Three dimensional high speed drop impact onto solid surfaces at arbitrary angles

Contents lists available at ScienceDirect

International

Journal

of

Multiphase

Flow

journal homepage: www.elsevier.com/locate/ijmulflow

Three-dimensional

high

speed

drop

impact

onto

solid

surfaces

at

arbitrary

angles

Radu

Cimpeanu

∗

,

Demetrios T.

Papageorgiou

DepartmentofMathematics,ImperialCollegeLondon,LondonSW72AZ,UnitedKingdom

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received3August2017 Revised20March2018 Accepted18June2018 Availableonline19June2018

Keywords: Dropimpact Stagnation-pointflow Spreading

Splashing Volume-of-fluid

Directnumericalsimulations

a

b

s

t

r

a

c

t

Therichstructuresarisingfromtheimpingementdynamicsofwaterdropsontosolidsubstratesathigh velocities areinvestigated numerically. Current methodologies in the aircraftindustry estimating wa-tercollectiononaircraftsurfaces arebasedonparticletrajectorycalculationsand empiricalextensions thereofinordertoapproximatethecomplexfluid-structure interactions.We performdirectnumerical simulations(DNS)usingthevolume-of-fluid methodinthreedimensions,foracollectionofdropsizes andimpingementangles.Thehighspeedbackgroundairflowiscoupledwiththemotionoftheliquidin theframeworkofobliquestagnation-pointflow.Qualitativeandquantitativefeaturesarestudiedinboth pre-and post-impactstages.One-to-onecomparisonsaremadewithexperimentaldataavailablefrom the investigationsofSorand García-Magariño(2015), whilethemain bodyofresultsis createdusing parametersrelevanttoflightconditionswithdropletsizesintherangesfromtenstoseveralhundredsof microns,aspresentedbyPapadakisetal.(2004).Dropdeformation,collision,coalescenceandmicrodrop ejectionanddynamics,alltypically neglectedorempiricallymodelled,areaccuratelyaccountedfor.In particular,weidentifynewmorphologicalfeaturesinregimesbelowthesplashingthresholdinthe mod-elledconditions.Wethenexpandonthevariationinthenumberanddistributionofejectedmicrodrops as afunctionoftheimpactingdropsizebeyondthisthreshold.The presenteddropimpact model ad-dresseskeyquestionsatafundamentallevel,howevertheconclusionsofthestudyextendtowardsthe advancementofunderstandingofwaterdynamicsonaircraftsurfaces,whichhasimportantimplications intermsofcompliancetoaircraftsafetyregulations.Theproposedmethodologymayalsobeutilisedand extendedinthecontextofrelatedindustrialapplicationsinvolvinghighspeeddropimpactsuchasinkjet printingandcombustion.

© 2018TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Since the days ofWorthington (1876), the problemof droplet

impacthasofferedthefluiddynamicsresearchcommunityexciting

opportunitiesandchallengesoverthecourseofitshistory.Forthe

firsttime ina systematicmanner,in hisbook entitled Astudy of

splashes(Worthington,1908),Worthingtonmakesuseofearly

pho-tographic technology (alongside carefulsketchwork) to provide a

comprehensivevisualinterpretation ofsplashingphenomena. The

framework has since captivated the interest of theoreticians and

experimentalistsalike,asitincorporatesoneofthemostinvitingly

simplegeometrical configurations, whileatthe same time giving

risetodiverseandrichphenomenaofimmensescope.

∗ _{Correspondingauthor.}

E-mailaddress:[email protected](R. Cimpeanu). URL:http://www.imperial.ac.uk/people/radu.cimpeanu11(R.Cimpeanu)

Aplethoraofapplicationareasbenefitfromunderstandingthe

outcomesofdropletimpactevents.Weemphasiseinparticularthe

roleofdropletsplashing(orabsencethereof)inprinting

technolo-gies(vanDamandLeClerc,2004;JungandHutchings,2012), com-bustion (Moreira etal., 2010), granularmaterialinteractions atall

scales(ThoroddsenandShen,2001;Marstonetal.,2012),

electron-ics(Kim, 2007) andspray-coolinginnuclearreactors (Sawanand

Carbon,1975).Thedesignofsuperhydrophobiccoatingsinrelation

todropletimpactdynamics(Tsaietal.,2009;Dengetal.,2009)is

yetanotherprimeexampleofthewidespreadapplicability ofthis

canonicalproblem.

Recent reviews provide an excellent insight into the

state-of-the-art in the field within each decade (Rein, 1993 in the

1990’s,Yarin,2006inthe2000’sandmorerecentlyJosserandand

Thoroddsen,2016).Thearea haswitnessedaverystrongsurgein

the past decade, fuelled in part by the development of

progres-sivelymorepowerfulimagingtechnologies,withboth framerates

andresolutions capableof capturing details beyondthe scope of

https://doi.org/10.1016/j.ijmultiphaseflow.2018.06.011

previous equipment (see Thoroddsen et al., 2008 as well).

Fur-thermore, the improvement of numerical algorithms and usage

of highperformance computinghas enabledcomputational

stud-iesthat complementandinformbothexperimental andanalytical

work. We focus particularly on the volume-of-fluid package

Ger-ris(Popinet,2003;2009),whichisoneofthemostpopular

open-source toolsdueto its strengths in dealingwithinterfacialflows

onarangeofverydifferentscales.Comparisonswithexperiments,

aswell asanalytical workhave beenconsistently robust,be itin

casesofliquid-liquidimpact(Thoravaletal.,2012;Agbaglahetal.,

2015)orimpacts ofliquidontosolid surfaces(Visseretal., 2015;

Philippietal.,2016;Wildemanetal.,2016).

In the case of normal (perpendicular) impact at

low-to-moderate velocities (and depending on specific fluid

proper-ties), an axisymmetric assumption can be used in analytical

and computational investigations. The reduction in

dimensional-ity is a significant advantage that has led to very efficient

(ax-isymmetric)computationsandgoodagreementwithexperiments.

Visser et al. (2015) for example, while innovating experimental

technology enabling the time-resolved investigation of

micron-sized dropimpacts, havemanagedtoconduct successful

compar-isons with direct numerical simulations at impact speeds of up

to 50m/s, aregime whichiscommonplace incombustion,inkjet

printingoraircraft-relatedapplications.Intherespectivescenario,

thesmalldropsspreadontothesurfaceinwhatisknownas

pan-caking motion, with the axisymmetric approximation remaining

validintheabsenceofsplashingevents.

In cases where spreading and later retraction rather than

splashing occurs,thevast majorityofeffortshavebeendedicated

towards identifying quantitiessuch asthe maximal spreading

ra-dius (Stow and Hadfield, 1981; Clanet et al., 2004; Fedorchenko et al., 2005; Roisman, 2009; Schroll et al., 2010) and most

re-cently Wildeman et al., 2016), as well as the resulting

mini-malfilm thickness, retractiondynamicsandthe roleofthe

inter-nal boundary layer - see Bartolo et al. (2005) and in particular

Eggersetal.(2010)foracomprehensiveinvestigationoftheabove.

Athigherspeedshowever,thereisstillanongoingdebateasto

howthesplashingphenomenaarefirstinitiated,andthesplashing

thresholdinparticular. Upuntil thegroundbreakingexperimental

investigation of Xu et al. (2005), there have been numerous

at-tempts to characterise the transitionfromspreading to splashing

dynamicsintheclassicalimpactproblemintermsofdrop-related

parametersonly(size,density,viscosity,surfacetensioncoefficient,

impactvelocity).TheChicagogroup discovered,however,that

de-creasingtheambientairpressuremaycompletelysuppress

splash-ing.Assuch,ahostofadditionalmodelling,experimental and

nu-merical efforts havebeen initiated,with thework of Riboux and

Gordillo(2014)proposingamodeldeducingathresholdsplashing

velocity asa functionofageneralised setofkeyparameters

con-tainingthe liquiddensityandviscosity, thedropradius,gas

den-sityandviscosity,theinterfacialtensioncoefficient,aswellasthe

nanometricmeanfreepathofthegasmolecules.

Oncethedropsplashes,thereisvery littleattentiondedicated

totheensuingdynamics,withthesizesandvelocitiesofsecondary

drops being prohibitively small experimentally and

computation-ally,althoughadvanceshavetakenplacerecentlyintermsof

sim-plifiedmodels.Inparticular,RibouxandGordillo(2015)have

pro-posed a one-dimensionalapproach to predictingsizesand

veloc-ities of ejecteddroplets for_O

(

1

)

mm sized impacting drops and

lowspeeds,findingreasonableagreementwithexperiments.

