Backyard Fluid Dynamics Revisited

Back in July, I did a post looking at how the fountain in our ornamental backyard pond shoots higher when the level of the pond drops. I set up a simple model of the process, which worked surprisingly well, but I said at the time that I really needed more data to say whether that agreement was real or accidental. Well, yesterday, I got some extreme data:

The leak in the pond has gotten worse, I think, and the water was barely covering the top of the pump box at all. A very rough calibration of this image, using the fact that the brick is 2in high, gives a height of the spray of about 62in, or 98.4% of Kate’s height.

So, how does this fit with our toy model?

The toy model I used last time had as its final result this expression for the “extra” mass of water above the fountain nozzle in terms of the base mass and the heights at two different fill levels:

Using the best-measured point from the earlier post, with a height of 27 cm, and yesterday’s picture with a height of 157 cm, we end up with a mass ratio of -0.83. The negative sign is because we need to take mass away from the top of the nozzle to get it to shoot up much higher, and this is a fraction of the initial mass, so this says that the depth of the water above the nozzle has decreased by 83%.

The initial depth in the July picture was about 1.5 in, or 3.8 cm. Reducing that by 83% leaves just 0.65 cm, or a little over a quarter-inch of water above the nozzle.

Which, again, is awfully good. This could be refined a little– my calibration of heights assumed that the brick was completely exposed, which would change the height a little, but that’s only about a 10% error, which isn’t really within the accuracy of the quasi-measurements I’ve done here. There could also be some distortion of the height from the angle of the camera,as the stream in yesterday’s picture takes up a good fraction of the CCD, but again, this is a very crude measurement at best, so I’m not worried about it.

What I am worried about is what I’m going to do about the fact that the pond is leaking water faster and faster. Repairing the leak is going to be a big pain in the ass, and I don’t need equations to tell me that.