2. Introduction
➢Open channel flow is a flow which has a free surface and
flows due to gravity.
➢Pipes not flowing full also fall into the category of
open channel flow
➢In open channels, the flow is driven by the slope of the
channel rather than the pressure
2
5. Types of flows
1. Steady and Unsteady Flow
2. Uniform and Non-uniform Flow
3. Laminar and Turbulent Flow
4. Sub-critical, Critical and Super-critical Flow
SGU/Civil/ADK/FM 5
6. Types of flows
❖ If the conditions (flow rate, velocity, depth etc) do not change with time, it is steady flow.
❖ Otherwise it is unsteady flow.
❖ If for a given length of 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 non uniform flow.
❑ Gradually varied flow
❑ Rapidly varied flow
❖ Based on Froud number, flows are classified as, Sub-critical, Critical and Super-critical Flow
SGU/Civil/ADK/FM 6
7. Laminar and Turbulent Flow
Both laminar and turbulent flow can occur in open channels depending on the Reynolds
number (Re).
Reynolds number in open channel is given by,
𝑅𝑒 =
𝜌𝑉𝑅
𝜇
Where,
ρ= density of fluid (for water ρ = 1000 kg/m3)
µ = dynamic viscosity
R = Hydraulic Mean Depth = Area / Wetted Perimeter
SGU/Civil/ADK/FM 7
Note the difference in Reynolds number for open channel flow and flow through pipe.
9. Geometric elements
The geometric elements are the physical properties of a channel section which can be defined
by the flow depth and other dimensions of the channel section.
The depth of flow is ‘y’ is vertical distance of lowest point of a channel from free surface
‘T’ is top width of the free surface
‘A’ is wetted area or c/s area in direction normal to flow
‘P’ is wetter perimeter
Hydraulic radius ‘R’ is ratio of wetter area to wetted perimeter.
Section factor for critical flow computations is ‘Z’,
SGU/Civil/ADK/FM 9
𝑅 =
𝐴
𝑃
𝑧 = 𝐴 𝐷
11. Velocity Distribution in open channel flow
➢Velocity is always vary across channel because of friction along the boundary
➢The maximum velocity usually found just below the surface
SGU/Civil/ADK/FM 11
12. Discharge through open channel
Discharge though open channel is calculated by
•The Chezy’s equation
•The Ganguillet-Kutter formula
•The Bazin formula
• The Manning’s formula
SGU/Civil/ADK/FM 12
13. Chezy’s equation
It is used to determine velocity in open channel flow for uniform flow.
SGU/Civil/ADK/FM 13
14. Chezy’s equation
The main features of the uniform flow in en channles is as follows:
▪ The depth of flow, wetted area, velocity of flow and discharge are constant at every section
along the channel reach.
▪ The total energy line and water surface and the channel bottom are parallel to each other.
▪ Newton’s second law of motion is to be applied for fluid in motion.
SGU/Civil/ADK/FM 14
15. Chezy’s equation
Forces acting on element are:
1. The force of hydrostatic pressure f1 and f2 on two ends of free body. As the depth is same, f1
and f2 are same and cancel each other.
2. The component of weight of water in a direction of flow, which is γ𝐴𝐿 sin 𝜃
3. The resistance to flow is exerted by wetted surface of the channel. It is given by, 𝑃𝐿𝜏0, τ0 is
average shear stress along boundary
SGU/Civil/ADK/FM 15
16. Chezy’s equation
For equilibrium of element, γ𝐴𝐿 sin 𝜃- 𝑃𝐿𝜏0 = 0
SGU/Civil/ADK/FM 16
But it is known that, 𝜏0 =
𝑓
8
𝜌𝑉2 Therefore, equating two equations,
𝑓
8
𝜌𝑉2
= 𝑟
𝐴
𝑃
sin 𝜃 OR 𝑣 =
8𝑟
𝜌𝑓
𝑅𝑆 =
8𝑔
𝑓
𝑅𝑆
The above equation can be written as,
𝑣 = 𝐶 𝑅𝑆 This equation is known as Chezy’s equation and C is known as Chezy’s constant.
