Wingtip Vortices and Free Shear Layer Interaction in The

Home , Lift (force), Transition point, Wingtip vortices

WINGTIP VORTICES AND FREE SHEAR LAYER INTERACTION IN THE

VICINITY OF MAXIMUM LIFT TO DRAG RATIO LIFT CONDITION

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Doctor of Philosophy in Engineering

Muhammad Omar Memon, M.S.

UNIVERSITY OF DAYTON

Dayton, Ohio

May, 2017

WINGTIP VORTICES AND FREE SHEAR LAYER INTERACTION IN THE

VICINITY OF MAXIMUM LIFT TO DRAG RATIO LIFT CONDITION

Name: Memon, Muhammad Omar APPROVED BY:

______Aaron Altman Markus Rumpfkeil Advisory Committee Chairman Committee Member Professor; Director, Graduate Aerospace Program Associate Professor Mechanical and Aerospace Engineering Mechanical and Aerospace Engineering

______Jose Camberos Wiebke S. Diestelkamp Committee Member Committee Member Adjunct Professor Professor & Chair Mechanical and Aerospace Engineering Department of Mathematics

______Robert J. Wilkens, PhD., P.E. Eddy M. Rojas, PhD., M.A., P.E. Associate Dean for Research and Innovation Dean, School of Engineering Professor School of Engineering

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ABSTRACT

WINGTIP VORTICES AND FREE SHEAR LAYER INTERACTION IN THE

VICINITY OF MAXIMUM LIFT TO DRAG RATIO LIFT CONDITION

Name: Memon, Muhammad Omar University of Dayton Advisor: Dr. Aaron Altman Cost-effective air-travel is something everyone wishes for when it comes to booking flights. The continued and projected increase in commercial air travel advocates for energy efficient airplanes, reduced carbon footprint, and a strong need to accommodate more airplanes into airports. All of these needs are directly affected by the magnitudes of drag these aircraft experience and the nature of their wingtip vortex. A large portion of the aerodynamic drag results from the airflow rolling from the higher pressure side of the wing to the lower pressure side, causing the wingtip vortices. The generation of this particular drag is inevitable however, a more fundamental understanding of the phenomenon could result in applications whose benefits extend much beyond the relatively minuscule benefits of commonly-used winglets. Maximizing airport efficiency calls for shorter intervals between takeoffs and landings. Wingtip vortices can be hazardous for following aircraft that may fly directly through the high-velocity swirls causing upsets at vulnerably low

iv speeds and altitudes. The vortex system in the near wake is typically more complex since strong vortices tend to continue developing throughout the near wake region.

Several chord lengths distance downstream of a wing, the so-called fully rolled up wing wake evolves into a combination of a discrete wingtip vortex pair and a free shear layer. Lift induced drag is generated as a byproduct of downwash induced by the wingtip vortices. The parasite drag results from a combination of form/pressure drag and the upper and lower surface boundary layers. These parasite effects amalgamate to create the free shear layer in the wake. While the wingtip vortices embody a large portion of the total drag at lifting angles, flow properties in the free shear layer also reveal their contribution to the aerodynamic efficiency of the aircraft. Since aircraft rarely cruise at maximum aerodynamic efficiency, a better understanding of the balance between the lift induced drag

(wingtip vortices) and parasite drag (free shear layer) can have a significant impact.

Particle Image Velocimetry (PIV) experiments were performed at a) a water tunnel at ILR Aachen, Germany, and b) at the University of Dayton Low Speed Wind Tunnel in the near wake of an AR 6 wing with a Clark-Y airfoil to investigate the characteristics of the wingtip vortex and free shear layer at angles of attack in the vicinity of maximum aerodynamic efficiency for the wing. The data was taken 1.5 and 3 chord lengths downstream of the wing at varying free-stream velocities. A unique exergy-based technique was introduced to quantify distinct changes in the wingtip vortex axial core flow.

The existence of wingtip vortex axial core flow transformation from wake-like (velocity

v less-than the freestream) to jet-like (velocity greater-than the freestream) behavior in the vicinity of the maximum (L/D) angles was observed. The exergy-based technique was able to identify the change in the out of plane profile and corresponding changes in the L/D performance.

The resulting velocity components in and around the free shear layer in the wing wake showed counter flow in the cross-flow plane presumably corresponding to behavior associated with the flow over the upper and lower surfaces of the wing. Even though the velocity magnitudes in the free shear layer in cross-flow plane are a small fraction of the freestream velocity (~10%), significant directional flow was observed. An indication of the possibility of the transfer of momentum (from inboard to outboard of the wing) was identified through spanwise flow corresponding to the upper and lower surfaces through the free shear layer in the wake. A transition from minimal cross flow in the free shear layer to a well-established shear flow in the spanwise direction occurs in the vicinity of maximum lift-to-drag ratio (max L/D) angle of attack. A distinctive balance between the lift induced drag and parasite drag was identified. Improved understanding of this relationship could be extended not only to improve aircraft performance through the reduction of lift induced drag, but also to air vehicle performance in off-design cruise conditions.

DEDICATION

Dedicated to my parents

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ACKNOWLEDGEMENTS

First of foremost, all praise and thanks to Almighty Allah (God) for blessing me throughout this journey and giving me strength to undertake the graduate program and the dissertation research. This accomplishment would not have been possible without His desire, guidance and help.

I would like to thank my advisor, Dr. Aaron Altman, for constantly guiding and supporting me during the course of my PhD program. I have learned a great deal from Dr.

Altman whether it is academics, research or valuable life lessons. He has set an example of excellence as an advisor, researcher, and role model. I am ever thankful to him for being very genuine, caring, and supportive in all aspects of our relationship.

I would like to extend my sincere thanks to the committee members Dr. Markus

Rumpkeil, Dr. Jose Camberos, and Dr. Wiebke Diestelkamp for sharing their knowledge and wisdom when it comes to research. I am thankful to them for being on my PhD committee and evaluating my research. I would also like to thank my former and present colleagues Kevin Wabick, Sidaard Gunasekaran, and Saad Qureshi for their immense support and help throughout this project. From spending countless hours in the lab to preparing for conferences, these people have always provided unconditional help and support.

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I owe my deepest gratitude to my amazing family for their love, support, and prayers for my health and success. This work would not have been possible without their love and encouragement. Last but not the least, I would like to thank Remah Alshinina for her love and support throughout the course of my degree.

TABLE OF CONTENTS

ABSTRACT ...... iv DEDICATION ...... vii ACKNOWLEDGEMENTS ...... viii LIST OF FIGURES ...... xii NOMENCLATURE ...... xix CHAPTER 1 INTRODUCTION ...... 1 CHAPTER 2 LITERATURE REVIEW ...... 4 2.1 Introduction ...... 4 2.2 Wingtip Vortices...... 4 2.3 Free Shear Layer ...... 9 2.4 Exergy Based Analysis ...... 10 2.5 Challenges in Cross-stream PIV ...... 12 CHAPTER 3 EXPERIMENTAL SETUP ...... 15 3.1 Water Tunnel – ILR Aachen, Germany ...... 15 3.2 Wind Tunnel – University of Dayton Low Speed Wind Tunnel (UD-LSWT)...... 19 CHAPTER 4 ANALYTICAL PERSPECTIVE...... 25 4.1 Error Analysis ...... 27 4.2 Vortex Identification ...... 29 4.3 Vortex Wandering Correction ...... 30 4.4 Thickness of the Laser-sheet in the Cross-stream Plane ...... 32 4.5 Spherical Aberration and Distortion ...... 34 CHAPTER 5 RESULTS ...... 38 5.1 Investigation of Wingtip Vortex Core Axial Flow – Water Tunnel ...... 39

5.2 Investigation of Wingtip Vortex in Cross-stream Flow – Wind Tunnel ...... 59 5.3 Multi Scale PIV to Resolve Wingtip Vortex Inner Core – Wind Tunnel...... 78 5.4 Investigating Behavior of the Wingtip Vortex Inner Core at 1.5 Chord Lengths Downstream of the Wing – Wind Tunnel (UD-LSWT) ...... 87 5.5 Investigation of Wingtip Vortex at a Higher Freestream Velocity – Wind Tunnel (UD-LSWT) ...... 93 5.6 Investigation of the Free Shear Layer in the Cross-stream – Wind Tunnel (UD-LSWT)...... 99 CHAPTER 6 CONCLUSIONS ...... 123 CHAPTER 7 RECOMMENDATIONS FOR FUTURE WORK ...... 126 REFERENCES ...... 130

LIST OF FIGURES

Figure 1 Circulating water tunnel at ILR, RWTH Aachen University, with the coordinate system definition ...... 16 Figure 2 Schematic of the test section with half wing installed. All dimensional units are in millimeters (mm) ...... 19 Figure 3 Representation of the axial vorticity during the vortex creation at the wingtip . 19 Figure 4 Test section with half wing installed ...... 20 Figure 5 Schematic of the test setup at UD-LWST to acquire wingtip vortex and free shear layer data ...... 21 Figure 6 Schematic of the wing configurations - a) without the boundary layer trip, b) with a forced boundary layer trip at 10% chord, and c) with a forced boundary layer trip at 20% chord ...... 23 Figure 7 Velocity contour and profiles before and after the vortex wandering correction ...... 32 Figure 8 Laser beam expanded vertically to form a laser-sheet - A) laser-sheet thickness of 3 mm and B) laser-sheet thickness of 9.5 mm...... 33 Figure 9 Top: Calibration grid with locations marked on the x and y axes. Bottom: Distance between pixels (using camera without extension ring) shows linear behavior across all sections on both x and y axes...... 35 Figure 10 Contour of time-averaged u component velocity. Contour values are given in %U with the DPIV exposure time, Δt of 2000 µs showing optical (non-physical) distortion [46]...... 36 Figure 11 Contour of stream-wise (u) component velocity averaged over 2040 image pairs showing optical distortion field...... 37 Figure 12 Circulating water tunnel at IRL used to acquire wingtip vortex data. All dimensional units are in millimeters (mm)...... 39 Figure 13 Contribution from each term showing skewing (C) as most significant while stretching (A and B) demonstrates symmetry around the vortex...... 40 Figure 14 Normalized v distribution showing similarity across the range of angles of attack ...... 41 Figure 15 Normalized w distribution showing similarity across the range of angles of attack ...... 41

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Figure 16 Circulation (obtained from the Stokes’ method) for 4° alpha showing continuous variations in magnitude of circulation as a function of vortex radius even in the region surrounding the core ...... 43 Figure 17 Comparison of circulation determined from the area integral method and the Stokes’ method. Sensitivity analysis of circulation showing similarity in the trends irrespective of the vortex core radius chosen...... 44 Figure 18 Comparison of normalized circulation profiles with theoretical models showing deviation from concave (4º) to convex profile (7º) ...... 47 Figure 19 Comparison of normalized circulation profiles with theoretical models showing consistency of the convex profiles ...... 47 Figure 20 Comparison of normalized circulation profiles with theoretical models showing significant deviation (from 14° to 15°) ...... 48 Figure 21 u distribution shows a distinct difference with a switch-over from wake like (4°) to jet like (7o) profile ...... 50 Figure 22 Integrated total exergy and L/D showing steady increase before a crossover point at 7°. The shaded area emphasizes the region of significant changes ...... 51 Figure 23 Vorticity distribution shows groupings at 11° and 12° and at 14° and 15° ...... 52 Figure 24 Exergy distribution shows profile groupings at 12° and 13° and at 14° and 15° ...... 53 Figure 25 Vorticity normalized by its maximum value shows coincident behavior ...... 54 Figure 26 Normalized exergy distribution shows divergent behavior at the crossover. Shaded area indicates the vortex core location ...... 55 Figure 27 Velocity contours comparison for various angles of attack showing visible difference in integrated magnitude ...... 56 Figure 28 Vorticity contours comparison for various angles of attack showing visible difference in integrated magnitude ...... 57 Figure 29 Exergy contours comparison for various angles of attack showing visible difference in integrated magnitude ...... 58 Figure 30 Schematic of the PIV test setup to investigate the wingtip vortex core three chords length downstream at freestream velocity of 10 m/s...... 59 Figure 31 Vortex inner and outer core diameter showing growth in the overall vortex as a function of angle of attack ...... 61 Figure 32 Vortex inner and outer core diameter normalized by the total diameter of the vortex showing a crossover at 4° angle of attack ...... 61 Figure 33 v-component velocity distribution showing linear increase in tangential velocity as a function of angle of attack ...... 63 Figure 34 Maximum tangential velocity as a function of angle of attack showing linear increase ...... 63 Figure 35 Normalized v-component velocity distribution showing slight asymmetry in 2° case in an otherwise similar profile across the range of angles of attack ...... 64 Figure 36 Vorticity distribution showing significant increase in (-) vorticity from 4° to 6º angle of attack ...... 65

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Figure 37 Minimum vorticity as a function of angle of attack showing significant change in the slope between 4° and 6º angle of attack ...... 66 Figure 38 Vorticity normalized by its minimum value showing bulging in 4° case in the inner core in an otherwise coincident shape across the range of angle of attack .... 66 Figure 39 Exergy destruction rate distribution showing gradual increase in the absolute value across the range of angle of attack ...... 67 Figure 40 Maximum exergy destruction rate distribution showing significant change in the slope somewhere between 4° and 5° angle of attack ...... 68 Figure 41 Exergy distribution normalized by the maximum absolute showing broadening of the shape in the inner core region for 4° angle of attack ...... 68 Figure 42 Vorticity normalized by its maximum value showing uniform distribution in the vortex inner and outer core across the range of angles considered ...... 70 Figure 43 Exergy distribution normalized by the maximum absolute exergy showing a difference in the shape of the vortex outer core at 2° angle of attack ...... 70 Figure 44 Vorticity normalized by its maximum value showing slight changes in the vortex inner core across the angles of attack ...... 71 Figure 45 Exergy distribution normalized by the maximum absolute exergy showing opposite redistribution of exergy from the outer to the inner core of the vortex with increasing angle of attack...... 72 Figure 46 Vorticity distribution comparison of the Baseline, BLT 10% and BLT 20% cases showing significant changes near the vortex core boundary at 6° angle of attack ...... 73 Figure 47 Exergy destruction rate distribution comparison of the Baseline, BLT 10% and BLT 20% cases showing significant changes in the shape of the vortex due to forced trip at 4° angle of attack ...... 73 Figure 48 Circulation for the inner core and the outer core of the vortex as a function of angle of attack showing linear increase in circulation [17] of the vortex except from 7° to 8° angle of attack ...... 75 Figure 49 Comparison of normalized circulation profiles with theoretical models [18] showing deviation in 2º case in comparison to the other cases ...... 76 Figure 50 Integrated total exergy and L/D showing linear increase in exergy as a function of angle of attack for all configurations. Maximum L/D point for the given airfoil is at 5.5° angle of attack ...... 77 Figure 51 Schematic of the PIV test setup to perform multiscale PIV to resolve wingtip vortex core three chords length downstream at freestream velocity of 10 m/s ... 78 Figure 52 u velocity comparison for each angle of attack showing well-defined vortex inner core ...... 79 Figure 53 Cross-sectional profiles through the u velocity contours showing similar behavior in comparison to the wingtip vortex profiles obtained for the entire vortex ..... 80 Figure 54 Velocity comparison for each angle of attack showing well-defined vortex inner core. Boundary of the vortex inner core is shown in circular regions for each angle of attack ...... 81

