topics telecon - dec 18

Dissertation Outline

I. Introduction

II. Background

III. Model Description

IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline

2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN ELECTRON TRANSPORT IN A HALL THRUSTER

Stanford UniversityPlasma Physics Lab

TopicsTelecon - Dec 18

Review Cheryl’s dissertation proposal slides: ignore “Results” section for now (will replace this with more organized results set, primarily based on IEPC 09 run) Need help with slide 6 – relevant Hall thruster simulations Additional comments on slides 5 and 7?

Wall loss model – review notes sent by Eduardo

Upwind discretization – turned “on/off” in the code?

Research plan: slides 45-52 – seems ambitious, need help with prioritization/goals

IEEE TPS paper edits – Thanks, Eduardo!


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



AXIAL-AZIMUTHAL HYBRID FLUID-PIC SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN ELECTRON TRANSPORT IN A HALL

THRUSTER

Cheryl M. Lam

Advisor: Mark A. Cappelli

Stanford Plasma Physics Laboratory

Mechanical Engineering Department

Dissertation Proposal Meeting

December 20, 2013


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




I. Introduction

II. Hall Thruster Simulations (Background)

III. Model Description: Hybrid Fluid-PIC z-θ Model

IV. Model Sensitivities

V. Simulation Results

VI. Discussion

VII. Conclusions and Future Work


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Motivation

Hall thruster anomalous electron transport Super-classical electron mobility observed in experiments1

Theory: Correlated (azimuthal) fluctuations in ne and uez induce super-classical electron transport

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

Quantify impact on electron transport

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.


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Hall Thruster Simulations

2D radial-axial (r-z) simulations Fife Michelle Eunsun? – ongoing

2D axial-azimuthal (z-θ) Aaron Mine French (Garrigues, Bouchoule, Adam, et al.) – not full azimuth Italian (Taccogna, Cappitelli?)

3D Cite recent work – fully-kinetic (PIC) computationally expensive:

smaller geometry (smaller-scale thruster) and/or shorter runs

Review IEPC 2013 papers


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Relevance of Hybrid z-θ Simulations

Thruster geometry Scale (thruster size) Resolve full azimuth

No artificial introduction of periodicity

Time scales of interest Hybrid approach enables longer (~100s μs) simulations Low to mid-frequency waves (~10 kHz – 100 MHz?? More like ~MHz

because inertialess?)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



2D in z-θ No radial dynamics

E x B + θ

Br: purely radial

(measured from SHT laboratory discharge)

Imposed operating voltage

(based on operating condition)

Geometry

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

Anode Exit Plane

G


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Hybrid Fluid-PIC Model

Ions: Particle-In-Cell approach Non-magnetized No particle-particle collisions; Wall collisions modeled in some cases

Neutrals: Collisionless particles (Particle-In-Cell approach) Injected at anode per mass flow rate No particle-particle collisions; Wall collisions modeled in some cases Ionized per local ionization rate

Based on fits to experimentally-measured collision cross-sections, assuming Maxwellian distribution for electrons

Electrons: Fluid continuum 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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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 field:

PIC Ions & Neutrals

rNW

rSE

rNE

rSW

FNW

FSE

FNE

FSW

Interpolation:Particle Grid

Interpolation:Grid Particle

BuqEqamFLorentz

≈ 0

neglect


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Ionization rate Nu_en nedot

Neutral injection Injection velocity: slow vs “normal” (thermalized) Wall collisions

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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Electron Fluid Equations

Species Continuity

Current Continuity

eeee nunt

n

)(

0

Jt

0

ni = ne


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




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

)1( 2

2

en

ceenm

e

Classical Mobility

e

kTD e

uez Ez Dne

ne

z D

Te

Te

z 1

1 en

ce

2

EBr

1

1 en

ce

2

kTe

eneBrrne

Classical Diffusion


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline





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

2

EBr

1

1 en

ce

2

kTe

eneBrrne

θ fluctuations/dynamics

classical E x B diamagnetic

Classical Diffusion



I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




Combine current continuity and electron momentum to get convection-diffusion equation for Φ:

A1

2 2 A2

A3

2z2 A4

z A5 0, where

E

where (φ is electric potential)

A1 ne

r2, A3 ne ,

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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




Energy (Temperature) Equation 1D in z

wallionizjouleeeeeeee

e SSSqukTnTut

Tkn

)(23

eeneejoule unmS

eeieiioniz kTnEnS2

3

ewallwallwall kTL

S2

312

where

with ionization cost factor αi = 1 (simplest model)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Model Sensitivities

Grid – current non-conservation

ICs and BCs

Numerical stability/sensitivity of energy (Te) equation

Source/Sink terms Ionization cost factor

– Constant factor

– Dugan model Energy loss to wall


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Sample Results

IEPC 2009 runs

Build upon these

More recent – with inclusion of wall collisions?? IEPC2013 – spoke, do not understand Higher voltage – do not understand


