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"Did GB need more high-altitude fighters at the time ... or were the Typhoons and Tempests going to be used as fighter-bombers and ground attack aircraft at low altitudes in any case?"
It was estimated that a 1000hp engine needed ABOUT 10 cubic ft of space for a turbo, inter-cooler and ducting. There may be some economy of scale but a Sabre is going to need at closer to 20 cubic feet.
Turbos were also rather sensitive to the temperature of the incoming exhaust gases. A few extra feet of piping may pay big dividends in turbine life. The blade type turbines could fail in a rather spectacular fashion. Some P-38s had 'scatter' shields in line with the pilot and a few other aircraft also had 'scatter' shields between turbos and crew positions.
There is no doubt that a turbo could improve the performance at high altitudes (sea level power at 25,000ft) however it may come at the price of 15-30mph in speed and less climb rate at low altitude.
The space between pilot's seat and empenage looks rather ... empty?
The Tempest that never was would be my THE Tempest: 2 stage Griffon, paired with LE radiators.
There would need to be some extensive redesign to cater for all of the ducting, exhaust wastegates etc (a' la P-47) by which time the weight and cg would be out of whack, requiring further redesign and lengthening etc.
Would it really have been worth it?
There was a Fury tested with a two stage Griffon, except it stuck with the chin...
vs the more elegant Sabre VII version
There ain't such thing as a free lunch.
If you take a look at the Fw-190 Kangaroo (the one I've mentioned already), you will notice that exhaust ducting is an external affair. The compressor -> intercooler ducting is both short and would not present a cahllenge re. drag and space needed. Then we have an intercooler -> carburetor ducting, that one being behind the chin radiator, ie. nothing too draggy.
OTOH, the 'external turbo' setup, as a complete sytem, is a draggy thing, I've covered the possible evolution of the turboed Typhoon/Tempest.
Tempest with LE radiators look to me as if it was designed by Pininfarina, the chin radiator Griffon version is plain ugly. Hence my humble request
I cannot think of a single successful British designed WW2 era plane that was fitted with a turbo, nor can I think of a single successful turbo equipped fighter that was not designed around the turbo system. Happy to be educated on both points.
The P-47 , the successful WW2 single engined fighter fitted with a turbo was hugely designed around the system using experience gained with the AP-4 and the P-43.
The R2800 also offered a reliable engine to bolt the turbo system to, something that could not be claimed of early Sabres.
Why would the British pursue fitting a turbocharger system to the Sabre engine?
Why did they need a Tempest that performed well at high altitude?
The British had a technological advantage in their supercharger technology as evidenced by the two stage 60 series Merlin and if a high altitude Tempest was the objective this technology (or a two stage Griffon) should have been transferred to the Sabre (after it was made reliable) earlier than was done.
I cannot think of a single successful British designed WW2 era plane that was fitted with a turbo, nor can I think of a single successful turbo equipped fighter that was not designed around the turbo system.
The P-47 , the successful WW2 single engined fighter fitted with a turbo was hugely designed around the system using experience gained with the AP-4 and the P-43.
The R2800 also offered a reliable engine to bolt the turbo system to, something that could not be claimed of early Sabres.