10 km water tower

10 km water towers & depressurizing aircraft/spacecraft: choked flow

This is in the spirit of Randall Munroe's excellent book,"What if?" (John Murray, 2014) whose sub-caption says it all: 'serious scientific answers to absurd hypothetical questions'.

I initially wanted to check on a cloudburst (> 100 mm/hr rainfall in a localized area): can you actually get a continuous sheet of water, with one raindrop almost touching the adjacent (or preceding) raindrop? Anyway, it is easy to calculate that even in a cloudburst it is nowhere near that limit.

So, the next question was what happens if I somehow make a (Mt.Everest-type) water tower 10 kms high, fill it with water, and then open a small hole at the bottom?

Equating potential energy to kinetic energy, you get the velocity of the water as it leaves:

                                                   v = (2gh)^1/2

       Plug in the numbers and you find that the exit velocity is greater than the speed of sound s (340 m/s).

Nitpicking #1: Yes, g will decrease a bit from mean sea level to a height of 10 kms, but not by a huge amount; so it will still go above s.

So the question is: do we get a sonic boom when the water rushes out?

Nitpicking #2: would a child ask such a question? Dunno. maybe I should pose as a child, and ask him!
Anyway, since I have not done that yet, let me partially answer that question.  
It turns out that there is a phenomenon in fluid mechanics called choked flow, because of which the water can only get out an orifice at a maximum speed, which is equal to the speed of sound s.
So, no sonic boom, I think...
The sudden doubt occurs because the pressure could build up sufficiently even if you do not actually cross the speed of sound, but just touch it. Anyway, I do not know enough to give an exact answer.

One of the best links to explain the phenomenon of choked flow is the NASA site:
            http://www.grc.nasa.gov/WWW/k-12/airplane/mflchk.html

This explains the math. I am not sure about the physics, but I guess that the reason the flow cannot exceed the speed of sound has to do with the very formula for the speed of sound:
            v = (E/r)^1/2

where E is the bulk modulus of elasticity and r is the density. I would argue that the pressure needs a certain continuity of medium for it to be transmitted - and that will not happen if the speed exceeds the speed of sound. The problem with this argument is that the NASA site uses a different formula for the speed of sound:

s = (gRT)^1/2

where g is the ratio of specific heats C_p/C_v and R is the gas constant. So, I leave it to the reader to draw his/her own conclusions.

Another example of choked flow is  rather disconcerting and dangerous: if you get a hole in a plane flying high up (because of a bomb, meteor etc), and the plane has a pressurized air (as they all do, if they fly above about 8,000 ft), the air will all get sucked out - but the exit speed is fixed at the speed of sound. A link that describes this phenomenon, but for the depressurizing time of a spacecraft, instead of a plane:
 http://www.spaceacademy.net.au/flight/emg/spcdp.htm