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U NSTEADY FLUID DYNAMICS OVER A LOW ASPECT RATIO PITCHING PLUNGING FLAT PLATE By ADAM B. HART A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 201 3
2 201 3 Adam Hart
3 To Amy
4 ACKNOWLEDGMENTS This project is only possible due to the help and continuing guidance of my advisor also like to extend my gratitude to the rest of my committee: Dr. Louis Cattafesta, Dr. Peter Ifju, and Dr. James Liao. Funding for this research was graciously provided by the University of Florida M echanical and A erospace E ngineering department the Florida Center for Advanced Aero Propulsion (FCAAP) Additionall y I am appreciative of the Science, Mathematics, and Research for Transformation ( SMART ) Scholarship Program which provided the majority of the financial resources to complete this research. I would also like to extend my thank s to all the colleagues I have worked with through o ut Erik Sllstrm Diego Camp o s, Amory Timpe, Jon Dudley Mike Sytsma and Judson Babcock.
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 ABSTRACT ................................ ................................ ................................ ................... 16 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 18 1.1. Unsteady Fluid Structures ................................ ................................ .............. 19 1.1.1. Leading Edge Vortex ................................ ................................ ........... 19 1.1.2. Tip Vortex ................................ ................................ ............................ 21 1.2. Low Aspect Ratio Wings ................................ ................................ ................ 22 1.3. Unsteady Aerodynamics ................................ ................................ ................ 26 1.3.1. Quasi Steady Analysis ................................ ................................ ........ 27 1.3.2. Wagner Effect ................................ ................................ ...................... 28 1.3.3. K ................................ ................................ .................... 28 1.3.4. Virtual Mass ................................ ................................ ......................... 29 1.3.5. Inviscid Unsteady Aerodynamic Flow Estimations ............................... 30 1.4. Periodic Kinematic Motions ................................ ................................ ............ 33 1.4.1. Instantaneous Angle of Attack and Angle of Attack Rate .................... 33 1.4.2. Non Dimensional Parameters ................................ .............................. 34 220.127.116.11. Reduced frequency ................................ ................................ 34 18.104.22.168. Strouhal number ................................ ................................ ..... 35 22.214.171.124. Feathering parameter ................................ ............................. 36 1.4.3. Engineered Investigations ................................ ................................ ... 36 1.5. Objectives ................................ ................................ ................................ ...... 40 1.6. Contrib utions ................................ ................................ ................................ .. 41 2 PARTICLE IMAGE VELOCIMETRY ................................ ................................ ....... 45 2.1. Particle Image Velocimetry Introduction ................................ ......................... 45 2.2. Development of Particle Image Velocimetry ................................ ................... 46 2.2.1. Scheimpflug Criterion ................................ ................................ .......... 48 2.2.2. Calibration ................................ ................................ ........................... 48 2.2.3. Ste reo Vector Field Calculation ................................ ........................... 52 2.2.4. Cross Correlation Algorithms ................................ ............................... 53 2.3. Particle Image Velocimetry Measurement Accuracy ................................ ...... 57 2.3.1. General PIV Error Analysis ................................ ................................ .. 57 126.96.36.199. Velocity measurement error ................................ .................... 58 188.8.131.52. Particle displacement error ................................ ..................... 59 184.108.40.206. Magnification error ................................ ................................ .. 60
6 220.127.116.11. Temporal error ................................ ................................ ........ 60 2.3.2. Two Dimensional Two Component Perspective Error ......................... 61 2.3.3. Two Dimensional Three Component Perspective Error ...................... 62 18.104.22.168. Symmetric viewing angles configuration ................................ 63 22.214.171.124. General setup ................................ ................................ ......... 64 2.3.4. Cross Correlation Accuracy ................................ ................................ 66 126.96.36.199. Peak locking error ................................ ................................ ... 66 188.8.131.52. Image reconstruction error ................................ ...................... 67 3 FACILITY AND CHARACTERIZATION EXPERIMENTS ................................ ........ 74 3.1. Low Speed Wind Tunnel Overview ................................ ................................ 74 3.1.1. ACF Entrance/Contraction Section ................................ ...................... 75 3.1.2. ACF Test Section ................................ ................................ ................ 75 3.1.3. ACF Diffuser/Fan Section ................................ ................................ .... 76 3.2. ACF Validation Experiments ................................ ................................ .......... 77 3.2.1. Mean Flow Uniformity ................................ ................................ .......... 77 3.2.2. Turbulence Intensity ................................ ................................ ............ 79 3.3. Dynamic Motion Rig ................................ ................................ ....................... 82 4 AE RODYNAMIC FORCES CALCULATED FROM FLOW FIELD MEASUREMENTS ................................ ................................ ................................ 94 4.1. Two Dimensional Unsteady Lift Estimations ................................ .................. 94 4.2. Static Lift Estimation Technique ................................ ................................ ..... 95 4.3. Unsteady Lift Estimation Technique ................................ ............................. 101 4.3.1. Moving Control Volume ................................ ................................ ..... 102 4.3.2. Unsteady Wing Control Volume ................................ ......................... 104 4.4. Summary ................................ ................................ ................................ ...... 105 5 INVESTIGATIONS AND INSTRUMENTATION ................................ .................... 109 5.1. Experimental Investigations ................................ ................................ ......... 109 5.1.1. Aerodynamic Parameter Selection ................................ .................... 110 5.1.2. General LEV and TV Development Experiments ............................... 111 5.1.3. Quasi steady LEV Response Experiments ................................ ........ 115 5.2. PIV Measurements ................................ ................................ ....................... 117 5.2.1. PIV Hardware ................................ ................................ .................... 117 5.2.2. PIV Setups ................................ ................................ ......................... 118 184.108.40.206. Streamwise two dimensional, two component PIV ............... 118 220.127.116.11. Streamwise two dimensional, three component PIV ............. 119 18.104.22.168. Spanwise two dimensional, three component PIV ................ 121 5.3. Aerodynamic Lift Measurements ................................ ................................ .. 123 5.3.1. Sting Balance Overview ................................ ................................ .... 123 5.3.2. Sting Balance Measurement Techniq ue ................................ ............ 124
7 6 FLOW FIELD RESULTS ................................ ................................ ....................... 136 6.1. Effect of Angle of Attack on Unsteady Vortex Development ......................... 136 6.1.1. Leading Edge Vortex Development ................................ ................... 136 22.214.171.124. General response ................................ ................................ 137 6. 1.1.2. Quasi steady response ................................ ......................... 139 6.1.2. Tip Vortex Development ................................ ................................ .... 141 6.1.3. Leading Edge Vortex/ Tip Vortex Interaction ................................ ..... 144 6.2. Effect of Angle of Attack Rate and Rotation Rate on Unsteady Vortex Development ................................ ................................ ................................ 146 6.2.1. Leading Edge Vortex Development ................................ ................... 146 126.96.36.199. General development ................................ ........................... 147 188.8.131.52. Quasi steady response ................................ ......................... 149 6.2.2. Tip Vortex Development ................................ ................................ .... 151 6.2.3. Leading Edge Vortex/Tip Vortex Interaction ................................ ...... 153 6.3. Summary ................................ ................................ ................................ ...... 156 7 AERODYNAMIC LIFT RESULTS ................................ ................................ ......... 191 7.1. Static Lift ................................ ................................ ................................ ...... 191 7.2. Quasi Steady Lift ................................ ................................ .......................... 197 7.3. Dynamic Lift Response ................................ ................................ ................ 202 7.4. Summary ................................ ................................ ................................ ...... 208 8 CONCLUDING REMARKS ................................ ................................ ................... 233 8.1. Summary ................................ ................................ ................................ ...... 233 8.2. Future Wo rk ................................ ................................ ................................ 23 9 8.2.1. Oscillation Frequency Effects ................................ ............................ 239 8.2.2. Wing Planform ................................ ................................ ................... 240 8.2.3. Flexible Wing ................................ ................................ ..................... 240 APPENDIX A VELOCITY MEASUREMENT ERROR CALCULATIONS ................................ ..... 242 B QUASI STEADY MODEL PARAMETER AND KINEMATIC PROFILES ............... 244 REFERENCE LIST ................................ ................................ ................................ ...... 248 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 259
8 LIST OF TABLES Table page 3 1 Turbulence Intensity as a function of mean velocity. ................................ .......... 91 5 1 Model Parameter Design Space ................................ ................................ ....... 127 5 2 RMS error associated with normalized plunge and pitch kinematic motions. ... 128 5 3 Error associat ed with model parameters for each kinematic motion. ................ 130 5 4 0.10 0.375 Specifications ................................ ......... 135
9 LIST OF FIGURES Figure page 1 1 Schematic of plunge and pitching kinematic motion s. ................................ ........ 44 2 1 Schematic for Scheimpflug condition ................................ ................................ .. 69 2 2 LaVision two level calibration target ................................ ................................ ... 69 2 3 Evaluation of PIV recordings using double framed images and a multiple interrogation window technique to calculate velocity vectors .............................. 70 2 4 Schematic of FFT cross correlation algorithm ................................ .................... 70 2 5 Adaptive multi pass correlation technique with a constant interrogation window size ................................ ................................ ................................ ........ 71 2 6 Schematic of 50% i nt errogation window o verlap ................................ ................ 71 2 7 Schematic of particle trajectory over a time interval ( t). ................................ .... 72 2 8 Schematic of 2D2C PIV setup. ................................ ................................ ........... 72 2 9 Schematic of 2D3C PIV setup. ................................ ................................ ........... 73 2 10 The peak locking mechanism for adding bias towards the nearest high data point. ................................ ................................ ................................ ................... 73 3 1 ACF flow entrance section of wind tunnel ................................ ........................... 84 3 2 Schematic of f low e ntrance s ection ................................ ................................ .... 84 3 3 Schematic of h oneycomb/screen pack section ................................ ................... 85 3 4 Test section wall schematic ................................ ................................ ................ 85 3 5 Flow entrance to the test section ................................ ................................ ........ 86 3 6 Flow exit from the test section. ................................ ................................ ........... 86 3 7 ACF diffuser/fan section of wind tunnel ................................ .............................. 87 3 8 Schematic of ACF diffuser/exit section ................................ ............................... 87 3 9 Flow Uniformity Pressure Rake Setup ................................ ................................ 88 3 10 Normalized, mean streamwise velocity profile f or a freestream velocity of 2.0 m/s at various streamwise locations. ................................ ............................. 89
10 3 11 Normalized, mean streamwise velocity profile f or a freestream velocity of 4.0 m/s at various streamwise locations. ................................ ............................. 90 3 12 Normalized centerline velocity profiles for various streamwise loca tions with a uniform core of 2. m/s.. ................................ ................................ ....................... 90 3 13 Power spectral density. ................................ ................................ ...................... 91 3 14 Coherence between hot wire and pressure signals ................................ ............ 92 3 15 Dynamic Motion Rig ................................ ................................ ........................... 92 3 16 Sting Mechanism ................................ ................................ ................................ 93 4 1 Cont rol volume for computation of total lift. ................................ ...................... 107 4 2 Parabolic Fit Profile ................................ ................................ .......................... 107 4 3 Schematic of moving control volume in the x y plane ................................ ....... 108 4 4 Schematic of fixed control volume with oscillating wing in the x y plane .......... 108 5 1 Angle of attack profiles for pure plunge (PP) and pitch plunge kinematic motions. ................................ ................................ ................................ ............ 125 5 2 Angle of attack rate profiles for pure plunge (PP) and pitch plunge kinematic motions. ................................ ................................ ................................ ............ 125 5 3 Pitch rate profiles for pure plunge (PP) and pitch plunge kinematic motions. ... 126 5 4 Plunging and pitching kinematic motions defined for pure plunge (PP) and pitch plunge motions with phase lags of 90 75 and 30 .............................. 127 5 5 Measured kinematic motions compared with commanded motions for pure plunge and pitch plunge motions.. ................................ ................................ .... 128 5 6 Angle of attack profiles estimated from the encoder feedback compared to the commanded profile for each kinematic motion. ................................ .......... 129 5 7 Angle of attack rate profiles estimated from the encoder feedback compared to the commanded profile for each kinematic motion ................................ ....... 129 5 8 Pitch rate profiles estimated from the encoder feedback compared to the commanded profile for each kinematic motion. ................................ ................ 130 5 9 Experimental aerodynamic parameter design space for the general (black) and quasi steady (red) investigations. ................................ .............................. 131
11 5 10 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 10.0 and 40 /s ................................ ................... 131 5 11 Schematic of streamwise 2D2C PIV setup ................................ ....................... 132 5 12 Schematic of image overlapping and weighting ................................ ............... 132 5 13 Streamwise of streamwise 2D3C PIV Setup ................................ .................... 133 5 14 Schematic of spanwise 2D3C PIV setup. ................................ ......................... 133 5 15 Schematic of PIV planes where spanwise 2D3C measurements are acquired. 134 5 16 PIV wake field of view schematics.. ................................ ................................ .. 134 5 17 Schematic of flat plate and tare mounting configurations.. ............................... 135 6 1 The normalized, mean streamwise velocity for various angles of attack at z/c=0.50.. ................................ ................................ ................................ .......... 159 6 2 The normalized, mean vertical velocity for various angles of attack at z/c=0.50.. ................................ ................................ ................................ .......... 160 6 3 Normalized, mean spanwise vorticity at various angles of attack at z/c=0.50. 161 6 4 1 criteria at various angles of attack at z/c=0.50. ................................ ............ 161 6 5 Normalized streamwise velocity at z/c=0.50 for various angles of attack at an angle of attack rate equal to 40.0 /s. ................................ ............................... 162 6 6 Normalized vertical velocity at z/c=0.50 for various angles of attack at an angle of attack rate equal to 40.0 /s. ................................ ............................... 163 6 7 Normalized spanwise vorticity at z/c=0.50 for various angles of attack at an angle of attack rate equal to 40.0 /s. ................................ ............................... 164 6 8 1 criterion at z/c=0.50 for various angles of attack at an angle of attack rate equal to 40.0 /s. ................................ ................................ .............................. 165 6 9 Normalized, mean velocity and streamwise vorticity associated with TV at x/c=0.6 for a pitch rate equal to 0.0 rad/s ranked as a function of angle of attack. ................................ ................................ ................................ ............... 166 6 10 Normalized, mean velocity and streamwise vorticity associated with TV at x/c=0.6 for a pitch rate equal to 0.0 rad/s rank ed as a function of angle of attack rate. ................................ ................................ ................................ ........ 167 6 11 Q criterion of tip vortex core for an angle of attack equal to 8.0 ..................... 168
12 6 12 Schematic of vortex core displacement from wing surface ( ). ........................ 168 6 13 Tip vortex core location with respect to surface of wing throughout pure plunge kinematic motion. ................................ ................................ .................. 169 6 14 Schematic of LEV/TV structures visualized through iso surfaces of the normalized freestream velocities equal to 1 (yellow) and 1.2 (red).. ................. 170 6 15 Normalized streamwise and spanwise components of velocities at various angle of attack rates at x/c=0.1 and 0.2. ................................ ........................... 171 6 16 Normalized mean spanwise velocity at z/c=0.50 with a pitch rate of 0.0 rad/s. 172 6 17 Normalized spanwise helicity at z/c=0.50 with a pitch rate equal to 0.0 rad /s. .. 173 6 18 Normalized streamwise and vertical velocity components along with 1 criterion at z/c=0.50. ................................ ................................ ......................... 174 6 19 Normalized streamwise and vertical velocity components of pitching, flat plate. ................................ ................................ ................................ ................. 175 6 20 The n ormalized streamwise and vertical velocity along with the 1 criterion at z/c=0.50 at angles of attack 12.9 and 0.7 ................................ ..................... 175 6 21 T he normalized streamwise and vertical components of velocity and 1 criterion at a spanwise location of z/c=0.50 ................................ ..................... 176 6 22 Normalized streamwise velocity at an angle of attack and angle of attack rate equal to 10.0 and 40.0 /s respectively. ................................ .......................... 177 6 23 1 criterion at an angle of attack and angle of attack rate equal to 10.0 and 40.0 /s respectively. ................................ ................................ ......................... 178 6 24 Normalized streamwise velocity at an angle of attack and angle of attack rate equal to 16.0 and 40.0 /s respectively. ................................ .......................... 179 6 25 1 criterion at an angle of attack and angle of attack rate equal to 16.0 and 40 /s respectively. ................................ ................................ ............................ 180 6 26 Normalized streamwise velocity for varying angles of attack and angle of attack rates for a constant pitch rate equal to 0.0 rad/sec. ............................... 181 6 27 1 criteria for varying angles of attack and angle of attack rates at a constant pitch rate equal to 0.0 rad/sec. ................................ ................................ ......... 182 6 28 Normalized spanwise circulation magnitude associated with the LEV for each quasi steady kinematic motion. ................................ ................................ ........ 183
13 6 29 Normalized flow field velocities and spanwise vorticity associated with the TV at x/c=0.6. ................................ ................................ ................................ ......... 184 6 30 Tip Vortex development at x/c=0.6 for a pitch rate equal to 0.9rad/s. ............... 185 6 31 Normalized displacement of the TV core from the wing surface at a nominal angle of attack equal to 13.0 ................................ ................................ .......... 185 6 32 Normalized displacement of the TV core from the wing surface at a nominal angle of attack equal to 8.0. ................................ ................................ ............ 186 6 33 Contour plots of normalized streamwise and spanwise velocities two opposite sign angle of attack rates at x/c=0.1 and 0.2. ................................ ..... 186 6 34 Contour plots of normalized streamwise and spanwise velocity at chordwise locations x/c=0.1 and 0.2. ................................ ................................ ................. 187 6 35 Normalized, spanwise helicity for varying angles of attack and angle of attack rates for a constant pitch rate equal to 0.0 rad/sec. ................................ .......... 1 88 6 36 Normalized spanwise helicity at an angle of attack and angle of attack rate equal to 10.0 and 40/s respectively. ................................ ............................. 189 6 37 Normalized spanwise helicity at an angle of attack and angle of attack rate equal to 16.0 and 120/s respectively. ................................ ........................... 190 6 38 Schematic of inward and outward spiraling LEV structure ................................ 190 7 1 Static lift measured and estimated for an aspect ratio two, flat plate at various ................................ ................................ .......................... 212 7 2 C L vs static angle of attack. ................................ ................................ ............... 212 7 3 Normalized, mean streamwise and vertical wake velocities at a static angle of attack equal to .. ................................ ................................ .................. 213 7 4 Normalized, mean streamwise and vertical wake velocities at a static angle of attack equal to ................................ ................................ ...................... 214 7 5 Normalized, mean streamwise and vertical wake velocities at a static angle of attack equal to ................................ ................................ ...................... 215 7 6 Normalized, mean streamwise and vertical wake velocities at a static angle of attack equal to ................................ ................................ .................... 216 7 7 Normalized, mean streamwise and vertical wake velocities at a static angle of attack equal to ................................ ................................ .................... 217 7 8 Schematic of the tip vortex velocity components ................................ .............. 218
14 7 9 Normalized, mean streamwise and vertical wake velocities at a static angle of attack equal to ................................ ................................ ..................... 219 7 10 C l,y vs static angle of attack. ................................ ................................ ............. 220 7 11 Mean coefficient of lift as a function of angle of attack for one kinematic cycle. ................................ ................................ ................................ ................ 220 7 12 Mean coefficient of lift for varying angle of attack rates at constant angles of attack. ................................ ................................ ................................ ............... 221 7 13 Quasi steady (dashed) lift estimation and measured lift (solid) profiles. ........... 221 7 14 Measured (solid line) and lifting line approximation (dash dot lines) of temporal lift. ................................ ................................ ................................ ...... 222 7 15 Coefficient of lift throughout a pure plunge cycle. ................................ ............. 222 7 16 Wake velocity fields at time 0.00 ................................ ................................ ...... 223 7 17 Wake velocity fields at time 0.14 ................................ ................................ ...... 224 7 18 Wake velocity fields at time 0.26 ................................ ................................ ...... 225 7 19 Wake velocity fields at time 0.46 ................................ ................................ ...... 226 7 20 Wake velocity fields at time 0.58. ................................ ................................ ..... 227 7 21 Wake velocity fields at time 0.65 ................................ ................................ ...... 228 7 22 Wake velocity fields at time 0.71 ................................ ................................ ...... 229 7 23 Wake velocity fields at time 0.82 ................................ ................................ ...... 230 7 24 C l,y profiles at various times in the pure plunge kinematic motion. .................... 231 7 25 C l,y profiles for static angles of attack and instantaneous angles of attack. ..... 231 7 26 Pure plunge kinematic motion dynamic force response ................................ ... 232 B 1 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 10.0 and 80 /s ................................ ................... 244 B 2 Aerodynamic parameter and kinematic profiles at an angle of atta ck and angle of attack rate equal to 10.0 and 120 /s ................................ ................. 244 B 3 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 13.0 and 40 /s ................................ ................... 245
15 B 4 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 13.0 and 80 /s. ................................ .................. 245 B 5 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 13.0 and 120 /s ................................ ................. 246 B 6 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 16.0 and 40 /s. ................................ .................. 246 B 7 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 16.0 and 80 /s. ................................ .................. 247 B 8 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 16.0 and 120 /s ................................ ................. 247
16 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy UNSTEADY FLUID DYNAMICS OVER A LOW ASPECT RATIO PITCHING PLUNGING FLAT PLATE By Adam B. Hart May 2013 Chair: Lawrence Ukeiley Major: Aerospace Engineering Natural fliers have shown the capability to utilize unsteady fluid phenomena to increase their flight performance. As engineered vehicles operating in equivalent flight regime s are developed, a m ethod to predict the development and aerodynamic response to the se unsteady fluid phenomena is desired. This dissertation utilizes various aerodynamic parameters such as the instantaneous angle of attack, angle of attack rate, and pitch rate to classify the leading edge vortex (LEV) and tip vortex (TV) development o ver a low aspect ratio, flat plate driven through various pitch plunge kinematic motions. Particle Image Velocimetry (PIV) measurements are utilized assess the vortex structures along the leading edge and wing tip. The aerodynamic response to the vortex structures are measured and computed utilizing a sting balance and control volume technique respectively. The LEV development is dominated by the angle of attack whereas t he angle of attack rate alter s the development of the LEV at constant angle s of atta ck The pitch rate response was minimal at angle s of attack greater than 10.0 Similarly, t he TV development is shown to be a function of angle of attack and angle of attack rate. The LEV structure formed an inward spiraling structure at angles of attack below 10.0 With the increase in angle of attack or decrease in angle of attack
17 rate, the LEV transitioned to an outward spiral. The aerodynamic lift associate with the outward spiraling LEV is more responsive to changes in the angle of attack rate due to additional momentum drive n into the TV structure thereby increasing the three dimensional flow on the aft of the wing Furthermore, a control volume lift estimation technique suggests the unsteady vortex structures dynamically alter the bound circulation over the wing Th e spanwise distribution of this circulation will have to be accounted for to increase the accuracy of current lifting line, unsteady models.
18 CHAPTER 1 INTRODUCTION Natural fliers have unique flight capabilities which have not been fully realized in current engineered flight vehicles This is a consequence of an absence of the full understanding of the flow physics associated with low Reynol ds number unsteady aerodynamics. Biologist have been studying small winged flapping vehicles for decades and have observed the ability of natural fliers to utilize unsteady vortex structures to enhance their flight characteristics. As the need for designi ng smaller flight platforms rises, fluid dynamists have become more involved which has culminat ed in increased experimental and computational studies on the aerodynamics and flow features associated with natural flier flight on various engineered wing plan forms. Though significant strides have been made, accurately modeling the fluid dynamics associated with flapping wing ed flight has proven to be a challenge. This study provides further insight into the dynamic development of the unsteady vortex structure s and their unsteady aerodynamic response The unsteady vortex structure development is characterized as a function of various aerodynamic parameters. Furthermore, t he unsteady aerodynamic response to these parameters is assessed This provides insight into the aerodynamic parameters driving the unsteady flow physics such that they can be incorporated into theoretical models in the future to achieve higher accuracies. The remainder of this chapter contains a review of research pertaining to unsteady fluid phenomena and unsteady aerodynamic s. Finally, it establishes the goals and final outcomes of this work. This chapter is followed by Chapter 2 which describes the
19 Particle I mage V elocimetry (PIV) measurement technique utilized to obtain the flow field information. Chapter 3 presents the A erodynamic C harac terization F acility (ACF) and characterization experiments performed on the low speed wind tunnel utilized in this study. Chapter 4 describes a control volume (CV) technique developed to compute the lift on a static and oscillating wing. Chapter 5 d iscusses the aerodynamic parameters investigated and the unsteady kinematic motions utilized to achieve these parameters Chapter 6 and Chapter 7 analyze the unsteady vortex development and unsteady lift respectively. Finally, Chapter 8 summarizes the work that has been accomplished and describes future investigations to build upon the results obtained in this study 1.1. Unsteady Fluid Structures Un steady flow structures such as the leading edge vortex and tip vortex have been shown to develop over various natural flier s and engineered planforms driven by dynamic kinematic motions. This section review s some notable f eatures of these vortical structures present over various wing types. Understanding the characteristics of each structure and its aerodynamic response is pertinent in developing a full understanding of the governing flow physics. 1.1.1. Leading E dge V ortex The leading edge vortex (LEV) was realized by Elli ngton [ 1 ] as an unsteady vortex structure cause d by flow separation at the leading edge of a wing. The LEV extended along the span of the wing. Natural fliers have been shown to utilize LEVs to increase their aerodynamic performance. In 2010, an investigation by Tobalske [ 2 ] measured the unsteady fluid structures over a hummingbird in flight. He concluded the downstroke of the humming bird account ed for 75% of the total lift generation during one flapping stroke while the upstroke only produces the remaining 25%. LEVs were shown to be
20 generated over the wing in the middle of the downstroke. Tobalske concluded that LEVs provided a large contribution o f the total bound circulation over the wing, which Delayed stall is the mechanism by which the LEV extends the lift to larger angles of attack greater than the angle of attack at which stall occurs on a st eady wing. This mechanism can improve the aerodynamic performance of natural fliers [ 3 4 5 ] Wu et al. [ 6 ] showed that the lift of a model fruit fly wing at a Reynolds number between 100 1500 was primarily generated by a LEV and delayed stall. McCroskey [ 7 ] classified th e delayed stall mechanism into two regimes: light and deep stall. Light stall occurs at angles of attack slightly greater than the static stall angle of attack where the scale of the viscous fluid is on the order of the airfoil thickness. Deep stall occurs a t angles of attack much greater than the static stall angle of attack. The initial transition to stall occurs when the LEV is shed downstream away from the boundary layer. In this deep stall regime, Freymuth [ 8 ] showed the delayed stall mechanism is responsible for large thrust coefficients. Dickinson et al. [ 9 ] investigated the ability of unsteady fluid mechanisms to provide the additional aerodynami c forces necessary for insect flight. Particularly, they investigated the delayed stall mechanism Dickinson impulsively started a model fruit fly wing at a high angle of attack in a low Reynolds number flow regime. The lift was enhanced by dynamic stall d ue to the generation of a LEV. After 2 3 chord lengths of travel, the LEV was shown to slowly shed from the wing, resulting in a slow decrease in lift. High lift values were attained during this shedding which occurred throughout the travel chord lengths of 3 5. Th is result lead Dickinson to postulate that the transient
21 development of the LEV system might contribute to the unsteady lift associated with insect flight. Rapid wing rotation occurs when natural fliers transition between the downstroke to upstr oke. Dickinson [ 10 ] assessed t he aerodynamic consequences of wing rotation about a constant chord location The lift on the wing was increased significantly when the vort icity generated by the wing throughout the downstroke was maintained over the wing throughout the wing rotation. This contribution of lift was minimal when averaged over the complete wing stroke ; however this stu dy does show the ability of utilizing wing kinematics to capture unstea dy vortex structures to increase the overall aerodynamic performance. 1.1.2. Tip V ortex The tip vortex (TV) is a nother vortical flow structure that has been investigated extensively The TV is typically a spiraling structure along the wing tip that develops on th e suction side of the wing A Batchelor vortex model which is indicative of the TV relates the rotational velocity of the TV to the overall circulation by (1 1) where is the vortex circulation, is the radial distance, and is the effective size of the vortex core with Gaussian axial vorticity distribution [ 11 12 ] The vortex circulation is proportional to ; therefore, increase s in cir culation result in larger downwash onto the surface of the wing introducing significant three dimensional spanwise flow effects. T he influence of the TV on the aerodynamic lift is typically minimized through the utilization of large aspect ratio wings oper ating in large Reynolds number flow regimes. As flight platform s become smaller and low aspect ratio wings are utilized, the influence of the
22 TV on the aerodynamics is increased Furthermore, with the development of the LEV, there exists an interaction bet ween these two unsteady vortex structures. I t is desired understand the flow physics associated with the interaction of the LEV and TV ; therefore, the ne xt section discusses low aspect ratio wing investigations which pertain to the development of unsteady vortical structures. 1.2. Low Aspect Ratio Wings Many biological fliers utilize low aspect ratio wings which result in significantly different flow characteristics than that of large aspect ratio wings [ 13 ] It is of interest to understand the influence of the three dimensional flow generated by TVs o ver low aspect ratio wings. The consequences of the three dimensional effects have a capability to significantly alter the develo pment of unsteady vortical stru ctures generated over the wing. Ellington et al. [ 14 ] provides insight into the stability of the LEV on the hawkmoth Manduca sexta They showe d the development of a LEV wa s maintained throughout th e downstroke of a flapping kinematic motion S moke rake visualizations showed the LEV resembling a conical spiral along the leading edge of the wing t hat enlarged as it was advected in the s tream wise direction along the wing. They postulated that the outwa rd spanwise flow was essential for the stability of the LEV by preventing the flow to accumulate into a large vortex that would be unstable under a two dimensional analysis. The LEV remained stable and attached to the wing until just after the middle of th e downstroke. At this moment, the LEV began to breakdown at 60% 70% of the wing length. The wing tip region was shown to completely separate from the surface of the wing at the bottom half of the downstroke. They postulated that separation occurred from th e deceleration of the wing at the bottom of the downstroke. Their investigation
23 provides insight into the ability of three dimensional effects to alter the stability and spanwise development of the LEV. Experiments by Pelletier et al. [ 15 ] and Torres et al. [ 16 ] provid e a fundamental understanding on the influence of aspect ratio on the static aerodynamic lift in a low Reynolds number flight regime. Pelletier et al. [ 15 ] showed the lift slope ( ) decreas ed as th e semi aspect ratio (sAR) of a rectangular wing is decrease d Furthermore, Torres showed large nonlinearities present in the li ft curves as a function of angle of attack especially at aspect ra tios below 1.25. Low aspect ratio wings were also indicative of large coefficients of lift at high angles of attack. The flows induced by these wings were shown to be significantly different f rom that of large aspect ratio, high Reynolds number flight regimes due to viscous effects and three dimensionality effects induced from the TVs Torres presented indications of this at large angles of attack where the mechanism for generating lift is the TVs Carmichael [ 17 ] characterized separation regions similar to the TVs and showed their ability to decrease the local of the wing decreased as the strength of the TV increased as a function of downstream distance along the wing tip ft on the wing in a highly nonlinear fashion. Birch et al. [ 18 ] investigated two theories thought to stabilize the LEV during hover These investigations were based on observations of natural fliers hav ing ability to maintain a LEV throughout their downstrokes [ 19 20 ] The first theory evaluated a reduction in energy through the utilization of a spiral ing vortex [ 14 21 ] postulated by Ellington in 1996. They performed P article I mage V elocimetry (PIV) measurements on
24 the model of a Drosophila flapping wing at Reynolds numbers between 100 and 250. Their results showed the spanwise velocity through the LEV to be only 2% 5% of the average wing tip velocity. Therefore, it was concluded that a significant spiral ing LEV was not developed This findi ng was contradictory to the spiral vortex observed on 3 It was postulated that a Reynolds number discrepancy between the two investigations might be the consequence for the con tradictory results. The second theory tested by Birch et al. was the ability of the spanwise flow resulting from the TVs having the ability to hinder the growth of the LEV. Fences were added on the wing to hinder axial flow along the core of the LEV. This resulted in minimal change s to the structure of the LEV. Alternatively, rearward facing fences blocking axial flow along the aft end of the wing resulted in a decrease in the LEV vortex strength which culminated in a 25% drop in the net lift force. T he re ar ward fences were shown to have an insignificant effect on the timing of the force generation. This is yet another investigation providing evidence of a flow mechanism existing between the LEV and TV structures. Numerical investigations by Cosyn et al. [ 22 ] provide s further insight into the three dimensional development of the LEV. They conducted numerical simulations on various aspect ratio wings at different static angles of attack. For an aspect ratio two flat plate, they showed that the separated flow over the leading edge of the wing is entrained into the TV The added momentum entrained into the TV resulted in strengthening the tip vortex and decreased the local pressure on the surface of the wing a long the chordwise direction. The separation of the TV from the wing was shown to reduce the overall vortex lift generated on the wing.
