49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2011, Orlando, Florida
AIAA 2011-1165
The Effects of Wind Tunnel Walls on the Near-field Behavior of a
Wingtip Vortex
Hirofumi Igarashi, Paul A. Durbin and Hui Hu ()
Department of Aerospace Engineering, Iowa State University, Ames, Iowa, 50011
Email: huhui@iastate.edu
and
Scott Waltermire and Joe Wehrmeyer
Arnold Engineering Development Center, Arnold AFB, TN
Abstract
An experimental study was conducted to investigate the behavior of the wing-tip
vortex structures generated by a square-tipped, rectangular NACA0012 wing. A
Stereoscopic Particle Image Velocimetry (SPIV) system was used to conducted
detailed flow field measurements to elucidate the key features of the wing-tip
vortex structures in the near field. One of the great advantages of the present
SPIV measurements over the classical measurement technique is that the vortex
wandering can be removed directly by tracking the center of the wingtip vortex
in the instantaneous measurement frames. By tracking the center of the wingtip
vortex, the wandering and turbulence in the vortex can be decoupled completely.
This method was applied to investigate the effects of the angle of attack of the
test wing and wind tunnel wall on the evolution of the wingtip vortex in the near
field. In order to decouple the effects of vortex wandering, Devenport et al.
(1996) suggested an analytical method to predict the wandering free velocity
profile. The velocity profile predicted by the Devenport et al. (1996) method was
compared with the SPIV re-centered velocity profile quantitatively.
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.
Vortex wandering is the slow side-to-side movement of the wing-tip vortex core. It has been
found to be a universal feature of wind-tunnel-generated wing-tip vortex structures. 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
Copyright © 2011 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
center are compared with the actual measured values. The wandering amplitudes are then
adjusted iteratively until the calculated and measured values converge. Using the wandering
correction theory, they concluded that the vortex core in their experiments was laminar and that
velocity fluctuations in the vortex core region were mainly due to wandering. 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 spatially-resolved flow field measurements are highly
desirable in order to elucidate underlying physics. 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 re-centering 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.
It should be noted the PIV measurement results reported in Heyes et al. (2004) was obtained by
using a conventional 2-DPIV 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 flows like wingtip vortex flows, 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.
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 12.0 × 12.0 in (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,
measured by using a hotwire anemometer.
Figure 1 shows the schematic of the experimental setup used in the present study. 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 5 Hz. The thickness of the laser sheet in
the measurement region is about 2.0 mm. Two high-resolution 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. In the present study, the angle the angle between the view axial of the two
cameras is set to about 60 degrees. For such an arrangement, the size of the stereoscopic PIV
measurement window is about 80mm by 60mm. The CCD cameras and double-pulsed Nd:YAG
lasers were connected to a workstation (host computer) via a Digital Delay 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).
Fig. 1: Schematic of the experimental setup
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 half-span
b/2=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
airfoil’s aerodynamic centre (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 . ) and the results from 2 to 12 degrees are shown in the present study. During the
experiments, the wind-tunnel speed was adjusted from 7.9 m/s ~ 17.3 m/s, which corresponds to
a chord Reynolds number Re C = 0.52 × 10 5 ~ 1.15 × 10 5.
(a). Without additional walls
(b). With additional wall (a=2.0, 4.0 and 5.0in)
Fig. 2: Cross section view of the test section
In order to investigate the effects of the wind tunnel wall on the evolution of the wing tip vortex,
additional U-shaped hollow walls are inserted in the test section to reduce the distance between the
wing tip and the tip-side wall. The width of the wall, as shown in figure 2, is varied to 2.0, 4.0 and
5.0 inch so that the distance between the wing tip and wall is set to 4.0, 2.0 and 1.0 in respectively.
These additional walls are made of optically transparent acrylic plates with the thickness of 1/8 in.
This wall is thick enough to stand rigid in the test section; yet, thin enough to cause no noticeable
tunnel blockage issues. Since the index of refraction may vary due to the additional walls, in-situ
calibration procedures for stereoscopic PIV measurements were performed each time to
compensate the optical distortion.
Experimental Results and Discussions
A. SPIV Method
In this section, the method used to extract the wandering free vortex velocity field is
described; hereafter, referred to as the ”SPIV method” to distinguish from the Devenport et al.
