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4. Static Measurements Performed with Three Component Hot Film Anemometry

Introduction

The hot wire/film probes are very useful instruments used in the wind tunnels to

measure the velocity components in the wing-tip vortex flow field. This is mainly

due to the small size which results high spatial resolution and little interference

with the flow. Commonly, measurements are performed by traversing the probe

and sampling in fixed points, thus, the obtained data are affected by wandering. In

the previous part of this work, data carried out with the rapid scanning technique

were used to estimate the wandering amplitudes and the wandering smoothing ef-

fects on mean velocity profiles. In order to achieve this aim, the rapid scanning

data corrected for wandering were compared with uncorrected data, both obtained

from the same measurements. In this part of the work, static measurements were

performed using a three sensor hot film probe for the same flow conditions already

tested with the rapid scanning. This measurements were carried out in order to

assess the interpretation of the rapid scanning data affected by wandering as data

obtained from static measurements. This validation allows to generalize the con-

clusions. Moreover, the large number of samples acquired for each measurement

allows to analyze the statistics of the velocity signals and to apply the Devenport

et al [12] method to evaluate the wandering amplitude. Furthermore, the high

frequency resolution of the hot film probe allowed to perform a spectral analysis of

the acquired signals.

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4.1. Three Sensors Hot-Film Probe and Calibration

Figure 4.1. Three sensors hot-film probe.

The hot-film anemometer is a probe that senses the changes in heat transfer from a small, electrically heated, sensor exposed to fluid motion. Its operation relies on the variation of the electrical resistance of the sensor material with varying the temperature.

The probe used in the present experimental campaign is a TSI 1299-20-18 (Fig. 4.1), see Tab. 4.1 for its specifications. It is a triple sensor hot-film probe (denoted as 3HFP in the following), whose significant dimensions are reported in Fig. 4.2. The probe design minimizes both the thermal wake interference between the sensors and the flow disturbances created by the prongs and the probe stem, keeping its size small to achieve 2.5 mm spatial resolution. To minimize the thermal wake interference of each sensor, the sensor and their active length portions were offset with respect to each other. Thus, when the velocity vector is in the domain centred around the probe axis, the thermal wake of any sensor is not seen by any of the active length portions of its downstream sensors. The three sensors are nominally orthogonal.

Each of the hot-film sensors was operated separately using a IFA AN-1003 made

by A.A. Lab-Systems. Its specifications are summarized in Tab. 4.2.

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4.1. Three Sensors Hot-Film Probe and Calibration

Figure 4.2. 3HFP shape and geometry, length [mm], angles [ ].

Sensor Res. 0 C Res.100 C- Res.0 C Racc. Racc. Int.

Oper. Res. Oper. Temp. Probe Res.

[Ω] [Ω] [Ω] C [Ω]

1 5.74 1.42 9.29 250 0.26

2 5.72 1.4 9.22 250 0.25

3 5.72 1.41 9.25 250 0.25

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

The interpretation of triple sensor signals requires a method to relate the anemome-

ter output voltage of each sensor to the magnitude and direction of the instanta-

neous velocity vector. This was targeted through a probe calibration based on the

method of effective cooling velocities. As extensively described in Appendix A,

this method allows to decouple the directional response and the velocity magni-

tude response of the probe according to Jørgensen hypothesis [20]. The velocity

magnitude calibration is an operation performed in order to establish a relation

between output voltages (E i , where i = 1, 2, 3 according to the sensor) from the

3HFP and effective cooling velocities (U ef f

i

). It consists in fixing the coefficients

in a fourth order polynomial law by a least square fitting of the experimental data,

as it is exhibited in Fig. 4.3. This operation takes changes in probe response due to

atmospheric conditions into account, and it was performed before each wind tunnel

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Number of channels 4

CTA Bridge Non Linear, Constant Temperature type

Bridge Ratios 1 : 1

Sensor Resistance Range 0.5 − 99.9 Ω Maximum Closed Loop Bandwidth DC − 120 kHz Equivalent input noise 1.6 nV

Typical Output noise 135 V rms Stability - Typical Input Drift 0.5 V / C

Output Voltages Range ±12 V

Amplifier Gain 1 ÷ 20

Gain accuracy ±0.5 %

DC Offset 0 ÷ 10 V

Output Impedance 100 Ω

Input Impedance 10 kΩ

Typical Input White Noise 30 nV

Frequency Range DC − 100 kHz

LOW-PASS FILTER Lower band 300Hz ÷ 5kHz

Upper band 7Hz ÷ 16kHz

Output Voltages ±5 V

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

run.

The angular calibration aims to establish a relationship between the effective cooling velocities and the pitch (α) and yaw (β) angles between the probe axis and the flow direction in x–z and x–y planes, respectively.

In the present work the analytical method developed by Lekakis [27] was cho- sen. This method is based on a representation of the directional response by the Jørgensen law [20], written afterwards for a single wire:

U ef f 2 = u 2 N + k 2 u 2 T + h 2 u 2 B (4.1)

where:

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4.1. Three Sensors Hot-Film Probe and Calibration

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

2 4 6 8 10 12 14 16 18

Ei [Volt]

U∞ [m/s]

experimental data 4th degree polynomial fitting

Figure 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.

u N = velocity component normal to the sensor and lying on the plane of its sup- porting prongs;

u T = velocity component tangential to the sensor;

u B = velocity component normal to the plane of sensor supporting prongs.

The mathematical calculations to achieve the velocity module and the pitch and

yaw angles and, consequently, the 3 components of the velocity vector (u, v, w)

are extensively explained in Appendix B where the calculations are completely

reviewed because some errors were found in the development of the mathematical

algorithm. Basically, the system of equations which yield the direction of the flow

can be solved once the probe geometrical dimensions and the coefficients h i and

k i are known. The purpose of the angular calibration is to determine h i and k i

through an extensive experimental campaign. These coefficients depend on the

non-exact orthogonality between the hot-film sensors, the prongs interference and

the thermal wake effects that significantly influence the sensors response. Thus,

they are highly probe dependent. It is important here to note that k i and h i have

been assumed constant over the full angular and velocity range, as suggested by

several authors, for instance Lekakis et al [27].

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The movement rig already used for the calibration of the 5HP was also used for the 3HFP calibration, in order to allows the α and β probe displacements required by the angular calibration procedure. The movement rig was mounted on an adequate support structure in the 2mWT, so that the 3HFP sensors were located in the centre of the test section.

