Latest / Mechanical Engineering Made Simple / (#146) Lesson 4: Scale Models and the Supersonic Paradox

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(#146) Lesson 4: Scale Models and the Supersonic Paradox · Mechanical Engineering Made Simple

• 0:00
  Welcome back to mechanical engineering Made Simple legends.
• 0:03
  Lesson 4. Fluid mechanics.
• 0:04
  I'm your host. Let's uncover what the textbooks
• 0:07
  only hint at in reality. Fluids reveal the universe's
• 0:11
  quiet rules. Bernoulli's principle doesn't
• 0:13
  just keep planes aloft, it shows us how pressure and speed dance
• 0:16
  together in every breath we take.
• 0:18
  Viscosity teaches patients too thick and nothing moves.
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  Too thin and everything slips away.
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  Reynolds number marks the boundary between order and
• 0:26
  beautiful chaos. Misjudge it and Water Hammer
• 0:29
  reminds you that even the gentlest flow can strike like
• 0:32
  Thunder when ignored. School gives you the formulas.
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  Life shows you why they matter. In the shower that suddenly
• 0:38
  feels alive, and blood that carries life without freezing,
• 0:41
  and honey that moves at its own deliberate pace.
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  The difference between memorizing and understanding is
• 0:46
  the difference between surviving and thriving.
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  Stay with me. We'll turn the invisible forces
• 0:50
  into clear insight before life teaches the lesson the expensive
• 0:54
  way. That is, that is really the
• 0:57
  perfect framing for today's mission because we are, you
• 0:59
  know, we're really taking a hammer to the foundational text
• 1:01
  today. Yeah, exactly.
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  We're looking at the big stuff. Dimensional analysis.
• 1:06
  Flow measurement. The the seemingly paradoxical
• 1:11
  world of ideal fluids. Oh, the ideal fluids, right?
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  Yeah, and of course the absolute mind Bender that is compressible
• 1:17
  flow. But you know, we're not here to
• 1:21
  just read equations of a chalkboard.
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  No, nobody wants that. Exactly.
• 1:25
  The goal of this deep dive is to decode them.
• 1:28
  We want to explain not just what these concepts are, but but why
• 1:33
  they physically dictate whether your design flies or floats or
• 1:38
  just, you know, fails catastrophically.
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  Right, because if you spend like any time on engineering forums,
• 1:43
  you see the exact same pain points surfacing daily.
• 1:47
  Oh yeah. Constantly.
• 1:48
  You see these young engineers and they're just drowning in
• 1:50
  Greek letters, right? And partial derivatives.
• 1:52
  And they're desperately asking for the real world link.
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  Yeah, they know the formula for drag, but they don't know how to
• 1:57
  trust the data coming out of a wind tunnel.
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  Exactly. So to you, listening right now,
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  today is about bridging that specific gap.
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  We are translating abstract textbook math into basically
• 2:10
  survival skills for the real world.
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  We're going to answer the most popular questions engineers ask
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  online about fluid dynamics. And, you know, it's a
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  translation that really matters. Yeah, a misunderstanding of
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  these principles. It doesn't just mean a failed
• 2:24
  midterm, right in the industry. I mean, it means multi
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  $1,000,000 testing errors. It means malfunctioning jet
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  engines. Structural collapse.
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  Yeah, catastrophic structural collapses.
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  The stakes are incredibly high. O, let's ground this
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  immediately. Say you are tasked with building
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  a colossal hydroelectric dam. Or, you know, a nextgeneration
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  passenger jet. OK, Yep, you can't just build
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  the fullscale thing, turn it on and cross your fingers right?
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  Yeah. I mean, you could try, but your
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  investors wouldn't be very happy.
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  Right. You have to know it works before
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  you pour a billion dollars into manufacturing.
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  So you build a scale model. But here's the massive trap that
• 2:59
  so many people fall into. How do you actually scale the
• 3:02
  physics? Yeah, that's the hard part.
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  Like you can scale the geometry down, make a tiny plastic plane,
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  but you can't, you know, scale down the air.
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  And that dilemma right there is the entire foundation of
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  dimensional analysis and the law of similarity.
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  OK, so let's unpack that. Well, nature doesn't care about
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  our human measurement systems, right?
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  Doesn't care if you measure your pipe in inches or meters or
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  nautical miles. Right, nature just does its.
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  Thing exactly. Nature operates entirely on
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  fundamental physical dimensions. So for mechanical systems, those
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  bedrock dimensions are mass, length, and time.
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  OK, so mass, length, and time, let's let's apply this.
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  If I'm an aerodynamicist and I'm looking at the drag on a sphere,
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  I have this massive list of variables.
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  You do? Yeah, I have the drag force
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  itself. I have the diameter of the
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  sphere, I have the velocity of the wind hitting it, the density
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  of the air and the viscosity of the air.
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  That's what 5 different variables I have to juggle. 5
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  variables and juggling 5 variables experimentally is an
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  absolute nightmare. Well, think about it.
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  If you want to graph how drag changes, you'd have to run
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  thousands of wind tunnel tests. You had to change 1 variable at
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  a time while holding the other four constant.
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  Oh wow, yeah. You would be testing forever.
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  It's just not feasible. And this is where the Buckingham
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  Π theorem comes in as an absolute lifesaver.
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  OK, the Buckingham Π theorem. I remember this from school.
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  Most engineers do. And it always felt, to me at
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  least, like a mathematical parlor trick.
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  A lot of people feel that way. Right, because you just like,
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  mash variables together. You multiply them, you divide
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  them until all the units magically cancel out and you are
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  left with a dimensionless number.
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  Yep, that's the process. But how does a racing units
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  actually keep an engineer from making $1,000,000 mistake?
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  It feels like you're losing information.
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  That skepticism is incredibly common, and it's a fair push
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  back. It really does feel like you are
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  deleting information by deleting the units, but you're actually
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  revealing the hidden ratios that govern the universe.
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  Let's look at your sphere example.
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  You have your five variables. Let's call that number of
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  variables N. So n = 5.
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  OK, And earlier we said the fundamental dimensions making up
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  those variables are mass, length, and time.
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  Let's call the number of basic dimensions K.
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  So K = 3. Perfect.
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  So the Buckingham Π theorem simply states that the entire
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  complex interaction can be completely described by a
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  smaller number of dimensionless groups, specifically n -, K
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  groups. So 5 -, 3 is 2.
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  Exactly 2. That means those thousands of
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  wind tunnel tests you thought you had to run?
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  They can be collapsed. Wow.
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  The entire physical reality of a sphere experiencing drag is
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  governed by just two dimensionless numbers.
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  When you do the algebra grouping, those five variables
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  so they're combined dimensions cancel out.
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  You get the drag coefficient and the Reynolds number.
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  The legendary Reynolds number density times velocity times
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  length divided by viscosity. That is the one.
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  It is arguably the most important of what I call the
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  BIG4 non dimensional numbers in fluid mechanics.
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  The big four. I like that.
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  Yeah, and you have to remember, dimensionless numbers are always
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  ratios of competing forces. The Reynolds number is the ratio
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  of inertia forces to viscous forces.
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  Inertia to viscosity. I always try to visualize this
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  competing forces. Like inertia is the fluids
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  momentum, right? Yes, exactly.
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  The mass of the fluid wanting to just keep barreling forward in a
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  straight line like a freight train.
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  And viscosity is the fluid's internal friction, it's
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  thickness, trying to pull everything together and slow it
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  down like a like a web. That is a brilliant way to
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  visualize it. I love that.
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  So if you have a very high Reynolds number, inertia is
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  dominating the web. The freight train is winning.
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  Exactly. The flow is fast, chaotic and
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  turbulent. Think of water violently rushing
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  out of a fire hydrant. The inertia is so high literally
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  tears the flow apart into swirling eddies.
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  OK yeah, but if you have a low Reynolds number, viscosity is
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  winning. The flow is thick, slow,
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  orderly, laminar flow. Laminar, right?
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  So think of cold honey pouring off a spoon.
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  Perfect example. The internal friction keeps all
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  the fluid particles moving in perfectly parallel smooth lines.
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  OK, so the Reynolds number dictates everything when you are
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  dealing with fluid completely enclosed by boundaries, like
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  pumping crude oil through a pipeline or when an object is
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  totally submerged like a submarine.
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  But you mentioned a BIG4. What happens when we aren't
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  totally submerged? What if we are like a ship on
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  the surface of the ocean? Then you enter the domain of the
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  Froude number. The Froude number.
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  Yeah, this is the ratio of inertia forces to gravity
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  forces. Gravity is interesting here
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  because in a completely full enclosed pipe, gravity doesn't
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  really dictate the flow pattern the same way, right?
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  Not really, no. So why does it matter so much on
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  the surface? Because of the free surface,
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  anytime you have a fluid interacting with the open air,
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  gravity can act on that boundary.
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  Oh, I see. Think about it, when a ship
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  pushes through the ocean, it's hull physically displaces water,
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  lifting it up against the force of gravity, and that creates a
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  wave. Right waves.
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  Lifting thousands of tons of water requires a massive amount
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  of energy, and that energy is robbed directly from the ship's
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  engines. We call this wave making
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  resistance. So if I'm a naval architect and
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  I'm designing a new destroyer and I test a tiny scale model in
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  a towing tank, I have to care about the waves the tiny model
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  makes. You have to obsess over them if
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  you want the wave resistance measured on your little model.
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  To accurately scale up to the real full size destroyer, you
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  must guarantee that the Froude number of the model matches the
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  Froude number of the real ship. Or else what?
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  If the ratio of inertia to gravity is different, the wave
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  patterns will be totally different and your drag
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  calculations will be completely useless.
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  You'll build a ship that is vastly underpowered.
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  Wow OK that makes perfect sense. Reynolds is for viscosity and
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  enclosed flows. Froude is for gravity and
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  surface waves. What is #3?
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  The Weber number. This is the ratio of inertia
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  forces to surface tension forces.
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  Surface tension. But surface tension is usually a
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  weak force, right? Like we were talking about
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  microscopic scales here. Usually, yes.
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  Surface tension is what keeps a raindrop spherical.
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  It's the cohesive force pulling the liquid together.
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  But the Weber number becomes the absolute controlling factor when
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  you were dealing with the formation of droplets or bubbles
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  or incredibly thin films of liquid.
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  Give me a real world scenario where an engineer is actually
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  sweating over the Weber number. Jet engine fuel injectors.
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  Oh wow. OK.
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  You have liquid aviation fuel being sprayed into a combustion
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  chamber. You need that fuel to atomize
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  perfectly, to break apart into a very specific mist of
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  microscopic droplets so it burns evenly and efficiently.
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  And if it doesn't? If the droplets are too big, the
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  fuel doesn't burn completely, you lose thrust and you create
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  soot. If the engineer doesn't balance
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  the Weber number correctly, predicting exactly how the
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  inertia of the spray will overcome the surface tension of
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  the fuel, the engine will fail. The stakes are incredibly high.
