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
Motivation
Penn State Computation Day
Computation Flowsheet
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.
Governing Equations
Conservation of Momentum
Conservation of Mass
d
 u  0
dt
Some industrial applications:
• Polymer processing
• Gas absorption in bio-reactors
• Liquid-liquid extraction
 u
 1
   u  u    p    τ   F
 t
 Re
Calculate
density and
viscosity for
each a
Pressure
Velocity
Stress tensor

τ    u  uT
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
Input initial
shape a

Use a to
obtain
surface force
via level set
Calculate
intermediate
velocity
Calculate new
pressure using
Poisson equation
Inside fluid
density
Surface normal
 in
h
0
Thick line is
interface
shape
u  un
1
n
n
 u u  n [  τ n ]
t
 Re
The stream function
diagram displays the flow
fields inside and outside the
drop
Simulation results for Re 1, Ca 1 and
Re 50, Ca 10 cases
un1  u 1
 n [p n1 ]
t

Update velocity
and use it to
move the fluid
-h
Simulation calculates
velocity fields along with
shape
1
1
n 1
  [ n p ]    u
t

Outer fluid
 out
Velocity Fields
Grid values of VOF that correspond
to initial shape
Force
1
1
F
 n n 
g
Re
Fr
Center line
Radial direction
position
Update a from
new velocities
Computational Method
Computational Setup
Volume of Fluid (VOF) Method*:
• VOF function a 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
Repeat
with new a
Tube Wall (r = R)
Computational grid
for axisymmetric
U
motion of a drop in
a cylindrical tube
z
Full Cell
VOF 1
Partial Cell
VOF a
Initial drop shape
Simulations run on
Atipa 20-node
Linux cluster
Axis of symmetry (r = 0)
N cells (r = R/N)
Evolution of drop shapes toward breakup of drop (Re 10 Ca 1)
Final solution
Ca 1
Ca 5
Ca 10
Ca 20
Ca 50
Re
Drop breakup
Re 1
Staggered Mesh
Level Set Method*:
• Level Set function  is the signed normal
distance from the interface
2 1 0
•   0 defines the location of the interface
• Advection of  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
Converges?
Yes
Ca
Requires inhibitively small cell sizes for accurate surface topology
* 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.
No
Computational Results for Drop Shape
(Buoyancy-Driven Motion)
5N cells (z = R/N)
r
Empty Cell
VOF 0
Computational Results for Drop Shape
(Pressure-Driven Motion)
Re 10
-1
Future Studies:
(i, j+1)
(i+1, j+1)
v(i, j)
v
z
• Obtain Level Set from VOF values
• Compute surface normals using Level Set function
• Move interface using VOF method of volumes
(i, j)
r(i)
• Application to Non-Newtonian two-phase systems
• Application to non-axisymmetric (three-dimensional) motion
of drops and bubbles in confined domains
pressure
Re 20
p(i, j)
u(i, j)
Conservation of mass not assured in advection step
New algorithm combining the best
features of VOF and level-set methods:
radial velocity, u
axial velocity, v
(i+1, j)
v(i, j-1)
Use time-splitting with
cell-centered differences
Acknowledgements:
Penn State Academic Computing Fellowship
u u(i, j )  u(i  1, j)

r
r
Thesis advisor: Dr. Ali Borhan, Chemical Engineering
v v(i, j )  v(i, j  1)

z
z
Former group members: Dr. Robert Johnson (ExxonMobil
Research) and Dr. Kit Yan Chan (University of Michigan)
Re 50
Test new algorithm on drop motion in a tube
• Frequently encountered flow configuration
• Availability of experimental results for comparison
• Existing computational results in the limit Re = 0
* 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.
Increasing deformation
ga2
o ga3
Ca 
Re 
2

o