Diffusion and accurate hydrodynamics - PowerPoint Presentation

Slide1

Diffusion and accurate hydrodynamics in SPH and grid codes

James Wadsley (McMaster) Tom Quinn (Washington), Fabio Governato (Washington), Hugh Couchman (McMaster)

Disks 2012, HeidelbergSlide2

Test

Agertz et al 2007Code comparison paper from the proto AstroSim conference (2004)

Gasoline

(

Wadsley

,

Stadel

& Quinn 2004)

Gadget(Springel)

FLASH

ENZO

ARTSlide3

Test

Agertz et al 2007Code comparison paper from the proto AstroSim conference (2004)

PPM

Piecewise

Parabolic

Method

(

Collela

& Woodward

1987)

SPHSmoothed

ParticleHydrodynamics(Monaghan 1992,

Springel

&

Hernquist

2002)Slide4

Basic Result: SPH blobs don’t break up

Quantitative measure: Fraction of cloud remaining above 64% of initial densityAs cloud fragments – size R halves every Kelvin-Helmholtz time τKH ~ R/v Effective Kelvin-Helmholtz time halves repeatedly until cloud catches up with flow (v

 0)

SPH: Kelvin-Helmholtz time staticSlide5

Basic Result: SPH blobs don’t break up

Immediate SPH issue: Surface Tension present in arithmetic sum Pressure force (e.g. Monaghan 1992, Gasoline, Gadget, …)Suppresses Kelvin Helmholtz instabilities Issue first identified byRitchie & Thomas (2001)Slide6

SPH Kelvin Helmholtz fixed

Ritchie & Thomas (2001) – smooth pressure not density and Geometric Density Average in Force: remove surface tension (pressure spike at density jump)Price (2008) -- smear density jumpsRead, Hayfield & Agertz (2010), Read & Hayfield (2011), Abel (2011), Murante et al (2011) … modified SPH

Solutions typically expensive (more accurate)Slide7

Key to alleviating SPH surface tension:

Geometric Density Average in Force (GDForce): Morris (1996), Monaghan (1992) Ritchie and Thomas (2001),

Can be derived from a

Lagrangian

:

Monaghan &

Rafiee (2012)see also Abel (2011)

Standard Force

Geometric Density Force

Merging Cluster TestSlide8

Blob Test in Entropy (T3/2/ρ)

Hi-Res ENZO

Hi-Res

Standard

SPHSlide9

Blob Test in Entropy (T3/2/ρ)

Hi-Res ENZO

Hi-Res

GDForce

SPHSlide10

Blob Test in Entropy (T3/2/ρ)

ENZOGDForce

SPH

Standard

SPH

t = 1.25

τ

KH

t = 3.75

τ

KH

t = 2.5

τ

KH Slide11

Blob Test in Entropy (T3/2/ρ)

ENZOGDForce

SPH

Standard

SPH

t = 1.25

τ

KH

t = 3.75

τ

KH

t = 2.5

τ

KH

Low Entropy Blobs Indestructible!Slide12

The second issue: Entropy mixing

Cluster Comparison (Frenk et al 1999)Grid codes have entropy cores, SPH codes don’t (because they don’t mix)

ENZO

SPH

Wadsley

,

Veeravalli

&

Couchman (2008)Slide13

How to get entropy cores?

Shocks (while

c

s

<v)

Mix

hot & cold cluster gas SPH can’t:

Eulerian codes can (accidentally):Slide14

Subgrid Turbulent Mixing

Fluid elements on a fixed (resolved) physical scale do exchange energy/entropy due to unresolved (turbulent) motions

Turbulent diffusive heat fluxSlide15

Ways to model turbulent diffusion:

Lowest-order turbulent diffusion model: Turb

has units of velocity x length

Smagorinksy

model (1963):

Assumes Prandtl number ~ 1 Sij = strain tensor of resolved flow, lS

Smagorinsky lengthIncompressible grid models set ls 2

~ 0.02 x 2 (Lilly 1967)

For SPH we can try Turb

= C h2

S C ~ 0.1

Wadsley

,

Veeravalli

&

Couchman

(2008)

Shen

,

Wadsley

& Stinson (2010)Slide16

Ways to model turbulent diffusion:

Lowest-order turbulent diffusion model:

Wadsley

,

Veeravalli

&

Couchman

(2008)