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Figure 55 Vorticity comparison for each angle of attack showing changes in the vortex inner core as a function of angle of attack. Boundary of the vortex inner core is shown in circular regions for each angle of attack ...... 82 Figure 56 Cross-sectional profiles through the vorticity contours showing well-defined vortex inner core for each angle of attack ...... 83 Figure 57 Exergy comparison for each angle of attack showing changes in the vortex inner core as a function of angle of attack. Boundary of the vortex inner core is shown in circular regions for each angle of attack ...... 84 Figure 58 Cross-sectional profiles through the exergy contours showing similar behavior for each angle of attack ...... 85 Figure 59 Circulation comparison showing much higher circulation as a function of angle of attack for the multiscale PIV results ...... 86 Figure 60 Schematic of the PIV test setup to investigate the behavior of the wingtip vortex roll up 1.5 chords length downstream of the wing at freestream velocity of 10 m/s ...... 87 Figure 61 u velocity comparison between z/c = 1.5 and z/c = 3 for each angle of attack showing signs of asymmetry in z/c = 1.5 in the wingtip vortex core contributed from the feeding shear layer ...... 88 Figure 62 Comparison of cross-sectional profiles of u velocity between z/c = 1.5 and z/c = 3 for each angle of attack showing asymmetry in z/c = 1.5 cases contributed from the feeding shear layer ...... 89 Figure 63 Vorticity contours comparison showing the vortex inner core is not fully developed for the z/c = 1.5 cases for each angle of attack...... 90 Figure 64 Cross-sectional profiles of vorticity showing non-uniformity of the vortex inner core for z/c = 1.5 cases for each angle of attack...... 91 Figure 65 Exergy contours comparison showing the vortex inner core is not fully developed for the z/c = 1.5 cases for each angle of attack...... 92 Figure 66 Comparison of cross-sectional profiles of exergy between z/c = 1.5 and z/c = 3 for each angle of attack showing asymmetry in z/c = 1.5 cases...... 92 Figure 67 Schematic of the PIV test setup to investigate the wingtip vortex core three chords length downstream at a higher freestream velocity of 20 m/s...... 93 Figure 68 v velocity evolution showing vortex core is not developed at -3° (zero lift angle), fully developed vortex core beyond 0° angle of attack...... 94 Figure 69 Left: v velocity profile comparison showing asymmetry for -3° and symmetric behavior for 0° and 6° angle of attack. Right: v velocity profiles normalized by the peak v showing large difference for -3° angle of attack ...... 95 Figure 70 v-velocity component contours contributing to the pure shear showing similarities in the wingtip vortex and the mid semi-span free shear layer for the -3° case...... 96 Figure 71 Velocity evolution showing development of the vortex core as a function of angle of attack...... 97 Figure 72 Left: Vorticity profiles comparison showing gradual increase in the magnitude with angle of attack. Right: vorticity profiles normalized by negative

xv peak vorticity showing large differences in the shape of the inner core boundary for each of the angles of attack ...... 97 Figure 73 Exergy evolution showing development of the vortex core as a function of angle of attack...... 98 Figure 74 Left: Exergy profiles comparison showing gradual increase in the magnitude with angle of attack. Right: exergy profiles normalized by the peak exergy showing large differences in the inner core boundary for -3° angle of attack...... 98 Figure 75 Schematic of the PIV test setup to investigate the free shear layer in the cross-stream direction 3 chords length downstream of the wing at freestream velocity of 20 m/s...... 99 Figure 76 u and v-component velocity contours showing a lack of clear definition of flowfield behavior due to insufficient seed particle displacement in the shear layer cross-stream...... 100 Figure 77 v-component velocity contour showing traces of the free shear layer forming past the wingtip vortex up to the mid semi-span and beyond. The positive and negative v-component velocities indicate the direction of the flow on either side of the free shear ...... 101 Figure 78 Comparison of u and v-velocity component contour (averaged over 170 image pairs) between 64/32-pixel and 16/16-pixel interrogation regions. The comparison shows more prominent evidence of the free shear layer in the u-component however the freestream in ...... 102 Figure 79 Comparison of the u and v-component velocity contour between 170 image pairs and 1020 image pairs showing smoother and less ambiguous contour obtained by averaging 1020 image pairs...... 103 Figure 80 Comparison of u and v-component cross-sectional line between 170 image pairs and 1020 image pairs showing improved uniformity obtained by averaging 1020 image pairs...... 104 Figure 81 v-velocity component contour comparison as a function of number of image pairs averaged for the -3° and 0° angles of attack cases showing much smoother contours when 4000 or more image pairs are averaged...... 105 Figure 82 v-velocity component profile comparison as a function of number of image pairs for the -3° and 0° angle of attack cases showing uniform profiles for data averaged over 4000 or more image pairs...... 106 Figure 83 v (spanwise) velocity distribution for three locations showing mean, variance, and standard deviation. Standard deviation for locations 1 and 3 are similar while for location 2 (in the free shear layer) is much higher as expected...... 107 Figure 84 u (downwash) velocity distribution contours for various angles of attack showing increase in the downwash with increasing angle of attack...... 108 Figure 85 u (downwash) velocity component profiles as a function of spanwise location showing linear (negative) increase in the magnitude with angle of attack. Sinusoidal shape shift for 0° up to 3° is an indication of the non-uniformity of the shear layer as a function of spanwise location...... 109

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Figure 86 u (downwash) velocity component profiles showing linear (negative) increase in the magnitude with angle of attack. Noticeable changes in the profile shapes are seen at 0° up to 3° before a much consistent shape past 4° angle of attack. . 111 Figure 87 u (downwash) velocity component profiles normalized by u-minimum showing distinct changes in 0° and 1° angle of attack profiles in an otherwise consistent behavior across other angles of attack...... 112 Figure 88 v (spanwise) velocity distribution contours for various angles of attack showing velocity in opposing sense for each of the angle of attack...... 113 Figure 89 v (spanwise) velocity component profiles as a function of spanwise location showing gradual increase in the magnitude with angle of attack...... 114 Figure 90 Cross-sectional lines through the v component velocity contours showing gradual increase in the v component (in the lower surface of the wing) with increasing angle of attack...... 116 Figure 91 v (spanwise) velocity component normalized by the peak showing distinct changes in 0° and 1° angle of attack profiles in an otherwise consistent behavior across other angles of attack...... 117 Figure 92 Velocity vector plots for each angle of attack showing significant changes (red circles) in the stratified wake of the upper surface of the wing and relatively constant behavior (green squares) in the lower surface wake of the wing. .... 119 Figure 93 Evolution of shear distribution as a function of spanwise location showing occurrence of non-uniform shear at -3° up to 1°, and uniform shear across the span past 2° angle of attack...... 120 Figure 94 Cross-sectional profiles of shear showing non-uniformity at lower angles of attack and uniform shear past 2° angle of attack...... 121 Figure 95 Cross-sectional profiles of shear normalized by the peak shear showing similarities in the overall shape past 2° angle of attack...... 122 Figure 96 Ideal elliptical lift distribution in comparison with the realistic lift distribution showing large differences due to lower aspect ratio, AR and taper ratio, λ, and higher sweep angles, ΛLE [50] ...... 128

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LIST OF TABLES

Table 1 Temperature variations for various time steps during a representative run ...... 16

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NOMENCLATURE

AR – Aspect Ratio

푏 – Span

푐 – Chord u – Component of velocity in x- direction v – Component of velocity in y- direction w – Component of velocity in z- direction

V∞ - Freestream velocity

푠 – Semi-span of the wing

휂 – Normalized vortex radius

푟 – Radius of the vortex

푟c – Radius of the vortex core

Γ – Circulation

Γc – Vortex core circulation

Γ1/Γ2 – Vortex Identification constants

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CHAPTER 1

INTRODUCTION

When lift is produced through the pressure difference between the upper and lower surface of the aircraft wing the flow wraps around the edges of the wing creating wingtip vortices. Lift induced drag is generated as a byproduct of the downwash from the vortices, that affects the aerodynamic efficiency of the aircraft. Depending upon the aircraft type and phase of flight, approximately 35 to 40 percent of the total drag can be associated with lift induced drag [1]. In addition to that, wingtip vortices can cause large rolling moments on following aircraft, especially in the vicinity of airports during takeoff and landing. The vortex system in the near wake is typically more complex since strong vortices tend to continue developing in the near wake region. The strength of these vortices is proportional to the total circulation [2]. Since some lift induced drag caused by wingtip vortices is the inevitable by-product of three-dimensional lift generation, there is much interest among researchers to optimize the aerodynamic performance of aircraft by minimizing the drag due to lift.

The relationship between the lift induced drag and the parasite drag determines the aerodynamic efficiency of the aircraft. Much can be learned by exploring this relationship in the wake of the wing. This research begins with a comprehensive literature review on the background of the problem, theoretical perspective, and review of the previous work

1 on wingtip vortices, the free shear layer, and the interaction between the two. Moreover, an introduction to the exergy-based technique, used for analyzing wingtip vortex data, and its advantages in light of the literature is shown. Furthermore, the challenges of cross- stream PIV including correction methods for capturing sufficient particle displacement are discussed.

A detailed overview of two experimental setups used in this research; a water tunnel test setup at the Institute of Aeronautics and Astronautics (ILR) in Aachen, Germany, and the open jet test setup at the University of Dayton Low Speed Wind Tunnel, is shown. The biggest motivation behind this research came from intriguing wingtip vortex results obtained from the water tunnel at ILR. Analyzing the wingtip vortex data (obtained from the water tunnel) from the exergy based approach enabled identification of the change in the out of plane velocity profiles (at the crossover point between the wake-like and jet-like profile). These changes were also related to the maximum lift-to-drag-ratio [max (L/D)] lift conditions. These initial results added to the curiosity of the researchers who were keen to not only re-run some of the experiments in the UD-LSWT but also investigate the wing wake in the vicinity of max (L/D). Design, geometry, and configurations of the tested wings, test parameters, and initial test conditions are presented in detail along with the schematics of the test setup.

The theoretical perspective including the governing equations and the methodologies behind the analysis of the data are presented in detail. Governing equations were based on the theories and models from the literature which were expressed in terms of the initial conditions for this research. Deviation and uncertainty analysis associated with the experimental data is presented based on the published models. Additionally, the

2 vortex identification technique implemented, vortex wandering corrections, optimization of the light sheet and corrections for spherical aberration and distortion are also presented.

A wide variety of results are presented for the wingtip vortex data obtained at ILR, wingtip vortex data obtained in the UD-LSWT, and the free shear layer data obtained in the UD-LSWT are shown. All of these results lead to identifying the behavior of the wingtip vortex, the free shear layer, and the interaction between the two in the cross-stream plane. Initial conditions such as the camera field of view (FOV) and freestream velocity were altered to accurately identify the relationship between the lift induced drag and the parasite drag. Each section within the results introduces the objective and test parameters at the beginning of the section while referring to a specific schematic of the setup. The conclusions are then presented in addition to the future directions of the research work.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

The rapid increase in air travel advocates for a strong need to accommodate more airplanes into airports. This calls for shorter intervals between takeoffs and landings in an effort to maximize airport efficiency. Wingtip vortices formed as a result of airflow rolling from the higher pressure side to the lower pressure side can be hazardous for following aircraft that may traverse the high-velocity swirls. The vortex system in the near wake is typically more complex since strong vortices tend to be in a transitional stage of development in this region. Several chord lengths distance downstream of a wing, the so- called fully rolled up wing wake evolves into a discrete wingtip vortex pair and a free shear layer. While the wingtip vortices embody a large portion of the total drag at lifting angles, flow properties in the free shear layer also expose their contribution to the aerodynamic efficiency of the aircraft. Since aircraft rarely cruise at their maximum aerodynamic efficiency conditions, a better understanding of the wingtip vortex rollup process and free shear layer interaction is of high significance.

2.2 Wingtip Vortices

Wingtip vortices cause large rolling moments on following aircraft, especially in the vicinity of airports during takeoff and landing. The vortex system in the near wake is

4 typically more complex since strong vortices tend to continue developing in the near wake region. The strength of these vortices is proportional to the total circulation [2]. Since some lift induced drag caused by wingtip vortices is the inevitable of three-dimensional lift generation, there is much interest among researchers to optimize the aerodynamic performance of the aircraft by minimizing the drag due to lift.

There is ample literature present on the complexity of vortex dynamics and wingtip vortex formation at different levels of flight. Batchelor [3] presented a valuable theoretical description of the wingtip vortex roll-up process by describing a relationship between the azimuthal velocity and the wingtip vortex core axial velocity. Viscous effects in the wing wake reduce the azimuthal velocity (mostly of the outer core) which lead to a positive pressure gradient, consequently resulting in the loss of axial momentum in the wingtip vortex core. The asymptotic variation of the axial velocity defect at the center is given by,

푤 = 푥−1 log(푥) (1) where 푤 is the core axial velocity of the wingtip vortex and 푥 is the distance downstream.

Batchelor also introduced the drag associated with the core of the trailing vortex. The drag was expressed as an integral over the cross-stream plane independent of the downstream distance.

Brown [4] provided a functional dependency between the axial core velocity and the relationship between the profile drag and the lift induced drag. The development of the turbulent core as a function of downstream distance depends on the axial flow associated with the profile drag of the wing and that of the swirling flow. Brown translated pressure equations of pertinence for axisymmetric swirling flows as,

휌푣2 ∆푝 = −휌푈푢 − − ∆퐻 (2) 2

∞ 휌푣2 ∆푝 = − ∫ 푑푟 + 푐 (3) 푟 푟

Where ∆푝 is the pressure increment above ambient pressure, 휌 is the fluid density,

∆퐻 is the total head loss, and c is a zero to first order constant. Brown related ∆퐻 and profile drag through the axial momentum assuming that the entire wing wake rolls up with the wingtip vortex. When employing this relation, an evident direction for axial core flow results. Brown concluded that the vortex axial flow formed at the vortex center may exhibit wake-like (less-than the freestream) or jet-like (greater-than the freestream) core axial flow depending on the ratio of profile drag to the induced drag for a given aircraft under a given set of conditions.

Devenport et al. [5] conducted experiments on the vortex shed from the wingtip of a rectangular NACA 0012 half-wing. Using hot-wire probes, the authors took measurements between 5 and 30 chord lengths downstream of the wing. The authors showed that the vortex boundary (outside of the vortex core) was dominated by the wing wake which rolls up in an increasing spiral. While the shape of the spiral remains relatively constant with downstream distance, the size of the spiral grows as the square root of the streamwise distance [5]. Additionally, velocity fluctuations measured in the center of the vortex showed that the core flow remains laminar. Greater velocity fluctuations, observed when moving from spiral wake to the core, were due to buffeting caused by the spiral motion of the wingtip vortex. Turbulence structure outside of the rotating spiral appears as a two-dimensional self-similar wake, an assumption initially made by Batchelor [3] for the inviscid vortex model.

On the basis of the self-similarity of trailing vortices, Birch [6] discussed a number of factors which contribute to and promote such behavior. The author showed that the amplitude of vortex wandering (random modulations of the vortex) in experimental measurements can cause an axisymmetric flow structure to collapse with an idealized trailing vortex when scaled on inner parameters. Moreover, Birch argued that the similarity in the outer core region of an incompletely developed wingtip vortex may be an artefact of the rate of the roll-up [6]. Chow and Zilliac [7] on the other hand showed that vortex wandering was so small that it did not contribute to the turbulence measurements. While the flow was turbulent in the near wake, the turbulence decayed downstream due to the stabilizing effect of the rotation of the vortex core mean flow. Moreover, measurements of the Reynolds stresses showed that the radial normal stress was larger than the circumferential component [7].

Moore and Saffman [8] investigated trailing vortices behind a finite wing on which the boundary layer was laminar. They showed that the viscosity removes the singularity and the structure of the viscous core. Axial velocities produced by streamwise pressure gradients were calculated through pressure in the viscous core. The authors noted that the perturbation of axial velocity (away from the wing or towards the wing) is dependent on the distribution of the tip loading on the wing. For an ideal elliptic distribution, the perturbation was towards the wing. Conversely, Phillips [9] considered wingtip vortices rolling up behind finite wing on which the boundary layer is turbulent. He reviewed the effects of viscous and turbulent diffusion regions in the roll up vortex sheet. He also found a multi-structured core where the inner core of the merged spirals has reached equilibrium and the outer core regions shows evidence of discrete turns. The distributions of circulation

7 and Reynolds stress were seen to be dependent on the initial spanwise circulation on the wing except for a small region near the peak tangential velocity. Similar behavior was described earlier by Hoffmann & Joubert [10] in a logarithmic relationship where the circulation distribution limits at peak core radius (where tangential velocity reaches its peak).

Phillips and Graham [11] increased the level of turbulence in the trailing vortices by imposing axisymmetric wakes and jets on the vortex while keeping the total circulation constant. They found that, concurrent with higher turbulent intensities and Reynolds stresses, the radial dispersion of vorticity was hastened by the forced flow. Furthermore, no change in the velocity field and very little change in the turbulence field were observed downstream. The authors also highlighted that the radial and axial velocity terms in mean- momentum equations cannot be ignored since linear theory may not provide an adequate description of the vortex.

The lift to drag ratio plays a vital role in aerodynamic systems and is closely related to the formation of the wingtip vortex. Lee and Pereira [12] tested square and round wingtips to investigate the mechanisms responsible for the wake-like and jet-like axial flows in the core of a wingtip vortex. For smaller angles of attack, they postulated it was a pocket of jet-like fluid entrained by the shear layers and the wing wake. For higher angles of attack, they proposed that the jet-like fluid pocket was surrounded by the shear layers.

This corresponds to the velocity lower than the freestream (wake-like) at smaller angles of attack and velocity higher than the freestream (jet-like) at higher angles.