Each sensor was calibrated separately in three steps:

• a preliminary velocity magnitude calibration, performed with the tested sen- sor normal to the free-stream and its supporting prongs parallel to the free-stream;

• an angular calibration to obtain h i , performed by placing the tested sensor in a reference position parallel to the z axis and by yawing the probe around the y axis, in order to maintain the u B component of the cooling velocity equal to zero, see Fig. 4.4 (a);

• an angular calibration to obtain k i , performed by placing the tested sensor in a reference position parallel to the y axis and by yawing the probe around the y axis, in order to maintain the u T component of the cooling velocity equal to zero, see Fig. 4.4 (b).

In both data sets the yaw angle was varied from −60 to 60 with a step of 2 . For each sample point of the angular calibration, the U ef f is determined from the measured voltages using the velocity calibration curve and the components u N , u T and u B are calculated from the known position of the probe. Finally, k i and h i were obtained through a least square fitting of the two data sets carried out with the angular calibration using the Eq. A.2.

In Fig. 4.5 (a) and (b) are shown the data acquired with the probe displaced as it is exhibited in Fig. 4.4 (a) and (b), respectively, and the fitting of the experimental data obtained using the Jørgensen law. In particular, Fig. 4.4 (a) presents the fitting that determined h for a wire, whereas Fig. 4.4 (b) presents the fitting that determined k.

The coefficients resulted from the calibration are reported in Tab. 4.3 for all the

sensors.

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4.1. Three Sensors Hot-Film Probe and Calibration

(a)

(b)

Figure 4.4. Sketch of the procedure for the angular calibration of the 3HFP, shown

for a single wire: arrangement for h i determination tests (a) and ar-

rangement for k i determination tests (b).

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−60 −40 −20 0 20 40 60 0.7

0.8 0.9 1 1.1 1.2 1.3 1.4

Yaw [°]

(Ueff/U∞)2

Data h=1.169

−40 −20 0 20 40 60

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Yaw [°]

(Ueff/U∞)2

Data k=.5k=.2 k=.1 k=0

(a) (b)

Figure 4.5. Example of the calibration data for a wire of the 3HFP. U ef f registered with varying the yaw angle and fitting of the data with the Jørgensen law with h = 1.169 (a); U ef f registered with varying the yaw angle and fitting of the data with the Jørgensen law with several k (b).

Sensor h k

1 1.078 0

2 1.169 0

3 1.013 0

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

4.2. Tests Execution

After the calibration phase the model wing was mounted on its support as for the rapid scanning campaign. The 3HFP was fixed on a suitable holder placed on the tip of a forward-swept wing to minimize the interference effects of the support. The forward-swept wing was placed on the same traversing apparatus that was used for the rapid scanning campaign and described in Sec. 3.2. This traversing apparatus allows to displace the probe in a generic (x,y,z) location. A picture of the set-up is reported in Fig. 4.6

The test matrix was determined taking the rapid scanning campaign into account.

Indeed, the three main test series performed with the 3HFP are analogous of the

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4.2. Tests Execution

Figure 4.6. Set-up for 3HFP measurements.

ones described in Sec. 3.3:

• downstream evolution of the vortex: α = 8 , U = 10 m/s, x/c = 1 ÷ 5;

• effects of angle of attack variation: α = 4 ÷ 12 , U = 20 m/s, x/c = 3, 5;

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

The test matrix is reported in Tab. 4.4. Traverses were executed in the spanwise direction to carry out data comparable to the ones acquired by the rapid scanning.

Indeed, as it was pointed out by Iungo et al [24] grid measurements, the wing- tip vortex is roughly anisotropic. Moreover, its size and intensity slightly change according to the direction of the traverse. The sampling frequency was fixed at 2 kHz, the maximum allowable frequency that did not introduce any buffer issue.

The sampling time was set to 33 s to achieve at least 2 16 samples for each signal in order to maximize the higher detectable frequency according to the Nyquist theorem.

The velocity magnitude calibration of the three hot-film sensors was performed

before each run. However, the three sensors were calibrated simultaneously, by

placing the probe in the unperturbed flow with its axis parallel to the free-stream

direction. In this position, each wire is influenced by all the three components of

the effective cooling velocity. For each wire, the Eq. A.2 allows to determine the

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α [deg] U [m/s] x/c

8 10 1

8 10 2

8 10 3

8 10 4

8 10 5

4 20 3

6 20 3

8 20 3

10 20 3

12 20 3

4 20 5

6 20 5

8 20 5

10 20 5

12 20 5

8 30 5

Table 4.4. Test matrix of 3HFP measurements.

coefficients of the velocity magnitude calibration from the probe geometry (writing u N , u T and u B as a function of U ) and the h i and k i coefficients.

For each condition a preliminary test was performed to detect the mean position

of the vortex centre by several fast traverses in the cross-plane. This procedure was

carried out using a real-time display of the velocity components.

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4.3. Data Analysis

4.3. Data Analysis

This section describes the procedure to analyze the velocity signals acquired from each 3HFP spanwise traverse. Three velocity signals (one for each velocity compo- nent) were acquired at each sampling position of the traverse. A statistical analysis of all the signals was performed by calculating the following quantities referred to the velocity components:

• mean value;

• RMS value;

• standard deviation;

• skewness (the third order statistic moment);

• kurtosis (the fourth order statistic moment);

• cross-correlation coefficient.

4.3.1. Mean Flow Field

Considering the mean flow field, a typical result for the cross-plane velocity com- ponents v and w measured by spanwise traverses is shown in Fig. 4.7. As expected and confirmed by present measurements, a traverse across the mean vortex centre is characterized by a mean value of the spanwise velocity component v almost equal to zero in each acquisition point. Consequently, the normal velocity component w coincides to the tangential velocity component V θ . However, the tangential com- ponent is the most appropriate velocity component in order to compare the vortex velocity profile obtained by varying the test condition. Indeed, it is in general im- possible to guarantee that v is exactly equal to zero because of the random errors done in the centre finding and the small changes in the external conditions, thus, the value of V θ is more reliable than the value of w.

The tangential and axial velocity profiles obtained from 3HFP static measure- ments were compared with the ones obtained from 5HP rapid scanning technique.

Fig. 4.8 reports an example of this comparison performed for one condition. This

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−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

R/c

Vθ/U w/U v/U

Figure 4.7. Tangential, spanwise and normal velocity components measured by 3HFP traverse at the condition α = 8 , U = 10 m/s and x/c = 1.

check assumes a significant relevance because it validates the rapid scanning re- sults about the wandering smoothing effects on conventional static measurements.

