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A pipe 200-meters in diameter comes out of a cliff face and turns downward, like a bathtub faucet. A torrent of water is coming out: 10 meters from exit of faucet to surface of lake. The lake is 40 meters deep.

  1. How high are the waves created by the water continuously dumping into the lake?
  2. How far away from the faucet would a dock have to be for people to reasonably board boats tied up at the dock?

The lake is in an old quarry with steep edges, so the depth is 40 meters as soon as you step off shore. I don't know if the size of the lake matters, but it is roughly oval with major axis 20 km and minor axis 5 km. Faucet is at one end of the 20km with a spillway into a river all the way at the far end of the quarry.

I have no idea how to model this, so if you can provide info on how you calculated your answer, I'd appreciate that, just in case I have to adjust some aspect of this setup later for story reasons.

Note: I have posted a version of this question to the Physics Stack Exchange. The answer given there is very complete.

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    $\begingroup$ As far as I know, at t the present level of understanding of fluid dynamics, the best you can do is model the system as accurately as you can and hope that a simulation would be somewhat true to reality. Important elements: How smooth is the flow of water out of the pipe? How much water is flowing out of the pipe per second? What is the shape of the bottom of the lake under the pipe? $\endgroup$
    – AlexP
    Aug 14, 2021 at 11:41
  • $\begingroup$ You need to know one of three (equivalent) things to begin to model it: how high the head of pressure is thats driving the water, how fast it moves when it exits the pipe, or how far away from the cliff the jet hits the lake. Any of these will give you rough info on mass of water per second, and horizontal velocity. $\endgroup$
    – Stilez
    Aug 15, 2021 at 12:54
  • $\begingroup$ You may also want to ask on Stack Exchange Physics, how to model this situation: "Water continually exits an extremely large horizontal pipe at V m/s. The pipe is diameter D and its centre is located at height H metres above an infinitely wide lake of depth d. Which of these factors impacts the size of the resulting waves? In principle, how would you go about calculating the "order of magnitude" height of the resulting wave train, at varying distances from the pipe?" $\endgroup$
    – Stilez
    Aug 15, 2021 at 13:01
  • $\begingroup$ @stilez I have and have a bounty on it. No answers. Oh well. I may just have to do empiric testing. physics.stackexchange.com/questions/659407/… $\endgroup$
    – SRM
    Aug 23, 2021 at 4:00
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    $\begingroup$ Okay, that river thing. Hm, okay now I get it - the whole river gets in the pipe. So water flow is limited by how big is the river, how big is it? I think current answer is quite good illystration to your problem, as definetly it(niagara) does not spill over more than a good river does, anything less, less effects. So diameter of the pipe turns out to be irrelevant. Okay, now I see. $\endgroup$
    – MolbOrg
    Sep 2, 2021 at 17:58

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Niagara falls may be a solid low end starting point of the water volume you are going to see, 2,400 m3/s. Most of the white water rapids only go 800 meters or so down stream. However from one photo I saw a tour boat was getting much closer than that.

So as a solid approximation I would say 200 m to 1000 m up depending on how calm you want the water.

Edit: As commented below a 200 m diameter pipe has a cross-sectional area of 31,000 m2 Multiply that by Niagara flow rate of 10 m/s, giving you a much higher 310,000 m^3/s. Or 100 times Niagera's water flow.

Some of the Docks at Niagara for reference:

enter image description here

Tour boat getting rather close to Horseshoe Falls:

enter image description here

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    $\begingroup$ A pipe 200 meters in diameter has a cross-sectional area of about 30000 m^2. Multiply that by the rate of water flow through Niagara, which is 10 m/s, and you obtain 300000 m^3/s, 100 times greater than Niagara. This pipe's rate of flow would actually be much faster than Niagara because Niagara is only 50m high, and this pipe is at least 400m tall if it's shaped like a faucet. $\endgroup$
    – causative
    Aug 14, 2021 at 11:48

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