Open channel hydraulics for engineers. Chapter 1 introduction
This lecture note is written for undergraduate students who follow the training programs in the fields of Hydraulic, Construction, Transportation and Environmental
Engineering. It is assumed that the students have passed a basic course in Fluid Mechanics and are familiar with the basic fluid properties as well as the conservation laws of mass, momentum and energy. However, it may be not unwise to review some important definitions and equations dealt with in the previous course as an aid to memory before starting.......
OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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Chapter INTRODUCTION
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1.1. Review of fluid mechanics
1.2. Structure of the course
1.3. Dimensional analysis
1.4. Similarity and models
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Summary
This introductory chapter briefly reviews the previous course, in order to remind the
students of some basic fluid properties and equations before starting this course on Open
Channel Hydraulics. Next, dimensional analysis, similitude and model studies are dealt
with and described.
Key words
Fluid mechanics; open channel flow; dimensional analysis; similitude; Reynolds number;
hydraulic model
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1.1. REVIEW OF FLUID MECHANICS
This lecture note is written for undergraduate students who follow the training
programs in the fields of Hydraulic, Construction, Transportation and Environmental
Engineering. It is assumed that the students have passed a basic course in Fluid Mechanics
and are familiar with the basic fluid properties as well as the conservation laws of mass,
momentum and energy. However, it may be not unwise to review some important
definitions and equations dealt with in the previous course as an aid to memory before
starting.
1.1.1. Fluid mechanics
Fluid mechanics, which deals with water at rest or motion, may be considered as
one of the important courses of the Civil Engineering training program. It is defined as the
mechanics of fluids (gas or water). This course will mostly deal with the liquid water. The
following properties then are important:
(a). Density
The density of a liquid is defined as the mass of the substance per unit volume at a
standard temperature and pressure. It is also fully called “mass density” and denoted by the
Greek symbol (rho). In the case of water, we generally neglect the variation in mass
density and consider it at a temperature of 4C and at atmospheric pressure; then = 1,000
kg/m3 for all practical purposes. For other specific cases, the densities of common liquids
are given in tables in most fluid mechanics books.
(b) Specific weight
The specific weight of a liquid is the gravitational force per unit volume. It is given
by the Greek symbol (gamma) and sometimes briefly written as sp.wt. In SI units, the
specific weight of water at a standard reference temperature of 4C and atmospheric
pressure is 9.81 kN/m3.
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(c) Specific gravity
Specific gravity is defined as the ratio of the specific weight of a given liquid to the
specific weight of pure water at a standard reference temperature. Specific gravity, or sp.
gr., is presented as:
Specific weight of liquid
Sp.gr. =
Specific weight of pure water
Specific gravity is dimensionless, because it is a ratio of specific weights.
(d) Compressibility
The compressibility of a fluid may be defined as the variation of its volume, with
the variation of pressure. All fluids are compressible under the application of an external
force, and when the force is removed they expand back to their original volume exhibiting
the property that stress is proportional to volumetric strain. In the case of water as well as
other liquids, it is found that volumes are varying very little under variations of pressure,
so that compressibility can be neglected for all practical purposes. Thus, water may be
considered as an incompressible liquid.
(e) Surface tension
The surface tension of a liquid is its property, which enables it to resist tensile
stress in the plane of the surface. It is due to the cohesion between the molecules at the
surface of a liquid. Looking at the upper end of a small-diameter tube put into a cup of
water, we can easily see the water risen in the tube with an upward concave surface, as
shown in Fig. 1a. However, if the tube is dipped into mercury, the mercury drops down in
the tube with an upward convex surface as shown in Fig. 1b. If the adhesion between the
tube and the liquid molecules is greater than the cohesion between the liquid molecules, we
will have an upward concave surface. Otherwise, we get an upward convex surface. The
surface tension of water and mercury at 20 ºC is 0.0075 kg/m and 0.0520 kg/m,
respectively.
Fig. 1.1a. Capillary tube in water Fig. 1.1b. Capillary tube in mercury
The phenomenon of rising water in a small-diameter tube is called capillary rise.
(f) Viscosity
The dynamic or absolute viscosity of a liquid is denoted by the Greek symbol
(mu) and defined physically as the ratio of the shear stress to the velocity gradient du/dz:
(1-1)
z du
u dz
z where u = velocity in x direction
dz
du u
Fig. 1.2: Velocity distribution
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Viscosity is its property which controls the rate of flow. In the same tube, the flow of
alcohol or water is much easier than the flow of syrup or heavy oil.
