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It has often been said on this site that induced drag on wings is the inevitable result of producing lift over a finite length. Converting this to the realm of propulsion, I assume that propellers have "induced drag" or something similar, as they produce thrust over a finite length.

With a ducted fan, the airstreams above and below the airfoil can't just "slip around the side," because they're blocked by the duct. My intuition thus suggests that ducted fans would not have induced drag (or its equivalent), assuming that the space between the blades and the duct is negligible.

Is this accurate?

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EDIT: I'll add this ahead of the answer because it will necessarily come up: This answer is based on the model described by Prandtl, which is only that: A model, a simplified description for what happens that works to explain some phenomena. There is some disagreement about this model, so a separate explanation using a differen theory might help. The problem is that an explanation with pressure differences doesn't help much here, at least I wouldn't know how to answer the question without working with Helmholtz/Prandtl.

Fundamentally, what we are talking about are vortex rings that are created because air is moved in some direction somewhere and not moved everywhere else. That's independent of how you move it, and fundamentally, a duct or winglet or something similar can't change that. You can look at the vortex itself as the thing that actually creates lift or thrust, so you want and need that vortex.

Then there's Helmholtz's theorems. Quick quote from that article:

  • Helmholtz’s first theorem:
    The strength of a vortex filament is constant along its length.

  • Helmholtz’s second theorem:
    A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path.

In the case of a normal wing, the vortex ring ("closed path") consists of one side along the length of the wing, two sides trailing behind the wingtips and the fourth side is somewhat lost, or far enought away that it can mostly be ignored. The three sides that are not along the wing itself are the problem, they are what the term "induced drag" refers to. Because there is no surface at the end of the wing for the vortex to end at, it has to continue into infinity. And there is some friction involved in creating long fast-spinning vortices, so you get drag. (Any help in explaining this part more eloquently is appreciated)

Because the amount of lift/thrust that you want is given, the vortex strength isn't something that can be changed either. What can be changed, however, is the distribution/speed/radius. If the lift distribution is elliptical (see: Spitfire), the lift gradient is minimal across the wing so there is no strong and fast vortex at the end, it is gradually created along the whole wingspan and the induced drag is minimal. Winglets do something similar by forcing the vortex to go all the way around the winglet.

In the case of a propeller, the inner side of the vortex ends at the spinner or shaft, so the ring is broken and you are only left with a linear vortex, but it is still trailing infinitely into the air at the tips of the blades. You can look at a wing in the same way if you interrupt the ring at the fuselage.

In the case of a ducted fan, you have two fluid boundaries, both on the inside and on the outside, which appears to solve the problem. You can have a vortex around the blade only and no vortices trailing into infinity creating drag. The problem is that the fluid doesn't really end at the duct because the duct has a finite length. So what you get is a large vortex around the whole duct. Air is moving in at the front and out at the back, and somewhere, that air moves back to the front far away from the engine.

In the case of a ducted fan, the vortex ring is created in the place where the moving air meets the static air at the beginning and end of the duct. Instead of a very small vortex around the tip of the wing (and along the wing, to not mess up Helmholtz's theorems), you get a larger one around the whole duct.

Therefore, I wouldn't say that the duct can eliminate induced drag. The effect is fundamentally comparable to a wingtip device: It increases the inner radius of the vortex and thereby reduces the induced drag significantly. It's far more effective at that than a wingtip device, but it has to be because ducted fans are usually relatively small so the drag would otherwise be enormous.

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  • $\begingroup$ Friction is what dissipates the vortex. The energy for creating it simply goes to kinetic energy of the air that is now moving in circles (drag is simply loosing energy). $\endgroup$ – Jan Hudec Apr 10 '15 at 18:17
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    $\begingroup$ In fact, the energy argument allows deriving the efficiency without considering vortices at all. Simply take the amount of air accelerated (down in case of wing, aft in case of propeller or fan) and it's speed and that gives you the impulse and energy per unit of time and those two give you the lifting/propulsion efficiency. $\endgroup$ – Jan Hudec Apr 10 '15 at 18:24
  • $\begingroup$ The energy argument allows you to not consider the vortices at all, but that's (in my opinion) exactly the problem with it here: To understand the effects of vortices, an approach that removes the vortices from the picture doesn't help. They are still there, they are just hidden inside of the control volume. $\endgroup$ – JulianHzg Apr 10 '15 at 18:30
  • $\begingroup$ What's the vortex ring you keep talking about? What shape is it/where on the wing is it? $\endgroup$ – raptortech97 Apr 11 '15 at 0:21
  • $\begingroup$ @JanHudec I'd really appreciate an energy/impulse argument. I've tried doing it out myself, but I've never been able to get something I'm satisfied with. $\endgroup$ – raptortech97 Apr 11 '15 at 1:13
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You are absolutely right, a propeller suffers from induced drag just as a wing does, and there are methods to minimize this induced drag. Simply put, you should distribute thrust elliptically over the prop blade's span and accelerate all air flowing through the prop disc by an equal amount. This was first found by A. Betz in 1919 and later refined by Larabee.

However, adding a duct will not change the basic reason why induced drag occurs. We had this discussion already here and here. A duct helps to equalize and decelerate the flow and is used in fan engines, because they prefer to operate in air flowing at a speed around Mach 0.5. But it will not help to reduce induced drag.

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