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  Prairie and the final of the BIG4.
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  The Mach number, the ratio of inertia forces to elasticity
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  forces. Yeah, which is just another way
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  of saying the fluids compressibility.
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  It is effectively the flow velocity divided by the local
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  speed of sound. Yeah, we're going to dedicate a
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  massive portion of our discussion to this later, but
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  Simply put, it governs high speed flight, right?
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  Push air fast enough and it stops acting like water and
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  starts acting like a highly coiled spring.
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  OK, so this brings us to a massive, painful reality check
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  for engineers. Let's go back to the idea of
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  model testing. I want to construct a
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  hypothetical scenario for you. OK.
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  What's here? I'm an aerodynamicist for an
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  automaker. I've designed a sleek new sports
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  car and I want to know it's aerodynamic drag before we
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  build. The factory tooling makes sense.
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  So I 3D print a beautiful, perfectly scaled one to 10 model
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  of the car. I put it in my wind tunnel.
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  I know the real car will drive at 60 mph on the highway, so I
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  turn the wind tunnel fan on, blast the little model with 60
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  mph of wind, and read the drag force off my sensors.
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  Am I getting good data? You are getting garbage data.
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  Garbage data. Absolute garbage and if you
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  trust that data you are walking straight into a catastrophic
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  design failure. But.
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  Why? I scaled the shape perfectly,
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  It's an exact geometric replica. You achieved geometric
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  similarity, yes, but you completely failed to achieve
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  dynamic similarity. Dynamic similarity.
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  OK, explain. Let's look back at the Reynolds
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  number formula. Density times velocity times
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  length divided by viscosity. Let's assume the air in the wind
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  tunnel is the exact same as the air outside.
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  Right. Slandered air.
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  So the density, viscosity are unchanged, but your model is
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  110th the length of the real car.
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  So my length variable just plummeted by a factor of 10.
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  Exactly. Which means your Reynolds number
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  just plummeted by a factor of 10.
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  I see it. The ratio of inertia to
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  viscosity is totally broken. For your 110th scale model,
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  sitting in a 60 mile per hour, wind viscosity is playing a
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  much, much larger role than it would for the full size car.
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  So the air literally feels different to the model.
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  Yes, the air will feel relatively thicker to the tiny
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  model. The boundary layer of air
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  hugging the surface of the car will behave differently.
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  The point where the airflow separates from the rear bumper
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  will be entirely wrong. The swirling wake behind the car
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  will be the wrong shape. Any drag numbers you pull from
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  that test or fiction? Wow, so the math demands that I
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  fix the Reynolds number. If my length dropped by a factor
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  of 10, I need to increase another variable in the
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  numerator by a factor of 10 to compensate.
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  That's the math. So option 1I crank up the
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  velocity. I take the real car speed, 60
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  mph and multiply it by 10. I blast my little model with 600
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  mph of wind. You could try that, but you just
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  traded one problem for a much, much worse one.
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  What's that? By trying to fix the Reynolds
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  number, you just shattered the Mach number.
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  Oh, because 600 mph is getting uncomfortably close to the speed
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  of sound. Exactly.
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  Mach One at sea level is roughly 760 mph, so your little model
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  car is now experiencing transonic airflow.
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  The air is compressing violently.
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  Shock waves might be forming over the windshield.
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  The air is acting like a spring. The real car driving at 60 mph
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  will never experience compressible flow, so again,
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  your data is completely useless. You cannot infinitely scale
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  velocity without breaking the physics of compressibility.
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  OK, I can't increase the length because that defeats the whole
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  purpose of a scale model, right? I can't increase the velocity
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  because I break the sound barrier.
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  The only other variable in the numerator is the density of the
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  fluid itself. I have to increase the density
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  of the air by a factor of 10, but how on earth do you make air
• 13:47
  10 times thicker? You build a variable density
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  wind tunnel. A what?
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  A variable density wind tunnel. This is an incredible feat of
• 13:54
  engineering. You take the entire wind tunnel
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  facility, the fan, the test section, the sensors, and you
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  seal it inside a colossal steel pressure vessel.
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  You're kidding. No, I mean it a submarine hall
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  basically. Then you use massive industrial
• 14:09
  compressors to pump air into the tunnel until the ambient
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  pressure inside is 10 atmospheres.
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  You literally pressurize the room until the air molecules are
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  packed 10 times denser. Yes, now your 110th scale model
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  can be tested at a normal 60 mph.
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  The velocity is realistic, so the Mach number is safe.
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  But because the air is 10 times tensor, the Reynolds number
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  perfectly matches the full size car.
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  Driving in normal air, you have achieved dynamic similarity.
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  The forces you measure on the model will scale up perfectly to
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  the real world. That is a breathtaking brute
• 14:43
  force solution to a math equation.
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  It really is, and it explains why aerospace companies and F1
• 14:49
  teams spend hundreds of millions of dollars on their wind tunnel
• 14:52
  facilities. You aren't just beating a fan,
• 14:55
  you are building a pressurized ecosystem.
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  It highlights it's the extreme lengths engineers must go to in
• 15:01
  order to respect the fundamental rules of scale.
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  And, you know, Speaking of recycling those rules, the
• 15:06
  source material includes A phenomenal historical dive into
• 15:09
  Ernst Mach himself. The namesake of the Mach number.
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  The very same he was an Austrian physicist in the late 1800s and
• 15:16
  he obviously didn't have variable density wind tunnels.
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  He had to rely purely on his analytical mind to diagnose the
• 15:22
  invisible. He used these exact principles
• 15:25
  of fluid dynamics to solve 2 massive military mysteries.
• 15:29
  Yeah, the first story here is gripping.
• 15:31
  It takes place during the Franco Prussian War around 1870.
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  The French army had deployed a brand new infantry weapon, the
• 15:38
  Chassapot rifle. A devastating weapon.
• 15:41
  And the Prussians were horrified by what they were seeing on the
• 15:43
  battlefield. When a Prussian soldier was hit,
• 15:46
  the wounds weren't small puncture holes.
• 15:48
  They were a massive crater, like explosive traumas.
• 15:52
  It caused an international diplomatic crisis.
• 15:55
  The international community, led by the Prussians, accused the
• 15:59
  French of secretly hollowing out their bullets and filling them
• 16:02
  with high explosives. Which was illegal.
• 16:04
  Highly illegal. This was a direct violation of
• 16:07
  the recently signed declaration of Saint Petersburg, which
• 16:09
  banned explosive projectiles under a certain weight.
• 16:12
  The French were basically being accused of heinous war crimes.
• 16:16
  But the French military brass swore up and down that the
• 16:18
  bullets were solid lead, no explosives whatsoever.
• 16:21
  It was a standoff. So they brought in Ernst Mock as
• 16:24
  an impartial scientific investigator to figure out what
• 16:27
  was tearing these bodies apart. And Mock approached it not as a
• 16:30
  weapons expert, but as a fluid dynamicist.
• 16:34
  Interesting. He realized that the Chassepot
• 16:36
  rifle was firing its solid LED bullets at an unprecedented
• 16:40
  velocity for the era. It was firing them significantly
• 16:44
  faster than the speed of sound. This is the 1870s.
• 16:47
  People barely understood aerodynamics, let alone
• 16:50
  supersonic flight. Exactly.
• 16:52
  But Mock deduced that the devastation wasn't chemical, it
• 16:55
  was pneumatic. Because the bullet was traveling
• 16:58
  faster than the speed of sound, the air sitting in front of the
• 17:01
  bullet couldn't receive any acoustic warning to move out of
• 17:03
  the way. It couldn't get out of the way.
• 17:05
  Right. The air was violently impacted
• 17:07
  and compressed into a highly pressurized, super dense cone, a
• 17:11
  bow shockwave that rode continuously on the tip of the
• 17:14
  bullet. So when the bullet struck a
• 17:15
  target, the target wasn't just getting hit by a piece of lead,
• 17:19
  it was getting hit by a solid wall of hyper compressed air.
• 17:23
  A blast wave mock proved mathematically and
• 17:27
  experimentally that the explosive style wounds were
• 17:30
  caused by this violent pneumatic shockwave, shattering the tissue
• 17:34
  upon impact. It was entirely independent of
• 17:36
  the solid projectile. He completely exonerated the
• 17:40
  French government of the war crimes using nothing but fluid
• 17:43
  dynamics. It is an incredible application
• 17:45
  of physics to global politics, and the sources outline a second
• 17:49
  mystery he solved, this time for the artillery.
• 17:53
  Men. Yes, the double bang.
• 17:54
  Right gun crews had noticed a bizarre auditory phenomenon.
• 17:58
  When they fired an older low velocity cannonball, a spotter
• 18:03
  standing way down range would hear a single loud bang.
• 18:06
  Standard stuff. But when they fired a modern
• 18:09
  high velocity rifled artillery shell, the downrange observer
• 18:13
  heard two completely distinct bangs separated by a couple of
• 18:16
  seconds. They thought the guns were
• 18:18
  malfunctioning. And mock apply the exact same
• 18:20
  logic. He analyzed the speed of the
• 18:21
  projectile versus the speed of the sound waves.
• 18:24
  He explained to the Artillerman that the first bang the observer
• 18:27
  heard had absolutely nothing to do with the cannon fire.
• 18:29
  It didn't, no. The first bang was the bow
• 18:33
  shockwave, the Sonic boom generated by the supersonic
• 18:36
  shell tearing through the air directly past their heads.
• 18:41
  The shell arrived first, bringing its own explosion of
• 18:43
  air with it. Yes, and the second bang was the
• 18:46
  actual acoustic noise of the gunpowder combusting at the
• 18:49
  muzzle, which you know, was traveling at the normal slower
• 18:52
  speed of sound and therefore arrived much later.
• 18:55
  Maquette essentially diagnosed and explained the Sonic boom
• 18:59
  decades before an airplane ever took flight.
• 19:01
  He understood what the invisible fluid was doing just by thinking
• 19:04
  about the physics of compression, which is honestly
• 19:07
  the perfect pivot for us. Oh yeah, we know the rules of
• 19:10
  scale, and we know how fluids behave conceptually.
• 19:13
  But how do we actually pull numbers out of thin air?
• 19:16
  How do we diagnose the flow in real time?
• 19:19
  You can't put a mathematical theorem inside a jet engine.
• 19:21
  We need physical tools. We must transition from the
• 19:24
  theoretical physicist chalkboard to the working engineers
• 19:28
  physical toolbox. We have to purposefully interact
• 19:31
  with the fluid to measure it. And the most ubiquitous tool for
• 19:34
  measuring velocity, A tool still proudly mounted on the nose of
• 19:38
  every single commercial airliner flying today, is the Pitot Tube.