(1996) correction method. SPIV data presented in this study is the result of 500 instantaneous
SPIV frames, taken at the rate of 4 frames per second. Since the SPIV data acquisition rate was
far below the frequency of the vortex motion, the velocity fields shown in this study is the time
average of 500 instantaneous frames, or about 125 seconds. To extract the wandering free vortex
velocity field, it is necessary to keep track of the vortex center at every instantaneous SPIV
frame. First, the vortex center is estimated by finding the maximum negative vorticiy around the
middle of the measured plane. Then, the first 7.0mm × 7.0mm grid with 8 × 8 nodes are formed
with maximum vorticity in the center and the cross stream velocities are interpolated to the
minimum values. After determining the minimum cross stream velocity with the 7.0mm ×
7.0mm grid, the new grid is formed twice to a 5.0mm × 5.0mm grid with 51 × 51 nodes and
further down to a 1.0mm × 1.0mm grid with 101 × 101 nodes. As the program generates new
grids, the grid size gets smaller and the grid gets finer. The velocities on the new grid nodes are
linearly interpolated from nearby node values in the old grid since the velocity profile is close to
linear in the vortex core. Then a 50 to 40mm square velocity field, depending on the case, is
extracted with the vortex center at (0, 0) as shown in Figure 3(a). By repeating this method for all
the frames, these square velocity fields can be considered as the instantaneous “wandering
corrected” velocity fields. Then, the core radius of the wingtip vortex was determined from the
measured tangential velocity profiles of the vortex, as shown in Fig. 3(b).
(a).Velocity field with a vortex at center
(b). core radius and maximum tangential velocity
Fig. 3: Extracted velocity field and a typical tangential velocity profile.
By tracking the vortex center for all 500 frames, the vortex wandering behavior is
quantitatively described with time averaged mean vortex center at (0,0). Figure 4(a) shows the
projection of the vortex centers in y-z plane for α = 2.0 o case and Figures 4(b) and 4(c) show the
vortex trajectory in y and z directions, respectively. From the statistical analysis, the amplitude
of the vortex wandering is described by the standard deviation of the instantaneous vortex center
location. The wandering amplitude of the vortex in y-direction is denoted σ y and z-direction is σ z .
They are normalized by the core radius, rc , as in Figure 3(b). The correlation of two is shown
as ρ to indicate the orientation of the vortex wandering. Figures 5(a) and 5(b) show the histogram
of the vortex location which validates one of the assumptions made by Devenport et al. (1996)
that the vortex wanders in a Gaussian distribution manner.
1.0
0.8
0.6
0.4
Z/rc
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1.0
Y/rc
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Z/rc
Y/rc
(a) Vortex center projection in y-z plane
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
0
20
40
60
80
100
Time (sec)
(b) Vortex center trajectory in y-direction
120
-0.8
0
20
40
60
80
100
120
Time (sec)
(c) Vortex center trajectory in z-direction
Fig. 4: Vortex center projection and trajectory for α = 2.0 o ( x = 4.0c , Re C = 1.15 × 10 5. )
(a) y-direction
(b) z-direction
Fig. 5: Histogram of vortex center at α = 2.0 o ( x / c = 4.0 , Re C = 1.15 × 10 5. )
B. The effects of the angle of attack of the test wing
Figure 6a shows the angle of attack effect on the wandering amplitude. The circle symbol
indicates the wandering amplitude in y-direction, σ y , and the square symbol indicates the
amplitude in z-direction, σ z . As shown in the Figure, the both amplitudes, σ y and σ z , were
found to decrease with the angle of attack increases until α ≈ 8O. After α > 8O, the wandering
amplitudes was found almost stay constant. This result was found to closely agree with the
findings of Devenport et al. (1996). They reported the wandering amplitudes (σ / c) of 0.009,
0.006, 0.006, and 0.005 at α = 2.5, 3.9, 5.0 and 7.5⁰ respectively, where the present study shows
that the wandering amplitude of 0.009, 0.005, 0.004, and 0.003 at α = 2.0, 4.0, 6.0 and 8.0 O ,
respectively. Also, as shown by Devenport et al. (1996), the vortex wanders 0.002c or 0.001c
more in y-direction (span-wise direction). This trend can be seen in the present study as the
vortex wanders 0.002c or 0.004c more in the span-wise direction. As shown in the Table 1, the
wandering is found to have the largest amplitude when the vortex had the lowest peak tangential
velocity, Vθmax. When the maximum tangential velocity Vθmax ≈ 2.0m/s at α = 2.0 O, the vortex
wanders about 0.9mm in both y and z directions. However, when the angle of attack is increased
to 12.0 O, the tangential velocity is increased to Vθmax ≈ 8.60m/s and the vortex wandering was
found to be reduced by 70%. For the same flow condition, even thought the wandering is
constant at the higher angle of attack, the increase in the tangential velocity will have some effect
on decreasing the vortex wandering which agree with the finding by Bailey and Tavoularis
(2008).
