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While doing an image search, I found these images of propellers: prop1

and

prop2

Some appear to be straight-ish on the leading edge and curved on the trailing, some curved on the leading edge and straght-ish on the trailing, and symmetrically curved and some pretty squared off.

It seems like, well, a propeller is a propeller. They all need to push the air back in a way as to propel (hence the name?) the plane forward.

Wouldn't there be a mathmatically 'best' design?

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The best design depends on what your goal is. This changes with operating conditions:

Best power/thrust ratio

For lightly loaded propellers this would be minimum induced loss. These are the propellers on human-powered aircraft, windmills or motor gliders. Minimum induced loss is achieved by distributing circulation elliptically across the propeller disc, and a typical consequence is a small chord near the tips and a low number of propeller blades. If you want to research this more, I recommend to play around with XRotor by Mark Drela (and maybe reading this). Mark used it to design the MLE (below, NASA photo EC87-0014-8) and Dedalus propeller.

MIT MLE in flight

Maximum thrust at high speed

Highly loaded propellers, which can be found on large turboprop aircraft, are less concerned with induced loss, but need to put out as much thrust as possible with a limited prop area. The diameter is restricted to limit tip speeds, and the tips typically are mildly supersonic. To limit the suction peaks on the blades, the local lift coefficient needs to be low, which is achieved by increasing chord and the number of blades. Increasingly, now tip sweep is also used, but this drives up the torsional loads on the blades and leads to warpage in operation. An extreme example is the P-27 prop on the D-27 turboshaft of the An-70 transport (below, picture by Marina Lystseva).

An-70 with P-27 propeller

Ship propellers

They operate in a medium which is 800 times denser than air, and the main concern is to avoid cavitation. This again means to limit suction peaks and leads to very high blade chords. To limit draft, ship propellers need to have extremely small diameters.

Five-bladed ship propeller

Submarine propellers

For submarines, add minimum noise emission to the design goals. Minimum draft is not a concern, so the propeller blades are less stubby. Their sweep distributes the cutting through the wake of the rudders over time, which helps a lot to reduce noise.

Submarine propeller

The fancy shapes of some model aircraft propellers have more to do with marketing or unscientific voodoo beliefs by some hobbyists. Generally, a straight shape works best.

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    $\begingroup$ Re: Submarine propellers, there's actually a scale model of one on display at the Smithsonian Institution. They went through a lot of trouble to get the thing, too. $\endgroup$ – voretaq7 Oct 3 '14 at 20:19
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    $\begingroup$ "For obvious reasons, I cannot post a submarine propeller photo here" This is not obvious. Why? $\endgroup$ – Vortico Oct 3 '14 at 20:27
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    $\begingroup$ @Vortico: Well, I could post an old photo. State-of-the-art submarine propellers are top secret. They are even hung with covers when the boat is in the dry dock, so satellite photos wouldn't reveal anything. $\endgroup$ – Peter Kämpf Oct 3 '14 at 20:42
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    $\begingroup$ Article at this link has a photo of an uncovered sub in drydock, visible on Bing Maps, and comments on the controversy that happened when this image was discovered. $\endgroup$ – Ross Presser Oct 3 '14 at 20:53
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    $\begingroup$ @RossPresser 's link is dead and 404s, but is on the wayback machine: web.archive.org/web/20150423005349/http://… $\endgroup$ – Baldrickk Jul 9 at 13:35
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A couple of extra points for piston engine aircraft:

The size and amount of propeller blades mostly depends on the power output of the engine. A giant prop on a tiny engine won't be good, because it consumes too much energy just to spin it. Conversely, a small prop on a big engine would waste a lot of the engine output, as the prop would overspeed before the engine reached peak performance.

Matching prop size to engine is one of the challenges for designers. As mentioned in the previous answer, it depends on what the aircraft will be used for.

Also, most propellers do not actually move the aircraft by 'pushing' the air behind. If that were the case it would not make any sense to have them on the leading edge of the wing or on the nose - you should place them with clear air behind. The propeller is actually an aerofoil, producing lift using the same principle as a wing. Negative pressure is created on the 'upward' side of the prop, so the propeller 'pulls' the aircraft forward.

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    $\begingroup$ A wing also pushes air down. Looking at the pressure is describing the same physical process in a different way. So propellers do push air behind, and they do this by accelerating it by means of suction on the forward-facing part of the blade. And pusher propellers are indeed the more efficient way of mounting a prop, but have problems of their own. $\endgroup$ – Peter Kämpf Oct 5 '14 at 10:19
  • $\begingroup$ Ben, in the AN-70 picture above, how do the second sets of props not interfere with the first? I know, you said most. I'm only an enthusiast, to me, a lot of flying is magic, but I'm learning. Talking about how a piston-driven prop works is somewhat akin to how electricity works.... hole flow? or electron flow? $\endgroup$ – CGCampbell Oct 5 '14 at 13:25
  • $\begingroup$ Unfortunately I am by no means an expert - I am still learning every day. The AN-70 has a different type of engine known as a Propfan, which I understand to be almost a hybrid between a turbofan and a turboprop engine. I could therefore make a guess as to how the two props interact but I think I will leave that to the experts :) $\endgroup$ – Ben Oct 6 '14 at 10:52

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