equations I I'll show you just to tell you that we have exactly like you could write down equations within your newtonian framework of single particle you know Newton's first law or second law force is equal to mass times acceleration you play the same game and you take a little bit of fluid at time X and T do a bit of algebra and you can come up with equations of this sort and the thing to keep in mind is this equation is non linear it has terms which are U squared that's a hard nut to crack when you have something which is nonlinear all right so and this is where sort of setting will end and then lo and behold Osborne Reynolds a little later said that different flows suppose you take this water right so in principle what I wrote down should be valid I mean can you see what I'm doing with this water I'm just tilting it a little and the surface sort of moves a little I mean if it was half empty it would help so maybe I'll just use up the water a little okay that's better if I move it around a little it moves in a nice way right if I shake it around it looks very different all right so the equation that I wrote down all over navier-stokes wrote an actually not me sort of should be valid for both these motions so how is that how is this sort of motion where you see all these sort of structures everywhere very different from when I pour things gently it's the same equation so that's where you know comes in the genius of a Reynolds actually Stokes himself suggested this before Reynold's was to define non-dimensional number called the Reynolds number which is a simple formula so it takes the velocity of the fluid it takes the length of the fluid so if it's a wind tunnel maybe the cross section of the wind tunnel or the length by span of the wind tunnel divided by the viscosity of the flow all right and what Reynolds suggested was that when this number becomes large so when I do this there's more velocity that's sitting in the numerator so this number becomes large and the flow is weird when this number is small the flow is what's known as a nice gentle a laminar flow all right so far are you guys with me I mean how you sort of encountered the word reynolds number before oh good so you should have just up shut up and move on because this is at for you all right okay so essentially what is so if I now come up with the more sophisticated definition of turbulent give some starting point so most sophisticated definition would be the turbulent flows are solutions of this equation when the reynolds number is very large or often when this viscosity this new is a measure of you know viscosity it really tells you whether you're using honey or air and for air that apparently is small okay so that's the sort of working definition that we use that is great all right good and then you know this is an example of a flow coming out of I actually don't remember what but probably a candle or something and you do see that the flow here is very laminar without any structure like the water that I'm holding now the flow here has a lot of structure and if you actually go ahead and measure this object here you find it has a very large value and that corresponds to our naive' idea of turbulent flows and if you measure the same chap here you will find that it it has a much smaller value and that corresponds to a sort of naive idea of being non turbulent or laminar all right okay so I'll ignore this for the moment okay so what is the problem the problem is the following this sort of subject comes under the broad classification this is just for nomenclature so you can google or go to Wikipedia it comes under the broad class problem which are known as non-equilibrium statistical physics so what it says is that if you have flows with a very large Reynolds number then there are likely to be turbulent and they remain in a non equilibrium steady state I'm using my words carefully if it was suppose you were to measure the energy versus time in an equilibrium system where things I've all equilibrated typically the energy versus time will be a flat curve nothing changes alright in turbulance if you were to measure the same energy versus time and I'll show you some results from experiments then you'll see there are huge fluctuations in the energy versus time profile but on average if you if you average it over one hour one day god knows what then you will find that roughly the values remains the same so it's an example of a system which is in some sort of a steady state but non equilibrium and that and just steady state is maintained because again on average the amount of energy that I'm pumping to the fluid by shaking is dissipated as heat by viscosity alright so that's the problem that we are going to study and here's an example so for example this is the velocity versus time taken in a wind tunnel by sticking presumably a hotter eater and you see that there are these huge fluctuations that if I take the series to be very long would see that on average if I make an average let's say from here to here and then from here to here then from here to here roughly the value remains the same in spite of these big jumps alright and also you know the fact that these flows for example if you were to measure how much energy you are using as a function of time then you'd see that there are times when you lose a lot of energy you see these spikes there are times when you lose absolutely nothing all right this is what are known as an intermittent signal alright in and I will come to this in a bit it's exactly what happens when you are trying to make a phone call where the the sort of the signal is weak so you end