Figure 6b shows the angle of attack effect on the vortex core radius. Initially the axialaveraged vortex core radius is 5.37mm at α = 2.0 o . When the angle of attack is increased
to α = 4.0 o , the vortex core radius was found to become smaller to 4.85mm. As the angle of
attack gets larger toward α = 12.0 o , the core radius became bigger and bigger. At α = 6.0 o , the
vortex core radius is 5.0mm and for the larger angle of attack of α = 12.0 o , the vortex radius was
found to increase to 6.19mm. This trend, a small decrease in the vortex core before the
increasing core radius at high angle of attack, also can be seen in the study of Gerontakos and
Lee (2006). By using a sweptback and tapered NACA0015, Gerontakos and Lee (2006) showed
the sharp decrease in the core radius up to α = 6.0 o and the gradual increase as the angle
approaches to α = 9.0 o .
0.20
7.0
0.18
σZ
0.16
6.5
σY
0.14
6.0
σ/r
c
rc (mm)
0.12
0.10
5.5
0.08
5.0
0.06
0.04
4.5
0.02
0
0
2
4
6
8
α (deg)
(a). Wandering amplitude
10
12
14
4.0
0
2
4
6
8
10
12
14
α (deg)
(b). Axial averaged vortex core radius
Fig 6: Angle of attack effect ( x / c = 4.0 , Re C = 1.15 × 10 5. )
Table1: The measured wandering amplitude vs. the estimations by using Devenport’s method.
Figures 7 show the normalized turbulence kinetic energy (T.K.E.) distributions of the wingtip
vortex with and without the wandering correction. The left column of the figures show the
distributions without the wandering correction and the right column show the distributions with
wandering correction. At α = 2.0 o , the highest normalized T.K.E. value of about 1.60 × 10 −3 is
observed at the center of the vortex (Figure 7a). After tracking the vortex center in every frame
to eliminate the vortex wandering effect, as shown in Figure 7b, the normalized T.K.E. at the
vortex center was found to be reduced to less than 1.0 × 10 −4 . By comparing the two plots, it can
be seen that the higher T.K.E. values in the center of the vortex is mainly due to the vortex
wandering rather than turbulence. As the angle of attack increases, the spiral wake becomes
larger, while the maximum T.K.E. value in the wake stays about 1.0 × 10 −3 . As the angle of attack
increasing from 8.0 o to 12.0 o , the T.K.E. level increases significantly. At α = 8.0 o , the maximum
T.K.E. is about 1.60 × 10 −3 at the center but for α = 12 .0 o , the T.K.E. goes above 2.0 × 10 −2 and fills
most of the vortex core. Even after removing the wandering, the T.K.E. in the vortex core
increases from 3.0 × 10 −3 to 6.0 × 10 −3 (Figure 7e and 7f). This suggests that the wingtip vortex is
much more turbulent at higher angles of attack.
(a). α = 2.0 o , before wandering correction
(b). α = 2.0 o , after wandering correction
(c). α = 6.0 o , before wandering correction
(d). α = 6.0 o , after wandering correction
(e). α = 8.0 o , before wandering correction
(f). α = 8.0 o , after wandering correction
(g). α = 12 .0 o , before wandering correction
(h). α = 12 .0 o , after wandering correction
Fig 7: Normarized T.K.E. distributions ( x = 4.0c , Re C = 1.15 × 10 5. ).
Figure 8 shows the Reynolds Stress distributions of the wingtip vortex before and after
wandering correction. The left column of the plots shows the distributions without wandering
correction and the right column shows the distribution with wandering correction. Before
correcting the center, the vortex core has two pairs of positive and negative Reynolds Stress
except for at α = 2.0 o .These pairs of stresses become stronger as the angle of attack increases.
For α = 8.0 o (Figure 8e), the maximum stresses are about ±2.0×10-4 and when the angle of
attack is increased to α = 12 .0 o , as shown in Figure 8g, the stress level exceeds ±3.0×10-4. These
two pairs of positive and negative stresses are found in the study done by Chow et al. (1997).
