1. 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 Mass
Conservation of Momentum
Velocity
Stress tensor
Force
Pressure
Surface normal
Migration Velocities
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Ca 1
Ca 5
Ca 10
Ca 20
Ca 50
Capillary number
Migration velocity ratio(U/U HR )
Re 1
Re 10
Re 20
Re 50
Thinning of drop leads to increased velocity
Pressure Driven Flow
Re 5
Re 10
Ca 0.1
Ca 0.5
Ca 0.7
Ca 1
Re < 1
Re = 50
Experimental results from A. Borhan and J. Pallinti, “Breakup of drops and bubbles translating through cylindrical capillaries”, Phys of Fluids 11, 1999 (2846).
Motivation
Some industrial applications:
Polymer processing
Gas absorption in bio-reactors
Liquid-liquid extraction
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
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
Computation Flowsheet
Grid values of VOF that correspond to initial shape
Input initial shape a
Calculate density and viscosity for each a
Use a to obtain surface force via level set
Calculate intermediate velocity
Calculate new pressure using Poisson equation
Update velocity and use it to move the fluid
Converges?
Yes No
Repeat with new a
Final solution
Update a from new velocities
Streamfunctions
Ca 5
Ca 10
Ca 20
Ca 50
Ca 1
Re 1
Re 10
Re 20
Re 50
Computational Method
Empty Cell
VOF 0
Full Cell
VOF 1
Partial Cell
VOF 
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
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.
Computational Results for Drop Shape (Pressure-Driven Motion)
Evolution of drop shapes toward breakup of drop (Re 10 Ca 1)
Drop breakup
Ca 5
Ca 10
Ca 20
Ca 50
Ca 1
Re 1
Re 10
Re 20
Re 50 Ca
Increasing deformation Re
Computational Results for Drop Shape
(Buoyancy-Driven Motion)
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
Level Set Method*: 2 1 -1
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
New algorithm combining the best
features of VOF and level-set methods:
Obtain Level Set from VOF values
Compute surface normals using Level Set function
Move interface using VOF method of volumes
* 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.
Conservation of mass not assured in advection step
Acknowledgements:
Penn State Academic Computing Fellowship
Thesis advisor: Dr. Ali Borhan, Chemical Engineering
Former group members: Dr. Robert Johnson (ExxonMobil
Research) and Dr. Kit Yan Chan (University of Michigan)
Future Studies:
Application to Non-Newtonian two-phase systems
Application to non-axisymmetric (three-dimensional) motion
of drops and bubbles in confined domains
Power Law Suspending Fluid
Shapes
Streamfunctions
Power index 0.5
Power index 1.5
Power index 1.5 Re = 1
Size ratio 0.7
0.9
1.1
Re = 10