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(#118) Introduction to Fluid Mechanics - Lesson 3 · 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
Thanks for having me back. Oh, absolutely.
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So imagine for a second that you just, you know, turn off your
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kitchen faucet and the pipes behind your wall just suddenly
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explode. Like with the force of a hand
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grenade. Which is a terrifying thought,
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honestly. Right.
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Or picture yourself flying a jet so incredibly fast that the
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invisible thin air in front of you just instantaneously turns
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into a solid, impenetrable wall of force.
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A wall that can literally RIP metal apart.
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Yeah, that's the sound barrier for you.
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Exactly. So today we are bringing you
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into a very special master class.
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We are talking directly to you, the student of the physical
0:45
world. And our mission today is
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ambitious, but I promise it is going to be incredibly
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rewarding, going to be a fun one.
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We are going to build a comprehensive mental model of
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fluid dynamics from the absolute ground up.
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We're talking compressible gases at supersonic speeds, then
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moving into the erratic nature of unsteady liquid flows right,
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and finally arriving at the cutting edge computer
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simulations that allow us to actually see and predict all
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this chaos. It really is a profound journey.
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You know, when you look at the history of how we understand
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fluids, which by the way means anything that flows liquids or
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gases, you realize that for a long time we were just kind of
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guessing. Just trial and error.
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Exactly. And we're going to trace the
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path of how we stopped guessing and started manipulating the
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very fabric of pressure and temperature to do things like
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we'll go to the moon. Yeah, you really can't
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appreciate why we need to build multi $1,000,000 supercomputers
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to simulate like a simple splash of water until you first
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understand the sheer mathematical headache of
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supersonic gas. Oh, it's a massive headache.
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I love that framing. So let's start at the very
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beginning. Let's talk about the fundamental
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nature of the stuff we were moving around.
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The fluids themselves. Right.
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When we talk about fluids in engineering, we're usually
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categorizing them based on one major physical property, which
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is how they handle being squeezed.
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I always thought of water and air as fundamentally different
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things, but mechanically they're both just fluids, right?
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That is the perfect starting point.
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Yes, both are fluids, but they sit on opposite sides of a very
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important line, which is compressibility.
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We define fluids broadly as either compressible or
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incompressible. Let's look at liquids first.
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Like water or hydraulic oil, for almost all engineering purposes
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liquids are treated as completely incompressible.
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So if I take like a plastic medical syringe, fill it
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entirely with water, put my thumb over the opening and push
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down on the plunger as hard as I can.
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The plunger doesn't move. It doesn't budget.
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At all, right? I mean, it barely yields on a
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microscopic level, but practically speaking it does not
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move. The density of that water, the
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amount of mass packed into that specific volume, remains
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essentially constant regardless of how much pressure your thumb
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is applying. Because the molecules are
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already packed close closely together.
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Exactly. Gases, however, are a completely
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different story. They are highly compressible
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because the molecules have vast amounts of empty space between
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them. So if you empty that same
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syringe, fill it with air, cap it and push.
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You can easily compress that volume of air down to a fraction
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of its original size. You're forcing those air
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molecules closer together, increasing the density.
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Like squeezing a massive volume of atmospheric air into a thick
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aluminum scuba tank. Precisely.
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But there is a massive caveat to this rule, and it's actually one
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of the most useful shortcuts in all the fluid mechanics.
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I love a good shortcut. What is it?
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Even though gas is highly compressible, we can actually
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treat it mathematically as if it were an incompressible liquid,
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provided the conditions are right.
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Wait, really? How can we just ignore the fact
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that air squishes? It all depends on how fast the
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air is moving relative to the speed of sound.
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If a gas is moving slowly, specifically below Mach 0.3,
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which is roughly 230 mph, the change in its density as it
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flows over an object is remarkably tiny.
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How tiny are we talking? It's actually less than 5%.
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Oh wow. And in the grand scheme of
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designing, say, an air conditioning duct, a 5% change
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in density just doesn't matter enough to complicate the math.
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Exactly. For practical everyday
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calculations, a 5% density change is totally negligible.
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So the air blowing out of the desk fan in your office behaves,
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mathematically speaking, almost exactly like water flowing
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through a wide pipe. That's wild.
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The equations are literally the same.
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They are. It is only when we start pushing
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gas to extreme high velocities or dealing with massive altitude
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and pressure changes that we cross a threshold.
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And suddenly that compressibility cannot be
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ignored. Right.
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The air stops acting like water and starts acting like a highly
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volatile spring. OK, so when we cross that
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threshold, when we start dealing with air moving at hundreds of
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miles an hour, we have to rely on foundational thermodynamics,
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right? We do.
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And the big one, the classic rule that governs all of this,
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is the perfect gas law. Yes, the absolute cornerstone of
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thermodynamics, the relationship where PV equals RT.
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Right. Pressure times volume equals the
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gas constant times temperature. Now I want to make sure I'm
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visualizing this correctly rather than just memorizing a
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formula. What is this law physically
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telling us about the error? It tells us that for a perfect
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gas, these core properties, pressure, volume and
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temperature, are inextricably LinkedIn this delicate balancing
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act. Let's break it down.
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Sure. Pressure is just the physical
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force of gas molecules bouncing off the walls of a container.
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Temperature is a measure of how fast those molecules are
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vibrating and zipping around, and volume is simply the space
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they have to move in. So if I take a sealed box of air
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and I crush it down, decrease in the volume, I am forcing the
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same number of molecules into a tighter space.
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Right. So they're going to hit the
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walls more often. Which means the pressure goes
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up. Exactly.
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But the wall also dictates that if you compress it and hold the
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pressure steady, somehow the temperature has to change.
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OK, how so? If you do work on a gas by
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compressing it quickly, you're adding energy to the system and
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the temperature spikes. This is why you know the bottom
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of a bicycle pump gets hot when you're vigorously feeling a
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tire. Oh, yeah, I've totally felt
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that. That makes total sense.
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Now this brings up something really fascinating about how
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gases actually store heat, Right.
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Yeah. The specific heats.
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Right. Depending on how you hold a gas
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while you heat it up, it behaves differently.
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There's specific heat at a constant volume and specific
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heat at a constant pressure. Why does the container matter so
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much? Let's use a thought experiment.
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Imagine you have a rigid heavy steel box filled with air.
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It is sealed completely shut. OK, so the volume cannot change.
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Exactly. Now you put a blowtorch to the
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bottom of that box. You are adding heat energy
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because the box can't expand. Every single ounce of that heat
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energy goes directly into raising the internal energy of
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the gas. It makes the molecules vibrate
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and bounce around faster and faster.
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Right, so the temperature rises rapidly.
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This is heating at a constant volume.
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We denote that specific heat as C sub V.
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OK, that is straightforward. Heat goes in, temperature goes
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up. But now imagine a different
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container. Imagine a hollow cylinder with a
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movable piston sliding inside it, kind of like the cylinder in
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your car's engine. OK, the gas is trapped under the
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piston. Yep, if you hold a steady weight
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on top of the piston, the pressure inside remains
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constant. Now apply the blowtorch again.
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Well, as the gas heats up, the molecules get energized, they
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hit the piston harder, and because the piston can move, the
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gas expands, it pushes the piston upward.
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Right. The gas is physically doing
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mechanical work to push that heavy piston against gravity.
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It's expending energy to expand. Oh, I see where this is going.
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Because it is spending some of that incoming blowtorch heat to
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do the physical work of lifting the piston, there is less energy
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leftover to simply raise the temperature of the gas itself.
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Oh wow. So if I want to raise the
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temperature of the gas in the cylinder by 10°, I have to apply
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the blowtorch for a much longer time than I did for the rigid
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steel box. You have to add more heat
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Exactly. The specific heat at a constant
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pressure, which is C sub P, is always higher than the specific
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heat at a constant volume C sub V.
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And I saw in the reading that the mathematical difference
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between those two heat values is really elegant.
