Accepted
ManuscriptAmassconservinglevelsetmethodfordetailednumericalsimulationofliquidatomizationKunLuo,ChangxiaoShao,YueYang,JianrenFanPII:
S0021-9991(15)00395-2DOI:
http://dx.doi.org/10.1016/j.jcp.2015.06.009Reference:
YJCPH5945Toappearin: Journal of Computational PhysicsReceiveddate:
17November2014Reviseddate: 2June2015Accepteddate:
5June2015Pleasecitethisarticleinpressas:K.Luoetal.,Amassconservinglevelsetmethodfordetailednumericalsimulationofliquidatomization,J.
Comput. Phys.(2015),http://dx.doi.org/10.1016/j.jcp.2015.06.009This
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Pleasenotethatduringtheproductionprocesserrorsmaybediscoveredwhichcouldaffectthecontent,
andalllegaldisclaimersthatapplytothejournalpertain.1 A mass
conserving level set method for detailed numerical simulation of
liquid atomization By Kun Luo1, Changxiao Shao1, Yue Yang2, Jianren
Fan1* 1State Key Laboratory of Clean Energy Utilization Zhejiang
University, Hangzhou 310027, P. R. China 2State Key Laboratory of
Turbulence and Complex Systems Peking University, Beijing 100871,
P. R. China Submitted to Journal of Computational Physics
*Corresponding author, E-mail: [email protected], Tel:
86-0571-87951764 2
Abstract:Animprovedmassconservinglevelsetmethodfordetailednumerical
simulations of liquid atomization is developed to address the issue
of mass loss in the existing level set method. This method
introduces a mass remedy procedure based on
thelocalcurvatureattheinterface,andinprinciple,canensuretheabsolutemass
conservation of the liquid phase in the computational domain. Three
benchmark cases,
includingtheZalesaksdisk,adropdeforminginavortexfield,andthebinarydrop
head-oncollision,aresimulatedtovalidatethepresentmethod,andtheexcellent
agreementwithexactsolutionsorexperimentalresultsisachieved.Itisshownthat
thepresentmethodisabletocapturethecomplexinterfacewithsecond-order
accuracyandnegligibleadditionalcomputationalcost.Thepresentmethodisthen
appliedtostudymorecomplexflows,suchasadropimpactingonaliquidfilmand
theswirlingliquidsheetatomization,whichagain,demonstratestheadvantagesof
mass conservation and the capability to represent the interface
accurately.
Keywords:Liquidatomization;Interfacecapturing;Levelsetmethod;Mass
conservation; Swirl atomization 3 1 Introduction
Problemsinvolvingmovingboundariesandinterfacesexistinawiderangeof
applications, such as multiphase flow, image processing and
premixed combustion. In
multiphaseflow,theliquidatomizationoccursinmostofpropulsiondevices.The
liquidfuelisinjectedintocombustionchamberandsubsequentlyundergoesthe
processofatomization,evaporationandcombustion.Sincetheatomizationprocess
governstheresultantliquiddropletfeatures,itstronglyaffectsboththecombustion
efficiencyandpollutantemissions.Therefore,itiscrucialtoaccuratelypredictthe
interface in the process of liquid atomization.
Ingeneral,twoclassesofmethodsareusuallyusedtolocatetheinterface:the
interfacetrackingmethodandtheinterfacecapturingmethod.Interfacetracking
schemestypicallyuseeitherthearbitraryLagrangian-Eulerian(ALE)methodbased
on a mesh that deforms with the interface or the marker and cell
(MAC) method that
advectsLagrangianparticlestodefineafluidflowfieldbytheirlocations[1][2][3].
The main drawback of the interface tracking method is that topology
changes are not
handledautomatically,whichpreventstheapplicationofthismethodfromdirect
numerical simulation (DNS) of primary atomization [4]. Interface
capturing methods include the volume of fluid (VOF) method, the
level set method, and the phase field method. Comprehensive reviews
of these methods can be found in literature [5][6][7]. The VOF
method captures the liquid volume fraction
ineachgridcellandhasexcellentmassconservationproperty.However,itsuffers
fromthechallengeofaccuratelyreconstructinginterfacebasedonlyontheliquid
volume fraction. Moreover, a specific advection scheme is required
and the interface
normalandcurvaturemaynotbeevaluatedaccurately,whichaddsmoreconstraints
on the accuracy of the VOF method. The level set method represents
the interface as
aniso-surfaceofascalarfunctionthatisalsoreferredasthelevelsetfunction.The
interfaceissharpanditsnormalandcurvaturecanbeeasilycalculated.Inaddition,
thismethodisabletohandletopologychanges.However,mostoftheexistinglevel
setmethodslackmassconservationproperty.Thephasefieldisdefinedtobeeither
the difference between or the fraction of one of the concentrations
of the two mixtures
[8],andsharpfluidinterfacesarereplacedbythinbutnonzerothicknesstransition
regions where the interfacial forces are smoothly distributed. The
phase field method is similar to the level set method to some
extent, but it suffers from more constraints 4
[9][10].Asaresult,thelevelsetmethodhasbecomemoreandmorepopularin
simulationsofgas-liquidmultiphaseflowsowingtoitssimplicityandefficiencyin
the recent years. Despite of the inherent advantages, the level set
method suffers from unphysical loss of the fluid mass as time
evolves. There are two main reasons for the mass loss in
theexistinglevelsetmethod.Firstly,thediscretizationofthelevelsetequationmay
lead to significant numerical dissipation that usually manifests
itself as a loss of mass
inareasofhighcurvatureorotherunder-resolvedregions[11].Secondly,inthe
re-initialization process for keeping the level set function as a
signed distance function, the zero level set can be altered owing
to numerical artifacts, which leads to additional mass loss. Many
attempts have been made to improve the mass conservation property
ofthelevelsetmethod.Theimprovementscanbeclarifiedintofourcategoriesas
follows. (1) High-order discretization of the level set equation As
the numerical error comes from the discretization of the level set
equation, it
isstraightforwardtoimprovetheaccuracybyincreasingtheorderofdiscretization
schemes. Nourgaliev et al. [12] used 5th-order linear WENO scheme
to discretize the level set equation and pointed out mass losses
were reduced compared to lower order
scheme.Salihetal.[13]examinedthefirst-orderupwindscheme,theMacCormack
method,thesecond-orderENOschemeandthefifth-orderWENOscheme,and
demonstratedthatthelevelsetmethodperformedbetterwhenhigh-orderschemes
wereusedforsolvingtheadvectionequation.Inaddition,Sheuetal.[14][15]
proposed a dispersion relation preserving advection scheme that is
able to sharpen the
interfaceandthenumericalresultsshowedgoodagreementwiththeexperimental
results. Discontinuous Galerkin method was first proposed by Reed
and Hill [16] and
hasbeenusedwildlyinthediscretizationofthelevelsetequationbecauseitisa
compactschemewiththehigh-orderaccuracy[17][18][24].Moreover,the
semi-Lagrangian approach is a discretization scheme that overcomes
the problems of
thesevereCFLconditionandthedistortedmesh.Ithasbeenappliedtothe
discretizationofthelevelsetfunctionbyusinganefficient,first-orderaccurate
semi-Lagrangianadvectionscheme[19][20]andahigh-orderschemein[21].
However, the mass conservation is still an issue for simulations of
some shearing and vortical velocity fields even with the
above-mentioned high-order schemes. 5 (2) Extension velocity level
set methods Instead of the high-order discretization of the level
set equation, Adalsteinsson et al. [22] believed that the level set
equation can be transported by an extension velocity rather than
the fluid velocity. This idea was based on the fact that the
gradient of the level set function grows very rapidly with time in
shearing velocity fields. In order to
maintainasigneddistancefunction,twoapproacheshavebeenapplied.Oneis
allowedtodeviatefromthesigneddistancefunctionandthencorrectedvia
re-initialization.Theotheristhatthevelocityfieldisconstructedspeciallysothat
1 =is preserved, here is the signed distance function. The
extension velocity that satisfies0extF = would maintain the signed
distance function for all time,
whereFextistheextensionvelocity.Withtheextensionvelocity,Ovsyannikovetal.
[23]maintainedthesigneddistancefunctionbyintroducingthesourcetermdirectly
intothelevelsetequation.However,theextensionvelocityisonlythefirst-order
approximationtothevelocityfieldneartheinterface,anditmayleadtounexpected
numericalartifactsduetothestrictrequirementofthelocalsolutionforthe
characteristicsoftheflowneartheinterface[24].Recentlytheapproachof
Ovsyannikov et al. [23] has been extended to higher-order accuracy
and its numerical assessment has been conducted [25]. (3)
Hyperbolic tangent level set methods Instead of the signed distance
function, Olsson et al. [26][27] firstly proposed the hyperbolic
tangent function as the level set function. It has the second-order
accuracy
andisofgoodmassconservationintheregionboundedbytheinterface.This
approach has been used in many studies [28][29][30][31][32], but it
can lead to small
piecesoffluidcalledflotsamandjetsamnon-physicallybreakingofffromthe
interfaceinunder-resolvedregionsandmovingaroundwitherroneousvelocity.In
addition,Kohnoetal.[33]presentedanovelnumericalmethodforsolvingthe
advection equation of the level set function using the
hierarchical-gradient truncation and remapping technique. (4)
Improvement of the re-initialization The re-initialization is
necessary in general to maintain the level set function as a
signeddistancefunction.Thepartialdifferentialequation(PDE)based
re-initializationprocedurewasoriginallyproposedin[34].Theequation (
)( )1tsign = issolvedsothatitisadistancefunctionwithoutchangingits
6
zerolevelset.Theimprovementscanbefoundin[35][36][37][38][39].Changetal.
