Latest / Mechanical Engineering Made Simple / (#116) Introduction to Fluid Mechanics - Lesson 1

Transcript

(#116) Introduction to Fluid Mechanics - Lesson 1 · Mechanical Engineering Made Simple

0:00
Welcome back design junkies, to Mechanical Engineering Made
0:02
Simple. If you enjoy the podcast, show
0:04
your support with a like, a follow or a comment.
0:07
So I want you to imagine being a soldier on a 19th century
0:12
battlefield. OK, setting the seam.
0:14
I like it. Right, so you're there and
0:16
suddenly you hear this deafening explosive crack of a bullet
0:20
striking the earth literally right next to you.
0:23
Terrifying. Yeah, completely.
0:25
But here's the crazy part. A moment later, you hear the
0:29
distant sound of the actual rifle that fired it.
0:32
Yes, the bullet physically arrived before the sound of the
0:35
gunshot. Exactly.
0:37
It's wild to think about or picture this.
0:39
You're trying to design A modern commercial airliner, right?
0:42
Like a 400 ton leviathan of the skies.
0:44
A massive engineering undertaking.
0:46
Huge. And you realize that the only
0:48
way to calculate how it'll actually fly is to use imaginary
0:51
numbers. I mean that literally do not
0:53
exist on any physical number line.
0:55
It sounds like science fiction, honestly, or perhaps magic, but
0:59
it is the absolute foundation of the modern world.
1:01
It really is, and today we are embarking on what I'm calling a
1:04
master class. I think that's the perfect word
1:07
for it. Yeah, because if you have ever
1:09
looked out the window of a commercial flight and wondered,
1:11
you know, how the air is actually holding you up.
1:13
Right. Or if you've ever just watched
1:14
water flow around a rock in a stream and wondered if that
1:18
chaotic movement could be perfectly predicted.
1:22
Exactly. This conversation is going to
1:24
fundamentally change the way you see the invisible forces around
1:28
you. We are going to completely
1:30
decode fluid dynamics today. It's a massive topic, but so
1:35
incredibly fascinating. It is, and we're breaking this
1:38
deep dive down into 3 distinct phases.
1:41
First, we're going to explore how engineers scale reality.
1:45
Which is way harder than it sounds.
1:46
Oh, absolutely. Because if you want to build a
1:48
brand new jetliner or a Revolution car design, you can't
1:52
just build a full size prototype out of the gate.
1:55
We have to look at how we shrink the physical world without, you
1:58
know, breaking the laws of physics.
1:59
Right. And then second, we're going to
2:02
look at the ingenious, almost mind bending tools engineers use
2:06
to measure the invisible. Like how we actually track the
2:09
movement of air and water when you can't even see it.
2:12
Exactly. And finally, we will dive into
2:15
the pure, elegant mathematics of flight.
2:19
We'll explore how theoretical impossible fluids and abstract
2:22
geometry miraculously dictate the reality of lift.
2:26
It is a phenomenal progression. I mean, we are going to travel
2:29
from incredibly practical, gritty engineering challenges.
2:33
Including a genuine medical mystery from the Franco Prussian
2:36
War, which is a great story. Yes, I can't wait to get to
2:39
that, all the way to the breathtakingly abstract
2:42
mathematics that literally govern our physical universe.
2:44
It's. Going to be quite the journey.
2:46
OK, so let's unpack this phase, one-dimensional analysis and the
2:50
law of similarity, or as I think of it, the incredible headache
2:54
of scaling reality. That is a very accurate title
2:57
for it. Right, so let's ground this
2:58
right away. Suppose you were designing a
3:00
brand new, highly aerodynamic electric vehicle.
3:03
OK, an EV. Got it.
3:04
And you want to test the aerodynamic drag, but building a
3:08
full size functioning prototype is prohibitively expensive.
3:11
So naturally you decide to build a one to 10 scale model of the
3:15
car and put it in a small wind tunnel.
3:18
Which is the logical, intuitive approach.
3:20
Everybody does that, right? They try to, but fluid dynamics
3:23
is rarely intuitive. Yeah.
3:25
I mean, the instinctive thought is that you can just take the
3:27
physical geometry of the car, shrink every single dimension by
3:31
a factor of 10, blow wind over it, and then just scale your
3:34
results back up. But the universe absolutely
3:37
refuses to let you do that it. Flat out says no.
3:39
I always compare this dilemma to baking.
3:42
If you are listening to this right now and you've ever baked,
3:44
you know this exact. Pain.
3:46
Oh, the baking analogy is perfect for this.
3:48
Right, let's say you have a family recipe for a standard
3:50
sized cupcake and it is just flawless delicious, and you
3:54
decide you want to bake a giant wedding cake size version of
3:58
this exact cupcake. You can't just multiply all the
4:01
ingredients by 10, pour it into a massive pan, and bake it at
4:05
the exact same temperature for 10 times as long.
4:08
You would end up with a complete.
4:10
Exactly. You'd end up with a cake that is
4:11
entirely incinerated on the outside and like a raw soupy
4:15
liquid on the inside. Because the physics of heat
4:17
transfer scale differently based on the ratio of volume to
4:21
surface area. Yeah, thermal mass completely
4:24
changes, and the same exact logic applies to fluids like air
4:29
and water. Just shrinking the physical
4:31
shape of the car isn't enough. No, it's not.
4:34
We need to achieve what engineers call dynamical
4:36
similarity. Dynamical similarity.
4:39
OK, break that down for us. Well, the baking analogy is
4:41
brilliant because it highlights that geometry is only one piece
4:45
of the puzzle. To achieve dynamical similarity,
4:48
meaning the fluid behaves exactly the same way around the
4:51
small model as it does around the prototype, engineers have to
4:55
use a mathematical framework called dimensional analysis.
4:59
And dimensional analysis has some pretty strict rules, right?
5:01
Very strict. The absolute bedrock rule of
5:04
dimensional analysis is that any equation that describes a
5:07
physical reality must be dimensionally consistent.
5:10
Meaning you cannot add mass to time.
5:13
Precisely, you cannot add 2 kilograms to five seconds.
5:15
It is a meaningless sentence in the language of physics.
5:18
It's like adding apples in Tuesday.
5:20
It just doesn't compute. Exactly.
5:21
You can multiply them, sure, or you can divide them, but you
5:24
absolutely cannot add or subtract them.
5:27
Every term in a valid physics equation must ultimately boil
5:31
down to the exact same fundamental dimensions.
5:33
OK, so how does that help us scale our car?
5:37
By utilizing this strict rule, engineers can take incredibly
5:41
complex, chaotic, messy problems and reduce them into much
5:46
simpler, non dimensional groups. And the key to this is a
5:49
mathematical magic trick, right? Buckingham's Π theorem.
5:53
Yes, buckingham's Π theorem. I want to spend some time on
5:56
this because when you first encounter buckingham's Π
5:59
theorem, it literally feels like a cheat code for the universe.
6:02
How does it actually work? It is incredibly elegant.
6:05
Basically, the theorem states that if a physical phenomenon is
6:08
influenced by a certain number of physical variables, let's
6:11
call that total number N. OK, so N is the number of
6:14
variables. Right.
6:14
And those variables can be described using a certain number
6:17
of basic fundamental dimensions like length, mass and time.
6:20
We will call that number pay. So K is the basic dimensions,
6:23
length, mass, time. Exactly.
6:26
Then the entire complex phenomenon could be
6:28
mathematically expressed by exactly n -, K non dimensional
6:32
groups. OK, we need to map this onto
6:34
something real so people can visualize it.
6:36
Let's take the classic example of calculating the aerodynamic
6:40
drag on a sphere moving through a uniform flow of fluid.
6:44
Like a baseball flying through the.
6:46
Air. Yeah, baseball or a submarine
6:48
dropping deep into the ocean. Perfect.
6:50
Let's list every single physical variable that dictates how much
6:53
drag force that sphere experiences.
6:55
OK, let's do it. We will call the drag force
6:57
capital D That's our first variable.
7:00
Now what else matters? Well, the physical size of the
7:02
sphere definitely matters, so we have its diameter.
7:05
That's two. The speed at which it is moving
7:07
matters, so we have flow velocity.
7:09
That's three. The heaviness of the fluid
7:11
matters too, so we have fluid density 4.
7:13
And finally, the stickiness or internal friction of the fluid
7:16
matters, which is the fluid viscosity that makes 5.
7:20
So counting them up, drag force, diameter, velocity, density and
7:24
viscosity. That is, 5 total variables
7:26
governing this scenario are n = 5.
