List of Tables v

Contents

List of Tables v

List of Figures viii

1. Introduction 1

2. The Facility 9

2.1. The Wind tunnel . . . . 9

2.1.1. The model . . . 12

3. Rapid Scanning 14 3.1. Five Hole Probe & Calibration . . . 15

3.1.1. Preliminary Tests . . . 17

3.1.2. Five Hole Probe Calibration Method . . . 21

3.1.3. Calibration Tests . . . 26

3.1.4. Calibration Accuracy . . . 31

3.2. Setup . . . 35

3.3. Test Procedure . . . 40

3.4. Data Analysis . . . 44

3.4.1. Evaluation of Vortex Centre Locations . . . 44

3.4.2. Fitting of the Experimental Probability Density Function of the Vortex Centre Locations . . . 53

3.4.3. Study of the Velocity Profiles . . . 63

3.4.3.1. Consideration about the Error in the Mean Velocity Profile Evaluation . . . 72

3.5. Results and Discussion . . . 83

iii

3.5.1. Downstream Variation of the Wandering . . . 83

3.5.2. Effects of the Variation of the Angle of Attack on the Wandering102 3.5.3. Reynolds Dependency of the Wandering . . . 123

3.5.4. Wandering Effects on Standard Deviation of Mean Velocity Profiles . . . 141

3.5.5. Fitting of Velocity and Circulation Profiles . . . 154

3.5.6. Secondary Vorticity Structures . . . 157

4. Static Measurements Performed with Three Component Hot Film Anemometry 169 4.1. Three Sensors Hot-Film Probe and Calibration . . . 170

4.2. Tests Execution . . . 176

4.3. Data Analysis . . . 179

4.3.1. Mean Flow Field . . . 179

4.3.2. Wandering Characterization from Static Measurements . . . 183

4.4. Results . . . 187

4.5. Time-frequency analysis . . . 195

5. Comparison of Rapid Scanning and Static Hot Wire Measure- ments 205 5.1. Downstream Distance Variation . . . 205

5.2. Effects of the Variation of the Angle of Attack . . . 211

5.3. Effects of the Variation of the Reynolds Number . . . 215

6. Conclusions 219

A. Hot Wire Anemometry Overview 226

B. Lekakis Calculation 236

Bibliography 242

iv

List of Tables

0.1. List of Symbols . . . xx

2.1. Specifications of the DAQ NI PXI 6052E. . . 12

3.1. Tangential reference system definition. . . 22

3.2. Calibration 5HP - Test matrix. . . 27

3.3. Calibration 5HP - Validation tests. . . 28

3.4. Roll scan - Statistical values. . . 28

3.5. Standard deviation of absolute errors applying different calibration matrices. . . 32

3.6. NI PXI 6602 specifications. . . 37

3.7. Main test matrix. . . 41

3.8. Test matrix to investigate about the secondary vorticity . . . 43

3.9. Sensitivity of fitting parameters with respect to the spacing of the space domain grid. . . 55

3.10. Sensitivity of fitting parameters with respect to the minimum num- ber of the vortex centre occurrences. . . 56

3.11. Sensitivity of fitting parameters with respect to the number of scans. 57 3.12. Wandering parameters for the analyzed locations. . . 85

3.13. Vortex parameters inherent to the mean tangential velocity profile for the analyzed locations. . . 91

3.14. Gradient of the mean tangential velocity profile at the vortex centre for the analyzed locations. . . 92

3.15. Non-dimensional axial velocity defect at the vortex centre for the analyzed locations. . . 99

3.16. Wandering parameters for the tested angles of attack. . . 105

v

3.17. Vortex parameters inherent to the mean tangential velocity profile

for the tested angles of attack. . . 112

3.18. Gradient of the mean tangential velocity profile at the vortex centre for the tested angles of attack. . . 113

3.19. Non-dimensional axial velocity defect (for negative values) or excess (for positive values) at the vortex centre for the tested angles of attack.120 3.20. Wandering parameters for each tested free-stream velocity. . . 123

