Investigation of the Ground Effect on Wingtip Vortex Generated by a Rectangular NACA0012 Wing

Anan Lu Supervisor: Prof. Tim Lee

Department of Mechanical Engineering McGill University Montreal, Quebec April 2020

A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Engineering-Thesis ©Anan Lu, 2020

Acknowledgement

The key to a successful graduate study is without doubt the right supervisor. I would like to thank Prof. Tim Lee, my supervisor, for his strong guidance, continuous supports and valuable lessons throughout this research project. His patience, motivation, enthusiasm, and immense knowledge have helped me in all the time of the research and writing of this thesis.

I would also like to thank our group members Vincent Tremblay-Dionne and Joseph Qiu, along with other lab mates Alais Hewes, Anushka Goyal, etc. for their support and discussion. My sincere thanks also go to the machine lab of the Department of Mechanical Engineering for the technical support.

Last but not least, I am grateful to my parents and fiancée, as well as the rest of my family and friends for their support and love.

Abstract

The two stream-wise counter-rotating wingtip vortices generated behind aircraft and their interactions with the runway have always been a concern to aviation, airport traffic control, and aircraft manufacturers alike. For aircraft flying at constant distance from ground (i.e. Ground Effect Vehicles), the behavior of wake vortices and the lift-induced drag are being investigated as well. It was observed that lift increase and drag reduction are experienced by wings in ground effect. To further understand the phenomenon both qualitatively and quantitatively, the ground effect on the aerodynamics and tip vortex flow of a rectangular wing was investigated experimentally at 푅푒 = 2.71 × 105. The results showed that there was a large lift increase with reducing ground distance. By contrast, a small drag increase was observed in ground effect except in close ground proximity for which a significant drag decrease appeared. The tip vortex also moved further outboard and upward with reducing ground distance. Near the ground, there was the presence of a corotating ground vortex (produced by the rolling up of the boundary layer developed on the ground surface), leading to an increased vortex strength. In extreme ground proximity, a counterrotating secondary vortex (SV) (induced by the crossflow of the tip vortex), relative to the tip vortex, appeared, which caused a reduced vortex strength and a lowered lift-induced drag, as well as a vortex rebound. The impact of ground effect on the vortex flow properties was also discussed. The lift-induced drag, computed based on the crossflow measurements via the Maskell wake integral method, in ground effect was compared against the inviscid flow predictions and wind tunnel total drag force measurements.

Abstract

Les deux tourbillons marginaux contrarotatifs générés derrière les avions et leurs interactions avec la piste ont toujours été une préoccupation pour l'aviation, le contrôle du trafic aéroportuaire et les avionneurs. Pour les avions volant à distance constante du sol (c'est-à-dire les véhicules à effet de sol), le comportement des tourbillons de sillage et la traînée induite par la portance sont également étudiés. Il a été observé que l'augmentation de la portance et la réduction de la traînée sont ressenties par les ailes en effet de sol. Pour mieux comprendre le phénomène à la fois qualitativement et quantitativement, l'effet du sol sur l'aérodynamique et l'écoulement tourbillonnaire d'une aile rectangulaire a été étudié expérimentalement à 푅푒 = 2.71 × 105. Les résultats ont montré qu'il y avait une augmentation importante de la portance avec la réduction de la distance au sol. En revanche, une légère augmentation de la traînée a été observée dans l'effet du sol, sauf à proximité du sol pour laquelle une diminution significative de la traînée est apparue. Le tourbillon marginal s'est également déplacé plus loin vers l'extérieur et vers le haut avec une distance au sol réduite. Près du sol, il y avait la présence d'un vortex de sol corotatif (produit par l'enroulement de la couche limite développée à la surface du sol), conduisant à une augmentation de la force du vortex. À proximité extrême du sol, un vortex secondaire (SV) contrarotatif (induit par l'écoulement transversal du vortex), par rapport au vortex de pointe, est apparu, ce qui a provoqué une force de vortex réduite et une traînée induite par la portance, ainsi qu'un vortex rebond. L'impact de l'effet du sol sur les propriétés d'écoulement du vortex a également été discuté. La traînée induite par la portance, calculée sur la base des mesures de l'écoulement transversal via la méthode intégrale de sillage de Maskell, en effet de sol a été comparée aux prévisions d'écoulement non visqueux et aux mesures de la force de traînée totale en soufflerie.

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Table of Contents Acknowledgement ...... i Abstract ...... ii Abstract ...... iii List of symbols ...... v List of tables...... vi List of Figures ...... vii 1 Introduction ...... 1 1.1 Static Wing Tip Vortex ...... 2 1.2 Interaction of aircraft trailing vortices with the ground ...... 8 1.3 Ground effect vehicles...... 12 1.4 Previous experimental work on ground effect ...... 18 1.5 Objectives ...... 24 2 Experimental Procedures ...... 25 2.1 Flow Facilities ...... 26 2.2 Wing model set-up ...... 28 2.3 Seven-Hole Pressure Probe ...... 30 2.4 Traverse Mechanism ...... 32 2.5 Two-component Force Balance ...... 33 2.6 Data Acquisition and Analysis ...... 36 2.7 Experimental Uncertainty ...... 39 2.7.1 Force Balance Measurements ...... 39 2.7.2 Seven-Hole Pressure Probe ...... 41 3 Ground Effect on Aerodynamic Characteristics ...... 46 4 Ground Effect on Wingtip Vortex Flow ...... 51 4.1 Effect of ground clearance on vortex flow properties in the near wake region ...... 53 4.2 Impact of ground proximity on vortex flow properties ...... 59 4.3 Spatial Evolution of the Vortex ...... 62

5 Ground Effect on 푪푫풊 ...... 71 6 Conclusion ...... 73 References ...... 75 Appendix A: Calibration Procedures...... 78 Appendix B: 푪푫풊 Calculation ...... 89

List of symbols

퐴푅 = aspect ratio

퐴푅푒푓푓 = effective aspect ratio 푏 = wing span 푏’ = distance between the center of vortices 푐 = chord

퐶퐷 = total drag coefficient 퐶퐷푖 = lift-induced drag coefficient 퐶퐷푝 = profile drag coefficient

퐶퐿 = total lift coefficient 퐷 = total drag

퐷푖 = lift-induced drag ℎ = ground distance 퐿 = total lift 푟 = vortex radius

푟푐 = vortex core radius 푅푒 = chord Reynolds number, = 푐푈/휗 푆 = wing surface area 푢, 푣, 푤 = mean axial, transverse and spanwise velocity

푢푐 = mean core axial velocity 푈 = freestream velocity 푥, 푦, 푧 = streamwise, transverse and spanwise direction

푦푐, 푧푐 = vertical and spanwise vortex core location 훼 = angle of attack

훼푠푠 = static-stall angle Γ = circulation or vortex strength

Γ푐 = core circulation Γ표 = total circulation 휙 = velocity potential 휈 = kinematic viscosity

푣휃 = tangential velocity 푣휃,푝 = peak tangential velocity 휓 = stream function 휁 = mean streamwise vorticity

휁푝 = peak 휁

List of tables

Table 1 Uncertainty of force balance measurement ...... 39 Table 2 Uncertainty of seven-hole pressure probe measurement ...... 41 Table 3 Uncertainty of data acquisition and calculation ...... 43

List of Figures

Figure 1-1 Details of formation and development of a wingtip vortex. (Chow, J. S., Zilliac, G. G., and Bradshaw, P.,, 1997) ...... 3 Figure 1-2 Tip vortex control. a) endplate; b) winglet; c) decelerating spline; d) spoiler/tab; e) flap; f) sheared tip; g) cascading tip sails; h) delta planform tip sail; i) movable tip strake (Pereira, 2011) ...... 6 Figure 1-3 The development of secondary vortex and vortex rebound (reproduced from (Harvey, J.K., Perry, F.J., 1971)) ...... 8 Figure 1-4 Vorticity contour with no cross wind. a) t”=0.00; b) t”=5.38; c) t”=8.03; d) t”=15.05, where t” is normalized time [ref] ...... 9 Figure 1-5 Illustration of an “ekranoplan”, the Capsian Sea Monster (Rozhdestvensky, 2006) ...... 12 Figure 1-6 Illustration of Lippisch type WIG (Rozhdestvensky, 2006) ...... 13 Figure 1-7 Illustration of TAF VIII-1 tandem vehicle (Rozhdestvensky, 2006) ...... 14 Figure 1-8 Selected iso-vorticity contours of a delta wing in ground effect at α=16° (Lee, T., Ko, L. S., 2018) ...... 20 Figure 1-9 Iso-vorticity contours of NACA 0012 airfoil in ground effect: a) OGE; b) h/c=5%; c) h/c=10%, and velocity distribution around the airfoil: d) OGE; e) h/c=5%; f) h/c=10% ...... 21 Figure 1-10 Simulated velocity profile around NACA 4412 airfoil under different boundary conditions, h/c=2.5%, Re = 8.2 × 106 (Barber, T., Hall, S., 2006) ...... 22 Figure 1-11 Lift coefficient, drag coefficient, lift/drag distribution of NACA 4412 airfoil, under different ground boundary conditions (Barber, T., Hall, S., 2006) ...... 23 Figure 2-1 Illustration of J.A. Bombardier wind tunnel. (a) schematic diagram (b) wind tunnel inlet (c) wind tunnel outlet (Pereira, 2011) ...... 27 Figure 2-2 Schematics of (a) seven-hole pressure probe measurement set-up, (b) wing model with coordinates, (c) boundary-layer over the elevated flat plate ...... 29 Figure 2-3 Schematic of seven-hole pressure probe ...... 30 Figure 2-4 Traverse mechanism (Pereira, 2011) ...... 32 Figure 2-5 Illustration of two-component force balance (Pereira, 2011) ...... 33 Figure 2-6 Schematic diagram of force balance and wing model set-up ...... 34 Figure 2-7 Data aquisition schematic diagram (Pereira, 2011) ...... 36 Figure 3-1 Impact of ground distance on the aerodynamics of the NACA 0012 wing. OGE denotes out of ground effect...... 47

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Figure 3-2 Ground effect of aerodynamic properties and surface pressure distribution of unflapped NACA 0015 airfoil. BA denotes base line airfoil in a free stream out of ground effect. (Tremblay-Dionne, 2018) ...... 49 Figure 4-1 A sample of adaptive grid plotted over iso-vorticity contour at h/c=100% . 52 Figure 4-2 Comparison of iso-vorticity contours out of ground effect: a) current experiment at h/c=100%; b) previous experiment, reproduced from ( Lee, T. and Choi, S., 2015) ...... 54 Figure 4-3 Ground effect on (a)-(k) iso-c/U and (l)-(v) iso-u/U contours at  = 10o and x/c = 2.5. (a) h/c = 100% or OGE, (b) h/c = 60%, (c) h/c = 50%, (d) h/c = 40%, (e) h/c = 30%, (f) h/c = 20%, (g) h/c = 15%, (h) h/c = 12.5%, (i) h/c = 10%, (j) h/c = 7.5%, and (k) h/c = 5%. OGE denotes out of ground effect. TV, GV, PV and SV denote tip vortex, ground vortex, primary vortex, and secondary vortex, respectively. BL rollup denotes boundary-layer rollup...... 57 Figure 4-4 Normalized tangential and axial velocity distributions across the vortex center at selected h/c ...... 58 Figure 4-5 Ground effect on wingtip vortex flow parameters at x/c=2.5 ...... 59 Figure 4-6 Zoom-in iso-vorticity contours of the tip vortex at x/c=2.5: (a) h/c=100%, (b) h/c=20%, (c) h/c=10%, and (d) h/c=5% ...... 61 Figure 4-7 Spatial progression of iso-fc/U contour of the tip vortex both along the tip and in the near field at a510 deg. (a) h/c=100% ...... 64 Figure 4-8 Spatial progression of the iso-u/U∞ contour of the tip vortex in the near field at selected h/c...... 68 Figure 4-9 Conceptual sketch of the existence of the spanwise ground vortex filament (SGVF) and its downstream development into the ground vortex, and the formation of SV, and the evolution of MV, SLV and TV ...... 70 Figure 5-1 Impact of ground distance on lift-induced drag coefficient ...... 72 Figure A-1 Flowchart of force balance calibration process ...... 80 Figure A-2 Typical force balance calibration curve-fits ...... 80 Figure A-3 Flowchart of (a) seven-hole probe calibration process and (b) velocity extraction process...... 86 Figure A-4 Seven-hole probe coordinate systems...... 87 Figure A-5 Seven-hole probe sectors...... 87 Figure A-6 Typical direction coefficients for low flow angle (Hole 7)...... 88

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1 Introduction

With the ever-growing power of the environmental movement, the increasing pressure for cleaner vehicles has stimulated a need for greener technologies and more efficient designs. For the aviation industry, the main focus has been on finding ways to minimize drag in order to improve aerodynamic efficiency. As a significant part of the total drag, the lift-induced drag has been studied heavily, which brings the study of wingtip vortex in the spotlight.

One of the many findings was the L/D increase occurred when the trailing vortices interact with the ground, which was first acknowledged by pilots during take-off and landing. The increase in lift was found to be a result of “ram pressure”, high pressure produced by the reduction of flow passage underneath the wing when ground distance is small. The decrease in drag mainly comes from the reduction in lift-induced drag. The combination of these phenomena led to a strong increase in aerodynamic efficiency.

A unique type of vehicle is thus developed, namely the ground effect vehicle (GEV) or wing- in-ground effect vehicle (WIG-vehicle). From the famous Russian military aircraft, the Capsian Sea Monster to the reactional Airfish-8 certified in Singapore recently, this type of vehicle was designed to take advantage of the L/D increase due to ground effect. Research groups over the world have worked on studying the details of ground effect on airfoil and finite wings, looking for ways to improve the design.

This chapters provides a brief review of previous publication on wingtip vortex and ground effect, which leads to the objectives of the current research project.

1.1 Static Wing Tip Vortex

As Spalart stated in his famous survey of airplane trailing vortices (Spalart, Phillipe R., 1998), “the time has long passed since scientists could write that the rolling up of the trailing vortices is of little practical importance”. Extensive experimental and theoretical studies of wingtip vortices have been carried out since then, while numerous results have been contributed to characterizing the structure, dissipation and control of tip vortices.

Wingtip vortex forms mainly due to the difference in pressure between the top and bottom surfaces of the wing, as illustrated in Figure 1-1. Because of this pressure difference, the fluid from the bottom boundary layer (high pressure side) separates and is sucked into the top flow (suction side). Such motion would initiate a circulation, which grows in size and strength as the rotational field moves downstream. The rolling-up is also aided by the vorticity in the boundary layers themselves. Eventually, a vortex emerges and detaches from the wing surface near the trailing edge of the wing. Afterwards, the vortex continues to capture the shear layer generated by the boundary layer of the wing and the wake created by the flow separation near trailing edge, until the circulation is nominally equal to that of the wing. The vortex would slowly diffuse and decay, but it could persist for hundreds of chord-lengths, which results in their significance in flight safety and drag (Pereira, 2011).

Some experimental works have been conducted to explain the formation of wingtip vortex along the tip and in near-wake region. Chow et. al (Chow, J. S., Zilliac, G. G., and Bradshaw, P.,, 1997) performed experiments with a round-tipped NACA 0012 wing at a Reynolds number of 4.6 × 106. The flow properties in terms of vorticity, axial flow velocity, and turbulence level were reported in great details. It showed that high crossflow velocity circumventing the tip starts to appear at about 40% chord-length from the leading edge, and it rolled up into a vortex at around 60% chord-length. The development of the crossflow velocities with chordwise distance induced a favorable axial pressure gradient in the core. The resultant core axial velocity at the trailing edge appeared to be 1.77 times the freestream velocity. The vortex was also aided by the local separation resulted from skin friction. The vortex development and structure are illustrated in Figure 1-1.

Figure 1-1 Details of formation and development of a wingtip vortex. (Chow, J. S., Zilliac, G. G., and Bradshaw, P.,, 1997)

Once the tip vortex passes the wing’s trailing edge, it began to interact with the separated flow from the wing’s wake. The wake continues to feed vorticity to the tip vortex, which could take over a hundred chord length to develop. However, it is in the near field where large changes in the vortex flow characteristics occur. The interaction and development happened in this region would determine the behavior of the fully developed tip vortex and have significant impact on the aerodynamic characteristics of the wing. Thus, there has been a great amount of studies focused on the complex dynamics and roll up of the tip vortex in the near wake region.