7:29
Exactly. Now, how many basic fundamental
7:32
dimensions do we need to describe those five variables?
7:35
Let's see. We need length to describe the
7:37
diameter, obviously. We need mass to help describe
7:40
the density. And we need time to help
7:42
describe velocity, because velocity is just length divided
7:45
by time. Spot on.
7:46
So our basic dimensions are length, mass and time.
7:50
Our K = 3. And here is where the Buckingham
7:52
Π theorem steps in. We take our five variables, we
7:56
subtract our three basic dimensions, and the theorem
7:59
guarantees that we are left with exactly 2 non dimensional groups
8:03
5 - 3 = 2. Yes, through a specific
8:06
algebraic process, engineers multiply and divide these five
8:10
variables against each other until all the units of mass,
8:13
length, and time completely cancel out.
8:15
It's just so clean. And for the sphere, what are
8:18
those two resulting groups? They are the drag coefficient
8:20
and the Reynolds number. Wow.
8:22
OK, think about how powerful this is instead of an engineer
8:25
trying to plot a wildly complex graph with five different axis
8:29
to track all those changing variables.
8:31
Which will be impossible to read by the way.
8:33
Right. They only need a single
8:34
two-dimensional graph comparing the drag coefficient to the
8:38
Reynolds number. It simplifies the chaos of
8:41
reality into a clean, predictable line.
8:43
That is just phenomenal. It takes all the noise out of
8:46
the system. Let's look at another common
8:48
engineering problem. Just to cement this, what about
8:50
the pressure you lose when you pump a fluid through a long
8:53
pipe? Oh, pipe friction is a great
8:55
example. If you are designing an oil
8:58
pipeline or a city water grid, you desperately need to know how
9:02
much pressure you are going to lose due to friction.
9:05
Because otherwise you can't size your pumps correctly, right?
9:08
Exactly, and for pressure loss in a pipe you have 6 variables
9:12
to juggle. OK, leave them out for me.
9:13
You have the fluid velocity, the pipe diameter, the fluid
9:16
density, the fluid viscosity, the physical roughness of the
9:19
inside wall of the pipe, and the actual pressure loss per unit of
9:22
length. So our N is 6.
9:24
Got it. Then we subtract our three basic
9:27
dimensions of mass, length, and time.
9:30
So 6 -, 3 leaves us with three nondimensional groups.
9:33
And those three groups turn out to link the friction factor of
9:37
the pipe to the Reynolds number and the relative roughness of
9:40
the pipe wall. That's amazing.
9:43
We have once again taken 6 messy interacting real winnowed
9:47
variables and boiled them down to three pure dimensionless
9:52
relationships. It's why dimensional analysis is
9:54
essentially the secret weapon of mechanical engineering.
9:57
OK, so this brings us right back to our electric vehicle scale
10:00
model. We have our clean, dimensionless
10:03
relationships, but how do we actually guarantee dynamical
10:06
similarity between the tiny one to 10 scale model sitting in the
10:10
wind tunnel and the real full size car cruising down the
10:14
highway? The actual physical test to
10:16
achieve true dynamical similarity you have to match the
10:19
force ratios. Match the force ratio.
10:21
Yes, the ratios of the various physical forces acting on the
10:24
tiny model must be mathematically identical to the
10:27
ratios of the forces acting on the giant prototype.
10:29
OK, and the most famous, the absolute king of these force
10:33
ratios is the Reynolds number, which we just discovered using
10:36
buckingham's Π theorem. It is the absolute king.
10:39
You cannot do fluid dynamics without the Reynolds number.
10:41
Let's define the Reynolds number, because like you said,
10:44
it is impossible to talk about fluids without.
10:46
Simply put, the Reynolds number is the ratio of inertia force to
10:51
viscous force. Inertia being the tendency of
10:53
the fluid to just keep barreling forward in a straight line.
10:57
Exactly, and viscosity is the fluids internal friction, its
11:01
thickness, its tendency to stick to itself and the object moving
11:05
through it. Like honey versus water.
11:07
Exactly. The Reynolds number tells you
11:09
which of those two forces is winning the battle.
11:11
Is the fluid rushing past smoothly because of high inertia
11:15
or is it dragging and sticking because of high viscosity?
11:18
And it's crucial for specific types of flows, right?
11:22
It is the crucial governing metric for anything that is
11:24
wholly submerged in a fluid without a free surface.
11:28
So a submarine navigating deep underwater, Yes.
11:32
Or an airship flying through the sky.
11:33
Or water flowing through a closed municipal pipe system.
11:37
But here is where my brain always hits a roadblock, and I
11:39
think it's the great headache of model testing.
11:42
Let's just follow the math for a second.
11:43
OK, let's do it. The formula for the Reynolds
11:45
number is fluid density times velocity times the
11:49
characteristic lengths like the physical length of the car, all
11:52
divided by the fluids viscosity. Oh, it's exactly correct.
11:55
OK, so if we are testing our car model in the exact same air that
11:59
the real car drives in, the density and the viscosity are
12:03
exactly the same. Right, it's just normal
12:05
atmospheric air. But our models length is 10
12:08
times smaller, so to keep the final Reynolds number
12:11
mathematically equal between the tiny model and the real car,
12:15
doesn't the wind speed and the tunnel need to be insanely and
12:19
possibly fast? It absolutely does.
12:21
This is the brutal, harsh reality of scaling.
12:24
I knew it. The math demands it.
12:26
Let's put real numbers to it so you can see how crazy it gets.
12:29
Let's say your full size prototype car is 3 meters long
12:32
and you want to test how it performs at a standard highway
12:34
cruising speed of 60 kilometers per hour.
12:37
OK, pretty normal scenario. To match the Reynolds number on
12:41
a one to 10 scale model, which is only 30 centimeters long, the
12:44
wind velocity inside your testing tunnel has to be blasted
12:47
at 167 meters per second. Wait 167 meters per second.
12:53
That is over 370 mph. Exactly.
12:57
It's a localized hurricane. But that introduces a completely
13:01
fatal flaw to the experiment, doesn't it?
13:03
It does, because 167 meters per second gives you a Mach number
13:08
of 0.49. You are traveling at nearly half
13:13
the speed of sound. Wow.
13:14
And at that velocity, the air fundamentally changes its
13:17
behavior. It begins to suffer from
13:19
compressibility. OK, let's explain
13:21
comressibility. Because air is a gas, it acts
13:25
kind of like a spring, right? Exactly like a spring.
13:27
At normal everyday highway speeds, the air simply parts and
13:31
flows around the car. It acts as if it is
13:33
incompressible. Right, it just moves out of the
13:35
way. But when you hit those massive
13:37
wind tunnel speeds, the air literally smashes into the front
13:40
of your tiny model car and compresses, packing tightly
13:43
together and physically changing its density.
13:46
Exactly. And the real car driving at 60
13:48
kilometers per hour does not experience that compressibility
13:51
at all. So the test is completely
13:53
invalid. Completely.
13:55
The general rule of thumb in aerodynamics is that the maximum
13:58
tolerable threshold for treating air as incompressible is around
14:02
Mach 0.3. Anything faster than that and
14:05
the physics of your wind tunnel no longer match the physics of
14:08
the highway. Your test is ruined.
14:10
Yep. You are testing a phenomenon
14:12
that the real car will never actually experience.
14:15
So you are trapped in a physics paradox.
14:18
You have to spin the wind tunnel up to 370 mph to match the
14:22
Reynolds number, but the moment you do that you ruin the air
14:25
density through compressibility. How on earth does an engineer
14:28
solve this well? You have two incredibly clever
14:32
workarounds. The first option is to just
14:34
completely abandoned Air as you're casting medium.
14:37
Wait, really? What do you use instead?
14:38
Instead of putting the car model in a wind tunnel, you submerge
14:41
it in water. You put the aerodynamic car in a
14:45
circulating water flume or a massive towing tank.
14:47
Yes, because water has a vastly different density and a vastly
14:51
different viscosity than air. The math completely shifts in
14:55
your favor. That is so smart.
14:56
If you run the Reynolds number equation using the properties of
14:59
water instead of air, you find that to match the full scale
15:03
car, the water velocity over the scale model only needs to be
15:07
11.1 meters per second. Only 11.1?
15:11
That is roughly 25 mph. That is incredibly manageable.
15:16
Very manageable and water is essentially incompressible under
15:19
these conditions, so you completely bypass the Mach
15:21
number problem. That is a brilliant hack.
15:24
But let's say you are an aeronautical engineer and you
15:27
really, truly want to use air. You have to use a wind tunnel
15:30
for some reason. What's the second option?
