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Titles of related interest:

J A Fox: An Introduction to Engineering Fluid Mechanics
B Henderson-Sellers: Reservoirs
N T Kottegoda: Stochastic Water Resources Technology


Essentials of Engineering
Hydraulics
JONAS M. K. DAKE
B.Se (Eng.) (London); M.Sc.Teeh. (Man.); Se.D. (M.LT.)

M
ANSTI


© Jonas M. K. Dake 1972,1983

All rights reserved. No part of this publication may
be reproduced or transmitted, in any form or by any means,
without permission.
First edition 1972
Reprinted with corrections 1974
Second edition 1983

Published by
THE MACMILLAN PRESS LTD
London and Basingstoke
Companies and representatives throughout

the world.
In association with;
African Network of Scientific and Technological Institutions
P.O. Box 30592
Nairobi
Kenya
ISBN 978-0-333-34335-7

ISBN 978-1-349-17005-0 (eBook)
DOI 10.1007/978-1-349-17005-0

Typeset by MULTIPLEX techniques ltd


Contents
Foreword to the First Edition
Preface to the Second (Metric) Edition
Preface to the First Edition
List of Principal Symbols

IX

X
Xl
XIII

PART ONE ELEMENTARY FLUID MECHANICS

1. Fundamental concepts of fluid mechanics
1.1

1.2
1.3
1.4
1.5
1.6
1.7
1.8

2

Introduction
The Continuum
Units of Measurement
Some Important Fluid Properties
Transfer Phenomena
Types of Flow
Boundary Layer Concepts and Drag
Fluids in Static Equilibrium

Methods of analysis
2.1 Control Volume Concepts
2.2 The Basic Physical Laws of Mass, Energy and Momentum
Transport

2.3 Conservation of Mass
2.4 The Linear Momentum Principle
2.5 The Principle of Conservation of Energy: First Law of
2.6

3

3
3
4
8
13
18
20
25

Thermodynamics
The Moment of Momentum Concept

40

41
42
45
52
60

Steady incompressible flow through pipes
3.1
3.2
3.3
3.4
3.5

Introduction
Enclosed Flow at a Low Reynolds Number

Momentum and Energy Correction Factors
Pipe Flow at a High Reynolds Number
Analysis of Pipe Systems

63
64
70
71
87


Contents

vi

4

Flow in non-erodible open channels
401
402
403
404
405
406

5

95
107
113

123
133
137

Experimental fluid mechanics
501
502
503
504
505

6

Introduction
Momentum Concepts
Energy Concepts
Gradually Varied Flow
Open Channel Surges
Miscellaneous Information

Introduction
Dynamic Similarity
Physical Significance of Modelling Laws
Models of Rivers and Channels
Dimensional Approach to Experimental Analysis

142
143
146
160

165

Water pumps and turbines
601
602
603
604
605
606

Introduction
The Pelton Wheel Turbine
Reaction Machines
Selection and Installation of Pumps and Turbines
Cavitation
Pumping from Wells

173
175
178
190
196
205

PART TWO SPECIALIZED TOPICS IN CIVIL
ENGINEERING

7

Flow in erodible open channels

701
702
703
704

8

Properties of Sediments
Mechanics of Sediment Transport
Design of Stable Alluvial Channels
Moveable Bed Models

213
219
229
236

Physical hydrology and water storage
801
802
803
804
805
806
807
808

Introduction
Precipitation
Evaporation and Transpiration

Infiltration
Surface Run-off (Overland Flow)
Stream Run-off
Storage and Streamflow Routeing
Design Criteria

242
244
249
252
254
258
266
276


Contents

9

vii

Groundwater and seepage
9.1
9.2
9.3

Introduction
Fundamentals of Groundwater Hydraulics
Some Practical Groundwater Flow Problems

282
285
298

10 Sea waves and coastal engineering
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8

Introduction
Wave Generation and Propagation
Small Amplitude Wave Theory
Finite Amplitude Waves
Changes in Shallow Water
Wave Reflection and Diffraction
Coastal Processes
Coastal Enginee ring

312
315
319
326
327
331

334
340

11 Fundamental economics of water resources
development
11.1 Introduction
11.2 Basic Economic and Technological Concepts (Decision
Theory)