The commonly reported vortex properties include core centerline axial velocity (푢푐), peak tangential velocity (푣휃푝), core radius (푟푐), peak vorticity (휁푝), core circulation(Γ푐) and total

3 circulation (Γ표). The results of axial velocity, however, has shown great controversy. Although by the definition of vortex an extremum of axial velocity should occur at the core centerline, whether the extremum appears at maximum or minimum was debated. The early experimental works demonstrated a range of 푢푐/푢∞ results, from 0.6 to 1.8. The vortex structures with minimum 푢푐/푢∞ at center was defined as wake like, while those with maximum 푢푐/푢∞ defined as jet like. Such distinctive behavior of wing tip vortex flow was a combined result of a number of parameters, including the distance downstream from the leading edge, angle of attack, Reynold’s number, tip condition and skin roughness.

Corsiglia et. al (Corsiglia, V., Jacobsen, R., and Chigier, N., 1970) studied the formation and development of a trailing vortex within 5 chord lengths of the leading-edge. Their experiments 5 with a blunt tipped NACA 0015 wing at 푅푒 = 9.53 × 10 and 훼 = 12° showed that 푢푐/푢∞ increased from 1.1 to 1.2 as the vortex moved from 푥/푐 = 1 to 5. The highest axial velocity actually occurred at 푥/푐 < 1, where it reached 1.4 푢∞. Thus, it was suggested that the interaction between tip vortex and wing wake would decrease the strength and rotation of the trailing vortex. The vortex, surprisingly, reached axi-symmetric within a couple of chord lengths, which was further confirmed by Birch et al. (Birch, D., Lee, T., Mokhtarian, F., and Kafyeke, F., 2004). They tested a rectangular, square tipped NACA 0015 wing at 푅푒 = 2.01 × 105 and took measurement at cross flow planes located between 0.5 to 3 chords downstream of the leading- edge. With the increase in downstream distance along the wing tip, the main vortex grew in magnitude and size, which could be attributed to the rolling up of the shear layer feeding vorticity to the main vortex. Past the trailing edge, the shear layer vortices merged with the main vortex, the strength of which began to decay while the vortex moved further downstream. Such diffusion was evidenced by the decrease in peak tangential velocity, peak axial velocity, peak vorticity and the increase in total radius of the tip vortex.

Orloff (Orloff, 1974) focused on the effect of angle of attack on a NACA 0015 wing with blunt tip at fixed downstream location from the leading edge (푥/푐 = 3) and 푅푒 = 7 × 105. The experiments were conducted with laser velocimetry that enabled more refined grid resolution and provided more detailed velocity distribution. It reported that 푣휃푝/푢∞increased from 0.35 to 0.56

4 with 훼 from 8° to 12°, while 푟푐/푐 grew from 6.8% to 8%. Most importantly, the growth in 푢푐/푢∞ from 0.85 to 1.15 suggested a potential crossover point between wakelike and jetlike flow. This idea of a crossover point was further studied by Brown (Brown, 1974), who showed that the core axial velocity is a function of the lift and profile drag. The maximum 퐿/퐷 point should correspond to 푢푐/푢∞ approximately equals 1.

Birch et al. (Birch, D., Lee, T., Mokhtarian, F., and Kafyeke, F., 2004) examined the tip vortex characteristics behind a cambered wing at a selection of Reynold’s number (푅푒 = 6.7 × 103, 1.63 × 105, 3.25 × 105). With the study of the related parameters, it reported that with increasing Reynold’s number, the vortex appeared weaker and more diffused, indicated by a drop in circulation, peak tangential velocity, peak vorticity and core axial velocity. Anderson and Lawton (Anderson, E., Lawton, T., 2003) examined 푅푒 effects over the range 푅푒 = 7.5 − 12.5 × 105 on a NACA 0015 wing with both square and round tips using triple wire scans. The results showed little variation in core velocities with Re. The differences in axial velocity were found to be correlated with the wing circulation. Most importantly, it was suggested that the velocities were sensitive to tip condition, which brought up the importance of tip design and skin roughness.

The most significant source of error in the measurement of peak values of wing tip vortex in near field is meandering. The vortices have a tendency to wander in both space and time downstream of the wingtip. Numerous experimental studies have recognized this meandering, which could be attributed to freestream turbulence that causes the instability in the vortex itself. As the vortex travels in time and space, packets of fluid are ejected from the core as a consequence of the large centrifugal force generated by its rotation. As the fluid packets are ejected, the core must adjust by moving in the opposite direction. Nevertheless, if a long time average is taken, the effect of wandering could be smoothed, resulting in a Gaussian distribution. Devenport et al. (Devenport, W., Rife, M., Liapis, S., and Follin, G., 1996) used a Gaussian distribution to analyze a laminar q-vortex. It showed that wandering of the vortex equivalent to 50% of core radius could lead to a discrepancy of up to 64% in the measurement of peak tangential velocities. In the cases that the measurement of vortex core position is crucial, such as for stability purposes, correction methods should be developed in order to achieve accurate

5 results. Nevertheless, the intensity of meandering outside the vortex core is negligible. In regions close to the wing’s trailing edge (푥/푐 < 1), the major source of turbulence is the wake rather than the vortex itself.

Apart from the parameters discussed above, the most significant impact of wing tip vortex is the induced drag. The formation of wing tip vortex along the wing tip and behind the wing’s trailing edge results in a downwash velocity that altered the flow pattern surrounding the wing. Therefore, an induced angle of attack occurs that shifts the effected angle of attack, leading to a tilted vector of lift force. The horizontal component of this resultant force is named as lift induced drag. The induced drag has become a major concern in aircraft design, as it contributes to approximately 30% of the total drag.

Figure 1-2 Tip vortex control. a) endplate; b) winglet; c) decelerating spline; d) spoiler/tab; e) flap; f) sheared tip; g) cascading tip sails; h) delta planform tip sail; i) movable tip strake (Pereira, 2011)

The main purpose of wing tip vortex control thus focused on minimizing induced drag. As illustrated in Figure 1-2, the common control methods include endplate, winglet, decelerating spline, spoiler/tab, flap, sheared tip, cascading tip sails, delta planform tip sail and movable tip strake. S. Choi et al. ( Lee, T. and Choi, S., 2015) examined the control of the tip vortex generated by a NACA 0012 wing, via tip-mounted half-delta wings (HDWs). These HDWs could generate leading edge vortices similar to delta wings, which showed impressive advantage in aerodynamic efficiency. The experiments at 푅푒 = 2.45 × 105 showed that regardless of the shapes and sizes, the addition of HDWs consistently led to a diffused tip vortex. The degree of diffusion, however, increases with decreasing swept angle and root chord of the HDWs.

1.2 Interaction of aircraft trailing vortices with the ground

The interaction of a descending vortex pair (generated behind large airplanes during taking off) with the runway or ground leads to the formation of secondary vortices, including vortex rebound, in close ground proximity. Apart from the altered vortex structure and trajectory, it also results in change in aerodynamic characteristics, such as lift and drag coefficients. More specifically, pilots observed that when aircrafts are flying close enough to the ground, an increase in lift and a decrease in drag are observed. However, such ground effect also results in instability and flight hazard. Thus, quite a few research works have been conducted to investigate such phenomenon.

Figure 1-3 The development of secondary vortex and vortex rebound (reproduced from (Harvey, J.K., Perry, F.J., 1971))

One of the earliest and fundamental work was provided by Harvey and Perry (Harvey, J.K., Perry, F.J., 1971), who investigated the interaction of a descending single wingtip vortex generated by a half span rectangular wing with a moving ground in a wind tunnel at a chord Reynolds number 푅푒 = 3.47 × 105. They observed that the vortex moves laterally as the ground is approached and that the vortex rises again after having descended close to ground. There also exhibits a vortex rebound originated from the formation and separation of the spanwise boundary layer, as a result of the cross flow induced by the wingtip vortex as it descends toward the ground surface (Figure 3). As illustrated in Figure 1-3, a bubble, containing vorticity of opposite sense to main vortex, forms as the vortex is sufficiently near the ground. Progressing

8 downstream, this bubble grows rapidly and detaches from the ground or wall as a counter- rotating secondary vortex (SV) which causes the primary vortex to rise (Figure 1-3). An increase of the strength, represented by circulation Reynolds number (Γ/휈), of the main vortex was also observed. Finally, it was mentioned that moving and fixed ground boundary conditions provided similar results qualitatively, but quantitative discrepancies occurred.

Figure 1-4 Vorticity contour with no cross wind. a) t”=0.00; b) t”=5.38; c) t”=8.03; d) t”=15.05, where t” is normalized time [ref]

The creation of secondary vortices and their rebound in close ground proximity are also demonstrated numerically by Corjon and Poinsot (Corjon, A., Poinsot, T. , 1997). Direct numerical simulation of a vortex pair embedded in a stable atmospheric boundary layer under the effects of various crosswind conditions were presented. A Navier-Stokes solver was used to simulate the interaction between a vortex and ground with a range of circulation-based Reynolds 3 5 numbers (3.77 × 10 < 푅푒Γ < 1.131 × 10 ). It proved that the presence of secondary vortex and vortex rebound was produced regardless of crosswind, while the height of rebounded vortex center is a strong function of crosswind velocity. The vorticity contour with no cross wing is

9 shown in Figure 1-4, which matches well with the illustration provided by Harvey and Perry. It confirmed the presence of counter-rotating SV and offered profound mathematical explanation of the phenomenon.

Puel and Victor (Puel, P., Victor, X. D. S. , 2000) further predicted the creation of a boundary layer at the wall perpendicular to the axis of wake vortices which leads to the formation of a counter-rotating secondary vortex and the ensuing vortex rebound near the ground or wall. The size of the secondary vortex increases quickly, thus pushing up the primary vortex; indicative of the mechanism of rebound. The pair of vortices was realized by means of summation of the velocity field of two contra rotating vortices of Lamb-Oseen. It was shown that the overall descending motion of the vortices was not influenced significantly by the rebound, and lower dissipation rate was observed with higher Reynolds number. As for the strength of secondary vortices, it was reported that it represented 25% of the intensity of the main vortices. The 3-D simulation of the rotating vortices also indicated that the mechanism was more complicated than that suggested by Harvey and Perry. The trajectory appeared to be loop-like, with the ascendant velocity generated by the transverse boundary layer slowing down the descent of the vortices and making them rebound, and formation of secondary vortices stopping the lateral translation of the vortices and making them go back on themselves.

Kliment and Rokhsaz (Kliment, L.K., Rokhsaz, K., 2008) investigated experimentally the motion of a pair of corotating vortices under the influence of a stationary splitter plate in a water tunnel with a water speed of 8 푐푚/푠. The vortices were generated by two flat blades and the splitter plate was placed to model the fixed ground boundary condition, and their trajectories near the ground were compared to OGE result and numerical simulation. The vortices were found to undergo a rebound possibly due to cross flow boundary-layer separation on the ground plate, and a lateral motion induced by ground effect as well. A lateral leapfrogging also caused cyclic changes in the vortex span. These experimental results were proven to agree well with the potential-flow theory, while the presence of ground did not alter the preferred direction of motion of the vortices or the vortex spiraling rate.

Most recently, the merging and rebound of a corotating vortex pair in ground effect, caused by the emergence of a secondary vortex, was also observed by Wang et al. (Wang, Y., Liu, P., Hu,

T., and Qu, Q., 2016) Experimental work was conducted in a water tunnel at 10.5 푐푚/푠, and an analytical model was introduced to compare the result. Their dye visualization of the vortex flow showed clearly the formation of secondary vortex and its interaction with the main vortex. The opposite sign vortical structure seemed to be induced by the vortex pair, developed and eventually detached to form SV. This work was especially focused on the uniqueness of the corotating vortex pair, whose merger was proved to be promoted by the presence of ground effect. The lateral motion and vortex rebound were also observed. Their simulation model showed that vortex rebound only existed if the secondary vortex was included in the model, which proved the link between the two phenomena.

1.3 Ground effect vehicles

As discussed in previous section, for aircraft flying close to the ground, especially during landing and take-off phase, there is always a large lift increase and lift-induced-drag reduction, resulting in a better lift-to-drag ratio as compared to the out of ground effect (OGE) case. The ground effect-induced lift increase can be attributed to the so-called chord dominated ground effect (CDGE) or the ram effect. The lift-induced drag reduction is caused by the so-called spanwise dominated ground effect (SDGE) as a result of the suppression of the wingtip vortices and their outboard displacement in ground effect. The SDGE-caused underdevelopment of the trailing vortices also gives rise to an increased lift-to-drag ratio as compared to the OGE case. Ground effect vehicles (GEVs) or wing-in-ground effect (WIG) craft, such as the rectangular- wing-planform Ekranoplans, have, therefore, been developed taking advantage of these beneficial ground effects. The behavior of wake vortices and the lift-induced drag of GEVs operating at fixed ground distances have been investigated subsequently.

Ground effect vehicles are designed to attain sustained flight over ground or water surfaces. With the discovery of ground effect by pilots and engineers in the 1920s, a great amount of attention was drawn toward the phenomenon, searching for possibilities of new transportation methods. In the 1960s, the technology matured with the contribution of the independent works by Rostislav Alexeyev in Soviet Union and a German engineer, Alexander Lippisch, working in the United States.

Figure 1-5 Illustration of an “ekranoplan”, the Capsian Sea Monster (Rozhdestvensky, 2006) The vehicle developed in the Soviet Union was names as an “ekranoplan”. The first operational model found was referred to as the Capsian Sea Monster by the US intelligence experts, due to

12 its gigantic shape and ghost-like speed (Figure 1-5). Later, it was revealed that the 92 meters long vehicle had a total take-off weight of 550 tons, with its best performance around 20 meters high from the ground at the speed of 300-400 kn. A number of similar models were prototyped afterwards and served for military purposes, before the end of the program around 1985.

Figure 1-6 Illustration of Lippisch type WIG (Rozhdestvensky, 2006)

In Germany, Alexander Lippisch was asked to design a fast boat, which he succeeded in 1963 with a model named X-112, a craft with reverse delta wing (RDW) and a T-tail (Figure 1-6). The vehicle was a stable and efficient masterpiece. Some prototypes were designed to fly out of ground effect as well, in order to maneuver over islands. Such configuration was commercialized by Hanno Fischer with a 3-seat Airfisch 3 model and a 6-seat FS-8 model. The company was sold to Wigetworks in Singapore, who developed an 8-seat model renamed as AirFish 8. The vehicle was certified and classified as a ship in 2010.

Figure 1-7 Illustration of TAF VIII-1 tandem vehicle (Rozhdestvensky, 2006)

Another German engineer, Günther Jörg, designed a tandem-airfoil flairboat, named “Skimmerfoil” (Figure 1-7). A number of prototypes were developed under this configuration, due to the impressive stability and efficiency it provided. However, they were never commercialized and put into production.

These designs and prototypes could be summarized in three categories: rectangular wing with ekranoplan as an example; reverse delta wing also known as the Lippisch type; tandem wing such as the Skimmerfoil. Due to its characteristic of flying at fixed low distance from water surface, there exists an ongoing debate on whether it should be characterized as airplanes or ships.

The advantage of ground effect vehicles is obviously the improved fuel efficiency due to reduced lift induced drag, which also results in higher speed to certain extent. However, many drawbacks and problems still exist in the design. The high speed that the vehicles travel at means higher risks in maneuvering. In the case of emergency, it would be difficult for pilots to react and avoid crashing. Besides, the possibility of evacuation is also limited. The taking-off of the vehicles is also highly dependent on wind conditions. With strong wind, it requires higher power to overcome the waves, leading to high stress exerted on the craft and discomfort to passengers

14 and pilots. In the case of mild wind, on the other hand, the directions of waves are more random and complicated, and thus it is hard to control the movement. Generally, a great amount of power is needed while taking off. Therefore, many techniques have been incorporated to assist it. For example, extra air cushion system could be added to the bottom of the craft. The bottom of the vehicle also requires special design to deal with the high pressure related to take-off and landing, but without sacrificing too much lateral stability.

