In order to maintain a constant water deep in canal, how much water must flow trought the pipe ?
As shown on picture, canal have a rectangular shape.
I don't know if canal length have an influence.
EDIT : to simplify things let's consider there is no turbulence, no viscosity, and that water falling from pipe do not disturb water in canal.
I tried to solve the problem by myself (i'm a physics beginner so it could be totally wrong, please do not downvote the question if you think this is not correct) :
Area of canal section : $A = w \, h$
If I calculate water velocity $v$ in canal, using this and surface $A$, I can calculate how much water $Q$ will flow :
$$Q = A \, v$$
and solve the problem…
So only thing left is to calculate $v$.
Let's say the canal have no inclination $Z = 0$, I think water velocity for a given water height can be calculated like this (I'm not sure about this) :
$$ v = \sqrt{ 2 \, g \, h } $$
$$ \delta Q = A \, v = w \, \delta h \, \sqrt{ 2 \, g \, h } $$
integrating h from 0 to H and gives :
$$ Q = w \, \sqrt{ 2 \, g } \int_0^H h^{1/2} \, dh$$
so discharge for a given height and width :
$$ Q = \frac{2}{3} \, w \, \sqrt{ 2 \, g } \, H ^{3/2} $$
Could anyone tell me if the above is correct (assuming there is not inclination), and try to answer my initial question ?
Best Answer
From the potential application of the question, I assume that question is to provide design parameters for a water slide. Because the depth of the water will be reduced from acceleration, the real question is how to add resistance to the canal. Resistance can be provided through the disruption through the use of twists and turns. I would generate empirical data of an actual set up for model. A water flow meter could show how various degree turns affect the speed reduction of the fluid. The use of a lower viscosity fluid could help to reduce the size of the pipe for modeling purposes. Because the fluid is not strictly contained within the diameter of a pipe, common equations are not applicable, however someone on this forum may provide something of use in the modeling area.