FMHM-4th UNIT-R20-JNTUA.pptx

Fluid Mechanics and Hydraulic
Machines
4th Unit
Flow in Open Channels
(Common to Civil & Mechanical)
Course Code :20A01302T
Mr.P.MADHU RAGHAVA
Assistant Professor
SVREC Engineering College
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1

UNIT - IV Flow in Open Channels
 Open Channel Flow-Comparison between open channel flow and
pipe flow,
 geometrical parameters of a channel,
 classification of open channels, classification of open channel
flow,
 Velocity Distribution of channel section.
 Uniform Flow-Continuity Equation,
 Energy Equation and Momentum Equation,
 Characteristics of uniform flow,
 Chezy’s formula, Manning’s formula.
 Computation of Uniform flow.
 Specific energy, critical flow, discharge curve,
 Specific force, Specific depth, and Critical depth.
 Measurement of Discharge and Velocity – Broad Crested Weir.
 Gradually Varied Flow Dynamic Equation of Gradually Varied Flow.
 Hydraulic Jump and classification - Elements and characteristics-
Energy dissipation.
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Flow-Comparison between
OPEN CHANNEL FLOW
• Open channel flow has a
free water surface
• Open channel flow is
subjected to atmospheric
pressure
• Open channel flow is not
completely enclosed by
boundaries,
• Open channel is always
under the action of gravity
AND PIPE FLOW
• opposite
• while pipe flow is not (when
pipe is full).
• unlike pipe flow.
• , while pipe can be under
gravity or may flow due to
some external pressure.

Aspects Open channel flow Pipe flow
Condition
It is usually uncovered, has a free surface
at the top and atmospheric pressure at a
free surface.
It is covered and has no free
surface.
Cross-section
It may have any shape. E.g. rectangular,
parabolic, triangular, trapezoidal, circular,
irregular etc.
Generally, pipe flow has a circular
cross-section.
Cause of flow
Flow is due to gravity in open channel
flow.
Flow is due to pressure in pipe flow.
Surface
roughness
Its roughness varies between wide limits
and also varies from place to place.
Its roughness depends upon the
material of the pipe.
Velocity
distribution
Velocity is maximum at a little distance
below the water surface in open channel
flow. The shape of the velocity profile of
the open channel depends on the
channel roughness.
Velocity is maximum at the center
of the flow and reduces to zero at
the pipe wall. Velocity distribution is
symmetrical about the pipe axis in
pipe flow.
Piezometric
head
Z+y where y = depth of flow; HGL
coincides with the water surface in open
channel flow.
Z+P/y, where p= pressure in the
pipe. HGL does not coincide with
the water surface in pipe flow.
Surface head Negligible Dominant for small diameter
3. Difference between open channel flow and pipe flow

Flow in Open Channels
• INTRODUCTION
• A channel may be defined as a passage through which
water flows under atmospheric pressure.
• As such in channels the flow of water takes place with a
free surface which is subjected to atmospheric pressure.
• Figure 15.1 shows a comparison between the flow through
a pipe and a channel.
• In the case of flow through a pipe there is no free surface
as in the case of flow through a channel.
• This is so because the water flowing in a pipe is under
pressure, which at any section of a pipe is indicated by the
level of water column in a piezometric tube installed on the
pipe at the section.

• The water levels in the piezometric tubes installed at
different sections of a pipe indicate the hydraulic grade
line.
• On the other hand in the case of channel flow the water
surface itself is the hydraulic grade line.
• However, the total energy lines in both the cases lie at a
distance of (V2/2g)
• above the hydraulic grade line at every section, where V is
the mean velocity of flow at the section.
• Since the pressure on the top surface of water in a channel
is constant, no pressure difference can be built up between
any two sections along the channel as in the case of pipes.

• Moreover, as the water flows in a channel, it
will always be subjected to some resistance.
• As such in order to overcome the resistance
and to cause the flow of water in a channel, it
is constructed with its bottom slopping
towards the direction of flow,
• so that a component of the weight of the
flowing water in the direction of flow is
developed which causes the flow of water in
the channel.

• The channels may be classified according to
different considerations as described below.
• On the basis of the cross-sectional form of the
channel
• it may be classified as natural channel or
artificial channel.
• A natural channel is the one which has irregular
sections of varying shapes, which is developed
in a natural way.
• The examples of natural channels are rivers,
streams etc.

• On the other hand an artificial channel is the one which is
built artificially for carrying water for various purposes.
• Obviously artificial channels have their cross-sections
with regular geometrical shapes, which usually remain
same throughout the length of the channel.
• The artificial channels may be further classified
according to the shape of the cross-section as,
 rectangular channel,
 trapezoidal channel,
 triangular channel,
 parabolic channel, and
 circular channel.