1.1.2. Hydrostatics
Hydrostatics means study of pressure as exerted by a liquid at rest. Since the fluid
is at rest, there are no shear stresses in it. The direction of such a pressure is always at right
angles to the surface, on which it acts (Pascal’s law).
(a) The total force F on a horizontal, a vertical or surface
an inclined immersed surface is expressed as:
F = .A.hgc [kN] (1-2)
liquid hgc
where = g = specific weight of the liquid [N/m ]; 3
A = area of the immersed surface [m2];
hgc = depth of the gravity center of the
horizontal immersed surface from
the liquid level [m] (see Fig. 1.3).
(b) The pressure center of an immersed surface is the point through which the resultant
pressure force acts (see Fig. 1.4):
90 surface
hgc
liquid hpc
hgc hpc
G G
P G
F = .A.hgc
P P
area A
Fig. 1.4. Vertical and inclined surface
(c) The depth of pressure center of an immersed surface from the liquid level, hpc, (see
Fig. 1.4) reads:
h gc
IG
hpc = [m] (for vertical immersed surface) (1-3)
A.h gc
I G . sin 2
hpc = h gc [m] (for inclined immersed surface) (1-4)
A.h gc
where IG = moment of inertia of the surface about the horizontal axis through its gravity
center [m4];
= angle of the immersed surface with respect to the horizontal
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(d) The pressure center of a composite section is found as follows:
first, by splitting it up into convenient sections;
then, by determining the pressures on these sections;
then, by determining the depths of the respective pressure centers; and
finally, by equating:
F.h pc Fi h pci
n
(1-5)
i 1
where F = (total) pressure force;
n = number of sections;
i = subscript denoting the ith section.
1.1.3. Continuity equation
The continuity principle is based on the conservation of mass as applying to the
flow of fluids with invariant, i.e. constant, mass density. The continuity equation of a
liquid flow is a fundamental equation stating that, if an incompressible liquid is
continuously flowing through a pipe or a channel (the cross-sectional area of which may or
may not be constant), the quantity of liquid passing per time unit is the same at all sections
as illustrated in Fig. 1.5.
Now consider a liquid flowing through a tube. Q2 Q3
Let Q = flow discharge [m3/s]; V3A3
V = average velocity of the liquid [ms-1]; Q1
V2A2
A = area of the cross-section [m2];
and i = the number of section.
V1A1
We get:
Q1 = Q2 = Q3 = … (1-6) Fig.1.5. Continuity of a liquid flow
or V1A1 = V2A2 = V3A3 = … (1-7)
1.1.4. Types of flow
A flow, in which the velocity does not change from point to point along any of the
streamlines, is called a uniform flow. Otherwise, the flow is called a non-uniform
flow.
A flow, in which each liquid particle has a definite path and the paths of
individual particles do not cross each other, is called a laminar flow. This flow is
void of eddies. But, if each particle does not have a definite path and the paths of
individual particles also cross each other, the flow is called turbulent.
A flow, in which the quantity of liquid flowing per second, Q, is constant with
respect to time, is called a steady flow. But if Q is not constant, it is called an
unsteady flow.
A flow, in which the volume and thus the density of the fluid changes while
flowing, is called a compressible flow. But if the volume does not change while
flowing, it is called an incompressible flow.
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A flow, in which the fluid particles also rotate about their own axes while flowing,
is called a rotational flow. But if the particles do not rotate about their own axes
while flowing, it is called an irrotational flow.
A flow, whose streamlines may be represented by straight lines, is called a one-
dimensional flow. If the streamlines are represented by curves, the flow is called
two-dimensional. A flow, whose streamlines can be decomposed into three
mutually perpendicular directions, is called three-dimensional.
1.1.5. Bernoulli’s equation
It states: “For a perfect incompressible liquid, flowing in a continuous stream, the
total energy of a particle remains the same, while the particle moves along a streamline
from one point to another”. This statement is based on the assumption that there are no
losses due to friction. Mathematically it reads
V2
z
p
= Constant (= energy head) (1-8)
2g g
where z = elevation, i.e. the height of the point in question above the datum; z
represents the potential energy;
V2
= energy head, representing the kinetic energy, V is the flow velocity along
2g
the streamline at the point in question;
p
and = pressure head, representing the pressure energy; p is the pressure at the
g
point in question and is the liquid density.
1.1.6. Euler's equation
Euler’s equation for steady flow of an ideal fluid along a streamline is based on
Newton’s second law (Force = Mass Acceleration). It is based on the following
assumptions:
The fluid is inviscid, homogeneous and incompressible;
The flow is continuous, steady and along the streamline;
The flow velocity is uniformly distributed over the section; and
No energy or force, except gravity and pressure force, is involved in the
flow.
