Table Of ContentSpringer Tracts in Civil Engineering
Sandro Longo
Maria Giovanna Tanda
Luca Chiapponi
Problems in
Hydraulics
and Fluid
Mechanics
Springer Tracts in Civil Engineering
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Sandro Longo Maria Giovanna Tanda
(cid:129) (cid:129)
Luca Chiapponi
Problems in Hydraulics
and Fluid Mechanics
123
SandroLongo Maria Giovanna Tanda
DepartmentofEngineeringandArchitecture DepartmentofEngineeringandArchitecture
(DIA) (DIA)
University of Parma University of Parma
Parma, Italy Parma, Italy
Luca Chiapponi
DepartmentofEngineeringandArchitecture
(DIA)
University of Parma
Parma, Italy
ISSN 2366-259X ISSN 2366-2603 (electronic)
SpringerTracts inCivil Engineering
ISBN978-3-030-51386-3 ISBN978-3-030-51387-0 (eBook)
https://doi.org/10.1007/978-3-030-51387-0
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Preface
The exercises of this textbook, fully worked, can be a useful complement to the
Lessons and are suitable to accustom the students to perform the calculations
applied to situations of technical interest. In some exercises, we propose two dif-
ferent methods of solving, to highlight that the level of complexity of the calcu-
lations is often related to the choice of method. In general, we have preferred the
simplestmethod.Chapters1and2dealwithexercisesonforcesonflatandcurved
surfaces and humps. Chapter 3 is entirely dedicated to floating bodies. Chapter 4
deals with some classic exercises that require the application of balance of linear
and angular momentum, in inertial and non-inertial references. Chapter 5 analyzes
pipeline systems, with particular applications to industrial plants in Chap. 6.
Chapter7dealswithhydraulicsystemswithmachines(pumpsandturbines).Thisis
followedbyChap.8,dedicatedtotransientphenomenainclosedpipelines.Chapter
9 deals with flows in open channels. The Appendices contain some data and for-
mulas of practical interest.
ThebookisaddressedtoundergraduatesandgraduatesinEngineeringSciences.
In many exercises, some parameters are given intermsof C and C , that are,for
u pu
example, the last and second-last digit of the registration number. This diversifies
the calculations of the students during the written tests.
Parma, Italy Sandro Longo
March 2020 Maria Giovanna Tanda
Luca Chiapponi
vii
Introduction
FluidMechanicsdealswiththebehaviouroffluidundertheactionofforces,atrestor
inmotion.Fluidsareclassifiedintoliquids,gasesandvapours.Liquidsaredifficultto
compress and generally the range of pressure is limited so as to consider them as
incompressible, although in some practical applications their compressibility is
invoked to explain the behaviour of elastic waves in water hammer. Gases can be
easilycompressed,althoughthechangeofpressureduringflowisoftensomodestthat
nosignificantchangeofvolumetakesplace,andthemotionisdefinedisochoric:in
isochoricflow,gasesbehavelikeliquids.Vapoursaregaseswhichcanbecondensed
by increasing the pressure without lowering temperature. Hence, in isothermal or
almostisothermalflows,vapourscanaffordtophasetransitionwiththecoexistenceof
aliquidandagasphase.Theiranalysisbelongstomulti-phasesystems.
Hydraulics deals with water, although concepts derived for water are broadly
adopted for the Fluid Dynamics of liquids similar to water, like oil and kerosene.
In a fluid at rest the internal stresses reduce to pressure, a normal force per unit
surface acting normally to any surface, and the shear stress is null. In a fluid in
motiontheinternalstressesincludealsoshearstress,butnotorqueperunitsurface
is considered. A special category of fluids, called polar fluids, is described by
includingalsotorqueperunitsurface.Apolarfluidiscapableoftransmittingstress
couples and being subjected to body torques.
Fluids are continua and differ from solid since in solids a stress determines a
strain proportional (linearly or not) to the stress, whereas in fluids a stress deter-
mines a rate of strain proportional (linearly or not) to the stress. Linear propor-
tionality between strain rate and stress is a characteristic of Newtonian fluids.
