COMPARISON OF RAPID SCANNING AND STATIC HOT WIRE MEASUREMENTS

DIPARTIMENTO DI INGEGNERIA AEROSPAZIALE L. Lazzarino

Corso di Laurea in Ingegneria Aerospaziale

Matteo Biancalana Luca Dell’Osso

WING-TIP VORTEX WANDERING:

COMPARISON OF RAPID SCANNING AND STATIC HOT WIRE MEASUREMENTS

UNIVERSITÀ DI PISA 1 343

IN

SU

PREMÆ DI_GN ITTA IS

UNIVERSIT ` A DEGLI STUDI DI PISA

FACOLT ` A DI INGEGNERIA

Corso di Laurea in Ingegneria Aerospaziale

WING-TIP VORTEX WANDERING:

COMPARISON OF RAPID SCANNING AND STATIC HOT WIRE MEASUREMENTS

I Relatori:

Prof. Ing. Giovanni Lombardi Ing. Giacomo Valerio Iungo Ing. Peter Skinner

I candidati:

Matteo Biancalana

Luca Dell’Osso

Wandering is a universal feature of wind-tunnel generated vortices and it consists of abrupt displacements of the vortex core location. Rapid scanning measurements on the tip vortex generated from a N ACA 0012 half-wing model were performed in order to achieve data not affected by wandering effects. This measurement technique allows to evaluate the instantaneous vortex centre locations. Wandering was characterized by a bi-variate gaussian function fitting the probability density function evaluated from experimental vortex centre locations. Wandering is surely attenuated with increasing vortex strength, consequently, this phenomenon can not be considered a self-induced one. For vortices with a moderate strength, wandering might be strongly dependent on flow conditions and measurement locations. It is found that wandering amplitude increases roughly linearly with increasing stream- wise distance up to 116% of the vortex core radius, whereas it is fairly invariant with increasing angle of attack. Furthermore, wandering amplitude slightly de- creases with increasing free-stream velocity. The wandering smoothing effects on mean velocity profiles were estimated comparing rapid scanning data with static hot wire measurements. In extreme circumstances wandering is responsible to 30%

underestimate of the peak tangential velocity and 85% overestimate of the vor-

tex core radius and, consequently, the measured vortex appears more diffuse and

weaker than in reality. The shape of the axial velocity profiles corrected for wander-

ing effects exhibit a significant velocity defect in the vortex core up to 20% of the

free-stream velocity, and an overshoot at the core border. At high angles of attack

a shift between wake flow and jet flow was observed at the vortex core. Wandering

leaded to 70% underestimate of the axial velocity deficit and to 30% underestimate

of the velocity excess. Finally, secondary vorticity structures were singled out from

rapid scanning data in proximity to the main vortex. Furthermore, from a spectral

analysis performed on the hot wire measurements, it is found a general increase of

the fluctuating energy at low frequencies with approaching the vortex core.

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

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

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

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

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

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

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

| 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

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

(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

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

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 (a) and Z

/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

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 (a) and Z

/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

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

/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

N oW

(a) and σ

W

(b) at α = 8

, U

= 10 m/s. . . 144 3.102.Effect of the angle of attack variation on σ

N oW

(a) and σ

W

(b) at x/c = 3, U

= 20 m/s and σ

N oW

(c) and σ

W

(d) at x/c = 5, U

= 20 m/s. . . 145 3.103.Effect of the free-stream variation on σ

N oW

(a) and σ

W

(b) at α = 8

, x/c = 3 and σ

N oW

(c) and σ

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

. . . 149

∞

tween the wandering amplitude obtained from the fitted PDF and the amplitude obtained from the mono-dimensional fitting of σ

