Modelling Spray Modelling Spray ImpingementImpingement

the importance of mutual the importance of mutual

droplet-droplet interactiondroplet-droplet interaction

Frank Bierbrauer and Tim PhillipsCardiff University, UK

Sprays in IndustrySprays in IndustryDirect fuel injection in Diesel enginesDirect fuel injection in Diesel engines

Spray cooling of steel sheetsSpray cooling of steel sheets

Spray coating and paintingSpray coating and painting

Agricultural: insecticide sprays, irrigationAgricultural: insecticide sprays, irrigation

Fire quenchingFire quenching

Spray CharacteristicsSpray CharacteristicsFirst Stage (spray injection)First Stage (spray injection)

– A liquid jet is injected into an ambient A liquid jet is injected into an ambient gaseous medium such as airgaseous medium such as air

– Cavitation within the injector causes Cavitation within the injector causes initial break-upinitial break-up

– The high speed flow is further broken up The high speed flow is further broken up into liquid sheets, ligaments and dropletsinto liquid sheets, ligaments and droplets

Spray CharacteristicsSpray Characteristics Second Stage (dispersed liquid phase)Second Stage (dispersed liquid phase)– Individual droplets are further broken up Individual droplets are further broken up

through aerodynamic forces producing a through aerodynamic forces producing a range of droplet sizesrange of droplet sizes

– Multiple droplets of varying diameters and Multiple droplets of varying diameters and shapes travel through the ambient gasshapes travel through the ambient gas

Third Stage (Impact)Third Stage (Impact)– Single droplet impact behaviourSingle droplet impact behaviour

• Kinematic and spreading phase, crown splashKinematic and spreading phase, crown splash

– Multiple droplet impact behaviourMultiple droplet impact behaviour• Generation of secondary droplets, liquid film Generation of secondary droplets, liquid film

accumulation on wallaccumulation on wall

Single DropletSingle Droplet• Droplets may stick, bounce or break up into Droplets may stick, bounce or break up into

smaller onessmaller ones• Impact behaviour depends on: inertial, viscous Impact behaviour depends on: inertial, viscous

and surfaces forcesand surfaces forces• Droplet break-up consists of an initial Droplet break-up consists of an initial

compressible/kinematic phase (very early on) compressible/kinematic phase (very early on) followed by approximately incompressible followed by approximately incompressible behaviourbehaviour

• Spreading phase: thin film lamella spreads Spreading phase: thin film lamella spreads outwards from the impact point bounded by a outwards from the impact point bounded by a rim for rim for tt* < 0.1 (t* = tU* < 0.1 (t* = tUdd/D/Ddd))

• Surface forces restrict spread after Surface forces restrict spread after tt* > √We* > √We • Splash crown formed, crown height and radius Splash crown formed, crown height and radius

directly dependent on Weber number, secondary directly dependent on Weber number, secondary droplets expelled depending on splashing droplets expelled depending on splashing thresholdthreshold

The Main AssumptionThe Main Assumption

• Prevailing models assume and extrapolate Prevailing models assume and extrapolate the results of single droplet impact onto the results of single droplet impact onto solid walls to the case of spray-wall solid walls to the case of spray-wall interaction by the superposition of many interaction by the superposition of many individual dropletsindividual droplets

• How accurate is this assumption ? When does it How accurate is this assumption ? When does it break down ? How important is mutual droplet-break down ? How important is mutual droplet-droplet interaction ?droplet interaction ?