Indeed, the 3HFP velocity profiles are really in good agreement with the rapid scanning profiles affected by wandering, viz. the rapid scanning profiles evaluated without the re-centring operation. Thus, the smoothing effects of wandering on static measurements are easily appreciable by considering the difference between the conventional profiles (denoted as 3HFP Traverse in Fig. 4.8) and the rapid scanning re-centred profiles (denoted as 5HP RS No-Wandering in Fig. 4.8). These effects, already highlighted in Section 3.5.1, are principally a reduction in the eval- uation of:

• the peak tangential velocity and the tangential velocity gradient in the vortex core, see Fig. 4.8 (a);

• the axial velocity defect at the vortex centre, see Fig. 4.8 (b).

The similarity between the 3HFP velocity profiles and the rapid scanning pro-

files affected by wandering is also very significant in order to check the reliability

of present measurements because these profiles were obtained both with different

measurement techniques and with different probe types.

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4.3. Data Analysis

−0.1 −0.05 0 0.05 0.1 0.15

−0.15

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

R/c V

θ

/U

5HP RS No−Wandering 5HP RS Wandering 3HFP Traverse

−0.1 −0.05 0 0.05 0.1 0.15

−0.15

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02 0 0.02

R/c (u−U

)/U

5HP RS No−Wandering 5HP RS Wandering 3HFP Traverse

(a) (b)

Figure 4.8. Comparison between velocity profiles measured by 5HP rapid scanning and 3HFP static measurements. Tangential velocity component (a) and axial velocity component (b) for the condition α = 8 , U = 10 m/s and x/c = 3.

Mean tangential velocity profiles were used to estimate the circulation through Eq. 3.28 assuming the hypothesis that the flow field induced by the vortex is axis- symmetric, thus the tangential velocity was assumed constant along a circular path around the vortex. Following the same procedure described in Section 3.4.3, the characterization of the vortex core was carried out using the model proposed by Hoffmann & Joubert in [22]. From the shape of the circulation profile, three regions were distinguished by moving outwards from the centre of the vortex: the vortex core, the logarithmic region and a defect region. The circulation was then fitted separately in the three regions with the laws of Eq. 3.31 containing the fitting constants A ÷ E. A good agreement of this method with the mean circulation obtained from the traverses was found. An example of the fitting result is shown in Fig. 4.9 (a), while in Fig. 4.9 (b) it is reported the tangential velocity profile corresponding to the fitted circulation profile.

The gradient of the tangential velocity profile evaluated at the vortex centre

is proportional to the coefficient A. The circulation Γ 1 and the radial position

r 1 at the point of maximum tangential velocity were determined from Eq. 3.31

since the coefficient B represents the percentage of the theoretical circulation at

the wing root, Γ 0 , rolled-up in the vortex core radius. The vortex core radius

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−0.10 −0.05 0 0.05 0.1 0.1

0.2 0.3 0.4 0.5 0.6

R/c Γ / Γ

0

Core region Logarithmic region Defect region Data

−0.1 −0.05 0 0.05 0.1

−0.6

−0.4

−0.2 0 0.2 0.4 0.6

R/c V

θ

/U

Core region Logarithmic region Defect region Data

(a) (b)

Figure 4.9. Hoffmann&Joubert fitting: example of the fitted circulation profile (a) and the tangential velocity profile corresponding to the fitted circula- tion profile (b) for the condition α = 10 , U = 20 m/s and x/c = 3.

was assumed equal to the value of r 1 as obtained from the Hoffmann & Joubert model. Consequently, the value of the peak tangential velocity V θ1 was evaluated from Γ 1 and r 1 values. Finally, the spanwise coordinate of the vortex centre Y c is determined from the location of the minimum modulus of the tangential velocity component.

Fig. 4.10 reports the typical trend of the skewness (a) and the kurtosis (b) eval- uated for the axial and the normal velocity components. As proposed by Iungo

& Skinner in [24], the skewness can be very useful in order to evaluate V θ1 and r 1 when velocity profiles are highly affected by wandering. Indeed, the skewness modulus of the normal velocity component presents a maximum in correspondence of the vortex core radius, R/r 1 = 1, and it is roughly zero at the core centre, as shown in Fig. 4.10 (a).

The skewness of the axial velocity component can be useful in order to analyze the shape of the wake cross-section. Indeed, Iungo & Skinner [24] suggested to define the wake centerlines from the skewness of the axial velocity. In particular, they observed that each local maximum, see Fig. 4.10 (b), of the modulus of this parameter correspond to a point of the wake centerline.

The behavior of the kurtosis is strictly linked to the one of the skewness. Indeed,

the local peaks of the kurtosis were found in correspondence of the peaks of the

skewness both in the case of the tangential and the axial velocity component,

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4.3. Data Analysis

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

R/r

1

Sku Skw

Core region

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

0 2 4 6 8 10 12 14 16

R/r

1

Kuu Kuw Core

region

(a) (b)

Figure 4.10. Skewness (a) and kurtosis (b) evaluated for the axial and the normal velocity components for the condition α = 8 , U = 10 m/s and x/c = 4.

compare Fig. 4.10 (b) with Fig. 4.10 (a). Furthermore, a characteristic trend of the kurtosis of the tangential velocity was found by Iungo & Skinner [24] as a function of the wandering amplitude. However, the scattering of the kurtosis does not enable this parameter to be a predictor of wandering amplitudes.

4.3.2. Wandering Characterization from Static Measurements A characterization of the wandering from static velocity measurements was per- formed by Iungo & Skinner in [24] through a numerical simulations of the wander- ing of a Lamb-Oseen vortex. The vortex wandering was simulated by representing the vortex centre locations through the bi-variate gaussian function of Eq. 3.21, as proposed by Devenport et al in [12] and confirmed both by Heyes et al in [21]

and by the the rapid scanning data reported in the present work (see Sec. 3.4.2).

The wandering amplitude in the spanwise direction σ y (in the normal direction σ z ) may be evaluated as the ratio of the RMS value and the gradient of the normal velocity w (spanwise velocity v), measured at the mean vortex centre, as proposed by Devenport et al [12]. From the numerical simulations Iungo & Skinner [24]

found that wandering amplitudes are accurately predicted with this method if they are smaller than 60% of the core radius, over this value the error increases with increasing the wandering amplitude.

