Open Channel

Hydraulics:
Fundamentals of open channel flow
Content
OPEN CHANNELS: INTRO & BASIC DEFINITIONS

FLOW STATE & CLASSIFICATION
REYNOLDS No, LAMINAR & TURBULENT FLOW
CHEZY & MANNING EQUATIONS

VELOCITY DISTRIBUTIONS
COMPOUND CHANNELS
PARTIALLY FULL CIRCULAR PIPES
WORKED EXAMPLE
OPEN CHANNELS: INTRODUCTION:

WHAT IS OPEN CHANNEL FLOW?
Open channel flow:

characterised by having
free water surface:
at which the
pressure
is atmospheric
&
across which
shear forces
are negligible
OPEN CHANNELS: BASIC DEFINITIONS

If we look @ the Longitudinal profile:
free surface defines the
hydraulic gradient
(shown below)
Q
Energy Gradient
Water surface &
Hydraulic gradient

Commonly used geometric properties of open channel flow are:
Water surface
h1
y1
y2
Datum
X-section 1
h2
X-section 2
Depth y :
The vertical distance from free surface to

lowest point of the channel bed at any given channel section.
Stage h :
Vertical distance from the free surface to an arbitrary datum

SURFACE WIDTH, B
AREA, A
P
Area A :
The cross sectional area of flow
(i.e. perpendicular to the direction of flow)
Wetted Perimeter P : The length of the wetted channel surface

(i.e. perpendicular to the direction of flow)
Surface Width B :
Width of the channel section,
(i.e. perpendicular to the direction of flow at free surface)

SURFACE WIDTH, B
AREA, A
Hydraulic Radius
R= A/P
Hydraulic Mean Depth
Dm = A / B
Content


COMPOUND CHANNELS
WORKED EXAMPLE
FLOW STATE - REVISION

In Previous Lectures:
You looked at flow classification
Vd
Re
Where Reynolds Number, Re, for pipe flow

Laminar flow : Re(pipe) < 2000
Turbulent flow : Re(pipe) > 4000
Laminar flow in a pipe:

http://uk.youtube.com/watch?v=KqqtOb30jWs&feature=related
Turbulent flow in a pipe:

http://uk.youtube.com/watch?v=NplrDarMDF8&feature=related

TYPES OF FLOW STATE
Most open
channel
flow
-turbulent
FLOW CLASSIFICATION OPEN CHANNEL

How are these two flows different?
Constant inflow
ow
s
ta
Con
s
tan
t
infl
s
Con
w
nflo
nt i
OUTFLOW
Steady, uniform flow
OUTFLOW
Steady non-uniform flow

4 classifications used for open channel flows:
DEPTH
1.) Steady
Uniform
Flow:
2.) Unsteady Uniform
Flow:
3.) Steady Non-uniform
Flow:
4.) Unsteady Non-Uniform Flow:

1.) Steady Uniform Flow:
Depth is constant w.r.t. both time & distance.
Gravity forces are in equilibrium with resistance forces.
Steady =
Constant
Uniform =
Constant

2.) Unsteady Uniform Flow:
Very rare.
Unsteady =
Not constant
Uniform =
Constant

3.) Steady Non-uniform Flow:
Depth varies with distance but not with time.
infl
ow
s
tan
Steady =
constant
Con
s
tan
t
s
Con
flow
t in
OUTFLOW
Non uniform =
not constant

3.) Steady Non-uniform Flow:
Flow may be gradually varied or rapidly varied.
u/s
d/s
Pier
DAM
Stilling Basin

4.) Unsteady Non-uniform Flow:
Depth varies with both time and distance
Unsteady =
Not constant
Non uniform =
not constant
The Severn Bore. http://www.youtube.com/watch?v=Ol9ogW-4bPQ&feature=fvw

It is a large tidal surge wave that can be seen in the estuary of the River Severn,
where the tidal range is the 2nd highest in the world,
being as much as 50 feet (approx. 15.4m).
FLOW CLASSIFICATION SUMMARY
Steady Uniform Flow:

Depth is constant with respect to
both time & distance.
Steady Non-uniform Flow:

Depth varies with distance but not with time.
Unsteady Non-uniform Flow:

Depth varies with both time and distance.
Unsteady Uniform Flow:

Vary rare.
Content


COMPOUND CHANNELS
WORKED EXAMPLE

REVISON: Reynolds Number, Re
We know that for pipe flow
To calculate Reynolds Number for open channels:
Re Channel
Where:
RV
R is the hydraulic radius

is the absolute coeff of viscosity
V is velocity
is density of water (1000 kgm3)
R= A/P
LAMINAR & TURBULENT FLOW