• 19:42
  The pitot tubes. They look like little metal
• 19:44
  spikes or antennas jutting out from the dock pit window, right?
• 19:47
  Yeah, that's them. The underlying principle of a
• 19:50
  pitostatic tube is an elegant application of Bernoulli's
• 19:54
  equation. It is essentially a hollow tube
• 19:57
  with a hole pointing straight forward into the oncoming wind
• 20:01
  and tiny secondary holes drilled into the side of the tube
• 20:05
  perfectly perpendicular to the wind.
• 20:06
  I always like to visualize the pedo tube like driving down the
• 20:09
  highway. Imagine you are driving at 70
• 20:11
  mph. If you roll down the window, the
• 20:14
  ambient air pressure you feel just sitting in the cabin, the
• 20:17
  pressure pushing on your eardrums is the static pressure.
• 20:19
  It's just the weight of the atmosphere.
• 20:21
  Correct. The holes on the side of the
• 20:23
  Pidot tube measure that static pressure because the air slips
• 20:27
  smoothly past them without ramming into them.
• 20:30
  Now if you stick your arms straight out the window and hold
• 20:33
  your palm flat against the wind, you feel a massive force pushing
• 20:37
  your hand backward. That is the air violently
• 20:40
  ramming into your skin and coming to a dead stop.
• 20:43
  And that dead stop is crucial. In fluid dynamics.
• 20:46
  We call that the stagnation point.
• 20:48
  All of the kinetic energy of the moving air is instantly
• 20:52
  converted into pressure energy. OK, kinetic to pressure.
• 20:55
  The hole pointing straight forward on the pitot tube
• 20:59
  captures this. It measures the total pressure,
• 21:02
  which is the sum of the ambient static pressure plus the dynamic
• 21:05
  pressure caused by the ramming effect.
• 21:07
  So the math is incredibly simple.
• 21:09
  You take the total pressure from the front hole, you subtract the
• 21:12
  static pressure from the side holes, and whatever is leftover
• 21:15
  is the dynamic pressure exactly. And since dynamic pressure is
• 21:18
  directly tied to the velocity of the fluid and its density, you
• 21:21
  can calculate exactly how fast you are flying.
• 21:24
  It is brilliant, robust, and it has absolutely no moving parts.
• 21:28
  Engineers even modify it for complex swirling environments.
• 21:31
  The text mentions A5 hole spherical plateau tube.
• 21:35
  Instead of just pointing forward, it has a central hole
• 21:38
  and four holes arranged in a cross pattern around it.
• 21:42
  By comparing the minute pressure differences across all five
• 21:45
  holes simultaneously, a computer can calculate not just the
• 21:48
  speed, but the exact 3 dimensional angle and vector of
• 21:52
  the airflow hitting the probe. That's really clever, but the
• 21:55
  source material flags a massive, deadly misconception here, a
• 21:59
  trap that catches young engineers and has caused real
• 22:03
  world aviation disaster. Impressibility issue.
• 22:05
  Yeah, we talked about compressibility earlier.
• 22:08
  A lot of people treat the pitot tube formula as absolute gospel.
• 22:11
  They assume air is just air and the simple subtraction math
• 22:14
  always works. That assumption is only valid if
• 22:17
  the fluid is strictly incompressible.
• 22:19
  For water, the standard Pitot formula is flawless.
• 22:23
  For a small Cessna flying at 100 mph, the formula is perfectly
• 22:26
  fine because the air isn't compressing significantly.
• 22:28
  But what happens when you put a Petot tube on an Airbus flying
• 22:31
  at Mach 0.7? The air ramming into that front
• 22:35
  stagnation hole isn't just stopping, it's violently
• 22:38
  squishing. The density is spiking.
• 22:41
  Right, exactly. The density of the air inside
• 22:44
  the tube is now much higher than the density of the air outside.
• 22:47
  The fundamental assumption of Bernoulli's simple equation is
• 22:50
  completely broken. So the math is wrong.
• 22:52
  Yes. If an aerospace engineer or a
• 22:55
  flight computer forgets to apply a specific complex mathematical
• 22:59
  adjustment called the compressibility correction
• 23:01
  factor, the Putaho tube will over read the pressure at Mach
• 23:05
  0.7. Uncorrected data will yield a
• 23:09
  velocity error of roughly 6%. 6% That might sound like a rounding
• 23:13
  error on a math test, but in an airliner cruising at 35,000
• 23:17
  feet, you were operating in a very narrow band between your
• 23:20
  stall speed, where you fall out of the sky, and your overspeed
• 23:23
  limit, where the aerodynamic forces literally RIP the wings
• 23:26
  off. A 6% hallucination in your
• 23:28
  airspeed data can plunge a plane into the ocean.
• 23:31
  It is a critical pain point. You must always know the limits
• 23:33
  of your physical tools. Now, Pinto tubes are phenomenal
• 23:37
  for steady, continuous flow, but they have a severe limitation.
• 23:41
  They are sluggish. Very sluggish.
• 23:43
  Right, If you are researching extra stream turbulence, where
• 23:46
  the velocity of the wind is randomly spiking and dropping
• 23:49
  thousands of times per second, the pressure inside a pedo tube
• 23:52
  physically cannot equalize fast enough to capture that data.
• 23:56
  The readings just blur into an average.
• 23:58
  And to capture high frequency turbulence, we abandoned
• 24:01
  pressure entirely and turn to thermodynamics.
• 24:04
  We use a hot wire anemometer. OK, this sounds like a piece of
• 24:07
  mad science. You are literally sticking a
• 24:09
  burning wire into the wind. Conceptually, it is incredibly
• 24:13
  intuitive. If you take an impossibly thin
• 24:15
  wire, usually made of tungsten or platinum.
• 24:18
  We're talking micrometers thick, far thinner than a human hair.
• 24:22
  You run an electrical current through this wire to heat it up
• 24:26
  until it's glowing hot. Then you place this delicate
• 24:29
  wire directly into the flowing fluid.
• 24:31
  And as the wind blows past it, it cools the wire down.
• 24:34
  Just like blowing on a hot spoonful of soup, the harder you
• 24:37
  blow, the faster it cools. Exactly, and here's the physics
• 24:40
  trick. As the metal wire cools down,
• 24:43
  it's electrical resistance changes.
• 24:46
  If you constantly monitor the electrical resistance of the
• 24:48
  wire, you can mathematically reverse engineer exactly how
• 24:52
  fast the fluid must be moving to 'cause that specific amount of
• 24:56
  cooling. OK, I followed that, but the
• 24:58
  text outlines 2 very different ways to run this system and one
• 25:02
  of them has a fatal flaw. Let's talk about the constant
• 25:05
  voltage method first. In a constant voltage setup, you
• 25:08
  supply a steady, unchanging electrical voltage to the wire,
• 25:12
  and you just passively watch the resistance fluctuate as the wind
• 25:15
  blows. It sounds simple, but it suffers
• 25:18
  from a massive physical bottleneck, thermal inertia.
• 25:22
  Thermal inertia. So like how an electric stove
• 25:23
  top takes a few minutes to actually turn red after you turn
• 25:26
  the dial, and takes 10 minutes to cool down after you turn it
• 25:29
  off? Yeah, precisely.
• 25:30
  The metal physically can't change temperature instantly.
• 25:33
  That is the exact problem. Even a microscopic tungsten wire
• 25:37
  takes a tiny fraction of a millisecond to heat up or cool
• 25:40
  down. If you are trying to measure a
• 25:43
  turbulent Eddie that is vibrating 10,000 times a second,
• 25:46
  the wire's physical temperature simply cannot keep up.
• 25:49
  It's just too slow. The thermal inertia acts as a
• 25:52
  damper. The high frequency data gets
• 25:54
  smeared, lagged and eventually lost.
• 25:57
  So how do engineers bypass the physical limits of metal?
• 26:00
  They flip the paradigm entirely. They use a constant temperature
• 26:04
  anemometer or CTA. This is the gold standard in
• 26:06
  turbulence research. Instead of passively watching
• 26:09
  the temperature change, you connect the wire to a hyper
• 26:12
  responsive closed loop electronic feedback circuit.
• 26:16
  This circuit constantly monitors the wires resistance and if the
• 26:19
  wind blows and tries to cool the wire down even a fraction of a
• 26:22
  degree, the circuit instantly violently spikes the electrical
• 26:26
  current to force the wire to stay at the exact same
• 26:28
  temperature. Oh, so the temperature of the
• 26:31
  wire never actually changes? You are measuring the effort the
• 26:34
  circuit exerts to fight the wind.
• 26:35
  You measure the current being pumped in.
• 26:38
  Because the circuit reacts electronically at the speed of
• 26:41
  light, you completely bypass the thermal inertia of the metal
• 26:45
  wire. A high quality CTA system can
• 26:49
  accurately capture violent turbulent velocity fluctuations
• 26:53
  up to 100 kHz. Wow.
• 26:56
  That is 100,000 distinct velocity changes every single
• 27:00
  second. That is mind boggling precision,
• 27:02
  but there is an obvious physical drawback to both pitot tubes and
• 27:07
  hot wires. You are shoving a piece of metal
• 27:09
  into the very fluid you are trying to measure.
• 27:11
  Yes, that's true. If you are studying the delicate
• 27:14
  boundary layer over a wing, sticking a probe in there acts
• 27:17
  like a speed bump. It disturbs the flow and ruins
• 27:19
  the very phenomenon you are trying to observe.
• 27:22
  If you want to measure the invisible without touching it,
• 27:24
  you have to use light. Which brings us to the laser
• 27:26
  Doppler anemometer, or LDA. There's optical wizardry based
• 27:30
  on the Doppler effect. The Doppler effect is something
• 27:32
  everyone experiences. When an ambulance drives towards
• 27:35
  you, the sound waves get bunched together so the siren sounds
• 27:38
  high pitched. When it drives away, the waves
• 27:40
  stretch out and the pitch drops lower.
• 27:43
  Light waves do the exact same thing, just to vastly higher
• 27:46
  frequencies. In an LDA setup, an engineer
• 27:49
  points a highly focused laser ream directly into the fluid.
• 27:54
  Now the fluid itself is transparent, so the laser would
• 27:56
  just pass right through. So how does it measure anything?
• 27:59
  You have to seed the fluid with microscopic tracer particles,
• 28:02
  like a fine mist of oil droplets in air or tiny reflective beads
• 28:06
  and water. These particles are so small
• 28:09
  that they flow perfectly along with the fluid.
• 28:12
  The laser beam hits these moving particles.
• 28:14
  The light hits the particles and scatters in all directions.
• 28:17
  Because the particle is physically moving through space
• 28:20
  at the exact moment the light hits it, the scattered light
• 28:23
  experiences a tiny dockler shift in its frequency.
