brookhaven science associates u.s. department of energy 1 axisymmetric mhd simulation of pellet...

Brookhaven Science AssociatesU.S. Department of Energy 1

Axisymmetric MHD Simulation of Pellet Ablation

Roman SamulyakComputational Science Center

Brookhaven National Laboratory

CPPG Seminar,Princeton Plasma Physics LaboratoryJuly 26, 2006, Princeton, NJ

Tianshi LuCSC/BNL, modeling, software development, fusion applications

Paul ParksGeneral Atomics, MHD theory, fusion applications

Hydrodynamic and MHD Multiphase Flows:James Glimm, Stony Brook University / BNL, modeling, numerical algorithmsJian Du, Stony Brook University, software development, accelerator applicationsZhiliang Xu, Xiaolin Li, CSC/SBU, front tracking methods


Pellet Ablation in the Process of Tokamak Fueling

Detailed studies of the pellet ablation physics (local models)

ITER schematic

Studies of the tokamak plasma in the presence of an ablating pellet(global models)


Previous Studies: Local Models

• Transonic Flow (TF) (or Neutral gas shielding) model, P. Parks & R. Turnbull, 1978• Provides the scaling of the ablation rate with the pellet radius and the plasma temperature and density• 1D steady state hydrodynamics model, monoenergetic electron distribution• Neglected effects: MHD, geometric effects, atomic effects (dissociation, ionization)

• 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• Non-spherical ablation flow (axial symmetry), proper treatment of scattering• Kinetic calculation of the electron heat deposition, atomic physics processes• MHD effects not considered


• Simulations using MH3D code, H. Strauss & W. Park, 1998• Finite element version of the MH3D full MHD code• Details of the ablation are not considered• Pellet is given as a density perturbation of initial conditions• Smaller values of density and larger pellet radius (numerical constraints)

• Simulations using MHD code based on CHOMBO AMR package, R. Samtaney, S. Jardin, P. Colella, D. Martin, 2004

• Analytical model for the pellet ablation: moving density source• 8-wave upwinding unsplit method for MHD• AMR package – significant improvement of numerical resolution

Previous Studies: Global Models (examples)


Improved model is needed:

• Studies of the local pellet ablation physics were still missing • MHD• 3D effects• Charging models for the ablation cloud

• Global plasma simulations in the presence of an ablating pellet need a better local model as input

• Studies of striation instabilities, observed in all experiments, are not possible without a 3D detailed physics model

• We are working on building and validations of such models


Pellet Ablation Model Based on Front Tracking

• Explicitly tracked interfaces: resolution of material properties and multiple scales

• MHD equations in the low magnetic Reynolds number approximation and numerical methods for free surface flows

• Equation of state with atomic processes

• Kinetic model for the interaction of hot electrons with the ablated gas

• Surface ablation model


2

2

1

1

4

( , ), 0

t

Pt c

e Pt

c

t

P P e

u

u u J B

u u q J

Bu B B

B

ext ext

1

1,

1with ( )

( , ), 0

c

c

c

x t

J u B

u B

u B nn

B B B

MHD equations and approximations

Full system of MHD equations Low magnetic Re approximation

Assumptions that will be verified later.

Near the pellet

* * *0 1m

BR r v

B

02

1/ 2

P

B

Downstream in the ablation channel


Simplification of the elliptic equation for the pellet problem

Axial symmetry:

Pellet cloud charging, polarization currents, and the axial rotation of the ablation channel are neglected in this study (implementation of the charging and rotation of the channel is in progress and will be discussed later):

0

0u

Magnetic field is constant: B = (0,0,Bz)

Therefore,

and the only non-zero component of the current is zJ u B

,const

Brookhaven Science AssociatesU.S. Department of Energy

Hyperbolic step

nijF

1/ 2,ni j

1/ 21/ 2,

ni j

1nijF

FronTier-MHD numerical scheme

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)


Stencil and equations for the interface point propagate


Schematic of the interface point propagate algorithm


Embedded Boundary Elliptic Solver

Main Ideas

• Based on the finite volume discretization

• Domain boundary is embedded in the rectangular Cartesian grid, and the solution is treated as a cell-centered quantity

• The discretized operator is centered in centroids of partial cells

• Using finite difference for full cell and linear interpolation for cut cell flux calculation


Validation of the Elliptical Problem

2D problem

3D problem

f

gn

Let be a solution of 2 2 21 2 3exp k x k y k z

Derive the R.H.S. and B.C., solve numerically and compare with the exact solution.

Mesh size Error Conv. Rate CPU time Iterations

64x64 9.09e-05 N/A 0.087 44

128x128 2.01e-05 2.175 0.389 98

256x256 4.80e-06 2.122 2.223 264

512x512 1.78e-06 1.893 15.445 500

Mesh size Error Conv. Rate Iterations

32x32x32 1.32e-03 N/A 42

64x64x64 3.18e-04 2.050 76

128x128x128 8.05e-05 2.016 144


Validation of the MHD Code

A free mercury jet travels longitudinally along the z axis in a magnetic field (0,By,0)

A satisfactory perturbation theory and experimental data are available (S. Oshima et al., JCME Int. J., 30 (1987), No. 261.