Recent development of GEVs mainly focuses on vehicles for recreational purposes or as small ferries. Possibilities of save and rescue purposes and horizontal launching of spacecrafts are also under consideration. However, limited utility has led to low production and insufficient study in various aspects of the design. For example, although it is acknowledged conventionally that ground effect would lead to an increase in lift and a decrease in lift-induced drag, experimental and numerical studies in the past has shown different trends and results under different assumptions and ground boundary conditions.

Chun et al. (Chun, H., Park, I., Chung, K., and Shin, M.,, 1996) carried out experimental and computational studies on wing in ground effect crafts, and found that in ground effect the trailing vortices are weakened due to the limited space for the vortices to be fully developed, leading to a decreased lift-induced drag.

Hsiun and Chen (Hsiun, C. M., and Chen, C. K., 1996) applied Navier-Stokes equations under steady and incompressible assumptions to simulate aerodynamic characters of an airfoil in ground proximity. The system of equations formulated for an airfoil with small to moderate angle of attack and a ground clearance between ℎ/푐 = 0.05 to infinity were solved numerically by flowflield discretization and application of the finite volume technique. Steinbach (Steinbach, 1997) later corrected the study by pointing out that the non-slip boundary condition at ground should be replaced with a slip condition due to the relative motion between WIG crafts and ground surface in reality. In fact, a tremendous amount of effort was done for wind tunnel experimental works to model the slip boundary condition, such as suction systems or moving belt. It was also suggested that the primary reason for the marked reduction in drag for a finite wing near the ground can be attributed to the decrease of the lift-induced drag. The down wash

15 field of the trailing vortices behind wing is diminished by the interference of the upwash field of the reflected (or image) wing.

Joh and Kim (Joh, C. Y., and Kim, Y. J.,, 2004) compared the total pressure of the vortex core in ground effect at ℎ/푐 = 30% and the OGE cases and found that the vortex strength grows with the ground proximity, while the induced drag is reduced. Tuck and Standingford (Tuck, E., and Standingford, D., 1996,, 1996), however, observed that lift and lift-induced drag increase dramatically as the ground clearance is reduced.

The lift increase produced by ground effect is generally disputable. One explanation is that a high-pressure field under the airfoil occurs because of the hindered air flow underneath the wing. Numerical simulation using image wings shows that the circulation of the reflected airfoil induces upstream-directed velocities, which diminish or even compensate the oncoming freestream, and thus the induced velocities of both airfoil circulations are directed primarily upstream (Steinbach, 1997). For ground clearance small enough, the mass flow between airfoil and ground is eliminated and the freestream flow passes completely over the suction side of the airfoil. Such phenomenon is commonly referred to as the air cushion effect or ram-wing effect, which is utilized for the design of most wing-in-ground vehicles.

The drag reduction observed in experimental or numerical studies mainly results from the decrease in lift induced drag. The downwash velocity field at the trailing edge of the wing produced by wingtip vortices is diminished by interaction with the upward velocity vectors generated by the image wing (Steinbach, 1997). Besides, the effective aspect ratio also increases due to the lateral motion of the trailing vortices. On the other hand, the pressure drag in ground effect remains unaffected or increase. Without the presence of flow separation, pressure drag coefficient is not strongly influenced by the negative pressure on the upper surface of the wing. However, due to the lower local velocities induced by decreased ground distance, boundary layer parameters usually increase, leading to the increase in pressure drag. Overall, in extreme ground proximity, the decrease in lift induced drag overpowers the increase in pressure drag, which results in the reduction in total drag.

As for the trajectory of trailing vortices, it is generally believed that ground effect would push wing tip vortex laterally outboard and vertically downward until in extreme ground proximity where it rebounds. Han and Cho (Han, C., and Cho, J.,, 2005) studied the unsteady evolution of vortices behind a WIG craft using a discrete vortex method. The ground effect is included by image method. They found that for a lifting line with an elliptic loading, the ground has the effect of moving the wingtip vortex laterally outward and suppressing the development of the vortex, and that an increase in the wing loading has the effect of moving the wingtip vortex more laterally and downward than wings that are outside the ground effect.

As a summary, ground effect vehicles are designed to take advantage of the increase in aerodynamic efficiency in ground effect. Detailed study in flow field of wingtip vortices is necessary to better understand the phenomenon.

1.4 Previous experimental work on ground effect

During the past few years, our lab has focused on the study of ground effect on airfoil, finite wing and different wing planforms. The most significant results are summarized here to illustrate the necessity and importance of the present research.

Due to the popularity and impressive efficiency of Lippisch type ground effect vehicles, a great amount of attention was drawn toward the research of reverse delta wing planform. A 50 degree sweep non-slender reverse delta wing was investigated for its vortex flow and lift force at 푅푒 = 11,000 (Lee, 2017). Particle image velocimetry was utilized for analysis together with flow visualization and force balance measurement. The non-slender reverse delta wing produced a delayed stall but lower lift compared to non-slender delta wing. The lower lift was due to the fact that the leading edge vortices (LEVs) are outboard of the wing, and are thus excluded from lift generation. The vortex flow field was characterized by the co-existence of reverse delta wing vortices and multiple shear-layer vortices. Therefore, the lift force is mainly generated by the wing lower surface while the upper surface acts as a wake generator. The stalling mechanism was found to be triggered by the disruption of multiple spanwise vortex filaments developed over the upper wing surface.

For slender reverse delta wing, inspired by the Lippisch-type RFB X-114 WIG (wing-in- ground effect) vehicle, the lift and drag forces and vortices generated by same wing model with different anhedrals was investigated experimentally. The results show that, by positioning the trailing edges of the anhedraled reverse delta wing parallel to the ground, the lift and drag coefficients were found to increase persistently with increasing anhedral as the ground was approached, especially for ground distances within 40% chord-length. The observed lift augmentation was also accompanied by an ever-increasing rotational speed and total circulation of the vortices generated by the anhedraled wing. The vortices were also found to be displaced more outboard as the ground was approached, which further proved their little relevance in lift generation of reverse delta wing.

Due to the inefficiency of lift generation observed in these experiments, some methods were invented for further taking advantage of ground effect. For instance, a cropped reverse delta wing

18 could bring the wing surface closer to the ground. The ground effect on the aerodynamic coefficients of a cropped slender reverse delta wing equipped with anhedral and Gurney flaplike side-edge strips was investigated experimentally at 푅푒 = 3.82 × 105 (Lee, T., Huitema, D., and Leite, P., 2018). In a free stream, the 30% cropping was found to cause a minor reduction in lift and drag coefficients, but a promoted stall compared to the non-cropped wing. The application of side-edge strips produced a significantly increased 퐶퐿 and 퐶퐷 with a minor change to the 퐶퐿/퐶퐷 ratio as compared to the baseline wing. Therefore, it could compensate the lift reduction due to cropped wing planform, together with anhedral. As expected, the cropped wing provided more lift compared to non-cropped wing in ground effect, resulting from the fact that the effective ground distance for majority part of the wing planform decreased for cropped wing. The 퐶퐿 and

퐶퐷 increase produced by anhedral and SES was not significantly affected by ground effect. The larger the side-edge strips’ height the larger the increase in 퐶퐿, but it is limited by ground distance.

The possibilities of incorporating delta wings in WIG craft designs were investigated as well. Experimental work was conducted to study the ground effect on the aerodynamic loading and leading-edge vortex (LEV) flow generated by a slender delta wing (Lee, T., Ko, L. S., 2018). Both the lift and drag forces were found to increase with reducing ground distance (up to 50% of the wing chord). The lift increment was also found to be the greatest at low angles of attack and decreased rapidly with increasing ground distance and alpha. The ground effect-caused earlier wing stall was also accompanied by a strengthened LEV with an increased rotational speed and size compared to the baseline wing. The smaller the ground distance, the stronger the LEV and the earlier vortex breakdown became. Meanwhile, the vortex trajectory was also found to be located further inboard and above the delta wing in ground effect compared to its baseline-wing counterpart. By comparing these parameters obtained at different ground distance, it was concluded that for wing-in-ground effect (WIG) craft with delta wing planform the most effective in-ground-effect flight should be kept within 10% of wing chord.

In order to explain the strengthened LEV due to ground effect, the vortex flow was studied in detail at different cross-sections along wing chord. Figure 1-8 illustrates the iso-vorticity contours near the trailing edge at selected h/c. It suggested that at h/c around 10%, a small co-

19 rotating vortex formed at the ground outboard of the wing, which strengthens and finally merges with LEV. Such phenomenon could be explained by the interaction between the wing and the ground boundary layer, which will be discussed in detail in later sections.

Figure 1-8 Selected iso-vorticity contours of a delta wing in ground effect at α=16° (Lee, T., Ko, L. S., 2018)

With substantial experimental works related to reverse delta wings and delta wings in ground effect, our focus has moved to rectangular wing planform, inspired by the Russian “ekranoplan”. A NACA 0012 airfoil was selected for its simplicity and abundance of data. Some experimental work was conducted with PIV to study the flow around the airfoil with a changing ground distance, as illustrated in Figure 1-9. It shows that at small ground clearance (especially around h/c=5%), a recirculation region occurs beneath the leading edge of the airfoil. It is believed that such phenomenon is a unique characteristic of fixed boundary condition that provides a streamwise boundary layer at the ground due to non-slip boundary condition. The ground effect produces a high ram pressure, which pushes the streamwise boundary-layer flow backwards and towards its leading edge, rolling up into the observed recirculation region. This recirculation region would speed up the flow velocity, creating a lower pressure on the airfoil’s lower surface and leading to a diminished enhancement in lift generation.

Figure 1-9 Iso-vorticity contours of NACA 0012 airfoil in ground effect: a) OGE; b) h/c=5%; c) h/c=10%, and velocity distribution around the airfoil: d) OGE; e) h/c=5%; f) h/c=10%

Such discrepancy has been observed and suggested by some previous experimental and numerical studies. For example, Harvey and Perry (Harvey, J.K., Perry, F.J., 1971) pointed out that the flow field measurement obtained with fixed ground is similar to moving ground results, but the quantified results showed noticeable differences. Afterwards, several experimental and numerical works have been provided by groups over the world regarding descending vortex or WIG studies. Therefore, Barber et al (Barber, T., Hall, S., 2006) compared these studies and summarized that the causes of discrepancies result from the dispute over moving and fixed ground boundary conditions, the difference between water surface and rigid ground, and the validation of potential flow simulation with negligible consideration of the viscosity of fluids. In order to find out the actual impact of ground boundary conditions, numerical simulation and experimental studies with PIV was carried out. As illustrated in Figure 1-10, the simulated velocity profile with image and stationary ground boundary condition showed the recirculation region clearly, while it is absent in the cases of slip and moving boundary condition. It matches the PIV results obtained in our lab, shown in Figure 1-9.

Figure 1-10 Simulated velocity profile around NACA 4412 airfoil under different boundary conditions, h/c=2.5%, 푅푒 = 8.2 × 106 (Barber, T., Hall, S., 2006)

In addition, the aerodynamic properties obtained by the simulation was plotted against ground clearance h/c and summarized in Figure 1-11. It showed that for moving ground 퐶푙 continues to increase as ground clearance gets smaller, while the fixed ground results match with moving ground results up to ℎ/푐 = 25% but departs afterwards and drops slightly. For 퐶푑, both boundary conditions showed a decrease until ℎ/푐 = 10% when the drag in the case of stationary ground drops dramatically further but increased slightly in the case of moving ground. Due to

22 equipment constraints, only fixed ground boundary condition was tested for the purpose of this research project. The results will be compared to these computational and experimental data in great detail in later chapters.

Figure 1-11 Lift coefficient, drag coefficient, lift/drag distribution of NACA 4412 airfoil, under different ground boundary conditions (Barber, T., Hall, S., 2006)

1.5 Objectives

Despite the seemingly large past investigations, the details of the vortex flow properties of the vortex and the lift-induced drag as a function of fixed ground distance are still needed. The objectives of this study are to investigate the impact of ground proximity on the wingtip vortices generated by a rectangular NACA 0012 semi-wing both along the tip and in the near field in a wind tunnel at various fixed ground distances. Special emphasis is focused on the changes in the vortex flow properties and the lift-induced drag with the ground distance at a fixed downstream location (2.5 chord length from leading edge), the results of which are obtained from seven-hole pressure probe measurement. Wind tunnel force-balance measurements are also obtained at the same location with a change of ground distances and angle of attack to supplement the seven- hole pressure probe results. The discussion will also include the interaction of the tip vortex with the ground, vortex rebound and the spatial evolution of the vortex structure.

2 Experimental Procedures

A summary of experimental set-up and data analysis procedure will be provided in this chapter. The major part of the experiment was the measurement of flow properties through a seven-hole pressure probe with the semi-wing model placed in a wind tunnel. The structure of the wind tunnel, the schematics of the model set-up, and the measurement mechanism will be introduced first. The study of the aerodynamics properties was backed up with force balance measurement of total lift and drag. The experiment devices will also be introduced in this chapter. Finally, the data analysis procedure will be briefly discussed.

2.1 Flow Facilities

The experimental work was conducted in the J.A. Bombardier low-speed suction-type wind tunnel (Figure 2-1). The wind tunnel is located in the Aerodynamics Laboratory of McGill University. It consists of a contraction section, a test section, a diffuser section, and a power section. The 3푚 contraction section that includes 10푚푚 honeycomb straighteners and four 2 푚푚 screens are responsible for flow conditioning. The contraction ratio of the wind tunnel is approximately 10: 1, and the freestream turbulence intensity is less than 0.005% at 35푚/푠. The test section has a rectangular cross section and measures 0.9푚 (vertical, 푦 axis) × 1.2푚 (horizontal, 푧 axis) × 2.7푚 (streamwise, 푥 axis). The test section is then connected to a 9 푚 long two-stage diffuser section. Finally, a 16 −blade, 2.5푚 diameter vibration isolated fan is located in the power section. The fan, which is controlled by a variable-speed AC motor equipped with an acoustic silencer, powers the wind tunnel. The structure of the wind tunnel is illustrated in Figure 2-1 (Pereira, 2011).

The wind tunnel has the capability to operate at quite high speed, but for our purposes the freestream was set at 푈 = 15 푚/푠, with a turbulence intensity of 0.3%. It gives a Reynold’s number of 2.71 × 105. The flow speed is monitored through a pitot tube connected to a Honeywell DRAL 501-DN differential pressure transducer with a maximum water head of 50 푚푚. The transducer was first calibrated with a water column to verify that the resolution was 97 Pascal/Volt and its response was linear to within 1%.

Figure 2-1 Illustration of J.A. Bombardier wind tunnel. (a) schematic diagram (b) wind tunnel inlet (c) wind tunnel outlet (Pereira, 2011)

2.2 Wing model set-up

For this experiment, a semi-wing model of NACA 0012 airfoil profile was used. It was NC- machined with a chord of 푐 = 28 푐푚 and a semi-span 푏/2 = 50.8 푐푚, yielding an 퐴푅 = 3.6, with a trailing-edge thickness of 1.5 푚푚. Dimensional tolerances on the model were 250 푢푚 on the chord, span and model thickness. The pitch axis is located at the 1/4 -chord location. The origin of the coordinates is located at the wing’s leading edge, with 푥, 푦, 푧 aligned with the streamwise, vertical and horizontal directions respectively. The NACA 0012 wing is selected because of its simplicity and the abundance of data from our lab and other groups.

The semi-wing model is mounted horizontally above an elevated flat plate which simulates the flat ground. The aluminum flat plate is 1.4 푚 long, 1.2 푚 wide and 2.5 푐푚 thick. It is positioned 15 푐푚 above the tunnel floor to eliminate the effects of tunnel boundary layer. The leading edge of the flat plate has an elliptical profile with a major-to-minor axis ratio of 5: 1. Ideally, in order to study flow conditions at extreme ground proximity, a moving ground should be used. However, it is difficult to incorporate such equipment for our experiment, especially due to the long scan hours (6 to 11 hours), which would cause a safety hazard for operating high speed moving belt. Therefore, the elevated plate was designed to minimize the effect of ground boundary conditions. The boundary layer profile over the flat ground is measured with a hot-wire probe shown in Figure 2-2. It indicates that the boundary layer thickness is around 3 푚푚 (1% chord length). The interaction between the boundary layer and the vortex structure in ground effect will be discussed in the following sections.