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Plasma Density

Time-Averaged Plasma PropertiesElectron Temperature

Axial Ion Velocity Electric Potential


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Electron Temperature


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Runaway Ionization


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Axial Ion Velocity


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Wave Propagation


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Correlated ne and uez fluctuations generate axial electron current

Correlated fluctuations generate axial current

Uncorrelated


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Current non-conservationIEPC13 100V run


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Electron Transport

Axial Electron Mobility:ze

ez

Eqn

J


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Electron Transport

Preliminary Simulation:

Spoke does not lead to anomalous transport


ez

Eqn

J


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Rotating Spoke – IEPC13 100V case

Near anode (z ≤ 0.01 m)

Primarily azimuthal m = 2 vph = ~ 1 km/s f = 10-20 kHz


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Fluctuations in θ – IEPC09?? Or 13?

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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Streak Plots – IEPC09?? Or 13?

E x B

E x B


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



160V SimulationRotating Spoke (m = 1)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



160V SimulationElectron Transport

Spoke does not lead to anomalous transport


ez

Eqn

J


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline





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 Diffusion



I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Correlated ne and uez fluctuations generate axial electron current

Correlated fluctuations generate axial current

Uncorrelated


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Fluctuations

Compare to experiments

Linearized dispersion relations?

Future results: include dispersion analysis/maps


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Anomalous current Contributions (from various terms) to electron current Axial variation Relate to shear, other gradients?

Electron transport / anomalous current – trends with operating conditions (e.g., increased voltage)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Anomalous electron transport

Suggestions for future work FV? Fully kinetic simulations


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



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


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Additional Simulations – 100V

Establish stable long-running simulation (~600 μs – 1 ms) for low voltage (100V) case Start from (continue) IEPC 2009 simulation

Ionization cost factor = 1 No wall collisions; Slow neutral injection velocity Zero-slope BC for Te

Increase number of particles (ionizspc) to enable longer simulation

Grid refinement study Finer grid in z: current non-conservation, structure near anode Finer, varied grid in theta: impact periodicity, azimuthal wavelength?

Initial Conditions Increased neutral density spoke at anode? Shape of neutral density profile (peaked/sloping, by how much) More realistic plasma (electron/ion) density profile and/or magnitude


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Additional Simulations – Higher Voltage

Higher voltage runs Incrementally increase operating voltage Look for trends (in frequency/wavelength/direction of fluctuations,

electron transport, anomalous contribution to transport)

Initial Conditions – Waves Smooth initial profiles (based on prescribed profile or experiment) –

allow fluctuations to evolve (as before) If needed, of interest, “seed” with particular waves/modes


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Development Process

Establish stable low voltage (100 V run)

Increase operating voltage – with all other conditions same as for 100V case Adjustments as needed to establish stable higher voltage run

Changes to 100V case – model improvements, additional physics, etc. Establish stable 100V run Increase operating voltage – with all other conditions same as for 100V

case Adjustments as needed to establish stable higher voltage run

REPEAT


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Model Improvements

Te stability and BC/IC impacts Stability and sensitivity analysis – contribution of source/sink terms,

esp. wall loss and ionization cost Enforcement of (IC/return to) experimental profile and/or experimental-

based limits Prescribed (fixed value) vs. zero-slope condition at domain boundaries Ionization cost factor

Dugan model Tuned constant factor?

Improved/tunable wall loss model Introduction of diffusive damping term? Effect of spline smoothing Implicit solve?

Improve stability – consider more global changes to model “External” power supply circuit model (potential BC) Hyperviscous damping (for potential equation)


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Model Improvements

Incremental changes (additions) to model – additional physics Introduction of wall collisions (w/ higher thermal injection velocity) Revisit ionization rate implementation?

Electron transport/mobility – sustain/generate waves (also help stability) Additive “baseline” mu_perp or nu_en Experimental mobility (in lieu of or in addition to mu_perp) Experimental or additional (or Bohm-like) mobility for electron fluid

equations only


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Analysis

Plasma properties (time-averaged and movies) compared to experiment – axial profiles – give sense of overall simulation performance (how well to we simulate/approximate experimental) Plasma density, axial ion velocity, electron temperature, etc.

Discharge current Axial profile Time evolution

Axial electron mobility – compare to experiment (axial profile)

Characterization of waves By “eye” – velocity, frequency, wavelength, direction Dispersion map/analysis Changes in direction, “Break up” of wave structure Correspond to any known/expected waves?? Experiment or theory?

Anomalous contribution to current Anomalous vs. classical terms Relate to shear, ne/Br gradient, etc.


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Publications

Conference Papers IEPC 2009 GEC 2012? IEPC 2013

Journal Papers IEEE TPS Special Ed. – Submitted

Planned additional publications Journal paper: waves and transport


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Questions?


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Back-up and Throw Away

topics telecon - dec 18

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