15:32
Your second option is to build a variable density wind tunnel.
15:35
OK, what does that look like? You take the entire wind tunnel
15:38
facility and you encase it inside a massive, heavily
15:42
reinforced steel shell, and then you crank up the atmosphere
15:46
pressure inside that shell. You literally pressurize the
15:49
room. Severely, Yeah.
15:51
If you increase the pressure inside that wind tone to 10
15:53
atmospheres, meaning 10 times the normal pressure of the
15:56
Earth's atmosphere at sea level, the density of the air increases
15:59
proportionally. Oh, I see the math now.
16:02
Because the air is now 10 times denser, you can equalize the
16:05
Reynolds number on your tiny model without increasing the
16:08
wind speed at all. You get to keep the wind speed
16:10
slow, avoiding the compressibility problem, but the
16:14
sheer crushing density of the air gives you the dynamic
16:17
similarity you need. Exactly.
16:18
It's an engineering marvel. But imagine standing next to a
16:21
wind tunnel that is pressurized to 10 atmospheres.
16:25
The structural engineering required just to keep the
16:27
building from loading outward is staggering.
16:30
It is a massive, expensive undertaking, but necessary for
16:33
accurate scaling. OK.
16:34
So the Reynolds number dictates the balance of inertia versus
16:37
viscosity, but there are other critical force ratios depending
16:41
on what you were trying to build, right?
16:43
Absolutely. If your domain is naval
16:45
architecture, so designing ships, boats, or harbor
16:47
structures, your primary concern actually shifts away from the
16:51
Reynolds number and toward the Froude number.
16:53
The Froude number and what forces is that comparing?
16:56
The Froude number is the ratio of inertia force to gravity
16:59
force. Why does gravity suddenly take
17:01
center stage when you are designing a boat?
17:04
I mean, submarines use Reynolds, why do surface ships use fruit?
17:08
Because a surface ship operates exactly at the interface of two
17:11
vastly different fluids, water and air.
17:15
Ah, the surface. Right as the ship's hull pushes
17:18
through the water, it physically displaces it upward into the
17:21
air, creating a surface wave. And gravity is the constant
17:25
relentless force trying to pull that wave back down to a flat
17:29
level surface. Exactly.
17:31
Therefore, the Froude number governs wave resistance, which
17:34
is the massive amount of energy a ship loses just by pushing
17:37
waves outward. If you look at a massive
17:39
commercial container ship, say 245 meters long, sailing across
17:44
the Pacific at 28 knots, it is dragging an enormous wake behind
17:48
it. And if you were an engineer
17:50
tasked with testing A1 to 25 scale model of that container
17:54
ship to optimize its whole shape, you cannot just drag it
17:57
through a towing pank at random speed.
17:58
Because the waves wouldn't match.
18:00
Right. Froude's law of similarity
18:02
dictates the precise, mathematically exact towing
18:05
velocity you must use so that the wave patterns generated by
18:09
the tiny wooden model perfectly mimic the towering ocean waves
18:13
generated by the real ship. That's fascinating.
18:15
Then we have the Weber number, which is a bit more niche but
18:19
incredibly important for a specific industries.
18:21
Very niche, yes. The Weber number is the ratio of
18:24
inertia force to surface tension force.
18:27
When do engineers actually use that one?
18:29
You use this when you are analyzing how liquids behave
18:33
when they are poured, how droplets form, or the behavior
18:36
of very tiny capillary surface waves.
18:38
So things like fuel injectors in car engines.
18:41
Exactly. Or industrial sprayers, where
18:43
breaking a fluid into a fine mist of tiny droplets is the
18:46
entire goal of the design. Makes sense.
18:49
And finally, we arrive at the Mach number, named after the
18:52
physicist Ernst. The legendary name in physics,
18:55
the Mach number, is the ratio of inertia force to elasticity or
18:59
compressibility force. And the mathematical formula for
19:02
that is surprisingly simple, isn't it?
19:04
It is. It is simply the velocity of the
19:06
fluid divided by the local speed of sound in that fluid.
19:09
That's it. That's it.
19:11
But it is the absolute defining metric for high speed
19:14
aerodynamics. It dictates whether a flow
19:16
regime is subsonic, Sonic, supersonic, or hovering
19:20
dangerously in the middle as transonic.
19:23
Right, like when you look at a Boeing 747 cruising at altitude
19:27
is flying at roughly Mach 0.82. It is moving at 82% of the speed
19:33
of. Sound which is subsonic, yeah.
19:34
Subsonic, but it is pushing right up against that transonic
19:37
boundary where the air over the top of the wings might actually
19:40
locally accelerate past speed of sound, creating shock waves.
19:44
Exactly. The local airflow can be
19:46
supersonic even if the plane itself is subsonic.
19:49
And this leads us to one of the most fascinating historical
19:51
detours in this whole subject. Ernst Mach, the man who gave his
19:55
name to the Mach number, was a true visionary.
19:58
He was brilliant. He used these exact principles
20:00
of high speed, flow and compressibility to solve 2
20:03
bizarre, seemingly inexplicable mysteries in the late 19th
20:07
century. I love these historical examles.
20:10
If you are a history buff listening to this, this is where
20:13
physics literally changes the historical record.
20:16
OK, mystery #1 the artillery bangs.
20:20
Yes, the dual explosion mystery. Right back in the day, artillery
20:24
crews and military observers noticed a truly bizarre acoustic
20:28
phenomenon when they fired I highly advanced high speed
20:32
artillery projectile and observer position far down range
20:36
near the target would hear 2 completely distinct loud bangs.
20:41
One and then another shortly after.
20:43
Exactly, but if they fired an older low speed projectile from
20:47
a mortar, the down range observer would only hear one
20:50
single bang. The military establishment was
20:52
totally mystified by this. Completely baffled.
20:55
Why did the fast shell make 2 explosion sounds and the slow
20:58
shell only make one? Mach tackled this and he
21:01
realized it was entirely a manifestation of the speed of
21:03
sound. How did he figure it out?
21:05
He realized that for the high speed projectile, the very first
21:08
bang the observer heard had absolutely nothing to do with
21:11
the cannon firing. It wasn't the muzzle blast at
21:13
all. Not at all.
21:14
The first bang was the bow shock, a literal Sonic boom
21:18
generated by the physical nose of the projectile violently
21:22
tearing through the air and breaking the sound barrier.
21:24
Oh wow, so because the projectile itself was traveling
21:27
faster than the speed of sound, that compressed shockwave of air
21:30
physically arrived at the observer's ear before the actual
21:34
acoustic sound of the cannons muzzle blast?
21:37
Exactly. The sound of the muzzle blast
21:40
had to travel at the slower normal speed of sound, arriving
21:43
moments later as the second bang.
21:46
The bullet literally outran its own gunshot.
21:48
It perfectly explains the opening scenario we talked about
21:51
at the top of the show. The sound of the impact arrives
21:54
before the sound of the trigger being.
21:56
Pulled. It's a perfect real world
21:57
example of the Mach number in action.
21:59
But the second mystery mock solved, is even Wilder and much
22:03
more gruesome. It takes place during the Franco
22:05
Prussian War, yes. A very dark but fascinating
22:09
application of fluid dynamics. The French infantry were
22:12
equipped with a revolutionary new weapon called the Chassapot
22:14
rifle. It fired a bullet at an
22:16
unprecedentedly high velocity for the era.
22:19
Unprecedented speed. And the Prussian soldiers on the
22:22
receiving end were suffering massive, devastating crater
22:26
shaped wounds that looked nothing like standard bullet
22:28
holes. The trauma was so severe that
22:31
the Prussian government was actually outraged.
22:33
Yeah, they formally accused the French military of committing
22:35
war crimes, claiming the French were illegally stuffing the
22:39
bullets with explosive charges that detonated on impact.
22:42
They firmly believed the bullets contained tiny bombs.
22:45
They couldn't explain the trauma any other way.
22:48
But Ernst Moff investigated the Ballistics and he proved
22:51
mathematically and physically that the French were not using
22:54
explosives at. All no explosives whatsoever.
22:57
The massive crater wounds were a pure, horrifying demonstration
23:01
of fluid dynamics. Because the Chasipov bullet was
23:04
moving at such extreme supersonic velocity, it was
23:07
creating an intensely pressurized bow wave of highly
23:11
compressed air violently pushed in front of the metal bullet.
23:14
Compressibility we talked about. Earlier, exactly.
23:17
Mach proved that when air is compressed by sheer,
23:19
unadulterated speed, it becomes a physical weapon.