Euler's equation in a differential-equation form can be written as:
dz V 0
dV dp
(1-9)
g g
After integrating the above equation, we easily come to Bernoulli's equation in the form of
energy per unit weight of the flowing fluid.
1.1.7. Flow through orifices, mouthpieces and pipes
An orifice is an opening (in a vessel) through which the liquid flows out. The
discharge through an orifice depends on the energy head, the cross-sectional area of
the orifice and the coefficient of discharge. A pipe, the length of which is generally
more than two times the diameter of the orifice, and which is fitted externally or
internally to the orifice is known as a mouthpiece. When a liquid is flowing through
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a mouthpiece, the energy head is declining due to wall friction, change of cross
section or obstruction in the flow.
A pipe is a closed conduit used to carry fluid. When the pipe is running full, the
flow is under pressure. The friction resistance of a pipe depends on the roughness
of the pipe inside. Early experiments on fluid friction were conducted, among
others, by Chezy: the frictional resistance varies approximately with: (a) the square
of the liquid velocity, and (b) the bed slope.
Frictional resistance per
Frictional resistance = wetted area (velocity)
2
unit area at unit velocity
Reminder:
Re
VD
+ Reynolds number: (1-10)
where = kinematic viscosity [m2/s]
V = characteristic flow velocity [m/s];
D = characteristic length, e.g. diameter of the pipe [m]
L V2
+ Darcy–Weisbach’s formula for head loss hf in pipes: h f f . . [m] (1-11)
D 2g
where f = friction coefficient according to Darcy–Weisbach;
L = length of the pipe
+ Chezy's formula for flow velocity V in pipe: V C Ri [m/s] (1-12)
where C = Chezy's coefficient [m½ s-1];
cross section area A
R = hydraulic radius [m] defined as: R
wetted perimeter P
i = loss of energy head per unit length (= bed slope in uniform flow).
1.1.8. Flow through open channel
An open channel is a passage, through which the water flows due to gravity with
atmospheric pressure at the free surface. The flow velocity is different at different points in
the cross-section of a channel due to the occurrence of a velocity distribution, but in
calculations, we use the mean velocity of the flow. In the course on Fluid Mechanics, we
have assumed that the rate of discharge Q, the depth of flow h, the mean velocity V, the
slope of the bed i and the cross-sectional area A remain constant over a given length L of
the channel (see Fig. 1.6).
L
Q h Q
i VA
Fig. 1.6. Uniform flow in open channel
Discharge through an open channel: Q VA AC Ri (1-13)
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1.2. STRUCTURE OF THE COURSE
1.2.1. Objectives of the course
Open Channel Hydraulics is an advanced course required for all students who
follow the field-study of water resources engineering. The subject is rich in variety and of
interest to practical problems. The content is focused on the types of problems commonly
encountered by hydraulic engineers dealing with the wide fields covered by open channel
hydraulics. Due to space and lecturing-time limitations, however, the lecture note does not
extend into the specialist fields of mathematical natural flow networks required, for
example, for river engineering computations.
The course aims to present the principles dealing with water flow in open channels and to
guide trainees to solve the applied problems for hydraulic-structure design and water
system control. The main objectives of the course are:
To supply the basic principles of fluid mechanics for the formulation of open
channel flow problems.
To combine theoretical, experimental and numerical techniques as applied to open
channel flow in order to provide a synthesis that has become the hallmark of
modern fluid mechanics.
To provide theoretical formulas and experimental coefficients for designing some
hydraulic structures as canals, spillways, transition works and energy dissipators.
1.2.2. Historical note for the course
Fluid mechanics and open channel hydraulics began at the need to control water for
irrigation purposes and flood protection in Egypt, Mesopotamia, India, China and also
Vietnam. Ancient people had to record the river water levels and got some empirical
understanding of water movements. They applied basic principles on making some fluid
machinery, sailing boats, irrigation canals, water supply systems etc. The Egyptians used
dams for water diversion and gravity flow through canals to distribute water from the Nile
River, and the Mesopotamians developed canals to transfer water from the Euphrates river
to the Tigris river, but there is no recorded evidence of any understanding of the theoretical
flow principles involved. The Chinese are known to have devised a system of dikes for
protection from flooding several thousand years ago. Over the past 2,000 years, many
dikes and canal systems have been built in the Red River delta in the North of Vietnam to
contain the delta and drain off its flood water that has always been serious problems.