As underlinedby Josserandand Thoroddsen (2016), there are

severalexciting challengeslyingahead,two ofwhichareofgreat

importance in the contextof the presentwork. First ofall,

gain-ing an improved understanding of splashing,particularly in

diffi-culthighspeedconditionsofindustrialrelevance,ismovingmore

and more within reach, and further detailed investigation using

theavailabletoolsisneeded. Secondly,obliqueimpactsare rarely

analysed(exceptionsbeingMundoetal.,1995;Sikaloetal.,2005;

Birdetal.,2009)duetotheadditionalflowcomplexity.Mostoften,

qualitativeratherthanquantitativephenomenaareexploredin

de-tail.Theexceptionstendtofocusonlargescaleeffectsatthelevel

oftheentiredrop,asopposedtodetailsatthelevelofthe

splash-ingitselfandtheinterestinglocalstructures arising.Bothofthese

themeslieattheheartofthepresentwork,whichfocusesonthe

modellingandcomputationofobliquethree-dimensionaldrop

im-pactinaerodynamicconditions.

Inaircraft-orientedresearch anddesign involvingdropimpact,

therelevantscales areoftendictatedbythesizeofthepartsthat

aremostaffectedby phenomenasuch aswaterimpingement,

re-tentionandfinally icinganditsprevention.Thewingsornacelles

areseveralmetreslong,whilecomputingaccurateairflowsaround

themrequiresdomainsthatspantensofmetresinalldimensions.

Thisbecomeshighlyprohibitive intermsofaccurateresolutionof

theintricate andsensitive physicaleffects pertainingto drop

im-pact,whichoftenhappenatsub-micronscalesintheorderoftens

tohundreds of microseconds.As such, particle-trajectory

calcula-tionsofvariousdegreesofcomplexityhavethus farproventobe

theonlytractablesolutioninindustrialsetting.

There are several important limitations of current models, as

pointedoutbyGentetal.(2000)inarelativelyrecentreview:

• droplets are assumed to be spherical and non-deformable as

they approach the solid surface, hence topological transitions

such astheemergence ofsecondarydropseitherbeforeor

af-terimpactarenotconsidered;

• phenomena related to multiple drops such as collisions are

completelyignored;

• aerodynamicdrag,gravityandbuoyancyareassumedtobethe

soleforcesaffectingthedroptrajectories;

• whereasthelocalvelocityoftheairflowisembeddedintothe

ordinary differential equations governing the updates in drop

trajectories, the liquidmass is assumed not to affectthe

sur-roundingairflow;

• onceonthesurface,empiricalmodelstranslatethedrop

contri-butiontowardsliquidfilmformationanditsmovementfurther

downstreamalongthesurfaceofinterest.

Manyof these assumptions become inaccurate inthe context

ofthelarge supercooled droplets(largerthan severaltens of

mi-crons) found in the atmosphere. The difficulties outlined above

haveyet to be overcome, andmost modelling isperformed at a

highly coarse-grained level (Potapczuk et al., 1993; Bragg, 1996;

Rutkowskietal.,2003;WrightandPotapczuk,2004;Wright,2005; 2006; Honsek et al., 2008; Bilodeau et al., 2015), with

semi-empirical relations of varying complexity being proposed in

or-der to match with the rich but ultimately limited experimental

dataavailable byNASAexperimentsconductedbyPapadakisetal.

(2003,2004).Thefocushereisprimarilyonthefinalwater

reten-tionvaluesratherthan themorefundamentalproblemofthe

de-tailedimpactprocess, makingitidealfroman engineering

stand-point but offering limited insight into the underlyingphysics. In

thepastfewyears,thegroup atINTA/Madrid(Vargasetal.,2012;

SorandGarcía-Magariño,2015)havelookedinmoredetailintothe

deformationoflarge-scale dropsprior toimpact,withresultsthat

indicate regimes far more complex than captured by the typical

assumptions mentionedabove. Severalstudies focusingon recent

numericaladvancesinthehighspeedregime(>50m/simpact

ve-locity)haveemerged,particularlyforimpactsontoliquid,butalso

ontosolidsurfaces(MingandJing,2014;ChengandLou,2015;Guo

etal., 2016;Cherdantsevetal.,2017;Xieetal., 2017).Theseoffer

excitingopportunitiestostudyshorttimescalephenomenabeyond

thereachoftraditionalparticlemethods,howeveruptothispoint

processintoaframeworkthat includesamorerealisticmodelfor

themovementandeffectoftheairflowaround thebodiesof

in-terest.

Thepresentworkbridgestherelevantscalesintheproblemof

dropimpactonto aircraftsurfacesandproposesa suitable model

forthe air flow around the solid bodies of interest inwhich we

thenaccurately resolvethe dropimpingementprocess. Whilethe

drops are initialised as spherical sufficiently far away from the

body,we characterise their deformation prior to impact andthe

spreading/splashingthereafter,dependingondropsizesandangles

ofimpingement.Wefocusontheasymmetricfeaturesofthedrop

spreading whendroplets are very small(less than a few tens of

microns),phenomenawhichtoourknowledgehaveyetto be

ob-served.Asthedropsizeincreases,wequantifythesizesand

posi-tionsofthesecondarydropsemergingasaresultofthe

impinge-mentand provide usefulmetrics for practitioners looking to

im-provewater retentioncalculation methodology anda deeper

un-derstandingof the physics involved in the impact process under

challenging conditions. All flow parameters have been carefully

chosen to match with previous experimental studies or known

flight-specificvalues,whilemanyofthequantifiedmetricsarealso

comparedtoclassicaltheoreticalresultswhereapplicable.

Theinvestigationisstructuredasfollows.Weintroducethe

pro-posedmathematicalmodelinSection2,followedbyadetailed

de-scriptionof the computational framework in Section 3. We then

analyseourfindingsinSection4,focusingonbothpre-impact

dy-namics in Section 4.1 and post-impact dynamics in Section 4.2.

Theseresults are discussedand placed intocontext in Section 5,

followedbyconcludingremarks.

2. Mathematicalmodel

Inthepresentsectionwe elaborateonhowweadaptthe

clas-sical problem of drop impact to the highspeed flow conditions

of interest around aircraft surfaces. First we discuss some

use-fulassumptions allowing usto reduce geometrical complexity in

theprobleminSection 2.1,after whichwe expand onthe

math-ematicalmodelitself,outlining therelevantequations,initial and

boundaryconditions.

2.1.Scaletransition

Thefullmodelgeometrydiscussedinpreviousparagraphs

(air-craftwings/fuselagecomponents)isfartoocomplicated-and

spe-cific - from many points of view. To begin with, our aim is to

presentageneralmethodology,applicabletoanumberofsurfaces

rather than a specific specialised geometry. Secondly, the

multi-scalemodellingofboththebackgroundairflow aroundthelarge

scalebody andthe splashing dynamicswithin the muchsmaller

dropimpact regions is beyondreach in terms of theoretical and

currentcomputational resources.We thus employ several

simpli-ficationstoenable acloser inspectionof a muchmore amenable

problem,whichstillpreservesthemainphysicalcharacteristicswe

wishtoaddress.

Based onthedisparitybetweenthetwoscales intheproblem

(the impacting droplet diameter and the solid body it impinges

upon),weassumethecurvatureofthebodytohavenegligible

ef-fects.Tojustifythisapproximation,theradiusofcurvature ofthe

leadingedgeofa typicalNACAairfoilornacellelipskin,themost

sensitiveregions to water retentionandicing, is estimatedto be

of R_b=O

(

10−1

₎

_{m for standard commercial aircrafts. For a}

rea-sonablylarge dropletofradiusR=100

μ

m,we findR/Rb≈10−3.

Thus,fromthe perspectiveofmodellingthe localdroplet impact,

thesurfacecanbeconsideredasapproximatelyflat.Froma

differ-entviewpoint,we zoominsufficientlyclosetothesurfaceofthe

solid body,such that in therespective region thedroplet

diame-ter is therepresentative lengthscale andhence the details ofthe

impactcanbecarefullyexamined.

