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Turbulence in Fluid Dynamics
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0:08
hi everyone welcome back today i'm really excited to tell you about one of my favorite topics in fluid dynamics specifically turbulence and i've been looking forward to this video for a long time because it's one of the kind of most interesting topics in fluids that has captivated people for really for millennia okay so um great art has been made to try to capture the nature of turbulence not just da vinci we have van gogh and we have the great wave
0:41
turbulence is a part of our literature it's a part of our very language some of my favorite words uh roiling churning uh turbulent these mean so much to us and so today what we're going to talk about specifically is what is turbulence okay and so you know even though we have a very intuitive feeling of what in turbulent flow is uh and is not it's actually very hard to pin down a single kind of unique definition that we can all agree on
1:12
and so instead of trying to do that i'm going to bring up some kind of key features of turbulence and walk you through some examples of turbulent flows how they're useful and how we might analyze them as engineers so uh turbulent flows generally are three-dimensional that's one of the the hallmark features of turbulence and in principle you can technically have approximately 2d flows that exhibit turbulence but that's like a special case so for example weather on the surface
1:43
of earth we have a thin atmosphere and so the fluid dynamics are approximately two-dimensional but you get weird things like small vorticity adds up to make bigger and bigger vortices and form hurricanes which you don't see in kind of a classical three-dimensional turbulence so when i say turbulence i really mean kind of the three-dimensional turbulence where things are spinning and roiling in three dimensions turbulent flows are multi-scale okay so they're multi-scale
2:12
in space and time which means that there are big eddies and medium eddies and small eddies and smaller ones that infinite okay and you'll notice that all of these features that i'm going to list are actually in this davinci painting so this is clearly kind of his attempt at three-dimensionality it is highly multi-scale you can see kind of self-similar patterns it's unsteady even though this is a fixed image it clearly evokes motion okay so turbulent flows
2:42
are not fixed they're not stationary they are evolving in time they are unsteady and they also induce mixing in the flow so you can kind of imagine if you dropped some red food coloring into this turbulent flow it would very very rapidly start mixing and diffusing and kind of turn pink and then dissipate away so these are four of the canonical features of turbulence that we're going to investigate more and i'm going to have a few videos in this series about what is turbulence
3:11
and some examples of canonical flows how we can analyze them things like that i'll even do some kind of by hand sketches and analysis in a few of these but this is kind of the big idea we're going to explore turbulent fluid dynamics which is literally all around us okay and i'll point out at the outset this has a tremendous amount of value for us as engineers when we design aircraft and ships and jet engines more fuel-efficient cars and trucks and trains
3:42
absolutely you know tremendous applications for turbulent fluid dynamics in engineering but i really do think that our fascination with turbulence goes far beyond just its usefulness and you know is often motivated by its beauty and by the fact that it's all around us good okay so when i was preparing this material i went back to a lot of the course notes where i learned turbulence and i was fortunate to learn turbulence from lex smiths at princeton and you can download
4:12
these course notes for mae 553 viscous flows and turbulence by by lex i highly recommend it this is absolute gold and a lot of what i'm going to pull from uh in these videos was you know out of this this book so i highly encourage you to check this out because this really is a unique perspective from one of the absolute foremost leaders in the field okay good so we're going to start talking
4:39
about about turbulence and i want to just start off with some some videos and some pictures just to kind of you know get you thinking about what we're talking about. This is one of my favorite videos by colleagues at KTH in Sweden and so this is going to be a simulation of a boundary layer and some of you have seen this in my videos before but this is a very very high fidelity numerical simulation
5:06
of a developing boundary layer flow... so basically think of the flow over a wing as that wing is going through the air that's what this is trying to simulate... or maybe the flow over the top of your car or train and what you see here this has all of the features: it's three-dimensional; it's unsteady, it's evolving in time Highly multi-scale: look at the vast vast range of scales that are captured by this turbulent flow
5:30
and clearly it's mixing... if you started out as a little upstream particle you'd be bouncing around between these eddies and after long enough you wouldn't know which way you end up So this is a great kind of visualization of a turbulent flow that we can have in our mind for the kinds of things we might care about and simulating and understanding and modeling these flows is very important.