• The channels may also be classified as open channels and
closed channels.
• The channels without any cover at the top are known as
open channels.
• On the other hand the channels having cover at the top are
known as closed channels.
• However, the closed channels will always be running partly
full of
• water, in order that the flow may be a channel flow with
atmospheric pressure prevailing over its entire top surface.
• Some of the common examples of closed channels are
• closed conduits or pipes flowing partly full of water,
• underground drains,
• tunnels etc., not running full of water.

• A channel having the same shape of various
sections along its length and laid on a constant
bottom slope is known as prismatic channel,
otherwise the channel is non-prismatic.
• In this chapter only prismatic channels have
been dealt with.

TYPES OF FLOW IN CHANNLES
• The flow in channels can be classified into
following types depending upon the change in
the depth of flow with respect to space and
time.

(a) Steady Flow and Unsteady Flow.
• Flow in a channel is said to be steady if the flow
characteristics at any point do not change with time
that is (∂V/∂t)= 0, (∂y/∂t) = 0 etc.
• However, in the case of prismatic channels the
conditions of steady flow may be obtained if only the
depth of flow does not change with time, that is
(∂y/∂t) = 0.
• On the other hand if any of the flow characteristics
changes with time the flow is unsteady.
• Most of the open channel problems involve the study
of flow under steady conditions and therefore in this
chapter only steady flow has been considered.

(b) Uniform and Non-uniform (or Varied) Flow.
• Flow in a channel is said to be uniform if the
depth, slope, cross-section and velocity remain
constant over a given length of the channel.
• Obviously, a uniform flow can occur only in a
prismatic channel in which the flow will be
uniform if only the depth of flow y is same at
every section of the channel, that is (∂y/∂s) = 0.
• Flow in channels is termed as non-uniform or
varied if the depth of flow y, changes from
section to section, along the length of the
channel, that is (∂y/∂s) is not equal to zero.

• Varied flow may be further classified as rapidly
varied flow (R.V.F.) and gradually varied flow
(G.V.F.).
• If the depth of flow changes abruptly over a
comparatively short distance, the flow is
characterized as a rapidly varied flow.
• Typical examples of rapidly varied flow are
hydraulic jump and hydraulic drop.
• In a gradually varied flow the change in the
depth of flow takes place gradually in a long
reach of the channel.

(c) Laminar Flow and Turbulent Flow.
• Just as in pipes, the flow in channels may also be characterized as
laminar, turbulent or in a transitional state, depending on the
relative effect of viscous and inertia forces and alike pipes, Reynolds
number Re is a measure of this effect in channel flow also.
• However, the Reynolds number for flow in channel is commonly
defined as
• where V is the mean velocity of flow,
R is the hydraulic radius (or hydraulic mean depth) of the channel
section
• and ρ and μ are respectively the mass density and absolute
viscosity of water.

• On the basis of the experimental data it has
been found that up to Re equal to 500 to 600
the flow in channels may be considered to be
laminar and for Re greater than 2000 the flow
in channels is turbulent.
• Thus for Re in between 500 to 2000 the flow
in channels may be considered to be in
transitional state.
• Since most of the channel flows are turbulent,
in this chapter only the turbulent flow has
been dealt with.

(d) Subcritical Flow, Critical Flow and
Supercritical Flow.
• As stated earlier gravity is a predominant force in the
case of channel flow.
• As such depending on the relative effect of gravity and
inertia forces the channel flow may be designated as
subcritical, critical or supercritical.
• The ratio of the inertia and the gravity forces is
another dimensionless parameter called Froude
number Fr which is defined as Fr=V gD ;
• where V is the mean velocity of flow,
• g is acceleration due to gravity and
• D is hydraulic depth of channel section (as defined in
Section 15.3).

• As indicated later, when Fr = 1, that is V = gD ,
• The channel flow is said to be in a critical state.
• If Fr < 1, or V < gD , the flow is described as
subcritical or tranquil or streaming.
• If Fr >1, or V > gD , the flow is said to be
supercritical or rapid or shooting or torrential.

15.3 GEOMETRICAL PROPERTIES OF
CHANNEL SECTION
• The geometrical properties of a channel section can be defined
entirely by the geometry of the section and the depth of flow.
• Some of the geometrical properties of basic importance are defined
below.
• The depth of flow y, is the vertical distance of the lowest point of a
channel section from the free surface.
• The top width T is the width of the channel section at the free
surface.
• The wetted area A (or water area or area of flow section) is the
cross-sectional area of the flow normal to the direction of flow.
• The wetted perimeter P is the length of the channel boundary in
contact with the flowing water at any section.