• There is a need to model the impact of more than There is a need to model the impact of more than one droplet to answer these questionsone droplet to answer these questions

Multiple DropsMultiple Drops• Individual droplet splashes generate Individual droplet splashes generate

secondary droplets, multiple droplets interact secondary droplets, multiple droplets interact in their splash behaviourin their splash behaviour

• Multiple droplets interact through their Multiple droplets interact through their spreading lamella as well as intervening spreading lamella as well as intervening gaseous mediumgaseous medium

• Multiple droplets accumulate a liquid film Multiple droplets accumulate a liquid film which influences the splashing threshold, which influences the splashing threshold, ejected mass and number of secondary ejected mass and number of secondary droplets, creation of central jets and splash droplets, creation of central jets and splash typetype

• In splashing the crown radius and height no In splashing the crown radius and height no longer depend directly on the impact Weber longer depend directly on the impact Weber numbernumber

(D. (D. Kalantari, C. Tropea, Kalantari, C. Tropea, Int. J. Multiphase FlowInt. J. Multiphase Flow, , 3333 (2007), 525-544 (2007), 525-544.).)

Mathematical ModelMathematical Model

1

gd

gd

/μC1Cμ

ρC1Cρ

FrρρWe

ρκμ

Reρ

1p

ρ

1

t

0Ct

C

0

u

u

Duuu j

0C 0,μ 0,ρ ,ΩΩΩΩ

nnn0u

Characteristic Impact BehaviourCharacteristic Impact Behaviour• Characteristic parameters for the droplet (Characteristic parameters for the droplet (dd) and ) and

the ambient gas (the ambient gas (gg))

• DDdd = 0.001 m, = 0.001 m, dd = 1000 kg/m = 1000 kg/m33, , dd = 0.001 kg/ms, = 0.001 kg/ms,

gdgd = 0.072 N/m, = 0.072 N/m, gg = 1 kg/m = 1 kg/m33, , gg = 1 = 1××1010-5-5 kg/ms, kg/ms,

gg = 9.81 m/s = 9.81 m/s22

d

2d

d

ddd

gd

d2dd

gD

UFr,

μ

DUρRe,

σ

DUρWe

The Multi-Droplet Impact The Multi-Droplet Impact ProblemProblem

No-slip conditions

Numerical ModelNumerical Model

• Multiphase flowMultiphase flow: : One-Field modelOne-Field model

• Solution TypeSolution Type: : Eulerian-Lagrangian, Eulerian-Lagrangian, mesh-particle methodmesh-particle method

• IncompressibilityIncompressibility: : Godunov approximate Godunov approximate projection methodprojection method

• Interface Tracking Interface Tracking

AlgorithmAlgorithm: : Marker-Particle Marker-Particle MethodMethod

(F. Bierbrauer, S.-P. Zhu, Comput. Fluids, 36 (2007), 1199-1212)

Godunov Projection Method: Godunov Projection Method: Algorithm 1Algorithm 1

1/2n1/2n1/2n1/2n1/2n

n1/2nμ

1/2n1/2nμ

1/2n

1nn1/2n1nn1/2n

1

gd1n1n1n

gd1n1n1n

1/2nn

σpσΔt

Lσ2Re

ΔtILσ

2Re

ΔtI

/2μμμ/2ρρρ

/μC1CμρC1Cρ

C

Fuu

uu

u

uuu

velocity teintermedia for the solve 3.

and obtain to

,

thatso method particle-marker theusing update 2.

calculate ,For 1. 1/2nnnn p,C,μ,ρ

Godunov Projection Method: Godunov Projection Method: Algorithm 2Algorithm 2

1/ρσ

,

ψ,σψL

μLFr

ρ

ρρWe

ρκρ

φσL2Re

Δtφpp

φΔtσΔt

1φL

1/2nσ

1/2nμ

1n1/2n1/2nμ

1n1/2n1/2n

1n1/2n1n1n1/2nσ

and

where

gradient pressure theupdate 5.

by followed

result eproject th 4.

