I recall reading that the Germans adopted fuel injection after BMW reported a 6% improvement in fuel economy by fitting fuel injection to a BMW VI engine (try searching on the web).
The WWII era literature differs as to the benefits of fuel injection. Heron criticised it on the grounds that it did not provide as much charge cooling as could be obtained with a carburettor (he noted that US tests of captured DB engines showed no increase in achievable boost/power when the engines were adjusted to run rich). This may have been why the arrangement of spraying additional fuel into the supercharger of the BMW 801 engine was used. On the other hand, direct fuel injection had some important benefits:
- reduction or elimination of problems with uneven air/fuel ratios (a major issue with early R-3350s - supposedly early versions of this engine that used fuel injection may have had reduced power, but had better reliability);
- elimination of the need for a carburetor choke (and the accompanying presure drop);
- elimination of backfiring problems.
The Soviets adopted the German technology on their later engines - presumably they found it worthwhile.
Modern automotive experience seems to favor direct fuel injection ("solid fuel injection" seems to be the current term).
I'm still not on top of all of these issues but it's noteworthy that in 1944 two separate boost systems were introduced on Fw 190A8 and F equipped with the 801D2 engine.
In one type of system the manifold pressure was simply raised about 10% at sea level from 1.42 atmospheres to 1.57 (in fact that boost was usually 1.68 as the engine quickly changed to second gear with the ratios then employed). I don't know if rich mixture was employed to prevent preignition to exploit the C3 fuels relatively high rich mixture response, I seems likely it was. This system continued to use the direct in cylinder injectors to add the fuel. The power increase came from the increased amount of air/fuel compressed into the cylinders due to the higher pressure. Exactly why this was possible would be likely a result of strengthened components in the engine, better fuels, better spark plugs, better lubricants. I'm not sure what the enablers were but it wasn't allowed in 1942 but was in 1944 (having been demonstrated in mid 1943)
The second system, which was initially seen only on ground attack variants of the Fw 190 instead added the fuel into the eye of the supercharger, thereby cooling the mixture, contracting it in volume, thereby inducing more air/fuel mix into the engine and also reducing supercharger work.
A similar system was added to the Jumo 213A of the Fw 190D-9 thereby increasing power from 1750 to 1900 metric horsepower, interestingly this system used the 87 octane grade B4 fuel.
In effect the Germans added a throttle body type injection carburetor for use only during WEP.
I imagine when engaged the fuel that was injected via the injectors was reduced to some low value, equal to engine idle, which the fuel flow was transferred and increased to injectors into the eye of the supercharger.
This sounds complicated but the key of any fuel injection system whether throttle body, plenum or direct injection is an mass air flow sensor since the basic requirement is to add 14 parts air to one part fuel by weight. Once this sensor is in existence it can be used to modulate the cam of any kind of pump whether it be a swashplate for central throttle body type or direct injection.
One way of doing this is to have a venturi in the intake to create a pressure difference and then used a pressure sensor (eg an aneroid capsule or a spring loaded piston) whose deflection is proportional to the air flow which is then used to operate a small hydraulic servo.
Famously the latter Merlin eliminated a venturi entirely to avoid the losses it might cause and calculated the air mass flow out of a combination of engine RPM, exhaust back pressure (essentially ambient), throttle valve position and temperature. It is possible the German engines used a similar concept as I have never seen an airflow sensor for these systems. I would have been foolish not to copy this elegant system which would be readily adaptable.
Allied engineers then were, to an extent, right in avoiding direct injection. Nevertheless the Germans still benefited. Direct injection allows a more radical profiling of intake/exhaust valve overlap for more complete scavenging without fear of sweeping out perfectly usable fuel. It allows slightly more fresh air to be drawn in and gets rid of end gasses that can cause pre-ignition. When using rich mixture injection to obtain WEP this possible extra loss of fuel in the scavenge compared to allied engines was probably considered tolerable. These engines seem to have had very good high speed cruise characteristics and fuel efficiency so there was a benefit. Also the Germans, like the Soviets, wanted to avoid the problems of strictly controlling the vaporization characteristics of their fuels. I imagine starting an engine in a Russian winter may not have been to easy either.
One other factor is that British and American companies controlled all of the good carburetor patents which provided an incentive to the Germans to avoid Royalty patents and to exploit their high speed diesel injection technology.
Ultimately the German direct injection engines headed in the direction of water injection to obtain charge precooling. Water injection was added to the BMW 801TS around Dec 44/Jan 45 and the Jumo 213A of the Fw 190D9 perhaps a month earlier as a field modification. The motivation was at least partially that this was more fuel efficient. Every liter was scrupulously accounted for.
The attraction of rich mixture injection was that it didn't require the inconvenience of a water tank and its C of G issues and additional plumbing and seems to have been less aggressive on the engine. It took a while to solve the C of G issues an MW50 tank created. The idea in German aircraft seem to have been to add a 115L (25 gallon) multipurpose tank in the Fw 190 and Me 109 that could be used for MW50, extra fuel or GM-1 as required though in reality few if any were plumbed so as to allow this multi purpose mission.
The ADI water tank in the P-47 was only a tiny 15 gallons and couldn't have lasted long.
The DB605 received MW-50 in April 1944 (in Me 109 squadron missions) in the form of the DB605AM and DB605ASM used on Me 109G6AM and Me 109G6ASM. They never used rich mixture injection.
I do not know if water injection was used in parallel or instead of supercharger fuel injection it seems they could complement each other.
German engines seem to have been held back by lubrication issues.
Having looked around it seems to me that non of the German fuel injected engines bothered with an airflow sensor. The pumps and injectors was linked to crankshaft speed mechanically. Air pressure and temperature would estimate air density while throttle valve position would provide air flow.