VELOCITY DISTRIBUTION IN A CHANNEL
SECTION
• The velocity of flow at any channel section is not uniformly
distributed.
• The non-uniform distribution of velocity in an open channel
is due to the presence of a free surface and the frictional
resistance along the channel boundary.
• The velocity distribution in a channel is measured either
with the help of a PITOT tube or a current meter, described
later.
• The general patterns for velocity distribution as
represented by lines of equal velocity, in some of the
common channel sections are illustrated in Fig. 15.2.
• A velocity distribution curve along a vertical line of the
channel section is also shown in Fig. 15.2.

• In a straight reach of a channel maximum
velocity usually occurs below the free surface at
a distance of 0.05 to 0.15 of the depth of flow.
• The velocity distribution in a channel section
depends on the various factors such as the
shape of the section, the roughness of the
channel and the presence of bends in the
channel alignment.

• The mean velocity of flow in a channel section
can be computed from the vertical velocity
distribution
• curve obtained by actual measurements.
• It is observed that the velocity at 0.6 depth
from the free surface is very close to the mean
velocity of flow in the vertical section.
• A still better approximation for the mean
velocity of flow is obtained by taking the
average of the velocities measured at 0.2
depth and 0.8 depth from the free surface.

• On account of non-uniform distribution of
velocities over a channel section, correction
factors will have to be applied while computing
the kinetic energy and the momentum.
• The kinetic energy correction factor α (Greek
‘alpha’) is also known as CORIOLIS coefficient in
honour of G.
• CORIOLIS who first proposed it.
• The momentum correction factor β (Greek
‘beeta’) is also called Boussinesq coefficient
after J. Boussinesq who first proposed it.

• As mentioned in previous chapters the values
of these correction factors can be obtained if
the actual velocity profile at a section is
known.
• Experimental data indicate that for turbulent
flow in fairly straight prismatic channels, the
value of α varies from 1.03 to 1.36 and that of
β from about 1.01 to 1.12.
• However, for the sake of simplicity, both α and
β are assumed to be unity in the present
analysis.

UNIFORM FLOW IN CHANNLES
• When water flows in an open channel
resistance is offered to it, which results in
causing a loss of energy.
• The resistance encountered by the flowing
water is generally counteracted by the
components of gravity forces acting on the
body of the water in the direction of motion
[Fig. 15.3 (b)].

A uniform flow will be developed if the
resistance is balanced by the gravity forces.
The magnitude of the resistance, when other
physical factors of the channel are kept
unchanged, depends on the velocity of flow.
When water enters the channel, the velocity and
hence the resistance are smaller than the gravity
forces, which results in an accelerating flow in
the upstream reach of the channel.
The velocity and the resistance increase
gradually until a balance between the resistance
and gravity forces is reached.

From this point onwards the flow becomes
uniform.
Several uniform-flow formulae have been
developed which correlate the mean velocity
of uniform flow in open channels with the
hydraulic radius, energy
line slope and a factor of flow resistance.
The most widely used uniform-flow formulae
are the Chezy and Manning formulae which
are described in the following paragraphs

• The main features of uniform flow in a
channel can be summarized as follows:
• (1) The depth of flow, wetted area, velocity of
flow and discharge are constant at every
section along the channel reach.
• (2) The total energy line, water surface and
the channel bottom are all parallel, that is,
their slopes are all equal, or Sf = Sw = S0 = S,
• where Sf is energy line slope, Sw is water
surface slope and S0 is channel bottom slope.

• The depth of a uniform flow is called the normal depth and
it is generally represented by yn.
• The fundamental equation for uniform flow in channels
may be derived by applying Newton’s second law of
motion.
• In uniform flow since the velocity of flow does not change
along the length of the channel, there is no acceleration.
• Hence the sum of the components of all the external
forces in the
direction of flow must be equal to zero.
• Consider a short reach of channel of length L in which
uniform
• flow occurs, as shown in Fig. 15.3 (b).

The forces acting on the free-body of water
ABCD in the direction of flow are as follows:
•
• (1) The forces of hydrostatic water pressure F1 and F2 acting on the
two ends of the free body.
• As the depths of water at these two sections are the same, the
forces F1 and F2 are equal and hence they cancel each other.
• (2) The component of weight of the water in the direction of flow,
which is (wAL sin θ), where w is specific weight of water, A is the
wetted cross-sectional area of channel and θ is the angle of
inclination of the channel bottom with the horizontal.
• (3) The resistance to the flow is exerted by the wetted surface of
the channel.
• If P is the wetted perimeter of the channel and τ0 is the average
shear stress at the channel boundary, the total resistance to flow
will be (PLτ0).