[40]proposedare-initializationprocedureinvolvingsolvingaperturbed
Hamilton-Jacobi equation to a steady state. With the development of
the conservative
levelsetmethodwithhyperbolictangentlevelsetfunction,thecorresponding
reinitializationequationissolvedinvariouscases[26][27][30].McCaslinand
Desjardins[41]furtherimprovedthere-initializationequationtoaccountfor
significant amount of spatial variability in level set transport.
To reduce computational cost in the re-initialization, the fast
marching method (FMM) was developed to solve the re-initialization
equation at the interface and marches outwards to create a signed
distancefunctioninasinglesweepthroughthedata,insteadofsolvingthe
re-initializationequationforanumberoftimesteps.Itisstraightforwardtoextend
thismethodtoachievehigher-orderaccuracy[30][42][43].Somealternative
re-initializationmethodshavealsobeenproposedrecently,suchasthegeometric
mass-preservingredistancingscheme[44]andthevolume-reinitializationscheme
including the effect of local curvature [45]. (5) Hybrid method
Taking advantages ofmassconservation property ofthe VOFmethod,
Sussman
etal.[46][47]developedacoupledlevelset/volume-of-fluid(CLSVOF)method.
AlthoughtheCLSVOFmethodanditsvariantshavebeenappliedinvarious
applications [48][49][50][51][52][53], they suffer from unphysical
flotsam and jetsam. In addition, procedures for the advection of
both the level set function and the volume fraction, the piecewise
linear interface construction, and the subsequent reconstruction
oflevelsetscalarbasedonthispiecewiselinearinterfacearequitecomplicated.
Recently, several improved coupled level set and VOF methods have
been developed,
suchastheVOSETmethod[54],theconservationcorrectionequationmethod[55],
and the level set-volume constraint method [56]. Since the major
issue for the hybrid
methodsisthecomplexityinimplementation,itisnotedthatPijletal.[57][58]
proposedamass-conservingmethod,inwhichlevelsetandvolumefractionwere
usedwithoutthecomplicatedinterfaceconstructions.Additionally,theparticlelevel
setmethodwasdevelopedasanumericalschemethatcombinestheaccuracyofthe
Lagrangianfronttrackingwithsimplicityandtheefficiencyofthelevelsetmethod
[11].Inthismethod,Lagrangianmarkerparticlesarepassivelyadvectedwithflow
andareusedtorebuildthelevelsetfunctioninunder-resolvedregions 7
[59][60][61][62]. (6) Spatially adaptive level set method
Ingeneral,sincethelevelsetfunctiononlyneedstoberesolvednearthe
interface,thecomputationalcostcanbegreatlyreducedbyusingthenarrowband
method.Butitstillsuffersthemasslossintheunder-resolvedregions.Several
spatiallyadaptivemethodsweredevelopedtoefficientlyresolvetheinterfaceand
greatlyincreasetheaccuracyofthelocationoftheinterface,includingtheadaptive
levelsetapproach[64],theoctreebasedmethods[11][65],thestructuredadaptive
meshrefinement[12],therefinedlevelsetgrid(RLSG)method[66],thespectrally
refined interface approach [43], and the adaptive level set method
[67].
Inthepresentstudy,anovelmassconservinglevelsetmethodisdeveloped
basedonthespectrallyrefinedinterface(SRI)approach[43].Nevertheless,the
notablemasslosswasstillreportedinprevioussimulationsofatomization[68].
Therefore, we propose an improved mass remedy procedure to
eliminate the mass loss.
DifferentfromtheVOFscalartransportation[58][59],theabsolutemasslosswithin
the computational domain is redistributed to the interface cells
according to the local
curvatureofthegas-liquidinterface.Thistreatmentisabletoavoidthemassloss
caused by the numerical diffusion. The rest of the paper is
organized as follows. In section 2, the SRI approach for
thelevelsetmethodisintroduced.Insection3,acurvature-basednovelmass
conservinglevelsetmethodisproposedandthecorrespondingnumerical
implementations are presented in detail. This is followed by
section 4, in which three benchmarkcasesare numerically simulatedto
validatethemassconservinglevel set
method.Insection5,theproposedapproachisfurtherappliedtocomplexflows
includingthedropimpactingonliquidfilmandtheswirlingatomization.Some
conclusions are drawn in section 6. 2 Numerical methods 2.1
Governing equations for incompressible two phase flow The
incompressible Navier-Stokes equations for gas and liquid phases
read ( )1 1 tpt ( + = + + + uu u u u g,(1) 0t+ =u ,(2)
whereuisthevelocity,isthedensity,pisthepressure,gisthegravitational
8 acceleration and is the dynamic viscosity. The material
properties of gas and liquid
areconstant,i.e.,=l,=linliquidphaseand=g,=gingasphase.The velocity
at the interface is continuous. The material properties are
subjected to jump at theinterface,thatis,[ ]l g = and[ ]l g =
.Thepressureacrossthe interface can be expressed as [ ] [ ] 2tp = +
n u n,(3)
whereisthesurfacetension,istheinterfacecurvatureandnistheinterface
normal. The code used can only handle structured meshes and
staggered grid. The spatial
discretizationoftheNavier-Stokesequationsisperformedusingthesecond-order
finite difference schemes. The pressure p is stored at the cell
centers, while velocity is stored at the face centers. The
second-order semi-implicit iterative procedure [69] for time
integration is utilized, which is economical, stable and accurate.
The iteration can be expressed as ( ) ( )1 1 1 11 11 12 2n n n n n
nk k k kfu u tf u u t u uu+ + + ++ +( (= + + + (( ,(4) where f is
the right hand side of Navier-Stokes equations, and / f u is the
Jacobian. Thelowandupperindexnotationsk and
ndenotethekthNewton-Raphson sub-iterative step and the time step,
respectively.
ThePoissonequationissolvedbyusingafractionalstepprojectionmethodto
enforcecontinuity.Firstly,thevelocityfieldisadvancedbysolvingEq.(1)without
pressure gradient. Then the velocity is projected by solving the
Poisson equation with the Ghost Fluid Method (GFM) described in
section 2.7. Finally, the velocity field is corrected to ensure
continuity using the pressure gradient. The time stept at time nt
is determined by the CFL conditions 2 2 2CFL CFL CFL CFLCFL CFLmin
, , , , ,4 4 4dy dy dx dzdx dztu v w | | < |\ .(5)
whereCFListheCFLnumber.Thenewtime-stepisfinallydeterminedby ( ) min
, t t dt = ,wheredtisgivenasaninputparametertoensuretheintegration
stable. 2.2 Signed distance level set 9 The gas-liquid interface is
represented as the zero iso-surface of a signed distance function
in the level set approach, i.e., ( ) , G t= x x x ,(6) where t is
time and xis the location on the interface that is closest to x.
The level set function is defined to be positive for liquid and
negative for gas. The motion of the interface is simulated by
solving a simple advection equation for the level set function
0GGt+ =u,(7) where u is the fluid velocity. The state-of-the-art
for level set transport typically relies on WENO-type schemes [43]
to combine accurate transport and numerical robustness.
However,theWENO-typeschemetendstobeoverlydiffusiveforthetransportof
small-scale scalar structures which are essential to the physics of
multiphase flows.
Sincetheimprovementofinterfacerepresentationsinnumericalsimulations
requiresanaccuratedescriptionofthesmallestscales,asub-celldescriptionofthe
interfaceisusuallyneeded.Inthepresentstudy,weuseapseudo-spectralsub-cell
reconstruction method with a semi-Lagrangian transport scheme [43].
2.3 Sub-cell reconstruction A polynomialreconstruction of thelevel
set function isappliedineachcell ina narrow band aroundtheinterface
by introducing asetof quadrature pointstoenable sub-cell
resolution. These points correspond to the locations where the
nodal values of
thelevelsetfunctionGarespecified[43].Inordertoimprovethecontinuityofthe
levelsetfunctionacrosscells,theGauss-Lobattoquadratureisusedsuchthatsome
quadrature points are located on the cell faces.
LettheGvalueofthequadraturepoint(l,m,n)oftheflowsolvercell(i,j,k)be
denoted , ,, ,l m ni j kGanditspositionvector , ,, ,l m ni j
kx.Thenumberofquadraturepointsper
directionismarkedtobepforthesakeofsimplicity.Thelevelsetfunction
reconstruction within cell (i,j,k) is then written as ( ) ( ) ( ) (
), ,, , , ,1 1 1p p pl m n l m ni j k i j kl m nG L x L y L z G= =
==x.(8) This reconstruction is of order p-1. The expression can be
further simplified owing to the independence of the directions and
the simple form of L. Consider a cell of unit
sizeinonedirectionandletrlwith[ ] 1, l p
representthepositionofthelth
quadraturepointinthatdirection,forwhichthenr1=0andrp=1.Thebasis 10
polynomials are written as ( )( )( )1,1,ppr rL rr r = = =.(9) The
reconstruction of the level set function for cell (i,j,k) is then
expressed as ( ), ,, , , ,1 1 1 1 1 1, ,p p pj l m n l m n i ki j k
i j ki i j j k k l m ny yx x z zG x y z L L L Gx x y y z z+ + + = =
= | | | | | |= ||| \ . \ .\ . .(10) The computational cost of this
sub-cell reconstruction is relatively low, because some
ofthequantitiesrelatedtothepolynomialreconstructioncanbepre-computedand
stored.Tofurtherreducethecomputationalcostassociatedwiththissub-cell
polynomial reconstruction, the polynomials are created only in a
narrow band around the interface. 2.4 Semi-Lagrangian transport and
discretization
Thesemi-Lagrangian(SL)transport,anefficientandaccuratescalartransport
scheme, is adopted to avoid severe CFL restrictions due to rapidly
decreasing distance
betweenquadraturenodes.InsteadofdiscretizingEq.(6),thesemi-Lagrangian
transport scheme is based on the fact that G should be constant
along the trajectory of
thematerialpointsevolvingatvelocityu.Thetrajectorythatpassesthroughxn+1at
time tn+1 can be traced backward in time to 1 n nt =t t+ to obtain
the old location xn.
