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TWO-DIMENSIONAL (z- θ ) SIMULATIONS OF HALL THRUSTER ANOMALOUS TRANSPORT

TWO-DIMENSIONAL (z- θ ) SIMULATIONS OF HALL THRUSTER ANOMALOUS TRANSPORT. Cheryl M. Lam, Aaron K. Knoll, and Mark A. Cappelli Plasma Physics Laboratory Stanford University, Mechanical Engineering Department Eduardo Fernandez Eckerd College, Department of Mathematics and Physics

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TWO-DIMENSIONAL (z- θ ) SIMULATIONS OF HALL THRUSTER ANOMALOUS TRANSPORT

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  1. TWO-DIMENSIONAL (z-θ) SIMULATIONS OF HALL THRUSTER ANOMALOUS TRANSPORT Cheryl M. Lam, Aaron K. Knoll, and Mark A. Cappelli Plasma Physics Laboratory Stanford University, Mechanical Engineering Department Eduardo Fernandez Eckerd College, Department of Mathematics and Physics IEPC-2009-102 Session 31 31st International Electric Propulsion Conferenc University of Michigan, Ann Arbor, MI September 24, 2009 Research supported by Air Force Office of Scientific Research (Program Manager: Dr. Mitat Birkan) Tuition and stipend support for C.M. Lam from Sandia National Laboratories

  2. Motivation • Hall thruster anomalous electron transport • Super-classical 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 • Hybrid Fluid-PIC • Fluid • 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.

  3. 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

  4. Channel Diameter = 9 cm Channel Length = 8 cm Anode Cathode Anode Exit Plane extends 4 cm past channel exit Geometry • 2D in z-θ • No radial dynamics • E x B  + θ • Br: purely radial (measured from SHT) • Imposed Ez (based on operating condition) • Computational Grid • z: 40 points, non-uniform • θ: 50 points, uniform

  5. Electron Fluid Equations • Momentum: Drift-Diffusion • Neglect inertial terms • Combine current continuity and electron momentum to get convection-diffusion equation for Φ:

  6. LEAP FROG  Time Advance Particle Positions & Velocities Neutrals & Ions (subject to F=qE) EGRID  EPART Ionize Neutrals Inject Neutrals Calculate Plasma Properties ni-PART, vi-PART, nn-PART, vn-PART  ni-GRID, vi-GRID, nn-GRID, vn-GRID QUASINEUTRALITY: ne = ni = nplamsa  Spline  RK4 Time Advance Te=Te(ne, ve) DIRECT SOLVE  2nd-order F-D Iterative Solve Φ Calculate Φ=Φ(ne, vi-GRID) ↔ EGRID Calculate ve=ve(Φ, ne, Te) r < ε0 CONVERGED r = Φ – Φlast-iteration Calculate vi-GRID-TEST= vi-GRID(EGRID) Solution Algorithm Boundary Conditions: • Dirichlet in z (Φ) • Periodic in θ

  7. Simulation Parameters

  8. Plasma Density Electron Temperature Axial Ion Velocity Potential Time-Averaged Plasma Properties

  9. E x B Plasma Density • Breathing Mode • ~10 KHz • Experiment: 20 KHz

  10. E x B Axial Electron Velocity Distinct wave behavior observed: • Throughout channel (upstream) • Tilted: - z, + E x B • Lower frequency, slow moving, longer wavelength • Near exit plane • Peak Br, High shear (∂ueθ/∂z) • Tilted: + z, - ExB • Higher frequency, faster moving, shorter wavelength • Outside exit plane (downstream) • Purely axial: + z • Same structure (in θ) as exit plane waves

  11. E x B Anode Cathode E x B E x B Fluctuations in θ f = 40 KHz λθ = 5 cm vph = 4000 m/s f = 700 KHz λθ = 4 cm vph = 40,000 m/s

  12. Electron Transport • Simulation predicts super-classical electron mobility Axial Electron Mobility:

  13. Channel Diameter = 9 cm Channel Length = 8 cm Anode Cathode extends 2 cm past channel exit Fluid Model • Ions, neutrals, and electrons modeled as a dynamic fluid continuum • 2D in z-θ • No radial dynamics • E x B  + θ • Br: purely radial (measured from SHT) • Imposed Ez (based on operating condition) • Computational Grid • z: 25 points, uniform • θ: 70 points, uniform

  14. Fluid Equations • System of 10 equations in 10 unknowns: • Plasma potential • Electron density • Neutral density • Electron velocity (axial/azimuthal) • Ion velocity (axial/azimuthal) • Neutral velocity (axial/azimuthal) • Electron temperature • Conservation of electrons, ions and neutrals: Quasineutrality: ni = ne

  15. Fluid Equations • Momentum equations for electrons, ions and neutrals: • Electron energy equation: Inertia terms 2D: • Convective & diffusive fluxes • Joule heating, Ionization losses, Effective wall loss where

  16. Define initial and boundary conditions Perform axisymmetric computation t = t + 100ns If time > 400μs Perform full computation t = t + 1ns If time > 10μs Solution Algorithm Boundary Conditions: • Dirichlet in z (Φ) • Periodic in θ IMPLICIT PREDICTOR-CORRECTOR • 1st order in space • 2nd order in time

  17. Simulation Parameters

  18. Plasma Density Electron Temperature Axial Ion Velocity Potential Time-Averaged Plasma Properties

  19. + E x B, flutter (+/-) in z Higher frequency Faster moving Shorter wavelength - E x B, flutter (+/-) in z Lower frequency Slower moving Longer wavelength E x B Axial Electron Velocity 2 distinct wave modes observed:

  20. Anode Cathode E x B Fluctuations in z-θ f = 1.5 MHz λθ = 7 cm vph = 100,000 m/s f = 5 MHz λθ = 6 cm vph = 300,000 m/s Axial “flutter” f = 5 MHz λz = 2 - 6 cm vph = 75,000 -300,000 m/s

  21. Electron Transport • Simulation predicts super-classical electron mobility Axial Electron Mobility:

  22. HYBRID FLUID f = 40 KHz λθ = 5 cm vph = 4000 m/s - E x B + E x B + E x B - E x B f = 700 KHz λθ = 4 cm vph = 40,000 m/s f = 1.5 MHz λθ = 7 cm vph = 100,000 m/s +/- z - z + z f = 5 MHz λz = 2 - 6 cm vph = 75,000 -300,000 m/s f = 5 MHz λθ = 6 cm vph = 300,000 m/s Predicted Waves

  23. Hybrid Fluid Electron Transport • Both simulations predict super-classical electron mobility Axial Electron Mobility:

  24. Questions?

  25. Back-up and Throw Away

  26. Unlike fully PIC codes, the electric potential is not obtained from a Poisson equation:

  27. Streak Plots E x B E x B

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