anisotropic heat conduction and frc translation and...
TRANSCRIPT
Testing and Benchmarking SELAnisotropic Heat Conduction and FRC Translation
and Neutral Fluid Effects
E.T. Meier, V.S. Lukin, R. Milroy, U. Shumlak
PSI-Center Annual MeetingJune 24, 2008
Resources used:• PSI-Center SGI Altix 350 cluster• NERSC IBM p575 POWER 5 system, BassiThis research is funded by DOE.
• Anisotropic heat conduction• FRC translation problem• Visco-resistive MHD model 1
+ Atomic physics 2+ Spitzer / Chodura variable resistivity 3+ Two-fluid effects
• SEL-NIMROD results comparison– MHD with uniform resistivity
• Comparison of SEL results with 1 , 2 , and 3 • Future work
Outline
Anisotropic heat conduction
Anisotropic heat conduction goals.
• Assess the ability of high-order finite elements to accurately model anisotropy with attention to:– Grid alignment– Polynomial degree– Number of cells
• Perfect alignment results in zero error– The effects of “real” perpendicular heat conduction
are measured and compared to numerical error.
2D heat equation with anisotropy is solved.
Interior equation:
( ) 0T T St
∂+∇• •∇ = =
∂D
where ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2// //
2 2//
cos sin sin cos. sin cos
D D D Dsymm D D
⊥ ⊥
⊥
Φ + Φ − Φ Φ= Φ + Φ
D .
Boundary equation:
( )ˆ 0T• •∇ =n D (insulator)
Initial condition is Gaussian temperature profile.
Constant temperature contours of the Gaussian profile are parallel to the principle direction of anisotropic heat conduction.
Φ
Assess error by measuring change in peak T.
Results are converted to effective transverse conductivity values.
Slump with non-zero transverse conductivity
y = 285.65x0.9631
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00
transverse conductivity
T0-T
max
DOF per unit length = 24
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
0 10 20 30 40 50 60 70
phi
effe
ctiv
e D
_per
p
np=2np=3np=4np=5np=6
Accuracy improves with grid alignment and with increasing polynomial degree.
As phi 45 degrees, spectral representation of the Gaussian profile is improved.
L1L2=L1*sqrt(2)
phi=45
phi=0
Alignment gives monotonic improvement if an “effective” DOF per unit length is considered.
effective DOF per unit length = 24
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
0 10 20 30 40 50 60 70phi
effe
ctiv
e D
_per
p
np=2np=3np=4np=5np=6
Grid size reduction also improves accuracy.
phi = 30 deg.
1.0E-08
1.0E-07
1.0E-06
1.0E-05
1.0E-04
1.0E-03
0.06 0.11 0.16 0.21 0.26 0.31 0.36
dx
effe
ctiv
e D
_per
p
np=2np=3np=4np=5np=6
DOF/L = 24
DOF/L = 30
DOF/L = 18
FRC translation problem
Initial condition is generated by a separate equilibrium solver.
• FRC translation is a valuable code development problem.– Various boundary conditions are necessary.– High gradients and velocities test code capability.
Normalization
0ψψ ψ= 4
0 3.9 10 Wbψ −= ×
FRC is compressed and translated by injecting flux with a series of mirror coils.
CL
Four mirror field coils are fired sequentially.
t = 0 µsec t = 10 µsec t = 18 µsec
Magnetic Flux (ψ0 = 3.9e-4 Wb)
extrema=(-5.5e-01,5.8e-02) extrema=(-3.1e+00,4.7e-02) extrema=(-3.1e+00,4.0e-02)
(Visco-resistive MHD result shown)
Visco-resistive MHD model
Visco-resistive MHD model
( ) 2Dtρ ρ ρ∂+∇ = ∇
∂vi
( )r jt φψ η φ∂ = − × ∂
v B i
( )
( )
// //
2
11 1
:
p p T Tt
p jφ
γ κ κ κγ γ
η µ
⊥ ⊥
∂+∇ − − ∇ + ∇ − ∂ −
= ∇ + + ∇ ∇ + ∇
v
v v v v
i
i T
( ) ( )2
02Bp
tρ
ρ µ ∂ +∇ + + − − ∇ + ∇ = ∂
vvv I BB v vi T
( ) ˆjφ φ= ∇×B i where
1 1ˆ ˆ, r zr z r r
ψ ψ∂ ∂= −
∂ ∂B , ˆ ˆ, r zv r v z=v ,
and axisymmetry is assumed ( 0φ∇ → ).
