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: 
[![enter image description here](https://i.stack.imgur.com/TYq0t.png)](https://i.stack.imgur.com/TYq0t.png)

A large part of this is provided by the turbofan inlet design. Above pic is from Aircraft Gas Turbines by C.J.Houtman and was used earlier in [this answer](https://aviation.stackexchange.com/a/73029/21091): the inlet shape slows the incoming air down at cruise speed, and trades velocity in for pressure.

The variables:

* At TakeOff thrust needs to be maximal, the fan blades need to suck all the air in, and the fan does so at maximum RPM.
* At cruise thrust is lower, the RPM is lower, and the air inflow is slowed down.

Indeed the variation in thrust requirements and airspeed create quite a difficult set of operational circumstances to tune, care must be taken to prevent stalling of the blade tips, and stalling of the inflowing air-stream around the inlet nose at lower velocities, which is why the inlet is optimised for these lower airspeeds.

Propellers operate at lower airspeeds than turbofans, and operate at constant RPM with variable blade AoA for optimal thrust efficiency. Turbofans would also benefit from variable blade AoA, and efforts have been made to develop them, but the cost/gain ratio seems to be prohibitive for now.
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.