Wandering Smoothing Effects on Wing-tip Vortex

Eulerian Measurements

Giacomo Valerio Iungo∗

Airspace Engineering Dep., University of Pisa, Pisa, 56122 Pisa, Italy

Peter Skinner†

Defence, Peace, Safety and Security, CSIR, Pretoria, 0001 Pretoria, Rep. of South Africa

Wandering is a universal feature of wind-tunnel generated vortices and it consists oferratic fluctuations of the vortex core. Vortices measured by static measuring techniquesappear more diffuse than in reality, a correction method is needed. Experiments were per-formed on a half-wing model, unswept , with NACA 0012 cross-section. Preliminary flowvisualization was conducted using laser sheet and smoke injected to qualitatively character-ize the vortex shape and trajectory. Further visualization showed that probe interferencewas not an issue, when the probe was aligned with the mean flow. The overall shape of thevortex and the roll-up process is surveyed by five hole probe (5HP) measurement grids.Most of the measurements were performed at a Reynolds number about 4.22 × 105 andangle of attack of 8◦. More detailed velocity profiles were achieved by traverses throughthe vortex core using both 5HP and a three sensor hot film anemometer (3HWA). Wakecenterlines are always spiral shaped. The distance between the vortex centre and the lowestpart of the centerline was defined through a spiral scale ξ and it grows as the square root ofthe streamwise distance. Centerlines become roughly self-similar when normalized with ξ.The vortex core radius increases proportionally to the square root of the streamwise dis-tance, whereas the tangential velocity peak is inversely proportional to the same quantity.Rapid scanning was performed by mounting a 5HP on the tip of a rotating arm. Thesemeasurements are theoretically not affected by wandering effects and, were then comparedwith static 5HP and 3HWA data. The static measurements were shown to indicate a 50%increase in core radius and a 30% decrease in tangential velocity peak. The method intro-duced by Devenport et al.1 to remove wandering effects from data was implemented, butthe wandering amplitudes were calculated from Rapid Scanning measurements rather thanusing statistical parameters determined by static measurements. The level of accuracy forboth criteria is comparable. The wandering amplitude increases proportionally to stream-wise distance and to angle of attack. Fitting of velocity profiles seems to be a prevalentsource of error in the correction method. Velocity profiles achieved by Rapid Scanning arein good agreement with 3HWA static measurements corrected with the Devenport method.

I. Introduction

Tip-vortices spread from a large aircraft represent a significant hazard for an aircraft that follows in itswake. This phenomenon affects the separation distance between transport aircrafts and, consequently, itremains a limiting factor on airport capacity. Furthermore, the flow close to the wing-tip is significant for aproper 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 todesign of the ogee tips, winglets and wing-tip sails.

A feature of trailing vortices that make them challenging for most conventional experimental measuringtechniques is the Wandering. Trailing vortices in a wind tunnel meander in space, the core location fluctuateserratically in time at a specific down-stream position. This motion seems to be a universal feature of wind

∗PhD Student, Pisa, 56122 Pisa, Italy.†Low-speed Facilities manager of Defence, Peace, Safety and Security, CSIR, Pretoria, 0001 Pretoria, Rep. of South Africa

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tunnel generated vortices. Wandering may be self-induced by the shear layers which wrap around the vortexcore or a consequence of free-stream turbulence. This meandering implies that any time-averaged Eulerianmeasurements, carried out by static experimental techniques, are actually a weighted average in both timeand space. Thus, the measured vortex appears more diffuse than what it should be in reality. Suitablemeasuring techniques are PIV and Rapid Scanning, which attempt to achieve a fixed vortex for the wholemeasuring time. This objective is reached for each snapshot with PIV and with Rapid Scanning by traversinga probe sufficiently fast through the vortex core.

