Fly in Elevator: Does Compression Affect Hovering?

[SlowThinker]

So I found this article: https://en.wikipedia.org/wiki/Thrust but so far it raised more questions than it answered (sorry, I keep doing that).

The article is somewhat misleading, they use a formula for zero intake speed and then use is as correct, but obviously, if there is no air going in, the engine has no air to burn fuel. Same with helicopter, you can't have air at zero speed above the propeller, because then there couldn't be any air going down below it. (This doesn't apply to a rocket engine because air/gas is "created" inside).

I'll try to repeat my derivation, others are welcome to check it... I'm not an expert on this, just using common sense.
v...speed of air going into rotor
v+$\Delta$v...speed of air going out
S...area of the rotor
$\rho$...density of air going in
F...force (lift) generated
P...power used by the rotor
m...mass of air passing through the rotor during an arbitrary time T
a...acceleration of air
s...length that air spends inside propeller

$$m=\rho S v T$$
$$F=m a=m \Delta v/T=\rho SvT\Delta v/T=\rho Sv\Delta v$$
$$s=\frac{1}{2}aT^2+vT=\frac{1}{2}\frac{\Delta v}{T}T^2+vT=\frac{1}{2}\Delta v T+v T$$
$$P=F s/T=F(\frac{1}{2}\Delta v T+v T)/T=F(v+\frac{1}{2}\Delta v)$$
Speed $v$ that minimizes power:
$$P=F(v+\frac{1}{2}\Delta v)=F(v+\frac{1}{2}F/(\rho S v))$$
$$0=P'=1-\frac{1}{2}F/(\rho Sv^2)$$
$$2\rho Sv^2=F$$
$$v=\sqrt{\frac{F}{2\rho S}}$$
$$P=F(\sqrt{\frac{F}{2\rho S}}+\frac{1}{2}\frac{F}{\rho S \sqrt{\frac{F}{2\rho S}}})=F^{3/2}(\sqrt{\frac{1}{2\rho S}}+\frac{1}{2}\sqrt{\frac{2}{\rho S}})=\sqrt{2}\frac{F^{3/2}}{\sqrt{\rho S}}$$
My constant factor is $\sqrt{2}$ instead of Wiki's $1/2$, not sure if I made a mistake or what. But then, they are doing something different.

 

[rcgldr]

Same with helicopter, you can't have air at zero speed above the propeller, because then there couldn't be any air going down below it.

The idealized case is that you have air at zero speed at some point above the rotor. The air accelerates and decreases in pressure until it reaches the plane of the rotor, where there is a pressure jump (from below ambient to above ambient) and little change in speed. The air continues to accelerate until it's pressure returns to ambient, and it's velocity at that point is called the "exit" velocity. The velocity at the rotor would be 1/2 the exit velocity. Nasa has an article about this for propellers:

https://www.grc.nasa.gov/WWW/K-12/airplane/propanl.html

 

[SlowThinker]

Nasa has an article about this for propellers:
https://www.grc.nasa.gov/WWW/K-12/airplane/propanl.html

The good thing is that I seem to have the same result for F as they do (their $v_p$ is my $v+\Delta v/2$).
I find the explanations of wing or propeller based on pressure needlessly complicated. It's really just pushing air down or behind that generates lift or thrust.

 

[rcgldr]

I find the explanations of wing or propeller based on pressure needlessly complicated. It's really just pushing air down or behind that generates lift or thrust.

or "pulling" air down or behind from above or from in front ...