The Pennsylvania State University The Graduate School INVISCID UNSTEADY AERODYNAMIC MODELS OF AIRFOILS MOVING NEAR BOUNDARIES A Thesis in Mechanical Engineering by Nilanjan Sen (cid:176)c 2008 Nilanjan Sen Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2008 The thesis of Nilanjan Sen was reviewed and approved∗ by the following: Christopher D. Rahn Professor of Mechanical Engineering Thesis Advisor Laura L. Pauley Professor of Mechanical Engineering Karen A. Thole Professor of Mechanical Engineering Department Head of Mechanical and Nuclear Engineering ∗Signatures are on file in the Graduate School. Abstract There is considerable interest in the development of flapping wing Nano Air Vehicles (NAVs) for military applications. These NAVs operate in the Reynolds number regime 102 −104 which is similar to the Reynolds regime of small insects. It is important to understand the aerodynamics of flapping wing flight in this regime in order to build high performance NAVs. Aerodynamic models that can predict the flow and associated forces are a useful tool to understand flapping wingmechanisms. TheWeis-Foghclap-and-flingmechanism,isusedbycertaininsectsoperating in the low Reynolds number range (102). The Weis-Fogh mechanism generates additional lift compared to conventional flapping wing mechanisms and is therefore advantageous to use in NAVs. This thesis presents a two-dimensional, non-linear, unsteady aerodynamic model of an airfoil inaboundeddomain. Keyconceptsofflappingwingssuchasleadingedgevorticesandwing-wake interaction are included in the model to provide a better prediction of the flow and aerodynamic forces. The model predicts the flow field and the lift and drag forces on the airfoil. The non- linear governing equations for the flow are solved using numerical methods. The Weis-Fogh clap-and-fling mechanism is studied and compared with previous CFD results. iii Table of Contents List of Figures vi List of Tables vii List of Symbols viii Acknowledgments x Chapter 1 Introduction 1 1.1 Aerodynamic Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 2 Airfoil in a Bounded Domain 5 2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Potential Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Sign Conventions and Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Quasi-Steady Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Unsteady Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 3 Numerical Implementation 13 3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Numerical Scheme Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Discretization of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.3 Vortex Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.3.1 Fast Multipole Method . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.4 Vortex Amalgamation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 iv 3.1.5 Limiting the Vortex Strength . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.6 Placement of New Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.7 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 4 Clap and Fling 20 4.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1.1 Implementation of S-C mapping . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.1 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 5 Conclusions and Future Work 27 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Appendix A Method of Images 29 Appendix B Dragonfly Flapping Simulation Using Ansari’s Model 30 Appendix C Fling Results 32 Appendix D Program Description 34 Bibliography 69 v List of Figures 2.1 Flow components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Joukowski mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 The S-C transformation used to map the airfoil and wall (D) into the annular region (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Finite domain for evaluation of fluid forces. . . . . . . . . . . . . . . . . . . . . . 12 3.1 Flowchart depicting numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Vortices shed from an airfoil separated into domains A, B1, B2 and D. . . . . . 17 3.3 Computation time versus number of vortices . . . . . . . . . . . . . . . . . . . . 18 4.1 Simplification of symmetrical 2-airfoil problem to airfoil-wall problem . . . . . . 21 4.2 Dimensionless translational (v/v ) (solid) and angular (ω/ω ) (dashed) ve- max max locity of airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Clap-and-fling numerical results: . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Clap-and-fling numerical results: . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Lift coefficient versus stroke fraction for clap-and-fling . . . . . . . . . . . . . . . 26 4.6 Drag coefficient versus stroke fraction for clap-and-fling . . . . . . . . . . . . . . 26 B.1 Lift and drag per unit span for four strokes. . . . . . . . . . . . . . . . . . . . . . 30 B.2 Vorticity plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 C.1 Lift coefficient versus α during ’fling’ . . . . . . . . . . . . . . . . . . . . . . . . . 33 vi List of Tables 4.1 Table of ’clap-and-fling’ motion variables . . . . . . . . . . . . . . . . . . . . . . 23 vii List of Symbols A stroke amplitude (m), p. 30 0 C inner cylinder corresponding to airfoil, p. 8 1 C outer cylinder corresponding to wall, p. 8 2 F fluid force/length (kgs−2) in inertial frame of reference (ζ), p. 11 Fˆ fluid force/length (kgs−2) in translating frame of reference (ζˆ), p. 11 R radius of cylinder (m), p. 6 Re Reynolds number S outer control surface, p. 12 S inner body surface, p. 12 b V control volume in which fluid forces are calculated, p. 12 Z coordinates of plane where airfoil is mapped to a 2-D cylinder, p. 6 c chord length (m), p. 6 f frequency of stroke (s−1), p. 30 h heave of airfoil (m), p. 6 l lunge of airfoil (m), p. 6 nˆ unit normal, p. 12 q square of ratio of radii of outer and inner cylinder, p. 9 r radial coordinates in ω-plane r radius of inner cylinder in ω-plane, p. 8 1 r radius of outer cylinder in ω-plane, p. 8 2 t time(s) viii u fluid velocity on inner body (ms−1), p. 12 b u fluid velocity on outer control surface (ms−1), p. 12 s v velocity at the edge of airfoil (ms−1), p. 18 edge v maximum translational velocity of airfoil (ms−1), p. 22 max α pitch of airfoil (rad), p. 6 δ distance between two vortices i and j, p. 17 ij δ time step (s) t (cid:178) size factor used during vortex amalgamation, p. 17 γ vorticity (ms−1) γ quasi-steady component of bound vorticity on airfoil (ms−1) 0,af γ unsteady component of bound vorticity on airfoil (ms−1) 1,af γ unsteady component of bound vorticity on wall (ms−1) 1,wall γ vorticity of leading edge wake (ms−1) lv γ vorticity of trailing edge wake (ms−1) wk ω coordinates of plane where airfoil and wall are mapped to an annulus, p. 8 ω rotational constant (rads−1), p. 22 max φ complex potential, p. 5 φ phase lag in wing rotation (rad), p. 30 t ψ stream function, p. 5 ρ density of fluid (kgm−3) θ angle (angular coordinates) in Z or ω plane (rad) ∆θ total angle of rotation of airfoil, p. 22 τ dimensionless time, p. 22 τ , τ dimensionless start of acceleration and deceleration of airfoil, p. 22 accel decel τ dimensionless start of rotation of airfoil, p. 22 turn ∆τ , ∆τ dimensionless duration of acceleration and deceleration of airfoil, p. 22 accel decel ∆τ dimensionless duration of rotation of airfoil, p. 22 rot ζ airfoil frame of reference, p. 6 ζˆ translating frame of reference, p. 6 ζ˜ inertial frame of reference, p. 6 ix Acknowledgments I would like to express my gratitude to all those who gave me the possibility to complete this thesis. IwanttothankmyadvisorDr. ChrisRahnforgivingmethefreedomtoworkonthisproject andprovidevaluableinputthroughoutmyresearch. IwouldalsoliketothankmylabmatesAmir, Deepak and Harish for their support and suggestions. I am deeply indebted to Dr. Chenglie Hu for providing me the software DSCPACK. x