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

. . . 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

. . . 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

. . . 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

. . . 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

/U

,

(u − U

)/U

, Γ/Γ

. . . 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

and Donaldson distribution. . . 161

∞

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

determination tests (a) and arrangement for k

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

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

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

, f

, f

. . . 235

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

b Wing span

c Mean geometric chord c

Mean aerodynamical chord c

Root chord

c

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

Centrifugal acceleration

p Static pressure P 5HP Port pressure C

Pressure coefficient C

Wing lift coefficient

q Dynamic pressure ρ Air density

µ Air dynamic viscosity

ν Air kinematic viscosity =

Re Reynolds number =

U

Free stream velocity

V

Tangential velocity component V

Tangential velocity component peak

Γ Circulation

Γ

Theoretical circulation at the wing root Γ

Circulation at maximum tangential velocity U

Axial velocity at the mean vortex centre

U

= U

− U

Axial velocity deficit at the mean vortex centre U

Maximum value of the axial velocity

U

= U

− U

Axial velocity deficit with respect to U

(x, y, z) 2mWT frame of reference

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

(u

, v

, w

) Velocity components due to the rotation of the probe (x

, y

, z

) 5HP frame of reference

(y

, z

) Vortex centre cross-plane coordinates for each scan

(Y

, Z

) Mean vortex centre cross-plane coordinates for each test condition (Y

, Z

) Cross-plane coordinates of the probe tip in the tunnel frame of reference

R Distance from mean vortex centre in the cross-plane

c|

p(y − y

)

+ (z − z

)

R

Nodes vector for no-wandering velocity profiles averaging

y

Nodes vector for wandering velocity profiles averaging (y

, z

) Coordinates of the bi-variate gaussian function centre

σ

Wandering amplitude along the spanwise direction σ

Wandering amplitude along the normal direction σ

= q

σ

+ σ

Non-directional wandering amplitude e Anisotropy parameter

Θ Direction of principal axes of wandering

Σ Covariance matrix

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

L

Distance between the centre of the rotor and the 5HP’s axis L

Distance between the 5HP’s tip and the rotating arm’s axis

E

3HFP output voltages U

Effective cooling velocities

h

, k

Coefficients of the Jørgensen directional response law

(u

, u

, u

) 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

Problem Definition

Tip-vortices spread from a large aircraft represent a significant hazard for an aircraft that follows in its wake. This phenomenon affects the separation distance between transport aircrafts and, consequently, it remains a limiting factor on airport capac- ity. Furthermore, the flow close to the wing-tip is significant for a proper evaluation of aerodynamic loads, of the flight mechanics characteristics (i.e. ailerons control moment) and of the induced drag. In addition, a correct assessment of tip-vortex velocity profiles is fundamental to design of the ogee tips, winglets and for the wing-tip sails.

A feature of trailing vortices that make them challenging for most conventional experimental measuring techniques is the Wandering. Trailing vortices in a wind tunnel meander in space, the core location fluctuates erratically in time at a specific down-stream position. This motion seems to be a universal feature of wind tunnel generated vortices. Wandering may be self-induced by the shear layers which wrap around the vortex core or a consequence of free-stream turbulence. This meander- ing implies that any time-averaged Eulerian measurements, carried out by static experimental techniques, are actually a weighted average in both time and space.

Thus, the measured vortex appears more diffuse than what it should be in real-

ity. Suitable measuring techniques are PIV and Rapid Scanning, which attempt to

achieve a fixed vortex for the whole measuring time. This objective is reached for

each snapshot with PIV and with rapid scanning by traversing a probe sufficiently

fast through the vortex core.

State of the Art

There is a large amount of literature describing experimental studies of tip vortices, but only few of them take wandering smoothing effects into account.

