Today we're doing Reynolds number.
Famous number, named after Reynolds, talks about fluid mechanics, to do with Navier-Stokes equations. - (Brady: But it's not one number, it's not like pi?)
It's not, no no no. This is, this is like the superhero
version of pi. Because pi is fixed. Pi has like one superpower, like circles and things. This is Reynolds number,
this guy can morph and change to be whatever value it wants to be. But,
depending on what sort of size it is, whether it's really big or really small, you get different behaviour with your fluid.
So it's more like a label that you give to a particular situation.
You have a fluid moving in some set up and that's like, high Reynolds number,
it does one thing. But if you have the same fluid
moving in a slightly different setup where it's now got a low Reynolds number it does something completely different.
(Brady: Like choosing your superhero from the Avengers) - If it needs to be invisible,
it can be invisible. If it needs to reverse time, it can reverse time. It can do whatever it wants. Starting point Navier-Stokes equations.
So these guys model the
motion, the flow of every fluid on earth, in the universe in fact. So we have these equations,
they come from sort of universal laws of physics. We have mass is conserved,
we have Newton's second law - we get our equations. So we'll start by writing those guys down.
We talked about these guys before and we did the full sort of what everything means but we got the top one, the little
guy, which is conservation of mass. And then we have Newton's second law.
I've made another slight simplification from the standard Navier-Stokes and I've said there are no body forces. Normally
there's a big plus F here, we're going to pretend there's not - that's the simple case. So here's our Navier-Stokes equations.
Now we want to get the Reynolds number. So we need to define what the Reynolds number is. We label it with our e,
and it's going to be our density rho, times a length scale L times a velocity u,
divided by
viscosity mu. - (Brady: Tom why do we even need a Reynolds number?)
(These Navier-Stokes equations)
(which you've explained to us before, it's - serviceable things for figuring out how fluids work, what was missing in the first place?)
Here; so equation one is nice and simple,
so we actually, that guy doesn't change with the Reynolds number discussion. But equation two, we've got like three bits.
So we've got - use a different colour. So we've got this guy is
acceleration, because it's a density times a change in velocity.
This guy is a pressure. And then this guy is viscosity.
So viscosity is like an internal friction, your layers of fluid sliding over one another and interacting.
So we've got three bits and the point is, in different
situations,
some of these bits will be more important than others. Practical example is, if
viscosity was important, when you go fast,
you could clean your car by driving really quickly. - (Brady: What the air would just blast off?) - Exactly.
So if you have loads of little bits of mud all over your car and
you just grow really fast
the air would actually wash it off, because the friction in the air would grab onto the dirt particles, clean your car for you.
So viscosity doesn't matter if you're going fast, it turns out, as we're going to see because of the Reynolds number. In some situations
they're all equally important,
but in certain situations, like the extreme cases,
you can get that one of them doesn't actually matter. Or one of them is so important that it dominates the other two. And
then you've got a nice simpler equation, which we can then solve. Because as we know, we can't always solve Navier-Stokes.
So if we can get our hands on this Reynolds number, allows us to simplify them and actually solve them and get somewhere.
(Brady: So the Reynolds number is what apportions different weightings to the different factors?) - Exactly, exactly.
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We can either have a really big Reynolds number or a really small Reynolds number. And the sort of, the standard thing
we do in maths is consider if it's really big or it's really small. - (Brady: And just so I've got some idea of)
(what numbers you're talking about, are we talking like billions and negative numbers like - tell us where they - )
So the Reynolds number has to be positive, because it's it's a density times a length times a velocity
over a viscosity. These are all positive, physical things you can measure. And you plug those values in you'll get a number,
So it can be as small as, sort of, you can go right down towards zero so, you know can't be zero
but can be just above zero. Or it can go as big as you want.
But generally we're talking sort of 10 to 1000,
most everyday situations. If it's over a thousand its turbulent.
So turbulence is this chaotic, crazy motion of fluids crashing into each other.
So we have two cases of the Reynolds number. We've got big Reynolds number, so we say it's not just greater than one
it's much greater than one, so sort of 1000 and above turbulence. And your equations,
so we've got rid now of
all of the sort of, these guys like the mu and the rho they've sort of - they're in their Reynolds number now,
so they're gone.
And so what that means is you could start to compare your terms, and you end up with
this guy: 1 over Reynolds...
that's squared.