Density diffusion is needed when resolution is
insufficient.
This model only allows for toroidal current.
( )rAφψ = −
+ atomic physics
Atomic physics model, I.C., and B.C.
( ) .2
.ion recombDtρ ρ ρ +∂+∇ = ∇ Γ −
∂Γvi
2. .
nn ion recombD
tρ ρ∂
= ∇ −Γ +Γ∂
( )r jt φψ η φ∂ = − × ∂
v B i
( ) . . .1 ... ...
11
1ion i ion recomb ip m mt
pγ ργ
−Γ Φ − Γ
∂+∇ =
− ∂ − i
( ) ( ) .... recomb itm
ρ−
∂+∇
∂Γ= v
vi
( ) ˆjφ φ= ∇×B i
where .ionΦ is the specific ionization potential.
. .ion ion e i nn nσΓ = ⟨ ⟩v , 2. .recomb recomb e inσΓ = ⟨ ⟩v 1/ 213
, 3 -1
, ,
2 10 13.6exp m s6.0 /13.6 13.6
e eVion e
e eV e eV
TT T
σ− ×
⟨ ⟩ = − + v
1/ 2
19 3 -1
,
13.60.7 10 m srec ee eVT
σ − ⟨ ⟩ = ×
v .
Ionization and recombination rates for hydrogen*
*Ref. R. J. Goldston, P. H. Rutherford, Introduction to Plasma Physics, IOP Publishing Ltd., 1995
I.C. for neutral density is ( ),max0.1n iρ ρ= .
B.C. for neutral density is . .n
ion recombtρ∂
= −Γ +Γ∂
.
(B.C. for other MHD variables are unchanged.)
Neutral gas is ionized and absorbed by FRC during translation.
Magnetic Flux (ψ0 = 3.9e-4 Wb)
Neutral Density (n0 = 2.5e-7 kg/m3)
t = 0 µsec
t = 9 µsec t = 14 µsec
Some aspects of the atomic physics model have been verified.
Verify accuracy of energy terms• Static 0-D problem with no
magnetics.• Add an energy source term, e.g. P60
• Set temperature to 55 eV.• Model correctly predicts 60 eV final
temperature.
Power loss vs. Tev
0.000E+00
5.000E-05
1.000E-04
1.500E-04
2.000E-04
0.0 20.0 40.0 60.0 80.0 100.0 120.0
T, eV
pow
er, W
P60=1.5e-4 W
Analytical power loss vs. Tev(Γion.=Γrecomb.)
Verify temporal convergence• Static 0-D problem with no magnetics.
– Γion. ≠ Γrecomb.
– energy loss terms = zero.• At large dt/τionization, initial overshoot is seen but steady-state solution is
correct.
+ Spitzer / Chodura variable resistivity
Chodura resistivity was developed to represent “anomalous” resistivity.
• ,2 1 exp eep iChodura C
s
m vCne fv
η ω
= − −
where CC and f are empirical constants: 0.1CC ≅ ; 3f ≅ .
• Choduraη is significant when the ratio of electron drift speed to ion sound
speed e
s
vv
is order unity or greater.
• The anomalous resistivity is attributable to LHDI*. • Choduraη and Spitzerη are added to yield a combined Spitzer / Chodura
resistivity model.
* Ref. A. Hakim, U. Shumlak, Two-Fluid Physics and Field-ReversedConfigurations, Vol. 14, Phys. Plasmas, 055911, 2007
Chodura resistivity is high in quasi-vacuum region, allowing magnetic diffusion.
ηSpitzer (η0 = 6.3e-3 Ω-m)
ηChodura*t = 0 µsec t = 8 µsec t = 14 µsec
* The Chodura constant, Cc, was set to Cc=0.01. With Cc=0.1 (the empirically determined value), magnetic dissipation was excessive for this particular FRC.
+ Two-fluid effects.