The ground distance h is defined as the distance between the wing’s trailing edge and the ground surface. It was adjusted by mounting the wing model on a vertical support that has fixing holes for a set of h (from 5% to 100% chord length). The vertical support is covered by a beveled endplate to minimize the effect of wall boundary layer.

The schematic of the experimental set-up is depicted in Figure 2-2, along with an illustration of the wing model including origin location of the coordinate system.

Figure 2-2 Schematics of (a) seven-hole pressure probe measurement set-up, (b) wing model with coordinates, (c) boundary-layer over the elevated flat plate

2.3 Seven-Hole Pressure Probe

The flow field and vortex development were obtained using mean velocity measurements from a miniature seven-hole pressure probe, which has the ability to determine velocity magnitude and direction from a measured pressure differential. The device used for this experiment was designed and manufactured by previous students. As shown in Figure 2-3, the probe consists of a tip, a shaft and a sting. The tip was made from brass and machined to a 30° cone angle, with an outer diameter of 2.7 푚푚. A total of seven holes were drilled on the tip, with one in the center and six arranged on a 2.4 푚푚 diameter circle. The probe shaft is 130 푚푚 long and has the same diameter as the tip. The probe sting has a diameter of 12 푚푚 and a length of 400 푚푚. The sting contains tygon tubing that allow flow to travel from tip through shaft to the pressure transducers.

Figure 2-3 Schematic of seven-hole pressure probe

The tygon tubing has a diameter of 1.6 푚푚 and a length of 550 푚푚, connected to an aluminum transducer array box. The transducer box contained seven Honeywell DC002NDR5 differential pressure transducers with a maximum head of 50 푚푚. The reference pressure is

30 ambient atmospheric pressure measured from inside a fibreglass covered damping unit. The output signals pass through a custom-built signal conditioner, which consists of a seven-channel analogue signal differential amplifier with gain of 5:1 and with external DC offset. The resolution of the pressure transducer was 61 푃푎푠푐푎푙푠/푉표푙푡 on average and their response was linear to 2%. The signal conditioner output was connected to a data acquisition system and monitored with an oscilloscope.

The calibration of the seven-hole pressure probe are provided in detail in Appendix A.

2.4 Traverse Mechanism

The seven-hole pressure probe is mounted on a two degree of freedom custom made traverse, shown in Figure 2-4. The z direction was powered by a Biodine model 2013MK2031 stepper motor and y direction was powered by Sanyo-Denki model 103-718-0140 stepper motor. These two directions of movement were controlled through Labview by a NI PCI-7344 4-axis motion controller which was synchronized with the data acquisition. Such set-up enabled automatic scanning at fixed x location, which was sufficient to capture vortex flow. The measurement of different x locations along the tip and in the near wake of wing was carried out by manually setting the location of traverse mechanism.

Figure 2-4 Traverse mechanism (Pereira, 2011)

2.5 Two-component Force Balance

For the total lift and drag measurement, a force balance set-up was used. It incorporated a different set-up of the wing model due to the structure of the device. The device was located at the bottom of the wind tunnel test section, mounted on a turntable that was installed in the test section floor. The sensor plate of the force balance was supported by two sets of flexures: one parallel to the wing chord (T-direction) and one normal to the wing chord (N-direction). Each cantilever-type spring flexure had a maximum deflection of 4 푚푚. The deflections were independently measured using two Sanborn 7DCDT-1000 linear variable differential transformers (LVDT) whose responses were linear to within 1% in the calibration range used. The resolution of the LVDTs was 120, 93 and 0.6 푁푒푤푡표푛푠/푉표푙푡 in the +N, -N and +T directions, respectively. The calibration of the force balance was summarized in Appendix A. The illustration of the force balance structure is shown in Figure 2-5.

Figure 2-5 Illustration of two-component force balance (Pereira, 2011)

Figure 2-6 Schematic diagram of force balance and wing model set-up

The wing model was mounted vertically on the force balance set-up, above a 0.45 푐푚 × 60푐푚 × 60푐푚 aluminum endplate with a sharp leading-edge in order to eliminate the effects of the wing root. A gap between wing model and endplate of less than 1 mm was used to reduce leakage flow. The endplate was fixed to the tunnel floor and an aerodynamic fairing placed around the shaft in order to isolate it from the flow. The angle of attack was adjusted by the turntable. A flat plate was placed vertically on the test section to mimic the flat ground, with the distance from the wing model adjusted manually. The detailed schematic is shown in Figure 2-6.

2.6 Data Acquisition and Analysis

For both force balance and seven-hole pressure probe measurement, data was acquired using a 16-channel, 16-bit NI-6259 A/D board powered by a Dell Dimension E100 PC. Sensor outputs were connected to the A/D board via a NI BNC -2110 connector box. Given the linearity of the calibration curves, only the mean voltages were recorded and later processed. A schematic diagram for data acquisition method is shown in Figure 2-7.

Figure 2-7 Data aquisition schematic diagram (Pereira, 2011)

Static flow field measurements were obtained at cross-stream planes downstream of the model leading-edge, located between 푥/푐 = 0.5 and 2.5. The size and boundaries of the scan grid were varied to accommodate the growth and trajectory of the vortex. For larger scan areas, an adaptive 1 grid was used. It includes a spacing of ∆푦 = ∆푧 = ” for the vortex region and a spacing of 8 1 ∆푦 = ∆푧 = ” for wake and outer flow. The grid was generated with a MatLab program and 4 imported to the lab computer for automatic scanning. A sample of grid set-up and the corresponding data plot will be presented in later sections. For each grid point, output voltage was sampled at 2000퐻푧 for 5 seconds in order to obtain a reliable average of flow velocity in

36 three directions (u, v, w). Other sampling frequencies and longer sampling times were also tested to ensure convergence, but the differences were less than 0.1 푚푉. This was due to the fact that the tubes connecting to the orifices in the seven-hole pressure probe were quite long and as a result any fluctuations were damped out.

The axial velocity was directly plotted as iso-u contour plot for analysis. The v and w velocity components were used to calculate vorticity with the equation:

휕푣 휕푤 푣푗+1,푖 − 푣푗−1,푖 푤푗,푖+1 − 푤푗,푖−1 휁 = − ( − ) ≈ − ( − ) (1) 푖,푗 휕푧 휕푦 2∆푧 2∆푦 where i = 1, 2, 3, … m and j=1, 2, 3, … n. The m and n are the number of measurement points in z and y directions. A central difference was used on interior data points, while forward and backward differences were used on points on the edges of the measurement grid.

The vorticity values were plotted as iso-vorticity contour plots, and further used to calculate circulation. The circulation was found through Stoke’s theorem by numerically integrating the product of vorticity and area. The core and outer circulation were calculated as follows:

Γ푐 = ∑ ∑ 휁푖,푗 × ∆푦∆푧 , 푟푖,푗 < 푟푐 (2)

Γ푐 = ∑ ∑ 휁푖,푗 × ∆푦∆푧 , 푟푖,푗 < 푟표 (3) where

2 2 푟푖,푗 = (푧푗 − 푧푐) + (푦푗 − 푦푐) (4)

푟표 = 푟(휁 = 0.01휁푖,푗푚푎푥) (5) and the origin of the polar coordinates was set to the vortex center (푧푐, 푦푐) and the core radius, 푟푐 was the radius at which the tangential velocity was maximum. The tangential velocity, 푣휃 was also calculated from polar coordinates as follows:

푣휃푖,푗 = (푣푖,푗 − 푣푐)푠𝑖푛휃 − (푤푖,푗 − 푤푐)푐표푠휃 (6) where 휃 is the polar angle relative to the vortex center.

The 푣 and 푤 are also used to calculate the induced drag coefficient, which will be discussed in detail in later sections. The calculation was based on Maskell’s method (Maskell, 1973) and the procedure is included in Appendix B.

All the numerical and graphic analysis was carried with MatLab programs.

2.7 Experimental Uncertainty

The uncertainty for the quantities reported in this thesis is mostly dependent on the interaction of several parameters. A jitter program as described by Moffat (Moffat, 1982) was used to account for this fact. The sources of error and their associated experimental uncertainty of related parameters, such as the vortex center location, radius, streamwise, vertical, horizontal and tangential velocity, vorticity, circulation and 퐶푑푖, are summarized as follows.

2.7.1 Force Balance Measurements Table 1 Uncertainty of force balance measurement

Quantity Uncertainty (dim) (non-dim) Notes

Experimental parameters

Free-stream velocity (15 m/s) ±0.05 m/s 0.29% (1)

Free-stream turbulence intensity (35 m/s) ±1.75 m/s 0.05% (2)

Model profile (11” NACA 0012) ±0.06 mm 0.03% (3)

Maximum Normalization Uncertainty 0.30% (4)

Force Balance Uncertainty

Resolution in N-direction ±0.09 N ±0.0037 (5)

Resolution in T-direction ±0.05 N ±0.0020 (5)

Angular position (_alpha) ±0.25⁰

Transducer calibration linearity (N-direction) 0.10%

Transducer calibration linearity (T-direction) 0.70%

A/D conversion ±1 mV

Transducer uncertainty in N-direction ±0.120 N ±0.0049 (6)

Transducer uncertainty in T-direction ±0.006 N ±0.0003 (6)

Total uncertainty in N-direction (e_N) ±0.18 N ±0.0061 (7)

Total uncertainty in T-direction (e_T) ±0.051 N ±0.0021 (7)

Maximum C_L Uncertainty: ±0.0061* (8)

Maximum C_D Uncertainty: ±0.0030* (8)

Notes: 1. Based on the pitot probe uncertainty analysis of Barlow et al. (Barlow, J., Rae, W., and Pope, A., 1999) 2. As measured by a hot-wire anemometer 3. Based on 0.0025” accuracy in machining profile 4. Using constant odds combination (Moffat, 1982) 5. Minimum mass sensed by transducer 6. Determined from A/D conversion error and calibration linearity. Details of analysis are as follows:

• A/D error: ±1 푚푉

• Linearity of calibration curve-fit: N-direction = 99.9%, T-direction = 99.3%

• Calibration sensitivity: N-direction = 120 푁/푉 (worst-case scenario), T-direction = 0.6 푁/푉 It results in the C_N and C_T errors as follows:

• C_N error = 0.0049

• C_T error = 0.0003 7. Total uncertainty is determined using constant odds combination of the resolution and transducer uncertainties. 8. Using constant odds combinations error procedure (Moffat, 1982),

퐶퐿 = 퐶푁푐표푠훼 + 퐶푇푠𝑖푛훼 (7) Such that the absolute error is:

2 2 2 휕퐶퐿 휕퐶퐿 휕퐶퐿 Δ퐶퐿 = √( 훿퐶푁) + ( 훿퐶푇) + ( 훿훼) 휕퐶푁 휕퐶푁 휕훼

2 2 2 = √(푐표푠훼휖푁) + (푠𝑖푛훼휖푇) + (퐶푇푐표푠훼휖훼 − 퐶푁푠𝑖푛훼휖훼) (8)

Similarly,

퐶퐷 = −퐶푁푠𝑖푛훼 + 퐶푇푐표푠훼 (9) Such that the absolute error is:

2 2 2 휕퐶퐷 휕퐶퐷 휕퐶퐷 Δ퐶퐷 = √( 훿퐶푁) + ( 훿퐶푇) + ( 훿훼) 휕퐶푁 휕퐶푁 휕훼

2 2 2 = √(−푠𝑖푛훼휖푁) + (푐표푠훼휖푇) + (−퐶푁푐표푠훼휖훼 − 퐶푇푠𝑖푛훼휖훼) (10)

2.7.2 Seven-Hole Pressure Probe

Table 2 Uncertainty of seven-hole pressure probe measurement

Quantity Uncertainty (dim) (%) Notes

Position Measurement

Angular position ±0.0⁰ 0.20

Traverse y position ±0.254 mm 0.09

Traverse z position ±0.762 mm 0.27

Traverse x position ±0.254 mm 0.09

Maximum Normalization Uncertainty: 0.36 (1)

Grid resolution

y coordinate (spanwise) ±3.175 mm 1.14 z coordinate (transverse) ±3.175 mm 1.14

Total Uncertainty in vortex center location: (2) y coordinate ±3 .241 mm 1.16 z coordinate ±3.325 mm 1.19

Total Uncertainty in vortex radius: 2.319 mm 0.83 (3)

Equipment parameters

Free-stream velocity (15 m/s) ± 0.05 m/s 0.29 (4)

Free-stream turbulence intensity (35 m/s) ± 1.75 m/s 0.05 (5)

Model profile (11” NACA 0012) ± 0.06 mm 0.03 (6)

Maximum Normalization Uncertainty 0.30 (1)

Probe Measurement

Reference pressure - 0.13

Transducer accuracy - 0.25 (7)

Transducer sensitivity ±0.032 mm H2O 0.18 (8)

Transducer calibration (linearity) - 0.02 (8)

Table 3 Uncertainty of data acquisition and calculation

Quantity Uncertainty (dim) (%) Notes

Signal Processing

Amplifier reference voltage 0.05

A/D Conversion

16 bit A/D conversion of 3.5V signal ±1 mV 0.02 (9)

Total Pressure Uncertainty: ±0.06 mm H2O 0.34 (1)

Data Reduction

2nd order calibration grid interpolation - 1.46 (10)

Filtering 0.5 (11)

Total Data Reduction Uncertainty: 1.54 (1)

Total Uncertainty in Velocity Fields: u velocity ±0.475 m/s 2.8 (12) v velocity ±0.254 m/s 1.5 (12) w velocity ±0.254 m/s 1.5 (12) v_theta velocity ±0.359 m/s 2.1 (13)

Vorticity Calculation

2nd order finite differences ±0.762 Hz 3.85 (14)

Uncertainty in Vorticity Field 8.21 (15)

Total Uncertainty in Vorticity Field: 9.07 (1)

Total Uncertainty in Circulation: 11.14 (16)

Notes: 1. Using constant odds combination (Moffat, 1982) 2. Total vortex center uncertainty is determined using constant odds combination of grid resolution, traverse position and angular position. 3. Total core radius uncertainty is determined using one-half (radius) of constant odds combination of y and z coordinate uncertainties. Since both the core radius and outer radius are dependent on the same parameters, it is assumed that this also represents the uncertainty of the outer radius. 4. Based on the pitot probe uncertainty analysis of Barlow et al. (Barlow, J., Rae, W., and Pope, A., 1999) 5. As measured by a hot-wire anemometer 6. Based on 0.0025” accuracy in machining 7. Taken from manufacturer specifications 8. Determined from pressure transducer calibration and transducer sensitivity. Details of analysis are as follows:

• Transducer sensitivity (rated): ±5 푚푉

• Linearity of calibration curve-fit: 99.8%

• Calibration sensitivity (worst-case scenario): 62.2 푃푎/푉 9. Based on standard deviation of instantaneous readings taken from a 3.5 푉 input at a sampling frequency of 500 퐻푧 over 10 푠. 10. Calculated from a sample calibration data set, from the average difference between adjacent measurement points. 11. Determined from the average difference found in using a 25 − 푝푡 Gaussian smoothing field.

12. Determined using jitter approach with data reduction uncertainty, pressure uncertainty and normalisation uncertainty as variables. 13. Total tangential velocity uncertainty is determined using constant odds combination of 푣 and 푤 velocity uncertainties. 14. Taken as the average error incurred while using a 2nd order finite differences scheme as opposed to integration or cubic spline curve-fitting 15. Determined using jitter approach with velocity field (v and w) and positional (y and z) uncertainties as variables. 16. Determined using jitter approach with vorticity field, vortex center and core radius uncertainty as variables.

3 Ground Effect on Aerodynamic Characteristics

The force balance set-up were utilized to measure normal forces exerted on the wing model within a range of angle of attack (0° < 훼 < 16°, with an increment of 1°) and ground distance (5% < ℎ/푐 < 100%, where ℎ is the distance from the wing’s trailing edge to the ground and 푐 is the chord length). Detailed experimental set-up and schematics were explained in Section 2.2.

The normal forces measured with force balance set-up were used to calculate 퐶퐿(lift coefficient) and 퐶퐷(drag coefficient), which are plotted with respect to ℎ/푐 and 훼 as summarized in Figure 3- 1. These plots could be used to explain the impact of ground clearance on aerodynamic characteristics of the wing model.