up a lot of your phone conversation becomes a low can you hear me all right and and and there are periods when you can hear each other and there are periods when you can't hear each other so that's an example of a process which is sort of intermittent okay so how do we understand turbulance? Jonathan Swift who you know who was a great satirist in the 17th century had a nice little poem which was an inspiration to as you will see for people who were in this field so Jonathan Swift talked it was really a satire on how poets live and the example he gave that of fleas so suppose you have an animal and there are ticks which is sucking blood from that animal often there are other ticks which attach themselves to the first stick which were sucking blood and essentially you have a process in which the blood from the big animal lets say dog or whatever gets transferred from tick to tick okay its a beautiful poem which I won't even read out to you but this sort of by this great fluid dynamicists Lewis fry Richardson who sort of I ll first read the poem and then I'll explain who sort of said that in such flows there are big sort of fluid structures so if this auditorium for example had a big fan then this fan would move and it would generate a flow but thats a big world all right but there are small little worlds which sucks energy from the big world and then again just like the ticks in the example of Jonathan Swift it then transfers that energy to smaller and smaller world and that was Richardson's way of understanding without complicated mathematics how energy gets transferred from the very large scales where they're injected to the very small scales where viscosity takes over and you lose the energy as heat okay so what it so again schematically what it means is if you have a large length scale L let's say the size of this water bottle you're inject energy here me by shaking this bottle then there is a sort of range of smaller scales which are smaller than this bottle where all it does is it takes energy and just passes it along to smaller and smaller length scales till it hits length scales which are so small that viscosity or friction can take over and dissipate that energy as heat because if things rub they're happening at very small scales right because either small particles all right and of course this is a this diagram is a little counterintuitive because normally and x-axis you have small numbers here large numbers here but this is drawn in the inverse way where L is much larger than R is much larger than eta okay so far all of you are with me at some broad basic level if you want we could go over this cascade picture a little more if that helps just just shout out or just catch me after the talk of you're shy right now okay fine all right we saw all these rather fancy videos of Astrophysical settings of golf balls of I forgotten what others were smoke etc now those are very complicated things to actually start a theoretical understanding so like all good physicists one simplifies things and one considers that the turbulence is isotropic what I mean by that is suppose I go close to this wall all right this I can use suppose I'm close to this pillar I'm no longer isotropic because isotropy means I want to turn in various directions because if right I see this pillar and I'm stuck all right if I am in the middle of the room far away from the walls or the boundaries then each direction should look very similar to each other right similarly with homogeneity it's just roughly the same concept which is again if I'm far away from any boundary so I don't see this table if I take a step here all right take a steep step backwards if there were no obstacles in principle statistically things should be the same right and the reason I end up having this is because I want to look at this problem far from boundaries where the direction in which I look doesn't matter where if I take a small step it doesn't matter from where I was before okay good so the question that we ask is what is the physics of when I am looking at this small region of the flow so again students from mechanical engineering or aerospace if you go to a wind tunnel what I'm saying is go to the middle of that wind tunnel well dont go yourself but you know look at the flow in the middle of wind tunnel where it's far away from the grid I mean you go to the wind tunnel as long as it is not running it's perfectly fine you go to the middle of the wind tunnel far away from the boundary far away from the grid and you are roughly in a setting which is homogeneous isotropic okay and we want to understand what is the physics of the flow there and whether it's universal what I mean by that is in your wind tunnel if you use SF6 if you you know in a water tank if you use water if you use some other gas the measurements that I'll get are they all going to be the same that's what I really mean by universality or if you go to another Institute and use their wind tunnel you should expect you should hope to get the same answer all right otherwise you know we are in trouble okay so this understanding of what happens at this scale starts with one of the great heroes in mathematics and physics called Kolmogrov who I should point out I mean you know in turbulence needs rightly venerated but yeah he has contributions to many other areas of specially mathematics and physics so Kolmogorov came up with this that the physics at this inertial scale you remember this this is what I mean by the inertial scale where energy is getting transferred which is length scales which are far from boundaries far from viscosity and hence probably