FLUID FLOW IN OPEN
CHANNELS
OPEN CHANNEL FLOW
• An open channel is the one in which stream is not
complete enclosed by solid boundaries and therefore
has a free surface subjected only to atmosphere
pressure.
• The flow in such channels is not caused by some
external head, but rather only by gravitational
component along the slope of channel. Thus open
channel flow is also referred to as free surface flow or gravity
flow.
• Examples of open channel are
Rivers, canals, streams, & sewerage system etc
OPEN CHANNEL FLOW
Thal Canal
Indus river
COMPARISON BETWEEN OPEN
CHANNEL FLOW AND PIPE FLOW
Aspect
Open Channel
Cause of flow
Pipe flow
Gravity force (provided
by sloping bottom)
Cross-sectional Open channels may have
shape
any shape, e.g., triangular,
rectangular, trapezoidal,
parabolic or circularetc
Surface
Varies with depth of flow
roughness
Pipes run full and flow takes place
under hydraulic pressure.
Pipes are generally round in
cross-section which is
uniform along length
Piezometric
head
(z+h), where h is
depth of channel
(z+P/γ) where P is the pressure
in pipe
Velocity
distribution
Maximum velocity occurs
at a little distance below the
water surface.The shape of
the velocity profile is
dependent on the channel
roughness.
The velocity distribution is
symmetrical about the pipe axis.
Maximum velocity occurs at the
pipe center and velocity at pipe
walls reduced to zero.
4
Varies with type of pipe material
TYPES OF CHANNELS
Natural Channels:
It is one with irregular sections of varying shapes, developing in natural
way. .e.g.,rivers, streamsetc
Artificial Channels:
It is the one built artificially for carrying water for various purposes. e.g.,
canals,
Open Channel:
A channel without any cover at the top. e.g.,canals, rivers streamsetc
Covered Channels:
A channel having cover at the top. e.g.,partially filled conduits carryingwater
Prismatic Channels:
A channel with constant bed slope and cross-section along its length.
DEFINITIONS
Depth of Flow:
It is the vertical distance of the lowest point of a channel section(bed of the
channel)from the free surface.
Depth of Flow Section:
It is depth of flow normal to bed of the channel.
Top Width:
It is the width of channel section at the free surface.
Wetted Area:
It is the cross-sectional area of the flow section of channel.
Wetted Perimeter:
It is the length of channel boundary in contact with the flowing water
at any section.
Hydraulic Radius:
It is ratio of cross-sectional area of flow to wetted perimeter.
TYPES OF FLOW IN OPEN CHANNELS
Steady and unsteady flow
Uniform and non-uniform flow
Same definition
with pipe flows
Laminar andTurbulent flow
Subcritical, critical and supercriticalflow
Laminar and Turbulent Flow: For open channels, it is defined with Reynolds No. as;
Re  VRh

Therefore,
For laminar flow:
Re < 500
For Turbulent flow: Re >1000
For transitionalflow: 500 < R e < 1000
Where,
A D
Rh = hydraulic radius = ≅
P
4
If width/depth > 10 then R ≅deptℎ
Remember in pipe flows
Re 
VD

For laminar flow: Re < 2000
For Turbulent flow: Re > 4000
Transitional 2000 < Re < 4000
TYPES OF FLOW IN OPEN CHANNELS
Subcritical, Critical and Supercritical Flow:
These are classified with Froude number.
Froude No. (Fr) is ratio of inertial force to gravitational force of flowing fluid.
Mathematically, Froude no.is
Fr 
V
gh
Where, V is average velocity of flow, h is depth of flow and g is gravitational
acceleration
Fr. < 1 Flow is subcriticalflow
Fr. = 1 Flow is critical flow
Fr. > 1 Flow is supercriticalflow
OPEN CHANNEL FORMULAE FOR
UNIFORM FLOW
For uniform flow in open channels, following formulae are widely used
1. Chezy’sFormula: Antoine de Chezy (1718-1798), a French
bridge and hydraulic expert, proposed his formula in 1775.
V  C RSo
C= Chezy’sconstant
2. Manning’s Formula: Robert Manning (An Irish engineer)
proposed the following relation for Chezy’s coefficient
1
R6
C=
n
According to which Chezy’s equation can be written as
V
1 2/3 1/2
R So
n
n= Manning’s Roughnesscoefficient
Here,
V=Average flowvelocity
R=Hydraulic radius
So=Channel bedslope
DERIVATION OF CHEZY’S FORMULA
In uniform flow the cross-sectional through which flow occurs is constant along the
channel and so also is the velocity. Thus y1=y2=yo and V1=V2 =V and the channel
bed, water surface and energy line are parallel to one another.
According to force balance along the direction of flow; we
can write,
F1  F2  ALsin   o PL
F1= Pressure force at section 1
F2= Pressure force at section 2
W = Weight of fluid between
section 1 and 2= 𝛾𝐴𝐿
So= slope of channel
θ= Inclination ofchannel
with horizontal line
τo= shearingstress
P= Wetted perimeter
L= length betweensections
V= Avg. Flow velocity
yo = depth offlow
DERIVATION OF CHEZY’S FORMULA
 o  ALsin    A sin   R sin 
PL
P
For channels with So<0.1, we can safely assume that
S o  S w  S  sin
Therefore;
 o  RS o
11
z1 z2
x
z  y  z2  y 2 
Sw  1 1
x
So 
S
S
z1  y1 v12 / 2g  z2  y2  v22 / 2g 
x
hL
x
DERIVATION OF CHEZY’S FORMULA
τo (shearing stress) can also be expressed as
2
V
o  Cf 
2
Comparing both equations of τo weget;
2
V
Cf 
 RSo
2
8g
2g
V
RS o 
RS o
f
Cf
V  C RS o
Cf  f / 4
C
8g
f
Where C is Chezy’s Constant whose value depend upon the type of channel surface
As f and C are related, the same consideration
that are present for determination of friction
factor, f, for pipe flows also applies here.
EMPIRICAL RELATIONS FOR
CHEZY’S CONSTANT, C
Although Chezy’s equation is quite simple, the selection of acorrect value of C is rather
difficult. Some of the important formulae developed for Chezy’s Constant C are;
1. Bazin Formula: A French hydraulic engineer H. Bazin (1897) proposed the
following empirical formula for C
C
157.6
181 K / R
R= HydraulicRadius
K=Bazin Constant
The value of K
depends upon
the type of
channel surface
13
EMPIRICAL RELATIONS FOR
CHEZY’S CONSTANT, C
2. Kutter’s Formula: Two Swiss engineers GanguilletandKutter
proposed following formula for determination of C
R= Hydraulic Radius n=Manning’s roughness coefficient
3. Manning’s Formula:Robert Manning (An Irish engineer) proposed
the following relation for Chezy’s coefficient C
• C  1/ n R1/ 6
V
1 2/3 1/2
R So
n
n= Manning’s Roughness coefficient
The values of n depends upon nature of channel surface
EMPIRICAL RELATIONS FOR
CHEZY’S CONSTANT, C
15
CHEZY’S AND MANNING’S EQUATIONS
Chezy’s Equation
Manning’sEquation
1 2/3 1/2
R So
n
1
1/ 2
3
Q m / s  AR 2/3 S o
n
V  C RSo
V
Q  CA RS o
C
A
= Chezy’sConstant
= Cross-sectional area offlow