Chigier & Corsiglia (1972) [9] and Corsiglia et al (1973) [10] compared mea- surements carried out by a fixed three sensors hot wire anemometer with tests performed with a rapid scanning. The latter consists of traversing an anemometer fixed on a rotating arm through the vortex core, to enable the latter to be consid- ered roughly fixed during each scan. They found that static measurements are very susceptible to wandering. Fluctuations of the axial velocity signals were already observed by Green & Acosta (1991) [17]. They found oscillation amplitudes of the axial velocity as large as the free-stream velocity at the vortex centreline, and these fluctuations fell rapidly with increasing distance from the centreline. For an angle of attack of 10 the fluctuations consisted of both ”fast” and ”slow” components, for 5 only of ”fast”. The unsteadiness in tangential velocity was less than for the axial component, and it became larger moving downstream. Shekarriz et al (1992) [34] observed by LDV measurements that the vortex seems to fluctuate primarily in the spanwise direction and less in the normal one. Also Yeung & Lee (1999) [37] evaluated wandering characteristics based on PIV data; they concluded that the wandering amplitude was comparable with the core radius and the maximum rate of wandering was roughly 4% of the free-stream velocity. Regarding delta wings, Gursul & Xie (2000) [18] attributed the random displacements of the vor- tex to the non-linear interaction of several small-scale vortices, generated by the Kelvin-Helmotz instability, with the primary vortex core.

Jaquin et al (2001) [19] proposed four possible causes for wandering: the vortex could be un-stabilized by wind-tunnel unsteadiness, turbulence in the surround- ing shear layer, co-operative instabilities or propagation of unsteadiness from the model. They showed that wandering was apparently insensitive to the free-stream unsteadiness.

Surely, the strong point of the survey on wing-tip vortex wandering is the work

of Devenport et al (1996) [12]. They described wandering motion by a bi-variate

normal probability density function, even though the authors did not support this

hypothesis with any experimental data. For this method the vortex is assumed to be

axisymmetric, the wandering independent of any turbulent motion, and velocities associated with the wandering itself negligible in comparison with those generated by the vortex. Obviously, all these hypothesis are not generally confirmed except for particular circumstances. With these assumptions, the mean velocity compo- nents and the mean Reynolds stresses, which correspond to the experimental data measured by static techniques, were expressed as the convolution of the actual field of those quantities with the bi-variate probability density function which repre- sents the wandering. Furthermore, to solve analytically the convolution integrals, the axial velocity profiles and the axial vorticity are fitted by sums of Gaussian functions; this is not always experimentally possible (due, e.g., to the presence of secondary vorticity structures) and, in addition, it is known that the azimuthal velocity profile of a fully rolled-up vortex is better represented by different models as, e.g., the Hoffmann & Joubert’s model. In summary, the fitting of the measured average velocity profiles by gaussian functions may be a non-negligible error source, and possible flow asymmetries are not taken into account.

The bi-variate normal probability density function, which represents wandering,

is characterized by two wandering amplitudes (σ

and σ

, for the spanwise and

normal direction, respectively), and a non-isotropy parameter e, which represents

the principal axes orientation of the vortex oscillation with respect to the frame of

reference. σ

and σ

are evaluated by dividing the root mean square value of the

normal and spanwise velocity, respectively, with the tangential velocity gradient

measured at the mean vortex centre. Obviously, those quantity are a good index

of the wandering amplitudes in both directions, but the authors did not support

this hypothesis with any explanation or numerical simulation. The anisotropy

parameter e is evaluated through the cross-correlation coefficient between the v and

w velocity components, measured at the mean vortex centre. These authors did not

explain how this quantity could be correlated with the directions of the principal

axes of the motion of the vortex and how the latter were evaluated. The preferred

direction of wandering was observed by Devenport et al to be between 53

and 69

in all cases, measured from the normal to the spanwise direction. The wandering

amplitude was found to increase roughly linear with increasing the downstream

distance; from 10% of the core radius up to 35% moving downstream from 5 to 35

chordlengths. Wandering was responsible for 12% and 15% errors in the measured

core radius and peak tangential velocity, respectively. The wandering amplitudes grew with increasing the free-stream velocity, probably due to the increased wake turbulence, but it decreased with growing the angle of attack. They assessed that the most important source of wandering is wind-tunnel unsteadiness, consequently, wandering decreases as the strength of the vortex is increased.

Conversely, Rokhsaz et al (2000) [33] showed that wandering amplitudes grew by increasing the angle of attack, which is opposite to the finding of Devenport et al. The flow separation at the higher angles of attack contributed to the increasing in wandering.

