Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab TWO-DIMENSIONAL SIMULATIONS OF COHERENT FLUCTUATION-DRIVE TRANSPORT IN A HALL THRUSER Cheryl M. Lam and Mark A. Cappelli Stanford Plasma Physics Laboratory Stanford University, Mechanical Engineering Department Eduardo Fernandez Eckerd College, Department of Mathematics and Physics 33 rd 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 University Plasma Physics Lab Hall Thruster oElectric (space) propulsion device n Demonstrated high thrust efficiencies u Up to 60% (depending on operating power) n Deployed production technology u Design Improvements u Better physics understanding oBasic Premise: Accelerate heavy (positive) ions through electric potential to create thrust n E x B  azimuthal Hall current u Radial B field (r) u Axial E field (z) n Ionization zone (high electron density region) u Electrons “trapped” u Neutral propellant (e.g., Xe) ionized via collisions with electrons  Plasma n Ions accelerated across imposed axial potential (E z / Φ z ) & ejected from thruster

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Motivation oAnomalous electron transport n Super-classical electron mobility observed in experiments 1 n Theory: Correlated fluctuations in n e and u ez induce super-classical electron transport oRenewed interest in rotating spoke (near anode) 1 Meezan, N. B., Hargus, W.A., Jr., and Cappelli, M. A., Physical Review, Vol. 63, No. 2, , 2001.

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab “Low” Frequency Mode (~700 kHz) *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 University Plasma Physics Lab AnodeExit Plane G extends 4 cm past channel exit z: 40 points, non-uniform θ: 50 points, uniform Anode Cathode Channel Diameter = 9 cm Channel Length = 8 cm oFirst fully-resolved 2D z-θ simulations of entire thruster 2  Predict azimuthal (E x B) fluctuations oHybrid Fluid-PIC  Ions: Non-magnetized particles  Neutrals: Particles (Injected at anode; Local ionization rate)  Electrons: 2D Fluid u Continuity (Species & Current) u 2D Momentum: Drift-Diffusion u 1D Energy (in z) 2D (z-θ) Simulation 0 n i = n e Quasineutrality 2 Lam, C. M., Knoll, A. K., Cappelli, M. A., and Fernandez E., IEPC

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Electron Fluid Equations oMomentum: Drift-Diffusion n Neglect inertial terms  Correlated azimuthal fluctuations induce axial transport: Classical Mobility Previous models under-predict J ez =qn e u ez θ fluctuations/dynamics 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 University Plasma Physics Lab Electron Fluid Equations oCombine current continuity and electron  - momentum to get convection-diffusion equation for Φ: oEnergy (Temperature) Equation n 1D in z where (φ is electric potential)

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Solution Algorithm Iterative Solve Φ Time Advance Particle Positions & Velocities Neutrals & Ions (subject to F=qE) Ionize Neutrals Inject Neutrals Calculate Plasma Properties n i-PART, v i-PART, n n-PART, v n-PART  n i-GRID, v i-GRID, n n-GRID, v n-GRID QUASINEUTRALITY: n e = n i = n plamsa Time Advance T e =T e (n e, v e ) Calculate Φ=Φ(n e, v i-GRID ) ↔ E GRID Calculate v e =v e (Φ, n e, T e ) r = Φ – Φ last-iteration r < ε 0 CONVERGED Calculate v i-GRID-TEST = v i-GRID (E GRID ) E GRID  E PART LEAP FROG   RK4 DIRECT SOLVE  2 nd -order F-D  Spline Boundary Conditions: Dirichlet in z (Φ,T e ) Periodic in θ

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Recent Progress & Challenges oAddition of particle collisions with thruster walls n Neutral particles reflected upon collision with anode or inner/outer radial channel walls n Ions recombine (with donor electron) to form neutral upon collision with inner/outer radial channel walls n Particles still otherwise collisionless, i.e., we do not model particle- particle collisions oFiner axial (z) grid resolution near anode oStability challenges n Sensitivity to Initial Conditions and Boundary Conditions n Strong fluctuation in T e n Current conservation u Finite Difference – present model u Finite Volume

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Numerical Grid 40 points non-uniform in z 50 points uniform in θ Previous 100V (IEPC 2009) 160V simulation (new) 61 points uniform in z 25 points uniform in θ 100V simulation (new)

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Simulation Parameters Initial Conditions Neutrals: neutral only run to establish profile Ions: uniform # particles per cell w/ Maxwellian velocity distribution T e : based on experiment Boundary Conditions Te (z = 0) = 3.2 eV Te (z = 0.12 m) = 3.0 eV Operating Voltage100V (160V) Neutral Injection2 mg/s (Xe propellant) Timestep Run Length dt = 1 ns ~187 μs Computational Performance ~7 days on Intel Xeon GHz (64-bit single core)

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Plasma Density Time-Averaged Plasma Properties Electron Temperature Axial Ion VelocityElectric Potential

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

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

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

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Fluctuations Distinct wave behavior observed: oNear exit plane (as before) n Tilted: + z, - ExB n Higher frequency, faster moving, shorter wavelength n 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 oNear anode n Rotating spoke n 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 University Plasma Physics Lab Wave Propagation

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Rotating Spoke oNear anode (z ≤ 0.01 m) oPrimarily azimuthal n m = 2 n v ph = ~ 1 km/s n f = kHz AnodeCathode E x B