25 These investigations begin to address the question of why natural fliers have low aspect ratio wings. Smaller aspect r atio wings offer a means of transferring momentum to and from the upper surface of the wing which results in an ability to strengthen, maintain, or hind er the development of unsteady vort ical structures Furthermore, the TVs provide localized low pressure regions that add to the overall lift on the wing. With the knowledge of general fundamentals associated with vortex structure development over low aspect ratio wings, it is desirable to discuss investigat ions pertaining to the unsteady development of the LEV and TV structures and their consequence on the aerodynamics of a dynamically driven wing. Multiple investigations have been conducted in an effort to understand the dynamic development and interaction between the LEV and TV over a dynamically driven wing. One such investigation is that by Freymuth et al. [ 23 ] in 1987. They provided flow visualizations for an impulsively started, low aspect ratio wing at a constant angle of attack equal to 40. An impulsive kinematic motion is characteristic of large wing acceleration from rest to a constant velocity of travel over a short period of time. The freestream velocity is associated with the final velocity of travel. Freymuth et al. showed the development of an arching LEV An arching LEV is representative of the LEV structure remaining anchored to the leading edge wing tips while the LEV lifts off the surface of the wing at the interior of the wing T he LEV is shown to develop a closed cannot end in a fluid, but only end at a boundary or form a closed path. These results are consistent with the numerical investigations by Taira et al. [ 24 ] whom explored various impulsively s tarted, low aspect ratio flat plates. Taira et al. showed a large
26 increase in the lift as the LEV develops. Th is was a consequence of a low pressure region present on the win g surface due to the development of the LEV. The transient development of the LEV was s trongly dependent on the aspect ratio of the wing. The flat plate with an aspect ratio equal to unity was dominated by counter rotating, TVs which occupied the entire downstream surface of the wing These TVs provided downwash onto the upper surface of the wing, resulting in the LEV remaining attached to the surface of the wing. The aspect ratio two flat plate also developed TVs which were sh own to be weaker than those generated over a unity aspect ratio wing Consequently, these weaker TVs provide d in sufficient downwash onto the surface of the wing to keep the LEV attached to the surface The ability of the TV s to stabilize the LEV was furthe r reduced using a wing with an aspect ratio equal to four These findings lead Taira et al. to suggest the LEV structure could be stabilized from the addition of spanwise perturbations induced by the TVs. These investigations over both static and dynamic ally driven wings show that three dimensional effects have a strong influence of the aerodynamic lift generation. Low aspect ratio wings are highly susceptible to spanwise flow on the aft end of the wing due to the development of the TV which ultimately in fluences the development of the LEV. Understanding mechanisms driving this development is essential in order to estimate the unsteady aerodynamic lift. 1.3. Unsteady Aerodynamics The need to develop capable, engineered flight vehicles which operate in a low Re ynolds number flight regime requires a n ability to model the aerodynamics associated with the development of unsteady fluid structures. This section first discusses the investigations pertaining to the usage of a quasi steady analysis with
27 respect to angle of attack to estimate of unsteady aerodynamic forces. The next three sections present various unsteady aerodynamic effects discussed in the literature. Lastly, various inviscid, unsteady theoretical techniques utilized to predict the unsteady aerodynamics forces are presented. 1.3.1. Quasi S teady A nalysis I n general, early biologists assumed that quasi steady aerodynamics were sufficient to produce the forces required in the fast, forward flight of insects. Quasi steady aerodynamics applies the coefficient of li ft at static angles of attack to that of a dynamic motion such as flapping. This assumption was based primarily on the results of Jensen [ 25 ] in 1956 for tethered locust. Jenson concluded that the downstroke is primarily responsible for the lift and thrust production. His calculations relied upon approximate calculations of effective angles of incidence otherwise known as the instantaneous ang le of attack which incorporates the induced velocity from the wing kinematic motion studies, ultimately showing the limitations of quasi steady m odels. Early investigations by Osborne [ 26 ] into bumblebee flight quantifiably demonstrated that the mean coefficient of lift required for flight was in excess of quasi steady estimations. Similarly while investigating tethered locusts, Cloupeau et a l. [ 27 ] showed high advanced ratios produced vertical forces well in excess of those predicted by quasi steady theory. Later investigations by Ennos [ 28 ] Dudley et al. [ 29 ] Zanker et al. [ 30 ] all provided additional evidence on the limitation of the quasi steady theory. In 2001, Sane et al. [ 31 ] argued that quasi steady estimates for the mean cyclic lift were reasonably accurate; however, the quasi steady estimates fail ed to capture the time course of the measured lift for nearly all kin ematic patterns. Also, their investigation consistently underestimated the
28 mean drag coefficient. The authors speculated the estimate errors were incurred from rotational effects for which quasi steady estimates did not account for. These studies all show the inadequacies of utilizing a quasi steady estimation of the aerodynamic forces with respect to the angle of attack for various kinematic motions. However, incorporating multiple aerodynamic parameters which account for the rotational effects could provi de more accurate estimates. Therefore, f urther insights into the aerodynamic parameters governing the underlying physics associated with unsteady vortex phenomena is required in order to utilize more accurately quasi steady models to predict the unsteady a erodynamics 1.3.2. Wagner E ffect Many investigations have shown the utilization of unsteady fluid phenomena to augment the aerodynamic response of a wing. Wagner [ 32 ] provided one of the first major breakthroughs in unsteady airfoil theory. He analyzed the growth of circulation around an airfoil section when the quasi steady circulation changes abruptly. His analysis applied directly to the growth of circulation around an airfoil section suddenly [ 33 ] confirmed wing at an angle of attack well below stall. Ellington [ 34 ] argue how lift is not simply proportional to the circulation for unsteady motions. Circulation and lift were postulated to lag the quasi steady value because o f wake vortices. 1.3.3. E ffect W ing rotation has the capability of delaying stall for a translating wing. If a wing in steady, transverse motion is given a slight positive rotation, stall is delayed until larger angles of incidence than those of steady state measurements. Enhanced values of
29 max imum coefficient of lift (C L max ) are obtained with larger rotational rates. This [ 35 ] Farren [ 36 ] investigated this phenomena and showed that negative rotation rates resulted in smaller incidence angles of C L,max This is clear evidence that C L,max measured under steady state conditions cannot be applied to dynamic kinematic motions that incorporate wing rotation and translation. Bennett [ 37 ] investigated the effect of large, rotational velocities on model flapping wings. He generated a large increase in lift when the wing was rotated about a chordwise axis rapidly in the middle of the downstroke. Lift was shown to be generated is shown to be capa ble of enhancing or retarding unsteady aerodynamic effects and must be fully understood in order to apply these effects to various rotational and translational kinematic motions. 1.3.4. Virtual M ass Additional unsteady forces on a wing arise in unsteady aerodynam ic theory based on the virtual wing mass [ 26 38 ] These forces arise from the fact that when a wing or immersed body is accelerated, it must set the surrounding air into motion. The inertia of the wing is increased by the mass of the air that is accelerated; therefore, there is an [ 39 ] showed the contribution of the virtual mass forces to the section coefficient of lift for small angles of attack is given by (1 2) where is the distance measured perpendicular to the flight path is the chord, is chordwise location about which the wing is pitched and is the angular velocity in units of radians per second The first two terms of this equation are functions of the
30 acceleration of the virtual mass perpendicular to the fligh t path and rotational motion respectively. The third term is representative of a circulatory term produced from virtual mass moving about the wing section. For a periodic kinematic motion, the first two terms cannot contribute to the mean wing forces. Howe ver, the instantaneous effects have the possibility of being significant. 1.3.5. Inviscid U nsteady A erodynamic F low E stimations Many important features of unsteady aerodynamic behavior can be described by linearized thin airfoil theory [ 7 ] Theodorsen [ 40 ] investigated an oscillating airfoil. The theory he utilized was potential flow theory and the Kutta condition. Theodorsen divided the flow into two regions. The first region is and sinks which define the boundary conditions of the oscillating plate. This term is depende nt on the instantaneous kinematic mot ion. The second term i component which includes the bound vortices and wake vortices. This term is utilized to enforce the Kutta condition by matching the non circulatory component. The circulatory term is dependent on the history of the m otion by means of the time dependent vorticity in the wake. Theodorse (1 3) Remarkably, the term outside the brackets is simply the quasi steady solution for a flat plate at a n instanta neous a ngle of attack ( ). All of the unsteady effects are a consequence of the bracketed term. The complex solutions to this equation have been solved by various authors including Fung [ 39 ] and McCroske y [ 41 ] McCroskey solved a problem similar to Theodors e unsteady viscous effects on oscillating airfoils are less important than unsteady potential
31 fl ow effects. However, in the case of dynamic stall, the inv iscid theory correctly indicates the trends associated with large scale boundary separation, but grossly underestimate s the angles of attack at which stall occurs Another approach to solving unste ady aerodynamic problems is through the utiliz ation of conformal mapping techniques. Choi [ 42 ] analyzed a two dimensional airfoil utilizing the K rm n Trefftz transformation to map the wing with a nearly circular section in a new frame of reference. This procedure is consistent with that of Theodorsen [ 43 ] Choi showed the lift for a sinusoidal plunging motion was out of phase with the circulation. The cir culation was shown to lead the plunge velocity by approximately 90 This finding agrees with experiments by Krmn and Sears [ 44 ] The phase lead is postulated to occur as consequence of a high redu ced frequency which results in the virtual mass becoming a dominant effect. Choi presents t his model as a good technique that is well suited for understanding the change in circulation behind an unsteady airfoil, but the model is limited to a two dimension lack of ability to estimate the circulation generated at the leading edge In contrast to [ 45 ] utilized conformal mapping to investigate the vortex structures generated at the leading edge and trailing edge of a pitching plunging flat [ 46 ] vortex model which estimated the dynamics of the vortex structures generated in the roll up of the shear layer. Brown and showed promise of computation efficiency; howeve r, computational accuracy required the use of models capable of predicting the development and separation of the viscous boundary layer along the surface of the wing.
32 Many researchers have iterated upon quasi state models in an effort to accurately estimat e the unsteady aerodynamics generated from various kinematic motions. In particular, Sane et al. [ 47 ] characterized the effect of wing rotation on the production of aerodynamics forces on a flapping ai rfoil. Their results showed good agreement between the measured force coefficients and estimated theoretical results Their inv iscid model is focused on a two dimensional airfoil rotating around an arbitrary axis while translating for small magnitudes of a ngular rotation. Their model predicts that the aerodynamic coefficients are independent of the magnitude of angular velocity This contradicts experimental investigations performed on similar kinematic motions. The reason for this discrepancy is most likel y a consequence of the three dimensional nature of flows, such as the TV which could influence the section circulation and downwash along the span [ 48 ] These investigations provide insight s into the ch allenges associated with unsteady aerodynamics. Particularly, a quasi steady analysis with respect to the instantaneous angle of attack under predicts the m easured lift of natural fliers. This analysis does not incorporate any of the observed unsteady aerodynamic effects, and therefore lacks the ability to account for the unsteady flow physics. T heoretical models though constrained to a two dimensional wing, provide insight into the ability of unsteady vortex structures hav ing the capability of leading or lagging the quasi steady aerodynamic response These models utilize various unsteady aerodynamic parameters to estimate the lift. Therefore, it can be reasoned that these parameters account for the underlying flow physics associated with unsteady aerod ynamics generated over an oscillating wing Therefore, i ncorporating unsteady aerodynamic parameters into quasi steady models
33 could improve the overall accuracies unsteady aerodynamic models and define the unsteady vortex development 1.4. Periodic Kinematic Motions Natural fliers utilize periodic flapping motions to develop and maintain unsteady vortex structure which increase their flight capabilities. Periodic motions often investigated in the literature are that of a plunge, pitching, or combination pitch-plunge kinematic motions. A pitch-plunge motion is comprised of a plunge and pitching kinematic equation respectively defined by (1 4) (1 5) A schematic of these kinematic motions is presented in Figure 1-1 The plunge motion is defined by a plunge amplitude ( ) and a frequency of oscillation ( ). For the purposes of this study, the plunge motion is constrained in the vertical direction Similarly, the pitch kinematic motion oscillates at the same frequency as the plunge motion. The pitch motion is defined by a constant pitch angle ( ) and a pitch amplitude ( ). The pitch motion is constrained to a rotation about a constant chordwise rotation point. A phase lag ( ) defines the timing of the pitch kinematic motion with respect to the plunge kinematic motion. A pure plunge kinematic motion is indicative of a constant pitch angle throughout the entire plunge cycle, thereby eliminating the oscillatory term in the pitch kinematics. 1.4.1. Instantaneous Angle of Attack and Angle of At tack Rate The unsteady kinematic motion imparts an induced velocity at the leading edge of the wing; therefore, the instantaneous angle of attack is distinctive from the pitch. Figure
34 1-1 presents the induced velocity vector incurred from the plunging kinematic motion and freestream velocity. Also presented is the instantaneous angle of attack ( ) T he instantaneous angle of attack is defined by (1 6) where is the freestream velocity. The second term accounts for the angle between the plunging velocity and freestream velocity vectors. Differentiating with respect to time yields the angle of attack rate given by (1 7) It can be seen that the angle of attack rate is not a function of the constant pitch angle ; thereby, altering the constant pitch angle allows for the generation of numerous angle of attack profiles for a particular angle of attack rate profile. 1.4.2. N on-Dimensional Parameters Various non-dimensional parameters are utilized to characterize the unsteady vortex development and aerodynamics generated from period ic kinematic motions. Discussed in this section are the reduced frequency, Strouhal number, and the feathering parameter. 184.108.40.206. Reduced frequency Periodic kinematic motions generate flow structures that can be characterized by the reduced frequency, defined by (1 8) Natural fliers which typically operate in forward flight, such as birds, bats, large insects, have a range of reduced frequencies ranging from 0.05 to 0.3 [ 49]. Rival et al. [ 50 ]
35 investigated the effect of the reduced frequency on the development and shedding of vortex structures from an SD7003 airfoil. For a pure plunge motion, Rival et al. showed that the reduced frequency has a large effect on the shedding present in the wake of the wing At lower values of the reduced frequency (k<0.15), the dynamic stall process and its corresponding shedding is shown to be typical of bluff body von K rm n type shedding. Higher values of the reduced frequency were shown to result in the kinematic forcing being dominant; therefore, the resulting shedding was a single leading edge and trailing edge vortex pair. Rival et al. found nearly equivalent flow fields when comparing pitching and plunging kinematics at con stant reduced frequencies The variations in lift between these motions were found to be a consequence of the position of the shed vortex. 220.127.116.11. Strouhal n umber The utilization of a p itch plunge kinematic motion introduce s a phase lag between the respective plu nge and pitch kinematic p rofiles which allows for the development of different flow shedding patterns. Von Ellenrieder et al. [ 51 ] showed that the flow structures developed behind a low aspect ratio, fla t plate could be characterized by the Strouhal number. The Strouhal number is defined by (1 9) As the Strouhal number is increased, a general trend of the spanwise vortex size increases while the streamwise separation of the shed vortices decreases. Ultimately, the Strouhal number provided insight into the unsteadiness of the flow and the unsteady v ortex structure development.
36 18.104.22.168. Feathering p arameter Anderson et al. [ 13 ] investigated various pitch plunge kinematic motions to gain insight into their thrust generation. They quantified the relationship between the plunge and pitch kinematic motions by a feathering parameter defined by (1 10) Higher values of this parameter result in a kinematic motion where the instantaneous angle of attack induced by the plunge velocity is partially removed from the pitching kinematic motion. For instance, a feathering value of zero results in the pure plunge kinematic motion where the plunging velocity dominates the angle of attack. Anderson et al. performed an extensive parametric study and concluded that the optimal thrust is produced if four criteria were met: a Strouhal number between 0.25 0.40, a large plunge amplitude, a large maximum angle of attack (15 25 ), and a phase lag of appro ximately 75 which corresponds to a pivot point at 1/3 of the chord length These high thrust values are shown to correspond with the development of moderately strong LEV. It was demonstrated that the phase lag and pivot location play a significant role in the LEV development as they determine the timing of the development and shedding of the LEV. 1.4.3. Engineered I nvestigations Numerous studies have been conducted utilizing periodic kinematic motions. This section focus es on a subset of these investigations whic h provide 1) a un ique understanding of the vortex f low structures generated 2) the dominant parameters driving the development of unsteady structures, and 3) the response of these structures
37 on the aerodynamics generated on the wing. The insight gained from these studies provide s the ground work for future studies. Visbal [ 52 ] utilized Computational Fluid Dynamics (CFD) to investigate the LEV and TV development on a plunging low aspect ratio wing at a fixed pitch angle of 8 .0 At the top of the downstroke, a LEV was shown to be present over the upper surface of the flat plate. As the wing plunged downward and increased its angle of attack, the size of the LEV increased. The size LEV was uniform o ver the entire span of the wing with the exception of near the wing tips where it was pinned to the leading edge of the wi ng. He showed highly three dimensional flow with particularly strong axial with in the LEV here the core axial flow was measured to be minimal. At the bottom of the downstroke, Visbal showed the lifting of the LEV region from the surface wing while ends remain ed pinned to the wing tips. The TVs were shown to develop as the plate begins to accele rate downward at the start of the downstroke. At the middle of the downstroke, h e concluded that the TV s began to breakdown due to the reversal of the axial flow along the aft of the wing. During the initial stages of the upstroke, the TV was non existent as it transitioned to the underside of the wing. Visbal simulated the development of the LEV and TVs and discovered it was remarkably consistent over a Reynolds number range of 1x 10 3 to 2x10 4 His simulations are in good agreement with water tunnel experim ents performed by Yilmaz et al. [ 53 ] Visbal performed spanwise lift calculations and discovered that the TVs provide increased lift due to their development of a low pressure region on the surface of the wing. The TVs were shown to be crucial in attaining the maximum lift on the wing
38 at the middle of the downstroke, which corresponds to when the TVs were the strongest. Furthe r insight into the effects of the kinematic motion on the unsteady vortex development is presented by Trizila et al. [ 54 ] CFD computations were computed over a wide range of pitching amplitudes, plunge amplitudes, and phase lags. They provide a sensitivity analysis of these parameters on the lift generated for a two dimensional and an aspect ratio four wing. The lift associated with finite wing was shown to be more sensitive to pitch amplitude. However, the wing with an aspect ratio equal to four was dominated by the phase lag and angular amplitude. Thi s is a consequence of the three dimensional effects incurred from a finite wing. Therefore, it can be reasoned that these effects are incurred from the inf luence of the TV. If the aspect ratio were to be further reduced, the phase lag and pitch amplitude parameters will most likely become more important as the influence from the TV becomes more prominent across the span of the wing. Von Ellenrieder et al. [ 51 ] investigated a pitch in g plunging, low aspect ratio wing. They showed how variation s in pitch rate altered the time and pitch angle corresponding to the LEV shedding throughout the pitch plunge cycle. The phase lag parameter was the most dominant parameter in changing the timing of the vort ex shedding. The phase angle variations provide d large deviation s in th e time history of the angle of attack throughout the kinematic motion. Therefore, altering the angle of attack has the potential to drive the flow development of the unsteady vortex structures. O l et al. [ 5 5 ] investigated the aerodyn amic lift on a pitch plunge and pure plunge, two dimensional SD7003 airfoil. They performed extensive experimental, computational,
39 and theoretical estimations to gain insight into the lift throughout the kinematic motion. La rge phase lags existed between the fluid structures generated from the airfoil and the kinematic motion However, these lags did not translate to large phase lags in the force measurement data. This sugges ts that the fluidic structures e ffe ct on the phase response of the lift is not dominant. Overall, the measured lift values exceed the stall value of lift which is evidence that the fluid structures over the wing act ing to delay stall. As the Reynolds number was varied between 1 .0 x10 4 and 6 .0 x10 4 the size of the laminar separation bubble associated with the LEV was altered when the angle of attack is well below stall. Okamoto et al. [ 56 ] investigated the dynamic force production on a pitching plunging fla t plate with an aspect ratio equal to six They concluded that a maximum thrust can be achieved from the selection of an optimal phase shift between the pitching and plunging kinematic motions. A sharp leading edge was shown to increase the overall lift, unlike that of static aerodynamic data wher e the airfoil leading edge configuration has little effect. The se engineered investigations provide an understanding of unsteady aerodynamics and the flow physics associated with the development of unsteady vortex structures. Quasi steady aerodynamics pro ves to be insufficient for calculating the lift for various kinematic motions in the low Reynolds number flight regime. This is most likely due to its inability to accurately predict the temporal development and shedding of unsteady vortex structures. The insights gained from the studies previously mentioned are a small subset of the low Reynolds number investigations in the literature, but provide the fundamental knowledge needed to progress to a further understanding of
40 the dynamic development of unstead y fluid structures and their influence on unsteady aerodynamic performance. 1.5. Objectives This work leverages the understanding gained from the aforementioned studies to provide a unique insight into the ability of unsteady vortex structures to alter the aero dynamic forces generated over a dynamically driven wing. Unsteady vortex structures have been shown to increase the aerodynamic performance of natural fliers and engineered planforms. However, a means to accurately predict the aerodynamic consequences due to the development of u nsteady vortex structures is still lacking. Ultimately, it is des ired to model the aerodynamic response of unsteady vortex phenomena independent from kinematic motions such that these models incorporate the governing flow physics. Th is investigation is intended to identify the unsteady vortex development and the response of these structures on the aerodynamic lift. First, t his investigation characterize s the development of the leading edge vortex (LEV) a nd tip vortex (TV) structures with respect to various aerodynamic parameters. This characterization provides insight into the fl ow mechanisms driving the development of the unsteady vortex structures independent from a particular kinematic motion The aerodynamic parameters utilized in this study are chosen from parameters employed in various theoretical unsteady inviscid models. This allows for the assessment of these models ability to accurately define the flow physics associated LEV and TV. Th is study captures the development of the LEV and TV s tructures with p article i mage v elocimetry flow field measurements acquired at discrete times throughout unique pitch plunge and pure plunge kinematic motions A low aspect ratio flat plate is utilized to gain insigh t i nto the three dimensionality of the unsteady vortex structures Particularly, it is desired to
41 develop further insight into the ability of spanwise flow to stabilize the LEV Similarly, the spanwise flow incurred from the TV on the downstrea m portion of the wing is investigated to determine its ability to reattach the highly accelerated streamwise flow over the LEV. Secondly, a erodynamic force measurements are utilized to investigate the response of the unsteady vortex structure s on the lift Direct force measurements are acquired from sting balance measurements utilizing a tare to remove the inertial effects from the dynamic kinematic motion These force measurements are assessed as a function of the aerodynamic parameters such that the forc e is directly related to the unsteady vortex development. A quasi steady model with respect to angle of attack and a corrected lifting line model are compared to the measured lift to assess their viability in accurately modeling the unsteady aerodynamic li ft. A control volume technique to compute the lift is also investigated to ascertain its ability to compute the unsteady lift from wake flow field measurements. This technique eliminates the need to account for the inertial forces on the wing. Furthermore, it provides a direct relationship between the wake flow field generated from the LEV and TV on the overall lift developed on the wing. 1.6. Contributions The primary contribution of this research is the identif ication of various flow mechanisms associated with the development of unsteady vortex structures and the characteriz ation of the aerodynamic response of these mechanisms. Previous studies have investigated the development of the LEV and TV as a function of discrete time s throughout various kinematic motion s [ 52 53 ] Similarly, there ha ve been investigations focused on the aerodynamic response of these unsteady vortex structure developed
42 from various unsteady kinematic motions [ 57 58 59 ] While all of these studies provide unique insight s into unstea dy vortex phenomena and unsteady aerodynamics a direct relationship linking the development of unst eady vortex structures with their aerodynamic response is still lacking. This study bridge s this gap by characteriz ing these unsteady flow phenomena as a function of quasi steady, aerodynamic parameters such as the angle of attack, angle of attack rate, and pitch rate. A quasi steady analysis allows for the governing flow physics to be characterized independent of kinematic motion. Ultim ately, the analysis utilized in this current study prov ide s an initial attempt to uncover the underlying flow physics associated with unsteady flow phenomena such that engineers will have a means to achieve a desired aerodynamic response by prescrib ing a d esired kinematic motion Observations of natural fliers have shown the existence of significant spanwise flow th rough the leading edge vortex which has been postulated to stabilize the leading edge vortex [ 14 ] This spanwise flow is unaccounted for in two dimensional inviscid, unsteady aerodynamic models [ 40 60 ] The experimental investigations and analysis techniques utilized in the current investigation provide insight into the influence of this spanwise flow on the LEV development The thre e dimensionality of the leading edge vortex is characterized as a function of aerodynamic parameters which define the spanwise fluid mechanisms responsible for the three dimensional development of the LEV. Though it is not in the scope of this study, the influence of the three dimensional flow incurred from the utilization of a finite wing will have to be incorporated into existi ng two dimensional, inviscid aerodynamic models to achieve increased accuracies. The
43 analysis utilized in the current study takes significant strides to evaluate the three dimensional flow present over the wing and the extent of its interaction with the de velopment of leading edge and tip vortex structures. However, a ccurately modeling the spanwise circulation and lift distributions due to the dynamic development of spanwise flow is essential to achieving more accurate unsteady aerodynamic models over finit e wings. Lastly, an application of a control volume technique is postulated and utilized to compute the lift generated on an o scillating wing. Th e benefit of such a technique is its ability to measure the aerodynamic lift independent of wing inertial forc es The wing inertial forces incurred from dynamic kinematic motions corrupt direct force measurements and ultimately, t hese inertial forces have to be estimated and removed from the measurement. Current techniques utilize various tare and filtering scheme s. However, a s passive and active means to control the wing are investigated, estimating the inertial forces of the wing becomes increasingly difficult. The current study modifies a control volume technique utilized by Orloff to compute the lift from wake flow field measurements [ 61 62 ] Orloff was limited to discrete velocity measurements at a single point because particle image velocimetry had yet to be developed to the extent it is currently. Therefore, his technique was limited to a particular s panwise location The control volume technique developed in the current study leverages modern p article i mage v elocimetry to compute the lift over an oscillating, finite wing. To this end, the current investigation provides an initial assessment and utilization of a control volume technique to compute the lift generated on a finite, oscillating wing in the presence of unsteady vortex phenomena.
