Yaw-Roll Coupled Oscillations of a Slender Delta Wing
Except where reference is made to the work of others, the work described in this
thesis is my own or was done in collaboration with my advisory committee. This
thesis does not include proprietary or classi ed information.
John C. Worley
Certi cate of Approval:
Gilbert Crouse Jr.
Associate Professor
Aerospace Engineering
Anwar Ahmed, Chair
Associate Professor
Aerospace Engineering
Brian Thurow
Assistant Professor
Aerospace Engineering
Joe F. Pittman
Interim Dean
Graduate School
Yaw-Roll Coupled Oscillations of a Slender Delta Wing
John C. Worley
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful llment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 9, 2008
Yaw-Roll Coupled Oscillations of a Slender Delta Wing
John C. Worley
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
John Worley, son of John Allen and Susan (Cantrell) Worley, was born on April
23, 1984, in Easley, South Carolina. He graduated from D. W. Daniel High School in
2002 before entering Auburn University in the fall. In 2006, he graduated magna cum
laude with a Bachelors of Aerospace Engineering. After working as a Wind Tunnel
Technician in Auburn University?s Aerospace Engineering department, he entered
Graduate School at Auburn University in August 2006.
iv
Thesis Abstract
Yaw-Roll Coupled Oscillations of a Slender Delta Wing
John C. Worley
Master of Science, August 9, 2008
(B.S., Auburn University, 2006)
52 Typed Pages
Directed by Anwar Ahmed
The delta wing has been a highly studied aerial planform, from counter rotating
vortices to a rolling ?wing rock? motion. The research done at Auburn University is
an extension of those studies. This research studies not only the inherent roll motion
of the delta wing but also the yaw motion that is coupled with that rolling motion.
Using a specially designed model, a wind tunnel, and a dynamic vortex laboratory,
the aerodynamics of a slender delta wing could be analyzed. The delta wing?s motion
could be tracked using an accelerometer and optical encoder and the shedding vortices
could be seen through various  ow visualization techniques. The time histories and
time averaged angle sweep data were also collected using a force and moment balance.
This model also allowed for the ability to study one type of motion at a time, which
allowed for comparison with previous purely rolling studies. Below are the results of
both the roll and yaw motion on the slender delta wing as well as the coupled e ect
of the two.
v
Acknowledgments
The author would like to thank Dr. Anwar Ahmed for providing him with this
opportunity, guidance, and support throughout this work. The author would also like
to thank the committee for this opportunity given to him, Mr. Bryan Recktenwald for
his assistance, Mr. Andy Weldon for making all the experimental setups that made
this work possible, and the Aerospace Engineering Department of Auburn University
for the opportunity and  nancial support. The author would like to address most of
his thanks to his parents for their unlimited support.
vi
Style manual or journal used Journal of Approximation Theory (together with
the style known as \aums"). Bibliography follows van Leunen?s A Handbook for
Scholars.
Computer software used The document preparation package TEX (speci cally
LATEX) together with the departmental style- le aums.sty. All graphs were made
using Tecplot.
vii
Table of Contents
List of Figures ix
List of Tables xi
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Experimental Setup 9
2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Model Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Wind Tunnel Model . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Water Tunnel Model . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Test Facility Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Wind Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Water Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Results and Discussion 19
4 Conclusions 34
Bibliography 36
Appendices 38
Appendix A: Additional Plots . . . . . . . . . . . . . . . . . . . . . . . . . 39
viii
List of Figures
1.1 Di erent delta wing planforms: a.) Slender[1] b.) Non-slender[2]. . . . 2
1.2 Detailed  ow schematic over a delta wing[4]. . . . . . . . . . . . . . . 3
1.3 Theoretical (potential  ow) and experimental lift coe cient curves for
a delta wing in subsonic  ow[7]. . . . . . . . . . . . . . . . . . . . . . 4
1.4 Delta wing experiencing both spiral (top) and bubble (bottom)
breakdown[11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Negative pressure distributions over a delta wing after ?bursting? has
occurred: a.)Upper surface b.)Rear surface. . . . . . . . . . . . . . . . 7
1.6 Diagram of lift and weight vectors pre- and post- ?bursting.? . . . . . 7
2.1 Planform views of the delta wing model. . . . . . . . . . . . . . . . . 10
2.2 Bearing locations on the model setup. . . . . . . . . . . . . . . . . . . 10
2.3 Close up of wind tunnel model. . . . . . . . . . . . . . . . . . . . . . 12
2.4 Close up of the acrylic plates used to lock the yaw motion. . . . . . . 12
2.5 Close up of water tunnel model. . . . . . . . . . . . . . . . . . . . . . 13
2.6 Wind tunnel experimental setup. . . . . . . . . . . . . . . . . . . . . 16
2.7 Water tunnel experimental setup. . . . . . . . . . . . . . . . . . . . . 17
3.1 Variation in lift coe cient versus time. . . . . . . . . . . . . . . . . . 20