2.2. Governingequations

The framework of studying these fluids as incompressible in

laminarflow conditions is anatural choice inthe context ofour

problem,astheprimarytargetflightregimesoftake-off and

land-ingarecharacterisedbyrelativelylowvelocitiescomparedtothose

reached at higher altitudes. Furthermore, most droplet

impinge-menteventsare concentratedclosetotheleadingedgeofthe

ge-ometries ofinterest, where the flow has yet to enter the

transi-tionfromlaminartoturbulentstate.Eveninsuchcircumstances,a

complexandlikelyempiricalturbulencemodelwouldpreventthe

inspectionofthedetailedliquiddynamics,whichisthemaingoal

ofthepresentinvestigation.

Themodelfluidsareassumedtobeincompressible,immiscible

andviscous.Subscript1isusedtorefertothefluidinsidethedrop

(taken to be water), whereas subscript 2 decorates quantities in

thesurrounding(air)flow.Let

ρ

1,2 and

μ

1,2 denotetheconstant

densitiesanddynamicviscositiesofthetwofluidsinthe system.

Theconstantsurfacetensioncoefficientattheinterfaceisgivenby

σ

.VelocityvectorsU1,2=

(

U1,2,V1,2

)

andpressures P1,2 are used

inthe formulationofthedimensional momentumandcontinuity

equations

ρ

1

(

U 1t+

(

U 1·

∇)

U 1

)

=−

∇

P1+

μ

1

U 1, (1)

ρ

2

(

U 2t+

(

U 2·

∇)

U 2

)

=−∇P2+

μ

2

U 2, (2)

∇

·U 1,2=0. (3)

Gravitational forces are assumed to be negligible. There are two

lengthscales in the problem: the droplet diameter D, the

natu-ral choice for the reference lengthscale, and the size of the

(fi-nite)computationaldomainL.WescalelengthsbyD,velocitiesby

a referencebackground velocity U_∞ and pressures by

ρ

1U_∞2. The

emergingnon-dimensionalparametersare

Re=

ρ

1U∞D/

μ

1, We=

ρ

1U_∞2D/

σ

,

K=We√Re=

ρ

3

1D3U∞5/

(

σ

2

μ

1

)

. (4)

The Reynoldsnumber Re andWeber number Weappear directly

from the non-dimensionalisation procedure, while the splashing

parameter K is introduced as an intrinsic element of a drop

im-pact problem. The expression, originally introduced by Stow and

Hadfield(1981), hasbeen used to classify the possibleoutcomes

oftheimpact.Thisparameterhasbeencontroversialinthe

litera-tureandcannotindependentlyaccountfortheclassificationofthe

complicatedimpactprocess(seeXuetal.,2005;Mandreand

Bren-ner, 2012), howeverit serves as an indicator of the force of the

splashingandpermitscomparisonswithpreviousinvestigations.

Wealsointroducedensityandviscosityratios

r=

ρ

1/

ρ

2, m=

μ

1/

μ

2, (5)

andnon-dimensionalisethegoverningEqs.(1)–(3),resultingin

u 1t+

(

u 1·

∇

)

u 1=−

∇

p1+Re−1

u 1, (6)

u 2t+

(

u 2·

∇

)

u 2=−r

∇

p2+rm−1Re−1

u 2, (7)

Thenon-dimensionaltimescaleisD/U_∞.Typicalfluidpropertiesin

the case of waterand air atnear freezingtemperature (close to

0°C)aregivenasfollows.Waterhasdensity

ρ

1=999.8kg/m3and

dynamicviscosity

μ

1=1.16×10−3 kg/ms,whiletheairdensityis

ρ

2=1.21kg/m3anditsdynamicviscosity

μ

2=1.81×10−5kg/ms.

Theconstantsurfacetensioncoefficientis

σ

=7.2×10−2_N/mand

a representative value forthe velocity of the backgroundflow is

U_∞=78.44 m/s. This value has been selected to coincide with

classical experimental investigations (Papadakis et al., 2004), as

well assubsequent numericalinvestigations in theaerospace

en-gineeringcommunity(e.g.Bilodeauetal.,2015).Weunderlinethe

large density(r=826.28) andviscosity(m=64.09) ratios, which

posesignificantnumericalchallenges-thesearetoucheduponin

Section3.Tables1and2intheresultsdiscussionindicatethe

val-uesofthekeydimensionlessgroupsintheproblemandhighlight

theviolenthighspeedimpactregimeinvestigatedhere.

Todefinetheinterfacialconditionsgoverningthemotionofthe

drop, we assume a sharp interface y=S

(

x,t

)

; subsequently this

isrelaxedinthecontextofthe volume-of-fluidmethodology

em-ployed in the direct numericalsimulations. The prescribed

inter-facial conditionsare,in order,thekinematiccondition, the

conti-nuityofnormalandtangentialstresses,andcontinuityofvelocity

components:

wi=St+uiSx+

v

iSy, i=1,2, (9)

[n ·T ·n ]12=We−1

κ

, (10)

[t ·T ·n ]1₂=0, (11)

[u ]12=0, (12)

where[

(

·

)

]1

2=

(

·

)

1−

(

·

)

2representsthe jumpacross theinterface,

n,andtaretheunitnormalandtangenttotheinterface,

respec-tively, and

κ

is the interfacial curvature. The stress tensor T is

givenby

Ti j=−p

δ

i j+

μ

∂

ui

∂

xj+

∂

uj

∂

xi

, (13)

wherethe appropriatesubscript isusedindifferentfluid regions.

The initial and boundary conditions for the finite computational

domainaredescribedinthefollowingsubsection.

2.3. Backgroundflow

One ofthemostimportantfeatures ofthe modelisthe

inter-action between the liquid drop and the air around it. In typical

experimental conditions, dropletsare formed atthe tip ofan

in-jectiondeviceandfallundergravity,withtheheightofthedevice

beingvariedinordertoadjusttheterminalvelocityandhencefix

therelevantdimensionlessparameters.Inordertoreachvelocities

beyond O

(

1

)

m/s it is necessary to have some form of ejection

mechanismthatensuresnotonlyreproducibilityoftheshapes,but

also a stabilityof the dynamicsin early stagesas thedrop

trav-els through thequiescent airflow andmay become immediately

sheared andviolently deformedandbroken up. As such,mostof

theinvestigationsconcerningvelocitiesabove10m/sarerestricted

toverysmalldrops(wellbelow100μm),suchthatsurfacetension

isstrongenoughtopreservetheapproximatelysphericalshapeof

thedrop.

In flight conditions, leading edge droplet impact can be

lo-cally embedded in a stagnation-point flow which develops into

boundary layers on either side of the geometry. As such, most

droplets encounter a developing boundary layerstructure witha

strongshearcomponent. Inan efforttoreproduce thesametype

ofairflow environmentwhilepreservinggenerality, weproposed

anoblique-stagnation pointflow modelfortheairflow, withthe

liquiddrop being seededsufficiently far away fromthe body on

thedividingstreamlineoftheflow.Thereasonsbehindthischoice

aretwofold:

1.Farawayfromthesurfacethedropshouldretainitsshapeand

settingauniformvelocityfieldinsidethedropwithazero(as

inmostdesktopexperimentsetups)orpurelyhorizontal(along

the body)airmotionwouldproduceinstantaneous breakupof

thedrop.Thechoiceforstagnation-pointbackgroundflowand

theinitialpositionofthedropensuresthattheairflow

under-goes onlysmallchanges until sufficientlycloseto thesurface,

which is whenwe expect the drop tostart deforming in real

lifeconditions.

2.The stagnation-point flow has the same characteristics in the

near vicinity of the point ofzero velocities ason the aircraft

surface, in that boundarylayers are developingon eitherside

ofitandgrowingaswemovefurtherdownstream.Assuchthe

liquid drop is subjected to the shearflow naturallyoccurring

above the solid surface. Thisisbest representedin thehighly

obliqueimpactcases,inwhichtheairflowstreamlinesnearthe

surface havestrong deviationsfrom their far-fieldorientation.

Oncomingdropsdepartfromtheirhoststreamlinesclosetothe

surfaceandtheirfinal impingementpointsarewellwithin the

boundarylayergrowthregion.Thechoiceininitialpositioning

ofthedropthusretainsgenerality,whileatthesametime

pro-vidingsuitable conditions fortheearly stagesof thedrop

dy-namics.

