Numerical Simulation of the Confined Motion of Drops and Bubbles Using a Hybrid VOF-Level Set Method

1. -1 Partial Cell VOF  Full Cell VOF 1 Empty Cell VOF 0 2 1 0 Governing Equations Motivation Migration Velocities Pressure Driven Flow Conservation of Momentum Conservation of Mass 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 Ca 0.1 Ca 0.5 Ca 1 Ca 0.1 Ca 0.5 Ca 1 0.18 Re 5 0.16 Thinning of drop leads to increased velocity Pressure Velocity 0.14 Re 5 ) Stress tensor HR • Some industrial applications: • Polymer processing • Gas absorption in bio-reactors • Liquid-liquid extraction Force 0.12 Re 10 0.1 Ca 0.1 Ca 0.5 Ca 0.7 Ca 0.1 Ca 0.5 Ca 0.7 Migration velocity ratio(U/U 0.08 Re 1 Re 10 Re 20 0.06 Re 50 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 Surface normal Re 10 0.04 Re < 1 0.02 0 Ca 1 Ca 5 Ca 10 Ca 20 Ca 50 Experimental results from A. Borhan and J. Pallinti, “Breakup of drops and bubbles translating through cylindrical capillaries”, Phys of Fluids11, 1999 (2846). Capillary number Re = 50 Computation Flowsheet Streamfunctions Input initial shapea Grid values of VOF that correspond to initial shape Computational Method Ca 1 Ca 5 Ca 10 Ca 20 Ca 50 Computational Results for Drop Shape (Pressure-Driven Motion) Volume of Fluid (VOF) Method*: Use a to obtain surface force via level set Calculate density and viscosity for each 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 Re 1 Evolution of drop shapes toward breakup of drop (Re 10 Ca 1) Calculate intermediate velocity Calculate new pressure using Poisson equation Re 10 Update velocity and use it to move the fluid Requires inhibitively small cell sizes for accurate surface topology Update a from new velocities * 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. Drop breakup Re 20 Repeat with new a No Yes Final solution Converges? Computational Results for Drop Shape (Buoyancy-Driven Motion) 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 Ca Re 50 Ca 50 Ca 10 Ca 5 Ca 20 Ca 1 Re Re 1 Conservation of mass not assured in advection step Power Law Suspending Fluid New algorithm combining the best features of VOF and level-set methods: Re 10 Future Studies: Shapes Streamfunctions • Obtain Level Set from VOF values • Compute surface normals using Level Set function • Move interface using VOF method of volumes Power index 1.5 Power index 0.5 • Application to Non-Newtonian two-phase systems • Application to non-axisymmetric (three-dimensional) motion • of drops and bubbles in confined domains Re = 10 Re 20 Test new algorithm on drop motion in a tube Acknowledgements: • Frequently encountered flow configuration • Availability of experimental results for comparison • Existing computational results in the limit Re = 0 Power index 1.5 Re = 1 Penn State Academic Computing Fellowship * 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. Thesis advisor: Dr. Ali Borhan, Chemical Engineering Former group members: Dr. Robert Johnson (ExxonMobil Research) and Dr. Kit Yan Chan (University of Michigan) Size ratio 0.7 0.9 1.1 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