3.2 Variation in lift coe cient and roll angle for the Roll Free case. . . . 21
3.3 LIF  ow visualization of the Roll Free case (left to right from top). . 21
3.4 Variation in lift coe cient and roll angle for the Yaw Free case. . . . 22
ix
3.5 Depiction of a yawing delta wing: (a) Pre ?burst? (b) At ?burst? (c)
Post ?burst.? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6 LIF  ow visualization of the Yaw Free case (left to right from top). . 24
3.7 Variation in lift coe cient and roll angle for the All Free case. . . . . 25
3.8 Roll angle versus roll rate for the Roll Free case. . . . . . . . . . . . . 25
3.9 Roll angle versus roll rate for the All Free case. . . . . . . . . . . . . 26
3.10 LIF  ow visualization of the All Free case (left to right from top). . . 26
3.11 LIF  ow visualization of the All Free case (left to right from top). . . 27
3.12 Variation in pitching moment coe cient versus time. . . . . . . . . . 28
3.13 Variation in rolling moment coe cient versus time. . . . . . . . . . . 29
3.14 Variation in yawing moment coe cient versus time. . . . . . . . . . . 30
3.15 Lift coe cient versus  . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.16 Rolling moment coe cient versus  . . . . . . . . . . . . . . . . . . . 31
3.17 Yawing moment coe cient versus  . . . . . . . . . . . . . . . . . . . 32
3.18 Accelerations and roll angle for the Roll Free case. . . . . . . . . . . . 32
A.1 Drag force coe cient versus time. . . . . . . . . . . . . . . . . . . . . 39
A.2 Side force coe cient versus time. . . . . . . . . . . . . . . . . . . . . 40
A.3 Side force coe cient versus  . . . . . . . . . . . . . . . . . . . . . . . 40
A.4 Lift coe cient versus drag coe cient. . . . . . . . . . . . . . . . . . . 41
x
List of Tables
2.1 Pyramidal balance reduction matrix. . . . . . . . . . . . . . . . . . . 15
2.2 Instrumentation samples and sampling rates. . . . . . . . . . . . . . . 18
xi
Chapter 1
Introduction
1.1 Background
Delta wing planforms and the aerodynamics surrounding them have been a sub-
ject of investigation for the past several decades; however, a full picture of the  ow
and its e ects on the aerodynamics of the wing has not yet emerged. Previous studies
must therefore be improved upon in order to fully understand the  ow over a delta
wing.
Delta wing planforms can be categorized as either slender or non-slender. The
distinguishing factor between the two is the sweep angle,  . This sweep angle is
measured between a line perpendicular to the center line that is attached to the nose
of the wing and the leading edge of the wing. As long as the sweep angle is less than
 = 55 the delta wing is considered to be non-slender, whereas a delta wing with
an angle larger than  = 55 is considered to be slender. Figure 1.1 shows aircraft
with slender and non-slender delta wing planforms. Both non-slender and slender
delta wings have been utilized on various aircraft; however, slender delta wings, with
a leading edge sweep angle greater than the Mach angle, have been more attractive
due to their good performance in supercritical  ow regimes. For a leading edge sweep
angle greater than the Mach angle, the shockwave is only attached to the nose of
the wing, yielding high subsonic  ow traveling over the wing. Thus a slender delta
wing in supersonic  ow experiences subsonic  ow on the wing and has the same
performance characteristics as a slender delta wing that is experiencing a purely
1
subsonic  ow. However, tests have shown that slender delta wings have a number of
stability problems at these lower speeds which generally result in an uncontrollable
aircraft. In order to fully comprehend the reasons behind this stability loss, a detailed
understanding of the  ow over the slender delta wing is needed.
Figure 1.1: Di erent delta wing planforms: a.) Slender[1] b.) Non-slender[2].
The dominant  ow structure over a highly swept (slender) delta wing consists of
two counter rotating vortices that span the length of the wing. These counter rotating
vortices are created when the shear layer of the oncoming  ow separates and rolls up
along the leading edge of a delta wing[3]. A detailed schematic of this  ow is seen in
Figure 1.2. This type of  ow structure is desirable due to the fact that  ow from the
underside of the wing will curl up onto the upper surface feeding the vortices which in
turn produce more lift[5, 6]. The amount of lift generated by the vortices can be seen
in Figure 1.3 which shows the theoretical lift that a delta wing should have without
the vortices and the experimental lift[7]. This vortex lift is even more desirable in
modern combat aircraft and unmanned aerial vehicles (UAVs) since the increase in
lift also gives an increase in maneuverability in high angle of attack regimes. However,
several studies have shown that at very high angles of attack these vortices begin to
2
experience one of two types of degradation: spiral breakdown or bubble breakdown. A
spiral breakdown occurs when the cores of one or both of the primary vortices detach
from the surface of the wing but maintains its structure, forming a helix pattern
downstream of the delta wing[8]. A bubble breakdown, on the other hand, occurs
when the cores of one or both of the primary vortices expand and lose some of their
axial  ow velocity which eventually results in regions of recirculation[3, 8, 9, 10]. A
depiction of both types of breakdown can be seen in Figure 1.4.
