1. Mechanical Engineering Department, KNUST, MARCH 2012
FLUID DYNAMICS (ME 252)
Kwame Nkrumah University of Science and Technology,
Kumasi, Ghana
Department of Mechanical Engineering
INSTIT UTE OF DISTANCE LEARNING,
BSc. MECHANICAL ENGINEERING, BRIDGING
C.K.K. SEKYERE

2. UNIT : FLOW IN OPEN CHANNELS
OUTLINE
2
Flow in channels
 Classification of Flows in Channels
 Discharge through Open Channels by Chezy’s Formula
 Empirical Formulae for the value of Chezy’s Constant
 Most Economical Sections of Channels
Energy of liquid in open channels
 Non Uniform Flow through Open Channels
 Specific Energy and Specific Energy Curve
 Hydraulic Jump or Standing Wave
 Gradual Varied Flow
C.K.K. SEKYERE; MARCH 2012

3. FLOW IN OPEN CHANNELS
3
CLASSIFICATION OF FLOW IN CHANNELS
The flow in open channel is classified into the following types:
1. Steady flow and unsteady flow,
2. Uniform flow and non-uniform flow,
3. Laminar flow and turbulent flow, and
4. Sub-critical, critical and super critical flow
Steady Flow and Unsteady Flow: If the flow characteristics such as
depth of flow, velocity of flow, rate of flow at any point in open
channel flow do not change with respect to time, the flow is said to
be steady flow. Mathematically, steady flow is expressed as
…………………………………………….(4.1)
Where V = velocity, Q = rate of flow and y = depth of flow Unit 3

4. If at any point in open channel flow, the velocity of flow, depth of
flow or rate of flow changes with respect to time, the flow is said to
be unsteady flow. Mathematically, unsteady-flow means
4
…………………………….(4.2)
Uniform Flow 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
Unit 13

5. On the other hand, if for a given length of the channel, the velocity
of flow, depth of flow, etc. do not remain constant, the flow is said
to be non-uniform flow
5
Mathematically, uniform and non-uniform flow are written as:
…………………….(4.3) and
…………………………………..(4.4)
Non-uniform flow in open channels is also called varied flow, which is
classified as:
•Rapidly varied flow (R.V.F), and
•Gradually varied flow (G.V.F).
Unit 4

6. Rapidly Varied Flow (R.V.F):
Rapidly varied flow is defined as that flow in which depth of flow
6
changes abruptly over a small length of the channel
 As shown in Fig. 4.1 when there is any obstruction in the path of
flow of water, the level of water rises above the obstruction and
then falls and again rises over a short length of the channel
Fig. 4.1
Unit 4

7. Gradually Varied Flow (G.V.F)
if the depth of flow in a channel changes gradually over a long length
of the channel, the flow is said to be gradually varied flow and is
7
denoted by G.V.F
Laminar Flow and Turbulent Flow
The flow in open channel is said to be laminar if the Reynolds
number (Re) is less than 500
 Reynolds number in case of open channel is defined as:
………………………………..….(4.5)
R = Hydraulic radius or Hydraulic mean depth
Cross  sec tional area of flow normal to the direction of flow
R
Wetted perimeter
Unit 4

8. If the Reynolds number is more than 2000, the flow is said to be
turbulent in open channel flow
 If Re lies between 500 and 2000, the flow is considered to be in
8
transition state
Sub-critical, Critical and Super Critical Flow
The flow in open channel is said to be sub-critical if the Froude
number (Fe) is less than 1.0
Sub-critical flow is also called tranquil or streaming flow
 The flow is called critical if Fe = 1.0
if Fe > 1.0 the flow is called super critical or shooting or rapid or
torrential.
………………………………………..(4.6)
D = Hydraulic depth of channel and is equal to the ratio of wetted
Unit 4
area to the top width of channel = A/T................................(4.7)

9. Where T = Top width of channel
9 DISCHARGE THROUGH OPEN CHANNEL BY CHEZY’S FORMULA
Consider flow of water in a channel as shown in Fig. 4.2.
As the flow is uniform, it means the velocity, depth of flow and area
of flow will be constant for a given length of the channel
Unit 4