Memon & Altman [13] investigated the wingtip vortex roll up process as a function of angle of attack three chord lengths downstream of a Clark-Y wing. The authors found a

8 discontinuity in the behavior of the vortex in terms of vorticity and dissipation (exergy) around max (L/D) angles. The discontinuity was attributed to the transformation of the wingtip vortex core axial flow from wake-like to jet-like profile around those angles of attack [14]. The wake-like axial core flow was observed at angles less than the angle of attack associated with max (L/D) whereas the jet-like axial core flow was observed at angles greater than those associated with max (L/D).

McAlister & Takahashi [15] also determined that the transformation of the wingtip vortex core axial velocity from wake-like to jet-like occurs in the vicinity of the maximum lift-to-drag ratio (max L/D) angles depending on the airfoil geometry. The authors indicated that the transition between the wake-like and jet-like axial velocity core profile occurs between 4 and 12 degrees angle of attack. For the NACA 0015 wing tested by the authors, the transition occurred at 8 degrees angle of attack. Chigier and Corsiglia [16] took velocity measurements 9 chord lengths downstream of the wing’s trailing edge at 4, 8, and

12 degrees angle of attack. Their results showed that the wake-like profile was obtained for angles less than 9 degrees and the jet-like profile for angles greater than 9 degrees. The small ambiguity in the transition point can be related to the wing geometry, downstream location, and the measurement technique. Despite these factors, it is important to consider that the transition point from wake-like to jet-like core profile occurs in the vicinity of the maximum lift to drag ratio for a given airfoil.

2.3 Free Shear Layer

Research has also focused on the formation of the turbulent free shear layer in the wake to better understand the balance between lift induced drag and parasite drag.

Wygnanski et al. [17] were first to show that the free shear layer wake is unique and

9 changes as a function of the drag coefficient of the turbulence generator as well as several other factors. Devenport et al. [5] showed that the wing wake was indeed rolled up in the vortex by observing fluctuating quantities associated with the spiral motion 5 to 30 chord lengths downstream of the NACA 0012 rectangular wing. Additionally, as mentioned in the previous section, Devenport et al. also showed that the turbulence structure outside of the rotating spiral appears as a two-dimensional self-similar wake. Gunasekaran & Altman

[22] quantified the changes in the turbulence character in free shear layer of the wake of a flat plate across the span. The authors showed that the wingtip vortex moves above the free shear layer with increasing angle of attack. At higher angles of attack, the wingtip vortex is fully separated from the free shear layer. Moreover, a transfer of momentum from the free shear layer to the wingtip vortex was evident when transition was forced using a boundary layer trip [22].

The concept behind the fluid dynamic analogy for the balance between the lift induced drag and the parasite drag can be explored further by studying the interaction between the wingtip vortex and the free shear layer. Cross-stream PIV was used to obtain velocity components in the wingtip vortex and wing wake free shear layer several chord lengths downstream of the wing and for varying angles of attack.

2.4 Exergy Based Analysis

Exergy-based approaches for aircraft design and integration have begun to attract some interest in an effort to achieve even higher efficiency. The term ‘exergy’ can be defined as the potential work that could be achieved from any given system. With lightweight and aerodynamically efficient structures, the most efficient designs could be obtained by minimizing the total exergy destruction of the system. This destruction of

10 exergy is proportional to the corresponding entropy generation [23]. A number of analytical methods have been used by researchers to perform exergy based analysis.

Exergy analysis could be applied to the aerodynamics of the wing to identify the maximum capability of the energy system in problems associated with wingtip vortices.

Alabi et al. [24] compared empirical and CFD based exergy calculation procedures for modeling the airframe subsystem of an aircraft (B747-200). The exergy-based approach supported the viability of using CFD for realistic airframe calculations in a system-level analysis and design optimization. Various researchers [25-28] have applied exergy-based methods to aerodynamic problems. Li et al. [27] investigated the impact of exergy on aerodynamic designs by applying exergy-based methods to various two-dimensional airfoils and three-dimensional wings under turbulent flow conditions. The authors performed exergy-based optimization to minimize entropy for the airfoils under test. They found that the results from the ‘exergy-based method’ were in much better agreement with the semi-empirical value than the ‘surface integration method’ of obtaining drag values

[27]. In addition, the exergy-based method resulted in a greater lift/drag ratio and less exergy destruction in comparison to the conventional method. Figliola and Tipton [28] compared the application of a traditional energy-based approach versus the exergy-based approach to the design of the environmental control system (ECS) of an advanced aircraft.

The objective of the research was to minimize the gross takeoff weight by the applied design approaches. The authors concluded that the results from the two analyses provide solutions that are similar although not exactly the same. However, the exergy-based method provides a ready estimate for efficiency on a component and system basis [28].

Analyzing the wingtip vortex from the perspective of entropy generation (exergy destruction rate) is one of the approaches to optimizing the lifting system. This destruction of exergy is proportional to the corresponding entropy generation [29]. Herwig and

Schmandt [30] showed that the second law of thermodynamics can be applied to the flow field to characterize the losses in the flow. Their results indicate minimum absolute entropy in the vortex core with peaks corresponding to the boundaries of the vortex. For higher

Reynolds numbers, asymmetry is seen at the vortex boundary. In addition, the absolute entropy generation rate at the vortex center is increased at higher Reynolds number. The authors suggested that the viscous dissipation rate is a fundamental quantity that is determined by the entropy generation rate [30]. While applying the exergy-based perspective to the wingtip vortex problem in the present research, it is hypothesized that the minimum exergy destruction of the system would be obtained under the conditions where the lift to drag ratio is maximum.

In the present research, exergy-based technique for the analysis of wingtip vortex data in the wing wake was implemented. The exergy approach enabled accurate identification for the change in the out of plane profile (at the crossover point between the wake-like and jet-like profile). Moreover, distinct changes in the lift-to-drag ratio performance were identified on the basis of the used approach.

2.5 Challenges in Cross-stream PIV

Optical distortion due to inaccurate alignment of the camera and spherical aberration (focusing aberration) can limit the reliability of the PIV data [31]. Soloff et al.

[32] showed that the accuracy of the PIV data depends on focusing aberrations (which cause the particle images to be enlarged) and image distortion (creates a nonlinear

12 relationship between the location of a particle and the location of its image). Variable magnification across the field of view causes the inaccuracy especially with low scattering cross-section. Blurring of the image in any region within the field of view can affect the accuracy of the measurements. Soloff et al. also showed the generation of an approximate mapping function to quantify the magnitude of the differences. The authors corrected for the distortion using a mapping function when determining particle displacements [32].

In the cross-stream plane, small cross-stream velocities relative to the freestream velocity place additional strain on the experimental accuracy of PIV. Representative time between laser pulses (Δt) for the experiment is important in determining the accuracy of the resulting data. In the present experiment, if the ideal ∆t is selected to capture cross- stream velocities, the resolution had to be compromised. With the ideal Δt, only a 2-3 pixel- displacement was obtained in the cross-stream within the free shear layer. It was necessary to either compromise on a sub-par Δt and compensate for distortion, or use two separate lasers at different downstream locations. Another possible solution was to design an optical setup with a combination of convex and concave lenses to increase the thickness of the laser sheet while trying to maintain sufficient laser energy to provide ample scattering. An adequate laser sheet thickness has to be guaranteed to minimize the out-of-plane particle loss [31]. Soloff et al. [32] increased the thickness of the laser-sheet based on the particle displacement for stereo PIV experiments. Adrian [33] showed that the out-of-plane particle displacement between image pairs should be less than one quarter of the laser-sheet thickness. This ensures sufficient increase in the representative time between the laser pulses (Δt) while achieving high correlation peaks. Mean pixel displacements were kept

13 constant by altering the Δt based on the laser-sheet thickness. This eliminates any errors due to the likelihood of enhanced turbulent distortion in the cross-stream plane.

CHAPTER 3

EXPERIMENTAL SETUP

3.1 Water Tunnel – ILR Aachen, Germany

The experiments were conducted in an enclosed circulating water tunnel with a test section of 1000 mm x 540 mm x 540 mm at the Institute of Aeronautics and Astronautics

(ILR) Aachen in Germany [34]. The tunnel contraction ratio is 1/1.8 and the freestream velocity (U) has a continuous range from 0 to 4 m/s. Figure 1 provides a schematic of the water tunnel. A Reynolds number of 200,000 is maintained throughout the tests cited in the present study. The freestream velocity was adjusted to account for kinematic viscosity variations due to rising water temperature while carrying out the measurements. The temperature variation across a given run was in the tenths of degrees Celsius as shown in

Table 1. This small variation had negligible (2-3 orders of magnitude smaller) effect on the absolute entropy and hence the effects of the miniscule temperature changes were neglected.

Figure 1 Circulating water tunnel at ILR, RWTH Aachen University, with the coordinate system definition

Table 1 Temperature variations for various time steps during a representative run. Also showing min, max, mean and the temperature ratio

Time Step T (°C) 1 26.1 2 26.1 3 26.3 4 26.4 5 26.5 6 26.6 7 26.8 T min (°C) 26.1 T max (°C) 26.8 T mean (°C) 26.4 T diff min-max (°C) 0.7 T diff ratio to mean 2.65%

All measurements were performed at freestream velocities ranging between 2.2 and

2.5 m/s. The measurements used in the current paper represent the x/c = 3 plane 16 downstream. Vortex core axial velocity is referenced to the freestream value U, which is gathered in separate measurements in undisturbed flow. A function f(rpm)= U is obtained, which delivers the freestream value for post processing the data.

Two CCD cameras were used to view the suction side of the wingtip where the primary and secondary vortices were expected to form. Water filled containers were positioned between the test section’s glass windows and the cameras to avoid the optical refraction resulting from viewing across several refractive indices at an angle not normal to the image plane. The Clark-Y airfoil wing was mounted vertically from the test-section ceiling. It is important to note that all data included here was obtained for a cross-stream plane three chords length downstream. The wake was determined to be fully rolled up at this downstream location in Buffo et al. [35]. At 15° angle of attack, flow over the wing is expected to be separated. When following the core circulation across the complete range of angles of attack the trend is linear since the local core creation mechanism at the side edge is not affected by local separation in the wing-root area. At x/c = 3 at very high angles of attack most of the vorticity is already rolled up and in the field of view. Therefore, also the trend of maximum circulation should allow for assessing how much flow is still attached. Theoretical maximum circulation may also be calculated when assuming an elliptical lift distribution which is reasonable at AR = 5.

The laser beam was expanded to a 2 mm thick light-sheet. In order to minimize the systematic errors, a small aperture of f = 8 was used to ensure minimum depth of focus with a maximum light-sheet spread-angle and minimum operational laser power. Every set of data was recorded with 500 image pairs at frequency of 1 Hz and an interframe delay

17 ranging from 100 − 117 μs. Using a 100 mm lens, the size of the viewing area was 55 mm x 70 mm and was divided into correlation windows of 32 x 32 pixels with an overlap of

75%. This yielded an overall resolution of 17 vectors per mm. This resolution was specific to the wingtip vortex PIV and not the cross-stream shear. The blockage ratio at 10° angle of attack was 0.95% with a turbulence intensity of 2.5%. The turbulence intensity results from a very modest water tunnel inlet contraction ratio of 1.8.

To account for vortex wandering, a method for aligning the single vorticity fields according to a common overall vortex center is implemented. This method provides velocity profiles which are not stretched and do not suffer from diminished peaks. As opposed to the vortex development phase in the near-wake behind a wing, the path of the center of the azimuthal velocity field (vortex center) during the vortex creation phase at the wing-tip is highly 3-dimensional. The vortex spirals around the tip and the axis and thus the axial and azimuthal (circulation) field permanently change their orientation. Therefore, the velocity fields undergo a Euler transformation according to the angle of the predetermined vortex axis. Figure 2 shows the corresponding coordinate axis definition

[34]. It is noteworthy that the y and z axes define the cross-plane in the experiments where the x-axis represents the out of plane component.

Figure 2 Schematic of the test section with half wing installed. All dimensional units are in millimeters (mm)

Figure 3 Representation of the axial vorticity during the vortex creation at the wingtip

3.2 Wind Tunnel – University of Dayton Low Speed Wind Tunnel (UD-LSWT)

Most experiments are conducted in the University of Dayton Low Speed (Open Jet)

Wind Tunnel (UD-LSWT) with a test section cross-section of 30 in x 30 in. The tunnel inlet includes 6 anti-turbulence screens and a contraction ratio of 16:1. The freestream

19 velocity was set to 10 m/s and 20 m/s for the tests. The Clark-Y airfoil semi-span wing of semi-span AR = 3, was mounted vertically on a Griffin Motion SN: 1651 rotary stage to ensure precise angle-of-attack increments of 1° and accuracy within several arc-seconds.

A Quantel 200 mJ/pulse laser (Twins) was used to create a light sheet in the Trefftz plane.

A PCO 1600 camera (1600 X 1200 pixels) with a 180 mm lens was used to capture the PIV images. Figure 4 shows the wind tunnel test section with the semi-span wing installed.

Figure 5 shows the schematic of the tests performed at UD-LSWT.

Z X

Light Sheet Collector Clark-Y wing Data Inlet Recording and Analysis

Freestream x/c = 3 PCO 1600 Camera 10 m/s

Griffin Motion Splitter Plate

Laser

Figure 4 Test section with half wing installed

Figure 5 Schematic of the test setup at UD-LWST to acquire wingtip vortex and free shear layer data

3.2.1 Acquiring Wingtip Vortex Data

The laser beam was expanded to a light-sheet of approximately 2 mm thickness. A

Cooke Corporation PCO 1600 camera (1600 x 1200 pixel array) with a 180 mm (Nikon f/2.8 AF-D) lens was used to capture images. The field of view (FOV) was 75 mm x 55 mm and was divided into the correlation windows in a two-pass 64 pixel/32 pixel interrogation region size with a 50% overlap. This yielded an overall resolution of 21.3 vectors per millimeter. The data was taken 1.5 and 3 chord lengths downstream of the trailing edge of the Clark-Y wing from 0° up to 8° angle of attack in 1° increments. The representative time between laser pulses (delta-t’s) for the experiments range from 95 to

115 microseconds. A pulse generated from a Quantum Composer (Model 9614) was used to synchronize the time between laser pulses with those of the PCO Camera. The uncertainty of 5 ns and 0.1 m/s was used for delta-t and freestream velocity respectively

[14]. For each angle of attack, the number of image pairs recorded varied between 3000 and 6000 images. Image cross-correlation was performed using DPIV from ISSI.

3.2.2 Boundary Layer Trip (BLT)

An ideal laminar flow is almost impossible to achieve over the entire chord length of the upper surface of an airfoil at practical airplane Reynolds numbers. In an effort to change the upper surface boundary layer characteristics and hence the balance-point between lift induced and parasite drag in the wingtip vortex formation process, a boundary layer trip is added to the wing upper surface to anchor the location of transition from a laminar to turbulent boundary layer. At cruise lift coefficients, the transition occurs at approximately 10% of the chord for most cambered airfoils [36]. While the exact location of the separation is difficult to control, a transition point (boundary layer trip) can be generated near the airfoil leading edge. This allows the viscous drag to increase due to forced transition to turbulent flow, but can reduce the pressure drag at higher angles of attack by delaying or eliminating the boundary layer separation. Mineck and Vijgen [37] fixed the location of the boundary-layer transition trip on the fore-body of the wing at 7.5% of the chord all the way out to the wingtip. The authors showed that a higher lift-curve slope and Oswald efficiency factor of the wing having the boundary layer trip was more efficient [37].

3.2.3 Wing Configurations

Three wing semi-span configurations were tested. A standard Clark-Y wing of semi-span aspect ratio 3 was tested as a baseline case. The other two configurations used the same Clark-Y but boundary layer transition was forced using a trip placed at 10% (BLT

10%) and 20% chord (BLT 20%) respectively. Figure 6 shows the schematic of the three different configurations tested.

Figure 6 Schematic of the wing configurations - a) without the boundary layer trip, b) with a forced boundary layer trip at 10% chord, and c) with a forced boundary layer trip at 20% chord

3.2.4 Acquiring Free Shear Layer Data

Exceptionally for the cross-stream shear layer PIV, the laser beam was expanded into a sheet of approximately 9.5 mm thickness (see Thickness of the Laser Sheet). A Cooke

Corporation PCO 1600 camera (1600 x 1200 pixel array) with a 180 mm (Nikon f/2.8 AF-

D) lens was used to capture images. The field of view (FOV) was 88 mm x 66 mm and was subdivided into correlation windows in a two-pass 64 pixel/32 pixel interrogation region size with a 50% overlap. This yielded an overall resolution of 17.3 vectors per millimeter.

The data was taken three chord lengths downstream of the trailing edge of the Clark-Y

23 wing across a range of angles of attack. Representative time between laser pulses (Δt) for the experiments range from 140 to 220 microseconds. A pulse generator from Quantum

Composer (Model 9614) was used to synchronize the time between laser pulses with those of the PCO Camera. The uncertainty of 5 ns and 0.1 m/s was used for Δt and freestream velocity respectively [14]. The number of image pairs recorded per condition varied between 3000 and 6000 images. Image cross-correlation was performed using DPIV from

ISSI.