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It is C sub P -. C sub V is exactly equal to R,
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the gas constant. That difference in heat energy
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is literally a direct measurement of the mechanical
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work the gas performed during its expansion.
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That is an amazing way to visualize the math.
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So we have pressure, volume, temperature and heat all dancing
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together, trading energy back and forth.
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The thermodynamic dance, Yeah. But how does this dance dictate
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something like the speed of sound?
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I mean, when I think of sound, I just think of noise.
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Someone talking, a dog barking. Physically, what is it?
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If we strip away biology and just look at the physics, a
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sound wave is nothing more than a microscopic disturbance in a
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fluid. It is a tiny temporary pressure
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discontinuity. Like a ripple, yeah.
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When you speak, your vocal cords push a layer of air molecules.
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Those molecules bump into their neighbors, transferring their
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momentum, and then bounce back. Those neighbors bump into the
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next layer, and so on. It's like a microscopic domino
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effect. Exactly.
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It is a longitudinal wave of compression propagating through
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the fluid. The actual air molecules don't
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travel from my mouth to your ear, they just wiggle back and
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forth in place. But the wave of pressure
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travels. Right, and the velocity at which
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that specific pressure wave propagates forward is what we
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call the speed of sound. And what governs how fast that
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domino effect can happen? Because I was looking at the
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equations for this and it totally upended my assumptions.
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What did you assume? I figured that the speed of
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sound would depend on how thick or pressurized the air is, like
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denser air faster sound, but it doesn't.
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It does not. The propagation velocity of a
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sound wave, usually denoted as A, is calculated by taking sqrt
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K * r * t. OK, let's break that down.
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K is a specific ratio of heats, R is the gas constant, and T is
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the absolute temperature. Now for a given gas, like the
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air we breathe, K&R are fixed numbers.
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They just don't change. Which means the only variable
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left in the equation is T temperature.
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That is the beautiful, isolating truth of it.
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The speed of sound in a perfect gas is strictly entirely
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proportional to the square root of its absolute temperature.
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Nothing else. Matters nothing else matters.
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Not the ambient pressure, not the density, just the
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temperature. Let me make sure I grasp this.
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If I am in a room at sea level and it's a comfortable 16°C, the
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speed of sound is roughly 340 meters per second.
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Correct. If I take a highly pressurized
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chamber at sea level, pump it up to 10 times atmospheric
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pressure, but keep the temperature exactly at 16°C, the
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speed of sound is exactly the same.
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Exactly the same, because while you increase the pressure, the
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density increase proportionally, and those two effects cancel
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each other out perfectly. The only thing that determines
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how fast those molecules can pass the bump of a sound wave to
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their neighbor is how fast they're already vibrating, and
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temperature is just a measure of that vibration.
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Hotter molecules are zipping around faster, so they transmit
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the wave faster. Colder molecules are sluggish,
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so the wave travels slower. So when a commercial airliner
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climbs up to 30,000 feet where the air pressure is incredibly
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thin, it's not the lack of air pressure that changes the speed
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of sound. No, it's the fact that it is
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freezing cold. Up there.
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In the freezing upper atmosphere, the speed of sound
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drops significantly. This is why engineers don't
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typically measure high speed flight in purely mph.
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Right, they use the Mach number named after Ernst Mock.
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Exactly, the Mach number is incredibly simple.
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It's a ratio m = U / a. You take the objects actual
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velocity U and you divide it by the local speed of sound a in
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the exact environment it is flying through.
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So if the object's velocity is equal to the speed of sound, the
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Mach number is exactly 1. So a plane flying at a constant
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500 mph on the deck right above the ocean might be flying at
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Mach 0.6 because the air is warm and sound travels fast, right?
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But if that exact same plane maintains 500 mph and climbs to
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40,000 feet where it's 30 below 0.
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Because the local speed of sound has dropped due to the cold,
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that same 500 mile per hour speed might now be Mach 0.8.
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The plane is much closer to breaking the sound barrier, even
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though it hasn't actually accelerated.
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OK, this brings us to a phenomenon that is absolutely
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crucial to understand before we can talk about building jet
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engines or rockets. The sound barrier itself.
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Right, What physically happens to the air as an object
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accelerates toward Mach one I want to visualize.
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This Let's use the Doppler effect to visualize it.
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Imagine a stationary object like a speaker resting on the ground,
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emitting A continuous tone. OK, it is radiating sound waves
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outward in perfect concentric spheres, like dropping a Pebble
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in a perfectly still pond. The ripple spread out evenly in
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all directions. I picture the ripples.
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Now put that speaker on a cart and start driving it forward,
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but drive it slower than the speed of sound so the Mach
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number is less than one. The speaker emits a wave, moves
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forward a bit, emits another wave, moves forward and so on.
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Exactly. It's essentially chasing its own
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sound waves. The waves in front of the cart
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get brunched up, squeeze closer together.
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The waves behind the card get stretched out.
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That bunching up is the Doppler effect.
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Because the waves in front are squeezed tighter, the frequency
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increases. Right, which is why when an
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ambulance is driving towards you, the siren sounds higher
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pitched. As it passes you and drives
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away, the waves stretch out and the pitch drops noticeably.
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I have a specific way I've always liked to think about
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this, to ground it in reality. I think of it like a traffic jam
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on a highway. I like this, walk me through it.
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Imagine those sound waves radiating forward from the
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object are like the red brake lights of cars.
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If a car hits its brakes, the red lights come on, the driver
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behind them sees it and hits their brakes, and their lights
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come on. It creates a chain reaction, a
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warning signal traveling backward up the highway, telling
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all the cars, hey, there's an obstacle ahead, slow down.
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Adjust your path. That's a great visual.
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When an airplane is flying subsonic, slower than sound, the
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pressure waves it emits act exactly like those brake lights.
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They travel forward faster than the plane, warning the
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undisturbed air molecules miles ahead.
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They are essentially communicating with the air.
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They are sending a physical signal that says a plane is
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coming. And because the air gets that
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advance note, notice the molecules have time to smoothly
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part and flow gracefully around the shape of the wings and the
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fuselage. The air is prepared, but now
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let's accelerate our cart. What happens when the Mach
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number equals exactly 1? The object is now moving at the
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exact propagation velocity of its own sound.
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Waves. It's keeping pace perfectly, so
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it emits a wave, but it doesn't fall behind.
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It rides right on top of it. Yes, the waves can no longer
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escape out in front. Every single pressure wave the
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object emits forward stacks up perfectly on top of the previous
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one, right at the leading edge of the object.
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It creates A formidable, intense wall of compressed air.
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Returning to the traffic analogy, if you are driving your
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car faster than the brake light warning can physically travel up
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the highway, you outrun your own warning signals.
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You slam into the stop traffic ahead of you without giving them
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any prior notice to move out of the way.
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You violently crash into the undisturbed air, and if you push
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even faster, if your Mach number goes greater than one, you
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completely leave your waves behind.
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You are constantly plunging into completely undisturbed,
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completely unwarned air. The air molecules don't have
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time to gently part, they are violently shoved aside.
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And this abrupt, violent collision is what creates a
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shockwave. Exactly.
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And what happens to all the sound the object is making?
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Well, it can't go forward. Right, because you are out
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running your own waves. All those overlapping pressure
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spheres form a strict geometric shape behind you.
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It creates a cone. We call this the mock cone.
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The mock cone. The tip of the cone is pinned to
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the nose of the aircraft, and the cone flares out behind it.
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All of the sound, all of the pressure disturbances are
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completely confined to the inside volume of this cone.
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That is wild. So if I am standing in a quiet
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desert and a supersonic fighter jet flies directly over my head
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at Mach 2, I literally will not hear it coming.
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You will hear absolute silence as it approaches, and even as it
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is directly overhead, it isn't until the edge of that invisible
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mock cone physically drags across your body on the ground
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that the sound finally reaches you.
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And. Because all those compressed
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sound waves are piled up on the edge of that cone.