Solid line: theoryDots: simulations


Muon Collider target: a brief summary of modeling and simulation

Jet surface instabilities

Cavitation in the mercury jet and thimble

Simulation of the mercury jet target interacting with a proton pulse in a magnetic field

• Studies of surface instabilities, jet breakup, and cavitation • MHD forces reduce both jet expansion, instabilities, and cavitation


Saha equation for the dissociation (ionization) fraction

2/

2/

1

1

d

d

i

i

e Tdd

d t

Tii

i t

f TN e

f n

f TN e

f n

2 /t g a in n n n m

2

+

Let's define:

is the total number density of nuclei

is the number density gas molecules D

is the number density atoms D

is the number density of ions D

t

g

a

i

n

n

n

n Dissociation (ionization) fractions:

/ , /d a i t i i tf n n n f n n

Equation of State with Atomic Processes.

d

For deuterium:

e = 4.48 eV

= 13.6 eVie24

21

1.55 10 , 0.327

3.0 10 , 3 / 2

d d

i i

N

N


1 1

2 2

1 1

2( 1) 1 2

d i

d d i d id i

m

kTP f f

m

f f f ke kekTE f f

m m m

EOS with Atomic Processes

Incomplete EOS(known from literature):

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.

Using the second law of thermodynamics T dS dE PdV

we found the complete EOS and showed that the compatibility with the second law of thermodynamics requires:

1 33 ,

1 2d im


Complete EOS with Atomic Processes

11

2 2

1

2

d dd

d

i ii

i

f f

f

f f

f

Notations:

dd d

ii i

e

Te

T

1

2d

d i

fm

a f f

We will define the sound speed in a form typical for the polytropic gas:2 *c pV

2

*

2 2

( ) 11

1 31 2

d d i i

d i

d d i im

m am a

m am a

2 21 3

1 2

d d i i

d d i im

m a

m a

where the effective gamma is the Gruneisen coefficient is

and the entropy is ln 11ln ln1 ln 1

2( 1) 2 2 2dd

d i i im

fS T Vf f f

R


For better numerical efficiency, FronTier operates with three pairs of independent thermodynamic variables:

Numerical Algorithms for EOS

, , , , ,E P T

• For the first two pairs of variables, solve numerically nonlinear algebraic equation, and find T. Using , find the remaining state.

• Such an approach is prohibitively slow for the calculation of Riemann integrals (involves nested nonlinear equations).

• To speedup calculations, we precompute and store values on Riemann integrals as functions of the density and entropy. Two dimensional table lookup and bi-linear interpolation are used.

,T


Influence of Atomic Processes on Temperature and Conductivity

3

3/ 2 0.059

9 3/ 2

1/ 2

9.675 10,

ln [ ] 0.54 [ ] 1/ 1

3.6 10where

e e i

e

e

mho

m T eV T eV f

T

n


Redlich-Kwong EOS for the cold and dense gas

Real gas EOS (work in progress)

1/ 2

2 5.2

where0.42748

0.08664

m m m

crit

crit

crit

crit

RT aP

V b T V V b

R Ta

P

RTb

P

• We have derived an extension of the Redlich-Kwong EOS to include atomic processes (dissociation and ionization)• The EOS contains three terms; the partial pressure/energy of the molecular gas is written in the Redlich-Kwong form, and the partial pressure/energies of the dissociated and ionized components is written in the ideal EOS form.• Complete EOS has been derived (expressions for entropy, sound speed etc.)• The numerical implementation is in progress


Electron Energy Deposition

( , ') '( , )

z n r z dzu r z

( , ') '

( , )z

n r z dzu r z

2 / 2q q uK u

1

( , )( ) ( )

( ) / 4

q n r zq g u g u

g u uK u

In the cloud: On the pellet surface:

2

48 ln ( , , )e

e i

T

e n T f


Physics Models for Pellet Studies :Surface Ablation

Features of the pellet ablation: • 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

p p u u qu c c u c

t n t n z


Main Simulation Parameters

Pellet radius rp 2 mm

Pellet density 0.2 g/cm3

Plasma electron temperature Te 2 keV

Plasma electron density ne

1014 cm-3(standard)1.6x1013 cm-3(electrostatic shielding)

Length of the ablation channel Lc 15 cm

Warm-up time 5 – 20 microseconds

Magnetic field B 2 – 6 Tesla


Three types of simulation

1D hydrodynamic model: spherically symmetric, no JxB force

2D hydrodynamic model: axially symmetric, directional heating along magnetic field lines, no JxB force

2D MHD model: axially symmetric, directional heating along magnetic field lines, JxB force is applied


We will compare results with

TF model: P. Parks and R. Turnbull, Phys. Fluids, 21 (1978), 1735.

Kuteev: B. V. Kuteev, Sov. J. Plasma Phys, 11 (1985), 236.

MacAulay: A. K. MacAulay, Nucl. Fusion, 34 (1994), 43.