23:22
So it was this invisible wall of compressed air hitting the flesh
23:26
A microsecond before the bullet itself that caused the explosive
23:30
style trauma. The air itself acted as the
23:33
explosive. It is a profound, terrifying
23:36
realization about what fluid dynamics actually means in the
23:40
physical world when things move fast enough.
23:43
That is just incredible. OK, let's pull back from the
23:45
battlefield. This perfectly sets up our
23:47
second phase. Transitioning to measurement.
23:49
Right, let's say we have navigated the headache of
23:52
scaling. We have our dynamically similar
23:54
model, We have placed it in our ten atmosphere pressurized wind
23:57
tunnel or our giant water flume. How do we actually read what the
24:01
fluid is doing? This is a major engineering
24:03
hurdle. Because air is totally
24:05
invisible, water is entirely transparent.
24:08
We cannot just stick a standard mechanical car speedometer out
24:11
the window into the wind tunnel and expect it to tell us
24:13
anything useful. No, the wind would just blow
24:15
right past it. We have to invent tools that can
24:18
measure the invisible. This brings us to the engine is
24:21
tools of the trade, and engineers generally divide these
24:24
measuring devices into two broad categories.
24:27
OK, what are the 2 categories? First you have point measurement
24:30
tools which tell you the exact velocity of the fluid at one
24:33
tiny specific spot in space. Like tracking a single particle.
24:37
Exactly. And second, you have volume
24:39
measurement tools which tell you the massive overall flow rate of
24:43
liquid moving through an entire system.
24:45
Let's start with measuring velocity at a single pinpoint
24:48
spot. The absolute reigning champion
24:51
of this is the Pitot Tube, right?
24:53
Without a doubt, if you look at the nose of any commercial
24:56
airliner, you will see these little metal spikes sticking
24:58
out. Those are pitot tubes.
25:00
Yes, specifically the Pitot static tube.
25:02
Like the classic National Physical Laboratory type
25:05
mentioned in the text, it is a remarkably simple, completely
25:09
mechanical, but profoundly brilliant device.
25:12
It really is. At its core, it consists of a
25:14
small metal tube pointing directly straight ahead into the
25:17
oncoming flow of fluid. And as the air or water Rams
25:21
straight into the open front hole of that tube, it has
25:24
nowhere to go. It hits a dead end.
25:25
Exactly. The fluid Rams into the tube and
25:28
is forced to come to a complete dead stop.
25:30
And in physics, when moving fluid stagnates like this, all
25:35
of its kinetic energy, its speed, is instantly converted
25:38
into pressure. We call this the total pressure
25:41
or the stagnation pressure. But just knowing that total
25:44
pressure reading at the dead end of the tube isn't enough to
25:46
calculate speed, right? Because what if the ambient
25:49
environment is just highly pressurized to begin with, Like
25:52
deep underwater? Right, you can't just take the
25:54
raw number, you have to subtract the ambient background pressure
25:58
to isolate the speed. And that background pressure is
26:00
called the static pressure, the pressure of the the fluid just
26:03
sitting there calmly. The genius of the pitostatic
26:05
tube is how it finds that background pressure.
26:08
It has tiny microscopic holes drilled into the side of the
26:11
tube, perfectly perpendicular to the flow.
26:14
So the wind blows directly over these holes without ramming into
26:17
them, allowing the instrument to measure the cure static
26:20
pressure. Exactly.
26:21
So you have two pressure readings, the massive total
26:25
pressure from the front hole and the calm static pressure from
26:28
the side holes. And by comparing the two you
26:30
extract the velocity. Yes, the formula is a beautiful
26:35
derivative of Bernoulli's equation.
26:37
The velocity of the fluid equals sqrt 2 times the difference
26:41
between the total and static pressure, completely divided by
26:45
the density of the fluid. You are literally taking the
26:47
physical brute force pressure pushing against a piece of metal
26:51
and mathematically converting it into speed.
26:54
It's brilliant, and because engineers love to iterate, there
26:57
are some incredibly specialized variations of this simple tube.
27:00
Oh yeah, like if you were trying to measure a complex 2
27:04
dimensional flow where you aren't exactly sure which way
27:06
the wind is blowing, you use a cylinder type tube, often called
27:10
a yameter. The Yameter is fascinating.
27:12
It's a cylinder with two pressure holes drilled at
27:15
precise angles. You insert it into the flow and
27:18
you physically rotate the cylinder on its axis until the
27:21
pressure reading from both holes is perfectly identical.
27:24
They balance perfectly. Exactly.
27:26
And the moment they balance, you know the cylinder's pointing
27:28
exactly perfectly dead center into the oncoming flow
27:31
direction. It physically finds the wind for
27:33
you. There is also a 5 hole spherical
27:36
pedo tube for mapping chaotic 3 dimensional flow directions.
27:40
Yes, for very complex flows. And if an engineer needs to
27:44
measure the velocity of the fluid right up against the solid
27:46
boundary wall of a pipe or a car hood where the boundary layer
27:50
gets microscopically thin, then they use flattened total
27:54
pressure tubes. Right, they squash the opening
27:56
down so it can fit in that tiny space.
27:58
Or they use surface pitot tubes that sit perfectly flush,
28:02
integrated right into the wall itself so they don't disturb the
28:05
very flow they are trying to measure.
28:06
Now all of those pitot variations rely on pressure, but
28:10
the next tool on our list takes a completely different approach.
28:13
This one is wild. The hot wire anemometer.
28:16
It measures velocity by measuring temperature.
28:18
This is one of my absolute favorite concepts and it brings
28:21
me right back to food analogies. Let's hear the food analogy.
28:24
OK, imagine you were sitting at a table with a bowl of scalding
28:28
hot soup. You scoop up a spoonful and you
28:30
blow on it to cool it down. A universal experience.
28:33
Right. The harder and faster you blow
28:35
on the spoon, the more rapidly the soup loses its heat.
28:39
The hot wire anemometer is exactly that, just weaponized
28:42
with electricity. That is precisely the underlying
28:44
principle. The instrument consists of two
28:47
tiny prongs, and suspended between them is an incredibly
28:50
fine, almost microscopic wire. Usually made of platinum or
28:54
tungsten, right? Yes, this tiny wire is heated
28:57
electrically until it is quite hot.
29:00
The probe is in place directly into the rushing wind or flowing
29:03
liquid. As the cold fluid rushes past
29:06
the hot wire, it forcefully strips the heat away, cooling
29:09
the metal down. And here's the trick of physics
29:11
we exploit. When the temperature of a
29:14
specific metal wire changes, it's electrical resistance, its
29:18
ability to let electricity flow through it changes
29:20
proportionally. So as the wire gets colder, the
29:22
resistance drops. Exactly.
29:24
So you are actually measuring the wind physically pushing
29:27
against a tube. You are measuring the electrical
29:29
resistance of this tiny wire, which tells you exactly how much
29:33
heat was stripped away, which tells you exactly how fast the
29:36
fluid is moving. But I mean, early versions of
29:38
this had a problem, didn't they? They did thermal lag.
29:42
Right, because if the wind wind speeds up, the wire cools down,
29:45
but then it takes a fraction of a second for the wire to heat
29:48
back up when the wind slows down again.
29:50
Yes, this is called thermal inertia and it makes the
29:53
readings lag behind reality. To solve this, modern engineers
29:58
develop the constant temperature anemometer or CTA.
30:01
How does the CTA fix the lag? By not letting the temperature
30:06
dry at all. Wait, what?
30:07
The CTA features a highly advanced, ultra fast electronic
30:10
feedback circuit. The instant the wing speeds up
30:13
and begins to cool the wire even a fraction of a degree, the
30:16
circuit instantly floods the wire with higher electrical
30:19
voltage. Oh, forcing the temperature to
30:21
remain perfectly steady. Exactly.
30:23
So the temperature never changes.
30:25
The instrument measures how much electrical power it has to
30:27
violently pump into the wire just to maintain that constant
30:31
heat. Wow.
30:32
And because the wire never physically changes temperature,
30:35
thermal inertia is virtually eliminated.
30:37
It reacts instantaneously to changes in wind speed.
30:41
These CTA instruments are so fast they have an incredible
30:44
frequency response. They can measure turbulent,
30:47
chaotic fluctuations in the wind up to 100 kHz.
30:51
That is 100,000 distinct velocity measurements every
30:55
single second. You can map the tiniest, most
30:59
violent micro ID's in a turbulent storm with that.
31:01
It's an incredibly sensitive instrument.
31:03
But if we want to talk about the absolute pinnacle of high tech
31:06
measurement, we have to talk about lasers.
31:09
The laser Doppler anemometer literally feels like it was
31:11
pulled straight out of science fiction.