So if we compare this one to our original equation, we have the same terms. We've still got acceleration,
we've still got pressure and we've still got viscosity. But now
we've sort of got rid of all the other little bits that were floating around and now we can compare everything. Because by introducing the
Reynolds number what we've done is we've non-dimensionalised the system, which means get rid of the units. If I say
I have 10 apples and I have 9 bananas. Then if we consider apples,
I have more fruit with my 10 apples than I do with my 9 bananas. But with my 9 bananas
I have more potassium than I do with my apples - the units matter. It depends, you can't say I have more apples than bananas.
It depends whether I'm talking about fruit or potassium. The units are important.
So now by introducing the Reynolds number what we've done is a non-dimensionalisation and we've got rid of units. So now we can compare
acceleration, pressure and viscosity. Three very different things with different units,
but now they have no units. If the number associated with the acceleration term is bigger than the number associated with pressure,
that means acceleration is bigger than pressure. It makes sense to compare them because they have no units anymore.
This is, this is the power of the Reynolds number really. - (Brady: Does the Reynolds number itself have units then?)
The Reynolds number does not have units. The Reynolds number is just a number and we can actually see that. So the Reynolds number itself is
density times length times speed,
divided by viscosity.
So let's think about the units. - (Brady: That's a real hodgepodge of units)
It is. So density, that's going to be kilograms per metre cubed, 'cause we're in 3D.
So we use the square bracket notation to denote units.
For our length, well, that's just in metres.
Our velocity, u, that's in metres per second and then our viscosity,
so this one is possibly less well-known, but this guy is kilograms per metre per second.
And now if we plug them all together in the Reynolds number formula we get the Re, the units,
so on the top, we've got a density so it's a kg m minus 3.
Then we've got a length, that's an m. Then we've got a speed (ms) minus one. And then we divide through by
viscosity. And now hopefully we've done this right. Kilograms cancel,
Metres, metres, metres minus 3 gives us metres minus one, cancels. Seconds minus 1, seconds minus one.
So the Reynolds number has no units. - (Brady: Brilliant!) - So it is just a number. That's why it's called Reynolds number. It does,
it's just a number. It's why we can think about what happens when it's big and get this equation.
So now the Reynolds number is really big, we've got acceleration equals pressure; plus 1 over a really big number
times viscosity.
So 1 divided by a really big number is really small.
So what's that telling us, is that that guy doesn't matter.
So in the situation of really large Reynolds numbers,
which could be very, very light - very, very high density.
Very, very long fluids or very, very fast fluids. So make any - just make this thing big, however you have to.
Viscosity doesn't matter. So you're driving your car, you're going really really fast,
that's why the viscosity of the air doesn't clean the dirt from your car. Because you're in this large Reynolds number and your equations say
viscosity doesn't matter. So it doesn't, it doesn't work.
So what if the Reynolds number is now really really small?
We say it's much less than 1. It's not just smaller than 1, it's much less than 1. And so what happens now
with our equations, is when we substitute this in we do our non-dimensionalisation in the same way as before, but now
we've got Reynolds number times acceleration
equals the pressure term plus the viscosity term.
So now it's slightly different to the one we had.
So we've got Reynolds number, just a number, times acceleration is pressure plus
viscosity. You might be wondering, why is the Reynolds number now here?
Why is it on acceleration when before we had it over on viscosity?
Now we have a small Reynolds number. And so in a small Reynolds number flow
we sort of know, we've done experiments, to say that the pressure term is actually balanced by
viscosity.
Whereas in the big Reynolds number case
the pressure term is balanced by the acceleration. So that there is some slight difference in how you
non-dimensionalise, but it still works in the same way, it still has units of pressure
you just have this choice between how to balance your pressure.
So when the Reynolds number is small we have this guy,
so it's Reynolds times acceleration is pressure plus viscosity, but the Reynolds number was really, really small.
So now I've got a tiny number times acceleration,
so acceleration doesn't matter,
because Reynolds number's really small. So that now goes. So this is the term that doesn't matter in this small Reynolds number regime.
So now just we've got pressure and viscosity, and that's it.
Our governing equation is just pressure gradient plus viscosity. My favourite thing,
I'm gonna say in all of fluids, is getting this result. Because
we no longer have time in our equations.
So we've started with Navier-Stokes which we know are true,
mass is conserved, Newton's second law,
that's a given. Then we've looked at small Reynolds number and it turns out that in that situation
time vanishes from our equations. - (Brady: Because the only thing bringing time to the table was acceleration?)
Exactly. The only possible T in our equations was in the acceleration term. So now that's gone, it doesn't matter.
I mean it's still there, but it doesn't matter.