3.21. Vortex parameters inherent to the mean tangential velocity profile for each tested free-stream velocity. . . 132

3.22. Gradient of the mean tangential velocity profile at the vortex centre for each tested free-stream velocity. . . 132

3.23. Non-dimensional axial velocity defect at the vortex centre for each tested free-stream velocity. . . 139

3.24. Amplitude parameters of wandering oscillation for the downstream variation. . . 147

3.25. Amplitude parameters of wandering oscillation for the angle of attack variation. . . 148

3.26. Amplitude parameters of wandering oscillation for the Reynolds num- ber variation. . . 148

3.27. Fitting of the circulation: coefficients for the Hoffmann & Joubert model of the tested conditions. . . 154

3.28. Fitting of the tangential velocity profile: coefficients for the series method of the tested conditions. . . 155

3.29. Fitting of the axial velocity profile: coefficients for the series method of the tested conditions. . . 156

4.1. Specifications of the TSI 1299-20-18 triple sensor hot-film probe. . . 171

4.2. Specifications of the AN-1003 hot-wire anemometry system. . . 172

4.3. h and k coefficients for the 3HFP. . . 176

4.4. Test matrix of 3HFP measurements. . . 178

4.5. Wandering parameters for the analyzed streamwise locations. . . 187

4.6. Wandering parameters for the tested angles of attack. . . 188

4.7. Wandering parameters for the tested free-stream velocities. . . 188

vi

4.8. Fitting of the circulation: coefficients for the Hoffmann & Joubert

model of the tested conditions. . . 193

4.9. Vortex parameters for the analyzed locations. . . 193

4.10. Vortex parameters for the tested angles of attack. . . 194

4.11. Vortex parameters for the tested free-stream velocities. . . 194

vii

List of Figures

2.1. Sketch of the wind tunnel and frame of reference. . . . 9

2.2. 2mWT vortex generator. . . 10

2.3. Half wing model positioning. . . 13

3.1. 5HP shape and geometry, length [mm], angles [ ^◦ ]. . . 15

3.2. 5HP tip geometry, port numbering convention and axes. . . 16

3.3. Picture of the 5HP positioning in the Calibration Wind Tunnel. . . 17

3.4. Arrangement for transducers calibration. . . 18

3.5. Relative Errors averaged on velocities and on channels for different pitch angles. . . 19

3.6. Relative Errors averaged on pitch angles and on channels for different velocities. . . 19

3.7. Division of angular space. . . 21

3.8. Low flow angle reference system. . . 22

3.9. Flow over the probe at high angles of attack. . . 24

3.10. Distribution of data points in α-β plane used for the 5HP calibration. 26 3.11. Example of 5HP calibration tests at U _∞ = 20 m/s. . . 29

3.12. Comparison between transducer N ^◦ 2 and N ^◦ 3 of the 5HP. . . 30

3.13. Roll scan - Without air flow through the wind tunnel . . . 30

3.14. Examples of errors in tests data . . . 34

3.15. General scheme of the rotating unit. . . 35

3.16. Optical encoder. . . 37

3.17. Geometrical quantity involved in software correction. . . 39

3.18. Sketch of quantities involved in the transformation from the probe frame of reference to the tunnel frame of reference. . . 44

viii

3.19. Conceptual sketch of the linear method to find the y component of the vortex centre. . . 46 3.20. Conceptual sketch of the linear method to find z component of vortex

centre. . . 46 3.21. Conceptual sketch of Corsiglia method to find the vortex centre. . . 48 3.22. Example of |V θ | profiles in two scans: (a) core crossed, (b) core

missed. Condition α = 8 ^◦ , U _∞ = 20 m/s, x/c = 5. . . 49 3.23. Example of Corsiglia method applied for a scan close to the vortex

centre . . . 51 3.24. Example of Corsiglia method applied for a scan far from the vortex

centre . . . 51 3.25. The vortex centres distribution evaluated by linear (a) and Cor-

siglia (b) methods for the condition α = 8 ^◦ , U _∞ = 20 m/s, x/c = 5. . 52 3.26. The vortex centres distribution for the condition α = 8 ^◦ , U _∞ =

