It takes heat to turn water into steam. If you inject cold water at the top of a compression stroke, some may turn into steam, but the process of turning cold water into steam absorbs the heat from the compressed air, and the pressure will DROP, not rise. It will consume mechanical power to do this. As the piston starts down, the cooled air cools even more as the pressure drops, which can cause water to condense, further reducing the pressure against the piston going down. It will take far more power to push the piston up, inject the water, than will be recovered on the way down.
There is nothing to gain here. NOTHING. It will consume water, fuel, and time, and you'll have a severe loss in fuel economy. You could, in theory, inject superheated water under high pressure, but that consumes large amoutns of fuel to heat the water and keep it compressed (not flash to steam). There is no efficiency to be gained here. None.
How does the pressure drop as water expands into steam? I agree that the pressure will decrease as the piston travels down, but it does with burning fuel as well. The water would not condense, as the piston and cylinder walls will still be hot. Water expanding into steam can create extremely high pressure. Are you telling me that steam engines don't work? The only difference here is that I will be using the heat from the previous combustion to make the steam.
Ok, think carefully about this. What turns the water into steam? Heat. How much heat is in the air? How much steam can you create with the heat contained in the small amount of air that's compressed inside the cylinder? At what pressure? You can figure all this out by calculating the amount of air that's in the cylinder, and how much you compress it. From this, you can determine the btu's of heat that the air contains, and how much water you can evaporate with the btu's avialable. At this point, it's a downhill slope.
What I suggest you do, is to figure out the pressure / temperature slope for water turning into steam, remember, the ONLY thing steam is, is evaporated water. And, the heat from the air compressed in the cylinder is absorbed evaporating the water. The only other heat that's available, is to suck the heat out of the pistons and cylinders and combustion chamber, heat that will have to be absorbed the next combustion stroke, instead of being converted to pressure to drive the piston.
The efficiency loss involved in a cold engine is NOT just from "running rich". It's from large amounts of heat absorption from the combustion, and it is from reduced combustion efficiency, as the lesser amount of heat vaporizes the fuel less well. The reason it runs rich is because it is difficult to ignite the mixture - and that's becuase there's not enough heat to vaporize the fuel well.
Now, as your air is cooled by evaporating water, you now have the piston begin to travel downward, relieving the pressure, but as your pressure falls, so does the temperature. As the temperature falls, the steam reconverts back to liquid. You may not see it like fog, or vapor, but it has recondensed, dropping the pressure even faster.
I'm sorry, if there was a "magical perpetual motion" machine, it might work. But there is no energy created by shoving water into the cylinder. There is simple physics of heat being absorbed by water. This can't possibly produce MORE mechanical energy than you put into the system = the only energy input is going to be the force requird to shove the water in and the generated kenetic energy from the previous combustion stroke shoving the piston up due to the flywheel effect. There has to be an energy input, and that energy input is GOING to be the fuel you burn.