Everything else is bigger than it so they dominate. So what that tells us is that when you have a small Reynolds number
time doesn't matter.
So if you go forwards in time, and then you do the exact same thing backwards in time,
and it has to be the same thing because the pressure and the velocity have to still be the same.
But if you go forwards and then you go backwards,
you should get back to where you started.
You can literally time travel. You can time travel in a fluid
because your Reynolds number is small and because time has vanished from your equation.
So how do we get small Reynolds number? We've got density times length times velocity divided by viscosity,
let's make viscosity big.
If we divide by a really big number, Reynolds number become small. A large viscosity means they're very thick, sticky fluid.
Let's take corn syrup - sugar syrup, your favourite type of syrup. It's very thick, it's very sticky - very large viscosity.
So now we have a small Reynolds number so let's see if we can do an experiment
and get the same thing as we go forwards in time and then backwards in time, because the equations say
it should work. Here is a cylindrical container filled with sugar syrup,
we're going to add three drops of food colouring and
then if we slowly turn this stirrer to mix the fluids together,
we go forwards in time, so we'll do several rotations.
And then stop. And as you can see the blobs of food dye have mixed together - as you might expect.
Now, if we do the exact same thing in reverse,
so the same speed, the same pressure - everything the same, but we reverse the direction of our stirrer,
we're going backwards in time.
And as you can see, it's magic.
Voila, we have gone back to the very beginning,
we have gone - mixed them together forwards in time and then gone backwards in the exact same way and returned to our original state. And
this
works because time is not in our governing equations. So without Navier-Stokes, without Reynolds number, without this knowledge,
we wouldn't know to even try this experiment. You wouldn't know to pick a really sticky fluid and see this awesome thing happening
but it is explained by the fact that time disappears from our equations and that came from
Reynolds number. - (Brady: What's the bonus or the cool thing about a really big Reynolds number?)
- Big Reynolds numbers? So big Reynolds numbers just means super
turbulent flow.
So, so large Reynolds number is any, any kind of aerodynamics - flow around a Formula One car, flow around a jet;
that's large Reynolds number flow. And the issue with large Reynolds number flow is,
well, turbulence. And
turbulence, turbulence we don't understand. Turbulence is one of the reasons we don't understand Navier-Stokes and one of the,
possibly one of the ways of getting that million dollar millennium problem prize is by trying to actually come up with a theory
for turbulence. So large Reynolds number flows,
turbulence; very, very difficult.
But fortunately, small Reynolds number flows
make a bit more sense. And the sort of, the intricate maths to it, or the key bit of the maths, is that
when you have the small Reynolds number, as well as losing time,
we lose time because we lose the acceleration term. But the acceleration term is
nonlinear. And so what that means is that's the difficult bit of Navier-Stokes. The reason Navier-Stokes are
such a hard set of equations to solve is because they're nonlinear.
So that means if you have one solution, solution A, and you have a second solution B,
for a linear equation A plus B will also satisfy that equation.
But Navier-Stokes is nonlinear, so when you do A plus B, you don't get a solution anymore.
And that's - because that's how we generally solve difficult
partial differential equations, difficult differential equations in general,
we find one solution; find one that's similar to it and keep adding them all together and you get all of the solutions.
But we can't do it because Navier-Stokes is nonlinear.
But it's only nonlinear for big Reynolds number. If we take small Reynolds number we've lost acceleration,
we've lost time, we've lost non-linearity and then we can actually solve it and get interesting solutions.
(Brady: Unfortunately, that's not a very practical thing, that doesn't happen very often in the real world.) - It doesn't, no, so
Reynolds numbers
generally in the real world, like most things, are turbulent.
If you just think the wind, the water in a river, in an ocean.
It's - most of our interactions with fluids are turbulent or are at least a hundred plus on the Reynolds number scale. This kind of thing,
getting rid of the nonlinear bit, getting rid of time, like, you need less much less than 1. So you don't really see that in
most of our world. You would on a tiny little, like, bacteria scale. If you think about you know,
what's going on on the really small cell, like individual cell scale,
you might start to get small enough length scales in your Reynolds number that it'll become small, but in our world,
unfortunately, we live in a turbulent world
and we don't understand turbulence.
(Brady: We live in a turbulent world, ain't that the truth?)
[Video Preview] Essentially means you need a bunch of positive charges here, and a bunch of negative charges here, and then the things sort of fairly catastrophically
re-equalise and the charges all balanced out by a
spark of electricity, essentially a current flying from one to the other to make the thing all equalise out.