don't care about the fact whether the experiment is being done with SF6 or water or whether you're using a turbine with a blade which is curved or straight so in these sort of scales Kolmogorov sort of asks the following question or propose the following that the physics here should not depend on viscosity as we've argued should not depend on the structure of your experiment it should be something more intrinsic to the material so an easy way to understand this is the following you have a velocity field so you can sort of stick your favorite measuring device at every point in this room and measure the velocity of the air in this room all right but Kolmogorov asked a more subtle question which is following i measure the velocity difference at this point and at this point so I'm measuring the difference of the velocity at a point X plus R and at point X so I'm going to look at the velocity difference over scale r all right ignore the projecting I mean that's just mathematical accuracy and then he says that things are homogeneous and isotropic and things should be independent of viscosity and how you the measure the flow so in that case this guy how many of you are familiar with dimensional analysis you must have done it in school at some point right or even now that's the easiest way to check in your exam whether the answer is wrong or not typically it sort of works okay so let's pretend to be Kolmogorov let's go back in 1941 and pretend we look like that and then sort of Kolmogorov argues that this delta u can really depend on two things it can depend on 'r' I'm measuring the velocity difference over the scale 'r' so it has to depend on it and it will depend up how much energy there is in the surrounding air reasonable sort of expectation to have so so so he said like you would do dimensional analysis he said let's write down Delta u r as being epsilon which is the amount of energy there is in the system to the power alpha and to the power beta so any of you can tell me what alpha and beta will be the dimension of velocity is meter per second the dimension of R will be meter let's say length and epsilon is the amount of energy you're putting in actually it is the time derivative of that so it's kinetic energy divided by time so that's meter squared over seconds squared over second don't know if I got the factors right meter square per second cube and there is a length there so I will wait for 10 second in the interest of time any of you can shout out to me what alpha and beta what the values are any number is good enough okay all right so I I'll let you do it in in the comfort of your hostel room you work that out and it's really a trivial calculation you see that Delta u should go as epsilon one-third epsilon being the energy injected and the length to the power one-third and then Kolmogorov went a bit further as he's a serious man and he said let's calculate all powers of this guy now that's easy to do you had something which was R to the power one-third if you take the p'th power of that then you end up so this guy take the power P so I'm ignoring the epsilon term it's not that interesting and then you end up with R to the power one-third to the power P which is R to the power by three right so this is so again just as an aside because it all appears very mystic that you know in physics we are fascinated with what objects such as this these are known as and these are sort of known as correlation functions it sort of tells you how corelated an event is with some other event so that's why Kolmogoov was really after this and he said that this guy which is known as Equal-time structure function should scale as R to the power zeta P where zeta P is just 'p' by 3 most of you see this right it's not it's not that non-transparent ok are all of you with me good ok so one of the consequence of this is what happens if you substitute the value P equal to 2 in this so put instead of 'p' put 2 to then you get the the exponent power law of second order structure function R to the power two third a consequence of that and I'll sort of gloss over this a little but this is really what you measure in a wind tunnel a consequence of that is when you measure the energy spectrum in a wind tunnel which is the distribution of kinetic energy across various scales typically done in Fourier space it's okay if you sort of for now if you don't know what the Fourier space is but the consequence of this is that the energy spectrum should be K minus 5/3 lo and behold here's a man in 1941 postulated this and then over a period of time people such as yourselves go ahead and measure this energy spectrum and they plot it and they find that in this inertial scale remember inertial scale where the physics is supposed to be universal they get so this sort of line shows that this really goes as K minus 5/3 again look at large scales where energy is getting injected in different experiments there is a scatter of points so that bit is not universal this bit is not universal that as Kolmogorov has predicted the form of the energy spectrum in this middle length scales are universal and seem to agree with Kolmogorov again so the other thing of course is if you look at the full spectrum of these exponents then Kolmogorov suggested that they go as P by 3 here's the red line however when people measured this these numbers so they they go to an experiment they take the velocity difference between two points they plot it then they take the power of p of it so they take the first path second path red powerful power so on and so forth and they see what is the power law R to the power what for the p'th power