Anton Smolianski Numerical Modeling of Two-Fluid Interfacial Flows

109 Pages·2012·2.55 MB·English

Anton Smolianski
Numerical Modeling of Two-Fluid
Interfacial Flows
UNIVERSITYOFJYVA¤SKYLA¤
JYVA¤SKYLA¤ 2001

ABSTRACT
Smolianski, Anton
Numerical ModelingofTwo-Fluid InterfacialFlows
Jyva¤skyla¤: University ofJyva¤skyla¤, 2001,109p.
(Jyva¤skyla¤ Studies inComputing
ISSN1456-5390;8)
ISBN951-39-0929-8
Finnishsummary
Diss.
The present work is devoted to the study on unsteady (cid:3)ows of two immiscible
viscous (cid:3)uids separated by free moving interface. The goal of the present work
istoelaborate auni(cid:2)edstrategy fornumericalmodeling ofallkindsoftwo-(cid:3)uid
interfacial(cid:3)ows,havinginmindpossibleinterfacetopologychanges(likemerger
orbreak-up)andrealisticallywiderangesforphysicalparametersoftheproblem.
The presented computational approach essentially relies on three basic com-
ponents: (cid:2)nite element method for spatial approximation, operator-splitting for
temporal discretization and level-set method for interface representation. Finite
element discretization is based on variational formulation of the problem and,
thus, allowsto naturally incorporate discontinuous materialcoef(cid:2)cients andsin-
gular interface-concentrated forces. The use of (cid:2)nite elements permits to local-
ize the interface precisely, without introduction of any arti(cid:2)cial parameters like
interface thickness. We also show that interface normal and curvature can be
recovered with the second-order accuracy after applying a gradient averaging
technique; that allows us to compute accurately the surface tension force. For
temporaldiscretizationweemployanoperator-splitting, thus,separatingallma-
jor dif(cid:2)culties of the problem. This approach enables us, in particular, to imple-
ment equal-order interpolation for the velocity and pressure. In order to model
the phenomena involving interface topology changes we make use of the level-
setapproach,the(cid:2)nite elementimplementationofwhichbringssomeadditional
bene(cid:2)ts as compared to the standard (cid:2)nite difference level-set realizations. We
introduce also a simple mass-correction procedure allowing to maintain an opti-
mal,second orderaccurate massconservation.
Diverse numerical examples including simulations of bubble dynamics, bi-
furcating jet (cid:3)ow and Rayleigh-Taylor instability are presented to validate the
proposed computational method.
Keywords: two-(cid:3)uidinterfacial(cid:3)ow,Navier-Stokesequations,discontinuousco-
ef(cid:2)cients,singularforce,freemovingboundary,(cid:2)niteelementmethod,operator-
splitting, level-setapproach
Author’sAddress Anton Smolianski
Department ofMathematical
Information Technology
University of Jyva¤skyla¤
P.O. Box35,FIN-40351Jyva¤skyla¤
Finland
E-mail: [email protected].(cid:2)
Supervisors Docent Heikki Haario
Department ofMathematics
University of Helsinki
Finland
Professor PekkaNeittaanma¤ki
Department ofMathematical
Information Technology
University of Jyva¤skyla¤
Finland
Professor TimoTiihonen
Department ofMathematical
Information Technology
University of Jyva¤skyla¤
Finland
Reviewers Professor OlivierPironneau
Laboratory of NumericalAnalysis
University Paris 6
France
Professor Sergey Repin
V.A.Steklov Institute of Mathematics
inSt.-Petersburg
RussianAcademyofSciences
Russia
Opponent Doctor Bertrand Maury
Laboratory of NumericalAnalysis
University Paris 6
France
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to Prof. Pekka Neittaanma¤ki and
Prof. Timo Tiihonen (University of Jyva¤skyla¤, Finland) for their support and for
giving me the opportunity to work at the Laboratory of Scienti(cid:2)c Computing,
where I have enjoyed a friendly atmosphere and access to excellent research fa-
cilities. Iamalsodeeplyindebtedtomylatesupervisor,Prof. ValeryRivkind,for
specifying thegeneral direction ofmyresearch.
I am extremely grateful to my adviser, Doc. Heikki Haario (University of
Helsinki, Finland), for his continuous support and encouragement. I would also
like to acknowledge a fruitful collaboration with Dr. Dmitri Kuzmin (University
ofDortmund, Germany) andthank himfor thorough reading ofmymanuscript.
Iamverythankfulto Prof. OlivierPironneau(UniversityParis6,France)and
Prof. Sergey Repin (V.A. Steklov Mathematical Institute, Russia) for reviewing
themanuscript andgiving encouraging feedback.
Thiswork was(cid:2)nanciallysupported byCOMASGraduateSchooloftheUni-
versity of Jyva¤skyla¤, by the Academy of Finland and by TEKES Technology De-
velopmentCenter.
Finally, I would like to express my deepest appreciation to my parents for
their support throughout my life and to my wife Tanya for her patience and un-
derstanding.
Jyva¤skyla¤, February 2001
AntonSmolianski