• According to Newton’s second law of motion, in
the direction of flow

• Equation 15.2 is known as Chezy’s formula named after the French
Engineer Antoine Chezy who developed this formula in 1775.
• The term C is known as Chezy’s coefficient.
• It may, however, be observed that the Chezy’s coefficient C varies
inversely as the square root of the Darcy-Weisbach resistance
coefficient f.
• Further it may be noted that Chezy’s coefficient is not a
dimensionless coefficient but it has the dimensions [L1/2 T–1].
• Although Chezy’s equation is quite simple the selection of a correct
value of C is rather difficult.
• As such many empirical formulae have been developed to determine
• the value of Chezy’s C.
• Some of the important formulae developed for this purpose are as
noted below.

a) The Ganguillet-Kutter Formula.
• On the basis of a series of flow measurements taken in
channels of various types, two Swiss engineers Ganguillet and
Kutter proposed an empirical formula in 1869,expressing the
value of Chezy’s C in terms of the slope S, hydraulic radius R
and a roughness coefficient n.

In SI or metric system the formula is
The roughness coefficient n is known as Kutter’s n.
The value of n varies widely depending upon the channel surface
and its condition.
Some typical values of n for different types of surfaces commonly
encountered in practice are given in Table 15.3.

(b) The Bazin Formula.
• In 1897, the French hydraulician H. Bazin
proposed another empirical formula for
Chezy’s C.
• According to this formula Chezy’s C is
expressed in terms of hydraulic radius R and
another roughness, factor m, known as Bazin’s
roughness coefficient.
• In metric system Bazin’s formula may be
expressed as

• The manning formula :In 1889 an Irish
engineer Robert Manning presented a formula
according to which the mean velocity V of
uniform flow in a channel is expressed in terms
of a coefficient of roughness n, called
Manning’s n, hydraulic radius R and channel
bottom slope S.
• The Manning’s formula expressed in SI or
metric units is written as

• Owing to its simplicity of form and to the
satisfactory result it yields in practice, the
Manning formula has now becomes the most
widely used of all the empirical formula for the
computation of uniform flow in open channels.
• If the Manning formula is compared with the
Chezy formula it can be seen that

• The Eq. 15.6 provides an important
relationship between Chezy’s C and Manning’s
n.
• The values of Manning’s n are found to be
approximately equal to those of Kutter’s n for
the normal ranges of slope and hydraulic
radius.
• Some of the typical values which hold good
for both Kutter’s n and Manning’s n are given
in Table 15.3.

Continuity Equation Derivation
• Continuity equation represents that the product of
cross-sectional area of the pipe and the fluid speed at
any point along the pipe is always constant.
• This product is equal to the volume flow per second or
simply the flow rate.
• The continuity equation is given as:
• R = A v = constant
• where,
• R is the volume flow rate
• A is the flow area
• v is the flow velocity

Assumption of Continuity Equation
• Following are the assumptions of continuity
equation:
• The tube is having a single entry and single
exit
• The fluid flowing in the tube is non-viscous
• The flow is incompressible
• The fluid flow is steady

Derivation

• now, consider the fluid flows for a short interval of
time in the tube.
• So, assume that short interval of time as Δt.
• In this time, the fluid will cover a distance of Δx1 with a
velocity v1 at the lower end of the pipe.
• At this time, the distance covered by the fluid will be:
• Δx1 = v1Δt
• Now, at the lower end of the pipe, the volume of the
fluid that will flow into the pipe will be:
• V = A1 Δx1 = A1 v1 Δt

• It is known that mass (m) = Density (ρ) × Volume (V).
So, the mass of the fluid in Δx1 region will be:
• Δm1= Density × Volume
• => Δm1 = ρ1A1v1Δt ——–(Equation 1)
• Now, the mass flux has to be calculated at the lower end.
• Mass flux is simply defined as the mass of the fluid per unit time passing through
any cross-sectional area.
• For the lower end with cross-sectional area A1, mass flux will be:
• Δm1/Δt = ρ1A1v1 ——–(Equation 2)
• Similarly, the mass flux at the upper end will be:
• Δm2/Δt = ρ2A2v2 ——–(Equation 3)