They describe that the contours of the Reynolds stress in the wake planes had a four-leaf clover
pattern with alternately changing sign of stresses in each leaf. Each leaf was roughly aligned at
±45⁰ off the y and z axis, with positive levels of stress found in the first and third quadrants (+/+
and - values of z and y, respectively) and negative levels of stress found in the second and fourth
quadrants. This four-leaf clover pattern is explained by converting the Reynolds stress equation
from a Cartesian coordinate to the cylindrical coordinate system and the detailed explanation is
described on Chow et al. (1997). The right column of Figure 8 shows the Reynolds stress
distribution after the wandering correction. As shown in the figures, the Reynolds Stresses in the
core region cannot be seen after the wandering correction. For the lower angles of attack, the
stress level is near before the correction but after the wandering correction, the stress is
completely neutralized to less than ±1.5×10-5 . For the higher angles of attack, especially at
α = 12 .0 o , some stress is still remaining even after the wandering correction; however, the stress
level is decreased more than 50%, from ±3.0×10-4 or greater to about ±1.5×10-4 . This reduction
in the Reynolds stresses tells that the four-leaf clover pattern in the vortex core is mainly due to
the vortex wandering. Interestingly enough, the wandering correction diminished the stress levels
within the core and only within the core. For example, at α = 6.0 o , only the stress pairs within
the core disappear and the stresses within the spiral wake, especially the square shaped positive
stress region, stay almost the same in the terms of the magnitude and distribution pattern.
(a). α = 2.0 o before wandering correction
(b). α = 2.0 o , after wandering correction
(c). α = 6.0 o before wandering correction
(d). α = 6.0 o after wandering correction
(e). α = 8.0 o before wandering correction
(f). α = 8.0 o after wandering correction
(g). α = 12 .0 o before wandering correction
(d). α = 12 .0 o after wandering correction
Fig 8: Normarized Reynolds stress distributions ( x = 4.0c , Re C = 1.15 × 10 5. ).
C. Comaprsions of the SPIV measurements and the predictions of the Devenport’s Method
The Devenport’s method was applied to process the measurement date at two different angles
of attack, α= 2.0⁰ and 4.0⁰. Figure 9a shows the tangential velocity profiles of the vortex along
the positive z-axis at α= 2.0⁰ and Figure 9b shows the tangential velocity profile with the same
flow conditions at α= 4.0⁰. For both figures, the blue solid curve represents the velocity profile
with vortex wandering. The black solid curve represents the velocity profile with wandering
correction, i.e., instantaneously corrected wandering free velocity profile, and the red symbols
represent the velocity profile estimated by Devenport’s method for wandering correction, i.e.,
numerically derived wandering free velocity profile. At α = 2.0⁰, as shown in Figure 9a, the
SPIV method and Devenport’s method agree very well with each other from the vortex core
center to the core radius. After passing the core radius, the Devenport’s correction curve starts to
diverge from the SPIV curve and converging to the non-corrected curve. This is because the
equations used to derive the Devenport’s curve are formulated to converge to the non re-centered
profile at the outer region of the vortex. For this case, the SPIV corrected curve is more like the
off–set profile from the non re-centered profile, so that there will be some discrepancy between
those two curves at the outer region of the vortex. Figure 9b shows the velocity profile at α= 4.0⁰.
For this case, the SPIV corrected profile and Devenport’s corrected profile within the vortex core
do not match to each other, or the Deveport’s method recovered less tangential velocity than
SPIV method. After passing the vortex radius, all three velocity profiles converged to each other
indicating that there is not much wandering effect occurring at this particular outer region of the
vortex core. By observing the profiles on Figures 9a and 9b, the Devenport’s method seems to be
not consistently predicting the correct velocity profile in radial direction. For α= 2.0⁰ case, they
have a good agreement within the core but for α= 4.0⁰ case they have a good agreement at the
outer core region. This inconsistency happens because the Devenport method uses only one
velocity profile on the positive z axis. This is not sufficient to fully recover the wandering free
velocity profile accurately. One of the assumptions made by Devenport et al. (1996) was to
assume that a wingtip vortex would be axisymmetric, however, in reality, a wing-tip vortex is
not perfectly axisymmetric, as shown in Figure 10, and has the distinctive profile shape
depending on the profile extracting locations.
