Are the fastest moving pieces of a jet engine supersonic? A jet engine is composed of fast moving pieces. Those pieces are moving into an airmass different form the ambient airmass the aircraft is flying into.

The turbine pieces are spinning quite fast, thus its blade tips may be moving at a supersonic speed. But, the airmass may also be spinning modifying the speed of the blades into the airmass. Moreover, the temperature of the air at this stage may modify the speed of sound.

The first stage of compressor is spinning at a much familiar air temperature and pressure, thus it may be easier to find if those compressor's blades' tips are moving faster than the speed of sound. But pressure (and thus temperature) is not the same before and after each compressor stage. It may be hard to define the speed of sound for this airmass going through the compressor.

Given the airmass inside a jet-engine is different from the ambient airmass, are there any moving piece inside a jet-engine whose velocity (frame of reference: the airmass in which it is moving) is greater than the speed of sound?
Yes, the tips of the fan blades.

Let's look at an example: The GE-90 has a fan diameter of 3124&nbsp;mm and a maximum rotational speed of 2475&nbsp;RPM. Their circumferential velocity is d·π·41.25 = 405&nbsp;m/s or Mach&nbsp;1.19 at sea level and ISO atmosphere. Now you need to add the flow speed of the air, but this calculation alone should be sufficient to prove the point.

The inner part of the fan and all the compressor stages belonging to the core engine experience flow at subsonic speed, even if the whole engine is flying at supersonic speed (see picture below). To make ignition as complete as possible, the flow speed is continuously reduced the closer you move to the fuel injectors. Since the air is heated in the combustion chamber, the hotter gas has a higher speed of sound, and all the flow through the turbine is subsonic, too. This includes the low-pressure turbine. Even the 1,600&nbsp;ft/s shown below at the turbine is subsonic, because at 1100°C the speed of sound in air is 720&nbsp;m/s or 2,362&nbsp;ft/s.



Overview of parameters in a typical jet engine.

Only in the rear part of a convergent-divergent nozzle with reheat switched on will supersonic flow again be attained. This is exactly aft of the narrowest section, called the throat, where airspeed is equal to the speed of sound, and the petals of the nozzle are the only moving parts here. And they move rather slowly.