Flow in Open Channels
(Please print the notes for yourself)
Significance
essential for water resources development;
designs of irrigation, navigation, spillways, sewers,
culverts and drainage ditches.
Rivers and creeks management
Aims
flow capacity, water depth
channel design
Characteristics of open channel flows
Existence of free surface
Driven force: gravity
More complex and more challenging than pipe flow
Pressure distribution: hydrostatic distribution
Type and Geometry of Channels
Natural channels
such as rivers and streams: normally irregular geometry
Artificial channels
developed by human, such as navigation channels, power canals,
irrigation and drainage channels
regular geometric and hydraulic properties
Horizontal
HGL
Fundamentals
Energy line (EL)
(p/+z)1
Hydraulic gradient line (HGL)
Basic Eq: Bernoullis equation
hL1-2
EL
V12/2g
V22/2g
y1
S0
Datum
(p/+z)2
2
y2
Main Contents of Open Channel Flows
Classification of Open Channel flows
(Week 5)
Part I: Uniform Flow
Equations of motion
(Weeks 6 & 7)
Part II: Non-uniform Flow
Gradually varied flow
Rapidly varied flow
Coefficient of friction
Compound channels
Section of maximum discharge
Changing conditions
take place over a long
distance
Changing conditions
take place over a short
distance and hydraulic
jumps
Classification of Open Channel flows
Steady uniform flow-constant depth in time and distance
Steady non-uniform flow-depth varies with distance but not time
Unsteady flow-depth varies with both time and distance
Constant depth
Steady nonuniform
Steady uniform Flow
Varying depth
y
Unsteady Uniform Flow
Unsteady Flow
4
In one flow, the flow type may change:
Some commonly used terms
Depth (y): vertical distance from free surface to channel bottom
Stage (h): vertical distance of free surface from an arbitrary datum
Area (A): cross sectional area of flow normal to the flow direction
Wetted perimeter (P): length of wetted surface measured normal
to the direction of flow
Hydraulic radius (Rh): ratio of area to wetted perimeter A/P
Surface width (B): channel width at the free surface
Discharge (Q): volume of flow per unit time
Specific discharge (q): the discharge per unit width
Hydraulic mean depth: ratio of area to surface width Dh = A / B
Geometric properties of some common prismatic channels
Laminar and turbulent channel flows
In channel flow, Reynolds number is defined as:
Re channel = VRh /
(5.1)
For laminar channel flow:
Re channel < 500
For turbulent channel flow:
Re channel < 1000
Note: Theoretically, a relation for friction loss in channel flow can
be developed similar to the Darcy-Weisdach formula. In practice,
we always use much simpler formulae to relate looses to velocity
and channel shape as will be discussed later.
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Part I: Uniform Flow
Flow depth is constant (termed as normal depth) both in
time and space;
Gravity force is balanced by friction forces
Truly uniform flow rarely found in reality;
Many flows can be treated as uniform flows;
Uniform flow is considered as the base (reference) for all
other types of flows;
What do we want to know in open channel flow?
Velocity and pressure distributions, discharge, water
level ...
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1.1 Equations of motion
L
Resistance = Driving Force
Channel
slope S0
gALsin
gAL
0 PL = gAL sin
sin tan = S0
(Since is very small)
0 = gA sin / P = gRh S0
0
Use the definition of the friction coefficient: f =
and let
2
V / 8
( f / 8) V 2 = 0 = gRh S0
V =
8g
f
Rh S0
Or V = C Rh S0
Weisbach-Darcy
(5.2)
Chzy equation
(5.3)
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Where C in Eq. (5.3) is the resistance coefficient of Chzy.
In uniform regime, the flow depth h is defined as the normal
flow depth, h = hn.
Some common formulae for friction coefficient have been
elaborated over the years:
1) coefficient of Weisbach - Darcy
2) coefficient of Chzy.
3) coefficient of Manning-Strickler
4) coefficient of friction for mobile bed
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1.2 Coefficient of friction
The Weisbach-Darcy equation (5.2)
Useful for laminar and turbulent flow with circular
cross section and standard roughness
Chzy equation Eq. (5.3)
for truly turbulent flow (often the case);
Accuracy of the two formulas is strongly dependent on
the choice of the friction coefficient, f or C;
Artificial and particularly natural channels have all types
of form of the cross section. No parameter exists which
would take care of the variability in forms;
For open channel flows, Eq(5.3) is widely used.
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1.2 Coefficient of Chzy
For rough turbulent flow, the formula of Chzy can be used:
V = C
Rh S0
(5.3)
The coefficient C [m1/2/s] is a dimensional expression and can be
calculated using various empirical formulas. In practice, the socalled Manning (Manning-Strickler) formula is often used:
1 1/ 6
C=
R
n h
(5.4)
Where n is the coefficient of Manning.