Two-fluid MHD model
( ) 2i D
tρ ρ ρ∂+∇ = ∇
∂vi
( )2 ˆie ez e
dr r v z ptψφ φ η ν
ρ ∂ = = − × + ∇ −∇ ∂
E j v Bi i
t∂
= −∇×∂B E
( ) ( ) ( )2
ˆ 02
ii i i i ez
Bp v ztρ
ρ µ ν ∂ +∇ + + − − ∇ + ∇ − ∇ = ∂
vv v I BB v vi T
e i iv v d jφ φ φρ ρ= −
( )
// //
2 2
11 1
:
i
i i i i ez
p p T Tt
p v
γ κ κγ γ
η µ ν
⊥ ⊥
∂+∇ − ∇ − ∇ − ∂ −
= ∇ + + ∇ ∇ + ∇ + ∇
v
v j v v v
i
i T
where
0i
pi
cdLω
= , ˆ ˆ, r zv r v z=v ,
and axisymmetry is assumed ( 0φ∇ → ).
Two-fluid decaying equilibrium
Two-fluid FRC translation
SEL-NIMROD comparison(MHD with uniform resistivity)
SEL-NIMROD simulations have been run with identical I.C. and parameters.
SEL shows 2.0% lower axial speed at t = 20 µsec.
SEL shows 4.8% less trapped flux at t = 20 µsec.
“Enclosed” refers to the region within the FRC separatrix.
Vz_avg. (enclosed) vs. time
0.00E+00
2.00E+04
4.00E+04
6.00E+04
8.00E+04
1.00E+05
1.20E+05
1.40E+05
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
Vz, m
/s
NIMRODSEL
Trapped flux vs. time
1.00E-04
1.10E-04
1.20E-04
1.30E-04
1.40E-04
1.50E-04
1.60E-04
1.70E-04
1.80E-04
1.90E-04
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
flux,
Wb
SEL-NIMROD comparison continued…
SEL shows 8.8% less retained particles at t = 20 µsec.
SEL shows 2.3% less KE at t = 20 µsec.
Number of particles enclosed vs. time
4.00E+17
4.50E+17
5.00E+17
5.50E+17
6.00E+17
6.50E+17
7.00E+17
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
Nto
t
NIMRODSEL
KE (enclosed) vs. time
0.00E+00
2.00E+00
4.00E+00
6.00E+00
8.00E+00
1.00E+01
1.20E+01
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-
time, sec.
KE,
joul
es
Comparison of SEL results with• Visco-resistive MHD
+ Atomic physics+ Spitzer / Chodura variable resistivity(using Cc=0.01)
SEL multi-fluid physics models produce physically consistent results.
Vz_avg. (enclosed) vs. time
0.00E+00
2.00E+04
4.00E+04
6.00E+04
8.00E+04
1.00E+05
1.20E+05
1.40E+05
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
Vz, m
/s
VR-MHD (uniform eta)with atomic physicswith Spitzer / Chodura
Trapped flux vs. time
6.00E-05
7.00E-05
8.00E-05
9.00E-05
1.00E-04
1.10E-04
1.20E-04
1.30E-04
1.40E-04
1.50E-04
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
flux,
Wb
SEL multi-fluid physics results continued…
Number of particles enclosed vs. time
0.00E+00
1.00E+17
2.00E+17
3.00E+17
4.00E+17
5.00E+17
6.00E+17
7.00E+17
8.00E+17
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
Nto
t
VR-MHD (uniform eta)with atomic physicswith Spitzer / Chodura
KE (enclosed) vs. time
0.00E+00
2.00E+00
4.00E+00
6.00E+00
8.00E+00
1.00E+01
1.20E+01
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
KE,
joul
es
Future Work
• Develop two-fluid MHD modeling via 2D FRC translation simulation.
• Develop and implement more sophisticated boundary conditions.– “open” boundary outflow.– Self-consistent magnetic boundary conditions.– Implement in MHD and for multi-fluid MHD.
• Extend multi-fluid modeling and boundary conditions to 3D.
SEL multi-fluid physics…extra slides…
Vol. averaged beta vs. time
0
1
2
3
4
5
6
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
Bet
a with Spitzer / Chodura
SEL multi-fluid physics…extra slides…
Vol. avg. resistivity (enclosed) vs. time
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
1.20E-02
1.40E-02
0.00E+00 5.00E-06 1.00E-05 1.50E-05 2.00E-05
time, sec.
eta with Spitzer / Chodura