Asymptotic Accuracy of the Equilibrium Diffusion Approximation and Semi-Analytic Solutions of Radiating Shocks

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Asymptotic Accuracy of the Equilibrium Diffusion Approximation and Semi-Analytic Solutions of Radiating Shocks ASYMPTOTIC ACCURACY OF THE EQUILIBRIUM DIFFUSION APPROXIMATION AND SEMI-ANALYTIC SOLUTIONS OF RADIATING SHOCKS A Dissertation by JIM MICHAEL FERGUSON Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, George Kattawar Co-Chair of Committee, Jim Morel Committee Members, Vitaly Kocharovsky Marv Adams Alexey Zheltikov Head of Department, George Welch May 2014 Major Subject: Applied Physics Copyright 2014 Jim Michael Ferguson ABSTRACT Radiation hydrodynamics (RH) provides a theoretical description for many astro- physical events spanning a wide range of observable phenomena. It is the goal of high-energy-density laboratory astrophysics (HEDLA) to reproduce some of these events in terrestrial settings. Computational models exist to aide our understanding by simulating astrophysical observations and laboratory experiments, and by mak- ing predictions which guide both experiments and observations. It is the goal of this dissertation to contribute to our understanding of computational models in RH. Two problems are solved that aide this understanding: 1) we showed that the asymptotic equilibrium diffusion limit (EDL) of the RH equations reproduces the equilibrium diffusion approximation (EDA), through first-order in the asymptotic expansion, in agreement with previous transport models that neglect material motion; and, 2) we produced semi-analytic radiative shock solutions using grey, angularly discretized (Sn) transport. The first problem establishes the asymptotic limits of the EDA in RH, which will have direct applications to discretizations of RH models. The second problem extends previous semi-analytic solution methods for radiative shocks to in- clude grey Sn transport. Previous semi-analytic methods relied on nonequilibrium diffusion theory to describe the radiation, which assumes that while the radiation is out of equilibrium with the material, the angular dependence of the radiation field is isotropic over the extended spatial domain of the radiative shock, and that the radiation energy density is monotonic over the shock's spatial domain. The purpose of using grey Sn transport is to determine the angular dependence of the radiation field. It is shown that the anisotropy of the radiation field can cause the radiation energy density to be nonmonotonic and exhibit a local maximum if a spike in the ii material temperature, called a Zel'dovich spike, near the shock discontinuity exists. This local maximum of the radiation energy density is termed \anti-diffusive" ra- diation because the radiation flux and the gradient of the radiation energy density may have the same sign in this region, which is in stark contrast to diffusion theory wherein the radiation flux is proportional to the negative gradient of the radiation energy density. iii DEDICATION This thesis is dedicated to my wife, Kimberly Rae Ferguson. You have been my constant champion, and every aspect of my life is better because you are in it. You are the wife of my youth, and I love You! iv ACKNOWLEDGEMENTS This dissertation is the end-product, over many years, and with the help of many different people, of a single pursuit: a Ph.D. in the applied sciences. Jon Hartgraves and Gaye-Lynn Seawright personally ensured that I received the best education they could provide, and just as importantly, they provided patient support and encouragement when I was still a child. Gaye-Lynn and Jimmy have provided an anchor to my past that is invaluable, and I cannot sufficiently express my gratitude. Christine Ehlig-Economides has become a true friend, and I am grateful for ev- erything that she has taken the time to teach me. Bruce Freeman introduced me to research, and true scientific curiousity. He was the first to teach me that saying \I don't know", can be the right answer. Jim Morel taught me to learn what \I don't know" before I got to his office. He has had the largest affect on how I think about physics, always pushing me to understand the de- tails, and their consequences. George Kattawar showed me a quiet forcefulness that impressed on me how much power can be wielded by silent contemplation. While George has retired, Marv Adams similarly shows all of his students this same quiet forcefulness. I am humbled, and grateful, to have been guided by these men along my scientific path. My committee has been exceedingly patient and helpful: George, Jim, Marv, Vi- v taly, and Alexey, thank you for your support and patience! Rob Lowrie, Bryan Johnson, and Nick Gentile made various contributions to the work in this dissertation, and my understanding. I look forward to continuing our collaborations. I have enjoyed many day-to-day conversations, and sundry distractions, with friends and colleagues alike. Jason Cezeaux, Erich Fruchtnicht, and Brent