Figure 3-1 a) shows that for 훼 > 3.5° and ℎ/푐 < 60% there is a clear increase in 퐶퐿 with reducing ℎ/푐 represented by the shifting up of the whole curve, which is mainly a result of chord dominated ground effect. Chord dominated ground effect results from the reduction of flow passage under the wing. As ℎ/푐 gets smaller, higher ram pressure is obtained, resulting in higher

퐶퐿. At ℎ/푐 = 60%, the 퐶퐿 versus 훼 curve matches the result of out of ground effect, which means that ground effect has not taken into effect at this stage. Below ℎ/푐 = 40%, ground effect starts to grow stronger. A 32% and 9% 퐶퐿 increase, for example, is achieved at 훼 = 10° for ℎ/푐 = 10% and 40%, respectively.

The ground proximity also leads to an increased lift-curve slope and a promoted wing stall, which is due to the increase in the adverse 푑푝/푑푥 gradient (pressure gradient) produced by ground effect. As ground effect gets stronger, the adverse pressure gradient on the suction side of the wing is increased, promoting boundary layer separation that leads to wing stall. Although the stalling angle was promoted, from 14° OGE to 9° at h/c=5% for example, the maximum 퐶퐿 increased due to the steeper lift-curve slope. The high ram pressure underneath the wing at extreme ground proximity overpowers the flow separation, leading to the persistent enhancement of lift. It also explained the fact that in ground effect 퐶퐿 did not drop dramatically post stall or even continued to rise.

Figure 3-1 Impact of ground distance on the aerodynamics of the NACA 0012 wing. OGE denotes out of ground effect.

However, for 훼 < 3.5 deg, the presence of the ground produces a smaller 퐶퐿 than its OGE counter part, which can be attributed to the converging-diverging flow passage exhibits underneath the NACA 0012 wing and the resulting unusual suction pressure develops on its lower surface. The lower-than-OGE 퐶퐿 can be avoided by the use of, for example, planar wings.

Figure 3-1 b) shows the drag polar as a function of ℎ/푐. At same 퐶퐿, the ground proximity causes a lowered 퐶퐷. 퐶퐷 here is a combination of 퐶퐷푝 (profile drag coefficient) and 퐶퐷푖 (lift- induced drag coefficient). Smaller ℎ/푐 leads to lower 퐶퐷. For example, a 25% reduction in 퐶퐷 at

퐶퐿=0.665 (corresponding to 훼 = 10°) for ℎ/푐 = 10% is obtained. The observed reduction in 퐶퐷 in ground effect could be attributed to the reduction in 퐶퐷푖 originating from the span dominated ground effect. The span dominated ground effect describes the phenomenon where the wingtip vortices move outboard and diminish in strength and size. 퐶퐷푖, the major contribution to the reduction of 퐶퐷, will be calculated from flowfield measurement and discussed in detail in later sections.

However, as shown in Figure 3-1 c), at same 훼 (<훼푠푠), no significant change in 퐶퐷 as compared to its OGE counterpart was observed for ℎ/푐 > 7.5%. A dramatic increase in 퐶퐷 is observed for all curves post stall, and the promotion of stalling angle matches the result shown in

Figure 3-1 a). A large 퐶퐷 increase exhibited for ℎ/푐 = 5%.

This result is compared to previous results regarding a NACA 0015 airfoil (Tremblay-Dionne,

2018). The 퐶푙, 퐶푑, 퐶푝 and 퐶푙/퐶푑 curves are summarized in Figure 3-2. The trend generally matches the current experiment. However, a few discrepancies occurred. For example, Figure 3- 2 (a) presented that for airfoil flow separation is delayed with decreasing ground distance, which is in contrary to the promoted stall observed with wing model. It could be explained by the fact that finite wings are in strong impact of tip effect. The presence of wingtip vortex and its interaction with the ground likely provided a vibrant flow environment around the wingtip, which results in a disturbance of flow on top of the wing that promoted flow separation. Such phenomenon was not observed for airfoil as the wingtip vortex was absent. In fact, flow separation was delayed by 2° for ℎ/푐 = 5%, which could be a result of complete flow blockage at bottom of the airfoil in extreme ground proximity. For drag, Figure 3-2 (c) showed that the sectional drag coefficient 퐶푑 always increases with reducing ℎ/푐, due to the chord dominated ground effect induced increase in 푑푝/푑푥 > 0 gradient which encourages flow separation and increase the 퐶퐷푝. Therefore, the reduction of 퐶퐷 obtained in this experiment could only be explained by the decrease in 퐶퐷푖.

The lift-to-drag ratio is plotted over angle of attack in Figure 3-1 d). The increase in aerodynamic efficiency could be clearly observed. For example, 퐶퐿/퐶퐷 at 훼 = 10° increased from 10 at OGE to 13.5 at ℎ/푐 = 10%. For ℎ/푐 = 5%, the large 퐶퐷 increase however

퐶퐿 overwhelms the 퐶퐿 increase, rendering a ( )푚푎푥 lower than the ℎ/푐 = 10% case. For angle of 퐶퐷 attack post stall, the aerodynamic efficiency decreases reasonably. Nevertheless, the increase in lift-to-drag ratio is less significant than results shown by ground effect vehicle designers

(Rozhdestvensky, 2006), which could be explained by the wing model design and uncertainties of measurement.

The force balance measurements verified the ground effect produced lift increase and drag reduction. The lift increase can be explained by the high ram pressure due to reduction of flow passage underneath the wing. The majority of drag reduction results from the decrease in 퐶퐷푖. In order to better understand the physics behind the impact of ground effect on aerodynamic characteristics, detailed study of flow filed is necessary and was conducted by experiments with seven-hole pressure probe.

4 Ground Effect on Wingtip Vortex Flow

As described in detail in section 2.4, a seven-hole pressure probe was used to measure velocity components in the vortex flow field at selected locations downstream from the wing’s leading edge, at fixed angle of attack (훼 = 10°) and Reynolds number (푅푒 = 2.71 × 105), within a range of ground distances (5% < ℎ/푐 < 100%).

The angle of attack was set at 10° based on the analysis of aerodynamic characteristics of the wing in ground effect, as summarized in chapter 3. The 퐶퐿 푣푠 훼 plot (Figure 3-1 a)) showed that the increase in lift coefficient was most significant at 훼 = 9° or 10° for ℎ/푐 = 10% to OGE. For

ℎ/푐 = 5%, however, the 퐶퐿 curve showed that stall has occurred, which led to a drop in 퐶퐿 but still persistent at the level of ℎ/푐 = 10%. 퐶퐷 obtained at 퐶퐿 = 0.65, corresponding to 훼 = 10° also showed the maximum decrease in drag (Figure 3-1 c)), resulting in the significant 퐿/퐷 increase at the same angle of attack (Figure 3-1 d)). Therefore, setting angle of attack at 10° would depict the impact of ground effect on aerodynamic characteristics at its maximum for the majority of ground distances, while allowing the observation of the vortex flow generated by the a wing post stall at extreme ground proximity.

Reynold’s number was selected at 2.71 × 105, corresponding to a freestream flow velocity

푈∞ = 15푚/푠. As reviewed in section 1.1, previous experimental and computational analysis on the impact of Reynold’s number on wing tip vortex or flow around airfoil showed that the phenomenon is not strongly dependent on Reynold’s number with 푅푒 > 2 × 105. Although at low to medium Reynold’s numbers the aerodynamic properties and characteristics are sensitive of tip condition, surface roughness and boundary layer flow separation, at reasonably high Reynold’s number, distinctive parameters used to study wing tip vortex strength, size and trajectory seem stabilized. Therefore, the freestream velocity was selected to best simulate the flight condition of ground effect vehicles, under the constraints of operation safety and endurance.

The minimum ground distance tested was ℎ/푐 = 5%, which is representative of extreme ground proximity. Lower ground clearance would cause some mechanical difficulties for both wing set-up and seven-hole pressure probe measurement. The maximum ground distance tested

51 was ℎ/푐 = 100%, as previous research has shown that the impact of ground effect at this height is negligible, so it could be identified as out of ground effect case (OGE).

An adaptive grid was used for scanning each 푥/푐 location, as illustrated in Figure 4-1. A range of resolution was used for the complete scan: ∆푦 = ∆푧 = 1/16" at vortex core; ∆푦 = ∆푧 = 1/8" at main vortex and wake regions; ∆푦 = ∆푧 = 1/4" at regions outside the vortex structure. The grid size was selected to maximize accuracy of measurement and minimize scanning time and energy consumption.

휁푐

푈∞ 푦/푐

푧/푐 Figure 4-1 A sample of adaptive grid plotted over iso-vorticity contour at h/c=100%

The three velocity components (푢, 푣, 푤) were used to calculate parameters that describe the vortex flow, which are discussed in detail in this chapter.

4.1 Effect of ground clearance on vortex flow properties in the near wake region

In order to understand the impact of ground effect on vortex structure and further explain the change in aerodynamic characteristics, the first set of experiment was done at 2.5 chord length downstream from the leading edge of the wing model at ℎ/푐 from 5% to 100%. The three components of the velocity vectors were analyzed and plotted as iso-vorticity and iso-axial flow plots, which will be discussed in this section.

The ground effect on the normalized iso-vorticity (휁푐/푈∞) and iso-axial flow (푢/푈∞) contours of the wingtip vortex at 푥/푐 = 2.5 for ℎ/푐 = 5% − 100% is displayed in Fig. 4-3. The Δ푤 Δ푣 streamwise vorticity (휁 = − ) is calculated from the crossﬂow 푣, 푤-measurements by using Δ푦 Δ푧 a central differencing scheme to evaluate the derivatives.

At h/c =100% (Fig.4-3 (a)), the iso- 휁푐/푈∞ contour is almost identical to the free stream results of Lee and Choi ( Lee, T. and Choi, S., 2015) and is, therefore, considered as the OGE case. A direct comparison of current experiment and previous lab results of the same wing model is included in Figure 4-2, which shows that the structure and strength of the vortex matches well. Figure 4-4 shows the normalized tangential and axial velocity distributions across the vortex center at selected ℎ/푐. It could be observed from both the iso-vorticity contour plots and the tangential velocity distribution that at ℎ/푐 = 100% that the inner ﬂow of the wingtip vortex becomes nearly axisymmetric at 푥/푐 = 2.5 (i.e., with the maximum and minimum tangential velocity matches, i.e. 푣휃푚푎푥 ≈ |푣휃푚푖푛| ; see Fig.4-4 (a)). The axial velocity distribution shows that the wingtip vortex has a jet-like flow out of ground effect and for most ℎ/푐’s, which also confirms that the vortex is axisymmetric for OGE case.

The circulation is obtained by using the Stokes Theorem by summing the vorticity multiplied with the incremental area of the measuring grid. The ratio of core circulation to total circulation

(i.e.,Γ푐/Γ표 ) attains a value of 0.698 as compared to 0.71 of Lamb’s vortex. The total circulation

Γ표 of the tip vortex at ℎ/푐 = 100% also translates into a C퐿 of 0.656 (via the 퐿 = 휌푈푏’Γ표 expression where b’ is the center distance between the two trailing vortices) as compared to 0.665 measured directly with the force balance. Note also the presence of a separated wake region behind the wing which continues to feed a small amount of vorticity to the tip vortex. This

53 wake ﬂow region (Fig.4-3 (l)) also imposes a non-negligible impact on the vortex behavior as it moves toward the ground. By contrast, no wake vorticity is presented in the study of a descending vortex pair.

휁푐

푈∞

Figure 4-2 Comparison of iso-vorticity contours out of ground effect: a) current experiment at h/c=100%; b) previous experiment, reproduced from ( Lee, T. and Choi, S., 2015)

Figures 4-3 (b)–4(k) show that in ground effect the vortex remains concentrated (except for the ℎ/푐 = 5% case) but moves outboard with reducing ℎ/푐 (indicative of an increased effective aspect ratio AR eff and presumably a reduced C퐷푖). The spanwise movement of the vortex center z푐 is denoted by the red dashed line. The iso-휁푐/푈 contours also remain basically unchanged as compared to the OGE case for ℎ/푐 > 40% (Figs. 4-3 (b)–4(d)) but have a slightly reduced core strength. Meanwhile, only a minor change in the vertical position of the vortex center y푐 is observed for ℎ/푐 > 40%. For ℎ/푐 < 40%, a descending movement and a rebound was observed, which will be explained in detail in later sections. The vortex center (i.e., z푐 and y푐) is identiﬁed by the location of peak vorticity 휁푝.

In near ground and extreme ground proximity, the presence of the stationary ground, however, leads to a more complicated vortex system as compared to those observed in Harvey and Perry’s work. An illustration of Harvey and Perry’s experimental result is shown in Figure 1-3. In

54 ground effect, there is an increase in the longitudinal boundary layer (developed over the elevated flat plate), as can be seen from the spanwise view of the boundary layer (as shown in Figs. 4-3 (m) – 4-3(v)) in comparison with the boundary layer thickness measured as 4 푚푚 with hot wires, shown in chapter 2. The boundary layer thickens due to the ram pressure produced 푑푝/푑푥 gradient, which further complicates the vortex development close to the ground. It should be noted that in Harvey and Perry’s experiment a small remnant of the wind tunnel boundary layer also exhibits above the moving floor due to the insufficient bleed. Although the remnant of longitudinal boundary layer in their study was negligible, the thickened boundary layer in this experiment and its interaction with the wing model and wake flow result in a unique phenomenon.

For 12.5% < ℎ/푐 < 40% (Figs. 4-3(e) – 4-3(h)), there exhibits a like- sign or corotating ground vortex (GV), relative to the primary vortex (PV) or tip vortex. The formation of ground vortex could be traced back to the interaction of wing model and the longitudinal boundary layer, which will be discussed in detail in section 4.4. The corotating GV moves closer to the PV and merges with it as the ground is approached, leading to an increased vortex strength compared to the OGE case. Note that the wake ﬂow generated behind the wing also continues to feed vorticity to the boundary layer and, subsequently, to the GV and PV. The existence of the spanwise boundary and the formation of the ground vortex and its interaction with the primary vortex for

12.5% < ℎ/푐 < 40% can be further illustrated from the iso-푢/푈∞ contours presented in Figure 4-3 (m) – 4-3(s).

Figures 4-3(h) – 4-3(k) and 4-3(s) – 4-3(v) further reveal that for 5% <= ℎ/ 푐 < 12.5% there is the presence of an opposite-sign or counter-rotating SV, which could be caused by the spanwise boundary layer (induced by the crossﬂow of the primary vortex, as observed by Harvey and Perry). The impact of the presence of the opposite-sign SV in close ground proximity on vortex flow parameters and trajectory will be discussed in later sections. In short, in ground effect the vortex strength decreases slightly (compared to the OGE value) with reducing ℎ/푐 for ℎ/푐 > 40%, followed by a sharp increase for 12.5% < ℎ/푐 < 40% (as a result of the merger of the primary vortex and the co-rotating ground vortex), and starts to drop drastically

55 with a further decrease in ground distance for h/ c < 12.5% (due to the presence of the counter- rotating SV).

(a) (l) 0.2 TV Wake 푦/푐 푦/푐 0 Vorticity -0.2 휁푐 Wake -0.4 푈 -0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 - (b) (m) 푧/푐 푧/푐

TV Wake 푦/푐 0 푦/푐 0 Vorticity -0.2 -0.2 -0.4 fixed ground -0.4 BL Wake -0.6 -0.6

-0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 (c) 푧/푐 (n) 푧/푐

TV 0 Wake 0 푦/푐 Vorticity 푦/푐 -0.2 -0.2 푢

푈 -0.4 -0.4 - 0.6 - 0.6 -0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐 (d) (o) Wake TV 0 푦/푐 Vorticity 푦/푐 0 -0.2 -0.2

-0.4 -0.4

-0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐

Figure 4-3 Ground effect on (a)-(k) iso-c/U and (l)-(v) iso-u/U contours at  = 10o and x/c = 2.5. (a) h/c = 100% or OGE, (b) h/c = 60%, (c) h/c = 50%, (d) h/c = 40%, (e) h/c = 30%, (f) h/c = 20%, (g) h/c = 15%, (h) h/c = 12.5%, (i) h/c = 10%, (j) h/c = 7.5%, and (k) h/c = 5%. OGE denotes out of ground effect. TV, GV, PV and SV denote tip vortex, ground vortex, primary vortex, and secondary vortex, respectively. BL rollup denotes boundary- layer rollup.

