How fast does rushing water travel through a tunnel?

Factual Questions
1

Weird water-related physics-type question:

Assume an infintely large swimming pool filled with water. At the base of one of its walls is a large hatch, 20’ wide and 20’ high.

The hatch opens onto a stone tunnel that is also 20’ high by 20’ wide. The tunnel slopes gently downward, directly away from the pool, at a 10 degree grade.

You open the hatch. Water goes spilling out of the pool and into the tunnel. After that water has traveled a quarter mile through the tunnel, with more pouring in from the swimming pool all the time, how fast is the front of the “wave” going? 10 MPH? 20 MPH? 100 MPH?

Informed answers appreciated. WAGs also appreciated. :slight_smile:

Thanks!

-P

2

This isn’t a homework question, is it?

3

Homework? Naw. Perhaps worse, it’s a Dungeons & Dragons-related question. :smiley:

-P

4

You’ll probably have to specify a depth for the pool feeding the tunnel as the pressure will affect the rate of flow (I assume that you only specified ‘infinite’ in terms of it not running our of water before the flow speed stabilises.

5

Hm. In the case in question, we think the “pool” is actually a huge underground lake. For purposes of this experiment, say the pool is a quarter-mile in depth, with the tunnel opening onto its bottom.

I would be interested in knowing how much the depth of the pool affects the answer. If if it were only 100 feet deep, for instance, how much would that reduce the speed?

Thanks!

-P

6

The emergency release penstocks at Hoover Dam are 68 inches in diameter, so they are similar to your example. The jet flow gates that control the emergency outflow from Lake Mead were recently tested, and was documented here:

JET FLOW GATE TESTING BRINGS A CROWD

7

The depth of the water above the hatch is critical; the resulting flow rate will depend very strongly on that. I don’t have any reference information here with me, and don’t feel like deriving equations right now, but this question could be answered reasonably precisely in symbolic form - even using empirical data to estimate the effects of the opening’s shape.

I don’t have time at the moment, but if no one else does it I’ll give this a shot tomorrow (hopefully).

8

Fear, while I’m sure the speed of the water right at the opening was phenomenally fast, I wonder if it would retain that speed through a long length of tunnel. I’d think that as the tunnel filled up, the speed up at the front of the wave would slow down, since the incoming water would be meeting more and more resistance (provided by the water already gone through).

-P

9

I don’t follow you Parthol. That speed is at the exit, so how could the water behind it in the tunnel move slower? If there is 28kgps flowing at the exit, it stands to reason that there must be 28kgps moving past any given point in the penstock at the same time. The only way it could be moving slower is if the penstock was larger in diameter than the jet flow gate, allowing the same volume to pass at a slower speed.

10

Depending on how much the depth of the pool matters, there’s also the question of the smoothness of the tunnel – is it assumed to be well-smoothed, or rough-hewn rock? The effects of friction on the sides of the tunnel may end up being important. (For a pool of infinite depth, I believe that & the viscosity of the water would be the only thing limiting the flow rate).

11

Yeah, there would be a conservation-of-mass problem if the flowrate through different sections of the tunnel weren’t the same.

12

Well, my thinking is that when the floodgate is first opened, there is no resistance or pressure on the tunnel side; it’s just full of air, which is easily pushed forward down the tunnel. But after a while, the water in the pool would be resisted by the water that has already entered the tunnel.

I’ll readily admit that my reasoning is probably flawed, me not being any kind of physicist. But I’d assume that the rate of flow would be governed by the pressures on both sides of the floodgate. (To consider the extreme case, if the floodgate opened up onto another equally-sized pool that was already filled with water, the rate of flow would be zero, right? And if it opened onto a filled pool that was only half the size of the first, the water would come through but not as fast as it would into empty space.) If that much is correct, then wouldn’t the water at the opening get slower and slower, as it had to push more and more water in front of it?

-P

13

Assume that the tunnel is smooth rock.

-P

14

A quick search on Google has come up with the “Manning Formula for Hydraulic Flow” which seems to be used pretty extensively in sewer design.

