48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2010, Orlando, Florida
AIAA 2010-1029
A Stereoscopic PIV Study of a Near-field Wingtip Vortex
Hirofumi Igarashi, Paul A. Durbin, Hongwei Ma and Hui Hu ()
Department of Aerospace Engineering, Iowa State University, Ames, Iowa, 50011
Email: huhui@iastate.edu
Abstract
An experimental study was conducted to investigate the wing-tip vortex generated from a
rectangular NACA0012 airfoil model. A high-resolution Stereoscopic Particle Image
Velocimetry (SPIV) system was used to conduct detailed flow field measurements to quantify
the transient behavior of the wing-tip vortex in the near field. The characteristics of the
vortex wandering phenomena, i.e., the slow side-to-side movement of the wing-tip vortex core,
were revealed in great detail based on the SPIV measurement results.
Introduction
The behavior of strong, coherent wingtip vortex structures is of great importance to various
commercial and military applications. One well-known problem is the hazardous effect of
trailing wingtip vortices on flight safety and airport capacity. From a military standpoint, there
are many issues associated with the effects of wingtip vortex structures on dynamics of towed
vehicles, tail buffeting, and icing arrays. Blade/vortex interaction on helicopter blades can impact
performance and cause undesirable noise and vibration. The study of wingtip vortices is not only
of great engineering importance but also of great scientific interest. It continues to be a
perplexing problem for computational scientists because of the presence of large gradients of
velocity and pressure in all three dimensions, especially in the near field, at high Reynolds
numbers.
A good physical understanding is essential in order to guide a variety of applications. Thus, a
detailed knowledge of the transient behavior of wing-tip vortex dynamics in the near field is
highly desired. Numerous experimental, theoretical and computational investigations have been
conducted in recent years to improve the basic understanding of wing-tip vortex structures, as
well as its control. Devenport et al. (1996) investigated vortex structure in the range of X/C = 4
to 29 downstream of a NACA 0012 airfoil with a blunt tip, at Re = 5.3 ×105. These authors used
a miniature, four-sensor hot-wire probe, and showed a deficit profile of approximately 84% of
the free-stream velocity. They reported that the flow outside the vortex core was dominated by
the remainder of the wing wake, which wound into an ever increasing spiral, and the turbulence
stress levels varied along the wake spiral in response to the varying rate of strain imposed by the
vortex. Chow et al. (1997) investigated the wing-tip vortex flow of a NACA 0012 airfoil model
with a rounded tip at Re = 4.6×106 and α =10o making measurements with a seven-hole pressure
probe and with a triple-hot-wire probe. They indicated a high level of axial velocity, in excess of
1
Copyright © 2010 by Hirofumi Igarashi, Paul A. Durbin, Hongwei Ma and Hui Hu . Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
1.7 U∞, at all measurement locations. They also reported that the turbulence intensity in the
vortex can be as high as 24%, but it decayed quickly with streamwise distance because of the
stabilizing effect of the nearly solid-body rotation in the vortex-core. Ramaprian & Zheng (1997)
observed no axial velocity excess for a tip vortex generated by a rectangular, square-tipped
NACA 0015 wing with Re=1.8×105 at α =10o. They used a three-component laser Doppler
anemometer. The inner part of the vortex was, however, found to be nearly axisymmetric within
X/C = 2.0 and exhibited a universal structure of asymptotic trailing vortices. More recently,
Birch & Lee (2004) examined the flow structure both along the tip and in the near field (up to
X/C =2.5) behind a square-tipped, rectangular NACA 0015 wing at Re = 2.1×105 for angles of
attack ranging from 2 to 19 degrees. Their measurements were produced from a miniature,
seven-hole pressure probe and a triple-hot-wire probe. The circulation was observed to have a
local maximum at X/C = 0.05 and remained virtually unchanged up to X/C = 2.5. The vortex
flow was self-similar and axisymmetric for X/C ≥ 0.5. The lift-induced drag was also computed
and compared with the wind-tunnel force-balance data.
Vortex wandering, the slow side-to-side movement of the wing-tip vortex core, has been
found to be a universal feature of wind-tunnel-generated wing-tip vortex structures. Jaquin et al.
