Flooding and Stream area and speed

[fog37]

TL;DR Summary

flooding in a stream with varying cross-sectional area

Hello,

I am thinking about a real-life problem: the flooding of a stream in may area of town.

A stream discharge, $Q=A v$, represents the volume of water passing through the cross-sectional area $A$ in one second as the water moves with speed $v$.

Let's assume that the stream has 2 connected sections with different cross-sectional areas $A_1 <A_2$ . As precipitations increase, more water is dumped into the stream. Let's say that, at one point, the section of the stream with cross-sectional are $A_1$ gets fully occupied. To accommodate that increased discharge and prevent flooding (water outpouring out of the stream section), the water speed $v_1$ increases since the cross-sectional cannot further increase.
Why would flooding occur if the water speed $v_1$ can naturally and adequately adjust itself and increase?

I am not clear on the mechanism of flooding. When the water transits from the stream section with area $A_1$ to the section with larger area $A_2$ the water speed slows down since $A_2 >A_1$ according to Bernoulli's principle.

To prevent flooding, do the two stream sections have to have the same maximum discharge?
Is the maximum discharge for each section calculated as $Q_{max} = A v_{max}$ implying that there is $v_{max}$ for each stream section (water speed cannot increase forever)?

For example, $Q1_{max} = A_1 v1_{max}$ and $Q2_{max} = A_2 v2_{max}$ with $Q1_{max} = Q2_{max}$

Thanks!

 

[fog37]

Just to elaborate further, discharge is $$Q = v A = v W D$$

Assuming a rectangular cross-section, the area $A=WD$.

All 3 factors (speed, width W, depth D) increase with an increased discharge due to strong precipitations.
Not sure which factor increases faster. Flooding occurs where W and D exceed the stream geometry...

 

[Lnewqban]

Water has inertia and viscosity; therefore huge volumes of it are unable to adapt to geometric changes.
There is also friction and turbulence that consume energy downstream.

 

[Baluncore]

Why would flooding occur if the water speed can naturally and adequately adjust itself and increase?

Because where the stream narrows, the water speed and therefore kinetic energy must increase. That KE comes from the potential energy as water falls into the narrow channel. That requires a drop from the wide channel into the narrow channel. Since the entry to the narrow channel is fixed, the water level in the section above the step change must increase, which causes the flooding.

 

[fog37]

Because where the stream narrows, the water speed and therefore kinetic energy must increase. That KE comes from the potential energy as water falls into the narrow channel. That requires a drop from the wide channel into the narrow channel. Since the entry to the narrow channel is fixed, the water level in the section above the step change must increase, which causes the flooding.

I see. Thank you Baluncore.

In summary:

1) let's consider first an oversimplified stream of rectangular cross-section $A$, width $W$, depth $D$. The channel cross-section is constant. The channel is initially half full of water. Precipitations increase and the channel get fuller. As it gets fuller, the water level rises and the water speed increases to adjust for the larger discharge. As it rains even more, eventually the stream cross-section becomes full and the speed cannot increase any further (due to friction, etc.) and flooding, i.e. spilling of water outside the banks, happens.

2) When the channel has 2 joined sections with different cross-sectional areas (large to small area), the weak link for flooding becomes the section with the smaller cross-section because the water needs to speed up as it transitions into the smaller cross-section and the water transferring from the large cross-section into the smaller cross-section causes the spilling...

But why does the "spilling" have to occur when the water transfers from a large cross-section into the smaller cross-section? Can it not just flow into it as it accelerates?

 

[Baluncore]

Can it not just flow into it as it accelerates?

Conservation of energy.
If the velocity doubles, the KE is four times greater, so the drop at the entrance that releases potential energy must be four times higher.

 

[fog37]

Because where the stream narrows, the water speed and therefore kinetic energy must increase. That KE comes from the potential energy as water falls into the narrow channel. That requires a drop from the wide channel into the narrow channel. Since the entry to the narrow channel is fixed, the water level in the section above the step change must increase, which causes the flooding.

I guess my confusion has been that, in a closed pipe, going from a large cross-section to a smaller one, there are no issues of spilling out. But when the channel is open air, and water goes from a larger channel to a narrower one, even with tapering, the water speed certainly increases but the variation in cross-section unavoidably leads to water overflowing out of the banks of the smaller cross-section channel...

 

[jack action]

You may be interested in the study of open-channel flow, particularly the shape of the free surface relative to the solid boundary, and the hydraulic jump.

 

[jack action]

Hydraulic jump video:

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[fog37]

Hydraulic jump video:

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Thank you! I am certainly interested in learning about this cool phenomenon which I have never heard about.

At high level, how would it play a role to the transition of flowing water from a wide stream section to a narrower stream section?

 

[jack action]

In the following video, they close a gate to restrict the flow downstream (effectively setting a narrower section). This moves the hydraulic jump upstream. One could imagine increasing the flow upstream would have the same effect. This could explain how the level of a river could raise where it used to be low with a lower flow rate.

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[jrmichler]

I guess my confusion has been that, in a closed pipe, going from a large cross-section to a smaller one, there are no issues of spilling out. But when the channel is open air, and water goes from a larger channel to a narrower one, even with tapering, the water speed certainly increases but the variation in cross-section unavoidably leads to water overflowing out of the banks of the smaller cross-section channel...

When a full flowing pipe changes diameter through a smooth transition, Bernoulli's equation gives the change in pressure. When an open channel flow flows through a change in channel shape, any pressure change shows up as a change in the surface gradient. Open channel flow is more complex than full pipe flow because of the large variations in channel shapes and roughness. The effect of channel shape is summarized in the Manning formula by the hydraulic radius, and roughness by a friction factor. See the Wikipedia article about the Manning formula: https://en.wikipedia.org/wiki/Manning_formula.

And I got my exercise this morning climbing under a bridge that will be replaced. The US infrastructure bill apparently has money to replace all narrow bridges in the US, and this is one such bridge. Just as the project manager asked about daily traffic over the bridge, we watched a string of vehicles pass, and a truck that had to stop so oncoming traffic could pass. We told her that was a common occurrence on this bridge.