But, I’m guessing that that formula isn’t applicable in this case because the OP is talking about a pressurized conduit and the Manning formula is for unpressurized, gravity fed sewer systems and the like.

Any hydraulic engineers out there?

15

if you have two lakes connected by a tunnel, then you can think of the lakes as one system, and one lake will drain to the other lake until water level of both lakes are at the same altitude. This will always happen whether one lake is bigger than the other or not as long as all of the air gets removed from the tunnel (which should result from gravity if I correctly understand you description of the question). The rate it goes through is proportional to the pressure, which results from the depth of water and not the volume of the lake.

16

Only if the water level of the second pool is the same as the first. If the second pool is below the first, the water would flow into it.

Nope. The size of the pool has no effect, only the relative levels of the two pools (assuming the same air pressure above each pool).

There’s your flaw. Water flowing downward through a pipe does not “push” the water in front of it, it “pulls” the water behind it. See Cecil’s explanation of the siphon principle.

17

Here’s a reasonably informed guess: The equation governing steady flow in a pipe is the following:

h[sub]L[/sub] = f(L/D)(V[sup]2[/sup]/2g)

where

h[sub]L[/sub] = head loss
f = friction factor
L = length of pipe
D = diameter of pipe
V = velocity
g = gravitational constant (32.2 ft/sec[sup]2[/sup])

The head loss, h[sub]L[/sub], is the vertical distance from the top of the “swimming pool” to the point where the water front is (in other words, the two points where there’s an air/water interface). In your case, this is L*sin([symbol]q[/symbol]) + h, where h = height of the swimming pool, and [symbol]q[/symbol] is the angle of the pipe.

The friction factor, f, in general depends on the relative roughness and Reynold’s number (which adjusts for laminar/turbulent flow). For gigantic pipes like you’re talking about, the flow is turbulent and the Reynold’s number dependence disappears. Relative roughness is defined as e/D, where e is the average size of a surface protuberence. For example, if you have 1" of roughness in a 20’ tunnel, e/D = 1/240 = 0.0042. The friction factor can be determined from a Moody diagram. In this case, use this chart:



e/D       f
0.01     0.036
0.004    0.027
0.001    0.019
0.0002   0.014
0.00005  0.011

Now, to simplify:

V = SQRT[(2gh[sub]L[/sub]/f)*(D/L)]

For a 20’ deep swimming pool, and a 1/4 mile, 10[sup]o[/sup] passage, L = 1320 ft and h[sub]L[/sub] = 20 + 5280/4*sin(10) = 229 ft. With a surface roughness of 1", the 20’ channel has a friction factor of 0.027, so

V = SQRT[(232.2229/0.027)*(20/1320)] = 90.1 ft/sec = 62mph.

Tah-dah! Better check my math, though.

There’s some approximations involved, of course. Square channel vs. round pipe is one. The most important is that we’re using the equation for fully developed flow, where in reality the flow at the end of the “pipe” is in transition. However, for a fairly long pipe like you describe I think the error would be small. In particular, I would think the error is not much larger than the erroe introduced by approximating the surface roughness, e.

18

[nitpick]
brad_d mentioned that the flowrate through all sections of the tunnel must be constant.

That’s the case during steady flow; under unsteady flow conditions the flowrate is dynamic.

Here I’m also making the distinction between velocity and flowrate. Velocity = feet per second; Flowrate = gallons per minute. Flowrate has a volume associated with it.

Velocity is not constant across the cross-section. Velocity distribution is more uniform for turbulent flow than for laminar flow, but still will vary across the cross section of the tunnel. In other words, water in the center of the tunnel will be traveling faster than that against the tunnel wall.

For laminar flow, the maximum velocity is at the center of the pipe and is twice the average velocity.

BTW the formula used by zut is the Darcy-Weisbach Formula.

19

Have I mentioned that I love this board? :slight_smile:

Thanks to everyone who responded, especially to zut who did the calculations. Consider a small part of my ignorance to have been successfully fought.

-P