10. Consider sections 1-1 and 2-2.
Let
10 L = Length of channel,
A = Area of flow of water,
i = Slope of bed,
V = Mean velocity of flow of water,
P = Wetted perimeter of the cross-section,
f = Frictional resistance per unit velocity per unit area
The weight of water between sections 1-1 and 2-2
W = Specific weight of water x volume of water
= w x A x L........................(4.8)
Component of W along direction of flow =W x sin i = wAL sin i ......(4.9)
Unit 4

11. ……..(4.10)
The value of n is found experimentally to be equal to 2 and the
surface area =P x L
11
……………………(4.11)
The forces acting on the water between sections 1-1 and 2-2 are
1. Component of weight of water along the direction of flow,
2. Frictional resistance against flow of water,
3. Pressure force at section 1-1,
4. Pressure force at section 2-2,
As the depths of water at the sections 1-1 and 2-2 are the same, the
pressure forces on these two sections are the same and acting in
opposite directions
Hence they cancel each other
Unit 4

12. In case of uniform flow, the velocity of flow is constant for the given
length of the channel
12
Hence there is no acceleration acting on the water
 Hence the resultant force acting in the direction of flow must be
zero
Resolving all forces in the direction of flow, we get
………………………(4.12)
………………………(4.13)
………………………(4.14)
Unit 4

13. ……………………………………….(4.15)
13
(4.14),
V  C m sin i …………………………………….(4.16)
……………………(4.16)
……………………(4.17)
Unit 4

14. EMPIRICAL FORMALAE FOR THE VALUE OF CHEZY’S
CONSTANT
14
Equation (4.16) is known as Chezy’s formula after the name of a
French Engineer, Antoine Chezy who developed this formula in 1975
C is known as Chezy’s constant; C is not dimensionless
The dimension of C is
Hence the value of C depends upon the system of units
The following are the empirical formulae, after the name of their
inventors, used to determine the value of C:
Unit 4

15. 1.Bazin formula (in MKS units): ………………………..(4.18)
15
where K = Bazin’s constant and depends upon the roughness of
the surface of channel, whose values are given in Table 4.1
m = Hydraulic mean depth or hydraulic radius.
2. Ganguillet-Kutter Forumula. The value of C is given by (in MKS
unit) as
……………………………………(4.19)
where N = Roughness co-efficient which is known as Kutter’s constant,
whose value for different surfaces are given in Table 4.2.
i = Slope of the bed
m = Hydraulic mean depth Unit 4

16. 16
Unit 4

17. Manning’s Formula
The value of C according to this formula is given as
17
……………………………………………………….(4.20)
N = Manning’s constant which is having same value as Kutter’s
constant for the normal range of slope and hydraulic mean depth.
The values of N are given in Table 4.2.
Unit 4

18. MOST ECONOMICAL SECTION OF CHANNELS
A section of a channel is said to be most economical when the cost
18
of construction of the channel is minimum
 But the cost of construction of a channel depends upon the
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
Most Economical Rectangular Channel
Fig. 4.3 Rectangular channel
Unit 4

19. Let b = width of channel, d = depth of flow, area of flow,
A,= bxd……………………….(4.21)
19
……………………………….(4.22)
From equ. (4.21), equ. (4.22) become
…………………………………………..(4.23)
…………………………..(4.24)
Unit 4

20. ……………………………(4.25)
20
….(4.26)
…………(4.27)
……………………………..(4.28)
From equations (4.25) and (4.28), it is clear that rectangular
channel will be most economical when:
1. either b = 2d
2. Or m=d/2 Unit 4

21. Most Economical Trapezoidal Channel
21
Fig. 4.4 Trapezoidal channel
Fig. 4.4The side slope is given as 1 vertical to n horizontal
……………(4.29)
Unit 4

22. ……………………………………..………….…(4.30)
22
……………………………………..………….…(4.31)
……………………………………..………….…(4.32)
…………………(4.33)
………………………………(4.34)
Substituting the value of b from (4.32) into (4.34)
Unit 4

23. …………………(4.35)
23
…………………………………………(4.36)
……………………………………………….(4.37)
Substituting the value of A from equ. (4.31)
...............(4.38)
Unit 4