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When it sweeps over you, you don't just hear the engines, you
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experience a massive explosive Sonic boom.
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Exactly. And the faster the jet flies,
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the sharper the angle of that cone becomes, right?
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It sweeps back tighter against the jet.
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Right, there's math for that too.
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The angle of the Mach cone, which we call alpha, is defined
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mathematically by sine. Alpha equals 1 / m so at exactly
17:19
Mach 1, the cone is flat. It's a wall.
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But at Mach 2 it sweeps back to a 30° angle.
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And at Mach 5, it's an incredibly tight needle shape.
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So let's pivot here. If pushing an object through the
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air at high speeds creates these massive, violent pressure
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differences in shock waves, how on earth do aerospace engineers
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design things like rockets? That's the $1,000,000 question.
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How do you take hot chaotic gas and intentionally force it past
17:49
that Sonic limit without it just blowing the engine apart?
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That is one of the great triumphs of mechanical
17:54
engineering. To understand it, we have to
17:56
leave the comfort of standard, everyday fluid dynamics behind.
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We have to look at how pipes and nozzles handle supersonic flow,
18:03
because the rules are entirely reversed.
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OK, let's explore this. We are moving into the bizarre,
18:08
paradoxical geometry of supersonic nozzles.
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To set the baseline, we have to imagine a theoretically perfect
18:15
scenario. We assume the flow of gas
18:17
through our nozzle is isentropic.
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What does that mean practically? It means we are assuming a
18:22
reversible adiabatic process. Adiabatic means no heat is being
18:26
transferred in or out of the pipe walls.
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The gas isn't losing heat to the metal.
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And reversible. Reversible means there is 0
18:32
friction. The entropy of the system, the
18:35
fundamental thermodynamic measure of disorder and wasted
18:37
energy remains completely constant.
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The change in entropy is 0 OK. A magical, frictionless, perfect
18:45
pipe. In this perfect system, we have
18:47
an unbreakable rule called the continuity equation.
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It simply states that the mass flow rate must remain constant.
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Matter cannot magically appear or disappear.
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So whatever amount of mass of gas enters the front of the
18:58
pipe, that exact same amount of mass must exit the back of the
19:02
pipe every single second. Exactly.
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The mass flow rate M is calculated by multiplying 3
19:07
things, the density of the gas rho, the velocity it's moving U,
19:11
and the cross-sectional area of the pipe.
19:13
Rho times U * a right? If you change one of those 3
19:16
variables, the other two must automatically adjust to keep the
19:19
total mass flow rate identical. OK, I can visualize this using
19:22
normal everyday subsonic flow, like water coming out of a
19:25
garden hose. Let's use that.
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The water is moving relatively slowly, so the Mach number is
19:29
far less than one. It behaves just how your
19:32
intuition expects. If I have water flowing out of
19:35
the hose and I want it to spray faster to reach the other side
19:37
of the yard, I put my thumb partially over the opening.
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You are decreasing the cross-sectional area of the
19:43
pipe. Right, and because water is
19:45
incompressible, its density doesn't change.
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So to force the exact same amount of mass through a much
19:50
smaller hole every second, the water has no choice but to
19:53
dramatically increase its velocity.
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Perfect for subsonic flow. A decrease in area causes an
20:00
increase in velocity. It is highly intuitive, but here
20:05
is the massive counterintuitive twist.
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If the gas inside that pipe is already moving at supersonic
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speeds. If the Mach number is greater
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than one, the physical rules of the universe flip completely
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upside down in a supersonic flow.
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If you put your thumb over the end of the hose.
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If you decrease the area of the pipe.
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The velocity of the gas actually decreases, it slows down.
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Let me pause you there. That makes 0 logical sense.
20:29
Why would making the hole smaller slowest speeding gas
20:32
down? It all comes back to our first
20:34
topic, extreme compressibility. When a gas is moving that
20:38
incredibly fast, faster than the speed of sound, and you suddenly
20:42
force it through a tighter bottleneck, the gas doesn't
20:44
smoothly speed up. It compresses.
20:46
Exactly. The intense collision with the
20:48
narrowing walls causes the gas to compress violently.
20:52
The molecules just slam into each other.
20:54
So the pressure and the density absolutely skyrocket.
20:57
In fact, the math dictates that in a supersonic flow, the
21:00
density of the gas increases at a much faster rate than the
21:03
physical area of the pipe decreases.
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Oh wait, let me think about the continuity equation.
21:08
Mass flow equals density times velocity times area, right?
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If the area shrinks a little bit, but the density absolutely
21:16
explodes to a huge number, then to keep the overall mass flow
21:20
equation balanced, the velocity variable is mathematically
21:24
forced to drop. You got it.
21:26
The immense spike in density essentially chokes the speed.
21:29
Squeezing A supersonic flow slows it down.
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Which means if shrinking the pipe slows A supersonic gas
21:35
down, to make a supersonic gas go even faster, you have to do
21:39
the exact opposite. That is the astonishing reality.
21:42
To accelerate supersonic flow, to make it go faster, the pipe
21:46
must actually widen. You have to increase the
21:49
cross-sectional area. That is so deeply
21:51
counterintuitive to my brain. If I take my thumb off the
21:54
garden hose, the water slows down to a trickle.
21:57
But for a supersonic gas, making the exit wider is the
22:01
accelerator pedal. It's wild, isn't it?
22:04
How did engineers ever figure out how to harness this?
22:07
It was figured out in the late 19th century by an engineer
22:10
named Gustav de Laval working on steam turbines.
22:13
He invented what we now call the convergent divergent nozzle.
22:17
The nozzle. Right.
22:19
It is arguably one of the most elegant, pivotal pieces of
22:21
engineering in human history. It is the iconic hourglass bell
22:26
shaped noggle. You see on the back of it every
22:28
rocket engine that has ever gone to space.
22:30
OK, I want to walk through exactly how this bell shape
22:32
works from front to back, step by step, because the geometry
22:35
feels like a magistrate. We start at the top right at the
22:38
combustion Chamber of the rocket.
22:39
Inside that chamber you have incredibly hot, highly
22:43
pressurized gas from burning rocket fuel.
22:47
But crucially, inside the massive chamber, the gas is
22:51
moving relatively slowly. It is subsonic.
22:54
OK, so as it flows out of the chamber, it enters the first
22:57
half of the hourglass shape. The pipe starts to narrow.
23:00
This is the convergence section. And because the gas is currently
23:03
subsonic, it behaves normally just like your garden hose.
23:07
The narrowing area causes the gas to accelerate.
23:09
It gets faster and faster and faster as the pipe squeezes
23:13
tighter. Until it hits the absolute
23:15
narrowest point in the middle of the hourglass Engineers call
23:17
this the throat. Right.
23:19
And if the pressure inside the combustion chamber is high
23:22
enough, something incredible happens precisely at that
23:24
microscopic slice of the. Throat.
23:26
An absolute physical limit is reached, yes.
23:29
The gas accelerates to exactly Mach one, the local speed of
23:32
sound. We call this the critical
23:34
velocity. And when a gas hits Mach 1,
23:36
specifically at the narrowest throat of a pipe, it triggers a
23:39
condition engineers call choked flow.
23:42
What does it mean for a pipe to be choked?
23:44
It means the flow has mathematically maxed out.
23:47
You cannot push any more mass through that hole at the throat.
23:51
The pressure of the gas drops to a very specific fraction of the
23:55
original chamber pressure. We call this the critical
23:57
pressure or P star, right? Yes, for a standard gas like
24:00
air, that critical pressure ratio P star divided by P
24:03
naughty is a fixed constant 0.528.
24:08
So the pressure at the throat is always just over half of
24:11
whatever the pressure is in the main chamber.
24:13
Exactly. Once you hit that critical
24:15
pressure ratio and the gas achieves Mach one at the throat,
24:18
the nozzle is completely choked. And here here's the crazy part.
24:22
Once it's choked, you could alter the environment completely
24:25
and the mass flow rate won't change.