of this object Kolmogorov said that it should as p by three so if you take the fourth power of delta u then should cut to the power 4/3 if you take the fifth power it should go like a 5/3 you can actually go ahead and measure this in your lab and when you do it you find that these black dots which are the measured dots and red is the line that Kolmogorov says so it's a line with slope p by 3 you find absolutely well for atleast certainly for large values of P you find absolutely no agreement with Kolmogorov ok it was going good so far till we were still at energy spectrum we saw that Kolmogorov is pretty pretty much correct but when we went further we find there's a problem okay so just to summarize take the velocity field take the velocity difference on a scale R take the p'th power so this is what I am plotting Sp versus R on a log-log scale power laws show up best when you plot things on log log right Exponentials show up best when you plot things on a semi-log scale etc and you and you will find that there's a power law which goes around zeta P but this number zeta p which Kolmogorov said was P by 3 is actually not correct so Kolmogorov the Kolmogorov theory goes as K41 okay that's the common nomenclature because it came up in 1941 and K is the first letter of his name okay so what goes wrong and here we are going to tread and make our way very slowly okay so so before we understand what goes wrong let us accept the following statement and you know you can come to me after the talk if you are if you want to proof of this statement with the following that the moments of the velocity increment which is what Kolmogorov was calculating is intrinsically related to the probability distribution of these increments how many of you are familiar with probability distributions amongst the students one hand up okay good few more hands going up so what is the probability distribution I will explain this in a bit actually in some detail in it you make repeated measurements of some and once you find the number one once you find the number two once you find the number three and you draw a histogram how many times have you found number three how many times have you found number two etc okay it turns out that objects such as this are related to the probability distribution of the Delta r u so if you've measured 1 million times Delta r u in your experiment in your lab then you can sort of just put in all the values that you've got and how many times you've got you get a distribution or you can calculate the moment and average them okay and the two are intrinsically related all right so what goes wrong is the fact that the distribution of increments must hold the key to this story okay and this brings me back to my naive introduction where I said the turbulent flows are probably random so we can ask ourselves if there is a process which is truly random then what should a distribution look like okay let's think about coin toss hum you know you would have done the coin tosses right or you know rolled a dice at some point suppose I begin tossing a coin four times so if I do it four times I can do it in my head the number of times you will get all four to be heads should be one the number of times you'd get all four to be tails should be again one and you'll have a distribution so the number of ways you can get zero heads is one that means all have come out to be tails number of ways you can get four heads is one all have turned out some things in between now I do this experiment eight times and you have a different distribution I mean same this distribution starts to take some shape I do it 32 time actually forgot to know me 16 maybe okay and I continue doing this I I continue repeating this experiment and slowly you start seeing a rather nice looking curve this curve is what's known as a Gaussian distribution and you see Gaussian distributions everywhere for example look at that Galton board so essentially you start with a lot of beads small particles which fall and you have these sort of obstacles I mean you can play this game with your friends actually you have this obstacle so the beads are entering in different sort of channels now if you wait long enough you again get a distribution in which they fill up is very similar to the distribution in the coin toss problem you can do something else which might be more relevant for you a few years time from now so this is the distribution of scores from the SAT exam I must be the same for GRE etcetera but I found the SAT distribution and you see again Gaussian so the number of people who get a score of 1500 so that's where the peak is you have outliers who are really bright and thought of students are working like crazy and they get very high scores but the number of them who do that is much less so in general distributions for a sequence of truly random events is Gaussian okay you can you can take a random walk and you'll find that the distribution is Gaussian and all moments of such distributions are easily calculated but what about the distribution of velocity increments in the turbulent flow we saw what happens when you toss a coin we saw what happens in SAT scores we saw what happens when you take particles falling randomly you can go ahead and see for yourself in the afternoon what happens when you click I mean it's a very simple game stand somewhere with a friend of yours who will count toss a coin if it's head you go right and if its tail you go left you do it for one hundred times write down how far you travel from where you started repeat the process continue repeating it till you or your friend gets bored and then plot