• 28:27
  The color of the light actually shifts by an imperceptible
• 28:29
  amount. And the detector catches that
• 28:31
  scattered light, compares its new frequency to the original
• 28:34
  frequency of the laser, and calculates the speed of the
• 28:37
  particle. Yes, it is completely non
• 28:40
  intrusive. Nothing physically touches the
• 28:43
  fluid but photons. You can map out incredibly
• 28:46
  complex swirling 3 dimensional vortices inside a glass pipe
• 28:51
  without ever disturbing a single drop of water.
• 28:54
  OK, so PIA tubes, hot wires, and lasers are fantastic tools for
• 28:58
  finding the velocity of a fluid at one microscopic specific
• 29:02
  point in space. Right point measurements.
• 29:04
  But in heavy industry, say a massive petrochemical refinery
• 29:07
  or municipal water plant, engineers usually don't care
• 29:10
  about the velocity city at a single coordinate.
• 29:12
  They want to know the bulk flow rate, how many thousands of
• 29:15
  gallons of fluid are moving through this massive pipe every
• 29:18
  minute. To measure bulk flow, we have to
• 29:20
  manipulate the entire pipeline. We use flow restrictions.
• 29:23
  Yeah, we purposefully choke the flow.
• 29:25
  Choke the. Flow.
• 29:26
  The three primary devices discussed in the source text are
• 29:29
  the orifice plate, the flow nozzle, and the Venturi tube.
• 29:33
  They all exploit the continuity equation and Bernoulli's
• 29:35
  principle. Let's visualize this.
• 29:38
  Imagine a massive 4 lane highway packed with cars moving at a
• 29:41
  steady 50 mph. Suddenly, construction forces
• 29:45
  the highway to choke down to just two lanes.
• 29:47
  OK, yeah. To get the same number of cars
• 29:49
  through that bottleneck without traffic backing up forever, the
• 29:53
  cars in the two lanes have to accelerate.
• 29:56
  They have to speed up to 70 mph to clear the restriction.
• 29:59
  The fluid acts exactly like the cars.
• 30:02
  When you install a restriction in a pipe, the area decreases.
• 30:05
  To maintain the constant mass flow, the fluid must accelerate
• 30:09
  through the narrow opening. Bernoulli's principle tells us
• 30:12
  that when a fluid accelerates, its static pressure drops.
• 30:16
  The kinetic energy goes up, so the pressure energy must go
• 30:19
  down. South, an engineer simply
• 30:21
  measures the pressure of the fluid just before the
• 30:23
  restriction and measures the pressure right at the narrowest
• 30:27
  point. You measure that differential
• 30:28
  pressure. The larger the pressure drop,
• 30:31
  the faster the fluid must be accelerating, which means a
• 30:33
  higher overall volume of flow is moving through the pipe.
• 30:36
  It is a very reliable mathematical relationship.
• 30:38
  Sounds simple enough. But this is where the textbook
• 30:41
  theory collides violently with corporate budgets.
• 30:44
  The source material highlights the stark differences between an
• 30:47
  orifice plate and a Venturi tube.
• 30:49
  Choosing between them is a massive real world Ain oint for
• 30:53
  lant managers. OK, let's set the stage.
• 30:56
  I'm a project engineer. I'm building a massive natural
• 30:58
  gas pipeline. My boss tells me to keep
• 31:01
  construction costs down. Classic.
• 31:03
  So I look at the catalog and I choose the orifice plate.
• 31:05
  Why? Because an orifice plate is
• 31:07
  literally just a flat piece of steel with a sharp hole drilled
• 31:10
  in the center. It costs almost nothing.
• 31:12
  You just unbolt 2 sections of pipe and sandwich it in between.
• 31:15
  It is incredibly cheap capital expenditure.
• 31:18
  It is the cheapest option upfront, but you will pay for it
• 31:21
  every single day for the next 30 years and operating costs.
• 31:24
  You have to visualize what forcing a massive column of
• 31:27
  fluid through a sharp edged hole actually does.
• 31:30
  The fluid slams into the flat steel plate.
• 31:32
  It has to make a violent 90° turn to squeeze through the
• 31:35
  hole. As it exits the hole, the flow
• 31:38
  violently rips apart from the edges, creating a massive,
• 31:41
  chaotic, swirling wake of vortices downstream.
• 31:44
  It's pure turbulence. And turbulence is energy being
• 31:48
  converted into heat. All those swirling eddies are
• 31:51
  grinding against each other dissipating the fluids kinetic
• 31:54
  energy due to viscosity. We call this a permanent
• 31:57
  pressure loss. Meaning it doesn't bounce back.
• 32:00
  Exactly. Once the fluid passes an orifice
• 32:03
  plate, it never recovers its original pressure.
• 32:07
  If you have a cross country pipeline with 50 orifice plates
• 32:10
  installed along it, the pressure drops so severely that you have
• 32:14
  to install massive multi $1,000,000 pumping stations just
• 32:18
  to keep the fluid moving. The electricity cost to overcome
• 32:21
  that permanent pressure loss will dwarf the savings of buying
• 32:25
  the cheap plate. So the orifice plate is a budget
• 32:27
  trap. It's CapEx versus OpEx.
• 32:30
  And now let's look at the premium option, the Venturi
• 32:33
  tube. A Ventura tube is a masterpiece
• 32:35
  of fluid dynamics. Instead of a flat plate, a
• 32:37
  venturi is a large section of pipe that has been beautifully,
• 32:40
  meticulously machined to gradually taper down to a smooth
• 32:43
  throat and then incredibly gently diverge back out to the
• 32:47
  original diameter. Because it tapers so smoothly
• 32:49
  the the fluid doesn't slam into anything.
• 32:51
  Exactly. The float doesn't separate.
• 32:53
  There are almost no swirling vortices.
• 32:55
  The fluid accelerates smoothly into the throat, drops its
• 32:58
  pressure so you can take your measurement, and then expand
• 33:00
  smoothly out the other side. Because the expansion is so
• 33:03
  gentle, the kinetic energy is clearly converted back into
• 33:06
  pressure energy. A venturi tube might recover up
• 33:09
  to 95% of the original pressure. Your permanent pressure loss is
• 33:13
  virtually 0. Your pumps barely have to work.
• 33:16
  But the upfront cost? The upfront cost is
• 33:18
  astronomical. They are massive.
• 33:21
  Heavy, difficult to install and require precision machining.
• 33:25
  The pain point for the engineer is justifying that massive
• 33:29
  initial capital expenditure to an accountant by proving the
• 33:32
  long term energy savings. It is the eternal struggle of
• 33:35
  engineering reality versus financial reality.
• 33:38
  It never ends. Before we leave the physical
• 33:40
  world, let's look at some of the more exotic bulk flow meters the
• 33:43
  text covers. Because they solve some very
• 33:45
  specific nasty industrial problems.
• 33:47
  Let's. Start with the rotometer.
• 33:48
  This is a wonderfully simple, entirely mechanical device.
• 33:52
  I've seen these in hospitals connected to oxygen tanks.
• 33:54
  It's a clear vertical tube with a little floating bead inside.
• 33:59
  Yes, the tube is slightly tapered wider at the top than at
• 34:02
  the bottom. The fluid flows vertically
• 34:04
  upward through the tube. As the fluid hits the float, it
• 34:08
  creates aerodynamic drag, lifting the float upwards.
• 34:11
  But gravity is pulling it down. Right, but gravity is pulling
• 34:15
  the float downwards. The faster the fluid flows, the
• 34:18
  higher it pushes the float until the drag force perfectly
• 34:21
  balances gravity and buoyancy. The engineer literally just
• 34:25
  looks at where the float is hovering and reads the number
• 34:28
  printed on the side of the glass.
• 34:30
  No electricity, no complex math, just a perfect equilibrium of
• 34:34
  physical forces. That's elegant, but a rotameter
• 34:37
  only works if you can see through the tube.
• 34:40
  What if I am trying to measure the flow of raw corrosive sewage
• 34:43
  or a slurry of rocks and water in a mining operation?
• 34:46
  Yeah, that's a problem. An orifice plate would get
• 34:49
  instantly clogged. A venturi tube would get eroded
• 34:51
  away by the rocks. I need a meter that doesn't
• 34:54
  obstruct the pipe at all. For that you deploy a magnetic
• 34:57
  flow meter. This device operates on
• 34:59
  Faraday's law of electromagnetic induction.
• 35:02
  Which states that if you move a conductive material through a
• 35:05
  magnetic field, you generate an electrical voltage.
• 35:08
  Precisely, and we apply that to fluid.
• 35:11
  You install 2 powerful Electro magnets on the outside of the
• 35:13
  pipe, creating a strong constant magnetic field cutting straight
• 35:17
  through the interior. As the fluid flows through that
• 35:20
  field, it acts like a moving wire.
• 35:22
  A voltage is induced inside the fluid itself.
• 35:25
  You place two small electrodes flush with the inner wall of the
• 35:28
  pipe to measure that voltage. And the faster the fluid moves,
• 35:31
  the higher the voltage generated.
• 35:34
  The beauty of the magnetic meter is that there is absolutely zero
• 35:37
  physical obstruction in the pipe.
• 35:39
  The pressure loss is absolutely zero.
• 35:41
  It is immune to changes in fluid density, viscosity or
• 35:44
  temperature. It doesn't care if there are
• 35:45
  rocks or sludge in the water. But wait, Faraday's law requires
• 35:49
  the material to be electrically conductive.
• 35:52
  Water with minerals in it conducts electricity.
• 35:55
  But what if I am pumping millions of barrels of pure
• 35:58
  refined diesel? Fuel oil is a fantastic
• 36:01
  electrical insulator. That is the hard limit.
• 36:04
  A magnetic flow meter on an oil pipeline is an expensive
• 36:07
  paperweight. It will read 0 no matter how
• 36:09
  fast the oil moves. For non conductive fluids you
• 36:12
  have to pivot to acoustics. You use an ultrasonic flow
• 36:15
  meter. Using sound to measure speed,
• 36:18
  how does that work in a pipe? You strap 2 ultrasonic
• 36:21
  transducers to the outside of the pipe position diagonally
• 36:24
  across from each other. They act like both speakers and
• 36:26
  microphones. Transducer A fires a high
• 36:29
  frequency sound pulse diagonally across the fluid to transducer
• 36:33
  B. Then transducer B fires a pulse
• 36:36
  back to transducer A and. Because they are firing
• 36:38
  diagonally, one pulse is fighting upstream and one pulse
• 36:42
  is surfing downstream. The downstream sound wave gets a
• 36:45
  speed boost from the flowing fluid.
• 36:47
  The upstream sound wave gets slowed down, fighting the
• 36:49
  current. The computer measures the
• 36:51
  microscopic difference in transit time between the two
• 36:53
  pulses. It is entirely non intrusive, 0
• 36:56
  pressure drop and it works perfectly on non conductive
• 36:59
  fluids like oil or ultra pure distilled water.
• 37:02
  And finally, the text mentions vortex shutting meters.
