1. Governing Equations Conservation of Mass Conservation of Momentum Velocity Stress tensor Force Pressure Surface normal Computation Flowsheet Grid values of VOF that correspond to initial shape Input initial shape  Calculate density and viscosity for each  Use  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  Final solution Penn State Computation Day Numerical Simulation of the Confined Motion of Drops and Bubbles Using a Hybrid VOF-Level Set Method Anthony D. Fick & Dr. Ali Borhan Computational Results for Drop Shape (Buoyancy-Driven Motion) Ca 5Ca 10Ca 20Ca 50Ca 1 Re 1 Re 10 Re 20 Re 50 Ca Re Increasing deformation 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. Computational Method Empty Cell VOF 0 Full Cell VOF 1 Partial Cell VOF  Volume of Fluid (VOF) Method * : 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 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. 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 * : 201 Level Set function  is the signed normal distance from the interface  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 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. Velocity Fields Simulation calculates velocity fields along with shape 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 Thick line is interface shape Radial direction Center line 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) Conservation of mass not assured in advection step Computational Setup Axis of symmetry (r = 0) Tube Wall (r = R) N cells (  r = R/N) Initial drop shape 5N cells (  z = R/N) Computational grid for axisymmetric motion of a drop in a cylindrical tube Simulations run on Atipa 20-node Linux cluster r z U Staggered Mesh (i, j)(i+1, j) (i, j+1) (i+1, j+1) p(i, j) v(i, j) u(i, j) r(i) radial velocity, u axial velocity, v pressure Use time-splitting with cell-centered differences v(i, j-1) Future Studies: Application to Non-Newtonian two-phase systems Application to non-axisymmetric (three-dimensional) motion of drops and bubbles in confined domains Update  from new velocities Computational Results for Drop Shape (Pressure-Driven Motion) Evolution of drop shapes toward breakup of drop (Re 10 Ca 1) Drop breakup