5:55
If i'm designing an aircraft, or i'm designing a wing for an aircraft, I need to know what the drag coefficient is for that wing i need to know what the skin friction as i go through the air or through the water is going to be and i need to be able to know how that changes as I increase the speed of the flow, decrease the speed of the flow if I change the curvature of the airfoil... things like that. So it's very very hard to model these things.
6:23
This is currently an open challenge in engineering, finding better ways of modeling, optimizing, and controlling these turbulent flows so another cool example and again i kind of got this this idea from from lex smith's you know when you see a turbulent flow you'll know it because it's going to have this multi-scale structure that the more you zoom in the more you see any piece you zoom into you're going to see more turbulent structure and more kind of fine details and resolution
6:55
another cool example is this this volcano erupting you pick any spot and you zoom in and you're going to see more roiling and churning and turbulence and actually one of the things i remember lex saying was you can kind of tell how high budget a hollywood movie is based on how many scales are in their turbulent explosions okay so and you know now i can't watch any action movie without looking for that multi-scale structure and i bet you're not going to be able to either
7:25
okay another example i really like this ocean next project you can google and download data and kind of follow what they're doing but you know climate the earth's climate the ocean currents and the atmospheric patterns are highly turbulent they mix and kind of transport material across the globe and this is hugely hugely important for us to better understand mixing and transport and dynamics in those turbulent systems as well good
7:59
now i also want to give an example one of my favorite examples is this rayleigh bernard convection flow where you have a hot plate on the bottom and a cold plate on the top and you can see that just from this thermal instability you get this kind of rising of the warm air and dropping of the cool air and eventually you're going to see this start to mix in a pretty chaotic fashion now this is not three-dimensional so but it is unsteady multi-scale and and mixing
8:30
and this is one of the original problems that motivated chaos theory so uh lorenz in 1963 was studying this kind of rayleigh bernard convection flow trying to model the chaotic
8:43
churning and overturning of these large large eddies in this domain and so i think there's always a very deep connection between turbulent fluids and chaotic dynamical systems and that's kind of where this mixing idea comes from i started with this red blob that was coherent after a very short amount of time in this chaotic system it's going to mix and spread around and that's what happens to particles in turbulent fluids as well okay and i'll
9:13
talk a lot more about chaos in future in future videos so now i want to get a little bit more into the mathematics of kind of where this comes from and how do we analyze these flows and so we're going to start with the incompressible navy or stokes equations
9:30
and i'll point out that this bold u that i'm dealing with here is really a three component velocity field so u v and w are the x the y and the z or z component of the velocity field and that's defined everywhere in space and in time okay so u is a is a vector field and these are the governing navier-stokes equations now from these equations you get a whole range of interesting flow phenomena just by speeding up or slowing down the flow so we have slow flows up here and
10:05
fast flows down here and you get this rich range of dynamics as you increase the flow velocity
10:12
i took this from the the feynman lectures and it turns out that this this range of complexity is controlled by this number here called the reynolds number so 1 over the reynolds number and the reynolds number is essentially a product of the fluid density the velocity of the flow the diameter of the of the sphere in this case or the cylinder divided by this uh dynamic viscosity quantity and so you you multiply and divide all of this up and this one number tells
10:44
you if you're going to have a simple flow or a very very complex flow okay so increasing either the diameter d or the flow velocity will make the flow more turbulent okay and decreasing those makes it more laminar or less turbulent and this turns out to be an extremely useful concept that increasing the reynolds number generally corresponds to an increase in complexity and turbulence and multi-scaleness of the flow and in fact the reynolds number can somehow be
11:14
thought of as the ratio of the biggest scales in the flow to the teeny tiniest scales in the flow so if i have a reynolds number of a million i'm going to expect eddie's uh you know that are a million times bigger than the teeny tiniest eddies or or swirls or vortices in the flow okay that's just a kind of a rule of thumb but a very very useful one okay good
11:41
you can also plot the drag coefficient versus this increasing reynolds number so this is a curve i believe from nasa and you can see that as the flow field changes as you get this kind of increased complexity from the turbulence in this flow field as you increase the reynolds number or the flow velocity you can see that you get a wide wide range of drag forces on that cylinder so you get you know huge range of drag forces just by changing the flow velocity
12:13