Figure 1.2: Detailed  ow schematic over a delta wing[4].
So far, there is no general agreement as to what causes either of these breakdowns[12].
Although, what has been determined is that both types of breakdowns happen at
some axial distance from the leading edge of the wing, usually in only one of the
two primary counter rotating vortices. Since only one of the vortices has experienced
this breakdown or ?burst,? there is an uneven lift distribution over the wing causing a
rolling motion. As the delta wing rolls to one side, the vortex that has ?burst? begins
to reform and simultaneously the other vortex ?bursts? causing the wing to roll back
3
Figure 1.3: Theoretical (potential  ow) and experimental lift coe cient curves for a
delta wing in subsonic  ow[7].
4
in the other direction. This rolling motion is known as a phenomenon called ?wing
rock?[3].
Figure 1.4: Delta wing experiencing both spiral (top) and bubble (bottom)
breakdown[11].
The ?wing rock? phenomenon is a major focus of research. Previous investigations
were conducted by mounting a slender delta wing to a wind tunnel balance either
from the bottom[13] or internally from the rear[14, 15, 16]. Both types of mounting
systems allowed the delta wing to undergo the self induced rolling motion that is
characteristic of the ?wing rock? phenomenon. Data from these studies has shown a
sinusoidal oscillation in the roll angle of the model[14, 15, 16]. This motion was also
detected by the force balance, showing the same sinusoidal motion in the normal and
side force coe cients. Time averaged studies have also been performed that illustrate
the e ect of the rolling motion on the lift curve[14, 17]. Results from these studies
showed that the rolling motion usually increased the CLmax while the  CLmax remained
the same. These studies also showed that there was a much larger hysteresis loop in
the lift curve when the model was allowed to roll freely. Additionally, water tunnel
5
 ow visualization results showed how the vortices formed, ?burst,? and interacted with
each other[8, 14, 16, 17, 18].
The ?wing rock? phenomenon by itself does not reveal a full picture of the  ow
that is a ecting the slender delta wing?s motion. As one of the primary vortices
?bursts,? the pressure distribution over the wing becomes uneven. This uneven pres-
sure distribution causes the center of pressure to shift which results in the ?rocking?
motion since the lift force acting at the center of pressure produces a rolling moment.
This uneven pressure distribution also imparts a side force resulting in a yawing mo-
ment. The side force is produced from the tilted upper surface that forms the sharp
leading edge needed to produce the counter-rotating vortices. A depiction of the
negative pressure distribution over the upper surface of the wing can be seen in part
(a) of Figure 1.5. This side force vector combines with the weight vector of the delta
wing, resulting in a larger sideslip force that when applied at the center of pressure
produces a yawing moment, as depicted in Figure 1.6. Part (b) of Figure 1.5 shows
the negative pressure distribution across the rear surface, which also applies some
yawing moment but to a lesser extent than that of the upper surface and weight.
These unstable yawing motions coupled with the rolling motion is what causes an
aircraft to go into an uncontrollable spin.
In contrast with the ?wing rock? motion, the yawing motion has not been explored
due to the complex nature of the motion. The few studies that have been performed
have only been on a delta wing with a  xed yaw, or sideslip ( ), angle[14, 18, 19, 20].
The results of those investigations were compared with the data for the  = 0 cases
and showed that the more the delta wing was yawed, the more detrimental the e ect
on the lift curve. Also, the  ow visualization showed that the secondary vortex, seen
in Figure 1.2, increased in size to match that of the primary vortex as the sideslip
angle increased[18].
6
Figure 1.5: Negative pressure distributions over a delta wing after ?bursting? has
occurred: a.)Upper surface b.)Rear surface.
Figure 1.6: Diagram of lift and weight vectors pre- and post- ?bursting.?
7
1.2 Research Objectives
In order to fully understand the dynamics of a slender delta wing, every aspect
must be studied in detail. Thus, the objectives of the present research are to design
and manufacture two slender delta wing models capable of the full range of motions
of \free  ight" and to investigate the yaw-roll coupled motions using both a wind
tunnel and water tunnel.