Vietnamese, under Ngo Quyen, have also known to apply the tidal law in Bach Dang river
battles in 939 A.D, which has become famous in Vietnamese history.
It was not until 250 B.C. that Archimedes discovered and recorded the principles of
hydrostatics and flotation. In the 17th and 18th centuries, Isaac Newton, Daniel Bernoulli
and Leonhard Euler formulated the greatest principles of hydrodynamics. The work of
Chezy on flow resistance began in 1768, originating from an engineering problem of sizing
a canal to deliver water from the Yvette River to Paris. The Manning-equation for open-
channel-flow resistance has a complex historical development, but was based on field
observations. Julius Weisbach extended the sharp-crested weir equation and developed the
elements of the modern approach to open channel flow, including both theory and
experiment. William Froude, an Engish engineer, collaborated with Brunel in railway
construction and in the design of the steamer “Great Eastern”, the largest ship afloat at that
time. He contributed to the study of friction between solids and liquids, to wave mechanics
and to the interpretation of ship model tests.
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The work of Bakhmeteff, a Russian émigré to the United States, had perhaps the most
important influence on the development of open channel hydraulics in the early 20th
century. Of course, the foundations of modern fluid mechanics were laid by Prandtl and his
students, including Blasius and von Kàrmàn, but Bakhmeteff’s contributions dealt
specifically with open channel flow. In 1932, his book on the subject was published, based
on his earlier 1912 notes developed in Russia. His book concentrated on “varied flow” and
introduced the notion of specific energy, still an important tool for the analysis of open-
channel flow problems. In Germany at this time, the contributions of Rehbock to weir flow
also were proceeding, providing the basis for many further weir experiments and weir
formulas.
By the mid-20th century, many of the gains in knowledge in open channel flow has been
consolidated and extended by Rouse (1950), Chow (1959, 1973) and Henderson (1966), in
which books extensive reference can be found. These books set the stage for applications
of modern numerical analysis techniques and experimental instrumentation to open-
channel flow problems.
1.2.3. Structure of the course
The lecture note is divided into three parts of increasing complexity.
(a) Part 1 introduces to the basic principles: course introduction (Chapter 1),
uniform flow (Chapter 2) and hydraulic jump phenomena (Chapter 3). This part
will take 15 teaching hours.
(b) Part 2 includes non-uniform flow (Chapter 4) and design application as
Spillways (Chapter 5) and Transitions and Energy dissipators (Chapter 6). This part
will take 20 teaching hours.
(c) Part 3 deals with unsteady flow (Chapter 7). This chapter will take 10 teaching
hours.
The course approach chart is presented in Fig. 1.7. on the next page.
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OPEN CHANNEL HYDRAULICS FOR ENGINEERS
+ Chapter 1: INTRODUCTION
1.1. Review of fluid mechanics. 1.2. Structure of the course.
1.3. Dimensional analysis. 1.4. Similarity and models
Chapter 2: UNIFORM FLOW
+ 2.1. Introduction. 2.2.Basic equations in uniform open channel flow.
2.3. Most economical cross-section. 2.4. Channel with compound cross-
section. 2.5. Permissible velocity against erosion and sedimentation.
Chapter 3: HYDRAULIC JUMP
+ 3.1. Introduction. 3.2. Specific energy. 3.3. Depth of hydraulic jump.
3.4. Types of hydraulic jump. 3.5. Hydraulic jump formulas in terms of
Froude-number. 3.6. Submerged hydraulic jump
+ Chapter 4: NON-UNIFORM FLOW
4.1. Introduction. 4.2. Gradually-varied steady flow.
4.3. Types of water surface profiles.
4.4. Drawing water surface profiles.
+ Chapter 5: SPILLWAYS
5.1. Introduction. 5.2. General formula. 5.3. Sharp-crested weir
5.4. The overflow spillway. 5.5. Broad-crested weir
Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS
+ 6.1. Introduction. 6.2. Expansions and Contractions.
6.3. Drop structures 6.4. Stilling basins.
6.5. Other types of energy dissipators
Chapter 7: UNSTEADY FLOW
+ 7.1. Introduction. 7.2. The equations of motion
7.3. Solutions to the unsteady-flow equations
7.4. Positive surge and negative waves; Surge formation
Fig.1.7. Course structure chart
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1.3. DIMENSIONAL ANALYSIS
Most hydraulic engineering problems are solved by applying a mathematical
analysis. In some cases they should be checked by physical experimental means. The
approach of such problems is considerably simplified by using mathematical techniques
for dimensional analysis. It is based on the assumption that the phenomenon at issue can be
expressed by a dimensionally homogeneous equation, with certain variables.
