Smoothed Particle Hydrodynamics Techniques for the Physics Based - - PowerPoint PPT Presentation

Smoothed Particle Hydrodynamics

Techniques for the Physics Based Simulation of Fluids and Solids

Part 3 Multiphase Fluids

Dan Koschier Jan Bender Barbara Solenthaler Matthias Teschner

Eurographics19 Tutorial - SPH 3

Mo Moti tivati tion

Complex mixing phenomena Fluid Interfaces

Gissler et al. 2019 Yang et al. 2015

Eurographics19 Tutorial - SPH

• Particles carry attributes individually

– Mass, rest density – Concentration, temperature, viscosity, ...

• Two fluids a and b, with
• Buoyancy emerges from individual rest densities

4

My My Fi First st M Mul ulti-fluid SP fluid SPH So H Solver lver

ma ρ0

a = mb

ρ0

b

be solved

Eurographics19 Tutorial - SPH 5

My My Fi First st M Mul ulti-fluid SP fluid SPH So H Solver lver

Lavalamp Boiling

Müller et al. 2005 Lenaerts & Dutre 2009

Switching densities

Eurographics19 Tutorial - SPH 6

Hig High D h Densit ensity R Rat atio ios

Eurographics19 Tutorial - SPH 7

Hig High D h Densit ensity R Rat atio ios

Solenthaler & Pajarola 2008

Eurographics19 Tutorial - SPH

• 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

8

Int Inter erfac face D e Disc iscont ntinuit inuities ies

Eurographics19 Tutorial - SPH

• Problems near interfaces where rest densities and

masses vary

• Falsified smoothed quantities

9

Int Inter erfac face D e Disc iscont ntinuit inuities ies

h ρ0=100 ρ0=1000

ρi = ∑

j

mjWij

Color-coded density

Eurographics19 Tutorial - SPH

• Problems near interfaces where rest densities and

masses vary

• Falsified smoothed quantities

10

Int Inter erfac face D e Disc iscont ntinuit inuities ies

h ρ0=100 ρ0=1000

• r pi = k1

✓⇣

ρi ρ0

⌘k2 1 ◆ . particle deficiency problem

500 1000 500 100

pi < 0

pi > 0

Eurographics19 Tutorial - SPH

• Use number density
• Adapted density of particle i given by

11

Ad Adapte ted De Density and and Pr Pressure

density δi = ∑jWij were adapted accordingly

˜ ρi = miδi.

˜ pi = k1 ✓ ˜ ρi ρ0 ◆k2 1 !

• Pressure computation using adapted density

Eurographics19 Tutorial - SPH

• Derive adapted forces
• Substitute adapted density and pressure into the NS pressure term
• Apply SPH derivation to get adapted pressure force
• Similarly derivation of viscosity force

12

Ad Adapte ted For

• rce

ces

• formalism. The

Fp = r ˜

p δ . ˜ p ˜ p

Fp

i = ∑ j

˜ p j δ j

2 + ˜

pi δi

2

! rWi j.

Fv

i = 1

δi ∑

j

µi +µj 2 1 δ j (vj vi)r2Wij

Eurographics19 Tutorial - SPH 13

Ad Adapte ted SPH SPH - Ob Observation

• ns
• 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 14

Ad Adapte ted SPH SPH - Re Results ts

Solenthaler & Pajarola 2008

Eurographics19 Tutorial - SPH

• Diffusion equation
• SPH equation

15

Di Diffusion

• n Effect

cts

Fluid mixing

∂C ∂t = αr2C,

SPH, this equation

∂Ci ∂t = α∑

j

mj Cj Ci ρj r2Wij,

Color diffusion

Müller et al. 2005

Temperature diffusion (and phase changes)

Lenaerts & Dutre 2009 Keiser et al. 2005

Eurographics19 Tutorial - SPH

• 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]) – Drift velocities: Phase velocities relative to mixture average

16

Co Complex x Mixi xing Effects

Ren et al. 2014

Eurographics19 Tutorial - SPH

• Phase:

– Volume fraction , – Phase velocity v_k

• Mixture:

– Mixture density (f( )) – Mixture velocity

• Continuity and momentum equations of the phases and mixture

17

Mi Mixtu ture re Mo Model

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

• n αk of

X

k αk = 1, αk 0.

• n αk of

and vm are computed

Eurographics19 Tutorial - SPH

• Continuity equation of the mixture model
• Momentum equation for the mixture

mixture density volume fraction of phase mixture velocity (avg over all phases)

18

Mi Mixtu ture re Mo Model

Dρm Dt = ∂ρm ∂t +r·(ρmvm) = 0,

D(ρm,vm) Dt = rp+r·(τm +τDm)+ρmg,

where ρm elocity, av

and vm are computed

• n αk of

., ρm = for the ∑k αkρk mixture is fraction αk of a phase and vm =

1 ρm ∑k αkρkvm .

given as

where τm tensors, respecti

and τDm respectively

viscous stress tensor of the mixture diffusion tensor of the mixture (convective momentum transfer between phases)

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 19

Al Algor gorith thm

• 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
• 1. Compute density and pressure with SPH

3 loops over all particles:

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Immisc Immiscible and M ible and Misc iscible L ible Liquids iquids

Ren et al. 2014

Immiscible Miscible, diffusion disabled Miscible, diffusion enabled Red / green miscible, immiscible with blue

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Mo More re Results ts

Ren et al. 2014

Eurographics19 Tutorial - SPH

• [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

22

Limit Limitat atio ions an and Ex Extensio ions

Yan et al. 2016 Yang et al. 2015

• [Yang et al. 15] Energy-based model using Cahn-

Hilliard equation that describes phase separation

• > incompressible flows
• [Yan et al. 16] Extension to fluid-solid interaction
• > dissolution of solids, flows in porous media,

interaction with elastics