44 Figure 1 1 Schematic of plunge and pitching kinematic motions.
45 CHAPTER 2 PARTICLE IMAGE VELOC IMETRY One of the goals of this w ork is to understand the aerody namic consequences of unsteady flow phenomena deve loped on a low aspect rat io oscillating flat plate Therefore, accurately measuring features of the flow are essential. Particle I mage Velocimetry (PIV) is a non intrusive, optical method for measuring the velocity distribution in a two dimensional plane which is extensively used in this study Th is chapter begins with a short historical review of PIV and then discuss es t he theory behind this technique and various sources of error associated with it. 2.1. Particle Image Velocimetry Introduction The most rudimentary form of PIV could possibly be traced back in history to the first observations of small debris moving on the surface of a stream and relating that to the concept of velocity [ 63 ] Th is simple observation has been modernized into a measurement technique that is capable of achieving highly accurate, vel ocity measurements throughout a plane This capability is inherently associated with a series of challenges and complexit ies Before PIV flow field measurements were typically achieved by acquiring multiple, single point measurements throughout an entire flow field The disa dvantage o f a technique such as this is that it lacks the ability to resolve instantaneous spatial flow features throughout an entire flow field simultaneously On the other hand, f low visualizations were utilized to provide knowledge of the existence of flow structures, but d id not provide an analytical means of quantifyin g them. The development of PIV resulted f ro m the need to fill the gap between multiple point measurements and instantaneous flow visualization images. PIV provides a quantitative
4 6 and instantaneous measurement of the velocity field and has significantly advanced the experimental study of flowing flu ids 2.2. D evelopment of Particle Image Velocimet ry The techniques employed in the application of PIV to fluid flows were first developed in the field of experimental solid mechanics. These techniques are based on l aser speckle photography which utilize s a coh erent light source to illuminate a speckle pattern on a t ransparent solid. In 1977, Barker and Fourney [ 64 ] show ed this techniques ability to measure displacements on a rigid body, ultimately resulting in a displacement contour map with resolution of approximately ed this measurement technique in to one that can be applied to seeded flow fields by illuminating a plane in the flow with a light source Co rrespondingly photographic film was double exposed which captured the scattered light speckle pattern. Analyzing this photograph consisted In 1984, Adrian [ 65 ] proposed the adoption of the name PIV to distinguish this technique from that of laser speckle photography. This naming convention was proposed because f luid flows illuminated by a light sheet would hardly ever produce a speck le pattern due to inherent turbulence mixing. Therefore the images contain ed individual particles rather than that of a speckle pattern. As PIV further developed, r esearchers began to realize PIV s potential as a tool to measure the velocity in highly tur bulent flow fields. But to accomplish such a measurement, a wide variety of length scales would have to be resolved being that turbulence is characterized by a broad spectrum of spatial scales. Also, t urbulence is a highly three dimensional random process; therefore it became desirable to measure all the velocity components The utilization of micron sized, tracer particles i ncreased the
47 resolution of the turbulence structures ; however, th is resulted in a need for a light source of high intensity i llumination Small exposure times to freeze the particles in high velocity flows compounded the need for more illumination The illumination possibilities included but were not limited to continuous wave (CW) lasers, pulsed lasers, CW illumination with shu ttered cameras, or xenon flash lamps [ 66 ] Due to the large advances in solid state lasers, PIV currently utilizes double pulsed, solid state Nd:Yag lasers equipped with cooling and dual oscillator pac kages. With the cap ability of acquiring highly seede d flow field images, a methodolo gy for analyzing double exposed images of random particle fields still had to be refined before PIV would become a useful measurement technique. Camera technology has als o significantly increase d since the inception of PIV. Before the development of digital cameras, photographic film was used to capture the images of the tracer particles. With the early photographic systems both laser pulse s w ere captured on one image. Auto correlation algorithms, much like those shown by Soo et al. [ 67 ] and Kovasznay et al [ 68 ] in 1957, were used to calculate the particle displacements One of t he drawbacks with such an approach was that it wa s direction ambiguous due to the lack of insight into which particle images were related to each light pulse However, i mage shifting techniques [ 69 ] were later developed to reduce this ambiguity. With t he development of digital cameras to replace film, Digital Particle Image Velocimetry (DPIV) eliminated the need for image shifting techniques because two separate images are acquired as opposed to one double exposed image Cross correlation algo rithms developed by Leese et al. [ 70 ] and Sutton et al [ 71 ] could now be implemented on two independent images, eliminating the ambiguity associated with
48 auto-correlation algorithms. Small particle displacements became measureable, and consequently the dynamic range of DPIV greatly increased. DPIV is currently the standard for PIV measurements. For the remainder of this dissertation, PIV refers to digital particle image velocimetry. A significant disadvantage of the two-component PIV method discussed above is that it is only capable of measuring the projection of a velocity vector into the plane of the light sheet. Specifically, the out of plane velocity cannot be measured in the twodimensional domain. Stereoscopic PIV recovers the out of plane velocity by utilizing a second camera such that the particle displacement is record ed from two oblique viewing angles Utilizing Stereoscopic PIV a ll three components of velocity are capable of being measured on a two dimensional plane. 2.2.1. Scheimpflug Criterion One of the issues associated with an oblique viewing angle is a limited depth of field. This can be accommodated by tilting the image plane with respect to the orientation of the camera lens and light sheet. The Scheimpflug criterion states that the image plane, lens plane, and object plane must intersect in a common line, shown in Figure 2-1 However, this configuration introduces a change in the magnification factor and a large perspective distortion for which the calibration must account for [ 72 ] 2.2.2. Calibration The calibration of the physical, world coordinate system to the image plane coordinate system is accomplished coordinate system is three-dimensional and represented by engineering units at the laser sheet plane. However, the image plane is a two-dimensional coordinate system randomly oriented from the world coordinate system.
49 empirically calculate th e relationship between these two coordinate systems, thereby mapping the world coordinate system to each camera [ 73 74 ] Th e camera mappings are represented by (2 1) where is the pixel location, is the real world coordinate, and is the mapping function for each camera Note the subscripts w refers to the world coordinate system and the superscript refers to a particular camera ( ) while bold script is utilized to represent vector quantities This notation will be consistent through out this chapter. The pa rameter model [ 75 ] Six external camera parameters are given by the rotation ( ) and translation ( ) of the world coordinates where (2 2) (2 3) The subscript c represents the coordinates of the object plane oriented parallel to the image plane for a particular camera. The undistorted and distorted camera position s are defined as and respectively. These values are computed by: (2 4) (2 5) with (2 6) (2 7) and
50 (2 8) where and are the 1 st and 2 nd order radial distortion terms respectively. Usually, for good lenses, is sufficient. However, for wide angle lenses or consumer cameras, additional terms may be required. is the distance between the principle point and the camera pinhole. Provided the calibration plate is in focus at a distance is theoretical ly related to by (2 9) For good lenses, the theoretical distance between the principle points deviat es from the theoretical value o n the order of 2 5% [ 74 ] Finally, the true pixel location on the CCD is given by (2 10) (2 11) where is an optional skew factor which is set equal to unity for square pixel sizes. is the known pixel size. The principal point ( is usually close to the center of the CCD chip unless a Scheimpflug adapter is used. The set of parameters ( The world coordinates are typically defined by physical units (e.g. millimeters). Therefore, i t is necessary to convert the world coordinates to pixel units. For two component PIV in air with the cameras mounted perpendicular to the measurement plane, a single length scale is sufficient to relate the size of the pixels to a real world coordinate frame. This can be accomplished by simply photographing a ruler. However, to perform a proper stereoscopic PIV calibration, the off axis viewing angles must be
51 corrected. This is accomplished in two steps. First, a two-level calibration target, seen in Figure 2-2 is photographed. The two-level plate removes error associated with traversing a one-dimensional target in the out of measurement plane direction. Secondly, the origin of the coordinate system is determined by the position of one of the marks on the calibration plate. For two cameras, approximately 22 parameters are nee ded to specify the pinhole model, thereby reducing the number of terms from a thirdorder polynomial fit which would require approximately 90. Various corrections can be utilized to increase the accuracy of the Pinhole model. Ideally, the alignment of the laser sheet with the calibration target is perfect; however, any disparities can lead to errors in the resultant velocity vectors. The alignment can be checked by dewarping the double framed images from both cameras to the world coordinate system. The corresponding discrepancies between the two dewarped This error can occur from the calibration plate being offset or tilted from the laser plane providing significant error in the three-component vector reconstruction. Fournel et al. [ 76 ] developed a technique to correct this disparity error First, the computation of the disparity vector map is computed from the first images of the two cameras by standard PIV cross-correlation techniques. An ensemble-averaged correlation is used by summing the correlation planes of many image pairs to reduce the error [ 77] Errors in the disparity map are indicative of two projected lines from the cameras not coinciding at a particular world coordinate point. This is corrected by fitting a plane through the world coordinate points in three-dimensional space and correcting the mapping functions of each camera. This is accomplished by replacing with
52 and with where is the rotation of the fitted plane to the relative calibration plate and is the distance of the laser plane to the calibration plate. The new origin is constrained to the point project ed onto the measurement plane from the previous origin in camera 1. Similarly, the x axis of camera 1 coincides with the x axis of the measurement plane. This procedure can be repeated as necessary to obtain more accurate ca libration fits. This correction scheme assumes that the 5 internal camera parameters, as well as the orientation of camera 2 to camera 1 are constant. In the 22 parameters of the stereo scopic Ts a i model one can substitute the external parameters an d by the relative transformation and The self calibration procedure is equivalent to newly fitting and while keeping and fixed. 2.2.3. Stereo V ector F ield C alculation Various methods for stereo scopic vector field computations have been proposed by Prasad [ 72 ] and Call u aud et al. [ 78 ] to name a few DaV is software which is the one being used for this work, computes the respective velocities on each camera in the real world coordinate s and then proceeds to calculate the three components of velocity in the real world coordinate system Prasad [ 72 ] derived this t echnique by first relating the camera to real world coordinate system by (2 12) Following the analysis of Soloff et al [ 79 ] the particle displacement given by (2 13) can be estimated as
53 (2 14) where (2 15) where = 1,2 for each camera and =1,2,3 for each coordinate axis. Each of the camera displacements are related to the real world displacements by (2 16) This system of equations is over determined; therefore, a fourth equation is utilized to distribute the error evenly over all three components of velocity. DaVis 7.2 utilizes a modified approach to this technique by iteratively processing a vector field and using the initial vector field to dewarp the image for the next iteration. This effectively removes spurious vectors. With a good calibration and two-component vector errors of less than 0.1 pixels, the DaVis documentation states that the reconstruction error is well below 0.5 pixels. 2.2.4. Cross-Correlation Algorithms Various correlation techniques are utilized to estimate the particle displacement. Modern PIV utilizes a cross-correlation algorithm to evaluate the tracer particle displacements between two independently acquired images. Each image is divided into what is termed interrogation windows. Figure 2-3 displays a representation of a double framed image acquired at an initial time (t) and a later time (t+dt). Each image has been divided up into a series of interrogation windows.
54 The correlation function operates on the intensities of each window ultimately resulting in one velocity vector for each interrogation window. The cross correlation function is defined by (2 17) where and are the image intensities of the 1 st and 2 nd interrogation windows. C represents the two-dimensional correlation array that contains the strength for all integer displacements between the two interrogation windows. The size of the interrogation window (n) defines the maximum displacement computed; the result is a correlation array the size of 2n-1 square. To increase the computational efficiency, Westerweel [ 80] and Wilbert et al. [ 81 ] proposed the utilization of the Fast Fourier Transform (FFT) to calculate the cross-correlation algorithm as opposed to direct summation of the correlation terms. Figure 2-4 shows a schematic of this processing procedure. The FFT reduces the each operation from O[ ] to O[ ] [ 82 ] Figure 2-4 shows the tedious, two-dimensional cross-correlation algorithm can be reduced to computing two, twoequal sized samples of the image. This is then preceded by a complex multiplication of the resulting Fourier coefficients. An inverse transform is then carried out to produce the cross-correlation array which has the same spatial units of the two input images. The FFT procedure introduces a weighting of the correlation coefficients with emphasis on displacement (0,0) with decreasing weighting toward larger displacements. Because of this weighting, the correlation function should only be used when the displacement is less than 1/3 of the interrogation window size [ 83]. The end result of the image processing is a two-dimensional correlation map for the interrogation window of
55 interest. The largest correlation peak determines the displacement of the particles. This information together with the time interval between the images defines the velocity vector for the interrogation region of interest. The correlation calculation is then repeated over the entire image ultimately resulting in a velocity field with one vector per interrogation region. The disadvantage with weighting functions used in the FFT algorithm can be corrected with multi-pass algorithms which enhance spatial resolution while reducing overall noise when calculating the cross-correlations. A schematic of a multi-pass algorithm with a constant interrogation window size is presented in Figure 2-5 Th e vector field is calculated by an arbitrary number of iterations on the same image. In each pass, a reference vector field is processed for each interrogation window. This new information is used to provide a window shift on the corresponding pass. This helps correlate the correct particles and improves the signalto -noise ratio. Similarly, this technique can be applied to a decreasing window size. The first iteration uses the largest interrogation region to calculate a reference vector field. On the next pass, the window size is reduced and the window shift from the previous pass is used to provide an image shift. Ultimately, this results in an adaptive window shift that improves the vector calculation as the interrogation window size is decreased. This ensures that the same particles are correlated as smaller window sizes are used. The overall advantage is smaller vector resolution with less erroneous vectors. The interrogation window size on the last pass of a multi-pass algorithm determines the grid size of the overall vector field. However, window overlapping is a technique to increase the overall grid resolution. Overlap defines the overlap among
56 neighboring interrogation windows. The bigger the overlap value, the closer the grid of computed velocity vectors. A schematic of overlapping is presented in Figure 2-6 For example, window sizes of 32x32 pixels with 0% overlap results in grid spacing of 32 pixels whereas window sizes of 32x32 pixels with a 50% overlap results in grid spacing of 16 pixels. Sub-pixel accuracy is desirable to enhance the resolution of the displacement vector to below the resolution of the camera acquiring the image. Both image reconstruction techniques and correlation peak-fits are utilized to obtain sub-pixel accuracy. Before one can estimate the sub-pixel location of a tracer particle, the nominal diameter of the tracer particle on the image plane must be greater than the size of a pixel. This will ensure that a Gaussian fit can be applied to the correlation. Adrian et al [ 84 ] estimated the nominal image diameter ( ) of a particle on the image plane by (2 18) where the diameter of the point response function of a diffraction-limited lens at the first dark ring of the Airy disk intensity distribution is defined by (2 19) The particle diameter is the wavelength is the magnification is M, and the focal length of the lens divided by the aperture diameter is Equation 218 represents the combined effects of magnification and image blurring on the particle image diameter. However, if the point response function and geometric image distribution were both Gaussian, equation 219 would be exact.
57 If the nominal image diameter of the particle is less than one pixel, the Gaussian estimation is impossible. Kahler et al. [ 85 ] estimated the probability of a single particle with illuminating only a single pixel on the CCD by (2 20) Therefore, it can be reasoned that is not required to achieve sub-pixel accuracies, however achieving such a condition is a guarantee of sub-pixel accuracy. 2.3. Particle Image Velocimetry Measurement Accuracy PIV has proven to be a highly useful technique for acquiring velocity measurements in many different flow fields. However, it is important quantify various sources of error associated with PIV such that these errors can be quantified and ultimately minimized 2.3.1. General PIV Error Analysis Simply stated, PIV is a measure of the displacement of a particle in a fluid medium over a known time interval. This can be expressed as (2 21) where is the position of the particle, is time, and is the time interval over which the two locations are acquired Figure 2-7 shows an example trajectory of a particle over an interval of time. Velocity gradients, particle lag, and acceleration of the particle can all occur along this trajectory. These terms are quantified when expressing the velocity of the particle as an Eulerian fluid velocity utilizing particle dynamics [ 86 ]. This results in the velocity being defined as (2 22)
58 where (2 23) and (2 24) The first term is the velocity measured using PIV. Therefore, it can be deduced that PIV measurement s assume th at velocity gradients, particle lag, and particle acceleration is small over the measured time. These conditions must be met to achieve accurate flow field measurements using PIV. Equation 2 22 shows that PIV measurements require the ability accurately measuring particle location ( ) and the time ( ) at which the particle location is acquired Errors associated with these two measurements directly result in velocity measurement error s 22.214.171.124. Velocity m easurement e rror The particle position is measured by a Charged Coupled Device (CCD) in the camera. The physical particle location on the CCD is related to the measured particle location by (2 25) where is the magnification from the lens on the camera. It should be noted that is utilized in classical lens/camera mapping techniques; therefore it is utilized in the following derivations for the purpose of consistency with the literature However with modern mapping techniques such as the pinhole meth od, the mapping function
59 ultimately replace s the magnification term Substituting equation 2 2 5 into equation 2 22 and dif ferentiating about each term yields the uncertainty in the measured velocity with respect to error in the displacement, magnificati on, and time measurements This expression is shown by (2 26) Squaring each side yields (2 27) T he first term is the particle displacement error. It is a random quantity based upon the fluid flow. The remaining t erms are the magnification error and temporal error. These are deterministic forms of error. These three components of error are further discus sed below 126.96.36.199. Particle d isplacement e rror The particle displacement error occurs from the PIV systems in ability to accurately interpret the particle position and corresponding displacement The error is represented by (2 28) where is the correlation of the location of the particle, is the diameter of the particle, and is the displacement of the particle. Typically, the max displacement of a particle ( ) is much less than 10 particle dia meters and the correlation error is on the order of 0.05 0.20 [ 86 87 ] t herefore this expressions can be rewritten as
60 (2 29) Factors influencing are the detector fill factor, background signal to noise ratio (SNR) pixel resolution, correlation algorithm, image overlap and truncation. The noise contributions to the signal correlation peak are included in Invalid disp lacements occur when the correlation peak is a noise peak rather than the true particle displacement peak. Techniques utilized to reduce the correlation error are discussed later. 188.8.131.52. Magnification e rror The magnification is defined by (2 30) where the distance between the image plane and the lens and is the distance between the lens and the object plane [ 88 ] Therefore using a method of similar to the particle displacement error, the error associated with these variables is defined by (2 31) Modern mapping techniques outline in the section 2.2.2 are utilized to empirically calculate the magnification However, any movement between the camera and object plane after the initial calibratio n can increase the error in the calculated mappi ng function 184.108.40.206. Temporal e rror The temporal error is composed of the error of the timing between the first and second particle positions This error is typically very small for any given PIV system
61 (2 32) However, an exception to this is misfires by the PIV system and high speed flow applications, neither of which is pertinent for this study. 2.3.2. Two-Dimensional Two-Component Perspective Error One of the major sources of error in two-dimensional, two component (2D2C) PIV occurs from out of plane movement by seed particles. A schematic of a 2D2C setup is presented in Figure 2-8 In this figure, a seed particle moves from an initial point x i to a final out of plane position x f Prasad et al. [ 88 ] calculated the error associated with the out of plane particle displacements. They showed the error between the measured displacements ( ) and the true in-plane displacements ( ). The apparent in-plane displacement ( ) is related to the measured position by (2 33) where is the nominal magnification on the object plane defined by (2 34) The final magnification ( ) due to the particle moving closer to the image plane is defined by (2 35) This results in recorded displacements equal to (2 36) The error between the true and apparent in-plane displacement is defined by (2 37)
62 where the subscripts t and a represent the true and apparent values. With this definition, the error for each component of velocity is (2 38) This is identical to the result provided by Jacquot et al. [ 89 ] for the displacement of a solid surface. The error is minimized when the in-plane displacement is much greater than the out of plane displacement. The equation also states that the error can be minimized by increasing angle subtended by the particle to the camera axis. The out of plane displacement at these large angles exaggerate the particle displacement on the measurement plane due to perspective error. 2.3.3. Two-Dimensional Three-Component Perspective Error The advantage of stereoscopic PIV is that the velocity associated with the out of plane motion of the particles can be measured and therefore, the third component of velocity is quantified. A schematic of the setup for a Scheimpflug, stereoscopic PIV setup is presented in Figure 2-9 Let and represent the displacement of the particle in the world coordinate system. and are measured displacements with respect to each camera. and are the projected displacement of the particle at the object plane. Wang et al. [ 90 ] utilized a ray tracing analysis similar to Prasad et al. [ 88 ] for the two-dimensional two component setup; however, he utilized a two-dimensional three component PIV setup. Wang showed the world displacement of a particle defined by (2 39)
63 (2 40) (2 41) Utilizing the separation distance between the lens axis ( ) yields (2 42) These equations relate the particle displacement to the apparent displacement on each camera. However, a relationship still has to be developed between the measured displacement of each particle and the mapping point on the CCD 220.127.116.11. Symmetric v iewing a ngles c onfiguration Prasad et al. [ 91 ] derived the equations relating the measure d displacements to the particle displacement for cameras symmetric about the center axis and equal distance from the object plane For the simplest case it was assumed that a magnification factor of unity existed for each camera and the Scheimpflug criterion was met resulting in (2 43) An initial particle position of x=y=z=0 was chosen This resulted in the relationship between the displacement of the particle and the measured displacement on each camera as derived by Wang et al. [ 90 ] : (2 44) (2 45)
64 (2 46) where is the object distance is the distance from the center of the lens to the center of the film, and is the distance between the two lens centers. These equations are only a function of and as and M can be deduced from them. This allows for a sensitivit y analysis to be performed on these variables to determine their contribution to the overall velocity measurement uncertainty The functionality of these equations is limited du e to the highly specific setup. A general setup analysis is required to further this analysis 18.104.22.168. General s etup The derivation by Prasad et al. is restricted by the assumptions of the PIV setup; however it is more desirable to achieve a generic relationship relating the measured displacements on each camera to the particle displacemen t. This will allow one to determine the minimal velocity component detectable by any 2D3C PIV setup. Following the work of Brucker [ 92 ] the three components of the particle displacement vector are reconstructed by (2 47) (2 48) (2 49) w here is the angle between the x z plane and each camera axis while is the angle between the object plane and th e camera axis in the y z plane. when the image
65 plane is parallel to the object plane. Therefore, it is desirable to rewrite the vertical displacement as (2 50) These values are equivalent to those derived by Wang and are purely a function of the locations of each camera with respect to the object plane. Therefore, utilizing a nominal magnification and the resolution of each camera, the minimal measureable velocity vector can be estimated for any 2D3C PIV setup. For example, the x component resolution is represented by (2 51) where and are the respective mapping function and resolution of each camera. Sub pixel correlation techniques can result in resolution distances less than a pixel [ 93 ] A sensitivity analysis on the above equation s is utilized to compute the veloc ity error associated with the corresponding two dimensional th r ee component setup. The displacement vectors are directly related to velocity by dividing by the time between the image pairs. The velocity error sources include the resolution and mapping error. The resolution is defined as the minimal detectable displacement vector which is estimated from sub pixel accuracies of a correlation algorithm utilized. Error in the calibration fit is another source of uncertainty that needs to be quan tified The calibration mappings influence on the reconstructed velocity vector is determined by differentiating each component of velocity with respect to the measured displacement in the x and y directions for each camera an d evaluating at particle displ acements and The mapping function errors on camera s 1 and 2 are quantified by the calibration and
66 represented by and respectively. This ultimately results in three expressions that are utilized to estimate the velocity uncertainty in a 2D3C setup: (2 52) (2 53) (2 54) 2.3.4. Cross-Correlation Accuracy The cross-correlation algorithm is utilized to estimate the particle displacements at each interrogation region. Therefore, it is important to estimate the accuracy of this algorithm to obtain to overall accuracy of the velocity measurements. Two sources of uncertainty are presented such that their influence can be minimized. These error sources are peak-locking and image reconstruction. 22.214.171.124. Peak-locking error The Gaussian fit utilized to achieve sub-pixel accuracy is susceptible to an error known as peak-locking. This is a relative error associated with estimating a continuous function from discretely-sampled approximations. If the discrete measurements capture or symmetrically bound the true peak, the error in estimating such a peak is minimal. Figure 210A and Figure 210B provides examples of scenarios where the discrete
67 measurements capture the true peak with minimal peak locking bias error. However, Figure 210C and Figure 210D display an anti-symmetric bounding of the true peak location which in turn leads to a bias error that estimates the peak towards the closest measurement point. The amplitude of the bias error is strongest at a true pixel shift of 1/4 or 3/4. Peak-locking, also termed mean bias error in the literature, can be large compared with other random components of error associated with PIV [ 83]. This error is typically combated with various interpolating or peak-fitting functions [ 81, 94 95 96 ]; however, little agreement exists on the best one. Center of brightness and centroid techniques are generally disregarded due to their sensitivity to arbitrary choices in the cut-off threshold [ 94 ]. The presence of peak-locking appears in any PIV technique where subpixel accuracy is attempted. The position of the correlation peak can be measure to sub-pixel level accuracies to the order of 0.1-0.05 pixels [ 97, 98 ]. This is dependent upon the particle image diameter and the interrogation window size. If the particle image diameter is two small, the correlation displacement tends toward integer values. The optimal particle image sizes are approximately 1.5 pixels; however, various techniques can be utilized to the accuracy of smaller particle diameters. DaVis employs a Gaussian peak fit as a threepoint estimator. Two-independent Gaussian functions are fitted to the peak in the x and y directions. The correlation peak can be estimated by a rotationally symmetric Gaussian bell working on 3x3 or 5x5 pixel range. 126.96.36.199. Image reconstruction error When using the multi-pass algorithm to enhance the accuracy of the correlation algorithm, the estimator shifts the images based upon an initial or given displacement
68 field. This operation utilizes image reconstruction to provide the needed image shift. The raw image is mapped pixel wise according to the given flow field and the pixel intensity is recalculated through the use of a bi linear interpolation. This interpolation technique results in a s uperior signal to noise ratio of the correlation function and suppression of peak locking [ 99 100 ] The accuracy of the correlation peak is i mprove d utilizing a Whittaker image reconstruction technique This reconstruction technique avoids smoothing of the image if the particle diameter is abov e 1 pixel. Therefore, the original image can be restored without any loss of information. This technique has been developed in signal theory. The underlying purpose of this technique is to rebuild a signal of limited bandwidth from sampling points. For PI V purposes, this is equivalent to tracer particles greater than or equal to 1 pixel measured over multiple pixels [ 101 ] This reconstruction results in enhancing sub pixel accuracies.
69 Figure 2 1 Schematic for Scheimpflug condition Figure 2 2 LaVision two level calibration target Photo courtesy of author.
70 Figure 2 3 Evaluation of PIV recordings using double framed images and a multiple interrogation window technique to calculate velocity vectors Figure 2 4 Schematic of FFT cross corre lation algorithm
71 Figure 2 5 Adaptive multi pass correlation technique with a constant interrogation window size Figure 2 6 Schematic of 5 0% i nterrogation w indow o verlap
72 Figure 2 7 Schematic of particle trajectory over a time interval ( t) Figure 2 8 Schematic of 2D2C PIV setup
73 Figure 2 9 Schematic of 2D3C PIV setup A B C D Figure 2 10 The peak locking mechanism for adding bias towards the nearest high data point.