8
Chapter 2
Experimental Setup
2.1 Model Description
As previously mentioned, the ?wing rock,? or purely rolling motion, has been
studied extensively in the past. This motion was achieved in these previous studies
simply by placing the slender delta wing on a shaft and then mounting that shaft on
a set of ball bearings which allowed it to spin freely. However, free yawing motion has
not been studied in detail due to mechanical complexities. This leads to the primary
challenge of studying the yaw-roll coupled e ect on a slender delta wing, which is the
design and development of the research models. The only way to achieve this yawing
motion without disturbing the  ow is to have a delta wing with an internal bearing
system. The delta wing for this experiment was then designed just thick enough to
accommodate such a yaw bearing system. The yaw bearing was placed at the center
of mass of the delta wing and hard mounted to the roll shaft. With both sets of
bearings in place, the models had the ability to move in both the yaw and roll planes.
The yaw plane is de ned here as a plane that remains parallel to the upper surface
of the model regardless of the roll and pitch angles. In the same fashion, the roll
plane is de ned as a plane that remains perpendicular to the upper surface regardless
of the yaw and pitch angles. This con guration is presented in Figures 2.1 and 2.2.
With simple set screws and acrilic plates, either of the motions could be locked out
allowing the models to be con gured to exhibit either motion independently.
9
Figure 2.1: Planform views of the delta wing model.
Figure 2.2: Bearing locations on the model setup.
10
2.2 Model Geometry
2.2.1 Wind Tunnel Model
The wind tunnel model was constructed from a solid block of aluminium. The
slender criteria was met by designing the model with a leading edge sweep angle of  
= 75 . The model had a 17 in (43.18 cm) chord, a 6 in (15.24 cm) base span, and a
1 in (2.54 cm) thickness. The center of the model was hollowed out to accommodate
the yaw bearing and the surrounding aluminium collar. A slot was then cut into the
base to allow the 1/4 in (0.64 cm) stainless steel roll shaft to connect to the yaw
bearing collar. A close up of the wind tunnel model can be seen in Figure 2.3. An
acrylic plate was used to close the top of hollow chamber where the yaw bearing was
placed. The acrylic plate served dual purposes:  rst, to close the chamber and keep
the delta wing surface uniform, so as to not adversely e ect the  ow; second, to allow
a view port to the yaw bearing, in case the roll shaft had to be disconnected. Thin
acrylic plates were then made to  t on the base and around the roll shaft in order to
lock the yaw motion but not impede the roll motion (Figure 2.4). The other end of
the roll shaft was then slid into a brass collar that contained two stainless steel ball
bearings. These bearings were placed such that the shaft would roll smoothly even
with the weight of the model or a wind load being applied to one end. The distance
between the base of the delta wing and the brass collar was adjustable to ensure that
the  ow was not a ected by the mounting linkage. The brass collar also had two set
screws in the rear end used to lock the roll motion in place. The ball bearings used
in the brass collar and the central yaw location had 1/4 in (0.64 cm) inner diameter,
a 1 in (2.54 cm) outer diameter, and a 1/4 in (0.64 cm) thickness.
11
Figure 2.3: Close up of wind tunnel model.
Figure 2.4: Close up of the acrylic plates used to lock the yaw motion.
12
2.2.2 Water Tunnel Model
The water tunnel model was designed and constructed to match the general
shape and motion of the wind tunnel model. However, due to the size limitations of
the water tunnel, the water tunnel model needed to be scaled down by a factor of 2.06
based on area. The same leading edge sweep angle  = 75 was kept while the chord
and the base span were reduced to 11 in (27.94 cm) and 4.5 in (11.43 cm) respectively.
The thickness of the delta wing was also reduced to 3/4 in (1.91 cm), which in turn
reduced the size of the roll shaft diameter and yaw bearing thickness to 1/8 in (0.32
cm) each. The same type of stainless steel ball bearings were used on the roll shaft
except the inner diameter was decreased to 1/8 in (0.32 cm) to accommodate the
shaft. The central yaw bearing on the other hand, had to be changed to a glass ball
bearing to prevent rusting while submerged in the water. The outer surface of the
model was also painted black to minimize spurious re ections of laser light during
 ow visualization. A close up of this model can be seen in Figure 2.5.
Figure 2.5: Close up of water tunnel model.
13
2.3 Test Facility Setup
2.3.1 Wind Tunnel
Once the model was constructed, a detailed study of the motions was conducted
using the Auburn University?s 3 ft x 4 ft (0.91 m x 1.22 m) subsonic, closed circuit,
atmospheric, closed section wind tunnel. Three di erent instruments were used to
acquire the data: a six component, force and moment, pyramidal, strain gage balance;
an Endevco model 23-R triaxial piezoelectric accelerometer; and a HB6M US Digital
hollow bore optical encoder. The pyramidal balance was calibrated using a series of
weights that ranged from 1 lb to 30 lbs[21]. A reduction matrix was then created
to determine the coupling e ects between the di erent components of the balance.
