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

Problem Definition

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

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

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

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

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

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

fast through the vortex core.

State of the Art

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

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

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

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

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

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

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

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

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

is characterized by two wandering amplitudes (σ

and σ

, for the spanwise and

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

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

reference. σ

and σ

are evaluated by dividing the root mean square value of the

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

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

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

this hypothesis with any explanation or numerical simulation. The anisotropy

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

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

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

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

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

and 69

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

amplitude was found to increase roughly linear with increasing the downstream

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

applied to correct wandering averaging effects. The corrections performed were

very accurate for the simulations with small wandering amplitudes whereas errors

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

experimental campaign was performed regarding the tip vortex generated from a

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

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

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

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

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

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

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

Main Objective of the Present Work

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Appendix B.