24. But from fig. 4.1
Equ. 4.11 is the required condition for a trapezoidal section to be
24
most economical, which can be expressed as half of the top width must
be equal to one of the sloping sides of the channel
Hydraulic mean depth
………………..(4.39)
From eq. (4.31)
…….(4.39)
…………………(4.40)
Unit 4

25. Hence for a trapezoidal section to be most economical hydraulic mean
depth must be equal to half the depth of flow
25
Best side slope for most economical trapezoidal
section
…………………(4.41)
From eq.(4.41) …………………(4.42)
…………………(4.43)
Wetted perimeter of channel
Substituting the value of b from (4.42)
Unit 4

26. ………………………..(4.44)
26
For the most economical trapezoidal section, the depth of flow, d and
area A are constant
Then n is the only variable. Best side slope will be when section is
most economical or in other words, P is minimum
Hence differentiating equation (4.44) with respect to n,
…………………………..(4.45)
Unit 4

27. 27
Squaring both sides
…………….(4.45)
If the sloping sides make an angle θ, with the horizontal, then we have
…………….(4.46)
Unit 4

28. Hence best side slope is at 60ᵒ to the horizontal or the value of n for
best side slope is given by equation (4.45)
28
For the most economical trapezoidal section, we have
Half of top width = length of one sloping side
………………………………(4.47)
Substituting the value of n from equation (4.45), we have
Unit 4

29. ……………………………………..(4.48)
29
……………………….(4.49)
Unit 4

30. Flow through Circular Channel
The flow of a liquid through a circular pipe, when the level of
30 liquid in the pipe is below the top of the pipe is classified as an
open channel flow
The rate of flow through circular channel is determined from the
depth of flow and angle subtended by the liquid surface at the
centre of the circular channel
Fig. 4.5 Circular Channel Unit 4

31. ……………………………..(4.50)
31
……………………………….(4.51)
Unit 4

32. 32
Most Economical Circular Section
 Incase of circular channels, the area of flow cannot be maintained
constant
With the change of depth of flow in a circular channel of any radius,
the wetted area and wetted perimeter changes
Thus in case of circular channels, for most economical section, two
separate conditions are obtained
They are:
1. Condition for maximum velocity, and
2. Condition for maximum discharge Unit 4

33. Condition for Maximum Velocity for Circular
Section
33
The velocity of flow according to Chezy’s formula
Unit 4

34. The velocity of flow through a circular channel will be maximum
when the hydraulic mean depth m or A/P is maximum for a given
34 value of C and i. In case of circular pipe, the variable is θ only
……………………………….(4.52)
where A and P both are functions of θ
The value of wetted area, A is given by equation (4.51) as
……………………………(4.52)
The value of wetted perimeter, P is given by equation (4.50) as
……………………………(4.53)
Unit 4

35. Differentiating equation (4.52) with respect to θ, we have
35
…………………(4.54)
From equ. (4.52)
into (4.54)
Unit 4

36. 36
Unit 4

37. 37
ately
The depth of flow for maximum velocity from fig. 4.5 is
Unit 4

38. 38
…………………………(4.55)
Unit 4

39. Thus for maximum velocity of flow, the depth of water in the circular
channel should be equal to 0.81 times the diameter of the channel.
39
Hydraulic mean depth for maximum velocity is
Unit 4

40. 40
…………………..(4.56)
Thus for maximum velocity, the hydraulic mean depth is equal to 0.3
times the diameter of circular channel
Unit 4

41. 41
Unit 4

42. …………………………………..(4.57)
42 But from equ. (4.53)
(4.58)
Unit 4

43. 43
Unit 4

44. Depth of flow for maximum discharge (see fig. 4.5)
44
…………………………….(4.59)
Thus for maximum discharge through a circular channel the depth of
flow is equal to 0.95 times its diameter
Unit 4

45. PROBLEM 1
Find the velocity of flow and rate of flow of water through a
45
rectangular channel of 6 m wide and 3 m deep, when it is running
full. The channel is having bed slope as 1 in 2000. Take Chezy’s
constant C = 55.
SOLUTION
Unit 4