31:14
You are pointing laser beams at dust.
31:16
It is an incredibly elegant, non intrusive application of
31:19
physics. It relies entirely on the
31:21
Doppler effect. OK, let's explain the Doppler
31:23
effect. If you are driving down the
31:25
street and an ambulance is speeding towards you with its
31:28
sirens blaring, you hear the Doppler effect.
31:31
The classic example. As the ambulance races toward
31:34
you, it is chasing its own sound waves, compressing them, making
31:38
the pitch of the sirens sound higher.
31:40
As it races past you and drives away, the sound waves are
31:43
stretched out behind it and the pitch noticeably drops.
31:47
Exactly. The frequency of the wave
31:49
changes based on the speed of the object emitting it.
31:52
But with the laser Doppler anemometer, instead of using
31:55
acoustic sound waves, we are using the electromagnetic waves
31:58
of light, a highly focused coherent laser beam.
32:02
To make this work, the fluid we are studying has to be seeded
32:05
with tiny microscopic tracer particles.
32:08
Think of incredibly fine dust or microscopic oil droplets that
32:11
are so small they perfectly follow every twist and turn of
32:14
the fluid flow. Yes, the tracer particles are
32:17
key. When you fire the laser beam
32:19
into the transparent fluid, the light eventually strikes one of
32:22
these microscopic particles moving with the flow.
32:24
And as the laser light bounces or scatters off that moving
32:27
particle, the frequency of the scattered light changes based on
32:30
how fast the particle is moving, just like the ambulance siren.
32:33
The instrument then collects that scattered light.
32:36
By mathematically measuring the microscopic difference in
32:39
frequency between the original pure laser beam and the altered
32:43
scattered light, you calculate the exact absolute velocity of
32:47
the particle and therefore the fluid.
32:49
The math behind this is wild. The velocity of the flow equals
32:53
the wavelength of the laser multiplied by the Doppler
32:56
frequency difference, completely divided by two times the sign of
33:00
the angle of the intersecting beams.
33:02
It's a beautiful equation. You are measuring the speed of
33:05
microscopic dust by analyzing how it bends laser light.
33:09
And engineers use several different optical setups to
33:11
achieve this right? They do primarily the reference
33:14
beam method, the single beam method, and the interference
33:17
fringe method. The fringe method is especially
33:19
cool. It is in the fringe method.
33:21
You actually split the laser into two beams and cross them
33:25
where they intersect. They create microscopic
33:27
alternating bands of light and dark fringes.
33:29
Like a barcode of light. Exactly as a particle flies
33:32
through these bright and dark bands, it flashes light back at
33:35
the sensor. The speed of the flashing tells
33:37
you the speed of the particle. The massive advantage here is
33:40
that it is totally not intrusive.
33:42
Unlike a metal pitot tube or a delicate hot wire, you are not
33:47
plunging a physical object into the flow that might disrupt the
33:50
delicate aerodynamics you were trying to measure.
33:52
You are just shining light through a glass window.
33:55
It is the perfect tool for highly sensitive, highly
33:59
turbulent flows. Now everything we just discussed
34:02
is for measuring velocity at one specific pinpoint location.
34:05
Right point measurement. But what if you don't care about
34:08
a single point? What if you were an engineer
34:10
managing a massive oil pipeline or a municipal water treatment
34:15
plant and you need to know the massive overall volume flow
34:19
rate? You need to know exactly how
34:20
many thousands of gallons or cubic meters are moving through
34:23
the entire pipe every minute. Exactly.
34:25
You don't need a point velocity, you need a flow meter.
34:28
And looking at the options, the most common rugged industrial
34:31
method is forcing the fluid to squeeze through a restriction.
34:34
You POTUS fully put a metallic obstacle inside the pipe.
34:38
Yes. The simplest version of this is
34:41
the orifice plate. Imagine a solid, flat, circular
34:45
steel plate with a precise hole drilled right in the center.
34:50
You insert this plate directly into the cross section of the
34:52
pipe. So as the massive volume of
34:54
fluid hits this barrier, it is forced to squeeze through that
34:58
smaller central hole. And according to Bernoulli's
35:00
principle of fluid dynamics, whenever a fluid is forced to
35:03
accelerate through a narrower space, its pressure drops
35:07
proportionally. Right.
35:08
So by placing a pressure sensor just upstream of the metal plate
35:11
and another pressure sensor just downstream of the plate, you can
35:14
measure the exact pressure difference.
35:16
Yes, the total volume flow rate of the fluid is directly
35:20
proportional to the square root of that pressure difference.
35:22
It is a highly reliable, mathematically sound way to
35:26
measure massive volumes. But there is a catch, isn't
35:28
there? There is always a catch.
35:30
If you just shove a flat plate into a high pressure water main,
35:33
the fluid violently smashes into the flat surface, squeezes
35:36
through the hole, and then violently expands on the other
35:38
side. It creates massive chaotic
35:40
turbulence. And turbulence is just wasted
35:44
energy. That violent expansion behind
35:47
the orifice plate results in a permanent pressure loss in your
35:50
pipeline system. Which is bad.
35:52
Very bad if you are managing A pipeline stretching hundreds of
35:56
miles. Every ounce of lost pressure
35:58
means your massive industrial pumps have to work harder,
36:01
burning more electricity and costing massive amounts of
36:04
money. So how do you measure the flow
36:06
restriction without creating the chaotic expensive turbulence?
36:10
You use, I say, 1932 nozzles or much more commonly Venturi
36:15
tubes. Venturi tubes, how are they
36:18
different? Instead of a harsh flat plate, a
36:20
venturi tube is a beautifully machined piece of pipe that's
36:23
smooth smoothly, gradually tapers down to a narrow throat,
36:26
and then gently, gradually expands back out to the full
36:29
diameter of the. Pipe so it guides the water
36:31
instead of blocking it. Yes, the physical principles are
36:34
identical to the orifice plate. The fluid accelerates in the
36:37
narrow throat, the pressure drops, and you measure the
36:39
difference. But without the violence.
36:41
Exactly. The flow coefficients differ
36:43
slightly in the math, but the massive advantage is that a
36:46
Venturi tube has a remarkably low permanent flow loss.
36:50
The gentle curves prevent the violent turbulent from forming,
36:53
saving massive amounts of pumping energy.
36:56
That makes a lot of sense. Let's shift away from pressure
36:59
based meters to area flow meter, specifically rotameters.
37:02
Protameters are fantastic. If you have ever been in a
37:05
hospital room with an oxygen line on the wall or a chemistry
37:08
laboratory, you've almost certainly seen a rotameter.
37:11
It is the vertical transparent glass or plastic tube with a
37:16
little floating Bob inside it. The rhodometer is a brilliant
37:20
tactile balancing act of physical forces.
37:23
The fluid, whether it is oxygen, gas, or liquid water, flows
37:27
vertically upward from the bottom of the tube to the top
37:29
right. As the fluid hits the solid
37:31
little float sitting at the bottom, it creates upward
37:34
pressure drag and friction drag. The rushing fluid literally
37:38
tries to push the float up into the air.
37:40
But gravity is constantly pulling the heavy float back
37:42
down. Exactly the float rises upward
37:46
until the pushing force of the fluid perfectly exactly balances
37:50
the downward weight of the float minus whatever natural buoyancy
37:54
the float has. Now if the glass tube was
37:56
perfectly straight, like a straw, the float would either
37:59
shoot all the way to the top or sit at the bottom.
38:01
Right, there would be no graduation.
38:03
The genius of the rhodameter is that the glass tube is tapered.
38:07
It is narrower at the bottom and wider at the top.
38:10
So as the float is pushed higher up the tube, the annular gap,
38:14
the empty space between the float and the glass wall, gets
38:17
wider and wider. Which means more and more fluid
38:20
can slip past the float. Yes.
38:22
Because of this expanding area, the upward drag force decreases
38:26
the higher the float goes. Eventually the float finds an
38:29
exact stable resting height where the drag perfectly matches
38:32
the weight. That physical resting height is
38:34
directly proportional to the flow rate.
38:37
You just look at the float, read the printed number on the side
38:39
of the glass tube next to it, and you know exactly how much
38:41
fluid is moving. It is so beautifully visual.
38:44
But let's say we are dealing with something incredibly thick,
38:47
sticky and valuable like crude oil or heavy syrup.
38:51
You don't want to rely on balancing a tiny float.
38:54
No, a float wouldn't work at all there.
38:56
Right, you use positive displacement meters like Oval
38:59
gear meters or roots type meters.
39:01
These feel much more like heavy machinery.
