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Numerical Simulation of Multiphase Jet Fragmentation Using Smoothed Particle Hydrodynamics Numerical simulation of multiphase jet fragmentation using Smoothed Particle Hydrodynamics Thomas Chun Long Yue, BSc., MSc. Thesis submitted to The University of Nottingham for the degree of Doctor of Philosophy, February 2015 Abstract This thesis is devoted to the study of multiphase jet fragmentation us- ing Smoothed Particle Hydrodynamics (SPH). The theoretical aspects of three hydrodynamic instabilities, namely the Kelvin-Helmholtz instability (KHI), Rayleigh-Taylor instability (RTI), and Rayleigh Plateau instability (RPI) are reviewed. The linear growth rate of the combined KHI and RTI are derived by means of linear perturbation in chapter 2. The linear growth rate of the multiphase RPI is presented in chapter 7. An overview of the Smoothed Particle Hydrodynamics is given in chapter 3. A pseudo-consistent SPH scheme is presented for the simulation of multiphase flow problems. Additionally, two interface stabilisation models are presented: quasi-buoyancy model and gas-repulsion model. When used in combination with the pseudo-consistent SPH scheme, these models are found to be superior than those presented in the weakly-compressible SPH literature and allows for the simulations for density ratio up to three-magnitudes. The development of an idealised KHI and a KHI subjected to constant grav- itational acceleration (stratified shear instability) is examined in chapter 5. The extracted linear growth rate are compared with the theoretical growth rate pre- sented both in the literature and in chapter 2 for the purpose of validation. The development of a single- and multi-mode RTI are studied by means of SPH in chapter 6. Chapter 7 presents the results for the three-dimensional RPI occur- ring between two fluids. Based on the knowledge acquired in chapter 5-7, the multiphase jet fragmentation driven by the previously mentioned hydrodynamic instabilities are presented in chapter 8. Finally, the major research findings and recommendations are summarised in chapter 9. First Supervisor: Dr. Frazer R. Pearce Second Supervisor: Dr. Arno C.H. Kruisbrink Internal Examiner: Dr. Mike Swift (University of Nottingham) External Examiner: Prof. Justin Read (University of Surrey) Submitted: Oct 15 2014 Examined: Feb 4 2015 Final Version: Feb 9 2015 1 Declaration I declare that the thesis is the result of my own work which has been mainly undertaken during the period of registration as a PhD candidate at the University of Nottingham unless otherwise referenced. Chapters 3 (sections 4.4.2, 3.11.3, 3.11.4), 4, 5, 6, 7 are based on the following articles. Journal articles 1. Numerical Simulations of two-dimensional Kelvin-Helmholtz Instability us- ing weakly compressible Smoothed Particle Hydrodynamics, T. Yue, F. Pearce, A.C.H. Kruisbrink, H. Morvan, International Journal for Numeri- cal Methods in Fluids, Date of submission: June 13 2014, status: accepted for publication. 2. Numerical Simulations of single and multi-mode Rayleigh-Taylor Instabili- ties using weakly-compressible Smoothed Particle Hydrodynamics, T. Yue, F. Pearce, A.C.H. Kruisbrink, H. Morvan, International Journal for Nu- merical Methods in Fluids, Date of submission: February 6 2015, status: submitted. 3. Numerical simulation of a stratified shear instability using Smoothed Par- ticle Hydrodynamics, T. Yue, F. Pearce, A.C.H. Kruisbrink, H. Morvan, International Journal for Numerical Methods in Fluids, status: drafted and ready for submission. 2 Conference proceedings 1. SPH concepts for continuous wall and pressure boundaries, A.C.H. Kruis- brink, F.R. Pearce, T. Yue, K.A. Cliffe and H.P. Morvan, Proceedings of 6th international SPHERIC workshop, Hamburg, Germany, 2011, 298-304. 2. SPH multi-fluid model with interface stabilization based on quasi-buoyancy correction, A.C.H. Kruisbrink, F.R. Pearce, T. Yue, K.A. Cliffe and H.P. Morvan, Proceedings of 7th international SPHERIC workshop, Prato, Italy, 2012, 285-302. 3. Numerical simulation of jet fragmentation in multi-fluid medium using Smoothed Particle Hydrodynamics, T. Yue, A.C.H Kruisbrink, F.R. Pearce, H.P. Morvan, Proceedings of 9th international SPHERIC workshop, Paris, France, 2014, 80-86. 4. Momentum conserving methods that reduce particle clustering in SPH, Proceedings of 9th international SPHERIC workshop, S. Korzilius, A.C.H. Kruisbrink, T. Yue, W. Schilders and M. Anthonissen, Paris, France, 2014, 268-274. Journal articles in preparation 1. SPH particle collisions for multi-fluid applications and walls, A.C.H. Kruis- brink, S.P. Korzilius, T. Yue, F.R. Pearce and H.P. Morvan, intended jour- nal for submission: Journal of Computational Physics. 2. Generalization of SPH multi-fluid models with interface stabilization based on pressure gradient correction, A.C.H Kruisbrink, F.R. Pearce, T. Yue, K.A. Cliffe, H.P. Morvan, intended journal for submission: Journal of Com- putational Physics. 