Dams and water pressure

[VoxRationis]

This is a very low-tech question compared to much of the rest of this subforum's discussion, but it hinges on principles of fluid physics that I don't know off the top of my head, so I thought I'd float the question.

Does the pressure in a connected body of water on its container exert based on all the potential energy of the water, even if the connections are scant?

For example, take two adjacent bodies of water, as in the diagram below:

Spoiler: Diagram 1; water is marked W

Show

|WWWBWWWA
|WWWBWWWA
|WWWBWWWA
|WWWBWWWA

Presume that the wall or dam marked A is sufficient to withhold the pressure of its body of water, but would not be sufficient to hold the two bodies together if dam B were removed. However, if only the topmost segment of B were removed, such that there were a connection between the two bodies at the top (as in the diagram below), would that result in a collapse?

Spoiler: Diagram 2

Show

|WWWBWWWA
|WWWWWWWA
|WWWBWWWA
|WWWBWWWA

Also, though I suspect I know the answer already, would there be a collapse if a connection were made at the bottom of dam B?

[Radar]

Interestingly, it does not matter in a static situation as the pressure on dam A depends only on the height of the water behind it. Hydrostatic pressure is a simple function of depth, so connection or lack thereof with another body of water is irrelevant as long as the water level stays the same.

[halfeye]

This is a very low-tech question compared to much of the rest of this subforum's discussion, but it hinges on principles of fluid physics that I don't know off the top of my head, so I thought I'd float the question.

Does the pressure in a connected body of water on its container exert based on all the potential energy of the water, even if the connections are scant?

For example, take two adjacent bodies of water, as in the diagram below:

Spoiler: Diagram 1; water is marked W

Show

|WWWBWWWA
|WWWBWWWA
|WWWBWWWA
|WWWBWWWA

Presume that the wall or dam marked A is sufficient to withhold the pressure of its body of water, but would not be sufficient to hold the two bodies together if dam B were removed. However, if only the topmost segment of B were removed, such that there were a connection between the two bodies at the top (as in the diagram below), would that result in a collapse?

Spoiler: Diagram 2

Show

|WWWBWWWA
|WWWWWWWA
|WWWBWWWA
|WWWBWWWA

Also, though I suspect I know the answer already, would there be a collapse if a connection were made at the bottom of dam B?

The static pressure in a column of liquid depends only on the depth. You could remove wall B entirely, and it wouldn't make any difference. The pressure is highest at the bottom, which is why dams are thicker there.

Ninja'd.

[VoxRationis]

So the glass wall of a 200-meter-tall, 1-cm wide graduated cylinder full of water could also hold plug Lake Mead? That seems... counterintuitive. I would not have thought that.

[Chronos]

No, because you couldn't make a 200-meter tall cylinder full of water that's only 1 cm wide. At least, not if you're talking about the outside diameter. You could make the inside a constant diameter, but the bottom would have to be extremely thick (like, comparable in thickness to Hoover Dam itself) to contain the pressure.

[Imbalance]

No, because you couldn't make a 200-meter tall cylinder full of water that's only 1 cm wide. At least, not if you're talking about the outside diameter. You could make the inside a constant diameter, but the bottom would have to be extremely thick (like, comparable in thickness to Hoover Dam itself) to contain the pressure.

Or engineered pressure piping of greater diameter, like the riser in a 60+ story building.

[Fat Rooster]

So the glass wall of a 200-meter-tall, 1-cm wide graduated cylinder full of water could also hold plug Lake Mead? That seems... counterintuitive. I would not have thought that.

The fact you are neglecting is that the curvature of the cylinder is where the strength is coming from. The curvature can turn the pressure into pure (relatively moderate) tensile forces, and transmitted sideways. A very wide flat wall cannot do that, needing to transmit all the force into the ground somehow, usually a bending moment. At the base of the wall you have a bending moment that has to deal with all the force the water is applying.

In a static system with nothing accelerating the forces on each 'thing' must add up to zero. For a body of water, that means that the pressure on all sides must add up to zero. Containing pressure essentially boils down to connecting an inward force on each side to an equal inward force on the opposite side. The difficulty is not in creating the force, it is connecting that force to the other side. For a 1cm pipe, it is pretty easy. The pressure is high, but the area is not, so the forces are not particularly high, and the other side is close. For a dam, things are not nearly so easy. If you consider an area maybe 20m below the waterline in the middle of the dam, that area is contributing a force that needs to be balanced, and it will be dozens or hundreds of meters from the nearest anchor. It is like suspending that same force in mid air. The difficulty of building a bridge is dependent on the span of a bridge, as well as how much weight it has to hold, and a similar thing happens here.

The reason pipes are so effective is that they are pure tensile structures, like suspension bridges, while dams are typically built like arch bridges. Pipes are also usually small, where dams are big.

As for the original question, all the potential energy is sort of available, but the forces only depend on the pressure. If you run a pipe down a hill and connect a turbine at the bottom the pipe doesn't care how big the reservoir at the top is. It only cares about the pressure (related to the level of the reservoir). How long the turbine can run for does depend whether it is a lake or a bathtub at the top though.

As another example, have you seen the difference what happens when a high pressure gas cylinder ruptures compared to a high pressure water one? Water at high pressure doesn't 'contain' energy the same way air does, so the compressed air has far more energy. The pressure vessel doesn't care though, all it sees is pressure, unless that energy is allowed to escape somehow.

[Tvtyrant]

A good example of the power of relatively small pressures.

[factotum]

A very wide flat wall cannot do that, needing to transmit all the force into the ground somehow, usually a bending moment. At the base of the wall you have a bending moment that has to deal with all the force the water is applying.

Just to note, dam walls can do the same thing. A concrete dam like Hoover is made curved precisely to redirect some of the water pressure into the abutments on either side and thus allow the dam wall itself to be somewhat thinner than it otherwise would be.

A good example of the power of relatively small pressures.

Great video, but the description is arrant nonsense. A vacuum of -27psi, out in the open, at sea level, on this planet? Nope, even with absolutely zero pressure inside there the difference would be a bit less than 15psi between inside and outside.

[Tvtyrant]

Great video, but the description is arrant nonsense. A vacuum of -27psi, out in the open, at sea level, on this planet? Nope, even with absolutely zero pressure inside there the difference would be a bit less than 15psi between inside and outside.

Yeah I don't know what the internal pressure was, I just know it is tiny compared to a lake and crumples that truck like a tin can.