Many rockets designed after the V2 but prior to Saturn V, do not show the typical "Bell" shaped diverging nozzle. Why is it so? Do they achieve supersonic speeds for the exhaust gases in any other way?


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You don't need a "bell-shaped" nozzle for a rocket -- a simple cone (often with an included angle of 30 degrees, it seems) is sufficient.

Cones are easier to design (they only have one design parameter, which isn't even that sensitive) and easier to construct (they can be made with the simplest machining methods). For this reason, many early rockets used plain cone nozzles, and some missiles and small thrusters that don't need especially high performance still use them. It also seems like they sometimes crop up in experimental rockets where the nozzle is not the interesting part, such as some early NERVA prototypes.

Cone nozzles don't use a separate principle from bell nozzles -- it's the same supersonic/compressible-flow expansion and acceleration of gas, in which the cross-sectional area of a channel gradually and monotonically increases. In fact, any converging-diverging duct will accelerate compressed gas to supersonic speed as long as there is enough chamber pressure relative to ambient pressure to make it get supersonic in the throat. Bell nozzles, however, must be carefully designed to tailor the expansion and make compression and expansion shock waves (created by the curvature) line up, and the parameters are more sensitive. You can read about the design of both kinds of nozzles (as well as a bit about aerospikes) in these university lecture slides from Purdue -- there's a fair amount of math.

There are two big benefits of bell nozzles. First, compared to a cone nozzle, they're significantly smaller and therefore lighter. Second, the bell shape focuses the exhaust stream into a narrow, low-divergence jet, while the cone nozzle produces a jet with the same divergence angle as the width of the cone (at least when the engine is operating at its design altitude). While this wastes some Isp, the loss isn't disastrously high.

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    $\begingroup$ The nozzle bell shape acts similar to a parabolic mirror of a reflector, deflecting particles of the exhaust bouncing off it at angles that keep the exhaust going as straight out as possible; length and scale of the curve differ between engines but the principle is identical. A simpler 'reflector' shape will dissipate a larger part of the exhaust at sideways angles, but the losses aren't very big. $\endgroup$
    – SF.
    Jun 3, 2020 at 15:55

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