1. CHAPTER 6: Water Flow in Open Channels
University of Palestine
Engineering Hydraulics
2nd semester
CHAPTER 6: Water Flow in Open Channels

2. Content Introduction. Type of Open Channels.
Types of Flow in Open Channels.
Flow Formulas in Open Channels.
Most Economical Section of Channels.
Energy Principles in Open Channel Flow.
Non-uniform Flow in Open Channels.
Hydraulic Jump.

3. Introduction
Open channel hydraulics, a subject of great importance to civil engineers, deals with flows having a free surface in channels constructed for water supply, irrigation, drainage, and hydroelectric power generation; in sewers, culverts, and tunnels flowing partially full; and in natural streams and rivers.
An open channel is a duct in which the liquid flows with a free surface.
This is in contrast with pipe flow in which the liquid completely fills the pipe and flow under pressure.
The flow in a pipe takes place due to difference of pressure (pressure gradient), whereas in open channel it is due to the slope of the channel bed (i.e.; due to gravity).

4. Introduction
It may be noted that the flow in a closed conduit is not necessarily a pipe flow. It must be classified as open channel flow if the liquid has a free surface.
for a pipe flow:
The hydraulic gradient line (HGL) is the sum of the elevation and the pressure head (connecting the water surfaces in piezometers).
The energy gradient line (EGL) is the sum of the HGL and velocity head.
The amount of energy loss when the liquid flows from section 1 to section 2 is indicated by hL.

5. Introduction
Pipe system

6. Introduction For open channel flow :
The hydraulic gradient line (HGL) corresponds to the water surface line (WSL); the free water surface is subjected to only atmospheric pressure which is commonly referred to as the zero pressure reference in hydraulic engineering practice.
The energy gradient line (EGL) is the sum of the HGL and velocity head.
The amount of energy loss when the liquid flows from section 1 to section 2 is indicated by hL. For uniform flow in an open channel, this drop in the EGL is equal to the drop in the channel bed.

7. Introduction
Open Channel

8. Type of Open Channels
Based on their existence, an open channel can be natural or artificial :
Natural channels such as streams, rivers, valleys , etc. These are generally irregular in shape, alignment and roughness of the surface.
Artificial channels are built for some specific purpose, such as irrigation, water supply, wastewater, water power development, and rain collection channels. These are regular in shape and alignment with uniform roughness of the boundary surface.

9. Type of Open Channels

10. Type of Open Channels

11. Type of Open Channels
Based on their shape, an open channel can be prismatic or non-prismatic:
Prismatic channels: a channel is said to be prismatic when the cross section is uniform and the bed slop is constant.
)Non-prismatic channels: when either the cross section or the slope (or both) change, the channel is referred to as non-prismatic. It is obvious that only artificial channel can be prismatic.
The most common shapes of prismatic channels are rectangular, parabolic, triangular, trapezoidal and circular.

12. Type of Open Channels
The most common shapes of prismatic channels are rectangular, parabolic, triangular, trapezoidal and circular.

13. Types of Flow in Open Channels
The flow in an open channel can be classified into the following types :
A).Uniform and non-uniform flow:
If for a given length of the channel, the velocity of flow, depth of flow, slope of the channel and cross-section remain constant, the flow is said to be uniform.
Otherwise it is said to be non-uniform.
Non-uniform flow is also called varied flow which can be further classified as:
Gradually varied flow (GVF) where the depth of the flow changes gradually along the length of the channel.
Rapidly varied flow (RVF) where the depth of flow changes suddenly over a small length of the channel. For example, when water flows over an overflow dam, there is a sudden rise (depth) of water at the toe of the dam, and a hydraulic jump forms.

14. Types of Flow in Open Channels

15. Types of Flow in Open Channels
Uniform Flow

16. Types of Flow in Open Channels
B). Steady and unsteady flow: :
The flow is steady when, at a particular section, the depth of the liquid and other parameters (such as velocity, area of cross section, discharge) do not change with time. In an unsteady flow, the depth of flow and other parameters change with time.
C). Laminar and turbulent flow:
The flow in open channel can be either laminar or turbulent. In practice, however, the laminar flow occurs very rarely. The engineer is concerned mainly with turbulent flow. In the case of open channel Reynold’s number is defined as:

17. Types of Flow in Open Channels
Recall that Reynold’s number is the measure of relative effects of the inertia forces to viscous forces.

