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A Discrete Vortex Method Application to Low Reynolds Number Aerodynamic

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A Discrete Vortex Method Application to Low Reynolds Number Aerodynamic A DISCRETE VORTEX METHOD APPLICATION TO LOW REYNOLDS NUMBER AERODYNAMIC FLOWS Thesis Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Master of Science in Aerospace Engineering By Patrick R. Hammer University of Dayton Dayton, OH August 2011 i A DISCRETE VORTEX METHOD APPLICATION TO LOW REYNOLDS NUMBER AERODYNAMIC FLOWS Name: Hammer, Patrick Richard APPROVED BY: ____________________________ __________________________ Aaron Altman, PhD Greg Reich, PhD Advisory Committee Chairperson Committee Member Associate Professor Adaptive Structures Team Lead Department of Mechanical and Aerospace Eng. Air Vehicles Directorate ____________________________ Frank Eastep, PhD Advisory Committee Chairperson Professor Emeritus Department of Mechanical and Aerospace Eng. ____________________________ __________________________ John G. Weber, PhD Tony E. Saliba, PhD Associate Dean Dean, School of Engineering School of Engineering & Wilke Distinguished Professor ii ABSTRACT A DISCRETE VORTEX METHOD APPLICATION TO LOW REYNOLDS NUMBER AERODYNAMIC FLOWS Name: Hammer, Patrick R. University of Dayton Advisor: Dr. Aaron Altman Although experiments and CFD are very powerful tools in analyzing a niche of fluid dynamics problems relevant to developing Micro Aerial Vehicles (MAVs), reduced order methods have shown to be very capable in helping researchers achieve a basic understanding of flow physics with application to highly iterative design processes due to the less computationally expensive nature of the low order models. The current study used one low order method, the Discrete Vortex Method, to model the aerodynamic flow fields and forces around a thin airfoil undergoing a variety of flows, as well as parametric studies to determine the important factors that had to be adjusted to make the results more representative of the physical phenomenon being modeled. Initial investigations validated the code’s use in steady flow and low amplitude unsteady flow cases by comparing it with circulation distributions of various airfoil shapes, the Wagner function, and Theodorsen’s function. The results showed a strong dependency on bound vortex number and time step size. The code was then used to capture the flow behavior around the airfoil for various AIAA Fluid Dynamics Technical Committee Low Reynolds iii Number Working Group (FDTC-LRWG) canonical cases. Implementing the Uhlman method in the Discrete Vortex Method allowed for the calculation of the pressure at the airfoil surface and in the flow field during high angle attack maneuvers. This method proved very capable in calculating the pressures, forces, and force coefficients around the airfoil post-flow separation in the canonical cases where other methods (such as the Unsteady Bernoulli Method) fall short. The code was also tuned with respect to the results with respect to vortex size, leading edge separation strength factor, desingularization function, wake radius size factor, and in the Uhlman method itself to yield an optimal comparison with experimental and CFD results. The study found a bound vortex number of 30, a leading edge separation strength factor of 1.0, the planetary desingularization function, a wake radius size factor of 1.0, and using just the volume integral term on the RHS of the Uhlman method gave the best results for the geometry analyzed. An investigation then determined the dependency of reduced frequency on the lift and drag coefficients for the canonical cases. Finally, the code was used to model a “true perch” by implementing a curve fit function which caused the horizontal free stream velocity to decrease to zero. In this context, the forces were of more interest than the force coefficients since the coefficients experienced anomalous behavior as the free stream velocity approached zero. It was also interesting to find that the code modeled behavior very similar to shear layer instabilities in the LE and TE shear layers, caused by a rippling effect as the bound circulation changed in strength and sign as the LEV and TEV interacting with it. Recommendations were then made to apply the code to airfoils with either fixed or variable camber since camber acts as a high lift device and could prove very beneficial in the design and development of MAVs iv ACKNOWLEDGEMENTS I would first like to thank Dr. Aaron Altman for his tremendous leadership as both an academic, an advisor, and in the aerospace field. Without him, I would not have been brought to this very interesting project with respect to MAV perching, nor would I have developed my interest in aerospace engineering to a passion. I would also like to acknowledge his wife, Servane, his son, Samuel, and his twin daughters, Eloise and Melodie. I would next like to thank my committee members, Dr. Greg Reich and Dr. Frank Eastep. I would especially like to thank Dr. Eastep for his introducing me to the Discrete Vortex Method, which was subject of this thesis. I would next like to thank Dr. Darrel Robertson for his tireless help throughout all of my coding issues, as well as teaching me how to implement various aerodynamic concepts into MATLAB. Without his help, this thesis would not have been completed. I would next like to thank Dr. James Joo for his help as UDRI liaison before joining AFRL as a civilian employee. I would like to thank Dr. Michael Ol for giving me data for which I could compare my low order code with. I would also like to thank my fellow students, Ethan Harper, Ben Hager, Frank Semelmayer, Matt Geyman, John Puttmann, and Danielle Christenson. v I would like to thank my family for their financial and emotional support during all of my academic years. I would finally like to thank Michigan State University for preparing my intellect for the coursework and research that I completed at the University of Dayton. Go Spartans! vi TABLE OF CONTENTS ABSTRACT.......................................................................................................................iii ACKNOWLEDGEMENTS................................................................................................v LIST OF FIGURES............................................................................................................ix LIST OF TABLES.............................................................................................................xx LIST OF SYMBOLS/ABBREVIATIONS......................................................................xxi CHAPTER 1 - INTRODUCTION......................................................................................1 1.1 Background........................................................................................................1 1.2 Literature Review...............................................................................................5 CHAPTER 2 – THIN AIRFOIL THEORY AND THE DISCRETE VORTEX METHOD..........................................................................................................................21 2.1 Potential Flow Theory and Thin Airfoil Theory..............................................21 2.2 Discrete Vortex Method ..................................................................................33 vii CHAPTER 3 – STEADY FLOW AND LOW ANGLE OF ATTACK UNSTEADY AERODYNAMIC VALIDATION....................................................................................47 3.1 Steady Flow Validation...................................................................................47 3.2 Classical Unsteady Aerodynamics..................................................................51 3.3 Unsteady Flow Validation...............................................................................59 CHAPTER 4 – HIGH ANGLE OF ATTACK CANONICAL CASES...............................................................................................................................67 4.1 High Angle of Attack Flow Field Validation...................................................67 4.2 The Uhlman Method.........................................................................................87 4.3 Pressure, Force, and Force Coefficient Calculations Using the Uhlman Method..................................................................................................................92 4.4 Reduced Frequency Dependency..................................................................120 4.5 Application of DVM Code and Uhlman Method to Perching Maneuver......127 4.6 Vortex/Shear Layer Instabilities....................................................................141 CHAPTER 5 – CONCLUSIONS AND RECOMMENDATIONS................................148 REFERENCES...............................................................................................................152 APPENDICES................................................................................................................156 A-1 Obstacles Encountered.................................................................................156 viii LIST OF FIGURES Figure 1-1: Vorticity plots for 40o ramp-hold-ramp case with reduced frequency k of 0.7 shows a good comparison between Ol (left), Lian (middle), and Eldredge (right)1. All three methods show a well defined trailing edge vortex during the pitch up while the leading edge vortex grows. As the plate pitches down, a counter-rotating vortex is shed from the trailing edge while the leading edge vortex continues to grow.....................................................................................................................................6
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