Twwwj

F

uuu

Marker-Particle TrackingMarker-Particle Tracking• Initial particle configuration (e.g. 4 particles per cell)Initial particle configuration (e.g. 4 particles per cell)

• Allocation of fluid colour Allocation of fluid colour CC within a computational cell within a computational cell containing two fluid phases: 1 and 2. Two sets of marker containing two fluid phases: 1 and 2. Two sets of marker particles are required, one for each fluid involvedparticles are required, one for each fluid involved

• Use Lagrangian tracking of particles by solving Use Lagrangian tracking of particles by solving dxdxpp/dt = u/dt = up p

where where uupp is a particle velocity interpolated from nearby grid is a particle velocity interpolated from nearby grid velocitiesvelocities

• Interpolate particle colour data back to gridInterpolate particle colour data back to grid• Particles permanently maintain fluid identity throughout the Particles permanently maintain fluid identity throughout the

simulationsimulation

Test CasesTest Cases

• UUdd = 1 = 1 m/sm/s, We = 13.8, Re = 1000, Fr =102, We = 13.8, Re = 1000, Fr =102

• UUdd = 10 = 10 m/sm/s, We = 1388, Re = 10000, , We = 1388, Re = 10000,

Fr = 10193Fr = 10193

We = 13.8We = 13.8

Single drop Two Isolated dropletsIsolated droplets

We = 13.8We = 13.8

Reference Case Larger central dropLarger central droplet

We = 13.8We = 13.8

Reference case Smaller central dropSmaller central droplet

We = 1388We = 1388

Two isolated dropsSingle droplet Isolated Droplets

We = 1388We = 1388

Larger central dropReference case Larger central droplet

We = 1388We = 1388

Reference case Smaller central droplet

ConclusionsConclusions

Surface Forces Dominant,Surface Forces Dominant, WeWe = 13.8 = 13.8– Provided that two impacting droplets are far enough Provided that two impacting droplets are far enough

apart their individual impact behaviour appears apart their individual impact behaviour appears independentindependent

– When three neighbouring droplets of equal size impact When three neighbouring droplets of equal size impact a solid surface some of the fluid from the two a solid surface some of the fluid from the two neighbouring droplets is shunted into the formation of a neighbouring droplets is shunted into the formation of a greater crown height and larger secondary droplets of greater crown height and larger secondary droplets of the central dropletthe central droplet

– If the central droplet is larger than the two neighbours If the central droplet is larger than the two neighbours most of the expelled droplets are of equal sizemost of the expelled droplets are of equal size

– If the central droplet is smaller than the two neighbours If the central droplet is smaller than the two neighbours most of the expelled droplets form larger fluid massesmost of the expelled droplets form larger fluid masses

ConclusionsConclusions

Inertial Forces Dominant,Inertial Forces Dominant, WeWe = 1388 = 1388– At higher kinetic energies two individual droplets must At higher kinetic energies two individual droplets must

be further apart, than the be further apart, than the WeWe = 13.8 case, in order for = 13.8 case, in order for their impacts to appear independenttheir impacts to appear independent

– When three neighbouring droplets of equal size impact When three neighbouring droplets of equal size impact a solid surface at high kinetic energy much of the a solid surface at high kinetic energy much of the expelled mass is distributed above the surface in a mist-expelled mass is distributed above the surface in a mist-like configurationlike configuration

– If the central droplet is larger than the two neighbours If the central droplet is larger than the two neighbours most of the combined droplet mass is centrally most of the combined droplet mass is centrally distributed with a large crown height and radiusdistributed with a large crown height and radius

– If the central droplet is smaller than the two neighbours If the central droplet is smaller than the two neighbours most of the combined droplet mass is spread along the most of the combined droplet mass is spread along the wall with a small central crown radiuswall with a small central crown radius

Future WorkFuture Work

• The current qualitative work is only a first stage The current qualitative work is only a first stage in an investigation of multi-droplet impact in an investigation of multi-droplet impact behaviourbehaviour

• Future work will involve detailed quantitative Future work will involve detailed quantitative measures frequently used in spray measurements measures frequently used in spray measurements such as the temporal variation of deposited fluid such as the temporal variation of deposited fluid mass, accumulated fluid layer thickness, crown mass, accumulated fluid layer thickness, crown height and radius as well as the distribution of height and radius as well as the distribution of secondary droplet sizes secondary droplet sizes