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

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 | N −p | ¹⁰⁰ p | 100

 (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 _T is taken to be the

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

probe axis. And β _T 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 _T

b b _T

V

Figure 3.8. Low flow angle reference system.

CONVENTIONAL TANGENT

u = V cos α cos β

α T = arctan ^w _u v = V sin β

β _T = arctan _u ^v 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

reference. The suitable transformation is reported below.

( α = arctan(tan θ cos φ)

β = arctan(tan θ sin φ) (3.2)

Where θ is the pitch angle and φ is the roll angle.

For low angles it is desirable to define dimensionless pressure coefficients which utilize all five measured probe pressures because all of them are sensitive to any changes in flow angularity with respect to the probe axis. From Fig. 3.2 a pressure coefficient sensitive to changes in angle of attack is defined as:

C α = P 4 − P ² P ₁ − P 2,3,4,5

(3.3) The numerator is function only of the measured pressures at taps 4 and 2 that are the only ones influenced by varying the angle of attack, and the denominator non-dimensionalizes the term with the apparent dynamic pressure. This pseudo- dynamic pressure is obtained from the difference between the central port pressure, P ₁ , which approximates the total pressure at low angles, and the average of the four surrounding pressures, P _2,3,4,5 , which approximates the static pressure.

Analogously, the pressure coefficient sensitive to the sideslip angle is defined as:

C _β = P 3 − P ⁵ P ₁ − P 2,3,4,5

(3.4) Once the two angular pressure coefficients are defined, other two pressure coef- ficients, C ₀ and C _q , are then evaluated:

C 0 = P 1 − P ⁰ P ₁ − P 2,3,4,5

(3.5)

C _q = P ₁ − P 2,3,4,5

P 0 − P ∞

(3.6) C ₀ is the non-dimensional apparent total pressure coefficient, C _q is the ratio between pseudo dynamic pressure and the actual one.

At high angles of attack, as illustrated in Fig. 3.9, pressure ports lying in this

separated region are insensitive to changes in flow angularity. Indeed, the pressure

coefficients for high flow angle must be defined so that they include only the pres-

sure measured from ports in attached flow. Therefore these coefficient are slightly

different from the ones defined for low angles of attack.

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

The velocity vector is defined by the quantities α _T , β _T , C ₀ and C _q . A fourth order polynomial expansion in two variables, viz. the angular coefficients, is used to solve these quantities. Using matrix notation the expansion takes the following form:













 A 1

A ₂ .. . A _n















=















1 C α

C _β

C _α ²

C α

C _β

C _β ²

. . . C _β ⁴

1 C _α

C _β

C _α ²

C _α

C _β

C _β ²

. . . C _β ⁴

. . . . . . . . . . . . . . . . . . . . . . . . 1 C _α

C _β

C _α ²

C _α

C _β

C _β ²

. . . C _β ⁴



























 K ₁ ^A K ₂ ^A .. . K ₁₅ ^A















(3.7)

The Eq. 3.7 can be rewritten as:

{A} = [C]{K ^A } (3.8)

where:

• n is the data points number of a particular sector;

• {A} is a n × 1 vector that represents one of the quantity among α T , β _T , C ₀

and C _q . During the calibration process, the quantities within the {A} vectors

are constituted by the known tunnel conditions;

• [C] is a n × 15 matrix which terms are determined in the calibration phase from the measured probe pressures;

• {K ^A } is a 15 × 1 vector constituited by calibration coefficients referred to a particular A quantity,

The calibration procedure involves the calculation of the unknown {K ^A } terms.

This calculation is performed by rearranging Eq. 3.8 to solve for the unknown calibration coefficients and the solution is:

{K ^A } = [C ^T C] ⁻¹ [C] ^T {A} (3.9) In this way the calibration coefficients are obtained with a least square curve fitting to the experimental data.

In measurements phase the desired flow properties in the A vectors are deter-

mined explicitly by Eq. 3.8, since the {K ^A } terms are already known, whereas the

angular pressure coefficients that fill the C − matrix are calculated from pressure

measurements.

3.1.3. Calibration Tests

In order to obtain, as described in 3.1.2, the entire set of calibration coefficients, a group of tests was performed:

• provide an adequate data set to solve the analytical calculations;

• be representative for all free-stream velocities considered in the present ex- perimental campaign.

The tests were performed at free-stream velocity of 5, 10, 20, 30 m/sec, and 20 samples for each measure point were acquired. The movement rig allows to change pitch (θ) and roll (φ) angle of the probe, these angular parameters, for each run of the tunnel, were set as follows:

• −50 ^◦ ≤ θ ≤ 0 ^◦

• −180 ^◦ ≤ φ ≤ 180 ^◦

The probe is moved in pitch and roll with angular steps of 5 ^◦ and 10 ^◦ respectively.

The motion was performed first in φ angle and then in θ. The distribution of data points in a α − β plane is shown in Fig. 3.10.

−60 −40 −20 0 20 40 60

−50

−40

−30

−20

−10 0 10 20 30 40 50

α

β

Figure 3.10. Distribution of data points in α-β plane used for the 5HP calibration.