Wake (f) PV (q) 0 Vorticity 0 GV 푦/푐 BL rollup

푦/푐- 0.2 -0.2

-0.4 -0.4 -0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐 (g) 0.2 (r) 0.2 GV 휁푐

푦/푐 0 푦/푐 0 푈 SV -0.2 -0.2

-0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐

(h) (s) 0.2 0.2 SV 푦/푐 0 푦/푐 0 SV -0.2 -0.2

-0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐

(i) (t) 0.2 Wake 0.2 Vorticity SV 푦/푐 0 푦/푐 0 SV 푢 -0.2 -0.2 푈 -0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐 (u) (j) 0.2 PV 0.2 푦/푐 0 SV 푦/푐 0

-0.2 -0.2

-0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐

(k) (v) 0.2 0.2

푦/푐 0 푦/푐 0

-0.2 -0.2 -0.8 0.4 0 -0.4 -0.8 0.4 0 -0.4 -0.8 푧/푐 푧/푐

Figure 4-4 Normalized tangential and axial velocity distributions across the vortex center at selected h/c

4.2 Impact of ground proximity on vortex flow properties

The ground proximity on the peak tangential velocity (푣휃푝), core radius (푟푐), peak vorticity

(휁푝), core circulation (Γ푐), total circulation (Γ표), core axial velocity (u푐) and vortex center (y푐, z푐), of the vortex is summarized in Figs. 4-5 (a)–4-5 (d).

Figure 4-5 Ground effect on wingtip vortex flow parameters at x/c=2.5

Figure 4-5 (a) shows that the 푣휃푝 decreases ﬁrst for ℎ/푐 > 40%, followed by a rise for 40% < ℎ/푐 < 12.5% (due to the presence of the corotating GV and its vorticity addition to the vortex core of the primary vortex), and begins to decrease sharply for ℎ/푐 < 12.5% (due to the negative vorticity cancelation from the counter-rotating SV in close ground proximity). Note

59 that for asymmetric vortices, circumferentially averaged 푣휃푝 is used. The variation in 푟푐 (Fig. 4-5

(a)) and the change of vortex strength with ℎ/푐 also generally follows that of 푣휃푝. The core diameter 2푟푐 is determined as the distance between 푣휃푚푎푥 and 푣휃푚푖푛 (see Fig. 4-4(a)). The observed 푟푐 reduction can be attributed to the fact that in ground effect the lifting line is closer to the ground which leads to a smaller vortex size. As ℎ/푐 gets smaller, the appearance of GV and

SV causes the observed increase (or decrease) in 푟푐 in near (or extreme) ground condition.

The addition and cancellation of the vorticity to the tip vortex from the GV and SV can also be reﬂected from the peak vorticity presented in Fig. 4-5(c). The increase (or decrease) in 휁푝 in near

(or extreme) ground condition can be clearly seen. A slight decrease in 휁푝 is observed from

ℎ/푐 = 100% to ℎ/푐 = 20%, while the curves of Γ푐 and Γ표 show a slight increase. The decrease in 휁푝 could be explained by the interaction of wake region with ground boundary layer, obstructing the vorticity feeding from wake region. The circulation on the other hand is a more representative of vortex strength and lift. The slight increase observed could be explained by the presence of ground vortex, which aided the size and strength of the whole vortex region. From ℎ/푐 = 20% to ℎ/푐 = 10%, both peak vorticity and circulation grow sharply, by 40% for instance for 휁푝. As shown in Figure 4-3, at this ground distance the strength of GV is at its maximum, resulting in significant increase in the vortex strength. As the counter rotating secondary vortex starts to join the vortex structure below ℎ/푐 = 10%, the strength of the vortex begins to decrease sharply. At extreme ground proximity, GV has already merged with primary vortex and SV is the dominating influence on peak vorticity and circulation. At ℎ/푐 = 5%, the fact that the wing already stalled at 훼 = 10° also means a diffused vortex center, leading to further diminished vortex strength. Note that as shown in chapter 3, the lift increased continuously as the wing model approaches the ground, which is contradictory to the circulation plot. It indicates that the direct relationship between circulation to lift is not valid in ground effect due to the change of boundary conditions. At extreme ground proximity, the dominating source of lift comes from ram pressure, which is not shown in the strength of wing tip vortex.

At last, the axial core velocity 푢푐 of the tip or primary vortex ﬂow as a function of ℎ/푐 is provided in Fig. 4-5(d). The normalized axial velocity distribution across the vortex center is also

60 given in Fig. 4-4(b). The 푢푐 of the tip or primary vortex was always found to be jet-like for ℎ/푐 ≥ 7.5%. At ℎ/푐 = 5%, the core axial velocity become wake-like as a result of the interaction with the boundary layer ﬂow generated on the ﬁxed ground.

The presence of the opposite-sign SV in close ground proximity not only renders a reduced strength of the primary or tip vortex (Fig. 4-5 (c)) but also causes the primary vortex to rebound from the ground. The 푧푐 and 푦푐 of the vortex center are summarized in Fig. 4-5 (b), which demonstrates that the vortex moves further outboard with reducing ℎ/푐 (indicative of an increased AR eff which implies that their inﬂuence on the wing would be reduced) and also vertically downward for ℎ/푐 > 10%. For ℎ/푐 < 10%, the vortex center is, however, undergoing an upward bouncing. The vortex rebound as well as the change in vortex strength are depicted more clearly in Figure 4-6. The mechanism of vortex rebound is believed to result from the presence of the counter-rotating secondary vortex. However, due to the presence of ground vortex, the rebound was suppressed and thus not as significant as that observed by previous studies in descending vortex, summarized in chapter 1.

Figure 4-6 Zoom-in iso-vorticity contours of the tip vortex at x/c=2.5: (a) h/c=100%, (b) h/c=20%, (c) h/c=10%, and (d) h/c=5%

4.3 Spatial Evolution of the Vortex

To further examine the ground proximity on the tip vortex, the streamwise formation and development of the vortex both along the tip (for 푥/푐 < 1) and in the near ﬁeld (for 1 < 푥/푐 ≤ 2.5) for ℎ/푐 = 100%, 20%, 12.5%, and 7.5% are presented in Figures 4-7 and 4-8. Figure 4-7 shows the iso-vorticity contours plotted over the semi-wing model in a 3D view, with detailed 2D plots for each 푥/푐 location indicating peak vorticity values. Note that only part of the wing model was shown for better illustration of the vortex. Main vortex (MV), trailing vortex (TV) and shear-layer vortices (SLV) could be identified clearly and were labelled accordingly. Ground vortex (GV) was the smaller and irregular-shaped vortex that grew besides MV and TV, which typically has a normalized peak vorticity from 2 to 4. Secondary vortex (SV) was the vortex with negative vorticity that grew later than GV. The negative vorticity, typically with a normalized peak value from -3 to -6, indicated that it was counter-rotating. GV and SV could be identified clearly on 2D plots in Figure 4-7 and were labelled accordingly on both 2D and 3D plots. Figure 4-8 presents iso-axial velocity plots in the near wake region.

The formation and growth of the shear-layer vortices (SLV, resulting from the rolling up of the shear layers separate from the wing’s lower surface) and main vortex (MV) along the tip, and their development into a tip vortex in the near ﬁeld persistently exhibit in ground effect except for their associated strength. As can been seen, the normalized peak vorticity (휁푝) decreases slightly with increasing 푥/푐 for 푥/푐 > 1. Overall, the ground proximity leads to a higher vorticity level of the SLV, attributing to the chord dominated ground effect (CDGE) contribution. That is, the smaller ℎ/푐 the higher 휁푝 of the SLV becomes. By contrast, the MV has a slightly lower value with reducing ℎ/푐. The presence of GV in the near ﬁeld for ℎ/푐 = 20% and 12.5% persists, leading to a higher 휁푝 of the tip vortex. The smaller h/c the higher 휁푝 becomes. It is of interest to note that in close ground proximity (h/c = 7.5%; Fig. 4-7 (d)) the ground vortex continues to exist, contributing to the growth of the tip vortex for 푥/푐 < 2.5. At

푥/푐 = 2.5, the counterrotating secondary vortex dominates and results in a lower 휁푝 and subsequently a reduced vortex strength as well.

As shown in Figure 4-7 (c) and (d), the ground vortex continuously grows starting from 푥/푐 = 0.5. It appears as bubbles or recirculation regions in the ground boundary layer beside the wing tip, which is believed to result from the recirculation region at leading edge of the airfoil, shown in chapter 1. The formation and growth of the TV, GV, and SV, including the presence of the separated wake ﬂow behind the trailing edge of the rectangular wing and the rolling up of the spanwise boundary layer, can also be elucidated by the near-ﬁeld iso-푢/푈∞ (Figure 4-8).

Comparing the experimental results along the wingtip and previous data regarding NACA 0012 airfoil in ground effect, it is suggested that for a wingtip vortex travelling downstream at fixed ground distances (like in the case of wing-in-ground effect vehicle) the formation of the ground vortex was originated from the downstream progression of the so-called “spanwise ground vortex filament (SGVF)” exhibited along the leading-edge region of the wing. The existence of the SGVF can be demonstrated from the PIV measurement of the flow field around a NACA 0012 airfoil positioned at 훼 = 9° with ℎ/푐 = 5% (Fig. 1-9). It reveals that for the stationary ground there was the appearance of a recirculation region or ground vortex beneath the airfoil’s leading edge region in close ground proximity, which persisted in the spanwise direction, leading to the formation of the SGVF. The appearance of the recirculation region developed beneath the 2-D airfoil’s leading edge can be attributed to the ground effect-produced ram pressure, which pushed the streamwise boundary-layer flow backwards and towards its leading edge and rolling up into the observed recirculation region. Preliminary results with moving ground showed that the recirculation region was absent for the moving ground condition.

(a)

Figure 4-7 Spatial progression of iso-fc/U contour of the tip vortex both along the tip and in the near field at a510 deg. (a) h/c=100%

(b)

Figure 4-7 Spatial progression of iso-fc/U contour of the tip vortex both along the tip and in the near field at a510 deg. (b) h/c=20%

(c)

Figure 4-7 Spatial progression of iso-fc/U contour of the tip vortex both along the tip and in the near field at a510 deg. (c) h/c=12.5%

(d)

Figure 4-7 Spatial progression of iso-fc/U contour of the tip vortex both along the tip and in the near field at a510 deg. (d) h/c=7.5%

ℎ = 100% 푐

푢푐 = 0.6 푢푐 = 0.7 푢푐 = 0.6 푢푐 = 0.6

푢푐 = 0.7 푢 = 1.1 푢 푢푐 = 1.2 푐 푈 ℎ = 20% 푐

Ground

ℎ = 12.5% 푐

푢푐 = 0.7

ℎ = 7.5% 푐 푢푐 = 1.1 푢푐 = 1.1 푢푐 = 0.7

Figure 4-8 Spatial progression of the iso-푢/푈∞ contour of the tip vortex in the near field at selected h/c

The spatial evolution of the vortex structure from MV and SLVs to TV with GV and SV was shown most clearly at h/c=5% (Figure 4-7 (d)). It showed that GV appeared as early as x/c=0.4, which could hypothetically be related to SGVF along the wing’s leading edge discussed previously. According to such mechanism, at the wing’s tip, the SGVF was swept downstream, developing into the co-rotating ground vortex over the stationary ground surface. In reality, the interaction between the streamwise boundary layer and the ground vortex distorted the ground vortex, leading to an irregular shape as shown in Figure 4-7 (a-d). Figure 4-7 (d) also depicts the escapement of the ground effect-produced ram-pressure fluid flow from the region beneath the wing and its rollup into the counter-rotating secondary vortex, as well as the formation of the main vortex and shear-layer vortices along the tip for 푥/푐 ≤ 1. The shear-layer vortices fed vorticity to the main vortex and eventually merged with it. Downstream the trailing edge of the rectangular wing, the main vortex evolved into the tip vortex, while the ground vortex and secondary vortex became identifiable. As the vortex structure moves further downstream, GV starts to co-rotate with TV and SV start to move upward pushing TV and leading to a vortex rebound. Figure 4-9 shows a conceptual sketch of this complete mechanism, with SGVF developed into GV, spanwise boundary layer rolling into SV and the progression of MV+SLVs to TV.

Figures 4-7 and 4-9 further indicate that, regardless of the ground boundary conditions, along the wing tip the presence and behavior of the main vortex and shear-layer vortices and their interaction were found to remain largely similar to the OGE case (except for the ground effect produced larger 휁푝), implying the minimization of the flow separation over the wing’s upper surface at the wing tip region in close ground proximity. In other words, away from the wing tip there exhibited a larger flow separation and thus wake region, resulting from the ground proximity-caused increase in the adverse 푑푝/푑푥 gradient. The minimization of flow separation at the wing tip region may lead to the generation of a larger lift at the tip, affecting the lateral control of the WIG craft or aircraft flying close to the ground.

Figure 4-9 Conceptual sketch of the existence of the spanwise ground vortex filament (SGVF) and its downstream development into the ground vortex, and the formation of SV, and the evolution of MV, SLV and TV

5 Ground Effect on 푪푫풊

퐷푖 In this section, the lift-induced drag coefficient 퐶퐷푖 = 1 is computed by using the Maskell 휌푈2푆 2 wake integral model and is also compared with inviscid-flow predictions as well as the total drag coefficient 퐶퐷 from the force balance. Equation (11) expresses the lift-induced drag 퐷푖 calculation based on the Maskell method

1 1 1 퐷 = 휌 ∬ 휓휁푑푦푑푧 − 휌 ∬ 휙휎푑푦푑푧 − 휌 ∬ (1 − 푀2 )(Δ푢)2 푑푦푑푧 (11) 푖 2 ∞ 2 ∞ 2 ∞ ∞ 푆휉 푆1 where 휁 is obtained from 푣, 푤-crossﬂow measurements, 휓 and 휙 are stream function and 휕푣 휕푤 velocity potential, and 휎 = + . Detailed calculation procedure is included in Appendix B. 휕푦 휕푧

Figure 5-1 shows that there is a small rise in 퐶퐷푖 compared to the OGE value, reaching a local peak at around ℎ/푐 = 30% (with a 17% 퐶퐷푖 increase), and begins to drop for ℎ/푐 < 30%. For

ℎ/푐 < 12.5%, the 퐶퐷푖 becomes smaller than the OGE value (due to the presence of the counterrotating SV). A 43% 퐶퐷푖 reduction at ℎ/푐 = 5% is obtained. The increased 퐶퐷푖 for

12.5% < ℎ/푐 < 40% is caused by the corotating or like-sign GV. The slight increase in 퐶퐷푖 for ℎ/푐 < 40% can be attributed to the ground effect-produced increase in the ram pressure and the subsequent stronger shear layers separating from the wing’s lower surface. Note that only a small increase in AR eff and/or outward movement of 푧푐 for ℎ/푐 > 40% exhibits. The 퐶퐷푖 computed also contributes to 28.5%, 55%, and 66% of the 퐶퐷 measured with a force balance for ℎ/푐 = 5%, 10%, and 20%, respectively. As a comparison, Tuck and Standingford (Tuck, E., and Standingford, D., 1996,, 1996) found that both lift and induced drag increase dramatically as the ground clearance is reduced. On the other hand, Barber et al. (Barber, T., Hall, S., 2006) report that at 훼 = 4 ° the induced drag is reduced by 75% for ℎ/ 푐 = 5%.

