a hybrid particle-mesh method for viscous, incompressible, multiphase flows jie liu, seiichi...
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A Hybrid Particle-Mesh Method for Viscous, Incompressible, Multiphase Flows
Jie LIU, Seiichi KOSHIZUKAYoshiaki OKA
The University of Tokyo,
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AbstractAbstract
• A hybrid method to simulate unsteady multiphase flows
– Moving particles
– Finite volume stationary mesh
• Continuum Surface Force (CSF) model
– surface tension
– wall adhesion
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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
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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
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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
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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
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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
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Solution Algorithm
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Description of Multiphase Flow by Particle and Mesh
• MPS method
– Particle : liquids
• Mass, position
• Interface tracking
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Governing Equations
• Conservation of mass
• Conservation of momentum
Identity matrix volume surface area normal
volume force
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Governing Equations (Con’t)
• Stress tensor
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Surface Tension Model
• The interfacial particles
– Determine by the particle number density
– Defined originally in the MPS method
• particle number density n
– weight function
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Surface Tension Model (Con’t)
• Surface tension force
– The Continuum Surface Force model (CFS)
• Surface force
– Curvature
– Normal vector
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Surface Tension Model (Con’t)
• Gradient vector between two particles i and j
• Neighboring particles j with the kernel function
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Surface Tension Model (Con’t)
• Divergence of unit normal vector
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Surface Tension Model (Con’t)
• Surface force be transferred to volume force
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The Continuum Surface Force modelThe Continuum Surface Force model
Interpolation
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Boundary Conditions: Wall Adhesion
• Wall interface normal
– With static contact angle
: fluid material propertyassume to be a constant
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Mesh Calculation by Finite Volume Method
• pressure, density,
viscosity
– center of cell
• velocity
– cell faces
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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
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Mesh Calculation by Finite Volume Method (Con’t)
• Surface force
• Fractional areas
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Mesh Calculation by Finite Volume Method (Con’t)
• finite-volume discretization
Conservation of momentum
Conservation of mass
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Mesh Calculation by Finite Volume Method (Con’t)
• Fluxes
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Mesh Calculation by Finite Volume Method (Con’t)
• Solved by projection method
– momentum equation is split
temporal velocity
Pressure term,
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Mesh Calculation by Finite Volume Method (Con’t)
• mass conversion equation
• pressure equation as follow
– Poisson solver : use Successive Over Relaxation
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Particle Calculation
• Particles move with the fluid velocities
– Velocity founded by area-weighted interpolating
• New position of particles
• New Particle number density
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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
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Particle Calculation (Con’t)
• position of particle is modified
• After this step, particle’ velocity is omitted
– Only the velocities defined on mesh remain
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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
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Equilibrium Rod
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Equilibrium Rod (Con’t)
• Mean pressure of the liquid rod
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Non-equilibrium Rod
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Equilibrium Contact Angle
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Flow Induced by Wall Adhesion
• wall adhesion in the wetting case
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Flow Induced by Wall Adhesion (Con’t)
• non-wetting case
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Rayleigh-Taylor Instability
• Tow-phase flow phenomenon
– equilibrium state is perturbed
– when a heavy fluid is
put upon a lighter one
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Rayleigh-Taylor Instability (Con’t)
• With Surface tension
– interface as flat as possible
– near one sidewall of tank
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Kelvin-Helmholtz Instability
• Fundamental instability of incompressible fluid flow
– different densities moving at different velocities
– be evaluated by Richardson’s number (Ri)
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Kelvin-Helmholtz Instability (Con’t)
• saltwater flows down
• freshwater flows upward
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Kelvin-Helmholtz Instability (Con’t)
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Kelvin-Helmholtz Instability (Con’t)