Or 𝜏0 = 𝑟
𝐴
𝑃
sin 𝜃
17. Chezy’s constant
C is known as Chezy’s constant
SGU/Civil/ADK/FM 17
𝑐 =
8𝑔
𝑓
French scientist Antonnie Chezy derived this formula in 1775.
It varies inversely with fsquare root of f 9Darcy Weisbach friction
factor)
C has dimension of [L1/2T-1]
18. Formulae for Chezy’s C
• The Ganguillet-Kutter formula
SGU/Civil/ADK/FM 18
• The Bazin formula
19. Manning’s formula
In 1889, Irish scientist engineer Robert Manning presented a formula according to which the
mean velocity of flow in c channel is expressed in terms of coefficient of roughness n, called
Manning’s n, hydraulic radius R and bottom slope S.
It is given as,
SGU/Civil/ADK/FM 19
𝑉 =
1
𝑛
𝑆2/3𝑅1/2
If we compare manning’s n and Chezy’s C, we get,
𝐶 =
1
𝑛
𝑅1/6
20. Most efficient channel section
➢ A channel section is considered most economical or most efficient when it can pass maximum
discharge for given cross sectional area, resistance factors, and bottom slope.
➢ From Continuity equation, we can say that, discharge will me maximum when area will be
maximum , for constant c/s area.
➢ From Chezy’s and Manning;s equation, it can be said that, for given value of slope, and
surface roughness, velocity will be maximum when hydraulic radius R will be maximum.
SGU/Civil/ADK/FM 20
21. Most economical rectangular section
SGU/Civil/ADK/FM 21
Substituting value of B in equation (ii)
𝑃 =
𝐴
𝑦
+ 2𝑦 (𝑖𝑖𝑖)
Assuming area A to be constant, eq (iii) can be differentiated with respect to y and equated to zero for
obtaining condition for minimum P.
.....(i)
2 .....(ii)
A By
P B y
=
= +
From equation (i)
A
B
y
=
22. Most economical rectangular section
SGU/Civil/ADK/FM 22
But, B = 2y, we get,
𝑅 =
2𝑦2
2𝑦+2𝑦
=
𝑦
2
Thus, it can be stated that, the rectangular section will be most economical
when
1. Depth of flow is half the total width
2. Or hydraulic radius is equal to half the depth of flow
2
2 0
dP A
dy y
= − + =
OR
2
2
2
A y By
B y
= =
=
2
B
y =
23. Most economical triangular section
For a triangular channel of section, if θ is the angle of inclination with depth of flow as y, We can
write
SGU/Civil/ADK/FM 23
2
tan ... (i)
tan
(2 )sec .... (ii)
A
A y or y
P y
= =
=
Substituting the value of y from eq (i) in eq (ii)
2
(sec )
tan
A
P
=
Assuming area A to be constant, above equation can be differentiated with respect to θ and
equated to zero for obtaining the condition for minimum P.
24. SGU/Civil/ADK/FM 24
Most economical triangular section
3
3/2
sec sec
2 0
2(tan )
tan
dP
A
d
= − =
45 or z = 1
o
= 45 or z = 1
o
=
Hence, a trianguart section will be most economical when its sloping sides make an angle of 45o with
vertical.
25. The hydraulic radius R of the triangular section can be expressed as,
SGU/Civil/ADK/FM 25
2
tan
2 sec
A y
R
P y
= =
Substituting θ = 45o in above equation,
2 2
y
R =
Thus it can be stated that, for most economical triangular section, triangular section will be half square
described on a diagonal and having equal sloping sides.
Most economical triangular section
26. Most economical trapezoidal section
Consider a trapezoidal section of bottom width B, and depth of flow y, and side slope z horizontal to 1
vertical. Following expressions for wetted area A and wetted perimeter P can be written,
SGU/Civil/ADK/FM 26
2
( ) ...... (i)
2 1 ...... (ii)
A B zy y
P B y z
= +
= + +
From equation (i),
A
B zy
y
= −
Substituting the value of B in equation (i),
2
2 1
A
P zy y z
y
= − + +
27. Assuming area A and side slope z to be constant, above equation can be differentiated with
respect to y and equated to zero for obtaining the condition for minimum P.