20 m/s, x/c = 5. . . 52 3.27. (a) Experimental PDF map, 1 mm grid spacing. (b) Fitted PDF

map, 1 mm grid spacing. . . 53 3.28. (a) Vortex centre locations, the cross-plane is divided by a grid. (b)

Representation of the occurrences matrix. . . 54 3.29. Experimental PDF map, 0.5 mm grid spacing. (a) Low wandering

amplitude. (b) High wandering amplitude. . . 56 3.30. Scheme of the iterative cycleof the least sqauare fitting of the PDF

with a bi-variate gaussian function. . . 60 3.31. Comparison between experimental PDF and fitted PDF. (a) Z =

const sections. (b) Y = const sections. . . 61 3.32. (a) Fitted PDF iso-contours. (b) Preferential direction Θ of the

vortex wandering. . . 62 3.33. Definition of R coordinate. . . . 63 3.34. Overlapping of original velocity signals as obtained from rapid scanning 64 3.35. Overlapping of velocity signals scaled by R and re-centred . . . 64 3.36. Average procedure on velocity profile . . . 65 3.37. Example of the average algorithm on tangential velocity data cor-

rected for wandering effects . . . 66

ix

3.38. Examples of the standard deviation . . . 67

3.39. Example of the individuation of the vortex core, the logarithmic and the defect regions according to Hoffmann and Joubert theory . . . . 69

3.40. Example of Hoffmann&Joubert fitting of the tangential velocity com- ponent . . . 70

3.41. Example of gaussian series fitting of V _θ (a) and u (b) profile . . . 70

3.42. Standard deviation of the tangential velocity profiles corrected for wandering effects. . . 73

3.43. Number of samples into each averaging window: example for the condition α = 8 ^◦ , U _∞ = 10 m/s, x/c = 5. . . 74

3.44. Zoom in the vortex core region of a tangential velocity profile for the condition α = 8 ^◦ , U _∞ = 10 m/s, x/c = 5. . . 75

3.45. Comparison between the procedure that involves the linear interpo- lation of the velocity signals and the procedure that use the averaging window . . . 76

3.46. Example of averaging window in the outboard part of the vortex core 77 3.47. Example of V _θ profile with a distinction between the two centre find- ing methods for the condition α = 8 ^◦ , U _∞ = 10 m/s, x/c = 5. . . 78

3.48. Simulated effects of the average error due to the gradient of the reference velocity profile. . . 80

3.49. Simulated effects of the centre finding algorithm error. . . 81

3.50. Simulated effects of the measurement error. . . 82

3.51. Simulated effects of the total error. . . 82

3.52. Downstream variation of the experimental and fitted PDF of vortex centre locations. . . 84

3.53. Wandering amplitude as a function of the streamwise distance. . . . 86

3.54. Anisotropy parameter e (a) and direction of the principal axes Θ (b) of wandering as a function of the streamwise distance. . . 87

3.55. Trajectory of the mean vortex centre as a function of the streamwise distance. . . 88

3.56. Downstream variation of the wandering smoothing effect on mean tangential velocity profiles. . . 89

3.57. Downstream variation of mean tangential velocity profiles. . . 90

x

3.58. Wandering smoothing effect: peak tangential velocity (a) and vortex core radius (b) as a function of the streamwise distance. . . 92 3.59. Wandering relative errors on peak tangential velocity and vortex core

radius as a function of the streamwise distance. . . 93 3.60. Downstream variation of mean circulation profiles. . . 94 3.61. Wandering smoothing effect: circulation evaluated at peak tangen-

tial velocity as a function of the streamwise distance. . . 95 3.62. Downstream variation of the wandering smoothing effect on mean

axial velocity profiles. . . 97 3.63. Downstream variation of mean axial velocity profiles. . . 99 3.64. Wandering smoothing effect: axial velocity defect (a) and relative

error on the axial velocity defect (b) as a function of the streamwise distance. . . 100 3.65. Rosby number (a) and vortex Reynolds number (b) as a function of

the streamwise distance. . . 101 3.66. Experimental and fitted PDF of vortex centre locations evaluated