346
349

Problems
Appendix: Notes on Flow Measurement
A.l Velocity-Area Methods
A.2 Direct Discharge Methods

401
404

Index

412


Foreword to the First Edition
by

J. R. D. Francis, B.Sc. (Eng.), M.Sc., M.I.C.E., F.R.MeLS.
Professor of Fluid Mechanics arid Hydraulic Engineering,

Imperial College of Science and Technology, London

It is a pleasure to have the opportunity of commending this book. The author, a
friend and former student of mine, has attempted to bring out the principles of
physics which are likely to be of future importance to hydraulic engineering
science, with particular reference to water resources problems. With the greater
importance and complexity of water resource exploitation likely to occur in the
future, our analysis and design of engineering problems in this field must become
more exact, and there are several parts of Dr. Dake's book which introduce new
ideas. In the past half-century, the science of fluid mechanics has been largely
dominated by the demands of aeronautical engineering; in the future it is not
too much to believe that the efficient supply, distribution, drainage and re-use
of the world's water supply for the benefit of an increasing population will
present the most urgent of problems to the engineer.
I feel particularly honoured, too, in that this book must be among the first
technical texts to come from a young and flourishing university, and is, I think,
the first in hydraulic engineering to come from Africa. Over many years,
academics in Britain and elsewhere have attempted, with varying success, to help
the establishment of degree courses at Kumasi, and to produce skilled technological manpower. That a book of this standard should now come forward is a
source of pleasure to all those who have helped, and an indication of future
success.

J. R. D. FRANCIS
1972

ix


Preface to the Second Edition

The Second Edition of Essentials ofEngineering Hydraulics has retained the
primary objectives and structure of the original book. However, the rational
metric system of units (Systerne International d'Unites) has been adopted
generally although a few examples and approaches have retained the imperial
units.
The scope of the book has been increased by inclusion of section 1.8, 'Fluids
in Static Equilibrium' and sub-sections 8.7.3 and 9.3.5 'Routeing of Floods in
River Channels' and 'The Transient State of the Well Problem', respectively.
There has been general updating.
A guide to the solution of the tutorial problems at the end of the book is
available for restricted distribution to lecturers upon official request to the
publisher.
Jonas M. K. Dake
Nairobi 1982

x


Preface to the First Edition

Teaching of engineering poses a challenge which, although also relevant to the
developed countries, carries with it enormous pressures in the developing
countries. The immediate need for technical personnel for rapid development
and the desire to design curricula and training methods to suit particular local
needs provide strong incentives which could, without proper control, compromise
engineering science and its teaching in the developing world.
The generally accepted role of an engineering institution is the provision of the
scientific foundation on which the engineering profession rests. It is also recognized that the student's scientific background must be both basic and environmental. In other words, engineering syllabuses must be such that, while not
compromising on basic engineering science and standards, they reflect sufficient
background preparation for the appropriate level of local development.

This text has been written to provide in one volume an adequate coverage of
the basic principles of fluid flow and summaries of specialized topics in hydraulic
engineering, using mainly examples from African and other developing countries.
A survey of fluid mechanics and hydraulics syllabuses in British universities
reveals that the courses are fairly uniform up to second year level but vary widely
in the final year. This book is well suited to these courses. Students in those
universities which emphasize civil engineering fluid mechanics will also find this
book useful throughout the whole or considerable part of their courses of study.
Essentials ofEngineering Hydraulics can be divided into two parts. Part I,
Elementary Fluid Mechanics, emphasizes fundamental physical concepts and
details of the mechanics of fluid flow. A good knowledge of general mechanics
and mathematics as well as introductory lectures in fluid mechanics covering
hydrostatics and broad definitions are assumed. Coverage in Part I is suitable up
to the end of the second year (3-year degree courses) or third year (4-year degree
courses) of civil and mechanical engineering undergraduate studies. Part lIon
Specialized Topics in Civil Engineering is meant mainly for final-year civil
engineering degree students. Treatment is concentrated on discussions of the
physics and concepts which have led to certain mathematical results. Equations
are generally not derived but discussions centre on the merits and limitations of
the equations.
The general aim of the book is to emphasize the physical concepts of fluid
flow and hydraulic engineering processes with the hope of providing a foundation
which is suitable for both academic and non-academic postgraduate work. Toxi