46. 46
Unit 4

47. 47
PROBLEM 2
Find the slope of the bed of a rectangular channel of width 5 m when
depth of water is 2 m and rate of flow is given as 20 m3/s. Take
Chezy’s constant, C=50.
Unit 4

48. 48
Unit 4

49. ENERGY OF LIQUID IN OPEN CHANNELS
49
Non Uniform (Varied ) Flow through Open Channels
A flow is said to be uniform if the velocity of flow, depth of flow,
slope of the bed of the channel and area of cross section remain
constant for a given length of the channel
if velocity of flow, depth of flow area of cross-section and slope of
the bed of channel do not remain constant for a given length of pipe,
the flow is said to be non-uniform
Non-uniform flow is further divided into Rapidly Varied Flow (R.V.F)
and, gradually varied flow (G.V.F) depending upon the depth of flow
over the length of the channel
If the depth of flow changes abruptly over a small length of
channel, the flow is said to be rapidly varied flow
if the depth of flow in a channel changes gradually over a long
length of channel, the flow is said to be gradually varied flow
Unit 4

50. EQUATION OF GRADUALLY VARIED FLOW
50
Before deriving an equation for gradually varied flow, the
following assumptions are made:
•the bed slope of the channel is small
•the flow is steady and hence discharge Q is constant
•accelerative effect is negligible and hence hydrostatic pressure
distribution prevails over channel cross-section
•the kinetic energy correction factor, α is unity
•the roughness co-efficient is constant for the length of the channel
and it does not depend on the depth of flow
•the formulae, such as Chezy’s formula, Manning’s formula, which
are applicable, to the uniform flow are also applicable to the
gradually varied flow for determining the slope of energy line
•the channel is prismatic
Unit 4

51. 51
Fig. 4.6 Specific energy
Unit 4

52. The total energy of a flowing liquid per unit weight is given by,
52 …………………………………..(4.58)
Let L = Length of channel
If the channel bottom is taken as the datum as shown in Fig.4.6, then
the total energy per unit weight of the liquid will be,
………………………………………….(4.59)
The energy given by equation (4.58) is known as specific energy.
Unit 4

53. Hence specific energy of a given liquid is defined as energy per
unit weight of the liquid with respect to the bottom of the channel.
53
Specific Energy Curve
It is defined as the curve which shows the variation of specific energy
with depth of flow. It is obtained as follows:
From equation (4.59), the specific energy of a flowing liquid,
………………………………….(4.60)
Unit 4

54. Consider a rectangular channel in which a steady but non-uniform flow
is taking place
54
Let Q=discharge through the tunnel
b=width of the channel
h=depth of flow, and
q=discharge per unit width
Q Q
Then q    cons tan t ……………………………(4.61)
width b
Velocity of the fluid, V  Disch arg e  Q  q
………..(4.62)
area bh h
……………………………….(4.63)
Unit 4

55. Specific Energy and Specific Energy Curve
55
Fig. 4.7 Specific energy
……..(4.64)
Unit 4

56. equation (4.63) gives the variation of specific energy (E) with the
depth of flow(h)
56
hence for a given discharge Q, for different values of depth of flow,
the corresponding values of E may be obtained
then a graph between specific energy (along x- axis) and the depth
of flow, h (along y-axis) may be plotted
the specific energy curve may be obtained by first drawing a curve
for potential energy (i.e. Ep=h) which will be a straight line passing
through the origin, making an angle of 45° with the X-axis as shown in
fig 4.7
drawing another curve for kinetic energy (i.e. Ek = q2/(2gh2)) which
will be a parabola as shown in fig 4.7
by combining these two curves, we can obtain the specific energy
curve
in Fig 4.8, curve ACB denotes the specific energy curve
Unit 4

57. Critical Depth (hc)
the critical depth, denoted by hc, is defined as that depth of flow of
57
water at which the specific energy is minimum
in Fig 4.7, the curve ACB is a specific energy and point C
corresponds to the minimum specific energy
the depth of flow of water at C is known as the critical depth
The mathematical expression for critical depth is obtained by
differentiating the specific energy equation (4.63) with respect to
depth of flow and equating the same to zero
……………………………………………………….(4.65
Unit 4

