Macroscopic shape of rocket exhaust

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.)

(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.)

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