yes I'm aware, but the two stage supercharger which was used on the F4F while helpful at altitude, was notoriously weak and inefficient. Is that correct?
The impellers were with straight blades, so indeed the efficiency left a lot to be desired from. OTOH, as a complete package, they were considerably better than the 1-stage superchargers from the P&W (that were also with the straight blades), and were still bette than the 1-stage S/Cs as installed on the V-1710s before the late 1942.
I want to plunge a bit deeper into the wartime F4Us and F6F, as these are more confusing to me. Were the wartime production R-2800 as installed on these fighters both two stage and two-speed as I am reading, and was the two-stage supercharger efficient / effective?
A 2-stage S/C driven just via the 1-speed drive would've been horribly ineficcient - depending on the gearing chosen, it will be either lacking the power at low altitudes, or at high altitudes.
The engine-stage S/C (here being the 2nd stage of compression) on the US-designed engines with 2-stage S/Cs was driven via a 1-speed gearing.
The auxiliary S/C (here being the 1st stage of compression) was driven via the gearbox with high, low and neutral gearing on the P&W engines (neutral setting allowed that the aux stage can be bypassed at low altitudes, so it a) does not use any power from the engine, and b) does not heat the compressed air too much); the V-1710 used the variable-speed drive for the aux S/C.
The 2-stage S/Cs as installed on some (but important) RR engines was with both impellers being driven by the same shaft, via a 2-speed drive; both impellers were turning the same RPM. Huge advantage was the compactness and simplicity of the set-up - a major thing when someone wants to up-engine the existing aircraft type. That 2-stage system was adopted by both Jumo and DB for their (very late) 2-stage supercharged engines, that were driven via a 3-speed gearbox (Jumo), or via the variable-speed drive (DB).
RR went with the 3-speed drive for the post-war Griffon 100 series for the Spiteful/Seafang, to gain extra power down low.
The 2-stage R-2800 gave a lot of power at all altitudes, powering the heavy and draggy US fighters at around 400 mph, and at 440-450 mph by the late ww2. That means that it scores high as far as I'm concerned.
General consensus seems to be that these fighters are not as effective at altitude above 25,000' as a P-47 or a P-51 or a Spit IX etc.
How much a fighter is effective is a sum of many factors; there is also a thing of 'horses for courses'.
The big & heavy F6F and F4U were designed around the heaviest and most powerful engines available, while also supposed to be good at low speed, with longer range than the previous designs. All of that drove the size and weight up, both of the things that will tone down the performance. The P-51 was a much smaller aircraft, powered initially by a V12 engine with a modest S/C, with next-gen aerodynamics and without the carrier suitability for the designers to take into account. Stick the modern V12 on it, and it becames an over-performer.
The P-47 didn't have the carrier requirement, but the R-2800, the associated turbo, big fuel tankage and 8 guns battery resulted with a fighter that was better above 25000 ft than the Navy birds. It was not much of a climber, though, due to the excessive weight.
F4U was of the performance similar to the Spitfire IX - not a mean feat considering the thin wing and light weight of the Spitfire. F6F was a bit worse. Both of the US fighters were also the much better carrier birds than the Spitfire/Seafire versions. F4U-4, with the much improved R-2800-18W, was comparable with the Spit XIV.
I'm also unclear as to the timeline of various versions of the R-2800 in these fighters.
The A series -4 powered the XF4U; 1850 HP max.
The B series -8 powered the F4U-1; the -10 powered the F6F-3 and -5, as well as the P-61. The -8 and -10, as well as the 8W and 10W, were basically the same engines, apart fro the carb and some small bits & pieces.
The C series -18W powered the F4U-4; took part in ww2 in 1945.
The E series -32W powered the F4U-5, a post-war model; that engine was with two small superchargers (so-called 'sidewinder' layout) feeding together the engine-stage S/C.
'W' denotes water-alcohol injection.
That is indeed interesting. It may be that it's just one of those things that works in prototype but is very tricky to get reliable and effective enough for broad military service.
Press on with development and testing - same as for any piece of kit that ups the state of art.
For the US it was extra tempting to go to turbo because they had developed commercial air-liners which used them. So they knew they could work, and they did pretty early on for bombers. It just turned out fitting them into a fighter was tricky.
Military application of turbochargers predates the commercial application of turbochargers.
Indeed, fitting the turboes on the fighters, even the bespoke ones made around the turboes was not easy.