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Brookhaven Science AssociatesU.S. Department of Energy 1
General Atomics July 14, 2009
Multiphase MHD at Low MagneticReynolds Numbers
Tianshi Lu
Department of MathematicsWichita State University
In collaboration with
Roman Samulyak, Stony Brook University / Brookhaven National Laboratory
Paul Parks, General Atomics
Brookhaven Science AssociatesU.S. Department of Energy 2
Tokamak (ITER) Fueling
• Fuel pellet ablation• Striation instabilities• Killer pellet / gas ball for
plasma disruption mitigation
Laser ablated plasma plume expansion
Expansion of a mercury jet in magnetic fields
MotivationMotivation
Brookhaven Science AssociatesU.S. Department of Energy 3
• Equations for MHD at low magnetic Reynolds numbers and models for pellet ablation in a tokamak
• Numerical algorithms for multiphase low ReM MHD
• Numerical simulations of pellet ablation
Talk OutlineTalk Outline
Brookhaven Science AssociatesU.S. Department of Energy 4
Equations for MHD at low magnetic Reynolds numbers
Full system of MHD equations Low ReM approximation
0
)(
)(
),( ),,(
1)(
)(
0)(
ext0
ext2
B
JJB
BE
BuEJ
qJuu
BJuuu
u
t
TppTee
pet
e
pt
t
0)(
,ext
BuJ
EBB
Maxwell’s equations without wave propagation
Ohm’s law
Equation of state for plasma / liquid metal/ partially ionized gas
)()( 02
0 BuBBB
t
01Re 0M
t
uLB
Elliptic
Parabolic
Brookhaven Science AssociatesU.S. Department of Energy 5
Models for pellet ablation in tokamak
• Full MHD system• Implicit or semi-implicit discretization• EOS for fully ionized plasma• No interface• System size ~ m, grid size ~ cm
Tokamak plasma in the presence of an ablating pellet
Pellet ablation in ambient plasma
Global Model Local Model
• MHD system at low ReM
• Explicit discretization• EOS for partially ionized gas• Free surface flow• System size ~ cm, grid size ~ 0.1 mm
Courtesy of Ravi Samtaney, PPPL
Brookhaven Science AssociatesU.S. Department of Energy 6
Schematic of pellet ablation in a magnetic field
Schematic of processes in the ablation cloud
i||
ep||
ep||
ec||
ep||
ep||
eth||
Cloud Plasma
Sheath boundary
(z)
Sheath Fluxes
Brookhaven Science AssociatesU.S. Department of Energy 7
Local model for pellet ablation in tokamak
1.Axisymmetric MHD with low ReM approximation
2.Transient radial current approximation
3.Interaction of hot electrons with ablated gas
4.Equation of state with atomic processes
5.Conductivity model including ionization by electron impact
6.Surface ablation model
7.Pellet penetration through plasma pedestal
8.Finite shielding length due to the curvature of B field
Brookhaven Science AssociatesU.S. Department of Energy 8
1. Axisymmetric MHD with low ReM approximation
z
qJJJpe
t
er
uuBJ
z
uu
r
uu
t
uz
p
z
uu
r
uu
t
ur
uBJ
r
p
z
uu
r
uu
t
u
t
hzr
rrzr
zz
zr
z
rz
rr
r
2
||
22
2
1)(
1)(
)(
0)(
)(
0)(
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u
0)]()[ˆˆ()]()[ˆˆ(0]]ˆ[[
0][)]([0)(
||||||
||
sheci
hh
eez
Bur
Jzz
Bur
rrr
nznrnJ
JJ
Centripetal force
Nonlinear mixedDirichlet-Neumann boundary condition
Brookhaven Science AssociatesU.S. Department of Energy 9
2. Transient radial current approximation
2
),(),( ,
)()( ,38.1
2)
221(
),(),(
)exp( ),erfc()1( ,
)1( :conditionty Ambipolari
shsh
432
sh||||shsh||||||||
||||||
zruzrw
Tk
rer
m
m
dtt
ewwwwe
zrzr
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e
w
twh
eepeecei
epheci
0)( ,0 ||||||||
shecish
h eez
Jz
BuEJ rr 0
r,z) depends explicitly on the line-by-line cloud opacity u.