at different values of the angle of attack, location x/c = 3. . . 103 3.67. Experimental and fitted PDF of vortex centre locations evaluated

at different values of the angle of attack, location x/c = 5. . . 104 3.68. Wandering amplitude as a function of the angle of attack. . . 105 3.69. Anisotropy parameter e (a) and direction of the principal axes Θ (b)

of wandering as a function of the angle of attack. . . 107 3.70. Mean coordinates of the vortex centre Y _c /c (a) and Z _c /c (b) as a

function of the angle of attack. . . 108 3.71. Wandering smoothing effect on mean tangential velocity profiles

evaluated at different values of the angle of attack. . . 109 3.72. Mean tangential velocity profiles evaluated at different values of the

angle of attack for the location x/c = 3. . . 110 3.73. Mean tangential velocity profiles evaluated at different values of the

angle of attack for the location x/c = 5. . . 111 3.74. Wandering smoothing effect: peak tangential velocity (a) and vortex

core radius (b) as a function of the angle of attack. . . 111

xi

3.75. Wandering relative errors on peak tangential velocity and vortex core radius as a function of the angle of attack. . . 114 3.76. Mean circulation profiles evaluated at different values of the angle

of attack for the location x/c = 3. . . . 115 3.77. Mean circulation profiles evaluated at different values of the angle

of attack for the location x/c = 5. . . . 115 3.78. Wandering smoothing effect on mean axial velocity profiles evaluated

at different values of the angle of attack. . . 116 3.79. Mean axial velocity profiles evaluated at different values of the angle

of attack for the location x/c = 3. . . . 119 3.80. Mean axial velocity profiles evaluated at different values of the angle

of attack for the location x/c = 5. . . . 119 3.81. Comparison between the axial velocity defect of re-centred profiles

evaluated with respect to the free-stream velocity and with respect to the maximum value of the axial velocity. . . 121 3.82. Wandering smoothing effect: axial velocity defect (a) and absolute

error on the axial velocity defect (b) as a function of the angle of attack. . . 122 3.83. Experimental and fitted PDF of vortex centre locations evaluated

with different free-stream velocities, location x/c = 3. . . . 124 3.84. Experimental and Fitted PDF of vortex centre locations evaluated

with different free-stream velocities, location x/c = 5. . . . 125 3.85. Wandering amplitude as a function of the free-stream velocity. . . . 126 3.86. Anisotropy parameter e (a) and direction of the principal axes Θ (b)

of wandering as a function of the free-stream velocity. . . 127 3.87. Mean coordinates of the vortex centre Y _c /c (a) and Z _c /c (b) as a

function of the free-stream velocity. . . 128 3.88. Wandering smoothing effect on mean tangential velocity profiles

evaluated with different free-stream velocities. . . 130 3.89. Mean tangential velocity profiles evaluated with different free-stream

velocities for the location x/c = 3. . . 131 3.90. Mean tangential velocity profiles evaluated with different free-stream

velocities for the location x/c = 5. . . 131

xii

3.91. Wandering smoothing effect: peak tangential velocity (a) and vortex core radius (b) as a function of the free-stream velocity. . . 133 3.92. Wandering relative errors on peak tangential velocity and vortex core

radius as a function of the free-stream velocity. . . 134 3.93. Mean circulation profiles evaluated with different free-stream veloc-

ities for the location x/c = 3. . . 135 3.94. Mean circulation profiles evaluated with different free-stream veloc-

ities for the location x/c = 5. . . 135 3.95. Wandering smoothing effect on mean axial velocity profiles evaluated

with different free-stream velocities. . . 137 3.96. Mean axial velocity profiles evaluated with different free-stream ve-

locities for the location x/c = 3. . . 138 3.97. Mean axial velocity profiles evaluated with different free-stream ve-

locities for the location x/c = 5. . . 138 3.98. Wandering smoothing effect: axial velocity defect (a) and absolute

error on the axial velocity defect (b) as a function of the free-stream velocity. . . 139 3.99. Standard deviation evaluated for V _θ /U _∞ , re-centred (a) and affected

by wandering (b) for downstream variations at α = 8 ^◦ , U _∞ = 10 m/s. 142 3.100.Standard deviation evaluated for u/U _∞ re-centred (a) and affected

by wandering (b) for downstream variation at α = 8 ^◦ , U _∞ = 10 m/s. 143 3.101.Downstream variation of σ _∇ ^W