• 37:05
  This one feels like harnessing chaos.
• 37:07
  It connects beautifully back to the strohole number from our
• 37:10
  dimensional analysis discussion. If you place a blunt objects
• 37:14
  like a simple rectangular bar vertically in the middle of a
• 37:17
  flowing pipe, the fluid cannot glide smoothly around it.
• 37:21
  The flow separates, forming a swirling vortex behind the bar.
• 37:25
  A localized tornado. But these tornadoes don't just
• 37:28
  sit there, they detach and float downstream.
• 37:31
  We call them von Karman vortices, and they shed in a
• 37:34
  perfectly alternating rhythmic pattern.
• 37:37
  One peels off the left side of the bar, the one peels off the
• 37:40
  right side, like a flag flapping in the wind.
• 37:43
  And the math shows that the frequency of that flapping, the
• 37:46
  speed at which the vortices are shed, is directly linearly
• 37:50
  proportional to the velocity of the fluid.
• 37:52
  The meter just uses a small pressure sensor downstream to
• 37:55
  feel the pulses and count how many vortices hit it every
• 37:58
  second. It uses the fluids inherent
• 38:00
  instability as a highly precise metronome.
• 38:03
  It's brilliant engineering, but everything we just discussed,
• 38:05
  the Pyto tubes, the lasers, the Venturas, these are tools for
• 38:09
  observing reality after it has already been built.
• 38:11
  If I want to design the aerodynamic profile of a brand
• 38:14
  new airplane wing from scratch, I can't just guess a shape,
• 38:17
  build it, and laser it. I need to model the flow
• 38:20
  mathematically before a single piece of metal is cut.
• 38:23
  And to do that, we must leave the physical world and enter
• 38:26
  Part 3 of the text. The Mathematicians.
• 38:29
  Utopia. Ideal Fluid Dynamics.
• 38:32
  Let me stop you right there, because this is a topic that
• 38:34
  caused me and thousands of other engineering students absolute
• 38:38
  existential frustration. I know, I know exactly where
• 38:40
  you're going with this. Every single advanced fluid
• 38:43
  dynamics textbook begins its heavy mathematical chapters by
• 38:47
  defining an ideal fluid, and the strict definition of an ideal
• 38:51
  fluid is a fluid that possesses exactly 0 viscosity.
• 38:54
  It has absolutely no internal friction.
• 38:56
  That is the definition, yeah. Why?
• 38:58
  Why do universities force students to spend hundreds of
• 39:01
  hours solving parifying differential equations for a
• 39:04
  magical, frictionless fluid that literally does not exist
• 39:08
  anywhere in the physical universe?
• 39:09
  It feels completely detached from reality.
• 39:12
  It is the ultimate source of cognitive dissonance for an
• 39:14
  engineer. You spend your early years
• 39:17
  learning how friction ruins everything, and then suddenly
• 39:20
  the textbook tells you to pretend friction doesn't exist.
• 39:23
  It feels like academic self indulgence.
• 39:25
  It really does. But there is a profoundly
• 39:27
  pragmatic, brilliant justification for it.
• 39:32
  Let's revisit our trusty Reynolds number, the ratio of
• 39:35
  inertia forces to viscous forces.
• 39:37
  OK, inertia over viscosity. In many of the most critical
• 39:41
  real world engineering applications, like a Boeing 777
• 39:44
  cruising at 500 mph or a nuclear submarine moving rapidly deep
• 39:48
  underwater, the Reynolds number is incredibly high.
• 39:52
  We're talking about values in the 10s of millions.
• 39:55
  Meaning the force of inertia is 10s of millions of times
• 39:58
  stronger than the force of viscosity.
• 40:00
  Exactly. Viscosity still exists.
• 40:02
  The air still has friction, but because the massive inertia of
• 40:05
  the high speed flow is so overwhelmingly dominant, the
• 40:08
  physical effects of that viscosity are squashed down and
• 40:11
  confined to an impossibly thin layer of air located right
• 40:14
  against the microscopic surface of the airplane wing.
• 40:17
  Boundary layer. Yes, inside that boundary layer,
• 40:21
  which might only be a few millimeters thick, friction is a
• 40:23
  violent reality. The air is shearing aggressively
• 40:26
  against the metal skin, but step up just one inch away from the
• 40:31
  wing into the vast bulk of the airflow surrounding the plane,
• 40:34
  and the velocity gradients are so incredibly gentle that the
• 40:37
  effects of viscosity become statistically negligible.
• 40:41
  The bulk of the air behaves, for all practical mathematical
• 40:44
  intents and purposes, exactly as if it had 0 viscosity.
• 40:47
  So it's a strategic lie. It is a mathematically necessary
• 40:50
  simplification. If you tried to calculate the
• 40:53
  exact true viscous friction of every single air molecule
• 40:57
  interacting around a whole airplane, the nonlinear math
• 41:00
  would bring the world's most powerful supercomputers to their
• 41:02
  knees. By strategically pretending
• 41:04
  viscosity is 0 in the bulk flow, the math becomes solvable, and
• 41:08
  the aerodynamic lift answers we get from that fake math are
• 41:11
  stunningly accurate for 99% of the flow field.
• 41:13
  OK, I will accept the premise. Let's unpack the math of this
• 41:16
  utopia. The text introduces 2 heavy
• 41:20
  hitters right away, the velocity potential and the stream
• 41:23
  function. What are these and how should a
• 41:27
  visual thinker imagine them? The best analogy is a
• 41:30
  topographical map of a mountain. OK, if you look at a topo map,
• 41:33
  you see contour lines. If you hike along a single
• 41:36
  contour line, your elevation never changes.
• 41:39
  You're walking perfectly flat around the mountain.
• 41:42
  Now, if you drop a boulder on the side of that mountain, which
• 41:44
  way does it roll? It rolls straight down, taking
• 41:47
  the steepest possible path. Mathematically, the steepest
• 41:50
  path is always perfectly perpendicular to the contour
• 41:53
  lines. The boulder crosses every
• 41:56
  elevation line at exactly 90°. OK, I see the mountain.
• 41:59
  How does this map to fluid? The velocity potential, denoted
• 42:03
  by the Greek letter Phi is the mountains elevation.
• 42:06
  It represents the invisible field of pressure or energy
• 42:09
  potential that is driving the fluid, and equal potential line
• 42:12
  is a line where that driving energy is constant.
• 42:14
  And the other one. The stream function denoted by
• 42:17
  the Greek letter CI is the path the boulder takes.
• 42:20
  A streamline is the actual physical path that a particle of
• 42:24
  fluid will follow. And just like the boulder
• 42:26
  crossing the elevation lines, the streamlines must cross the
• 42:29
  potential lines at perfect right angles, Yes.
• 42:33
  For an ideal irritational flow, the fundamental equations of
• 42:36
  physics dictate that the streamlines and equipotential
• 42:39
  lines form a perfectly orthogonal grid.
• 42:43
  Wherever they intersect, they cross at exactly 90° angles.
• 42:47
  This means if you can somehow map out the invisible energy
• 42:50
  potential, you can mathematically derive the exact
• 42:53
  physical path the fluid will take.
• 42:55
  They are two sides of the same mathematical coin, firmly linked
• 42:58
  together by the Kochi Riemann equations and governed by the
• 43:01
  absolute overlord of ideal fluid math, Laplace's equation.
• 43:05
  Let's talk about Laplace's equation, because just hearing
• 43:07
  the name makes engineers sweat. The text says Laplace's equation
• 43:10
  dictates that the divergent of the gradient of the potential is
• 43:13
  zero. That is pure mathematical
• 43:15
  jargon. Translate that into English.
• 43:17
  What does divergent of a gradient actually physically
• 43:20
  mean? Let's break it down into two
• 43:22
  words, gradient and divergent. We know what a gradient is.
• 43:26
  It's a slope. It's how steep the energy
• 43:27
  mountain is. A steep gradient means high
• 43:30
  velocity. Divergent is a measure of
• 43:32
  whether fluid is spreading out from a specific point or
• 43:35
  converging it to a specific. Point so spreading out.
• 43:38
  Right. If the divergent of a fluid was
• 43:40
  a positive number, it would mean fluid is spontaneously appearing
• 43:43
  out of nowhere and expanding outwards.
• 43:46
  If the divergent was negative, it would mean fluid is
• 43:48
  spontaneously disappearing, vanishing from existence at that
• 43:51
  point. But physics says you can't just
• 43:53
  magically create or destroy mass.
• 43:55
  Exactly the conservation of mass.
• 43:58
  The place's equation stating that the divergent is exactly 0
• 44:01
  everywhere is just the mathematical translation of the
• 44:04
  physical law that mass cannot be created or destroyed.
• 44:08
  It means that whatever amount of fluid enters a specific tiny box
• 44:11
  of space, that exact same amount of fluid must exit the box.
• 44:15
  The flow is continuous, smooth and unbroken.
• 44:18
  OK, that makes it much less terrifying.
• 44:20
  It just means the math is behaving itself.
• 44:22
  And because ideal fluid flow is governed by the places
• 44:25
  beautifully behaved equation, it possesses what is arguably the
• 44:29
  most powerful mathematical property in existence,
• 44:32
  linearity. Linearity means you can just add
• 44:35
  things together using simple addition.
• 44:37
  Yes, the principle of superposition.
• 44:39
  If you have a mathematical formula that perfectly describes
• 44:42
  flow pattern A, and you have another mathematical formula
• 44:45
  that perfectly describes flow pattern B because of linearity,
• 44:49
  you don't have to invent new math to combine them.
• 44:52
  You literally just take formula A, add formula B, and the result
• 44:56
  is a perfectly varied, physically accurate formula for
• 44:59
  a new, highly complex flow pattern C This is how engineers
• 45:03
  modeled complex aerodynamics for decades before digital computers
• 45:07
  were invented. Let's walk through the textbook
• 45:09
  example of this, because it shows how something incredibly
• 45:11
  abstract builds reality. They start with the most basic
• 45:14
  cartoonish building blocks imaginable, a source and a sink.
• 45:18
  Right. Imagine a perfectly flat
• 45:20
  infinite 2 dimensional plane. A source is a microscoic
• 45:24
  mathematical point where fluid is magically welling U from
• 45:27
  nowhere and flowing radially outward in all directions, like
• 45:31
  a erfect sunburst. It violates mass conservation at
• 45:34
  that one singular dot, but nowhere else.
• 45:37
  And a sink. A sink is the exact opposite.
• 45:40
  It's a microscopic drain hole where fluid is rushing inward
• 45:44
  from all directions and disappearing from the universe.
• 45:46
  OK, so we have a mathematical hose and a mathematical drain.
• 45:50
  What happens when you use superposition to add them
• 45:52
  together? If you place a source and a sink
• 45:54
  a few inches apart on your mathematical plane, the
• 45:57
  streamlines become beautiful arcs.