18. Types of Flow in Open Channels

19. Types of Flow in Open Channels
D). Sub-critical, critical, and supercritical flow:
The criterion used in this classification is what is known by Froude number, Fr, which is the measure of the relative effects of inertia forces to gravity force:

20. Types of Flow in Open Channels
D). Sub-critical, critical, and supercritical flow:
The criterion used in this classification is what is known by Froude number, Fr, which is the measure of the relative effects of inertia forces to gravity force:

21. Flow Formulas in Open Channels
In the case of steady-uniform flow in an open channel, the following main features must be satisfied:
The water depth, water area, discharge, and the velocity distribution at all sections throughout the entire channel length must remain constant, i.e.; Q , A , y , V remain constant through the channel length.
The slope of the energy gradient line (S), the water surface slope (Sws), and the channel bed slope (S0) are equal.
S = Sws = S0 D
Water Surface
T.E.L
channel bed

22. Flow Formulas in Open Channels
The depth of flow, y , is defined as the vertical distance between the lowest point of the channel bed and the free surface.
The depth of flow section, D , is defined as the depth of liquid at the section, measured normal to the direction of flow. D
Water Surface
T.E.L
channel bed
Unless mentioned otherwise, the depth of flow and the depth of flow section will be assumed equal. For uniform flow the depth attains a constant value known as the normal depth, yn

23. Flow Formulas in Open Channels
Many empirical formulas are used to describe the flow in open channels
The Chezy formula is probably the first formula derived for uniform flow. It may be expressed in the following form
1.The Chezy Formula(1769)
C is the Chezy coefficient (Chezy’s resistance factor), m1/2/s, a dimensional factor which characterizes the resistance to flow .

24. Flow Formulas in Open Channels
2. The Manning Formula: (1895)
where n = Manning’s coefficient for the channel roughness, m-1/3/s.
Substituting manning Eq. into Chezy Eq, we obtain the Manning’s formula for uniform flow:

25. Flow Formulas in Open Channels

26. Flow Formulas in Open Channels
3. The Strickler Formula:
where kstr = Strickler coefficient, m1/3/s Comparing Manning formula and Strickler formulas, we can see that

27. Flow Formulas in Open Channels
Example 1
open channel of width = 3m as shown, bed slope = 1:5000, d=1.5m find the flow rate using Manning equation, n=0.025.

28. Flow Formulas in Open Channels
Example 2
open channel as shown, bed slope = 69:1584, find the flow rate using Chezy equation, C=35.

29. Flow Formulas in Open Channels
Example 2 cont.

30. Flow Formulas in Open Channels
Example 3: Group work
The cross section of an open channel is a trapezoid with a ottom width of 4 m and side slopes 1:2, calculate the discharge if the depth of water is 1.5 m and bed slope = 1/1600. Take Chezy constant C = 50.

31. Most Economical Section of Channels
During the design stages of an open channel, the channel cross-section, roughness and bottom slope are given.
The objective is to determine the flow velocity, depth and flow rate, given any one of them. The design of channels involves selecting the channel shape and bed slope to convey a given flow rate with a given flow depth. For a given discharge, slope and roughness, the designer aims to minimize the cross-sectional area A in order to reduce construction costs

32. Most Economical Section of Channels
A section of a channel is said to be most economical when the cost of construction of the channel is minimum.
But the cost of construction of a channel depends on excavation and the lining. To keep the cost down or minimum, the wetted perimeter, for a given discharge, should be minimum.
This condition is utilized for determining the dimensions of economical sections of different forms of channels.

33. Most Economical Section of Channels
Most economical section is also called the best section or most efficient section as the discharge, passing through a most economical section of channel for a given cross sectional area A, slope of the bed S0 and a resistance coefficient, is maximum.
Hence the discharge Q will be maximum when the wetted perimeter P is minimum.

34. Most Economical Section of Channels
The most ‘efficient’ cross-sectional shape is determined for uniform flow conditions. Considering a given discharge Q, the velocity V is maximum for the minimum cross-section A. According to the Manning equation the hydraulic diameter is then maximum.
It can be shown that:
the wetted perimeter is also minimum,
the semi-circle section (semi-circle having its centre in the surface) is the best hydraulic section

35. Most Economical Rectangular Channel
Most Economical Section of Channels
Most Economical Rectangular Channel
Because the hydraulic radius is equal to the water cross section area divided by the wetted parameter, Channel section with the least wetted parameter is the best hydraulic section
Rectangular section

