Brookhaven Science Associates U.S. Department of Energy 1 General AtomicsJuly 14, 2009 Multiphase MHD at Low Magnetic Reynolds Numbers Tianshi Lu Department

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


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


• 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


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


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


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


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


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)(

uu

u

0)]()[ˆˆ()]()[ˆˆ(0]]ˆ[[

0][)]([0)(

||||||

||

sheci

hh

eez

Bur

Jzz

Bur

rrr

nznrnJ

JJ

Centripetal force

Nonlinear mixedDirichlet-Neumann boundary condition


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

eBi

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.


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


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


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


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


Influence of Atomic Processes on Temperature and Conductivity

Temperature Conductivity


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.


7. Pellet penetration through plasma pedestal

locitypellet ve

widthpedestal timeup-warm


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~



Numerical algorithms for multiphase low ReM MHD

• Numerical simulations of the pellet ablation in a tokamak



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


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


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.


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


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


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


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


• 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.


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



• Numerical algorithms for multiphase low ReM MHD

Numerical simulations of the pellet ablation in a tokamak



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


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


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


2. Axially symmetric hydrodynamic simulation

Temperature, eV Pressure, bar Mach number

Steady-state ablation flow


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


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


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.


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


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


cm15

6T

30

keV4

cm10

mm4

sh

314

0

L

B

st

T

n

R

w

e

e

p


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


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


• 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


Plasma disruption mitigation

Pressure distribution without rotation

Gas ballR = 9 mm

Killer pelletR = 9 mm


Plasma disruption mitigation

Mach number distributions in the gas shell


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

Documents

Brookhaven Science Associates U.S. Department of Energy 1 General AtomicsJuly 14, 2009 Multiphase MHD at Low Magnetic Reynolds Numbers Tianshi Lu Department