Data-driven Closure for Fluid Models of Hall Thrusters Benjamin - PowerPoint PPT Presentation

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Data-driven Closure for Fluid Models of Hall Thrusters Benjamin Jorns University of Michigan Princeton University ExB Workshop

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory The Hall effect thruster for space propulsion 𝐶 𝑪 𝐹 𝑭 𝐹

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport Ion continuity 𝜖𝑜 𝑗 𝜖𝑢 + 𝛼 ∙ 𝑜 𝑗 𝒗 𝑗 = 0 Ion momentum 𝜖 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 + 𝛼 ∙ 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 𝒗 𝑗 = 𝑟 𝑜 𝑗 𝑭 − 𝜉 𝑗 𝑛 𝑗 𝒗 𝑗 − 𝒗 𝑓 𝜖𝑢 Ohm’s Law 𝜉 𝑓 𝑛 𝑓 𝑜 𝑓 𝒗 𝒇 = −𝑟𝑜 𝑓 𝐹 − 𝛼𝑄 𝑓 − 𝑟𝑜 𝑓 𝒗 𝒇 × 𝐶 Electron Energy 3 𝜖𝑈 𝜖𝑢 = −𝑟𝑜 𝑓 𝑭 ∙ 𝒗 𝑓 − 𝛼 ∙ 5 + 𝑹 𝒇 + 3 𝑓 2 𝑜 𝑓 2 𝑜 𝑓 𝑈 𝑓 𝒗 𝑓 2 𝑈 𝑓 𝛼 ∙ 𝑜 𝑓 𝒗 𝑓 Current conservation Closed set of classical equations that can 0 = 𝛼 ∙ 𝑟𝑜 𝑓 𝒗 𝑓 − 𝒗 𝑗 be evaluated with standard techniques

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport I e / I d ~ 0.1% Ion continuity 𝜖𝑜 𝑗 𝜖𝑢 + 𝛼 ∙ 𝑜 𝑗 𝒗 𝑗 = 0 Ion momentum 𝜖 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 + 𝛼 ∙ 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 𝒗 𝑗 = 𝑟 𝑜 𝑗 𝑭 − 𝜉 𝑗 𝑛 𝑗 𝒗 𝑗 − 𝒗 𝑓 𝜖𝑢 Ohm’s Law 𝜉 𝑓 𝑛 𝑓 𝑜 𝑓 𝒗 𝒇 = −𝑟𝑜 𝑓 𝐹 − 𝛼𝑄 𝑓 − 𝑟𝑜 𝑓 𝒗 𝒇 × 𝐶 Electron Energy 3 𝜖𝑈 𝜖𝑢 = −𝑟𝑜 𝑓 𝑭 ∙ 𝒗 𝑓 − 𝛼 ∙ 5 + 𝑹 𝒇 + 3 𝑓 2 𝑜 𝑓 2 𝑜 𝑓 𝑈 𝑓 𝒗 𝑓 2 𝑈 𝑓 𝛼 ∙ 𝑜 𝑓 𝒗 𝑓 Current conservation Electron cross-field current from 0 = 𝛼 ∙ 𝑟𝑜 𝑓 𝒗 𝑓 − 𝒗 𝑗 evaluating classical equations

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport I e / I d ~ 0.1% Ion continuity 𝜖𝑜 𝑗 𝜖𝑢 + 𝛼 ∙ 𝑜 𝑗 𝒗 𝑗 = 0 Ion momentum 𝜖 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 + 𝛼 ∙ 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 𝒗 𝑗 = 𝑟 𝑜 𝑗 𝑭 − 𝜉 𝑗 𝑛 𝑗 𝒗 𝑗 − 𝒗 𝑓 𝜖𝑢 Ohm’s Law 𝜉 𝑓 𝑛 𝑓 𝑜 𝑓 𝒗 𝒇 = −𝑟𝑜 𝑓 𝐹 − 𝛼𝑄 𝑓 − 𝑟𝑜 𝑓 𝒗 𝒇 × 𝐶 Electron Energy 3 𝜖𝑈 𝜖𝑢 = −𝑟𝑜 𝑓 𝑭 ∙ 𝒗 𝑓 − 𝛼 ∙ 5 + 𝑹 𝒇 + 3 𝑓 2 𝑜 𝑓 2 𝑜 𝑓 𝑈 𝑓 𝒗 𝑓 2 𝑈 𝑓 𝛼 ∙ 𝑜 𝑓 𝒗 𝑓 Current conservation Actual cross-field current from 0 = 𝛼 ∙ 𝑟𝑜 𝑓 𝒗 𝑓 − 𝒗 𝑗 evaluating equations 1000 x higher!