The procedure to evaluate the wandering amplitudes from static measurements

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was applied to the data obtained from the 3HFP traverses. The RMS of the nor- mal and spanwise velocity components was investigated together with the standard deviation of the same quantities. The typical trend of both the RMS and the stan- dard deviation of the spanwise and the normal velocities are shown in Fig. 4.11 (a) and (b), respectively.

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6

R/r

1

σv/U

RMSv/U

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8

0 0.1 0.2 0.3 0.4 0.5 0.6

R/r

1

σw/U

RMSw/U

(a) (b)

Figure 4.11. RMS and standard deviation evaluated for the spanwise (a) and nor- mal (b) velocity components for the condition α = 10 , U = 20 m/s and x/c = 3.

Considering the spanwise velocity component v, the RMS and the standard devi- ation are substantially coincident in each sampling point, and both present a peak in correspondence of the vortex centre, see Fig. 4.11 (a). This result is typical for a traverse that crosses the mean vortex centre very precisely because this behavior of RMS v/U

and σ v/U

assesses that the mean value of v is roughly zero in all the samples. Indeed, from the definition of the RMS and the standard deviation it is clear that these parameters assumes an equal value when they are referred to a quantity characterized by a mean value of zero. During the test execution, to check the behavior of these two parameters can be very useful in order to verify if a traverse crossed the mean vortex centre exactly.

Comparing Fig. 4.11 (a) with Fig. 4.11 (b), it is evident that the standard de-

viation profile of the normal velocity is roughly equal to the one of the spanwise

velocity since the standard deviation does not depend on the respective mean ve-

locity values. Conversely, the RMS profile of the normal velocity is completely

different than the one of the spanwise velocity because it roughly follows the trend

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4.3. Data Analysis

of the normal velocity modulus. Indeed, RMS w/U

presents two maxima in corre- spondence of the vortex core radius, R/r 1 = ±1, and a minimum at the core centre, where it assumes the same value of the peak of σ w/U

because in the vortex centre the mean value of w is zero.

Concluding, the correspondence between the RMS and the σ values of v and w ensures that the RMS values at the mean vortex centre location evaluated from this data allow to achive an accurate evaluation of the wandering amplitude param- eters σ y and σ z . Moreover, the wandering amplitudes evaluation needs the slope of the tangential velocity at the mean vortex centre. As previously anticipated, this parameter was obtained from the mean circulation profile fitted through the Hoffmann & Joubert method.

Furthermore, the parameter e of the bi-variate gaussian function, which repre- sents the anisotropy of the wandering with respect to the actual frame of reference, was found by Iungo & Skinner [24] to be equal to the opposite value of the cross- correlation coefficient between the spanwise and the normal velocity components, measured at the mean vortex centre location. The reliability of such estimate of e is inadequate because the scattering of the cross-correlation data is fairly rele- vant. Fig. 4.12 reports an example of the cross-correlation coefficients evaluated for one condition. Apart from the scattering, the analysis of the cross-correlation coefficients between the velocity components (uv, uw and vw) pointed out typi- cal trends of these parameters observed for the majority of the conditions inves- tigated. The cross-correlation coefficient between the spanwise and the normal velocity component vw is negative in the vortex core region and positive elsewhere, see Fig. 4.12 (a). It reaches the minimum in the vicinity of the mean vortex centre and it presents also a sudden decrease in correspondence of the secondary vortic- ity structure (see Sec. 3.5.6) that is located at a radial coordinate of about 5 r 1 outboard from the main vortex for the condition investigated. Finally, both the cross-correlation coefficients uv and uw follow an analogous trend characterized by crossing the zero through the vortex core, see Fig. 4.12 (b).

As suggested in Section 3.4.2, the estimate of the direction of the principal axes of

the wandering Θ can be performed through the evaluation of the covariance matrix

Σ corresponding to the bi-variate gaussian function, see Eq. 3.25. The direction of

the principal axes of the wandering may be then evaluated from the eigenvectors of

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−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

R/r

1 vw

Core region

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

R/r

1 uv uw

Core region

(a) (b)

Figure 4.12. Cross-correlation coefficients vw (a), uv and uw (b) between the three velocity components evaluated for the condition α = 10 , U = 20 m/s and x/c = 3.

the covariance matrix Σ, and the wandering amplitudes along these peculiar axes may be evaluated from the square root of its eigenvalues. However, this procedure is affected by an high uncertainty because it was found that small errors on the estimate of the anisotropy parameter e produce large errors on the estimate of Θ.

Moreover, it was previously observed that the estimate of the anisotropy parameter

e, based on the analysis of the cross-correlation coefficient between v and w, is not

reliable.

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4.4. Results

4.4. Results

In this section the results obtained by 3HFP traverses are reported for the condi- tions itemized in Tab. 4.4. The attention is focused on the wandering parameters (σ yz and e), on the mean vortex centre locations (Y c and Z c ), and on the vortex core parameters (V θ1 , r 1 , Γ 1 and U D ). The review of the results will be extensively discussed in the section dedicated to the comparison between data carried out with the rapid scanning and data carried out with the 3HFP (Sec. 5). This choice is in agreement with the purpose of the anemometer tests, that is the assessment of the interpretation of the rapid scanning data affected by wandering as data obtained from static measurements.

Moreover, in the present section the mean tangential and axial velocity profiles are plotted for downstream distance, angle of attack and Reynolds number varia- tions separately.

Wandering Characterization

Tab. 4.5 presents the wandering parameters of the vortex as a function of the downstream distance at x/c = 1, 2, 3, 4, 5 for an angle of attack of α = 8 and a free-stream velocity of U = 10 m/s.

In Tab. 4.6 the wandering parameters of the vortex are reported as a function of the angle of attack. The tests are performed at x/c = 3, 5 for the angles of attack of α = 4, 6, 8, 10, 12 and a free-stream velocity of U = 20 m/s.

The wandering parameters with varying the free-stream velocity are reported in Tab. 4.7. The analyzed conditions ware performed at α = 8 , U = 10, 20, 30 m/s

Condition Amplitude Mean center Direction/Angle

α [

] Re x/c σ

yz

/c σ

y

/c σ

z

/c Y

c

/c Z

c

/c e Θ [

] 1 0.012 0.009 0.008 -0.090 0.057 0.56 40 2 0.015 0.011 0.010 -0.131 0.069 0.52 39 8 149000 3 0.020 0.014 0.014 -0.163 0.049 0.56 45 4 0.031 0.021 0.023 -0.210 0.041 0.56 51 5 0.032 0.023 0.022 -0.245 0.024 0.59 42

Table 4.5. Wandering parameters for the analyzed streamwise locations.

