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Smoothed Particle Hydrodynamics
Outline •Non-relativistic hydrodynamics.• SPH equations.• Applications.
• Relativistic hydrodynamics.• Relativistic SPH.• High energy nuclear physics.
Carlos Eduardo AguiarInstituto de Física - UFRJ
Fluid Dynamics
Hydrodynamic Equations
vtdt
d
velocity fluidv
density mass
pressureP
Pdt
ddt
d
1v
vContinuity equation
Euler’s equation
Ideal fluid
0dt
ds
massunit entropy / s
),( sPP
Entropy Equation
• no viscosity• no thermal conduction
v
P
dt
de
massunit energy / thermale
),( ePP
Pvdt
d
v
1)( 2
21
)(1
)( 221 vPev
dt
d
dP
TdsPdVde2
Energy Equation
Ideal Gas
))(1(
0
0 0)( ssen
PsK
enenP )1(),(
)1/( Te
TnP
nsKsnP )(),(
mn /
)(
])[()(
)()(
)(
221
221
v
v
vvIv
v
t
Pvt
v
Pt
t
Conservation Laws
umeenergy/vol thermal e
lumeentropy/vo s
SPH
- L.Lucy, Astron.J. 82, 1013 (1977)- R.Gingold, J.Monaghan, MNRAS 181, 378 (1977)
• Developed to study gas dynamics in astrophysical systems. • Lagrangian method.• No grids.• Arbitrary geometries.• Equally applicable in 1, 2 and 3 space dimensions.
Reviews:- J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543 (1992)- L. Hernquist, N. Katz, Ap. J. Suppl. 70, 419 (1989)
Smoothing
xxxxxx dhWAAA S ),()()()(
)()()( 2hOAAS xx
h
x0 1);( xx dhW
)()()()()]()([ xxxxxx SSS BABABA
kernel smoothing),( hW x
Error:
Particles
b
bb
bbSPS WA
mAA )()(
)()()( xx
x
xxx
xxxxx dWS )()()(
xxxxx
xx
dWAAS )()()(
)()(
b
bbbSPS Wm )()()()( xxxxx
N
bbbm
1
)()( xxx
"Monte-Carlo" sampling
b
bb
bb Wm )(
)(
)()( xx
x
xvxv
b
bbb Wm )()()()( xxxvxvx
)()()()()]()([ xxxxxx SPSPSP BABABA
)()(0, xx AAhN SP
Different ways of writing SP estimates(we omit the SP subscript from now on):
b
bbb Wm )()()(
1)( xxxv
xxv
Derivatives
b
bb
bb WA
mA )()( xxx
b
bb
bb WA
mA )()( xxx
No need for finite differences and grids:
211 ii
i
AAA
i-1 i+1i
bab
b
bba Wm
vv
b
abab
bba Wm
vv)(
vvv )(
b
abaabbaa Wm )()( vvv
Exact Galilean invariance
More than one way of calculating derivatives:
AAA )(
b
baaabba WAAmA )(][)( xx
)( vdt
d
b
abaabba Wmdt
d)( vv
)]()([ ttWm bb
aba xx
b
abababa Wmdt
d)( xx
aa
dt
dv
x
Moving the Particles
baba
ba
bb
a
WP
mP
2
PPP
b
abaa
a
b
bb
a
WPP
mP
22
Euler's Equation
a
a P
dt
d
v
b
abaa
a
b
bb
a WPP
mdt
d22
v
Exact momentum conservation
Entropy
bbbb
bb
b
bb
Wsm
Wm
)(
)()(
xx
xxx
)()()( xxx s
0dt
dsa
),( aaa sPP
Energy
a
a P
dt
de
v
),( aaa ePP
babaabb
a
a
a
WmPP
)(2
vvv
PPPv
vv
babaab
b
bb
a
WP
mP
)(2
vvv
bababa
b
b
a
ab
a WPP
mdt
de)(
2
122
vv
2
)( vvv PPP
a
aa P
dt
evd
)()( 2
21 v
baba
b
bb
a
aab
aa WPP
mdt
evd22
221 )( vv
Alternatively:
SPH Equations
babaab
b
b
a
ab
a WPP
mdt
dev
222
1
baba
b
b
a
ab
a WPP
mdt
d22
v
aa
dt
dv
x
b
abba Wm ),( aaa ePP
Smoothing Kernels
q
qqq
hhW
2;0
21;4/)2(
10;4/32/31
),( 3
32
x
1,
7
10,
3
2hq /x
dimensions
)/exp(1
),( 222/
hh
hW xx
Gaussian:
Spline:
Shock Waves
shock wave
x
numerical calculation
Artificial Viscosity
babaab
b
b
a
ab
a WPP
mdt
d22
v
babaabab
b
b
a
ab
a WPP
mdt
dev
222
1
ndissipatioab
0;0
0;2
abab
ababab
ababab
ab
c
rv
rv
22
ab
ababab
h
r
rv• Galilean invariant.• Vanishes for rotations.• Conserves linear and angular momentum.