39:04
Because they are positive displacement meters do not try
39:07
to infer the flow rate by measuring pressure drops or drag
39:10
forces. How do they work them?
39:12
They literally physically trap a very specific, known volume of
39:17
fluid in a mechanical chamber transported from the inlet to
39:20
the outlet and dump it out. It is exactly like a revolving
39:25
door at the entrance of a busy hotel.
39:26
That's a great way to picture it.
39:28
Every time the revolving door turns one complete circle, you
39:31
know exactly 4 people walked inside.
39:32
Precisely in an Oval gear meter, 2 interlocking Oval shaped gears
39:37
rotate as the fluid pushes against them.
39:40
Every time those gears complete one full revolution, a
39:42
mathematically exact pocket of fluid has been moved to the
39:45
pipe. So to calculate the mass of flow
39:47
rate, you don't measure pressure, you simply hook up a
39:50
sensor to the physical revolutions of the gears.
39:53
And closely related to that is the turbine flow meter.
39:56
You just drop a multi bladed propeller turbine into the
40:00
middle of the pipe. The rushing water catches the
40:03
angled blades and spins the turbine.
40:05
The faster the water, faster it spins.
40:07
But how do you accurately count the speed of a spinning blade
40:11
inside a high pressure sealed steel pipe without leaking water
40:15
everywhere? You.
40:16
Use electromagnetism. Of course.
40:18
You embed a small, powerful magnet into the tip of one of
40:21
the of the turbine blades, and on the outside of the pipe,
40:24
completely sealed away from the fluid, you mount a magnetic
40:27
pickup coil. Every single time that
40:29
magnetized blade spins past the coil, it generates a tiny
40:33
microscopic electrical pulse. A computer simply counts the
40:37
frequency of those electrical pulses.
40:39
More pulses equals more flow. Which perfectly transitions us
40:42
into what I consider absolute physics magic magnetic flow
40:46
meters. I want you to explain this
40:47
slowly, because when I first read how this works, it sounded
40:50
entirely impossible. It does sound like magic.
40:52
It relies on Michael Faraday's law of electromagnetic
40:56
induction. Oh, OK, Faraday's law.
40:58
Faraday discovered in the 19th century that if you take a
41:01
conductive material like a copper wire and you physically
41:04
move it through a magnetic field, the invisible magnetic
41:08
forces will induce a voltage, an electromotive force, directly
41:12
inside the wire. It is how every electrical
41:15
generator on Earth works. Exactly, but a magnetic flow
41:18
meter doesn't have any moving copper wires in it.
41:21
No it doesn't. So how does it generate voltage?
41:23
In a magnetic flow meter, the pipe itself is lined with a non
41:26
conducting material like rubber or Teflon, but the fluid flowing
41:30
through the pipe, say municipal wastewater or acidic chemicals
41:34
or liquid metal, is electrically conductive.
41:36
Wait, so the fluid itself acts as the moving wire?
41:39
Yes, the river of wastewater is the wire.
41:42
That is insane. The flow meter uses massive
41:44
coils to project a strong constant magnetic field straight
41:48
across the diameter of the pipe. As the conductive fluid flows
41:52
rapidly through this invisible magnetic field, the physics of
41:55
Faraday's law activate. An electromotive force, a
41:58
distinct electrical voltage, is generated directly within the
42:02
flowing water. And how do you capture that
42:03
voltage? The meter has two small metallic
42:06
electrodes mounted perfectly flush against the inside wall of
42:09
the pipe, directly in contact with the water.
42:12
These electrodes detect the voltage.
42:14
And the mathematical formula governing this is pure elegance.
42:18
The voltage generated equals the magnetic flux density multiplied
42:21
by the diameter of the pipe multiplied by the mean velocity
42:25
of the flow. Since the magnetic field and the
42:27
pipe diameter are constant, the voltage you measure is perfectly
42:32
linearly proportional to how fast the water is moving.
42:34
And what is the massive world changing engineering advantage
42:38
of doing it this way? Zero pressure loss.
42:41
Absolutely 0. That's huge.
42:43
There's no orifice plate violently squeezing the water.
42:46
There's no heavy turbine blade sitting in the way waiting to
42:49
get jammed. The pipe is a completely open,
42:52
hollow, unobstructed tube. You're measuring the massive
42:55
rush of millions of gallons of water purely by feeling it's
42:58
invisible electromagnetic pulse. It is the ultimate tool for
43:02
handling slurries, heavy sewage, or any fluid thick with solid
43:06
debris that would instantly shatter or jam a mechanical
43:09
turbine. It really.
43:10
Is that is incredible. Next we have ultrasonic flow
43:13
meters. These sound incredibly high
43:16
tech, but the concept is actually very straightforward.
43:19
It is the meter sends acoustic sound pulses diagonally across
43:23
the diameter of the pipe. It fires 1 high frequency sound
43:26
wave downstream, completely traveling with the flow of the
43:29
water, and it fires a second sound wave upstream, fighting
43:32
against the flow of the water. Because the entire body of water
43:35
is physically moving, the sound wave traveling downstream is
43:39
carried along and arrives at the opposite slightly faster than
43:42
normal. Conversely, the sound wave
43:44
fighting its way upstream is slowed down.
43:46
It is exactly like swimming with a river current versus swimming
43:49
against it. The flow meter's internal
43:51
computer measures the microscopic fraction of a
43:54
millisecond time difference between the two arrivals.
43:57
That minute time delta tells you exactly how fast the fluid is
44:00
moving. The major advantage here is that
44:02
ultrasonic meters can measure non conducting fluids like ultra
44:06
pure hydrocarbons or distilled water where the magnetic flow
44:09
meter would completely fail. Because there's no conductivity,
44:12
right? However, they are highly complex
44:15
and notoriously expensive. And we have one last major pipe
44:19
meter, the vortex shedding flow meter.
44:23
This takes us right back to the deep aerodynamics we discussed
44:26
earlier. Yes, back to vortices.
44:28
Imagine an island in the middle of a fast moving ocean current.
44:32
As the water flows around the island, it doesn't just reform
44:35
cleanly behind it, it creates alternating swirling whirlpools
44:39
that detach from the island and trail behind it in a very
44:42
specific alternating zipper pattern.
44:44
That beautiful repeating pattern of swirling eddies is called a
44:48
Carmen Vortex St. You can actually see them in
44:51
satellite photos of clouds parting around mountain peaks.
44:54
It's mesmerizing to look at. A vortex shedding meter
44:56
intentionally places a non aerodynamic obstacle, usually a
45:00
blunt cylinder, directly into the center of the pipe to
45:03
forcefully create these alternating vortices.
45:06
The text mentions A dimensionless number controlling
45:08
this the scroll hole number. Yes, the stroll hole number.
45:12
The stroll number governs oscillating flow mechanisms, and
45:15
for a standard cylinder, the stroll number remains almost
45:18
perfectly constant at approximately 0.2 / a massive
45:22
range of different fluid velocities.
45:24
Because that number is constant, it dictates that the frequency
45:27
at which these swirling vortices detached from the cylinder is
45:30
completely directly proportional to the velocity of the fluid.
45:34
The formula is beautifully simple.
45:36
Velocity equals the vortex shedding frequency multiplied by
45:39
the diameter of the cylinder completely divided by 0.2.
45:43
So the meter just contains a tiny sensor that detects the
45:47
physical pulsing of the vortices hitting it like a heartbeat.
45:50
It counts the pulses to count the flow.
45:53
The engineering text also touches briefly on specialized
45:56
applications like fluidic oscillators, which utilize
45:59
complex wall attachment feedback loops to create a rapidly
46:02
oscillating jet of fluid whose frequency tracks the flow.
46:05
Rate And for massive open channel flows, like monitoring
46:08
an entire river or a concrete aqueduct, engineers use weirs.
46:12
Weirs are essentially controlled waterfalls.
46:15
You build a barrier across the river, let the water Crest over
46:18
it, and measure how high the water gets.
46:20
Yes, there are triangular, rectangular and full width
46:23
weirs. By measuring the physical height
46:26
of the waterhead as it spills over the notch, engineers use
46:29
specific British Standard formulas to calculate the
46:32
massive volumetric flow of the entire river.
46:35
It is an absolutely staggering array of physical tools.
46:39
From counting interlocking gears to bouncing laser beams off
46:42
microscopic dust, to measuring the electromagnetic voltage of a
46:45
moving river to counting the heartbeat of a vortex, St.
46:49
engineers have found a way to measure fluid in almost every
46:52
conceivable physical state. It is a triumph of applied
46:55
physics. But all of these tools, as
46:58
incredible as they are, are reactionary.
