Non-Uniform Flow 2

Non-Uniform Flow 2


Now you are the engineer, and you have been
given this design and your task is to analyze the channel to determine the flow depth in
the channel from start to finish. This is a single channel with a constant geometry
and roughness. The only change in the channel is the slope. Water enters the channel through
a sluice gate at the top of the channel. At the bottom of the channel the water is backed
up by the water ponding behind a dam. The first step is to complete the steady / uniform
analysis. Start by calculating the critical depth. Since
this is a rectangular channel with a base width of 15 feet and a design flow rate of
500 cfs, then we can calculate the critical depth directly using this formula.
I get a critical depth of 3.26 feet. Next calculate the normal depth for each segment
of the channel. The normal depth is found using the iterative approach to Manning’s
equation. For the mild channel I get 4.58 feet for the
normal depth, and for the normal depth in the steep channel
I get 1.47 feet. It helps to visualize the solution better if you
sketch in the normal and critical depth to your design drawing.
Completing this step allows us to classify each segment of the channel as mild or steep.
The next step is to qualitatively assess the actual channel flow depth. We do this by sketching
how we predict the flow will change as it navigates the changes in slope. For example,
at the downstream end, just before the dam, the flow depth
will have to increase in order to match the ponded water depth behind the dam. This is
a gradually varied flow section. Because the flow depth is supercritical, and the dam is
ponding the water to a depth above critical depth, then just before the GVF profile there
will be a hydraulic jump. When the slope transitions from Mild to Steep
we should expect to see a GVF profile in the mild and steep sections.
We assume that the flow will pass through the critical depth at the grade break.
Finally, as the water enters the channel, the sluice gate has an opening less than the
normal depth, so it is possible that there may be a gradually varied flow section followed
by a hydraulic jump. For this to happen the depth of flow in the mild channel can not
be so deep that it buries the jump. We can use the conjugate depth equation to determine
if the jump will occur or not. To organize our calculations it is important
to classify each of the GVF profiles. Starting with the GVF just downstream of the
sluice gate we know that it is an M profile because it is in a Mild channel. Because the
orange transition line is below the critical depth and normal depth, then we state that
it is a Zone 3 profile as well. This makes our first
GVF profile an M3. Moving downstream we are still in the mild channel and the depth is
decreasing as the velocity begins to increase. A decreasing flow depth is
A zone 2 profile. Making this an M2. Entering the steep channel this becomes an
S2 for the same reasoning that we called the last one an M2. Finally, just after the hydraulic
jump, and just before the dam we have a flow depth that is above the normal and critical
depth. This means this GVF is classified as an S1 profile.
Next it is always helpful to identify all of the unknowns remaining in the channel.
For example, just before the S1, We do not know the depth. It just so happens
that we can calculate this depth using the hydraulic jump conjugate depth equation.
Once that is done we can calculate the length of the S1.
Since we already know the normal depth in the mild and steep channel sections, we just
need to calculate the length of the M2 and S2.
Finally, just downstream of the sluice gate we do not know the depth at the end of the
M3. To find this we use the hydraulic jump conjugate depth formula using the normal depth
and velocity in the mild channel as the downstream input data. By the way, if you calculate the
upstream depth and find that it is greater than the critical depth, then there will be
no hydraulic jump – there will be no M3 – because the normal depth in the channel
will be so great that it will bury the jump. If there is a jump, then the last unknown
is the length of the M3. With the channel flow assessed and the unknowns
identified you are ready to start your calculations.

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