[AR] Re: Nozzle shapes
- From: Henry Spencer <henry@xxxxxxxxxxxxxxx>
- To: Arocket List <arocket@xxxxxxxxxxxxx>
- Date: Mon, 18 Nov 2013 00:50:55 -0500 (EST)
On Sun, 17 Nov 2013, Norman Yarvin wrote:
> > ...I've never been able to go from Rao's paper to an actual set of
> > equations to design a nozzle...
>
> 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...
Note that there are *two* Rao papers on thrust-optimized nozzles -- the
first (Jet Propulsion aka ARS Journal, June 1958) uses MoC, the second
(ARS Journal, June 1960) is polynomial approximations to the resulting
shapes. Beware of discussing one when somebody else means the other...
Note also that the Rao nozzle is optimal only in certain ways under
certain assumptions. There is at least one other major design approach,
the "truncated perfect nozzle" (see, e.g., Ahlberg et al, ARS Journal May
1961), used in the RL10, Viking, and RD-0120 in particular. See also
Knuth, ARS Journal Oct. 1960, on approximations to different kinds of
"optimal".
> ... (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.)
Rao (1960) nozzles are not *paraboloids*, formed by truncating a complete
parabola and then rotating it around its axis of symmetry. However, the
wall shape *is* a *parabola*, truncated more drastically and then rotated
around an axis which is *not* its axis of symmetry.
Mathematically going from Rao's 1960 end conditions -- positions and
slopes at the endpoints of the parabolic contour -- to the equation of
the parabola is doable but not simple.
Henry Spencer
henry@xxxxxxxxxxxxxxx
(hspencer@xxxxxxxxxxxxx)
(regexpguy@xxxxxxxxx)
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