47:00
They are designed to measure what a fluid is already doing.
47:03
Which leads us to the final and absolutely most profound phase
47:06
of this master class. We can scale our models, we can
47:09
measure the air in our tunnels. But what happens before any of
47:12
that? How does an aeronautical
47:14
engineer mathematically predict how a fluid will behave before
47:17
they ever build a single piece of metal?
47:19
That is the Holy Grail of fluid dynamics, to predict the
47:22
chaotic, swirling nature of reality using only a pencil and
47:26
paper. And to achieve this, the early
47:29
pioneers of engineering had to make a massive, counterintuitive
47:33
leap. They had to invent a perfect,
47:36
entirely impossible fluid. Welcome to the realm of ideal
47:40
fluid flow and the breathtaking math of flight.
47:43
Let's go back to the Reynolds number, the battle between
47:45
forward inertia and sticky viscosity.
47:48
We established that for aircraft flying at high speeds or ships
47:52
moving through water, the Reynolds number is incredibly,
47:55
massively high. Right, the forward inertia
47:57
absolutely dominates the viscosity.
47:59
And because the inertia is so dominant, a strange physical
48:02
phenomenon occurs. The effects of the fluid sticky
48:05
viscosity become entirely confined to an unimaginably thin
48:08
layer of air right against the solid surface of the object.
48:11
We call that the boundary layer. And outside of that
48:14
microscopically thin boundary layer, the vast overwhelming
48:17
majority of the air flowing over the giant wing of an airplane
48:19
simply ignores viscosity entirely.
48:22
It flows as if the stickiness doesn't exist.
48:24
So mathematically, engineers make a terrifyingly bold,
48:27
simplifying assumption. They say let's treat the entire
48:31
ocean of air as an ideal fluid. An ideal fluid is defined by two
48:36
impossible conditions. First, it is perfectly invisid,
48:40
meaning it has absolutely 0 viscosity, 0 internal friction.
48:44
And second, it is irritational, meaning there is no swirling
48:47
vorticity, no microscopic whirlpools twisting the fluid
48:50
particles. The particles only glide
48:52
smoothly. But wait, if you just drop
48:55
viscosity, one of the fundamental properties of the
48:57
universe, from your math, isn't that just cheating?
49:00
Doesn't the math break? It sounds like cheating, but
49:02
it's actually a masterful stroke of reduction.
49:05
By dropping the viscosity, we get to discard the incredibly
49:08
brutal, unsolvable, viscous terms of the Navier Stokes
49:12
equations. Which are famously impossible to
49:14
solve analytically. Exactly.
49:16
We can replace them with Euler's equations of motion, and the
49:19
profound beauty of this ideal fluid assumption is that it
49:22
cleanly splits the physics into two manageable independent
49:26
parts. Explain how it splits the
49:27
physics. Well, the momentum equation is
49:29
now solely responsible for determining the physical
49:31
pressure distribution. But the actual geometric shape
49:35
of the flow, the beautiful curving pathways the air takes
49:39
as it flows around the wing, is completely and entirely dictated
49:43
by the continuity equation governed by the geometry of the
49:46
solid object itself. OK, I need to visualize this
49:49
because we are entering pure geometry now.
49:51
The concepts here are the velocity potential and the
49:54
stream function. I try to picture this like a
49:57
vast, perfectly planned city grid, like Manhattan.
50:00
I like this visualization. Walk me through it.
50:02
Imagine the fluid flow as a massive city grid.
50:06
The streamlines, what the math calls the stream function, are
50:08
the wide avenues. They dictate exactly where the
50:11
traffic or the fluid particles physically flow.
50:14
A particle cannot cross a streamline, it must flow
50:16
perfectly down the Ave. OK, that tracks.
50:18
Now intersecting those avenues are the cross streets.
50:21
Those are the equal potential lines, what the math calls the
50:24
velocity potential. And here is where the strict
50:27
rules of our ideal fluid create magic.
50:30
Because we assume the flow is perfectly irrotational, we
50:33
mathematically prove that a velocity potential which we
50:36
denote with the Greek letter Phi must exist.
50:39
And because we assume the flow is incompressible, we
50:41
mathematically prove that a stream function PSI must exist.
50:45
And the profound mathematical reality that governs this entire
50:48
system is that both Phi and PSI perfectly satisfy Laplace's
50:53
equation. And when you map out Laplace's
50:55
equation onto our theoretical city grid, a beautiful geometric
50:59
law emerges at every single microscopic point on that
51:03
massive map. Wherever an Ave. streamline
51:05
intersects with a cross street equipotential line, they cross
51:09
at exact perfect 90° right angles.
51:11
Always, without exception, they form an orthogonal net.
51:15
You have constructed a perfect mathematical grid of fluid flow.
51:18
But here is where the mathematics take an even Wilder,
51:20
almost spiritual leap. This is where we transition into
51:23
the true beauty of potential flow.
51:25
To manipulate this grid, we use complex variables.
51:28
Complex numbers. We're talking about imaginary
51:29
numbers. Yes, we define a single unified
51:33
complex variable which we call Z&Z equals the X coordinate plus
51:38
the imaginary unit I multiplied by the E coordinate.
51:40
For anyone who needs a quick refresher, the imaginary unit I
51:44
is sqrt -1. A number that literally cannot
51:48
exist in standard arithmetic because no real number
51:51
multiplied by itself can be negative, but it creates a
51:54
second vertical dimension to our number line.
51:56
Yeah, it creates A2 dimensional mathematical plane, and in high
52:00
level mathematics any regular function of a complex variable
52:04
automatically satisfies an incredibly strict set of rules
52:07
known as the Kochi Riemann equations.
52:09
But why does an aeronautical engineer trying to build a
52:12
physical metal wing care about a rule for imaginary numbers?
52:17
Because of an unbelievable coincidence in the structure of
52:19
the universe. I love when that happens.
52:21
If a complex math function satisfies the Cauchy Riemann
52:23
equations, it's real part automatically perfectly obeys
52:26
every single rule for our velocity potential Phi, and it's
52:30
imaginary part automatically perfectly obeys every single
52:33
rule for our stream function size.
52:35
Let me make sure I am grasping the magnitude of this.
52:38
You can literally just write down a random algebraic function
52:41
using an imaginary number, differentiate it on a
52:44
chalkboard, and it will perfectly map out the real
52:47
physical movement of invisible air and water.
52:50
Yes, the imaginary math seamlessly translates into
52:54
physical fluid mechanics. We call this unified
52:57
mathematical tool the complex potential.
52:59
And by manipulating this complex potential, theoretical engineers
53:03
can build predictive models of any flow pattern imaginable.
53:07
To do this, we use a set of basic mathematical building
53:10
blocks. Think of them as the fundamental
53:12
Lego bricks of fluid dynamics. OK, let's open up the Lego box.
53:15
If we are building a theoretical flow from scratch, what is the
53:19
absolute simplest mathematical brick we can start?
53:22
With block number one parallel flow, this represents a massive
53:26
uniform wind blowing steadily in one straight direction.
53:29
The mathematics are wonderfully simple.
53:31
The complex potential is simply the uniform forward velocity U
53:35
multiplied by our complex variable Z.
53:37
The streamlines are perfectly straight horizontal avenues.
53:41
The equal potential lines are perfectly straight vertical
53:43
cross streets. That is our blank canvas.
53:46
What is block number 2? Block number 2 is the source in
53:48
the sink. Imagine a mathematical pinpoint
53:51
in space where fluid miraculously appears out of
53:54
nothing, flowing equally outward in every possible direction.
53:58
That is a source. The opposite is a sink, a
54:00
pinpoint where fluid drains endlessly out of the universe.
54:04
The complex potential for this involves the natural logarithm
54:07
of Z. If you map it, the echo
54:09
potential cross streets are perfectly concentric circles
54:13
expanding outward. The streamline avenues are
54:15
straight radial lines shooting outward from the center, exactly
54:18
like the metal spokes of a bicycle wheel.
54:20
OK, we have wind and we have bursting sources.
54:24
What is block number 3? Block number 3, The free vortex.
54:28
Imagine fluids spinning in perfect infinite concentric
54:31
circles around a central origin point.
54:33
It is a mathematical Whirlpool. The velocity of the fluid is
54:36
incredibly high near the dead center and drops off
54:39
exponentially the further out you travel.
54:41
This block introduces the vital concept of circulation.
