Fluid Mechanics Research Laboratory Vibration Induced Droplet Ejection Ashley James Department of Aerospace Engineering and Mechanics University of Minnesota

Fluid Mechanics Research Laboratory

Vibration Induced Droplet Ejection

Ashley JamesDepartment of Aerospace Engineering and Mechanics

University of Minnesota

Marc K. SmithGeorge W. Woodruff School of Mechanical Engineering

Georgia Institute of Technology

Supported by NASA Microgravity Research Division

and Hoechst Celanese Corp.


Outline

• Problem definition

• Project overview

• Transducer-drop interaction

• Numerical simulations

• Conclusions and future work


• Vertical vibration induces the formation of capillary waves on the free surface.

• When the forcing amplitude is large enough secondary droplets are ejected from the wave crests.

PrimaryDrop

VibratingSurface

SecondaryDroplets

Ejection Schematic


Literature

• Faraday (1831) - wave formation due to vibration

• Benjamin & Ursell (1954) - stability analysis

• Sorokin (1957) - vibration induced droplet ejection

• Woods & Lin (1995) - stability on an incline, ejection

• Lundgren & Mansour (1988) - vibration of an unattached drop

• Wilkes & Basaran (1997,1999) - vibration of an attached drop

• Goodridge et al. (1996, 1997) - vibration induced droplet ejection


Applications

• Fuel atomization and injection for engine combustors

• Thermal management and control

• Electronic cooling

• Mixing processes

• Material processing

• Encapsulation

• Emulsification


Heat Transfer Cellfor high power electronic cooling (100 W/cm2)

Printed Circuit BoardIntegrated Circuit

CondensationSurface

Fins

Resonance Atomizer


Low Frequency Forcing

• Axisymmetric motion

• Single drop ejected from center

• 0 to 100 Hz

• Driver is a rigid piston

• Experiments performed to determine ejection behavior

• Focus of simulations

Photographs courtesy of Kai Range


High Frequency Forcing

• Chaotic motion

• Multiple droplet ejection across drop surface

• ~ 1 kHz

• Driver is a flexible diaphragm

• Coupling between driver and ejection dynamics

• Experimental investigation of spray characteristics

unforced ejection atomization


Close-up of High Frequency Ejection

• A crater forms on the drop surface.

• As the crater collapses an upward jet is created.

• One or more secondary droplets are ejected from the end of the jet.

crater

Photographs courtesy of Bojan Vukasinovic


Transducer-Drop Interaction Model

x

DamperNonlinear

spring

Piezo-electricdriver

Liquid dropletDiaphragm

dr

cc

cd

mm

xxxx

xxm

0

),,( rdt mmmfm

tfaVxxkxf

xcxm 2cos


Amplitude ResponseUnloaded Transducer

0.16 V1.85 V4.06 V

0

50

100

150

200

250

800 850 900 950 1000 1050 1100 1150

Acc

eler

atio

n A

mpl

itud

e (g

)

Frequency (Hz)


Effect of Drop Size on Response

0 L100 L200 L

Driving Voltage:0.74 V

0

10

20

30

40

50

700 750 800 850 900 950 1000 1050

Acc

eler

atio

n A

mpl

itud

e (g

)

Frequency (Hz)


Response of System to f = 0.99 Forcing

5.91 V6.20 V6.50 V

a

f0

50

100

150

200

0 1 2 3 4 5Liq

uid

Vol

ume

(L

)

Time (s)

100

150

200

250

300

0 1 2 3 4 5

Acc

eler

atio

nA

mpl

itude

(g)


Response of System to f = 1.04 Forcing

5.91 V6.20 V6.50 V6.79 V

f

a

100

150

200

250

300

0 1 2 3 4 5

Acc

eler

atio

nA

mpl

itude

(g)

0

50

100

150

200

0 1 2 3 4 5Liq

uid

Vol

ume

(L

)

Time (s)


100

150

200

250

300

0 1 2 3 4 5

Acc

eler

atio

nA

mpl

itude

(g)

Time (s)

Comparison of Model to Experiment

5.91 V6.20 V6.50 V6.79 V

f

a

Model

Experiment

100

150

200

250

300

0 1 2 3 4 5

Acc

eler

atio

nA

mpl

itude

(g)


Response Behavior

0 L100 L200 L

0

10

20

30

40

50

700 750 800 850 900 950 1000 1050

Acc

eler

atio

n A

mpl

itud

e (g

)

Frequency (Hz)

f < fr f > fr


Computational Method

• Transient, axisymmetric, incompressible governing equations.

• Forcing is an oscillating body force in inertial reference frame.

• Finite volume discretization on a uniform, staggered grid.

• Explicit projection method for Navier Stokes solver.

• Incomplete-Cholesky conjugate gradient method for solution of pressure-Poisson equation.


Volume of Fluid Method

• The position of the interface is tracked via a volume fraction, F.

• The evolution of the volume fraction is governed by a convection equation.

• The interface is approximated by a straight line in each cell.

• To prevent false smearing of the interface the volume fraction flux is computed from the straight line approximation.

0

z

Fv

r

Fu

t

F


Continuum Surface Force

• The surface tension forces are incorporated as a source term in the momentum equation.

