What they meant here is the static pressure at point of contact inside the control volume. Assuming you can shrink yourself small enough and can be inside the control volume. That is the pressure that you see or feel if the propeller is accelerated
P1-P2 is the unknown variable here and.the speed is the known variable Just to show that rotaion create vacuum therefore it function as vacuum pump!
It is terminology because the -1/2 RhoV2 is called velocity pressure and it is negative
First of all P1-P2 doesn't matter. What matters is that the static pressure at the most cavitation prone part of the propeller, which I called P2, is greater than the vapor pressure of water. This is what the video you posted said.
The other thing is that according to this image you posted yourself
When the water moves around the propeller at an increased speed, it reduces the internal pressure of the water, which can cause cavitation. Notice P2 <P1, thats what the picture says.
Admittedly this is a simplification like we've both said. In my example, I'm not going into the minutiae of how bubbles form, but thinking of the propeller as a system that has an input of water at a speed of V1=0 and pressure P1=rho*g*depth+P_atmosphere, and an output of an accelerated jet of water.
Inside the propeller "system" there exists a maximum water speed, which I called V2. It depends on propeller design and rpm. At that point, according to Bernoulli, there is the lowest static pressure P2, which must not be below P_vaporpressure for cavitation not to occur.
Now if we plug in these values into the formula in the picture above you gave and additionally substitute h1=h2, because there are no height changes, and do some algebra we get:
V2=sqrt (2g*depth+2P_atm/rho-2P_vaporpressure/rho)
Which clearly is a function of depth, rest of the stuff on the right is constant.
Now if you want to be super pedantic, technically the input water speed V1 is the speed of the sub which depends on V2. The equation would be a lot messier, but I left it at 0 for simplicity. The equation now shows the trend how fast a sub at a standstill can rev its propeller without cavitation in various depths. This is assuming V2 is proportional to propeller rpm, which is imho a fair estimate to the first order.
Hendrik, I appreceate your criticisms of my approach, but could you specify what exact things you think the individual variables in Bernoulli's equation describe when applied to a ship's propeller.