Thereare howeverseveralpointsto bemadeprior to

advanc-ingtothe mathematicaldescriptionofthemodel.Firstofall, the

dynamics ofdrops in uniform flow has been extensively

investi-gated andthe deformation characteristicsfor large enoughdrops

areveryrich(seeJalaalandMehravaran,2012fora recent

compu-tational study).Therefore,even inthecaseof tailoringtheinitial

positionofthedroptoaregionofuniformairflow,thedropis

an-ticipatedtosuffersignificantdeformationsasitmovestowardsthe

solid body.Thesize ofthe finitecomputational domain canthen

beusedtoalleviate(orenhance)thiseffect.Secondly,itshouldbe

notedthattherestillremainsafundamentaldifferencetothe

prac-ticalscenarioinwhichasolidbodyismovingthroughhighliquid

watercontentclouds(withstationarywaterdropsofvaryingsizes)

asopposedtodropsimpingingontoastaticsolidsurface,asinthe

presentcase. Here we are enhancing the inertial contribution in

thepre-impactdropdynamicsandourchoiceininitialpositionof

thedropdoesultimatelyaffecttheliquidvolume impinging onto

thesurface.Previousexperimentalresultshavehoweverbeenused

asguidanceinordertobestaccountforthecomplexflow

dynam-ics,whileretainingasuitablewell-controlledflowenvironment.

Withtheabovepropertiesinmind,weunderlinethatthe

back-ground air flow poses its own non-trivial challenges. The

his-tory of the problem dates back to Hiemenz (1911), who was

the first to present a solution for the two-dimensional normal

stagnationpoint flow. Howarth (1951) then extended the

formu-lation to three dimensions. The oblique case was first touched

upon by Stuart (1959), then later rediscovered independently by

Tamada(1979) and Dorrepaal (1986).There havebeen a number

of corrections, extensions and generalisations on the main

prob-lem, for example by Wang (2008) and Tooke and Blyth (2008),

includingextensiontotwo-fluid systems(airflowimpinging onto

liquidfilmsabovea solidsurface),asstudiedbyTilleyand

Weid-man(1998) and Blythand Pozrikidis (2005). As far as we know

there is no general analytical solution to the three-dimensional

stagnation-pointflow problemat an arbitraryangle. Assuch, we

combina-Fig. 1. Snapshotsoftheconverged backgroundvelocityfieldobtainedasaresultofimposinguniformflowboundaryconditionsatanangleofθi=60◦ onthe upper boundary,hittingaflatsolidsurfaceatthebottom,withoutflowconditionsonalllateralboundaries.Thetwocross-sectionsthrough(a)thecentralx−yplaneand(b)the y−zplaneillustratetheverticalvelocityfield(negativeabove,zeroduetono-sliponthesurface),aswellasstreamlinesoftheflow.Thewaterdrop,showninwhite,is initialisedonthedividingstreamlineneartheupperborderofthegeometry.

tionofsuitableboundaryconditionsthatpreserveitsmain

charac-teristicsintheinteriorofthedomain.

The three-dimensionalcomputational box isselected to be of

size4L/D×L/D×2L/Din

(

x,y,z

)

−directions,whereL/D istakento

beofsize20,i.e.20dropdiameters.Theflatsolidsurfaceistaken

tobein the

(

x_,z

)

₋plane,withno-slipandimpermeability

condi-tionsprescribedinthisregion,suchthatatu1,2=0aty=0.

In order tomodel theoncoming flow atan arbitraryangle of

incidence

θ

iweprescribeinflowconditionsgivenby

u2

(

x,L/D,t

)

=cos

(

θ

i

)

,

v

2

(

x,L/D,t

)

=−sin

(

θ

i

)

aty=L/D. (14)

Laterally we impose typical free outflow conditionson all

re-maining foursides ofthe box. The main reasonfor doing so

re-latestothemovement ofthesecondarydropsresultingasa

con-sequenceofthesplashwhichcauseperturbationsintheflowfield,

making it difficult to fix velocities at the boundaries. The initial

conditionsaresetto

u2

(

x,y,0

)

=cos

(

θ

i

)

,

v

2

(

x,y,0

)

=−sin

(

θ

i

)

, (15)

prompting the need forthe convergence of thebackground flow

toasteadystate prior totheinclusion oftheliquiddroplets into

thecomputationaldomain.Forall casesconsideredinthepresent

investigation,anevolutionoftheflowspanning100dimensionless

timeunitsprovedmorethansufficientforthispurpose,witha

tol-eranceof10−6 _{inthecomponentsofthevelocityfieldselectedto}

verifyflow convergenceto steadystate. We have confirmed this

forallangles ofincidenceusinga rootmeansquare normof the

velocityfield,presentedinFig.2(b).Timet=0istakentobe the

time atwhich thedrop isseeded insidethe domain andby this

conventionthedirectnumericalsimulationsbeginatt₌₋100.

Focusing on the mid-(x, y)-plane (at z=0, see Fig. 1(a)), we

findsimilarflowpropertiestotheclassicalcaseofoblique

stagna-tionpoint-flowintwodimensions.Usingthetypicaldefinitionfor

thestreamfunction

ψ

(x,y),wherethehorizontalvelocity

compo-nent u=

ψ

y and the vertical velocity component

v

=−

ψ

x,

suffi-cientlyfarawayfromthewalltheflowtakestheform

ψ

(

x,y

)

=kxy+1

2

ζ

y

2_{. (16)}

Thisis effectively a superposition ofirrotational stagnation-point

flowof strengthk anda uniform shearflow parallel to thesolid

surface (in the x-direction), where k and

ζ

are scale constants

(seeBlythandPozrikidis,2005forarecentexpositiononthis

sce-nario).

ψ

=0denotesthedividingstreamlineontowhichthe

liq-uiddropisinitialisedjustbelowtheupperboundaryofthe

three-dimensional domain, with its centre at y=19.25 and z=0 and

withx varying as a function of the angle of incidence

θ

i of the

backgroundflow.Fig.1(a)providesavisualisationoftheconverged

flow field atthe instanceof theinitialisation of thedrop forthe

casewhen

θ

_i=

π

_/3.

We underline that, despite the flow being essentially

two-dimensional in the upper part of the domain (below the inflow

boundary),duetothepresenceofthesolidsurfaceandthelateral

outflowcondition, itdevelops itsfull three-dimensional structure

closetotheimpingementregion,withasinglestagnationpoint

be-ingpresentintheflowirrespectiveoftheimpingementangle.This

isbestobservedinFig.2(a),butalsoinFig.1,wherestreamlines

are drawn on top ofvelocity fields (illustrated incolour) plotted

indifferenttwo-dimensionalcross-sectionalplanestoindicatethe

deflectionintheairflow.

Oncethebackgroundairflowhasreachedits steadystate,the

initially sphericalliquiddropis prescribedto enterthe

computa-tionaldomainatadesiredlocation

(

x_i,y_i=19_.25_,z_i=0

)

.Thedrop

then inherits the local velocity field of the background air flow

whichisan approximatelyuniformflowdirectedtowardsthe

sur-faceatanangle

θ

_i,andisadvectedtowardsthesolidsurface.The

droplet shape issubject to physicaldeformations up to the time

ofitsimpact.Fullhydrodynamiccouplingdeterminesitstrajectory

andshape, withno furtherassumptions being madebeyondthis

point.

3. Numericalmethodology

The numerical computations in the present study have been

carried out usingthe open-source package Gerris (Popinet, 2003;

2009) (http://gfs.sourceforge.net/), which has been used

exten-sivelywithgreatsuccessbythemulti-phaseflowcommunityover

thelastdecade.Thepackage isidealforourpurposessinceit

ac-curatelysolves theincompressibleNavier–Stokesequations(anda

varietyofadditionalmulti-physicsextensions)usingthefinite

vol-umemethodandavolume-of-fluidapproachtoaccountfor

fluid-fluid interfaces. The schemes are second order accurate in both

space and time, with strong adaptive mesh refinement

capabili-tiesensuringthecomputationalcostremainsrelativelylowevenin

challengingmulti-scalecontextssuchasthoseinthepresent

prob-lem.Inthefollowingparagraphweelaborateonsomeofthe

spe-cificmeasures usedtoensureagoodnumericalperformance,and

alsounderlinetheoverall featuresofourextensivecomputational

effort.