CHAPTER 4

ANALYTICAL PERSPECTIVE

Traditionally, the viscous dissipation rate at constant temperature can be derived by the laws of mechanics. A less conventional yet equally valid manner of applying the viscous dissipation rate through the second law of thermodynamics was chosen. While this approach has traditionally been limited to use in thermal systems, there is tremendous potential for application in fluid systems as well. Analyzing the wingtip vortex from the perspective of entropy generation provides for another approach to optimizing the lifting system. From the perspective of the 2nd law of thermodynamics, given two systems that generate the same lift, will the one with lower viscous dissipation in the wingtip vortex have a higher aerodynamic efficiency? The results are thus placed in the context that the viscous dissipation rate can be used to quantify the entropy generation involved in the process [38] following accepted practice. This entropy generation can also be described as work potential within the system and is directly related to the exergy destruction rate. The exergy destruction rate can be used as an objective parameter to improve any system in its capacity to quantify the losses.

According to the second law of thermodynamics, only a portion of energy at a temperature above the environment temperature can be converted into work. The maximum useful work is produced by passing energy transfer through a reversible system.

Expressing this concept through the laws of thermodynamics, the first law of thermodynamics can be written as,

퐸푖푛 − 퐸표푢푡 = 퐸̇푠푦푠 (8)

The second law of thermodynamics is expressed as

푆푖푛̇ − 푆표푢푡̇ + 푆푔푒푛̇ = 푆푠푦푠̇ (9)

Where 푆푔푒푛̇ defines the entropy generation rate and T0 is the temperature. In order to express equation 9 in terms of exergy destruction rate, the equation can be written as,

푋̇푖푛 − 푋̇표푢푡 − 푋̇푑푒푠푡 = 푋̇푠푦푠 (10)

Where 푋̇푑푒푠푡 is exergy destruction rate (lost work potential) which provides a sagacious manner to compare lost available work across different scenarios, and can be defined as

[39];

푋̇푑푒푠푡 = 푇0 푆푔푒푛̇ (11)

Equation 11 can alternatively be written as [39];

푋̇ 푑푒푠푡 푆푔푒푛̇ = (12) 푇0

In this application the viscous dissipation rate Φ is the primary contributor to the entropy generation rate per unit volume [39], finally,

훷̇ 푆̇ = (13) 푔푒푛 푇

Entropy generation is one way to identify potential sources of reduced efficiency when comparing wingtip vortices. It provides a good metric to compare across many

26 scenarios (different wings, wingtip shapes, Reynolds numbers etc.). The equation for the viscous dissipation rate [40] in the Trefftz plane can be written as;

휕푣 2 휕푤 2 휕푣 휕푤 2 휕푇 2 휕푇 2 Φ̇ = 휇 [2 ( ) + 2 ( ) + ( + ) ] + 푘 [( ) + ( ) ] (14) 휕푦 휕푧 휕푧 휕푦 휕푦 휕푧

where Φ is the viscous dissipation rate, 푣 is the velocity component in the 푦 direction, 푤 is the velocity component in the 푧 direction, 휇 is the dynamic viscosity and T is temperature. As mentioned in the previous section, the local temperature gradients were relatively small and therefore had negligible effect on the entropy generation. In concert with this assumption, equation (14) simplifies to;

휕푣 2 휕푤 2 휕푣 휕푤 2 Φ̇ = 휇 [2 ( ) + 2 ( ) + ( + ) ] (15) 휕푦 휕푧 휕푧 휕푦

4.1 Error Analysis

In order to estimate the errors in the velocity gradients (exergy terms), the Kline-

McClintock (K-M) Uncertainty method [41] was applied. In the single-sample uncertainly analysis shown in [41], each term (having partial derivative of R with respect to Xi) represents the uncertainly associated with that term, contributing to the overall uncertainty of the resulting variable. The underlying assumptions are [41],

1. Each measured variable was independent of the other.

2. Any repetitions in each measurement would result in a Gaussian distribution.

3. The initial uncertainty associated with each measurement had same odds.

The governing equation is written as;

휕푅 2 휕푅 2 휕푅 2 푊푅 = √[( 푤1) + ( 푤2) + ⋯ + ( 푤푛) ] (16) 휕푥1 휕푥2 휕푥푛

Where WR is the uncertainty of the resultant, x1, x2, …xn are the variables and w1, w2, …wn represent the uncertainty associated with those variables. Using the K-M method, the uncertainty in the absolute velocity is given by;

휕푣 2 휕푣 2 푊 = √[( 푤 ) + ( 푤 ) + ⋯ ] (17) 푣 휕푥 푥 휕푡 푡

Where wx is the spatial uncertainty of 0.16 pixel/pixel [42] and wt is the temporal uncertainty of 5 ns. Knowing that v=dx/dt, substituting v and differentiating equation 17 yields

1 2 푥 2 푊 = √[( 푤 ) + (− 푤 ) ] 푣 푡 푥 푡2 푡

Substituting the values and solving for the uncertainty in the velocity 푊푣,

1 2 5.86 × 10−5 2 푊 = √[( 9.77 × 10−6) + (− 5 × 10−9) ] 푣 5.86 × 10−4 (5.86 × 10−4)2

m 푊 = 0.017 푣 s In order to calculate the uncertainty in the exergy terms, the velocity gradient from the viscous dissipation rate (equation 15) can be expressed as;

휕푣 2 A = 휇 [2 ( ) ] (18) 휕푦

Subsequently, the uncertainty in the gradient (equation 18) is calculated by;

휕퐴 2 휕퐴 2 푊 = √[( 푤 ) + ( 푤 ) ] (19) 퐴 휕푥 푥 휕푣 푣

Where wx is the uncertainty in the spatial distance and wv is the uncertainty

in the velocity. Substituting equation 11 into 12 and differentiating results in

2 2 2 2 2 −8휇 (푣2 − 푣1) 8휇 (푣2 − 푣1) 푊 = √[( 푤 ) + ( 푤 ) ] 퐴 푥 푥 푥3 푣

Substituting the values and solving for the uncertainty in the exergy term 푊퐴,

−8 × (1.5 × 10−5)2(10.7 − 10.1) 2 8 × (1.5 × 10−5)2(10.7 − 10.1)2 2 푊 = √[( 9.77 × 10−6) + ( 0.017) ] 퐴 5.86 × 10−5 (5.86 × 10−5)3

W 푊 = 0.09 퐴 m3

The error in the exergy terms is several orders of magnitude less than the

absolute value of the exergy. The overall error is dominated by the uncertainty in

the spatial distance.

4.2 Vortex Identification

There are a number of vortex identification techniques found in the literature [43].

While swirling phenomena can easily be visualized through experimentation, translating it into a formal, systematic, rigorous and repeatable description is vital. The vortex identification was accomplished by determining the vortex center (Г1) and vortex core

29 boundary (Г2). These capital Greek letters Gamma are not to be confused with the circulation. Graftieaux et al. [44] derived the scalar functions (Г1 and Г2) to characterize the locations of center and boundary of the vortex, by considering the topology of the velocity field. The scalar functions for Г1 and Г2 at a fixed point (P) in the measurement domain are shown in equations 20 and 21 respectively [44].

1 (PM Λ 푈푀) z 1 Γ1(P) = ∫ dS = ∫ sin(휃푀) d푆 (20) 푆 푀휖푆 ‖PM‖.‖푈푀‖ 푆 푆

Where S is a two dimensional area surrounding P, M lies in S and z is the unit vector normal to the measurement plane; θM represents the angle between the velocity vector UM and the radius vector PM; Г1 is a dimensionless scalar, with | Г1| bounded by 1. The equation for Г2 is derived by equation 4 and takes into account a local convection velocity

1 UP around P where 푈 = ∫ 푈 d푆 [12]. 푃 푆 푆

1 [PM Λ (푈푀−푈̃푃] z Γ2(P) = ∫ d푆 (21) 푆 푀휖푆 ‖PM‖.‖푈푀−푈̃푃‖

Based on the local maximum detection, the vortex center (Г1) values typically range from 0.9 to 1.0. Conversely, the vortex core (Г2), depending on the rotation, is typically in the range 0.6 to 0.7 [44]. The core and boundary of the vortex are identified through this technique to obtain circulation and integrated exergy destruction rate within the vortex.

4.3 Vortex Wandering Correction

Vortex wandering is a term commonly used to describe behavior in which the vortex core moves in an apparently random fashion preventing reliable conclusions obtained from temporally averaged measurements. The flow structure in the vortex core region is of high importance which can be biased by vortex wandering behavior. It is therefore essential to

30 correct for vortex wandering. Devenport et al [5] proposed a now widely accepted method of calculating the probability density function (PDF) of the location of the vortex center and subsequent remapping of the images. Similarly, Iunga et al. [45] conducted wandering simulations of a Lamb-Oseen vortex using a bi-variate normal PDF. The authors found that the ratio between the RMS of the mean velocity and slope at the center of the vortex can be predicted for wandering amplitudes less than 60% of the core radius. Correcting for the wandering through deconvolution (remapping) showed smoothing effects in the mean velocity field.

In the present research, for each of the image pairs, the location of the vortex core was identified. The mean, variance and the standard deviation of the core were calculated.

These parameters were used to calculate the PDF. At the most probable location of the center of the vortex core, the deconvolution was performed for each of the images. The wander-corrected average across the entire range of the image pairs was used to obtain velocity components. Figure 7 shows an example of the comparison of the uncorrected and corrected velocity contours and profiles. In figure 7, the wandering corrected contour is smoother with a better defined core than the one seen in the uncorrected contour. Even minor inaccuracy due to vortex wandering, as seen from the velocity plot, has to be corrected.

Figure 7 Velocity contour and profiles before and after the vortex wandering correction

4.4 Thickness of the Laser-sheet in the Cross-stream Plane

The thickness of the laser-sheet plays an important role in achieving the requisite pixel displacement for the seed particles. It is critical to put typical wingtip vortex cross-stream velocity magnitudes in perspective to understand the need to alter the laser-sheet thickness better. Generally, around the maximum lift to drag ratio [max (L/D)] angles of attack the wingtip vortex velocity in the cross-stream plane is about 50% of the freestream velocity [13]. Typical particle displacements

(8-10 pixels, targeting 32 pixel interrogation regions) at such magnitudes can be achieved with a laser-sheet thickness of approximately 1-2 mm without particles convecting out of plane at too high

32 a rate. The velocity magnitudes contained within the free shear layer in the cross-stream plane are considerably smaller (about 10% of the freestream velocity), resulting in an unacceptable number of particles leaving the plane during the time intervals between the two images if the same laser sheet thickness were maintained. Reducing the Δt to maintain higher correlation peak ratios results in a mere 2-3 pixel-displacement. In order to achieve the optimal particle displacement in the free shear layer, the laser-sheet was thickened to 9.5 mm using a combination of plano-convex and plano-concave lenses (without diffusing the laser energy adequately to sacrifice sufficient scattering to obtain good correlation peak ratios). Figure 8 shows a visual representation (at the same magnification) of the thin and thick laser-sheets next to each other for the purposes of easy comparison.

Figure 8 Laser beam expanded vertically to form a laser-sheet - A) laser-sheet thickness of 3 mm and B) laser-sheet thickness of 9.5 mm.

4.5 Spherical Aberration and Distortion

Spherical aberration and optical distortion associated with the camera lens can affect the accuracy of the particle displacement recorded by the camera. Spherical aberration exists to some extent in all lenses especially those set for high magnification factors. The PIV data in the cross- stream plane was obtained with and without the camera lens extension ring (14 mm) to reduce the chance of errors due to magnification factor. A simple, linear, first attempt at a correction map was obtained to correct for these aberrations throughout the field of view. Plot Digitizer, an open-source program to extract numerical data accurately from plots and images, was used to obtain precise distances between the pixels. Figure 9 shows the correction map with three locations marked for each of the x and y axes. On the x-axis, top, middle, and bottom sections were selected in this example. Figure 8 also shows the plots for each of the three locations with linear trend-lines. The equations associated with each of the sections show true focus across all regions when using the camera without extension. The slopes show that deviations from the top to the bottom are on the order of a tenth of a percent.

Figure 9 Top: Calibration grid with locations marked on the x and y axes. Bottom: Distance between pixels (using camera without extension ring) shows linear behavior across all sections on both x and y axes.

Spedding et al. [46] experimentally described magnitudes of distortion that exceeded the likely value of the turbulence levels in the wind tunnel. The authors showed that the difference in the lens distortion characteristics increases in significance with increasing Δt. Figure 10 shows coherent patterns of u component velocity contour averaged over 10 velocity fields captured during 1 s [46]. These non-physical distortions have to be corrected for in the cross-stream PIV of the free shear layer to obtain minimally distorted velocity magnitudes.

Figure 10 Contour of time-averaged u component velocity. Contour values are given in %U with the DPIV exposure time, Δt of 2000 µs showing optical (non-physical) distortion [46]

In order to obtain magnitudes of optical distortion in the present research, open tunnel freestream PIV data was obtained in the stream-wise plane using the same combination of focal length, magnification factor, depth of field and f# as that used in the cross-stream plane. This method was chosen to precisely identify averaged distortion (if any) in an otherwise streamline flow. Data was averaged over a range of 170 to 6000 image pairs with Δt ranging from 140 to 200 µs. Figure 11 shows the contour of streamwise (u) velocity component showing optical aberration. To correct for this optical distortion field when processing the free shear layer data in the cross-stream plane, this averaged distortion field was subtracted from each velocity field obtained before any averaging or similar post- processing was performed.

Figure 11 Contour of stream-wise (u) component velocity averaged over 2040 image pairs showing optical distortion field.

CHAPTER 5

RESULTS

The unique balance between lift induced drag and parasite drag in the vicinity of maximum lift-to-drag ratio angles is verified through analyzing the wingtip vortex (in the water tunnel at ILR and UD-LSWT) and the free shear layer (at UD-LSWT). Analysis of wingtip vortex axial core velocity to determine the location of the transition from wake- like (less than freestream) to jet-like (greater than freestream) is shown in much detail in section 5.1 of this chapter. Significant changes in the behavior of the vortex around maximum (L/D) angles is evident from analysis of derivative quantities. More precise investigation of the wingtip vortex around those angles of attack was done at the UD-

LSWT (section 5.2). The results not only verified the findings from the water tunnel, but also provided more insight into the vortex behavior pertaining to increase in size of the vortex and core circulation. For further insight into the wingtip vortex roll up process, multi-scale PIV was conducted (section 5.3), downstream distance was decreased to 1.5 chord lengths (section 5.4), and freestream velocity was increased to 20 m/s (section 5.5).

The overall findings from the wingtip vortex was tied to the development of the free shear layer in the cross-stream plane at maximum (L/D) angles (section 5.6). Independent analysis of the wingtip vortex, the free shear layer, and the interaction between the two all

38 point towards the unique balance between the lift induced drag and the parasite drag corresponding to the maximum aerodynamic efficiency of the aircraft.

5.1 Investigation of Wingtip Vortex Core Axial Flow – Water Tunnel

Figure 12 Circulating water tunnel at IRL used to acquire wingtip vortex data. All dimensional units are in millimeters (mm).

The total exergy, exergy distributions, and the means of exergy transport in a wingtip vortex were examined from 4° to 15° angle of attack. The experiments were conducted in an enclosed circulating water tunnel at ILR with a continuous freestream velocity (U) ranging from 0 to 4 m/s three chord lengths downstream of a Clark-Y half wing. Each term (A, B and C) in the viscous dissipation rate equation (8) was plotted to understand its contribution. From the equation, it is noteworthy that the radial core growth, which is stretching (A and B), and skewing (C) contribute to exergy loss whereas pure rotation about the axis does not. Figure 13 shows the contribution of the stress terms as a function of angle of attack. A second order finite difference technique was used. Each of the three terms was compared to the total integrated exergy obtained. The largest

39 contribution per unit volume was achieved from the skewing term. Errors shown for each of the terms are several orders of magnitude less than the absolute values of exergy.

휕푣 2 휕푤 2 휕푣 휕푤 2 Ф [ ( ) ( ) ( ) ] 푣푤 = 휇 2 휕푦 + 2 휕푧 + 휕푧 + 휕푦

A B C

x10 Exergy Destruction Rate Terms

12 ) 3 A 10 B (W/m C

dest Total

8 푋

ExergyDestruction Rate, 0 4 6 8 10 12 14 16 Angle of Attack (degrees°) Figure 13 Contribution from each term showing skewing (C) as most significant while stretching (A and B) demonstrates symmetry around the vortex.

