Modelling Spray Impingement the importance of mutual droplet-droplet interaction Frank Bierbrauer and Tim Phillips Cardiff University, UK

Sprays in Industry Direct fuel injection in Diesel engines Spray cooling of steel sheets Spray coating and painting Agricultural: insecticide sprays, irrigation Fire quenching

Spray Characteristics First Stage (spray injection) A liquid jet is injected into an ambient gaseous medium such as air Cavitation within the injector causes initial break-up The high speed flow is further broken up into liquid sheets, ligaments and droplets

Spray Characteristics Second Stage (dispersed liquid phase) Individual droplets are further broken up through aerodynamic forces producing a range of droplet sizes Multiple droplets of varying diameters and shapes travel through the ambient gas Third Stage (Impact) Single droplet impact behaviour Kinematic and spreading phase, crown splash Multiple droplet impact behaviour Generation of secondary droplets, liquid film accumulation on wall

Single Droplet Droplets may stick, bounce or break up into smaller ones Impact behaviour depends on: inertial, viscous and surfaces forces Droplet break-up consists of an initial compressible/kinematic phase (very early on) followed by approximately incompressible behaviour Spreading phase: thin film lamella spreads outwards from the impact point bounded by a rim for t* < 0.1 (t* = tUd/Dd) Surface forces restrict spread after t* > √We Splash crown formed, crown height and radius directly dependent on Weber number, secondary droplets expelled depending on splashing threshold

The Main Assumption Prevailing models assume and extrapolate the results of single droplet impact onto solid walls to the case of spray-wall interaction by the superposition of many individual droplets How accurate is this assumption ? When does it break down ? How important is mutual droplet-droplet interaction ? There is a need to model the impact of more than one droplet to answer these questions

Multiple Drops Individual droplet splashes generate secondary droplets, multiple droplets interact in their splash behaviour Multiple droplets interact through their spreading lamella as well as intervening gaseous medium Multiple droplets accumulate a liquid film which influences the splashing threshold, ejected mass and number of secondary droplets, creation of central jets and splash type In splashing the crown radius and height no longer depend directly on the impact Weber number (D. Kalantari, C. Tropea, Int. J. Multiphase Flow, 33 (2007), 525-544.)

Mathematical Model

Characteristic Impact Behaviour Characteristic parameters for the droplet (d) and the ambient gas (g) Dd = 0.001 m, rd = 1000 kg/m3, md = 0.001 kg/ms, sgd = 0.072 N/m, rg = 1 kg/m3, mg = 1×10-5 kg/ms, g = 9.81 m/s2

The Multi-Droplet Impact Problem No-slip conditions

Numerical Model Multiphase flow: One-Field model Solution Type: Eulerian-Lagrangian, mesh-particle method Incompressibility: Godunov approximate projection method Interface Tracking Algorithm: Marker-Particle Method (F. Bierbrauer, S.-P. Zhu, Comput. Fluids, 36 (2007), 1199-1212)

Godunov Projection Method: Algorithm 1

Godunov Projection Method: Algorithm 2

Marker-Particle Tracking Initial particle configuration (e.g. 4 particles per cell) Allocation of fluid colour C within a computational cell containing two fluid phases: 1 and 2. Two sets of marker particles are required, one for each fluid involved Use Lagrangian tracking of particles by solving dxp/dt = up where up is a particle velocity interpolated from nearby grid velocities Interpolate particle colour data back to grid Particles permanently maintain fluid identity throughout the simulation

Test Cases Ud = 1 m/s, We = 13.8, Re = 1000, Fr =102

We = 13.8 Two Isolated droplets Single drop Isolated droplets

We = 13.8 Larger central drop Reference Case Larger central droplet

We = 13.8 Smaller central drop Reference case Smaller central droplet

We = 1388 Two isolated drops Single droplet Isolated Droplets

We = 1388 Larger central drop Reference case Larger central droplet

We = 1388 Reference case Smaller central droplet

Surface Forces Dominant, We = 13.8 Conclusions Surface Forces Dominant, We = 13.8 Provided that two impacting droplets are far enough apart their individual impact behaviour appears independent When three neighbouring droplets of equal size impact a solid surface some of the fluid from the two neighbouring droplets is shunted into the formation of a greater crown height and larger secondary droplets of the central droplet If the central droplet is larger than the two neighbours most of the expelled droplets are of equal size If the central droplet is smaller than the two neighbours most of the expelled droplets form larger fluid masses

Inertial Forces Dominant, We = 1388 Conclusions Inertial Forces Dominant, We = 1388 At higher kinetic energies two individual droplets must be further apart, than the We = 13.8 case, in order for their impacts to appear independent When three neighbouring droplets of equal size impact a solid surface at high kinetic energy much of the expelled mass is distributed above the surface in a mist-like configuration If the central droplet is larger than the two neighbours most of the combined droplet mass is centrally distributed with a large crown height and radius If the central droplet is smaller than the two neighbours most of the combined droplet mass is spread along the wall with a small central crown radius

Future Work The current qualitative work is only a first stage in an investigation of multi-droplet impact behaviour Future work will involve detailed quantitative measures frequently used in spray measurements such as the temporal variation of deposited fluid mass, accumulated fluid layer thickness, crown height and radius as well as the distribution of secondary droplet sizes