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Title: University of Palestine Engineering Hydraulics 2nd semester


1
CHAPTER 6 Water Flow in Open Channels
University of Palestine Engineering
Hydraulics 2nd semester 2010-2011
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
Uniform Flow
Types of Flow in Open Channels
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 Reynolds number is defined as

17
Types of Flow in Open Channels
Recall that Reynolds 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

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.

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 (Chezys 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 Mannings coefficient for the channel
roughness, m-1/3/s.
Substituting manning Eq. into Chezy Eq, we
obtain the Mannings 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
Example 1
Flow Formulas in Open Channels
open channel of width 3m as shown, bed slope
15000, d1.5m find the flow rate using Manning
equation, n0.025.
28
Example 2
Flow Formulas in Open Channels
open channel as shown, bed slope 691584, find
the flow rate using Chezy equation, C35.
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 12, 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
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
Example 4
Most Economical Section of Channels
Circular open channel as shown d1.68m, bed slope
15000, find the Max. flow rate the Max.
velocity using Chezy equation, C70.
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 Q10m3/s,
velocity 1.5m/s, for most economic section. find
wetted parameter, and the bed slope n0.014.
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 Q10m3/s if the
bed slope 0.0016, n0.014. 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
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
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.
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 _at_ 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 _at_
    y1 and y2 These two depths for a given specific
    energy are called the alternate depths.
  • If the flow depth y gt yc , the flow is said to be
    sub-critical (Fr lt 1.0). In this case Es
    increases as y increases.
  • If the flow depth y lt yc , the flow is said to be
    super-critical (Fr gt 1.0). In this case Es
    increases as y increases.

53
Froude Number (Fr)
Energy Principles in Open Channel Flow
54
Critical Flow
Energy Principles in Open Channel Flow
55
Rectangular Channel
Energy Principles in Open Channel Flow
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,
qQ/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 _at_ yc .
and
therefore for critical flow Fr 1.0
If the depth flow y gt yc , the flow is said to be
sub-critical. In this case Es increases as y
increases. For this type of flow, Fr lt 1.0 . If
the depth flow y lt yc , the flow is said to be
super-critical. In this case Es increases as y
decreases. For this type of flow, Fr gt 1.0 .
57
Energy Principles in Open Channel Flow
Rectangular Channel
d) Minimum Specific Energy in terms of critical
depth At (Emin , yc ) ,
58
Other Sections
Energy Principles in Open Channel Flow
at critical flow Fr 1 where
Rectangular section
Trapezoidal section
Circular section
Triangle section
59
Example 1
Energy Principles in Open Channel Flow
Determine the critical depth if the flow is
1.33m3/s. the channel width is 2.4m
60
Example 2
Energy Principles in Open Channel Flow
Rectangular channel , Q25m3/s, bed slope 0.006,
determine the channel width with critical flow
using manning n0.016
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.

63
Non-uniform Flow in Open Channels
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 lt
    Sc or yn gt yc .
  • 3) Steep Slope the bottom slope of the channel
    is greater than the critical slope. In this case
    S0 gt Sc or yn lt 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.

65
Non-uniform Flow in Open Channels
Classification of Channel-Bed Slopes
66
Non-uniform Flow in Open Channels
Classification of Flow Profiles (water surface
profiles)
67
Non-uniform Flow in Open Channels
Classification of Flow Profiles (water surface
profiles)
68
Non-uniform Flow in Open Channels
Classification of Flow Profiles (water surface
profiles)
69
Non-uniform Flow in Open Channels
Classification of Flow Profiles (water surface
profiles)
70
Non-uniform Flow in Open Channels
Classification of Flow Profiles (water surface
profiles)
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.
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.
73
Hydraulic Jump
Hydraulic Jump in Rectangular Channels
But for Rectangular section
74
Hydraulic Jump
Hydraulic Jump in Rectangular Channels
75
Hydraulic Jump
Hydraulic Jump in Rectangular Channels
76
Hydraulic Jump
Hydraulic Jump in Rectangular Channels
77
Hydraulic Jump
78
Hydraulic Jump
79
Example 1
Hydraulic Jump
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
Example 2
Hydraulic Jump
dn Depth can calculated from manning equation
81
Hydraulic Jump
a)
b)
82
Hydraulic Jump
c)
83
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