1.3.1. Fundamental dimensions
We know that all physical quantities are measured by comparison. This comparison
is always made with respect to some arbitrarily fixed value for each independent quantity,
called dimension (e.g. length, mass, time, temperature etc). Since there is no direct
relationship between these dimensions, they are called fundamental dimensions or
fundamental quantities. Some other quantities such as area, volume, velocity, force etc,
cannot be expressed in terms of fundamental dimensions and thus may be called derived
dimensions, derived quantities or secondary quantities.
There are two systems for fundamental dimensions, namely FLT (i.e. force, length, time)
and LMT (i.e. length, mass, time). The dimensional form of any quantity is independent of
the system of units (i.e. metric or English). In this course, we shall use the LMT-system.
The following table gives the dimensions and units for the various physical quantities,
which are important form the hydraulics point-of-view.
Table 1.1: Dimensions in terms of LMT
No. Quantity Symbol Dimensions in terms of LMT-system
1. Length L L
2. Area A L2
3. Volume Vol L3
4. Time t T
5. Velocity V LT-1
6. Acceleration a LT-2
7. Gravitational acceleration g LT-2
8. Frequency N T-1
9. Discharge Q L3 T-1
10. Force/weight F, W LMT-2
11. Power P ML2T-3
12. Work/Energy E ML2T-2
13. Pressure p ML-1 T-2
14. Mass m M
15. Mass density ML-3
16. Specific weight ML-2 T-2
17. Dynamic viscosity ML-1 T-1
18. Kinematic viscosity L2 T1
19. Surface tension MT-2
20. Shear stress ML-1 T-2
21. Bulk modulus ML-1 T-2
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All variables used in science or engineering are expressed in terms of a limited number of
basic dimensions. For example, we can designate the dimensions of velocity as:
LT 1
Dis tan ce L
[V] = (1-14)
Time T
Here the brackets [x] mean "dimension of". So, equation (1-14) reads as follows: "the
dimension of velocity V equals the ratio of the distance to the time". In this case, L
represents the dimension of distance and T that of time.
Note: There are four systems of units, which are commonly used and universally adopted.
These are known as:
SI Units (International System of Units or Système International d'unités in
French): a unified and systematically constituted system of fundamental and
derived units for international use have been recommended by the 11th General
Conference of Weights and Measures (CGPM). SI is widely used in Vietnam as an
official unit system. The fundamental units of LMT (length, mass and time) are
meter, kilogram and second, respectively.
CGS Units: the fundamental units of LMT are centimeter, gram and second,
respectively.
MKS Units: the fundamental units of LMT are meter, kilogram and second,
respectively.
English Units: the fundamental units of LMT are foot, pound and second,
respectively
1.3.2 Dimensional homogeneity
Let us consider the common equation of hydraulics.
Q =A.V
We can write: L T = L2 x LT-1 = L3T-1
3 -1
(1-15)
Above example goes without saying that all equations must balance in magnitude.
However, all rational equations (those developed from basic laws of physics) must also be
dimensionally homogeneous. An equation is called dimensionally homogeneous, if the
fundamental dimensions have identical powers of LMT on both sides. That is, the left-hand
side (LHS) of the equation must have the same dimensions as the right-hand side (RHS).
Moreover, every term in the equation must have the same dimensions. Such an equation
would essentially be independent of the system of measurement (i.e. English or SI).
Note: Two dimensionally homogeneous equations can be multiplied or divided without
affecting the homogeneity. But the two dimensionally homogeneous equations cannot be
added or subtracted, as the resulting equation may not be dimensionally homogeneous.
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1.3.3. Principles of Dimensional Homogeneity
The principle of dimensional homogeneity has a number of applications. The
following issues are important from the point of view of the subject.
(a) Determining the dimension of a physical quantity
The dimensions of any physical quantity may be easily determined with this principle, e.g.
the dimension of energy:
Energy = Work = Force Distance (1-16)
= [LMT ] [L]
-2
(Force, [F ]= [LMT-2])
= [ML2T-2]
Example 1.1: Determine the dimension of the following quantities in the LMT-system:
(i) Force (ii) Pressure, (iii) Power, (iv) Specific weight, and (v) Surface tension
Solution:
We know the dimension of ‘force‘ in the LMT-system:
(i) Force = Mass x Acceleration
= [M] 2 [MLT 2 ]
Length [ML]
2
Ans.
Time [T ]
Force [MLT 2 ]
Similarly, (ii) Pressure = 2
[ML1T 2 ] Ans.