58. 58
………………………………(4.66)
Hence, the critical depth is ………………………………(4.67)
Critical Velocity (Vc)
the velocity of flow at the critical depth is known as the critical
velocity (Vc)
 The mathematical expression for critical velocity is obtained from
the equation (4.67) as
………………………………(4.68)
Unit 4

59. Taking the cube of both sides, we get
..(4.69)
59
….......(4.70)
Substituting (4.70) into (4.69) ……………….(4.71)
………………………………….(4.72)
Unit 4

60. Minimum Specific Energy in terms of Critical Depth
60 Specific energy equation is given by equation (4.72)
When specific energy is minimum depth of flow is critical depth and
hence above equation becomes
………………………………..(4.73)
But from the equation (4.67)
Substituting the value of into equ. (4.73)
………………………..(4.74)
Unit 4

61. Hydraulic Jump or Standing Wave
61
Fig. 4.8 Hydraulic jump
Unit 4

62. consider the flow of water over a dam as shown in Fig. 4.8
 the height of water at the section 1-1 is small as we move towards
62 downstream, the height or depth of water increases rapidly over a
short length of the channel
this is because at the section 1-1, the flow is a shooting flow as the
depth of water at section 1-1 is less than the critical depth
shooting flow is an unstable type of flow and does not continue
on the downstream side
then this shooting flow will convert itself into a streaming or tranquil
flow and hence depth of water will increase
this sudden increase of depth of water is called a hydraulic jump
or a standing wave
thus, hydraulic jump is defined as: the rise of water level, which
takes place due to the transformation of the unstable shooting flow
(Super-critical) to the stable streaming flow (sub- critical flow)
Unit 4

63. Expression for depth of hydraulic jump
63
Assumptions:
the flow is uniform and the pressure distribution is due to
hydrostatic before and after the jump
losses due to friction on the surface of the bed of channel are
small and hence neglected
the slope of the bed of the channel is small, so that the
component of the weight of the fluid in the direction of flow is
negligibly small
Consider two sections 1-1 and 2-2 before and after a hydraulic
pump as shown in fig. 4.9
Unit 4

64. 64
Fig. 4.9 Hydraulic jump
Unit 4

65. 65
Consider unit width of the channel
The forces acting on the mass of water between section 1-1 and 2-2
are:
Unit 4

66. (iii) Frictional force on the floor of the channel, which is assumed to be
66
negligible
……………………..(4.75)
Unit 4

67. 67
Unit 4

68. 68
……………………………………(4.76)
But from the momentum principle, the net force acting on a mass of
fluid must be equal to the rate of change of momentum in the same
direction
Rate of change of momentum in the direction of the force
= mass flow rate of water x change of
velocity in the direction of force
Unit 4

69. …(4.77)
Therefore, according to the Newton’s 2nd law, equ. (4.76) = equ.
69
(4.77)
…………….(4.78)
But from the equation (4.75),
Dividing by ρ
Dividing by (d2-d1)
…………………………(4.79)
Unit 4

70. 70
………………………………..(4.80)
Equation (4.80) is a quadratic equation in d2 and hence its solution
is
Unit 4

71. 71
………………………………(4.81)
…………………………(4.82)
………………………….(4.83)
Unit 4

72. Expression for loss of Energy due to Hydraulic Jump
when hydraulic jump takes place, a loss of energy due to eddies
72
formation and turbulence occurs
this loss of energy is equal to the difference of specific energies at
sections 1-1 and 2-2
Unit 4

73. …(4.84)
73
But from (4.84) (4.85)
Substituting (4.85) into (4.84)
Unit 4

74. ……………………….(4.86)
74
Expression for depth of Hydraulic Jump in terms of
upstream Froude Number
…………………………(4.87)
Unit 4

75. (4.83)
75
…………………………(4.88)
But from the equation (4.87),
Unit 4

76. Substituting this value in the equation (4.88) we get
76
(4.89)
LENGTH OF HYDRAULIC JUMP is defined as the length between two
sections where one section is taken before the hydraulic jump and the
second section is taken immediately after the jump
 for a rectangular channel from experiments, it has been found equal
to 5 to 7 times the height of the hydraulic jump.
Unit 4