The large density ratio (recall that for water-air flows

r=826.281) between the fluids may cause convergence

is-sues for multi-grid Poisson solvers as the one used in Gerris

(Tryggvasonetal.,2011),causingslowconvergenceorleadingtoa

breakdownofthenumericalsolutionaltogether.Asmoothing

Fig. 2. (a)Magnitudeofvelocityvectoronanx−zplaneimmediatelyabovethesurface,aty=0.01.Asinglestagnationpointisvisibleinthecentreofthecomputational box,withstreamlinesaidingthevisualisationoftheflowasitincreasesinvelocitytowardsthelateralboundaries.(b)Root-mean-squarenormofthevelocityvectorfor differentanglesofincidence30°≤θi≤90°ofthebackgroundairflow.

toalleviatethis.Spatialfilteringconsistsofaveragingoverthe

cor-ners ofa computational cell(four in 2D andeight in3D),which

are in turn obtainedby averaging the centredvalues of the

cor-ner neighbours. Applying the filter effectively smoothes the

rep-resentationoftheinterfaceover alargernumberofcellsandcan

be appliedanynumberoftimes,althoughprevious investigations

on dropimpactarguethat asingleiteration ofthefiltering

oper-atorissufficient(Thoraval,2013).Asaresultofthismanipulation,

theerrorsare maintainedata reasonable(andcontrollable)

mag-nitude, while the convergence propertiesof the solver are much

improved.

Thequalitiesofthepackage, inparticularintermsofadaptive

meshrefinement(AMR),becomeevidentinthestudyofthe

prob-lemofdropimpactathighvelocities.Thebackgroundairflow

re-quiresastronglevelofrefinementclosetothesurfaceofthesolid

body toaccountforthepresenceofthedevelopingboundary

lay-ersaroundthestagnationpointoftheflow.Atthesametime,

cap-turingtheevolutionofthefluid-fluidinterfacedemandsan

appro-priate resolution,enablingpossibletopologicaltransitions.

Splash-ingentailsthecreationandsubsequenttrackingofalargenumber

ofsecondarydroplets,whichmayormaynotcoalescewithother

bodiesoffluid.Inaddition,suitablechoicesforrefinementwith

re-specttosharpchangesinvorticitywerealsoimplemented.Wealso

note themorestringenttreatmentrequiredduringtouchdown,in

whichareducedtimestepandanextendedlocalrefinementregion

is necessarytoavoid numericalartefacts.The computational gain

whencomparingtothecaseofauniformmeshisremarkable.The

largedomainwouldrequireO

(

1010−11

₎

_{gridcellsatthefinest}

reso-lution,howeverwiththeuseoftheadaptivemeshrefinementthis

isdecreasedseveralordersofmagnitudedownto O

(

106

₎

_degrees

of freedom, whichbecomessignificantly more tractable. Manyof

the results presented would have reached considerable runtimes

(as well as challenging memory and data storage requirements)

withouttheusageofadaptivemeshrefinement,andpossibly

mak-ingmanyofthecalculationspresentedhereunrealisable.

We also employed the functionality to selectively eliminate

droplets and bubbles whose dimensions are below a threshold

numberof gridcells(chosento be 16),thus fixing theminimum

lengthscale that computations can account for. Note that this is

already well within the sub-micron scale. This feature becomes

usefulwhensecondarydropletbreak-off isviolent andcausesthe

fragmentationofthefluidintodropletsofaverysmallsizewhich

suffer fromgeometrical reconstruction errorsas aresult ofthem

spanningasmallnumberofgridcellsineachdimension.In

prac-ticethetechniqueworksbyreplacingtheconnectedvolumes

(un-deraspecifiedsize)containingthedropfluidphase(waterinour

case) withthebackground fluid (air). Furthermore,inour

imple-mentationcompletedropletremovaltakesplaceifthedropletsare

found within one spatial unit of lateral boundaries, in order to

limitnumericalartefactswhenencounteringtheoutflowregion,or

sufficientlyhighabove thesurfaceofthe solidbody (y_>10.0), to

avoidhighspeedsecondarydropletsreachingtheinflowboundary

andcausingnumericalinstabilities.Inpractice,thementioned

situ-ationcanbeavoidedbyprescribingalargercomputationaldomain

thatdemandsincreasedcomputationalcosts.Theselectiveremoval

ofdropletsensuresthatageometryofmanageablesizecanstillbe

usedreliably.Theflow inthevicinityofthe impactregionis

un-affectedby this treatment restrictedto thenear-lateral boundary

regions,hencenoflowinformationisartificiallylost.

Many of the problems of interest require the treatment of a

triple contactpoint betweenthe solid surface andthe liquid-gas

interface. We note that the default mesh-dependent static

con-tact angle model witha selected value of 90° is used here. The

limitations of this basic method, as well as proposed

improve-ments have been recently discussed by Afkhami et al. (2009),

who introduced a versatiledynamiccontact anglemodel

(imple-mentedinanextensionofGerris).Inarelatedcontext,

Pasandideh-Fardetal.(1996)note thatthe inertiallydominatedstagesofthe

flow are unaffected by changes in the contact angle, which had

beenalteredwiththe useof surfactantsintheir investigation. In

general, the suitability of the static contact angle model in the

inertia-dominated spreadingregime hasbeen studied extensively

(Yokoietal.,2009;Guoetal.,2016)andthepresentchoiceisnot

restrictive.Wehaveexperimentednumericallywithbothgridsizes

anddifferent imposed static contact anglevalues in two

dimen-sions,confirmingthatintheearly stagesoftheimpactwe arein

aregimewhichisinsensitivetothechoiceofcontactangleatthe

wall.

The runsinthe presentstudyhave beenperformed at

multi-pleresolutionlevels,varyingfrom210 _to212_{gridcellsperspatial}

dimension ineach computational box. As the interfacialshape is

setto be resolved atthislevel,thiswould translateto up to

ap-proximately 200 cellsper diameter forthe initial spherical drop.

Before impingementwe do not requiresuch levels of refinement

awayfrominterfaces.Ontheother hand,rightbefore,duringand

afterimpingement,theentireliquidvolumedemandsastrong

re-finementlevel.Despitethesestringentrequirements,withthe

cho-sensettingsanddropsizes,each finestresolutioncellspansfrom

0.097μm forthesmallest impingingdrops studied to1.15μm in

the caseof thelargest drops of initial diameterof justover 230

μm.Theselevelshavebeen selectedtoprovide asmuch detailat

themicronandsub-micronlevelsaspossible.Manyfeatures,such

asfor example the minimal film thickness arising as a result of

whichare useful guidelines forwhat scales need to be captured

andareusedforcomparisonsandvalidation.Fromamoregeneral

standpoint,forthedecisionon asuitable levelofrefinement and

meshadaptivity settingwehavereliedon threemaincriteria:(a)

massconservation;(b) changes indefinedmetrics such as

veloc-ityandvorticitynorms, secondary dropsize distributionetc. and

(c)comparisonstoavailableanalyticalpredictionsand

experimen-taldataintheliterature.Onceallthreecriteriahavebeenmet,the

configurationin questionwas propagated towards full parameter

studies overthe variables of interest. We emphasise that forthe

top two levels of refinement, volume conservationis accurate to

within1% acrosstheentiresetoftestsinthepresentwork,with

onlythemostchallengingoftestcases(thelargestinitialdrop

di-ameterprescribed)causingerrorsoftheorderof3–4%a

combina-tionofthedifficultconditions(highRe/We)andtheselectivedrop

removalmechanismintroducedabove,withsmallerscalefeatures

beingmorefrequentinthisscenario.Atypicalcomputationunder

theseconditions requires in excess of 2×103 _{CPU hours for the}

lowestresolutiontestedandapproximately104 _{CPU hoursforthe}

morerefinedcases.

Inthenextsection wedescribe,inturn,ourcomputational

re-sultsforpre-impact deformationandpost-impact dynamics,

con-centratingonbothfundamentalphenomenaandaspectsrelatedto

thelargerscalesystemitself.

4. Results

Once the drop is initialised inthe steady background flow, it

travelstowardsthesurfaceguidedbyaninitiallyuniform(but

θ

_i

-dependent)velocityfield, withstreamlinesdeforming asthesolid

surfaceis approached. Analysing the deviation from the initially

sphericalshapeasafunctionoftimeisoneoftheprimarygoalsof

thepresentwork, since,asnotedearlier,thiseffectisoften

over-lookedinstandardwaterretentioncalculationmodels.

In order to provide a suitable validation framework for the

presentresults,we havetailoredtheparameters to coincidewith

asubsetofthe dataofthe onlyexperimentalinvestigation ofthe

pre-impact deformation and break-up phenomena we are aware

of-seeVargasetal.(2012) andSorandGarcía-Magariño(2015).

