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.
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Fluid Mechanics Research Laboratory Vibration Induced Droplet Ejection Ashley James Department of Aerospace Engineering and Mechanics University of Minnesota.
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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.
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.
PrimaryDrop
VibratingSurface
SecondaryDroplets
Ejection Schematic
Fluid Mechanics Research Laboratory
Literature
• Faraday (1831) - wave formation due to vibration
• 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.
0
z
Fv
r
Fu
t
F
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.
F
Fluid Mechanics Research Laboratory
• 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
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
2
3
2
gLBo
L
LaA
LRe
Fluid Mechanics Research Laboratory
Initial and Boundary Conditions
Symmetryline
Outlet
No-slipwalls
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 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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
Fluid Mechanics Research Laboratory
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
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.