To provide a context for the factors influencing the exergy distribution, the normalized v and w velocity components were plotted as shown in Figures 14 and 15 respectively. Figures 14 and 15 show the distributions of normalized velocity components in the Trefftz plane. The y-direction velocity component distributions in Figure 5 are normalized by the maximum value for the associated angle of attack, v*=v/vmax and likewise for the z-direction velocity component (w), w*=w/wmax in Figure 15. The spatial axis in each of the two figures is normalized by the radius of the vortex core. In each of the

40 two figures, the shaded region indicates the location of the boundary of the vortex inner core. These boundaries were determined from the peak azimuthal velocities for the given direction. For all angles of attack, dissimilarities can be observed in the normalized velocity distributions near the boundary of the inner core in an otherwise consistent distribution.

Figure 14 Normalized v distribution showing similarity across the range of angles of attack

Figure 15 Normalized w distribution showing similarity across the range of angles of attack

In order to calculate the circulation at each angle of attack, two separate methods were compared. First, the area integral of the vorticity was calculated. As a first step, the vortex identification [36] was accomplished by determining the vortex center (Г1) and vortex core boundary (Г2). These capital Greek letters Gamma are not to be confused with the circulation. As a second method, circulation was calculated from Stokes theorem as the path integral of velocity along a closed streamline. Application of circulation obtained from the Stokes’ Method relies on the presence of an irrotational region surrounding the core.

Circulation, determined as the line integral of velocity, can be calculated through;

Г푆푡표푘푒푠 = 2휋푣푟

Circulation determined by invoking Stokes’ theorem predicts that for a circular core flow the circulation remains constant. As can be clearly seen in Figure 16, however the circulation calculated in this manner varies across an unacceptably wide range as a function of vortex radius. Whilst this exercise provides a measure of circulation calculated from a more-conventional method, it is not considered suitably robust in this application. As a result of this somewhat ambiguous determination of circulation, a more robust method was employed that has found strong favor in vortex identification circles.

Circulation (Stokes' Theorem) -0.35 4 deg -0.3

-0.25

/s) 2

(m -0.2 Γ -0.15

-0.1

Circulation -0.05

0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Normalized radius, r/r c Figure 16 Circulation (obtained from the Stokes’ method) for 4° alpha showing continuous variations in magnitude of circulation as a function of vortex radius even in the region surrounding the core

Calculation of circulation using Stokes’ theorem shows similarity in the trends irrespective of the radial location relative to the vortex core the tangential velocity is sampled from. Unfortunately, though, the value of circulation was sensitive to the choice of radial location regardless of the region sampled and more robust and repeatable methods were employed to determine the circulation. Figure 17 shows the comparison of circulation obtained from both methods as well as the sensitivity analysis as a function of angle of attack. The graph shows sensitivity with Г1 at 0.87, 0.9, 0.96 and 0.99 and Г2 at 0.57, 0.63,

0.66, and 0.69 relative to the baseline values, t (See Vortex Identification). The most important characteristic to note is that regardless of the definition of the vortex center or boundary used, the trends remain same with decay present in circulation from 14° to 15°.

While the circulation obtained from the line integral method differs slightly but was obtained on the same order of magnitude as expected.

Circulation (Sensitivity) -0.12 Baseline G1=0.9 G2=0.6 -0.1 G1=0.87 G2=0.57

/s) G1=0.96 G2=0.66 2

-0.08 G1=0.99 G2=0.69 (m

Γ Stokes Method -0.06

-0.04

Circulation -0.02

0 4 6 8 10 12 14 16 Angle of Attack (degrees°) Figure 17 Comparison of circulation determined from the area integral method and the Stokes’ method. Sensitivity analysis of circulation showing similarity in the trends irrespective of the vortex core radius chosen.

In an effort to better characterize the nature of the changes in the vortex topology, a number of traditional vortex radial circulation distribution models were employed. It was hoped that any deviations from the traditional models would better isolate possible sources for any differences in predicted behavior.

The circulation was normalized on the basis of the viscous inner core for each angle of attack. The values for the Г1 and Г2 used were 0.93 and 0.63 respectively. Birch [6] compared a number of models; Hoffmann and Joubert [10], Phillips [9] and Batchelor [3], to reveal universal inner-scaled circulation profiles. The calculated circulation was normalized by its inner core radius where the boundary of the inner core was defined by the location of the maximum velocity tangential to the vortex rotation. Equations 22-24 represent the models found in the literature, where Г is the circulation, η = r/rc is the non-

44 dimensional radial coordinate, and rc and Гc are the vortex core radius and core circulation, respectively [38].

Hoffmann-Joubert Model [10]

Γ(η) 2 = 퐴0휂 0 < 휂 < 0.4 (22) Γ푐

Γ(η) = 1 + 퐴1ln(휂) 0.5 < 휂 < 1.4 (23) Γ푐

Where A0 = 1.83 and A1 = 0.929.

Phillips Model [9]

Γ(η) 푛 2푘 = ∑푘=1 퐵푘휂 0 < 휂 < 1.3 (24) Γ푐

Where B1 = 1.7720, B2 = -1.0467, B3 = 0.2747

Batchelor Model [3]

2 Γ(η) 1−푒−훼휂 = −훼 (25) Γ푐 1−푒

Where α = 1.26543.

Figures 9, 10 and 11 show the normalized circulation profiles for three different groupings of angle of attack:

1. 4º and 7º,

2. 10º, 11º and 12º, and

3. 14º and 15º

The experimentally determined profiles are compared to several theoretical models found in the literature. In Figure 18, the change in the profiles from 4º to 7º is intriguing.

The 4º curve deviates slightly from the models with a concave profile while at 7º a convex profile can be seen. This implies that the core circulation and hence the lift is less at 4º when compared to the 7º case (given that the outer cores are the same between the 4° and

7° cases). Similar differences in the core circulation were seen in the results obtained from a completely different facility (UD-LSWT) at different freestreams conditions. A concave profile was evident for 2° angle of attack showing much less circulation in comparison to higher angles of attack (Figure 47). This change is thought to be related to the transition from a wake-like to a jet-like wingtip vortex axial core flow profile. In some sense, a difference in behavior between these two angles of attack was expected. The curves in

Figure 19 show convex profiles with the distribution of circulation remaining along the path of that observed from the cited models. Significant deviation is seen in Figure 20 with the convex profiles for both the 14 º and 15 º cases. At present there is no obvious explanation for this change in behavior. It was perhaps once again possible to anticipate this result given the change in integrated circulation seen above in Figure 17. This difference should motivate further research in an effort to explore cases involving higher angles of attack.

Normalized Circulation 1.4

H-J (0-0.4) 1.2

c H-J (0.5-1.4)

Γ / 1 Phillips Batchelor 0.8 4 deg. 7 deg. 0.6

0.4 Normalized Circulation, Γ 0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Normalized radius, r/rc Figure 18 Comparison of normalized circulation profiles with theoretical models showing deviation from concave (4º) to convex profile (7º)

Normalized Circulation 1.4

H-J (0-0.4) 1.2

H-J (0.5-1.4)

Γ / 1 Phillips Batchelor 0.8 10 deg. 12 deg. 0.6 13 deg.

0.4 Normalized Circulation, Γ 0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Normalized radius, r/rc Figure 19 Comparison of normalized circulation profiles with theoretical models showing consistency of the convex profiles

Normalized Circulation 1.4

H-J (0-0.4) 1.2

c H-J (0.5-1.4)

Γ / 1 Phillips Batchelor 0.8 14 deg. 15 deg. 0.6

0.4 Normalized Circulation, Γ 0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Normalized radius, r/rc

Figure 20 Comparison of normalized circulation profiles with theoretical models showing significant deviation (from 14° to 15°) The most significant outcome from comparing circulation profiles with accepted profile models from the literature lies in the change in shape of the curve from concave (at

4º) to weakly convex (at 7º) which happens to correspond to the region in which the axial core flow crossover from jet-like to wake-like behavior occurs. It is intriguing that the profiles continue to exhibit a convex shape at higher angles of attack. This emphasizes the point that there is a change in the nature of the profile after the crossover point as expected.

Transitioning from the vortex core circulation behavior, the focus will now be placed on the out of plane component and correlating the behavior of derivative quantities to these changes. Figure 21 shows the x-direction velocity (core axial flow) distribution for each angle of attack. A distinct change in behavior can be seen in the distribution. The switch from wake-like to jet-like out of plane velocity profile from 4° to7o angle of attack, as described in [7], is noteworthy. This profile switch coincides with the crossover point in

48 the exergy versus the angle of attack and maximum lift-to-drag ratio versus angle of attack plot in Figure 22. The L/D ratio was obtained from XFoil for the Clark-Y airfoil under the same conditions as the experiment. The L/D data obtained from XFoil was extended to three-dimensions via the Helmbold equation [47]

퐶퐿 푎0 = (26) 훼 푎 2 푎 √{1+( 0 ) }+ 0 휋퐴푅 휋퐴푅

Where 퐶퐿is the three dimensional coefficient of lift, α is the angle of attack, 푎0 is the effective angle of attack and AR is the aspect ratio of the wing. The XFoil results were restricted to the angles of attack for which the experimental cross-stream data was obtained.

The L/D increases in the range from 4° to 7° before reaching a plateau at approximately 8° angle of attack. The significance of the L/D curve for Clark-Y airfoil presented here lies in the general trends and not the absolute values.

It is important to note that the peak could be anywhere in the range from 5° to 6° angle of attack, which would coincide with the axial switchover between 4° and 7° angle of attack. This transition corresponds to the switchover from a wake-like (at 4°) core axial flow profile to a jet-like profile (at 7°). This aforementioned switchover is manifested in a profile cut of axial velocity through the vortex in Figure 21 at 4° and 7° as well as for the remaining angles of attack tested.

Integrated total exergy destruction in the vortex (Figure 22) was calculated by area integration. The exergy destruction rate changes slightly up to 12° degrees until the slope increases significantly between 12° and 13° angle of attack. In the range from 13° to 15° there is a substantial increase in the slope of the integrated exergy curve. Correspondingly,

49 there is a large drop in aerodynamic efficiency from 12° to 14° angle of attack after which the ratio settles down. Thus, the change in macroscopic integrated total exergy shows little indication of the switchover condition at maximum L/D. However, it does appear to capture the changes in performance at the higher angles of attack. Wingtip vortex data obtained from the UD-LSWT also showed a linear increase in the exergy destruction rate as a function of angle of attack around max (L/D) angles (Figure 48).

u distribution (Normalized by the freestream) 1.4 4 deg. 7 deg. 1.3 10 deg. 12 deg. 13 deg. 1.2 7° peak 14 deg. 15 deg. 1.1 4° blurb

Profile Switch u velocity (m/s)

1.0

0.9 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 Nomrmalized radius, r/rc Figure 21 u distribution shows a distinct difference with a switch-over from wake like (4°) to jet like (7o) profile

Figure 22 Integrated total exergy and L/D showing steady increase before a crossover point at 7°. The shaded area emphasizes the region of significant changes

Greater insight into the behavior of the vorticity and exergy distributions can be gained from Figures 22 and 23 respectively. Figure 23 shows the vorticity distribution across all angles of attack tested. There is no difference in the general shape of the profiles.

Immediately noticeable, however, is the grouping of the 12° and 13° cases and separately, the 14° and 15° cases. Referring back to the total exergy plotted in Figure 22 above, it is noted that the exergy also experienced a large increment in slope around these two pairs of angle of attack as well.

The profiles through the wingtip vortices of absolute exergy as a function of angle of attack are shown in Figure 24. Similar to the vorticity profiles, the exergy profiles also group at 12° and 13° and 14° and 15° as would be expected from the total exergy results from Figure 22. It is important to notice that these two groups of most similar vorticity and absolute exergy distribution demonstrate overlapping of the results at high angles of attack.

From the integrated total exergy variation with angle of attack in Figure 22, it is perhaps unsurprising that the exergy distributions are clustered at 12° and 13°, and again around

14° and 15°. The clustering of the profiles is clearly visible in the overall trend in Figure

23. This behavior is most likely related to changing vortex topology at the wing-tip. The results for the square edge tip maintains lift at high angles of attack resulting from small regions of concentrated vorticity that remain attached to the sharp corners in combination with massive downwash over the wing. This serves to keep the flow attached to the wing’s upper surface longer than with other wingtip shapes tested and published elsewhere.

Vorticity Distribution 0

-2

2 4 deg. -4 7 deg.

(1/s)x10 -6 10 deg. ω Increasing α 12 deg.

-8 13 deg. Vorticity -10 14 deg. 15 deg. -12 -3 -2 -1 0 1 2 3 Normalized radius, r/r c Figure 23 Vorticity distribution shows groupings at 11° and 12° and at 14° and 15°

Exergy Distribution

18 2 4 deg. 16 7 deg. 14

(W/m) x10 10 deg. 12

dest 12 deg. 푋 10 13 deg. 8 14 deg. 6 15 deg. 4

2 Increasing α

Exergy Exergy DestructionRate 0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Normalized radius, r/r c Figure 24 Exergy distribution shows profile groupings at 12° and 13° and at 14° and 15°

The normalized vorticity and exergy distributions are shown in Figures 25 and 26 respectively as a function of angle of attack. The plot of normalized vorticity shows no significant change in overall shape regardless of the angle of attack. The interpretation of the exergy distributions across the vortices as a function of angle of attack becomes intriguing since angles of attack 10° through 15° exhibit essentially the same distribution of normalized exergy across the vortices. However, between 4° and 7° angle of attack

[collectively] and the remainder of the cases evaluated, there is a palpable difference in the exergy distributions across the viscous inner core of the vortex. Figure 26 shows a slice through the spatially normalized distribution which demonstrates growth in the inner core region (shaded region indicates the approximate boundary of the inner core). Most notably, between 4° and 7° there is a substantial change in the absolute inner core exergy contour.

What is even more surprising is that the out of plane change in axial core behavior from

53 wake-like to jet-like is identified via the exergy profile. This out of plane change is not identified by any of the other parameters more traditionally used to evaluate wingtip vortex characteristics. This difference in exergy distribution coincides with the crossover point

(from wake-like to jet-like in the axial flow/out of plane vortex core velocity) discussed earlier, indicating the attainment of the maximum lift to drag ratio angle of attack. Thus, despite the fact that the macroscopic evaluation of total integrated exergy was unable to identify these changes, the actual profile of the distribution of normalized exergy very clearly indicates this transition phase and likely merits further inquiry.

Normalized Vorticity Distribution 0

max 0.1 ω

/ 0.2 ω 0.3 4 deg. 0.4 7 deg. 0.5 10 deg. 12 deg. 0.6 13 deg. 0.7 14 deg. 0.8 Normalized Vorticity, 15 deg. 0.9 1 -3 -2 -1 0 1 2 3 Normalized radius, r/r c Figure 25 Vorticity normalized by its maximum value shows coincident behavior

Figure 26 Normalized exergy distribution shows divergent behavior at the crossover. Shaded area indicates the vortex core location

Figures 27, 28 and 29 show the velocity, vorticity and exergy contour distribution respectively for various angle of attack cases. Several lower and higher angle of attack cases are shown to virtually visualize the flow for each of the calculated components.

Continuous distribution of the growth in the viscous inner core in the vorticity and exergy contours can be seen. Various contours show the visual representation of a wake-like profile (at 4° and 7°) as well as the jet-like profile (at 14° and 15°) angles of attack. The figures highlight the similarities and differences within these two different regions. The first case (4°) indicates a wake like profile which can be compared across other cases at increasing angles of attack. Comparing the u velocity contours, there is a distinct difference. There is no distinction or definition in the core at 4° angle of attack. The defined core seen at the other angles of attack can be seen as an indication of a jet like profile.

Across the vorticity contours, a clear increase in the radial distribution of vorticity can be

55 observed through the core of the vortex. This was previously visualized in the difference between the 4° and 7° cases in Figure 21.

Figure 27 Velocity contours comparison for various angles of attack showing visible difference in integrated magnitude

Figure 28 Vorticity contours comparison for various angles of attack showing visible difference in integrated magnitude

Figure 29 Exergy contours comparison for various angles of attack showing visible difference in integrated magnitude

The most intriguing about this result is that although only 2-d Trefftz plane data was used to obtain the exergy distribution across the plane of the wingtip vortices, the crossover point for the out of plane change from wake-like to jet-like core axial flow (which corresponds to the attainment of the maximum lift to drag ratio angle of attack) can be identified by the in-plane exergy distribution. This result has potential implications for the reduction of lift induced drag through manipulation of the wingtip vortex formation and dissipation processes. Exergy distribution can be used as an indicator of overall integrated

58 wing performance or as an indicator of the manner in which a wingtip vortex would dissipate under a given set of conditions.

