Some of the later engines had supercharger
systems that were approaching the size and weight of
power section (engine block/cylinders).
Technically very interesting but the increasing complication was a headache.
Multi-stage compressors (or pumps) were nothing new, they had been used in industrial applications of decades before WW II. But they usually ran at a fixed speed ( or small speed variation) and little or no change in intake air pressure or density. Getting stages to
play nicely with each other in the widely changing conditions of aircraft use was a real challenge.
AS an illustration here is a compresser map from a modern turbo:
Note that while it
can reach a pressure ratio of 3:1 it is only at one specific air flow.
The curved (or wavy) lines that run from left to right with the big numbers (74990, 948?0, 135018 ) are the rpm of the impeller.
Notice that the pressure ratio falls off considerable as the airflow goes past about 22.5lbs/min. The impeller is 'choking' and cannot supply much more air.
Also note the left most line that runs from about 5lb/min and PR of about 1.4 up to about 18lb/min and just under a 3.0 pressure ratio. ANYTHING to left and the impeller is "stalling", the impeller is trying to pull a vacuum and the airflow "stalls" or slows down/stops until enough air builds up in the intake to continue flowing again, this can happen multiple times per second and if bad enough can send small shock waves back through the intake duct/s. This is the cause of the "rumbling" some F4F pilots complained about.
Rolls Royce got
lucky in that the Vulture supercharger was actually a good match to the Merlin supercharger right from the start and only needed minor tweaking. Allison lost time trying to use a second same sized impeller to feed the engine supercharger. P W spent a lot of time working out the impellers and housings to eliminate the problems. Somebody at R-R was reluctant to start work on the 2 stage system because they were afraid the second stage would multiply any mistakes/problems in the first stage.
Please remember that these guys were working though
ALL the problems and tricks concerning airflow though engines that we almost take for granted today.