1. Chapter 13: Open Channel Flow
Eric G. Paterson
Department of Mechanical and Nuclear Engineering
The Pennsylvania State University
Spring 2005

2. Note to Instructors
These slides were developed1, during the spring semester 2005, as a teaching aid for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of Mechanical and Nuclear Engineering at Penn State University. This course had two sections, one taught by myself and one taught by Prof. John Cimbala. While we gave common homework and exams, we independently developed lecture notes. This was also the first semester that Fluid Mechanics: Fundamentals and Applications was used at PSU. My section had 93 students and was held in a classroom with a computer, projector, and blackboard. While slides have been developed for each chapter of Fluid Mechanics: Fundamentals and Applications, I used a combination of blackboard and electronic presentation. In the student evaluations of my course, there were both positive and negative comments on the use of electronic presentation. Therefore, these slides should only be integrated into your lectures with careful consideration of your teaching style and course objectives.
Eric Paterson
Penn State, University Park
August 2005
1 This Chapter was not covered in our class. These slides have been developed at the request of McGraw-Hill

3. Objectives
Understand how flow in open channels differs from flow in pipes
Learn the different flow regimes in open channels and their characteristics
Predict if hydraulic jumps are to occur during flow, and calculate the fraction of energy dissipated during hydraulic jumps
Learn how flow rates in open channels are measured using sluice gates and weirs

4. Classification of Open-Channel Flows
Open-channel flows are characterized by the presence of a liquid-gas interface called the free surface.
Natural flows: rivers, creeks, floods, etc.
Human-made systems: fresh-water aqueducts, irrigation, sewers, drainage ditches, etc.

5. Classification of Open-Channel Flows
In an open channel,
Velocity is zero on bottom and sides of channel due to no-slip condition
Velocity is maximum at the midplane of the free surface
In most cases, velocity also varies in the streamwise direction
Therefore, the flow is 3D
Nevertheless, 1D approximation is made with good success for many practical problems.

6. Classification of Open-Channel Flows
Flow in open channels is also classified as being uniform or nonuniform, depending upon the depth y.
Uniform flow (UF) encountered in long straight sections where head loss due to friction is balanced by elevation drop.
Depth in UF is called normal depth yn

7. Classification of Open-Channel Flows
Obstructions cause the flow depth to vary.
Rapidly varied flow (RVF) occurs over a short distance near the obstacle.
Gradually varied flow (GVF) occurs over larger distances and usually connects UF and RVF.

8. Classification of Open-Channel Flows
Like pipe flow, OC flow can be laminar, transitional, or turbulent depending upon the value of the Reynolds number
Where
 = density,  = dynamic viscosity,  = kinematic viscosity
V = average velocity
Rh = Hydraulic Radius = Ac/p
Ac = cross-section area
P = wetted perimeter
Note that Hydraulic Diameter was defined in pipe flows as Dh = 4Ac/p = 4Rh (Dh is not 2Rh, BE Careful!)

9. Classification of Open-Channel Flows
The wetted perimeter does not include the free surface.
Examples of Rh for common geometries shown in Figure at the left.

10. Froude Number and Wave Speed
OC flow is also classified by the Froude number
Resembles classification of compressible flow with respect to Mach number

11. Froude Number and Wave Speed
Critical depth yc occurs at Fr = 1
At low flow velocities (Fr < 1)
Disturbance travels upstream
y > yc
At high flow velocities (Fr > 1)
Disturbance travels downstream
y < yc

12. Froude Number and Wave Speed
Important parameter in study of OC flow is the wave speed c0, which is the speed at which a surface disturbance travels through the liquid.
Derivation of c0 for shallow-water
Generate wave with plunger
Consider control volume (CV) which moves with wave at c0

13. Froude Number and Wave Speed
Continuity equation (b = width)
Momentum equation

14. Froude Number and Wave Speed
Combining the momentum and continuity relations and rearranging gives
For shallow water, where y << y,
Wave speed c0 is only a function of depth

15. Specific Energy
Total mechanical energy of the liquid in a channel in terms of heads
z is the elevation head
y is the gage pressure head
V2/2g is the dynamic head
Taking the datum z=0 as the bottom of the channel, the specific energy Es is

16. Specific Energy For a channel with constant width b,
Plot of Es vs. y for constant V and b

17. Specific Energy This plot is very useful
Easy to see breakdown of Es into pressure (y) and dynamic (V2/2g) head
Es   as y  0
Es  y for large y
Es reaches a minimum called the critical point.
There is a minimum Es required to support the given flow rate.
Noting that Vc = sqrt(gyc)
For a given Es > Es,min, there are two different depths, or alternating depths, which can occur for a fixed value of Es
A small change in Es near the critical point causes a large difference between alternate depths and may cause violent fluctuations in flow level. Operation near this point should be avoided.