SGU/Civil/ADK/FM 27
Most economical trapezoidal section
2
2
2
2
2 1 0
2 1
dP A
z z
dy y
A
OR z z
y
= − − + + =
+ = +
Substituting value of A from eq (i), above equation becomes,
2
2
( )
2 1
B zy y
z z
y
+
+ = + 2
2
1
2
B zy
z y z
+
+ = +
This is condition for most economical
trapezoidal section.
28. The hydraulic radios R can be expressed as ,
SGU/Civil/ADK/FM 28
2
( )
2 1
A B zy y
R
P B y z
+
= =
+ +
Substituting the value of B from equation for most economical condition in above eq,
( )
2
2 2
2 1 2
2
2 1 2 2 1
y z zy zy y y
R
y z zy y z
+ − +
= =
+ − + +
2
y
R = This is condition for most economical trapezoidal section.
Most economical trapezoidal section
29. Further let O be the centre of top width of
trapezoidal channel section. Drop Perpendicular OA
from O to the sloping side of channel, in Fig.
If θ is angle made by sloping side with the
horizontal, then from right angled triangle OAC,
OA=OC Sin θ
But,
SGU/Civil/ADK/FM 29
Most economical trapezoidal section
2
2
B zy
OA
+
=
30. And,
SGU/Civil/ADK/FM 30
Most economical trapezoidal section
2
2 2
Sin
1
2 2
2 1 2 1
y
y z
B zy y B zy
OA
y z z
=
+
+ +
= =
+ +
But,
2
( )
2 1
A B zy y
R
P B y z
+
= =
+ +
Substituting for (B+2zy) OA y
=
Thus, if semi circle is drawn with O as centre, and radius equal to depth of flow y, the three sides of
most economical section, will be tangential to semi circle.
31. These conditions have been derived by assuming that the side slope is constant.
But, if due to site limitations, if it is not possible to maintain these conditions, we can derive in
other forms as,
SGU/Civil/ADK/FM 31
Most economical trapezoidal section
1) When bottom width is constant,
2
y
A
z
=
2) When depth of flow y is constant,
1
or =60
3
o
z
=
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How it works:
1 Start a new presentation by going to File > New > Blank
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2 On the very first slide, add a picture: Go to Insert > Pictures or
Insert > Online Pictures and choose the picture.
Hint: You need to be online when you add the picture.
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select Let's Go.
4 Choose a design you like from the Design Ideas task pane.
SGU/Civil/ADK/FM 32
33. Morph
Morph makes smooth animations and object movements in your
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Play the video on the right to see a quick example.
Morph is a subscription-only feature. If you have an Office 365
subscription, you can try it yourself with the steps on the next
slide.
SGU/Civil/ADK/FM 33
34. Setting up Morph
Try it yourself with these two simple “planets”:
1 Duplicate this slide: Right-click the
slide thumbnail and select
Duplicate Slide.
2 In the second of these two identical
slides, change the shapes on the
right in some way (move, resize,
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Transitions > Morph.
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select Play to see your circle morph!
Hint: Effect Options gives you even more
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SGU/Civil/ADK/FM 34
35. Working together in real time
When you share your presentation with others, you’ll see them
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How it works:
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a marker shows who is on which slide…
3 …and the part of the slide they're
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SGU/Civil/ADK/FM 35
36. You’re an expert with Tell Me
The Tell Me box finds the right command when you need it,
so you can save time and focus on your work.
Try it:
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2 Type animation in the Tell Me box, and then
choose Add Animation.
3 Choose an animation effect,
like Zoom, and watch
what happens.
SGU/Civil/ADK/FM 36
37. Explore without leaving your slides
Smart Lookup brings research directly in to PowerPoint.
Try it:
1 Right-click in the word office in the
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SGU/Civil/ADK/FM 37
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