Moreover, Devenport et al found, by a spectral analysis, that velocity spectra collapsed when normalized with velocity and length parameters defined on the wake characteristics, concluding, supported by all the velocity spectra, that the vortex core is free of any turbulent motion. Furthermore, in Devenport et al (1998) [13] it was found that outside the core the azimuthal velocity spectra contain a single maximum at a non-dimensional frequency of about 20 (corresponding to a wavelength almost equal to the core radius), which was attributed to the passage frequency of large spanwise oriented eddies. Analogous spectral status was found by Bandyopadhyay et al (1991) [1] and Beninati & Marshall (2005) [4]. Devenport et al imputed to wandering the high spectral levels at non-dimensional frequencies lower than 20. Frequencies greater than 200 revealed the laminar nature of the flow in the vortex core. The spectral range between 20 and 200 was attributed to the buffeting of the core produced by the surrounding wake turbulence. An analogous analysis was performed by Beninati & Marshall, but the threshold frequencies were 0.2 and 2 for the wandering domain, the latter corresponding to a wavelength roughly equal to the vortex core diameter. For the buffeting generated by the surrounding wake the spectral range was defined to be between 2 up to 20 .

Heyes et al (2004) [21] evaluated wandering effects by re-centering PIV data.

They assessed that the Devenport et al assumption of using a bi-variate normal probability density function could be valid, and their corrections were in good agreement with these predicted by the Devenport et al method. They found 12.5%

over prediction of the core radius and 6% under prediction of the peak tangential

velocity. The errors were greater for lower angles of attack. They found that the

wandering magnitude increases linearly with streamwise distance; a linear reduc-

tion was found by increasing the angle of attack, hence they concluded that the mechanism responsible for wandering is not self induced, as proposed by Rokhsaz et al, but rather the vortex is responding to an external influence, for example the background turbulence level, and the tip vortex becomes less susceptible as the vortex strength is increased.

Previous works developed for the Department of Aerospace Engineering were focused on the analysis on the tip vortex dynamic and morphology. A series of experimental campaigns were performed with the same facility to carry out grid measurements both using a vorticimeter and an hot wire anemometer: Binni &

Raco (2003) [6] and even more Iungo (2003) [23] confirmed that wandering has the effect to modify the measurements of the vorticity that appears more diffuse than in reality. Barbaro (2005) [2] performed a frequency analysis on signals carried out using a five hole pressure probe. He assessed that wandering produces low frequency oscillations and the amplitude of these oscillations increases moving downstream.

Iungo [24] performed several numerical simulations of the wandering of a Lamb-

Oseen vortex in order to evaluate the possibility to characterize wandering from

static measurements. Wandering was simulated representing the vortex centre lo-

cations through a bi-variate probability density function. It was found that small

wandering amplitudes are well evaluated from the ratio between the RMS value of

the azimuthal velocity and its slope measured at the mean vortex centre. Further-

more, Iungo found that the principal axes of wandering are well predicted from

the opposite of the crosscorrelation coefficient between the spanwise and the nor-

mal velocities measured at the mean vortex centre. Several algorithms were then

applied to correct wandering averaging effects. The corrections performed were

very accurate for the simulations with small wandering amplitudes whereas errors

become larger with increasing the wandering amplitudes. In the same work, an

experimental campaign was performed regarding the tip vortex generated from a

NACA 0012 half-wing model. Preliminary flow visualizations, with laser sheet and

smoke injected, were carried out to characterize the vortex wandering. From the

videos it was observed that the wandering was confirmed to be not a regular oscil-

lation but to be characterized by abrupt displacements, and to be more evident for

angles of attack close to the wing-stall where, probably, the flow separated at high

incidences generates an increased instability. Successively, the whole procedure to

evaluate wandering from static measurements and to correct mean velocity field from wandering effects was applied to the data carried out experimentally through static five hole probe measurements. The correction methods highlight the large effects of wandering on the mean flow field measured with static techniques; in extreme circumstance the actual peak azimuthal velocity was 70% greater than the measured value. Iungo also found that the wandering amplitude increases roughly as the square root of the streamwise distance, conversely it is always reduced with increasing the strength of the vortex (i.e. with increasing the angle of attack and the free-stream velocity).