A preliminary test was performed before each calibration run in order to check

if the probe is properly set. The tested conditions are reported in Tab. 3.2.

V elocity pitch range pitch step roll range roll step N ^◦

[m/s] [deg] [deg] [deg] [deg] samples

5 −50 ≤ θ ≤ 0 5 −180 ≤ φ ≤ 180 10 20

10 −50 ≤ θ ≤ 0 5 −180 ≤ φ ≤ 180 10 20

20 −50 ≤ θ ≤ 0 5 −180 ≤ φ ≤ 180 10 20

30 −50 ≤ θ ≤ 0 5 −180 ≤ φ ≤ 180 10 20

Table 3.2. Calibration 5HP - Test matrix.

In Fig. 3.11 is reported an example of the data acquired for the 5HP calibration ¹ . The pressure values are plotted against roll angle for each pitch condition, fixed free-stream velocity, and for each channel.

All calibration curves present the common sinusoidal shape varying roll angle.

An approximatively 30 ^◦ shift between the probe holes and the reference surface on its support it is evident from the plots.

All acquired data are the input for the analytical calibration procedure described in Sec. 3.1.2. Several matrices were created combining data obtained at different wind tunnel velocities ² .

Other two tests series were conducted (summarized in Tab. 3.3) to verify and to validate the entire calibration procedure. The first is constituted of three wind tunnel runs at some intermediate velocities, 8, 12, 15 m/s and at the same pitch and roll angles of the calibration tests; the second series was done varying the free-stream velocity.

Analyzing the whole calibration data it was highlighted an anomalous behavior of the tap number 2 of the probe, as shown in Fig. 3.12, compared with transducer 3 for the same tunnel run at 5 m/s. It is clear a bigger scatter of data corresponding to the transducer 2.

In order to demonstrate that the scatter was not produced by any aerodynamical effect another test was carried out doing a roll scan without air flow through the

1

All the figures relative to the 5HP calibration are stored in Archives [D:Archives\5HP Calibration\Plots] in the folders: vel5, vel10, vel20, vel30.

2

All the matrices are contained in Archives [D:Archives\5HP Calibration\Calibration Matrix]

both in text format and in Excel format

V elocity pitch range pitch step roll range roll step N ^◦

[m/s] [deg] [deg] [deg] [deg] samples

8, 12, 15 −50 ≤ θ ≤ 0 5 −180 ≤ φ ≤ 180 30 20

5, 6, 7, 8, 9

10, 11, 12, 13 −30 ≤ θ ≤ 30 30 −180 ≤ φ ≤ 180 90 20 14, 15, 20, 30

Table 3.3. Calibration 5HP - Validation tests.

wind tunnel. The test confirmed a non-regular behavior of the channel 2 data (see Fig. 3.13). In the following Tab. 3.4 are reported the mean, maximum and minimum pressure values registered by each channel and the relative standard deviation, σ.

Channel 1 Channel 2 Channel 3 Channel 4 Channel 5

P mean [P a] -0.1 -0.2 -0.3 0.1 -0.4

P max [P a] 0.7 1.8 0.2 0.5 -0.1

P min [P a] -0.9 -1.8 -0.7 -0.2 -0.6

σ [P a] 0.4 0.9 0.2 0.2 0.1

Table 3.4. Roll scan - Statistical values.

The conclusion is that the irregular behavior of channel 2 is originated from a

defect of its transducer that can be considered as an intrinsic characteristic of the

probe and it cannot be corrected.

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−200

−150

−100

−50 0 50 100 150 200

Channel 1 U_∞= 20 [m/s]

Roll [deg]

Pressure [Pa]

Pitch [deg]

−50°

−45°−40°

−35°

−30°−25°

−20°

−15°−10°

−5°

0°

(a)

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−200

−150

−100

−50 0 50 100 150 200

Channel 2 U_∞= 20 [m/s]

Roll [deg]

Pressure [Pa]

Pitch [deg]

−50°

−45°−40°

−35°

−30°−25°

−20°−15°

−10°

−5°0°

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−200

−150

−100

−50 0 50 100 150 200

Channel 3 U_∞= 20 [m/s]

Roll [deg]

Pressure [Pa]

Pitch [deg]

−50°

−45°−40°

−35°

−30°−25°

−20°−15°

−10°

−5°0°

(b) (c)

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−200

−150

−100

−50 0 50 100 150 200

Channel 4 U_∞= 20 [m/s]

Roll [deg]

Pressure [Pa]

Pitch [deg]

−50°−45°

−40°

−35°−30°

−25°

−20°−15°

−10°

−5°0°

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−200

−150

−100

−50 0 50 100 150 200

Channel 5 U_∞= 20 [m/s]

Roll [deg]

Pressure [Pa]

Pitch [deg]

−50°−45°

−40°

−35°−30°

−25°

−20°−15°

−10°

−5°0°

(d) (e)

Figure 3.11. Example of 5HP calibration tests at U _∞ = 20 m/s.