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Condition Amplitude Mean center Direction/Angle α [

] Re x/c σ

yz

/c σ

y

/c σ

z

/c Y

c

/c Z

c

/c e Θ [

]

4 0.020 0.015 0.014 -0.122 0.069 0.54 42

6 0.023 0.018 0.015 -0.159 0.069 0.17 22

8 298000 3 0.021 0.017 0.013 -0.167 0.057 0.34 26

10 0.016 0.012 0.010 -0.176 0.045 0.31 29

12 0.019 0.013 0.013 -0.171 0.024 0.45 45

4 0.028 0.022 0.018 -0.180 0.069 0.38 31

6 0.027 0.019 0.018 -0.229 0.053 0.20 37

8 298000 5 0.024 0.016 0.017 -0.208 0.033 0.39 48

10 0.021 0.014 0.015 -0.220 0.012 0.62 47

12 0.021 0.014 0.016 -0.220 -0.008 0.72 51

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

Condition Amplitude Mean center Direction/Angle

α [

] Re x/c σ

yz

/c σ

y

/c σ

z

/c Y

c

/c Z

c

/c e Θ [

] 149000 0.032 0.023 0.022 -0.245 0.024 0.59 42 8 298000 5 0.024 0.016 0.017 -0.208 0.033 0.39 48 447000 0.030 0.019 0.023 -0.224 0.004 0.51 54

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

and x/c = 5.

In the following part of this section the attention is focused on the mean velocity profiles. Fig. 4.13, Fig. 4.14 and Fig. 4.15 present the mean tangential velocity profiles measured by traversing the 3HFP for the tested streamwise distances, angles of attack and free-stream velocities, respectively.

The wandering smoothing effects on the tangential velocity profiles are significant

with proceeding downstream, especially for the location x/c = 5. This trend is in

agreement with the increase of the wandering amplitude, reported in Tab. 4.5,

with increasing the streamwise distance. The comparison between the profiles

shown in Fig. 4.13 and the analogous ones carried out with rapid scanning, shown

in Fig. 3.57, assesses the previous observation. The same feature is detectable, for

each incidence of the wing, comparing the tangential velocity profiles carried out

at x/c = 3 and x/c = 5 shown in Fig. 4.14 (a) and (b), respectively. Indeed, the

wandering amplitude increases with proceeding from x/c = 3 to x/c = 5 and the

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4.4. Results

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

R/c Vθ/U∞

x/c=1 x/c=2 x/c=3 x/c=4 x/c=5

Figure 4.13. Downstream variation of the mean tangential velocity profiles for the condition α = 8 and U = 10 m/s.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

R/c Vθ/U∞

α=4°

α=6°

α=8°

α=10°

α=12°

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

R/c Vθ/U∞

α=4°

α=6°

α=8°

α=10°

α=12°

(a) (b)

Figure 4.14. Mean tangential velocity profiles evaluated at different values of the angle of attack for the condition U = 20 m/s, x/c = 3 (a) and x/c = 5 (b).

diffusion of the vorticity is not relevant travelling downstream up to x/c = 5, as

concluded in Sec. 3.5.1. The increase of the tangential velocity magnitude with

increasing the angle of attack is due to the increase of the vortex strength, and it

acts with the same rate for the two tested downstream locations. The effects of the

increasing vortex strength are also detectable in Fig. 4.15, where the evolution of

the tangential velocity profiles is shown as a function of the free-stream velocity.

(22)

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

R/c Vθ/U∞

U=10m/s U=20m/s U=30m/s

Figure 4.15. Mean tangential velocity profiles evaluated with different free-stream velocities for the condition α = 8 and x/c = 5.

From the analysis of the mean tangential velocity profiles showed in Fig. 4.13, Fig. 4.14 and Fig. 4.15, a weak asymmetry of the vortex is pointed out, according to rapid scanning measurements (see Sec. 3.5.1) and to Iungo & Skinner findings [24]. The slight difference between the two sides of vortex is not constant for all the configurations, and it has neither the same sign. Consequently, only the average of the peak tangential velocity and of the vortex core radius will be reported in the following.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

R/c (u−U∞)/U∞

x/c=1 x/c=2 x/c=3 x/c=4 x/c=5

Figure 4.16. Downstream variation of the mean tangential velocity profiles for the

condition α = 8 and U = 10 m/s.

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4.4. Results

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

R/c (u−U∞)/U∞

α=4°

α=6°

α=8°

α=10°

α=12°

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

R/c (u−U∞)/U∞

α=4°

α=6°

α=8°

α=10°

α=12°

(a) (b)

Figure 4.17. Mean axial velocity profiles evaluated at different values of the angle of attack for the condition U = 20 m/s, x/c = 3 (a) and x/c = 5 (b).

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

R/c (u−U∞)/U∞

U=10m/s U=20m/s U=30m/s

Figure 4.18. Mean axial velocity profiles evaluated with different free-stream ve- locities for the condition α = 8 and x/c = 5.

The axial velocity profiles present an evident defect region at all the tested loca- tions, as for the rapid scanning profiles, compare Fig. 4.16 with Fig. 3.63. Moreover, at x/c = 1, 2 the axial velocity shows two distinguished minima in the defect region.

This feature was found by Ghias et al and also by rapid scanning measurements

for x/c = 2, see Fig. 3.63 (a). It is probably correlated to the early roll-up phase

of the wing-tip vortex. The evolution of the defect region of the axial velocity in

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the vortex core with increasing the angle of attack is exhibited in Fig. 4.17 (a) and (b). The shape of the profiles is very similar to the ones obtained from the rapid scanning data affected from wandering, shown in Fig. 3.79 (b) and in Fig. 3.80 (b), respectively. The defect region, detectable at α = 4 , 6 , 8 , reduces with increas- ing the angle of attack, and it changes from a deficit to an excess over an angle of attack of 8 , both for x/c = 3 and x/c = 5. This feature was already found in the analysis of the rapid scanning data and by Chigier & Corsiglia [9] and it is more evident at x/c = 3 than at x/c = 5, because of the wandering smoothing effects.

In Fig. 4.18 a reduction of the axial velocity defect with increasing the free-stream velocity is evident, and for U = 30 m/s the defect disappears.

The overshoot at the core border is not evident in any tested condition at U = 10 m/s, conversely to what found with rapid scanning measurements, both re- centred and affected by wandering. However, the overshoot is detectable in the outboard side of the vortex core border for U = 20 m/s at α = 12 , x/c = 3, see Fig. 4.17 (a), and at α = 8 , 10 , x/c = 5, see Fig. 4.17 (b).