Area [L ]
Work done Force Distance
(iii) Power =
Time Time
[MLT -2 ][L]
= [ML2T -3 ] Ans.
[T]
Weight Force
(iv) Specific weight = (Weight = Force)
Volume Volume
[MLT 2 ]
= 3
[ML2 T 2 ] Ans.
[L ]
Force [MLT -2 ]
and (v) Surface tension = = [MT-2] Ans.
Length [L]
Example 1.2. Determine the dimension of the following quantities in the LMT-system
(i) Discharge, (ii) Torque and (iii) Momentum.
Solution:
We know the dimension of ‘discharge’ in the LMT-system:
Volume [L3 ]
(i) Discharge = [L3T 1 ] Ans.
Time [T]
Similarly (ii) Torque = Force Distance = [LMT-2][L] = [ML2T-2] Ans.
and (iii) Momentum = Mass Velocity = [M] [LT-1] = [LMT-1] Ans.
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(b) Checking the dimensional homogeneity of an equation
The dimensional homogeneity of an equation may be easily checked with this principle;
e.g. let us consider Darcy - Weisbach’s formula for loss of energy head in pipes:
L V2
hf f . . (1-17)
D 2g
The dimension of 2 in the denominator is not considered. The dimension of f, being
constant, is taken as 1. Now substituting the dimensions on the LHS and RHS of the
equation, we get:
[1] x [L] x [LT -1 ] 2
[L] = = [L] (1-18)
[LT 2 ][ L]
Example 1.3. Check the dimensional homogeneity of the following common equations in
the field of hydraulics: (i) Q C d A 2gH and (ii) V C Ri
Solution
(i) Given equation, Q = C d A 2gH
Substituting the dimensions on the LHS and RHS of the equation (the dimension of Cd,
being a discharge coefficient, is taken as 1):
[L3T-1] = [1] [L2] [[1] ½ [LT-2 x L]½ = [L3T-1]
Since the dimensions on both sides of the equation are the same, the equation is
dimensionally homogeneous. Ans.
(ii) Given equation, V = C Ri
Substituting the dimensions on the LHS and RHS of the equation (the dimension of i,
being dimensionless is taken as 1):
LT-1 = C [L 1]1/2 = C [L]1/2
Since the dimensions on both sides of the equation are not the same, the equation is not
dimensionally homogeneous. Ans.
From the above equation, we find that:
[ LT 1 ]
C [ L1 / 2 T 1 ]
[L]1 / 2
(c) Changing the coefficient of an equation while using an other system of units
The coefficient of an equation may be easily changed, while using the same equation in an
other system of units, for example from English to MKS or vice versa.
Let us consider Manning’s formula for the velocity
1 2 3 12
V= .R .i (1-19)
n
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where n is a resistance coefficient, called Manning's constant. Now substituting the
dimensions on the LHS and RHS of the equation, we get:
[L]2/3 [1]1/2 = [L]2/3
1 1
[LT-1] =
n n
(The dimension of the bed slope i is 1).
Since the dimensions of both sides are not the same, the equation is dimensionally non-
homogeneous. From the above equation, we find that:
1
[LT ] 13 1
L .T
1
[L] 3
= 2
n
Now, in order to make the above equation applicable to English units, the coefficient of M
has to be changed. We know that 1m = 3.281 ft, and the unit of time is the same in both the
systems. Therefore the new constant is:
L1/3 = 3.2811/3 = 1.486
1.486 2 3 1 2
It is obvious, that the equation for English units will be V= .R .i (1-20)
n
(d) Using the dimensional analysis methods
There are several methods that may be used to carry out the process of dimension analysis,
such as the Step-by-Step method, the Exponent method, .... Students can find them in
reference books. In this course, Buckingham's -theorem will be introduced shortly in the
next section.
1.3.4. Buckingham’s - theorem
Buckingham’s -theorem states, “If there are n variables in a dimensionally
homogeneous equation and if these variables contain m fundamental dimensions such as
(L, M, T), they may be grouped into (n-m) non-dimensional independent -terms.”
Mathematically, if a variable X1 depends on the independent variables X2, X3, X4, ..., Xn
the function may be written as:
X1 = k (X2, X3, X4, ..., Xn)
The equation may be written in its general form as:
f(X1, X2, X3, X4, ..., Xn) = C
where C is a constant, and f represents the functional relationship. In this equation, there
are n variables. If there are m fundamental dimensions, then according to Buckingham’s -
theorem
f1 (1,2,3,...,n-m) = Constant
where is a dimensionless term.
Students can read for understanding details and how to apply in reference hydraulics
books.