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−20

−15

−10

−5 0 5 10 15 20

Channel 2 U_∞= 5 [m/s]

Roll [deg]

Pressure [Pa]

Pitch [deg]

−50°−45°

−40°

−35°−30°

−25°

−20°−15°

−10°

−5°0°

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−20

−15

−10

−5 0 5 10 15 20

Channel 3 U_∞= 5 [m/s]

Roll [deg]

Pressure [Pa]

Pitch [deg]

−50°−45°

−40°

−35°−30°

−25°

−20°−15°

−10°

−5°0°

(a) (b)

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

−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

Roll [deg]

Pressure [Pa]

Channel 1 Channel 2 Channel 3 Channel 4 Channel 5

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

3.1.4. Calibration Accuracy

The data set was used to generate several calibration matrices following the pro- cedure explained in Sec. 3.1.2. A preliminary check on errors made by applying the calibration method was carried out ³ : absolute errors made in velocity mag- nitude (| − →

V |), α and β computations were evaluated and statistically analyzed for each data point. Indeed, in order to have an indicative parameter of the method accuracy, for each analyzed quantity among | − →

V |, α, β was calculated the standard deviation σ of absolute errors, as written in Eq. 3.10, 3.11 and 3.12; where abso- lute error meants the difference between the values achieved from the application of the calibration matrix to the 5HP pressure data (indicated with the subscript

”calculated”) and the known values imposed in the calibration procedure fixing the asymptotic flow velocity and probe position (indicated with the apex). The bar indicates the average of the quantities overlined.

σ | − → V | =

v u u t

X

i

 

| − →

V | calculated − | − → V | ⁰ 

−



| − →

V | calculated − | − → V | ⁰

 2

(3.10)

σ α = s

X

i

(α calculated − α ⁰ ) − α ^calculated − α ⁰
 2

(3.11)

σ _β = s

X

i

(β _calculated − β ⁰ ) − β calculated − β ⁰
 2

(3.12) Firstly a matrix was calculated from the data obtained by tests executed at all wind tunnel velocity (5, 10, 20, 30 m/s). The standard deviation values relative to that matrix, shown in Tab. 3.5, were too high to satisfy the requirements of the present calibration work.

The data carried out at 5 m/s, the minimum wind tunnel velocity, degradate the calibration matrix performances, thus

it was decided to not implement this data in the calibration matrix. How can be observed in Tab. 3.5, the standard deviation of errors gets less using a calibration matrix that does not involve measurements corresponding to 5 m/s and this matrix was called matrix A ⁴ . To use matrix A to analyze flow field with a free-stream

3

Complete sheet of calculation is reported in Archives [D:Archives\5HP Calibration\Calibration Matrix\5HP Cal matrix 200511 new10.xls]

4

The matrix A is reported in Archives [D:Archives\5HP Calibration\Calibration

Matrix\matrix 200511 V30V20NewV10plusrecalc.txt]

velocity of 5 m/s introduces too high errors. However, it should be considered that the present calibration was performed for flows with a free-stream velocity higher than 10 m/s.

Similar discussion needs to be done about data carried out at 10 m/s; indeed, the calibration matrix B ⁵ , constructed without these data, can more accurately analyze flows within a range of asymptotic velocities between 20 and 30 m/s, whereas its precision decreases if the same matrix is used to study pressure data in a flow of 10 m/s or less, see again Tab. 3.5.

Free-stream velocity, U

5 m/s 10 m/s

Velocities involved

in matrix construction σ

σ

σ

σ

σ

σ

5, 10, 20, 30 m/s 0.66 4.42 5.00 0.42 2.38 2.39 (matrix A) 10, 20, 30 m/s N aN 24.36 21.11 0.39 2.05 2.02 (matrix B) 20, 30 m/s N aN 9.33 11.53 0.27 1.27 1.26

Free-stream velocity, U

20 m/s 30 m/s

Velocities involved in matrix construction σ

V|

σ

σ

σ

V|

σ

σ

5, 10, 20, 30 m/s 0.90 2.14 2.28 1.51 2.26 2.34 (matrix A) 10, 20, 30 m/s 0.21 0.40 0.40 0.30 0.37 0.43 (matrix B) 20, 30 m/s 0.27 0.72 0.72 0.54 0.86 0.80

Table 3.5. Standard deviation of absolute errors applying different calibration ma- trices.

The two matrices A and B, with and without data relative to 10 m/s, were used for reducing measurements for different tests conditions in order to reduce errors:

matrix A for the tests performed at U _∞ = 10 m/s; matrix B for tests at U _∞ = 20 and 30 m/s.

In Fig. 3.14 are reported, as an exemple, the errors made evaluating flow char- acteristics using the appropriate matrix for U _∞ = 10 m/s and U _∞ = 20 m/s.

These errors are absolute for α and β quantities and relative for | − →

V | quantity, non-dimensionalizing the latter one with U _∞ .

5

For matrix B consultation, see Archives [D:Archives\5HP Calibration\Calibration

Matrix\matrix 200511 V30V20plusrecalc.txt].

The tested expected pitch angles are much less than the maximum tested value of 45 ^◦ , thus, the evaluated errors are surely conservative. Indeed errors reported in Tab. 3.5 take all the angular configurations into account, whereas in operative situations the maximum angle between the velocity vector and the probe axis that will be found in regions close to the tip vortex will not overtake 30 ^◦ .