xii

Preface to the First Edition

wards this end, serious efforts have been made to steer a middle course between

the thorough mathematical approach and the strictly down-to-earth empirical
approach.
Chapter 11 gives an introduction to the fundamental economics of water
resources development which is a very important topic at postgraduate level. I
feel that economics and decision theory must be given more prominence in
undergraduate engineering curricula especially in countries where young graduates
soon find themselves propelled to positions of responsibility and decision
making.
In an attempt to make this book comprehensive and yet not too bulky and
expensive, I have resorted to a literary style which uses terse but scientific words
with the hope of putting the argument in a short space. I have also followed
rather the classroom 'hand-out' approach than the elaborate and sometimes longwinded approach found in many books.
'The author of any textbook depends largely upon his predecessors' - Francis.
Existing books and other publications from which I have benefited are listed at
the end of each chapter in acknowledgement and as further references for the
interested reader. The tutorial problems have been derived from my own class
exercises, homework and class tests at M.I.T. and from other sources, all of which
are gratefully acknowledged. In the final chapter, problems 3.18, 4.23, 4.24,
5.18,5.19,6.3,6.14,8.1,8.2 and 8.3 have been included with the kind permission of the University of London. All statements in the text and answers to
problems, however, are my responsibility.
I wish to thank Prof. J. R. D. Francis of Imperial College, London and
Dr. J. O. Sonuga of Lagos University and other colleagues who read the manuscript in part or whole and made many useful suggestions. The encouragement of
Prof. Francis, a former teacher with continued interest in his student and the
external examiner in Fluid Mechanics and Hydraulics as well as the moderator
for Civil Engineering courses at U.S.T., has been invaluable. Mr. D. W. Prah of
the Department of Liberal and Social Studies, U.S.T., made some useful comments on the use of economic terms in Chapter 11. The services of the clerical
staff and the draughtsmen of the Faculty of Engineering, U.S.T., especially of
Messrs. S. K. Gaisie and S. F. Dadzie during the preparation of the manuscript
and drawings are also gratefully acknowledged.
Finally, I wish to express my sincere gratitude to the University of Science

and Technology, Kumasi, whose financial support has made the production of
this book possible.

University ofScienceand Technology, Kumasi, Ghana

Jonas M. K. Dake
1972


list of Principal Symbols
cross-section area of a jet (L 2)
area (L 2)
acceleration (L/t 2 ) , area (L 2), wave amplitude (L)
amplitude of wave beat envelope (L)
B
b

top width of a channel (L)
bottom width of a channel (L)

C

Chezy coefficient (L 2/t); wave velocity (L /t)
group velocity (waves) (L/t)
specific heat at constant pressure (L 2 /Tt 2 )
Capital Recovery Factor
specific heat at constant volume (L 2 /Tt 2 )
concentration of mass, surge wave speed (L/t) , speed of sound (L/t)
coefficient of drag
coefficient of discharge

coefficient of drag (friction)
control surface
control volume
coefficient of velocity

CG

Cp

C.R.F.
Cv

C

Co
Cd
Cf

c.s.
c.v.

ev

I

molecular mass conductivity (M/Lt), pipe diameter (L), drag (ML/t 2 )
sieve diameter which pass N% of soil sample (L)
median sand particle size (L)
geometric mean size (sand) (L), depth (L), drawdown (L)
geometric mean size (sand) (L)

diameter of a nozzle (L)
E

energy (ML 2/t 2 ) , specific energy (L), Euler number, rate of evaporation (L/t);
wave energy (M/t 2 ) , modulus of elasticity (M/Lt 2 )
rate of transmission of wave energy (ML/t 3)
thermal eddy diffusivity (L 2 /t )
mass edd y diffusivity (L 2/t)
kinetic energy (ML 2 /t 2 )
potential energy (ML 2 /t 2 )
exponential constant (= 2.71828)
vapour pressure (mmHg), void ratio