5.2 Investigation of Wingtip Vortex in Cross-stream Flow – Wind Tunnel (UD-

LSWT)

Figure 30 Schematic of the PIV test setup to investigate the wingtip vortex core three chords length downstream at freestream velocity of 10 m/s

5.2.1 Vortex Radius

To begin with, it is important to visualize the growth in the inner core and the

outer core regions of the wingtip vortex as the angle of attack increases. The size of the

vortex grows in general as a function of angle of attack. However, around some angles

of attack this growth appears to stagnate, especially in the inner core. Figure 31 shows

the vortex inner core, outer core and the total diameter for the baseline configuration

as a function of angle of attack. The values shown were calculated from the maximum

59 tangential velocity peaks and using a vortex identification technique respectively. The increase in the area of the outer core diameter from less than the inner core to greater than the inner core past 4° is noteworthy (especially in light of its proximity to max

L/D and in the context of the Lee and Pereira result [8]). The diameter of the vortex inner core remains constant for a number of angles, for example from 4° to 5°, annotated in Figure 30. When comparing the vortex outer core growth at those angles, it seems to grow linearly. This behavior would play a key role in understanding velocity distributions and circulation in the inner and outer core of the vortex. The regression lines with their associated equations precisely quantify the slope of the overall curves.

This slope represents the relative rates of growth of the inner and outer cores. Similar trends were observed for the BLT 10% and the BLT 20% wing configurations (see

Wing Configurations). Figure 32 shows the vortex inner and outer core diameter normalized by the total diameter. The crossover point where the two diameters are equal happens to be at 4° angle of attack which lies in the vicinity of the maximum L/D ratio angle of attack. The difference is within the band of uncertainty and the crossover could at any of those angles of attack.

Vortex Diameter - Baseline 1.8

1.6 y = 0.0826x + 1.0472 R² = 0.9943 1.4

1.2 (mm)

An Increase from 4° to 5° y = 0.0555x + 0.4671 Ø 1 R² = 0.9465

0.8

Diameter, Diameter, 0.6 y = 0.0271x + 0.5801 Constant from 4° to 5° R² = 0.8577 0.4 1 2 3 4 5 6 7 8 9 Angle of Attack (degrees°) Inner Outer Total Linear (Inner) Linear (Outer) Linear (Total)

Figure 31 Vortex inner and outer core diameter showing growth in the overall vortex as a function of angle of attack

Vortex Core Diameter - Baseline 0.6

Crossover

vortex Ø

/ 0.55 Ø

0.5

0.45

Inner Outer 0.4

Normalized Diameter, NormalizedDiameter, 1 2 3 4 5 6 7 8 9 Angle of Attack (degrees°)

Figure 32 Vortex inner and outer core diameter normalized by the total diameter of the vortex showing a crossover at 4° angle of attack

5.2.2 Wing Configuration A – Without the Boundary Layer Trip (Baseline)

Equation 5 demonstrates how changes in the individual velocity components greatly influence the net exergy destruction rate. The v-velocity component normalized by the freestream velocity is shown in Figure 33 as a function of angle of attack. The velocity

61 profiles are shown for 2°, 4°, 6° and 8° angle of attack. There is a gradual increase in the maximum tangential velocity as a function of angle of attack. This is shown in Figure 34 where the maximum tangential velocity increases linearly with the angle of attack. The linear increase in the maximum azimuthal velocity indicates that at angles where the vortex inner core did not grow in size (Figure 31), velocity is apportioned differently between the inner core and the outer core. The saturation of the vortex inner core allows the momentum to increase in the outer core of the vortex. The v-velocity component normalized by the maximum tangential velocity for the associated angle of attack, is shown in Figure 35. The spatial axis is normalized by the radius of the vortex core. The boundaries of the vortex core lie at -1 and 1 on the spatial axis as highlighted in the shaded region. These boundaries were determined from the peak tangential velocity from a vertically oriented slice through the vortex. The normalized velocity distributions align well except for a slight asymmetry observed for the 2° angle of attack case. This could be attributed to the inclusion of the feeding shear layer or the vortex not being fully rolled up at a 2° angle.

Figure 33 v-component velocity distribution showing linear increase in tangential velocity as a function of angle of attack

Maximum Azimuthal Velocity - Baseline 0.65

0.6 freestream 0.55 y = 0.0365x + 0.2963 R² = 0.9967 0.5 0.45 0.4 0.35 0.3 Normalized Velocityv/v 1 2 3 4 5 6 7 8 9 Angle of Attack (degrees°)

Figure 34 Maximum tangential velocity as a function of angle of attack showing linear increase

Figure 35 Normalized v-component velocity distribution showing slight asymmetry in 2° case in an otherwise similar profile across the range of angles of attack

Figure 36 shows the vorticity distribution across all angles of attack tested. The overall shape of the profiles remains constant with an increase in vorticity as a function of angle of attack. Note the large drop in absolute vorticity from the 4° to 6° case. A similar trend was observed from 4° to 7° angle of attack in the data obtained from the water tunnel at ILR (Figure 23). Figure 37 sheds more light on this intriguing behavior by showing minimum vorticity as a function of angle of attack. There is a clear discontinuity in the behavior of the profile between 4° and 7° angle of attack. As the derivative quantity, this trend suggests the existence of the transformation point between the wake-like profile and the jet-like profile between 4° and 6° angle of attack. Figure 38 shows the vorticity profiles normalized by the peak vorticity associated with each angle of attack. The spatial axis is normalized by the radius of the vortex inner core. The overall shape of the profiles remains constant except for the 4° angle of attack case. A broadening in the vortex inner core is 64 noteworthy in the 4° case which crosses over to align with other cases in the outer core region. This implies a redistribution of vorticity due to shear originating from the boundary layer. These differences seen in the 4° to 6° range could also be accounted for by an increase in axial momentum in the vortex. This behavior could play a significant role if the transformation from wake-like to jet-like core profile occurs in the neighborhood of 4° angle of attack.

Figure 36 Vorticity distribution showing significant increase in (-) vorticity from 4° to 6º angle of attack

Minimum Vorticity - Baseline

-3.5 x x 10000 -4

-4.5 (1/s)

ω -5

-5.5 Vorticity,

-6 1 2 3 4 5 6 7 8 9 Angle of Attack (degrees°)

Figure 37 Minimum vorticity as a function of angle of attack showing significant change in the slope between 4° and 6º angle of attack

Figure 38 Vorticity normalized by its minimum value showing bulging in 4° case in the inner core in an otherwise coincident shape across the range of angle of attack

Profiles of absolute exergy destruction rate as a function of angle of attack through the wingtip vortices are shown in Figure 39. There was a gradual increase in the absolute exergy destruction with an increase in the angle of attack. Any differences in the profiles are seen at the transition between the vortex inner core and outer core (from inner core normalized radius 1 to 2 in either direction). The possibility of a change in dissipation in

66 the outer core of the vortex is noticeable from these profiles. Figure 40 shows maximum absolute exergy destruction as a function of angle of attack. Annotated representation in

Figure 39 shows a significant change in slope somewhere between 4° and 5° angle of attack. Referring back to the absolute vorticity profiles in Figure 35, a drastic increase in the negative vorticity from 4° to 6° angle of attack was observed. Figure 41 shows the exergy profiles normalized by the maximum absolute exergy. The overall shape of the profile remains same with all cases overlapping each other at the peak. However, broadening for the 4° case in the inner core region is evident in Figure 41. It is supposed that the broadening in the inner core and then crossing back over represents an exchange in dissipation between the inner and outer core of the vortex.

Figure 39 Exergy destruction rate distribution showing gradual increase in the absolute value across the range of angle of attack

Maximum Exergy Destruction - Baseline

0.35 x x 10000

(W/m) 0.3

dest 푋 0.25

0.2

0.15

1 2 3 4 5 6 7 8 9 Exergy Destruction Rate, Destruction Exergy Angle of Attack (degrees°)

Figure 40 Maximum exergy destruction rate distribution showing significant change in the slope somewhere between 4° and 5° angle of attack

Figure 41 Exergy distribution normalized by the maximum absolute showing broadening of the shape in the inner core region for 4° angle of attack

5.2.3 Wing Configuration B – Boundary Layer Trip (BLT) at 10% Chord

Boundary layer transition was forced in this wing configuration using a trip placed at 10% chord of the Clark-Y wing, as shown in the schematic in Figure 6 (b). The v-velocity

(azimuthal) profile across the vortex as a function of angle of attack followed a similar

Stokes’ vortex trend as seen for the baseline case. The peak tangential velocity increased linearly as a function of angle of attack with slight asymmetry seen in the 2° case in the normalized v-component distribution. The absolute vorticity distribution for the BLT-10% configuration showed comparable trends to the baseline configuration with a gradual increase in the negative vorticity as a function of angle of attack.

Figure 42 shows the vorticity distribution normalized by the peak vorticity associated with each angle of attack. The normalized vorticity distributions showed no signs of broadening at lower angles of attack when compared to the baseline configuration.

This indicates a constant inner and outer core vorticity partition in the vortex resulting from the delay in the boundary layer separation forced by the trip. While the absolute exergy profiles showed similar behavior when benchmarked to the baseline configuration, the normalized exergy profiles, shown in Figure 43, are of much interest. In particular, the change in the shape of the vortex for the 2° case is noteworthy. The boundary layer trip has the greatest effect on the boundary layer at small angles of attack and hence this change is expected. The viscous drag is increased tremendously due to the forced trip. This, in turn, causes a disruption in the vortex roll-up process precipitating a change in the shape and orientation of the vortex. For angles of attack from 4° up to 8°, the overall shape of the profiles remains constant. Comparing the normalized exergy profiles for the 10% boundary layer trip case to the baseline configuration, there is no broadening of the vortex inner core across the range of angles of attack seen earlier. In general, significant changes in the shape and behavior of the vortex were found at small angles for the 10% boundary layer tripped configuration. Beyond 4° angle of attack, no disruption in the behavior of the vortex was seen.

Figure 42 Vorticity normalized by its maximum value showing uniform distribution in the vortex inner and outer core across the range of angles considered

Figure 43 Exergy distribution normalized by the maximum absolute exergy showing a difference in the shape of the vortex outer core at 2° angle of attack

5.2.4 Wing Configuration C – Boundary Layer Trip (BLT) at 20% Chord

A third wing configuration, one with a forced boundary layer trip placed at 20% chord of the Clark-Y wing, as shown in the schematic in Figure 6 (c), was tested. This

70 configuration was deliberately considered as a sensitivity on the placement of the boundary layer trip to insure poor placement would not dramatically alter the results. Once again, the absolute velocity, vorticity and exergy distributions show similar behavior when compared to the other two wing configurations. The normalized vorticity profiles are plotted as a function of angle of attack as shown in Figure 44. The vortex inner core widens with increasing angle of attack up until the 6° case. The vortex narrows significantly at 8° angle of attack. This deviation in the behavior of the 8° angle case also transfers to the normalized exergy distribution shown in Figure 45. From the normalized exergy distribution, the changes in the outer core region for the 2° angle of attack case are also apparent. In comparison to the BLT 10% configuration, the change in shape for the 2° angle case is much more prominent in the outer core.

Figure 44 Vorticity normalized by its maximum value showing slight changes in the vortex inner core across the angles of attack

Figure 45 Exergy distribution normalized by the maximum absolute exergy showing opposite redistribution of exergy from the outer to the inner core of the vortex with increasing angle of attack

5.2.5 Baseline, BLT 10% and BLT 20% Comparison

The absolute magnitudes of vorticity and exergy destruction rate are compared for a two distinct angles of attack for better visualization of the differences between them.

Figure 46 shows the comparison of vorticity for 4° and 6° angle of attack. These angles of attack were chosen for visualization since they lie in the vicinity of max (L/D) for the

Clark-Y airfoil. The boundaries of the vortex core lie at -1 and 1 on the spatial axis as highlighted in the shaded region. The 4° angle of attack case shows minor changes in the vortex core and broadening of the profile near the vortex core boundary for BLT 20%. On the other hand, the 6° case shows significant changes in the BLT 10% and BLT 20% cases when compared to the baseline case. While the absolute (negative) magnitude increases in both tripped configurations, the broadening of the profile near the boundary is significant.

The comparison of exergy destruction rate profiles, shown in figure 47, suggest that most changes in the shape of the vortex are seen for the BLT 10% case at 4° angle of attack. For

6° angle of attack, the changes in the shape of the vortex due to the forced boundary layer trip are minimal. A similar phenomenon was noticed in the normalized profiles for BLT

10% and BLT 20% that minimum or no disruption in the behavior of the vortex was seen beyond 4° angle of attack.

Figure 46 Vorticity distribution comparison of the Baseline, BLT 10% and BLT 20% cases showing significant changes near the vortex core boundary at 6° angle of attack

Figure 47 Exergy destruction rate distribution comparison of the Baseline, BLT 10% and BLT 20% cases showing significant changes in the shape of the vortex due to forced trip at 4° angle of attack

The circulation is calculated here through the area integral of vorticity within both the isolated vortex inner core and the combined inner and outer core. Figure 48 shows the

73 circulation as a function of angle of attack for the baseline wing configuration. It is obvious that the circulation in the isolated outer core (total minus inner core) is much less than that in the isolated inner core of the vortex. While there are differences in the slope of the two curves, the circulation for both cases increases as a function of angle of attack. The change in size of the vortex inner and outer core seen in Figure 31 can be used to rationalize the apportionment of vorticity between the inner and outer cores. Even though the size of the vortex inner core remained constant across some angles of attack for example from 4° to

5°, the circulation at those angles increases. Quantified differences between the slopes of the inner core and the outer core indicate that at the angles of attack tested, the generation of lift is much more dominant in the inner core of the vortex. The vortex lift becomes much more prevalent after the transformation to the jet-like profile. Moreover, even though the area of the vortex outer core was larger than that of the inner core, the circulation in the outer core was much lower. This shows a difference in the distribution of circulation from the inner core to the outer core. The vortex grows at a reasonably constant rate as a function of angle of attack (Figure 31) resulting in azimuthal velocities increasing with the angle of attack (Figure 34). With the increase in vorticity as a function of angle of attack, the total circulation in the vortex increases relatively linearly with the angle of attack, as predicted by Prandtl’s Lifting Line theory [42]. Furthermore, corresponding to a significant increase in the area of the vortex inner core from 7° to 8°, the circulation distribution is much higher in the inner core from 7° to 8°, and decreases in the outer core of the vortex. The other two wing configurations (BLT 10% and BLT 20%) showed similar trends with generally linear increases in the circulation for the inner and the outer core of the vortex.

Circulation - Baseline 0.01 0.009 0.008 y = 0.0008x + 0.0029

0.007 R² = 0.9991 /s)

2 0.006 (m

Г 0.005 y = 0.0006x + 0.002 0.004 R² = 0.9733 0.003 0.002 y = 0.0002x + 0.0009

Circulation, 0.001 R² = 0.8776 0 1 2 3 4 5 6 7 8 9 Angle of Attack (degrees°) Inner Outer Total Linear (Inner) Linear (Outer) Linear (Total)

Figure 48 Circulation for the inner core and the outer core of the vortex as a function of angle of attack showing linear increase in circulation [17] of the vortex except from 7° to 8° angle of attack

Birch [6] compared a number of models; Hoffman and Joubert [10], Phillips [9] and Batchelor [3], to reveal universal inner-scaled circulation profiles. The calculated circulation was normalized by its inner core radius where the boundary of the inner core was defined by the location of the maximum velocity tangential to the vortex rotation. The experimentally determined circulation profiles are compared to these theoretical and semi- empirical models as shown in Figure 49. The shape of the profile at a 2º angle of attack is intriguing. The curve deviates slightly from the models showing a concave profile while the other cases (4º up to 8º angle of attack) coincide well with the theoretical and semi- empirical models. Similar changes in behavior were identified earlier in the water tunnel results, and documented also in [14]. The graphs in Figure 49 imply that the inner core circulation is proportionally less at 2º when compared to the other cases. It is due to the interaction between the free shear layer and the wingtip vortex shown in section 5.6.

Velocity components in and around the free shear layer showed counter flow associated

75 with the upper and lower surfaces of the wing in the wake. Increasing velocity transfering momentum inboard (mid semi-span) to outboard (wingtip) with increasing angle of attack is evident. This change in shape happens to correspond to the region in which the axial core flow crossover from jet-like to wake-like behavior occurs.

