I found a link that helps to answer part of this.
http://antoine.frostburg.edu/chem/senese/101/gases/faq/wet-air-dry-air-density.shtml
It states simply that
at the same temperature and pressure, air that is more humid is less oxygen-dense than air that is less humid.
The underlined point is important as that is where I'm unsure of the answer. So it's safe to say that on a day that is more humid the air going into the turbo is less dense than the air that goes into the turbo on a day that is less humid. That makes sense. We all know engines make more power on cool, dry days (what they always hope for in a race so we can see higher speeds). That would be due to the density of the air that the engine is ingesting on that day, and if the amount of fuel is increased proportionately so the air/fuel ratio stays the same, then you're making more power (generally speaking). That would be true for forced induction or naturally aspirated, diesel or gasoline.
Now I'm thinking of the space between the turbocharger and the intake valve as being a container of a specific volume - especially if a single cylinder engine is used for the example for simplicity's sake. Now if you compress air that has a known temperature and humidity with a turbocharger you have air with the same density ratio in that space, just more of it. Unfortunately heat was also added to the mix due to the nature of compressing a gas. That would increase the pressure of the air because the air is expanding due to the addition of heat. That could make it harder for the turbo to compress the air because the air inside is pushing back more due to the heat. This in my mind is where the water injection has its benefit - by injecting the water in the space between the turbo and intake valve, it cools the air back down. This cooling makes the air contract, which on its own makes the air more dense, but then it gives more space for the turbo to push more air into the space, which also increases density. So now there is more oxygen in the same space that you had before and next time the intake valve opens to allow air in it there would be more oxygen entering the combustion chamber. However, if you're forcing water vapor into this same pressurized space, I think it could also add a push that could prevent the turbocharger from forcing air in like it wants because you're trying to add more "stuff" into this same finite space that is trying to stay at the same pressure. But, I think it's safe to say that the cooling benefits of adding the water outweigh the negatives of forcing more stuff in the space.
A second thing that water injection can do is deter detonation. Humid air would have that same affect on a gasoline engine, naturally aspirated or forced induction. It would cool the combustion event, which would stave off preignition which would increase power. It could even allow more ignition advance which could add more power. As I understand it, water injection will have some of the same effects on our forced induction diesel engines because cooling the combustion will make our EGTs stay lower so we can keep our foot in the throttle longer and keep making a higher level of power longer. Humid air would do the same thing, yes, but at the expense of oxygen, so is that a good trade-off? That's what I don't know. My gut feel is that it isn't a fair trade, but I don't know that for sure.
Simplified:
1) humid air at the same temperature has less oxygen per volume than air that is less humid
2) does the vapor in the air overcome the lesser oxygen by cooling the charge enough for it to be benefit? I don't know the answer to that, but my gut feel is that it doesn't.
Here's their website:
