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What if the water jet is placed at the center of the compressor wheel? the radial speed is at it lowest and also give the the water droplets to fan outwards rather than hitting the tips randomly.
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That has been used successfully. I've mentioned in the past a guy that injects directly onto the impeller hub. When the liquid hits the rapidly spinning impeller shaft it is instantly atomized.
NACA also did some experiments along these lines developing a drilled compressor hub that channeled the ADI fluid directly into the impeller it self.
(see NACA Memorandum Report E6C25 May 1946 --- "Hydraulic Characteristics of the NACA Injection Impeller" & NACA Memorandum Report E4L23a "Effect of the NACA Injection Impeller on the Mixture Distribution of the Wright R-3350 Engine" Available as NACA TN 1069 1946, also Report 821 "Effect of NACA Injection Impeller on Mixture Distribution of Double-Row Radial Aircraft Engine)
Unfortunately these sort of investigations ground to a halt when the transition to turbojet engines began in the late 40's early 1950's
Larry