those numbers and you still have a nice Gaussian distribution if you do it for a sufficiently large number of times all right so most things a lot of things that we are familiar with as being truely random end up with such Gaussian distributions what about velocity increments so this is the distribution of Delta U our favourite Delta U that's the only thing I'm gonna talk about and and when you plot that you see and this is plotted for different R so I can take out to be 1 centimeter 2 centimeter 3 centimeter I plot the distribution and they don't look Gaussian I can plot the distribution in time and I'm I can sit I can follow a fluid particle and I can calculate Delta u not in space but in time and I plot them for different time intervals I calculate the velocity difference at one second two second three second what have you doesn't look Gaussian I can plot the velocity gradient gradient is just a fancy word for saying taking a derivative okay I plot it again it doesn't look Gaussian I can plot the acceleration I mean that's something that's easy to calculate again the sort of damn thing doesn't look Gaussian at all so this is the problem really that how do we understand a problem where everything is non non Gaussian everything that we can think of gaussians we understand I mean we understand gaussians we understand how to calculate correlation functions of gaussians but when you have non Gaussian distributions they're somehow related to the fact that the ideal Kolmogorov scaling tends to break down and people in the trait use the word intermittency to somehow push it under the rug right I I did give a sort of feeling of what intermittency is but but and I mean to my taste at least it's only strongly linked with the fact that whatever you measure turns out to be non Gaussian and here is the problem with turbulent flows and then you go ahead and I'm not going to dwell too much on this because I've talked about this and you go ahead and measure different things in turbulence this is just a flow visualization exactly what you would do in your courses and you'll find that there are you know black regions are somehow regions with low vorticity Green are with high vorticity but don't worry about specifics but just the sheer patchiness if you like of signals from a turbulent flow it all appears very patchy all right and the question is do we understand the origins of this patchiness and I'm not going to go over the theory but this is just for the sake of completeness so people like you know in the 1980s Uriel frisch and Georgio Parisi extremely distinguished they sort of came up with a rationalization called the multifractal formalism for those you know it could be something you could do for your summer for example to understand this formalism and now it's used widely in other areas such as art etc they came up with a way to rationalize it then Bob Kraichnan came up with what's known as a simpler problem again to rationalize it the purpose of me showing this is not because I want you to know what these are but to sort of convince that people in the field have not been highly lazy and there has been lots of interesting work which has tried to sort of understand what is at the heart of such behavior but the bottom line is that the origin of intermittency is still up for grabs we have absolutely no idea starting with the equations of motion where all this come from so in my sort of abstract I said it's an unsolved problem so if you again Google you'll find several or various great scientists perhaps rightly or misreported over a period I'm saying that this is really the last great unsolved problem so you can ask is this the unsolved problem I'm talking about the answer if you ask me is yes this is the unsolved problem how do you get non Gaussian distributions how do you why and then why'd you get such multi scaling behavior however the answer to this will only depend on whom you ask if you ask an engineer the answer to what is the unsolved problem is different and then you ask the mathematician the answer will be very different that's because you know if there's nothing else that you remember which is very likely what you should remember from this lecture is the fact that turbulence is a collection of many problems that here you all are from different departments and you will in your courses will be introduced to these aspects but in very different ways and that is because really it's a collection of many different problems and what I sort of discussed at length is just one of those problems all right so at this point I'll take a break and just check about time and I'm almost out of time okay fine so so I I won't sort of go into some of the other problems which are sort of people have looked at which has to do with with a very famous problem in mathematics called finite time blow-up and if you actually do it there's a million-dollar prize waiting for you but these I will be happy to discuss with you later so again maybe I have five minutes okay good so I think we have reached a stage where a lot of us are wondering it's a complicated problem but why do we care about this problem how does it affect us absolutely valid question so very often the flow itself doesn't affect us but it affects things which are more tangible for example the motion of airplanes and strong winds is a rather scary example YouTube videos of ships caught in storms and airplanes caught in storms I guarantee you'll have a sleepless night and never ride either of them for a very long time okay that's a very tangible prop where we really have to care about understanding what turbulence does so this