Main Objective of the Present Work

The main objective of the present investigation is to characterize wandering from a comparison between rapid scanning and static measurements of the flow field of a wing-tip vortex, and then to determine the effects produced by wandering on the velocity signals carried out by static measurement techniques.

Firstly, a measurements campaign of the tip vortex generated from a N ACA 0012 half-wing model was performed using the rapid scanning measurement technique.

For each conditions 1400 scans across the mean vortex centre were executed. The velocity signals were used to find the instantaneous vortex centre location at each scan, in order to determine their distribution in the cross plane for each condition.

Consequently, an experimental probability density function (PDF ) of the vortex centre locations can be evaluated. The experimental PDF was then fitted with a bi-variate gaussian function (using a purpose-written least squares algorithm), in order to obtain the wandering amplitude from the standard deviations of the spanwise and normal vortex centres coordinates. The direction of the principal axis of wandering was inferred from the geometrical shape of the fitted PDF.

The wandering amplitude as obtained from the the fitted PDF of the centre locations was compared to the same parameter obtained applying the method pro- posed by Devenport et al [12] to the rapid scanning uncorrected data, founding a substantial agreement between the results of the two methods.

The determination of the vortex centre location for each scan allowed to re-

centre all the instantaneous velocity profiles by a radial coordinate and consequently

to obtain mean velocity profiles corrected from wandering effects. These time- averaged velocity profiles were compared to the ones generated from the rapid scanning not re-centred data, averaged both in time and in space and consequently affected by wandering. This comparison allowed to investigate the smoothing effects of the wandering on the axial and the tangential velocity signals.

In addition, a qualitative investigation was conducted on the secondary vortic- ity structures detected from the rapid scanning data. The localization and the characterization of the secondary vorticous structures were achieved by founding clear evidences in the velocity and in the circulation profiles and in the standard deviation of the velocity signals.

Then, a further experimental campaign was performed measuring the tip vortex flow field by traversing a three sensors hot film probe for the same flow conditions already tested with the rapid scanning. This Eulerian measurements were carried out in order to assess the interpretation of the rapid scanning data affected by wandering as data obtained from static measurements. The substantial agreement between these two data sets allows to generalize the conclusions achieved from the analysis of the rapid scanning data.

From the experimental measurements it was found that wandering amplitudes increase roughly linearly with increasing the streamwise distance, whereas they are fairly invariant by increasing the angle of attack and they decrease by increasing the free-stream velocity. However, even though the origin of wandering is still unknown, the different response of the wandering to the variation of the flow conditions depending on the vortex initial strength. If the vortex strength is weak enough, wandering is surely attenuated by increasing the vortex strength.

Finally, a spectral analysis of the 3HFP velocity signals was conducted in corre- spondence of the most significant sampling points along the traverse.

This paper is organized as follows. Sec. 1 is the introduction. The facility and the wing model used for the experimental campaign are described in Sec. 2.

The description of the probe used for the rapid scanning measurements and its

calibration is reported in Sec. 3.1. Then, the whole rapid scanning equipment

(Sec. 3.2) and all the tested conditions and locations are presented (Sec. 3.3). The

methodology followed to process the rapid scanning data is report in Sec. 3.4,

whereas the discussion of the results is presented in Sec. 3.5. Furthermore, the

discussion about the secondary vorticity structures is presented in that section.

The 3HFP and its calibration is described in Sec. 4.1, thus, all the conditions and locations tested with the 3HFP are presented (Sec. 4.2). The analysis procedure of the 3HFP data is reported in Sec. 4.3, followed by the results overview (Sec. 4.4) and the spectral analysis(Sec. 4.5). The comparison between the results obtained with the rapid scanning data and the 3HFP traverses data is pointed out in Sec. 5.

Finally, the conclusions are highlighted in Sec. 6. In addition, an overview over

the calibration methods for the hot wire-film probe is reported in Appendix A

and all the corrected coefficients of the Lekakis calibration method are reported in

Appendix B.

