1. A Hybrid Particle-Mesh Method for Viscous, Incompressible, Multiphase Flows Jie LIU, Seiichi KOSHIZUKA Yoshiaki OKA The University of Tokyo,

2. Abstract A hybrid method to simulate unsteady multiphase flows – Moving particles – Finite volume stationary mesh Continuum Surface Force (CSF) model – surface tension – wall adhesion

3. 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

4. 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

5. 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

6. 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

7. 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

8. Solution Algorithm

10. Description of Multiphase Flow by Particle and Mesh MPS method – Particle : liquids Mass, position Interface tracking

11. Governing Equations Conservation of mass Conservation of momentum Identity matrixvolumesurface areanormal volume force

12. Governing Equations (Con’t) Stress tensor

13. Surface Tension Model The interfacial particles – Determine by the particle number density – Defined originally in the MPS method particle number density n – weight function

14. Surface Tension Model (Con’t) Surface tension force – The Continuum Surface Force model (CFS) Surface force – Curvature – Normal vector

15. Surface Tension Model (Con’t) Gradient vector between two particles i and j Neighboring particles j with the kernel function

16. Surface Tension Model (Con’t) Divergence of unit normal vector

17. Surface Tension Model (Con’t) Surface force be transferred to volume force

18. The Continuum Surface Force model Interpolation

19. Boundary Conditions: Wall Adhesion Wall interface normal – With static contact angle : fluid material property assume to be a constant

20. Mesh Calculation by Finite Volume Method pressure, density, viscosity – center of cell velocity – cell faces

21. 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

22. Mesh Calculation by Finite Volume Method (Con’t) Surface force Fractional areas

23. Mesh Calculation by Finite Volume Method (Con’t) finite-volume discretization Conservation of momentum Conservation of mass

24. Mesh Calculation by Finite Volume Method (Con’t) Fluxes

25. Mesh Calculation by Finite Volume Method (Con’t) Solved by projection method – momentum equation is split temporal velocity Pressure term,

26. Mesh Calculation by Finite Volume Method (Con’t) mass conversion equation pressure equation as follow – Poisson solver : use Successive Over Relaxation

27. Particle Calculation Particles move with the fluid velocities – Velocity founded by area-weighted interpolating New position of particles New Particle number density

28. 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

29. Particle Calculation (Con’t) position of particle is modified After this step, particle’ velocity is omitted – Only the velocities defined on mesh remain

30. 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

31. Equilibrium Rod

32. Equilibrium Rod (Con’t) Mean pressure of the liquid rod

33. Non-equilibrium Rod

34. Equilibrium Contact Angle

35. Flow Induced by Wall Adhesion wall adhesion in the wetting case

36. Flow Induced by Wall Adhesion (Con’t) non-wetting case

37. Rayleigh-Taylor Instability Tow-phase flow phenomenon – equilibrium state is perturbed – when a heavy fluid is put upon a lighter one

38. Rayleigh-Taylor Instability (Con’t) With Surface tension – interface as flat as possible – near one sidewall of tank

39. Kelvin-Helmholtz Instability Fundamental instability of incompressible fluid flow – different densities moving at different velocities – be evaluated by Richardson’s number (Ri)

40. Kelvin-Helmholtz Instability (Con’t) saltwater flows down freshwater flows upward

41. Kelvin-Helmholtz Instability (Con’t)