With the knowledge required, could the pump jet be created for use in ships in the given timeline? If not, what other forms of engines can be made in said timeline that can power ships without the use of sails?
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Yes, a pump jet or a screw propeller or a paddle wheel could be built.
No, they would not be efficient, friction would be high, shafts would leak, gearing (if any) would wear out quickly.
If there was a need to do something other than oars or sails, perhaps because the ship has to be armored, my favorite would be paddle wheels either between the hulls of a catamaran (if armored) or on the sides of a monohull vessel.
- The shafts would be above the waterline.
- Relatively easy shape, less strain on it than on a propeller.
- Relatively low RPM. There could be oxen walking in circle with a few gears.
But my gut feeling is that this would be less efficient than oars. The Royal Navy experimented with muscle-powered paddle wheels to move ships-of-the-line without or against the wind, but speed was limited to a few knots.
Yes, as a novelty.
Pump jet is essentially an Archimedes' screw enclosed in a pipe that is used to propel watercraft. Conceptually, this can certainly be implemented in 15-16 century. However, making this design practical would require centuries of progress.
Pump jet is not an engine. It is a propelling device that requires an engine. At the given time, humanity simply did not have any engines that could be used to drive this jet. A practical steam engine was developed only in mid-18 century. Hand-cranking pump jet may work as "proof of concept", but would be much less efficient than paddling.
Second issue is materials and engineering. Modern pump jets are a result of decades and even centuries of refinement. An inventor in 15-16 centuries would not have access to the steels and alloys that had become available only in industrial era, and probably would had to stick with brass.
A Voith Schneider -like propeller might work: Vertical blades mounted on a spinning disc, with a mechanism to periodically alter the direction the blades are facing.
Nice: Thrust vector can be produced in any horizontal direction, and changed in a second -the disc only ever needs to spin in one direction, one speed. Vertical axle lends itself to various walk-in-circles human/cattle engine schemes.
Bad: Bladelength determines maximum Thrust, and is itself bounded by the competency and available materials of your shipwright. Shallow water is bad.