[AR] Re: Nozzle shapes
- From: Norman Yarvin <yarvin@xxxxxxxxxxxx>
- To: arocket@xxxxxxxxxxxxx
- Date: Sun, 17 Nov 2013 21:38:10 -0500
On Sun, Nov 17, 2013 at 09:54:19AM -0800, Doug Jones wrote:
>I don't have TDK, the Two Dimensional Kinematic software package, but
>the nozzle shapes it creates are logarithmic, not parabolic. (A licensed
>TDK user shared an output file with me.) I've never been able to go from
>Rao's paper to an actual set of equations to design a nozzle, but I
>strongly suspect that the best fluid dynamic shape is indeed
>logarithmic. I don't know what method TDK uses, but a brute force finite
>element analysis with an evolutionary algorithm will probably converge
>on the same shape.
The way the Rao paper gets its curves is by (1) doing the precise
calculation, via the method of characteristics, and (2) fitting a
second-order polynomial curve to the result. The idea was to make it
easy for people to build nozzles without having access to a computer
and code to do the full calculations. Judging by your remarks, he was
less than perfectly successful at this. If he were writing the paper
these days, he'd probably just put the method-of-characteristics code
on github and not bother about polynomial approximations. Output
could consist of closely-spaced (r,z) coordinate pairs.
I doubt that the optimum is any common mathematical function
(parabola, logarithm, whatever), but it's a smooth function that can
be approximated well using polynomials or any of a variety of other
smooth functions. (Rao nozzles are not, strictly speaking, parabolas:
if you build a nozzle by his recipe it won't focus light to a point,
for instance, no matter how shiny you polish it. He uses a
second-order polynomial to approximate the curve, but there's an
additional degree of freedom implicit in that: a parabola has the
focus at radius r=0, whereas he might have the focus at any r.)
--
Norman Yarvin http://yarchive.net/blog
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