In particular, uncertainties in velocity magnitude evaluation, especially at high pitch angles, rise proportionally to the free-stream speed. Consequently, relative errors remain fairly invariant or at least decrease with increasing U _∞ , as shown in the three plots relative to that adimensionalized quantity, see Fig. 3.14. Values of these uncertainties stay below 8% for U _∞ = 10 m/s, and below 5% for higher velocities.

In terms of accuracy in velocity vector angles calculation can be lined out that, whereas condition U _∞ = 10 m/s uncertainties are ±3 ^◦ , both in α and in β, for higher velocity tests, errors are ±1.5 ^◦ if pitch angles over 40 ^◦ (red markers in figure) are not taken into account.

In Fig. 3.14 is clear that, especially for low angles configurations, the points distribution is not random, but it is smoothly sinusoidal with a non-zero mean value. This behavior can be due to the residual bias error made in probe positioning.

In experimental rig a misalignment of the rolling axis with the tunnel axis can be held responsible for the function error oscillations, that anyway should have the same value in α and in β. On the other hand, an imperfection in parallelism between probe axis and rolling axis can originate a constant error, both in angle of attack and of sideslip, but it should have different values for the two parameters on the entire roll rotation. Another possible reason of the error behavior is that the transducers work at the lower measuring limit, and one of them present a noisy response, as already lined out in Sec. 3.1.3.

As a guide line, Matrix A was applied in case of free-stream velocity between 5

and 15 m/s, whereas matrix B was used for higher velocities. Statistical study of

these data assesses that the standard deviation of errors remains unchanged with

respect to the values reported in Tab. 3.5, that guarantees the consistency of the

approach.

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−0.1

−0.08

−0.06

−0.04

−0.02 0 0.02 0.04 0.06

Roll [deg]

Module Error / U∞

−50°

−45°−40°

−35°

−30°−25°

−20°

−15°−10°

−5°0°

Pitch

−180−150−120 −90 −60 −30 0 30 60 90 120 150 180

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

Roll [deg]

Module Error / U∞

−50°

−45°−40°

−35°

−30°−25°

−20°

−15°−10°

−5°0°

Pitch

(a) U _∞ = 10 m/s (d) U _∞ = 20 m/s

−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180

−4

−3

−2

−1 0 1 2 3 4 5

Roll [deg]

Alpha Error [deg]

−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180

−4

−3

−2

−1 0 1 2 3

Roll [deg]

Alpha Error [deg]

(b) U _∞ = 10 m/s (e) U _∞ = 20 m/s

−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180

−4

−3

−2

−1 0 1 2 3 4 5

Roll [deg]

Beta Error [deg]

−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180

−5

−4

−3

−2

−1 0 1 2 3 4

Roll [deg]

Beta Error [deg]

(c) U _∞ = 10 m/s (f ) U _∞ = 20 m/s

Figure 3.14. Examples of errors in tests data: (a), (b), (c) errors on | − →

V |, α and β respectively, data carried out in test condition with U _∞ = 10 m/s and processed applying matrix A. (d), (e), (f ) errors on | − →

V |, α and

β respectively, data carried out in test condition with U _∞ = 20 m/s

and processed applying matrix B .

3.2. Setup

Rotating unit

The rotating unit, sketched in Fig. 3.15, is composed by a central rotor to which are joined, from opposite sides, two arms that support at one tip the 5HP and at the other one a dummy probe for inertial balancing. The arms are equal with a diameter of 16 mm and a length of about 1 m. The arms were connected to the rotor through a gear that allows the angular rotation. The holder that fix the probe to the rotating arm was made in the DPSS workshop.

Figure 3.15. General scheme of the rotating unit.

In addition to the 2mWT reference system, fixed with the tunnel (see Sec. 2), it was defined another reference system fixed with the probe and depicted in Fig. 3.15:

• Origin at the probe tip;

• x ^p axis directed along the probe axis from the tip to the holder;

• y p axis in the direction of the rotating arm from the centre of the rotor to the outside;

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

One of the most important geometrical dimension of the rotating unit is the distance between the centre of the rotation and the probe axis L _arm , that is 991 mm.

Two requirements were fundamental for the rotating apparatus design: to be as long as possible in order to achieve a roughly straight probe path trough the vortex core, and to be adequately small for interference issues. L arm determines the peripheral velocity, V _p , of the probe tip at a fixed rotation frequency of the rotor: V _p = 2πL arm ω. Probe velocity is 2.49 m/s for the ω value of 0.4 Hz chosen for the experiments, see the following Sec. 3.3. Another fundamental dimension is the distance between the probe tip and the arm axis, L _probe , that is 182.5 mm. The last important geometrical characteristic is the angular rotation, γ, of 6.3 ^◦ of the probe around the arm axis. This rotation is performed in order to compensate, as more as possible, the misalignment between the free-stream flow and the probe axis due to the peripheral velocity induced by the rotation. Indeed the composition of a typical free-stream velocity, U _∞ , of 20 m/s and a V _p of 2.49 m/s would give a γ = arctan( _U ^V

∞

) of 7.1 ^◦ and 6.3 ^◦ is the nearest value permitted by the gear-wheel of the arm.