Therefore,thevalueofthelevelsetfunctionGn+1atxn+1canbeobtainedby (
) ( )1 1 n n n nG G+ += x x
.LargertimestepsizecanbeusedowingtotheLagrangian nature of this
method. The only requirement is the computation of xn from xn+1,
which involves solving an ordinary differential equation (ODE). In
addition, this approach is efficient and easy to implement using
the fourth-order Runge-Kutta method. 2.5 Re-initialization level
set function
Inthepresentstudy,there-initializationsimplyinvolvesinterpolatingthe
velocityuandGvalueatthequadraturepointsbyusingthetri-linearinterpolation
from the eight closest grid points of the flow solver. This
re-initialization of the level set functionis found tobemostly
superfluous, and itis only performed typically for every 100 time
steps. 2.6 Curvature computation and density and viscosity
computation
Theinterfacecurvatureisessentialtothecomputationofthesurfacetension
force, but it is challenging to obtain the curvature in the present
sub-cell structures. In 11
thiswork,thecurvatureiscomputedbytwostepsasfollows[43].(1)The
reconstructionofasigneddistancefunctionfortheinterfacebycombininga
marching-cubes(MC)algorithmwithaparallelFMM.(2)Thethird-order
least-squares computation of curvature from the reconstructed
distance function. The density and viscosity fields are computed as
( )g l g = + (11) ( )g l g = + (12) where ( ) ( )1tanh 12 2G= +
isthehyperbolictangentfunctionand isa parameter to set the
thickness of the interface. 2.7 Ghost fluid method
Thediscretizationofthepressuregradienttermisimportantinthelevelset
method. In this work, the discontinuity in pressure that arises in
the pressure gradient
termofEq.(2)istreatedusingtheghostfluidmethod(GFM)[43].Thismethod
providesasharpandrobustdiscretizationofthediscontinuityacrosstheinterface.
Consideraninterfacelocatedat x betweenthetwogridlocationsxiandxi+1,
wherexi+1iswithintheliquidphase.ThepressurejumpinthepressurePoisson
equation is then written as ( ) ( )[ ], 1 , , , 1 *2 * 2,1 11l i g
i g i g igg ip p p pppx xx x+ | |= | \ .,(13) where( )*1g l = +
and( ) /ix x x = . The ghost fluid method is challenging in the
presence of the viscous term in the
Equation3,asdemonstratedinDesjardinsetal.[43].Thecontinuumsurfaceforce
(CSF) approach by Brackbill et al. [44] is used to deal with the
viscous term. 3 Curvature-based mass conserving level set method
Althoughthesub-cellresolutionforthesmall-scalestructurescanbeobtained
using the numerical approach mentioned above, the mass conservation
still cannot be ensured owing to the unavoidable numerical
dissipation in the level set method. Pai et
al.[68]performeddetailednumericalsimulationsofprimaryatomizationofliquid
jetsincrossflowandshowedthatthemasslosscanreachupto20-30%evenwith
sufficient computationalcells, e.g., 36-110 million cells.
Therefore, it is necessary to 12 develop a new method to overcome
the mass loss problem. (a) (b) (c)Figure 1. Layout of the
curvature-based mass conserving level set method. The grey color
represents the liquid phase. The thick line and dash line represent
the interface and updated interface, respectively. (a) The
interface location before mass correction at a new time step; (b)
Mass correction based on the local curvature at the interface; (c)
Smooth the interface.
Inthepresentstudy,inordertoconservemasswiththelevelsetmethod,
corrections to the level set function based on the curvature of
each cell at the interface
areproposedbyconsideringthevolumefractionGvolofacertainfluidwithina
computationalcell.Figure1showsthelayoutofthepresentmethod.Thecurvature
and the level set scalar are stored in the center of each cell. The
interface before mass correction at a new time step is shown in
Figure 1(a), in which mass loss occurs. Once
theoveralllostmassisevaluated,itisdistributedtothecellsattheinterfacebythe
interfacelocalcurvature,asshowninFigure1(b).Afterthisstep,theupdated
interfacemaybenotsmoothacrosstheconjointcells.TheFMMisthenappliedto
obtain the smooth interface, as shown in Figure 1(c). The regular
level set advection is
performedusingthesemi-Lagrangianscheme.Sincetheobtainedlevelsetfunction
Gn+1,*cannotconservemass,correctionstoGn+1,*areappliedtomaintainthemass
conservation by three steps as follows.
(1)Therelativevolumeofliquidphaseinacomputationalcell,i.e.,thevolume
13 fraction Gvoln+1,*, is computed from the level set function
Gn+1,*;
(2)ThevolumefractionGvoln+1,*iscorrectedformassconservationbasedonthe
curvature of each cell at the interface within a time step towards
Gvoln+1 ; (3) With the new volume fraction Gvoln+1, corrections to
Gn+1,* are obtained by solving ( )1 1 1,n n nvolf G G G+ + + =.
Thecomputationalcostofthemassremedyprocedureisnegligiblecomparedwith
that of the original level set method. Next, this correction
procedure will be described in detail. 3.1 Calculation of volume
fraction from the level set function
ArelationshipbetweenthelevelsetfunctionGandthevolumefractionGvolis
found by considering the fractional volume of a certain fluid in a
computational cell.
ThevolumefractionGvoliscalculatedusingananalyticalformuladevelopedbyvan
derPijletal.[58][59].Thisrelationisobtainedbyassumingpiecewiselinear
interfaces within a computational cell, and can be written as ( )
,volG f G G = .(14) After some algebraic manipulations, the volume
fraction Gvol is obtained as , 01 , 0volvolvolG GGG G = >,(15)
where 2 23 3 3 3 36212, , ,, , ,, , ,0, , , , 0, , , , 0volA B C D
ED D DA CD DADD D DD D DD D DGD D D GD D D G +> + > +
>> + > + > + + = + + + + + + =,(16) with ( )1( ) , 02A
max D D D G = + + ,(17) ( )1( ) , 02B max D D D G = + ,(18) ( )1( )
, 02C max D D D G = + ,(19) 14 ( )1( ) , 02D max D D D G = + +
,(20) ( )1( ) , 02E max D D D G = ,(21) with ( ), ,x y zD max D D D
=,( ), ,x y zD min D D D=, x y zD D D D D D = + + ,(22) and xGD xx=
, yGD yy= , zGD zz= .(23) 3.2 Volume fraction remedy In general,
the volume fraction can be transported at time step n to n + 1 by
the method used in the VOF method. However, this method usually
involves the interface
reconstructionandincreasesthecomplexityofthelevelsetmethod.Moreover,it
cannot ensure realizability of Gvol, i.e., unphysical values Gvol
< 0 or Gvol > 1 could be
obtained.Therefore,wedevelopanovelmassremedyproceduretoensurethemass
conservationandrealizabilityinthenumericalschemewiththesimple
implementation.
ThevolumelossinacomputationaldomainisestimatedasdGvoltot
=Gvoltot,init- Gvoltotn+1,*, where Gvoltot,init and Gvoltotn+1,*
are the volume at the initial time and at time
stepn+1,respectively.Inprinciple,Gvoltot,initisthetheoreticalvolumeinthe
computationaldomain.Ifliquidphaseflowsinacrosstheinletboundary,Gvoltot,init
=
Gvoltot,init0+UAdt,whereGvoltot,init0isthevolumeattheinitialtime,Uistheliquid
phasevelocityattheinletboundary,Aistheareaofinletboundary,dtisthetime
duration up to n + 1 time step. This part of liquid volume loss is
then distributed to the
computationaldomain,typicallytotheinterface,andthecorrespondingvolume
fraction is 0 < Gvoln+1,*< 1. A simple approach is to
distribute the lost volume equally to the interface, i.e.,
,voltotvol pdGdGN= ,(24)
wheredGvol,pandNarethevolumeaddedtoeachcellpattheinterfaceandthe
number of cells at the interface, respectively. However, this
treatment does not take into account any local property of the
level set function to maintain the mass conservation. Here, we
propose another distribution
methodbasedonthelocalcurvatureoftheinterface,becausethemasslosstendsto
15
happeninthevicinityoflargecurvatureregions[60]ratherthaninsmoothregions.
