Shortround6
Lieutenant General
Wright R-1820 engine used a two-stage supercharger with an 11in impeller and two gears. Low = 7.14:1 and High 10.0:1. Granted the R-1820 only turned 2500rpm but with the size of impeller and the gear ratio the tip speed was 1,199fps. R-1820 was rated at 1000hp at 13,600ft (?). So using the existing Wright R-1820 supercharger was not going to work on the Allison.Sure but both the Merlin and domestic P&W etc. types were potentially available to Allison in 1941-42. So we aren't necessarily talking about designing from scratch so much as adapting something.
Idea being to gain power, not lose it
Perhaps there was room in the R-1830 supercharger. 11 in rotor, 8.57 supercharger gear, 2700rpm, tip speed 1,097fps. R-1830 was rated at 1050hp at 13,100ft.
Maybe it they could have (would have?) used a higher tip speed with the higher rpm engine but it doesn't look like a lot room.
P-W upgraded the supercharger used in the later R-2000 engines (C-54 engines) to 11.3in impeller and used a 9.52:1 high gear ratio.
At 2700rpm they got 1100hp at 16,000ft using a 1266fps tip speed. An awful lot work to pretty much match what the Allison could do with a set of 9.60 gears.
Wright got the late R-1820s up to 1000hp at 17,000ft in the FM-2s, in part by getting the engine up to 2600rpm and later up to 2700rpm. Not sure what, if anything they did to supercharger.
Now maybe you could have tried using the supercharger off of an R-2600 but that really seems to be like a lot of work. It also doesn't offer much change in pressure even if a lot more volume and with changing things you are going to trading volume for pressure and quite possibly pushing the supercharger into a less efficient operating area.
Using a two-stage supercharger, in theory, offered better efficiency and with either the P&W system (3 speeds) or the Allison system (variable speed) better matching of the airflows through the two supercharger stages.
This was the advantage of the GE turbos. In theory, constant pressure and volume to the carburetor inlet on the engine supercharger.
The 'miracle' of the two-stage Merlin was that Hooker and the rest of RR managed to get two stages that were fixed in rotation to each other (both impellers were operated off the same shaft) to cover a wide range of pressures and air flows without either choking each other or limiting each other to any great extent.
For Allison in 1941 (test labs or drawings) and 1942 was teh two speed supercharger seen as 1938-1940 technology and the two-stage seen as the future?
If Allison had limited resources for R&D which path do they take?
Now it turns out that two stage superchargers are a lot harder to work out in practice than in theory. How far (or how long) did Bristol and Napier get? How far did Wright get?
How far did Continental get (even using a turbo)?
France, Germany and the Soviet Union, for bombers, all tried putting an engine in the fuselage driving a large compressor to supply high pressure air to the wing engines.
By the time the US got any working Japanese two speed engines back to the US it was too late? Japanese did pretty well with two speed engines.