Hi Faro, Buzz here. I'm familiar with the inner workings of some of the high power piston engines as well as modern turbofans. I think you're mixing up the 2... some of the reasons that one works well don't really apply to the other kind of engine. Now if it makes economic sense, or technical sense... I'll let somebody else figure that out. (grin)
Water Injection was used on some of the high power piston engines, and older turbojet engines. In the piston engines the water / alcohol mix was used to reduce the temperature of the compressed air charge going into the cylinders - there was a slim margin before detonation would occur. Some smart people figured that cooling the incoming fuel-air mixture gave a good margin of (internal engine) safety before the fuel-air mixture would detonate while being compressed by the moving piston - before the sparkplug ignites it.
Maybe the idea of "higher octane kerosene" had confused me. Octane is used to measure piston engine resistance to pre-ignition / detonation. It's really not applicable to turbine engines... although there is a compression ratio of 27:1 in a 757 / Pratt 2000 engine which is similar to what a diesel does. Higher energy fuel?
Water Injection was used on some of the older turbojet engines to reduce the EGT margin... the difference between the EGT and the temperature where the turbine starts to suffer damage. When a turbine engine gets tired, you can only put so much fuel into it before the EGT gets to the normal max limit. If you were able to cool the EGT, then you could pour in more fuel... and hopefully get more thrust before you're maxed out in the EGT department.
Sometimes we'd get DC-10's that were EGT limited taking off on a nice warm afternoon in HNL
(Honolulu Hawaii). The Flight Engineer had to reduce power a little to keep one of the engine's EGT from going over red-line. But you park it on a nice cool night at PDX
(Portland Oregon) and we have no problem getting the normal Thrust / N1 before we get to max EGT. In short, the turbines were tired and leaking, not able to turn all the hot exhaust gasses into rotary motion to turn the big fan in the front. Maybe water injection would have helped... but they still would have had a tired engine and written it up.
When turbofan engines became common, water injected turbojet engines were retired. I recall hearing of JT3C-6 engines called "water wagons", some DC-8's had them. I don't know which model 707 used water injection. And I recall seeing diagrams of B-52's with water injection. When the TF33 / JT3-7 fan engine was installed they didn't need the extra thrust: it went from 12,000 - 14,000 lbs to 17,000 lbs.
So to add extra thrust to the airplane... most of the time it's not needed, most of the takeoffs are de-rated takeoffs: you "tell" the fuel control that's it's hot outside (how about 40 degrees C?) and the fuel flow is reduced, EGT is reduced, thrust is reduced. And there's still no problem getting off the runway.
I've been involved with a few bird strikes / mangled engines that happened just after takeoff, and the pilots had not prayed for more thrust... besides, it might make them fly in a big circle (grin).