(2001) proposed four possible causes for vortex wandering: the vortex could be un-stabilized by
wind-tunnel free-stream unsteadiness, turbulence in the surrounding shear layer, co-operative
instabilities, or propagation of unsteadiness from the model. In either case its most significant
effect is to obscure point-wise measurements by smearing the vortex core. Corsigilia et al. (1973)
reported that vortex wandering could be overwhelming, with the wandering amplitude being
several times of vortex core diameter. Devenport et al. (1996) developed an analytical technique
to correct mean-velocity profiles measured with a fixed hot-wire probe, thereby providing
quantitative estimates of wandering amplitude and its contributions to Reynolds stress. This
process involves forming a “corrected” mean velocity field from the measured version, using
guessed levels of wandering amplitude. The corrected field is then artificially subjected to
wandering, described by a bi-variate Gaussian probability density function (PDF) for vortex
position, and the resulting Reynolds stresses at the vortex center are compared with the actual
measured values. The wandering amplitudes are then adjusted iteratively until the calculated and
measured values converge. Using their wandering correction theory, they found that the
amplitude of wandering varied linearly between 0.1 and 0.4 vortex core radii at stream-wise
distances of 5 to 36 chord-lengths, and at its highest level, wandering was responsible for a 12%
and 15% error in the measured core radii and peak tangential velocity respectively. They
concluded that the vortex core in their experiments was laminar and that velocity fluctuations in
the vortex core region were entirely due to wandering. Rokhsaz (2000) investigated wandering of
a tip-vortex from a rectangular flat plate airfoil in a water tunnel and showed that wandering
increased with angle of attack and therefore vortex strength, the opposite to the finding of
Devenport et al. (1996) who showed a 15% reduction in wandering amplitude with an increase in
vortex strength of 85%.
These studies have uncovered useful information, but some of the inconsistencies noted
above raise questions. The majority of the previous studies used point-wise flow measurement
techniques such as pressure probes, hot-wire anemometry, and laser Doppler anemometer. A
common shortcoming of such point-wise measurements is the inability to provide spatial
structure of the unsteady vortices. Full field measurements are needed to effectively reveal the
transient behavior of the wing-tip vortex structures. Temporally-synchronized and spatiallyresolved flow field measurements are highly desirable in order to elucidate underlying physics.
2
Advanced flow diagnostic techniques such as high-resolution stereoscopic Particle Image
Velocimetry (SPIV) used in the present study are capable of providing such information.
Based on the flow field measurements with a conventional two-dimensional PIV system, Heyes
et al. (2004) evaluated wandering effects by recentering PIV data. They found that the Devenport
et al. (1996) assumption of using a bi-variate normal probability density function could be valid,
and their corrections were in good agreement with those predicted by the Devenport et al. (1996)
method. They found a 12.5% over-prediction of the core radius and a 6% under-prediction of the
peak tangential velocity. The errors were larger for lower angles of attack. They also found that
the wandering amplitude increases linearly with streamwise distance; a linear reduction was
found by increasing the angle of attack, so that they concluded that the mechanism responsible
for wandering is not self-induced, as had been proposed by Rokhsaz et al. (2000), but rather that
the vortex is responding to an external perturbation, as for instance the background turbulence
level, to which the tip vortex becomes less susceptible as the vortex strength is increased.
It should be noted the PIV measurement results reported in Heyes et al. (2004) were obtained by
using a conventional 2-D PIV system. It is well known that a conventional 2-D PIV system is
only capable of recording the projection of velocity into the plane of the laser sheet. That means
the out-of-plane velocity component is lost while the in-plane components may be affected by an
unrecoverable error due to the perspective transformation (Prasad & Adrian, 1993). For the
highly three-dimensional flow fields like wingtip vortex, the two-dimensional measurement
results may not be able to reveal their three-dimensional features successfully. In the present
study, a high-resolution Stereoscopic Particle Image Velocimetry (SPIV) system, which is
capable of achieving quantitative measurements of all three components of instantaneous flow
velocity vectors, is used to conduct detailed flow field measurements to quantify the transient
behavior of the wing-tip vortex generated from a rectangular NACA0012 airfoil. Because the
field is instantaneous, wandering does not smear the vortex. The trajectory of the wing-tip vortex
structures in the cross planes at different downstream locations will be revealed directly as a time
sequence of the instantaneous SPIV measurements. The underlying physics and characteristics of
vortex wandering is revealed in great detail.
Experimental Setup and Stereoscopic PIV System
The experimental study was conducted in a closed-circuit low-speed wind tunnel located in the
Aerospace Engineering Department of Iowa State University. The tunnel has a test section with a
1.0 × 1.0 ft (30 × 30 cm) cross section and the walls of the test section are optically transparent.
The tunnel has a contraction section upstream of the test section with honeycombs, screen
structures and a cooling system installed ahead of the contraction section to provide uniform low
turbulent incoming flow into the test section. The standard deviation of velocity fluctuations at
the test section entrance was found to be about 0.8% of the free-stream velocity, as measured by
a hotwire anemometer.