74 CHAPTER 3 FACILITY AND CHARACTERIZATION EXPERIMENTS All the investigations presented in this dissertation are a cquired at the Aerodynamic Characterization Facility (ACF) located at the University of Florida Research and Engineering Education Facility (UF REEF) in Shalimar, FL. T he facility is designed for the purpose of perform ing aerodynamic investigations in low Reynolds number flight regimes. The ACF consists of a low speed wind t unnel designed to operate with minimal freestream turbulence intensi t ies Another unique capability at the ACF is a dynamic motion rig which allows for a wing to be driven through various dynamic kinematic motions The following sections provide det ailed description s of the low speed wind tunnel and its flow quality along wit h a full description of the dynamic motion rig. 3.1. Low Speed Wind Tunnel Overview The ACF is a unique facility designed for the purpose of investigating unsteady fluid phenomena, and one of its mos t important elements is the low speed wind tunnel. The wind tunnel is of Eiffel type design which corresponds to the fan located on the downstream side of the test section such that air is pulled through the tunnel The wind tunnel can be broken up into three regions. The f irst region is the flow entrance /contraction section where stagnant air is drawn into the wind tunnel where the large scale turbulent structures are broken down before the flow is accelerated. The second region is the test section where flow from the entrance region is used for testing. Lastly, the flow exits the test section and enters the diffuser/fan section of the win d tunnel. These three regions of the wind tunnel are further described in the subsequent sections.
75 3.1.1. ACF Entrance/Contraction Section T he flow entrance consists of a bell mouth inlet, a honeycomb/screen pack section a settling chamber, and ultimately a contraction region to accelerate the flow. Figure 3-1 and Figure 3-2 present actual and schematic views of the flow entrance section respectively. The bell mouth entrance reduces the flow entrance size to approximately 10 foot square. A honeycomb/screen pack section exists downstream of the bell mouth entrance to reduce any large coherent structures existing in the flow. A schematic of the honeycomb/screen pack section is presented in Figure 3-3 The flow first enters t he precision 0.5 inch, hexagonal cell, aluminum honeycomb which is sandwiched between two high porosity stainless steel, wire mesh screens with mesh densities of 24. Four wire mesh screens with densities 24, 32, 46, and 56 are located downstream of the honeycomb at equal distance of 3.375 inches. A settling chamber which is fluidic structures generated by the flow conditioning to decay. The settling chamber is followed by a square contraction section utilized to accelerate the flow. This 8:1 contraction section was designed using the tools discussed by Mathew [ 102] Finally, an aluminum lip is attached to the fiberglass contraction exit. All seams joining the various components are attached with flanged joints which are sealed with resilient, closed cell, urethane foam gaskets. 3.1.2. ACF Test Section The open jet test section is utilized to place experimental equipment and apparatuses. The enclosure around the working test section is 11ft high, 12ft wide, and 15ft long. The walls are comprised of a laminate and Styrofoam composite and joined by an extruded aluminum structure. Figure 3-4 shows a picture of the composite wall
76 sections being joined by the extruded aluminum plates. The small pressure drop within the test section has the ability to exert a significant inward force on the composite wall sections due to their large area. To reduce the loading from the composite wall sections, an 8020 extruded aluminum, cage-like structure was constructed inside the test section. Figure 3-4 shows one of the vertical 8020, 20x20 beams running along the interior of the test section. The beams reside in the middle of the composite walls to reduce the overall area being supported by the aluminum joining supports. The flow enters the test section from the contraction section in the form a 42 inch square column of air. This can be seen in Figure 3-5 The column of air then proceeds to exit through the diffuser which incorporates a bell mouth entrance section, as seen in Figure 3-6 3.1.3. ACF Diffuser/F an Section The actual and schematic views of the ACF diffuser/fan section is presented in Figure 3-7 and Figure 3-8 respectively. The entrance to the diffuser, located at the downstream end of the test section, is a 45.6 inch square opening into the diffuser section. The diffuser section transitions from a square cross section to a 60 inch diameter circle over an 8.35 foot axial distance. This results in total expansion included angle of 8.27. The fan is connected to the diffuser section with a flexible coupling to isolate any motor vibrations from the rest of the wind tunnel structure. The axial fan is manufactured by Howden Buffalo. The fan blades have a constant pitch and are driven by a 50hp, Totally Enclosed Air Over (TEAO), constant speed, 1185 RPM, AC induction motor. The fan motor is controlled by the ToshibaVF -AS1 inverter, otherwise known as Variable Frequency Drive (VFD). The flow finally exits from the fan into the surrounding room through an inline flow silencer.
77 3.2. ACF Validation Experiments The low speed wind tunnel at the ACF is designed to operate at low Reynolds numbers using a length scale on the order of typical M icro Air Vehicle (~6 in.) Another design constraint is the air flow having minimal levels of turbulence intensity. The following sections detail the mean flow uniformity and turbulent intensity characterization investigations performed to assess the flow quality of the wind tunnel. 3.2.1. Mean F low Uniformity It is desired to quantify the flow uniformity throughout the test section at various freestream velocities. To this end, a 32 port, total pressure probe rake with a 1 inch spatial resolution is manually traversed in a single measurement plan e to acquire the mea n flow on 2x 2 inch grid. The freestream velocity at each probe location i s governed by a simplified form of (3 1) which is a function of the differential pressure ( ) and the density ( ). The differential pressure is the difference in pressure between the static and total pressure measured by the total pressure probe rake. A Hesie model ST 2H equipped with a 0.5 inWC pressure transducer is used to measure the differential pressure with a n accuracy of 0.07% of the full scale pressure rating. The Heise is used to acquire 100 samples at a rate of 2 Hz f or each location. The density is calculated from the ideal gas law. The ideal gas l aw is defined by (3 2)
78 where P is the static pressure, R is the universal gas constant (287 J/(kg K)), and T is the static temperature of the air. The static pressure of the air in the test section is measured by a GE Druck DPI 142 Barometric Indicator with a range between 10.9 and 16.7 psia and a resolution of 0.01% FS. The static temperature is measured with an Omega P-M-1/10 RTD sensor. It is rated as 1/10 DIM accuracy which corresponds to an error of 0.04C at a reference temperature of 20C. Due to static pressure fluctuations that can occur from atmospheric conditions throughout an experiment on a single measurement plane (~400 Pa), a mean density is calculated at each port and used in the final velocity calculation. Appendix A presents an analysis of the velocity measurement uncertainty. The analysis shows the uncertainty in the overall measurement system produces a relative error in the velocity measurements of 1.8% and 0.48% at a mean velocity of approximately 2 .0 m/s and 4 .0 m/s respectively. Figure 3-9 is a photograph of total-pressure probe rake mounted at the entrance to the test section. It can be seen that the pressure rake is mounted to a rigid 8020 structure that is traversed in the streamwise direction. The mean flow is acquired at four streamwise locations normalized by the length of the test section: 0.0% (test section entrance), 0.5%, 40% and 62.5% and is repeated for the mean flow velocities of 2 .0 m/s and 4 .0 m/s. Figure 310 and Figure 311 present the normalized mean flow fields at a freestream velocity of 2 .0 m/s and 4 .0 m/s respectively. For both velocities, it can be seen that the core of the open jet remains uniform at the inlet in Figure 310 A and Figure 311A respectively. The size of the uniform jet core reduces as a function of downstream distance due to the development of the shear layer between the freestream
79 flow and the static air inside the test section. A clear example of this is shown in Figure 310D A similar trend exists for the uniform flow velocity of 4.0m/s presented in Figure 311. The size of the uniform cores for both freestream velocities at each downstream location is very consistent. Normalized, centerline vertical and horizontal velocity profiles for the four axial locations are shown in Figure 312A and Figure 312 B respectively. Both the vertical and horizontal profiles are similar due to the symmetric nature of the flow. In Figure 312A it can be seen that the core velocity at the exit plane of the contraction deviates by less than 1% of the normalized flow velocity throughout the entire test section entrance. This uniform flow region exists for each downstream measurement location but the size of the region reduces. At a normalized streamwise distance equal to 40% of the test section length, the uniform flow region size has been reduced by approximately 35%. A similar decrease in size is seen for a mean freestream velocity of 4 .0 m/s. Ultimately, the mean flow uniformity experiments provide adequate validation of the flow uniformity achieved by the low-speed wind tunnel. A deviation of less than 1% exists throughout the core region of flow. However, mean flow uniformity provides no information into the background levels of disturbances that might exist in the core region. 3.2.2. Turbulence Intensity The uniformity experiments discussed above provide sufficient mean flow information; however, it is still desired to understand the fluctuating component of velocity. Therefore, a single component, hot-wire probe with a constant temperature bridge is used to measure the time resolved streamwise velocity component. The sensing element of the probe, supplied by Auspex Corp., has a diameter of 5 microns
80 and a length of nominally 1 mm. The sensing element is connected to a Dantec 55M10 bridge. The output of the bridge is digitized with the National Instruments 4472 card containing a sigma delta analog to digital converter with 24 bit resolution. For these experiments, 256 blocks of 4096 samples are acquired at a rate of 2048 Hz. These conditions result in a spectral frequency bin width of 0.5 Hz and a normalized random error of 6.3%. Calibration of the hot wire probe is accomplished by placing a pitot static probe in close proximity to the hotwire and collecting data for a series of wind tunnel velocities (typically greater than 25) that extended over the full range of velocities for e valuation. For each of the free stream velocities acquired, mean values of both the hot wire voltage and velocity (calculated from the pitot static probe) are determined, and a fourth order polynomial is used to relate the two [ 103 104 ] The calibration curve fit result ed in errors of the mean velocities to be less than 1% for all velocities greater than 1 m/s. It is common practice to high pass filter the velocity fluctuations when calculating turbulence intensity in a wind tunnel [ 105 106 ] This insures that fluctuations associated with scales larger than the facility are not taken into account. T his is accomplished by calculating a cut off frequency ( based on the time it would take a disturbance to propagate across the entire test section length ( ) at the mean centerline velocity ( ) The cut off is defined by (3 3) After filter ing out these large scale contributions the turbulence intensity values are listed in the last column of Table 3 1 and are all less than 0.25% for fre estream
81 velocities greater than 1 m/s. The spectra of the hot-wire signal is utilized to assess the frequency distribution of the fluctuations associated with the turbulence intensities. Figure 313 presents the power spectral density of the hot-wire signal with respect to the frequency and non-dimensional Strouhal number. The Strouhal number is defined by (3 4) where is the frequency, is the diameter of the inlet, and is the freestream velocity. In Figure 313A no significant spikes in the power spectral density are prese nt at the baseline case of 0.0 m/s, as this value represents the noise floor measurement of the system. As the freestream velocity is increased, a spike at the lower frequencies between 1 and 10 Hz develops and increases in magnitude as the corresponding velocity is increased. The frequencies of 15 and 30 Hz are also indicative of large power spectral density spikes which become more significant at the larger mean velocities. Figure 313B presents the power spectral density as a function of the Strouhal number. The high power spectral density amplitude associated with 15 and 30 Hz do not collapse onto one another; therefore, it is reasoned that these spikes are more indicative of noise rather than a flow phenomenon introduced on the hotwire signal. In contrast, the lower frequency spikes collapse into a Strouhal number range of 0.3 to 0.5. This Strouhal number range corresponds to what is generally expected for a jet column mode [ 107 ]. The jet column modes are further investigated with the addition of a Kulite pressure transducer in an attempt to obtain coherence measurements The experiments were conducted with a Kulite LQ125-5SG pressure transducer placed on the bell
82 mouth of the wind tunnel diffuser and a hot-wire probe located in the center of the entrance to the test section. The Kulite and hot-wire signals are sampled simultaneously using a National Instruments 4472 card. The ordinary coherence is calculated between the signals and a significant spike is seen at lower frequency values. Figure 314 displays the magnitude of the coherence between the ho t-wire probe and pressure transducer for the Strouhal range associated with the jet column modes mentioned previously. High values of the ordinary coherence function are indicative of the hotwire and pressure signal measuring the same phenomenon. Therefore, it is concluded that power spectral density spikes associated with frequencies between 1 and 10 Hz are indeed a result of the jet column modes. The ACF has shown to provide uniform flow to within 1% of the normalized free stream velocity at turbulence intensities lower than 0.25% for all velocities greater than 1 .0 m/s The jet column mode interacting with the bell mouth of the diffuser is seen to introduce spike in the power spectral density at frequencies between 1 and 10 Hz, ultimately resulting in higher turbulence intensities. If lower turbulence intensity values are desired in the future, a porous collector could be installed on the entrance of the diffuser to alleviate the turbulence intensity associated with the jet column modes. 3.3. Dynamic Motion Rig There is a desire to understand the role of unsteady fluid features on the aerodynamics generated on a wing driven through a dynamic kinematic motion. Performing such maneuvers requires an apparatus to perform these various motions. The dynamic motion rig (DMR) at the ACF was built for this purpose It is comprised of two Parker Ironless linear positioners (Model: T4DB53-6NPB1-B73-1BA) capable of 767mm of travel which are presented in Figure 315 Both motors are mounted vertically
83 on a support structure built out of 8020 beams. Each motor is equipped with a 5 resolution linear encoder along with limit and homing sensors for consistent positioning. Power is applied to the motors through Parker AR-08AE drivers. A Galil DMC 2030 controller is used to communicate the desired linear motions. PID compensation is incorporated into the controller to accurately achieve the desired motion. The PID control is tuned by altering the proportional and differential gains to achieve an acceptable rise time with minimal overshoot. The integral term is left constant such that the system remained stable. The controller uses the operation of motion termed contour mode This mode of operation synchronizes each linear motion in time by specifying a fixed time step for a desired motion displacement. These two time-synced, linear motions result in the wing being driven through various kinematic motions utilizing the sting mechanism presented in Figure 316 These motions include static angles of attack from approximately 0, pitching motions, plunging motions, and combined pitch-plunge motions. The two vertical beams shown in Figure 316 are rigidly attached to each corresponding vertical linear motor. The connection between the linear motor and the sting closest to the wing has a single rotational degree of freedom. The second linear motor is connected with both a rotational degree of freedom and linear degree of freedom in the sting axis direction. These degrees of freedom allow the linear kinematic motions to be converted into various pitching and plunging motions.
84 Figure 3 1 ACF flow entrance section of wind tunnel. Photo courtesy of author. Figure 3 2 Schematic of f low e ntrance s ection
85 Figure 3 3 Schematic of h oneycomb/screen pack se ction Figure 3 4 Test s ection w all s chematic Photo courtesy of author.
86 Figure 3 5 Flow entrance to the test section Photo courtesy of author. Figure 3 6 Flow exit from the test section. Photo courtesy of author.
87 Figure 3 7 ACF diffuser/fan s ection of wind tunnel. Photo courtesy of author. Figure 3 8 Schematic of ACF diffuser/exit section
88 Figure 3 9 Flow Uniformity Pressure Rake Setup Photo courtesy of author.
89 A B C D Figure 3 10 Normalized, mean streamwise velocity profile f or a freestream velocity of 2.0 m/s at various streamwise locations A ) 0.0%, B) 0.5%, C) 40%, and D ) 62.5%.
90 A B Figure 3 11 Normalized, mean streamwise velocity profil e for a freestream velocity of 4.0 m/s at various streamwise locations A ) 0.0% B ) 40 %. A B Figure 3 12 Normalized centerline velocity profiles for various streamwise locat ions with a uniform core of 2.0 m/s. A) Vertical. B) Horizontal
91 Table 3 1 .Turbulence In tensity as a function of mean velocity [m/s] Turbulence Intensity [ / ] 0.42 1.16% 1.13 0.08% 1.85 0.06% 2.58 0.07% 3.68 0.08% 7.78 0.11% 9.67 0.15% 11.56 0.21% 13.58 0.16% 15.37 0.10% 17.24 0.07% 19.00 0.05% A B Figure 3 13 Power spectral density A) Presented with respect to frequency B ) Presented with respect to Strouhal number.
92 Figure 3 14 Coherenc e between h ot wire and p ressure s ignals Figure 3 15 Dynamic Motion Rig Photo courtesy of author.
93 Figure 3 16 Sting Mechanism Photo courtesy of author.
94 CHAPTER 4 AERODYNAMIC FORCES C ALCULATED FROM FLOW FIELD MEASUREMENTS When a body is immersed into a fluid with relative motion, it experiences a force. The force exerted by the body can be derived from fluid dynamic equations and evaluated using measured flow field quantities. PIV makes this intrinsic measurement possible for unsteady aerodynamics; however, the fluid dynamic equations governing the unsteady forces must be developed. This chapter investigate s various techniques for computing aerodynamic forc es from f low field measurements. The advantage of such technique s is the elimination of inertial forces from direct force measurements Secondly, the ability to relate discreet components of momentum to the overall aerodynamic force provide s insight into the unstea dy fluid structures ability to alter the aerodynamic forces. 4.1. Two Dimensional Unsteady Lift Estimations Noca [ 108 109 ] comprised of surface integrals and is constrained to a divergent free flow. The FE e quation is derived from the integral momentum equation bounding an internal body and is represented by (4 1) where
95 (1 1) is the unit normal 3 component vector, is the three component velocity vector, is the three component position vector, is the identity matrix, is the viscous tensor, is the dynamic viscosity, and is the number of dimensions (N=2 for two dimensions, N=3 for three dimensions). and denote the arbitrary control surfaces enclosing the body and body bo undary respectively. The FE yield s time dependent forces from experimental measurements collected on an arbitrary surface surround ing the wing system However, measuring the required flow field quantities can be difficult and error prone. The FE equation has been utilized to compute t he temporal forces over a two dimensional wing where the spanwise flow on the interior of the wing is minimal [ 59 ] However, low aspect ratio wings are associated with large spanwise velocities and therefore the three dimensional form of the FE equation must be utilized. Implementing this equation requ ires the knowledge of the velocity derivate in the third dimension which require s multiple two dimensional PIV measurement planes Therefore, application of the FE equation becomes sufficiently more challenging Therefore, a new technique is needed for lo w aspect ratio wings. 4.2. Static Lift Estimation Technique Orloff [ 61 62 ] investigated the utilization of the incompressible three dimensional momentum integral over a finite wing. Ultimately, h e developed a technique to compute
96 the section al coefficient of lift on a static wing from velocity surveys behind the wing. The following derivation is based on the analysis performed by Orloff. is utilized to relate the extensive momentum property into a control volume (CV) system. This system relates the change of momentum inside the CV to that leaving or entering the CV. The integral form of the momentum equation for a control volume is defined as (4 2) The steady momentum integral is defined as (4 3) where is the shear stress tensor on the surface surrounding a control volume; is th e outward normal to the surface(S) ; is the static pressure; the fluid density, and is the Eulerian fluid velocity. Figure 4-1 represents the CV around a section of the wing. Equation 4-3 can be rewritten in terms of surface integrals resulting in (4 4)
97 Some simplifications are utilized based on the nature of the problem. The velocity normal to the wing surface ( ) is equal to zero resulting in (4 5) The shear forces on the surface of the CV are assumed to be negligible when the Reynolds number ( is much larger than unity; therefore, (4 6) It is also assumed that the lift generated over the wing surface ( ) by shearing forces is negligible compared to the lift due to pressure forces. Therefore, t he lift is equivalent to the pressure force acting in the y direction resulting in the expression (4 7) Substituting the following assumptions into e quation 4 4 the lift component is defined by (4 8) It is useful to u se the definition of the total pressure in order to relate the pressure field to the velocity field. The definition of the total pressure is given by
98 (4 9) where is the static pressure and is the total press ure. The total pressure is constant in the flow field where viscous losses are minimal When S 2 and S 4 are placed sufficiently far from the wing, the freestream velocity is the only significant velocity component as th e vertical and spanwise velocities app roach zero Therefore, the contributions from th ese surfaces on the overall lift are zero. To obtain zero net force in the z direction, the surfaces S 5 and S 6 are symmetric about the centerline of the wing. Furthermore, a s the distance between these surfac es and the wing increases and approach zero thereby eliminating the contributions to the lift from S 5 and S 6 Expanding the CV surfaces in such a manner allows the lift to be rewritten as (4 10) N on dimensionalizing the equations by the dynamic pressure ( yields (4 11) where is defined as (4 12) S 1 is normal to the freestream velocity, therefore reducing the lift to (4 13) The coefficient of lift is defined by (4 14)
99 where is the span of the wing and the chord length is defined by Another important characteristic of a three dimensional wing is the aspect ratio (AR) which is defined by (4 15) Combining equations 4 1 3 4 1 4 and 4 15 yields (4 16) where and are normalized streamwise and vertical velocit ies respectively and Y and Z are the normalized vertical and spanwise directions respectively. Equation 4 1 6 represents the coefficient of lift estimated from the measured velocity field behind a wing. Computing the lift from th e control volume technique is dependent on highly accurate velocity measurements over a two dimensional surface with significant spatial resolution. PIV has a unique capability to provide such a measurement at a single instance in time However, PIV has its own inherent sources of error which can propagate into the lift calculation through th e integration of the measured velocity components over S 3 One major source of error is the alignment of the laser sheet plane normal to the oncoming freestream velocity. This can result in small velocity error s at each interrogation region The significan ce of these velocity errors is compounded by integrating over S 3 which could result in a significant lift contribution This error is investigated by rewriting equation 4 10 as (4 17)
100 where the bracketed term ( ) represents the sum of the integral in the vertical direction o n S 1 and S 3 A bias error ( ) is introduced to each velocity component resulting in (4 18) With the assumption that =0 on S 1 the lift is defined by (4 19) where it can be seen that the bias error h as the ability significantly alter the computed lift. The bias error can be eliminated from the computed lift through the measurement of the velocity component on the surface S 1 Assuming the freestream velocity is constant throughout the test section, the measurement of the flow field on S 1 is accomplished by removing the wing from the test section. Therefor e, the PIV setup remains constant resulting in equivalent measurement bias errors on S 1 and S 3 Equation 4 18 shows that the bias errors cancel each other out when the flow field information on S 1 is utilized. Therefore, using equation 4 1 7 to compute the lift provide s a more accurate estimation of the aerodynamic lift on the wing. The control volume technique is further simplified by a ssuming symmetry across the mid span location. This allows for a single measurement plane to be acquired ove r half the span of the wing This reduc es t he number of measurement planes, thereby improving the overall accuracy in calculating the lift on the wing A t the midspan of the wing, a sting is mounted to the wing which increases the difficulty in acquiring v elocity measurements due to blooming and shadow regions. Therefore, t he lift contribution from this region of flow is estimated through the use of a 2 nd order polynomial fit on
101 in equation 4-17. The profile of the polynomial fit is presented in Figure 4-2 and is defined by (4 20) where and are the polynomial coefficients. These coefficients are solved for using the slope as boundary conditions is known from the measured velocity field away from the centerline of the wing, therefore its slope at the two closest points to the midspan can be estimated by At the midspan of the wing a symmetry condition is applied resulting in This results in the expression (4 21) for which the coefficients and are solved for. The coefficient is solved from (4 22) 4.3. Unsteady Lift Estimation Technique Further analysis of the static lift estimation technique derived in the previous section is required such that the lift can be estimated for a dynamically driven wing. Therefore, the following sections present a derivation of the control volume technique utilizing an unsteady wing. This is accomplished in two ways. First, the CV is fixed to the
102 wing frame of reference which eliminates the unsteady term. Secondly, the CV can be held static while the wing surface boundary is allowed to move inside the CV. Each technique provides an equivalent solution with unique insights into the nature of the problem. 4.3.1. Moving Control Volume The control volume utilized in this derivation is the same as in Figure 4-1 with the exception that the CV moves with the wing. Figure 4-3 shows a schematic of the CV moving relative to the wing such that the wing is stationary relative to the control volume. The integral form of the linear momentum equation is (4 23) The first term represents the linear momentum stored or released from inside the CV. With a fixed control volume and an incompressible fluid, this term can be simplified as (4 24) At various instances throughout the kinematic motion, there possibly exists a lead/lag response due to the unsteady vortex structures ability to accumulate and release linear momentum from inside the CV. However, because the kinematic motion is repeatable from cycle to cycle, the contribution of this term to mean lift per cycle is zero. The movement of the control volume is defined by (4 25)
103 for a plunging wing. The absolute velocity in a fixed reference system is Therefore, the relative fluid velocity ( as seen by an observer moving with the CV is (4 26) Substituting this velocity into equation 4 2 yield s (4 27) Exp anding out the left hand side of equation 4 27 and substituting equation 4 26 above yields (4 28) S 2 and S 4 along with S 5 and S 6 cancel each other out due to equal and opposite lift contributions The wing surface is fixed to the control volume and therefore (4 29) Finally, relating the pressure forces to the lift results in the lift being equal to (4 30)
104 This result is consistent with the steady solution in that it is of the same form; however it was assumed that momentum could not be accumulated or released from inside the CV. This derivation also restricted the kinematic motion to a plunging wing. 4.3.2. Unsteady W ing C ontrol V olume T he derivation of an unsteady control volume technique for a wing pe rforming a generic unsteady kinematic motion is presented in this section. Th e control volume utilized is represented in Figure 4 4 The dashed lines r epresent the control surfaces around the wing and the plane cutting into the CV such that the wing control volume is represented. The integral form of the momentum equation is (4 31) Like the previous derivation, t he first term represents the momentum accumulated or released from within the CV. This term is removed with the assumption that momentum inside the CV is constant ; however, the influence of the unsteady vortex structures on this t erm is later investigated The momentum added to the CV from the unsteady wing is add ed through the control surface around the wing, (4 32) From the fixed control volume frame of reference, the fluid on the surface of the wing is moving at the velocity of the wing surface ( ). Therefore, the relative velocity is
105 (4 33) Substituting into equation 4 3 2 yields (4 34) Therefore, the problem is reduced to similar form as the steady case. Furthermore, t his derivation relaxes the assumption of a vertical dynamic motion utilized in the previous section Equation 4 29 states that for any dynamic kinematic motion, the wing surface integral will be equal to zero Therefore, the lift can be estimated at discrete times throughout a kinematic motion with the underlying assumption that linear momentum cannot be stored of released from inside the CV 4.4. Summary This chapter provides insight into investigations that att empt to compute the unsteady aerodynamic lift from flow field measurements. A control volume ( CV ) technique adopted from Orloff is derived such that it can be applied to compute the lift behind an unsteady wing from wake flow field measurements. This tec hnique is ideally suited for PIV which provides a means to non intrusively measure the flow field with high spatial resolution. This CV technique provides a direct relationship between the wake flow field velocities associated with the leading edge vortex and tip vortex on the overall lift. Therefore, insight into the ability of these structures to alter the unsteady aerodynamics is achieved. This dissertation provide s a validation of the steady and unsteady forms of the CV technique. First, the static lif t on a low aspect ratio wing is computed technique Secondly th e unsteady CV technique is utilized on wing driven by a pure
106 plunge motion to assess its accuracy at computing the mean and temporal lift profile throughout one kinematic cycle. Comparing the measured and computed lift provides insight into the unsteady vortex structures ability to accumulat e and release lin ear momentum from within the CV.
107 Figure 4 1 Control v olume for computation of total lift. Figure 4 2 Parabolic Fit Profile
108 Figure 4 3 Schematic of moving control volume in the x y plane Figure 4 4 Schematic of fixed control volume with oscillating wing in the x y plane
109 CHAPTER 5 INVESTIGATIONS AND INSTRUMENTATION This chapter defin es the experimental investigations utilized to ascertain the quasi steady development of the leading edge vortex ( LEV ) and tip vortex ( TV ) structures and their i nfluence on the unsteady aerodynamics The first section provide s details in to the experimental i nvestigation s utilized to assess the development of the LEV and TV structures with respect to various aerodynamic parameters The second and third sections describe details into the PIV flow field and aerodynamic force measurements used t o evaluate the res ponse of the LEV and TV structures 5.1. Experiment al Investigations The ultimate goal of this dissertation is to develop a n understanding of the underlying physics that drive the development of unsteady vortex structures. T he s e flow physics are directly r elated to the unsteady aerodynamic response of the wing. Natural fliers are capable of utiliz ing unsteady vortex structures to enhance their flight characteristics thereby making them highly efficient and agile fliers. However, current engineered vehicles are limited by their designers understanding of the se unsteady vortex phenomena thereby limiting their capabilities. This study investigate s the flow field and aerodynamic responses generated from the development of unsteady vortex structures generated over a pitching plunging, low aspect ratio flat plate. These responses are characterized as a function of static and rate dependent aerodynamic parameters. Section 5.1.1 discusses the se lection of the se parameters based upon various experimental investigations and unsteady aerodynamic theory. The experiments utilized to assess the general development of the unsteady
110 vortex structures are discussed in section 5.1.2 Furthermore, Section 5.1.3 defines a quasi steady investigation to determine the response of the unsteady vortex structures to each aerodynamic parameter. 5.1.1. Aerodynamic P arameter S election Ol et al. [ 55 ] showed the lift response from a two dimensional, plunging wing to be nearly quasi steady with respect to angle of attack. This occurred in the presence of a LEV being maintained over the wing throughout the downstroke. It can be reasoned that the lift response from the LEV is quasi steady with respect to angle of attack ( ) Theodorse n [ 110 ] analytically showed the angle of attack being the dominant term for a two dimensional, oscillating wing. Therefore, t he angle of attack, defined in section 1.4.1 is utilized to characterize the LEV and TV development and the lift re sponse. The angle of attack rate ( ) and pitch rate ( ) are also utilized to define the unsteady vortex and lift response. These aerodynamic parameters incorporate the temporal development of the plunging and pitching motions. The angle of attack rate is a function of the plunge velocity and pitch rate of the wing. This is a consequence of the plunge velocity imparting an induced velocity at the leading edge of the wing. However, the pitch rate is independent from the plunge kinematic motion. Fung [ 39 ] showed the virtual mass, which is a function of the angle of attack rate and pitch rate as defined in section 1.3.4 t o alter the unsteady aerodynamics on a dynamically, oscillating wing. function of the plunge rate an d pitch rate. The lift response due to these parameters could account for the small discrepancies in the lift measured by Ol et al. [ 55 ] from the quasi steady model with respect to angle of attack.