• Here, v2 is the velocity of the fluid through the upper end of
the pipe i.e.
• Through Δx2 , in Δt time and A2, is the cross-sectional area
of the upper end.
• In this, the density of the fluid between the lower end of
the pipe and the upper end of the pipe remains the same
with time as the flow is steady.
• So, the mass flux at the lower end of the pipe is equal to
the mass flux at the upper end of the pipe i.e. Equation 2 =
Equation 3.
• Thus,
• ρ1A1v1 = ρ2A2v2 ——–(Equation 4)

• This can be written in a more general form as:
• ρ A v = constant
• The equation proves the law of conservation of mass in fluid
dynamics.
• Also, if the fluid is incompressible, the density will remain constant
for steady flow. So, ρ1 =ρ2.
• Thus, Equation 4 can be now written as:
• A1 v1 = A2 v2
• This equation can be written in general form as:
• A v = constant
• Now, if R is the volume flow rate, the above equation can be
expressed as:
• R = A v = constant
• This was the derivation of continuity equation.

COMPUTATION OF UNIFORM FLOW

• Equation 15.34 applies to a channel section when the flow is uniform.
• The right hand side of this equation contains the terms n, Q and S, but the
left hand side contains the terms which depend only on the geometry of
the wetted area.
• Therefore it indicates that for given values of n, Q and S, there is only one
possible depth of flow at which the uniform flow will be maintained in any
channel section, provided the value (AR2/3) always increases with increase
in depth, which is true in most of the cases.
• This depth of flow is known as normal depth which is represented by yn.
Similarly if n, S and the depth of flow are given for a channel section, it can
be seen from Eq. 15.34 that there can be only one discharge for
maintaining a uniform flow through the channel section.

• This discharge is known as normal discharge,
which is represented by Qn.
• Furthermore when Q, n and depth of flow are
given for a channel section, Eq. 15.34 can be used
to determine the bottom slope which will
maintain the uniform
• flow at a given depth of flow y in the channel.
• The bottom slope thus determined is known as
normal slope which is represented by Sn.
• Equation 15.34 is quite useful for computation
and analysis of uniform flow.

SPECIFIC ENERGY AND CRITICAL
DEPTH
• Since the free surface in the case of a channel
flow represents the hydraulic gradient, the
Bernoulli’s equation applied between the
sections (1) and (2) L distance apart may be
written as

• It can be seen from Eq. 15.35 that, for a given channel section and
discharge Q, the specific energy is a function of the depth of flow only.
Thus for a given channel section and discharge, Eq. 15.35 may be
• represented graphically in which specific energy is plotted against the
depth of flow.
• The curve so obtained is known as specific energy curve and is shown in
Fig. 15.9. This curve has two limbs AC and BC.
• The lower limb AC approaches the horizontal axis (or the specific energy
axis) asymptotically towards the right.
• The upper limb BC approaches asymptotically to the line OD which passes
through the origin and has an angle of inclination equal to 45°.
• At any point P on this curve the ordinate represents the depth of flow and
the abscissa represents the specific energy, which is equal to the sum
• of the depth of flow and the velocity head.

• It can be seen from the specific energy curve that, there is one
point C on the curve which has a minimum specific energy, thereby
indicating that below this value of the specific energy the given
discharge cannot occur.
• The depth of flow at which the specific energy is minimum is called
critical depth yc.
• Similarly the velocity of flow at the critical depth is known as critical
velocity Vc.
• For any other value of the specific energy there are two possible
depths, one greater than the critical depth and the
• other smaller than the critical depth, at which a given discharge can
occur with the same specific energy.
• These two depths for given specific energy are called the alternate
depths–y1 smaller than yc and y2 greater than yc as shown in Fig.
15.9.

• Apparently at point C the two alternate depths become one,
• equal to critical depth yc.
• For any depth of flow greater than the critical depth, the
specific energy increases with the increase in the depth.
• Moreover when the depth of flow is greater than the critical
• depth, the velocity of flow is less than the critical velocity for
the given discharge.
• Hence the flow at the depths greater than the critical depth is
known as subcritical flow or tranquil flow.
• On the other hand when the depth of flow is less than the
critical depth, the specific energy increases as the depth of
flow decreases.

• Furthermore for given discharge when the depth
of flow is less than critical depth, the velocity of
flow is greater than the critical velocity.
• Hence the flow at the depths less than the critical
depth is known as supercritical flow or rapid flow.
• As shown in Fig. 15.9 y1 is the depth of a
supercritical flow, and y2 is the depth of a
subcritical flow.

Manning’s formula.
One the most commonly used equations
governing Open Channel Flow is known as the
Mannings’s Equation.
• It was introduced by the Irish Engineer Robert
Manning in 1889 as an alternative to the Chezy
Equation.
• The Mannings equation is an empirical equation
that applies to uniform flow in open channels and
is a function of the channel velocity, flow area
and channel slope.

