1. If there is no change in friction or slope as we move down stream
OPEN CHANNEL FLOW: In a channel if the flow is steady (no changes in time)
and uniform (no changes down stream) we have normal flow
Through a balance of the gravity (weight) to friction shear forces-for wide channels h B S or
where
Manning's Formula mks system
cf is dimensionless BUT n is not L
If there is no change in friction or slope as we move down stream
Then the depth of the flow remains constant –the normal depth
This value can be calculated for any stream
as a stream characteristic—the value, however
will only coincide with the stream surface
under the Normal flow condition
using the definition of discharge Q = (B x h)u (the volume flow per time)

2. Derivation:--once more As a force balance
consider fluid element with unit area base
and height of depth h moving down slope
Shear force acting on fluid element
Up along slope is =
Gravity force down stream = h B S L

3. Energy in a fluid: In stream settings We have seen how useful balances of
Mass (Volume) and Momentum (Manning) can be in understanding basic behavior-.
In addition to these conservations we can also consider the balance of Energy
Engineers (Civil) like to measure energy as a vertical height above a datum A
Consider a large fluid filled tank with a hole near the bottom.
If the level h (height) of the tank is maintained, the flow near A
Is essential still, and the fluid speed at the exit is u. The potential
energy per unit volume at A (relative to the datum) is h
This is converted into the Kinetic Energy
Per unit volume at B B
These energy forms can be expressed in terms of height (length) dimensions, by
dividing by
Then an Energy balance between A and B can be expressed as
friction head loss

4. Derive Normal flow with Energy balance
consider fluid element with unit area base
and height of depth h moving down slope a h B S =
friction head loss
friction force per unit mass X length /r g b L

5. Applying this idea to a stream channel under-normal flow conditions
Average Velocity
Bed elevation h B S a
Specific Energy b
So In uniform flow energy
Balance becomes
But by definition of slope S same as
So under normal flow Energy balance will recover our previous result

6. Specific Energy:
Energy Line h
The specific energy
Energy (head) relative to channel bed
Changes Non-linearly with stream depth, h
See Excell worksheet
(make sure to download “open.xls” along with this power point

7. Critical Depth: The specific energy Is non-linear
Can show that, for a given discharge and
stream width E will reach a minimum when E
The critical depth
At critical depth
Sub-critical Flow
Sp Energy increases
with increase in depth Q=
0.5 B= 3
And
hcr=
Super-critical Flow
Sp Energy increases
with decreases in depth

8. Why we want to know about this
Normal flow is all well and good….
???
???
Or slope?
What about changes in depth?
The Question: How does a disturbance in the flow propagate up and down stream
How far is the effect felt

9. The form of the flow response to changes such as this depends on a number called the Froude Number
Flow speed
Max wave speed

10. Flow Over an Object: What happens as a sub-critical normal flow
Goes over a low object? Recognize that this an important SRES question ?
h =0.3
B=3
Q=.5

11. We can construct an Energy Balance between A and B—neglecting
slope changes and bed friction (small distance) A B
h =0.3
B=3
Q=.5
Implies that

12. We can construct an Energy Balance between A and B—neglecting
slope changes and bed friction (small distance) A B
h =0.3
B=3
Q=.5
implies that
See Excell worksheet

13. What happens when we increase the “bump” height
The level of the flow over the bump will continue to decrease---
until the critical height hcr is reached. A B
h =0.3
B=3
Q=.5
See Excell worksheet
If the bump height increases more: The level over the bump is caught between
A “rock and a hard place” going both up or down will only increase the energy
--so with an increasing bump the the level over the bump is fixed at hcr

14. --so with an increasing bump the the level over the bump is fixed at hcr
The only way to establish an energy balance is for the height at A to
Increase
Back
water A B
This will mean that down-stream of the bump the flow
Will not be normal
As we move further back friction will reestablish
Normal flow—the distance to reestablish normal flow is the
Back Water Distance—will be calculated

15. What about down stream of our high bump ?
Since fluid has lost the pot. energy
The energy is more easily recovered
By the level dropping below critical A B
Further down stream
The bottom friction
Will drive it back up to the normal
Depth

16. If normal depth is sub-critical hn > hcr something interesting happens
The flow level can not
Pass smoothly
Thorough the critical depth
And you get a hydraulic jump

17. How Far down stream in an other wise normal flow is an obstacle felt ?
This elevation change is important ?
The Stress slope
See derivation on Board

18. Now consider head balance behind an obstacle-not in uniform flow
Average Velocity
Bed elevation h B S a b

19. Now consider head balance behind an obstacle-not in uniform flow
Average Velocity h B S a b
For a small distance deltax
Result follows on subbing for u
and noting that

20. Behavior of water surface near an object depends on normal
flow level in relation to critical flow level
The Stress slope
Assumed to be
Slope at normal flow
critical
Fr < 1, Sf<S, A > 0
normal
S=Sf
normal
Fr < 1, Sf>S, A < 0
Fr < 1, Sf<S, A > 0
critical
Fr >1, Sf<S, A < 0
Fr > 1, Sf>S, A > 0
Fr=1
Fr > 1, Sf>S, A > 0
Fr=1
S=Sf
Mild slope, normal flow depth > critical flow depth
Steep slope, normal flow depth < critical flow depth
Homework: find a web page that shows examples of water level change for obstacles
In streams/channels—send web-address to Voller