Therein, an experimental setup consisting of a monosize droplet

dispenser, a rotatingarm witha model wingfixed at its end, as

wellasassociatedmotorandcameraequipmentare usedto

cap-turethe drop dynamics asthe solid body approachesthe liquid

droplets at velocities of up to 100 m/s. As a result of the very

violent regime, the size variation for the drops is restricted to

D=300

μ

mandabove.Veryfewpixelsperdropdiameterare

vis-iblebelow thisthresholdandthe resultingimagescan nolonger

becomprehensivelyanalysed.Asaconsequence,intheresultsthat

followwe haveselectedthree valueswithinthe respectiverange,

aswellasone smallerdrop,typicalofthesizesfoundinthehigh

liquidwatercontentregionsaircrafttravelthrough.Thedropsizes,

aswellasallotherassociateddimensionlessparametersare

sum-marisedinTable1,wherewe underlinethat wehaveuseda

ref-erencevelocity ofU∞=90 m/s(the same as in the main series

ofexperiments Sor andGarcía-Magariño, 2015) andthe physical

propertiesofwaterandairatrelevantnearfreezingtemperatures.

Following impactitself,dependingontherelevantparameters,

thedropisanticipatedtoeitherspreadduetoitsmomentumand

subsequentlyrecedeundersurfacetensioneffects,or,inthecases

ofthelargerdrops,tosplashandbreakupintosecondarydroplets

which move away from the surface but may later re-impinge.

Dropletsfound intheatmospheretypicallyliewithin theinterval

of20–250μmindiameterandasaconsequencewatercatch

stud-iesreportedintheliterature(Papadakisetal.,2003;2004;Wright, 2005;2006;Honseketal.,2008;Bilodeauetal.,2015)arefoundin thisregime.Weconsiderfourtestcases(D=20,52,111,236

μ

m)

forcomplete analysisofpre- andpost-impactdynamics,inorder

tofacilitatecomparisonswithresultsinthefieldandprovide

fur-therinsightunderflight conditionsofpracticalinterest.The

com-plete list of parameters is provided in Table 2, where the same

water-airconfigurationis used,howeverthistime withreference

velocityU_∞=78.44m/s, inagreementwithdatasetsdiscussedin

previouslymentionedstudies.

Forthesmallest20μmdropsweconsideranextensive

param-eterstudyintermsofimpingementangles10°≤

θ

i≤90°in

incre-mentsof10°.Thisenablesadetailedanalysisoftheeffectsrelated

to the competition betweeninertial and capillaryregimes, while

notingtheinfluenceofthebackgroundflowonthedropdynamics.

Forthemorechallenginglargerdropletswe focusontwo specific

cases,namely

θ

i=60◦ and

θ

i=90◦,guidingustowardsresultsin

bothsymmetricandasymmetricimpact,describedinfulldetailin

Section4.2.

4.1. Pre-impactdynamics

Inthepresentsubsectionwedescribequalitativeand

quantita-tive features relatedto the motionofdroplets prior to them

im-pacting the solid surface. Intuitively we expect the most

defor-mation and possible break-up to happen close to the solid

sur-face as the airflow slows down andthe droplet encounters

de-veloping boundary layers. We note however that, particularlyfor

largedrops,arichdynamicscharacterisedbyso-calledbag

break-upandrupturecanbeobservedeveninthecaseofsimpleuniform

flow and in the absence of any streamlinedeflection (Jalaal and

Mehravaran,2012). Thesestrongly time-dependentmorphological

changesunderlinetheimportanceofoneoftheparametersinthe

presentedmodel,namelytheinitialpositionofthedroprelativeto

thesolid surface. Ifprescribedtoofar awayfromthesurface, the

initial spherical dropmay become completelyfragmented by the

time itreachesthesurface, whileseeding it tooclosetothe

sur-facemaynotallowsufficienttimeforitsnaturaldynamicstooccur

before impingement. As such, the comparison to the

experimen-tal results from INTA/NASA (Vargas et al., 2012; Sor and

García-Magariño, 2015) serves asan important validation step. The

au-thorsfocusedondescribingandmodellingthechangeinshape,as

well asthe consequencesthereof interms ofpredictingthedrag

coefficient of the evolving shape. They found that for

moderate-sized droplets (with diameters in the hundreds of microns) the

approximation of the shape as an oblate spheroid proves to be

reasonablyaccurate,quantifyingthisdeformation asa(t)/R,where

a(t) denotes the evolving major semi-axis of the spheroid,

nor-malised bythe initialdropradius. Thisvalue wasreportedto

in-creasesmoothly from1.0asthedropissufficientlyfarawayfrom

the surfaceto values inthe range of1.3 forD₌362

μ

m, to 2.0

forD≈1 mm, increasing monotonically asa function of the size

oftheinitial drop.As theyapproach thesurface,the largerdrops

suffer considerable deformations inwhich the symmetric

frame-workpostulatedbeforeisnolongerapplicable.Finally,whenclose

towithin10mmofthesolidsurface,thedropsviolentlybreakup

into a cloud of secondary droplets which can only be described

qualitativelyintheexperiments.

Example evolutions of the droplet shapes are shown in

pan-els (a) and (b) of Fig. 3, in which we analyse the deformation

of a relatively smalldrop (D₌362

μ

m), as well asa large drop

(D=1048

μ

m) alongsidetheir experimental counterparts. In the

formercase, we findthat theproposed mild deformationinto an

oblate spheroidal shape is recovered and good qualitative

agree-ment with the experiments is found. The same applies for the

latter larger drop case, in which the flattening of the shape is

muchmorepronouncedandasymmetricfeaturesariseinthelatter

stages.Notehowthecentreofgravityoftheshapeshiftstowards

Fig. 3. Pre-impactdropdeformationvisualisationforsphericaldropsofdiameter(a)D=362μmand(b)D=1048μm.Insideeachpaneltheleftimagesareexperimental resultsbySorandGarcía-Magariño(2015),whiletherightimagesarethecorrespondingDNSresults.TheimagesarereproducedwithpermissionbyInstitutoNacionalde TécnicaAeroespacial.(c)QuantificationofthedropdeformationintermsofthedropsemiaxisanormalizedbytheinitialradiusR=D/2,withthecorrespondingparameters describedinTable1.Thetimestepatwhichthedropfirsttouchesthesolidsurfaceisalsohighlightedwithanopencircle.

Table 1

Relevantdimensionlessparametersinthecaseofpre-impactdeformationstudiesinhighspeedconditions, matchingindropdiametertoasubsetofthestudiesperformedbySorandGarcía-Magariño(2015).

D[m] Re=ρlU∞ D/μl We=ρlU∞ 2D/σ Oh=

√

We/Re Ca=μlU∞ /σ St=μg/(ρlDU∞ )

128×10−6 _{8653.717 10936.183 0.012 1.263 1.803×10}−6

362×10−6 _{24473.794 30928.893 0.007 1.263 6.376×10}−7

634×10−6 _{42862.943 54168.282 0.005 1.263 3.640×10}−7

1048×10−6 _{70852.309 89539.999 0.004 1.263 2.202×10}−7

secondary structures around the edges which ultimately rupture

fromthe mainshape andbreakoff intosmaller dropletsprior to

impact.Itshould benotedthatthereisadifference intimescales

when comparing the experimental and computational results; in

theexperimentaldatathedeformationtakesplaceoveradistance

of severalhundred dropdiameters, whereas in all computational

resultsthisevolutiontakesplacewithintheprescribeddistanceof

roughly20initialdropdiameters.Theflowfieldanditsextensional

natureiseffectivelyscaleddown tothesizeofthecomputational

box.

From a quantitative perspective, for comparison purposes we

usethesamesemiaxisdeformationmetrica(t)/RinFig.3(c)to

un-coveranexcellentagreementwiththeexperimentaldata.Wemark

thetimeofimpactwithanopencircleandnotethattheobtained

valuesarewithin10%oftheirexperimentalcounterparts,whilethe

evolution ofthis measurementin time also showsthesame

fea-tures. Notably, for thelarger dropwe plotthe full extent ofthe

liquidvolume(accountingforthesheddingofsecondarydroplets).

Ifthesearetobe excluded,atadistanceofhalfadiameterabove

thesolidsurface,thedeformationisfoundtobe1.36,1.72and1.85

forthe362μm,634μmand1048μmdrops,respectively,with

ap-proximately1.3,1.7and1.94beingtheequivalentvaluesinthe

ex-periment.Theinclusionofsecondarydropsbecomesvisiblearound

t≈15in both casesandcauses an increase inthis metricto just

below2.0and3.0forthetwolargestdroplets,indicatingthe

com-plexityoftheflowintherespectiveregimesasthedropsapproach

thesurface.

