Ok, the external shape is fixed (has to be scale). The issue once you get inside the duct is how to taper away the cross section area of the spike without messing up the flow. The problem is that you now have increasing cross section area as the flow goes downstream, and so its going from low pressure, high speed flow, to high pressure low speed flow, and thus it wants to separate. Then the flow gets to the engine. A bunch of the air goes into the compressor, while the rest goes around the engine. BUT, a lot of the duct area is now filled with engine and the net cross section is small again.

So the issue is that you have this area of large cross section area just ahead of the engine, and thats likely to cause flow separation on the centerbody unless the centerbody is really long with a gradual taper.

The way around this is to "area rule" the duct. Namely, put a constriction on the outside of the duct, such that the cross section area stays pretty constant in the area between the inlet throat and the engine intake. As long as the cross section area is constant, the flow can handle some pretty tight bends without separating.

Note, in general, you want ANY afterbody to come to a point, IF the taper is gradual enough for the flow to stay attached. Its the same as a wing trailing edge. However, if you cannot have it that long, then you taper it as much as you reasonably can and then round it off. If the engine has an electric starter in front, you can blend the back of the spike into the front of the starter.

However, our turbines with the centrifugal compressors are pretty tolerant about turbulent inlet air. Chances are that regardless of what you do , the engine will still run. A good duct design will have a bit less drag and a bit more thrust, but it wont be dramatic. (A bad duct like this would totally kill the performance of a ducted fan, since its a low pressure ratio, high mass flow device.)

The attached drawing is just a sketch. I didnt calculate any cross section areas.

One thing to remember.. ANY airplane that uses a movable spike inlet had it designed for supersonic flow. The rules for our low subsonic flow are very different. The designers also had to make the inlet work over a wide range of throttle, altitude, and mach number. So dont just blindly copy what they did!

Bob parks