There is a large amount of literature describing experimental studies of tip vortices, but only few ofthem take wandering smoothing effects into account. Chigier and Corsiglia2,3 compare data acquired bya fixed three sensors hot wire anemometer with measurements achieved with rapid scanning; they foundaveraged Eulerian point measurements to be very sensitive to vortex wandering. In 1996, Devenport etal.1 executed static three components measurements. They proposed a fundamental statistical method tocorrect velocity profiles from wandering effects, in which the vortex motion is described by a non-isotropicgaussian probability density function. The most critical elements are the evaluation of wandering oscillationamplitude and the fitting of velocity profiles. The former is established by statistical parameters from staticmeasurements at the vortex center. For the latter we found that fitting can introduce an error of up to22% for the axial velocity minimum and 38% for the axial vorticity peak. Heyes et al.4 compared themethod introduced by Devenport et al. with a method based on PIV measurements. They found that coreoscillations are well represented by a non-isotropic gaussian function and that the Devenport corrections arein good agreement with PIV velocity profiles. Also Yeung5 evaluated wandering characteristics based on PIVdata; they concluded that the wandering amplitude is comparable with the core radius and the maximumrate of wandering is roughly 4% of the free-stream velocity.

The objective of the present investigation was to perform an overall survey of the roll-up process of thevortex, in particular its shape and trajectory, by flow visualization, 5HP and 3HWA static measurements.In addition the velocity profiles were measured by Rapid Scanning, which consisted of traversing a 5HPon a rotating arm at frequency of 0.5Hz through the vortex core, to enable the latter to be consideredroughly fixed during each scan. These data was then compared with static 5HP and 3HWA measurementsto quantify wandering smoothing effects on Eulerian measurements. The correction method proposed byDevenport et al.1 was implemented. Wandering amplitudes were evaluated using the Rapid Scanning data.For each scan the vortex center location was found by the method suggested by Corsiglia et al.,3 so that aprobability density function could easily be evaluated. This was fitted with a non-isotropic gaussian that gavethe wandering amplitudes. This method is supposed to be less affected by errors than that one suggestedby Devenport (amplitude was found from the rms of a velocity component divided the tangential velocitygradient at the core center), given that wandering amplitudes were evaluated by more than 2000 independentscans.

This paper is organized as follows. Section 1 is the introduction. The facility and all instrumentsused are described in Section 2. We summarize all flow visualization performed in Section 3. The roll-upprocess is analyzed by 5HP measurement grids in Section 4. In Section 5 velocity profiles achieved bystatic measurement techniques are analyzed. Rapid Scanning results are shown in Section 6. The wanderingcorrection method is presented in Section 7 and finally the conclusions and reccomendations are highlightedin Section 8.

II. Apparatus and Instrumentation

The tests were performed in the Two Meter Wind Tunnel at the Defence, Peace, Safety and Security(DPSS) operating unit of the CSIR in Pretoria (RSA). This facility is an open-circuit low-speed wind tunnelwith a test section diameter of 1.7 m and a length of 2.55 m. The speed range was between 3 and 33 m/s.The free-stream turbulence level was less than 0.75% and a small negative axial pressure gradient was present(dCP /dx = −0.9% m−1).

The model was a zero-swept, untwisted, half-wing with NACA 0012 cross-section, bluff tip, aspect ratioof 5.7 and taper ratio 0.4. The wing semi-span was 0.7 m and the mean geometric chord c was 0.245 m. Inall tests no boundary layer trip was used on the model. The model was mounted in a vertical position on asupport allowing the variation of the angle of attack. The origin of the reference frame was located as thewing-tip trailing edge; the x -axis was chosen as the free-stream direction and the spanwise direction as they-axis, with positive direction moving from the root to the tip. The z direction was consequently defined,

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producing a clockwise frame of reference.Flow visualization was conducted using a laser sheet generated by a cylindrical lens from a 8 Watt Argon

Ion Laser. To mark the vortex we injected smoke into the flow upstream of the wing, from the high-pressureside. Both still photographs and videos were taken.

The 5HP used was built by Aeroprobe Corporation Blacksburg, Virginia, USA. The probe externaldiameter was 3 mm and the distance between two opposite taps was about 1.5 mm. The pressure transducerswere connected to each orifice with 30 mm length tubes. The probe was calibrated in the Calibration tunnelat DPSS, which is an open-circuit low-speed tunnel, using a method described by Gerner et al.6 BIAS errorwas always less than 1◦ for the velocity vector direction and about 3% for velocity magnitude.

