A STUDY OF SOLVING NAVIER-STOKES EQUATIONS WITH A FINITE VOLUME METHOD BASED ON POLYGONAL UNSTRUCTURED GRIDS AND THE COMPUTATIONAL ANALYSIS OF GROUND VEHICLE AERO- DYNAMICS by JIANNAN TAN, B.S. A Dissertation In MECHNICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Siva Parameswaran, Ph.D. Chair of the Committee Jingzhou Yang, Ph.D. Zhaoming He, Ph.D. Sukalyan Bhattacharya, Ph.D. Xinzhong Chen, Ph.D. Fred Hartmeister, Dean of the Graduate School December, 2010 Copyright 2010, Jiannan Tan Texas Tech University, Jiannan Tan, December2010 ACKNOWLEDGMENTS Although the following dissertation is an individual work, I could never complete it without the help from a lot of people. Firstly I’d like to express my sincere gratitude to my advisor, Dr. Siva Parameswaran for his consistent guidance and encouragement in the past five years. His knowledge, expertise and enthusiasm on CFD have always been the driving force throughout my graduate research career at Texas Tech Univer- sity. I want to thank my committee professors, Dr. Jingzhou Yang, Dr. Zhaoming He, Dr. Sukalyan Bhattacharya and Dr. Xinzhong Chen for their kind support and help in the past few years. My thanks also extend to my lab mates: Dongdae Lee, Zixi Chen, Tengxiao Liu, Di- vya R. and Suranga Dharmarathne. Working with them has always been a pleasure, which makes my research progressive and my life cheerful and colorful. Finally, my very special thanks go to my family, especially my wife, Ou Zhang. This piece of work would never exist without her supervision and support on a daily basis. ii Texas Tech University, Jiannan Tan, December2010 TABLE OF CONTENTS ACKNOWLEDGMENTS………………………………………………………………………... ii ABSTRACT……………………………………………………………………………………….. v LIST OF TABLES……………………………………………………………………………….. vi LIST OF FIGURES……………………………………………………………………………... vii 1. INTRODUCTION .................................................................................................1 1.1 The Development of Computational Fluid Mechanics .............................1 1.2 Objectives ...............................................................................................2 1.3 Contents of Dissertation ..........................................................................3 2. THE DISCRETIZATION OF N-S EQUATIONS ......................................................4 2.1 Derivation and Description of the Navier-Stokes (N-S) Equations ..........4 2.2 N-S Equations under Cartesian Coordinating System in A 2-D Case .......5 2.3 Finite Volume Method ............................................................................5 2.4 The Discretization of 2-D N-S Equations ................................................6 2.4.1 Gauss’s Theorem in a 2-D Polygonal Cell Case ............................................... 6 2.4.2 Discretization of 2-D N-S Equations by casting Gauss’s Theorem ..................... 7 2.4.3 Important Variables of the Discretized N-S Equations ....................................... 8 3. GRID STRUCTURE AND GRID GENERATION ....................................................9 3.1 Structured and Unstructured Grids/Mesh ................................................9 3.2 Triangular, Quadrilateral and Polygonal Grids/Mesh ............................ 10 3.3 Cartesian Grids ..................................................................................... 11 3.4 Grid Generation .................................................................................... 12 3.5 The Data Structure of the Mesh ............................................................ 12 3.6 An Example: The Data Structure of a 2-D Mesh File from ANSYS® FLUENT® ............................................................................................................. 14 3.6.1 Point Information Block .................................................................................. 14 3.6.2 Face Information Block ................................................................................... 16 3.6.3 Cell Information Block .................................................................................... 18 3.6.4 User Defined Information Block ...................................................................... 18 3.6.5 The Internal Code and the Corresponding Boundary Condition ....................... 19 3.6.6 The Demonstrated Data structured of the Mesh in Figure 3.5 ......................... 19 iii Texas Tech University, Jiannan Tan, December2010 4. SOLVING THE DISCRETIZED N-S EQUATIONS ............................................... 21 4.1 The Goals of Solving the N-S Equations ............................................... 21 4.2 The Positions of the Flow Variables ...................................................... 21 4.3 The Iterative Method ............................................................................ 22 4.4 Handling Diffusive Term, Convective Term and Pressure Term ........... 23 4.4.1 The Diffusive Term ......................................................................................... 24 4.4.2 The Pressure Term and Least Square Method ............................................... 25 4.4.3 Convective Term and Momentum Interpolation for Face Velocity .................... 27 4.4.4 A New Momentum Interpolation Method ......................................................... 27 4.5 Conclusion............................................................................................ 39 5. CFD VALIDATION ............................................................................................ 40 5.1 Internal Flow in a Straight Channel ....................................................... 40 5.2 Internal Flow in a Z-Pipe ...................................................................... 42 5.3 Internal Flow In a Jet ............................................................................ 45 5.4 Laminar Flow Past a Square Cylinder ................................................... 49 5.5 Conclusion............................................................................................ 54 6. EFFECTS OF CROSS WIND ON SPORT UTILITY VEHICLES (SUV): A COMPUTATIONAL STUDY....................................................................................... 56 6.1 INTRODUCTION ................................................................................ 56 6.2 Aerodynamic Forces and coefficients.................................................... 57 6.2.1 Aerodynamic forces and coefficients .............................................................. 57 6.2.2 Method of determining yaw angle and the resultant wind velocity ................... 58 6.3 CAD models and cfd simulation ........................................................... 59 6.3.1 Vehicle CAD model ........................................................................................ 59 6.3.2 The wind tunnel geometry and boundary configuration ................................... 59 6.3.3 Blockage ratio ................................................................................................ 60 6.3.4 Method of dealing with different angles of attack ............................................. 61 6.3.5 Mesh generation and simulation ..................................................................... 62 6.4 Results and discussion .......................................................................... 63 6.5 Concluding remarks .............................................................................. 65 REFERENCES ........................................................................................................... 68 iv Texas Tech University, Jiannan Tan, December2010 ABSTRACT Navier-Stokes (N-S) equations describe the motion of fluid flow in the nature and they are called the governing equations of fluid flows. Solving Navier-Stokes equations is of great interest to the scientists and researchers. Due to the high nonlinearity, achiev- ing the analytical solutions for the N-S equations is extremely difficult, if not impossi- ble. Thus, people have to switch to numerical solutions with putting on certain restric- tions on the N-S equations. This leads to the development of Computational Fluid Dy- namics (CFD). This dissertation contains two major sections. The first section is about theoretical study of CFD. We go through the whole process that a CFD analysis normally re- quires: generating mesh, setting boundary conditions and achieving numerical solu- tions of N-S equations, and post-processing to achieve flow field plots. An in-house 2- D CFD code based on unstructured polygonal mesh is presented, in which a new mo- mentum interpolation method is developed and implemented to calculate the flow flux on the cell faces. The 2-D code is also validated by comparing the numerical results with widely-known analytical results, if available, or by benchmarking with the results produced by commercial CFD software packages. The second section of this disserta- tion is about one of the applications of CFD in modern auto industry – ground vehicle aerodynamics. The cross wind effect on a sport utility vehicle (SUV) is studied and analyzed using CFD methods and compared with available wind tunnel experimental results. The first section contributes to the philosophy of mechanical physics and the second section aims to fulfill the purpose of engineering. v Texas Tech University, Jiannan Tan, December2010 LIST OF TABLES Table 3.1 Table of Internal Code