The rotor has a cylindrical body with a diameter of 120 mm and it is divided in a static and a rotating part. The static part of the rotor holds a stepper motor that rotates the probe with a frequency between 0.1 and 2Hz. A digital encoder is positioned inside the rotor to measure the angular displacement of the arms. A flat plate on the bottom allows to assemble the rotor to the traversing apparatus.

Electrical signals and power supplies of the 5HP and the reference pressure trans- ducer from the rotating to the static part were transferred by using 24 different sliding contacts inside the rotor.

The incremental optical encoder is a device to convert the rotation into a digital

signal. It measures incremental angular steps, but it does not produce neither the

absolute position of the arm neither the spin direction. The encoder consists of a

rotating disk, a light source, and a photodetector (light sensor). A series of opaque

and transparent sectors are coded into the disk, which is mounted on a rotating

shaft (see Fig. 3.16). As the disk rotates, these patterns interrupt the light emitted

onto the photodetector, generating a digital signal output. The angular resolution

of the encoder is 0.036 ^◦ , since the full circle angle is divided in 10 ⁴ stations. It should

be noted that the discrete way of working of the encoder makes the sampling points

not random displaced along the circular path of the probe, but evenly spaced at fixed angular stations.The length of the arc corresponding to the encoder angular resolution ( ^2πL ₁₀

) yields the minimum detectable distance between two several samples: 0.0025 c = 0.6125 mm.

Digital data, generated by the encoder, were acquired by a NI PXI 6602, whose characteristics are summarized in Tab. 3.6.

Counter/Timer 8

Size 32 bits

Max Sampling Rate 333kS/s

Max Source Frequency 80M Hz

Digital I/O up to 32

Table 3.6. NI PXI 6602 specifications.

In order to obtain accurate pressure measurements with the five differential trans-

Light Sensor(s)

Shaft

Code Track

Light Source

Rotating Disk

Figure 3.16. Optical encoder.

ducers of the 5HP it is fundamental to know very precisely the pressure acting in the reference chamber of each transducer. This necessity leaded to place the refer- ence pressure transducer on the rotating part of the rotor. It was placed in a slot obtained on the rotor frontal part. The reference pressure was the one acting in a 1 m tube connected from one side to the reference chambers of the 5HP, and from the other side to the reference pressure transducer. In such a way the tube itself became an isolated closed chamber in which the pressure remained almost constant.

The effect of the gravity in the tube was not relevant because during measurements the probe crossed the vortex when the arm and so the tube were almost horizontal.

The effect of the centrifugal force: ∆p = ρa _c L _arm = ρ ^V

2 p

L

L _arm = ρV _p ² was esti- mate to be less than 6 P a, for V _p = 2.49 m/s (ω = 0.4 Hz). This value is negligible for the 5HP accuracy.

Transducers calibration

Setup had been completed by the calibration in situ of all pressure transducers.

Five hole probe transducers were calibrated as explained in Sec. 3.1. Output of this procedure was the slope of the curve of pressure values of each tap against the raw signal.

Velocity vectors analysis

The calibration matrices were applied to data signals from 5HP to achieve local velocity vectors in each sample. The software automatically corrects the three local velocity components in order to epurate themselves from velocity vector due to probe rotation. The correction of the velocity vector acquired from the 5HP was separated from the components due to the rotation of the probe. The latter are evaluated as follows:



 



 



u r = V p cos(δ) sin(γ) = 2πωL arm sin(γ)

v _r = V _p sin(δ) sin(γ) = 2πωL _probe cos(γ) sin(γ) w _r = V _p cos(δ) cos(γ) = 2πωL _arm cos(γ)

(3.13)

Where:

tan(δ) = L _probe · sin(γ)

L _arm

Figure 3.17. Geometrical quantity involved in software correction.

V _p = 2πωL ⁰ _arm L ⁰ _arm = q

L ² _arm + L ² _probe sin ² (γ) The geometrical parameters are sketched in Fig. 3.17.

The velocity due to the probe rotation was subtracted from the velocity vector

in the 5HP reference system and then transformed in the tunnel reference system,

using the relations that involve both probe parameters (i.e. L _arm , L _probe , γ and

rotation frequency ω) and the relative position between the two reference system

indicated from Φ.

3.3. Test Procedure

Conditions and Locations

The test matrix, was determined taking previous works, as [11] and especially [2], into account.

Three main test series were performed to study the tip vortex wandering:

• downstream evolution of the vortex: α = 3 ^◦ , U _∞ = 10 m/s, x/c = 2 ÷ 5.5;

• effects of angle of attack variation: α = 6 ÷ 14 ^◦ , U _∞ = 20 m/s, x/c = 3, 5;

• Re-dependency: α = 8 ^◦ , U _∞ = 10, 20, 30 m/s, x/c = 3, 5.

The choice of the locations where the data were acquired was heavy influenced by the wind tunnel geometrical dimensions and by the overall dimensions of the rotating arm.

The angles of attack range is limited for high values by the wing stall and for low values by the necessity of a development of a vortex with a relevant cross-plane velocity component.

The Re numbers range is determined by the velocity range of the wind tunnel.