• 45:59
  Fluid erupts from the source, curves through space, and dives
• 46:02
  straight into the sink. It looks exactly like the
• 46:05
  magnetic field lines looping between the North and South
• 46:08
  poles of a bar magnet. And here is where the
• 46:10
  mathematician's brain takes over.
• 46:12
  What happens if you take that source and that sink and you
• 46:14
  start sliding them closer and closer together while
• 46:17
  simultaneously turning up their strength?
• 46:19
  You slide them closer and closer, turning the strength to
• 46:22
  Infinity until the distance between them is 0.
• 46:26
  They occupy the exact same coordinate.
• 46:28
  The hose and the drain are stacked on top of each other.
• 46:31
  That sounds like it should just cancel out to nothing.
• 46:33
  Intuition says they cancel, but the math says they create a
• 46:37
  singularity called a doublet. A doublet has an incredibly
• 46:40
  specific, mesmerizing flow pattern.
• 46:43
  The fluid erupts and immediately loops back on itself.
• 46:47
  The streamlines form a series of perfect nested circles that all
• 46:51
  pass through the origin point and are tangent to the axis.
• 46:54
  It is a pure abstract mathematical construct, a
• 46:58
  circular flow of energy. But why?
• 47:00
  Why does an engineer designing an airplane care about stacking
• 47:03
  a fake hose in a fake drain to make circles?
• 47:05
  Because of the next step of superposition, watch the magic
• 47:08
  trick. We take the complex equation for
• 47:09
  that double its circular flow, and we're going to add it to the
• 47:12
  simplest equation in the book, uniform flow.
• 47:15
  Uniform flow is just wind blowing perfectly straight left
• 47:17
  to right at a constant speed. Straight parallel lines.
• 47:20
  When you add the mathematical equation of the straight uniform
• 47:23
  flow to the equation of the circular doublet flow, the
• 47:26
  geometries collide. The straight lines of the wind
• 47:29
  push against the expanding circles of the doublet and the
• 47:32
  resulting mathematical equation perfectly.
• 47:34
  Florida State describes the complex airflow going around a
• 47:37
  solid 2 dimensional circular cylinder.
• 47:40
  You take a fake hose, a fake drain, blow some fake straight
• 47:43
  wind at it, and the maths spits out the exact blueprint for how
• 47:47
  real wind interacts with a real smokestack, or a real pipe, or a
• 47:51
  real wire. You are seeing the matrix now.
• 47:54
  By stacking incredibly simple abstract mathematical building
• 47:57
  blocks we can construct the geometry of physical reality.
• 48:01
  We bypassed solving terrifying non linear differential
• 48:04
  equations from scratch. We just use basic addition to
• 48:07
  synthesize a cylinder. OK, but cylinders don't fly.
• 48:10
  Airplanes have wings. They have sleep teardrop shaped
• 48:12
  airfoils. Adding a source and a sink gets
• 48:15
  me a circle. How do we get from a circle to a
• 48:17
  Boeing 747? To jump from a cylinder to a
• 48:21
  wing, we need one more mathematical tool.
• 48:24
  It is arguably one of the most brilliant insights in the
• 48:27
  history of aerodynamics. Conformal mapping, specifically
• 48:31
  the Jaikowski transformation developed by a Russian
• 48:34
  mathematician. I always struggled with
• 48:35
  conformal mapping. The text says it maps points
• 48:38
  from the Z plane to the Zeta plane.
• 48:40
  Let's use a very grounded analogy.
• 48:42
  Think about a globe of the Earth.
• 48:44
  It's a sphere. Now think about a flat paper map
• 48:47
  of the Earth. Like the Mercator projection,
• 48:50
  the map maker had to use a mathematical transformation to
• 48:52
  stretch, pull and warp the surface of the 3D sphere onto a
• 48:56
  2D piece of paper. Greenland looks massive on the
• 48:59
  paper, but small on the globe. The geometry is warped, but the
• 49:03
  mathematical rules linking them are absolute.
• 49:05
  OK, so mapping is just stretching geometry according to
• 49:08
  a strict rule. Yes.
• 49:10
  Now imagine a sheet of incredibly stretchy mathematical
• 49:13
  rubber. On this rubber sheet we have
• 49:15
  drawn the perfect beautiful streamlines for flow around our
• 49:18
  cylinder. We know that math works.
• 49:20
  Conformal mapping applies a specific formula to stretch that
• 49:24
  rubber sheet, but it is a very special type of stretching.
• 49:28
  Conformal mapping preserves local angles.
• 49:30
  Meaning what? Remember our mountain topo map?
• 49:33
  The streamlines and equipotential lines must cross
• 49:36
  at exactly 90° to satisfy Laplace's equation.
• 49:39
  When you stretch the rubber sheet, the gridlines bend and
• 49:41
  warp, but wherever they intersect, they still cross at
• 49:44
  exactly 90°. The orthogonal grid is
• 49:46
  preserved. Which means Laplace's equation
• 49:48
  is still satisfied, the math is still physically valid.
• 49:51
  It hasn't broken the laws of physics, it just changed shapes.
• 49:55
  Is the genius of it. So Nikolai Jakowski found a
• 49:58
  specific transformation formula. When you apply his formula, the
• 50:02
  solid circle of the cylinder stretches, warps, and flattens
• 50:05
  out into the elegant teardrop shape of an airplane wing.
• 50:09
  And all those beautiful simple streamlines we calculated for
• 50:12
  the cylinder automatically stretch and wrap themselves
• 50:15
  perfectly around the new wing shape.
• 50:18
  We trick the math into giving us the complex aerodynamics of a
• 50:21
  wing by starting with a simple circle.
• 50:23
  It's mathematical alchemy, but the source text points out a
• 50:26
  massive catastrophic problem here, a point where the elegant
• 50:30
  math completely betrays physical reality unless the engineer
• 50:33
  steps in and forces a patch. Let's talk about the sharp
• 50:37
  trailing edge of the wing and the CUDA condition.
• 50:40
  This is where theory violently meets reality.
• 50:43
  If you just trust the pure math of the Jakowski transformation,
• 50:46
  you get a flay pattern around the wing.
• 50:48
  The air flows smoothly over the top and smoothly under the
• 50:51
  bottom. But look at the trailing edge,
• 50:53
  the razor sharp back edge of the wing.
• 50:55
  Remember our foundational assumption?
• 50:58
  We assume the fluid was ideal 0 viscosity 0 friction.
• 51:03
  It can slip and slide infinitely perfectly over any surface.
• 51:07
  O What does the pure math say the air flowing along the bottom
• 51:10
  of the wing does when it hits that shar back edge?
• 51:13
  Well, without friction to slow it down, the math dictates that
• 51:16
  the air hits the sharp point and whips around it.
• 51:19
  It makes an impossibly tight hairpin turn, flowing from the
• 51:23
  bottom surface, curling around the razor edge and flowing
• 51:25
  forward up the top surface. Exactly, and to make a zero
• 51:28
  radius turn without friction, the math demands that the
• 51:31
  velocity of the air at that sharp edge must be infinite.
• 51:34
  Infinite velocity, which is physically impossible.
• 51:37
  It is a singularity. If this happened in reality,
• 51:40
  airplanes would never generate lift.
• 51:42
  The flow would just curl around the back and cancel everything
• 51:44
  out. But in the real world, airplanes
• 51:45
  do fly. Why?
• 51:46
  Because our initial assumption was a lie.
• 51:49
  Viscosity does exist. That microscopic boundary layer
• 51:52
  of friction, yes. That tiny, almost negligible
• 51:55
  amount of friction is just strong enough to prevent the air
• 51:59
  from making that impossible. Infinite velocity.
• 52:01
  Hairpin turn. The friction grabs the fluid.
• 52:05
  In reality, the air flowing over the top and the air flowing
• 52:08
  under the bottom meet at the sharp trailing edge and flow off
• 52:11
  smoothly together straight off the back.
• 52:14
  The trailing edge is a stagnation point.
• 52:17
  So the math is predicting infinite curling velocity, but
• 52:20
  reality is predicting smooth finite departure.
• 52:23
  How do engineers fix the math? They manually hack the equation.
• 52:28
  Wilhelm Kuta, a brilliant German aerodynamicist, realized the
• 52:31
  flaw. He said we must artificially
• 52:33
  constrain the math so it mimics the friction we ignored.
• 52:36
  He introduced the CUDA condition.
• 52:38
  The CUDA condition mandates that the mathematical circulation of
• 52:41
  air around the wing must be exactly strong enough to force
• 52:43
  the rear stagnation point, the point where the flow is separate
• 52:46
  to sit exactly on the sharp trailing edge.
• 52:48
  You mathematically pin the flow to the back edge, forbidding it
• 52:51
  from curling around. You force the velocity at the
• 52:53
  trailing edge to remain finite and smooth by manually inserting
• 52:58
  this one artificial rule into the ideal equations.
• 53:01
  Suddenly the math perfectly predicts the actual lift
• 53:04
  generated by a real wing. The CUDA condition is the bridge
• 53:08
  between the mathematicians frictionless fantasy and the
• 53:11
  physical reality of a 747 taking off.
• 53:14
  It is the sole reason why airplane wings are designed with
• 53:17
  sharp trailing edges. If a wing had a rounded back,
• 53:22
  the CUDA condition wouldn't lock the flow in place.
• 53:24
  The flow would become chaotic and stable flight would be
• 53:26
  impossible. That is the ultimate lesson.
• 53:29
  You learn the pure theory, you realize its physical limits, and
• 53:33
  you use your engineering intuition to patch the
• 53:35
  difference. And that realization is the
• 53:37
  perfect transition to part four of the source material.
• 53:40
  Breaking the rules. Compressible flow and shock
• 53:42
  waves. Oh, this is the good stuff.
• 53:44
  Up until this exact moment, whether we were calculating
• 53:47
  Reynolds numbers, calibrating pitot tubes, or mathematically
• 53:50
  stretching cylinders into wings, we have relied on one massive,
• 53:53
  comforting assumption. We assume the fluid cannot be
• 53:56
  squeezed. We assume density is a rock
• 53:58
  solid constant. Water is famously
• 54:01
  incompressible. But air?
• 54:03
  We've treated air like water this whole time.
• 54:06
  When does that assumption fundamentally shatter?
• 54:08
  When do we push the limits of speed so far that intuition
• 54:11
  completely fails? This is the realm of high speed
• 54:14
  aerodynamics, Ballistics and rocket nozzles.
• 54:17
  Gas is inherently compressible. We all know you can squeeze air
• 54:20
  into a stuba tank, but the physics dictate that if the gas
• 54:23
  is moving slowly compared to the speed of sound, the actual
• 54:27
  physical squishing of a molecules against an object is
• 54:30
  so minor we can ignore it. But there is a threshold.
• 54:33
  The source material is very explicit about this threshold.
