1. MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 10: OPEN CHANNEL FLOWS
Instructor: Professor C. T. HSU

2. 10.1 General Concept of Flows in Open Channel
Open channel flows are flows with free surface that have many applications
Rivers, Streams, Aqueducts, Canals, Sewers, Irrigation
Water Channels
Pipe Flow vs. Channel Flow
Pipe Flow
Open Channel Flow
Closed with solid boundary
Open with free surface
Fixed crossed-section
Variable depth
Driven by pressure gradient
Driven by gravity
Mostly circular
All kind of shapes

3. 10.1 General Concept of Flows in Open Channel
Hydraulic radius:
Hydraulic depth:

4. 10.1 General Concept of Flows in Open Channel
Based on the property of yh, the open channel flow can be classified as:
a) Constant depth:
1-D model can be used
b) Gradually varying:
Depth changes slowly so that 1-D model remains as a good approximation
c) Rapidly varying:
Need a 2-D model to treat the problem

5. 10.1 General Concept of Flows in Open Channel
For most practical cases involving large and deep
channels, the Reynolds number, is high
Hence, most open channel flows are turbulent
With a free surface for open channel flow, gravity is important. The important parameter is the Froude number:

6. 10.1 General Concept of Flows in Open Channel
The open channel flows as classified by Froude number Fr are:

7. 10.2 Propagation of Surface Waves
If the free surface initially calm is perturbed by a vertical displacement , there will be an associated velocity perturbation in fluid.
If the water depth is small compared with the length scale of the displacement, the displacement perturbation will have a fixed form that propagates with a velocity C, which is called phase velocity.
We now consider the flow in a frame moving with the phase velocity. Then, in the moving frame the shape of the displacement is fixed and steady

8. 10.2 Propagation of Surface Waves

9. 10.2 Propagation of Surface Waves
For small displacement disturbances,
Hence, the displacement can propagate upstream and downstream with a speed equal to
If the fluid moves at a velocity U, then the Froude number,

10. 10.2 Propagation of Surface Waves
This phenomenon can easily be demonstrated by a boat moving at a constant speed U on an initial clam water where the disturbance are generated by the vertical oscillating of the boat.
The wave patterns behind the boat for Fr> 1 are called the ship wakes

11. 10.3.1 Friction Loss Bernoulli’s with friction loss,
Since the flow is uniform,
For free surface, we also have P1-P2 since the fluid depths are the same. Therefore,
where Sb is the hydraulic slope

12. 10.3.1 Friction Loss Similar to pipe flow, so,
Empirical values of C were determined by Manning, who suggested that

13. Friction Loss
Now,
For a rectangular channel,

14. Specific Energy
Define specific energy, E, at a single section in the channel as,
Let q be volume flow rate per unit width, Q=qb, so for rectangular channel:

15. Specific Energy
The variation of depth as a function of specific energy for a given flow rate are summaries in the specific energy diagram
For a given flow rate (Q>0) and specific energy, there are 2 possible values of depth, y. These are called alternate depths.
Fr<1
E=y
Fr>1

16. Specific Energy
For any value of E, the horizontal distance from the vertical axis to the line, y=E, gives the depth
And the distance from the line, y=E, to the Q curve is then equal to the K.E., U2/2g.
For each curve representing a given flow rate, there is a value of depth that gives a minimum E. This depth is the critical depth obtained by:

17. 10.3.1 Specific Energy The velocity at the critical condition:
Since , we have
Continuity
Then

18. Specific Energy
For non-rectangular channels, the channel depth varies across the width. At the minimum specific energy,

19. Specific Energy
Consider,

20. 10.4 Gradually Varies Flow
The energy equation for the differential C.V. is:
For a rectangular channel U=Q/by, so:

21. 10.4 Gradually Varies Flow
Since dz=-sbdx and similarly, we can define dhL= sfdx, the energy equation now becomes,
For flow at normal depth,
The sign of the slope of the water surface profile depends on whether the flow is subcritical or supercritical, and on sf and sb

22. 10.4 Gradually Varies Flow
To calculate the surface profile, rewrite the equation as
As dE/dx= sf – sb, and in finite difference form,
m denotes the mean properties over a channel length ,
sf can be obtained from the Manning correlation, since sb for flow at normal depth equal sf

23. 10.5 Hydraulic Jump
For subcritical flow, disturbances cause by a change in bed slope or flow cross section may move upstream and downstream. The result is a smooth adjustment of the flow
However, when the flow is supercritical, disturbances cannot be transmitted upstream. Thus, a gradual change is not possible.
The transition from the supercritical to subcritical flow occurs abruptly through the hydraulic jump

24. 10.5 Hydraulic Jump
The abrupt change in depth involves a significant loss of mechanical energy through turbulent mixing
The extent of a hydraulic jump is short, so friction is negligible. Assuming horizontal surface, gravitational effect of bottom elevation can be neglected

25. 10.5.1 Depth Change Across a Hydraulic Jump
Eliminate U2 from the momentum equation by using the continuity equation to get:

26. 10.5.2 Head Loss Across a Hydraulic Jump
Head loss through a jump is just the difference in specific energy,

27. 10.6 Flow Over a Bump
Consider frictionless flow in a horizontal rectangular channel of constant width, b, with a bump of height, h(x). The flow is assumed uniform.
Since the flow is steady, incompressible and frictionless, applying Bernoulli’s equation along the free surface gives:

28. 10.6 Flow Over a Bump
Along the free surface, p1=p=patm, thus:

29. 10.6 Flow Over a Bump