The main test matrix is reported in Tab. 3.7.

Preliminary Tests to Set Sampling Parameters

In this section will be explained the reasons of choosing several parameter like the sampling frequency, the arm rotational speed and the spin way of the rotor.

The sampling frequency was set to 1kHz, to reach an adequate number of points through the vortex core. That value was acceptable for D.A.Q. characteristics, as already explained in Sec. 3.2.

The angular speed and the spin way of the arm rotation were chosen with the support of a preliminary tests series. During these tests the bias error on the arm rotation angle was also determined.

The arm rotation frequency was set as a compromise between two opposite re-

quirements: on one hand the probe must crosses the vortex core as fast as possible

to enable the vortex to be considered roughly fixed during the scan, on the other

side, high velocity may arises structural issues of the rotating arm and signal noise.

α [deg] U _∞ [m/s] x/c

8 10 2

8 10 2.5

8 10 3

8 10 3.5

8 10 3.88

8 10 5

8 10 5.5

6 20 3

8 20 3

10 20 3

12 20 3

14 20 3

6 20 5

8 20 5

10 20 5

12 20 5

14 20 5

8 30 3

8 30 5

Table 3.7. Main test matrix.

Some acquisitions were carried out spinning the arm at 0.1, 0.3, 0.4, 0.5, 0.7 Hz in the condition of α = 8 ^◦ , U _∞ = 20 m/s and x/c = 3. The frequency of 0.4 Hz, that allows to sample on average 5 data points in the vortex core for each scan, was considered the best choice.

The spin way was chosen after another test series carried out making the rotator

spin in both directions at several downstream stations and for several values of the

angle of attack. The tests were performed with both clockwise and anticlockwise

rotation. The comparison of the data exhibited that there was not any appreciable

difference between the two cases. Several tests showed the presence of a secondary

vorticity structure placed between 0.1 c and 0.2 c outboard of the vortex centre.

Spinning the probe from outboard to inboard this feature was more clearly detected.

This is probably due because in the opposite spin way the sensor crosses firstly the main vortex, and then the secondary structure. Indeed, the strong velocity gradient of the main vortex may alter measurements in correspondence to the secondary vorticity structure.

Further tests series was carried out in order to determine the bias error on the arm rotation angle. Error might be due to the delay between analogical data acquisition and the digital signal from the encoder. To this end a thin plate (thickness ' 1 mm) was fixed with a zero angle of attack. The probe was spin crossing its wake immediately downstream of the body. Bias error can be evaluated as the difference between φ _plate ,corresponding to the known position of the plate, and the angle with the most negative pressure value generated by the plate wake. After several tests, the bias error was estimated as:

err _φ = φ _plate − φ measured = 1.65 ^◦ (3.14)

Tests Execution

For each measurement a preliminary test was performed to detect the mean position of the vortex centre moving the rotating unit in the cross-plane. This procedure was carried out through a real-time displaying of the velocity vectors acquired from each scan. Each test condition was performed in two subsequent runs of about 700 rotations to avoid any temperature drift. The angular sector of measurement was set to 35 ^◦ , centred with the vortex centre.

Tests to investigate about the secondary vorticity

Several tests were performed in order to better understand the features and the

evolution of the secondary vorticity structures observed few centimeters outboard

from the vortex core. The tested configuration was α = 8 ^◦ , x/c = 3, U _∞ = 20 m/s

and the tests were performed shifting the probe path upward and downward with

respect to the mean vortex centre, see Tab. 3.8. Other measurements were carried

out at locations slightly downstream and upstream with respect to the analyzed

condition.

α [deg] U _∞ [m/s] x/c z [mm]

8 20 3 -20

8 20 3 -15

8 20 3 -10

8 20 3 -5

8 20 3 0

8 20 3 5

8 20 3 15

8 20 3 20

8 20 3 25

8 20 3 30

8 20 3 35

8 20 3 40

8 20 2.9 10

8 20 2.9 25

8 20 2.9 35

8 20 3.1 12

8 20 3.1 27

8 20 3.1 37

12 20 3 3

12 20 3 18

12 20 3 28

Table 3.8. Test matrix to investigate about the secondary vorticity

3.4. Data Analysis

3.4.1. Evaluation of Vortex Centre Locations

In this Section it will be explained the technique used to find the vortex centre location at each scan. Firstly the velocity components were transformed to the fixed coordinate system of the wind tunnel. The transformation is based on the Eq. 3.15 which involved quantities are defined in the sketch of Fig. 3.18.

( y = Y pos + L arm sin(Φ) + L probe sin(γ)(1 + cos(Φ))

z = Z _pos − L arm (1 + cos(Φ)) + L _probe sin(γ) sin(Φ) (3.15)

(Y _pos , Z _pos ) is the position of the probe tip, corresponding to Φ = 180 ^◦ , that is the vortex mean centre.