24:27
What do you mean? You can take the exit of that
24:28
nozzle and expose it to the vacuum of deep space.
24:31
You could drop the outside pressure to absolute zero and
24:34
the amount of gas moving through that throat will not increase by
24:37
a single solitary gram. I am struggling to understand
24:40
why if I hook a vacuum cleaner up to a pipe, it sucks more air
24:45
through it. Why doesn't the immense infinite
24:48
vacuum of space physically suck the gas out of the rocket
24:51
faster? It goes back to how fluid knows
24:54
what the environment is doing, the information that there is a
24:57
vacuum downstream, that suction is physically transmitted
25:02
backward through the gas in the form of a pressure waste, a
25:06
sound wave. OK, so the vacuum tries to send
25:08
a signal upstream saying hey, it's empty down here, flow
25:11
faster, yes. But that signal can only travel
25:14
through the gas at the local speed of sound.
25:16
Meanwhile, the gas itself is physically flowing downstream
25:19
through the throat at the exact speed of sound.
25:22
Oh my God, it's like a salmon trying to swim upstream against
25:24
a waterfall. If the waterfall is falling at
25:27
20 miles an hour and the salmon can only swim at 20 miles an
25:29
hour, the salmon stays perfectly stationary relative to the
25:33
shore. It is exactly like that.
25:35
The pressure signal trying to report the vacuum conditions can
25:38
never physically swim past the throat.
25:40
It is swept downstream as fast as that tries to propagate
25:43
upstream. That's brilliant.
25:45
Therefore, the combustion chamber is completely blindly
25:48
isolated from whatever is happening past the throat.
25:51
The chamber has no idea the vacuum is there.
25:54
The throat limits the flow absolutely.
25:56
That is a phenomenal piece of physics.
25:58
But here's the problem. A rocket needs its exhaust to
26:02
leave the engine much, much faster than Mach 1 to generate
26:05
enough thrust to overcome gravity.
26:07
If the throat limits it to Mach one, how do we get more speed?
26:11
This is where Gustav de la Valle's genius comes in.
26:14
Immediately after the narrow throat, the nozzle begins to
26:17
gently flare outward. The pipe lightens.
26:20
This is the divergent section, the bottom half of the
26:22
hourglass, the big bell shape you see on a space shuttle.
26:25
And because the gas is already traveling at exactly Mach 1 when
26:28
it enters this widening section, our paradoxical supersonic rule
26:32
kicks. In exactly the gas is now
26:33
supersonic, so the gradually widening area of the bell
26:37
doesn't slow the gas down like it would in a garden hose.
26:39
Instead, the widening area allows the gas to rapidly,
26:42
violently expand. The density plummets, and to
26:46
satisfy the continuity equation, as the density drops, the
26:49
velocity is forced to absolutely skyrocket.
26:51
The gas expands and accelerates to Mach 2, Mach 3, Mach 10,
26:55
shooting out the back and generating massive world shaking
26:59
thrust. When you step back and look at
27:01
the bigger picture, you have to admire the profound elegance of
27:04
this. We are not using complex moving
27:07
mechanical pumps to force gas past the sound barrier.
27:11
There are no moving parts in the nozzle.
27:13
Just geometry. We are simply manipulating the
27:16
stationary physical geometry of a piece of metal to trick a hot
27:19
gas into breaking the laws of intuition.
27:22
We are exploiting the mathematical quirks of density
27:25
and pressure ratios to send humanity into orbit.
27:28
It is absolute design perfection, but nature usually
27:31
doesn't like it when we push things to the absolute limit.
27:34
No it doesn't. What happens if our perfect
27:36
geometry meets the messy real world?
27:38
Like a rocket engine is designed to expand gas to a specific
27:42
pressure, what if the atmospheric pressure outside the
27:44
rocket doesn't perfectly match what the nozzle is expecting?
27:47
That leads to highly violent consequences.
27:49
The gas hits a wall and this is where we have to explore what
27:52
happens when supersonic flows fail and how shock waves are
27:55
formed inside machinery. Let's dive into that.
27:58
We've got this beautifully smooth supersonic acceleration
28:01
happening in the divergent bell of our nozzle.
28:04
But what if the back pressure, the actual pressure of the
28:07
atmosphere pushing back against the exit of the engine, is too
28:09
high? Say the rocket is still on the
28:12
launchpad at sea level where the air is thick.
28:14
If the back pressure from the atmosphere is too high, the gas
28:17
inside the nozzle overextends. It drops its pressure too low
28:21
during its acceleration, and suddenly it finds itself pushing
28:25
against the wall of thick atmospheric air.
28:27
So it hits a literal wall of air.
28:29
Yeah, the supersonic velocity simply cannot be maintained
28:32
against that heavy resistance. So it has to slow down.
28:35
Yes, but fluid dynamics dictates that a supersonic flow cannot
28:40
just gently, smoothly decelerate back down to subsonic speeds.
28:44
It can't just ease off the gas pedal, it has to slam on the
28:47
brakes. Oh boy.
28:48
And the way a fluid slams on the brakes is by spontaneously
28:51
generating a shockwave right inside the divergent bell of the
28:55
nozzle. Wait, it generates a Sonic boom
28:57
inside the metal engine itself? It does.
28:59
The gas is cruising along at Mach 2.
29:02
It hits this invisible threshold of atmospheric resistance and
29:05
instantaneously, across a microscopic distance, it drops
29:09
to subsonic speeds. That sounds incredibly
29:11
destructive. It can be.
29:13
As a rocket climbs through the atmosphere and the outside air
29:15
pressure changes, this internal shockwave will actually migrate
29:19
up and down the inside of the bell.
29:21
Let's clearly define what this shockwave is physically, because
29:24
it's not just a loud noise. Earlier we said a normal sound
29:28
wave is a tiny gentle pressure disturbance.
29:32
What is the shockwave? The shockwave is a macroscopic
29:35
pressure discontinuity. It is a massive, incredibly
29:38
violent step change in the state of the gas.
29:40
Imagine a wall roughly a few micrometers thick.
29:43
OK, picture a wall. On the upstream side of that
29:45
invisible wall, the gas is moving at supersonic speed with
29:48
low pressure and relatively low temperature.
29:50
The instant the gas molecules pass through that micrometer
29:53
thin wall, the velocity plummets to subsonic, the pressure spikes
29:57
violently, and the temperature absolutely skyrockets.
29:59
It's like the gas is running full speed into a brick wall of
30:02
its own making. And does a shockwave travel at
30:05
the speed of sound like a normal noise?
30:08
No, this is another key difference.
30:09
A normal sound wave always travels at exactly the local
30:13
seed of sound, regardless of how loud it is.
30:15
But a shockwave is different. A shockwave actually travels
30:18
faster the larger the pressure rises across it.
30:22
So bigger explosion creates a faster shockwave.
30:25
Exactly. If a bomb goes off, the initial
30:27
shockwave rips outward much, much faster than Mach one, but
30:31
as it expands, it loses energy, the pressure difference drops,
30:35
the wave slows down, and eventually it decays into a
30:38
regular Sonic sound wave that just sounds like a distant boom.
30:41
I was looking at the math that governs this brutal transition.
30:45
There is a set of relationships called the rank and Hugoniot
30:48
equations. Now, I am not going to pretend I
30:49
can solve nonlinear differential equations in my head, but
30:52
conceptually, what do these equations fundamentally prove
30:55
about how shockwaves behave? The rank and Hugoniot equations
30:59
are the definitive map of a shockwave.
31:02
They precisely correlate exactly how pressure, density and
31:05
temperature on the front side of the shock relate to the
31:07
backside, right? But the most profound revelation
31:10
hidden inside these equations comes when you look at the
31:13
entropy calculations. Entropy being the measure of
31:16
disorder or wasted heat in a system.
31:18
Right. The math lays out an absolute
31:20
ironclad rule of physics. A shockwave can only develop
31:24
when the upstream flow of the gas is supersonic.