2.1. The Wind tunnel

The rapid scanning tests were carried out in the 2 meter Wind Tunnel (2mWT), a low speed wind tunnel at the Defence, Peace, Safety and Security (DPSS) Operative Unit of the South Africa Council for Scientific and Industrial Research (CSIR) in Pretoria (RSA). In Fig. 2.1 is reported a sketch of the 2mWT.

2500 mm

1700mm 2000 mm

U

x y z

Wing Model

g

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

The speed range of the wind tunnel is between 3 and 33 m/s. The open test

section has a length of 2500 mm. Further dimensions are depicted in Fig. 2.1. The

flow is generated by a drive system equipped with 30 kW power electric motor and

velocity is set by a dedicated control panel. The free-stream turbulence level is

about 0.75 % and the pressure gradient along the tunnel axis (

) is −0.9 % m

.

The 2mWT is equipped with a vortex generator, shown in Fig. 2.2, in order to

reduce vibrations generated from the diffuser (see [6] and [23]).

Figure 2.2. 2mWT vortex generator.

Maintaining the same convention of [6], [23] and [2] the frame of reference was defined as follows:

• origin at the tip of the wing model;

• x axis as the free-stream direction;

• y axis as the spanwise direction of the model from the root to the tip of the wing;

• z axis was consequently defined, producing a clockwise frame of reference.

The reference system is depicted in Fig. 2.1.

The facilities used in the 2mWT include the basic instrumentations to measure static and dynamic pressure and static temperature. The instruments characteris- tics are reported below:

Dynamic pressure transducer

Brand: Setra system Model: 239

Range: 0 to 0.1 PSI Output: 0 to 5 Volts DC

Static pressure transducer Brand: Huba Control Model: 15T80

Range: 800 to 1200 mbar P abs max 2 bar Output: 4 to 20 mA

Temperature probe

Brand: Wika Instruments LTD Model: 10-1100

Range: 0 to 100

C Output: 0 to 10 Volts DC

All wirings to connect the transducers with the acquisition system and to the power supply units were executed and a preliminary check to verify all instruments was performed. The pressure transducers were calibrated using the procedure explained in Sec. 3.1.1.

The 2mWT has not a permanent traversing apparatus. The movement rig, used for previous works (see [6], [23] and [2]), and adapted for the present campaign, is equipped by three stepper motors that allows displacements along the x, y and z axes of the tunnel frame of reference. The traversing apparatus is controlled in remote.

The data acquisition used is NI PXI 6052E, and its characteristics are summa-

rized in Tab. 2.1.

Analog inputs 16 SE/8 DI

Input Resolution 16 bits

Max Sampling Rate 333kS/s

Input Range ±0.05 to ±10V

Output Rate 333kS/s

Output Range ±10V

Digital I/O 8

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

2.1.1. The model

The model used for the present experimental campaign was an unswept half wing with constant cross-sections NACA 0012 and bluff tip. No boundary layer trips were applied on the wing model during tests. Its geometrical characteristics were:

Wing semi-span (b/2): 700 mm Mean geometric chord (c): 245 mm Mean aerodynamical chord (c

): 260 mm Root chord (c

): 350 mm

Tip chord (c

): 140 mm Taper ratio (λ): 0.4 Aspect ratio (AR): 5.7

Projected surface (S): 171500 mm

The model was mounted on a base plate upon which are assembled two alu- minium splitter plates from both sides. A sketch of this set-up is reported in Fig. 2.3.

Half W

ing Model

Base Plate

Splitter Plates

Figure 2.3. Half wing model positioning.

The whole base was fixed to a support which allows to set the angle of attack of

the wing by a manual rotating gear. The structure was designed in order to place

the wing tip at the centre of the test section. Once the model was positioned, the

horizontality of the base plate was checked with an inclinometer and the verticality

of the wing was controlled using a theodolite.