Simplified equations for non-transient radial current has been implemented.
Brookhaven Science AssociatesU.S. Department of Energy 10
3. Interaction of hot electrons with ablated gas
In the cloud On the pellet surface
)5.7
516.2ln()1(
)1035.1
ln(ln10
ei
i
ei
Tf
nf
Tf
)(|| eph hh
)(2|| uKuTkq epeBh
)]()()[,(2
/
)(
11||
2||
uKuuKuzrnTk
zq
uKuTkq
epeB
epeBh
z dzzrn
zru
')',(),(
z
dzzrnzru
')',(
),(
ln8 4
2
e
Te
Brookhaven Science AssociatesU.S. Department of Energy 11
Saha equation for the dissociation and ionization
4. Equation of state with atomic processes (1)
m
ef
m
ef
m
Tkff
fE
m
Tkff
fp
ii
dd
Bid
m
d
Bid
d
2
1)
2
3
2
3
)1(2
1(
)2
1(
2
3
1
13
i
md
DeuteriumEd=4.48eV, Nd=1.55×1024,d=0.327Ei=13.6eV, Ni=3.0×1021,i=1.5
)exp(1
)exp(1
2
2
Tk
e
n
TN
f
f
Tk
e
n
TN
f
f
B
i
ti
i
i
B
d
td
d
d
i
d
mnnnn
nnfnnnf
iagt
tiitiad
/2
/ ,/)(
Dissociation and ionization fractions
Brookhaven Science AssociatesU.S. Department of Energy 12
High resolution solvers (based on the Riemann problem) require the sound speed and integrals of Riemann invariant type expressions along isentropes. Therefore the complete EOS is needed.
• Conversions between thermodynamic variables are based on the solution of nonlinear Saha equations of (,T).
• To speedup solving Riemann problem, Riemann integrals pre-computed as functions of pressure along isentropes are stored in a 2D look-up table, and bi-linear interpolation is used.
• Coupling with Redlich-Kwong EOS can improve accuracy at low temperatures.
4. Equation of state with atomic processes (2)
)1ln(2
)1ln()1()1(
22
ln
)1(2
lni
d
B
iii
B
dd
d
m
ff
Tk
ef
Tk
efVT
R
S
Brookhaven Science AssociatesU.S. Department of Energy 13
5. Conductivity model including ionization by impact
geeg
eae
eaeea
dec
D
i
eBe
ei
eieaege
e
eieaege
e
nT
TnT
TnT
rd
d
nTkm
e
m
en
m
en
74.07
059.07
39129.07
min
max
3
4
2
||
2
1074285.1
eV45937.0 ,1057.1
eV45937.0 ,1022856.2
),max(lnlnln
)(3
ln24
51.0
1
1
)(
)(
)(2
1
cloudrecomb
hotighothote
hoteitei
idta
hote
dtg
T
Enn
nfnnn
ffnn
nf
nn
Ionization by Impact
HHeH
eeHeH hh
2
22
eHeH
eeHeH hh
2
Brookhaven Science AssociatesU.S. Department of Energy 14
Influence of Atomic Processes on Temperature and Conductivity
Temperature Conductivity
Brookhaven Science AssociatesU.S. Department of Energy 15
6. Surface ablation model
Some facts: • The pellet is effectively shielded from incoming electrons by its ablation cloud• Processes in the ablation cloud define the ablation rate, not details of the phase
transition on the pellet surface• No need to couple to acoustic waves in the solid/liquid pellet• The pellet surface is in the super-critical state • As a result, there is not even well defined phase boundary, vapor pressure etc.
This justifies the use of a simplified model: • Mass flux is given by the energy balance (incoming electron flux) at constant
temperature• Pressure on the surface is defined through the connection to interior states by
the Riemann wave curve • Density is found from the EOS.