It can be shown that:
(For further details see blue module book &
Chadwick and Morfett, Chapter 5):
Re (Channel ) Re ( Pipe ) / 4
For Pipe flow
Hence, for open channels:

Laminar flow : Re(channel) < 500
Turbulent flow : Re(channel) > 1000
(In practice, the lower limit for turbulent
flow is taken as Re > 2000)
Content


COMPOUND CHANNELS
WORKED EXAMPLE
FOR STEADY UNIFORM FLOW

It is possible to adapt
equations such as:
Darcy-Weisbach
&/or
Colebrook-White
for use with
open channel flows

However these approaches are:
complex &
open to question
Due to the effect that:

free surface has on
velocity distributions

BUT most open channel flow occurs
in the rough turbulent zone:
it is possible to develop
easier formulae
NB: Where we assume
Turbulent flow : Re > 2000
THE CHEZY EQUATION

For steady uniform flow in open channels,
Chezy developed the following equation:
V C ( RS o )
So is the bed slope
C is the chezy coefficient:
(i.e. which depends on Reynolds number Re
& Boundary roughness -> friction factor)
R is hydraulic radius
R= A/P
Mannings Equation
Many equations have been developed to

derive the chezy coefficient, C
The most significant approach is that derived by

Manning (an Irish engineer / accountant)
16
where :
n = Mannings n,
R
C
n
which depends on surface roughness & channel alignment
.
R= A/P
Mannings Equation cont.

Combining the Chezy equation
with Manning
Gives
V C ( RS o )
R1 6
C
n
1 2 / 3 1/ 2
V R So
n
Mannings Equation cont.

Since:
1 2 / 3 1/ 2
V R So
n
5/3
1 A
1/ 2
Q
S
2/3 o
nP
Then:
This is known as the Manning equation
This equation can be applied to any
channel shape So
n values have been developed for a range of situations
** APPLICATION OF MANNINGS EQUATION **

- Most applications of Mannings Eq
are w.r.t. steady uniform flow
& therefore quite straight-forward
However, several points are worth considering:
Content


COMPOUND CHANNELS
WORKED EXAMPLE
Velocity Distributions
In Earlier Lectures:
when using velocity to calculate pipe flows :
we assumed an average velocity
despite the actual velocity distribution being quite complex
In the case of a circular pipeline:

the velocity distribution is complex
but relatively predictable
(i.e. being of a symmetrical form)
Open channel flow,

with its free surface, is much less predictable.
In a rivers cross-section
where would you find
the max velocities?
Unpredictability of open channel flow
It might be assumed that
that max velocity would be at the surface
i.e. due to lack of significant shear forces across the free surface
But this, is not the case:

the surface velocity affected by secondary currents
that circulate from the channel sides
The actual velocity distribution is likely to be

of the form shown below.
Content


COMPOUND CHANNELS
WORKED EXAMPLE
Compound Channels
Some open channels:
may vary considerably across its section
For example:
where a river overflows to a contained flood plain.
For compound Channels

Figure needed
divide the River cross-section into several sub-sections

and apply different Mannings n values to each.
NB:
the exact liquid interface between different sections is fairly arbitrary

as shear stresses between them are very small compared
to the shear stresses at the channel boundaries
Compound Channels - REVISION
In Lecture 1 you looked at :
coefficient
the energy coefficient &
the momentum
used with the:
energy equation:
& momentum equation:
P V 2
z constant
g 2 g
F Q (V2 x V1x )
to account for irregular velocity distributions across a channels section

x
e.g. For turbulent pipe flow:
(energy coefficient) =1.06
(momentum coefficient) = 1.02
For turbulent flow in regular channel :

(energy coefficient) rarely exceeds 1.15
(momentum coefficient) rarely exceeds 1.05
And hence generally ignored (considered to =1)

Further details Section 2.4 Hydraulics in Civil & Environmental Eng, Chadwick & Morfett
BUT for irregular channels:

the value of the energy coefficient
cant be assumed to approx = 1
It may well approach (or even exceed) 2

& must be included in when using the energy equation
(Bernoulli equation)
For compound channels
To calculate
(overall channel section)
Assume that for each smaller section of channel equals 1

Then:
(overall channel section)
Where :
V13 A1 V23 A2 V33 A3

3
V ( A1 A2 A3 )
Q V1 A1 V2 A2 V3 A3
V
A
A1 A2 A3
Content


COMPOUND CHANNELS
WORKED EXAMPLE
Partially Full Circular Pipes

Another common problem is:
when circular pipes run partially full, thus
acting as an open channel rather than
as a pipeline
Consider a full pipe with: Velocity Vfull

