Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
TWO-DIMENSIONAL SIMULATIONS OF COHERENT FLUCTUATION-DRIVE TRANSPORT
IN A HALL THRUSER
Cheryl M. Lam and Mark A. CappelliStanford Plasma Physics Laboratory
Stanford University, Mechanical Engineering Department
Eduardo FernandezEckerd College, Department of Mathematics and Physics
33rd International Electric Propulsion Conference
Washington, DC
October 6-10, 2013
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Hall Thruster
Electric (space) propulsion device Demonstrated high thrust efficiencies
Up to 60% (depending on operating power)
Deployed production technology Design Improvements Better physics understanding
Basic Premise:
Accelerate heavy (positive) ions through electric potential to create thrust E x B azimuthal Hall current
Radial B field (r) Axial E field (z)
Ionization zone (high electron density region)
Electrons “trapped” Neutral propellant (e.g., Xe) ionized
via collisions with electrons Plasma
Ions accelerated across imposed axial potential (Ez / Φz) & ejected from thruster
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Motivation
Anomalous electron transport Super-classical electron mobility observed in experiments1
Theory: Correlated fluctuations in ne and uez induce super-classical electron transport
Renewed interest in rotating spoke (near anode)
1Meezan, N. B., Hargus, W.A., Jr., and Cappelli, M. A., Physical Review, Vol. 63, No. 2, 026410, 2001.
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
“Low” Frequency Mode (~700 kHz)
z = - 6.2 cm z = - 3.2 cm
z = + 0.5 cm
cathode-directedaxial wave 13o tilt
weak tilted -35o
wave
cathode-directed+15o tilted wave
weak cathode-directedaxial wave
anode-directedaxial wave with symmetric azimuthal spread
E x B
Cathode
E x B
Cathode
E x B
Cathode
E x B
Cathode
E x B
Cathode
E x B
Cathode
z = - 6.2 cm z = - 3.2 cm
z = + 0.5 cm
cathode-directedaxial wave 13o tilt
weak tilted -35o
wave
cathode-directed+15o tilted wave
weak cathode-directedaxial wave
anode-directedaxial wave with symmetric azimuthal spread
E x B
Cathode
E x B
Cathode
E x B
Cathode
E x B
Cathode
E x B
Cathode
E x B
Cathode
*A. Knoll, Ph.D. Thesis, Stanford University, 2010
Moderate Motivation
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Anode Exit Plane
G
extends 4 cm past channel exitz: 40 points, non-uniform
θ: 50 points, uniform
Anode Cathode
Channel Diameter = 9 cm
Channel Length = 8 cm
First fully-resolved 2D z-θ simulations of entire thruster2
Predict azimuthal (E x B) fluctuations
Hybrid Fluid-PIC Ions: Non-magnetized particles Neutrals: Particles (Injected at
anode; Local ionization rate) Electrons: 2D Fluid
Continuity (Species & Current)
2D Momentum: Drift-Diffusion 1D Energy (in z)
2D (z-θ) Simulation
eeee nunt
n
)( 0
Jt
0
ni = ne Quasineutrality
2Lam, C. M., Knoll, A. K., Cappelli, M. A., and Fernandez E., IEPC-2009-102.
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Fluid Equations
Momentum: Drift-Diffusion Neglect inertial terms
Correlated azimuthal fluctuations induce axial transport:
ue E Dner
ne
1
1 en
ce
2
Ez
Br 1
1 en
ce
2kTe
eneBr
ne
z 1
1 en
ce
2k
eBr
Te
z
)1( 2
2
en
ceenm
e
Classical Mobility
e
kTD e
uez Ez Dne
ne
z D
Te
Te
z 1
1 en
ce
2EBr
1
1 en
ce
2kTe
eneBrrne
Previous modelsunder-predict
Jez=qneuezθ fluctuations/dynamics
eeinducede unJ~~
,
classical E x B diamagnetic
Classical Diffusion
classical E x B diamagnetic
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Fluid Equations
Combine current continuity and electron - momentum to get convection-diffusion equation for Φ:
Energy (Temperature) Equation 1D in z
A1
2 2 A2
A3
2z2 A4
z
A5 0, where
wallionizjouleeeeeeee
e SSSqukTnTut
Tkn
)(23
E
where (φ is electric potential)
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Solution Algorithm
Iterative Solve Φ
Time Advance Particle Positions & VelocitiesNeutrals & Ions (subject to F=qE)
Ionize Neutrals
Inject Neutrals
Calculate Plasma Propertiesni-PART, vi-PART, nn-PART, vn-PART ni-GRID, vi-GRID, nn-GRID, vn-GRID
QUASINEUTRALITY: ne = ni = nplamsa
Time Advance Te=Te(ne, ve)
Calculate Φ=Φ(ne, vi-GRID) ↔ EGRID
Calculate ve=ve(Φ, ne, Te)
r = Φ – Φlast-iterationr < ε0
CONVERGED
Calculate vi-GRID-TEST= vi-GRID(EGRID)
EGRID EPART
LEAPFROG
RK4
DIRECT SOLVE 2nd-order F-D
Spline
Boundary Conditions:
• Dirichlet in z (Φ,Te)
• Periodic in θ
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Recent Progress & Challenges
Addition of particle collisions with thruster walls Neutral particles reflected upon collision with anode or inner/outer
