Transcript Document

Numerical Simulation of the Confined Motion of Drops
and Bubbles Using a Hybrid VOF-Level Set Method
Anthony D. Fick & Dr. Ali Borhan
Governing Equations
Conservation of Momentum
d
 u = 0
dt
 u
 1
   u  u  =  p    τ   F
 t
 Re
Shape of the interface between the
two phases affects macroscopic
properties of the system, such as
pressure drop, heat and mass transfer
rates, and reaction rate

τ =   u  u
T
HR )
Force

1
1
F=
f n   n  g
Re Ca
Fr
Outer fluid
 out
Inside fluid
Surface
normal
0.14
0.12
Ca 0.1
0.08
Re 1
Re 10
Re 20
Re 50
0.06
1.8
Ca 1
Ca 10
Ca 20
Ca 50
Ca 0.7
Re 10
1
0.8
Experimental results from A. Borhan and J.
Pallinti, “Breakup of drops and bubbles translating
through cylindrical capillaries”, Phys of Fluids 11,
1999 (2846).
Re = 50
0.4
0.2
0
0.5
Grid values of VOF that
correspond to initial shape
u  un
1
n
n
= u u  n [  τ n ]
t
 Re
Calculate
intermediate
velocity
1.5
2
Streamfunctions
Ca 1
Use  to
obtain
surface force
via level set
1
Size ratio
Computational Method
Ca 5
Ca 10
Ca 20
Ca 50
Computational Results for Drop Shape
(Pressure-Driven Motion)
Re 1
Evolution of drop shapes toward breakup of drop (Re 10 Ca 1)
1
1
n 1
  [ n p ] =   u
t

Calculate new
pressure using
Poisson equation
un1  u 1
= n [p n1 ]
t

Update velocity
and use it to
move the fluid
Re 10
Update 
from new
velocities
* C. W. Hirt and B. D. Nichols, “Volume of Fluid (VOF) Method for the Dynamics of Free
Boundaries,” Journal of Comp. Phys. 39 (1981) 201.
Drop breakup
Repeat
with
new 
No
Yes
Converges?
Final
solution
Re 20
Computational Results for Drop Shape
(Buoyancy-Driven Motion)
Level Set Method*:
• Level Set function f is the signed normal
distance from the interface
-1
2 1 0
• f = 0 defines the location of the interface
• Advection of f moves the interface
• Level Set needs to be reinitialized each time step to
maintain it as a distance function
• Advantage: Accurate representation of surface topology
Ca 5
Computation Flowsheet
Requires inhibitively small cell sizes for accurate surface topology
Ca 0.5
Re < 1
0.6
Partial
Cell
VOF 
Ca 0.1
1.4
0
shape 
Full
Cell
VOF 1
Ca 0.7
1.2
-h
Input
initial
Empty
Cell
VOF 0
Ca 0.5
2
Capillary number
• VOF function  equals fraction of cell filled with fluid
• VOF values used to compute interface normals and curvature
• Interface moved by advecting fluid volume between cells
• Advantage: Conservation of mass automatically satisfied
Ca 1
0.1
position
Calculate
density and
viscosity
for each 
Ca 0.5
Re 10
0.02
0
Ca 0.1
Re 5
1.6
h
Ca 1
Re 5
Thinning of drop leads
to increased velocity
0.04
 in
Volume of Fluid (VOF) Method*:
Ca 0.5
0.18
Pressure
Stress tensor
density
Some industrial applications:
• Polymer processing
• Gas absorption in bio-reactors
• Liquid-liquid extraction
Ca 0.1
0.16
Velocity
Pressure Driven Flow
Cacr
Deformation of the interface between two immiscible fluids
plays an important role in the dynamics of multiphase flows,
and must be taken into account in any realistic computational
model of such flows
Migration Velocities
Migration velocity ratio(U/U
Motivation
Conservation of Mass
o ga
Re =
o2
3
Ca

Re 50
Ca 1
Re
Ca =
ga2
Ca 5
Ca 10
Ca 20
Ca 50
Re 1
Future Studies:
Conservation of mass not assured in advection step
Power Law Suspending Fluid
Re 10
New algorithm combining the best
features of VOF and level-set methods:
Shapes
• Obtain Level Set from VOF values
• Compute surface normals using Level Set function
• Move interface using VOF method of volumes
Power index 0.5
Streamfunctions
Power index 1.5
Re = 10
• Application to Non-Newtonian two-phase systems
• Application to non-axisymmetric (three-dimensional)
motion
of drops and bubbles in confined domains
Acknowledgements:
Penn State Academic Computing Fellowship
Thesis advisor: Dr. Ali Borhan, Chemical Engineering
Re 20
Test new algorithm on drop motion in a tube
Former group members: Dr. Robert Johnson (ExxonMobil
Research) and Dr. Kit Yan Chan (University of Michigan)
• Frequently encountered flow configuration
• Availability of experimental results for comparison
• Existing computational results in the limit Re = 0
Power index 1.5 Re = 1
* S. Osher and J. A. Sethian, “Fronts Propagating with Curvature-Dependent Speed: Algorithms based
on Hamilton-Jacobi Formulations,” Journal of Comp. Phys. 79 (1988) 12.
Size ratio 0.7
Re 50
Increasing deformation
0.9
1.1