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport I e / I d ~ 0.1% Ion continuity 𝜖𝑜 𝑗 𝜖𝑢 + 𝛼 ∙ 𝑜 𝑗 𝒗 𝑗 = 0 Ion momentum 𝜖 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 + 𝛼 ∙ 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 𝒗 𝑗 = 𝑟 𝑜 𝑗 𝑭 − 𝜉 𝑗 𝑛 𝑗 𝒗 𝑗 − 𝒗 𝑓 𝜖𝑢 Ohm’s Law 𝜉 𝑓 𝑛 𝑓 𝑜 𝑓 𝒗 𝒇 = −𝑟𝑜 𝑓 𝐹 − 𝛼𝑄 𝑓 − 𝑟𝑜 𝑓 𝒗 𝒇 × 𝐶 −𝑜 𝑓 𝑛 𝑓 𝜉 𝐵𝑂 𝒗 𝒇 , Electron Energy Need to introduce ad hoc factor 3 𝜖𝑈 𝜖𝑢 = −𝑟 𝜍 𝑗 𝑭 ∙ 𝒗 𝑓 − 𝛼 ∙ 5 + 𝑹 𝒇 + 3 𝑓 2 𝑜 𝑓 2 𝑜 𝑓 𝑈 𝑓 𝒗 𝑓 2 𝑈 𝑓 𝛼 ∙ 𝑜 𝑓 𝒗 𝑓 𝑛 𝑗 “Anomalous collision frequency ” Current conservation Actual cross-field current from 0 = 𝛼 ∙ 𝑟𝑜 𝑓 𝒗 𝑓 − 𝒗 𝑗 evaluating equations 1000 x higher!

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport I e / I d ~ 10% Ion continuity 𝜖𝑜 𝑗 𝜖𝑢 + 𝛼 ∙ 𝑜 𝑗 𝒗 𝑗 = 0 Ion momentum 𝜖 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 + 𝛼 ∙ 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 𝒗 𝑗 = 𝑟 𝑜 𝑗 𝑭 − 𝜉 𝑗 𝑛 𝑗 𝒗 𝑗 − 𝒗 𝑓 𝜖𝑢 Ohm’s Law 𝜉 𝑓 𝑛 𝑓 𝑜 𝑓 𝒗 𝒇 = −𝑟𝑜 𝑓 𝐹 − 𝛼𝑄 𝑓 − 𝑟𝑜 𝑓 𝒗 𝒇 × 𝐶 −𝑜 𝑓 𝑛 𝑓 𝜉 𝐵𝑂 𝒗 𝒇 , Electron Energy Need to introduce ad hoc factor 3 𝜖𝑈 𝜖𝑢 = −𝑟 𝜍 𝑗 𝑭 ∙ 𝒗 𝑓 − 𝛼 ∙ 5 + 𝑹 𝒇 + 3 𝑓 2 𝑜 𝑓 2 𝑜 𝑓 𝑈 𝑓 𝒗 𝑓 2 𝑈 𝑓 𝛼 ∙ 𝑜 𝑓 𝒗 𝑓 𝑛 𝑗 “Anomalous collision frequency ” Current conservation Anomalous friction term promotes 0 = 𝛼 ∙ 𝑟𝑜 𝑓 𝒗 𝑓 − 𝒗 𝑗 additional cross-field current

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport I e / I d ~ 10% Ion continuity 𝜖𝑜 𝑗 𝜖𝑢 + 𝛼 ∙ 𝑜 𝑗 𝒗 𝑗 = 0 Ion momentum 𝜖 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 + 𝛼 ∙ 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 𝒗 𝑗 = 𝑟 𝑜 𝑗 𝑭 − 𝜉 𝑗 𝑛 𝑗 𝒗 𝑗 − 𝒗 𝑓 𝜖𝑢 Ohm’s Law 𝜉 𝑓 𝑛 𝑓 𝑜 𝑓 𝒗 𝒇 = −𝑟𝑜 𝑓 𝐹 − 𝛼𝑄 𝑓 − 𝑟𝑜 𝑓 𝒗 𝒇 × 𝐶 −𝑜 𝑓 𝑛 𝑓 𝜉 𝐵𝑂 𝒗 𝒇 , Electron Energy Need to introduce ad hoc factor 3 𝜖𝑈 𝜖𝑢 = −𝑟 𝜍 𝑗 𝑭 ∙ 𝒗 𝑓 − 𝛼 ∙ 5 + 𝑹 𝒇 + 3 𝑓 2 𝑜 𝑓 2 𝑜 𝑓 𝑈 𝑓 𝒗 𝑓 2 𝑈 𝑓 𝛼 ∙ 𝑜 𝑓 𝒗 𝑓 𝑛 𝑗 Problem: introducing ad-hoc term opens set of Current conservation equations (too many unknowns) Anomalous friction term promotes 0 = 𝛼 ∙ 𝑟𝑜 𝑓 𝒗 𝑓 − 𝒗 𝑗 additional cross-field current

