Macroscopic shape of rocket exhaust

[Alonso Quixano]

Greetings to everyone. I would like to ask how the shape of a rocket exhaust plume changes with distance, when the rocket operates in a vacuum. What I'm mainly looking for, is to see how large the diametre of the plume would be at a distance of ~20km from the nozzle. We're assuming an ordinary LH2/LOX fuel mix.

I took a look at Rocket Propulsion Elements, but I didn't manage to find the answer. It mainly concerns itself with how the plume changes depending on the surrounding atmosphere, and when it does mention a shape, it does so in a scale of a few metres. So, either it does not mention what I'm after, or I missed it completely.
(What I did find is this: In a small corner of a figure (p. 646, figure 18-4) there is a note that says "Vacuum limit 0.1-10 m dia.". I'm not sure if that answers my question, however, mainly because I'm worried that the diametre will increase with distance.)

Many thanks in advance.

(Edit: I'm posting this in "General Physics" rather than "Homework Questions", mainly because I don't know the equations that govern what I'm asking. Please inform me if that was the wrong forum.)

 

[mfb]

You can get a rough estimate if you know the temperature of the exhaust gas once it left the rocket. With an exhaust velocity of ~4km/s, the gas had ~5 seconds time to expand. Typical velocities of the gas follow from the gas mixture and its temperature, that allows to calculate a typical length scale. There is no fixed plume width, of course, it just gets thinner the more far away you are, with this typical length scale calculated before.

 

[Alonso Quixano]

Oooooooh! Neat!

Still, I will need one more nudge. I tried searching for the rate of expansion of a gas during a free expansion, but I didn't find any relevant equations. A little more help? (Maybe it has something to do with the mean speed of the molecules in it?)

 

[mfb]

Typical velocities = thermal velocities. The velocity a molecule has with the energy kT where k is the Boltzmann constant and T is the temperature. T will go down as the gas expands, but if you are just interested in a rough estimate it should be fine to take the temperature of the exhaust after leaving the rocket.