Heh. A far from perfect analogy, true - indeed, greatly oversimplified.
Yet, just good enough to elicit a more precise yet still entirely
non-mathematical description of the problem. For which I expect many of
us whose eyes were glazing over are grateful.
Henry
On 12/4/2016 9:36 PM, Henry Spencer wrote:
On Fri, 2 Dec 2016, Henry Vanderbilt wrote:
The rocket nozzle is also acting as an aligning reflector, but there
is a significant difference from the searchlight case. The rocket
throat will be too wide in relation to the nozzle size to be treated
as a point source of high-velocity gas molecules without losing some
efficiency. (I expect the gas molecules' bulk interaction with each
other, decreasing as expansion increases but still present, is also a
complicating factor.)
Alas for an appealing analogy, chemical-rocket exhausts are firmly in
the "continuum flow" regime, with gas density high enough that the
molecules bump into each other much more frequently than they bump into
the walls, so they flow as a compressible fluid rather than as
independent molecules. The average exhaust molecule never even comes
close to the nozzle wall; it's constantly jostling back and forth among
its neighbors, which in turn jostle their neighbors, which jostle
theirs, etc. etc. until you finally reach molecules that are bumping
along the wall. So force transmission between the average molecule and
the wall goes via many intermediary molecules and many jostlings, rather
than a single simple bounce off a surface, and the resulting behavior is
very different. This is a much more significant difference from the
searchlight case.
(In fact, even if the gas *were* thin enough for the molecules to move
pretty much independently of each other -- the "molecular flow" regime
-- a parabolic reflector still wouldn't work very well. On the
molecular scale, essentially any solid surface is rough and sticky,
rather than smooth and reflective. A gas molecule which hits the
surface tends to stick for a moment and then wander off again at lower
velocity, having transferred much of its momentum and kinetic energy to
the surface. This complicated, messy, poorly understood interaction
makes it very difficult to, e.g., predict the drag coefficient of a LEO
satellite.)
Parabolas appear in Rao's approximate form not because of any analogy to
searchlights, but just because they are one of the
mathematically-simplest curves. The true Rao nozzle is optimal -- for
one particular definition of "optimal" -- at expanding the flowing gas
to convert the thermal energy into bulk kinetic energy. However, its
curve is complicated to compute, and in common cases it's *roughly* a
parabola, so Rao came up with a quick method that approximates it as a
parabola.
Alas, it's still not entirely trivial to compute, because not only is
the parabola's axis not along the engine axis, it's typically not
parallel to the engine axis either. The two sides of the parabola are
not just moved outward, they are separately rotated inward or outward as
well. So you need to work with the equation of a rotated parabola,
which is not as simple as the usual axis-parallel-to-the-Y-coordinate
form. Hence this discussion.
Henry
