Does shrouding a propeller minimize induced drag by equalizing the downwash velocity along its blades? EDIT: It's not a duplicate of Are ducted fans more efficient? That question and the answers doesn't address the reason for the higher theoretical efficiency, it is more about efficiency in practice (drag on the duct, weight, etc) and hence why they aren't used despite the higher theoretical efficiency

I was told shrouded propellers are more efficient becuase tip vorticies are eliminated by the wall which would imply no induced drag but apparently that is wrong Do ducted fans eliminate induced drag? therefore I've been trying to figure out why they are still more efficient than an open propeller despite still having induced drag. It must have less induced drag.

The vortex around an unshrouded propeller:



At first I thought the wall somehow increases the effective wingspan, moving the vorticies to the top of the wall much like winglets do and thus reducing the induced drag that way



However unlike winglets the walls don't have a pressure difference on either side (it's not an airfoil) therefore the vortex can't be there.

So the vortex must be around the whole wall because above the wall the pressure is low and below it the pressure is high. 



But this doesn't change the effective wingspan so why does it have less induced drag?

My explanation is that the existence of the wall causes the inside of the vortex to "straighten out" with the flow through the duct which makes the downwash velocity constant along the length of the blades (since the flow is irrotational). This means that this condition is satisfied:

 

from page 7 of http://naca.central.cranfield.ac.uk/reports/1923/naca-report-121.pdf

Therefore induced drag is minimized. Is this correct?
Basically, yes. The difference between shrouded and unshrouded propeller is that the shrouded one can produce uniform thrust across the diameter, while for the unshrouded one the thrust decreases near the tips.

That way a shrouded propeller accelerates more air than an unshrouded one of the same diameter. This air therefore needs to be accelerated to lower speed, and therefore carries away less kinetic energy, requiring less induced power¹.

However, diameter can be varied, so the efficiency comparison is not that straightforward. When the propeller spins relatively slowly, making it larger is better, similarly to how increasing wing span is better, aerodynamically, than adding winglets.

However increasing the speed of the tips increases parasite drag, especially if it becomes supersonic. And since increasing size while maintaining angular speed increases the orbital speed of the tips, increasing size only helps to a certain point. That's when shrouds become useful.



¹ In propellers and rotors it is called induced power rather than induced drag, because it counts directly against the engine power. It also describes the physics better, since in both cases it is the work that is done on the air by the reaction to the generated lift/thrust.