Shen, Wadsley & Stinson (2010)Cluster Entropy Cores easily obtained in SPH with thermal diffusion

included – solved in 2008Slide17

Bottom Line on Diffusion & SPH

Physical diffusion in needed in all simulations e.g. Metals should mix in galactic outflows (Shen, Wadsley & Stinson 2010)With thermal diffusion in SPH:Get entropy cores in Galaxy ClustersNecessary to model entropy of mixing (

Springel

2010)

Better Kelvin-Helmholtz

Better Blob results … (but not quite there)

… Need Geometric Density Force AND DiffusionSlide18

Blob Test in Entropy (T

3/2/ρ)

Hi-Res ENZO

GDForce

+ Turbulent Diffusion

SPHSlide19

Blob Test in Entropy (T3/2/ρ)

ENZOGDForce

+

Turbulent

Diffusion

SPH

Standard

SPH

t = 1.25

τ

KH

t = 3.75 τKH

t = 2.5

τ

KH Slide20

Is grid (PPM) the right answer?No: Numerical Diffusion Approximate,

e.g. is not Galilean InvariantENZOMoving flow

t = 1.25

τ

KH

t = 3.75

τ

KH

t = 2.5

τKH

ENZO

1/2 – 1/2

ENZO

Moving

blobSlide21

Blob’s falling apart…

t / τKH

Dense Cloud Remaining

GDForce

+ Diffusion

Coefficient 0.1,0.03,0.01

Standard SPH

Standard SPH + Diffusion

GDForce SPHSlide22

Blob’s falling apart… (Log)

t / τKH Dense Cloud Remaining

GDForce

+ Diffusion

Coefficient 0.1,0.03,0.01

Standard SPH

Standard SPH + Diffusion

GDForce

SPHSlide23

Blob’s falling apart… (Log)

t / τKH

GDForce

+ Diffusion

Coefficient 0.1,0.03,0.01

Standard SPH

Standard SPH + Diffusion

GDForce SPH

ENZO (1/2 each)ENZO Moving blobENZO Moving flow (

Agertz et al.)

Dense Cloud RemainingSlide24

Diffusion coefficient…

Smash galaxy clusters together: Clean version of

Frenk

et al. ( 1999) with no substructure

Results look the same but ENZO mixes more in the core (see left) than

SPH+GDForce+Diffusion

Gasoline diffusion coefficients:

Also: Mass

metallicity + IGM Metals Inner 100 kpc

ENZO @ 4.5

Gyr

GASOLINE @ 4.5

Gyr

ENZO

1.0

0.1

,

0.03

,

0.01

,

no Diffusion

500 pc

Gasoline

Entropy at 20

GyrSlide25

Diffusion coefficient…

Smash smooth galaxy clusters together: Clean version of

Frenk

et al. ( 1999) with no substructure

Results look the same but ENZO mixes more in the core (see left) than

SPH+GDForce+Diffusion

Gasoline diffusion coefficients:

Also: Mass

metallicity + IGM Metals Inner 100 kpc

ENZO @ 4.5

Gyr

GASOLINE @ 4.5

Gyr

ENZO

1.0

0.1

,

0.03

,

0.01

,

no Diffusion

500 pc

Gasoline

Entropy at 20

GyrSlide26

Galaxy Formation:Kaufmann et al (2008) blobs no more

Toy Galaxy Model (cf. Kaufmann et al.), 20

kpc

wide edge on view

Conclusion: Blobs product of SPH surface tension effects

(see also:

Joung et al 2011)

Standard SPH

GDForce+Diffusion SPHSlide27

Galaxy Formation:

Smoother disks & spiral structure

Toy Galaxy Model (cf. Kaufmann et al.), ~ 12

kpc

wide face on view

Smoother Structure

Standard SPH

GDForce+Diffusion SPHSlide28

Galaxy Formation

10

11

Solar Mass Galaxy, z=0.6 (latest output)

Initial indications: Cold flow mass only ~ 2% different

Increased star formation. Need to understand how marginally resolved cooling works

Standard SPH

GDForce+DiffusionSlide29

ConclusionsGeometric Density Average Force alleviates surface tension affects in SPH

Well-characterized diffusion treatments necessary for all codes (SPH & Grid)Careful attention to resolution scale behaviour important (e.g. Instabilities: KH, Jeans, Thermal/Cooling…)Gasoline public release: including corrected force (blob free!), Stinson et al. star formation and feedback, cooling, planetesimal collisions – coming to google

code this summer…Slide30

Sedov Test GDForce