Fluid Mechanics and the SPH Method: Theory and Applications

611 Pages·2012·3.917 MB·English

by  Violeau D.

FLUIDMECHANICSANDTHESPHMETHOD
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Fluid Mechanics and the
SPH Method
Theory and Applications
D. VIOLEAU
EDFR&D
1
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
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Preface
AmongallthebookswhichIhavereadinthefieldofphysics,especiallyfluid
mechanics, only the unrivalled volumes of the L.D. Landau and E. Lifshitz’s
course left me fully unfrustrated. This is presumably an effect of my irre-
pressiblerigour,whichwasundoubtedlyinheritedfrommyeducationinwhich
mathematicsprevailedoverphysics.AsIbecamemoreknowledgeableinfluid
mechanics, I found that the gaps which had been left unfilled in my general
approach to that construction were more and more flagrant, giving a feeling
of incompleteness. Scientific books and papers too often referred to a sector
in physics which I was not familiar with. Intuitively, however, they allowed
metoconsideranswerswhich,ultimately,Imanagedtocomeacrossinsome
other book, at the cost of painstaking effort. Since I became more aware of
that issue when I worked as a teacher, I contemplated writing myself a text
thatwouldmeetmyownexpectations;thus,Ihadtogainbasicknowledgeof
thesemissingelements,whichresultedinthefirstpartofthisbook.Itoccurs
in the simple, sometimes rather ingenuous form, of a theoretical fluid flow
physicscourseinwhichtheadmittedprincipleshavebeenasmuchreducedas
possible.
IbeganworkingontheSPH(smoothedparticlehydrodynamics)methodat
a complete loss. As I discovered its leading principles, I luckily noticed, as
I browsed through literature, that such a numerical approach had close links
totheoreticalmechanics;thatmademeevenkeenertounderstandthelatter’s
fundamentals, while becoming gradually more confident in the former. The
desire to gather my skills in SPH in a pedagogic text slowly gained ground;
thus,Ihadtomakeconstantreferencestothetheoreticallawsgoverningfluids,
whichitisbasedon.Sincetheideaofabookonhydraulicswassimultaneously
germinating, it became obvious to me that I had to deal with both aspects of
my research work as jointly as possible. That idea gave birth to this book.
Nevertheless, because of the huge corpus of fluid mechanics literature, along
withtheeverincreasingnumberofpublicationsabouttheSPHmethod,Ihad
to make a number of choices as regards the topics being dealt with herein.
Predictably enough, I have decided to give prevalence to those which I am
familiar with. That is why I will only discuss the weakly compressible or
incompressible,possiblyturbulentflows.
This book comprises two parts that refer to each other. The first, which
deals with the fundamentals of hydraulics, is composed of four chapters. It
is based on the elementary principles of both Lagrangian and Hamiltonian
mechanics. The specific laws governing a system of macroscopic particles
are built, then the major systems (statistical physics) involving the dissipa-
tive processes are explained. The continua are then discussed; lastly, a fairly
vi Preface
exhaustiveaccountofturbulenceisgiven.Thesecondpartcomprisesthelast
fourchapters.ItdisclosesthebasesoftheLagrangiannumericalmethodSPH
fromthecontinuousequationswhichareintroducedinthefirstpart,aswellas
fromdiscretevariationalprinciples,settingoutthemethod’sspecificproperties
of conservativity and invariance. Various numerical schemes are compared,
permanentlyreferringtophysicsasdealtwithinthefirstpart.Applicationsto
schematicinstancesarethendiscussed;ultimately,practicalapplicationstothe
dimensioningofcoastalandfluvialstructuresareconsidered.
This book is intended for scientists, doctoral students, teachers and engi-
neers who want to enjoy a rather unified approach to the theoretical bases
of hydraulics or who want to get more skills in the SPH method in the light
of these principles. In this form, it will probably look too complex to those
