Looking for Jumo 222 powered aircraft, I came across this picture of a Junker Ju 288, with this strange spinning cowling and propeller setup, what are the considerations behind this design, how does it work?
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The pictures in your question show the Ju-288 V5, the first of its type to be powered by the Jumo 222 engine. Previous Versuchsmuster (V-types) used the less powerful, but larger BMW 801 (see below, source).
Ju 288 V3 after its 100th test flight
The Ju-288 was designed to cruise at an altitude of 11 km (36,000 ft) where low density requires a high volume of cooling air. The very compact Jumo 222, which was basically three improved Jumo 210, shortened to 8 cylinders each, in a radial arrangement, had a much increased power density and required low cooling drag to make a top speed of 690 km/h (430 MPH) at that altitude possible. By using what is effectively a jet engine intake, without the disturbance of rotating propeller blades, it was hoped to maximize pressure recovery and optimize the effectivity of the cooling system.
Later Versuchsmuster used a more conventional design. The high wetted surface of the ducted spinner causes more friction drag and the gain in effectivity could not outweigh this. Besides, the ducted spinner also has four propeller blades spinning inside, although not visible from the outside.
Ju-288 V14 in flight (picture source)
The Ju-288 was planned to be for Junkers what the B-29 did for Boeing: It swallowed huge amounts of development expenses and would had given Junkers an unassailable technological advantage over all other German airplane companies. Although 35% larger, it would require fewer man-hours to build than the Junkers 88. But the quickly changing priorities of the ongoing war and an inept government bureaucracy which did not have the patience for finishing the development were its undoing.
Note that the first prototype of the FW-190 used the same type of air intake for its cooling air.
FW-190 V1 (picture source)