31:27
It is physically mathematically impossible to generate a
31:30
shockwave in a subsonic flow. Why is it impossible?
31:33
Because the equations show that if a shockwave were to form in a
31:36
subsonic flow, it would cause the total entropy of the system
31:39
to decrease. And according to the second law
31:42
of thermodynamics, in an isolated system entropy can only
31:45
increase or remain constant, it can never go backward.
31:48
The universe forbids order from spontaneously forming from chaos
31:52
without adding outside energy. Therefore, the universe forbids
31:55
subsonic shock waves. That is mind bending.
31:58
The physical geometry of our universe requires you to be
32:02
breaking the sound barrier to create this phenomenon.
32:05
Now these shock waves aren't just flat walls, right?
32:08
They take on different shapes depending on what they are
32:11
crashing into. Yes, the reference is
32:13
distinguished between normal and oblique shock waves.
32:18
Let's talk about that. A normal shockwave is the
32:20
simplest type. It stands at exactly 90°
32:23
perpendicular to the flow direction.
32:25
It's a flat wall. You typically only see normal
32:28
shocks contain inside pipes or hovering right in the throat of
32:31
a jet engine intake. OK, what about an oblique?
32:34
Shock An oblique shockwave happens at an angle.
32:37
Like the wake behind a boat, but in the air.
32:39
That's a great visual. Imagine a supersonic fighter
32:42
jet. The nose of the jet is usually
32:44
shaped like a sharp wedge or a needle.
32:47
As that sharp point slices into the supersonic air, it deflects
32:50
the air at an angle. It can't compress straight back.
32:53
Exactly. The air compresses diagonally,
32:55
creating an oblique shockwave that sweeps back from the nose.
32:58
The shape of the object doing the piercing really matters
33:01
here, doesn't it? If the wedge angle of The Jets
33:03
nose is sharp enough, the shockwave stays perfectly
33:06
attached to the very tip of the metal.
33:08
It slices the air clean. It does.
33:11
But what if you try to push a blunt fat object to supersonic
33:15
speeds? If the physical angle of the
33:17
object is too blunt. If the metal curves away too
33:20
steeply, the air simply cannot physically turn fast enough to
33:24
get out of the way. The mathematical limit is
33:26
exceeded. What happens then?
33:28
When that happens, the shockwave actually detaches from the
33:31
vehicle entirely. It detaches.
33:32
Where does it go? It stands off in front of the
33:34
object. It's called a bow shock, where a
33:36
detached shockwave. It hovers a few inches or
33:39
sometimes feet directly in front of the blunt nose, moving along
33:43
with the craft. Wow.
33:44
This creates a massive, violent zone of highly pressurized,
33:47
extremely hot subsonic air trapped right between the
33:51
shockwave and the front of the vehicle.
33:53
Which creates incredible aerodynamic drag.
33:55
This is why supersonic planes like the Concorde or fighter
33:58
jets look like sharp needles. They want to keep the shockwave
34:01
cleanly attached to minimize drag.
34:04
Yes, but on the other hand, look at the Apollo space capsules
34:07
returning from the moon. They hit the atmosphere at Mach
34:09
25 and their heat Shields were completely flat and blunt.
34:13
And that was a deliberate genius engineering choice.
34:17
Because they wanted to stop. Right, The Apollo capsule wanted
34:20
to slow down. They wanted massive drag.
34:23
But more importantly, by using a blunt shape to intentionally
34:26
create a detached bow shock, that hovering layer of
34:30
incredibly hot compressed air acted like a thermal buffer.
34:34
Oh, that's smart. The most intense heat of re
34:36
entry was concentrated in the shock wave standing a foot in
34:39
front of the capsule rather than directly touching the metal
34:42
surface of the heat shield. They used a wall of invisible
34:45
compressed air to shield themselves from burning up.
34:48
That is brilliant. OK, so we've established that a
34:50
wedge or a concave curve compresses supersonic air and
34:55
creates a shockwave. But what happens if a supersonic
34:58
flow goes around a convex curve? What do you mean?
35:01
Like what if the air is flowing fast over a flat surface and the
35:04
surface suddenly drops away like a ledge?
35:06
Does it create a reverse shockwave?
35:08
It creates something entirely different and quite beautiful,
35:12
called a Prantlemeyer Expansion fan.
35:14
An expansion fan. Remember, a shockwave is a
35:16
sudden violent compression. But when supersonic flow goes
35:20
around a convex corner when it has more room to expand, it
35:24
doesn't violently snap. Instead, it smoothly spreads
35:28
out. How does a fan work?
35:30
From the very point of the corner, the flow generates a fan
35:33
made of countless infinitely thin pressure waves called Mach
35:37
lines radiating outward. OK, so it curves around.
35:40
As the supersonic air turns around the curve and passes
35:43
through this fan, it's smoothly and continuously accelerates,
35:46
expands and drops in pressure. There is no violent shock, no
35:50
sudden wall, no loss of total energy, just a beautiful
35:54
continuous acceleration around the bend.
35:56
Wow. OK.
35:57
So far we've been talking about these flows in a highly
35:59
theoretical way. We assumed earlier that the flow
36:02
in our nozzle was isentropic, perfect, frictionless and
36:06
retaining all its heat. Which is never true in the real
36:08
world. Exactly.
36:09
Real pipes are made of rough metal that causes friction.
36:12
Real jet engines have intense fire inside them.
36:15
How do we model reality? That is where engineers rely on
36:19
2 highly specific models, Fanoflow and really flow.
36:23
Let's start with Fano. Fanoflow is the mathematical
36:26
model for a pipe where the flow is still adiabatic, meaning we
36:31
assume no heat is passing through the walls, but we
36:34
introduced the irreversible reality of friction.
36:37
The gas is physically rubbing against the rough interior of
36:40
the pipe, creating drag. OK, so Fano flow is the math of
36:43
friction. What is Rayleigh Flow?
36:45
Rayleigh flow approaches the problem from the opposite side.
36:48
Rayleigh flow assumes the pipe is perfectly smooth and
36:50
frictionless, but we add intense, deliberate heat
36:53
transfer. Like in an engine.
36:55
Yes, this is the model used to understand what happens inside
36:59
the combustion Chamber of a jet engine, where burning jet fuel
37:02
is constantly dumping massive amounts of thermal energy
37:05
directly into the flowing gas. Friction and heat and the most
37:08
mind blowing thing about these two models is what happens when
37:11
you plot them on a graph. Oh yes, the entropy diagram.
37:14
If you chart both Fanno flow and Rayleigh flow on a graph
37:18
measuring entropy, they both do something completely bizarre.
37:21
They do. Whether you were dealing with
37:23
subsonic flow or supersonic flow, whether you are applying
37:26
friction or applying heat, both the Fanno line and the Rayleigh
37:30
line naturally inevitably curve and converge toward exactly Mach
37:35
1. The speed of sound, yes.
37:38
If you have a subsonic flow in a pipe and you add friction, the
37:41
friction causes the pressure to drop and the flow to accelerate
37:44
toward Mach. One, and if it's supersonic.
37:46
If the flow is supersonic, friction causes it to decelerate
37:49
toward Mach 1. The same is true for heat.
37:52
In Rayleigh flow, adding heat drives the gas toward the speed
37:55
of sound. Mach 1 represents the exact
37:58
state of maximum entropy for that specific pipe.
38:01
It is like the thermodynamic destiny of the system.
38:04
No matter what you do, the universe is trying to drive the
38:06
flow toward the speed of sound, and once it hits that bottleneck
38:09
of maximum entropy, it physically cannot go any further
38:12
without changing the pipe itself.
38:14
It is a profound bottleneck in nature.
38:16
The physics simply converge. I want to pivot drastically here
38:20
because this raises a major question for me.
38:22
We have spent this entire time talking about gases.
38:25
Compressible, squishy, high speed gases.
38:27
But what happens when we look at liquids?
