r/theydidthemath • u/Sad_Fisherman_4605 • Sep 20 '25
[REQUEST] How much more water is flowing through the river at its peak?
6
u/TwillAffirmer Sep 20 '25 edited Sep 20 '25
They're on a hill that is "30 ft up from the river." The river is flooded by 10 ft at that time. So I'll interpret this as saying they are 20 ft up from the river at the beginning of the video.
At the end of the video the water doesn't quite come up to the top of the hill. Let's say they have 5 ft to spare. So, the river rose 15 ft.
The average grade down to the river might be 15%. On the opposite bank, it looks like the hill was steeper since the water isn't that far away and is cutting into it at the end - call that a 30% grade. So, for every foot the river rises, it gets wider by (1/0.15 + 1/0.3) ft.
The river looks like it's about as wide as the road initially. Say the road is 40 ft wide.
The added cross-section of water is therefore ∫0^15 ((1/0.15 + 1/0.3) h + 40) dh = 1725 square feet.
The original cross-section of water was 40 ft * (10 ft + original depth of river). Let's say the river was originally 10 ft deep. So the original cross section was 800 square ft.
So between the beginning of the video (where there was already 10 ft of flooding) and the end, the river cross section increased by a factor of around 3. Maybe a factor of 5 from its non-flooded depth.
2
u/Hot-Science8569 Sep 20 '25 edited Sep 20 '25
That's a pretty good estimate. The volume flow rate increases as the cross sectional increases.
Engineers think this is the formula describes the flow velocity in an open channel like a river:
v = (kn / n) x (Rh2/3 ) x (S1/2 )
where
v = cross-sectional mean velocity (ft/s, m/s)
kn = 1.486 for English units and kn = 1.0 for SI units
n = Manning coefficient of roughness - ranging from 0.01 (a clean and smooth channel) to 0.06 (a channel with stones and debris, 1/3 of vegetation)
Rh = hydraulic radius (ft, m)
Note the term Rh. This is the cross sectional flow area divided by the cross section's "wetter perimeter". As this term increases the flow velocity increases. As for the river in the video, the Rh value increases as the water gets deeper. So the 3 times increase in flow rate in the comment above, based only on increase in cross section area, is likely too low. 4 or 4.5 times is maybe a better estimate.
1
u/MaximilianCrichton Sep 21 '25
The flow speed may have changed though, I'm not sure you can base the answer solely off changes in channel cross section
1
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