I'm not seeing anything that forms a "pump" to move the air. It needs something with vanes and enough holes to let the air pushed out and away to leave in order to generate any airflow. If it'll be an inrunner motor the core vanes are shrouded in by the magnets on the bell. So they only stir up the air without moving it by any significant amount. If it's an outrunner again the bell doesn't have anything of a shape to actually move any air. Some airflow may occur but it would be due to pressure differences occuring from the main fan and any such airflow would be a matter of good luck more than good design.

Certainly at some points on the housing there will be areas of higher and lower pressure that occur due to the air movement and fan spinning. Typically and most likely they would be found on the "backside" of the tail cone fairing. But the question is will there be a positive pressure at the gap between the rotor and the housing or will there be a pressure drop at that point caused by the rotor pulling the air out onto the faster spinning area of the blades? If both areas are seeing a pressure reduction then the net flow may be very low or even zero.

Stealling a technique from the old days the team could build one unit which has multiple pickup tubes which run to water U shaped manometer tubes and then run the setup to see which spots have a pressure rise and reduction over normal atmospheric pressure. With that knowledge you could then arrange for some airflow to occur between those points and incorporate this in the final version. Otherwise it's all just guesswork. Suitable sensing tubing would be small bore stainless steel hypodermic needle tube. I believe that Small Parts sells such tubing in 12 or 24 inch lengths.

The modern day alternate would be a suitable CFD program which can accurately model the spinning fan along with pressures in the EDF housing and around the center body. But that's pretty high brow stuff. I doubt it would be OK to ignore the spinning of the fan and just model the EDF interior with air being pushed through it. The spinning fan would significantly alter the nature of any such flow.