The cross-plane component of the the velocity vector is generated by w and

Figure 3.18. Sketch of quantities involved in the transformation from the probe

frame of reference to the tunnel frame of reference.

v: in the vicinity of the tip vortex the cross-component is assumed to be equal to the tangential component, since the radial component is negligible. Hence, no distinction will be made between the cross-component of the velocity and the tangential velocity component in the following. Thus the tangential velocity is defined as:

V _θ = w

|w|

p v ² + w ² (3.16)

V _θ corresponds to the cross-plane velocity component with the sign of w. In the rest of the present work the module of the tangential velocity will be defined as

|V ^θ | = √

w ² + v ² .

The vortex centre position was found using a procedure that involves two subse- quent steps:

• Linear method;

• Corsiglia method.

Linear Method

The so-called linear method is based on the assumption of a linear tangential velocity profile in the vortex core. Hence, the vortex centre in the spanwise direction is individuated from the V _θ component profile, that is where a change in the sign of V _θ occurs where w changes its sign. This feature is independent on the distance between the traverse and the actual vortex centre. For each scan, the two sampling points that surround the sign change were individuated, and these points were called P _i and P _i+1 , as sketched in Fig. 3.19. The line interpolating w between P _i and P i+1 crosses the zero in a point whose y coordinate (y) is roughly the vortex centre coordinate y _c .

From the hypothesis of the linearity and the axis-symmetry of the vortex tan- gential velocity ensues that:

y _c = y + y _i − w(P _i )

slope(w) (3.17)

Figure 3.19. Conceptual sketch of the linear method to find the y component of the vortex centre.

The two above-mentioned hypothesis are a good representation of the real veloc- ity field for scan sufficiently close to the vortex centre.

Figure 3.20. Conceptual sketch of the linear method to find z component of vortex centre.

The z coordinate of the vortex centre z _c was also evaluated involving V _θ . As

explained in Fig. 3.20, the distance between the probe tip path and the vortex

centre is evaluated using the following relation:

d _z = v(P _i )

|v(P ⁱ )|

s

 V _θ (P _i ) slope(V θ )

 2

−

 w(P _i ) slope(w)

 2

(3.18)

where:

slope(w) = tan δ(w) slope(V _θ ) = tan δ(V _θ ) and the quantities δ(w) and δ(V _θ ) are defined in Fig. 3.20.

The following hypothesis is introduced to define slope(V _θ ):

slope(V _θ ) = slope(w) (3.19)

Then:

z c = z i + d z (3.20)

The linear method easily yields the vortex centre coordinates (y _c , z _c ), but it presents several disadvantages:

• for P i far from the vortex centre in z direction the Eq. 3.19 is unrepresentative of the real velocity field;

• if the scan overpasses the vortex core, then the hypothesis of linearity, that bases the method, is meaningless;

• the procedure to determine the vortex centre is based on the quantity recorded in only two samples for y _c and one for z _c . Thus little measurements error can lead to big uncertainties.

In conclusion, the linear method is reliable, especially for y _c evaluation, for scans

that roughly cross the actual vortex centre.

Corsiglia Method

The Corsiglia method was proposed by Corsiglia et al in [10] for rapid scanning data analysis.

This method is valid for each point of the measured velocity field. And is based on the assumption of perpendicularity between the velocity vector in the cross- plane − →

V θ and the distance of the sampling point from the vortex centre of the vortex ⁶ . Consequently, the vortex centre of the vortex may be identified by only two samples: its location is the intersection of the lines perpendicular to two − → V _θ vectors, how is sketched in Fig. 3.21. Generally, many couple of samples for each

Figure 3.21. Conceptual sketch of Corsiglia method to find the vortex centre.

scan can be used to evaluate the vortex centre location. However, better results are obtained with choosing higher amplitude samples. Finally, the vortex centre is determined by the average of all intersections, this operation minimizes the mea- surement errors. In order to choose the samples of highest amplitude signals two cases were distinguished:

a) the probe crosses the vortex core;

b) the probe misses the vortex core.

6

This assumption proceeds from the hypothesis of radial velocity component equal to zero.

−110 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0.5

1 1.5 2 2.5 3 3.5 4 4.5

Y [mm]

|Vθ| [m/s]

−110 −1001 −90 −80 −70 −60 −50 −40 −30 −20 −10 1.5

2 2.5 3 3.5

Y [mm]

|Vθ| [m/s]

(a) (b)

Figure 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.

For the former case (Fig. 3.22 (a)), the |V θ | peaks are located approximatively in correspondence of the core radius, which is crossed twice.

Six points in total were chosen to apply the method: the two points relative to the peaks, each one together with the previous and the following points. For the second case (Fig. 3.22 (b)) there is only one peak of |V θ |, approximatively at the same y of the vortex centre. In this case the point relative to the peak and the nearest six ones were chosen to apply the method. Furthermore, this case consists also of all the scans in which there are two relative maxima, but the difference between one of two relative maxima and the minimum is less than 10% of the maximum of |V θ |.

Each selected point is associated with the corresponding V θ vector: the intersec- tions generated by the lines perpendicular to tangential velocity vectors individuate 15 or 21 different points, depending on a) or b) case. The average of these values individuates the vortex centre.

This method is reliable especially when the traverse does not passes too close from

the vortex centre. Indeed if the samples are displaced along a straight line that

contains the vortex centre,that is theoretically axialsymmetric, all the tangential

velocity vectors are perpendicular to the trajectory of the probe. In this case the

lines perpendicular to the tangential velocity vectors have not any intersections.