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Example 1.4: Flow through a closed conduit with rectangular cross-section. Let us
determine the wall friction as a dependent quantity.
Solution:
Let be the average wall shear stress over the full perimeter
depends on: b = internal breadth [m]
h = internal height [m]
k = dimension of wall roughness [m]
= specific mass density of fluid [kgm-3]
= dynamic viscosity of fluid [kgm-1s-1]
V = fluid velocity, averaged over cross-section [ms-1]
= f1(b, h, k, , , V)
[] = ML-1T-2
[b] = L
[h] = L
[k] = L
[] = ML-3
[] = ML-1T-1
[V] = LT-1
We have totally 7 quantities and 3 basic dimensions, viz. M. L and T
There are 4 independent dimensionless parameters.
b k
f2 , ,
V 2
h h Vh
= cfV2
cf
V2
Therefore,
cf is a friction factor, i.e. dimensionless
b k Vh
c f f 3 , , Re h , with Re h
h h
h k Vb
c f f 4 , , Re b , with Re b
or
b b
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1.3.5. Limitations of dimensional analysis
Some problems may be met when applying dimensional analysis:
In order to use dimensional analysis, we must first decide which variables are
significant. If we do not understand the problem well enough to make a good initial
choice of variables, dimensional analysis seldom provides clarification.
One error might be the inclusion of variables whose influence is already accounted
for. For example, one might tend to include two or three length variables in a scale-
model test, where only one may be sufficient.
Another serious error might be the omission of a significant variable. If this is done,
one of the significant dimensionless parameters will likewise be missing.
How do we know whether a variable is significant for a given problem? Probably the
proper answer is from experience. After working in the field of fluid mechanics and open-
channel hydraulics for several years, one develops a feeling for the significance of
variables to certain kinds of application.
1.4. SIMILARITY AND MODELS
Since the beginning of the twentieth century, the engineers engaged on the creation
or design of hydraulic structures (such as dams, spillways or large hydraulic machines)
have developed a new and scientific method to predict the performance of their structures
and machines. This is done by preparing physical scale models and testing them in a
laboratory; so as to form some opinion, about the working and behaviour of the proposed
hydraulic structures, after their completion or actual installation. The structure, of which
the model is prepared, is known as prototype and the model is known as scale model or
simply physical model.
1.4.1 Advantages of model analysis
Though there are numerous advantages of model testing, yet the following is to be
mentioned:
1. The behaviour and working details of a hydraulic structure or a machine can be
easily predicted from its physical model. The smooth and reliable working of a
hydraulic structure or a machine can be ascertained by spending a relatively
small sum of money, which is a negligible fraction of the total cost to be spent
on the prototype.
2. If the hydraulic structure or machine is made directly, then in case of its failure,
it is very difficult to change its design. Moreover, it is very costly. Laboratory
tests can result in saving human labour and material.
3. With the help of model testing, a number of alternative designs can be studied.
Finally, the most economical, accurate and safe design may be selected.
4. When the existing hydraulic structure is not functioning properly, then model
testing can help us in detecting and rectifying the defects.
5. Sometimes, it is difficult to design a particular portion of a complex hydraulic
structure or machine. In such a case, model testing is very essential in order to
ascertain the safety and reliability of that particular portion of the prototype.
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1.4.2. Hydraulic similarity
If we look at a photograph of a man, very carefully, we can have an idea of the
proportion of various parts of his body. The photograph will also give an idea of the
features of each part of the man. Similarly, to know the complete working and behaviour
of the prototype from its model, there should be a complete similarity between the
prototype and its scale model. This similarity is known as hydraulic similitude or
Hydraulic Similarity. Three types of hydraulic similarity are important, viz.:
1. Geometric similarity,
2. Kinematic similarity, and
3. Dynamic similarity
1.4.3. Geometric similarity
Geometric similarity is said to exist between the model and the prototype, if both of
them are identical in shape, but differ only in size. Or in other words, geometric similarity
is said to exist between the model and the prototype, if the ratios of all corresponding
linear, geometrical dimensions are equal (see Fig. 1.8).
Lp
(a) Bp
Lm
(b)
Bm
Fig.1.8. Geometric similarity: (a) Prototype and (b) Model
Let Lp = length of the prototype
Bp = breadth of the prototype,
Dp = depth of the prototype, and
Lm, Bm and Dm = corresponding values for the model.
