0. Lecture Notes 3: (Chapter 12) Energy principles in Open-Channel
Energy generated at an overfall (Niagara Falls).
prepared by Ercan Kahya

1. Total & Specific Energy
Specific Energy: the energy per unit weight of water measured from
the channel bottom as a datum
► Note that specific energy & total energy are not generally equally.
At section 1: Specific Energy Total Energy

2. Total & Specific Energy
► Specific energy varies abruptly as does the channel geometry
► Velocity coefficient (α) is used to account nonuniformity of the
velocity distribution when using average velocity.
► It varies from 1.05 (for uniform cross-sections) to 1.2 (nonuniform sections).
► For natural channels, a common method to estimate α:
Weighted mean velocity:
A channel section divided into three sections

3. Specific Energy Diagram (SED)
Assuming α equal to 1, it is convenient to express E in terms of Q
for steady flow conditions
f(E, Q, y) = 0
Specific Energy Diagram (SED)
SED is a graphical representation for the variation of E with y.
Let`s write E equation in terms of static & kinetic energy:
where
and

4. Specific Energy Diagram
Es varies linearly with y
Ek varies nonlinearly with y
Horizontal sum of the line OD & the curve kk` produces SED
For given E: alternate depths (y1 & y2)
They are two depths with the same specific energy and conveying the same discharge
Emin vs critical depth
The specific energy diagram

5. Specific Energy Diagram
An increase in the required Emin yields bigger discharges.
Fn : Froude number
equals to V square / gD
The specific energy diagram
for various discharges

6. Critical Flow Conditions
General mathematical formulation for critical flow conditions:
Assume dA/dy = B

7. Critical Flow Conditions
At the critical flow conditions, specific energy is minimum:
Then,
which can also be
expressed as -->
Then,
In wide or rectangular section, D = y
at critical depth

8. Critical Velocity The general expressions for
Used to determine the state of flow
Critical state condition:
Critical velocity for the general cross section:
Velocity head at critical conditions:
In wide or rectangular section, D = y

9. Critical Depth For a certain section & given discharge:
Critical depth is defined as the depth of flow requiring minimum specific energy
This equation should be solved …
For the trapezoidal cross section:
Solve this by trial & error …
For the rectangular
cross section:
Critical depth trapezoidal and circular sections

10. Critical Energy
Critical Energy is the energy when the flow is under critical conditions.
Recall for any cross section:
Then,
For wide or rectangular section, D = y

11. Critical Slope
Critical slope is the bed slope of the channel producing critical conditions.
► depends discharge; channel geometry; resistance or roughness
For Chezy equation:
Then,
For Manning equation:
In English unit:
For direct computation:

12. Critical Slope
Critical slope is very important in open-channel hydraulics. WHY?
The summary given above encompasses much of the important concepts
of the energy & resistance principles as applied to open channels.

13. Discharge-Depth Relation for Constant Specific Energy
Now assume Eo constant, then evaluate Q-y relation:
For the condition of the Qmax:
It reduces to
Then substitute this into Q equation at the top:
implies that the Qmax is encountered at the critical flow condition for given E.

14. Discharge-Depth Relation for Constant Specific Energy
Q-y relation
for constant specific energy
For wide or rectangular section, D = y
can be written as
Differentiating this w.r.t. y and equating to zero:

15. Transitions in Channel Beds
Consider an open-channel with a small drop ∆z in its bed
Assume that friction losses and minor losses due to drop are negligible
The method provides a good first approximation of the effects of the transition
First step: compare the given conditions to critical conditions to determine
the initial state of flow.
A small drop in the channel bed (subcritical flow): (a) change in water levels, and (b) steps for solution.

16. Transitions in Channel Beds
Consider an abrupt rise ∆z in the open-channel bed
Assume that upstream conditions are subcritical & initial E1
Note that ∆z should be subtracted from E1 & While TEL unchanged, E reduced

17. Transitions in Channel Beds
Consider an abrupt rise ∆z in the open-channel bed
Assume that upstream conditions are supercritical & initial E1
Note that ∆z should be subtracted from E1 & While TEL unchanged, E reduced
RESULT : Water depth must rise after the step

18. Chokes
Chokes can only occur when the channel is constricted, but will not occur where the flow area expanded such as drops or expansions.
In designing a channel transition that would tend to restrict the flow, engineer wants to avoid forcing a choke to occur if at all possible.

19. Chokes
Figure 12.16: Rise in a channel bed: (a) a small step-up, (b) a bigger step-up

20. Chokes
Figure 12.16: Rise in a channel bed: (c) a still bigger step-up, and (d) changes in the specific energy.

21. Enlargements and constructions in channel widths
(b)
(c)
(d)
(a) A contracted channel. (b) Water levels in a contracted channel. (c) SED for a contracted channel. (d) Water level in a contracted channel-supercritical flow.

22. EXAMPLE
A 6.0 m rectangular channel carries a discharge of 30 m3/s at a depth of 2.5m. Determine the constricted channel width that produces critical depth.

23. EXAMPLE: s o l u t i o n
b2 = Q/ q2 = 30 / 7.56 = 3.07 m

24. Weirs & Spillways y2=0 To control the elevation of the water
Functions as a downstream choke control
Classified as sharp crested or broad crested
depending on critical depth occurrence on the crest
y2=0
Head on the weir crest
Orifice equation:

25. Weirs & Spillways Assume V1=0 Immediate region of weir crest
Discharge through the element:
Integrate across the head (0 - H):
Total discharge across the weir:

26. Coefficient of Discharge
Losses due to the advent of the drawdown of the flow immediately upstream of the weir as well as any other friction or contraction losses;
To account for these losses, a coefficient of discharge Cd is introduced.
(Henderson, 1966)
where, H is the head on the weir crest, Z is the height of the weir.
Use this equation up to H/Z = 2

27. Discharge Measurements
Weirs
Flume
Orifices
Weirs and flumes not only require a simple head reading to measure discharge but they can also pass large flow without causing the upstream level to rise significantly and causing flooding.
Discharge Control
- Orifices are rather cumbersome for discharge measurements, but
they are very useful for discharge control
Practical Hydraulics by Melvyn Kay
Copyright © 1998 by E & FN Spon . All rights reserved.

28. Discharge Control Practical Hydraulics by Melvyn Kay
Copyright © 1998 by E & FN Spon . All rights reserved.

29. WEIRS

30. WEIRS

31. FLUMES Practical Hydraulics by Melvyn Kay
Copyright © 1998 by E & FN Spon . All rights reserved.

32. Class Exercises: