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Fluid mechanics for civil egineers

FLUID MECHANICS
FOR CIVIL ENGINEERS

Bruce Hunt
Department of Civil Engineering
University Of Canterbury
Christchurch, New Zealand

? Bruce Hunt, 1995



Table of Contents

Chapter 1 – Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Review of Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1
1.2

1.4
1.9

Chapter 2 – The Equations of Fluid Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Momentum Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1
2.1
2.4
2.9

Chapter 3 – Fluid Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1
Pressure Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1
Area Centroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6
Moments and Product of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8
Forces and Moments on Plane Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8
Forces and Moments on Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . 3.14
Buoyancy Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19
Stability of Floating Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23
Rigid Body Fluid Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.36
Chapter 4 – Control Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1
Extensions for Control Volume Applications . . . . . . . . . . . . . . . . . . . 4.21
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.27
Chapter 5 – Differential Equation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1
Chapter 6 – Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1
Circulation and the Velocity Potential Function . . . . . . . . . . . . . . . . . . 6.1
Simplification of the Governing Equations . . . . . . . . . . . . . . . . . . . . . . 6.4
Basic Irrotational Flow Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7
Stream Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15
Flow Net Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.20
Free Streamline Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.28
Chapter 7 – Laminar and Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1
Laminar Flow Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13
Turbulence Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.24


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Chapter 8 – Boundary-Layer Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1
Boundary Layer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2
Pressure Gradient Effects in a Boundary Layer . . . . . . . . . . . . . . . . . . 8.14
Secondary Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.23
Chapter 9 – Drag and Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1
Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1
Drag Force in Unsteady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7
Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.12
Oscillating Lift Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.19
Oscillating Lift Forces and Structural Resonance . . . . . . . . . . . . . . . . 9.20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.27
Chapter 10 – Dimensional Analysis and Model Similitude . . . . . . . . . . . . . . . . . . . . . . . . 10.1
A Streamlined Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5
Standard Dimensionless Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6
Selection of Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.22
Chapter 11 – Steady Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1
Hydraulic and Energy Grade Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3
Hydraulic Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8
Pipe Network Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13
Pipe Network Computer Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.24
Chapter 12 – Steady Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1
Rapidly Varied Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1
Non-rectangular Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.12
Uniform Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.12
Gradually Varied Flow Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 12.18
Flow Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.22
Flow Profile Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.23
Numerical Integration of the Gradually Varied Flow Equation . . . . . 12.29
Gradually Varied Flow in Natural Channels . . . . . . . . . . . . . . . . . . . 12.32
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.32
Chapter 13 – Unsteady Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1
The Equations of Unsteady Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . 13.1
Simplification of the Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3
The Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7
The Solution of Waterhammer Problems . . . . . . . . . . . . . . . . . . . . . . 13.19
Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.23
Pipeline Protection from Waterhammer . . . . . . . . . . . . . . . . . . . . . . . 13.27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.27

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Chapter 14 – Unsteady Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1
The Saint-Venant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1
Characteristic Form of the Saint-Venant Equations . . . . . . . . . . . . . . . 14.3
Numerical Solution of the Characteristic Equations . . . . . . . . . . . . . . 14.5
The Kinematic Wave Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 14.7
The Behaviour of Kinematic Wave Solutions . . . . . . . . . . . . . . . . . . . 14.9
Solution Behaviour Near a Kinematic Shock . . . . . . . . . . . . . . . . . . . 14.15
Backwater Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.18
A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.29
Appendix I – Physical Properties of Water and Air
Appendix II – Properties of Areas
Index

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iv


Preface
Fluid mechanics is a traditional cornerstone in the education of civil engineers. As numerous
books on this subject suggest, it is possible to introduce fluid mechanics to students in many
ways. This text is an outgrowth of lectures I have given to civil engineering students at the
University of Canterbury during the past 24 years. It contains a blend of what most teachers
would call basic fluid mechanics and applied hydraulics.
Chapter 1 contains an introduction to fluid and flow properties together with a review of vector
calculus in preparation for chapter 2, which contains a derivation of the governing equations of
fluid motion. Chapter 3 covers the usual topics in fluid statics – pressure distributions, forces on
plane and curved surfaces, stability of floating bodies and rigid body acceleration of fluids.
Chapter 4 introduces the use of control volume equations for one-dimensional flow calculations.
Chapter 5 gives an overview for the problem of solving partial differential equations for velocity
and pressure distributions throughout a moving fluid and chapters 6–9 fill in the details of
carrying out these calculations for irrotational flows, laminar and turbulent flows, boundary-layer
flows, secondary flows and flows requiring the calculation of lift and drag forces. Chapter 10,
which introduces dimensional analysis and model similitude, requires a solid grasp of chapters
1–9 if students are to understand and use effectively this very important tool for experimental
work. Chapters 11–14 cover some traditionally important application areas in hydraulic
engineering. Chapter 11 covers steady pipe flow, chapter 12 covers steady open channel flow,
chapter 13 introduces the method of characteristics for solving waterhammer problems in
unsteady pipe flow, and chapter 14 builds upon material in chapter 13 by using characteristics
to attack the more difficult problem of unsteady flow in open channels. Throughout, I have tried
to use mathematics, experimental evidence and worked examples to describe and explain the
elements of fluid motion in some of the many different contexts encountered by civil engineers.
The study of fluid mechanics requires a subtle blend of mathematics and physics that many
students find difficult to master. Classes at Canterbury tend to be large and sometimes have as
many as a hundred or more students. Mathematical skills among these students vary greatly, from
the very able to mediocre to less than competent. As any teacher knows, this mixture of student
backgrounds and skills presents a formidable challenge if students with both stronger and weaker
backgrounds are all to obtain something of value from a course. My admittedly less than perfect
approach to this dilemma has been to emphasize both physics and problem solving techniques.
For this reason, mathematical development of the governing equations, which is started in
Chapter 1 and completed in Chapter 2, is covered at the beginning of our first course without
requiring the deeper understanding that would be expected of more advanced students.
A companion volume containing a set of carefully chosen homework problems, together with
corresponding solutions, is an important part of courses taught from this text. Most students can
learn problem solving skills only by solving problems themselves, and I have a strongly held
belief that this practice is greatly helped when students have access to problem solutions for
checking their work and for obtaining help at difficult points in the solution process. A series of
laboratory experiments is also helpful. However, courses at Canterbury do not have time to
include a large amount of experimental work. For this reason, I usually supplement material in
this text with several of Hunter Rouse's beautifully made fluid-mechanics films.

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This book could not have been written without the direct and indirect contributions of a great
many people. Most of these people are part of the historical development of our present-day
knowledge of fluid mechanics and are too numerous to name. Others have been my teachers,
students and colleagues over a period of more than 30 years of studying and teaching fluid
mechanics. Undoubtedly the most influential of these people has been my former teacher,
Hunter Rouse. However, more immediate debts of gratitude are owed to Mrs Pat Roberts, who
not only encouraged me to write the book but who also typed the final result, to Mrs Val Grey,
who drew the large number of figures, and to Dr R H Spigel, whose constructive criticism
improved the first draft in a number of places. Finally, I would like to dedicate this book to the
memory of my son, Steve.

Bruce Hunt
Christchurch
New Zealand

vi


vii


Chapter 1
Introduction
A fluid is usually defined as a material in which movement occurs continuously under the
application of a tangential shear stress. A simple example is shown in Figure 1.1, in which a
timber board floats on a reservoir of water.

Figure 1.1 Use of a floating board to apply shear stress to a reservoir surface.
If a force, F, is applied to one end of the board, then the board transmits a tangential shear stress,
-, to the reservoir surface. The board and the water beneath will continue to move as long as F
and - are nonzero, which means that water satisfies the definition of a fluid. Air is another fluid
that is commonly encountered in civil engineering applications, but many liquids and gases are
obviously included in this definition as well.
You are studying fluid mechanics because fluids are an important part of many problems that a
civil engineer considers. Examples include water resource engineering, in which water must be
delivered to consumers and disposed of after use, water power engineering, in which water is
used to generate electric power, flood control and drainage, in which flooding and excess water
are controlled to protect lives and property, structural engineering, in which wind and water
create forces on structures, and environmental engineering, in which an understanding of fluid
motion is a prerequisite for the control and solution of water and air pollution problems.
Any serious study of fluid motion uses mathematics to model the fluid. Invariably there are
numerous approximations that are made in this process. One of the most fundamental of these
approximations is the assumption of a continuum. We will let fluid and flow properties such as
mass density, pressure and velocity be continuous functions of the spacial coordinates. This
makes it possible for us to differentiate and integrate these functions. However an actual fluid
is composed of discrete molecules and, therefore, is not a continuum. Thus, we can only expect
good agreement between theory and experiment when the experiment has linear dimensions that
are very large compared to the spacing between molecules. In upper portions of the atmosphere,
where air molecules are relatively far apart, this approximation can place serious limitations on
the use of mathematical models. Another example, more relevant to civil engineering, concerns
the use of rain gauges for measuring the depth of rain falling on a catchment. A gauge can give
an accurate estimate only if its diameter is very large compared to the horizontal spacing between
rain droplets. Furthermore, at a much larger scale, the spacing between rain gauges must be small
compared to the spacing between rain clouds. Fortunately, the assumption of a continuum is not
usually a serious limitation in most civil engineering problems.


1.2

Chapter 1 — Introduction

Fluid Properties
The mass density, ', is the fluid mass per unit volume and has units of kg/m3. Mass density is
a function of both temperature and the particular fluid under consideration. Most applications
considered herein will assume that ' is constant. However, incompressible fluid motion can
occur in which ' changes throughout a flow. For example, in a problem involving both fresh and
salt water, a fluid element will retain the same constant value for ' as it moves with the flow.
However, different fluid elements with different proportions of fresh and salt water will have
different values for ', and ' will not have the same constant value throughout the flow. Values
of ' for some different fluids and temperatures are given in the appendix.
The dynamic viscosity, µ , has units of kg / ( m s )
N s / m 2 * and is the constant of
proportionality between a shear stress and a rate of deformation. In a Newtonian fluid, µ is a
function only of the temperature and the particular fluid under consideration. The problem of
relating viscous stresses to rates of fluid deformation is relatively difficult, and this is one of the
few places where we will substitute a bit of hand waving for mathematical and physical logic.
If the fluid velocity, u , depends only upon a single coordinate, y , measured normal to u , as
shown in Figure 1.2, then the shear stress acting on a plane normal to the direction of y is given
by
-
µ

du
dy

(1.1)

Later in the course it will be shown that the velocity in the
water beneath the board in Figure 1.1 varies linearly from
a value of zero on the reservoir bottom to the board
velocity where the water is in contact with the board.
Together with Equation (1.1) these considerations show
that the shear stress, -, in the fluid (and on the board
surface) is a constant that is directly proportional to the
board velocity and inversely proportional to the reservoir
depth. The constant of proportionality is µ . In many
problems it is more convenient to use the definition of
kinematic viscosity

µ /'

(1.2)

Figure 1.2 A velocity field in
which u changes only with the in which the kinematic viscosity, , has units of m2/s.
coordinate measured normal to Values of µ and  for some different fluids and
the direction of u .
temperatures are given in the appendix.

*

A Newton, N, is a derived unit that is related to a kg through Newton's second law, F
ma .
Thus, N
kg m / s 2 .


Chapter 1 — Introduction

1.3

Surface tension, ), has units of N / m
kg / s 2 and is a force per unit arc length created on an
interface between two immiscible fluids as a result of molecular attraction. For example, at an
air-water interface the greater mass of water molecules causes water molecules near and on the
interface to be attracted toward each other with greater forces than the forces of attraction
between water and air molecules. The result is that any curved portion of the interface acts as
though it is covered with a thin membrane that has a tensile stress ). Surface tension allows a
needle to be floated on a free surface of water or an insect to land on a water surface without
getting wet.
For an example, if we equate horizontal pressure and
surface tension forces on half of the spherical rain droplet
shown in Figure 1.3, we obtain

p %r 2
) 2%r

(1.3)

in which
p = pressure difference across the interface.
This gives the following result for the pressure difference:
Figure 1.3 Horizontal pressure and
surface tension force acting on half
of a spherical rain droplet.


p


2)
r

(1.4)

If instead we consider an interface with the shape of a half circular cylinder, which would occur
under a needle floating on a free surface or at a meniscus that forms when two parallel plates of
glass are inserted into a reservoir of liquid, the corresponding force balance becomes

p 2r
2)

(1.5)

which gives a pressure difference of

p


)
r

(1.6)

A more general relationship between
p and ) is given by

p
)

1
1

r1 r2

(1.7)

in which r1 and r2 are the two principal radii of curvature of the interface. Thus, (1.4) has
r1
r2
r while (1.6) has r1
r and r2
 . From these examples we conclude that (a)
pressure differences increase as the interface radius of curvature decreases and (b) pressures are
always greatest on the concave side of the curved interface. Thus, since water in a capillary tube
has the concave side facing upward, water pressures beneath the meniscus are below atmospheric
pressure. Values of ) for some different liquids are given in the appendix.
Finally, although it is not a fluid property, we will mention the “gravitational constant” or
“gravitational acceleration”, g , which has units of m/s2. Both these terms are misnomers because


1.4

Chapter 1 — Introduction

g is not a constant and it is a particle acceleration only if gravitational attraction is the sole force
acting on the particle. (Add a drag force, for example, and the particle acceleration is no longer
g . ) The definition of g states that it is the proportionality factor between the mass, M , and
weight, W, of an object in the earth's gravitational field.
W
Mg

(1.8)

Since the mass remains constant and W decreases as distance between the object and the centre
of the earth increases, we see from (1.8) that g must also decrease with increasing distance from
the earth's centre. At sea level g is given approximately by
g
9.81 m / s 2

(1.9)

which is sufficiently accurate for most civil engineering applications.

Flow Properties
Pressure, p , is a normal stress or force per unit area. If fluid is at rest or moves as a rigid body
with no relative motion between fluid particles, then pressure is the only normal stress that exists
in the fluid. If fluid particles move relative to each other, then the total normal stress is the sum
of the pressure and normal viscous stresses. In this case pressure is the normal stress that would
exist in the flow if the fluid had a zero viscosity. If the fluid motion is incompressible, the
pressure is also the average value of the normal stresses on the three coordinate planes.
Pressure has units of N /m 2
Pa , and in fluid mechanics a positive pressure is defined to be a
compressive stress. This sign convention is opposite to the one used in solid mechanics, where
a tensile stress is defined to be positive. The reason for this convention is that most fluid
pressures are compressive. However it is important to realize that tensile pressures can and do
occur in fluids. For example, tensile stresses occur in a water column within a small diameter
capillary tube as a result of surface tension. There is, however, a limit to the magnitude of
negative pressure that a liquid can support without vaporizing. The vaporization pressure of a
given liquid depends upon temperature, a fact that becomes apparent when it is realized that
water vaporizes at atmospheric pressure when its temperature is raised to the boiling point.
Pressure are always measured relative to some fixed datum. For example, absolute pressures are
measured relative to the lowest pressure that can exist in a gas, which is the pressure in a perfect
vacuum. Gage pressures are measured relative to atmospheric pressure at the location under
consideration, a process which is implemented by setting atmospheric pressure equal to zero.
Civil engineering problems almost always deal with pressure differences. In these cases, since
adding or subtracting the same constant value to pressures does not change a pressure difference,
the particular reference value that is used for pressure becomes immaterial. For this reason we
will almost always use gage pressures.*

*

One exception occurs in the appendix, where water vapour pressures are given in kPa absolute. They
could, however, be referenced to atmospheric pressure at sea level simply by subtracting from each
pressure the vapour pressure for a temperature of 100C (101.3 kPa).


Chapter 1 — Introduction

1.5

If no shear stresses occur in a fluid, either because the fluid has no relative motion between
particles or because the viscosity is zero, then it is a simple exercise to show that the normal
stress acting on a surface does not change as the orientation of the surface changes. Consider, for
example, an application of Newton's second law to the two-dimensional triangular element of
fluid shown in Figure 1.4, in which the normal stresses )x , )y and )n have all been assumed
to have different magnitudes. Thus (Fx
m ax gives
)x
y )n
x 2 
y 2 cos 


'

x

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