Stagnation point and a water dam

[fog37]

Hello Forum,

I am clear on what the pressure at a point $P$ is on the wall of heigh of a dam at a certain depth $d$ below the water free surface: $\rho g d+P_atm$. The deeper we go the higher the pressure.

Now let's consider a different scenario: there is initially no water and water starts flowing at a speed $v$ and rushes against the wall of the dam. The point of impact becomes a stagnation point since the fluid is brought to rest and or/ diverted upward: as time $t$ goes by, the water level increases since the water has nowhere else to go except upward.

Question: as water comes in (the water level will eventually reach the height of the dam wall) is the pressure $p$ on the wall at point $P$ the same, larger or smaller than the hydrostatic pressure at the same point $P$ when the fluid is instead completely at rest?

This is clearly an application of Bernoulli's equation but I am not sure how to use the principle properly.

I had some flooding recently and the fence was pushed down so I wonder if it is due to the pressure due to the amount of water (height of the water volume) that started accumulating against the wall or to the impact/momentum of the water rushing in...

Thank you!

 

[anorlunda]

If you think of a dam and its reservoir, then the stagnation point is where the incoming stream enters the reservoir. That is typically far from the dam.

But if you are thinking of a fence washed away by a flood, then both the vertical head and velocity of the water can be significant. It depends on the numbers, how high and how fast. Also remember that as soon as the fence is overtopped, it will have water behind it too, so the static head difference upstream/downstream goes away.

To help point you to equations that may be helpful, consider this https://en.wikipedia.org/wiki/Pelton_wheel#Design_rules
Although Pelton wheels read maximum efficiency at relatively high heads, you are not concerned with maxima, but rather the forces needed to knock down a fence.

 

[fog37]

Thank you anorlunda!

For the fence case, I guess, as you mention, both the vertical head and velocity of the water can be significant. If the water stopped rushing in, the pressure at a point $p$ would become solely hydrostatic and due to the vertical head. My dilemma was about the pressure when the fluid was still rushing in against the fence wall.

Just as a personal clarification, the pressure $p$ in Bernoulli equation is termed "static" because it would be measured by an instrument not moving relative to the flowing fluid. If the fluid is not flowing, the static pressure becomes the hydrostatic pressure of the fluid at rest. If we measured the pressure of a flowing fluid with an instrument moving at the same speed as the fluid, the recorded pressure would be null. This is what think...

Thank you for the link.

 

[anorlunda]

My dilemma was about the pressure when the fluid was still rushing in against the fence wall.

Absolutely. Think how easily a fire hose can knock down your fence with zero water backing up like a dam.