SPIV re-centered
Devenport re-centered
Not re-centered
1.0
SPIV re-centered
Devenport re-centered
Not re-centered
1.00
0.8
0.6
v/vθ
v/vθ
0.75
0.50
0.4
0.25
0.2
0
0
1
2
3
0
4
0
1
2
z/rc
3
4
z/rc
(a). α = 2.0 deg .
(b). α = 4.0 deg .
Fig. 9: Tangential velocity profile for non re-centered data and re-centered data using SPIV
method and Devenport’s method ( Re C = 1.15 × 10 5 , c = 4.0 ).
2.5
Positive y-axis
Positive x-axis
Negative y-axis
Negative x-axis
2.0
vθ (m/s)
1.5
1.0
0.5
0
0
10
20
30
r (mm)
Fig. 10: Tangential velocity profiles along different axes (α=2.0⁰, Rec=1.15×105, x/c=4.0)
D. Effects of the Wind Tunnel Walls
As described in the set-up section, an additional wall was inserted in the wind tunnel test
section to change the wall-to-tip distance from 6.0in to 4.0, 2.0, and 1.0in in order to investigate
the effects of the wind tunnel walls on the evolution of the wingtip vortex. Figure 11 shows the
projection of the vortex centers and the average vortex center with different wall lengths. When
there is no additional wall, i.e., 6.0in wall-to-tip distance, the vortex center lies around the
location of z /c= −0.052, which is near the trailing edge height. As the tunnel wall gets closer to
the wing tip, the vortex starts moving upward. When the wall exists, it acts as the mirror vortex
image which exists on the other side of the wall. Therefore, there will be a stronger influence of
the vortex up-wash, which drives the vortex upward. The correlation between the wall to the
wing-tip distance and the vortex up wash driven height can be shown in log-log scale plot as in
Figure 12. By fitting the curve with the power equation, the equation is determined as ln y =
−1.69 log x + 8.61. These coefficients may vary with the flow condition and wing angle of attack,
as the equation is depending on the vortex radius.
0.15
Tip to Wall = 0.25c
0.1
z/c
0.05
Tip to Wall = 0.50c
Wind tunnel center
+
0
Tip to Wall = 1.0c
Tip to Wall = 1.5c
NACA0012
Trailing Edge
at α=5.0°
-0.05
-0.1
-0.05
0
0.05
0.1
0.15
0.2
y/c
Fig. 11: Mean vortex location with different tip-to-wall distance.
( α = 5.0 deg, x = 4.0c )
Fig 12: Tip-to-Wall distance versus measn vortex center location in z-direction.
Z=0.0mm at airfoil trailing-edge. ( α = 5.0 deg, x = 4.0c )
The Figure 13 shows the normalized T.K.E. and Reynolds stresses distributions of the
wingtip vortex with the different wingtip-to-wall distances. As the wall gets closer from 4.0in to
1.0in, the T.K.E. values get higher. After removing the wandering effect, which is shown in
Figures 13a and 13c, it is clearly shown that moving the inner wall closer to the wing-tip will
increase the turbulence level within the vortex core more than twice. When the wall is further
pushed up to the 1.0in tip-to-wall distance, the both non re-centered and re-centered distribution
look the same. That suggests that the T.K.E., at 1.0in distance, is dominated by the turbulence
within the vortex itself and the turbulence by the wandering has little or no effect impact on the
total T.K.E. distributions. Table 2 shows the wandering statistics of the wing-tip vortex with the
wall effect. The introduction of the additional wall increased the wandering amplitude about
twice from σ ≈ 0.4 to σ ≈ 0.7 , and the center location distribution is less axisymmetric, e.g.,
σ y = 0.203rc and σ z = 0.296rc with 2.0in wall-to-tip distance. For the 1.0in tip-to-wall distance
case, the vortex wanders 50% more in the y-axis direction. The vortex radius tends to get larger
from 4.9mm to 6.33mm as the wall get closer from thirty to less than ten radii away from the tip;
however, at 1.0in wall-to-tip, the radius gets smaller because the physical wall is getting closer to
less than five vortex radii away. While the vortex behaves differently with the additional walls,
the vorticity is not affected by the wall effect and stays around ω ≈ −3.1 at up to 8.0rc. When the
wall is moved closer to less than 5 radii away from the tip, the vorticity gets smaller by 10%.
Observing from the vortex structure and behavior, the vortex seems to interact with the mirror
image vortex on the other side of the wall up to 8.0rc away; however, if the wall gets to within
5.0 or 4.0 radii way from the wing-tip, vortex would have interaction with the physical wall more
and behaves more in a chaotic fashion.