The three sensor hot film anemometer was a TSI 1299 probe, driven by an A.A. Lab Systems Ltd IFA100.The calibration proposed by Lekakis et al.7 was employed. Probes were mounted on their holder, then fixedto a vertical support.

Rapid scanning was performed by mounting the 5HP on the tip of a rotating arm 991 mm in length. Toincrease measurement accuracy, the probe was pitched to an angle of 6◦ in order to set measured velocityvector approximately parallel to the probe axis while the arm was rotating. The arm was rotated with afrequency of 0.5Hz by a stepper motor and the azimuthal angle was measured by a digital encoder.

III. Flow Visualization

The objective of the present flow visualization is to analyze the roll-up process of the vortex and toachieve qualitative knowledge about the vortex shape and trajectory. In the first tests series the laser sheetwas moved from the wing trailing edge up to six chord-lengths downstream of the trailing edge (6c). Figure 1shows two vortex cross-sections at a condition of U∞ = 10 m/s and angle of attack α = 8◦. The vortex radius

(a) (b)

Figure 1. Flow Visualization at U∞ = 10m/s, α = 8◦. a) x/c = 0.02; b) x/c = 1.5.

increases moving downstream as a consequence of the roll-up process. From several videos we observed thatthe vortex trajectory changed abruptly at a distance of about 0.5c. Upstream of this position the vortexmoved slightly upwards and downstream of this position it reverses moving downwards. Some authors8

suggest that this behavior could be due to the merging of different vorticity structures. The vortex alwaysmoves inboard in the spanwise direction .

The second tests series was executed using the laser sheet in a fixed position, orthogonal to the free-stream direction, and changing the wing angle of attack from 0◦ to 16◦. From Fig. 2 we can assess that thevortex radius increased with increasing incidence. Close to the wing stall the wake became thicker and theshear layers could not be wrapped easily into the vortex core; furthermore, we observed that the vortex coreposition was more unstable at this condition.

Some visualization was carried out to assess probe interference using a dummy probe. It was concluded

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(a) (b)

Figure 2. Flow Visualization at U∞ = 15m/s, x/c = 0.66. a) α = 2◦; b) α = 12◦.

that the probe interference was negligible for probe pitch and yaw angle of 0◦ (Fig. 3). These visualizationswere performed at all the conditions tested in the present experimental campaign. Other visualizations were

Figure 3. Flow Visualization to assess that probe interference in not an issue for used probes.

carried out, even though not strictly related to the tip-vortex survey, for instance the interaction of thevortex with a winglet was visualized (Fig. 4). All the flow visualization photos and videos may be accessedon the World Wide Web at http://dottorato.dia.ing.unipi.it/Dottorandi/schedaDott.asp?id=10.

IV. Five Hole Probe Measurement Grids

The purpose of the 5HP measurement grids was to perform a survey of the wake roll-up process intothe vortex, as well as of the vortex trajectory and shape. These measurements were performed in planesperpendicular to the direction of the free-stream velocity with a constant space-step between each measure-ment point, from 2 mm up to 6 mm depending on the stream-wise position. Most of the measurementswere conducted with a wing angle of attack of 8◦, free-stream velocity of 10 m/s and stream-wise positionsfrom 0.1c up to 6c. Others were performed at different free-stream velocities (20 and 30 m/s) or differentincidences (4◦ and 12◦). The data sampling rate was 2kHz and the total sampling time was 5s at eachmeasurement point.

In the present paper some results for the condition of U∞ = 10 m/s, angle of attack α = 8◦ and x/c = 1

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Figure 4. Interaction of tip vortex with a winglet

are presented, however, they are representative of all conditions analyzed. Figure 5a highlights the typicalreduction of the axial velocity component in the flat part of the wake and in the core center. An overshootof this quantity was found at the edge of the vortex core and it reaches a magnitude of 105% to 115% ofthe free-stream velocity. From the map of normalized variance of the axial velocity u2/U2

∞ (Fig. 5b) and ofthe turbulence kinetic energy (Fig. 5c) oscillations in the vortex core can be observed, which are consideredattributable to the wandering phenomenon. From this data we evaluated the non-dimensional axial vorticityωxc/U∞ obtained by differentiating the grid data with a second order accuracy scheme on circular paths(Fig. 5d). For all conditions and locations analyzed in the present experimental campaign, no secondaryvorticity structures were observed, neither co-rotating or counter-rotating with respect to the main vortex.