F
F'

force (ML/t 2 ) , Froude number, fetch (L)
densimetric Froude number
friction factor
infiltration capacity (L/t or L3/t)
silt factor

f

J:
g

go

acceleration due to gravity (L/t 2 )

constant - 32.174 Ibm/slug
enthalpy (L 2/ t2), total head (L), wave height (L)
head developed or consumed by a rotodynamic machine (L)
pum p head (L)
theoretical head of a rotodynamic machine (L)
xiii


xiv

n,

H sv
HT
h

hf

I

List of Principal Symbols
static lift (L)
net positive suction head (NPSH) (L)
turbine head (L)
head of water above spillway crest (L), hydraulic head (L)
friction head loss (L)

io

moment 0 f inertia (ML"), infiltration amount (L 3), rate of interest

seepage (hydraulic) gradient (LIL)
rainfall intensity (Lit)

J

mechanical equivalent of heat, (ML"It 2 )

K

thermal molecular conductivity (MLITt 3 ) , coefficient of hydraulic resistances,
modulus of compressibility (MILt")
nozzle (loss) coefficient
coefficient of wave refraction
coefficient of permeability (superficial) (Lit); wave number (21TIL)
size of roughness (L)

Kn
Kr

k

ks

L
Lo
1

Ibm
lbf
M

MB
Me
MP
MRS
MRT

m

N
Ns
Nu

n

ns

o

l5

OMR
P
Pu

p
ppm
Pat
Pv

Q

Qu

q

q

lib
qs

length (L), wavelength (L)
wavelength in deep water (L)
length (L)
pound mass (M)
pound force (MLlt")
mass (M), Mach number
marginal benefit
marginal cost
marginal productivity
marginal rate of substitution
marginal rate of transformation
mass (M), mass rate of flow (Mit), hydraulic mean depth or radius (L)
speed of rotation (rev/min)
specific speed (turbines)
unit speed (rotodynamic machines)
porosity, ratio of wave group velocity to phase velocity (CalC)
specific speed (pumps)
outflow (L 3 It)
average outflow (L 3 It)
operation, maintenance and repairs
force (MLlt") , wetted perimeter (L), power (ML"lt 3 ) , principal investment,

precipitation (rainfall)
unit power
pressure (MILt")
parts per million
atmospheric pressure (MILt")
vapour pressure (MILt 2 )
discharge rate (L 31t) , heat (ML 2It 2 )
unit discharge
discharge per unit width (L" It)
velocity vector (Lit)
rate of bed load transport per unit width (L 2 It)
rate of suspended load transport per unit width


list of Principal Symbols

xv

universal gas constant (L 2 1t 2 T), Reynolds number, rainfall amount (L 3 )
Reynolds number based on shear velocity (v'd/v)
Richardson number
degrees Rankine
radius (L), (suffix) ratio of model quantity/prototype quantity

S

specific gravity, storage (L 3 ) , degree of saturation, storage constant
slope of energy grade line (LIL)
shape factor (sand grains)
flow net shape factor

bed slope (LIL)
specific gravity of solids

Sf

SF
SF
So

s,

temperature (1), wave period (t), torque (ML 2/t 2), transmissibility (L 2/t)
time (t), wind duration (t)
time of concentration (t)
recurrence interval (t)
duration of rainfall (t)

u

internal energy (ML 2 It 2 ) , wind speed (Lit)
specific internal energy (L 2 I t 2 )

u

volume (L 3 )
volume of voids (L 3)
velocity (Lit)
time average velocity (turbulent flow) (Lit) (sections 1.5.4,4.1.3,7.2.3);
sectional average velocity (sections 3.1, 3.2, 3.3, 3.4.1, 3.4.2)
radial (flow) component of velocity in a rotodynamic machine (Lit)

velocity of nozzle jet (Lit)
seepage velocity (ch, 9) (Lit)
shear velocity V(To / p) (Lit)
absolute surge wave speed (Lit), rotodynamic whirl component of velocity (Lit)
weight (MLlt 2 ) , Weber number, work (ML 2It2 )
work against pressure (ML 2 1t2 )
work against shear (ML 2 1t 2 )
shaft work (ML 2It 2 )
settling velocity (sand particles) (Lit)
Y
Yc