will be like two minutes I won't go into this at all but just to tell you a little is for example what is the role of turbulence when it comes to small particles so where do you find these small particles by the way this again goes under a broad area called turbulent transport for example forget about the slides right now for example think about a cloud so in high school what were we told water evaporates it then condenses and then the cloud rains here's a challenge when water evaporates and it condenses it makes micron sized particles micron is 10 power minus 6 of meter then through condensation because they're super saturated water vapor this grows but it grows to give or take 8 microns 10 microns for it however when it starts raining these water droplets are about 50 microns ten times larger how does it grow to something which is ten times larger now if you actually work it out and try to use just condensation as a mechanism of growth then you will find just like your coffee which would never get sweetened if you don't stir it you have to wait for an enormously long time before your clouds can actually precipitate so now there's a general consensus that because when you go up there they're strong winds these strong turbulent airflow is really kicking around these small water droplets as they get kicked around they tend to coalesce and so for example here is a very practical case you want rains you possibly it's still a reasonably open problem but I think there's consensus that you possibly need to understand what turbulent flow does to these small water droplets which are zigzagging around crowds clouds and again you know just for this and I think I won't go into details for example here again you write down the same equations as I did but something more for the particles Newton's laws but what I wanted to emphasize is once you have a mastery of these equations they are applicable not only to your clouds but if someone wants to understand how small microorganisms flock you must have seen big birds flocking right in the evening when they go home as do small organisms and you can use the same sort of model to you understand how the flock you can sort of try to understand it's not very visible how chains and filaments coil and uncoil in a flow so I think in the interest of time I will sort of just I won't go over this but I will end by I try to make a fancy talk which makes the transition a little long I will leave you with some lasting images this is because I think there is a certain wonder and amazement when you look at flows and flow structures which has excited not only scientists but people you know people have looked at such objects so I thought it might be nice to leave you with some lasting images this is from da Vinci's notebook who was no mean scientist him says where he sketched you see its very similar in its shape to that of actual flows that I was showing you this is from the great Japanese artists which sort of looks at turbulent waves you see all these structures it's not very different to what happened shaking the bottle of water I mean if nothing else these images hopefully will stick to your mind this is Van Gog's famous starry nights and you see these huge vortices the stuff that dreams are made of where you know flow circulate that's from 1889 and in the end while you know if there any questions I'll be happy to take I'll ask you to focus on this video it goes on for a bit so I can take questions where at the end you'll find something dramatic so it's just bubbles surfactants so you can play while you are trying to wash your clothes then you put your detergent in water and you stir it a nice way and you'll see at the end it's able to recreate something which I won't tell you unless we wrap up before the movie runs and also at the end you know I should certainly like to acknowledge various people including anupam and gobhinath who are here and there who sort of helped in this talk and and a bunch of people from whom I've learned some of the things that is with you particularly my thesis advisor Rahul Pandit and my mentors Uriel Frisch and Jeremie Beck ok and a whole host of students okay so I will leave it at this and I'll be free to take questions and you know meet you guys after the talk without the formal set up thank you but keep an eye on this it's more interesting than anything that I have to say okay so yeah yes good we can I just keep this spot switched off because I'd like to this is really nice it's getting there okay while he gets there does it start to resemble something that I showed you about a few seconds ago that van gock but done through a fluid experiment of bubbles interact with the flow that's what you should take and feel excited about the subject more explain the act of rationalizing you talked about the intermittency is an unsolved problem and yes many people are your rational Isaac formalism doing a formula for that what does it mean rationalizing the intermittency can you just explain this sure that's the only question okay I'll explain it in brief here but we can continue offline afterwards okay so what I mean is unsolved is that we when I say something of solved do I understand it from the basic equations of motion okay by the way they do I understand it from the basic equations of motion I don't all right so I'll go for the first that means sort of talked a little about the fresh Polizzi multifractal formalism so do you remember this poem I showed about how energy is transferred from scale to scale so there eventually it goes to small enough scales where did you get this so within any of these models so let's