Non-Newtonianfluidsshowamorecomplexrelationbetweenstrainrateandstress,
in some cases involving also stress varying with strain and in time.
Thefundamentalequationsadoptedforsolvingproblemsaremassconservation,
linear momentum and angular momentum balances, energy conservation. These
equations are the consequence of some principles (e.g. linear momentum balance
equation is derived from the hypothesis of homogeneity of space), and require
boundary and initial conditions. In several real cases, some material properties are
necessarytomodel,e.g.thereactionofafluidtoastressstateortheheatfluxdueto
ix
x Introduction
animposedthermalfield.Thesearedescribedbyconstitutiveequations,mechanical
and thermal, which contain some material parameters, like viscosity, which is
evaluated through experiments. The constitutive equations must have a
non-dimensional formulation, satisfy coordinate indifference (have a tensorial for-
mulation), frame-reference indifference (material indifference) and satisfy the sec-
ond principle of thermodynamics.
SI Units and Dimensions
The Système International d’Unités (SI) has been introduced with the aim of a
common language for units, although in several countries former systems are still
used. The SI is based on seven fundamental quantities: mass, length, time, electric
current, thermodynamic temperature, amount of substance and luminous intensity.
Fundamental quantities are independent and are sufficient to describe all other
quantities,definedasderivedquantities.ThefundamentalunitsarelistedinTable 1.
Table1 FundamentalquantitiesandunitsofmeasurementsintheSI
Fundamentalquantities Symbol Denominationoftheunit Symboloftheunit
Length L metre m
Mass M kilogram kg
Time T second s
Thermodynamictemperature H kelvin K
Electriccurrent I ampere A
Luminousintensity C candela cd
Amountofsubstance mol mole mol
However, recently (in 2019), the fundamental units in SI have been defined in
terms of seven dimensional constants. The conversions have been introduced once
therequiredaccuracyinmeasuringtheseconstantshasbeenachieved.Althoughthe
seven dimensional constants could be adopted as fundamental, the past seven fun-
damentalunitsarestillinuse,buttheyarenowdefinedintermsof(1)thecaesium
hyperfine frequency,Dm ;(2) thespeed oflightinvacuum,c;(3) the Planck con-
Cs
stant,h;(4)elementarycharge,e;(5)Boltzmannconstant,k;(6)Avogadroconstant,
N ,and(7)theluminousefficacyofadefinedvisibleradiation,K .
A cd
For instance, the caesium-133 hyperfine frequency is equal to
Dm ¼Dmð133CsÞ ¼9192631770Hz;
Cs hfs
and the time unit is expressed as
Introduction xi
Fig.1 LogooftheSIcon-
stants(internalring),inforce
since20May2019todefine
thefundamentalunits(exter-
nalring)
9192631770
1s¼ :
Dm
Cs
We could say that the time of rotation of the Earth about its axis is
0:79(cid:2)1015Dm(cid:3)1, but it is a lot more in everyday life to say that it is 86400s.
Cs
Figure 1 shows the logo used to disseminate the new way by the Bureau
International des Poids et Mesures (BIPM). The BIPM is an international orga-
nization established by the Metre Convention, through which Member States act
together on matters related to measurement science and measurement standards.
Units are necessary to express the size of a measured quantity with respect to a
knownquantity:ifamanis1.75mtall,wearesimplysayingthattheratiobetween
hisheightandthefundamentalunit(metre)is1.75.Theuseofadifferentsystemof
units can modify the number, but not the intrinsic height of the man, in the sense
that the same man is defined as 5 3/4 inches tall in the Imperial System, not being
different from the 1.75 m tall man in SI units.
In order to avoid misunderstanding, it is convenient to think in terms of mass,
length, time, force, etc., instead of thinking in terms of the units. The use of
dimensions instead of the units renders the analysis independent on the systems
used, and is helpful in checking the correctness of the equations. Thus
distance
velocity¼ ;
time