Introduction

Wandering implies that trailing vortices in a wind tunnel meander in space and the core location fluctuates erratically in time. Thus, any time-averaged Eulerian mea- surements, carried out by static experimental techniques, are actually a weighted average in both time and space.

Wing-tip vortices were measured using the Rapid Scanning technique described in Corsiglia et al [10]. Rapid scanning consists of traversing a probe fixed on a rotating arm through the vortex core. The aim of the rapid scanning is to perform a scan as fast as possible through the vortex core in order to consider the vortex fixed during each scan. Measurements carried out with this dynamic measuring technique are theoretically not affected by wandering.

Rather than using a linear traversing mechanism that may arise structural issues, a rotating arm apparatus was built which allows a constant traversing speed. A five hole pressure probe was mounted on the tip of the rotating arm.

Making many scans along a fixed path, a series of essentially instantaneous ve-

locity profiles in proximity to the vortex core were obtained.

3.1. Five Hole Probe & Calibration

The probe chosen for the rapid scanning tests is a five hole pressure probe (denoted as 5HP in the following) built by Aeroprobe Corporation (Blacksburg, Virginia, U.S.A.). The main dimensions of the probe are sketched in Fig. 3.1. The instru- ment sensitive part is disposed on the hemispheric probe tip which has a diameter of 3 mm. The five steel taps (see Fig. 3.2) are linked with the relative transducer by rubber tubes. The transducers are placed inside the probe stem with a distance between the transducers and the probe tip less than 150 mm to achieve an ade- quate frequency resolution (150 Hz suggested by Iungo in [24]). The differential transducers use static pressure as reference pressure.

Figure 3.1. 5HP shape and geometry, length [mm], angles [

].

Signals from transducers are monitored by Agilent / HP 3456A Digital Voltmeter 6 digit meter. Its characteristics are: 300 readings per second; 100nV resolution;

ranges: 0.1 volt to 1000 volts DC; 1.0 volt to 1000 volts AC RMS; resistance:100Ω to 1GΩ. The Data Acquisition Control Unit (DAQ) is an Agilent / HP 3497A. This DAQ device works in a sequential way. It is equipped of an automatic optimizer that maximize the sampling rate according to the number of samples and channel required.

The calibration procedure was carried out at CSIR Calibration Tunnel, a sub-

sonic, open circuit, low turbulence, closed test section wind tunnel. The calibration

A

A

x

5 1 2

3 4

y 3

z

Sec. A-A

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

tunnel presents an octagonal working section, 610 mm wide and 740 mm long and it is equipped with an electric fan able to give an air flow speed range between 5 and 35 m/s.

The permanent facilities of this wind tunnel include the basic instrumentation for measuring static pressure, static temperature and humidity of the testing chamber.

In agreement with previous works and with the software developed on purpose, the reference system, fixed with the wind tunnel, is set in the following way:

• origin at the probe tip;

• x axis as the free-stream direction;

• y axis directed vertically up;

• z axis consequently defined to produce a clockwise frame of reference.

The probe was fixed to the holder by a shaft directly joint with the roll stepper

motor. That shaft has a plane slot where was possible to match the proper reference

surface of the probe, obtained in the hexagonal section part at the back. The

fastener was realized using steel clamps and aluminium tape. In Fig. 3.3 is reported

a picture of the probe set-up. The correct positioning of the measuring instrument,

according to the wind tunnel axis, was checked using a theodolite, whereas the zero

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

roll angle was reached using a digital inclinometer referred to the ground plane of the working section .

The movement apparatus is controlled by two stepper motors whit their relatives drivers; the first, placed outside of the working section, provides the rotation of the probe in the horizontal plane, the second one allows the rolling motion of the probe.

The software for both control the movement rig and manage the data acquired was developed in Labview environment by Eng. Peter Skinner. The positioning error was checked with an inclinometer for the roll angle, whereas the pitch angle was checked through the projected position of a lead tied on the probe support then measured with a protractor.