Figure 6a shows the map of the cross-correlation coefficient between the horizontal and the verticalvelocity components. For all conditions a particular direction with predominantly negative values of thisquantity is observed. The latter always goes through the average vortex center location and the anglebetween this direction and the orthogonal to the span-wise direction reaches an asymptotical value about35◦. This characteristic is strictly related to the main oscillating direction of the vortex, as will be shown inthe Section VII.

Figure 7 shows the overall shape of the vortex in the baseline flow in terms of contours of axial normalturbulence stress u2/U2

∞ at U∞ = 10 m/s and α = 8◦ where the red line represents the position of thewing tip, the green line is the trailing edge and the scale is constant. The wake flow is characterized by asmall concentrated vortex surrounded by shear layers which wrap around the vortex itself and the wake isstretched into an ever-increasing spiral. Moving downstream the vortex core expands and it moves inboardand downwards, as expected, proportional to the square root of the stream-wise position.

For each vortex cross-section, the wake centerline could be defined by the locus of u2/U2∞ peaks and it

found to always be spiral shaped. A specific scale of each spiral ξ can be evaluated as the distance betweenthe vortex center and lowest part of the wake centerline (Fig. 5 b) and this grows approximately as the squareroot of the streamwise distance from the trailing edge. Interestingly, the centerlines of each cross-sectionbecome roughly self-similar when normalized with ξ (Fig. 8), as already reported by Devenport et al.1

V. Five Hole Probe and Three Sensor Hot Film Anemometer Traverses

Traverses through the vortex core were performed with 5HP and 3HWA in order to achieve more detailedvelocity profiles. The conditions analyzed were the same as in Section IV, but the space-step between eachmeasurement point was 0.5 mm. The sampling rate was always 2 kHz but the total sampling time was 33 sfor each point. A longer sampling time was necessary to achieve a good characterization of the wanderingphenomenon at low frequencies.

The vortex is roughly axisymmetric and it is evident that at the vortex core an abrupt reduction in the

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(a) (b)

(c) (d)

Figure 5. 5HP velocity measurements at U∞ = 10 m/s, α = 2◦, x/c = 1 in the baseline flow. a) U/U∞; b)

Normalized axial normal stress u2/U2∞; c) Turbulence kinetic energy k/U2∞; d) Mean streamwise vorticityωxc/U∞ obtained by differentiating grid data.

axial velocity is present. The core radius, which is the distance between the vortex center and the location ofthe maximum tangential velocity, grows approximately as the square root of the streamwise distance, whereasthe tangential velocity peak itself is inversely proportional to the same law. As was reported in Section IV,velocity profiles become roughly self-similar when the radius is normalized with ξ and the velocities withtheir respective tangential peak velocities. It was found that vortex circulation is well represented by theHoffmann and Joubert model.9

VI. Rapid Scanning

The aim of the Rapid Scanning is to perform a scan as fast as possible through the vortex core in order toconsider the vortex itself fixed during the whole sampling time. Thus these measurements are theoreticallynot affected by wandering. The 5HP was mounted on the tip of a rotating arm. The probe was rotatedat a frequency of 0.5 Hz which was assumed to be adequately high to consider the vortex core stationarywhile the probe was scanning. Sampling rate was set at 1 Khz. Before each run the probe was set in thetime-averaged vortex core position previously identified by the static measurements. Obviously, just a few

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(a) (b)

Figure 6. a)Cross-correlation coefficient between the horizontal and vertical velocity components, U∞ = 10m/s, α = 8◦, x/c = 1; b) Preferential direction of oscillation U∞ = 10 m/s, α = 8◦.

