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University of Oklahoma Graduate College UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE CONSIDERATION OF LIE SYMMETRY GROUPS IN COMPUTATIONAL FLUID DYNAMICS A DISSERTATION SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY By ERSIN OZBENLI Norman, Oklahoma 2018 CONSIDERATION OF LIE SYMMETRY GROUPS IN COMPUTATIONAL FLUID DYNAMICS A DISSERTATION APPROVED FOR THE SCHOOL OF AEROSPACE AND MECHANICAL ENGINEERING BY Dr. Prakash Vedula, Chair Dr. M. Cengiz Altan Dr. Wilson E. Merchán-Merchán Dr. Jivtesh Garg Dr. Dimitrios Papavassiliou © Copyright by ERSIN OZBENLI 2018 All Rights Reserved. This dissertation is dedicated to my parents and grandparents for their unconditional love and support: Mehmet Özbenli (Father) Ayşe Özbenli (Mother) Remzi Özbenli (Grandfather) Züleyha Özbenli (Grandmother) iv Acknowledgements I would like to express my sincere gratitude to my committee chair and adviser Dr. Prakash Vedula, who has set an example of excellence as a mentor and role model, for his invaluable guidance and support throughout the development of this dissertation. I will be forever thankful not only for the extra hours he took from his busy schedule just to help me overcome problems related to my research, but also for his persistent effort in preparing me for the life in academia. Without his guidance, this dissertation would have not been possible. I would also like to express my deepest appreciation to my committee member Dr. M. Cengiz Altan for his invaluable contributions to my development as a researcher. His dedication to research and persistent efforts to mentor graduate students were truly inspirational to me. Further, I would like to thank my outside committee member Dr. Dimitrios Papavassiliou and other committee members Dr. Wilson E. Merchán-Merchán and Dr. Jivtesh Garg not only for participating in my dissertation committee but also for their valuable comments and feedback. I will be forever grateful. I also want to thank the Ministry of National Education of Turkey for providing me with a scholarship that simply funded me throughout my graduate life. And last but not least, I would like to express my deepest appreciation and sincere gratitude to my parents, for their unconditional love and support, and my wife for her persistent encouragement, support, and love. v Contents Acknowledgements v List of Tables ix List of Figures xi Abstract xvi 1 Introduction 1 1.1 Motivation . .1 1.2 Historical Preface of Group Theory . .2 1.3 Lie Symmetry Groups . .4 1.4 Symmetry Preservation in Numerical Schemes . .6 1.5 Scope of the Dissertation . 11 2 Lie Symmetry Analysis of Differential Equations 16 2.1 Scope of the Chapter . 16 2.2 Introduction to Lie symmetry Analysis . 16 2.3 Applications and Discussion . 23 2.3.1 Chemically Reactive Boundary Layer flow . 23 2.3.2 Boundary Layer flow over a Wedge with Slip . 29 2.3.3 Analysis of Stagnation Point Conditions in an Inviscid, Com- pressible Blunt-Body Flow . 33 3 Numerical Solution of Modified Differential Equations based on Symmetry Preservation 40 3.1 Scope of the Chapter . 40 3.2 Mathematical Formulation and Moving Frames . 41 3.3 Method of Modified Equations . 44 3.3.1 Linear Advection Equation . 44 3.3.2 Inviscid Burgers’ Equation . 47 3.4 Construction of Invariant Modified Schemes . 48 3.4.1 Linear Advection Equation in 1D . 48 3.4.2 Linear Advection Equation in 2D . 52 3.4.3 Inviscid Burgers’ Equation . 54 3.5 Numerical Experiments . 57 3.5.1 Linear Advection Equation in 1D . 57 3.5.2 Linear Advection Equation in 2D . 61 vi 3.5.3 Inviscid Burgers’ Equation . 63 3.6 Chapter Summary . 67 4 High Order Accurate Finite Difference Schemes based on Symme- try Preservation 70 4.1 Scope of the Chapter . 70 4.2 Construction of High Order Invariant Schemes . 72 4.2.1 Linear Advection-Diffusion Equation in 1D . 73 4.2.2 Inviscid Burgers’ Equation . 77 4.2.3 Viscous Burgers’ Equation . 84 4.2.4 Linear Advection-Diffusion Equation in 2D . 86 4.3 Numerical Experiments . 91 4.3.1 Linear Advection-Diffusion Equation in 1D . 91 4.3.2 Inviscid Burgers’ Equation . 94 4.3.3 Viscous Burgers’ Equation . 98 4.3.4 Linear Advection-Diffusion Equation in 2D . 100 4.4 Chapter Summary . 102 5 Construction of Invariant Compact Finite Difference Schemes 106 5.1 Scope of the Chapter . 106 5.2 Mathematical Formulation . 107 5.2.1 Construction of Compact Schemes . 107 5.2.2 Invariantization of Compact Schemes . 109 5.3 Development of Invariant Compact Schemes . 110 5.3.1 Inviscid Burgers’ Equation . 110 5.3.2 Linear Advection-Diffusion Equation in 1D . 114 5.3.3 Viscous Burgers’ Equation . 116 5.3.4 Linear Advection-Diffusion Equation in 2D . 