38:29
We established the very beginning that liquids, like
38:32
Water Act completely incompressible.
38:35
They don't stretch, they don't squeeze.
38:36
They are stubborn. So what happens if a heavy,
38:39
stubborn, incompressible liquid suddenly undergoes an extreme
38:42
pressure change? When you subject an
38:45
incompressible liquid to extreme dynamic forces, the liquid stops
38:49
acting so polite it becomes highly destructive.
38:53
This brings us to the chaotic, often violent world of unsteady
38:57
flow and a phenomenon known as the water hammer.
39:00
Let's get into it. We are shifting our perspective
39:03
from steady state flow, which is smooth, constant predictable
39:06
movement, to unsteady flow where velocities and pressures are
39:10
rapidly changing over time. Where do engineers actually deal
39:13
with this in the real world? Unsteady flow is everywhere.
39:16
It is the core of any system that transmits power or provides
39:19
automatic precise control using liquids hydraulics.
39:23
Like construction equipment. The heavy machinery on a
39:25
construction site lifting tons of dirt, the flight control
39:28
surfaces, moving the flaps on a commercial airliner, even the
39:31
hydraulic brakes in your car. All of these rely on unsteady
39:34
fluid dynamics. The foundation of this usually
39:36
starts with a very simple, almost peaceful thought
39:39
experiment. Imagine a clear plastic tube
39:43
bent into AU shape and partially filled with liquid.
39:47
Right, a simple U2 manometer. The water rests evenly in both
39:51
vertical columns, perfectly balanced by gravity.
39:54
Now imagine you blow gently into one side of the tube, pushing
39:58
the liquid column down on the left, which forces the liquid on
40:01
the right side to rise. And then you let go.
40:03
You stop blowing. What happens?
40:05
The liquid column immediately falls back down, but its
40:08
momentum carries it past the center resting point.
40:11
It pushes the left side back up. Exactly.
40:13
The liquid will oscillate back and forth, sloshing from side to
40:17
side, exactly like a pendulum swinging on a grandfather clock.
40:20
And what's amazing is that you can calculate the exact timing
40:23
of that slosh using the same physics as a pendulum.
40:27
The cycle time t = 2 * π * sqrt l / 2 * G It is pure harmonic
40:35
motion. It is, however, unlike a perfect
40:37
pendulum in a vacuum, the water doesn't slosh forever.
40:40
The liquid is rubbing against the walls of the plastic tube.
40:43
Right friction. This laminar frictional
40:45
resistance steals a tiny bit of kinetic energy with every swing.
40:49
It acts as a dampener. Eventually the oscillations get
40:52
smaller and smaller until the fluid settles perfectly back to
40:55
rest. OK, so that's a gentle
40:57
oscillation, but let's scale that up to industrial sizes.
40:59
Let's look at what happens in a straight pipe.
41:01
Imagine a massively long straight pipe connected to a
41:05
huge water reservoir. There's a closed valve at the
41:08
end. When you suddenly crank that
41:10
valve open, the water doesn't just instantly blast out at its
41:14
absolute maximum speed. You know it takes time.
41:17
In the very first fraction of a second after opening the valve,
41:21
the entire head of pressure, the massive weight of the water
41:25
sitting in the reservoir, is dedicated entirely to overcoming
41:28
inertia. Overcoming inertia.
41:30
It is applying force to accelerate a very heavy
41:33
stationary mass of water residing in the pipe.
41:35
So it starts accelerating, yes. But as the water speeds up, the
41:39
dynamic changes. Fluid friction against the inner
41:41
walls of the pipe kicks in. The faster the water goes, the
41:44
harder the friction fights back, stealing energy from the driving
41:47
pressure. It creates this smooth
41:50
mathematical curve on a graph, a logarithmic curve.
41:53
Yes, extremely fast acceleration at first.
41:57
When the water is slow and friction is low, then the
41:59
acceleration tapers off gently. It curves until the driving
42:03
pressure from the reservoir perfectly balances the friction
42:06
loss of the pipe and the water achieves a final steady
42:09
velocity. OK, so opening a valve is
42:12
mathematically quite elegant. It is a smooth transition to a
42:15
steady state. Closing a valve, however, is an
42:18
absolute nightmare. The water hammer.
42:20
The water hammer. I think almost everyone has
42:23
experienced this in an old house or an apartment building.
42:26
You wash your hands, you turn the tap off quickly, and
42:28
suddenly the pipes behind the drywall violently bang like
42:31
someone hit them with a literal hammer.
42:33
Yeah. It's startling.
42:34
I want to break down the exact mechanics of a water hammer.
42:37
Let's go millisecond by millisecond.
42:39
You've got water flowing steadily down a pipe and someone
42:42
violently slams a lever valve shut.
42:45
What exactly happens in that first instant right at the valve
42:48
face? In that very first millisecond,
42:51
the physical momentum of the fluid right at the valve face
42:53
drops to absolute zero. It hits a solid wall.
42:58
But water is heavy, it has mass, and it was moving with velocity.
43:03
It possessed kinetic energy. That energy cannot just vanish.
43:06
And because water is essentially incompressible, it cannot squish
43:10
like a pillow to absorb the blow.
43:12
So where does the energy go? It converts instantaneously into
43:16
a massive localized pressure spike right against the metal
43:19
face of the closed valve. Exactly.
43:22
The formula for this spike is terrifyingly simple and linear.
43:26
DP equals rho times V * A. So the change in pressure equals
43:31
the density of the fluid multiplied by its original
43:34
flowing velocity, multiplied by the speed of sound in that
43:37
specific fluid. And here is where the chaos
43:39
begins. That pressure spike doesn't just
43:41
sit there. The water directly behind the
43:43
valve has stopped, but the water 100 feet back up the pipe has no
43:48
idea the valve is closed yet. Because the signal hasn't
43:50
reached. It it is still rushing forward
43:52
at full speed. Yeah.
43:53
So that trailing water violently crashes into the stopped water.
43:56
It compresses the water by a microscopic fraction of a
43:59
millimeter and instantly stops itself.
44:01
It's a microscopic high speed pileup.
44:03
Car after car crashing into the back of the jamb.
44:06
Exactly, and as each layer of water crashes and stops, the
44:10
high pressure zone expands backward.
44:12
This creates a high pressure shockwave that travels
44:14
physically upstream, moving backward against the original
44:17
flow direction at the speed of sound through water.
44:19
OK, so I am picturing this invisible wall of intense
44:23
destructive pressure traveling at thousands of miles an hour
44:27
all the way back up the pipe. What happens when it finally
44:30
reaches the open reservoir at the other end?
44:33
When the shockwave reaches the open tank, the highly
44:36
pressurized water inside the pipe suddenly finds itself open
44:39
to the much lower normal resting pressure of the reservoir.
44:42
Nature hates an imbalance. Oh man, so the highly
44:45
pressurized pipe literally vomits the water backward into
44:48
the tank? The flow actually reverses
44:50
direction. It reverses, but this sudden
44:53
release of water drops the pressure inside the pipe
44:55
significantly below normal. Now you have a low pressure
44:58
wave, and that low pressure wave travels all the way back
45:01
downstream, back toward the closed valve.
45:03
And when that low pressure wave finally hits the closed valve,
45:06
the water is trying to pull away from the valve face.
45:09
It creates a suction. A powerful suction.
45:11
That suction then reaches back to the tank, pulls fresh water
45:14
from the tank back into the pipe, and the entire high
45:17
pressure slam cycle starts all over again.
45:20
It bounces high pressure traveling upstream, low pressure
45:23
traveling downstream, over and over again, bouncing back and
45:27
forth. That loud, violent banging noise
45:30
in your walls is the physical shockwave ricocheting off the
45:32
ends of the plumbing, vibrating the metal until friction finally
45:36
dampens the energy away. Now, you mentioned earlier that
45:38
the speed of this destructive wave, the Sonic velocity,
45:42
depends on the fluid. But we have to factor in the
45:45
real world here. Water isn't flowing through a
45:48
magical rigid vacuum, it's flowing inside a real, physical
45:52
pipe. Right, a pipe made of steel or
45:54
copper or concrete. And under that immense violent
45:57
pressure spike of a water hammer, even solid steel is not
46:00
perfectly rigid. The metal actually stretches.