111 The angle of attack, angle of attack rate, and pitch rate parameters are utilized throughout this study to characterize the unsteady vortex development and aerodynamic lift response. The general and quasi-steady responses of the unsteady vortex structures to the aerodynamic parameters are investigated in this dissertation. 5.1.2. General LEV and TV Development Experiments Figure 5-1 Figure 5-2 and Figure 5-3 present the angle of attack, angle of attack rate, and the pitch rate achieved from the kinematic motions utilized in this investigation A pure plunge, or heaving motion, constricts the parameter design space by eliminating the rotation rate throughout the entire motion The remaining design space is generated through the use of pitch-plunge kinematic motions. Equations 1-4 and 1-5 define the plunging and pitching kinematics respectively. For all the investigations in this dissertation, the pitch amplitude ( is equal to 0.5 which is similar to the root kinematic motion of a fruit bat [ 111 ] This results in the wing being driven vertically one chord length for the downstroke and upstroke respectively. The pitch kinematic motion defines the geometric angle of the wing with respect to the freestream velocity throughout plunge stroke. For the general LEV and TV development investigations, a pitch amplitude ( and constant pitch angle ( of 8.42 and 8.00 are utilized. These parameters result in positive and negative pitch angles along with angles of attack which are associated with an induced velocity from the plunging kinematic motion, that extend throughout light and deep stall regimes defined by McCroskey [ 7 ] By definition of the pure plunge kinematic motion, the pitch amplitude is set to zero resulting in a constant pitch angle throughout a kinematic cycle. The wing is pitched about the 0.25 chord location which corresponds to the aerodynamic center in thin airfoil theory. Furthermore, this location is similar to that of Anderson et al. [ 13 ] whom suggested high
112 propulsive forces are generated when pitching the wing about x/c=0.3. The Reynolds number and reduced frequency are 4x10 4 and 0.25 respectively which is consisten t with forward flying natural fliers where the wing kinematics drives the development of the vortex structures. This results in a chord length and oscillation frequency of 0.156 meters and 2.02 Hz respectively The wing itself is an aspect-ratio two, flat plate. An aspect-ratio two wing is utilized to ensure three-dimensional flows are incurred along the interior of the wing. Furthermore, utilizing a flat plate allows for a direct comparison with inviscid, ideal flow models developed for oscillating wings. Lastly, the phase lag ( ) between the plunge and pitch kinematic motions is varied upon to provide unique developments on the unsteady vortex structures. The investigation of multiple phase lag values result in distinctive aerodynamic parameter profiles being that these profiles are directly affected from the timing between the plunging and pitching kinematic motions. Figure 5-1 displays the angle of attack profiles that resulted from the pure plunge (PP) and pitch-plunge kinematic motions with phase lags 30, 75, and 90. The angle of attack profiles corresponding to the downstroke between normalized times 0.00 thru 0.50 are unique. It should be noted that t he normalized time is referenced to the start of the downward, plunging motion and normalized by the period of one cycle. The pure plunge motion provides a symmetric al development of the angle of attack. This kinematic motion also achieves the largest angle of attack of approximately 22 degrees. Pitch-plunge motions with phase lags equal to 30 and 75 provide angle of attack profiles such that the maximum and minimum values are offset from time 0.25 and 0.75 thereby altering the phase response from that of the pure plunge kinematic motion. These motions are also characteristic of unique maximum angles of attack of
113 20 and 14 respectively. The motion associated with a phase lag equal to 90 is characteristic of an angle of attack profile similar to the pure plunge kinematic motion; however, this profile provides angle of attack values that are much smaller than the pure plunge Figure 5-2 and Figure 5-3 present the angle of attack rate and pitch rate profiles for each kinematic motion respectively The angle of attack rate profiles have unique magnitude and phase responses due to the phase lag between the pitching and plunging kinematic motions. However, the magnitude of the pitch rate is equivalent for each motion. The various motions provide a wide range of the aerodynamic parameters (angle of attack, angle of attack rate, and pitch rate ) for which the unsteady vortex response is characterized. The response of the unsteady fluid structures to the aerodynamic parameters is realized through experimental measurements. Flow field measurements are utilized to measure the general development of the unsteady flow structures. To this end, PIV m easurements have been acquired at normalized times equal to 0.00, 0.25, 0.33, 0.42, and 0.50. These times represent various times throughout the downstroke where high lift values are typically achieved. Two-dimensional, two-component and threecomponent PIV is utilized to measure the flow field associated with the LEV and TV structures Details pertaining to each PIV setup are provided in section 5.2.2 Table 5-1 presents the values of each aerodynamic parameter where PIV flow measurements are acquired. The aerodynamic response of the unsteady fluid structures is acquired through sting balance measurements acquired throughout the entire cycle. The responses of the flow field and aerodynamic measurements to the various aerodynamic
114 parameters further the understanding of the influence of the aerodynamic parameters on unsteady fluid dynamics. The accuracy of the kinematic motion directly affects the aerodynamic parameter profiles. Therefore, it is desirable to quantify the error between the commanded and ac hieved pitch and plunge motions. The individual plunge and pitch kinematic motions employed to achieve the desired aerodynamic parameter profiles are presented in Figure 5-4 Deviations from the commanded motions are quantified by measuring the linear motor displacements driving the pitch and plunge motions. Encoder feedback from the dynamic motion right (DMR) is used to quantify the error between the measured and commanded positions. The encoder signal is acquired on a National Instruments NI-6250 high speed multifunctional data acquisition card. This data acquisition card is equipped with two counter channels wi th a 32 bit resolution. Encoder information is sampled for each linear motor at a rate of 3000 Hz to obtain 60,000 samples. This results in capturing the start of the motion, approximately 30 cycles, and the end of the motion. The first three full cycles are ignored to eliminate any transitory effects. To estimate the error associated with the motion, the encoder feedback is averaged over eight cycles. The mean encoder positions are used to calculate the measured pitch and plunge profiles and ultimately the aerodynamic parameter profiles. The root mean square (rms) of the error between the measured and commanded values is calculated by (5 1)
115 where is the variable of interest [ 112 ] Figure 5-5 presents the measured kinematic motions for the plunge and pitching motions associated with the pure plunge and pitchplunge kinematic motions with phase lags 30, 75, and 90. The most significant error occurs at the peaks of the sinusoidal motions due to the linear motors overshooting their commended responses. PID tuning iterations were conducted to minimize the overshoo t and reduce the overall error Table 5-2 shows the root mean square error each of the kinematic motions utilized in the LEV/TV development experiments outlined previously. The plunge and pitch kinematic motions are achieved to within a root mean square value of 0.0002 normalized units and 0.03 respectively. The propagation of the linear motor errors into the model parameters can be visualized in Figure 5-6 Figure 5-7 and Figure 5-8 Figure 5-6 presents the commanded and measured angle of attack profiles for each kinematic motion. An overall good agreement exists for all the kinematic motions. Similarly Figure 5-8 shows that the measur ed pitch rate is consistent with the desired pitch rate. The angle of attack rate profile presents the most significant error. This is clearly evident in the rms error for each model parameter presented in Table 5-3 Minimizing the angle of attack rate error is essential if it is discovered that this parameter has a significant influence on the development of the LEV or TV structures. 5.1.3. Quasi-steady LEV Response Experiments Due to the large range of aerodynamic parameters associated with the general development of the LEV and TV structures presented in section 5.1.2 a quasi-steady analysis is utilized to isolate the LEV development from the influence of each aerodynamic parameter. Figure 5-9 presents the aerodynamic parameter de sign space for the general development of the LEV and TV experiments, as indicated by the black
116 squares. Th ese experiments provide sufficient understanding into the overall development of the unsteady vortex structures as a function of angle of attack and angle of attack rate, but lack insight into the quasi-steady nature of these structures. To this end, a quasi-steady design space is developed and is represented by the red squares in Figure 5-9 Th e angle of attack range was chosen due to large coefficient of lifts being present at these angles of attack. Each quasi-steady design point utilizes between two to four independent kinematic motions to achieve the desired aerodynamic parameters. This allows for the assessment of the LEV development to be purely a function of the aerodynamic parameters rather than similar kinematic motions. This analysis implies that the temporal history of the flow field development does not affect the instantaneous flow field development. At lower Reynolds number flight regimes where the LEV has been shown to dynamically shed from the wing, this analysis is not valid. The motions for the quasi-steady investigations are developed from the pitch-plunge and pure plunge kinematic motions outlined in the previous section. However, the constant pitch angle ( is varied for each motion to achieve the desired angle of attack and angle of attack rate at each design point. The influence of rotation rate is realized comparing each kinematic motion at a particular design point. Figure 510A presents the angle of attack and angle of attack rate profiles at the design point where and It can be seen that the angle of attack and angle of attack rate profiles are highly unique for a respective cycle The pitch and plunge profiles used to generate the aerodynamic parameter profiles are presented in Figure 510B It can be seen that the desired aerodynamic parameters occur from time 0.19 to 0.35 in the plunge stroke. If the LEV
117 or TV is in fact a function of the plunge stroke, this will be realized when relating the flow fields at these various times. Figures of all the aerodynamic parameter and kinematic profiles utilized at each design point are located in the Append ix B S tereoscopic PIV is utilized to measure the development of the LEV at each design point. Further details into the PIV setup are described in section 188.8.131.52 This setup allows for the spanwise velocity through the LEV to be captured which provides insig ht into influence of this flow on the stability of LEV. The spanwise flow induced from the TV behind the LEV is also captured such that the ability of this spanwise flow to retard the shedding of the LEV structures or enabling the flow to remain attached to the wing surface is realized. Similarly, the interaction between the LEV and TV structures is assessed. 5.2. PIV Measurements Flow field measurements are utilized in this study to realize the development and interaction of the LEV and TV along with estimating the lift on the wing. Both t wo component and three-component PIV measurements are used to measure various flow fields. This section outlines the hardware and measurement setups for the PIV measurements acquired in this dissertation. 5 .2.1. PIV Hardware A LaVision two-dimension al three component (2D3C) PIV system is used to measure the flow field. The PIV system is provided by LaVision and consists of a Litron Nano L 135-15 laser which is used to create pairs of 532nm wavelength pulses at a maximum of 13 5 mJ/pulse. The laser pulses are expanded into a light sheet and aligned with respect to the freestream. A pair of Imager ProX-4M cameras comprises the stereoscopic PIV system. Each camera has nominally a 2048x2048 pixel resolution
118 CCD chip which is capable of acquiring images at 14 frames per second. The 512 mb of onboard, camera memory is capable of storing 36 full resolution images before reaching capacity. The camera and laser systems are integrated though a computer running DaVis 7.2 software. 5.2.2. PIV Setups Three different PIV setups are utilized to measure the flow fields presented in this dissertation Streamwise t wo -dimensional, two component (2D2C) PIV is used to capture the unsteady flow features of the LEV on the midspan of the wing. This setup allows for a large field of view but lacks the ability to capture the spanwise component of velocity. Therefore, streamwise two-dimensional, three component PIV focused at the leading edge is utilized such that the spanwise flow through the LEV system can be quantified. The TV structure is highly three dimensional and requires the use of t wo dimensional, three component (2D3 C) PIV oriented in the spanwise direction A similar setup is used to measure the wake flow field such that the lift on the wing can be estimated The following sections outline each setup. 184.108.40.206. Streamwise t wo -dimensional two component PIV The 2D2C PIV setup utilizes one or two ProX-4M cameras perpendicular to the xy plane of the wing. A schematic of this setup is presented in Figure 511. The twodimensional laser plane is located at the 25% spanwise location. Two cameras are position ed side by side such that higher spatial resolution is ac hieved while still resolving the flow over the entire wing. The cameras are nominally centered on the LE and TE to reduced shadow regions incurred by the lenses parallax effects. This results in an overlap region of approximately one inch in the streamwise direction. An average resolution of 16.9pix/mm is achieved for each camera. The two images are stitched
119 together using common calibration points on each image. The overlapping vector fields from the corresponding images are combined using a weighted mean from each image to calculate the total mean. Figure 512 shows a schematic of the weights chosen for each image. The weights are linearly chosen from 0 being the edge of the particular image and 1 being the edge of the overlapped region in that image. A value of 0 is chosen for the edge because PIV data is inherently noisey around the edges due to seed particles entering or leaving the field of view. DaVis 7.2 PIV software is used to analyze the PIV images with a multipass, cross-correlation algorithm. The first pass consists of an interrogation region of 64x64 pixels with 50% overlap. Two more passes are conducted using a 32x32 pixel interrogation window with 50% overlap. DaVis software finally post processes each single vector field to remove vectors greater than three times the root mean squared (rms) value of its neighbors. This procedure is completed for 420 double frame images at each time in the motion. After using the DaVis software to calculate velocity vector field, the mean velocity field is calculated by averaging each interrogation region over the 420 ensembles. Any velocity vector lying outside of three standard deviations throughout the ensembles is removed. This process typically eliminated 12% of the vectors at any given location. 220.127.116.11. Streamwise t wo -dimensional, three component PIV Streamwise two-dimensional, three component PIV is utilized to realize the interaction between the LEV and TV at the 25% spanwise location. This setup requires that the cameras view the two-dimensional laser sheet from oblique angles while remaining out of the flow. Each camera utilizes a Sigma DG MACRO 105mm lens with an f-number of 4.0. A schematic of the setup is shown in Figure 513 The front and
120 back cam eras are rotated about the vertical axis 15 and 30 respectively These angles are chosen to minimize any large intensity laser blooming off the wing leading edge from being visible to the cameras. A consequence of these angles is the wing block ing the observation of the flow field just upstream of the leading edge such that the velocit y cannot be measured Similarly, s ignificant laser blooming is observed by the cameras off the upper surface of the wing when viewing from such oblique angles. This is minimized by rotating the cameras about the spanwise axis such that the wing appears lev el at each angle of attack After the vector fields are computed in this rotated coordinate system, a coordinate rotation on the velocity vectors is completed such that the velocity vectors are in the original wind tunnel coordinate system. An in house Mat lab script computes the camera spanwise angle of rotation by iterative rotating the measured freestream velocity vectors until the angle of the freestream velocity is less than 0.01 This procedure is validated utilizing a digital inclinometer to measure the angle of rotation. The vector fields are computed from the double frame d images utilizing DaVis 7.2. A multi pass algorithm makes a single pass with an interrogation region of 64x64 pixels and is s ubsequently followed by two passes of a 32x32 pixel interrogation window. Each pass utilizes a 50% overlap. The final vector field has a scaling of 16.64pix/mm Erroneous vectors due to insufficient seeding are remove d by a median filter This filter remov es any vectors outside three standard deviations of the nearest neighbors A total of 60 ensembles are acquired and averaged to obtain the mean flow field. Any vectors outside three standard deviations of the ensemble mean are removed.
121 The uncertainty in the velocity measurements are determined from the equations outlined in section 18.104.22.168 The particle image diameter is estimated to be nominally 1.56 and 1.55 normalized pixel diameters for cameras 1 and 2 respectively. Therefore, the sub-pixel accuracy for each camera can be estimated as 0.04 pixels for a 32x32 interrogation window utilized on double framed image [ 98 ]. However, it is desirable to use a more conservative estimate for the sub-pixel accuracy due to the presence of noise; therefore, a sub-pixel accuracy of 0.1 pixels is chosen. The resulting systematic PIV uncertainty for the x, y, and z components of velocity are 0.058m/s, 0.055m/s, and 0.1 54 m/s respectively. 22.214.171.124. Spanwise t wo -dimensional, three component PIV The tip vortex and wake flow fields are measured using a two-dimensional, three component (2D3C) PIV setup with the laser sheet oriented in the spanwise direction normal to the freestream velocity. This setup is shown in Figure 514 Cameras 1 and 2 are rotated about the vertical access by 30.1 and 61.1 respectively Laser blooming is alleviated by aligning the laser sheet along the spanwise axis resulting in larger laser blooming at the wing tip while minimizing the blooming over the upper and lower surfaces of the wing. Rhodamine paint along with green color optical filters further reduces the influence of the laser blooming DaVis 7.2 software is used to analyze the PIV images. A multi-pass correlation algorithm is performed reducing the interrogation window size from 64x64 pixel to 32x32 pixel window with a 50% overlap, ultimately resulting in a 13. 92 pix/mm resolution. The a freestream particle displacement of approximately 1mm which is under the 2mm laser sheet thickness. The particle image diameters on camera 1 and 2 are 1.54 and 1.51
122 normalized pixel units respectively. The mapping error from the calibration is 0.2088 and 0.3361 for camera 1 and 2 respectively. Assuming a sub-pixel accuracy of 0.1 pixels, the uncertainty in the x, y, and z velocity components are 0.099m/s 0.058m/s and 0.087m/s respectively The tip vortex structure is obtained through multiple, two-dimensional spanwise planes acquired at normalized chord locations of x/c=0.10, 0.20, 0.30, 0.40, 0.60, and 0.80. It should be noted the centerline and tip of the flat plate corresponds to z/c=0 and z/c=1. A schematic of these locations is presented in Figure 515 The mean velocity field is calculated from 250 instantaneous, phase averaged velocity fields such that statistical convergence is achieved. The resulting statistical uncertainty of the mean streamwise velocity is 0.272mm/s assuming a 95% confidence interval. The control volume technique utilized to estimate the static lift outlined in Chapter 4 requires the measurement of flow field in the wake of wing. The spanwise 2D3C setup is utilized with the laser sheet spaced approximately 1 inch from the trailing edge ( TE ) of the wing. This spacing is chosen such that it was far enough from the wing to remove glare off the wing surface Figure 516 presents schematics of each measurement setup. Figure 516 A shows one field of view (FOV) utilized to measure the wake behind a static wing. The FOV size is roughly 5x5 inches with the wing tip centered from left to right. The vertical placement is varied with angle of attack such that the boundary conditions are satisfied. For angles of attack 1725, the boundary conditions are unsatisfied; therefore, two FOVs are utilized to capture the flow field. This setup is shown in Figure 516B A n approximate 1 inch overlap is utilized between the FOVs to ensure the entire flow field is measured. For each of these FOVs, measurements are
123 acquired at 3 Hz until 2500 image pairs are obtained. The 2500 instantaneous flow fields are utilized to calculate the mean flow field. The control volume technique utilized to compute lift developed on a plunging flat plate requires the measurement of the wake flow field velocity. The spanwise 2D3C setup is again utilized with an increased number of fields of view such that entire wake region behind the oscillating wing is captured. To this end, nine PIV FOVs are utilized and are presented in Figure 516C Each FOV is approximately 5x5 inches and overlapped by an inch resulting in a total FOV of 13x13 inches or roughly two chord lengths. The wing travels vertically one chord length throughout a kinematic cycle Th erefore, ample room is present to measure the flow field such that the boundary conditions remain satisfied throughout the entire kinematic motion. The PIV system is triggered at the middle of the plunge stroke from the Galil controller. Delays introduced from the trigger provide a temporal resolution of 0.065. At each time 120 image pairs are acquired and used to compute the instantaneous, mean wake flow field. 5.3. Aerodynamic Lift Measurements Aerodynamic lift measurements throughout the unsteady motions are essential to understand the aerodynamic consequences of the development of unsteady fluid structures. The following sections provide detai ls into the sting balance utilized to acquire these measurements and the tare technique employed to remove inertia effects induced by the kinematic motion from the sting balance measurements. 5.3.1. Sting Balance Overview The aerodynamic force measurements are acquired from a 6-component sting balance although only the resulting lift is presented in this dissertation The sting balance is an -0.10-0.375 6-component, strain gauge balance.
124 The balance is calibrated to obtain a 6x39 matrix that resolves the second and third order force interacti ons. Table 5 4 presents the maximum loading, resolution, and accuracy of the balance. 5.3.2. Sting Balance Measurement T echnique All aerodynamic measurements are acquired from an initial start trigger produced from the Galil controller This trigger ensures consistent timing between consecutive ensembles. The trigger simultaneously initiates the sampling of the encoder positions equipped on each linear motor. The balance signals are acquired on a National Instru ments SCXI 1520. A built in 4 th order lowpass Butterworth filter having a cutoff frequency of 1 000 Hz is utilized to prevent aliasing. Balance and encoder measurements are acquired at a sampling frequency of 3000 Hz for the duration of a 30 cycle period. The inertial effects induced from the motion are remove d through the utilization of a tare. The tare is designed with equivalent mass and moment properties of the flat plate. Specifically, the tare is comprised of a steel rod with the diameter of 0.615 inc hes and an exposed length of 4.40 inches. Figure 5 17 presents the configuration of the flat plate and tare attached to the sting balance. The tare is used to measure the inertial forces with a mean free stream velocity of 0.0 m/s and 4.0 m/s. The maximum coefficient of lift achieve d between these two measurements is less than 0.1 when normalized with equivalent scaling parameters as the wing Therefore the tare produce s minimal aerodynamic lift while effectively removing the inertial forces The sting balance lift measurements presented in this dissertation are the measured aerodynamic lift force with the tare force measurement subtracted out. This lif t value is the resulting aerodynamic lift.
125 Figure 5 1 Angle of attack profiles for pure plunge (PP) and pitch plunge kinematic motions. Figure 5 2 Angle of attack rate profiles for pure plunge (PP) and pitch plunge kinematic motions.
126 Figure 5 3 Pitch rate profiles for pure plunge (PP) and pitch plunge kinematic motions.
127 Table 5 1 Model Parameter Design Space Case [ ] [ /s] [rad/s] 1 0.7 126.6 0.9 2 5.8 76.8 1.8 3 6.4 153.3 0.0 4 8.0 180.0 0.0 5 8.0 73.1 1.9 6 8.0 73.1 1.9 7 8.0 180.0 0.0 8 9.0 78.4 1.3 9 10.2 76.8 1.8 10 10.7 61.8 1.6 11 12.1 57.3 0.4 12 12.9 31.3 0.9 13 13.0 138.1 1.0 14 13.5 0.5 0.0 15 13.8 27.6 0.5 16 14.8 155.5 0.0 17 15.3 126.6 0.9 18 17.7 92.5 1.6 19 20.2 82.7 0.0 20 21.9 1.4 0.0 Figure 5 4 Plunging and pitching kinematic motions defined for pure plunge (PP) and pitch plunge motions with phase lags of 90 75 and 30
128 A B C D Figure 5 5 Measured kinematic motions compared with commanded motions for pure plunge and pitch plunge motions A) pure plunge, B ) C ) and D ) Table 5 2 RMS error associated with normalized plunge and pitch kinemat ic motions. [ ] [ ] PP 0.0002 00 0.01 2 = 90 0.000212 0.03 0 = 75 0.000239 0.009 = 30 0.000092 0.018
129 Figure 5 6 Angle of attack profiles estimated from the encoder feedback co mpared to the commanded profile for each kinematic motion. Figure 5 7 Angle of attack rate profiles estimated from the encoder feedback compared to the commanded profile for each kinematic motion
130 Figure 5 8 Pitch rate profiles estimated from the encoder feedback co mpared to the commanded profile for each kinematic motion Table 5 3 Error associated with model parameters for each kinematic motion. [ ] [ /sec] [rad/sec] PP 0.0875 95. 6 0.000887 0.0679 34.6 0.0064 2 0.150 41. 2 0.00245 0.0434 36. 6 0.00224
131 Figure 5 9 Experimental aerodynamic parameter design space for the general (black) and quasi steady (red) investigations. A B Figure 5 10 A erodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 10.0 and 40 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles
132 Figure 5 11 Schematic of streamwise 2D 2C PIV setup Figure 5 12 Sche matic of image overlapping and w eight ing
133 Figure 5 13 Streamwise of streamwise 2D3 C PIV Setup Figure 5 14 Schematic of spanwise 2D3C PIV setup
134 Figure 5 15 Schematic of PIV planes where spanwise 2D3C measurements are acquired. A B C Figure 5 16 PIV wake field of view schematics A) 1 FOV B) 2 FOVs and C) 9 FOVs.
135 Table 5 4 0.10 0.375 Specifications Normal Force Axial Force Side Force Roll Mom e nt Pitch Moment Yaw Moment [N] + 15% [N] + 15% [N] + 15% [N cm] + 15% [N cm] + 15% [N cm] + 15% Max Load 44.4982 22.2491 22.2491 45.2101 203.4456 56.5127 Resolution 0.1112 0.0089 0.1112 0.0565 0.5651 0.5651 Accuracy 0.0334 0.0167 0.0167 0.0565 0.1526 0.0424 Figure 5 17 Schematic of flat plate and tare mounting c onfigurations Photo courtesy of author.