• 54:36
  An engineer can safely treat air as an incompressible fluid with
• 54:39
  constant density only up to a Mach number of 0.3.
• 54:42
  Mach 0.3 at sea level that is roughly 230 mph power.
• 54:47
  A fast race car, but a very slow airplane.
• 54:50
  Right at Mach 0.3, the air ramming against the nose of the
• 54:54
  plane compresses enough that it's density changes by about
• 54:57
  5%. In the gritty reality of
• 55:00
  engineering, a 5% margin of error is generally accepted as
• 55:03
  the boundary of good enough. But once you push the throttle
• 55:07
  past MOG 0.3, the density changes scale exponentially.
• 55:12
  The air stops behaving like a gentle stream of water and
• 55:15
  starts behaving like a massive, heavy, highly tensioned slinky.
• 55:19
  The rules of incompressible flow do not just bend the invert.
• 55:23
  And this leads us to what is easily the biggest mind Bender
• 55:26
  in the entire source text. It is a concept that
• 55:29
  aggressively violates common sense.
• 55:31
  Let's talk about pipe flow and the effect of sectional change.
• 55:34
  The nozzle, the subsonic versus supersonic nozzle.
• 55:36
  This is where we separate the intuition of a plumber from the
• 55:39
  intuition of a rocket scientist. Let's start with the plumbers
• 55:42
  intuition, which we discussed earlier with the orifice plate.
• 55:45
  Imagine a subsonic flow, water flowing from a garden hose.
• 55:48
  If you take your thumb and pinch the end of the hose, making the
• 55:51
  exit area narrower, what happens to the water?
• 55:54
  It speeds up, it sprays out much faster and further across the
• 55:57
  yard. We rely on this every day.
• 56:00
  It is dictated by the fundamental continuity equation.
• 56:03
  Mass flow rate must remain constant.
• 56:06
  Mass flow equals density multiplied by area multiplied by
• 56:09
  velocity. If the fluid is water, the
• 56:11
  density is locked. It can't change.
• 56:13
  So if I decrease the area with my thumb, the velocity
• 56:16
  absolutely must increase to squeeze that same volume of
• 56:20
  water through the smaller gap. That is flawless classical
• 56:23
  physics. A narrowing pipe.
• 56:25
  A convergent pipe forces subsonic fluid to accelerate.
• 56:29
  Conversely, A widening pipe, a divergent pipe, allows subsonic
• 56:32
  fluid to spread out and slow down.
• 56:34
  That makes total sense. Now let us step through the
• 56:37
  looking glass into the supersonic regime.
• 56:39
  Imagine you have a pipe, and you have air blasting through it at
• 56:42
  Mach 2, twice the speed of sound.
• 56:44
  If that supersonic air enters A narrowing section of pipe, what
• 56:48
  happens? My brain screams that it should
• 56:49
  speed up even more. It's a bottleneck.
• 56:51
  But the math and the brood reality of physics dictate the
• 56:53
  exact opposite. If a supersonic flow enters A
• 56:56
  narrowing pipe, it violently slows down.
• 56:58
  OK. And if the pipe widens if the
• 57:00
  supersonic flow enters a divergent section?
• 57:02
  If a supersonic flow enters A widening pipe, it accelerates.
• 57:06
  It goes even faster. Mach 2 becomes Mach 3.
• 57:08
  Let's stop and unpack this. This is the pain point.
• 57:11
  Why? Why does a widening pipe make a
• 57:14
  supersonic gas go faster when we know for a fact that a widening
• 57:18
  pipe makes a subsonic gas go slower?
• 57:21
  How can both statements be true in the same universe?
• 57:24
  It all comes back to the variable.
• 57:25
  We finally unlocked density. Let's look at your continuity
• 57:28
  equation again. Constant mass flow equals
• 57:31
  density times area times velocity.
• 57:33
  In subsonic flow, density was locked.
• 57:35
  It was a constant, so area and velocity were locked.
• 57:37
  In a simple seesaw relationship, one goes up, the other goes
• 57:40
  down. But in supersonic flow we cross
• 57:42
  the Mach 0.3 threshold. Density is no longer locked.
• 57:45
  Is highly variable. Not only is it variable, it
• 57:49
  becomes the absolute dictator of the equation.
• 57:52
  As a highly compressed supersonic gas moves forward and
• 57:55
  tries to accelerate, its pressure drops rapidly.
• 57:58
  Because it is a compressible gas, when the pressure drops,
• 58:01
  the density plummets. It expands violently.
• 58:05
  Here's the secret to the paradox.
• 58:07
  In supersonic flow, the density of the gas decreases at a much,
• 58:11
  much faster rate than the velocity increase.
• 58:13
  Oh, let. Me run that through the
• 58:14
  equation. Constant mass flow equals
• 58:16
  density times area times velocity.
• 58:19
  If the velocity is trying to inch up but the density is
• 58:22
  crashing downwards at 10 times the speed, the right side of the
• 58:25
  equation is shrinking, the total mass flow is dropping.
• 58:27
  But the laws of physics state mass flow must be constant.
• 58:30
  You cannot destroy mass in the pipe.
• 58:32
  So to balance the equation, 3rd variable has to step up.
• 58:34
  If density is crashing, the area must increase to compensate and
• 58:38
  keep the mass moving. You see it, you physically have
• 58:41
  to widen the pipe to provide more spatial volume for the
• 58:44
  rapidly expanding low density gas, just so it has the room to
• 58:48
  accelerate. If you tried to narrow the pipe,
• 58:50
  the expanding gas would have nowhere to go.
• 58:52
  It would choke on itself, build up massive back pressure and
• 58:55
  violently decelerate. A widening pipe gives the
• 58:58
  expanding gas room to breathe and speed up.
• 59:01
  That is incredibly counterintuitive but
• 59:03
  mathematically perfect, and this specific paradox is the
• 59:07
  fundamental engine behind every space launch in history.
• 59:11
  It is the physics of the convergent divergent nozzle,
• 59:14
  famously known as the de la Valle nozzle, invented by Gustaf
• 59:17
  de la Valle in the late 19th century.
• 59:19
  If you look at the business end of a SpaceX Falcon 9 or the old
• 59:23
  Space Shuttle main engines, you see those massive iconic bell
• 59:27
  shaped nozzles. They are narrow at the top and
• 59:29
  incredibly wide at the bottom. Why?
• 59:31
  Because they are manipulating this exact paradox to achieve
• 59:34
  supersonic exhaust. Let's trace a particle of rocket
• 59:37
  fuel through that engine. It starts in the combustion
• 59:40
  chamber. We mix fuel and oxygen and
• 59:42
  ignite it. Inside the combustion chamber
• 59:45
  you have created a hellscape of incredibly high pressure and
• 59:48
  high temperature gas. But the gas isn't moving very
• 59:51
  fast yet. It is highly energized, but it's
• 59:54
  velocity is subsonic. So step one, we need to speed
• 59:58
  the subsonic gas up. What do we do?
• 1:00:01
  It's subsonic, so we use plumbers Intuition.
• 1:00:03
  We narrow the pipe. Right, the top of the engine
• 1:00:05
  nozzle converges. It pinches aggressively inward.
• 1:00:08
  As the area decreases, our subsonic exhaust gas accelerates
• 1:00:11
  faster and faster and it's pressure begins to drop.
• 1:00:14
  If the aerospace engineers designed the curve of that metal
• 1:00:17
  perfectly, the gas will accelerate until it reaches
• 1:00:19
  exactly yes Mach one the speed of sound at the absolute
• 1:00:23
  narrowest point of the pinch. This narrowest point is called
• 1:00:26
  the nozzle throat. And reaching Mach one at the
• 1:00:28
  throat triggers A phenomenon the text heavily emphasizes.
• 1:00:32
  Choked flow is a critical thermodynamic limit.
• 1:00:35
  When the gas hits Mach one at the throat, it is physically
• 1:00:39
  impossible to push any more mass through that specific area.
• 1:00:42
  Under those specific upstream pressure conditions, the throat
• 1:00:46
  is fundamentally bottlenecked by the speed of sound.
• 1:00:48
  Even if you took the exit of the rocket nozzle and shoved into
• 1:00:51
  the deepest, emptiest vacuum of deep space, dropping the
• 1:00:54
  downstream pressure to absolute 0, the mass flow rate through
• 1:00:58
  the engine would not increase by a single oz.
• 1:01:00
  The flow is choked for air, the text notes.
• 1:01:03
  This choked condition occurs when the pressure at the throat
• 1:01:06
  drops to exactly 0.5 to 8 times the chamber pressure.
• 1:01:10
  So we have maxed out the bottleneck.
• 1:01:12
  We have Mach one gas at the throat, but Mach 1 exhaust isn't
• 1:01:15
  going to get a rocket to orbit. We need massive supersonic
• 1:01:18
  thrust. We need Mach 3, Mach 4.
• 1:01:20
  So now we cross the threshold. The gas at the throat is Mach
• 1:01:23
  one. To make it go faster, to push it
• 1:01:25
  into the supersonic regime, we have to flip our engineering
• 1:01:27
  logic. We must widen the pipe so
• 1:01:29
  immediately past the throat, the rocket nozzle flares outwards
• 1:01:33
  into that massive divergent bell shape.
• 1:01:35
  As the area expands, the supersonic gas violently expands
• 1:01:39
  with it. Its density plummets, its
• 1:01:41
  pressure drops to match the outside atmosphere, and its
• 1:01:44
  velocity rockets to Mach 3, Mach 4, Mach 5 at the back.
• 1:01:48
  Converge to get to Mach 1, diverge to go beyond it.
• 1:01:51
  The shape of the nozzle is a physical graph of the math, but
• 1:01:54
  rockets operate in the real world.
• 1:01:56
  What happens if your math doesn't match the environment?
• 1:01:59
  What if you designed your divergent belt perfectly for the
• 1:02:02
  vacuum of space, but you fire the engine at sea level where
• 1:02:05
  the thick heavy atmospheric pressure is pushing back against
• 1:02:08
  the exhaust? Then that beautiful smooth
• 1:02:11
  supersonic acceleration shatters.
• 1:02:13
  If the expanding gas inside the wide bell drops to a pressure
• 1:02:16
  that is significantly lower than the heavy atmospheric pressure
• 1:02:19
  pushing from the outside, the supersonic flow physically
• 1:02:23
  cannot smoothly adjust to the higher pressure.
• 1:02:27
  Supersonic flows cannot smoothly feel what is ahead of them, so
• 1:02:30
  the flow violently collapses in on itself.
• 1:02:32
  This collapse creates a shockwave.
• 1:02:35
  We mentioned shockwaves earlier with Ernst Marks bullets.
• 1:02:37
  Let's define it properly. It's not just a loud sound, it
• 1:02:40
  is a physical wall in the air. A shockwave is a massive,
• 1:02:44
  sudden, violent discontinuity in the fluid over a distance of a
• 1:02:48
  few micrometers, literally the thickness of a bacterium.