Figure 7. Contours of normalized axial normal turbulence stress in the baseline flow for the condition U∞ = 10m/s and α = 8◦. Contours start from the value of 4× 10−4 and they are at intervals of 4× 10−4

scans cross the centre of the vortex core. Figure 9 highlights typical discrepancies that were found betweenvelocity profiles measured by Rapid Scanning and static 3HWA due to the wandering smoothing effects.A vortex measured by static experimental techniques always appears more diffuse and weaker than RapidScanning data. The indicated core radius was up to 50% greater than the actual value and peak tangentialvelocity was reduced by 30% on average.

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Figure 8. Wake centerlines become self-similar when normalized on ξ. U∞ = 10 m/s and α = 8◦.

−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

∞

Static 3HWA

Devenport

Corrected

Rapid Scannning

Figure 9. Tangential velocity profiles at U∞ = 10 m/s, α = 8◦, x/c = 4. Comparison between static measure-ments, Rapid Scanning and velocity profiles corrected.

The vortex center for each scan was found with the method proposed by Corsiglia et al.3 Consequently, aprobability density function of core locations could be easily evaluated for each condition. These experimentalprobability density functions were well represented by a non-isotropic Gaussian function:

p(y, z) =1

2πσyσz

√1− e2

exp

{− 1

2(1− e2)

[(y − y0

σy

)2

+(

z − z0

σz

)2

− 2e(y − y0)(z − z0)σyσz

]}(1)

Vortex core coordinates in a cross-plane are (y0, z0), σy and σz are wandering amplitudes and e is thecross-correlation coefficient between horizontal and vertical velocity components. All those parameters were

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evaluated through a least squares fitting routine. Devenport et al.1 suggest that the wandering amplitudebe evaluated by the rms of a velocity cross-component divided by the tangential velocity gradient at thevortex core location. It seems that the accuracy levels are comparable for both criteria. Table 1 outlinesthat wandering amplitude increases proceeding downstream, which is in agreement with previous studies.1,4

The wandering amplitudes increase slightly with increasing angle of attack, which was also supported byflow visualization (Section III).

x/c σ/c

2.5 0.023 0.0214 0.025

α[deg] σ/c

8 0.01310 0.01612 0.017

(a) (b)

Table 1. Table I. Wandering amplitudes. a)U∞ = 10 m/s, α = 8◦ as function of x/c. b) U∞ = 20 m/s, x/c = 3 asfunction of α.

VII. Wandering smoothing effects correction

The significant influence of wandering on Eulerian measurements has been outlined in the previoussection. Consequently, a method to correct velocity profiles was deemed necessary. The method proposedby Devenport et al.1 was implemented. This method can reverse wandering smoothing effects on Eulerianmeasurements by representing the motion of the vortex core with a non-isotropic gaussian probability densityfunction (Eq. 1). Two features are fundamental for this method: fitting of velocity profiles and evaluationof wandering amplitude. The former was executed by a sum of gaussian functions centered at the meanvortex core location. Typical errors from 15% up to 40% of the fitted value for the axial vorticity peak werefound. Consequently, this operation seems to be one of weakest steps of the method. Wandering amplitudeswere evaluated by both criteria presented in Sec. VI. We evaluated wandering amplitudes from the fittingof probability density functions of the Rapid scanning data with a non-isotropic gaussian. The accuracylevel was improved by considering more than 2000 independent scans for each condition, even though fittingis always a tricky operation. With the Devenport method the accuracy is based on statistics of staticmeasurements, hence optimized by adequate frequency response of hot wire probes.

Table 2 outlines typical errors on the evaluation of core radius in terms of a percentage of the valuemeasured by Rapid Scanning.

x/c Static 3HWA Deven. Fit. Gaussian2.5 11% 11.16% 6.17%3 50.19% 9.66% 20.4%4 51% 3.5% 12.41%

α[deg] Static 3HWA Deven. Fit. Gaussian8 33.31% 1.31% 2.29%10 36.23% 15.04% 15.04%12 53.22% 25.34% 11.41%

(a) (b)

Table 2. Core radius errors in terms of core radius measured by Rapid Scanning measurements. a) U∞ = 10m/s, α = 8◦ as function of x/c. b) U∞ = 20 m/s, x/c = 3 as function of α.

