Table Of ContentDIRECT NUMERICAL SIMULATIONS
OF GAS–LIQUID MULTIPHASE FLOWS
Accurately predicting the behavior of multiphase flows is a problem of immense
industrial and scientific interest. Using modern computers, researchers can now
studythedynamicsingreatdetail,andcomputersimulationsareyieldingunprece-
dentedinsight. Thisbookprovidesacomprehensiveintroductiontodirectnumer-
icalsimulationsofmultiphaseflowsforresearchersandgraduatestudents.
After a brief overview of the context and history, the authors review the gov-
erning equations. Aparticular emphasisis placedon the “one-fluid” formulation,
where a single set of equations is used to describe the entire flow field and in-
terface terms are included as singularity distributions. Several applications are
discussed,suchasatomization,dropletimpact,breakupandcollision,andbubbly
flows,showinghowdirectnumericalsimulationshavehelpedresearchersadvance
both our understanding and our ability to make predictions. The final chapter
givesanoverviewofrecentstudiesofflowswithrelativelycomplexphysics,such
as mass transfer and chemical reactions, solidification, and boiling, and includes
extensivereferencestocurrentwork.
GRE´TAR TRYGGVASON is the Viola D. Hank Professor of Aerospace and
MechanicalEngineeringattheUniversityofNotreDame,Indiana.
RUBEN SCARDOVELLI isanAssociateProfessorintheDipartimentodiInge-
gneriaEnergetica,NucleareedelControlloAmbientale(DIENCA)oftheUniver-
sita` degliStudidiBologna.
STE´PHANE ZALESKI is a Professor of Mechanics at the Universite´ Pierre et
MarieCurie(UPMC)inParisandHeadoftheJeanLeRondd’AlembertInstitute
(CNRSUMR7190).
DIRECT NUMERICAL SIMULATIONS
OF GAS–LIQUID MULTIPHASE FLOWS
GRE´TAR TRYGGVASON
UniversityofNotreDame,Indiana
RUBEN SCARDOVELLI
Universita`degliStudidiBologna
STE´PHANE ZALESKI
Universite´PierreetMarieCurie,Paris6
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
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(cid:1)C CambridgeUniversityPress2011
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Contents
Preface pageix
1 Introduction 1
1.1 Examplesofmultiphaseflows 3
1.2 Computationalmodeling 7
1.3 Lookingahead 18
2 Fluidmechanicswithinterfaces 21
2.1 Generalprinciples 21
2.2 Basicequations 22
2.3 Interfaces: descriptionanddefinitions 30
2.4 Fluidmechanicswithinterfaces 36
2.5 Fluidmechanicswithinterfaces: theone-fluidformulation 41
2.6 Nondimensionalnumbers 42
2.7 Thinfilms,intermolecularforces,andcontactlines 44
2.8 Notes 47
3 NumericalsolutionsoftheNavier–Stokesequations 50
3.1 Timeintegration 51
3.2 Spatialdiscretization 55
3.3 Discretizationoftheadvectionterms 59
3.4 Theviscousterms 61
3.5 Thepressureequation 64
3.6 Velocityboundaryconditions 69
3.7 Outflowboundaryconditions 70
3.8 Adaptivemeshrefinement 71
3.9 Summary 72
3.10 Postscript: conservativeversusnon-conservativeform 73
4 Advectingafluidinterface 75
4.1 Notations 76
v
vi Contents
4.2 Advectingthecolorfunction 77
4.3 Thevolume-of-fluid(VOF)method 81
4.4 Fronttracking 84
4.5 Thelevel-setmethod 87
4.6 Phase-fieldmethods 90
4.7 TheCIPmethod 91
4.8 Summary 93
5 Thevolume-of-fluidmethod 95
5.1 Basicproperties 95
5.2 Interfacereconstruction 98
5.3 Testsofreconstructionmethods 106
5.4 Interfaceadvection 108
5.5 Testsofreconstructionandadvectionmethods 122
5.6 Hybridmethods 128
6 Advectingmarkerpoints: fronttracking 133
6.1 Thestructureofthefront 134
6.2 Restructuringthefronts 143
6.3 Thefront-gridcommunications 145
6.4 Advectionofthefront 150
6.5 Constructingthemarkerfunction 152
6.6 Changesinthefronttopology 158
6.7 Notes 160
7 Surfacetension 161
7.1 Computingsurfacetensionfrommarkerfunctions 161
7.2 Computingthesurfacetensionofatrackedfront 168
7.3 Testingthesurfacetensionmethods 177