• Surface cells and interior cells are treated the same.

• The source term is nonzero only near the interface.

• The surface tension is distributed over a small region near the computed interface.

• The curvature is calculated directly from the volume fraction.

F


• Continuity:

• Radial momentum:

• Vertical momentum:

• Volume fraction:

Governing Equations

0

1

z

v

r

ru

r

r

F

z

u

r

v

zr

ur

rrRer

p

z

uv

r

uu

t

u

21

z

FtABo

z

v

zz

u

r

vr

rrRez

p

z

vv

r

vu

t

v

sin2

11

0

z

Fv

r

Fu

t

F


Verification

• Translation of a fluid region.

• Exact solution of Poisson equation.

• Poiseuille flow.

• Transient Couette flow in an annular region.

• Stability of a drop in equilibrium.


Parameters Range

0 - 500 Viscous effects

0 - 100 Forcing amplitude

0 - 5 Forcing frequency

0 - 5 Gravity effects

Ejection Simulations

2

3

2

gLBo

L

LaA

LRe


Initial and Boundary Conditions

Symmetryline

Outlet

No-slipwalls

80 cells

30 cells


Video Cases

Re = 475 Re = 10 Re = 10 Re = 10 Re = 10

A = 8.7 A = 18 A = 20 A = 25 A = 30

= 1.2 = 1 = 1 = 1 = 1

Bo = 1.3 Bo = 0 Bo = 0 Bo = 0 Bo = 0


Comparison of Simulation and Experiment Re = 475, A = 8.7, = 1.2, Bo = 1.3

Scale:

1 cm

Forcing stepped on

Forcing slowly ramped up


Ejection Simulation - Case 2 Re = 10, A = 18, = 1 , Bo = 0

t = 2.8 t = 3 t = 3.2 t = 3.4 t = 3.6 t = 3.8 t = 4


Ejection Simulation - Case 3 Re = 10, A = 20, = 1 , Bo = 0

t = 1.8 t = 2 t = 2.2 t = 2.4 t = 2.6 t = 2.8 t = 3


Ejection Simulation - Case 4Re = 10, A = 25, = 1 , Bo = 0

t = 0.6 t = 0.8 t = 1 t = 1.2 t = 1.4 t = 1.6 t = 1.8


Ejection Simulation - Case 5Re = 10, A = 30, = 1 , Bo = 0

t = 0.8 t = 1 t = 1.2 t = 1.4 t = 1.6 t = 1.8


Effect of Forcing Amplitude on EjectionBo = 0, Re = 10, = 1

0

0.2

0.4

0.6

0.8

1

10 20 30 40 50A

Eje

cted

Dro

p V

olum

e

-1

0

1

2

3

4

5

6

7

Eje

cted

Dro

p V

eloc

ity

Tim

e of

Eje

ctio

n (p

erio

ds)

Volume

Velocity

Time


Effect of Bond Number on EjectionRe = 10, A = 25, = 1

0

0.2

0.4

0.6

0.8

1

-8 -6 -4 -2 0 2 4Bo

Eje

cted

Dro

p V

olum

e

-2

0

2

4

6

8

10

12

Eje

cted

Dro

p V

eloc

ity

Tim

e of

Eje

ctio

n (p

erio

ds)

Volume

Velocity

Time


Effect of Reynolds Number on EjectionBo = 0, A = 25, = 1

0

0.2

0.4

0.6

0.8

1

0 20 40 60Re

Eje

cted

Dro

p V

olum

e

-1

-0.5

0

0.5

1

1.5

2

2.5

Eje

cted

Dro

p V

eloc

ity

Tim

e of

Eje

ctio

n (p

erio

ds)

Volume

Velocity

Time


Effect of Forcing Frequency on EjectionRe = 10, Bo = 0, A = 25

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

Eje

cted

Dro

p V

olum

e

-2-1012345678

Eje

cted

Dro

p V

eloc

ity

Tim

e of

Eje

ctio

n (p

erio

ds)

Volume

Velocity

Time


1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-08 1.E-06 1.E-04 1.E-02* /Re

a* =

A/R

e

3

4Ejection Threshold

Ejection

No ejection

SimulationsRange et al.Goodridge et al.low viscosityGoodridge et al.high viscosity


Conclusions

• Although the forcing frequency has a dramatic effect on the response, ejection may occur when a crater collapses to form a spike in both the low and high frequency regimes.

• The bursting behavior is explained by the coupling of the diaphragm vibration with the changing drop mass.

• The single degree-of-freedom model with linear droplet ejection is sufficient to describe the system dynamics.

• Low-frequency ejection is promoted by increasing A, decreasing Bo, increasing Re, or decreasing .

• The simulated drop behavior and the ejection threshold compare well with experiments.


Future Work

• Extend simulations to three dimensions.

• Improve computational methodology.

• Investigate the formation of satellite drops.

• Determine effect of contact line condition.

• Simulate the vibration of a liquid layer.

• Improve understanding of high-frequency atomization.

• Design systems involving high-frequency spray formation.

Documents

Fluid Mechanics Research Laboratory Vibration Induced Droplet Ejection Ashley James Department of Aerospace Engineering and Mechanics University of Minnesota