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A Parametric Study of Formation Flight of a Wing Based on Prandtl's Bell

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A Parametric Study of Formation Flight of a Wing Based on Prandtl's Bell A PARAMETRIC STUDY OF FORMATION FLIGHT OF A WING BASED ON PRANDTL’S BELL-SHAPED LIFT DISTRIBUTION A Thesis presented to the Faculty of California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the Requirements for the Degree Master of Science in Aerospace Engineering By Kyle Sean Lukacovic June 2020 © 2020 Kyle Sean Lukacovic ALL RIGHTS RESERVED ii COMMITTEE MEMBERSHIP TITLE: A Parametric Study of Formation Flight of a Wing Based on Prandtl’s Bell-Shaped Lift Distribution AUTHOR: Kyle Sean Lukacovic DATE SUBMITTED: June 2020 COMMITTEE CHAIR: Aaron Drake, Ph.D. Professor of Aerospace Engineering COMMITTEE MEMBER: Paulo Iscold, Ph.D. Associate Professor of Aerospace Engineering COMMITTEE MEMBER: David Marshall, Ph.D. Professor of Aerospace Engineering, Department Chair COMMITTEE MEMBER: Russell Westphal, Ph.D. Professor of Mechanical Engineering iii ABSTRACT A Parametric Study of Formation Flight of a Wing Based on Prandtl’s Bell-Shaped Lift Distribution Kyle Sean Lukacovic The bell-shaped lift distribution (BSLD) wing design methodology advanced by Ludwig Prandtl in 1932 was proposed as providing the minimum induced drag. This study used this method as the basis to analyze its characteristics in two wing formation flight. Of specific interest are the potential efficiency savings and the optimal positioning for formation flight. Additional comparison is made between BSLD wings and bird flight in formation. This study utilized Computational Flow Dynamics (CFD) simulations on a geometric modeling of a BSLD wing, the Prandtl-D glider. The results were validated by modified equations published by Prandtl, by CFD modeling published by others, and by Trefftz plane analysis. For verification, the results were compared to formation flight research literature on aircraft and birds, as well as published research on non-formation BSLD flight. The significance of this research is two part. One is that the BSLD method has the potential for significant efficiency in formation flight. The optimal position for a trailing wing was determined to be partially overlapping the leading wing vortex core. For a BSLD wing these vortices are located inboard from the wingtips resulting in wingtip overlap and have a wider impact downstream than the elliptical lift distribution (ELD) wingtip vortices. A second aspect is that avian research has traditionally been studied assuming the ELD model for bird flight, whereas this study proposes that bird flight would be better informed using the BSLD. Keywords: [Bell Shaped Lift Distribution, BSLD, Formation Flight, Computational Fluid Dynamics, CFD, Prandtl, Prandtl-D, Avian Flight, Birds, Vortex, Optimization, Induced Drag, Efficiency] iv ACKNOWLEDGMENTS First and foremost, I would like to thank my Dad, John Lukacovic. A father’s love is the compass by which his children learn to navigate, to sail, and to reach their dreams. I would not be here without you. From the bottom of my heart I send my appreciation for all that you do. I would like to thank Dr. Graham Doig for giving me the opportunity to pursue graduate education at Cal Poly. Without this opportunity and his guidance, I would not be at NASA Ames working as a wind tunnel test director. It is a step or two up from the lab in Building 41 but in the grand scheme of things it operates the same. I would also like to thank my advisor Dr. Aaron Drake for his patience and time. Their expertise and professionalism in the field of aerospace engineering and knowledge of aerodynamics has propelled my drive to shoot for excellence. His acceptance of the advisor position at an advanced stage in the project reflects his dedication to his students. I send my deepest gratitude to my distinguished committee members. The feedback I received from Dr. Paulo Iscold, Dr. David Marshall, and Dr. Russell Westphal helped push my thesis. I especially appreciate their willingness to step in and help with my extended thesis process. I thank Albion Bowers and Oscar Murillo for the experience of a lifetime out in the Mojave Desert where I learned of a little thing called the Bell-Shaped Lift Distribution. My internship sending the Prandtl-D wing soaring into the sky led me to the hypotheses I present herein for the wing in formation flight. This research would not have been published to the world without turning me into a believer of the impossible. I would also like to acknowledge my science and math teachers at Sehome High School; Ms. Hankinson, Mr. Toney, Mr. Swets, and Mr. Criez. From the catalytic chemistry, basswood bridge building, composite canoe manufacturing, and robotic design, their enthusiasm had my young mind always striving for more to soak up. They inspired me to pursue a degree in engineering. v A big shout out also goes Dr. Nancy Squires at Oregon State University. Her classes in aerospace engineering and many long office hour talks persuaded me to reach towards the stars and fly towards a masters in aeronautics. I also send thanks to two of my best friends and roommates from my time at Cal Poly, Sabir Utamsing and Matt Nguyen. From the countless hours of watching movies with popcorn, to the uncountable hours sitting in the lab trying to determine how to get the schlieren to work, my fondest memories will always be the experiences I shared with them. I can’t wait to see what they do in the future. To the rest of my family, friends, coworkers, and everyone else who has touched me throughout this journey we call life, I thank you. A whole book can be written for how they have shaped me into the person I am today. Lastly, I want to send love to my Mom for all the support you have given me from day one. This thesis is dedicated to you. vi TABLE OF CONTENTS SECTION PAGE LIST OF TABLES .......................................................................................................................... ix LIST OF FIGURES ......................................................................................................................... x NOMENCLATURE ...................................................................................................................... xii 1. INTRODUCTION ................................................................................................................. 1 1.1. Hypotheses ....................................................................................................................... 2 1.2. Methodology .................................................................................................................... 2 2. PREVIOUS WORK ............................................................................................................... 4 2.1. The Bell-Shaped Lift Distribution ................................................................................... 4 2.1.1. Discovering the Minimum Induced Drag Solution .................................................. 4 2.1.2. A Simplification of Prandtl’s Equations ................................................................ 12 2.1.3. Additional Minimum Induced Drag Studies .......................................................... 14 2.2. Far-Field Induced Drag .................................................................................................. 15 2.3. Prandtl-D Research ........................................................................................................ 17 2.4. Birds and Avian Formation Flight ................................................................................. 20 2.4.1. Beneficial Positioning vs Position Frequency........................................................ 24 2.4.2. Induced Drag Reduction ........................................................................................ 25 2.4.3. Optimal Bird Position ............................................................................................ 25 2.4.4. Beneficial Range .................................................................................................... 26 2.4.5. Vortices .................................................................................................................. 30 2.4.6. Reported Findings .................................................................................................. 32 2.4.7. Summary – Bird Formation Flight ......................................................................... 33 2.5. Formation Flight of Aircraft .......................................................................................... 34 2.5.1. Induced Drag Reduction ........................................................................................ 35 2.5.2. Optimal Wing Position .......................................................................................... 35 2.5.3. Beneficial Range .................................................................................................... 36 2.5.4. Vortices .................................................................................................................. 37 2.5.5. Summary – Aircraft Formation Flight ................................................................... 37 3. THREE-DIMENSIONAL MODELING.............................................................................. 39 3.1. Prandtl-D Geometry ....................................................................................................... 39 3.2. Three-Dimensional Point Manipulation ......................................................................... 41 3.3. CAD Creation ...............................................................................................................
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