• Under the assumption of uniform flow conditions the
bottom slope is the same as the slope of the energy
grade line and the water surface slope.
• The Manning’s n is a coefficient which represents the
roughness or friction applied to the flow by the
channel.
• Manning’s n-values are often selected from tables, but
can be back calculated from field measurements.
• In many flow conditions the selection of a Manning’s
roughness coefficient can greatly affect computational
results.

MANNING FORMULA FOR DETERMINING OPEN
CHANNEL FLOWS
• One way of calculating open channel flow in
pipes and channels without using a flume or
weir is the Manning Formula.
• Although not as accurate as a hydraulic
structure, the formula can provide a sufficient
level of accuracy in some applications.

Measurement Conditions
• For best results with when applying the Manning
formula:
• The channel should be straight for at least 200
feet (and preferrably 1,000 feet)
• The channel should be uniform in cross-section,
slope, and roughness
• There sould be no rapids, dips, sudden
contractions / expansions, or tributary flows
• The flow should not backup or be submerged

Manning Formula
• The Manning formula uses water surface
slope, cross-sectional area, and wetted
perimeter of a length of uniform channel to
determine the flow rate.

• The cross-section area (A) and the hydraulic radius (R)
are calculated for the given depth of the liquid in the
channel at the moment of measurement (and not at
some arbitrary maximum or minimum condition).
• The slope (S) often has to be estimated based upon
previous installation drawings of the channel or pipe,
but true site measurements will provide a more
accurate flow rate.
• The roughness coefficient (n) is selected from standard
reference roughnesses based upon the channel / pipe
material and its condition.

Roughness Coefficients (n)
• Nummerous n-values have been calculated for a variety of streams,
channels, and pipes. Usuallly the values are given as a range (minimum -
normal - maximum) for a particular channel type or material.
• There are numerous factors that affect n-values, including:
• Surface roughness
• Vegetation
• Silting / scouring
• Obstruction
• Size / shape of channel
• Seasonal change
• Suspended material
• Bed load
• Stage (depth of flow)
• Discharge

• Tables for various n-values can be found
online at the Corvallis Forestry Research
Community site (from Chow, 1959 and AISI,
1980). A guide to selecting Manning's
roughness coefficients for natural channels
and flow plains is available from the U.S.
Department of Transportation here.

Chezy’s formula
• The Chezy’s formula was given by the French
hydrologist Antonie Chezy.
• He related the discharge (Q) of the water flowing
through an open channel with the respective
channel dimensions and slope of the bed.
• The Chezy's formula for determination of
discharge of water in open channels and methods
to determine the Chezy's constant are briefly
explained in this article.

Accuracy
• Under ideal conditons, the Manning formula
can acheive accuracies of +/- 10-20%.
• However, variances in the above
measurement conditons means that
accuracies of +/- 25-30% are more likely, with
errors of 50% or more possible if care is not
taken.

Chezy’s Formula for Open Channel
Flow
• For a steady uniform flow of liquid through an open
channel, the rate of flow of fluid or the discharge is
given by the formula:
This equation is called as Chezy’s Formula. Where, A is the area of flow of water,
m is the hydraulic mean depth or hydraulic radius, ‘i' is the slope of the bed and
‘C’ is the Chezy’s Constant.
In the above equation, the velocity is given by,
If ‘P’ is the wetter perimeter of the cross-section, then hydraulic mean depth
or hydraulic radius is given by the formula, m = A/P;

• The chezy formula is directly related to the:
• Frictional Coefficient (F)
• Reynolds Number
• Hydraulic Mean Depth

Determination of Chezy’s Constant by
Empirical Formula
• The chezy’s constant ‘C’ is a dimensionless
quantity which can be calculated by three
formulas, namely:
• Bazin Formula
• Ganguillet -Kutter Formula
• Manning’s Formula

1. Bazin Formula

2. Ganguillet -Kutter Formula

Manning’s Formula

APPLICATION OF SPECIFIC ENERGY AND DISCHARGE DIAGRAMS
TO CHANNEL TRANSITIONS
• In the case of long channels often it becomes necessary to
provide transitions.
• A transition is the portion of a channel with varying cross-
section, which connects one uniform channel to another,
which may or may not have the same cross-sectional form.
• The variation of a channel section may be caused either by
reducing or increasing the width or by raising or lowering
the bottom of the channel.
• Various channel transitions may be broadly classified as
sudden transitions and gradual transitions.
• Sudden transitions are those in which the change of cross-
sectional dimensions occur in a relatively short length.