The last note regards the secondary vorticity structure, detectable from a small deficit of the axial velocity (Fig. 4.16), especially at x/c = 2 and x/c = 4, placed at R/c = 0.17 and R/c = 0.2, respectively, in agreement with the findings reported in Sec. 3.5.6.

A more exhaustive analysis of the tangential and the axial velocity profiles was conducted investigating the main vortex parameters: the peak tangential velocity (V θ1 ), the vortex radius (r 1 ), the circulation at the maximum tangential velocity (Γ 1 ), the profile gradient at the vortex centre and the axial velocity defect (U D ). As explained in Sec. 4.3.1 the parameters referred to the tangential velocity component were deducted from the Hoffmann & Joubert [22] fitting of the circulation. The fitting constants are reported in Tab. 4.8 for each analyzed condition.

The results of this analysis are listed in Tab. 4.9, Tab. 4.10 and Tab. 4.11 with

varying the streamwise distance, the angle of attack and the Reynolds number,

respectively.

(25)

4.4. Results

Conditions Outboard profile Inboard profile

α[

] Re · 10

5 xc

A B C D E A B C D E

1 399 0.226 1.047 -4.109 -0.232 336 0.262 1.176 -5.019 -0.211 2 447 0.288 1.296 -1.743 -0.401 283 0.267 1.146 -3.698 -0.240 8 149000 3 490 0.267 1.246 0.718 -0.708 240 0.266 1.106 -4.277 -0.182 4 196 0.318 1.286 3.674 -0.940 290 0.265 1.141 -0.422 -0.512 5 176 0.253 1.048 -2.765 -0.218 289 0.247 1.082 -2.863 -0.350 4 278 0.314 1.332 -0.380 -0.706 440 0.123 0.611 -2.046 -0.223 6 456 0.268 1.224 -0.576 -0.705 323 0.317 1.369 -4.402 -0.342 8 298000 3 401 0.430 1.820 -2.000 -0.698 439 0.258 1.183 -2.574 -0.406 10 425 0.315 1.385 -1.773 -0.547 456 0.224 1.051 -2.600 -0.443 12 324 0.349 1.457 -2.477 -0.537 338 0.270 1.181 -1.242 -0.525

4 234 0.317 1.315 1.085 -0.822 359 0.167 0.784 -4.064 -0.121

6 265 0.261 1.111 -1.992 -0.413 366 0.280 1.244 -4.611 -0.250 8 298000 5 344 0.216 0.982 1.060 -0.559 249 0.218 0.958 -0.989 -0.423 10 243 0.244 1.047 -1.958 -0.319 394 0.185 0.846 -3.050 -0.179 12 292 0.216 0.952 0.978 -0.560 268 0.221 0.953 -0.442 -0.408 149000 176 0.253 1.048 -2.765 -0.218 289 0.247 1.082 -2.863 -0.350 8 298000 5 344 0.216 0.982 1.060 -0.559 249 0.218 0.958 -0.989 -0.423 447000 272 0.370 1.524 -6.700 -0.199 250 0.198 0.874 -3.942 -0.166

Table 4.8. Fitting of the circulation: coefficients for the Hoffmann & Joubert model of the tested conditions.

Condition Peak tangential Radius Grad(V

θ

/U

) Circulation Axial velocity

velocity at vortex deficit

α [

] Re · 10

5 xc

V

θ1

/U

r

1

/c centre Γ

1

0

U

D

/U

1 0.4375 0.031 18.1 0.271 -0.162

2 0.397 0.033 18.0 0.263 -0.122

8 149000 3 0.332 0.039 18.0 0.261 -0.104

4 0.310 0.041 12.0 0.257 -0.129

5 0.333 0.045 11.5 0.302 -0.130

Table 4.9. Vortex parameters for the analyzed locations.

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Condition Peak tangential Radius Grad(V

θ

/U

) Circulation Axial velocity

velocity at vortex deficit

α [

] Re · 10

5 xc

V

θ1

/U

r

1

/c centre Γ

1

0

U

D

/U

4 0.213 0.029 8.9 0.245 -0.129

6 0.335 0.031 14.5 0.276 -0.056

8 298000 3 0.462 0.033 20.7 0.305 -0.049

10 0.549 0.033 26.7 0.296 0.062

12 0.561 0.037 23.2 0.294 0.163

4 0.184 0.031 7.4 0.227 -0.124

6 0.263 0.033 11.7 0.231 -0.098

8 298000 5 0.355 0.035 14.7 0.249 -0.068

10 0.430 0.037 19.3 0.261 0.016

12 0.496 0.039 19.6 0.274 0.053

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

Condition Peak tangential Radius Grad(V

θ

/U

) Circulation Axial velocity

velocity at vortex deficit

α [

] Re · 10

5 xc

V

θ1

/U

r

1

/c centre Γ

1

0

U

D

/U

149000 0.333 0.045 11.5 0.302 -0.168

8 298000 5 0.355 0.035 14.7 0.302 -0.068

447000 0.367 0.039 12.9 0.304 -0.054

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

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4.5. Time-frequency analysis

4.5. Time-frequency analysis

The time-frequency analysis was performed through the evaluation of the wavelet power spectra. The wavelet spectra aim to evaluate the energy contributions on the velocity signals carried out with the 3HFP and to investigate on the frequency associated to these contributions. The study of the energy contribution was per- formed on the signals of the axial and the tangential velocity components obtained by varying the radial position with respect to the mean vortex centre, the stream- wise distance, the angle of attack and the free-stream velocity.

The wavelet power spectra were determined by using the program Wavelet 8.0 developed at the Department of Aerospace Engineering (DIA) which performs the wavelet transform of the velocity signals. The reference function used for the wavelet transform is the Morlet function (complex):

ψ(t) = e

0

t · e

−t

2

2

(4.2)

The value of the parameter ω 0 was assumed equal to 6π in order to achieve an ade- quate frequency resolution, as suggested by Iungo in [23]. The software Wavelet 8.0 was configured to work in L 2 space (rather than in L 1 space) in order to maintain constant the energy of the wavelet functions with varying the frequency, although in L 2 the spectra are slightly shifted towards the low frequencies. The number of samples for each signal is greater than 2 16 , acquired with a sampling frequency of 2 kHz. Thus, the frequency range of the wavelet spectra was chosen from 1 to 401 Hz and it was divided into 201 scales allocated following a logarithmic law.