The test model used in the present study was a zero-sweep, untwisted half wing, with a
NACA0012 symmetric profile. The wing had a rectangular planform with a semi-span
b=0.1524m and a chord c=0.1016m and a plane tip with sharp edges. Its root was mounted on
one side of the wind tunnel wall. The angle of attack of the tested wing could be changed using a
rotary table mounted outside the wind tunnel with the axis of rotation passing through the
3
airfoil’s aerodynamic center (quarter-chord point). The wing was mounted through the wind
tunnel wall such that its quarter-chord axis was along the centerline of the wind tunnel.
Following the work of Bailey & Tavoularis (2008), a boundary layer trip wire was attached on
the suction surface of the airfoil, 0.10c away from the leading edge, to induce transition, thus
reducing the possibility of separation and sensitivity to free-stream conditions. For the present
study, the angle of attack of the tested wing was changed from 0 to 20 degrees (i.e.,
α = 0 ~ 20 deg . ). During the experiments, the wind-tunnel speed was adjusted from 5.0m/s ~ 35
m/s, which corresponds to a chord Reynolds number Re C = 0.33 × 105 ~ 2.33 × 105.
Figure 1: Schematic of the experimental setup
Figure 1 shows the schematic of the experimental setup used in the present study for the
stereoscopic PIV measurements. The flow was seeded with 1~5 μm oil droplets. Illumination
was provided by a double-pulsed Nd:YAG laser (NewWave Gemini 200) adjusted on the second
harmonic and emitting two pulses of 200 mJ at the wavelength of 532 nm with a repetition rate
of 10 Hz. The laser beam was shaped to a sheet by a set of mirrors, spherical and cylindrical
lenses. The thickness of the laser sheet in the measurement region is about 1.0 mm. Two highresolution 12-bit (1600 x 1200 pixel) CCD camera (PCO1600, CookeCorp) were used to perform
stereoscopic PIV image recording. The two CCD cameras were arranged in an angular
displacement configuration to get a large overlapped view. With the installation of tilt-axis
mounts, the lenses and camera bodies were adjusted to satisfy the Scheimpflug condition
(Prasad&Jensen, 1995). In the present study, the distance between the illuminating laser sheet
and image recording planes of the CCD cameras is about 1000mm, and the angle between the
view axial of the two cameras is about 60 degrees. For such an arrangement, the size of the
stereoscopic PIV measurement window is about 80mm by 60mm. The CCD cameras and doublepulsed Nd:YAG lasers were connected to a workstation (host computer) via a Digital Delay
4
Generator (Berkeley Nucleonics, Model 565), which controlled the timing of the laser
illumination and image acquisition.
A general in-situ calibration procedure was conducted in the present study to obtain the
mapping functions between the image planes and object planes Soloff et al. (1997). The
mapping function used in the present study was taken to be a multi-dimensional polynomial,
which is fourth order for the directions (X and Y directions) parallel to the laser sheet plane and
second order for the direction (Z direction) normal to the laser sheet plane. The coefficients of
the multi-dimensional polynomial were determined from the calibration images by using a “least
square” method. The two-dimensional particle image displacements in each image plane were
calculated separately by using a frame-to-frame cross-correlation technique involving successive
frames of patterns of particle images in an interrogation window 32×32 pixels. An effective
overlap of 50% of the interrogation windows was employed in PIV image processing. By using
the mapping functions obtained by the in-situ calibration and the two-dimensional displacements
in the two image planes, all three components of the velocity vectors in the illuminating laser
sheet plane were reconstructed (Hu et al. 2002).
After the instantaneous velocity vectors ( u , v , w ) were determined, instantaneous spanwise
vorticity ( ω x ) could be derived. The time-averaged quantities such as mean velocity ( U ,V ,W ),
ensemble-averaged streamwise vorticity( Ω x ), turbulent velocity fluctuations ( u ' , v' , w' ) and
normalized turbulent kinetic energy distributions, i.e., T .K .E. =
2
2
2
1
(u ' + v' + w' ) , were
2
2U ∞
obtained from a cinema sequence of about 500 frames of instantaneous velocity fields. The
measurement uncertainty level for the velocity vectors is estimated to be within 1.0%, and that of
the turbulent velocity fluctuations ( u ', v' , w' ) and T .K .E. are about 5.0%.