Parks 2000: P. Parks, W. Sessions, L. Baylor, Phys. Plasmas., 5 (2000), 1968

Ishizaki: R. Ishizaki, P. Parks, N. Nakajiama, M. Okamoto, Phys. Plasmas, 11 (2004), 4064


Spherically symmetric 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

4/3

51.8898 for 7 / 5

pG r

M


Axially Symmetric Hydrodynamic Simulation

Temperature, eV Pressure, bar Mach number

Distributions of temperature, pressure, and Mach number of the ablation flow near the pellet at 20 microseconds.


Axially Symmetric Hydrodynamic Simulation

Temperature, pressure, and Mach number of the ablation flow near the pellet in the longitudinal (solid line) and radial (dashed line) directions at 20 microseconds.


• The ablation rate, 90 g/s, agrees with Kuteev and MacAulay, disagrees with Ishizaki

• The reduction of the ablation compared to the spherically symmetric case is ~ 18%.

• This disagrees with prevailing expectations of the factor of 2 reduction (Kuteev and Ishizaki)

• However Kuteev did not calculate the 1D ablation rate: he compared with the FT model that used the monoenergetic electron distribution, and found a 2.2 reduction

• Our explanation of the factor of 2.2 reduction:• Maxwellian heat flux increases the 1D ablation rate by 2.75 (Ishizaki) • Directional heat flux in 2D reduces the ablation rate by 0.82 (this work)• 2.75 x 0.82 = 2.25

Reduction of the ablation rate in 2D


Velocity distribution of the ablation flow near the pellet in 6 Tesla magnetic field. Warm up time is 20 microseconds.

1 microsecond 2 microseconds 3 microseconds

Axlally symmetric MHD simulation


Mach number distribution of the ablation flow near the pellet in 6 Tesla magnetic field. Warm up time is 20 microreconds.

3 microseconds

5 microseconds

9 microseconds


Temperature (eV) distribution of the ablation flow near the pellet in 6 Tesla magnetic field. Warm up time is 20 microseconds.

3 microseconds

5 microseconds

9 microseconds


Pressure (bar) distribution of the ablation flow near the pellet in 6 Tesla magnetic field. Warm up time is 20 microseconds.

3 microseconds

5 microseconds

5 microseconds


Pressure along the z-axis of steady state ablation channel.

No shielding Electrostatic shielding

Solid line: 2 Tesla, dashed line: 4 Tesla, dotted line: 6 Tesla. Warm up time is 10 microseconds.


Temperature along the z-axis of steady state ablation channel.




Mach number along the z-axis of steady state ablation channel




Normalized temperature and pressure in the ablation channel

Solid lines: Simulation in 2 Tesla field with electrostatic shielding.Dashed lines: Theoretical parallel flow model for the ablation channel (Parks 2000)


Radius of the ablation channel

Solid line: 10 microseconds warm up time, ne = 1.0e14 cm-1

Dashed line: 10 microseconds warm up time, ne = 1.6e13 cm-1

Dotted line: 5 microseconds warm up time, ne = 10e14 cm-1


Density along the axis of symmetry and the ablation rate

Solid line: tw = 10, ne = 1.0e14 cm-1

Dashed line: tw = 10, ne = 1.6e13 cm-1

Dotted line: tw = 5, ne = 10e14 cm-1

Solid line: MHD model, B = 6 Tesla, ne = 1.0e14 cm-1

Dashed line: MHD model, B = 2 Tesla, ne = 1.6e13 cm-1

Dotted line: 1D spherically symmetric model


Justification of the Electrostatic Approximation

B

ne

2 Tesla 4 Tesla 6 Tesla

1014cm-3 0.530 0.822 0.128 0.223 0.051 0.100

1.6x1013cm-3 0.110 0.180 0.029 0.055 0.015 0.026

near near nearfar far far

The ratio of induced magnetic field to the toroidal magnetic field.

0far 2

1

2

P

B

ind 0far ind 3

0

( ),

4

B r j rB dr

B r


Current work: implementation of the pellet charging model

2 ||

2z2

Jhotz

Bˆ z [(u)] z

||z

||z J||,sheath()

(zˆ n ) uB ˆ n 0


Other Low Magnetic Reynolds Number Flows in Tokamaks

• Laser driven pellet acceleration• Gyrotron driven pellet acceleration• Liquid jet for plasma disruption mitigation• High density, low temperature gas jets for tokamak fueling (inefficient method?)

Laser driven pellet accelerationFueling using a high speed gaseous jet


Conclusions and future work

• Developed MHD code for free surface low magnetic Re number flows• Front tracking method multiphase/multimaterial flows• Elliptic problems in geometrically complex domains• Phase transition models

• Performed numerical simulation of tokamak fueling through the injection of frozen deuterium - tritium pellets

• Computed ablation rates in hydro and MHD case• Explained the factor of 2 reduction of the ablation rate• Performed first systematic studies of the ablation in magnetic fields

• Future work• 3D simulations of the pellet ablation• Studies of striation instabilities• Coupling our pellet ablation model as a subgrid model with a tokamak plasma simulation code• Laser -- plasma interaction model with sharp absorption front, laser acceleration of pellets (possible)

brookhaven science associates u.s. department of energy 1 axisymmetric mhd simulation of pellet...

Documents