Thus, the lost volume can be distributed depending on the local
curvatures as ,1pmaxvol p voltotmpmax pdG dG== ,(25)
wherepandmarethepthcellattheinterfaceandthetotalcellsattheinterface,p
and max are the local curvature of the interface and the maximum
value of curvature within a narrow band around the interface,
respectively. For every cell at the interface,
thefinalaccurateliquidvolumefractionGvol,pn+1=Gvol,pn+1,*+dGvol,p.Furthermore,
this linear dependence on local curvature is chosen in view of
motion involving mean curvature [83], which will be validated in
Section 4.1. 3.3 Reverse calculation of the level set function
Afterthevolumefractionremedy,thepredictedlevelsetfunctionGn+1*is
corrected to ensure that the mass is conserved within the whole
computational domain. In other words, level set function (G1,G2,)
at the interface is obtained implicitly by ( )1 1 n 1l,, , 1,
2,...n nk k vo kf G G G k + + + =(26) where= 10-8 is theerror
tolerance and Gn+1,* is used for the initial guess. It can be
expectedthat correctionto Gn+1,*is minor, because themass error
inGn+1,* from the
truncationerrorofthediscretizationofthelevelsetfunctionissmall.Therefore,a
simple Newton iteration is used to compute Gn+1 as ( )( )n 1 1,
1l1, 1 1,1, 1,,n m nvo k kn m n mk kn m nk kG f G GG GfG GG+ + ++ +
++ + = +,(27) where the derivative fGcan be computed analytically
and m denotes the sub step in theiteration.ThesmoothnessofJacobian
fGinEq.27toensuresystematic convergence is validated in Section
4.3. Insummary,aschematicdiagramofourproposedmassconservinglevelset
method is shown in Figure 2. 16 level set
advectionreinitializationstep1: ( )volG f G =nG1, nG+ step2: volume
fraction remedy1, nvolG+ 1 nvolG+step3: ( )1volG f G=1 nG+ Figure
2: The schematic diagram of the proposed mass conserving method 4
Numerical validations
Inthissection,thenumericalapproachpresentedinsections2and3willbe
validated through three benchmark cases. 4.1 Zalesaks disk - rigid
body rotation of a slotted disk
First,wetestthemassconservinglevelsetmethodfortheproblemoftherigid
body rotation of Zalesaks disk in a constant velocity field [56].
The advection test of Zalesak is often used to demonstrate the
ability of interface-advection algorithms. The
computationaldomainis11andthediskisaslottedcircleinitiallycenteredat(0,
0.25)withradiusof0.15,andslotlengthof0.25andwidthof0.05.Theconstant
velocity field is given by 2 U y =(28) and 2 V x = ,(29) and the
disk completes one revolution within every one time unit.
Figure3showstheinterfacelocationafteronerevolutionfordifferentmeshes
andthecomparisontovanderPijletal.[44].Withincreasingthemeshresolution,
17
bothmethodsleadtobetterresults.Withthesamemeshsize,itisobviousthatthe
results from the present method considering mass remedy are
qualitatively better than that of van der Pijl et al.[44]. XY-0.2
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0.20.050.10.150.20.250.30.350.40.45XY-0.2 -0.15 -0.1 -0.05 0 0.05
0.1 0.15 0.20.050.10.150.20.250.30.350.40.45 (a) 5050(b) 100100
XY-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
0.20.050.10.150.20.250.30.350.40.45XY-0.2 -0.15 -0.1 -0.05 0 0.05
0.1 0.15 0.20.050.10.150.20.250.30.350.40.45 (c) 150150(d) 200200
Figure 3: Computed interface after one full revolution for Zalesaks
disk of different mesh (Black line for initial interface; Red line
for Pijl et al.[44]; Blue line for the present method) 18
timerelativevolumeloss0 0.2 0.4 0.6 0.8 110-710-610-510-410-310-2no
mass remedy, 5050no mass remedy, 100100no mass remedy, 150150no
mass remedy, 200200mass remedy, 5050mass remedy, 100100mass remedy,
150150mass remedy, 200200 Figure 4: Relative volume loss for
Zalesaks disk test for different meshes. The calculations without
mass remedy is compared with the present method considering mass
remedy Table 1: Relative volume loss after one revolution for
Zalesaks disk for different meshes MeshNo mass remedy error (%)Mass
remedy error (%) 50501.69920.0583 1001000.53270.0715
1501500.04620.0240 200200-0.04870.0151 XY0.01 0.02 0.03 0.04 0.05
0.060.090.10.110.120.130.140.15exact interface locationwith mass
remedy, 100100without mass remedy, 100100 Figure 5: The interface
location for mass remedy and no mass remedy cases with the same
mesh
Figure4illustratestherelativevolumelossforZalesaksdiskfordifferent
meshes.Itisapparentthatthepresentmethodconsideringmassremedyhasless
volumelosscomparedwiththemethodwithoutmassremedy.Wecomparethe
relativevolumelossafteronerevolutionofZalesaksdiskandtheresultislistedin
19
Table1.Withincreasingthemeshresolution,bothmethodsleadtosmallmassloss.
With the same mesh size, the mass loss is greatly reduced using the
method with mass
remedy.Tobetterillustratetheaccordanceoftheinterfacelocationafterone
revolution and the initial interface location, the corners of the
notch for the cases with
andwithoutmassremedyareshowninFigure5,bothofwhichhave100100grid
points. It is obvious that the result with mass remedy is closer to
the exact location.
Furthermore,thenumericalerrorisquantitativelyestimatedwiththerelative
added volume in the present method. Since G0 field is a distance
function, |G-G0| is a
measurefortheshiftoftheinterfaceafteronerevolution.Theone-normand
two-norm for variable G are defined as 111x yN Nnumerical exactk kx
y kL G GN N== (30) and ( )2211x yN Nnumerical exactk kx y kL G GN
N== ,(31) where Nx and Ny are the total mesh points in x and y
direction, respectively. The upper
subscriptsnumericalandexactrepresenttheGvalueafteronerevolutionandthe
initialGvalue,respectively.Therelativeaddedvolume,one-normerror,and
two-normerrorareshowninFigure6.Therelativeaddedvolumeisreduced
quadraticallyasdecreasingthemeshsize.Itisnotedthattheaddedvolumetoeach
cell at the interface is almost one percent of the liquid volume or
even smaller in each
cellattheinterface.Theerrorsofone-normandtwo-normaredecreasedalmost
quadraticallyaswell.Thisconfirmsthatthepresentmethodhasthesecond-order
accuracy. 20
x/Lnormerror10-310-210-110-410-410-310-310-210-2relativeaddedvolumeVadded/VinitL1((L2((xx2xx2
Figure 6: Convergence of relative volume loss after one revolution
As demonstrated in Section 3.2, the mass loss dependence on local
curvature (Eq. 25) is illustrated. Noting that in Figure 3, the
mass loss mainly occurs in the notch of
thediskandtwohornsatthebottomoftheslot,wherethecurvatureislarge.The
interfacemovesinthedirectionofconcavityoflargecurvature.Similarwiththe
motionbymeancurvature,thevelocityinthenormaldirectioncanbeexpressedas
V a n =
, wherea is a positive constant. Therefore, the mass loss can be
written as volG V S a nS = =
,whereSisthelocalunitinterfacearea.Itcanbeshownthat the mass loss
is proportional to the local curvature. XY0 0.2 0.4 0.6 0.8
100.20.40.60.81XY0.4 0.60.50.550.60.650.70.750.80.850.90.951 (a1)
6464, t = 4(a2) 6464, t = 8 21 XY0 0.2 0.4 0.6 0.8
100.20.40.60.81XY0.4 0.60.50.550.60.650.70.750.80.850.90.951 (b1)
128128, t = 4(b2)128128, t = 8 XY0 0.2 0.4 0.6 0.8
100.20.40.60.81XY0.4 0.60.50.550.60.650.70.750.80.850.90.951 (c1)
256256, t = 4(c2) 256256, t = 8 Figure 7: The interface location at
t = 4 and t = 8 for different meshes (Blue line for no mass remedy;
Red line for mass remedy) 4.2 Two-dimensional deformation Since the
Zalesak's disk is not able to validate the accuracy of methods
proposed
intheconditionsoflargestrainrateduetoitslinearvelocityfield,weconsideran
unsteadyvorticalfieldtovalidatetheabilityofthepresentmethodtocapturethin
filaments during the circle's deformation. This case was proposed
by Bell et al. [57] to test a second-order projection method for
incompressible Navier-Stokes equations. In
adomainof11,acircleofradius0.15isinitiallyat(0.5,0.75).Thevelocityis
obtained by the stream function: ( ) ( ) ( )( )2 2 1,tx t sin x sin
y cosT =,(32) where T = 8 is the period of a reversing vortical
flow. The velocity field stretches the circle into an thinner
filament that wraps around 22 the center of the domain and then
slowly reverses and pulls the filament back into the initial
circular shape at time t = 8. The results of the present method at
t = 4 and t = 8
fordifferentmeshesareshowninFigure7.Resultsarecomparedwiththemethod
withoutmassremedy.Withincreasingthemeshresolution,bothmethodsleadto
betterresults.Withthesamemeshsize,itisobviousthatthepresentmethodwith
mass remedy is capable of accurately capturing the interface of
thin filament, and the results are better than those without mass
remedy qualitatively. Figure 8 shows the relative volume loss in
the two-dimensional deformation case for differentmeshes. It is
apparent that the presentmethodconsideringmass remedy has less
volume loss compared with themethod withoutmass remedy. We compared
therelativevolumelossinthiscaseatt=8forthemethodswithmassremedyand
withoutmassremedy,andtheresultislistedinTable2.Theresultshowsthatthe
volumelossisreducedsignificantlybyusingthepresentmethodconsideringmass
remedy and the error is reduced by two orders of magnitude.
timerelativevolumeloss0 2 4 6 8-0.0200.020.040.06no mass remedy,
6464no mass remedy, 128128no mass remedy, 256256mass remedy,
6464mass remedy, 128128mass remedy, 256256 Figure 8: Relative
volume loss for two dimensional deformation tests for different
meshes. No mass remedy is compared with the present method
considering mass remedy Table 2: Relative volume loss at T = 8 for
two dimensional deformation for different meshes MeshNo mass remedy
error (%)Mass remedy error (%) 64645.26600.6486 1281283.26300.0636
2562561.49410.0181 The one-norm and two-norm errors defined in Eqs.
(30) and (31) for this case are
showninFigure9.Theerrorsofone-normandtwo-normaredecreasedalmost 23
quadraticallywithdecreasingmeshsize.Thisdemonstratesthatthesecond-order
accuracy is achieved in the present case as well.