Vortex Center Identification
To extract the wandering statistics of the wingtip vortex, it is necessary to keep track of the vortex
center at every instantaneous SPIV frames. The center of the vortex is usually interpreted in two
ways. (1) the point where both tangential and radial velocities are equal to zero, i.e. vθ = vr = 0 , or
(2) the point where the streamwise vorticity is maximum (positively or negatively, depending on
the rotating direction of the vortex). It is natural to think that conditions (1) and (2) can be met
simultaneously. It is true for a steady vortex with pure swirl motion. However, it is not true for an
unsteady vortex where the vortex core is moving up and down just like the wingtip vortex
considered in the present study.
In the present study, condition (2) is used to identify the vortex center to characterize the vortex
wandering phenomena. Following the work of Cohn and Koochesfahani(2000), the streamwise
vorticity, ω x , is calculated from the measured velocities using a 4th order accurate central finite
difference scheme:
5
⎛ ∂w ⎞
⎛ ∂v ⎞
ω x = ⎜⎜ ⎟⎟ − ⎜ ⎟
⎝ ∂y ⎠ i , j ⎝ ∂z ⎠ i , j
(1)
Where
− wi + 2, j + 8wi +1, j − 8wi −1, j + wi −2, j
⎛ ∂w ⎞
⎜⎜ ⎟⎟ =
12Δy
⎝ ∂y ⎠ i , j
(2)
and
− vi + 2, j + 8vi +1, j − 8vi −1, j + vi −2, j
⎛ ∂v ⎞
⎜ ⎟ =
12Δz
⎝ ∂z ⎠ i , j
(3)
Using Eq.(2) and (3) alone, the location of the identified vorticity peak falls on to the grid point. A
2-dimensional 2nd order polynomial function was used to fit the measured vortex field in the
neighborhood the vorticity peak location in order to improve the accuracy to indentify the center of
the vortex at the sub-grid interpolation level, which shown below.
R ( y, z ) = Ay 2 + Bz 2 + Cyz + Dy + Ez + F
(4)
where six unknown coefficients were determined by taking the six nearest grid pints near the
measured vorticity peak. The partial derivatives of Eq(4) were taken and set equal to zero in order
to derive the location of the vorticity peak.
∂R
= 2 Ay + Cz + D = 0
∂y
(5)
∂R
= 2 Bz + Cy + E = 0
∂z
(6)
and
The accurate vortex center can be determined by solving the system of equations (5) and (6).
6
Experimental Results and Discussion
a). 1.0 chord length downstream
d). 2.0 chord length downstream
c). 3.0 chord length downstream
d). 4.0 chord length downstream
Figure 2: Stereoscopic PIV measurements to reveal the evolution of the wingtip vortex at 4
different downstream locations ( Re C = 0.52 × 105 , α = 5.0 deg . )
Figure 2 shows the evolution of the wingtip vortex at four different downstream locations where
the color contour represents the normalized streamwise velocity. The interaction of the wake of the
wing and the tip vortex is clearly revealed. At 1.0 chord length downstream (Figure 2a), the vortex
center is observed around z = −0.5mm and the wake of the airfoil is located about z = −20.0mm .
As the vortex progresses downstream, the vortex center location stays fairly the same, whereas the
wake of the wing decays rapidly. The contour indicates that the vortex core has the velocity deficit
of more than 20%. Also, this deficit region becomes smaller as the vortex moves downstream.
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(a). α = 0.0 deg .
(b). α = 2.0 deg .
(c). α = 4.0 deg .
(d). α = 6.0 deg .
(e). α = 8.0 deg .
(f). α = 10.0 deg .
Figure 3: Stereoscopic PIV measurements to reveal the normalized T.K.E. evolution at 6 different
angles of attack ( Re C = 1.15 ×10 5 ,X/ c = 4.0 ).
8
Figure 3 shows the evolution of the tip-vortex with angles of attack with the measurement plane at
4.0 chordlength downstream. As the angle of attack increased, the wake from the wing moves
down and the T.K.E. within the wake increases. For all positive angles of attack, the vortex center
seems to have the largest T.K.E. value and the turbulent region increase as the angle of attack.
2.0
-1
σz
σy
ρ
1.5
-2
1.0
Vorticity
-3
0.5
-4
0
-5
-0.5
-1.0
0
2
4
6
8
10
12
-6
14
0
2
4
6
8
10
12
14
Angle of Attack (deg)
Angle of Attacke (deg)
(a). Wandering amplitude and correlation
(b).Vorticity
Figure 4: Wandering amplitude and vorticity with angle of attack ( Re C = 1.15 × 105 , c = 4.0 ).
