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Reduced Order Modelling of Streamers and Their Characterization by Macroscopic Parameters by Colin A

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Reduced Order Modelling of Streamers and Their Characterization by Macroscopic Parameters by Colin A Reduced order modelling of streamers and their characterization by macroscopic parameters by Colin A. Pavan BASc, University of Waterloo (2017) Submitted to the Department of Aeronautical and Astronautical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Aeronautical and Astronautical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019 ○c Massachusetts Institute of Technology 2019. All rights reserved. Author................................................................ Department of Aeronautical and Astronautical Engineering May 21, 2019 Certified by. Carmen Guerra-Garcia Assistant Professor of Aeronautics and Astronautics Thesis Supervisor Accepted by........................................................... Sertac Karaman Associate Professor of Aeronautics and Astronautics Chair, Graduate Program Committee 2 Reduced order modelling of streamers and their characterization by macroscopic parameters by Colin A. Pavan Submitted to the Department of Aeronautical and Astronautical Engineering on May 21, 2019, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautical and Astronautical Engineering Abstract Electric discharges in gases occur at various scales, and are of both academic and prac- tical interest for several reasons including understanding natural phenomena such as lightning, and for use in industrial applications. Streamers, self-propagating ioniza- tion fronts, are a particularly challenging regime to study. They are difficult to study computationally due to the necessity of resolving disparate length and time scales, and existing methods for understanding single streamers are impractical for scaling up to model the hundreds to thousands of streamers present in a streamer corona. Conversely, methods for simulating the full streamer corona rely on simplified models of single streamers which abstract away much of the relevant physics. This disconnect highlights the need for a simplified model of individual streamers which captures the core dynamics but is scalable to ensembles of many mutually interacting streamers. In this work, several such models are developed. First, a 1.5D model of a single streamer was created wherein particles are treated one dimensionally and electric fields two dimensionally (axisymmetric). This model incorporates developments in modelling streamer processes such as photoionization that were not available in the days when 1.5D models were first invesitgated. Next, a 1.5D model was created with the governing equations solved in the reference frame of the streamer. The existence of such a quasi-steady frame has previously been hypothesized; this work gives a thorough evaluation of the validity of a steady-state streamer model and finds it to be a reasonable approximation on the time scale of electron motion. Based on the success of the quasi-steady model, a further simplification is made wherein streamers are characterized by a small set of macroscopic parameters: tip electric field, velocity, radius and background electric field. A simple model is developed relating these various properties and an efficient graphical representation of their interdependencies is presented. Thesis Supervisor: Carmen Guerra-Garcia Title: Assistant Professor of Aeronautics and Astronautics 3 4 Acknowledgments The author wishes to acknowledge following groups which have financially supported his education and the research presented in this work: The Boeing Company, through the Strategic Universities for Boeing Research and Technology Program; the MIT- Spain La Caixa Foundation Seed Fund through the MISTI Global Seed Funds grant program; and the MIT AeroAstro Vos fellowship. He would also like to thank Pro- fessor Manuel Martinez-Sanchez for many helpful discussions related to this work. 5 6 Contents 1 Introduction 21 1.1 Overview of electric discharges in gases . 21 1.1.1 Background . 21 1.1.2 Different discharge regimes and their transitions . 22 1.2 Streamers . 24 1.2.1 Description . 24 1.2.2 Past modelling efforts . 25 1.3 This work . 31 1.3.1 Motivation . 31 1.3.2 Outline . 32 2 1.5 dimensional model monstruction 33 2.1 General fluid model . 33 2.1.1 Governing equations and non-dimensionalization . 33 2.1.2 Transport coefficients . 36 2.1.3 Sources . 37 2.2 Modifications for 1.5D model . 41 2.2.1 Particle equations . 41 2.2.2 Electric field . 42 2.2.3 Photoionization . 49 2.2.4 Radial flux . 51 2.3 Anode mounted model . 52 2.3.1 Modifications to base 1.5D model . 52 7 2.3.2 Numerical methods considerations . 54 2.4 Quasi-steady streamer . 56 2.4.1 Modifications to base 1.5D model . 56 2.4.2 Solution scheme . 