Numerical simulations of droplet impact on a dry surface, and

droplet coalescence and separationKensuke Yokoi

School of Engineering, Cardiff UniversityUnited Kingdom

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A numerical result of droplet splashing on a dry surface

Liquid (distilled water) Density ratio of air: liquid = 1.25:1000Diameter = 2.28mmImpact speed 3.0m/s Re=6840, We=285,

Oh=0.0025 Max. dynamic advancing angle 160 Equilibrium angle 40 Min. dynamic receding angle 30No roughnessPerturbation (numerical errors, small random noise)Mesh: 200x200x100

Yokoi, Soft Matter 2011

AgendaIntroduction Background

Numerical method for free surface flows Surface force model

Validations and numerical results Single bubble rising Droplet collisions Droplet impacting on dry surface Droplet splashing on super hydrophobic substrate Preliminary results

Multiple droplet impacting, liquid jet, swirl atomization

IntroductionDroplet plays an important role in many industrial and scientific applications such as Energy

Combustion (fuel droplet, atomization) Spray cooling

Steel products, turbine blades Spray coating Ice accumulations on aircraft and power cable Inkjet

Previous works of droplet splashing on a dry surface and droplet collision

Numerical simulations of droplet splashing on dry surface 3D numerical simulations

Bussman et al, 1999, Yokoi, 2011 2D numerical simulations

Some since Harlow, Shannon 1967Josserand, Zaleski et al. 2005

Droplet collisions (3D) Nobari, Tryggvason 1996, Rieber, Frohn 1997,

Inamuro, Tajima, Ogino 2004, Tanguy, Berlemont 2005, etc.

Methodology Numerical method consists of CLSVOF (Coupled Level Set and Volume of Fluid ) Sussman, Pucket

Level set method Osher, Sethian Interface capturing method

THINC/WLIC method Xiao/Yokoi Interface tracking method (VOF type method)

CIP-CSL (Constraint Interpolated Procedure-Conservative Semi-Lagrangian) method Yabe et al.

Conservation equation solver VSIAM3 (Volume/Surface Integrated Average Multi-Moment Method) Xiao

Finite Volume based fluid solverA kind of fractional step method Kim, Moin

CSF (Continuum Surface Force ) model Brackbill et al.Surface tension force modelDensity scaling + curvature interpolation using level set

Contact angle implementation Sussman

GridRegular Cartesian fixed gridThe interface is represented by level set function and VOF function

CLSVOF

VOF deals with interface motion.Level set is generated from the interface indicated by VOF and used for surface force computation (not only curvature).

VOF method（Interface motion）THINC/WLIC (Tangent of Hyperbola for Interface Capturing / Weighted Line Interface Calculation) method Yokoi JCP 2007

Using interface reconstructionNo diffusion at the interfaceNo volume error

air for the 0liquid for the 1

),(

0)(

yx

uut

Interface reconstruction

Two reconstruction algorithms

WLIC (Weighted line interface calculation)

||||||

||||||

yx

yy

yx

xx

nnn

nnn

THINC method

position. andsmoothing direction, interface therepresent x~ and ,

~tanh121 2/1

ii

x xxxx

Governing equations

Continuity equationNavier-Stokes equation

with surface tension force and gravity

gravity:g force, tension Surface:F tensorstress viscous:pressure:

velocity:density:

)(

0

s

,

,u,

p

gFpuutu

u

s

Governing equations

gFpuutu

u

s

)(

0

gdVFndSpndSdSnuuudVt

ndSu

s

11)(

0

A fractional step method

Advection step -> CIP-CSL method

Non-advection stepA standard pressure Poisson based method

0)(

dSnuuudVt

pndSudVt

ndSu

gndSpndSudVt

1

0

11

tup

n

n

**

1

1

1

1

n

nptu

Density scaling CSF model with curvature interpolation using level set

Standard CSF model based on level set

else 0

if cos121

)(

ls

ls

lss

n

n

nF

α is the smoothing length of the interface.

Density scaling CSF modelDensity scaling Brackbill, et al. 92, Kothe, et al. 96, Bussmann et al. 99.

The Original CSF is not robust enough.Although balanced force is robust and better in terms of spurious current, may not be better for droplet splashing.

function Heaviside : 2)(

HH

nFscaling

lsscaling

s

Curvature interpolation

Standard curvature interpolation

Curvature interpolation using level set

jijiji ,,1,2/1 21

sf

lssf

t

nt

0

Bilinear interpolation

Droplet impacting onto dry surfacesImplementation of contact angle Extrapolate level set and VOF functions into

the solid wall.

Single bubble rising

64x64x128 Cartesian grid20 meshes for the initial diameter

Experiment by Hnat and Buckmaster

Drop-drop collision

We=23X=0.0564x64x64

Ashgriz, Poo, JFM, 221, 183 (1990)

Drop-drop collision

We=４０X=0.1128x128x128

Ashgriz, Poo, JFM, 221, 183 (1990)

Drop-drop collision

Liquid: Distilled waterWe=1000X=0.5256x256x256

Numerical result of droplet impact on homogeneous substrate

2D simulation !!!

Parameters Liquid

Distilled water U=1.0m/s D=2.28mm Contact angle model

Dyn. advancing = 115Equilibrium = 90Dyn. receding = 55δa= 0.03m/sδr= 0.3m/s

Grid 100x100

Comparison with experiment

Yokoi, Vadillo, Hinch, Hutchings, POF, 09

Comparison with experiment

Time evolution of contact diameterD

iam

eter

[mm

]

Time [ms]

A numerical result of droplet splashing

Liquid (distilled water) Density ratio of air: liquid = 1.25:1000Diameter = 2.28mmImpact speed 3.0m/s Re=6840, We=285,

Oh=0.0025 Max. dynamic advancing angle 160 Equilibrium angle 40 Min. dynamic receding angle 30No roughnessPerturbation (numerical errors, small random noise)Mesh: 200x200x100

Comparison with an experimentExperiment Tsai et al. 2009

Water droplet

Re=5568

We=230

Contact angle 163 °

192x192x48

Numerical results using different grid resolutions

s

128x128x32 256x256x64

Comparison of curvature interpolation techniques

jijiji

,,1

,2/1

21

sf

lssf

t

nt

0

Curvature interpolation ←|→using level set

Standard curvature interpolation

Level set based density scaling vs. Balanced force

sF

xHH

H

HF

jiji

x

s

,,,1,

)(

Balanced force, α=1.5 Francois et al. JCP 2006

Level set based density scaling α=1.5

←|→

Level set based density scaling vs. Balanced force

Balanced force, α=1.0Level set based density scaling, α=1.5

←|→

Multiple droplet splashing

200x200x100

Liquid jet breakup

32x32x256

Swirl atomization

192x192x192

Summary

Numerical framework using CLSVOF (with THINC/WLIC), CIP-CSL, VSIAM3, Level

set based density-scaled CSF with curvature interpolation using level set

The framework was validated through Single bubble rising Drop-drop collision Droplet impacting and splashing on solid surface

Level set based density-scaled CSF model with the curvature interpolation can capture droplet splashing.

Particles-liquid interface interaction

100x80air: rigid: liquid=1.25:500:1000