The errors of the 3HWA static measurements are shown in the second column. In the third column errorsare reported related to wandering amplitudes evaluated by the Devenport method. In the last column errorsare reported relative to the method of using fitted experimental probability density functions. Even if errorsof the correction method are not negligible there was always an improvement realized over the results fromstatic measurements. Furthermore, we should recall that a significant proportion of the final correction erroris introduced by the fitting of the velocity profiles.

For peak tangential velocity, errors of static measurements are on average 30% of the value measuredwith Rapid scanning. The error of the final corrected peak tangential velocity is always less than 10%.

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VIII. Concluding Remarks

Static measurements (5HP and 3HWA) have shown that a wing wake flow is characterized by a smallconcentrated vortex surrounded by shear layers which wrap around the vortex itself, and the wake is stretchedinto an ever-increasing spiral. The vortex core radius increases proportional to the square root of the stream-wise distance from the trailing edge, whereas the tangential peak velocity is inversely proportional to thesame quantity. Distance between the vortex center and the lowest point of the wake centerline was defined asξ and this increases as square root of the streamwise distance. Wake centerlines become roughly self-similarwhen normalized with ξ. Similarly, velocity profiles become self-similar when velocities were normalized withthe tangential peak velocity and radial distance with ξ.

Rapid Scanning measurements have highlighted that static measurements errors could be up to 50%of the actual values for the core radius and, on average, 30% for the tangential peak velocity. Therefore,the correction method proposed by Devenport et al.1 was deemed necessary. Accuracy on evaluationof wandering amplitudes is comparable using both statistics of static 3HWA data or fitting experimentalprobability density functions obtained by Rapid Scanning tests. Error of corrected static measurements werealways less of 10%, even though a significant part of the errors is introduced by the fitting of the velocityprofiles.

Acknowledgements

The authors would like to thank Mauro Morelli who made a useful contribution to plan the tests atCSIR. Thanks are due to Giuseppe Barbaro, Matteo Biancalana and Luca Dell’Osso whose contributionswere fundamental to the execution of the tests and to analyze the data. Furthermore, thanks are due toProf. Franco Flandoli and Prof. Massimiliano Gubinelli for their essential suggestions on statistics.We wouldlike to thank Prof. Guido Buresti for all suggestions.

References

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2N.A. Chigier, and V.R. Corsiglia, “Wind-tunnel Studies of wing wake Turbulence”, J. of Aircraft, vol. 9, pp. 820–825,1972.

3V.R. Corsiglia, R.G. Schwind, and N.A. Chigier, “Rapid Scanning, Three-Dimensional Hot-Wire Anemometer Surveys ofWing-Tip Vortices”, J. of Aircraft, vol. 12, pp. 752–757, 1973.

4A.L. Heyes, S.J. Hubbard, A.J. Marquis, and D.A. Smith, “On the roll-up of a trailing vortex sheet in the very near field”,Proceedings of the institution of mechanical engineers part G-Journal of aerospace engineering, vol. 217, pp. 217–269, 2003.

5A.F.K. Yeung, and B.H.K. Lee, “Particle image velocimetry study of wing-tip vortices”, J. of Aircraft, vol. 36, pp. 482–484,1999.

6A.A. Gerner, and C.L. Maurar“Calibration of seven-hole probes suitable for high angles in subsonic compressible flow”,USAFA-TR-81-4.

7I.C. Lekakis, R.J. Adrian, and B.G. Jones“Measurement of velocity vectors with orthogonal and non-orthogonal triple-sensor probes”, Exp. in fluids, vol. 7, pp. 228–240, 1989.

8S.I. Green, and A.J. Acosta“Unsteady flow in trailing vortices”, J. Fluid Mech., vol. 227, pp. 107–134, 1991.9E.R. Hoffmann, and P.N. Joubert“Turbulent line vortices”, J. Fluid Mech., vol. 16, pp. 395–411, 1963.

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