Circulation Comparison - Baseline 1.4

c 1.2

/ Г 1 2° 0.8 4° 6° 0.6 8° 0.4 H-J (0-0.4) H-J (0.5-1.4) Normalized Normalized Circulation, 0.2 Phillips Batchelor 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Normalized radius, r/rc

Figure 49 Comparison of normalized circulation profiles with theoretical models [18] showing deviation in 2º case in comparison to the other cases Figure 50 shows the integrated total exergy destruction in the vortex for each of the three wing configurations as a function of angle of attack. The total exergy in the vortex is calculated by area integration. The integrated exergy destruction rate shows linear increase with angle of attack. The absolute differences in the integrated exergy across the three wing configurations are within the 10% uncertainty bounds. On the secondary vertical axis, Lift- to-Drag ratio (L/D) for Clark-Y airfoil is plotted as a function of angle of attack. The L/D ratio was obtained from XFoil for the Clark-Y airfoil for a range of angles of attack with a

0.5° increment. The L/D data obtained from XFoil was again extended to three-dimensions

76 using the Helmbold equation (equation 26). The L/D significantly increases in the range from 0° to approximately 5°, reaches a maximum at 5.5° before leveling out. The experimental data from the University of Illinois at Urbana–Champaign (UIUC) shows maximum L/D occurs in the vicinity of 5° angle of attack for the Clark-Y wing [48].

Referring back to the maximum exergy profile (Figure 40), the change in the behavior of the curves (for the water tunnel tests at ILR) was noticed in the range from 4° to 6° angle of attack. The location of this change, if attributed to the transformation between the wake- like and jet-like core profile, corresponds to the maximum efficiency point (maximum L/D ratio) for the Clark-Y wing.

x 106 Exergy Destruction Rate Vs. L/D

3 16 ) 3 14 2.5 Max L/D 12 2

10 L/D 1.5 Baseline 8 BLT - 10% 1

BLT-20% 6 ExergyDestruction (W/m Rate L/D 0.5 4 0 1 2 3 4 5 6 7 8 9 10 Angle of Attack (degrees°)

Figure 50 Integrated total exergy and L/D showing linear increase in exergy as a function of angle of attack for all configurations. Maximum L/D point for the given airfoil is at 5.5° angle of attack

5.3 Multi Scale PIV to Resolve Wingtip Vortex Inner Core – Wind Tunnel (UD-

LSWT)

Figure 51 Schematic of the PIV test setup to perform multiscale PIV to resolve wingtip vortex core three chords length downstream at freestream velocity of 10 m/s

Multiscale PIV was performed to obtain desired particle resolution in the wingtip vortex inner core. The camera was placed much closer to the light sheet which resulted in a FOV of 44 mm by 32 mm. Figure 52 shows the contours of u velocity component for various angles of attack. Adjusting the ∆t to resolve the vortex core shows symmetry and uniformity in the u velocity component profiles, as shown in Figure 53. The cross-sectional profiles for multiscale PIV shows similarity in the vortex inner core. The vortex inner core was defined by the maximum tangential velocity in each direction. The increase in velocity magnitude as a function of angle of attack reflects the growth of the vortex inner core with increasing angles.

Figure 52 u velocity comparison for each angle of attack showing well-defined vortex inner core

Figure 53 Cross-sectional profiles through the u velocity contours showing similar behavior in comparison to the wingtip vortex profiles obtained for the entire vortex

Figure 54 Velocity comparison for each angle of attack showing well-defined vortex inner core. Boundary of the vortex inner core is shown in circular regions for each angle of attack

Figure 55 shows the vorticity contours for various angles of attack. The vortex inner core is indicated with the circular regions. It is important that sufficient particle resolution was obtained only in the inner core region. The outer core region (region outside of the circular boundary) was not well-resolved for this particular set of experiments. Vorticity, a derivative quantity, reveals greater uniformity in the vortex inner core as shown in Figure

56. Due to sufficient resolution in the core, higher more precise magnitudes in comparison to the baseline experiments were obtained.

Figure 56 Cross-sectional profiles through the vorticity contours showing well-defined vortex inner core for each angle of attack

Figure 57 shows the second derivative, exergy destruction rate contours for various angles of attack. The vortex inner core is indicated with the circular regions. Once again, it is noticeable that sufficient particle resolution was obtained only in the inner core region.

The outer core region (region outside of the circular boundary) was not resolved for this particular set of experiments. In Figure 58, cross-sectional exergy profiles show similarity in the profiles in comparison to the baseline experiments.

Figure 57 Exergy comparison for each angle of attack showing changes in the vortex inner core as a function of angle of attack. Boundary of the vortex inner core is shown in circular regions for each angle of attack

Figure 58 Cross-sectional profiles through the exergy contours showing similar behavior for each angle of attack

Circulation was obtained as the area integral of vorticity for the vortex inner core.

Figure 59 shows the circulation of the vortex inner core as a function of angle of attack in comparison to that obtained for the baseline results. Well-defined resolution of the vortex core as observed from the velocity and vorticity profiles results in much higher circulation in the multiscale PIV results. The highlighted region shows the area of the largest differences between the circulation obtained through multigrid PIV and the baseline. Even though the differences in absolute circulation are close to the bounds of uncertainty, it is important to note that biggest differences (also seen in absolute vorticity plots in Figure

53) lie in the vicinity of the maximum lift-to-drag ratio angles (4° to 6°). Additionally, the

85 linearity as a function of angle of attack is in much better agreement with the results obtained from poorer resolution.

Figure 59 Circulation comparison showing much higher circulation as a function of angle of attack for the multiscale PIV results

5.4 Investigating Behavior of the Wingtip Vortex Inner Core at 1.5 Chord Lengths

Downstream of the Wing – Wind Tunnel (UD-LSWT)

Figure 60 Schematic of the PIV test setup to investigate the behavior of the wingtip vortex roll up 1.5 chords length downstream of the wing at freestream velocity of 10 m/s

In order to identify the downstream distance where the wingtip vortex is fully rolled up, experiments were conducted at 1.5 chord lengths downstream of the wing. Other parameters were kept similar to the previous experiments as shown in the schematic in

Figure 60. Figure 61 shows the u component velocity contours for 1.5c in comparison to the velocity contours seen in the 3c case. As shown in Figure 62, the contributions from the feeding shear layer that is manifested in the upper azimuthal velocity profile of the wing to the wingtip vortex for the 1.5c cases show asymmetrical behavior. At the 1.5c downstream distance, the u component contours indicate that the vortex is not fully rolled

87 up. While the magnitudes are roughly similar for both cases, the asymmetry is evident in the 1.5c cases from the u velocity profiles as shown in Figure 62.

Figure 61 u velocity comparison between z/c = 1.5 and z/c = 3 for each angle of attack showing signs of asymmetry in z/c = 1.5 in the wingtip vortex core contributed from the feeding shear layer

Figure 62 Comparison of cross-sectional profiles of u velocity between z/c = 1.5 and z/c = 3 for each angle of attack showing asymmetry in z/c = 1.5 cases contributed from the feeding shear layer

Non-uniformity in the vorticity contours for 1.5c cases is evident in comparison to the 3c cases for the contours in Figure 63. The derivative quantity, vorticity further shows that the vortex inner core is not fully developed for the 2° and 4° cases. However, at higher angles of attack (6° and 8°), there are very minor differences in the vortex core between the 1.5c and 3c cases. The non-uniformity in the derivative quantity is more evident from the cross-sectional profiles shown in Figure 64. It can hence be concluded that at lower angles of attack, the vortex is not fully developed at 1.5 chords length downstream distance.

However, at higher angles of attack, the vortex seems to be fully rolled up. Since velocity increases as a function of angle of attack, it requires less time for a single vortex rotation

89 at higher angles and thus the vortices are more highly evolved at shorter downstream distances.

Figure 63 Vorticity contours comparison showing the vortex inner core is not fully developed for the z/c = 1.5 cases for each angle of attack.

Figure 64 Cross-sectional profiles of vorticity showing non-uniformity of the vortex inner core for z/c = 1.5 cases for each angle of attack.

Exergy destruction rate contours are shown in Figure 65 for the two cases. Even in the second derivative quantity, the non-uniformity of the vortex inner core is seen at small angles of attack. The cross-sectional profiles in Figure 66 show asymmetry for 2° and 4° angles of attack. At higher angles however, symmetric behavior is seen for both the 1.5c and 3c cases.

Figure 65 Exergy contours comparison showing the vortex inner core is not fully developed for the z/c = 1.5 cases for each angle of attack.

Figure 66 Comparison of cross-sectional profiles of exergy between z/c = 1.5 and z/c = 3 for each angle of attack showing asymmetry in z/c = 1.5 cases

5.5 Investigation of Wingtip Vortex at a Higher Freestream Velocity – Wind Tunnel

(UD-LSWT)

Figure 67 Schematic of the PIV test setup to investigate the wingtip vortex core three chords length downstream at a higher freestream velocity of 20 m/s

In order to understand the effects of free shear layer contributions to the wingtip vortex, three angles of attack were tested at the same freestream as the free shear layer data acquired previously. A zero lift angle (-3°), 0° angle where the wingtip vortex is supposed to have been fully formed, and a higher angle (6°) where a much larger physical scale vortex is expected. Figure 68 shows the v component velocity contours for each of three angles of attack tested. The contour for the -3° case is on a different contour scale (to enable visibility) whereas the 0° and 6° contours share the same contour scale (on the far right). It is visible that the vortex core is not developed at the zero-lift angle (-3°). However, at higher (positive) angles, the vortex inner core is fully developed and the size of the vortex grows linearly as a function of angle of attack. Figure 69 shows the v component velocity 93 profiles for each angle of attack in terms of the magnitudes and also normalized by the u velocity peaks. For the -3° case, there is much asymmetry observed in the area that is composed by the feeding shear layer. It was discerned that since there is nearly zero lift produced at -3°, the flow is dominated by the parasite drag. Very little or no asymmetry is observed in the u velocity profiles for the 0° and 6° angles of attack. Large differences exist between the -3° angle and the other angles. This is evidenced in the normalized profiles in

Figure 69.

Figure 68 v velocity evolution showing vortex core is not developed at -3° (zero lift angle), fully developed vortex core beyond 0° angle of attack

Figure 69 Left: v velocity profile comparison showing asymmetry for -3° and symmetric behavior for 0° and 6° angle of attack. Right: v velocity profiles normalized by the peak v showing large difference for -3° angle of attack

It is noteworthy that the -3° case resembles a pure shear field compared to the other positive (0° and 6°) angle of attack cases. In order to better visualize it, the rotation field in the v-velocity component in the -3° case is subtracted and compared to the velocity field obtained from the mid semi-span free shear layer in the wake. Figure 70 shows the comparison of the velocity components contributing to pure shear in the wingtip vortex and the mid semi-span free shear layer.

Figure 70 v-velocity component contours contributing to the pure shear showing similarities in the wingtip vortex and the mid semi-span free shear layer for the -3° case.

The development and evolution of the wingtip vortex is clearly seen in the vorticity contours in Figure 71. The contour for the -3° case is on a different contour scale whereas the 0° and 6° contours share the same contour scale (on the far right). The magnitudes of the vorticity profiles seen in Figure 72 show that the vortex core is not fully developed at the -3° angle. The difference in shape in the normalized -3° plot is evident of that behavior.

The vortex is fully developed at 0° angle of attack as seen from the contour in Figure 71 and consequent profiles in Figure 72.

Figure 71 Velocity evolution showing development of the vortex core as a function of angle of attack.

Figure 72 Left: Vorticity profiles comparison showing gradual increase in the magnitude with angle of attack. Right: vorticity profiles normalized by negative peak vorticity showing large differences in the shape of the inner core boundary for each of the angles of attack

A similar story is told through the second derivative (dissipation) quantity, exergy in Figures 73 and 74. The contour for the -3° case is on a different contour scale whereas the 0° and 6° contours share the same contour scale (on the far right). While -3° angle shows highly asymmetric behavior, little asymmetry is also observed for the 0° case in the

97 absolute magnitude exergy profiles in Figure 74. The exergy destruction rate highlights small differences in such quantities.

Figure 73 Exergy evolution showing development of the vortex core as a function of angle of attack.

Figure 74 Left: Exergy profiles comparison showing gradual increase in the magnitude with angle of attack. Right: exergy profiles normalized by the peak exergy showing large differences in the inner core boundary for -3° angle of attack.

5.6 Investigation of the Free Shear Layer in the Cross-stream – Wind Tunnel (UD-

LSWT)

Figure 75 Schematic of the PIV test setup to investigate the free shear layer in the cross-stream direction 3 chords length downstream of the wing at freestream velocity of 20 m/s

5.6.1 Initial Exploratory Experiments

The first set of experiments was conducted using the 3 mm laser-sheet thickness and representative time between pulses (Δt) of 100 µs. This Δt was initially selected based on experience obtaining sufficient resolution for wingtip vortex cross-stream PIV behind the same model under the same conditions [8]. With this Δt, only a 2-3 pixel-displacement was obtained in the free shear layer. Figure 76 shows the u and v component velocity contours

99 averaged over 170 image pairs. The cross-stream u component responsible for flow in the shear layer cannot be identified due to the poor resolution. While the v velocity component does suggest the presence of the free shear layer, the likelihood of averaging distortions at low pixel displacement is very high. Increasing the Δt to increase pixel displacement resulted in poor or no correlation of the moving particles.

Figure 76 u and v-component velocity contours showing a lack of clear definition of flowfield behavior due to insufficient seed particle displacement in the shear layer cross-stream.

5.6.2 Second Iteration of the Experiments

To accurately track the free shear layer, the PIV field of view (FOV) was expanded in the spanwise direction by stitching multiple smaller FOVs. This time, the velocity data was obtained from the wingtip vortex up to the mid semi span. Figure 77 shows the v- velocity component contour from the wingtip vortex inboard along the span to the mid semi-span station of the Clark-Y wing. It is important to note that the Δt was adjusted to ensure optimal seed particle displacement in and around the free shear layer. The v-velocity component contour clearly indicates the presence of the free-shear layer. The expanded

100 view of the mid semi-span region shows positive and negative (directional) v-velocities, maintaining zero v-velocity in the middle of the free shear layer as expected.

Figure 77 v-component velocity contour showing traces of the free shear layer forming past the wingtip vortex up to the mid semi-span and beyond. The positive and negative v-component velocities indicate the direction of the flow on either side of the free shear

In order to determine the sensitivity of the result to the choice of interrogation region size, the images were also correlated using a 16/16 interrogation region. This allowed better correlation in the areas where the pixel displacement was much lower.

Figure 78 shows the comparison of u and v-velocity components analyzed using a multi- pass 64/32-pixel interrogation window size and a 16/16-pixel interrogation window size.

101

While the free shear layer is much better defined in the u-component contour, the freestream area on either side of the free shear layer was still not accurately resolved.

Figure 78 Comparison of u and v-velocity component contour (averaged over 170 image pairs) between 64/32-pixel and 16/16-pixel interrogation regions. The comparison shows more prominent evidence of the free shear layer in the u-component however the freestream in

5.6.3 Third Iteration of the Experiments

In this iteration of the experiments, the thickness of the laser sheet was increased to 9.5 mm to increase the particle residence time within the laser-sheet during the interval between the two images. This allowed for a much higher Δt which was increased from 100

µs to 240 µs. With the thickened laser-sheet, the increased Δt resulted in the required 8-10 pixel-displacement (for a 32 pixel interrogation window) in the free shear layer with much

102 improved correlation peak ratios. With improved resolution, the data was averaged first over 170 and later over 1020 image pairs to investigate the differences between the two cases. Figure 79 shows the comparison of u and v-velocity components between 170 image pairs and 1020 image pairs at the longer Δt. It is instantly conspicuous that the velocity contours averaged over 1020 image pairs capture the character of the flow better in the free shear layer. Additionally, data averaged over 1020 image pairs minimized the minor spatial fluctuations in the velocity components evident from the more uniform distribution in and around the free shear layer. A cross-sectional line through the contour shows the comparison of the u and v-component velocities averaged over 170 and 1020 images in

Figure 80. The sectional plots show more uniformity in the free shear layer region when averaged over 1020 images.

Figure 79 Comparison of the u and v-component velocity contour between 170 image pairs and 1020 image pairs showing smoother and less ambiguous contour obtained by averaging 1020 image pairs. 103

Figure 80 Comparison of u and v-component cross-sectional line between 170 image pairs and 1020 image pairs showing improved uniformity obtained by averaging 1020 image pairs.

While velocity components obtained by averaging 1020 images showed relative uniformity, derivative quantities showed much more distortion in the shear layer.