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Correlated n e and u ez 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 University Plasma Physics Lab Discharge current is low and decreases with time Experiment: ~2 A (for 100V)

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Discharge current is low and decreases with time Experiment: ~2 A (for 100V)

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Electron Transport Axial Electron Mobility:

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Electron Transport  Preliminary Simulation: Spoke does not lead to anomalous transport Axial Electron Mobility:

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab 160V Simulation Rotating Spoke (m = 1)

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab 160V Simulation Electron Transport  Spoke does not lead to anomalous transport Axial Electron Mobility:

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Summary oRotating spoke observed n First simulations to predict spoke: important to resolve full azimuth n Model: added particle wall collisions (neutral reflection, ion recombination) n Consistent with theory and experimental observations n Preliminary simulations: Spoke generates current, but does NOT lead to anomalous transport. oRemaining challlenges n Low voltage (100V) case: plasma cooling/quenching? n Stability: T e instability, ICs, BCs n Current conservation  Finite Volume discretization

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

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Back-up and Throw Away

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

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Rotating Spoke oNear anode (z ≤ 0.01 m) oPrimarily azimuthal n m = 2 n v ph = ~ 1 km/s n f = kHz

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

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Azimuthal Fluctuations induce Axial Transport Consider Induced Current Induced current depends on phase shift ξ t ξ E θ = E 0 cos(ωt) n e = n 0 cos(ωt + ξ)

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Motivation oPrimary Design Concern: Thruster Lifetime n Wall (ceramic insulator) erosion n Typical Lifetime: ~1000 hours (m propellant ≈ m system ) oPredictive Modeling & Simulation for Design Optimization Design Objective: Keep (fast) ions from hitting walls n Thruster geometry & operating voltage: fixed n Design parameter: B field (shape & strength) u Imposed B-field ↔ E z oUnderlying plasma physics n Electron transport u Plasma density & E field fluctuations n Ionization (via collisions) n Plasma-surface interactions (e.g., sputtering, electron damping, recombination at walls) oCertain physical phenomena observed in experiment not well understood  Numerical experiments oResearch 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 University Plasma Physics Lab Motivation oHall thruster anomalous electron transport n Super-classical electron mobility observed in experiments 1 n Correlated (azimuthal) fluctuations in n e and u e o2D r-z models use tuned mobility to account for azimuthal effects 2,3 o3D model is computationally expensive oFirst 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 1 Meezan, N. B., Hargus, W.A., Jr., and Cappelli, M. A., Physical Review, Vol. 63, No. 2, , Fife, J. M., Ph.D. Dissertation, Massachusetts Inst. of Technology, Cambridge, MA, Fernandez 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 University Plasma Physics Lab Hybrid Fluid-PIC Model oIons: Collisionless particles (Particle-In-Cell approach) n Non-magnetized n Wall collisions not modeled oNeutrals: Collisionless particles (Particle-In-Cell approach) n Injected at anode per mass flow rate u Half-Maxwellian velocity distribution based on r-z simulation (w/ wall effects) n Ionized per local ionization rate u Based on fits to experimentally-measured collision cross-sections, assuming Maxwellian distribution for electrons oElectrons: Fluid n Continuity (species & current) n Momentum u Drift-diffusion equation u Inertial terms neglected n Energy (1D in z) u Convective & diffusive fluxes u Joule heating, Ionization losses, Effective wall loss Quasineutrality: n i = n e

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab o2D in z-θ n No radial dynamics oE x B  + θ n B r : purely radial (measured from SHT) n Imposed operating (based on operating condition) Geometry AnodeExit Plane extends 4 cm past channel exit z: 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 University Plasma Physics Lab oParticle-In-Cell (PIC) Approach n Particles: arbitrary positions u Force  Particle acceleration Interpolate: Grid  Particle n Plasma properties evaluated at grid points (Coupled to electron fluid solution) u Interpolate: Particle  Grid n Bilinear Interpolation oIons subject to electric force: PIC Ions & Neutrals r NW r SE r NE r SW F NW F SE F NE F SW Interpolation: Particle  Grid Interpolation: Grid  Particle ≈ 0 neglect

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Electron Fluid Equations oSpecies Continuity oCurrent Continuity 0 n i = n e

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Electron Fluid Equations oMomentum: Drift-Diffusion n Neglect inertial terms Classical Mobility Previous models under-predict J ez =qn e u ez θ fluctuations/dynamics

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Electron Fluid Equations oMomentum: Drift-Diffusion n Neglect inertial terms oCorrelated azimuthal fluctuations induce axial transport: Previous models under-predict J ez =qn e u ez θ fluctuations/dynamics

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Unlike fully PIC codes, the electric potential is not obtained from a Poisson equation:

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Fluctuations in θ AnodeCathode E x B f = 40 KHz λ θ = 5 cm v ph = 4000 m/s f = 700 KHz λ θ = 4 cm v ph = 40,000 m/s

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Streak Plots E x B

Introduction Model Description Results Summary 2D SIMULATIONS OF COHERENT FLUCTUATION- DRIVEN TRANSPORT IN A HALL THRUSTER Stanford University Plasma Physics Lab Future Work oNumerical Stability n Alternative solution algorithms n Timestep and grid refinement oGoverning physics n Enhanced electron mobility n Wall model n Potential BC u Power supply circuit model n Recombination n Magnetized ions oModel validation against experiments