This reduction matrix was then used to calculate the loads experienced on the balance
during testing. The reduction matrix that was used for this study is seen in Table 2.1.
To insure that the calibration of the balance remaind true, test loads were applied
before any testing was performed. If the test load readings di ered from the true
weight by more than 1 % then the calibration process was repeated. The triaxial
accelerometer was connected to its own signal conditioner which provided an output
of micro-g?s and had a charged response frequency of 20 Hz to 10 kHz[22]. The optical
encoder was able to track the roll shaft up to 6000 RPM at a maximum sampling
frequency of 10 KHz[23]. Since the output from the optical encoder was in volts, a
calibration curve was applied to the signal to obtain the roll angle. The calibration
curve was generated by placing a digital protractor on the roll shaft and obtaining
the voltage readings from the optical encoder every 2 for a range of -60 to 60  1 .
The model was connected to the pyramidal balance via a specially designed
mounting system (Figure 2.6). This mounting system was designed to extend the
angle of attack range of the tunnel by 10 since the standard mounting systems have
14
Table 2.1: Pyramidal balance reduction matrix.
Force or Moment Normal Pitching Axial Side Rolling Yawing
Normal 17.7596 0.5705 0.2023 0.0082 -0.0734 0.1968
Pitching 0.0738 -4.140 0.001 0.0589 -0.0216 0.0095
Axial 0.0499 -0.2768 3.7944 -0.2085 0.037 0.1518
Side -0.2857 -0.066 0.0193 10.3528 0.6452 -0.0425
Rolling 0.0164 0.0312 -0.0137 0.0067 4.4716 -0.017
Yawing 0.0227 -0.0339 -0.0051 0.2168 -0.0154 -7.0653
an angle of attack limit of 20 and the higher angle regimes are of more interest in
this study. For this study the mounting system was also articulated in such a way as
to keep the model in the center of the test section so that wall boundary e ects could
be neglected. A shroud was then placed around the mounting system to reduce its
e ects on the pyramidal balance during wind loading. The triaxial accelerometer was
mounted at the center span on the rear surface of the model with the x, y, and z axes
oriented in the normal, span wise, and axial directions respectively. The wires to the
accelerometer were then run down and through the shroud and secured so that the
tension would not e ect the delta wing?s motion during testing. Lastly, the optical
encoder was slid onto the roll shaft behind the brass sleeve. The internal bearing of
the optical encoder was attached to the roll shaft via two set screws, while a bracket
was attached to the outer casing and the brass sleeve of the mounting system to insure
that the optical encoder?s calibration stayed true.
2.3.2 Water Tunnel
Flow visualization tests were conducted in the Auburn University?s 18 in x 18 in
(0.46 m x 0.46 m) water tunnel. The delta wing was mounted in such a way that the
angle of attack could be adjusted without submerging the roll bearings in the water.
This was done so that only one glass ball bearing was needed since a metal bearing
15
Figure 2.6: Wind tunnel experimental setup.
would rust with prolonged exposure to the water. Video data was collected using
a camera at the rear of the water tunnel test section. The camera captured images
of laser-induced  uorescence (LIF) of Sodium Fluorescein salt dye as it passed over
the delta wing model. A 5 watt Argon Ion laser beam was passed through an OZ
Optics  ber coupler to generate the laser light sheets that illuminated the Sodium
Fluorescein salt dye. A diagram of this setup can be seen in Figure 2.7. The laser
sheet was  rst placed at the mid chord and then at 1/4 chord past the trailing edge.
This allowed for visualization of the vortices both during and well after breakdown.
2.4 Data Acquisition
In both the wind and water tunnels, the experiments were performed for four
di erent cases: \none free," \roll free," \yaw free," and \all free." For the none free
case, both the rear set screws on the roll shaft and the acrylic plates were screwed in
16
Figure 2.7: Water tunnel experimental setup.
place to lock the roll and the yaw motion respectively. This also resulted in a static
baseline case to which all the other cases were compared. The roll free case had the
acrylic plates in place to restrict the yaw motion, but the roll shaft was completely
free to move. This allowed for the direct comparison with previously published ?wing
rock? data. Finally the last two cases had the acrylic plates removed but one had the
roll shaft set screws in place and the other did not (yaw free and all free respectively).