• On the other hand in the case of a gradual
transition the change of cross-sectional area
takes place gradually in a relatively long length of
the channel.
• Some of the functions which channel transitions
are made to serve are metering of flow,
dissipation of energy, reduction or increase of
velocities, change in channel section or alignment
with a minimum of energy dissipation etc.

• Some of the devices commonly used for measuring
discharge through channels are venturi flume, standing
wave flume and Parshall flume which are provided with
transition as described later.
• The dissipation of energy may be caused by providing a
sudden drop in the channel bottom.
• In the case of irrigation channels, in order to prevent
scouring, velocities are required to be reduced, while for
navigation channels in order to prevent shoaling higher
velocities are required to be developed.
• Such reduction or increase of velocities may be made
possible by expansion or contraction of channel section.

• The change in channel section with a minimum of
energy dissipation is obtained by providing a
gradual expansion or a gradual contraction with
well rounded corners.
• Apart from the above mentioned transitions,
often channel cross-section may be reduced if a
bridge is constructed across it and bridge piers
are constructed.
• Some of the channel transitions described above
are shown in Fig. 15.12.

• For a rectangular channel, transitions of decreasing
channel cross-section may be analysed with the help of
specific energy and discharge diagrams.
• The decrease in cross-section or a channel contraction
may be obtained either by a decrease in width or by a
decrease in depth or by a combination of both.
• In the following paragraphs transitions for rectangular
channels with reduced width and raised bottom have
been discussed separately.
• The combination can, however, be analysed by using
the same principles in combination.

Gradually Varied Flow Dynamic
Equation of Gradually Varied Flow
• As explained in Chapter 15 the non-uniform or varied flow in a channel is the one
in which depth of
• flow changes from section to section along the length of the channel. It may be
further classified as
• gradually varied flow (G.V.F.) and rapidly varied flow (R.V.F.). The gradually varied
flow is a steady nonuniform
• flow in which the depth of flow varies gradually. Many cases of gradually varied
flow are of
• practical interest to engineers such as flow upstream of a weir or a dam, flow
downstream of a sluice
• gate, flow in channels with break in bottom slopes etc., wherein study of back
water and the location
• of hydraulic jump is of major importance. In a rapidly varied flow the depth of flow
changes abruptly
• over a comparatively short distance. Typical examples of rapidly varied flow are
hydraulic jump and
• hydraulic drop. In this chapter both the types of non-uniform flows have been
discussed.

GRADUALLY VARIED FLOW
• The problem of gradually varied flow is that of predicting overall
flow pattern, or in other words prediction of the water surface
profile to be expected in a given channel with given steady
discharge.
• Such problems can be solved by writing the differential equation for
the water surface profile and then integrating it.
• 1. Dynamic Equation of Gradually Varied Flow. The dynamic
equation for gradually varied flow
• can be derived from the basic energy equation with the following
assumptions:
• (a) The uniform flow formulae (such as Manning’s or Chezy’s) may
be used to evaluate the energy slope of a gradually varied flow and
the corresponding coefficients of roughness developed primarily
• for uniform flow are applicable to the gradually varied flow also.

Hydraulic Jump and classification - Elements
and characteristics- Energy dissipation.
• The hydraulic jump is defined as the sudden and turbulent
passage of water from a supercritical state to subcritical
state.
• It has been classified as rapidly varied flow, since the
change in depth of flow from rapid to tranquil state is in an
abrupt manner over a relatively short distance.
• The flow in a hydraulic jump is accompanied by the
formation of extremely turbulent rollers and there is a
considerable dissipation of energy.
• A hydraulic jump will form when water moving at a
supercritical velocity in a relatively shallow stream strikes
water having a relatively large depth and subcritical
velocity.

• It occurs frequently in a canal below a regulating
sluice, at the foot of a spillway, or at the place where a
steep channel bottom slope suddenly changes to a flat
slope.
• In order to study the conditions of flow before and
after the hydraulic jump the application of the energy
equation does not provide an adequate means of
analysis, because hydraulic jump is associated with an
appreciable loss of energy which is initially unknown.
• As such in the analysis of hydraulic jump the
momentum equation is used by considering the
portion of the hydraulic jump as the control volume.

• The following assumptions are, however, made in this
analysis:
• (1) It is assumed that before and after jump formation
the flow is uniform and the pressure distribution
• is hydrostatic.
• (2) The length of the jump is small so that the losses
due to friction on the channel floor are small
• and hence neglected.
• (3) The channel floor is horizontal or the slope is so
gentle that the weight component of the water
• mass comprising the jump is negligibly small.