radial channel walls Ions recombine (with donor electron) to form neutral upon collision with
inner/outer radial channel walls Particles still otherwise collisionless, i.e., we do not model particle-
particle collisions
Finer axial (z) grid resolution near anode
Stability challenges Sensitivity to Initial Conditions and Boundary Conditions Strong fluctuation in Te
Current conservation Finite Difference – present model Finite Volume
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Numerical Grid
40 points non-uniform in z50 points uniform in θ
Previous 100V (IEPC 2009)160V simulation (new)
61 points uniform in z25 points uniform in θ
100V simulation (new)
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Simulation Parameters
Initial Conditions
Neutrals: neutral only run to establish profile
Ions: uniform # particles per cell w/ Maxwellian velocity distribution
Te: based on experiment
Boundary ConditionsTe (z = 0) = 3.2 eV
Te (z = 0.12 m) = 3.0 eV
Operating Voltage 100V (160V)
Neutral Injection 2 mg/s (Xe propellant)
Timestep
Run Length
dt = 1 ns
~187 μs
Computational Performance
~7 days on Intel Xeon 5355 2.66 GHz (64-bit single core)
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Plasma Density
Time-Averaged Plasma PropertiesElectron Temperature
Axial Ion Velocity Electric Potential
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Runaway Ionization
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Temperature
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Axial Ion Velocity
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Fluctuations
Distinct wave behavior observed:
Near exit plane (as before) Tilted: + z, - ExB Higher frequency, faster moving,
shorter wavelength Transition to standing wave
(purely +z) downstream of exit plane (z = 0.1 m)
Mid-channel
Tilted: - z, + E x B Lower frequency, slow moving,
longer wavelength “More tilted” (stronger/faster θ
component) – compared to previous
Near anode Rotating spoke m = 2 (100V)
E x B
Axial Electron Velocity
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Wave Propagation
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Rotating Spoke
Near anode (z ≤ 0.01 m)
Primarily azimuthal m = 2 vph = ~ 1 km/s f = 10-20 kHz
Anode Cathode
E x B
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Correlated ne and uez fluctuations generate axial electron current
Correlated fluctuations generate axial current
Uncorrelated
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Discharge current is low and decreases with timeExperiment: ~2 A (for 100V)
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Discharge current is low and decreases with timeExperiment: ~2 A (for 100V)
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Transport
Axial Electron Mobility:ze
ez
Eqn
J
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Transport
Preliminary Simulation:
Spoke does not lead to anomalous transport
Axial Electron Mobility:ze
ez
Eqn
J
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
160V SimulationRotating Spoke (m = 1)
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
160V SimulationElectron Transport
Spoke does not lead to anomalous transport
Axial Electron Mobility:ze
ez
Eqn
J
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Summary
Rotating spoke observed First simulations to predict spoke: important to resolve full azimuth Model: added particle wall collisions (neutral reflection, ion
recombination) Consistent with theory and experimental observations Preliminary simulations: Spoke generates current, but does NOT lead to
anomalous transport.
Remaining challlenges Low voltage (100V) case: plasma cooling/quenching? Stability: Te instability, ICs, BCs Current conservation Finite Volume discretization
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Questions?
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Back-up and Throw Away
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Rotating Spoke
Near anode (z ≤ 0.01 m)
Primarily azimuthal m = 2 vph = ~ 1 km/s f = 10-20 kHz
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Motivation
Develop predictive lifetime/erosion in Hall thrusters
Thruster Life/Erosion Simulations*
Computed erosion behavior over 2500 hours:
Ion density in the Hall thruster simulation domain
Plasma properties are evolved over the life of the thruster
Erosion rate on the inner wall
Erosion rate on the outer wall
r - z
*E. Sommier, M. K. Scharfe, N. Gascon, M. A. Cappelli, and E. Fernandez, IEEEITransactions on Plasma Science, 35 (5), October 2007, pp. 1379-1387.