x/Lnormerror10-310-210-110-510-410-310-210-1L1((L2((xx2xx2 Figure
9: Convergence of the relative volume loss at t = 8 4.3
Three-dimensional deformation
SimilarwiththeworksbyLeVeque[83]andEnrightetal.[60],the
three-dimensional deformation test case is used. The velocity field
is given by ( ) ( ) ( ) ( ) ( )2, , 2sin sin 2 sin 2 cos / u x y z
x y z t T = (33) ( ) ( ) ( ) ( ) ( )2, , sin sin 2 sin 2 cos / v x
y z y x z t T = (34) ( ) ( ) ( ) ( ) ( )2, , sin sin 2 sin 2 cos /
w x y z z x y t T = (35)
whereT=3.Asphereofradius0.15isplacedat(0.35,0.35,0.35)inaunit
computationaldomain.The643and1283gridcellsareused.Snapshotsofthe
interface at T = 0, 0.3, 0.6, 1.0, 1.5, 2.0, 2.5, 3.0 are shown in
Figure 10. The sphere is entrained by two vortices and is then
compressed to stretch out. Parts of the interface
stretchestoformthinfilmatt=T/2.Afterfullperiodofdeformation,thesphere
recovers from the thin film. This test cases on different grids and
for different methods are investigated at t =
T.Figure11showstheinterfacesatt=Twithoutmasscorrection,withmass
correction based on the redistribution of volume on the local
curvature and with mass
correctionbasedontheuniformredistributionofvolume.For643grid,theresultof
the original method [43] loses significant amount of volume and the
results with mass correction can avoid the volume lost despite of
the low resolution. For 1283 grid, the 24 results with mass
corrections almost recover the original sphere shape in Figure 12
(b).
Itisnotedthattheresultwiththemasscorrectionbasedonlocalcurvatureisbetter
thanthatwiththemasscorrectionbasedontheuniformredistributionofvolume.
Figure12showsthevolumelossforthree-dimensionaldeformationtestondifferent
gridsandfordifferentmethods.Itisshownthatthevolumelossisgreatlyreduced
with the mass correction. (a) 643 (b) 1283 Figure 10. Interface
evolution with mass correction based on the redistribution of
volume on the local curvature with different grids at T = 0, 0.3,
0.6, 1.0, 1.5, 2.0, 2.5, 3.0 (a) 643 (b) 1283 25 Figure 11. The
interface comparisons at t = T on different grids without mass
correction (left), with mass correction based on the redistribution
of volume on the local curvature (middle) and with mass correction
based on the uniform redistribution of volume (right) TV/Vo0 0.5 1
1.5 2 2.5 30.750.80.850.90.951643, (a)643, (b)643, (c)1283,
(d)1283, (e)1283, (f) Figure 12. Volume loss for the
three-dimensional deformation test on different grids and for
different methods. (a) and (d) for the method without mass
correction; (b) and (e) for the method with mass correction based
on the redistribution of volume loss on the local curvature; (c)
and (f) for the method with mass correction based on the uniform
redistribution of volume loss 4.3 Three-dimensional interactions
Inthethirdtestingcase,thesimulationofthebinarydropcollisionisusedto
highlightthecapabilityofthemassconservinglevelsetmethod.Thissimulationis
motivated by the experiments of Ashgriz and Poo [58] and by the
level set simulations of Tanguy and Berlemont [59]. The dynamics of
binary drop collision is dependent on the dimensionless number,
namely, the Weber number 2l cU DWe=,(36) where D is the smaller
drop diameter, Uc is the droplet relative velocity, l is the liquid
density, andis the surface tension coefficient. Table 3: The
physical properties of gas and liquid phases Density of liquid
(kg/m3)1000 Density of gas (kg/m3)1.226 Dynamic viscosity of liquid
(kg/m/s)1.13710-3 Dynamic viscosity of gas (kg/m/s)1.78010-5
Surface tension coefficient (N/m)0.0728 26
Initially,thedropletvelocityisimposedforagivenWebernumber.Thegas
phaseisairandthe liquid phaseiswater. Thephysical properties of
bothphasesare given in Table 3. 4.3.1 Coalescence followed by
separation for head-on collision
CollisionandseparationhavebeenobservedbyAshgrizandPoo[58]fortwo
identicalwaterdropletswithaWebernumberof23.Simulationsarepresentedhere
for water droplets with the same Weber number, and with a droplet
radius of 400 m.
Thecomputationaldomainis2mm4mm2mmandthegridsizeis6412864ina
three-dimensional configuration. (a) (b)Figure 13: (a) Head-on
collision, We = 23, from Ashgriz and Poo (1990) [58]; (b) Head-on
collision, We = 23, from the present simulation (a) (b) 27 Figure
14: (a) Head-on collision, We = 40, from Ashgriz and Poo (1990)
[58]; (b) Head-on collision, We = 40, the present simulation
Comparisonsbetweensimulationandexperimentalvisualizationareshownin
Figure 13. The qualitative agreement between our simulation and
experimental results
isexcellent.Atearlytimes,thetwodropletscandevelopintoaflyingsaucershape.
Then it evolves into a shape of torus when the inertia force
decreases. At later times, the bounce between two droplets appears
and they separate from each other. Here, the
separationoccurswhentheinertiaforceovercomesthesurfaceforceduetosurface
tension. 4.3.2 Coalescence followed by separation with formation of
one satellite droplet With the same computational configuration, we
only change the droplets relative
velocityfortheWebernumberof40intheexperimentofAshgrizandPoo[58]to
study the coalescence followed by the separation of a satellite
droplet in the head-on collision. Comparisons between simulation
and experimental visualization are shown
inFigure14.Astimeincreases,theligamentformsbetweenthetwodroplets.Then,
theligamentisseparatedfromthetwodropletsandthenevolvesintoasatellite
droplet.Itisobservedthatthenumericalresultsareingoodqualitativeagreement
with the experimental results.
Themasslossesofdropletcollisionforthemethodswithoutandwithmass
correctionareshowninFigure15.Itshowsthatthepresentmethodwithmass
correction can greatly reduce the mass loss. TV/Vo0 2 4 6 8 10
120.90.920.940.960.9811.02Without mass correctionWith mass
correctionTV/Vo0 5 10 150.90.920.940.960.9811.02Without mass
correctionWith mass correction (a)(b) Figure 15: Mass loss with and
without mass correction for different Weber numbers. (a) We = 23
(b) We = 40 28 AsdemonstratedinSection3.3,tovalidatetheJacobian
fGinEq.27is
smoothenoughtoensuresystematicconvergence,therelationshipbetweenliquid
volumeinacellGvolandlevelsetscalarGforthe3Ddropletcollisiondiscussed
below is plotted in Figure 16. It is shown that the data are seated
in a smooth line that can ensure the Jacobian has a valid value in
the whole domain. Figure 16: The relationship between liquid volume
in a cell Gvol and level set scalar G for the 3D droplet collision
4.4 CPU costs
Typically,theCPUcostsofvariousnumericalproceduresareinvestigated
throughconstantvelocityfield,2Dshearvelocityfieldand3Dactualcases,
correspondingtoZalesaksdisk,2Ddeformationand3Ddropletcollision,
respectively. Figure 17 shows the time spent per time step in the
multiphase solver for
thescalartransport,thevelocitysolver,thepressuresolverforthePoissonequation
andthemasscorrectionsolver.Themasscorrectionsolverisembeddedinthe
multiphase solver. It can be seen that multiphase solver accounts
for the most of the cost of one time
stepforthemethodwithmasscorrectioninthesethreecases.Thetimecostinthe
multiphase solver of the method with mass correction is two times
larger than that of the method without mass correction. This
indicates that mass correction solver greatly increases the cost of
themultiphasesolver. Thecosts of velocity solver and pressure
solver are almost the same for Zalesaks disk and 2D deformation.
However, the cost 29 of pressure solver is greatly larger than that
of velocity solver in 3D droplet collision.
Thisisattributedtothehighperformanceofparallelpolydiagonalsolversforthe
implicit formulation in the velocity solver [30]. TimeCPUcost(s)0
0.2 0.4 0.6 0.8
100.10.20.3MultiphaseVelocityPressureMultiphaseVelocityPressureMass
correction TimeCPUcost(s)0 2 4 6
800.511.522.53MultiphaseVelocityPressureMultiphaseVelocityPressureMass
correction (a) CPU costs of Zalesaks disk(b) CPU costs of 2D
deformation TCPUcost(s)0 2 4 6 8 10
1200.511.522.53MultiphaseVelocityPressureMultiphaseVelocityPressureMass
correction (c) CPU costs of 3D droplet collision Figure 17: CPU
costs of three cases (Solid line and dashed line represent the
methods without and with mass correction, respectively)
TheswirlingatomizationsimulationpresentedaboveisconductedonNational
SupercomputerCenterinTianjin.Typically,the3Ddropletcollisionistakenfor
example. This case uses eight CPUs and about three seconds per time
step are needed.