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Problem of anomalous electron transport I e / I d ~ 10% Ion continuity We need a functional form for 𝜉 𝐵𝑂 (𝑈 𝜖𝑜 𝑗 𝑓 , 𝑜 𝑓 , . . ) 𝜖𝑢 + 𝛼 ∙ 𝑜 𝑗 𝒗 𝑗 = 0 that depends on classical fluid parameters Ion momentum 𝜖 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 + 𝛼 ∙ 𝑛 𝑗 𝑜 𝑗 𝒗 𝑗 𝒗 𝑗 = 𝑟 𝑜 𝑗 𝑭 − 𝜉 𝑗 𝑛 𝑗 𝒗 𝑗 − 𝒗 𝑓 𝜖𝑢 Ohm’s Law 𝜉 𝑓 𝑛 𝑓 𝑜 𝑓 𝒗 𝒇 = −𝑟𝑜 𝑓 𝐹 − 𝛼𝑄 𝑓 − 𝑟𝑜 𝑓 𝒗 𝒇 × 𝐶 −𝑜 𝑓 𝑛 𝑓 𝜉 𝐵𝑂 𝒗 𝒇 , Electron Energy Need to introduce ad hoc factor 3 𝜖𝑈 𝜖𝑢 = −𝑟 𝜍 𝑗 𝑭 ∙ 𝒗 𝑓 − 𝛼 ∙ 5 + 𝑹 𝒇 + 3 𝑓 2 𝑜 𝑓 2 𝑜 𝑓 𝑈 𝑓 𝒗 𝑓 2 𝑈 𝑓 𝛼 ∙ 𝑜 𝑓 𝒗 𝑓 𝑛 𝑗 Problem: introducing ad-hoc term opens set of Current conservation equations (too many unknowns) Anomalous friction term promotes 0 = 𝛼 ∙ 𝑟𝑜 𝑓 𝒗 𝑓 − 𝒗 𝑗 additional cross-field current

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Closures for anomalous collision frequency: first-principles Ԧ 𝐺 𝐵𝑂 = −𝑜 𝑓 𝑛 𝑓 𝜉 𝐵𝑂 𝒗 𝑓 *N. Gascon, M. Dudeck, and S. Barral, PoP , vol. 10, no. 10, 2003 † J. M. Fife and M. Martinez-Sanchez/ IEPC-95-24 ‡ M. A. Cappelli, C. V. Young, E. Cha, and E. Fernandez, PoP , vol. 22, no. 11, 2015. T. Lafleur, S. D. Baalrud, and P. Chabert, PoP , vol. 23, no. 5, 2016 .

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Closures for anomalous collision frequency: first-principles Ԧ 𝐺 𝐵𝑂 = −𝑜 𝑓 𝑛 𝑓 𝜉 𝐵𝑂 𝒗 𝑓 Instabilities ‡ Wall Interactions* 2 𝜉 𝐵𝑂 = 1 v 𝑒𝑓 𝜉 𝐵𝑂 = 𝛾 𝑈 𝐿 𝜕 𝑑𝑓 𝑓 𝑑 𝑡 𝜉 𝐵𝑂 = 𝛼 ∙ 𝑣 𝑗 𝑜 𝑓 𝑈 𝑓 𝑛 𝑓 𝑑 𝑡 𝑜 𝑓 v 𝑒𝑓 Bohm Diffusion † 𝜉 𝐵𝑂 = 1 𝐿 𝜕 𝑑𝑓 *N. Gascon, M. Dudeck, and S. Barral, PoP , vol. 10, no. 10, 2003 † J. M. Fife and M. Martinez-Sanchez/ IEPC-95-24 ‡ M. A. Cappelli, C. V. Young, E. Cha, and E. Fernandez, PoP , vol. 22, no. 11, 2015. T. Lafleur, S. D. Baalrud, and P. Chabert, PoP , vol. 23, no. 5, 2016 .

University of Michigan – Plasmadynamics and Electric Propulsion Laboratory Closures for anomalous collision frequency: first-principles Ԧ 𝐺 𝐵𝑂 = −𝑜 𝑓 𝑛 𝑓 𝜉 𝐵𝑂 𝒗 𝑓 Closure models from first-principles are potentially predictive Instabilities ‡ Wall Interactions* 2 𝜉 𝐵𝑂 = 1 v 𝑒𝑓 𝜉 𝐵𝑂 = 𝛾 𝑈 𝐿 𝜕 𝑑𝑓 𝑓 𝑑 𝑡 Models have to date have had limitations, yielding qualitative 𝜉 𝐵𝑂 = 𝛼 ∙ 𝑣 𝑗 𝑜 𝑓 𝑈 agreement over only limited range of conditions 𝑓 𝑛 𝑓 𝑑 𝑡 𝑜 𝑓 v 𝑒𝑓 Bohm Diffusion † Possible that reality is too complicated or models or too reduced fidelity 𝜉 𝐵𝑂 = 1 𝐿 𝜕 𝑑𝑓 *N. Gascon, M. Dudeck, and S. Barral, PoP , vol. 10, no. 10, 2003 † J. M. Fife and M. Martinez-Sanchez/ IEPC-95-24 ‡ M. A. Cappelli, C. V. Young, E. Cha, and E. Fernandez, PoP , vol. 22, no. 11, 2015. T. Lafleur, S. D. Baalrud, and P. Chabert, PoP , vol. 23, no. 5, 2016 .