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(1)Wandering is a universal feature of wind-tunnel generated vortices and it consists of abrupt displacements of the vortex core location

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Wandering is a universal feature of wind-tunnel generated vortices and it consists of abrupt displacements of the vortex core location. Tip vortices in a wind tunnel meander in space, the core location fluctuates erratically in time. 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.

Experiments have been performed regarding the tip vortex generated from a NACA 0012 half-wing model. A rapid scanning tests series has been carried out using a five hole pressure probe. This dynamic measuring technique returns a snap- shot of the velocity components for each scan. An algorithm has been elaborated to find the instantaneous vortex centre location at each scan from the acquired ve- locity signals. This algorithm is an hybrid procedure between the method proposed by Corsiglia [10] and a method based on the hypothesis of linearity of the tangen- tial velocity component in the vortex core. Indeed, it has been found that the first method is highly error affected if the scan crosses the vortex at a distance from its centre lower than ∼ 1/4 of the vortex core radius. Conversely, in this range the second method has revealed an adequate reliability. For each condition the vortex centre locations found at each scan generate a distribution in the cross-plane. Con- sequently, an experimental probability density function (PDF ) of the vortex centre locations has been evaluated for each tested condition. The experimental PDF of the vortex centre locations has been found to be well represented by a bi-variate gaussian function. An algorithm to perform a least squares fitting of the exper- imental PDF with a bi-variate gaussian function has been developed in order to obtain the wandering amplitude from the standard deviations of the spanwise and normal vortex centres coordinates and the anisotropy parameter e. The direction of the principal axis of wandering has been inferred from the geometrical shape of

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the fitted PDF.

The determination of the vortex centre location for each scan has allowed to re-centre all the instantaneous velocity profiles by a radial coordinate and, con- secutively, to obtain mean velocity profiles corrected for wandering effects. These time-averaged velocity profiles have been 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.

Then, further measurements of the tip vortex flow field have been performed by traversing a three sensors hot film probe for the same flow conditions already tested with the rapid scanning.

A good agreement has been found between the mean velocity profiles obtained with Eulerian measurements carried out with the 3HFP and the mean velocity profiles obtained from the rapid scanning data affected by wandering. This finding has allowed to consider the non re-centred data from rapid scanning measurements equivalent to static measurements.

The method proposed by Devenport et al [12] to evaluate the wandering am- plitudes has been applied both to the rapid scanning uncorrected data and to the 3HFP data. A substantial agreement has been singled out from the results ob- tained both from the Devenport method applied to these two data sets and from the the PDF standard deviations of the fitted PDF, although the Devenport et al [12] method is reliable to evaluate wandering amplitudes smaller than 60% of the core radius.

Regarding the static measurements, the direction of the principal axes of wan- dering has been evaluated from the eigenvectors of the covariance matrix Σ, where the wandering anisotropy parameter e has been calculated as the opposite value of the cross-correlation coefficient between the spanwise and the normal velocity components, measured at the mean vortex centre location.

From the experimental measurements it has been found that wandering ampli- tudes increase roughly linearly with increasing the streamwise distance, from 43%

of the actual core radius up to 116% proceeding downstream from 2 to 5.5 chord- lengths, suggesting that the streamwise distance is a primary variable affecting the wandering amplitude. However, the wandering amplitudes are fairly invariant with increasing the angle of attack and they slightly decrease with increasing the free-

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stream velocity. Thus, the mechanism responsible of wandering is not self-induced because in this case it would be expected that wandering would increase with vor- tex strength. Comparing data carried out in the present work with data obtained by Iungo & Skinner [24] it has been shown 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, the wandering is surely attenuated with increasing vortex strength. Moreover, the reduction of wandering amplitude with increasing Reynolds number have suggested to ascribe the source of wandering to the wind tunnel flow unsteadiness. Finally, a weak anisotropy of the wandering characterized by an higher amplitude in the spanwise direction than in the normal one (σy > σz) has been observed.

The characterization of the vortex core has been carried out using the model proposed by Hoffmann & Joubert in [22], and a good agreement of this method with the mean circulation obtained from experimental measurements was found.

The wandering smoothing effects on the tangential velocity profiles were signifi- cant for all the tested conditions, they leaded to:

• a decrease of the peak tangential velocity;

• an increase of the vortex dimension;

• a change in the velocity profile shape due to a decrease of the tangential velocity gradient at the vortex core.

The errors induced by the wandering smoothing effects on static measurements roughly followed the trend of the wandering amplitude with varying the test con- dition. However, when the variation of the vortex core radius is significant, a more suitable parameter seems to be the ratio between the wandering amplitude and the vortex core radius. In extreme circumstance, the errors leaded to 30% under- estimate of the peak tangential velocity and 85% overestimate of the vortex core radius evaluated at a streamwise location of 5.5 chord-lengths with a wandering amplitude equivalent to 116% of the actual vortex core radius.

A roughly linear decay of both the peak tangential velocity and linear increase of the vortex core radius was found with increasing streamwise distance, both for data corrected and uncorrected for wandering. However, the rate was considerably

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lower for the re-centred data because of the lack of wandering smoothing effects.

Moreover, a roughly linear increase of the peak tangential velocity and the vortex core radius was found with increasing the angle of attack, with the same rate for data corrected and uncorrected for wandering. Finally, an increase of the free- stream velocity leaded to an increase of the peak tangential velocity and a reduction of the vortex core radius.

