If a concrete stream bed were 120" wide and the water depth was 6" on the average ( street runoff) . The slope was 5ft per mile. The sea is about 6 miles away from this point.
How many gallons a minute might be flowing
If the stream bed had a constant slope and demenisons and all the water came in from a point upstream of anything measured. Would the water depth remain constant regardles of altitude? And would the rate of flow also be constant regardless of altitude at any point along the stream all the way to the sea. ( ignore tides)?
I come up with about 15,000 gallons a minute and 5.5 miles per hour. I tried this with a kinetic energy/velocity method. Curious what you all might have.
Is 2 yes because of friction? Because, like, I’m envisioning a “trivial case” of a waterfall, where the speed definitely increases, at least up to a point. In a long, gently sloping streambed (idealized for smoothness) would there at least be some very initial small acceleration of flow, right at the source?
(I’m not actually meaning to nitpick; what I’m wondering about is the mechanism by which the flow would be “regulated” to be constant.)
(North of Los Angeles, there is – or used to be – a huge, very steep flume, racing right down a hillside – visible from the freeway – where the bottom has been roughened by placement of vertical concrete slabs – sort of like a forest of very large tombstones – which slows the water, keeping it from being a virtual waterfall – and which also aerates the water by forcing it to foam heavily. Without these obstacles, wouldn’t the water accelerate as it flowed?)
I’m coming from a river running perspective, where wider or flatter equals slower current and constricted or steeper equals faster. Consistent cross section of the streambed, consistent gradient and no obstacles means an even flow.
Trinopus actually has a good point. To put it simply, would those numbers be different if the inlet of this stream bed were a waterfall or some other source with a higher-than-usual water pressure/flow rate?
Well, if the water came down a 50 ft waterfall it would have dropped the same amount that would have taken it 10 miles. Here is where I get confused. If the water had 10 miles to drop 50 ft would it be carrying the same energy as if it dropped off of a 50 ft falls. I think it would but the energy would be delivered much faster, in about 2 seconds as opposed to almost 2 hours. Narrowing would create an increase in pressure but also cause the water behind it to slightly back up depending on how much it narrowed. I think with a constant smooth slope the pressure and speed are regulated by the feed of water which establishes a pressure and flow responding to the slope.
This summer, we turned off of I-80 in California, onto highway 20 toward Grass Valley (north-westward). After some ten or fifteen miles, we passed over an aqueduct that the map identifies as Drum Canal. The descent of the road made it look irresolvably like the water was flowing slightly uphill. It was simply not possible to look at it and see the slightest bit of downslope to the flow. Is it ever possible for an open water channel to flow uphill for a short distance without overtopping the lowest stretch of its banks?
To estimate open channel flow, you’d usually use either the Manning or Chézy equation. These open channel flow methods incorporate both the roughness and the shape of the channel. Those two features have a pretty big impact on the flow rate, especially with a channel as wide and shallow as the one you’re talking about.
With Manning, I get an average velocity of 1.2 mph and a discharge of just over 4000 gpm. Chézy returns 0.83 mph for a discharge around 2700 gpm.
If you want to move more water, you have three options: change the shape of the channel, give it a smoother finish, or increase the slope. The velocity is not uniform within the channel’s cross section, and the farther from the walls you get the faster the water flows. This is why we typically use round pipes - a round cross section minimizes the influence of the channel walls on the flow. Your slope is also very, very low - you’d seldom see designed drainage channels with slopes less than 1% (more than 10 times the slope you specified).
The flood control channel runs next door to my house. I can tell the impact of drag on the sides and bottom is significant just by watching the flow but ignored that for the initial figure. I believe the actual slope is probably about what I guessed based on how the tides affects the flow. Would a cork flow at the same speed as the current if I wanted to time it?
Your figures actually look pretty close to what is actually happening in the drain ditch in real life. I just walked out and timed some debri and it appears to be going about 1 mph.
If you get a cork floating in the center of the channel, it would go a little faster than the average velocity (the floating cork would be as far away from the influence of the channel walls as possible) but in a channel that wide I doubt it would be that much of a difference. Give it a shot and see what you get?
I guess you did - it’s always nice when experiments confirm theory!
What actually got me started on this was a couple of things. I started taking math classes on the Khan Academy on line. I was also curious if I built a cabin next to a small stream if I could use the stream to generate enough power for a fridge and my basic power tools. Not so much that I plan to do this as much as I am looking for real life ways to use math. I can’t seem to grasp it if I can’t apply it to something I can relate to. I had the same problem as a kid. I find energy to be a facinating source for creating basic math problems.
Yeah, I often found it much easier to learn math (or, perhaps more honestly, to care about learning math) when I saw a practical application of the material. It helps me to think about the problem first, then consider the math as a tool for analyzing that problem.
Friction is exactly right. What you have is a system with a neutral energy balance - the energy added to the water by gravitational acceleration is equal to the energy dissipated by friction in the channel bed.
Those blocks are energy dissipation structures. In steep channels, water tends to flow at shallow depths with high velocity (called super-critical flow); in gently-sloped channel, it flows more slowly at deeper depths (sub-critical). The transition from super-critical to sub-critical involves a hydraulic jump, which has a large amount of energy to dissipate. In a straight transition with no additional controls, a lot of that energy is directed toward the channel bed and sides, causing scour and erosion of the channel itself.
The concrete slabs begin the energy dissipation upstream from the transition and direct the energy at the slabs themselves, which are easier to repair or replace when needed than the channel bed.
If you put more water into the channel, either the depth or velocity (or both) would increase. Unless HoneyBadger is very close to the ocean, the flow at his location is controlled by the upstream discharge (the flow rate) and channel geometry. Change the flow rate and you’ll change the depth of flow and velocity. A high-pressure inlet would affect the flow only between the inlet and the point at which the channel reaches an energy equilibrium; beyond that, everything is at atmospheric pressure. Same thing with a waterfall - there are local effects at the inlet, but ultimately it’s all discharge and channel geometry.
The channel flows about 12 or 14 miles to the ocean from where I am. I am about 80ft above sea level. The ocean is less than 6 miles from me in one direction and about 12 in another direction, the route the channel takes. The channel is a natural stream bed that was concreted over in the 1960’s.
(In nature, I’ve seen creeks splash water up over rocks, for instance, but on a very small scale.)
Another cool thing I’ve seen is a small, meandering creek, where the center of channel is flowing normally, but alongside the banks, eddying water actually is flowing backward, uphill. I watched a piece of floating trash make its way ten yards upstream in about ten minutes.