Is the well known textbook-formula for net-thrust oversimplified or do I have a misinterpretation? In order to understand how Jet-Engines works I came across the &quot;famous&quot; formula for net-thrust:
https://en.wikipedia.org/wiki/Turbojet#Net_thrust
$F_{thrust} = Q_{out} \cdot v_{out} - Q_{in}\cdot v_{in}$
From well known physical law about momentum this sound quite reasonable for me.
But now assume an empty cylindrical chamber with nozzle at the end, mounted statically in a wind channel:

Regardless of any details, just by continuity equation we must have
$Q_{out} = Q_{in}$
and
$v_{out} \gt v_{in}$
Nevertheless, this construction would not produce a net propulsion in left direction. It it were true, we just had to put this &quot;empty&quot; construction on a plane...
I was thinking over that quite a time and become already a little crazy.
Where is my mistake? Or is the formula above not the whole truth?
Anyway, I'm not sure if
$v_{in} = v_{air}$
although it sounds reasonably for me.
But even when I take the formula
$F_{thrust} = Q_{out} \cdot v_{out} - Q_{in}\cdot v_{air}$
a left thrust would be produced, whenever
$v_{out} \ge v_{air}$
Is this the key-factor where I'm wrong?
You forgot to take pressure into account. This formula works assuming your nozzle is well adapted and the exit flow is at atmospheric pressure. In your case the velocity is increased but the pressure drops as stated in Bernoulli's equation.
$$P_0+ 0.5\rho V_0^2 = P_1+ 0.5\rho V_1^2$$
This means that the pressure force on your nozzle will counteract the thrust provided by the fluid acceleration.
See this explanation  your general thrust equation if the pressure are different is
$$\text{Thrust} = (Q * V)_e - (Q * V)_0 + (P_e - P_0) * A_e $$
Given $A_e$ is the exit aera of the nozzle.