Can anyone explain to me what the Velocity Area method for measuring river or water flow is?
My guess is that the product of the cross sectional area and the velocity of water flowing in a pipe is always constant. If the Cross sectional area of the pipe increases at a particular point, then the velocity decreases so that the product $AV$ is a constant. Am I right?
If so, how can we extend this to pipes where the water is accelerating & does not have a constant velocity? For example, the system may be under the action of gravity & hence the acceleration of the water is $g$, the acceleration due to gravity?
Answer
What you refer to is conservation of mass under some assumptions:
- Constant density
- A steady state flow
I'll bring us back to your equation by starting with the very fundamental mass accounting for a given fluid flow. To be comprehensive, we need to recognize that velocity isn't constant over the entire area, but we will assume that it is. Take the flow rate to be $\dot{m}$.
$$\dot{m} = \rho V A$$
Now, if we have a steady state flow along a single flow path, then this quantity will be constant over the entire path, $\dot{m}=const$. Water in the cases you are concerned about is sufficiently incompressible so $\rho = const$. This results in your conclusion that $VA$ is constant.
Gravity may or may not shift the balance from $V$ to $A$ or vice versa. It depends on if there are rigid boundaries to the flow. If you have a flow fall freely in air or flow downward in a trench (like a river) then the boundary of the fluid may change freely. If you have a pipe with a given flow area, then the velocity is fully determined from that. Anyway, there are laws that conserve other things - like energy. So in a rigid pipe flowing downward (absent friction) the pressure will increase as you go down in elevation, which results directly from gravity.
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