Figure 4 shows the wandering amplitude and vorticity with angle of attack at four chord lengths
downstream. As described in Birch (2004), the vorticity increases as the angle of attack (Figure 4b).
The vorticity magnitude increases about five times as the angle of attack increased from 2 to 12
degrees. The corresponding wandering amplitude is shown in Figure 4a. From 2 to 6 deg, the
wandering amplitude in both y and z-directions decreases. At higher angles of attack, from 6 to 12
degrees, the wandering amplitude increases in both directions; however, the amplitude in ydirection increases sharply, whereas the z-direction amplitude increases rather gradually. The two
figures show that, at lower angle of attack, 50 % increase in the vorticity results in a 70% decrease
in wandering amplitude. And for the higher angles of attack, the wandering amplitude increases
50% with the 25% increase in vorticity. Rokhsaz (2000) and Devenport et al. (1996) reported that
wandering amplitude increases with increasing angle of attack and decreasing angle of attack,
respectively; however, figure 4 shows that the wandering could increase or decrease with
increasing angle of attack.
9
-1.0
1.00
0.75
-1.2
0.50
-1.4
Vorticity
Wandering Amplitude
σz
σy
ρ
-1.6
0.25
-1.8
0
-2.0
0
1
2
3
4
5
0
1
2
3
4
5
Downstream (Chord Length)
Downstream (chord length)
1.0
1.0
0.5
0.5
z/r
z/r
(a). Wandering amplitude and correlation
(b).Vorticity
Figure 5: Wandering amplitude and vorticity with downstream ( Re C = 0.53 ×10 5 , α = 5.0 ).
.
Figure 5 shows the downstream effect on the wandering amplitudes and maximum vorticity. The
general trend indicates that the maximum vorticity decreases by about 25% as the distance
increasing from 1 to 4 chord lengths downstream. The amplitudes of the wandering (figure 5a),
stay constant and they are about the same for both y and z-directions. Small change, about 20%, in
vorticity seems to have a little or no effect on the wandering amplitudes.
0
-0.5
-0.5
-1.0
-1.0
0
-0.5
0
0.5
1.0
-1.0
-1.0
-0.5
0
y/r
y/r
(a). vortex center projection in y-z plane.
10
0.5
1.0
1.00
0.75
0.75
0.50
0.50
0.25
0.25
y/r
y/r
1.00
0
0
-0.25
-0.25
-0.50
-0.50
-0.75
-0.75
-1.00
-1.00
0
20
40
60
80
100
120
0
20
40
60
80
100
120
Time (sec)
Time (sec)
1.00
0.75
0.75
0.50
0.50
0.25
0.25
z/r
z/r
(b). History of vortex center in y-direction.
1.00
0
0
-0.25
-0.25
-0.50
-0.50
-0.75
-0.75
-1.00
0
20
40
60
80
100
-1.00
120
0
20
40
60
80
100
120
Time (sec)
Time (sec)
(c). History of vortex center in z-direction.
(A). α = 2.0 deg
(B). α = 6.0 deg
Figure 6: Vortex wandering trajectories ( Re C = 1.15 × 105 , c = 4.0 ).
From figure 4a, the highest and lowest wandering amplitude cases are selected and the details are
show in figure 6. In the highest wandering amplitude case of α = 2.0 deg (Figure 6A), the
amplitude stays within the 75% of a core radius in y-direction, and 50% of a core radius in zdirection. For the lowest amplitude case, the wandering amplitude stays within 25% of a core
radius in both y and z-direction. In anyway, the wandering amplitude is relatively small and cannot
be several times of vortex core diameter in the near field, as described in Corsigilia et al. (1973).
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Conclusion
In the present study, a high-resolution Stereoscopic Particle Image Velocimetry (SPIV) system,
which is capable of achieving quantitative measurements of all three components of instantaneous
flow velocity vectors, was used to conduct detailed flow field measurements to quantify the
transient behavior of the wing-tip vortex generated from a rectangular NACA0012 airfoil.
By tracking of the vortex center at every instantaneous frames, the characteristic of the vortex
wandering phenomena was described quantified in detail. While the present SPIV measurement
results confirmed the findings of previous studies about the characteristics of the wingtip vortex in
the near, the direct relationship between the vorticity and wandering amplitude was not observed
based on the present SPIV measurement results.
Acknowledgments
The authors want to thank Mr. William Rickard of Iowa State University for his help and support in conducting
the wind tunnel experiments. This work is sponsored by the AFOSR, grant FA9550-08-1-0294.
12
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