Lindeburg knew me before I knew myself. Jianwei Mei and Chris Bertinato helped me learn every- thing I needed to know about black-holes, and high-energy theoretical physics: That studying these subjects wouldn't allow me to support a family! Adam \Eeyore" Hetzler, Hayes Stripling, Jarrod Edwards, and Paul and Jenny Bruillard have gone before me; Don Bruss, Marco Delchini, Akansha Kumar, Alex Long, Peter Maginot, Carolyn McGraw, Andrew Till, I look forward to seeing you finish. I am grateful for these relationships! My wife has shown me exceeding patience and support; I'm still trying to learn. Ken and LouEllen have been similarly patient, and have provided me with many examples I didn't have before. I have learned, and will continue to learn, from you both, the most important lessons in life. My sweet Evelyn Grace will no longer ap- preciate being called \baby-girl" long before she can understand the words written here. You are a Blessing, and I give thanks to my God every time I think of you. vi TABLE OF CONTENTS Page ABSTRACT . ii DEDICATION . iv ACKNOWLEDGEMENTS . v TABLE OF CONTENTS . vii LIST OF FIGURES . x CHAPTER I INTRODUCTION . 1 I.1 Background . 3 I.2 Equilibrium diffusion in radiation hydrodynamics . 5 I.3 Radiative shocks . 7 I.4 State-of-the-art computational methods . 10 CHAPTER II RADIATION TRANSPORT AND RADIATION HYDRODY- NAMICS . 12 II.1 Radiation transport without material motion . 15 II.1.1 The P1 approximation . 16 II.1.2 The diffusion approximation . 19 II.1.3 The equilibrium diffusion approximation . 20 II.2 Invariants and transformations . 20 II.3 Radiation transport in the lab-frame with material motion . 23 II.4 Radiation hydrodynamics . 25 CHAPTER III ASYMPTOTIC ACCURACY OF THE EQUILIBRIUM DIF- FUSION APPROXIMATION IN RADIATION HYDRODY- NAMICS . 27 III.1 Example: asymptotic analysis of the neutron transport equation . 27 III.2 Nondimensional variables and scaling ratios . 37 III.3 Asymptotic expansion of the radiation hydrodynamic equations . 40 III.4 P1 and diffusion models in the lab- and comoving-frames in the equi- librium diffusion limit . 45 III.4.1 The comoving- and lab-frame P1 models . 48 vii Page III.4.2 The comoving- and lab-frame grey-diffusion models . 52 CHAPTER IV PREVIOUS ANALYTIC AND SEMI-ANALYTIC RADIA- TIVE SHOCK SOLUTIONS . 59 IV.1 Nonequilibrium diffusion analytic results . 60 IV.1.1 Derivation of the nonequilibrium diffusion equations . 61 IV.1.2 Subcritical radiating shocks . 65 IV.1.3 Supercritical radiating shocks . 70 IV.2 Analytic work of Drake, and McClarren and Drake . 74 IV.2.1 The three-layer model . 75 IV.2.2 Anti-diffusive radiative shocks . 81 IV.3 Previous semi-analytic solutions and code-verification . 85 IV.3.1 Review of work before 2007 . 85 IV.3.2 Semi-analytic solutions by Lowrie and collaborators . 86 IV.3.3 Code-verification of KULL's Implicit Monte Carlo radiation transport package using Lowrie's semi-analytic solutions . 97 CHAPTER V RADIATIVE SHOCK SOLUTIONS WITH GREY SN TRANS- PORT . 100 V.1 Governing equations and nondimensionalization . 103 V.2 Problem statement . 109 V.3 Reduced equations . 110 V.3.1 The ordinary differential equations for the radiation hydrody- namic solve . 111 V.3.2 The ordinary differential equations for the radiation transport solve . 114 V.4 Solution procedure . 115 V.4.1 The Rankine-Hugoniot jump conditions . 116 V.4.2 The radiation hydrodynamic solve . 116 V.4.3 The radiation transport solve . 119 V.5 Results and analysis . 120 V.5.1 Comparison of nonequilibrium diffusion and Sn radiative shock solutions . 121 V.5.2 Code-verification of KULL's Implicit Monte Carlo radiation transport package using grey Sn transport . 124 V.5.3 Comparison of the material and radiative temperatures near the embedded hydrodynamic shock to the final temperature and anti-diffusive radiation . 131 viii Page V.5.4 Onset of supercritical shocks . 135 CHAPTER VI SUMMARY . 137 REFERENCES . 141 APPENDIX A DERIVATION OF THE FOUR LORENTZ INVARIANTS . 147 APPENDIX B DERIVATION OF THE TRANSFORMED IN-SCATTER SOURCE . 152 APPENDIX C DERIVATION OF THE LAB-FRAME RADIATION TRANS- PORT EQUATION THROUGH O(β2) . 154 APPENDIX D ASYMPTOTIC DERIVATION OF THE RADIATION VARI- ABLES THROUGH O(2) . 162 APPENDIX E DERIVATION OF THE O(β2) CORRECTIONS TO THE RADIATION TRANSPORT EQUATION . 169 APPENDIX F DERIVATION OF THE WAVE EQUATION AND NONDI- MENSIONALIZATION OF THE IDEAL GAS EQUATION OF STATE . 173 APPENDIX G DERIVATION OF THE LINEARIZATION PROCEDURE USED FOR THE RADIATION HYDRODYNAMIC SOLVE IN THE FIRST ITERATION . 180 ix LIST OF FIGURES Page Figure I.1 Simeis 147, the remnant of SN185. A neutron star is in the center of the figure. The red filaments represent color enhancement of hydrogen recombination at the front of the expanding shock. Adapted from the March 24, 2005, Astronomy Picture of the Day [16]. 7 Figure I.2 SN1987A was viewed from earth on February 23, 1987. Only 168,000 light-years away, it is one of the most well-studied super- novae.
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