now i can't help myself i have to relate this back to the galileo's famous ball drop experiment from the leaning tower of pisa you can now picture dropping this ball from this tower as it accelerates uh from gravity and speeds up it's going to transition through all of these different flow regimes and experience massively different drag forces at every stage of its fall and so technically its acceleration is not going to be you know
12:43
exactly a constant due to gravity because it's going to have this very unsteady uh drag force due to these turbulent forces and it turns out you know this is actually pretty consistent with the older aristotelian view of uh you know a ball's trajectory is going to depend on its mass or its density or its ability to resist these unsteady turbulent forces so just kind of an interesting connection if you think about that ball drop experiment it's actually not so obvious that
13:15
gravity would be a constant if you did it you know naively now if you drop a heavy enough ball like a bowling ball maybe these are kind of negligible forces okay good uh so i want to go back why do we call this the reynolds number um so that number that reynolds number that controlled the turbulence from laminar or smooth solutions laminar kind of means layered and so that means not roiling not unsteady or turbulent and it turns out that reynolds
13:44
osborne reynolds in 1883 was doing experiments in a pipe so pipe flow is one of the most important and oldest engineering applications of fluids is literally just pumping water through pipes makes sense you you know how big of a motor do i need to pump all of that water through all of my city streets okay and to pump all of the sewage out and so pumping water through pipes has always been important and osborne reynolds was studying you know the flow through these pipes
14:14
and made a very interesting observation that as he increased the flow velocity he had these very different regions or or characteristics of flow from laminar flow again layered or very kind of steady flow all the way down to these turbulent flow scenarios where the flow is kind of roiling
14:35
you know what you picture as turbulence now i don't think this picture does does this experiment justice because in this picture he looks very calm and composed but you can imagine if you were trying to do an experiment of pipe flow and you were trying to get a smooth solution to collect whatever numbers you cared about and it was getting turbulent and roiling you might have thought that something was wrong with the experiment that you weren't doing it carefully
15:02
enough or that maybe your flow wasn't well conditioned going in so i think in this picture he really should have been biting his nails and pulling his hair out but you know this is a little bit more composed and actually uh you can go and see this uh original apparatus at the university of manchester so it's still there and it's still kind of on display which i think is kind of cool but so a long time ago we've known that there were these transitions as you increase the flow
15:29
velocity you can trip the flow into these very complex disordered turbulent solutions
15:37
okay good so let's go back to uh kind of this picture here so as we increase the reynolds number we get more and more complex turbulent flows but what i want to point out here is that uh this this flow here in the middle this kind of reynolds number 100 flow is laminar so it's not turbulent it is unsteady but it's very predictable so it does the same thing over and over and over again it's not kind of statistically unreliable like these turbulent flows in the bottom but the
16:09
the middle and the next one are actually very similar okay so this is the laminar flow and this is a turbulent flow but they have a lot of the same patterns and structures and so what i want to point out is that turbulence is not random a lot of people think that chaotic systems or turbulent systems are random and it's true that they have an element of randomness or stochasticity to them but they are not random and so in fact there are what are known as
16:36
coherent structures or patterns that emerge even in these very very complex turbulent flow fields so this is a picture i pulled from a review paper by sam tyra and collaborators in 2017 i absolutely love this figure it shows the flow over richiri island in japan it's literally you know the flow over an island this is an atmospheric flow and then the very very simple laminar low reynolds number flow past a cylinder in the bottom and you can see that even though the
17:08
reynolds number you know this one's probably in the millions or tens or hundreds of millions
17:14
you get the same qualitative patterns emerging in these flows okay and so that's really important is trying to understand and characterize those big patterns in turbulent flows
17:26
now i recently saw this video on on twitter and i absolutely love it so this is a live video of the atmospheric flow over guadalupe island and again clear as day you can see the emergence of this carmen vortex street forming in this flow even though it's you know astronomically high reynolds number extreme complexity but these patterns these coherent structures emerge
17:50
now this was observed early on uh by brown and roshko in 1974 when they were looking at these mixing layers so these are compressible mixing layers in the laminar low reynolds number and turbulent high reynolds number regime and you can see that you know even though the turbulent flow has a lot more complexity and kind of multi-scale behavior there are a lot of similarities between the laminar and the turbulent case given by these coherent structures so brown
