Question: A fixed pitch propeller is only good for aircraft that do around 100kts because the angle of attack on the propeller decreases as the forward airspeed of the aircraft increases, limiting the thrust available. Variable pitch propellers overcome this allowing the angle of attack to remain optimal at speed thus allowing faster cruise speeds.

However the pitch angle of the bypass stage of a turbofan is fixed but yet they do not seem limited by the same problem affecting a fixed pitch propeller. Turbofans provide sufficient thrust at take-off (very low forward speed) as well as much higher cruise speeds.

I'm trying unsuccessfully to understand the physics of this or more specifically: If the pitch of the turbofan is set optimally for cruise, why does it not stall at take-off due to a very high angle of attack? Conversely, if the pitch was set for high thrust at take-off (or some lower speed setting), shouldn't the angle of attack reduce as speed increases thereby limiting cruise speed attainable?

Answer: 
Compressor's (or fan or turbine) blades are designed to properly work in a quite ample range of angles of attack, more or less 15° around their design point.

The speed $W$ seen by the blade (and the relevant angle of attack $\alpha$) is given by the sum of two terms:

1. the speed $U$ due to the rotation of the blades;
2. and the speed of the incoming airflow $V$.

This can be seen for example in the following plot (slightly modified by me) taken from [this report](https://www.researchgate.net/publication/330392171_Design_of_a_Flux-Switching_PM_Machine_with_Axial_Fan_Capability) (blades are seen from above and rotate to left; flight direction is upward):

[![velocity triangle](https://i.stack.imgur.com/DOtb7.png)](https://i.stack.imgur.com/DOtb7.png)

The rotating speed $U$ can be more or less controlled via the throttle while the speed $V$ of the incoming flow is driven toward the blades (of the first compressor's stage) by an extremely important component that lies in front of the engine and that most of the time passes unnoticed: the inlet.

The main job of an inlet is to reduced the speed of the incoming air to some Mach 0.4, whatever high the speed of flight is. The range from Mach 0 to 0.4 is the limit within which the compressor's blades (and especially their tips) work in a proper way.

If the speed of flight is high subsonic (Mach 0.4 to 1), then the inlet simply works as a divergent inlet (diffuser): its section gets bigger and bigger, the pressure gets higher and higher and the speed gets lower and lower (the compressor thanks Bernoulli).

Finally, if the speed of flight is supersonic, in front of the inlet one or more shockwaves form, across which the speed reduces to high subsonic and afterward a divergent portion of the inlet finishes the work reducing again the airspeed till Mach 0.4.

This is how inlets of a modern jetliner, the [Concorde](https://en.m.wikipedia.org/wiki/Concorde#/media/File:Concordeintake.svg) or the [Blackbird](https://en.m.wikipedia.org/wiki/Lockheed_SR-71_Blackbird#/media/File:SR71_J58_Engine_Airflow_Patterns.svg) work(ed) to supply their compressors with a relative slow and stable airflow.
The AoA of the fan blades varies throughout the speed range, very similar in principle to a fixed pitch propeller. However, the change in AoA also depends on the propulsive efficiency of the engine.

This variation in AoA is determined by the change in **"down" wash** at the fan blades (think of the fan blade like a wing). This downwash is for to two reasons: forward speed (TAS), and the inherent downwash (jet-wash) due to the production of lift (thrust) itself. On a turboprop engine with high propulsive efficiency, this latter downwash is low, but is relatively high on turbofans (and is even higher on turbojets). The net downwash as experienced by the fan blades is a sum of the two.

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When a turbofan is operating on ground at zero TAS, the effective TAS at the fan blades is quite high. This is due to the aforementioned downwash at the fan blades due to the production of thrust itself. This downwash reduces the effective AoA of the blades enough to prevent it from stalling. As aircraft TAS increases, the net downwash increases, causing the effective AoA of the blades to decrease.

However, on a turboprop with a much higher propulsive efficiency, the initial downwash (due to thrust production) is relatively low, and it's influence on the effective AoA is much less. When the aircraft TAS increases through the same speed range as in the turbofan case, the percentage increase in the net downwash is much higher, and so is the change in AoA. The only way to maintain a reasonable AoA is through the use of variable pitch propellers.

In the turbofan case, the change in AoA as the speed increased was significant, but still low enough - a variable pitch mechanism would be able to provide much less improvement, meaning that it's use is not feasible.

If you find this difficult to accept, think of the extreme case: if the initial downwash was infinite (blade pitch would be 90°), an increase in TAS through any speed range would have no effect on the AoA, and so a variable pitch system will be absolutely useless in this case.