Yn
Yo

y

Y'
6.Z or
~

am

{3

'Y
'Ys
6

e

€s

8

z;

distance measured from wall (L)
critical depth (L)
uniform (normal) depth (L)
depth (generally) in an open channel (L)
centroid of section measured from water surface (L)
centre of pressure measured from water surface (L)
height of weir (L)
summation of
approaches (equivalent or equal to)
thermal molecular diffusivity (L 2 It), angle
mass molecular diffusivity (L 2 It)
angle, constant of proportionality in es =(3e
specific (unit) weight (MIL 2t2 )
specific weight of solid matter (MIL 2 t 2 )
boundary layer thickness (L)
eddy kinematic viscosity (L 2 It)
eddy diffusity for suspended load (L 2 It)
angle, temperature (1)


xvi
11
11h
11m

J..L

v
P
Ps
(J

ac
ag

"Tc
"0worn
t;

list of Principal Symbols
efficiency, small amplitude wave form (L)
hydraulic efficiency
mechanical efficiency
dynamic molecular viscosity (MILt), discharge factor
kinema tic molecular viscosity (L 2 It)
density (MIL 3)
density of solid matter (MIL 3 )
surface tension (Mlt 2 ) , standard deviation, wave number (2n/T)
critical cavitation number
geometric standard deviation
shear stress (MILt 2 )
critical shear stress (MILt 2 )
wall shear stress (MILt 2 )
angular velocity (rad/t)
Cauchy number

PART ONE:
Elementary Fluid Mechanics


1 Fundamental Concepts of
Fluid Mechanics

1.1 Introduction
Matter is recognized in nature as solid, liquid or gas (or vapour). When it exists in
a liquid or gaseous form, matter is known as a fluid. The common property of all
fluids is that they must be bounded by impermeable walls in order to remain in
an initial shape. If the restraining walls are removed the fluid flows (expands)
until a new set of impermeable boundaries is encountered. Provided there is
enough fluid or it is expandable enough to fill the volume bounded by a set of
impermeable walls, it will always conform to the geometrical shape of the
boundaries. In other words, a fluid by itself offers no lasting resistance to change
of shape. The essential difference between a liquid and a gas is that a given mass
of the former occupies a fixed volume at a given temperature and pressure whereas
a fixed mass of a gas occupies any available space. A liquid offers great resistance
to volumetric change (compression) and is not greatly affected by temperature
changes. A gas or a vapour, on the other hand, is easily compressed and responds
markedly to temperature changes.
The above definitions and observations indicate that the ultimate shape and
size of a fixed mass of a fluid under a deforming force depend on the geometry of
the container and on the compressibility of the fluid. The rate at which a fluid
assumes the new shape is governed by the property known as viscosity. Viscosity
is a molecular property of a fluid which enables it to resist rapid deformation and
this is discussed more fully in Section 1.4.

1.2 The Continuum
Fluids are composed of discretely spaced molecules in constan~ motion and
collision. An exact analysis of fluid motion would therefore require knowledge
of the behaviour of each molecule or group of molecules in motion. In most
3


4

Essentials of Engineering Hydraulics

engineering problems, however, average measurable indications of the general
behaviour of groups of molecules are sufficient. These indications can be conveniently assumed to arise from a continuous distribution of molecules, referred
to as the continuum, instead of from the conglomeration of discrete molecules
that exists in reality.
Thus in fluid mechanics generally, the terms density, pressure, temperature,
viscosity, velocity, etc. refer to the average manifestation of these quantities at a
point in the fluid as opposed to the individual behaviour of individual molecules
or particles. The adoption of the continuum model implies that all dimensions in
the fluid space are very large compared to the molecular mean free path (the
average distance traversed by the molecules between collisions). It also implies
that all properties of the fluid are continuous from point to point throughout a
given volume of the fluid.
1.3 Units of Measurement
Two main systems of measurement are used in engineering; the metric system
and the Imperial system. The metric system mainly will be used in this book but
examples based on the Imperial system are also included. The general analytical
principles are the same and the student should be familiar with both systems.
The SI system is basically a refined metric (mks) system.