The wind tunnel tests were later categorized into two types: angle  xed and
angle sweep. For the angle  xed tests the angle of attack ( ) of the delta wing was
held constant at  = 28 . This allowed for the instrumentation to acquire data for
an extended amount of time (4 seconds) for time histories and mean values. The
pyramidal balance data consisted of the forces and moments at each time whereas
the triaxial accelerometer output was integrated twice to obtain displacements. The
optical encoder was used to collect the roll angle ( ) at each time to correlate the forces
and moments with the roll angle and to verify the triaxial accelerometer readings.
17
In addition, an angle sweep was also conducted whereby the delta wing model
was swept through an angle of attack range of  = -3 to  = 28 . At each angle
of attack, data was acquired from the pyramidal balance for 3 seconds and then
averaged to obtain the forces and moments for that angle. The triaxial accelerometer
and optical encoder were not used during this set of tests.
For all of the wind tunnel testing the test section velocity was set to 110 ft/s  2
ft/s (33.53 m/s), which corresponds to a Reynolds number of 9.93 x 105 based on the
model?s root chord. All of the data was acquired using a National Instruments 16 bit
resolution analog to digital converter board. The data was then saved and processed
using a Lab View virtual instrument and Tecplot. The number of samples and the
sampling rates of each of the instruments used are presented in Table 2.2.
Table 2.2: Instrumentation samples and sampling rates.
Number of Samples Sampling Rate (Hz)
Pyramidal Balance 400 100
Triaxial Accelerometer 4000 1000
Optical Encoder 200 50
The water tunnel tests were conducted for  ow visualization at the mid chord
and 1/4 chord downstream of the trailing edge of the model. These tests utilized a
test section velocity of 1.98 ft/s (0.61 m/s) which corresponds to a Reynolds number
of 1.70 x 103. Video images of the Sodium Fluorescein salt dye as it passed over the
mid chord were recorded for each of the four cases. The laser light sheet was recorded
at 250 frames per second using a JVC BR-S622DXU professional video recorder.
18
Chapter 3
Results and Discussion
The wind tunnel data was plotted in Tecplot so that the di erent cases could be
compared and conclusions about each case could be made. The water tunnel video
was digitized so that screen shots could be captured and the video could be analyzed
one frame at a time.
The lift was observed to have a sinusoidal characteristic when the model was free
to roll. The motion started at the same level as the yaw free case (Figure 3.1) but then
increased to a point and returned to the starting position. This oscillatory motion
could be explained by the fact that while one of the vortices was present over one side
of the wing the lift was increased and the wing was rolled to that side, but when the
vortex ?burst? the lift decreased and the wing rolled back in the other direction. This
motion can also be seen in Figure 3.2 which shows the lift force and roll angle versus
time. It should be noted that the rolling motion should be symmetric yielding in an
average roll angle of  = 0 . The reason for the positive average angle in Figure 3.2
is possibly due to a slight weight imbalance in the model, causing the model to have
a larger weight vector on one side resulting in a increased roll angle on that side.
The water tunnel  ow visualization also showed self induced rolling oscillation
due to the vortex ?bursting.? This can be seen in Figure 3.3 which shows the still
images from the LIF  ow visualization video. Here the self induced rolling motion can
be seen by the model body rolling in a counterclockwise direction and the ?bursting?
is seen by a lack of  uorescent dye on one side. The dye is injected into the  ow
via the dye tube (Figure 2.7), it then  ows over the surface of the model and is
19
Figure 3.1: Variation in lift coe cient versus time.
drawn toward the part of the  ow that has a higher velocity. Thus the lack of dye
represents the ?bursting? phenomenon because the ?un-burst? vortex will have higher
core velocity drawing more of the dye to that side. These results were very similar to
other studies that have focused purely on the rolling delta wing (Ref [14, 15]). Both
the roll angle and the lift coe cient versus time presented also matched with other
studies (2 roll angle periods per second), however the amplitude was observed to be
less than the present work (peaks at 25 - 30 instead of 35 - 40 ). This di erence
can be attributed to an increase in test section velocity to 110 ft/s (33.53 m/s) from
25 m/s (82.02 ft/s) [15] and 15 m/s (49.21 ft/s) [14].
The yaw free case showed very little oscillations in the lift (Figure 3.4), indicating
that the sinusoidal motion was a consequence of the rolling motion. The average lift
was also decreased from that of the roll free case indicating that the amount of vortex
20
Figure 3.2: Variation in lift coe cient and roll angle for the Roll Free case.
Figure 3.3: LIF  ow visualization of the Roll Free case (left to right from top).
21
lift was less for the yaw free case than for the roll free case. A diagram of a yawing
delta wing can be seen in Figure 3.5, where part (a) depicts the delta wing prior to
?bursting? and part (c) shows the result after ?bursting? where the model has slid under
the vortex due to the yawing moment imposed by the uneven pressure distribution
over the surface of the delta wing as discussed in Chapter 1. This process yields a
portion of the delta wing without a vortex to produce lift.