• 1:02:51
  The physical properties of the gas snap as a supersonic flow
• 1:02:54
  crashes through a shockwave. The pressure skyrockets
• 1:02:57
  instantly, the temperature spikes thousands of degrees, the
• 1:03:00
  entropy, the thermodynamic disorder of the universe,
• 1:03:02
  increases massively, and the velocity instantly drops from
• 1:03:05
  supersonic down to subsonic. It is an incredibly violent,
• 1:03:09
  entirely irreversible thermodynamic collision.
• 1:03:11
  The source material categorizes them.
• 1:03:14
  It draws a hard line between a normal shockwave and an oblique
• 1:03:17
  shockwave. Yes.
• 1:03:18
  A normal shockwave stands perfectly perpendicular, exactly
• 1:03:22
  90° to the oncoming flow. It is a flat wall.
• 1:03:26
  It is the most brutal energy destroying type of shock.
• 1:03:29
  It decelerates the flow straight from supersonic to subsonic in
• 1:03:33
  one catastrophic hit. You might see a normal shockwave
• 1:03:36
  forming inside the bell of a rocket engine if it is severely
• 1:03:39
  over expanded at sea level, which can cause intense
• 1:03:41
  vibrations that tear the engine apart.
• 1:03:43
  And an oblique shockwave. And a weak shock forms at an
• 1:03:46
  angle. Think of a supersonic fighter
• 1:03:47
  jet like an F22. The nose of the jet is a sharp
• 1:03:51
  wedged point driving into the air at Mach 2.
• 1:03:53
  The air can't get out of the way, so it compresses into a
• 1:03:56
  shockwave. But because the nose is angled,
• 1:03:58
  the shockwave attaches to the tip and sweeps backward at an
• 1:04:01
  angle, exactly like the V shaped wake behind a speedboat.
• 1:04:05
  Because the shock is at an angle, the collision isn't head
• 1:04:07
  on, it's a glancing blow. So it's less violent.
• 1:04:10
  Considerably. It still causes a sudden jump in
• 1:04:13
  pressure and temperature, but the velocity drop isn't as
• 1:04:16
  severe. In fact, the flow behind an
• 1:04:19
  oblique shockwave usually remains supersonic.
• 1:04:21
  It has just been slightly slowed down and deflected in the new
• 1:04:25
  direction. Supersonic aircraft use
• 1:04:28
  carefully designed oblique shocks on their engine intakes
• 1:04:31
  to gently step the air down to subsonic speeds before it hits
• 1:04:35
  the delicate turbine blades. There is a detail in the text
• 1:04:37
  regarding shock waves that I find absolutely fascinating.
• 1:04:41
  A shockwave cannot propagate upstream into a supersonic flow,
• 1:04:44
  it can only be swept downstream. Why is the wave trapped?
• 1:04:48
  Think about what a shockwave, or any pressure wave, or any sound
• 1:04:51
  wave fundamentally is. It is a physical disturbance
• 1:04:54
  communicating information through the molecules of the
• 1:04:56
  fluid. The speed at which that
• 1:04:58
  information travels is, by definition, the speed of sound.
• 1:05:01
  If the fluid itself is flowing toward the source of the sound
• 1:05:04
  faster than the speed of sound, the way physically cannot swim
• 1:05:07
  upstream. It's like a salmon trying to
• 1:05:09
  swim up a waterfall, but the water is falling faster than the
• 1:05:11
  salmon can swim. The fish just gets swept
• 1:05:14
  backward. That is the exact mechanism.
• 1:05:17
  This is why when an airplane flies supersonically, the air 1
• 1:05:20
  inch in front of the nose has absolutely zero physical warning
• 1:05:23
  that a plane is about to hit it. The pressure waves trying to
• 1:05:26
  warn the air cannot propagate forward.
• 1:05:29
  Instead, all those pressure ways, the sound of the engines,
• 1:05:32
  the displacement of the air, get swept backwards, piling up on
• 1:05:35
  top of each other, bunching into an incredibly dense conical
• 1:05:37
  envelope, dragging behind the plane.
• 1:05:39
  The mock cone. Yes, the tip of the cone is the
• 1:05:42
  nose of the plane. The angle of this cone, the mock
• 1:05:44
  angle, get sharper and tighter the faster the plane flies, and
• 1:05:48
  this creates A surreal physical boundary.
• 1:05:50
  Inside the cone, trailing behind the plane, you have the chaotic
• 1:05:53
  turbulence and the deafening roar of the jet engines.
• 1:05:56
  But outside the cone, ahead of that boundary line, there is
• 1:05:59
  absolute, pristine silence. The air is completely
• 1:06:02
  undisturbed. Until that invisible cone sweeps
• 1:06:04
  over your house on the ground and that massive pile up of
• 1:06:07
  pressure hits you all at once as a Sonic boom.
• 1:06:10
  It is a terrifying display of physics.
• 1:06:13
  Now, before we wrap up, the text introduces 2 final highly
• 1:06:17
  specialized concepts regarding compressible flow inside pipes,
• 1:06:21
  Phanoflow and Rayleigh flow. They sound like obscure corner
• 1:06:25
  cases. They are specialized, but they
• 1:06:27
  represent the grim, unavoidable realities of engineering.
• 1:06:30
  Everything we just discussed about the perfect smooth
• 1:06:33
  acceleration in a rocket nozzle assumed the flow was isentropic,
• 1:06:37
  a theoretical condition meaning perfect frictionless adiabatic
• 1:06:41
  expansion. But real metal pipes have
• 1:06:43
  friction, and real jet engines generate massive amounts of
• 1:06:46
  heat. Enter Fano and Rayleigh.
• 1:06:48
  Fanoflow is the study of a compressible gas moving through
• 1:06:51
  a pipe with a constant area where the pipe is perfectly
• 1:06:53
  insulated, so no heat enters or leaves, but there is significant
• 1:06:57
  friction against the rough walls of the pipe.
• 1:06:59
  Real world example pumping highly pressurized natural gas
• 1:07:02
  through a pipeline spanning hundreds of miles across Alaska.
• 1:07:05
  Exactly. The friction of the steel walls
• 1:07:08
  creates a constant drag on the gas.
• 1:07:10
  What is fascinating and highly counterintuitive is how that
• 1:07:14
  friction affects the speed of the gas.
• 1:07:16
  If the gas enters the pipeline at a subsonic speed, the
• 1:07:20
  friction causes the pressure to drop over distance.
• 1:07:23
  But remember, a pressure drop means a density drop to maintain
• 1:07:27
  mass flow. The subsonic gas actually
• 1:07:29
  accelerates along the pipe, moving faster and faster toward
• 1:07:32
  Mach One as it travels. Friction makes the gas speed up.
• 1:07:35
  That is another massive mind. Bender.
• 1:07:37
  It accelerates because it is expanding.
• 1:07:39
  However, if you somehow forced supersonic gas into that same
• 1:07:43
  pipe, the friction would act as a violent brake, causing the
• 1:07:45
  velocity to decrease, decelerating down toward Mach
• 1:07:48
  one. Notice the pattern?
• 1:07:50
  Phano flow dictates that friction always drives the
• 1:07:53
  velocity of a compressible fluid toward the speed of sound.
• 1:07:56
  It's a thermodynamic trap. If your Alaskan pipeline is long
• 1:07:59
  enough, the friction will eventually accelerate subsonic
• 1:08:01
  gas all the way to Mach One, which point the flow will choke
• 1:08:05
  and you cannot pump any more gas through the line no matter how
• 1:08:07
  hard you push. And Rayleigh flow.
• 1:08:09
  How does heat change the game? Rayleigh flow strips away the
• 1:08:12
  friction, assuming a perfectly smooth pipe, but focuses
• 1:08:15
  entirely on heat transfer. Imagine the core combustion
• 1:08:18
  Chamber of a scramjet engine. You have air flowing through a
• 1:08:21
  constant area, and you are dumping phenomenal amounts of
• 1:08:24
  heat energy into it by spraying and burning aviation fuel.
• 1:08:28
  The thermodynamics dictate a similar trap.
• 1:08:31
  If the air entering the chamber is subsonic, dumping heat into
• 1:08:34
  it causes the air to violently expand and accelerate toward
• 1:08:37
  Mach one. If the air is supersonic and
• 1:08:40
  heat causes it to decelerate toward Mach 1.
• 1:08:42
  So whether it is the friction of the pipe pulling on it or the
• 1:08:46
  heat of combustion expanding it, the fundamental laws of
• 1:08:48
  thermodynamics force compressible flow to constantly
• 1:08:51
  seek the Sonic barrier. It is a physical limit that
• 1:08:54
  aerospace engineers are fighting against in every single design
• 1:08:56
  meeting. And that leads to a final,
• 1:08:58
  provocative thought for you to ponder long after we sign off.
• 1:09:02
  Today. We have mapped out the bizarre,
• 1:09:04
  counterintuitive rules that govern subsonic and supersonic
• 1:09:07
  flow. We've seen how density changes
• 1:09:10
  dictate the necessity of weirdly shaped, diverging nozzles, and
• 1:09:13
  how shock waves form impenetrable physical walls in
• 1:09:16
  the sky. But ask yourself, what happens
• 1:09:20
  when we push even further? What happens when we transcend
• 1:09:23
  supersonic and enter the hypersonic regime?
• 1:09:25
  Mach 5, Mach 10, Mach 25. When a space capsule re enters
• 1:09:29
  the Earth's atmosphere at orbital velocity, the friction
• 1:09:32
  and compression ahead of the heat shield are so intense that
• 1:09:34
  the air doesn't just act like a spring anymore.
• 1:09:37
  The heat is so extreme, thousands of degrees, that the
• 1:09:39
  air itself fundamentally breaks down.
• 1:09:42
  It turns into a searing, glowing plasma.
• 1:09:44
  The nitrogen and oxygen molecules physically tear apart,
• 1:09:47
  changing the fundamental chemistry of the fluid.
• 1:09:50
  The phantom lines, the Rayleigh flows, the ideal equations we
• 1:09:53
  discussed today, they all disintegrate.
• 1:09:55
  They're just the anti chamber to the wildest forces in the
• 1:09:58
  universe. At Mach 25, you weren't just
• 1:10:00
  flying through air, you were flying through a chemical
• 1:10:02
  inferno of your own making, relying on material science to
• 1:10:06
  survive the fluid dynamics. If this helped you even a
• 1:10:09
  little, say something. We're putting this out for free.
• 1:10:12
  No paywall, no subscription, nothing locked behind a card
• 1:10:15
  that only works if we know it's actually reaching people and
• 1:10:18
  doing what it's supposed to to do.
• 1:10:20
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• 1:10:22
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• 1:10:25
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• 1:10:29
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• 1:10:31
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• 1:10:33
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