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A Dissertation Entitled Towards Improvement of Numerical A Dissertation entitled Towards Improvement of Numerical Accuracy for Unstructured Grid Flow Solver by Qiuying Zhao Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mechanical Engineering _________________________________________ Dr. Chunhua Sheng, Committee Chair _________________________________________ Dr. Abdollah A. Afjeh, Committee Member _________________________________________ Dr. Glenn Lipscomb, Committee Member _________________________________________ Dr. Ray Hixon, Committee Member _________________________________________ Dr. Terry Ng, Committee Member _________________________________________ Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo December 2012 Copyright 2012, Qiuying Zhao This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of Towards Improvement of Numerical Accuracy for Unstructured Grid Flow Solver by Qiuying Zhao Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Doctor of Philosophy Degree in Mechanical Engineering The University of Toledo December 2012 An effort to improve the numerical accuracy of a three dimensional unstructured grid finite volume scheme is pursued in the present work. Unstructured grid methods have been widely used in computational fluid dynamics for the convenience of modeling complex geometries in realistic applications. In the present work, improvements towards high order unstructured grid schemes are proposed using high order flux formula for spatial discretizations. The Riemann variables on the left and right sides of the interface are reconstructed using quadratic and quartic polynomials composed of both flow variables and their gradients. The high order flux is then calculated using the concept of MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) approach. In order to maintain the accuracy for the finite volume scheme, an innovative method based on Radial Basis Function interpolation is introduced as a substitution to Gaussian quadrature to achieve the higher order surface integration on mixed element unstructured grids. iii The proposed high order improvements for unstructured grid schemes have been tested for a wide range of flows from very low Mach number to supersonic speeds. The observed accuracy for the improved schemes is verified using a benchmark case about an inviscid vortex transporting in a free stream flow. In addition, the ability to capture the tip trailing vortex, which is a major challenge in computational fluid dynamics today, is extensively verified on two vortex dominated viscous flows, a fixed NACA0015 wing at a subsonic Mach number and a rotating NACA0012 hovering rotor at a transonic tip speed. The numerical validations are also performed on two realistic industrial applications including a marine propeller P5168 and a Bell Helicopter aircraft 427 main rotor. Computational results indicate that the methods proposed in the present work can significantly improve numerical accuracy in predicting the strong vortical flows in smooth regions, while maintaining the stability of the schemes in discontinuous regions such as shockwaves. iv To my parents Acknowledgements First, I sincerely thank my advisor, Dr. Chunhua Sheng, for his support and guidance involved in this thesis. His patient supervision, encouragement, and inspiration have helped me as a PhD student over the years at the University of Toledo. His challenging and rigorous research approach has impressed and trained me as a research assistant, as well as, shown me the importance and responsibility of being an engineer. I would like to express my appreciation to Dr. Abdollah A. Afjeh, Chair of Mechanical, Industrial, and Manufacturing Engineering, Dr. Glenn Lipscomb, Chair of Chemical and Environmental Engineering, Dr. Ray Hixon, and Dr. Terry Ng. for serving on my thesis committee. Their advice and suggestions are very valuable to my dissertation work. I thank Dr. Hixon’s systematic tutoring on the CFD course, from which the knowledge laid the foundation of my research work afterwards. I would like to especially thank Dr. Christian Allen from the Department of Aerospace Engineering, University of Bristol, UK, for his patient explanation of the Radial Basic Function method and longtime discussion on my research work. His practical suggestions have gone a long way in shaping my ideas and helped me conquer the difficulties of developing a robust and efficient CFD tool. I am very grateful to Dr. Li Wang from the Graduate School of Computational Engineering, the University of Tennessee at Chattanooga, for discussions about the higher order methods and suggestions on code verifications. vi I would like to thank my colleagues and friends, Kenneth Miller, Ahmed Magdy, and Jingyu Wang in Dr. Sheng’s CFD group, for the good atmosphere of research discussion and friendship. I express my gratitude to my great colleagues and friends from MIME, Adrian Sescu, Daniel Ingraham, Nima Mansouri, Chuanbo Yang, Steven Koester, Boya Zhang, Hao Jiang, Yue Hou, Ting Wen, Yaoying Wang, and Xiaotong Li for the friendships and memories we shared. I truly thank my friend Qi Zhao for her continuous friendship since high school. I would like to express my appreciation to Aaron Kirgesner for his valuable help in correcting this dissertation. Their encouragement and support cheered me up to face challenges from both daily life and research work. I am deeply indebted to my parents for their endless love, encouragement, and understanding throughout my life. Their selfless affection is my great source of power and gives me the confidence to face anything in my life fearlessly. I express my true love and gratitude and dedicate this dissertation to them. vii Table of Contents Abstract ............................................................................................................................. iii Acknowledgements ............................................................................................................ vi Table of Contents ............................................................................................................. viii List of Tables .................................................................................................................... xii List of Figures .................................................................................................................. xiii List of Abbreviations ....................................................................................................... xix List of Symbols ................................................................................................................ xxi 1 Introduction ...................................................................................................................... 1 1.1 Motivation ............................................................................................................ 1 1.2 Literature Review about Higher Order Schemes ................................................. 3 1.2.1 Finite Difference Schemes ............................................................................ 3 1.2.2 Compact Schemes ......................................................................................... 4 1.2.3 Spectral Difference Schemes ........................................................................ 5 1.2.4 Finite Volume Schemes ................................................................................ 5 1.2.5 Spectral Volume Schemes ............................................................................ 7 1.2.6 END and WEND Schemes ........................................................................... 7 1.2.7 Finite Element Schemes ................................................................................ 8 1.2.8 Discontinuous Galerkin Method ................................................................... 9 1.3 Current State of CFD .......................................................................................... 10 viii 1.4 Research Objectives and Scope .......................................................................... 10 1.5 Organization of Dissertation .............................................................................. 12 2 Computational Fluid Dynamics Overview .................................................................... 14 2.1 Historical Perspective of CFD............................................................................ 15 2.2 The Importance of CFD in Virtual Prototyping ................................................. 20 2.3 Procedures of Applying CFD ............................................................................. 22 3 Computational Methodology ......................................................................................... 24 3.1 The CFD Solver ................................................................................................. 24 3.2 Mathematical Formulation ................................................................................. 25 3.2.1 Governing Equations in an Absolute Frame ............................................... 25 3.2.2 Governing Equations in an Rotating Relative Frame ................................. 28 3.2.3 Non-Dimensionalization of Governing Equations ...................................... 29 3.2.4 Preconditioned Governing Equations ........................................................
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