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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?

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Yes, the tips of the fan blades.

Let's look at an example: The GE-90 has a fan diameter of 3124 mm and a maximum rotational speed of 2475 RPM. Their circumferential velocity is d·π·41.25 = 405 m/s or Mach 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 ft/s shown below at the turbine is subsonic, because at 1100°C the speed of sound in air is 720 m/s or 2,362 ft/s.

Jet engine parameters

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.

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    $\begingroup$ Does that mean there is a sonic boom every time you turn on the engine? If so, why dont we hear it? Is it just small/muffled by everything else? $\endgroup$ – Reinstate Monica Oct 23 '15 at 14:08
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    $\begingroup$ @DavidGrinberg: A combination of this answer and this answer should help: Yes, those tips make a lot of noise, but the engine shroud helps to keep it manageable. $\endgroup$ – Peter Kämpf Oct 23 '15 at 14:14
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    $\begingroup$ @DavidGrinberg Keep in mind that a sonic boom isn't produced when going from subsonic to supersonic ('breaking the sound barrier'), but is rather a continuous shock wave produced by objects going faster than the speed of sound. It's the loud whirring you hear when taking off in certain jetliners. $\endgroup$ – Sanchises Oct 26 '15 at 18:58
  • $\begingroup$ I just wanted to add that flow through the Stage 1 High Pressure Turbine (HPT) Nozzles (Vanes) is typically choked (i.e. Mach = 1). Of course, this is only the airflow, not any moving parts. One thing that would make the chart better is if it showed the M=1 line throughout the engine, since speed of sound is proportional to temperature, the speed of sound is much higher in the HPT than at standard sea level. $\endgroup$ – OSUZorba Apr 29 '16 at 2:01
  • $\begingroup$ Just noticed that your image could use some attribution, if you recall where it came from. (It's only been about 8 months...) $\endgroup$ – FreeMan May 12 '16 at 15:07
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In a turbofan engine, the fastest moving parts are usually the bypass fan tips. This is because at the same rotational speeds, the outer tips are at more distance and have higher linear velocities. In some engines, the tips reach supersonic speeds, usually at high thrust conditions. This is the reason for having a swept blade (like swept wing) in high bypass turbofans.

Swept fan

Source: sheffield.ac.uk

In case of turbine blades (and to a lesser extent, compressors), the local temperature is high (due to combustion and compression); this increases the speed of sound (as it is proportional to $\sqrt{T}$ for an ideal gas). Due to this effect, the local flow is subsonic. Also, the blade size is very small compared to the bypass fan.

The next part where the flow goes supersonic is the divergent section of the convergent-divergent nozzle used in military jets (Note that there are no moving parts here).

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  • $\begingroup$ It may be true that the bypass fan blade tips are fastest, but your answer does not offer supporting evidence. If N1 is sufficiently below N2, other blade tips might exceed the bypass fan blade tips. $\endgroup$ – J Walters Mar 11 '17 at 20:58

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