Why are adjustable area jet engine nozzles mostly limited to military use?
A variable area nozzle helps to adjust the exit pressure of the combustion gases to ambient pressure. When they leave the turbine, the combustion gases have still some remaining pressure which is converted into speed by a convergent contour of the nozzle.

If the combustion gases have enough pressure to be accelerated to supersonic speed, the nozzle is actually first convergent and then divergent to achieve the best acceleration of the flow. The convergent, subsonic flow accelerates until in the section with the smallest area, called the throat, the speed of sound is reached, and the following divergent section accelerates the now supersonic flow further until its pressure drops to ambient pressure. This con-di-nozzle needs to adjust both the cross section of the throat and the cross section of the exit area. Failure to get especially the throat area right will mean substantial thrust loss in operation.



Afterburning engines need adaptable nozzles due to the different operating conditions in dry and wet mode, so they profit the most from adjustable nozzles. Reheating the exhaust gas means to increase its volume, so the nozzle needs to become wider to be properly matched. In general, if the exit speed of the combustion gases is supersonic, an adjustable nozzle is required. Even some early jets without afterburners had an adjustable nozzle: The central cone of the Jumo-004 nozzle could be moved fore and aft, which changed the location of the bulbous aft section, helping to get the throat area right. Note in the picture below that the cross section widens again in the last part of the nozzle, which shows that the exhaust stream of the Jumo engine was mildly supersonic.



Airliner engines would only gain little from an adjustable nozzle, and considering the mass increase of a fully adjustable nozzle, would in total have lower efficiency. Short-range aircraft usually have nozzle geometries optimized for field performance while long-range aircraft prefer to optimize for best cruise efficiency. Their core flow exit speed is still subsonic or only mildly supersonic - please note that the speed of sound in the heated combustion gas is much higher. At 500°C the speed of sound is almost 560 m/s.

The variable area fan nozzle tries to do for the fan flow what the regular nozzle does for the core flow of a jet. So this is indeed a related concept and helps to optimize performance both at low and high speed. The picture below is copied from the US patent 2011/0302907A1 and shows an adjustable end part of the cowling (crosshatched, 54). The exit speed of the fan flow is subsonic at take-off and mildly supersonic in cruise, and since its mass flow is much higher than that of the core in airliner engines, making the nozzle variable pays off earlier. By adjusting the exit pressure by varying the exit area it can squeeze out some performance gains.