54:44
OK, we have our basic Lego bricks, parallel flow sources
54:47
and sinks, and free vortices. Now we start synthesizing
54:50
physical reality. Because Laplace's equation is
54:53
linear, you could literally just take the mathematical equations
54:56
for these different blocks and add them together to build
54:59
complex shapes. Let's build a physical object,
55:02
take a source and a sink and place them on our mathematical
55:05
grid separated by a distance. Now mathematically move them
55:09
closer and closer together, decreasing the distance while
55:12
increasing their strength, until they are infinitely close to one
55:15
another, practically overlapping.
55:17
You have just forged a new advanced building block called a
55:21
doublet. OK, I have an infinitely dense
55:23
doublet in my hand. Now take that doublet and drop
55:26
it right into the middle of our first block, the uniform
55:29
parallel flow wind. Mathematically, you just add the
55:32
equation of the doublet to the equation of the parallel flow
55:35
and. When you map the resulting
55:36
streamlines, the math miraculously parts the uniform
55:40
wind perfectly around a solid physical cylinder.
55:44
It does. It is breathtaking to see on
55:46
paper the mathematical streamlines part around the
55:49
microscopic origin exactly Florida State as real air parts
55:53
around a physical steel pipe in a wind tunnel.
55:56
Using this equation, you can perfectly calculate the exact
55:59
stagnation points at the absolute front and absolute rear
56:03
of the cylinder where the fluid smashes into it and its velocity
56:06
drops to 0. You can calculate the highest,
56:08
fastest velocity points where the air accelerates over the top
56:11
and bottom, all derived purely from adding a theoretical
56:14
doublet to a uniform wind. OK, this is beautiful.
56:17
The geometry is elegant. But here is the ultimate
56:21
practical pushback. Airplanes do not fly on giant,
56:24
blunt, circular cylinders. They fly on highly specialized,
56:28
flat, asymmetrical teardrop airfoils.
56:32
That is a very valid point. How does perfectly calculating
56:35
the wind around a circular pipe help Boeing or Airbus get a
56:38
massive jetliner off the ground? This is the master stroke of
56:41
fluid dynamics. This is where abstract
56:43
mathematical genius meets the gritty reality of engineering.
56:47
We use a mind bending technique called conformal mapping.
56:50
Conformal mapping? Yes, Specifically an equation
56:53
known as Jukowski's transformation.
56:55
It is essentially A mathematical warp drive.
56:57
Oh, warp drive. You take the entire complex
57:00
grid, the original Z plane, where we just build our perfect
57:03
symmetrical flow around a cylinder, and you mathematically
57:07
warp, stretch and map that entire space onto a brand new
57:11
dimensional plane. The Jakowski transformation
57:14
formula itself is incredibly compact.
57:18
The new warped variable Zeta equals the old variable Z + a
57:21
constant squared divided by Z. If I apply that warp drive to
57:25
our cylinder, what happens to it?
57:27
If you take a standard perfect circle centered dead on the
57:29
mathematical origin, Jukowski's transformation violently crushes
57:34
it, flattening the circle completely into a single flat 2
57:37
dimensional plate. But, and this is the absolute
57:39
genius of the technique, if you offset the original circle just
57:42
a little bit, sliding its center slightly up and slightly left
57:45
away from the origin before you apply the mathematical warp.
57:48
The transformation stretches, pulls and warps the perfect
57:51
cylinder into the exact, unmistakable, asymmetrical
57:54
teardrop shape of an airplane wing.
57:56
An airfoil, yes. And here is the true power of
57:59
conformal mapping. It doesn't just warp the solid
58:02
shape of the wing, it warps the entire invisible grid of the
58:05
fluid. The Streamline avenues and the
58:07
Echo potential cross streets perfectly elastically bend and
58:11
stretch along with the solid shape, all while strictly
58:14
maintaining their perfect 90° intersections.
58:17
The theoretical flow pattern around a simple cylinder is
58:20
Florida State translated into the highly complex flow pattern
58:23
around an aircraft wing. It is mathematical alchemy.
58:27
You turn a pipe into an airplane wing.
58:29
But there is one final critical physics glitch.
58:33
Because nature hates a paradox, we have to talk about the CUDA
58:37
condition. We do.
58:38
If you just naively map the cylinder flow directly to the
58:41
wing shape using the Drukowski transformation, the math
58:44
dictates something physically, violently impossible.
58:47
And Airfoil by design has a sharp, tapered trailing edge at
58:50
the back. The pure, unmodified
58:52
mathematical mapping demands that the air flowing under the
58:55
bottom of the wing has to whip around that razor sharp trailing
58:58
edge, curling up to the top at an infinite velocity.
59:01
And nature absolutely, categorically forbids infinite
59:04
velocity. It does in the real world.
59:07
Nature resolves this impossible paradox through viscosity, the
59:11
stickiness of the air tearing and creating turbulence at the
59:14
edge. But remember, we assumed our
59:17
ideal fluid has 0 viscosity. So mathematically, to force our
59:21
perfect theoretical fluid to obey the harsh rules of physical
59:25
reality, we manually introduce an overriding law called the
59:28
CUDA condition. Which states what?
59:31
The CUDA condition simply states that the sharp trailing edge of
59:33
the wing must be a stagnation point.
59:36
The air flowing over the top of the wing and the air flowing
59:39
under the bottom of the wing must meet and leave the trailing
59:42
edge smoothly, beautifully, at the exact same time, without
59:45
ever whipping around the sharp corner.
59:47
But how do we physically force the mathematical equations to
59:50
obey the CUDA condition? How do we stop the infinite
59:53
velocity? We reach back into our box of
59:55
mathematical Lego bricks. We take our third block, the
59:58
free vortex circulation. Before we map the cylinder to
1:00:01
the wing, we mathematically force a highly specific,
1:00:04
precisely calculated amount of spinning circulation into the
1:00:08
cylinders equation. We basically add a spinning
1:00:10
Whirlpool to the flow. And this forced circulation
1:00:13
mathematically required purely to stop the impossible infinite
1:00:17
velocity at the trailing edge. Is literally the force that
1:00:20
generates lift. The circulation accelerates the
1:00:23
air over the top of the wing, lowering the pressure, and
1:00:26
decelerates the air under the wing, raising the pressure.
1:00:29
The high pressure pushes up into the low pressure.
1:00:32
Lift is born. By adding a spinning vortex to a
1:00:35
cylinder and using imaginary numbers to stretch that cylinder
1:00:39
into a teardrop, we unlocked the mathematical secret to human
1:00:42
flight. It is, without exaggeration, one
1:00:45
of the greatest intellectual triumphs in the history of
1:00:48
theoretical engineering. You start with the absurd
1:00:51
assumption of an impossible frictionless fluid.
1:00:54
You calculate it using imaginary numbers that possess sqrt -1.
1:00:58
You warp dimensional space using conformal mapping and you end up
1:01:01
with an elegant equation that perfectly Florida State predicts
1:01:05
the massive real world physical lift required to wrench a 400
1:01:09
ton aircraft off the runway and carry it safely across an ocean.
1:01:13
It leaves you with such a profound, almost unsettling,
1:01:16
lingering thought. Look at the journey of this
1:01:18
master class. We covered so much ground.
1:01:20
We started this deep dive with very physical, tangible, gritty
1:01:25
things. We talked about dragging wooden
1:01:27
model ships through massive water towing tanks.
1:01:30
We talked about building heavily reinforced wind tunnels
1:01:33
pressurized to 10 atmospheres. We talked about firing Sonic
1:01:37
boom inducing artillery shells and shooting actual lasers into
1:01:40
water to bounce off microscopic dust particles.
1:01:44
Immensely physical real world interactions.
1:01:46
And yet we ended the journey entirely in the realm of pure,
1:01:50
formless, abstract mathematics. Exactly.
1:01:52
We ended up relying on imaginary number numbers that we literally
1:01:56
invented, which cannot be measured on any physical ruler
1:01:59
or scale in the universe, to perfectly absolutely predict the
1:02:03
massive brute force physical forces that dictate our reality.
1:02:07
It really leaves you staring into the void, wondering, does
1:02:10
mathematics simply describe the universe we live in as a
1:02:12
convenient tool, or is the physical universe at its most
1:02:16
fundamental invisible level, actually made of mathematics?
1:02:18
That is the ultimate question. That is the paradox that keeps
1:02:22
theoretical physicists staring at the ceiling at night.
1:02:24
It's certainly going to keep me awake tonight looking at the
1:02:27
ceiling. That concludes our master class
1:02:30
on the invisible forces that move the world.
1:02:33
If you haven't already, give us a like and follow.
1:02:35
And please leave us a comment if you enjoy the podcast.
1:02:38
We love to hear from the folks who make this possible.