1. Chapter 7 continued Open Channel Flow
Specific Energy, Critical Flow,
Froude Numbers, Hydraulic Jump

2. Bernoulli Return to the Bernoulli equation for open channels:
Return to the Bernoulli equation for open channels:
H is the total head. The units are length.
Pressure , kinetic energy, and potential energy head

3. Depth and height above datum
As you know, we can separate z into two components: the depth of water h and the depth to some lower datum z0, maybe to sea-level.

4. Specific Energy
We can define a component of the total energy that only contains the flow depth and the velocity term.
*Note, no P/g
This is called the specific energy. Notice we changed h to y, and H (energy head) to E for energy
* this is appropriate for open channels, since nearby areas have about the same pressure

5. Specific Energy for Flow Rate
Now redefine specific energy in terms of discharge Q instead of velocity V.
Substitute V = Q/A
The velocity in kinetic energy was squared in the previous slide, so we will get a discharge term squared Q2/A2 .

6. Flow Rate per unit width
Let’s take the simple example of a rectangular channel, and then define q = Q/width. The area for a rectangle is A = base (i.e. width) x height, so only the height part is left after we divide by the width (the width of a unit width is 1. )
We changed from Q to q to show that it’s discharge per unit width.
On the right hand side of the equation, q2/2gy2 is the specific kinetic energy, and y is the specific potential energy. Notice that all terms have units of length, for the depth.

7. Plot of Specific Energy vs. Depth
Lets plot E, the specific energy, against y, for a particular flow rate (discharge) per unit width, q.
At the red line specific energy is minimum, so the exact slope at that point is zero, dE/dy = 0. For higher energy (blue line) there are two possible depths for the same specific energy.

8. Same plot with depth vertical
Let’s turn the graph on its side, as in the text. We can again graph how flow depth y changes for any change in Specific Energy E. For some constant q:
For the energy line shown, there are two possible depths where it crosses the blue plot of some flow per unit width q. The upper one is mostly potential energy (the water is elevated) , and the velocity is small; the lower one has greater velocity and is not as high

9. Critical Depth
There’s also one specific depth, yc, the critical depth, for which energy E in the system is minimized. This is the lowest specific energy for a given discharge q. If the flow is deeper (higher on the graph) than this, the velocity drops, but if the flow is shallower than this, the velocity increases.

10. Solving for Critical Depth
This critical point occurs where the derivative (slope) dE/dy is 0. So, take the derivative of E with respect to y. Only E and y are variables.
which is
Then or
Setting this equal to zero
gives

12. At the minimum specific energy
the ratio of velocity squared to depth times the gravitational acceleration is one.

13. Froude Number
V2/gy is the Froude Number, squared. Notice that it is dimensionless, i.e. all the units cancel. It is the ratio of kinetic to potential energy, and is used to characterize open channel flow.
And so, returning to the text, at the minimum specific energy the dimensionless Froude Number is:

14. Flow deeper than a Froude Number of Fr=1 (large depth in denominator so Fr <1) is called subcritical flow. It is higher and slower.
Flow shallower than Fr=1 (Fr>1) is called supercritical or shooting flow. It is lower and faster.
Never design a channel on a slope that is near critical (Fr = 1) because of the unpredictable water surface.

15. Hydraulic Jump
What happens if the Froude number crosses from Fr>1 (shallow, fast) to Fr < 1 (deep, slow)? At the transition, the flow has to suddenly change from one flow depth to the other. It forms a jump between one and the other. The two regions are separated by a continuously collapsing wall of water referred to as a hydraulic jump.

16. The Depth Ratio for a Hydraulic Jump
The ratio of the depths is:

17. Momentum Balance for a Rectangular Channel
Again, Bedient skips the derivation. Here it is.

19. Example 7-6: Calculation of a Hydraulic Jump
A sluice gate is constructed across an open channel. Water flowing under it creates a hydraulic jump. Determine the depth just downstream of the jump (point b) if the depth of flow at point a is meters and the velocity at point a is 5.33 meters/sec.