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

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

3. 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

4. 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

5. 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

6. 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

7. 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), )

8. Mathematical Model

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

10. The Multi-Droplet Impact Problem
No-slip conditions

11. 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), )

12. Godunov Projection Method: Algorithm 1

13. Godunov Projection Method: Algorithm 2

14. 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

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

16. We = 13.8
Two Isolated droplets
Single drop
Isolated droplets

17. We = 13.8
Larger central drop
Reference Case
Larger central droplet

18. We = 13.8
Smaller central drop
Reference case
Smaller central droplet

19. We = 1388
Two isolated drops
Single droplet
Isolated Droplets

20. We = 1388
Larger central drop
Reference case
Larger central droplet

21. We = 1388
Reference case
Smaller central droplet

22. 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

23. 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

24. 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