Has anyone actually fired a Rao nozzle, its parabolic approximation, and
parabolas a degree or two of exit angle on either side of it and compared
the real world results? Cf efficiency is such a small part of overall
thrust that it seems like anything that is vaguely correct is going to give
you a high nineties solution. We're not building SSME nozzles.
Look at some of the guys doing solids; for a lot of them dogma is to make a
nozzle by connecting two conical sections with a not-insignificant straight
section drilled through. The loss from that straight part is way worse than
any slightly off-nominal parabola, yet they seem to keep flying them.
To the original question, my approach has generally been to draw a parabola
in Solidworks, make it a set percentage shorter than a 15 degree cone, set
the end constraints, and sweep it around the axis. The idea of getting
points out of some other software seems like it would lead to excessive
dicking around when you realize you want to change some geometry, unless
you can do it by design tables. The focus of the parabola is definitely not
on the axis of the nozzle.
Alexander, does RPA do the full Rao method?
Armadillo did some engines with conical nozzles because the engines were
film cooled and you can roll a cone out of sheet steel a hell of a lot
cheaper than turning it from a block of metal. They worked.
Keep in mind that the 15% half angle that the parabola approximation is 80%
of is itself a compromise. A smaller half angle would be higher
performance, but becomes excessively long.
Ben
On Sunday, December 4, 2016, Norman Yarvin <yarvin@xxxxxxxxxxxx> wrote:
On Fri, Dec 02, 2016 at 01:34:09PM -0700, Henry Vanderbilt wrote:
(I expect the gas molecules' bulk interaction with each other,
decreasing as expansion increases but still present, is also a
complicating factor.)
It's not just a complicating factor; it's an overwhelming factor. The
mean free path -- the average distance a gas molecule moves before
hitting another gas molecule -- is microscopic; that's true for any
gas concentration that's large enough to exert any pressure that's
useful for propulsion. So as annoyingly complicated as fluid dynamics
is :-), it's what one has to use here.
Anyway, probably the best way to think of the Rao approximation is as
its creator did: a second-order polynomial approximation. Even a
first-order polynomial approximation (a straight line, which when
turned into a surface of revolution becomes a cone) loses very little
efficiency (something like 1%). A second-order approximation, again
turned into a surface of revolution, recovers most of that 1%. It
wasn't worth doing a third-order approximation.
(As others have mentioned, although the second-order formula gives a
parabola, when you chop out the proper piece of that parabola and
sweep it around an axis to turn it into a surface of revolution, the
resulting surface is not a parabola.)
These days, with ubiquitous computers, one might skip the
approximations and just go to Rao's papers and implement the full
method-of-characteristics solution -- the one that this polynomial
approximation is an approximation to. In his era, publishing a method
by which computerless engineers could work out a good approximate
solution with pen and paper was a worthwhile thing to do; in our era,
we can just put source code on Github for the exact solution.
