A Hybrid Particle-Mesh Method for Viscous, Incompressible, Multiphase Flows

Jie LIU, Seiichi KOSHIZUKAYoshiaki OKA

The University of Tokyo,

AbstractAbstract

• A hybrid method to simulate unsteady multiphase flows

– Moving particles

– Finite volume stationary mesh

• Continuum Surface Force (CSF) model

– surface tension

– wall adhesion

IntroductionIntroduction

• 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 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 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 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

Identity matrix volume surface area normal

volume force

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 modelThe Continuum Surface Force model

Interpolation

Boundary Conditions: Wall Adhesion

• Wall interface normal

– With static contact angle

: fluid material propertyassume 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 momentum

Conservation of mass

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

temporal velocity

Pressure term,

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

• Poisson equation of correction pressure

Soved Cholesky conjugate gradient method

Dirichlet boundary condition

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)