The general shape of the axial velocity profiles corrected for wandering effects exhibited a significant velocity defect in the vortex core and a velocity overshoot at the vortex core border. The maximum velocity deficit, that is associated to the vortex centre, increased linearly with the streamwise distance up to the 20% of the free-stream velocity at 5.5 chord-lengths downstream, whereas the overshoot always remained below the 10% of the free-stream velocity.

The linear increase of the axial velocity deficit evaluated for the re-centred axial velocity profile with proceeding downstream, together with the very slow decay of both the peak tangential velocity and the vortex radius, suggested that the viscosity action is not relevant travelling downstream up to 5.5 chord-lengths. Consequently, the wandering is responsible to an under prediction of the vortex strength and to an over prediction of the diffusion effects.

The wandering smoothing effects on the axial velocity profiles were significant for all the tested conditions. Indeed, an apparent decay of the velocity deficit at the vortex centre is singled out from the downstream evolution of the mean axial velocity profiles affected by wandering.

At high angles of attack a shift between wake flow and jet flow was observed in the vortex core. The shift between velocity defect and excess occurred in the range between the angles of attack of 8 and 10 at the streamwise location x/c = 3 and in the range between the angles of attack of 10 and 12 at the streamwise location x/c = 5. The maximum velocity excess reached the 15% of the free-stream velocity at the angle of attack of 14.

At 5 chord-lengths downstream, the wandering produced drastic changes in the axial velocity profiles aspect at the vortex centre at the highest values of the angle of attack (12 and 14). The re-centred velocity profiles retained a well defined defect region in correspondence of the vortex centre, exactly in the middle of the jet flow that interest all the vortex core region. This defect region disappears

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if profiles affected by wandering are considered. Moreover, the velocity deficit is roughly constant with the angle of attack (' 15% of the free-stream velocity) if evaluated as the difference with respect to the maximum value of the axial velocity associated to the jet flow region.

The wandering smoothing effects on the axial velocity profiles leaded to a reduc- tion of both the velocity defect and excess. In extreme circumstance, the errors leaded to 70% underestimate of the axial velocity defect and 30% underestimate of the velocity excess.

The presence of secondary vorticity structures have been revealed by the analysis of the mean velocity profiles and of the standard deviation of the velocity signals.

Although a systematic investigation has been not feasible, a secondary vorticity structure located outboard and upward with respect to the main vorticous structure has been found. The abrupt decreasing in the circulation profiles exhibits that this structure is a co-rotating vortex. The comparison of the mean axial velocity component, its standard deviation and the circulation have revealed the shift of the secondary vorticity structure towards higher radial distance from the vortex centre with increasing the angle of attach. Moreover, at high incidence and for high free-stream velocities the secondary structure divides into two different vorticous structures.

The wavelet spectra evaluated in correspondence of the most significant sampling points along the traverse path showed that the energy contribution of the signals is mostly concentred at low frequency scales (large turbulent structures), f c/Uless than about 2·10−1; whereas, for f c/Ugreater than about 2·10−1, the energy level falls because of the influence of small turbulent structures. The vortex core is the region characterized by the highest energy in the flow. Indeed, the wavelet spectra pointed out an increase of the signals energy with proceeding towards the vortex centre location due to wandering effects, suggesting that the spectra are scaling with a parameter highly dependent on the radial distance from the vortex centre.

Moreover, this energy increase is stronger for the tangential velocity component than for the axial one. A good self-similarity of the wavelet spectra measured at the core centre was highlighted with varying the streamwise distance and the angle of attack, excluding a dependence of the spectra energy from the main vortex parameters.

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Further Work

The rapid scanning measurement technique has allowed to achieve an high precision level in estimating the wandering features and effects.

Anyway, some fields need to be improved and deeply investigated. Firstly, in a further research it is necessary to perform rapid scanning measurements at a greater downstream distance locations with respect to the tested ones, in order to improve the knowledge of the wandering behavior with varying the most important parameter in wandering evolution.

Moreover, further work is needed to establish the wandering source reliably, for instance varying the turbulence level of the wind tunnel using grids with different cell size in correspondence of the nozzle. Furthermore, can be helpful to individuate the threshold in the vortex strength that regulates the sensitivity of the wandering behavior to the flow parameters, in order to investigate if this threshold is a constant feature of the wandering or it depends from the wind tunnel characteristics.

A possible future research direction is the use of the PIV technique in order to achieve a wider characterization of the velocity field rather than the one allowable using the rapid scanning traverses.

In a further research, it is necessary elaborate methods of correction that implies reliable estimation of the vortex wandering from static measurements, even though the wandering amplitude exceed the 60% of the vortex core radius.

Acknowledgements

The authors would like to thank Eng. Mauro Morelli, who made a useful contri- bution to plan the tests and all the CSIR staff, whose assistance were fundamental to the execution of the experimental campaign. Thanks are due to the staff of the Department of Aerospace Engineering for the kindness and affability exhibited.

The authors are grateful to the respective families for the support.

A special thanks is due to all the friends who attended the authors during this experience, especially to Gd, who shared with the authors a large part of the road.

In addiction, Matte thanks his black cockroach for its ability in spreading en- thusiasm and Luca would like to address a special thanks to his lady for the love

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shown through these years.

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