The 퐶퐷푖 values in ground effect predicted by Eq. (12), developed by Prandtl for inviscid ﬂow (Mantle, 2016), are also included in Fig. 5-1 to serve as a comparison

(1 − 휎)퐶2 퐶 = 퐿 (12) 퐷푖 휋퐴푅

ℎ ℎ where 휎 = (1 − 1.32( ))/(1.05 + 7.4( )) is a modifier called the influence coefficient which 푏 푏 accounts for the effect of ground proximity, and ℎ is the height of the wing (at the quarter chord point) above the ground, and 푏 is span of the wing. The 퐶퐷푖 computed via the Maskell Method has a similar trend of the inviscid-flow prediction (i.e., Eq. (11)) but has a higher magnitude than its inviscid-flow counterpart. The 퐶퐷푖 in ground effect is also estimated via

2 퐶퐿,푂퐺퐸 퐶퐷푖 = (13) 휋푒퐴푅푒푓푓 where 퐶퐿.푂퐺퐸 the OGE 퐶퐿 value at 훼 = 10 °, e is known as the span efﬁciency representing the 푏′ non-ellipticity of the spanwise lift distribution and is set at 0.9, 퐴푅 = at each h/c and 푏′ is 푒푓푓 푐 the distance between the two vortices. The 퐶퐷푖 determined via Eq. (13) follows more the trend of those predicted by Eq. (12) but has a higher value than that computed by Eq. (11). In close ground proximity, the 퐶퐷푖 predicted by Eq. (13) also exhibits a much higher value that those computed by using Eq. (11).

Figure 5-1 Impact of ground distance on lift-induced drag coefficient

6 Conclusion

The objective of this research project is to study the impact of ground vortex on wing tip vortex generated by a rectangular NACA 0012 semi-wing, in order to better understand the vortex flow and aerodynamic characteristics that could aid the design of new ground effect vehicles.

Force balance measurement were conducted to obtain 퐶퐿 and 퐶퐷 at a range of angles of attack and ground distances. Lift and drag coefficients were plotted against 훼 and ℎ/푐. 퐿/퐷 was also calculated and plotted against 훼 at selected ℎ/푐’푠. The detailed results could be found in section 4.1. It was suggested that a clear increase in lift was observed as the wing model approached the ground in general, although a slight decrease was observed at very small angle to attack that could be avoided by using wings with planer bottom. The increase in lift was mostly due to chord dominated ground effect, namely the ram pressure resulted from limited flow passage. A decrease in drag was observed as h/c decreases, which was the result of reduction in 퐶퐷푖. The aerodynamic efficiency (퐿/퐷) also showed an impressive increase at the wing gets closer to the ground, proving that ground effect offers an advantage for the flight.

The vortex flow was measured at selected downstream locations from the leading edge (푥/푐) with a seven-hole pressure probe. At 푥/푐 = 2.5, the main vortex appeared to be axi-symmetric, showing that it was fully developed. Therefore, a study of impact of ground effect on vortex properties was carried out at first at this location. Measurement was taken from ℎ/푐 = 100% to ℎ/푐 = 5%, at fixed angle of attack at 10°.

The iso-vorticity contours at 푥/푐 = 2.5 showed that the vortex structure underwent significant change with different ground clearance. A co-rotating ground vortex appeared starting at ℎ/푐 = 20%, which is a result of the interaction between the wing model and ground boundary layer. It was believed this ground vortex could be related to the recirculation region at the leading edge of an airfoil in ground effect. A counter-rotating secondary vortex was also observed at extreme ground proximity, similar to the results from studies related to descending vortex.

The vortex strength was strongly affected by the change of vortex structure and ram pressure. The ground vortex added vorticity to the main vortex, leading to the significant increase in peak vorticity and circulation from ℎ/푐 = 100% to ℎ/푐 = 10%. For ℎ/푐 < 10%, the impact of

73 secondary vortex was no longer negligible. It diminished the vortex strength and led to a phenomenon referred to as vortex rebound.

The center of main vortex moved downward from ℎ/푐 = 100% to ℎ/푐 = 15%, as the wing approached the ground. Afterwards, the vertical location of the vortex centered rose significantly, which represented the vortex rebound. It was believed that the presence of secondary vortex pushed the main vortex upward, preventing its destruction by the ground. In the horizontal direction, the vortex continuously moved outboard, leading to a higher effective AR as ground distance gets smaller.

Seven-hole pressure prove measurements were also taken along the tip and in the near wake of the wing model at ℎ/푐 = 100%, 20%, 12.5%, and 7.5%. Along the tip in ground effect, stronger shear layer vortices were observed as a result of ram pressure, which also aided the increase in peak vorticity of the main vortex in near wake. Besides, the presence of ground vortex was observed as early as 푥/푐 = 0.5 for ℎ/푐 = 7.5%, linking it to the recirculation region at the leading edge of airfoil in ground effect.

퐶퐷푖 was calculated from flow field measurement and compared to theoretical curves. The decrease in 퐶퐷푖 observed as ground distance decreased was a combined result of secondary vortex, horizontal movement of vortex center and limited space for the rolling up mechanism.

The immediate future work is to include the moving ground boundary condition. It was indicated clearly from the vortex flow study that the fixed ground induced unique vortex structure (with GV), leading to discrepancies in aerodynamic characteristics, vorticities and circulation values. A moving ground boundary condition could be realized with fixed wing model on a moving belt. Accurate force balance measurement should be conducted to find out exact differences between the two boundary conditions, in order to offer profound knowledge for future design of ground effect vehicles.

References

Lee, T. and Choi, S. (2015). Wingtip vortex control via tip-mounted half-delta wings of different geometric configurations. ASME Journal of Fluids Engineering, 137(12):1-9. Anderson, E., Lawton, T. (2003). Correlation between vortex strength and axial velocity in a trailing vortex. Journal of Aircraft, 40(4):699–704. Barber, T., Hall, S. (2006). Aerodynamic ground effect: a case-study of the integration of CFD and experiment. International Journal of Vehicle Design. Barlow, J., Rae, W., and Pope, A. (1999). Low Speed Wind Tunnel Testing. Wiley-Interscience, 3 edition. Birch, D., Lee, T., Mokhtarian, F., and Kafyeke, F. (2004). Structure and induced drag of a tip vortex. Journal of Aircraft, 41(5):1138-1145. Birch, D., Lee, T., Mokhtarian, F., and Kafyeke, F. (2004). The structure and induced drag of a tip vortex. Journal of Aircraft 41(5), 1138-1145. Brown, C. (1974). Aerodynamics of wake vortices. AIAA Journal, 11(4):531-536. Chow, J. S., Zilliac, G. G., and Bradshaw, P.,. (1997). Mean and Turbulence Measurement in the near field of a wingtip vortex. AIAA J., 35(10), 1561-1567. Chun, H., Park, I., Chung, K., and Shin, M.,. (1996). Computational and Experimental Studies on Wings in Ground Effect and a WIG Effect Craft. Workshop Proceedings of Ekranoplans and Very Fast Craft, (pp. 38-60). University of New South Wales, Sydney, Australia. Corjon, A., Poinsot, T. . (1997). Behavior of wake vortices with the ground. AIAA Journal, 35(5):849-855. Corsiglia, V., Jacobsen, R., and Chigier, N. (1970). An experimental investigation of trailing vortices behind a wing with a vortex dissipator. Aircraft Wake Turbulence, 229-242. Devenport, W., Rife, M., Liapis, S., and Follin, G. (1996). The structure and development of a wing-tip vortex. Journal of Fluid Mechanics, 312:67-106. Han, C., and Cho, J.,. (2005). Unsteady Trailing Vortex Evolution Behind a Wing in Ground Effect. Journal of aircraft, 42(2) 429-434. Harvey, J.K., Perry, F.J. (1971). Flowfield Produced by trailing vortices in the vicinity of the ground. AIAA Journal, 9(8): 1659-1660. Hsiun, C. M., and Chen, C. K. (1996). Aerodynamic Characteristics of a Two-Dimensional Airfoil with Ground Effect. Journal of Aircraft, 33(2) 386-392.

Joh, C. Y., and Kim, Y. J.,. (2004). Computational Aerodynamic Analysis of Airfoil for WIG (Wing-in-Ground Effect)) Craft. J. Korean Soc. Aeronaut, 32(8): 37-46. Kliment, L.K., Rokhsaz, K. (2008). Experimental investigation of pairs of vortex filaments in ground effect. Journal of Aircraft, 45(2) 622-629. Lee, T. a.-D. (2017). Impact of ground proximity on a slender reverse delta wing. International Journal of Aerodynamics. Lee, T., Huitema, D., and Leite, P. (2018). Ground effect on a cropped reverse delta wing equipped with Gurney flaplike side-edge strips and anhedral. Journal of Aerospace Engineering. Lee, T., Ko, L. S. (2018). Ground effect on the vortex flow and aerodynamics of a slender delta wing. Journal of Fluids Engineering. Mantle, P. J. (2016). “Induced Drag of Wings in Ground Effect. Aeronautical Journal, 120(1234) 1867-1890. Maskell, E. (1973). Progress towards a method for the measurement of the components of the drag of a wing of finite span. RAE TR 72232. Moffat, R. (1982). Contributions to the theory of single-sample uncertainty analysis. Transactions of the ASME. Journal of Fluids Engineering, 104:250-258. Orloff, K. (1974). Trailing vortex wind-tunnel diagnostics with a laser velocimeter. Journal of Aircraft, 11(8):477-482. Pereira, J. L. (2011). Experimental investigation of tip vortex control using a half delta shaped tip strake. Montreal: McGill University. Puel, P., Victor, X. D. S. . (2000). Interaction of Wake Vortices with the ground. Aerospace science and technology, 4, 239-247. Rozhdestvensky, K. (2006). Wing-in-ground effect vehicles. Progress in Aerospace Sciences, 42, 211-283. Spalart, Phillipe R. (1998). Airplane trailing vortices. Annual Review of Fluid Mechanics, 30(1), 107-138. Steinbach, D. (1997). Comment on “Aerodynamic Characteristics of a Two-Dimensional Airfoil With Ground Effect. Journal of aircraft, 34(3):455-456. Tremblay-Dionne, V. a. (2018). Ground Effect on the Aerodynamics. Fluid. Mech. Res. Int. J.,, 2(1) 6-12.

Tuck, E., and Standingford, D., 1996,. (1996). Lifting Surface in Ground Effect. Workshop Proceedings of Ekranoplans and Very Fast Craft (pp. 230-243). Sydney, Australia: University of New South Wales. Wang, Y., Liu, P., Hu, T., and Qu, Q. (2016). Investigation of Co-Rotating Vortex Merger in Ground Proximity. Aerospace Science and Technology, 53 116-127.

Appendix A: Calibration Procedures

The calibration procedures of force balance and seven-hole pressure probe measurement is summarized in this appendix, referenced from documentation of previous research studies carried out using the same equipment (Pereira, 2011).

Force Balance Calibration

The force balance used in this study consisted of two orthogonally set linear velocity differential transformers (LVDT). The LVDTs were set in directions normal (N-direction) and tangential (T-direction) to the wing chord. Figures 2-5 and 2-6 show the set-up of the force balance below the wind tunnel. From these figures, it is apparent that any force acting on the wing was imparted through the mounting shaft, which was mounted on the sensor plate. This sensor plate was supported by two sets of cantilever-type spring steel flexures that deflected as force was applied to the sensor plate. In the N-direction, the flexures had a thickness of 0.9 mm, while the flexures in the T-direction were much thinner at 0.3 mm. These two sets were chosen to account for the fact that the normal force is generally one or two orders of magnitude higher than the tangential force. The sensor plate was also connected to the LVDTs which converted the deflection into a voltage. The advantage of using an LVDT is its precision and high resolution as it is able to sense very small deflections. The LVDT itself consists of three coils and a ferromagnetic core. The central (or primary) coil induces a magnetic field in the core. As the core moves along an axis, it induces a voltage in the secondary coils, much in the same way a solenoid works. The difference in the voltages induced by each secondary coil is the output, which varies linearly with displacement. Accordingly, calibration of the force balance required accurate calibration of each LVDT through its linear range. For the force balance measurements, the wing was mounted at its quarter chord. This unfortunately did not correspond to the center of gravity on the wing. In order to compensate for this imbalance, the wing was mounted on the force balance during all calibrations.

Figure A-1 is a flowchart of the steps involved in the force balance calibration. For the calibration phase, known weights were used to create a tension force along either the normal or tangential directions. The weights were hung over a pulley system and attached to the wing

78 mounting shaft via aluminum wire. The aluminum wire was found to provide sufficient strength in order to support the various weights and remain rigid, while having a minimal weight so as to not overwhelm for low weights. During calibration, careful attention was given to ensuring that the tension was perfectly aligned with the LVDT being calibrated. In order to achieve this, the pulleys were attached to posts which were screwed into holes aligned with the freestream on the tunnel floor. The height of the pulley was first set using both a height gauge and bubble type level to ensure that the tensioned string was level. The pulley height was finely adjusted until the output in the particular direction was maximized. In the yaw direction, the force balance was rotated on the turntable until the output along the axis being calibrated was maximized, and the output along the orthogonal axis remained unchanged as weight was added. This portion of the calibration procedure was found to be the most time consuming. However, once set-up, weights ranging from 10g to 1kg (0.1N to 9.8N) were used to calibrate each LVDT. It was also found that the calibration curve was similar, but not exactly the same in the positive and negative N- directions. Therefore, three calibration curves (two in the N-direction, one in the T-direction) were obtained and are plotted in Figure A-2, showing that the curves are linear to within 1%. The sensitivity of the curves was 120, 93 and 0.6 Newtons/Volt in the positive N-direction, negative N-direction and positive T-direction, respectively.

Figure A-1 Flowchart of force balance calibration process

Figure A-2 Typical force balance calibration curve-fits

Seven-Hole Probe Calibration

Calibration of the seven-hole probe was done, in situ, following the calibration procedures of Wenger and Devenport. Figure A-3 presents flowcharts of the seven-hole probe calibration process and velocity extraction process. In order to reduce interference effects, the probe was calibrated prior to installation of both the wing and three-degree of freedom traverse. The probe was mounted on a custom-built two degree of freedom traverse. The traverse was powered by two Sanyo Denki micro-stepping motors for independent motion in the pitch and yaw directions. During calibration, the probe was positioned at known angles by pitching the probe at an angle α in the x-y plane and rotating the probe about the stem at an angle β. The probe was calibrated over its angular range, −70◦ ≤ α ≤ 70◦ and −70◦ ≤ β ≤ 70◦, in increments of 5◦.

Typically, as shown in Figure 2-3(b), the probe tip consists of one tap at the probe tip (hole 7) and six equally spaced taps (holes 1 to 6) arranged in a ring around this center hole. In a seven- hole probe it is assumed that the hole which registers the highest pressure is most closely aligned with the direction of flow. Accurate flow direction is then determined through comparison of the normalized, velocity invariant pressure coefficients on two faces of the probe. Since the flow is assumed independent of velocity, its direction in 2-D space will then be dependent on only two directional coefficients. Once these directional coefficients are determined, the velocity magnitude can be determined much in the same way it is obtained from a pitot tube. As such, a relative total pressure coefficient (dependent on the maximum pressure sensed), and a relative static pressure coefficient (based on the pressures at the surrounding holes) are used in conjunction with Bernouilli’s equation to determine the velocity magnitude. As a result of this independence in flow velocity, calibration need only be obtained at one flow speed (usually at the expected speed used for the experiment). The advantage of using a seven-hole probe is its ability to provide mean, three-component velocity measurements up to high flow angles (70◦).

In order to extend the probe’s angular range, a sectoring scheme must be used where the form of the four coefficients (two directionality, static pressure coefficient and total pressure coefficient) depends on the holes which have the highest sensitivity to the flow. At low flow angles, the flow is assumed attached over the entire probe tip. The pressure is highest at the center port and is thus used to determine the relative total pressure. The six remaining pressures

81 are then used to determine the average static pressure. The directionality coefficients are assumed to vary with pitch and yaw and are determined from the pressure differences of holes 1 to 6. The difference in pressure over the top of the probe (holes 3 and 4) and the bottom of the probe (holes 1 and 6) determines the pitch angle coefficient, while the pressure difference between hole 2 and hole 5 is used to determine the yaw angle coefficient. The coefficients are expressed accordingly:

1 1 (푝 + 푝 ) − (푝 + 푝 ) 2 3 4 2 1 6 퐶훼 = 푝𝑖푡푐ℎ 푎푛푔푙푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (14) 푝7 − 푝̅

푝2 − 푝5 퐶훽 = 푦푎푤 푎푛푔푙푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (15) 푝7 − 푝̅

(7) 푝7 − 푝푡표푡 퐶푡 = 푡표푡푎푙 푝푟푒푠푠푢푟푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (16) 푝7 − 푝̅

(7) 푝7 − 푝푠푡푎푡 퐶푠 = 푠푡푎푡𝑖푐 푝푟푒푠푠푢푟푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (17) 푝7 − 푝̅ 6 푝 푝̅ = 푎푣푒푟푎푔푒 푠푡푎푡𝑖푐 푝푟푒푠푠푢푟푒 = ∑ 푖 (18) 6 푖=1 where 푝푛 is the pressure sensed at port n. At high flow angles (> 30◦), the flow becomes separated on the lee side of the probe. The pressure sensed by ports located in the wake of the separated flow, will then be insensitive to changes in pitch and yaw. As such, including their values in the calculation of the four coefficients would lead to skewed and erroneous results.