The frequency f was then non-dimensionalized dividing it by the ratio between the free-stream velocity U and the chord-length c.

The analysis of the 2mWT free-stream flow did not reveal any peculiar energy contribution contrarily to the findings of Barbaro [2] who observed a characteristic frequency of 16 Hz for all the tested free-stream velocities. An example of the free- stream wavelet spectra, performed at U = 20 m/s with varying the streamwise location, is given in Fig. 4.19 for the axial (a) and the tangential (b) velocity component, respectively.

A systematic analysis was performed for each condition by evaluating the wavelet

spectra in correspondence of the most significant sampling points along the traverse

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10−1 100 10−5

10−4 10−3

u Po

fc/U

x=0 mm (x/c=0) x=275 mm (x/c=1.122) x=550 mm (x/c=2.245)

10−1 100

10−5 10−4 10−3

fc/U

v+w Po

x=0 mm (x/c=0) x=275 mm (x/c=1.122) x=550 mm (x/c=2.245)

(a) (b)

Figure 4.19. Wavelet spectra of the free-stream flow evaluated at U = 20 m/s.

path in order to characterize the frequency spectra of the velocity signals induced by the wing-tip vortex. These radial locations with respect to the vortex centre were individuated considering the shape of each tangential velocity profile, see Fig. 4.20 as example for one condition. In particular the sampling points were chosen:

• in the vicinity of the traverse path limits, where the vortex induced velocity is negligible (points A and I);

• in the vicinity of the vortex core, at the beginning of the defect region (points B and H);

• in correspondence of the peak tangential velocities, in the middle of the log- arithmic region (points C and G);

• inside of the core region (points D and F);

• in correspondence of the mean vortex centre, in the middle of the core region (point E);

The wavelet spectra evaluated in these sampling points for the axial and the

tangential velocity components are reported as an example in Fig. 4.21 for one

condition. The energy contribution of each signal is mostly concentred at low

frequencies (large vorticity structures), f c/U less than about 2 · 10 −1 ; whereas,

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4.5. Time-frequency analysis

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

V

θ

/U

R/r

1

A B

C

D E

F

G H I

Figure 4.20. Example of the individuation of the most significant radial locations where the wavelet spectra were analyzed for the condition α = 10 , U = 20 m/s and x/c = 5.

for f c/U greater than about 2·10 −1 , the energy level falls because of the influence of smaller structures. This trend was observed for all the analyzed conditions since it is a common feature of fluid dynamics velocity signals. Both spectra of the axial and the tangential velocity component pointed out an increase of the signals energy with proceeding towards the vortex centre location, according to Devenport et al findings, see [12]. However, this energy increase is stronger for the tangential velocity component than for the axial one. Indeed, inside the vortex core the energy of the tangential velocity signals is generally two orders of magnitude greater than the energy of the axial velocity signals, compare the spectra evaluated at the radial locations D, E and F in Fig. 4.21 (a) with the ones of Fig. 4.21 (b). Devenport et al [12] suggested that the rise of spectral levels as the vortex centre is approached is due to wandering and other inactive motion of the vortex core. Concluding, the vortex core is the region characterized by the highest energy in the flow and this energy depends on the vortex induced velocity and, consequently, on the wandering, thus it is more relevant considering the tangential component than the axial one.

Finally, the wavelet spectra evaluated from the two sides of the mean vortex centre

shown a strong self-similarity. Indeed, comparing the outboard side (signals A, B,

C and D) with the inboard side (signals I, H, G and F), the energy of the signals

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is fairly constant for the radial locations characterized by an analogous distance with respect to the mean vortex centre. This statement suggests that the spectra might be scaled with a parameter highly dependent on the radial distance from the vortex centre.

10−1 100

10−5 10−4 10−3 10−2 10−1 100 101

u Po

fc/U

(R/r1)A=8.11 (R/r1)B=2.44 (R/r1)C=1 (R/r1)D=0.33 (R/r1)E=0 (R/r1)F=−0.22 (R/r1)G=−1 (R/r1)H=−2.11 (R/r1)I=−8.33

(a)

10−1 100

10−5 10−4 10−3 10−2 10−1 100 101

fc/U

v+w Po

(R/r1)A=8.11 (R/r1)B=2.44 (R/r1)C=1 (R/r1)D=0.33 (R/r1)E=0 (R/r1)F=−0.22 (R/r1)G=−1 (R/r1)H=−2.11 (R/r1)I=−8.33

(b)

Figure 4.21. Wavelet spectra of the axial, (a), and tangential, (b), velocity compo- nents evaluated for the condition α = 10 , U = 20 m/s and x/c = 5.

The wavelet spectra referred to the axial velocity component reveal a peak fre-

quency contribution at f c/U ' 10 −1 visible at the furthest radial locations with

respect to the vortex core, see the spectra of signals A (outboard side) and I (in-

board side) reported in Fig. 4.21 (a). This frequency contribution on the axial

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4.5. Time-frequency analysis

velocity was found at different free-stream velocities, mostly for high angles of at- tack (α ≥ 8 ) and at a streamwise distance x/c ≥ 3. For all these conditions, it was observed at the radial locations A and I only. The non-dimensional value of this frequency contribution is constant with varying the free-stream velocity, see Fig. 4.22, whereas the dimensional value is ' 4, 8, 12 Hz for U ∞ = 10, 20, 30 m/s, respectively. These findings indicate that the source of the energy contribution at f c/U ' 10 −1 is fluid dynamical. This statement is also supported by the observa- tion of the wavelet time-frequency map reported in Fig. 4.23. The map highlights that the peak energy is not constant during the acquisition time and it has the aspect of several spots that cover a fairly wide frequency range.

10−1 100

10−5 10−4 10−3

u Po

fc/U

U= 10 m/s U= 20 m/s U= 30 m/s

10−1 100

10−5 10−4 10−3

u Po

fc/U

U= 10 m/s U= 20 m/s U= 30 m/s

Inboard Outboard

Figure 4.22. Wavelet spectra of the axial velocity component in the vicinity of the traverse path limits, radial locations I (Inboard) and A (Outboard), evaluated with different free-stream velocities for the condition α = 8 and x/c = 5.