• Consider a hydraulic jump formed in a prismatic channel
with horizontal floor carrying a discharge Q as shown in Fig.
16.8.
• Let the depth of flow before the jump at section 1 be y1
and the depth of flow after the jump at section 2 be y2.
• The depth y1 is known as initial depth and y2 is known as
sequent depth.
• The symbols A1, V1 and z1 represent the area of cross-
section, mean velocity of flow and the depth of the centroid
of area A1 below the free surface respectively at section 1
before the jump and A2, V2 and z2 are the corresponding
quantities at section 2 after the jump.

• The specific force F is a function of the depth of flow y and hence it
can be plotted against the depth of flow y to obtain specific force
curve as shown in Fig. 16.8.
• It may be noted from the specific force curve that alike specific
energy the specific force also attains a minimum value at critical
depth yc.
• It is obvious from Eq. 16.15 that the specific force F1 corresponding
to y1 and the specific force F2 corresponding to y2 are same which
may also be seen from the specific force curve shown in Fig. 16.8.
• The initial depth y1 and the sequent depth y2 are commonly known
as the conjugate depths, which indicate the same specific force (in
order to distinguish them from the alternate depths which indicate
• the same specific energy).

• Equation 16.14 enables the determination of y2, if y1 is given or
vice versa for a known discharge Q
• flowing in a given channel section. Alternatively from the specific
force curve also, knowing y1 and
• hence F1, y2 can be determined or vice versa, since both y1 and y2
correspond to same specific force.
• Knowing the conjugate depths y1 and y2 for a known discharge Q in
a given channel section the specific energies E1 and E2 at the
sections 1 and 2 respectively may be computed from which the loss
of energy in the jump can be determined.
• Alternatively the loss of energy in the jump may also be determined
by using the specific energy curve in combination with specific force
curve as indicated below.

• Knowing y1 the corresponding specific force F1 can be
found from specific force curve as indicated by the point P1.
• A vertical through point P1 will cut the curve at point P2.
• Since F1 = F2, the ordinate of point P2 will indicate the
depth y2.
• Horizontal lines drawn through points P1 and P2 will cut the
specific energy curve at points P´1 and P´2 respectively,
indicating the values of the specific energies before and
after the jump, i.e., E1 and E2 respectively.
• The horizontal distance between points P´1 and P´2 is the
energy loss ΔE, due to hydraulic jump.

(b) Types of Hydraulic Jump.
• Equation 16.23 emphasizes the importance of
the Froude number Fr1 of
• the incoming supercritical flow, as a parameter
describing the phenomenon of hydraulic jump. As
such
• according to the studies of U.S. Bureau of
Reclamation, depending upon the value of
Froude number Fr1 of
• the incoming flow, there are five distinct types of
the hydraulic jump which may occur on a
horizontal floor.

(c) Applications of Hydraulic Jump.
• The phenomenon of hydraulic jump has many practical
• applications as listed below.
• (1) It is a useful means of dissipating excess energy of water flowing over
spillways and other hydraulic structures or through sluices and thus
preventing possible erosion on the downstream side of these structures.
• (2) It raises the water level in the channels for irrigation etc.
• (3) It increases the weight on an apron of a hydraulic structure due to
increased depth of flow and
• hence the uplift pressure acting on the apron is considerably
counterbalanced.
• (4) It increases the discharge through a sluice by holding back the tail
water.
• (5) It may be used for mixing chemicals in water and other liquids, since it
facilitates thorough
• mixing due to turbulence created in it.

• Example 8.10. Two different liquids which are miscible are mixed in a
device shown in Fig. Ex. 8.10. The
• device consists of a pipe of diameter 10 cm with a bend at one end. A
smaller pipe of diameter 5 cm and negligible
• wall thickness is introduced into it as shown in the figure and a liquid of
specific gravity 0.8 is pumped through
• it at a constant rate so that it issues out at section 1 with a uniform
velocity of 6 m/s. The other liquid, whose
• specific gravity is 0.9, is pumped through the larger pipe and it has a
uniform velocity of 3 m/s at section 1. The
• pressure is the same in both the fluid streams at section 1. At section 2 the
mixed stream has the same density,
• velocity and pressure at every location. The net resistive force acting along
the pipe wall between the sections 1
• and 2 is estimated to be 5 kg(f).

THANK YOU
References by
Hydraulics &
Fluid Mechanics including
Hydraulics Machines
Dr. P.N Modi Dr. S.M Seth
Text book in Pdf you may download from given link
Downloaded From : www.EasyEngineering.net

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