# 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.

## 3 Replies to “Non-Uniform Flow 2”

You are awesome Kenneth

this is perfect

Thank you very much for these awesome videos