18:21
was actually the other person i learned fluids from at princeton gary brown really fantastic lectures uh rosh go this is not the roscoe of um chicken and waffles although this roscoe also was in l.a i love this picture i mean this really this changed the way that everybody thought about fluids when this picture came out it's one of the most famous pictures in all of fluid mechanics because it showed that even though you have you know this incredible complexity it has
18:50
a lot of the same patterns of much much simpler flows and that really got people thinking about these coherent structures how to characterize them how to quantify them how to model them and so decades of work maybe the next you know 50 years up until now a lot of it's been motivated by kind of this seminal picture very cool okay good so coherent structures exist uh we want to model them and characterize them and understand them uh just a couple of other things i want to point out
19:20
there's also something called intermittency i pulled this again directly from lex smith's book and intermittency you can kind of picture this if you have clouds that are kind of developing on the planes as uh you know as the the ground heats up and the sun is up and you have this wind and this boundary layer forming and convecting over earth some areas of it are going to be more turbulent than others and some patches you're going to see are actually pretty still there's going to
19:47
be patches of still and turbulent and still and turbulent and so turbulent flows are often what is called intermittent and so you can define this intermittency function which essentially turns on and turns off and turns on and turns off and it tells you when it's turbulent or not turbulent or turbulent or not turbulent either in space or in time that often happens you can kind of picture
20:12
if you have a stick of incense or a cigarette and the smoke is rising you know sometimes it's going to be pretty laminar sometimes it's going to be very turbulent it might actually kind of settle down again and then get more turbulent again and this kind of intermittency is a hallmark feature
20:28
okay so turbulent flows are everywhere they're extremely complex they're very important for us as engineers and scientists and they have all of these different kind of complexities and features
20:44
so how do we get a handle on this complexity how do we start to model this now what we have done for you know the last hundred years since reynolds and probably before is we've tried to abstract the turbulent fluid dynamics problem into a few canonical flows simple example flows that isolate a couple of features you know one or two features that are that add to the complexity of turbulence so we can really drill down and start understanding those constituent pieces now
21:16
real engineering flows like you know a boeing jet flying through turbulence is going to have a lot of these kind of aspects you know there's going to be a jet there's going to be a mixing layer and a boundary layer and isotropic turbulence there's gonna be all of these different pieces working together but before we tackle those huge problems we kind of break it down and distill it into you
21:39
know isolated kind of idealized problems there is a gorgeous book by milton van dyke called an album of fluid motion where he walks through you know beautiful flow visualizations of some of these canonical flows i think you can find pdfs online it is really really cool you should check this out but here is just some of my favorite canonical flows um you know wall-bounded flows like the boundary layer mixing flows so um you know what does the flow behind a wing look like uh pipe
22:10
flows again pumping you know water or oil through pipes very economically important and so on and so forth and so here i'm just going to go through a couple of these and i'm going to have a whole lecture the next lecture is going to be all about zooming into these canonical flows why they're interesting what are some of the questions we want to ask and answer okay but this is how we've approached fluids uh you know essentially for the last century and this is how we approach most
22:37
problems in mathematical engineering where it's too complex is we try to abstract and boil down to the simplest piece that kind of exemplifies our problem now calling these toy problems is a little bit uh maybe a little too unfair these are still quite complex and challenging themselves
22:58
so just some example wake flows tons and tons of effort has gone into studying wake flows um if you want to reduce the drag behind a train or a plane or a truck or a car you need to understand the wake behind behind that body also hugely important for missile design bullets and ballistics baseballs you know lots and lots of flows create wakes and those wakes contribute to the drag and unstudy forces on those bodies another really important example are mixing layers so again
23:32
that brown and rosco example i showed is a mixing layer many of you have seen these wave makers where you have kind of oil and water in a rectangle rotating and it creates these waves and these mixing layers are really important canonical flows to understand things like ocean waves the flow behind a sharp trailing edge on a wing often creates a mixing layer and these are very very important another very important application of fluid dynamics and
24:02
