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A Comparison of Euler Finite Volume and Supersonic Vortex Lattice Methods Used During the Conceptual Design Phase of Supersonic Delta Wings

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A Comparison of Euler Finite Volume and Supersonic Vortex Lattice Methods used during the Conceptual Design Phase of Supersonic Delta Wings THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Daniel Guillermo-Monedero, B.S. Graduate Program in Aeronautical and Astronautical Engineering The Ohio State University 2020 Master’s Examination Committee: Dr. Clifford A. Whitfield, Advisor Dr. Richard J. Freuler Dr. Matthew H. McCrink 1 Copyrighted by Daniel Guillermo-Monedero 2020 2 Abstract This thesis uses two different methods to analyze wings in supersonic flows with a focus on preliminary design. The primary goals of this study are to compare the Euler equations finite volume method and supersonic vortex lattice method in predicting surface pressure on wings, and to develop a low-order supersonic vortex lattice method as a baseline tool that can be extended for further wing design and analysis. The supersonic vortex lattice method uses vortical sources to model the flow on the boundary, which replicates the aerodynamic shape of interest, in an inviscid and irrotational flow field and obtain solutions. The Euler equations of flow can be discretized using finite volume methods and can be integrated over the volume of interest and solutions can be obtained over the surface. These mathematically similar methods have a lot of differences in their numerical formulations, which can be critical in the design and analysis of wings. Hence, it is important to understand the key differences of these in order to develop a reliable baseline low-order design tool. To compare these two methods, a flat plate subsonic leading-edge delta wing with a leading-edge factor (βcot(ΛLE)) of 0.6 will be modeled at the same conditions. The lattice method will be coded using MATLAB and the Euler equations will be solved using ANSYS® Fluent. The differential pressure at the camber, aerodynamic coefficients, time to solve, effort to discretize, and other mathematical considerations for both methods are ii compared. As expected, pressure results shown good congruency between both methods. The lift coefficient and moment coefficient show around a 10% difference, while the drag shows the most difference at 30%. Convergence for the supersonic vortex lattice method happens immediately, on-the-order of seconds, while ANSYS® Fluent takes significantly longer, on-the-order of hours. In general, the Euler method is much harder to set up and to discretize (mesh) the domain, but it can show flow structures at any point of the domain. With these results it can be concluded that the supersonic vortex lattice method, when used appropriately, has significant advantages and potential to be used as an effective baseline tool for linear modeling of wings during the early stages of design. The fast convergence, setup, and the few amounts of elements are the most notable strengths of the lattice method. Furthermore, non-linearities can be superimposed in the method and could be used to analyze, for example, active and passive flow control or any geometry of wings. iii Dedication To my family, friends, advisor, instructors, and everyone else that made this thesis possible. iv Acknowledgments I would like to express my deep gratitude to Dr. Clifford Whitfield for his constant guidance and education through my undergraduate and graduate careers. I’m very grateful for his dedication, enthusiasm, and knowledge about design, fluid dynamics, and teaching. He is always an excellent mentor and an exceptional educator for me and many other students at the Ohio State University. I would like to thank my committee members, Dr. Rick Freuler and Dr. Matthew McCrink, for their time and help on the final stages of my thesis. I also would like to express my gratitude to the Mechanical and Aerospace Department at the Ohio State University for the opportunity to be a teacher assistant and cover my first year expenses for my master’s degree. Thanks to my parents, Conchita Monedero and Ernesto Guillermo, for the economic support in my last semester of the degree. And thank you to the Office of International Affairs for the grant that allowed me to finish this degree. A huge thanks to my colleague and friend Rodrigo Auza-Gutierrez for his frequent help on computational fluid dynamics. A very special recognition goes to my family and girlfriend whose constant support and motivation helped me achieve my very best every single day. v Lastly, I would like to acknowledge all my school, undergraduate, and graduate instructors. All of you have made invaluable marks in my life, which make me be the best student and engineer I can be. vi Vita May 13, 1995 ........................................................ Born, Caracas, Venezuela May 2017 .............................................................. B.S. Aeronautical and Astronautical Engineering, The Ohio State University August 2018 to May 2019 .................................... Graduate Teaching Assistant, The Ohio State University Fields of Study Major Field: Aeronautical and Astronautical Engineering vii Table of Contents Abstract ............................................................................................................................... ii Dedication .......................................................................................................................... iv Acknowledgments............................................................................................................... v Vita .................................................................................................................................... vii List of Tables ...................................................................................................................... x List of Figures .................................................................................................................... xi Chapter 1: Introduction ....................................................................................................... 1 1.1 Introduction ............................................................................................................... 1 1.2 Background ............................................................................................................... 6 1.2.1 Overview of Steady, Inviscid Supersonic Flows and Solution Methods for Wings .......................................................................................................................... 6 1.2.2 Singularity Distribution Methods Modeling and the Supersonic Vortex Lattice Method ........................................................................................................................ 7 1.2.3 Computational Fluid Dynamics Modeling....................................................... 10 1.3 Motivation and Main Objectives. ........................................................................... 11 Chapter 2: Singularity Distribution Method: The Supersonic Vortex Lattice Method ... 13 2.1 Governing Equations and Boundary Conditions .................................................... 13 viii 2.2 Numerical Approach ............................................................................................... 17 Chapter 3: CFD Modeling ................................................................................................ 23 3.1 Governing Equations of Fluid Flow ....................................................................... 23 3.2 Computational Modeling and Boundary Conditions .............................................. 23 3.3 Fluid Domain and Grid ........................................................................................... 26 Chapter 4. Results ............................................................................................................. 31 4.1 Supersonic Vortex Lattice Method ......................................................................... 31 4.2 Euler equations CFD ............................................................................................... 37 4.3 Comparison of Methods .......................................................................................... 41 Chapter 5. Conclusions ..................................................................................................... 45 5.1 Conclusions ............................................................................................................. 45 5.2 Future Work Recommendations ............................................................................. 46 References ......................................................................................................................... 48 ix List of Tables Table 1: Mesh Independence Study Results ..................................................................... 28 Table 2: Aerodynamics Coefficients for Delta Wing with Leading Edge βcot(ΛLE)=0.6 at Mach=1.5 and AoA=2.5 deg. ............................................................................................ 33 Table 3: Aerodynamics Coefficients for Delta Wing of Subsonic LE at Mach=1.5 and AoA=2.5 deg. with Euler formulations............................................................................. 37 Table 4: Percentage Difference of Aerodynamic Coefficients for Delta Wing obtained with Euler Formulations and SVLM Methods.................................................................. 41 x List of Figures Figure 1: Types of Flow on Highly Swept
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