Transcript Slide 1
A Hybrid Particle-Mesh Method for Viscous, Incompressible, Multiphase Flows Jie LIU, Seiichi KOSHIZUKA Yoshiaki OKA The University of Tokyo, Abstract • A hybrid method to simulate unsteady multiphase flows – Moving particles – Finite volume stationary mesh • Continuum Surface Force (CSF) model – surface tension – wall adhesion Introduction • Needed Effects – Capillarity phenomena, wetting effect, droplet , bubble • Marker-And-Cell – With a regular, stationary mesh • Volume-Of-Fluid – With marker function to identify the interface • CIP & Phase field method – Capture fluid interfaces Introduction (Con’t) • Adaptive (moving) grid methods – Interface is well-defined, – Continuous curve – Sharp resolution • Front tracking – To restructure the interface grid – Merged into one interface or eliminated – Ex solution : Level-set Introduction (Con’t) • Numerical algorithms – Eulerian particle method (Particle-In-Cell) • explicitly associated with different materials • interfaces can be easily followed • pressure and fluid velocity are computed in Cell • Lagrangian particle method – Smooth Particle Hydrodynamics (SPH) • approximation of spatial derivatives – Moving Particle Semi-implicit (MPS) method • represented by a finite number of moving particles • analyze incompressible flows Introduction (Con’t) • In this paper, – Hybrid method • coupling MPS method with mesh method – Incompressible, viscous, multiphase flows – Without specific front tracking algorithm • automatically determined by the distribution of particles – Continuum Surface Force (CSF) model • surface tension force Numerical Method • Solution Algorithm • Description of Multiphase Flow by Particle and Mesh • • • • • Governing Equations Surface Tension Model Boundary Conditions: Wall Adhesion Mesh Calculation by Finite Volume Method Particle Calculation Solution Algorithm Description of Multiphase Flow by Particle and Mesh • MPS method – Particle : liquids • Mass, position • Interface tracking Governing Equations • Conservation of mass • Conservation of momentum volume force Identity matrix volume surface area normal Governing Equations (Con’t) • Stress tensor Surface Tension Model • The interfacial particles – Determine by the particle number density – Defined originally in the MPS method • particle number density n – weight function Surface Tension Model (Con’t) • Surface tension force – The Continuum Surface Force model (CFS) • Surface force – Curvature – Normal vector Surface Tension Model (Con’t) • Gradient vector between two particles i and j • Neighboring particles j with the kernel function Surface Tension Model (Con’t) • Divergence of unit normal vector Surface Tension Model (Con’t) • Surface force be transferred to volume force The Continuum Surface Force model Interpolation Boundary Conditions: Wall Adhesion • Wall interface normal – With static contact angle : fluid material property assume to be a constant Mesh Calculation by Finite Volume Method • pressure, density, viscosity – center of cell • velocity – cell faces Mesh Calculation by Finite Volume Method (Con’t) • Procedure : Conservation of momentum Eq. – (1) the cell that encloses the center of the interfacial particle is found – (2) the neighbors of the cell are found – (3) the fractional areas that the particle occupied on the neighbor cells are computed – (4) these fractional areas are used to distribute the surface force Mesh Calculation by Finite Volume Method (Con’t) • Surface force • Fractional areas Mesh Calculation by Finite Volume Method (Con’t) • finite-volume discretization Conservation of mass Conservation of momentum Mesh Calculation by Finite Volume Method (Con’t) • Fluxes Mesh Calculation by Finite Volume Method (Con’t) • Solved by projection method – momentum equation is split Pressure term, temporal velocity Mesh Calculation by Finite Volume Method (Con’t) • mass conversion equation • pressure equation as follow – Poisson solver : use Successive Over Relaxation Particle Calculation • Particles move with the fluid velocities – Velocity founded by area-weighted interpolating • New position of particles • New Particle number density Particle Calculation (Con’t) • Particle’s mass conservation equation • Correction pressure gradient term Dirichlet boundary condition • Poisson equation of correction pressure Soved Cholesky conjugate gradient method Particle Calculation (Con’t) • position of particle is modified • After this step, particle’ velocity is omitted – Only the velocities defined on mesh remain Computational Examples • Standard static and dynamic problems – Equilibrium Rod – Non-equilibrium Rod – Equilibrium Contact Angle – Flow Induced by Wall Adhesion – Rayleigh-Taylor Instability – Kelvin-Helmholtz Instability Equilibrium Rod Equilibrium Rod (Con’t) • Mean pressure of the liquid rod Non-equilibrium Rod Equilibrium Contact Angle Flow Induced by Wall Adhesion • wall adhesion in the wetting case Flow Induced by Wall Adhesion (Con’t) • non-wetting case Rayleigh-Taylor Instability • Tow-phase flow phenomenon – equilibrium state is perturbed – when a heavy fluid is put upon a lighter one Rayleigh-Taylor Instability (Con’t) • With Surface tension – interface as flat as possible – near one sidewall of tank Kelvin-Helmholtz Instability • Fundamental instability of incompressible fluid flow – different densities moving at different velocities – be evaluated by Richardson’s number (Ri) Kelvin-Helmholtz Instability (Con’t) • saltwater flows down • freshwater flows upward Kelvin-Helmholtz Instability (Con’t) Kelvin-Helmholtz Instability (Con’t)