Discharge Qfull
It is worth noting:
1.) At a proportional depth of 0.5:
V0.5 = Vfull
Q0.5 = 0.5 Qfull
2.) At proportional depths slightly below D:

both proportional velocity & discharge
Are higher than Qfull & Vfull
Partially Full Circular Pipes
The depth of flow is referred to as the proportional depth, d

Hence,
d is expressed as the ratio d / D
Content

FLOW STATE AND CLASSIFICATION

COMPOUND CHANNELS
WORKED EXAMPLE
LAB (Some pointers)
* Further Study:
Complete Tutorial Sheet Number 8.
Tutorial Sheet Number 8

Question 1)
Open channel of trapezoidal section:
is 2.5m wide at base &
slides inclined at 60 degrees to horizontal &
bed slope of 1 in 500
It is found that when :
the flow is 1.24 m3/s the corresponding
water depth (in the channel) is 350mm
a.) Is the flow laminar or turbulent?

(take coeff of viscosity as 1.14 x 10-3 kg/ms)
b.) Assuming the validity of Mannings equation,
calculate the flow rate when the depth is 500mm
a.) Is the flow laminar or turbulent? (NEED TO CHECK REYNOLDS No)
We know that for open channels Reynolds Number is given by:
Re Channel
Where:
RV
R is the hydraulic radius

V is velocity
is density of water (1000 kgm3)
And, for open channels:

Laminar flow : Re(channel) < 500
Turbulent flow : Re(channel) > 1000
(In practice, the lower limit for turbulent
flow is taken as Re > 2000)
Open channel of trapezoidal section:

is 2.5m wide at base &
slides inclined at 60 degrees to horizontal &
bed slope of 1 in 500
When: the flow is 1.24 m3/s the corresponding
water depth (in the channel) is 350mm
a.) Is the flow laminar or turbulent?
(take coeff of viscosity as 1.14 x 10-3 kg/ms)
Re Channel
What terms do we already know?
RV
is density of water = 1000 kgm3

is absolute coeff of viscosity = 1.14 x 10-3 kg/ms
So we need to find values for:

R=
V=
hydraulic radius
Velocity
We know Hydraulic Radius R = A / P

So we must consider the geometry of trapezoidal section
i.e. in order to calculate A and P
x
350 mm
30o
y
60o
2.5 m
Tan (30o) = opp / adj

= x / 0.35
x = 0.35 * Tan (30o)

= 0.2 m
Sin (30o) = opp / hyp

=
x/ y
y = 0.2 / Sin (30o)

= 0.4 m
To find R (hydraulic radius)

2.9 m
0.2m
60o
350 mm
m
0.4
30o
2.5 m
A=
We know
Hydraulic Radius R = A / P
2.5 2.9 0.35 0.945m 2

2
P = 2.5 + (2 x 0.4) = 3.3 m

So
R = A / P = 0.945 / 3.3 =
0.286 m
Re Channel
RV
is density of water = 1000 kgm3

is the absolute coeff of viscosity = 1.14 x 10-3 kg/ms

R=
V=
hydraulic radius = 0.286 m

Velocity
To find Velocity
0.2m
0.4
350 mm
A = 0.945 m2
m
2.5 m
We know that when flow is 1.24 m3/s

the corresponding water depth in the channel is 350mm
Q = AV
V=Q/A
V = 1.24 / 0.945
V = 1.312 m /s
Re Channel
RV
We now know:
is density of water
V is Velocity
Re Channel
=
=
=
=
1000 kgm3
1.14 x 10-3 kg/ms
0.286 m
1.312 m/s
1000 0.286 1.312
329150
3
1.14 10
(In practice, the lower limit for turbulent flow is taken as Re > 2000)
Open channel of trapezoidal section

is 2.5m wide at base and slides inclined at 60 degrees to horizontal
and has bed slope of 1 in 500
It is found that when the flow is 1.24 m3/s
the corresponding water depth in the channel is 350mm
calculate the flow rate (Q) when the depth is 500 mm
We need to use the

Manning equation:
5/3
1 A
1/ 2
Q
S
2/3 o
nP
So = bed slope = 1 in 500

n=
Mannings coeff
A (Area) & P (Wetted perimeter)
(N.B. due to new depth of 0.5m instead of 0.35m now different channel geometry)
To establish a value for Mannings coeff, n
We know from lecture:
1 2 / 3 1/ 2
V R So
n
Rearrange to calculate mannings number
1 2 / 3 1/ 2
n R So
V
To establish a value for Mannings coeff, n
1 2 / 3 1/ 2
n R So
V
We know:
So (bed slope) = 1 in 500 = 0.002

