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Computational Flui d Dynamics in Astrophysics
James Murray Sw inburne University jmurray@sw in.e du.au


Computational Flui d Dynamics: how boring is that? What is CFD? What are the various approaches? If I were starting my PhD now


Matthew Bate, University of Exeter, UKAFF


Matthew Bate, University of Exeter, UKAFF


A ra diatively warpe d accretion disk in a low mass x-ray binar y

Steve Foulkes, Open University


Si de view of an a diabatic accretion disk surroun ding a neutron star / black hole

Jim Stone et al., Princeton


Turbulence in a molecular clou d

Jim Stone, Princeton


for mation of the Mo on after a g iant impact

Eiichiro Kokubo, National Astronomical Obser vator y of Japan, HARP


the effect of a nuclear weapon detonating on an astero i d

Erik Asphaug, UCSC


Dam breaking in the Rocky Mountains

Joe Monaghan, Monash & Paul Clear y, CSIRO


Water filling a bath

me


Drops into a "square" vase


What is computational flui d dynamics?
The numerical solution of the equations of motion an d state of a flui d


What is computational flui d dynamics?


What sorts of astrophysical problems can we solve?
· · · · · Shocks Explosions Collisions For mation Destruction


Possible forces that nee d be consi dere d
· Gas pressure · Gravity · Viscosity · Turbulence · Magnetic fields · Ra diation


Equations of state
· Isother mal · Adiabatic · Polytropic · Gas w ith co oling · Soli d!


Discrete equations
· The computer can only store quantities such as velocity, density, pressure an d temperature at a finite number of places. · How do we cho ose where those places are? · How do we calculate spatial derivatives?


The two approaches to CFD


Smo othe d Particle Hydro dynamics
Flui d is represente d by a collection of particles that move w ith the flow The key to SPH is a metho d for interpolating flui d quantities from a "disordere d" set of po ints


Kernel interpolation
For any flui d property A, we can write an interpolant W is the interpolating "kernel"

AI (rI ) = A(r ) W (r - rI , h) dV
V


Interpolating from particles
AI (rI ) =
V



A(r ) W (r - rI , h ) dm

The integral is approximate d by a summation

n

AI (ri )


j =1

m

A
j

j j



W (ri - r j , h )


Interpolating flui d properties
The interpolant for the density
n

I (ri )


j =1

m j W (ri - r j , h )

Spatial derivatives also become a simple summation
n

AI (ri )


j =1

m

A
j

j j



W (ri - r j , h )


Kernel properties
For the mag ic to work we require

limW (r, h ) = (r )
h 0

an d

V



W (r, h )dV = 1

hence toy SPH equations usually use a gaussian kernel

1 h



e

r2 - h2


Computational Issues
1 I (ri ) = h

m e
j j =1

n

( ri -r j ) 2 - h2

With a gaussian kernel ever y particle is neighbour to ever y other.


The equations of motion
dv =- dt i
n


j =1

Pi Pj m j 2 + 2 + µij W (ri - r j , h ) j i

du = dt i

n


j =1

Pi Pj m j 2 + 2 + µij ( i - v v j i

j·

)

W (ri - r j , h )


Gri ds an d Particles
· Gri ds split the do main up into a set of regular boxes or cells. Differential equations become finite difference equations. (Eulerian) · A secon d approach is to sample the volume w ith a set of po ints that move. This is the particle or Lagrang ian approach.


Don't write your own co de
· To write an d test your own flui d dynamics co de takes months or years... · Writing a co de is "dea d time" · Several public domain co des are available · These co des have un dergone years of testing on a range of problems. · One new piece of physics = one PhD thesis
· Stone & Nor man, 1992, ApJS, 80, 753


ATHENA
· Finite difference magnetohydro dynamics · Written by Jim Stone, John Hawley an d collaborators (base d upon ZEUS) · The gri ds are regular an d nona daptive · Perhaps the easiest co de to a dapt to your own nefarious porpo ises · http://www.astro.princeton.e du/~jstone/ athena.html


ENZO
Mike Nor man, Greg Br yan Adaptive Mesh Refinement hydro dynamics + N-bo dy
http://cosmos.ucsd.e du/enzo/

Designe d for supercomputers (difficult to install)


Smo othe d Particle Hydro dynamics
· Monaghan 1992, Annual Reviews of Astronomy an d Astrophysics · Several hydro+self-gravity SPH co des available · GADGET ( Volker Springel)
· http://www.mpa-garching.mpg.de/ga dget/

· HY DRA (Hugh Couchman)
· http://hydra.mcmaster.ca/hydra/


Which co de do I use?
· What physics is important (hydro dynamics, magnetic fields, ra diation)? · What is the geometr y of the problem? · What are the boun dar y con ditions? · Am I lo oking for gross features or fine details?


An d then?
· Having obtaine d a co de you must learn how to use it · ATHENA for example comes bun dle d w ith a few test problems. · Expect to spen d "some" time learning the co de. · Refer to "Numerical Recipes"


What to lo ok out for
· Accuracy. Do the answers lo ok anything like what is expecte d? · Stability. Do the answers converge when you improve resolution? · Resolution. Particularly in three dimensions extra resolution is costly.


Ancient Proverb

Do not base your PhD upon any simulation that takes more than 5 minutes to run


John Dubinski, University of Toronto (Blue Gene)

His philosophy was a mixture of three famous schools -- the Cynics, the Stoics and the Epicureans -- and summed up all three of them in his famous phrase, "You can't trust any bugger further than you can throw him, and there's nothing you can do about it, so let's have a drink." Dydactylos the philosopher (Terry Pratchett, Small Gods)