The computational framework developedhere can be used to

accessinformation onthe flow field anddropshape atdistances

very closeto thesurfacethat arebeyondtheframe-restricted

ca-pabilitiesof currentpowerfulvideo technology.Consequently, we

considerthecaseofasmallerdropofinitialdiameterD=128

μ

m

and find very small deviations from the imposed shape during

its entireevolution. Asmallinitialflatteningoftheshape intoan

oblate spheroidsufferscorrectionsprior toimpact andultimately

impingesalmostundeformed.

For completeness, all four cases are illustrated in Fig. 4 at

thelast computedtimestep before touchdown, withthe last D=

1048

μ

m case being placed side-by-side with its experimental

counterpart. For the smallest drop, deformation is hardly visible

(as confirmed by Fig. 3(a)), with an approximatelyspherical

liq-uidvolume impingingontothesurface.Astrongflatteningofthis

shape with the beginning of breakup features becoming visible

aroundtheedgestakesplaceforslightlylargerdropsandthis

ulti-matelyleadstoprogressivelysmallerliquidfragments/dropsbeing

shedfromthesides.Inthelargestdropvolumecase,thecloudof

droplets behindthe main liquid volume becomes visibleand

re-semblestheexperimentalresult.

Inwhatfollowswefocusontheimpingementprocessitselfand

in particular on the spreading or splashing characteristics of the

flow,aswell astheassociated secondarydropformationand

dy-namics.

4.2.Post-impactdynamics

Oncethedropapproachestheregionveryclosetothewall,the

gradually thinnerair filmbelow isforcedto move away laterally

and the pressure underneath the droplet continues to grow

un-tilimpacttakesplace.Wenote thepresence ofeitherasingleor

multipleair bubbles entrained underthe surface. In theclassical

context witha quiescent airflow and small to moderate impact

velocities,thesizeandevolutionoftheairbubbleiswellstudied,

anditseffectonthesplashingprocessitselfhasbeenshowntobe

negligible(RibouxandGordillo,2014).Inthepresentcasehowever,

therearetwofundamentaldifferencesfromthetraditionalimpact

problemduetotheveryhighimpactvelocity,aswellasthestrong

pre-impactdropdeformation,particularlyintheoblique

impinge-mentcases.InthefirstinstanceandonthebasisofFig.5,wewill

provideaqualitativeassessmentoftheresults.Thestudied

param-eterspaceconsistsofthetwocasesoutlinedinTable2ofnormal

impact andoblique impact at60°,andfour differentdrop

Fig. 4. Initiallysphericaldropletsofdiameter(a)D=128μm,(b)D=362μm,(c)D=634μm,and(d)D=1048μm,atthemomentofimpactontoaflatsolidsurface, havingbeendeformedbythebackgroundstagnationpointflow.Thesmallestdropretainsitsshape,whiletheedgesofthelargestdropbreakupintoalargenumberof secondarydropletsevenbeforeimpact.Thiscomparesfavourablytoe)previousexperimentalinvestigationsofdropdeformationpriortoimpactingamovingsolidbody (Dexp=1048μmaswell)byVargasetal.(2012).ThelastimageisreproducedwithpermissionbyInstitutoNacionaldeTécnicaAeroespacial.

Fig. 5. SplashingdynamicsfordropsofsizesD=20, 52, 111and 236μm(eachrowrepresentsadifferentdropsize,withthecompletelistofparametersdefinedin Table2)atanangleofincidenceofθi=60◦.Theleftcolumnillustratesthedropshapesastheircenterofmassisaty=Dabovethesurface,theimagesinthesecond columnareplottedatthedimensionlesstimestepatwhichthedropillustratedonthetoprow(smallestdrop,withinitialdiameterD=20μm,and relativelyregular spreadingbehaviour)reachesitsmaximumspreadtsmax,whilethethirdcolumnshowsthedropshapestentimeunitslater,onceeitherretractionormorepronouneced splashinghasoccurred.Therightmostcolumnisusedtovisualisethesplashingfortheθi=90◦impactcaseattsmax.

Table 2

Relevantdimensionlessparametersinthe caseoflong-timedropimpactdirectnumericalsimulationsin highspeedconditions,matchinginmedianvolumetricdiametertoasubset ofthestudies performedby Papadakisetal.(2003).ThesplashingparameterK=We√Revariesbetween6.283×104_and2.547×107_.

D[m] Re=ρlU∞ D/μl We=ρlU∞ 2D/σ Oh=

√

We/Re Ca=μlU∞ /σ St=μg/(ρlDU∞ )

20×10−6 _{1352.143 1708.779 0.031 1.263 1.154×10}−5

52×10−6 _{3515.573 4442.824 0.019 1.263 4.438×10}−6

128×10−6 _{8653.717 10936.183 0.012 1.263 2.079×10}−6

viewsoftheliquiddroponthelefthandside ofFig.5,illustrates

thedropshapeatthreekeytimesinitsevolution,namely:

(i) whenthecenterofmassofthedropliesoneinitialdiameter

abovethesurface(leftcolumn);

(ii) when the drop reaches its maximum spread on the

sur-faceandbeforeretractionundercapillaryforcestakesplace

(middlecolumn);

(iii)ten time units later, which serves as an indicationof how

the longertimescaleofthe impactdevelops intoeither

re-tractionforthesmallerdropsorviolentruptureand

splash-ingforthelargerdrops(rightcolumn).

Forthe 90°impact casewe concentrateon the second ofthe

above time instances,namelywhen thesmallest dropreachesits

maximumdiameter– resultsareshownintherighthandside

col-umnof Fig. 5.In each image a referencelengthscale of 20μmis

addedasavisualaidtotheextentofthedropatomisation(orlack

thereof).

The smallest dropsize (initial diameterD₌20

μ

m)

impinge-ment is characterisedby inconsequential pre-impact deformation

with theapproximately sphericalshape retained up to very near

the time of impact, followed by a strong spreading motion in a

highlyinertialregime,finallyfollowedbyretractionduetosurface

tension.Intriguingcorner-typefeaturesemergeparticularlyforthe

obliqueimpactcasesduetothedirectionalityoftheimpact,which

will bediscussed indetailinsubsequentparagraphs. Referringto

theobliqueimpactscenario,theasymmetrybecomesmorevisible

formedium sized drops(at theorderof 100μmininitial

diame-ter)priortoimpactandparticularlyafterimpactasfluid volumes

havesufficient momentumtoovercome surfacetensionandpush

outsidethetypicalnearlycircularcontour,insteadspreading

later-allyoutwardtowardsthefrontofthedrop.Thedynamicsis

how-everstill dominated by one largefluid volume fromwhichsmall

secondarydropsare ejectedasthedropincreasesinsize.We

un-derline thatthe imposed angleofincidencehasa clearinfluence

on theangleand extentoflateral spreadofthe liquidmass.The

largestdrops(D=236

μ

m)experienceviolentsplashing,with

vis-ibleliquidthreadsformingintheforwardandlaterallyoutward

di-rections asthefluid massdisintegratesintohundredsofdroplets.

Similar featuresare observedinthe normalimpactcaseinterms

of fragmentation, withtraditional spreading motion transitioning

to azimuthal instabilities,followedby a ruptureoftheliquid rim

into smalldrops,butwitha mainfluid massstill intactnearthe

impact site.Ultimately aviolent fragmentationbreaksup the

liq-uidvolumeintothinfilamentsnearthesurface,andnumerous

sec-ondarydropsareadvectedawayfromtheimpactregionunderthe

influenceofthebackgroundflow.

Conductingasystematicanalysisofthedrop’smorphology

dur-ing the early and intermediate stages of the impact is most

ac-cessible for the smallest drops (below several tens of microns

in initial diameter, top row of Fig. 5, when no splashing

oc-curs), whereearly andvery recentanalytical resultsare available

for comparison when

θ

i=90◦. Following this baseline, the

gen-eralisation to the predominantly three-dimensional effectsof the

asymmetric impact are bestconstructed. Evenin thenormal

im-pact case however, the presence of the non-quiescent air flow

at high speeds is anticipated to produce some modifications in

the standard metrics surrounding the characterisation of the

im-pingement process, which will be emphasised in the following

paragraphs.

