Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics.

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



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



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



“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



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



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




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



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



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



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



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



Plasma Density

Time-Averaged Plasma PropertiesElectron Temperature

Axial Ion Velocity Electric Potential

Introduction

Model Description

Results

Summary



Runaway Ionization

Introduction

Model Description

Results

Summary



Electron Temperature

Introduction

Model Description

Results

Summary



Axial Ion Velocity

Introduction

Model Description

Results

Summary



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



Wave Propagation

Introduction

Model Description

Results

Summary



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



Correlated ne and uez fluctuations generate axial electron current

Correlated fluctuations generate axial current

Uncorrelated

Introduction

Model Description

Results

Summary



Discharge current is low and decreases with timeExperiment: ~2 A (for 100V)

Introduction

Model Description

Results

Summary



Discharge current is low and decreases with timeExperiment: ~2 A (for 100V)

Introduction

Model Description

Results

Summary



Electron Transport

Axial Electron Mobility:ze

ez

Eqn

J

Introduction

Model Description

Results

Summary



Electron Transport

Preliminary Simulation:

Spoke does not lead to anomalous transport


ez

Eqn

J

Introduction

Model Description

Results

Summary



160V SimulationRotating Spoke (m = 1)

Introduction

Model Description

Results

Summary



160V SimulationElectron Transport

Spoke does not lead to anomalous transport


ez

Eqn

J

Introduction

Model Description

Results

Summary



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



Questions?

Introduction

Model Description

Results

Summary



Back-up and Throw Away

Introduction

Model Description

Results

Summary



Introduction

Model Description

Results

Summary



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



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



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



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



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



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



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 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



Anode Cathode

G

Introduction

Model Description

Results

Summary



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




Species Continuity

Current Continuity

eeee nunt

n

)(

0

Jt

0

ni = ne

Introduction

Model Description

Results

Summary





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


Jez=qneuez

θ fluctuations/dynamics

Introduction

Model Description

Results

Summary





Correlated azimuthal fluctuations

induce axial transport:eeinducede unJ

~~

,


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



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



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



Streak Plots

E x B

E x B

Introduction

Model Description

Results

Summary



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

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics.

Documents

d z simulations

thruster slide

axial transport

electrons plasma n ions

n theory

n e quasineutrality

d momentum

d energy