Figure 3.4: Variation in lift coe cient and roll angle for the Yaw Free case.
Flow visualization of the yaw free case showed the presence of a very unstable
set of vortices. Figure 3.6 shows that the ?un-burst? primary vortex moved up from
the surface of the model while the secondary vortex increased in size. Both vortices
then yaw with the model until ?bursting? of both vortices has occurred. The move-
ment of these vortices were similar to the depiction in Figure 3.5, however the  ow
22
Figure 3.5: Depiction of a yawing delta wing: (a) Pre ?burst? (b) At ?burst? (c) Post
?burst.?
visualization shows that both the model and the vortex yaw to one side before ?burst-
ing? occurs in that vortex, yawing the model back in the other direction. Since the
previous work on a yawing delta wing is limited there are no comparisons for the lift
force vs time, however in some  ow visualization studies of a statically sideslipped
delta wing, the same unstable set of vortices could be seen (Ref [18, 19]).
With the combined motion (the all free case) there was a more drastic e ect
on the lift. Figure 3.7 shows that the all free case has a similar lift decrease and
roll angle change to that of the roll free case. However, from visual observations
of the model during testing, the delta wing yawed to one side while the roll angle
continued to increase in the direction of the yaw until the wing began to yaw back in
the other direction. This was the reason for an overlying frequency on the roll angle
versus time for the all free case (Figure 3.7). This also indicated that the roll angle
did not complete a full cycle until the wing had yawed back to the other direction
which is further shown in Figures 3.8 and 3.9. Here Figure 3.8 shows that the rolling
motion was cyclic and repetitive whereas Figure 3.9 shows that the coupled motion
was cyclic but not repetitive. This indicates that the roll angle for the roll only motion
23
Figure 3.6: LIF  ow visualization of the Yaw Free case (left to right from top).
is symmetric whereas for the coupled motion the roll angle is shifted to one side due
to the yawing motion.
Figure 3.10 and Figure 3.11 show the  ow visualization of the all free case. Here
Figure 3.10 has an initial clockwise roll direction and a left to right yaw direction,
whereas Figure 3.11 has an initial counterclockwise roll direction and a right to left
yaw direction. These  gures show the same unstable vortex set that the yaw free case
showed but with larger vortices similar to the ones seen in the roll free case. These
images also indicated that the ?un-burst? vortex moved to one side similar to the yaw
free case, but the vortex was larger in size causing an increase in the amount of vortex
lift experienced by the wing.
The same oscillatory motion can be seen in the pitching, rolling, and yawing
moment plots (Figures 3.12 - 3.14). The roll free case again exhibited a sinusoidal
pattern for the pitching, rolling, and yawing moments. The highest peaks occurred
24
Figure 3.7: Variation in lift coe cient and roll angle for the All Free case.
Figure 3.8: Roll angle versus roll rate for the Roll Free case.
25
Figure 3.9: Roll angle versus roll rate for the All Free case.
Figure 3.10: LIF  ow visualization of the All Free case (left to right from top).
26
Figure 3.11: LIF  ow visualization of the All Free case (left to right from top).
with the yawing moment since the lift, which was the largest force acting on the
balance, was rotated to the side with the roll angle causing a twisting motion on the
balance. The pitching moment was less a ected since the lift, which was reduced
by the maximum e ect of the weight of the model, was only aligned to twist in the
pitching direction when the roll angle approached zero. The yaw free case showed
that all three moments remained nearly constant with the largest  uctuations coming
in the yawing moment. This was again caused by a shifted lift, but not as much as
seen in the roll free case. The largest  uctuations in all three moments came for the
all free case. As in the lift versus time graph (Figure 3.1), the all free case showed a
large non repetitive oscillation in the moment plots. The only variation was in the
yawing moment (Figure 3.14) where an overlying sinusoidal oscillation was present.
The e ect that all three cases had on lift coe cient for an angle sweep ( = -3 
to  = 28 ) was also studied. Figure 3.15 shows that the roll free case was in line with
27
Figure 3.12: Variation in pitching moment coe cient versus time.
the none free case until the rolling motion started at  = 23 . After this transition
angle the lift curve starts to decrease. The same trend is seen in the yaw free and all
free cases, however the detriment to the lift curve is increased. The decrease in the
lift curve is greatest for the yaw free case, meaning that the rolling motion in the all
free case helps recover some of the lost lift from the yawing motion. This seems to
contradict Figure 3.1 since the lift force is a ected by the all free case more than the
others. However, since the data taken during the angle sweep tests was averaged, the
large oscillations were canceled out giving the all free case a higher overall lift when
compared to the yaw free case. The large hysteresis loop observed in other works
(Ref [17]) could not be observed since 28 angle of attack was not steep enough to
completely stall the model.