As shown in Table 1.1 both the metric system of measurement and the imperial
system of measurement have two subdivisions. The difference lies in the units
used for measuring mass and sometimes length. In the metric cgs (centimetregramme-second) system the gramme is used as a unit of mass and the centimetre
as a unit of length. In the mks (metre-kilogramme-second) system the kilogramme and the metre are used respectively. Both use the second as the basic

Table 1.1
Units of measurement

The metric system
cgs
Mass
Length
Time
Force
Ternperature

gramme (g)
centimetre (em)
second (s)
dyne (dyne)
degree Kelvin (K)

SI (mks)
kilogramme (kg)
metre (m)
second (s)
newton (N)
degree Kelvin (K)

The Imperial system

British absolute
Mass
Length
Time
Force
Temperature

pound mass (Ibm)
foot (ft)
second (s)
poundal (pdl)
degree Rankine (0 R)

Engineers'
slug (slug)
foot (ft)
second (s)
pound force (lbf)
degree Rankine (0 R)


Fundamental Concepts of Fluid Mechanics

5

unit of time. In the Imperial system the pound mass is the unit of mass when the
so-called British absolute system is adopted and the slug is the unit in the so-called
engineers' system. In both cases the unit of length is the foot and the unit of
time is the second.
Newton's second law of motion states that a mass moving by virtue of an

applied force will accelerate so that the product of the mass and acceleration
equals the component of force in the direction of acceleration. In symbols,

F=ma

(1.1)

where F is force, m is mass and a is acceleration.
By definition if the mass is 1 gramme and the acceleration is 1 cm/s 2 , the
force is 1 dyne. Similarly 1 newton of force produces an acceleration of 1 m/s 2
in 1 kg mass; 1 poundal of force accelerates a pound mass, 1 ft/s 2 and a mass of
1 slug will require 1 lbf to produce an acceleration of 1 ft/s 2 •
Supposing one slug is equivalent to go pound mass. From equation (1.1)
l Ibf
l Ibf

or

= 1 slug x 1 ft/s 2
=go Ibm x 1 ft/s 2

(1.2)

But the pound force is defined in terms of the pull of gravity, at a specified
(standard) location on the earth, on a given mass of platinum. One pound mass
experiences a pull of 1 lbf due to standard gravitational acceleration of
g = 32·174 ft/s 2 . Thus
1 lbf = 1 Ibm x 32·174 ft/s 2

(1.3)

Since equations (1.2) and (1.3) define the same quantity, l lbf, it is obvious
that go = 32·174 Ibm/slug. This development shows that go is a constant and is
not necessarily equal to g which varies from location to location. This should be
expected since go relates two mass units and a given mass is the same anywhere
in the universe. The gravitational pulling force on the same mass however varies
from place to place. For instance, a 10 Ibm will weigh 10 lbf under standard
gravitational pull. The weight of the same 10 Ibm under any other gravitational
field, g, is given by
_

10 (Ibm)

2

W(Ibf) - go (Ibm/slug) g(ft/s )
where

g

= 31·0 ft/s 2
10

W =32.174 x 31·0 = 9·6351bf
Table 1.2 lists the SI equivalents of some of the units commonly used in fluid
mechanics.


6

Essentials of Engineering Hydraulics
Table 1.2
Sf equivalents of other Units

Physical
quantity
Length

Area

Volume

Mass
Density
Force

Pressure

Energy

Energy
Power
Temperature

Unit
angstrom
inch
foot
yard
mile

nautical mile
square inch
square foot
square yard
square mile
cubic inch
cubic foot
U.K. gallon (imperial)
U.S. gallon
pound
slug
pound/cubic foot
dyne
poundal
pound-force
kilogramme-force
atmosphere
torr
pound(f)/sq. in.
erg
calorie (LT.)
calorie (15° C)
calorie (thermochemical)
Bit.u.
foot poundal
foot pound (f)
horse power
degree Rankine
degree Fahrenheit

Equivalent
10- to m
0·0254m
0·3048 m
0·9144m
1·60934km
1·853 18 km
645·16 mm '
0.092903 m?
0·836127m 2
2·58999 km 2
1·63871 x 10- 5 m"
0·028 316 8 rrr'
0·004 546 092 m 3
0·003 785 4 m"
0·453 592 37 kg
14·5939 kg
16·0185 kg m?
10- 5 N
0·138 255 N
4·44822 N
9·80665 N
101·325, kN rn?
133·322 N m?
6894· 76 N m?
10- 7 J
4·1868 J
4·1855 J
4·184 J
1055·06 J

