Fluid Mechanics Research Laboratory
Vibration Induced Droplet Ejection
Ashley James
Department of Aerospace Engineering and Mechanics
University of Minnesota
Marc K. Smith
George W. Woodruff School of Mechanical Engineering
Georgia Institute of Technology
Supported by NASA Microgravity Research Division
and Hoechst Celanese Corp.
Fluid Mechanics Research Laboratory
Outline
•
Problem definition
•
Project overview
•
Transducer

drop interaction
•
Numerical simulations
•
Conclusions and future work
Fluid Mechanics Research Laboratory
•
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.
Ejection Schematic
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
Applications
•
Fuel atomization and injection for engine combustors
•
Thermal management and control
•
Electronic cooling
•
Mixing processes
•
Material processing
•
Encapsulation
•
Emulsification
Fluid Mechanics Research Laboratory
Heat Transfer Cell
for high power electronic cooling (100 W/cm
2
)
Printed Circuit Board
Integrated Circuit
Condensation
Surface
Fins
Resonance Atomizer
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
Transducer

Drop Interaction Model
Fluid Mechanics Research Laboratory
Amplitude Response
Unloaded Transducer
0.16 V
1.85 V
4.06 V
Fluid Mechanics Research Laboratory
Effect of Drop Size on Response
0
m
L
100
m
L
200
m
L
Driving Voltage:
0.74 V
Fluid Mechanics Research Laboratory
Response of System to f = 0.99 Forcing
5.91 V
6.20 V
6.50 V
a
f
Fluid Mechanics Research Laboratory
Response of System to f = 1.04 Forcing
5.91 V
6.20 V
6.50 V
6.79 V
f
a
Fluid Mechanics Research Laboratory
Comparison of Model to Experiment
5.91 V
6.20 V
6.50 V
6.79 V
f
a
Model
Experiment
Fluid Mechanics Research Laboratory
Response Behavior
0
m
L
100
m
L
200
m
L
f < f
r
f > f
r
Fluid Mechanics Research Laboratory
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.
Fluid Mechanics Research Laboratory
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.
Fluid Mechanics Research Laboratory
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.
Fluid Mechanics Research Laboratory
•
Continuity:
•
Radial momentum:
•
Vertical momentum:
•
Volume fraction:
Governing Equations
Fluid Mechanics Research Laboratory
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.
Fluid Mechanics Research Laboratory
Parameters
Range
0

500
Viscous effects
0

100
Forcing amplitude
0

5
Forcing frequency
0

5
Gravity effects
Ejection Simulations
Fluid Mechanics Research Laboratory
Initial and Boundary Conditions
Symmetry
line
Outlet
No

slip
walls
80 cells
30 cells
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
Ejection Simulation

Case 4
Re = 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
Fluid Mechanics Research Laboratory
Ejection Simulation

Case 5
Re = 10, A = 30,
= 1 , Bo = 0
t = 0.8
t = 1
t = 1.2
t = 1.4
t = 1.6
t = 1.8
Fluid Mechanics Research Laboratory
Effect of Forcing Amplitude on Ejection
Bo = 0, Re = 10,
= 1
Fluid Mechanics Research Laboratory
Effect of Bond Number on Ejection
Re = 10, A = 25,
= 1
Fluid Mechanics Research Laboratory
Effect of Reynolds Number on Ejection
Bo = 0, A = 25,
= 1
Fluid Mechanics Research Laboratory
Effect of Forcing Frequency on Ejection
Re = 10,
Bo = 0, A = 25
Fluid Mechanics Research Laboratory
Ejection Threshold
Ejection
No ejection
Simulations
Range et al.
Goodridge et al.
low viscosity
Goodridge et al.
high viscosity
Fluid Mechanics Research Laboratory
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.
Fluid Mechanics Research Laboratory
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.
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