TheperformanceoftheparallelismhasbeeninvestigatedbyDesjardinsetal.[82],
showing that the code will perform efficiently at least up to 109
cells on 104 cores. 30 5 Applications to complex flows Despite of
the excellent results obtained by using simple flows in section 4,
it is necessary to evaluate the accuracy of the present method for
more complex flows. In this section, two typical cases, i.e., a
drop impacting on a liquid film and the swirling liquid
atomization, are simulated to demonstrate the capability of the
present method. 5.1 Drop impacting on a liquid film A drops impact
on a liquid film can be observed in many technical applications,
suchasturbineblade,sprayinjectionininternalcombustion(IC)engine,painting,
surface cooling. When a drop impacts on a liquid film, drop
splashing and spreading occur and the lamella and crown form. To
examine the accuracy of the present method with mass remedy, the
case reported in the experiment conducted by Cossali et al. [60]
issimulatednumerically.Thedimensionlessnumbersthataffectthebehaviorofthe
drop impact are the Weber number and the Ohnesorge number 2lU DWe=,
llOhD =,(37) where D is the drop diameter, U is the drop terminal
velocity, l is the liquid density, l is the dynamic viscosity of
liquid, andis the surface tension coefficient. Table 4: The
physical properties of gas and liquid phases Density of liquid
(kg/m3)1000 Density of gas (kg/m3)1.226 Dynamic viscosity of liquid
(kg/m/s)1.13710-3 Dynamic viscosity of gas (kg/m/s)1.78010-5
Surface tension coefficient (N/m)0.0728 In the experiment of
Cossali et al. [60], a water drop of diameter 3.82mm falls on a
liquid film. The time is set to t = 0 when the drop reaches the
film. The experimental conditionofWe=297,
Oh=0.0019andthedimensionlessfilmthickness/ h D == 0.29 is applied.
Therefore, the velocity at t = 0 is 2.389m/s and the film thickness
is h=1.1078mm.Thecomputationaldomainof8D3.6D8Disusedtoavoidtothe
effect of the domain size on the result. The grid size is
256128256. The no-slip wall
boundaryconditionisappliedtothebottomwall,andtheperiodicconditionisused
for the other boundaries. The y-coordinate is aligned with the
direction of impact. The physical properties of liquid and gas are
listed in Table 4.
Thedynamicsoftheimpactingdropisillustratedinasequenceofsnapshotsin
31 Figure 18. The dimensionless time is= Ut/D. Since the drop
impacts the liquid film,
thekineticenergyofthedropistransferredtotheliquidfilmandacrown-like
structure forms. Then the crown grows outward and upward due to
inertia. In Figure
19(h),asthelamellarise,therimofthecrownbecomesunstableandthecrown
exhibitsthree-dimensionalcharacteristics.Evolutionoftheinterfaceintheplane-cut
at z = 0 is shown in Figure 20. The interface gradually evolves
from a simple shape to a complex structure. (a)(b)(c) (d)(e)(f)
(g)(h)(i) Figure 18: Evolution of the interface for drop impacting
on liquid film ((a)-(i) for the dimensionless time= Ut/D = 0-8)
Themasslossofdropimpactingonaliquidfilmisadded.Similarly,themass
loss in Figure 19 is reduced with the mass correction. 32 TV/Vo0 5
10 150.920.940.960.981Without mass correctionWith mass correction
Figure 19: Mass loss of drop impacting on a liquid film As shown in
Figures 21 and 22, the time series of the crown height and diameter
fromthepresentsimulationarecomparedwithexperimentalresultsofCossalietal.
[60]andnumericalresultsofLeeetal.[61].Intheexperiment,thecrownheightis
defined as the distance between the crown base and the rim base. It
is obvious that the crown height in the present simulation agrees
well with the experimental results. The
two-dimensionalnumericalresultsofLeeetal.[61]showgoodagreementwith
experimentalresultsintheearlystageofcrownformation,butsomediscrepancies
appearinthelaterstageofthecrownspreading,whichindicatesthatthecrown
displaysthree-dimensionalcharacteristicsinthelatertime.Thepresent
three-dimensionalnumericalresultsshowexcellentagreementwiththeexperimental
resultsinthelaterstageofthecrownspreading,whichisdisplayedasaveragein
Figure21.Thethree-dimensionalnon-uniformcrownstructurescanbeobservedin
Figure 19(h) and marked as low and high in Figure 21. Figure 20:
Evolution of the interface in the plane-cut at z = 0 33
Non-dimensional time ()Non-dimensionalcrownheight(H/Do)0 2 4 6 8 10
12 14 16 18 2000.20.40.60.811.21.41.61.82 = 0.29, Cossali et al.
(2004), experiment = 0.29, Lee et al. (2004), simulation = 0.29,
low, the present simulation = 0.29, high, the present simulation =
0.29, average, the present simulation Figure 21: Comparison of
crown height evolution between experiment and simulation (low
stands for the mean crown height of z = 0 and x = 0 plane; high
stands for the mean crown height of two diagonal planes of the
computational domain; average stands for the average crown height
of low and height) Non-dimensional time
()Non-dimensionalcrowndiameter(D/D0)0 2 4 6 8 10012345678=0.29,
Cossali et al. (2004), experiment=0.29, Lee et al. (2011),
simulation=0.29, the present simulation Figure 22: Comparison of
the evolution of the crown diameter between experiment and
simulations The crown diameter is defined as the external diameter
at the base of the rim as in Cossali et al. [60]. We compare the
present numerical result with the experimental
andnumericalresultsofLeeetal.[61]inFigure22.Itisshownthatthetwo
numerical results are very similar and both predicted diameters are
slightly larger than the experimental result. 5.2 Swirling
atomization
Theprimaryatomizationexhibitsarichphysicalphenomenologythatisstill
34
poorlyunderstood.Withtheincreaseofthecomputationalcapability,thenumerical
simulation of primary atomization has been developed rapidly in
recent years, such as jet atomization (in Menard et al. [37],
Desjardins et al. [23][62], Shinjo et al.[63], and
Herrmann[64])andcrossflowatomization(inHerrmann[65],Paietal.[66],and
Muppidi et al. [67]). However, the mass loss becomes an issue using
the existing level
setmethod,asshownbyPaietal.[66].Furthermore,thereislackofnumerical
studiesonswirlingliquidatomizationpartiallybecauseofitsthinthicknessofsheet
and long period of existence of dispersed droplets. Therefore, the
mass losses maybe increase in the sheet atomization.
Inthissection,theswirlingliquidsheetatomizationisinvestigatedusingthe
presentmassconservinglevelsetmethod.Thesketchoftheswirlingliquidsheet
atomization is shown in Figure 23. The gas is injected at the
center with the nozzle of
diameterDin.Theswirlingliquidflowscoaxiallywiththeannulargapthickness
(Dout-Din)/2. The detailed parameters in this case are listed in
Table 5. Figure 23: Sketch of the swirling sheet atomization Table
5: Summary of parameters for swirling liquid sheet atomization
Outer diameter Dout (mm)0.4 Inner diameter Din (mm)0.2 Liquid
swirlyes Density of liquid (kg m-3)800 Density of gas (kg m-3)40
Dynamic viscosity of liquid (kg m-1 s-1)1.610-3 Dynamic viscosity
of gas (kg m-1 s-1)1.610-4 Axial velocity of liquid (m s-1)15 Swirl
velocity of liquid (m s-1)15 Velocity of gas (m s-1)30 Surface
tension (N m-1)0.036 Rel (based on Dout)3000 35 We (based on lip
thickness 0.1 mm)500 Dynamic viscosity ratio10 Kinematic viscosity
ratio0.5 Density ratio20 Ohnesorge number0.015
Domain5Dout5Dout5Dout Numbers of CPU666=256 Numbers of
cells256256256= 16.78 million x (m) 7.8 Time step size (s)110-8 T =
0T = 1.125T = 1.875T = 2.25T = 2.625 T = 3.75T = 4.875T = 5.625
Figure 24: Interface evolution of swirling liquid sheet atomization
Fig.24showstheinterfaceevolutionoftheswirlingliquidsheetatomization.
The initial liquid film is set as a semi-sphere membrane with
thickness 1/2(Dout-Din).
Theliquidfilmejectingfromtheannulusisatthelaminarstate.Thedimensionless
time is defined as T = tVl/Dout, where V1 is the axial velocity of
liquid. At an early time
ofT=1.125,thetipmushroomshapeisobserved.Arecirculationregionappears
behind the jet front and instantaneous breakup occurs from the
mushroom edge. With
theevolutionofatomization,aliquidsheetisobserved.Withoutdisturbancesadded
36
upstream,theliquidsheetmaintainssmoothwithinacertaindistance.Itisobserved
that transverse azimuthal perturbations occur at the tip of liquid
sheet at T = 4.875. It is expected that transient acceleration in
the direction normal to the liquid at the rims
triggerstheRayleigh-Taylorinstability,whichproducestheazimuthalperturbation.
Withthegrowingamplitudeoftransverseazimuthalperturbations,liquidligaments
areobservednearthewavecrestsatT=5.625.Then,theliquidligamentsare
stretched and droplets are produced at the tip of the ligaments.