36. Most Economical Section of Channels
Most Economical Rectangular Channel

37. Most Economical Section of Channels
Most Economical Trapezoidal Channel or

38. Most Economical Section of Channels
Other criteria for economic Trapezoidal section k
The best side slope for Trapezoidal section

39. Most Economical Section of Channels Most Economical Circular Channel
Circular section
Maximum Flow using Manning
Maximum Velocity using Manning or Chezy
Maximum Flow using Chezy

40. Most Economical Section of Channels

41. Most Economical Section of Channels
Example 4
Circular open channel as shown d=1.68m, bed slope = 1:5000, find the Max. flow rate & the Max. velocity using Chezy equation, C=70.
Max. flow rate

42. Most Economical Section of Channels
Example 4 cont.
Max. Velocity

43. Most Economical Section of Channels
Example 5
Trapezoidal open channel as shown Q=10m3/s, velocity =1.5m/s, for most economic section. find wetted parameter, and the bed slope n=0.014.

44. Most Economical Section of Channels
Example 5 cont.
To calculate bed Slope

45. Most Economical Section of Channels
Example 6:
Use the proper numerical method to calculate uniform water depth flowing in a Trapezoidal open channel with B = 10 m, as shown Q=10m3/s if the bed slope , n= k = 3/2. to a precision 0.01 m, and with iterations not more than 15.
Note: you may find out two roots to the equation.

46. Most Economical Section of Channels
Example 6 cont.

47. Variation of flow and velocity with depth in circular pipes

48. Energy Principles in Open Channel Flow
Referring to the figure shown, the total energy of a flowing liquid per unit weight is given by D
Water Surface
T.E.L
channel bed
Where:
Z = height of the bottom of channel above datum,
y = depth of liquid,
V = mean velocity of flow.
If the channel bed is taken as the datum (as shown), then the total energy per unit weight will be.
This energy is known as specific energy, Es. Specific energy of a flowing liquid in a channel is defined as energy per unit weight of the liquid measured from the channel bed as datum

49. Energy Principles in Open Channel Flow
The specific energy of a flowing liquid can be re-written in the form: D
Water Surface
T.E.L
channel bed

50. Energy Principles in Open Channel Flow
Specific Energy Curve (rectangular channel)
It is defined as the curve which shows the variation of specific energy (Es ) with depth of flow y. It can be obtained as follows:
Let us consider a rectangular channel in which a constant discharge is taking place.
But Or

51. Energy Principles in Open Channel Flow
Specific Energy Curve (rectangular channel)
The graph between specific energy (x-axis) and depth (yaxis)
may plotted. H G E

52. Energy Principles in Open Channel Flow
Specific Energy Curve (rectangular channel)
Referring to the diagram above, the following features can be observed:
The depth of flow at point C is referred to as critical depth, yc. It is defined as that depth of flow of liquid at which the specific energy is minimum, Emin, i.e.; Emin @ yc . The flow that corresponds to this point is called critical flow (Fr = 1.0).
For values of Es greater than Emin , there are two corresponding depths. One depth is greater than the critical depth and the other is smaller then the critical depth, for example ; Es1 @ y1 and y2 These two depths for a given specific energy are called the alternate depths.
If the flow depth y > yc , the flow is said to be sub-critical (Fr < 1.0). In this case Es increases as y increases.
If the flow depth y < yc , the flow is said to be super-critical (Fr > 1.0). In this case Es increases as y increases.

53. Energy Principles in Open Channel Flow
Froude Number (Fr) T T
Flow Fr
Sub-critical
1 > Fr
Critical
1 = Fr
Supercritical
1 < Fr

54. Energy Principles in Open Channel Flow
Critical Flow
Subcritical
Super critical
critical
Hydraulic Jump

55. Energy Principles in Open Channel Flow
Rectangular Channel
For rectangular section
At critical Flow
a) Critical depth, yc , is defined as that depth of flow of liquid at which the specific energy is minimum, Emin,
q=Q/B
b) Critical velocity, Vc , is the velocity of flow at critical depth.