3 Articles published but not do not form part of the discus- sion in this thesis 1. SPH surface tension model without need for calibration or evaluation of curvature, A.C.H. Kruisbrink, F.R. Pearce, T. Yue, K.A. Cliffe and H.P. Morvan, Proceedings of 8th international SPHERIC workshop, Trontheim, Norway, 2013, 31-37 2. Smoothed particle hydrodynamics simulations of flow separation at bends, Computers & Fluids, Jan 2014:90:138-146. 3. Subhaloes gone Notts: Spin across subhaloes and finders, Monthly Notices of the Royal Astronomical Society 12/2012; 429(3) 4 Acknowledgements Foremost, I would like to express my sincere gratitude to my supervisors Prof. Frazer Pearce and Dr. Arno Kruisbrink, and collaborators Prof. Andrew Cliffe, Prof. Herve Morvan, Dr. Hanni Lux and Dr. Claudio Dalla Vecchia for everything I learned from them and for their encouragement during my studies. In particular, I am thankful for Prof. Frazer Pearce for granting me the opportunities to attend annual CosmoComp conferences, and Dr. Claudio Dalla Vecchia for organising the visitor position at the Max-Planck Institute for Extraterrestrial Physics. These opportunities enable an once astronomer enthusiasts like myself to transform from a never-was to a want-to-be astronomer. I sincerely hope that the knowledge and experience acquired would enable me to conduct further research in numerical astrophysics. I would like to thank the School of Physics and Astronomy, University of Nottingham High Performance Computing center and Rolls Royce Plc. for funding this interdisciplinary research project. This research project has boarden my scope by applying state-of-the-art computational methods in physics to simulate fluid flows in aero-engineering related topics. I am forever grateful to my family for their support and love throughout the years. Particularly my father for paving my way through an adventurous journey into Mathematics and Sciences. I would like to thank friends I made during my studies in U.K., namely Jordi, Marco and Jeremy for their support in my journey in academia. Finally, I would like to thank my brothers for their unconditional support, especially to those who reached out their hands during the darkest mo- ments of my life. Last but not least, a special thanks to Dr. M.X. Chen for his mentorship and encouragement since my undergraduate studies. Contents 1 Introduction 1 1.1Motivation............................... 1 1.2 Outline of present work ........................ 7 2 Theory 10 2.1Introduction.............................. 10 2.2Background.............................. 11 2.3Mathematicalmodel......................... 13 2.4Lineargrowthrates.......................... 18 2.4.1 Sharp interface fluid instabilities ............... 18 2.4.2 Smoothed interface fluid instabilities ............ 25 2.5 Summary ............................... 31 3 Smoothed Particle Hydrodynamics 34 3.1Introduction.............................. 34 3.2Literaturereview........................... 35 3.3SPHinterpolation........................... 37 3.4Kernelfunctions............................ 39 3.5AccuracyofSPHinterpolation.................... 45 3.5.1 spatialdiscretisationerror.................. 45 3.5.2 Consistency.......................... 45 3.6 Smoothing length . ......................... 48 ii 3.7SpatialderivativesinSPH...................... 49 3.7.1 SPHdivergenceformulaI .................. 49 3.7.2 SPHdivergenceformulaII.................. 50 3.8 SPH Lagrangian fluid dynamics ................... 51 3.9SPHviscosity............................. 55 3.9.1 Standardartificialviscosity................. 55 3.9.2 Balsaraswitch......................... 57 3.9.3 Monaghanrealviscosity................... 58 3.9.4 State-of-the-artviscositymodels............... 58 3.10EquationofState........................... 62 3.11 Multiphase SPH ............................ 64 3.11.1 Survey of multiphase SPH literature . ......... 64 3.11.2Pseudo-consistentSPH.................... 68 3.11.3Quasi-buoyancycorrectionmodel.............. 77 3.11.4 Gas repulsion model ..................... 79 3.12 Multiphase pseudo-consistent SPH ................. 84 3.13 Summary ............................... 86 4 Software Implementation 87 4.1Introduction.............................. 87 4.2Literaturereview........................... 88 4.3SPHcode-Draco........................... 90 4.3.1 Developmenthistory..................... 90 4.3.2 Softwarearchitecture..................... 91 4.3.3 Cell linked list ......................... 92 4.4 Boundary Conditions in SPH .................... 96 4.4.1 Physical interpretation of boundary conditions ....... 96 4.4.2 Wall boundaries ........................101 iii 4.5Timeintegration...........................104
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