N oW

(a) and σ ^W _∇

W

(b) at α = 8 ^◦ , U _∞ = 10 m/s. . . 144 3.102.Effect of the angle of attack variation on σ _∇ ^W

N oW

(a) and σ ^W _∇

W

(b) at x/c = 3, U _∞ = 20 m/s and σ _∇ ^W

N oW

(c) and σ _∇ ^W

W

(d) at x/c = 5, U _∞ = 20 m/s. . . 145 3.103.Effect of the free-stream variation on σ ^W _∇

N oW

(a) and σ _∇ ^W

W

(b) at α = 8 ^◦ , x/c = 3 and σ ^W _∇

N oW

(c) and σ _∇ ^W

W

(d) at α = 8 ^◦ , x/c = 5. . . 146 3.104.Amplitude of the vortex wandering by varying the downstream dis-

tance at α = 8 ^◦ U _∞ = 10 m/s: comparison between the wandering amplitude obtained from the fitted PDF and the amplitude obtained from the mono-dimensional fitting of σ _∇ ^W

. . . 149

xiii

3.105.Amplitude of the vortex wandering by varying the angle of attack at U _∞ = 10 m/s x/c = 3 (a) and x/c = 5 (b): comparison be- tween the wandering amplitude obtained from the fitted PDF and the amplitude obtained from the mono-dimensional fitting of σ ^W _∇

N oW

(a). . . 150 3.106.Amplitude of the vortex wandering by varying the angle of attack at

α = 8 ^◦ x/c = 3 (a) and x/c = 5 (b): comparison between the wan- dering amplitude obtained from the fitted PDF and the amplitude obtained from the mono-dimensional fitting of σ ^W _∇

. . . 150 3.107.Amplitude of the vortex wandering by varying the downstream dis-

tance at α = 8 ^◦ U _∞ = 10 m/s: comparison between the wandering amplitude obtained from the fitted PDF and the amplitude obtained from the peaks of σ _∇ ^W

. . . 151 3.108.Amplitude of the vortex wandering by varying the angle of attack

at U _∞ = 10 m/s x/c = 3 (a) and x/c = 5 (b): comparison be- tween the wandering amplitude obtained from the fitted PDF and the amplitude obtained from the peaks of σ _∇ ^W

. . . 152 3.109.Amplitude of the vortex wandering by varying the free-stream at α =

8 ^◦ x/c = 3 (a) and x/c = 5 (b): comparison between the wandering amplitude obtained from the fitted PDF and the amplitude obtained from the peaks of σ _∇ ^W

. . . 152 3.110.Condition α = 8 ^◦ , U _∞ = 20 m/s, x/c = 3: map of the tangential

velocity field (a), map of the axial velocity field (b), map of the standard deviation of the tangential velocity (c), map of the standard deviation of the axial velocity (d). . . 158 3.111.Condition α = 8 ^◦ , U _∞ = 20 m/s, x/c = 3: mean profiles of V _θ /U _∞ ,

(u − U ∞ )/U _∞ , Γ/Γ ₀ . . . 159 3.112.Condition α = 8 ^◦ , U _∞ = 20 m/s, x/c = 3: example of traces of the

tangential and the axial velocity components as obtained for a single scan. . . 160 3.113.Condition α = 8 ^◦ , U _∞ = 20 m/s, x/c = 3: averaged profile of Γ/Γ ₀

and Donaldson distribution. . . 161

xiv

3.114.Circulation in the outboard proximity of the vortex core evaluated at different downstream distances for α = 8 ^◦ , U _∞ = 20 m/s. . . 162 3.115.Condition α = 8 ^◦ , U _∞ = 10 m/s: standard deviation evaluated for

the axial velocity with increasing the downstream distance. . . 163 3.116.Circulation in the outboard proximity of the vortex core with in-

