Going back 40 years puts a strain on the old memory cells but I am fairly certain that the effect on the Avon of water/meth injection was to increase the max RPM at the limiting JPT. The throttles were fully advanced and the RPM in dry power was automatically limited to the max JPT. On selecting water/meth the RPM increased by some 2 to 3 % with the JPT remaining the same, a noticeable increase in thrust and significantly for stream take-offs, a much smokier exhaust. Certainly in that application the cooling of the turbine would seem to have been the raisin d'etre.