0·0421401 J
1·35582 J
745·700W
5/9 K
t/oF = ~TrC+ 32

There are two convenient ways of converting units of measurement from one
system to another. One is the method of dimensional representation and the
other a technique of forming the ratio of a unit and the proper numerical value
of another unit such that there is physical equivalence between the quantities.
There are four basic dimensions and in fluid mechanics the properties of fluids
can be expressed in terms of these basic dimensions of mass (M), length (L),
time (t) and temperature (T). In Table 1.3 some important properties commonly
met in fluid mechanics are listed together with their appropriate dimensions and
the relevant engineers' units and SI units.


7

Fundamental Concepts of Fluid Mechanics
Table 1.3
Dimensions and units of physical quantities

Property

Length

Mass
Time
Temperature

Velocity
Acceleration
Force

)

Symbol

Primary
dimensions

v
a
F

Dimensions
L
M
t
T
L/t
L/t 2
ML/t 2
M
Lt 2

Imperial
(engineers ')
units
ft

slug
s
of or oR

ft/s
ft/s 2
lbf
Ibf/ft 2

SI units
m
kg
s
K

m/s
m/s 2
N
N/m 2

Pressure

p

Density

p

Specific weight

'Y

Viscosity (dynamic)

JJ.

Viscosity (kinematic)

v

Modulus of elasticity

E

Surface tension

a

Universal gas constant

R

.s:
t

ft lbfl (mass ° R)

J /(kg K)

Specific heat at:

constant pressure
constant volume

cCp }

L2
t2T

Btu/ (mass ° R)

J/(kg K)

Specific internal energy

M

V

M
Lt

v

u

M
L 2t 2

L2
t

M
Lt 2

M
(2

2T

L

or lbf/in"

slug/It?
lbf/ft?
slug/It s
or lbf s/ft 2
ft2 Is
lbf /ft?
or lbf/in"
lbf/ft

kg/m"
kg/(m 2 s2

)

N s/m 2
m 2 /s

N/m 2
N/m

2

(2

Btu/mass

J/kg

As an example, the conversion factor between the engineers' units and SI units
of pressure will be derived both ways. First the conversion factor is obtained by
writing pressure dimensionally, substituting basic units of the Imperial system
and changing these units to equivalent metric units. Using Tables 1.2 and 1.3

_ ( M ) _ (Slug) = 32·174 (Ibm)
1
1
(P) - Lt2 - fts 2 - 2-20 (Ibm/kg) x 0-305 (m) x ~
Hence

1(lbf)=47.9 !L =47.9(newton)
ft 2
rns2
rn2

Alternatively

(P) =(lbf) == (lbf) (4·45 newton/lbf)

ft 2
(ft 2) (0·305 rn/ft)2


Essentials of Engineering Hydraulics

8
Hence

1 (lbf)
== 47-9 (newton)
ft 2
m2

The latter method is generally more intuitive and simpler to apply.

1.4 Some Important Fluid Properties
I

It has been stressed in discussing the concept of the continuum in Section 1.2
that the properties of materials normally employed in solving engineering

~surfoce
force

Centre of gravity

,

/' Normal component on area

,/ 8A at A (p8A)

"./

Tangential component on
area 8A at A (r8A)

Weight= yx volume

Fig. 1.1 Forces on an isolated free body of matter

problems are average manifestations of general molecular activity within the
matter and these are considered to be continuous in material space. If a
volume of matter is isolated as a free body (see Fig. 1.1) the force system
acting on the volume is made up of a body force' due to the gravitational pull on
the mass contained in the volume and surface forces over every element of area
bounding the volume. The gravitation pull per unit volume of the matter is
known as the specific (unit) weight 'Y (force/unit volume). Thus specific weight
depends on the gravitational field in contrast to density or specific mass p (mass/
unit volume) which is invariant so long as the volume of the given mass is not
increased or reduced.
Surface forces, in general, will have components normal and tangential to the
bounding surface. The normal component per unit of area is called the normal
stress. In fluids the stress is always considered compressive and is called pressure.
Pressure is a scalar quantity but the force associated with it is a vector quantity
which is always directed normal to the surface over which it acts. Pressure when
measured relative to atmospheric pressure is called gauge pressure but relative to
absolute zero it is called absolute pressure. The tangential component of the
surface force per unit area makes up what is known as shear stress.

