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.
One solution I haven’t seen addressed in the existing answers is compressor bleed valves. The speed with which the air passes over the compressor blades is a function of the pressure differences ahead of, and behind, the blades. The whole point of multi-stage compressors is for the pressure inside the engine to keep increasing as the air flows from inlet to combustion chamber.

At low airspeeds, and increasing pressures as the engine RPM increases, this means pressure is building up behind the blades faster than the plane is accelerating to create increased inlet pressure. This is what typically causes compressor stall… pressure increasing behind the blade reduces the speed of the flow over the blades.

Multi-stage engines will often have a bleed valve in the rear compressor stages that opens to let some air escape until airspeed increases. This alleviates the pressure building up behind the compressor blades, allowing faster flow, reducing blade AOA, alleviating stall.

Thus the fixed blade angle can be optimized for cruise performance and the engine manages effective AOA by managing pressure gradients inside the compressor. The bleed valves are pneumatic and operate autonomously with no crew intervention.