136 CHAPTER 6 FLOW FIELD RESULTS This chapter presents a discussion of the leading edge vortex ( LEV ) and tip vortex ( TV ) development as a function of the aerodynamic parameters discussed in section 5.1.1 such as angle of attack, angle of attack rate, and pitch rate. First, v el ocity field measurements are utilized to examine the effects of the angle of attack on the LEV and TV. This is followed by a more detailed analysis examin ing the effects of the angle of attack rate and pitch rate on the vort ical flow features The results presented provide insight into the development of the vortical structures as a function of the various aerodynamic parameters. 6.1. Effect of Angle of Attack on U nsteady V ortex D evelopment Th e development of the LEV and TV structure with respect to the angle of attack is discussed in this section. The LEV and TV responses to the angle of attack are examined in the presence at minimal pitch rates such that the flow field response incurred from t he pitch rate is isolated ; however, various angle of attack rates are present ed Minimizing the influence of the pitch rate is accomplished by extracting the flow fields at various times in the kinematic motions, presented in section 5.1.2 where the pitch rate is minimal. Finally, the interaction between the LEV and TV is discussed to gain insight into the unsteady fluid mechanisms responsible for their development 6.1.1. L eading E dge V ortex D evelopment The response of the LEV to the an gle of attack is presented in this section. First, the general response of the LEV throughout the downstroke of the pure plunge motion outlined in section 5.1.2 is discussed The pure plunge kinematic motion is analyzed because it isolates the response due to changes in pitch rate. Flow fields at normalized
137 times 0.00, 0.25, and 0.50 are presented an d characterized as a function of the angle of attack. Secondly, a quasi-steady analysis with respect to angle of attack is utilized to determine the LEV development from unique kinematic motions with constant aerodynamic parameters This analysis provides insight into the independence of the LEV development from the kinematic motion suggesting the physical mechanisms driv ing the LEV development are a function of the aerodynamic parameters. The various kinematic motions along with their corresponding angle of attack profiles are defined in section 5.1.3 and A ppendix B 6.1 .1.1. General response Figure 6-1 and Figure 6-2 present the normalized streamwise and vertical velocit y components respectively at different angles of attack throughout the pure plunge downstroke. Comparing the streamwise velocity flow fields of Figure 6-1 A and Figure 6-1 C a distinct difference is observed for equivalent angles of attack equal to 8.0. Figure 6 -1 A shows a small region of accelerated flow over the leading edge of the wing. The flow remains attached to upper surface of the wing. At an equivalent angle of attack Figure 6-1 C shows a larger region of recirculating flow over the wing surface. The size difference between these two structures is clearly observed in Figure 6-2 A and Figure 6-2 C The downward velocities behind each LEV stagnate on the wing at approximately x/c=0 .1 0 and 0.55 respectively. The large discrepancy in the development of the LEV at a constant angle of attack of 8.0 suggests another para meter is responsible for the LEV development. One such parameter is the temporal derivative of the plunge amplitude as these cases are indicative of the top and bottom of the plunge stroke. The angle of attack rate is inclusive of the plunge rate; therefore, the
138 influence of the plunge rate is observed in section 6.2.1 where the LEV flow field response to the angle of attack rate is discussed. Figure 6-1 B presents the normalized, mean streamwise velocity for an angle of attack of 21.9. At this larger angle of attack, the flow field shows the development of a large LEV region on the upper surface of the wing. This development is consistent with the LEV formed in Figure 6-1 C At an angle of attack equal to 21.9, the region of accelerated flow above the shear layer is larger in size than that in Figure 6-1 C. It can also be seen that the recirculation region at the higher angle of attack is concentrated closer to the LE of the wing. The spanwise component of mean vorticity for these cases is presented in Figure 6-3 The vorticity is calculated from the mean streamwise and vertical velocity components Each case is associated with large negative values of vorticity in the shear layer dividing the highly accelerated flow due to the curvature of the streamlines and the recirculating flow close to the wing surface. These large values of vorticity are a result of large velocity gradients existing between the LEV and accelerated flow Therefore, it is useful to utilize the criterion to isolate shear from rotational effects such that unsteady vortex structures are more clearly identified. Graftieaux et al. [ 113 ] utilized the criterion to identify the LEV core locations. The algorithm is ideal for discrete spatial data locations. The vortex core identification technique is defined by (6 1) where S is a rectangular domain of fixed size, N is the number of points inside of S. P is the current point of interest in the measurement field. The angle between the velocity vector at a point and the radius vector ( ) is defined by A value of uni ty
139 yields an ideal vortex core. While the technique is not Galilean invariant, it provides a robust way to identify vortex core locations. Figure 6-4 displays the criteri on for the flow fields presented in Figure 6-1 and Figure 6-2 The criterion indicates a well-defined vortex structure is not present at an angle of attack and angle of attack rate equal to 8.0 and 180/s respectively. However, a LEV is shown to form at an angle of attack and angle of attack rate equal to 8.0 and 180/s. At an angle of attack of 21.9, the LEV core exists and resides in close proximity to the leading edge of the wing. This is an indication that response of the LEV is coupled between the angle of attack and angle of attack rate parameters. Th e investigation of the general LEV development with respect to angle of attack is inconclusive. Large deviations in the LEV development occur; however, isolating the response from the angle of attack and angle of attack rate is nearly impossible due to the large changes in these parameters throughout the downstroke of the pure plunge kinematic motion. Therefore, a quasi-steady analysis is utilized to isolate each of the corresponding LEV response to the aerodynamic parameters such that the parameters driving the unsteady flow physics can be ascertained 126.96.36.199. Quasi-steady response The quasi-steady responses of the unsteady vortex structures to the various aerodynamic parameters are utilized to assess the flow field response to each parameter independently. The flow fields presented in this section are extracted from the various motions at equivalent angle of attacks, independent of their temporal development. Similarly, constant angle of attack rates and pitch rates are presented to isolate the flow field effects incurred by their interactions. Nearly equivalent flow fields
140 indicate the LEV development being purely a function of angle of attack, whereas unique flow fields are indicative of other parameters defining the LEV development. Figure 6-5 presents the LEV development at z/c=0. 50 for angles of attack 10.0, 13.0 and 16.0 at a constant angle of attack rate equal to 40.0 /s. These flow fields are extracted from the pure plunge kinematic motion to eliminate the flow field response from various pitch rates. Similar to the previous results, the streamwise velocity associated with this angle of attack range is consistent with a highly accelerated flow over a recirculating region at the leading edge of the wing. It can be seen that the accelerated region of flow increase in size over wing as the angle of attack is increased. Similarly, t he region of reverse flow under the shear layer tends to increase in size as well. The angles of attack equal to 13 and 16 present what appears to be reverse flow regions that are separated into two distinct regions Figure 6-5 B shows a large negative streamwise velocity region close to the leading edge at x/c =0.08 and another region roughly centered at x/c=0.25. Increasing the angle of attack to 16.0 re sults in these regions moving to x/c=0.12 and x/c=0.32 respectively as can be seen in Figure 6-5 C The normalized vertical velocity, presented Figure 6-6 shows a large negative velocity dividing the two reverse flow regions. This figure also shows regions of strong, downward flow scattered along of the x-axis which corresponds to the large negative streamwise velocity regions mention previously. These structures, though varying in size across the flow fields, suggest a vortex shedding phenomenon from behind the LEV which is located at the leading edge of the wing The spanwise vorticity i n Figure 6-7 indicates a large amount of negative vorticity within the shear layer developed at the leading edge of the wing. At an angle of attack of 10.0 th e spanwise vorticity extends
141 downstream to approximately x/c=0.233. Increasing the angle of attack to 13 .0 and 16.0 results in the vorticity developed off the leading edge extending to x/c=0.144 and 0.152 respectively There also exists a strong, vortical structure present further downstream at the larger angles of attack This shedding-like response does not appear to be solely a function of the angle of attack. The deviation in the vortex core locations with respect to angle of attack is determined utilizing the criterion Figure 6-8 presents the criterion calculated from the streamwise and vertical velocities in Figure 6-5 and Figure 6-6 respectively. The LEV core is elongated along the wing surface at angle of attack equal to 10 .0 A significant LEV core is not realized due to the lack of flow field resolution close to the wing surface. The increase in angle of attack results in the LEV core moving off the surface of the wing and downstream. The contours at angles of attack equal to 13.0 and 16.0 indicate the existence of a LEV core in close proximity to the leading edge of the wing. However, at an angle of attack equal to 13.0, an additional vortex core is present just aft of the LEV. These two vortex core locations are consistent with the regions of recirculating flow shown in the streamwise velocity. Th e downstream vortex core is shed from the LEV and is not quasi-stea dy with respect to the angle of attack. 6.1.2. Tip Vortex Development The flow fields associated with the TV development for various angles of attack and angle of attack rates are presented in Figure 6-9 and Figure 610 respectively. The flow fields are acquired at x/c=0.6 and are presented as a streamwise slice viewed from an observer downstream looking upstream. This streamwise location represents the general development of the TV which is generally consistent with the TV upstream and downstream of this location.
142 In Figure 6-9 it can be seen that the growth of the TV is clearly dependent upon the angle of attack As the angle of attack is increased, the TV structure increases in size resulting in a significant dow nw ard velocity impinging on the wing surface. This increase in radial velocity indicates an increase in overall circulation associated with the TV [ 11 12 ]. A further indication of the increased strength of the TV is indicated by definition of circulation which is (6 2) where is the TV cross sectional area. This relationship shows that the circulation is proportional to the increase in streamwise circulation for equivalent cross sectional areas. Therefore, as the TV streamwise vorticity increases, the circulation of the TV increases and consequently th e downwash onto the surface of the wing increases. This is realized at an angle of attack equal to 20.2 and 21.9 where the TV has large values of streamwise vorticity. Subsequently, a large downwash onto the upper surface of the wing is present. This downwash results in inward spanwise velocities greater than 50% of the freestream velocity over the interior surface of the wing Figure 610 presents equivalent flow fields shown in Figure 6-9 ranked as a function of angle of attack rate which varies from 180 to -180/s No apparent trend is present in the development of the TV. Therefore, it can be reasoned that the angle of attack rate is not a dominant parameter in the formation of the TV. However, the angle of attack rate has the ability to alter the TV development at equivalent angles of attack which is evident at an angle of attack equal to 8.0 shown in Figure 6-9 This development is further investigated in section 6.2
143 The streamwise development of the TV for the plunge kinematic motion is presented in Figure 611 at equivalent angles of attack Figure 611 presents the Qcriterion which is defined by (6 3) The Q-criterion is used to quantify the vortical nature of the flow field [ 114 ]. The angular rotation and rate of strain tensor are defined by R ij and S ij respectively, such that the velocity gradient tensor is defined by their corresponding sum. These values are calculated in each measurement plane and are then interpolated in the streamwise directions to give an estimation of the development of the TV structure. Figure 611 presents the TV at an angle of attack equal to 8.0 for streamwise locations x/c=0.3 thru 0.8. The resulting TV structure presented in Figure 611A and Figure 611B have angle of attack rates equal to 180 /s and 180/s respectively. Clearly the angle of attack rate has an influence on the development of the TV because a TV is present on the upper side of the wing at -180/s and is nonexistent at 180/s. The development of the TV is more identifiable when plotting displacement of the center of the vortex core from the wing surface ( ) as a function of downstream distance over the wing. A schematic of this displacement is presented in Figure 612 The core location is determined from the center the criterion greater than 0.7. Figure 613 shows as a function of streamwise location for angles of attack equal 8.0, 14.8, 20.2, and 21.9 In general, the slope of this displacement increases as the angle of attack increases. Physically, this corresponds in a stronger TV structure with increasing angles of attack which ultimately provides larger spanwise flow on the aft end
144 of the wing. The increased three dimensional nature of the flow inevitably alters the aer odynamic response of the wing. 6.1.3. Leading Edge Vortex/ Tip Vortex Interaction Flow field measurements at various streamwise planes in close proximity to the leading edge are examined to gain insight into the LEV structure dynamics and its interaction with the TV. A schematic of the flow structures visualized at these streamwise planes is presented in Figure 614 This figure presents iso-surfaces of the nor malized streamwise velocities equal to 1 .0 (yellow) and 1.2 (red). This schematic is obtained through the interpolation of 2D2C PIV measurements at various spanwise locations ranging from z/c=0.5 to 0.9. This schematic indicates that streamwise slices of the flow fields acquired at x/c=0.1 and 0.2 provide insight into the interior structure of the LEV and the interaction between the LEV and TV at the wing tip. Figure 615 presents the mean, normalized streamwise and spanwise velocities measured at chordwise locations x/c= 0.1 and 0.2 for various angles of attack and angle of attack rates. The development of the LEV at the wing tip can be realized in the normalized streamwise velocity contour plots. At x/c = 0.1, it is shown that the LEV and TV are pinned at the intersection of the leading edge and the wing tip. Increasing the angle of attack increases the size of the recirculation region associated with the LEV. This is indicated by increasing magnitudes of negative and positive streamwise flow inside and above the LEV structure respectively. However, the LEV development is not purely a function of angle of attack. In Figure 615, it is shown that the flow fields at a constant angle of attack equal to 8 .0 with corresponding angle of attack rates equal to 180/s and -180/s are quite unique Therefore, the angle of attack rate plays a role in the development of the LEV structure.
145 The spanwise flow developed within the vortex core has been postulated in the literature to alter the development of the LEV [ 14] The results plotted in Figure 615 gives insight into the spanwise flow present in the core of the LEV structure. An outward spanwise velocity greater than 50% of the freeestream velocity exists within the core of the LEV structure. Further downstream of the leading edge at x/c=0.2, the outward spanwise flow from the LEV can be seen to feed into TV. This in turn adds momentum to the TV structure thereby strengthening it. This increase in strength results in larger spanwise momentum driven from the TV on to the aft surface of the wing. The spanwise flow inside the LEV core is plotted in Figure 616 for various angles of attack equal to 10 .0 13 .0 and 16 .0 At an angle of attack equal to 10, the LEV is a highly two-dimensional structure as indicated by minimal spanwise flow through the core. Significant inward, spanwise flow is present on the aft end of the wing due to the downwash incurred from the TV. Also present is outward spanwise flow in the shear layer bounding the LEV and accelerated freestream flow above the LEV. With the increase of the angle of attack to 16 .0 the outward spanwise flow increases in the core of the LEV structure to approximately 40% of the freestream velocity. This is further evidence of the LEV development being a function of angle of attack. Increasing the angle of attack also increases the magnitude of the outward spanwise flow in the shear layer above the LEV structure and the inward spanwise velocity from the TV on the aft of the wing. The three dimensionality of the LEV structure is realized by plotting the spanwise helicity. The normalized, spanwise helicity is presented in Figure 617 and is defined by
146 (6 4) where and are the spanwise velocity and vorticity respectively [ 115] The normalizing velocity and length scales are the freestream velocity and chord respectively It can be seen in Figure 617 that the LEV region is associated with negative values of spanwise helicity. The size of these helical structures increases with angle of attack. Positive values of spanwise helicity exist behind the LEV as a consequence of increased inward spanwise flow fr om the TV. These positive helical structures have larger intensities at an angle of attack rates equal to -40/s rather than 120/s. A further discussion of the LEV spanwise helicity as a function of the angle of attack rate and pitch rate is presented in section 6.2.3 6.2. Effect of Angle of Attack Rate and Rotation Rate on Unsteady Vortex Development The previous section identified the response of the LEV and TV to variations in angle of attack. However, inconsistencies in the flow fields indicated that the state of the LEV and TV development analyzes various flow fields as a function of angle of attack rate and pitch rate. Similar to the previous section, t he first two sections discuss the development of the LEV and TV structures respectively, followed by a discussion on the interaction of the LEV and TV structures. 6.2.1. Leading Edge Vortex Development Section 6.1.1 suggested an interaction between the angle of attack and angle of attack rate defining the development of the LEV. Therefore, the general re sponse of the LEV to a variety of angle of attack rates and pitch rates is discussed This is followed by an analysis of the quasi-steady development of the LEV. Th e quasi-steady analysis
147 alters one of the model parameters to assess the flow field response to that particular parameter while the other parameters are held constant. This isolates the LEV flow field response to the aerodynamic parameters of interest. 188.8.131.52. General development The general development of the LEV is investigated from flow field measurements throughout the downstroke of various pitch-plunge and pure plunge motions which were outlined in section 5.1.2 These flow fields have been characterized by the angle of attack, angle of attack rate, and pitch rate. The LEV development is discussed based upon these parameters. Figure 618 presents mean, normalized streamwise and vertical velocities at z/c=0.50. The criterion is also presented to identify the LEV vortex core location The cases presented represent discrete increases in the pitch rate such that the LEV dependency on the pitch rate can be evaluated. It can be seen that a general trend in the development of the LEV does not exist The LEV development is nearly equivalent at with pitch rates equal to -1.9 and 0.0 rad/sec. Each flow field at has minimal flow separation at the leading edge of the wing. Figure 619 presents the mean normalized streamwise and vertical velocity developed over a pitching wing pivoted about the leading edge which was investigated by Hart et al. [ 116 ]. This flow field is developed over an aspect-ratio four flat plate at an angle of attack and pitch rate of 7.2 and 2.52 rad/sec. At nearly double the pitch rate, a significant LEV is not formed. This is an indication of the LEV development being independent of pitch rate at an angle of attack equal to 8.0. The importance of the angle of attack parameter compared to the pitch rate is again shown in Figure 620 which presents the flow fields with constant pitch rate
148 values of 0.9 rad/sec at angles of attack 12.9 and 0.7. The two flow fields are drastically different. At constant pitch rates, a LEV structure is present at an angle of attack equal to 12.9, however this does not occur at 0.7. The criterion estimates the presence of a vortex core located at x/c=0.4 for an angle of attack equal to 12.9. This is an indication of another physical mechanism driving the LEV development which is not dependent on the pitch rate. The response of the LEV structure with respect to the angle of attack rate for a few discrete conditions is shown in Figure 621 This figure presents the normalized streamwise and vertical velocities associated with a nominal angle of attack equal to 13.0 Each set of conditions sh ow s the development of a LEV structu re in the flow field However, the criterion indicates the vortex core streamwise location being a function of the angle of attack rate. At angle of attack rates less than -27.6/s, the core of the LEV moves further downstream along the surface of the wing. The streamwise velocity indicates that this movement downstream results from the increase in the size or the recirculation region. Furthermore, the vertical velocity at angle of attack rates greater than -155.5/s indicate an oscillatory developme nt as the flow moves downstream behind the LEV. This is indicative of a shedding phenomenon. However, the flow re mains fully attached to the wing surface at large negative angle of attack rates This is most likely a result of strong spanwise momentum on the aft end of the wing generated from the TV. Therefore, it could be argued that large negative angle of attack rates result in different flow mechanisms present in the flow field that drive the development of the LEV and ultimately the unsteady aerodynamics
149 184.108.40.206. Quasi-steady response The LEV development with respect to the angle of attack rate and pitch rate is discussed from a quasi-steady analysis of the flow fields as described in section 5.1.3 This discussion presents discrete flow fields while varying only one aerodynamic parameter such that its influence on the LEV development can be ascertained. Figure 622 and Figure 623 presents the streamwise velocity and criterion for an angle of attack and angle of attack rate equal to 10 .0 and -40/s respectively. Each subfigure has a unique rotation rate varying from 1.11 to -1.84 rad/sec. The flow fields are consistent for a rotation rate range of 2.95 rad/sec. Each flow field consists of large regions of accelerated and recirculating flow. The largest recirculation region is present at a pitch rate equal to 0.0 rad/sec where it can be seen that large negative streamwise velocity exists along approximately 40% of the wing. This reverse flow region remains close to the wing surface where it elongates and contracts as the magnitude of the pitch rate increases. The contours indicate a highly elongated vortex core along the wing surface. However, a concentrated LEV core exists at The response of the LEV to angle of attack at constant pitch rates is presented in Figure 624 and Figure 625. Figure 624 and Figure 625 presents the flow fields associated with equivalent aerodynamic parameters as Figure 622 and Figure 623 with th e exception of the angle of attack being increased from 10 .0 to 16 .0 When comparing Figure 622 with Figure 624, it is shown that the changes in the flow field due to altering the pitch rate appears to be minimal as opposed to changes in angle of attack which has a significant effect on the LEV development The increase in angle of attack results in increase accelerated flow above the shear layer. The LEV acts as a
150 mechanism to increase the camber o f the wing, thereby further accelerating the flow over the wing surface. Highly cambered wings have been shown to achie ve higher coefficient of lift values [ 15 ] The ability of the LEV to accelerate this flow while prevent ing separation enables a dynamic ability to increase the aerodynamic performance of the wing. The region of reverse flow associated with the recirculation region also increases with the angle of attack. As this region increase s in size t he criterion shows the development of con centrated vortex cores at the leading edge of the wing, indicating the formation of a highly developed LEV structure. The development of the LEV with respect to various angles of attack and angle of attack rat es can be seen in Figure 6 26 and Figure 6 27 The pitch rate is equal to 0.0 rad/sec for each flow field. Increasing the angle of attack from 10 .0 to 16 .0 over each angle of attack rate consistently presents increased accelerated flow above the recirculation region and a concentration of the rever se streamwise flow at the leading edge of the wing. The criterion indicates the transition of the elongated vortex core at low angle s of attack transitioning to c oncentrated LEV structure s at the leading edge for higher angle of attack The LEV transitions to a more coherent structure at the leading edge of the wing as the angle of attack rate is decreased from 40/s to 120/s. Particularly, t his is shown at an angle of attack equal to 10 .0 where an elongated recirculation region form s a disc rete LEV core. At an angle of attack 16 .0 the LEV at an angle of attack rate of 40/s is further developed as the angle of attack rate increases to 120/s. Figure 6 28 presents the magnitude of the normalized spanwise circulation of the LEV computed from the flow field at z/c =0.50. The spanwise circulation associated with
151 the LEV is presented for each quasi-steady kinematic motion and plotted as a function of angle of attack and angle of attack rate. This analysis provides insight into the bound circulation associated with the LEV as a function of the aerodynamic parameters. The uncertainty bars represent the deviation of the mean normalized spanwise circulation with a 95% confidence interval due to the various pitch rates at each angle of attack and angle of attack rate. Overall, the LEV spanwise circulation increases with the angle of attack. As the angle of attack rate is decreased from 40/s to -120/s, the circulation associated with the LEV increases as angle of attack is increased. The LEV circulation is less responsive to changes in pitch rate at an angle of attack rate equal to -120/s as indicated by smaller uncertainty bars. The deviation in the circulation due to the pitch rate is more significant and angle of attack rates -40/s and -80/s where the uncertainty bars are larger The circulation growth between angles of attack 13.0 and 16.0 and angle of attack rates less than 80/s begin to coincide with one another. This is indicative of the fluid mechanisms driving the development of the LEV being constant. Section 6.1.3 suggests that the spanwise flow through the LEV could be responsible for such a development. Further analysis of the LEV/TV interaction as a function of the angle of attack rate is discussed in section 6.2.3 6.2.2. Tip Vortex Development The TV development is assessed from the examination of flow fields measured normal to the streamwise direction along the wing tip. Figure 629 presents the normalized velocity components associated with the TV at a chordwise location of x/c=0.6. The normalized velocity components and streamwise vorticity are presented as a function of angle of attack for various rotation rates. The normalized streamwise velocity and streamwise vorticity associated with the TV have similar structures at an
152 angle of attack equal to 8.0. This TV development occurs at equal and opposite rotation rates of 1.9 rad/sec; therefore, the response of the TV is likely independent of rotation rate. At larger angles of attack, accelerated streamwise velocity exists above regions of slower streamwise flow on the inward portion of the wing. The influence of the angle of attack on the development of the TV is shown in Figure 630. Figure 630 presents the flow fields associated with angles of attack equal to 0.7 and 12.1 at a constant pitch rate of 0.9 rad/sec. A t an angle of attack equal to 0.7, a TV is not present whereas at an angle of attack 12.1 a TV is present. The larger values of angle of attack drastically alter the development of the TV. This is further shown in Figure 631 which presents the displacement of the TV core from the surface of the wing ( ) as a function downstream distance for equivalent angles of attack equal to nominally 13 .0 Each case has a varying value of angle of attack rate and pitch rate The displacement of the vortex core is consistent for the various aerodynamic parameters except for one outlier. Generally, t he core separation distances tend toward x/c=0.25 as the separation distance tends toward zero. This location coincid es with the ax is of pitch rotation on the wing. The outlier TV core displacement profile is generated by a pure plunge kinematic motion where no such rotation point exists. This implies that the kinematic rotation point is important in the development of the TV structure. Figure 632 presents the TV core displacement for an angle of attack equal to 8 .0 A similar outlier is shown for the pure plunge kinematic motion. The displacement of the TV from the wing forms a slight concave shape as one moves downstream from x/c=0.25 to 0.80. This is the result of the rotation rate of the wing adding an induced velocity at the wing tip. This induced velocity increases as a function of chordwise location giving the
153 displacement curve this concave shape. This is not apparent at larger angles of attack where the velocity over the wing tip is dominated by the angle of attack. The TV development at smaller angles of attack is more susceptible to the wing tip induced velocity and therefore the pitch rate parameter. This induced velocity alters the instantaneous angle of attack realized along the wing tip. The influence of this wing tip velocity can be estimated by adding the contribution from induced rotational velocity of the wing to the definition of the instantaneous angle of attack. For small angles of attack, the angle of attack can be redefined as (6 5) where is the chordwise location an d is the chordwise location at which the wing is pivoted about This definition of the instantaneous angle of attack incorporates both the induced plunging and rotational velocities due to the kinematic motion This angle of angle of attack definition will be more appropriate at defining the development of the TV at lower angles of attack when the induced rotational velocity associated with the pitch rate becomes dominant over the freestream velocity and induced plunge velocity. 6.2.3. Leading Edge Vortex/Tip Vort ex Interaction This section discusses the interaction between the LEV and TV as a function of the aerodynamic parameters. Figure 633 presents the normalized streamwise and spanwise velocities at streamwise planes x/c = 0.1 and 0.2. The se flow fields are developed at angle of attack rate values approximately equal and opposite of one another These cases show the two unique LEV/TV interactions. At /s, a TV is developed on the wing tip. Evidence of this structure is the strong inward spanwise flow above the wing tip and a reduced streamwise velocity present near the wing tip
154 where the core of the TV resides. The spanwise flow generated from the TV is directed toward the midspan of the wing. Physically this is representative of flow from the TV injecting itself into the LEV core. At an angle of attack rate equal to -138.2/s, the spanwise velocity transitions to an outward spanwise flow This outward spanwise flow is present within the core of the LEV structure Ultimately, this spanwise momentum is driven into the TV system increasing its overall strength downstream. Figure 634 shows the development of the LEV and TV at the leading edge of the wing as a function of angle of attack rates. The angle of attack is approximately 13.0 while the angle of attack rate is va ried The influence of the pitch rate on the vortex structure development should be minimal as the measurement planes are located in close proximity to the rotation point which in turn minimizes the induced velocity on the wing. There appears to be a transition in the development of the LEV/TV structures when the angle of attack rate is less than -57.3/s The physical phenomenon associated with this condition is the development of the TV structure At angle of attack rates equal to 0.5/s, -27.6 /s and -57.3 /s, the recirculation region associated with the LEV extends to approximately z/c=0.93, 0.95, and 0.98 respectively at x/c=0.1. This is a result of the TV structure being highly developed along the leading edge/wing tip intersection This development allows the TV to drive momentum into the interior of the wing by means of the LEV structure Th e added momentum in to the LEV core is eventually shed from the LEV structure as described in section 6.2.1 and presented in Figure 621. However at the LEV recirculation region extends to the wing tip at x/c=0.1. This is a consequence of the LEV being the dominant structure along the leading edge/wing tip intersection In this case, the flow is driven from the LEV into the
155 TV structure This mechanism is characteristic of additional momentum from the LEV structure to be transferred to the TV, thereby eliminating the need for the LEV to expel momentum through shedding. Th e TV strength is increased on the aft end of the wing due to the added momentum from the LEV structure, thereby increasing the downwash and ultimately the three dimensional flow on the aft of the wing. This ultimately allows for accelerated streamwise flow over the LEV to remain attached to the wing surface. The spanwise momentum exchange between the LEV/TV systems is shown by investigating the spanwise helicity at z/c=0.5. The spanwise vorticity of the LEV is consistently negative at the leading edge of the wing. Therefore, positive or negative spanwise helicity represents a spiraling outward or inward structure respectively. Figure 635 presents the normalized spanwise helicity for various angles of attack and angle of attack rates. The spiraling inward LEV structure exists at an angle of attack and angle of attack rate equal to 10 and 40/s respectively This structure is consistent of an inward spiraling LEV core driving spanwise momentum in to the interior of the wing through the LEV. As the angle of attack is increased or angle of attack rate is decreased, this inward, spiraling structure transitions to an outward spiraling structure. These results ar e consistent with the spanwise velocity measurements at x/c=0.1 and 0.2 presented previously in Figure 634 This outward, spiraling structure is present at an angle of attack equal to 10 while the angle of attack rate is -120/s. Therefore, the angle of attack rate dominates this response from the LEV. Figure 636 and Figure 637 present the spanwise helicity at constant angles of attack equal to 10 and 16 respectively. Figure 636 has an angle of attack rate equal to 40/s and various pitch rates. Each flow field is consistent with a spiraling inward LEV structure. At an angle of attack and
156 angle of attack rate equal to 16 .0 and 120 /s, the LEV stru cture is a spiraling outward structure for various rotation rates. This is a clear indication of the angle of attack and angle of attack rate being the aerodynamic parameters defining the formation of the spiraling LEV structure. 6.3. Summary Th e results pres ented in th is chapter identify the development of the leading edge vortex (LEV) and tip vortex (TV) as a function of the angle of attack, angle of attack rate, and pitch rate. Various flow fields were analyzed to assess the development of each structure. This analysis provide d insight into the quasi steady nature of the se unsteady vortex structures. The LEV ha d a unique development with respect to the angle of attack and angle of attack rate parameters. The LEV development w a s not solely a function of the angle of attack which wa s evident from highly unique flow fields exist ing at equivalent angle of attack s The quasi steady investigations utilized equivalent angle of attack and angle of attack rate parameters and showed a s imilar LEV development for a pitch rate range of 2.94rad/sec. Furthermore, the normalized spanwise circulation associated with the LEV indicated a consistent LEV development at larger angle of attacks and large negative angle of attack rates. The criter ion showed a transition of the LEV core from a n elongated structure along the wing surface at low angles of attack to a concentrated vortex core at larger angle of attacks. The results indicated the angle of attack and angle of attack rates defined the dev elopment of the LEV structure at angles of attack greater than approximately 10 and angle of attack rates less than approximately 4 0 /s. The TV development ha s been analyzed from flow field measurement acquired at various chordwise locations. At x/c=0.6 the TV development wa s shown to be a
157 function of the angle of attack and angle of attack rate. This was reasoned from highly unique flow fields associated with the TV at angles of attack equal to 8.0 and angle of attack rates equal to 180/s and 180 /s However, as the angle of attack wa s increased, the levels of streamwise vorticity in the TV increased which wa s indicative of a highly developed TV structure. Increasing the angle of attack from 8 to 21, the vertical velocity associated with the TV provided significant downwash onto the aft surface of the wing which drove spanwise flow toward the midspan of the wing. Evaluating the separation distance of the TV core from the surface of the wing along the chordwise direction, it was shown that the TV lifts further off the surface of the wing at larger angles of attack. Furthermore, the pitch rate had little effect on this separation distance at angles of attack equal to 8.0 and 13.0. This was an indication of the pitch rate having negligible influence on the TV development at larger angles of attack. However, the rotation point about which the wing was pitched could possibly alter the streamwise development of the TV. The interaction between the LEV and TV has been shown to provide a highly three dimensional LEV structure that was a function of the angle of attack and angle of attack rate. At an angle of attack equal to 13, flow field measurements centered about the wing tip at x/c=0.1 and x/c=0.2 indicated an inward spanwise flow at angle of attack rates greater than approximately -80/s where the TV was shown to develop over the wing tip. Decreasing the angle of attack rate further, the LEV develop ed along the entire leading edge all the way out to the wing tip. This correspond ed to outward spanwise flow through the LEV structure. Figure 638A and Figure 638B present a schematic of the inward and outward spiraling LEV structure. These figures utilize iso-surfaces of the
158 freestream velocity to define the size of the LEV and TV structures where the yellow and red contours represent the normalized streamwise velocities of 1.0 and 1.2. In Figure 638A the LEV develops a inward spiraling LEV structure due to the formation of the TV on the wing tip which in turn drives momentum into the LEV structure as indicated by the black arrow. Quasi-steady investigations utilizing spanwise helicity contours indicated the development of an inward spiraling structure which wa s consistent with angles of attack below 13 and angle of attack rates greater than -80/s. The additional momentum driven into the LEV ultimately develop ed a shedding like phenomena behind the LEV. This provided the LEV a mechanism to alleviate this additional momentum; otherwise the LEV would be unstable. As the angle of attack increased or the angle of attack rate decreased, the LEV transition ed to an outward spiraling LEV structure as presented in Figure 638B The spanwise helicity contours indicated an outward spiraling LEV structure which was indicative of spanwise momentum being driven through the LEV to the wing tip. This momentum was entrained into the TV where it provided further downwash onto the aft end of the wing. With the increase in spanwise flow on the aft end of the wing, the accelerated flow over the LEV wa s shown to remain attached to the surface of the wing. The shedding-like phenomenon was not present with an outward spiraling LEV.