To avoid this situation, at high incidences, only the four highest pressures are used to calculate each of the four coefficients. In this case, the hole at which pressure is maximum, 푝푖 is first determined and used to calculate the total and static pressure coefficients. The flow angle is then determined from the difference between the pressures of the 3 holes adjacent to the maximum 푡ℎ pressure port, namely, the center port, 푝7, the port next to the 𝑖 port in the clockwise direction, pcw, and the port next to the 𝑖푡ℎ port in the counter-clockwise direction, pccw. The average static pressure is determined from pcw and pccw. In this case, because the four ports lie on a segment of a cone, it is easier to use spherical coordinates and cone and roll angles to describe the geometry, as shown in Figure A-4. The difference between pcw and pccw is used to determine

82 the roll angle, while the pressure difference between the center tap and the 𝑖푡ℎ port determines the cone angle. The four coefficients then take on the following form:

(푖) 푝푖 − 푝7 퐶휃 = 푐표푛푒 푎푛푔푙푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (19) 푝푖 − 푝̅

(푖) 푝푐푤 − 푝푐푐푤 퐶ϕ = 푟표푙푙 푎푛푔푙푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (20) 푝푖 − 푝̅

(푖) 푝푖 − 푝푡표푡 퐶푡 = 푡표푡푎푙 푝푟푒푠푠푢푟푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (21) 푝푖 − 푝̅

(푖) 푝푖 − 푝푠푡푎푡 퐶푠 = 푠푡푎푡𝑖푐 푝푟푒푠푠푢푟푒 푐표푒푓푓𝑖푐𝑖푒푛푡 = (22) 푝푖 − 푝̅ 푝 + 푝 푝̅ = 푎푣푒푟푎푔푒 푠푡푎푡𝑖푐 푝푟푒푠푠푢푟푒 = 푐푤 푐푐푤 (23) 2 From these equations, it is evident that there are seven sets of coefficients that may be used, depending on which port senses the highest pressure. The calibration procedure thus involves first positioning the probe at known pitch and yaw angles relative to the flow and measuring the seven pressures as well as the tunnel reference pressure (used for 푝푡표푡 and 푝푠푡푎푡 values). The eight pressures are recorded and the four non-dimensional coefficients are calculated at each measurement point. By calibrating over the range of the probe, the seven sets of coefficients may be obtained.

Figure A-5 illustrates the sectors chosen by the calibration scheme where each symbol corresponds to the hole which senses the highest pressure. From this figure along with Figure A-

6, it is apparent that each set of coefficients varies smoothly with 퐶훼 and 퐶훽. Indeed, as a result of the velocity independence, the velocity vector as well as the four coefficients are solely dependent on α and β (or θ and ϕ). These four coefficients should be a continuous function of these two variables such that:

(푖) (푖) (푖) (푖) 퐶푡 = 퐶푡 (퐶휃 , 퐶ϕ ) (24)

(푖) (푖) (푖) (푖) 퐶푠 = 퐶푠 (퐶휃 , 퐶ϕ ) (25)

(푖) (푖) (푖) (푖) 휃 = 휃 (퐶휃 , 퐶ϕ ) (26)

(푖) (푖) (푖) (푖) ϕ = 휙 (퐶휃 , 퐶ϕ ) (27)

For high flow angles and for low flow angles,

(7) (7) 퐶푡 = 퐶푡 (퐶훼, 퐶훽) (28)

(7) (7) 퐶푠 = 퐶푠 (퐶훼, 퐶훽) (29)

훼 = 훼(퐶훼, 퐶훽) (30)

훽 = 훽(퐶훼, 퐶훽) (31) To determine these functions, one may use a two-dimensional polynomial least squares curve- fit, a quintic piecewise surface or a simple look-up table. While a least squares fit is simple to implement, in areas where the calibration points deviate from the polynomial form, the polynomial function is likely to oscillate between calibration points, where it should be more linear. In contrast, a look-up table circumvents this problem by using the actual calibration points. However, this method also involves interpolation which may lead to error in regions with large gradients and is also more time consuming. Wenger and Devenport (1999) show that a combination of these two schemes offers the greatest accuracy. Their method is a two-step process. First, the data is fitted to a third order least-squares surface, f(퐶푎,퐶푏). It may then be assumed that:

퐶(퐶푎, 퐶푏) = 푓(퐶푎, 퐶푏) + 푒(퐶푎, 퐶푏) (32) where e(퐶푎,퐶푏) is the error in the polynomial surface fit at each calibration point. Each error value is recorded and this is used as the look-up table. This entails four sets of look-up tables for each sector (28 in total). This method uses the simplicity of the curve-fit, while providing a more accurate result by linearly interpolating over the smaller magnitudes in the error look-up table.

Thus at each measurement point, 퐶푎 and 퐶푏 are first calculated from the seven pressures. The third-order surface fits are then used to obtain a first estimate of each of the values in either equations (14)-(18) or equations (19)-(24). The corresponding look-up table is then used to interpolate the error and this value is added to the first estimate to determine the final value. A first order triangulation scheme is used to perform the 2D error interpolation in the 퐶푎 – 퐶푏 plane. Once both directionality values (either α and β or θ and ϕ) and both pressure coefficients

(퐶푡 and 퐶푠) are calculated, the three velocity components may be obtained from the following set of equations:

2 |푉⃑ | = √ (푝 − 푝̅)(1 + 퐶 − 퐶 ) (33) 휌 푛 푠 푡

푢 = |푉⃑ |푐표푠훼푐표푠훽 = |푉⃑ |푐표푠휃 (34)

푣 = |푉⃑ |푠𝑖푛훼푐표푠훽 = |푉⃑ |푠𝑖푛휃푠𝑖푛휙 (35)

푢 = |푉⃑ |푠𝑖푛훽 = |푉⃑ |푠𝑖푛휃푐표푠휙 (36) where 푝푛 is the maximum pressure sensed by all seven holes.

Figure A-3 Flowchart of (a) seven-hole probe calibration process and (b) velocity extraction process

Figure A-4 Seven-hole probe coordinate systems.

Figure A-5 Seven-hole probe sectors.

Figure A-6 Typical direction coefficients for low flow angle (Hole 7).

Appendix B: 푪푫풊 Calculation

The following is a brief overview of the procedure referenced from previous publication (Pereira, 2011) but further details may be obtained from Maskell (Maskell, 1973).

Derivation of Drag integrals

The drag due to a lifting surface can be calculated from a momentum balance subject to the following assumptions:

• steady, incompressible flow

• solid walls

• constant area control surface at control volume inlet and outlet

• neglect viscous, shear stress (acceptable at low Re)

Given these assumptions, conservation of mass reduces to:

0 = 휌푉2퐴2 = 휌푉1퐴1

푉2퐴2 = 푉1퐴1 (37) where V is the total velocity, A is the cross-sectional area, and 1 and 2 refer to the upstream and downstream planes of the control volume. Conservation of momentum in the freestream direction reduces to:

2 2 퐷 = ∬ (푃∞ − 푃) + 휌(푢∞ − 푢 ) 푑푆 (38) 퐶푆 where P is the static pressure and u is the velocity component in the freestream direction. If the static pressure is replaced with the total pressure such that,

1 푝 = 푃 + 휌푢2 (39) ∞ ∞ 2 ∞ then equation (38) reduces to

1 2 2 1 2 2 퐷 = ∬ (푝1 − 푝2) + 휌(푣 + 푤 ) + 휌(푢∞ − 푢 ) 푑푆 (40) 퐶푆 2 2

The first term in this equation represents the pressure drag, the second term is the vortex drag and the third is a profile and induced drag combination. The integral however must be taken over the entire wake.

In order to limit the measurement area to the viscous wake alone, Betz (1925) introduced an artificial axial velocity to describe the viscous wake velocity profile. The artificial velocity correlates the axial velocity with the local pressure profile,

2 푢∗2 = 푢2 + (푝 − 푝 ) (41) 휌 1 2

The perturbation velocity was also introduced where u′ = u∗ −u∞. Substituting the artificial and perturbation velocities into equation (38), the drag simplifies to:

1 ∗ ∗ 퐷 = ∬ (푝1 − 푝2) 푑푆 + 휌 ∬ (푢 − 푢)(푢 + 푢 − 2푢∞) 푑푆 퐶푆 2 퐶푆 1 + 휌 ∬ (푣2 + 푤2 − 푢′2)푑푆 (42) 2 퐶푆 and can be rearranged into profile and induced drag components.

1 ∗ ∗ 퐷푝 = ∬ (푝1 − 푝2) 푑푆 + 휌 ∬ (푢 − 푢)(푢 + 푢 − 2푢∞) 푑푆 퐶푆 2 퐶푆 1 + 휌 ∬ (푣2 + 푤2 − 푢′2)푑푆 (43) 2 퐶푆

Maskell interpreted the velocity perturbation term in the induced drag relation as a wake blockage correction factor. However, if the wind tunnel walls are far away, the u′2 term becomes negligibly small and can be eliminated such that the induced drag simplifies to:

1 2 2 퐷푖 = 휌 ∬ (푣 + 푤 )푑푆 (44) 2 퐶푆 where the integral becomes small but does not vanish outside of the vortical wake region. The v and w components can be re-written in terms of a stream function, ψ, and velocity potential, ϕ,

휕휙 휕휓 푣 = + (45) 휕푦 휕푧

휕휙 휕휓 푤 = − (46) 휕푧 휕푦 resulting in

1 휕휙 휕휓 2 휕휙 휕휓 2 퐷푖 = 휌 ∬ (( + ) + ( − ) ) 푑푆 (47) 2 퐶푆 휕푦 휕푧 휕푧 휕푦

Maskell showed that by using Green’s theorem and divergence this can be further simplified to:

1 퐷푖 = 휌 ∬ (휓휁 − 휙휎)푑푆 (48) 2 푤푎푘푒 where ζ is the vorticity and σ is the source term. Equation (48) is subject to the following boundary conditions:

• Tunnel walls are streamlines, ψ(wall) = 0

• No flow through the walls, ∂ϕ/∂n = 0

Thus to calculate the induced drag, one need only determine ψ, ζ, ϕ and σ at each point (i,j) in the vortical wake. For the present measurements, vorticity and source terms were calculated from the velocity vectors using centered-difference formula:

Δ푤 Δ푣 푤푖−1,푗 − 푤푖+1,푗 푣푖,푗+1 − 푣푖,푗−1 휁 = − = − (49) 푖,푗 Δ푦 Δ푧 2휂 2휂

Δ푣 Δ푤 푣푖−1,푗 − 푣푖+1,푗 푤푖,푗+1 − 푤푖,푗−1 휎 = + = − (50) 푖,푗 Δ푦 Δ푧 2휂 2휂 where η = Δy = Δz, i = 2, 3, . . . n − 1, j = 2, 3, . . .m − 1

The stream function and velocity potential were then inferred using Poisson’s equation and centered-difference formula:

1 휁 = −∇2휓 ≈ − (휓 + 휓 − 4휓 + 휓 + 휓 ) (51) 푖,푗 푖,푗 휂2 푖+1,푗 푖−1,푗 푖,푗 푖,푗+1 푖,푗−1

1 휎 = ∇2휙 ≈ (휙 + 휙 − 4휙 + 휙 + 휙 ) (52) 푖,푗 푖,푗 휂2 푖+1,푗 푖−1,푗 푖,푗 푖,푗+1 푖,푗−1

Along the tunnel walls the boundary conditions were imposed resulting in the following relations on the left wall, right wall, ceiling and floor, respectively:

1 휁 ≈ − (휓 + 휓 − 4휓 + 휓 ) (53) 푖,2 휂2 푖+1,2 푖−1,2 푖,2 푖,3

1 휎 ≈ (휙 + 휙 − 3휙 + 휙 ) (54) 푖,2 휂2 푖+1,2 푖−1,2 푖,2 푖,3

1 휁 ≈ − (휓 + 휓 − 4휓 + 휓 ) (53) 푖,푚−1 휂2 푖+1,푚−1 푖−1,푚−1 푖,푚−1 푖,푚−2

1 휎 ≈ (휙 + 휙 − 3휙 + 휙 ) (54) 푖,푚−1 휂2 푖+1,푚−1 푖−1,푚−1 푖,푚−1 푖,푚−2

1 휁 ≈ − (휓 − 4휓 + 휓 + 휓 ) (53) 푛−1,푗 휂2 푛−2,푗 푛−1,푗 푛−1,푗+1 푛−1,푗−1

1 휎 ≈ (휙 − 3휙 + 휙 + 휙 ) (54) 푖,2 휂2 푛−2,푗 푛−1,푗 푛−1,푗+1 푛−1,푗−1

1 휁 ≈ − (휓 − 4휓 + 휓 + 휓 ) (53) 푖,2 휂2 3,푗 2,푗 2,푗+1 2,푗−1

1 휎 ≈ (휙 − 3휙 + 휙 + 휙 ) (54) 푖,2 휂2 3,푗 2,푗 2,푗+1 2,푗−1

This then formed a system of (n - 2) × (m - 2) equations and (n - 2) × (m -2) unknowns, expressed as 퐴푥⃗ = 푏⃑⃗ where A is the (n - 2) × (m - 2) by (n - 2) × (m -2) matrix of coefficients, 푥⃗ is the vector of unknowns (ψ or ϕ) and 푏⃑⃗ is the vector of knowns (σ or ζ). To solve for the unknowns, was simply a matter of computing the inverse of A and multiplying by 푏⃑⃗. However, for reasonable resolution (larger than 3 mm), the A matrix quickly grew to [100,000 × 100,000] and could not be inverted without the use of a supercomputer. The A matrix however was sparse, containing at most five non-zero entries in each row. As a first measure to reduce computing time the A matrix was packed into an (n − 2) × (m − 2) by 5 array containing the indices of the non-zero elements. To further reduce computing time, an iterative successive over-relaxation (SOR) technique based on the Gauss-Seidel method was used. For any iterative method, an initial estimate must be provided. To reduce the number of iterations required to get to the

92 solution, an initial guess was calculated at a coarser resolution of 25.4 cm resulting in a [1632 × 1632] A matrix. This A matrix could be inverted and multiplied by 푏⃑⃗ to obtain a close initial approximation. Finally, in order to speed up the rate of convergence, the over-relaxation parameter used was:

4 푤 = (55) 휋 휋 2 √2 + 4 − cos ( ) + cos ( ) [ 푚 푛 ]

The values of ⃗x could then be solved using the following equation:

푖−1 푛×푚 푤(− ∑푗=1 푎푖,푗푥푗 − ∑푗=푖+1 푎푖,푗푋푂푗 + 푏푖) 푥푖 = (1 − 푤)푋푂푖 + (56) 푎푖,푖 where 푥푖 is the value of 푥⃗ being computed, 푋푂푖 is the initial estimate, 푎푖,푗 is a coefficient in the A ⃑⃗ matrix, and 푏푖 is a value in the known 푏 vector. This process was repeated until the difference between successive iterations was less than 0.01. The tolerance was varied between 0.0001 and 0.01 but the results showed a difference of less than 0.5%, the difference being mainly attributed to round-off error. As such, in order to reduce computing time the tolerance of 0.01 was selected. Once the 푥⃗ values were computed, these were substituted into equation (48) and an approximation to the induced drag was obtained as:

푛−1 푚−1 휌 퐷 = ∑ ∑ (휓 휁 − 휙 휎 )휂2 (57) 푖 2 푖,푗 푖,푗 푖,푗 푖,푗 푖=2 푗=2