creasing the angle of attack. . . 164 3.117.Condition U _∞ = 20 m/s, x/c = 3: standard deviation evaluated for

the axial velocity with increasing the angle of attack. . . 165 3.118.Circulation in the inboard proximity of the vortex core with increas-

ing the angle of attack. . . 165 3.119.Condition U _∞ = 20 m/s, x/c = 5: standard deviation evaluated for

the axial velocity with increasing the angle of attack. . . 166 3.120.Circulation in the outboard proximity of the vortex core with in-

creasing the free-stream velocity. . . 167 3.121.Condition α = 8 ^◦ , x/c = 3: standard deviation evaluated for the

axial velocity with increasing the free-stream velocity. . . 168 3.122.Condition α = 8 ^◦ , x/c = 5: standard deviation evaluated for the

axial velocity with increasing the free-stream velocity. . . 168 4.1. Three sensors hot-film probe. . . 170 4.2. 3HFP shape and geometry, length [mm], angles [ ^◦ ]. . . . 171 4.3. Example of velocity magnitude calibration of an hot-film sensor by a

least square fitting of the experimental data through a fourth order polynomial law. . . 173 4.4. Sketch of the procedure for the angular calibration of the 3HFP,

shown for a single wire: arrangement for h i determination tests (a) and arrangement for k _i determination tests (b). . . 175 4.5. Example of the calibration data for a wire of the 3HFP . . . 176 4.6. Set-up for 3HFP measurements. . . 177 4.7. Tangential, normal and spanwise velocity components measured by

3HFP traverse. . . 180 4.8. Comparison between velocity profiles measured by 5HP rapid scan-

ning and 3HFP static measurements. . . . 181

xv

4.9. Example of Hoffmann&Joubert fitting of the tangential velocity com- ponent . . . 182 4.10. Skewness (a) and kurtosis (b) evaluated for the axial and the normal

velocity components. . . 183 4.11. RMS and standard deviation evaluated for the spanwise and normal

velocity components. . . 184 4.12. Cross-correlation coefficients vw (a), uv and uw (b) between the

three velocity components. . . 186 4.13. Downstream variation of the mean tangential velocity profiles. . . . 189 4.14. Mean tangential velocity profiles evaluated at different values of the

angle of attack. . . 189 4.15. Mean tangential velocity profiles evaluated with different free-stream

velocities. . . 190 4.16. Downstream variation of the mean axial velocity profiles. . . 190 4.17. Mean axial velocity profiles evaluated at different values of the angle

of attack. . . 191 4.18. Mean axial velocity profiles evaluated with different free-stream ve-

locities. . . 191 4.19. Wavelet spectra of the free-stream flow. . . 196 4.20. Example of the individuation of the most significant radial locations

where the wavelet spectra were analyzed. . . 197 4.21. Wavelet spectra of the axial, (a), and tangential, (b), velocity com-

ponents. . . 198 4.22. Wavelet spectra of the axial velocity component in the vicinity of

the traverse path limits. . . 199 4.23. Wavelet time-frequency map: energy of the axial velocity signal eval-

uated at the radial location I (inboard side). . . 200 4.24. Wavelet spectra of the axial velocity component evaluated at several

radial locations by proceeding outside the vortex core both in the inboard and in the outboard direction. . . 201 4.25. Wavelet spectra of the tangential velocity component evaluated at

several radial locations by proceeding outside the vortex core both in the inboard and in the outboard direction. . . 202

xvi

4.26. Downstream variation of the wavelet spectra of the axial and tan-

gential velocity components evaluated in the mean vortex centre. . . 203

4.27. Wavelet spectra of the axial and tangential velocity components eval- uated in the mean vortex centre at different angles of attack. . . 204

5.1. Downstream variation of the non-directional wandering amplitude and of the anisotropy parameter . . . 206

5.2. Downstream variation of the trajectory of the mean vortex centre . . 208

5.3. Downstream variation of the peak tangential velocity and the core radius . . . 209

5.4. Downstream variation of the axial velocity deficit at the core centre 211 5.5. Non-directional wandering amplitude and anisotropy parameter of wandering (b) as a function of the angle of attack . . . 212