58 3 Analysis of 1.5D Model 67 3.1 Detailed analysis of model outputs . 67 3.1.1 Transient model . 67 3.1.2 Steady state model . 73 3.2 Applicability of quasi-steady model . 75 3.2.1 Existence of a quasi-steady frame . 77 3.2.2 Comparison of quasi-steady and transient solutions . 80 3.3 Comparison to published data . 81 3.4 Discussion . 85 3.4.1 Relative importance of diffusion and photoionization . 85 3.4.2 Current . 87 4 Macroscopic quasi-steady streamer modelling 91 4.1 Derivation of model . 91 4.1.1 Electric field and characteristic dimension . 91 4.1.2 Radius . 92 4.1.3 Background field . 96 4.1.4 Graphical representation . 100 4.2 Comparison with 1.5D model . 102 4.2.1 Steady state . 102 4.2.2 Transient . 104 4.2.3 Comparison of all 3 models . 106 4.3 Comparison with published data . 106 4.3.1 Data sources . 106 4.3.2 Comparison and discussion . 111 8 5 Model extensions 115 5.1 Macroscopic propagation . 115 5.2 Negative streamers . 118 6 Conclusions 123 6.1 Summary of contributions . 123 6.2 Recommendations for future work . 125 A Additional derivations 127 A.1 Integrated absorption function . 127 A.2 Mirror charge . 128 B Additional model results 135 B.1 Transient model parameter variation . 135 B.2 Numerical accuracy analysis . 137 9 10 List of Figures 2-1 Electron transport coefficients: (L) experimental data from [1] (R) data from BOLSIG+. 37 2-2 Peclet number based on ionization length for electrons as a function of normalized electric field. 38 2-3 Comparison of kernel functions for different calculation methods. 48 2-4 Comparison of modified uniform cross-section to shell of charge. .48 2-5 Electric field magnitude with flux streamlines overlayed; adapted based on data from [2]. 51 2-6 Geometry for anode-mounted streamer. Subscript a is the anode, sub- script s is the streamer. d is the diameter of streamer or anode. 53 2-7 Region of grid refinement; electric field of streamer shown for reference 55 2-8 Effect of truncating electric field . 59 2-9 Sparsity pattern of matrix . 60 2-10 Velocity calculation; note and on the axis correspond to 0 ST;calc ST ST and 0 respectively in the text. 62 ST 2-11 Steady-state model iteration scheme . 65 3-1 Example output for transient model. The left two figures have been plotted at t=1.7, 4.9, 8.3 and 11.8 (non-dimensional time). Input pa- rameters are: a0 = 0:15mm, Ra = 5a0, 휑a=Ra = 4 . 68 11 3-2 Transient model output varying initial conditions. a0=0.25mm, Ra=5a0 Colours refer to 휑a=Ra: red=2.5, green=3, black=3.5, blue=4. Linestyle refers to maximum of Gaussian initial seed. solid line=2, dashed line (- - -) = 20, broken dash (− · −·) = 50, dotted (···)=100 . 70 3-3 Transient model output varying streamer radius and anode size with fixed initial seed n0=20 and surface field 휑a=Ra = 3. Colours refer to Ra, measured in streamer radii: green=10, black=7.5, blue=5, ma- genta=2.5. Linestyle refers to streamer radius: solid line=0.15mm, dashed line (- - -) = 0.25mm, broken dash (− · −·) = 0.35mm . 71 3-4 Transient model output varying background charge density. a0=0.25mm, Ra=10a0. Colours refer to 휑a=Ra: red=4,green=3.5, black=3, blue=2.5. Linestyle refers to charge density magnitude (C in 휌 = C=r). solid line=0, dashed line (- - -) = 1, broken dash (− · −·) = 2, dotted (···)=4. 74 3-5 Example Output for Quasi-Steady Model. Model inputs: a0=0.15mm, E1=0.14 . 75 3-6 Example output for quasi-steady model; same case as figure 3-5 with enlargement of head area and particle densities on linear scale . 76 3-7 Comparison of quasi-steady velocity (calculated by equation 3.2) to transient velocity (measured based on tip movement between time steps) 78 3-8 Comparison of velocity calculated by equation 3.2 (dashed lines) to true velocity (solid lines) for two cases. (A) a0 = 0:35mm; Ra = 5a0 (B)a0 = 0:25mm; Ra = 10a0 ....................... 79 3-9 Comparison of transient model (black) to steady state model at various locations (chosen based on matching tip field). For transient model, a0 = 0:15mm, Ra = 5a0, 휑a=Ra = 4 ................... 82 3-10 Same as figure 3-7, with the effect of diffusion and photoionization selectively neglected . 86 3-11 Comparison of electron source term magnitudes . 86 3-12 Example current analysis applied to transient model for streamer with a0 =0.25mm and Ra = 5a0, streamer tip at 35 . 89 12 3-13 Current ratio in streamer channel (10 radii behind head) and acceler- ation of streamer. Ra = 5a0 for all . 90 4-1 Analysis of equation 4.4. See text for description of plots . 95 4-2 Control volume for charge conservation calculation . 96 4-3 Comparison of ratio of LHS to RHS for equation 4.11 ("Full Equation") and equation 4.12 ("Simple Equation") . 99 4-4 Evaluation of the value of 휅 . 101 4-5 Graphical representation of the solution of equation 4.4 and 4.15. Solid lines are constant effective radius (mm), dashed lines are constant back- ground field (normalized by breakdown field) .
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