CONTENTS
1 INTRODUCTION 9
1.1 Numerical methods forinterfacial (cid:3)ows . . . . . . . . . . . . . . . . 10
1.2 Computational strategy andthesis outline . . . . . . . . . . . . . . . 20
2 MATHEMATICALMODEL 22
2.1 Physical assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Equations andinterfacial conditions . . . . . . . . . . . . . . . . . . 23
2.3 Weak andclassical formulations . . . . . . . . . . . . . . . . . . . . 29
3 DEVELOPMENTOFTHECOMPUTATIONALMETHOD 33
3.1 Discretization ofthe Navier-Stokes equations . . . . . . . . . . . . . 33
3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 Operator-splitting approach . . . . . . . . . . . . . . . . . . . 35
3.1.3 Navier-Stokes convection step . . . . . . . . . . . . . . . . . 43
3.1.4 Viscous diffusion step . . . . . . . . . . . . . . . . . . . . . . 47
3.1.5 Projection step . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Approximation of theinterface . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Level-setapproach . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2 Level-setconvection step . . . . . . . . . . . . . . . . . . . . 55
3.2.3 Reinitializationstep . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.4 Level-setcorrection step . . . . . . . . . . . . . . . . . . . . . 61
3.2.5 Approximation of theinterfacenormal andcurvature . . . . 63
3.2.6 Evaluating theinterfacialforce and density/viscosity (cid:2)elds 67
3.3 Summary of thealgorithm . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Stability issuesandtimescales . . . . . . . . . . . . . . . . . . . . . 70
4 NUMERICALRESULTS 72
4.1 Static bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Risingbubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Breaking bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Merger of twobubbles . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 Rayleigh-Taylor instability . . . . . . . . . . . . . . . . . . . . . . . . 86
4.6 Bifurcating jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 CONCLUSIONS 92
BIBLIOGRAPHY 94
YHTEENVETO(FINNISHSUMMARY) 109

