OPEN CHANNEL FLOW: In a channel if the flow is steady (no changes in time)

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

5. h h S B Applying this idea to a stream channel under-normal flow conditions Average Velocity Bed elevation 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 And 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 A B 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. h h S B Now consider head balance behind an obstacle-not in uniform flow Average Velocity Bed elevation a b

19. h h S B Now consider head balance behind an obstacle-not in uniform flow Average Velocity 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 Fr > 1, Sf>S, A > 0 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