3.1.1. Preliminary Tests

Last step of the set-up was the transducers calibration. The calibration was per-

formed applying different pressure values to the reference pressure of the probe

while the taps were disconnected. A sketch of this arrangement is reported in

Fig. 3.4. Voltage values from the 5HP were then stored for the imposed pressure

values from 100 P a to 1000 P a with a step of 100 P a. Consecutively, the slope of

Free edge

p

patm

p1

p

p

p

Output signals Voltmeter

Trasducer

Trasducers

Water Manometer

Figure 3.4. Arrangement for transducers calibration.

the calibration line was evaluated for each transducer.

Some preliminary tests were carried out to determine the minimum number of samples for each measurement in order to achieve steady statistical signals. Roll angle was kept at 0

, while for each value of the free-stream velocity within the range of interest, 10, 20, 30 m/s, three pitch angles were tested: −40

, 0

, 40

. This series of test was repeated for 10, 20, 30, 50, 100 samples, whereas the sampling frequency was maximized by the data acquisition system.

The tests goal was to analyze the behavior and the values of relative errors calculated comparing the data carried out at 100 samples with data carried out with less samples, in the following way:

err =   

p |

−p |

p |

 (3.1)

where N is the number of samples used; this calculation was repeated for each condition of speed, yaw, angle of attack and for each channel. The tested conditions were defined combining the following parameters:

- Velocity: 10, 20, 30 m/s - Pitch angle: −40

, 0

, 40

- Roll angle: 0

Relative errors were then processed in two way:

10 20 30 50 0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

N° of Samples

Relative errors

Statistical Steadiness

−40°0°

40°

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

10 20 30 50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

N° of Samples

Relative errors

Statistical Steadiness

10m/s 20m/s 30m/s

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

• averaged on velocities and on channels;

• averaged on pitch angles and on channels;

Results of the first process are shown in Fig. 3.5, whereas results of the second one are shown in Fig. 3.6.

As expected, relative errors decrease with increasing the number of samples.

Furthermore, errors with the probe aligned with the free-stream velocity vector are always less than in the other two cases.

Tests performed with more than 10 samples produce a roughly adequate error.

However, errors slightly decrease by increasing the number of samples.

Finally, 20 samples were chosen for each data point, which assure a relative error

less than 2.5%, while sampling time remains substantially the same as using less

points. Indeed, during some preliminary tests it was noticed a non-linear increasing

of the sampling time varying the samples number. In particular this non-linear

behavior, over a threshold value of about 20 samples per channel, leads to a very

long acquisition time.

3.1.2. Five Hole Probe Calibration Method

The procedure to calibrate the five hole probe derives from the calibration method described in [15], applied to a seven hole probe. The adaptation of this method to a five hole probe does not need any particular effort.

The calibration method provides a differentiation depending on the angle of attack of the flow with respect to the probe axis: α < 30

is denoted as low flow angle, whereas α > 30

is denoted as high flow angle. In this Section it will be described only the method developed for low flow angles, considering that the procedure for high flow angles proceeds from the previous one and the present work is not focused on the calibration philosophy of this kind of probes.

More precisely the method performs a division of the probe tip in five sectors which division lines are based on the isobars depicted in Fig. 3.7.

1

5 3

2

4 Sector 1 Sector 2

Sector 3

Sector 4 Sector 5

Figure 3.7. Division of angular space.

Velocity vectors are measured adopting the tangential frame of reference illus-

trated in Fig. 3.8 and in Tab. 3.1. In this frame of reference, α

is taken to be the

projection on the vertical plane of the angle between the velocity vector and the

probe axis. And β

is defined as the projection on the horizontal plane of the angle

between the probe axis and the relative wind. This system is slightly different with

z x y

w

v u

a=a

b b

V

Figure 3.8. Low flow angle reference system.

CONVENTIONAL TANGENT

u = V cos α cos β

α

= arctan

v = V sin β

β

= arctan

w = V sin α cos β

Table 3.1. Tangential reference system definition.

respect to the α − β system, where α is the angle of attack and β the angle of sideslip.

The traversing apparatus allows rolling and pitching motion of the probe, whereas

the calibration procedure requires the angular quantities in the α − β frame of