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