forget about it right let's start with something simpler suppose you have big bulls then you produce things which is tightly smaller scale these could be space-filling so your big world we have had a volume v then you produce two or three daughter ladies daughters which can which typically in the old picture in Colma girls it would be something if you add up those small daughter ad also have the volume D alright it could be that they don't have a volume V they have a slightly smaller volume it's like how you build up a candle set how you build up a fractal structure and so this energy cascade process the scale to scale transfer of energy may not be as efficient everything along so what it means is that regions in the floor which are not active at all and you have regions in the active which are carrying energies of carrying energy and these adhesion and and the rest of it you know we can talk later and these Eddie's really live on a fractal set so that is nice to see how this intermittent of multi scaling behavior can image alright the problem was a slightly different problem but that is really the so what Crichton said forget navier-stokes for a minute but think about the problem of how a blob of ink in a fluid moves and let's look at that problem right so that's that's roughly what's known sustain a problem what it does is and I'm you know we outside on the black it reduce full nonlinear problem to a slightly manageable linear problem and then and you can see how this multi scaling emerges okay but yeah more of the you know round meet a little more I mean beyond that I'll need a bit of blackboard and then in paper yeah it's in dispute gets ready to a Christian good afternoon sir yes it was nice talk about turbulence how it will be turbulent flows in real life and biological way so I'm very curious about explain or turbulent flow means so can I just explain me how Dublin's flow in porous medium in porous medium poor is okay so this champ so this porous medium do you have in mind I mean because there are porous medium where the flow need not be turbulent are you talking about things which are sedimenting and porous media means as a porous in a tree so if you consider earth itself is a porous medium mmm so if you take it for example sponge yes if you consider that sponge well me I must in the water yes that would be flow yes so can we expect a turbulent flow in it instead of laminar flow in a porous medium it depends on how you are stirring up your water tank right so in principle sure you could make it turbulent for example that was this with the gauls ball which had these holes on top of it so what the porous medium will do is complicated the boundary conditions around the flow it will have an obstacle so if I'm thinking of putting us in a tank of water which is already stirred up and turbulent then that you know that sponge Jumma be will change the boundary conditions around it and it'll they might be Carmen sweets etc but what I'm maybe you can talk afterwards because what I'm not completely understanding is whether you want to use a porous medium to trigger tip so that is how the flow inside a porous well I mean for example if the falls are very small right typically you can have capillary motion as well so if yeah I think it's really how you would set it up so maybe I'm not I mean you're here maybe with a piece of pen paper you'd be thank you so I hope you all enjoyed the talk on turbulent flows did you should you give a huge round of applause [Music] see you can already see there are a number of students who are already working with cert on baby it's your interest if you are interested in contributing something to this particular field or UK you are interested to know whether something really interesting is happening and you would like to collaborate with sir you want to meet sir he is always available we can share his mail ID and the number with you you can see there are many students who are engineering students only so maybe any small project also or any in small internship at the end of this semester you would like to take up it sir he is I'm sure he will really appreciate yes so I'm sure he would really appreciate it sir thank you very much for giving us this and lightning girl talk I also appreciated the mathematics part behind it especially the Gaussian curve which we have been teaching to the students and I'm sure when they learn they forget it the minute they complete the semester I hope now they will remember it for the life thing so thank you very much self on behalf of the management of DSi Ganon's our College of Engineering kindly accept a small memento as a token of our appreciation I request professor ignition sir to do the honor [Applause] so we have come to the end of this talk and if you have really liked this talk we would continue to arrange such kind of talks which are very much relevant to your subjects in future also I would like to thank the organizers especially the ICTs team and mr. Anupam who has been coordinating throughout five I think first more than two months he has been coordinating with one of our faculty dr. Sanjay Olli so I would like to thank dr. Sanjay also and mr. Anupam also for this wonderful talk which has been arranged today the stock will be available on the YouTube in few days time will let you know you can have a look at it if you have any further confusions or doubts you can always revert back to sir thank you so much [Applause]

Channel: International Centre for Theoretical Sciences

Views: 2,414

Rating: 5 out of 5

Keywords:

Id: EIeUoQZm5yo

Length: 86min 7sec (5167 seconds)

Published: Fri Jan 10 2020