Adam asks "can someone point me to any drawings or papers which spell out the optimum shape for a bellmouth? Just has we have specifications for NACA scoops and airfoils I imagine someone did the research for air intakes."

There is an optimal design for ducted fan inlets (turbofan engines) because the duct inlet contributes about a quarter of the total engine thrust. The engine is accelerating air over a curved shape, just like what happens with the wing.

Induction air inlets are not as sensitive as all you are trying to do is maximize the diameter of the vena contracta and to do that all you need is a nice fat radius, say about a quarter the inlet tube radius. So for a 3" diameter, use a 3/8" radius minimum.

Brent Regan