46:02
It does. When that shockwave hits, the
46:04
massive internal pressure causes the pipe to expand radially.
46:07
It literally bulges outward slightly as the wave passes.
46:11
Right. And because the walls of the
46:13
pipe yield and give way under the pressure, the volume of the
46:16
pipe increases slightly. This makes the incompressible
46:19
fluid appear to be compressible. It suddenly has extra room to
46:23
give. So the environment itself
46:25
changes the fundamental math of the fluid.
46:27
We can't just use the standard stiffness or bulk modulus of
46:30
water anymore. We have to calculate a modified
46:34
bulk modulus K prime. Right AK prime.
46:37
That actually includes the elasticity, the Young's modulus,
46:39
of the solid pipe material itself.
46:41
We have to treat the water in the pipe as a single combined
46:44
flexible system. Correct.
46:46
And because the pipe absorbs a significant portion of the shock
46:49
energy by physically expanding, the velocity of the shockwave
46:52
actually drops. If you have an ordinary steel or
46:55
concrete water pipe, the wave velocity drops from the pure
46:58
speed of sound and water down to roughly 600 to 1200 meters per
47:02
second. Which is still incredibly fast,
47:04
over 2000 miles an hour. So how do civil engineers
47:07
prevent this phenomenon from just destroying municipal water
47:10
grids and industrial plants? The simplest, most effective way
47:13
is to just not slam the door. You use slow closing valves.
47:18
Engineers use a complex formula called Olivais equation to
47:22
calculate the exact maximum pressure spike based on the
47:25
closure time of the valve. The goal is to stretch the
47:28
physical closure time out to be longer than the time it takes
47:31
the shockwave to make a full round trip up to the reservoir
47:34
and back down the pipe. Exactly.
47:36
So if you close it slowly, the low pressure reflection wave
47:40
returning from the reservoir meets the high pressure wave
47:42
still being generated by the closing valve and they
47:45
destructively interfere. Yes, the waves essentially
47:48
cancel each other out, bleeding off the kinetic energy safely
47:51
and gradually, rather than in one catastrophic spike.
47:55
I want to make sure we understand the real world stakes
47:58
of this. This isn't just about avoiding
47:59
an annoying banging noise in your apartment walls.
48:02
Think about a massive industrial hydropower station built inside
48:06
a mountain. OK, picturing it.
48:08
The stakes are absolute life and death.
48:10
A large hydropower station uses a giant, heavily reinforced pipe
48:15
called a penstock. This pipe funnels thousands of
48:18
tons of water down a massive vertical drop directly into a
48:22
spinning turbine. Wow, OK.
48:24
Now imagine a worst case scenario.
48:26
The electrical grid suddenly drops its load.
48:29
Maybe a substation blows out. The turbine suddenly has no
48:33
resistance and it starts over speeding.
48:35
If it spins too fast, it will tear itself apart.
48:38
So the safety systems have to immediately choke off the water
48:41
supply to save the turbine. But if those massive multi ton
48:45
penstock valves at the bottom of the mountain are commanded to
48:47
close too fast, the resulting water hammer wouldn't just make
48:51
a noise. It would be a bomb.
48:52
The kinetic energy of thousands of tons of falling water
48:55
instantly converting to pressure would easily exceed the yield
48:58
strength of the steel. The pressure spike would
49:00
literally explode the Penn stock right out of the side of the
49:03
mountain, flooding the facility and causing a catastrophic
49:06
failure. That is genuinely terrifying.
49:09
The invisible forces hidden in moving water are immense.
49:13
OK, I want to take a step back and look at where we are.
49:15
We have equations for perfect gases in perfect nozzles.
49:19
We have equations for single shock waves traveling up
49:22
straight pipes. Nimple shapes.
49:24
But real world fluid dynamics is almost never a straight pipe.
49:27
It's turbulent, chaotic air tumbling over a swept jet wing,
49:32
or chaotic water violently spinning in three dimensions
49:35
through a curved turbine blade. It's messy.
49:38
Trying to calculate valve leakage, irregular pipe friction
49:42
and three-dimensional interacting shock waves by hand.
49:45
It's just impossible, right? It is fundamentally
49:48
mathematically impossible able to solve by hand using the
49:51
governing physics equations, which are known as the Navier
49:53
Stokes equations. Why are they unsolvable?
49:55
What makes them so uniquely difficult?
49:58
Because the Navier Stokes equations are deeply non linear,
50:00
in almost all practical intermediate flow regimes you
50:03
have two competing forces. You have the inertia term, which
50:06
is the forward momentum of the fluid, it's desire to keep
50:10
moving in chaotic directions. OK, inertia and you have the
50:13
viscosity term, which is the internal friction of the fluid,
50:16
the stickiness that tries to smooth the flow out.
50:19
Because inertia and viscosity fight each other in incredibly
50:22
complex non linear ways, a change in one intimately changes
50:27
the other in unpredictable loops.
50:29
There are no analytical pencil and paper solutions for complex
50:32
shapes. Which perfectly sets up the
50:34
final act of our master class, teaching computers to see the
50:37
flow. Because we could not solve the
50:40
math by hand, humanity had to invent an entirely new field of
50:43
engineering. We had to invent computational
50:45
fluid dynamics, or CFD. Since we cannot calculate a
50:49
single elegant equation that solves the entire volume of
50:52
fluid holistically at once, we had to find a workaround.
50:55
And what's the? Workaround The solution is
50:56
something called the finite difference method.
50:59
Instead of looking at the hole, we discretize the fluid.
51:01
We break the continuous flowing mass of air or water down into a
51:05
highly structured mathematical grid or a mesh made of millions
51:08
of tiny individual boxes. I love comparing this to the
51:11
evolution of photography. Back in the day, theoretical
51:15
fluid dynamics was like an analog photograph.
51:18
You were trying to capture the entire continuous image in one
51:22
single holistic chemical exposure, one giant math
51:26
equation. That is a highly accurate
51:28
analogy. But CFD is like a modern digital
51:31
camera sensor. Instead of looking at the whole
51:33
picture, CFD breaks the universe down into millions of tiny
51:37
independent pixels or cells. And the magic happens inside
51:41
each individual pixel. To figure out what the fluid is
51:44
doing inside one specific box, we use a mathematical trick
51:47
called Taylor series expansion. Explain that simply, if
51:50
possible. Essentially, Taylor series
51:52
expansion allows us to take those impossible curvy non
51:55
linear partial differential equations and approximate them
51:58
into simpler straight line algebraic equations that a
52:01
computer's processor can actually solve.
52:04
We calculate the basic velocity and pressure in one single cell,
52:07
and then we pass that result across the boundary to the
52:10
neighboring cell. The computer solves every cell
52:13
and then refreshes the entire frame thousands of times a
52:16
second to simulate the passage of time.
52:18
And there are different mathematical strategies for how
52:20
you pass that information between the cells, right?
52:23
The literature mentions forward difference, backward difference
52:26
and central difference. Yes, let's look at central
52:29
difference. If a computer wants to know the
52:31
rate of change inside a target cell, Central difference looks
52:34
at the cell immediately behind it, in the cell immediately in
52:37
front of it, and averages them. It looks at both neighbors.
52:40
That seems like a smart way to do it.
52:42
It provides what we call 2nd order accuracy, meaning it is
52:46
highly precise and captures fine details.
52:49
However, it has a fatal flaw, which is if you're simulating
52:52
flows with very high Reynolds numbers, meaning very fast,
52:56
highly turbulent, chaotic flow, using central difference can
53:00
actually cause the entire simulation to become numerically
53:03
unstable. Unstable how?