7.4 Moresophisticatedsurfacetensionmethods 181
7.5 Conclusiononnumericalmethods 186
8 Dispersebubblyflows 187
8.1 Introduction 187
8.2 Homogeneousbubblyflows 189
8.3 Bubblyflowsinverticalchannels 194
8.4 Discussion 201
9 Atomizationandbreakup 204
9.1 Introduction 204
9.2 Thread,sheet,andrimbreakup 205
9.3 High-speedjets 214
9.4 Atomizationsimulations 219
Contents vii
10 Dropletcollision,impact,andsplashing 228
10.1 Introduction 228
10.2 Earlysimulations 229
10.3 Low-velocityimpactsandcollisions 229
10.4 Morecomplexslowimpacts 232
10.5 Corolla,crowns,andsplashingimpacts 235
11 Extensions 243
11.1 Additionalfieldsandsurfacephysics 243
11.2 Imbeddedboundaries 256
11.3 Multiscaleissues 266
11.4 Summary 269
AppendixA Interfaces: descriptionanddefinitions 270
A.1 Two-dimensionalgeometry 270
A.2 Three-dimensionalgeometry 272
A.3 Axisymmetricgeometry 274
A.4 Differentiationandintegrationonsurfaces 275
AppendixB Distributionsconcentratedontheinterface 279
B.1 Asimpleexample 281
AppendixC Cube-choppingalgorithm 284
C.1 Two-dimensionalproblem 285
C.2 Three-dimensionalproblem 286
AppendixD Thedynamicsofliquidsheets: linearizedtheory 288
D.1 Flowconfiguration 288
D.2 Inviscidresults 288
D.3 ViscoustheoryfortheKelvin–Helmholtzinstability 293
References 295
Index 322
Preface
Progressisusuallyasequenceofeventswhereadvancesinonefieldopenupnew
opportunities in another, which in turn makes it possible to push yet another field
forward, and so on. Thus, the development of fast and powerful computers has
led to the development of new numerical methods for direct numerical simula-
tions(DNS)ofmultiphaseflowsthathaveproduceddetailedstudiesandimproved
knowledgeofmultiphaseflows. WhiletheoriginofDNSofmultiphaseflowsgoes
back to the beginning of computational fluid dynamics in the early sixties, it is
only in the last decade and a half that the field has taken off. We, the authors of
this book, have had the privilege of being among the pioneers in the development
ofthesemethodsandamongthefirstresearcherstoapplyDNStostudyrelatively
complexmultiphaseflows. Wehavealsohadtheopportunitytofollowtheprogress
of others closely, as participants in numerous meetings, as visitors to many labo-
ratories,andaseditorsofscientificjournalssuchastheJournalofComputational
PhysicsandtheInternationalJournalofMultiphaseFlows. Tous,thestateofthe
artcanbesummarizedbytwoobservations:
• Even though there are superficial differences between the various approaches
beingpursuedforDNSofmultiphaseflows, thesimilaritiesandcommonalities
oftheapproachesareconsiderablygreaterthanthedifferences.
• As methods become more sophisticated and the problems of interest become
morecomplex,thebarrierthatmustbeovercomebyanewinvestigatorwishing
todoDNSofmultiphaseflowskeepsincreasing.
Thisbookisanattempttoaddressbothissues.
The development of numerical methods for flows containing a sharp interface,
as fluids consisting of two or more immiscible components inherently do, is cur-
rentlya“hot”topicandsignificantprogresshasbeenmadebyanumberofgroups.
Indeed, for a while there was hardly an issue of the Journal of Computational
Physics that did not contain one or more papers describing such methods. In the
present book we have elected to focus mostly on two specific classes of methods:
volume of fluid (VOF) and front-tracking methods. This choice reflects our own
background,aswellasthefactthatbothtypesofmethodhavebeenverysuccessful
and are responsible for some of the most significant new insights into multiphase
flow dynamics that DNS has revealed. Furthermore, as emphasized by the first
bullet point, the similarities in the different approaches are sufficiently great that
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