Governing Equations

1. -1 Full Cell VOF 1 Partial Cell VOF  Empty Cell VOF 0 2 1 0 Penn State Computation Day Staggered Mesh Computational Setup Tube Wall (r = R) radial velocity, u (i, j+1) (i+1, j+1) Computational grid for axisymmetric motion of a drop in a cylindrical tube v(i, j) axial velocity, v U Computational Results for Drop Shape (Buoyancy-Driven Motion) pressure 5N cells (Dz = R/N) u(i, j) p(i, j) Use time-splitting with cell-centered differences Initial drop shape Ca (i, j) (i+1, j) v(i, j-1) r(i) Ca 1 Ca 5 Ca 10 Ca 20 Ca 50 Simulations run on Atipa 20-node Linux cluster Re Axis of symmetry (r = 0) N cells (Dr = R/N) Re 1 Re 10 Re 20 Conservation of mass not assured in advection step z r Re 50 Increasing deformation 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 Momentum Conservation of Mass 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 • 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 Yes 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.