who fully rely on empirism; on the other hand, it is perhaps insufficiently
anchoredtomathematicalformalismforthehardcoreoftheoreticians.Ithink
it will nevertheless inspire many readers with a feeling of unity, answering
manyquestionswithoutanydetrimentalabstruseformalism.Ihopeitwillbe
helpful to young researchers and doctoral students involved in the increasing
developmentofinterestabouttheSPHmethod.
February2012 D.V.
Acknowledgements
Writingabookonone’sownisastrenuousandawkwardtask,andismislead-
ing, since the author is actually indebted to an endless list of people. Such is
thecasewiththistextbook;IamonlyafraidImightignoresomeoftheindirect
contributors.
I would like to express my gratitude first to the students and researchers
who have taken part in some of my works that this project has stemmed
from. I must emphasize two names: Réza Issa and Eun-Sug Lee; I enjoyed
somuchsupervisingtheirdoctoralworksintheSPHmethod.Thediscussions
developedinthesecondpartofthisbookowequitealottothem,aswellasto
severaltraineesfromhighschools,namelyDrSylvainFaure,HélèneFlament,
AdrienFoucart,AsvenGariah,AntoineJoly,ChristopheMagnin-Decugis,Dr
JilalHammouch,BastienPelliccioliandSandrinePiccon.
Thefirstpartofthisbookhasbeengreatlyinfluencedbothbymyteaching
activities and my confident colleagues Prof. Pierre-Louis Viollet and Prof.
MichelBenoit.
A number of scientists and engineers from various industry or university
organizationsprovidedmewitheffectiveassistanceinmyresearchworkabout
the SPH method. I must therefore mention the names of Dr John Biddis-
combe,DrClémentBuvat,DavidLatino,Prof.DominiqueLaurence,DrJean-
ChristopheMarongiu,DrCharlesMoulinec,DrEtienneParkinson,DrDamien
Pham Van Bang, Stéphane Ploix, Dr Ben Rogers and Prof. Peter Stansby.
I also want to express my thanks to Dr Kamel Abed-Meraïm, Dr Sofiane
Benhamadouche, Jacques Chorda, Dr Laurent David, Jean-Michel Hervouet,
DrMarie-MadeleineMaubourguet,DrEmmanueldeNanteuil,KhaledSaleh,
DrJuanUribeandDrPascalVezolles.
I want to express particular thanks to the valiant proofreaders: Martin
Ferrand, Prof. Francis Lebœuf, Prof. Pierre Ferrant, Prof. Stefano Sibilla
and Omar Mahmood. I am also grateful to Prof. Philippe Fraunié, Jean-Paul
Chabard and Prof. Mathieu Mory, who assisted them assessing this work,
withintheframeofanofficialapplicationfortheallocationofanaccreditation
tosuperviseresearch,aswellasThierryPain,whotookonthetranslationfrom
FrenchtoEnglish.
ThisbookwaswritteninthecourseofmyresearchactivitiesatEDFR&D,
withthesupportoftheEuropeanCommunitythroughtheMarieCurieproject
TransferofKnowledgeAgreementESPHI(EuropeanSPHInitiative).Alarge
part of that work was done at Manchester University (United Kingdom).
OwingtothefruitfulexchangeswhichIusedtoenjoywithmanyresearchers,
I am also much indebted to the SPHERIC working group (SPH European
ResearchInterestCommunity).
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Contents
Globalintroduction xiii
PARTI PHYSICSOFWEAKLYCOMPRESSIBLEFLUIDS
1 LagrangianandHamiltonianmechanics 3
1.1 Introduction 3
1.2 Leastactionprinciple 4
1.2.1 Inertialframesandprincipleofrelativity 4
1.2.2 Generalizedcoordinates.Lagrangianofasystem 6
1.2.3 Leastactionprinciple 8
1.2.4 Lagrangeequations 10
1.3 Mechanicsofasystemofparticles 14
1.3.1 Lagrangianofanisolatedparticle 14
1.3.2 Lagrangianofaparticlesystem 16
1.3.3 Newton’ssecondlaw 20
1.4 Conservationlaws 22
1.4.1 EnergyandHamiltonian 23
1.4.2 Hamiltonequations 27
1.4.3 Linearmomentumandangularmomentum 33
1.4.4 Investigatingagenericcase:thesimplependulum 37
1.5 Compositesystems 41
1.5.1 Centreofmass.Inertiatensor 41
1.5.2 Internalandexternalforces 49
1.5.3 Internalenergy 51
1.5.4 Pressureanddensity 55
2 Statisticalmechanics 60
2.1 Introduction 60
2.2 Statisticalbehaviouroflargesystems 61
2.2.1 Probabilitydensities 61
2.2.2 Liouville’stheorem.Boltzmann’sequation 63
2.2.3 Entropy 70
2.3 Thermodynamicalquantities 75
2.3.1 Boltzmanndistribution 75
2.3.2 Heatandwork 82
2.3.3 Stateequation 87
2.4 Dissipativesystems 91
2.4.1 Kineticcoefficients 91