1 INTRODUCTION
Fluid (cid:3)ows with free moving surfaces or interfaces can be roughly divided into
four general classes: bubbles/drops, jets, waves and (cid:2)lms. Each class encom-
passesalargenumberofreal-lifephysicalphenomenahavingagreatimportance
in diverse industrial applications. For example, bubble dynamics is of particu-
lar interest for chemical engineering, as bubbly (cid:3)ows are the core of bubble col-
umn chemical reactors; propulsion of liquid-metal jets constitutes the main part
ofmetalformingprocesses;oceanwavesareunderthoroughinvestigationinma-
rineand coastal engineering,andliquid(cid:2)lm(cid:3)owsarefrequently encounteredin
coating anddrying processes during paperorpolymer production.
Itisworthnotingthatallabovementionedclassesof(cid:3)uid(cid:3)owsare,inessence,
two-(cid:3)uid (cid:3)ows, since even in the case when the second (cid:3)uid is a gas (e.g., air) its
dynamics cannot be neglected, with only few exceptions. Thus, in general, we
have to deal with the (cid:3)ows of two immiscible (cid:3)uids separated by their natural
interfacerather thanwithone-liquidfree-surface (cid:3)ows.
Experimental results are usually supposed to bethe major source of informa-
tion on the behaviour of the physical process at hand. However, in many cases
of free-surface/interfacial (cid:3)uid (cid:3)ows the physical time and length scales are so
small that any reliable experimental observations become extremely expensive
or impossible. Then, numerical modelling turns out to be the only tool allow-
ing to investigate the physical phenomenon qualitatively and, sometimes, even
quantitatively.
The goal of the present work is to elaborate a uni(cid:2)ed strategy for numeri-
cal modelling of all kinds of two-(cid:3)uid interfacial (cid:3)ows, having in mind possi-
ble interface topology changes (like merger or break-up) and realistically wide
rangesforphysicalparametersoftheproblem. Thereareseveralintrinsicdif(cid:2)cul-
ties, a correct treatment of which essentially determines the success of the entire
method. First, large jumps of (cid:3)uid density and viscosity across the interface are
tobeproperlytakenintoaccountinordertosatisfythemomentumbalanceinthe
vicinity of theinterface. Sincethe surfacetension force playsvery important role
in the interface dynamics, the in(cid:3)uence of this force should be accurately evalu-
atedandincorporatedintothemodel. Next,asharpinterfaceresolutionhastobe
maintained, including the cases of interface folding, breaking and merging. Fi-
nally, mass conservation is of primary importance for any (cid:3)uid (cid:3)ows, especially
forinterfacial ones.
10
All these issues are addressed, and special techniques are proposed for their
treatment, whichenableto construct the desiredcomputational method.
1.1 Numerical methods for interfacial (cid:3)ows
There is a vast amount of literature devoted to numerical methods for free sur-
face/interface (cid:3)uid (cid:3)ows. As the comprehensive overviews containing a large
number of references we would mention the papers by Anderson et al. [3], Cu-
velier and Schulkes [32], Floryan and Rasmussen [56], Hou [86], Scardovelli and
Zaleski[165],Tsai andYue [197]andthebook by Shyyet al.[171].
In order to systematize the knowledge on existing methods their clear clas-
si(cid:2)cation is de(cid:2)nitely required. Without loss of generality, we may say that the
most popular way is to divide allnumerical algorithms for (cid:3)uid(cid:3)ows into Eule-
rian, Lagrangian and mixed Eulerian-Lagrangian. Eulerian methods are charac-
terized by a coordinate system that is stationary in the laboratory frame of refer-
ence. The (cid:3)uid travels between different computational cells, in contrast to the
Lagrangian methods, where each computational cell always contains the same
(cid:3)uidelements. Thus,Lagrangian methodsarecharacterizedbyacoordinate sys-
tem that moves with the (cid:3)uid. The mixed Eulerian-Lagrangian methods rely on
both Lagrangian and Eulerian concepts. This classi(cid:2)cation is very reasonable to
describethewayofmodelingof(cid:3)uid(cid:3)ow,butdoesnot containanyinformation
on approaches to modeling the interface motion. In this respect, there exists an-
other commonly-used classi(cid:2)cation which treats all methods as either interface-
tracking or interface-capturing. In the interface-tracking method the interface
(freesurface)isexplicitlytrackedalongthetrajectoriesof(cid:3)uidparticlesinpurely
Lagrangian manner, which gives rise to the frequent use of interface-tracking in
combination with Lagrangian or with mixed Eulerian-Lagrangian methods. The
interface-capturing method is characterized by a reconstruction of the interface
from the properties of appropriate (cid:2)eld variables, e.g., (cid:3)uid fraction or density.
The latter classi(cid:2)cation clari(cid:2)es the geometrical part of interfacial-(cid:3)ow model-
ing,thatis theissuesrelatedtotheinterfacemotion,butleavesunclearotherkey
points ofanalgorithm.
It seems that any computational method for free-surface (cid:3)ow consists of the
following main ingredients: (i) (cid:3)ow modeling, (ii) interface modeling, and (iii)
modeling of (cid:3)ow(cid:150)interface coupling. This information is already suf(cid:2)cient to
gain an insight into particular method, but there are still two important compo-
nentsofanalgorithmtobeincludedinthelistof(cid:147)principleclassi(cid:2)cationfactors(cid:148).
First of them is the spatial discretization, which strongly in(cid:3)uences the interface
representationand,toalargeextent,determinesthelastsigni(cid:2)cantalgorithmical
component: (cid:3)ow equations solver. Under the latter we do not mean a method
of resolving a linear algebraic system but a strategy for the treatment of intrinsic
dif(cid:2)culties (nonlinearities,constraints) inherentto the(cid:3)uid(cid:3)owequations.
Collecting together the main parts of a numerical modeling procedure for in-
terfacial(cid:3)ows,wearriveat theclassi(cid:2)cation:
(1) flowmodeling: Eulerian,Lagrangian,mixedEulerian-Lagrangian,mapping
method