And for the previous part of the question we had calculated:
V = 1.312 m/s
R = 0.286 m
NOTE:
We are assuming that mannings coeff has not changed
(i.e. we are calculating n for a channel flow depth of 0.35m
And will reapply it to the same channel with a new flow depth of 0.5m
Plug in values
1
2/3
1/ 2
0.286 0.002
n
1.312
n = 0.01483

We need to use the

Manning equation:
5/3
1 A
1/ 2
Q
S
2/3 o
nP
n=
Mannings coeff = 0.01483
A (Area) & P (Wetted perimeter)

(N.B. due different depth there is now different channel geometry)
To find new values for A & P

Recalculate channel geometry for a flow depth of 500mm
we need to reconsider the geometry of trapezoidal section
x
60o
500 mm
30o
2.5 m
From trig:
Tan (30o) = x / 0.5
Sin (30o) = x / y
x = 0.5 * Tan (30o)

= 0.286 m
y = 0.286 / Sin (30o)
= 0.571 m
To find R (hydraulic radius)

3.071 m
0.286m
60o
500 mm
m
71
0.5
30o
2.5 m
We know
Hydraulic Radius R = A / P
A = 2.5 3.071 0.5 1.393m 2
P = 2.5 + (2 x 0.571) = 3.642 m

We need to use the

Manning equation:
5/3
1 A
1/ 2
Q
S
2/3 o
nP
we now know:
n=
Mannings coeff = 0.01483
A (Area) = 1.393 m2
P (Wetted perimeter) = 3.642 m
1
1.393
1/ 2
3
0.002 2.2m / s
Q
2/3
0.01483 3.642
5/3

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Hydraulics:

Fundamentals of open channel flow

OPEN CHANNELS: INTRO & BASIC DEFINITIONS

CHEZY & MANNING EQUATIONS

OPEN CHANNELS: INTRODUCTION:

Open channel flow:

OPEN CHANNELS: BASIC DEFINITIONS

OPEN CHANNELS: BASIC DEFINITIONS

The vertical distance from free surface to

Vertical distance from the free surface to an arbitrary datum

OPEN CHANNELS: BASIC DEFINITIONS

Wetted Perimeter P : The length of the wetted channel surface

OPEN CHANNELS: BASIC DEFINITIONS

Hydraulic Mean Depth

OPEN CHANNELS: INTRO & BASIC DEFINITIONS

REYNOLDS No, LAMINAR & TURBULENT FLOW

FLOW STATE - REVISION

Where Reynolds Number, Re, for pipe flow

FLOW STATE - REVISION

Laminar flow in a pipe:

Turbulent flow in a pipe:

FLOW STATE - REVISION

FLOW CLASSIFICATION OPEN CHANNEL

Steady, uniform flow

Steady non-uniform flow

FLOW CLASSIFICATION OPEN CHANNEL

2.) Unsteady Uniform

3.) Steady Non-uniform

4.) Unsteady Non-Uniform Flow:

FLOW CLASSIFICATION OPEN CHANNEL

FLOW CLASSIFICATION OPEN CHANNEL

FLOW CLASSIFICATION OPEN CHANNEL

FLOW CLASSIFICATION OPEN CHANNEL

FLOW CLASSIFICATION OPEN CHANNEL

The Severn Bore. http://www.youtube.com/watch?v=Ol9ogW-4bPQ&feature=fvw

FLOW CLASSIFICATION SUMMARY

Steady Uniform Flow:

Steady Non-uniform Flow:

Unsteady Non-uniform Flow:

Unsteady Uniform Flow:

OPEN CHANNELS: INTRO & BASIC DEFINITIONS

REYNOLDS No, LAMINAR & TURBULENT FLOW

REYNOLDS No, LAMINAR & TURBULENT FLOW

To calculate Reynolds Number for open channels:

R is the hydraulic radius

LAMINAR & TURBULENT FLOW

Hence, for open channels:

OPEN CHANNELS: INTRO & BASIC DEFINITIONS

REYNOLDS No, LAMINAR & TURBULENT FLOW

FOR STEADY UNIFORM FLOW

FOR STEADY UNIFORM FLOW

Due to the effect that:

FOR STEADY UNIFORM FLOW

THE CHEZY EQUATION

Many equations have been developed to

The most significant approach is that derived by

which depends on surface roughness & channel alignment

Mannings Equation cont.

Mannings Equation cont.

APPLICATION OF MANNINGS EQUATION