159 A B C Figure 6 1 The normalized, mean streamwise velocity for various angles of attack at z/c=0.50. A ) 8.0 B ) 21.9 and C ) 8.0
160 A B C Figure 6 2 The normalized, mean vertical velocity for various angles of attack at z/c=0.50. A ) 8.0 B ) 21.9 and C ) 8.0
161 A B C Figure 6 3 Normalized, mean spanwise vorticity at various angles of attack at z/c=0.50. A ) 8.0 B ) 21.9 and C ) 8.0 A B C Figure 6 4 1 criteri a at various angles of attack at z/c=0.50. A ) 8.0 B ) 21.9 and C ) 8.0
162 Figure 6 5 Normalized streamwise velocity at z/c=0.50 for various angles of attack at a n angle of attack rate equal to 40.0 /s.
163 Figure 6 6 Normalized vertical velocity at z/c=0.50 for various angles of attack at a n angle of attack rate equal to 40.0 /s.
164 Figure 6 7 Normalized spanwise vorticity at z/c=0.50 for various angles of attack at a n angle of attack rate equal to 40.0 /s.
165 Figure 6 8 1 criterion at z/c=0.50 for various angles of attack at a n angle of attack rate equal to 40.0 /s.
166  [/s] 8.0 180.0 8.0 180.0 14.8 155.5 20.2 82.7 21.9 1.4 Figure 6 9 Normalized, mean velocity and streamwise vorticity associated with TV at x/c=0.6 for a pitch rate equal to 0 .0 rad/s ranked as a function of angle of attack
167  [/s] 8.0 180.0 21.9 1.4 20.2 82.7 14.8 155.5 8.0 180.0 Figure 6 10 Normalized, mean velocity and streamwise vorticity associated with TV at x/c=0.6 for a pitch rate equal to 0.0 rad/s ranked as a function of angle of attack rate.
168 A B Figure 6 11 Q criterion of tip vortex core for an angle of attack equal to 8.0 Figure 6 12 Schematic of vortex core displacement from wing surface ( ).
169 Figure 6 13 Tip vortex core location with respect to surface of wing th roughout pure plunge kinematic motion.
170 A B C Figure 6 14 Schematic of LEV/TV structures visualized through iso surfaces of the normalized freestream velocities equal to 1 (yellow) and 1.2 (red). A) isometric view B) right view and C ) top view
171 [ ] [/s] x/c = 0.1 x/c = 0.2 8.0 180.0 21.9 1.4 20.2 82.7 14.8 155.5 8.0 180.0 Figure 6 15 Normalized streamwise and spanwise components of velocities at various angle of attack rates at x/c=0.1 and 0.2.
172 Figure 6 16 Normalized mean spanwise velocity at z/c=0.50 with a pitch rate of 0.0 rad/s
173 Figure 6 17 Normalized spanwise helicity at z/c=0.50 with a pitch rate equal to 0 .0 rad /s.
174 [ ] [rad/s] 8.0 1.9 13.0 1.0 8.0 0.0 12.9 0.9 8.0 1.9 Figure 6 18 N ormalized streamwise and vertical velocity components along with 1 criterion at z/c=0.50
175 Figure 6 19 N ormalized streamwise and vertical velocity components of pitching flat plate [rad/s]  0.9 12.9 0.9 0.7 Figure 6 20 The normalized streamwise and vertical velocity along with the 1 criterion at z/c=0.50 at angles of attack 12.9 and 0.7
176 [ ] [/s] 13.5 0.5 13.8 27.6 12.1 57.3 14.8 155.5 Figure 6 21 The normalized streamwise and vertical components of velocity and 1 criterion at a spanwise location of z/c=0.50
177 Figure 6 22 Normalized streamwise velocity at a n angle of attack and angle of attack rate equal to 10.0 and 40.0 /s respectively.
178 Figure 6 23 1 criterion at a n angle of attack and angle of attack rate equal to 10.0 and 40.0 /s respectively.
179 Figure 6 24 Normalized streamwise velocity at a n angle of attack and angle of attack rate equal to 16.0 and 40.0 /s respectively
180 Figure 6 25 1 criterion at a n angle of attack and angle of attack rate equal to 16.0 and 40 /s respectively.
181 [ /s] Figure 6 26 Normalized streamwise velocity for varying angles of attack and angle of attack rates for a constant pitch rate equal to 0.0 rad/sec.
182 [ /s] Figure 6 27 1 criteria for varying angles of attack and angle of attack rates at a constant pitch rate equal to 0.0 rad/sec.
183 Figure 6 28 No r malized spanwise circulation magnitude associated with the LEV for each quasi steady kinematic motion
184 [ ] [rad/s] 8.0 1.9 13.0 1.0 8.0 0.0 12.9 0.9 8.0 1.9 Figure 6 29 Normalized f low field velocities and spanwise vorticity associated with the TV at x/c=0.6
185  12.9 0.7 Figure 6 30 Tip Vortex development at x/c=0.6 for a pitch rate equal to 0.9 rad/s Figure 6 31 Normalized displacement of the TV core from the wing surface at a nominal angle of attack equal to 13.0
186 Figure 6 32 Normalized d isplacement of the TV c ore from the wing surface at a nominal angle of attack equal to 8.0 [ ] [/sec] x/c = 0.1 x/c = 0.2 15.3 126.6 13.0 138.1 Figure 6 33 Contour plots of normalized streamwise and spanwise velocities two opposite sign angle of attack rates at x/c=0.1 and 0.2.
187 [ ] [/sec] x/c = 0.1 x/c = 0.2 13.5 0.5 13.8 27.6 12.1 57.3 13.0 138.1 14.8 155.5 Figure 6 34 Contour plots of normalized streamwise and spanwise velocity at chordwise locations x/c=0.1 and 0.2.
188 [ /s] Figure 6 35 Normalized spanwise helicity for varying angles of attack and angle of attack rates for a constant pitch rate equal to 0.0 rad/sec.
189 Figure 6 36 Normalized spanwise helicity at a n angle of attack and angle of attack rate equal to 10 .0 and 40/s respectively.
190 Figure 6 37 Normalized spanwise helicity at a n angle of attack and angle of attack rate equal to 16.0 and 12 0/s respectively. A B Figure 6 38 Schematic of inward and outward spiraling LEV structure. A) Inward spiraling. B) Outward spiraling
191 CHAPTER 7 AERODYNAMIC LIFT RES ULTS This chapter discusses the aerodynamic lift generated o n a low aspect ratio flat plate The first section discusses investigations pertain ing to t he static lift on the wing to 1) validate the lift estimation technique outline d in Chapter 4 and 2) to provide a baseline for which the in fluence of the unsteady fluid structures can be assessed T he second sectio n present s the aerodynamic lift as a function of the angle of attack and the angle of attack rate to assess the response of the unsteady vortex structure s presented in Chapter 6 Lastly, the control volume lift estimation technique is utilized to compute the lift generated by the pure plunge kinematic motion This analysis provid e s i nsight into the ability to alter the lift profile. Overall, this discussion offers significant insight into the response of unsteady vortex structu res on the unsteady aerodynamic s 7.1. Static Lift The static lift of the aspect ratio tw o flat plate wing at a Reynolds number of 4 .0x10 4 is measured utilizing a sting balanc e and is presented in Figure 7 1 The figure also presents r esults by Torres et al. [ 16 ] and Okamoto et al. [ 117 ] which are acquired at Reynolds number s 1x10 5 and 1.15x10 4 respectively. An overall good agreement is shown for the linear increase in the coefficient of lift with respect to the static angle of attack ( ) At a Reynolds number of 4.0 x10 4 the measured maximum coefficient of lift is 0. 827 and the stall angle of attack is app ro x imately 18 3 The results presented show an influence the stall angle of attack. Torres et al. showed the coefficient of lift in the stall regime plateauing whereas Okamoto et al. show ed a rapid decrease in the coefficient of lift. At a Reynolds number of 4 .0x10 4 where the current investigations are acquired, a rapid
192 drop off in coefficient of lift past the stall angle of attack occurs. These results provide the baseline coe fficient of lift profile for the aspect ratio two flat plate as a function of angle of attack. The measured coefficient of lift is assessed as a function of theoretical models. Figure 7 1 presents t he lift slopes for a two dimensio nal theoretical flat plate and aspect ratio two wing [ 118 ] The two dimensional lift for a thin, finite wing is defined a s (7 1) based on lifting line theory. This model significantly over predicts the measured lift curves from the current and published results Lifting line theory is utilized to develop a relationship to cor rect for influence of the three dimensional flow effects occurred from the TV and is defined by (7 2) where AR is the aspect ratio [ 118 ] This model provides a more accurate estimation of the measure d slope. The deviation in for a two dimensional and finite wing result s in significant errors at large angles of attack. These results make apparent the discrepancies that can occur when applying a two dimensional unsteady lift model to a three d imensional body Furthermore, the three dimensional effects due to unsteady kinematic motions are increased due to the dynamic TV development and therefore, further corrections to the lifting line theo ry are inevitably required for oscillating wings. The coefficient of the lift is computed utilizing the control volume technique to assess its validity Figure 7 2 presents the measured ( ) a nd computed ( ) coefficient of lift values as a function of angle of attack. The computed coefficient of lift
193 uncertainty is representative of the resolution and random uncertainty of the mean to with in a 95% confidence interval However, because 2500 vector fields are utilized to calculate the mean velocity components, the PIV resolution accounts for the majority of the uncertainty in the mean computed lift. This is due to the random uncertainty of the mean is a function of the inverse square root of the number of samples for which 2500 samples are acquired There is a strong agreement between the calculated and measured coefficient of lift from angles of attack between -5.0 and 12.5 The difference between the measured and computed coefficient of lift values are presented in Figure 7-2 and are defined by (7 3) does not exceed 0.036 for angles of attack between -5.0 and 12.5 Significant discrepancies occur around the stall angle of attack and 15.0. The bias error defined in section 4.2 is also presented in Figure 7-2 The bias error is a function of the PIV measurement plane being misaligned from the wind tunnel x-y coordinate plane. It was proposed in section 4.2 to remove the bias error from the overall lift utilizing measurements of the freestream plane such that the bias error is eliminated when the freestream is subtracted from the wake flow field contributions. The coefficient of lift contribution associate with the bias error is approximately 0.072 and 0.108 for the single and double field of views, defined in section 220.127.116.11 utilized to measure the wake respectively. This error, if not accounted for, would dominate the measurements providing inaccurate lift computations. The wake velocity fields are analyzed to provide insight into the physical location attributing to the computed lift Figure 7-3 and Figure 7-4 present the normalized
194 streamwise and vertical velocity components in the wake of a static wing at -5.0 and 5.0 respectively. The trailing edge (TE) wing location is also plotted to provide a point of reference. A weak TV is formed as indicated by minimal vertical velocity at the wing tips. The mean streamwise velocity behind the TE of the wing is sufficiently slower due to the wake deficit. At an angle of attack equal to 2.5, these flow features become less significant as seen in Figure 7-5 The increase in angle of attack above 5.0 shows an increase in the TV as indicated by strong er downward velocities at the wing tip. Figure 7-6 and Figure 7-7 present the normalized streamwise and vertical velocity components behind static angles of attack equal to 10.0 and 12.5. A region of negative vertical velocity begins to appear over the interior of the wing, whereas at lower angles of attack this did not occur. The region of slower streamwise flow once developed behind the trailing edge begins to increase in size at the interior of the wing at z/c<0.6. This is an indication of the flow beginning to separate on the interior of the wing. The helical axis of the TV up until angle 12.5 has been oriented on a constant x-y plane along the wing tip as indicated by a uniform streamwise velocity component. However, at an angle of attack of 12.5 this axis bends towards the interior of the wing. Figure 7-8 A presents the rotational component of velocity ( ) associated with a TV axis normal to the z-y plane. As the helical axis bends toward the interior of the wing at an angle as seen in Figure 7-8 B the measured streamwise component of velocity is comprised of positive or negative contributions due to the rotational velocity above or below the vortex core respectively. The consequence of this is an antisymmetric streamwise flow associated with the TV structure. The antisymmetric streamwise flow is present angles of attack ranging from 15.0, as seen in Figure 7-9 through the stall angle of attack equal to
195 18.2 The negative vertical velocity present on the interior of the wing is further increased with the angle of attack. Similarly, the downwash due to the TV increases indicating a stronger TV with increased three-dimensional effects occurring over the wing. The flow fields presented provide insight into the wake flow field development as a function of angle of attack. The response of the lift with respect to the flow field development will need to be further assessed. Figure 710 presents profiles as a function of angle of attack. as defined by equation 4.17, is the integral of the normalized vertical and streamwise velocity components multiplied together along the spanwise direction It is desirable to discuss th e development of as a function of angle of attack because relates the fluid structures developed over the wing to the computed lift quantities. Positive and negative values correspond to positive and negative contributions to the overall computed lift. The solid and dashed lines correspond to the measured and estimated contributions. For details pertaining to the estimation techniques utilized, please refer to section 4.2 At angles of attack -5.0, 0.0, and 5.0, a nearly constant value of exists on the interior of the wing from approximately z/c<.9. Recalling the flow field, this is indicative of the development of a small TV with minimal negative vertical velocity on the interior of the wing. Essentially, the flow remains fully attached through these angles of attack. The profile at an angle of attack equal to 12.5 presents a plateau at =0.3 from z/c=0.8 to 0.5. This most likely occurs from the TV providing sufficient momentum to keep the flow attached to the wing. From z/c<.5 is estimated to decrease to a value of 0.25. This is indicative of the flow starting to separate on the interior of the wing. As the angle of attack is further increased, the spike in due to
196 the TV increases; however a significant decrease of exists on the interior of the wing unlike the plateau present at lower angles of attack. The computed coefficient of lift accuracy is assessed by comparing it with sting balance measurements Figure 7-2 presents approximately a 10% discrepancy between the measured and computed lift at angles of attack equal to 15.0 and in the stall region of angle of attack between 20.0 and 25.0. A discussion of these large differences is warranted such that they can be eliminated when utilizing the CV technique behind an unsteady kinematic motion. The uncertainty bars for each measurement indicate the difference is not occurring from measurement uncertainties. Therefore, the error is most likely occurring from a violation of the assumptions defining th e CV technique. At an angle of attack equal to 15.0, the analysis of the wake flow field in Figure 7-9 indicates the upper boundary condition is not satisfied. The negative velocity region on the interior of the wing is not fully captured and ultimately results in an under prediction of the profile on the interior of the wing. Larger values of along the interior of the wing would ultimately increase the lift and most likely account for the discrepancy between the computed and measured lift values. At angles of attack between 20.0 and 25.0, another violation of the CV assumptions is incurred; however this time it occurs at the TV. The estimation technique utilized to predict the decay from the TV outboard of the wing is chosen to be linear. At small angles of attack where the influence of the TV is small, this estimation technique is sufficient to produce accurate lift values. With the increase in size of the TV, the linear estimation provides a more aggressive slope in as the full size of the TV is not captured. For a nearly uniform streamwise velocity at x/c>1.1, an exponential decay would be more appropriate according to the Batchelor
197 model defined in section 1.1.2 An exponential decay profile would gradually decrease the computed lift providing a better agreement with the measured lift. The discrepancies at angle of attack 15.0 and greater than 20.0 highlight the importance of utilizing a sufficiently large measurement field of view such that the boundary conditions are satisfied. 7.2. Quasi-Steady Lift The quasi-steady aerodynamic lift is discussed to assess the lift response from the angle of attack and angle of attack rate. Figure 711 presents the measured mean coefficient of lift as a function of angle of attack throughout a single cycle for each motion defined in section 5.1.2 The error bars represent an estimate of the random error in the mean value to within a 95% confidence interval. The measured static coefficient of lift profile is presented as a function of angle of attack such that the influence of the unsteady vortex structures is realized It can be seen that the unsteady fluid structures developed from periodic kinematic motions can significantly increase the coefficient of lift over that of the static lift profile The pure plunge motion presents a coefficient of lift at an angle of attack equal to 18 .0 that is approximately 40% greater than that of the static angle of attack. For the pure plunge case at angles of attack greater than approximately 13.0, the deviation in the coefficient of lift is small. This is indicative of a parameter other than the angle of attack driving the development of the unsteady vortex structures. Flow field measurements presented in Chapter 6 have shown the ability of the o f the angle of attack rate to alter the development of the LEV/TV structures. Figure 712 presents the mean coefficient of lift for various angle of attack rates at nearly constant angles of attack. Clearly, the angle of attack rate has a significant effect on the mean
198 coefficient which is incurred from the development of the unsteady vortex structures presented in section 6.2 Generally, the coefficient of lift range for various angle of attack rates increases as the angle of attack is reduced. For example, at an angle of attack equal to 6.0, the coefficient of lift ranges from approximately -0.10 to 0.75. This is a consequence of the induced velocities from the angle of attack rate and pitch rate al tering the development of the unsteady vortex structures. For an angle of attack equal to 17.0, the coefficient of lift deviates between approximately 1.1 and 0.9. This range is significantly smaller than the range at an angle of attack equal to 6.0. This is evidence of the angle of attack parameter being responsible for the unsteady vortex development and consequently the lift. Flow field measurement of the LEV and TV structure presented in Chapter 6 agree with the angle of attack rate being responsible for their development. Therefore, one can reason that the unsteady vortex structure development provides the deviations present in the measured lift curves. The LEV is shown to develop with outward and inward spanwise flow; however the aerodynamic consequence of these various formations has yet to be realized. At an angle of attack equal to 13 .0 outward spanwise flow through the LEV is shown to exist at angle of attack rates less than approximately -80/s Figure 712 indicates the coefficient of lift having a sharp increase as the angle of attack rate is decreased below -80/s at an angle of attack equal to 13.0 Inward spanwise flow through the LEV is present at angle of attack rates greater than approximately -80/s at a constant angle of attack of 13 .0 This corresponds to an increase in the coefficient of lift; however, this increase is more gradual than at lower angle of attack rates Utilizing the flow field information presented in sections 6. 1 and 6.2 the angle of attack rate parameter space
199 can be divided into two regions: an inward spiraling LEV and an outward spiraling LEV. The dashed line in Figure 712 represents an estimated division between the developments of the inward versus outward spiraling LEV structure. This dashed line has a slight slope due to the influence of the angle of attack parameter. The inward spiraling vortex is consistent with a gradual growth in the coefficient of lift with respect to the angle of attack rate. Therefore, it can be reasoned that lift response of a wing in the presence of a LEV core with an inward spanwise flow is less susceptible to fluctuations in the angle of attack rate. However, small changes in the angle of attack rate with an outward spiraling LEV results in a significant change in the coefficient of lift. This is likely a consequence of a further development of the TV structure associated with increased spanwise flow on the aft end of the wing due to additional momentum transported from the LEV to the TV. Multiple theoretical models have been utilized to estimate the dynamic development of the lift generated by various kinematic motions. Theodorsen [ 110 ] Sears et al. [ 44 ], and Katz et al. [ 60] utilized unsteady potential flow theory to estimate the coefficient of lift on a two-dimensional, unsteady flow around an oscillating flat plate. These theories assume the flow is inviscid, time-dependent, and the flow can be modeled with a series of singularities as in steady, potential flow. The sectional lift is ultimately defined by (7 4) where the first term represents the quasi-steady lift and the second term represents the unsteady lift is the total instantaneous bound circulation around the body. The unsteady term is comprised of various influences such as the angle of attack rate, pitch
200 rate, and plunge acceleration. For a three dimensional wing this becomes further complicated through the introduction of spanwise flow over the wing surface. Integrating the sectional lift over the span of the wing results in the total lift on the wing as described by (7 5) where is the span of the wing. For a steady wind speed, the freestream velocity is constant with respect to time. Therefore, it is reasonable to assume that the temporal lift response could be adequately modeled by the quasi-steady response. To account for the spanwise effects, the measured slope is utilized to predict the lift. This lift slope is concurrent with an elliptic spanwise circulation distribution. Figure 713 presents the quasi-steady estimation of the lift with respect to the angle of attack utilizing the measured lift slope from the static wing. The maximum lift for the pure plunge and phase lags 30 and 90 are under predicted by 28 .0 %, 27.8%, and 29.5%. The phase lag of 75 is under predicted by 16.6%. The lag between the measured and estimated profiles are approximately 14, 49 65, and 17 for the pure plunge and pitch-plunge kinematic motion with phase lags 90, 75, and 30 respectively. The corresponding phase lags between the angle of attack and pitch rate profiles are 97 57.6 and 7.2 for the phase lags 90, 75, and 30 respectively. These phase lags are nearly double that of the phase lags between the quasi-steady and force response with the exception of phase lag 75. This relationship indicates the pitch rate altering the timing of the lift response These results show that the inclusion of the unsteady circulation term is required to achieve higher accuracies.
201 Katz et al. [ 60 ] utilized the work of Glauer t [ 119 ] to express the unsteady circulation term as a function of Fourier series coefficients for a two-dimensional wing. After applying time-dependent boundary conditions and the definition of angle of attack and the plunge profile, Katz et al. defined the sectional lift coefficient by (7 6) where (7 7) and (7 8) The expression is called the wake deficiency factor which is related to wake potential and is a function of the reduced frequency ( ) The first two bracketed terms in A are representative to the instantaneous angle of attack whereas the first bracketed term in B is related to the angle of attack rate. To account for a finite wing, an elliptic spanwise circulation profile is assumed to estimate the total lift on an aspect-ratio two wing similar to the results presented in section 7.1 Figure 714 presents the measured and lifting line estimations of the lift profiles for the pure plunge (PP) and pitch-plunge kinematic motions with phase lags 30, 75 and 90. The phasing of the pitch-plunge kinematic motions at the maximum coefficient of lift is better predicted by the lifting line theory. However, the three-dimensional flow physics induced from the unsteady vortex structures ultimately have to be accounted for in order to achieve improved accuracies. Reissner et al. [ 120 ] has shown an elliptical lift assumption is invalid for plunging, lowaspect-ratio airfoils as the spanwise circulation distribution changes as a function of the
202 reduced frequency. Future work will need to focus on understanding the development of the circulation distribution over the wing to improve the overall accuracies on lifting line theory. 7.3. Dynamic Lift Resp o nse The development of the u nsteady vortex structures inherently alters the temporal lift profile as indicated in the previous section It is desired to understand the direct relationship between the development of the unsteady vortex structure s and their influence on aerodynamic performance. As with the previous results, d irect force measurements through the use of a sting balance or load cell are typically utilized to measure the loads on oscillating wing s [ 121 ] The dynamics of the kinematic motion are inherently coupled into this measurement and have to be removed with the utilization of tares and signal filters. However, this can be problematic in two ways. First, utilizing a tare requires the knowledge of the wing mass and moment properties which can be difficult to measure when investigating flexible wings or comp lex shaped wing planforms Secondly, signal filte rs require knowledge of the force signal to appropriately select a filter cut off frequency. Water tunnels have the advantage of filtering the force signals at rates well above the kinematic motion frequency of oscillation and flow characteristic frequency ( ) For example, Bernal et al. [ 58 ] filters the force signals with the low pass filter utilizing a cut off frequency of 6.5 Hz for a typical kinematic frequency of oscillation equal to 0.2Hz Wind tunnels utilizing air as their medium are associated with higher oscillation f requenc ies to maintain equivalent reduced frequency and Reynolds number s Similarly t he larger flow characteristic frequenc ies are closer to sting balance and load cell resonant frequencies. Therefore, it is desirable to compute the lift indirectly from flow field measurements such that structural frequencies of the
203 sting balance and kinematic motion is isolated. The control volume ( CV ) technique outline in Chapter 4 is utilized to compute the lift from the measured flow field behind a pure plunge, unsteady motion to evaluate th is techniques accuracy and gain insight into the ability of unsteady flow structures to alter the lift. Figure 715 presents the measured and computed coefficient of lift profiles for a pure plunge kinematic motion outlined in section 5.1.2 Results from the University of Michigan (UM) and Technical University of Braunschweig (TUBS) for equivalent parameters are also presented [ 57 ] The measured lift profile shows consistent agreement with the lift profiles from UM and TUBS. The maximum peak and minimum peaks occur at times 0.25 and 0.75 respectively. The mean maximum and minimum coefficient of lift values are 1.33 and -0.34 with a standard deviation of 0.023 and 0.117. A phase lag of approximately 4 exists between the measured and UM/TUBS minimum lift values. Baik et al. [ 59 ] presented lift profiles generated from an equivalent kinematic motion at various Strouhal and Reynolds numbers. Their results showed the lift profile to be nearly sinusoidal for a Strouhal number less than 0.16 and Reynolds numbers greater than 1x10 4 This trend is concurrent with this study for an aspect-ratio two, flat plate at a Strouhal and Reynolds number of 0.078 and 4x10 4 Figure 715 also presents the computed lift from the CV computation. From time 0.00 thru 0.32, the CV estimation under predicts the lift by approximately 50% There also exists an apparent 20 phase lag between the maximum computed a nd measured coefficient of lift However, the minimum coefficient of lift deviated from the UM/TUBS results by a standard deviation of only 0.04. To ascertain the discrepancy between the measured and computed coefficient of lift values, the wake flow fields are discussed.