56. Energy Principles in Open Channel Flow
Rectangular Channel
c) Critical, Sub-critical, and Super-critical Flows:
Critical flow is defined as the flow at which the specific energy is minimum or the flow that corresponds to critical depth. Refer to point C in above figure, Emin @ yc .
and
therefore for critical flow Fr = 1.0
If the depth flow y > yc , the flow is said to be sub-critical. In this case Es increases as y increases. For this type of flow, Fr < 1.0 .
If the depth flow y < yc , the flow is said to be super-critical. In this case Es increases as y decreases. For this type of flow, Fr > 1.0 . 56

57. Energy Principles in Open Channel Flow
Rectangular Channel
d) Minimum Specific Energy in terms of critical depth:
At (Emin , yc ) , 57

58. Energy Principles in Open Channel Flow
Other Sections
at critical flow Fr =1 where:
Rectangular section
Trapezoidal section
Circular section
Triangle section

59. Energy Principles in Open Channel Flow
Example 1
Determine the critical depth if the flow is 1.33m3/s. the channel width is 2.4m

60. Energy Principles in Open Channel Flow
Example 2
Rectangular channel , Q=25m3/s, bed slope =0.006, determine the channel width with critical flow using manning n=0.016

61. Energy Principles in Open Channel Flow
Example 2 cont.

62. Non-uniform Flow in Open Channels:
Non-uniform flow is a flow for which the depth of flow is aried. This varied flow can be either Gradually varied flow (GVF) or Rapidly varied flow (RVF).
Such situations occur when control structures are used in the channel or when any obstruction is found in the channel
Such situations may also occur at the free discharges and when a sharp change in the channel slope takes place.
The most important elements, in non-uniform flow, that will be studied in this sectionare:
Classification of channel-bed slopes.
Classification of water surface profiles.
The dynamic equation of gradually varied flow.
Hydraulic jumps as examples of rapidly varied flow. 62

63. Non-uniform Flow in Open Channels:63

64. Non-uniform Flow in Open Channels:
Classification of Channel-Bed Slopes
The slope of the channel bed can be classified as:
1) Critical Slope: the bottom slope of the channel is equal to the critical slope. In this case S0 = Sc or yn = yc .
2) Mild Slope: the bottom slope of the channel is less than the critical slope. In this case S0 < Sc or yn > yc .
3) Steep Slope: the bottom slope of the channel is greater than the critical slope. In this case S0 > Sc or yn < yc .
4) Horizontal Slope: the bottom slope of the channel is equal to zero (horizontal bed). In this case S0 = 0.0 .
5) Adverse Slope: the bottom slope of the channel rises in the direction of the flow (slope is opposite to direction of flow). In this case S0 = negative .
The first letter of each slope type sometimes is used to indicate the slope of the bed. So the above slopes are abbreviated as C, M, S, H, and A, respectively. 64

65. Non-uniform Flow in Open Channels:
Classification of Channel-Bed Slopes 65

66. Non-uniform Flow in Open Channels:
Classification of Flow Profiles (water surface profiles): 66

67. Non-uniform Flow in Open Channels:
Classification of Flow Profiles (water surface profiles): 67

68. Non-uniform Flow in Open Channels:
Classification of Flow Profiles (water surface profiles): 68

69. Non-uniform Flow in Open Channels:
Classification of Flow Profiles (water surface profiles): 69

70. Non-uniform Flow in Open Channels:
Classification of Flow Profiles (water surface profiles): 70

71. Hydraulic Jump
A hydraulic jump occurs when flow changes from a supercritical flow (unstable) to a sub-critical flow (stable). There is a sudden rise in water level at the point where the hydraulic jump occurs. Rollers (eddies) of turbulent water form at this point. These
rollers cause dissipation of energy. 71

72. Hydraulic Jump General Expression for Hydraulic Jump:
In the analysis of hydraulic jumps, the following assumptions are made:
(1) The length of hydraulic jump is small. Consequently, the loss of head due to friction is negligible.
(2) The flow is uniform and pressure distribution is due to hydrostatic before and after the jump.
(3) The slope of the bed of the channel is very small, so that the component of the weight of the fluid in the direction of the flow is neglected. 72

73. Hydraulic Jump Hydraulic Jump in Rectangular Channels
But for Rectangular section 73

74. Hydraulic Jump
Hydraulic Jump in Rectangular Channels 74

75. Hydraulic Jump
Hydraulic Jump in Rectangular Channels 75

76. Hydraulic Jump
Hydraulic Jump in Rectangular Channels 76

77. Hydraulic Jump 77

78. Hydraulic Jump 78

79. Hydraulic Jump Example 1
A 3-m wide rectangular channel carries 15 m3/s of water at a 0.7 m depth before entering a jump. Compute the downstrem water depth and the critical depth

80. Hydraulic Jump Example 2 d2 d1=dn
dn = Depth can calculated from manning equation

81. Hydraulic Jump a)
d1=dn d2 b)

82. Hydraulic Jump c)
d1=dn d2