Todemonstratetheeffectofthemassremedyapproachinthepresentmass
conservinglevelsetmethod,theresultsfromthemethodswithmassremedyand
without mass remedy at T = 5.625 are compared in Figure 25. During
the formation of
theliquidsheet,anumberofdropletsspreadduetothecentrifugalforce.These
dropletscanbecapturedusingthepresentmethodwithmassremedy.Ontheother
hand,thesedisperseddropletscannotberesolvedusingthemethodwithoutmass
remedy. It confirms that the new approach with the mass remedy" is
able to resolve
detachmentofligamentsanddropletscomparedtothestandardapproachwiththe
same grid resolution. If the grid resolution is further increasing
so that the mass loss is negligible, the advantage of the new
approach will disappear. As shown in Figure 26, the mass loss is
about 40% of the liquid mass present at T=5.625 in the
computational domain, and this discrepancy can be corrected by
using the mass remedy method. (a) (b) 37 (c)Figure 25: The
interface at T = 5.625 from the results without the mass remedy
method on 2563 grid (a) with the mass remedy method on 2563 grid
(b) and without the mass remedy method on 5123 grid (c) TV/Vo0 0.5
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 600.20.40.60.81No mass remedyMass
remedy Figure 26: The comparison of the mass loss between methods
without and with mass remedy (V0 refers to the theoretical volume
of the liquid until T=5.625, which can be computed from the initial
semi-sphere membrane and the temporal integration of boundary
fluxes) 6 Conclusions An improved mass conserving level set method
for detailed numerical simulation of liquid atomization is
developed to overcome the mass loss in the existing level set
methods.Thismethodisbasedonthefactthatthemasslosstendstohappeninthe
vicinity of large curvature regions rather than in smooth regions.
The mass loss within every time step owing to the numerical
dissipation and re-initialization is remedied to
theinterfacebasedonthecurvatureattheinterface.Inprinciple,theabsolutemass
38 conservation can be obtained via this method. In the benchmark
cases of Zalesaks disk and drop deformation in a vortex field, the
excellent agreement between simulation results and theoretical
results is obtained.
Itisshownthatgridsizesignificantlyinfluencesthenumericalresultsandthemass
losscanbegreatlyreducedbythepresentmethodwithsecond-orderaccuracyand
negligible additional computational cost. Additionally, in the
simulations of the binary
dropcollision,qualitativegoodagreementswithexperimentalresultsareshownin
visualizations of the droplet evolution.
Formorecomplexcases,thepresentmethodisdemonstratedtosuccessfully
reproducetheevolutionofthedropimpactingonliquidfilm,withbetteragreement
comparedtoexperimentaldatathanthatofLeeetal.[61].Finally,thedetailed
numerical simulation of a complicated swirling liquid sheet
atomization is conducted
usingthepresentmethod.Thecomplexatomizationinterfacescanbecaptured,and
the mass loss is greatly reduced compared with the existing level
set method. Acknowledgement
ThisworkisfinanciallysupportedbytheZhejiangProvincialNaturalScience
FoundationforDistinguishedYoungScholars(No.LR12E06001)andtheNational
NaturalScienceFoundationofChina(Nos.51222602,51276163).YueYang
acknowledges the support by the Young Thousand Talent Program of
China. References
[1]F.H.Harlow,J.E.Welch,Numericalcalculationoftime-dependentviscous
incompressibleflowoffluidwithafreesurface, Physicsof Fluids
8(1965)21822189
[2]B.J.Daly,ANumericalstudyoftwo-fluidRayleigh-Taylorinstability,
Phys. Fluids 10(1967)297-307
[3]B.J.Daly,W.E.Pracht,Numericalstudyofdensity-currentsurges,Physicsof
Fluids 11(1968)15-30 [4]M. Gorokhovski, M. Herrmann, Modeling
primary atomization, Annu. Rev. Fluid Mech., 40(2008)343-366 [5]J.
A. Sethian, P. Smereka, Level set methods for fluid interfaces,
Annu. Rev. Fluid Mech., 35(2003)341-372 [6]F. Denner, D. R. van der
Heul, G. T. Oud, M. M. Villar, A. da S. Neto, B. G. M. 39
vanWachem,Comparativestudyofmass-conservinginterfacecapturing
frameworksfortwo-phaseflowswithsurfacetension,Int.J.MultiphaseFlow,
61(2014)37-47 [7]H. A. A. Amiri, A. A. Hamouda, Evaluation of level
set and phase field methods
inmodelingtwophaseflowwithviscositycontrastthroughdual-permeability
porous medium, Int.J.Multiphase Flow, 52(2013)22-34
[8]J.Kim,Phase-fieldmodelsformulti-componentfluidflows,Commun.Comput.
Phys., 12(3)(2012)613-661 [9]Y. Sun, C. Beckermann, Sharp interface
tracking using the phase-field equation, J. Comput.
Phys.,200(2007)626-653 [10]
P.H.Chiu,Y.T.Lin,Aconservativephasefieldmethodforsolving
incompressible two-phase flows, J. Comput. Phys.,230(2011)185-204
[11]
F.Losasso,R.Fedkiw,S.Osher,Spatiallyadaptivetechniquesforlevelset
methods and incompressible flow, Computers & Fluids,
35(2006)995-1010 [12]
R.R.Nourgaliev,S.Wiri,N.T.Dinh,T.G.Theofanous,Onimprovingmass
conservationoflevelsetbyreducingspatialdiscretizationerrors,Int.J.
Multiphase Flow,31(2005)1329-1336 [13] A. Salih, S. G. Moulic, Some
numerical studies of interface advection properties of level set
method, Sadhana, 34(2009)271-298 [14]
T.W.H.Sheu,C.H.Yu,P.H.Chiu,Developmentofadispersivelyaccurate
conservativelevelsetschemeforcapturinginterfaceintwo-phaseflows,J.
Comput. Phys.,228(2009)661-686 [15] T. W. H. Sheu, C. H. Yu, P. H.
Chiu, Development of level set method with good
areapreservationtopredictinterfaceintwo-phaseflows,Int.J.Numer.Meth.
Fluids, 67(2011) 109-134 [16]
W.H.Reed,T.R.Hill,Triangularmeshmethodsfortheneutrontransport
equation,Tech.ReportLA-UR-73-479,LosAlamosScientificLaboratory,Los
Alamos, NM, 1973 [17]
P.Rasetarinera,M.Y.Hussaini,Anefficientimplicitdiscontinuousspectral
Galerkin method, J. Comput. Phys. 172(2001)718-738 [18]
J.F.Remacle,N.Chevaugeon,A.Marchandise,C.Geuzaine,Efficient
visualization of high-order finite elements, Int. J. Numer. Methods
Eng. 69(2007) 750771 40 [19] D. Enright, F. Losasso, R. Fedkiw, A
fast and accurate semi-Lagrangian particle level set method,
Computers and Structures, 83(2005)479-490 [20]
J.Strain,Semi-Lagrangianmethodsforlevelsetequations,J.Comput.Phys.,
151(1999)498-533 [21]
D.Xiu,G.E.Karniadakis,Asemi-Lagrangianhigh-ordermethodfor
Navier-Stokes equations, J. Comput. Phys., 172(2001)658-684 [22]
D.Adalsteinsson,J.A.Sethian,Thefastconstructionofextensionvelocitiesin
level set methods, J. Comput. Phys.,148(1999)2-22 [23] A.
Ovsyannikov, V. Sabelnikov, M. Gorokhovski, A new level set
equation and its numerical assessments, CTR Proceedings of the
Summer Program,2012 [24]
D.L.Chopp,Anotherlookatvelocityextensionsinthelevelsetmethod,SIAM
Journal on Scientific Computing, 31(2009)3255-3273 [25] V.
Sabelnikov, A. Y. Ovsyannikov, M. Gorokhovski, Modified level set
equation and its numerical assessment, J. Comput. Phys.,
278(2014)1-30 [26]
E.Olsson,G.Kreiss,Aconservativelevelsetmethodfortwophaseflow,J.
Comput. Phys., 210(2005)225-246 [27]
E.Olsson,G.Kreiss,S.Zahedi,Aconservativelevelsetmethodfortwophase
flow II, J. Comput. Phys.,225(2007)785-807 [28] F. Xiao, Y. Honma,
T. Kono, A simple algebraic interface capturing scheme using
hyperbolic tangent function, Int. J. Numer. Meth. Fluids,
48(2005)1023-1040 [29]
C.Walker,B.Muller,Aconservativelevelsetmethodforsharpinterface
multiphase flow simulation, ECCOMAS CFD 2010 [30]
O.Desjardins,V.Moureau,H.Pitsch,Anaccurateconservativelevelset/ghost
fluidmethodforsimulatingturbulentatomization,J.Comput.Phys.,227(2008)
8395-8416 [31]
M.Owkes,O.Desjardins,AdiscontinuousGalerkinconservativelevelset
schemeforinterfacecapturinginmultiphaseflows,J.Comput.Phys.,
249(2013)275-302 [32]
T.Nonomura,K.Kitamura,K.Fujii,Asimpleinterfacesharpeningtechnique
with a hyperbolic tangent function applied to compressible
two-fluid modeling, J. Comput. Phys., 258(2014) 95-117 [33]
H.Kohno,J.C.Nave,Anewmethodforthelevelsetequationusinga
hierarchical-gradienttruncationandremappingtechnique,Comput.Phys.
41 Commun.,184(2013)1547-1554 [34] M. Sussman, P. Smereka, S.
Osher, A level set approach for computing solutions to
incompressible two-phase flow, J. Comput. Phys., 114(1994)146-159
[35] M. Sussman, E. Fatemi, P. Smereka, S. Osher, An improved level
set method for incompressible two-phase flows, Computers and
Fluids, 27(1998)663-680 [36]
M.Sussman,E.Fatemi,Anefficient,interface-preservinglevelsetredistancing
algorithm and its application to interfacial incompressible fluid
flow, SIAM J. Sci. Comput. 20(1999)1165-1191 [37]
G.Russo,P.Smereka,Aremarkoncomputingdistancefunctions,J.Comput.
Phys., 163(2000)51-67 [38]
D.Hartmann,M.Meinke,W.Schrder,Differentialequationbasedconstrained
reinitialization for level set methods, J. Comput. Phys.,
227(2008)6821-6845 [39] D. Hartmann, M. Meinke, W. Schrder, The
constrained reinitialization equation for level set methods, J.
