Can an airfoil generate enough lift, if it is contained in an enclosed space, to lift the enclosed space? Assume that there would be sufficient airflow over the foil to generate lift if it was not in an enclosed space to lift the airfoil and the enclosed space if it were externally attached.
Can an airfoil generate enough lift, if it is contained in an enclosed space, to lift the enclosed space?
If this were possible, the enclosed structure would rise up through the atmosphere without creating any downwash at all. It's intuitively clear that this would not be possible.
In fact, some upwash would be created by drag on the exterior walls of the enclosed structure as it rises up through the atmosphere. This would be inconsistent with the principle of conservation of momentum.
If you had the enclosed structure supported above the ground on a stand, and someone came and took away the stand at exactly the same instant the fan inside the enclosed structure started blowing air over the airfoil and (theoretically) creating lift exactly equal to the weight of the enclosed structure ( + airfoil), so that the vertical speed of the enclosed structure remained at zero even with the stand removed, the conservation of momentum argument would no longer apply. Still, it's intuitively clear that this enclosed structure could not hover above the ground without creating any downwash.
You need to first understand what is lift in this situation.
In all 3 diagrams. Lift is the force between the airfoil and the enclosed space. The lift will generate a force that pulls the airfoil towards the ceiling of the space and away from the floor. It has nothing to do with the relationship of the enclosed space and the outside world.
This situation is exactly the same as you pulling a rope on the floor:
Is it ever possible to lift the floor you are standing on by pulling on a rope attached to it?
Once you see it this way it is obvious that you cannot lift yourself by pulling on a rope attached to the floor. The only way for it to work is to step outside the floor onto the ground around it - only then can you lift the floor because you are now generating a force (lift) against the ground instead of against the floor.
Similarly, if you fully enclose the air it will not lift the space around it. You need to let the air have access to the space outside the enclosed space in order for it to work because you need to generate lift relative to the outside instead of relative to the enclosed space.
Another way to get to the correct answer of "no" is that all your designs have wing sections in a tube. Any low pressure created on the upper surface of these sections is replicated on the bottom of the tube just above, cancelling any lift generated. What you have made is a venturi, not a lifting device.
btw In option 2 no airflow beyond leakage is possible as there is no outlet. You just pressurize your container.
In short, no. A aerofoil works by creating a region of high pressure below the wing and low pressure above, this provides lift.
In the enclosed system, we still have high pressure below and low pressure above the aerofoil but countering that we have normal atmospheric pressure above and below the contining tube with both providing a net downward force which equals and cancels the upward force experienced by the aerofoil.
Sure, depending how you define "Lift"
Option 1, no way. You could possibly make it leap into the air by sloshing the air around violently enough, but that's unlikely, unsustainable, and too much of a stretch of the definition.
Option 2, sure. If your wing is near enough to the exit, it could redirect the air down, which will push the system up. It's hard to imagine that providing enough lift to fly.
Option 3, maybe. If you implemented it in a way that drives significant pressure changes, you might be able to get a pressure differential that causes the diffusing structure to lift up.
In both Option 2 & 3 though, the wings basically don't matter. The wind tunnel is acting as an overly-elaborate airfoil.
So. Does that count?
Expanding on @pilothead's answer, remember that the airfoil generates lift by creating a low pressure area above the wing. This low pressure area isn't just lifting the wing up, it is pulling in air and whatnot from every direction. The airfoil lift's because it is part of the "whatnot". In your enclosed space, roof of your enclosure is also part of the "whatnot" and the exact same amount of force placed on the upper surface of the wing is placed on the enclosure above the low pressure area, pulling it downward.
Basically, it means that since the low pressure area is inside the enclosure, it can't lift the enclosure. You would need a low pressure area above the enclosure in order to lift the enclosure.
Option one is a fully closed system so its center of gravity will not (in steady state) be affected of the internal fans and wings
Since all flow is inward bound in option two it would either go to zero or pressure would go to infinity, so no lift.
But for option three, since the geometry produces circulation around the upwards facing tube-bends, it might actually lift off. Not perhaps because of the wings but rather because of the geometry of the tubing and the fans.