118 5.4 Results and Discussion . 120 5.4.1 Inviscid Burgers’ Equation . 121 5.4.2 Linear Advection-Diffusion Equation in 1D . 123 5.4.3 Viscous Burgers’ Equation . 124 5.4.4 Linear Advection-Diffusion Equation in 2D . 129 5.5 Chapter Summary . 131 6 Construction of Invariant Schemes for Euler Equations 135 6.1 Scope of the Chapter . 135 6.2 Introduction . 136 6.3 Mathematical background and Symmetry Methods . 138 6.3.1 Euler Equations in 1D . 138 6.3.2 Euler Equations in 2D . 140 6.4 Invariant schemes . 141 6.4.1 Invariant Lax-Friedrichs Scheme . 142 vii 6.4.2 Invariant van Leer Flux Splitting scheme . 145 6.5 Results and Discussion . 146 6.6 Chapter Summary . 157 7 Summary and Future Work 158 7.1 Scope of the Chapter . 158 7.2 Research Highlights . 158 7.2.1 Chapter Summaries . 159 7.3 Future Research . 162 7.3.1 Extensions of Scope to Address Limitations . 162 7.3.2 Extensions to More General Problems . 163 Bibliography 168 Appendix A An Invariant MUSCL Scheme for Solution of Euler Equa- tions in 1D 181 Appendix B Lie Symmetry Groups of Euler Equations in 3D 185 Appendix C Lie Symmetry Groups of Incompressible Navier-Stokes Equations 188 viii List of Tables 3.1 Root mean square error (RMSE) and L∞ error in numerical solutions of advection equation obtained from various numerical schemes for τ = 0.01, h = 0.2, and α = 1....................... 60 3.2 Root mean square error (RMSE) and L∞ error in numerical solutions of inviscid Burgers’ equation obtained from various schemes, with τ = 0.01, and h = 0.05 .......................... 63 3.3 Root mean square error (RMSE) and L∞ error in numerical solutions of inviscid Burgers’ equation (in the case of a Riemann problem) obtained from various schemes for τ = 0.01, and h = 0.05 ....... 66 4.1 Root mean square error (RMSE) and L∞ error associated with nu- merical solutions for linear advection-diffusion equation. 92 4.2 Root mean square error (RMSE) and L∞ error associated with nu- merical solutions for inviscid Burgers’ equation. 95 4.3 Root mean square error (RMSE) and L∞ error associated with nu- merical solutions for the viscous Burgers’ equation. 99 5.1 Root mean square error (RMSE) and L∞ error associated with nu- merical solutions (based on compact schemes) for the inviscid Burg- ers’ equation. 121 5.2 Root mean square error (RMSE) and L∞ error associated with nu- merical solutions (based on compact schemes) for the one-dimensional linear advection-diffusion equation. 124 5.3 Root mean square error (RMSE) and L∞ error associated with nu- merical solutions (based on compact schemes) for the viscous Burgers’ equation. 126 5.4 Variation of RMSE and L∞ errors associated with numerical solutions presented in figure 5.7 (left) with respect to the Galilean parameter c. 127 5.5 Variation of RMSE and L∞ errors associated with numerical solutions presented in figure 5.7 (right) with respect to the Galilean parameter c.127 5.6 Root mean square error (RMSE) and L∞ error associated with nu- merical solutions (based on compact schemes) for two-dimensional linear advection-diffusion equation. 129 6.1 Initial configurations for one-dimensional Euler equations. 147 6.2 L∞ and RMSE errors (based on velocity u) for Euler equations in 1D. 148 6.3 Two-dimensional Euler equations. Initial data for case 1........ 153 6.4 Two-dimensional Euler equations. Initial data for case 2........ 154 ix 6.5 Two-dimensional Euler equations. Initial data for case 3........ 155 6.6 Two-dimensional Euler equations. Initial data for case 4........ 156 A.1 Initial configurations for one-dimensional Euler equations. 182 x List of Figures 2.1 Mapping of points on a curve from a source space to a target space. 18 2.2 Stream function (F ), velocity (F 0), and concentration (G) versus the distance from the surface (α). Parameter settings: left : S = 0, γ = 0, β = 0.1, Sc = 0.7, right : S = 0.1, γ = 0.1, β = 0.1, Sc = 0.7 .... 28 2.3 Boundary layer flow over a Wedge. 30 2.4 Representation of stagnation point in a blunt-body flow. 34 3.1 Advection equation (1D). Comparison of profiles of u (at t = 2) ob- tained from the exact solution, first order upwind scheme, and pro- posed invariant scheme is shown in the left figure. Spatial distribu- tions of percentage errors are shown in the right figure. Parameter settings: τ = 0.05, h = 0.2, α = 1, and f = mach............ 58 3.2 L∞ errors associated with numerical solutions of advection equation (1D) versus number of grid points for a fixed Courant number, C = 0.5. 59 3.3 Advection equation (1D). Comparison of profiles of u (at t = 2) ob- tained from the exact solution, and the numerical solutions is shown in the left figure.
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