Comput. Phys., 229(2010)1514-1535 [40]
Y.C.Chang,T.Y.Hou,B.Merriman,S.Osher,Alevelsetformulationof
Eulerianinterfacecapturingmethodsforincompressiblefluidflows,J.Comput.
Phys., 124(1996)449-464 [41]
L.O.McCaslin,O.Desjardins,Alocalizedre-initializationequationforthe
conservative level set method, J. Comput. Phys.,262(2014)408-426
[42] J.A.Sethian,Levelsetmethodsandfastmarchingmethods,Cambridge
University Press,1999 [43]
O.Desjardins,H.Pitsch,Aspectrallyrefinedinterfaceapproachforsimulating
multiphase flows, J. Comput. Phys.,228(2009)1658-1677 [44]
J.U.Brackbill,D.B.Kothe,C.Zemach,Acontinuummethodformodeling
surface tension. J. Comput. Phys., 100(1992)335-354 [45]
R.F.Ausas,E.A.Dari,G.C.Buscaglia,Ageometricmass-preserving
redistancingschemeforthelevelsetfunction,Int.J.Numer.Meth.
Fluids,65(2011)989-1010 [46] A. Salih, G. Moulic, A mass
conservation scheme for level set method applied to
multiphaseincompressibleflows,InternationalJournalforComputional
Methods in Engineering Science and Mechanics,14(2013)271-289 [47]
M.Sussman,E.G.Puckett,Acoupledlevelsetandvolume-of-fluidmethodfor
computing3Dandaxisymmetricincompressibletwo-phaseflows,J.Comput. 42
Phys., 162(2000)301-337 [48]
M.Sussman,Asecondordercoupledlevelsetandvolume-of-fluidmethodfor
computinggrowthandcollapseofvaporbubbles,J.Comput.
Phys.,187(2003)110-136 [49]
G.Son,Efficientimplementationofacoupledlevel-setandvolume-of-fluid
methodforthreedimensionalincompressibletwo-phaseflows,NumericalHeat
Transfer,PartB:Fundamentals:AnInternationalJournalofComputationand
Methodology, 43(2003)549-565 [50] X. Yang, A. J. James, J.
Lowengrub, X. Zheng, V. Cristini, An adaptive coupled
level-set/volume-of-fluidinterfacecapturingmethodforunstructuredtriangular
grids, J.Comput.Phys.,217(2006)364-394 [51] T. Menard, S. Tanguy,
A. Berlemont, Coupling level set/VOF/ghost fluid methods:
validation and application to 3D simulation of the primary break-up
of a liquid jet, Int. J. Multiphase Flow, 33(2007)510-524 [52] X.
Lv, Q. Zou, Y. Zhao, D. Reeve, A novel coupled level set and volume
of fluid
methodforsharpinterfacecapturingon3Dtetrahedralgrids,J.Comput.Phys.,
229(2010)2573-2604 [53]
Z.Wang,J.Yang,F.Stern,Anewvolume-of-fluidmethodwithaconstructed
distancefunctionongeneralstructuredgrids,J.Comput.Phys.,231(2012)
3703-3722 [54]
Z.Wang,A.Y.Tong,Asharpsurfacetensionmodelingmethodfortwo-phase
incompressible interfacial flows, Int. J. Numer. Meth. Fluids,
64(2010)709-732 [55] D. L. Sun, W. Q. Tao, A coupled
volume-of-fluid and level set (VOSET) method
forcomputingincompressibletwo-phaseflows,Int.J.HeatMass
Tran.,53(2010)645-655 [56]
C.E.Kees,I.Akkerman,M.W.Farthing,Y.Bazilevs,Aconservativelevelset
methodsuitableforvariable-orderapproximationsandunstructuredmeshes,J.
Comput. Phys.,230(2011)4536-4558 [57]
Y.Wang,S.Simakhina,M.Sussman,Ahybridlevelset-volumeconstraint
methodforincompressibletwo-phaseflow,J.Comput.Phys.,
231(2012)6438-6471 [58]
S.P.vanderPijl,A.Segal,C.Vuik,P.Wesseling,Amass-conservinglevel-set
methodformodelingofmulti-phaseflows,Int.J.Numer.Meth.Fluids, 43
47(2005)339-361 [59] S. P. van der Pijl, A. Segal, C. Vuik, P.
Wesseling, Computing three-dimensional
two-phaseflowswithamass-conservinglevelsetmethod,Comput.VisualSci,
11(2008)221-235 [60] D. Enright, R. Fedkiw, J. Ferziger, I.
Mitchell, A hybrid particle level set method for improved interface
capturing, J. Comput. Phys.,183(2002)83-116 [61] S. E. Hieber, P.
Koumoutsakos, A Lagrangian particle level set method, J. Comput.
Phys., 210(2005)342-367 [62] P. Trontin, S. Vincent, J. L.
Estivalezes, J. P. Caltagirone, A subgrid computation
ofthecurvaturebyaparticle/level-setmethod.Applicationtoa
front-tracking/ghost-fluidmethodforincompressibleflows,J.Comput.Phys.,
231(2012)6990-7010 [63]
P.H.Chiu,Y.T.Lin,Aconservativephasefieldmethodforsolving
incompressible two-phase flows, J. Comput. Phys.,230(2011)185-204
[64] M. Sussman, A. S. Almgren, J. B. Bell, P. Colella, L. H.
Howell, M. L. Welcome,
Anadaptivelevelsetapproachforincompressibletwo-phaseflows,J.Comput.
Phys.,148(1999)81-124 [65] D. Fuster, A. Bague, T. Boeck, L. L.
Moyne, A. Leboissetier, S. Popinet, P. Ray, R.
Scardovelli,S.Zaleski,Simulationofprimaryatomizationwithanoctree
adaptivemethrefinementandVOFmethod,Int.J.MultiphaseFlow,
35(2009)550-565 [66] M. Herrmann, A balanced force refined level
set grid method for two-phase flows on unstructured flow solver
grids, J. Comput. Phys.,227(2008)2674-2706 [67] H. Kim, M. S. Liou,
Accurate adaptive level set method and sharpening technique
forthreedimensionaldeforminginterfaces,Computers&Fluids,
44(2011)111-129 [68]
M.G.Pai,H.Pitsch,O.Desjardins,Detailednumericalsimulationsofprimary
atomizationofliquidjetsincrossflow,47thAIAAAerospaceSciencesMeeting
IncludingtheNewHorizonsForumandAerospaceExposition,2009,Orlando,
Florida [69]
C.D.Pierce,P.Moin.Progress-variableapproachforlargeeddysimulationof
turbulentcombustion.TechnicalReportTF80,FlowPhysicsandComputation
Division, Dept. Mech. Eng., Standford University, 2001. 44 [70]
S.T.Zalesak,Fullymultidimensionalflux-correctedtransportalgorithmsfor
fluids, J. Comput. Phys., 31(1979)335-362 [71]
J.B.Bell,P.Colella,H.M.Glaz,Asecondorderprojectionmethodforthe
incompressible navier stokes equations, J. Comput.
Phys.,85(1989)257-283 [72]
N.Ashgriz,J.Y.Poo,Coalescenceandseparationinbinarycollisionsofliquid
drops, J. Fluid Mech., 221(1990)183-204 [73]
S.Tanguy,A.Berlemont,Applicationofalevelsetmethodforsimulationof
droplet collisions, Int. J. Multiphase Flow, 31(2005) 1015-1035
[74] G. E. Cossali, M. Marengo, A. Coghe, S. Zhdanov, The role of
time in single drop splash on thin film, Experiments in fluid,
36(2004)888-900 [75] S. H. Lee, N. Hur, S. Kang, A numerical
analysis of drop impact on liquid film by
usingalevelsetmethod,J.Mechanicalscienceandtechnology,
25(2011)2567-2572 [76]
O.Desjardins,H.Pitsch,Detailednumericalinvestigationofturbulent
atomization of liquid jets. Atomization and Sprays, 20(2010)311-336
[77]
J.Shinjo,A.Umemura,Simulationofliquidjetprimarybreakup:dynamicsof
ligament and droplet formation. Int. J. Multiphase Flow,
36(2010)513-532 [78] M. Herrmann, On simulating primary atomization
using the refined level set grid method. 21st ILASS, Orlando,(2008)
[79]
M.Herrmann,Detailednumericalsimulationsoftheprimaryatomizationofa
turbulentliquidjetincrossflow.ProceedingsofASMETurboExpo,Orlando,
(2009) [80]
M.G.Pai,H.Pitsch,O.Desjardins,Detailednumericalsimulationsofprimary
atomizationof liquid jetsincrossflow. 47thAIAA Aerospace
SciencesMeeting, Orlando, (2009) [81]
S.Muppidi,K.Mahesh.Directnumericalsimulationofroundturbulentjetsin
crossflow. J. Fluid Mech., 574(2007)59-84 [82]
O.Desjardins,J.O.McCaslin,M.Owkes,P.Brady,Directnumericaland
large-eddysimulationofprimaryatomizationincomplexgeometries,
Atomization and Sprays, 23(2013)1001-1048 [83]
S.Osher,R.Fedkiw,Levelsetmethodsanddynamicsimplicitsurfaces,Pages
41-46, Springer-Verlag. ISBN 0-387-95482-1, 2002 [84]
R.J.LeVeque.High-resolutionconservativealgorithmsforadvectionin 45
incompressible flow. SIAM Journal on Numerical Analysis,
33(1996)627-665