In this way, the Corsiglia method is unapplicable. In real cases the probe path is not a straight line, and it never crosses the vortex centre exactly. However the error increases with decreasing the distance between the probe path and the vortex centre.

It was decided to use an algorithm composed by both Corsiglia and linear meth- ods in order to reduce the uncertainties introduced by each method. Indeed for all scans the vortex centre is firstly calculated with the linear method, then the intersections generated from different couples of V _θ vectors are evaluated. Then a control window of 30 mm wide in spanwise direction is introduced; it is centred on the center coordinate y _c obtained with the linear method. Corsiglia method is ap- plied only if at least 12 points are located inside the control window and only these points are used to calculate the average location of the vortex centre. Otherwise, the (y c , z c ) coordinates of the vortex centre remains the ones evaluated using the linear method.

The control window takes only the y coordinate into account because the linear method is less precise in z c individuation. However, the excursion between z c | linear

and z of all points individuated by the intersections, as proposed by Corsiglia et al, is very small, less than 5 mm.

Most of the points usually falls inside the control window and scans where less than 12 points stays inside of the control window limits are only 10% of the total.

In general, it happens when the probe passes very close to the vortex centre and the lines that generate the intersections are almost parallel, as shown in Fig. 3.23.

As an example, in the reference condition α = 8 ^◦ , U _∞ = 20 m/s, x/c = 5, the 87%

of the scans that sampled in a point distant from the vortex centre not more than 1.7 mm were processed with the linear method to find the vortex centre.

At the opposite, when the probe passes far from the vortex centre, the only task of the control window is to discard bad points, because the intersection points are usually concentrated in a region much smaller than the one limited by the control window, as shown in Fig. 3.24. The vortex centres as evaluated with the two methods are not so different: in general the difference between the vortex centres individuated by linear method and by the Corsiglia method is less than 5 mm for more than ≈ 85% of the scans and less than 3 mm for more than ≈ 75%.

From all the scans for each condition a distribution of vortex centre locations

−100 −80 −60 −40 −20 0 20 40 60 80

−150

−100

−50 0 50 100 150 200

Y [mm]

Z [mm]

−40 −20 0 20 40 60

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

Z [mm]

Y [mm]

Linear method centre Intersections Y_pos,Z_pos Centre Control window limits

(a) (b)

Figure 3.23. Example of Corsiglia method applied for a scan close to the vortex centre: tangential velocity vector representation of the samples along the scan(a) and intersections generated by lines perpendicular to the V _θ vectors (b).

−100 −50 0 50 100 150

−100

−50 0 50 100 150

Y [mm]

Z [mm]

Y−Z Plane Velocity Vectors Scan=101 Flag=0

(a) (b)

Figure 3.24. Example of Corsiglia method applied for a scan far from the vortex centre: tangential velocity vector representation of the samples along the scan (a) and intersections generated by lines perpendicular to the V _θ vectors (b).

is obtained on the cross-plane. The comparison between the two methods vortex centre distributions and the final result for a condition are shown in Fig. 3.25 and Fig. 3.26, respectively.

For each condition, once the vortex centres for each scan were evaluated, the

mean centre of the vortex (Y c , Z c ) was found averaging (y c ) i and (z c ) i where i is

referred to the scan number.

−110 −100 −90 −80 −70 −60 −50 −40 −30 −20

−20

−10 0 10 20 30

Y [mm]

Z [mm]

−110 −100 −90 −80 −70 −60 −50 −40 −30 −20

−20

−10 0 10 20 30

Y [mm]

Z [mm]

(a) (b)

Figure 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.

−110 −100 −90 −80 −70 −60 −50 −40 −30 −20

−20

−10 0 10 20 30

Y [mm]

Z [mm]

Figure 3.26. The vortex centres distribution for the condition α = 8 ^◦ , U _∞ =

20 m/s, x/c = 5.

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

The procedure described in the previous section allows to find the vortex centre location for each scan thus to obtain the distribution of vortex centre locations, as shown in Fig. 3.26 (a). Consequently, an experimental probability density function (denoted as PDF in the following) of the vortex centre locations can be evaluated for each condition. An example of the experimental PDF is given by the map in Fig. 3.27 (a). It is clear from the map that the PDF has a typical bell-shaped form.

(a) (b)

Figure 3.27. Condition α = 8 ^◦ , U _∞ = 10 m/s and x/c = 2: (a) ex- perimental PDF map, 1 mm grid spacing, (b) fitted PDF map [N occurrences /(N _scans · mm ² )].

The experimental PDF of the vortex centre locations can be well represented, according to [12], by a bi-variate gaussian function:

p(y, z) = 1

2πσ y σ z √ 1 − e ² ·

· exp (

− 1

2(1 − e ² )

"

 y − y 0

σ _y

 2

+  z − z 0

σ _z

 2

− 2e(y − y 0 )(z − z 0 ) σ _y σ _z

#) (3.21)

where:

(y ₀ , z ₀ ) are the coordinates of the bi-variate gaussian function centre, i.e. the

mean vortex centre (Y c , Z c ) in a cross plane;