Therefore, in a later iteration, the velocity fields were averaged over a range of 4000 to

6000 images depending on the changing character of the wake across the range of different angles of attack. Figure 81 shows the comparison of v-velocity component contours as a function of number of images (2000 – 6000 with an increment of 2000 images) for -3° and

0° angles of attack. It is unmistakable that the velocity contours averaged over 4000 and more image pairs better represent the character of the flow in the free shear layer. There are fewer fluctuations in the velocity in the free shear layer as a function of increasing the

104 sample size. A cross-sectional line through the contour compares the v-velocity component averaged over the range of images in Figure 82. When averaged over 4000 or more images the flow is more consistent throughout the free shear layer region. This can be more easily visualized in the plots of the cross-sectional lines. Additionally, comparatively little improvement is seen beyond 4000 image pairs.

Figure 81 v-velocity component contour comparison as a function of number of image pairs averaged for the -3° and 0° angles of attack cases showing much smoother contours when 4000 or more image pairs are averaged.

105

Figure 82 v-velocity component profile comparison as a function of number of image pairs for the -3° and 0° angle of attack cases showing uniform profiles for data averaged over 4000 or more image pairs.

5.6.4 Statistical Analysis

While the data averaged over 4000 plus image pairs showed diminished distortion and higher consistency, statistical analysis on the sample was essential to identifying the magnitude of the deviation from the mean velocity components. A negative, a positive, and a near-zero location were selected within the v-component velocity matrix for each of the

4000 image pairs. Mean, variance, and standard deviation were calculated for each of the locations to identify any relative changes across the field of view. Figure 83 shows the v- velocity contour along with the plots of temporal variation for the three points. The standard deviation in the regions on either side of the shear layer are much lower than in the free shear layer and it is reassuring the standard deviation is similar to two significant digits for locations 1 and 3. The standard deviation at location 2 (in the free shear layer) is substantially higher than the other two locations due to higher turbulent intensity in the

106 turbulent near-zero velocity region in the free shear layer.

Figure 83 v (spanwise) velocity distribution for three locations showing mean, variance, and standard deviation. Standard deviation for locations 1 and 3 are similar while for location 2 (in the free shear layer) is much higher as expected.

After running a sensitivity study on the number of image pairs and performing statistical analysis on the results, a relatively robust method of obtaining reliable PIV data in the free shear layer was established. The u (downwash) component velocity contours for each of the angles is shown in Figure 84. As shown in Figure 84, the magnitude of the downwash velocity increases with increasing angle of attack. The contour scales are purposely kept independent for each case to better visualize the flow behavior. Figure 85 shows the spanwise profiles for each of the angles of attack. The magnitude for -5° to -2° angles are all positive, thus indicating no shear at those angles. The flow behavior in the spanwise direction for 0° up to 3° angle is intriguing. While there is a visible shear layer at those angles, sinusoidal shifts in magnitude across the span indicates non-uniformity of

107 shear. This behavior appears fully developed at a 4° angle of attack and up where there is uniform shear in the spanwise direction in the wake.

Figure 84 u (downwash) velocity distribution contours for various angles of attack showing increase in the downwash with increasing angle of attack.

108

Figure 85 u (downwash) velocity component profiles as a function of spanwise location showing linear (negative) increase in the magnitude with angle of attack. Sinusoidal shape shift for 0° up to 3° is an indication of the non-uniformity of the shear layer as a function of spanwise location.

Figure 86 shows the cross-sectional profiles of the u (downwash) velocity component for each angle of attack. The u velocity component plot shows (negative) increase in the magnitude with increasing angle of attack. The shape of the profiles starts to change at positive angles of attack (from 0° to 3°) and progresses to become much more 109 significant at higher angles (4° to 12°). It is also noteworthy that the u velocity component is symmetric on either side of the free shear layer for the 6°, 8° and 12° angles. This behavior could be indicative of the total segregation of the free shear layer from the wingtip vortex at higher angles. In Figure 87 the downwash component is normalized by the peak

(hump) indicating similarity in shape from 2° up to 12° angle of attack. Also curious is that the downwash in the lower surface varies much less as a function of angle of attack as the downwash associated with the wing upper surface.

110

0.18 -5°

0.12 -4° -3°

0.06 -2° 0°

0 1°

x/(b/2) 2°

-0.06 3° 4°

-0.12 5° 6° Increasing α -0.18 8° -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04

u/w∞

111

u (downwash) velocity normalized by u-min 0.18

0.12 0° 1° 0.06 2°

3° 0

x/(b/2) 4°

-0.06 5°

6° -0.12 8°

-0.18 0.4 0.8 1.2 1.6 2 2.4 2.8

u/uminSL

Figure 87 u (downwash) velocity component profiles normalized by u-minimum showing distinct changes in 0° and 1° angle of attack profiles in an otherwise consistent behavior across other angles of attack.

The v (spanwise) velocity component contours for each of the angles is shown in

Figure 88. The contours show an increase in magnitude only above the upper surface of the wing. From -5° up to -2° angle of attack, the magnitude of the shear in the shear layer is not easily observed. The shear layer begins to develop at 0° up to 3° angle but it is not uniform across the wing span. Beyond 3° angle (4° to 8°), a uniform shear layer is observed across the span. The sectional profiles show similar behavior across the wingspan for all angles of attack with a gradual increase (positive and negative) in magnitude, as shown in

Figure 89. The arrows show the direction of the increase in magnitude for a set of angles of attack.

112

Figure 88 v (spanwise) velocity distribution contours for various angles of attack showing velocity in opposing sense for each of the angle of attack.

113

Figure 89 v (spanwise) velocity component profiles as a function of spanwise location showing gradual increase in the magnitude with angle of attack.

Figure 90 shows cross-sectional v component profiles for each angle of attack tested. The results are plotted for two negative lift angles (-5° and -4°) and the remainder of the angles shown are positive lift angles (0° to 12°). The profiles for the two negative

114 lift angles show little variation in upper and lower surfaces of the wing. Results for the positive angles are intriguing where the (negative) magnitude increases only in the portion of the profile aligned above the stratification and associated with the upper surface of the wing with increasing angle of attack. The contribution from the lower surface is relatively constant. There is evidence of increasing velocity transferring momentum from inboard

(mid semi-span) to outboard (wingtip) with increasing angle of attack [15]. The 12° angle is close to the stall angle for the Clark-Y airfoil and thus saturation in the upper surface is observed. The arrow shows the direction of the increase in magnitude for a set of angles of attack. Figure 91 shows peak normalized v component profiles. The results are only plotted for 0° to 12° angle of attack for visual clarity. The 0° and 1° cases are different compared to the other cases where the shear layer is mostly uniform.

115

Figure 90 Cross-sectional lines through the v component velocity contours showing gradual increase in the v component (in the lower surface of the wing) with increasing angle of attack.

116

Figure 91 v (spanwise) velocity component normalized by the peak showing distinct changes in 0° and 1° angle of attack profiles in an otherwise consistent behavior across other angles of attack.

To verify whether or not there is actually a net flow moving from inboard to outboard and vice versa, vector plots were used to highlight the direction of the flow. Figure

89 shows velocity vector plots for each of the four (2°, 4°, 6° and 8°) angles of attack. The u velocity component shows the mean flow occurring in the downward direction as would be expected due to downwash in the cases of positive lift. The v velocity component shows flow in opposite directions in the stratified shear layer resulting from the upper and lower surface flow in the wake of the wing. Figure 92 illustrates the vectors resulting from the upper and lower surface flows of the wing along with the location of the wing root and tip upstream. As seen from the pressure side of the wing (annotated in green squares), the magnitude and direction of the velocity vectors are relatively constant with increasing

117 angle of attack. This shows (in complement to Figure 90) that the lower surface wing wake miniscule does not contribute to transfer of momentum. On the other hand, the magnitude and direction of the velocity vectors change significantly on the upper surface side of the wing wake as a function of angle of attack, as shown in the red circles in Figure 92. When obtaining the net contribution of the wing wake downstream in the crossflow, there is a net momentum transfer from inboard (from the wing root) to outboard (towards the wingtip).

Similar behavior was evident in the streamwise flow of a flat plate in a recent research [15].

It is noteworthy that despite much smaller velocity magnitudes in the cross flow compared to the streamwise flow, the net transfer of momentum from inboard to outboard can still be observed. This momentum transfer is responsible, at least in part, for altering the balance between the parasite drag and the lift induced drag. It is also noted that this transfer of momentum occurs in the vicinity of maximum lift to drag ratio angles of attack.

118

Figure 92 Velocity vector plots for each angle of attack showing significant changes (red circles) in the stratified wake of the upper surface of the wing and relatively constant behavior (green squares) in the lower surface wake of the wing.

Figure 93 shows contours of the evolution of the shear distribution as a function of spanwise location for each angle of attack. It is evident that the [measurable] shear begins at a -3° angle and is not uniform until a 2° angle of attack. At around 2° angle of attack, the shear distribution starts to become uniform across the entire wingspan. Beyond 2°, the uniformity across the wingspan is observed across all angles of attack with a gradual increase in the magnitude. Figure 94 shows the cross-sectional profiles of shear for each angle of attack. The non-uniformity is observed at lower angles and a uniform shear past

2° angle of attack. Cross-sectional profiles of shear normalized by the peak shear are shown

119 in Figure 95. The normalized profiles point out the differences in the shape for lower angles up to 2° after which the shape of the profiles are roughly consistent. The uniformity of the shear distribution across the wingspan indicates the bifurcation of the free shear layer from the wingtip vortex at higher angles of attack. This bifurcation at higher angles of attack was observed previously in the streamwise flow past a flat plate [15]. It is astounding that the derivative quantities in an already fractional freestream velocity in the cross flow are representative of the transition from minimal cross flow in the free shear layer to the well- defined uniform shear.

Figure 93 Evolution of shear distribution as a function of spanwise location showing occurrence of non-uniform shear at -3° up to 1°, and uniform shear across the span past 2° angle of attack.

120

Figure 94 Cross-sectional profiles of shear showing non-uniformity at lower angles of attack and uniform shear past 2° angle of attack.

121

Figure 95 Cross-sectional profiles of shear normalized by the peak shear showing similarities in the overall shape past 2° angle of attack.

122

CHAPTER 6

CONCLUSIONS

The adverse effects of lift induced drag on the aerodynamic efficiency of aircraft are significant. Addressing the lift induced drag problem through the use of winglets on the tips of the aircraft wings has a minuscule effect compared to the sheer magnitude of the drag. Results from the wingtip vortex and the free shear layer data obtained in the present research has provided a deeper understanding of the drag problem. Particle Image

Velocimetry experiments were performed in the University of Dayton Low Speed Wind

Tunnel in the near wake of a Clark-Y airfoil to investigate the flow physics in the wing wake. An exergy-based technique is established to investigate distinct changes in the behavior of the wingtip vortex core. Additionally, the exergy-based technique was able to identify distinct changes in the out of plane profile.

To change the upper surface boundary layer characteristics and hence the balance- point between lift induced and parasite drag in the wingtip vortex formation process, a boundary layer trip is added to the wing upper surface to anchor the location of transition from a laminar to turbulent boundary layer. Three wing configurations were tested, a baseline (no trip), a 10% chord location boundary layer trip (BLT 10%) and a 20% chord location boundary layer trip (BLT 20%). Results from the azimuthal velocity component for all three wing configurations showed a gradual increase in the absolute peak velocity

123 as a function of angle of attack as expected from theory. For each of the three wing configurations, the asymmetrical behavior at 2° angle of attack was attributed to the interference from the free shear layer. This corresponds to the increase in the isolated diameter of the vortex outer core from less than that of the inner core (at 2°) to greater than the inner core (beyond 4°). Furthermore, the velocity distribution was apportioned differently between the inner core and the outer core of the vortex between 4° and 6° angle of attack. Vorticity and exergy profiles for all three wing configurations revealed discontinuity in the behavior of the vortex in the range from 4° to 6° angle of attack. This behavior was indicative of the existence of the transformation from wake-like (less-than the freestream) to jet-like (greater-than the freestream) profile around those angles of attack. This transformation occurs in the vicinity of the maximum aerodynamic efficiency point (max L/D ratio). The vortex inner core circulation was found to be much more prominent compared to that in the outer core of the vortex, regardless of the size of the vortex outer core.

Aside from the similarities in the results across the three wing configurations, the

BLT 10% configuration revealed some intriguing differences when compared to the baseline configuration. There was a constant inner and outer core vorticity and exergy distribution due to the delay in the boundary layer separation forced by the trip. Moreover, at 2° angle of attack, a discontinuity in the vortex roll-up process causing a change in the shape of the vortex is observed. Since the forced trip has the greatest effect on the boundary layer at small angles of attack, no discontinuity in the behavior of the vortex was seen beyond 4° angle of attack. The results from the BLT 20% configuration showed an opposite redistribution of exergy from the outer to the inner core of the vortex with

124 increasing angle of attack. While this configuration was least effective in terms of the change in shape of the vortex, it does serve the purpose of sensitivity on the placement of the boundary layer trip.

The resulting velocity components in and around the free shear layer showed flow in opposite directions in the wing wake corresponding to the upper and lower surfaces of the wing. The opposing directional velocity was evident through the positive and negative vectors on either side of the free shear layer. An indication of transfer of momentum from inboard (wing root) to outboard (wingtip) of the wing was seen in the cross flow where the velocity is a fraction of the freestream. A transition from minimal cross flow in the free shear layer to a well-established shear flow aligned with the spanwise direction occurs in the vicinity of maximum lift-to-drag ratio (max L/D) angle of attack. The momentum transfer is indicative of the balance between the parasite drag and the lift induced drag.

125

CHAPTER 7

RECOMMENDATIONS FOR FUTURE WORK

Changes associated with the wingtip vortices and the free shear layer in the vicinity of maximum (L/D) angles have shown tremendous potential for future research. The strong possibility of the transfer of momentum from inboard to outboard of the wing calls for further investigation into the balance between lift induced drag and parasite drag. The relationship between the parasite drag in the free shear layer and the azimuthal velocity of the wingtip vortex can be explored further using the Batchelor model for a laminar vortex

[3] and the Hoffmann-Joubert model for a turbulent vortex [10]. Batchelor’s laminar vortex model for the distribution of azimuthal velocity and the distribution of axial velocity is given by equations 27 and 28

푉 (휂) 1 1 2 휃 = (1 + ) (1 − 푒−훼휂 ) (27) 푉0 2훼 휂

푉 (휂) 2 2 푧 = 푒−훼푙 휂 (28) 푊0

where 푉휃 (휂) is the azimuthal velocity, 푉0 is the maximum azimuthal velocity, 휂 = 푟/푟푐 is the radial coordinate with 푟푐 being the core radius, 훼 is Lamb’s constant of 1.256, 푉푧(휂) is the axial velocity, 푊0 is the peak axial velocity, and 푙 is a rescaling parameter for the axial

126 profile [3]. The Hoffmann-Joubert (H-J) turbulent model for the azimuthal velocity distribution in the wingtip vortex is given by equation 29

푉 (휂) 1 퐴 휃 = + 1 ln(휂) (29) 푉0 휂 휂

Where 퐴1 is a constant factor and was given a value of 1. As noted from the above equation, the normalized azimuthal velocity is directly proportional to the natural log of the normalized radius. While both Batchelor’s and H-J models for circulation were compared in this research, understanding the origins behind the differences in the velocity models is expected to reveal key differences in determining interaction between the free shear layer and the wingtip vortex in the cross-stream flow. With increasing viscous effects on the wing, for example using a force boundary layer trip, it is hypothesized that the azimuthal velocity profiles would show much better agreement with the Hoffmann-Joubert model than the Batchelor’s laminar model. This is due to the possible change in the vortex inner core and the vortex rollup process. A closer look at this relationship based on the underlying theory and numerical models is recommended.

The desired elliptical lift distribution along the wingspan is difficult to achieve in airplanes due to limited aspect ratio of the wings and wing sweep. Figure 96 shows the comparison of the ideal lift distribution and realistic lift distribution along the wingspan.

127

Based on these differences, some recommendations for the future work are listed here.

Figure 96 Ideal elliptical lift distribution in comparison with the realistic lift distribution showing large differences due to lower aspect ratio, AR and taper ratio, λ, and higher sweep angles, ΛLE [50]

1. Investigation of the velocity field across different locations of the wingspan to

identify lift distribution and the magnitudes and directions of the momentum

transfer at those locations. Computational modeling and simulations must also be

performed to determine the sensitivity associated with the magnitudes of

momentum and circulation along the wingspan. This exercise would reveal key

differences when determining cross-stream flow in the wake of a wing.

2. Wings with various aspect ratios must be tested to study the variations as a function

wing aspect ratio.

3. Wings with different sweep angles must also be investigated to determine if the

directional flow associated with the transfer of momentum changes.

128

4. PIV experiments in the water tunnel must also be done to investigate free shear

layer in the cross-stream direction. At higher density of fluid (water) compared to

that of air, an expanded range of flow visualization and testing can be performed.

129

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