204 Figure 716 through Figure 723 presents the streamwise and vertical components of velocity measured in the wake of the plunging wing viewed from behind the wing looking upstream. Also presented is the UV contour which provides insight into the lift contribution as this value is integrated over the measurement plane to compute the lift. Negative values indicated by the color blue are representative of positive lift contributions. At time 0.00, Figure 716 shows the retarding of the streamwise flow over the interior of the wing. The vertical velocity at the wing tip indicates the formation of a small TV. Minimal downward vertical velocity is observed on the interior of the wing and provides a positive contribution to the lift. This lift contribution is significantly increased at times 0.14 and 0.26 as indicated by Figure 717 and Figure 718 respectively. A more developed TV is indicated by larger velocity magnitudes at the wing tip and a reduction in the streamwise velocity inside the TV core. At time 0.26, the streamwise velocity is equal to that of the freestream near the TV along the aft surface of the wing. This is an indication of the TV applying sufficient momentum to the flow such that it reattaches the flow along the aft end of the wing. Figure 719 presents the flow field at time 0.46 when the plunge velocity begins to decelerate. The streamwise velocity flow field indicates the TV core moving closer to the surface of the wing. It can also be seen that highly accelerated streamwise flow is present under the vortex core. This accelerated streamwise velocity is likely and the wing surface, thereby accelerating it. Furthermore, a region of high accelerated streamwise flow is present over the interior of the wing. Visbal [ 52] utilized CFD to visualize the vortical structures developed over an equivalent wing and kinematic motion. He showed the development of an arch-type vortex structure forming at the
205 leading edge of the wing at this time. Therefore, the accelerated flow present in the wake flow field is most likely a consequence of the accelerated flow around this archtype structure. The UV contours at time 0.46 show significant positive contributions of lift present on the interior of the wing due to the downwash from the TV and accelerated streamwise flow. By time 0.58, the UV contour resembles that of time 0.00 as can be seen in Figure 720 The wing has now started to accelerate upward in the plunging motion. The TV is still present on the upper surface of the wing though minimal in size. The UV contours indicate a positive lift contribution on the interior of the wing due to the presence of the negative vertical velocities. The streamwise flow on the interior of the wing presents a large shear layer as one moves from y/c=0.5 to 0.0. This is most like ly due to the advection the arch-type vortex. The unsteady vortex structures on the upper surface of the wing are fully advected downstream by time 0.65 which can be seen in Figure 721. A weaker, less defined TV is formed along the wing tip. The CV contour indicates minimal negative contribution to lift on the interior of the wing. By the middle of the upstroke as presented in Figure 722 and Figure 723, the TV provides minimal contributions to lift unlike that of the downstroke. When comparing the UV contours between the downstroke and upstroke, it is clear that the development of the unsteady vortex structure on the underside of the wing is hindered by an adverse pressure gradient. The flow fields utilized to compute the lift is quantitatively examined from profiles at various times throughout the kinematic motion which demonstrate the influence of the fluid structures on the lift. Figure 724 presents the measured and estimated profiles indicated by the solid and dashed lines respectively. rapidly
206 increases at spanwise locations just inside of z/c<1.0. This increase is a consequence of added momentum from the TV. As one continues to move further towards the interior of the wing, continually increases. Overall, an increase in at the interior of the wing exists throughout the plunge stroke until time 0.58. At time 0.58, has a concave downward type profile. As shown by Visbal [ 52 ] the change in continuity occurs when the LEV and TV structures generated on the downstroke are fully broken down and transition to the underside of the wing. With the formation of the LEV on the underside of the wing, the concave downward profile remains until the LEV begins to break down at time 0.88. Therefore it can be reasoned that the concavity of the curves on the interior of the wing are a direct response to the development of the LEV structure. Though the profile as a function of spanwise location is not directly related the spanwise coefficient of lift, some arguments can be made. The profile shows a unique temporal development of the spanwise lift distributions which violate the spanwise elliptic circulation distribution profile assumed for three-dimensional lifting line theory. Therefore, this technique might provide a basis to estimate the spanwise lift profiles based on wake flow field measurements. The influence of the unsteady vortex structures on the lift is realized by comparing the profile at nearly equivalent static and instantaneous ( ) angles of attack. Figure 725 presents at angles of attack approximately -5, 5 and 17. The peak due to the TV is increase by approximately 200%, 381%, and 71%, for angles of attack -5, 5, and 17 respectively This is primarily due to the interaction between the LEV and TV structures. Furthermore, the LEV has the ability to accelerate the streamwise flow while keeping it attached to the wing surface. This is realized when comparing the static and
207 instantaneous angle s of attack equal to 17 The static angle of attack has been shown to separate along the interior of the wing resulting in the reduction of However, t he LEV /TV structure s developed at the instantaneous angle of attack provides the ability to accelerate the flow and maintain attachment of the flow to the surface of the wing as indicated by the concave upward profile Theref ore, the unsteady vortex structures enable the wing to achieve sufficient lift at larger angles of attack. One can think of the LEV as a dynamic cambering which accelerates the flow. Additional spanwise momentum induced from the TV allows for the accelerat ed flow over the LEV to remain attached to the surface of the wing T here still exists a large discrepancy in the temporal measure d and compute d lift values. However, the mean lift for one cycle of the measured and computed lift is 0.461 and 0.356 respectively. This represents a 5.8% error from the measured coefficient of lift range equal to 1.80. This error is an indication of the temporal term being responsib le for the large deviation in the computed lift. Reexamining the assumptions in the derivation of the CV technique accumulated or destroyed inside the CV. The coefficient of lift error due to the volume integral of the momentum inside the CV is defined by (7 9) T he term represents the volume momentum deficit/accumulation inside the CV such that the measured coefficient of lift is obtained. Differentiating with respect to time represents the change of the vertical linear momentum inside the CV. If the unsteady vortex structure s are capable of accumulating momentum, this change in CV
208 momentum will be directly related to the growth of the unsteady vortex structures. This results in a direct relationship relating the ability of unsteady fluid structures to alter the momentum inside a CV and ultimately the lift on the wing. Figure 726 presents the temporal differentiation of the coefficient of lift difference Positive values represent linear momentum accumulation inside the CV whereas negative values are indicative of linear momentum release. Comparing the accumulation/deficit values with flow fields computed by Visbal [ 52], this accumulation agrees with the qualitative growth of the LEV structure. The LEV structure is a closed vortex structure inside the CV and therefore its contribution to lift cannot be accounted for other than its influence on the bounding control surface. The dynamic increase in the bound circulation over the wing provides additional lift. Utilizing the quasi-steady, measured LEV circulation values at z/c=0.5 presented in 18.104.22.168 it has been shown the circulation associated with the LEV increases with angle of attack. Therefore, the coefficient of lift contribution due to the LEV circulation increases when the angle of attack is increased from 10 to 16. Therefore, it is quite plausible that the discrepancy in the CV method is due to the evolving circulation contained within the LEV structure. Utilizing a similar CV technique over a two-dimensional wing, Jardin et al. [ 122 ] showed significant discrepancies in the measured and computed lift with the existence of vortical structures inside the CV. Further investigations are required to fully resolve the bound circulation associated with the LEV; however, this technique has provided significant insight into the influence of the unsteady vortex structures on the aerodynamic lift. 7.4. Summary Th e results presented in this chapter demonstrated the influence of unsteady vortex structures on the aerodynamics generated on an oscillating wing. This was
209 accomplished by first quantifying the lift over a static aspect ratio two, flat plate. Secondly, utilizing insight from Chapter 6 the quasi steady assessment of the lift wa s investigated to identify the aerodynamic parameters governing the underlying flow physics Lastly, a control volume technique wa s utilized to inve stigate the influence of the unsteady vortex structures to the temporal and mean aerodynamic lift responses Each of the following analysis provide d a unique perspective on the ability of unsteady vortex phenomena to alter the aerodynamic lift. The aerody namic lift on a static wing at various angles of attack wa s utilized to provide a baseline to which the influence of leading edge vortex (LEV) and tip vortex (TV) were analyzed. The low aspect ratio wing ha d a lift slope which concurred with an elliptic sp anwise circulation distribution The flat plate had a stall angle of attack and maximum coefficient of lift equal to 18.3 and 0.827 respectively. The steady state control volume technique described in Chapter 4 computed the lift from the wake flow field to within 0.05 of the measured lift at angles of attack less than 12.5 This error increase d at larger angles of attack due to the PIV flow field not capturing the entirety of the flow features around the wing However, the error in the coefficient of lift remained less than 0.10 even at these larger angles of attack Therefore, a sufficiently large field of view must be utilized to accurately compute the coefficien t of lift especially as one applies this technique to an oscillating wing which incorporate s a plunging kinematic motion Furthermore, it was shown that the bias error associated with the PIV laser plane being misaligned from the normal freestream velocit y results in a significant error in the computed lift. This bias error can amount to a 12% error in the computed coefficient of lift at the stall angle of attack ; however, th e bias error wa s eliminated by
210 measuring the freestream velocity components utiliz ing an equivalent PIV setup The insight gained from these investigations provide d the knowledge required to progress into an analysis of the unsteady aerodynamics generated over an oscillating wing. A quasi stead y analysis wa s utilized to assess the inf luence of the unsteady vortex structures on the aerodynamic lift. First, an analysis of the coefficient of lift with respect to th e quasi steady angle of attack was discussed. It wa s shown that th e unsteady vortex structures ha d a significant influence on the aerodynamic lift. At angles of attack below 10 large deviations in the lift were realized at constant angles of attack. This wa s a consequence of the governing flow physics not being fully defined by the angle of attack parameter, but rather multipl e aerodynamic parameters. Larger angles of attack were indicative of smaller deviations in the coefficient of lift. This wa s an indicati on of a physical mechanism which wa s a function of the angle of attack driving the unsteady vortex development. A nalyzin g discrete lift contributions as a function of angle of attack and angle of attack rate, the angle of attack rate at angles of attack greater than 17 were shown to have little influence on the lift. However, at angles of attack less than approximately 13 the angle of attack rate significantly alter ed the coefficient of lift. An inflection point in the lift profile at an angle of attack equal to 13 occurred at an angle o f attack rate of approximately 8 0 /s. Utilizing the insight from Chapter 6 this inflection point occurred as the LEV transitions from an inward to out ward spiraling structure. This wa s a clear indication of the unsteady vortex structure development altering the aerodynamic lift. A quasi steady model with respect to angle of attack and a lifting line model utilizing a spanwise elliptical circulat ion distribution over the wing were analyzed to determine their accuracies at predict ing the measured lift
211 profiles. Both models significantly under predicted the lift. However, the lifting line theory more accurately predicted t he temporal lift history. This wa s due to the fact that the lifting line theory incorporat ed both the angle of a ttack and angle of attack rate parameters which have been shown to drive the development of the LEV and TV structures The inaccuracies in the lift magnitude were likely a consequence of the assumed elliptical circulation distribution along the span of the wing Lastly, a control volume technique has been utilized to compute the temporal and mean cycle lift of a plunging wing. There exist ed a significant difference between the measured and computed lift throughout the downstroke of the wing from time 0.0 0 to 0.50. However, the mean cyclic lift wa s computed to within 5.8% of the measured lift. At angles of attack greater than 15 t he wake flow fields utilized to compute the lift indicate d large contributions of lift on the interior of the wing due to acce lerated flow over LEV and spanwise flow incurred from the TV This wa s unique from the static wake flow fields where large angles of attack were indicative of flow separation on the interior of the wing. This wa s evidence of the LEV providing a dynamic cam bering effect which accelerate d the flow over the LEV which ultimately provid e d large contributions to the overall lift. An analysis of the temporal difference between the measured and computed lift indicate d the LEV having the ability to accumulate or rel ease momentum. The spanwise circulation associated with the LEV presented in Chapter 6 agree d with the increase in momentum associated with the LEV throughout the downstroke. Therefore, it wa s reasoned that the LEV wa s a closed vortex structure residing on the surface of the wi ng which dynamically alter ed the bound circulat ion of the wing.
212 Figure 7 1 Static lift measured and estimated for an aspect ratio two, flat plate at Figure 7 2 C L vs static angle of attack.
213 A B Figure 7 3 Normal ized, mean streamwise and vertical wake velocities at a static angle of attack equal to A) Streamwise velocity. B) Vertical velocity.
214 A B Figure 7 4 Normal ized, mean streamwise and vertical wake velocities at a static angle of attack equal to A) Streamwise velocity. B) Vertical velocity.
215 A B Figure 7 5 Normal ized, mean streamwise and vertical w ake velocities a t a static angle of attack equal to A) Streamwise velocity. B) Vertical velocity.
216 A B Figure 7 6 Normal ized, mean streamwise and vertical wake velocities at a static angle of attack equal to A) Streamwise velocity. B) Vertical velocity.
217 A B Figure 7 7 Normalized, mean streamwise and vertical wake velocities at a static angle of attack equal to A) Streamwise velocity. B) Vertical velocity.
218 A B Figure 7 8 Schematic of the tip vortex velocity components A ) H elical axis along the wing tip B ) H elical axis at an angle in the z x axis.
219 A B Figure 7 9 Normal ized, mean streamwise and vertical wake velocities at a static angle of attack equal to A) Streamwise velocity. B) Vertical velocity.
220 Figure 7 10 C l,y vs static angle of attack. Figure 7 11 Mean c oefficient of lift as a function of angle of attack for on e kinematic cycle
221 Figure 7 12 Mean c oefficient of lift for varying angle of attack rates at constant angles of attack Figure 7 13 Quasi steady (dashed) lift estimation and measured lift (solid) profiles.
222 Figure 7 14 Measured (solid line) and lifting line approximation (dash dot lines) of temporal lif t Figure 7 15 Coefficient of lift throughout a pure plunge cycle
223 A B C Figure 7 16 Wake velocity fields at time 0.00. A) normalized streamwise velocity, B) normalized vertical velocity, and C) normalized UV contour
224 A B C Figure 7 17 Wake velocity f ields at time 0.14. A) normalized streamwise velocity, B) normalized vertical velocity, and C) normalized UV contour
225 A B C Figure 7 18 Wake velocity fields at time 0.26. A) normalized streamwise velocity, B) normalized vertical velocity, and C) normalized UV contour
226 A B C Figure 7 19 Wake velocity fields at time 0.46. A) normalized streamwise velocity, B) normalized vertical velocity, and C ) normalized UV contour
227 A B C Figure 7 20 Wake velocity fields at time 0.58. A) normalized streamwise velocity, B) normalized vertical velocity, and C) normalized UV contour
228 A B C Figure 7 21 Wake velocity fields at time 0.65. A) normalized streamwise velocity, B) normalized vertical velocity, and C) normalized UV contour
229 A B C Figure 7 22 Wake velocity fields at time 0.71. A) normalized streamwise velocity, B) normalized vertical velocity, and C) normalized UV contour
230 A B C Figure 7 23 Wake velocity fi elds at time 0.82. A) normalized streamwise velocity, B) normalized vertical velocity, and C) normalized UV contour
231 Figure 7 24 C l,y profile s at various time s in the pure plunge kinematic motion. Figure 7 25 C l,y profiles for static angles of attack and instantaneous angles of attack.
232 Figure 7 26 Pure p lunge kinematic motion dynamic force response
233 CHAPTER 8 CONCLUDING REMARKS Th is study investigat es the development of the leading edge vortex (LEV) and tip vortex (TV) s tructures and their unsteady aerodynamic response. The LEV and TV are analyzed as a function o f angle of attack, angle of attack rate, and pitch rate to assess the aerodynamic parameters driving their development This allows for the physical flow mechanisms associated with the unsteady vortex struct ures to be defined independent f rom the kinematic m otions The response of these unsteady flow mechanisms on the aerodynamic lift is uncovered by analyzing the measured lift as a function of the aerodynamic pa rameters. The following section summarize s the findings presented in this dissertation This is fo llowed by a discussion of future investigations which build upon the results presented in this study 8.1. Summary The current investigation focused on the general and quasi steady development of the LEV and TV structures. The general development of the LEV and TV was investigated over a large aerodynamic parameter space which occurs throughout the downstroke o f pure plunge and pitc h plunge kinematic motions This parameter space was chosen because 75% of the lift obtained throughout a natural flier kinematic motion is generated from the downstroke through the utilization of unsteady vortex structures [ 29 ] However, d ifficulties ar ose discerning the unsteady vortex response from each aerodynamic parameter due to th e large parameter space incurred throughout the downstroke. There fore, a quasi steady analysis was utilized to determine the response of the unst eady vortex structures to each aerodynamic parameter generated from
234 unique kinematic motion s This is accomplished by acquiring the flow fields at particular angles of at tack and angle of attack rates. The LEV development wa s shown to be primarily a functi on of the angle of attack and angle of attack rate. The pitch rate provide d minimal in fluence on the LEV development. At an angle of attack and angle of attack rate equal to 10 and 40 /s, the LEV structure is quite similar over a pitch rate range of 2.95 rad/s which bound s positive and negative pitch rates. Large regions of accelerated streamwise flow are present above the LEV. As the angle of attack is increased or the angle of attack rate is decreased, the s ize of the LEV increases and consequently the streamwise velocity above the LEV is accelerated Therefore, the development LEV provides a dyna mic cambering effect to which the incoming freestream velocity is accelerated over the wing The spanwise circulat ion and criterion indicated a more coherent LEV forming at angles of attack above 13 and angle of attack rates below 80 /s. Furthermore, the deviation in the LEV spanwise circulation due to the pitch rate is reduced in this parameter regime. This indic ate d a change in the flow structure development at large positive and negative angles of attack and angl e of attack rates respectively. T wo unique flow fields were shown to develop behind the LEV structure. At lower angles of attack and higher angle of att ack rates, the vertical velocity component present ed a shedding like phenomenon behind the LEV structure. With the i ncreas e in the angle of attack above 13.0 and decrease in the angle of attack rate below 80 /s, the flow behind the LEV reattache d to the wing surface These two flow mechanisms are independent from the kinematic motion and are solely a function of the angle of attack and angle of attack rate.
235 The literature suggests that three dimensional flow through the LEV is responsible for stabilizin g the LEV resulting in increased lift capabilities [ 14 ] To date, t he flow mechanisms responsible for the three dimensional flow are not fully understood. T his study investigate d the three dimensionality and the influence of the TV on the development of the LEV structure Specifically, the spanwise helicity is utilized to quantify the three dimensional spiraling LEV structure. The measured LEV spanwise helicity indicate d the direct ion of spanwise flow through the core of the LEV changes as the angle of attack and angle of attack rates are altered. At lower angle s of attack and higher angle of attack rate s the LEV has a positive value of spanwise helicity indicating an inward, spira ling structure. With the development of an inward spiraling structure, the shedding like flow phenomenon described previously exists on the downstream end of the LEV. T his shedding like phenomenon is the mechanism by which the LEV releases momentum obtaine d from the TV in to the surrounding flow. Increasing the angle of attack or decreasing the angle of attack rate results in the reversal of the spanwise flow within the LEV This results in an outward spiraling LEV structure which is associated with a larger spanwise circulation Spanwise momentum is driven through the LEV core to the wing tip where it is e ntrained into the TV. The addition of the momentum from the LEV increases the strength of the TV which in turn provides increased downwas h and spanwise flow on the aft end of the wing. Because the outward spiraling LEV transports spanwise momentum from the interior of the wing to the wing tip the shedding like phenomenon is not present but rather the flow reattaches to the wing surface downstream of the LEV.
236 The aerodynamic lift response is quantified as a function of the aerodynamic parameters to assess the influence of the LEV and TV on the lift Overall, the vortex structures were shown to have the capability to significantly alter the lift generated over a dynamically oscillating wing. The development of the spiraling LEV structures h a d various consequences on the lift response. The lift associated with an outward spiraling LEV wa s shown to be more responsive to changes in the angle of attack rate. T his is a result of increase d bound circulation associated with the LEV Furthermore, additional momentum is driven onto the aft end of the wing from the TV such that the accelerated streamwise flow over the LEV remains attached to the wing surface The lif t associate with t he inward spiraling LEV structure wa s less responsive to changes in the angle of attack rate. It is reasoned that is primarily a result of the shedding phenomenon realized behind the LEV. The shedding phenomenon r elease s momentum behind t he LEV structure thereby reducing the influence of the accelerated streamwise flow developed over the LEV. Similarly, t he shedding phenomena d id not further the development of any coherent structures such as the TV as wa s observed for an outward spiraling LEV structure Ultimately, it is desirable to predict the temporal lift profiles for an unsteady kinematic motion such that engineers can utilize specific kinematic motions to fit their desired flight characteristics. The current study investigated various theoretical mo dels to assess their ability to accurately predict the lift over an oscillating wing. Two unsteady aerodynamic models were utilized to predict the temporal lift development A quasi steady model with respect to angle of attack provide d minimal agreement wi th the temporal development of the lift forces throughout the various kinematic motions. This
237 agrees with the flow field information which sho w ed the development of the vortex structures being heavily influenced by the angle of attack rate. A two dimension al lifting line theory corrected by an elliptical spanwise circulation distribution for a finite wing present ed reasonable agreement with the temporal development of the measured lift. The phasing of the maximum and minimum lift values are more accurately modeled for the pitch plunge kinematic motions. To achieve greater accuracies, this model requires accurate estimations of the temporal and spanwise bounded circulation distribution over the wing. A control volume (CV) technique was utilized to compute th e static and unsteady lift from wake flow field measurements. This technique utilize d modern PIV to accurately measure the wake velocity field with high spatial resolution A major advantage of the CV technique is that it isolates any inertial effects that are incurred on direct force measurements from dynamic kinematic motions The computed lift for a static wing was shown to provide good agreement with the measured lift. Utilizing this technique behind a plunging wing, the temporal development of the com puted lift ha d significant errors at discrete times throughout a kinematic cycle. An a nalysis of this error suggest ed the ability of the LEV to accumulate and release linear momentum from within the CV, thereby violating one of the major assumptions associ ated with the CV technique. However, this insight indicates the LEV having the ability to increase the bound circulation of the wing which increases the lift on the wing. T he mean lift for one cycle was computed to within 5.8%. Furthermore, the flow field contributions to the overall lift at e ach time throughout a cycle suggest ed a non elliptic spanwise circulation profile.
238 Th e CV technique may provide a means to characterize the circulation distribution over the wing which would increase the accurac y of th e current lifting line models. This study provides insight s into the various developments of the unsteady vortex structures and their response on the unsteady aerodynamics Below lists the key points realized in this study. The LEV is nearly quasi steady with respect to the angle of attack and angle of attack rate for various oscillating kinematic motion s Increasing the angle of attack or decreasing the angle of attack rate accelerates the streamwise flow over the LEV providing a dynam ic cambering effect. Furthermore, these changes in aerodynamic parameters increase the bound circulation of the LEV. The TV is primarily a function of the angle of attack and angle of attack rate. However, the TV is less responsive to changes in the angle of attack rate unlike the LEV. As the angle of attack is decreased to 8.0 slight deviations in the separation distance of the TV core from the surface of the wing indicate the induced rotational velocity of the wing beginning to influence the chordwise d evelopment of the TV. An inward spiral ing LEV develop s at low angles of attack and high angle of attack rates where the TV impinges on the development of the LEV a t the leading edge / wing tip intersection Spanwise momentum is transported through the LEV t o the interior of the wing where it is released through a shedding like phenomenon behind the LEV. An outward spiraling LEV develops at large positive angles of attack and large negative angle of attack rates. This structure drives spanwise momentum from the LEV into the TV which increases the downwash incurred from the TV on the aft end of the wing. The streamwise flow downstream of the LEV reattaches to the wing surface. A q uasi steady lift mode l with respect to angle of attack provide s minimal agreeme nt with the measured lift on a finite wing Two dimensional lifting line models corrected with an elliptic spanwise circulation distribution provide a more accurate agreement with the measured lift profile at various times throughout the motion
239 A contro l volume technique accurate ly comput es the lift of a static wing from wake flow field measurements. Th is technique utilized behind a plunging wing computed the mean lift over one kinematic cycle to within 5 .8 %. However, large errors in the lift computation at various times throughout the motion result from the LEV accumulating and releasing li near momentum from within the control volume 8.2. Future Work 8.2.1. Oscillation Frequency Effects This study constrains the frequency of oscillation to 2.02 Hz whi ch is associated with a reduced frequency and Strouhal number of 0.25 and 0.076 respectively. Though this study is not based upon pure bio mimicry, it is i nspired by the development of unsteady vortex structures and aerodynamic forces generated by natural fliers. Birds, bats, and large insects have been shown to utilize reduced frequencies ranging from 0.05 to 0.3 0 [ 49 ] Von Ellenri e der [ 51 ] and Tr i ant afyllou et al. [ 123 ] have shown the consequences of deviation s in the wing oscillation frequency to alter the shedding of the LEV It is desirable to furthe r understand the unsteady aerodynamic s associat ed with this vortex shedding and if it is indeed a function of the aerodynamic parameters utilized in this study. The increase in the oscillation frequency will result in an increase in the angle of attack and angle of attack rates throughout the kinematic cycles. The results presented in the current investigation suggest the pitch rate will provide minimal influence on the development of the LEV and TV structures. Furthermore, th e aerodynamic parameter range associated with larger oscillation frequencies s uggests the formation of an outward spiraling LEV structur e.
240 8.2.2. Wing Planform Alterations in t he wing planform can vary the three dimensional effects incurred from the interaction of the LEV and TV structures. Larger aspect rati o wings have the capability of reduc ing the influence of spanwise flow incurred from TV on the interior of the wing. Variations in the wing planform shape have been shown to change the interaction between the LEV and TV. Yilmaz et al. [ 12 4 ] showed a merging of the LEV and TV for an elliptic planform with an aspect ratio two wing driv en through a pitching maneuver. had a large radius of curvature at the wing tip due to the low a spect ratio wing. However as the aspect ratio is increase d, thereby r educ ing the radius of curvature at the wing tip / leading edge intersection the elliptical wing planform resembles a large aspect ratio, rectangular wing for which the LEV and TV interaction is similar to the results presented in this study. The three dimensional effects on the aft end of the wing present ed in this study are not present for a low aspect ratio, elliptic wing due to the absence of downwash p roduced from the TV o nto the aft wing surface As a consequence of these unique results, further investigations are needed to provide sufficient insight into the effects of the cir culation, and unsteady aerodynam ics 8.2.3. Flexible Wing Many natural fliers utilize various active and passive means of control to alter their aerodynamic performance. Wing torsional flexibility allows for a passive pitching motion due to the inertial forces in duced from the kinematic motion [ 125 126 ] Dickinson et al. [ 19 ] defined three mechanisms similar to that of rigid wing model experiments: delayed pitching, synchronized pitching, and advanced pitching. These relationships are shown
241 to be a function of the ratio between the flapping frequency and the natural frequency of the wing [ 127 ] Spanwise flexibility has been shown to alter the performance of plunging wings by introducing a phase lag between the root and tip position of the wing. If the wing flexibility is prop erly timed throughout the kinematic cycle, increases in aerodynamic performance are achievable [ 128 ] Longitudinal and torsional flexibility provide a passive means to change the angle of attack and angl e of attack rates as a function of spanwise location. The findings from the current study suggest the change in these parameters as a function of the spanwise location will result in a unique development of the LEV along the span of the wing. Therefore, th e wing shape could be utilized to provide a specific LEV response.
242 APPENDIX A VELOCITY MEASUREMENT ERROR CALCULATIONS It is desirable to understand the error in the freestream velocity measurements due to errors in each measurement. The velocity is measure from a static pitot tube utilizing a Heise differential pressure transducer, RTD temperature sensor, and Druck atmosp heric pressure transducer. The following section presents an error analysis to estimate the error in the velocity measurements due to each measurement device. A static (A 1) where is the local density, is the differential pressure between the static and total pressure ports, and is the local velocity. The density is calculated from the ideal gas law represented by (A 2) where is the static pressure, is the universal gas constant (287 J/kgK), and is the local temperature. Therefore, three measurements are required to obtain the local velocity using a static pitot tube: differential pressure ( ), static pressure ( ), an d temperature ( ). It is desirable to understand the consequences of these errors on the overall velocity measurements. Therefore, each equation is differentiated with respect to the measurable quantities yielding four equations: ( A 3) (A 4)
243 (A 5) (A 6) The final velocity error can be calculated using the sum of squares for all the errors associated with each measurement. This results in the following equation (A 7) Each term represents a measurement made in determining the velocity. The values with are evaluated at each measured value. At approximately 4 .0 m/s, the nominal and are measured to be 101 946 [Pa], 295.9[K], and 8.64 [Pa]. This results in the following equation (A 8) Substituting each measurements respective error of 11.5 [Pa] for 0.4 [K] for and 0.087[Pa] for yields (A 9) This analysis shows the measurement uncertainty in the system produces a relative error in the velocity measurements of 0.48% at a mean velocity of approximately 4 .0 m/s. A similar analysis can be conducted on an approximate mean velocity of 2 .0 m/s and the relative error is 1.8 %.
244 APPENDIX B QUASI STEADY MODEL PARAMETER AND KINEMA TIC PROFILES This section presents the angle of attack and angle of attack rate profiles at each design point generated from the various pitch plunge and pure plunge kinematic motions. A B Figure B 1 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 10.0 and 80 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles A B Figure B 2 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 10.0 and 120 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles
245 A B Figure B 3 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 13.0 and 40 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles A B Figure B 4 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 13.0 and 80 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles
246 A B Figure B 5 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 13.0 and 120 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles A B Figure B 6 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 16.0 and 40 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles
247 A B Figure B 7 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 16.0 and 80 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles A B Figure B 8 Aerodynamic parameter and kinematic profiles at an angle of attack and angle of attack rate equal to 16.0 and 120 /s. A) Aerodynamic parameter profiles. B) Kinematic Profiles
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259 BIOGRAPHICAL SKETCH Adam B. Hart earned a degree in aerospace engineering and mechanical engineering in 2008 from the University of Florida. In 2010, h e received a Master of Science degree in aerospace engineering from the University of Florida. In 2010, Adam was awarded the highly competitive SMART Scholarship, a collaborative effort of the American Society for Engineering Education (ASEE) and the Naval Postgraduate School (NPS). Under the SMART Scholarship, Adam is working full time on a doctorate at the University of Florida, with an expected graduation dat e of May 2013.