18. Continuity and Energy Equations
1D steady continuity equation can be expressed as
1D steady energy equation between two stations
Head loss hL is expressed as in pipe flow, using the friction factor, and either the hydraulic diameter or radius

19. Continuity and Energy Equations
The change in elevation head can be written in terms of the bed slope 
Introducing the friction slope Sf
The energy equation can be written as

20. Uniform Flow in Channels
Uniform depth occurs when the flow depth (and thus the average flow velocity) remains constant
Common in long straight runs
Flow depth is called normal depth yn
Average flow velocity is called uniform-flow velocity V0

21. Uniform Flow in Channels
Uniform depth is maintained as long as the slope, cross-section, and surface roughness of the channel remain unchanged.
During uniform flow, the terminal velocity reached, and the head loss equals the elevation drop
We can the solve for velocity (or flow rate)
Where C is the Chezy coefficient. f is the friction factor determined from the Moody chart or the Colebrook equation

22. Best Hydraulic Cross Sections
Best hydraulic cross section for an open channel is the one with the minimum wetted perimeter for a specified cross section (or maximum hydraulic radius Rh)
Also reflects economy of building structure with smallest perimeter

23. Best Hydraulic Cross Sections
Example: Rectangular Channel
Cross section area, Ac = yb
Perimeter, p = b + 2y
Solve Ac for b and substitute
Taking derivative with respect to
To find minimum, set derivative to zero
Best rectangular channel has
a depth 1/2 of the width

24. Best Hydraulic Cross Sections
Same analysis can be performed for a trapezoidal channel
Similarly, taking the derivative of p with respect to q, shows that the optimum angle is
For this angle, the best flow depth is

25. Gradually Varied Flow
In GVF, y and V vary slowly, and the free surface is stable
In contrast to uniform flow, Sf  S0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effects
To derive how how the depth varies with x, consider the total head

26. Gradually Varied Flow Take the derivative of H
Slope dH/dx of the energy line is equal to negative of the friction slope
Bed slope has been defined
Inserting both S0 and Sf gives

27. Gradually Varied Flow
Introducing continuity equation, which can be written as
Differentiating with respect to x gives
Substitute dV/dx back into equation from previous slide, and using definition of the Froude number gives a relationship for the rate of change of depth

28. Gradually Varied Flow
This result is important. It permits classification of liquid surface profiles as a function of Fr, S0, Sf, and initial conditions.
Bed slope S0 is classified as
Steep : yn < yc
Critical : yn = yc
Mild : yn > yc
Horizontal : S0 = 0
Adverse : S0 < 0
Initial depth is given a number
1 : y > yn
2 : yn < y < yc
3 : y < yc

29. Gradually Varied Flow
12 distinct configurations for surface profiles in GVF.

30. Gradually Varied Flow
Typical OC system involves several sections of different slopes, with transitions
Overall surface profile is made up of individual profiles described on previous slides

31. Rapidly Varied Flow and Hydraulic Jump
Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distance
Sluice gates
Weirs
Waterfalls
Abrupt changes in cross section
Often characterized by significant 3D and transient effects
Backflows
Separations

32. Rapidly Varied Flow and Hydraulic Jump
Consider the CV surrounding the hydraulic jump
Assumptions
V is constant at sections (1) and (2), and 1 and 2  1
P = gy
w is negligible relative to the losses that occur during the hydraulic jump
Channel is wide and horizontal
No external body forces other than gravity

33. Rapidly Varied Flow and Hydraulic Jump
Continuity equation
X momentum equation
Substituting and simplifying
Quadratic equation for y2/y1

34. Rapidly Varied Flow and Hydraulic Jump
Solving the quadratic equation and keeping only the positive root leads to the depth ratio
Energy equation for this section can be written as
Head loss associated with hydraulic jump

35. Rapidly Varied Flow and Hydraulic Jump
Often, hydraulic jumps are avoided because they dissipate valuable energy
However, in some cases, the energy must be dissipated so that it doesn’t cause damage
A measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio

36. Rapidly Varied Flow and Hydraulic Jump
Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr

37. Flow Control and Measurement
Flow rate in pipes and ducts is controlled by various kinds of valves
In OC flows, flow rate is controlled by partially blocking the channel.
Weir : liquid flows over device
Underflow gate : liquid flows under device
These devices can be used to control the flow rate, and to measure it.

38. Flow Control and Measurement Underflow Gate
Underflow gates are located at the bottom of a wall, dam, or open channel
Outflow can be either free or drowned
In free outflow, downstream flow is supercritical
In the drowned outflow, the liquid jet undergoes a hydraulic jump. Downstream flow is subcritical.
Free outflow
Drowned outflow

39. Flow Control and Measurement Underflow Gate
Schematic of flow depth-specific
energy diagram for flow through
underflow gates
Es remains constant for idealized gates with negligible frictional effects
Es decreases for real gates
Downstream is supercritical for free outflow (2b)
Downstream is subcritical for drowned outflow (2c)

40. Flow Control and Measurement Overflow Gate
Specific energy over a bump at station 2 Es,2 can be manipulated to give
This equation has 2 positive solutions, which depend upon upstream flow.

41. Flow Control and Measurement Broad-Crested Weir
Flow over a sufficiently high obstruction in an open channel is always critical
When placed intentionally in an open channel to measure the flow rate, they are called weirs

42. Flow Control and Measurement Sharp-Crested V-notch Weirs
Vertical plate placed in a channel that forces the liquid to flow through an opening to measure the flow rate
Upstream flow is subcritical and becomes critical as it approaches the weir
Liquid discharges as a supercritical flow stream that resembles a free jet

43. Flow Control and Measurement Sharp-Crested V-notch Weirs
Flow rate equations can be derived using energy equation and definition of flow rate, and experimental for determining discharge coefficients
Sharp-crested weir
V-notch weir
where Cwd typically ranges between 0.58 and 0.62