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Azimuthal Fluctuations induce Axial Transport
Consider
Induced Current
r
ce
en
ez B
Eu
xBE
2
1
1
xBExBE ezeez uqnJ
cos2
1
)cos(
1
1)cos(
200
0
020
T
v
En
B
qJ
dttB
EtnqJ
ce
enr
eez
T
tr
ce
en
eez
xBE
xBE
Induced current depends on phase shift ξ
t
ξ
Eθ = E0cos(ωt)
ne = n0cos(ωt + ξ)
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Motivation
Primary Design Concern: Thruster Lifetime Wall (ceramic insulator) erosion Typical Lifetime: ~1000 hours (mpropellant ≈ msystem)
Predictive Modeling & Simulation for Design Optimization
Design Objective: Keep (fast) ions from hitting walls Thruster geometry & operating voltage: fixed Design parameter: B field (shape & strength)
Imposed B-field ↔ Ez
Underlying plasma physics Electron transport
Plasma density & E field fluctuations Ionization (via collisions) Plasma-surface interactions
(e.g., sputtering, electron damping, recombination at walls)
Certain physical phenomena observed in experiment not well understood
Numerical experiments
Research focus:
Azimuthal (θ) dynamics Axial (z) electron transport
Erosion rate on the inner wall
Erosion rate on the outer wall
** Movie courtesy of E. Sommier
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Motivation
Hall thruster anomalous electron transport Super-classical electron mobility observed in experiments1
Correlated (azimuthal) fluctuations in ne and ue
2D r-z models use tuned mobility to account for azimuthal effects2,3
3D model is computationally expensive
First fully-resolved 2D z-θ simulations of entire thruster
** Initial development by E. Fernandez
Predict azimuthal (ExB) fluctuations
Inform r-z model
Motivate 3D model
Channel Diameter = 9 cm
Channel Length = 8 cm
1Meezan, N. B., Hargus, W.A., Jr., and Cappelli, M. A., Physical Review, Vol. 63, No. 2, 026410, 2001. 2Fife, J. M., Ph.D. Dissertation, Massachusetts Inst. of Technology, Cambridge, MA, 1999. 3Fernandez et al, “2D simulations of Hall thrusters,” CTR Annual Research Briefs, Stanford Univ.,1998.
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Hybrid Fluid-PIC Model
Ions: Collisionless particles (Particle-In-Cell approach) Non-magnetized Wall collisions not modeled
Neutrals: Collisionless particles (Particle-In-Cell approach) Injected at anode per mass flow rate
Half-Maxwellian velocity distribution based on r-z simulation (w/ wall effects)
Ionized per local ionization rate Based on fits to experimentally-measured collision cross-sections,
assuming Maxwellian distribution for electrons
Electrons: Fluid Continuity (species & current) Momentum
Drift-diffusion equation Inertial terms neglected
Energy (1D in z) Convective & diffusive fluxes Joule heating, Ionization losses, Effective wall loss
Quasineutrality:ni = ne
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
2D in z-θ No radial dynamics
E x B + θ
Br: purely radial
(measured from SHT) Imposed operating
(based on operating condition)
Geometry
Anode Exit Plane
extends 4 cm past channel exitz: 40 points, non-uniform
θ: 50 points, uniform
Channel Diameter = 9 cm
Channel Length = 8 cm
Anode Cathode
G
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Particle-In-Cell (PIC) Approach Particles: arbitrary positions
Force Particle acceleration
Interpolate: Grid Particle Plasma properties evaluated at grid points
(Coupled to electron fluid solution) Interpolate: Particle Grid
Bilinear Interpolation
Ions subject to electric force:
PIC Ions & Neutrals
rNW
rSE
rNE
rSW
FNW
FSE
FNE
FSW
Interpolation:Particle Grid
Interpolation:Grid Particle
BuqEqamFLorentz
≈ 0
neglect
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Fluid Equations
Species Continuity
Current Continuity
eeee nunt
n
)(
0
Jt
0
ni = ne
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Fluid Equations
Momentum: Drift-Diffusion Neglect inertial terms
ue E Dner
ne
1
1 en
ce
2
Ez
Br 1
1 en
ce
2kTe
eneBr
ne
z 1
1 en
ce
2k
eBr
Te
z
uez Ez Dne
ne
z D
Te
Te
z 1
1 en
ce
2
EBr
1
1 en
ce
2
kTe
eneBrrne
)1( 2
2
en
ceenm
e
Classical Mobility
e
kTD e
Previous modelsunder-predict
Jez=qneuez
θ fluctuations/dynamics
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Electron Fluid Equations
Momentum: Drift-Diffusion Neglect inertial terms
Correlated azimuthal fluctuations
induce axial transport:eeinducede unJ
~~
,
Previous modelsunder-predict
Jez=qneuez
θ fluctuations/dynamics
uez Ez Dne
ne
z D
Te
Te
z 1
1 en
ce
2
EBr
1
1 en
ce
2
kTe
eneBrrne
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Unlike fully PIC codes, the electric potential is not obtained from a Poisson equation:
A1 ne
r2, A3 ne ,
A1
2 2 A2
A3
2z2 A4
z
A5 0, where
A2 1r
( ne
r
rne
1
1 en
ce
2
z
ne
Br
ne
Br
z
1
1 en
ce
2 )
A4 1
1 en
ce
2
1rBr
ne
ne
z
ne
z
ne
rBr
1
1 en
ce
2
A5 f (ne ,Te ,, en ,ce ) ne
rui
ui
rne
ne
uiz
z uiz
ne
z
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Fluctuations in θ
Anode Cathode
E x B
E x B
E x B
f = 40 KHzλθ = 5 cmvph = 4000 m/s
f = 700 KHz
λθ = 4 cmvph = 40,000 m/s
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Streak Plots
E x B
E x B
Introduction
Model Description
Results
Summary
2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN TRANSPORT IN A HALL THRUSTER
Stanford UniversityPlasma Physics Lab
Future Work
Numerical Stability Alternative solution algorithms Timestep and grid refinement
Governing physics Enhanced electron mobility Wall model Potential BC
Power supply circuit model Recombination Magnetized ions
Model validation against experiments
Top Related