A

= Cross-sectional area offlow
MOST ECONOMICAL SECTION
From Manning’s formula, we can write that
1
Q  AR hS o
n
For agiven channel of slope, So, area of cross-section, A, and roughness, n,
we can simplify above equation as
A
1
Q  Rh  Q   Q 
P
P
It emphasis that discharge will be maximum, when Rh is maximum and for a
given cross-section, Rh will be maximum if perimeter isminimum.
Therefore, the most economical section (also called best section or most
efficient section) is the one which gives maximum discharge for a given
area of cross-section.
17
MOST ECONOMICAL RECTANGULAR
SECTION
Let’sconsider a rectangular channel as shown in figure in which width of channel
is b and depth of flow is h.
Cross - sectional area of flow  A  bh
h
Wetted Perimeter = P  b  2h  A / h  2h
b
For most economical section, perimeter should be minimum. i.e
dP/dh  0
d
dP/dh 
b  2h  d A/ h  2h  0
dh
dh
18
Hence for most economical rectangular section,
width is twice the depth of channel

A
 2  0  A  2h 2
2
h
bh  2h 2
b  2h or
h  b/ 2
MOST ECONOMICAL TRAPEZOIDAL
SECTION
b+2Sh
Let’sconsider atrapezoidal channel
having bottom width, b,depth of
flow is d, and side slope, S, as
shown in figure
Sh
1
h S 1
Sh
h
s
θ
2
b
Cross - sectional area of flow  A  bh  Sh 2
Wetted Perimeter  P  b 2h S2  1 
A  bh  Sh 2
A/ h  Sh2h
S2  1
For most economical section, perimeter should be minimum. i.e.,
dP
dh
19
 0
d
dh
A/ h  Sh  2h S  1  0
2
b
A
 Sh
h
MOST ECONOMICAL TRAPEZOIDAL
SECTION


d
A/ h  Sh  2h S2 1  0
dh
 
A
2
S

S

2
1 0
2
h
2
A
bh

Sh
2
2
S
S

S

2
1


S

2
1
2
2
h
h
b  Sh
b  Sh Sh
 S  2 S2 1 

 2 S2 1
h
h
h
b  2Sh
 2 S2 1
h
b  2Sh
 h S2 1  b  2Sh  2h S2 1
2
Hence for most economical trapezoidal section, top width is twice the length of one
sloping side or half of top width is equal to length of one sloping side
20
MOST ECONOMICAL TRAPEZOIDAL
SECTION
For given width, b,and depth, h, perimeter becomes only the function of
side slope, S,.So if we estimate value of S that provide minimum P then
we have;

dP
 0  d A / h  Sh  2
dS
dS

S2 h 2  h 2  0
1 2
1/ 21


  h  2h S 1  2S   0
2




h S2 1  2Sh


A/ h  Sh  2h
dS
d
  h  2Sh S2 11/ 2  0
S 1 2S

2
Squaring both sides of equation, weget
S2 1  4S 2
 S21
3
S
1
3
If sloping sides make an angle θ with the horizontal than S=tanθ
S  tan 
1
3
21
   60o

S2 1  0
PROBLEM # 1
Water is flowing in a 2-m-wide rectangular, brick channel (n=0.016) at a depth of 120 cm.
The bed slope is 0.0012. Estimate the flow rate using the Manning’s equation.
Solution: First, calculate the hydraulicradius
Manning’s equation (for SI units) provides
22
PROBLEM # 2
Solution
y
For SI units


1
1/ 2
Q m / s  AR 2 / 3 S o
n
23
3
4
PROBLEM # 3
For SI units


1
1/ 2
2/3
AR
S
Q m /s 
o
n
Q  Q1  Q2  Q3
24
3
PROBLEM # 4
Solution:(a)
For SI units
Qm3 / s
25
1
1/ 2
AR 2/ 3 So
n
(b) For SIunits
26
1
1/ 2
2/3
AR
S
Qm / s
o
n
3