A more detailed analysis of this frequency contribution was carried out for the

condition α = 8 , U = 10 m/s and x/c = 5 by evaluating the wavelet spectra

at several radial locations outside the vortex core (R/r 1 > 1). The locations were

individuated by proceeding with constant radial step of 3 mm both in the inboard

and in the outboard direction. Fig. 4.24 and Fig. 4.25 report these spectra for

the axial and the tangential velocity components, respectively. The observation

of these figures leads to suppose that the contribution at f c/U ' 10 −1 is a

characteristic of the 2mWT flow in the backward region of the test chamber (x/c ≥

(32)

Figure 4.23. Wavelet time-frequency map: energy of the axial velocity signal eval- uated at the radial location I (inboard side) for the condition α = 8 , U = 10 m/s and x/c = 5.

3). Indeed, both spectra confirm that the contribution at f c/U ' 10 −1 effects the axial velocity component only which is more influenced by the wind tunnel flow. Moreover, Fig. 4.24 reveals that the energy of this contribution is detectable only in correspondence to the higher radial distances from the vortex core. This contribution disappears moving towards the vortex core because of the increase in the flow energy due to the presence of the surrounding shear layers. However, the peak frequency at f c/U ' 10 −1 was not present in the wavelet spectra of the free-stream flow because these spectra were performed at a streamwise distance of x/c ≤ 2.245.

Fig. 4.26 shows the evolution with distance downstream of the wavelet spectra of the axial and tangential velocities measured at the core centre, where the energy levels of the signals are maxima for each condition. Fig. 4.27 reports the analogous spectra obtained by varying the angle of attack.

Apart from the condition x/c = 1 in Fig. 4.26, the spectra collapse one each

other proceeding towards the highest frequency scales, especially for the tangential

velocity component. These figures highlight only a very little shift of the spectra

towards higher energy at low frequency, contrarily to Devenport et al [12] that

found a systematic shift in the spectra towards low frequency and energy with

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4.5. Time-frequency analysis

10−1 100

10−5 10−4 10−3 10−2

fc/U

u Po

R/r1=−0.91 R/r1=−1.18 R/r1=−1.45 R/r1=−1.73 R/r1=−2 R/r1=−2.27 R/r1=−2.55 R/r1=−2.82 R/r1=−3.1 R/r1=−3.36 R/r1=−3.64 R/r1=−4.18 R/r1=−4.73 R/r1=−5.27 R/r1=−5.82 R/r1=−6.36 R/r1=−6.91 R/r1=−7.45 R/r1=−8

−2.3

Inboard

10−1 100

10−5 10−4 10−3 10−2

fc/U

u Po

R/r1=1 R/r1=1.27 R/r1=1.55 R/r1=1.82 R/r1=2.09 R/r1=2.36 R/r1=2.64 R/r1=2.91 R/r1=3.18 R/r1=3.45 R/r1=3.82 R/r1=4.36 R/r1=4.91 R/r1=5.45 R/r1=6 R/r1=6.55 R/r1=7.09 R/r1=7.64

−2.3

Outboard

Figure 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, for the condition α = 8 , U = 10 m/s and x/c = 5.

varying the streamwise distance. This result also excludes a dependence of the

spectra energy from the vortex parameters like V θ1 , r 1 or U D as one would expect,

see again Devenport et al [12]. Moreover, the slope of the linear region (in bi-

logarithmic scale) of the spectra visible at high frequencies is reported on Fig. 4.26

(34)

10−1 100 10−5

10−4 10−3 10−2 10−1

fc/U

v+w Po

R/r1=−0.91 R/r1=−1.18 R/r1=−1.45 R/r1=−1.73 R/r1=−2 R/r1=−2.27 R/r1=−2.55 R/r1=−2.82 R/r1=−3.1 R/r1=−3.36 R/r1=−3.64 R/r1=−4.18 R/r1=−4.73 R/r1=−5.27 R/r1=−5.82 R/r1=−6.36 R/r1=−6.91 R/r1=−7.45 R/r1=−8

−3.3

−0.9

Inboard

10−1 100

10−5 10−4 10−3 10−2 10−1

fc/U

v+w Po

R/r1=1 R/r1=1.27 R/r1=1.55 R/r1=1.82 R/r1=2.09 R/r1=2.36 R/r1=2.64 R/r1=2.91 R/r1=3.18 R/r1=3.45 R/r1=3.82 R/r1=4.36 R/r1=4.91 R/r1=5.45 R/r1=6 R/r1=6.55 R/r1=7.09 R/r1=7.64

−3.3

−1.2

Outboard

Figure 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, for the condition α = 8 , U = 10 m/s and x/c = 5.

and Fig. 4.27. The slope of the spectra measured at the core centre is useful in

order to investigate the turbulence level of the flow inside the vortex core. In

particular, the diffusion of the turbulence structures increases with increasing the

slope of the spectra. According to what presented above, the slope of the tangential

(35)

4.5. Time-frequency analysis

10−1 100

10−4 10−3 10−2 10−1 100

u Po

fc/U

fc/U

fc/U

fc/U

x/c=1 x/c=2 x/c=3 x/c=4 x/c=5

−2

10−1 100

10−4 10−3 10−2 10−1 100

fc/U

v+w Po

x/c=1 x/c=2 x/c=3 x/c=4 x/c=5

−3.5

(a) (b)

Figure 4.26. Downstream variation of the wavelet spectra of the axial (a) and tan- gential (b) velocity components evaluated in the mean vortex centre (radial location E) for the condition α = 8 and U = 10 m/s.

velocity spectra is much greater than the one of the axial velocity spectra for all

the analyzed conditions, since the energy fall is greater for the tangential velocity

spectra. According to what presented above, the slope of both axial and tangential

velocity spectra is fairly constant with varying the streamwise distance (Fig. 4.26)

and the angle of attack (Fig. 4.27). Conversely, a considerable reduction of the

slope of the linear portion of the spectra was observed for the tangential velocity

with increasing the radial distance with respect to the mean vortex centre, see

Fig. 4.25. As expected, this result indicates that the diffusion of the turbulence

structures reduces with proceeding outside from the vortex core. However, for the

same condition the slope of the axial velocity spectra is fairly constant, see Fig. 4.24.

(36)

10−1 100 10−3

10−2 10−1 100 101

fc/U

u Po

α=4°

α=6°

α=8°

α=10°

α=12°

−2.3

10−1 100

10−3 10−2 10−1 100 101

fc/U

v+w Po

α=4°

α=6°

α=8°

α=10°

α=12°

−3.5

(a) (b)

Figure 4.27. Wavelet spectra of the axial (a) and tangential (b) velocity compo-

nents evaluated in the mean vortex centre (radial location E) at dif-

ferent angles of attack, for the condition U = 20 m/s and x/c = 5.

Riferimenti

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