Smoothed Particle Hydrodynamics Techniques for the Physics Based Simulation of Fluids and Solids Part 3 Multiphase Fluids Dan Jan Barbara Matthias Koschier Bender Solenthaler Teschner
Mo Moti tivati tion Fluid Interfaces Complex mixing phenomena Gissler et al. 2019 Yang et al. 2015 Eurographics19 Tutorial - SPH 3
My My Fi First st M Mul ulti-fluid SP fluid SPH So H Solver lver • Particles carry attributes individually – Mass, rest density – Concentration, temperature, viscosity, ... m a a = m b • Two fluids a and b, with ρ 0 ρ 0 b be solved • Buoyancy emerges from individual rest densities Eurographics19 Tutorial - SPH 4
My My Fi First st M Mul ulti-fluid SP fluid SPH So H Solver lver Switching densities Boiling Lavalamp Lenaerts & Dutre 2009 Müller et al. 2005 Eurographics19 Tutorial - SPH 5
Hig High D h Densit ensity R Rat atio ios Eurographics19 Tutorial - SPH 6
Hig High D h Densit ensity R Rat atio ios Solenthaler & Pajarola 2008 Eurographics19 Tutorial - SPH 7
Inter Int erfac face D e Disc iscont ntinuit inuities ies • Standard SPH (SESPH) – Cannot handle discontinuities at interfaces – Results in spurious and unphysical interface tension – Large density differences lead to instability problems • Adapted SPH – Capture density discontinuities across interfaces – Stable simulations despite high density ratios – We need full control over behavior Eurographics19 Tutorial - SPH 8
Int Inter erfac face D e Disc iscont ntinuit inuities ies • Problems near interfaces where rest densities and masses vary h • Falsified smoothed quantities ρ 0 =100 ρ 0 =1000 ρ i = ∑ m j W ij j Color-coded density 9 Eurographics19 Tutorial - SPH
Inter Int erfac face D e Disc iscont ntinuit inuities ies • Problems near interfaces where rest densities and masses vary h • Falsified smoothed quantities ρ 0 =100 ρ 0 =1000 500 p i < 0 1000 ✓⇣ ⌘ k 2 � 1 ◆ ρ i or p i = k 1 . ρ 0 particle deficiency problem 500 p i > 0 100 10 Eurographics19 Tutorial - SPH
Ad Adapte ted De Density and and Pr Pressure • Use number density density δ i = ∑ j W ij were adapted accordingly • Adapted density of particle i given by ρ i = m i δ i . ˜ ✓ ˜ ! ◆ k 2 ρ i • Pressure computation using adapted density p i = k 1 ˜ � 1 ρ 0 Eurographics19 Tutorial - SPH 11
Ad Adapte ted For orce ces • Derive adapted forces • Substitute adapted density and pressure into the NS pressure term formalism. The F p = � r ˜ p δ . p ˜ p ˜ • Apply SPH derivation to get adapted pressure force ! p j ˜ 2 + ˜ p i F p i = � ∑ r W i j . 2 δ j δ i j • Similarly derivation of viscosity force µ i + µ j i = 1 1 F v ( v j � v i ) r 2 W ij δ i ∑ δ j 2 j Eurographics19 Tutorial - SPH 12
Ad Adapte ted SPH SPH - Ob Observation ons • For a single phase fluid equations are identical to SESPH • For multi-fluid simulations interface problems are eliminated • No performance overhead • Extended with incompressibility condition [Akinci et al. 12, Gissler et al. 19] Gissler et al. 2019 Eurographics19 Tutorial - SPH 13
Ad Adapte ted SPH SPH - Re Results ts Solenthaler & Pajarola 2008 Eurographics19 Tutorial - SPH 14
Di Diffusion on Effect cts Fluid mixing • Diffusion equation ∂ C ∂ t = α r 2 C , SPH, this equation ∂ C i C j � C i • SPH equation r 2 W ij , ∂ t = α ∑ m j ρ j j Color diffusion Temperature diffusion (and phase changes) Keiser et al. 2005 Lenaerts & Dutre 2009 Müller et al. 2005 Eurographics19 Tutorial - SPH 15
Co Complex x Mixi xing Effects • Previous work – Mixture is only caused by diffusion effects – Different phases move at the same bulk velocity as the mixture • SPH based mixture model [Ren et al. 2014] – Mixing and unmixing due to (relative) flow motion and force distribution – Dynamics of multi-fluid flow captured using mixture model – Spatial distribution of phases modeled using volume fraction (similar to [Müller et al. 05]) Ren et al. 2014 – Drift velocities: Phase velocities relative to mixture average Eurographics19 Tutorial - SPH 16
Mi Mixtu ture re Mo Model • Phase: – Volume fraction , X on α k of k α k = 1 , α k � 0 . – Phase velocity v_k • Mixture: – Mixture density (f( )) on α k of – Mixture velocity and v m are computed • Continuity and momentum equations of the phases and mixture Ø The nonuniform distribution of velocity fields will lead to changes in the volume fraction of each phase Ø The drift velocities play a key role in this interaction mechanism Eurographics19 Tutorial - SPH 17
Mixtu Mi ture re Mo Model • Continuity equation of the mixture model D ρ m = ∂ρ m ∂ t + r · ( ρ m v m ) = 0 , D t where ρ m mixture density . , ρ m = ∑ k α k ρ k volume fraction of phase elocity, av mixture is on α k of for the fraction α k of a phase 1 and v m mixture velocity (avg over all phases) and v m = ρ m ∑ k α k ρ k v m . are computed given as • Momentum equation for the mixture D ( ρ m , v m ) = �r p + r · ( τ m + τ Dm )+ ρ m g , D t viscous stress tensor of the mixture where τ m diffusion tensor of the mixture (convective momentum transfer between phases) tensors, respecti and τ Dm respectively Ø The nonuniform distribution of velocity fields will lead to changes in the volume fraction of each phase Ø The drift velocities play a key role in this interaction mechanism Eurographics19 Tutorial - SPH 18
Al Algor gorith thm 3 loops over all particles: 1. Compute density and pressure with SPH 2. Compute drift velocity of each phase / particle Analytical expression of drift velocity, three terms defining - Slip velocity due to body forces - Pressure effects that cause fluid phases to move from high to low pressure regions - Brownian diffusion term representing phase drifting from high to low concentration Update diffusion tensor, advect volume fraction (using drift velocity) 3. Compute total force, advect particle Eurographics19 Tutorial - SPH 19
Immisc Immiscible and M ible and Misc iscible L ible Liquids iquids Immiscible Miscible, Miscible, Red / green miscible, diffusion disabled diffusion enabled immiscible with blue Ren et al. 2014 Eurographics19 Tutorial - SPH 20
Mo More re Results ts Ren et al. 2014 Eurographics19 Tutorial - SPH 21
Limitat Limit atio ions an and Ex Extensio ions • [Ren et al. 14] Uses WCSPH; a divergence-free velocity field cannot be directly integrated since neither the mixture nor phase velocities are zero, even if the material is incompressible • [Yang et al. 15] Energy-based model using Cahn- Hilliard equation that describes phase separation -> incompressible flows Yang et al. 2015 • [Yan et al. 16] Extension to fluid-solid interaction -> dissolution of solids, flows in porous media, interaction with elastics Yan et al. 2016 Eurographics19 Tutorial - SPH 22
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