28
Figure 3.13: Variation in rolling moment coe cient versus time.
A similar trend could be seen with the rolling and yawing moment coe cients
(Figures 3.16 and 3.17 respectively). All of the cases follow the trend of the none free
static case until the transition angle of attack (  23 ). Once this transition angle
was reached, the moments began to decrease, with the largest decrease observed for
the all free case.
A triaxial accelerometer was also used on the roll and all free cases to record
the trajectory of its mounting location over time. The results of this sensor where
brought into question when they were compared to the results produced by the optical
encoder. Figure 3.18 shows the accelerations from the triaxial accelerometer plotted
with the roll angle from the optical encoder. Since the rolling motion produces a
sinusoidal roll angle versus time then the accelerations should be zero when the roll
angle peaks and approach a maximum when the roll angle approaches zero. This trend
29
Figure 3.14: Variation in yawing moment coe cient versus time.
however is not observed in Figure 3.18. One possible reason for this di erence was the
proximity of the triaxial accelerometer to the roll axis causing very low accelerations
on the sensor which where damped out of the signal by noise. This rendered the
triaxial accelerometer unreliable and therefore the motion of the delta wing model
was tracked solely by the optical encoder.
The remaining forces (drag, Figure A.1, and side force, Figure A.2-A.3) along
with the drag polar (Figure A.4) have been plotted in Appendix A. These plots show
similar trends to the ones seen in the lift and moment curves. The only di erence arose
in the side force versus time plot. This plot showed that the side force experienced
the same sinusoidal oscillations that the lift did, however the phase of the curve was
shifted. This indicates that when the delta wing model rolled to one side the lift force
30
Figure 3.15: Lift coe cient versus  .
Figure 3.16: Rolling moment coe cient versus  .
31
Figure 3.17: Yawing moment coe cient versus  .
Figure 3.18: Accelerations and roll angle for the Roll Free case.
32
would be taken out of the normal direction and tilted to one side causing the balance
to register the load on the side force component.
33
Chapter 4
Conclusions
The aerodynamics of a slender delta wing capable of two-degree of freedom mo-
tion was investigated using both a wind tunnel and water tunnel. A strong yaw-roll
coupling was observed. There existed a transition angle (  23 ) at which the the
yaw-roll coupling began to a ect the aerodynamic characteristics of the delta wing.
This change was seen in the moment curves as well as the lift curve.
The  ndings presented here mirrored those of previous studies conducted on one-
degree of freedom slender delta wings. The only di erences in the results came in
increased amplitudes which could be attributed to an increased test section velocity.
The yaw-roll coupling caused a sinusoidal oscillation in the moment versus time
curves (Figure 3.12 - 3.14). These oscillations were similar to the oscillations seen
in the roll free case, but had a larger amplitude and also had an overlying oscilla-
tion causing the entire curve to shift. The lift versus time curves exhibited similar
trends with the yaw-roll coupling producing larger amplitude oscillations than that
of the other cases. The only exception came from the yaw free case that produced
a near constant lift line versus time. This showed that the oscillations were purely
a consequence of the rolling motion. Also the yaw free case had a lower average lift
indicating that it did not have the increased vortex lift that the other cases did.
The lift curve versus angle of attack showed that yaw-roll coupling decreased the
amount of lift produced by the delta wing when compared to that of the none free
and roll free cases. The yaw free case however, showed an even greater decrease in
34
lift. This further indicated that the amount of vortex lift generated by the yaw free
case was less than the other cases.
The motions of the slender delta wing observed here show that the aerodynamic
forces have a greater e ect than the body forces. This means that the weight of the
model did a ect the yawing motion, however since the model was able to return to
it original position and then yaw in the other direction, the weight was counter acted
by the uneven pressure distribution over the surfaces of the model.
Recommendations for future work would include measuring the yaw angle in
order to correlate with the roll angle and normal force time histories. Also the sweep
angle could be altered to see how the yaw-roll coupling is a ected. In order to reduce
the contribution of the weight vector on the yawing motion, a lighter model could
be designed and manufactured. Particle image velocimetry (PIV) studies could also
be performed to observe the speed and strengths of the primary vortices during the
motion. Addition of control surfaces could be made to the model to study their e ects
on the coupled motion.
35
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37
Appendices
38
Appendix A
Additional Plots
Figure A.1: Drag force coe cient versus time.
39
Figure A.2: Side force coe cient versus time.
Figure A.3: Side force coe cient versus  .
40
Figure A.4: Lift coe cient versus drag coe cient.
41