9.22.2006

Random Randomness

For me, the hard part about coming up with blog content is getting past the urge to think, "no one would find that interesting." It's too easy to dismiss things you take for granted that other people may find interesting.

For example... One if the things I learned while working on my degree is that if you wanted to model traffic flow like a gas, you'd be better off using a supersonic model. A subsonic model would be a waste of time.

Why is that? It has to do with the way information propogates through a medium. The speed of sound in a gas is limited by how quickly vibrations and motion in one particle can be transmitted to its neighbors. This is information, information about what is happening elsewhere in the gas. It can be heating, movement, just about any disturbance in the medium.

Now, imagine you have a gas flowing slowly through a large diameter pipe. Some distance down the pipe, the diameter suddenly decreases. The law of conservation says that any mass entering the pipe must leave the pipe at the same rate. The gas accelerates as the pipe narrows to maintain the mass flowrate. This is possible because the low velocity of the gas allows information about the decreasing diameter to flow upstream to other particles. They all start accelerating when required. If the pipe later increases back to its original diameter, the particles all decelerate in an orderly fashion.

What happens if we push this gas through the pipe beyond its sonic speed? Think about what happens in heavy traffic. When two lanes merge (say for a construction zone), you get a back up. Humans aren't capable of instantaneous reaction. We don't want to hit one another, and you can't see very well around the car in front of you. You're not well informed about what's happening far in front of you. The same is true of the particles in our pipe. We're aproaching the diameter change faster than information about the change can travel upstream. In traffic you get people slamming on the brakes (and if someone isn't paying attention, an accident). In the gas, you get a standing shock.

Inside a shock all kinds of nasty stuff is going on. Particles are smacking into one another and generally getting pissed off. The result is an increase in density, pressure, and temperature. Sounds remarkably like a traffic jam doesn't it?

The flow behind the shock is slower than in front, and our traffic through the construction zone is also slower.

What happens when we finally get to the other side? A driver that's just spent a half-hour in one-lane gridlock looks at a five lane highway in glee. Everyone stomps on the gas and spreads backout. The same happens to our particles. An expanding supersonic flow accelerates. This is why rocket engines have gigantic nozzles. The burning propellants are forced to accelerate past their sonic speed inside the engine, then they are expanded in the nozzle to even higher speeds. The nozzle's exit diameter determines the final speed of the exhaust gas. It also affects the pressure of the exhaust gas. The expansion causes a decreas in pressure.

Ideally, you want the pressure of the exhaust to match the pressure of the outside air. If your nozzle is limited to only one size (or a small range of sizes), there's only one or a small range of pressures where you are getting optimal performance. If the pressure outside is higher than your expanded flow (at lower altitudes and take off), the flow is "over expanded" and the result is the nifty shock diamonds you see coming out of jet afterburners. If you're at very high altitude, your nozzle may not expand the flow enough the equalize with the outside pressure, in this case, the flow is "under expanded" and in some cases you can also get shock diamonds. However, if you're exteremely high, the miniscule pressure may only just turn the expanding flow back in on itself (if ever). This is what you see in extremely long range shots of the Saturn V at high altitude.

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