5.6. Trajectory of the mean vortex centre as a function of the angle of attack . . . 213

5.7. Peak tangential velocity as a function of the angle of attack . . . 214

5.8. Radius of the vortex core as a function of the angle of attack . . . . 215

5.9. Axial velocity deficit at the core centre as a function of the angle of attack . . . 216

5.10. Non-directional wandering amplitude as a function of the free-stream velocity . . . 217

5.11. Axial velocity deficit at the core centre as a function of the angle of attack . . . 218

A.1. Four sensor probe prong geometry of Auspex AVOP-4-100 . . . 231

A.2. Iso-contour of f 1 , f 2 , f 3 . . . 235

B.1. Triple sensor probe geometry and coordinate system . . . 237

xvii

List of Symbols

b Wing span

c Mean geometric chord c ^∗ Mean aerodynamical chord c r Root chord

c _t Tip chord λ Taper ratio AR Aspect ratio

Λ Sweep angle

S Wing planar surface α Angle of attack β Angle of sideslip

θ Pitch angle φ Roll angle

σ Standard Deviation

∇ Gradient

g Gravity acceleration a _c Centrifugal acceleration

p Static pressure P 5HP Port pressure C _p Pressure coefficient C _L Wing lift coefficient

q Dynamic pressure ρ Air density

µ Air dynamic viscosity ν Air kinematic viscosity = ^µ _ρ Re Reynolds number = ^ρU _µ

^c U _∞ Free stream velocity

xviii

r 1 Vortex core radius

| − →

V | Velocity magnitude

V _θ Tangential velocity component V _θ1 Tangential velocity component peak

Γ Circulation

Γ 0 Theoretical circulation at the wing root Γ ₁ Circulation at maximum tangential velocity U c Axial velocity at the mean vortex centre

U _D = U _c − U ∞ Axial velocity deficit at the mean vortex centre U _max Maximum value of the axial velocity

U _D

= U _c − U max Axial velocity deficit with respect to U _max (x, y, z) 2mWT frame of reference

(u, v, w) Velocity components along (x, y, z)

(u _r , v _r , w _r ) Velocity components due to the rotation of the probe (x _p , y _p , z _p ) 5HP frame of reference

(y _c , z _c ) Vortex centre cross-plane coordinates for each scan

(Y _c , Z _c ) Mean vortex centre cross-plane coordinates for each test condition (Y _pos , Z _pos ) Cross-plane coordinates of the probe tip in the tunnel frame of reference

R Distance from mean vortex centre in the cross-plane _|y−y ^y−y

c

| p(y − y c ) ² + (z − z c ) ² R m Nodes vector for no-wandering velocity profiles averaging

y _m Nodes vector for wandering velocity profiles averaging (y ₀ , z ₀ ) Coordinates of the bi-variate gaussian function centre

σ _y Wandering amplitude along the spanwise direction σ _z Wandering amplitude along the normal direction σ _yz = q

σ _y ² + σ _z ² Non-directional wandering amplitude e Anisotropy parameter

Θ Direction of principal axes of wandering Σ Covariance matrix

xix

V _p Peripheral velocity of the 5HP’s tip ω Rotation frequency of the rotor

γ Angular rotation of the 5HP around the arm’s axis Φ Angular rotation of the arm

L _arm Distance between the centre of the rotor and the 5HP’s axis L _probe Distance between the 5HP’s tip and the rotating arm’s axis

E _i 3HFP output voltages U _{ef f}

Effective cooling velocities

h _i , k _i Coefficients of the Jørgensen directional response law

(u N , u T , u B ) Velocity components of the Jørgensen directional response law Sk Skewness (the third order statistic moment)

Ku Kurtosis (the fourth order statistic moment) RM S Root mean square

uv Cross-correlation coefficient between u and v uw Cross-correlation coefficient between u and w vw Cross-correlation coefficient between v and w

f Frequency

P o Wavelet power spectrum

Table 0.1. List of Symbols

xx