The average velocity and discharge are:
1 2 / 3 1/ 2
V =
Rh S0
n
(5.5)
1 2 / 3 1/ 2
Q = A Rh S0
n
(5.6)
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Manning coefficient n
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Example 1
A channel is to be built of medium-quality concrete to convey a
discharge of 80 m3/s. The channel should have a trapezoidal cross
section with a bottom width of 5 m and side slopes of 3. The channel
slope is S0 =0.1%. Suppose the flow is uniform, calculate the flow
depth using Manning coefficient.
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Example 2.
The normal depth of flow in a trapezoidal concrete lined channel is 2 m.
The channel has a base width of 5 m and side slopes of 1:2. The bed
slope S0 is 0.001. Determine the discharge Q and the mean velocity V.
Solution:
B=b+2mh=5+8=13
Area:
A=(b+mh)h=18m2
P = b + 2h 1 + m = 13.9m
2
1
m=2
h
b
Rh = A / P = 1.29m
For concrete, the Manning coefficient n=0.013
1 2 / 3 1/ 2
Q = Rh S 0 A = 52m 3 / s
n
V = Q / A = 52 / 18 = 2.9m / s
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Example 3
The discharge in a trapezoidal concrete lined channel is 30 m3/s. The
channel has a base width of 5 m and side slopes of 1:2 while the bed
slope S0 is 0.001. Determine the depth y of the flow.
B
B=b+2my
Area: A=(5+2y)y
y=?
m=2
P = b + 2 y 1 + m = 5 + 2 5y
2
Rh = A / P
1 2 / 3 1/ 2
1
(5 + 2 y ) y
(5 + 2 y ) y
Q = Rh S 0 A =
0.013
n
5 + 2 5y
Using trial and error to solve for y=1.509m.
2/3
(0.001)1/ 2 = 30
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1.3 Compound channels
A3
A1
A2
p3
If the entire channel has an
uniform roughness, eqs. (5.5)
and (5.6) can be applied as
they are.
In reality, channel sections normally have different roughness. The use
of (5.5) and (5.6) in such cases will lead to large errors. One procedure
for this is to separate the cross section into several parts. It is assumed
there is no resistance along the dashed vertical lines. The total discharge
can be calculated using the following formula:
1 2 / 3 1/ 2
1 2 / 3 1/ 2
1 2 / 3 1/ 2
Q = A1
Rh1 S0 + A2
Rh 2 S0 + A3
Rh 3 S0
n1
n2
n3
Where Rh1 = A1 / P1
Rh2 = A2 / P2
(5.6)
Rh3 = A3 / P3
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Example 4: The bed slope of a compound channel is 0.001. Other
conditions are as in the figure. Find the discharge of the channel.
3m
Q = Qi
2m
3m
i =1
n=0.02
Ai 2 / 3 1/ 2
Qi = Rni S 0
ni
n=0.03
n=0.015
Ai
Pi
Rni
ni
Qi
4.5
4.5
0.02
1.25
0.015
4.5
4.5
0.03
1.5m
1m
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1.4 Section of Maximum Discharge
The construction of a channel with a given discharge, a given slope
and a given roughness will be less expensive if the cross section is
the smallest.
1 2 / 3 1 / 2 1 1 / 2 A5 / 3
Take the formula of discharge: Q = A Rh S 0 = S 0
n
n
P2/3
For a constant A, if Rh is maximal or P is minimal Q maximal.
Amongst all geometrical forms possible, the cross section of a
semi-circular form will give a Pmin for a given constant A.
For a channel in an alluvium, one should take into account the angle
of repose as well as various constraints due to construction.
Consequently a trapezoidal form may be the most reasonable one.
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yn
yn
yn
L
b
b=2yn
What is the best trapezoidal form?
We can prove that for a trapezoidal cross-section channel,
we have the following optimised dimensions:
A=
3 yn2 ,
2 yn
b=
3
and L = b
(5.7)
For a rectangular channel, we have
b = 2 yn
(5.8)
The optimum cross-sections for some channels are given below:21
Properties of optimum open-channel sections (From Fox and McDonald)
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Example 5
Find the dimensions of the most-efficient cross section for a
rectangular channel that is to convey a uniform flow of 10 m3/s if
the channel is lined with gunite concrete and is laid on a slope of
0.001.
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Example 6
Find the dimensions of the most-efficient cross section for a
trapezoidal channel that is to convey a uniform flow of 10 m3/s if
the channel is lined with gunite concrete and is laid on a slope of
0.001. The side slopes of the channel cross section are 2.
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