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.
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.