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I have seen "engine skirt" and "engine nozzle" both used for the bell-shaped end of a rocket engine. Is there a difference?

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The terminology is even less rigorous than that! Skirt, nozzle, and bell can informally refer to the same thing.

Bell seems to be shorthand for bell nozzle, a common shape for a nozzle.

One chapter in a book (preview) confusingly does not distinguish between skirt, skirts, and skirt structures. "Skirts" suggests Victorian hoop skirts, which indeed look like the loud end of a rocket. But in clothing at least the plural usage has become archaic, so that may be where the singular "skirt" came from.

Two special cases distinguish nozzle from skirt, though:

  • An expanding nozzle has two skirts. Here the nozzle is (as usual) the structure that converts high pressure gas into high speed gas, but it does so by means of two concentric skirts, one at a time.

    A plug nozzle as in a garden hose sprayer again converts pressure into speed, but with a blockage in the middle of the flow. So here the nozzle is the entire structure, while the skirt is just the outer wall, excluding the central blockage.

Lijat's answer is a third example of a skirt being part of a nozzle.

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    $\begingroup$ +1, I've heard "bells" quite a bit in my work experience when talking about nozzles, "skirts" never. In shuttle a "skirt" was a (usually) flared region but not of an engine. The aft most structural part of an SRB (not the nozzle, but the outward flaring bottom end of the tube) was called the "aft skirt". Confusingly the SRB "forward skirt" was a cylinder! $\endgroup$ Commented Sep 15, 2019 at 19:22
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    $\begingroup$ More cylindrical "skirts" are on the Saturn V, namely the shrouds around the engine nozzles for all stages except the first. $\endgroup$ Commented Sep 15, 2019 at 19:26
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The engine skirt is a nozzle extension of the original bell. For example, SpaceX's Merlin 1D engine and its vacuum variant (the Merlin 1D vacuum) which I'll refer to as the Mvac engine, the engine 'skirt' is the part of a nozzle extension for the vacuum variant (the part that glows red when firing) as for other engines, if they are NOT nozzle extensions, (for example the nozzle extension on the Saturn V's J2 engine) they are generally refered to as the 'Bell'.

The best way to differentiate 'Bells' and 'Skirts' is still to check if they are part of the bell (meaning made together) or are a nozzle extension (meaning made separately or are for variants).

However, there are some exceptions, for example: usually, for SpaceX engines, vacuum variants have their nozzle extensions called 'Skirts' (e.g. Mvac, Raptor Vacuum....) but Aerojet Rocketdyne which makes the RL-10, RS-25, and RS-68 engines (just a few to list) , don't call them 'Skirts', just bells. So, the line is not really clear cut, it's what the manufacturer calls them. So, just follow whatever the manufacturer calls them.

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