another canonical flow is a jet and so there are supersonic jets you can actually see the acoustic radiation in this turbulent supersonic jet uh here you can actually see kind of the mock lines forming in this jet and jets are extremely important for you know producing thrust and power also we want to minimize the noise given high thrust and high power so you know any of you who live near lax or heathrow no i mean there's there's 10 million people
24:35
in the top two or three airports alone near the top two or three airports in the world alone that are directly impacted by jet noise and decreasing that jet noise would improve the quality of life for tens of millions of people directly okay so huge huge problem here okay so canonical flows are very important it's one of the ways that we study
24:58
study fluid dynamics we try to break it down so uh here again are just some of the canonical flows wakes and jets and mixing layers and we use those canonical flows to study much more complex engineering systems like trains and boats and planes and engines and things like that okay so many of these flows are going to have lots of elements of these canonical flows and i'll talk about that later so for example you know a wind turbine blade there's going to be boundary
25:26
layers and mixing layers and wake flows all of those interacting in a pretty complex system and at the end of the day we want to understand these systems we want to be able to model them and understand what they will do when we change different things about our engineering designs that gives us the ability to optimize for example design better blades for turbines but also to control the system to actively manipulate the behavior of the fluid for better engineering
25:55
outcomes to increase lift decrease drag enhance mixing things like that okay and this is something i'm very passionate about and one of the reasons i love turbulence as you know an important field of fluid mechanics okay good um so i think one of the last things i want to point out here is again uh we go back to this we're always going to go back to the the navy or stokes equations that's kind
26:20
of our entry point as a mathematical engineers and we know that we have this kind of increasing complexity with the reynolds number so one of the the kind of important things that you can do
26:34
to study these flows and i'm going to have a whole series of lectures on this this is going to be the basis of what's known as turbulence closure modeling which is one of our approaches for modeling these systems more efficiently on computers one of the things you do if we zoom into a turbulent case is something called the reynolds decomposition so we're going to take this flow field u which is you know very complicated function of space and time
27:00
and what we're going to do is we're going to break it apart into an average field u bar and all of the fluctuating components u prime and so u bar we get by literally taking this flow u and just averaging in time we just take flow snapshot think of a movie of this flow evolving i'm just going to average all of those frames of the movie in time and get the average frame that's the mean flow
27:24
and then the turbulent fluctuations are just my original flow minus the mean flow okay and so what we're doing is we're taking our flow and we're decomposing it into a function that is the average flow which is no longer a function of time it's just a function of space and then all these little fluctuations all of the turbulent fluctuations and when i take this flow this decomposition and plug it into the navier-stokes equations i get lots of interesting things happening i get
27:53
an equation i a navier-stokes equation for the mean flow and for the fluctuating components and i can model things more efficiently this way okay so huge field of reynolds averaging but this is a very useful perspective where we decompose our flow into the average flow and the fluctuations
28:11
and i'll point out this is a largely historical why we do this and it's kind of fortuitous is that historically we didn't have computers to measure these flow fields accurately in space and time we didn't have laser technology until the 60s 50s and 60s so that we could build laser sheets to image the full flow field and experiments and so back in the olden days you were stuck with single point
28:38
measurements hot wires where you would put a hot wire in the flow and i could maybe measure that for a long time and figure out what its statistics were and average and build the fluctuations and then i could move it one delta x or delta y down and then i could move it again and move it again and in the course of a day or a few days i might be able to measure the entire wake point by point
29:01
in a time series fashion but i can't tell what every point is doing at the same time and so this idea of averaging your flow is actually you know goes back to the days when we didn't have enhanced measurement techniques and computers but it turns out that it's a very very useful perspective for modeling today so that's pretty cool okay good so i'm going to have a whole lecture on reynolds
29:25
averaging soon this is going to be kind of a pen and paper or like lightboard video lecture and i hope that i have convinced you in this lecture that turbulence is interesting it's everywhere it's extremely important and it has these kind of four canonical attributes that's how these these are like the fingerprint of turbulence when you see three-dimensional multi-scale unsteady mixing flows you know it's turbulence okay stay tuned there's going to be a lot more to come thank you
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