Inorderto aidfuturecomparisons,inFig.6we define several

quantitiesofinterest,namelythetime-dependentdropdiameterin

thex−directionDx(t)(the directionofimpactforthenon-normal

incidence cases),the dropdiameterin thetransverse z−direction

Dz(t),aswellastheheightofthedropnearitscenterofmasshf.

The first two metrics are bestobserved fromthe top view (x−z

plane)presented inthe top part ofpanel a, while a cut through

the x−y plane provides information on the minimum thickness

ofthe film. The entrapment ofa smallair bubbledue to impact

cushioning,resultsinasmallvariationinthedrop’scurvaturejust

above thisfeature,which is whyforthe relevantlocal minimum

we select a point where this localadjustment is negligible.

Sev-eralnotablestudies(seeIntroduction)haveaddressedthetopicof

themaximumspreadDmofthedropinnormalimpactconditions,

with the recent investigation of Wildeman et al. (2016) chosen

asreferencehere. Pluggingourparametersintotheir mainresult,

wefindDm≈4.112,whichcomparesverywell withthecomputed

valuefor

θ

i=90◦ inFig.6(c). Symmetryinthiscaseispreserved

andwefindDmaxx =Dmaxz ≈3.9,whichalongsidethegood

agree-mentalsoindicatesthatthesurroundingflowhasalimited

influ-enceonthemaximumspread.

As the angle of incidence is decreased down to

θ

i=30◦, the

flowspeedinconjunctionwiththeincreasinglypronounced

direc-tionalityoftheimpactenablestheliquidmasstoadvancetowards

thefrontside(in thex−direction)oftheimpacting drop,pushing

morestrongly towards the front edge and increasingly distorting

it inthis direction. Fig. 6(c)indicates thismonotonic increase in

Dmaxx and decreasein Dmaxz as

θ

i is reduced, withthe final

as-pectratiobeingmeasuredatalmostafactoroftwo.Wepointout

thatinthisregimethedropisalsosubjectedtoastrongerairflow

asitliesfurtherawayfromthedividingstreamlineandthe

back-groundflow velocity has an increasedmagnitude. For illustrative

purposes,inFig.6(b)we expandonhow themaximumdiameter

valuesareobtainedinthe asymmetriccases,withthetwo

diam-eters Dx(t) andDz(t) beingshown throughouttheir evolution for

an angle of incidence

θ

i=60◦. The dynamics in the x−direction

ischosenasreference,asthisisthe dominantmotionduetoour

choiceinimpactdirectionality.ThevalueofDmaxz isthendefined

asthe value of Dz

(

txmax

)

, where txmax it the timestep at which

Dx reachesitsmaximum,despiteitnotnecessarilybeingthe

high-estabsolutevalue inthez−direction. Thefigure showsnegligible

deformationuptothetimeofimpactt_≈20.0,followedbyasharp

increaseindiameterinbothdirectionsbutmorestronglyinx,with

therimfinallyretractingundertheeffectofsurfacetensionfrom

all directions. The referencevalues (see vertical dashed line)

de-rivedfromsimilarstudiesofeachincidenceanglearethenusedto

constructFig.6(c).

Anotherkeymorphologicalmetricweconsideristheminimum

filmheight,asextractednearthedropcenter,sufficientlyfaraway

fromtheentrappedbubble.Inthenormalimpactcaseandinthe

stronginertialregimedescribedhere,Eggersetal.(2010)estimate

this thin film height to reach a minimum h_f/R_≈Re−2/5_, which

wouldgivehf≈0.028inourcase. Thisistheheight atwhichthe

thinningfilm reaches theliquid boundarylayers within the drop

itself andceases its decrease. The resultobtained in our

investi-gationishf≈0.033andwe foundno evidenceofsignificant

vari-ation as a result of modifying the angle of incidence. The very

slightoverestimationisperhapscounterintuitivegiventhatthefast

air flow pushing from above would be expected to enhance the

thinningeffect.Wenote thatevenatthesesmalllengthscalesthe

meshissufficientlyfinewithseveralgridpointsspanningthethin

filmregion;changesintheresolutiondidnotresultinmeaningful

changesofthisvalue.

Oneofthemostsalientfeaturesofthedropimpactinthe

mod-elledhighspeedregimeistheemergenceofacorner-type feature

neartheadvancingfrontofthespreadingliquidmass;thisfeature

becomeshighlyprominent, particularly asthe angleof incidence

θ

i is 60°or lower.Above the respective angle, normal impact is

characterisedby approximately axisymmetric behaviour, while in

slightlyoblique impacts (

θ

i≈70◦−80◦) the footprint can be

de-scribed as elliptical, although a slight symmetry-breaking tilt to

Fig. 6. SpreadingdynamicsofmicrodropletswithaninitialdiameterofD=20μmatanglesofincidencerangingfrom30°to90°.(a)Topviewschematicofthespreading diameterinthex−direction(theimpingementdirection)andthez−direction,aswellastheminimalfilmthicknesshf(insideviewbelow),foranangleofincidenceof θi=60◦.(b)Evolutionintimeofthespreadingandlaterretractingliquiddropforthe60°impingementanglecase.(c)Summaryofthemaximumspreadinbothxandz foracollectionofanglesofincidence,indicatingthetransitionfromsymmetricspreadingtoastronglyasymmetricfinalshapeinthedirectionofimpact.

Fig. 7. (a)Characterisationofthegeometricfeaturearisingattheleading(front)sideofthedropduetotheobliqueimpact.Angleϕtismeasuredfromthemostadvanced pointofthedropinthedirectionofimpacttothemaximuminthespreadintheperpendiculardirectionofthesameplane,whileϕnrepresentsthemorelocalfeature arisingat0.25R0behindthefront,whereR0representstheinitialdropradius.Bothanglesaredefinedinpanel(b),whilethethreeinsetspresenttopviewsofthedrop

shapeatthemomentofmaximumspreadinthex−direction,thetimestepatwhichalltheanglesinthefigurearecalculated.

comprehensiveanalysisofthecorner-typepropertyhasbeen

per-formed for angles varying in the range 10°≤

θ

i≤90° and small

dropsize (initialdiameterD=20

μ

m), andthe resultsare given

inFig.7.Hereinwetracktheevolvingdropshapefromabove,and

concentrateonthemomentwhereitsspreadingdiameterreaches

its maximumvalue foreach of the particularcases.Two angular

metrics are then definedas illustrated on theright hand side of

Fig. 7:

ϕ

t (angle from the tip of the advancing front in the

im-pingementx₋directiontothetoppartofthedrop,themaximum

inthez−direction)is a moreglobalmeasure ofthedeformation,

whereas

ϕ

n is a local measure of the corner angle near the tip

oftheadvancing front, definedby a trianglewhosebase isfixed

to be a quarter ofthe initial radius R/4, asshown in the figure.

Weemphasise thatwhile thediscussed featureiscalleda corner

(or of corner-type) throughout this subsection, the shape would

bemore accuratelydescribedasan apparent corner,since locally

nearthetipoftheadvancingfrontsurfacetensionalwaysinduces

asmoothingoftheshape.

The progressivelymorestretched shapeofthedrop,aswell as

the evolution near the tip of the advancing front capturing the

corner-type feature itself are both embeddedin the above

quan-tities, which are presented at the bottom of Fig. 7, with

exam-plesofthe underlyingdropshapesdepictedintherowabovefor

θ

i=40◦, 60◦ and 80°. In the intermediate casesmall distortions

of the liquid rim are already visible, while at 40° a pronounced

outgrowth near the advancing front selected by the direction of

impact is clearlyidentifiable. Due to the preservedaxisymmetry,

at

θ

i=90◦ we compare the numerical results with simple

pre-dictions.We naturallyexpect

ϕ

t≈45°andbased onthe maximal

spreadingradiusdescribedinFig.6(c),thedefinitionoftheangle

ϕ

n,aswellasusingbasictrigonometry,weestimate

ϕ

n≈75°.We

recover

ϕ

t=44.93◦ and

ϕ

t=74.76◦ by analysing thedata,which

iswell alignedwiththeanticipatedaxisymmetric evolution.Both

angle measurements are expected to decrease in oblique impact

scenarios,withtheelongationoftheliquidshapegradually

reduc-ing their values as

θ

i decreases. This is indeed the case, with a

smooth monotonic variationin

ϕ

t finalisingatapproximately22°

forthe

θ

i=30◦impingementcase.Thelocalangle

ϕ

nnaturally

be-ginsatamuchhighervalue,butagain, astheimpingementangle