Local Resolution Length
aa
dt
dv
x
babaab
b
b
a
ab
a WPP
mdt
d ~22
v
babaabab
b
b
a
ab
a WPP
mdt
de ~
2
122
v
babaabb
a
aa Wmh
dt
dh ~v
)],(),([2
1~babaabab hWhWW rr
dt
dh
dt
hd a
a
aa
babaabb
a Wmdt
d ~v
SPH Simulation of the Hubble Volume
Mass density in a thin slice (100x100x20Mpc/h)at the present epoch. This is a view one would
observe if the speed of light were infinite.
SPH Simulation of Galaxy Formation
Density of gas and dark matter in a group of galaxies.
Gas Dark Matter
SPH Simulation of Supernova Explosion
Herant et al., Ap.J. 435, 339 (1994)
75 ms after bounce
x (km)
z (k
m)
SPH Simulation of Supernova Explosion
Herant et al., Ap.J. 435, 339 (1994)
SPH Simulation of Stellar Collision
Disruption of a main sequence star by a close encounter with a high velocity neutron star.
Colors represent log (density).
SPH Simulation of Colliding Asteroids
An 8 m radius rock strikes the 1.6 km long asteroid Castalia at 5 km/s. Red is totally
fractured rock, blue is intermediate fractured rock, and white particles represent the impactor.
SPH Simulation of Projectile Impact on Sand
Particles Temperature
Projectile: Aluminum cylinder 30x10 cm (2d). Initial velocity: 3 km/sec
Relativistic Hydrodynamics
PguuPT )(
)1,1,1,1(
),(
frame)(rest density baryon
frame)(rest density energy
pressure
g
u
n
P
v
unn
0 T
0 n
Energy-momentumconservation
Baryon-numberconservation
Continuity equation:
n
vdt
d
Entropy equation:
0)(0
uTuv
vdt
d )(
0dt
dsns /
entropy density (rest frame)
0)(
Tuug v
th
B
emne
TsnPew
0/
/
u
d
d
Pn
uwd
d
1
Relativistic Euler equation:
w = enthalpy per baryon
Momentum equation:
Pn
wdt
d
1)( v
t
P
nw
dt
d
1)(
Energy equation:
)(1
vPndt
dE
n
PwE
202
10 vv meE
Relativistic SPH
b
abba W)(x
aa particle ofnumber baryon
vdt
d
b
abaabba W
dt
dv
aa
dt
dv
x
b
abab
bb
a
aab
a WPP
dt
dE22
vv
baba
b
b
a
ab
a WPP
dt
d22
p
Pn
wdt
d
1)( v
aaaa w vp
)(1
vPndt
dE
aa
aaaa n
PwE
Energy (per baryon number)
Momentum (per baryon number)
Particle Velocity
),(,,, aaaaaaa enPE vp ?
nenPeenw /),(),(
)1(),( 222 enww pvp
1|| 2
pEen
PwE
)1()]1(||,/[ 2222 pp Ew
v,),(,,, enPwen
RSPH Equations
b
abab
bb
a
aab
a WPP
dt
dE22
vv
baba
b
b
a
ab
a WPP
dt
d22
p
aa
dt
dv
x
b
abba W
)1(])1(||,/[ 2222 aaaaaaaa Ew pp
Baryon-Free System
TPB 0
dT
TdPT
)()(
0 T
vdt
d
PTdt
d
1)( v
(Rest frame)
(Lab frame)
entropy density:
b
abab
b
a
ab
a WPP
sdt
d22
p
aa
dt
dv
x
b
abba Ws
1)/( 2 aaaa Tp
Baryon-Free RSPH
Ultrarelativistic Pion Gas
42
30TP
32
15
2T
PPT 3
27766.02
)3(152
n
3
1
d
dPcs
- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2
x
0
0.2
0.4
0.6
0.8
1
1.2
entr
opy
dens
ity
exactSPH
N /L = 80h = 0.1dt = 0.05
R arefaction w aveP = (15/1282)1/3 4 /3
Ultrarelativistic Pions
Rarefaction Wave
SHASTA et al.
0 2 4 6 8 10 12
x
0
0.1
0.2
0.3
0.4
0.5
entr
opy
dens
ity
exactSPH
Landau-Kalatn ikov so lutionP = (15/128 4 /3
N /L = 400h = 0.05dt = 0.02
Ultrarelativistic Pions
Landau Solution
Artificial Viscosity
• E.Chow and J.Monaghan, Journal of Computational Physics 134, 296 (1997)
See also:• S.Siegler and H.Riffert, astro-ph/9904070
babaab
b
bb
a
aab
a WPP
dt
dEG
vv22
babaab
b
b
a
ab
a WPP
dt
d22
p
Shock Tube
-40 -20 0 20 40
x
0
0.4
0.8
1.2
bary
on d
ensi
ty
N /L = 8h = 1dt = 0.48t = 48
= 1.4m 0 = 1
Ideal nucleon gas
SHASTA et al.
Shock Tube
Ideal nucleon gas
-40 -20 0 20 40
x
0
2
4
6
8
10
12
bary
on d
ensi
ty
N /L = 8h = 1dt = 0.36t = 36
= 5/3m 0 = 1
SHASTA et al.