Table2: Numerical data for SPIV method.
(a). 4.0in tip-to-wall, T.K.E.
(b). 4.0in tip-to-wall, Reynolds stress.
(c). 2.0in tip-to-wall, T.K.E.
(d). 2.0in tip-to-wall, Reynolds stress.
(e). 1.0in tip-to-wall, T.K.E.
(f). 1.0in tip-to-wall, Reynolds stress.
Fig 13: T.K.E. and Reynolds stress distributions with different tip-to-wall distance (α=2.0⁰,
Re C = 1.15 × 10 5 , x / c = 4.0 ).
Concluding Remarks
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. One of the great advantages of SPIV over the classical technique is that the vortex
wandering can be removed by tracking the center of the vortex in every SPIV frame. By tracking
the center of the vortex, the effects of the wandering and turbulence on the evolution of the
wingtip vortex can be separated. The results show that after wandering correction, the T.K.E. and
Reynolds stress distributions become lower by more than 2 factor of two at 4.0 chord length
downstream. This suggests that the wingtip vortex itself is laminar after the rollup and the higher
turbulence intensity in the vortex core, reported in past studies, is mainly due to vortex
wandering.
The Devenport’s method under and/or over predicts the tangential velocity depending on the
radial distance from the center and the prediction is not consistently accurate. This is because the
calculation is depending on only one tangential velocity profile. This is not sufficient to fully
recover the accurate wandering free velocity profile. Therefore, the Devenport’s method should
only be limited to approximate the velocity profile and with reasonably larger wandering
amplitude, perhaps σ > rc / 5 , for satisfactory results since the smaller wandering amplitude
means less smearing of the velocity by the wandering. If the accurate wandering free velocity
profile is necessary, the spatially-resolved instantaneous non-intrusive technique like PIV or
SPIV is desirable.
When the wind tunnel wall gets closer to the wing-tip vortex, it will increase the T.K.E.
levels. When the wall is set further apart, the T.K.E. comes from the wandering and the vortex
turbulence itself, but when the wall gets closer enough to about 4.5 vortex core radius, then the
T.K.E. is mainly coming from the turbulence level of the vortex itself and there is a little or no
impact from the wandering. As the wall get closer to the wing tip, the vortex wanders more and
the vortex radius gets bigger hence the image vortex on the other side of the wall effect is big.
But as the wall gets even closer to within the four or five vortex radii away, the vortex starts to
wander in z-direction more with the radius and vorticity becoming smaller. When the wall exists,
it acts as the mirror vortex image exists on the other side of the wall, therefore there will be a
stronger influence of the vortex up-wash, which drives the vortex upward. The correlation
between the tip-to-wall distance and the vortex up-wash driven height is found to be fitted well
by using a power equation.
Acknowledgments
The authors also want to thank Mr. Bill Rickard of Iowa State University for his help in
conducting the experiments. This work is sponsored by the AFOSR, grant FA9550-08-1-0294.
Reference
Bailey and Tavoularis (2008). Measurements of the velocity field of a wing-tip vortex, wandering in grid
turbulence. J. Fluid Mech., 601, 281-315.
Devenport, W. J., Rife, M. C., Liapis, S. I., and Follin, G. J. (1996). The structure and development of a
wing-tip vortex. J. Fluid Mech., 312, 67–106.
Gerontakos and Lee (2006). Near-field tip vortex behind a swept wing model. Exp. Fluids, Vol.
4, No.1, 141–155.
Heyes, A. L., Hones, R. F., and Smith, D. A. R. (2004). Wandering of wing-tip vortices. In Proc. 12th Intl
Symp. On the Applications of Laser Techniques to Fluid Mechanics, Lisborn, Portugal.
Hu, H., Saga, T., Kobayashi, T., and Taniguchi, N. (2001). A Study on a Lobed Jet Mixing Flow by
Using Stereoscopic PIV Technique. Phy. Fluid, Vol. 13, No.11, 3425–3441.
Prasad A. K., and Adrian R. J. (1993). Stereoscopic Particle Image Velocimetry Applied to Fluid Flows.
velocimetry in liquid flows. Exp. Flows, 15, 49–60.
Soloff, S. M., Adrian, R. J., and Liu, Z-c. (1997). Distorsion compensation for generalized stereoscopic
particle image velocimetry. Meas. Sci. Technol., 8, 1441–1454.