53:05
The tiny rounding errors between the cells amplify, bounce back
53:08
and forth, and the math literally explodes to Infinity.
53:12
The simulation crashes. So how do software engineers fix
53:15
a crashing simulation? They switched to a technique
53:18
called the upwind difference method.
53:20
It relies on a very simple physical truth in a highly fast
53:24
rushing flow. Almost all the relevant physical
53:27
information is coming from upstream, blowing down toward
53:31
you. What happened is downstream
53:32
doesn't really affect you. Exactly so.
53:34
The mathematical formula intentionally weights the
53:37
upstream cells much heavier than the downstream cells.
53:41
By favoring the upwind data, it suppresses those chaotic
53:44
mathematical oscillations and keeps the simulation perfectly
53:47
stable even at extreme speeds. OK, so that is how we build the
53:51
grid and keep it stable. Now let's look at how we
53:54
actually simulate physical events inside that digital
53:57
space, starting with incompressible fluids like
54:00
water. There is a historic landmark
54:03
breakthrough in CFD called the MAS method or Marker and Sell
54:07
method. Why was it so revolutionary?
54:09
The Mac method was developed specifically to solve a problem
54:12
that plagued early computers simulating free surfaces.
54:15
Imagine trying to simulate water sloshing in 1/2 empty tank or a
54:19
wave crashing on a beach. That sounds incredibly
54:21
complicated. It is the boundary where the
54:24
water meets the empty air is incredibly complex to track
54:27
mathematically. In the Mac method, the computer
54:31
generates the background mesh, but then it drops 10s of
54:34
thousands of weightless marker particles into the simulated
54:37
fluid. What do you mean by weightless?
54:39
They don't have mass, they don't affect the physics equations,
54:42
they're essentially digital dye. They're just dropped into the
54:45
mathematical current, and they simply track exactly where the
54:49
fluid physically exists within the grid as the velocity and
54:52
pressure equations push them around over time.
54:54
OK so if a cell has markers in it, it's water.
54:56
If it's empty it's air, right? I remember seeing the visual
54:59
they used to prove this method worked and it is iconic.
55:02
They use used the Mac method to simulate a single liquid drop
55:06
falling into a shallow puddle. We have all seen that famous
55:10
high speed macro photograph where a milk drop hits a surface
55:14
and the splash violently forms a perfect symmetrical little crown
55:17
of liquid spikes shooting upward.
55:19
Yes, the coronet splash. It is a highly complex chaotic 3
55:23
dimensional physical event. The MSC method, using a
55:26
primitive computer from decades ago, was able to perfectly
55:30
simulate the physics of that crown forming spike by spike,
55:33
droplet by droplet, purely by tracking those weightless marker
55:37
particles through a grid. It proved that computers could
55:40
replicate the chaotic beauty of fluid nature.
55:42
It was a triumph. But as impressive as the MSC
55:46
method is, what is truly fascinating is how much more
55:49
difficult the math becomes when we switch back to our first
55:52
topic, Compressible fluids. Supersonic air.
55:56
You can't just track velocity and pressure with digital die
55:59
anymore. This is the shockwaves.
56:00
Exactly. Shock waves are violent
56:02
discontinuities. To simulate them, the computer
56:05
is forced to solve the equation of state and the energy equation
56:08
simultaneously with mass and momentum.
56:11
The temperature and density are wildly, violently fluctuating
56:14
from cell to cell. So how do supercomputers compute
56:17
a shockwave without crashing? We use what are called time
56:19
marching methods. A prominent example is the
56:21
McCormack method. It is incredibly useful because
56:25
it uses conservation forms for mass, momentum, and energy.
56:29
But to save massive amounts of computational power, it
56:32
temporarily ignores the discussity of the air.
56:35
It treats the air as frictionless.
56:37
So it's a simplification. Yes, but this method is
56:40
brilliant and highly efficient for predicting macro level
56:43
aerodynamic events. Like calculating exactly where a
56:47
detached bow shockwave will form a foot in front of the nose cone
56:50
of a supersonic rocket. It paints the big picture.
56:53
But what if the big picture isn't enough?
56:55
What if you need to calculate the viscosity?
56:58
What if the microscopic friction of the air rubbing against the
57:01
metal is the whole point of the engineering problem?
57:03
When you need absolute granular reality, you have to step up to
57:07
the heavy artillery. You use something like the IAF
57:10
method, the Implicit Approximate Factorization method, which is
57:13
also commonly known as the Beam warming method.
57:16
OK, the beam warming method. This is heavy duty mathematics.
57:18
It does not take shortcuts. It solves the full complete
57:22
viscous compressible Naver Stokes equations.
57:25
Where would an engineer ever need that much punishing
57:27
mathematical detail? Look inside a modern commercial
57:30
jet engine. A turbofan.
57:33
Imagine the massive titanium fan blades at the front of that
57:36
engine. They are rotating at thousands
57:38
of revolutions per minute. At those rotational speeds, the
57:41
tips of the blades are actually breaking the sound barrier,
57:44
while the base of the blades near the hub are moving
57:47
subsonic. So the air flowing through the
57:49
fan is this nightmare chaotic mixture of subsonic and
57:53
supersonic flow at the exact same time.
57:56
Transonic flow. Precisely, and in that transonic
57:59
environment, microscopic shock waves are constantly forming,
58:03
detaching and interacting with the turbulent friction boundary
58:07
layer right on the curved surface of the metal plates.
58:10
That is intensely complicated. It is the beam warming method
58:14
running on parallel supercomputers can physically
58:16
map that incredibly complex density distribution.
58:20
It allows engineers to literally see the friction in the
58:22
microscopic shock waves rubbing against the titanium, allowing
58:25
them to tweak the curve of the blade by a fraction of a
58:28
millimeter to save millions of dollars in jet fuel.
58:30
It is almost God like the ability to take the invisible,
58:34
chaotic, destructive forces of nature, forces that would RIP a
58:38
building apart and map them out and perfect digital color on a
58:41
monitor. It's an incredible time to be an
58:43
engineer. OK, as we wrap up this master
58:45
class deep dive, I want you to give us something to chew on.
58:48
We have covered a massive amount of ground.
58:51
What is the ultimate destination of all this physics and computer
58:55
science? I think you just have to look
58:57
back at the incredible journey we just took.
58:59
We started with scientists in the 1800s playing with sealed
59:02
boxes, realizing that pressure times volume equals temperature,
59:07
just the basic rudimentary relationship.
59:10
And we traced that intellectual lineage through supersonic
59:13
rockets, through exploding water pipes, all the way to
59:16
supercomputers calculating 3 dimensional friction on a
59:19
rotating jet turbine. Here's the provocative thought
59:22
for the future. As computing power marches
59:25
relentlessly toward quantum levels, the mathematical pixels
59:28
in our CFD meshes will become infinitely small.
59:31
We might soon reach a point where we can run fluid dynamic
59:35
models so perfectly precise, so indistinguishable from reality,
59:40
that physical wind tunnels and metal prototypes become entirely
59:44
historical relics. You mean never building a test
59:46
model again? Imagine an aerospace engineer
59:49
designing A next generation supersonic aircraft.
59:52
They can test every single physical parameter, stress the
59:55
metal against virtual shock waves, and fly the plane for
59:58
thousands of simulated hours in every weather condition
1:00:01
imaginable, entirely inside a silicon chip, before a single
1:00:04
piece of actual metal is ever cut on a factory floor.
1:00:08
We will transition from reacting to the physical world to
1:00:11
generating it perfectly in code. That is profound, designing and
1:00:15
perfecting reality in the digital realm before it ever
1:00:17
physically exists. Well designed junkies.
1:00:19
That brings us to the end of our master class on the invisible
1:00:21
forces around us. If you haven't already, give us
1:00:23
a like and follow. And please leave us a comment if
1:00:25
you enjoy the podcast. We love to hear from the folks
1:00:27
who make this possible.