Brookhaven Science AssociatesU.S. Department of Energy 16
7. Pellet penetration through plasma pedestal
locitypellet ve
widthpedestal timeup-warm
Brookhaven Science AssociatesU.S. Department of Energy 17
8. Finite shielding length due to the curvature of B field
The grad-B drift curves the ablation channel away from the central pellet shadow. To mimic this 3D effect, we limit the extent of the ablation flow to a certain axial distance.
Without MHD effect, the cloud expansion is three-dimensional. The ablation rate reaches a finite value in the steady state.
With MHD effect, the cloud expansion is one-dimensional. The ablation rate would goes to zero by the ever increasing shielding if a finite shielding length were not in introduced.
cmcmmRRL chtorsh 15~12~
Brookhaven Science AssociatesU.S. Department of Energy 18
• Equations for MHD at low magnetic Reynolds numbers and models for pellet ablation in a tokamak
Numerical algorithms for multiphase low ReM MHD
• Numerical simulations of the pellet ablation in a tokamak
Talk OutlineTalk Outline
Brookhaven Science AssociatesU.S. Department of Energy 19
Multiphase MHD
Solving MHD equations (a coupled hyperbolic – elliptic system) in geometrically complex, evolving domains subject to interface boundary conditions (which may include phase transition equations)
Material interfaces:• Discontinuity of density and physics properties (electrical conductivity) • Governed by the Riemann problem for MHD equations or phase transition equations
Brookhaven Science AssociatesU.S. Department of Energy 20
Front Tracking: A hybrid of Eulerian and Lagrangian methods
Two separate grids to describe the solution:1. A volume filling rectangular mesh2. An unstructured codimension-1
Lagrangian mesh to represent interface
Major components:1. Front propagation and redistribution2. Wave (smooth region) solution
Main ideas of front tracking
Advantages of explicit interface tracking:• No numerical interfacial diffusion• Real physics models for interface propagation• Different physics / numerical approximations
in domains separated by interfaces
Brookhaven Science AssociatesU.S. Department of Energy 21
Level-set vs. front tracking method
5th order level set (WENO)
4th order front
tracking (Runge-Kutta)
Explicit tracking of interfaces preserves geometry and topology more accurately.
Brookhaven Science AssociatesU.S. Department of Energy 22
FronTier is a parallel 3D multi-physics code based on front tracking Physics models include
Compressible fluid dynamics MHD Flow in porous media Elasto-plastic deformations
Realistic EOS models Exact and approximate Riemann solvers Phase transition models Adaptive mesh refinement
Interface untangling by the grid based method
The FronTier code
Brookhaven Science AssociatesU.S. Department of Energy 23
Targets for future accelerators
Supernova explosion
Tokamak refuelling through the ablation
of frozen D2 pellets
Liquid jet break-up and atomization
Main FronTier applications
Richtmyer-Meshkov instability
Rayleigh-Taylor instability
Brookhaven Science AssociatesU.S. Department of Energy 24
Hyperbolic step
nijF
1/ 2,ni j
1/ 21/ 2,
ni j
1nijF
Elliptic step
1/ 2nijF
• Propagate interface• Untangle interface• Update interface states
• Apply hyperbolic solvers• Update interior hydro states
• Generate finite element grid• Perform mixed finite element discretizationor• Perform finite volume discretization• Solve linear system using fast Poisson solvers
• Calculate electromagnetic fields • Update front and interior states
Point Shift (top) or Embedded Boundary (bottom)
FronTier – MHD numerical scheme
Brookhaven Science AssociatesU.S. Department of Energy 25
Hyperbolic step
• Complex interfaces with topological changes in 2D and 3D• High resolution hyperbolic solvers
• Riemann problem with Lorentz force• Ablation surface propagation• EOS for partially ionized gas and conductivity model• Hot electron heat deposition and Joule’s heating• Lorentz force and saturation numerical scheme• Centripetal force and evolution of rotational velocity
Interior and interface states for front tracking
Brookhaven Science AssociatesU.S. Department of Energy 26
• Based on the finite volume discretization
• Domain boundary is embedded in the rectangular Cartesian grid.
