Why would this inlet for the induction system get wider as the air approaches the filter?
What net effect does this have on intake pressure, and how would an inlet that starts wide and then narrows down perform differently?
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It's simply a NACA duct inlet. On a fully parallel to flow flush surface it has some (but not all) of the effectiveness of a protruding scoop, without the drag. The curved sides of the ramp act like vortex generators, with the rotation of the vortices on each side pulling air down into the ramp.
On a very clean airplane like one of the RV series homebuilts, that kind of duct can have a measurable benefit. On a plane like the one pictured, there is a theoretical benefit drag wise, but it would make little detectable difference in that sort of a light plane (with its unfaired nose gear) and is mostly for looks.
In fact since the surface isn't really parallel to the airflow but curves in, the opening is presented more or less directly into the airstream for ram effect and there really isn't a need to create vortices to pull air down the ramp. A straight sided ramp would work just as well in that particular instance. Looks cool though, which I suspect is the main reason it's being used there.
The typical shape as pointed out in the question, a diverging nozzle, is for "pressure recovery". For engine efficiency or combustion efficiency, higher static pressure is required. The shape is selected to convert the dynamic pressure (pressure due to velocity) to static pressure, which improves the engine performance.
The opposite shape would be a converging nozzle, that would increase the velocity and decrease the static pressure, which is not what is required for combustion.
What do you see in the image is a typical NACA duct. It recovers pressure at the same time creates a minimum increase in drag (or less total pressure loss). A well designed NACA duct (just the shape is not enough, the air simply won't enter the duct then) ensures a good amount of air intake into the engine.