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Lagrangian Coherent Structures in Vortex Ring Formation

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Lagrangian Coherent Structures in Vortex Ring Formation Lagrangian Coherent Structures in Vortex Ring Formation Master’s Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Braxton N. Harter, B.S. Graduate Program in Aeronautical and Astronautical Engineering The Ohio State University 2019 Thesis Committee: Dr. James W. Gregory, Advisor Dr. Mohammad Samimy Dr. Matthew H. McCrink Copyrighted by Braxton N. Harter 2019 Abstract Vortex rings are among the only consistent natural feature to achieve effective transport of mass, momentum, and energy in fluids. For most scenarios where each of these transport mechanisms are utilized, efficiency of the vortex generator is parameterized by how much vorticity is entrained by the created vortex ring—the more vorticity that a vortex ring entrains during the formation process, the more efficient each of the transport mechanisms will be. They cannot grow indefinitely, however. Vortex ring formation is known to be limited by a dimensionless ratio that compares the length of an ejected slug of fluid L to its diameter D. This ratio is termed formation time, Ţ = L/D, and has a limiting value of approximately 4. The upper limit of ring growth has been justified by the Kelvin-Benjamin variational principle, which states that vortex ring formation is limited due to an energy deficit of the vortex generator in comparison to the translating vortex ring. This principle forms an elegant analytical theory for predicting vortex ring pinch-off; however, a key limitation of findings reported in the literature is the inability to accurately quantify vortex characteristics early in the formation process (especially before pinch-off). In this thesis, laminar vortex rings are experimentally studied with a piston-cylinder device for formation times well beyond the limiting ratio of vortex ring pinch-off at Ţ ≈ 4. To answer the question of what the physical limiting processes are, the Lagrangian ii coherent structures in vortex ring formation are analyzed using both particle image velocimetry and the background-oriented schlieren technique. Utilizing the finite-time Lyapunov exponent calculation, a presently-developed vortex-identification technique is outlined, allowing for the accurate and consistent quantification of vortex formation parameters throughout the entire generation process. A comprehensive examination of the vortex ring limiting process is presented using modern optical measurement and analysis techniques—a study that has previously been absent from literature. The dynamics of the system indicate that the process is a highly coupled problem; however, findings show that the process is dependent on the spatial characteristics of the vortex system, as opposed to a temporal parameter that quantifies the volume of fluid ejected (i.e., Ţ). Early in the formation process, the irrotational region of the lead vortex accelerates the trailing jet of fluid, entraining it into the ring. After the vortex has advected beyond a noteworthy axial-location zD-1 ≈ 1, the vortex has advected to a location where the irrotational region can no longer accelerate newly ejected fluid. As such, the lead vortex diverges away from the ejected fluid, leading to a bifurcation in the slug. The jet—and its associated shear layer—stretches at this bifurcation location until it reaches a fatal inflection point and breaks down. The proposed limiting process in vortex ring formation is consistent with existing analytical descriptions of pinch-off; present findings reveal the physical phenomena that initialize the process. These insights provide a clear path forward for researchers to maximize vortex ring formation for the effective transport of mass, momentum, and energy in fluids. iii Dedication To those who have touched my life. iv Acknowledgments There are many people who have contributed to my completion of this checkpoint in my life. First and foremost, I must acknowledge Dr. Gregory and Dr. McCrink for their support and guidance through this process. Dr. Samimy’s expertise in shear layers led to insightful discussions when forming this Thesis. I must also acknowledge my lab mates for providing positive diversions throughout. I am grateful for financial support from the Army Research Office though a NDSEG fellowship. Additionally, this work has benefitted from equipment acquired under an Army Research Office DURIP grant (W911NF-12-1-0248). v Vita 2015............................................. Mentor, Student Athlete Enhancement Center, Middle Tennessee State University; Murfreesboro, Tennessee 2016............................................. Supplemental Instruction Leader, Office of Student Success, Middle Tennessee State University; Murfreesboro, Tennessee 2017............................................. B.S. Aerospace, Middle Tennessee State University; Murfreesboro, Tennessee 2016 to present............................ Graduate Researcher, Mechanical and Aerospace Engineering, Aerospace Researcher, The Ohio State University; Columbus, Ohio 2017............................................. University Fellow, The Graduate School of The Ohio State University; Columbus, Ohio 2018 to present............................ NDSEG Fellow, Army Research Office, Department of Defense vi Publications Harter, B., McCrink, M., and Gregory, J., "Identification of Lagrangian Coherent Structures using the Background Oriented Schlieren Method," AIAA Aviation 2019 Forum, 2019, pp. 2810. Harter, B., McCrink, M., and Gregory, J., "Visualization of Lagrangian Coherent Structures in Vortex Formation and Advection," Bulletin of the American Physical Society, 2018. Wang, Z., Henricks, Q., Pandey, A., Sutkowy, M., Harter, B., McCrink, M., Gregory, J., and Zhuang, M., "Impact of Rotor–Airframe Orientation on the Aerodynamic and Aeroacoustic Characteristics of Small Unmanned Aerial Systems," Drones, Vol. 3, No. 3, 2019, pp. 56. Zhu, W., Harter, B., Gregory, J.W., "Characterizing Wave Propagation in an Unsteady Transonic Wind Tunnel," 2018 AIAA SciTech 2018 Forum, 2018, pp. 3568. Sutkowy, M., Harter, B., McCrink, M., and Gregory, J., “Impact of Wake Structure Characteristics on Small-Scale Rotor Performance over a Range of Reynolds Numbers,” American Helicopter Society 74th Annual Forum, 2018. Wang, Z., Pandey, A., Sutkowy, M., Harter, B., McCrink, M., Gregory, J., and Zhuang, M., “A Comprehensive Approach to Study Aerodynamics and Aerocoustics around Small Multicopter Unmanned Aerial Systems,” 2018 AIAA Aerospace Sciences Meeting, 2018, pp. 268. vii Field of Study Major Field: Aeronautical and Astronautical Engineering viii Table of Contents Abstract ...........................................................................................................................ii Dedication ...................................................................................................................... iv Acknowledgments ........................................................................................................... v Vita ................................................................................................................................ vi Publications ................................................................................................................... vii Field of Study ...............................................................................................................viii Table of Contents ........................................................................................................... ix List of Tables ............................................................................................................... xiv List of Figures ............................................................................................................... xv Chapter 1. Introduction to Vortex Rings .......................................................................... 1 Chapter 2. Vortex Ring Formation ................................................................................... 7 2.1 Model for Vortex Ring Formation.......................................................................... 9 2.2 A Limiting Process .............................................................................................. 11 2.3 Constraints on Vortex Ring Growth ..................................................................... 16 2.4 Manipulation of Vortex Ring Formation Number................................................. 17 Chapter 3. Experimental Methodology .......................................................................... 21 ix 3.1 Vortex Ring Generator ......................................................................................... 21 3.1.2 Generator Geometry ...................................................................................... 21 3.1.1 Slug Ejection Control .................................................................................... 23 3.2 Particle Image Velocimetry .................................................................................. 25 3.2.1 Imaging Parameters ...................................................................................... 25 3.2.2 Expression of Vortex Ring Advection Data ................................................... 26 3.3 The Background-Oriented Schlieren Method ....................................................... 29 3.3.1 Imaging Parameters ...................................................................................... 30 3.3.2 The BOS Method Applied
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