• The solution is always treated as a cell-centered quantity.
• Using finite difference for full cell and linear interpolation for cut cell flux calculation
• 2nd order accuracy
Elliptic step
Embedded boundary elliptic solver
For axisymmetric pellet ablation with transient radial current, the elliptic step can be skipped.
Brookhaven Science AssociatesU.S. Department of Energy 27
High Performance Computing
Software developed for parallel distributed memory supercomputers and clusters
• Efficient parallelization• Scalability to thousands of processors• Code portability (used on Bluegene Supercomputers and various clusters)
Bluegene/L Supercomputer (IBM)
at Brookhaven National Laboratory Chip
(2 processors)
Com pute Card(2 ch ips, 2x1x1)
Node Board(32 ch ips, 4x4x2)
16 Com pute C ards
System(64 cabinets, 64x32x32)
Cabinet(32 Node boards, 8x8x16)
2.8/5.6 G F/s4 M B
5.6/11.2 G F/s0.5 G B DDR
90/180 G F/s8 G B DDR
2.9/5.7 TF/s256 G B DDR
180/360 TF/s16 TB D DR
Brookhaven Science AssociatesU.S. Department of Energy 28
• Equations for MHD at low magnetic Reynolds numbers and models for pellet ablation in a tokamak
• Numerical algorithms for multiphase low ReM MHD
Numerical simulations of the pellet ablation in a tokamak
Talk OutlineTalk Outline
Brookhaven Science AssociatesU.S. Department of Energy 29
Previous studies
• Transonic Flow (TF) (or Neutral Gas Shielding) model, P. Parks & R. Turnbull, 1978•Scaling of the ablation rate with the pellet radius and the plasma temperature
and density•1D steady state spherical hydrodynamics model•Neglected effects: Maxwellian hot electron distribution, geometric effects, atomic
effects (dissociation, ionization), MHD, cloud charging and rotation•Claimed to be in good agreement with experiments
• Theoretical model by B. Kuteev et al., 1985•Maxwellian electron distribution•An attempt to account for the magnetic field induced heating asymmetry
• Theoretical studies of MHD effects, P. Parks et al.
• P2D code, A. K. MacAulay, 1994; CAP code R. Ishizaki, P. Parks, 2004•Maxwellian hot electron distribution, axisymmetric ablation flow, atomic
processes•MHD effects not considered
Brookhaven Science AssociatesU.S. Department of Energy 30
1. Spherical model• Excellent agreement with TF model and Ishizaki
2. Axisymmetric pure hydro model• Double transonic structure• Geometric effect found to be minor
3. Plasma shielding• Subsonic ablation flow everywhere in the channel• Extended plasma shield reduces the ablation rate
4. Plasma shielding with cloud charging and rotation• Supersonic rotation widens ablation channel and increases ablation rate
Our simulation results
Spherical model Axis. hydro model Plasma shielding
Brookhaven Science AssociatesU.S. Department of Energy 31
1. Spherically symmetric hydrodynamic simulation
Normalized ablation gas profiles at 10 microseconds
Polytropic EOS Plasma EOS
Poly EOS Plasma EOS
Sonic radius 0.66 cm 0.45 cmTemperature 5.51 eV 1.07 eVPressure 20.0 bar 26.9 barAblation rate 112 g/s 106 g/s
• Excellent agreement with TF model and Ishizaki.• Verified scaling laws of the TF model
5
7for8898.1
5
~ 3/4
M
rG p
Brookhaven Science AssociatesU.S. Department of Energy 32
2. Axially symmetric hydrodynamic simulation
Temperature, eV Pressure, bar Mach number
Steady-state ablation flow
Brookhaven Science AssociatesU.S. Department of Energy 33
Velocity distribution Channeling along magnetic field lines occurs at ~1.5 μs
3. Axially symmetric MHD simulation (1)
Plasma electron temperature Te 2 keV
Plasma electron density ne
1014 cm-3(standard)1.6x1013 cm-3(el. shielding)
Warm-up time tw 5 – 20 microseconds
Magnetic field B 2 – 6 Tesla
Main simulationparameters:
st 1 st 2 st 3
Brookhaven Science AssociatesU.S. Department of Energy 34
3. Axially symmetric MHD simulation (2)
Mach number distribution
Double transonic flow evolves to subsonic flow
st 3
st 5
st 9
cm15
2T
20
keV2
cm10
mm2
sh
314
0
L
B
st
T
n
R
w
e
e
p
Brookhaven Science AssociatesU.S. Department of Energy 35
Dependence on pedestal properties
-.-.- tw = 5 s, ne = 1.6 1013 cm-3
___ tw = 10 s, ne = 1014 cm-3
----- tw = 10 s, ne = 1.6 1013 cm-3
Critical observationFormation of the ablation channel and ablation rate strongly depends on plasma pedestal properties and pellet velocity.