Now, if geometric similarity exists between the prototype and the model, then the linear
ratio of the prototype and the model (also called scale ratio) reads as:
L B D
Lr = p = p = p
L m Bm D m
Similarly, it hold for the area ratio of the prototype and the model:
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L B D
2 2 2
Ar = p = p = p
L m Bm D m
And for the volume ratio:
L B D
3 3 3
Vr = p = p = p
L m Bm D m
1.4.4. Kinematic similarity
Kinematic similarity is said to exist between the model and the prototype, if both of
them have corresponding motions or velocities. Or in other words, kinematic similarity is
said to exist between the model and the prototype, if the ratio of the corresponding
velocities at corresponding points are equal.
Let V1p = velocity of liquid in prototype at point 1,
V2p = velocity of liquid in prototype at point 2,
V1m, V2m = corresponding values for the model.
Now, if kinematic similarity exists between the prototype and the model, then the velocity
ratio of the prototype and the model reads as:
V1p V2 p V3 p
Vr = ...
V1m V2 m V3 m
1.4.5. Dynamic similarity
Dynamic similarity is said to exist between the model and the prototype, if both of
them have corresponding forces. Or in other words, dynamic similarity is said to exist
between the model and the prototype, if the ratios of the corresponding forces acting at
corresponding points are equal:
Let F1p and F1m = force acting in prototype and model at point 1;
F2p and F2m = force acting in prototype and model at point 2.
Now, if dynamic similarity exists between the prototype and the model, then the force ratio
of the prototype and the model reads as:
F F F
Fr = 1p = 2p = 3p =...
F1m F2m F3m
Consider the flow over the spillway shown in Fig. 1.9. Here corresponding masses of fluid
in the model and the prototype are acted on by corresponding forces. These forces are the
force of gravity Fg, the pressure force Fp, and the viscous resistance force Fv. These forces
add vectorially in Fig. 1.9 to yield a resultant force FR, which will in turn produce an
acceleration of the volume of fluid in accordance with Newton’s second law.
FRm M m a m
FRp Mpa p
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forces on fluid element prototype force polygon
FPp FPp
Fgp
Fvp
FRp = Mp.ap
(a)
Fvp
Fgp
model force polygon
FPm
FPm
Fgm
Fvm FRm = Mm.am
(b) Fvm
Fgm
Fig. 1.9: Dynamic similarity (a) Prototype and (b) Model
1.4.6. Technique of hydraulic modelling
The technique of hydraulic modelling involves the following steps:
a. Selection of suitable scale,
b. Operation of the hydraulic model, and
c. Correct prediction
a. Selection of suitable scale
This depends on many factors. But the following is important concerning this issue:
Availability of funds;
Availability of time;
Availability or space for accommodating the model; and
Availability of employees.
The usual practice is to make the model geometrically similar to the prototype. But in some
cases distorted models are also employed.
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b. Operation of the hydraulic model
After selecting the type, the scale and the materials of the model, the next step is to
construct the model accurately according to the plan. High-tech instruments can be
essential for precisely measuring the hydraulic quantities in the experiments. Great care
and patience are required for correctly interpreting the model results.
c. Correct prediction
After obtaining the precise measurements of the required hydraulic quantities in an
experiment, the next step is to predict the correct working of the prototype. We shall study
the correct prediction of prototypes in the following pages.
1.4.7. Developments in hydraulic model testing
The model testing is the most scientific and common feature of the design and
successful working of hydraulic structures and machines. Two types of facility are
important and need to be dealt with: 1) the wind tunnel, and 2) the water tunnel.
1. Wind tunnel
A wind tunnel is a standard equipment for aircraft design. It provides a steady flow of air
around the model which is suspended in the stream. Though the walls of the tunnel will
interfere, to some extent, with the stream of air, yet its effect is generally neglected.
In a wind tunnel, the air is set in motion by means of a compressor. The model under
investigation is mounted in the path of the wind stream. Sometimes, the compression of air
in the wind tunnel produces an appreciable rise in temperature, which must be dissipated
by a cooling device.
2. Water tunnel
A water tunnel is a standard equipment for the design of turbines, pumps and ships. In
water tunnels, a uniform stream of water is produced and the model under investigation is
mounted in the path of the water.
The size of the water tunnel is, usually, expressed as the diameter of its best section. The
existing water tunnels range as size from 10 cm to 150 cm.
1.4.8. Undistorted models
All the hydraulic models may be broadly classified into the following two types:
1. Undistorted models, and
2. Distorted models.
A model, which is geometrically similar to its prototype is known as an undistorted model.
The prediction from an undistorted model is comparatively easy and the results obtained
from the model, can be easily transferred to the prototype, if the basis condition (of
geometric similarity) is satisfied. A distorted model will be discussed in Section 1.4.10.
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