'Trompe' (see Wikipedia) is a water-powered air compressor, commonly used before the advent of the electric-powered compressor. A trompe is somewhat like an airlift pump working in reverse, used to provide compressed air for bloomery furnaces in Catalonia and USA. By sending a fast water flow through a duct, a pull effect on air is obtained. I wonder if a pure Turborreactor working inside a larger Duct, with no Low Pressure Fan, would have same efficiency, with less parts, as a high Bypass Ratio Turbofan. Operation inside duct, or even inside fuselage, would result in reduction of sound and thermal prints. Salut +
You are basically describing an ejector pump. This would not be very efficient, I'm afraid (I'll try to dig up some comparative figures).
The arrangement in this case would be a jet engine used to accelerate an amount of air inside a duct. With a long enough duct, the end result would be a single airmass exiting at (somewhat) uniform speed. Unfortunately, because energy must be conserved, the kinetic energy of the air exiting the apparatus would not even match the kinetic energy of the air exiting the jet engine, because of turbulence, skin friction of the duct etc.