Simulations suggest that novel pellet acceleration technique (laser or gyrotron driven) are necessary for ITER.
Brookhaven Science AssociatesU.S. Department of Energy 36
Supersonic rotation of the ablation channel
4. MHD simulation with cloud charging and rotation (1)
Isosurfaces of the rotational Mach number in the pellet ablation flow
Density redistribution in the ablation channel
Steady-state pressure distribution in the widened ablation channel
2TB
Brookhaven Science AssociatesU.S. Department of Energy 37
0 50 100 150 2000
50
100
150
200
250
300
350
t, s
G,
g/s
finite spin-upinstantaneous spin-upJ
r=0 & shrinking pellet
no rotationno rotation & induction
G, g
/s
Pellet ablation rate for ITER-type parameters
4. MHD simulation with cloud charging and rotation (2)
cm15
6T
30
keV4
cm10
mm4
sh
314
0
L
B
st
T
n
R
w
e
e
p
Brookhaven Science AssociatesU.S. Department of Energy 38
Channel radius
Ablation rate
|ΔB/B|
Non-rotating 2.3 cm 195 g/s 0.079
Rotating 2.8 cm 262 g/s 0.088
Channel radius and ablation rate
4. MHD simulation with cloud charging and rotation (3)
Normalized potential along field lines
Grot is closer to the prediction of the quasisteady ablation model Gqs = 327 g/s
Magnetic β<<1 justifies the static B-field assumption
Potential in the negative layer
Brookhaven Science AssociatesU.S. Department of Energy 39
• Current work focuses on the study of striation instabilities
• Striation instabilities, observed in all experiments, are not well understood
• We believe that the key process causing striation instabilities is the supersonic channel rotation, observed in our simulations
Striation instabilities: Experimental observation
(Courtesy MIT Fusion Group)
Striation instabilities
Brookhaven Science AssociatesU.S. Department of Energy 40
Plasma disruption mitigation
Pressure distribution without rotation
Gas ballR = 9 mm
Killer pelletR = 9 mm
Brookhaven Science AssociatesU.S. Department of Energy 41
Plasma disruption mitigation
Mach number distributions in the gas shell
Brookhaven Science AssociatesU.S. Department of Energy 42
Conclusions and future work
• Developed MHD code for free surface low magnetic Re number flows• Front tracking method for multiphase flows• Elliptic problems in geometrically complex domains• Phase transition and surface ablation models
• Axisymmetric simulations of pellet ablation• Effects of geometry, atomic processes, and conductivity model• Warm-up process and finite shielding length• Charging and rotation, transient radial current• Ablation rate, channel radius, and flow properties• Tracking of a shrinking pellet
• Future work• 3D simulations of pellet ablation and striation instabilities• Asymptotic ablation properties in long warm up time• Natural cutoff shielding length• Magnetic induction• Systematic simulation of plasma disruption mitigation using killer pellet / gas ball• Coupling with global MHD models