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.
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.