What water injection does [on supercharged engines] is lower the temperature of the boosted compressed air. This shrinks the air allowing more air in the cylinder. This lower temps allowes one to increase boost and make more power....
OK, we have to also recognize the other big effects of water injection that are equally important, and in some engines, the dominant reasons for using it.
Yes, the water does cool the charge, IF and only IF the water evaporates before it gets to the cylinder. in many WWII aircraft engines, most of the water went into the cylinders in liquid form because of poor atomization. So charge cooling was not the dominant effect in that era. It was late in the war that Pratt & Whitney and Curtis-Wright figured out that injecting the air upstream of the compressor would help mechanically vaporize it, but they were more concerned with even distribution to each cylinder.
Whether the water is evaporated or not, once it is in the cylinder it will also do several other things. First, it cools the piston crown itself as it evaporates. As HP/in2 of piston area increases, cooling often becomes critical and pistons start to melt. We've all seen it. Rings weld, holes melt straight through the crown, etc. This isn't as common in modern engines (especially imports) that have dedicated cooling oil jetted right at the bottom of the pistons. As we boost up modern engines, we use TBC's (thermal barrier coatings) to keep the heat flux under control, water is a last resort. Among other things, proper cooling from whatever combination of oil, TBC, and water prevents pre-ignition. Pre-ignition is often called detonation, but it is a different effect and cooling the intake charge has a limited effect on it compared to cooling the piston.
Second, water also retards the burn rate of the end gas; this is detonation prevention equal to cooling the intake. It is why cooling the intake with water injection has more effect than lowering the As the flame moves across the piston from the point of ignition, the remaining gasses compress to far higher levels than where the spark went off. This effectively raises the compression ratio for the end gas and leads to detonation. Water in the mix slows down the burn rate. This actually
absorbs power early in the burn and reduces engine output. In exchange, it takes energy out of the end gas burn, often lowering its energy below the threshold for detonation.
Last, and most important for nitrous engines, small amounts of water provide a large increase in the working gas. "Working gas" is what we call the nitrogen, CO2, H2O, and Argon that makes up the un-burnable atmosphere and the combustion products from the fuel and O2 in the air (more CO2 and H2O). If there is more heat than the engine can handle, adding water increases the working gas and creates more pressure and less heat from a given amount of combustion. For this effect to be dominant, the water actually
needs to get into the cylinder in liquid form.
When using nitrous, we are adding fuel and oxidizer and not adding anywhere near as much working gas. This adds heat. So an engine making a given amount of power with nitrous will run hotter and pre-ignite/detonate/melt sooner than the same engine on boost, all other things being equal. In the case of nitrous, more than any other, water is an important tool to provide this working gas.
Before it is used, TBC's and proper cooling MUST be used. Otherwise, water is just a band-aid. About half of the air racing crowd has finally figured this out, the other half is still burning engines down no matter how much water they use.