Fundamental Concepts of Fluid Mechanics

9

1.4.1 Viscosity and Shear Stress
Shear stress is evident in the structure of any substance whose successive layers
are shifted laterally over each other. The existence of velocity changes in a particular direction, therefore, produces shear in a direction perpendicular to the
direction of velocity changes. In certain fluids commonly known as Newtonian
fluids, the shear stress on the interface tangential to the direction of flow is
proportional to the gradient (rate of change with distance) of velocity in a direction normal to the interface. In mathematical terms,
(1.4)
where
Tns

= the shear stress acting in the s direction in a plane normal to the

n direction
v = velocity
J.1 = the coefficient of proportionality called dynamic viscosity.

In Fig. 1.2 the relationship expressed in equation (1.4) is illustrated. The use
of the partial derivative avian emphasizes the point that the velocity v may be
variable in all directions of space and with time but it is its gradient normal to a
particular plane which produces shear on that plane. The direction of the velocity

y

Velocity,v
Direction of flow measurement, x

(a) Definition of shear stress. Ty x =fl-

if

~

c
d
,...-----------. ~v+8v
d

a

Ty x

d'

b

Shape at t
Shape at (,+8t)
(b) Angular deformation of an element located at point A

Fig. 1.2

c'


10

Essentials of Engineering Hydraulics

determines the direction of the shear. Shear is, therefore, a vector quantity with
magnitude and direction. Figure I.2{a) explains the relationship expressed in
equation (I.4) for a two-dimensional fluid flow in the xy plane.
Figure 1.2{b) which represents a magnification of the distortion of a fluid
element located at A of Fig. 1.2{a) may be used to show that avlay is equal to
the time rate of angular deformation or displacement. A shear stress T acts 'on the
top and the bottom of an infinitesimally small element of fluid, abed, in the
directions shown. The relative velocity between the top cd and the bottom ab is
ov. In a small interval of time ot, abed is distorted into a'b' c' d', The angular
distortion is given by

8e
for a small displacement. Thus

= dd'
8y

ov

8e=-8t
8y
dd'

since
In the limit as bt and by

~

0,

ae

=ov8t
av

at - ay

which is the rate of angular deformation.
The dynamic viscosity J.1 is dependent on temperature and pressure. The pressure dependence is negligible for liquids and small or negligible for most gases
and vapours except in cases of very high pressures. Figure 1.3 shows the variation
of dynamic viscosity with temperature for various fluids. The curves show that
the viscosity of a liquid decreases with increasing temperature but that of a gas
increases with increasing temperature. The ratio of dynamic viscosity to density
J.11 p is known as kinematic viscosity v because it has kinematic dimensions and
units only (L2 It). It is shown in Fig. 1.4 as a function of temperature. Kinematic viscosity appears quite frequently in fluid flow problems.
There are non-Newtonian fluids which do not exhibit direct proportionality
between shear stress and rate of angular deformation. Such behaviour is illustrated
in Fig. 1.5. These fluids include various types of plastics, dilatants and blood.
Most engineering fluids such as water, petroleum, kerosene, oils, air and steam
can however be considered Newtonian. The study of the behaviour of plastics
and non-Newtonian fluids is included in the discipline of rheology which is
beyond the scope of this book.

1.4.2 Surface Tension
The ability of the surface film of liquids to exert a tension gives a property
known as surface tension. It is commonly observed that small objects such as

\

\

\

-

-

,1-0

I-f---+----+-~~f-+---+----+--+--~

50

~~I- 30

~~~~~

~~ i,..oo"~20

0·4
Saturated
steam
100 kN/m 2

0·3

~f-_~~_C_H~_-r:::~~~~

0·2
0·15
- 10

L-L--~

_ 15
10.0
7.5

4

540

H2 1.00 kN/m 2

40

100

Temperature, (Oe)

Fig. 1.3 Dynamic molecular viscosity (s is the specific gravity at IS .SoC relative to water at
IS.SoC) (After Daily and Harleman)