Governing Equations

1
Numerical Simulation of the Confined Motion of
Drops and Bubbles Using a Hybrid VOF-Level Set
Method
Anthony D. Fick Dr. Ali Borhan
Computation Flowsheet
Motivation
Velocity Fields
Center line
Governing Equations
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.
Simulation calculates velocity fields along with
shape
Input initial shape a
Conservation of Mass
Conservation of Momentum
Grid values of VOF that correspond to initial
shape
Thick line is interface shape
Use a to obtain surface force via level set
Calculate density and viscosity for each a

• Some industrial applications
• Polymer processing
• Gas absorption in bio-reactors
• Liquid-liquid extraction

The stream function diagram displays the flow
fields inside and outside the drop
Pressure
Velocity
Stress tensor
Force
Calculate intermediate velocity
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
Simulation results for Re 1, Ca 1 and Re 50, Ca
10 cases
Calculate new pressure using Poisson equation
Surface normal
Radial direction
Update velocity and use it to move the fluid
Computational Method
Update a from new velocities
Computational Results for Drop Shape
(Pressure-Driven Motion)
Volume of Fluid (VOF) Method
Evolution of drop shapes toward breakup of drop
(Re 10 Ca 1)
No
Repeat with new a
Yes

• 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

Converges?
Final solution
Requires inhibitively small cell sizes for
accurate surface topology
Drop breakup
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.
Level Set Method

• Level Set function f is the signed normal
• distance from the interface
• 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

Future Studies

• 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
New algorithm combining the best features of VOF
and level-set methods
Thesis advisor Dr. Ali Borhan, Chemical
Engineering

• Obtain Level Set from VOF values
• Compute surface normals using Level Set function
• Move interface using VOF method of volumes

Former group members Dr. Robert Johnson
(ExxonMobil Research) and Dr. Kit Yan Chan
(University of Michigan)
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.